Modern Mechanics and Applications: Select Proceedings of ICOMMA 2020 (Lecture Notes in Mechanical Engineering) 9811632383, 9789811632389
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Contents
A Dual Approach for Modeling Two- and One-Dimensional Solute Transport
1 Introduction
2 Establishing a 2DV Solute Transport Equation Based on the Dual Approach
3 Comments
4 Results
5 Comments
6 Conclusions
References
Rayleigh Quotient for Longitudinal Vibration of Multiple Cracked bar and Application
1 Introduction
2 Rayleigh Quotient for Multiple Cracked Bar
3 Application for Calculating Natural Frequencies
4 Numerical Illustration and Discussion
5 Conclusion
References
Vibroacoutic Behavior of Finite Composite Sandwich Plates with Foam Core
1 Introduction
2 Theoretical Study
2.1 Plate Geometry and Assumptions
2.2 Flexural Motion of the Plate
2.3 Boundary Conditions of the Plate-Porous Coupling
2.4 Determination of Sound Transmission Loss
2.5 Modelling the Poroelastic Material
3 Validation
4 Numerical and Experimental Results
4.1 Test Setup
4.2 Sample Description
4.3 Results and Discussion
5 Conclusions
Appendix A. Expressions of the Matrix Equation (58)
References
Crystal Structure and Mechanical Properties of 3D Printing Parts Using Bound Powder Deposition Method
1 Introduction
2 Experimental
2.1 Material
2.2 Fabrication Method
2.3 Analysis Methods
3 Result and Discussion
3.1 Dimension Accuracy
3.2 Morphology
3.3 Porosity
3.4 Tensile Test
3.5 Hardness Test
4 Conclusion
References
Experimentally Investigating the Resonance of the Vibration of Two Masses One Spring System Under Different Friction Conditions
1 Introduction
2 Design and Setup the Model of Apparatus
2.1 Design and Setup Experimental Apparatus
2.2 Experimental Operation
3 Results and Discussion
4 Conclusion
References
Nonlinear Bending Analysis of FG Porous Beams Reinforced with Graphene Platelets Under Various Boundary Conditions by Ritz Method
1 Introduction
2 Material Properties
3 Formulations
4 Ritz Method
5 Numerical Results and Discussions
6 Conclusions
References
Self-vibrational Analysis of a Tensegrity
1 Introduction
2 Self-vibration of Axially Prestressed Elements
3 Equation of Axial Motion in Vibration
4 Spectral Element Formulation
4.1 Governing Equations in the Frequency Domain
4.2 Spectral Nodal DOFs and Forces
5 Dynamic Shape Function
5.1 Weak Form of Governing Equation
6 Spectral Element Equation
7 Solving the Spectral Element Equation
8 Vibration of Triplex Tensegrity
9 Concluding Remarks
References
Free Vibration of a Bi-directional Imperfect Functionally Graded Sandwich Beams
1 Introduction
2 Problem and Formulation
3 Numerical Results
4 Conclusions
References
Free Vibration of Prestress Two-Dimensional Imperfect Functionally Graded Nano Beam Partially Resting on Elastic Foundation
1 Introduction
2 Problem and Formulation
2.1 Grade Functionally Beam
2.2 Government Equations
2.3 Nonlocal Continuum Beam
2.4 Numerical Formulation
3 Numerical Result
4 Conclusions
References
Design and Hysteresis Modeling of a New Damper Featuring Shape Memory Alloy Actuator and Wedge Mechanism
1 Introduction
2 Configuration of the SMA Damper
3 Modeling of the SMA Damper
3.1 Characterization of the SMA Springs
3.2 Design of the SMA Damper
4 Experimental Test
5 Hysteresis Model of the SMA Damper
6 Conclusions
References
A High-Order Time Finite Element Method Applied to Structural Dynamics Problems
1 Introduction
2 Variational Formulation for Structural Dynamics
3 First-Order Time Finite Element
4 High-Order Time Finite Elements
5 Numerical Example
6 Conclusions
1 The Proof of Equations (24) and (25)
2 Element ``Stiffness'' Matrix of p-TFE for SDOF System
3 MATLAB Code for Obtaining ``Stiffness'' Matrix of bp-TFE for SDOF Systems
References
A Multibody Dynamics Approach to Study an Insect-Wing Structure
1 Introduction
2 Multibody Dynamics Approach
2.1 Hencky-Bar Chain Model
2.2 Equations of Motion
2.3 Validation
3 Beam Model of Hawkmoth Wings
3.1 FEM Model
3.2 Mass and Stiffness Distributions of the Multibody System
4 Results and Discussion
5 Conclusions
References
Natural Ventilation and Cooling of a House with a Solar Chimney Coupled with an Earth – To – Air Heat Exchanger
1 Introduction
2 Problem Formulation and Numerical Method
2.1 The Coupled Solar Chimney and Earth – to – Air Heat Exchanger System
2.2 Numerical Model
3 Results and Discussions
3.1 Changing the Applied Heat Flux in the Solar Chimney
3.2 Changing the Chimney Gap
3.3 Changing the EAHE Pipe Length
3.4 Changing the EAHE Pipe Diameter
4 Conclusions
References
Dynamic Analysis of Curved Beam on Elastic Foundation Subjected to Moving Oscillator Using Finite Element Method
1 Introduction
2 Finite Element Simulation and Dominant Equations
2.1 The Beam Element on Elastic Foundation Subjected to Moving Oscillator
2.2 Governing Differential Equations for Total System
3 Numerical Analysis
3.1 Model Validation
3.2 Calculation Results
4 Conclusions
Appendix
References
Buckling Analysis of FG GPLRC Plate Using a Naturally Stabilized Nodal Integration Meshfree Method
1 Introduction
2 Mathematical Formulations
2.1 Multilayer Functionally Graded Graphene Platelet Reinforced Composites
2.2 Kinematics of FG GPLRC Plate
2.3 Moving Kriging Interpolation Shape Functions
2.4 Nodal Integration Formulations for Plates
2.5 Naturally Stabilized Nodal Integration
3 Numerical Results
3.1 Comparison Results
3.2 Square Plate with a Heart Hole
4 Conclusion
References
Low Temperature Effect on Dynamic Mechanical Behavior of High Damping Rubber Bearings
1 Introduction
2 Experiments
2.1 Specimens and Test Conditions
2.2 Static Equilibrium Hysteresis
2.3 Instantaneous State
2.4 The Temperature Dependence of the Viscosity Property
3 A Calculation to Determin the Viscosity of HDRB from SR Tests
4 Conclusion
References
Effects of Clayey Soil and Hemp Fibers on the Flexural Behavior of Soil Concrete
1 Introduction
2 Materials and Specimens
2.1 Materials
2.2 Specimens
3 Flexural Behavior of Soil Concrete
3.1 Three-Point Bending Test
3.2 Load – Displacement Curve
3.3 Load – Crack Width Curve
4 Conclusions
References
Optimal Design of Functionally Graded Sandwich Porous Beams for Maximum Fundamental Frequency Using Metaheuristics
1 Introduction
2 Frequency Maximization for FG-SWP Beams
2.1 Design Problem
2.2 Analytical Solution for Natural Frequency
3 Implemented Metaheuristics
3.1 Parameter Setting
3.2 Constraint Handling
4 Numerical Results
4.1 Comparison Among Meta-heuristics
4.2 Comparison Among Beam Theories
5 Conclusion
References
Nonlinear Buckling Behavior of FG-CNTRC Toroidal Shell Segments Stiffened by FG-CNTRC Stiffeners Under External Pressure
1 Introduction
2 Fundamental and Established Formulations
3 Validation, Numerical Investigations and Discussions
4 Conclusion
References
Nonlinear Vibration of Shear Deformable FG-CNTRC Plates and Cylindrical Panels Stiffened by FG-CNTRC Stiffeners
1 Introduction
2 Smeared Stiffener Technique and Fundamental Equations
3 Numerical Investigations
4 Conclusion
References
Vibration Characteristics of Functionally Graded Carbon Nanotube-Reinforced Composite Plates Submerged in Fluid Medium
1 Introduction
2 Theoretical Formulation
2.1 Geometry and Material Properties
2.2 Kinematics
2.3 Constitutive Relations
2.4 Formulation of Fluid
2.5 Governing Equations
2.6 Solution Procedure
3 Results and Discussion
3.1 Validation Study
3.2 Parametric Study
4 Conclusions
References
Large Amplitude Free Vibration Analysis of Functionally Graded Sandwich Plates with Porosity
1 Introduction
2 Theoretical Formulation
2.1 FG Sandwich with Porosity Properties and Geometric Configuration
2.2 Displacement Field and Strains
2.3 Constitutive Relations
2.4 Strain Energy of the FG-SPP
2.5 Kinetic Energy
2.6 Finite Element Formulation
3 Results and Discussion
3.1 Convergence and Validation Study
3.2 Parametric Study
4 Conclusions
References
Dynamic Analysis of a Functionally Greded Sandwich Beam Traversed by a Moving Mass Based on a Refined Third-Order Theory
1 Introduction
2 Theoretical Formulations
2.1 The FGSW Beam Model
2.2 Mathematical Model
3 Finite Element for Mulation
4 Numerical Results
4.1 Accuracy and Convergence Studies
4.2 Parametric Study
5 Conclusions
References
Thermal Bending Analysis of FGM Cylindrical Shells Using a Quasi-3D Type Higher-Order Shear Deformation Theory
1 Introduction
2 Theoretical Formulation
3 Solution Procedure
4 Numerical Analysis and Discussion
4.1 Validation
4.2 The Effect of the Boundary Condition
4.3 The Effects of Thickness
5 Conclusions
References
Experimental and Finite Element Analysis of High Strength Steel Fiber Concrete – Timber Composite Beams Subjected to Flexion
1 Introduction
2 Experimental Campaign
2.1 High Strength Steel Fiber Concrete in Compression and Flexion
2.2 Properties of Oak and Beech Timber
3 Constitutive Laws
3.1 Concrete Damaged Plasticity
3.2 Hill Criterion
4 Modelling of Timber-Concrete Composite Beam
4.1 Verification of the Selected Model
4.2 Modelling of the Timber-Concrete Composite Beam
4.3 Parametric Study: Influence of the Steel Connection Diameter on Shear Resistance
5 Conclusions
References
Free Vibration of Stiffened Functionally Graded Porous Cylindrical Shell Under Various Boundary Conditions
1 Introduction
2 FGP Cylindrical Shell
3 Theory and Formulation
4 Analytical Solution
5 Numerical Results and Discussions
5.1 Comparison Research
5.2 Numerical Investigations
6 Conclusion
Appendix A
References
Effects of Design Parameters on Dynamic Performance of a Solenoid Applied for Gas Injector
1 Introduction
2 Simulation Modeling
2.1 Mathematical Modeling
2.2 Solenoid Model in Maxwell
2.3 Solenoid Model in Simplorer
3 Simulation Results
3.1 Effects of Plug and Sleeve Shapes on Electromagnetic Force
3.2 Dynamic Response Under the Effects of Plug and Sleeve Shapes
4 Conclusions
References
Practical Method for Tracking Fatigue Damage
1 Introduction
2 Description of the Method
2.1 Variogram
2.2 Smooth Orthogonal Decomposition
3 Numerical Simulation
4 Experimental Validation
5 Discussion
6 Conclusion
References
An Assessment for Fatigue Strength of Shaft Parts Manufactured from Two Phase Steel
1 Introduction
2 Theoretical Basis
2.1 Fatigue
2.2 Theoretical Basis of Heat Treatment
3 Experiment and Results
3.1 Experimental Equipment
3.2 Recommended Sample Parts
3.3 Fabrication of Sample Parts
3.4 Heat Treatment
3.5 Fatigue Test Process
4 Conclusions
References
Advanced Signal Decomposition Methods for Vibration Diagnosis of Rotating Machines: A Case Study at Variable Speed
1 Introduction
2 Advanced Signal Decomposition Methods
3 Experimental Results
4 Conclusion
References
Nonlinear Effects of the Material in the Pier Dam Experiment Under Cyclic Loading
1 Introduction
2 Experimental Investigation
2.1 Material
2.2 Test Specimens
2.3 Test Set-Up
3 Compare the Experimental Results with the Results Obtained from the Numerical Model
3.1 3D FEM Element Modeling
3.2 Visco-plastic Regularization on the CDP
3.3 Isotropic Hardening
4 Results and Discussion
4.1 Failure Characteristics
4.2 Measurement Data
5 Conclusions
References
Experimental and Numerical Investigations on the Fracture Response of Tubular T-joints Under Dynamic Mass Impact
1 Introduction
2 Fracture Tests
2.1 Dimensions and Material Properties of the Test Models
2.2 Experimental Set-Up
2.3 Test Results
3 Numerical Analysis of Fracture Tests
3.1 Finite Element Modelling
3.2 Material Definition
3.3 Numerical Results
4 Disscussions
4.1 Collision Tests
4.2 Numerical Simulations
5 Conclusions
References
Assessment of Reduced Strength of Existing Mooring Lines of Floating Offshore Platforms Taking into Account Corrosion
1 Introduction
2 Methodology to Assess the Strength of Mooring Lines of Floating Offshore Structures
2.1 Line Dynamic Response Analyses
2.2 Safety Factor
2.3 Minimum Breaking Loads of Mooring Lines from Catalogues of Manufacturers [4]
3 Numerical Method for Chain Testing Simulation
3.1 Plastic Models
3.2 Small-Sliding Interaction Between Bodies [9]
3.3 Contact Formulation [9]
3.4 Contact Pressure in Case of Hard Contact [9]
3.5 Stress Analysis by FEM
4 Strength Assessment of Mooring Line System of DH01 FPU
4.1 Input Data
4.2 Chain Testing Simulation
4.3 Numerical Testing Results
4.4 Comparison to Calculation of MBL According to Vicinay [4]
4.5 Strength Checking Results
5 Discussion
References
An Investigation on Behaviors of Mass Concrete in Cua-Dai Extradosed Bridge Due to Hydration Heat
1 Introduction
2 Hydration Heat analysis
2.1 Heat Transfer Analysis
2.2 Thermal Stress Analysis
2.3 Boundary Condition of Heat Transfer in Concrete Block
3 Behaviors of Tower Footing of Cua-Dai Extradosed Bridge Due to Hydration Heat
3.1 Modelling of Concrete Block
3.2 Hydration Temperature in Tower-Footing Concrete
3.3 Stress in Concrete Block Due to Heat of Hydration
3.4 Displacement Due to Heat of Hydration
4 Conclusions
References
A Nonlocal Dynamic Stiffness Model for Free Vibration of Functionally Graded Nanobeams
1 Introduction
2 Nonlocal Dynamic Stiffness Model of a FGM Timoshenko Nanobeam Element
3 Numerical Results and Discussion
3.1 Comparison of Calculation Results and Published Results
3.2 Effects of Neutral Axis Position and Nonlocal Parameter on the Nondimensional Frequency
3.3 Effect of Boundary Conditions on Nondimensional Frequencies
3.4 Effect of Nanobeam Length on the Nondimensional Frequencies
4 Conclusion
References
Dynamic Performance of Highway Bridges: Low Temperature Effect and Modeling Effect of High Damping Rubber Bearings
1 Introduction
2 Input Data for the Analysis
2.1 Model of the Bridge
2.2 Model of the HDRB
2.3 Earthquake Ground Motions and Solution Algorithm for Motion Equations
3 Output Data of the Analysis
3.1 Low Temperature Effect
3.2 Modeling Effect
4 Conclusion
References
Nonlinear Vibration Analysis for FGM Cylindrical Shells with Variable Thickness Under Mechanical Load
1 Introduction
2 Governing Equations
3 Solution Method
4 Numerical Result
4.1 Validation
4.2 Natural Vibration Frequencies
4.3 Nonlinear Response of Variable Thickness FGM Cylindrical Shell
5 Conclusions
References
Nonlinear Dynamic Stability of Variable Thickness FGM Cylindrical Shells Subjected to Mechanical Load
1 Introduction
2 Governing Equations
3 Solution Method
4 Results and Discussion
4.1 Validation
4.2 Nonlinear Dynamic Stability of Variable Thickness FGM Shell
5 Conclusions
Appendix 1: Stiffness Coefficients Aij ,Bij ,Dij in Eq. (6, 7)
Appendix 2: The Coefficients Lij (i,j = 1,3 ),Pk (k = 1,5 ) in Eq. (9)
Appendix 3: The Coefficients Iij (i,j = 1,3 ),Rk (k = 1,6 ) in Eq. (11)
References
Vibration and Dynamic Impact Factor Analysis of the Steel Truss Bridges Subjected to Moving Vehicles
1 Introduction
2 Computational Models and Assumptions
2.1 Computational Models and Assumptions
2.2 The Differential Equation of Motion
2.3 Analysis Vibration of the Steel Truss Bridge
3 Conclusion
References
An Experimental Study on the Self-propelled Locomotion System with Anisotropic Friction
1 Introduction
2 Experimental Setup with Varied Anisotropic Friction
2.1 Working Principle of the System
2.2 Experimental Implementation
3 Results and Discussion
4 Conclusion
References
Investigation of Maneuvering Characteristics of High-Speed Catamaran Using CFD Simulation
1 Introduction
2 Governing Equations and Turbulence Modeling
3 Case Study and Coordinate System
4 Virtual Captive Model Test
4.1 Static Drift Test
4.2 Harmonic Test
5 Numerical Modeling
6 Results and Discustions
6.1 Validation
6.2 Hydrodynamic Forces
7 Conclusions
References
Optimum Tuning of Tuned Mass Dampers for Acceleration Control of Damped Structures
1 Introduction
2 Methods
2.1 Dynamic Magnification Factor for Steady State Acceleration
2.2 Optimization Process for TMD Parameters
3 Results and Discussion
3.1 New Optimum TMD Parameters
3.2 Empirical Expressions for New TMD Parameters
3.3 Evaluating the Efficiency of the Proposed Tuning Solutions
4 Conclusions
References
Optimal Electrical Load of the Regenerative Absorber for Ride Comfort and Regenerated Power
1 Introduction
2 Mathematical Modelling
3 Optimal Solution of Electrical Damping Ratio Under Harmonic Excitation
4 Numerical Investigation for Harmonic Excitation
5 Optimal Solution of Electrical Damping Ratio Under White Noise Excitation
6 Conclusion
References
Aerodynamic Responses of Indented Cable Surface and Axially Protuberated Cable Surface with Low Damping Ratio
1 Introduction
2 Wind Tunnel Test Investigation
2.1 Experimental Setup
2.2 Wind Attack Angles
2.3 Experimental Model Fabrication
2.4 Experiment Parameters
3 Results and Discussions
3.1 Wind-Induced Vibration of Indented Cable Surface in Low Damping Region
3.2 Wind-Induced Vibration of Axially Protuberated Cable in Low Damping Region
3.3 Axial Flow Near the Wake of Cable
4 Conclusions
References
The Equation Connecting Boyle’s and Bernoulli’s Laws
1 Introduction
2 Connecting Equation
2.1 The Volume Flow Rate of Air Passing Through an Orifice Between Two Compartments
2.2 The Internal Pressure Change of the System Due to Changing External Pressure
2.3 The System with N Compartments
3 Illustrative Example
4 Discussion and Conclusions
References
Effects of the Computational Domain Sizes on the Simulated Air Flow in Solar Chimneys
1 Introduction
2 Numerical Model
2.1 Assumptions for the Flow and Heat Transfer
2.2 Governing Equations
2.3 Computational Domain and Boundary Conditions
2.4 Computational Mesh and Numerical Setup
2.5 Model Validation
3 Results and Discussions
3.1 Flow and Temperature Fields
3.2 Length of the Reverse Flow
3.3 Induced Flow Rate
3.4 Thermal Efficiency
4 Summaries
References
A Solar Chimney for Natural Ventilation of a Three – Story Building
1 Introduction
2 Problem Formulation and Numerical Method
2.1 The Solar Chimney
2.2 Assumptions and Governing Equations
2.3 Validation of the Numerical Model
3 Results and Discussions
3.1 Base Configuration (Base Case)
3.2 Testing Configurations (Case 2 to 4)
3.3 Selected Configuration (Case 5)
3.4 Changing Inlet Size for Similar Flow Rates at All Inlets (Case 6)
4 Conclusions
References
Solar Chimneys for Natural Ventilation of Buildings: Induced Air Flow Rate Per Chimney Volume
1 Introduction
2 Numerical Method
2.1 Studied Solar Chimney
2.2 Numerical Model
2.3 Validation of the Numerical Model
3 Results and Discussions
3.1 Induced Flow Rate Versus Channel Width and Gap
3.2 Induced Flow Rate Versus Channel Volume
3.3 Thermal Efficiency
3.4 Proposed Chimney Configurations for Maximizing Induced Flow Rate
4 Conclusions
References
Enhancing Ventilation Performance of a Solar Chimney with a Stepped Absorber Surface
1 Introduction
2 Description of the Problem and Numerical Modelling
2.1 Numerical Model
2.2 Model Validation
3 Results and Discussions
3.1 Effects of the Step on the Flow Field and Heat Transfer
3.2 Effects of the Step Length and Height
4 Conclusions
References
The Effect of Ground Surface Geometry on the Wing Lift Coefficient
1 Introduction
2 Panel Method and Calculation Model
2.1 Panel Method
2.2 Model Calculation
2.3 Validations
3 Numerical Results
3.1 The Influence of the Ground Surface Geometry Shape
3.2 The Influence of Wind
4 Conclusions
References
Design a Small, Low-Speed, Closed-Loop Wind Tunnel: CFD Approach
1 Introduction
2 Wind Tunnel Design
3 Numerical Method
4 Results and Discussion
5 Conclusion
References
A Size-Dependent Meshfree Approach for Free Vibration Analysis of Functionally Graded Microplates Using the Modified Strain Gradient Elasticity Theory
1 Introduction
2 Basic Equations
2.1 Problem Description
2.2 Modified Strain Gradient Elasticity Theory
2.3 Kinematics of FG Microplate
3 FG Microplate Formulation Based on Moving Kriging Interpolation Shape Functions
3.1 Moving Kriging Interpolation Shape Functions
3.2 A MKI-Based Formulation Based on the sFSDT and Modified Strain Gradient Theory
4 Numerical Examples and Discussions
4.1 FG Square Microplate
4.2 FG Circular Microplate
5 Conclusions
References
Static Behavior of Functionally Graded Sandwich Beam with Fluid-Infiltrated Porous Core
1 Introduction
2 Theory and Formulations
2.1 Sandwich Beam and Material Modelling
2.2 Displacement and Strain Fields
2.3 Constitutive Equation
2.4 Energy Expressions
2.5 Hamilton’s Principle and Governing Equations
2.6 Analytical Solutions
3 Results and Discussion
3.1 Validation
3.2 Comprehensive Study
4 Summary and Conclusions
References
Failure Analysis of Pressurized Hollow Cylinder Made of Cohesive-Frictional Granular Materials
1 Introduction
2 Second Gradient Model Formulation
2.1 Variational Principle
2.2 Equations with Lagrange Multipliers
2.3 Finite Element: First and Second Quadratic Elements
2.4 Constitutive Relations
3 Failure Analysis of Pressurized Hollow Cylinder
3.1 Pressurized Hollow Cylinder Problem
3.2 Failure Analysis
4 Conclusion
References
Studying the Strength of an Acidic Soil-Cement Mixing in Laboratory
1 Introduction
2 Method
2.1 Literature Review on the Effects of pH on the Soil Strength
2.2 Model and Factors for a Soil-Cement Samples in the Model
2.3 Strategy for Virtual Experiment
3 Results
4 Discussion
5 Conclusion
References
Effects of Test Specimens on the Shear Behavior of Mortar Joints in Hollow Concrete Block Masonry
1 Introduction
2 Experimental Program
2.1 Materials
2.2 Shear Pull-Out Test of Masonry
3 Conclusions and Perspectives
References
Nonlinear Buckling and Postbuckling of ES-FG Porous Cylindrical Shells Under External Pressure
1 Introduction
2 ES-FG Porous Cylindrical Shells
3 Fundamental Equations and Solution of the Problem
4 Numerical Results
4.1 Comparisons
4.2 Effect of Porosity Coefficients e0 and Foundation
4.3 Effect of Geometric Parameters (Effects of hcore/h,R/h, L/R Ratios)
4.4 Effects of Stiffeners and Volume Fraction Index
5 Conclusions
Appendix
References
Bulk Modulus Prediction of Particulate Composite with Spherical Inclusion Surrounded by a Graded Interphase
1 Introduction
2 Model of Multi-coated Sphere Assemblage and Limiting Case
3 Differential Substitution Scheme and the Coating with Radially Variable Moduli
4 Numerical Results
5 Conclusion
References
Post-buckling Response of Functionally Graded Porous Plates Rested on Elastic Substrate via First-Order Shear Deformation Theory
1 Introduction
2 Functionally Graded Porous Plates
3 Theoretical Formulation
4 Buckling and Postbuckling Analysis
5 Numerical Results and Discussion
5.1 Validation Examples
5.2 Parametric Study
6 Conclusions
References
Elastic Buckling Behavior of FG Polymer Composite Plates Reinforced with Graphene Platelets Using the PB2-Ritz Method
1 Introduction
2 Laminated FG-GPLRC Plates
3 Theoretical Formulation
4 Pb2-Ritz Method Based on the FSDT
5 Numerical Results and Discussions
5.1 Convergence Study
5.2 Validation Studies
5.3 Effect of Number of Laminae NL
5.4 Effect of GPL Distribution Type and GPL Weight Fraction
5.5 Effect of Boundary Conditions
5.6 Effect of Plate Length-To-Thickness Ratio
5.7 Effect of Plate Aspect Ratio (b/a)
6 Conclusions
References
Design and Fabrication of Mecanum Wheel for Forklift Vehicle
1 Introduction
2 Methodology
3 Results and Discussion
4 Conclusion
References
Muti Object Prediction and Optimization Process Parameters in Cooling Slope Using Taguchi-Grey Relational Analysis
1 Introduction
2 Experimental Procedure
2.1 Materials
2.2 Cooling Slope Casting Process
2.3 Analysis of Samples
2.4 Taguchi’s Experimental Design
3 Grey Relational Analysis
4 Results and Discussion
4.1 Microstructural Investigations
4.2 Analysis of Variance for Grey Relational Grade
4.3 Predicting Optimal Value and Confirmation Experiments
5 Conclusions
References
The Effect of Porosity on the Elastic Modulus and Strength of Pervious Concrete
1 Introduction
2 The Strain and Stress Fields
3 Calculation of the Elastic Modulus of Porous Concrete
4 Calculation of the Compressive Strength of Porous Concrete
5 Conclusions
References
Predicting Capacity of Defected Pipe Under Bending Moment with Data-Driven Model
1 Introduction
2 Finite Element Model
3 Developing a Data-Driven Model with Random Forest Regression
4 FEA Database and the Developed Model
5 Conclusion
References
Image Recognition Using Unsupervised Learning Based Automatic Fuzzy Clustering Algorithm
1 Introduction
2 Some Distance Measurements for Discrete Data
2.1 Measure Distance Between Points
2.2 Clustering Evaluation Criteria
3 The Proposed Technique
3.1 Methods to Extract Image Data
3.2 The Proposed Algorithm
4 Applying in Image Recognition
5 Conclusion
References
The Effect of the Non-uniform Temperature and Length to Height Ratio on Elastic Critical Buckling of Steel H-Beam
1 Introduction
2 Finite Element Model
3 Results
4 Conclusions
References
On the Thin-Walled Theory’s Application to Calculate the Semi-enclosed Core Structure of High-Rise Buildings
1 Introduction
2 Working Characteristics and Deformation of the Semi-enclosed Core Structure Subjected to Torsion
3 Investigation of the Behavior of the Semi-enclosed Core Wall Structure Subjected to Horizontal Load
4 Conclusion
References
Numerical Simulation of Full-Scale Square Concrete Filled Steel Tubular (CFST) Columns Under Seismic Loading
1 Introduction
2 FEA Modeling Program
2.1 Modeling Procedure
2.2 Numerical Validation
2.3 Parametric Studies
3 Numerical Results and Discussions
3.1 Numerical Results
3.2 Effect of Axial Load Level
3.3 Effect of B/t Value
3.4 Effect of Concrete Strength
4 Conclusions
References
Numerical Modeling of Shear Behavior of Reinforced Concrete Beams with Stirrups Corrosion: Finite Element Validation and Parametric Study
1 Introduction
2 Materials Law for Modeling Corroded RC Beams
2.1 Concrete Material Law
2.2 Steel Reinforcement Law
2.3 Constitutive Law of the Steel - Concrete Bond
3 Validation of FE Models for Shear-Critical Corroded RC Beams
3.1 Presentation of the Tested Beam
3.2 Modeling of the RC Corroded Beam with Stirrups Corrosion in Shear Span
3.3 Validation of FE Model
4 Parametric Study
4.1 Effect of Concrete Compressive Strength
4.2 Effect of Stirrup Ratio
4.3 Effect of Bond Strength
5 Conclusions
References
Characterization of Stress Relaxation Behavior of Poscable-86 High-Strength Steel Wire
1 Introduction
2 Methodology
3 Experimental Procedure
4 Results and Discussion
5 Conclusions
References
Seismic Performance of Concrete Filled Steel Tubular (CFST) Columns with Variously Axial Compressive Loads
1 Introduction
2 Experimental Program
2.1 Specimen Design
2.2 Material Properties
2.3 Test Procedure
3 Test Results and Discussions
3.1 Test Results and Observations
3.2 Lateral Load vs. Lateral Displacement Curves
3.3 Lateral Strength and Deformation Capacity
3.4 Lateral Stiffness Degradation
4 Conclusions
References
Damage Simulation Based on the Phase Field Method of Porous Concrete Material at Mesoscale
1 Introduction
2 Finite Element Discretization from the Phase-Field Problem Framework
3 Arrangement of Aggregate Particle Based on Monte-Carlo Simulation
4 Calculation of the Compression Strength of Porous Concrete
5 Conclusions
Referencess
Constitutive Response and Failure Mechanism of Porous Cement-Based Materials Under Triaxial Stress States
1 Introduction
2 Porous Cement-Based Materials Under Triaxial Tests
2.1 DEM Simulation of Triaxial Tests
2.2 The Influence of Confining Stress
2.3 Influence of Intermediate Principal Stress on Octahedral Plane
3 Conclusion
References
Development of Automatic Landing Control Algorithm for Fixed-Wing UAVs in Longitudinal Channel in Windy Conditions
1 Introduction
1.1 Effects of Turbulence on UAV’s Motion in the Longitudinal Channel
2 Development of Automatic Landing Control Algorithm for Fixed Wing UAVs in Longitudinal Channel in Windy Conditions
3 Simulation Results and Conclusion
References
An Assessment of Terrain Quality and Selection Model in Developing Landslide Susceptibility Map – A Case Study in Mountainous Areas of Quang Ngai Province, Vietnam
1 Introduction
2 Methodology
2.1 Study Area
2.2 Data Collection
2.3 DEM Analysis
2.4 Landside Susceptibility Model
2.5 Accuracy Assessment
3 Result and Discussion
3.1 DEM Selection
3.2 Landslide Susceptibility Mapping
3.3 Landslide Susceptibility Assessment
4 Conclusion
References
Analysis of the Impact of Ponds Locations in Flood Cutting from a Developing City in Vietnam
1 Introduction
2 Methodology and Materials
2.1 Study Area
2.2 Scenarios
2.3 Methodology
3 Result and Discussion
4 Conclusion
References
Turning Electronic and Optical Properties of Monolayer Janus Sn-Dichalcogenides By Biaxial Strain
1 Introduction
2 Methodology
3 Results and Discussion
4 Conclusions
References
A Meshfree Method Based on Integrated Radial Basis Functions for 2D Hyperelastic Bodies
1 Introduction
2 Constitutive Equations of Hyperelastic Material
3 IRBF-Based Meshfree Method for Hyperelasticity
3.1 Integrated Radial Basis Function Approximation
3.2 The Weak Form and Meshfree Method
4 Numerical Examples
4.1 Inhomogeneous Compression Problem
4.2 Curved Beam Problem
5 Conclusion
References
Structural Damage Identification of Plates Using Two-Stage Approach Combining Modal Strain Energy Method and Genetic Algorithm
1 Introduction
2 Modal Strain Energy and Genetic Algorithm-Based Damage Identification Approach
2.1 Finite Element Method for Free Vibration Analysis of Plate Structures
2.2 Modal Strain Energy Method
2.3 Genetic Algorithm
2.4 Flow Chart
3 Numerical Verification
3.1 Properties of Plate
3.2 Modal Analysis
3.3 Identification of the Damage’s Location
3.4 Identification of the Damage’s Extent
4 Conclusions
References
Auxetic Structure Design: A Multi-material Topology Optimization with Energy-Based Homogenization Approach
1 Introduction
2 Homogenization with Periodic Boundary Conditions
3 Alternating Active Phase Algorithm (AAPA) of Topology Optimization
4 Optimization Model
5 Numerical Examples
5.1 Single-material Auxetic Patterns
5.2 Multi-material Auxetic Patterns
5.3 Benchmark Verification
6 Conclusions
References
Fuzzy Structural Identification of Bar-Type Structures Using Differential Evolution
1 Introduction
2 Fuzzy Fe Model Updating
2.1 FE Model Updating Procedure
2.2 Fuzzy Concept of Uncertainty
2.3 Updating Fuzzy Model Parameters
3 Differential Evolution with Nearest Neighbor Comparison
3.1 Basic of Differential Evolution
3.2 Nearest Neighbor Comparison Method
4 Numerical Examples
4.1 2D Truss
4.2 2D Frame
5 Conclusion
References
Application of Artificial Intelligence for Structural Optimization
1 Introduction
2 Problem Description
3 Building the Deep Neural Network Classification Model
4 Structural Optimization Using Differential Evolution
5 Conclusions
References
A Sliding Mode Controller for Force Control of Magnetorheological Haptic Joysticks
1 Introduction
2 The Proposed 3D Haptic Joystick
3 Controller Design for the 3D Haptic Joystick
3.1 Analysis of the 3D Haptic System
3.2 Close Loop Control of the Feedback Force
3.3 Design of PID Controller for the Feedback Force
3.4 Design of SMC Controller for the Feedback Force
4 Results and Discussions
5 Conclusions
References
Experimental and Numerical Investigations on Flexural Behavior of Retrofitted Reinforced Concrete Beams with Geopolymer Concrete Composites
1 Introduction
2 Research Significance
3 Experimental Work
4 Finite Element Modeling
5 Comparison Experimental and Analytical Results
6 Conclusion
References
A Study of Fluid-Structure Interaction of Unsteady Flow in the Blood Vessel Using Finite Element Method
1 Introduction
2 Governing Equations and the Finite Element Method
2.1 Governing Equations
2.2 Finite Element Formulation for FSI Coupling
3 Results and Discussions
3.1 Validation
3.2 Blood Flow in a Carotid Bifurcation
3.3 Blood Flow Through the Aortic Valve
4 Conclusion
References
Stable Working Condition and Critical Driving Voltage of the Electrothermal V-Shaped Actuator
1 Introduction
2 Theoretical Model of EVA
2.1 Configuration of EVA
2.2 Heat Transfer Model
2.3 Thermal Expansion Force
3 Critical Voltages of Mechanical and Thermal Stability
3.1 Critical Voltage for Beam-Buckling Condition
3.2 Critical Voltage for Thermal Stability Condition
4 Safe Working Condition of the V-Shaped Actuator
5 Conclusion
References
Electromechanical Properties of Monolayer Sn-Dichalcogenides
1 Introduction
2 Computational Methods
3 Results and Discussion
4 Conclusions
References
Author Index
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Lecture Notes in Mechanical Engineering
Nguyen Tien Khiem Tran Van Lien Nguyen Xuan Hung Editors
Modern Mechanics and Applications Select Proceedings of ICOMMA 2020
Lecture Notes in Mechanical Engineering Series Editors Francisco Cavas-Martínez, Departamento de Estructuras, Universidad Politécnica de Cartagena, Cartagena, Murcia, Spain Fakher Chaari, National School of Engineers, University of Sfax, Sfax, Tunisia Francesco Gherardini, Dipartimento di Ingegneria, Università di Modena e Reggio Emilia, Modena, Italy Mohamed Haddar, National School of Engineers of Sfax (ENIS), Sfax, Tunisia Vitalii Ivanov, Department of Manufacturing Engineering Machine and Tools, Sumy State University, Sumy, Ukraine Young W. Kwon, Department of Manufacturing Engineering and Aerospace Engineering, Graduate School of Engineering and Applied Science, Monterey, CA, USA Justyna Trojanowska, Poznan University of Technology, Poznan, Poland Francesca di Mare, Institute of Energy Technology, Ruhr-Universität Bochum, Bochum, Nordrhein-Westfalen, Germany
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Nguyen Tien Khiem Tran Van Lien Nguyen Xuan Hung •
•
Editors
Modern Mechanics and Applications Select Proceedings of ICOMMA 2020
123
Editors Nguyen Tien Khiem Vietnam Academy of Science and Technology Hanoi, Vietnam
Tran Van Lien National University of Civil Engineering Hanoi, Vietnam
Nguyen Xuan Hung Ho Chi Minh City University of Technology (HUTECH) Ho Chi Minh, Vietnam
ISSN 2195-4356 ISSN 2195-4364 (electronic) Lecture Notes in Mechanical Engineering ISBN 978-981-16-3238-9 ISBN 978-981-16-3239-6 (eBook) https://doi.org/10.1007/978-981-16-3239-6 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore
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Contents
A Dual Approach for Modeling Two- and One-Dimensional Solute Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Hung Nguyen
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Rayleigh Quotient for Longitudinal Vibration of Multiple Cracked bar and Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nguyen Tien Khiem, Nguyen Minh Tuan, and Pham Thi Ba Lien
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Vibroacoutic Behavior of Finite Composite Sandwich Plates with Foam Core . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tran Ich Thinh and Pham Ngoc Thanh
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Crystal Structure and Mechanical Properties of 3D Printing Parts Using Bound Powder Deposition Method . . . . . . . . . . . . . . . . . . . . . . . . Do H. M. Hieu, Do Q. Duyen, Nguyen P. Tai, Nguyen V. Thang, Ngo C. Vinh, and Nguyen Q. Hung Experimentally Investigating the Resonance of the Vibration of Two Masses One Spring System Under Different Friction Conditions . . . . . . Quoc-Huy Ngo, Ky-Thanh Ho, and Khac-Tuan Nguyen Nonlinear Bending Analysis of FG Porous Beams Reinforced with Graphene Platelets Under Various Boundary Conditions by Ritz Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dang Xuan Hung, Huong Quy Truong, and Tran Minh Tu Self-vibrational Analysis of a Tensegrity . . . . . . . . . . . . . . . . . . . . . . . . Buntara S. Gan
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Free Vibration of a Bi-directional Imperfect Functionally Graded Sandwich Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 Le Thi Ha
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Free Vibration of Prestress Two-Dimensional Imperfect Functionally Graded Nano Beam Partially Resting on Elastic Foundation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 Le Thi Ha Design and Hysteresis Modeling of a New Damper Featuring Shape Memory Alloy Actuator and Wedge Mechanism . . . . . . . . . . . . . . . . . . 125 Duy Q. Bui, Hung Q. Nguyen, Vuong L. Hoang, and Dai D. Mai A High-Order Time Finite Element Method Applied to Structural Dynamics Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 Thanh Xuan Nguyen and Long Tuan Tran A Multibody Dynamics Approach to Study an Insect-Wing Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 Vu Dan Thanh Le, Anh Tuan Nguyen, Ngoc Thanh Dang, and Binh Van Phung Natural Ventilation and Cooling of a House with a Solar Chimney Coupled with an Earth – To – Air Heat Exchanger . . . . . . . . . . . . . . . . 158 Viet Tuan Nguyen, Y Quoc Nguyen, and Trieu Nhat Huynh Dynamic Analysis of Curved Beam on Elastic Foundation Subjected to Moving Oscillator Using Finite Element Method . . . . . . . . . . . . . . . . 169 Nguyen Thai Chung, Duong Thi Ngoc Thu, and Nguyen Van Dang Buckling Analysis of FG GPLRC Plate Using a Naturally Stabilized Nodal Integration Meshfree Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 Chien H. Thai, P. Phung-Van, and H. Nguyen-Xuan Low Temperature Effect on Dynamic Mechanical Behavior of High Damping Rubber Bearings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 Nguyen Anh Dung and Le Trung Phong Effects of Clayey Soil and Hemp Fibers on the Flexural Behavior of Soil Concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 Duc Chinh Ngo and Ngoc Tan Nguyen Optimal Design of Functionally Graded Sandwich Porous Beams for Maximum Fundamental Frequency Using Metaheuristics . . . . . . . . 229 Hoang-Anh Pham, Truong Q. Huong, and Hung X. Dang Nonlinear Buckling Behavior of FG-CNTRC Toroidal Shell Segments Stiffened by FG-CNTRC Stiffeners Under External Pressure . . . . . . . . . 240 Tran Quang Minh, Vu Minh Duc, and Nguyen Thi Phuong Nonlinear Vibration of Shear Deformable FG-CNTRC Plates and Cylindrical Panels Stiffened by FG-CNTRC Stiffeners . . . . . . . . . . 256 Dang Thuy Dong, Tran Quang Minh, and Vu Hoai Nam
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Vibration Characteristics of Functionally Graded Carbon Nanotube-Reinforced Composite Plates Submerged in Fluid Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 Duong Thanh Huan, Tran Huu Quoc, Vu Van Tham, and Chu Thanh Binh Large Amplitude Free Vibration Analysis of Functionally Graded Sandwich Plates with Porosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 Tran Huu Quoc, Duong Thanh Huan, and Ho Thi Hien Dynamic Analysis of a Functionally Greded Sandwich Beam Traversed by a Moving Mass Based on a Refined Third-Order Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 Le Thi Ngoc Anh, Tran Van Lang, Vu Thi An Ninh, and Nguyen Dinh Kien Thermal Bending Analysis of FGM Cylindrical Shells Using a Quasi-3D Type Higher-Order Shear Deformation Theory . . . . . . . . . 316 Tran Ngoc Doan, Tran Van Hung, and Duong Van Quang Experimental and Finite Element Analysis of High Strength Steel Fiber Concrete – Timber Composite Beams Subjected to Flexion . . . . . 331 Ngoc Tan Nguyen, Van Dang Tran, Viet Duc Nguyen, and Dong Tran Free Vibration of Stiffened Functionally Graded Porous Cylindrical Shell Under Various Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . 347 Tran Huu Quoc, Vu Van Tham, and Tran Minh Tu Effects of Design Parameters on Dynamic Performance of a Solenoid Applied for Gas Injector . . . . . . . . . . . . . . . . . . . . . . . . . . 362 Nguyen Ba Hung and Ocktaeck Lim Practical Method for Tracking Fatigue Damage . . . . . . . . . . . . . . . . . . 373 Nguyen Hai Son An Assessment for Fatigue Strength of Shaft Parts Manufactured from Two Phase Steel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382 Tang Ha Minh Quan and Dang Thien Ngon Advanced Signal Decomposition Methods for Vibration Diagnosis of Rotating Machines: A Case Study at Variable Speed . . . . . . . . . . . . . 393 Nguyen Trong Du and Nguyen Phong Dien Nonlinear Effects of the Material in the Pier Dam Experiment Under Cyclic Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401 N. Van Xuan, N. Canh Thai, N. Ngoc Thang, and T. Van Toan Experimental and Numerical Investigations on the Fracture Response of Tubular T-joints Under Dynamic Mass Impact . . . . . . . . . 416 Quang Thang Do, Sang-Rai Cho, and Van Dinh Nguyen
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Assessment of Reduced Strength of Existing Mooring Lines of Floating Offshore Platforms Taking into Account Corrosion . . . . . . . 431 Pham Hien Hau and Vu Dan Chinh An Investigation on Behaviors of Mass Concrete in Cua-Dai Extradosed Bridge Due to Hydration Heat . . . . . . . . . . . . . . . . . . . . . . 446 Nguyen Van My and Vo Duy Hung A Nonlocal Dynamic Stiffness Model for Free Vibration of Functionally Graded Nanobeams . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458 Tran Van Lien, Ngo Trong Duc, Tran Binh Dinh, and Nguyen Tat Thang Dynamic Performance of Highway Bridges: Low Temperature Effect and Modeling Effect of High Damping Rubber Bearings . . . . . . . 476 Nguyen Anh Dung and Nguyen Vinh Sang Nonlinear Vibration Analysis for FGM Cylindrical Shells with Variable Thickness Under Mechanical Load . . . . . . . . . . . . . . . . . . . . . 489 Khuc Van Phu, Dao Huy Bich, and Le Xuan Doan Nonlinear Dynamic Stability of Variable Thickness FGM Cylindrical Shells Subjected to Mechanical Load . . . . . . . . . . . . . . . . . . 506 Khuc Van Phu, Dao Huy Bich, and Le Xuan Doan Vibration and Dynamic Impact Factor Analysis of the Steel Truss Bridges Subjected to Moving Vehicles . . . . . . . . . . . . . . . . . . . . . . . . . . 522 Toan Nguyen-Xuan, Loan Nguyen-Thi-Kim, Thao Nguyen-Duy, and Duc Tran-Van An Experimental Study on the Self-propelled Locomotion System with Anisotropic Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537 Van-Du Nguyen, Ky-Thanh Ho, Ngoc-Tuan La, Quoc-Huy Ngo, and Khac-Tuan Nguyen Investigation of Maneuvering Characteristics of High-Speed Catamaran Using CFD Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546 Thi Loan Mai, Myungjun Jeon, Seung Hyeon Lim, and Hyeon Kyu Yoon Optimum Tuning of Tuned Mass Dampers for Acceleration Control of Damped Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 560 Huu-Anh-Tuan Nguyen Optimal Electrical Load of the Regenerative Absorber for Ride Comfort and Regenerated Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573 La Duc Viet, Nguyen Cao Thang, Nguyen Ba Nghi, and Le Duy Minh Aerodynamic Responses of Indented Cable Surface and Axially Protuberated Cable Surface with Low Damping Ratio . . . . . . . . . . . . . 584 Hoang Trong Lam and Vo Duy Hung
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The Equation Connecting Boyle’s and Bernoulli’s Laws . . . . . . . . . . . . 596 Nguyen Ngoc Tinh and Nguyen Linh Ngoc Effects of the Computational Domain Sizes on the Simulated Air Flow in Solar Chimneys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 606 Trieu Nhat Huynh and Y. Quoc Nguyen A Solar Chimney for Natural Ventilation of a Three – Story Building . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617 Tung Van Nguyen, Y Quoc Nguyen, and Trieu Nhat Huynh Solar Chimneys for Natural Ventilation of Buildings: Induced Air Flow Rate Per Chimney Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 631 Y Quoc Nguyen and Trieu Nhat Huynh Enhancing Ventilation Performance of a Solar Chimney with a Stepped Absorber Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 641 Y Quoc Nguyen and Trieu Nhat Huynh The Effect of Ground Surface Geometry on the Wing Lift Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653 Dinh Khoi Tran and Anh Tuan Nguyen Design a Small, Low-Speed, Closed-Loop Wind Tunnel: CFD Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663 Phan Thanh Long A Size-Dependent Meshfree Approach for Free Vibration Analysis of Functionally Graded Microplates Using the Modified Strain Gradient Elasticity Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673 Lieu B. Nguyen, Chien H. Thai, and H. Nguyen-Xuan Static Behavior of Functionally Graded Sandwich Beam with Fluid-Infiltrated Porous Core . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 691 Tran Quang Hung, Do Minh Duc, and Tran Minh Tu Failure Analysis of Pressurized Hollow Cylinder Made of Cohesive-Frictional Granular Materials . . . . . . . . . . . . . . . . . . . . . . . 707 Trung-Kien Nguyen Studying the Strength of an Acidic Soil-Cement Mixing in Laboratory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 716 Tham Hong Duong, Huan NguyenPhu Vo, and Danh Thanh Tran Effects of Test Specimens on the Shear Behavior of Mortar Joints in Hollow Concrete Block Masonry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 731 Thi Loan Bui Nonlinear Buckling and Postbuckling of ES-FG Porous Cylindrical Shells Under External Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 743 Le Kha Hoa, Pham Van Hoan, Bui Thi Thu Hoai, and Do Quang Chan
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Bulk Modulus Prediction of Particulate Composite with Spherical Inclusion Surrounded by a Graded Interphase . . . . . . . . . . . . . . . . . . . 755 Nguyen Duy Hung and Nguyen Trung Kien Post-buckling Response of Functionally Graded Porous Plates Rested on Elastic Substrate via First-Order Shear Deformation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 761 Le Thanh Hai, Nguyen Van Long, Tran Minh Tu, and Chu Thanh Binh Elastic Buckling Behavior of FG Polymer Composite Plates Reinforced with Graphene Platelets Using the PB2-Ritz Method . . . . . . 780 Xuan Hung Dang, Dai Hao Tran, Minh Tu Tran, and Thanh Binh Chu Design and Fabrication of Mecanum Wheel for Forklift Vehicle . . . . . . 795 Thanh-Long Le, Dang Van Nghin, and Mach Aly Muti Object Prediction and Optimization Process Parameters in Cooling Slope Using Taguchi-Grey Relational Analysis . . . . . . . . . . . 811 Anh Tuan Nguyen, Dang Giang Lai, and Van Luu Dao The Effect of Porosity on the Elastic Modulus and Strength of Pervious Concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 823 Viet-Hung Vu, Bao-Viet Tran, Viet-Hai Hoang, and Thi-Huong-Giang Nguyen Predicting Capacity of Defected Pipe Under Bending Moment with Data-Driven Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 830 Hieu C. Phan, Nang D. Bui, Tiep D. Pham, and Huan T. Duong Image Recognition Using Unsupervised Learning Based Automatic Fuzzy Clustering Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 841 V. V. Tai and L. T. K. Ngoc The Effect of the Non-uniform Temperature and Length to Height Ratio on Elastic Critical Buckling of Steel H-Beam . . . . . . . . . . . . . . . . 857 Xuan Tung Nguyen, Myung Jin Lee, Thac Quang Nguyen, and Jong Sup Park On the Thin-Walled Theory’s Application to Calculate the Semi-enclosed Core Structure of High-Rise Buildings . . . . . . . . . . . 866 Nguyen Tien Chuong and Doan Xuan Quy Numerical Simulation of Full-Scale Square Concrete Filled Steel Tubular (CFST) Columns Under Seismic Loading . . . . . . . . . . . . . . . . . 875 Hao D. Phan, Ker-Chun Lin, and Hieu T. Phan Numerical Modeling of Shear Behavior of Reinforced Concrete Beams with Stirrups Corrosion: Finite Element Validation and Parametric Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 890 Tan N. Nguyen and Kien T. Nguyen
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Characterization of Stress Relaxation Behavior of Poscable-86 High-Strength Steel Wire . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 905 Ngoc-Vinh Nguyen and Viet-Hung Truong Seismic Performance of Concrete Filled Steel Tubular (CFST) Columns with Variously Axial Compressive Loads . . . . . . . . . . . . . . . . 915 Hao D. Phan and Ker-Chun Lin Damage Simulation Based on the Phase Field Method of Porous Concrete Material at Mesoscale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 926 Hoang-Quan Nguyen, Ba-Anh Le, and Bao-Viet Tran Constitutive Response and Failure Mechanism of Porous Cement-Based Materials Under Triaxial Stress States . . . . . . . . . . . . . . 935 Thang T. Nguyen, Lien V. Tran, Tung T. Pham, and Long N. Tran Development of Automatic Landing Control Algorithm for FixedWing UAVs in Longitudinal Channel in Windy Conditions . . . . . . . . . . 945 Hong Son Tran, Duc Cuong Nguyen, Thanh Phong Le, and Chung Van Nguyen An Assessment of Terrain Quality and Selection Model in Developing Landslide Susceptibility Map – A Case Study in Mountainous Areas of Quang Ngai Province, Vietnam . . . . . . . . . . . 959 Doan Viet Long, Nguyen Chi Cong, Nguyen Tien Cuong, Nguyen Quang Binh, and Vo Nguyen Duc Phuoc Analysis of the Impact of Ponds Locations in Flood Cutting from a Developing City in Vietnam . . . . . . . . . . . . . . . . . . . . . . . . . . . . 971 Quang Binh Nguyen, Duc Phuoc Vo, Hoang Long Dang, Hung Thinh Nguyen, and Ngoc Duong Vo Turning Electronic and Optical Properties of Monolayer Janus Sn-Dichalcogenides By Biaxial Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . 981 Vuong Van Thanh, Nguyen Thuy Dung, Le Xuan Bach, Do Van Truong, and Nguyen Tuan Hung A Meshfree Method Based on Integrated Radial Basis Functions for 2D Hyperelastic Bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 990 Thai Van Vu, Nha Thanh Nguyen, Minh Ngoc Nguyen, Thien Tich Truong, and Tinh Quoc Bui Structural Damage Identification of Plates Using Two-Stage Approach Combining Modal Strain Energy Method and Genetic Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1004 Thanh-Cao Le and Duc-Duy Ho Auxetic Structure Design: A Multi-material Topology Optimization with Energy-Based Homogenization Approach . . . . . . . . . . . . . . . . . . . 1018 T. M. Tran, Q. H. Nguyen, T. T. Truong, T. N. Nguyen, and N. M. Nguyen
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Fuzzy Structural Identification of Bar-Type Structures Using Differential Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1033 Ba-Duan Nguyen and Hoang-Anh Pham Application of Artificial Intelligence for Structural Optimization . . . . . . 1052 Tran-Hieu Nguyen and Anh-Tuan Vu A Sliding Mode Controller for Force Control of Magnetorheological Haptic Joysticks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1065 Diep B. Tri, Le D. Hiep, Vu V. Bo, Nguyen T. Nien, Duc -Dai Mai, and Nguyen Q. Hung Experimental and Numerical Investigations on Flexural Behavior of Retrofitted Reinforced Concrete Beams with Geopolymer Concrete Composites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1078 Do Van Trinh and Khong Trong Toan A Study of Fluid-Structure Interaction of Unsteady Flow in the Blood Vessel Using Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 1089 S. T. Ha, T. D. Nguyen, V. C. Vu, M. H. Nguyen, and M. D. Nguyen Stable Working Condition and Critical Driving Voltage of the Electrothermal V-Shaped Actuator . . . . . . . . . . . . . . . . . . . . . . . 1102 Kien Trung Hoang and Phuc Hong Pham Electromechanical Properties of Monolayer Sn-Dichalcogenides . . . . . . 1113 Le Xuan Bach, Vuong Van Thanh, Hoang Van Bao, Do Van Truong, and Nguyen Tuan Hung Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1121
A Dual Approach for Modeling Two- and One-Dimensional Solute Transport The Hung Nguyen(B) Faculty of Water Resources Engineering, University of Science and Technology, The University of Danang, Da Nang, Vietnam [email protected]
Abstract. The solute transport equation of one-dimensional (1D) or twodimensional vertical (2DV) flow is normally constructed by the classic average method. These solute transport equations are integrated from the right to the left river bank; the average values received by this approach therefore do not generalize by means of dual approach. This paper presents the application of a dual approach to establish the 2DV solute transport equation. In particular, the concentration in a 2DV flow is obtained by twice integrals: (i) integration from the right river bank to the intermediate vertical surface layer between the right bank and the left bank, and then (ii) integration from the right bank to the left bank. From the twodimensional horizontal (2DH) [7] and 2DV flow constructed by a dual approach, the researcher receives the 1D flow equation. The average concentration obtained from the dual approach is better than the classical approach, particularly, in the case of mixed solute transport, stratification, and etc. The basic equation obtained is based on the dual approach that describes the solute transport is more accurate than the classical method. In other words, it provides some flexible parameters to adjust based on the field or experimental data. A case study of solute transport (salinity transport) in Huong river system is illustrated. Keywords: Dual approach · Two-dimensional solute transport equation · Two-dimensional vertical average concentration · One-dimensional solute transport equation
1 Introduction The solute transport of the 2DV flow equations is usually obtained by integrating the three-dimensional (3D) flow equation from the right to the left bank of a river cross-session [9]; in the 2DV equations, the average values are known as the global average values (GAV). Based on the document [1], the global-local average values (GLAV) obtained using the dual approach are more general than the GAV. Therefore, in this paper, the 2DV solute transport equation is established based on the dual approach [1, 2], the 3D solute transport equation is integrated twice; the first integration is from the right bank to an intermediate vertical surface layer between the right and the left river
© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 1–12, 2022. https://doi.org/10.1007/978-981-16-3239-6_1
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T. H. Nguyen
bank, then the second integration is from the right bank to the left bank. The average concentration value obtained from this 2DV solute transport equation is more accurate than the classical approximation. It can be described many complex physical phenomena of solute transport such as the stratification flow, mixed solute transport, etc. The numerical calculation results compared with the field data of the 1D salinity intrusion problem are also illustrated to further clarify the research problem.
2 Establishing a 2DV Solute Transport Equation Based on the Dual Approach The 3D solute transport equation in the form of conservation reads [9]: ∂(uc) ∂(vc) ∂(wc) ∂ ∂c ∂ ∂c ∂ ∂c ∂c (D + εx ) (D + εy ) (D + εz ) + + + = + + ∂t ∂x ∂y ∂z ∂x ∂x ∂y ∂y ∂z ∂z (I )
(J )
(K)
(L)
(M )
(N )
(1)
(P)
Where t is the time; x, y, z are the Cartesian coordinates system; u, v, w are the velocity components in the x, y, and z directions respectively; c is the solute transport conservative scalar variable; D is the molecular diffusion coefficient; εx , εy , εz , are the turbulent diffusion coefficients. Firstly, integrating the term (I ) from the right bank Y1 to the intermediate vertical surface layer Ym located between the right bank Y1 and the left bank Y2 and at a distance b1 from the right bank Y1 (see Fig. 1), we get: Ym TI = Y1
∂ ∂c dy = ∂t ∂t
Ym
Ym cdy −
Y1
c Y1
∂ ¯ ∂(dy) ¯ 1 . ∂b1 = (Cb 1) − C ∂t ∂t ∂t
(2)
In Fig. 1 b is the width of the river section at the free surface; b1 is the width at the free surface from at which the concentration Cm is almost equal to zero; Y1 , Ym and Y2 are the Cartesian coordinates corresponding.
Fig. 1. Sketch of the dual approach of the 2DV solute transport equations
A Dual Approach for Modeling Two- and One-Dimensional Solute
3
Where C¯ and C¯ 1 are the average concentrations at the width b and b1 respectively is given: ⎛ ⎞ Y2 Ym Y2 Ym 1⎜ 1 ⎟ 1 ¯ cdy = ⎝ cdy + cdy ⎠ = cdy (3a) C= b b b Y1
Y1
Ym
Y1
(Because, from Ym to Y2 , C ≈ 0). Posing C1 is average concentration, such that: Ym Y1
∂(dy) = C1 c. ∂t
C1 =
Y1
∂(dy) ∂b1 = C1 ∂t ∂t
Ym
∂b ∂t Y 1
c.
∂(dy) ∂t
(3b)
C¯ 1 = α1 .C¯
(3c)
Y1 1 C¯ 1 ∂(dy) = ∂b c. α1 = ¯ ¯ ∂t C C ∂t
(3d)
Posing:
Where
1
Ym
Y1
Then, the term I in Eq. (1) is integrated the second time from the right bank Y1 to the left bank Y2 , we get: Y2 T 2I = Y1
Y2 Y2 ∂ ¯ ∂ ¯ ∂b ∂b1 1 ¯ (C.b1 ) − C1 . db1 = (Cb1 ) db1 − C¯ 1 . db1 ∂t ∂t ∂t ∂t Y1 Y1 I1
Y2 T I1 = 2
Y1
I2
Y2 Y2 ∂ ¯ ∂ ¯ ¯ 1 . ∂ (db1 ) (Cb1 ) db1 = Cb1 .db1 − Cb ∂t ∂t ∂t Y1 Y1 I11
(4a)
(4b)
I12
Where: I11 =
1 ∂ ¯ 2 . C(Y2 − Y12 ) 2 ∂t
Y2 ∂ 2 ∂ ∂b1 1 ¯ 2 ∼ ¯ ¯ = Cb1 . (db1 ) = I2 = C1 . db1 = .C12 (Y − Y1 ) ∂t ∂t 2 ∂t 2 Y2 Y1
(4c)
Y2
I12
I12
I2
(4d)
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T. H. Nguyen
¯ C¯ 12 = α12 .C.
Let:
(4e)
Y2
∂ C¯ 1 . (ydy) ∂t Y1 ⎧ ⎫ ⎪ Y2 ⎪ Y2 ⎨ 1 2 ∂(dy) ⎬ ∂ = ∂ 2 . (ydy) . c. ∂b ⎪ ∂t ⎪ C¯ ∂t (Y2 − Y12 ) ⎩ ∂t ⎭ ∂t
Where: α12 =
2
C¯ ∂t∂ (Y22
− Y12 )
.
Y1
Y1
With: T 2 I = I1 − I2 = (I11 − I12 ) − I2 = I11 − 2I12 T 2I =
(4f)
∂ 2 1 ∂ ¯ 2 . C(Y2 − Y12 ) − α12 C¯ (Y2 − Y12 2 ∂t ∂t
(4g) (5a)
In the non-conservative form: 1 1 ∂ ¯ ¯ ∂ (Y22 − Y12 ) − α12 C. T 2I ∼ = (Y22 − Y12 ). (C) 2 ∂t 2 ∂t
(5b)
Similarly, the second term J is integrated the first time from the right bank Y1 to the intermediate vertical surface layers Ym , we get: Ym TJ = Y1
∂ ∂uc .dy = ∂x ∂x
Ym
Ym ucdy −
Y1
uc Y1
∂ ∂ ∂ (dy) = (b1 .UC) − β1 .UC. (b1 ) (6a) ∂x ∂x ∂x
Where: UC =
β1 =
1
1 h1
Ym
Y1 Ym
.
uc
UC ∂b ∂x Y 1
(uc).dz
(6b)
∂ (dy) ∂x
(6c)
The term J of Eq. (1) is integrated the second time from the right bank to the left bank: T 2J =
Y2 Y1
∂ (b .UC)dy − ∂x 1
J1
Where: J1 =
Y2 Y1
∂ ∂x (b1 .UC)dy
β1 .UC.
Y1
∂ (b )dy ∂x 1
(7a)
J2
Y2 Y2 ∂ ∂ = b1 .UCdy − β1 b1 UC (dy) ∂x ∂x Y1 Y1 J11
J1 =
Y2
J12
∂ 1 1 ∂ β2 UC (Y22 − Y12 ) − β1 β3 UC (Y22 − Y12 ) ∂x 2 2 ∂x J11
J12
(7b)
A Dual Approach for Modeling Two- and One-Dimensional Solute
5
Noticed that: J2 = J12 .
(7c)
So: T2J = J1 − J1 = (J11 − J12 ) − J2 = J11 − 2J12
(8a)
J1 =
1 ∂ ∂ β2 UC.(Y22 − Y12 ) − β1 β3 UC (Y22 − Y12 ) 2 ∂x ∂x Where: β2 =
β3 =
Y2
1
UCb1 dy
2
UC b2
− Y12 )
(8c)
Y1
Y2
1 ∂ UC ∂x (Y22
(8b)
UC Y1
∂ (ydy) ∂x
(8d)
Again, the fourth term L is integrated on the left-hand side of the Eq. (1) the first time from Y1 to Ym , we get: Ym TL = Y1
Ym Ym ∂ ∂ ∂ (wc)dy = (wc)dy − (wc) (dy) ∂z ∂z ∂z Y1 Y1 L1
∂ Where: L1 = ∂z Ym L2 =
(wc) Y1
(9a)
L2
Ym (wc)dy = Y1
∂ (WC.b1 ) ∂z
(9b)
∂ ∂ ∂ (dy) = δ1 .(WC). (Ym − Y1 ) = δ1 .(WC). (b1 ) ∂z ∂z ∂z
(9c)
∂ ∂ (WC.b1 ) + δ1 .(WC). (b1 ) ∂z ∂z
(9d)
Thus: TL =
1 Where: WC = b1
δ1 =
1
Ym (wc)dy Y1
Ym
∂ (WC). ∂z (b1 ) Y
(WC) 1
(10a)
∂ (dy) ∂z
(10b)
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T. H. Nguyen
Similarly, the term L of the Eq. (1) is integrated the second time, we obtain: Y2
Y2 ∂ ∂ T L= (WC.b1 )dy − δ1 .(WC). (b1 )dy ∂z ∂z Y1 Y1 2
L11
Y2 L11 = Y1
L22
Y2 Y2 ∂ ∂ ∂ (WC.b1 )dy = WC.b1 .dy − (WC).b1 (dy) ∂z ∂z ∂z Y1 Y1 L111
L11
(11a)
L112
∂ 1 2 1 ∂ 2 δ2 .WC. (Y2 − Y1 ) − δ3 .WC. . (Y22 − Y12 ) = ∂z 2 2 ∂z
(11b)
L112
L111
We have: L22 ≈ L112 . Therefore: T2L = L11 − L22 = L111 − L112 − L22 = L111 − 2L112
(11c)
The correction coefficients δ2 , δ3 are calculated as follows: 2 δ2 = 2 (Y2 − Y12 ) δ3 =
2 ∂ 2 ∂z (Y2
− Y12 )
Y2 bdy
(11d)
Y1
Y2 Y1
∂ (bdy) ∂z
(11e)
The third term K is integrated on the left-hand side of Eq. (1), the first time, we obtain: Ym TK = Y1
∂(vc) dy = 0 (noted that v ≈ 0) ∂y
(12)
The first term M is integrated on the right-hand side of Eq. (1), the first time, we get: Ym ∂ ∂ ∂c ∂ C¯ (D + εx ). dy = γ1 TM = (13a) (D + εx ). .(Ym − Y1 ) ∂x ∂x ∂x Y1 ∂x
A Dual Approach for Modeling Two- and One-Dimensional Solute
7
Where: γ1 =
∂ ∂x
∂ ∂c (D + ε dy . ). x ¯ ∂x (D + εx ). ∂∂xC .(Ym − Y1 ) Y ∂x Ym
1
(13b)
1
The term M of Eq. (1) is integrated the second time, we receive: Y2 T 2M = Y1
∂ ∂ ∂ C¯ 1 ∂ C¯ γ1 (D + εx ). .b1 dy = .γ1 .γ2 (D + εx ). .(Y22 − Y12 ) ∂x ∂x 2 ∂x ∂x (14a)
The coefficient γ2 is calculated by: 2 γ2 = 2 . (Y2 − Y12 ) Thus:
Y2 b.dy
(14b)
Y1
¯ ∂ 1 ∂ C T 2 M = .γ1 .γ2 (D + εx ). .(Y22 − Y12 ) 2 ∂x ∂x
(14c)
Similarly, the third term P on the right-hand side of Eq. (1) is integrated, the first time, we get: Ym TP = Y1
∂ ∂ ∂c ∂ C¯ (D + εz ). dy = θ1 (D + εz ). .(Ym − Y1 ) ∂z ∂z ∂z ∂z
(15a)
Where: θ1 =
∂ ∂z
Ym
1 ¯
(D + εz ). ∂∂zC .(Ym − Y1 )
. Y1
∂ ∂c (D + εz ). dy ∂z ∂z
(15b)
The term P of Eq. (1) is integrated, the second time, we obtain: Y2 T P= 2
Y1
∂ ∂ ∂ C¯ 1 ∂ C¯ θ1 (D + εz ). .b1 dy = .θ1 .θ2 (D + εz ). .(Y22 − Y12 ) ∂z ∂z 2 ∂z ∂z (15c)
The coefficient θ2 is calculated as: 2 θ2 = 2 . (Y2 − Y12 )
Y2 bdy Y1
(15d)
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T. H. Nguyen
Therefore:
∂ 1 ∂ C¯ T P = .θ1 .θ2 (D + εz ). .(Y22 − Y12 ) 2 ∂z ∂z 2
(15e)
The second term N on the right-hand side of Eq. (1) is integrated, the first time, we get: Ym TN = Y1
∂ ∂c ∂c ∂c (D + εy ). dy = (D + εy ). y − (D + εy ). y ∂y ∂y ∂y m ∂y 1
(16a)
The term N is integrated the second time, we obtain: Y2 T N= 2
Y1
In case:
∂c ∂y y Z
= m
Y2 ∂c ∂c (D + εy ). y dy − (D + εy ). y dy ∂y m ∂y 1
(16b)
Y1
∂c ∂y y Z
, then the term: T 2 N = 0 1
Therefore, the 2DV solute transport equation based on the dual approach can be expressed as follows: T 2I + T 2J + T 2K + T 2L = T 2M + T 2N + T 2P 1 ∂ ¯ 2 ∂ 2 1 ∂ 2 2 ¯ . C(Y2 − Y1 ) − α12 C (Y2 − Y1 ) + β2 UC.(Y22 − Y12 ) 2 ∂t ∂t 2 ∂x ∂ 1 ∂ ∂ WC.(Y22 − Y12 ) − δ1 δ3 .WC (Y22 − Y12 ) − β1 β3 UC (Y22 − Y12 ) + δ2 . ∂x 2 ∂z ∂z 1 ∂ ∂ C¯ = .γ1 .γ2 (D + εx ). .(Y22 − Y12 ) 2 ∂x ∂x 1 ∂ ∂ C¯ + .θ1 .θ2 (17) (D + εz ). .(Y22 − Y12 ) 2 ∂z ∂z From Eq. (17), we received the 1D solute transport equation based on the dual approach as follows: ∂ 2 1 ∂ ¯ 2 2 2 ¯ . C(Y2 − Y1 ) − α12 C (Y − Y1 ) 2 ∂t ∂t 2 ∂ 1 ∂ UC.(Y22 − Y12 ) − β1 β3 UC (Y22 − Y12 ) + β2 (18) 2 ∂x ∂x 1 ∂ ∂ C¯ = .γ1 .γ2 (D + εx ). .(Y22 − Y12 ) 2 ∂x ∂x
A Dual Approach for Modeling Two- and One-Dimensional Solute
9
3 Comments It is easy to see that the 2DV solute transport equation averaged by the classic method is a special case of the general form Eq. (17). In fact, if Y2 − Y1 = b ∼ = const, so: ∂ 2 − Y 2 ) ∼ 0, ∂ (Y 2 − Y 2 ) ∼ 0, ∂ (Y 2 − Y 2 ) ∼ 0 (the bank width changes (Y = = = 1 1 1 ∂t 2 ∂x 2 ∂z 2 ∼ insignificantly over the time and space) and the coefficients (αi , βi , γi , δi , θi ) = 1, ∂c ∼ ∂c ∂y y m = ∂y y 1 then Eq. (17) becomes the 2DV solute transport equation averaged by the classic method Eq. (19): ∂ ¯ 2 ∂ ∂ C(Y2 − Y12 ) + UC.(Y22 − Y12 ) + WC.(Y22 − Y12 ) ∂t ∂x ∂z ¯ ∂ ∂C ∂ ∂ C¯ 2 2 = (D + εx ). .(Y2 − Y1 ) + (D + εz ). .(Y22 − Y12 ) ∂x ∂x ∂z ∂z From Eq. (19), we receive the 1D Eq. (20) as follows: ¯ ∂ ¯ 2 ∂ ∂ ∂ C C(Y2 − Y12 ) + UC.(Y22 − Y12 ) = (D + εx ). .(Y22 − Y12 ) ∂t ∂x ∂x ∂x
(19)
(20)
If this equation is combined with Eq. (20) in [6], we have the 1D solute transport equation based on the dual approach more generally as follows: ∂ ¯ 2 ∂ ¯ 2 ∂ ∂ C(Zs − Zb2 ) + C(Y2 − Y12 ) + UC.(Zs2 − Zb2 ) + UC.(Y22 − Y12 ) ∂t ∂t ∂x ∂x ¯ ∂ ∂ C¯ ∂ ∂ C = (D + εx ). .(Zs2 − Zb2 ) + (D + εx ). .(Y22 − Y12 ) ∂x ∂x ∂x ∂x (21) In the case Zs − Zb = h ∼ = const and Y2 − Y1 = b ∼ = const, Eq. (21) is simplified to: ∂ ¯ ∂ ∂ ∂ C¯ C + UC = (D + εx ). (22) ∂t ∂x ∂x ∂x Equation (22) is the familiar classical 1D solute transport equation [4, 9].
4 Results To illustrate the generality of the solute transport Eqs. (17) or (21) established by the dual approach in this paper, a case study of the salinity intrusion of Huong river system in the dry season is presented [5]. There are two branches: Ta-Trach and Huu-Trach confluencing at Nga-Ba-Tuan, upstream of the Huong river; the water flows along the mainstream of Huong river to the Thao-Long barrage at downstream [5], see Fig. 2.
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T. H. Nguyen
Fig. 2. A simplified sketch of the Huong river system
Boundary conditions: (i) Upstream boundary conditions: QBinhÐieu + QDu,o,ngHoa + QKL = (0.98 + QKL ) m3 /s; (ii) Downstream boundary condition: tidal water level (frequency P = 25%) at ThaoLong barrage, see Fig. 3.
Fig. 3. Water level boundary condition at downstream
To determine the reasonable size of Khe-Lu reservoir used for salinity repelling on the Hu,o,ng river system, all step discharge from Khe-Lu reservoir are assumed in the document [5].
A Dual Approach for Modeling Two- and One-Dimensional Solute
11
Fig. 4. Salinity at La-Y in a tidal period corresponding with the Khe-Lu discharge QKL = 10.00 m3 /s
Algorithm and Program: We apply the finite difference method using the four-point Preissmann scheme, the weight θ = 0.66, to solve the 1D Saint -Venant equations system [3–6, 8, 9], then solve the solute transport Eq. (21) and (22) using fractional steps: transport equation and diffusion equation. The characteristic method was applied to solve the transport equation and the weighted finite difference scheme for the diffusion equation [5]. The calculated results of the salinity at La-Y corresponding with the Khe-Lu discharge QKL = 10.00 m3 /s, are plotted in Fig. 4.
5 Comments Figure 4 shows that the numerical solution results based on the dual approach Eq. (21) are more accurate than the ones of the classical method Eq. (22). In addition, in the general Eq. (18) based on the dual approach, the calculation results can be better by changing the coefficients (αi , βi , γi , δi , θi ). Due to the complicated changing of the river-bed topography and the surface roughness of Huong river-bed, so the simulation results based on the dual approach still cannot capture all the observed data.
6 Conclusions The solute transport Eqs. 2DV (17) and 1D (21) derived from the dual approach are more general than those received from the classic approach with the appearance of new terms in the Eqs. (17) and (21); they could be used to describe the complex physic phenomena of solute transport better than the equations established by classical method. On the other hand, the calculated results can be better obtained by changing corrected coefficients (αi , βi , γi , δi , θi ).
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References 1. Anh, N.D.: Dual approach to averaged values of functions. Vietnam J. Mech. 34(3), 211–214 (2012) 2. Anh, N.D.: Dual approach to averaged values of functions: advanced formulas. Vietnam J. Mech. 34(4), 321–325 (2012) 3. Cunge, J.A., et al.: Practical Aspects of Computational River Hydraulics. Pitman Publishing INC (1980) 4. Nguyen, T.D.: One-dimensional mathematical model of salinity intrusion in rivers network, Physical Mathematical Dissertation, Hanoi (1987) 5. Nguyen, T.H.: Salinity intrusion in Huong river network and the measure of hydraulic construction, J. Sci. Technol. (Five University of Technology), (2), 17–21 (1992) 6. Nguyen, T.H.: A dual approach to modeling solute transport. In: The International Conference on Advances in Computational Mechanics, ACOME 2017 August 02–04, Phu quoc, Vietnam, Lecture Notes in Mechanical Engineering, pp. 821–834 (2017) 7. That, T.T., Nguyen, T.H., Nguyen, D.A.: A dual approach for model construction of twodimensional horizontal flow. In: Viet, N.T., Xiping, D., Tung, T.T. (eds.) APAC 2019: Proceedings of the 10th International Conference on Asian and Pacific Coasts, 2019, Hanoi, Vietnam, pp. 115–119. Springer Singapore, Singapore (2020). https://doi.org/10.1007/978-98115-0291-0_17 8. Preissmann, A.: Propagation des intumescences dans les canaux et Les Rivieres. Congres de l’Association Francaise de Calcule, Grenoble, France (1961) 9. Wu, W.: Computational River Dynamics. Taylor & Francis Group, London, UK (2008)
Rayleigh Quotient for Longitudinal Vibration of Multiple Cracked bar and Application Nguyen Tien Khiem1,2(B) , Nguyen Minh Tuan1 , and Pham Thi Ba Lien3 1 Institute of Mechanics, Vietnam Academy of Science and Technology, Hanoi, Vietnam
{ntkhiem,nmtuan}@imech.vast.vn
2 CIRTECH Institute, HUTECH, Ho Chi Minh City, Vietnam
[email protected]
3 University of Transport and Communications, Hanoi, Vietnam
[email protected]
Abstract. The Rayleigh quotient is an attractive relationship between natural frequency and mode shape of structural vibration. However, it would be useful only for approximately calculating natural frequencies of a structure by using the properly chosen trial functions of the mode shapes in case when both the modal parameters are unknown. This is completely appropriate for the case of cracked structures when an explicit expression of natural frequencies is needed for some purpose such as structural health monitoring. The present report is devoted first to establish Rayleigh quotient for longitudinal vibration in multiple cracked bars and then to involve it for calculating natural frequencies of the cracked structure by using mode shapes of uncracked one. Numerical examples are accomplished for accessing usability of various approximate expressions of the obtained Rayleigh quotient. Keywords: Rayleigh quotient · Natural frequencies · Multiple cracked bar · Structural health monitoring
1 Introduction The frequency-based method has proved to be an efficient technique for solving the structural damage detection problem. This is because natural frequencies are the most typical characteristic of a structure that can be easily and accurately measured in the practice. One of the drawbacks of the method is that the characteristic equations established for determining natural frequencies of damaged structures are usually in implicit form. The lack of explicit expression of natural frequencies through damage parameters obstructs solving the inverse problem of damage detection. Thus, seeking explicit expression of the characteristic equation or the natural frequencies themself with purpose to apply for solving the damage detection problem is vitally needed. A lot of studies were devoted to establish explicit expression of characteristic equation of cracked structures, but there are a few publications concerned with finding explicit expression for natural frequencies of the structures. The first formula for calculating fundamental frequency of a mechanical system was proposed by Rayleigh and it has been named Rayleigh quotient. The © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 13–25, 2022. https://doi.org/10.1007/978-981-16-3239-6_2
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N. T. Khiem et al.
Rayleigh quotient was first applied for calculating fundamental frequency of cracked beam [1] and then it has been developed for multiple cracked beam [2, 3]. The present study is devoted to establish Rayleigh quotient for multiple cracked bars and apply it for calculating natural frequencies of longitudinal vibration in the structures. Longitudinal vibration of bar with single crack was first studied in [4–8] and then it has been developed for multiple cracked bars in [9–12]. General procedure for crack detection in vibrating bars was theoretically proposed by Morassi and his coworkers [7, 8, 13], who revealed that using only natural frequencies cannot uniquely resolve the problem of crack detection even in case of single crack. The authors proposed to use additionally anti-resonant frequencies for obtaining unique solution of the crack detection problem [14–16]. General theory and application of anti-resonant frequencies for multi-crack detection in bars were proposed and discussed recently in [17] by the authors of this study. Nevertheless, as shown below, the Rayleigh quotient established in this work would be helpful to solve the non-uniqueness problem of crack detection in bars without involving the anti-resonant frequencies but using only natural frequencies. The structure of this report is arranged as follows: First, general Rayleigh quotient of multiple cracked bars is established (Subsect. 2). Then, the established formula is applied for calculating natural frequencies of multiple cracked bar (Subsect. 3) using mode shapes of uncracked one. Finally, numerical illustration and discussion are provided in Subsect. 4 for singly cracked bars with fixed-free and free-free ends.
2 Rayleigh Quotient for Multiple Cracked Bar Let’s consider an uniform bar with material and geometrical parameter E, ρ, A, L that contains n cracks at positions e1 , ..., en modeled by translational spring of stiffness K j , j = 1, ..., n, calculated from the crack depth a1 , ..., an by using the formulas [18] (Fig. 1). γ = EA/KL = 2π(1 − ν 2 )(h/L)f (z), z = a/h, f (z) = z 2 (0.6272 − 0.17248z + 5.92134z 2 − 10.7054z 3 + 31.5685z 4 − 67.47z 5 + 139.123z 6 − 146.682z 7 + 92.3552z 8 ),
Fig. 1. Model of a multiple-cracked bar
Rayleigh Quotient for Longitudinal Vibration
For the bar, natural mode of vibration is governed by the equation (x) + λ2 (x) = 0, x ∈ (0, 1), λ = ωL ρ/E,
15
(1)
that is solved together with boundary conditions at the bar ends x = 0, x = 1 and compatibility conditions at the cracked sections
Kj [(ej + 0) − (ej − 0)] = EA (ej ) = EA (ej − 0) = EA (ej + 0), j = 1, ..., n or (ej + 0) = (ej − 0) + γj (ej ), γj = EA/LKj , j = 1, ..., n.
(2)
Let solution of Eq. (1) in the segment (ej−1 , ej ), j = 1, ..., n + 1, e0 = 0, en+1 = 1 be denoted by j (x), j = 1, ..., n + 1, that must satisfy conditions (2) rewritten as. j+1 (ej + 0) = j (ej − 0) = j (ej ), j+1 (ej + 0) = j (ej − 0) + γj j (ej ), j = 1, ..., n.
(3)
Since every function j (x) satisfies Eq. (1) in the segment (ej−1 , ej ) one has got j (x) + λ2 j (x) = 0, x ∈ (ej−1 , ej ).
(4)
Multiplying both sides of Eq. (1) by j (x)and taking integration along the segment (ej−1 , ej ) lead to ej
ej
ej
j (x)j (x)dx + λ2 ej−i
ej
2j (x)dx = B ej−1 , ej − j2 (x)dx + λ2 2j (x)dx = 0 ej−i ej−i ej−i
(5)
or ej λ
ej 2j (x)dx
2
=
ej−1
j2 (x)dx − B(ej−1 , ej ),
(6)
ej−1
where B(ej−1 , ej ) = j (ej − 0)j (ej − 0) − j (ej−1 + 0)j (ej−1 + 0) . Now, summing both sides of Eq. (3) yields 1 λ
1 (x)dx =
2 0
2 (x)dx −
2
0
n+1 j=1
B(ej−1 , ej ).
(7)
16
N. T. Khiem et al.
Taking account of conditions (3) the last sum in Eq. (4) can be calculated as. n+1
B(ej−1 , ej ) = 1 (e1 − 0)1 (e1 − 0) − 1 (0)1 (0) + 2 (e2 − 0)2 (e2 − 0)
j=1
− 2 (e1 + 0)2 (e1 + 0) + 3 (e3 − 0)3 (e3 − 0) − 3 (e2 + 0)3 (e2 + 0)
..
+ ... + n (en − 0)n (en − 0) − n (en−1 + 0)n (en−1 + 0)
+ n+1 (1)n+1 (1) − n+1 (en + 0)n+1 (en + 0) = n+1 (1)n+1 (1) − 1 (0)1 (0) −
n
γj 2 (ej ).
j=1
So that we obtain 1 1 n 2 2 (x)dx = 2 (x)dx + γj 2 (ej ) − (1)(1) + (0)(0) λ 0
or
⎡
λ =⎣
1
2
(x)dx +
2
0
j=1
0
n
⎤ 2
γj (ej ) + (0)(0) − (1)(1)⎦/
j=1
1 2 (x)dx, (8) 0
that is so-called Rayleigh quotient for multiple cracked bars. Obviously, the obtained quotient is general for arbitrary boundary conditions that is represented by the term B(0, 1) = (0)(0) − (1)(1) It’ easily to verify that the latter term would be disappeared for the conventional boundary conditions (free-free ends; fixed-free ends or fixed-fixed ends), so that the Rayleigh quotient is reduced to ⎡ 1 ⎤ 1 n γj 2 (ej )⎦/ 2 (x)dx. (9) λ2 = ⎣ 2 (x)dx + j=1
0
0
In case of uncracked bar, the latter quotient becomes 1 λ20
=
2
1
0 (x)dx/ 0
20 (x)dx,
(10)
0
where ω0 , 0 (x) are natural frequency and mode shape of the uncracked bar that are provided below (Table 1) for further application. It can be verified that the quotient (10) is consistently fulfilled with the exact natural frequencies and mode shapes given in Table 1. Therefore, the Rayleigh quotient would be meaningless if both the exact frequency and mode shape are available. But it will be greatly useful for calculating natural frequencies by using trial functions of unknown mode shapes. Obviously, the natural frequencies are more accurately calculated as the trial functions are chosen more closed to the exact mode shapes. This technique is applied below for calculating natural frequencies of cracked bars.
Rayleigh Quotient for Longitudinal Vibration
17
Table 1. Natural frequencies and mode shapes of uncracked bar with conventional boundary conditions Boundary conditions
Natural frequencies
Natural mode shapes
Free-Free Ends
λ0k = kπ, k = 1, 2, 3, ....
0k (x) = Ck cos kπ x
Fixed-Free Ends
λ0k = (2k + 1)π/2, k = 0, 1, 2, ....
0k (x) = Ck sin(k + 1/2)π x
Fixed-Fixed Ends
λ0k = kπ, k = 1, 2, 3, ....
0k (x) = Ck sinkπ x
3 Application for Calculating Natural Frequencies In this section, natural frequencies of multiple cracked bars are calculated by properly choosing trial mode shapes. Namely, the mode shapes are chosen in the form j (x) = 0 (x) + Aj x + Bj , x ∈ (ej−1 , ej ), j = 1, ..., n + 1
(11)
with 0 (x) being mode shape of uncracked bar satisfying given boundary conditions and constants Aj , Bj . Substituting expression (8) into conditions (3) leads to
Aj+1 = Aj ; Bj+1 = Bj + γj 0 (ej ), j = 1, ..., n The latter relationships allow one to have got Aj = A1 ; Bj = B1 +
j−1
γk 0 (ek ), j = 1, ..., n
(12)
k=1
and, as a particularity, An+1 = A1 ; Bn+1 = B1 +
n
γk 0 (ek ), j = 1, ..., n.
(13)
γk 0 (ek ), x ∈ (ej−1 , ej ), j = 1, ..., n + 1
(14)
k=1
Consequently, j (x) = 0 (x) + A1 x + B1 +
j−1
k=1
and 1 (x) = 0 (x) + A1 x + B1 ; n+1 (x) = 0 (x) + A1 x + B1 +
n
γk 0 (ek ), x ∈ (ej−1 , ej ).
(15)
k=1
Substituting expressions (12) into boundary conditions allow the constants A1 and B1 to be determined, for example, for free-free (a), fixed-free (b) and fixed-fixed (c) end bar one gets respectively (a) A1 = 0, B1 = 0; (b)A1 = B1 = 0; (c) A1 = −
n k=1
γk 0 (ek ), B1 = 0 .
(16)
18
N. T. Khiem et al.
Now, using expressions (11) we can calculate the following integrals 1
2
(x)dx =
e n+1 j
j2 (x)dx
j=1 ej−1
0
=
e n+1 j
2
1
[0 (x) + 2A1 0 (x) + A21 ]dx
=
j=1 ej−1
02 (x)dx + 2A1 [0 (1) − 0 (0)] + A21 ; 0
(17) 1 (x)dx = 2
e n+1 j
2j (x)dx
j=1 ej−1
0
=
e n+1 j
[20 (x) + A21 x2 + Bj2 + 2A1 x0 (x) + 2Bj 0 (x) + 2A1 Bj x]dx
j=1 ej−1
1 =
+
n+1
20 (x)dx − 2 A1 /λ20 ej 0 (ej ) − 0 (ej ) − ej−1 0 (ej−1 ) + 0 (ej−1 )
0 n+1
j=1 n+1 Bj2 ej − ej−1 − 2/λ20 Bj 0 (ej ) − 0 (ej−1 )
j=1
j=1
n+1 n+1 3 2 ej3 − ej−1 + A1 Bj ej2 − ej−1 + A21 /3
j=1
1 = 0
j=1
20 (x)dx + 2λ−2 0
n
−2 γj 02 (ej ) + λ−2 0 0 + λ0 1 + 2 ,
(18)
j=1
where 0 = A21 /3 + (A1 + B1 )B1 λ20 − 2A1 0 (1) − 0 (1) + 0 (0) − 2B1 0 (1) − 0 (0) ; 1 =
n
bj γj 0 (ej ); bj = 2 λ20 2B1 + A1 − 2B1 ej − A1 ej2 − 0 (1) , j = 1, ..., n;
j=1
2 =
n j,k=1
ajk γj γk 0 (ej )0 (ek ); aij = 1 − ej , ajk = 1 for j = k
(19)
Rayleigh Quotient for Longitudinal Vibration
19
So, we obtain λ
2
/λ20
1 0
=
n
2
0 (x)dx + 1
λ20
0
γj
2
j=1
20 (x)dx
+2
(ej ) + 2A1 [0 (1) − 0 (0)] + A21
n j=1
. 2
γj 0 (ej ) + 0 + 1 + λ20 2
Recalling Eq. (13) it can be assumed that γj 2 (ej ) ≈ γj 02 (ej ), the latter equation can be rewritten as n 1 2 2 2 0 (x)dx + γj 0 (ej ) + 2A1 [0 (1) − 0 (0)] + A1 λ2 /λ20 =
j=1
0
λ20
1
20 (x)dx
0
+2
n j=1
.
(20)
2
γj 0 (ej ) + 0 + 1 + λ20 2
On the other hand, from Eq. (7) one gets
1 0
02 (x)dx = λ20
intact mode shape 0 (x) normalized by condition λ20
1 0
1 0
20 (x)dx and in case of
20 (x)dx = 1, the quotient (20)
would be simplified to 1+ λ2 /λ20 =
n j=1
γj 02 (ej ) + 2A1 [0 (1) − 0 (0)] + A21
1+2
n j=1
.
(21)
γj 02 (ej ) + 0 + 1 + λ20 2
Particularly, the obtained Rayleigh’s quotient for conventional boundary conditions: fixed-free and free-free end bars gets the form 1+ λ2 /λ20 =
n j=1
1+2
n j=1
γj 02 (ej ) n
2
γj 0 (ej ) + λ20
j,k=1
(22)
ajk γj γk 0 (ej )0 (ek )
with natural frequencies and mode shapes of uncracked bars given in Table 1. This formula is similar to that obtained by Khiem et al. [2, 3] for multiple cracked beams and provides an efficient tool for modal analysis and identification of cracked bars. Also, expanding the function in the right-hand side of Eq. (19) in Taylor’s series of crack magnitude vector γ = (γ1 , ..., γn ) and keeping first three order terms allow one to obtain λ2 /λ20 = 1 −
n j=1
γj 02 (ej ) − (λ20 /2)
n j,k=1
ajk γj γk 0 (ej )0 (ek ),
(23)
20
N. T. Khiem et al.
Fig. 2. Three lowest frequency ratios (cracked/intact) of fixed-free end bar calculated by Rayleigh quotient and first approximation compared to the ones obtained from exact characteristic equation (solid lines – exact solution of frequency equation; dash line - first approximation; dash-dot lines – second approximation; dot lines – Rayleigh quotient)
Rayleigh Quotient for Longitudinal Vibration
21
Fig. 2. (continued)
Moreover, the first order asymptotic approximation of the formula (19) λ2 /λ20 = 1 −
n
γj 02 (ej )
(24)
j=1
was obtained earlier by Morassi in [7] and Shifrin in [11, 12].
4 Numerical Illustration and Discussion For illustration of the proposed above theory, natural frequencies in longitudinal vibration of singly cracked bar with fixed-free and free-free ends are computed by using the conducted Rayleigh quotient mutually with its first and second approximations. The approximately computed frequencies are compared with those obtained by solving the exact frequency equations that allows evaluating the usefulness of the quotient for cracked bar. Results shown in Figs. 2 and 3 are ratio of natural frequencies of cracked bar to those of intact (uncracked) one computed as function of crack position along the bar length in various crack depth. For a given crack depth, there are depicted in the Figures four curves that represent the frequency ratio computed from exact frequency equation [17] and Eqs. (19)–(21) respectively. Figure 2 and Fig. 3 show the first three dimensionless frequencies of fixed-free and free-free end bar respectively. The graphs given in the Figures demonstrate clearly that discrepancy between natural frequencies computed from the Rayleigh quotient and the exact ones increases with
22
N. T. Khiem et al.
Fig. 3. Three lowest frequency ratios (cracked/intact) of free-free end bar calculated by Rayleigh quotient and first approximation compared to the ones obtained from exact characteristic equation (solid lines – exact solution of frequency equation; dash line - first approximation; dash-dot lines – second approximation; dot lines –Rayleigh)
Rayleigh Quotient for Longitudinal Vibration
23
Fig. 3. (continued)
crack depth and mode number. Namely, for crack depth less than 20% bar thickness fundamental natural frequency computed from either Rayleigh quotient or exact frequency equation are almost identical and all the frequencies computed from the first approximation (Eq. 21) of Rayleigh quotient are most closed to the exact ones in comparison with those computed from the second approximation (Eq. (20)) and even with the original Rayleigh quotient (Eq. (19)). However, the frequencies computed from Eq. (19) and (20) abolish the symmetrical effect of the cracks located symmetrically about the middle of bar with symmetric boundary conditions (Fig. 3). This is helpful for solving the non-uniqueness problem in identification of crack in bar with symmetric boundary conditions [7]. All the discussions allow one to make a concluding remark as follows: The first approximation of Rayleigh quotient is best employed for calculating natural frequencies of cracked bar, while the second approximation is more useful for crack detection in bar from natural frequencies.
5 Conclusion In the present report, so-called Rayleigh quotient is established for longitudinal vibration in multiple cracked bar that provides a simple tool for engineers to calculate natural frequencies of the damaged structures. This is then used for obtaining an explicit expression of the natural frequencies in terms of crack parameters such as position and depth of the cracks. The obtained expression not only simplifies calculating natural frequencies of cracked bars but also offers a useful tool for crack detection in bar by using natural
24
N. T. Khiem et al.
frequencies. First and second asymptotic approximations of the Rayleigh formula with respect to crack depth have been derived for more effective use in both analysis and identification of the cracked structures. Comparison of natural frequencies computed by using different approximations with those computed from the well-known exact frequency equation shows that first approximation of Rayleigh formula is well appropriate for calculating natural frequencies of cracked bar while the second one would be more useful for crack localization from measured natural frequencies. The use of the established Rayleigh quotient for the crack detection problem has not been investigated in this work, but it will be subject of further study of the authors. Acknowledgement. This work was completed with support from VAST under Grant of number NCVCC03.04/20–20.
References 1. Fernandez-Saez, J., Rubio, L., Navarro, C.: Approximate calculation of the fundamental frequency for bending vibration of cracked beams. J. Sound Vib. 225(2), 345–352 (1999) 2. Khiem, N.T., Toan, L.K.: A novel method for crack detection in beam-like structures by measurements of natural frequencies. J. Sound Vib. 333, 4084–4103 (2014) 3. Khiem, N.T., Tran, H.T., Ninh, V.T.A.: A closed-form solution to the problem of crack identification for a multistep beam based on Rayleigh quotient. Int. J. Solids Struct. 150, 154–165 (2018) 4. Adams, R.D., Cawley, P., Pye, C.J., Stone, B.J.: A vibration technique for non-destructively assessing the integrity of structures. J. Mech. Eng. Sci. 20, 93–100 (1978) 5. Narkis, Y.: Identification of crack location in vibrating simply supported beams. J. Sound Vib. 172, 549–558 (1994) 6. Gladwell, G.M.L., Morassi, A.: Estimating damage in a rod from changes in node positions. Inverse Probl. Eng. 7, 215–233 (1999) 7. Morassi, A.: Identification of a crack in a rod based on changes in a pair of natural frequencies. J. Sound Vib. 242(4), 577–596 (2001) 8. Rubio, L., Fernandez-Saez, J., Morassi, A.: Crack identification in non-uniform rods by two frequency data. Int. J. Solids Struct. 75–76, 61–80 (2015) 9. Ruotolo, R., Surace, C.: Natural frequencies of a rod with multiple cracks. J. Sound Vib. 272, 301–316 (2004) 10. Khiem, N.T., Toan, L.K., Khue, N.T.L.: Change in mode shape nodes of multiple cracked rod. Vietnam J. Mech. 35(3), 175–188 (2013) 11. Shifrin, E.J.: Inverse spectral problem for a rod with multiple cracks. Mechanical Systems and Signal Processing 56–57, 181–196 (2015) 12. Shifrin, E.I.: Identification of a finite number of small cracks in a rod using natural frequencies. Mech. Syst. Signal Process. 70–71, 613–624 (2016) 13. Rubio, L., Fernandez-Saez, J., Morassi, A.: Identification of two cracks in rod. J. Sound Vib. 339, 99–111 (2015) 14. Rubio, L., Fernandez-Saez, J., Morassi, A.: Identification of two cracks in a rod by minimal resonant and antiresonant frequency data. Mech. Syst. Sig. Process. 60–61, 1–13 (2015) 15. Dilena, M., Morassi, A.: Structural health monitoring of rods based on natural frequency and antiresonant frequency. Struct. Health Monit. 8(2), 149–173 (2009)
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16. Dilena, M., Morassi, A.: Reconstruction method for damage detection in beam based on natural frequency and anti-resonant frequency measurements. J. Eng. Mech. 136(3), 329–344 (2010) 17. Lien, P.T.B., Khiem, N.T.: Resonant and anti-resonant frequencies of multiple cracked bar. Vietnam J. Mech. VAST 41(2), 157–170 (2019) 18. Chondros, T.G., Dimarogonas, A.D., Yao, J.: Longitudinal vibration of a continuous cracked rod. Eng. Fract. Mech. 61, 593–606 (1998)
Vibroacoutic Behavior of Finite Composite Sandwich Plates with Foam Core Tran Ich Thinh1(B) and Pham Ngoc Thanh2 1 Hanoi University of Science and Technology, 1 Ðai Co Viet, Ha Noi, Viet Nam
[email protected] 2 Viettri University of Industry, Viet Tri, Phu Tho, Viet Nam
Abstract. In this study, based on a modal superposition method and Biot’s theory, an analytical model on vibroacoustic behavior of clamped and simply supported orthotropic rectangular composite sandwich plates with foam core has been derived. Theoretical predictions of sound transmission loss (STL) across finite composite sandwich plates with poroelastic material agree well with the experimental results in most frequency ranges of interest. Basing on the numerical results obtained, the influence of different parameters of the two thin laminated composite sheets and polyurethane foam core layer on STL of a sandwich plate is quantitatively evaluated and discussed in detail. Keywords: Composite sandwich plate · Poroelastic material · Sound transmission loss · Sound insulation
1 Introduction The composite sandwich plate with a foam core is widely used in many industries, including aircraft, building, motor vehicle and ships because of the advantages like high strength, light weight, low cost and good sound insulation. The sound transmission loss (STL) of sandwich structures has been the subject of many studies. As far as sound insulation improvement is concerned, poroelastic materials among various sandwich cores have been widely applied in double-wall structures due to their excellent sound absorption capability. By employing Biot’s theory [1] on wave propagation in a poroelastic medium, Bolton et al. [2, 3] modelled analytically the problem of sound transmission through infinite double-wall sandwich panels lined with poroelastic materials and validated the theoretical model against their experimental results. Panneton and Atalla [4] studied the sound transmission loss through multilayer structures made from a combination of elastic, air, and poroelastic materials. The presented approach is based on a three-dimensional finite element model. It uses classical elastic and fluid elements to model the elastic and fluid media and uses a two-field displacement formulation derived from the Biot theory [1]. Kropp and Rebillard [5] investigated the possibilities of the sound insulation of double-wall constructions at low frequencies. The expression ‘low frequencies’ implies a frequency range where the fundamental resonance frequencies of double wall constructions or the critical frequency of stiff single leaves (e.g. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 26–53, 2022. https://doi.org/10.1007/978-981-16-3239-6_3
Vibroacoutic Behavior of Finite Composite Sandwich Plates
27
concrete walls) are normally situated. Both effects lead to a reduced sound insulation which can cause problems to fulfill requirements. Sgard et al. [6] researched the lowfrequency diffuse field transmission loss through double-wall sound barriers with elastic porous linings. The studied sound barriers are made up from a porous-elastic decoupling material sandwiched between an elastic skin and a septum. The prediction approach is based on a finite element model for the different layers of the sound barrier coupled to a variation boundary element method to account for fluid loading and calculated efficiently using a Gauss integration scheme. Lee and Kondo [7] investigated the analytical and experimental studies of noise transmission loss of a three-layered viscoelastic plate. An improved equation of motion for sandwich plates with simply supported boundary condition is devisedand implemented to the finite plate noise transmission problem by using Rayleigh integral. Antonio et al. [8] studied the acoustic insulation provided by infinite double panel walls, when subjected to spatially sinusoidal line pressure loads, is computed analytically. The model calculation does not involve limiting the thickness of any layer, as the Kirchhoff or Mindlin theories require. Carneal √ and Fuller [9] proposed to increase of the panel stiffness by an empirical factor of 2 for the analytical solution of the simply supported boundary condition in order to approximate the experimental results of the clamped boundary condition. Wang et al. [10] presented theoretical modeling of the sound transmission loss through double-leaf lightweight partitions stiffened with periodically placed studs by assuming that the effect of the studs can be replaced with elastic springs uniformly distributed between the sheathing panels, a simple smeared model is established. Tanneau et al. [11] proposed a method for optimizing acoustical linings is described and applied to multilayered panels including solid, fluid, and porous components. This optimization is based on an analytical simulation of the insulation properties and a genetic algorithm. Lee et al. [12] simplified Bolton’s formulation by retaining only the energetically strongest wave among those propagating in the poroelastic material because the contribution of the shear wave to the transmission loss was found to be always negligible. Chazot and Guyader [13] studied sound transmission loss through double panels used a patch-mobility approach. The model is excitation by blocked patch pressures that take into account room geometry and source location. In recent years, D’Alessandro et al. [16] reviewed the most significant works in the literature about the acoustic response of sandwich panels. The focus is on presenting an exhaustive list of dedicated and validated models, which are able to predict the sound transmission through sandwich panels according to their specific configuration. Xin and Lu [17] adopted the equivalent fluid model to study sound transmission through an infinite double-wall panel with and without rib-stiffened core, respectively, filled with porous sound absorptive materials. Enhanced sound insulation properties of the double-wall sandwich panels have been observed in all these previous works. Shojaeifard et al. [18] studied power transmission through infinite double-walled laminated composite panels using the classical laminated plate theory and Biot’s theory [1] for wave propagation in the porous sandwich layer. Petrone et al. [19] investigated numerical and experimental on the acoustic power, radiated by Aluminium Foam Sandwich panels used a Finite Element model and formulation of the Rayleigh integral. Liu [21] extended the theoretical model of Bolton et al. [3] to address sound transmission across
28
T. I. Thinh and P. N. Thanh
triple-wall panels with poroelastic linings. Zhou et al. [14, 15] investigated the effects of an external mean ow and extended the vibroacoustic problem to a three-dimensional one, and Liu and Sebastian and Liu [22] further considered the effects of an internal gap mean flow. Hassan et al. [20] studied the acoustic behavior of double-walled panels, with sandwiched layer of porous materials is presented within Classical laminated plate theory for laminated composite panels. Equations of wave propagation are firstly extracted based on Biot’s theory for porous materials, then the transmission loss of the structure is estimated in a broadband frequency. Liu and Daulin [23, 24] studied the vibroacoustic problem of sound transmission across a rectangular double-wall sandwich panel clamp mounted on an infinite acoustic rigid baffle and lined with poroelastic material based on Biot’s theory [1] and the coupling methods between the poroelastic core and the panel determine the various configurations and associated boundary conditions. In this study, an analytical model on vibroacoustic behavior of clamped and simply supported orthotropic rectangular composite sandwich plates with foam core has been derived. Biot’s theory is employed to describe wave propagation in elastic porous media. The two face composite plates are modeled as classical thin plates. By using the modal superposition theory, a double series solution for the sound transmission loss of the structure is obtained with the help of the Galerkin method. Theoretical predictions of sound transmission loss (STL) across finite composite sandwich plates with poroelastic material agree well with the experimental results in most frequency ranges of interest. Basing on the numerical results obtained, the influence of different parameters of the two thin laminated composite sheets and polyurethane foam core layer on STL of sandwich plate is quantitatively evaluated and discussed in detail. Specifically, we have compared sound transmission losses between theoretical and experimental by influencing factors such as: the core density, the surface plates thickness and the core thickness for the composite sandwich plate with clamped boundary condition and consider the influence of the porous layer thickness, the faceplate thickness and the incident angles of the composite sandwich plate with simply supported boundary condition. Thereby, we consider the effect of two clamped and simply supported conditions on the sound transmission loss.
2 Theoretical Study 2.1 Plate Geometry and Assumptions A double-laminated composite plate lined with poroelastic materials consists of two rectangular, orthotropic, homogeneous and sufficiently thin plates commonly made of laminated composite, as illustrated in Fig. 1. The two elastic plates are fully clamped along their edges to an infinite rigid acoustic baffle. This double-composite plate is filled with porous materials. The bottom plate is excited by √a plane harmonic sound wave with ω being the angular frequency and the symbol j = −1. Without loss of generality, an incident sound wave of unit amplitude is assumed, and its incident direction is determined by the elevation angle ϕ and the azimuth angle θ (see Fig. 1). The vibrations of the upper plate are transmitted through the structure via the poroelastic medium to the bottom plate which induces pressure variations and thus a transmitted sound wave. The incidence field and transmission field separated by the infinite
Vibroacoutic Behavior of Finite Composite Sandwich Plates
29
Fig. 1. Schematic of a clamped double- laminated composite plate lined with poroelastic materials: (a) global view, (b) side view.
rigid baffle are assumed to be semi-infinite with identical air properties, i.e. air density ρ 0 and speed of sound in ambient air c0 . The panel dimensions are chosen as follows: the length and width of the plate are a, b; the bottom and upper plates thickness are h1 , h2 , and H is the core thickness of the composite sandwich (porous materials) in between the two plates. 2.2 Flexural Motion of the Plate The vibro-acoustic response of an orthotropic symmetric double- composite plate with poroelastic materials (Fig. 1) induced by sound excitation for the bottom and upper plates, respectively: D11 ∂
+ 2(D12 + 2D66 ) ∂
4 w (x,y;t) 1 ∂x4 4 ∂ w1 (x,y;t) 22 ∂y4
+D
∗ ∂ D11
+ m∗1 ∂∂tw21 = jωρ0 1 + σzs + σz 4 ∗ ∗ ∂ w2 (x,y;t) + 2 D12 + 2D66 ∂x2 ∂y2
4 w (x,y;t) 2 ∂x4 4 ∗ ∂ w2 (x,y;t) 22 ∂y4
+D
4 w (x,y;t) 1 ∂x2 ∂y2
2
(1)
f
+ m∗2 ∂∂tw22 = −jωρ0 3 − σzs − σz 2
f
(2)
where: w1 , w2 are the transverse displacements of the upper and bottom plates, respectively. m∗i = ρp hi (i = 1, 2) is the mass per unit area of the bottom and upper plates. ρ p is the density of the upper or bottom plates; ρ 0 is the density of air. F1 and F3 are the f acoustic velocity potential in the incident sound field and the transmission field. σzs ; σz are the normal stresses in the z-direction in the exterior fluid field and solid field. Dij and Dij∗ are the flexural rigidities (see any textbook of Mechanics of composite materials): 1 k 3 Qij zk+1 − zk3 For the bottom plate : Dij = 3 n
k=1
(3)
30
T. I. Thinh and P. N. Thanh
For the upper plate : Dij∗ =
1 ∗k ∗3 Qij zk+1 − zk∗3 3 n
(4)
k=1
2.3 Boundary Conditions of the Plate-Porous Coupling Boundary conditions representing the continuity of the normal velocity and displacement are applied at the surfaces of the porous layer and the facing plate, dependent on the coupling method. Since the plates are assumed to be homogeneous, orthotropic and sufficiently thin compared with the lateral dimensions, the in-plane deformations (and shear stresses) are commonly neglected according to the classical Kirchhoff-Love plate theory. For the coupling method as shown in Fig. 2, three boundary conditions must be satisfied at the bonded interface of the porous layer and facing plate, i.e., one normal velocity condition and two normal displacement conditions: v*z = jωw, usz = w, ufz = w
(5)
where w is the transverse (normal) displacement of the elastic plate that contains the convention eiωt , usz and ufz are the solid and fluid displacement vectors and v*z is the normal acoustic particle velocity in the exterior fluid field defined in Eq. (16).
Fig. 2. Different configurations of the double-laminate composite plate
Moreover, the boundary conditions of the bonded coupling in the Eq. (5) must be applied at the interface of the plate and the porous lining: at z = h1 −
∂1 = jωw1 , usz = w1 , ufz = w1 ∂z
atz = H + h1 −
∂3 = jωw2 , usz = w2 , ufz = w2 ∂z
(6) (7)
Under the excitation of harmonic sound waves, the transverse deflection of the upper and bottom plates can be expressed in a form of modal decomposition: w1 (x, y, t) =
∞ m,n=1
φmn (x, y)α1,mn ejωt ; w2 (x, y, t) =
∞ m,n=1
φmn (x, y)α2,mn ejωt
(8)
Vibroacoutic Behavior of Finite Composite Sandwich Plates
with φmn is the modal function for the clamped boundary can be written as: 2nπ y 2mπ x 1 − cos φmn (x, y) = 1 − cos a b and the simply supported boundary takes the form: mπ x nπ y ϕmn = sin sin a b
31
(9)
(10)
and α 1,mn ; α 2,mn are the modal coefficients of the plate displacement and will be determined by applying the clamped and simply supported boundary conditions. The fully clamped boundary of the bottom and upper plates, the transverse displacement and the rotation moment must be equal to zero along the plate edges, i.e. the following clamped boundary conditions should be satisfied: x = 0, a, ∀ 0 < y < b, w1 = w2 = 0,
∂w1 ∂w2 = =0 ∂x ∂x
(11)
y = 0, b, ∀ 0 < x < a, w1 = w2 = 0,
∂w2 ∂w1 = =0 ∂y ∂y
(12)
and the simply supported boundary conditions should be satisfied: x = 0, a, ∀ 0 < y < b, w1 = w2 = 0,
∂ 2 w1 ∂ 2 w2 = =0 ∂x2 ∂x2
(13)
y = 0, b, ∀ 0 < x < a, w1 = w2 = 0,
∂ 2 w1 ∂ 2 w2 = =0 2 ∂y ∂y2
(14)
2.4 Determination of Sound Transmission Loss The acoustic velocity potential in the incident sound field (field 1 in Fig. 1) consists of an incident wave and a reflective wave with the amplitudes I and β, respectively, and it can be expressed in a harmonic form as: 1 = Ie−j(kx x+ky y+kz z−ωt ) + βe−j(kx x+ky y−kz z−ωt )
(15)
The transmission field (field 3 in Fig. 1) with the amplitude γ in this field and its velocity potential can be written as: 3 = γ e−j(kx x+ky y+kz z−ωt )
(16)
The poroelastic medium is excited by an incident plane harmonic sound wave. For a given incidence direction (ϕ, θ ), the components of the acoustic wavenumber are defined as: kx = k0 sin φ cos θ ; ky = k0 sin φ cos θ ; kz = k0 cos φ
(17)
32
T. I. Thinh and P. N. Thanh
where k x , k y , k z are wave number in the x, y, and z directions and k0 = ω/c0 is the acoustic wave number in air and c0 is the acoustic speed in the air. The amplitudes of the reflected and transmitted waves, β(x,y) and γ (x,y) are dependent on the local positions on the upper and bottom plates, respectively. These acoustic velocity potentials are related to the acoustic pressure p and the normal acoustic particle velocity v*z in the fluid field and solid field: p = ρ0
∂ ∂ = jωρ0 ; v*z = − ∂t ∂t
(18)
The sound power of the incident or transmitted wave per unit area (i.e. acoustic intensity) is defined as p = Re(pv ∗ ) 2, and the acoustic particle velocity is related to the sound pressure through p = p (ρ0 c0 ) for harmonic waves, ρ 0 is the density air and c0 is the acoustic speed in the air. The sound power can be expressed as [26] 1 = Re 2
b a 0
pvz∗ dA
0
1 = 2ρ0 c0
b a p2 dA 0
(19)
0
Considering the relation (18) and the velocity potential definitions (15) and (16), the sound power of incident and transmitted are determined by. 1 =
ω2 ρ0 2c0
b a 21 dxdy; 3 = 0
0
ω2 ρ0 2c0
b a 23 dxdy 0
(20)
0
The power transmission coefficient for a single incident wave with fixed direction angles is defined as τ (φ, θ ) =
3 1
(21)
For diffuse incident sound, the averaged transmission coefficient can be derived by integration as [3, 5, 9, 17]: 2π φ lim
τdiff =
0
τ (φ, θ ) sin φ cos θ d φd θ
0 2π φ lim 0
(22) sin φ cos θ d φd θ
0
where ϕlim is the limiting angle defining the diffuseness of the incident field. The sound transmission loss is defined as: STL(φ, θ ) = 10 log[1/τ (φ, θ )]
(23)
Vibroacoutic Behavior of Finite Composite Sandwich Plates
33
2.5 Modelling the Poroelastic Material According to Biot’s theory [1], an elastic porous material is assumed statistically isotropic and has the solid phase (elastic frame) as well as the fluid phase (air contained in pores). The model of Bolton [3] allows three waves in the porous material, two longitudinal waves, and one shear wave. Meanwhile, Lee et al. [7] shown that the shear wave is always negligible compared to the longitudinal waves. Hence, there are four wave components within the porous material, i.e. positive and negative propagating components of the two most energetically waves along the z-axis, which are characterised by the complex amplitudes D1 -D4 . Thus, the displacements of the solid and fluid phase are obtained as:
k1z −jk1z z k1z jk1z z k2z −jk2z z k2z jk2z z s −j(kx x+ky y−ωt ) D1 2 e − D2 2 e + D3 2 e − D4 2 e uz = je k1 k1 k2 k2 (24) ⎛ ⎞ k1z −jk1z z k1z − c1 D2 2 ejk1z z ⎟ ⎜ c1 D1 k 2 e k1 ⎜ ⎟ f 1 uz = je−j(kx x+ky y−ωt ) ⎜ (25) ⎟ k2z −jk2z z k2z jk2z z ⎠ ⎝ +c2 D3 2 e − c2 D4 2 e k2 k2 The expressions of the normal stresses in the z-direction are derived as σzs = e−j(kx x+ky y−ωt ) ε1s D1 e−jk1z z + D2 ejk1z z + ε2s D3 e−jk2z z + D4 ejk2z z
(26)
f f f σz = e−j(kx x+ky y−ωt ) ε1 D1 e−jk1z z + D2 ejk1z z + ε2 D3 e−jk2z z + D4 ejk2z z
(27)
f
f
for the solid and fluid phases, respectively, where the parameters ε1s , ε2s , ε1 , ε2 are in the form: ε1s = 2N f ε1
2 k1z
k12
+ A + c1 Q, ε2s = 2N
= Q + c1 S,
f ε2
2 k2z
k22
+ A + c2 Q
(28)
= Q + c2 S
where, the first Lame coefficient A and the elastic shear modulus N are defined as A=
νs Es Es ; N= 2(1 + νs ) (1 + νs )(1 − 2νs )
(29)
here, E s is the real static Young’s modulus of solid phase, ν s is the Poision’s ratio. The coefficients Q and S represent the coupling between the volume changes of the solid and fluid phases and are defined as: Q = (1 − β)Ef and S = βEf
(30)
where β is the porosity of the porous material and E f denotes the bulk modulus of the −1 elasticity of the fluid in the pores (assumed to be air), Ef = E0 1 + 2(χ − 1)Tc (λ) λ
34
T. I. Thinh and P. N. Thanh
where E0 = ρ0 c02 is the bulk modulus of air, and χ is the ratio of specific heats; √ 1 2 2 / λ = P r λc −j, with Pr the Prandtl number, λc = 8ωρ 0 ϑ βψ, ϑ the geometrical structure factor and ψ the flow, resistivity. Tc (λ) = J1 (λ) J0 (λ), with J0 and J1 the zero- and first-order Bessel functions of the first kind. 2 2 − k 2 + k 2 , and the = k1,2 The wavenumber component in the z-direction, k1,2z x y coefficients are c1 = b1 − b2 k12 , c2 = b1 − b2 k22 ∗ ∗ ∗ ∗ b1 = ρ11 S − ρ12 Q ρ22 Q − ρ12 S ∗ ∗ b2 = PS − Q2 ω2 ρ22 Q − ρ12 S
(31)
Under the assumption of plane harmonic waves, their wavenumber are expressed as A1 ± A21 − 4A2 2 (32) k1,2 = 2 where, the parameters are defined as ∗ ∗ ρ∗ − ρ∗ 2 ∗ Q + ρ∗ P ω4 ρ11 S − 2ρ12 ω2 ρ11 22 12 22 A1 = , A2 = 2 2 PS − Q PS − Q with P = A + 2N, where the complex equivalent densities are ∗ ρ11 = ρ11 + a jω, ρ11 = ρs + ρa ∗ ρ12 = ρ12 − a jω, ρ12 = −ρa ∗ ρ22 = ρ22 + a jω, ρ22 = ρ2 + ρa
(33)
(34)
With ρ s and ρ 2 = β.ρ 0 being the bulk densities of the solid and fluid phases, respectively; ρa = ρ2 (ψ − 1) is an inertial coupling coefficient between the fluid and solid, and a is viscous coupling factor: −1 a = iωψρ2 ρc∗ ρ0 − 1 with ρc* = ρ0 1 − 2 λc −j Tc λc −j
(35)
Besides, two auxiliary parameters can be defined as f
f
1 = ε1s + ε2s and 2 = ε1 + ε2
(36)
which will be used in the subsequent derivations. f Substituting the expressions of F1 , F3 in Eqs. (15) and (16), uzs , uz in Eqs. (24) and (25) and w1 , w2 in Eq. (8) into the boundary condititions (9) and (10), one obtains: atz = h1 e−j(kx x+ky y) kz (I − β) = ω1
k1z k1z k2z k2z je−j(kx x+ky y) D1 2 − D2 2 + D3 2 − D4 2 k1 k1 k2 k2
(37a)
= 1
(37b)
Vibroacoutic Behavior of Finite Composite Sandwich Plates
je
−j(kx x+ky y)
k1z k1z k2z k2z c1 D1 2 − c1 D2 2 + c2 D3 2 − c2 D4 2 k1 k1 k2 k2
35
= 1
atz = H + h1 e−j(kx x+ky y+kz (H +h1 )) kz γ = ω2
(37c) (38a)
⎛
⎞ k1z −jk1z (H +h1 ) k1z −jk1z (H +h1 ) D e − D e + 2 2 ⎜ 1 k2 ⎟ k1 ⎜ ⎟ 1 je−j(kx x+ky y) ⎜ ⎟ = 2 k2z −jk2z (H +h1 ) ⎠ ⎝ k2z −jk2z (H +h1 ) D3 2 e − D4 2 e k2 k2 ⎛ ⎞ k1z −jk1z (H +h1 ) k1z −jk1z (H +h1 ) − c1 D2 2 e +⎟ ⎜ c1 D1 k 2 e k1 ⎜ ⎟ 1 je−j(kx x+ky y) ⎜ ⎟ = 2 k2z −jk2z (H +h1 ) k2z −jk2z (H +h1 ) ⎠ ⎝ c2 D3 2 e − c2 D4 2 e k2 k2 where the parameters G1 , G2 are in the form: 1 = φmn (x, y)α1,mn and 2 = φmn (x, y)α2,mn m,n
(38b)
(38c)
(39)
m,n
Combine Eqs. (37a) and (38a), it is straightforward to solve the amplitudes of the reflected and transmitted wave as. ω ω β(x, y) = I − 1 ej(kx x+ky y) ; γ (x, y) = 2 ej(kx x+ky y+kz (H +h1 )) (40) kz kz Substituting the expressions of β and γ back into Eqs. (19) and (20), the acoustic velocity potentials F1 and F3 on the surface of the upper and bottom plates can be rewritten as. 1 (x, y, 0, t) = 2Ie−j(kx x+ky y−ωt ) −
ω ω 1 ejωt ; 3 (x, y, H , t) = 2 ejωt kz kz
(41)
Combining Eqs. (37b), (37c), (38b) and (38c) yields a set of four algebraic equations for four unknown (1.e. D1 – D4 ) which can be rearranged into a matrix equation: MD = R
(42)
where M is a 4 x 4 transfer matrix: ⎤ ⎡ −kI kII −kII kI ⎥ ⎢ c1 kI −c1 kI c2 kII −c2 kII ⎥ M=⎢ ⎣ e−jk1z (H +h1 ) kI −ejk1z (H +h1 ) kI e−jk2z (H +h1 ) kII −ejk2z (H +h1 ) kII ⎦ (43) c1 e−jk1z (H +h1 ) kI −c1 ejk1z (H +h1 ) kI c2 e−jk2z (H +h1 ) kII −c2 ejk2z (H +h1 ) kII with the auxiliary coefficients kI = jk1z k12 and kII = jk2z k22
(44)
36
T. I. Thinh and P. N. Thanh
T the 4 x 1 unknown vector D = D1 D2 D3 D4 , and the forcing vector T R = ej(kx x+ky y) 1 1 2 2
(45)
The unknown vector D can be soved simultaneously from the matrix Eq. (42) as D = V.R, where V = M −1 is the inverse of the transfer matrix M. Substituting the solutions of D1 – D4 into Eqs. (26) and (27), the expressions of the normal stresses in the solid f and fluid phases, σzs , σz can be obtained as ! −jk1z z " e [(V11 + V12 )1 + (V13 + V14 )2 ] s,f = ε1 ejωt +ejk1z z [(V21 + V22 )1 + (V23 + V24 )2 ] ! −jk2z z " [(V31 + V32 )1 + (V33 + V34 )2 ] s,f jωt e + ε2 e +ejk2z z [(V41 + V42 )1 + (V43 + V44 )2 ] s,f
σz
(46)
where V ij (i, j = 1, 2, 3, 4) are the elements of the inverse matrix V. In order to determine the unknown α 1,mn and α 2,mn , applying the Galerkin method to the plate motion Eqs. (1) and (2) leads to: ⎛ ⎞ ∂ 4 w1 (x, y; t) ∂ 4 w1 (x, y; t) + 2(D12 + 2D66 )
b a ⎜ D11 ⎟ ∂x4 ∂x2 ∂y2 ⎜ ⎟ ⎜ ⎟.φmn (x, y)dxdy = 0 2w ⎝ ∂ 4 w1 (x, y; t) ∂ 1 f⎠ ∗ s +D22 + m1 2 − jωρ0 1 − σz − σz 0 0 ∂y4 ∂t (47) ⎛ ⎞ 4 4 ∗ ∗ ∂ w2 (x, y; t) ∗ ∂ w2 (x, y; t) + 2 D + 2D
b a ⎜ D11 ⎟ 12 66 ∂x4 ∂x2 ∂y2 ⎜ ⎟ ⎜ ⎟.φmn (x, y)dxdy = 0 4 w (x, y; t) 2w ⎝ ⎠ ∂ ∂ 2 2 f ∗ ∗ s +D22 + m2 2 + jωρ0 3 + σz + σz 0 0 ∂y4 ∂t (48) f
After substituting the expressions of w1 , w2 , F1 , F3 and σzs , σz , i.e. Equations (10), (40) and (46) into Eqs. (47) and (48) and integrating over the entire plate surface with laborious algebraic manipulations, one obtains two sets of infinite simultaneous equations: at z = 0, !# m 4 n 4 m 2 n 2 $ α1,mn + 3D22 + 4(D12 + 2D66 ) 4π 4 ab 3D11 a b a b % n 4 m 4 + 2D22 α1,kn + 2D11 α1,ml b a k l (49) 9ab 3ab + R1,mn + R1,kn 4 m,n 2 k 3ab + R1,ml + ab R1,kl = 2jωρ0 Ifmn kx , ky , (k = m, l = n) 2 l
k,l
Vibroacoutic Behavior of Finite Composite Sandwich Plates
37
at z = H + h1 , $ !# 4 4 m 2 n 2 ∗ ∗ m ∗ n ∗ α2,mn + 3D22 + 4 D12 + 2D66 4π 4 ab 3D11 a b a b % 4 4 ∗ n ∗ m + 2D22 α2,kn + 2D11 α2,ml b a k
l
3ab 3ab 9ab R2,mn + R2,kn + R2,ml + ab R2,kl = 0, (k = m, l = n) + 4 m,n 2 2 k
l
k,l
(50) where the coefficients Qi,mn (i = 1, 2) are in the form: R1,mn = −α1,mn R11 − α2,mn R12 , R2,mn = α1,mn R21 + α2,mn R22
(51)
ρ0 ∗ R11 = 1 (V11 + V12 + V21 + V22 ) + 2 (V31 + V32 + V41 + V42 ) + m1 − j ω2 , kz R12 = 1 (V13 + V14 + V23 + V24 ) + 2 (V33 + V34 + V43 + V44 ), R21 = 1 (V11 + V12 )e−jk1z (H +h1 ) + (V21 + V22 )ejk1z (H +h1 ) + + 2 (V31 + V32 )e−jk2z (H +h1 ) + (V41 + V42 )ejk2z (H +h1 ) , R22 = 1 (V13 + V14 )e−jk1z (H +h1 ) + (V23 + V24 )ejk1z (H +h1 ) + + 2 (V33 + V34 )e−jk2z (H +h1 ) + (V43 + V44 )ejk2z (H +h1 ) , (52) The constants f mn are produced through the integration process combined with clamped boundary condition fmn kx , ky =
b a 2mπ x 2nπ y −j(kx x+ky y) 1 − cos 1 − cos e dxdy a b 0
(53)
0
with simply supported boundary condition fmn kx , ky =
b a sin 0
mπ y a
sin
nπ y e−j(kx x+ky y) dxdy b
(54)
0
The infinite algebraic equation system specified in Eqs. (47) and (48) has the modal coefficients α 1,mn and α 2,mn as the unknowns. This set of infinite equations can be sovle numerically by taking an appropriate truncation: 1 ≤ m, n ≤ M, where the maximum
38
T. I. Thinh and P. N. Thanh
mode number M is a compromise between accuracy and computational cost. Therefore, the simultaneous equation system can be rearranged as a 2M 2 × 2M 2 matrix equation: $ ! " # " ! α1,mn T11,mn T12,mn Fmn = (55) T21,mn T22,mn 2M 2 x2M 2 α2,mn 2M 2 x1 0 2M 2 x1 where the elements of the displacement coefficient vectors α i,mn (i = 1, 2) and the generalized force vector Fmn take the form: & ' αi,mn M 2 x1 = αi,11 αi,21 ... αi,M 1 (56) T α1,12 α1,22 ... αi,M2 ... αi,MM (i = 1, 2) T {Fmn }M 2 x1 = 2iωρ0 f11 f21 ... fM 1 f12 f22 ... fM 2 ... fMM (57) and the matrix elements are expressed as 4 4 3 ∗i T11,mn M 2 xM 2 = 4π 4 ab ∗i − R ; T = −R ∗i 11 12,mn M 2 xM 2 12 1 2 2; i=1
i=1
i=1
4 4 3 4 ∗∗i T21,mn M 2 xM 2 = R21 ∗i ; T = 4π ab − R ∗i 22,mn M 2 xM 2 11 2 1 2; i=1
i=1
i=1
(58) ∗i where the expressions of the M 2 x M 2 submatrices ∗i 1 (i = 1, 2, 3), 2 (i = 1, 2, 3, 4) ∗∗i and 1 (i = 1, 2, 3) are given in Appendix A. After sovling this matrix Eq. (55), the unknown modal coefficients α i,mn (i = 1, 2) of the plate displacements will be determined and therefore all the dependent variables such as w1 , w2 , β, γ will be known.
3 Validation For the first validation study, the sound transmission loss (STL) across a finite clamped double-aluminium plate with poroelastic material is calculated and compared with result of Bolton et al. [3]. The properties of aluminium plates and poroelastic material as follows: For aluminium plate: length × width of plate, a × b = 1.2 m × 1.2 m; upper plate thickness, h1 = 1.27 mm; bottom plate thickness, h2 = 0.762 mm; porous layer thikness, H = 27 mm; Young’s moduluas, E p = 70 GPa; Bulk density, ρ p = 2700 kg/m3 ; Poisson’s ratio, ν p = 0.33. For poroelastic material: Bulk density of the solid phase, ρ s = 30 kg/m3 ; In vacuo bulk Young’s modulus, E s = 8.105 Pa; Bulk Poisson’s ratio, ν s = 0.40; flow resistivity, ψ = 5.103 MKS Rayls m−1 ; Porosity, β = 0.9; geometrical structure factor, ϑ = 0.78; density, ρ 0 = 1.21 kgm−3 , speed of sound, c0 = 343 ms−1 , ratio of specific heats, χ = 1.4; Prandtl number, Pr = 0.71. It is obvious from Fig. 3 that there is an acceptable agreement between present analytical and theoretical results shown in [3].
Vibroacoutic Behavior of Finite Composite Sandwich Plates
39
Fig. 3. Comparison between the predicted STL curve and result of Bolton et al. [3].
For the second validation study, the STL across a finite simply supported doublecomposite plate with poroelastic material is calculated and compared with result of Lee et al. [7]. The properties of aluminium plates and poroelastic material as follows: For aluminium plate: Length × width of plate, a × b = 0.303 m × 0.203 m; upper plate thickness, h1 = 0.5 mm; bottom plate thickness, h2 = 0.5 mm; porous layer thikness, H = 1 mm; Young’s moduluas, E p = 73.2 GPa; Bulk density, ρ p = 2720 kg/m3 ; Poisson’s ratio, ν p = 0.33. For poroelastic material are: Bulk density of the solid phase, ρ s = 1600 kg/m3 ; In vacuo bulk Young’s modulus, E s = 4.12. GPa; Poisson’s ratio, ν s = 0.40; flow resistivity, ψ = 25.103 MKS Rayls m−1 ; Porosity, β = 0.9; geometrical structure factor, ϑ = 0.78; density, ρ 0 = 1.21 kgm−3 , speed of sound, c0 = 343 ms−1 , ratio of specific heats, χ = 1.4; Prandtl number, Pr = 0.71.
Fig. 4. Comparison between the predicted STL curve and results of Lee and Kondo [7].
Figure 4 shows a comparison between the predicted STL curve and the experimental curve of Lee and Kondo in [7]. The curves are presented in third-octave bands in the frequency range of 100–10000 Hz. The agreement between the present analytical method and experimental curves is good. The results illustrated the capability of the proposed method to predict STL curves for sandwich plates. It is seen that the minimal STL is
40
T. I. Thinh and P. N. Thanh
obtained in the low frequency range where the modal behavior of the plate is important. The first mode of the plate produces an important resonant dip in the STL.
4 Numerical and Experimental Results 4.1 Test Setup In this section, to measure the sound transmission loss through the sandwich structure, we proceed through a system of two rooms (Source room and Receiver room) meeting ISO 3741–1988 [24] as shown in Fig. 5. Measurement experiments were conducted at the Experimental Testing Center – Nha Trang University in cooperation with Vietnam Metrology Institute. Sound transmission loss is determined by the sound pressure level measurement method, in accordance with ASTM E2249–02 (E) [25, 27].
Fig. 5. The transmission suite
The test samples are clamped between two reverberation rooms: the source room is (2,06 m × 3,7 m × 2,1 m) and the receiving room is (1,54 m × 3,7 m × 2,1 m) [26, 27], as shown in Fig. 6. The sound transmission loss is given by [26]: STL = L1 − L2 + 10 log(
AP T ) 0.161 V2
(59)
where L 1 and L 2 are the average sound pressure levels in the source and receiving room, respectively. Ap is the test-specimen area; V 2 is the volume of the receiving room; T is the reverberation time of the receiving room when the sandwich plate is clamped between the two rooms. To conduct the experiment, first of all, it is necessary to determine the position of the speaker (sound source) as described in Fig. 6. Omnidirectional speaker 4292L B&K with a large capacity will create a strong sound source. Microphone 4189 (3) is placed 0.2 m from the face of the sample board to measure the sound pressure level in the source room. Microphone 4189 (4) is placed at a distance of 0.2 m from the test surface attached to the partition wall to measure the sound pressure level in the receiving room. The reverberation time (T60) is calculated by the average decay data from 16 points equidistant from the sample.
Vibroacoutic Behavior of Finite Composite Sandwich Plates
41
Fig. 6. Test arrangement for samples.
The composite sandwich plates have an area (1.2 m × 1.2 m) are divided into 16 squares (0.3 m × 0.3 m), corresponding to 16 discrete measurement points. According to experimental, the distance between the measuring head and the center of the sound source must be 2 to 3 times the distance between the two microphones to ensure the measurement error is less than 1 dB. After a number of tests, the distance of the microphone (3) was selected to be 0.2 m and the probe (4) in the studio was located 0.17 m from the sample surface. Sound transmission loss (STL) through composite sandwich plates has been calculated and measured according to ASTM E 1289–91 (E) [25, 27]. Sound leaks on the side of the samples were detected at some high frequencies and treated by sealing with a sealant. Test equipment has been calibrated and reinstalled for subsequent measurements. Then, to assess the role of component materials for sound transmission loss (STL), we compare STL values across three structural groups. 4.2 Sample Description According to the standards of Vietnam Register: QCVN 56–2013/BGTVT. The upper and lower skin layers of the sandwich composite panel are made of fiberglass (WR800)/polyester with a symmetrical configuration in the 0°/90° direction; core is polyurethane foam - PU.
42
T. I. Thinh and P. N. Thanh
Fig. 7. Fabrication of sandwich composite materials.
Configuration and geometry parameters of composite sandwich plates are given in Table 1. Table 1. Configuration and geometry parameters of composite sandwich plates Plate
Configuration
a
b
tf
tc
tf
h
ρs
Surface mass
m
m
m (×10–3 )
m (×10–3)
m (×10–3 )
m
kg/m3
kg/m2
A
3WR800 + PU1 + 3WR800
1,2
1,2
2,53
30
2,53
0,0351
46,88
9,502
B
3WR800 + PU2 + 3WR800
1,2
1,2
2,53
30
2,53
0,0467
57,87
9,832
C
3WR800 + PU3 + 3WR800
1,2
1,2
2,53
40
2,53
0,0451
79,86
11,290
D
4WR800 + PU3 + 4WR800
1,2
1,2
3,37
40
3,37
0,0467
79,86
13,978
E
4WR800 + PU3 + 4WR800
1,2
1,2
3,37
30
3,37
0,0359
79,86
13,173
4.3 Results and Discussion Firstly, the effects of core density, skin thickness and core thickness on STL of a clamped composite sandwich plates are investigated theoretically and experimentally. The geometry and material properties of plates are given in Table 2.
Vibroacoutic Behavior of Finite Composite Sandwich Plates
43
Table 2. Geometric parameters and mechanical properties of the face plate and core materials Plate
Dimension (m)
Thickness t f /t c /t f (x103 m)
Density of skin ρ f (kg/m3 )
Core density ρ s (kg/m3 )
E1 (GPa)
E2 (GPa)
G12 (GPa)
v12
Ec (GPa)
vc
Surface mass kg/m2
A
1.2 x 1.2
2.53/30/2.53
1600
46.88
10.58
2.64
1.02
0.17
0.0570
0.25
B
1.2 x 1.2
2.53/30/2.53
1600
57.87
10.58
2.64
1.02
0.17
0.0580
0.25
9.832
C
1.2 x 1.2
2.53/40/2.53
1600
79.86
10.58
2.64
1.02
0.17
0.0585
0.25
11.290
D
1.2 x 1.2
3.37/40/3.37
1600
79.86
10.58
2.64
1.02
0.17
0.0585
0.25
13.978
E
1.2 × 1.2
3.37/30/3.37
1600
79.86
10.58
2.64
1.02
0.17
0.0585
0.25
13.173
9.502
a. Effect of the Core Density In this section, we consider the effect of core density to STL when they have the same skin layer thickness and core thickness for material pair (A – B). Pair (A - B) changes the density of core layer ρ s = 46.88 kg/m3 and ρ s = 57.87 kg/m3 , the remaining parameters are the same in Table 2. In Fig. 8, the STL curves across clamped plate A and D are plotted. In the lowfrequency region (f < 125 Hz), at 100 Hz, the STL value of the theoretical STL curve of plate B increases compared to plate A is 0.35 dB, the STL value of the experimental STL curve of plate B increased compared to plate A is 0.13 dB. In the medium-frequency region (125 Hz < f < 2000 Hz), at 1000 Hz, the STL value of the theoretical STL curve of plate B increases compared to plate A is 0.34 dB, the STL value of the experimental STL curve of plate B increased compared to plate A is 0.84 dB. In the high-frequency region (f > 2000 Hz), at 10000 Hz, the STL value of the theoretical STL curve of plate B increases compared to plate A is 2.07 dB, the STL value of the experimental STL curve of plate B increased compared to plate A is 2.58 dB.
Fig. 8. Comparision of sound transmission loss through clamped plates A and B.
The density of the core layer has the greatest effect on the sound transmission loss, as the density of the core layer increases, the theoretical and experimental STL values increase. The theoretical average increase is 2.124 dB, which is empirically 1,988 dB for most of the frequency band in 1/3 of the octave band.
44
T. I. Thinh and P. N. Thanh
b. Effect of the Surface Plates Thickness In this section, we consider the effect of skin thickness to STL when they have the same skin layer thickness and core thickness for material pair (C – D). Pair (C – D) increased skin thickness from t f = 2.53.10−3 m to t f = 3.37.10−3 m and core thickness t c = 40.10−3 m, the remaining parameters are the same in Table 2.
Fig. 9. Comparision of sound transmission loss through clamped plates C and D.
Figure 9 show, the STL curves across clamped plate C and D are plotted. In the low-frequency region (f < 125 Hz), at 80 Hz, the STL value of the theoretical STL curve of plate D increases compared to plate C is 0.09 dB, the STL value of the experimental STL curve of plate D increased compared to plate C is 0.83 dB. In the medium-frequency region (125 Hz < f < 2000 Hz), at 1000 Hz, the STL value of the theoretical STL curve of plate D increases compared to plate C is 1.60dB, the STL value of the experimental STL curve of plate D increased compared to plate C is 2.19 dB. In the high-frequency region (f > 2000 Hz), at 5000 Hz, the STL value of the theoretical STL curve of plate D increases compared to plate C is 5.26 dB, the STL value of the experimental STL curve of plate D increased compared to plate C is 6.37 dB. The thickness of the skin layer has a significant effect on the sound transmission loss through the composite sandwich panels. For example, when t f increases from t f = 2.53.10−3 m to t f = 3.37.10−3 m, the other parameters do not change, the STL theoretically increases by 1.476 dB, empirically the STL increases 1.774 dB corresponding to the 1/3 Octave frequency range. c. Effect of the Core Thickness In this section, we consider the effect of core thickness to STL when they have the same skin layer thickness and the density of the core for material pair (D – E). Group (D – E) increased core thickness from t c = 30.10−3 m to t c = 40.10−3 m and skin thickness t f = 3.37. 10−3 m, the remaining parameters are the same in Table 2. In Fig. 10, the STL curves across clamped plate D and E are plotted. In the lowfrequency region (f < 125 Hz), at 80 Hz, the STL value of the theoretical STL curve of plate D increases compared to plate E is 0.32 dB, the STL value of the experimental STL curve of plate D increased compared to plate E is 0.13 dB. In the medium-frequency
Vibroacoutic Behavior of Finite Composite Sandwich Plates
45
Fig. 10. Comparision of sound transmission loss through clamped plates D and E.
region (125 Hz < f < 2000 Hz), at 1000 Hz, the STL value of the theoretical STL curve of plate D increases compared to plate E is 1.64 dB, the STL value of the experimental STL curve of plate D increased compared to plate E is 2.19 dB. In the high-frequency region (f > 2000 Hz), at 3125 Hz, the STL value of the theoretical STL curve of plate D increases compared to plate E is 3.14 dB, the STL value of the experimental STL curve of plate D increased compared to plate E is 4.13 dB. The core layer thickness does not significantly affect sound transmission loss through composite sandwich plates. Specifically, when t c increases from 30.10−3 m to t c = 40.10−3 m, other parameters do not change, STL theoretically only increases 0.560 dB, experimental STL curve increases 0.567 dB corresponding to 1/3 Octave frequency range. In this next subsection, the effects of the porous layer thickness, the faceplate thickness and the incident angle on STL of a simply supported composite sandwich plates are investigated. The bottom and upper Glass/Epoxy orthotropic laminated composite plates configuration is [90/0/0/90]s . The geometry and material properties of plates are given in Table 3. Table 3. Composite materials properties and the geometrical dimensions Composite
E1 (GPa) E2 (GPa) G12 (GPa) ν 12
Glass/Epoxy 40.851
10.097
3.788
ρ a x b h1 (m) (Kg/m3 ) (m2 )
0.27 1946
h2 (m)
H (m)
1.2 x 1.27.10–3 0.762.10–3 27 1.2
d. Effect of the Porous Layer Thickness In order to explore the effects of the core cavity thickness on sound transmission loss through a finite simply supported double-composite plate calculated with changed values of the porous layer thickness: H = 17, 27, 37 and 47.10−3 m, as shown in Fig. 11.
46
T. I. Thinh and P. N. Thanh
From Fig. 11, increasing the thickness of the core layer leads to a reduction the coincidence frequency because the porous layer plays a major role in reducing the noise of the sound. On the other hand, we also see that the sound transmission loss increases as the core porous layer thickness increases and also shows the soundproofing capacity of the double-plates filled poroelastic material.
Fig. 11. Effect of the porous layer thickness on STL of a simply supported composite sandwich.
e. Effect of the Faceplate Thickness To quantify the effects of the faceplate thickness, the STL versus frequency curve is presented in Fig. 12 for a finite system. Four pairs values of the faceplate thickness are selected: (h1 , h2 ) = (1.0,1.0), (1.0, 2.0), (2.0, 1.0) and (2.0, 2.0).10−3 m. According to Fig. 12, the effect of the plate thickness parameter on the sound transmission loss is considered. It is observed that increasing the thickness of the plate will increase the sound transmission loss, which means that sound waves transmitted through thicker sheets will be less spreading and more reflective, and will also lead to reduced frequency coincidence.
Fig. 12. Effect of the skin thickness on STL a simply supported composite sandwich.
Vibroacoutic Behavior of Finite Composite Sandwich Plates
47
f. Effects of the Incident Angles The influence of sound incident angles (elevation angle and azimuth angle) on the sound transmission loss of a finite simply supported orthotropic laminated double-composite plate is considered. Shown in Fig. 13 for the double-plate excited by sound waves incident at a series of incident angles ϕ = 0°; 30°; 45° and 60° and a fixed azimuth angle θ = 45°. Generally, the STL spectra of the oblique incidence cases drop considerably as the elevation angle increases and this STL reduction grows at higher frequencies. In other words, more oblique incident waves are easier to transmit through the double-wall sandwich panel especially in the high-frequency range. This is because constructive interference between the incident sound waves and structural bending waves may occur at oblique incidence.
Fig. 13. Effect of the incident angle on STL of a simply supported composite sandwich.
Similarly, Fig. 14 shows the effect of the azimuth angle on the STL spectra for the double-plate by sound waves incident at a series of azimuth angles θ = 15°; 30°; 45° and 60° and a fixed incident angle ϕ = 45°. As shown in Fig. 14, the azimuth angle has a limited influence on STL behaviors. In fact, the STL spectra of all azimuth angle (θ) values coincide with one another at frequencies below 1000 Hz, i.e. a negligible effect of the azimuth angle. The effect of the azimuth angle (θ) on STL, however, become more obvious and complicated above this frequency. The overall trend of the STL levels increases with larger azimuth angles despite the complex modal behaviors at high frequencies. The limited effect of the azimuth angle on the sound transmission characteristics of the finite clamped plate may be attributed to the fairly symmetrical system of the square panel, which is in accordance with the observation in previous work. In the final study, we investigated the effect of boundary conditions on STL of the composite sandwich plates. g. Effects of Boundary Conditions In this section, we evaluate the influence of boundary conditions (clamped and supported) on sound transmission loss.
48
T. I. Thinh and P. N. Thanh
Fig. 14. Effect of the azimuth angle on STL of a simply supported composite sandwich.
Fig. 15. Effect of the clamped and simply supported boundary conditions on STL of composite sandwich plate
From Fig. 15, other dips determined mainly by the natural vibration of the radiating panel are significantly shifted when the boundary conditions are changed. Additionally, it can be seen that the STL values of the clamped system are distinctly higher than those of the simply supported system in the lower frequency range. This is attributed to the more rigorous constraint provided by the clamped condition than that by the simply supported condition, which is equivalent to increasing the plate stiffness.
5 Conclusions An analytical model on sound transmission through a finite composite sandwich plates with foam core has been derived on the basis of previous theories [3, 24]. The vibroacoustic problem is formulated and solved by employing a modal decomposition to account for the clamped and simply supported boundary condition, and the Galerkin method for the plate motion equations. We draw some of the following conclusions: • Confirm the correctness, reliability and excellent agreement between current research and theories [3, 7].
Vibroacoutic Behavior of Finite Composite Sandwich Plates
49
• The sound transmission loss through a clamped composite sandwich is highly dependent on the density of the core layer, or in other words, the core density is the strongest factor affecting the sound insulation of the composite sandwich structure. The experimental data above is a useful scientific basis for simulation studies, numerical calculations of sound transmission loss through the composite sandwich, and an important contribution in the calculation, design, and fabrication of the composite sandwich structures has the best sound insulation. • The sound transmission loss across a simply supported composite sandwich increases as the core porous layer thickness increases. This is one of the parameters that greatly affect the sound insulation of double-plates. • The sound transmission loss across a simply supported composite sandwich increase as the thickness of the faceplate increase while greatly affecting the resonant frequency. • When the elevation angle of the incident sound waves increases, the sound transmission loss decreases. On the other hand, the incident azimuth angle has negligible influence on the STL of a finite simply supported composite sandwich plate. • Obtained results at oblique sound incidence for the two different boundary condition cases suggest that, the sound insulation of the double composite-plate with a clamped boundary condition is better than that of the double-plate with a simply supported boundary condition.
Appendix A. Expressions of the Matrix Equation (58) ∗i The expressions of the M 2 x M 2 submatrices ∗i 1 (i = 1, 2, 3), 2 (i = 1, 2, 3, 4) and ∗∗i 1 (i = 1, 2, 3), as appearing in the matrix elements T ij,mn (i, j = 1, 2; m, n = 1, 2, …, M), are as flows: ⎡ ⎤ ⎡ ⎤ λ∗1 C1 1,1i ⎢ ⎥ ⎢ ⎥ λ∗1 C2 ⎢ ⎥ 1,2i ⎢ ⎥ ∗1 ⎢ ⎥ 1 = ⎢ ; C = (i = 1,2,...,M); ⎥ i . . ⎢ ⎥ .. .. ⎣ ⎦ ⎣ ⎦
λ∗1 1,mn = 3D11
m 4
CM
a
M 2 xM 2
+ 3D22
n 4 b
+ 4(D12 + 2D66 )
λ∗1 1,Mi MxM m 2 n 2 a
b (A.1)
⎡ ⎢ ⎢ ⎢ = ∗2 1 ⎢ ⎣
λ∗2 1,1
⎡
⎤ λ∗2 1,2
..
.
λ∗2 1,M
⎥ ⎥ ⎥ ⎥ ⎦
; λ∗2 1,n M 2 xM 2
0 ⎢ ⎢ n 4 ⎢ 1 ⎢ .. = 2D22 ⎢ b ⎢. ⎢. ⎣ ..
⎤
1 ··· ··· 1 .. ⎥ . 0 .. .⎥ ⎥ . . . . . . .. ⎥ . . . .⎥ ⎥ ⎥ .. . 0 1⎦
1 ··· ··· 1 0
MxM
(A.2)
50
T. I. Thinh and P. N. Thanh
⎡
∗3 1
⎤ λ∗3 1 .. ⎥ . ⎥ ⎥ .. ⎥ . ⎥ ⎥ ⎥ ∗3 λ ⎦
0 λ∗3 1 ··· ··· ⎢ ∗3 .. ⎢λ . ⎢ 1 0 ⎢ .. . . . . . . =⎢ . . . . ⎢ ⎢ . .. ⎣ .. . 0 ∗3 λ1 · · · · · · λ∗3 1
⎡ ; λ∗3 1 = 2D11
m 4 ⎢ ⎢ ⎢ a ⎣
⎤
14 24
..
. M4
1
0
⎥ ⎥ ⎥ ⎦ MxM
M 2 xM 2
(A.3) ⎤
⎡
∗1 2
1 ⎢ 9ab ⎢ 1 = ⎢ .. 4 ⎣ .
⎥ ⎥ ⎥ ⎦ 1
⎡ ∗2 2 =
9ab ⎢ ⎢ ⎢ 4 ⎣
⎡
λ∗2 2
..
.
⎥ ⎥ ⎥ ⎦ λ∗2 2
, λ∗2 2 M 2 xM 2
∗3 ⎤
λ∗3 2
(A.4)
M 2 xM 2
⎡
⎤ λ∗2 2
;
⎤ 0 1 ··· ··· 1 ⎢ .. ⎥ ⎢1 0 ... .⎥ ⎢ 3ab ⎢ . . . . . ⎥ ⎥ = ⎢ .. . . . . . . .. ⎥ ⎢ ⎥ 2 ⎢. ⎥ . . . ⎣. . 0 1⎦ 1 ··· ··· 1 0
(A.5)
MxM
0 · · · · · · λ2 ⎤ ⎡ ⎢ ∗3 .. ⎥ .. 1 ⎢λ ⎥ . 0 . ⎥ ⎢ 2 ⎥ 3ab ⎢ ⎥ ⎢ .. . . . . . . .. ⎥ ⎢ 1 ∗3 ∗3 = ; λ = ⎢ ⎥ ⎢ . . . . ⎥ .. ⎥ 2 2 ⎢ . ⎦ 2 ⎣ . ⎢ . ⎥ .. ⎣ .. ⎦ . 0 λ∗3 1 MxM 2 ∗3 0 λ∗3 · · · · · · λ 2 2 2 2 M xM ⎡ ⎡ ⎤ ⎤ ∗4 ∗4 0 λ2 · · · · · · λ 2 0 1 ··· ··· 1 ⎢ ∗4 ⎢ .. ⎥ .. ⎥ .. ⎢λ ⎢1 0 ... . .⎥ . ⎥ ⎢ 2 0 ⎢ ⎥ ⎥ ⎢ .. . . . . . . .. ⎥ ⎢ .. . . . . . . .. ⎥ ∗4 ∗4 2 = ⎢ . ; λ = ab ⎢ ⎥ ⎥ . . . . ⎥ 2 ⎢ ⎢. . . . .⎥ ⎢ . ⎢ ⎥ ⎥ . .. .. ⎣ .. ⎣ .. . 0 1⎦ . 0 λ∗4 ⎦ 2
(A.6)
(A.7)
∗4 0 1 · · · · · · 1 0 MxM λ∗4 2 · · · · · · λ2 M 2 xM 2 ⎡ ⎤ ⎤ ⎡ ∗ λ∗∗1 C1 1,1i ⎢ ⎥ ⎥ ⎢ λ∗∗1 C2∗ ⎢ ⎥ 1,2i ⎥ ⎢ ∗∗1 ∗ ⎢ ⎥ 1 = ⎢ ; Ci = ⎢ (i = 1,2,...,M); ⎥ . . ⎥ . . ⎦ ⎣ . . ⎣ ⎦ ∗ CM λ∗∗1 1,Mi MxM M 2 xM 2 m 4 n 4 2 n 2 m ∗ ∗ ∗ ∗ λ∗∗1 + 3D22 + 4 D12 + 2D66 1,mn = 3D11 a b a b (A.8)
Vibroacoutic Behavior of Finite Composite Sandwich Plates
⎡ ∗∗2 1
⎢ ⎢ =⎢ ⎢ ⎣
λ∗∗2 1,1
..
.
λ∗∗2 1,M
⎤ 0 1 ··· ··· 1 ⎢ .. ⎥ .. ⎢ . .⎥ ⎥ n 4 ⎢ 1 0 ⎢ .. . . . . . . .. ⎥ ∗ = 2D22 ⎢. . . . .⎥ ⎥ b ⎢ ⎥ ⎢. .. ⎣ .. . 0 1⎦ ⎡
⎤ λ∗∗2 1,2
51
⎥ ⎥ ⎥ ⎥ ⎦
; λ∗∗2 1,n M 2 xM 2
1 ··· ··· 1 0
MxM
(A.9) ⎡
∗∗3 1
∗3 ⎤
0 λ∗3 1 ··· ··· ⎢ ∗3 .. ⎢λ . ⎢ 1 0 ⎢ .. . . . . . . =⎢ . . . . ⎢ ⎢ . .. ⎣ .. . 0 λ∗3 · · · · · · λ∗3 1 1
λ1 .. . .. .
⎡
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ∗3 λ ⎦
; λ∗∗3 1 = 2D11
m 4 ⎢ ⎢ ⎢ a ⎣
⎤
14 24
..
. M4
1
0
⎥ ⎥ ⎥ ⎦ MxM
M 2 xM 2
(A.10) The expression of the constants f mn (k x , k y ) is in the form: fmn kx , ky =
b a 0
ϕmn (x, y)e−j(kx x+ky y) dxdy
0
b a 2mπ x 2nπ y −j(kx x+ky y) 1 − cos 1 − cos e = dxdy a b 0 0 ⎧ ⎪ ab for kx = 0, ky = 0 ⎪ ⎪ ⎪ 2 π 2 a 1−e−jbky ⎪ 4in ⎪ ⎪ for kx = 0, ky = 0 ⎪ ⎪ ⎨ ky ky2 b2 −4n2 π 2 = 4im2 π 2 b 1−e−jakx ⎪ for kx = 0, ky = 0 ⎪ kx (kx2 a2 −4m2 π2 ) ⎪ ⎪ ⎪ −jbk ⎪ ⎪ 16m2 n2 π 4 1−e−jakx 1−e y ⎪ ⎪ ⎩ − k k (k 2 a2 −4m2 π 2 ) k 2 b2 −4n2 π 2 for kx = 0, ky = 0 x y x y fmn kx , ky =
b a 0
0
b
a
=
ϕmn (x, y)e−j(kx x+ky y)
sin 0
0
mπ x a
sin
nπ y e−j(kx x+ky y) dxdy b
52
T. I. Thinh and P. N. Thanh
=
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
ab { 1−(−1)m −(−1)n +(−1)m+n } if kx mnπ 2 ' & mab 1−(−1)m e−jakx −(−1)n +(−1)m+n e−jakx 2 mnπ , nab 1−(−1)m −(−1)n e−jbky +(−1)m+n e−jbky
= 0 and ky = 0 if kx = 0 and ky = 0
if kx = 0 and ky = 0 ⎪ ⎪ mnπ 2 , ⎪ ⎪ 2 1−(−1)m e−jakx −(−1)n e−jbky +(−1)m+n e−j (akx +bky ) ⎪ mnabπ ⎪ ⎪ ⎪ if kx = 0 and ky = 0 ⎩ (k 2 a2 −m2 π 2 ) k 2 b2 −n2 π 2 x
y
References 1. Biot, M.A.: Theory of propagation of elastic waves in a uid-saturated porous solid I. lowfrequency range. II. higher frequency range. J. Acoust. Soc. Am. 28(2), 168–191 (1956) 2. Bolton, J.S., Green, E.R.: Normal incidence sound transmission through double-panel systems lined with relatively stiff, partially reticulated polyurethane foam. Appl. Acoust. 39, 23–51 (1993) 3. Bolton, J.S., Shiau, N.M., Kang, Y.J.: Sound transmission through multi-panel structures lined with elastic porous materials. J. Sound Vib. 191(3), 317–347 (1996) 4. Panneton, R., Atalla, N.: Numerical prediction of sound transmission through finite multilayer systems with poroelastic materials. J. Acoust. Soc. Am. 100(1), 346–354 (1996) 5. Kropp, W., Rebillard, E.: On the air-borne sound insulation of double wall constructions. Acta Acust. 85(5), 707–720 (1999) 6. Sgard, F.C., Atalla, N., Nicolas, J.: A numerical model for the low frequency diffuse field sound transmission loss of double-wall sound barriers with elastic porous linings. J. Acoust. Soc. Am. 108(6), 2865–2872 (2000) 7. Lee, C., Kondo, K.: Noise transmission loss of sandwich plates with viscoelastic core. American Institute of Aeronautics & Asreonautics AIAA-99–1458, pp. 2137–2147 (1999) 8. Antonio, J., Tadeu, A., Godinho, L.: Analytical evaluation of the acoustic insulation provided by double infinite walls. J. Sound Vib.. 263(1), 113–129 (2003) 9. Carneal, J.P., Fuller, C.R.: An analytical and experimental investigation of active structural acoustic control of noise transmission through double panel systems. J. Sound Vib. 272(3), 749–771 (2004) 10. Wang, J., Lu, T.J., Woodhouse, J., Langley, R.S., Evans, J.: Sound transmission through lightweight double-leaf partitions: theoretical modelling. J. Sound Vib. 286, 817–847 (2005) 11. Tanneau, O., Casimir, J.B., Lamary, P.: Optimization of multilayered panels with poroelastic components for an acoustical transmission objective. J. Acoust. Soc. Am. 120(3), 1227–1238 (2006) 12. Lee, J.S., Kim, E.I., Kim, Y.Y., Kim, J.S., Kang, Y.J.: Optimal poroelastic layer sequencing for sound transmission loss maximization by topology optimization method. J. Acoust. Soc. Am. 122(4), 2097–2106 (2007) 13. Chazot, J.D., Guyader, J.L.: Prediction of transmission loss of double panels with a patchmobility method. J. Acoust. Soc. Am. 121(1), 267–278 (2007) 14. Zhou, J., Bhaskar, A., Zhang, X.: Sound transmission through a double-panel construction lined with poroelastic material in the presence of mean ow. J. Sound. Vib. 332(16), 3724–3734 (2013) 15. Zhou, J., Bhaskar, A., Zhang, X.: Optimization for sound transmission through a double-wall panel. Appl. Acoust. 74(12), 1422–1428 (2013) 16. D’Alessandro, V., Petrone, G., Franco, F., De Rosa, S.: A review of the vibroacoustics of sandwich panels: Models and experiments. J. Sandw. Struct. Mater. 15(5), 541–582 (2013)
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17. Lu, T.J., Xin, F.X.: Vibro-acoustics of Lightweight Sandwich, Science Press Beijing and Springer-Verlag Berlin Heidenberg (2014) 18. Shojaeifard, M.H., Talebitooti, R., Ranjbar, B., Ahmadi, R.: Power transmission through double-walled laminated composite panels considering porous layer-air gap insulation. Appl. Math. Mech. 35(11), 1447–1466 (2014). https://doi.org/10.1007/s10483-014-1877-7 19. Petrone, G., DAlessandro, V., Franco, F., De Rosa, S.: Numerical and experimental investigations on the acoustic power radiated by aluminium foam sandwich panels. Compos. Struct.118, 170–177 (2014) 20. Hassan, S., Roohollad, T., Reza, A., Behzad, B.: A study on acoustic behavior of poroelastic media bonded between laminated composite panels. Latin Am. J. Solid Struct. 11, 2379–2407 (2014) 21. Liu, Y.: Sound transmission through triple-panel structures lined with poroelastic materials. J. Sound Vib. 339, 376–395 (2015) 22. Liu, Y., Sebastian, A.: Effects of external and gap mean flows on sound transmission through a double-wall sandwich panel. J. Sound Vib. 344, 399–415 (2015) 23. Daudin, C., Liu, Y.: Vibroacoustic behaviour of clamped double-wall panels lined with poroelastic materials. In: Proceedings of the 23rd International Congress on Sound and Vibration, vol. 341. Athens, Greece (2016) 24. Liu, Y., Daudin, C.: Analytical modeling of sound transmission through finite clamped doublewall sandwich panels lined with poroelastic materials. Compos. Struct. 172, 359–373 (2017) 25. Acoustics-Determination of sound power levels of noise sources—Precision methods for broad-band sources in reverberation rooms, International Standard ISO 3741–88(E), International Organization of Standardization, Geneva, Switzerland (1998) 26. ASTM standard method for laboratory measurement of airborne sound transmission loss of building partitions and elements using sound intensity, American Standard ASTM 2249– 02(E) (2002) ´ truyên ` âm qua tâm ´ composite sandwich và u´,ng du.ng vào 27. Tien. D.D.: Nghiên cu´,u tôn thât ` tàu thuy. Ða.i ho.c Nha Trang (2019) giam ôn ij
ij
ij
Crystal Structure and Mechanical Properties of 3D Printing Parts Using Bound Powder Deposition Method Do H. M. Hieu, Do Q. Duyen, Nguyen P. Tai, Nguyen V. Thang, Ngo C. Vinh(B) , and Nguyen Q. Hung(B) ` Mô.t, Binh Duong, Vietnam Faculty of Engineering, Vietnamese-German University, Thu Dâu {hieu.dhm,duyen.dq,tai.np,thang.nv,vinh.nc,hung.nq}@vgu.edu.vn ij
Abstract. This research focuses on introduction and evaluation of machine parts manufactured by bound powder deposition method, a new approach in metal 3D printing. After a review of metal 3D printing technology, a new approach for metal printing, called bound powder deposition, is introduced. Several specimens manufactured by the bound powder deposition are then conducted for experimental works. In the experimental works, the crystal structure of the printed part is investigated. In addition, typical mechanical proprieties of the printed part produced by the bound powder deposition are tested and compared with other those produced by other 3D printing and metallurgy methods. Keywords: Metal 3D printing · Additive manufacturing · Bound powder deposition · Bound powder extrusion · Binder jetting
1 Introduction Additive manufacturing (AM), commonly called as 3D printing, has become increasingly popular in various industrial sectors such as aerospace, medical applications, transportation, consumer products. This AM has also been considered as a significant factor in development of the fourth industrial revolution. The AM processes are classified into seven groups, which is defined by the standard terminology of ISO/ASTM 52900 [1], a cooperation between ISO/TC 261 and ASTM Committee F42. The groups include directed energy deposition (DED), powder bed fusion (PBF), material extrusion, material jetting, binder jetting, sheet lamination, and VAT photopolymerization process. These processes can produce complex shapes, high-strength lightweight designs, scalable rapid prototypes on various types of materials such as metals, ceramics, composites, polymers [2]. Among these processes, PBF processes such as electron beam melting (EBM) or selective laser melting (SLM), and DED processes like laser engineering net shape (LENS) are the most common approaches in industry to create 3D printed metallic parts [3, 4]. Such processes employ electron beams or high laser energy to fuse metallic powders or wires resulting in layer by layer of 3D objects. Although the electron beams or high-power lasers can provide parts with high density (99%), which results in superior © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 54–62, 2022. https://doi.org/10.1007/978-981-16-3239-6_4
Crystal Structure and Mechanical Properties of 3D Printing
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properties to castings; the two common approaches of metallic additive manufacturing have low production volumes with slow processing speed, high material cost, and high machining cost [4]. Moreover, outer shapes, uniform sizes, and chemical compositions of metal powders must be controlled well, which makes them become costly. Therefore, the systems using PBF or DED approach are commonly costly, approximately in a region of $250,000 to more than $1.5 million [5]. Additionally, PBF processes exist some limitations in design related to enclosed cavities. After 3D printing, unprocessed powders or trapped powders, which are remained inside the printed component cavities, needs to be removed by additionally mechanical operations [6]; as a result, PBF processes do not work effectively with the enclosed cavities. Moreover, metallic powders at micro sizes are hazardous to human health, and need to be manipulated correctly with proper safety protections [7]. Recently, a new metal 3D printing system based on material extrusion approach, called bound powder deposition (BPD) or bound powder extrusion (BPE), has been developed with a base cost of $99,500 [8]. A filament which is made of a combination between metallic powders and polymer is thermally extruded through a nozzle to deposit layer by layer of desired 3D objects. Therefore, the process does not lose any metal powders. Additionally, filament costs are more economic than metal powder costs [9]. This 3D printing system is an advanced fused filament fabrication (FFF) 3D printer, which is ranked in number one of additive manufacturing techniques according to their utility function values [10]. The presented system can reduce capital costs, compared to the ones of SLS and EBM. Moreover, during BPE procedure, metal powders kept by polymer are not easy to be diffused into the ambient air. Therefore, toxicity and flammability risks can be minimally decreased. In our experiment, several 17-4 PH stainless steel specimens were first fabricated by the BPE technique. Several typical mechanical proprieties of the 3D printed stainless steel (17-4 PH) parts are then investigated and are compared to the properties of 17-4 PH stainless steel parts, which are produced by conventional manufacturing techniques. Such examination approaches like dimension accuracy, morphology, surface roughness, porosity test, tensile test and hardness test were carried out. The obtained results can open feasibility for practical applications of 17-4 PH stainless steel metal, which is produced by the new, economic, and friendly 3D metal printing system.
2 Experimental 2.1 Material A filament comprised of metal powder and plastic binder (17-4 PH stainless steel, Markforged, USA) was used in this research. The composition of 17-4 PH stainless steel includes chromium (15–17.5%), nikel (3–5%), copper (3–5%), silicon (1% max), manganese (1% max), niobium (0.15–0.45%), carbon (0.07% max), phosphorus (0.04%), sulphur (0.03% max), and iron (the rest).
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2.2 Fabrication Method Bound powder extrusion includes several steps: 3D model design, slicing the model into digital layers using a conversion to STL file, set-up of printing parameters, layer-bylayer printing, debinding, sintering, and post processing. Most of the steps are similar to other additive manufacturing processes, except the layer-by-layer printing, debinding and sintering, which are unique characteristics of BDE. A Metal X system (Markforged, USA) used in this research consists of three machines, which can represent for the three main unique steps of BPE technique, as shown in Fig. 1. The printer, with a maximum printing volume of 300 × 220 × 180 mm3 , has a heated print chamber, a heated bed with vacuum-sealed print sheet and two extruders, one for supplying metal powders mixed with wax polymer and one for supplying ceramic as release material. The washing machine with a washing volume of 18.356 cm3 is employed to dissolve the binding material using Opteon SF79. The sinter machine is a high-performing furnace with a sintering workload of 3.020 cm3 to transform lightly bound-metal-powder part to a full
Fig. 1. A bound powder extrusion system.
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metal part. An argon gas is inserted into the chamber of the sinter machine during the sintering process. In the beginning of printing steps, a filament consisting metal powders enclosed in a thermoplastic polymer is heated and softened to create layers of 3D object by the printer. Depending on design of the printed part, supports can be printed to prevent any deformation or any damages of the main printed parts. A ceramic layer is deposited between the supports and the printed parts during the construction at 1:10 ratio to metal spools. This layer supports detaching and removing the support structures easily. After that, a debinding process is carried out by using the washing machine, following by using a sintering process to create the final 3D parts. The printed specimens are exanimated to evaluate typical mechanical properties. The fabrication parameters (recommended by the manufacturer) are shown in Table 1. Table 1. Fabrication parameters (recommended by the manufacturer) Mass of printed part
58.99 g
Post-sintered layer height 0.125 mm Printing time
4 h 16 m
Debinding time
12 h
Sintering time
27 h
Material cost
7.62 USD
2.3 Analysis Methods An optical microscope (BX53M, Olympus, Japan) was used to observe morphology of the printed part. To measure surface roughness, a portable surface roughness tester (SJ210, Mitutoyo, Japan) was used. Porosity of the printed part was tested by observation of the printed part’s cross-section surface. The cross section was prepared by the following steps: the printed part was cut into a small pieces by an abrasive cutter (Metacut 251, Metkon, Turkey); an automatic mounting press (Ecopress 100, Metkon, Turkey) was then used to hold the small piece for the next polishing process, which employed a grinding–polishing machine (Forcipol 1V, Metkon, Turkey). Moreover, tensile test was carried out on printed specimens, which had been produced with ASTM E8 Standard [11], by a tensile tester (AGX-Plus 50 kN, Shimadzu, Japan). Additionally, hardness test was performed on printed samples with ASTM E18 standard [12], by a Rockwell hardness testing machine (250 MRS, Affri, USA).
3 Result and Discussion 3.1 Dimension Accuracy The measured dimensions (length × width × thickness) of the 3D printed part are 151.32 mm × 20.13 mm × 3.04 mm while the nominal ones are 150 mm × 20 mm ×
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3 mm. The actual dimensions are greater than the design’s ones, with errors less than 1–2%. This can prove for the high accuracy of the BPD system. Additionally, the printed part with these dimensions only costs approximately 7.62 USD. This is cost-effective compared to other 3D metal printing approaches like PBF or DED, which is in good agreement with the IDTechEx report [9]. 3.2 Morphology Figure 2 illustrates morphology of the printed specimen, which includes not only the appearance but also the surface morphology. The design model was shown in Fig. 2a with a taken photo of the specimen during the printing process attached in the top right corner. After printing, debinding, and then sintering, top view and front view of the final specimen were shown in Fig. 2b and 2c. In the top view, the metallic filament and its printing direction could be observed by naked eyes. The boundaries around the inside metallic filaments, which created the shape of an object, were also seen. The inside metallic filament played an important role in the printed object’s mechanical properties. In this research, the printing direction had a 45°-tilted angle (red arrow in Fig. 2b), which could strongly enhance for the strength of the specimen. Each filament after heating and depositing on the support has a width of approximately 500 µm, as shown in Fig. 2d. Line by line was deposited on each layer of the printed specimen resulting in surface roughness, which is approximately 2.68 µm. In the front view, layer-by-layer structure of the printed specimen can be clearly observed. The thickness of each layer was approximately 125 µm. The connections between layers were sufficiently good. Additionally, bending issue almost did not appear on the printed specimen.
Fig. 2. Morphology of the printed specimen.
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3.3 Porosity Open porosity was not observed on the outer surface of the 3D printed specimen. Therefore, appearance of the 3D printed product is acceptable compared to the one made by conventional techniques. However, enclosed porosity can be observed on the cross section, as shown in Fig. 3. The porosity of the printed one, roughly 0.50–5 µm in diameter, was quite large compared to that of conventionally fabricated 17-4 PH stainless steel, which is approximately 0.45–1.80 µm in diameter. The largest porosity, which appeared in BPD technique, was roughly 2.8 times in diameter to the one appeared in the traditional technique. The conventionally fabricated 17-4 PH stainless steel showed small sizes of porosities because it was produced by top-down approach, where the initial material was a bulk metal with optimized fabrication parameters like heat treatment time. Conversely, the 3D printed specimen was fabricated by bottom-up approach, where the starting material was a metal filament. The connection between filaments could cause large gaps, which would become porosities after sintering. To reduce the quantities and the sizes of porosities, several fabrication parameters like sintering times, which have not been optimized yet in this research, should be investigated. Although the obtained results showed large sizes of porosities, BPD technique is still advantageous compared to other 3D printing techniques using fused filament fabrication approaches. For instance, in Thompson’s research [13], the sizes of porosities were up to 10 µm in diameter or even greater than 30 µm in diameter. The samples also existed more porosities than the ones using this BPD technique.
Fig. 3. Porosity of: (a) conventionally fabricated 17-4 PH stainless steel, and (b) 3d printed specimen.
3.4 Tensile Test A comparison in ultimate tensile strength (UTS) between 3D printed specimen and conventionally fabricated specimen is presented in Fig. 4. The UTSs are 985 MPa in the 3D printed specimen and 1163 MPa in the conventionally fabricated specimen. The strength of the printed one was reduced roughly 15%. Sintering time and porosity
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may cause difference between the two specimens. The insufficient sintering time or the large sizes of porosities made the connection between filaments be not highly strong. Therefore, during tensile test, the samples could be broken at a lower value of UTS, compared to those using conventional techniques. Interestingly, the 3D printed specimen had no necking, a mode of tensile deformation. After its ultimate tensile strength, the 3d printed specimen was immediately broken, which might represent for a brittle property. On the other hand, the conventionally fabricated specimen still existed its typical necking after applying the ultimate tensile strength. The specimen clearly showed a deformation in the neck position, which can present for a ductile property. The disappearance of necking in the 3D printed specimen made its strains at the applied ultimate tensile strength moment and at the fracture moment same; whereas, the strain at the fracture point was greater than that at the applied ultimate tensile strength point in the conventionally fabricated specimen. Although the fabrication parameters have not been optimized in this research, the obtained UTSs are sufficiently high, similar as 17-4 PH stainless steel H1150 [14]. This makes the printed specimens satisfy for applications required good strengths such as end-of-arm tools, lightweight brackets, and pump shafts. The BPD technique demonstrates not only for its advantages like other 3D printing techniques, such as complexity, simple operation steps; but also for its ability to produce similar mechanical properties as the ones made by conventional techniques.
Fig. 4. Tensile test of the conventionally fabricated specimen and of the 3D printed specimen.
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3.5 Hardness Test The 3D printed specimen showed a hardness value of 29.5 HRC, as shown in Fig. 5. Although this value is not remarkable compared to the value of conventionally fabricated 17-4 PH stainless steel (37.1 HRC), the hardness of the printed specimen using BPE approach is sufficiently strong. The measured results of the 3D printed specimen were in good agreement with the printed specimen’s porosity. When the porosity of the printed specimen increases, its hardness can be decreased. Additionally, in this primarily research, the printed specimen was just as sintered, and the sintering time has not been optimized yet.
Fig. 5. Hardness test of the 3D printed part: (a) before the test, and (b) after the test.
4 Conclusion The new and low-cost approach in metal 3D printing, bound powder deposition, is introduced. Several 17-4 PH stainless steel specimens were prepared to investigate typical mechnical properties. The printed dimensions have small errors compared to desired ones. The morphology was also obsersed. The porosity of 3D printed specimens has its diameters larger than the one of conventionally fabricated specimens. The ultimate tensile strengths between the printed one and the conventional one were compared. The 3D printed specimen showed a low value of the strength with the interesting disappearance of the neck, a typical phenomenon of metal material during tensile test. Additionally, the hardness number of the 3D printed specimen was smaller than that of the traditional specimen. The studies in this research will open a new way for applications of 17-4 PH stainlesss steel with a low 3D printing fabrication cost and a friendly approach. Acknowlegement. This work was supported by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant no. 107.01–2018.335.
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References 1. A. Standard. ISO/ASTM 52900: 2015 Additive manufacturing-General principlesterminology. ASTM F2792-10e1 (2012) 2. Gibson, I., Rosen, D.W., Stucker, B.: Additive Manufacturing Technologies, vol. 17. Springer, Boston, MA (2014) 3. Gu, D.D., Meiners, W., Wissenbach, K., Poprawe, R.: Laser additive manufacturing of metallic components: materials, processes and mechanisms. Int. Mater. Rev. 57(3), 133–164 (2012) 4. Frazier, W.E.: Metal additive manufacturing: a review. J. Mater. Eng. Perform. 23(6), 1917– 1928 (2014) 5. Garcia-Colomo, A., Wood, D., Martina, F., Williams, S.W.: A comparison framework to support the selection of the best additive manufacturing process for specific aerospace applications. Int. J. Rapid Manuf. 9(2–3), 194–211 (2020) 6. Hunter, L.W., Brackett, D., Brierley, N., Yang, J., Attallah, M.M.: Assessment of trapped powder removal and inspection strategies for powder bed fusion techniques. Int. J. Adv. Manuf. Technol. 106(9–10), 4521–4532 (2020). https://doi.org/10.1007/s00170-020-0430-w 7. Bours, J., Adzima, B., Gladwin, S., Cabral, J., Mau, S.: Addressing hazardous implications of additive manufacturing: complementing life cycle assessment with a framework for evaluating direct human health and environmental impacts. J. Ind. Ecol. 21(S1), S25–S36 (2017) 8. Campbell, R.I., Wohlers, T.: Markforged: Taking a different approach to metal additive manufacturing. Metal AM 3(2), 113–115 (2017) 9. Richard, C., Jonathan, H., Ivan De, B.: Metal Additive Manufacturing 2020–2030. IDTechEx (2020) 10. Sgambatia, A., et al.: URBAN: conceiving a lunar base using 3D printing technologies. In: Proceedings of the 69th International Astronautical Congress, pp. 1–5. Bremen, Germany (2018) 11. Astm, E.: Standard test methods for tension testing of metallic materials. Annual Book of ASTM Standards, ASTM (2001) 12. Astm, A.: ASTM E18-03: Standard test methods for rockwell hardness and rockwell superficial hardness of metallic materials. Annual Book of ASTM Standards (2003) 13. Thompson, Y., Gonzalez-Gutierrez, J., Kukla, C., Felfer, P.: Fused filament fabrication, debinding and sintering as a low cost additive manufacturing method of 316L stainless steel. Additive Manuf. 30, 100861 (2019) 14. Steel, A.K.: 17-4 PH Stainless Steel: Product Data Bulletin. AK Steel Corporation, West Chester, OH, USA (2015)
Experimentally Investigating the Resonance of the Vibration of Two Masses One Spring System Under Different Friction Conditions Quoc-Huy Ngo1 , Ky-Thanh Ho1 , and Khac-Tuan Nguyen2(B) 1 Faculty of Mechanical Engineering,
Thai Nguyen University of Technology, Thai Nguyen, Vietnam {ngoquochuy24,hkythanh}@tnut.edu.vn 2 Faculty of Automotive Engineering, Thai Nguyen University of Technology, Thai Nguyen, Vietnam [email protected]
Abstract. This report presents some experimental results of the effect of the Coulomb friction on the resonant frequency of the two-mass system connected by a nonlinear leaf spring. The experimental apparatus designed and built has an ability to vary the excitation frequency, the excitation force and friction force. Experimental data show that the resonance frequency of the system tends to decrease when increasing the friction force. The resonant frequency of the system can be expressed as the functions depending both on the amplitude of excitation force and on the Coulomb friction force. The experimental results serve as the basic premise for the development of studies applied in self-movement structure operating under different resistant environments. Keywords: Resonance frequency · 2DOF · Vibration-driven locomotion · Friction force · Excitation force
1 Introduction The nonlinear oscillator consists of two masses connected by one spring is the basic model used to theoretically study and applied to a variety of research fields, such as vibrations, multi-body systems, structural dynamics and transportation… especially in the operability of a self-moving mechanism (new locomotion system) [1–6]. Generally, the mathematical model of the system contains two ordinary conjugate differential equations with cubic non-linearity. Some researchers have presented several techniques for solving analytically a second order differential equation with various strong non-linear characteristic. S.K. Lai and C.W. Lim [5] used an analytical approach developed for non-linear free vibration of a conservative, two degrees of freedom mass-spring system having linear and non-linear stiffness (with model in Fig. 1). Max–Min Approach (MMA) is applied to obtain an approximate solution of three practical cases in terms of a non-linear oscillation system [3]. Recently, many researchers based on two-mass one-spring model developed the mobile devices employing vibration for motion, also © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 63–71, 2022. https://doi.org/10.1007/978-981-16-3239-6_5
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called vibration-driven locomotion systems; such as designing, modeling and experimental validation; dynamical analysis; optimal progression and motion control [7–13]. Figure 2 is the physical model of the system employing vibration for motion. However, most of these studies have not yet evaluated the effect of friction on the mechanical movement. In previously theoretical studies, the friction usually is not fully considered. In fact, the self-moving models, such as biomedical applications of capsule and rehabilitation robots in medical, pipe capsule robots… have to work in different environments, where the environmental resistance is different. Therefore, the study of self-movement under different resistance conditions is very important. Recently, Christoph Kossack et al. [14] reported frequency response function (FRF) determined by either experiment or simulation for a dynamic oscillator with a sliding friction contact (Coulomb friction). The results of this study have been just used for single degree of freedom (SDOF) system, thus haven’t applied to capsule robots.
Fig. 1. Basic system of the two masses connected by linear or non-linear spring [2, 3, 5]
Fig. 2. Physical model of the system
With vibration-driven locomotion systems based on two masses one spring, phenomenon resonance plays an important role. When an oscillating force is applied at a resonant frequency of a dynamical system, the system will oscillate at a higher amplitude than when the same force is applied at the others, i.e. non-resonant frequencies. Mathematically, when a system operates at resonant frequency, there is a 90° phase between the exciting force and the system’s response. Therefore, it’s possible to say that the force is in phase with the speed response of the system. It means that the sense of the force is the same as the sense of the motion [15, 16]. Consequently, the exciting force applies permanently in the direction and does positive work-done on the system body. This can enhance the effectively changing energy of the system. This study uses the model of two masses connected by a nonlinear spring to evaluate oscillation of mechanical system, specifically in terms of vibration resonance. The research results are very significant in assessing the ability of vibration-driven system under the different resistance conditions.
2 Design and Setup the Model of Apparatus 2.1 Design and Setup Experimental Apparatus Design and prototype are two cornerstone aspects in studying vibration-driven locomotion systems. Generally, the design of vibration-driven locomotion systems is based
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on the mechanics of the interaction between the inertial mass and the system body. In such system, the inertial mass is excited and controlled to periodically move inside the system body, creating the inertial force when changing either its direction of motion, or acceleration, or velocity. The locomotion thus can be generated with the presence of inertial force inside and resistant force from surrounding environment.
Fig. 3. (a) The experimential diagram and (b) experimential apparatus
This experimental study uses an apparatus based on the model of two masses and a non-linear leaf spring system, as shown in Fig. 2 and Fig. 3. A mini electro-dynamical shaker (1) is placed on a slider of a commercial linear bearing guide (3), providing a tiny friction force. An additional mass (2) was clamped on the shaker shaft. Generally, a sinusoidal current applying to the shaker leads to relative linear oscillation of the shaker shaft with the inertial mass added on, creates the excitation force F e . This force depends on the current supplied and can be adjusted the sinusoidal voltage supplying to the amplifier. The moveable mass, combined by the addition mass and the shaker shaft, is assigned as inertial mass m1 , playing the role of the internal mass of the capsule robot. A non-contact position sensor (6), model Kaman KD-2306, was used to measure the relative motion of the inertial mass and the shaker body, i.e. measuring X 1 –X 2 . The movement of the shaker body was recognized by a linear variable displacement transformer (LVDT), i.e. absolute distance X 2 . A carbon tube (4) is connected with the shaker body by means of a flexible joint, avoiding any misalignment when moving. As shown on Fig. 4(a), the carbon tube is able to slide between two aluminum pieces in the form of a V-block (5). The two V-blocks are fixed on two electromagnets (7). The body shaker, including the sensors LVDT and the carbon tube, was referred as the mass m2 of the mass-spring model. The total weight of additional mass and the shaker shaft was considered as the inertial mass m1 . A set of tests was carried out to determine the stiffness of leaf spring connecting the movable shaft with the shaker body. It is noted that the shaker body was fixed on the slider during these tests. A string was attached to the shaker shaft and rode over a pulley while a series of certain masses were hung on the other end of the string. The gravitational force of the masses pulled the shaft moving forward. The displacement of the shaker shaft, corresponding to each level of masses, was measured by using the non-contact position sensor. A nonlinear curve fitted with the cubic function was then applied to determine the relationship between the gravitational force and the movement. The experiment tests revealed that the spring is a nonlinear spring with hard characteristics (the cubic
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term is positive). In addition, the damping coefficient, c was determined by logarithmic decrement method. 2.2 Experimental Operation Supplying a certain value of electrical current to the electromagnets provides a desired clamping force on the tube and thus receiving a corresponding value of sliding friction. The friction force was measured by pushing or pulling the body to move at a steady speed V s . Experimental data revealed that the friction force F f depends on the applied voltage V by fitted curve in the following relationship: Ff = 1.52889 + 0.13591 ∗ e1.25642∗V
(1)
This is configuration allows varying the friction force without changing the body mass and thus can experimentally check with the locomotion capacity in different friction levels. The signals from the sensors were captured by a data acquisition system (DAQ) and then stored and analyzed.
Fig. 4. Varying the friction force: (a) apparatus structure and (b) the dependency of friction force on supplied voltage
A supplementary experiment was implemented to determine the relationship of the magnetic force and the supplied current. A load-cell was used as an obstacle resisting the shaker movement and thus to measure the magnetic force induced. A DC voltage was supplied to the shaker to generate the magnetic force. Varying the voltage, several pairs of the current passing the shaker and the force were collected. Experimental data revealed that the force is proportional to the current supplied to the shaker (see Nguyen et al. 2017 [7] for detailed information of how to determine this relationship). In this study, the amplitude of supplied current was setup with three values, as 0.5 A, 0.75 A and 1.0 A, providing the magnitude of the exciting force, as 5.4 N, 8.1 N and 10.8 N, respectively. During each experiment, these values were kept without changing. All of the experimental parameters are summarized on Table 1.
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Table 1. Parameters of experiments. Parameter
Notation
Value
Unit
Internal mass
m1
0.518
Kg
Body mass
m2
1.818
Kg
Linear stiffness
k1
2988.388
N/m
Cubic stiffness
k2
52158700
N/m3
Damping
c
8.893542
Ns/m
Friction force
Ff
0; 3.56; 8.31; 16.63; and 28.5
N
Excitation force
Fe
5.4; 8.1; and 10.8
N
Excitation frequency
f exc
[2–30]; swept by 1 Hz step in 2 s
Hz
3 Results and Discussion Theoretically, resonance occurs when the phase difference between the excitation force F e and the X 1 –X 2 shift is 90 degrees. At that time, the excitation current is minimum and the swept amplitude of oscillation X 1 –X 2 is maximum. Figure 5 shows an experimental data received by sweeping from 2 Hz to 30 Hz with 3.56 N of friction force and 5.4 N of excitation force. The parameters included sweeping step as 1 Hz and sweeping time as 3 s each step. The experimental data was firstly conducted in time regime (right figure) then changed into frequency regime (right figure). The maximum amplitude of oscillation X 1 –X 2 was revealed at 14 Hz. At this frequency, the excitation force is the smallest. Others experimental data were carried out similarly by changing the excitation forces and friction forces.
Fig. 5. Determining the experimental frequency resonance
Figure 6 is the experimental data showing the relationship of resonance frequency and friction force at three levels of excitation force 5.4 N, 8.1 N and 10.8 N, respectively. Firstly, as can be seen in all sub-plots, the amplitude relative displacement of the internal mass and the shaker body increases when raising the excitation frequency to maximum,
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then decreases if continuously raising. The excitation frequency, where the relative displacement reaches maximum value, is the resonant frequency. Secondly, at each level of excitation force, when increasing the friction force, the resonance frequency tends to decrease. Thirdly, without changing the level of resistant force, the resonance frequency increases as the amplitude of excitation force F e increases from 5.4 N to 10.8 N. Theoretical studies of the dynamics of the mechanical system with linear springs show that the natural frequency f n of this system can be determined by following: 1 k1 ∗ (m1 + m2 ) 1 2988.388 ∗ (0.518 + 1.818) = 13.703Hz = fn = 2π m1 ∗ m2 2π 0.518 ∗ 1.818 The results showed that the resonance frequency of the system was different from the natural frequency f n . The causes of this difference may be from the nonlinearity of the system, the friction force and damping applied to the system.
Fig. 6. The variation of resonance frequency under different friction force without changing the exciting force, respectively: (a) F e = 5.4 N; (b) F e = 8.1 N; and (c) F e = 10.8 N
Another view of relationship of resonance frequency and friction force is shown on Fig. 7 to support the above mentioned ideas. The amplitude of oscillation of the mechanical system at resonance increases with increasing the amplitude of excitation force at all investigated resistance levels F f . Besides, with the same value of the excitation force, the magnitude of oscillation (i.e. amplitude of X 1 –X 2 ) at resonance decreases when increasing the resistant force F f . As shown on Fig. 8(a), the resonant frequency f res of the system at each amplitude of excitation force can be represented by a quadratic polynomial function depending on the Coulomb friction force as following: fres = A + B ∗ Ff + C ∗ Ff2
(2)
By fitting the plot data (Fig. 8(a)), the detailed expressions describe the relationship between resonant frequency and friction force as following: fres1 = 12.98197 − 0.38692 ∗ Ff + 0.01123 ∗ Ff2 fres2 = 13.98197 − 0.38692 ∗ Ff + 0.01123 ∗ Ff2
(3)
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Fig. 7. The variation of resonant frequency under the same Coulomb friction when varying the excitation force, respectively: (a) F f = 3.56 N; (b) F f = 16.63 N; and (c) F f = 28.5 N
fres3 = 15.03625 − 0.32791 ∗ Ff + 0.00898 ∗ Ff2 Where f res1 , f res2 , f res3 were the resonant frequency of the system depending on friction force at exctitation force as 5.4 N, 8.1 N and 10.8 N, respectively. The experimental data also show that the coefficients of Eq. (2) depend on excitation force, as shown on the Fig. 8(b). This relation can be expressed as following equations: y = a + b ∗ Fe A = 10.91864 + 0.38042 ∗ Fe B = −0.45577 + 0.01093 ∗ Fe
(4)
C = 0.01386 − 4.167 ∗ 10−4 ∗ Fe
Fig. 8. Fitted curves of the Frequency Response Function with the Coulomb friction force and excitation force.
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4 Conclusion The effects of the Coulomb friction on the resonant frequency of the two-mass system connected by a nonlinear leaf spring were experimentally investigated by sweeping in the range from 2 Hz to 30 Hz. Experimental data show that the resonant frequency of the system seemed to decrease when increasing the resistant force. Under the different friction, the resonant frequencies have been changed quite in comparing with the natural frequency of the system. The resonant frequency of the system can be expressed as the functions depending both on the amplitude of excitation force and on the Coulomb friction force. The experimental results serve as the basic premise for the development of studies applied in self-movement structure operating under different resistant environments. Acknowledgement. This research was funded by Vietnam Ministry of Education and Training, under the grant number B2019-TNA-04. The authors would like to express their thank to Thai Nguyen University of Technology, Thai Nguyen University for their supports. Technical guidances and supports from Dr. Van-Du Nguyen from Thai Nguyen University of Technology are very much appreciated.
References 1. Cveticanin, L.: Vibrations of a coupled two-degree-of-freedom system. J. Sound Vib. 247(2), 279–292 (2001) 2. Cveticanin, L.: The motion of a two-mass system with non-linear connection. J. Sound Vib. 252(2), 361–369 (2002) 3. Ganji, S.S., Barari, A., Ganji, D.D.: Approximate analysis of two-mass–spring systems and buckling of a column. Comput. Math. Appl. 61(4), 1088–1095 (2011) 4. Guo, S., et al.: Development of Multiple Capsule Robots in Pipe. Micromachines (Basel) 9(6) (2018) 5. Lai, S.K., Lim, C.W.: Nonlinear vibration of a two-mass system with nonlinear stiffnesses. Nonlinear Dyn. 49(1–2), 233–249 (2006) 6. Xu, J., Fang, H.: Improving performance: recent progress on vibration-driven locomotion systems. Nonlinear Dyn. 98(4), 2651–2669 (2019). https://doi.org/10.1007/s11071-019-049 82-y 7. Nguyen, V.-D., et al.: The effect of inertial mass and excitation frequency on a Duffing vibro-impact drifting system. Int. J. Mech. Sci. 124–125, 9–21 (2017) 8. Liu, L., et al.: Reliability of elastic impact system with Coulomb friction excited by Gaussian white noise. Chaos Solitons Fractals 131, 109513 (2020) 9. Liu, Y., et al.: Bifurcation analysis of a vibro-impact experimental rig with two-sided constraint. Meccanica 55(12), 2505–2521 (2020). https://doi.org/10.1007/s11012-020-011 68-4 10. Nguyen, V.-D., et al., A new design of horizontal electro-vibro-impact devices. J. Comput. Nonlin. Dyn. 12(6), 061002 (2017) 11. Safaeifar, H., Farshidianfar, A.: A new model of the contact force for the collision between two solid bodies. Multibody Syst. Dyn. 50(3), 233–257 (2020). https://doi.org/10.1007/s11 044-020-09732-2
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12. Yan, Y., Liu, Y., Manfredi, L., Prasad, S.: Modelling of a vibro-impact self-propelled capsule in the small intestine. Nonlinear Dyn. 96(1), 123–144 (2019). https://doi.org/10.1007/s11 071-019-04779-z 13. Duong, T.-H., et al.: A new design for bidirectional autogenous mobile systems with two-side drifting impact oscillator. Int. J. Mech. Sci. 140, 325–338 (2018) 14. Christoph Kossack, J.Z., Schmitz, T.L.: The sliding friction contact frequency response function. Procedia Manufacturing 34, 73–82 (2019) 15. Polesel-Maris, J.R.M., et al.: Experimental investigation of resonance curves in dynamic force microscopy. Nanotechnology 14(9), 1036–1042 (2003) 16. Bleck-Neuhaus, J.: Mechanical resonance 300 years from discovery to the full understanding of its importance (2018)
Nonlinear Bending Analysis of FG Porous Beams Reinforced with Graphene Platelets Under Various Boundary Conditions by Ritz Method Dang Xuan Hung, Huong Quy Truong(B) , and Tran Minh Tu National University of Civil Engineering, 55 Giai Phong Road, Hai Ba Trung District, Hanoi, Vietnam {hungdx,truonghq,tutm}@nuce.edu.vn
Abstract. This paper deals with the nonlinear bending response of functionally graded porous beams reinforced by graphene platelets (GPLs) with various boundary conditions using the Ritz method. Based on the trigonometric shear deformation beam theory and the von Kárman type of geometrical nonlinearity strains, the system of nonlinear governing equations is derived using the minimum total potential energy principle. This system of nonlinear equations is then solved by the Newton–Raphson method. The comparison with the available published results validates the obtained results. The effects of the porosity distribution patterns, the porosity coefficient, the GPL reinforcements, the slenderness ratios and the boundary conditions on the nonlinear deflection of the FGP porous beam are also investigated. Keywords: Porous beam · Nonlinear bending · Ritz method · Trigonometric shear deformation beam theory · Graphene platelet reinforcement
1 Introduction Functionally graded materials (FGMs) are a class of composites that are firstly invented by Japanese scientists in 1984. FGMs are characterized by a continuous variation in both composition and material properties in one or more directions, thus eliminating interface problems and diminishing the stress concentration that normally exists in conventional laminated composites [1, 2]. Porous materials (metal foams) are a new class of FGMs characterized by low density, lightweight, good stiffness and excellent energy absorption, Porosities inside metal foams can be distributed in different manners. Their mechanical properties are significantly influenced by the amount of porosities and their pattern of distribution. To archive desire material properties, graded non-uniform porosities are introduced to produce FG metal foams which have been proved to have better structural response than the normal uniform foams and are investigated by the increasing number of researchers [3–10]. Introducing nanofillers into porous materials is a practical way to strengthen their mechanical properties and maintain their potential for lightweight structures simultaneously. The properties of such materials can be significantly improved with the addition of © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 72–86, 2022. https://doi.org/10.1007/978-981-16-3239-6_6
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nanofillers in the form of carbon nanotubes (CNTs) and graphene platelets (GPLs). Thus, carbon nanotube reinforced beams and plates have been extensively studied to examine their static and dynamic response [11–15]. Until now, static and dynamic analysis of GPL reinforced structures is still at the beginning stage [16–20]. A nonlinear static analysis is required for any static application in which the stiffness of the entire structure changes during the loading scenario. Many works focused on nonlinear behaviours of beams and plates are reported in the existing literature. Feng et al. [21] studies the nonlinear bending behavior of Timoshenko multi-layer polymer nanocomposite beams reinforced with GPLs using Ritz method. Shen et al. [22] studied the large amplitude vibration of functionally graded graphene-reinforced composite laminated plates resting on an elastic foundation and in thermal environments based on a higher-order shear deformation plate theory and perturbation technique. Wu et al. [12] analyzed nonlinear vibration of imperfect FG-CNTRC beams is based on the firstorder shear deformation beam theory and von Kármán geometric nonlinearity. Ke et al. [15] investigated the nonlinear free vibration of FG nanocomposite beams reinforced by single-walled carbon nanotubes (SWCNTs) based on Timoshenko beam theory. Barati et al. [20] used a refined beam model to investigate the post-buckling behavior of geometrically imperfect porous beams reinforced with graphene platelets. Yas and Rahimi [23] presented thermal buckling of FG porous nanocomposite beams subjected to a thermal gradient using the generalized differential quadrature method. This paper deals with the geometrically nonlinear bending analysis of functionally graded porous beams reinforced by graphene platelets with various boundary conditions using the Ritz method. GPLs are distributed in the thickness direction with uniform and nonuniform patterns. Uniform, symmetric, and asymmetric distributions of porosity have been considered. After conducting the validate example, the effects of the porosity distribution patterns, the porosity coefficient, the GPL reinforcements, the slenderness ratios and the boundary conditions on the nonlinear deflection of the FGP porous beam are investigated in detail.
2 Material Properties Consider a beam with thickness h, width b, and length L defined in the Cartesian coordinate system (x, z) as shown in Fig. 1. The porosity distributes within the thickness according to the symmetric, asymmetric and uniform laws below [24]. ⎧ ⎨ ψ(z) = cos(π z/h) (1) ψ(z) = cos π z/2h + π4 ⎩ ψ(z) = ψ0 This porosity distribution leads to a variety of material properties as follows. ⎧ ⎪ ⎨ Ez = Ec [1 − e0 ψ(z)] Gz = Ez /[2(1 + νz )] ⎪ ⎩ ρz = ρc [1 − em ψ(z)]
(2)
Where Ec and ρc denote the Young modulus and the mass density of the material without porosity, respectively; E1 , E2 (ρ1 , ρ2 ) denote the maximum, minimum Young’s
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Porosity distribution 1
Porosity distribution 2
Uniform distribution
Fig. 1. Porosity distribution: symmetric, asymmetric and uniform laws
modulus (mass density) without GPLs of the non-uniform porosity distribution and that of the uniform law is denoted by E ; The porosity coefficient e0 and the mass density coefficient em are defined by e0 = 1 − E2 /E1 and em = 1 − ρ2 /ρ1 . Based on the relationship between the mechanical properties of closed-cell cellular solids (3) that is fitted from statistical Gaussian Random Fields model [25], Ez = Ec
2.3 ρz + 0.121 /1.121 ρc
ρz 1. There is an intersection between the frequencies of different porosity volume fractions. Comparison between Fig. 4 (nx = 0.5) and (nx = 2), shows that when nx is low, the first frequency parameter of Fig. 4 (nx = 0.5) is higher than Fig. 4 (nx = 2). The effect of the layer thickness ratio on the frequency parameter can also be seen from Fig. 5, it shows the first frequency parameter of imperfect beam for different values of the layer thickness ratio versus the FG (nz) power indexes. It is observed that increasing nz decrease the frequency parameter of the sandwich beam. It is seen that the first frequency parameter of imperfect FG beam, increases with the increment of the layer thickness ratio value. But the effect of the layer thickness ratio on the first frequency parameter of the imperfet FG beam is complicated. When nz is low, increasing the layer thickness ratio decreases the frequency parameter of beam, while when nz is more, the frequency parameter increases with the increment of the layer thickness ratio.
4 Conclusions The vibration of bi-dimensional imperfect FG sandwich beam has been investigated in the present work. The beam consists of three layers, the upper face of the sandwich beam is made of porous 2D-FG, the lower face is made of ceramic and the core is made of one-dimensional functionally graded (1D-FG) porous. The material properties
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are assumed to vary in both the thickness and longitudinal directions by the power-law functions. Based on a third-order shear deformation theory, a beam element was formulated and employed in computing the vibration characteristics. The obtained numerial result reveals that the variation of the material properties plays an important role in the frequencies. A parametric study has been carried out to highlight the effects of the material distribution and the layer thickness ratio on the frequencies of the sandwich beam. Moreover, the effect of porosity volume fraction vp is studied on the vibrational behavior of the sandwich beam. The influence of the aspect ratio on the vibration behaviour of the beam has also been examined and discussed.
References 1. Zenkour, A.M., Allam, M.N.M., Sobhy, M.: Bending analysis of FG viscoelastic sandwich beams with elastic cores resting on Pasternak’s elastic foundations. Acta Mech. 212(3–4), 233–252 (2010) 2. Vo, T.P., Thai, H.-T., Nguyen, T.-K., Maheri, A., Lee, J.: Finite element model for vibration and buckling of functionally graded sandwich beams based on a refined shear deformation theory. Eng. Struct. 64, 12–22 (2014) 3. Vo, T.P., Thai, H.-T., Nguyen, T.-K., Inam, F., Lee, J.: A quasi-3D theory for vibration and buckling of functionally graded sandwich beams. Compos. Struct. 119, 1–12 (2015) 4. Zhu, S., Jin, G., Wang, Y., Ye, X.: A general Fourier formulation for vibration analysis of functionally graded sandwich beams with arbitrary boundary condition and resting on elastic foundations. Acta Mech. 227(5), 1493–1514 (2016) 5. Simsek, M., Al-Shujairi, M.: Static. Free and forced vibration of functionally graded (FG) sandwich beams excited by two successive moving harmonic loads. Compos. B: Eng. 108(3), 18–34 (2017) 6. Vo-Duy, T., Ho-Huu, V., Nguyen-Thoi, T.: Free vibration analysis of laminated FG-CNT reinforced composite beams using finite element method. Frontiers Struct. Civil Eng. 13(2), 324–336 (2018) 7. Nemat-Alla, M., Noda, N.: Edge crack problem in a semi-infinite FGM plate with a bidirectional coefficient of thermal expansion under two-dimensional thermal loading. Acta Mech. 144(3–4), 211–229 (2000) 8. Lu, C.F., Chen, W.Q., Xu, R.Q., Lim, C.W.: Semi-analytical elasticity solutions for bi-directional functionally graded beams. Int. J. Solids Struct. 45(1), 258–275 (2008) 9. Simsek, M.: Bi-directional functionally graded materials (BDFGMs) for free and forced vibration of Timoshenko beams with various boundary conditions. Compos. Struct. 133, 968–978 (2015) 10. Wang, Z.-H., Wang, X.-H., Xu, G.-D., Cheng, S., Zeng, T.: Free vibration of twodirectional functionally graded beams. Compos. Struct. 135, 191–198 (2016) 11. Karamanlı, A.: Bending behaviour of two directional functionally graded sandwich beams by using a quasi-3D shear deformation theory. Compos. Struct. 174, 70–86 (2017) 12. Nguyen, D., Nguyen, Q., Tran, T., Bui, V.: Vibration of bi-dimensional functionally graded Timoshenko beams excited by a moving load. Acta Mech. 228(1), 141–155 (2016) 13. Tran, T.T., Nguyen, D.K.: Free vibration analysis of 2D-FGM beams in thermal environment based on a new third-order shear deformation theory. Viet. J. Mech. 40(2), 121–140 (2018) 14. Ha, L.T.: Free vibration of sandwich beams with bi-directional functionally graded core. Transport Commun. Sci. J. 68, 9–16 (2019)
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15. Ha, L.T., Tram, T.T.: Vibration analysis of sandwich beam with functionally graded (FG) skins subject to a moving load in high temperature environment. Transport Commun. Sci. J. 58, 34–40 (2017) 16. Bourbié, T., Coussy, O., Zinszner, B.: Acoustics of Porous Media (1987) 17. Sallica-Leva, E., Jardini, A., Fogagnolo, J.: Microstructure and mechanical behavior of porous Ti–6Al–4V parts obtained by selective laser melting. J. Mech. Behav. Biomed. Mater 26, 98–108 (2013) 18. Ebrahimi, F., Ghasemi, F., Salari, E.: Investigating thermal effects on vibration behavior of temperature-dependent compositionally raded Euler beams with porosities. Meccanica. https://doi.org/10.1007/s11012-0150208-y 19. Wattanasakulpong, N., Chaikittiratana, A.: Flexural vibration of imperfect functionally graded beams on Timoshenko beam theory: Chebyshev collocation method. Meccanica 50, 1331– 1342 (2015) 20. Ha, L.T., Khue, N.T.K.: Free vibration of functionally graded (FG) porous nano beams. Transport Commun. Sci. J. 70(2), 98–103 (2019). https://doi.org/10.25073/tcsj.70.2.32 21. Shafei, N., Mirjavadi, S.S., MohaselAfshari, B., Rabby, S., Kazemi, M.: Vibration of twodimensional imperfect functionally graded (2D-FG) porous nano-/micro-beams. Comput. Method Appl. Mech. Eng. 322, 615–632 (2017) 22. Simsek, M.: Buckling of Timoshenko beams composed of two-dimensional functionally graded material (2D-FGM) having different boundary conditions. Compos. Struct. 149, 304–314 (2016) 23. Shi, G., Lam, K.Y.: Finite element formulation vibration analysis of composite beams based on higher-order beam theory. J. Sound Vib. 219, 707–721 (1999) 24. Nemat-Alla, M.: Reduction of thermal stresses by developing two-dimensional functionally graded materials. Int. J. Solids Struct. 40, 7339–7356 (2003) 25. Reddy, J.N.: A simple higher-order theory for laminated composite plates. ASME-J. Appl. Mech. 51(4), 745–752 (1984) 26. Touratier, M.: An efficient standard plate theory. Int. J. Eng. Sci. 29(8), 901–916 (1991) 27. Soldatos, K.P.: A transverse shear deformation theory for homogeneous monoclinic plates. Acta Mech. 94(3–4), 195–220 (1992) 28. Karama, M., Afaq, K.S., Mistou, S.: Mechanical behaviour of laminated composite beam by the new multi-layered laminated composite structures model with transverse shear stress continuity. Int. J. Solids Struct. 40(6), 1525–1546 (2003) 29. Aydogdu, M.: A new shear deformation theory for laminated composite plates. Compos. Struct. 89, 94–101 (2009)
Free Vibration of Prestress Two-Dimensional Imperfect Functionally Graded Nano Beam Partially Resting on Elastic Foundation Le Thi Ha(B) University of Transport and Communications, No.3 Cau Giay Street, Lang Thuong Ward, Dong Da District, Hanoi, Vietnam [email protected]
Abstract. In this paper, the free vibration of prestress two-dimensional imperfect functionally graded (2D-FG) nano beam partially resting on a Winkler foundation is investigated by finite element method. The material properties of 2D-FG nano beam are assumed vary in both axial and thickness directions according to a power law. Based on Eringen nonlocal elasticity theory, the governing equations of motion are derived. A parametric study in carry out to show the effect of material distribution, nonlocal effect, prestress and elastic support on the natural frequencies of the beams. The finite element method is employed to establish the equations and compute the vibration characteristics of the beam. Keywords: Two-dimensional imperfect functionally graded · Nanobeams · Nonlocal model · Elastic foundation · Free vibration · Finite element method
1 Introduction Functionally graded materials (FGMs) have wide applications in modern industries including aerospace, mechanical, electronics, optics, chemical, biomedical, nuclear and civil engineering. Besides, nanotechnology is primarily concerned with fabrication of FGMs and engineering structures at a nanoscale. The structures such as nanoplate and nanobeam used in systems and devices such as nanowires, nano-probes, atomic force microscope (AFM), nanoactuators and nanosensors. Investigations on mechanical behavior of nanostructures in general and nanobeams in particular have received much attention from researchers recently. The nonlocal field theory, one of the talented continuum models in nanomechanics considering size-dependent effects, was first developed by Eringen [1–3]. Most classical continuum theories are based on hyperelastic constitutive relations which assume that the stress at a point is functions of strain at the point. On the other hand, the nonlocal continuum mechanics assumes that the stress at a point is a function of strains at all points in the continuum. By using this theory, the equilibrium differential and motion equations for nanostructures can be derived. Several investigations on vibration of nanobeams have been reported in the literature. Reddy [4] studied the bending, bending, buckling and © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 112–124, 2022. https://doi.org/10.1007/978-981-16-3239-6_9
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vibration of homogenous nanobeams. A compact generalized beam theory is used by Aydogdu [5] to analyze the bending, buckling and vibration of nanobeams. Simsek and Yurtcu [6] derived analytical solutions for bending and buckling of FG nanobeams. Based on the finite element method, Eltaher et al. [7, 8] studied free vibration of FG sizedependent nanobeams. On the basis of the nonlocal differential constitutive relations of Eringen, Zemri, Houari, Bousahla and Tounsi [9] proposed a nonlocal shear deformation beam theory to study the bending, buckling, and vibration of FG nanobeams. In addition to the functionally graded materials, porous materials are also a new type of material with emerging applications. Thus, these kinds of materials have been the focus of many mechanical studies [10, 11]. Besides, similar to the functionally graded materials, the dynamic and vibration of the structures that are made of porous materials are interesting for scientists. Ebrahimi et al. [12] studied thermo-mechanical vibration of functionally graded (FG) beams made of porous material subjected various thermal loadings are carried out by presenting a Navier type solution and employing a semi analytical differential transform method (DTM) for the first time. Wattanasakulpong [13] studied flexural vibration of imperfect functionally graded beams based on Timoshenko beam theory by using the Chebyshev collocation method. Based on Bernoulli beam theory, the free vibration of functionally graded (FG) porous nano beams is studied by Ha and Khue [14]. The finite element method is used to discretize the model and to compute the vibration characteristics of the beams. A parametric study in carry out to show the effects of the nonlocal parameter and porous parameter, material distribution on the natural frequencies of the beams are examined and discussed. Besides, similar to the functionally graded materials, the dynamic and vibration of the structures that are made of porous materials are interesting for scientists. Guastavino and Goransson [15] studied the vibration of anisotropic porous foam materials. Takahashi and Tanaka [16] investigated on a method of theoretical approach of acoustic coupling due to flexural vibration of porous plates made of elastic materials. In addition, two-dimensional functionally graded (2D-FG) porous nanobeam was studied by Shafiei [17]. Based on Timoshenko beam theory, Shafiei studied the vibration behavior of the two-dimensional functionally graded (2D-FG) nano and microbeams which are made of two kinds of porous materials for the first time. In this paper, the free vibration of prestress two-dimensional imperfect functionally graded (2D-FG) nanobeam partially resting on a Winkler is studied. Euler–Bernoulli beam theory incorporated with nonlocal differential equation of Eringen is used to derive the nonlocal differential equations of motion and the finite element method is employed to compute the frequencies of the beam. The effect of nonlocal parameter, a porosity volume fraction, FG power indexes and the foundation support on the vibration characteristics is examined and discussed.
2 Problem and Formulation 2.1 Grade Functionally Beam Figure 1 shows a simply supported beam with length L subjected to axial compression force Pe , width b and height h, resting on a Winkler elastic foundation with ratio stiffness k W and length L e . The 2D-FG nanobeams is composed of metal and ceramic with varying
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z Em, Ec, ρm , ρc, L, h Pe
porous 2D- FG
Pe
h x
kw, Le Fig. 1. 2D-FG nanobeam partially resting on a Winkler elastic foundation.
material volume fraction along x and z directions. So the mechanical properties of the nanobeams such as Young’s modulus ‘E’, mass density ‘ρ’ vary along axial (x-axis) and the thickness directions (z-axis). The porous 2D-FG nanobeam is considered to have even porosity distributions across the beam thickness. Consider the imperfect 2D-FG beam with a porosity volume fraction, α (α 1 ), distributed equally in the ceramic and metal phases, the modified rule of mixture is proposed as [17]: α α + Pc Vc − (1) P(x, z) = Pm Vm − 2 2 the ()m and ()c subscripts respectively denote the ceramic and metal, now, the total volume fraction of the metal and ceramic is 1 and the power law of volume fraction of the ceramic is given as [17] Vm + Vc = 1 nz x nx Vc (x, z) = 21 + hz L
(2)
where z is measured from the mid-plane; Vc and Vm are the volume fractions of ceramic and metal, respectively; ‘nx’ and ‘nz’ are the FG power indexes (along the length and thickness), respectively, are ascribed to the volume fraction change of the material composition. The material properties can be evaluated by a simple rule [17] z nz x nx α 1 + P(x, z) = Pm + (Pc − Pm ) − (Pc + Pm ) (3) 2 h L 2 When the beam is perfect (α = 0), the material of the beam is 100% ceramic when nx and nz are set to be zero. Thus, Young’s modulus ‘E’, mass density ‘ρ’ equations of the imperfect nanobeam can be expressed as: nz x nx α − (E + Em ) E(x, z) = Em + (Ec − Em ) 21 + hz 1 z nz xLnx α2 c (4) ρ(x, z) = ρm + (ρc − ρm ) 2 + h − L 2 (ρc + ρm ) 2.2 Government Equations The axial displacement u and transverse displacement w at any point based on Euller Bernoulli beam theory are given by 0 u(x, z, t) = u0 (x, t) − z ∂w ∂x , w(x, z, t) = w0 (x, t)
(5)
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where u0 and w0 are the axial and transverse displacements at the mid plane, and t is the time. By using the small deformation, the axial strain reads εxx =
∂u0 ∂ 2 w0 ∂u 0 = − z 2 = εxx − zκ 0 ∂x ∂x ∂x
(6)
0 = ∂uo , κ 0 = ∂ w0 are the extensional strain and the bending strain, where εxx ∂x ∂x2 respectively. Hamilton’s principle has the form 2
t2 (δU + δV − δT )dt = 0
(7)
t1
where δU , δT , δV are the virtual strain and kinetic energies, and virtual potential. L 0 N δεxx δU = − M δκ 0 dx,
(8)
0
in which, N b
h/2 −h/2
=
b
h/2 −h/2
σxx (x, z) dz is the axial normal force and M
=
zσxx (x, z) dz is the bending moment. L ∂w0 ∂δw0 δV = −b Pe + kW w0 δw0 dx ∂x ∂x L
δT =
I11
0
0
∂u0 ∂δu0 ∂w0 ∂δw0 + ∂t ∂t ∂t ∂t
− I12
∂δu0 ∂ 2 w0 ∂u0 ∂ 2 δw0 + ∂t ∂x∂t ∂t ∂x∂t
(9)
+ I22
∂ 2 w0 ∂ 2 δw0 dx ∂x∂t ∂x∂t
(10) where I11 , I12 , I22 are the mass moments, defined as h/2 (I11 , I12 , I22 ) = b
ρ(x, z) 1, z, z 2 dz
(11)
−h/2
Substituting Eqs. (8), (9) and (10) into Eq. (7), we obtained the following equations of motion ∂N ∂ 2 u0 ∂ 3 w0 = I11 2 − I12 ∂x ∂t ∂x∂t 2
(12)
∂ 2 w0 ∂ 3 u0 ∂ 4 w0 ∂ 2 w0 ∂ 2M = I + I − I + k w − P 11 12 22 W 0 e ∂x2 ∂t 2 ∂x∂t 2 ∂x2 ∂t 2 ∂x2
(13)
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2.3 Nonlocal Continuum Beam According to Eringen [1, 2], the nonlocal stress tensor components σij at point x are expressed as (14) σ = K x − x, τ t(x ) d x V
where t(x) are the components x at point ofthe classical macroscopic stress tensor and the kernel function K x − x, τ represents the nonlocal modulus, x − x is the distance (in Euclidean norm) and τ is a material constant that depends on internal and external characteristic lengths (such as the lattice spacing and wavelength, respectively). The macroscopic stress t at a point x in the Hookean solid is related to the strain at the point by the generalized Hooke’s law t(x) = C(x) : ε(x) tij = Cijmn εmn
(15)
where C is the fourth-order elasticity. The integral constitutive relation in Eq. (14), however makes the elasticity problems difficult to solve, in addition to possible lack of determinism. It is possible [2] to represent the integral constitutive relations in an equivalent differential form as 1 − τ 2 l 2 ∇ 2 σij = Cijmn εmn (16) τ = e0l a where e0 is a material constant, a, l are the internal and external characteristic lengths, respectively. Equation (16) can be expressed as L(σij ) = Cijmn εmn L = 1 − μ∇ 2 μ = e02 a2
(17)
Nonlocal stress resultants for beams is determined by using (17). The nonlocal theory results in differential relations involving the stress resultants and the strains, and for 2D-FG beam, stress resultants can be written by. L(σxx ) = E(x, z)εxx , where L = 1 − μ
∂2 ∂x2
(18)
and E(x, z) is elastic modulus. If μ = 0, the relations of the local theories are resumed. For Euler - Bernoulli theory beam, Eq. (18) can be written as σxx − μ
∂ 2 σxx = E(x, z)εxx ∂x2
(19)
From Eq. (19), we have the constitutive relations N −μ
∂ 2N 0 = A11 εxx − A12 κ 0 ∂x2
(20)
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M −μ
∂ 2M 0 = A12 εxx − A22 κ 0 ∂x2
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(21)
where A11 , A12 and A22 are respectively the axial, axial-bending coupling and bending rigidities, and they are defined as (22) (A11 , A12 , A22 ) = E(x, z) 1, z, z 2 dA A
Substituting Eq. (12) into Eq. (20) and Eq. (13) into Eq. (21) we obtain the expressions for axial force and moment of the beam ∂u0 ∂ 2 w0 ∂ 3 u0 ∂ 4 w0 (23) − A12 2 + μ I11 − I12 2 2 , N = A11 ∂x ∂x ∂x∂t 2 ∂x ∂t ∂u0 ∂ 2 w0 ∂ 2 w0 ∂ 3 u0 ∂ 4 w0 ∂ 2 w0 M = A12 + μ I11 2 + I12 −I22 2 2 + kW w0 − Pe . − A22 ∂x ∂x2 ∂t ∂x∂t 2 ∂x ∂t ∂x2
(24) Finally, we can obtain the differential equations of motion for the beam in the forms ∂ 2 u0 ∂ 3 w0 ∂ 4 u0 ∂ 5 w0 ∂ 2 u0 ∂ 3 w0 A11 2 − A12 3 + μ I11 2 2 − I12 3 2 = I11 2 − I12 (25) ∂x ∂x ∂x ∂t ∂x ∂t ∂t ∂x∂t 2 3 4 4 5 6 4 2 A12 ∂∂xu30 − A22 ∂∂xw40 + μ I11 ∂x∂ 2w∂t02 + I12 ∂x∂ 3u∂t02 − I22 ∂x∂ 4w∂t02 −Pe ∂∂xw40 + kW ∂∂xw20 ∂ u0 ∂ w0 ∂ w0 = I11 ∂∂tw20 + I12 ∂x∂t 2 − I22 ∂x 2 ∂t 2 + kW w0 − Pe ∂x 2 2
3
4
2
(26)
2.4 Numerical Formulation This section is devoted to drive the finite element model for nonlocal Euler–Bernoulli beam. Based on the Hamilton principle, by substituting Eq. (23, 24) into Eq. (8) and the results produced is substituted into Eq. (7), the following variational statement of nonlocal Euler–Bernoulli beam can be deduced: t2 L ∂ 2 w0 ∂ 2 δw0 ∂ 2 w0 ∂δu0 ∂u0 ∂ 2 δw0 0 ∂δu0 A11 ∂u ∂x ∂x + A22 ∂x2 ∂x2 − A12 ∂x2 ∂x − A12 ∂x ∂x2 t1 0 2 2 ∂ 2 δw0 ∂ 2 w0 ∂ 2 δw0 0 ∂δw0 − I11 μ ∂∂tw20 ∂ ∂xδw2 0 + k1 w0 δw0 + Pe ∂w − μk w + μP w 0 e 2 2 2 ∂x ∂x ∂x ∂x ∂x ∂ 3 u0 ∂δu0 ∂ 4 w0 ∂δu0 ∂ 3 u0 ∂ 2 δw0 ∂ 4 w0 ∂ 2 δw0 − μI12 ∂x∂t + μ I11 ∂x∂t 2 ∂x − I12 ∂x 2 ∂t 2 ∂x 2 ∂x 2 + μI22 ∂x 2 ∂t 2 ∂x 2 2 ∂w0 ∂δw0 ∂δu0 ∂ 2 w0 ∂ 2 w0 ∂ 2 δw0 0 ∂δu0 0 ∂ δw0 + I12 ∂u − I11 ∂u ∂t ∂t + ∂t ∂t ∂t ∂x∂t + I12 ∂t ∂x∂t −I22 ∂x∂t ∂x∂t dxdt = 0 (27) The finite element method is employed herein to solve the equations of motion. To this end, the beam is assumed being divided into a number of two-node beam elements
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with length of l. The vector of nodal displacements (d) for the element considering the transverse shear rotation γ0 as an independent variable contains six components as d = {u1 , w1 , θ1 , u2 , w2 , θ2 }
(28)
where u1 , w1 , θ1 , u2 , w2 , θ2 are the values of u0 , w0 and θ0 at the node 1 and at the node 2. In Eq. (28) and hereafter, a superscript ‘T ’ is used to denote the transpose of a vector or a matrix. u0 = NuT d, w0 = NwT d.
(29)
where Nu, Nw denote the matrices of shape functions for u0 , w0, respectively. In the present work, using linear and cubic Hermite polynomials to interpolate the axial and transverse displacements Nu = Nu1 0 0 Nu2 0 0 (30) Nw = 0 Nw1 Nw2 0 Nw3 Nw4 with Nu1 = l−x l , x Nu2 = l , 2 Nw1 = 1 − 3xl 2 +
2x2 l + 3 3x2 − 2xl 3 , l2 −x2 x3 l + l2 .
Nw2 = x −
Nw3 = Nw4 =
2x3 , l3 x3 , l2
(31)
Putting Eq. (29) into the variational statement form Eq. (27), performing integration, the following element equation is obtained: ¨ + KD = 0 (M + Mnl )D
(32)
where D is the total vector displacement, M and K are the structural mass and stiffness matrices assembled from the element mass and stiffness matrices over the total elements, respectively; and Mnl is the nolocal mass over the total element. When the nonlocal parameter μ = 0, Eq. (32) returns to the free vibration equation for the conventional beams. In addition, Eq. (32) leads to an eigenvalue problem for determining the frequency ω as ¯ =0 K − ω2 (M + Mnl ) D (33) ¯ is the vibration amplitude. Equation (33) leads with ω is the circular frequency and D to an eigenvalue problem, and its solution can be obtained.
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3 Numerical Result The 2D-FG nanobeams is composed of steel (SUS304) and alumina (Al2 O3 ) where its properties vary along axial (x-axis) and the thickness directions (z-axis) with (SUS304, Em = 210 Gpa, ρm = 7800 kg/m3 , alumina (Al2 O3 ), Ec = 390 Gpa, ρc = 3960 kg/m3 ). The bottom surface of the beam is pure steel, whereas the top surface of the beam is pure alumina. The beam geometry has the following dimensions: L (length) = 10, b (width) = 1 and h (thickness). The frequency is normalized (λi ) as [7] 2 ρc A (34) λi = ωi L Ec I where ωi is the i th natural frequency of the beam and A = b.h, 3 I = bh 12 .
(35)
In (35), I is the moment of inertia of the cross section of the beam, A is the cross section area of the beam. The foundation support parameter Le (0 ≤ Le ≤ 1) is defined as a ratio of the supported length to the total beam length, and the foundation stiffness parameter k 1 is introduced as. k1 = kW
L4 Em I
(36)
and axial compression force Pe [18] Pe =
−π 2 Em I L2
(37)
Table 1. Comparison of fundamental frequency paraneter for simply supported 1D-FG nanobeams (nx = 0, k 1 = 0, Pe = 0, α = 0). μ n=0 Present [7]
n=1
n=5
Present [7]
Present [7]
1 9.4062 9.4238 6.6669 6.7631 5.6639 5.7256 2 9.0102 9.0257 6.3863 6.4774 5.4255 5.4837 3 8.6604 8.6741 6.1384 6.2251 5.2148 5.2702 4 8.3483 8.3607 5.9172 6.0001 5.027
5.0797
5 8.0678 8.0789 5.7184 5.7979 4.858
4.9086
The validation of the derived formulation is necessary to confirm before computing the vibration characteristics of the beam. In Table 1, the fundamental frequencies
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Table 2. Comparison of fundamental frequency parameter for porous 2D-FG nanobeams [17] (α = 0.1; μ = 0.15, k 1 = 0, Pe = 0). nz
Source
L/h = 12 nx = 0
nx = 1
nx = 5
nx = 0
nx = 1
nx = 5
1
Present
13.5034
11.5727
10.3789
13.7486
11.6084
10.4099
[17]
13.5487
11.5981
10.4459
13.8728
11.8787
10.6968
Present
11.4050
10.6151
10.0909
11.4385
10.6460
10.1205
[17]
11.3922
10.7048
10.2581
11.6787
10.9691
10.5079
Present
10.8555
10.3432
10.0001
10.8865
10.3731
10.0294
[17]
10.8931
10.4660
10.1917
11.1672
10.7241
10.4399
5 10
L/h = 18
parameter of a simply supported nanobeam with an aspect ratio L/h = 20 and nx = 0, k 1 = 0, Pe = 0, α = 0 obtained in the present paper are compared with the results by Eltaher et al. [7]. A good agreement can be noted from Table 1, irrespective of the nolocal parameter. The frequency factor in the Table 2 is obtained by setting k1 and Pe to zero, α = 0.1, μ = 0.15 and using the geometric and material data of the corresponding references [17]. A very good agreement between the results of the present work with those of the references [17] is noted from Table 2.
Fig. 2. Non-dimensional natural frequency parameter of imperfect 2D-FGM nanobeams when L/h = 20, α = 0.3, k1 = 50, Le = L/3.
Figure 2 shows the non-dimensional frequency of simply supported FG, porous nanobeam for different values of nonlocal parameter (μ) versus the FG power indexes when L/h = 20, α = 0.3, k 1 = 50, L e = L/3, respectively. It is observed that increasing nx
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and nz decrease the frequency parameter of the nanobeam which is because increasing the nz and nx power indexes decrease the stiffness of the nano beam. It is seen that the non-dimensional frequency of FG nanobeam, decreases with the increment of the nonlocal value. In addition, the decrease of the fundamental frequency parameter is more significant for nx < 1 and nz < 1. The influence of the foundation stiffness on the fundamental frequency parameter of the nanobeam can be seen from Fig. 3, where the grading index nx and nz versus the fundamental frequencies is depicted for various values of the foundation stiffness parameter k 1 when L/h = 20, α = 0.3, μ = 0.3, L e = L/2. Regardless of the index nx and nz, the fundamental frequency parameter is increased with the increase of the foundation stiffness k 1 which is because increasing the foundation stiffness k1 increase the stiffness of the nanobeam.
Fig. 3. Grading index nx and nz versus on fundamental frequency parameter with different foundation stiffness parameter k1 for imperfect 2D-FG nano beam when L/h = 20, α = 0.3, μ = 0.3, Le = L/2.
Figure 4 shows the non-dimensional frequency of simply supported nano beam made of porous materials versus the nz and nx power indexes. Figure 4 shows that increasing the nz power indexes decrease the frequency which is because increasing the nz decrease the stiffness of the nano beam. In addition, it is seen in Fig. 4 that the dependency of the frequency on the nz power indexes increase with the porosity volume fraction (α), and this is more intense for porous, as it is seen that the decrement of the frequency due to the FG power indexes is so high that when nz 1, there is an intersection between the frequencies of different porosity volume fractions. In fact, the effect of the porosity volume fraction on the frequency of the porous beams highly depends on nz as in higher values of nz, the frequency decreases with the increment of α but when nz are low, the increment of α increases the frequency.
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Fig. 4. Non-dimensional natural frequency of imperfect 2D-FGM nanobeams when L/h = 20, μ = 0.5, Pe = 0, Le = L/2, nx = 0.5.
Fig. 5. Effect of foundation parameter Le on fundamental frequency parameter of imperfect 2DFGM nano beam when L/h = 20, μ = 0.5, α = 0.3, k1 = 50, nx = 0.5
The effect of the foundation support on the fundamental frequency parameter can be seen from Fig. 5, where the frequencies of the nano beam are given for various values of the foundation parameter Le . The frequencies are clearly seen to be increased by raising the foundation parameter Le , regardless of the axial compression forces (Pe = 2 0 or Pe = −πL2Em I ) which is because increasing the foundation parameter Le increase the stiffness of the nano beam. Besides, the frequency parameter is increased with the increase of the axial compression force Pe . Similar to Fig. 3, it is seen that increasing nz decreases the non-dimensional frequency of the nano beam.
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The grading index nz versus the fundamental frequency parameters of the nanobeams is illustrated in Fig. 6 for L/h = 20, α = 0.3, L e = L/2, nx = 0.5, respectively. As seen from the figures, for a give value of the nonlocal parameter, the fundamental frequencies decreases when the nonlocal parameter μ increase, regardless of the axial compression force and the foundation stiffness parameter. It is observed that as the value of Pe increases and k1 decreases, the fundamental frequencies increase, regardless of the nonlocal parameter.
Fig. 6. Grading index nz versus fundamental frequency parameter with different parameter μ of porous 2D-FG nano beam when L/h = 20, α = 0.3, Le = L/2, nx = 0.5.
4 Conclusions The vibration characteristics of porous 2D-FG Euler-Bernoulli nanobeam partially resting on a Winkler elastic foundation are evaluated. The nonlocal Eringen model considering the scale effect is taken into account in the derivation of the equations of motion. The finite element method is employed to discretize the model and to compute the fundamental frequency parameters. A parametric study has been carried out to highlight the effect of nonlocal parameter, foundational support, axial compression force and the foundation stiffness on fundamental frequency parameters of the nano beam. The obtained numerical results show that, the nonlocal parameter plays an important role in the frequencies of nanobeam, and the frequencies increase with increasing the nonlocal parameter. Besides, the fundamental frequency parameters increase with increasing the axial compression force and the foundation stiffness.
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References 1. Eringen, A.C., Edelen, D.: On nonlocal elasticity. Int. J. Eng. Sci. 10(3), 233–248 (1972) 2. Eringen, A.C.: Nonlocal Continuum Field Theories. Springer-Verlag, New York (2002) 3. Eringen, A.C.: On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J. Appl. Phys. 54, 4703–4710 (1983) 4. Reddy, J.N.: Nonlocal theories for bending, buckling and vibration of beams. Int. J. Eng. Sci. 45, 288–307 (2007) 5. Aydogdu, M.: A general nonlocal beam theory: Its application to nanobeam bending, buckling and vibration. Phys. E 41, 1651–1655 (2009) 6. Simsek, M., Yurtcu, H.H.: Analytical solutions for bending and buckling of functionally graded nanobeams based on the nonlocal Timoshenko beam theory. Compos. Struct. 97, 378–386 (2013) 7. Eltaher, M.A., Emam, S.A., Mahmoud, F.F.: Free vibration analysis of functionally graded size-dependent nanobeams. Appl. Math. Comput. 218, 7406–7420 (2012) 8. Eltaher, M.A., Alshorbagy, A.E., Mahmoud, F.F.: Vibration analysis of Euler-Bernoulli nanobeams by using finite element method. Appl. Math. Model. 37(7), 4787–4797 (2013) 9. Zemri, A., Houari, M.S.A., Bousahla, A.A., Tounsi, A.: A mechanical response of functionally graded nanoscale beam: an assessment of a refined nonlocal shear deformation theory beam theory. Struct. Eng. Mech. 54, 693–710 (2015) 10. Bourbié, T., Coussy, O., Zinszner, B.: Acoustics of Porous Media (1987) 11. Sallica-Leva, E., Jardini, A., Fogagnolo, J.: Microstructure and mechanical behavior of porous Ti–6Al–4V parts obtained by selective laser melting. J. Mech. Behav. Biomed. Mater 26, 98–108 (2013) 12. Ebrahimi, F., Ghasemi, F. Salari, E.: Investigating thermal effects on vibration behavior of temperature-dependent compositionally raded Euler beams with porosities. Meccanica., https://doi.org/10.1007/s11012-015-0208-y 13. Wattanasakulpong, N., Chaikittiratana, A.: Flexural vibration of imperfect functionally graded beams on Timoshenko beam theory: Chebyshev collocation method. Meccanica 50, 1331– 1342 (2015) 14. Ha, L.T., Khue, N.T.K.: Free vibration of functionally graded (FG) porous nano beams. Transport and Communications Science Journal 70(2), 98–103 (2019). https://doi.org/10. 25073/tcsj.70.2.32 15. Guastavino, R., Göransson, P.: Vibration dynamics modeling of anisotropic porous foam materials. In: Proceedings of Forum Acusticum, Budapest Hungary, pp. 123–128 (2005) 16. Takahashi, D., Tanaka, M.: Flexural vibration of perforated plates and porous elastic materials under acoustic loading. J. Acoust. Soc. Am. 112, 1456–1464 (2002) 17. Shafei, N., Mirjavadi, S.S., MohaselAfshari, B., Rabby, S., Kazemi, M.: Vibration of twodimensional imperfect functionally graded (2D-FG) porous nano-/micro-beams. Comput. Method Appl. Mech. Eng. 322, 615–632 (2017) 18. Gere, J.M., Timoshenko, S.P.: Mechanics of materials. Chapman & Hall, Third SI Edition (1989)
Design and Hysteresis Modeling of a New Damper Featuring Shape Memory Alloy Actuator and Wedge Mechanism Duy Q. Bui1,2 , Hung Q. Nguyen3(B) , Vuong L. Hoang2 , and Dai D. Mai4 1 Faculty of Civil Engineering, HCMC University of Technology and Education,
Ho Chi Minh City, Vietnam [email protected] 2 Faculty of Mechanical Engineering, Industrial University of Ho Chi Minh City, Ho Chi Minh City, Vietnam {buiquocduy,hoanglongvuong}@iuh.edu.vn 3 Faculty of Engineering, Vietnamese-German University, Thu Dâu ` Mô.t, Binh Duong, Vietnam [email protected] 4 Faculty of Mechanical Engineering, HCMC University of Technology and Education, Ho Chi Minh City, Vietnam [email protected] ij
Abstract. This study aims at design and hysteresis modeling of a novel damper featuring shape memory alloy (SMA) to mitigate structural vibrations. The damper consists of SMA springs for actuation and a wedge mechanism to amplify and convert the actuating force into friction against the inner cylindrical face of the damper housing. From the friction between the wedges and housing, the damping force is archived. Following an introduction of SMA spring actuators and SMA dampers, the proposed SMA damper is configured. Experiments are then conducted on SMA springs to obtain their performance characteristics such as transformation temperature, heating time and actuating force. Based on the experimental data, design of the proposed SMA damper is performed and a damper prototype is fabricated. Experimental tests are then conducted to evaluate the damper performance. From the experimental results, hysteresis phenomenon of the SMA damper is presented and investigated. To predict the damper behavior, several hysteresis models are adopted and validated with comparisons and discussions. Keywords: SMA actuator · SMA spring · SMA damper · Hysteresis model · Structural vibrations
1 Introduction Material science has attained considerable improvements over the past decades. From the demand for light weight, powerfulness and controllability, smart materials have entered society. Smart materials additionally supply functions such as actuating and sensing to address strict engineering problems by convert reciprocally between mechanical (e.g., © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 125–136, 2022. https://doi.org/10.1007/978-981-16-3239-6_10
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force, displacement) and non-mechanical (e.g., voltage, temperature) responses [1]. Due to these unique characteristics, smart materials have received numerous interest in the field of structural vibration control. It is noticeable that conventional suspension systems can help mitigate most engineering oscillations at low resonant frequency. However, they possess unchangeable damping level causing the exciting force to be transmitted greater to contiguous parts and floor at high frequencies, which may lead to noises, uncomfortableness and step–by– step failure of the devices using them. Therefore, semi–active suspension systems that can reasonably adjust damping coefficient corresponding to frequency excitations are desired. A type of smart material that has been widely explored in semi-active suspension systems is magneto-rheological fluid (MRF). MRF contains microscopic magnetic particles suspended within carrier oil. A change in the magnetic field applied to the MRF associates with a change in the fluid viscosity, which leads to damping ability. There are many study works of MRF dampers to solve engineering vibration problems [2–6]. As compared with conventional passive dampers, MRF dampers have remarkable advantages of high damping force, easy control and high reliability. However, the off–state force of the dampers is to some extent high resulting in unwanted formidable oscillations at high frequency excitations. In addition, sealing up the MRF during operating process is a challenging issue as it gives rise to structural complexity of the dampers. High cost of MRF is also a major impediment for the broad application of such dampers in practice. Another smart material that has been recently investigated to develop damper is shape memory alloy (SMA). SMA has two phases possessing different crystal structures and properties called martensite M (lower temperature) and austenite A (higher temperature). When subjected to particular excitations, SMA experiences a reversible transformation between the two phases to dissipate or absorb energy, which respectively provides applicability for actuating or sensing. In actuating function, SMA can return to its memorized original shape as heated over Af point (austenite finish temperature), generating a useful force. Inspired by this unique characteristic, several scholars have designed SMA dampers for seismic isolation of engineering structures such as Graesser and Cozzareli [7], Clark et al. [8], Wilde et al. [9], Han et al. [10] and Zuo et al. [11]. The satisfactory research results have shown the promise and potential of such SMA damper kind. Aiming at simplicity and compactness for installability in small spaces (e.g., suspension systems of washing machine, vehicle), this paper deals with a new damper featuring SMA spring actuator that can produce a high damping force to suppress structural oscillations at low resonant frequency while keeping off-state force small to lessen the force transmission at high frequencies. The characteristics of the SMA springs are first determined through experimental works to serve as an input for calculation, then a damper prototype is designed, fabricated and evaluated. Hysteresis behavior of the proposed damper is analyzed and simulated by some idealized models with comparisons and discussions.
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2 Configuration of the SMA Damper Figure 1 shows configuration of the proposed SMA damper. As shown in the figure, a sleeve shaft slides along a cylindrical housing. Inside the sleeve shaft, a wedge mechanism including an actuating head and four wedges is used to generate friction force against the housing.
Fig. 1. Schematic configuration of the SMA damper.
When the SMA springs are heated, they extend and push the actuating head to the right. This makes the four wedges move outward in contact with the inner face of the housing. A damping force is produced from the friction between the housing and wedges. Noteworthily, the SMA spring actuator has an inherent disadvantage of low actuating force. In this study, by using a wedge mechanism actuated by an appropriate number of SMA springs, the actuating force from the wedges to the inner face of the housing can be magnified to yield a required damping force. When the SMA springs are cool down, they are contracted to their original shapes and a return spring is employed to push the actuating head back to initial position. The four wedges then move inward and the friction between the wedges and the cylindrical housing is significantly reduced. The damping force and off-state force (the force when the SMA is at ambient temperature of 25 °C) of the SMA damper can be adjusted by using the adjustor.
3 Modeling of the SMA Damper 3.1 Characterization of the SMA Springs Figure 2 shows the experimental setup to specify characteristics of SMA springs. Two heat insulating pads are used to prevent direct contact between the tested SMA spring and adjacent parts. A DC power supply is employed to provide the spring with a high enough current so that the austenite phase transformation occurs to the finish. Data of force and temperature are transfered from a force sensor and a thermo sensor to a computer through an A/D converter.
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Fig. 2. Experimental setup to specify characteristics of SMA springs.
In this work, three specimens of the SMA springs made by SAES® Getters Group (SmartFlex ® SMA spring) are considered for testing. The geometric dimensions of the springs are given as follows: – Spring 1: mean diameter 9 mm, wire diameter 0.8 mm, length 15 mm – Spring 2: mean diameter 6 mm, wire diameter 1.2 mm, length 20 mm – Spring 3: mean diameter 10 mm, wire diameter 2 mm, length 25 mm
(a) spring 1
(b) spring 2
(c) spring 3
Fig. 3. Experimental responses of the three SMA spring specimens.
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The experimental force–temperature–time responses of the three SMA springs are presented in Figs. 3(a–c). As shown in the figures, the actuating force increases with the induced temperature, which is time transient response. At steady state, the maximum actuating forces of the springs 1–3 are respectively about 8.3, 13.8 and 28.1 N. The force saturation of the three springs are almost reached at 20, 26 and 53 s (response time), indicating that the Af points are approximately 50, 60 and 80 °C, respectively. It is noted that the low response of the SMA spring actuators come from both the time for phase transformation of the SMA material and the time transient response of the excitation temperature. From the results, it can be realized that the spring 1 exhibits the fastest response ability but lowest actuating force while the spring 3 is contrary. In regard to damper size, the spring 2 is the best choice. Considering these data, the design process of the SMA damper is next implemented. 3.2 Design of the SMA Damper In this section, the SMA damper is designed based on the equilibrium of forces acting on the damper components when the SMA springs are active, as shown in Fig. 4.
Fig. 4. Equilibrium of forces acting on the actuating head (left) and wedges (right).
As the SMA springs are heated, the actuating force F SMA is generated making the atuating head slide to the right. The sleeve shaft is lubricated on the inner cylindrical face to minimize the friction against the actuating head. By neglecting this friction force, the axial equilibrium equation of the actuating head is given as following FSMA − Fsp − FW sin α = 0
(1)
where F W is the total force caused by the four wedges and α is the cone angle of each wedge. The restoring force of the return spring F sp is obtained by Fsp = ksp = k
w tgα
(2)
where k is the return spring stiffness, Δsp and Δw are respectively the displacements of the return spring and wedges. As the four wedges are pushed outward, the reaction force F N at the contact between the wedges and housing is generated. By neglecting the very small friction between the four wedges and sleeve shaft, the radial equilibrium equation of the four wedges is then obtained by FN − FW cos α = 0
(3)
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The friction force between the four wedges and housing is determined by Ff = μFN
(4)
in which μ is the friction coefficient. This friction force is also the on-state damping force F d of the damper which can be derived by combining the Eqs. (1–3) as follows w Fd = μ FSMA − k ctgα (5) tgα In this study, the material of the housing and four wedges is commercial C45 steel, the cone angle of each wedge α is 10°, the return spring stiffness k is 5 N/mm and the initial thickness of the gap between the wedges and housing is 0.2 mm. With respect to devices owning small assembly spaces, the desired damping force is about 80 to 100 N. From the view of balance between the actuating force, state changing time and damper size, the SMA spring 2 is chosen for our design. To achieve the desired damping force, two SMA springs should be used, which is able to produce a damping force up to 80.8 N.
4 Experimental Test To verify the proposed SMA damper, a damper prototype is fabricated and its performance is experimentally evaluated. The SMA damper prototype and its components are shown in Fig. 5.
Fig. 5. SMA damper prototype and its decomposed components.
Figure 6 shows the test rig to assess the SMA damper performance. In the system, the linear motion of the shaft is created from the turning motion of a motor through a crank–slider mechanism. The servo motor is controlled by a computer and provides the crank shaft with different constant angular velocities. Once the experiment process is stated, a current of 4 A is applied to the SMA springs. A linear variable differential transformer (LVDT) is employed to record the displacement and a force sensor is used to measure the damping force. From the sensors, the output signals are then sent to the computer via a data acquisition (DAQ).
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Fig. 6. Test rig to assess the SMA damper performance.
Figure 7(a) represents the responses of the SMA damper prototype in damping force-time relation when the motor rotates constantly at an angular velocity of 4π rad/s (2 Hz). This is also the frequency that the resonance usually occurs. From the figure, it is observed that when the SMA damper is not heated (no current is applied to the SMA springs), the off-state force is about 8 N. This is mainly the friction force between the housing and the shoulders of the sleeve shaft. When the SMA springs are electrically powered, the average damping force is 76.5 N at the steady state, which is about 95% of the theoretical one (80.8 N). Thus, a good correlation is achieved. Moreover, the figure indicates that the rising time to change from the off-state force to the steady value of the damping force is around 25 s, which is consistent to the experimentally measured data of the spring 2 (26 s).
(a) force – time
(b) force – displacement
Fig. 7. Experimental step responses of the SMA damper under 2 Hz frequency.
To clearly describe the damping characteristics of the SMA damper at steady state (at which the temperature is around 50 °C), the relation between the damping force and the displacement of the damper shaft over two cycles under the same frequency excitation is obtained, as presented in Fig. 7(b). The hysteresis of the SMA damper, particularly at
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the traveling stroke ends, is clearly shown in the figure. It is noted that there is a sudden change of damping force when the shaft motion changes its direction. This is obvious because the damping force of the proposed SMA damper is generated by the friction between the cylindrical housing and the wedges which is almost Coulomb friction.
(a) 3 Hz
(b) 5 Hz
Fig. 8. Experimental step responses of the SMA damper under higher frequencies.
The same remarks can be derived from Figs. 8(a–b) as testing the damper under higher frequency excitations of 3 and 5 Hz. Minor differences in the off–state force and maximum damping force are admitted as they tend to raise a little with the motor spindle speed, which can be basically explained by the inertial effect of the damper shaft.
5 Hysteresis Model of the SMA Damper In this research, three hysteresis models including Bingham model [12], Bouc–Wen model [13, 14] and Bui’s model [15] are adopted to predict the hysteresis phenomenon of the SMA damper. The three models are in turn mathematically expressed as follows. Bingham: Fd = c˙x + fc sgn(˙x) + f0
(6)
Fd = c˙x + kx + αz
(7a)
z˙ = −γ |˙x|z|z|(n−1) − β x˙ |z|n + A˙x
(7b)
Bouc–Wen:
Bui’s model:
Fd = c˙x + kx + f0 + D sin C arctan B(1 − E)z + E arctan(Bz) + H arctan7 (Bz) (8a)
Design and Hysteresis Modeling
z=
x˙ + Sa x for x¨ ≥ 0 x˙ + Sb x for x¨ < 0
133
(8b)
where x is the displacement, x˙ is the velocity, c and k are respectively the damping and stiffness coefficients, f 0 is the constant bias force, z is an independent variable, and f c , α, β, γ , n, A, S a , S b , B, C, D, E, H are the factors characterizing the hysteresis curve shape. Based on the experimental data, the model parameters are estimated using the curve fitting combined with least square method. The objective function OBJ is to minimize the sum of squared errors between the experimental and predicted damping forces, which is represented by OBJ = min
p
Fm.i − Fexp.i
2
(9)
i=1
where F m.i , F exp.i are respectively the ith predicted and experimental forces, and p denotes the number of calculation points. The estimated parameters of the three models for the 2 Hz frequency excitation are listed in Table 1. Table 1. Estimated parameters of the three models for the 2 Hz frequency excitation. Model
Parameter
Bingham model
c = 142.1 N.s/m, f c = 41.5 N, f 0 = 0 N
Bouc–Wen model
c = 67.6 N.s/m, k = 277.6 N/m, α = 1041.5 N/m, β = 0 m–2 , γ = 278.7 m–2 , n = 2, A = 0.88
Bui’s model
c = 74.7 N.s/m, k = 308.2 N/m, S a = 4.63 s−1 , S b = 6.64 s−1 , B = 13.7 s/m, C = 0.93, D = 57.1 N, E = –0.6, H = 2.11
Fig. 9. Comparisons between the predicted and experimentally measured responses under a frequency of 2 Hz.
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Using the above optimized parameters, the damper responses predicted by the three models are obtained and compared with the experimental data. Figure 9 show the comparisons between the experimental and predicted damping forces in relation to displacement under 2 Hz frequency. From the figure, it is observed that the three models well adapt the experimentally measured data. Against the others, the Bingham model cannot thoroughly depict the hysteresis behavior of the SMA damper at the two stroke ends; however, it possesses structural simplicity which is beneficial in cases of requiring quick modeling with relative precision such as design for an expected damping force or initial estimation of damper characteristics. On the contrary, the Bouc–Wen model and Bui’s model can track the damping force better but are more complicated at the same time; they are thus appropriate to strict cases such as feedback or control designs. More evidences can be found in Figs. 10(a–b) for the higher frequency excitations.
(a) 3 Hz
(b) 5 Hz
Fig. 10. Comparisons between the predicted and experimentally measured responses under higher frequencies.
To further assess performance of the three models, errors between the experimentally measured data and each model are quantitatively analyzed. The normalized error of damping force in displacement domain E x is given by
T
2
Fexp − Fm dx dt dt
0
(10) Ex = 2 T
Fexp − μexp dx dt dt 0
where μexp are the average experimental forces over the cycle T. The results under different frequency excitations are shown in Table 2. It can be realized that the Bouc–Wen model and Bui’s model have higher accuracy in capturing the hysteresis phenomenon of the SMA damper, which mainly comes from the effective and close control of the model physical parameters.
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Table 2. Normalized errors between the experimental responses and each model. Model
2 Hz
3 Hz
5 Hz
Bingham model
0.262 0.268 0.261
Bouc–Wen model 0.074 0.086 0.06 Bui’s model
0.03
0.029 0.034
6 Conclusions This research work contributed to a novel SMA damper for structural vibration control featuring SMA spring actuator and wedge mechanism. After a description of the damper configuration, three SMA springs were tested to specify their characteristics. Modeling of the proposed damper was then conducted based on the experimental data and a damper prototype was designed, manufactured and validated. The results showed that the damper behaves almost like a Coulomb friction damper. The average damping force could reach up to 95% of the calculated value while the off–state force is only about 8 N, which is small enough to prevent most force transmissibility at high frequencies. However, with the state changing time of about 25 s, the damper should be further investigated for effective applicability in closed-loop control systems. The proposed SMA damper also exhibited the hysteresis behavior in damping force– displacement relation, especially at the stroke ends. To capture this phenomenon, three idealized models consisting of Bingham model, Bouc–Wen model and Bui’s model were adopted. Although the Bingham model has lowest accuracy, its simple structure facilitates the modeling which is advantageous to design or premilinary characterization. The Bouc–Wen model and Bui’s model control the hysteresis curve more effectively relying on more physical parameters and hence usually prove their abilities in feedback or control designs. In the next stage of this research, many types of SMA springs and various heating methods will be studied to reduce the actuating time of the damper. In addition, the hysteresis with temperature variable will be investigated for dynamic modeling of the SMA damper. Acknowledgement. This work was supported by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant no. 107.01–2018.335.
References 1. Lagoudas, D.C. (ed.): Shape Memory Alloys, Modeling and Engineering Applications. Springer, New York (2008). https://doi.org/10.1007/978-0-387-47685-8 2. Carlson, J.D.: Low–cost MR fluid sponge devices. J. Intell. Mater. Syst. Struct. 10, 589–594 (1999) 3. Nguyen, Q.H., Nguyen, N.D., Choi, S.B.: Optimal design and performance evaluation of a flow–mode MR damper for front–loaded washing machines. Asia Pac. J. Comput. Eng. 1, 3 (2014)
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4. Nguyen, Q.H., Choi, S.B., Woo, J.K.: Optimal design of magnetorheological fluid–based dampers for front–loaded washing machines. Proc. Inst. Mech. Eng. C-J. Mech. Eng. Sci. 228, 294–306 (2014) 5. Bui, D.Q., Hoang, V.L., Le, H.D., et al.: Design and evaluation of a shear–mode MR damper for suspension system of front–loading washing machines. In: Proceedings of the International Conference on Advances in Computational Mechanics, Phu Quoc Island, Vietnam, pp. 1061– 1072 (2017) 6. Bui, Q.D., Nguyen, Q.H., Nguyen, T.T., et al.: Development of a magnetorheological damper with self–powered ability for washing machines. Appl. Sci. 10, 4099 (2020) 7. Graesser, E.J., Cozzarelli, F.A.: Shape memory alloys as new materials for seismic isolation. J. Eng. Mech. 117, 2590–2608 (1991) 8. Clark, P.W., Aiken, I.D., Kelly, J.M., et al.: Experimental and analytical studies of shapememory alloy dampers for structural control. In: Proceedings of SPIE 2445, San Diego, CA, USA, pp. 241–251 (1995) 9. Wilde, K., Gardoni, P., Fujino, Y.: Base isolation system with shape memory alloy device for elevated highway bridges. Eng. Struct. 22, 222–229 (2000) 10. Han, Y.L., Li, Q.S., Li, A.Q., et al.: Structural vibration control by shape memory alloy damper. Earthq. Eng. Struct. Dyn. 32, 483–494 (2003) 11. Zuo, X.B., Li, A.Q., Chen, Q.F.: Design and analysis of a superelastic SMA damper. J. Intell. Mater. Syst. Struct. 19, 631–639 (2007) 12. Stanway, R., Sproston, J.L., Stivens, N.G.: Non-linear modeling of an electrorheological vibration damper. J. Electrost. 20, 167–184 (1987) 13. Bouc, R.: Modele mathematique d’hysteresis. Acustica 24, 16–25 (1971) 14. Wen, Y.K.: Method of random vibration of hysteretic systems. J. Eng. Mech. 102, 249–263 (1976) 15. Bui, Q.D., Nguyen, Q.H., Bai, X.X., et al.: A new hysteresis model for magneto–rheological dampers based on Magic Formula. Proc. Inst. Mech. Eng. C–J. Mech. Eng. Sci. (2020)
A High-Order Time Finite Element Method Applied to Structural Dynamics Problems Thanh Xuan Nguyen1(B) and Long Tuan Tran2 1
National University of Civil Engineering, Hanoi, Vietnam College of Urban Works Construction, Hanoi, Vietnam
2
Abstract. The article proposes a high-order time finite element method based on the well-posed variation formulation that is equivalent to the conventional strong form of governing equations in structural dynamics. Three cases related to the term “high-order” include: the time finite element that is analogous to the spatial second-order beam element; the p-power of the time-to-go (T − t) in the formulation of “stiffness” matrix and “nodal force” vector; and the combination of both of them. In each case, the element “stiffness” matrix and “nodal force” vector are established and shown in details with notes on practical implementations.
Keywords: High-order time finite element Variation formulation
1
· Structural dynamics ·
Introduction
Dynamic response of a structure to an external excitation is of great concern in practical analysis and design. When dealing with a dynamic problem, usually the finite element method or a modal superposition approach is used to spatially discretize the structure, hence to reduce the problem to a set of ordinary differential equations in time that can be solved with one of many time stepping approaches [1,2]. This kind of procedure is widely used in practice and fairly well understood. Generally, for solving a set of ordinary differential equations in time, there are mainly two classes of direct time integration methods: explicit and implicit. Implicit methods (such as ones in β-Newmark family, Houbolt’s method, and Runge-Kutta method) possess unconditional stability, but may require much more computations than that needed for an explicit method. On the other hand, explicit methods are conditionally stable. When coupled with the conventional finite element (FE) computation, the step size in any explicit method depends on the FE spatial mesh size and thus requires more computational effort. For both implicit and explicit methods, a priori error analysis is often not easily available, since they are all derived in the spirit of finite difference. A review of implicit and explicit ones can be found in [3–5], just to name a few. c The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 137–148, 2022. https://doi.org/10.1007/978-981-16-3239-6_11
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A different approach for dealing with dynamic problem, being less popular than the methods mentioned above, is to use of the time (or temporal) finite element method (TFEM). The time finite element (TFE) formulation has several potential advantages as it can be applicable to both energy equation, and directly to the equations of motion. It is the straightforward derivation of higher order approximations in time. A key advantage of time finite element method, and the one often overlooked in its past applications, is the ease in which the sensitivity of the transient response with respect to various design parameters can be obtained. Usually, the TFEM approximation yields an accuracy superior to that of more conventional time stepping schemes at same computational cost [6]. Furthermore, the formulation is easy and convenient for computer implementation. The pioneer approaches, based on Hamilton’s Law of varying action, can date back to the research of Argyris and Scharpf, who employed Hermite cubic interpolation polynomials (akin to the beam finite element) to express the response over each time finite element [7]. The method, based on the Hamilton’s principle, was applied to a single-degree-of-freedom system but no numerical examples were considered. Fried [8] applied this approach to study the transient response of a damped system and transient heat conduction in a slab. Fried used a step by step approach to avoid storing and working with large matrices. Zienkiewicz and Parekh [9] used a time finite element approach to solve heat conduction problems. The formulation was based on Galerkin procedure over a time interval. In [10], Hulbert also employed the time-discontinuous Galerkin method and incorporates stabilizing terms having least-squares form. A general convergence theorem can be proved in a norm stronger than the energy norm. French and Peterson [11] proposed a time-continuous finite element method by transforming the second-order differential equations into first-order ones. Some other researchers have presented the variational formulation by allowing the TFE solution to be discontinuous at the end of each time element interval. Tang and Sun [12] introduced a unified TFE framework for the numerical discretization of ordinary differential equations based on TFE methods. In [6], Park used a bi-linear formulation for developing the time finite element method to obtain transient responses of both linear, nonlinear, damped and undamped systems. The sensitivity of the response with respect to various design parameters was also established. Results for both the transient response and its sensitivity to system parameters, when compared to a previously available approach that employs a multi-step method, are excellent. Recently, Wang and Zhong [13] proposed a time continuous Galerkin finite element method for structural dynamics. Its convergence property was proved through an a priori error analysis. To the best of our knowledge, the time finite elements already available until now are just of the first-order kind under the view of time discretization. Thus the authors will propose in this article some developments of high-order time finite elements based on the well-posed variational formulation. Three cases related to the term “high-order” include: the time finite element that is analogous to the spatial second-order beam element; the p-power of the time-to-go (T − t) in the formulation of “stiffness” matrix and “nodal force” vector; and the combination of both of them. The rest of the article is as follows. In Sect. 2, variational
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formulation for structural dynamics is presented, followed by the first-order time finite element in Sect. 3. In Sect. 4, we propose high-order time finite elements in the above mentioned three cases. A numerical example is shown in Sect. 5, to illustrate the use of proposed high-order time finite elements in solving an SDOF dynamic problem. The article finalize with conclusions in Sect. 6.
2
Variational Formulation for Structural Dynamics
Consider a structure having n degrees of freedom in the time interval LT = ]0, T [. The mass matrix M ∈ Rn×n , and stiffness matrix K ∈ Rn×n are, generally, symmetric and positive definite. The damping matrix C ∈ Rn×n is, generally, symmetric non-negative definite. The structure is subjected to initial conditions u0 and u˙ 0 , in addition to an external load F ∈ L2 (LT ), where L2 (LT ) is denoted for the Hilbert space on the time interval LT . The governing equations with the unknown u is u (0) = u0 , u˙ (0) = u˙ 0
L u ≡ M¨ u + Cu˙ + Ku = F,
(1)
Denote H (LT ) as the Sobolev space of order two. We introduce the two spaces as follows 2 (2) (LT ) = u ∈ H 2 (LT ) : u (0) = u0 , u˙ (0) = u˙ 0 H0p 2 2 H00 (LT ) = u ∈ H (LT ) : u (0) = 0, u˙ (0) = 0 (3) 2
Also, an energy norm vE : H 2 (LT ) → R of a vector v is defined as T 1 T 1 T 2 vE = v˙ Mv˙ + v Kv dt 2 2 0
(4)
The following two theorems were proved in [13]. Theorem 1 (Variational formulation for structural dynamics). The strong form of structural dynamics in the above equation is equivalent to the following for2 (LT ) such that mulation: find u ∈ H0p B (u, v) = (v) ,
2 ∀v ∈ H00 (LT )
(5)
where B (u, v) =
T
0
(v) =
0
T
s
v˙ T (M¨ u + Cu˙ + Ku) dt ds =
0 s 0
v˙ T F dt ds =
T
(T − t) v˙ T (M¨ u + Cu˙ + Ku) dt
(6)
0
T
(T − t) v˙ T F dt
(7)
0
Theorem 2 (Solution estimation). The following estimate for solution from the variational formulation uE ≤ C (F0 + |u0 | + |u˙ 0 |)
(8)
holds, where C denotes a positive constant (independent of mesh size or time step size), ·0 is the usual norm of L2 (LT ) and |·| is defined as the usual length of a vector.
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First-Order Time Finite Element
Let the interval [0, T ] be divided into a finite number N of non-overlap sub2 2 intervals each of which has the length of Ti . Denote H0p,τ (LT ) and H00,τ (LT ) as 2 2 the finite dimensional sub-spaces of H0p (LT ) and H00 (LT ), respectively. Usually, 2 2 (LT ) and H00,τ (LT ) are assumed to be of polynomial forms in each eleH0p,τ ment with degree p ≥ 2. Then, the time (Galerkin) finite element formulation is 2 (LT ) established by referring to the variational formulation (5): find uτ ∈ H0p,τ such that 2 ∀vτ ∈ H00,τ (LT ) (9) B (uτ , vτ ) = (vτ ) , Well-posedness of the time FEM (9) is directly implicated in Theorem 1 and 2. Generally, in structural dynamics, we consider both displacement and velocity responses, and moreover, uτ ∈ H 2 (LT ). Thus, the Hermitian interpolation functions are suitable for establishing the time finite element. In [13], the firstorder time finite element was obtained using Hermitian interpolation of degree p = 3, that is, for the ith element uτ = H1 ub + H2
Ti Ti u˙ b + H3 ue + H4 u˙ e = Hq, 2 2
τ ∈ [0, Ti ]
(10)
where ub , u˙ b and ue , u˙ e are, respectively, nodal (displacement and velocity) responses at the beginning and at the end of the time interval [0, Ti ] of that element. Vector q collects all the nodal responses in element i. The Hermitian interpolation functions - the shape functions - are as usual given as follows 1 (1 − ξ)2 (2 + ξ) 4 1 H3 (ξ) = (1 + ξ)2 (2 − ξ) 4 H1 (ξ) =
1 (1 + ξ)(1 − ξ)2 4 1 H4 (ξ) = − (1 + ξ)2 (1 − ξ) 4 H2 (ξ) =
(11) (12)
where ξ = (−1 + 2τ /Ti ) ∈ [−1, 1]. Then, element stiffness matrix K and nodal element load vector f are obtained as follows Ti ˙ T MH ¨ + CH ˙ + KH dt (13) K= (T − t) H 0
f=
Ti
˙ T F dt (T − t) H
(14)
0
Now, by usual assembly process as seen in conventional finite element method, a set of algebraic equations is established. Also, the initial displacement and velocity are taken into account and entered to the formulation as the boundary conditions of the problem. After these steps, the nodal displacement and velocity at each instant of time can be obtained by solving Ksys qsys = fsys
(15)
It is noteworthy that the global stiffness matrix Ksys is a block diagonal matrix and therefore, Eq. (15) can be efficiently solved by several usual algorithms, including the parallel method [13].
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High-Order Time Finite Elements
In this section, we propose three high-order time finite elements. The first one, denoted as 2-order b-TFE, is a direct extension of the first-order time finite element above, and adapted analogously to the second-order beam element with three equidistant nodes (see Fig. 1). Normally, the finite elements of order larger than two are not commonly used in practice since they might destroy the simple philosophy of the finite element method. In stimulating the concept of secondorder beam element in conventional method, the following Hermitian interpolation functions are chosen as H2 (ξ) = ξ − 6ξ 2 + 13ξ 3 − 12ξ 4 + 4ξ 5 Ti H4 (ξ) = −8ξ 2 + 32ξ 3 − 40ξ 4 + 16ξ 5 Ti H6 (ξ) = −ξ 2 + 5ξ 3 − 8ξ 4 + 4ξ 5 Ti
H1 (ξ) = 1 − 23ξ 2 + 66ξ 3 − 68ξ 4 + 24ξ 5 H3 (ξ) = 16ξ − 32ξ + 16ξ 2
3
4
H5 (ξ) = 7ξ 2 − 34ξ 3 + 52ξ 4 − 24ξ 5
(16) (17) (18)
where ξ = τ /Ti ∈ [0, 1], and Ti is the “length” of the ith element. The three nodes in this element might not be equidistantly located. However, under practical view, the use of elements having non-equidistantly located nodes are not common. Also, we could otherwise use hierarchical functions as interpolation ones. However, if the isoparametric element is intended, then the ξ-t relationship is complex enough so that the advantage of using hierarchical polynomials in saving computational efforts might be lost.
With the element nodal responses qT = uTb u˙ Tb uTm u˙ Tm uTe u˙ Te , the response uτ for the ith element can be interpolated from the above Hermitian functions. The formula for the matrix H in Eq. (10) varies case by case, since it depends on the number of degrees of freedom. For example, if the system is single-degree-of-freedom, then matrix H in Eq. (10) is defined as
(19) H = H 1 H2 H3 H4 H5 H6 meanwhile if the system has n degree-of-freedom, then ⎡ ⎢ ⎢ ⎢ H=⎢ ⎢ ⎣
H1
H2
H1 ..
H6
H2 ..
. H1
···
. H2
diag. matrix of size n diag. matrix of size n
⎤ H6 ..
. H6
⎥ ⎥ ⎥ ⎥ ⎥ ⎦
(20)
diag. matrix of size n
Now, we follow Eqs. (13) and (14) for all time elements in the temporal mesh. In these equations, the time derivatives of H are given as ˙ = 1 dH , H Ti dξ
2 ¨ = 1 d H H Ti2 dξ 2
(21)
The explicit formulae for the resulting element “stiffness” matrix and element “equivalent force” vector also vary case by case, since they are functions of the
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Fig. 1. (a) Conventional (spatial) three-node beam finite element; and (b) Corresponding three-node time finite element
M, C, and K matrices obtained from spatial discretization as well. We observe here that a large intermediate manipulations of matrices for the element matrices (stiffness matrix and equivalent force vector) should be conducted. However, in practical implementation, we can take the advantage of sparsity in matrix H in these computations. In addition, these computations are prepared just once. For an n-DOF system, the resulting element time “stiffness” matrix is 6n × 6n, and that of “equivalent force” vector is 6n × 1. When all element matrices are already determined, the global stiffness matrix and global equivalent force vector can be found by usual assembling procedure as in conventional finite element method. The second time finite element, denoted as p-TFE, is based on the following observations. As we can see from Theorem 1, Eq. (5) is simply posed to be the ˙ which is a residual L u − F weighted by a time dependent function (T − t) v, ˙ It raises to product of the time-to-go (T − t) and the test on-going velocity v. the idea that another time dependent function can be used as the weight. This idea is also supported by the fact (see proof at the Appendix) 0
T
s1
0
···
sp
0
p folds
f dt dsp . . . ds1 =
T
p
(T − t) f (t) dt
(22)
0
Thus, Eq. (5) in Theorem 1 holds with the bi-linear form B : H 2 (LT ) × H 2 (LT ) → R and linear functional : H 2 (LT ) → R expressed as below B (u, v) =
T
0
0
= 0
T
s1
···
p folds
0
sp
v˙ T (M¨ u + Cu˙ + Ku) dt dsp . . . ds1
p (T − t) v˙ T (M¨ u + Cu˙ + Ku) dt
(23)
(24)
High-Order Time Finite Element Method
T
(v) =
0
0
s1
···
p folds
0
sp
T
v˙ f dt ds =
T
p (T − t) v˙ T f dt
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(25)
0
Supported by this, as before, we have the element “stiffness” matrix and the element “equivalent force” vector where the time-to-go (T − t) is raised to the power of p as follows K=
Ti
p ˙ T ¨ + CH ˙ + KH dt MH (T − t) H
(26)
p ˙ T (T − t) H F dt
(27)
0
f=
Ti
0
We see that, in this formulation, the specific types of “shape” functions will give different variants of the element matrices. For the second type of time finite element proposed in this article, the common Hermitian interpolation polynomials of degree 3 are used. For SDOF system n = 1, the element “stiffness” matrix is shown in the Appendices section. When the six “shape” functions in Eqs. (16)–(18) are used, then we have the third type of time finite element, which is a combination of the second-order beam analogous element and the “p-power of the time-to-go” element. This type of element is denoted as bp-TFE for later reference. It is too lengthy to show the formulas of element “stiffness” matrix and “equivalent force” vector here. Instead, MATLAB code to obtain these quantities are given in the Appendices section.
5
Numerical Example
In this section, we will illustrate the use of proposed high-order TFEs in solving an undamped single-degree-of-freedom (SDOF) dynamic problem. Out of the accuracy aspect, we will not explore all other aspects of numerical performance of the proposed method through this very example. For comparison, the problem is also solved by Newmark-β method - the most popular numerical integration method for transient structural dynamics - with three different sets of parameters (γ, β), namely (1/2, 0), (1/2, 1/6), and (1/2, 1/4). The most accurate results obtained by using Newmark-β methods with these sets of parameters are shown in Table 1 and Table 2. The results offered by [13] are also taken into the comparison of accuracy. The statement of the problem is as follows. Given a SDOF system having the mass of m = 1, the stiffness of k = π 2 /4, and there is no damping c = 0. The system is at rest when it is suddenly enforced by a pulse force f (t) given below 1, 0 < t < 1 f (t) = 0, t ≥ 1
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The exact displacement and velocity responses of the system are found to be 4 (1 − cos πt/2) /π 2 , 0 σˆ¯ 3 , ˆ ˆ ˆ of where the effective stress tensor. It can be easily shown σ¯ 1 , σ¯ 2 and σ¯ 3 are eigenvalues 2 = 3 q¯ − p¯ and σˆ¯ max = 13 q¯ − p¯ , along the tensile and compressive that σˆ¯ max TM CM meridians, respectively.
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If σˆ¯ max < 0, the corresponding yield conditions are:
Let Kc = q¯ (TM ) /¯q(CM ) for any given value of the hydrostatic pressure p¯ with σˆ¯ max γ +3 < 0, then Kc = 2γ +3 . The fact that K c is constant does not seem to be contradicted by experimental c) evidence [13]. The coefficient γ is, therefore, evaluated as γ = 3(1−K 2Kc −1 . A value Kc = 2/3, which is typical for concrete, gives γ = 3. If σˆ¯ max > 0, the yield conditions along the tensile and compressive meridians reduce to:
Let Kt = q¯ (TM ) /¯q(CM ) for any given value of the hydrostatic pressure p¯ with σˆ¯ max β+3 > 0, then Kt = 2β+3 . σ¯ ) The plastic-damage model assumes non-associated potential flow, ε˙ pl = λ˙ ∂G( ∂ σ¯ . The flow potential G chosen for this model is the Drucker-Prager hyperbolic function as Eq. (9) where ψ is the dilation angle measured in the p-q plane at high confining pressure, σ t0 is the uniaxial tensile stress at failure, and ∈ is a parameter, referred to as the eccentricity, that defines the rate at which the function approaches the asymptote (the flow potential tends to a straight line as the eccentricity tends to zero). (9) G = (∈ σt0 tan ψ)2 + q¯ 2 − p¯ tan ψ
3.2 Hill Criterion It can be assumed that the anisotropic plasticity of timber progresses according to the Hill yield criterion (Hill R. 1948) [15]. The Hill’s potential function can be expressed in terms of rectangular Cartesian stress components as: f (σ ) =
2 + 2M σ 2 + 2N σ 2 F(σ22 − σ33 )2 + G(σ33 − σ11 )2 + H (σ11 − σ22 )2 + 2Lσ23 31 12
(10)
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Where: F, G, H, L, M and N are constants obtained by tests of the material in different orientations. They are defined as:
0 2 σ 1 1 1 1 1 1 1 F= + + + 2 − 2 = (11) 2 2 2 2 2 R222 R33 R11 σ22 σ33 σ11
0 2 σ 1 1 1 1 1 1 1 + + + 2 − 2 G= = (12) 2 2 2 2 2 R233 R11 R22 σ33 σ11 σ22
0 2 σ 1 1 1 1 1 1 1 + + + 2 − 2 H= = (13) 2 2 R211 R22 R33 σ2 σ2 σ2 11
22
33
τ0
2
3 = σ23 2R223 2 3 τ0 3 M = = 2 σ13 2R213 2 3 τ0 3 N= = 2 σ12 2R212 3 L= 2
(14)
(15)
(16)
Where, each σ¯ ij is the measured yield stress value when σij∞ is applied as the only √ nonzero stress component; σ 0 is the user-defined reference yield stress and τ 0 = σ 0 / 3. The six yield stress ratios such as R11 , R22 , R33 , R12 , R13 and R23 corresponding to σσ110 , σ22 σ33 σ12 σ13 , , , and στ230 , respectively, which have to be defined in the simulation. σ0 σ0 τ0 τ0 The flow rule is described in Eq. (17) as follows, with b from the definition of f above. ⎡ ⎤ −G(σ33 − σ11 ) + H (σ11 − σ22 ) ⎢ F(σ − σ ) − H (σ − σ ) ⎥ 22 33 11 22 ⎥ ⎢ ⎢ ⎥ d λ ∂f −F(σ − σ − σ + G(σ ) ⎢ 22 33 33 11 ) ⎥ (17) = b, with b = ⎢ d εpl = d λ ⎥ ⎢ ⎥ 2N σ12 ∂σ f ⎢ ⎥ ⎣ ⎦ 2M σ31 2Lσ23
4 Modelling of Timber-Concrete Composite Beam 4.1 Verification of the Selected Model The models have been constructed based on the experimental tests, 3-point bending test for the concrete and 4-point bending test for the timber. The Concrete Damaged Plasticity model and the Hill criterion have been considered for the concrete and the timber behavior, respectively. The selected models were previously used by many authors in the literature. The parameters of the selected models are adjusted by many calculations
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Dilation angle ψ (degree)
Eccentricity ∈ (non-unit)
σb0 /σc0 (non-unit)
K (non-unit)
Viscosity parameter (second)
35
0.1
1.16
0.667
0.007985
and assessing the difference between the experiment and the model, namely comparing the Force/Displacement curve from the model with experiment. The parameters of the Concrete Damaged Plasticity model are summarized in Table 4 as below. The evolution of the damage in tension and compression is shown in Fig. 6. The concrete grade in compression and tension is 45 MPa and 5.5 MPa, respectively.
Compression
Tension
Fig. 6. Damaged evolution of the concrete in compression and tension
For the timber, the elasto-plastic anisotropic model with Hill criterion has been applied. The elastic and plastic parameters of timber have been presented in Table 5 as below. The comparison between the experiment and the numerical model has been demonstrated in Fig. 7. It is observed that the numerical curve can meet effectively the experimental curves. A high precision in the elastic term and also in the damaged moment can be achieved. 4.2 Modelling of the Timber-Concrete Composite Beam In the context of timber-concrete composition, a timber-concrete composite T-beam has been considered under finite element analysis. The selected models have been integrated in the global model. The concrete flange and the timber web are connected by steel dowels. The steel dowels are considered to be 2D beam elements. The steel dowel gets the yield strength of 450 MPa at 0.2% of strain. The elastic modulus of the steel dowel is equal to 210 GPa.
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Table 5. Properties of timber model Timber elastic properties
Hill parameters of timber
Longitudinal Young’s Modulus (YM)
E L (MPa)
14788
σ c,90 (MPa)
7
Radial YM
E R (MPa)
1848
F
0.99
Tangent YM
E T (MPa)
1087
G
0.09
Shear Modulus (SM) in LR plane
GLR (MPa)
1260
H
0.01
SM in TL plane
GTL (MPa)
971
N
1.7
SM in RT plane
GRT (MPa)
366
L=M
1.5
RT plane Poisson’s Ratio
ν RT
0.67
LT plane Poisson’s Ratio
ν LT
0.46
LR plane Poisson’s Ratio
ν LR
0.39
Fig. 7. Comparison of the experimental curves with FE model: (a) Concrete; (b) Timber
These beam elements are placed within the 3D elements of the concrete and timber environment. The constraint called “embedded region” available in ABAQUS software has been used in this case of connection. That means the steel dowel is embedded in the concrete and the timber. The contact between the timber and the concrete is “hard” contact that does not allow each other to penetrate. The friction coefficient of the contact is 0.2. On haft of the composite beam is presented in Fig. 8.
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Fig. 8. 3D FE model of one half of the timber-concrete composite beam
Figure 9 presents the numerical results of the model such as the Mises Stress, the damaged status of concrete in tension, the Mises stress of steel dowels. This model may predict effectively the behavior of the composite beam. By the Mises stress distribution, the concrete and the timber are significantly affected in the middle area. But the visual deformation of steel dowels shows the important shear stress in the extreme zone of beam. That can be explained by the important shear force distributed around the supported zones. It is noted that the damage of concrete in tension is observed when the connection is deformed and the concrete is loaded in tension.
Damage of concrete in tension
Deformation of steel dowels
Fig. 9. Results of the numerical composite beam model
In this study, the different contacts between the timber – the steel dowels and the concrete – the steel dowels haven’t been distinguished because of a more complex calculation. Normally, the contact with timber is more flexible than with concrete. Consequently, the concrete is theoretically more loaded in tension. It is recommended to reinforce the steel dowels. It can be concluded that the role of steel dowels is important. In order to optimize the dimension of the steel dowel, a parametric study presented in the next section has been mentioned.
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4.3 Parametric Study: Influence of the Steel Connection Diameter on Shear Resistance In this extended section, the influence of the steel dowel diameter on shear resistance in the context of the composition is focused. Three different diameters of dowel such as 5 mm, 10 mm and 20 mm have been studied. Figure 10 presents the relative sliding between the timber and the concrete in the function of the different diameters of steel dowel. It is noted that the diameter of 5 mm is weak for maintaining the composition. Nevertheless, the diameter of Fig. 10. Relative sliding between timber and concrete in 20 mm presents a high shear the function of dowel diameter resistance. It can be observed that the dowel with a diameter larger than 20 mm may not necessary
5 Conclusions This research presented the experimental results with regard to the mechanical properties of the high strength steel fiber concrete and beech and oak timber subjected to flexion. The experimental results concerned the bending resistance and the elastic modulus. It can be concluded that the component of steel fiber can openly affect the bending resistance of the global concrete beam. With reference to the timber, the beech takes up a high bending resistance that can achieve around 71 MPa. In the context of concrete-timber composite beam, a finite element model has been constructed that allows predicting the behavior of the concrete-timber composite beam. The selected models have been verified by experiment for all timber and concrete. The results presented a high precision of the numerical model. An extended study has been carried out in order to study the influence of the dowel diameter on the shear resistance of the composite beam. Acknowledgement. This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 107.01–2019.11.
References 1. Holschemacher, K., Klotz, S., Weisse, D.: Application of steel fibre reinforced concrete for timber-concrete composite constructions. Leipz. Annu. Civil Eng. Rep. 7, 161–170 (2002)
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2. Holschemacher, K., Klotz, S., Klug, Y., Weisse, D.: Application of steel fibre reinforced concrete for the revaluation of timber floors. In: Proceedings of the Second International Structural Engineering and Construction Conference, Rome, Italy, pp. 1393–1398 (2003) 3. Heiduschke, A., Kasal, B.: Composite cross sections with high performance fibre reinforced concrete and timber. For. Prod. J. 53(10), 74–78 (2003) 4. Kieslich, H., Holschemacher, K.: Composite constructions of timber and high-performance concrete. Adv. Mater. Res. 133–134, 1171–1176 (2010) 5. Le Roy, R., Pham, H.S., Foret, G.: New wood composite bridges. Eur. J. Environ. Civil Eng. 13(9), 1125–1139 (2009) 6. Pham, H.S.: Optimization and fatigue behavior of wood-UHPC connection for new composite bridges. PhD thesis, Ecole Nationale des Ponts et Chaussées (2007). (in French) 7. Schafers, M., Seim, W.: Investigation on bonding between timber and ultra-high performance concrete (UHPC). Constr. Build. Mater. 25, 3078–3088 (2011) 8. Caldova, E., Blesak, L., Wald, F., Kloiber, M., Urushadze, S., Vymlatil, P.: Behaviour of timber and steel fibre reinforced concrete composite construction with screwed connections. Wood Res. 59(4), 639–660 (2014) 9. Auclair, S.C., Sorelli, L., Salenikovich, A.: A new composite connector for timber-concrete composite structures. Constr. Build. Mater. 112, 84–92 (2016) 10. Schanack, F., Ramos, Ó.R., Reyes, J.P., Low, A.A.: Experimental study on the influence of concrete cracking on timber concrete composite beams. Eng. Struct. 84, 362–367 (2015) 11. ISO 13061–1: 2014 Physical and mechanical properties of wood – test method for small clear wood specimens – Part 1: Determination of moisture content for physical and mechanical tests (2014) 12. European standard EN 408: Timber structures. Structural timber and glued laminated timber. Determination of some physical and mechanical properties. European Committee for Standardization (2010) 13. Lubliner, J., Oliver, J., Oller, S., Oñate, E.: A plastic-damage model for concrete. Int. J. Solids Struct. 25 (3): 299–326 (1989) 14. Lee, J., Fenves, G.L.: Plasticdamage model for cyclic loading of concrete structures. J. Eng. Mech. 124(8), 892–900 (1998) 15. Hill, R.: A theory of Yielding and Plastic Flow of Anisotropic Metals, p. 281. Royal Society Publishing, London (1948)
Free Vibration of Stiffened Functionally Graded Porous Cylindrical Shell Under Various Boundary Conditions Tran Huu Quoc, Vu Van Tham(B) , and Tran Minh Tu National University of Civil Engineering, Hanoi, Vietnam {quocth,thamvv,tutm}@nuce.edu.vn
Abstract. This paper presents an analytical approach to analyze the free vibration of stiffened cylindrical shells made of porous functionally graded (FGP) materials. The governing equations are derived by using Hamilton’s principle based on the first-order shear deformation theory (FSDT) in conjunction with Lekhnitsky smeared technique. Two types of porosity distributions including symmetric and non-symmetric are considered. Applying the Galerkin method and the axial displacement field functions, the natural frequencies of the cylindrical shell under different boundary conditions are determined. The accuracy of the present formulation is made by comparing the obtained results with those available in published reports. The influences of material parameters such as porosity coefficient, porosity distribution types, and shell geometrical parameters on the natural frequencies are investigated and discussed. Keywords: Cylindrical shell · Free vibration · Functionally graded porous material · FSDT
1 Introduction Functional graded porous materials (FGPMs) are new material, which is invented by the combination of the concept of FGMs and porous materials. In FGPMs, microstructure and porosities are graded along a specific direction according to a defined rule. Due to their excellent properties such as lightweight, heat-resistant property, crackresistant wealth, and energy-absorbing capability, FGPMs have been widely used in many engineering fields. There are numerous studies on the vibrational characteristics of FGP shells [1–10]. Among those, the free vibration response of functionally graded porous cylindrical shells (FGP-CS) is investigated by Wang and Wu [5] using sinusoidal shear deformation theory and the Rayleigh-Ritz method. Survey results indicate that the porosity distribution types, porosity coefficient considerably affect the natural frequency of the FGP-CS. Using the FSDT and Rayleigh-Ritz method, Li et al. [6, 7] predicted natural frequencies of FGP spherical and cylindrical shells subjected to elastic restraints. The free vibration of FGP doubly-curved shallow shell has been examined by Zhao et al. [8, 9] in the framework of FSDT. Li et al. [10] analyzed the vibration of FGP stepped CS in the thermal environment. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 347–361, 2022. https://doi.org/10.1007/978-981-16-3239-6_26
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One of the most effective solutions for increasing stiffness of the structures is to stiffen them with stiffeners. The stiffness of the structure is increased significantly, while the structural weight increases slightly when the stiffeners are appropriately strengthened. For this reason, many studies have been made on the dynamic behaviour of shells stiffened by various materials, e.g. [11–14]. Mustafa and Ali [15] determined natural frequencies of isotropic cylindrical shells stiffened by stringers and rings using the energy method. Lee and Kim [16] presented the optimization of the orthogonally stiffened cylindrical shell under axial compression according to the minimum weight criterion. At a later stage, when the FGMs are applied, free vibration characteristics of FG cylindrical shell continue to gain significant attention of researchers. Tu et al. [17, 18] examined free vibration of the rotating FG-CS in a thermal environment based on classical shell theory and FSDT. Through reviewing the literature and to the best of the authors’ knowledge, there are no studies on the free vibration of stiffened FGP-CS subjected to different boundary conditions. Thus, based on the FSDT and smeared stiffener technique, the free vibration of FGP-CS reinforced by both stringers and rings is studied in this paper. Using the Galerkin method with the axial displacement field of beam functions, the free vibration responses of the stiffened FGP-CS under different boundary conditions are investigated.
2 FGP Cylindrical Shell Considered a circular cylindrical shell (CS) with the reference cylindrical coordinate system (x, θ, z) is placed on the shell’s middle surface, as shown in Fig. 1. The cylindrical shell has length L, uniform thickness h, radius R, and made of FGPMs. The FGP-CS is stiffened by isotropic rings and stringers. The symbols (hr , br ) and (hs , bs ) denote the rings and stringers’ height and width, while ss and sr are the distance between two adjacent rings and stringers, respectively. Two porosity distribution patterns named symmetric (Fig. 1b) and non-symmetric (Fig. 1c) through the thickness direction are considered in this study. The effective material properties of the FGP-CS are defined by [5]: Type I (symmetric distribution): E(z) = E1 [1 − e0 cos(π z/h)], ρ(z) = ρ1 [1 − em cos(π z/h)] Type II (non-symmetric distribution): 1 1 1 1 z z E(z) = E1 1 − e0 cos π + , ρ(z) = ρ1 1 − em cos π + 2 h 2 2 h 2
(1)
(2)
in which, e0 is the porosity coefficient and em is the porosity coefficient for mass density: e0 = 1 −
E0 G0 =1− E1 G1
(0 < e0 < 1), em = 1 −
ρ0 = 1 − 1 − e0 ρ1
(0 < em < 1)
(3) where (E0 , ρ0 ) and (E1 , ρ1 ) are the minimum and maximum values of effective parameters that occur at the bottom and the top surfaces, respectively.
Free Vibration of Stiffened FGP Cylindrical Shell
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Fig. 1. Configuration of the stiffened FGP-CS: (a) Stiffened cylindrical shell; (b) symmetric porosity distributions; (c) non-symmetric porosity distributions.
3 Theory and Formulation According to the FSDT, the displacements of a point in the FGP-CS are assumed as [19]: u(x, θ, z, t) = u0 (x, θ, t) + zϕx (x, θ, t), θ (x, θ, z, t) = v0 (x, θ, t) + zϕθ (x, θ, t), w(x, θ, z, t) = w0 (x, θ, t)
(4)
where u0 , v0 and w0 are the displacements of a certain point located on the midsurface of the shell and ϕx (x, θ, t), ϕθ (x, θ, t) are the rotations of the normal to the mid-surface of the shell about the θ -and x-axis, respectively. The strain-displacement relation of the shell can be expressed as: ⎧ ⎫ ⎧ ⎫ ⎧ ⎫ 0 ⎪ ⎪ ⎪ εxx ⎪ ⎪ κxx ⎪ ⎪ ⎪ ⎪ ⎪ εxx ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎨ εθθ ⎨ κθθ ⎪ ⎬ ⎪ ⎬ ⎨ εθθ ⎪ ⎬ 0 + z κxθ (5) γxθ = γxθ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ γθz ⎪ ⎪ γθz ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩γ0 ⎪ ⎩ 0 ⎪ ⎭ ⎪ ⎭ ⎩γ ⎪ ⎭ xz xz where w0 ∂u0 v0 ∂u0 ∂v0 ∂v0 ∂w0 0 0 0 ; εθθ + ; γxθ + ; γθz − ; = = = ϕθ + ∂x R∂θ R ∂x R∂θ R∂θ R ∂ϕ ∂w ∂ϕ ∂ϕ ∂ϕ 0 x θ x θ γxz0 = ϕx + ; κxx = ; κθθ = ; κxθ = + (6) ∂x ∂x R∂θ R∂θ ∂x
0 εxx =
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The constitutive equations of FGP-CS are obtained by using Hooke’s law as follows: ⎫ ⎡ ⎫ ⎧ ⎤⎧ ⎪ εxx ⎪ Q11 Q12 0 0 0 ⎪ σxx ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ε ⎪ ⎪ ⎢Q Q ⎪ ⎪ ⎥⎪ ⎪ ⎨ θθ ⎪ ⎬ ⎢ 12 11 0 0 0 ⎥⎪ ⎬ ⎨ σθθ ⎪ ⎢ ⎥ (7) σxθ = ⎢ 0 0 Q66 0 0 ⎥ γxθ ⎥⎪ ⎪ ⎪ ⎪ ⎪ ⎢ ⎪ ⎪σ ⎪ ⎪ ⎪ ⎪ ⎪ ⎣ 0 0 0 Q44 0 ⎦⎪ γθz ⎪ θz ⎪ ⎪ ⎪ ⎪ ⎩γ ⎪ ⎭ ⎭ ⎩σ ⎪ 0 0 0 0 Q xz
xz
55
where Q11 = Q22 =
E(z) νE(z) E(z) ; Q12 = ; Q44 = Q55 = Q66 = 2 2 1−ν 1−ν 2(1 + ν)
(8)
The stresses resultants of the stiffened FGP-CS are obtained by using the smeared stiffener technique: ⎤⎧ ⎧ ⎫ ⎡¯ ¯ 0 ⎫ A11 A12 0 B¯ 11 B¯ 12 0 ⎪ εxx Nxx ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎢ A¯ A¯ ⎪ ⎥⎪ 0 ⎪ ⎪ Nθθ ⎪ εθθ ⎪ ⎪ ⎢ 12 22 0 B¯ 12 B¯ 22 0 ⎥⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎨ ⎬ ⎢ ⎬ ⎥ 0 ¯ ¯ 0 0 B 0 0 A ⎢ ⎥ Nxθ γxθ 66 66 =⎢¯ ¯ (9) ⎥ ¯ 11 D ¯ 12 0 ⎥⎪ ⎢ B11 B12 0 D ⎪ Mxx ⎪ κxx ⎪ ⎪ ⎪ ⎪ ⎪ ⎢ ⎪ ⎥ ⎪ ⎪ ⎪ ⎪κ ⎪ ⎪M ⎪ ⎪ ⎣ B¯ B¯ ⎪ ¯ 12 D ¯ 22 0 ⎦⎪ ⎪ 12 22 0 D ⎪ ⎪ ⎩ θθ ⎪ ⎩ θθ ⎪ ⎭ ⎭ ¯ 66 Mxθ κxθ 0 0 B¯ 66 0 0 D Qθθ 0 A¯ γθz = ks 44 ¯ (10) Qxx γxz 0 A55 where A¯ 11 = A11 + Ess As s ; A¯ 12 = A12 ; B¯ 11 = B11 + Es Asss zs ; B¯ 12 = B12 ; ¯ 11 = D11 + Es As Is ; D ¯ 12 = D12 ; D ss h/2 A¯ 44 = ks Q44 dz + Gr Ar /sr ; Gr = Gs =
E0 1−2ν
−h/2
A¯ 22 = A22 + ErsrAr ; A¯ 66 = A66 ; B¯ 22 = B22 + Er Asrr zr ; B¯ 66 = B66 ; ¯ 22 = D22 + Er Ar Ir ; D ¯ 66 = D66 ; D A¯ 55 = ks
h/2 −h/2
for the external stiffeners and Gr = Gs =
sr
Q55 dz + Gs As /ss ; E1 1−2ν
for the internal stiffeners
(11) with A11 = A22 = A12 = A66 =
h/2 −h/2
h/2
−h/2
h/2
h/2 h/2 E(z) E(z) E(z) 2 dz; B = B = zdz; D = D = z dz 11 22 11 22 2 2 2 1 − ν 1 − ν 1 −h/2 −h/2 −h/2 − ν h/2 h/2 νE(z) νE(z) νE(z) 2 dz; B = zdz; D = z dz; 12 12 2 2 1 − ν2 1 − ν 1 −h/2 −h/2 − ν h/2 h/2 E(z) E(z) E(z) 2 dz; B66 = zdz; D66 = z dz; 2(1 + ν) 2(1 + ν) 2(1 + ν) −h/2 −h/2
Free Vibration of Stiffened FGP Cylindrical Shell Is =
bs h3s br h3r hs + h hr + h + As zs2 ; Ir = + Ar zr2 ; zs = ± ; zr = ± 12 12 2 2
351
(12)
in which As , Ar and zr ,zs are the cross-sectional areas and distances from the midsurface of the shell to the centroid of each ring and stringer, respectively. k s = 5/6 is the shear correction factor. Using Hamilton’s principle, the equations of motion of the stiffened FGP-CS are obtained as: ∂Nxθ ∂Nxx + = J0 u¨ 0 + J1 ϕ¨x ∂x R∂θ ∂Nθθ Qθθ ∂Nxθ + + = J0 v¨ 0 + J1 ϕ¨θ ∂x R∂θ R ∂Qθθ Nθθ ∂Qxx + − = J0 w ¨0 ∂x R∂θ R ∂Mxθ ∂Mxx + − Qxx = J1 u¨ 0 + J2 ϕ¨x ∂x R∂θ ∂Mθθ ∂Mxθ + − Qθθ = J1 v¨ 0 + J2 ϕ¨θ R∂θ ∂x
(13)
in which, Ji are the moments of inertia: h/2 ρr Ar ρs As i + Jj = ρ(z)z ¯ dz (i = 0, 1, 2); ρ(z) ¯ = ρ(z) + ss h sr h −h/2
(14)
where ρs = ρr = ρ1 for the outer stiffeners and ρs = ρr = ρ0 for the inner stiffeners. Using Eqs. (5)–(7), and (9)–(10), the equations of motion (13) can be rewritten as χ11 u0 + χ12 v0 + χ13 w0 + χ14 ϕx + χ15 ϕθ χ21 u0 + χ22 v0 + χ23 w0 + χ24 ϕx + χ25 ϕθ χ31 u0 + χ32 v0 + χ33 w0 + χ34 ϕx + χ35 ϕθ χ41 u0 + χ42 v0 + χ43 w0 + χ44 ϕx + χ45 ϕθ χ51 u0 + χ52 v0 + χ53 w0 + χ54 ϕx + χ55 ϕθ
=0 =0 =0 =0 =0
(15)
where differential operators χij (i, j = 1, 2,…,5) are defined in Appendix A.
4 Analytical Solution The Galerkin technique is implemented to solve the system of differential equations to determine the natural frequencies of stiffened FGP-CS under various boundary conditions (BC). The displacement components are chosen as [5, 20]: u0 (x, θ, t) = w0 (x, θ, t) = ϕθ (x, θ, t) =
∞ ∞ m=1 n=1 ∞ ∞
Umn φ1m (x)φ1n (θ) cos(ωt), v0 (x, θ, t) =
m=1 n=1 ∞ ∞ m=1 n=1
∞ ∞
Vmn φ2m (x)φ2n (θ) cos(ωt),
m=1 n=1 ∞ ∞
Wmn φ3m (x)φ3n (θ) cos(ωt), ϕx (x, θ, t) =
m=1 n=1
m (x)φ n (θ) cos(ωt), mn φφx φx
m (x)φ n (θ) cos(ωt). mn φφθ φθ
(16)
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with n n φ1n (θ ) = φ3n (θ ) = φφx (θ ) = cos(nθ ), φ2n (θ ) = φφθ (θ ) = sin(nθ ).
(17)
in which: Umn , Vmn , Wmn , mn , mn are unknowns, m and n are the numbers of half-wave in the longitudinal and circumferential directions, respectively. It is noted that φkm (x) (k = 2, 3) is the function which satisfied boundary conditions: γ x γ x γ x γ x m m m m + H2 cos − μm H3 sinh + H4 sin φkm (x) = H1 cosh L L L L (18) ∂φ m (x)
k The remaining functions φim (x) (i = 1, φx) can be chosen such that φim (x) = ∂x . Coefficients H1 , H2 , H3 , H4 , m and µm in Eq. (16) for three types of boundary condition are listed in Table 1 [5]:
Table 1. Values of H i , γm and μm for several boundary conditions Hi (i = 1, 2, 3, 4)
γm
μm
Clamped–clamped (C–C)
H 1 = 1, H 2 = −1, H 3 = 1, H 4 = −1
(2m+1)π 2
coshγm −cosγm sinhγm −sinγm
Clamped–simply supported (C–S)
H 1 = 1, H 2 = −1, H 3 = 1, H 4 = −1
(4m+1)π 4
coshγm −cosγm sinhγm −sinγm
Simply supported (S–S)
H 1 = 0, H 2 = 0, H 3 = 0, H 4 = −1 mπ
BC
1
Next, substituting Eq. (16) into Eq. (15), and then applying the Galerkin method, one obtained the following equations [20]: (χ11 u0 + χ12 v0 + χ13 w0 + χ14 ϕx + χ15 ϕθ )u0 dxd θ dt = 0 t θ x (χ21 u0 + χ22 v0 + χ23 w0 + χ24 ϕx + χ25 ϕθ )v0 dxd θ dt = 0 t θ x (χ31 u0 + χ32 v0 + χ33 w0 + χ34 ϕx + χ35 ϕθ )w0 dxd θ dt = 0 (19) t θ x (χ41 u0 + χ42 v0 + χ43 w0 + χ44 ϕx + χ45 ϕθ )ϕx dxd θ dt = 0 t θ x (χ51 u0 + χ52 v0 + χ53 w0 + χ54 ϕx + χ55 ϕθ )ϕθ dxd θ dt = 0 t θ x
Integrating and rearranging terms in Eq. (19), the obtained equations can be expressed in the form: (20) [K] − ω2 [M ] {q} = {0} in which {q} = {Umn , Vmn , Wmn , mn , mn }T . The natural frequencies of the shell can be obtained by solving the eigenvalue problem of Eq. (20).
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5 Numerical Results and Discussions In the following examples, the geometric parameters of shell and stiffeners and material properties are listed in Table 2. Table 2. Geometrical and structural properties of stiffened shells Model
R1 R2 (stringers) R3 (rings) (unstiffened)
Number of stiffeners
R4 (stringers) R5 (orthogonal rings/stringers)
60
19
04
13/20
Radius (cm)
100
24.2
49.759
19.45
20.3
Thickness (cm)
5
6E-2
0.165
4.64E-2
0.204
Length (cm)
20E3
60.96
39.45
98.68
81.3
Height of stiffeners (cm)
0.702
0.5334
01.01
0.6/0.6
Width of stiffeners (cm)
0.2554
0.3175
0.104
0.4/0.8
E 0 (N/m2 )
6.9 × 1010
6.9E × 1010 2.00 × 1010
2.07 × 1011
v0
0.3
0.3
0.3
0.3
ρ o (kg/m3 )
2714
2762
7770
7430
Stiffeners type
External
External
Internal
Internal
5.1 Comparison Research The accuracy of the proposed model is validated through numerical examples for stiffened isotropic cylindrical shells, FGP cylindrical shells under some different boundary conditions. The present results are compared with the available data in the literature. 5.1.1 Example 1 Firstly, a simply-supported isotropic CS reinforced by various stiffeners (models R2, R3, R4, and R5) is considered. The comparison presented in Table 3 shows a good agreement between the present results and those of Mustafa and Ali [15]. The results also reveal that the cylindrical shell frequencies with stringers decrease. In contrast, those of the cylindrical shell with rings increase as the circumferential wave number increases, and the frequencies of the cylindrical shell with both stringers and rings first decrease and then increase.
354
T. H. Quoc et al. Table 3. The comparison of the natural frequency f (Hz) of the stiffened ICS
Mode number m
f (Hz) n R2 (Stringers) Ref. [15]
1
1 1141
R3
R4
R5
(Rings)
(Stringers)
(Orth. stiffened)
Present Ref. [15]
Present Ref. [15]
Present Ref. [15] Present
1144
1204
1226
778
778
942
936
2
674
673
1587
1585
317
317
439
437
3
427
425
4462
4396
159
159
337
331
4
296
295
8559
8328
100
102
482
477
5
225
223
13780
13227
92
91
740
738
5.1.2 Example 2 The second example is conducted for the unstiffened ICS under different boundary conditions. The shell parameters are set equal to h/R = 1/100 and L/R = 20. Five first non-dimensional circumferential frequencies ω¯ = ωR 1 − ν 2 ρ/E(m = 1) are calculated and compared with those in Ref. [5]. It can be seen that the results of this study match well with those obtained by Wang and Wu based on the sinusoidal shear deformation theory and the Rayleigh-Ritz method (Table 4). Table 4. Comparison of non-dimensional natural frequency for the unstiffened ICS (m = 1, R/h = 100, L/R = 20) n BC S-S [5]
C-C Present [5]
C-S Present [5]
Present
1 0.0161 0.0160 0.0340 0.0340 0.024830 0.024721 2 0.0094 0.0094 0.0119 0.0119 0.008411 0.008379 3 0.0221 0.0221 0.0072 0.0072 0.005897 0.005884 4 0.0421 0.0420 0.0090 0.0090 0.008717 0.008704 5 0.0680 0.0679 0.0137 0.0138 0.013682 0.013663
5.1.3 Example 3 The next validation is made for an un-stiffened cylindrical shell with √ different porosity coefficients. The dimensionless frequency parameter ωˆ = ωR ρ1 /E1 of simplysupported (S-S) FGP-CS with m = 1 and various n are listed in Table 5. The excellent
Free Vibration of Stiffened FGP Cylindrical Shell
355
agreement between the obtained results by the present approach and Wang and Wu [5] model confirms the accuracy of the current solution. Table 5. Comparison of natural frequencies for S-S unstiffened CS with different PC (m = l, R/h = 500, L/R = 20) e0
Model
n 1
2
3
4
5
6
7
8
0
Wang [5]
1.2429
1.2387
1.2325
1.2256
1.2195
1.2159
1.2165
1.2228
Present
1.2449
1.2407
1.2346
1.2277
1.2217
1.2183
1.2190
1.2256
0.2
Wang [5]
1.2155
1.2118
1.2064
1.2006
1.1958
1.1937
1.1959
1.2038
Present
1.2176
1.2139
1.2086
1.2028
1.1982
1.1962
1.1985
1.2068
0.4
Wang [5]
1.1893
1.1862
1.1818
1.1772
1.1740
1.1736
1.1777
1.1876
Present
1.1914
1.1883
1.1840
1.1796
1.1765
1.1763
1.1806
1.1909
0.6
Wang [5]
1.1677
1.1653
1.1620
1.1590
1.1577
1.1595
1.1660
1.1785
Present
1.1701
1.1677
1.1645
1.1617
1.1605
1.1626
1.1694
1.1823
0.8
Wang [5]
1.1633
1.1617
1.1599
1.1591
1.1601
1.1648
1.1745
1.1906
Present
1.1661
1.1647
1.1630
1.1623
1.1637
1.1689
1.1791
1.1958
5.2 Numerical Investigations In this section, some parametric investigations are conducted to explore the effects of boundary conditions, porosity coefficients, the number of stiffeners, length-to-radius ratio, and thickness-to-radius ratio on the free vibration response of stiffened FGP-CS (R5). 5.2.1 Natural Frequencies of FGP-CS with Several Boundary Conditions (BCs) The effect of BCs on the vibration characteristics of the FGP-CS under three types of BCs is studied here. The boundary conditions including simply supported at both ends (S-S), clamped at both ends (C-C), and the clamped-simply supported at two ends (C-S). Both symmetric and non-symmetric porosity distributions are investigated. In Table 6, the first eight natural frequencies of the orthogonal stiffened FGP-CS under three boundary conditions are calculated and presented. These frequencies are also shown graphically in Fig. 2. The results show that at the low value of n, the influence of boundary conditions is significant. The C-C FGPCS has the highest frequencies, followed by the corresponding C-S and S-S ones. The frequencies decrease when n increases from 0 to 3, and frequencies converge when n > 4. The convergence of the frequencies indicates that the effect of BCs is insignificant for the high circumferential modes of vibration of cylindrical shells.
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Table 6. Natural frequencies f (Hz) of the orthogonal stiffened (external) FGP-CS with for several boundary conditions (e0 = 0.4, m = l, L/R = 0.2, R/h = 100, nr = ns = 10) BCs
n 1
2
3
4
5
6
7
8
Type I S-S
823.399 345.535 268.500 418.564 663.924 971.432 1335.698 1755.440
C-C
1146.592 602.457 400.950 461.161 677.682 976.795 1338.228 1756.872
C-S
1032.318 487.669 331.101 435.650 669.027 973.353 1336.613 1755.990
Type II 1146.560 601.958 397.268 449.946 657.965 947.576 1298.072 1704.232
C-S
1032.302 487.176 326.805 423.873 649.115 944.075 1296.444 1703.356
f (Hz)
823.396 344.981 263.331 406.380 643.917 942.149 1295.551 1702.839
C-C
f (Hz)
S-S
Fig. 2. Natural frequencies of stiffened FGP-CS with S-S, C-C, and C-S boundary conditions
5.2.2 Effect of the Porosity Distribution and Porosity Coefficient The effects of the porosity coefficient e0 and two types of porosity distribution on the natural frequencies of the C-C orthogonal stiffened FGP-CS with different PC (m = l, L/R = 5, h/R = 0.01) are shown in Table 7 and Fig. 3. It is observed that the frequencies of the FPG-CS with symmetric porosity distribution are higher than those of the shell with non-symmetric porosity distribution. The results also indicate that the frequencies of FGP-CS with symmetric porosity distribution decrease at a lower rate than those of FGP-CS with non-symmetric porosity distribution. 5.2.3 Effect of the Stiffener An FGP-CS with various stiffeners is considered in this example. Material properties are set as in the model R5 (m = l, L/R = 5, R/h = 100). Natural frequencies of the unstiffened cylindrical shell and the CS reinforced by ten rings and ten stringers are shown in Fig. 4a for the symmetrical porosity distribution and Fig. 4b for non-symmetrical
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Table 7. Natural frequencies of the C-C orthogonal stiffened FGP-CS with different PC (m = l, L/R = 5, R/h = 100) e0
n 1
2
3
4
5
6
7
8
Type I 0
1237.125 646.994 417.680 453.253 652.461
936.461 1281.801 1682.552
0.2
1193.241 625.301 409.126 456.098 663.138
953.773 1306.095 1714.584
0.4
1146.592 602.457 400.950 461.161 677.682
976.795 1338.228 1756.872
0.6
1098.186 579.154 394.155 470.284 699.004 1009.763 1383.998 1817.003
0.8
1054.077 558.925 392.245 488.954 735.655 1065.101 1460.416 1917.268
Type II 1237.125 646.994 417.680 453.253 652.461
936.461 1281.801 1682.552
0.2
1193.229 625.101 407.651 451.485 654.953
941.619 1289.381 1692.665
0.4
1146.560 601.958 397.268 449.946 657.965
947.576 1298.072 1704.232
0.6
1098.117 578.175 386.945 448.956 661.878
954.868 1308.613 1718.234
0.8
1053.936 557.084 378.745 450.349 669.150
967.004 1325.827 1741.037
f (Hz)
f (Hz)
0
a)
b)
Fig. 3. Variation of fundamental frequencies versus porosity coefficients
distribution. These figures indicate that the frequencies of the shell increase as the number of stiffener increase. However, the increment is not significant when n < 3. It is getting more significant when n > 4. The effect of the width-to-thickness ratio of stiffeners on the stiffened FGP-CS is calculated and presented in Fig. 5. As expected, the natural frequencies of the FGP-CS increase as the width-to-thickness ratio increase.
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Fig. 4. Effect of the number of stiffener on the natural frequencies of the FGP-CS
Fig. 5. Effect of the geometrical parameter of stiffener on the natural frequencies of the FGP-CS
Fig. 6. Effect of the L/R ratio on the natural frequencies of the FGP-CS
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5.2.4 Effect of L/R Ratio The variation of fundamental frequencies versus length-to-radius ratio L/R is given graphically in Fig. 6a and b for type I and type II of the FGP-CS, respectively. The shell properties are set equal to R = 1m, h/R = 0.02. It is observed that the frequencies of the CS decrease as the L/R ratio increases and strongly decrease for L/R ≤ 5. Furthermore, the figures also show that the frequencies of the un-stiffened type I porous cylindrical shell and the stiffened type II porous cylindrical shell converge when L/R > 5 for different porosity coefficients.
6 Conclusion The present study developed an analytical solution for the free vibration analysis of the stiffened FGP-CS. The governing equation is derived by using the FSDT and smeared stiffener technique. The free vibration responses of the stiffened FGP-CS under different boundary conditions are determined by using the Galerkin method. The comparison results validate the accuracy of the present model. The numerical investigation shows that the porosity configuration, porosity coefficients, the number of the stiffener, the width-to-thickness of stiffener significantly affect the vibration behaviour of the FGP-CS. Furthermore, a few conclusion can be withdrawn as: (1) The boundary condition has a remarkable influence on the vibration response of FGP-CS at a low circumferential number. For all types of boundary conditions, the frequency first decreases and then increases as the circumferential number n increases. (2) The stiffness of the FGP-CS decreases as the L/R ratio increases. As shorter cylindrical shell as the fundamental frequency more sensitive to the L/R ratio. (3) The non-symmetric porosity distribution has a more considerable effect on the free vibration behaviour of the FGP-CS than the symmetric porosity distribution. (4) The stiffened FGP cylindrical shell always vibrate with higher frequency than the un-stiffened FGP cylindrical shell. Thus, by proper stiffener arrangement, the stiffness of the cylindrical shell increases significantly while the weight gain is insignificant.
Acknowledgements. This research is funded by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) (Grant No. 107.02-2018.322). The financial support is gratefully acknowledged.
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Appendix A 2 2 A¯ +A¯ ∂2 ; χ = A ¯ 12 ∂ ; χ14 = B¯ 11 ∂ 2 + B¯ 66 ∂ 2 ; χ11 = A¯ 11 ∂ 2 + A¯ 66 2∂ 2 ; χ12 = 12 R 66 ∂x∂θ 13 R∂x ∂x R ∂θ ∂x2 R2 ∂θ 2 ¯B12 +B¯ 66 ∂ 2 χ15 = R ∂x∂θ ; 2 2 A¯ χ21 = χ12 ; χ22 = A¯ 66 ∂ 2 + A¯ 22 2∂ 2 − ks 44 ; χ23 = A¯ 22 + ks A¯ 44 2∂ ; ∂x R ∂θ R2 R ∂θ ∂2 2 2 A¯ ; χ25 = B¯ 66 ∂ 2 + B¯ 22 2∂ 2 + ks R44 ; χ24 = B¯ 12 + B¯ 66 R∂x∂θ
∂x
R ∂θ
χ31 = −χ13 ; χ32 = −χ23 ; 2 2 k A¯ B¯ 12 ∂ B¯ 22 A¯ ∂ ¯ ¯ χ33 = ks A¯ 55 ∂ 2 + s 244 ∂ 2 − 22 2 ; χ34 = ks A55 − R ∂x ; χ35 = ks A44 − R R∂θ ; ∂x
R
∂θ
R
2 2 2 ¯ 11 ∂ − ks A¯ 55 + D ¯ 66 ∂ ; χ45 = D¯ 12 +D¯ 66 ∂ ; χ41 = χ14 ; χ42 = χ24 ; χ43 = −χ34 ; χ44 = D 2 2 2 R ∂x∂θ
∂x
2
R ∂θ
2
¯ 66 ∂ − ks A¯ 44 + D ¯ 22 ∂ χ51 = χ15 ; χ52 = χ25 ; χ53 = −χ35 ; χ54 = −χ45 ; χ55 = D ∂x2 R2 ∂θ 2
References 1. Tran, T.T., Pham, Q.-H., Nguyen-Thoi, T.: Static and free vibration analyses of functionally graded porous variable-thickness plates using an edge-based smoothed finite element method. Defence Technol. (2020) 2. Salehipour, H., Shahsavar, A., Civalek, O.: Free vibration and static deflection analysis of functionally graded and porous micro/nanoshells with clamped and simply supported edges. Compos. Struct. 221, 110842 (2019) 3. Salehipour, H., et al.: Static deflection and free vibration analysis of functionally graded and porous cylindrical micro/nano shells based on the three-dimensional elasticity and modified couple stress theories. Mech. Based Des. Struct. Mach. 2020, 1–22 (2020) 4. Ghadiri, M., SafarPour, H.: Free vibration analysis of size-dependent functionally graded porous cylindrical microshells in thermal environment. J. Therm. Stresses 40(1), 55–71 (2017) 5. Wang, Y., Wu, D.: Free vibration of functionally graded porous cylindrical shell using a sinusoidal shear deformation theory. Aerosp. Sci. Technol. 66, 83–91 (2017) 6. Li, H., et al.: Free vibration characteristics of functionally graded porous spherical shell with general boundary conditions by using first-order shear deformation theory. Thin-Walled Struct. 144, 106331 (2019) 7. Li, H., et al.: Vibration analysis of functionally graded porous cylindrical shell with arbitrary boundary restraints by using a semi analytical method. Compos. B Eng. 164, 249–264 (2019) 8. Zhao, J., et al.: A unified solution for the vibration analysis of functionally graded porous (FGP) shallow shells with general boundary conditions. Compos. B Eng. 156, 406–424 (2019) 9. Zhao, J., et al.: Vibration behavior of the functionally graded porous (FGP) doubly-curved panels and shells of revolution by using a semi-analytical method. Compos. B Eng. 157, 219–238 (2019) 10. Li, Z., et al.: The thermal vibration characteristics of the functionally graded porous stepped cylindrical shell by using characteristic orthogonal polynomials. Int. J. Mech. Sci. 182, 105779 (2020) 11. Bahmyari, E., et al.: Vibration analysis of thin plates resting on Pasternak foundations by element free Galerkin method. Shock. Vib. 20(2), 309–326 (2013) 12. Damnjanovi´c, E., Marjanovi´c, M., Nefovska-Danilovi´c, M.: Free vibration analysis of stiffened and cracked laminated composite plate assemblies using shear-deformable dynamic stiffness elements. Compos. Struct. 180, 723–740 (2017) 13. Duc, N.D.: Nonlinear thermal dynamic analysis of eccentrically stiffened S-FGM circular cylindrical shells surrounded on elastic foundations using the Reddy’s third-order shear deformation shell theory. Eur. J. Mech. A/Solids 58, 10–30 (2016)
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14. Qin, X., et al.: Static and dynamic analyses of isogeometric curvilinearly stiffened plates. Appl. Math. Model. 45, 336–364 (2017) 15. Mustafa, B., Ali, R.: An energy method for free vibration analysis of stiffened circular cylindrical shells. Comput. Struct. 32(2), 355–363 (1989) 16. Lee, Y., Kim, D.: A study on the optimization of the stiffened cylindrical shell. Trans. KSME 13(2), 205–212 (1989) 17. Tu, T.M., Loi, N.V.: Vibration analysis of rotating functionally graded cylindrical shells with orthogonal stiffeners. Latin Am. J. Solids Struct. 13(15), 2952–2969 (2016) 18. Tran, M.-T., et al.: Free vibration of stiffened functionally graded circular cylindrical shell resting on Winkler–Pasternak foundation with different boundary conditions under thermal environment. Acta Mechanica (2020) 19. Reddy, J.N.: Mechanics of Laminated Composite Plates and Shells: Theory and Analysis. CRC Press, Boca Raton (2004) 20. Lam, K., Loy, C.: Influence of boundary conditions for a thin laminated rotating cylindrical shell. Compos. Struct. 41(3–4), 215–228 (1998)
Effects of Design Parameters on Dynamic Performance of a Solenoid Applied for Gas Injector Nguyen Ba Hung1(B) and Ocktaeck Lim2 1 School of Mechanical Engineering, Hanoi University of Science and Technology, Hanoi,
Vietnam [email protected] 2 School of Mechanical Engineering, University of Ulsan, Ulsan, Republic of Korea [email protected]
Abstract. The development of fast response injectors is one of the potential research trends, which plays an important role to improving the performance and reducing exhaust emission of the internal combustion engine. A model-based study is conducted to examine the effects of design parameters on electromagnetic force as well as dynamic performance of a solenoid applied for gaseous fuel injectors. Operation of the solenoid is depicted by mathematical models including a mechanical model and an electrical model. A 2D model of the solenoid is created in Maxwell software to calculate electromagnetic force. This 2D model is created in a symmetric shape to reduce the computational cost. Afterwards, the 2D model of solenoid with the electromagnetic force calculated is imported into a Simplorer software to simulate the dynamic performance. The establishment of the solenoid model in Simplorer is based on the mathematical models presented previously. The shapes of plug and sleeve in the solenoid are changed as input variables to examine their effect on the electromagnetic force and dynamic response of the solenoid. The simulation results show that the change of plug and sleeve shapes can improve the electromagnetic force and minimize dynamic response of the solenoid, which can be a useful contribution for designing a high performance solenoid gas injector. Keywords: Solenoid · Gas injector · Electromagnetic force
1 Introduction Solenoid is an electromechanical device, which consists of an electromagnetic subsystem (coil, magnetic parts) and a mechanical subsystem (spring, plunger). When the coil of the solenoid is supplied an input voltage, an electromagnetic force is generated to attract the plunger. Then the plunger will move to the initial position by an elastic force of the spring as the supply of the input voltage is stopped. One of the applications of solenoid is fuel injector [1, 2]. For fuel injection systems, the solenoid plays an important role in controlling the valve opening time of an injector, as well as deciding the engine © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 362–372, 2022. https://doi.org/10.1007/978-981-16-3239-6_27
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combustion performance [3]. The valve opening time of a solenoid injector is decided by the electromagnetic force, which can be improved by changing designs parameters. Yujeong et al. [3] used Ansys Maxwell software combined with a commercial optimization software PIAnO to simulate electromagnetic dynamics and minimize the response time under the effects of dimension parameters of a direct-acting solenoid valve applied for a fuel pump. A study of a high-speed solenoid valve for a common rail injector [4] showed that the electromagnetic force was affected by drive current and the air gap between the armature and iron core. They showed that, the reduced air gap resulted in increasing the electromagnetic force. Besides, they showed that the increment of the electromagnetic force first increased and then decreased as the drive current increased. Hwang et al. [5] examined the influence of input voltage, winding numbers and wire diameters on the response time of a solenoid applied for diesel injector. Their results showed that the winding number of the wire had significant effect on the response time of the solenoid. Yin and Wu [6] established mathematical models to simulate operating characteristics of a solenoid valve used for the fuel injection system of a gas engine. They investigated the influences of voltage, air gap and number of the coil turns on the open and close response time of the solenoid valve. Their simulation results showed that when the number of coil turns increased, the electromagnetic force increased; however, they also indicated that an increase in coil turns number resulted in an increase in the close response time of the solenoid valve which were similar to simulation results shown in [1, 7]. Liu et al. (2014) analyzed the effects of key factors on the electromagnetic force of a high-speed solenoid valve. Cvetkovic and his research group [8] used a modelling approach to investigate the influence of various plunger pole shapes on the operating performance of a solenoid applied for a fuel injector. Their results showed that the initial size of the fuel injector could be reduced by 35%, the attraction force increased by 26% and the response time reduced by 76% by using the developed approach. Besides the studies presented above, some other studies also concentrated on improving the electromagnetic force as well as dynamic response of the solenoid based on the influence of design parameters such as number of coil turns, pole radius, plunger shape, armature radius [9–13]. Among design parameters, dimensions of plug and sleeve in a solenoid are considered key parameters affecting the electromagnetic force as well as dynamic response of the solenoid, which were rarely mentioned in the previous studies. This paper presents a model-based study to improve the electromagnetic force and dynamic response of a solenoid applied for a gaseous fuel injector. The solenoid consists of main components such as a plunger, a plug, a spring, a coil, a bobbin, a casing and a sleeve inserted between the plunger and the coil. Mathematical models are established to describe operation of the solenoid. A two-dimension (2D) symmetric model of the solenoid is built in Maxwell software to simulate the electromagnetic force under the effects of air gap between the plunger and plug. Then, the 2D model of the solenoid is imported into a Simplorer software to simulate dynamic response. The establishment of the solenoid model in Simplorer is based on the mathematical models presented previously. The electromagnetic force and dynamic response of the solenoid are investigated under the effect of changing the dimension parameters of the plug and sleeve.
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2 Simulation Modeling 2.1 Mathematical Modeling A free body diagram of the solenoid is depicted in Fig. 1. When the coil is provided an input voltage, an electromagnetic force is generated to attract the plunger. The plunger attracted by the electromagnetic force moves up and compress the spring. Then, the plunger will be returned to the initial position by the spring elastic force as the supply of input voltage is stopped. Based on the operating principle of solenoid, mathematical models are established to describe the operation of the solenoid, including mechanical and electrical models. In the mechanical model, the motion of plunger is obeyed the Newton’s second law:
F =m
d 2x dt 2
Fe − Fs − Fd − Fg = m
(1) d 2x dt 2
(2)
where, m is the mass of the plunger, x is the displacement of the plunger, t is the time, F e is the electromagnetic force, F s is the spring force, F d is the damping force and F g is the gravitational force. For the electrical model, the input voltage providing to the coil is defined by: v0 = Ri +
dλ dt
(3)
where vo is the input voltage, R is the resistance in the coil, i is the current, and λ is the total flux linkage of the winding. The λ is also defined by: λ = Nϕ
(4)
where N is the number of coil turns and Φ is the total magnetic flux. By combining Eqs. (3) and (4), the input voltage is rewritten as follow: v0 = Ri + N
dϕ dt
(5)
The λ is also defined as a function of the inductance and the current: λ = L(x)i
(6)
where L(x) is the inductance of the system. Based on Eqs. (3) and (6), the input voltage and current providing to the coil can be derived: v0 = Ri + L(x)
dL(x) dx di +i dt dx dt
(7)
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Fig. 1. Free body diagram of the solenoid
1 di dL(x) dx = v0 − Ri − i dt L(x) dx dt
(8)
The electromagnetic force in Eq. (2) is a variable connecting the mechanical model and the electrical model, which is derived based on the following equations [14]:
i
Wm (i, x) =
i λ(i, x)di =
0
L(x).idi =
1 2 i .L(x) 2
(9)
0
Fe =
1 dL(x) ∂Wm (i, x) = i2 . ∂x 2 dx
(10)
where Wm (i, x) is called the co-energy [14], which is defined as a function of the current in the coil and inductance. The inductance of the system is calculated by the following relationship [15]: L(x) =
N2 n i
(11)
1
where
n
i is the total reluctance of the system.
1
2.2 Solenoid Model in Maxwell In order to reduce the computational cost, a 2D symmetric model of the solenoid is created in Maxwell software shown in Fig. 2a to simulate the electromagnetic force, which
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uses the initial conditions shown in Table 1. The material properties of the components in the solenoid are described in Table 2, which are also utilized as input parameters for the simulation. The initial conditions and material properties are selected based on specifications of a real solenoid gas injector. A boundary condition is applied on the plunger, assigning it as a moving object by effect of the electromagnetic force. The coil is assigned as an excited element when provided by current to generate the electromagnetic force acting on the plunger. In this paper, the effects of plug and sleeve shapes on the electromagnetic force are investigated, which are considered through the variation of their dimension parameters, as shown in Fig. 2b.
Fig. 2. (a) Model of the solenoid injector in Maxwell and (b) design parameters of plug and sleeve
Table 1. Initial conditions of the solenoids Input voltage, V
12
Resistance of the coil,
8.5
Coil turns
412
Max. stroke of the plunger, mm 0.2
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Table 2. Material properties of the solenoid Components Material type Coil
Copper
Bobbin
Polyamide 66
Casing
Stainless steel S430
Sleeve
Stainless steel S410
Plug
Stainless steel S416
Plunger
Stainless steel S416
2.3 Solenoid Model in Simplorer The solenoid operates based on mechanical and electrical models as described in Fig. 1, in which the mechanical model depicts the motion of the plunger under the effects of forces applied on it, while the electrical model describes the input voltage and current providing to the coil to create the electromagnetic force acting on the plunger. Based on the mechanical and electrical models, a model of the solenoid is created in Simplorer, in which this model uses initial conditions shown in Table 1 and the simulated 2D model of the solenoid exported from Maxwell software as input parameters to simulate the dynamic response of the solenoid under the effects of design parameters as mentioned in Fig. 2b. Model of the solenoid in Simplorer is shown in Fig. 3
Fig. 3. Model of the solenoid in Simplorer
3 Simulation Results 3.1 Effects of Plug and Sleeve Shapes on Electromagnetic Force The shape of plug is varied by changing the dimension L1 , as illustrated in Fig. 2b. This implies that the shape of sleeve is also changed according to L1 . Besides, the shape of sleeve is also changed in dimension L2 and L3 , as described in Fig. 2b. Effects of these dimension parameters on the electromagnetic force are shown in Fig. 4. The simulation results show that the electromagnetic force increases when the air gap between the
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plunger and plug is reduced. The reduced air gap leads to increasing the magnetic flux density in the plunger, thus it causes the electromagnetic force to increase. When L1 is increased from 0 mm (original version) to 0.25 mm, the electromagnetic force is increased accordingly, as observed in Fig. 4a. This can be explained by the increased magnetic flux density in the plug as the dimension L1 is increased. The increased speed of electromagnetic force is clearly observed when the air gap is reduced from 0.2 mm to 0.02 mm, as shown in Fig. 4a. When L1 is increased to 0.25 mm, the electromagnetic force obtains the highest value, thus it is selected as an input parameter for studying the effects of L2 on the electromagnetic force. Figure 4b depicts the effects of L2 on the electromagnetic force, in which L2 is varied at 0 mm, 0.1 mm and −0.1 mm. It can be seen that the increase of L2 in the positive direction (L2 = 0.1 mm) results in decreasing the electromagnetic force. Conversely, when L2 is reduced in the negative direction (L2 = −0.1 mm), the electromagnetic force is increased due to the increased magnetic field strength in the air gap. Figure 4c shows the effects of L3 on the electromagnetic force, in which L1 = 0.25 mm and L2 = −0.1 mm are selected as input parameters which resulted in the highest electromagnetic force. It can be found that the increase of L3 in the positive direction (L3 = 2 mm) does not significantly affect the electromagnetic force. On the contrary, the electromagnetic force is significantly increased when L3 is reduced to − 3 mm in the negative direction, as observed in Fig. 4c. The magnetic field strength in the air gap increases when L1 , L2 and L3 are selected at 0.25 mm, −0.1 mm and −3 mm, respectively. This can be seen through a comparison of magnetic field strength between the original and developed versions, as shown in Fig. 5.
Fig. 4. Electromagnetic force under the effects of (a) L1 , (b) L2 and (c) L3
3.2 Dynamic Response Under the Effects of Plug and Sleeve Shapes Effects of dimension parameters L1 , L2 and L3 on the electromagnetic force of the solenoid in transient mode are described in Fig. 6. The simulation results show that the increased speed of the electromagnetic force is faster during 2 ms when L1 is increased from 0 mm to 0.25 mm, as shown in Fig. 6a. This is due to the increased electromagnetic force as presented in Fig. 4a. Similarly, the increase of electromagnetic force by increasing L2 described in Figs. 4b causes the electromagnetic force to increase faster
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Fig. 5. Magnetic field strength of (a) original version and (b) developed version of the solenoid
in the transient mode, as observed in Fig. 6b. For the case of changing L3, the variation of electromagnetic force in the transient mode is not significant as L3 is increased from 0 mm to 2 mm in the positive direction. Conversely, the electromagnetic force starts to show an increased trend when L3 is reduced from 0 mm to −2 mm in the negative direction, which is shown by blue dash line in Fig. 6c. Furthermore, by reducing L3 to −3 mm, the increased speed of the electromagnetic force is significantly improved, as shown by green dash line. This is explained by the increased electromagnetic force in Fig. 4c. As the results of increasing the electromagnetic force, the plunger displacement is increased faster in the transient mode, as shown in Fig. 7. The electromagnetic force increases faster within 2 ms, which causes the plunger to quickly moves to the maximum stroke.
Fig. 6. Effects of (a) L1 , (b) L2 and (c) L3 on electromagnetic force in transient mode
Based on the simulation results obtained in Fig. 7c, the response time of the plunger is derived under the effects of L3 . It can be found that the lowest response time (1.15 ms) is obtained at L3 = −3 mm, as can be seen in Fig. 8. By combining with the dimension parameters optimized previously, a developed version of the solenoid is created to compared with the original version, as shown in Fig. 9.
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Fig. 7. Effects of (a) L1 , (b) L2 and (c) L3 on plunger displacement in transient mode
Fig. 8. Effects of L3 on response time of the plunger
Figure 9 shows comparison results in electromagnetic force and plunger displacement of the original and developed versions of the solenoid, in which the initial conditions, such as resistance of the coil, input voltage and number of the coil, are retained at 8.5 , 12 V and 412, respectively. The simulation results show that the electromagnetic force of the developed version is much faster increased than that of the original version. As the result, the plunger displacement in the developed version is increased faster, which implies that the response time of the developed version is significantly improved when compared with the original version.
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Fig. 9. Comparison results between original version and developed version of the solenoid
4 Conclusions Operation of a solenoid applied for gas injector was described through the mathematical models. The solenoid was modelled and simulated in Maxwell and Simplorer to calculate the electromagnetic force and dynamic response. Effects of design parameters of plug and sleeve, including dimensions L1 , L2 and L3 , on the electromagnetic force and dynamic response of the solenoid were investigated. The simulation results indicated that the electromagnetic force and response time of the solenoid were strongly affected by L1 and L3 , while the effect of L2 was not significant. The dynamic performance of the developed solenoid was significantly improved when compared with the original solenoid, which was obtained at L1 = 0.25 mm, L2 = −0.1 mm and L3 = −3 mm. This study could be a useful reference for designing high-performance solenoid injectors and other applications related to the solenoid. Acknowledgement. This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 107.01-2020.17.
References 1. Hung, N.B., Lim, O.T.: A simulation and experimental study on the operating characteristics of a solenoid gas injector. Adv. Mech. Eng. 11(1), 1–14 (2018)
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2. Hung, N.B., Lim, O.T., Yoon, S.: Effects of structural parameters on operating characteristics of a solenoid injector. Energy Procedia 105, 1771–1775 (2017) 3. Yujeong, S., Seunghwan, L., Changhwan, C., Jinho, K.: Shape optimization to minimize the response time of direct-acting solenoid valve. J. Magn. 20(2), 193–200 (2015) 4. Zhao, J., Fan, L., Liu, P., Grekhov, L., Ma, X., Song, E.: Investigation on electromagnetic models of high-speed solenoid valve for common rail injector. Math. Probl. Eng. 2017, 1–10 (2017) 5. Hwang, J.W., Kal, H.J., Park, J.K., Martychenko, A.A., Chae, J.O.: A study on the design and application of optimized solenoid for diesel unit injector. KSME Int. J. 13, 414–420 (1999) 6. Yin, L., Wu, C.: The characteristic analysis of the electromagnetic valve in opening and closing process for the gas injection system. J. Electromagn. Anal. Appl. 8, 152–159 (2016) 7. De, X., Hong, F., Peng, L., Wei, Z., Yun, F.L.: Electromagnetic force on high-speed solenoid valve based on correlation analysis. Int. J. Smart Sens. Intell. Syst. 8, 2267–2285 (2015) 8. Cvetkovic, D., Cosic, I., Subic, A.: Improved performance of the electromagnetic fuel injector solenoid actuator using a modeling approach. Int. J. Appl. Electromagn. Mech. 27, 251–273 (2008) 9. Liu, P., Fan, L., Hayat, Q., Xu, D., Ma, X., Song, E.: Research on key factors and their interaction effects of electromagnetic force of high-speed solenoid valve. Sci. World J. 2014, 1–13 (2014) 10. Yoon, S., Hur, J., Chun, Y., Hyun, D.: Shape optimization of solenoid actuator using finite element method and numerical optimization technique. IEEE Trans. Magn. 33, 4140–4142 (1997) 11. Subic, A., Cvetkovic, D.: Virtual design and development of compact fast-acting fuel injector solenoid actuator. Int. J. Veh. Des. 46, 309–327 (2008) 12. Grekhov, L., Zhao, J., Ma, X.: Fast-response solenoid actuator computational dimulation for engine fuel systems. In: International Conference on Industrial Engineering, Applications and Manufacturing, Russia (2017) 13. Liu, H., Zhao, S., Wang, J.: Research on the influencing factors of dynamic characteristics of electronic injector. Appl. Mech. Mater. 44–47, 260–264 (2011) 14. Woodson, H.H., Melcher, J.R.: Electromechanical Dynamics. Wiley, New York (1968) 15. Giurgiutiu, V., Lysheyski, S.E.: Micromechatronics: Modeling, Analysis, and Design with MATLAB, 2nd edn. CRC Press, Boca Raton (2009)
Practical Method for Tracking Fatigue Damage Nguyen Hai Son(B) School of Mechanical Engineering, Hanoi University of Science and Technology, Hanoi, Vietnam [email protected]
Abstract. Estimating and tracking dynamics of fatigue damage is essential for fatigue failure prediction. Many recent developments in the area of damage diagnosis are mainly aimed at detecting damage in variable environmental or operating conditions. These methods do not provide a comprehensive framework that can be used to determine the time evolution in damage. In this paper, a reliable and practical methodology for tracking the evolution of the fatigue damage from the measurement of structural vibration is presented. The proposed approach is independent of any particular model of the system and is only based on the variogram of the measured data in conjunction with the smooth orthogonal decomposition. Validation of the method is demonstrated for both synthetic and experimental data. Keywords: Fatigue dynamics · Variogram · Smooth orthogonal decomposition
1 Introduction Fatigue is an omnipresent problem in all oscillating mechanical and structural systems that can lead to catastrophic injuries or loss of life. Fatigue process is driven by the structural dynamics and environmental factors. However, fatigue itself also affects these dynamics by altering structural parameters. The fundamental problem in developing effective damage predictive model is that it is very hard to measure the needed damage variable due to its hidden nature. Therefore, tracking fatigue damage is one of the critical tasks in the field of structural health monitoring (SHM). In general, SHM techniques can be classified into two classes: model-based [1] and data-based approaches [2]. The model-based methods require a priori knowledge of the system’s properties [3]. On the other hand, data-based (or model-free) approaches can be applied to the measured responses of the system without any prior information [4]. Most of the approaches are based on features that are sensitive enough to the damage, for example, the statistics in time or frequency domain [5–7]. Nonlinear time series methods tend to characterize dynamical systems through long-time invariant quantities such as Lyapunov exponents or fractal dimensions. The application of these metrics in SHM is presented in [8, 9]. Other interesting features that describe a change in the geometric properties of an attractor has been employed to identify damage [10, 11]. Although these quantities are able to detect sudden changes in a system caused by damage, they are ill-suited for continuous damage tracking. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 373–381, 2022. https://doi.org/10.1007/978-981-16-3239-6_28
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In [12], a simple nonlinear feature called characteristic distance (CD) was introduced to characterize changes in slow-time variability in the fast-time flow. A one-to-one relationship between CD-based tracking vectors and actual damage states causing these changes has been demonstrated [12]. However, this procedure requires the reconstruction of the phase space [13], which cannot process in real-time and is sensitive to noise. To address this problem, the metric that is fast to calculate and robust to noise, namely the variogram [14] of the structural responses is introduced as the feature vector. This metric is applicable in situations where fast online conputation is needed. In the next section, the variogram of a signal is introduced. The independence of the evolution of the variogram from a Gaussian white noise is demonstrated. Then the variogram coupled with the smooth orthogonal decomposition are used to track fatigue damage variable in both simulations and an experimental system.
2 Description of the Method In material, a fatigue crack usually grows over thousands of load cycles. Hence, the fatigue damage is viewed as a slow-time state variable evolving in a hierarchical dynamical system [17]: x˙ = f(x, t; φ),
φ˙ = g(φ, x),
and
y = h(x),
(1)
where x ∈ Rn is a dynamical state variable, f : Rn × R → Rn is a flow, over-dot indicates differentiation with respect to time t, and φ ∈ R is a hidden slow-time damage variable, which alters the system’s dynamics. 0 < 1 is a small rate constant defining the time scale separation. A time series y is a measurement of the fast time variable. In practical situations, the functions f and g are not provided, we only have access to the scalar time series of measurements {yi }N i=1 . In the following, the variograms of the observed fast time variable y are utilized to characterize the variations of the slow damage variable φ. 2.1 Variogram A variogram is a function describing the spatial continuity of the data. The variogram of a scalar time series {yi }N i=1 is defined [14] as γ (h) =
2 1 E y(i + h) − y(i) , 2
(2)
where E is the expected value function. By the Birkhoff ergodic theorem [15], the variogram of the scalar time series is an invariant measure of the system described by N Eq. (1). In practice, the measurements {yi }N i=1 is corrupted by noise {μi }i=1 . Assume N {μi }i=1 is a Gaussian white noise. Hence, the variogram becomes 2 1 E (y(i + h) + μ(i + h)) − (y(i) − μ(i)) 2 2 1 1 γ (h) = E y(i + h) − y(i) + E μ(i + h)2 + μ(i)2 − 2μ(i + h)μ(i) 2 2
γ (h) =
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+ E (y(i + h) − y(i))(μ(i + h) − μ(i))
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(3)
Since {μi }N i=1 is a Gaussian white noise, E[μ(i)] = 0, therefore γ (h) =
2 1 E y(i + h) − y(i) + E[μ(i)]2 . 2
(4)
Equation (4) shows that all variograms are shifted by the power of the noise. This component can be eliminated by subtracting the variogram with the mean of all variograms. It is assumed that the state variable φ changes slowly. Thus, φ is considered as approximately constant for a fast-time data set collected over an intermediate time interval. The variogram γ (h) is a function of φ. For tracking purposes, the variograms are calculated for each data recorded over intermediate time interval k and are assembled into a feature vector as: vk = [γ (1), γ (2), . . . γ (m)]. Then the feature vectors are constructed into a tracking matrix Y = [y1 ; y2 ; . . . ; yN ]. This matrix Y embeds the changes in the parameters of the fast-time subsystem and can be projected onto actual damage state. In particular, there is a projection q that yields a damage tracking coordinate ϕ = Yq. It is hypothesized that ϕ is a linear combination of components of φ. In practical situations, q and ϕ are not known. However, they can be estimated from Y by the smooth orthogonal decomposition (SOD) [16, 17] technique. 2.2 Smooth Orthogonal Decomposition SOD can be viewed as a generalization of the proper orthogonal decomposition (POD) [18, 19]. In a time-series reduction context, the problem is stated as decomposition of a multivariate time series arranged into a matrix Y ∈ Rm×n - where n states are measured/recorded at m time instances (m > n) into coordinates and modes: Y = QXT =
n
qi xiT ,
(5)
i=1
where the modal coordinates {qi }ni=1 ∈ Rm are the columns of Q ∈ Rm×n , and the corresponding modes {xi }ni=1 ∈ Rn are in X ∈ Rn×n . Before the decomposition, the mean is removed from each column of Y to make results unbiased. In POD, the modal matrix is orthogonal and the coordinates are capturing maximal (in the least squares k qi xiT . It is sense) signal energy in each successive k-dimensional approximation Y ≈ i=1
usually presented as an eigenvalue problem for an auto-covariance matrix Y = 1n YT Y: Y xi = λi xi .
(6)
Since the auto-covariance matrix is symmetric, the resulting eigenvectors are orthonormal. However, if this matrix is ill-conditioned than it is better to solve it by a singular value decomposition: Y = UXT ,
(7)
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where U is an orthogonal matrix and has the same size as Y, ∈ Rn×n is a diagonal matrix with nonnegative diagonal elements in decreasing order called singular values (square roots of eigenvalues), and Q = U. The square of a singular value is called a proper orthogonal value and corresponds to the variance or energy in the corresponding coordinate. Given the fact that proper orthogonal modes are orthonormal, the model reduction does not require pseudo-inverse of the modal subspace P = [x1 , . . . , xk ]: q˙ = PT f(Pq, t; φ).
(8)
The SOD generalizes the POD by adding a constraint on the variances in the velocity ˙ = DY ∈ Rp×n , where D ∈ Rp×m is a temporal discrete field of the original matrix: Y differential operator. This has an effect of imposing smoothness on the time coordinates q, and the corresponding modes are obtained from the generalized eigenvalue problem ˙ T Y: ˙ of the auto-covariance matrices Y and ˙ = 1 Y Y
n
Y zi = λi Y˙ zi ,
(9)
where z are smooth projective modes (SPMs), and the corresponding modal matrix Z = [z1 , z2 , . . . , zn ] is no longer orthogonal. The columns of Q = YZ now contain smooth orthogonal coordinates (SOCs), and smooth orthogonal modes (SOMs) are given by columns of Z−T . Alternatively, a generalized singular value decomposition can be used to obtain SOD: Y = UXT , Y˙ = VXT ,
(10)
where U ∈ Rm×n and V ∈ Rp×n are orthogonal, ∈ Rn×n and ∈ Rn×n are diagonal matrices, and the generalized singular values diag()/diag() are square root of the smooth orthogonal values (SOVs) λi from Eq. (9). The corresponding SOCs are given by Q = U, and the smooth orthogonal modes are X = Z−T . Larger SOVs correspond to smoother SOCs. Since the slow-time state variable is a product of a smooth deterministic process, these variables are expected to be embedded in the smoothest SOCs.
3 Numerical Simulation In the following, we will present an example with the time series derived from a Rössler equation to validate the proposed damage identification algorithm. In the simulations, a damage process is introduced by slowly changing a parameter in the equations x˙ = −y − z y˙ = x + ay z˙ = b + z(x − c),
(11)
where b is fixed at 0,6; c is fixed at 6,0 and a is varied sinusoidally from 0,1 to 0,4. Here, a is considered a damage variable. For each value of a, 100 000 steady-state is generated with time step 0,06. The bifurcation diagram is shown in Fig. 1. Then 300 variograms
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Fig. 1. Bifurcation diagram for Rössler equation
Fig. 2. (a) Parameter a versus time; (b) plot of the dominant SOC1 corresponding to the largest SOV; (c) plot of SOC1 versus parameter a (blue dots) with least square linear fit (red line)
of time series y are estimated. Finally, the 500 × 300 tracking matrix Y is assembled. The tracking result after applying SOD to this matrix is shown in Fig. 2. In Fig. 2c, the blue dots are almost on the red line which is the least square linear fit of the blue dots. This indicates that there is a strong linear relationship between the smoothest SOC and the parameter a. Now, we consider a two-well Duffing equation x¨ + γ x˙ + αx + βx3 = f cos t,
(12)
where the parameters and forcing frequency are fixed to γ = 0, 25; α = −1; β = 1; = 1. f is used as a slow damage variable, is varied sinusoidally from 0,373 to 0,4 (Fig. 3.a). In this range of f , the response of the system is in the chaotic region.
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In the calculations, the first 50 cycles of data are dropped, and 100,000 steady state points are recorded for each forcing amplitude using a sampling time t = π/100. The tracking result for 700 variograms of time series x is shown in Fig. 3 b and c. Again, we obtain a linear relationship between the smoothest SOC1 and the slowl variable f .
Fig. 3. (a) Parameter f versus time; (b) plot of the dominant SOC1 corresponding to the largest SOV; (c) plot of SOC1 versus parameter f (blue dots) with least square linear fit (red line)
4 Experimental Validation
Fig. 4. (a) Photograph of the system; (b) schematic of the apparatus. 1. Shaker; 2. Granite base; 3. Slip table; 4. Back mass; 5. Specimen supports; 6. Pneumatic cylinder supports; 7. Rails; 8. Front cylinder; 9. Front mass; 10. The specimen; 11; Linear bearings for the masses; 12. Central rail for the masses; 13. Linear bearings for the slip table; 14. Back cylinder; 15. Flexible axial coupling.
The experimental system was first introduced in [20]. A schematic of the test rig is shown in Fig. 4. The specimen is a single-edge notched beam, which is simply supported by pins on each end. The masses are kept in contact with the specimen by using two pneumatic cylinders. The main operating principle of the rig is that inertial forces of masses generated by the electromagnetic shaker provide dynamic loads to the specimen. In this experiment, the shaker is driven by random signals.
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Fig. 5. Tracking results for 3 tests (left); The corresponding fatigue damages (right)
The crack growth is monitored by an Alternating Current Potential Difference (ACPD) crack growth monitor. ACPD is a nondestructive testing technique which utilize the skin effect of current flow on the surface of metallic objects. First, a high-frequency AC voltage (20 kHz) is injected into the beam. Then a voltage (potential difference) across the notch is measured. If the crack grows, the current flowing along the beam’s surface will travel further and the potential difference will be greater. The structural response is measured using a single-axis accelerometer mounted to the front mass. All data from the sensors are low-pass filtered with 400 Hz cut off frequency and are recorded at 1 kHz sampling rate. In what follows, the variogram-based damage
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identification method is applied to the time series of the acceleration of the front mass to identify the hidden damage and to track its evolution. The acceleration time series is split into small records. Each record contains 20 000 points (20 s). 160 variograms are calculated for each record. Then the tracking matrix Y is assembled. The tracking results are shown in Fig. 5 for 3 different tests, where the right figures show the ACPD signals and the left figures show the corresponding tracking results. There is a strong linear relationship between the real fatigue crack and the smoothest SOC as seen in Fig. 6 where all the blue dots are close to the red lines.
Fig. 6. Plots of the dominant SOC1 versus actual fatigue damages (blue dots) with least square linear fit (red line)
5 Discussion Rössler oscillator was used to demonstrate how the variograms can identify and track linearly varying parameters that put the oscillator through complicated bifurcations. In Duffing simulation, the tracking result is not as clean as of Rössler simulation. However, we still have the linear relationship between the smoothest SOC and the actual damage variable. Experimental validation of the proposed method showed that the variograms can be used to track and identify slow damage changes in realistic scenarios (Fig. 5). While other metrics (i.e., characteristic distances [12]) can also track these parameter changes, the new metrics required several orders of magnitude fewer calculation time, making them suitable for online applications.
6 Conclusion The variogram originated from linear geostatistics is used for tracking fatigue damage variable in conjunction with the smooth orthogonal decomposition. We have shown that this feature is an invariant measure of a dynamical system. The main advantage of the variograms was that they were fast to calculate and did not require large data or computational resources, making them suitable for online, real-time applications. Numerical simulations and experiments were used to show how the slow fatigue damage variable can be continuously tracked and identified using the proposed metrics. Acknowledgment. This research is funded by the Hanoi University of Science and Technology (HUST) under project number T2018-PC-215.
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References 1. Park, K., Reich, G.W.: Model-based health monitoring of structural systems: progress, potential and challenges. In: Proceedings of the 2nd International Workshop on Structural Health Monitoring, pp. 82–95 (1999) 2. Farrar, C.R., Worden, K.: Structural Health Monitoring: A Machine Learning Perspective. Wiley, Hoboken (2012) 3. Doebling, S.W., Farrar, C.R., Prime, M.B., Shevitz, D.W.: Damage identification and health monitoring of structural and mechanical systems from changes in their vibration characteristics: a literature review. Technical report, Los Alamos National Lab., NM, United States (1996) 4. Doebling, S.W., Farrar, C.R., Prime, M.B., et al.: A summary review of vibration-based damage identification methods. Shock Vibr. Dig. 30(2), 91–105 (1998) 5. Luo, G., Osypiw, D., Irle, M.: Real-time condition monitoring by significant and natural frequencies analysis of vibration signal with wavelet filter and autocorrelation enhancement. J. Sound Vib. 236(3), 413–430 (2000) 6. Sampaio, R., Maia, N., Silva, J.: Damage detection using the frequency-response-function curvature method. J. Sound Vib. 226(5), 1029–1042 (1999) 7. Tian, J., Li, Z., Su, X.: Crack detection in beams by wavelet analysis of transient flexural waves. J. Sound Vib. 261(4), 715–727 (2003) 8. Ghafari, S., Golnaraghi, F., Ismail, F.: Effect of localized faults on chaotic vibration of rolling element bearings. Nonlinear Dyn. 53(4), 287–301 (2008) 9. Wang, W., Chen, J., Wu, X., Wu, Z.: The application of some non-linear methods in rotating machinery fault diagnosis. Mech. Syst. Sig. Process. 15(4), 697–705 (2001) 10. Todd, M., Nichols, J., Pecora, L., Virgin, L.: Vibration-based damage assessment utilizing state space geometry changes: local attractor variance ratio. Smart Mater. Struct. 10(5), 1000 (2001) 11. Hively, L.M., Protopopescu, V.A.: Channel-consistent forewarning of epileptic events from scalpeeg. IEEE Trans. Biomed. Eng. 50(5), 584–593 (2003) 12. Nguyen, S.H., Chelidze, D.: New invariant measures to track slow parameter drifts in fast dynamical systems. Nonlinear Dyn. 79(2), 1207–1216 (2014). https://doi.org/10.1007/s11 071-014-1737-y 13. Kantz, H., Schreiber, T.: Nonlinear Time Series Analysis, vol. 7. Cambridge University Press, Cambridge (2004) 14. Armstrong, M.: Basic linear geostatistics. Springer, Heidelberg (1998). https://doi.org/10. 1007/978-3-642-58727-6 15. Pollicott, M., Yuri, M.: Dynamical Systems and Ergodic Theory, vol. 40. Cambridge University Press, Cambridge (1998) 16. Chelidze, D., Zhou, W.: Smooth orthogonal decomposition-based vibration mode identification. J. Sound Vib. 292(3), 461–473 (2006) 17. Chelidze, D.: Identifying multidimensional damage in a hierarchical dynamical system. Nonlinear Dyn. 37(4), 307–322 (2004) 18. Chatterjee, A.: An introduction to the proper orthogonal decomposition. Curr. Sci. 78(7), 808–817 (2000) 19. Berkooz, G., Holmes, P., Lumley, J.L.: The proper orthogonal decomposition in the analysis of turbulent flows. Annu. Rev. Fluid Mech. 25(1), 539–575 (1993) 20. Falco, M., Liu, M., Chelidze, D.: A new fatigue testing apparatus model and parameter identification. In: International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, vol. 44137, pp. 1007–1012 (2010)
An Assessment for Fatigue Strength of Shaft Parts Manufactured from Two Phase Steel Tang Ha Minh Quan1(B) and Dang Thien Ngon2 1 Faculty of Engineering, Van Lang University, Ho Chi Minh, Vietnam
[email protected]
2 Faculty of Mechanical Engineering, University of Technology and Education, Ho Chi Minh,
Vietnam [email protected]
Abstract. Duplex stainless steel (Duplex) is a new material that has superior properties compared to other stainless steels in terms of avoiding corrosion stress and extremely good yield strength, high flow resistance and cheaper than austenite stainless steel series. However, fatigue strength that always leads to destruction occurs during the working process of axle-shaped machine parts under phase changes. In this paper, the evaluation of fatigue strength of Duplex steel to predict the longevity, maintenance plan for axial-shaped machine parts as well as the advanced mechanical heat treatment mode for Duplex steel are investigated. Samples are made from Duplex SAF 2205 (ISO 1143: 2009) to run fatigue tests by “Staircase” method to determine fatigue strength for Duplex 2205. In addition, this study also provides heat treatment mode for Duplex 2205: heating temperature 950 °C, heat retention time of 15 min and cooling down in the same furnace. The results indicate that supplied-stage Duplex 2205 steel will not be destroyed by fatigue under load under phase changes with 360 MPa stress, which is about 50% higher than AISI 304 (240 MPa) stainless steel. Duplex 2205 steel after heat treatment is enhanced mechanical properties and better durability than Duplex 2205 steel in the supply state. Keywords: Fatigue strength · Duplex stainless steel · DSS 2205 · Duplex heat treatment
1 Introduction Duplex stainless steel is also known as two-phase steel because it has a structure consisting of ferrite phases and austenite phases with approximately equal proportions and intermingling. With such a two-phase construction they will have the best performance combination of the two phases. In the chemical composition of steel contains a high content of chromium (21–23%), it has twice the strength of austenite stainless steel and has a significantly better ductility than ferrite stainless steel. This steel can be mentioned as LDX 2101, SAF 2304, 2205, 253MA [1].
© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 382–392, 2022. https://doi.org/10.1007/978-981-16-3239-6_29
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The axle-shaped machine parts are commonly used in the mechanical industry, it is often subjected to corrosion due to friction and stressed under load. During the working process, there is little downtime so machine parts may be damaged, especially shaft parts…. causing the situation of stopping production to carry out maintenance, repair, and replacement of machine parts. The replacement of shaft parts is inevitable, the shaft replacement process can be time consuming, costly, and reduces the machine’s performance resulting in low productivity. Therefore, we need to research more new materials such as two-phase steel (Duplex), with higher durability and longevity to put into the manufacture of machine parts, to meet the requirements of high-strength operation of machines, increasing stability and longevity of machines are essential in the development of science and technology. Duplex stainless steel solves the above problems, it has good corrosion resistance, high durability, and cheaper price than other common stainless steels. Stainless steel manufacturing technology often involves forming methods such as rolling, forging, casting, and machining. Materials such as Duplex stainless steel will have a different internal microstructure after forming processes due to the different properties of the two phases. Duplex steel is used in the paper industry, automobile, aviation, and ship industries… [2] to improve the durability of shaft-shaped machine parts that operate at high frequency and intensity, easily damaged and destroy. Cast Duplex Steel was first produced in Finland in 1930 and patented in France in 1936 [3]. Duplex stainless steel is gradually being used in Europe to replace traditional steels according to Japanese JIS standards such as: SUS 201, SUS 304, SUS 316… European countries are always at the forefront of research. and development of this steel. There have been studies on fatigue strength when changing the ferrite phase ratio of Duplex steel [4], the results showed that when changing the composition of the ferrite phase, the fatigue strength of steel is affected. To aid in predicting the fatigue life of a component, fatigue tests are carried out using coupons to measure the rate of crack growth by applying constant amplitude cyclic loading and averaging the measured growth of a crack over thousands of cycles. However, there are also a number of special cases that need to be considered where the rate of crack growth obtained from these tests needs adjustment. Such as: the reduced rate of growth that occurs for small loads near the threshold or after the application of an overload; and the increased rate of crack growth associated with short cracks or after the application of an underload [5]. This paper focuses on the fatigue strength of Duplex 2205 steel. In addition, the heat treatment mode to improve the mechanical properties for this steel is also studied.
2 Theoretical Basis 2.1 Fatigue Fatigue is the weakening of a material caused by cyclic loading that results in progressive and localized structural damage and the growth of cracks. Once a fatigue crack has initiated, it will grow a small amount with each loading cycle, typically producing striations on some parts of the fracture surface. The crack will continue to grow until it reaches a critical size, which occurs when the stress intensity factor of the crack
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exceeds the fracture toughness of the material, producing rapid propagation and typically complete fracture of the structure. The fatigue limit of a material (Sr ) under a given condition is the maximum value of the time-varying stress corresponding to a number of fundamental stress cycles without the material being destroyed. Each material has its own number of fundamental stress cycles (N0 ) [6]. The fatigue curve is based on the results of fatigue tests, the fatigue curve is designed to show the relationship between the varying stresses (σ) with the corresponding number of stress cycles (N) (Fig. 1). If the stress is reduced to a certain limit σr for some materials, the life of N can be greatly increased without the part of the machine being destroyed. The value σr is called the fatigue strength (long term) limit of the material.
Fig. 1. Wöhler fatigue curve
2.2 Theoretical Basis of Heat Treatment In theory, the metal is heated at a specified temperature, then the temperature is maintained for a certain period. After that, the metal is cooled down at a predetermined speed to achieve the required texture and properties of the metallic material. (Fig. 2) [7].
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Fig. 2. Characteristic parameters of the heat treatment process
3 Experiment and Results 3.1 Experimental Equipment Fatigue strength tests were performed on the machine of the research group of mechanical engineering and environment (REME Lab) of the Ho Chi Minh City University of Technology Education. Fatigue testing machine as shown in Fig. 3 (Table 1).
Fig. 3. Fatigue testing machine
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Value
Engine power (kW)
2
Spindle speed (rpm)
500–10000
Force exerted through the loadcell (N) 20–2000 Sample size (mm)
12–15
3.2 Recommended Sample Parts The standard specifies the part of a rotating bend fatigue test sample of nominal diameter 5–12,5 (mm) and specifies that the machining process does not residual stress concentration on the test sample. The purpose of fabricating a rotating bend fatigue test sample is to determine the fatigue properties of the material represented on an S-N (stress-cycle) curve. Based on ISO 1143: 1975 and actual fatigue testing machine conditions, sample parts of fatigue strength test proposed as shown in Fig. 5 (Fig. 4).
Fig. 4. Rotating bend fatigue test sample
3.3 Fabrication of Sample Parts The test sample is machined from two-phase stainless steel (Duplex), grade 2205. Chemical composition after being analyzed by Spectro Max spectrometer (Germany) at Nam Long Construction Engineering Co., Ltd is shown in Table 2. The parts are machined simultaneously on the Miyano LK-01 CNC lathe (Fig. 6) with the following cutting modes [7]: – Cutting depth: t = 0,5 mm – Feed rate: s = 0,2 mm/r – Cutting speed: Roughing turning v = 1200 rpm, Finishing turning v = 2000 rpm
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Fig. 5. (a) Machined parts on CNC lathes; (b) Check by clock comparison; (c) Check by panme Table 2. Specifications of fatigue resistance testing machine Steel grade Chemical composition (% mass) 2205
Cr
Ni
Mo
Mn
Si
P
S
C
Fe
21.48
5.54 3.57 1.63 0.529 0.0021 0.0053 0.045 66.5
Fig. 6. Test sample in supply status of steel
3.4 Heat Treatment – Metallographic specimen preparation The parts used for metallographic specimen is a cylinder with the size 12 × 20 (mm), this size is suitable for the actual furnace conditions for easy implementation during the inspection. The microscopic organization test sample in the supply state is checked at two location: RD (Rolling Direction) and TD (Transverse Direction) positions. The purpose of this double-sided inspection is to know what the grain structure of the steel in the supply state looks like, thereby determining the appropriate heat treatment regime to uniformize the grain structure to serve the process of running the steel fatigue test in the heat treatment state (Fig. 7).
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Fig. 7. (a) 3D sample design; (b) Metallographic specimen sample
– Etchant for Duplex steel 2205 Duplex 2205 steel has a high corrosion resistance so the use of etchanting solutions must be appropriate. The composition of the etchanting solution used includes distilled water 100 ml, ethanol 100 ml, HCl 100 ml, CuCl2 5 g, the etchanting time is 5 min [8]. – Metallographic specimen result of Duplex steel 2205 in supply state Duplex 2205 steel in the supply state after etchanting (surface corrosion) was examined for microscopic organization on an optical microscope IMS-300. The results were observed in two cross sections (Transverse Direction - TD) and vertical section (Rolling Direction - RD) as shown in Fig. 8.
Fig. 8. (a) Transverse Direction – TD; (b) Rolling Direction - RD
In the supply state in the cross-section, the particles are round and evenly distributed, in the vertical section, the grain structure is elongated, this affects the machining process as well as the steel’s cyclic load capacity. Therefore, processing the stretched grain structure by heat treatment, making the steel grain becomes more spherical so that the steel can be easily machined, improving the mechanical properties and durability of steel. – Metallographic specimen result of Duplex steel 2205 in heat treatment state
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The sample is subjected to heat treatment in many states, but in this study, the author proposes 3 states of heat treatment as follows: State 1: The sample is heated at 500 °C for 6 h, cooled with air and the microscopic structure of Duplex 2205 steel after heat treatment has the shape and structure as shown in Fig. 9.
Fig. 9. (a) Heat treatment diagram; (b) Microstructure in state 1
In state 1, there was a transformation of the steel grain to the shape, but some particles still did not achieve the desired shape. State 2: the sample is heated at 500 °C for 6 h, cooled with the furnace, the microstructure of Duplex steel 2205 after heat treatment in state 2 has the shape and structure as shown in Fig. 10.
Fig. 10. (a) Heat treatment diagram; (b) Microstructure in state 2
In state 2, the majority number of steel grains had a pronounced change in shape and were more evenly arranged than in state 1. But in this state, the degree of steel recrystallization has not been completely. some large seeds. State 3: the sample is heated at 950 °C for 15 min, cooled with the furnace, the microstructure of Duplex steel 2205 after heat treatment in state 3 has the shape and structure as shown in Fig. 11.
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Fig. 11. (a) Heat treatment diagram; (b) Microstructure in state 3
In state 3, all steel grains have been small, with nearly the same shape and size, evenly distributed in the organization of steel. After performing heat treatment in many states, we see that in state 3, the steel will achieve a stable state after restructuring the grain, small steel grains, more evenly distributed to improve the mechanical properties and durable for steel. From that result, we choose heat treatment mode in state 3 to heat treatment for the parts that needs to fatigue test. 3.5 Fatigue Test Process A machine, which is called the four-point bending fatigue testing machine, was designed to examine the fatigue strength of the specimens (Fig. 3). There are four clamps in the machine to keep rotating bending specimen fixed. While the two external clamps force the specimen’s movement vertically, the expected specimen’s bending was created by the vertical sinusoidal of the two inside clamps. Fatigue was tested by a measured force mode, which was calculated by a load cell. The number of cycles to failure was recorded by an encoder near the motor spindle. Fatigue tests were carried out under rotating bending conditions with a stress ratio R of −1 and a loading frequency of 50 Hz. The fatigue strength is determined when specimen fracture occured or the number of cycles reaches 107 . Fracture of specimens is defined as the increase in vibration amplitude above a preset level. Fatigue test process of Duplex steel 2205 are carried out in two states: supply state and heat treatment state. The parts are destroyed (Fig. 12), this is the weakening of a material caused by cyclic loading that results in progressive and localized structural damage and the growth of cracks. From the above results, we plot the fatigue curve according to the collected data. The fatigue curve is a representation of the relationship between stress and cycle shown in Fig. 13. The fatigue curve of the sample (Fig. 13) shows the cyclic load capacity of Duplex 2205 steel in the supply state and heat treatment state. In the supply state, when at the stress of 360 MPa the parts is not destroyed, reaching the value 107 . Thereby, we find that the Duplex 2205 steel in the supply state has about 50% higher strength than the other common stainless steels (AISI steels 107 at a stress level of 240 MPa). In the state
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Fig. 12. Parts are destroyed
Fig. 13. Wöhler fatigue curve
of heat treatment, the parts reach 107 with a stress level of 400 Mpa, so it has a higher strength than Duplex 2205 steel in the supply state about 11%.
4 Conclusions The results achieved in this study: Proposing sample parts for fatigue testing in accordance with ISO 1143: 1975 and proposed parts of tensile test specimens made in accordance with ISO 6892-1: 2009. The study has performed a mechanical review of Duplex steel 2205 in the heat treatment mode as follows: temperature 950 °C, time 15 min and cooling with the furnace.
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Determining the fatigue strength of shaft parts made from Duplex steel 2205 in the supply state with a value of 360 MPa, the parts reach 107 cycles, about 50% higher than the common stainless steels. Determining the fatigue strength of the shaft parts made from Duplex steel 2205 in the state of heat treatment, with a value of 400 MPa, the parts reach the cycle of 107 . The results show that, through heat treatment will make increased mechanical strength and fatigue strength by about 11% for Duplex 2205 steel, which is also worth considering in the part design process. Acknowledgements. This study is supported by the Research Group of Mechanical and Environmental Engineering (REME Lab) of Ho Chi Minh City University of Technology and Education in Viet Nam.
References 1. Ta´nski, T., Brytan, Z., Labisz, K.: Fatigue behaviour of sintered duplex stainless steel. Procedia Eng. 74, 421–428 (2014). https://doi.org/10.1016/j.proeng.2014.06.293 2. Propelling the boating world. International Molybdenum Association, The voice of the molybdenum industry 3. Dr. Sunil, D.K.: Duplex stainless steels-an overview. Int. J. Eng. Res. Appl. 07(04), 27–36 (2017). https://doi.org/10.9790/9622-0704042736 4. Johansson, J.: Residual Stresses and Fatigue in a Duplex Stainless Steel. Linkoping Studies in Science and Technology, Sweden (1999) 5. Suresh, S.: Fatigue of Materials. Cambridge University Press. ISBN 978–0–521–57046–6 6. Van Quyet, N.: Fatigue Theory Basis. Vietnam Education Publising House Limited Company, Vietnam (2000) 7. Dang, V.N.: Fatigue Theory Basis. Vietnam National University Press, Ho Chi Minh City 8. Co-ordinating working group “Classification Societies – Diesel”. Guidance for evaluation of Fatigue Tests. Conseil International Machines a Combustion (2009)
Advanced Signal Decomposition Methods for Vibration Diagnosis of Rotating Machines: A Case Study at Variable Speed Nguyen Trong Du(B) and Nguyen Phong Dien School of Mechanical Engineering, Hanoi University of Science and Technology, Hanoi, Vietnam [email protected]
Abstract. Effective signal processing methods are essential for machinery fault diagnosis. Most conventional signal processing methods lack adaptability, thus being unable to extract meaningful diagnostic information. Signal decomposition methods have excellent adaptability and high flexibility in describing arbitrary complicated signals. They can extract rich characteristic details and revealing the underlying physical nature. In recent years, several signal decomposition methods are applied to rotating machines operating steady rotating speed. However, it is challenging to use these methods in the case of variable rotation speed. This study proposes a novel signal analysis procedure that combines several advanced signal decomposition methods such as order tracking, spectrum analysis, and Hilbert transform to eliminate the influence of speed changing in vibration signals. It is convenient to create different feature vectors from the measured vibration signal. A classification algorithm is finally applied to detect gear faults automatically. The experimental test rig in the case of rotating machinery having cracked gear demonstrates the usefulness of the proposed signal processing procedure. Keywords: Signal decomposition · Fault diagnosis · EMD · Wavelet packet · Neural networks
1 Introduction The fault diagnosis of rotating machines has attracted increasing interest in the last decades due to their ubiquity and importance in the industry. Recent research trends focus on developing a diagnostic system that works automatically with the application of artificial intelligence. According to this, the diagnostic procedure usually involves the following steps: – Diagnostic data collection, – Diagnostic feature selection to determine condition indicators, – Evaluating the instant values of these indicators from measured diagnostic signals, corresponding to different technical conditions of the diagnostic object © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 393–400, 2022. https://doi.org/10.1007/978-981-16-3239-6_30
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– Implementation of an algorithm for pattern recognition and classification, or a classifier with supervised learning such as artificial neural networks (ANN) [1] and support vector machine (SVM) [2]. Reliable diagnostic systems often require a large enough set of diagnostic data to ensure accurate results. However, a low-cost and applicable diagnostic system should require only a limited amount of diagnostic data while ensuring acceptable diagnostic accuracy. The problem here is to find solutions that reduce the amount of diagnostic data needed for the system. It is believed that the diagnostic feature selection using signal processing methods is very promising to solve the problem. There are many methods for selecting a diagnostic feature, and the signal decomposition method is one of the most appreciated ways. However, most of these methods only give diagnostic features in steady operation condition of the rotating machine. This study evaluates existing signal decomposition methods and proposes a novel method for selecting diagnostic features that can apply for variable rotating speed. This novel method is called advanced signal decomposition. These selected parameters constitute a set of input data for training and implementing a classification algorithm to classify fault in the rotating machines in non-stationary conditions.
2 Advanced Signal Decomposition Methods The raw vibration signals are usually decomposed into different parts using various decomposition methods to extract the diagnostic features. The results of these methods allow a more in-depth insight into the signal components. Thus, it quickly realizes diagnostic symptoms, enhances damage detection, and selects features closely related to different types of damages when failures occur in the considered machine. Signal decomposition methods proposed for fault diagnosis can be divided into three groups: The conventional technique from the first group have been well verified and already in industrial applications such as spectral analysis, time-synchronous averaging, cepstrum analysis, etc. Other methods based on time-frequency transforms are proposed, published, and used for individual tasks, but not in common use, such as wavelet analysis. The last group includes approaches and methods that are relatively new, complicated, and not tested in large scale projects. Besides, machines often operate under fluctuating load conditions and varying rotational speeds (non-stationary conditions). The conventional analysis methods suitable for analyzing stationary vibrations cannot be well applied in such situations. A simple but effective decomposition method is the time-synchronous averaging (TSA) [3–5]. TSA has become a standard technique to detect gear faults in the time domain for many years. Using TSA, a vibration signal measured at a rotating machine can be divided into a periodic part and a non-periodic part, including the random noise components. However, TSA is just appropriate to rotational varying slowly, and it just seems a preprocessing step to reduce measured noise [6]. Wavelet packet decomposition (WPD) is a generalization of the wavelet transform that is firstly introduced by Coifman [7, 8]. In this method, the original signal is decomposed into several signals containing one approximation and some details. It divides the
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time-frequency space into various parts and allows better time-frequency localization of signals versus the wavelet transform. The generalized synchrosqueezing transform (GST) [9, 10] has been developed to overcome the limitation of WPD in analyzing signals containing close frequencies. However, a sophisticated signal processing tool such as GST requires a complex numeric algorithm with a higher computation cost. Cepstrum transform [11] is the inverse Fourier transform of the logarithm of the Fourier transform of the signal. It is readily decomposed into a primary source and noise source. The disadvantages of cepstrum analysis are several Fast Fourier transforms (FFTs) and Inverse Fast Fourier transform (iFFTs are used on each window, which can be computationally expensive [12]. Also, cepstrum analysis essentially low-pass filters the spectrum to get the spectral envelope, which potentially averages some of the spectrum’s spectral peaks. For fault diagnostics, this can be undesirable [13]. Empirical mode decomposition (EMD) is also called Huang trans-form, which can effectively separate frequency components of the signal in the form of intrinsic mode function (IMF) [2, 14] from high frequency to low frequency. And the original signal can be reconstructed by decomposed components. This method, which has been researched widely, is very suitable for nonlinear and non-stationary signal processing. The above methods are all effective with steady rotating speeds. However, there are still many limitations in the case of the variable rotation speed. Therefore, the novel analysis procedure is proposed as follows: Step 1. Taking computed order tracking of the raw vibration signal using a reference signal and a standard interpolation algorithm. After this step signal is called a resampled signal in domain angle to eliminate the effect of various rotating speed. Step 2. Calculation of IMFs of the resampled signal based on the EMD method. Step 3. Using the Hilbert spectrum to look for gear fault symptoms. If there are any symptoms, go to the next step. If not, go back to the previous step. Step 4. Calculation standard deviation of IMFs to generate feature vectors as inputs for neural networks to classify gear fault.
3 Experimental Results This example demonstrates the effects of the proposed method in a gearbox operating at variable rotating speeds. Before selecting diagnostic features, the vibration signal is needed preprocessing by many different techniques. Firstly, resampling the original signal with equal angle increment to eliminate the affection of variable rotating speed. Secondly, the resampled signal is analyzed by the EMD method to generate IMFs mode with the different frequency ranges. Finally, each IMFs signal’s standard deviation is chosen to create a feature vector that is an input parameter for an Artificial neural network. The experimental setup consists of a single-stage gearbox presented in the article [13]; as shown in Fig. 1, this gearbox has a gear ratio of 14/40. The input shaft is driven by a motor controlled by an AC inverter to change the rotating speed from 500 RPM to 1000 RPM. There is a tooth crack on the pinion of the gearbox. The vibration signal is measured by an accelerometer positioned at the closest position to the test gear. An
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Crack Tooth
Fig. 1. Gearbox test rig and pinion with a crack tooth
optical tachometer mounted in proximity to the input shaft is used to generate a onepulse-per-revolution signal called a tacho signal. The vibration signals were sampled at 10 kHz. The following equation determines the meshing frequency: GMF = 14 × f shaft (GMF is gear meshing frequency, and F shaft is input shaft frequency) [15]. Applying conventional methods such as spectral analysis (FFT) to measured vibration signal from this gearbox is given in Fig. 2a. It’s seen that the frequency spectrum is significantly obscured by spectral smearing. Thus, this is a significant problem with conventional analysis methods, and it has less effective with the vibration signal is sampled in the time domain under variable rotating speed conditions.
Fig. 2. The spectrum of the original signal (a) and order spectrum of the resampled signal in the angle domain (b)
The angular resampling technique is applied to the original vibration signal, then FFT analysis for the resampled signal to achieve the order spectrum as Fig. 2b. The order
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spectrum is dominated by the gear meshing frequency and its harmonics (shown 14, 28, 42 order). In the second step, the EMD algorithm is applied to decompose vibration signals into modes. Figure 3 displays the empirical mode decomposition in nine IMFs of the angle domain resampled signal from IMF1 to IMF9 represents the frequency components excited by the gear crack fault, IMF10 is the residual, respectively. Mode IMF1 contains the highest signal frequencies, mode IMF2 the next higher frequency band, and so on.
Fig. 3. IMFs of the resample signal with gear crack fault
The Hilbert-Huang transform is then applied to each of the above IMFs. The Hilbert spectrum is shown in Fig. 4. Many transient vibrations are repetitive with a 2π rad interval, corresponding signal vibration of the cracked gear. This symptom only appears when the signal is resampled in the angle domain. After finding symptoms of cracked gear as mentioned above, the last step of the proposed method, taking the standard deviation of each IMF mode, constitutes feature vectors. Those features are the input data sets for the classification of faults based on the neural network. Classification results between healthy gear and cracked tooth gear using selected features based on the proposed method, as in Fig. 5b. In which the correct classification and testing results are the same 100%. This result is a better classification of fault without it, as in Fig. 5a only with 83% correct classification and 92.3% correct testing. This good result has eliminated the effect of variable rotating speed by order tracking and using EMD to decompose resampled signal to achieve diagnostic features that sensitively responded with gear crack fault.
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2π (1 revolution)
Total revolutions Fig. 4. HHT spectrum of the resampled signal with gear crack fault
Fig. 5. Result of classification based on selected feature vectors by proposed method (b) and without the proposed method (a)
4 Conclusion The application of advanced and sophisticated signal decomposition tools such as the time-frequency analysis using GST or EMD allows us to get more information about the gear fault condition from vibration signals. Also, these decomposition methods
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are applicable for fault detection of machines operating under non-stationary conditions. However, these methods require a more complex numeric algorithm with a higher computation cost. This study proposes a novel signal analysis procedure that combines many signal decomposition methods such as order tracking, spectrum analysis, and Hilbert transform to eliminate the influence of speed changing in vibration signals. It is convenient to create different feature vectors from the measured vibration signal. A classification algorithm is finally applied to detect gear faults automatically. The gear fault classification results in a gearbox operating at a variable rotating speed are also presented using a gearbox test rig. The continued investigation on the application of these decomposition methods is therefore highly recommended. Acknowledgment. This research is funded by the Hanoi University of Science and Technology (HUST) under project number T2018-PC-214.
References 1. Rafiee, J., Arvani, F., Harifi, A., Sadeghi, M.H.: Intelligent condition monitoring of a gearbox using artificial neural network. Mech. Syst. Signal Process. 21, 1746–1754 (2007) 2. Dien, N.P., Du, N.T.: On a diagnostic procedure to automatically classify gear faults using the vibration signal decomposition and support vector machine. In: Fujita, H., Nguyen, D.C., Vu, N.P., Banh, T.L., Puta, H.H. (eds.) ICERA 2018. LNNS, vol. 63, pp. 425–432. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-04792-4_55 3. Bechhoefer, E., Kingsley, M.A.: A review of time synchronous average algorithms. Paper presented at the Annual Conference of the Prognostics and Health Management Society, San Diego, CA (2009) 4. Combet, F., Gelman, L.: An automated methodology for performing time synchronous averaging of a gearbox signal without speed sensor. Mech. Syst. Signal Process. 21, 2590–2606 (2007) 5. Braun, S.: The synchronous (time domain) average revisited. Mech. Syst. Signal Process. 25(4), 1087–1102 (2011) 6. Zhang, G., Isom, J.: Gearbox vibration source separation by integration of time synchronous averaged signals. Paper presented at the Proceedings of the annual conference of the prognostics and health management society, Montreal, QC, Canada 25–29 Sept 2011 7. Coifman, R.R.: Signal Processing and Compression with Wavelet Packets (1990) 8. Feng, Y., Schlindwein, F.S.: Normalized wavelet packets quantifiers for condition monitoring. Mech. Syst. Signal Process. 23, 712–723 (2009) 9. Li, C., Liang, M.: A generalized synchrosqueezing transform for enhancing signal time– frequency representation. Mech. Syst. Signal Process. 92, 2264–2274 (2012) 10. Oberlin, T., Meignen, S., Perrier, V.: The fourier-based synchrosqueezing transform. Paper presented at the 2014 IEEE International Conference on Acoustic, Speech and Signal Processing (ICASSP), France (2014) 11. Randall, R.B.: A history of cepstrum analysis and its application to mechanical problems. Mech. Syst. Signal Process. 97, 3–19 (2016) 12. Hao, T., Xiao-yong, K., Yong-jian, L., Jun-nuo, Z.: Fault diagnosis of gear wearing based on order cepstrum analysis. Appl. Mech. Mater. 543–547, 922–925 (2014)
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13. Dien, N.P., Du, N.T.: Fault detection for rotating machines in non-stationary operations using order tracking and cepstrum. In: Sattler, K.-U., Nguyen, D.C., Vu, N.P., Tien Long, B., Puta, H. (eds.) ICERA 2019. LNNS, vol. 104, pp. 349–356. Springer, Cham (2020). https://doi. org/10.1007/978-3-030-37497-6_41 14. Cheng, J., Yu, D., Tang, J., Yang, Y.: Application of frequency family separation method based upon EMD and local Hilbert energy spectrum method to gear fault diagnosis. Paper presented at the Mechanism and Machine Theory vol. 43 (2007) 15. Meltzer, G., Dien, N.P.: Fault diagnosis in gears operating under non-stationary rotational speed using polar wavelet amplitude maps. Mech. Syst. Signal Process. 18(5), 985–992 (2004)
Nonlinear Effects of the Material in the Pier Dam Experiment Under Cyclic Loading N. Van Xuan1(B) , N. Canh Thai2 , N. Ngoc Thang2 , and T. Van Toan2 1 Directorate of Water Resources, 2 Ngoc Ha, Ba Dinh, Hanoi, Vietnam 2 Thuyloi University, 175 Tay Son, Dong Da, Hanoi, Vietnam
{ncanhthai,nnthang}@tlu.edu.vn, [email protected]
Abstract. In recent years, the pillar dam is a built efficient water regulation facility in Vietnam. For the pillar dam using the radial gate, all the load and the impact from the radial gate to the pier trunnion are concentrated in the area of the mandrel wheel. During the operation and operation, the pier trunnion area has many potential dangers if not calculated properly. For the types of specialized structures, as the structure just stated, simulation work is most tested and is the basis for the numerical model [1–3]. In experiments, depending on each stage of loading, stress-strain values have differences compared with theoretical calculations [4–6]. In the scope of the paper, the authors will analyze the effect of the choice of concrete grades on the material nonlinearity in this area, the effect of the dynamic load with the fluctuation amplitude of the wave cycle, thereby determining the changes in stress versus strain allowed of the material on the basic cross-section. Keywords: Monotonic loading · Cyclic loading · Nonlinear material · Dynamic · Pillar dam
1 Introduction The impact of the circulating load has a great influence on the waterworks in general and the Pillar dam in particular. Due to the influence of wave vibrations, the pressure creates stress-strain relationships that change at the position of the load. With a Pillar dam using the Radial gate, all wave pressure is applied to the valve and transmitted to the pier trunnion. The change in the amplitude of the wave oscillation, characterized by the Jonswap wave spectrum, produces a cyclic load, affecting the strength at this site. With the cylindrical structure of the pier trunnion of the radial gate, the design and operation process has many potential risks if not paid adequate attention. The 1986 Dau Tieng hydroelectric pier trunnion accident has left many harmful consequences as proof [7]. Designing the structure of the reinforced concrete cellar pillar structure following TCVN 10400:2015, TCVN 8218:2009 standards often use the material to work on the linear elastic phase, not considering the nonlinear stress-strain relationship the actual material [8, 9].
© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 401–415, 2022. https://doi.org/10.1007/978-981-16-3239-6_31
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Some constructions in the 50s of the 20th century, such as the Folsom dam, USA, had a breakdown in 1995, due to the error of not considering the effect of friction on the stress distribution in the ear region, pin valve, which has a nonlinear effect on geometry and material, leads to poor design stress distribution, bending stresses in the compression member area of the valve gate system [10]. In the world, the problem of structural analysis in the nonlinear behavior has been studied and included in the standards in the form of additional formulas for determining internal force, such as EN 1992 1-1 of the European Union, JSCE No15 of Japan, ACI 318-95 of the United States,… but just stop at some basic forms [4, 11, 12, 13, 14]. Therefore, empirical research is very necessary. The effect of the cyclic oscillation of the wave creates a varying pressure on the radial gate. Depending on the geographical region, the Jonswap wave spectrum has different values, with different cycles and amplitudes. In Vietnam, wave fluctuations are usually measured by either the pressure method or the resistance method, see Fig. 1 and 2. The wave model in the study applies to the pillar dam. Pillar dams are built a lot in the estuary area, in addition to keeping fresh water, it also has a very important task to limit storm surge, rising water,… Therefore, the influence of waves, especially waves in the bays and estuaries, is very important.
Fig. 1. Wave spectrum at KimSon estuary, Ninh Binh, Vietnam
Fig. 2. Wave spectrum at Danang bay, Vietnam
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There are two solutions to design the trunnion area of the Pillar dam using the radial gate. The first design option is to select the reinforcement of the circular arch, the trunnion girder is located deep in the supporting pillar body. This is a commonly used solution in Vietnam today, as the newly built Van Phong dam also uses this solution (Fig. 3). The second design option is to select the reinforcement placed parallel, the trunnion girder is usually located on the outer edge of the concrete pillar body (Fig. 4).
r ei n f o r c emen t st eel
t r u n n io n g i r d er
pu t r ei n f o r c ed st eel i n c i r c u l a r sh a pe
a) Common steel laying
b) VanPhong Dam, BinhDinh, Vietnam
Fig. 3. Typical design in Vietnam
t r u n n i o n g i r d er
Ten d o n Gr o u ps
pu t r ei n f o r c i n g st eel i n pa r a l l el
a) Common steel laying
b) John Redmond Dam, Texas, US
Fig. 4. The trunnion is at the boundary of pier (EM 1110-2-2702, 2000)
In this paper, the effects of the trunnion girder design according to the first option are studied and discussed. The compressive stress of material is considered to be the main property suitable for concrete materials. Conducting studies of reinforced concrete structures under different load conditions and repeated cycles is often done by experimental methods. Feedback results are analyzed, compared to draw problems to overcome. At the same time, the results are also used as a basis for the correction of numerical models. For each type of specialized structure, a lot of research has been done, Mirmiran and Shahawy studied the behavior of specimens subjected to the static effects of compressive loads, and in terms of the load loop cycle – unloading [15]. Chen et al., studied the behavior of the reinforced concrete structure’s different joints [16]. Experimental results pointed out that the connective joints have the ability to efficiently dissipate energy. Tong et al., have studied the behavior of H-shaped reinforced concrete steel beams [17]. Chen et al., have studied with many samples to give different solutions for the combination of steel and concrete.
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Recently, Lam Thanh Quang Khai in his thesis, also analyzed the steel-fiber reinforced concrete dome structure [18]. Experimental research and analysis on numerical model linear and nonlinear values in static load. In this study, the experiment was performed on a cylindrical-dam body model using a bow valve. The results are evaluated based on analyzing the relationship of linear and nonlinear load-displacement, stress-strain. The test also considers that the difference in the load cycle affects the stress-strain distribution.
2 Experimental Investigation 2.1 Material Materials used are concrete M200, M300, M400 designed according to the natural conditions in Vietnam. Material quality meets TCVN 8219: 2009, ASTM C39. Concrete using PC40 cement. The composition of materials for 1 m3 of concrete is shown in Table 1. A comparison between grades of concrete according to Vietnamese and EU standards is shown in Table 2. Table 1. Material distribution norms for 1 m3 of concrete No.
Material
Unit
Concrete grade 200
300
400
1
Cement PC 40
kg
296
394
470
2
Yellow sand
m3
0.489
0.447
0.427
3
Crushed stone (0.5 × 1)cm
m3
0.888
0.87
0.86
4
Water
litre
195
195
186
5
Chemical additives
Chemical
Table 2. Converted concrete grade according to EU standards Durability level EC-2
Compressive strength level
Concrete grade
B15
M200
90
B20
M250
110
B22.5
M300
130
C20/25
B25
M350
155
150
C25/30
B30
M400
170
180
C16/20
Computational strength
Computational strength EC-2 120
Concrete was cast with 6 samples and maintained at its design strength after 28 days of age. The stress-deformation relationship of concrete is shown in Fig. 5. The physical and mechanical properties of the tested concrete sample are given in Table 3. Experimental cube samples and destructive material samples are illustrated in Fig. 6.
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Fig. 5. Stress - strain curve of the concrete sample to be tested
Fig. 6. Damage model of the experimental concrete
Table 3. Properties of the materials used in the specimens Concrete grade
Rn kN/m2
Rk kN/m2
E kN/m2
1c
1cu
f cm,tt kN/m2
f cm, LT kN/m2
200
9120
740
2.41E + 07
0.0018
0.0035
17000
20000
300
13680
1030
2.91E + 08
0.0019
0.0035
21500
30000
400
17630
1210
3.28E + 07
0.0022
0.0035
25500
40000
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2.2 Test Specimens The experimental Specimens are designed in a steel frame system, performed at the Structural mechanics and strength of materials Laboratory in Thuyloi University, Vietnam. In collaboration with The National Key Laboratory of River and Coastal Engineering (KLORCE), the address in the University area. Test and measurement equipment: using strain gauge, conductor, bonding glue, electronic displacement LVDT, load cell, data logger, of TML (Tokyo Measuring Instruments Laboratory Co., Ltd), Japan. Experimental models are performed according to the instructions of national standards of Vietnam [19–21]. The frame system used this test is designed with full capacity of up to 400 kN by the hydraulic jack. Loads can be applied in terms of force/time or displacement/time. Figure 7 shows the steel frame and test specimen in the laboratory. The design of the position of the steel frame and the load cell, hydraulic jack is shown in Fig. 8.
Fig. 7. Framework of the tests run in the laboratory
The research model is a hydraulic construction model built on the basis of the crosssection of the pier dam using the arch gate. The purpose of the study is load interaction and the impact on the structure. The wave load is converted to the basic form for analysis as the cyclic loading acts directly on the mandrel wheel. Designing cross-section model with scale of 1:35, similar to main cylindrical model of Van Phong dam, Phuoc Hoa dam built. The selection of the test Pillar dam size is based on a representative basic cross-section dimension of a Pillar dam using a radial gate. The goal is to analyze the difference in stress values when calculated analytically, linearly, and the actual results are nonlinear, experimentally. The load exerted at the valve ear position, simulating the working of the gate valve to beat the pillar. The designs comply with TCVN 10400: 2015, EM 1110-2-2702. The steel frame is designed to load the specimen in the direction of the effect of the valve gate pressure on the ear of the battery cylinder. The loading is performed in increments of increments and is carried out in different repetition cycles. Figure 10 shows the attachment of the sensors to the surface of a strain gauge test piece under load.
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Fig. 8. Loading framework
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Fig. 9. Loading - unloading step
Fig. 10. Paste the Strain Gauge sensor
Measure displacement by electronic device CDP-25 (accuracy: 0.002 × 10–6 mm), mechanical displacement meter (Indicator, accuracy: 0.001 mm); Deformation measurement with resistance sensor (Strain Gauge, Rosette paste); The devices are connected to the data logger and the computer records the load parameters in each step. 2.3 Test Set-Up The experimental model is built based on a sample segment of a miniature pillar dam with the size 800 × 600 × 80 mm, linking the hard clamp to the ground, and bearing concentrated load in the direction of transmission from the Radial arch valve. In the test of the equal load on both sides of the valve, the stages are performed in sequence as follows, see Fig. 9: A. The load increases gradually from 0–5 kN. Check the operation of load systems, sensors, gauges. Eliminate peripheral effects such as tilting,… B. Unloading gradually decreases to 0. Return to the initial state. C. The load increases gradually from 0–8 kN. Get experimental data, check calculation results. D. Unloading gradually decreases to 0. Return to the initial state. E. The load increases gradually to 10 kN, the load increases gradually until the sample fails. Monitor residual deformation, check destructive phenomena
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In the 1st load test procedure, the cyclic loading is simple. Exact determine the value of stress and strain corresponding to each load level, from there are experimental data compared with the linear calculation results. Specimens were tested under different kinds of cyclic loading to investigate the hysteretic behavior and constitutive characteristics. The cyclic load forms are simulating some basic wave phases, usually occurring in the waters of Vietnam. In Fig. 11, the load period is to simulate the wave action acting on the radial gate of the Pillar dam.
a ) load cycle type 1
b ) load cycle type 2
Fig. 11. Cyclic load patterns
3 Compare the Experimental Results with the Results Obtained from the Numerical Model 3.1 3D FEM Element Modeling The purpose of this paper is actual experimental results, numerical results are for reference only. Linear values can be calculated using algebraic formulas, by mechanics of continuous medium [22]. So these details are a brief introduction. The paper is done based on analysis by the finite element method (FEM), with Abaqus software. The model uses Concrete Damaged Plasticity (CDP). Destruction mechanism of the model that are based on tensile cracking and compression crushing. In the CDP model, the damage law is based on the formula of Lubliner and Lee/Fenves, which is the basis for the Abaqus plastic destruction model for concrete [23, 24]. The mentioned yield surface revised by Lee and Fenves (1998) to take into account the varying tensile and compressive strength of concrete. The present yield surface is determined with equation: ∼pl ∼pl ∼pl 1 (1) q − 3αp + β ε σmax − γ −σmax − σc εc F σ, ε = 1−α
Where p is the effective hydrostatic pressure stress and q is the Mises equivalent effective stress. These two parameters are stress invariants of the effective stress tensor which are used by the yield surface and plastic flow potential function [23], σmax is the algebraically
∼pl
maximum eigenvalue of the deviatory part of effective stress tensor (σ ), ε is equivalent plastic strains. The parameters α, β and γ are dimensionless material constants.
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The model is made with dimensions similar to the actual dam design. Concrete uses C3D8R element, which is a 3-dimensional, 8-node solid element. Steel uses T3D2 element, embedded in concrete with the assumption that adhesion between reinforcement and concrete is absolute. The displacement of the x, y, z directions of the bottom is zero. The input parameters and the stress measurement point are performed similarly to the experimental model. The element size is 4 mm at the main Pillar dam area and 1 mm around the pier trunnion area. 3.2 Visco-plastic Regularization on the CDP The process of adjusting viscosity-plasticity to the CDP model parameters is done through the continuous correction of finite elements that are closely observed in reality. The viscosity parameter was adjusted according to the behavior of the concrete material, and refer to Lee [25]. Viscosity parameters are also adjusted for non-linearity for each grade of concrete, see Wosatko et al., Weiren et al., [26, 27]. 3.3 Isotropic Hardening Isotropic hardening describes the variation of the size of yield surface σ0 , which is a function of equivalent plastic strain εp , and is shown in the following equation: σ0 = σ|0 + Q∝ (1 − e−biso ε ) p
(2)
where σ |0 is the yield stress at zero plastic strain and Q∞ and b are material parameters. Q∞ is the maximum change in the size of the yield surface, and b defines the rate at which the size of the yield surface changes as plastic straining develops. When the equivalent stress defining the size of the yield surface remains constant (σ 0 = σ |0 ), the model reduces to a nonlinear kinematic hardening model.
4 Results and Discussion 4.1 Failure Characteristics In the test on both sides of the valve, the phases are performed sequentially according to the load levels, gradually increasing until the sample is damaged. In all cases, cracks appear oblique as shown in Fig. 12. Alternating are small cracks, which begin to open when the load exceeds 6 kN. This phenomenon is relatively consistent with the process of pulling and breaking of the material. The area around the valve ear separates the tension and the compression zone, the process of deformation increases gradually, appears cracks. The forcedisplacement relationship in the load test 1 at procedure is illustrated in Fig. 13 and 14. In terms of linear behavior analysis, solving the problems can be done by analytic formulas in material strength, and continuous environmental mechanics. However, in practice, the linear relationship between stress and strain accounts for only a fraction of the entire working time of the material (linear elastic phase). In the nonlinear analysis
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Fig. 12. Failure patterns of the specimens and FEM result
problem, the dangerous stress value is different from the linear analysis problem, and this value reflects more accurately the actual work of the project. The force-displacement relationship under cyclic loads is depicted in Fig. 15. The peak point of the graph in the helices is similar to that of the graph under normal loads. With the wave spectrum tending to grow, the graph is broken into many short, continuous foot beats. With an equal wave spectrum, simulating transverse waves, the plot is broken into multiple long legs and tends to be relatively more evenly interrupte.
Fig. 13. Force-displacement relation at point I
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Fig. 14. Force-displacement relation at point II
Fig. 15. Compare the relationship between cyclic load types
4.2 Measurement Data Table 4, 5 and 6 are the results of experimental measurements, with comparison of calculated results from numerical models, where: – Phase 1: Elastic deformation phase – Phase 2: Stress starts to cause cracking – TT: linear result; Pt: numerical result, nonlinearity; TN: experimental results
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Phase 1 - σ Principal σ TT
σ Pt
σ TN
kN
kN/m2
kN/m2
kN/m2
%
I
1.602
3.881
4.051
4.08
4.88
II
2.628
70.508
67.493
66.8
−5.55
III
9.007
365.408
351.336
348.6
−4.82
IV
9.984
263.286
258.75
258.12
−2
Measuring point
Load
Phase 2 - σ Principal σ TT
σ Pt
σ TN
kN
kN/m2
kN/m2
kN/m2
%
I
2.045
4.955
6.294
6.2
20.08
II
2.935
78.745
65.913
64.7
−21.71
Measuring point
III
Does not form phase 2
IV
Does not form phase 2
Different rates
Different rates
Table 5. Measurement results ε, U Measuring point
Load
Phase 1 - ε εTT
Different rates εPt
εTN
kN
%
I
1.6
4.3762E-06
4.374E-06
0.00000459
4.66
II
2.63
5.0118E-06
4.64E-06
0.00000452
−10.88
Measuring point
Load
Phase 2 - ε εTT
Different rates εPt
εTN
kN I
2.05
% 5.607E-06
5.595E-06
0.00000588
4.64
4.53E-06
0.00000442
−26.75
II
2.94
5.6025E-06
Measuring point
Load
Phase 1 - U U TT
Different rates U Pt
U TN
kN
mm
mm
mm
%
I
1.6
0.00205763
0.002058
0.00216
4.74
II
2.63
0.00267112
0.0026545
0.00252
−5.997
Measuring point
Load
Phase 2 - U U Pt
U TN
U TT
Different rates
kN
mm
mm
mm
%
I
2.05
0.002636
0.0026401
0.00277
4.83
II
2.94
0.00298597
0.0029298
0.00273
−9.38
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Table 6. Measurement results at point I with grades of concrete Load
Phase 1 - σ Principal σ TT
σ Pt
σ TN
kN
kN/m2
kN/m2
kN/m2
%
M200
2.5
6.057
7.52
7.896
23.29
M300
2.5
6.057
7.106
7.461
18.82
M400
2.5
6.057
6.793
7.133
15.08
Concrete grade
Different rates
5 Conclusions When the concrete works in the elastic stage, the difference between the linearly calculated value and the experiment is not large, ≤6%, depending on the position near or far from the applied point. When the concrete works in the next stage, and when cracking begins to appear, there is a big change in the geometrical size of the cylinder, the difference between linear and experimental values is ~22%. This result is similar to the study of L.T.Q.Khai when studying the roof of concrete cover [21, 28–31]. The higher the grade of concrete, the lower the difference between the linear and experimental values. The cracks begin to crack at the end of the transition, and gradually expand to complete failure as the load continues. This stage of work while maintaining this partial crack is relatively common in old buildings that need upgrading today. The article has pointed out the need for experimentation, noting nonlinear analysis for new structural design, and upgrading and repairing old works effectively.
References 1. Xuan, N.V., Thai, N.C., Thang, N.N.: Anh huong cua mac be tong den tinh phi tuyen vat - tru do. - Hoi nghi Khoa hoc Thuy loi toan quoc; Ha noi, pp. 16–18 (2017). lieu tai tai van dap . ISBN 978-604-82-2273-4. (N.V. Xuan, N.C. Thai, N.N. Thang. Influence of concrete grade on material nonlinearity at the pier trunnion of Pillar dam. National irrigation science conference; Ha noi, pp. 16–18 (2017). ISBN 978-604-82-2273-4) 2. Sakai, J., Kawashima, K.: Unloading and reloading stress–strain model for confined concrete. J. Struct. Eng. 132(1), 112–122 (2006) 3. MacGregor, J.G., Wight, J.K.: Reinforced Concrete: Mechanics. Prentice-Hall, Hoboken (2005) 4. Ferguson, H.A., Blokland, P., et al.: The Haringvliet sluices. The Hague: RIJKSWATERSTAAT Communications (1970) 5. US Army Corps of Engineers. EM 1110-2-1604 Hydraulic Design of Navigation Locks. Washington, DC (2006) 6. The U.S Army Corps of Engineers. Ohio River & Tributaries Navigation System Five Year Development Plan. FY11-FY15 (2010)
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7. Phan, S.K.: Su co mot so cong trinh thuy loi tai Viet nam va cac bien phap phong tranh. Ha noi: Nha xuat ban nong nghiep; 2000 Phan Sy Ky. Breakdown of some water works in Vietnam and preventive measures. Agricultural Publishing Company, Hanoi (2000) 8. TCVN 10400:2015 Cong trinh thuy lo.,i-Dap tru do-Yeu cau ve thiet ke. Ha Noi: Bo Khoa ho.c va Cong nghe; 2015 (TCVN 10400:2015 Hydraulic Structures–Pillar Dam–Technical requirement for Design. Hanoi: Ministry of science and technology; 2015) 9. TCVN8218:2009. Tieu chuan quoc gia. Be tong thuy cong. Yeu cau ky thuat. Ha Noi: Bo Khoa ho.c va Cong nghe; 2009 (TCVN8218:2009. Hydraulic concrete–Technical requirements. Hanoi: Ministry of science and technology; 2009) 10. Ishii, N., Anami, K., Knisely, C.W.: Retrospective consideration of a plausible vibration mechanism for the failure of the folsom dam tainter gate. Int. J. Mech. Eng. Robot. Res. 3(4), 314–345 (2014) 11. EM 1110-2-2702. Engineering and Design: DESIGN OF SPILLWAY TAINTER GATES. U.S. Army Corps of Engineers, Washington, DC (2000) 12. EN 1992 1-1. Design of concrete structures. European Standard (2004) 13. Japan Society of Civil Engineers. JSCE No.15 Standard specifications for concrete structures. Tokyo (2007) 14. American Concrete Institute. ACI 318-95 Building Code Requirements for Structural Concrete (1995) 15. Mirmiran, A., Shahawy, M.: Behavior of concrete columns confined by fiber composites. J. Struct. Eng. 123, 583–590 (1997) 16. Chen, C.C., Lin, K.T.: Behavior and strength of steel reinforced concrete beam–column joints with two-side force inputs. J. Constr. Steel Res. 65, 641–649 (2009) 17. Tong, L., Liu, B., Xian, Q., Zhao, X.-L.: Experimental study on fatigue behavior of Steel Reinforced Concrete (SRC) beams. Eng. Struct. 123, 247–262 (2016) 18. Lam, T.Q.K.: Nghien cuu trang thai ung suat bien dang cua mai vo thoai be tong cot thep cong hai chieu duong nhieu lop. Chuyen nganh: Ky thuat xay dung cong trinh dan dung va cong nghiep. DH Kien truc Ha Noi ed.: Luan an Tien si ky thuat. 2019, p. 137–138. (L.T.Q.Khai. Stress strain study of multilayer positive two-dimensional curved reinforced concrete shell roof. Specialized: Civil construction and industry. Hanoi University of Architecture, pp. 137– 138. Ph.D. thesis (2019) 19. TCVN 8214:2009. Hydraulics physical model test of water headworks. Ministry of Science and Technology, Ha Noi (2009) 20. TCVN 12196:2018. Hydraulics structures-Physical model test of rivers. Ministry of Science and Technology, Ha Noi (2018) 21. Thang, N.N., Van Thuan, B., Khai, L.Q., Chuyen, N.V.: Lectures subjects experimental works. Thuyloi University, Ha Noi (2012) 22. Thu, D.V., Oanh, N.N.: Co hoc moi truong lien tuc. Hà nô.i: Nha xuat ban tu dien bach khoa. D. V. Thu, N. N. Oanh. Mechanics of continuous medium. The Publisher of the Encyclopedia, Hanoi (2007) 23. Lubliner, J., Oliver, J., Oller, S., Oñate, E.: A plastic-damage model for concrete. Solids Struct. 25(3), 299–326 (1989) 24. Lee, J., Fenves, G.L.: Plastic-damage model for cyclic loading of concrete structures. Eng. Mech. 124(8), 892–900 (1998) 25. Lee, J.: Theory and implementation of plastic-damage model for concrete structures under Cyclic and Dynamic Loading. Ph.D. thesis (in English). University of California, Berkeley, USA (1996) 26. Wosatko, A., Pamin, J., Polak, M.A.: Application of damage-plasticity models in finite element analysis of punching shear. Comput. Struct. 151, 73–85 (2015)
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27. Ren, W., Sneed, L.H., Yang, Y., He, R.: Numerical simulation of prestressed precast concrete bridge deck panels using damage plasticity model. Int. J. Concr. Struct. Mater. 9(1), 45–54 (2015) 28. Sarti, F., et al.: Development and testing of an alternative dissipative posttensioned rocking timber wall with boundary columns. J. Struct. Eng. 142(4), E4015011 (2016) 29. Kankam, C.K., Meisuh, B.K., Sossou, G., Buabin, T.K.: Stress-strain characteristics of concrete containing quarry rock dust as partial replacement of sand. Case Stud. Constr. Mater. 7, 66–72 (2017). https://doi.org/10.1016/j.cscm.2017.06.004 30. Abdulrezzak, B.A.K.I.S, ¸ Mesut, Ö.Z.D.E.M.˙IR., Ercan, I.SI.K., ¸ EL Alev, A.K.I.L.L.I.: The impact of concrete strength on the structure performance under repeated loads. Bitlis Eren Univ. J. Sci. Technol. 6(2), 87–87 (2016) 31. Dere, Y., Koroglu, M.A.: Nonlinear FE modeling of reinforced concrete. Int. J. Struct. Civil Eng. Res. 6(1), 71–74 (2017)
Experimental and Numerical Investigations on the Fracture Response of Tubular T-joints Under Dynamic Mass Impact Quang Thang Do1(B) , Sang-Rai Cho2 , and Van Dinh Nguyen3 1 Department of Naval Architecture, Nha Trang University, Nha Trang, Vietnam
[email protected]
2 School of Naval Architecture and Ocean Engineering, University of Ulsan, and UlsanLab Inc.,
Ulsan 44610, Korea [email protected] 3 Department of Mechatronics, Nha Trang University, Nha Trang, Vietnam [email protected]
Abstract. Tubular member structures are the major component structures of jacket of fixed floating offshore platforms, tension leg platforms (TLPs), or drilling jack-up rigs. During its operations, these structures are constantly serviced by support vessels. Collisions between them are unavoidable. One key considers during the design and safety of these structures is to make sure that they have good safety in the scenario of ship collisions. This research presents a series of fracture tests and numerical study results of T-joints tubular structures subjected to dynamic mass impact. The collision scenarios considered in this study were the collisions between T-joints tubular members and support vessels or floating objects. Eight T-joints of H-shaped tubular members were fabricated with different dimensions and tested under dynamic mass impact by applying the drop test machine. The detailed explanations of test setups and test results are reported. Finite element analyses (FEA) of the collision behaviors of the experimental models were performed using the ABAQUS software. For describing the fracture of T-joint tubular members, the Hosford-Coulomb fracture model was applied. A good agreement between the test results and numerical results was obtained. Furthermore, two ductile fracture modes of the T-joints tubular structures are also discussed. Keywords: Tubular T-joint · Impact test · Numerical simulation · Fracture response
1 Introduction Tubular member structures are widely used in FPSOs (Floating Production Storage and Offloading), jackets/legs of fixed offshore platforms and bracings of floating offshore installations. The main benefits of tubular structures are good resistance for axial compression/tension loadings and small drag force of passing fluid. Moreover, tubular member structures are also beneficial for fabrication and construction. During its operations, these structures are constantly serviced by support vessels. Collisions between © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 416–430, 2022. https://doi.org/10.1007/978-981-16-3239-6_32
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them are unavoidable. The collision between tubular members and support vessels is the most important reason for damaged offshore structures. It can lead to serious consequences for economic losses and even human life or loss of structural strength [1]. Thus, the reparations of these structures are very difficult and time-consuming [2]. Nowadays, collision accidents between ship and offshore tubular members are the key consideration for designing tubular structural performance and safety. Therefore, assessing the effect of collision damages on these structure’s strength is important for designing procedures. Several researchers have studied experiments on the collision of tubular structures. Furnes and Amdahl [3], Søreide et al. [4], Ellinas and Walker [5] reported the experimental investigations on quasi-static impact responses of tubular. Cho [2] presented the dynamic impact tests on small-scale single tubes. Jones et al. [6] performed quasi-static and dynamic impact experiments on fully fixed boundary conditions for mild steel pipes. Pendersen et al. [7] performed on the estimation of collision force between merchant vessels’ bow and gravity-supported offshore installations. Amdahl and Johansen [8] provided the collision of the ship’s bow and rigid-jacket tubular, while Amdahl and Eberg [9] studied a rigid ship’s bow collision with deformable tubular structures. Recently, Cerik et al. [10] and Cho et al. [11] presented extensive test data of the dynamic impact tests on H-shape tubular members for damage assessment at T-joint of bracing and chord. The impact locations were performed at mid-span and 200 mm away from mid-span. More recently, Do et al. [12–19] reported the test and numerical results on ring- and stringer-stiffened cylinder subjected to dynamic mass impact. Then, these models were tested under hydrostatic pressure for assessing their residual strength. It is noted that most of tests and numerical results have been reported in the literature, which was only considered the plastic deformation domain. In an actual case, with the large impact velocities, the fracture will occur. However, until now, there is a lack of test results for the fracture tubular members subjected to dynamic mass impact. Thus, it is necessary to provide several test results of this loading. This paper may useful for researchers to understand deeply the fracture behavior of T-joins tubular members under the dynamic mass impact. Nowadays, Recently, NFEM (Nonlinear Finite Element Methods) is a powerful tool to forecast the residual strength of offshore tubular structures during ship collisions. Because NFEM is convenient to use and the cheapest way to perform the full dimensions of actual ocean engineering structures [19, 21]. Thus, NFEM is the best-chosen way for assessing the collision responses of tubular structures during ship collision accidents. However, one of the main challenges when modeling the collisions between tubular structures and supporting vessels is the ductile fracture criterion to forecast the fracture propagation in the ductile material. Until recently, these problems are not yet answered accuracy and it is still an intensive research topic. When performing NFEM of collision, it is necessary to input a suitable fracture criterion because it is strongly affected by the membrane stretching, crushing, and tearing of shells [20]. These progressive failures would be influenced by global structural collapse modes [21]. In this paper, the HosfordCoulomb ductile fracture material models provided by Cerik et al. [22–25] were applied. The Hosford-Coulomb ductile fracture model has recently become quite popular because it was the ease of calibration, simplicity, flexibility, and accuracy.
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On this background, the objective of this study is to provide the test results of fracture T-joins tubular members under dynamic mass impact. Then, the numerical simulation was performed by using a Hosford-Coulomb ductile fracture model. Finally, the deformed shape and failure modes, and stress states obtained for various models were also discussed.
2 Fracture Tests 2.1 Dimensions and Material Properties of the Test Models The models are H-shape tubulars which consisted one brace and two chords joined to the brace ends by welding. In actual offshore tubular structures, the ranges of D/t ratio are generally used from 20 to 60. In this study, the dimensions of test models are also chosen based on this range of D/t ratio. There were eight H-shape tubular models which were fabricated. The dimensions of chords were the same for all models but different dimensions for braces. The experimental models were classified as E, G and H series following their bracing dimensions. The detail dimensions of each model are indicated in Table 1. It is noted that the design dimensions and fabrication procedures of test models were following the design code of API [26] and ABS [27]. In this table, L, D and t are the total length, outside diameter and thickness of tubular, respectively. It is noted that the thicknesses of chord and brace were carefully measured at several points on each chord and brace using ultrasonic measuring instruments and determined to be an average value, which is commonly less than nominally value. The material of test models was fabricated from the general-purpose structural steel. The material properties were achieved by carry out the quasi-static tensile tests. The average value of quasi-static tensile test results is summarized in Table 2.
Table 1. Measured scantlings of experimental models (unit: mm) Model Dimension of chord
Dimension of bracing tc
Lb
Db
L c /Dc Dc /t c L b /Db Db /t b
Lc
Dc
tb
E3
1300
114 4.05 886
89.1 2.10 11.40
28.15
9.94
42.43
G1
1300
114 6.02 866
76.0 1.79 11.40
18.94 11.39
42.46
G2
1300
114 6.02 866
76.0 1.80 11.40
18.94 11.39
42.22
G3
1300
114 6.00 866
76.0 1.80 11.40
19.00 11.39
42.22
G4
1300
114 6.05 866
76.0 1.80 11.40
18.84 11.39
42.22
G6
1300
114 6.05 866
76.0 1.79 11.40
18.84 11.39
42.46
H3
1300
114 6.04 886
90.0 2.08 11.40
18.87
9.84
43.27
H4
1300
114 6.07 886
90.0 2.08 11.40
18.78
9.84
43.27
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Table 2. Material properties for each series of experimental models. Model series
E-series G-series H-series
Yield strength, σY (MPa)
360.3
319.7
317.3
Ultimate tensile strength, σT (MPa) 413.5
418.7
391.1
Young’s modulus, E (GPa)
207
206
206
Hardening start strain, εHS
0.0279
0.0186
0.0245
Ultimate tensile strain, εT
0.1447
0.1723
0.1743
2.2 Experimental Set-Up The experiments were performed using a drop test machine which was successful in earlier dynamic mass impact tests [10–19]. The details of the experimental arrangements are indicated in Fig. 1. On the top of the drop test machine, there is a pulley which guides and keeps the striking mass by an electromagnet. The striking mass was attached to a rigid knife-edge striker. During the drop test, drop height can be controlled for obtaining the expected collision energy level. When electromagnet is disconnected, the striking mass falls freely due to its gravity. In this study, the different impact energy levels and striking mass types were applied for each model. The shapes and mass arrangement of strikers were designed to minimize the imbalances in free-fall process. Furthermore, to investigate the fracture response at T-joint of tubular, the impact locations were selected around 200 mm aside from the mid-length of bracing. Therefore, the shape of the striker was also required that it cannot touch the chord of tubular during the test. The radius value of knife edge header is 10 mm for the striker type I and II. The dimensions of striker and its indenter header shape are depicted in Fig. 2. The details of collision test conditions are shown in Table 3. For the experimental boundary conditions, the model was firmly clamped and inserted rubber pads grip at both ends of the chord using the support structures, as shown in Fig. 1. The support structures of clamps were bolted to a rigid foundation of the loading frames. It is guaranteed that the chords are fully fixed and not allowed any translations and rotations of both ends of chord. In addition, 2D strain gauge was attached in eight positions at both T-joint regions and impact areas. Among them, four strain gauges were attached in outer of the shell at the collision area, and the next four strain gauges were placed at both T-joint of the tubular model. The impact location was moved with various distance values away from the midlength of bracing members to investigate the behavior of T-joint. During the impact test, when the striking mass hit the brace of model, the rebounded phenomenon was commonly occurred. Therefore, for protecting the model from unexpected deformation after the striking mass rebounded, the rubber pads were used for covering all surfaces of model except impact region as indicated in Fig. 1.
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2.3 Test Results In this section, the results of the series dynamic mass impact tests were presented. The aim of these series test models was investigated the damages of H-shape tubular member under dynamic mass impact from initial fracture to complete fracture at T-joint. It means that the impact energies were increased gradually until complete fracture occur by changing the striking masses or drop highs. Therefore, the deformations of the tested models were classified into two groups: partial fracture and complete fracture damages.
Fig. 1. Dynamic mass impact test setup
Fig. 2. Dimensions of striker mass: (a) Type I; (b) Type II
When stiffness of bracing member is quite smaller compared to that of the chord members, the plastic deformation occurs only on bracing. In these cases, if the collision energy level was increased, the fracture will take account on bracing at T-joint location which near the impact line. Furthermore, if the length of bracing is short, the deformation angle α as in Fig. 3 is increased. It means that the overall bending damage on bracing causes to the increase of the rotational deformation angle on bracing member. It is a significant reason leading to the value of shear strain to be increased. When this shear value reached to the shear failure criterion of material properties, fracture damage was taken a place at T-joint location, as shown in Fig. 4. When the brace ruptures at the T-joint, it will not absorb the collision energy anymore. Owing to overall bending, high tension forces at the welded joints can reason to fracture on the bracing. It is noted that the large plastic deformation of bracings which were absorbed the impact energy, leading to decrease the damages on chord. It is improved the safety of entire structures. The partial fracture damages at T-joint locations are depicted in Figs. 5 and 6 for models E3, G6, H3 and H4, respectively. For all models, the fracture was occurred near the welded joint of bracing and chord. After obtained the partial fracture damages, the collision energies were increased to achieve the complete fracture. Figure 7 is illustrated
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Table 3. Impact test conditions for tubular members Model
Velocity (m/s)
Striker mass (Kg)
Kinetic energy (J)
Striker type
Impact location
E3
5.25
460
6339
I
200 mm aside from mid-length
G1
5.13
673
8856
II
200 mm aside from mid-length
G2
4.93
673
8179
II
200 mm aside from mid-length
G3
4.52
673
6875
II
200 mm aside from mid-length
G4
3.81
673
4885
II
200 mm aside from mid-length
G6
2.58
673
2240
II
200 mm aside from mid-length
H3
2.94
673
2909
II
200 mm aside from mid-length
H4
2.39
673
1922
II
200 mm aside from mid-length
the complete fracture damage of model G4 before releasing boundary condition. The complete fracture damages and each cross-section of models G1 and G2 are described in Fig. 8, respectively. It is noted that cross-section at fracture lines were not smooth and it has some tearing parts. The rotation of striker when it was released from the electromagnet, was one of the most important reason for tearing parts of the cross-section. The summary of experimental results is shown in Table 4. In this table, the measured permanent extents of damages including the dent depth (d d ) and overall damages (d 0 ) were presented. The failure mode of each model is also provided in this table.
Fig. 3. Idealised profiles of a damage on both of bracing and chord tubular
d0 = sin(α + β)
a2 + b2
tan α =
a b
(1) (2)
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tan β =
c d
(3)
Fig. 4. Idealised stress distribution in T-joint of tubular member after denting test
Fig. 5. Partial fracture damage of models: (a) E3; (b) G6
3 Numerical Analysis of Fracture Tests 3.1 Finite Element Modelling It is confirmed that the proposed numerical simulation method has been verified and assessed the accuracy by comparing it with available test data, which could be found in references [12–18]. Thus, in the current paper has only presented the summary of NFEA (Nonlinear Finite Element Analyses) for fracture tubular members. NFEA was carried out by applying the ABAQUS explicit method. The dynamic/explicit solution was used for collision analysis. The shell element S4R was used for all structural modeling. For the striker, R3D4 element type with the rigid body was applied. The general contact with penalty method was defined for the contact between striker mass outside surface and bracing of tubular. The friction coefficient at contact region was assumed with 0.3. Before performing the numerical simulations on test model, the convergence tests were checked for selecting the optimum mesh size. The mesh size of the contact area was 5 × 5 mm, while that for the outside of the collision region was 10 × 10 mm. The selected mesh size is suitable to capture the deformed shape and fracture behavior accurately. For the boundary conditions, the ends of two chords of the tubular member
Experimental and Numerical Investigations on the Fracture Response
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were fixed with all degrees by using the high tensile steel bolt as the test setups. The boundary conditions and finite element modelling were described in Fig. 9. Furthermore, the model was generally vibrated during the collision process. Thus, the Rayleigh damping was applied for reducing these vibrations and quickly obtain a static equilibrium state. Rayleigh damping equation was presented as Eq. (4): C = αM + βK
(4)
where M and K are the mass matrix and stiffness matrix, respectively. α is a coefficient, which can be calculated as the lowest natural frequency of model. The lowest natural frequency can be obtained by the first mode of a Subspace Eigen solver in ABAQUS. β is a coefficient of the stiffness proportional damping factor.
Fig. 6. Partial fracture damage of models: (a) H3; (b) H4
Fig. 7. Complete fracture damage of model G4 before releasing boundary condition
3.2 Material Definition For impact simulation, the equations suggested by Do et al. [12] were applied for determining the material definition. These equations were developed based on a large database of dynamic tensile test results of various marine steels. The true stresses and strains were determined as Eqs. (5) and (6) using the static tensile test results in Table 2. When considering the effects of the strain hardening and yield plateau, the equations from (7) to
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(11) were used. when considering the strain-rate hardening effects, the revised equations reported by Do et al. [12] were used. The dynamic yield strength (σ YD ), dynamic ultimate tensile strength (σ TD ), dynamic hardening start strain (εHSD ) and dynamic ultimate tensile strain (εTD ), which are calculated relation to the strain rate ε˙ as shown in Eqs. (12)–(15). In this paper, the strain rate was created with different values (10 s−1 , 20 s−1 , 50 s−1 , 70 s−1 , 100 s−1 and 150 s−1 ).
Fig. 8. Complete fracture damage of models: (a) G1; (b) G2
Table 4. Results of measured damage for each test models Model
Bracing
Damage mode
dd
do
E3
59.90
64.50
Partial fracture
G1, G2, G3, G4
–
–
Complete fracture
G6
50.0
45.90
Partial fracture
H3
56.0
33.20
Partial fracture
H4
49.50
23.20
Partial fracture
Fig. 9. Finite element model for collision analysis
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σtr = σ (ε + 1)
(5)
εtr = ln(ε + 1)
(6)
σtr = Eεtr when 0 < εtr ≤ εY ,tr
(7)
εtr − εY ,tr σtr = σY ,tr + σHS,tr − σY ,tr when εY ,tr < εtr ≤ εHS,tr εHS,tr − εY ,tr
(8)
σtr = σHS,tr + K(εtr − εHS,tr )n when εHS,tr < εtr
(9)
where n=
σT ,tr εT ,tr − εHS,tr σT ,tr − σHS,tr σT ,tr − σHS,tr K= n εT ,tr − εHS,tr
εTD εT
σYD = 1 + 0.3(E/1000σY )0.5 (˙ε )0.25 σY 0.35 σTD = 1 + 0.16(σT /σYD )3.325 (˙ε )1/15 σYD εHSD = 1 + 0.1(E/1000σY )1.73 (˙ε )0.33 εHSS = 1 − 0.117 (E/1000σT )2.352 (σT /σY )0.588 (˙ε )0.2
(10) (11) (12) (13) (14) (15)
It is noted that there is fracture occurred in the impact test. Therefore, when generating the true stress-plastic strain curves which were extended after the initiation of necking. For describing the fracture on T-joint tubular members, the Hosford-Coulomb fracture model was applied. The Hosford-Coulomb fracture model defines the ductility limit of structures with different stress states. In particular, this model can be understood as a mesh-independent. In this model, the initial crack was occurred by the onset of shear localization. The fracture stress criterion can be described as the Hosford equivalent stress σ¯ HF , and the normal stress reaches the maximum shear stress at critical value b. The detail of Hosford-Coulomb ductile fracture model formulations was presented as Eqs. (16) to (23). σ¯ HF + c(σ1 + σ3 ) = b σ¯ HF =
(1/a) 1 a a a (σ1 − σ2 ) + (σ1 − σ3 ) + (σ2 − σ3 ) 2
(16) (17)
Equation (17) can be rewritten into cylindrical coordinates as below:
b σ¯ f η, θ¯ = 1 a a a (1/a) + c(2η + f1 + f3 ) 2 (f1 − f2 ) + (f1 − f3 ) + (f2 − f3 )
(18)
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where 2 π 1 − θ¯ f1 θ¯ = cos 3 6 2 π f2 θ¯ = cos 3 + θ¯ 3 6
2 π f3 θ¯ = − cos 1 + θ¯ 3 6
(19) (20) (21)
The mixed stress-strain can be expressed by applying the inverse of isotropic powerlaw hardening as follows: ε¯ f
1
η, θ¯ = b(1 + c) nf g η, θ¯
pr
(22)
where
g η, θ¯ =
1/nf
(1/a) 1 a a a + c(2η + f1 + f3 (f1 − f2 ) + (f1 − f3 ) + (f2 − f3 ) 2 (23)
3.3 Numerical Results Figures 10 and 11 show the comparison between numerical deformed shape of models E3 and G4 with experimental results. The damage on model E3 is a partial fracture while that of model G4 is a complete fracture, respectively. The deformed shapes observed in tests are captured well by numerical analysis. As mention in the previous section, when the collision energy level was significantly increased, the fracture will take account on bracing at the T-joint location which near the impact line. The energy dissipation of axially restrained tubular is limited by fracture at joints due to excessive membrane straining. It is a typical example of tensile fracture at joint. In the numerical model, the shear fracture criteria were used as fracture criteria. When this shear value reached to the shear failure criterion of material properties, fracture damage occurred at T-joint location. The comparisons of test results and numerical results are summarized in Table 5. The mean of the uncertainty modelling factor X m is 1.027 for local shell denting and 1.059 for overall bending damage, respectively. The predicted local shell denting and overall bending damage are commonly larger than those of test results. It should be concluded that the numerical results are quite accuracy and reliable when compared to test results. Furthermore, it is noted that the complete fracture result is not included in this table because when complete a fracture occurs the bracing tube is released and it is not contributed to the whole structure’s strength.
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Fig. 10. Comparison between numerical result and experimental result for model E3
Fig. 11. Comparison between numerical result and experimental for model G4
4 Disscussions 4.1 Collision Tests A series of dynamic impact tests included four partial fracture models and four complete fracture models, were successfully provided in this research. The detailed explanations for test setup, test procedure and test results were also presented in this study. The main goal of these tests was to investigate the damage responses at T-joint tubular members. In general, the behavior of tubular members under dynamic mass impact is very complicated because the combination between local shell denting and the overall bending damage. Thus, it is difficult to propose to classify all deformation modes of these structures. The failure modes of experimental results are depicted in Fig. 12. Where the energy parameter λE is defined as the ratio of kinetic energy of striker (E k ) to energy absorption capacity of struck structure (E a ), as defined in Eq. (24). The kinetic energy of the striking mass and the static energy absorption capacity is described in Eq. (25) and (26), respectively. Where σ Y and σ T are yield strength and ultimate tensile strength,
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Q. T. Do et al. Table 5. Comparison of test results and numerical results
Model
Db (mm)
Experiment results
Numerical results
Bias (FEA/Test), X m
dd/Db (1)
d0/Db (2)
dd/Db (3)
d0/Db (4)
(3)/(1)
(4)/(2)
E3
89
0.67
0.72
0.71
0.70
1.060
0.972
G6
76
0.66
0.60
0.71
0.64
1.076
1.067
H3
90
0.62
0.37
0.57
0.40
0.919
1.081
H4
90
0.55
0.26
0.58
0.29
1.055
1.115
1.027
1.059
Mean
respectively. εT and V str are the ultimate tensile strain and structure volume, respectively. In this figure, the failure mode of tubular member was divided into two groups: Mode I-Partial fracture at T-joint and Mode II- Complete fracture at T-joint. When the ratio of energy (λE ) was small, the partial fracture was occurred. And when the ratio of energy (λE ) was then increased further, the complete fracture at T-joint was occurred. It means that the bracing tube is released and it is not contributed to the whole structure’s strength. Ek Ea
(24)
1 2 mv ; Kinetic energy 2
(25)
σY + σT εT Vstr ; Strain energy absorption capacity 2
(26)
λE = Ek = Ea =
Fig. 12. Plotting of failure mode of test models against basic parameters Ek /Ea and Lb /Db ,and Db /tb
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4.2 Numerical Simulations Based on the numerical results, it is concluded that a quite good accuracy was observed between the numerical results and test results. The mean of X m (the uncertainty modelling factor) is 1.027 and 1.059 for local shell denting and overall bending damage, respectively. The predicted local shell denting and overall bending damage are commonly larger than those of test results. The reason can be related to the sensitive of local damage due to the rotation angle and indenter direction during the collision tests. Another reason can be explained by the uncertainty of the dynamic material modelling, where the strain rate effects have the important role. Additionally, the Hosford-Coulomb ductile fracture model which applied in numerical analysis are reasonable and accuracy for predicting the fracture responses of tubular T-joint members under dynamic mass impact. Therefore, the suggested NFEM can be applied to perform further case studies for developing the design equations.
5 Conclusions The main goal of the present study was to provide the series of experimental and numerical results of H-shape tubular member under dynamic mass impact. These results were shown insights understanding of the fracture behavior at T-joint of tubular member during a ship collision. The numerical simulations have a good agreement with the experimental results. The predicted local shell denting and overall bending damage are commonly larger than those of test results. The rotation angle during the free fall of striking mass and the uncertainty of the dynamic material modelling were the main reason. The failure modes of tubular member under dynamic mass impact were very complex owning to the combination of local shell denting and the overall bending damage. It is not straightforward to suggest the simple parameter for classifying all deformation modes and responses of tubular members. Acknowledgement. This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 107.01-2019.333.
References 1. DNV. Accident statistics for fixed offshore units on the UK Continental Shelf 1980–2005: Research Report RR566. HSE Books (2007) 2. Cho, S.R.: Design approximations for offshore tubulars against collisions. Ph.D. thesis, Faculty of Engineering. University of Glasgow, Glasgow, UK (1987) 3. Furnes, O., Amdahl, J.: Ship collisions with offshore platforms, pp. 310–318. Intermaritec, Hamburg, Germany (1980) 4. Søreide, T.H., Moan, T., Amdahl, J., Taby, J.: Analysis of ship/platform impacts. In: 3rd International Conference on Behaviour of Offshore Structures, Boston, USA, pp. 257–278 (1982) 5. Ellinas, C.P., Walker, A.C.: Damage on tubular bracing members. In: Proceedings of IABSE Colloquium on Ship Collision with Bridges and Offshore Structures, Copenhagen, Denmark, pp. 253–261 (1983)
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6. Jones, N., Birch, S.E., Birch, R.S., Zhu, L., Brown, M.: An experimental study on the lateral impact of fully clamped mild steel pipes. Proc. Int. Mech. Eng. Part E J. Process. Mech. Eng. 206, 111–127 (1992) 7. Pedersen, P.T., Valsgard, S., Olsen, D.: Ship impact: bow collisions. Int. J. Impact Eng. 13, 163–187 (1993) 8. Amdahl, J., Johansen, A.: High-energy ship collision with jacket legs. In: Proceedings of the International Offshore and Polar Engineering Conference (2001) 9. Amdahl, J., Eberg, E.: Ship collision with offshore structure. In: Moan (ed.) Structural Dynamics – EURODYN 1993, Balkema, Rotterdam (1993) 10. Cerik, B.C., Shin, H.K., Cho, S.R.: A comparative study on damage assessment of tubular members subjected to mass impact. Mar. Struct. 46, 1–29 (2016) 11. Cho, S.R., Le, D.N.C., Jeong, J.H., Frieze, P.A., Shin, H.K.: Development of simple designoriented procedure for predicting the collision damage of FPSO caisson protection structures. Ocean Eng. 142, 458–469 (2017) 12. Do, Q.T., Muttaqie, T., Shin, H.K., Cho, S.R.: Dynamic lateral mass impact on steel stringerstiffened cylinders. Int. J. Impact Eng. 116, 105–126 (2018) 13. Cho, S.R., Do, Q.T., Shin, H.K.: Residual strength of damaged ring-stiffened cylinders subjected to external hydrostatic pressure. Mar. Struct. 56, 186–205 (2017) 14. Do, Q.T., Muttaqie, T., Park, S.H., Shin, H.K., Cho, S.R.: Predicting the collision damage of steel ring-stiffened cylinders and their residual strength under hydrostatic pressure. Ocean Eng. 169, 326–343 (2018) 15. Do, Q.T., Muttaqie, T., Park, S.H., Shin, H.K., Cho, S.R.: Ultimate strength of intact and dented steel stringer-stiffened cylinders under hydrostatic pressure. Thin-Walled Struct. 132, 442–460 (2018) 16. Do, Q.T., Le, D.L.C., Seo, B.S., Shin, H.K., Cho, S.R.: Fracture response of tubular T-joints under dynamic mass impact. In: ICCGS-2019, Lisbon, Portugal, pp. 75–84 (2019) 17. Do, Q.T., Huynh, V.V., Vu, M.T., Tuyen, V.V., Pham, T.N.: A new formulation for predicting the collision damage of steel stiffened cylinders subjected to dynamic lateral mass impact. Appl. Sci. 10, 3856 (2020) 18. Do, Q.T., Huynh, V.N., Tran, D.T.: Numerical studies on residual strength of dented tension leg platforms under compressive load. J. Sci. Technol. Civil Eng. 14, 96–109 (2020) 19. Do, Q.T.: Deriving formulations for forecasting the ultimate strength of locally dented ringstiffened cylinders under combined axial. Sci. Technol. Dev. J. 23, 640–654 (2020) 20. Moan, T.: Development of accidental collapse limit state criteria for offshore structures. Struct. Saf. 31, 124–135 (2009) 21. Calle, M.A.G., Alves, M.: A review-analysis on material failure modeling in ship collision. Ocean Eng. 106, 20–38 (2015) 22. Cerik, C.B., Lee, K., Park, S.J., Choung, J.: Simulation of ship collision and grounding damage using Hosford-Coulomb fracture model for shell elements. Ocean Eng. 173, 415–432 (2019) 23. Park, S.-J., Lee, K., Cerik, B.C., Choung, J.: Ductile fracture prediction of EH36 grade steel based on Hosford–Coulomb model. Ships Offshore Struct. 14(sup1), 219–230 (2019) 24. Cerik, B.C., Park, B., Park, S.J., Choung, J.: Modeling, testing and calibration of ductile crack formation in grade DH36 ship plates. Marine Struct. 66, 27–43 (2019) 25. Cerik, B.C., Choung, J.: Rate-dependent combined necking and fracture model for predicting ductile fracture with shell elements at high strain rates. Int. J. Impact Eng. 146, 103697 (2020) 26. API. Bulletin on Stability Design of Cylindrical Shells, API Bulletin 2U, 3rd edn, Washington (2004) 27. ABS, Guide for Buckling and Ultimate Strength Assessment for Offshore Structures (2014)
Assessment of Reduced Strength of Existing Mooring Lines of Floating Offshore Platforms Taking into Account Corrosion Pham Hien Hau(B) and Vu Dan Chinh National University of Civil Engineering, Hanoi, Vietnam {hauph,chinhvd}@nuce.edu.vn
Abstract. Strength of mooring lines of floating offshore platforms are specified by Minimum Breaking Loads (MBL). The MBL are determined by experimental formula depended on diameters of mooring lines in catalogues of manufacturer. For existing floating offshore platforms, after an operating duration the diameters are reduced due to corrosion. It means that strength of the mooring lines are reduced, too. There are some studies about the problem. In our previous researches, the theoretical basis and fundamental steps for evaluating and re-evaluating the durability of mooring lines of floating offshore structures were studied. However, the corrosion is not uniform for different locations of the lines, so it is not easy to estimate accurately the reduced strength. In this paper, we propose a method to assess reduced strength of the existing mooring chains based on numerical models. It will be applied to evaluate strength of mooring line system of DH-01 FPU Platform which is operating in Dai Hung field in Viet Nam sea. The research results can be used as a reference for similar projects in Vietnamese conditions. Keywords: Floating offshore platforms · Existing mooring lines · Reduced strength · Corrosion · Numerical models
1 Introduction During the exploitation process, the mooring system of floating offshore platforms have been degraded due to environmental factors (corrosion, marine growth…). When the platforms are changed exploitation position or extended life time, it is necessary to re-evaluate the strength of existing mooring lines, taking into account the influence of the environmental factors. One of the most important factors is corrosion. It causes the reduction of diameter, weight and strength of mooring chains. Strength condition of mooring lines are depended on the maximum tension force and the Minimum Breaking Loads (MBL) of the lines [1, 2]. The Minimum Breaking Loads can be simply determined through the relationship with sections of the segments from catalogues of manufacturers by experimental formula [4]. However, in case of lack of survey data, the breaking loads can be predicted by numerical testing simulations of mooring lines. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 431–445, 2022. https://doi.org/10.1007/978-981-16-3239-6_33
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There are some studies in the world about the problem, such as [5, 6]. In our previous researches [7, 8], the theoretical basis and fundamental steps for evaluating and reevaluating the durability of mooring lines of floating offshore structures were studied. In this paper, we establish different models using ABAQUS program to determine MBL of corroded chain links based on survey data. The results will be compared with MBL in Vicinay’s Catalogue to verify the accuracy of the models. Finally, we evaluate strength conditions of a mooring line system through the MBL results and tension forces in these lines due to external loading by ARIANE program.
2 Methodology to Assess the Strength of Mooring Lines of Floating Offshore Structures 2.1 Line Dynamic Response Analyses A methodology based on time domain simulation has been developed and proved by Bureau Veritas [3] for mooring lines design and assessment using ARIANE software. In our research, we have to perform the analyses in quasi-dynamic of the mooring line. All of the response simulations in the mooring line systems are executed in time domain. The calculation procedure was proposed in H.H Pham (2015) [7].
Fig. 1. Time domain simulation of tension force of mooring lines using ARIANE software
For any considered sea-state, n simulations in 3h (a short-term sea-state duration) should be performed (Fig. 1). The response signals are built up depended on time steps. For each mooring line, the maximum tension at fairlead (connection point between mooring line and floating structure) during each simulation is obtained by using different “seeds”, i.e. different sets of elementary waves, wind components. Means and the standard deviations of the maximum values in n simulation are obtained. A design line tension corresponding to n simulations is defined by Bureau Veritas (2004) [2]: n n 1 1 Tk ; TS = (1) TD = TM + aTS ; TM = (Tk − TM )2 n n−1 k=1
where : n is the number of simulation;
k=1
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Tk is the maximum tension at fairlead obtained during the k-th simulation; TM is the mean of Tk obtained by the n simulations; TS is the (n-1) standard deviation; TD is the design tension of the line. a is a factor, given in [2] as a function of n et the type of analysis. With quasi-dynamic method, n =5 => a=1.8; n=20 => a=0.5; n≥30 => a=0.4. 2.2 Safety Factor To assess strength condition of mooring lines of floating offshore structures, we have to evaluate the Safety Factor (SF) for line components. According to Bureau Veritas (2004) [2], SF for mooring lines is expressed by the formula as below: (2) SF = TBr TD ≥ [SF] where: TBr is the Minimum Breaking Load (MBL) of the mooring line; TD is the design tension of the line given by Eq. (1); [SF] is the minimum Safety Factor, depend on analysis conditions and analysis method. For quasi-dynamic method in the intact condition [SF] is 1.75 and in the survival condition (one mooring line is broken) [SF] is 1.25 [2]. 2.3 Minimum Breaking Loads of Mooring Lines from Catalogues of Manufacturers [4] According to Vicinay Mooring Lines Catalogue [4], Minimum Breaking Loads of mooring chain are expressed by formulas given in Tables of Mechanical properties, depends on the grade of chain (Fig. 2). For example, the MBL of chain links Grade R4S is: TBr = 0.0304 Z (kN) with Z = d2 (44 − 0.08d) where d is diameter of the chain (mm).
Fig. 2. Mechanical properties of chain from catalogues of manufacturers [4]
(3)
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The formula (3) can be used to determine MBL of uniform corroded chains with arbitrary diameters. However, in the next sections, authors will present a method to evaluate the MBL by numerical models, using Abaqus software, in case of non-uniform corroded chains for different locations of the lines.
3 Numerical Method for Chain Testing Simulation 3.1 Plastic Models
Fig. 3. Illustration of stress – strain curve of a metal sample belonging to over-stress power law [10]
Fig. 4. Slave nodes interacting with a two-dimensional master surface [9]
To determine breaking loads of chain mooring line, it is necessary to analyse chain links in plastic stage. The plastic models are usually used is over-stress power law with stress - strain relationship as below (Fig. 3) [9, 10]. n σ (4) ε˙ pl = D σyoy − 1 for σy ≥ σyo Where εpl is the equivalent plastic strain rate; σy is the yield stress at nonzero plastic strain rate; σyo is the static yield stress which may depend on the plastic strain εpl via isotropic hardening and other parameters; D and n are material parameters that can be functions of temperature and, possibly, of other predefined state variables. 3.2 Small-Sliding Interaction Between Bodies [9] An “anchor” point on the master surface, Xo, is computed for each slave node so that the vector formed by the slave node and Xo coincides with the normal vector N(Xo) (Fig. 4). Suppose that a search for the anchor point, Xo, of slave node 103 reveals that Xo is on segment 1–2. Then, we find that: Xo = X (uo ) = (1 − uo )X1 + uo X2
(5)
Where X1 and X2 are the coordinates of nodes 1 and 2, respectively, and uo is calculated so that Xo – X103 coincides with N(Xo). Moreover, the contact plane tangent direction, vo , at Xo is chosen so that it is perpendicular to N(Xo); i.e., ∂(Xo ) = T (X2 − X1 ) ∂u where T is a (constant) rotation matrix. vo = T
(6)
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3.3 Contact Formulation [9] At each slave node that can come into contact with a master surface we construct a measure of over closure h (penetration of the node into the master surface) and measures of relative slip. These kinematic measures are then used, together with appropriate Lagrange multiplier techniques, to introduce surface interaction theories for contact and friction. In three dimensions the over closure h along the unit contact normal n between a slave point XN+1 and a master plane M(ξ1 , ξ2 ), where ξi parametrize the plane, is determined by finding the vector (M− XN+1 ) from the slave node to the plane that is perpendicular to the tangent vectors v1 and v2 at M. Mathematically, we express the required condition as: h.n = M (ξ1 , ξ2 ) − XN +1
(7)
v1 .(M (ξ1 , ξ2 ) − XN +1 ) = 0 v2 .(M (ξ1 , ξ2 ) − XN +1 ) = 0
(8)
When
If at a given slave node h < 0, there is no contact between the surfaces at that node, and no further surface interaction calculations are needed. If h ≥ 0, the surfaces are in contact. 3.4 Contact Pressure in Case of Hard Contact [9] In this case. p = 0 for h < 0 (open), and h = 0 for p > 0 (close) The contact constraint is enforced with a Lagrange multiplier representing the contact pressure in a mixed formulation. The virtual work contribution is: δ = δp.h + p.δh
(9)
and the linearized form of the contribution is: d δ = δp.dh + dp.δh
(10)
3.5 Stress Analysis by FEM In case of two bodies contact together and have pressure interaction, stress will be analysed by FEM as similar as a rigid body when suffer loads [11].
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4 Strength Assessment of Mooring Line System of DH01 FPU The section presents a strength assessment of mooring lines of DH01 (Fig. 5) based on the numerical method and Vicinay’s formula as mentioned above. These results will be compared to clarify applicability and suitability of numerical models for the problem. The contents are detailed in items as below. 4.1 Input Data 4.1.1 DH01 FPU Data [12]
Fig. 5. Position and general layout of Dai Hung Field, Vietnam
DH-01 FPU Structural data are assumed in the Table 1: Table 1. Dai Hung FPU (DH-01) structural data Items
Unit Value
Main particulars pontoon length m
108.2
Moulded Breadth
m
67.36
Main deck length
m
68.6
Main deck elevation
m
36.6
Main deck breadth
m
60.92
Maximum draught
m
21.3
Lightship weight
T
10715
Displacement
T
20543
Draft - Operational
m
21.34
Draft – Survival
m
19.51
Draft – Transit
m
6.7
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4.1.2 Mooring Input Data Mooring lines configuration and arrangement are presented in Fig. 6.
Fig. 6. Dai Hung FPU mooring lines profile and arrangement
As original design, one mooring line includes two segments. Static segments are laid out on seabed with the length of 685m, Dynamic segments with the length of 465m. The mooring systems of DH-01 is modeling in ARIANE program (Fig. 7).
Fig. 7. Modeling of DH-01’s mooring systems in ARIANE software
The original and corroded configurations of chain links are presented in Fig. 8. Whereas, based on site survey report in 2015 [13], corroded diameters, Dc, include values of 91 mm, 90 mm, 88 mm, 87 mm, 86 mm, 84 mm, respectively. Chain link material has Grade R4S, with mechanical properties are listed in the table as below.
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a) Original Chain Link
b) Corroded Chain Links
Fig. 8. Configuration of chain links Table 2. Mechanical properties of Chains Yield strength (MPa)
Ultimate strength (MPa)
Elongation (%)
MBL (kN)
700
960
12%
10600
4.1.3 Environmental Data [14] Environmental data are assumed in the Table 3 as below: Table 3. Summaries of environmental data Items
Unit Value
Highest astronomical tide HAT
m
10 min. mean speed (1 year period)
m/s
111.5 23.4
10 min. mean speed (100 year period)
m/s
34.6
Significant wave height (1 year period)
m
4.8
Peak period (1 year period)
s
10.7
Significant wave height (100 year period)
m
10.0
Peak period (100 year period)
s
16.0
Surface total steady current (1 year period)
m/s
1.53
Surface total steady current (100 year period) m/s
1.97
4.2 Chain Testing Simulation The number of chain links should be odd to satisfy the symmetric. Then the models have more than 3 links to change corroded chain links at different locations. So in this paper we choose testing model of 5 links joined together using Abaqus as the Fig. 9 as below. Herein, two outermost chains are connected with reference points which are positions of tension forces.
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Fig. 9. Testing model in Abaqus
439
Fig. 10. Meshing of chain testing models
The first model uses original dimension for all chain links. In the next models, for one case of corrosion in relation with Dc = 91 mm, 90 mm, 88 mm, 87 mm, 86 mm, 84 mm, the corroded chain link is located at location numbers of 2, 3, respectively. Meshing models are shown in Fig. 10. Contact model between two chain links are hard contact with small sliding interaction (Fig. 11). Displacements are assigned at Reference Points RP1 and RP2 to make tension forces. The displacements are increased step by step. At the time the first chain link meets fully plastic condition, the related tension force is Minimum Breaking Load (MBL).
Fig. 11. Contact interaction model between chain links
4.3 Numerical Testing Results 4.3.1 Minimum Breaking Load of Original Model of Chain As the analysis result of numerical model using Abaqus, the Minimum Breaking Load of original model of chain is 10689 kN. The deviation from the designed MBL in Table 2 is 89 kN (0.84%). The stresses distribution result and tension force chart are expressed in the Figs. 12 and 13.
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Fig. 12. Stress distribution on original model of chain
Fig. 13. Tension force chart of original model
4.3.2 Minimum Breaking Loads of Corroded Chain Models As the input data, corroded diameters Dc = 91 mm, 90 mm, 88 mm, 87 mm, 86 mm, 84 mm are used to make numerical testing models of chain. For a case of Dc, there are 2 models in relation with the corroded chain link at location numbers of 2 and 3. So, total of testing models are 6x2 = 12. The results of MBL are summarized in the Table 4. Table 4. Minimum Breaking Load (MBL) analysis results Dc (mm)
MBL (kN) Corroded chain link at location No 2
Corroded chain link at location No 3
84
7995
8043
86
8352
8303
87
8572
8553
88
8674
8667
90
8948
8966
91
9201
9203
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The stresses distribution results, tension force charts and deformations of chain testing models in relation with Dc = 91 mm and 84 mm are expressed in the Figs. 14, 15, 16 and 17.
Fig. 14. Stress and deformation distribution on corroded chain model, Dc = 91 mm at location No. 2
Fig. 15. Tension force chart of corroded chain model, Dc = 91 mm at location No. 2
Fig. 16. Stress and deformation distribution on corroded chain model, Dc = 84 mm at location No.3
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Fig. 17. Tension force chart of corroded chain model, Dc = 84 mm at location No.3
4.4 Comparison to Calculation of MBL According to Vicinay [4] As the formula (3), according to Vicinay the Minimum Breaking Loads of corroded chain models are expressed in the table below. Table 5. Minimum Breaking Load (MBL) results according to Vicinay Dc (mm)
MBL (kN)
Deviation from the numerical model results (%)
84
7997
0.03 – 0.57
86
8346
0.07 – 0.51
87
8553
0 – 0.22
88
8667
0 – 0.08
90
8966
0 – 0.2
91
9203
0 – 0.02
The comparison results can be expressed in the chart as Fig. 18.
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Fig. 18. Chart of MBL results comparison
4.5 Strength Checking Results According to [8], the maximum tension force at Fairlead location of mooring lines of DH01 FPU is 5422kN for extreme condition and 6826 kN for survival condition. The strength checking results, follow the expression (2), are expressed in Tables 6, 7 as below. Table 6. Strength checking of corroded chain – Extreme Condition Dc (mm)
MBL (kN)
Tension force (kN)
SF
[SF]
Conclusion
84
7997
5422
1.47
1.75
Failed
86
8346
5422
1.54
1.75
Failed
87
8553
5422
1.58
1.75
Failed
88
8667
5422
1.60
1.75
Failed
90
8966
5422
1.65
1.75
Failed
91
9203
5422
1.70
1.75
Failed
As the results in Tables 6 and 7, according to Bureau Veritas (2004) [2], formula (2), safety conditions are not satisfied for almost case of corroded chains. So it is necessary to replace the mooring line by the new one. In case of re-using the mooring line, the owner have to increase platform draft or limit the allowable environmental parameters for operating.
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MBL (kN)
Tension force (kN)
SF
[SF]
Conclusion
84
7997
6826
1.17
1.25
Failed
86
8346
6826
1.22
1.25
Failed
87
8553
6826
1.25
1.25
Failed
88
8667
6826
1.27
1.25
Pass
90
8966
6826
1.31
1.25
Pass
91
9203
6826
1.35
1.25
Pass
5 Discussion Assessment of reduced strength of the mooring lines taking into account corrosion is an important recommend for owners to decide solutions to extend life or to re-use. In the recent, the experimental formulas of Vicinay are usually used to estimate strength of corroded mooring lines in approximately with assumption that corrosion is uniform. However, the corrosion in fact is non-uniform. The paper suggests a method to estimate strength of non-uniform corroded chain links by numerical testing simulations. Based on survey data of mooring lines of DH01 FPU, there are 12 numerical testing models are performed in relation with six different types of corroded chain links. The results of tension forces at the time of failure of chain links are compared together and compared with MBL according to Vicinay. So, it can be seen that: - MBL of corroded mooring chains of numerical models and Vicinay’s formulas are the same. The maximum deviation is only 0.57%. - In testing models, we can notice that there is a very small change of MBL when corroded chain links lay at different locations. However, the models only consider to 5 chain links. The effects of number of chain links to actual strength of mooring lines may be considered in the next studies. - The results show the suitability of numerical models for determination of reduced strength of mooring chains. So, in general the numerical models can be used instead of experimental testing models with high cost. Then, the method can be applied for actual projects in Vietnamese conditions. However, due to survey data only provide corrosion diameters at intersection location of two adjacent chain links, so the testing models are only simple. Moreover, in case of assessment of chain links with local corrosions, the numerical method can be used although Vicinay formulas were not available. Authors will publish studied results of the problem in the next papers. Acknowledgements. This research is funded by National University of Civil Engineering (NUCE) under grant number 35–2019/KHXD-TÐ. We would like to thank NUCE for this funding.
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References 1. American Petroleum Institute: Recommended practice for design and analysis of stationkeeping systems for floating structures. API RP 2SK 3rd edition (2005) 2. Bureau Veritas: Classification of mooring systems for permanent offshore units, Guidance Note NI 493 DTM R00 E, Paris (2004) 3. Bureau Veritas: Quasi-dynamic analysis of mooring systems using ARIANE software, Guidance Note NI 461 DTO R00 E, Paris (1998) 4. Vicinay: Mooring lines catalogue. http://www.vicinaycadenas.net/brochure/#/30/zoomed 5. Crapps, J., et al.: Strength assessment of degraded mooring chains. Offshore Technology Conference, USA (2017) 6. Gordon, R.: Considerations for Mooring Life Extension. Society of Naval Architects and Marine Engineers, USA (2015) 7. Pham, H.H.: FPSO - Fiabilité des lignes d’ancrage avec prise en compte de fatigue, ISBN: 978–3–8381–7928–5, p. 336. Presses Académiques Francophones (2015) 8. Pham, H.H., Vu, D.C.: Re-evaluation of mooring system durability for renewal and reuse of floating offshore structures. In: Proceeding of APAC 2019 - International Conference on Asian and Pacific Coasts, pp. 1115–1122 (2019) 9. Dassault Systèmes Simulia Corp. Abaqus Theory Manual (2011) 10. Keller, C., Herbrich, U.: Plastic instability of rate dependent materials a theoretical-approach in comparison to FE analyses. In: 11th Europan LS-Dyna Conference 2017, Austria (2017) 11. Zienkiewicz, O.C., Taylor, R.L., Zhu, J.Z.: The Finite Element Method: Its Basis and Fundamentals, Seventh Edition. Butterworth-Heinemann (2013) 12. PVEP. DH 01-FPU Structural and Mooring Systems Drawings (2006) 13. PVEP. DH 01-FPU Mooring Chain Survey Reports (2015) 14. PVEP: Final meteocean and environmental design criteria for Dai Hung fields, VSP.VN.2006, p. 157 (2006)
An Investigation on Behaviors of Mass Concrete in Cua-Dai Extradosed Bridge Due to Hydration Heat Nguyen Van My and Vo Duy Hung(B) University of Science and Technology, The University of DaNang, 54 Nguyen Luong Bang Street, Danang City, Vietnam [email protected]
Abstract. The effects of hydration heat can cause the potential of cracks in towerfooting. This paper presents a case study in which the construction of mass concrete bridge foundation of Cua-Dai Extradosed Bridge in Quang Ngai, Vietnam was investigated through FEM software. The 3D-simulation will be conducted to predict the thermal performance. The temperature development profile, temperature difference, tensile stress and displacement were predicted in detail. Results showed that the heat of hydration in Cua-Dai Bridge was very high, which can cause early cracks in concrete structure. The investigation also provided clear insight into the temperature development of concrete block with complicated compositions and ambient conditions. In addition, hydration heat induced tensile stress and displacement also investigated thoroughly. Finally, the critical comments will be given. Keywords: Mass concrete · Tower-footing · Cua-Dai Bridge · Temperature · Tensile stress · Displacement
1 Introduction The early-age temperature development in mass concrete structures has a significant impact on their durability and quality. A high temperature differential can result in large temperature-induced tensile stresses that can cause early cracking. [1–4] The large thermal differential is primarily caused by a large amount of heat generated due to hydration of cement in the core of the structure that is dissipated at a very slow rate. [5–8] The proper design and construction of mass concrete can help prevent disasters such as crack or over displacement. In Vietnam, many concrete dam and long span bridge projects are actively planned or in progress. Due to effect of hydration heat, mass concrete construction and maintenances have great concern from designers and operators. The main difference between mass concrete and other types is its heat of hydration behavior. Depending on thickness of concrete block, the temperature generates from the hydration of cement can reach a true adiabatic condition in the interior. At a high temperature, the internal concrete tends to expand, while the external concrete tends to shrink and resist the expansion of internal concrete. The temperature difference between the center and © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 446–457, 2022. https://doi.org/10.1007/978-981-16-3239-6_34
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the surface can cause thermal cracks in concrete block. Therefore, one of the challenges in mass concrete design and construction is analysis and control early crack due to hydration heat. Many thermal stress analyses in mass concrete have been performed by the finite element method (FEM). Midas Civil software is a multipurpose finite element analysis package that can perform 3-D analysis [9]. These components can accurately simulate the behavior of different kind of structures, from small structures to large and complex models. [10, 11] In this paper, concrete behavior due to hydration heat of Cua-Dai extradosed bridge will be examined. Firstly, 3D simulation of tower footing will be established by solid element with similar dimensions of Cua-Dai tower footing. The boundary condition of footing will be considered carefully at this stage. Then, heat convection and conduction will be simulated under considering the characteristic of concrete and ambient condition. Finally, behaviors of Cua-Dai tower footing due to hydration heat will be investigated thoroughly. The characteristic of temperature, stress, displacement due to hydration heat will be elucidated in detail.
2 Hydration Heat analysis Hydration heat analysis includes heat transfer analysis and thermal stress analysis. Heat transfer analysis is the process of calculating temperature changes over time related to heat generation, convection, thermal conductivity that occur during cement hydration. Thermal stress analysis provides stress calculations for mass concrete according to each construction phase based on changes in temperature distribution over time obtained from heat transfer analysis. Thermal stress analysis also considers changes in material properties as well as shrinkage and creep. Heat transfer analysis consists of two main parts: thermal conductivity analysis and thermal convection analysis [12]. 2.1 Heat Transfer Analysis Heat Transfer Analysis Relate to Below Definition
a) Heat Conduction Heat conduction is a form of heat transfer related to the exchange of energy from a high temperature area to a low temperature area. The heat conduction rate is proportional to the area perpendicular to the direction of conduction and the thermal gradient. According to Fourier’s law [13, 14]: Qx = −kA
∂T ∂x
Where: Qx : The amount of heat transferred, the unit is kcal / h.m.° C. k: The thermal conductivity coefficient, for saturated concrete, k = 1.21–3,11. A: The heat conductive area, perpendicular to the thermal conductivity direction. ∂T ∂x : Thermal gradient.
(1)
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b) Heat Convection The amount of heat transmitted by convection on a solid block area unit is calculated as follows [13, 14]: q = hc (T − T∞ )
(2)
Where. hc : Heat transfer coefficient between solids and liquids or air. hc depends on many factors such as the geometry of the surface, the physical properties of the liquid / air, the average temperature of the liquid surface, when calculating the concrete blocks in the air, hc can be calculated as follows: hc = 5.2 + 3.5v (v is wind speed; m/s) [13]. T: The solids surface temperature. T∞: The liquid / air ambient temperature. The amount of heat inside the concrete arises due to hydration in a unit of time and volume is [13, 14]: g=
1 rcKae−at/24 24
(3)
Where. K is the maximum temperature rise (° C). a is the reaction rate coefficient. r is the volume of concrete (kg / m3 ). t is time (day). 2.2 Thermal Stress Analysis Stress in mass concrete at each construction stage is calculated using the results of heat transfer analysis, node temperature distribution, as well as consideration of changes in material properties by time. These calculations relate to some concepts such as the equivalent age of concrete by temperature and time and cumulative temperature. The compressive strength of concrete is calculated according to the equivalent age and cumulative temperature, calculated by the following formula [14, 15]: fc (t) =
t f (28) a + bteq c
Where. fc (t) is the concrete strength at the time of calculation. fc (28) is concrete strength at age 28 days. a, b are coefficients depending on the type of cement. t is the calculation time (day). t eq is the equivalent age of concrete.
(4)
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2.3 Boundary Condition of Heat Transfer in Concrete Block The detail boundary condition of heat transfer in tower footing of Cua-Dai Bridge shown as Fig. 1. The construction of tower footing divide into two main stages. During the construction, steel formwork is used, and the concrete block put on ground with specific heat around 25°. In first stage of construction, heat transfer through steel formwork in horizontal direction. The heat convection between concrete block and ambient air through the top surface. For second construction stage, boundary condition is quite similar to the first construction stage. The only difference is that the bottom of second concrete block will transfer heat through first block in boundary of concrete layers. Ambient Steel formwork Ambient Ambient
Steel formwork
Boundary of concrete layers
Ambient Steel formwork
Ambient
Soil ground with specific heat
Fig. 1. Boundary condition of heat transfer in bridge tower footing
3 Behaviors of Tower Footing of Cua-Dai Extradosed Bridge Due to Hydration Heat 3.1 Modelling of Concrete Block Cua-Dai extradosed brige across the Tra-Khuc River in Quang Ngai Province. It is the biggest extradosed bridge ever built in Vietnam. The largest tower-footing has dimmension as Fig. 2 and Fig. 3. Therefore, the concrete for its tower footing is very huge, more than 3000 m3 . To figure out the characteristic of hydration heat in tower-footing of Cua-Dai bridge, the tower-footing with dimension 25 m × 25 m × 5 m will be simulated by Midas Civil. The heat hydration analysis of Midas Civil already proved its reliability. The temperature by field measurement and FEM analysis are very close in term of value and temperature development [16, 17]. Figure 2, Fig. 3 and Fig. 4 illustrate the tower footing simulation by Midas Civil software. In heat hydration analysis, concrete block is simulated by solid element of 0.5 × 0.5 × 0.5 m. Two stages of construction can be seen as Fig. 2 and Fig. 3. The first construction stage relate to tower footing while the second construction stage is for tower shaft. The detail of physical characteristic of concrete and ambient environment can be seen Table 1. [14, 18]. The cement contents is 463 kg/m3 and the strength of concrete of tower footing is 35 MPa [18]. The heat convection coefficient of steel formwork and air are 12 kcal/m2 °C and 15 kcal/m2 °C respectively [14]. The detail of other paramenters can be seen in Table 1.
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Fig. 2. First construction stage of tower footing
Fig. 3. Second stage of foundation blocks
Fig. 4. Location of typical nodes
3.2 Hydration Temperature in Tower-Footing Concrete According to the analysis results, the temperature in concrete blocks is a characteristic result of the hydrothermal analysis. Due to the hydration process, the hydration temperature in the first construction stage increases sharply and then decreases with time as shown in Fig. 5. The highest temperature in first construction stage is 72.7 °C at 72 h after pouring in the center of footing-block. The high temperature zone is concentrated in the center and gradually decreases around the concrete block.
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Table 1. Characteristic of concrete [14, 15, 18] Parameters
Value
Specific heat capacity (kcal.g/kg °C)
0,25
Specific weight kg/m3
2400
Heat convection coefficient of steel formwork (kcal/m2 °C) 12 Heat convection coefficient of air (kcal/m2 °C)
15
Ambient temperature (°C)
25
Temperature at concrete pouring (°C)
20
Concrete strength 28 days (kgf/m2 )
3.5×106
Elastic modulus of concrete (kgf/m2 )
2,8.109
Coefficient of thermal expansion
1,0.10–5
Poisson coefficient
0,18
Cement content (kg/m3 )
463
From the analysis results at the temperature of 5 nodes in Fig. 5, the highest temperature is concentrated in the core of concrete block (Node N2727 and N2745). Node N4759 and node N4772 exhibit similar heat distribution after pouring 130 h, then the temperature at the node N4772 started to increase when pouring concrete of tower. Node N4759 and N6801 are located on the open surface, therefore the temperature increases rapidly after 20 h pouring concrete and gradually decreases to the ambient temperature. On the block surfaces, the temperature increases a little, then quickly decreases to the ambient. The temperature difference between surface and the core of mass is up to 40 °C at 60 h after pouring. The finding absolutely agreed with field measurement in [17, 18]. Figure 6, Fig. 7, and Fig. 8 show the hydration temperature change during time in tower footing of Cua-Dai Bridge. At first, temperature was evenly distributed in the concrete block. Later, it increases strongly in the core of block. This could be because that when concrete pours, the concrete mortar is still in liquid form, the convection and heat transfer ability is good, so the temperature is evenly distributed. Later, the heat emits from the cement hydration reaction will accumulate in the block when the concrete begins to harden. Consequently, temperature increase highly around the core on mass concrete. 3.3 Stress in Concrete Block Due to Heat of Hydration Figure 9 shows that the stress in the concrete block. At pouring time, stress is mainly the compressive stress, while the tensile stress only appears mainly in the surface and around the edges of the formwork. In this period, temperature inside of concrete block increases, then the inner concrete tends to heat up, so the stress generated mainly compression. Whereas temperature at the block-surface was decreased rapidly. The reduction of temperature makes concrete tend to shrink, but this shrink will be kept by higher temperatures of the inner layers. This is the main cause of the tensile stress appearance.
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Fig. 5. Temperature distribution of five nodes
Fig. 6. Temperature at 10 h in first construction stage
Fig. 7. Temperature at 72 h in first construction stage
When the tensile stress exceeds the allowable value, early cracks will be occurred. After 80 h, the tensile stress appears nearly all the block surface (Fig. 10). The development of the tensile strength in center of block can be seen in Fig. 11, in which the green line is allowable stress and the red line is hydration heat stress. It was found that in the temperature increase phase (60–80 h after pouring) the compressive
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Fig. 8. Temperature after 140 h in second construction stage
stress at the center increased gradually. However, during the temperature decrease phase (after 90–100 h) the compressive stress reduces due to the balance with the tensile stress in the center of block. The value of tensile stress increases gradually depending on the difference of temperature between the center and the surface of the concrete block. In parallel, the stress distribution in the concrete surface is shown as Fig. 12, in which the green and yellow are allowable stress. It is realized that at 10 h, the stress at the surface starts exceeding the allowable stress, leading to cracks occurrence. This period lasts until about 820 h. The highest stress of 4.3 MPa coincides with the time when the temperature in the concrete block reaches the highest temperature (80 h). The maximum tensile stress occurs at the concrete surface because the temperature difference in this position is the highest. Nevertheless, it should pay attention that the reinforcement did not be considered in current model. Placement of reinforcement in mass concrete can reduce the stress on surface. Hence, the tensile stress in current model might be higher when compared to reality. 3.4 Displacement Due to Heat of Hydration The displacement due to heat of hydration shows in Fig. 13. Fig. 14. Fig. 15. Fig. 16 The largest displacement in the block occurs when the temperature in the concrete block is at the strongest heat development stage, then it will reduce by time. In detail, the largest value occurred at 50 h with 5.4 mm of displacement. Then, the displacement decreases with decreasing temperature in the concrete block. This proves that the temperature change in the concrete block greatly affects the displacement of the concrete block.
Fig. 9. Tensile stress after 10 h
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Fig. 10. Tensile stress after 80 h
Fig. 11. Stress in the center of block
Fig. 12. Stress in block surface
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Fig. 13. Displacement field at 10 h
Fig. 14. Displacement field at 50 h
Fig. 15. Displacement field at 140 h
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Fig. 16. Displacement field at 1030 h
4 Conclusions The impact of heat hydration on the mass concrete of Cua-Dai Bridge’s tower-footing has been discussed in detail. The hydration temperature can cause early crack on concrete surface, so it should be carefully considered during construction not only this bridge but also the similar structure. Current study allows the following conclusions: • The maximum temperature is caused by hydration heat of cement at the center of the block is over 70°C at around 80h after pouring concrete. The temperature difference between the core and the surface of block is more than 40°C. • The largest tensile stress appearing during pouring occurred at the tower-footing surface. This tensile stress exceeded the allowable stress in early age concrete. In addition, heat hydration also created quite large displacement in concrete block of Cua-Dai Bridge tower footing, especially in the conner of concrete block. • The heat of hydration in tower-footing of Cua-Dai Bridge can cause early cracks and displacement in concrete structure. Hence, it should pay attention during construction of mass concrete of tower. The mitigation countermeasures should be consider applying to reduce the effect of hydration heat.
References 1. Ballim, Y.: A numerical model and associated calorimeter for predicting temperature profiles in mass concrete. Cem. Concr. Compos. 26, 695–703 (2004). https://doi.org/10.1016/S09589465(03)00093-3 2. Nili, M., Salehi, A.M.: Assessing the effectiveness of pozzolans in massive high-strength concrete. Constr. Build. Mater. 24, 2108–2116 (2010). https://doi.org/10.1016/j.conbuildmat. 2010.04.049 3. Choktaweekarn, P., Tangtermsirikul, S.: Effect of aggregate type, casting, thickness and curing condition on restrained strain of mass concrete. Songklanakarin J. Sci. Technol. 32, 391–402 (2010) 4. Kolani, B., et al.: Hydration of slag-blended cements. Cem. Concr. Compos. 34, 1009–1018 (2012). https://doi.org/10.1016/j.cemconcomp.2012.05.007
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5. ACI (American Concrete Institute) Guide to Mass Concrete, ACI 207–06 (2006) 6. Riding, K.A., Poole, J.L., Schindler, A.K., Juenger, M.C.G., Folliard, K.J.: Statistical determination of cracking probability for mass concrete. J. Mater. Civ. Eng. 26, 04014058 (2013). https://doi.org/10.1061/(ASCE)MT.1943-5533.0000947 7. Folliard, K.J., et al.: MeadowsPrediction model for concrete behavior: final report, Texas Dep. Transp. (2002) 8. Riding, K.A., Poole, J.L., Schindler, A.K., Juenger, M.C.G., Folliard, K.J.: Evaluation of temperature prediction methods for mass concrete members. ACI Mater. J. 103, 357–365 (2006) 9. MIDAS Information Technology (2004), Heat of Hydration- Analysis Manual (2004) 10. Kim, J.Y.: Heat Transfer Analysis 4th (Ed.) Tae Sung Software Engineering. INC (2005) 11. Akin, J.E.: Finite Element for Analysis and Design, Academic Press (1994) 12. Gebhart, B.: Heat Condition and Mass Diffusion, McGraw-Hill (1993) 13. Midas IT. Analysis Reference- Midas Civil. (2005–2006) 14. Ngo, D.Q., et al.: Modeling and analysis of bridge structures with MIDAS Civil. Construction Publishing House, 2 (2016) 15. ACI (American Concrete Institute) Cement and Concrete Terminology, ACI 116R (2000) 16. Ho, N.K, Vu, C.C.: The analysis of temperature fields and thermal stresses in mass concrete structures, J. Constr. Sci. Technol. (2002) 17. Sargam, Y., et al.: Predicting thermal performance of a mass concrete foundation- A field monitoring case study. Case Stud.Constr. Mater. 11, e00289 (2019) 18. The Design of Cua-Dai Extradosed Bridge, Project management board of transportation construction investment, Quang Ngai province (2017)
A Nonlocal Dynamic Stiffness Model for Free Vibration of Functionally Graded Nanobeams Tran Van Lien1(B) , Ngo Trong Duc2 , Tran Binh Dinh1 , and Nguyen Tat Thang1 1 National University of Civil Engineering, Hanoi, Vietnam
{LienTV,dinhtb,thangnt}@nuce.edu.vn 2 Fujita Corporation Vietnam, Ho Chi Minh City, Vietnam
Abstract. This paper presents a nonlocal dynamic stiffness model (DSM) for free vibration analysis of Functionally Graded (FG) nanobeams. The nanobeam is investigated on the basis of the Nonlocal Elastic Theory (NET). The NET nanobeam model considers the length scale parameter, which can capture the small scale effect of nano structures considering the interactions of non-adjacent atoms and molecules. Material characteristics of FG nanobeams are considered nonlinearly varying throughout the thickness of the beam. The nanobeam is modelled according to the Timoshenko beam theory and its equations of motion are derived using Hamilton’s principle. The DSM is used to obtain an exact solution of the equation of motion taking into account the neutral axis position with different boundary conditions. The DSM is validated by comparing the obtained results with published results. Numerical results are presented to show the significance of the material distribution profile, nonlocal effect, and boundary conditions on the free vibration of nanobeams. It is shown that the study can be applied to other FG nanobeams as well as more complex of framed nanostructures. Keywords: DSM · Nonlocal · FG · Nanobeam · Nondimensional frequency
1 Introduction Functionally Graded Material (FGM) [1, 2] is a new generation composite material that is made up of two or more component materials with a continuous variation in the ratio of components in one or more directions. FGMs are employed in micro/nano electro-mechanical systems (MEMS/ NEMS) to archive high sensitivity and desired performance. Nano-sized structures as plates, sheets and beams are widely used in NEMS devices. Nanobeams are specially attracting more and more attention due to their various potential applications. Because of the size effect, classical elasticity theory cannot fully and accurately investigate the mechanical behaviours of nanostructures. Therefore, Nonlocal Elasticity Theory (NET) was first proposed by Eringen [3] assuming that the stress tensor at one point is not only a function of deformation but also includes all surrounding ones. Currently, NET is widely used to formulate differential equations of motion of nanostructures using homogeneous materials [4–6] and FGM materials [7]. Reddy [8] established © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 458–475, 2022. https://doi.org/10.1007/978-981-16-3239-6_35
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the equations of vibration and stability of homogeneous nanobeams according to the NET for Euler – Bernoulli, Timoshenko, Reddy and Levinson beam theories. Many other authors have developed analytical methods [9–12], Rayleight – Ritz method [13], Finite Element Method (FEM) [14–19], differential transform method [20], differential quadrature method [21],.. to consider the bending, stability and free vibration problems of nanobeams from homogeneous materials. Simsek and Yurtcu [22], Rahmani and Pedram [23] simultaneously studied bending and buckling of Timoshenko FGM beams using analytical methods. In addition, Mechab at el [24] studied on free vibration, while Uymaz [25] researched on forced vibration of nanobeams, both using the higher-order shear deformation theory. Ebrahimi and Salari [26] exploited a semi-analytical method to study the vibrational and buckling analysis of Euler – Bernoulli FGM nanobeams considering the physical neutral axis position. A spectral finite element formulation was indicated by Narendar and Gopalakrishnan [27] to investigate the vibration of nonlocal continuum beams. The analytical solutions found above are all in Navier’s series, therefore, they are limited for simply supported beams. For other boundary conditions, the authors applied FEM to analyze free vibration and buckling of FGM nanobeams according to Euler - Bernoulli beam theory [28, 29], and Timoshenko theory [30–32]. Recently, the authors of [33] solved for natural frequencies and mode shapes of microbeams under various boundary conditions using the state-space concepts. As the FEM is formulated on the base of frequency independent polynomial shape function, it could not be used to capture all necessary high frequencies and mode shapes of interest. An alternative approach called dynamic stiffness method fulfilled the gap of FEM by using frequency-dependent shape functions that are found as exact solution of vibration problem in the frequency domain [34, 35]. Although exact solutions of the vibration problem are not easily constructed for complete structures, but they, if were available, enable to study exact response of the beam in arbitrary frequency range. Adhikari et al. [4, 36] obtained the dynamic stiffness matrix of a nonlocal rod in closed form. The frequency response function obtained using the proposed DSM shows an extremely high modal density near the maximum frequency. To the best of the authors’ knowledge, the DSM based approach to the nonlocal nanobeams is a gap that has to be fulfilled. In the present work a nonlocal dynamic stiffness model is developed to investigate the free vibration characteristics of FGM nanobeams on the basis of NET and Timoshenko beam theory. Comparison between obtained results and published results shows the reliability of the method. The effect of actual neutral axis position, nonlocal size effects, material distribution profile and geometry parameters on the vibration frequency of nanobeams with different boundary conditions has been studied in detail.
2 Nonlocal Dynamic Stiffness Model of a FGM Timoshenko Nanobeam Element For a FGM nanobeam (Fig. 1), the material properties vary along the z thickness direction, and are assumed to take the form [1]:
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Fig. 1. A FGM beam
P(z) = Pb + (Pt − Pb )
z 1 + h 2
n
h h − ≤z≤ 2 2
(1)
where P stands for elasticity modulus E, shear modulus G and mass density ρ, respectively; the subscripts t and b refer to the corresponding values of top and bottom layer materials; n is volume fraction index, and z is the coordinates from the mid-plane of the beam. The displacement field of the Timoshenko beam is given by: u(x, z, t) = u0 (x, t) − (z − h0 )θ (x, t) ; w(x, z, t) = w0 (x, t)
(2)
where u0 (x,t), w0 (x,t) are axial displacement, deflection of a point on neutral axis, respectively; h0 is the distance from neutral axis to x axis; θ is the angle of rotation of the cross-section around the y axis. From there, we get the strain components: εxx = ∂u0 /∂x − (z − h0 )∂θ/∂x ; γxz = ∂w0 /∂x − θ = −ϕ
(3)
Using notations:
A33
(A11 , A12 , A22 ) = E(z) 1, z − h0 , (z − h0 )2 dA; A = η G(z)dA; (I11 , I12 , I22 ) = ρ(z) 1, z − h0 , (z − h0 )2 dA A
(4)
A
where η is the shear correction factor, η = 5/6 for a rectangular cross-section. Because the modulus of elasticity of FGM material changes with height, the neutral plane of the FGM beam may not coincident with its geometrical mid-plane. Neglecting the influence of axial displacement and the nonlocality effect, while using the relations σxx = Eεxx ; σxz = Gγxz [30], neutral axis position can be determined when the axial force at cross-section vanishes: ⎡ ⎤ ∂θ ∂θ = −A12 =0 σx dA = −⎣ E(z)(z − h0 )dA⎦ ∂x ∂x A
A
This leads to A12 = 0. The position of the neutral axis h0 and I 12 can be written by: h0 =
n(RE − 1)h nbh2 ρb Rρ − RE ρt Et ; I12 = . ; Rρ = ; RE = 2(n + 2)(n + RE ) 2(2 + n) n + RE ρb Eb
(5)
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Based on Hamilton’s principle: T (δT − δU )dt = 0
(6)
0
where T and U are kinetic energy and strain energy, respectively. They can be described as:
L ∂u0 ∂δu0 ∂u0 ∂δθ ∂w0 ∂δw0 ∂θ ∂δu0 ∂θ ∂δθ I11 + − I12 + + I22 dx δT = ∂t ∂t ∂t ∂t ∂t ∂t ∂t ∂t ∂t ∂t 0
(7) L δU =
N 0
∂δu0 ∂δθ ∂δw0 −M +Q − Qδθ dx ∂x ∂x ∂x
(8)
where N, M, Q are the axial normal force, the bending moment and the shear force, respectively: N = σxx dA ; M = (z − h0 )σxx dA ; Q = σxz dA (9) A
A
A
Substituting Eqs. (7)–(9) into Eq. (6), we obtained: ∂ 2 u0 ∂ 2 θ ∂Q ∂ 2 w0 ∂M ∂ 2 u0 ∂ 2θ ∂N = I11 2 − I12 2 ; = I11 2 ; − Q = I12 2 − I22 2 ∂x ∂t ∂t ∂x ∂t ∂x ∂t ∂t
(10)
and the boundary conditions are: u0 = 0 or N = 0; w0 = 0 or Q = 0; θ = 0 or M = 0
(11)
The nonlocal constitute equations for nanobeams can be written in form [3]: σxx − μ
∂ 2 σxx ∂ 2 σxz = Eε ; σ − μ = Gγxz xx xz ∂x2 ∂x2
(12)
where μ = e02 a2 is a nonlocal parameter; e0 is a constant associated with each material; a is an internal characteristic length. Taking the derivative Eq. (10) and using Eqs. (3),(12), we obtain: ∂w0 ∂u0 ∂θ ∂ 3 u0 ∂ 3θ ∂ 3 w0 − A12 + μ I11 − θ + μI11 N = A11 − I ; Q = A 12 33 2 2 ∂x ∂x ∂x ∂x∂t ∂x∂t ∂x∂t 2 ∂u0 ∂θ ∂ 2 w0 ∂ 3 u0 ∂ 3θ M = A12 − I22 − A22 + μ I11 2 + I12 2 ∂x ∂x ∂t ∂x∂t ∂x∂t 2
(13)
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Substituting Eq. (13) into Eq. (10) leads to the equations of free vibration of the FGM nanobeam:
A22
∂2θ ∂x2
∂ 2 u0 ∂2θ ∂ 2 u0 ∂2θ ∂ 4 u0 ∂4θ A11 2 − A12 2 − I11 2 + I12 2 + μ I11 2 2 − I12 2 2 = 0 ∂x ∂x ∂t ∂t ∂x ∂t ∂x ∂t 2 2 2 4 ∂w0 ∂ u0 ∂ θ ∂ u0 ∂ u0 ∂4θ − θ − I22 2 + I12 2 − μI12 2 2 + μI22 2 2 = 0 − A12 2 + A33 ∂x ∂x ∂t ∂t ∂x ∂t ∂x ∂t ∂ 2 w0 ∂ 2 w0 ∂ 4 w0 ∂θ − I11 2 + μI11 2 2 = 0 A33 − ∂x ∂x2 ∂t ∂x ∂t
and the corresponding boundary conditions (11) can be written in form: ∂u0 ∂θ ∂ 3 u0 ∂ 3θ =0 − A12 + μ I11 − I12 u0 = 0 or A11 ∂x ∂x ∂x∂t 2 ∂x∂t 2 ∂w0 ∂ 3 w0 w0 = 0 or A33 − θ + μI11 =0 ∂x ∂x∂t 2 ∂u0 ∂θ ∂ 2 w0 ∂ 3 u0 ∂ 3θ θ0 = 0 or A12 − I =0 − A22 + μ I11 2 + I12 22 ∂x ∂x ∂t ∂x∂t 2 ∂x∂t 2
(14)
(15)
When A12 = 0 and I 12 = 0, which means n = 0 or RE = Rρ , the first equation of (14) is independent to the others two equations. Meanwhile, the axial vibration of the FGM beam is uncoupled with the bending vibration similar to that of a homogeneous beam. The following two equations are similar in form to the bending vibration equation of the Timoshenko beam, but the coefficients depend on the parameters of FGMs. Setting: ∞ {U , , W } =
{u0 (x, t), θ (x, t), w0 (x, t)}e−iωt dt
(16)
−∞
Equation (14) in the frequency domain can be obtained as: d 2U d 2 − A12 − μI12 ω2 + I11 ω2 U − I12 ω2 = 0 A11 − μI11 ω2 dx2 dx2 d 2U 2 2 d + A dW − I ω2 U + I ω2 − A + A − μI ω =0 − A12 − μI12 ω2 22 22 33 12 22 33 dx dx2 dx2 d 2W d A33 − μI11 ω2 + I11 ω2 W = 0 − A33 dx dx2
(17)
Putting into the matrices and vectors are defined as following: ⎛ ⎞ 2 2 0 A11 − μI11 ω 2 − A12 − μI12 ω2 ⎠ A = ⎝ − A12 − μI12 ω A22 − μI22 ω 0 0 0 A33 − μI11 ω2 ⎧ ⎫ ⎛ ⎞ ⎞ ⎛ 0 0 0 −I12 ω2 0 I11 ω2 ⎨U ⎬ B = ⎝ 0 0 A33 ⎠ ; C = ⎝ −I12 ω2 I22 ω2 − A33 0 ⎠ ; {z} = ⎩ ⎭ 0 −A33 0 0 0 I11 ω2 W (18)
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Equation (17) now can be described in form of: A z + B z + C {z} = {0}
(19)
Choosing the solutions of Eq. (19) in the form of {z0 } = {d}eλx leads to λ2 A +λ B + C {d} = {0}
(20)
Equation (20) have non-trivial solutions when ! " ! " ! " ˜ =0 det λ2 A˜ + λ B˜ + D
(21)
We receive cubic equations of η = λ2 : η3 +aη2 +bη +c = 0. Using notations η1 , η2 , η3 are solutions of the cubic equations and λ1,4 = ±k1 ; λ2,5 = ±k2 ; λ3,6 = ±k3 ; kj =
√
ηj ; j = 1, 2, 3
Then the general solutions of Eq. (19) are in form as {z0 (x, ω)} =
(22)
6 # dj eλj x . From j=1
the first and the third equations of (20), we yield ⎡
α1 C 1 ⎢ {z0 (x, ω)} = ⎣ C1 β1 C 1
α2 C 2 C2 β2 C 2
α3 C 3 C3 β3 C 3
α4 C 4 C4 β4 C 4
α5 C 5 C5 β5 C 5
⎤ α6 C 6 'T & ⎥ C6 ⎦ · ek1 x ek2 x ek3 x e−k1 x e−k2 x e−k3 x β6 C 6
(23) where {C} = (C1 , ..., C 6 )T are independent constants and A12 − μI12 ω2 k12 + I12 ω2 A33 k1 α1 = = α4 ; β1 = = −β4 A11 − μI11 ω2 k12 + I11 ω2 A33 − μI11 ω2 k12 + I11 ω2 (24) Similarly, α2 = α5 ; β2 = −β5 ; α3 = α6 ; β3 = −β6 , Eq. (23) can be rewritten in the form: {z0 (x, ω)} = [G(x, ω) ]{C}
(25)
⎤ α1 ek1 x α2 ek2 x α3 ek3 x α1 e−k1 x α2 e−k2 x α3 e−k3 x [G(x,ω)] = ⎣ ek1 x ek2 x ek3 x e−k1 x e−k2 x e−k3 x ⎦ k x k x k x −k x −k x 1 2 3 1 2 β1 e β2 e β3 e −β1 e −β2 e −β3 e−k3 x
(26)
where ⎡
Let’s consider an FGM nanobeam element as shown in Fig. 2. Nodal displacements and forces of the element are introduced as & ' ˆ e = {U1 , 1 , W1 , U2 , 2 , W2 }T ; {P e } = {N1 , M1 , Q1 , N2 , M2 , Q2 }T U (27)
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where U1 = z1 (0, ω) ; 1 = z2 (0, ω) ; W1 = z3 (0, ω); U2 = z1 (L, ω) ; 2 = z2 (L, ω) ; W2 = z3 (L, ω) " " ! ! ; Q1 = − A33 − μI11 ω2 ∂x z3 − A33 z2 N1 = − A11 − μI11 ω2 ∂x z 1 − A12 − μI12 ω2 ∂x z2 x=0 x=0 ! " M1 = − A12 − μI12 ω2 ∂x z1 − A22 − μI22 ω2 ∂x z2 − μI11 ω2 z3 x=0 " " ! ! N2 = A11 − μI11 ω2 ∂x z 1 − A12 − μI12 ω2 ∂x z2 ; Q2 = A33 − μI11 ω2 ∂x z3 − A33 z2 x=L x=L ! " 2 2 2 M2 = A12 − μI12 ω ∂x z1 − A22 − μI22 ω ∂x z2 − μI11 ω z3 x=L
(28)
y Q2
Q1 L
N1
W1
W2 U1
Θ1
N2
j
i M1
x M2
U2 Θ2
Fig. 2. Node co-ordinates, nodal loads of a nanobeam element
Substituting Eq. (25) into Eq. (28), we get & ' [G(0, ω)] −B (G)x=0 · {C} Ue = .{C} ; {Pe } = F BF (G)x=L [G(L, ω)] (
with [BF ] is the matrix operator ⎤ ⎡ 2 ∂ − A − μI ω2 ∂ − μI ω 0 A 11 11 x 12 12 x 2 ⎦ [BF ] = ⎣ A12 − μI12 ω2 ∂x − A22 − μI22 ω2 ∂x −μI11 ω 2 A33 − μI11 ω ∂x 0 −A33
(29)
(30)
Eliminating constant vector {C} in Eq. (29) leads to {Pe (ω)} =
−1 & ' ! "& ' −BF (G)x=0 · [G(0, ω)] . Ue = Ke (ω) . Ue BF (G)x=L [G(L, ω)] (
(
(
(31)
! " where Kˆ e is dynamic stiffness matrix of the FGM nanobeam element. For a given structure that consists of a number of FGM nanobeam elements like above, by means of balancing all the internal forces at every nodes of the structure, there
A Nonlocal Dynamic Stiffness Model for Free Vibration (
465
(
will be obtained total dynamic stiffness matrix K(ω) and nodal load vector P. Letting U be the total DOF vector of the structure, equation of motion of which conducted by the dynamic stiffness method is ! "& ' & ' ˆ = Pˆ K (ω) . U (32) (
(
Therefore, natural frequencies {ω} = {ω1 ω2 ... ωn } can be found from of the equation ˆ det[K(ω)] =0
(33)
and mode shape related to the natural frequency ωj is
& ' 0 ϕj (x) = Cj G x, ωj Uj (
(34)
where ˆ [G(x, ω)] = [G(x, ω)] ·
[G(0, ω)] [G(L, ω)]
−1 (35)
& ' ˆ j is the normalized solution corresponding to ωj . Cj0 is an arbitrary constant and U
3 Numerical Results and Discussion 3.1 Comparison of Calculation Results and Published Results In this subsection, the numerical results are compared with the published results to validate the present study. Firstly, the first five natural frequencies of a simply supported FGM beam are compared with results of Simsek [37] for the following geometric and material properties: L = 0.5 m, h = 0.125 m, E t = 70 Gpa, ρ t = 2707 kg/m3 , E b = 210 Gpa, ρ b = 7850 kg/m3 ,vt = vb = 0.3 and volume fraction index n = 1. As seen from Table 1, the present frequencies are in good agreement with the results of Simsek and the difference between the frequencies of the two studies is very small. Table 1. Comparison of the first five natural frequencies (rad/s) of a simply supported FGM beam 1st freq 2nd freq
3rd freq
4th freq
5th freq
Simsek [37] 6443.08 21470.95 39775.55 59092.37 78638.36 Present
λ01
6443.09 21470.95 39775.55 59092.37 78637.78
The second * comparison is made for the nondimensional fundamental frequencies ) ) 2 = ω1 .L h. ρt Et with various boundary conditions for the following geomet-
ric and material properties: L/h = 5, E t = 70 GPa, ρ t = 2702 kg/m3 , E b = 280 GPa,
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ρ b = 3960 kg/m3 , vt = vb = 0.3. The results of the proposed model are compared to the results for five different higher order beam theories: parabolic shear deformation theory (PSDBT) [37], higher order shear deformation (HSDBT) [38], Reddy-Bickford beam theory (RBT) [39], a quasi-3D theory [40] and first order shear deformation theory (FSDBT) [32]. As seen from Table 2, the present dynamic stiffness model is in a good agreement with the two analytical [37, 38] and three finite element [32, 39, 40] models. The*third comparison is made for the nondimensional fundamental frequencies λ1 = ) ω1 .L2 . ρt A Et I of homegenous nonlocal beams with four nonlocal beam theories: the Euler-Bernoulli (EBT), the Timoshenko beam theory (TBT), the Reddy beam theory (RBT) and Levinson beam theory (LBT). The results from present nonlocal dynamic stiffness model are compared to analytical solutions of Reddy [8] in Table 3 with different nonlocal parameters. As seen from Table 3, the proposed model is very accurate estimates for the nondimensional fundamental frequencies of homegenous nonlocal nanobeams. Finally, consider a simply supported FGM nanobeam with geometric and material parameters as follows: L = 10 nm, b = h = L/10 = 1 nm; E t = 70 Gpa, Gt = 26 Gpa, 3 The calculated ρ t = 2700 kg/m3 ; E b = 393 Gpa, Gb = 157 Gpa, ρ b = 3960 kg/m * [30]. ) 2 results for three first nondimensional frequencies λi = ωi .L . ρt A Et I ; i = 1, 2, 3 with regard to the neutral axis (NA) and mid-plane axis (MA) are compared to the results published by Eltaher et al. [30] which used FEM with 100 elements shown in Table 4. It can be seen that the results are quite consistent, especially when taking into account the actual position of the neutral axis, the error is less than 0.5%. Comparisons of calculation results and published results show the reliability of the proposed nonlocal dynamic stiffness model. 3.2 Effects of Neutral Axis Position and Nonlocal Parameter on the Nondimensional Frequency Figure 3 illustrates the variation of nondimensional frequency of the simply supported beam considering the neutral axis and the mid-plane axis regarding different nonlocal coefficient μ. The first nondimensional frequency is greatly influenced by the volume fraction index n and the neutral axis position. Simutaneostly, the first nondimensional frequency calculated according to the actual neutral axis position is always lower than this in the mid-plane axis case. Besides, the relation (5) shows that the neutral axis position is independent from the nolocal coefficient μ. In later calculation, we only consider the case that the neutral axis is corresponding to the actual position. Figure 4 shows the variation of 15 first nondimensional frequencies of a simply supported FGM Timoshenko beam with different nonlocal coefficient μ with volume fraction index n = 5. It is obvious that the nondimensional frequencies decrease significantly as the non-local coefficient μ increases, especially for high nondimensional frequencies, and thus the small scale effect cannot be neglect. The same effect is showed for any volume fraction index. In sum, the nonlocal beam theory should be used if one needs accurate predictions of high frequencies of nanobeams.
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+ * ) Table 2. Comparison of the nondimensional fundamental frequencies λ01 = ω1 .L2 h. ρt Et of FGM beams with various boundary conditions and volume fraction index n Pinned – Pinned 1/n
PSDBT [37]
HSDBT [38]
RBT [39]
Quasi-3D [40]
FSDBT [32]
Present
0
5.15274
5.1528
5.1528
5.1618
5.1525
5.1524
0.5
4.41108
4.4102
4.4019
4.4240
4.2312
4.2449
1.0
3.99042
3.9904
3.9716
4.0079
3.9708
3.9903
2.0
3.62643
3.6264
3.5979
3.6442
3.7051
3.7248
5.0
3.40120
3.4009
3.3743
3.4133
3.3605
3.3722
10
3.28160
3.2815
3.2653
3.2903
3.1307
3.1362
∞
2.67732
-
-
-
2.6771
2.6772
Fixed- Fixed 1/n
PSDBT [37]
HSDBT [38]
RBT [39]
Quasi-3D [40]
FSDBT [32]
Present
0
10.07050
10.0726
10.0678
10.1851
9.9975
9.9974
0.5
8.74670
8.7463
8.7457
8.8641
8.4251
8.4251
1.0
7.95030
7.9518
7.9522
8.0770
7.8998
7.8998
2.0
7.17670
7.1776
7.1801
7.3039
7.3228
7.3228
5.0
6.49349
6.4929
6.4961
6.5960
6.5579
6.5579
10
6.16515
6.1658
6.1662
6.2475
6.0695
6.0695
∞
5.23254
-
-
-
5.1946
5.1947
Fixed- Free 1/n
PSDBT [37]
HSDBT [38]
RBT [39]
Quasi-3D [40]
FSDBT [32]
Present
0
1.89523
1.8957
1.8952
1.9055
1.8944
1.8942
0.5
1.16817
1.1612
1.6180
1.6313
1.5547
1.5548
1.0
1.46328
1.4636
1.4633
1.4804
1.4627
1.4628
2.0
1.33254
1.3328
1.3326
1.3524
1.3674
1.3674
5.0
1.25916
1.2594
1.2592
1.2763
1.2401
1.2402
10
1.21834
1.2187
1.2184
1.2308
1.1504
1.1540
∞
0.98474
-
-
-
0.9843
0.9843
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* ) Table 3. Comparison of the nondimensional fundamental frequencies λ1 = ω1 .L2 . ρt A Et I of a simply supported beam with various nolocal parameter (L/h = 10) μ (10−18 ) EBT [8] TBT [8] RBT [8] LBT [8] Present 0
9.8696
9.7454
9.7454
9.7657 9.7451
0.5
9.6347
9.5135
9.5135
9.5333 9.5132
1.0
9.4159
9.2973
9.2974
9.3168 9.2971
1.5
9.2113
9.0953
9.0954
9.1144 9.0951
2.0
9.0195
8.9059
8.9060
8.9246 8.9057
2.5
8.8392
8.7279
8.7279
8.7462 8.7277
3.0
8.6693
8.5601
8.5602
8.5780 8.5599
3.5
8.5088
8.4017
8.4017
8.4193 8.4015
4.0
8.3569
8.2517
8.2517
8.2690 8.2515
4.5
8.2129
8.1095
8.1095
8.1265 8.1093
5.0
8.0761
7.9744
7.9744
7.9911 7.9742
Table 4. Comparison of nondimensional frequencies of the beam with results of Eltaher et al. Frqs μ (10-18) ( )
0.0
2.0
4.0
MA (Present)
1 2 3
9.7075 37.0962 78.1547
1 2 3
17.0278 64.1874 136.4469
1 2 3
8.8713 27.7303 46.9034
1 2 3
15.5611 48.0080 81.9216
1 2 3
8.2196 23.0989 36.6272
1 2 3
14.4179 39.9980 63.9788
NA MA NA MA NA MA (Present) ([30]) ([30]) (Present) (Present) ([30]) n=0.0 n=0.5 9.7075 9.7032 9.7032 14.5998 13.6072 14.8669 37.0962 37.0382 37.0382 51.8360 52.1179 52.7163 78.1547 77.9135 77.9135 110.6544 110.1423 119.2343 n=5 n=10 16.7305 17.1057 16.7366 17.6526 17.5472 17.6845 64.2634 65.6963 64.3491 67.2760 67.3039 67.8750 136.2956 139.3094 136.6522 142.5265 142.4755 143.8072 n=0.0 n=0.5 8.8713 8.8674 8.8674 13.3416 12.4351 13.5863 27.7303 27.6870 27.6870 38.8463 38.9595 42.3969 46.9034 46.7622 46.7622 66.5345 66.1005 71.5567 n=5 n=10 15.2894 15.6323 15.2950 16.1321 16.0358 16.1613 48.0385 49.1097 48.1026 50.3002 50.3114 50.7383 81.7958 83.6044 82.0098 85.5480 85.5045 86.3037 n=0.0 n=0.5 8.2196 8.2160 8.2160 12.3611 11.5216 12.5883 23.0989 23.0628 23.0628 32.3883 32.4527 35.3159 36.6272 36.5169 36.5169 51.9778 51.6184 55.8791 n=5 n=10 14.1662 14.4840 14.1714 14.9470 14.8578 14.9741 40.0154 40.9076 40.0687 41.9022 41.9086 42.2642 63.8749 65.2873 64.0420 66.8071 66.7711 67.3952
NA ([30]) 13.6100 52.1595 110.3186 17.5542 67.4009 142.8763 12.4377 38.9905 66.2061 16.0421 50.3839 85.7451 11.5241 32.4785 51.7008 14.8637 41.9690 66.6590
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Fig. 3. Variation of first nondimensional frequency of the simply supported FGM Timoshenko beam with different volume fraction index n and nonlocal parameter μ
Fig. 4. Variation of nondimensional frequencies of the simply supported FGM Timoshenko beam with different nonlocal parameter μ in case volume fraction index n = 5
3.3 Effect of Boundary Conditions on Nondimensional Frequencies Tables 5–7 show the results of three first frequencies considering with the neutral axis for a beam with simply support - clamped (S-C) boundary condition (Table 5), clamped ends (C-C) boundary conditions (Table 6) and cantilever (C-F) boundary conditions (Table 7), respectively. It can be seen that cantilever nanobeam has smaller nondimensional frequencies than the simply supported nanobeam with the same geometric and material characteristics. Simultaneously, nondimensional frequencies of simply supported nanobeam are smaller than the nondimensional frequencies of clamped ends beam as well as normal macro-beams. Especially, the frequencies of FGM Timoshenko
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nanobeam decrease gradually when the nonlocal coefficient increases, except the case of the fundamental frequency of cantilever beam, which gradually increases. This finding was also mentioned in [14, 41] when considering the vibration of Euler-Bernoulli and Timoshenko homogenenous nanobeam, respectively. Table 5. Nondimensional frequencies of FGM Timoshenko beam with S-C boundary conditions μ Frqs n (10−19 ) (λi ) n = 0.0 0.0
1.0
2.0
3.0
4.0
n = 0.2
n = 0.5
n = 1.0
n = 2.0
n = 5.0
n = 10
1
14.8361
18.8361
20.8406
22.3482
23.7961
25.6493
26.8765
2
45.3006
57.5081
63.8553
68.6619
73.2147
78.8466
82.4600
3
87.7203 111.3415 124.1009 133.8410 142.9432 153.7930 160.5041
1
14.0555
17.8442
19.7415
21.1688
22.5404
24.2969
25.4604
2
37.9963
48.2260
53.5290
57.5488
61.3649
66.0974
69.1389
3
63.3329
80.3602
89.5092
96.5040 103.0665 110.9269 115.8056
1
13.3841
16.9911
18.7965
20.1549
21.4609
23.1341
24.2427
2
33.3665
42.3453
46.9929
50.5178
53.8679
58.0278
60.7035
3
52.0901
66.0861
73.5934
79.3369
84.7328
91.2057
95.2275
1
12.7989
16.2477
17.9731
19.2716
20.5204
22.1209
23.1815
2
30.1021
38.2000
42.3879
45.5654
48.5873
52.3424
54.7589
3
45.2911
57.4563
63.9763
68.9663
73.6574
79.2890
82.7897
1
12.2831
15.5925
17.2476
18.4934
19.6918
21.2281
22.2464
2
27.6430
35.0777
38.9206
41.8369
44.6118
48.0615
50.2821
3
40.6156
51.5227
57.3657
61.8386
66.0455
71.0976
74.2389
3.4 Effect of Nanobeam Length on the Nondimensional Frequencies Figures 5a–c illustrate the variation of three first natural frequencies of the simply supported FGM Timoshenko beam with different nonlocal coefficient μ in case of volume fraction index n = 5 and with the ratios: L/h = 10, 20, 30, 40, 50, respectively. As the nonlocal coefficient μ increases, there is a significant decrease in the natural frequencies, especially when the ratio L/h in a range of 10–30. It is obvious that non-local coefficient shows to be more influenced on the high frequencies compared to the lower ones.
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Table 6. Nondimensional frequencies of Timoshenko FGM beam with C-C boundary conditions μ Frqs n (10−19 ) (λi ) n = 0.0 0.0
1.0
2.0
3.0
4.0
n = 0.2
n = 0.5
n = 1.0
n = 2.0
n = 5.0
n = 10
1
20.9722
26.6388
29.5363
31.7150
33.7859
36.3901
38.0894
2
53.7468
68.2727
76.0277
81.9004
87.3889
94.0134
98.1736
3
97.1430 123.3937 137.9543 149.0651 159.3058 171.1992 178.3833
1
19.8080
25.1564
27.8870
29.9418
31.8977
34.3602
35.9682
2
44.6699
56.7189
63.1132
67.9657
72.5217
78.0500
81.5340
3
69.6280
88.3834
98.6814 106.5632 113.8842 122.4700 127.6931
1
18.8121
23.8890
26.4778
28.4272
30.2848
32.6257
34.1551
2
38.9900
49.4959
55.0549
59.2783
63.2530
68.0883
71.1408
3
57.1223
72.4901
80.9002
87.3453
93.3479 100.4092 104.7140
1
17.9486
22.7904
25.2570
27.1153
28.8876
31.1228
32.5836
2
35.0220
44.4527
49.4343
53.2218
56.7911
61.1397
63.8874
3
49.6190
62.9594
70.2486
75.8388
81.0521
87.1936
90.9413
1
17.1910
21.8268
24.1867
25.9655
27.6630
29.8052
31.2056
2
32.0537
40.6814
45.2339
48.6968
51.9633
55.9464
58.4647
3
44.4842
56.4391
62.9654
67.9730
72.6468
78.1571
81.5212
Table 7. Nondimensional frequencies of cantilever (C-F) Timoshenko FGM beam μ (10−19 ) Frqs (λi ) n 0.0
1.0
2.0
3.0
4.0
1
n = 0.0
n = 0.2
n = 0.5
n = 1.0
n = 2.0
n = 5.0
3.4893
4.4277
4.8875
5.2327
5.5679
n = 10
6.0059
6.3008
2
20.9127 26.5248 29.3562 31.4989 33.5595 36.1810
37.9040
3
54.4253 69.1555 77.2754 83.2275 88.8537 95.7577 100.1455
1
6.0307
6.3269
2
19.6114 24.8746 27.5311 29.5415 31.4744 33.9325
35.5475
3
45.4655 57.6334 64.0013 68.8667 73.4910 79.1785
82.7965
1
3.5037
18.4917 23.4547 25.9605 27.8567 29.6796 31.9971
33.5194
3
39.6347 50.2509 55.8058 60.0472 64.0769 69.0319
72.1833 6.3808
2
17.5142 22.2150 24.5888 26.3852 28.1118 30.3067
31.7483
3
35.6308 45.1771 50.1736 53.9879 57.6109 62.0641
64.8954
3.5489
4.5032
4.9711
5.2992
5.6146
6.0823
1
4.9494
5.2765
5.5910
2
4.4836
4.9282
5.2544
6.3535
3.5334
4.4645
4.9076
6.0562
1
3.5184
4.4459
5.3225
5.6388
5.6636
6.1090
6.4089
2
16.6503 21.1194 23.3763 25.0842 26.7258 28.8123
30.1827
3
32.6801 41.4369 46.0221 49.5222 52.8461 56.9298
59.5252
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Fig. 5. Variation of first three nondimensional frequencies of the simply supported FGM Timoshenko beam with different nonlocal parameter μ and ratios L/h in case of n = 5.
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4 Conclusion In this work, free vibration of FGM Timoshenko nanobeams is investigated. Based on Haminton’s principle and Nonlocal Elasticity Theory, the differential equations of vibration of a FGM nanobeam element is derived. A nonlocal dynamic stiffness model is developed to obtain the exact solution of these equations to taking into account the actual neutral axis. Comparison with published results of other authors shows the reliability of the proposed model. On that basis, the influence of nonlocal parameters, materials, geometric parameters and boundary conditions on the vibration frequencies of nanobeams is investigated. The analysis results show that the first frequency counting on the actual neutral axis position is always lower than the one counting on the mid-plane axis. The frequencies of Timoshenko FGM nanobeams decrease gradually as the nonlocal coefficient increases in all cases of boundary conditions except the first frequency of cantilever beams. Simutaneously, the effect of nonlocal coefficients on high frequencies is much greater than lower ones. In this regards, the studies can be extended to other types of FGM materials as well as more complex types of beam structures. Acknowledgement. This research is funded by National University of Civil Engineering under grant number 35–2020/KHXD-TÐ.
References 1. Shen, H.-S.: Functionally graded materials: nonlinear analysis of plates and shells. CRC press (2016) 2. Mahamood, R.M., Akinlabi, E.T.: Functionally graded materials. Springer (2017) 3. Eringen, A.C.: Nonlocal continuum field theories. Springer Science & Business Media (2002) 4. Karlicic, D., et al.: Non-local structural mechanics. John Wiley & Sons (2015) 5. Polizzotto, C.: Nonlocal elasticity and related variational principles. Int. J. Solids Struct. 38(42–43), 7359–7380 (2001) 6. Eltaher, M., Khater, M., Emam, S.A.: A review on nonlocal elastic models for bending, buckling, vibrations, and wave propagation of nanoscale beams. Appl. Math. Model. 40(5–6), 4109–4128 (2016) 7. Salehipour, H., Shahidi, A., Nahvi, H.: Modified nonlocal elasticity theory for functionally graded materials. Int. J. Eng. Sci. 90, 44–57 (2015) 8. Reddy, J.: Nonlocal theories for bending, buckling and vibration of beams. Int. J. Eng. Sci. 45(2–8), 288–307 (2007) 9. Wang, C., Zhang, Y., He, X.: Vibration of nonlocal Timoshenko beams. Nanotechnol. 18(10), 105401 (2007) 10. Li, C., et al.: Analytical solutions for vibration of simply supported nonlocal nanobeams with an axial force. Int. J. Struct. Stab. Dyn. 11(02), 257–271 (2011) 11. Aydogdu, M.: A general nonlocal beam theory: its application to nanobeam bending, buckling and vibration. Phys. E. 41(9), 1651–1655 (2009) 12. Wang, C., et al.: Beam bending solutions based on nonlocal Timoshenko beam theory. J. Eng. Mech. 134(6), 475–481 (2008) 13. Chakraverty, S., Behera, L.: Free vibration of non-uniform nanobeams using Rayleigh-Ritz method. Phys. E. 67, 38–46 (2015)
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14. Eltaher, M., Alshorbagy, A.E., Mahmoud, F.: Vibration analysis of Euler-Bernoulli nanobeams by using finite element method. Appl. Math. Model. 37(7), 4787–4797 (2013) 15. Eltaher, M., et al.: Coupling effects of nonlocal and surface energy on vibration analysis of nanobeams. Appl. Math. Comput. 224, 760–774 (2013) 16. Pradhan, S.: Nonlocal finite element analysis and small scale effects of CNTs with Timoshenko beam theory. Finite Elem. Anal. Des. 50, 8–20 (2012) 17. de Sciarra, F.M.: Finite element modelling of nonlocal beams. Phys. E. 59, 144–149 (2014) 18. Tuna, M., Kirca, M.: Bending, buckling and free vibration analysis of Euler-Bernoulli nanobeams using Eringen’s nonlocal integral model via finite element method. Compos. Struct. 179, 269–284 (2017) 19. Alotta, G., Failla, G., Zingales, M.: Finite element method for a nonlocal Timoshenko beam model. Finite Elem. Anal. Des. 89, 77–92 (2014) 20. Ebrahimi, F., Nasirzadeh, P.: A nonlocal Timoshenko beam theory for vibration analysis of thick nanobeams using differential transform method. J. Theor. Appl. Mech. 53(4), 041–1052 (2015) 21. Jena, S.K., Chakraverty, S.: Free vibration analysis of variable cross-section single layered graphene nano-ribbons (SLGNRs) using differential quadrature method. Frontiers Built Environ. 4, 63 (2018) 22. Sim¸ ¸ sek, M., Yurtcu, H.: Analytical solutions for bending and buckling of functionally graded nanobeams based on the nonlocal Timoshenko beam theory. Compos. Struct. 97, 378–386 (2013) 23. Rahmani, O., Pedram, O.: Analysis and modeling the size effect on vibration of functionally graded nanobeams based on nonlocal Timoshenko beam theory. Int. J. Eng. Sci. 77, 55–70 (2014) 24. Mechab, I., El Meiche, N., Bernard, F.: Free vibration analysis of higher-order shear elasticity nanocomposite beams with consideration of nonlocal elasticity and poisson effect. J. Nanomech. Micromech. 6(3), 04016006 (2016) 25. Uymaz, B.: Forced vibration analysis of functionally graded beams using nonlocal elasticity. Compos. Struct. 105, 227–239 (2013) 26. Ebrahimi, F., Salari, E.: A semi-analytical method for vibrational and buckling analysis of functionally graded nanobeams considering the physical neutral axis position. CMES: Comput. Model. Eng. Sci. 105(2), 151–181 (2015) 27. Narendar, S., Gopalakrishnan, S.: Spectral finite element formulation for nanorods via nonlocal continuum mechanics. J. Appl. Mech. 78(6), 061018 (2011) 28. Eltaher, M., Alshorbagy, A., Mahmoud, F.: Determination of neutral axis position and its effect on natural frequencies of functionally graded macro/nanobeams. Compos. Struct. 99, 193–201 (2013) 29. Eltaher, M., Emam, S.A., Mahmoud, F.: Free vibration analysis of functionally graded sizedependent nanobeams. Appl. Math. Comput. 218(14), 7406–7420 (2012) 30. Eltaher, M., et al.: Vibration of nonlinear graduation of nano-Timoshenko beam considering the neutral axis position. Appl. Math. Comput. 235, 512–529 (2014) 31. Eltaher, M., et al.: Static and buckling analysis of functionally graded Timoshenko nanobeams. Appl. Math. Comput. 229, 283–295 (2014) 32. Aria, A., Friswell, M.: A nonlocal finite element model for buckling and vibration of functionally graded nanobeams. Compos. B Eng. 166, 233–246 (2019) 33. Trinh, L.C., et al.: Size-dependent vibration of bi-directional functionally graded microbeams with arbitrary boundary conditions. Compos. B Eng. 134, 225–245 (2018) 34. Su, H., Banerjee, J.: Development of dynamic stiffness method for free vibration of functionally graded Timoshenko beams. Comput. Struct. 147, 107–116 (2015) 35. Van tran, L., Ngo, D.T., Nguyen, K.T.: Free and forced vibration analysis of multiple cracked FGM multi span continuous beams using dynamic stiffness method. (2018)
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36. Adhikari, S., Murmu, T., McCarthy, M.: Dynamic finite element analysis of axially vibrating nonlocal rods. Finite Elem. Anal. Des. 63, 42–50 (2013) 37. Sim¸ ¸ sek, M.: Fundamental frequency analysis of functionally graded beams by using different higher-order beam theories. Nucl. Eng. Des. 240(4), 697–705 (2010) 38. Nguyen, T.-K., et al.: Vibration and buckling analysis of functionally graded sandwich beams by a new higher-order shear deformation theory. Compos. B Eng. 76, 273–285 (2015) 39. Vo, T.P., et al.: Finite element model for vibration and buckling of functionally graded sandwich beams based on a refined shear deformation theory. Eng. Struct. 64, 12–22 (2014) 40. Vo, T.P., et al.: A quasi-3D theory for vibration and buckling of functionally graded sandwich beams. Compos. Struct. 119, 1–12 (2015) 41. Li, X.-F., Wang, B.-L.: Vibrational modes of Timoshenko beams at small scales. Appl. Phys. Lett. 94(10), 101903 (2009)
Dynamic Performance of Highway Bridges: Low Temperature Effect and Modeling Effect of High Damping Rubber Bearings Nguyen Anh Dung(B) and Nguyen Vinh Sang 175 Tay Son, Dong Da, Hanoi, Vietnam
Abstract. Use of high damping rubber bearings (HDRBs) in seismic isolation is a viable practice to protect bridges from earthquakes. The behavior of HDRB under cyclic load is dominated by nonlinear rate-dependent elasto-plastic responses and low temperatures. Rate-dependency effect due to viscosity depends strongly on the current strain but the effect is vividly different in loading/unloading phases. A rheology model, proposed recently for HDRB, can reproduce the behaviors of HDRB under cyclic loads. In this paper, the effects of incorporating the rheology model in simulating the seismic responses of a bridge superstructure-pier-foundation (S-PF) system are investigated. In addition, the change of the model parameters at low temepratures affacting to the dynamic responses of the bridge is also investigated in this study. To this end, nonlinear dynamic analysis of a six span continuous seismically isolated bridge is conducted for two different strong earthquake ground motions. The nonlinear hysteretic behaviors of bridge piers are considered using Takeda tri-linear model. The nonlinearity is restricted to be lumped in plastic hinges located at the bottom of piers. Three temperature conditions are considered in the analysis: the room temperature (+23 °C) and the two low temperatures (−10 °C and −30 °C) for checking the low temperature effect. Two models for the isolation bearings are considered for comparison: the conventional bilinear model [1, 2] used in design practice and the ratedependent rheology model [3]. To evaluate the time-history response of the bridge system, a solution algorithm has been developed to solve the equations of motion of the system and the first order differential equation governing the nonlinear rate-dependent behavior of HDRB. The solution algorithm is successfully implemented in general purpose finite element code. The implication of using the rheology model in response prediction of the S-P-F system is studied by comparing the rotation of pier and shear strain of the bearing obtained using the bilinear model and the rheology model. The comparison suggests that the modeling of HDRB considering the nonlinearities due to elasticity and viscosity effects is vital for rational prediction of the seismic response of highway bridges. Keywords: Dynamic analysis · Low temperature effect · Modeling effect · High damping rubber bearings
© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 476–488, 2022. https://doi.org/10.1007/978-981-16-3239-6_36
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1 Introduction Bridges are a crucial part of the overall transportation system as they play very important roles in evacuation and emergency routes for rescues, first-aid, firefighting, medical services and transporting disaster commodities to expatriates. In this regard, bridges serve as a transportation lifeline of modern society. In view of the importance of the bridge structure, it is a contemporary key issue to minimize as much as possible the loss of the bridge functions against earthquakes to enhance continued functioning of the community life. However, bridges may pose a serious threat to public safety during extreme events like earthquakes. In the recent major earthquakes such as the 1994 Northridge earthquake, the 1995 Kobe earthquake, the 1999 Chi-Chi earthquake, the 2008 Wenchuan earthquake, and the 2010 Chile earthquake, highway bridges suffered from a variety of structural damages [4–7]. To reduce the damage concentration at the piers of an ordinary bridge, an effective way is to adopt an isolated bridge, which utilizes a seismic dissipation device that limits forces transferred from the superstructure to the substructure while accommodating the concomitant displacement [8]. Rubber bearings is the one of most effective isolators, economical laminated-rubber bearings featuring alternate layers of rubber and steel sheets have been extensively adopted in highway bridges. The bearings possess the capacities of sustaining large vertical loads and accommodating thermal movements of the superstructure with low maintenance [9]. There are three types of laminated rubber bearings: natural rubber bearings, lead rubber bearings, and high damping rubber bearings (HDRB). More recently, high damping rubber bearings have been widely used in Japan due to their high flexibility and high damping capability. In the manufacture process of high damping rubber material, a large amount of fillers (about 30%) including carbon black, silica, oils and some other particles is added during the vulcanization process [10, 11] in order to improve the desirable material properties, such as the strength and damping capabilities. HDRB consist of alternative layers of high damping rubber and steel plates bonded by vulcanization. They have some similar behavior to other elastomeric bearings, such as being able to support vertical loads with limited or negligible deflection and horizontal loads with large deflections. However, HDRB are characterized by some special properties that are very different from standard elastomeric bearings. They present high damping capability, and bearings defined as HDRB should provide an equivalent viscous damping of at least 10% [12]. HDRB also present a very useful property for the application in the base isolation of structures. They are very stiff for small deformation and soft for large deformation. This property allows the structure to respond rigidly to small excitations like braking forces and provides high flexibility for large excitations like earthquakes. Moreover, high damping rubber material is very durable. Gu and Itoh (2006) [13] estimated that the change of the equivalent horizontal stiffness of HDRB is about 10–25% after 100 years. Therefore, there is little or no maintenance requirement for HDRB. However, even after years of practice, the mechanical behaviors of HDRB, such as the rate dependence and the low temperature dependence behaviors, are still difficult and unsolved issues in engineering practice.
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Experimental observations [14, 15] showed that mechanical properties of HDRB are dominated by the nonlinear rate dependence. Furthermore, experimental results also indicated that HDRB possess strong nonlinear strain hardening at high strain levels. These aspects are more obvious at low temperatures. The experimental investigations of Lion (1997) [16] indicated that the rate sensitivity of the reinforced rubber strongly depends on temperature, especially at low temperatures. In addition, the experimental observations in [17] showed that the low temperature dependence on the shear stress-strain responses of HDRB is more significant than that of natural rubber bearings and lead rubber bearings. However, the bilinear design model in some guide specifications [1, 2] omit the rate dependence behavior and the strain hardening behavior of HDRB, and this omission may cause major errors in predicting the seismic responses of the structures isolated by HDRB. Therefore, it is a great concern for engineers designing isolation structures in the areas where are not only very cold but also seismically active such as Hokkaido of Japan (1968 Tokachi Earthquake), Alaska of US (1964 Alaska Earthquake). In order to overcome the limitations of the current design model, Bhuiyan et al. (2009) [3] have developed a rate-dependent rheology model. The underlying key approach in this work is an additive decomposition of the total stress response into two contributions including rate-independent equilibrium stress and rate-dependent overstress. In this approach, the rate-independent behavior relating to the equilibrium response and the rate-dependent behavior relating to the overstress response can be obtained separately from experimental data then the equilibrium parameters and overstress parameters are determined directly from experiments. In this paper, a nonlinear dynamic analysis of a highway bridge isolated with HDRB is conducted to investigate the effect of the HDRB’s mechanical behavior on the dynamic performance of the bridge. In order to study about the temperature effect of HDRB, there are three temperature conditions considered in the analysis: the room temperature (+23 °C); and the two low temperatures (−10 °C and −30 °C). In order to investigate the modeling effect of HDRB, there are two analytical models of HDRB considered for comparison: the bilinear design model in [1, 2] and the rate-dependent rheology model in [3].
2 Input Data for the Analysis 2.1 Model of the Bridge The highway bridge used in this analysis is similar to the bridge in [18], the differece between them only is the kind of rubber bearings. Figure 1(a) shows the physical model of a five-span continuous reinforced concrete (RC) deck-steel girder bridge isolated by laminated rubber bearings provided at top of the RC piers. The laminated rubber bearings comprised is high damping rubber bearings. The isolation bearings are positioned at between the steel girders and top of the piers. The superstructure consists of continuous reinforced concrete slab, covered by 8 cm of asphalt with 0.26 × 18 m section, supported on two continuous steel girders. The substructures consist of RC piers and footings supported on pile foundations. The geometric and material properties of the bridge deck, piers with footings are given in Table 1.
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(a)
(b)
Fig. 1. Description of the target bridge (a) longitudinal view of a multi-span continuous highway bridge, (b) analytical model of the bridge; all dimensions are in mm
Table 1. Geometric and material properties of the piers Properties Cross-section of the pier cap (mm2 ) B1 × W1 Cross-section of the pier body (mm2 ) B2 × W2 Cross-section of the footing (mm2 ) B3 × W3 Number of piles/pier
Specifications Pier S1 & S2
Pier P1 & P4
3300 × 9600
2000 × 9600
3300 × 6000
2000 × 6000
5000 × 8000
5000 × 8000 4
Young’s modulus of concrete (MPa)
25000
Young’s modulus of steel (MPa)
200000
The analytical model of the bridge system is shown Fig. 1(b). The entire structural system is approximated as a continuous 2-D frame. A mathematical model of discrete form is used to approximate the continuous system. This form of modeling leads to a system with a finite number of degrees of freedom. Following the standard finite element procedure, these degrees of freedom are chosen as the nodal displacements of the discrete finite element model. The superstructure and substructure of the bridge are
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modeled as a lumped mass system divided into a number of small discrete segments. Each adjacent segment is connected by a node and at each node two degrees of freedom are considered: horizontal translation and rotation. The masses of each segment are assumed to be distributed between the two adjacent nodes in the form of point masses. The dynamic analysis has been successfully implemented in commercially available finite element software [19] in order to compute the seismic responses of bridges. 2.2 Model of the HDRB In order to characterize the mechanical behavior of isolation bearing two types of analytical models of the bearings are used in the study: the rate-dependent rheology model as developed by [3] and the bilinear design models as specified in [1, 2]. These two models are briefly discussed in the following sub-sections. 2.2.1 Rate-dependent Rheology Model The structure of the rate-dependent rheology model in [3] is shown in Fig. 2. The mathematical description of the model is briefly stated in Eqs. (1a)–(1e).
Fig. 2. The rate-dependent rheology model in [3] for HDRB in the analysis.
τ = τep (γa ) + τee (γ ) + τoe (γc ) γ˙s = 0 for τep = τcr = C1 γa with γ˙s = 0 for τep < τcr τee = C2 γ + C3 γ m sgn(γ )
(1a)
τep
n γ˙d τoe = C4 γc with τoe = A sgn(γ˙d ) γ˙o
(1b) (1c) (1d)
Dynamic Performance of Highway Bridges
and A =
1 1 (A1 exp(q|γ |) + Au ) + (A1 exp(q|γ |) − Au ) tanh(ξ τoe γd ) 2 2
481
(1e)
where C i (i = 1 to 4), τ cr , m, Al , Au , q, n, and ξ are parameters of the model to be determined from experimental data. Table 2. The rate-dependent rheology model parameters are determined from experimental data
In order to investigate the low temperature effect, the test data were carried out at three tempertures: 23 °C; −10 °C; −30 °C. The experimetal results were presented in [20]. These data are used to determine the parametarers in this analysis. The identification parameters of the rheology model are presented in Table 2. 2.2.2 Bilinear Design Model The bilinear design model of the bearings can be represented using a simplified structure of the rheology model. The simplified structure of the model is formed by using a linear elastic response (τee ) and an elasto-plastic response (τep ) only and hence the third branch (τoe ) of the model is discarded. In this case three parameters are required to represent the bilinear relationships of stress-strain responses of the bearings: C 1 parameter corresponds to the initial shear modulus, C 2 , the post yield shear modulus and τ cr , the characteristic yield strength of the bearings. The structure of this model is presented in Fig. 3.
Fig. 3. The bilinear design model in [1, 2] for HDRB in the analysis.
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The bilinear parameters are determined from sinusoidal loading tests with frequency of 0.5 Hz. The determination of these parameters at different temperatures are presented in Table 3. Table 3. The bilinear design model parameters are determined from experimental data
2.3 Earthquake Ground Motions and Solution Algorithm for Motion Equations 2.3.1 Earthquake Ground Motions The uncertainty characteristics of the earthquake ground motions regarding ground type, intensity and frequency contents have a great effect on nonlinear time history responses of members. In designing bridges [2], two levels of design ground motions should be considered, i.e. ground motion that is likely to occur during the service period of the bridge and destructive ground motion that is less likely to occur during the service period. These design ground motions are termed as Level 1 ground motion, and Level 2 ground motion, respectively. Level 2 ground motion contains the ground motion resulting from a large plate-boundary earthquake (type-I) and that from an inland earthquake (type-II). Type-I earthquake is characterized by a ground motion with large amplitude and long duration such as the Kanto earthquakes (Tokyo, 1923) and the type-II earthquake is characterized with low probability of occurrence, strong acceleration and short duration such as the Kobe earthquake (Kobe, 1995). Figure 4(a) and (b) show the time histories of type-I and type-II earthquake ground motions, respectively, used in the analysis. 2.3.2 Motion Equations and Solution Algorithm Equations that govern the dynamic response of the bridge can be derived by considering the equilibrium of all forces acting on it using the d’Alermbert’s principle. In this case, the internal forces are the inertia forces, the damping forces, and the restoring forces, while the external forces are the earthquake induced forces. To allow incorporation of nonlinear hysteretic models of the isolation bearings and the piers, the equations of motion of the the superstructure-pier-foundation (S-P-F) system can be written as ¨ − Ct U ˙ − t+t Fb − t Fs ¨ + CU ˙ + t KU = t+t P − Mt U MU
(2)
where M and C are the mass and damping matrices, respectively; t K is the tangent stiffness matrix at time t; U = t+t U − t U stands for the increment of displacement vector; t+t Fb the internal force vector derived from the isolation bearings at time t; t Fs ,
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(a)
Fig. 4. The ground acceleration histories used in the seismic response analysis bridge pier system (a) type-I, and (b) type-II earthquake ground motions; the acceleration histories correspond to the standard acceleration histories designed for level-2 earthquakes [2].
the restoring force of the bridge excluding isolation bearing at time t; and t+t P, the external force vector at time t + t. A single-step solution algorithm is developed to implement the rheology models in the nonlinear dynamic analysis for computing the responses of the bridge system. The solution algorithm includes the solution of equations of motion using the unconditionally stable Newmark’s constant-average-acceleration method and the solution of the differential equation governing the behavior of isolation bearings. Furthermore, Newton-Raphson iteration procedure consisting of corrective unbalanced forces is employed within each time step until equation condition is achieved. All calculations are implemented in a commercially available finite element software (Resp-T software) [19].
3 Output Data of the Analysis The dynamic analysis is implemented in commercially available Resp-T software [17]. The two designed earthquake waves: type-I and type-II specified by JRA (2004) are applied to the model bridge in the longitudinal direction. Due to symmetry of the bridge structure shown in Fig. 1(a) and due to space limitation, only one pier’s results P1 (=P4) using HDRB at three different temperature conditions are graphically presented and discussed herein. Seismic responses of the bridge system are presented in Figs. 5, 6, 7, and 8. 3.1 Low Temperature Effect Figure 5 shows the moment-rotation relations of the plastic hinges of the pier and Fig. 6 represents the shear stress-strain responses of the isolation bearings at room temperature (23 °C) and low temperatures (−10 °C, −30 °C). The low temperature effect is very clear in Fig. 6, the difference between the room temperature responses and the low temperature responses in Fig. 5 is quite large.
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Fig. 5. Moment-rotation responses of plastic hinge of the pier P1 (=P4) obtained using HDRB for Level-2 earthquake ground motions, (a) type-I earthquake and (b) type-II earthquake. Moment ratio (M/My) is the bending moment (M) at the level of plastic hinge divided by the yield moment (My) and rotation ductility is the rotation occurred at the plastic hinge level divided by the yield rotation
Fig. 6. Shear stress-strain responses of the HDRB at the top of the P1 (=P4) piers for Level-2 earthquake ground motions, (a) type-I earthquake and (b) type-II earthquake.
3.2 Modeling Effect In Figs. 7 and 8, the dynamic responses of the bridge are shown for a typical earthquake ground motion (level 2 type II earthquake ground motion). Figures 7 and 8 present the moment-rotation hysteresis at the plastic hinge level and the shear stressstrain responses of the HDR at three temperatures, respectively. Each figure presents the dynamic responses of the bridge obtained for the bilinear design model and the ratedependent rheology model of the isolation bearings. In general, the effect of modeling of isolation bearing on the seismic responses is clearly appeared in all figures. However, at low temperature (−10 °C, −30 °C), this effect is somewhat more significant than at the room temperature (+23 °C). The similar observations are also obtained for the type I earthquake ground motion but with different magnitudes.
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Fig. 7. Moment-rotation responses of plastic hinge of the pier P1 (=P4) obtained using HDRB for Type II earthquake ground modtion at (a) 23 °C (b) −10 °C (c) −30 °C; moment ratio (M/My ) is the bending moment (M) at the level of plastic hinge divided by the yield moment (My ) and rotation ductility is the rotation occurred at the plastic hinge level divided by the yield rotation
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Fig. 8. Shear stress-strain responses of the HDRB at the top of the P1 (=P4) piers for Type II earthquake ground modtion at (a) 23 °C (b) −10 °C (c) −30 °C.
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4 Conclusion A nonlinear dynamic analysis of a seismically isolated five-span continuous highway bridge was carried out. Commercially available software was used to implement the algorithm and to compute the seismic responses of the bridge. The seismic responses of the bridge were computed using the two analytical models of the isolation bearings at three different temperatures (at 23 °C, −10 °C, and −30 °C). The dynamics responses of the highway bridge, moment-rotation relationships of plastic hinges of the piers and shear stress-strain responses of the bearings, are compared at different temperatures. The effect of low temperatures on these responses is very clear. Moreover, the comparison of the dynamic responses of the bridge with different models for bearings is also conducted. In general, the effect of modeling of HDRB is clearly appeared in all earthquake ground motions for the isolation bearings in three temperatures. However, at low temperature, this effect is more significantly appeared in the responses than that at room temperature. This may be attributed to an overestimation of the second shear modulus of HRDB using the bilinear model in compared with the rheology mode in simulating the experimental data. Acknowledgements. The experimental works were conducted by utilizing the laboratory facilities and bearings-specimens provided by Japan Rubber Bearing Association. Author indeed gratefully acknowledge the kind cooperation extended by them in performing the tests. This work was supported by JSPS KAKENHI (Grants-in-Aid for Scientific Research) Grant Number 26289140 and Prof. Okui in Saitama University, Japan.
References 1. American Association of State Highways and Transportation Officials (AASHTO): Guide Specification for Seismic Isolation Design, 2nd edn. Washington D.C., USA (2010) 2. Japan Road Association: Specifications for Highway Bridges, Part V: Seismic Design, Tokyo, Japan (2004) 3. Bhuiyan, A.R., Okui, Y., Mitamura, H., Imai, T.: A rheology model of high damping rubber bearings for seismic analysis: identification of nonlinear viscosity. Int. J. Solids Struct. 46, 1778–1792 (2009) 4. Xiang, N., Goto, Y., Makoto Obata, M., Alam, S.: Passive seismic unseating prevention strategies implemented in highway bridges: a state-of-the-art review. Eng. Struct. 194, 77–93 (2019) 5. Di Sarno, L., da Porto, F., Guerrini, G., Calvi, O.M., Camata, G., Prota, A.: Seismic performance of bridges during the 2016 Central Italy earthquakes. Bull. Earthq. Eng. 17, 5729–5761 (2019) 6. Kilanitis, I., Sextos, A.: Impact of earthquake-induced bridge damage and time evolving traffic demand on the road network resilience. J. Traffic Transp. Eng. 6, 35–48 (2019) 7. Zhang, Y., Shi, Y., Liu, D.: Seismic effectiveness of multiple seismic measures on a continuous girder bridge. Appl. Sci. 10, 624–648 (2020) 8. Yi, J., Yang, H., Li, J.: Experimental and numerical study on isolated simply-supported bridges subjected to a fault rupture. Soil Dyn. Earthq. Eng. 127, 01–11 (2019) 9. Xiang, N., Shahria Alam, M., Li, J.: Shake table studies of a highway bridge model by allowing the sliding of laminated-rubber bearings with and without restraining devices. Eng. Struct. 171, 583–601 (2018)
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10. Kelly, J.M.: Earthquake Resistant Design with Rubber, 2nd edn. Springer, London (1997). https://doi.org/10.1007/978-1-4471-0971-6 11. Yoshida, J., Abe, M., Fujino, Y.: Constitutive model of high-damping rubber materials. J. Eng. Mech. 130(2), 129–141 (2004) 12. Marioni, A.: The use of high damping rubber bearings for the protection of the structures from the seismic risk. In: Jornadas Portugesas de Engenharia de Estruturas, Lisbon, Portugal (1998) 13. Gu, H., Itoh, Y.: Ageing effects on high damping bridge rubber bearing. In: Proceeding of the 6th Asia-Pacific Structural Engineering and Construction Conference, Kuala Lumpur, Malaysia (2006) 14. Dall’Asta, A., Ragni, L.: Experimental tests and analytical model of high damping rubber dissipating devices. Eng. Struct. 28, 1974–1984 (2006) 15. Hwang, J.S.: Evaluation of equivalent linear analysis methods of bridge isolation. J. Struct. Eng. 122, 972–976 (1996) 16. Lion, A.: On the large deformation behavior of reinforced rubber at different temperatures. J. Mech. Phys. Solids 45, 1805–1834 (1997) 17. Imai, T., Bhuiyan, A.R., Razzaq, M.K., Okui, Y., Mitamura, H.: Experimental studies of rate-dependent mechanical behavior of laminated rubber bearings. In: Joint Conference Proceedings 7CUEE & 5ICEE, 3–5 March 2010, Tokyo Institute of Technology, Tokyo, Japan, pp. 1921–1928 (2010) 18. Razzaq, M.K., Okui, Y., Bhuiyan, A.R., Amin, A.F.M.S., Mitamura, H., Imai, T.: Application of rheology modeling to natural rubber and lead rubber bearings: a simplified model and low temperature behavior. J. Jpn. Soc. Civ. Eng. 3(68), 526–541 (2012) 19. Resp-T: User’s Manual for Windows, Version 5 (2006) 20. Nguyen, D.A., Dang, J., Okui, Y., Amin, A.F.M.S., Okada, S., Imai, T.: An improved rheology model for the description of the rate-dependent cyclic behavior of high dampung rubber bearings. Soil Dyn. Earthq. Eng. 77, 416–431 (2015)
Nonlinear Vibration Analysis for FGM Cylindrical Shells with Variable Thickness Under Mechanical Load Khuc Van Phu1 , Dao Huy Bich1 , and Le Xuan Doan2(B) 1 Vietnam National University, Hanoi, Vietnam 2 Tran Dai Nghia University, Ho Chi Minh City, Vietnam
Abstract. The paper investigates nonlinear vibration of FGM cylindrical shells with varying thickness subjected to mechanical load. The novelty of the paper is to establish motional equations of the structure. Since then, the natural frequency of vibration and the dynamic responses of the structure have been determined. Effect of material (coefficient k) and geometric parameters on natural frequency and nonlinear dynamic responses of the shell are also examined. Keywords: Nonlinear vibration · Variable thickness · FGM cylindrical shell · Dynamic responses
1 Introduction Variable thickness structures help to reduce the weight of produces and save materials without losing the bearing capacity, there for, this is the most favorite structure of designers. Variable thickness FGM shells are special structures and frequently used in modern industries such as: aerospace, aircraft, space vehicles and in other important engineering structures. Recently, nonlinear vibration problems of FGM shell in general and in particular variable-thickness FGM shells are attracting attention of many scientists. On nonlinear vibration of constant thickness FGM shell, Loy et al. [1, 2] based on Love’s shell theory to study vibration of FGM cylindrical shells subjected to mechanical load in thermal environment. The governing equations were derived by using Rayleigh– Ritz method. Haddadpour et al. [3] cared about natural frequencies and vibration of FGM cylindrical shells subjected to mechanical load including thermal effects with four sets of in-plane boundary conditions. Equilibrium equations were derived based on Love’s shell theory and solved by using Galerkin method. Matsunaga [4] presented an analysis for nonlinear vibration of FGM circular cylindrical shells based on 2D higherorder deformation theory. The power series expansion method and Hamilton’s principle were employed to solve the problem. Sofiyev [5] found out fundamental frequencies of clamped FGM conical shells subjected to uniform pressures. The governing equations were established based on Donnell’s theory and solved by Galerkin’s method. Mollarazi [6] investigated free vibration of cylindrical shell made of FGM by using mesh less method. Weak form of equilibrium equation and squares approximation were © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 489–505, 2022. https://doi.org/10.1007/978-981-16-3239-6_37
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used to establish motion equations of structure. By using analytical method Bich and Nguyen [7] examined effects of axial and transverse loads on nonlinear vibration of FGM cylindrical shell. The governing equations are derived based on improved Donnell shell theory. Nonlinear dynamic responses of structure are obtained by using Galerkin method, the Volmir’s assumption and fourth-order Runge–Kutta method. Also based on Donnell shell theory, Avramov [8] analyzed nonlinear vibrations of FGM cylindrical shell. The nonlinear vibration modes were determined by using the harmonic balance method and Galerkin method. Malekzadeh et al. [9] investigated free vibration of rotating FGM truncated conical shells subjected to mechanical load with various boundary conditions. Dynamic equilibrium equations and motion equations of the shell were derived according to the first-order shear deformation theory (FSDT). Take into account effect of Coriolis forces and centrifugal forces. By the same way, effect of internal pressure on natural frequencies of structures was examined by these authors [10]. Kim [11] analyzed free vibration and natural frequencies of FGM shells partially resting on elastic foundations with an inclined edge by using analytical method. Equilibrium equations of structure were established according to the FSDT, variational approach and Rayleigh–Ritz method. Quan et al. [12] solved nonlinear vibration problems of imperfect FGM shallow shells surround by elastic foundations subjected to mechanical, thermal and damping loads. The Reddy’s third-order shear deformation theory was employed to establish governing equations. Natural frequencies and dynamic responses of structure were obtained by using Galerkin method and fourth-order Runge–Kutta method. Also using analytical method, Phu et al. [13, 14] studied on nonlinear vibration of sandwich-FGM and stiffened sandwich FGM cylindrical shell filled with fluid subjected to mechanical loads in thermal environment. The classical shell theory with geometrical nonlinearity in von Karman-Donnell sense and smeared stiffener technique were used to define motion equations of structure. Natural frequencies and dynamic responses of the shell were determined by using Galerkin’s method and Runge–Kutta method. Nonlinear vibration problems of FGM elliptical cylindrical shells reinforced by carbon nanotubes embedded in elastic mediums in thermal environments were dealt with by Dat et al. [15]. Research is carried out based on classical shell theory, nonlinear dynamic responses of structure were found out by using the Galerkin method, RungeKutta method and Airy stress function. Han et al. [16] predicted free vibration of FGM thin cylinder shells filled inside with pressurized fluid based on Flügge shell theory. In governing equations, internal static pressure is regarded as the pre-stress term. On vibration of variable thickness shell structures, there are some typical research as: Ganesan et al. [17] employed semi-analytical finite element to analyze vibration characteristic of variable thickness circular cylindrical shells made of steel based on Love’s first approximation shell theory. By the same way, natural frequencies and vibration of laminated conical shells and variable-thickness cylindrical shells made of composite were also considered by these authors [18, 19]. Abbas et al. [20] predicted fundamental frequencies and vibration mode of open circular cylindrical shells with the thickness varies in axial direction in high temperature environment. Transfer matrix method and the fourth-order Runge–Kutta method were used to solve problem. Kang et al. [21, 22] employed 3-D method and Ritz method to examine fundamental frequencies and vibration mode of spherical shell segments and variable thickness conical-cylindrical
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shells. Also study on variable thickness spherical shells, Efraim et al. [23] based on two-dimensional theory and dynamic stiffness method to determine natural frequencies and vibration mode shapes of structure with different boundary conditions. Ataabadi et al. [24] presented an analysis on vibration of variable thickness shells in parabolic or circular profile. The linear shell theory, energy method and spline functions were used to survey. Nonlinear vibrations investigation of complex-shape shallow shells with variable thickness under various boundary conditions has been proposed by Awrejcewicz et al. [25]. The classical theory and first-order shear deformation theory are used to establish governing equations. Natural frequencies of structure were obtained by using R-functions theory, Ritz’s method, Galerkin method and the fourth-order Runge–Kutta method. About variable thickness FGM structure, Ghannad et al. [26] based on the FSDT and the perturbation theory to analyze static problems of variable thickness FGM cylindrical shells subjected to internal pressure. The distribution of displacement and stress in axial and radial direction were determined by using inverse matrix method. Free vibration problems of variable thickness sandwich-FGM shells were solved by Tornabene et al. [27]. Higher-order shear deformation theory (HSDT) and local generalized differential quadrature method were used to determine natural frequencies of structure. By using finite element method, Golpayegani et al. [28] analyzed free vibrations of FGM cylindrical shells under different boundary conditions. The thickness of structure varies linearly along the axial direction. Effects of varying thickness, the length and radius of structure on frequencies were also considered. From above review, According to authors’ knowledge, there is no investigation on nonlinear vibration of FGM cylindrical shell with variable thickness subjected to axial compression load and external pressure published yet. In present paper, the governing equations of variable thickness structure are established based on classical shell theory, take into account the geometrical nonlinearity in von Karman-Donnell sense. Natural frequencies and nonlinear dynamic responses of the shell are obtained by using RungeKutta method and Galerkin method. Effect of material and geometric dimension on natural frequencies and nonlinear dynamic responses of the shell are also considered.
2 Governing Equations Consider the variable-thickness FGM cylindrical shell with the length L and the radius R subjected to uniform pressure q(t) and axial compression load p(t) as shown in Fig. 1. Assume that radius is much larger than the thickness of the shell (R >> h), Thickness 0 of shell h can be determined as: h(x) = ax + b in which: a = h1 −h L ; b = h0 . The effective properties of material can be expressed as follow: P(z) = Pm · Vm (z) + Pc · Vc (z) = Pm + (Pc − Pm )Vc (z)
(1)
In which Pm and Pc are material and ceramic properties, V c and V m are volume fractions of ceramic and metal, respectively, and are related by V c + V m = 1. Ceramic volume fractions in the structure are distributed as following law: z k 1 + (2) Vc (z) = 2 h(x)
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Fig. 1. Variable-thickness FGM cylindrical shell
Poisson’s ratio is assumed to be constant (ν = constant). According to classical shell theory [29], the strain-displacement relationship of the shell: εij = εij0 + zkij with (i, j = x, y) in which 0 = εxx
kxx =
∂u0 ∂v0 1 ∂w 2 0 w ∂x + 2 ∂x ; εyy = ∂y − R 2 2 − ∂∂xw2 ; kyy = − ∂∂yw2 ;
+
1 2
∂w ∂y
2
(3)
0 = ; γxy
∂u0 ∂y
+
∂ w kxy = −2 ∂x∂y
∂v0 ∂x
2
Applying Hooke’s law for FGM shell subjected to mechanical load: ⎫ ⎫ ⎧ ⎡ ⎤ ⎧ 1ν 0 ⎨ σxx ⎬ ⎨ εxx ⎬ E(z) ⎣ ⎦ · εyy = σ ν1 0 2 ⎭ ⎩ yy ⎭ 1 − υ(z) ⎩ τxy γxy 0 0 (1 − ν)/2
+
∂w ∂w ∂x ∂y
(4)
(5)
Integrating Stress-Strain relationship through the thickness of the shell, we obtain the governing equations of the variable thickness FG cylindrical shell: 0 ε N [A] [B] ij = (6) . ij with (i, j) = (xx, yy, xy) Mij [B] [D] kij Equation (6) can be rewritten as following: ⎫ ⎡ ⎧ N ⎪ ⎪ ⎪ xx ⎪ A11 ⎪ ⎪ ⎪ ⎢A ⎪ ⎪ Nyy ⎪ ⎪ ⎪ 21 ⎢ ⎪ ⎪ ⎪ ⎬ ⎢ ⎨N ⎪ xy ⎢ 0 =⎢ ⎢ B11 ⎪ Mxx ⎪ ⎪ ⎪ ⎢ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎣ B21 ⎪ ⎪ ⎪ Myy ⎪ ⎪ ⎪ ⎭ ⎩ 0 M xy
A12 A22 0 B12 B22 0
0 0 A66 0 0 B66
B11 B12 0 B21 B22 0 0 0 B66 D11 D12 0 D21 D22 0 0 0 D66
⎧ 0⎫ ⎤ ⎪ εxx ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ εyy ⎪ ⎪ ⎥ ⎪ ⎪ ⎪ ⎥ ⎪ ⎨ 0⎪ ⎬ ⎥ ⎪ γxy ⎥ ⎥· ⎪ ⎥ ⎪ ⎪ kxx ⎪ ⎪ ⎥ ⎪ ⎪ ⎪ ⎪ ⎦ ⎪ ⎪ ⎪ ⎪ ⎪ k yy ⎪ ⎪ ⎪ ⎩ ⎪ ⎭ kxy
in which: A11 =A22 =
E1 · h(x) ν · E1 · h(x) ; A12 = A21 = ; 2 1−ν 1 − ν2
(7)
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B11 =B22 = D11 = D22 = A66 =
E2 · h2(x) 1 − ν2
; B12 = B21 =
E3 · h3(x) 1 − ν2
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ν · E2 · h2(x)
; D12 = D21 =
E2 · h2(x) E1 · h(x) ; B66 = ; D66 2(1 + ν) 2(1 + ν)
1 − ν2 ν · E3 · h3(x)
; 1 − ν2 E2 · h2(x) = 2(1 + ν)
Ec − Em (Ec − Em )k ; E2 = ; (k + 1) 2(k + 1)(k + 2) 1 1 1 Em + (Ec − Em ) − + E3 = 12 k + 3 k + 2 4(k + 1)
E1 =Em +
Internal forces and moment resultants can be inferred from Eq. (7) as follows: ⎧ 0 + υ · ε0 + B ⎪ Nxx = A11 εxx 11 kxx + ν · kyy ⎪ yy ⎪ ⎪ ⎪ ⎪ 0 + υ · ε0 + B ⎪ Nyy = A11 εyy 11 kyy + ν · kxx ⎪ xx ⎪ ⎪ ⎨N = A · γ0 + B · k xy 66 xy 66 xy 0 0 +D ⎪ Mxx = B11 εxx + ν · εyy ⎪ 11 kxx + ν · kyy ⎪ ⎪ ⎪ ⎪ 0 + ν · ε0 + D ⎪ Myy = B11 εyy ⎪ 11 kyy + ν · kxx xx ⎪ ⎪ ⎩ 0 +D ·k Mxy = B66 · γxy 66 xy
(8)
Nonlinear motion equations of FGM cylindrical shell with variable thickness based on classical shell theory are [29]: ⎧ ∂Nxy ∂Nxx ∂2u ⎪ ⎪ ∂x + ∂y = ρ1 ∂t 2 ⎪ ⎪ ∂Nxy ∂Nyy ⎪ ∂2v ⎪ ⎪ ⎨ ∂x + ∂y = ρ1 ∂t 2 ∂ 2 Myy ∂ 2 Mxy ∂ 2 Mxx ∂ 2w ∂ 2 w Ny ∂ 2w (9) + + Nyy 2 + +2 + Nxx · 2 + 2Nxy ⎪ ⎪ 2 2 ∂x ∂x∂y ∂y ∂x ∂x∂y ∂y R ⎪ ⎪ ⎪ ⎪ ∂ 2w ∂w ∂ 2w ⎪ ⎩ −p · h(x) 2 + q = ρ1 2 + 2ερ1 ∂x ∂t ∂t −ρm in which: ρ1 = ρm + ρck+1 h(x) = ρ1∗ h(x) . Substituting Eq. (4) and Eq. (8) into Eq. (9) we obtain: ⎧ ∂2u ⎪ ⎪ ⎪ L11 (u) + L12 (v) + L13 (w) + P1 (w) = ρ1 ∂t22 ⎪ ⎪ ⎨ L21 (u) + L22 (v) + L23 (w) + P2 (w) = ρ1 ∂ 2v ∂t (10) L31 (u) + L32 (v) + L33 (w) + P3 (w) + P4 (u, w) + P5 (v, w) ⎪ ⎪ 2 2 ⎪ ⎪ ∂ w ∂w ∂ w ⎪ ⎩ +q + p · h(x, y) 2 = ρ1 2 + 2ερ1 ∂x ∂t ∂t in which: L11 (u) =A11
∂ 2 u ∂A11 ∂u ∂ 2u + A + . ; 66 ∂x2 ∂x ∂x ∂y2
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L12 (v) =A66 ·
∂A11 ∂v ∂ 2v ∂ 2v +ν· + ν · A11 ; ∂x∂y ∂x ∂y ∂x∂y
∂ 3w ∂ 3w ∂ 3w ∂B11 ∂ 2 w − ν · B11 − 2B66 · − 3 2 2 ∂x ∂x∂y ∂x∂y ∂x ∂x2 1 ∂w ∂B11 ∂ 2 w ∂A11 w − ν · A11 ; −ν · −ν· ∂x ∂y2 ∂x R R ∂x 2 ∂w ∂w ∂ 2 w ∂w 2 1 ∂A11 ∂w ∂ 2 w P1 (w) = +ν· + ν · + A11 2 ∂x ∂x ∂y ∂x ∂x2 ∂y ∂x∂y 2 ∂ w ∂w ∂w ∂ 2 w ; +A66 · + ∂x∂y ∂y ∂x ∂y2 L13 (w) = −B11
∂A66 ∂u ∂ 2u + ; ∂x∂y ∂x ∂y ∂ 2v ∂ 2 v ∂A66 ∂v L22 (v) =A66 · 2 + + A11 2 ; ∂x ∂x ∂x ∂y
L21 (u) =(A66 + A11 ν) ·
∂B66 ∂ 2 w 1 ∂ 3w ∂w ∂ 3w · − A11 ; − (νB11 + 2B66 ) · 2 − 2 3 ∂y ∂x ∂y ∂x ∂x∂y R ∂y ∂w ∂ 2 w ∂A66 ∂w ∂w ∂w ∂ 2 w ∂ 2 w ∂w ∂w ∂ 2 w + A + A + ; P2 (w) = A11 + ν · · 66 66 ∂y ∂y2 ∂x ∂x∂y ∂x2 ∂y ∂x ∂x∂y ∂x ∂x ∂y L23 (w) = −B11
I31 (u) = B11
∂ 3u ∂B11 ∂ 2 u ∂ 2 B11 ∂u ∂ 3u + + ν · B +2 + (2B ) 66 11 ∂x3 ∂x∂y2 ∂x ∂x2 ∂x2 ∂x ∂u 1 ∂B66 ∂ 2 u + ν · A11 ; +2 2 ∂x ∂y R ∂x
∂B11 ∂ 2 v ∂B66 ∂ 2 v ∂ 3v ∂ 3v + 2ν · + 2 + + ν · B (2B ) 66 11 ∂y3 ∂x2 ∂y ∂x ∂x∂y ∂x ∂x∂y 2 1 ∂v ∂ B11 ∂v + A11 ; +ν 2 ∂x ∂y R ∂y 4 ∂ w ∂ 4w ∂ 4w L33 (w) = −D11 + 4 − 2(D11 ν + 2D66 ·) 2 2 4 ∂x ∂y ∂x ∂y 3 2 3 2 ∂ D11 ∂ w ∂D11 ∂ w ∂ w ∂ 2w − −2 + ν · − ν · ∂x ∂x3 ∂x2 ∂y ∂x2 ∂x2 ∂y2
L32 (v) = B11
∂ 2 B11 w 1 2 w − A11 − ν · 2 ∂x R R R R ∂ 2w ∂ 2w 2 ∂D66 2 − ν · B11 2 − B11 2 − 4 R ∂x R ∂y ∂x −ν ·
∂B11 ∂w ∂x ∂x ∂ 3w · ; ∂x∂y2
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2 ∂w 2 w ∂ 2w 1 ∂w ∂ 2w A11 P3 (w) = +ν· −ν· 2 − A11 2R ∂y ∂x R ∂y2 ∂x 2 2 ∂w ∂w 1 ∂ 2 B11 ∂w ∂ 2 w ∂B11 ∂w ∂ 2 w + + ν · + ν · + 2 2 ∂x2 ∂x ∂y ∂x ∂x ∂x2 ∂y ∂x∂y 2 2 ∂w ∂ 3 w ∂w ∂ 3 w ∂w ∂ 3 w ∂w ∂ 3 w ∂ w +B11 2ν · + ν + + + ν · ∂x∂y ∂y ∂y3 ∂x ∂x3 ∂y ∂x2 ∂y ∂x ∂x∂y2 ∂ 2w ∂ 2w ∂B66 ∂ 2 w ∂w ∂w ∂ 2 w + −2B11 ν 2 +2 ∂y ∂x2 ∂x ∂x∂y ∂y ∂x ∂y2 2 ∂ 3 w ∂w ∂ 2 w ∂ 2 w ∂ 2w ∂w ∂ 3 w +2B66 · − + ; + ∂x2 ∂y ∂y ∂x2 ∂y2 ∂x∂y ∂x ∂x∂y2 2 2 ∂w 2 ∂w 2 ∂ 2w ∂ 2w 1 ∂w ∂w 1 P4 (w) = A11 2 +ν· +ν· + A11 2 2 ∂x ∂x ∂y 2 ∂y ∂y ∂x +2A66 ·
∂w ∂w ∂ 2 w ; ∂x ∂y ∂x∂y
∂u ∂ 2 w ∂u ∂ 2 w ∂u ∂ 2 w ; + ν · A11 + 2A66 · 2 2 ∂x ∂x ∂x ∂y ∂y ∂x∂y ∂v ∂ 2 w ∂v ∂ 2 w ∂v ∂ 2 w P6 (v, w) =A11 + ν · A11 + 2A66 · . 2 2 ∂y ∂y ∂y ∂x ∂x ∂x∂y
P5 (u, w) =A11
Equation (10) can be used to investigate nonlinear dynamic vibration of simply supported circular cylinder shell with variable thickness under mechanical load.
3 Solution Method This paper considers the variable thickness FGM cylindrical shell with simply supported at both ends, subjected to external uniformly distributed pressure q(t) and axial compression force in term of time p(t). The boundary conditions are: w = 0, Nxy = 0, Mxx = 0, Nxx = −p · h at x = 0 and x = L Displacement components of the cylindrical shell can be expanded as: u = Umn (t) cos αx sin βy; v = Vmn (t) sin αx cos βy; w = Wmn (t) sin αx sin βy (11) in which: α = respectively).
mπ L ;
β =
n R
(m, n - the half-waves number in x and y direction,
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Substituting Eq. (11) into Eq. (10) then applying Galerkin procedure yields ⎧ 2 ⎪ I11 U + I12 V + I13 W + R1 W 2 = ρ1∗∗ ddtU2 ⎪ ⎪ ⎪ ⎪ ∗∗ d 2 V 2 ⎪ ⎪ ⎨ I21 U + I22 V + I23 W + R2 W = ρ1 dt 2
pm2 π 3 (La + 2b)R 2 3 ·W I U + I V + I W + R W + R W + R U + R V + 32 33 3 4 6 5 ⎪ 31 ⎪ 8L ⎪ ⎪ ⎪ ⎪ 4δn δm RLq d 2W dW ⎪ ⎩ + = ρ1∗∗ 2 + 2ερ1∗∗ mnπ dt dt (12) in which: ρ1∗ L(La + 2b)π R (−1)n − 1 (−1)m − 1 ; δn = ; δm = ; 2 2 8 E1 π (ν − 1)n2 L2 − 2π 2 R2 m2 (La + 2b) ; = 16RL(1 − ν 2 )
ρ1∗∗ = I11
E1 mπ 2 n(La + 2b) ; 16(ν − 1) E1 νmπ 2 (La + 2b) E2 π 2 R2 m2 + L2 n2 ν E2 n2 ; = − + 24(1 + ν)mR 8 ν2 − 1 24L2 Rm ν 2 − 1 2E1 2π 2 R2 m2 − L2 n2 ν (La + 2b) E1 n(La + 2b) R1 = ; − 9(1 + ν)R 9L2 Rn ν 2 − 1 E1 π (La + 2b) 2L2 n2 − R2 m2 (ν − 1)π 2 I22 = ; 16(ν 2 − 1)LR
I12 =I21 = I13
E1 π Ln(La + 2b) 8R 1 − ν 2 E2 n 3L4 a2 n2 − n2 R2 m2 π 2 + L2 − 3L2 a2 + 3(2ν − 1)m2 π 2 L2 R2 a2 ; + 24(ν 2 − 1)R2 Lm2 π 2E1 2L2 n2 − π 2 R2 m2 ν (La + 2b) E1 mπ(La + 2b) R2 = − ; 9R2 Lmπ(ν 2 − 1) 9(1 + ν))L νE1 mπ 2 (La + 2b) E2 m2 π 2 R2 + L2 n2 ν − 3L2 a2 − 3L2 a2 E2 n2 ; =− − + 24m(1 + ν)R 8 1 − ν2 24L2 Rm ν 2 − 1 E2 nπ E1 Lnπ(La + 2b) E2 L − 6L2 a2 n3 − I32 = + ; 24m2 π R2 (ν 2 − 1) 24(1 + ν)L 8R ν 2 − 1 I23 = −
I31
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E1 L(La + 2b)π 8R ν 2 − 1 ! " E2 3L4 a2 n2 − νR2 m2 π 2 + 2n2 L2 − 3L2 a2 − 6m2 π 2 (2R − 3)νR2 L2 a2 I33 =
+
+ 12R2 Lπ m2 (ν 2 − 1) ! " ! " 4 4 2 2 2 2 2 2 ∗ 2 E3 (La + 2b) m π R R m π + 2n L ν − 3L R2 a2 + L4 n4 m2 π 2 ∗ + 6νR2 a2 − 3L2 a2 16π m2 (ν 2 − 1)R3 L3 −
E3 π(m2 ∗ π 2 + 3L2 a2 )(La + 2b) ; 8(1 + ν)LR
2E1 5L2 n2 − 3π 2 R2 m2 ν (La + 2b) 8a2 E2 L2 n2 ν − 11R2 m2 π 2 + R3 = − 9LR2 mnπ(ν 2 − 1) 27LRmnπ(ν 2 − 1) # 4 4 4 $ 4 4 2 2 ∗ 2 2 4E2 9 R m π + L n m π − 4L a 7R4 m4 π 4 − 10L4 n4 + 81L3 R3 m3 π 3 n(ν 2 − 1) 2 2 ∗ 8E2 nν 9m2 ∗ π 2 − 34L2 a2 4E2 nν( 9m π − 44L2 a2 + + 27LRmπ(ν 2 − 1) 81LRmπ(ν 2 − 1) 2E2 n( 4L2 a2 − 9m2 π 2 ∗ 16E2 a2 Ln + + 27Rmπ(1 + ν) 81LRmπ(1 + ν) 4 4 4 3E1 π π R m + 2L2 π 2 R2 m2 n2 ν + L4 n4 (La + 2b) E1 m2 π 3 n2 (La + 2b) + R4 = 256R3 L3 (ν 2 − 1) 128L(1 + ν)R 2 2 2 8E1 π R m + L2 n2 ν (La + 2b) 2E1 n(La + 2b) + ; R5 = − 9L2 Rn(ν 2 − 1) 9(1 + ν)R 2 2 2 8E1 νR m π + L2 n2 (La + 2b) 2E1 mπ (La + 2b) R6 = ; + 9L(1 + ν) 9R2 Lmπ 1 − ν 2 = 2π 2 m2 L2 a2 + 3Lab + 3b2 + 3L2 a2 ; ∗ = L2 a2 + 2Lab + 2b2 ; According to Volmir’s assume [30] by ignoring the inertial components along x and y axes (u ω0 ), the relationship between velocity and deflection of variable thickness FGM shell is complex curves. When increase the amplitude of excitation force to great value, the velocity-deflection relationship become disturbed curves (Fig. 10).
Fig. 10. The dw/dt-w relationship of the shell
5 Conclusions In present article, motion equations of FGM cylindrical shells with variable thickness is presented based on the classical shell theory, taking into account the nonlinear geometry of von Karman-Donnell. Nonlinear differential equation of the shell is obtained by
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using Galerkin method. Nonlinear dynamic responses of the shell are determined by the fourth-order Runge-Kutta method and Matlab software. Some conclusions can be inferred from results of present analysis: • Fundamental frequencies of variable thickness FGM shell depend on volume fraction index (k) and the vibration mode (m, n). The lowest natural frequencies of the shell corresponding to mode shape (m, n) = (1, 7). • Geometric parameters of the shell (L, R, h1 , h0 ) are factors effect on vibration amplitude of the structure. When geometric dimensions of the shell increase (L, R), vibration amplitude of the shell will increases, it means, the stiffness of structure decreases. • When the frequencies of external force are near to the natural frequencies of the shell, the beating phenomenon is observed, the deflection-velocity relationship is closed curve. • When the frequency and amplitude of excitation force are far from the natural frequency of the shell, the velocity-deflection relationship becomes disturbed curves.
Acknowledgments. This research is funded by National Foundation for Science and Technology Development of Vietnam (NAFOSTED) under Grant number 107.02-2018.324.
References 1. Loy, C.T., Lam, K.Y., Reddy, J.N.: Vibration of functionally graded cylindrical shells. Int. J. Mech. Sci. 41, 309–324 (1999) 2. Pradhan, S.C., Loy, C.T., Lam, K.Y., Reddy, J.N.: Vibration characteristics of functionally graded cylindrical shells under various boundary conditions. Appl. Acoust. 61, 111–129 (2000) 3. Haddadpour, H., Mahmoudkhani, S., Navazi, H.M.: Free vibration analysis of functionally graded cylindrical shells including thermal effects. Thin Walled Struct. 45, 591–599 (2007) 4. Matsunaga, H.: Free vibration and stability of functionally graded circular cylindrical shells according to 2D higher-order deformation theory. Compos. Struct. 88, 519–531 (2009) 5. Sofiyev, A.H.: On the vibration and stability of clamped FGM conical shells under external loads. J. Compos. Mater. 45, 771–788 (2011) 6. Mollarazi, H.R., Foroutan, M., Dastjerdi, R.M.: Analysis of free vibration of functionally graded material (FGM) cylinders by a meshless method. J. Compos. Mater. 46, 507–515 (2011) 7. Bich, D.H., Nguyen, N.X.: Nonlinear vibration of functionally grade circular cylindrical shells based on improved Donnell equations. J. Sound Vib. 331(25), 5488–5501 (2012) 8. Avramov, K.V.: Nonlinear modes of vibrations for simply supported cylindrical shell with geometrical nonlinearity. Acta Mech. 223, 279–292 (2012) 9. Malekzadeh, P., Heydarpour, Y.: Free vibration analysis of rotating functionally graded truncated conical shells. Compos. Struct. 97, 176–188 (2013) 10. Heydarpour, Y., Malekzadeh, P., Aghdam, M.M.: Free vibration of functionally graded truncated conical shells under internal pressure. Meccanica 49(2), 267–282 (2013) 11. Young-Wann, K.: Free vibration analysis of FGM cylindrical shell partially resting on Pasternak elastic foundation with an oblique edge. Compos. Part B Eng. 70, 263–276 (2015)
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12. Quan, T.Q., Duc, N.D.: Nonlinear vibration and dynamic response of shear deformable imperfect functionally graded double-curved shallow shells resting on elastic foundations in thermal environments. J. Therm. Stresses 39(4), 437–459 (2016) 13. Phu, K.V., Bich, D.H., Doan, L.X.: Analysis of nonlinear thermal dynamic responses of sandwich functionally graded cylindrical shells containing fluid. J. Sandwich Struct. Mater. 21(6), 1953–1974 (2017) 14. Phu, K.V., Bich, D.H., Doan, L.X.: Nonlinear thermal vibration and dynamic buckling of eccentrically stiffened sandwich-FGM cylindrical shells containing fluid. J. Reinf. Plast. Compos. 38(6), 253–266 (2019) 15. Dat, N.D., Khoa, N.D., Nguyen, P.D., Duc, N.D.: An analytical solution for nonlinear dynamic response and vibration of FG-CNT reinforced nanocomposite elliptical cylindrical shells resting on elastic foundations. ZAMM J. Appl. Math. Mech./Z. Angew. Math. Mech. 100(1), e201800238 (2020). https://doi.org/10.1002/zamm.201800238 16. Han, Y., Zhu, X., Li, T., Yu, Y., Hu, X.: Free vibration and elastic critical load of functionally graded material thin cylindrical shells under internal pressure. Int. J. Struct. Stab. Dyn. 18(11), 1850138 (2018). https://doi.org/10.1142/S0219455418501389 17. Ganesan, N., Sivadas, K.R.: Free vibration of cantilever circular cylindrical shells with variable thickness. Comput. Struct. 34(4), 669–677 (1990) 18. Sivadas, K.R., Ganesan, N.: Vibration analysis of laminated conical shells with variable thickness. J. Sound Vib. 148(3), 477–491 (1991) 19. Sivadas, K.R., Ganesan, N.: Asymmetric vibration analysis of thick composite circular cylindrical shells with variable thickness. Comput. Struct. 38(5–6), 627–635 (1991) 20. Abbas, L.K., Lei, M., Rui, X.: Natural vibrations of open-variable thickness circular cylindrical shells in high temperature field. J. Aerosp. Eng. 23(3), 205–212 (2010) 21. Kang, J.H., Leissa, A.W.: Three-dimensional vibrations of thick spherical shell segments with variable thickness. Int. J. Solids Struct. 37(35), 4811–4823 (2010) 22. Kang, J.H.: Three-dimensional vibration analysis of joined thick conical-cylindrical shells of revolution with variable thickness. J. Sound Vib. 331(18), 4187–4198 (2012) 23. Efraim, E., Eisenberger, M.: Dynamic stiffness vibration analysis of thick spherical shell segments with variable thickness. J. Mech. Mater. Struct. 5(5), 821–835 (2010) 24. Ataabadi, P.B., Khedmati, M.R., Ataabadi, M.B.: Free vibration analysis of orthtropic thin cylindrical shells with variable thickness by using spline functions. Lat. Am. J. Solids Struct. 11(12), 2099–2121 (2014) 25. Awrejcewicz, J., Kurpa, L., Shmatko, T.: Investigating geometrically nonlinear vibrations of laminated shallow shells with layers of variable thickness via the R-functions theory. Compos. Struct. 125, 575–585 (2015) 26. Ghannad, M., Rahimi, G.H., Nejad, M.Z.: Elastic analysis of pressurized thick cylindrical shells with variable thickness made of functionally graded materials. Compos. B 45(1), 388– 396 (2013) 27. Tornabene, F., Fantuzzi, N., Bacciocchi, M., Viola, E., Reddy, J.: A numerical investigation on the natural frequencies of FGM sandwich shells with variable thickness by the local generalized differential quadrature method. Appl. Sci. 7(2), 131 (2017). https://doi.org/10. 3390/app7020131 28. Golpayegani, I., Arani, E., Foroughifar, A.: Finite element vibration analysis of variable thickness thin cylindrical FGM shells under various boundary conditions. Mater. Perform. Charact. 8(1), 491–502 (2019) 29. Brush, D.O., et al.: Buckling of Bars, Plates and Shells. Mc Graw-Hill Inc., New York (1975) 30. Volmir, S.: Nonlinear Dynamics of Plates and Shells. Science Edition, Moscow (1972)
Nonlinear Dynamic Stability of Variable Thickness FGM Cylindrical Shells Subjected to Mechanical Load Khuc Van Phu1(B) , Dao Huy Bich1 , and Le Xuan Doan2 1 Vietnam National University, Hanoi, Vietnam 2 Tran Dai Nghia University, Ho Chi Minh city, Vietnam
Abstract. The main purpose of this paper is to study on nonlinear dynamic stability of FGM cylindrical shell with variable thickness under mechanical load. The governing equations of this structure are established based on the classical shell theory and take into account the geometrical nonlinearity in von KarmanDonnell sense. Nonlinear dynamic responses of the shell are obtained by using the fourth-order Runge-Kutta method and Galerkin method. Dynamic critical loads are determined according to Budiansky-Roth criterion. Effect of material and geometric dimension on dynamic critical load and nonlinear dynamic responses of the shell are also considered. Keywords: Dynamic stability · Variable thickness · FGM cylindrical shell · Dynamics responses
1 Introduction FGM shells are special structures and frequently used in modern industries such as: aerospace, aircraft, space vehicles and in other important engineering structures. Recently, Variable thickness structure becomes more and more popular in important industries due to its preeminent properties. It helps to reduce the weight of structure and saves materials while ensuring the bearing capacity. Study on the dynamic stability of FGM cylindrical shell is necessary to ensure the structure works efficiently and reliably. This topic has attracted the interest of many scientists in research and discussion. The nonlinear static and dynamic stability of FGM shell structure has been analyzed by several scientists such as Shen [1] studied the post-buckling of finite-length FGM cylindrical thin shell subjected to axial compression loads in thermal environments. Research was carried out based on the classical shell theory. Buckling critical loads were determined by using singular perturbation technique. Sofiyev et al. [2] solved dynamic stability problems of FGM cylindrical thin shells subjected to linear torsional load. Galerkin’s method and Lagrange–Hamilton principle were employed to determine dynamic critical load. Also by using Galerkin method, Sofiyev [3–5] investigated dynamic buckling of FGM truncated conical and cylindrical shells resting on Pasternak foundations under axial compressive load, axial tension and hydrostatic pressure. Dynamic critical loads were obtained by using Galerkin method and superposition © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 506–521, 2022. https://doi.org/10.1007/978-981-16-3239-6_38
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method. Golchi et al. [6] based on the first-order shear deformation theory (FSDT) and Hamilton’s principle to analyze thermal buckling of stiffened FGM truncated conical shells. The obtained governing equations were solved by using differential quadrature method (DQM). Dynamic buckling problem of suddenly-heated FGM cylindrical shell subjected to mechanically load was studied by Shariyat. [7]. Dynamic buckling load was detected according to modified Budiansky criterion. In this work, the effect of initial geometrical imperfection was taken into account. Darabi et al. [8] based on large deflection theory to investigate dynamic stability of FGM shells under periodic axial load using Galerkin method and Bolotin’s method. Sheng et al. [9] presented an analysis of dynamic stability of FGM cylindrical shells embedded in elastic media based on FSDT. Effect of transverse shear strains and rotary inertia were taken into account. Bagherizadeh et al. [10] studied the mechanical buckling of simply-supported FG cylindrical shell surrounded by elastic media and subjected to mechanical loads. The governing equations are established based on a higher-order shear deformation theory (HSDT). Nonlinear static and dynamic buckling problems of FGM cylindrical shells subjected to mechanical load were solved by Huang et al. [11–13]. The Donnell shell theory, large deflection theory, energy method and the four-order Runge–Kutta method are employed to investigate nonlinear dynamic responses of structures, the critical loads were determined according to Budiansky-Roth criterion. Najafizadeh et al. [14] presented an analysis which is concerned with elastic stability of stiffened FGM circular cylinder shells subjected to axial compression loading using the Sander’s assumption. Dung et al. [15–19] focused on using semi-analytical approach to investigate nonlinear dynamic buckling of stiffened FGM thin cylindrical shells surrounded by elastic foundations and subjected to mechanical load in thermal environments. Equilibrium equations were derived based on the classical shell theory, Donnell’s shell theory and smeared stiffeners technique. Nonlinear dynamic responses of structures were obtained by using Galerkin method and fourth order Runge–Kutta method. Nonlinear dynamic critical loads were determined according to Budiansky–Roth criterion. By similar approach, Nam et al. [20, 21] studied the nonlinear buckling of sandwich-FGM cylindrical shells reinforced by spiral stiffeners under mechanical-thermal load. Khoa et al. [22] studied the nonlinear dynamic buckling of imperfect piezoelectric S-FGM cylindrical shells subjected to electrical and mechanical loads, resting on elastic medium in thermal environment. The Reddy’s third-order shear deformation theory and Galerkin method were employed to solve problems. Heydarpour et al. [23] proposed an analysis into dynamic stability of rotating FGM shells reinforced by carbon nanotube under mechanical load based on FSDT and DQM. Phu et al. [24, 25] investigated dynamic buckling of sandwich-FGM shells and stiffened sandwich-FGM cylindrical shells filled with fluid, surrounded by elastic foundations in thermal environment based on the classical shell theory and the smeared stiffener technique. Dynamic responses of structures were obtained by using Galerkin method and Runge–Kutta method. The dynamic critical load was determined according to Budiansky-Roth criterion. Recently, Zhang et al. [26] analyzed the dynamic buckling of FGM cylindrical shells under thermal shock based on the Hamiltonian principle. Thermal buckling problem of FGM cylindrical shells with properties of material depending on temperature was solved by Huang et al. [27].
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Until this time, there are very little publications investigate on dynamic stability of variable thickness structures. Some typical authors such as Luong et al. [28, 29] based on the small deflection theory and the thin shell theory to investigate buckling of simply supported cylindrical panel with small changes in the shell thickness subjected to mechanical load. On dynamic stability of variable thickness shell made from FGM, Shariyat et al. [30] proposed an analysis of nonlinear thermal buckling of imperfect, variable thickness cylindrical shells made of bidirectional FGM subjected to thermal load. Governing equations were established by using the third-order shear-deformation theory and nonlinear finite element method. Buckling critical load was determined according to modified Budiansky’s criterion. Next, Thang et al. [31] presented an investigation on effects of variation of the thickness on buckling of imperfect S-FGM cylindrical panels under axial compressive load and external distributed pressure. The governing equations were derived by using classical shell theory and Galerkin procedure. From above reviews, according to authors’ knowledge, analysis on nonlinear dynamic stability of variable thickness FGM cylindrical shell subjected to mechanical load has no investigation published yet. In present article, the governing equations of this structure are established based on the classical shell theory and take into account the geometrical nonlinearity in von Karman-Donnell sense. Nonlinear dynamic responses of the shell are obtained by using Runge-Kutta method and Galerkin method. Dynamic critical loads are determined according to Budiansky-Roth. Effect of material, geometric dimension and loading speed on dynamic critical loads and nonlinear dynamic responses of the shell are also considered in detail.
2 Governing Equations Consider a variable thickness FGM cylindrical shell with geometric dimensions and bearing as shown in Fig. 1. Assume that the radius of shell is much larger than the thickness (R >> h), Thickness of the shell h can be determined as: h(x) = a.x + b. in which: a = (h1 −h0 )/L; b = h0 . Assumes that V c and V m are volume fractions of ceramic and metal, respectively, are related by V c + V m = 1. In which, the volume fraction of ceramic is expressed as: z k 1 + (1) Vc (z) = 2 h(x) Effective properties of materials (E, ρ, ν…etc.) can be expressed as: P(z) = Pm .Vm (z) + Pc .Vc (z) = Pm + (Pc − Pm )Vc (z);
(2)
According to classical shell theory [28], the strain-displacement relationship of the shell in form as: 0 + 2z.kxy εx = εx0 + z.kx ; εy = εy0 + z.ky ; γxy = γxy
in which 0 = εxx
kxx =
∂u0 1 ∂w 2 0 ∂x + 2 ∂x ; εyy 2 − ∂∂xw2 ; kyy
= =
∂v0 w ∂y − R 2 − ∂∂yw2 ;
+
1 2
∂w ∂y
2
0 = ; γxy
∂u0 ∂y
+
∂v0 ∂x
∂2w
kxy = −2 ∂x∂y
(3)
+
∂w ∂w ∂x ∂y
(4)
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Fig. 1. Variable thickness FGM cylindrical shell
Hooke’s law for FGM shell subjected to mechanical load is defined as: ⎫ ⎫ ⎧ ⎧ εx + νεy ⎬ ⎨ σxx ⎬ E(z) ⎨ = σ νεx + εy 2 ⎩ ⎭ ⎩ yy ⎭ 1 − υ(z) τxy (1 − ν).γxy /2
(5)
Integrating Stress-Strain relationship through the thickness of the shell, internal force and moment resultants of variable FGM shell can be obtained as follows: ⎧ 0 0 ⎪ ⎪ ⎨ Nxx = A11 εxx + υ.εyy + B11 kxx + ν.kyy 0 + υ.ε 0 + B (6) Nyy = A11 εyy 11 kyy + ν.kxx xx ⎪ ⎪ ⎩ 0 + B .k Nxy = A66 .γxy 66 xy ⎧ 0 0 +D ⎪ ⎪ Mxx = B11 εxx + ν.εyy 11 kxx + ν.kyy ⎨ 0 + ν.ε 0 + D (7) M ε = B yy 11 11 kyy + ν.kxx yy xx ⎪ ⎪ ⎩ 0 + D .k Mxy = B66 .γxy 66 xy Stiffness coefficients Aij , Bij , Dij in Eq. (6, 7) see Appendix 1. Nonlinear motion equations of FGM cylindrical shell with variable thickness subjected to external pressure q(t) and an axial compression p(t) based on the classical shell theory [33] are expressed as: ⎧ ∂Nxy ∂Nxx ∂2u ⎪ ⎪ ∂x + ∂y = ρ1 ∂t 2 ⎪ ⎪ ⎪ ⎨ ∂Nxy + ∂Nyy = ρ1 ∂ 22v ∂x ∂y ∂t (8) ∂ 2 Mxy ∂ 2 Myy ∂ 2 Mxx ∂2w ∂2w ⎪ + 2 + ⎪ 2 2 + Nxx . ∂x 2 + 2Nxy ∂x∂y ∂x∂y ⎪ ∂x ∂y ⎪ ⎪ ⎩ + N ∂ 2 w + Ny − p.h(x) ∂ 2 w + q = ρ ∂ 2 w + 2ερ ∂w yy ∂y2 1 ∂t 2 1 ∂t R ∂x2 −ρm h(x) = ρ1∗ h(x) . in which: ρ1 = ρm + ρck+1 Substituting Eq. (4), Eq. (6) and Eq. (7) into Eq. (8) we obtain: ⎧ 2 L11 (u) + L12 (v) + L13 (w) + P1 (w) = ρ1 ∂∂t 2u ⎪ ⎪ ⎪ 2 ⎨ L21 (u) + L22 (v) + L23 (w) + P2 (w) = ρ1 ∂∂t 2v (9) ⎪ L31 (u) + L32 (v) + L33 (w) + P3 (w) + P4 (u, w) ⎪ ⎪ ⎩ 2 2 ∂w + P5 (v, w) + q + p.h(x, y) ∂∂xw2 = ρ1 ∂∂tw 2 + 2ερ1 ∂t
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The coefficients Lij (i, j = 1, 3), Pk (k = 1, 5) in Eq. (9) see Appendix 2. Equation (9) are governing equations used to investigate nonlinear dynamic stability of variable thickness FGM circular cylinder shell subjected to mechanical load.
3 Solution Method The present article considers a variable thickness FGM cylindrical shell with simply supported at both ends, subjected to uniformly pressure q(t) and axial compression force p(t). The boundary conditions are: w = 0, Nxy = 0, Mxx = 0, Nxx = −p.h at x = 0 and x = L. Displacement components of the cylindrical shell can be expanded as follows: u = Umn (t)cosαx sin βy; v = Vmn (t) sin αxcosβy; w = Wmn (t) sin αx sin βy (10) n in which: α = mπ L ; β = R (m, n - the half-waves number in x and y direction, respectively). Substituting Eq. (10) into Eq. (9) then applying Galerkin procedure yields: ⎧ 2 ⎪ I11 U + I12 V + I13 W + R1 W 2 = ρ1∗∗ ddtU2 ⎪ ⎪ ⎪ ⎨ I21 U + I22 V + I23 W + R2 W 2 = ρ ∗∗ d 2 V 1 dt 2 2 3 2 ⎪ I31 U + I32 V + I33 W + R3 W + R4 W 3 + R5 U + R6 V + pm π (La+2b)R .W ⎪ 8L ⎪ ⎪ ⎩ 4δn δm RLq d 2W dW ∗∗ ∗∗ + mnπ = ρ1 dt 2 + 2ερ1 dt (11)
The coefficients Iij (i, j = 1, 3), Rk (k = 1, 6) in Eq. (11) see Appendix 3. According to Volmir’s assumption [34], by ignoring inertial components along x and y axes (u F f− ). It is contrasted to the experimental study in [10, 11], where F f+ < F f− . Keeping the preset friction force, F f0 unchanged, the anisotropic friction condition will increase when raising the value of inclined angle. In this study, θ was considered as an adjustable parameter. 2.2 Experimental Implementation The above model (Fig. 2(b)) was realized as shown in Fig. 3, as developing from the apparatus built at TNUT’s laboratory by Nguyen et al. [3, 8, 12].
Fig. 3. (a) Experimental diagram and (b) a photograph of the experimental apparatus
A mini electro-dynamical shaker (1) was placed on a slider of a commercial linear bearing guide (4), providing a tiny rolling friction force. Since the friction force inside the slider is much smaller than the preset friction, the effect of slider type on the experimental results is also small and can be neglected. The slider support can be adjusted to provide a certain inclined angle θ. An additional mass (2) was clamped on the shaker shaft with the support of sheet springs. Generally, applying a sinusoidal current with frequency f exc to the shaker leads to relative linear oscillation of the shaker shaft with the mass added on. Hereafter, the moveable mass, combined by the addition mass and the shaker shaft, is assigned as inertial mass, m1 , playing the role of the internal mass of the system. A noncontact position sensor (7) was used to measure the relative motion of the inertial mass m1 . The motion of the shaker body was recognized by a linear variable displacement transformer. The body shaker, including the sensors and the carbon tube, is referred as the mass m2 . A force sensor (3) was used as the obstacle block to measure the impact force. In order to preset the friction force when remaining the body mass m2 , a carbon tube (5) is connected with the shaker body by means of a flexible joint, avoiding any misalignment when moving. The detailed mechanism of providing preset friction is depicted in Fig. 4. As depicted in detail on Fig. 4a, the carbon tube is able to slide between two aluminium pieces in the form of a V-block (6). The two V-blocks are fixed on two electromagnets (7). Supplying a certain value of electrical current to the coupled electromagnets provides a desired clamping force on the tube and thus a corresponding value of sliding
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Fig. 4. Varying the friction force: (a) apparatus structure and (b) the dependency of friction force on the supplied voltage.
friction (Fig. 4b). The preset friction force was measured by pulling the body to move at a steady speed. By adjusting the voltage supplied to the coupled electromagnets, the corresponding friction force can be obtained (See [3] for more detailed). The shaker is powered by a sinusoidal current generated by a laboratory function generator and then amplified by a commercial amplifier. The application of a sinusoidal current to the shaker leads to oscillation of the shaker shaft and the mass attached on it (i.e. inertial mass m1 ). As provided by the shaker supplier, the magnetic force, F m is solely depended on the current supplied. Adjusting the sinusoidal supplying to the amplifier can provide a desired excitation force. A supplementary experiment was implemented to verify the relation of the magnetic force and the current supplied. A load cell was used as an obstacle resisting the shaker movement and thus to measure the magnetic force induced. A DC voltage was supplied to the shaker to generate the magnetic force. Varying the voltage, several pairs of the current passing the shaker and the force were collected. Experimental data confirmed that the excitation force is proportional to the current supplied to the shaker (See [8] for detailed information of how to determine this relation). For operational parameters, two levels of the preset friction force, F f0 were set as 6.8 N and 13.6 N. With respect to the total weight of the two masses as 2.336 kg, such levels of friction force correspond to two levels of friction coefficient as approximately be 0.3 and 0.6. In addition, two levels of the inclined angle θ were implemented at 2.5° Table 1. Parameters of the apparatus. Parameter
Notation
Value
Unit
Inertial mass
m1
0.518
kg
Body mass
m2
1.818
kg
Preset resistant force Ff0
6.8 and 13.6 N
Force ratio
α = Fm/Ff0 0.59
Inclined angle
θ
2.5 and 5.0
°
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and 5.0°. The force ratio, α is the ratio between excitation force amplitude F m and the preset resistant force F f0 . In this study, the force ratio was chosen, as 0.59, and unchanged during all experiments. This value was practically found to effectively drive the system moving. The overall experimental parameters are given in Table 1.
3 Results and Discussion Firstly, the progression velocities of the whole system are considered. Figure 5 presents the average velocities of the system body when varying the preset friction force F f0 , corresponding to two levels of the inclined angle θ.
Fig. 5. The average velocities of the locomotion system for: (a) θ = 2.5°; (b) θ = 5.0°.
As can be seen, when the excitation frequency f exc is less than 10 Hz, at each value of the inclined angle θ, i.e. each friction force ratio, the body mass m2 almost moves forward slowly. Also, larger preset friction force resulted in faster moving forward of the system. When increasing the excitation frequency larger than 10 Hz, the average velocity of the system significantly changed. The following interesting issues can be remarked: – In the case of applying the smaller friction ratio, corresponding to the inclined angle θ = 2.5° (Fig. 5a), with larger preset friction force (F f0 = 13.6 N), increasing excitation frequency in the range f exc ∈ [10, 17] Hz resulted in faster forward motion of the system. However, for smaller preset friction (F f0 = 6.8 N), this relationship was only true for the range of f exc ∈ [10, 13] Hz. Operation with larger excitation frequency reduced the moving forward velocity of the system. With the excitation larger than 19 Hz for large preset friction force, and 16 Hz for small preset friction force, the system started moving backward. – A similar phenomenon can be observed with higher inclined angle, θ = 5.0° (Fig. 5b). However, the range of the excitation frequency where the forward velocity increased with higher frequency become narrower compared to that in previous case. The highest forward velocity appeared at f exc = 15 Hz with high preset friction force (F f0 = 13.6 N). For smaller preset friction force (F f0 = 6.8 N), the system started moving backward if the excitation frequency is equal to or larger than 11 Hz.
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In order to give deeper insight into the system behavior, Fig. 6 illustrates the time history of the two masses’ motion for two cases: one for forward movement and another for backward movement. The two sub-plots have the same inclined angle (θ = 5.0°) and other parameters, except the preset friction force and the excitation frequency. In each sub-plots, the oscillation period of both masses m1 and m2 on each sub-plot were limitted by two impact instants. The oscillation of the mass m2 between two impact instants was divided into four areas: area (I) where the body mass m2 moved forward just after impact; area (II) where the body mass m2 moved backward; area (III) where body mass m2 may move forward or backward direction; and area (IV) where the body mass m2 moved backward. In Fig. 5, two impact instants are denoted by two red-dashed vertical lines; the areas between impact instants are separated by red-dot vertical lines. (a) Ff0 = 13.6 N; fexc = 15 Hz
(b) Ff0 = 6.8 N; fexc = 19 Hz
Fig. 6. Time history of motions of the internal mass X1 (black short dashed curve) and of the body X2 (blue solid curve) with θ = 5.0°: (a) Ff0 = 13.6 N; fexc = 15 Hz; and (b) Ff0 = 6.8 N; fexc = 19 Hz.
As can be seen on the Fig. 6, the impacts occurred at the instant of the peaks of the relative motion between the inertial mass m1 and the system body, m2 . The body mass m2 moved forward just after obtaining impact force from the inertial mass m1 . The both two masses started moving forward together (area I). Under the action of the preset friction force, the body mass m2 was quickly stopped. As a result of the gravity force and the inertial force of mass m1 , the system then started moving backward significantly in dynamic friction condition (area II). In the area III on the Fig. 5(b), the backward moving trend of the body mass applied by the smaller preset friction force seemed continuing. This could be suggested that at this instant, the inertial mass m1 would reached its limitted position of it amplitude, hence its velocity was slowdown, the inertial force of mass m1 acting on the mass m2 increased significantly. However, in the case of larger preset friction force, the velocity in backward direction of the inertial mass m1 increased, thus the inertial force acting body mass m2 decreased. This inertial force may be the reason make the body mass m2 moving forward in area III (Fig. 5(a)). In the first stage of area IV, both the gravity force and inertial force of mass m1 had the trend to pull the body mass m2 moving backward, so m2 moved very fast in backward direction. The width of the first stage of area IV is not the same, it is smaller on the Fig. 5(a) than that on the Fig. 5(b), maybe depending on the levels of preset friction force. In the second
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stage of area IV on the Fig. 5(a), the resultant of inertial and gravity force may be equal to the larger preset friction force. Thus it is seemed that the body mass m2 did not move. Even the body mass m2 tended to move forward because of the forward direction of the mass m1 . Decreasing the preset friction force, in the second stage of area IV, the mass m2 continued moving backward until the mass m1 moved forward (Fig. 6(b)). From these practical results, it could be concluded that the forward motion of the locomotion system in the case of anisotropic friction may strongly depend on the level of the preset friction force. The larger preset friction force is, the easier for the system to move forward. It is worth noting that, the mechanism behind of moving forward motion of the whole system in anisotropic frictional condition is still an open dynamic problem. The data obtained would be a good resource for further researches to explain why the system moves forward or backward direction in anisotropic frictional condition.
4 Conclusion This paper presented experimental results and several initial analyses on a vibrationdriven locomotion apparatus in anisotropic frictional condition. The experimental apparatus provided an ability of presetting the friction force and adjusting the difference between backward friction and forward friction by mean of an inclined angle of the sliding guide, while maintaining the weight of the whole system unchanged. A series of experimental tests were implemented, providing a deep insight of the locomotion system responses, both in progression velocity of the system and in the relative motions of the masses. The directional motion of the system in anisotropic friction condition was experimentally found to strongly depend on the level of the preset friction force. The larger initial friction force is, the easier and faster in forward direction of the locomotion system moves, even with high anisotropic friction condition, i.e. with larger inclined angle. The experimental results would be useful for further researches on the design and operation of vibration-driven locomotion systems. Acknowledgement. This research was funded by Vietnam Ministry of Education and Training, under the grant number B2019-TNA-04. The authors would like to express their thank to Thai Nguyen University of Technology, Thai Nguyen University for their supports.
References 1. Liu, P., Yu, H., Cang, S.: Modelling and analysis of dynamic frictional interactions of vibrodriven capsule systems with viscoelastic property. Eur. J. Mech. A. Solids 74, 16–25 (2019) 2. Yan, Y., et al.: Proof-of-concept prototype development of the self-propelled capsule system for pipeline inspection. Meccanica 53(8), 1997–2012 (2017). https://doi.org/10.1007/s11012017-0801-3 3. Nguyen, V.-D., La, N.-T.: An improvement of vibration-driven locomotion module for capsule robots. Mech. Based Des. Struct. Mach. 1–15 (2020) 4. Chernous’ko, F.L.: Analysis and optimization of the motion of a body controlled by means of a movable internal mass. J. Appl. Math. Mech. 70(6), 819–842 (2006)
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5. Pavlovskaia, E., et al.: Modelling of ground moling dynamics by an impact oscillator with a frictional slider. Meccanica 38(1), 85–97 (2003) 6. Nguyen, V.-D., Woo, K.-C., Pavlovskaia, E.: Experimental study and mathematical modelling of a new of vibro-impact moling device. Int. J. Non-Linear Mech. 43(6), 542–550 (2008) 7. Liu, Y., et al.: Modelling of a vibro-impact capsule system. Int. J. Mech. Sci. 66, 2–11 (2013) 8. Nguyen, V.-D., et al.: The effect of inertial mass and excitation frequency on a Duffing vibro-impact drifting system. Int. J. Mech. Sci. 124–125, 9–21 (2017) 9. Xu, J., Fang, H.: Stick-slip effect in a vibration-driven system with dry friction: sliding bifurcations and optimization. J. Appl. Mech. 81, 061001 (2014) 10. Xu, J., Fang, H.: Improving performance: recent progress on vibration-driven locomotion systems. Nonlinear Dyn. 98(4), 2651–2669 (2019). https://doi.org/10.1007/s11071-019-049 82-y 11. Du, Z., et al.: Experiments on vibration-driven stick-slip locomotion: a sliding bifurcation perspective. Mech. Syst. Sig. Process. 105, 261–275 (2018) 12. Nguyen, V.-D., et al.: Identification of the effective control parameter to enhance the progression rate of vibro-impact devices with drift. J. Vibr. Acoust. 140(1), 011001 (2017)
Investigation of Maneuvering Characteristics of High-Speed Catamaran Using CFD Simulation Thi Loan Mai, Myungjun Jeon, Seung Hyeon Lim, and Hyeon Kyu Yoon(B) Department of Naval Architecture and Marine Engineering, Changwon National University, Gyeongsangnam-do, Republic of Korea [email protected]
Abstract. Many studies have been done to estimate the maneuvering characteristics of commercial ships. Studies on a high-speed catamaran, there was a great number of theoretical, numerical studies, and experimental investigation, most of them have focused on the resistance performance. In addition, a CFD (Computational Fluid Dynamics) method has become a possible tool to predict hydrodynamics. This study focuses on predicting hydrodynamic maneuvering characteristics of high-speed catamaran by utilizing RANS (Reynold-Averaged Navier-Stokes) solver. The Delft 372 catamaran model was selected as the target hull to analyze hydrodynamic characteristics. Due to the high-speed condition and changeable attitude, the motion of the catamaran was complex. The comparisons of the obtained CFD results including the free surface effects in resistance performance with experimental data were shown a relatively good agreement and it could demonstrate that the presented method could be used for predicting hydrodynamic coefficients of high-speed catamaran. Then virtual captive model tests were performed to obtain hydrodynamic coefficients. The lower-order Fourier coefficients were applied to get the hydrodynamic coefficients in dynamic motion. The linear fitting to Fourier coefficients was observed very well using least square method in the harmonic motion. Keywords: Delft 372 catamaran · RANS solver · High-speed · Virtual captive model test · Hydrodynamic characteristics · Maneuvering simulation
1 Introduction The demand for high-speed catamaran has been increasing significantly in recent years for variety of purposes such as passenger, military, and commercial applications. Studies on high-speed catamaran, there was a great number of theoretical, numerical studies, and experimental investigation, however, most of them have focused on the resistance performance and seakeeping performance. Broglia et al. (2011) simulated to analyze the interference phenomena between monohull with focus on its dependence on the Reynold number using CFD numerical method. The flow around a high-speed vessel in both catamaran and monohull configuration was observed, in addition, the wave © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 546–559, 2022. https://doi.org/10.1007/978-981-16-3239-6_41
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patters, wave profiles, streamlines, surface pressure, and velocity fields were analyzed. In addition, they performed an experimental investigation of hull interference effects on the total resistance by changing the distance between monohull of a catamaran in calm water in 2014. On the other hand, the seakeeping investigation for high-speed catamaran has been done by this author in order to emphasize the influence of Froude on the maximum response of the vertical ship motions and the role of the nonlinear effects both on the ship motions and on the added resistance. Wave resistance for high-speed catamaran was investigated by Moraes et al. (2004), they applied two methods were the slender-body theory and the 3D panel method using Shipflow software to estimate the effects of catamaran hull spacing included the effects of shallow water on the wave resistance component. Regarding CFD simulation for estimating hydrodynamic characteristics, Su et al. (2012) utilized RANS numerical method to predict the motion and analyze the hydrodynamic performance of planning vessels at high speed. Liu et al. (2018) simulated a virtual captive model tests for KCS (KRISO Container Ship) using CFD to estimate linear and nonlinear hydrodynamic coefficients in the 3rd-order Abkowiz model. After that maneuvering simulation was predicted. Applying CFD in marine vehicles using STARCCM+ (Hajivand and Mousavizadegan 2015), OpenFOAM (Islam and Soares 2018), and Ansys FLUENT (Nguyen et al. 2018) have been done for predicting maneuvering characteristics. This paper focuses on estimating the maneuvering characteristics of Delft 372 catamaran at high-speed. Ansys FLUENT 20.1 is used for solving RANS equation. Resistance performance is conducted in order to validate the numerical method. The comparisons of obtained results in resistance performance with experimental data were demonstrated that the presented numerical method was appropriate. Therefore, virtual captive model tests as static drift, pure sway, pure yaw, and combined pure yaw with drift test is conducted in order to obtain hydrodynamic coefficients. The Fourier series is applied to analyze hydrodynamic coefficients in harmonic motion. Especially, the coupling coefficients are found at combined pure yaw with drift angle.
2 Governing Equations and Turbulence Modeling The homogenous multiphase Eulerian fluid approach is adopted in this study to describe the interface between the water and air, mathematically. Assumption, the flow around the ship is incompressible. The governing equations that need to be solved are the mass continuity equation and momentum equations, which are given in Eqs. (1) and (2) respectively. ∂ui =0 ∂xi
(1)
∂τij ∂ui ∂ui 1 ∂p ∂ 2 ui + uj =− +ν − + fi ∂t ∂xj ρ ∂xi ∂xi ∂xj ∂xj
(2)
where ui and uj are the average velocity components; p is the average pressure; ν is the kinematic viscosity; xi and xj are the ith and j th coordinates in the fluid domain
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respectively; and ρ is the water density; τij = ui uj is so-called the Reynolds stress tensor; ui and uj are the fluctuating components; fi is the external forces. In order to capture the wave pattern of the free surface, the volume of fluid (VOF) method is implemented. A transport equation in Eq. (3) is then solved for the advection of this scalar quantity, using the velocity files obtained from the solution of the NavierStokes equations at the last time step. ∂q u) = 0 + ∇(q ∂t 2
(3)
Equation (3) gives the volume fraction q for each phase in all computation cell where qk = 1; u is the velocity; ∇ is the gradient.
k=1
Furthermore, a k −ω SST (Shear Stress Transport) model turbulence model is applied to consider the viscous effects. In this turbulence model, k is the turbulence kinetic energy and ω is the dissipation rate of the turbulent energy. The two equation model of a k − ω SST is given by the following: ∂k ∂ ∂k ∂k ∗ (4) = Pk − β ωk + + ui (ν + σk νt ) ∂t ∂xi ∂xi ∂xi ∂ω ∂ω γ ∂ ∂ω σω2 ∂k ∂ω ∗ 2 = Pk − β ω + (5) + 2(1 − F1 ) + ui (ν + σω νt ) ∂t ∂xj μt ∂xi ∂xi ω ∂xi ∂xi
3 Case Study and Coordinate System The candidate ship presented in this study is Delft 372 catamaran model with symmetrical demi-hulls shape that originally used in TU-Delft by Van’t Veer (1988). The model hull of Delft 372 catamaran is depicted in Fig. 1 and the main particulars are given in Table 1.
Fig. 1. Delft 372 catamaran geometry
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Table 1. Main particular of Delft 372 catamaran Main particular
Symbol
Scale
Unit
Full scale
Model scale
–
1.00
33.33
Length perpendiculars
Lpp
m
100.00
3.00
Beam overall
BOA
m
31.33
0.94
Beam demi-hull
b
m
8.00
0.24
Distance between center of the demi-hull
H
m
23.33
0.70
Separation distance
s
m
15.33
0.46
Draft
T
m
5.00
0.15
Vertical center of gravity
KG
m
11.33
0.34
Longitudinal center of gravity
LCG
m
47.00
1.41
Wetted surface area
S
m2
–
1.945
Two right-hand coordinate system are adopted to define the kinematic and hydrodynamic forces acting on the Delft 372 catamaran as shown in Fig. 2. The earth-fixed coordinate system o0 –x 0 y0 is set up to define the ship motion. The body-fixed coordinate system o–xy is used for computing the hydrodynamic forces. The origin is located at the intersection of the waterline plane and the center-line plane at the mid-ship section. The equation of motion for maneuvering ship in 3DOF become: m(˙u − vr − xG r 2 ) = X m(˙v + ur + xG r˙ ) = Y I˙z r˙ − mxG (˙v + ur) = N
(6)
where m is ship mass; u, v, and r are the surge velocity, sway velocity, and yaw rate, respectively; u˙ , v˙ , and r˙ are the corresponding surge acceleration, sway acceleration, and angular acceleration; Iz is the moment of inertia about the z-axis; X , Y and N are the resultant forces acting on ship in surge force, sway force, and yaw moment, √ respectively; U is the ship speed defined as U = u2 + v2 ; β is the drift angle defined by β = tan−1 (−v / u). In addition, the hydrodynamic forces acting on the ship hull are expressed as the below equation: XH = X0 + Xu u + Xuu u2 + Xuuu u3 + Xvv v2 + Xrr r 2 + Xvr vr YH = Yv˙ v˙ + Yr˙ r˙ + Yv v + Yvvv v3 + Yr r + Yrrr r 3 + Yvvr v2 r + Yvrr vr 2 NH = Nv˙ v˙ + Nr˙ r˙ + Nv v + Nvvv v3 + Nr r + Nrrr r 3 + Nvvr v2 r + Nvrr vr 2
(7)
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Fig. 2. Coordinate system
4 Virtual Captive Model Test The maneuvering characteristics of the high-speed catamaran are estimated by performing a virtual captive model test in order to get the hydrodynamic force. The computational conditions are depicted in Table 2. Table 2. The computational conditions. Case
Froude number (-)
β(deg.)
vmax (m/s)
rmax (rad/s)
Resistance
0.1–0.8 *interval 0.05
–
–
–
Static Drift
0.45
−10°–10o *interval 2o
–
–
Pure Sway
0.45
–
0.179, 0.269, 0.358, 0.448
–
Pure Yaw
0.45
–
–
0.366, 0.448, 0.529, 0.610
Combined Pure Yaw-Drift
0.45
2°, 4°, 6°, 8o
–
0.366, 0.448, 0.529, 0.610
4.1 Static Drift Test The ship travels through the tank in oblique flow due to a given drift angle β in static drift test. The configuration in Fig. 3 and mathematical model Eq. (8) described static drift test: X = X0 + Xvv v2 Y = Y0 + Yv v + Yvvv v3 N = N0 + Nv v + Nvvv v3
(8)
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Fig. 3. Configuration of static drift test
4.2 Harmonic Test The harmonic test includes pure sway, pure yaw, and combined yaw with drift test. Pure sway test, the ship moves through the tank on straight-ahead course while it is oscillated from side to side. The pure sway motion expresses in terms of the velocities u = constant while r = 0 and v oscillates harmonically. The linear acceleration coefficients such as Yv˙ and Nv˙ will be estimated in this test. The configuration in Fig. 4 and hydrodynamic forces in Eq. (9) determined for pure sway motion: X = X0 + Xvv v2 Y = Y0 + Yv˙ v˙ + Yv v + Yvvv v3 N = N0 + Nv˙ v˙ + Nv v + Nvvv v3
(9)
Fig. 4. Configuration of pure sway test
In harmonic motion, the Frourier series method (Sakamoto et al. 2012) is ultilized to simplify the mathematical models Eq. (9). Since hamonic motion are prescribed by sine and cosine function, hence hydrodynamic forces can be rewritten as Fourier series with angular frequency ω: f = f0 +
3
fcn cos(nωt) +
n=1
3
fsn sin(nωt)
(10)
n=1
On the other hand, the harmonic forms are determined by replacing the motion equation (v = −vmax cos ωt, v˙ = v˙ max sin ωt) into Eq. (9). X = X0 + Xc2 cos 2ωt Y = Yc1 cos ωt + Ys1 sin ωt + Yc3 cos 3ω N = Nc1 cos ωt + Ns1 sin ωt + Nc3 cos 3ω
(11)
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Pure yaw test, the ship moves through the tank while it conducts a pure yaw motion, where it is forced to follow the tangent of the oscillating path. The term of velocities is expressed that v = 0, while r and u oscillate harmonically. The hydrodynamic coefficients as Xrr , Yr˙ , Yr , Yrrr , Nr˙ , Nr , and Nrrr are estimated in the test. The configuration Fig. 5 and hydrodynamic forces in Eq. (12) defined for pure yaw motion: X = X0 + Xrr r 2 Y = Y0 + Yr˙ r˙ + Yr r + Yrrr r 3 N = N0 + Nr˙ r˙ + Nr r + Nrrr r 3
(12)
Fig. 5. Configuration of pure yaw test
Combined yaw and drift is described the same as with pure yaw test, however, a drift angle is set on the motion in order to obtain a drift angle relative to the tangent of the oscillating path. The terms of velocity represents v = 0 and constant, while r and u oscillate harmonically. The coupling hydrodynamic coefficients as Xvr , Yvvr , Yvrr , Nvvr , and Nvrr are estimated in this test. The combined pure yaw with drift test is described in Fig. 6 and hydrodynamic forces are shown in the below equation: X = X0 + Xvv v2 + Xrr r 2 + Xvr vr Y = Y0 + Yv v + Yvvv v3 + Yr r + Yrrr r 3 + Yvvr v2 r + Yvrr vr 2 N = N0 + Nv v + Nvvv v3 + Nr r + Nrrr r 3 + Nvvr v2 r + Nvrr vr 2
(13)
Fig. 6. Configuration of combined yaw and drift
Corresponding to pure sway test, the harmonic forms of pure yaw and combined yaw and drift are obtained by substituting motion equation (r = rmax sin ωt, r˙ = r˙max cos ωt) into Eqs. (12) and (13). Finally, coefficient of hamonic test can obtain from Fourier seriers as shown in Table 3.
3 Ys1 = Yr r + Yrrr r 3 4 Yc1 = Yr˙ r˙ 1 Ys3 = − Yrrr r 3 4 1 Y0 = Y∗ + Yv v + Yvvv v3 + Yvrr vr 2 2 3 1 Ys1 = Yr r + Yrrr r 3 + Yvvr v2 r 4 2 Yc1 = Yr˙ r˙ 1 Yc2 = − Yvrr vr 2 2 1 Ys3 = − Yrrr r 3 4
1 X0 = X∗ + Xrr r 2 2 1 Xc2 = − Xrr r 2 2
1 X0 = X∗ + Xvv v2 + Xrr r 2 2 Xs1 = Xvr vr 1 Xc2 = − Xrr r 2 2
Pure yaw
Combined pure yaw with drift
1 N0 = N∗ + Nv v + Nvvv v3 + Nvrr vr 2 2 3 1 Ns1 = Nr r + Nrrr r 3 + Nvvr v2 r 4 2 Nc1 = Nr˙ r˙ 1 Nc2 = − Nvrr vr 2 2 1 Ns3 = − Nrrr r 3 4
3 Ns1 = Nr r + Nrrr r 3 4 Nc1 = Nr˙ r˙ 1 Ns3 = − Nrrr r 3 4
Ns1 = Nv˙ v˙ 1 Nc3 = − Nvvv v3 4
Ys1 = Yv˙ v˙ 1 Yc3 = − Yvvv v3 4
1 X0 = X∗ + Xvv v2 2 1 Xc2 = Xvv v2 2
Pure sway
N component 3 Nc1 = − Nv v + Nvvv v3 4
Y component 3 Yc1 = − Yv v + Yvvv v3 4
X component
Harmonic test
Table 3. Fourier coefficients for harmonic test Investigation of Maneuvering Characteristics of High-Speed Catamaran 553
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The obtained hydrodynamic forces are non-dimensionalized by ship speed, ship length and water density as follows, u v rL ; v = ; r = U U U X Y N X = ;Y = ; N = 2 2 2 2 0.5ρU L 0.5ρU L 0.5ρU 2 L3
u =
(14)
5 Numerical Modeling In order to simulate the virtual captive model tests, the CFD software ANSYS Fluent 2020R1 is applied. Incompressible unsteady RANS with k − ω SST turbulence model is used for simulating two-phase volume of the fluid technique. The volume of fluid (VOF) method is applied to capture the position of the free surface. In addition, a SIMPLE algorithm solves for pressure-velocity coupling. The y + value of 30 is estimated for Reynolds number of 7.3E+6. The catamaran is covered by a rectangular domain to simulate the captive model test. According to Practical Guidelines for ship CFD Application (ITTC 2011) the dimensions of the domain are chosen to be able to avoid backflow and side flow. Additionally, boundary conditions are needed to assign for ensuring the physical characteristics of a fluid problem. The wall boundary condition represents the object’s surface, it is so-called no-slip wall. Pressure-inlet with open channel flow is required to start the calculation. Pressure-outlet with open channel flow is usually treated as a far-field condition, where flow properties are almost unchanged and hydrostatic pressure is specified. The symmetry condition indicates that the normal velocity and gradients of all variables at the symmetry plane is zero. The mesh generation for calculation is shown in Fig. 7.
Fig. 7. Mesh generation
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6 Results and Discustions 6.1 Validation First at all, in order to estimate well the calculation of catamaran, a validation of the numerical method is conducted by comparing the results of CFD simulation with experimental results performed by Broglia et al. (2014). Resistance performance is simulated with a range of Froude numbers from 0.1–0.8. The comparison of the results in Figs. 8, 9, 10 and 11 shown the consistency between CFD and the experiment. Consequently, the CFD numerical method is appropriate to simulate the captive model tests.
Fig. 8. The total resistance
Fig. 9. The resistance coefficient
Fig. 10. The sinkage
Fig. 11. The trim
6.2 Hydrodynamic Forces After validation the virtual captive model test is performed. The Froude number of 0.45 is selected for simulating high-speed catamaran. The reason for this selection due to fast increasing resistance and trim angle at this point as illustrated in Figs. 8, 9, 10 and 11. Moreover, sinkage is largest at Froude number 0.45 in Fig. 10. The below figures show the results of hydrodynamic forces acting on catamaran for all the computed cases, which cover static drift, pure sway, pure yaw, and combined pure yaw with drift tests. The catamaran is symmetrical about the vertical center plane, the hydrodynamic forces acting on the hull in case of static drift are symmetrical in both positive and negative drift angles as shown in Fig. 12. The hydrodynamic forces in the case of harmonic motion are analyzed using the Fourier series. The added mass coefficients are obtained from
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Fig. 12. Hydrodynamic forces of static drift test
Fig. 13. Fitting in-phase values of hydrodynamic forces versus lateral acceleration in pure sway test
Fig. 14. Fitting in-phase values of hydrodynamic forces versus yaw rate in pure yaw test
in-phase values and damping coefficients are obtained from outphase values. On the other hand, the coupling coefficients are determined from the combined yaw with drift by subtracting forces acting on the hull in the single motions from the force acting on the hull in the combined motions. The obtained hydrodynamic forces are approximated using least square regression for each mathematical model in order to get hydrodynamic coefficients. Figures 13, 14, 15, 16 describe the hydrodynamic forces acting on the catamaran and fitting curves in the harmonic motion. It can be seen that the linear
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Fig. 15. Fitting out-phase values of hydrodynamic forces versus yaw rate in pure yaw test -2.5E-3
-5.0E-4
-3.0E-4 -2.0E-4 -1.0E-4 -0.0E+0
-2.0E-3 β= β= β= β=
-2.0E-3
Y'out-phase (-)
X'out (-)
-4.0E-4
2 deg. 4 deg. 6 deg. 8 deg.
2 deg. 4 deg. 6 deg. 8 deg.
-1.5E-3 -1.0E-3 -5.0E-4
0
0.1 0.2 0.3 0.4 Non-dimensional angular velocity (-)
0.0E+0
β= β= β= β=
-1.6E-3
N'out-phase (-)
β= β= β= β=
2 deg. 4 deg. 6 deg. 8 deg.
-1.2E-3 -8.0E-4 -4.0E-4
0
0.1 0.2 0.3 0.4 Non-dimensional angular velocity (-)
0.0E+0
0
0.1 0.2 0.3 0.4 Non-dimensional angular velocity (-)
Fig. 16. Fitting out-phase values of hydrodynamic forces versus yaw rate in combined yaw with drift test
Fig. 17. Wave pattern at various drift angle
reactions are fitting very well to the hydrodynamic forces. The wave pattern of static drift test in Fig. 17 illustrates cause asymmetry wave pattern around the ship which leads to sway force and yaw moment and as drift angle increases the asymmetry of the wave pattern is clearly observed. Finally, the hydrodynamic coefficients obtained from CFD simulation in this study are listed in Table 4.
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Coefficient X0 Xu Xuu Xuuu Xvv Xrr Xvr
Values
Coefficients
−2.41E−3
Yv˙ Yr˙ Yv Yvvv Yr Yrrr Yvvr Yvrr
6.60E−3 −5.42E−3 6.98E−4 −6.05E−2 3.07E−3 −9.15E−6
Values
Coefficients
Values
−4.92E−4
Nv˙ Nr˙ Nv Nvvv Nr Nrrr Nvvr Nvrr
2.71E−4
−2.45E−4 −1.36E−2 −4.18E−1 −3.01E−2 3.50E−2 5.72E−4 −7.64E−2
−8.33E−5 −5.84E−3 −6.82E−2 −1.34E−2 1.48E−2 5.81E−4 −1.04E−1
7 Conclusions In this paper, the CFD numerical method has been utilized for simulating the virtual captive model test of Delft 372 catamaran at the high-speed conditions. The straight and oblique motion was simulated in the stationary reference frame. The harmonic motions were implemented in the dynamic mesh motion using a user-defined function written by c programming. The resistance performance was conducted to validate the CFD numerical method, an agreement was observed between the CFD method and experiment results. Additionally, the fitting resistance result using the least square method was found in first-order, secondorder, and third-order terms. Harmonic motions were performed for obtaining the added mass coefficient in pure sway and pure yaw test. Especially, the coupling coefficients were estimated in combined yaw with drift test. The Fourier series was applied for predicting the hydrodynamic coefficients in the harmonic test. The linear fitting using the least square method was observed very well to the hydrodynamic forces in the harmonic test. The estimated coefficients will be used for predicting maneuvering characteristics in further study. Acknowledgement. Following are results of a study on the “Leaders in INdustry-university Cooperation + Project, supported by the Ministry of Education and National Research Foundation of Korea.
References Broglia, R., Bouscasse, B., Jacob, B., Olivieri, A., Zaghi, S., Stern, F.: Calm water and seakeeping investigation for fast catamaran. In: 11th International Conference on Fast Sea Transportation 2011, pp. 336–344 (2011) Broglia, R., Jacob, B., Zaghi, S., Stern, F., Olivieri, A.: Experimental investigation of interference effects for high-speed catamarans. Ocean Eng. 76, 75–85 (2014)
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Castiglione, T., Stern, F., Bova, S., Kandasamy, M.: Numerical investigation of the seakeeping behavior of a catamaran advancing in regular head waves. Ocean Eng. 38, 1806–1822 (2011) Ghadimi, P., Mirhosseini, S.H., Dashtimanesh, A.: RANS simulation of dynamic trim and sinkage of a planning hull. Appl. Math. Phys. 1, 6 (2013) Hajivand, A., Mousavizadegan, S.: Virtual simulation of maneuvering captive tests for a surface vessel. Int. J. Naval Archit. Ocean Eng. 7, 848–872 (2015) Islam, H., Soares, C.: Estimation of hydrodynamic derivatives of a container ship using PMM simulation in OpenFOAM. Ocean Eng. 164, 414–425 (2018) International Towing Tank Conference (ITTC): ITTC-Recommended Procedures and Guidelines Practical Guidelines for Ship CFD Application (2011) Jeon, M.J., Nguyen, T.T., Yoon, H.K.: A study on verification of the dynamic modeling for a submerged body based on numerical simulation. Int. J. Eng. Technol. Innov. 10, 107–120 (2020) Liu, Y., Zou, Z.J., Guo, H.P.: Predictions of ship maneuverability based on virtual captive model tests. Eng. Appl. Comput. Fluid Mech. 12, 334–353 (2018) Mai, T.L., Nguyen, T.T., Jeon, M.J., Yoon, H.K.: Analysis on hydrodynamic force acting on a catamaran at low speed using RANS numerical method. J. Navigat. Port Res. 2, 53–64 (2019) Milanov, E., Chotukova, V., Stern, F.: System based simulation of Delft372 catamaran maneuvering characteristics as function of water depth and approach speed. In: 29th Symposium on Naval Hydrodynamics (2012) Moraes, H.B., Vasconcellos, J.M., Latorre, R.G.: Wave resistance for high-speed catamarans. Ocean Eng. 31, 2253–2282 (2004) Nguyen, T.T., Yoon, H.K., Park, Y.B., Park, C.J.: Estimation of hydrodynamic derivatives of full-scale submarine using RANS solver. J. Ocean Eng. Technol. 32, 386–392 (2018) Su, Y.M., Chen, Q.T., Shen, H.L., Lu, W.: Numerical simulation of a planning vessel at high speed. J. Mar. Sci. Appl. 11, 178–183 (2012) Sakamoto, N.M., Carriace, P.M., Stern, F.: URANS simulations of statics and dynamic maneuvering for surface combatant: part 1. Verification and validation for forces, moment, and hydrodynamic derivatives. J. Mar. Sci. Technol. 17, 422–445 (2012) Van’t Veer, R.: Experimental results of motions, hydrodynamic coefficients and wave loads on the 372 catamaran model. Delft University Report 1129 (1998)
Optimum Tuning of Tuned Mass Dampers for Acceleration Control of Damped Structures Huu-Anh-Tuan Nguyen(B) Faculty of Civil Engineering, University of Architecture Ho Chi Minh City, Ho Chi Minh City, Vietnam [email protected]
Abstract. A tuned mass damper (TMD) is a type of auxiliary energy dissipation device mounted in a structure to mitigate the vibrations of the structure. Optimum TMD’s parameters including the natural frequency and damping ratio have been suggested in the literature, which normally focus on attenuating the displacement response of the primary structure. While excessive displacements can affect the safety and integrity of a structure, limiting acceleration response levels are more important at serviceability limit state as these relate to the functionality of nonstructural components and occupier comfort. The current paper aims at developing optimum TMD parameters for minimizing the steady state acceleration of structures such as building floors subjected to harmonic forces, especially considering the inherent damping present in the main structure. For each mass ratio and structure damping value considered, numerical simulations were performed to search for a combination of TMD tuning frequency and damping that minimize the peaks observed on the response spectra of the combined structure-TMD system. The new optimum TMD parameters were found for a range of mass ratios and primary structural damping ratios commonly used in practice. Empirical expressions for the tuning parameters were also derived by means of curve fitting to facilitate design work. The reliability of the fitting models was satisfactory with the calculated 95-th percentile error being as low as 0.2% and the correlation coefficient being as high as 0.999. Compared with various tuning formulas in relevant literature, the new proposal for TMD parameters was found to provide higher reduction in the structural acceleration response, hence more effective in terms of acceleration control. Keywords: Tuned mass damper · Optimum parameters · Harmonic excitation · Acceleration · Curve fitting · Empirical formula
1 Introduction A tuned mass damper (TMD) which conventionally consists of a mass, a spring, and a dashpot is a type of auxiliary energy dissipation device attached to a structure to attenuate the vibrations of the structure. The TMD’s frequency is tuned to a particular natural frequency of the primary structure. When that frequency is excited, the TMD will resonate out of phase with the structural motion and dissipate energy via its inertia force © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 560–572, 2022. https://doi.org/10.1007/978-981-16-3239-6_42
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acting on the structure. In the past few decades, TMDs and their variations have been widely applied in buildings, chimneys, bridges and other industrial facilities. Passive and active TMDs have been proved to be very effective in controlling horizontal motion of tall buildings under wind gust loadings and earthquake excitations [1–3]. Different types of TMDs ranging from conventional spring-mass-dashpot systems to pendulum TMD have also been explored for human-induced floor vibration applications with some degree of success [4, 5]. Regarding office floors whose occupants would even be sensitive to low level of vibrations, an innovative configuration of viscoelastic TMD in the form of grouped sandwich beams has recently been developed to reduce the floor response to human walking [6, 7]. A number of solutions for optimum TMD dynamic parameters have been suggested in the literature, which usually focus on attenuating the displacement response of the primary structure. The loading scenarios considered commonly include harmonic excitation or white noise random excitation applied as either input force or base acceleration on the primary structure [8–14]. This paper limits the investigation to structures subjected to harmonic forces such as building floors excited by unbalanced rotating machines or walking activities [15, 16]. Figure 1 shows a two degree of freedom (2DOF) system which is a simplified model of a primary structure attached to a TMD. The symbols m, k, c, x represent the mass, stiffness, damping coefficient and displacement with subscripts s and d referring to the primary structure and the TMD, respectively. Let f be the ratio of the TMD’s natural frequency to that of the main system and ζ d be the TMD’s damping ratio. The TMD design is essentially the process of specifying optimum values for f and ζ d to effectively reduce the motion of the primary structure.
Fig. 1. Simplified model of a structure with a TMD
Let m be the ratio of the TMD mass and the main mass. When the primary structure is undamped, the TMD parameters that minimize the peak displacement of the main mass can be determined using the classical expressions (1) and (2) developed by Den Hartog, which are functions of only the mass ratio [8]. f =
1 1+m
(1)
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ζd =
3m 8(1 + m)
(2)
In order to gain better damping for the primary structure and limit the relative motion of the damper mass, Krenk [9] suggested replacing the factor 3/8 in (2) by the factor 1/2 whilst retaining (1) for the tuning frequency. Their formula for optimum TMD damping ratio therefore becomes: m (3) ζd = 2(1 + m) When the primary structure has a certain amount of damping, a closed form solution for TMD parameters cannot be obtained analytically. Tsai and Lin [11] performed numerical simulations to find the optimum values of tuning frequency ratio and TMD damping ratio that minimize the structure’s displacement response. They then derived the fitted regression formulas for f and ζ d as given by Eqs. (4) and (5) where ζ s is the damping ratio of the main structure. √ √ 1 + 1 − 2ζs2 − 1 − 1.398 + 0.126 m − 2.004m ζs m f = 1+m √ √ (4) − 0.362 − 5.897 m + 8.533m ζs2 m ξd =
3m + 0.157ζs − 0.321ζs2 + 0.195ζs m 8(1 + m)
(5)
Another extension of the Den Hartog solution to minimize the maximum displacement of the damped main mass was introduced by Abubakar and Farid [12] with the fitted Eqs. (6) and (7) being functions of both m and ζ s . More recently, Salvi and Rizii [13] produced Eqs. (8) and (9) for best tuning of TMD with regard to displacement control as well. m 1 1 − 1.5906ζs (6) f = 1+m 1+m 0.1616ζs 3m ζd = + (7) 8(1 + m) (1 + m) √ 1√ f = 1 − 3m (8) m + ζs 2 ζd =
1 3√ m + ζs 5 6
(9)
While excessive displacements can affect the safety and integrity of a structure (so these need be controlled to acceptable limits in design), limiting acceleration response levels are more important at serviceability limit state as these relate to the functionality of non-structural components and occupier comfort. Current guidelines on floor vibrations
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normally specify acceptable vibration levels for human comfort, relating to various human activities in different environments, in terms of acceleration [16, 17]. A solution for the TMD parameters that could minimize the structure’s acceleration response was recommended by Warburton [14] with expressions (10) and (11) for undamped structure, i.e. ζ s = 0. 1 f =√ 1+m 3m ζd = 8(1 + m/2)
(10)
(11)
The present paper aims at developing optimum TMD parameters for minimizing the steady state acceleration response of damped structures subjected to harmonic forces. The numerical search for tuning parameters is followed by the derivation of empirical expressions that well fit the numerical results hence useful for practical design. Comparisons with various tuning formulas in the literature are also made to verify the efficiency of the newly proposed solution.
2 Methods 2.1 Dynamic Magnification Factor for Steady State Acceleration The governing equation of motion of the 2DOF system of Fig. 1 when the main mass is subjected to an external harmonic force F(t) can be written as: M¨x + C˙x + Kx = F
(12)
in which the mass matrix M, stiffness matrix K, damping matrix C, displacement vector x and forcing vector F are expressed as follows: M=
ms 0 0 md
K=
ks + kd −kd −kd kd
C=
cs + cd −cd −cd cd
x = {xs xd }T F = {F 0}T
(13)
The steady-state solution for Eq. (12) can be found by using exponential representation. The harmonic√force is expressed as F = F 0 ejωt in which ω is the forcing circular frequency and j = −1. Let the steady-state solution of x be of the form: x = Xejωt x˙ = jωXejωt x¨ = −ω2 Xejωt
(14)
where X = {X s X d }T is the vector of displacement amplitude for the combined 2DOF system. Moreover, the following conventional terms are used: ks kd cs cd ωd = ζs = ζd = ωs = ms md 2ωs ms 2ωd md md ωd ω m= f = r= (15) ms ωs ωs
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The solution of (12) can then be obtained in the form of (16) with associated coefficients defined by (17). Xs −(a1 + jb1 ) = F0 /ks a2 + jb2
(16)
a1 = f 2 − r 2 b1 = 2ζd rf a2 = r 2 − r 4 + f 2 r 2 (1 + m) − 1 + 4r 2 f ζs ζd
b2 = 2r ζs r 2 − f 2 + f ζd r 2 + mr 2 − 1
(17)
The dynamic magnification factor for steady state acceleration response, D, of the primary structure in the combined system is now defined as: 2 2 X¨ s 2 a1 + b1 D= =r (18) F0 /ms a22 + b22 in which X¨s is the magnitude (modulus) of the complex acceleration amplitude X¨s . 2.2 Optimization Process for TMD Parameters A Matlab code [18] was developed to perform numerical simulations that searched for optimal values of f and ζ d with regard to minimizing the maximum value of the steady state acceleration magnification factor D of a damped primary structure. Table 1 shows the investigated range for various parameters where each parameter would vary from its lower to upper values in the numerical search. Table 1. Investigated range of parameters Parameter
Lower Upper Increment
Mass ratio, m
0.0025 0.1
0.0025
Main structure damping ratio, ζ s 0
0.1
0.005
Tuning frequency ratio, f
1.2
0.00025
0.8
TMD damping ratio, ζ d
0
0.3
0.0005
Forcing frequency ratio, r
0.8
1.2
0.0005
For a given mass ratio m and primary structural damping ratio ζ s , the search procedure was as follows:
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(a) For a tuning frequency ratio value, f , generate the acceleration response spectra corresponding to different values of TMD damping ratios ζ d in the investigated range. Each response spectrum was calculated using Eq. (18), showing the relationship between the dynamic magnification and the forcing frequency ratio. An example of the generated response spectra is presented in Fig. 2 where various response spectra with different ζ d were plotted for the same m = 0.02, ζ s = 0.05 and f = 1. Two peaks can normally be observed in each response spectrum. Identify the ζ d value that minimized the peak response at the tuning frequency ratio being considered, f . (b) Repeat step (a) for the range of tuning frequency ratio specified in Table 1. (c) Select the combination of f and ζ d that minimized the peak acceleration response. This combination defined the optimum TMD parameters for the mass ratio and primary structural damping ratio under consideration. The search procedure was then repeated for another combination of m and ζ s from which the corresponding optimum combination of f and ζ d was found. The investigation covered a range of m and ζ s normally used in civil engineering structures.
Fig. 2. Different response spectra with varying ζ d for the same m = 0.02, ζ s = 0.05, and f = 1
3 Results and Discussion 3.1 New Optimum TMD Parameters The numerical search resulted in 840 combinations of optimum tuning frequency f and optimum TMD damping ratio ζ d for the range of mass ratio m and structure damping ζ s investigated. Examples of the newly proposed optimum TMD parameters attuned to minimum structure acceleration are presented in Tables 2 and 3. The lower the inherent damping present in the primary structure, the lower the required TMD’s damping value
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and the lower the tuning frequency ratio would be. Moreover, for the same value of the main structure damping, the tuning frequency ratio decreases and the TMD’s damping increases when the mass ratio increases. Table 2. Optimum tuning frequency ratio m
f for ζ s = 0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.10
0.01 0.9958
0.9968 0.9978 0.9993 1.0010 1.0028 1.0048 1.0070 1.0098 1.0125
0.02 0.9910
0.9923 0.9935 0.9953 0.9970 0.9990 1.0013 1.0040 1.0068 1.0100
0.03 0.9863
0.9875 0.9895 0.9913 0.9930 0.9953 0.9978 1.0008 1.0038 1.0065
0.04 0.9818
0.9830 0.9850 0.9870 0.9890 0.9913 0.9938 0.9968 0.9998 1.0028
Table 3. Optimum TMD damping ratio m
ζ d for ζ s = 0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.10
0.01 0.0630
0.0640 0.0670 0.0675 0.0680 0.0700 0.0720 0.0740 0.0740 0.0755
0.02 0.0885
0.0900 0.0925 0.0935 0.0955 0.0975 0.0995 0.1000 0.1020 0.1025
0.03 0.1080
0.1105 0.1100 0.1120 0.1150 0.1170 0.1185 0.1190 0.1205 0.1245
0.04 0.1235
0.1265 0.1270 0.1285 0.1310 0.1335 0.1360 0.1370 0.1395 0.1425
Figures 3 and 4 present a comparison of various solutions for optimum f and ζ d where the mass ratio was fixed at 2% and the main structure damping varied from 0 to 10%. Compared with the solutions for displacement optimization (Den Hartog, Krenk, Tsai & Lin, Abubakar & Farid, Salvi & Rizii), the ones developed for acceleration control (Warburton, this paper) suggested a higher value for the tuning frequency ratio. In response to an increase in the main structure damping, the tuning frequency ratio decreases for displacement control but increases for acceleration control. The new tuning frequency ratio can even be greater than 1.0 for a certain level of the main mass damping. Regarding the TMD damping, whilst the ζ d value proposed by this paper is bit higher than that suggested by Den Hartog and Tsai & Lin, the highest required value for ζ d comes from the solution of Krenk for ζ s less than about 7.5% as observed in Fig. 4.
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Fig. 3. Various proposals for tuning frequency ratio (m = 0.02)
Fig. 4. Various proposals for TMD damping ratio (m = 0.02)
3.2 Empirical Expressions for New TMD Parameters The relationship between the new optimum TMD parameters (f, ζ d ) and the input parameters (m, ζ s ), as found from the numerical search, is graphically illustrated in Fig. 5. Attempts were made to fit empirical models to the numerical data to facilitate the TMD
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design process. It was proposed that the fitting model consisted of three terms with the first one being a function of the mass ratio, m, as in the case of Warburton’s suggestion for undamped structures. The second and third terms would represent the effect of the damping of the main structure, ζ s , and the interaction between m and ζ s respectively. The empirical expressions for optimum tuning frequency and optimum TMD damping ratio were obtained by employing data fitting with Matlab, as given by Eqs. (19) and (20). f =√ ξd =
1 1+m
√ + 0.8860ζs1.8 + 0.3916ζs m
√ 3m + 0.0362ζs0.7 + 0.6493ζs m 8(1 + m/2)
(19)
(20)
Fig. 5. Optimum TMD parameters
To examine the quality of the proposed curve fitting functions, the TMD parameters acquired from the numerical search were plotted against those calculated using Eqs. (19) and (20), as shown in Fig. 6. The fitting model demonstrated very good accuracy with the scattered plot being well fitted on a straight line. The R-squared correlation coefficient was found to be as high as 0.999 for both of the proposed empirical formulas. Furthermore, Fig. 7 shows the cumulative distribution of error of the fitting models. The error defined here was the ratio of the absolute difference between the numerical search result and the fitted value to the former. The average error was only 0.03% for the fitted tuning frequency ratio and 0.60% for the fitted absorber damping ratio. The 95-th percentile error was also found to be very minimal at just 0.08% and 1.54% for the approximation of f and ζ d respectively, which indicated a high degree of accuracy of the fitting models.
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Fig. 6. Fitting model vs Numerical simulation
Fig. 7. Cumulative distribution of error of fitting models
3.3 Evaluating the Efficiency of the Proposed Tuning Solutions The dynamic magnification factor for the steady state acceleration response of the original structure (without TMD) can be calculated using the following equation [19]: r2 D0 = 2 1 − r 2 + (2ζS r)2
(21)
As an example of the optimized TMD’s efficiency, Fig. 8 presents a comparison of the acceleration response of the original structure and the structure with TMD for m = 2% and ζ s = 5% where the TMD parameters were selected using various formulas. It can be seen that, among the optimum tuning solutions examined, only the TMD formula developed by this paper resulted in the same response levels at the two peaks observed on the associated response spectrum. The spectra in accordance with the other absorber solutions all had maximum accelerations exceeding the response level resulted from the new tuning parameters. The percentage of reduction in acceleration response, Ra , due to the addition of TMD to the primary mass, can now be defined as Ra = (D0.max − Dmax )/D0.max where D0.max and Dmax are the maximum dynamic magnification factors of the original structure
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Fig. 8. Acceleration response spectra (m = 0.02; ζ s = 0.05)
and the structure with TMD respectively. Table 4 presents the Ra values derived from different tuning solutions when the mass ratio was taken as 2%. Compared with the popular Den Hartog’s method, the solutions offered by Tsai & Lin and Abubakar & Farid provided a little lower reduction whilst those suggested by Krenk and Warburton resulted in a higher reduction in the acceleration response. On the other hand, the new TMD parameters developed in this paper were proved to be most effective, resulting in the greatest reduction in acceleration. For instance, the maximum steady state response of a structure with a damping ratio of 5% can be alleviated by 47.4% by an optimally tuned TMD that has a mass ratio of 2%. Another observation was that the Ra value was higher in structures with lower inherent damping. Table 4. Acceleration response reduction (m = 2%) ζs
Percentage of reduction, Ra , in accordance with TMD solutions Den Hartog Krenk
Tsai & Lin
Abubakar & Farid
Salvi & Rizii
Warburton
This paper
0.01
81.6%
81.7%
81.3%
81.3%
81.6%
82.9%
83.1%
0.02
68.1%
68.5%
67.4%
67.4%
67.7%
70.0%
70.5%
0.03
57.8%
58.4%
56.7%
56.7%
57.0%
60.0%
60.9%
0.04
49.8%
50.5%
48.2%
48.3%
48.6%
52.0%
53.4%
0.05
43.3%
44.2%
41.3%
41.6%
41.8%
45.5%
47.4%
0.06
38.0%
38.9%
35.6%
36.1%
36.2%
40.0%
42.4%
0.07
33.5%
34.5%
30.9%
31.6%
31.6%
35.5%
38.3%
0.08
29.8%
30.8%
26.9%
27.8%
27.8%
31.6%
34.9%
0.09
26.5%
27.6%
23.5%
24.6%
24.5%
28.2%
31.9%
0.10
23.7%
24.8%
20.6%
21.8%
21.8%
25.3%
29.3%
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4 Conclusions The present paper discussed a numerical simulation procedure to search for optimum TMD parameters that can minimize the acceleration response of a damped structure subjected to harmonic excitations. New optimum values for the tuning natural frequency ratio and the absorber’s damping ratio were proposed for the mass ratio and the structure’s damping ratio of up to 10%. Empirical formulas for the optimum absorber parameters were developed and the accuracy of the regression equations was successfully evaluated. For a given mass ratio and primary structural damping ratio, the tuning frequency ratio for acceleration optimization was found to be higher than that for displacement control, i.e. the TMD’s natural frequency should be closer to the primary structure’s natural frequency. When the damping inherent in the main structure increases, the optimum tuning frequency ratio for acceleration control shows an increasing trend whereas that for displacement control shows a decreasing trend. The absorber’s damping, on the other hand, should be increased with the increase of the structure’s damping for either displacement or acceleration optimization. The new solution for optimum TMD parameters was benchmarked against the classical Den Hartog method and a number of its modifications. In the event that acceleration response is of concern, a TMD with parameters tuned to the newly proposed values can flatten the transfer function of the damped structure in the vicinity of resonance. Compared with various tuning formulas in relevant literature, the proposed solution was found to provide higher reduction in the structural acceleration response, thus more effective in terms of minimizing acceleration due to harmonic excitation.
References 1. Tuan, A.Y., Shang, G.Q.: Vibration control in a 101-storey building using a tuned mass damper. J. Appl. Sci. Eng. 17(2), 141–156 (2014) 2. Elias, S., Matsagar, V.: Research developments in vibration control of structures using passive tuned mass dampers. Ann. Rev. Control 44, 129–156 (2017) 3. Ikeda, Y., Yamamoto, M., Furuhashi, T., Kurino, H.: Recent research and development of structural control in Japan. Jpn. Archit. Rev. 2(3), 219–225 (2019) 4. Setareh, M., Ritchey, J., Baxter, A., Murray, T.: Pendulum tuned mass dampers for floor vibration control. J. Perform. Const. Facil. 20(1), 64–73 (2006) 5. Nyawako, D., Reynolds, P.: Technologies for mitigation of human-induced vibrations in civil engineering structures. Shock Vibr. Digest 39(6) (2007) 6. Nguyen, T.H., Saidi, I., Gad, E.F., Wilson, J.L., Haritos, N.: Performance of distributed multiple viscoelastic tuned mass dampers for floor vibration applications. Adv. Struct. Eng. 15(3), 547–562 (2012) 7. Nguyen, T., Gad, E., Wilson, J., Haritos, N.: Mitigating footfall-induced vibration in long-span floors. Aust. J. Struct. Eng. 15(1), 97–109 (2014) 8. Den Hartog, J.P.: Mechanical Vibrations, 4th edn. McGraw-Hill (1956) 9. Krenk, S.: Frequency analysis of the tuned mass damper. J. Appl. Mech. 72(6), 936–942 (2005) 10. Krenk, S., Høgsberg, J.: Tuned mass absorbers on damped structures under random load. Probab. Eng. Mech. 23(4), 408–415 (2008)
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11. Tsai, H.-C., Lin, G.-C.: Explicit formulae for optimum absorber parameters for force-excited and viscously damped systems. J. Sound Vib. 176(5), 585–596 (1994) 12. Abubakar, I., Farid, B.: Generalized Den Hartog tuned mass damper system for control of vibrations in structures. Earthq. Resistant Eng. Struct. 104, 185–193 (2009) 13. Salvi, J., Rizzi, E.: A numerical approach towards best tuning of Tuned Mass Dampers. In: Proceedings of the International Conference on Noise and Vibration Engineering ISMA2012, pp. 2419–2433 (2012) 14. Warburton, G.B.: Optimum absorber parameters for various combinations of response and excitation parameters. Earthquake Eng. Struct. Dynam. 10(3), 381–401 (1982) 15. Bachmann, H., Ammann, W.: Vibrations in Structures Induced by Man and Machines. IABSE, Zürich, Switzerland (1987) 16. Smith, A., Hicks, S., Devine, P.: SCI P354 - Design of floors for vibration: a new approach. The Steel Construction Institute, Ascot (2009) 17. ISO. ISO 10137:2007: Bases for Design of Structures - Serviceability of Buildings and Walkways Against Vibrations, 2nd edn. International Organization for Standardization, Geneva (2012) 18. The MathWork Inc.: Matlab Data Analysis. The MathWork Inc., Massachusetts, USA (2014) 19. Chopra, A.K.: Dynamics of Structures - Theory and applications to Earthquake Engineering, 4th edn. Prentice Hall (2012)
Optimal Electrical Load of the Regenerative Absorber for Ride Comfort and Regenerated Power La Duc Viet(B) , Nguyen Cao Thang, Nguyen Ba Nghi, and Le Duy Minh Institute of Mechanics, Vietnam Academy of Science and Technology, Hanoi, Vietnam [email protected]
Abstract. The automotive industry has started to show interest in harvesting energy from the motion of the suspension. Unlike conventional shock absorber which dissipates energy by passing a fluid, generally oil, through small orifices, thus generating heat, the regenerative shock absorber transforms the linear motion of the suspension into a rotation of an electric generator. The electrical load of the generator should be optimized because the too large value suppresses the useful electric current while the too small value gains too little energy. This paper presents the analytical solutions of the optimal electrical load of the generator for two objectives: ride comfort and regenerated power, under two cases of excitation: harmonic and random. Keywords: Vibration energy harvesting · Analytical optimization · Regenerative shock absorber · Regenerative damper
1 Introduction The traditional shock absorber of vehicle dissipates vibration energy as heat. This traditional design has the advantage of simplicity but the disadvantages are the overheat problem and the wasted energy. The regenerative absorber converts the wasted energy to the useful form. As stated in [1], there is only about 10–20% of the fuel energy used to ride the vehicle, while a significant amount of energy is wasted in the shock absorber, about 10% (Fig. 1). Base on the operation principle, the regenerative shock absorber can be divided into two groups: mechanical and electromagnetic. The mechanical regenerative shock absorber is constructed from the normal hydraulic or air absorber. However, the kinetic energy transferring to the absorber is converted to the potential energy of the fluid or air in the chamber. These types of absorber have many disadvantages including the complex pipe system, the additional weight and the remarkable installation space. The leak problem can affect to the system. The operation frequency range of the hydraulic system is narrow. Moreover, the usable ability of the potential energy of the fluid or air is limited because the modern automotive industry aims to the electric vehicles. Therefore, the studies on the regenerative hydraulic or air absorber are quite limited. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 573–583, 2022. https://doi.org/10.1007/978-981-16-3239-6_43
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Conversely, the electromagnetic regenerative absorber converts the impact energies to the electrical energy, which is more convenient in storing and reuse, has high efficiency, requires a small installation space and many more advantages [2]. In the recent years, electromagnetic regenerative absorber has attracted many interests. The motor in the electromagnetic absorber system can be used to provide the control force in the control task and concurrently provide the absorb force in the regenerative task.
Fig. 1. Energy distribution provided by the car engine
The regenerative absorber can use the ball screw to convert the translation to the rotation motion. The reference [3] proposed the regenerative absorber (Fig. 2) and analyzed its dynamic and efficiency in regenerating energy.
Fig. 2. Device regenerating energy by ball screw system
Another mechanism can be used is the rack and pinion mechanism to convert the translation to the rotation (Fig. 3) [4].
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Fig. 3. Electromagnetic regenerative absorber using rack and pinion
The simplest model for the regenerative absorber is the single-degree-of-freedom (SDOF) mass-spring-damper system integrated with the regenerative energy circuit through the electromagnetic transductions [5]. Because the mechanical and electrical systems are connected through the transduction, the mechanical and electrical parameters influence each other in the energy extraction. To optimize the extracted energy, the parameters of two systems should be considered simultaneously. The electrical load of the motor is a key parameter needs to be optimized. It affects both the absorptive and regenerative efficiencies. The too large value of the electrical load can suppress the useful electric current and also reduce the absorptive efficiency. The too small value of electrical load, however, extracts too small energy and results in the overdamped situation, which also reduce the absorptive efficiency. This paper presents the analytical optimal solution for the electrical load of the regenerative absorber. The optimal solution guarantees both the absorptive and regenerative efficiencies.
2 Mathematical Modelling We consider a simplest model of the regenerative absorber. Figure 4 shows a SDOF oscillator with a electromagnetic transduction. The system contains a mass-spring-damper system which is base excited by the ambient vibration.
Fig. 4. SDOF regenerative energy system under base excitation, electromagnetic transduction.
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The motion equation of this model can be easily obtained by the second Newton’s law as [6] ⎧ ⎨ m¨x + cm (˙x − y˙ ) + k(x − y) + fe = 0 (1) k k (˙x − y˙ ) ⎩ fe = t e RH In the Eq. (1), f e is the force provided by the electromagnetic transduction (electromotive force), RH is the resistance of the electric load, k t and k e respectively are the motor constant and the electromotive force coefficient. Moreover, cm is the inherent mechanical damping coefficient of the system, k is the stiffness of the leaf spring and m is the vehicle’s mass. At last, x and y respectively are the displacements of the mass and the base, respectively. Base on the equation, the electromotive force can be seen as a viscous damping force. Therefore, the equivalent electrical damping of the regenerative circuit can be defined as: ce =
kt ke RH
(2)
The acceleration of m represents the absorptive efficiency while the extracted energy of the electromagnetic transduction expresses the regenerative efficiency. Use the following normalized variables: natural frequency ωn = k m, mechanical damping ratio ζm = cm (2mωn ), electrical damping ratio ζe = kt ke (2mωn RH ), we change the Eq. (1) to: x¨ + 2(ζm + ζe )ωn (˙x − y˙ ) + ωn2 (x − y) = 0
(3)
The regenerative energy is the energy extracted through electrical load. The average energy can be written in the form: 1 P= T
T
2mωn fe (˙x − y˙ )dt = T
0
T ζe (˙x − y˙ )2 dt
(4)
0
in which T is a certain time.
3 Optimal Solution of Electrical Damping Ratio Under Harmonic Excitation Let us consider the rough road described as harmonic form and is represented in the complex form: y=
iωt −iωt e cos ωt = + e ω2 2ω2
(5)
in which is an amplitude independent of the excitation frequency ω. The amplitude of the rough (/ω2 ) is inversely proportional to the square of frequency. It implies the
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constant acceleration spectral, which is commonly available in daily lives. The stationary response has form: 1 iωt xp e + xp e−iωt (6) x= 2 where x p is the complex amplitude of the vehicle’s displacement. Substitute (5) and (6) into Eq. (3) yields:
i iω −ω2 iωt xp e + xp e−iωt + 2(ζm + ζe )ωn xp eiωt − xp e−iωt − eiωt − e−iωt 2 2 2ω
1 xp eiωt + xp e−iωt − eiωt + e−iωt =0 +ωn2 2 2ω2
(7) Let the term relating to eiωt be equal to zero and solve the resulting equation we obtain the complex amplitude of the vehicle’s displacement as: xp =
ω2 + i2(ζm + ζe )ωn ω n
ω2 ωn2 − ω2 + i2(ζm + ζe )ωn ω
(8)
Substitute the formula (8) into (6), then into the formula of average power (4), in which the time of the integration is one period, we have: mωωn P= π
2π/ω
ζe
i 2 iω iωt xp e − xp e−iωt − eiωt − e−iωt dt 2 2ω
0
Substitute the formula (8) into (9) and integrate, note that 2π/ω
(9)
2π/ω
eiωt dt
=
0
e−iωt dt = 0, after some manipulations, we have:
0
2 mωn ω2 ζe P=
2 ωn2 − ω2 + 4(ζm + ζe )2 ω2 ωn2
(10)
Moreover, the ride comfort is characterized as the acceleration transmissibility ratio as:
ω2 xp ωn2 + 4(ζm + ζe )2 ω2 = ωn
2 ω2 − ω2 + 4(ζm + ζe )2 ω2 ω2 n
(11)
n
We have some remarks: – When ζe is very large, the regenerative power in (10) tends to 0 and the transmissibility ratio in (11) tends to 1. That implies there is no regeneration of energy and there is also no absorptive efficiency (base acceleration is transmitted totally to the mass).
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– When ζe is very small, the regenerative power in (10) also tends to 0, which implies no regenerative efficiency. ωn . Espe– When ζe and ζm is very small, the transmissibility ratio in (11) tends to |ω−ω n| cially, when the resonance occurs (ω = ωn ), the transmissibility ratio tends to infinity and the absorber is totally broken down. The above remarks show that the electrical damping ζe (depending on the electrical load RH ) needs to be optimal calculated. The optimal problem at this time is multiobjective: maximize P in (10) and minimize transmissibility ratio in (11). To combine the objectives, we consider the following objective function: 2 ω2 xp Pωn J =γ − − γ (1 ) m2
ωn2 ωn2 + 4(ζm + ζe )2 ω2 γ ωn2 ω2 ζe = − (1 − γ ) (12)
2
2 ωn2 − ω2 + 4(ζm + ζe )2 ω2 ωn2 ωn2 − ω2 + 4(ζm + ζe )2 ω2 ωn2 in which the first term is the normalized power while the second term is the square of transmissibility ratio. The weighting γ (0 < γ < 1) defines the importance between two objectives: regenerate energy and shock absorption. When γ = 0 or 1, the objective is totally shock absorption or regenerating energy, respectively. The problem is to maximize J with respect to ζ e . Simplify J, we have:
γ ω2 ζe + (γ − 1) ωn2 + 4(ζm + ζe )2 ω2 J = ωn2 (13)
2 ωn2 − ω2 + 4(ζm + ζe )2 ω2 ωn2 Letting the derivative of J with respect to ζ e be equal to 0 results in the equation to calculate the optimal damping ratio ζ e :
γ ω2 ζe + (γ − 1) ωn2 + 4(ζm + ζe )2 ω2 γ + 8(ζm + ζe )(γ − 1) = (14)
2 2 2 2 2 2 8(ζm + ζe )ωn2 + 4(ζm + ζe ) ω ωn ωn − ω Simplifying (14) leads to a quadratic equation of ζe : 8α(ζm + ζe )(γ ζe + (γ − 1)(2 − α)) = (1 − α)2 + 4(ζm + ζe )2 α γ
(15)
where we define α as the square of frequency ratio: α=
ω2 ωn2
(16)
The optimal solution of ζ e depends on the ratio α and weighting value γ . Especially, when γ = 1 and at the resonance (α = 1), solving (15) gives: ζe = ζm
(17)
Therefore, if the objective is totally regenerate energy then the electrical damping ratio should be chosen as the mechanical damping ratio. This result was stated in [6, 7].
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4 Numerical Investigation for Harmonic Excitation We study the change of optimal electrical damping ratio versus the changes of α and γ. The plot of ζe versus α is show in Figs. 5, 6 with varying values of γ and ζm .
Fig. 5. Change of optimal value of ζe versus α, for ζm = 10%
The results lead to some following remarks: – When γ is near 1 implying the objectives is more important in regenerating energy, the optimal damping ratio changes not much. At the resonance, the optimal damping is equal to ζm as discussed above, Outside the resonance point (the frequency ration is smaller or larger than 1), the optimal damping ratio should be increased. – When γ decreases implying the objective is more important in shock absorption, the clear trend is that: the damping ratio needs to be large at low frequency and vice verse. That means the damping should be very small to isolate the vehicle at the high frequency. – The results have no remarkable change when the mechanical damping ratio is changed. – At small α, i.e. stiff dampers, the optimal ζe is very sensitive to γ.
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Fig. 6. Change of optimal value of ζe versus α, for ζm = 1%
It is important to note that there is a special point in Fig. 5 or 6, which is fixed when γ varies. This point can be found by substituting the extreme cases of γ = 0 and γ = 1 into (15). This yields: 8α(ζm + ζe )(α − 2) = 0 (18) 8α(ζm + ζe )ζe = (1 − α)2 + 4(ζm + ζe )2 α Solving (18) gives quadratic equation: 16(ζm + ζe )ζe = 1 + 8(ζm + ζe )2 which results in the value of ζe independent of γ: 2 + 16ζm2 ζe = 4
(19)
(20)
The value (20) is a good trade-off electrical damping regardless of the value of weighting γ. To demonstrate it, with ζm = 0.1, we plot the objective function (13) with two values of γ and with the trade-off damping (20) (=0.3674), low damping ζe = 0.1 and high damping ζe = 1. The resulting curves are shown in Fig. 7. It is seen that the trade-off value (20) has the good effective in raising the performance curves, regardless of the value γ as expected.
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Fig. 7. Objective function J versus α, for ζm = 1% and γ = 0.6 (left), γ = 0.9 (right)
5 Optimal Solution of Electrical Damping Ratio Under White Noise Excitation Let the acceleration y¨ is modeled as a white noise with intensities of σ. The stationary response of the system (3) has following form [8]:
(x − y)2
=
σ2 ; 4(ζm + ζe )ωn2
(˙x − y˙ )2
=
σ2 4(ζm + ζe )
(21)
where • denotes the stationary stochastic averaging operator. As mentioned above, the absolute acceleration of the vehicle represents the absorption efficiency. From (21) we calculate the average of square of acceleration as: 2 2 2 x¨ = ωn (x − y) + 2(ζm + ζe )ωn (˙x − y˙ ) = ωn4 (x − y)2 + 2(ζm + ζe )ωn3 (x − y)(˙x − y˙ ) + 4(ζm + ζe )2 ωn2 (˙x − y˙ )2 (22) Because the displacement and velocity are two independent stochastic processes, we have:
Then
(x − y)(˙x − y˙ ) = 0
(23)
x¨ 2 = ωn4 (x − y)2 + 4(ζm + ζe )2 ωn2 (˙x − y˙ )2
(24)
Using (21) in (24) gives:
1 ω2 σ 2 x¨ 2 = n + 4(ζm + ζe ) 4 ζm + ζe
(25)
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The acceleration transmissibility is: 2
x¨ 1 ωn2 = + 4(ζm + ζe ) σ2 4 ζm + ζe Use (21), the average power has form: mωn σ 2 ζe P = 2mωn ζe (˙x − y˙ )2 = 2(ζm + ζe )
(26)
(27)
The same combined objective function as (12) is considered here: 2
x¨ Pωn 1 ωn2 ωn2 ζe J =γ − (1 − γ ) − (1 − γ ) 2 = γ + 4(ζm + ζe ) mσ 2 σ 2(ζm + ζe ) 4 ζm + ζe
2γ ζe + (γ − 1) 1 + 4(ζm + ζe )2 (28) = ωn2 4(ζm + ζe ) The problem is to maximize J with respect to ζ e . Letting the derivative of J with respect to ζ e be equal to 0 results in the equation to calculate the optimal damping ratio ζ e:
2γ ζe + (γ − 1) 1 + 4(ζm + ζe )2 2γ + (γ − 1)8(ζm + ζe ) (29) = 4(ζm + ζe ) 4 Solving (29) leads to the simple solution of ζ e : 1 2γ ζm 1+ ζe = 2 1−γ
(30)
It is seen from (30) that the optimal electrical damping ratio increases with the weighting value γ . That means if the importance of the regenerated power increases, the electrical damping ratio should be large. In the limit case, when ζm = 0, the optimal solution is ζe√= 1/2. Return to (20), in the harmonic excitation case, the optimal solution is ζe = 1/ 8, which is a bit smaller.
6 Conclusion This paper investigates the optimal electrical damping ratio for a regenerative shock absorber. The studied model is a single-degree-of freedom vehicle model subjected to road excitation. The paper studies the cases of harmonic and random excitation and considers an index combining the regerative energy and the acceleration transmissbility ratio. The results show that the electrical damping, which is inversely proportional to the electrical load of the motor, should be optimized to achieve both two objectives. The optimal solution is calculated from a quadratic equation in the case of harmonic excitation or from a simple explicit formula in the case of random excitation. In the limit case, √ where the mechanical damping is zero, the optimal electrical damping should be 1/ 8 and 1/2 for the harmonic and random excitation, respectively. To verify the solution, in some specific cases, the optimal value returns to the known solutions in literature. The numerical investigation is also carried out to see the effects of the weighting value.
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Acknowledgement. This paper is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number “08/2018/TN".
References 1. Pei, S.Z.: Design of Electromagnetic Shock Absorbers for Energy Harvesting from Vehicle Suspensions. Master Degree thesis, Stony Brook University (2010) 2. Gysen, B.L.J., Tom, P.J., Paulides, J.J.H., et al.: Efficiency of a regenerative direct-drive electromagnetic active suspension. IEEE Trans. on Vehicular Technology 60(4) (2011) 3. Zuo, L., Scully, S., et al.: Design and characterization of an electromagnetic energy harvester for vehicle suspensions. Smart Material and Structure 19 (2010) 4. Graves et al.: Experimental verification of energy-regenerative feasibility for an automotive electrical suspension system. In: ICVES IEEE International Conference on Vehicular Electronics and Safety. IEEE, pp. 1–5 (2007) 5. Stephen, N.: On energy harvesting from ambient vibration. J. Sound Vib. 293(1), 409–425 (2006) 6. Tai, W.-C., Zuo, L.: On optimization of energy harvesting from base-excited vibration. J. Sound Vib. 411, 47–59 (2017) 7. Tang, X., Zuo, L.: Enhanced vibration energy harvesting using dual-mass systems. J. Sound Vib. 330(21), 5199–5209 (2011) 8. Lutes, L.D., Sarkani, S.: Random Vibration: Analysis of Structural and Mechanical Systems. Elsevier, Amsterdam (2004)
Aerodynamic Responses of Indented Cable Surface and Axially Protuberated Cable Surface with Low Damping Ratio Hoang Trong Lam and Vo Duy Hung(B) The University of DaNang – University of Science and Technology, 54 Nguyen Luong Bang Street, Danang city, Vietnam [email protected]
Abstract. Cable vibration due to wind is one of key issue in design of stayed-cable bridge. To control the rain wind induced vibration, indented surface and axially protuberated cables have been applied in Japan for many years. Nevertheless, it is also figured that those methods are still defective in mitigating cable dry galloping. Moreover, vibration characteristics of indented surface and axially protuberated cables with low damping ratio have not fully investigated yet. The aim of this study is to investigate the aerodynamic responses of low damping indented surface and axial protuberated cable. Firstly, the WTT will examine cables vibration in dry and rain condition for indented cables. Then, aerodynamic responses of axial protuberated cables will be elucidated fully. Finally, the axial flow in the wake of cables will be measured and discussed in detail. Keywords: Aerodynamic responses · Indented surface · Axially protuberated cables · Low damping ratio · Axial flow
1 Introduction Since stay cables can be vulnerable due to wind or rain-wind excitation [1–5], many control methods have been developed to mitigate cable vibration so far. Those methods can simply be categorized in two groups. The first is aerodynamic control methods, which related to modification the aerodynamic characteristic of stay cable. The other group is mechanical control methods, which typically install the external dampers or use crosstie system. Consequently, the damping of cable will be enhanced, or natural frequency be decreased and thus mitigate the vibration of cable. Aerodynamic control is a passive control method that modifies aerodynamic feature of the cable cross-section. It is most effective in controlling rain-wind-induced cable vibration by preventing accumulation of rain to form water stream on the surface of cable. In Japan, indented cable surface and axial protuberated cable surface were applied in some bridges. The use of axially protuberated cable was developed for Higashi-Kobe cable-stayed bridge. The idea was to have these deep axial protuberated cables along the cable, which would drive the water down without allowing any transverse movements [6]. In a preliminary investigation, © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 584–595, 2022. https://doi.org/10.1007/978-981-16-3239-6_44
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cable with parallel, spiral, wave, and other forms were examined. Consequently, axial protuberated cable were found as the most effective solution in restraining rain-wind induced vibration [7]. However, it was reported on site that this kind of cable still exhibited large vibration to some extent [8]. In parallel, for controlling rain-wind induced vibration, indented surface was firstly developed by Miyata et al. in 1994 [9]. This type of cable was applied firstly in Tatara Stayed Cable Bridge in Japan, which is the longest stay cable at completion time. This control method was effective in suppressing rain wind with low drag force coefficient as smooth surface. Its effectiveness in real scale confirmed as no rain wind induced vibration occurred after opening about one year [10]. However, this mitigation method still exhibited the large amplitude vibration in the original stage. Indented cable with Scruton number around 21, started vibrating with large amplitude from 10 m/s of wind speed and became divergent around 30 m/s of wind speed [11]. In addition, it was reported by Katsuchi and Yamada recently that dry galloping could occur for indented cable in low damping ratio [10]. Furthermore, the health monitoring of Tatara cablestayed bridge recently revealed that indented surface cable exhibited limited vibration [12]. In parallel, the detail of vibration of these type of cables in low damping region have not investigated fully yet. Above discussion have motivated this study, in which vibrations of two cable surface modification in low damping region were investigated by wind tunnel tests. One of cable model is a circular cylinder with indented surface. Another is axially protuberated surface. Rainfall was simulated by rain simulator system in wind tunnel. Finally, mechanism of axial flow in span-wise and stream-wise direction will be compared between smooth surface, indented surface, and axially protuberated surface. The paper is structured into three main sections. Section 2 shows the detail of wind tunnel test setup and aerodynamic responses of low damping indented-cable and axially protuberated-cable in both rain and dry condition. In addition, the discussion about axial flow mechanism near the wake of cable also elucidate clearly in this section. Finally, critical conclusions are provided in Sect. 3.
2 Wind Tunnel Test Investigation 2.1 Experimental Setup Tests were conducted in the open-circuit wind tunnel of the Yokohama National University. A cable model was supported by a single degree of freedom spring system in vertical plane and small wire system was used in horizontal plane to keep cable model unmoved laterally. Cable’s position can be changed by flow angle and inclined angle as shown in Fig. 1. A uniform flow was used with turbulence intensity of about 0.6%. In this experiment, rain can be simulated by an overhead nozzle system. Rain volume at the cable model position was 40–50 mm/h in the wind-speed range (around 8–15 m/s), which was adjusted to create the critical rain-wind induced vibration amplitude. Because of the limitation of wind tunnel capacity, maximum wind speed is up to 20 m/s. Cable diameters 158 mm with damping ratio ranges from 0.08 to 0.25%, which is very low damping region of stay cables compare to the on-site cables damping. The natural frequency is around 0.8–1 Hz.
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2.2 Wind Attack Angles Wind attack angles are determined through α, β, and β* defined as in Fig. 2. In this study, the inclined angle was fixed at 9° 25° and 40° in combination with the flow angle of 0°, 15°, 30°, 45° and 60°. These inclinations of 25° and 40°were examined to consider the longest and medium stay cables angles, while the inclined angle of 9° to consider the stay cables located in aviation restriction area. 2.3 Experimental Model Fabrication Two kind of models were examined, included a cable with axially protuberated surface and a cable with indented surface. The model samples were fabricated with same scale to real bridge cables. Cable diameters are 158 mm with an effective model length of 1.5 m and aspect ratio is 9.5. The indented model was fabricated same pattern of stayed cables of Tatara stayed cable Bridge as Fig. 3, while axially protuberated model was fabricated similarly to Higashi Kobe Bridge cables by adding the twelve rubber fillets axially. The detail of axially protuberated surface modification shown as Fig. 4.
β∗
β
α
Wind
Fig. 1. Cable model orientation in wind tunnel
Fig. 2. Wind attack angles
Fig. 3. Indented surface cable
2.4 Experiment Parameters Table 1 shows the detail of experiment parameters in which cable diameters are 110 mm and 158 mm. Damping ratio ranges from approximately 0.08% to 0.25% and natural
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υ
Fig. 4. Axially protuberated cables cable
frequency is around 0.77–1.02 Hz in considering typical stay cables values. Due to the limitation of wind tunnel capacity, maximum wind speed is up to 20 m/s equivalent to Reynolds number around 2.1 × 105 . Nevertheless, according to previous studies cable galloping of inclined cable could be observed in the subcritical Reynolds number regime as well as in the transition and critical Reynolds number regime. Therefore, above range of Reynolds number enable to reproduce the dry galloping. Table 1. Conditions of experiments Diameter: D (mm)
158
Effective length (mm)
1,500
m (kg/m)
14.55–16.13
Natural frequency (Hz)
0.82–1.02
Damping ratio
0.08%–0.25%
Scruton number (2mδ/ρD2 ) 5.1–15.6 Reynolds number
0–2.1 × 105
3 Results and Discussions 3.1 Wind-Induced Vibration of Indented Cable Surface in Low Damping Region Figure 5, Fig. 6 and Fig. 7 show the galloping amplitude versus wind speed in dry and rain condition of indented cable surface. In case of inclination angle at 40°, most of vibration amplitude is small except cases of 40°–45° and 40°–60° in rain condition. Large amplitude start occurring at wind speed around 15 m/s. In dry condition, large vibration only occur in case of 40°–60°, the highest amplitude occurred started at wind speed 12 m/s, the remain cases exhibits small amplitude ( P2t and ‘−’ when P1t < P2t . 2.2 The Internal Pressure Change of the System Due to Changing External Pressure Consider the system (consisting of only one compartment) with n orifices ventilating out, Ai is the area of each orifice ventilating out, i = 1..n, see Fig. 3.
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Fig. 3. The system has one compartment
During airflow, the atmospheric pressure change ventilates the air in and out of the system. Pressure Pt and volume V t are related by Eq. (1): Pt Vt = P0 V0
(8)
in which: P0 is the internal atmospheric pressure of the compartment at the beginning t = 0; V 0 = V is the volume of the system; Pt is the internal atmospheric pressure of the system at time t; V t is the atmospheric volume corresponding to Pt : Vt = V + Vt
(9)
where ΔV t is the volume of air circulating (into, out of) through the system per interval of time t. Thus: Vt =
n
Vit
(10)
i=1
in which: ΔV it is the volume of air circulating through orifice i. According to Eq. (7), ΔV it can be expressed as
t Vit = ±
Ai 0
2 | Pτ − piτ | d τ ρ
(11)
in which: piτ or pit is the external pressure at orifice Ai . From Eqs. (9), (10), (11), V t becomes: t n 2 | Pτ − piτ | d τ Vt = V + (12) ± Ai ρ i=1
0
Replace V t from Eq. (12) in Eq. (8): n P0 V = V + ± Ai Pt i=1
0
t
2 | Pτ − piτ | d τ ρ
(13)
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When external pressure changes the same (pit = pt for all orifices), Eq. (13) is represented as: t 2 P0 A | Pτ − pτ | d τ (14) = 1 ± Pt V ρ 0
in which A is the total area of all the orifices of the system: A = Replace pt from Eq. (2) in Eq. (14) P0 A = 1 ± Pt V
n i=1
Ai .
t 2 1 2 dτ P ρu + − C τ t ρ 2
(15)
0
Get the derivative of Eq. (15)
⇒
dPt 1 2 A Pt2 2 Pt + ρut − C − =± dt V P0 ρ 2
2
2 4 dPt 2 A Pt 1 2 P ρu = + − C t dt ρ V 2 P02 2 t
(16) (17)
in which C is the pressure of the field when the flow velocity is zero (ut = 0). 2.3 The System with N Compartments An isothermal system consists of N compartments and contains orifices, see Fig. 4, during airflow, the change in atmospheric pressure causes air to go in and out of the system and also circulate between compartments. Consider compartment e and name parameters of this compartment as follows: V e is the volume of compartment e, e = 1..N; Pet is internal atmospheric pressure of compartment e; ne is the number of orifices ventilating out; Aei is the area of orifice ventilating out, i = 1..ne ; Aej is the area of the orifice which ventilates with other compartments, j = 1..N, j = e; case of compartments e and j do not open to each other, then Aej = 0. Pressure P and volume V are related by Boyle-Mariotte: Pet Vet = P0 Ve
(18)
in which: P0 is the internal atmospheric pressure of compartment e at the beginning t = 0; Pet is the internal atmospheric pressure of compartment e at time t; V et is the atmospheric volume corresponding to Pet : Vet = Ve + Vet
(19)
where ΔV et is the volume of air circulating (into, out of) through compartment e per interval of time t. Vet = Vit + Vjt
(20)
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Fig. 4. Connection diagram of the N compartments which the system consists of
where: ΔVit is the volume of air circulating through orifice ventilating out Aei ; ΔVjt is the volume of air circulating through orifices which ventilate with other compartments Aej . Thus: Vit =
ne
Veit
(21)
Vejt
(22)
i=1
Vjt =
N j=1
in which: ΔV eit is the volume of air circulating through orifice i; ΔV ejt is the volume of air circulating within compartment j. According to Eq. (7), ΔV eit and ΔV ejt can be expressed as
t Veit =
Aei 0
Aej 0
(23)
t Vejt =
2 |Peτ − peiτ | d τ ± ρ 2 Peτ − Pjτ d τ ± ρ
(24)
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in which: Pjτ or Pjt is the internal pressure of compartment j; peiτ or peit is the external pressure at orifice Aei . From Eqs. (19), (20), (21), (22), (23), (24), Eq. (19) becomes: t t ne N 2 2 | Peτ − peiτ | d τ Vet = Ve + Peτ − Pjτ d τ + ±Aej ±Aei ρ ρ j=1
i=1
0
0
(25) Replace V et from Eq. (25) in Eq. (18): N P0 Ve = Ve + ±Aej Pet j=1
t
ne 2 ±Aei Peτ − Pjτ d τ + ρ i=1
0
t
2 | Peτ − peiτ | d τ ρ
0
(26)
When external pressure changes the same, Eq. (26) is represented as: t t N P0 Ve 2 2 | Peτ − pτ | d τ Peτ − Pjτ d τ ± Ae = Ve + ±Aej Pet ρ ρ j=1
0
0
(27) in which Ae is the total area of the orifices of the compartment e: Ae = Replace pt from Eq. (2) in Eq. (27): N P0 Ve = Ve + ±Aej Pet j=1
t
0
2 Peτ − Pjτ d τ ± Ae ρ
ne i=1
t 1 2 2 dτ + − C ρu P eτ ρ 2 t
Aei .
(28)
0
Get the derivative of Eq. (28) ⎞ ⎛ N 2 dPet Pet 2 2 1 Pet + ρu2 − C ⎠ Pet − Pjt ± Ae ⎝ = − ±Aej dt Ve P0 ρ ρ 2 t j=1
(29) Thus, the internal pressure of the system included N compartments is represented as a system of first-order differential equations with N unknowns Pet . In the case of the closed system included N + 1 compartments in which the pressure of the compartment N + 1 takes to change initiative (the gas flow field is the same as the compartment N + 1), the Eq. (27) is rewritten as below after getting the derivative: ⎞ ⎛ N 2 P 2 2 dPet = − et ⎝ Pet − Pjt ± Ae(N +1) Pet − P(N +1)t ⎠ ±Aej dt Ve P0 ρ ρ j=1
(30) or 2 P0 P˙ et = −Pet
N +1 j=1
±αej
2 Pet − Pjt ρ
(31)
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A
where αej = Veje ; e = 1..N. Likewise, in the case of the closed system included N + M compartments in which the pressure of M compartments take to change initiative, the internal pressure of the system is expressed: ⎛ ⎞ N N +M 2 dPet Pet 2 2 Pet − Pjt + Pet − Pjt ⎠ ⎝ = − ±Aej ±Aej dt Ve P0 ρ ρ j=1
j=N +1
(32) or 2 P0 P˙ et = −Pet
N +M j=1
±αej
2 Pet − Pjt ρ
(33)
The Eq. (33) is a system of first-order differential equations that describes the internal pressure of compartment e (e = 1..N) of the isothermal system included N + M compartments in which the pressure of M compartments take to change initiative (known parameter).
3 Illustrative Example An isothermal system (with orifices ventilating out) that is immersed in the velocity of the airflow field (abide by Bernoulli’ law). The airflow longitudinal velocity fluctuates between 10 m/s and 42 m/s (corresponding with umin and umax ) with a frequency of 1 Hz (T = 1), see Fig. 5. V is the volume of the system, A is the total area of the orifices of the system. Study differential pressure between inside and outside of the system, i.e. ΔPt .
Fig. 5. Graph of the airflow longitudinal velocity
The differential pressure, W = ΔPt is expressed. 1 Pt = Pt − pt or Pt = Pt + ρut2 − pA 2
(34)
here, pA is the pressure of the field when the airflow velocity equaled to zero, pA = 1013 hPa. Figure 6 shows the result for solving Eqs. (16) and (34) with different values of the A/V ratio, in which the differential pressure values also fluctuate with the
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corresponding frequency of 1 Hz. In the case of A/V = 0 the peak value of differential pressure in the T period, W p reaches the maximum W 0 = 350 N/m2 and does not change direction (W ≥ 0). When A/V = 0 the value of W p decreases rapidly as the A/V ratio increases, the value of W p is only 16.3%W 0 when the A/V = 8.10–4 that is still very small.
Fig. 6. Graph of the differential pressure between inside and outside of the system, Pt
4 Discussion and Conclusions In addition to the isothermal conditions, to apply the Boyle formula, the condition for the static state of air inside the system must be satisfied. In order to satisfy this, the total area of the orifices must be very small compared to the volume of the whole system (very small S/V ratio) so that a significant dynamic state of the air takes place only at the orifices. The inside volume of the system is considered large, resulting in a large circulation area, which greatly reduces the air movement within the system. Hence, the air velocity in it can be considered negligible. This is equivalent to the air pressure in the system (having one compartment) being the same at all points. In summary, the internal pressure of the system with N compartments is represented as a system of first-order differential equations with N unknowns. In practice, there are many processes very close to the isothermal process that needs to be controlled which this equation can solve. For example, controlling the pressure difference between the inside and outside of a building (loading) during a tornado by installing ventilation openings of sufficient area and located at appropriate locations [8], etc. The connecting equation has not been able to fully describe the physical phenomena occurring in nature, for specific situations it needs to be studied alongside other impact conditions. The authors hope that the equation connecting Boyle’s and Bernoulli’s laws will pave the way for more complex analyses in relevant fields.
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References 1. https://en.wikipedia.org/wiki/Boyle’slaw 2. https://en.wikipedia.org/wiki/Bernoulli’sprinciple 3. Betz, A.: Introduction to the Theory of Flow Machines, 1st edn. Elsevier Ltd., Rio de Janeiro (1966) 4. Loitsyanskii, L.G.: Mechanics of Liquids and Gases, 2nd edn. Elsevier Ltd., Rio de Janeiro (1966) 5. Landau, L.D., Lifshitz, E.M.: Fluid Mechanics, 2nd edn. Elsevier Ltd., Rio de Janeiro (1987) 6. https://en.wikipedia.org/wiki/Newton’slawsofmotion 7. Nguyen, N.T.: Force on building due to changing atmospheric pressure during tornadoes. J. Sci. Technol. Vietnam 44(5), 91–100 (2006) 8. Nguyen, N.T., Nguyen, L.N.: Tornado-induced loads on a low-rise building and the simulation of them. In: Huang, Y.-P., Wang, W.-J., Quoc, H.A., Giang, L.H., Hung, N.-L. (eds.) GTSD 2020. AISC, vol. 1284, pp. 543–555. Springer, Cham (2021). https://doi.org/10.1007/978-3030-62324-1_46
Effects of the Computational Domain Sizes on the Simulated Air Flow in Solar Chimneys Trieu Nhat Huynh(B) and Y. Quoc Nguyen Faculty of Engineering, Van Lang University, Ho Chi Minh City, Vietnam {trieu.hn,y.nq}@vlu.edu.vn
Abstract. Solar chimneys absorb solar energy to create stack effects which induce air flow for natural ventilation of buildings. Numerical models based on Computational Fluid Dynamics (CFD) have been increasingly utilized to simulate air flow and heat transfer in solar chimneys. One of the factors influencing the accuracy of the CFD models for solar chimneys is the size of the computational domain. In this study, effects of the sizes of the computational domain for a vertical solar chimney were investigated. Two sizes of the domain were tested, as suggested in the literature: A small domain which has the same physical size as a cavity inside the solar chimney (Domain S) and an extended one that covers both the cavity and the ambient air (Domain L). The flow structure, induced air flow rate, and thermal efficiency of the chimney were predicted and compared between the two domains. It is seen that Domain S offered identical results to those of Domain L at low gap – to – height ratios. At higher gap – to – height ratios, Domain S over – predicted the reverse flow region, and under – predicted the induced flow rate and the thermal efficiency. The critical gap – to – height ratios increased with the chimney height. The results can be used to determine whether Domain S, which requires less computational cost than Domain L, can be used to predict performance of a solar chimney with acceptable accuracy. Keywords: Natural ventilation · Solar chimney · Thermal effect · Computational domain · CFD
1 Introduction Solar heat gain on buildings can cause two opposite effects. For normal buildings, solar heat gain increases cooling load; hence the energy consumptions. For buildings with sustainable design, solar heat gain can be a benefit for natural heating or cooling of the building and help to reduce the energy usage. Solar chimney is one of the most common methods based on solar radiation for ventilating the building naturally. In solar chimneys, solar energy is converted into the flow energy. A typical solar chimney has an open cavity enclosed by surfaces which can absorb solar radiation. The absorbed heat warms the air in the cavity and induces the thermal, or stack effect which draws the air through the openings of the cavity. With suitable designs, the air flow can help to circulate air through the building for ventilation or cooling [1]. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 606–616, 2022. https://doi.org/10.1007/978-981-16-3239-6_46
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Studies of solar chimney have been conducted with experiments [2, 3], mathematical analysis [4] or numerical simulations [5–8]. Numerical models based on Computational Fluid Dynamics (CFD) have been increasingly utilized to simulate air flow and heat transfer in solar chimneys [5–8]. One of the factors influencing the accuracy of the CFD models for solar chimneys is the size of the computational domain. Employing smaller domains reduces the computational grid elements; hence lessening the computational cost. As the simulated flow variables are determined from the conditions applied at the boundaries of the domains, different selections of the computational domain can change the results significantly [5, 9, 10]. Zhang et al. [9] and Liu et al. [10] tried to reduce the computational domains for natural convection flow in a cavity [9] or for an external flow with special treatments of the flow variables at the boundaries. However, they did not report the benefits of using small domains over the additional costs due to the additional treatment at the boundaries. For simulating solar chimneys, Gan [5] tested two kinds of the computational domain. The smaller domain covered only the air cavity inside the chimney while the larger one included both the air cavity and surrounding air space. He showed that the air flow rate through the chimney predicted with the extended domain was closer to the experimental data than that obtained with the small domain. From the results, he suggested different minimum extensions of the domain from the chimney cavity. Gan [5] also claimed that for chimneys with low ratios of the cavity width (channel gap) and height, the small domain can offer acceptable results. However, he did not report how the computed flow variables changed with the size of the domain. In this study, we investigated effects of the size of the computational domain on the predicted air flow through a vertical solar chimney. The examined factor included the induced air flow rate and thermal efficiency through the chimney, flow structure, temperature fields, and particularly the reverse flow at the outlet of the chimney as the size of the domain changed.
2 Numerical Model 2.1 Assumptions for the Flow and Heat Transfer The following assumptions were assumed for the air flow and heat transfer in solar chimneys, as suggested in the literature [5–8]: • • • •
The air flow can be described in two dimensions in a vertical plane. The air flow and heat transfer are statistically steady. The air flow is incompressible. Convective heat transfer dominates in the air channel of the chimney. Conduction heat loss through the walls of the chimney and radiative heat transfer can be ignored. • Air properties follow the Boussinesq approximation where only the air density varies with the air temperature according to Eq. (1) [11]. ρ = ρ0 (1 − β(1 − T0 ))
(1)
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where ρ and ρ0 are the air densities at the flow temperature T and the reference temperature T0 , respectively; β is the air thermal expansion coefficient, β = 1/T . T0 is the ambient temperature, which is 293 K. 2.2 Governing Equations The equations governing the air flow and heat transfer in solar chimneys describe the conservations of mass, momentum, and energy. To obtained time – averaged quantities of the flow variables, the Reynolds – Averaged Navier – Stokes equations can be applied. Their two – dimensional forms for steady turbulent flows using vector notation are as follows. ∂uj =0 ∂xj
∂ ui uj ∂ui 1 ∂p ∂ ν =− − gi β(T − T0 ) + − ui uj ∂xj ρ ∂xi ∂xj ∂xj ∂ Tuj ν ∂T ∂ = − T uj ∂xj ∂xj Pr ∂xj
(2) (3) (4)
where u and u , and T and T indicate the time – averaged and fluctuating velocity, and temperature, respectively; p is the pressure; ρ, ν, and Pr stand for the air density, kinematic viscosity, and the Prandtl number, respectively; g is the gravitational acceleration; presents a time – averaged quantity. In the Eqs. (3) and (4), the turbulence model RNG k− ∈ was employed for solving the turbulence stress ui uj and the turbulent heat flux T uj , as suggested by Gan [5]. 2.3 Computational Domain and Boundary Conditions The considered solar chimney associated with two types of the computational domain is displayed in Fig. 1. The chimney was a typical one with a straight vertical air channel. Absorbed solar radiation was stored and transferred to the air in the channel on one side. The length of the channel was H while its gap was G. For the small domain (Domain S, Fig. 1a), the sizes of the domain coincided with that of the air channel. The large domain (Domain L, Fig. 1b) was extended from Domain S. We followed the findings from Gan [5] for the minimum requirement of the extension, as shown in Fig. 1b. He suggested that the domain should cover a space of 5.0 × G to the sides and the bottom, and 10.0 × G above the outlet of the air channel. On the domain S, the walls of the air channel were applied with no – slip conditions. At the inlet and outlet, atmospheric pressure conditions were employed. For thermal boundary conditions, the air entered the domain at the inlet at the ambient temperature which was assumed at 293 K. A uniform heat flux of 600 W/m2 was applied on the heated wall. The other wall of the air channel was assumed adiabatic. On the domain L, the whole boundary of the domain was exposed to the ambient air; hence applied with atmospheric pressure. Air entering the domain from the ambient was
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also at 293 K. The heated wall inside the air channel was also assigned with a uniform heat flux of 600 W/m2 . All other walls were adiabatic. With Domain S, the flow pressure, direction, and gradient were constrained right at the inlet and the outlet of the air channel. For example, the velocity vector must be normal to the inlet and outlet line in Fig. 1a. With Domain L, the flow was allowed to adapt freely to the flow dynamics at the inlet and outlet of the air channel. Consequently, the flow pressure, direction, and gradient may deviate from those applied on the Domain S.
Fig. 1. Computational domains and the according mesh: Domain S (a) and Domain L (b).
2.4 Computational Mesh and Numerical Setup The structures of the computational mesh for both domains are also presented in Fig. 1. Structured rectangular mesh cells were utilized for the whole domain. The mesh density was finer near the solid surface and at the inlet and outlet of the air channel, as the gradients of the flow quantities are strong at those locations. To check effects of the mesh resolution, we used the non – dimensional distance y+ of the first grid cell near the solid surfaces in the air channel. It was found that the induced flow rate and the temperature rise of the air through the chimney changed within 1.0% as the maximal y+ was less than 1.0. Details of the mesh – independent tests can be seen in [7, 8]. Finite Volume Method was utilized to discretize the governing equations, Eq. (2)–(4). The commercial CFD software ANSYS Fluent (Academic Version 2019R3) was used for
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the numerical setup. For the coupling between the continuity and momentum equations, the SIMPLEC scheme was selected. The PRESTO! method was used for interpolation of the pressure from the cell center to the cell faces. Other settings included second order upwind scheme for the momentum and energy equations and first order upwind scheme for k and ∈ equations. For the convergence criteria, the scaled residuals of all variables (velocities, pressure, temperature, turbulent kinetic energy, and turbulent dissipation rate) were set to 10−6 and achieved in all cases. 2.5 Model Validation The CFD model was validated against the CFD results by Gan [5] and the experimental data by Chen et al. [2]. The tested solar chimney was a vertical one whose height and width were 1.5 m and 0.62 m, respectively. The air gap changed from 0.1 m to 0.6 m. A uniform heat flux of 400 W/m2 was applied on one side of the air channel.
Fig. 2. Comparison of the air flow rate obtained with our CFD model, the CFD results by Gan [5], and the experimental data by Chen et al. [2].
Figure 2 compares the induced flowrate among the three methods. The CFD results by Gan [5] were obtained for both domains. Compared to the CFD results by Gan [5], our CFD results agreed well for the Domain S. Good agreement was also seen for the Domain L at G = 0.4 m and 0.6 m. At smaller gaps with the Domain L, our predicted data were lower than those by Gan [5]; however, they matched better with the measured data at G = 0.1 m. The difference between the two CFD results might be due to different boundary conditions of the turbulence which were not reported by Gan [5].
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Compared to the measured data, both CFD results agreed only at G = 0.1 m. As the gap increased, the CFD data with the Domain S decreased while both the measured data and the CFD results with the Domain L increased. The difference between the measured flow rate and the predicted flow rate with the Domain L was within 20.0% which should be within the measurement uncertainties [2, 5].
3 Results and Discussions It is observed in Fig. 2 that the flow rate predicted with the Domain S matched with that with the Domain L only at G = 0.1 m. As the gap increased, the CFD model with the Domain S underpredicted the induced flow rate. Therefore, it is deduced that both domains may offer similar results at small air channel gap, or in general at low gap – to – height ratio G/H. By changing that ratio for a given solar chimney, one can find the range of G/H where the Domain S can work acceptably. In this section, solar chimneys with different height, ranging from 0.5 m to 2.0 m, and gaps, ranging from G/H = 0.01 to 0.2 were examined with both domains. The performance in terms of the flow and temperature fields, induced flow rate, and thermal efficiency was then analyzed. In all cases, the heat flux was kept to 600 W/m2 . 3.1 Flow and Temperature Fields Examples of the predicted flow and temperature fields at the outlet of a solar chimney of H = 0.5m but with two different G/H ratios of 0.05 and 0.1 for both types of the
Fig. 3. Velocity and temperature fields at the outlet of a solar chimney with H = 0.5 m and two values of G/H predicted with both Domain S and Domain L. Lines inside the air channel indicate the regions of the reverse flow.
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computational domain. At G/H = 0.05 (Fig. 3a), Domain S predicted a reverse flow at the outlet. The reverse flow with Domain S brought air at the ambient temperature of 293 K into the air channel from the outlet and yielded a lower temperature area near the outlet. In contrast, a reverse flow was not observed with the domain L. The velocity and thermal boundary layers developed from the heated surface and occupied the whole channel gap at the outlet. In Fig. 3b, both domains predicted a reverse flow at the outlet of the chimney but with different sizes. The flow reversal region obtained with Domain S was about three times larger than that with Domain L. Though with a reverse flow, the thermal boundary layer with Domain L extended to the whole air gap at the outlet of the chimney. With Domain S, the velocity and thermal boundary layers for the case of G/H = 0.1 were even thinner than those for the case of G/H = 0.05. 3.2 Length of the Reverse Flow Figure 4 shows the length l where the reverse flow penetrated from the outlet of the chimney predicted with both domains. As sketched in Fig. 2a, the penetration area is the region with negative y – velocity component, as suggested by Khanal and Lei [12]. For comparison, the penetration length l was normalized by the channel height H.
Fig. 4. Length of the reverse flow normalized by the channel height predicted by Domain L and Domain S.
With the Domain S, a reverse flow appeared at G/H = 0.03 and H = 0.5 m. There was no reverse flow for other heights at G/H = 0.03. With the Domain L, flow reversals started at G/H ≈ [0.05 – 0.1]. At a given G/H ratio, the penetration length predicted with the Domain S was always higher than that with the Domain L.
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With the Domain L, the normalized penetration length l/H was the same for different heights at G/H = 0.1. At G/H = 0.15, l/H is seen to decrease with the height. In contrast, for the Domain S, a clear trend of l/H versus H is not observed. Khanal and Lei [12] used domain size similar to the Domain S in this study and reported the penetration length for a chimney with G/H = 0.137 and at different Rayleigh number, Ra, which is defined by Eq. (5). Ra =
gβqi H 4 vαk
(5)
where qi is the heat flux applied on the absorber surface; v, α, and k are the kinematic viscosity, thermal diffusivity, and thermal conductivity of air, respectively. They (Khanal and Lei 2012) [12] showed that l/H increased with Ra. In contrast, our results, for example at G/H = 0.05 with Domain S, showed that l/H decreased with H; hence decreased with the Rayleigh number according to Eq. (5). This difference may be due to the fact that in Khanal and Lei [12] simulations, the chimney geometries were fixed, and the heat flux changed, in contrast to our settings. 3.3 Induced Flow Rate The induced mass air flow rate through the tested solar chimneys with both domains is presented in Fig. 5. Figure 5a shows the variation of the mass flow rate as the chimney height and gap changed with both domains. For a given H, the flow rates obtained with both domains were identical when G/H was below a critical value. When G/H increased beyond that value, the flow rate predicted by the Domain S was significantly lower than that of the Domain L. Furthermore, the flow rate with the Domain S decreased while that of the Domain L increased as G/H increased.
Fig. 5. Induced air flow rate obtained with both domains for different chimney heights and gaps.
The flow rate obtained with the Domain L was non-dimensionalized by that with the Domain S and plotted in Fig. 5b. The critical values of G/H below which the difference between the two flow rates was within ±10.0% were 0.03, 0.05, 0.15, and 0.2 for H = 0.5 m, 1.0 m, 1.5 m, and 2.0 m, respectively. Therefore, the critical value increased with the chimney length, and according to Eq. (5), with the Rayleigh number.
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3.4 Thermal Efficiency Thermal efficiency of a solar chimney is defined by Eq. (6) which is the ratio of the heat absorbed by the air flow and the heat supplied to the air channel.
Fig. 6. Thermal efficiency obtained with both domains for different chimney heights and gaps.
η=
Qm cp (To − Ti ) Qi
(6)
where Qm is the mass flow rate; cp is the air specific heat capacity; Ti and To are the air temperature at the inlet and outlet of the air channel, respectively; Qi is the heat supplied to the air channel. Thermal efficiencies of the tested chimneys are plotted in Fig. 6. In Fig. 6a, it is seen that both domains predicted that the thermal efficiency decreased as the gap increased. At the lowest G/H ratio, 0.01, both domains offered identical values of η which was close to 1.0. This can be explained from the data in Fig. 4 and Fig. 5. Firstly, from Fig. 4, there was no flow reversal at G/H = 0.01 with both domains. At this very low G/H ratio, velocity and thermal boundary layer are expected to fully occupy the channel gap at the outlet [2, 5, 7, 8]. Therefore, both domains could result in identical temperature rises, T = To − Ti . Secondly, the data in Fig. 5 show similar air flow rate at G/H = 0.01. Consequently, Eq. (6) yielded identical thermal efficiencies at this gap – to – height ratio. As the gap increased further, the Domain S always offered lower η than the Domain L did. This fact again can be seen from the data in Figs. 4 and 5. The Domain S resulted in larger reverse flow regions at the outlet of the air channel (Fig. 4). As shown in Fig. 3, a larger reverse flow region led to lower average air temperature at the outlet; hence lower T. It also resulted in lower air flow rate, as seen in Fig. 5. Therefore, lower thermal efficiency obtained with the Domain S as the air gap increased can be expected. Figure 6b shows the ratio between the thermal efficiency predicted with Domain S and Domain L, ηL /ηS . All data of ηL /ηS for H = 1.5 m and 2.0 m were within [0.8–1.2], i.e. the difference of ηL and ηS was always within ±20.0%. For H = 0.5 m and 1.0 m, the ratio ηL /ηS within that range was when G/H < 0.03 and 0.05, respectively. Therefore, the critical G/H ratio at which the difference in the predicted thermal efficiency was within an acceptable range also increased with the chimney height, similar to the trend of the induced flow rate.
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4 Summaries Domain S and Domain L were utilized to compute the air flow and heat transfer in solar chimneys at different heights and gaps. It is seen that both domains offered similar results in terms of the flow structure, induced air flow rate, and thermal efficiency only at low gap – to – height ratios, which was below 0.03. As the gap increased, Domain S over – predicted the reverse flow at the outlet of the air channel. For a given chimney height, Domain S resulted in initiation of flow reversal at lower gap – to – height ratio. Once a flow reversal was observed with both domains, Domain S offered larger flow reversal regions. Both domains resulted in induced air flow rate which differed from each other below 10.0% when the gap – to – height ratio was below a critical value for a given height. The critical gap – to – height was 0.03, 0.05, 0.15, and 0.2 for H = 0.5 m, 1.0 m, 1.5 m, and 2.0 m, respectively; hence increased with the chimney height. Similarly, both domains also offered thermal efficiencies with difference below 20.0% when the gap – to – height ratio was below 0.03 and 0.05 for H = 0.5 m and 1.0 m, respectively. For H = 1.5 m and 2.0 m, the difference was always below 20.0%. In summary, the results from this study show that the small domain, which saves computational cost, can predict performance of solar chimneys at acceptable accuracy for low gap – to – height ratios. The critical gap – to – height ratios increased with the chimney height. A simple rule is that once there is no reverse flow predicted with the small domain, the results obtained with the small domain are identical to those with the large one.
References 1. Shi, L., Zhang, G., Yang, W., Huang, D., Cheng, X., Setunge, S.: Determining the influencing factors on the performance of solar chimney in buildings. Renew. Sustain. Energy Rev. 88, 223–238 (2018) 2. Chen, Z.D., Bandopadhayay, P., Halldorsson, J., Byrjalsen, C., Heiselberg, P., Li, Y.: An experimental investigation of a solar chimney model with uniform wall heat flux. Build. Environ. 38(7), 893–906 (2003) 3. Burek, S.A.M., Habeb, A.: Air flow and thermal efficiency characteristics in solar chimneys and Trombe Walls. Energy Build. 39(2), 128–135 (2007) 4. Al Touma, A., Ouahrani, D.: Performance assessment of evaporatively-cooled window driven by solar chimney in hot and humid climates. Sol. Energy 169, 187–195 (2018) 5. Gan, G.: Impact of computational domain on the prediction of buoyancy-driven ventilation cooling. Build. Environ. 45(5), 1173–1183 (2010) 6. Kong, J., Niu, J., Lei, C.: A CFD based approach for determining the optimum inclination angle of a roof-top solar chimney for building ventilation. Sol. Energy 198, 555–569 (2020) 7. Nguyen, Y.Q., Wells, J.C.: Effects of wall proximity on the airflow in a vertical solar chimney for natural ventilation of dwellings. J. Build. Phys. 44(3), 225–250 (2020) 8. Nguyen, Y.Q., Wells, J.C.: A numerical study on induced flowrate and thermal efficiency of a solar chimney with horizontal absorber surface for ventilation of buildings. J. Build. Eng. 28, 101050 (2020) 9. Zhang, T., Yang, H.: Flow and heat transfer characteristics of natural convection in vertical air channels of double-skin solar façades. Appl. Energy 242, 107–120 (2019)
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10. Liu, X., Xie, Z., Dong, S.: On a simple and effective thermal open boundary condition for convective heat transfer problems. Int. J. Heat Mass Transf. 151, 119355 (2020) 11. Tritton, D.J.: Physical Fluid Dynamics, 2nd edn. Oxford University Press, Oxford (1988) 12. Khanal, R., Lei, C.: Flow reversal effects on buoyancy induced air flow in a solar chimney. Sol. Energy 86(9), 2783–2794 (2012)
A Solar Chimney for Natural Ventilation of a Three – Story Building Tung Van Nguyen(B) , Y Quoc Nguyen, and Trieu Nhat Huynh Faculty of Engineering, Van Lang University, Ho Chi Minh City, Vietnam {tung.nv,y.nq,trieu.hn}@vlu.edu.vn
Abstract. Energy consumption of buildings can be significantly reduced with appropriate design for natural ventilation, particularly based on solar energy. Solar heat gained on the building envelope can be converted into thermal effects to induce air flow for natural ventilation or cooling of the building interior and the building envelope. For the ventilation application, the solar chimney should provide sufficient air flow rate. In this study, we investigated a solar chimney for ventilation of a three – story building. The chimney was assumed to be integrated into the building envelope and connected to all three stories of the building. To predict the air flow rate induced by the chimney through each story, a computational model was built based on the Computational Fluid Dynamics (CFD) method. By changing the design parameters (locations and dimensions of the air inlets) of the chimney, the ventilation rate through each story was computed. Temperature and velocity distributions in the chimney were also obtained for different design scenarios. From the results, the optimal design of the chimney which can provide nearly equal ventilation rate for all three stories was found. This study can serve as a demonstration for applications of the CFD technique in design of sustainable buildings. Keywords: Solar chimney · Flow rate · Thermal efficiency · Natural ventilation · CFD
1 Introduction With appropriate designs for natural ventilation, energy consumption of a building can be significantly reduced. Methods for natural ventilation can be based on effects induced by external wind flow or heat [1]. The methods based on heat usually employ structures absorbing solar radiation which can be converted into thermal effects to induce air flow for natural ventilation or cooling of the building interior and the building envelope. The structures creating such effects are classified as Solar Chimneys [2]. Many studies have reported effectiveness of solar chimneys in reducing energy consumption on buildings. In Japan, Miyazaki et al. [3] showed that a solar chimney integrated into an office building in Tokyo could save up to 50% of annual energy consumption. That number for a solar chimney attached to a test room in Qatar was 8.8% [4]. In China, Hong et al. [5] reported that a solar chimney for a two–story building saved up to 77.8% of energy for ventilation. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 617–630, 2022. https://doi.org/10.1007/978-981-16-3239-6_47
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When integrated into multi–story buildings, different configurations of solar chimneys have been examined [6–8]. Punyasompun et al. [6] compared two types of solar chimney for a three – story building: separate solar chimneys for each floor and a combined one connected to all floors. They reported that the combined solar chimney outperformed the separate ones with lower air temperature in each room and higher induced air flow rate. Asadi et al. [7] compared the induced air flow rate obtained with seven different configurations of solar chimneys for a seven–story building. Their results showed that the solar chimney on the southeast corner of the building offered the highest 24 h ventilation rate while the one in the middle of the south wall offered the highest flow rate in day time. Similar conclusions were also reported by Mohammed et al. [8] for a two – story building. One of the factors influencing most the performance of solar chimneys is the location of the air inlet [9, 10]. Effects of the inlet location of the air inlet of the solar chimney for a single room were reported by Shi et al. [10]. Those for multi – story buildings
Fig. 1. Skechmatic of a solar chimney for ventilation of a three – story building (H = 9.4 m, G = 0.2 m) with different configurations together with an example of the computational domain and mesh of Case 2.
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have not been reported thoroughly in the literature. Although Mohammed et al. [8] also considered different locations of the solar chimney inlet in their simulation, their model is for a small–scale building with dimensions of 1 m × 1 m × 1 m of each story. They did not report the scaling of the results for real size buildings. In this study, design of a solar chimney for a three – story building is examined. By changing the location, the number of inlets of each floor, and the size of inlet, it is aimed to find the best configurations of the solar chimney to achieve nearly equal ventilation rate for all floors. In the following sections, the problem is described together with the numerical method. Next, effects of the above factors are presented and discussed. Finally, main findings are summarized.
2 Problem Formulation and Numerical Method 2.1 The Solar Chimney A solar chimney is assumed to be integrated into a three – story building, as sketched in Fig. 1. The solar chimney consists of an air channel enclosed by an outer wall and a wall of the building. For typical designs of solar chimneys, the outer wall is usually a glass plate which receives solar radiation directly. Solar radiation is then transmitted through the outer wall and absorbed by the building wall. With proper insulation of the wall inside the building, the absorbed heat is not transferred into the building but supplied to the air inside the air channel of the solar chimney. Once heated, the air rises due to the stack effects and tends to induce an air flow through the chimney. If the solar chimney is connected to the rooms of the building and its upper end is open to the atmosphere, the induced air flow can help to ventilate the building naturally. In this study, the solar chimney was examined numerically under four configurations, as shown in Fig. 1. In Fig. 1a, the chimney has two openings at the lower and upper ends. An induced air flow enters and exits the air channel at the lower and upper openings, respectively. The solar chimney was then connected to the rooms by the inlets near the floor (lower inlet) in Fig. 1b while in Fig. 1c, the inlet of each floor was near the ceiling (upper inlet). In Fig. 1d, there were two inlets of each floor, both the lower inlet and the upper inlet. In Figs. 1b–d, two cases of the lower end of the solar chimney were considered: being open or closed. In all cases, the upper end of the solar chimney was open. The lower end of the air channel, in general, can be open to the atmosphere. Therefore, when it is open, ambient air can enter the air channel. It is assumed that the total height of each floor is 3 m. The total height H of the solar chimney is 9.4 m including the heights of three floors and the top ceiling. The width G of the air channel was fixed to 0.2 m. The inlets in Fig. 1b–d have the same height of 0.2 m while its length L is 0.5 m. 2.2 Assumptions and Governing Equations Performance of the solar chimney was assessed in term of the induced air flow rate induced through each inlet and the total flow rate at the outlet of the chimney. In practice, solar radiation approaching the chimney changes during daytime. However, for simplicity, a quasi–steady two–dimensional problem was considered in this study. Therefore, the
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air flow and heat transfer in the chimney were assumed to be steady and two – dimensional. In addition, as the air speed in typical solar chimneys have an order of about 1.0 m/s, the air flow was expected to be incompressible. For heat transfer, as the most important process is from the absorbed surface, which is the building wall in Fig. 1, to the air in the channel and the opposing glass plate, both convective and radiative heat transfer were considered. The convection and radiation are from the absorber surface to the air in the channel and the outer wall, respectively. Conduction loss through the building wall and the glass plate was not modeled for simplicity. For obtaining the induced air flow rate through the chimney together with the assumption of steady flow, time – averaged flow variables can be described with the Reynolds – Averaged Navier Stokes equations [9, 11]. The equations represent the conservation principles of the mass, momentum, and energy of the flow. Using the Einstein summation notation, the equations are as follows. ∂uj =0 ∂xj
∂ ui uj ∂ui 1 ∂p ∂ ν =− − gi β(T − T0 ) + − ui uj ∂xj ρ ∂xi ∂xj ∂xj ∂ Tuj ν ∂T ∂ = − T uj ∂xj ∂xj Pr ∂xj
(1) (2) (3)
In Eq. (1) to (3), the velocity u and temperature T are time – averaged while u and indicate their fluctuating components, respectively. p, ν, ρ, and Pr are the pressure, air kinematic viscosity, density, and the Prandtl number, respectively. β = 1/T0 and is the air thermal expansion coefficient. g is the gravitational acceleration; is the time – averaged operator. Air properties are taken at T0 . In Eq. (2), the Boussinesq approximation is used for variation of the air density with temperature. As the temperature rise of the air flow through typical solar chimneys is reported to be not high, this approximation has been widely used for simulations of air flows in solar chimneys in the literature [9, 11–13]. The turbulence tress ui uj in Eq. (2) and the turbulent heat flux T uj in Eq. (3) were solved with the standard k–ω model with modification for low – Reynolds number effects by a damping coefficient for the eddy viscosity. This turbulent model was also employed in previous studies of solar chimneys [9, 13]. For the solar chimney in Fig. 1, the gap – to – height ratio, G/H, is 4.25%. Gan [14] showed that for solar chimneys with very low G/H, a computational domain fitted with the air channel was sufficient. Accordingly, in this study, the computational domain covered only the air channel for Fig. 1a, and the air channel and the inlet sections for other cases. An example of the computational domain is presented in Fig. 1b. For discretizing the governing equations, Eqs. (1), (2) and (3), Finite Volume Method was used on a structured rectangular mesh with the ANSYS Fluent CFD Code (Academic version 2019R3). In the numerical setup, the SIMPLE method was employed for the coupling between the continuity and momentum equations; the PRESTO! Method was applied for interpolation of the pressure value from the cell center to the cell faces on T
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the staggered mesh; the second order upwind scheme was employed for the momentum and energy equations while first order upwind discretization was used for the k and ω equations. In all cases, the convergence criterion was set for the scaled residual to be lower than 10−6 . It was achieved in al tests. The mesh density was higher near the walls of the chimney where the gradients of the flow variables are strong. An example of the computational mesh is also displayed in Fig. 1b. To check effects of the mesh resolution, we ran the preliminary numerical model for the solar chimney in the experiment by Yilmaz and Fraser [15]. The number of elements in each direction was increased gradually and the according induced flow rate was plotted in Fig. 2. It is seen that a mesh with 20,000 elements is sufficient for this test. Denser meshes resulted in a change of the flow rate below 1.0%. The equivalent required non-dimensional distance of the first grid point from the solid surfaces, or y+ was below 1.0, which agrees with the findings by Zamora and Kaiser [9]. For the boundary conditions, atmospheric pressure was applied at the inlets of the rooms and at the outlet of the solar chimney. At the lower end of the chimney, section a–b in Fig. 1, when it was closed, a no–slip wall condition was applied. When it was open, the atmospheric pressured was assumed. Other walls were treated with no–slip conditions. The air temperature was set to 293 K at the inlets. A uniform heat flux of 600 W/m2 was assigned on the left wall (the absorber surface) in Fig. 1. The right wall was allowed to receive radiative heat transfer from the heated surface which was described by the S2S model in Fluent. When the lower end of the air channel was closed, it was treated as an adiabatic wall. At all inlets, turbulent kinetic energy and a length scale were used. As the turbulence should be low [9], it was assumed a turbulent intensity of 2.0% and a length scale of half of the inlet height, i.e. 0.1 m.
Fig. 2. Variation of the flow rate versus the number of elements for simulation of the solar chimney in the experiment by Yilmaz and Fraser [15] (Case of 100 °C). The measured flow rate by Yilmaz and Fraser [15] was also plotted. The selected mesh is indicated with the open marker.
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2.3 Validation of the Numerical Model The CFD model was applied to predict the induce flow rate through the solar chimney in the experiment by Yilmaz and Fraser [15]. The solar chimney is similar to the base case (Fig. 2a). The height and the gap of the chimney were 3.0 m and 0.1 m, respectively. In the experiment, uniform heat fluxes were applied on a surface of the air channel and the corresponding wall temperatures were in the range of 60–130 °C. Uniform wall tempratures were also applied in the CFD model. The predicted and measured flow rates are presented in Fig. 3 and Table 1. It is seen that the model slightly over–predicted the flow rate. However, the maximum discrepancy is less than 10.0% and should be within measurement uncertainty. Therefore, the model is acceptably accurate for predicting the induced flow rate.
Fig. 3. Comparison of the computed flowrate versus the measured data for the experiments by Yilmaz and Fraser [15]. Table 1. Comparison of the computed flowrate versus the measured data for the experiments by Yilmaz and Fraser [15] Wall temperature (°C)
Measurement (kg/s)
CFD (kg/s)
Difference (%)
60
0.0599
0.0603
0.79
75
0.0674
0.0736
9.16
90
0.0792
0.0846
6.76
100
0.0854
0.0912
6.76
130
0.1062
0.1091
2.71
3 Results and Discussions In this section, the induced flow rate through different cases of the solar chimney in Figs. 1a to 1d is presented. The heat flux applied on the heated wall was from 200 W/m2 to 800 W/m2 . From the results, the configuration which offers the highest flow rate was proposed and further examined.
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3.1 Base Configuration (Base Case) In the base case shown in Fig. 1a, the chimney had one inlet at the lower end of the air channel. This configuration has been studied widely in the literature [15–18]. The flow and velocity fields are plotted in Fig. 4. In Fig. 4a, the thermal boundary layers are seen to develop from both walls of the air channel. While the heat source was on the left wall, the right wall was heated due to radiative heat transfer from the left one. At the outlet, both the thermal and velocity boundary layers have occupied the whole channel gap. No flow reversal appears at the outlet. Gan [14] argued that with a reverse flow at the outlet, an extension of the computational domain outside the air channel is required to achieve reliable results. Furthermore, without a reverse flow, using a domain covering only the air channel can offer results as good as those of an extended one. As there is not a flow reversal at the outlet seen in Fig. 4a, using the computational domain consisting only the air channel in this study should be sufficient. The flow rate was also obtained with different heat fluxes on the heated wall and plotted in Fig. 4c. The flow rate increased with the applied heat flux. This trend can be physically expected, as the thermal effect; hence induced flow rate, is boosted with more heat input in the air channel.
Fig. 4. Temperature and flow fields, and the induced flow rate of the base case.
3.2 Testing Configurations (Case 2 to 4) The induced flow rate through the configurations 2 to 4 was predicted and presented in Fig. 5. For comparison, the flow rate in each case was normalized by that of the base
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case, Qtotal /Qbase . Though the data seem to increase with the heat flux, the variations are within only about 2.0% for each case. Except for the Case 3 (closed), all other cases have Qtotal /Qbase ≈ 1.0. The flow rate of Case 3 (closed) was about 94.0%–95.0% of the flow rate of the base case. Lower flow rate in Case 3 may be due to shorter elevation difference between the inlet of the first floor and the outlet of the air channel. Therefore, the configurations of Case 2 to Case 4 offered similar flow rate to that of the base case. Effects of the location and configurations of the inlets on the total flow rate through the chimney were seen negligible. Figures 6, 7 and 8 show the temperature and velocity fields in the air channel and the induced flow rate when the lower end was closed and open. The flow rate through each inlet is normalized by the total flow rate at the outlet of the air channel, Q/Qtotal , in each case. The temperature fields in Figs. 6a, 7a, and 8a show that when the lower end of the air channel was closed, heat was trapped below the inlet of the first floor. The portion of the air channel attached to the first floor had higher temperature. As heat transport took place only at the upper part of this portion, and moreover, the air flow only circulated inside that space, the trapped heat may result in such high temperatures. However, on the upper part of the channel, both cases of closed and open lower ends had very similar temperature distributions. Figure 6b, 7b, and 8b show difference only at the first floor inlet of the closed lower end cases where the velocity was higher than that of the open lower end cases. The flow fields on the upper part of the chimney in both cases of the lower end were identical.
Fig. 5. Comparison of the total flow rate of Case 2 to 4 to that of the base case, when the lower end of the channel was closed (closed) or open (open).
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Fig. 6. Temperature and flow fields, and the induced flow rate of Case 2.
Figures 6c, 7c, and 8c show that the Q/Qtotal of an inlet almost does not change with the heat flux. For both cases of the lower end, the induced flow rate at each inlet decreased from the first to the third floor. This point can be expected, as the higher floor has lower elevation difference between the inlet and the outlet of the air channel; hence weaker thermal effects and accordingly lower induced flow rate. For all configurations, at each inlet, Q/Qtotal of the closed lower end cases was always higher than that of the open lower end cases. For the open lower end cases, flow rates at all three inlets were below 30.0% of the total flow rate. About from 30% to 40.0% of the total flow rate was from the lower inlet of the air channel. In contrast, when the lower end was closed, the first floor inlet had the highest flow rate with more than 50% of the total flow rate. The other two inlets had flow rates similar to those of the open lower end cases. Particularly, in Case 3 with the closed lower end, the Q/Qtotal at the first floor inlet was up to about 70% (Fig. 7c). In contrast, those ratios of the third floor were only about 5.0%. This low flow rate is possibly because of short distance between the third floor inlet and the outlet of the air channel. 3.3 Selected Configuration (Case 5) From the induced flow rate obtained in Cases 2 to 4 in Fig. 6c, 7c, and 8c, it is seen that higher flow rates through the inlets in the floors are obtained when the lower end of the chimney was closed. In addition, the following configurations of the inlets offered the highest flow rate: • For the first floor: The inlet of Case 3 which yielded Q/Qtotal ≈ 70.0%.
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Fig. 7. Temperature and flow fields, and the induced flow rate of Case 3.
• For the second and third floor: The inlets of Cases 2 and 4 offered similar ratios of Q/Qtotal which are higher than those in Case 3. However, the ratios of Q/Qtotal of Case 4 are slightly higher than those of Case 2. Aimed at achieving both high and uniform flow rates at the inlets of all rooms, the selected configuration (Case 5) of the chimney has inlets similar to that of Case 3 for the first floor and those of Case 4 for the second and the third floor. Characteristics of the selected configuration are presented in Fig. 9. The total flow rate of Case 5 was similar to those of Cases 2 to 4 which was about 98.0% of that of the base case. The temperature and velocity fields in Figs. 9a and 9b show that on the second and third floors, among two inlets, more air was induced into the lower one. The flow rate through the upper inlet was about 20.0% of that of the lower one. The total flow rate through each room is plotted in Fig. 9c. The ratio of Q/Qtotal also changed negligibly with the heat flux. The highest flow rate is seen at the first floor with Q/Qtotal ≈ 41.0% and the lowest flow rate was at the third floor with Q/Qtotal ≈ 24.0%. Therefore, the flow rates among the floors were less different from each other, compared to those of Cases 2 to 4. As seen in Fig. 9, the flow rate through the first floor is about twice that through the third floor. In order to reduce the difference in the flow rates among the floors, different inlet sizes were tested (Case 6). Totally, there were five combinations of the inlet sizes which are denoted in Fig. 10a. Table 2 shows the total flow rates as the inlet sizes changed. The heat flux was kept to 600 W/m2 . In Table 2, except for the Case 6b, all other cases offered the same flow
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Fig. 8. Temperature and flow fields, and the induced flow rate of Case 4.
rate of 0.35 kg/s and about 6.0% higher than that of the base case. However, Case 6b resulted in 34.0% less flow rate compared to that of the base case. It is seen in Fig. 10b that the flow rates of Cases 6a and 6c–6d are similar to each other. Similar flow rates can be achieved for the first and the second floors. The flow rates through the third floor were always lower than those through the other floors. The flow rates of Case 6b differed from those of the others at the first and the third floors. It is seen that the main difference of Case 6b to the other cases is the inlet size i4. Between the inlet i4 and i5, the flow rate through i5 was minor, as seen from the flow fields in Figs. 7c, 8c, and 9c. Therefore, changing the inlet size i5 might not change the flow rate through the third floor much. However, reducing the inlet size i4 resulted in lower flow rate, as expected. From the pressure distributions of Cases 6a and 6b, it is seen that lower flow rate at the inlet i4 in Case 6b yielded higher pressure in the air channel around that inlet; hence lower pressure differences between that area and the inlets i1–i3, and consequently lower flow rate through those inlets in Case 6b. Therefore, the size of the lower inlet of the third floor is critical as it affects not only the total flow rate (Table 2) but also the flow rates through the floors 1 and 3. To check the ventilation performance of the proposed solar chimney, it was assumed that the chimney in Case 6 had a width of 1.0 m. The ACH (Air Changes per Hour) of that chimney for the building in Fig. 1 with a floor area of 8.0 m × 4.0 m is also plotted in Fig. 10b. ACH is defined as the number of times in one hour the air in a room is replaced, as seen in Eq. (4). ACH =
Q(kg/s) × 3600 ρ Vol.
where Vol. (m3 ) is the volume of the room.
(4)
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From Fig. 10b, except for Case 6b, all other cases offered ACH values greater than 3.0, which is the minimum requirement for ventilation of residential buildings [ASHREA 62.2].
Fig. 9. Temperature and flow fields, and the induced flow rate of Case 5.
3.4 Changing Inlet Size for Similar Flow Rates at All Inlets (Case 6) Table 2. Flow rates as the inlet sizes changes. Case
Qtotal (kg/s)
Case 6a (i1 = i2 = i3 = 0.2 m, i4 = 0.3, i5 = 0.1 m)
0.35
Case 6b (i1 = i2 = i3 = 0.2 m, i4 = 0.25, i5 = 0.15 m)
0.27
Case 6c (i1 = i2 = i3 = 0.2 m, i4 = 0.35, i5 = 0.05 m)
0.35
Case 6d (i1 = 0.2, i2 = i3 = 0.15, i4 = 0.35, i5 = 0.05 m)
0.35
Case 6e (i1 = 0.15, i2 = i3 = 0.1, i4 = 0.35, i5 = 0.05 m)
0.35
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Fig. 10. The induced flow rate at different inlet sizes.
4 Conclusions Different configurations of a solar chimney for natural ventilation of a three–story building have been examined. The results show that effects of the location and configurations of the inlets on the total flow rate through the chimney were seen negligible. While in all tested configurations, the induced flow rates varied within 6.0% around the value obtained with the base case, higher flow rates through the rooms were obtained when the lower end of the air channel was closed. The maximum difference was seen at the first floor where the flow rate with the closed lower end was up to about three times of that with the open lower end (Case 3). The best configuration for achieving the highest flow rates and the most similar flow rates through all three floors was with one inlet in the first floor and two inlets in the second and the third floors. By changing the inlet sizes, it was possible to achieve similar flow rates through the first and the second floors but the flow rate through the third floor was always less. The size of the lower inlet of the third floor is critical to both the total flow rate and the flow rates through the other floors. This study demonstrates possible applications of the CFD technique in design of sustainable buildings. In future works, optimization techniques may be applied to further optimize the inlet sizes to achieve more similar flow rates among the floors.
References 1. Awbi, H.: Ventilation of Buildings, 2nd edn. Spon Press, London (2003) 2. Bansal, N.K., Mathur, R., Bhandari, M.S.: Solar chimney for enhanced stack ventilation. Build. Environ. 28(3), 373–377 (1993)
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3. Miyazaki, T., Akisawa, A., Kashiwagi, T.: The effects of solar chimneys on thermal load mitigation of office buildings under the Japanese climate. Renew. Energy 31(7), 987–1010 (2006) 4. Al Touma, A., Ouahrani, D.: Performance assessment of evaporatively-cooled window driven by solar chimney in hot and humid climates. Sol. Energy 169, 187–195 (2018) 5. Hong, S., He, G., Ge, W., Wu, Q., Lv, D., Li, Z.: Annual energy performance simulation of solar chimney in a cold winter and hot summer climate. Build. Simul. 12(5), 847–856 (2019) 6. Punyasompun, S., Hirunlabh, J., Khedari, J., Zeghmati, B.: Investigation on the application of solar chimney for multi-storey buildings. Renew. Energy 34(12), 2545–2561 (2009) 7. Asadi, S., Fakhari, M., Fayaz, R., Mahdaviparsa, A.: The effect of solar chimney layout on ventilation rate in buildings. Energy Build. 123, 71–78 (2016) 8. Mohammed, H.J., Jubear, A.J., Obaid, H.: Natural ventilation in passive system of vertical two-stores solar chimney. J. Adv. Res. Fluid Mech. Therm. Sci. 69(2), 130–146 (2020) 9. Zamora, B., Kaiser, A.S.: Optimum wall-to-wall spacing in solar chimney shaped channels in natural convection by numerical investigation. Appl. Therm. Eng. 29(4), 762–769 (2009) 10. Shi, L., et al.: Interaction effect of room opening and air inlet on solar chimney performance. Appl. Therm. Eng. 159, 113877 (2019) 11. Kong, J., Niu, J., Lei, C.: A CFD based approach for determining the optimum inclination angle of a roof-top solar chimney for building ventilation. Sol. Energy 198, 555–569 (2020) 12. Khanal, R., Lei, C.: An experimental investigation of an inclined passive wall solar chimney for natural ventilation. Sol. Energy 107, 461–474 (2014) 13. Zavala-Guillén, I., Xamán, J., Hernández-Pérez, I., Hernández-Lopéz, I., Gijón-Rivera, M., Chávez, Y.: Numerical study of the optimum width of 2a diurnal double air-channel solar chimney. Energy 147, 403–417 (2018) 14. Gan, G.: Impact of computational domain on the prediction of buoyancy-driven ventilation cooling. Build. Environ. 45(5), 1173–1183 (2010) 15. Yilmaz, T., Fraser, S.M.: Turbulent natural convection in a vertical parallel-plate channel with asymmetric heating. Int. J. Heat Mass Transf. 50(13–14), 2612–2623 (2007) 16. Chen, Z.D., Bandopadhayay, P., Halldorsson, J., Byrjalsen, C., Heiselberg, P., Li, Y.: An experimental investigation of a solar chimney model with uniform wall heat flux. Build. Environ. 38(7), 893–906 (2003) 17. Burek, S.A.M., Habeb, A.: Air flow and thermal efficiency characteristics in solar chimneys and Trombe walls. Energy Build. 39(2), 128–135 (2007) 18. Khanal, R., Lei, C.: Flow reversal effects on buoyancy induced air flow in a solar chimney. Sol. Energy 86(9), 2783–2794 (2012)
Solar Chimneys for Natural Ventilation of Buildings: Induced Air Flow Rate Per Chimney Volume Y Quoc Nguyen(B) and Trieu Nhat Huynh Faculty of Engineering, Van Lang University, Ho Chi Minh City, Vietnam {y.nq,trieu.hn}@vlu.edu.vn
Abstract. Ventilation of buildings can be based on mechanical systems, such as fans, or natural driving forces, such as wind or heat. Of common natural ventilation methods, solar chimneys convert solar heat gained on the envelope of buildings into flow energy to ventilate or to cool the buildings. As solar chimneys are typically integrated into the building envelope, i.e. walls or roofs, architects determine shapes and sizes of a chimney mainly based on available space on the envelope. This raises a need for maximizing the ventilation performance of a solar chimney for a given space on a building envelope. In this study, ventilation performance of a typical vertical solar chimney was assessed in term of the induced flow rate that it can provide per volume. A three-dimensional numerical model based on the Computational Fluid Dynamics method was built to predict the induced air flow rate through the chimney as its dimensions changed. The tested dimensions included the height, the width, and the gap of the chimney. The induced air flow rate was obtained with different volumes of the chimney. The results show that the induced air flow rate nominated by the chimney volume changed with all three dimensions. Higher flow rate per volume was achieved with the chimneys with shorter heights and lower gap – to – height ratios. Therefore, it is suggested that to maximize the air flow rate per chimney volume, smaller chimneys are preferred. These findings agree with the results in the literature. Keywords: Natural ventilation · Solar chimney · Thermal effect · CFD · Flow rate per volume
1 Introduction Ventilation of buildings is to supply fresh air into the buildings to achieve desired levels of indoor air quality and thermal comfort. For driving the air flow, mechanical systems, such as fans, or natural driving forces, such as wind or heat can be used [1]. The methods based on thermal effects create air flows through a confined space with two conditions: temperature difference between the air inside the space and elevation difference between the inlet and the outlet of the space.
© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 631–640, 2022. https://doi.org/10.1007/978-981-16-3239-6_48
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Solar chimneys [2] are of the common systems for natural ventilation of building based on thermal effects. The main component of a solar chimney is the air channel which had open inlet and outlet at the bottom and the top, respectively. Solar radiation is absorbed on either the front or the rear wall of the channel. When the front wall is a glazing (glass) plate, solar radiation is transmitted through the front wall and absorbed on the rear wall of the channel. In contrast, when the front wall is opaque, it absorbs solar radiation directly. The absorbed heat is then transferred to the air in the channel. As the air temperature rises, the pressure inside the air channel deviates from that of the ambient and induces an air flow through the channel. When the inlet of the channel is connected to a room, stalled air inside the room is withdrawn. Fresh air is then supplied to the room through suitable arrangement of windows or doors. A typical solar chimney is illustrated in Fig. 1 where it is integrated into the front wall of the building.
Fig. 1. Illustration of a solar chimney for ventilation of a building.
As the thermal effects depend on the temperature rise in the air channel and the elevation difference between the inlet and the outlet, the absorbed heat and the height of the channel have been shown to influence most the performance of a solar chimney [3–9]. Those studies reported that the induced air flow rate increases with both the heat flux on the surface of the channel and the height of the channel. In addition, the flow rate also increases with the cross – sectional area of the channel which is the product of the channel gap and width (Fig. 2) [3, 4]. Therefore, increasing either the gap or the width results in higher flow rate. As solar chimneys are typically integrated into the building envelope, i.e. walls or roofs, architects determine shapes and sizes of a chimney based on not only the required induced flow rate sufficient for ventilation of the building but also the available space on the envelope. This raises the need for maximizing the ventilation performance of a solar chimney for a given space on a building envelope. For maximizing induced flow rate for a roof solar chimney, Hirunlabh et al. [10] compared different configurations with different lengths. They found that increasing the length resulted in higher induced air flow rate but decreased the ratio of the induced flow rate and the area of the absorber surface. Therefore, they suggested, for the same available length, to use multiple short chimneys instead of a long one.
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The research by Hirunlabh et al. [10] showed effects of the length of the chimney on the induced flow rate per surface area. However, as the chimney gap and width were fixed in their experiment, effects of the gap and the width have not been reported. In this study, we studied effects of all three main dimensions of the chimney, i.e. height, gap, and width, on the ratio of the induced flow rate and the volume of the chimney numerically. The configurations of the examined solar chimney together with the numerical model are described in section “Numerical Method”. Effects of the dimensions of the chimney on the induced flow rate are presented in section “Results and Discussions”. Finally, section “Conclusions” summarizes the main findings of this study.
2 Numerical Method 2.1 Studied Solar Chimney To focus on effects of the dimensions of the chimney, the examined chimney was a simple vertical one as depicted in Fig. 2. The air inlet and outlet are at the bottom and the top of the air channel, respectively. With this configuration, influences of other configurational factors can be eliminated. For example, the solar chimney illustrated in Fig. 1 has horizontal inlet and outlet. Zamora and Kaiser [11] and Shi et al. [12] showed that the size and the location of the inlet strongly affect the induced flow rate. By using straight vertical inlet and outlet, such undesired influences can be removed. As denoted in Fig. 2, three main dimensions of the chimney are the height H, the gap of the air channel G, and the width W. Besides the inlet and the outlet of the air channel, which are openings, other sides of the channel are rigid walls.
Fig. 2. The studied problem: a) Modeled solar chimney and b) Computational mesh.
The air flow and heat transfer were modeled inside the air channel. Consequently, the computational domain had the same size as the air channel. Absorbed solar radiation was transferred to the air in the channel from one wall. Accordingly, a heat source with uniform distribution was assumed on the left wall in Fig. 2a.
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2.2 Numerical Model The air flow and heat transfer in the air channel are described by conservation equations for the mass, momentum, and energy. As the air speed and the temperature rise through the air channel were not high [3], it was assumed that the air flow in the channel is incompressible; the air flow and heat transfer are statistically steady; and the Boussinesq approximation can be used for the change of air properties with temperature. Besides, convection heat transfer, which was reported to dominate in solar chimneys [3, 5], together with radiative heat transfer were considered. Conduction heat transfer and heat loss through the wall of the channel to ambient environment, was not modeled. As the flow in solar chimneys with practical sizes are in early turbulent regime, a suitable turbulence model was also required. The Reynolds – Averaged Navier – Stokes equations were employed in this study. Their forms for steady turbulent flows are as follows. ∂uj =0 ∂xj
∂ ui uj 1 ∂p ∂ ∂ui =− − gi β(T − T0 ) + − ui uj v ∂xj ρ ∂xi ∂xj ∂xj ∂ Tuj v ∂T ∂ = − T uj ∂xj ∂xj Pr ∂xj
(1) (2) (3)
In Eq. (1) to (3), time – averaged velocity and temperature are denoted as u and T. u and T indicates the fluctuating component of the velocity and temperature, respectively. p is the pressure. ν, ρ, and Pr are the air kinematic viscosity, density, and the Prandtl number, respectively. g stands for the gravitational acceleration; indicates a time – averaged quantity. For the turbulence tress ui uj and the turbulent heat flux T uj , the standard k–ω model with modifications for effects of low – Reynolds number was utilized. This turbulent model was also employed in previous studies of solar chimneys [11, 13]. Finite Volume Method was used to discretize the governing equations, Eq. (1) to (3) on the structured staggered mesh presented in Fig. 2b. The mesh was distributed more near the walls of the chimney where there were strong gradients of the flow variables. To find the most suitable mesh which offered mesh – independent solutions, the number of grid cells was gradually increased for different test cases. Plots of the induced flow rate versus the non-dimensional distance of the first grid point from the solid surfaces, or y+ , showed the range of y+ where the change of the induced flow rate was below 1.0%. It was found that the required range of y+ was below 1.0. Details can be seen in [8, 9]. For the boundary conditions, it was assumed that the flow enters and leaves the domain at the atmospheric pressure. Therefore, zero – gage pressure was applied at the inlet and the outlet of the domain. The walls were treated as no – slip and adiabatic ones except for the heated wall where a uniform heat flux of 600 W/m2 was assigned. For the coupling between the continuity and momentum equations, Eq. (1) and (2), the SIMPLE method was employed. In addition, to interpolate the pressure value from
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the cell center to the cell faces, the PRESTO! Method was applied. The above numerical schemes were conducted with the ANSYS Fluent CFD code (Academic version). Radiative heat transfer among the surfaces of the channel was modeled with the S2S model in ANSYS Fluent. 2.3 Validation of the Numerical Model For validation, the CFD model was applied to simulate the solar chimney in the experiment by Burek and Habeb [4]. The chimney was 1.0 m high and 0.92 m wide. The gap of the air channel changed in the range of 20 mm–110 mm. A uniform heat flux was applied on one wall of the channel whose dimensions were 1.0 m × 0.92 m. The measured induced flow rate through the chimney in the experiment and the one obtained with our CFD model at the total heat input of 600 W are plotted together in Fig. 3. It is seen that the numerical results are close to the measured ones. The CFD model slightly over – predicted the induced flow rate at the gaps of 100 mm and 110 mm. However, the maximum difference was only 6.5% which should be within the measurement errors.
Fig. 3. Comparison of the computed flow rate versus the measured data for the experiments by Burek and Habeb [4].
3 Results and Discussions In this section, the induced flow rate through solar chimneys with different heights, widths, and gaps obtained with the CFD model are presented. The chimney heights were 0.5 m, 1.0 m, and 1.5 m, which is the typical range in practical applications of solar chimney for natural ventilation of buildings [2]. The width and gap at each H changed and resulted in the gap – to – height ratio of 0.05, 0.1, and 0.15. Similarly, the width – to – gap ratio at each height was from 1.0 to 15.0. The heat flux was fixed at 600 W/m2 in all cases.
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3.1 Induced Flow Rate Versus Channel Width and Gap In Fig. 4, the induced flow rate is plotted as functions of the width and the gap of the examined chimneys. It is seen in Fig. 4a that the induced flow rate increased linearly with the width at any gap. In general, at a given width, the flow rate increased with the height of the chimney. However, at a given height, the trend of the flow rate with the gap was not consistent. At H = 1.5 m, the flow rate was enhanced as G/H increased from 0.05 to 0.1 and 0.15. At the other heights (H = 1.0 m and 0.5 m), the flow rate did not always increase with G/H.
Fig. 4. Induced flow rate as functions of the channel width and gap: (a) Flow rate vs. channel width, (b) Flow rate per volume vs. width – to – gap ratio.
To evaluate the effectiveness of increasing the width and the gap, the flow rate was normalized by the volume, as seen in Fig. 4b. The width was also non – dimensionalized by the gap. As the width – to –gap ratio, W/G, increased beyond a certain value, the flow rate per volume approached to a constant. The critical W/G was in the range of 5 to 7 for all tested cases. The flow rate per volume is plotted versus G/H for W/G = 15 in Fig. 5. For a given height, the flow rate per volume decreased with the gap – to – height ratio. As G/H increased from 0.05 to 0.15 for H = 0.5 m with W/G = 15, the flow rate per volume was reduced from 0.91 to 0.36. This trend is consistent with the experimental results by Chen et al. [3] and Burek and Habeb [4] which are also presented in Fig. 5. In Fig. 5, at G/H = 0.05 and 0.1, the flow rate per volume always decreased as the height increased from 0.5 m to 1.5 m. Hirunlabh et al. [10] reported that the flow rate per area of the absorber surface always decreased in their chimneys with the height up to 3.0 m and the gap of 0.14 m; hence G/H = 0.047. Therefore, the trend in Fig. 5 for G/H = 0.05 is consistent with that by Hirunlabh et al. [10] for G/H = 0.047. It is noticed in Fig. 5 that the trend at G/H = 0.15 is different from that of the other gap – to – height ratios. The flow rate per volume of H = 1.0 m dropped below that of H = 1.5 m. Examining the flow velocity at the outlet of this case reveals that the boundary layers of the H = 1.0 m case did not develop to the channel center and yielded an area with very low velocity. In contrast, in the cases of H = 0.5 m and 1.5 m, the boundary layers occupied the whole channel gap, as seen in Fig. 6. For the cases with G/H = 0.05
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Fig. 5. Induced flow rate per volume as functions of the channel gap for W/G = 15 together with the experimental results by Chen et al. [3] and Burek and Habeb [4].
and 0.1, the boundary layers always occupied the whole channel gaps at all heights. Accordingly, the trend of the flow rate per volume at G/H = 0.15 was different from the other cases.
Fig. 6. Velocity field at the outlet of the chimney with G/H = 0.15 and W/G = 15: a) H = 0.5 m, b) H = 1.0 m, c) H = 1.5 m.
3.2 Induced Flow Rate Versus Channel Volume The induced flow rate is plotted versus the volume of the chimney in Fig. 7. Figure 7a shows that the flow rate increased linearly with the volume with similar slopes for all chimneys. The experimental data by Chen et al. [3] and Burek and Habeb [4] also have similar trend. While the slope of the data by Burek and Habeb [4] is similar to that of our data, the slope of the data by Chen et al. [3] is much lower. This may be because of the same applied heat flux in our simulations and the experiments by Burek and Habeb [4], which was 600 W/m2 , and different heat flux in Chen et al. [3], which was 400 W/m2 . The trends of the data in Fig. 7a shows that at a given volume, the flow rate decreased as both the gap – to – height ratio and the chimney height increased. This observation
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Fig. 7. Induced flow rate as functions of the volume: a) Flow rate versus volume and b) Flow rate per volume versus volume.
applies to all cases, except for the case of H = 1.0 m and G/H = 0.15. The reason for this exception may be the same as that discussed in Sect. 3.1. To examine the effectiveness of increasing the chimney volume, the flow rate is normalized by the volume and plotted in Fig. 7b. Among three groups of G/H, for G/H = 0.05, the flow rate per volume increased with the volume. In contrast, for G/H = 0.1 and 0.15, the flow rate per volume decreased with the volume. In all cases, the flow rate per volume approached to constants as the volume increased. The flow rate per volume also decreased with the gap – to – height ratio and the chimney height. These trends are similar to those in Fig. 4b and 5. The experimental data by Chen et al. [3] and Burek and Habeb [4] are also plotted in Fig. 7b. In these experiments, the height and width of the chimneys were fixed while the gap changed. Therefore, in Fig. 7b, the increase of the volume of their chimneys is merely due to the increase of the air gap and associated with the decrease of the G/H ratio. The trends of variation of our results are seen to be similar to their data, as the flow rate per volume decreases with the gap – to – height ratio. 3.3 Thermal Efficiency Thermal efficiency of solar chimneys is the ratio between the heat absorbed by the air flow and the heat supplied to the flow from the wall of the channel [4, 8, 9], as defined by Eq. (4). η=
Qcp (To − Ti ) . Qi
(4)
where Q is the mass flow rate; cp is the air specific heat capacity; Ti and To are the air temperature at the inlet and outlet of the air channel, respectively; Qi is the heat supplied to the air channel. Thermal efficiencies of the examined chimneys decreased and approached constants as the volume increased, as seen in Fig. 8. Higher efficiencies were achieved with lower gap – to – high ratios for a given chimney height. The highest efficiency was obtained
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Fig. 8. Thermal efficiency.
with H = 0.5 m and G/H = 0.05. For this chimney height with W/G = 15, increasing G/H from 0.05 to 0.1 and 0.15 resulted in the reductions of the thermal efficiency from 0.96 to 0.56 and 0.34, respectively. However, a consistent trend of variation of the thermal efficiency with the chimney height was not seen in Fig. 8. 3.4 Proposed Chimney Configurations for Maximizing Induced Flow Rate From the data in Fig. 4, 5, 7, and 8, it is concluded that to achieve higher flow rate per volume, using shorter chimneys with lower gap – to – height ratios is preferred for an available volume. For example, for H = 0.5 m and W = 0.375 m, increasing the gap from 0.025 m (G/H = 0.05) to 0.05 m (G/H = 0.1) resulted in an increase of the flow rate from 4.3 g/s to 5.3 g/s. Therefore, the volume increases 200.0% but the increase of the flow rate is only 23.3%. For this case, if the architectural space is available, instead of utilizing single chimney with G = 0.05 m, one can design two chimneys with G = 0.025 m and get a total flow rate of 2 × 4.3 g/s = 8.6 g/s, which is 62.3% higher than that of the single chimney. Another example is considered for an available space where one can design a chimney with H = 1.5 m, W = 0.375 m, and G = 0.075 m and achieve a flow rate of 18.4 g/s. Instead, with the same volume, nine chimneys with the same W but H = 0.5 m and G = 0.025 m can be arranged. The total flow rate of nine smaller chimneys is 9 × 4.3 g/s = 38.7 g/s, which is 210% higher than that of the single chimney. Similar configurations for roof solar chimney was also proposed by Hirunlabh et al. [10]. They compared flow rate through chimneys for a given length of 3.5 m: a single one and three chimneys whose lengths were 1.0 m, 1.0 m, and 1.5 m. The total induced flow rate by three chimneys was up to 176% higher than that by the single one.
4 Conclusions The induced air flow rate through the chimneys with different heights, widths, and gaps have been predicted numerically. Performance effectiveness of the chimneys was
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assessed by the flow rate per volume. The results show that the flow rate per volume changes with the chimney width and approaches constant as the width – to – gap ratio is above about from 3 to 5. The flow rate per volume decreases as the gap – to – height ratio increases. When plotted as the function of the chimney volume, the flow rate per volume is higher for lower gap – to – height ratio of a given chimney height, and lower height for a given gap – to – height ratio. The thermal efficiency is also higher for lower gap – to – height ratio. From the results, it is proposed that to maximize the induced flow rate, with an available architectural space of a building, arrangement of many small chimneys with lower height and gap – to – height ratio is preferred over a big chimney which occupies the whole space. Acknowledgement. We appreciate Mr. N. V. Tung of CARE Research Group at Van Lang University for preparing the sketch in Figure 1.
References 1. Awbi, H.: Ventilation of Buildings, 2nd edn. Spon Press, London (2003) 2. Shi, L., Zhang, G., Yang, W., Huang, D., Cheng, X., Setunge, S.: Determining the influencing factors on the performance of solar chimney in buildings. Renew. Sustain. Energy Rev. 88, 223–238 (2018) 3. Chen, Z.D., Bandopadhayay, P., Halldorsson, J., Byrjalsen, C., Heiselberg, P., Li, Y.: An experimental investigation of a solar chimney model with uniform wall heat flux. Build. Environ. 38(7), 893–906 (2003) 4. Burek, S.A.M., Habeb, A.: Air flow and thermal efficiency characteristics in solar chimneys and Trombe walls. Energy Build. 39(2), 128–135 (2007) 5. Gan, G.: Impact of computational domain on the prediction of buoyancy-driven ventilation cooling. Build. Environ. 45(5), 1173–1183 (2010) 6. Al Touma, A., Ouahrani, D.: Performance assessment of evaporatively-cooled window driven by solar chimney in hot and humid climates. Sol. Energy 169, 187–195 (2018) 7. Kong, J., Niu, J., Lei, C.: A CFD based approach for determining the optimum inclination angle of a roof-top solar chimney for building ventilation. Sol. Energy 198, 555–569 (2020) 8. Nguyen, Y.Q., Wells, J.C.: Effects of wall proximity on the airflow in a vertical solar chimney for natural ventilation of dwellings. J. Build. Phys. 44(3), 225–250 (2020) 9. Nguyen, Y.Q., Wells, J.C.: A numerical study on induced flowrate and thermal efficiency of a solar chimney with horizontal absorber surface for ventilation of buildings. J. Build. Eng. 28, 101050 (2020) 10. Hirunlabh, J., Wachirapuwadon, S., Pratinthong, N., Khedari, J.: New configurations of a roof solar collector maximizing natural ventilation. Build. Environ. 36(3), 383–391 (2001) 11. Zamora, B., Kaiser, A.S.: Optimum wall-to-wall spacing in solar chimney shaped channels in natural convection by numerical investigation. Appl. Therm. Eng. 29(4), 762–769 (2009) 12. Shi, L., Cheng, X., Zhang, L., Li, Z., Zhang, G., Huang, D., Tu, J.: Interaction effect of room opening and air inlet on solar chimney performance. Appl. Therm. Eng. 159, 113877 (2019) 13. Zavala-Guillén, I., Xamán, J., Hernández-Pérez, I., Hernández-Lopéz, I., Gijón-Rivera, M., Chávez, Y.: Numerical study of the optimum width of 2a diurnal double air-channel solar chimney. Energy 147, 403–417 (2018)
Enhancing Ventilation Performance of a Solar Chimney with a Stepped Absorber Surface Y Quoc Nguyen(B) and Trieu Nhat Huynh Faculty of Engineering, Van Lang University, Ho Chi Minh City, Vietnam {y.nq,trieu.hn}@vlu.edu.vn
Abstract. Maximizing the utilization of renewable energy is one of the important points for designing sustainable buildings. Among the natural energy resources applied in buildings, solar radiation can be harnessed with solar chimneys. These devices absorb solar radiation for heating air in an enclosed channel. The thermal effects associated with the heated air can induce an air flow which can be used for ventilation, heating, or cooling of the connected buildings. This method can help to reduce the energy consumption of a building significantly. As solar chimneys have been attracting a number of studies in the literature, research interests in this topic have been focusing on enhancing the ventilation performance of typical solar chimneys by testing different shapes of the absorber surface. In this study, a novel type of a vertical solar chimney with a stepped absorber surface, unlike a straight one in typical chimneys, was studied with a numerical model. The air flow and heat transfer inside the air channel were computed with a CFD (Computational Fluid Dynamics) model. Performance of the chimney in terms of the induced air flow rate and thermal efficiency through the chimney, and the Nusselt number inside the air channel was investigated under different dimensions of the step and at different heat fluxes. The results show that the step strongly disturbed the distribution of the Nusselt number on the absorber surface and enhanced the induced air flow rate up to 11%, the air temperature rise through the chimney, and the thermal efficiency of the air flow up to 225% compared to those of a typical solar chimney. Therefore, the effectiveness of the proposed stepped absorber surface has been seen. Keywords: Computational fluid dynamics · Solar chimney · Natural ventilation · Stepped surface
1 Introduction According to the World Green Building Council [1], one of the main features of a green building is to maximize the usage of renewable energy resources to reduce negative impacts on the natural environment. Among the renewable energy resources available for buildings, solar energy has many applications, such as heating water, lighting, and particularly for ventilation with solar chimneys. Solar chimneys are referred to separate systems or structures of a building which can develop air flows based on solar radiation. A solar chimney consists of a confined channel © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 641–652, 2022. https://doi.org/10.1007/978-981-16-3239-6_49
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whose surfaces absorb solar radiation to heat the air in the channel. As the air is heated, thermal effects cause the internal pressure gradient to deviate from the ambient one which induces an air flow through the openings of the channel. With suitable arrangement of the openings of the air channel on the buildings, the induced air flow can be used for different purposes, such as heating the building, cooling the building facades, or ventilating the building naturally [2, 3]. For ventilation purpose, performance of a solar chimney is evaluated mainly by the speed of the air flow through the channel or the induced air flow rate [3]. As the main driving force of the air flow is buoyancy, the induced air speed is proportional to the heat supplied to the air channel, or the solar radiation flux that the air channel absorbs [3]. The induced air flow rate is reported to increase with the chimney height and the gap of the air channel [4, 5]. In addition, nearby walls also strongly influence performance of solar chimneys [6]. Several attempts have been made in the literature to improve ventilation performance of solar chimneys. Hirunlabh et al. [7] examined several configurations of a solar chimney with the same height to maximize the induced flow rate. They concluded that using several isolated chimneys offers more flow rate than a single one did. The flow rate is enhanced by special designs of the inlet of the air channel [8] or the absorber surface [9, 10]. Al-Kayiem et al. [8] reported that for a roof – top solar chimney, a vertical inlet of the air channel offers more flow rate up to 84% compared to that of a horizontal one. Higher flow rates are also achieved with a perforated absorber surface [9] compared to those of a smooth surface. Chorin et al. [10] examined flow field and heat transfer in a closed cavity with an obstacle on the heated surface. They showed that the heat transfer rate is strongly enhanced around the obstacle. However, as the cavity in their study was closed, no induced flow rate was reported. As an attempt to increase the heat transfer rate on the absorber surface, hence the induced flow rate through the channel, we examined a solar chimney with a stepped absorber surface in this study by numerical simulations. The problem is described in Sect. 2 together with the numerical model. Section 3 presents the results and discussions. Finally, the main findings are summarized in the section Conclusions.
2 Description of the Problem and Numerical Modelling A typical solar chimney for ventilation of a building is sketched in Fig. 1a. The chimney is integrated into a building wall. The air channel is confined between the building wall and a cover plate which absorbs solar radiation. With proper thermal insulation, the building wall can be adiabatic. The absorbed heat is transferred into the air in the channel. Heated air rises in the channel and escapes through the top of the channel. Air is sucked into the channel from the room through the opening at the bottom of the channel. Accordingly, the room is then ventilated. To study air flow and heat transfer, a two – dimensional model of the typical solar chimney is employed, as also presented in Fig. 1a. The modeled chimney is also confined by the cover and the building walls. The opening on the building wall is represented by a horizontal inlet. The absorbed heat is modeled as a heat source on the absorber wall. This type of model is similar to those in previous studies [11–13].
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The solar chimney in Fig. 1a has a smooth absorber surface while in Fig. 1b, the absorber surface has a step on the upper part. The main objectives of the step are to enhance heat transfer on the absorber surface and to suppress flow reversal at the chimney outlet. Khanal and Lei [14] reported that as the gap – to – height ratio of a chimney exceeds a certain value, a flow reversal exists at the chimney outlet and obstructs the increasing rate of the flow rate. They proposed an inclined wall to eliminate the flow reversal. Similar effects can also be expected with the step shown in Fig. 1b. 2.1 Numerical Model To assess ventilation performance of the solar chimney in Fig. 1, Computational Fluid Dynamics (CFD) technique was employed. It was assumed that the air flow and heat transfer inside the chimney are steady. Accordingly, time – averaged quantities of the flow field can be obtained with the Reynolds Averaged Navier – Stokes equations. Other assumptions of the governing equations are as follows. • The air flow is incompressible, as the average air speed in the air channel is typically quite low, which was about 1.0 m/s [4, 5, 17]. • The fluid properties follow the Boussinesq approximation [11, 12]. • Convective heat transfer dominates in the air channel. Radiative heat transfer was not modeled [12]. • Heat loss to the ambient through conduction of the channel walls was also not modeled. Besides, Chen et al. [4] reported that the air flow in typical solar chimneys with the heights of about from 1.0 m to 3.0 m was turbulent. Therefore, a turbulence model is also required. The governing equations for conservations of mass, momentum, and energy are described by Eq. (1) to (3), using the Einstein notation. ∂uj =0 ∂x
(1)
Fig. 1. Schematics of a typical solar chimney with a plane wall absorbing solar radiation (a) and the proposed chimney with a stepped absorber surface (b).
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ρuj
∂ui ∂ui ∂p ∂ μ =− − ρfi β(T − T0 ) + − ρui uj ∂xj ∂xi ∂xj ∂xj μ ∂T ∂T ∂ ρuj = − ρT uj ∂xj ∂xj Pr ∂xj
(2) (3)
In the above equations, u, p, and T indicate velocity vector, pressure, and temperature, respectively. and − stand for the fluctuating and time – averaged components, respectively. ρ and μ are respectively the fluid density and dynamic viscosity. f is the gravitational acceleration. Pr is the Prandtl number. In Eqs. (2) and (3), a turbulence model is required for the Reynolds tress − ρui uj
and the turbulent heat flux − ρT uj . Previous studies of solar chimneys have employed different turbulence models, such as the standard k – ε model [2, 15], standard k – ω model [16], and low Reynolds number k–ω model [11]. In this study, we tested the CFD model for the solar chimney in the experiment by Yilmaz and Fraser [17]. It is found that the low Reynolds number k–ω model offered the best agreement between the predicted and measured temperature distribution across the chimney gap, as seen in Fig. 3a. Therefore, it is adopted in this study. The computational domain is displayed in Fig. 2a which covers both the solar chimney and the ambient air. As suggested by Gan [12], the domain was extended to include the space surrounding the inlet and outlet of the chimney to allow the air flow adapt to local flow dynamics at those locations. For this chimney, we adopted the minimum extensions of ten times and five times of the air gap above and to the sides of the chimney, respectively. It was also found that including an extension at the inlet had negligible effects on the simulation results. Therefore, for saving computational cost, the domain was not extended at the inlet. The computational mesh is presented in Fig. 2b. Non-uniform rectangular mesh was used. The mesh density was higher near the solid surfaces where there were high gradients of the flow variables. To find the most suitable mesh density and pattern, the number of grid cells and different grow rates of the mesh size from the solid surface were tested for the solar chimney in the experiment by Yilmaz and Fraiser [17]. The induced flow rate and temperature rise through the chimney was plotted against the nondimensional distance of the first grid point from the solid surfaces, or y+ . The results show that for y+ ≤ 1.0, and the according mesh of more than 80 × 200 elements in the air channel, the results were mesh – independent. This agrees with the findings by Zamora and Kaiser [11]. For the boundary conditions, the atmospheric pressure was assigned at the open boundaries of the domain, as indicated in Fig. 2a. On the walls, no – slip conditions were applied for the velocity and adiabatic conditions for the heat transfer, except for the heated wall where a uniform heat flux was distributed. Finite Volume Method was used for discretizing the governing equations. For the coupling between the continuity and momentum equations, Eq. (1) and (2), the SIMPLE method was employed. In addition, to interpolate the pressure value from the cell center to the cell faces on staggered mesh, the PRESTO! Method was applied. The above numerical schemes were conducted with the ANSYS Fluent CFD code (Academic version 2019R3).
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Fig. 2. Schematics of the computational domain (a) and mesh (b). H and G are the height and the gap of the air channel, respectively. L and g are respectively the length from the outlet and the height of the step.
2.2 Model Validation The CFD model was validated against the experiment by Yilmaz and Fraser [17] and the CFD model by Zhang and Yang [15]. The experiment was conducted on a solar chimney whose height, gap, and width were 3.0 m, 0.1m, and 1.0 m, respectively. A uniform temperature of 100 °C was applied on one side of the air channel. Distributions of the flow velocity and temperature were measured across the channel gap at half the channel height and at the outlet, respectively. The CFD simulation by Zhang and Yang [15] was also conducted with a two – dimensional RANS model and the standard k – ε model.
(a) Temperature profile.
(b) Velocity profile.
Fig. 3. Comparison of the distributions of temperature and velocity profiles across the channel gap at H/2 among our CFD results, measured data by by Yilmaz and Fraser [17], and the CFD results by Zhang and Yang [15].
Comparisons of the temperature and velocity profiles are presented in Fig. 3a and 3b, respectively. Good agreement between our results and those by Zhang and Yang
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[15] for both profiles can be seen. Compared to the measured data, our results are closer than those by Zhang and Yang [15] for the temperature distribution. For the velocity distribution, both of the CFD results slightly underestimated the velocity near the heated surface and overestimated near the opposite wall. However, the difference of the induced flow rate between our data and the measured one was only less than 6.7%. Therefore, our CFD model is acceptably accurate.
3 Results and Discussions The validated CFD model was used to simulate the flow field and heat transfer inside the solar chimney sketched in Fig. 1 with and without the step on the absorber surface. In all cases, the chimney height H and the channel gap G were kept to 2.0 m and 0.2 m, respectively. The inlet height was also kept to 2G, as shown in Fig. 2a. Bassiouny and Koura [18] reported that effects of the inlet height on performance of a solar chimney were negligible when it was equal to or more than twice the channel gap. On the heated wall of the air channel, a heat flux of 600 W/m2 was assigned. This value is around the average solar radiation used in previous studies [4, 12]. The Rayleigh number based on the channel height and the heat flux is 4.1013 . Therefore, the flow in the air channel is expected to be turbulent [4]. For the step, the length L and the height g changed to yield the ratios of g/G and L/H up to 0.5. Effects of the step on the velocity and temperature fields in the air channel, the distributions and the averaged value of the Nusselt number on the heated wall, the induced flow rate, and the thermal efficiency were obtained and discussed in this section. 3.1 Effects of the Step on the Flow Field and Heat Transfer Figure 4 shows examples of the velocity and flow fields near the outlet of the solar chimney without a step on the heated surface (base case) and with the step with two different step heights. Without a step, the velocity and thermal boundary layers developed along the heated surface. At the outlet, both layers expanded to only about half of the channel gap. There existed a reverse flow at the outlet of the channel; hence the flow area was significantly reduced. The reverse flow also brought air from the ambient at lower temperature into the air channel and resulted in lower temperature distribution on the region of the reverse flow. With the step, in both cases of the step length, the reverse flow was successfully suppressed. The flow was guided toward the unheated wall by the step. Accordingly, the thermal layer almost occupied the whole channel gap. However, with the shorter step length (L/H = 0.025), a new reverse flow appeared near the outlet on the heated side. The flow area was reduced even more than that of the base case. With the longer step length (L/H = 0.1), the flow occupied the whole channel gap at the outlet. Distributions of the Nusselt number on the heated surface of the cases in Fig. 4 are plotted in Fig. 5. The local Nusselt number is defined based on the channel gap as in Eq. (4). Nu =
qw G (Tw − Ta )λ
(4)
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(a) Flow field.
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(b) Temperature distribution.
Fig. 4. Velocity and temperature distributions near the outlet of the air channel with the heated plane sureface (a), the stepped surface with L = 0.1 m (L/H = 0.025) (b) and L = 0.2 m (L/H = 0.1) (c), with g = 0.05 m (g/G = 0.25)
Fig. 5. Distributions of the Nusselt number on the heated surface without and with the step with L = 0.1 m and L = 0.2 m (g = 0.05 m).
Where qw = 600 W/m2 and Tw were the heat flux and temperature of the wall, respectively; Ta is the air temperature at the inlet of the air channel (293 K); and λ is the air thermal conductivity. It is seen in Fig. 5 that, compared to the base case, the presence of the step caused a significant drop of the local Nusselt number in front of the step. This may be because of a small flow separation zone at that location, which is not visible in Fig. 5a due to the scale of the plot. Higher wall temperature in that separation zone, as seen in Fig. 4b, resulted in lower Nusselt number (Eq. (4)). At further downstream of the step, the local Nusselt number gradually increased and achieved a peak in both cases of the step length. Chorin et al. [10] also reported an increase of the Nusselt number downstream of the circular cylinder in their experiment. However, a trend of variation similar to those
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with the step in Fig. 5 was not reported. It may be because the number of measurement points along the heated surface close to the obstacle in their experiment was not high enough to capture such a trend. 3.2 Effects of the Step Length and Height From the local Nusselt number distributions, such as those in Fig. 5, the averaged Nusselt number is obtained by Eq. (5).
Fig. 6. Normalized averaged Nusselt number as functions of the step length and height.
H Nuav =
(Nu dl)/H
(5)
0
In Fig. 6, the averaged Nusselt numbers for three cases of the step height g are normalized by the averaged Nusselt number of the base case and plotted as functions of the step length. It is seen that the ratio of Nuave /Nubase was less than 1.0 and varied slightly about 92.0% for all three cases of g and at any value of L/H. Therefore, with a step, the average Nusselt number on the heated surface decreased compared to that of the base case. Effects of the step height and length on the Nusselt number was also negligible. The induced flow rate was also normalized by that of the base case and plotted in Fig. 7. The flow rate changed with both the step height and length. At L/H < 0.3, compared to the flow rate of the base case, the case with g = 0.1 m always offered lower flow rates while the case with g = 0.025 m resulted in higher values. The maximum reduction of the flow rate for g = 0.1 m was up to 25.0% while the maximum increase for g = 0.025 m was about 11.0%. For the case with the intermediate step height, g = 0.05 m, the flow rate first decreased and then increased to a peak of about 9.0% higher than that of the base case.
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Fig. 7. Normalized flow rate as functions of the step length and height.
As L/H exceeded about 0.3, the induced flow rate was almost constant with the step length. The flow rate was about 5.0% higher than that of the base case only for g = 0.05 m. For two other step heights, the flow rates were similar to that of the base case. As observed in Fig. 6, the Nusselt number always slightly decreased with the presence of the step compared to that of the base case. However, the induced flow rate was enhanced in some cases. Therefore, the increase of the flow rate was not due to an increase of the heat transfer on the heated surface, which was the case reported by Lei et al. [9]. They showed that the increase of the flow rate was mainly due to better heat transfer with a perforated absorber surface. In this study, the increase of the flow rate was possibly due to the suppression of the reverse flow at the outlet. This point can be seen in Fig. 4a and 4b for g = 0.05 m and L/H = 0.1 and in Fig. 7 where with that step height and length, the ratio of Q/Qbase is equal to 1.09. This effect was similar to that reported by Khanal and Lei [13]. They suppressed the reversed flow at the outlet of their chimney by using a wall inclined toward the outlet of the air channel and obtained an enhancement of the induced flow rate. The reduction of the flow rate for g = 0.05 m and L/H < 0.1, and g = 0.1 m and L/H < 0.3 is expected due to severe contraction of the flow area. The steps in such cases served as obstacles to the flow; hence increased the flow resistance. Accordingly, the flow rate was less compared to that of the base case. The temperature rise (Eq. (6)) and the thermal efficiency (Eq. (7)) were compared to those of the base case and plotted in Fig. 8. T = To − Ti
(6)
In Eq. (6), To and Ti are the temperatures at the outlet and the inlet of the air channel, respectively. (7) η = Qcp T /Ii In Eq. (7), Q is the induced flow rate; cp is the specific thermal capacity of air; and I i is the total heat delivered from the heated surface.
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(a) Temperature rise.
(b) Thermal efficiency.
Fig. 8. Temperature rise of the air flow through the chimney and thermal efficiency as functions of the step length and height.
It is seen in Fig. 8 that the temperature rise and the thermal efficiency were always higher than those of the base case. T and η are seen to increase with the step height. The increase of the temperature rise was up to 2.4 times of that of the base case while the relative increase of the thermal efficiency was up to 2.25. As the L/H ratio increased, the ratios of T/Tbase and η/ηbase approached constants, which were always above 1.0. As seen in Fig. 4b, the enhancement of the temperature rise was due to the elimination of the reverse flow at the outlet of the channel with the step. As the flow rate did not always increase or increased only up to 11.0% with the presence of the step, the highly enhanced thermal efficiency in Fig. 8 is expected mainly due to the increase of the temperature rise, according to Eq. (7). This conclusion is further confirmed as the trends of variations of the data in Fig. 8a and 8b are very similar to each other.
(a) Induced flow rate.
(b) Temperature rise and thermal efficiency.
Fig. 9. Effects of the step height on the flow rate, temperature rise, and thermal efficiency for two different step lengths: L = 0.05 m (L/H = 0.025) and L = 0.2 m (L/H = 0.1).
To further examine effects of the step height, two cases of L/H with the peak flow rates in Fig. 7, i.e. L/H = 0.025 and 0.1, were selected to test with different step heights. The results are presented in Fig. 9 where the ratio of g/G changed from 0.05 to 0.5 (g
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= 0.01 m to 0.1 m). The Nuave /Nubase ratio slightly varied from about 0.93 to 0.89 while the ratio of Q/Qbase changed in the ranges of 0.95–1.08 and 0.75–1.1 for L = 0.2 m and 0.05 m, respectively. The peak flow rates were similar to those in Fig. 7. The ratio of g/G corresponding to the peak flow rates were 0.125 and 0.5 for L = 0.05 m and 0.2 m, respectively, as seen in Fig. 9a. In Fig. 9b, both the temperature rise and the thermal efficiency increased with the step height. Therefore, the trends of T /T base and η/ηbase are consistent with those in Fig. 8. For L = 0.05 m, the peak η/ηbase of 1.85 is seen at g/G = 0.3125. For L = 0.2 m, the η/ηbase seems to grow steadily. Similar points are also observed for the T/Tbase ratio in Fig. 9b.
4 Conclusions Effects of a step on the absorber surface of a solar chimney were investigated numerically. While the chimney height, air gap, and the heat flux were fixed, the step height g and length L changed. Two main effects are observed from the results. Firstly, the step guided the induced air flow toward the unheated wall; hence suppressed the reverse flow at the outlet. This effect is seen with low step height. Secondly, the higher step height also blocked the air flow and increased the flow resistance. While the presence of the step slightly reduced the averaged Nusselt number about 8.0% in average, the induced flow rate, and particularly the temperature rise and the thermal efficiency were strongly influenced. Compared to the base case, higher flow rate was achieved with g = 0.05 m and up to 11.0%, but lower flow rate is seen for g = 0.1 m and up to 25.0%. The highest Q/Qbase of about 1.1 was achieved with g/G = 0.0125 and L = 0.05 m. The temperature rise and thermal efficiency were always higher than those of the base case up to 2.4 and 2.25 times, respectively. For possible extensions of this study, three – dimensional steps or other types of the obstacles may be examined with three – dimensional CFD models. For verifications of the simulations, experiments can be conducted.
References 1. World Green Council: What is green building? (2020). https://www.worldgbc.org/whatgreen-building. Accessed 21 Sep 2020 2. Gan, G.: Simulation of buoyancy-induced flow in open cavities for natural ventilation. Energy Build. 38(5), 410–420 (2006) 3. Shi, L., Zhang, G., Yang, W., Huang, D., Cheng, X., Setunge, S.: Determining the influencing factors on the performance of solar chimney in buildings. Renew. Sustain. Energy Rev. 88, 223–238 (2018) 4. Chen, Z.D., Bandopadhayay, P., Halldorsson, J., Byrjalsen, C., Heiselberg, P., Li, Y.: An experimental investigation of a solar chimney model with uniform wall heat flux. Build. Environ. 38(7), 893–906 (2003) 5. Burek, S.A.M., Habeb, A.: Air flow and thermal efficiency characteristics in solar chimneys and Trombe walls. Energy Build. 39(2), 128–135 (2007) 6. Nguyen, Y.Q., Wells, J.C.: A numerical study on induced flowrate and thermal efficiency of a solar chimney with horizontal absorber surface for ventilation of buildings. J. Build. Eng. 28, 101050 (2020)
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7. Hirunlabh, J., Wachirapuwadon, S., Pratinthong, N., Khedari, J.: New configurations of a roof solar collector maximizing natural ventilation. Build. Environ. 36(3), 383–391 (2001) 8. Al-Kayiem, H.H., Sreejaya, K.V., Chikere, A.O.: Experimental and numerical analysis of the influence of inlet configuration on the performance of a roof top solar chimney. Energy Build. 159, 89–98 (2018) 9. Lei, Y., Zhang, Y., Wang, F., Wang, X.: Enhancement of natural ventilation of a novel roof solar chimney with perforated absorber plate for building energy conservation. Appl. Therm. Eng. 107, 653–661 (2016) 10. Chorin, P., Moreau, F., Saury, D.: Heat transfer modification of a natural convection flow in a differentially heated cavity by means of a localized obstacle. Int. J. Therm. Sci. 151, 106279 (2020) 11. Zamora, B., Kaiser, A.S.: Optimum wall-to-wall spacing in solar chimney shaped channels in natural convection by numerical investigation. Appl. Therm. Eng. 29(4), 762–769 (2009) 12. Gan, G.: Impact of computational domain on the prediction of buoyancy-driven ventilation cooling. Build. Environ. 45(5), 1173–1183 (2010) 13. Khanal, R., Lei, C.: A numerical investigation of buoyancy induced turbulent air flow in an inclined passive wall solar chimney for natural ventilation. Energy Build. 93, 217–226 (2015) 14. Khanal, R., Lei, C.: An experimental investigation of an inclined passive wall solar chimney for natural ventilation. Sol. Energy 107, 461–474 (2014) 15. Zhang, T., Yang, H.: Flow and heat transfer characteristics of natural convection in vertical air channels of double-skin solar façades. Appl. Energy 242, 107–120 (2019) 16. Zavala-Guillén, I., Xamán, J., Hernández-Pérez, I., Hernández-Lopéz, I., Gijón-Rivera, M., Chávez, Y.: Numerical study of the optimum width of 2a diurnal double air-channel solar chimney. Energy 147, 403–417 (2018) 17. Yilmaz, T., Fraser, S.M.: Turbulent natural convection in a vertical parallel-plate channel with asymmetric heating. Int. J. Heat Mass Transf. 50(13–14), 2612–2623 (2007) 18. Bassiouny, R., Koura, N.S.A.: An analytical and numerical study of solar chimney use for room natural ventilation. Energy Build. 40(5), 865–873 (2008)
The Effect of Ground Surface Geometry on the Wing Lift Coefficient Dinh Khoi Tran(B) and Anh Tuan Nguyen Le Quy Don Technical University, 236 Hoang Quoc Viet, Ha Noi, Vietnam [email protected]
Abstract. The studies of wing aerodynamics in ground effect generally consider the ground surface to be either absolutely flat or wavy in a form of a sinusoidal function. However, in practice, ground surfaces may have complex geometry, which affects the aerodynamic characteristics of wings flying over them. The aim of the paper is to analyze the influence of some basic ground geometry parameters on the lift force coefficient of a wing model. The mirror-image method is integrated into the unsteady panel code to solve the ground effect problem with the complex geometry. The program of the panel method, which was developed based on the potential-flow theory, is written in FORTRAN with the use of parallel computation techniques to reduce the run time. The calculation results show that the variation of the ground surface geometry may have profound effects on the unsteady lift coefficient properties of the wing. The phase, the shape and the amplitude of the lift coefficient variation may be altered significantly when we change the geometry of the ground surface. Moreover, the effect of the ground geometry in the presence of horizontal wind is also analyzed. This paper provides close insight into these phenomena with physical explanations, which have not been addressed by other researchers. Keywords: Ground effect · Unsteady panel method · Unsteady aerodynamics
1 Introduction Ground effect is of great importance for vehicles flying at low altitude [1]. This effect needs considering while studying wing-in-ground (WIG) craft or airplanes during takeoff and landing. In many previous studies of ground effect, the surface is normally considered to be flat [2]. At the same time, many researchers have investigated the ground effects of sinusoidal surfaces [3, 4]. In reality, ground surfaces can be much more complex; and therefore, their effect may change significantly as a function of the geometry. Tang and Wang [5] have shown that a wind field can be strongly affected by complex terrain. However, these researchers did not indicate to what degree the flight characteristics of an airplane may be altered while flying over terrain. There is the fact that the impact of hilly ground on the aerodynamics of an airplane may be insignificant. Nevertheless, this phenomenon could be magnified due to the presence of horizontal wind. Liu et al. [6] have obtained the flow around a 3D hill through large eddy simulations in the case © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 653–662, 2022. https://doi.org/10.1007/978-981-16-3239-6_50
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of horizontal wind. It should be noted that, this kind of Computational Fluid Dynamics (CFD)-based method requires large computer resources. Hence, the simulation for the combined model of a hilly surface and a flying vehicle is challenging. In our previous paper [7], we have introduced a novel model based on the unsteady panel method and the mirror-image technique for the problem of wing and curved ground surface interaction. The validity of the model was analyzed and confirmed for various cases. In here, the model is applied to answer the question to what extend the ground surface geometry influence the wing lift coefficient, which is regarded as an important indication of wing aerodynamic characteristics.
2 Panel Method and Calculation Model 2.1 Panel Method According to [8], the following assumptions are made during the development of the unsteady panel method: (1) the flow is attached on the surfaces of solid bodies; (2) viscous effects are negligible; (3) the flow is irrotational, which means there is no vorticity everywhere in the computational domain except for the surfaces of solid bodies and wake sheets. The panel method used in this paper is similar to that in [9] with the distribution of sources and doublets. More details of the present panel method can be found in [7]. 2.2 Model Calculation To study the effects of the geometry of the ground surface on the wing lift coefficient in the ground effect, the authors assume that the wing flies over an infinite flat surface, on which there is a curved surface located within ABCD. Then we can model the wing and curved surface located within ABCD using sources and doublets. The whole model will be mirrored symmetrically over the flat surface EFGH. The mathematical basis of this model is depicted in Fig. 1. The sources and doublets of the wings are taken in mirror symmetry in the usual way. The mirrored values of the sources and the doublets on the curved surface are σmirror = σ
Fig. 1. Modeling of curved limit surface
(1)
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μmirror = −μ
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(2)
As explained in [7], the sources on the flat plane ABCD have the zero strength according to the Neumann boundary condition. At the same time, the doublets and those on the mirrored surface A’B’C’D’ have the same strength but in the opposite direction. Therefore, when the surface ABCD is in proximity to its mirrored counterpart, singularities on these two surfaces will cancel out each other. This approach can eliminate computational errors due to the finite size of the ground surface, which happens in [10]. Figure 2 shows the wing model flying at a speed V over a stationary curved surface without wind.
Fig. 2. Computational model without wind
In the presence of horizontal wind, the calculation model will include the wing flying over a stationary curved surface and the incoming flow at a speed Vwind . In this case, the curved geometry of the ground surface will alter the velocity direction of the fluid around it; therefore affect the aerodynamic properties of the wing. 2.3 Validations To verify, the authors build a model with sinusoidal limit surface corresponding to the conditions as [3, 4]. The results of calculation and comparison are presented in detail in [7]. They have confirmed the validity of the model and computational program.
3 Numerical Results 3.1 The Influence of the Ground Surface Geometry Shape 3.1.1 Effects of Ground Curved in the Flight Direction Here, we study a rectangular wing with an aspect ratio AR of 4, the length of chord c = 0,1(m), the flight velocity V = 20(m/s), the curve amplitude a = 0.5c, the wavelength λ = 5c. The elevation of the ground surface is defined as for various shapes [11] x a tanh C sin 2π (3) z(x) = tanh C λ x is the longitudinal coordinate. As C approaches 0, function z(x) becomes sinusoidal; and as C increases, the function approaches the step form.
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Fig. 3. Dimension curved ground
Fig. 4. The wing model and its wake patterns over several types of ground surface geometry (a); and lift coefficient corresponding to various ground surface geometry parameters (b).
The ground surface has dimensions of 5c × 6c (Fig. 3). Results obtained with different C values are shown in Fig. 4. The calculation results show that the variation of the ground surface geometry may have profound effects on the unsteady lift coefficient properties of the wing. The phase, the shape and the amplitude of the lift coefficient variation may be altered significantly when we change the geometry of the ground surface. In order to obtain the details of these
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effects, the result is presented in the domain of the nondimensional spatial frequency F through the Fourier transform. As shown in Fig. 4, the variation of the lift coefficient can be given as a function of the nondimensional travelled distance, which is obtained through a division by the chord length. Therefore, the spatial frequency is normalized by 1/c. Figure 5 exhibits the amplitude spectrum of the variation shown in Fig. 4 for various values of C. It is seen that, all of the spectrums have a peak at a frequency of 0.2 that is considered to be a fundamental frequency. The magnitude of this peak increases with C. Similarly, the second peak is observed at a frequency of 0.6. However, the trend at this peak is not as apparent as the first one. When we compare the amplitude spectrum of the lift coefficient that that of the ground elevation (Fig. 5), we can find consistency at the first peak (the major one); the consistency at the second peak is not so clear. However, it is possible to state that the lift coefficient follows the elevation of the ground in terms of amplitude (Fig. 6). A similar analysis is conducted for the phase of the variation. While Fig. 7 shows the phase differences between cases at a frequency of 0.2, those are not observed for the ground elevation (Fig. 8). Hence, these phase differences cannot come from the ground elevation, rather they are due to the unsteadiness of the flow. When C increases from 0 to 3, the lift coefficient variation is delayed by 5° in phase. At a large C, the slope of the ground is more significant, and so is the unsteadiness. According to Wrights and Cooper [12], the unsteadiness causes a delay in the lift force development, which is seen in Fig. 7.
Fig. 5. Amplitude spectrum of the lift coefficient in the frequency domain
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Fig. 6. Amplitude spectrum of the ground elevation in the frequency domain
Fig. 7. Phase of the lift coefficient in the frequency domain
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Fig. 8. Phase of the ground elevation in the frequency domain
3.1.2 Effects of Laterally Curved Ground Figure 9 shows the definitions of longitudinally and laterally curved ground problems. In this subsection, we study sinusoidal curved surfaces with the number of waves N varying from 1 to 5 (Fig. 10).
Fig. 9. Longitudinally (a) and laterally (b) curved ground
Fig. 10. The wake patterns with different numbers of waves.
The wing and surface geometry parameters are similar to those above. Wing velocity of 20 m/s. The calculation results are obtained and shown in Fig. 11.
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Fig. 11. Change of wing lift coefficient when the ground surface has different horizontal number of waves
Fig. 12. Distribution of the lift coefficients along wing span with different values of N
There are significant differences between the cases in Fig. 11. It is found that in general, the lift coefficient increases with the number of waves. This trend can be attributed to the air trapped between waves, which plays a roll of cushions to augment the lift force of the wing. Figure 12 provides clear evident of this trend. 3.2 The Influence of Wind To study the effect of wind on the wing lift coefficient in the ground effect, we use the above wing model flying over a longitudinal curved surface. Several wind speed values are included in the computation and the results are presented in Fig. 13. The calculation results show that wind has a significant influence on the wing lift coefficient when it is combined with curved ground. For a positive wind value, a drop
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Fig. 13. The variation of the lift coefficient on the wings when flying over a limited surface with different wind speeds
in the lift coefficient when the wing above the leeward region is followed by an increase trend above the windward side of the wave. To provide more details of this phenomenon, the distributions of the pressure coefficient on the wing at the entrance (x = 0) and at the exist (x = 5c) of the wave are plotted and compared (Figs. 14 and 15).
Fig. 14. Pressure distribution on the wing when wind speed = −2 m/s
Fig. 15. Pressure distribution on the wing when wind speed = 2 m/s
4 Conclusions In this paper, the mirror image method is combined with the unsteady panel method to efficiently simulate the aerodynamic interaction between the wing and the curved ground surface. Furthermore, this method allows to study the effect of wind on the wing lift coefficient when it flies over a curved ground surface. The calculation results show the influence of the ground surface geometry on the change of the lift force coefficient. When analyzing the result in the frequency domain, it is observed that the amplitude spectrum of the lift coefficient follows that of the ground elevation. However, due to the unsteadiness of the flow, there is phase differences between cases. In general, the unsteadiness causes a delay in the phase of the lift force
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development. The effect of laterally curved ground is also studied in this paper. The results show that the lift coefficient tends to increase with the number of lateral waves. When there is a combination of curved ground and wind effects, the change in the lift coefficient of the wing becomes considerable. Further research is needed for this problem.
References 1. Federal Aviation Administration: Airplane Flying Handbook, p. 348. U.S. Dep. Transp. (2016) 2. Qu, Q., Jia, X., Wang, W., Liu, P., Agarwal, R.K.: Numerical simulation of the flowfield of an airfoil in dynamic ground effect. J. Aircr. 51(5), 1659–1662 (2014) 3. Im, Y.H., Chang, K.S.: Unsteady aerodynamics of a WIG airfoil flying over a wavy wall. In: 38th Aerospace Sciences Meeting and Exhibit, vol. 37, no. 4, pp. 690–696 (2000) 4. Zhi, H., Xiao, T., Chen, J., Wu, B., Tong, M., Zhu, Z.: Numerical analysis of aerodynamics of a NACA4412 airfoil above wavy water surface, no. June, pp. 1–29 (2019) 5. Tang, C., Wang, L.: Numerical simulation of wind field over complex terrain and its application in Jiuzhaihuanglong airport. Procedia Eng. 99, 891–897 (2015) 6. Liu, Z., Hu, Y., Fan, Y., Wang, W., Zhou, Q.: Turbulent flow fields over a 3D hill covered by vegetation canopy through large eddy simulations. Energies 12(19), 1–18 (2019) 7. Tran, D.K., Nguyen, A.T.: The application of the panel method to predict the unsteady aerodynamic coefficients of a wing model flying over bumpy ground. In: ICOMMA (2020) 8. Katz, J., Plotkin, A.: Low-Speed Aerodynamics From Wing Theory to Panel Methods, 2nd edn. Cambridge University Press, New York (2001) 9. Nguyen, A.T., Han, J.S., Han, J.H.: Effect of body aerodynamics on the dynamic flight stability of the hawkmoth Manduca sexta. Bioinspiration Biomimetics 12(1), 016007 (2017) - nh d˘ -ac tru,ng khí dô - ng lu,c cua cánh quay tru,c th˘ang xét d-ên ´ su., 10. Pham, T.D.: Nghiên cu´,u xác di . . . . . tu,o,ng tác vo´,i thân và m˘a.t gio´,i ha.n. Le Quy Don Technical University (2020) 11. Berman, G.J., Wang, Z.J.: Energy-minimizing kinematics in hovering insect flight. J. Fluid Mech. 582, 153–168 (2007) 12. Wright, J.R., Cooper, J.E.: Introduction to Aircraft Aeroelasticity and Loads, 2nd edn. Wiley, Hoboken (2015) ij
Design a Small, Low-Speed, Closed-Loop Wind Tunnel: CFD Approach Phan Thanh Long(B) Faculty of Transportation Mechanical Engineering, The University of Danang – University of Science and Technology, Danang, Vietnam [email protected]
Abstract. Closed-loop small wind tunnels, with many advantages, play an essential role in experimental research on aerodynamics in many fields. In this study, to design a small, low-speed, closed-loop wind tunnel for researching and teaching, the author carried out a systematic investigation of flow in the wind tunnel using Computational Fluid Dynamics (CFD). The required objective is a uniform flow in the test section of the wind tunnel and a low-pressure loss. The effect of guide vane configurations in the big corners on the flow quality in the test section was evaluated. This analysis showed the flow in the test section was more affected by the presence of upstream guide vanes. Finally, the simulation results demonstrated that uniformity of flow in test section is well obtained in the case of full configuration of wind tunnel. Keywords: Small wind tunnel · Aerodynamics · CFD
1 Introduction Experimental research on low-speed aerodynamics plays an important role in many fields such as studying the basic problems of fluid mechanics, automotive design, wind turbines, and building design. This research is usually carried out in a small wind tunnel, combined with measuring equipment to obtain data on drag force, lift force, and moment of the airflow acting on an object. In the case of the use of complex measuring equipment, these experiments can also reveal the velocity field and structure of flow in detail [1]. Although with the development of numerical simulation methods, especially the CFD (Computational Fluid Dynamics) method and the computational capacity of the computer, several studies involving low-speed aerodynamics can carry out using CFD simulation to achieve the required parameters [2]. However, the experimental studies in the wind tunnel still play an important role, helping to obtain detailed results and make final decisions in the design process [1]. The design of a wind tunnel is relatively complicated, while the use of CFD to support wind tunnel design is still limited. Several studies on evaluating the wind tunnel system using numerical methods have been performed. Moonen et al. [3] established a methodology of modeling the airflow in a closed-loop wind tunnel, using the CFD method and a standard k-ε turbulence model to determine pressure loss and flow rate in © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 663–672, 2022. https://doi.org/10.1007/978-981-16-3239-6_51
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the test section. Gordon and Imbali [4] used CFD to evaluate the flow conditions within the test section of a closed-loop wind tunnel at the University of Aberdeen. Besides, Calautit et al. [5] also performed a numerical study to investigate the effect of the guide vanes on flow quality in a small closed-loop wind tunnel at the University of Leeds. In this study, to build a small wind tunnel for research and teaching in the technical university of Vietnam, a design of a small closed-loop wind tunnel was introduced. Total pressure loss in the wind tunnel was determined from the analytical model, then used directly as a boundary condition in a numerical simulation. The quality of the flow in the wind tunnel, especially in the test section, was investigated, from which the performance of the designed wind tunnel was evaluated.
2 Wind Tunnel Design Small low-speed wind tunnels are generally divided into two types: open-loop and closed-loop. The advantages of open-loop wind tunnel are the low construction cost and the use of smoke for visualization of the flow. The disadvantages of the open-loop wind tunnel are that the flow quality, especially in the test section is not high, the required power of the fan is higher and therefore it tends to be noisy. On the other hand, the closedloop wind tunnel has the advantage of being able to create a uniform flow in the test section, the required power of the fan is lower, so less noise when operating. However, the fabrication cost of the closed-loop wind tunnel is higher than that of the open-loop wind tunnel [6]. A schematic view of the closed-loop wind tunnel is shown in Fig. 1.
Fig. 1. CAD model of a closed-loop wind tunnel 1 – Test section; 2 – Small diffuser; 3, 4 – Small corners; 5 – Fan; 6 – Big diffuser; 7, 8 – Big corners; 9 – Expansion; 10 – Setting chamber; 11 – Contraction;
The test section is the place where test models and the measuring equipment are installed. The size of the test section as well as the flow velocity determines the size of the other components and the overall dimensions of the wind tunnel. In this study, the test section has a rectangular shape, with a height/ width ratio of about 4/3 [1]. Figure 2 illustrates the geometric sizing of the test section.
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Fig. 2. Geometric sizing of the test section: (a) side view; (b) front view
In the closed-loop wind tunnel, the diffusers, with an increasing cross-section area along its length, are used to recover the static head of the flow by decreasing the flow velocity, therefore the energy loss of system reduces. The small diffuser is placed at the downstream end of test section and the big diffuser is at outlet of the fan. The small diffuser was designed with an inlet area of 0.6 × 0.32 m2 and an outlet area of 0.85 × 0.6 m2 . The length of small diffuser is 1.94 m. For the big diffuser, the sizes are 0.85 × 0.6 m2 ; 1.2 × 0.85 m2 and 2.76 m, respectively. The geometric sizing of small diffuser is shown in Fig. 3.
Fig. 3. Geometric sizing of small diffuser: (a) side view; (b) front view
The corners in closed-loop wind tunnel have an important task to turn the flow by 90° without separation and recirculation. Behind the small diffuser, the small corners have dimensions of 0.85 × 0.6 m2 in the inlet and outlet areas. This corner is integrated with 7 guide vanes to reduce the flow separation occurring it the turn (Fig. 4a). The big corner, with 8 guide vanes and 1.2 × 0.85 m2 cross-section area, is located after the second diffuser (Fig. 4b).
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Fig. 4. Geometric sizing of (a) small corner and (b) big corner
3 Numerical Method To evaluate the performance of the designed wind tunnel, the flow characteristics in the wind tunnel are considered and predicted by CFD numerical simulation method, using commercial software ANSYS Fluent [7]. This is a software used to solve fluid dynamics problems through the finite volume method. To simplify the calculation, the flow through the fan is ignored. Instead, the pressure loss in the wind tunnel is identified, and then is assigned to the intake fan boundary condition. The fan power required to maintain a steady flow in the wind tunnel is equal to the total pressure loss of flow while moving through the system. The pressure loss coefficients obtained at different components of wind tunnel are summarized in Table 1. The pressure loss of this wind tunnel was calculated at 105.22 Pa. Figure 5 shows the 3-D model of wind tunnel, in which an intake fan condition was applied to inlet surface and a zero gauge pressure was set for outlet surface. Table 1. Loss coefficients and pressure loss of wind tunnel components No.
Component
Loss coefficient k [Ref.]
Pressure loss p = 1 2 2 kρv [Pa]
1
Test section
0.023 [8]
12.25
2
Small diffuser
0.079 [8]
23.8
3
Big diffuser
0.079 [8]
4
Small corner
0.152 [8]
40.5
5
Big corner
0.158 [8]
10.52
6
Setting chamber
0.22 [1]
10.65
7
Contraction
0.008 [8]
1.54
5.96
The mesh used in this simulation is an unstructured type, in which a high mesh resolution was applied at guide vanes and setting chamber to capture exactly the flow passing
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Fig. 5. 3-D CAD model of wind tunnel: (1) Intake flow; (2) Pressure outlet
through. The number of elements was varied from 2,596,431 (coarse mesh), 4,950,075 (medium mesh) and 6,267,618 (fine mesh) respectively for grid-independence evaluation. Variation of the velocity of airflow along the centerline of the test section is observed and shown in Fig. 6. This figure shows that the difference in velocity is independent of grid elements beyond 4,950,075 elements. Medium mesh was then adopted for further study. For this mesh configuration, the average orthogonal quality and skewness were 0.717 and 0.28, respectively (Fig. 7). A approriate average value of 54.3 for y+ of wall bounded cells has been determined.
Fig. 6. Grid independence test
In the CFD method, the three-dimensional flow analysis is solved by approximating the governing equations of the flow. These equations represent the conservation of mass and conservation of momentum. For the steady, incompressible flow, the conservation of mass in a control volume is expressed in terms of the continuity equation as follows [9]: ∂u ∂v ∂w + + =0 (1) ∂x ∂y ∂z
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Fig. 7. Unstructured mesh of 3-D wind tunnel model
where ρ is the density of air, u, v, and w are the velocity components in the x, y, and z directions, respectively. The conservation of momentum, commonly known as the Navier-Stokes equation, in a three-dimensional, steady, incompressible flow, has the following form [9]: 2 ∂u ∂u ∂p ∂ u ∂ 2u ∂ 2u ∂u + ρfx +v +w =− +μ + + (2) ρ u ∂x ∂y ∂z ∂x ∂x2 ∂y2 ∂z2 2 ∂p ∂v ∂v ∂v ∂ v ∂ 2v ∂ 2v + ρfy =− ρ u + + (3) +v +w +μ ∂x ∂y ∂z ∂y ∂x2 ∂y2 ∂z2 2 ∂w ∂w ∂p ∂ w ∂ 2w ∂ 2w ∂w + ρfz +v +w =− +μ + + (4) ρ u ∂x ∂y ∂z ∂z ∂x2 ∂y2 ∂z2 Where p is the pressure, fx , fy , and fz are the body force per unit mass in the x, y, and z directions, respectively. To solve the governing equations applied to turbulent flow, CFD method is often used in combination with the turbulence models to close the Navier-Stokes equations. In this study, the two-equation model k-ω SST is used because it has the ability to model both near-surface and far-field flow [10]. This turbulence model utilizes the transport equations based on turbulent kinetic energy (k) and specific dissipation rate (ω), similar to the standard k-ω model. In addition, in this model, a blending function is also introduced to activate the turbulence model according to the proximity to the surface. This makes the k-ω SST turbulence model suitable and highly reliable for flows with adverse pressure gradients and separation. The transport equations for turbulence model are given by: ∂ ∂k ∂ ∂ k + Gk − Yk + Sk (ρk) + (5) (ρkui ) = ∂t ∂xi ∂xj ∂xj ∂ ∂ ∂ω ∂ ω + Gω − Yω + Sω + Dω (ρω) + (6) (ρωui ) = ∂t ∂xi ∂xj ∂xj where k and ω represent the diffusivity of turbulent kinetic energy and ω, Gk and Gω symbolize the generation rate of k and ω, Yk and Yω are the dissipation rates of k and ω, respectively; Sk and Sω are user-defined source terms, respectively.
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4 Results and Discussion This study investigated two different configurations of the wind tunnel. The first configuration is a wind tunnel without guide vanes in two big corners, and the other is a full configuration of the wind tunnel. The numerical results evaluated the influence of guide vanes in the big corners on the test section flow quality. The result in Fig. 8 shows that in the absence of the guide vanes, the flow separation zones appear in two big conners and therefore, the flow in test section is unsteady. The maximum velocity that can be reached in this section is about 22 m/s.
Fig. 8. Contours of velocity magnitude for wind tunnel without guide vanes in big corners
For a more detailed investigation of the flow inside the wind tunnel in this case, the streamline of flow throughout wind tunnel is investigated as presented in Fig. 9. The results show more clearly the vortices formed in the upper big corner. Meanwhile, in the small corner at the downstream of flow, because of the presence of guide vane, the flow through this section is very uniform.
Fig. 9. Streamlines of circulating flow in the wind tunnel without guide vanes in big corners
For the second configuration, in which the guide vanes are integrated for both small and big corners, the numerical results are shown in Fig. 10 and Fig. 11. Figure 10 presents the contours of velocity in the wind tunnel, and a uniform flow is obtained in the test
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section. However, the maximum flow velocity is lower (−21 m/s) as compared to the first configuration without guide vanes in the big corners. The improvement in the test section flow quality demonstrated the influence of the guide vanes on wind tunnel design. The disadvantage of the presence of the guide vanes is an increase in energy loss, therefore the maximum flow velocity slightly decreases.
Fig. 10. Contours of velocity magnitude for wind tunnel with guide vanes in corners
The streamlines of flow in wind tunnel are presented in Fig. 11, in which the vortices only appear at a few points in big corners, and the flow is steady throughout the wind tunnel.
Fig. 11. Streamlines of circulating flow in the wind tunnel with guide vanes in corners
In addition, the velocity flow distributions at inlet, middle, and outlet areas in the test section are considered as shown in Fig. 12. The results show that the velocity distribution at the center of the test section is very uniform.
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Fig. 12. Velocity flow distribution in test section of wind tunnel (a) inlet; (b) middle; (c) outlet
The velocity profiles of airflow at middle of test section of the wind tunnel with and without guide vanes are presented in Fig. 13. It is possible to observe that the importance of guide vanes in the corner to obtain a uniformity of airflow velocity in the test section of wind tunnel. This result also demonstrated the obtained quality of flow in the test section.
Fig. 13. Velocity profile of the flow at middle of the test section (a) width; (b) height
5 Conclusion A small, low-speed closed-loop wind tunnel was designed. The airflow characteristics were then evaluated by the numerical CFD simulation method using ANSYS Fluent software. The results demonstrated the importance of the guide vanes in the corners of wind tunnel. In the case of presence of the guide vanes, a good overall flow distribution
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was observed in the wind tunnel system, especially in the test section. These results showed that the design of the wind tunnel is suitable and may be further studied for perfection. Acknowledgment. This work was supported by The University of Danang, University of Science and Technology, code number of Project: T2019-02-08.
References 1. Barlow, J.B., Rae Jr., W.H., Pope, A.: Low-Speed Wind Tunnel Design. Wiley (1999) 2. Turkel, E.: Preconditioning techniques in computational fluid dynamic. Annu. Rev. Fluid Mech. 31(1), 385–416 (1999) 3. Moonen, P., Blocken, B., Roels, S., Carmeliat, J.: Numerical modelling of the flow conditions in a closed circuit low-speed wind tunnel. J. Wind Eng. Ind. Aerodyn. 94, 699–723 (2006) 4. Gordon, R., Imbabi, M.S.: CFD simulation and experimental validation of a new closed circuit wind/water tunnel design. J. Fluids Eng. 120(2), 311–318 (1998) 5. Calautit, J.K., Chaudhry, H.N., Hughes, B.R., Sim, L.F.: A validated design methodology for a closed-loop subsonic wind tunnel. J. Wind Eng. Ind. Aerodyn. 125, 180–194 (2014) 6. Mehta, R.D., Bradshaw, P.: Design rules for small low speed wind tunnels. Aeronaut. J. 83, 443–453 (1979) 7. https://www.ansys.com/. Accessed 10 Oct 2020 8. González, M.H.: Design Methodology for a Quick and Low-Cost Wind Tunnel. Polytechnic University of Madrid, Spain (2013) 9. Annderson, J.D.: Computational Fluid Dynamics: The Basics with Application. McGraw-Hill (1995) 10. Menter, F.R.: Two-equation eddy-viscosity turbulence models for engineering applications. AIAA 32, 1598–1605 (1994)
A Size-Dependent Meshfree Approach for Free Vibration Analysis of Functionally Graded Microplates Using the Modified Strain Gradient Elasticity Theory Lieu B. Nguyen1(B) , Chien H. Thai2,3 , and H. Nguyen-Xuan4 1 Faculty of Civil Engineering, Ho Chi Minh City University of Technology and Education,
Ho Chi Minh City, Vietnam [email protected] 2 Division of Computational Mechanics, Ton Duc Thang University, Ho Chi Minh City, Vietnam [email protected] 3 Faculty of Civil Engineering, Ton Duc Thang University, Ho Chi Minh City, Vietnam 4 CIRTech Institute, Ho Chi Minh City University of Technology, Ho Chi Minh City, Vietnam [email protected]
Abstract. In this work, we present a size-dependent numerical approach for free vibration analysis of functionally graded (FG) microplates based on the modified strain gradient theory (MSGT), simple first-order shear deformation theory (sFSDT) and moving Kriging meshfree method. The present approach decreases one variable compared with the original first-order shear deformation theory (FSDT). Moreover, it only uses three material length scale parameters to capture the size effects. The effective material properties as Young’s modulus, Poisson’s ratio and density mass are homogenized by a rule of mixture. Thanks to the principle of virtual work, the discrete system equations solved by the moving Kriging meshfree method, are derived. In addition, due to satisfying a Kronecker delta function property of the moving Kriging integration shape function, the essential boundary conditions are easily enforced similar to the standard finite element method. Rectangular and circular FG microplates with different boundary conditions, material length scale parameter and volume fraction are exampled to evaluate natural frequencies. Keywords: Moving Kriging meshfree method · Functionally graded microplate · Modified strain gradient theory · Simple first-order shear deformation theory
1 Introduction Experimental studies have recently been indicated that size effects are obviously occurred in micro/nano-structures [1], in which the effect of the strain-gradient tensor is significant compared with the strain tensor. To solve micro/nanostructures, there are several modelling approaches consisting of atomistic, hybrid atomistic continuum mechanics and continuum mechanics. In practical problems, the continuum mechanic model © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 673–690, 2022. https://doi.org/10.1007/978-981-16-3239-6_52
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is simply and easily implemented compared to those models. However, it cannot be explained mechanical behavior of micro/nanostructures due to ignoring material length scale parameters (MLSPs). For this reason, many various size dependent continuum theory models including the couple stress [2], modified couple stress [3], nonlocal elasticity [4], modified nonlocal elasticity [5], surface elasticity [6], strain gradient [7] and general strain gradient [8] have been proposed to investigate size effects of micro/nanostructures. By using the symmetric condition of the couple-stress tensor for equilibrium equation of couple moments, Lam et al. [1] proposed the modified strain gradient theory (MSGT) with three MLSPs instead of five as in [8]. Thus, it is suitable to use in practical computations. Moreover, the size-dependent model based on MSGT has been applied to analyze the behaviors of micro-beam, micro-plate and micro-shell structures. A micro-scale Kirchhoff model was firstly presented by Movassagh and Mahmoodi [9] for bending analysis of isotropic plate. This model was developed for analyzing of FG microplate [10] and multi-layer [11]. In addition, a size dependent model based on FSDT was presented by Ansari et al. [12, 13] to study bending, buckling and free vibration behaviors of FG microplates. Moreover, a size dependent higher-order shear deformation theory model for analyzing of FG microplates was reported in Refs. [14, 15]. Using this model, Thai et al. [16] studied for bending and free vibration analyses of functionally graded carbon nanotube-reinforced composite microplates. Zhang et al. [17, 18] presented a size dependent refined model by using the MSGT and a refined higher-order shear deformation theory (RPT). That model combining with isogeometric analysis (IGA) for the static bending, free vibration and buckling analyses of FG microplates was also developed by Thai et al. [19, 20]. After that, that model was applied for the thermal analysis of FG microplates by Farzam and Hassani [21]. In the other hand, Salehipour and Shahsavar [22] developed a size dependent three dimensional elasticity theory model for free vibration analysis of FG microplates. Moreover, a size dependent cylindrical thin shell model was presented by Zeighampour and Tadi [23]. As concluded in a review paper [24], it can be seen that the size-dependent models have been developed in the last five years, in which the couple stress and nonlocal elasticity theory models are larger than strain gradient theory models. In addition, a number of papers published based on the analytical solutions are very large compared to the numerical solutions. For reasons, the development of new size-dependent numerical models as finite element method, isogeometric analysis, meshfree method, etc. is really necessary. This paper concentrates on a moving Kriging interpolation (MKI) meshfree method. It was firstly applied for mechanical problems by Gu [25]. An advantage of the present meshfree method is satisfy the kronecker delta property of the shape functions. Therefore, the essential boundary condition is imposed similar to the finite element method. The meshfree method combined with modified couple stress theory was presented in Refs. [26, 27]. In this study, a new size-dependent meshless computational model based on the MSGT and sFSDT for analyzing of FG microplates is presented. The displacement field of the proposed model decreases one variable compared to FSDT. Moreover, it is simple and free of shear locking. Several examples are given in order to demonstrate the efficiency of proposed method.
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2 Basic Equations 2.1 Problem Description The FG plate is made of a mixture of ceramic and metal, in which ceramic-rich and metalrich surfaces are distributed at the top (z = h/2) and bottom (z = −h/2), respectively. The effective material properties are computed by the rule of mixture as follow z n 1 h h + ; Vm = 1 − Vc Vc (z) = ,z ∈ − , (1) 2 h 2 2 where the symbols c, m andn are the ceramic, metal and power index, respectively. The effective material properties based on the rule of mixture are defined by: Ee = Ec Vc (z) + Em Vm (z); νe = νc Vc (z) + νm Vm (z); ρe = ρc Vc (z) + ρm Vm (z) (2)
2.2 Modified Strain Gradient Elasticity Theory The MSGT which only contains three material length scale parameters, is proposed by Lam et al. [1]. The virtual strain energy U in an isotropic linearly elastic material according to the MSGT can be expressed by: 1 U = (3) (σ : ε + m : χ + p : ζ + τ : η)dV 2 V where ε, χ, ζ and η are the strain tensor, the symmetric rotation gradient tensor, the dilatation gradient tensor and the deviatoric stretch gradient tensor, respectively; σ is the Cauchy stress tensor; m, p and τ are high-order stress tensors corresponding with strain gradient tensors χ, ζ and η, respectively. The components of the strain, symmetric rotation gradient and dilatation gradient tensors are written as: 1 ¯ T (4) ε = ∇ u¯ + (∇ u) 2 1 χ = ∇θ + (∇θ)T (5) 2 ⎧ ⎫ ⎧ ⎫ ⎨ ζx ⎬ ⎨ εxx,x + εyy,x + εzz,x ⎬ ζ = ζy = εxx,y + εyy,y + εzz,y (6) ⎩ ⎭ ⎩ ⎭ ζz εxx,z + εyy,z + εzz,z where
∇=
∂ ∂ ∂x ∂y
⎫ ⎧ ⎧ ⎫ ⎧ ⎫ ⎪ 1 ∂ w¯ − ∂ v¯ ⎪ ⎪ ∂z ⎪ ⎨ u¯ ⎬ ⎨ θx ⎬ ⎨ 2 ∂y ⎬ ∂ 1 ∂ u ¯ ∂ w¯ ¯ ; u = = ; θ = − θ v ¯ y ∂z 2 ∂z ∂x ⎪ ⎩ ⎭ ⎩ ⎭ ⎪ ⎪ ⎭ ⎩ 1 ∂ v¯ − ∂ u¯ ⎪ θz w¯ 2
∂x
∂y
(7)
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The deviatoric stretch gradient components are expressed by ηxxx = εxx,x − 15 ζx − 25 εxx,x + εxy,y + εxz,z ; ηyyy = εyy,y − 15 ζy − 25 εxy,x + εyy,y + εyz,z ; ηzzz = εzz,z − 15 ζz − 25 εxz,x + εyz,y + εzz,z ; ηyyx = ηyxy = ηxyy = 13 ηxxx + 2εxy,y + εyy,x − εxx,x 1 1 ηzzx = ηzxz = ηxzz = 3 ηxxx + 2εxz,z + εzz,x − εxx,x ; ηxxy = ηxyx = ηyxx = 3 ηyyy + 2εxy,x + εxx,y − εyy,y ηzzy = ηzyz = ηyzz = 13 ηyyy + 2εyz,z + εzz,y − εyy,y ; ηxxz = ηxzx = ηzxx = 13 ηzzz + 2εxz,x + εxx,z − εzz,z ηyyz = ηyzy = ηzyy = 13 ηzzz + 2εyz,y + εyy,z − εzz,z ; ηxyz = ηyzx = ηzxy = ηxzy = ηzyx = ηyxz = 13 εyz,x + εxz,y + εxy,z
(8)
where symbols ‘,x’, ‘,y’ and ‘,z’ define the derivative of any function following x, y and z directions, respectively. The constitutive relations are given by σ = Cε; m = 2G 21 Iχ; p = 2G 22 Iζ and τ = 2G 23 Iη
(9)
where C, G, 1 , 2 , 3 , I are the stiffness tensor or elasticity tensor, shear modulus, three material length scale parameters and corresponded identity matrices, respectively. 2.3 Kinematics of FG Microplate Let us consider a plate of total thickness h. The mid-plane surface denoted by is a function of x and y coordinates and, the z-axis is taken normal to the plate. The displacement field of any points in the plate according to the first-order shear deformation theory is described by: u¯ (x, y, z) = u(x, y) + zβx (x, y) v¯ (x, y, z) = v(x, y) + zβy (x, y) w(x, ¯ y, z) = w(x, y)
(10)
where u, v, w, βx and βy are in-plane, transverse displacements and two rotation components in the y–z, x–z planes, respectively. To eliminate the shear locking phenomenon in FSDT, a hypothesis is introduced as follows b b ; βy = −w,y w = wb + ws ; βx = −w,x
(11)
Inserting Eq. (11) into Eq. (10), the displacement fields of the FSDT which is called the simple first-order shear deformation theory (sFSDT) are described by: ⎧ ⎫ ⎧ ⎫ ⎧ ⎫ b b u¯ = u − zw,x u ⎨ u¯ ⎬ ⎨ ⎬ ⎨ −w,x ⎬ b or u b ¯ = v¯ = + z −w,y = u1 + zu2 v¯ = v − zw,y v ⎩ ⎭ ⎩ b ⎭ ⎩ ⎭ s b s w¯ w +w w¯ = w + w 0
(12)
Inserting Eq. (12) into Eq. (4), strain components according to sFSDT are presented by b ; ε = v − zw b ; ε = 1 γ = 1 u + v b εxx = u,x − zw,xx yy ,y xy ,x − zw,xy ; ,yy 2 xy 2 ,y s ; ε = 1 γ = 1 ws εxz = 21 γxz = 21 w,x yz 2 yz 2 ,y
(13)
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Bending and shear strains can be described as: T T = ε1 + zε2 and γ = γxz γyz = εs ε = εxx εyy γxy
(14)
where T T T s b w b 2w b s ws ε1 = u,x v,y u,y + v,x ; ε2 = − w,xx ; ε = w ,yy ,xy ,x ,y
(15)
Similarly, the rotation vector is rewritten by inserting Eq. (12) into Eq. (7) by ∂ u¯ 1 1 b 1 1 ∂ v¯ s b s θx = 2w,y + w,y ; θy = −2w,x − w,x ; θz = − = v,x − u,y 2 2 2 ∂x ∂y 2 (16) Substituting Eq. (16) into Eq. (5), the rotation gradient components are described as 1 b + ws b − ws ; χ −2w χxx = 21 2w,xy = ,xy ,xy ,xy ; 2 yy ; b − wb + 1 ws − ws (17) χxy = 21 w,yy ,xx ,yy ,xx 2 χxz = 41 v,xx − u,xy ; χyz = 41 v,xy − u,yy ; χzz = 0 The rotation gradient components can be rewritten under a compact form by ⎧ ⎫ ⎫ ⎧ b + 1 ws w,xy ⎪ ⎪ 2 ,xy ⎨ ⎨ χxx ⎪ ⎬ ⎬ ⎪ χ 1 b s xz s −w,xy χb = χyy = − 2 w,xy ;χ = χ ⎪ yz ⎩ 1 wb − wb + 1 ws − ws ⎪ ⎩χ ⎪ ⎭ ⎭ ⎪ ,xx ,yy ,xx 4 xy 2 ,yy v − u,xy = 14 ,xx v,xy − u,yy
(18)
Substituting the strains in Eq. (13) into Eq. (6), the dilatation tensor is expressed by: b b b b ; ζy = v,yy + u,xy − z w,yyy ; ζx = u,xx + v,xy − z w,xxx + w,xyy + w,xxy (19) b + wb ζz = − w,xx ,yy The dilatation tensor can be rewritten under a compact form by: T = ζ1 + zζ2 ζ = ζ x ζy ζz
(20)
where T T b − wb b + wb b b ζ1 = u,xx + v,xy v,yy + u,xy −w,xx ; ζ2 = − w,xxx ,yy ,xyy w,yyy + w,xxy 0
(21)
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Similarly, substituting the strains in Eq. (13) into Eq. (8), the deviatoric stretch gradient components are defined as b b ; + 35 w,xyy ηxxx = 25 u,xx − 15 u,yy − 25 v,xy + z − 25 w,xxx b b ηyyy = 25 v,yy − 15 v,xx − 25 u,xy + z − 25 w,yyy ; + 35 w,xxy 3 4 8 3 b b ηyyx = ηyxy = ηxyy = − 15 u,xx + 15 u,yy + 15 v,xy + z 15 w,xxx − 12 w,xyy 15 3 4 8 3 b b v,yy + 15 v,xx + 15 u,xy + z 15 w,yyy − 12 w ηxxy = ηxyx = ηyxx = − 15 15 ,xxy ; 3 1 2 3 b 3 b ηzzx = ηzxz = ηxzz = − 15 u,xx − 15 u,yy − 15 v,xy + z 15 w,xxx + 15 w,xyy ; (22) 3 1 2 3 b 3 b ηzzy = ηzyz = ηyzz = − 15 v,yy − 15 v,xx − 15 u,xy + z 15 w,yyy + 15 w,xxy ; 1 4 b b + wb s s ηzzz = 15 w,xx ,yy − 5 w,xx + w,yy ; ηxxz = ηxzx = ηzxx = − 15 w,xx 1 b 4 s 1 s + 15 w,yy + 15 w,xx − 15 w,yy ; 4 b 1 b 4 s 1 s ηyyz = ηyzy = ηzyy = − 15 w,yy + 15 w,xx + 15 w,yy − 15 w,xx ; 1 b s ; ηxyz = ηyzx = ηzxy = ηxzy = ηzyx = ηyxz = − 3 w,xy + 13 w,xy
These components are also rewritten under compact forms by: T T η¯ = ηxxx ηyyy ηyyx ηxxy ηzzx ηzzy = η¯ 1 + z η¯ 2 ; η˜ = ηzzz ηxxz ηyyz ηxyz
(23)
where ⎫ ⎧ 2 b b ⎫ 2 1 2 − 5 w,xxx + 35 w,xyy ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 5 u,xx − 5 u,yy − 5 v,xy ⎪ ⎪ 2 b 3 b ⎪ ⎪ ⎪ ⎪ 2 1 2 − w + w ⎪ ⎪ ⎪ v − v − u ,yyy ,xxy ⎪ ⎪ ⎪ 5 5 5 ,yy 5 ,xx 5 ,xy ⎪ ⎪ ⎪ ⎬ ⎨ 12 3 b b ⎬ 3 4 8 w − w − 15 u,xx + 15 u,yy + 15 v,xy ,xxx ,xyy 15 15 ¯ ; η η¯ 1 = = 2 3 4 8 12 b ⎪; 3 b ⎪ − 15 ⎪ v,yy + 15 v,xx + 15 u,xy ⎪ ⎪ ⎪ ⎪ 15 w,yyy − 15 w,xxy ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 3 1 2 3 b ⎪ 3 b ⎪ ⎪ ⎪ − 15 u,xx − 15 u,yy − 15 v,xy ⎪ w,xxx + 15 w,xyy ⎪ ⎪ ⎪ ⎪ ⎪ 15 ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ 3 ⎩ 3 b 1 2 3 b ⎭ − 15 v,yy − 15 v,xx − 15 u,xy w + w ,yyy ,xxy 15 ⎧ 1 ⎫ 15 b + 1 wb − 1 ws − 1 ws w,xx ⎪ ⎪ ,yy ,xx ,yy ⎪ ⎪ 5 5 5 5 ⎪ 4 b ⎨ 1 b 4 s 1 s ⎪ − 15 w,xx + 15 w,yy + 15 w,xx − 15 w,yy ⎬ η˜ = 4 b 1 b 4 s 1 s ⎪ ⎪ − 15 w,yy + 15 w,xx + 15 w,yy − 15 w,xx ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ 1 b 1 s − 3 w,xy + 3 w,xy ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
(24)
The classical and higher-order stress elastic constitutive relations are expressed as: ⎧ ⎫ ⎡ ⎫ ⎤⎧ ⎪ σxx ⎪ εxx ⎪ Q11 Q12 0 0 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎢ ⎪ ⎪ ⎪ ⎪ ⎥⎪ ⎪ ⎨ σyy ⎪ ⎨ εyy ⎪ ⎬ ⎢ Q21 Q22 0 0 0 ⎥⎪ ⎬ ⎢ ⎥ (25) ςxy = ⎢ 0 0 Q66 0 0 ⎥ γxy ⎢ ⎪ ⎥⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎣ ⎦ ς 0 γ 0 0 0 Q xz xz 55 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ς ⎪ ⎭ 0 0 0 0 Q44 ⎩ γyz ⎭ yz ⎧ ⎫ ⎫ ⎡ ⎤⎧ ⎪ mxx ⎪ χxx ⎪ 10000 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎢ 0 1 0 0 0 ⎥⎪ ⎪ ⎨ myy ⎪ ⎨ χyy ⎪ ⎬ ⎬ ⎢ ⎥⎪ ⎢ ⎥ (26) mxy = 2Gl12 ⎢ 0 0 1 0 0 ⎥ χxy ⎪ ⎢ ⎥⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎣ ⎦ 0 0 0 1 0 m χ xz xz ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩m ⎪ ⎭ 0 0 0 0 1 ⎩ χyz ⎭ yz
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⎧ ⎫ ⎤⎧ ⎫ ⎡ 1 0 0 ⎨ ζx ⎬ ⎨ px ⎬ = 2Gl22 ⎣ 0 1 0 ⎦ ζy (27) p ⎩ ⎭ ⎩ y⎭ 001 pz ζz ⎫ ⎫ ⎧ ⎡ ⎤⎧ ⎪ ηxxx ⎪ τxxx ⎪ 100000 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎧ ⎫ ⎫ ⎪ ⎡ ⎤⎧ ⎪ ⎪ ⎪ ⎢ 0 1 0 0 0 0 ⎥⎪ ⎪ ⎪ ⎪ ⎪ τ η 1000 ⎪ ηzzz ⎪ yyy yyy ⎪ ⎪ ⎪ ⎪ ⎢ ⎥ ⎪ ⎪ τzzz ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ ⎬ ⎨ ⎢ ⎥⎨ ⎬ ⎨ ⎢ 0 1 0 0 ⎥ ηxxz ⎬ τyyx τxxz ⎢ 0 0 1 0 0 0 ⎥ ηyyx ⎥ = 2Gl32 ⎢ ; = 2Gl32 ⎢ ⎥ ⎣ 0 0 1 0 ⎦⎪ ηyyz ⎪ (28) ⎪ ⎢ 0 0 0 1 0 0 ⎥⎪ ⎪ τxxy ⎪ ηxxy ⎪ τyyz ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎢ ⎥ ⎪ ⎪ ⎪ ⎭ ⎭ ⎩ ⎪ ⎪ ⎪ ⎪ ⎪ ⎣ 0 0 0 0 1 0 ⎦⎪ τzzx ⎪ ηzzx ⎪ τxyz ηxyz 0001 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎭ ⎩ 000001 τzzy ηzzy where Ee , 1−νe2
Q11 = Q22 =
Q12 = Q21 = G=
νe Ee , Q66 1−νe2 Ee s 2(1+νe ) ; k
= =
Ee 2(1+νe ) ; 5 6
Q55 = Q44 =
k s Ee 2(1+νe ) ;
(29) in which Ee and νe are the effective Young module and Poisson’s ratio, respectively. The discrete Galerkin weak form for the free vibration analysis of the FG microplate are described by: $ h/2 $ σxx δεxx + σyy δεyy + ςxy δγxy + ςxz δγxz + ςyz δγyz d dz + . . .
−h/2
$ h/2 $ mxx δχxx + myy δχyy + 2mxy δχxy + 2mxz δχxz + 2myz δχyz d dz + . . .
−h/2
$ h/2 $ τxxx δηxxx + τyyy δηyyy + 3τyyx δηyyx + 3τxxy δηxxy + 3τzzx δηzzx + 3τzzy δηzzy d dz + . . .
−h/2
$ h/2 $
−h/2
+
τzzz δηzzz + 3τxxz δηxxz + 3τyyz δηyyz + 6τxyz δηxyz d dz + . . .
$ h/2 $ $ $ h/2 ¯ e u¨¯ d dz = 0 δ uρ px δζx + py δζy + pz δζz d dz +
−h/2
−h/2
(30) The Eq. (30) can split into two independent integrals following to middle surface and z-axis direction. Substituting Eqs. (25)–(28) into Eq. (30), the discrete Galerkin weak form can be rewritten as follows: T T δ χb Dbc cb χb d + δ χs Dsc cs χs d + ...
T de de T di T dev dev ¨ˆ ˆ ˆ ˆ ˆ ˆ ˜ ˆ ud
¯ + + δ ζˆ D ζˆ d + δ η¯ D ηd
δ η˜ D ηd
δ uˆ T m =0
ˆ εd + δ εˆ T Dˆ
T δ εs Ds εs d +
(31)
where T T T εˆ = ε1 ε2 ; ζˆ = ζ1 ζ2 ; ηˆ¯ = η¯ 1 η¯ 2 ; de de 0 Adi Bdi Ade Bde Ab Bb di de ˆ ˆ ˆ ˆ ;D = ;D = ; = D= Bb Db Bdi Ddi Bde Dde 0 de
(32)
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in which
⎡ ⎤ Q11 Q12 0 $ $ h/2 Q44 0 h/2 b b b 2 s ⎣ ⎦ dz; A , B , D = −h/2 1, z, z Q21 Q22 0 dz; D = −h/2 0 Q55 0 0 Q66 $ h/2 $ h/2 Dbc = −h/2 2Gl12 I3×3 dz; Dsc = −h/2 2Gl12 I2×2 dz; Adi , Bdi , Ddi $ h/2 ; = −h/2 2Gl22 1, z, z 2 I3×3 dz $ h/2 $ h/2 Ddev = −h/2 2Gl32 I4×4 dz; Ade , Bde , Dde = −h/2 2Gl32 1, z, z 2 I6×6 dz; cb = diag(1, 1, 2); cs = diag(2, 2); dev = diag(1, 3, 3, 6); de = diag(1, 1, 3, 3, 3, 3) 1 h/2 $ I 1 I2 u ˆ ; m = ρe 1, z, z 2 I3×3 dz; uˆ = (I1 , I2 , I3 ) = u2 I2 I3 −h/2 (33)
in which I2×2 , I3×3 , I4×4 , I6×6 are the identity matrices of size 2 × 2, 3 × 3, 4 × 4 and 6 × 6, respectively.
3 FG Microplate Formulation Based on Moving Kriging Interpolation Shape Functions 3.1 Moving Kriging Interpolation Shape Functions To construct the moving Kriging shape functions for interpolation kinematic variables, the domain of problem is discretized into a set of nodes xI (I = 1,…, NP), in which NP denotes the number of nodes in the problem domain. The nodal displacement vector is defined as uh (x) =
NP %
NI (x)uI
(34)
I =1
where uI and φI are respectively the unknown coefficient associated with node I and the moving Kriging shape function which is described as follows NI (x) =
m % j=1
pj (x)AjI +
n % k=1
rk (x)BkI or NI (x) = pT (x)A + rT (x)B
(35)
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681
in which n is a number of nodes in a support domain x ∈ and m is the dimension of a polynomial basis function space. In addition, A, B, p(x) and r(x) are expressed as: A = (PT R−1 P)−1 PT R−1 , B = R−1 (I − PA) T p(x)= 1 x y x2 xy y2 ... and r(x) = [ R(x1 , x) R(x2 , x) · · · R(xn , x) ]T
(36)
where I is the identity matrix of size n × n. The cubic polynomial functions are chosen for the present work by: T p(x)= 1 x y x2 xy y2 x3 x2 y xy2 y3 (m = 10) Besides, P and R are written as follows ⎡ ⎤ ⎡ ⎤ p1 (x1 ) ... pm (x1 ) R(x1 , x1 ) ... R(x1 , xn ) ⎦ P =⎣ ... ... ... ⎦ and R =⎣ ... ... ... p1 (xn ) ... pm (xn ) R(xn , x1 ) ... R(xn , xn )
(37)
(38)
2 is a correlation function, in which E denotes where R(xI , xJ )= 21 E uh (xI ) − uh (xJ ) an expected value of a random function. In this study, the Gaussian function is used and it is defined as R(xI , xJ ) = e
2 βr − a IJ 0
(39)
where rIJ = xI −xJ , the correlation parameter β is related to the variance σ 2 of the normal distribution function by β 2 = 1/2σ 2 . The scale factor a0 is used to normalize the distance. Regularly distributed nodes, a0 is taken as the length of two adjacent nodes, and it is chosen to be the maximum distance between a pair of nodes in the support domain for the irregularly distributed nodes. In addition, the correlation parameter β = 1 is chosen for the present work. The size of the support domain to build the shape functions can be defined as dm = αdc
(40)
where dc and α are an average distance between nodes and a scale factor, respectively. In this study, the scale factor is taken equal to 3.0 (α = 3.0). 3.2 A MKI-Based Formulation Based on the sFSDT and Modified Strain Gradient Theory According to the MK interpolation, the displacement field can be approximated as u (x, y) = h
n %
NI (x, y)qI
I =1
where qI =
uI vI wIb wIs
T
are degrees of freedom of node I.
(41)
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Substituting Eq. (41) into Eq. (15), the strain components can be rewritten as n n n % % T % 1 2 T εˆ = ε1 ε2 = BsI qI Bˆ I qI and εs = BI BI qI = I =1
I =1
in which, ⎡
⎡ ⎤ NI ,x 0 0 0 0 0 NI ,xx B1I = ⎣ 0 NI ,y 0 0 ⎦; B2I = −⎣ 0 0 NI ,yy NI ,y NI ,x 0 0 0 0 2NI ,xy
(42)
I =1
⎤ 0 0 0 0 NI ,x s ⎦ 0 ; BI = 0 0 0 NI ,y 0
(43)
Similarly, the curvatures are obtained by substituting Eq. (41) into Eq. (18) as follows χb =
n %
BbcI qI ; χs =
I =1
n %
BscI qI
(44)
I =1
in which, ⎡ ⎤ 1 00 NI ,xy 2 NI ,xy 1 −NI ,xy NI ,xx 0 0 b s 1 ⎣ ⎦ (45) ; BcI = BcI = 0 0 −NI ,xy − 2 NI ,xy 4 −NI ,yy NI ,xy 0 0 0 0 21 NI ,yy − NI ,xx 41 NI ,yy − NI ,xx Substituting Eq. (41) into Eq. (20), the dilatation components can be rewritten as n n T % T % ζ ζ1 ζ2 ζˆ = ζ1 ζ2 Bˆ I qI = qI = BI BI I =1
(46)
I =1
where ⎡
ζ1
BI
0 NI ,xx NI ,xy = ⎣ NI ,xy NI ,yy 0 0 0 −NI ,xx − NI ,yy
⎡ ⎤ 0 0 0 −NI ,xxx − NI ,xyy 2 ζ ⎣ ⎦ = ; B 0 0 0 −NI ,yyy − NI ,xxy I 0 00 0
⎤ 0 0 ⎦; 0
(47)
Substituting Eq. (41) into Eq. (23), the dilatation stretch gradient components can be rewritten as n n n T % % T % η¯ η˜ η¯ η¯ ηˆ¯ = η¯ 1 η¯ 2 = BI qI Bˆ I qI ; η˜ == BI 1 BI 2 qI = I =1
I =1
(48)
I =1
where ⎡
⎡ ⎤ 2N 1 0 0 − 25 NI ,xxx + 35 NI ,xyy − 25 NI ,xy 00 5 I ,xx − 5 NI ,yy 2 2 1 2 3 ⎢ ⎢ ⎥ − 5 NI ,xy ⎢ ⎢ 0 0 − 5 NI ,yyy + 5 NI ,xxy 5 NI ,yy − 5 NI ,xx 0 0 ⎥ ⎢ 3 ⎢ ⎥ 4 3 12 8 + 15 NI ,yy 0 0 ⎥ η¯ 2 ⎢ 0 0 15 NI ,xxx − 15 NI ,xyy η¯ ⎢− N 15 NI ,xy BI 1 = ⎢ 15 I ,xx ⎥ ; BI = ⎢ 8 N 3 N 4 ⎢ ⎢ 0 0 3 NI ,yyy − 12 NI ,xxy − 15 I ,yy + 15 NI ,xx 0 0 ⎥ 15 15 ⎢ 3 15 I ,xy1 ⎢ ⎥ 2 ⎣ − NI ,xx − NI ,yy ⎣ 0 0 3 NI ,xxx + 3 NI ,xyy ⎦ − N 0 0 15 15 15 I ,xy 15 15 2 N 3 N 1 3 N 3 − 15 0 0 15 − 15 I ,xy I ,yy − 15 NI ,xx 0⎤0 I ,yyy + 15 NI ,xxy ⎡ 1 1 1 1 0 0 5 NI ,xx + 5 NI ,yy − 5 NI ,xx − 5 NI ,yy ⎢ ⎥ 4 N 1 1 4 ⎢ 0 0 − 15 η˜ I ,xx + 15 NI ,yy 15 NI ,xx − 15 NI ,yy ⎥ BI = ⎢ ⎥ 4 N 4 N 1 N 1 N ⎣ 0 0 − 15 ⎦ + − I ,yy 15 I ,xx 15 I ,yy 15 I ,xx 1 1 00 − 3 NI ,xy N I ,xy 3
⎤ 0 0⎥ ⎥ ⎥ 0⎥ ⎥; 0⎥ ⎥ 0⎦ 0
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683
Substituting Eq. (41) into Eq. (12), u1 and u2 can be described as follows n n % T % 1 2 T ˆ I Nq = uˆ = u1 u2 NI NI qI = I =1
(49)
I =1
Substituting Eqs. (42), (44), (46) and (48) into Eq. (31), the weak form of the free vibration analysis of the FG microplate is rewritten as K − ω2 M q¯ = 0 (50) where K (K = Kε + Kχ + Kζ + Kη ) and F are the global stiffness matrix and force vector, respectively, in which $ $ T $ T ˆTD ˆ Bd
ˆ + (Bs )T Ds Bs d ; Kχ = Bbc Dbc bc Bbc d + Bsc Dsc sc Bsc d ; T de de η¯ T di ζ T $ $ $ η˜ ˆ B ˆ d ; Kη = ˆ ˆ Bˆ d + ˆζ D ˆ η¯ D Kζ = B Ddev dev Bη˜ d
B
B $ T iωt ˆ m ˆ ˆ Nd ; ¯ q = qe M= N
Kε =
$
B
(51)
where ω and q¯ in Eq. (51) are the natural frequency and corresponded mode shapes.
4 Numerical Examples and Discussions In this study, three length scale parameters which are assumed to be equal through the thickness of plate l1 = l2 = l3 = l = 15 × 10−6 m, are taken the same as in [17]. The FG microplate is made from the bottom metal surface (Aluminum-Al) to the top ceramic surface (alumina-Al2 O3 ). The material properties for Al are Em = 70 GPa, vm = 0.3, 3 ρm = 2702 kg/m3 and for Al2 O3 Ec = 380 GPa, vc = 0.3, & . The & ρc = 3800 kg/m non-dimensional natural frequency is normalized by ω¯ = ωh
ρc Ec
or ω¯ = ωR2
hρc Dc ,
in
Ec h3 ; a and R are the length of square plate or the radius of circular plate, 12(1−vc2 )
which Dc = respectively. The square and circular microplates are modeled as in Fig. 1, respectively.
a) Simply supported boundary condition (BC)
b) Roller/simply supported BC
c) Fully clamped BC.
Fig. 1. Distribution nodes of square and circular microplates.
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4.1 FG Square Microplate Let us consider a simply supported FG square microplate with a distribution of node as shown in Fig. 1. Different length-to-thickness ratios, power index and material length scale-to-thickness ratio are studied. Table 1 tabulates the first two non-dimensional natural frequencies of the simply supported isotropic FG square microplate. Obtained results are compared to those proposed by Zhang et al. [17] using the analytical solution based on the RPT (4 DOFs) and MSGT, Thai et al. [14] using the IGA based on the TSDT (5 DOFs) combined to MSGT and Thai et al. [19] using the IGA based on the RPT (4 DOFs) and MSGT. As in Table 1, it is observed that the non-dimensional natural frequencies predicted by the classical plate model (l/h = 0) are always smaller than those of the size dependent microplate model (0 < l/h ≤ 1). The non-dimensional natural frequencies decrease when increasing the length-to-thickness ratio and power index. In addition, for the case of l/h = 0, it can be seen that an excellent agreement is achieved when comparing with those results. In the case of l/h = 0, obtained results are smaller than those compared ones. It can be explained that the present model only considers FSDT instead of HSDT as those referenced solutions. Besides, an increase of the material length scale ratio l/h leads to a rise of the non-dimensional natural frequencies. For this reason, it can be concluded that the stiffness of FG microplate increases when considering the size-dependent effect. Table 1. Comparison of non-dimensional natural frequencies ω¯ of simply supported isotropic FG square microplate. a/h n
References
l/h 0.0
5
0
0.1
0.2
0.5
1.0
Exact-RPT [17] 0.2113 0.2273 0.2698 0.4688 0.7011 IGA-TSDT [14] 0.2113 0.2264 0.2666 0.4566 0.8344 IGA-RPT [19]
0.2113 0.2271 0.2687 0.4630 0.8477
Present
0.2112 0.2243 0.2566 0.3756 0.5433
0.5 Exact-RPT [17] 0.1804 0.1953 0.2344 0.4131 0.6290 IGA-TSDT [14] 0.1807 0.1948 0.2318 0.4032 0.7406
1
IGA-RPT [19]
0.1807 0.1955 0.2338 0.4085 0.7491
Present
0.1804 0.1931 0.2240 0.3346 0.4866
Exact-RPT [17] 0.1631 0.1774 0.2144 0.3823 0.5832 IGA-TSDT [14] 0.1631 0.1766 0.2118 0.3729 0.6867
2
IGA-RPT [19]
0.1631 0.1772 0.2136 0.3778 0.6948
Present
0.1630 0.1751 0.2046 0.3088 0.4504
Exact-RPT [17] 0.1472 0.1603 0.1944 0.3490 0.5270 IGA-TSDT [14] 0.1472 0.1596 0.1920 0.3402 0.6266 IGA-RPT [19]
0.1471 0.1601 0.1935 0.3449 0.6360
Present
0.1477 0.1586 0.1850 0.2789 0.4068 (continued)
A Size-Dependent Meshfree Approach for Free Vibration Analysis Table 1. (continued) a/h n
References
l/h 0.0
5
0.1
0.2
0.5
1.0
Exact-RPT [17] 0.1378 0.1470 0.1764 0.3117 0.4553 IGA-TSDT [14] 0.1358 0.1463 0.1740 0.3029 0.5540 IGA-RPT [19]
0.1357 0.1466 0.1753 0.3081 0.5656
Present
0.1379 0.1462 0.1669 0.2436 0.3524
10 Exact-RPT [17] 0.1358 0.1402 0.1666 0.2896 0.4150 IGA-TSDT [14] 0.1301 0.1394 0.1642 0.2807 0.5111
10
0
IGA-RPT [19]
0.1299 0.1397 0.1655 0.2861 0.5230
Present
0.1323 0.1393 0.1569 0.2240 0.3220
Exact-RPT [17] 0.0577 0.0619 0.0730 0.1258 0.2309 IGA-TSDT [14] 0.0577 0.0617 0.0725 0.1240 0.2268 IGA-RPT [19]
0.0577 0.0619 0.0729 0.1254 0.2297
Present
0.0577 0.0616 0.0719 0.1155 0.1790
0.5 Exact-RPT [17] 0.0489 0.0529 0.0632 0.1113 0.2057 IGA-TSDT [14] 0.0490 0.0528 0.0629 0.1098 0.2023
1
IGA-RPT [19]
0.0490 0.0529 0.0633 0.1110 0.2047
Present
0.0490 0.0527 0.0624 0.1026 0.1602
Exact-RPT [17] 0.0442 0.0480 0.0578 0.1028 0.1907 IGA-TSDT [14] 0.0442 0.0478 0.0573 0.1013 0.1873
2
IGA-RPT [19]
0.0442 0.0479 0.0577 0.1024 0.1896
Present
0.0442 0.0478 0.0569 0.0946 0.1483
Exact-RPT [17] 0.0401 0.0435 0.0524 0.0933 0.1731 IGA-TSDT [14] 0.0401 0.0434 0.0520 0.0918 0.1698
5
IGA-RPT [19]
0.0401 0.0435 0.0523 0.0930 0.1722
Present
0.0401 0.0433 0.0516 0.0855 0.1340
Exact-RPT [17] 0.0377 0.0405 0.0478 0.0825 0.1514 IGA-TSDT [14] 0.0377 0.0403 0.0474 0.0810 0.1482 IGA-RPT [19]
0.0377 0.0404 0.0477 0.0822 0.1508
Present
0.0379 0.0404 0.0469 0.0750 0.1161
10 Exact-RPT [17] 0.0364 0.0388 0.0453 0.0764 0.1390 IGA-TSDT [14] 0.0364 0.0387 0.0449 0.0750 0.1359 IGA-RPT [19]
0.0363 0.0387 0.0451 0.0761 0.1384
Present
0.0366 0.0387 0.0444 0.0692 0.1062
685
686
L. B. Nguyen et al.
4.2 FG Circular Microplate Let us consider a FG circular microplate of the radius R subjected to roller, simply supported and fully clamped BCs. Similarly, various values of power index and material length scale-to-thickness ratio l/h are also studied. Table 2 gives the first non-dimensional natural frequency of the FG circular microplate. Again, results are compared to those of given by Zhang et al. [18] based on the exact-RPT solution using MSGT, Thai et al. [14] based on the IGA-TSDT solution using MSGT and Thai et al. [19] based on the IGA-RPT solution using MSGT. As observed in Table 2, it can be seen that the non-dimensional natural frequency increases when changing boundary conditions from roller to fully clamped. Besides, the non-dimensional natural frequencies obtained from the present solution in the case of l/h = 0 are always smaller than those referenced ones because of using FSDT. Moreover, the difference of non-dimensional natural frequency between MSGT and classical theory is much significant as l/h = 1, however, this difference decreases when the thickness of the microplate becomes larger. The first six mode shapes of fully clamped FG circular microplate are plotted in Fig. 2. Table 2. Comparison of normalized frequency ω¯ of simply supported Al/Al2 O3 circular microplate with R/h = 5. n
Reference
l/h 0.0
0.1
0.2
0.5
1.0
Roller 0
0.5
2
Exact-RPT [18]
4.7369
5.1587
6.0483
10.3040
18.8251
IGA-TSDT [14]
4.7787
5.1089
5.9896
10.2008
18.6325
IGA-RPT [19]
4.7787
5.1112
5.9961
10.2134
18.6459
Present
4.7787
5.0945
5.9188
9.4435
14.4214
Exact-RPT [18]
3.9449
4.3098
5.1568
9.0789
16.7701
IGA-TSDT [14]
4.0578
4.3697
5.1915
9.0335
16.6167
IGA-RPT [19]
4.0569
4.3709
5.1964
9.0408
16.6140
Present
4.0557
4.3569
5.1341
8.3864
12.9085
Exact-RPT [18]
3.0603
3.3293
4.0844
7.5050
14.0837
IGA-TSDT [14]
3.3185
3.5868
4.2909
7.5573
13.9654
IGA-RPT [19]
3.3167
3.5864
4.2932
7.5649
13.9725
Present
3.3204
3.5780
4.2396
6.9873
10.7901 (continued)
A Size-Dependent Meshfree Approach for Free Vibration Analysis
687
Table 2. (continued) n
Reference
l/h 0.0
5
10
0.1
0.2
0.5
1.0
Exact-RPT [18]
2.9573
3.1333
3.7489
6.6380
12.3157
IGA-TSDT [14]
3.1228
3.3396
3.9174
6.6810
12.2095
IGA-RPT [19]
3.1206
3.3383
3.9185
6.6910
12.2299
Present
3.1346
3.3363
3.8641
6.1337
9.3556
Exact-RPT [18]
2.8568
3.1021
3.6326
6.1304
11.3096
IGA-TSDT [14]
3.0159
3.2046
3.7118
6.1833
11.1987
IGA-RPT [19]
3.0135
3.2030
3.7125
6.1940
11.2236
Present
3.0285
3.2008
3.6567
5.6608
8.5548
Exact-RPT [18]
4.7369
5.1587
6.0483
10.3041
18.8253
IGA-TSDT [14]
4.7787
5.1090
5.9898
10.2016
18.6339
Simply supported 0
0.5
2
5
10
IGA-RPT [19]
4.7787
5.1112
5.9961
10.2134
18.6459
Present
4.7787
5.0945
5.9188
9.4435
14.4214
Exact-RPT [18]
4.1906
4.5444
5.3535
9.1899
16.8296
IGA-TSDT [14]
4.1695
4.4740
5.2801
9.0861
16.6500
IGA-RPT [19]
4.1686
4.4753
5.2853
9.0936
16.6484
Present
4.1671
4.4604
5.2204
8.4299
12.9248
Exact-RPT [18]
3.8259
4.0600
4.7081
7.8705
14.2831
IGA-TSDT [14]
3.6642
3.9131
4.5742
7.7347
14.0839
IGA-RPT [19]
3.6622
3.9129
4.5770
7.7419
14.0932
Present
3.6677
3.9036
4.5160
7.1320
10.8457
Exact-RPT [18]
3.5844
3.7315
4.2734
6.9620
12.4965
IGA-TSDT [14]
3.4042
3.6086
4.1573
6.8400
12.3195
IGA-RPT [19]
3.4018
3.6073
4.1586
6.8501
12.3428
Present
3.4200
3.6061
4.0972
6.2606
9.4057
Exact-RPT [18]
3.1929
3.4342
3.9258
6.3122
11.4127
IGA-TSDT [14]
3.1729
3.3551
3.8468
6.2738
11.2620
IGA-RPT [19]
3.1704
3.3535
3.8476
6.2849
11.2893
Present
3.1872
3.3510
3.7868
5.7324
8.5833
Exact-RPT [18]
9.4248
10.2145
12.0955
20.9426
38.8453
IGA-TSDT [14]
9.2672
9.9484
11.7378
20.1819
36.9736
IGA-RPT [19]
9.2705
9.9680
11.7958
20.3440
37.2740
Present
9.2643
9.8476
11.3035
16.7967
24.4090
Fully clamped 0
(continued)
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L. B. Nguyen et al. Table 2. (continued)
n
Reference
l/h 0.0
0.5
2
5
10
0.1
0.2
0.5
1.0
Exact-RPT [18]
8.3094
9.0188
10.7161
18.6309
33.9181
IGA-TSDT [14]
7.9253
8.5595
10.2094
17.8361
32.8279
IGA-RPT [19]
7.9261
8.5766
10.2619
17.9658
33.0175
Present
7.9131
8.4752
9.8615
14.9591
21.8619
Exact-RPT [18]
7.4473
7.9817
9.3874
16.0942
29.4843
IGA-TSDT [14]
6.4726
7.0300
8.4722
15.0740
27.9396
IGA-RPT [19]
6.4711
7.0398
8.5083
15.1851
28.1224
Present
6.4934
6.9760
8.1615
12.4815
18.2847
Exact-RPT [18]
6.6452
7.2463
8.4811
14.3555
25.9336
IGA-TSDT [14]
5.9806
6.4559
7.6895
13.4247
24.7246
IGA-RPT [19]
5.9784
6.4620
7.7195
13.5515
24.9871
Present
6.0685
6.4405
7.3719
10.9099
15.8414
Exact-RPT [18]
6.2216
6.6676
7.7952
13.1556
23.9134
IGA-TSDT [14]
5.7278
6.1522
7.2532
12.4215
22.7102
IGA-RPT [19]
5.7254
6.1570
7.2824
12.5619
23.0236
Present
5.8192
6.1335
6.9290
10.0331
14.4730
a) Mode 1
b) Mode 2
d) Mode 4
e) Mode 5
c) Mode 3
f) Mode 6
Fig. 2. The first six mode shapes of fully clamped FG circular microplate with n = 10, l/h = 1.
A Size-Dependent Meshfree Approach for Free Vibration Analysis
689
5 Conclusions A size-dependent numerical approach based on MSGT, sFSDT and moving Kriging meshfree method was presented for analyzing free vibration of FG microplates. The advantages of the present approach are only having four unknowns and three material length scale parameters. Thus, it is suitable for computation of size-dependent practical problems. The rule of mixed is used to homogenize material properties as Young’s modulus, Poison’s ratio and density mass. In addition, when taking zero of all material length scale parameters, the classical sFSDT model was retrieved from the present model. Numerical results pointed out that the non-dimensional natural frequencies obtained from the present solutions are always smaller than those referenced ones in the case of l/h = 0. In addition, the stiffness of microplate can be increased as considering small scale effects leading to a rise of the natural frequencies. Acknowledgment. This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 107.02-2019.35.
References 1. Lam, D., Yang, F., Chong, A., Wang, J., Tong, P.: Experiments and theory in strain gradient elasticity. J. Mech. Phys. Solids 51, 1477–1508 (2003) 2. Mindlin, R.D., Eshel, N.N.: On first strain-gradient theories in linear elasticity. Int. J. Solids Struct. 4, 109–124 (1968) 3. Yang, F., Chong, A.C.M., Lam, D.C.C., Tong, P.: Couple stress based strain gradient theory for elasticity. Int. J. Solids Struct. 39, 2731–2743 (2002) 4. Eringen, A.C.: Nonlocal polar elastic continua. Int. J. Eng. Sci. 10, 1–16 (1972) 5. Salehipour, H., Shahidi, A.R., Nahvi, H.: Modified nonlocal elasticity theory for functionally graded materials. Int. J. Eng. Sci. 90, 44–57 (2015) 6. Gurtin, M.E., Weissmuller, J., Larche, F.: The general theory of curved deformable interfaces in solids at equilibrium. Philis. Mag. A 78, 1093–1109 (1998) 7. Aifantis, E.C.: Strain gradient interpretation of size effects. Int. J. Fract. 95, 1–4 (1999) 8. Mindlin, R.D.: Micro-structure in linear elasticity. Arch. Ration. Mech. Anal. 16, 51–78 (1964) 9. Movassagh, A.A., Mahmoodi, M.J.: A micro-scale modeling of Kirchhoff plate based on modified strain-gradient elasticity theory. Eur. J. Mech.-A/Solids 40, 50–59 (2013) 10. Mirsalehi, M., Azhari, M., Amoushahi, H.: Buckling and free vibration of the FGM thin micro-plate based on the modified strain gradient theory and the spline finite strip method. Eur. J. Mech.-A/Solids 61, 1–13 (2017) 11. Li, A., Zhou, S., Wang, B.: A size-dependent model for bi-layered Kirchhoff micro-plate based on strain gradient elasticity theory. Compos. Struct. 113, 272–280 (2014) 12. Ansari, R., Gholami, R., Shojaei, M., Mohammadi, V., Sahmani, S.: Bending buckling and free vibration analysis of size-dependent functionally graded circular/annular microplates based on the modified strain gradient elasticity theory. Eur. J. Mech.-A/Solids 49, 251–267 (2015) 13. Ansari, R., Hasrati, E., Shojaei, M., Gholami, R., Mohammadi, V., Shahabodini, A.: Sizedependent bending, buckling and free vibration analyses of microscale functionally graded mindlin plates based on the strain gradient elasticity theory. Latin Am. J. Solids Struct. 13, 632–664 (2016)
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14. Thai, S., Thai, H.T., Vo, P.T., Patel, V.I.: Size-dependant behaviour of functionally graded microplates based on the modified strain gradient elasticity theory and isogeometric analysis. Comput. Struct. 190, 219–241 (2017) 15. Sahmani, S., Ansari, R.: On the free vibration response of functionally graded higher-order shear deformable microplates based on the strain gradient elasticity theory. Compos. Struct. 95, 430–442 (2013) 16. Thai, H.C., Ferreira, A.J.M., Rabczuk, T., Nguyen-Xuan, H.: Size-dependent analysis of FG-CNTRC microplates based on modified strain gradient elasticity theory. Eur. J. Mech.A/Solids 72, 521–538 (2018) 17. Zhang, B., He, Y., Liu, D., Shen, L., Lei, J.: An efficient size-dependent plate theory for bending, buckling and free vibration analyses of functionally graded microplates resting on elastic foundation. Appl. Math. Model. 39, 3814–3845 (2015) 18. Zhang, B., He, Y., Liu, D., Lei, J., Shen, L., Wang, L.: A size-dependent third-order shear deformable plate model incorporating strain gradient effects for mechanical analysis of functionally graded circular/annular microplates. Compos. B Eng. 79, 553–580 (2015) 19. Thai, C.H., Ferreira, A.J.M., Nguyen-Xuan, H.: Isogeometric analysis of size-dependent isotropic and sandwich functionally graded microplates based on modified strain gradient elasticity theory. Compos. Struct. 192, 274–288 (2018) 20. Thai, C.H., Ferreira, A.J.M., Phung-Van, P.: Free vibration analysis of functionally graded anisotropic microplates using modified strain gradient theory. Eng. Anal. Bound. Elem. 117, 284–298 (2020) 21. Farzam, A., Hassani, B.: Size-dependent analysis of FG microplates with temperaturedependent material properties using modified strain gradient theory and isogeometric approach. Compos. B Eng. 161, 150–168 (2018) 22. Salehipour, H., Shahsavar, A.: A three dimensional elasticity model for free vibration analysis of functionally graded micro/nano plates: modified strain gradient theory. Compos. Struct. 206, 415–424 (2018) 23. Zeighampour, H., Beni, Y.T.: Cylindrical thin-shell model based on modified strain gradient theory. Int. J. Eng. Sci. 78, 27–47 (2014) 24. Thai, H.T., Vo, P.T., Nguyen, T.K., Kim, S.E.: A review of continuum mechanics models for size-dependent analysis of beams and plates. Compos. Struct. 177, 196–219 (2017) 25. Gu, L.: Moving Kriging interpolation and element-free Galerkin method. Int. J. Numer. Meth. Eng. 56, 1–11 (2003) 26. Thai, C.H., Ferreira, A.J.M., Lee, J., Nguyen-Xuan, H.: An efficient size-dependent computational approach for functionally graded isotropic and sandwich microplates based on the modified couple stress theory and moving Kriging-based meshfree method. Int. J. Mech. Sci. 142, 322–338 (2018) 27. Tran, D.T., Thai, C.H., Nguyen-Xuan, H.: A size-dependent functionally graded higher order plate analysis based on modified couple stress theory and moving Kriging meshfree method. Comput. Mater. Continua 57, 447–483 (2018)
Static Behavior of Functionally Graded Sandwich Beam with Fluid-Infiltrated Porous Core Tran Quang Hung1 , Do Minh Duc1(B) , and Tran Minh Tu2 1 The University of Da Nang - University of Science and Technology, Da Nang, Vietnam
[email protected]
2 National University of Civil Engineering, Hanoi, Vietnam
[email protected]
Abstract. In this report, the static response of sandwich beam with fluidinfiltrated porous core and two face sheets made of functionally graded materials is investigated. Variation in mechanical properties of the sandwich beam is assumed to be continuous along the thickness direction. Various beam theories for one-dimensional modelling of the beam are considered. The relationship between stress and strain obeys Biot’s theory of linear poroelasticity. The governing equations of the beam are derived by applying Hamilton’s principle and solved analytically by Navier’s solution. Comparative and comprehensive studies are conducted to examine both the accuracy and the effects of various parameters, such as power-law index, porosity and pore pressure coefficients, core-to-face thickness ratio, span-to-height ratio on the bending characteristics of the beam. Keywords: Functionally graded material · Fluid-infiltrated porous material · Porous sandwich beam · Navier’s solution · Beam theories
1 Introduction Due to the distribution of pores in their body, porous materials (i.e., metal foams, ceramic foams and polymer foams) own a series of excellent properties, such as low density, the efficient capacity of energy dissipation, low thermal conductivity. These advanced materials, therefore, is a reasonable selection for engineering applications of aerospace, automotive industry, civil construction, ship building, etc. [1–3]. In porous materials, the pores with different shapes and sizes are non-/uniformly distributed and can be either infiltrated with liquid or empty. Porous materials with liquid phase in their pores are classified as fluid-infiltrated/saturated porous materials. Due to the pores together with the liquid phase inside, mechanical behavior of structures made of porous materials may be different from that of conventional structures made of monolithic materials and needs much more research efforts to reveal. The stress-strain relation in porous materials is first developed by Biot [4]. This relation is an extension of Hooke’s law and called the linear theory of poroelasticity. This theory is commonly used to simulate the physical phenomena of porous media. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 691–706, 2022. https://doi.org/10.1007/978-981-16-3239-6_53
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Up to date, there have been many theoretical and experimental studies investigating the mechanical responses of porous structures. Some authors have focused on the effects of porosities without the liquid phase in the pores on static, vibration and buckling response of structures, such as beams [5, 6], plates [7, 8] and shells [9–11]. If the pores are infiltrated with liquid, the fluid pressure could appear in the pores. Many studies have considered this additional pressure to provide a more realistic prediction of structural behavior. For example, Galeban et al. [12] used Euler-Bernoulli assumptions to study free vibration of functionally graded (FG) thin beams made of saturated porous materials. Rad et al. [13] employed both first-order and third-order shear deformation theories to study elastic buckling of fluid-infiltrated porous plates. Ebrahimi and Habibi [14] predicted the deflection and vibration characteristics of rectangular plates made of saturated porous FG materials with the framework of third-order shear deformation plate theory. Based on 3D elasticity theory, Babaei et al. [15] investigated natural frequencies and dynamic response of thick annular sector plates and cylindrical panels made of saturated porous materials. Stress analysis of FG saturated porous rotating truncated cone by finite element modeling and 2-D axisymmetric elasticity theory was presented by Babaei and Asemi [16]. Rezaei and Saidi [17] attempted to study free vibration response of fluid-saturated porous annular sector plates based on Mindlin plate theory. Heshmati et al. [18] studied the vibration behavior of poroelastic thick curved panels with graded open-cell and saturated closed-cell porosities. Using classical plate theory, Panah et al. [19] examined pore fluid pressure and porosity effects on the bending and thermal postbuckling behavior of FG saturated porous circular plates. Sandwich structures with porous core, also called porous sandwich (PS) structures, is an ideal solution for lightweight structures while maintaining structural performance. The two face sheet layers are thin, but strong and stiff to withstand the bending and compression/tension in the plane. Meanwhile, the thick core mainly sustains the transverse shear and connects the face sheet layers. Moreover, the thick inner core with porosity reduces self-weight and plays roles as a heat diffuser, energy absorber, sound insulation sheet or vibration damper. With all the mentioned advantages, PS structures have become more and more popular in structural design. Mechanical behavior of PS structures has received lots of attention from researchers. Chen et al. [20] investigated nonlinear free vibration of shear deformable sandwich beams with an FG porous core based on the context of Timoshenko beam theory. Bamdad et al. [21] investigated free vibration and buckling of sandwich Timoshenko beam with porous core and composite face sheets subjected to electric and magnetic fields and resting on elastic foundation. Mu and Zhao [22] analyzed the natural vibration of composite sandwich beams with FG face sheets and a metal foam core. Jasion et al. [23] studied the global and local buckling-wrinkling of the face sheets of sandwich beams and sandwich circular plates with metal foam core. Using three modified Timoshenko hypotheses, Magnucka-Blandzi [24] presented the dynamic stability and static stress state of a simply supported sandwich beam with a metal foam core. Qin and Wang [25] studies the large deflections of fully clamped and simply supported slender sandwich beams with a metallic foam core subjected to transverse loading by a flat punch. Zhang et al. [26] investigated the large deflection response and the optimal design of slender multilayer metal sandwich beams with metal foam core subjected to low-velocity
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693
impact. Wang et al. [27] proposed a high-order shear deformation theory to investigate the transient response of a sandwich beam with FG face sheets and FG porous core under moving mass. Chinh et al. [28] studied the static flexural of sandwich beam with FG face sheets and FG porous core by third-order shear deformation beam theory. Based on the first-order shear deformation plate theory, Chen et al. [29] examined the buckling and bending behavior of sandwich plates with uniform and FG porous cores. Akbari et al. [30] employed differential quadrature method and third-order shear deformation theory to study the free vibration of sandwich cylindrical panels made of an FG saturated porous core and two similar homogenous face sheets. The literature reviews show that: (i) researches on PS structures taking pore pressure of fluid phase into account are extremely rare when compared with single-layer porous structure. There has been one work by Akbari et al. [30] for sandwich cylindrical panels available in the open literature; (ii) there has been a few reports, i.e., [20, 27–29], regarding the structural layers so that ensuring mechanical continuity at the interfaces, which is a requirement for designing sandwich structures in order to eliminate the stress concentration phenomenon at the contact between layers; (iii) the beam theories are employed almost individually in each of the studies. There has not been a comparison of their effects on the results of simulating the behavior of the sandwich structures with porous core yet. This issue is worth examining, particularly for shear deformation beam theories because shear stress distribution in the porous core may be different from that in monolithic one. This study attempts to analyze the static behavior of sandwich beams with FG face sheets and an FG fluid-infiltrated porous core. The material of the layers is tailored so that its mechanical properties vary continuously along the beam depth in order to avoid the stress concentration at the interfaces between layers. Different beam theories are considered for 1-D beam modeling and their effects on the analysis results are examined. The constitutive equation is based on Biot’s theory of linear poroelasticity. The governing equations of the beams are derived from Hamilton’s principle and solved analytically by Navier’s solution. Comparative studies are carried out to validate the analysis. Effects of power-law index, porosity and pore pressure coefficients, core-to-face thickness ratio, span-to-height ratio on the bending characteristics of the beam are investigated in detail.
2 Theory and Formulations 2.1 Sandwich Beam and Material Modelling A PS beam of dimensions L × b × h as shown in Fig. 1 is considered in this report. The z-axis is taken along the height (h) of the beam, and the x-axis coincides with the mid-plane. The two similar face sheet layers are made of an FG material, whereas a fluid-infiltrated FG porous material is adopted to construct the core. For convenience, the thickness ratio among the face sheet-core-face sheet is used to name the sandwich beam. For instance, 1-8-1 beam denotes the sandwich beam whose core thickness is eight times the face thickness (hc /hf = 8). Young’s modulus of two FG face sheets is assumed to alter according to the power-law form, whereas that of FG porous core varies according to a cosine rule as follows:
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z
FG material
q(x)
hf hc hf
x
o FG porous material
FG material
h4 = h/2 h3 h2
h
h1=-h/2
b
L Fig. 1. Geometric parameters and coordinate system of PS beams
⎧ h −z k ⎪ + Ec , z ∈ [h3 , h4 ] (top face sheet layer) E (3) (z) = (Em − Ec ) h 4−h ⎪ ⎪ 4 3 ⎨ z E (2) (z) = Em 1 − eo cos h π−h , z ∈ [h2 , h3 ] and h2 = −h3 (porous core layer) , 3 2 ⎪ ⎪ ⎪ (1) ⎩ z−h1 k E (z) = (Em − Ec ) h −h + Ec , z ∈ [h1 , h2 ] (bottom face sheet layer) 2
(1)
1
where E c and E m are, respectively, Young’s moduli of the constituents; k is the powerlaw index, k ≥ 0; e0 is the porosity coefficient (0 ≤ eo < 1). Equation (1) shows that Young’s modulus E varies smoothly along the thickness and is symmetric with respect to the mid-plane (z = 0); the super script (i) denotes the i-th layer. It should be noted that Poisson’s ratio v is assumed to be constant for each layer and shear modulus G(z) = E(z)/(2 + 2ν). 2.2 Displacement and Strain Fields Based on quasi-3D beam theory, the displacement components u(x, z), w(x, z), which are, respectively, along the x- and z-directions, can be expressed by the following equations [31]:
o (x) + Φ(z)ϕos (x) u(x, z) = uo (x) − z ∂w∂x (2) w w(x, z) = wo (x) + ∂Φ(z) oz (x) ∂z In Eq. (2), uo , wo are the axial and transverse displacements on the mid-plane (z = 0), respectively; ϕos is the transverse shear strain of any point on the x-axis; Φ is the shape function which characterizes the distribution of the shear strain through the beam depth, Φ = z − 4z3 / 3h2 is selected for this study. The strain-displacement relations can be given by ⎧ ∂ϕos ∂uo ∂ 2 wo ∂u ⎪ ⎪ ⎨ εx = ∂x = ∂x2 − z ∂x2 + Φ ∂x ∂ Φ εz = ∂w (3) ∂z = ∂z 2 woz ⎪ ⎪ ∂w ∂u ∂w ∂Φ oz ⎩ γxz = ϕos + + = ∂z
∂x
∂z
∂x
It is observed that Eqs. (2) and (3) are expressed for quasi-3D; they can, however, be simplified to other beam theories by adjusting functions Φ and woz . For example, Φ
Static Behavior of Functionally Graded Sandwich Beam
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= 0 for classical beam theory (CBT); Φ = z and woz = 0 for first-order beam theory (FBT); Φ = z − 4z3 / 3h2 and woz = 0 for third-order beam theory (TBT). In other words, quasi-3D theory could be considered as a general form of the four typical beam theories mentioned. 2.3 Constitutive Equation For fluid-filled porous materials, the stress-strain relations obey the linear poroelasticity theory of Biot, which can be express as [4] σij = 2Gεij + λθ δij − pαδij
(4)
in which σij and εij are the stress and strain components, respectively; δ ij is Kronecker delta; θ is the volumetric strain; α is the Biot coefficient of effective stress; p is the pore fluid pressure; λ is Lame parameter. The quantities p and α are defined as p = M (ξ − αθ ); α = 1 − GG0 , 2G(νn −ν) Eo ; M = α 2 (1−2ν ; νn = ν+αB(1−2ν)/3 Go = 2(1+ν) 1−αB(1−2ν)/3 )(1−2ν)
(5)
n
where ξ is the variation of fluid volume content; M is Biot’s modulus; ν n is undrained Poisson’s ratio; B is the Skempton coefficient; E o is the elastic modulus of the material with no porosity (eo = 0). For the undrained condition (ξ = 0) and quasi-3D theory with plane stress condition (σ y = 0), Eq. (4) can be written in the following matrix form: ⎧ ⎫ ⎡ ⎤⎧ ⎫ Q11 Q12 0 ⎨ εx ⎬ ⎨ σx ⎬ {σ } = σz = ⎣ Q21 Q22 0 ⎦ εz , (6) ⎩ ⎭ ⎩ ⎭ τxz γxz 0 0 Q33 where elastic coefficients are given as Q11 = Q22 =
2G(z) ; 1 − νn
Q12 = Q21 =
2νn G(z) = Q11 νn ; Q33 = G(z) 1 − νn
(7)
The other beams theories (CBT, FBT and TBT) can also use Eq. (6) by setting the coefficients as Q11 = 2(1 + νn )G(z);
Q12 = Q21 = 0;
Q33 = G(z)
(8)
If pore fluid pressure is neglected (p = 0), Eq. (4) reduces to conventional Hook’s law. Thus, Eqs. (4–8) can be used for the monolithic material of the face sheets by setting B = 0 or ν n = ν. 2.4 Energy Expressions The variation of the strain energy can be stated as L (σx δεx + σz δεz + τxz δγxz )dAdx
δU = 0 A
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L ∂δuo ∂ 2 δwo ∂δϕos ∂δwoz N − Mb + Rs δwoz + Q δϕos + dx, δU = + Ms ∂x ∂x2 ∂x ∂x 0
(9) where the stress resultants are defined as N = Q=
A A
σx dA;
Mb =
τxz ∂Φ ∂z dA
zσx dA; Ms =
A
Φσx dA; Rs =
A
A
∂2Φ σ dA; ∂z 2 z
(10)
In Eqs. (9) and (10), A is the area of the beam cross-section, and L is the length of the beam. By substituting Eqs. (3) and (6) into Eq. (10), the stress resultants can be expressed by matrix form ⎫ ⎡ ⎫ ⎧ ⎤⎧ ⎪ ⎪ ∂uo /∂x A1 B1 Bs C1 0 ⎪ N ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎢ ⎪ ⎪ ⎪ ⎥ 2 2 ⎪ ⎪ ⎪ ⎪ D Ds C2 0 ⎥⎨ −∂ wo /∂x ⎬ ⎨ Mb ⎬ ⎢ ⎢ ⎥ , (11) Ms = ⎢ ∂ϕos /∂x Hs C3 0 ⎥ ⎢ ⎥⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎣ ⎦ R 0 C w s ⎪ 4 oz ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎩Q ⎪ sym. As ⎩ (ϕos + ∂woz /∂x) ⎭ where ⎧ 2 A1 = Q11 dA; B1 = zQ11 dA; Bs = ΦQ11 dA; C1 = Q12 ∂∂zΦ2 dA ; ⎪ ⎪ ⎪ ⎪ A A A A 2 ⎪ ⎪ ⎪ ⎪ ⎨ D = z Q11 dA; Ds = zΦQ11 dA; A A 2 2 C2 = Q12 z ∂∂zΦ2 dA ; Hs = Φ 2 Q11 dA; C3 = Q12 Φ ∂∂zΦ2 dA; ⎪ ⎪ ⎪ ⎪ A A A ⎪ 2 2 ⎪ 2 ⎪ ⎪ ⎩ C4 = Q22 ∂∂zΦ2 dA; As = G ∂Φ dA ∂z A
(12)
A
The variation of the potential energy under a transverse load q acting on the top surface of the beam can be stated as L δV = −
qδwo dx
(13)
0
2.5 Hamilton’s Principle and Governing Equations Hamilton’s principle is used to derive the governing equations of the beam. This principle can be expressed for the case of static equilibrium as t2 (δU + δV )dt = 0 t1
(14)
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Substituting the expressions δU and δV from Eqs. (9) and (13) into Eq. (14) and then integrating by parts, and collecting the coefficients of δuo , δwo , δϕ os and δwoz , the governing equations of the beam are obtained: δuo :
∂N ∂ 2 Mb ∂Ms ∂Q = 0; δwo : + Q = 0; δwoz : Rz − =0 + q = 0; δϕos : − ∂x ∂x2 ∂x ∂x
(15)
Next, substituting Eq. (11) into Eq. (15), the governing equations can be expressed as
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
oz A1 ∂∂xu2o − B1 ∂∂xw3o + Bs ∂∂xϕ2os + C1 ∂w ∂x = 0 2
B1 ∂∂xu3o − D ∂∂xw4o + Ds ∂∂xϕ3os +C2 ∂ ∂xw2oz + q=0 ∂ 2 ϕos ∂ 3 wo ∂woz ∂woz − Ds ∂x3 + Hs ∂x2 + C3 ∂x − As ϕos + ∂x = 0 2 ∂ϕos ∂ 2 woz os =0 + C2 ∂∂xw2o − C3 ∂ϕ − C w + A + 4 oz s ∂x ∂x ∂x2 3
⎪ Bs ∂∂xu2o ⎪ ⎪ ⎪ ⎪ ⎩ −C1 ∂uo ∂x 2
2
3
3
4
2
(16)
2.6 Analytical Solutions Navier’s technique based on Fourier series is used to determine the analytical solution for a simply supported beam. Assuming each of the displacement components uo , wo , ϕ os and woz is trigonometric series, which satisfies the boundary conditions, as follows ⎧ ∞ ∞ ⎪ ⎪ ⎪ u U cos(λx); w Wn sin(λx) = = (x) (x) ⎪ n o ⎪ ⎨ o n=1 n=1 (17) ∞ ∞ ⎪ ⎪ ⎪ ⎪ ϕos (x) = Φn cos(λx); woz (x) = Wzn sin(λx) ⎪ ⎩ n=1
n=1
where U n , W n , Φ n and W zn are the unknown coefficients which need to be determined; (λ = nπ/L). The uniformly distributed load qo is also expanded in Fourier series as q(x) =
∞ n=1
Qn sin(λx); Qn =
4qo nπ
(n = 1, 3, 5, ...),
(18)
Substituting Eqs. (17) and (18) into Eq. (16) yields the following analytical solution: ⎤⎛ ⎞ ⎛ ⎞ ⎡ Un 0 s11 s12 s13 s14 ⎟ ⎜ ⎜ ⎟ ⎢ s22 s23 s24 ⎥ ⎥⎜ Wn ⎟ = ⎜ −Qn ⎟, ⎢ (19) ⎦ ⎠ ⎝ ⎝ ⎣ s33 s34 Φn 0 ⎠ sym. s44 Wzn 0 where ⎧ s11 ⎪ ⎪ ⎨ s22 ⎪ s ⎪ ⎩ 33 s44
= −λ 2 A1 ; s12 = λ 3 B1 ; s13 = −λ 2 Bs ; s14 = λC1 ; = −λ 4 D; s23 = λ 3 Ds ; s24 = −λ 2 C2 ; = −λ 2 Hs − As ; s34 = λC3 − λAs ; = −C4 − λ 2 As
(20)
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The unknown series coefficients are obtained by solving Eq. 19. Having known the coefficients, the displacements and stresses can be straightforwardly calculated by using Eqs. 6 and 17.
3 Results and Discussion Based on mathematical modelling, numerical examples for the simply supported beam subjected to uniform load qo are conducted to validate and comprehend the static responses. Four different beam theories (CBT, FBT, TBT and quasi-3D) are considered in the analysis. In the case of FBT, shear correction factor is taken to be 5/6 for the beam of rectangular cross-section. In this study, assuming that the FG face sheets are composed of ceramic (E c = 380 MPa, ν c = 0.3) and metal (E m = 70 MPa, ν m = 0.3) [32], whereas the porous soft-core are made of a metal foam (E m = 70 MPa, ν m = 0.3) - referred from [16]. Following non-dimensional parameters are used to present all numerical results: ⎧ ⎨ wˆ = 100Em4bh3 w L , 0 ; wˆ oz = 100Em4bh3 woz L ; uˆ = 100Em4bh3 u 0, − h ; 2 2 2 qo L qo L qo L (21) ⎩ σˆ x = bh σx L , h ; σˆ z = bh σz L , z ; τˆxz = bh τxz (0, 0) qo L 2 2 qo L 2 qo L 3.1 Validation Firstly, the validation is done for the specific case of the investigated beam having a hardcore without porosities by setting the porosity coefficient eo to zero and interchanging the roles of E m and E c in Eq. (1). This problem was studied by Vo and his co-workers [32] using the analytical solution and the finite element method, and by Sayyad et al. [33] using Navier’s solution. It is seen that the obtained results given in Table 1 are almost identical to those of Vo et al. [32] and agree very well with Sayyad et al. [33]. For further validation, another example is carried out for PS beam without pore fluid pressure by setting the Skempton coefficient B = 0 or ν n = ν for the porous core. The present results are compared with those of Chinh et al. [28] which are based on TBT and mesh-free method. The comparison of results in Table 2 shows an excellent agreement between them. Table 1. Comparison of the displacement, axial stress and shear stress for 1-2-1 hardcore beam with different beam theories (L/h = 5, eo = 0) Theory CBT FBT
Source
k=1
k=5
wˆ
σˆ x
τˆxz
wˆ
σˆ x
τˆxz
Present
5.0798
1.2192
–
8.1409
1.9538
–
Vo et al. [32]
5.0798
–
–
8.1409
–
–
Present
5.4408
1.2192
0.6268
8.5762
1.9538
0.7559
Sayyad et al. [33]
5.3807
1.2192
0.6183
8.5036
1.9539
0.7457
Vo et al. [32]
5.4408
1.2192
0.7507
8.5762
1.9538
0.9053 (continued)
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Table 1. (continued) Theory TBT Quasi-3D
Source
k=1
k=5
wˆ
σˆ x
τˆxz
wˆ
σˆ x
τˆxz
Present
5.4122
1.2329
0.8124
8.5136
1.9705
0.8926
Vo et al. [32]
5.4122
1.2329
0.8123
8.5137
1.9705
0.8925
Present
5.3611
1.2315
0.7993
8.4276
1.9672
0.8763
Vo et al. [32]
5.3612
1.2315
0.7993
8.4276
1.9672
0.8763
Table 2. Comparison of the displacement wˆ for 1-1-1 PS beam (k = 0.5, TBT, B = 0) Source
L/h eo 0
Present
Chinh et al. [28]
0.4
0.6
0.8
5
6.3807 6.4530 6.5342 6.6263 6.7316
20
5.3496 5.3586 5.3682 5.3785 5.3896
Chinh et al. [28] Present
0.2
6.3807 6.4530 6.5342 6.6263 6.7316 5.3496 5.3586 5.3682 5.3785 5.3896
3.2 Comprehensive Study Table 3 is used to present the displacement, axial stress and shear stress values of the 1-81 beam with fluid-infiltrated FG porous core. Two values of span-to-height ratio (L/h = 5, which represents for thick beams, and L/h = 20, which represents for thin ones), different values of porosity coefficient eo and different beam theories are considered. As expected, CBT, without shear deformation effect, gives the smallest displacements wˆ of all beam theories; the displacements (ˆu, w) ˆ calculated by quasi-3D are slightly smaller than those calculated by TBT. Those two effects are more pronounced for the thick beam (L/h = 5) than for the thin one (L/h = 20). Interestingly, CBT and FBT give the same values of uˆ as well as of σˆ x . In addition, due to the different shape functions Φ, the values of shear stress obtained by FBT are distinctly different from those obtained by TBT/quasi-3D. Moreover, as increasing the value of the porosity coefficient eo , both the displacements uˆ and wˆ increase. This is because it reduces the stiffness of the beam. Besides, the deviation of the calculated results among the beam theories increases with the increase in eo . For example, for eo = 0 and L/h = 5, the relative difference of wˆ between FBT and TBT is (wˆ TBT − wˆ FBT )/wˆ TBT × 100% = (10.1349 − 9.9604)/9.9604 × 100% = 1.75%, but for the same situation with eo = 0.9 that is (13.6431 − 11.4876)/11.4876 × 100% = 18.76% (the data are taken from Table 3). Unlike CBT, FBT and TBT, the displacement wˆ oz exists for quasi-3D theory, which including thickness stretching effect. This displacement versus L/h ratio for 1-1-1 and 1-8-1 schemes with different values of eo is illustrated in Fig. 2. It is seen that eo has little effect, whereas L/h and sandwich schemes have a strong effect on wˆ oz . The displacement
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Table 3. Values of displacements, axial stress and shear stress of 1-8-1 PS beam (B = 0.5, k = 0.5) L/h = 5
L/h = 20
eo
Theory
0
CBT
8.7560 2.8019 11.4078 –
FBT
9.9604 2.8019 11.4078 0.3852 8.8313
TBT
10.1349 2.8659 11.5866 0.6819 8.8423
0.7015 45.6760 0.6967
Quasi-3D 10.0671 2.8276 11.5907 0.6745 8.8383
0.7009 45.6770 0.6949
wˆ
0.3 CBT
uˆ
σˆ x
τˆxz
9.0211 2.8867 11.7532 –
wˆ
uˆ
8.7560
0.7005 45.6313 –
9.0211
σˆ x
τˆxz
0.7005 45.6313 0.3852
0.7217 47.0127 –
FBT
10.3866 2.8867 11.7532 0.3057 9.1064
0.7217 47.0127 0.3057
TBT
10.7751 2.9643 11.9702 0.6099 9.1309
0.7229 47.0669 0.6241
Quasi-3D 10.7062 2.9248 11.9685 0.6041 9.1265
0.7222 47.0303 0.6228
0.6 CBT
9.3092 2.9789 12.1285 –
9.3092
0.7447 48.5141 –
FBT
10.8855 2.9789 12.1285 0.2017 9.4077
0.7447 48.5141 0.2017
TBT
11.7392 3.0804 12.4134 0.4848 9.4614
0.7464 48.5853 0.4973
Quasi-3D 11.6691 3.0396 12.4110 0.4808 9.4563
0.7456 48.5187 0.4964
0.9 CBT
9.6235 3.0795 12.5380 –
9.6235
0.7699 50.1522 –
FBT
11.4876 3.0795 12.5380 0.0596 9.7400
0.7699 50.1522 0.0596
TBT
13.6431 3.2361 12.9802 0.2010 9.8754
0.7724 50.2627 0.2071
Quasi-3D 13.5721 3.1941 12.9903 0.1996 9.8697
0.7716 50.1808 0.2068
wˆ oz decreases and approaches zero when increasing L/h. Furthermore, 1-1-1 beam with higher stiffness has smaller wˆ oz than that of 1-8-1 one. 1-8-1 beam 1-8-1 beam 1-1-1 beam
1-1-1 beam
L/h
L/h
Fig. 2. The displacement wˆ oz of 1-1-1 and Fig. 3. The displacement wˆ of 1-1-1 and 1-8-1 1-8-1 beams with respect to L/h (B = 0.5, k = beams with respect to L/h (B = 0.5, k = 1) 1)
Variation of the non-dimensional displacement wˆ at the center of the beam with respect to L/h ratio is depicted in Fig. 3. Two schemes of the sandwich beam (1-1-1 and 1-8-1 beams), together with four different beam theories, are considered. As can be seen,
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701
1-8-1 beam
1-1-1 beam
eo
Fig. 4. The displacement wˆ of 1-1-1 and 1-8-1 beams with respect to eo (B = 0.5, k = 0.5, L/h = 5)
k
Fig. 5. The displacement wˆ of 1-8-1 beam with respect to k (quasi-3D, B = 0.5, L/h = 5)
B Fig. 6. The displacement wˆ of 1-8-1 beam with respect to B (quasi-3D, k = 0.5, L/h = 5)
the effect of the different beam theories on the obtained results is important, especially for the thick beams (small L/h). This effect, however, can be neglected for the thin beams. Figure 3 also shows that the results calculated using TBT are not significantly discrepant with those using quasi-3D, but clearly different from those using FBT. In other words, there is a significant difference in the outcome between FBT and TBT/quasi-3D. In Fig. 4, the displacement wˆ versus the porosity coefficient eo for 1-1-1 and 1-8-1 beams is displayed. The displacement increases with increasing eo . This trend is more pronounced in thick core beam (1-8-1 beam) than in thin core one (1-1-1 beam). It is also unveiled that for thick core beam, the computed results employing TBT/quasi-3D increases more rapidly with respect to eo than employing FBT/CBT. Figure 5 portrays the transverse displacement wˆ versus power-law index k with different values of eo . Only quasi-3D theory and 1-8-1 beam with L/h = 5 are performed for this investigation. It can be seen that for each of eo , the greatest displacement is obtained when k = 0. This is because the beam is full of metal and has the smallest stiffness. As k increases, the volume fraction of the ceramic phase constituting the face sheets, which has much higher Young’s modulus, increases. Thus, it reduces quickly the displacement.
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hc/hf
hc/hf b) L/h = 20
a) L/h = 5
Fig. 7. The displacement wˆ with respect to hc /hf (quasi-3D, k = 0.5, B = 0.5)
Effect of pore fluid pressure through the Skempton coefficient B on the transverse displacement wˆ is depicted in Fig. 6. It is revealed that increasing B results in a slight reduction of w, ˆ except for eo = 0. However, this effect is not significant. Effect of the layer thickness ratio (hc /hf ) on the transverse displacement wˆ of PS beams with different values of eo is displayed in Fig. 7. It is pointed out that the displacement increases with increasing hc /hf . The reason is that the beam becomes softer as the porous core is thicker. Besides, the gap between the two adjacent curves increases with the increase in L/h. It proves that the effect of eo on the response of the beam is more sensitive for the thick porous core beams (high hc /hf ) than for the thin core ones. It is also observed that this gap is greater for the thick beam (Fig. 7a) than for the thin one (Fig. 7b).
k
L/h
Fig. 8. Effect ofk and L/h on transverse displacement wˆ of 1-1-1 and 1-8-1 beams (quasi-3D, eo = 0.6, B = 0.5)
Effect of the interaction between k and L/h ratio on the displacement wˆ is plotted in 3 dimensions in Fig. 8. Again, the strong effect belongs to the region having small values of k or L/h.
Static Behavior of Functionally Graded Sandwich Beam
z/h
703
z/h
a) Axial stress
b) Shear stress
Fig. 9. Variation of stresses σˆ x , τˆxz over the beam depth (1-8-1 beam, eo = 0.6, B = 0.5, k = 2, L/h = 5)
Axial and shear stresses through the beam depth for different beam theories are illustrated in Fig. 9. As expected, the stresses are distributed continuously throughout the beam depth. It is clear that in Fig. 9, the influence of the beam theories considered on the axial stress is negligible. However, due to the difference of the shape functions Φ, the shear stress obtained by FBT is entirely different to that obtained by TBT/quasi-3D. It is interesting to note that the maximum shear stress belongs to the face sheets instead of the core.
z/h
Fig. 10. Variation of normal stress σˆ z over the beam depth (1-8-1 beam, B = 0.5, k = 2, L/h = 5)
For quasi-3D theory, the variation of the normal stress caused by the thickness stretching effect is shown in Fig. 10. As expected, this stress exists but is small when compared with the axial stress in Fig. 9a. Besides, this stress focuses on the face sheets and is anti-symmetrically distributed with respect to the mid-plane (z = 0) for the studied beam.
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4 Summary and Conclusions Analytical solution together with four different beam theories (i.e., classical, first-order, third-order and quasi-3D theories), which are unified in quasi-3D theory, is presented for static behavior analysis of porous sandwich beam. The beam is constructed of two FG face sheets and a fluid-infiltrated FG porous core so that mechanical properties vary smoothly in the thickness direction. Stress-strain relations obeys Biot’s theory of linear poroelasticity. Hamilton’s principle is used to derive the governing equations of the beam. The accuracy of the study is validated. Comprehensive studies are conducted to investigate the effects of parameters, including power-law index, porosity and pore pressure coefficients, core-to-face thickness ratio, span-to-height ratio on the bending characteristics of the beam. Some major conclusions can be reached from the study: (1) Sandwich beam with face sheets made of FG materials shows the effectiveness in both increasing the stiffness and satisfying the continuous condition of mechanical properties at the interface of layers; (2) Pore fluid pressure has no significant effect on the static response of the sandwich beam with fluid-infiltrated FG porous core; (3) For thick beams, the effects of different beam theories on the obtained results of the analysis of displacements and shear stress are significant. However, their effects on axial stress are negligible; (4) Static response predicted by third-order shear deformation beam theory and that by quasi-3D shear deformation beam theory is slightly different; (5) There is a considerable difference in the calculated outcomes between first-order and third-order/quasi-3D beam theories, especially for the thick beam with thick core and high porosity.
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27. Wang, Y., Zhou, A., Fu, T., Zhang, W.: Transient response of a sandwich beam with functionally graded porous core traversed by a non-uniformly distributed moving mass. Int. J. Mech. Mater. Des. 16(3), 519–540 (2019) 28. Chinh, T.H., Tu, T.M., Duc, D.M., Hung, T.Q.: Static flexural analysis of sandwich beam with functionally graded face sheets and porous core via point interpolation meshfree method based on polynomial basic function. Arch. Appl. Mech. 91(3), 933–947 (2020) 29. Chen, D., Yang, J., Kitipornchai, S.: Buckling and bending analyses of a novel functionally graded porous plate using Chebyshev-Ritz method. Arch. Civ. Mech. Eng. 19(1), 157–170 (2018) 30. Akbari, H., Azadi, M., Fahham, H.: Free vibration analysis of thick sandwich cylindrical panels with saturated FG-porous core. Mech. Based Des. Struct. Mach. 1–19 (2020) 31. Sayyad, A.S., Ghugal, Y.M.: Modeling and analysis of functionally graded sandwich beams: a review. Mech. Adv. Mater. Struct. 26(21), 1776–1795 (2019) 32. Vo, T.P., Thai, H.-T., Nguyen, T.-K., Inam, F., Lee, J.: Static behaviour of functionally graded sandwich beams using a quasi-3D theory. Compos. Part B: Eng. 68, 59–74 (2015) 33. Sayyad, A.S., Ghugal, Y.M.: A unified five-degree-of-freedom theory for the bending analysis of softcore and hardcore functionally graded sandwich beams and plates. J. Sandwich Struct. Mater. 23, 473–506 (2019)
Failure Analysis of Pressurized Hollow Cylinder Made of Cohesive-Frictional Granular Materials Trung-Kien Nguyen(B) Faculty of Civil Engineering and Industrial Construction, National University of Civil Engineering, 55 Giai Phong Road, Hanoi, Vietnam [email protected]
Abstract. An innovative approach that combines the Finite Element Method (FEM) and Discrete Element Method (DEM) has been applied for investigating the failure of a pressurized hollow cylinder made of cohesive-frictional granular materials. At the microscopic level, a Volume Element (VE) by granular assembly is used to describe the discrete nature of granular media. DEM-based model is defined through this VE. At the macroscopic level, we use FEM to simulate the boundary value problem. Two levels are then bridged through numerical homogenization. In this way, the mechanical problem could be studied at both macro-scale (finite element mesh) and micro-scale (grain interaction). It is well-known in continuum modeling by FEM that when bifurcation occurs, the numerical solutions lose their uniqueness if only first gradient model is used. To preserve the objectivity of the solution, a local second gradient is employed and shown to efficiently regularize the problem. The numerical results by the combined FEM×DEM modeling indicate that when the failure occurs, the shear band is reproduced numerically in the case of pressurized hollow cylinder. Moreover, the second gradient parameter gives an implicit internal length which is directly related to the width of the shear band. Keywords: FEM · DEM · Second gradient model · Pressurized hollow cylinder · Granular materials
1 Introduction An innovative approach that combines the Finite Element Method (FEM) and the Discrete Element Method (DEM) (FEM×DEM approach) has been developed since the last decade [1–4]. The main interest of this method lies in a numerical homogenization process that bridges the gap between macro- and micro-scales. At the microscopic level, DEM-based model is used to describe the constitutive relation of the material while at the macroscopic level, FEM model is used. By the way, continuum modeling can take into account the discrete nature of the materials. This method has been successfully applied in modeling the behavior of granular materials and real-scale boundary value problem made of granular materials. However, most of them used first gradient model at the macroscopic level, which shows the non-objectivity of the solution. Especially when strain localization is observed, the result shows a strong mesh dependency. This is © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 707–715, 2022. https://doi.org/10.1007/978-981-16-3239-6_54
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because the first gradient model does not consider the microstructure of the material. In this case, an enriched model such as the second gradient model has to be considered to overcome this limitation. Several second gradient approaches have been developed in the literature [5–8]. They are mainly classified into non-local and local families. Regarding the local approach, Chambon and co-workers [9, 10] developed a local second gradient model and applied to 2D finite element modeling. This local approach is believed to be coupled with various first gradient constitutive models. Pressurized hollow cylinder is a well-known problem that its basic solution could be found in every textbook. However, the numerical solution that accounts for the complex behavior of a hollow cylinder made of geomaterials remains an interesting question, which was not fully answered in the literature. The problem of a hollow cylinder can be related to borehole stability or gallery behavior in the field of geotechnical engineering. To demonstrate the capabilities of the FEMxDEM approach to reproduce the complexities of geomaterials in a real scale problem and the advantages of local second gradient in preserving the objectivity of the solution, this paper aims at simulating and analyzing such phenomenon through the bias of failure of a pressurized hollow cylinder made of cohesive-frictional granular materials. Moreover, as the strain localization is believed to suffers in both first and second gradient finite element models, the paper aim also to demonstrate the effect of the second gradient model in regularizing mesh dependency problems. The rest of the paper is structured as follows: After the introduction, Sect. 2 briefly introduces local second gradient model formulation, coupled with the FEM×DEM approach. Section 3 analyzes the failure of the hollow cylinder made of cohesive-frictional granular materials with an emphasis on the influence of the material-made density and second gradient parameters. Conclusion and remarks are given in the last section.
2 Second Gradient Model Formulation Before entering the text of second gradient model formulation, we briefly summarize the principle of FEM×DEM approach. At the microscopic level, a Volume Element (VE) by the granular assembly is used to describe granular materials and served as DEMbased constitutive law. At the macroscopic level, we use FEM to resolve the boundary value problem. Unlike conventional FEM, numerical resolution by FEMxDEM approach required two levels of numerical convergence. One is numerical convergence at microscale, i.e. convergence of DEM calculation. This DEM convergence criterion requires that the granular assembly is stable enough after being submitted to deformation applied. It is controlled through the total kinetic of the system. When the scale is bridged from the micro-scale (DEM) to macro-scale (FEM) through the numerical homogenization process, traditional convergence on residual force and displacement are required at the macro-scale (FEM). These two levels of convergence ensure that the whole problem (both macro- and micro-scales) reaches an equilibrium state at the end of each loading step. Local second gradient has been developed in laboratory 3SR, Grenoble over two decades. For more details about the formulation and model, the readers can refer to [9, 10]. We recall hereafter the principal elements with emphasis on the first gradient
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constitutive relation based on DEM. It can be noted that the current DEM model is fully developed for 3D cohesive-frictional granular materials and it can be used either in 3D FEM modeling or 2D FEM plane stress/strain problems. We limit this paper to the 2D problem at the macroscopic level. 2.1 Variational Principle We present the weak form of the balance equations for a given body as follows [10]. ∗ 2 ∗ t ∂ui t ∂ ui (1) σij t + ijk t t d t − Pe∗ = 0 ∂xj ∂xj ∂xk t ∂u∗ ρ t fit ui∗ d t + (2) Pe∗ = pit ui∗ + Pit i nk d t ∂xk t ∂t t are Cauchy and double stress tensors, respectively; P ∗ is the external Where σijt , ijk e ∗ virtual work. ui is a virtual displacement field; ρ t , fit are mass density and body force; pit , Pit are classical and double external forces; ∂ is the boundary of .
2.2 Equations with Lagrange Multipliers To avoid the difficulties while solving the boundary value problems by finite element with second gradient models, it commonly weakens the constraint by introducing Lagrange multipliers, noted λtij . Equation (1) is therefore re-written: ∗ ∗ ∗ 2 u∗ ∂u ∂u ∂ ∂u t i i λtij − it d t − Pe∗ = 0 (3) σijt it + ijk d t − ∂xj ∂xjt ∂xkt ∂xjt ∂xj t t Moreover, the fields ui and νij = ∂ui ∂xj have to meet the following weak form constraint: t ∂uit ∗ ∂ui λij − t d t = 0 (4) ∂xjt ∂xj t The external virtual power could be expressed as: t ∗ pi ui + Pit νik∗ ntk ds ρ t fit ui∗ d t + Pe∗ = t
∂t
(5)
This result leads to assuming that the constitutive relation of the first and second t function of ν . gradient is as: σijt function of ui and ijk ij 2.3 Finite Element: First and Second Quadratic Elements Finite elements used in this study are depicted in Fig. 1, including the first gradient (FG) and the second gradient (SG) quadrilateral elements. The FG element has eight nodes for ui while SG element has eight nodes for ui , four nodes for νij , and one node for λij . Both types of elements are used in the numerical simulations in the following section.
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Fig. 1. First and Second gradient quadrilateral elements (left) and transformed elements (right)
2.4 Constitutive Relations a. First gradient part Regarding the first gradient part, we used Numerical Homogenized Law based on Discrete Element Modeling (DEM-based model). Spherical particles are used as described in Fig. 2. The normal and tangential contact forces are noted fn and ft , respectively. They are calculated as follows: fn = −kn · δ + fc
(6)
ft = −kt · ut
(7)
Where δ is the overlap between two contacted particles; ut is the relative tangential displacement. To reproduce cohesive-frictional granular materials, a local cohesion fc is introduced. Tangential contact force is limited by Coulomb threshold as: ft ≤ μ · |fn | = tan ϕ · |fn | where μ is the intergranular coefficient of friction and ϕ is the friction angle. The stress tensor σ is determined by the homogenized formula: 1 σ = f ⊗l V
(8)
(9)
C
where f is contact force and l is the branch vector joining the centers of two particles in contact; V is the volume of granular assembly and C is the contact list; f ⊗ l denotes the tensor product of two vectors f and l.
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Fig. 2. Contact model
b. Second gradient part For the second gradient part, we use the particular case of isotropic linear relation obtained by [9, 11]. This latter involving only one parameter noted D. This parameter is linked implicitly to the internal length of the model [10]. ∇ ∂ v˙ ij (10) = [D] ∂xk ijk ∇
where is Jaumann derivative of double stress. ijk
3 Failure Analysis of Pressurized Hollow Cylinder 3.1 Pressurized Hollow Cylinder Problem The cross-section behavior of the pressurized hollow cylinder problem is simulated in this section. Thanks to the axisymmetric conditions, only one-fourth of the crosssection is considered. The problem is 2D plane strain, stress-controlled by external (pe ) and internal (pi ) pressures. In order to investigate the behavior of the hollow cylinder when increasing the inner pressure, in these simulations, the external pressure is kept constant and equal to initial isotropic stress pe = σ0 = const. The inner pressure is increasing from initial pressure (σ0 ) up to the failure of the specimen (Fig. 3). The inner and outer radius of the hollow cylinder are ri = 10 mm and re = 40 mm. Regarding finite element mesh, first gradient and second gradient quadratic elements are used. The domain is discretized in 870 elements (FG or SG element). At the microscopic level, a granular assembly of 1000 spherical grains is used to build the first gradient constitutive relation. These grains interact via contact model as described in Sect. 2. Microscopic parameters are chosen based on dimensional analysis: kn kt = 1; κ = kn σ0 = 500 and μ = 0.5. The influence of the internal length of the second gradient model is investigated thanks to the second gradient parameter D.
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Fig. 3. Pressurized hollow cylinder problem
3.2 Failure Analysis a. Influence of material-made density: Dense (DS) and loose samples (LS) In this section, first gradient finite element model is used to highlight the influence of material-made density on the macroscopic behavior of the hollow cylinder. The density of the material is characterized by the density of granular assembly (i.e. Volume Element at Gauss point). It is classified into dense and loose samples. Both samples have the same grain size distribution but different solid fractions. The dense sample has a solid fraction of 0.641 while the loose sample has a solid fraction of 0.560. Figure 4 illustrates the equivalent strain in DS and LS cases atfailure. The failure occurred earlier in the case of loose sample. It can be seen that at pi pe = 3.0, the shear band was formed in LS case. In DS case, as the internal pressure continued to increase, the failure developed and strain localization was also observed at pi pe = 4.0. In both cases, the shear bands were initiated and propagated from the inner surface. It can be noted that in first gradient modeling, as failure develops if strain localization occurs,
Fig. 4. Influence of density: dense and loose samples
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the width shear band shrinks to mesh size element. The shear band in spiral form is in agreement with numerical/experimental works performed by [12–16]. b. Influence of internal length Parametric studies have been done by varying the second gradient parameter D. This latter requires that D give an (implicit) internal length greater than the internal length of finite element mesh. On the contrary, the regularization will not take effect because the width band reduces to the mesh size. To highlight the effect of second gradient parameters, dense sample cases were chosen for performing the FEMxDEM calculations. Numerical parameters were similar for all simulations except the second gradient parameter that takes the value of D = 0.01 and D = 0.001. Shear band distributions of simulated cases are displayed in Fig. 5. The result shows that internal length correlates with the width of the shear band. By decreasing D, the width of the shear band decreases as well.
Fig. 5. Influence of internal length parameters on the width of shear band.
The result in Fig. 6 shows that the mesh dependency problem suffered in the first gradient model has been regularized by the second gradient model. Two finite element meshes of 780 SG and 1280 SG elements were used. Shear bands with similar width, orientation, and distribution were observed in both cases. This result demonstrates that the second gradient model has been preserved the objectivity of the problem. It means that the behavior of the problem is independent of finite element mesh (geometry characteristic) but representing the physics internal length of material-made (second gradient parameter). Careful calibration of internal length by the second gradient parameter is mandatory.
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Fig. 6. Second gradient model regularizes mesh dependency
4 Conclusion Failure analysis of pressurized hollow cylinder problems under 2D plane strain condition was presented in this paper. In order to represent the cohesive-frictional granular materials, a DEM-based model was used to describe the first gradient constitutive relation. To preserve the objectivity of the problem, a local second gradient was employed within the framework of FEM×DEM multiscale modeling. In all cases, failure of the pressurized hollow cylinder was presented in form of shear band distribution. The results of the numerical simulations showed a great influence of loose/dense sample and second gradient parameter on strain localization which occurred at the inner boundary of the hollow cylinder. The failure occurred earlier in the loose case than in the dense case. Regarding the effect of the second gradient model, the second parameter control implicitly the internal length of the problem. This latter is correlated to the width of the shear band. The result also demonstrated that the second gradient model preserves the objectivity of the solution when bifurcation occurs. Thus selecting an appropriate second parameter, which has a strong influence on the failure phenomenon of the problem, for such analysis should be carefully considered. Finally, to highlight the advantages of multi-scale continuum-discrete approach, future works should be placed in a detailed comparison of current pressurized hollow cylinder solutions by FEM×DEM computations with the use of other classics/advanced constitutive models for geomaterials or with other concurrences multi-scale modeling such as FEM2 .
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References 1. Nguyen, T.K., Combe, G., Caillerie, D., Desrues, J.: FEM×DEM modelling of cohesive granular materials: numerical homogenisation and multi-scale simulations. Acta Geophys. 62(5), 1109–1126 (2014) 2. Nguyen, T.K., et al.: FEM× DEM: a new efficient multi-scale approach for geotechnical problems with strain localization. In: EPJ Web of Conferences, vol. 140, p. 11007. EDP Sciences (2017) 3. Desrues, J., Argilaga, A., Dal Pont, S., Combe, G., Caillerie, D., Kein Nguyen, T.: Restoring mesh independency in FEM-DEM multi-scale modelling of strain localization using second gradient regularization. In: International Workshop on Bifurcation and Degradation in Geomaterials, pp. 453–457. Springer, Cham (May 2017) 4. Desrues, J., et al.: From discrete to continuum modelling of boundary value problems in geomechanics: an integrated FEM-DEM approach. Int. J. Numer. Anal. Meth. Geomech. 43(5), 919–955 (2019) 5. Pijaudier-Cabot, G., Bažant, Z.P.: Nonlocal damage theory. J. Eng. Mech. 113(10), 1512–1533 (1987) 6. Vardoulakis, I., Aifantis, E.C.: A gradient flow theory of plasticity for granular materials. Acta Mech. 87(3–4), 197–217 (1991) 7. De Borst, R.E.N.É.: Simulation of strain localization: a reappraisal of the Cosserat continuum. Eng. Comput. 8, 317–332 (1991) 8. Chambon, R., Caillerie, D., El Hassan, N.: One-dimensional localisation studied with a second grade model. Eur. J. Mech.-A/Solids 17(4), 637–656 (1998) 9. Chambon, R., Caillerie, D., Matsuchima, T.: Plastic continuum with microstructure, local second gradient theories for geomaterials: localization studies. Int. J. Solids Struct. 38(46–47), 8503–8527 (2001) 10. Matsushima, T., Chambon, R., Caillerie, D.: Large strain finite element analysis of a local second gradient model: application to localization. Int. J. Numer. Method Eng. 54(4), 499–521 (2002) 11. Mindlin, R.D.: Microstructure in linear elasticity. Columbia Univ New York Dept of Civil Engineering and Engineering Mechanics (1963) 12. François, B., Labiouse, V., Dizier, A., Marinelli, F., Charlier, R., Collin, F.: Hollow cylinder tests on boom clay: modelling of strain localization in the anisotropic excavation damaged zone. Rock Mech. Rock Eng. 47(1), 71–86 (2014) 13. Marinelli, F., Sieffert, Y., Chambon, R.: Hydromechanical modeling of an initial boundary value problem: studies of non-uniqueness with a second gradient continuum. Int. J. Solids Struct. 54, 238–257 (2015) 14. Van den Hoek, P.J.: Prediction of different types of cavity failure using bifurcation theory. In: DC Rocks 2001, The 38th US Symposium on Rock Mechanics (USRMS). American Rock Mechanics Association (2001) 15. Labiouse, V., Orellana, F.: A hollow cylinder testing device to study at small scale the damaged zone around underground openings in porous rocks. In: 13th ISRM International Congress of Rock Mechanics. International Society for Rock Mechanics and Rock Engineering (2015) 16. Crook, T., Willson, S., Yu, J.G., Owen, R.: Computational modelling of the localized deformation associated with borehole breakout in quasi-brittle materials. J. Petrol. Sci. Eng. 38(3–4), 177–186 (2003)
Studying the Strength of an Acidic Soil-Cement Mixing in Laboratory Tham Hong Duong(B) , Huan NguyenPhu Vo, and Danh Thanh Tran Ho Chi Minh City Open University, Ho Chi Minh City, Vietnam {tham.dh,huan.vnp,danh.tt}@ou.edu.vn
Abstract. This paper aims to study the control factors which contribute to the unconfined compression strength (UCS) qu of acidic soil mixing with cement in the laboratory. By selecting factors of physical properties (represented by plasticity index Ip), mechanical properties (undrained strength Su), chemical property of pore water (pH degree, or fine contents that stand partly for mineral components), and binder (by weight per m3 of improved soil), a design of experiments are preliminarily studied to rank the importance of each factor. It is necessary to govern the lab test by using a Plaxis model in which the lab test is simulated with data obtained from the real lab test. Two uncontrollable factors, namely E and F, are the undrained/drained condition of analysis and the scale of mixing, i.e. mixing with a small amount and mass mixing. There are 9 runs of virtual experiments, integrated with four possible cases of the uncontrollable factors. An analysis of variance (ANOVA) was conducted to identify the percentage of contribution of control factors in all the possible cases. The results indicate unconfined compression strength is affected mostly by the binder, then the pH concentration, the soil plasticity, and the strength of the original soil. The best performance of the mixture will be of the high percentage of binder, for high plasticity soil and soft soil. This study can be further to predict the effectiveness of soil-cement mixture in the laboratory before mixing the materials on the site. Keywords: Design of experiments · Orthogonal array · Unconfined compression strength · ANOVA · Percentage of contribution
1 Introduction Soil Cement Mixing is one of the semi-rigid solutions for improving soil strength and enhancing the bearing capacity of a weak soil foundation beside the pile foundation. Binder_ including Lime or Cement, be dependent on soil types_ is mixed at site with soil using shafts of augers with mixing paddles. Soil as the basic material with microparticles as fillers, with moisture and cohesion in situ, will be incorporated with cement under the pressure of compaction or jet grouting to perform a chemical-mechanical stabilized soil having higher engineering properties than that of undisturbed soil. Cement-based mixing was used popularly in Mekong Delta for different scales of the project, from the foundation of the embankment for highways to the foundation for residential buildings. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 716–730, 2022. https://doi.org/10.1007/978-981-16-3239-6_55
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Unfortunately, pore water in soil in many regions of Mekong Delta may be contaminated and classified as acidic or basic soils; nearly half of the total area in Mekong Delta is acidic polluted soil, more than the area of saline soil or alkalinized soil [1]. As such, studying soil-cement is relevant to stabilize the soft soil for construction purposes. The frequent question is that what is the effects of those chemical factors on the geoengineering properties of soil in polluted areas, especially in case of improvement method using soilcement mixing method; and how to determine the most essential factor in the mixture. This paper aims to solve the abovementioned problem. By approaching the problem experimentally, a design of experiment using a numerical model is firstly created. The model will be a governing tool for carrying out a parametric study on the mixture of the disturbed soil and cement as a binder. For the method of soil improvement using mixing, the mixture will have essential factors of the original soil, the binder, the water, time of mixing, the method of curing and etc. [2]. Many specific problems raised such as: In what condition of the pore water does the mixture use lime as binder, or cement as binder? How the plasticity, pH, and acidity concentration affect cement stabilized soil ? Does high or low acidity affect the stabilized soil performance of the mixture? etc. This paper aims to study the undrained strength of acidic soil mixed with cement. In the mixture including factors relating to different properties, the study focuses on determining which is the most important factor in the experiment of soil-cement mixing and its contribution to the unconfined compression strength (UCS) in lab condition. For ensuring a sufficient data set in statistical viewpoint, the Taguchi’s method is selected as an approach to design the experiment, find the influenced factor(s), and compute the percentage of contribution of each factor.
2 Method 2.1 Literature Review on the Effects of pH on the Soil Strength The effects of pH levels on soil strength of contaminated soil have still not yet been understood, especially in a mixture with a binder such as cement or lime. In conventional conditions (i.e. pH = 7), for granular soil and clay with low plasticity, cement proved to be more efficient than lime [5]. Regarding the effects of pH levels on strength and compressibility, it is not clearly consistent. As for the soil strength, the UCS qu decreases to the decrease of pH levels, in which pH range 4 to 9 [6]. During the increase of acidity (i.e. pH levels decrease), the plasticity also decreases, resulting in an indirect increase of strength [2]; but in some other specific condition, the cohesion does not decrease in some polluted soil [14]. In some specific conditions, UCS increases to the increase of acidity in pore water (i.e. pH is less than 4) [5]. As for the compressibility, the stiffness (i.e. modulus of compressibility) is generally a function of the UCS, however, some results show no variation the compression index [3]. For determining the strength, the unconfined compression tests are mainly used in a laboratory. Because of relating many factors together with different levels of values, it requires a relevant method to design the experiment and obtain the output data for analysis, including to determine the main factor(s), together with the extent of contribution of each factor. The main tests both in a laboratory and in the site are to determine at what level of governing factors in mixing products will give the highest performance, i.e. the
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highest value of unconfined compression strength (UCS) of soil-cement samples creating in lab condition. High-performance soil-cement UCS in laboratory will definitely result in good quality in the field. Besides, the Taguchi method was applied for determining the main factor involved, formulating the regression equation, and identifying the most effective factor(s). Besides, the question of the problem of soft soil improvement is how to mix efficiently the low soil shear strength (i.e. Su less than 20 kPa)? How does plasticity, pH concentration affect the soil-cement stabilized soil at site? Does dry or wet mixing affect the stabilized soil performance (Kitazume and Terashi [6, 9])? etc. For different kinds of soil, Curtin et al. 1976 [7] suggested some solutions for using binder (i.e. cement and lime) as in Fig. 1.
Fig. 1. Different alternatives for selecting stabilizer (Curtin et al., 1976) [7]
As lime is more suitable to improve the alkalinized soil [6], this study focuses on using cement as a binder and stabilizer to mix with original acidic soil. Giovanni Spagnoli et al. [8] studied the variability of the undrained shear strength concerning the pH level. To quantitatively study the mechanical properties and characteristics of the soil-cement mixture (e.g. percentage of contribution of each factor to the overall performance of the mixture etc.), it is necessary to someway quantify the effects of pH on strength to assign values (i.e. levels) into the software Plaxis as input data for the virtual experiment. According to the research, undrained shear strength would increase with the decrease of pH level, as in Fig. 2. The less pH value than 7, the more
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undrained strength increases; on the other hand, the research indicated, the more pH bigger than 8 or alkaline soil, the shear strength also increases. In Fig. 2, Ic as defined in German regulation DIN 18122-2, is the consistency of the soil. Ic is defined by the equation below: IC =
LL − ω LL − PL
(1)
in which LL is the liquid limit, PL is the plastic limit, and ω is the water content. This study focuses on the effects of acidity on the soil-cement compression strength. The physical property is temporarily classified in three levels of acidity, i.e. low acidity (pH = 5.5 to 7), medium acidity (pH = 4.5 to 5.5) and high acidity (pH = 3–4.5). However, the research suggested cement should be used for stabilization of a wide range of soil, especially the soil having pH lower than 5.3 and with low plasticity index [8].
Fig. 2. Undrained shear strength changes to pH variability (by Giovanni Spagnoli et al., 2012) [8]; b) The consistency of soil-cement mixture [5].
Nakamura et al. (1980) conducted experiments over many different kinds of clayey soil at various pH and different physical states [9]. The results from the study indicated a trend of increase qu to the F number that is defined as the dried weight (tons) of cement per each m3 original soil to be mixed, F as follows: qu = 32.5F − 1.625
(2)
Wc 9−pH
when pH < 8, and F = Wc when
in which qu in MPa, and F in t/m3 , with F = pH > 8.
2.2 Model and Factors for a Soil-Cement Samples in the Model The mechanical properties of the soil-cement mixture have uncertainties that depend on the original soil, water content, binder,and many other factors such as time of mixing, mixing mass, and time of curing. Statistical analysis for in-situ tests reduces significantly the discrepancy between the lab results of UCS and in-situ strength; besides, there are contrary findings obtained by different research works about the strength [4]. For reducing the uncertainties during mixing in a real lab or later in site, and govern the lab
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tests in the future, a virtual experiment using a numerical model is suggested as in this paper. A numerical model of the unconfined compression test (UCT) is developed to manipulate the data the soil properties subjected to acidic pollution. The trend of increase or decrease would be recognized during the test. The virtual experiment with the numerical model is a good base for optimizing the performance of the mixture, and preliminarily selecting the appropriate data set of ingredients. A process of cement treatment results in a variation of the soil plasticity and the soil plasticity is influenced by the water content and the soil mineral components [7]. For a high plasticity soil, the cementitious treatment will reduce the strength of the cementstabilized soil; for modeling this trend, a reduction in strength is suggested over the data of the experimental input; for the higher plasticity, a smaller value cohesion was used in Plaxis software. Besides, a higher amount of binder, particularly cement, the mixture will increase the plastic limit ωp and reduce the liquid limit ωL , resulting in lower plasticity of the mixture; this may imply that the percentage of the binder should not so high, roughly 6% to less than 15%. Four factors for the experimental model are A for soil plasticity (this is a typical physical property), B for soil strength being a mechanical property, C for the pH as a chemical property, and D stands for the amount of cement (stabilizer binder). The numerical model using Plaxis 2D axisymmetry one (cf Fig. 3) and Mohr-Coulomb analysis model will be used for virtually experimental design with the four abovementioned factors. The Mohr-Coulomb model is rather practicable and suitable to soil with two stages of response, elastic and plastic stage. Another reason for using the model is due to its simplicity with fewer parameters than other advanced models; it has obviously proved to be suitable for the early stage of bearing the pressure. A sequence of phases is set up, in which displacement increments are prescribed. The prescribed displacement is about 1% per step to ensure at least 100 steps are computed for a 1 mm of the vertical displacement.
Fig. 3. a) A numerical model for the unconfined compression test; b) normal stress and Plaxis outcome.
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As for the material aspect of analysis, this study assumes that the results of UCT will be valid to all the sizes of samples which have a wide range of the ratio diameter to height, d/h. And the model was calibrated with samples using diameter to height being 50/100. This is to make use of the results from real tests for the process of calibration. In the numerical model, for taking the soil plasticity and the effects of the pH on the soil strength, the cohesion for each combined case of plasticity and a specific pH level will be tentatively replaced by a slight modification in the cohesion and the internal friction angle (by literature review in the previous paragraph). For the condition of the virtual experiment using a numerical model, it is assumed that the effects of the time of curing, the temperature, and the massive characteristics of the mixture (i.e. mixing at site is a huge mass; mixing in a laboratory is a small mass), etc. is replaced by a correction factor, concerning the increase in strength when the time of mixing takes more than 10 min [9]. Two main uncontrollable factors are the time of mixing, the scale of mixing mass (time of curing is temporarily disregarded). They are named E and F, respectively; their levels are as follows: level = 1 for the time less than 10 min of mixing, = 2 for the time longer than 10 min of mixing, yielding a higher strength. The strength is assumed to be the undrained mode of analysis for level 1 if small mixing mass is conducted, and drained mode of analysis for level 2, or a bigger amount of mixing mass. For the strength of the sample concerning mass mixing (i.e. level 2), the data would be reasonably selected to be 20 to 70% of that in lab mixing [10]. This is to reduce the possible errors or discrepancies due to the estimation of the data, including cohesion and the E50 (also modulus Eoed ) stiffness of soil-cement samples in the model. Input data are the results of both DSS and UCT lab tests over soil-cement mixing samples. The properties of the original soil are given in Table 1. Table 1. Some technical properties of the original soil [12] Layer
Description
1
N-SPT
Unit weight (kN/m3 )
Index of plasticity Ip
Index of liquidity
C (kPa)
Organic grey 0–3 clay, liquid
14.78
28.7
1.68
6,9
2
Semi-stiff clay
18–23
19.67
18.6
0.06
40.6
3
Yellow Clay, 14–18 stiff
19.29
17.4
0.2
33.5
4
Coarse sand, 27–36 medium dense
19.70
–
–
3.3
32
5
Brownish grey clay, semi-stiff to stiff
19.87
19.6
0.02
44.6
15,5
19–37
Friction angle (o )
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T. H. Duong et al.
2.3 Strategy for Virtual Experiment The data for this study is obtained quantitatively by using real results of tests on soilcement samples [12]. They are assigned to the model after being multiplied by a modification factor. As such, this study quantifies the effects of the pH levels on the soil properties by applying a modification factor. A tentatively relevant modification which is less than 20% will be based on the literature review [7–10] concerning the trend of increase or decrease the soil strength, plasticity, and stiffness. For the compressibility, the cement-treated soil stiffness Eoed is taken from E50 value of the real UCTs multiplied by a factor; for the shear strength, the cohesion and friction angle are taken from real direct shear tests with factor. All these manipulated data would be used to collect data for determining the role of the individual factor and percentage of contribution. For not doing too many unnecessary experiments before carrying out real lab tests, it is necessary to design experiments with a numerical model. Taguchi method is a statistically based method in which orthogonal arrays are used to remove repeated experiments. About the data collection for this study, four factors will have commonly three levels (i.e. low, medium, and high level). For a statistical combination of all the possible cases, the total number is 34 = 81 cases. By applying the Taguchi method [11] of the design of the experiment, the total number will reduce to 9 (i.e. orthogonal array L9 (34 )). A strategic plan for solving 9 combinations as statistically representative trials for all the possible cases are described in Table 2. Table 2. Orthogonal array L9(3ˆ4) Noises Experiment A B C D E no.
F
1
1
1
1
1
1
1
2
1
2
2
2
1
2
3
1
3
3
3
1
1
4
2
1
2
3
1
2
5
2
2
3
1
2
1
6
2
3
1
2
2
2
7
3
1
3
2
2
1
8
3
2
1
3
2
2
9
3
3
2
1
1
2
Studying the Strength of an Acidic Soil-Cement Mixing in Laboratory
723
In the Table 2, the values of factor level, or 1, 2,3 are taken in Table 3 as below: Table 3. Level and its solution of modification for the input data Factor Description Level Level 1
Solution Level 2
Solution Level 3
A
Plasticity Index of soil
Low Increase Medium Neutral (Ip < 17) Su (17 < Ip < 30)
High (Ip > 30)
B
strength Su of original soil
15–20% (250 kg binder per 1 m3 soil to be improved). From this results, cohesion and the internal friction angle are obtained; as for the stiffness, Eoed is taken being E50 from the unconfined compression tests carried over such three kinds of binder amount (c.f. Fig. 4a, left). Figure 4b show the results of the numerical model in which the effective stress is the compression strength qu .
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T. H. Duong et al.
Fig. 4. a) UC for obtaining Eoed and DST for selecting shear strength parameters; b) A typical result of the normal stress and the stress-strain relationship in the UCS test (binder 6–8%, 7 days of curing) [12].
3 Results Data of nine combinations of experiments (first column of Table 4) are obtained, assigned, modified and assigned into the Plaxis model. The soil properties are prescribed in Table 1. The model was calibrated by real data, concerning the specific percentage of stabilizer. These nine sets of factors as detailed in Table 3, would be a representative set for 34 possibly combined cases [11]. Results from each line of combined factors are filled into Table 4, columns 6 to 9 were the last two columns, 8 and 9 will be half values
Studying the Strength of an Acidic Soil-Cement Mixing in Laboratory
725
of columns 6 and 7, respectively. The reason for this is explained by the assumption on the mass mixing mentioned in Sect. 2. Table 4. Orthogonal array L9(3ˆ4) for UCS of soil-cement mixture associated with different conditions E F
1
2
1
2
1
1
2
2
(*)
Experiment no.
A
B
C
D
Values of qu
Average
S/N (ηk )
1
1
1
1
1
303.21
301.10
242.4
245
272.93
48.5710
2
1
2
2
2
1620.00
793.00
1296.0
640
1087.25
59.0239
3
1
3
3
3
5360.00
1190.00
4288.0
960
2949.50
63.2780
4
2
1
2
3
4310.00
989.00
3448.0
800
2386.75
61.6704
5
2
2
3
1
245.93
394.00
200.0
320
289.98
48.4057
6
2
3
1
2
793.12
793.12
635.0
640
715.31
56.9353
7
3
1
3
2
792.45
792.51
620.0
610
703.74
56.7400
8
3
2
1
3
989.41
989.13
792.0
800
892.63
58.8603
9
3
3
2
1
389.58
387.44
311.0
310
349.51
50.7064
* Mass mixing (level 2 of the factor F) is assumed to reduce 20% out of q that is obtained by the u
numerical model over the unity level of the factor F, i.e. column 6 and 7 of Table 3)
For each factor, the total signal at a level m, values ηk in Table 4 is summed up and the average S/N values are described in Table 5. Table 5. Control factors and the order of importance Level
Factor A
Factor B Avg S/N
1
170.87
56.95 166.98
55.66 164.37
54.79 147.68
49.23
2
167.01
55.67 166.29
55.43 171.40
57.13 172.70
57.56
3
166.30
55.43 170.92
56.97 168.42
56.14 183.80
61.27
1.53
Avg S/N
1.54
Sum SC
Factor D
Sum SA
= Max – min(S/N) =
Sum SB
Factor C Avg S/N
2.34
Sum Avg SD, level ) S/N
12.04
With the biggest value of ‘Delta’ in Table 4, factor D (% binder of cement per 1 m3 soil to be improved) was ranked the first order of importance, then comes the factor C (pH level), and the two factor A (plasticity index of soil) and factor B (undrained
726
T. H. Duong et al.
shear strength of the original soil Su ). The output of Taguchi analyzing tool by Minitab software V.17 yields the same result, as follows: Taguchi Analysis: C5, C6, C7, C8 Versus Ip, Su, pH, %Binder
Response Table for Signal to Noise Ratios Larger is better Level 1 2 3 Delta Rank
Ip 56.96 55.67 55.44 1.52 4
Su 55.66 55.43 56.97 1.54 3
pH 54.79 57.13 56.14 2.34 2
%Binder 49.23 57.57 61.27 12.04 1
Trends of variation for the factors are plotted by Minitab V.17 as below:
Fig. 5. Signal-to-Noise plot by MiniTab V. 17
This plot in Fig. 5 will be used to design an optimum set of factors for the best performance of shear strength. The highest value of the unconfined compression strength qu , some basic consideration will be as follows: a) For the least amount of binder used: This is the case when factor D has the level 1. See in the Table 3, it is easy to find out the set A3B3C2D1. b) The usual objective for the soil improvement is the weak soil with small values of S: The factor B yields the highest value of undrained strength qu at level 1, then it requires the combined set of factors will be A2B1C2D3 (the 4th line in Table 3). c) The main purpose of the study in for an acidic soil (pH < 7), pH concentration is a priority: The outcome qu will attain the highest value at the level 2 of the factor C
Studying the Strength of an Acidic Soil-Cement Mixing in Laboratory
727
(i.e. pH level is between 4.5 to 6), as such the set of factors should be A3B2C1D3 (line 8 in the Table 3); d) The low Plasticity index is expected: The factor A yields the highest value of the unconfined compression strength qu at the level 1, so the combined set should be A1B3C3D3. Based on the abovementioned consideration from criteria a) to d), to the best performance, or qu has the highest value, there are four possible following sets required to mix is as below: A3B3C2D1 A2B1C2D3 A3B2C1D3. A1B3C3D3 The factor which has appeared more frequently in all sets will be selected for the optimum combination. About factor D, there are three times of appearance in different data set; about factor C, there are two times of appearance, so the value for C is 2. It is usually to choose the weakest soil for the improvement purpose, so the factor B1 is selected; about the factor A, there are two times of appearance for the level 3, therefore A3 will be chosen. So the relevant combination in which the unconfined qu will be best obtained is the A3B1C2D3. This set of factors will result in the highest performance of the soil-cement mixture in the laboratory. As such, for obtaining the highest value of qu in the laboratory, the data set should be as follows: acidic soil with pH levels between 4.5 to 6 (C2), with high plasticity index (A3), the weak soil with the lower strength Su (B1) and the high amount of the binder (D3) should be selected. The optimum formula for the best mixture is A3B1C2D3. For determining the percentage of contribution of each factor, an analysis of variance (ANOVA) is conducted with S/N ratios in the Table 4 and 5. The results are described in Table 6 as below: Table 6. ANOVA for evaluating the percentage of contribution Factor
Level
S/N ratio
Average S/N
S/NX
SSx
%p = SSx /SSx
Rank
A
1
56.95
18.98
56.02
0.45
1.6%
4
2
55.67
18.55 56.02
0.46
1.7%
3
56.02
0.93
3.3%
2
B
C
3
55.43
18.48
1
55.66
18.55
2
55.43
18.48
3
56.97
18.99
1
54.79
18.26
2
57.13
19.04
3
56.14
18.71 (continued)
728
T. H. Duong et al. Table 6. (continued)
Factor
Level
S/N ratio
Average S/N
S/NX
SSx
%p = SSx /SSx
Rank
D
1
49.23
16.4
56.02
25.35
93,2%
1
2
57.56
19.08
3
61.27
20.42
SS = SST
≈100%
The amount of binder as stabilizer proves to keep a vital role in the UCS of stabilized clayey soil. For the soil properties, the chemical factor (i.e. pH level) contributes nearly 50% in improving the mixture, the plasticity, and the undrained strength of the original soil appears to be equally important.
4 Discussion Results from lab tests were modified and assigned to the Plaxis model as a virtual experiment. This process might be tentatively cautious. Samples at 7, 14, and 28 days of curing have a logarithm relationship, but the stiffness, i.e. the Eoed which takes the same value as E50 taken from a triaxial compression test, is not reasonable; the fact is that two values of the soil modulus are approximately chosen. A slight modification in acidic soil strength which are not more than 20% as compared to that in the conventional condition is temporarily acceptable for the pilot experiment. Direct shear tests on real samples of improved soil provide the cohesion c and internal friction angle ϕ, then they are modified for concerning the interaction with acidic concentration; the modification is just a slight modification, not more than 20% as compared to that in the conventional condition. Hence, it requires some verification by tests over acidic soil samples with various pH degree, to determine how much the percentage of modification is. In general, the shear strength increased to the increase in acidity (i.e. decrease of pH) [13]. This will be further studied. The variation of the undrained shear resistance to pH, as in Fig. 1, should be verified by carrying out lab tests on a variety of the real soil subjected to different pH levels. This study is of the acidic soil, or pH less than 7, therefore within the range, it is not clear to recognize the effects of pH on the soil strength (e.g. alkaline or basic soil, pH > 8). The better procedure is to study to a wide range of pH concentration, when the effects may be more clearly determined. Besides the cement, as a stabilizer is an external factor, three internal factors such as plasticity index, undrained soil strength Su , and the pH level are the representative factors that require supplementation of other factors such as the water content, the modulus of deformation, time of curing, etc. [14]. This issue increases significantly the complication. The binder dominates the unconfined compression strength qu as its percentage of contribution is nearly 93%. This is a remarkable difficulty in determining the order of importance of the soil properties, which contributes only a totally 7%.
Studying the Strength of an Acidic Soil-Cement Mixing in Laboratory
729
For a specific original soil that is polluted by a specific kind of acid, a specific pH level, the optimum data set of the factors chosen from the abovementioned approach would be changed suitably to meet the requirements.
5 Conclusion The study on acidic soil is conducted by a numerical model. It plays the role of a pilot test in governing the real lab test in the future and determining the importance of each factor in the data set. A Plaxis model is calibrated with data from real tests, including both the unconfined compression test and the direct shear test. Some assumption on a 20% modification on the strength is made by prior research in the literature review, and the results from the virtual experiment are processed with the Taguchi orthogonal array L9 (34 ); some uncontrollable factors such as the time of curing which is replaced by undrained/drained mode of analysis, and the scale of mixing are prescribed in the model. The results indicate the binder is the first rank of the contribution to the unconfined compression strength qu of the soil-cement products mixed in the laboratory; the second role is the pH level as a chemical property of pore water of the soil, and the final properties are the plasticity and the strength of the original soil. Other useful results are that the best UCS qu made in a laboratory would be attained with the high plasticity soil, pH levels from 4.5 to 6, and a high percentage of the binder (e.g. 12% by weight). This result is reasonable for the practical purpose and will be verified by real lab tests. To have more applicable results, some suggestions are to develop a wide range of experimental studies on the different pH levels, to obtain the appropriate modification coefficients for the numerical model. That process of the virtual experiment using a numerical model can save much time preparing for the mixing at the site, reduce remarkably the cost of testing both in the laboratory and in the field, and can make use it for many practical purposes.
References 1. Formal website from Ministry of Agriculture and Rural Development, Department of Crop Production. http://www.cuctrongtrot.gov.vn/TinTuc/Index/3611. Accessed 12 Nov 2020 2. Jacobson, J.R., Filz, G.M., Mitchell, J.K.: Factors affectings strength gain in Lime-Cement Columns and development of a laboratory testing procedure. Final Contract Report No. 57565 Federal Highway Administration FHWA/VTRC 03-CR16 (2003). 69 p. 3. Tajnin, R., Abdullah, T., Rokonuzzaman, MD.: Study on the salinity and pH and its effect on geotechnical properties of soil in south-west region of Bangladesh. Int. J. Adv. Struct. Geotech. Eng. 03(02), 138–147 (2014). ISSN 2319–5347 4. Fan, J., Wang, D., Qian, D.: Soil-cement mixture properties and design considerations for reinforced excavation. J. Rock Mech. Geotech. Eng. 10(4), 791–797 (2018) 5. Bayat, M., Asgari, M.R., Mousivand, M.: Effects of cement and lime treatment on geotechnical properties of a low plasticity clay. In: International Conference on Civil Engineering Architecture & Urban Sustainable Development 27 & 28 November 2013, Tabriz, Iran (2013). https://www.researchgate.net/publication/267097580 6. Kitazume, M.: Deep Mixing Method, First Edition. CRC Press (2017). ISBN 9781138075795
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7. Muhunthan, B., Sariosseiri, F.: Interpretation of geotechnical properties of cement treated soil. FHWA RReport No WA-RD 715.1 (2008). 155 p. 8. Spanoli, G., Rubinos, D., Stanjek, H., Steeger, T.F., Feinendegen, M., Azzam, R.: Undrained shear strength of clays as modified by pH variations. Bull. Eng. Geol. Environ. 71(1) (2015). https://doi.org/10.1007/s10064-011-0372-9 9. Kitazume, M., Nakamura, T., Terashi, M., Ohishi, K.: Laboratory tests on long-term strength of cement treated soil. In: The 3rd International Conference on Grouting and Grout Treatment, vol. 471 (2003). https://doi.org/10.1061/40663(2003)31 10. Szymon Topolinski. Unconfined Compressive Strength Properties of a Cement- Organic Soil 11. Composite. IOP Conference Series in Material Science and Engineering, vol. 471, 042018 (2019). https://doi.org/10.1088/1757-899X/471/4/042018 12. Ozcan Tan, A., Zaimoglu, S., Hinislioglu, S., Altun, S.: Taguchi approach for optimization of the bleeding on the cement-based grouts. Int. J. Tunnell. Undergr. Space Eng. 20, 167–173 (2005) 13. Phi, D.N.: Nghien cuu giai phap xu ly Nen dat yeu bang coc Xi mang dat cho cong trinh ven song o tinh Long An. Master thesis in Vietnamese, HCM City University of Technology (2016) 14. Ma, Q., Hu, X., Yan, Y.-c., Jin, K., Zhang, X.-t.: Direct shear strength properties of clay polluted by Hidrochloric Acid. Electron. J. Geotech. Eng. (22.11), 4447–4458 (2017) 15. Tong, F., Ma, Q., Hu, X.: Triaxial shear test on hydrochloric acid-contaminated clay treated by lime, crushed concrete and super absorbent polymer. Hindawi Adv. Mater. Sci. Eng. (2019). https://doi.org/10.1155/2019/3865157.Article ID 3865157, 13 pages
Effects of Test Specimens on the Shear Behavior of Mortar Joints in Hollow Concrete Block Masonry Thi Loan Bui1,2(B) 1 Faculty of Construction Engineering,
University of Transport and Communications, N°3 Cau Giay, Lang Thuong, Dong Da, Hanoi, Vietnam [email protected] 2 Research and Application Centre for Technology in Civil Engineering (RACE), University of Transport and Communications, N°3 Cau Giay, Lang Thuong, Dong Da, Hanoi, Vietnam
Abstract. The shear characteristic of mortar joint is the most important factor which decides the behavior of masonry structures subjected to lateral in-plane loading. This article focuses on the experimental study of shear behavior of mortar joint in hollow concrete brickwork. In this experimental program, two types of specimen prepared according to two different standards (RILEM and European standard) were tested in order to study the influence of some formal parameters of test specimens to the shear behavior of mortar joint in hollow concrete brickwork. The test results showed that no matter whether there are horizontal joints in test specimens or not, an important dispersion is highlighted both in shear behavior and in shear failure criterion of mortar joint in hollow concrete block masonry assembly. However, the presence of horizontal joints may cause artifacts which have effect on the redistribution of stress and therefore on the correct transmission of lateral compression to the shear mortar joint. This effect tends to impact the damage and failure mechanism and therefore the failure criteria. In consequence, the Mohr-Coulomb criterion is only verified (with a coefficient 99% regression) for the case of test specimens without horizontal mortar joints although this verification is restrictive in terms of a low number of valid tests. Keywords: Hollow concrete masonry · Shear pull-out test · Shear behavior · Mortar joint
1 Introduction Hollow concrete masonry structure has been widely used worldwide. This structure can be subjected to in-plane shear loading cause by seismic, wind or by an accidental loading (settlement or car crush). Under this loading condition, the failure mode of masonry structures could be shear type characterized by cracks across mortar joints caused by shear stress. The comprehension of the shear behavior as well as the shear failure criteria © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 731–742, 2022. https://doi.org/10.1007/978-981-16-3239-6_56
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T. L. Bui
of mortar joint in hollow concrete brickwork is therefore necessary and that has been significantly scrutinized in several studies in the literature. However, the effect of test specimens on the shear behavior of mortar joint in hollow concrete brickwork which has not been addressed before will be focused on in the present study. For the study on the shear behavior of mortar joint in brickwork, numerous test configurations are proposed in the literature (couplet or triplet) to characterize the shear behavior of mortar joints. The direct shear tests on couplets were used in many studies [1–6]. Whereas, the shear tests on triplets in accordance with standard NF EN1052-3 [7] or according to standard RILEM TC-76 LUMB5 (1991) [8] were used in [9–15]. The configurations are distinguished by the assembly loading dispositive, by the quantity of bricks, by the number and position of mortar joints in the assemblies. These numerous configurations were based on the objective to obtain the compromise, difficult to find, between a simplicity of carrying out the tests and the desire to ensure uniform distribution of normal and shear stresses in the assemblies [6]. Among all the proposed configurations, a comparative study between two types of shear tests on triplet was carried out: one established according to the RILEM recommendation [8], the other based on the study of triplets, in accordance to standard (NF EN1052-3 (2003) – [7]). The main difference between these two test set-up bases on the difference in the test assemblies: there are horizontal mortar joints in the test specimens according to the first recommendation while they are absent in the test specimens manufactured according the other one.
2 Experimental Program 2.1 Materials The bricks used in the present study are halved lengthwise of hollow concrete brick; class B40 (the characteristic compressive strength is 4 MPa - according to the Eurocode 6 [16]) whose dimension is 500 × 200 × 75 mm3 . The dimension of one brick unit is therefore 250 × 200 × 75 mm3 . The uniaxial compressive test was performed at laboratory (according to the European standard EN 772-1:2000 [17]) and shows that the average compressive strength of these bricks is 6.5 MPa. The mortar used in our study is a Portland cement-based mortar (CEM I 52.5), of which the formulation is shown in Table 1. Table 1. Mortar formulation Type Mortar based on CEM I 52,5
Proportions by mass Cement
Sand 0/4
Water
1
3
0,5
The uniaxial compressive tests and flexural tests were performed at our laboratory with nine prismatic specimens measuring 40 × 40 × 160 mm3 manufactured according to the European standard (EN 1015-11) [18]. These tests were done on the 3 specimens after 31curing days with the result of average compressive strength equals to 48 MPa.
Effects of Test Specimens on the Shear Behavior of Mortar Joints
733
2.2 Shear Pull-Out Test of Masonry 2.2.1 Specimens In order to study the influence of test specimens to the shear behavior of mortar joint in hollow concrete brick masonry, this shear pull-out test is performed on specimens constructed according to both the European standard (NF EN1052-3(2003)) [7] and the RILEM recommendation (RILEM (1991d)) [8]. Both specimens are the type of triplet. The main difference between these two types of assembly is that there are horizontal mortar joints in the test specimens according to the RILEM recommendation while they are absent in the assemblies manufactured according to European standard. The average thickness of both horizontal and vertical mortar joints is 10 mm. The detail and the dimension of both specimens are represented in Fig. 1.
(a)
(b)
Fig. 1. Specimens of shear pull-out test: (a) – according to RILEM recommendation and (b) – according to European standard.
Rigid steel I shape Hydraulic actuator associated with a load cell
Hydraulic actuator associated with a load cell
Fig. 2. Shear pull-out test set-up
734
T. L. Bui
2.2.2 Test Set-Up The test set-up of these shear pull-out tests is presented in the Fig. 2. This test setup permits to apply simultaneously both lateral compressive stress and shear stress on the mortar joints. The device used to apply the lateral compressive stress consists of a hydraulic actuator associated with a loadcell, supported on a rigid frame. In addition, the rigid IPE profiles are positioned on either side of the assemblies. The vertical shear load is provided by a very rigid loading frame on which a hydraulic actuator with a capacity of 50 tones is supported, associated with a vertical loadcell. This vertical load is applied and distributed uniformly on the upper face of the central brick of the triplet using a very rigid steel I shaped beam. The instrumentations adopted aim to measure the relative displacement between two adjacent bricks during loading process: four LVDT sensors ±10 mm (referenced C1 to C4) are symmetrically positioned on both faces of the test specimens (two of them are glued in the front face and the two others are on the behind face), Fig. 3.
Fig. 3. Position of LVDT sensors on two faces (front face and behind face)
Regarding the testing procedure, first the lateral compressive load is applied up to the desired value, then this load is identically kept and the vertical shear load is gradually applied up to the failure by displacement control with the speed of 0.05 mm/min. A total of 32 specimens (17 specimens constructed according to the RILEM recommendation and 15 specimens constructed according to the European standard) were tested with different lateral compressive stress varied between 0 and 30% compressive strength of bricks (correspond to 0 and 1.2 MPa) in order to avoid the influence of bricks’ significant damage on the results. 2.2.3 Results and Discussions a) Failure modes The failure modes observed for both two cases of assembly in these shear pull-out tests are generally divided into 4 following modes, Fig. 5: – Mode a, fracture plane of the brick-mortar interfaces. – Mode b, fracture plane of brick-mortar interface accompanied by cracking of mortar joint;
Effects of Test Specimens on the Shear Behavior of Mortar Joints
735
– Mode c, fracture plane through the bricks. – Mode d, local crushing of the upper central brick where the vertical load is applied.
Mode a
Mode b
Mode c
Mode d
Mode a
Mode b
Mode c
Mode d
Fig. 4. Failure modes of hollow brick masonry assemblies in shear pull-out test
Regarding the relationship between the failure modes and the lateral compressive stress value (as marked in Table 2 and 3), it appears that for both two cases of specimens, the failure mode tends to be accompanied the mode c and (or) mode d (collapse of bricks) when the lateral compressive stress is upgrade. This result can be explained by the fact that in increasing the lateral compressive stress, the bricks have to be subjected by higher compressive and shear stress while the shear strength of mortar joint is upgrade. In consequence, the bricks are collapsed before the damage of mortar joints. Additionally, Table 2. Failure modes and shear strengths obtained in the shear tests according to the RILEM recommendation (specimens with horizontal mortar joints).
Table 3. Failure modes and shear strengths obtained in the shear tests according to European standard (specimens without horizontal mortar joints).
736 T. L. Bui
Effects of Test Specimens on the Shear Behavior of Mortar Joints
737
in comparison the results between two cases of assembly; it is worthy noted that the two latter failure modes (mode c and mode d) tend to appear at lower lateral compression levels (about 20% of compressive strength of masonry) for the specimens with horizontal mortar joints than for the specimens without horizontal mortar joints (about 35% of compressive strength of masonry). Indeed, it is likely that this multiplicity of failure modes in the case of assemblies with horizontal joints is also linked to the difficulties of filling these joints during the implementation of assemblies. Thus, the lack of filling of certain joints can cause artifacts in the distribution of the stresses having repercussions on the failure mode. b) Shear behavior of mortar joint The shear behavior of mortar joint is represented by the evolution of shear stress versus relative displacement recorded by the LVDT sensors. The shear stress along the shear mortar joint is calculated by the following equations: – For the case specimens with horizontal mortar joints: τ1 =
Fv1 2 × lf 1 × t
(1)
– For the case specimens without horizontal mortar joints: τ2 =
Fv2 2 × lf 2 × t
(2)
Where: Fv1 , Fv2 are vertical shear load recorded during the test for two cases with and without horizontal mortar joint respectively (see Fig. 1). lf 1 , lf 2 are the length of vertical shear mortar joint corresponding to the case with and without horizontal mortar joint respectively (see Fig. 1). t, is the thickness of specimens, equal to 75 mm for both cases. Figure 5 shows this evolution for some different lateral compression levels for both types of specimen. It is worthy of note that all the specimens whose failure modes contain mode (c), and mode (d) could not be considered because they are worthless in this shear behavior study of mortar joints. It is highlighted from these figures that there is dissymmetrical behavior of two vertical mortar joints. In addition, the shear failure of these hollow concrete bricks assemblies is not sudden but is affected by “jerks” which result in several peaks in the evolution curves. The dissymmetrical behavior and the “jerks” can correspond to a nonsimultaneous failure of the two vertical joints and to the different failure mechanisms described above in Fig. 4. Additionally, in considering the evolution of one vertical joint, three phases can be identified: the first elastic phase which is generally characterized by an extremely high rigidity, followed by a hardening phase until reaching the maximum stress. Then, if a
738
T. L. Bui σh = 0.3MPa
0.80
σh=0.2(MPa)
Shear stress (MPa)
0.60
Shear stress (MPa)
Left joint Right joint
0.70
0.50 0.40 0.30 0.20 0.10 0.00 0.10
-0.40 -0.90 -1.40 relative displacement (mm)
-1.90
1.00 0.90 0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.10 0.00
Left right joint
0.00
-0.50 -1.00 Relative displacement (mm)
-1.50
Fig. 5. Shear stress versus relative displacement for both two types of assembly
lateral compressive stress is applied, a softening post-peak phase appears, characterized by a residual plateau. In this study, this residual level is more or less marked or identifiable depending on the test specimens: this residual plateau is associated with a sliding mechanism between two adjacent bricks which was not always happened in this study (often observed in the case of test specimens without horizontal joints but only observed at low pre-compression levels for test specimens with horizontal joints). The complexity of damage mechanisms at higher lateral compressive stress results in more disordered post-peak phases. c) Evolution of shear strength in function of lateral compression stress The shear strength and the residual shear strength for different level of lateral compression stress (correspond to the maximum value and the residual value calculated from Eq. (1) and (2)) are represented in Table 2 (with horizontal joints) and Table 3 (without horizontal joints). It is useful to recall that only the results correspond to failure types of mode (a) and or mode (b) should be considered in this analysis. From these results, the evolution of shear strength (τmax ) in function of precompressive stress (σh ) was identified and presented in Fig. 6 (for the test specimens with horizontal mortar joints) and Fig. 7 (for the test specimens without horizontal mortar joints). Despite the high dispersion in the results, the failure curves established in the plane (τmax , σh ) show an increasing trend which means that the shear strength of mortar joint increases with increasing of the lateral compression stress and this result is consistent with the scientific literature. However, most authors in the literature noted that the mortar joint obeyed the Mohr-Coulomb failure criterion, establishing a linear relationship between the shear strength and the lateral compression stress. The results of the present study for both cases, admittedly very dispersed, do not tend to support this conclusion. Indeed, Fig. 6 and Fig. 7 plot the linear trend curves with the relative small of linear regression coefficient (0.53 and 0.56 respectively for specimens with and without horizontal joints). However, a refined analysis (the same analysis as previously) was performed in considering only the results correspond to the presence of residual shear strength which
Effects of Test Specimens on the Shear Behavior of Mortar Joints
739
1.40
Shear strength (MPa)
1.20
y = 0.76x + 0.46 R² = 0.53
1.00 0.80 0.60 0.40 0.20 0.00 0.00
0.20 0.40 0.60 0.80 Precompression stress (MPa)
1.00
Fig. 6. Evolution of shear stress in function of pre-compression stress (for the test specimens with horizontal mortar joints)
1.60
Shear strength (MPa)
1.40 1.20
y = 0.87x + 0.46 R² = 0.56
1.00 0.80 0.60 0.40 0.20 0.00 0.00
0.20 0.40 0.60 0.80 Precompression stress (MPa)
1.00
Fig. 7. Evolution of shear stress in function of pre-compression stress (for the test specimens without horizontal mortar joints)
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reflects a correct transmission of the lateral compressive stress on the shear joint studied. This refined analysis, admittedly restrictive in terms of test specimens (only in some test specimens without horizontal mortar joints), leads to a verification of the Mohr-Coulomb criterion with a coefficient 99% regression (Fig. 8). The analysis of the residual shear strength also leads to a linear evolution of the residual strength as a function of the lateral compression stress. These linear relations reflect the Mohr-Coulomb criterion by the expressions: τmax = σh . tan(φmax ) + cmax
(3)
τr = σh . tan(φ r ) + cr
(4)
and
Where: τmax , τr : ultimate and residual shear strength of mortar joint φmax , φr : ultimate and residual internal friction angle cmax , cr : ultimate and residual cohesion
1.60
Shear strength (MPa)
1.40
Ultimate shear strength Residual shear strength
y = 001x + 000 R² = 001
1.20 1.00 0.80 0.60
y = 001x + 000 R² = 001
0.40 0.20 0.00 0.00
0.20 0.40 0.60 0.80 Precompression stress (MPa)
1.00
Fig. 8. Evolution of ultimate shear strength and residual shear strength of mortar joint for the specimens without horizontal mortar joints.
The obtained values characterize the Mohr-Coulomb criterion of these mortar joints are represented in Table 4. The obtained values of tanφ (both ultimate and residual) seem to be acceptable as they range within the usual interval of variation found in the literature (this range varies
Effects of Test Specimens on the Shear Behavior of Mortar Joints
741
Table 4. Shear characteristics of mortar joints Characteristics
Ultimate Residual
Cohesion (c, MPa) 0.34
0.13
Tanφ
1.05
1.15
between 0.7 and 1.2 – [2]). Furthermore, it appears that the residual cohesion in this case is non-zero and it can be explained by the penetration of mortar in the holes, which avoids the separation of bricks. In addition, the residual cohesion value equal to about 38% the ultimate cohesion which is in line with the results found in the literature ([3]).
3 Conclusions and Perspectives This experimental study based on shear pull-out tests on triple with two types of assemblies according two different standards (RILEM and European) helps to point out several findings related to the effects of test specimens as follows: No matter whether the test specimens contain horizontal mortar joints or not, an important dispersion is highlighted both in shear behavior and in shear failure criterion (based on the evolution of shear strength in function of lateral compression stress) of mortar joint in hollow concrete block masonry assembly. The main effect of test specimens lays on that the presence of horizontal joints in the test specimens may cause artifacts which have influence on the redistribution of stress and therefore on the correct transmission of pre-compression (lateral compression) stress to the shear mortar joint. Consequently, the damage and failure mechanism and therefore the failure criteria might be impacted. Indeed, the refined analysis leads to a verification of the Mohr-Coulomb criterion with a coefficient 99% regression (with the shear characteristics are in range within the usual interval of variation found in the literature) only for the case of specimens without horizontal mortar joints. However, this verification is restrictive in terms of a low number of valid tests. This restriction proposes a continuous study in the future with numerous tests specimens without horizontal mortar joints (use the same test set-up according to the European standard) to verify the shear failure criterion of mortar joint for this type of masonry. Acknowledgements. This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 107.01-2018.19.
References 1. Pluijm, R.V.D.: Shear behavior of bed joints. In: Hamid, A.A., Harris, H.G. (eds.) North American Masonry Conference, Drexel University, Philadelphia, Pennsylvania, USA, pp. 25– 136 (1993) 2. Lourenco, P.B., Barros, J.O., Oliveira, J.T.: Shear testing of stack bonded masonry. Constr. Build. Mater. 18, 125–132 (2004)
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3. Abdou, L., Ami Saada, R., Meftah, F., Mebarki, A.: Experimental investigations of the jointmortar behaviour. Mech. Res. Commun. 33, 370–384 (2006) 4. Fouchal, F.: Contribution à la modélisation numérique des interfaces dans les structures maçonnées. PhD thesis, Université de Reims Champagne-Ardenne (2006) 5. Luccioni, B., Rougier, V.C.: Shear behaviour of brick–mortar interface in CFRP retrofitted or repaired masonry. Int. J. Mech. Sci. 52, 602–611 (2010) 6. Popal, R.: A new shear test method for mortar bed joints. Master thesis, University of Calgary, Calgary (2013) 7. EN 1052-3:2002 Methods of test for masonry - Part 3: Determination of initial shear strength (2002) 8. RILEM TC-76 LUMB5: Short term shear test for the interface between the masonry unit and mortar or moisture insulation interlayer (1991) 9. Gabor, A.: Contribution à la caractérisation et à la modélisation des maçonneries nonrenforcées et renforcées par matériaux composites. PhD thesis, University of Lyon (2002) 10. Mohammed, A.G.: Experimental comparison of brickwork behavior at prototype and model scales. PhD thesis (2006) 11. Zimmermann, T., Strauss, A.: Variation of shear strength of masonry with different mortar properties. In: Eleventh NAMC Minneapolis, MN, USA (2011) 12. Alecci, V., Fagone, M., Rotunno, T., De Stefano, M.: Shear strength of brick masonry walls assembled with different types of mortar. Constr. Build. Mater. 40, 1038–1045 (2013) 13. Bahaaddini, M.: Effect of boundary condition on the shear behaviour of rock joints in the direct shear test. Rock Mech. Rock Eng. 50(5), 1141–1155 (2017). https://doi.org/10.1007/ s00603-016-1157-z 14. Andreotti, G., Graziotti, F., Magenes, G.: Expansion of mortar joints in direct shear tests of masonry samples: implications on shear strength and experimental characterization of dilatancy. Mater. Struct. 52(4), 1–16 (2019). https://doi.org/10.1617/s11527-019-1366-5 15. Lan, G., Wang, Y., Xin, L., Liu, Y.: Shear test method analysis of earth block masonry mortar joints. Constr. Build. Mater. 264, 119997 (2020) 16. Eurocode 6: Design of masonry structures - part 1-1: general rules for reinforced and unreinforced masonry structures (2005) 17. EN 772-1:2000: Methods of test for masonry units. Determination of compressive strength (2000) 18. EN 1015-11: Methods of test for mortar for masonry - part 11: determination of flexural and compressive strength of hardened mortar (2007)
Nonlinear Buckling and Postbuckling of ES-FG Porous Cylindrical Shells Under External Pressure Le Kha Hoa1(B) , Pham Van Hoan1 , Bui Thi Thu Hoai2 , and Do Quang Chan3 1 Military Academy of Logistics, Hanoi, Vietnam
[email protected]
2 Institute of Mechanics, Hanoi, Vietnam 3 Faculty of Mechanical Engineering and Mechatronics, PHENIKAA University,
Hanoi 12116, Vietnam [email protected]
Abstract. By the analytical approach, this paper investigated the nonlinear buckling and post-buckling behavior of eccentrically stiffened functionally graded (ESFG) porous cylinder subjected to external pressure. The shells are reinforced by eccentrically stiffeners attached to the inside. The material properties of cylinders and stiffeners are assumed to be continuously graded in the thickness direction. Based on the smeared stiffeners technique and on the Donnell shell theory with von Kármán geometrical nonlinearity, Fundamental relations and equilibrium equations are derived. Using three-terms solution and Galerkin’s method, the expression used to determine the critical load and post-buckling are given. To validate the proposed method, the comparisons are made with available results and show good agreements. The effects of geometric parameter, porosity parameters, the thickness of the porous core, stiffeners, foundation and material parameters are investigated. Keywords: Porous cylindrical shells · Stiffener · Post-buckling · Analytical modeling
1 Introduction Cylinders are used as common structural load-bearing components in modern engineering. The buckling and post-buckling behavior are one of the important mechanical characteristics of composite cylinders and have attracted many researchers. The stability analysis of isotropic and FGM cylinders under external pressure is presented in references [1–8]. Recently, porous materials are an important genre of lightweight materials with excellent energy-absorbing capability. Thus, stability problem of these structures is an attractive topic. There is some research on porous cylinders. With analytical method, Nam et al. [9] investigated the nonlinear buckling and post-buckling of FG-porous cylinder under torsional load in thermal environment. Utilizing the Galerkin method and the © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 743–754, 2022. https://doi.org/10.1007/978-981-16-3239-6_57
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Runge–Kutta method, Foroutan et al. [10] presented the nonlinear static and dynamic of imperfect functionally graded porous cylinders under hygrothermal loading. With the same method, the nonlinear static and dynamic thermal buckling of imperfect multilayer FG cylinders with an FG porous core are shown by Ahmadi et al. [11]. They also studied the nonlinear static and dynamic buckling of imperfect FGP cylinders subjected to axial compression resting on nonlinear elastic foundation in reference [12]. Based on the FSDT and using the Rayleigh - Ritz method, Shahgholian et al. [13, 14] presented buckling of FGP nanocomposite cylinders reinforced by graphene platelets under uniform external lateral pressure and axially compressive load. A review of the literature shows that few studies have been done on buckling and post-buckling of FG-porous cylinders under external pressure, especially there is no work addressed on buckling and post-buckling the stiffened FGP cylinders. This is the research objective of this paper. The sandwich cylindrical shell is composed of FG porous core and FG face sheets reinforced by FG stiffeners subjected to external pressure is considered. The material properties of the cylinder and stiffeners are assumed to be continuously graded in the thickness direction according to a simple power-law distribution. The core layer is made of a metal foam characterized by a porosity coefficient which influences the physical properties of the shells in the form of the simple cosine function in the shell’s thickness direction. The material continuity of face sheets-core and face sheets-stiffeners is guaranteed. The Donnell shell theory and the improved Lekhnitskii’s smeared stiffeners technique in conjunction with the Galerkin method are applied to solve the nonlinear problem.
2 ES-FG Porous Cylindrical Shells We consider an eccentrically stiffened FG-porous cylinder subjected to external pressure as shown in Fig. 1. The face sheets are thin, while the metal foam core is relatively thick. The thickness of each FGM face sheets is hFG /2, the thickness of the core layer (metal foam) is hcore (h = hcore + hFG ). Young’s moduli of shell and inside FGM stiffener are expressed by ⎧ k ⎪ 2z+hFG +hcore ⎪ E + − E −(hFG + hcore )/2 ≤ z ≤ −hcore /2 (E ) ⎪ c m c hFG ⎨ Eshell =
Em [1 − e0 cos(π z/hcore )] −hcore /2 ≤ z ≤ hcore /2 ⎪ k ⎪ ⎪ −2z+h +h core FG ⎩ Ec + (Em − Ec ) hcore /2 ≤ z ≤ (hFG + hcore )/2 hFG
(1)
2z − h k2 , h/2 ≤ z ≤ h/2 + hs , k2 ≥ 0, Es = Ec + (Em − Ec ) 2hs 2z − h k3 Er = Ec + (Em − Ec ) , h/2 ≤ z ≤ h/2 + hr , k3 ≥ 0, 2hr h = hFG + hcore , k ≥ 0, N ≥ 0 where e0 is the porosity coefficient (0 ≤ e0 < 1),
(2)
Nonlinear Buckling and Postbuckling of ES-FG Porous Cylindrical Shells
745
y x z h bs zs
ds hs L
hr
br
dr
zr
h
Fig. 1. Stiffened FG-porous cylinder
hFG / 2, hcore are the thickness of each FGM layer and the porous core layer, hs , hr -the thickness of the stringer and the ring respectively, k, k 2 and k 3 are the volume fractions indexes of shell, stringer and ring, respectively. According to [9], the Poisson’s ratios are assumed to be constant νsh = νs = νr = ν = const.
3 Fundamental Equations and Solution of the Problem Based on the Donnell shell theory with von Karman geometrical nonlinearity and smeared stiffeners technique, the governing equations are given [9] α11 w,xxxx + α12 w,xxyy + α13 w,yyyy + α14 ϕ,xxxx + α15 ϕ,xxyy + α16 ϕ,yyyy
+ ϕ,xx /R + ϕ,xx w,yy + ϕ,yy w,xx − 2ϕ,xy w,xy + K2 w,xx + w,yy − K1 w + q = 0, (3) 2 β11 ϕ,xxxx + β12 ϕ,xxyy + β13 ϕ,yyyy + β14 w,xxxx + β15 w,xxyy + β16 w,yyyy − w,xy
+ w,xx w,yy + w,xx /R = 0
(4)
where the coefficients αij and βij can be found in Appendix. K 1 , K 2 are Pasternak’s foundation parameters and q is external pressure load. Assume that the cylinder is simply supported at the two edges x = 0 and x = L. The deflection of pressure shell is presented [5, 6] w = f0 + f1 sin αx. sin βy + f2 sin2 αx
(5)
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where α = mπ/L, β = n/R, and in which m, n are the half waves numbers and waves numbers along along x-axis and y-axis, respectively. Substituting Eq. (5) into Eq. (4) yields β11 ϕ,xxxx + β12 ϕ,xxyy + β13 ϕ,yyyy = B01 cos 2αx + B02 cos 2βy + B03 sin αx. sin βy + B04 sin 3αx sin βy where
2 1 1 B01 = 8β14 α 4 − α 2 f2 + α 2 β 2 f12 , B02 = α 2 β 2 f12 , R 2 2
α2 4 2 2 4 2 2 f1 + f1 f2 α β , B04 = f1 f2 α 2 β 2 . B03 = − β14 α + β15 α β + β16 β − R The solution of the equation is 1 ϕ = B1 cos 2αx + B2 cos 2βy + B3 sin αx sin βy + B4 sin 3αx sin βy − σ0y hx2 (6) 2 where B1 = a1 f2 + a2 f12 , B2 = a3 f12 , B3 = a4 f1 f2 + a5 f1 , B4 = a6 f1 f2
(7)
in which coefficients ai are given in references [6]. Introducing w and φ into the Eq. (3), then applying Galerkin’s method, lead to 2σ0y h − K1 (f2 + 2f0 ) + 2q = 0 R
σ0y hβ 2 − 2D01 + 2D04 f22 + 2D05 f2 + K2 α 2 + β 2 + K1 −
f12 =
2D03
D06 f2 + D07 f12 + D08 f12 f2 − 8
σ0y h − 8K2 α 2 f2 − 6K1 f2 − 8K1 f0 + 8q = 0 R
(8) (9) (10)
where D0i can be found in reference [6]. The circumferential closed condition is 2π RL 0
∗ v,y dxdy = 0 ⇒ −8C11 σ0y h +
4 (2f0 + f2 ) − β 2 f12 = 0 R
(11)
0
For eliminating circumferential stress σ0y , using Eq. (8) yields σ0y =
R 2q − K1 (f2 + 2f0 ) 2h
(12)
Substituting Eqs. (9) and (11) into Eq. (11) leads to f0 = L01 q + L02 + L03 f2 + L04 f22
(13)
Nonlinear Buckling and Postbuckling of ES-FG Porous Cylindrical Shells
747
Substituting Eq. (13) into Eqs. (11) and (9), we get
R 1 R f2 + L04 f22 σ0y = (1 − K1 L01 )q − K1 L02 + L03 + h h 2
(14)
f12 = L11 q + L12 + L13 f2 + L14 f22 where 1
L0 = − ∗ .RK + 8C11 1 L02 = L0
8 R
+
Rβ 4 2D03 K1
, L01 = −L0
β2 2D01 + K2 α 2 + β 2 + K1 , 2D03
L03 =
(15)
∗ .RD + Rβ 4 16C11 03 2D03
β 2 L0 D05 1 − , D03 2
L04 = L0
,
β 2 D04 , D03
β 2 RK1 L02 + 2D01 + K2 α 2 + β 2 + K1 Rβ 2 L11 = , (1 − K1 L01 ), L12 = − 2D03 2D03
1 1 1 2 2 L13 = − β RK1 L03 + + 2D05 , L14 = − β RK1 L04 + 2D04 . 2D03 2 2D03 Substituting expressions (13), (14) and (15) into Eq. (10)
⎡ ⎤ D07 L12 + D06 + D07 L13 + L12 D08 − 8K2 α 2 − 2K1 f2 1 ⎦ ×⎣ q=− (D07 L11 + D08 L11 f2 ) +(D L + L D )f 2 + L D f 3 07 14
13 08 2
14 08 2
(16) Expression (16) is used to determine the buckling loads and to analyze the postbuckling response. From Eq. (16), let f2 → 0, we have the expression of buckling external pressure
β 2 RK1 L02 + 2D01 + K2 α 2 + β 2 + K1 L12 qupper = − (17) = L11 Rβ 2 (1 − K1 L01 ) From Eq. (5), the maximal deflection of the shells is Wmax = f0 + f1 + f2
(18)
Introducting Eqs. (13) and (15) into Eq. (18), leads to 1/2 Wmax = L01 q + L02 + (L03 + 1)f2 + L04 f22 + L11 q + L12 + L13 f2 + L14 f22 (19) Combining Eq. (16) with Eq. (19), the effects of the geometric parameter, porosity parameters, the thickness of the porous core, stiffeners, foundation, and material parameters on the post-buckling load - maximal deflection curves of cylinder can be analyzed.
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4 Numerical Results 4.1 Comparisons Table 1 compares the buckling loads of an isotropic cylinder with Reddy and Starnes [1]; Shen [3]; Baruch and Singer [4]. This comparison shows that the research results are reliable. Table 1. Comparisons of buckling load q (Psi) for isotropic cylindrical shell under external pressure Shell
Reddy and Starnes [1]
Shen [3]
Baruch and Singer [4]
Present Equation (20)
Un-stiffened
93.5
100.7 (1,4)*
102
103.327 (1,4)
Stringer stiffened
94.7
102.2 (1,4)
103
104.494 (1,4)
Ring stiffened
357.5
368.3 (1,3)
370
379.694 (1,3)
Orthogonal stiffened
365
374.1 (1,3)
377
387.192 (1,3)
* buckling modes (m, n)
In the following sections, a eccentrically stiffened FG-porous cylinder is considered. The FGM of face sheets and stiffeners are mixture of Zirconia and Ti-6Al-4V. The core layer made of metal foam. The parameters are given as follows: h = 0.004 m, L/R = 2, R/h = 80, hcore /hFG = 3, e0 = 0.4, k st = k = 1, hs = 0.005 m, bs = 0.005 m, ns = 30, hr = 0.005 m, br = 0.005 m, nr = 30, K 1 = 2 × 107 N/m3 , K 2 = 6 × 104 N/m, ΔT = 0K. 4.2 Effect of Porosity Coefficients e0 and Foundation Based on Eqs. (16) and (17), the lower and upper load are determined. Table 2 describes the effect of porosity coefficient and foundation on the upper and lower load. It shows that the critical load decrease when the porosity coefficient increases. This point is also presented in Fig. 2 when illustration the load–deflection curves of ES porous sandwich cylinder. It indicates that the load-carrying of the sandwich cylinder is decreased when e0 is increased. Effects of foundation on the critical buckling load of FG porous cylinder are presented in Table 2 and Fig. 3. It can be observed that the critical external pressure increase when the foundation parameters K1 and K2 separately or together increase. The critical pressure load of shell without foundation is the smallest. For example, with the same value of porosity coefficient (e0 = 0.4), the upper critical load qupper = 2993.012 kPa (without foundation) with qupper = 3843.365 kPa (K 1 = 4 × 107 N/m3 , K 2 = 9 × 104 N/m), it can be seen that the critical load increases about 28.4%.
Nonlinear Buckling and Postbuckling of ES-FG Porous Cylindrical Shells
749
Table 2. Effect of porosity coefficients e0 and elastic foundation on critical load (inner stiffeners) h = 4 mm, R/h = 80, L/R = 2, hcore /hFG = 3, hs = 5 mm, hr = 5 mm, bs = 5 mm, br = 5 mm, ns = 30, nr = 30, k = 1, ΔT = 0K qcr (kPa)
e0=0
e0=0.4
e0=0.8
3202.736 (1,5) 2710.793L (1,4)
2993.012 (1,5) 2468.121 (1,4)
2715.348 (1,4) 2203.120 (1,4)
3470.288 (1,5)
3261.511 (1,5)
3018.885 (1,5)
3152.288 (1,5)
2920.901 (1,4)
2656.911 (1,4)
3676.982 (1,5)
3468.308 (1,5)
3225.825 (1,5)
3363.202 (1,5)
3142.228 (1,4)
2878.410 (1,4)
3780.328 (1,5)
3571.707 (1,5)
3329.295 (1,5)
3468.660 (1,5)
3252.892 (1,4)
2989.160 (1,4)
3739.550 (1,5) 3464.155 (1,5)
3531.979 (1,5) 3312.903 (1,5)
3290.839 (1,5) 3114.708 (1,4)
3946.931 (1,5)
3739.569 (1,5)
3498.714 (1,5)
3675.800 (1,5)
3524.770 (1,5)
3337.243 (1,4)
4050.622 (1,5) 3781.622 (1,5)
3843.365 (1,5) 3630.704 (1,5)
3602.651 (1,5) 3448.128 (1,5)
U
K1=0 N/m3
K2=0 N/m K2=0 N/m
K1=2×107 N/m3
K2=6×104 N/m K2=9×104 N/m K2=0 N/m
K1=4×107 N/m3
K2=6×104 N/m K2=9×104 N/m
Stiffener: hs=hr=0.005m bs=br=0.005m ns=nr=30 3676.982
R= 0.32m, R/h=80 L/R=2, hcore/hFG=3 k=1, (m,n)= (1,5) (1) (2) (3)
3843.365 3468.308 2993.012
3468.308 3225.835
Foundation: K1=2×107 N/m3 K2=6×104 N/m
Fig. 2. Effects of porosity cofficient e0 on q – W max /h curves
R= 0.32m, R/h=80 L/R=2, hcore/hFG=3 e0=0.4, k=1 (m,n)= (1,5)
(3) (2) (1)
Stiffener: hs=hr=0.005m bs=br=0.005m ns=nr=30
Fig. 3. Effects of foundation on q – W max /h curves
4.3 Effect of Geometric Parameters (Effects of hcore /h, R/h, L/R Ratios) With the thickness h constant, Table 3 shows the buckling loads of ES - porous sandwich cylinder for different core-to-face sheets thickness ratios hcore /hFG with corresponding buckling modes. Figure 4 also illustrates the effect of hcore /hFG ratio on the load–deflection paths of porous sandwich cylinder. It shows that when hcore /hFG increases, the critical buckling loads decrease. In the case of R/h = 80 and L/R = 2, the upper load
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L. K. Hoa et al.
decreases from qupper = 3997.851 kPa (with hcore /hFG = 0) to qupper = 3168.536 MPa (with hcore = h, hFG = 0). Table 3 and Figs. 5, 6, 7 and 8 present effects of R/h and L/R ratios on upper and lower buckling load and postbuckling behavior of the sanwich shell. It can be observed that the buckling load qcr decreases markedly with the increase of R/h ratio, i.e. the more the shell is thin the more the value of critical load is small. It is observed, the capacity of mechanical load q bearing of the FG-porous cylinder is considerably reduced with the increase of L/R. hcore =0 hcore/hFG=3 hFG=0 3997.851
Stiffener: hs=hr=0.005m bs=br=0.005m ns=nr=30
Stiffener: hs=hr=0.005m bs=br=0.005m ns=nr=30
4136.397
3468.308
q (kPa)
q (kPa)
(1) (2) (3)
3168.536
(1) 3468.308
R= 0.32m, L/R=2, hcore/hFG=3, e0=0.4 k=1, (m,n)= (1,5)
(2) 2691.578
R= 0.32m, R/h=80 L/R=2, e0=0.4 k=1, (m,n)= (1,5)
Foundation: K1=2×107 N/m3 K2=6×104 N/m
Wmax/h
Wmax/h
Fig. 4. Effects of hcore /hFG on q – W max /h curves 4675.133
Fig. 5. Effects of R/h on q – W max /h curves
(1)
q (kPa) 2257.576
(3)
Foundation: K1=2×107 N/m3 K2=6×104 N/m
R= 0.32m, R/h=80 hcore/hFG=3, e0=0.4 k=1, m=1
qupper (kPa)
(2)
Stiffener: hs=hr=0.005m bs=br=0.005m ns=nr=30
3468.308
Foundation: K1=2×107 N/m3 K2=6×104 N/m
(3)
(1)
R= 0.32m, R/h=80 hcore/hFG=3, e0=0.4 k=1, m=1
(2) (3)
Foundation: K1=2×107 N/m3 K2=6×104 N/m
Wmax/h
Fig. 6. Effects of L/R on q – W max /h curves lines (1, 2): n = 5; line (3): n = 4;
Stiffener: hs=hr=0.005m bs=br=0.005m ns=nr=30
R/h
Fig. 7. Effects of L/R on qupper – R/h curves lines (1, 2): n = 5; line (3): n = 4;
4.4 Effects of Stiffeners and Volume Fraction Index The effects of stiffeners are given in Table 4. It can be seen that, the critical load of unstiffened FGM shell is the smallest, the critical load of FGP cylindrical shell reinforced by rings is biggest. The effects of volume fraction index on the critical pressureof FGP shell are considered in Fig. 8 and Fig. 9. It is found that, the critical pressure decreases with the increase of k. This property is suitable to the real property of material, because the higher value of k corresponds to a metal-richer shell which usually has less stiffness than a ceramic-richer one.
Nonlinear Buckling and Postbuckling of ES-FG Porous Cylindrical Shells R= 0.32m, R/h=80 L/R=2, hcore/hFG=3 e0=0.4, (m,n)= (1,5)
qupper (kPa)
(1)
Stiffener: hs=hr=0.005m bs=br=0.005m ns=nr=30
k=0 k=1 k=∞
Stiffener: R= 0.32m, R/h=80 hs=hr=0.005m L/R=2, hcore/hFG=3 e0=0.4, (m,n)= (1,5) bs=br=0.005m ns=nr=30
3761.575
q (kPa)
(2) (3)
751
(1)
3468.308 3385.620
(2) (3)
k=0 k=1 k=∞
Foundation: K1=2×107 N/m3 K2=6×104 N/m
Foundation: K1=2×107 N/m3 K2=6×104 N/m
Wmax/h
R/h
Fig. 8. Effects of k on qupper – R/h curves
Fig. 9. Effects of k on q – W max /h curves
Table 3. Effects of geometric parameters on upper and lower critical pressure (inner stiffeners) R = 320 mm, hs = 5 mm, hr = 5 mm, bs = 5 mm, br = 5 mm, ns = 30, nr = 30, k = 1, ΔT = 0K, K 1 = 2 × 107 N/m3 K 2 = 6 × 104 N/m, e0 = 0.4 qcr (kPa)
R/h=70 U
L/R=1.5 hcore=0 hFG=h
L/R=2 L/R=3 L/R=1.5
hcore/hFG=1
L/R=2 L/R=3 L/R=1.5
hcore/hFG=3
L/R=2 L/R=3 L/R=1.5
hcore/hFG=5
L/R=2 L/R=3 L/R=1.5
hcore=h hFG=0
L/R=2 L/R=3
6713.093 (1,5) 5609.772L (1,5) 4789.178 (1,5) 4359.533 (1,5) 3010.036 (1,4) 2786.149 (1,4) 6028.873 (1,5) 5163.989 (1,5) 4421.291 (1,5) 3942.605 (1,4) 2791.288 (1,4) 2617.298 (1,4) 5565.780 (1,5) 4819.682 (1,5) 4136.379 (1,5) 3662.870 (1,4) 2625.273 (1,4) 2476.403 (1,4) 5385.410 (1,5) 4678.786 (1,5) 4019.537 (1,5) 3554.253 (1,4) 2557.580 (1,4) 2417.120 (1,4) 4977.757 (1,5) 4349.876 (1,5) 3746.285 (1,5) 3309.234 (1,4) 2399.846 (1,4) 2276.277 (1,4)
R/h=80
R/h=100
R/h=150
5632.807 (1,5) 4662.108 (1,5) 3997.851 (1,5) 3621.131 (1,5) 2566.530 (1,4) 2372.898 (1,4) 5052.500 (1,5) 4292.150 (1,5) 3693.065 (1,5) 3369.234 (1,4) 2387.142 (1,4) 2237.457 (1,4) 4675.133 (1,5) 4020.073 (1,5) 3468.308 (1,5) 3142.228 (1,4) 2257.576 (1,4) 2130.079 (1,4) 4530.736 (1,5) 3910.770 (1,5) 3377.796 (1,5) 3055.645 (1,4) 2205.710 (1,4) 2085.654 (1,4) 4208.402 (1,5) 3658.623 (1,5) 3168.536 (1,5) 2862.741 (1,4) 2086.247 (1,4) 1981.162 (1,4)
4348.823 (1,5) 3568.630 (1,5) 3084.262 (1,5) 2783.505 (1,5) 2058.137 (1,4) 1907.329 (1,4) 3895.664 (1,5) 3286.087 (1,5) 2851.277 (1,5) 2615.151 (1,5) 1923.556 (1,4) 1808.361 (1,4) 3616.725 (1,5) 3093.064 (1,5) 2691.578 (1,5) 2488.403 (1,5) 1833.209 (1,4) 1736.077 (1,4) 3512.767 (1,5) 3017.846 (1,5) 2629.156 (1,5) 2437.088 (1,5) 1798.109 (1,4) 1707.052 (1,4) 3285.131 (1,5) 2847.875 (1,5) 2487.681 (1,5) 2318.058 (1,5) 1718.843 (1,4) 1640.039 (1,4)
2973.191 (1,5) 2455.576 (1,5) 2149.996 (1,5) 1954.249 (1,5) 1530.278 (1,5) 1450.903 (1,4) 2665.383 (1,5) 2264.646 (1,5) 1990.090 (1,5) 1839.458 (1,5) 1453.967 (1,4) 1385.852 (1,4) 2488.029 (1,5) 2146.553 (1,5) 1890.458 (1,5) 1763.062 (1,5) 1399.082 (1,4) 1343.504 (1,4) 2424.223 (1,5) 2102.647 (1,5) 1853.229 (1,5) 1733.707 (1,5) 1378.661 (1,4) 1327.321 (1,4) 2288.321 (1,5) 2006.797 (1,5) 1771.566 (1,5) 1668.027 (1,5) 1333.950 (1,4) 1291.190 (1,4)
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Table 4. Effects of stiffeners and volume fraction index on on upper and lower critical pressure h = 4 mm, R/h = 80, L/R = 2, hcore/hFG = 3, hs = 5 mm, hr = 5 mm, bs = 5 mm, br = 5 mm, ns = 30, nr = 30, ΔT = 0K, K1 = 2 × 107 N/m3 K2 = 6 × 104 N/m, e0 = 0.4 qcr (kPa)
Unstiffened
Stringer (ns = 60)
Ring (nr = 60)
Orthogonal (ns = nr = 30)
k=0
1237.277 (1,6) 1182.008 (1,5)
1244.460 (1,6) 1204.537 (1,5)
4687.754 (1,4) 4114.811 (1,4)
3761.575 (1,5) 3283.865 (1,4)
k = 0.5
1343.330 (1,6) 1262.522 (1,5)
1350.237 (1,6) 1282.081 (1,5)
4529.226 (1,4) 3930.532 (1,4)
3521.886 (1,5) 3161.382 (1,4)
k=1
1393.725 (1,6) 1300.836 (1,5)
1400.241 (1,6) 1319.282 (1,5)
4516.512 (1,4) 3904.212 (1,4)
3468.308 (1,5) 3142.228 (1,4)
k=5
1485.786 (1,6) 1371.014 (1,5)
1491.131 (1,6) 1387.360 (1,5)
4534.340 (1,4) 3894.305 (1,4)
3407.016 (1,5) 3132.362 (1,4)
k=∞
1524.934 (1,6) 1401.016 (1,5)
1529.547 (1,6) 1416.316 (1,5)
4552.000 (1,4) 3898.026 (1,4)
3385.620 (1,5) 3117.052 (1,5)
5 Conclusions An analytical approach to analyze the nonlinear buckling and post-buckling behavior of eccentrically stiffened functionally graded porous cylinder under external pressure and surrounded by elastic foundations based on the classical shell theory and the smeared stiffeners technique with geometrical nonlinearity in von Karman sense is presented in this paper. The shell is reinforced by eccentrically rings and stringers attached to the inside and material properties of shell and stiffeners varying continuously graded in the thickness direction are considered. The obtained results show that the reinforced stiffeners, volume fraction index, the geometrical characteristics, and foundation parameters significantly influence the buckling and post-buckling behaviors of the ES-FG porous sandwich cylinders. Moreover, the study also shows the effects of the porosity coefficient and the core layer thickness on the critical buckling loads and load-deflection paths in the post-buckling response of the cylinder. Acknowledgements. This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 107.02–2018.324.
Appendix In Eqs. (3) and (4) α11 α14 β11 β14
∗ ,α = −D44 12 ∗ = D24 , α15 ∗, = C11 β12 ∗, β = −C24 15
∗ ∗ + D∗ , α = − D45 + 2D66 13 54 ∗ − 2D∗ + D∗ , α = D14 16 25 63 ∗ − 2C ∗ , = C33 β 13
∗ 12 ∗ ∗ , β16 = − C14 + C25 + C36
∗ , = −D55 ∗ = D15 , ∗, = C22 ∗. = −C15
Nonlinear Buckling and Postbuckling of ES-FG Porous Cylindrical Shells
753
where 2 , C ∗ = C / , C ∗
= C22 C11 − C12 22 22 12 ∗ = (C C − C C )/ , ∗ C15 C14 12 24 22 14 ∗ ∗ C16 = (C22 − C12 )/ , C11 ∗ ∗ C25 C24 = (C12 C14 − C11 C24 )/ , ∗ = (C − C )/ , ∗ C26 C33 11 12 ∗ C36 = C36 /C33 .
= C12 / , = (C12 C25 − C22 C15 )/ , = C11 / , = (C12 C15 − C11 C25 )/ , = 1/C33 ,
∗ = C C∗ − C C∗ , ∗ = C + C C∗ + C C∗ , D14 D44 14 22 24 12 44 24 24 14 14 ∗ ∗ = C C∗ + C C∗ + C , ∗ ∗ D24 = C24 C11 − C14 C12 , D45 14 15 24 25 45
∗ = C C ∗ + C C ∗ , D∗ = C C ∗ − C C ∗ , D46 14 16 24 26 15 22 25 12 15 ∗ = C C ∗ + C C ∗ + C , D∗ = C C ∗ − C C ∗ , D54 15 14 25 24 45 15 12 25
25 11 ∗ = C C ∗ + C C ∗ + C , D∗ = C C ∗ + C C ∗ , D55 15 15 25 25 55 15 16 25 26 56 ∗ = C − C C∗ , ∗ = C C∗ , D63 D66 63 33 66 63 36
in which E1 E1s bs νE1 E2 E2s bs νE2 + , C12 = , C14 = + , C15 = , 2 2 2 1−ν ds 1−ν 1−ν ds 1 − ν2 E1 E1r br νE2 E2 E2r br E1 , = + , C24 = , C25 = + , C33 = 1 − ν2 dr 1 − ν2 1 − ν2 dr 2(1 + ν) E2 E3 E3s bs νE3 E3 E3r br , C44 = = + , C45 = , C55 = + , 2 2 2 1+ν 1−ν ds 1−ν 1−ν dr E2 E3 , C66 = , = 2(1 + ν) 1+ν
C11 = C22 C36 C63
FGM – Porous Metal Core - FGM Cylindrical Shell h/2 E1 = −h/2
2hcore 1 , Eshell dz = Ec hFG + (Em − Ec )hFG + Em hcore − e0 k +1 π
h/2 E2 =
zEshell dz = 0, −h/2
h/2 E3 =
z 2 Eshell dz =
−h/2
⎡
Ec (hFG + hcore )3 − h3core 12
h3FG /(k + 3)
⎤
(Em − Ec ) ⎢ ⎥ ⎣ +hFG (hFG + hcore )2 /(k + 1) ⎦ 4 −2h2FG (hFG + hcore )/(k + 2) + Em h3core /12 − e0 h3core π 2 − 8 / 2π 3
+
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Inside Stiffener
Eis Eir
h/2+h s
= h/2
Es (z)
Er (z)
z i−1 dz
i = 1, 2, 3
References 1. Reddy, J.N., Starnes, J.H.: General buckling of stiffened circular cylindrical shells according to a Layerwise theory. Comput. Struct. 49, 605–616 (1993) 2. Shen, H.S., Zhou, P., Chen, T.Y.: Post-buckling analysis of stiffened cylindrical shells under combined external pressure and axial compression. Thin-Walled Struct. 15, 43–63 (1993) 3. Shen, H.S.: Post-buckling analysis of imperfect stiffened laminated cylindrical shells under combined external pressure and thermal loading. Int. J. Mech. Sci. 40(4), 339–355 (1998) 4. Baruch, M., Singer, J.: Effect of eccentricity of stiffeners on the general instability of stiffened cylindrical shells under hydro-static pressure. J. Mech. Eng. Sci. 5, 23–27 (1963) 5. Huang, H., Han, Q.: Research on nonlinear postbuckling of functionally graded cylindrical shells under radial loads. Compos. Struct. 92, 1352–1357 (2010) 6. Dung, D.V., Hoa, L.K.: Nonlinear buckling and post-buckling analysis of eccentrically stiffened functionally graded circular cylindrical shells under external pressure. Thin-Walled Struct. 63, 117–124 (2013) 7. Hieu, P.T., Tung, H.V.: Nonlinear buckling behavior of functionally graded material sandwich cylindrical shells with tangentially restrained edges subjected to external pressure and thermal loadings. J. Sandwich Struct. Mater. (2020).https://doi.org/10.1177/1099636220908855 8. Sun, J., Xu, X., Lim, C.W.: Buckling of cylindrical shells under external pressure in a hamiltonian system. J. Theoret. Appl. Mech. 52(3), 641–653 (2014) 9. Nam,V.H., Trung, N.T., Hoa, L.K.: Buckling and post-buckling of porous cylindrical shells with functionally graded composite coating under torsion in thermal environment. ThinWalled Struct. 144, 106253 (2019) 10. Foroutan, K., Shaterzadeh, A., Ahmadi, H.: Nonlinear static and dynamic hygrothermal buckling analysis of imperfect functionally graded porous cylindrical shells. Appl. Math. Model. 77, 539–553 (2020) 11. Ahmadi,H., Foroutan, K.: Nonlinear static and dynamic thermal buckling analysis of imperfect multilayer FG cylindrical shells with an FG porous core resting on nonlinear elastic foundation. J. Thermal Stresses 43 (2020). https://doi.org/10.1080/01495739.2020.1727802 12. Foroutan, K., Ahmadi, H.: Nonlinear static and dynamic buckling analyses of imperfect FGP cylindrical shells resting on nonlinear elastic foundation under axial compression. Int. J. Struct. Stab. Dyn. 20, 2050074 (2020) 13. Shahgholian, D., Rahimi, G., Khodadadi, A., Salehipour, H., Afrand, M.: Buckling analyses of FG porous nanocomposite cylindrical shells with graphene platelet reinforcement subjected to uniform external lateral pressure. Mech. Based Des. Struct. Mach. (2020). https://doi.org/ 10.1080/15397734.2019.1704777 14. Shahgholian, D., Safarpour, M., Rahimi, A.R., Alibeigloo, A.: Buckling analyses of functionally graded graphene-reinforced porous cylindrical shell using the Rayleigh-Ritz method. Acta Mech. 231, 1887–1902 (2020)
Bulk Modulus Prediction of Particulate Composite with Spherical Inclusion Surrounded by a Graded Interphase Nguyen Duy Hung1 and Nguyen Trung Kien2(B) 1 Department of Civil Engineering, University of Transport and Communications,
Campus in Ho Chi Minh City, Ho Chi Minh City, Vietnam [email protected] 2 Research and Application Center for Technology in Civil Engineering, University of Transport and Communications, Ha Noi City, Vietnam [email protected]
Abstract. The aim of this paper is to estimate the bulk modulus of composite containing spherical inclusion surrounded by an interphase whose elastic properties vary with radius. Based on the sphere assemblage model and the differential substitution scheme, the differential equation for the bulk modulus is established. The effective bulk modulus of the material are predicted by numerical integration. The effect of interphase zone is also investigated. Keywords: Particulate composite · Effective bulk modulus · Graded interphase
1 Introduction The effective moduli of materials is affected not only by the properties and microstructure of the components but also by the interface between the matrix and inclusion. The earliest studies often assumed that the components are perfectly bonded across a sharp interface. Later models considered the effect of sliding or debonding between the inclusion and matrix. In other cases, there exists an interfacial transition zone around inclusion as observed in mortar or cement concrete. Hashin and Rosen developed a model for composites in which a thin layer existed outside of each inclusion [1]. The properties were uniform within this layer, but different from those in the matrix or inclusion. Jayaraman and Reifsnider considered a transition zone outside of a cylindrical inclusion and the moduli vary according as r β , where β is some constant [2]. Lutz and Zimmerman developed a power-law model similar to Jayaraman for spherical inclusion but added a constant term [3, 4], thereby allowing a smooth transition between the interphase and the matrix. A closed-form solution for a media containing such an inclusion, under hydrostatic far-field loading, is derived by using the method of Frobenius series and thereby an expression for the effective bulk modulus of a material that contained a dispersion of such inclusions is found. In this paper, we will propose a simpler method to estimate the bulk modulus of composite containing spherical inclusion surrounded by a graded interphase. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 755–760, 2022. https://doi.org/10.1007/978-981-16-3239-6_58
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The paper is organized as follows: firstly the model of multi-coated sphere assemblage is presented. In Sect. 3, the ordinary differential equation for the bulk modulus is derived from differential substitution scheme. The numerical results and comparison are illustrated in Sect. 4. The conclusion completes the paper.
2 Model of Multi-coated Sphere Assemblage and Limiting Case Let us consider a representative volume element (RVE) of a statistically isotropic multicomponent material that occupies region V of Euclidean space and consists of n components occupying regions V α of volume proportions vα (α = 1,…,n) and having isotropic elastic moduli K α , μα . Here spheres of phase 1 are coated with spherical shells of phase 2, which in turn are coated with spherical shells of phase 3,…, ending with the last coating spherical shells of phase n, and the relative volume proportions and coating orders of the phases in all n-compound spheres are the same. The space of V is entirely filled by such compound spheres distributed randomly with diameters varying to infinitely small (Fig. 1).
Fig. 1. A multi-coated sphere assemblage
From the minimum energy and complementary energy principles, three-point correlation upper and lower bounds on the effective elastic moduli of isotropic multicomponent materials have been constructed, which converge to yield the exact value of the effective bulk modulus of the multi-coated sphere assemblage [5]. The effective bulk modulus obtained from the minimum energy principle for the RVE under the imposed averaged strain constraint has the particular expression [6]. K eff = KV − vK .A−1 K .vK
(1)
Bulk Modulus Prediction of Particulate Composite
757
Where vk = {v1 (K1 − KR ), . . . vn (Kn − KR )}T , vK = {v1 K1 , . . . , vn Kn }T n −1 n −1 KV = vα Kα , KR = vα Kα , AK = AK αβ α=1
AK αβ = vα Kα δαβ +
n γ =1
α=1
Aαβ γ − vα KR
n δ=1
(2)
Kδ−1 Aδβ 2μγ γ
βγ
δ αβ is the Kronecker symbol, Aα are the three-point correlation parameters. In the two-component case, Eq. (1) can be presented as K
eff
=
v1 K1 +
4 3 μ2
+
−1
v2 K2 +
4 3 μ2
4 − μ2 3
(3)
In which V 1 is referred to as the inclusion phase, while V 2 is the matrix phase. If we let v2 approaches 1 (i.e. v1 → 0) in Eq. (3), we obtain asymptotically the respective dilute solution for the spherical inclusion of phase 1 in the matrix phase 2 K eff = K2 + v1 (K1 − K2 )
K2 + 43 μ2 K1 + 43 μ2
+ O v12
(4)
Inversely, if we let v2 → 0 (thin coating) we get asymptotic expression K eff = K1 + v2 (K2 − K1 )
K1 + 43 μ2 K2 + 43 μ2
+ O v22
(5)
In the next section, Eq. (5) will be used in the differential substitution scheme to derive the ordinary differential equation for bulk modulus.
3 Differential Substitution Scheme and the Coating with Radially Variable Moduli The Hill substitution scheme is generalized to a differential substitution scheme for the coated sphere assemblage with the inner spherical inclusion having characteristics K 1 , μ1 , v1 , R1 embedded in the spherical coating shell having radially variable moduli K c (r), μc (r), volume proportion vc = 1 − v1 and outer radius Rc . The differential scheme construction process starts with the inclusion phase 1. At each step of the procedure, we add infinitesimal volume amount Δv of spherical coating shell of moduli K c (r), μc (r) into already constructed compound sphere of the previous step, which contains volume fraction v of the coating phase and has effective bulk modulus K (v increase from 0 to vc ).
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According to Eq. (5), we obtain the differential equation for the effective bulk modulus of the composite [7] dK dv
=
1 1−v
(Kc −K) K+ 43 μc
K(0) = K1 ,
Kc + 43 μc K eff = K(vc )
(6)
When the coating shell moduli is given as a function of radius K c = K c (r), we make a change of variables r 3 − R31 , r3
3R31 dr r4
(7)
3[Kc (r)−K] K+ 43 μc (r)
= r Kc (r)+ 43 μc (r) K(R1 ) = K1 , K eff = K(Rc )
(8)
v=
dv =
in Eq. (6) to get dK dr
The ordinary differential Eq. (8) determines the effective bulk modulus K eff of the coated sphere assemblage with the coating of radially variable moduli K c (r), μc (r) surrounding the inclusion phase 1.
4 Numerical Results For illustration of the approach, let us consider the composite in which the moduli in the coating, surrounding the inclusion of moduli K 1 , μ1 , are assumed to vary as −β Kc (r) = K2 + Kif − K2 Rr1 −β μc (r) = μ2 + μif − μ2 Rr1
R1 ≤ r ≤ Rc
(9)
where K 2 , μ2 are called the matrix moduli, K if = K c (R1 ), μif = μc (R1 ) are the coating moduli at the interface with the inner inclusion. For example, we take K 2 = 1, μ2 = 0.6, K 1 = 5, μ1 = 3, K if = 0.75, μif = 0.45 and β = 10. Using the Runge-Kutta method, the differential Eqs. (8) and (9) can be integrated numerically. Table 1 presents the numerical results for the effective bulk modulus of the composite. The results of Lutz and Zimmerman in [4] are also presented for comparison. A good agreement is observed.
Bulk Modulus Prediction of Particulate Composite
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Table 1. Comparison of the numerical results and those according to Lutz and Zimmerman (2005) v1
Differential Lutz and method Zimmerman [4]
0.1 1.117
1.117
0.2 1.250
1.250
0.3 1.407
1.403
0.4 1.592
1.581
0.5 1.814
1.791
0.6 2.095
2.040
Figure 2 shows the effective bulk modulus as a function of the volume fraction of inclusion with some values of K if (K 2 = 1, K 1 = 5 and the Poisson ratios of all phases are taken to be 0.25). Note that for the case K if = 1, the effective bulk modulus of the composite can be calculated directly from Eq. (3) and the result coincides with the Mori-Tanaka prediction [8], as well as with the Hashin-Strikman lower bound [9].
Fig. 2. Effective bulk modulus of composite containing the spherical inclusion surrounded by a grade interphase with some values of K if
5 Conclusion Based on the sphere assemblage model and the differential substitution scheme, an ordinary differential equation has been derived for the bulk modulus of the composite containing spherical inclusion surrounded by a coating of radially variable moduli.
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This equation is used to estimate the effective bulk modulus of matrix-composite with inhomogeneous interphase whose moduli is described by Eq. (9). Numerical results are compared with those of Lutz and Zimmerman [4] showing the effectiveness of the method. This approach can be applied to other problems such as thermal and electrical conductivity, dielectric and magnetic permeability. Acknowledgement. This research has been supported by the Vietnam National Foundation for Science and Technology Development – Project 107.02-2019.13.
References 1. Hashin, Z., Rosen, B.W.: The elastic moduli of fiber-reinforced materials. ASME J. Appl. Mech. 31, 223–228 (1964) 2. Jayaraman, K., Reifsnider, K.L.: Residual stresses in a composite with continuously varying Young’s modulus in the matrix/matrix interphase. J. Compos. Mater. 26, 770–791 (1992) 3. Lutz, M.P., Zimmerman, R.W.: Effect of the interphase zone on the bulk modulus of a particulate composite. ASME J. Appl. Mech. 8(63), 855–861 (1996) 4. Lutz, M.P., Zimmerman, R.W.: Effect of an inhomogeneous interphase zone on the bulk modulus and conductivity of a particulate composite. Int. J. Solids Struct. 42, 429–437 (2005) 5. Pham, D.C.: Bounds on the elastic moduli of statistically isotropic multicomponent materials and random cell polycrystals. Int. J. Solids Struct. 49, 2646–2659 (2012) 6. Pham, D.C., Vu, L.D., Nguyen, V.L.: Bounds on the ranges of the conductive and elastic properties of randomly inhomogeneous materials. Phil. Mag. 93, 2229–2249 (2013) 7. Pham, D.C., Nguyen, T.K., Tran, B.V.: Macroscopic elastic moduli of spherically symmetric inclusion composites and the microscopic stress-strain fields. Int. J. Solids Struct. 169, 141–165 (2019) 8. Mori, T., Tanaka, K.: Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metall. 21, 571–574 (1973) 9. Hashin, Z.: Analysis of composite materials: a survey. J. Appl. Mech. 50, 481–505 (1983)
Post-buckling Response of Functionally Graded Porous Plates Rested on Elastic Substrate via First-Order Shear Deformation Theory Le Thanh Hai1 , Nguyen Van Long2(B) , Tran Minh Tu2 , and Chu Thanh Binh2 1 Faculty of Civil Engineering, Vinh University, Vinh, Vietnam 2 Faculty of Industrial and Civil Engineering, National University of Civil Engineering,
Hanoi, Vietnam {longnv,tutm,binhct}@nuce.edu.vn
Abstract. This paper presents the post-buckling analysis of functionally graded porous (FGP) plates rested on the elastic substrate subjected to in-plane compressive mechanical loads. Based on the first-order shear deformation theory taking into account Von Karman nonlinearity, the governing equations are derived. The elastic modulus of FGP material is assumed to vary across the plate thickness following three various distribution patterns including uniform, symmetric, and asymmetric. Galerkin’s approach and stress function is utilized to obtain the loaddeflection relation for analyzing the post-buckling behavior of FGP plates. The theoretical formulation is verified by comparing the present results with those available in publications and found good agreement. Through the numerical results, the effect of porosity distribution pattern, porosity coefficient, geometrical configurations, elastic foundations, as well as mechanical loads on the post-buckling behavior of the FGP plate is indicated. Keywords: Post-buckling · Porous plates · Analytical approach · First-order shear deformation theory · Stress function
1 Introduction Functionally graded porous materials (FGPMs), such as metal foams, are ones of the category of lightweight materials and have potential applications in lightweight structures. Possesses many outstanding features such as excellent energy absorption capability, high specific strength, low thermal conductivity, etc., they are widely applied in aircraft, aerospace, ocean and civil engineering fields [1–3]. By adjusting the material composition, porosity patterns, pore size, and density the FGPMs can achieve the desire structural performance. Recently, the application of such materials tends to increase strongly, hence the studies on their mechanical behaviors have attracted considerable attention from scientist communities. Wattanasakulpong and Ungbhakorn [4] investigated the effect of porosities on the linear and nonlinear vibrational characteristic of FG beams under different types of elastic supports employing differential transformation method. Using Timoshenko beam theory and Ritz method, Chen et al. studied static and © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 761–779, 2022. https://doi.org/10.1007/978-981-16-3239-6_59
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buckling behaviors [5], free and forced vibration [6], nonlinear free vibration [7] of FGP beams. Jabbari et al. presented buckling analysis [8] and thermal buckling analysis [9] of thin saturated FGP plates. Rezaei and Saidi [10] used Carrera unified formulation and state-space method to explore the effect of porosities on free vibrational characteristics of thick saturated FGP plates with Levy-type boundary conditions. Using simple first-order shear deformation theory (FSDT), Rezaei et al. [11] studied free vibration of FG plates with porosities by an analytical approach. Akbas [12] presented the effect of porosities on free vibration and bending behavior of FG plates by using Navier technique. There are plenty researchers focusing on the post-buckling analysis of FG plates, i.e., Liew et al. [13] presented the post-buckling of FGM plates integrated with surfacebonded piezoelectric actuators subjected to in-plane force in the thermal environment applying the Galerkin-differential quadrature iteration algorithm; Bakora and Tounsi [14] studied post-buckling of thick FG plates subjected to thermomechanical loads; Feyzi and Khorshidvand [15] presented post-buckling of saturated porous circular plates subjected to mechanical load based on CLPT (classical plate theory); Tung and Duc [16] used classical plate theory and Galerkin’s procedure to examine nonlinear stability of FG plates under thermo-mechanical loads; Cong et al. [17] investigated the effect of porosity distribution patterns on the thermomechanical buckling and post-buckling behavior of FG plate resting on elastic foundations; Barati and Zenkour [18] studied postbuckling of imperfect FG nanoplates with porosities by employing general higher-order plate theory and nonlocal elasticity; Using isogeometric analysis, Phung-Van et al. [19] investigated hygro-thermo-mechanical effects on the porosity-dependent geometrically nonlinear transient analysis of FGP plates based on third-order shear deformation theory. In [20] Tu et al. examined the nonlinear buckling and post-buckling response FGP plates taking to account geometrical imperfect based on the CLPT. In this paper, we take a further step to analyze the post-buckling behavior of the FGP plates rested on Pasternak’s foundation and subjected to uni-axial and bi-axial compressive loading. The theoretical formulation is developed by using the FSDT and von-Karman nonlinearity including the initial geometrical imperfection. The effective material properties of FGPMs (open-cell metal foam) are assumed to vary across the plate thickness according to a simple cosine rule. Three porosity distribution patterns such as uniform, non-uniform symmetric, nonuniform unsymmetric are considered. An analytical approach using stress function and Galerkin procedure is employed to obtain buckling loads, post-buckling load-deflection relation. The effects of material’s properties, geometric parameters of the plates, elastic foundation stiffness on the post-buckling response of the plate are investigated through the numerical examples.
2 Functionally Graded Porous Plates An FGP plate resting on Pasternak’s elastic substrate with length a, width b and thickness h subjected to the bi-axial forces px , py is considered in this study. The coordinate system (x, y, z) is established in the middle plane as shown in Fig. 1. The Pasternak foundation with Winkler stiffness K w , shear stiffness K si (i = x, y).
Post-buckling Response of Functionally Graded Porous Plates
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Fig. 1. Functionally graded porous plate on elastic foundation
The open-cell metal foam with three porosity distribution patterns along the thickness direction is considered (see Fig. 2). The elastic moduli of each pattern can be determined as follow [21, 22]
(1)
π z Type 2 : Non - uniform symmetric distribution:{E(z), G(z)} = {E1 , G1 } 1 − e0 cos (2) h πz π + Type 3 : Non - uniform asymmetric distribution : {E(z), G(z)} = {E1 , G1 } 1 − e0 cos 2h 4
(3)
in which: E1 , G1 , E2 , G2 are maximum and minimum values of Young’s modulus, shear modulus respectively. Poisson ratio is assumed to be constant.
(b) Non-uniform symmetric
(a) Uniform
(c) Non-uniform asymmetric
Fig. 2. The three types of porosity distribution
The porosity coefficient e0 is defined as e0 = 1 −
E2 G2 =1− ; (0 < e0 < 1) E1 G1
(4)
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Herein, the physical neutral surface is chosen as the reference surface to vanish the stretching–bending coupling effect in constitutive equations. For the case of non-uniform asymmetric porosity distribution, the neutral surface position is defined as [23]: ⎤ ⎡ ⎤ ⎡ h/2 h/2 h/2 ⎥ ⎢ ⎥ ⎢ zE(z)dz ⎦/⎣ E(z)dz ⎦ (5) (z − C)E(z)dz = 0 ⇒ C = ⎣ −h/2
−h/2
−h/2
3 Theoretical Formulation Using the physical neutral surface concept, the displacements u˜ , v˜ , w˜ according to FSDT in the coordinate system (x, y, zns ) take the following form [24]: u˜ (x, y, zns ) = u0 (x, y) + zns ϑx (x, y); v˜ (x, y, zns ) = v0 (x, y) + zns ϑy (x, y); w(x, ˜ y, zns ) = w0 (x, y)
(6)
where u0 , v0 , w0 are displacements on the physical neutral surface along the directions x, y, and zns , respectively; and ϑx , ϑy are neutral surface rotations of transverse normal about y-, x- axes respectively. The nonzero strains at the neutral surface of the FGP plate, including von Kárman nonlinearity, and initial geometrical imperfection w0∗ , are given as follows [24]: ⎧ ⎫ ⎧ 0⎫ ⎧ ⎫ ⎨ εx ⎬ ⎪ ⎬ ⎨ εx ⎪ ⎨ kx ⎬ γxz0 γxz 0 = (7) εy = εy + zns ky ; ⎩ ⎭ ⎪ ⎩ ⎭ γyz γyz0 ⎭ ⎩γ0 ⎪ γxy k xy xy w2
w2
0,y ∗ ; k = θ ; ε0 = v ∗ in which: εx0 = u0,x + 20,x + w0,x w0,x x x,x 0,y + 2 + w0,y w0,y ; y ∗ ∗ 0 = u ky = ϑy,y ; γxy 0,y + v0,x + w0,x w0,y + w0,y w0,x + w0,x w0,y ; kxy = ϑx,y + ϑy,x ; 0 0 γxz = w0,x + ϑx ; γyz = w0,y + ϑy . The comma followed by x or y denotes the differentiation with respect to the x or y coordinates respectively. The stress-strain relationships for the FGP plate are given by: ⎫ ⎡ ⎫ ⎧ ⎤⎧ Q11 Q12 0 ⎨ εx ⎬ ⎨ σx ⎬ Q55 0 γxz σxz ⎦ ⎣ = (8) = Q21 Q22 0 ; σ ε ⎩ y ⎭ σyz ⎩ y⎭ 0 Q44 γyz σxy γxy 0 0 Q66
in which: Q11 = Q22 = Q66 =
1 2(1+v) E(zns ).
1 E(zns ), 1−ν 2
Q12 = Q21 =
ν E(zns ), 1−ν 2
Q44 = Q55 =
Post-buckling Response of Functionally Graded Porous Plates
765
The constitutive relations of FGP plate based on the physical neutral surface concept are described as follow: ⎧ ⎫ ⎧ ⎫ ⎡ ⎤⎪ ε0 ⎪ ⎧ M ⎫ ⎡ ⎤⎧ k ⎫ N ˜ ˜ ⎪ ⎪ x A11 A12 0 ⎪ C˜ 11 C˜ 12 0 ⎪ ⎨ x ⎪ ⎨ ⎬ ⎪ ⎬ ⎬ ⎬ ⎨ x ⎪ ⎨ x ⎪ 0 ⎣ ⎣ ⎦ ⎦ ˜ ˜ ˜ ˜ εy ; My = C12 C11 0 Ny = A12 A11 0 ky ; ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎭ ⎩ Mxy ⎭ ⎩γ0⎪ 0 0 A˜ 66 ⎪ 0 0 C˜ 66 ⎩ kxy ⎭ Nxy (9) xy s 0 γxz A˜ 44 0 Qxz = γyz0 Qyz 0 A˜ s44 where: A˜ ij =
h/2−C −h/2−C
Qij dzns ; C˜ ij =
h/2−C −h/2−C
s 2 dz ; A Qij zns ns ˜ 44 = ks
h/2−C −h/2−C
Q44 dzns ;
ij = 11, 12, 66; For metal foam, the shear correction factor ks = 5/6 is used. The equilibrium equations of FGP plate resting on the elastic foundation are derived from principle minimum total energy [25], its mathematical formulation has the form: 0 = δUP + δUF + δV
(10)
where δUP , δUF , δV are the variation of plate strain energy, of Pasternak’s foundation strain energy, and work done by external loads, respectively. The obtained equilibrium equations are expressed as [25]: ∂Nx /∂x + ∂Nxy /∂y = 0;
(11.1)
∂Ny /∂y + ∂Nxy /∂x = 0;
(11.2)
∂Qyz /∂y + ∂Qxz /∂x + Nx ∂ 2 w0 /∂x2 + 2Nxy ∂ 2 w0 /∂x∂y + Ny ∂ 2 w0 /∂y2 − Kw w0 + Ksx ∂ 2 w0 /∂x2 + Ksy ∂ 2 w0 /∂y2 = 0;
(11.3)
Qxz − ∂Mx /∂x − ∂Mxy /∂y = 0;
(11.4)
Qyz − ∂Mxy /∂x − ∂My /∂y = 0;
(11.5)
By introducing the stress functions as follow: Nx = ∂ 2 ϕ/∂y2 ; Ny = ∂ 2 ϕ/∂x2 ; Nxy = −∂ 2 ϕ/∂x∂y
(12)
Two Eqs. (11.1)–(11.2) are identically satisfied. Using the relations (7), (9) and (12), the system of equilibrium equations is rewritten in terms of displacements and stress function as follow A˜ s44 ∂ 2 w0 /∂y2 + A˜ s44 ∂ 2 w0 /∂x2 + A˜ s44 ∂ϑx /∂x + A˜ s44 ∂ϑy /∂y − Kw w0 + Ksx ∂ 2 w0 /∂x2 + Ksy ∂ 2 w0 /∂y2 + ∂ϕ/∂y2 ∂ 2 w0 /∂x2 + ∂ 2 w0∗ /∂x2 − 2∂ 2 ϕ/∂x∂y ∂ 2 w0 /∂x∂y + ∂ 2 w0∗ /∂x∂y + ∂ϕ/∂x2 ∂ 2 w0 /∂y2 + ∂ 2 w0∗ /∂y2 = 0; (13.1)
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C˜ 11 ∂ 2 ϑx /∂x2 + C˜ 66 ∂ 2 ϑx /∂y2 + C˜ 12 + C˜ 66 ∂ 2 ϑy /∂x∂y − A˜ s44 ϑx − A˜ s55 ∂w0 /∂x = 0; (13.2) C˜ 12 + C˜ 66 ∂ 2 ϑx /∂x∂y + C˜ 66 ∂ 2 ϑy /∂x2 + C˜ 11 ∂ 2 ϑy /∂y2 − A˜ s44 ϑy − A˜ s44 ∂w0 /∂y = 0 (13.3) The geometrical compatibility equation is written as [26]: 0 ∂ 2 εx0 /∂y2 + ∂2 εy0 /∂x2 − ∂ 2 γxy /∂x∂y = ∂ 2 w02 /∂x∂y − ∂ 2 w0 /∂x2 .∂ 2 w0 /∂y2
+ 2∂ 2 w0 /∂x∂y.∂ 2 w0∗ /∂x∂y − ∂ 2 w0 /∂x2 .∂ 2 w0∗ /∂y2 − ∂ 2 w0 /∂y2 .∂ 2 w0∗ /∂x2
(14)
From Eqs. (9) and (12), the strains can be expressed as εx0 = εy0 =
A˜ 11 Ny A˜ 211 −A˜ 212
−
A˜ 11 Nx A˜ 211 −A˜ 212
−
A˜ 12 Nx A˜ 211 −A˜ 212
=
A˜ 12 Ny A˜ 211 −A˜ 212 A˜ 11 2 A 11 −A˜ 212
=
∂2ϕ ∂x2
A˜ 11
∂2ϕ
−
2 ∂y 2 A˜ 211 − A 12 ∂2ϕ A˜ 12 ˜A2 −A˜ 2 ∂y2 11 12
−
A˜ 12 2 A 11 −A˜ 212
0 = ; γxy
∂2ϕ ; ∂x2
1 A66 Nxy
= − A166 ϕ,xy (15)
Substituting the Eq. (15) into geometrical compatibility Eq. (14), we obtain: ∇ 4 ϕ = D ∂ 2 w02 /∂x∂y − ∂ 2 w0 /∂x2 .∂ 2 w0 /∂y2 + 2w0 /∂x∂y.w0∗ /∂x∂y − ∂ 2 w0 /∂x2 .∂ 2 w0∗ /∂y2 − ∂ 2 w0 /∂y2 .∂ 2 w0∗ /∂x2
(16)
∂4 ∂4 ∂4 2 ˜ in which: ∇ 4 = ∂x 4 + 2 ∂x 2 ∂y 2 + ∂y 4 ; D = A11 1 − ν . Four Eqs. (13.1)–(13.3) and (16) are nonlinear equations with four unknowns w0 , ϑx , ϑy , ϕ. and used to analyze the buckling and post-buckling response of FGP plates taking to account initial geometrical imperfection.
4 Buckling and Postbuckling Analysis In this study, three types of boundary conditions (BCs.), referred to as four edges are simply supported and freely moveable (SSSS), four edges are clamped and freely moveable (CCCC), and two opposite edges are simply supported, the remaining edges are clamped and freely moveable (SCSC) are considered: – SSSS with associated boundary conditions as w0 = ϑs = 0; Nns = 0; Mn = 0; Nn = Nn0
(17)
– CCCC with associated boundary conditions as w0 = ϑn = ϑs = 0; Nns = 0; Nn = Nn0
(18)
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767
– SCSC with associated boundary conditions as at x = 0, a : w0 = ϑy = 0; Nxy = 0; Mx = 0; Nx = Nx0 at y = 0, b : w0 = ϑx = ϑy = 0; Nxy = 0; Ny = Ny0
(19)
in which Nx0 , Ny0 are in-plane compressive loads at movable edges along the directions x, y, respectively. a. Boundary conditions SSSS For this BC, the solutions are chosen as [16, 27, 28]: w0 = w0mn sin αx sin βy; ϑx = ϑxmn cos αx sin βy; ϑy = ϑymn sin αx cos βy; w0∗ = ξ h sin αx sin βy
(20)
nπ where: α = mπ a , β = b ; m, n are the number of half-waves in x and y directions; w0mn , ϑxmn , ϑymn are unknown coefficients to be determined; the coefficient ξ ∈ [0, 1] represents an FGP plate imperfection size. Putting Eq. (20) into Eq. (16), we obtain: 2 + 2ξ hw0mn α 2 β 2 D w0mn 4 ∇ ϕ= (21) (cos 2αx + cos 2βy) 2
The stress function may be assumed in the following form: ϕ = f1 cos 2αx + f2 cos 2βy + Nx0
y2 x2 + Ny0 2 2
(22)
2 Dβ 2 2 Dα 2 where: f1 = 32α 2 w0mn + 2ξ hw0mn ; f2 = 32β 2 w0mn + 2ξ hw0mn . Substituting the solutions (20) into two Eqs. (13.2) and (13.3), the rotations are determined as ϑxmn xmn = D1 w0mn ; ϑymn = D2 w0mn
(23)
A2 C1 −A1 C2 2 2 2 C1 ˜ ˜ ˜s in which: D1 = BA11CB22 −B −A2 B1; D2 = A1 B2 −A2 B1 ; A1= C11 α + C66 β + A44 ; A2 = B1 = C˜ 12 + C˜ 66 αβ; B2 = C˜ 66 α 2 + C˜ 11 β 2 + A˜ s44 ; C1 = A˜ s44 α; C2 = A˜ s44 β. By substituting Eqs. (22) and (23) into Eq. (13.1) and then applying Galerkin’s procedure, we get: 2 + 2ξ hw0mn (w0mn + ξ h) = 0. c1a w0mn + Nx0 α 2 + Ny0 β 2 (w0mn + ξ h) + c2a w0mn
(24)
where: c1a = A˜ s44 + Ksy β 2 + A˜ s44 + Ksx α 2 + (D1 α + D2 β)A˜ s44 + Kw ; c2a =
α 4 +β 4 D . 16
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Consider an FGP plate subjected to bi-axial compressive loads Nx0 = −γ1 N0 h, Ny0 = −γ2 N0 h in the edges x = 0, a and y = 0, b . From the Eq. (24) we have N0 =
¯ c1a c2a W ¯ h; W ¯ = w0mn ¯ 2 + 2ξ W + W ¯ + ξ h γ1 α 2 + γ2 β 2 γ1 α 2 + γ2 β 2 W h
(25)
It is the basic equations used to study the nonlinear post-buckling behavior of perfect and imperfect FGP plate including the critical load determination and load-deflection curve investigation. For perfect plates, ξ = 0; because w0mn = 0, the compressive load is derived in the form: c1a c2a ¯ 2h + N0∗ = (26) W 2 2 2 γ1 α + γ2 β h γ1 α + γ2 β 2 The buckling load obtained: c1a Nbl = 2 γ1 α + γ2 β 2 h
(27)
The buckling loads Nbl expressed by Eq. (27) belong to corresponding buckling modes (m, n). The critical buckling load Ncr is the minimal value of these buckling loads. Thus, for perfect plates ξ = 0, the function N0∗ reaches the minimum at w0mn = 0 ∗ and Nbl = N0 w =0 . 0mn
b. Boundary conditions CCCC For this type of boundary condition, the solutions are chosen as [27, 28]: w0 = w0mn sin2 αx sin2 βy; ϑx = ϑxmn sin 2αx sin2 βy ϑy = ϑymn sin2 αx sin 2βy; w0∗ = ξ h sin2 αx sin2 βy
(28)
Substituting Eq. (28) into Eq. (16), we get: 2 D w0mn + 2ξ hw0mn α 2 β 2 cos 2αx + cos 2βy − cos 4αx − cos 4βy ∇ ϕ= 2 −2 cos 2αx cos 2βy + cos 2αx cos 4βy + cos 4αx cos 2βy 4
(29) The stress function may be assumed in the following form: ϕ = f1 cos 2αx + f2 cos 2βy + f3 cos 4αx + f4 cos 4βy + f5 cos 2αx cos 2βy (30) y2 x2 + Ny0 2 2 2 Dα 2 w0mn + 2ξ hw0mn ; f3 = 64β 2 2 Dα 2 w0mn + 2ξ hw0mn ; 1024β 2
+ f6 cos 2αx cos 4βy + f7 cos 4αx cos 2βy + Nx0 Dβ 2 2 where: f1 = 64α = 2 w0mn + 2ξ hw0mn ; f2 2 Dβ 2 w0mn f4 + 2ξ hw0mn ; = 1024α 2
Post-buckling Response of Functionally Graded Porous Plates
2 Da2 β 2 2 w0mn + 2ξ hw0mn ; 64(α 2 +β 2 ) 2 Da2 β 2 2 w0mn + 2ξ hw0mn . 64(4α 2 +β 2 )
f5 =
f6 =
2 Da2 β 2 2 w0mn 64(α 2 +4β 2 )
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+ 2ξ hw0mn ; f7 =
Substituting Eq. (30) into Eqs. (13.2) and (13.3), applying Galerkin’s procedure, we obtain rotations in term of deflections: ϑxmn = D1 w0mn ; ϑymn = D2 w0mn ;
(31)
A2 C1 −A1 C2 2 C1 in which: D1 = BA11CB22 −B 12C˜ 11 α 2 + 4C˜ 66 β 2 + 3A˜ s55 ; −A2 B1 ;D2 = A1 B2 −A2 B1 ; A1 = A2 = 4 C˜ 66 + C˜ 21 αβ; B1 = 4 C˜ 12 + C˜ 66 αβ; B2 = 4C˜ 66 α 2 + 12C˜ 22 β 2 + 3A˜ s44 ; C1 = 3A˜ s55 α; C2 = 3A˜ s44 β. By substituting Eqs. (30) and (31) into Eq. (13.1) and then implementing Galerkin’s procedure, leads to: 2 w0mn c1b + (w0mn + ξ h) Nx0 α 2 + Ny0 β 2 + c2b w0mn + 2ξ hw0mn (w0mn + ξ h) = 0
(32) where: c1b = A˜ s55 αD1 + A˜ s44 βD2 + 34 Kw + α 2 A˜ s55 + Ksx + β 2 A˜ s44 + Ksy ; c2b
! 5 4 a4 β 4 a4 β 4 a4 β 4 4 α +β − =D 2 + 2 + 2 . 128 24 α 2 + β 2 48 α 2 + 4β 2 48 4α 2 + β 2
For an FGP plate subjected to bi-axial compressive loads Nx0 = −γ1 N0 h, Ny0 = −γ2 N0 h in the sides x = 0, a and y = 0, b , from the Eq. (32) we get: N0 =
¯ c1b W c2b ¯ h ¯ 2 + 2ξ W + W ¯ + ξ h γ1 α 2 + γ2 β 2 γ1 α 2 + γ2 β 2 W
(33)
Similar to the SSSS boundary conditions, from Eq. (33) we can determine the relationship between compressive loads and deflection in case of perfect plates: c1b c2b ¯ 2h + N0∗ = W 2 2 2 γ1 α + γ2 β h γ1 α + γ2 β 2
(34)
and buckling loads: c1b Nbl = γ1 α 2 + γ2 β 2 h
(35)
The buckling loads Nbl expressed by Eq. (35) belong to corresponding buckling modes (m, n). The critical buckling load Ncr is the minimal value of these buckling loads.
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c. Boundary conditions SCSC For this case, the solutions are chosen as [27, 28]: w0 = w0mn sin αx sin2 βy; ϑx = ϑxmn cos αx sin2 βy
(36)
ϑy = ϑymn sin αx sin 2βy; w0∗ = ξ h sin αx sin2 βy
Substituting Eq. (36) into Eq. (16), we have: 2 + 2ξ hw0mn α 2 β 2 D w0mn 4 ∇ ϕ= (cos 2αx + cos 2βy − cos 4βy − cos 2αx cos 2βy) 2 (37) The stress function may be assumed in the following form: ϕ = f1 cos 2αx + f2 cos 2βy + f3 cos 4βy + f4 cos 2αx cos 2βy + Nx0
y2 x2 + Ny0 2 2 (38)
2 Dβ 2 2 Dα 2 where: f1 = 32α = 32β = 2 w0mn + 2ξ hw0mn ; f2 2 w0mn + 2ξ hw0mn ; f3 2 2 2 Dα β Dα 2 2 − 512β 2 w0mn + 2ξ hw0mn ; f4 = − 2 w0mn + 2ξ hw0mn . 32(α 2 +β 2 ) Putting Eq. (38) into Eqs. (13.2) and (13.3), applying Galerkin’s procedure, we obtain rotations in term of deflections: ϑxmn = D1 w0mn ; ϑymn = D2 w0mn
B1 C2 −B2 C1 A2 C1 −A1 C2 ˜ s α; C2 = A˜ s β; A1 A1 B2 −A2 B1 ; D2= A1 B2 −A2B1 ; C1 = 3A55 44 4C66 β 2 + 3As44 ; A2 = C˜ 12 + C˜ 66 αβ; B1 = 4 C˜ 12 + C˜ 66 αβ; B2
in which: D1 = α2
(39) =
3C11 + = C˜ 66 α 2 + 4C˜ 11 β 2 + A˜ s44 . By substituting Eqs. (38) and (39) into Eq. (13.1) and then implementing Galerkin’s procedure, we get: 2 w0mn c1c + 3Nx0 α 2 + 4Ny0 β 2 (w0mn + ξ h) + c2c w0mn + 2ξ hw0mn (w0mn + ξ h) = 0
where: c1c
(40) = 3A˜ s44 αD1 + 4A˜ s44 βD2 + 3Kw + 3 A˜ s44 + Ksx α 2 + 4 A˜ s44 + Ksy β 2 ; c2c
! 17α 4 β4 α4 β 4 + + =D 2 . 64 4 8 α2 + β 2
Similar to boundary conditions SSSS and CCCC, in case of boundary condition SCSC, the relationship between compressive loads and deflection is determined in case of imperfect plates subjected to bi-axial compressive loads Nx0 = −γ1 N0 h, Ny0 = −γ2 N0 h at the edges x = 0, a and y = 0, b : N0 =
¯ c1c W c2c ¯ h. ¯ 2 + 2ξ W + W ¯ + ξ h 3γ1 α 2 + 4γ2 β 2 3γ1 α 2 + 4γ2 β 2 W
(41)
Post-buckling Response of Functionally Graded Porous Plates
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For the perfect plate: c1c c2c ¯ 2h + N0∗ = W 3γ1 α 2 + 4γ2 β 2 h 3γ1 α 2 + 4γ2 β 2
(42)
From there, the buckling and critical loads are determined as follow: c1c Nbl = 2 3γ1 α + 4γ2 β 2 h
⇒ Ncr = min{Nbl }
(43)
5 Numerical Results and Discussion Based on the above mentioned presented analytical solution, the Matlab’s code is built to implement numerical examples. Numerical results are presented for nonlinear analysis unless previously stated. For convenience, the nondimensional results are used in the form [29–31]: Ksy b2 a2 Kw a4 Ksx a2 = ; E0 = 1.0 GPa; ; K = ; J = N¯ = Ncr 0 0 E1 h2 E0 h3 E0 h3 ν E0 h3 ν
(44)
5.1 Validation Examples Example 1: Validation for isotropic plate. Consider a simply supported isotropic rectangular plate (e0 = 0, h = 0.1 in., a = 20 in., a/b = 2) subjected to in-plane uniaxial compressive load (x-direction, γ1 = 1, γ2 = 0). Table 1 presents the critical loads Pcr = bhNcr of isotropic plates made of different materials such as Aluminum (E = 10 × 106 psi, ν = 0.3), Titanium alloy (E = 15.1 × 106 psi, ν = 0.3) and Stainless steel (E = 30.1 × 106 psi, ν = 0.3). The obtained results are compared with those of Brush and Almroth [26] used the analytical method. Table 1. Critical loads Pcr of simply supported isotropic rectangular plate under uniaxial compression in the x-direction (γ1 = 1, γ2 = 0) Pcr [pound]
Brush and Almroth [26]
Present
Error (%)
E = 10 × 106 psi, ν = 0.3 Aluminum (Al)
3620 (2, 1)a
3613.20 (2, 1)
0.188
E = 15.1 × 106 psi, ν = 0.3 Titanium alloy (Ti-6Al-4V)
5459 (2, 1)
5455.93 (2, 1)
0.056
E = 30.1 × 106 psi, ν = 0.3 Stainless steel (SUS304)
10882 (2, 1)
10875.74 (2, 1)
0.058
a The numbers in brackets indicate the buckling mode (m, n)
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The post-buckling load-deflection curves Ncr /h- w0mn /h of simply supported isotropic square plate (e0 = 0, E1 = 380 GPa, ν = 0.3, b/h = 40) are illustrated in Fig. 3 for two cases: perfect plate (ξ = 0) and imperfect plate (ξ = 0.1). The results are compared with those given by Tung and Duc [16] using the classical plate theory.
Fig. 3. Post-buckling load-deflection curves of the isotropic square plate
C11
Table 2 presents the comparison of the nondimensional critical load Nˆ = hb2 Ncr of the isotropic square plate made of SiC (E = 420 GPa, ν = 0.3, a/h −B2 /A 11
11
= 10, b/a = 1, boundary conditions SSSS and SCSC) under bi-axial compressive loads (γ1 = γ2 = 1) with the results given by Thai và Choi [32] using analytical approach and refined four-variable shear deformation theory. Table 2. Nondimensional critical loads Pcr of simply supported isotropic rectangular plate subjected to bi-axial compressive loads (γ1 = γ2 = 1) Method
SSSS
SCSC
Thai and Choi [32] 18.6861 34.1195 Present Error (%)
18.6854 33.9600 0.004
0.467
Example 1: Validation for functionally graded porous plate. Table 3 shows the values of nondimensional critical load of square FGP plates under uniaxial in-plane compression in x-direction (a/h = 10, b/a = 1, γ1 = 1, γ2 = 0) with different porosity coefficients e0 and two porosity distribution patterns: uniform and non-uniform symmetric. The porous plate made of metal foam with material properties: G1 = 26, 293 GPa, ν = 0.3, E1 = 2G1 (1 + ν) [10]. The obtained results are compared with those of Thang et al. [31] using Navier’s solution and FSDT. From two above-mentioned validation examples, a good agreement can be found.
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Table 3. Nondimensional critical loads N¯ = Ncr a2 /E1 h2 of simply supported FGP plate under uniaxial in-plane compression (γ1 = 1, γ2 = 0, b/a = 1, a/h = 10) Porosity distribution
Sources
e0 0.1
Uniform
0.2
0.3
0.4
0.5
0.6
Thang et al. [31] 3.2109 2.9856 2.7549 2.5173 2.2710 2.0135 Present
3.2023 2.9777 2.7475 2.5105 2.2650 2.0081
Error (%)
0.268
0.265
0.269
0.270
0.264
0.268
Non-uniform symmetric Thang et al. [31] 3.3023 3.1729 3.0432 2.9130 2.7822 2.6506 Present
3.2933 3.1640 3.0343 2.9041 2.7733 2.6417
Error (%)
0.273
0.281
0.292
0.306
0.320
0.336
5.2 Parametric Study In this section, the effect of material properties, boundary conditions, elastic foundation parameters, and initial imperfection on buckling and post-buckling response of FGP plates is examined. Consider the metal foam rectangular plate resting in Pasternak’s elastic foundation with input data: h = 0.1 m, E 1 = 200 GPa, ν = 1/3. This plate subjected to bi-axial in-plane compression Nx0 = −γ1 N0 h, Ny0 = −γ2 N0 h at the edges x = 0, a and y = 0, b. Table 4 presents nondimenssional critical loads N¯ and corresponding buckling modes of perfect square FGP plates (h = 0.1 m, b/a = 1, a/h = 10; K 0 = J 0 = 0) with three porosity distribution patterns namely Type 1 (uniform), Type 2 (symmetric), and Type 3 (non-symmetric), and different porosity coefficients (e0 = 0.1; 0.3; 0.5 and 0.8). The plate under various BCs: SSSS, CCCC and SCSC, and subjected to uniaxial in-plane compression. Table 4. Nondimensional critical loads N¯ of square FGP plates with different porosity coefficients e0 and various porosity distribution patterns under three boundary condition types Boundary condition SSSS, γ1 = 1, γ2 = 0
CCCC, γ1 = 1, γ2 = 0
Porosity distribution types
e0 0.1
0.3
0.5
0.8
Type 1 (uniform)
3.2696(1,1)
2.8053(1,1)
2.3126(1,1)
1.4676(1,1)
Type 2 (symmetric)
3.3623(1,1)
3.0973(1,1)
2.8302(1,1)
2.4222(1,1)
Type 3 (non-symmetric)
3.2869(1,1)
2.8486(1,1)
2.3639(1,1)
1.4688(1,1)
Type 1 (uniform)
7.9758(1,1)
6.8431(1,1)
5.6413(1,1)
3.5800(1,1)
Type 2 (symmetric)
8.1825(1,1)
7.4943(1,1)
6.7949(1,1)
5.7078(1,1)
Type 3 (non-symmetric)
8.0148(1,1)
6.9442(1,1)
5.7701(1,1)
3.6273(1,1) (continued)
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Boundary condition SCSC, γ1 = 1, γ2 = 0
SCSC, γ1 = 0, γ2 = 1
Porosity distribution types
e0 0.1
0.3
0.5
0.8
Type 1 (uniform)
5.8520(2,1)
5.0209(2,1)
4.1391(2,1)
2.6267(2,1)
Type 2 (symmetric)
6.0004(2,1)
5.4887(2,1)
4.9679(2,1)
4.1562(2,1)
Type 3 (non-symmetric)
5.8801(2,1)
5.0943(2,1)
4.2343(2,1)
2.6689(2,1)
Type 1 (uniform)
5.1855(1,1)
4.4491(1,1)
3.6678(1,1)
2.3276(1,1)
Type 2 (symmetric)
5.3237(1,1)
4.8844(1,1)
4.4389(1,1)
3.7496(1,1)
Type 3 (non-symmetric)
5.2115(1,1)
4.5157(1,1)
3.7508(1,1)
2.3496(1,1)
a The number in brackets indicate the buckling mode (m, n)
The variation of nondimensional critical loads with respect to porosity coefficients and various porosity distribution patterns is illustrated in Fig. 4: (a) SSSS, γ1 = 1, γ2 = 0; (b) CCCC, γ1 = 1, γ2 = 0; (c) SCSC, γ1 = 1, γ2 = 0; (d) SCSC, γ1 = 0, γ2 = 1. The obtained results indicated that when porosity coefficients increase, the critical buckling (a)
(b )
(c )
(d )
Fig. 4. The variation of nondimensional critical loads of square FGP plates with respect to porosity coefficients e0 and porosity distribution patterns: (a) SSSS, γ1 = 1, γ2 = 0; (b) CCCC, γ1 = 1, γ2 = 0; (c) SCSC, γ1 = 1, γ2 = 0; (d) SCSC, γ1 = 0, γ2 = 1
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loads of FGP plates decrease for all patterns of porosity distribution. This is due to an increase in the porosity coefficient that results in a reduction of FGP plate stiffness. The critical buckling loads of Type 2 (non-uniform symmetric) are the highest. The critical buckling loads of Type 1 (uniform) and Type 3 (non-uniform symmetric) distributions are slightly different. The effect of boundary conditions is clearly illustrated: as expected, the critical buckling load of the CCCC plate is the highest and of the SSSS plate is the smallest. (b )
(a)
(c )
Fig. 5. The effect of in-plane compressive load types on the post-buckling load-deflection curves for the SSSS (a), SCSC (b) and CCCC (c) FGP plates
Figure 5 shows post-buckling load-deflection curves of perfect and imperfect rectangular FGP (h = 0.1m, a/h = 10, b/a = 2, e0 = 0.5, K 0 = J 0 = 0) with Type 2 porosity distribution under different types of in-plane compressive loads. As can be observed that the post-buckling curve of the bi-axial compressive load is the lowest, and the curve of uniaxial compressive loads along the shorter edge is the highest for all cases of boundary conditions. Besides, it is seen that better resistance to compression in the longer side than the shorter side. Effect of porosity distribution and porosity coefficient on the post-buckling loaddeflection curves of perfect and imperfect simply supported (SSSS) rectangular FGP plates ( h = 0.1m, a/h = 10, b/a = 2, e0 = 0.5, K 0 = J 0 = 0) under uniaxial compression (x-direction) is shown in Fig. 6 and 7. It can be seen that the post-buckling load-deflection curves for Type 2 distribution are higher than remain two distribution types; the higher
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porosity coefficient, the lower post-buckling load-deflection curve. This trend is similar for both perfect and imperfect plates.
Fig. 6. Effect of porosity distribution patterns on post-buckling of FGP plates
Fig. 7. Effect of porosity coefficients e0 on post-buckling of FGP plates
Figure 8 shows the influence of imperfection size on the post-buckling response of simply supported FGP plates (h = 0.1 m, a/h = 10, b/a = 2, K 0 = J 0 = 0,) under uniaxial (x-direction) compression. It is obvious that the post-buckling curves of perfect FGP plates are higher than those of imperfect FGP plates when deflection is small. The effect of Winkler and Pasternak elastic foundation stiffness on load-deflection curves of simply supported FGP plates (h = 0.1m, a/h = 10, b/a = 2, e0 = 0.5) under uniaxial (x-direction) compression is depicted in Fig. 9. It figure shows that the postbuckling load-deflection curves become higher as the linear Winkler foundation parameter K 0 and Pasternak foundation parameter J 0 increased. Furthermore, the effect of Pasternak foundation is larger than the Winkler foundation.
Fig. 8. The effect of the imperfection ξ on the Fig. 9. The effect of elastic foundation post-buckling of FGP plates parameters on the post-buckling of FGP plates
Post-buckling Response of Functionally Graded Porous Plates
Fig. 10. The effect of the side-to-thickness ratio a/h on post-buckling of FGP plates
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Fig. 11. The effect of aspect ratio b/a on post-buckling of FGP plate
The Figs. 10 and 11 depict the effect of side-to-thickness ratio a/h and aspect ratio b/a on the post-buckling behavior of perfect and imperfect simply supported rectangular FGP plates (h = 0.1 m, e0 = 0.5, K 0 = J 0 = 0) with Type 2 porosity distribution, respectively. It can be seen that the post-buckling load-deflection curves of imperfect (ξ = 0.1) and perfect (ξ = 0) plates move downward as the side-to-thickness ratio a/h and aspect ratio b/a increase. In other words, when the side-to-thickness ratio a/h and aspect ratio b/a increase, the bearing load capacity decreases.
6 Conclusions In this paper, the buckling and post-buckling analysis of FGP plates resting on Pasternak elastic foundation and subjected to in-plane compressive loads are presented based on the first-order shear deformation theory. The initial geometrical imperfection is taken into account. Three porosity distribution patterns namely uniform, non-uniform symmetric, non-uniform asymmetric are considered. By using the Galerkin method and Airy’s stress function, the analytical solution is developed for FGP rectangular plates under various types of boundary condition. The validate examples are conducted and the accuracy between obtained results and published ones is found. Numerical results indicate the significant effects of material parameters (porosity coefficient, porosity distribution patterns), geometric parameters (aspect and side-to-thickness ratio), initial geometrical imperfection, elastic foundation as well as in-plane boundary conditions on the critical buckling loads and the postbuckling response of FGP plates. Acknowledgements. This research is funded by the Ministry of Education and Training for Science and Technology Project under grant number: CT.2019.03.04.
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References 1. Smith, B.H., Szyniszewski, S., Hajjar, J.F., Schafer, B.W., Arwade, S.R.: Steel foam for structures: a review of applications, manufacturing and material properties. J. Constr. Steel Res. 71, 1–10 (2012) 2. Ashby, M.F., et al.: Metal foams: a design guide. Appl. Mech. Rev. 54(6), B105–B106 (2001) 3. Lefebvre, L.P., Banhart, J., Dunand, D.C.: Porous metals and metallic foams: current status and recent developments. Adv. Eng. Mater. 10(9), 775–787 (2008) 4. Wattanasakulpong, N., Ungbhakorn, V.: Linear and nonlinear vibration analysis of elastically restrained ends FGM beams with porosities. Aerosp. Sci. Technol. 32(1), 111–120 (2014) 5. Chen, D., Yang, J., Kitipornchai, S.: Elastic buckling and static bending of shear deformable functionally graded porous beam. Compos. Struct. 133, 54–61 (2015) 6. Chen, D., Yang, J., Kitipornchai, S.: Free and forced vibrations of shear deformable functionally graded porous beams. Int. J. Mech. Sci. 108, 14–22 (2016) 7. Chen, D., Kitipornchai, S., Yang, J.: Nonlinear free vibration of shear deformable sandwich beam with a functionally graded porous core. Thin-Walled Struct. 107, 39–48 (2016) 8. Jabbari, M., Mojahedin, A., Khorshidvand, A.R., Eslami, M.R.: Buckling analysis of a functionally graded thin circular plate made of saturated porous materials. J. Eng. Mech. 140(2), 287–295 (2014) 9. Jabbari, M., Hashemitaheri, M., Mojahedin, A., Eslami, M.R.: Thermal buckling analysis of functionally graded thin circular plate made of saturated porous materials. J. Therm. Stresses 37(2), 202–220 (2014) 10. Rezaei, A.S., Saidi, A.R.: Application of Carrera Unified Formulation to study the effect of porosity on natural frequencies of thick porous–cellular plates. Compos. Part B Eng. 91, 361–370 (2016) 11. Rezaei, A.S., Saidi, A.R., Abrishamdari, M., Mohammadi, M.P.: Natural frequencies of functionally graded plates with porosities via a simple four variable plate theory: an analytical approach. Thin-Walled Struct. 120, 366–377 (2017) 12. Akba¸s, SD.: ¸ Vibration and static analysis of functionally graded porous plates. J. Appl. Comput. Mech. 3(3), 199–207 (2017) 13. Liew, K.M., Yang, J., Kitipornchai, S.: Postbuckling of piezoelectric FGM plates subject to thermo-electro-mechanical loading. Int. J. Solids Struct. 40(15), 3869–3892 (2003) 14. Bakora, A., Tounsi, A.: Thermo-mechanical post-buckling behavior of thick functionally graded plates resting on elastic foundations. Struct. Eng. Mech. 56(1), 85–106 (2015) 15. Feyzi, M.R., Khorshidvand, A.R.: Axisymmetric post-buckling behavior of saturated porous circular plates. Thin-Walled Struct. 112, 149–158 (2017) 16. Van Tung, H., Duc, N.D.: Nonlinear analysis of stability for functionally graded plates under mechanical and thermal loads. Compos. Struct. 92(5), 1184–1191 (2010) 17. Cong, P.H., Chien, T.M., Khoa, N.D., Duc, N.D.: Nonlinear thermomechanical buckling and post-buckling response of porous FGM plates using Reddy’s HSDT. Aerosp. Sci. Technol. 77, 419–428 (2018) 18. Barati, M.R., Zenkour, A.M.: Analysis of postbuckling behavior of general higher-order functionally graded nanoplates with geometrical imperfection considering porosity distributions. Mech. Adv. Mater. Struct. 26(12), 1081–1088 (2019) 19. Phung-Van, P., Thai, C.H., Ferreira, A.J.M., Rabczuk, T.: Isogeometric nonlinear transient analysis of porous FGM plates subjected to hygro-thermo-mechanical loads. Thin-Walled Struct. 148, 106497 (2020) 20. Tu, T.M., Hoa, L.K., Hung, D.X., Hai, L.T.: Nonlinear buckling and post-buckling analysis of imperfect porous plates under mechanical loads. J. Sandwich Struct. Mater. 22(6), 1910–1930 (2020)
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21. Binh, C.T., Van Long, N., Tu, T.M., Minh, P.Q.: Nonlinear vibration of functionally graded porous variable thickness toroidal shell segments surrounded by elastic medium including the thermal effect. Compos. Struct. 255, 112891 (2020) 22. Barati, M.R., Zenkour, A.M.: Investigating post-buckling of geometrically imperfect metal foam nanobeams with symmetric and asymmetric porosity distributions. Compos. Struct. 182, 91–98 (2017) 23. Larbi, L.O., Kaci, A., Houari, M.S.A., Tounsi, A.: An efficient shear deformation beam theory based on neutral surface position for bending and free vibration of functionally graded beams#. Mech. Based Des. Struct. Mach. 41(4), 421–433 (2013) 24. Reddy, J.N.: Theory and Analysis of Elastic Plates and Shells. CRC Press, Boca Raton (2006) 25. Reddy, J.N.: Energy Principles and Variational Methods in Applied Mechanics. Wiley, Hoboken (2017) 26. Brush, D.O., Almroth, B.O.: Buckling of Bars, Plates, and Shells. Mc GrawHill, New York (1975) 27. Sobhy, M.: Buckling and free vibration of exponentially graded sandwich plates resting on elastic foundations under various boundary conditions. Compos. Struct. 99, 76–87 (2013) 28. Meziane, M.A.A., Abdelaziz, H.H., Tounsi, A.: An efficient and simple refined theory for buckling and free vibration of exponentially graded sandwich plates under various boundary conditions. J. Sandwich Struct. Mater. 16(3), 293–318 (2014) 29. Thai, H.T., Choi, D.H.: A refined plate theory for functionally graded plates resting on elastic foundation. Compos. Sci. Technol. 71(16), 1850–1858 (2011) 30. Zenkour, A.M.: The refined sinusoidal theory for FGM plates on elastic foundations. Int. J. Mech. Sci. 51(11–12), 869–880 (2009) 31. Thang, P.T., Nguyen-Thoi, T., Lee, D., Kang, J., Lee, J.: Elastic buckling and free vibration analyses of porous-cellular plates with uniform and non-uniform porosity distributions. Aerosp. Sci. Technol. 79, 278–287 (2018) 32. Thai, H.T., Choi, D.H.: An efficient and simple refined theory for buckling analysis of functionally graded plates. Appl. Math. Model. 36(3), 1008–1022 (2012)
Elastic Buckling Behavior of FG Polymer Composite Plates Reinforced with Graphene Platelets Using the PB2-Ritz Method Xuan Hung Dang, Dai Hao Tran(B) , Minh Tu Tran, and Thanh Binh Chu National University of Civil Engineering, Hanoi, Vietnam {hungdx,haotd,tutm,binhct}@nuce.edu.vn
Abstract. The buckling behavior of functionally graded polymer composite plates reinforced with graphene platelets (GPL) is investigated using first-order shear deformation theory. The rectangular functionally graded graphene platelets reinforced composite (FG-GPLRC) plate subjected to in-plane compression loads. The Halpin-Tsai micromechanics model is employed to estimate the effective material properties of the FG-GPLRC. Four GPL distribution types are considered. The pb2-Ritz method is implemented for developing theoretical modelling. The present results have been validated by comparison with those of available in published literature. The influence of material parameters such as GPL weight fraction, distribution type on the critical loads of the laminated FG-GPLRC plates is investigated. Besides, the effects of plate length-to-thickness, aspect ratio and boundary conditions are also studied through the numerical examples. Keywords: First-order shear deformation theory · Buckling behavior · Graphene platelet · pb2-Ritz method
1 Introduction In the era of the strong development of science and technology, the invention of new technologies and materials plays an important role. The finding advanced materials with outstanding properties compared to traditional materials has always attracted many domestic and foreign researchers. Since its invention in 1991, carbon nanotubes (CNTs) are chosen as the reinforcing nanofillers to improve the strength and of nanocomposite structures due to their excellent mechanical and chemical properties [1–5]. Many studies have shown that when graphene sheets are rolled-up to form CNTs may provide considerable improvement of polymeric matrix properties, especially when they are dispersed at low concentrations. Besides these advantages, some obstacles affect the intensive applications of CNT-reinforced polymer composites including high production cost, difficult to make the CNT uniform dispersions. Recently a novel type of carbon material - graphene nanoplatelet (GPL) is developed. GPLs are two-dimensional carbon materials, can be used as advanced reinforcement for polymer matrix to enhance their mechanical properties. Through the conducted experiment, Rafiee et al. [6] have compared the mechanical stiffness and strength of the © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 780–794, 2022. https://doi.org/10.1007/978-981-16-3239-6_60
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reinforced polymer composites in two cases: a) adding 0.1% weight fraction (wt.%) of GPLs; b) adding 1.0 wt.% of carbon nanotubes (CNTs). The obtained results are similar, and their elastic moduli increase by around 31%. Functionally graded graphene nanoplatelets reinforced composite materials (FG-GPLRCs) are a combination of FGMs and GPLCs and become a new concept attracting the attention of the scientific community since it can be widely used in many engineering applications. The numerous works reported about the response of FG-GPLRC shows the potential application of these types of nanocomposites in various industrial fields. Bending and buckling response of multi-layered FG-GPLRC plates based on firstorder shear deformation theory (FSDT) is investigated by Song et al. [7]. Using FSDT and von Kármán-type nonlinearity, Song et al. [8] examined the buckling and postbuckling behaviors of laminated FG-GPLRC plates under biaxial compressive loads. Kitipornchai et al. [9] examined the natural frequency and buckling behavior of Timoshenko FGP nanocomposite reinforced by graphene platelets using Ritz method. The buckling and post-buckling of FG laminated nanocomposite beams reinforced by GPLs rested on elastic foundation are presented by Yang et al. [10]. Based on higher-order deformation theory (HSDT), Shen et al. [11] presented buckling and post-buckling of laminated FG-GPLRC plates working in thermal environments. Mirzaei and Kiani [12] analyzed the thermal buckling response of laminated FG-GLRC plates based on isogeometric finite element method (FEM) and FSDT. Using the FSDT and FEM, Reddy et al. [13] studied the free vibration of thin, moderately thick and thick laminated FG-GLRC plates. Applying the element-free improved moving least-squares Ritz method, Guo et al. [14] studied free vibration of laminated FG-GLRC quadrilateral plates. Yang et al. [15] investigated free vibrational characteristic and buckling behavior of FGP nanocomposite plates reinforced with GPLs using Chebyshev-Ritz method. Thai et al. [16] presented the bending, buckling and free vibration response of laminated FG-GPLRC plates by using the NURBS approach in conjunction with the four-variable refined plate theory. Using the spectral-Chebyshev approach, Anamagh and Bediz [17] investigated the vibrational characteristic and buckling response of FGP plates reinforced with GPLs. Based on HSDT, Noroozi et al. [18] proposed the meshfree method to predict the natural frequency and buckling loads of laminated FG-GPLRC perforated plates under in-plane compression. An above-mentioned review shows that there is a lack of studies on buckling analysis of multilayer FG-GPLRC thick plates under various boundary conditions using a Rayleigh-Ritz method. Thus, in this study, the pb-2 Rayleigh-Ritz method is proposed based on the two-dimensional Ritz function is applied to analyze elastic buckling of multilayer FG-GPLRC plates with different boundary conditions. Convergence test is made to illustrate the reliability of the present approach. Furthermore, the effects of different GPLs distributions, the GPLs weight fraction, plate geometric parameters, and boundary conditions on the buckling parameters is investigated.
2 Laminated FG-GPLRC Plates Consider a rectangular laminated FG-GPLRC plate with length a, width b and thickness h as shown in Fig. 1. The plate is composed of NL laminae with equal thickness and
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perfectly bonded. Assume that the individual laminae are made of mixture polymeric matrix and GPLs. GPLs are supposed to be uniformly dispersed in each lamina, but GPLs weigh fraction varies along the thickness direction according to four GPL distributions patterns including either uniformly distributed (UD) or FG distributions (V, O, and X).
Fig. 1. Geometry and configuration of the laminated FG-GPLRC plate with four GPLs distribution patterns through the thickness direction (k) The k th layer GPL weight fraction gGPL (k = 1, 2, .....NL ) is expressed as: (k) gGPL (k) gGPL (k) gGPL (k) gGPL
∗ = gGPL (UD) ∗ /(1 + N ) = 2kgGPL (FG − V ) L ∗ = 2gGPL [NL + 1 − |NL + 1 − 2k|]/(2 + NL ) (FG − O) ∗ [1 + |N + 1 − 2k|]/(2 + N ) = 4gGPL (FG − X ) L L
(1)
∗ where gGPL is the plate GPLs weight fraction. NL is the plate number of laminae. (k) The GPLs volume fraction VGPL of the k th lamina is determined as (k)
VGPL =
(k)
(k)
gGPL
(k)
gGPL + (ρGPL /ρm )(1 − gGPL )
(2)
in which ρGPL is GPLs density and ρm is matrix density. The effective material properties of the GPL/polymer composite of the k th lamina can be determined according to the Halpin–Tsai micromechanical model and the rule of mixture, respectively: (k) (k) 3 1 + ξL ηL VGPL 5 1 + ξw ηw VGPL (k) Ec = Em + Em 8 1 − ηL V (k) 8 1 − ηw V (k) (3) GPL
GPL
(k) (k) νc(k) = νGPL VGPL + νm (1 − VGPL )
where ηL and ηw are defined as: ηL =
EGPL /Em − 1 EGPL /Em − 1 ; ηw = EGPL /Em + ξL EGPL /Em + ξw
(4)
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in which Em , EGPL and νm , νGPL are Young’s moduli, Poisson’s ratios of the polymer matrix and GPLs, respectively; ξL and ξw are the geometric characteristic parameters of GPLs, defined as: ξL = 2
lGPL wGPL ; ξw = 2 tGPL tGPL
(5)
where lGPL , wGPL and tGPL are the GPL length, width and thickness, respectively.
3 Theoretical Formulation According to the FSDT, the displacements can be assumed as [19]: u = u0 (x, y) + zθx (x, y); v = v0 (x, y) + zθy (x, y); w = w0 (x, y)
(6)
where u0 , v0 , w0 are displacement components of a point on the mid-plane of the plate along with the (x, y, z) axes; θy , θx are the mid-plane rotations of the transverse normal about the x- and y- axes, respectively. The strain-displacement relationship is expressed as: ⎫ ⎧
⎨ εxx ⎬ γyz = γ0 (7) = ε0 + zκ; ε ⎩ yy ⎭ γxz γxy in which
⎧ ⎧ ⎫ ⎫ ∂θx ∂u0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎫ ⎪ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂w0 ∂x ∂x ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ + θ ⎨ ⎨ ∂v0 ⎬ ⎬ ⎬ ⎨ y⎪ ∂θy ∂y ; γ0 = ; κ= ε0 = ∂y ∂y ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂w ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ 0 + θx ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂θ ∂v ∂u ∂θ 0 0 y x ∂x ⎪ ⎪ ⎪ ⎪ ⎩ ⎩ ⎭ ⎭ + + ∂y ∂x ∂y ∂x
The stress-strain relation of the k th lamina is given as: ⎫(k) ⎡ (k) (k) ⎫(k) ⎧ ⎤⎧ ˜ Q ˜ Q 0 0 0 ⎪ ⎪ εxx ⎪ σxx ⎪ 11 12 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (k) ˜ (k) ⎪ ⎪ ⎪ ⎪ ⎥⎪ ⎢Q ⎪ ⎪ ⎪ 0 0 0 ε ⎨ ⎬ ⎬ ⎨ σyy ⎪ ⎥ ⎢ ˜ 21 Q yy 22 ⎥ ⎢ (k) ˜ =⎢ 0 σxy γ ⎥ 0 0 0 Q 66 ⎥⎪ xy ⎪ ⎢ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ˜ (k) 0 ⎦⎪ ⎣ 0 σyz ⎪ γyz ⎪ 0 0 Q ⎪ ⎪ ⎪ ⎪ 44 ⎪ ⎪ ⎪ ⎩ ⎭ ⎩σ ⎪ (k) γxz ⎭ ˜ xz 0 0 0 0 Q
(8)
(9)
55
where ˜ (k) = Q ˜ (k) = Q 11 22
(k)
Ec
(k) 1 − (vc )2
˜ (k) = Q ˜ (k) = ;Q 12 21
(k) (k)
vc Ec
(k) 1 − (vc )2
˜ (k) = Q ˜ (k) = Q ˜ (k) = ;Q 66 44 55
(k)
Ec
(k)
2(1 + vc )
(10)
in which the elastic moduli in the principal material coordinate and Poisson’s ratios of the GPL/composite materials, respectively are determined from Eq. (2) to (5).
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Nx
Nx
Ny
y
Fig. 2. A rectangular laminated FG-GPL reinforced plate under biaxial compressions
A rectangular plate under in-plane biaxial compressive loading is considered (see Fig. 2). To obtain the equilibrium equations for buckling problems, Hamilton’s principle is applied. Hamilton’s principle with absence kinetic energy takes the form: δU + δV = 0
(11)
The variation of strain energy δU and the variation of potential energy δV are given by: 1 δU = 2
1 ε SεdA; δV = − 2
T
A
Nx
A
∂w ∂x
2
+ Ny
∂w ∂y
2 dA
(12)
where ⎧ ⎫ ⎤ ⎡ A˜ B˜ 0 ⎨ ε0 ⎬ ˜ 0 ⎦; Nx = N0 ; ε = κ ; S = ⎣ B˜ D ⎩ ⎭ γ0 0 0 A˜ s
Ny = λN0
(13)
For a laminated FG-GPLRC plate, the stiffness coefficients are defined as: NL hk+1 ˜ (k) (1, z, z 2 )dz ˜ ij ) = (A˜ ij , B˜ ij , D Q ij k=1 h k NL hk+1 ˜ (k) dz Q A˜ sij = Ks ij
(i, j = 1, 2, 6); (14)
(i, j = 4, 5)
k=1 h k
where Ks = 5/6 is the transverse shear correction factor; hk , hk−1 are the coordinates from the mid-plane of the plate to the lower and upper surface of k th layer;
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4 Pb2-Ritz Method Based on the FSDT According to the FSDT, the Pb2-Ritz method approximates the displacement components by a sum of a polynomial series that satisfies the boundary conditions as following: u0 (x, y) = w0 (x, y) = θy (x, y) =
n m
Uijmn fij (x, y)ψ u (x, y); v0 (x, y) =
i=1 j=1 n m i=1 j=1 n m
Wijmn fij (x, y)ψ w (x, y);
n m
Vijmn fij (x, y)ψ v (x, y)
i=1 j=1 n m
θx (x, y) =
θijxmn fij (x, y)ψ θx (x, y)
i=1 j=1
θij fij (x, y)ψ θy (x, y) ymn
i=1 j=1
(15) ymn
where Uijmn , Vijmn , Wijmn , θijxmn , θij are the unknowns; the complete set of twoi−1 j−1 (i = 1, 2, 3, ...m; j = dimensional polynomials fij(x, y) can be expressed as: X Y α 1, 2, 3, ...n); ψ (x, y) and α = u, v, w, θx , θy are the functions, which are chosen to satisfy the geometric boundary conditions (BCs) and are expressed as: ψ α (x, y) =
ne k (x, y) k
(16)
k=1
in which ne is the number of supporting edges; k is the equation of the k th supporting edge (Fig. 3); k are the representative parameters of the boundary conditions and are given in Table 1. In this research, three types of BCs are considered: clamped edge (C), simply supported edge (S) and free edge (F). The detail of these boundary conditions is followings. for clamped edge x = 0, a : u0 = v0 = w0 = 0; θx = θy = 0 y = 0, b : u0 = v0 = w0 = 0; θx = θy = 0
(17)
for simply supported edge x = 0, a : u0 = v0 = w0 = 0 y = 0, b : u0 = v0 = w0 = 0
(18)
x = 0, a : −;
(19)
for free edge y = 0, b : −
Substituting Eqs. (15) into Eqs. (11), the eigenvalue problem of stability of laminated FG-GPLRC plate takes the form: =0 (K + N0 M)X
(20)
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k F
S C
u, v, w 0
1 1
θx , θy
0 1
0
Fig. 3. A rectangular laminated FG-GPL reinforced plate
is the unknown where K is the stiffness matrix and M is the mass matrix. X ymn mn mn mn xmn . coefficients displacement vector Uij , Vij , Wij , θij , θij The eigenvalue problem is solved by using the standard eigenvalue algorithm provided in a Matlab code.
5 Numerical Results and Discussions Firstly, the convergence is conducted and next, the accuracy of the proposed Pb-2 method is validated by a comparison between the obtained results and those of available ones. Afterwards, the influences GPL distribution type, GPL weight fraction as well as plate geometric parameters on the non-dimensional critical loads of the laminated FG-GPLRC plate are examined through the numerical examples. Material properties GPLs and epoxy/polymer matrix are [8]: EGPL = 1.01 TPa; vGPL = 0.186; ρGPL = 1.06 g/cm3 ; Em = 3 GPa; vm = 0.34; ρm = 1.2 g/cm3 ; The GPLs dimensions are lGPL = 2.5 μm; lGPL = 2.5 μm; wGPL = 1.5 μm; tGPL = 1.5 nm. The formula of non-dimensional critical load is defined as: a2 N¯ cr = N0 Em h3
(21)
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5.1 Convergence Study Table 2 presents the convergence of non-dimensional critical loads of laminated (UD) FG-GPLRC square plates subjected to bi-axial compressive loads (λ = 1) under various types of BCs. Geometric parameters of the plate are: b/a = 1; a/h = 10. The GPL ∗ = 1% and number of layers NL = 10 are selected. It can be seen weight fraction gGPL that when the number of degree sets of polynomial terms increases, the non-dimensional buckling load converges to an upper-bound value. Thus, in the next numerical examples, the degree sets of polynomial terms m = n = 7 are used for all calculations to achieve accurate solutions. Table 2. Convergence study of non-dimensional critical load under various types of boundary conditions BCs
1×1
2×2
3×3
4×4
5×5
6×6
7×7
SSSS
134.7754
27.6614
8.2247
7.6880
7.1802
7.1087
7.0739
CCCC
134.7754
52.8162
18.7293
18.4518
18.1431
18.1425
18.1419
CSCS
134.7754
36.2967
13.9082
13.6167
13.2801
13.2796
13.2180
CFCF
134.7754
25.2243
10.2186
9.3780
9.2344
9.0139
9.0043
SFSF
134.7754
4.6987
4.2948
3.5585
3.5091
3.5064
3.4869
5.2 Validation Studies For the validation study, isotropic square plate under in-plane biaxial compressions and various BCs is considered. The following material properties are used: E = 210GPa; μ = 0.3; G = E/2(1 + μ). The calculated results are validated with existing results given by Xiang in [20] and are presented in Table 3. Xiang used the pb-2 Ritz method and the FSDT. It can be observed that the very good agreement is found with the small discrepancy between these results. 5.3 Effect of Number of Laminae NL Effect of the number of laminae on non-dimensional critical loads of laminated SSSS ∗ = square FG-GPLRC plates subjected to biaxial compressions with a/h = 10, gGPL 1% is presented in Table 4 and depicted graphically in Fig. 4. It is observed that the nondimensional critical loads of UD plate are not affected by the number of laminae due to the homogeneity of materials. The non-dimensional critical loads decrease significantly as the number of laminae increases up to 20 for FG-O and FG-V plate, while when the number of laminae increases, the non-dimensional critical loads of FG-X plate increase. With the number of laminae NL > 20 the non-dimensional critical loads are almost unchanged. Because of that, NL = 20 is selected for the remaining parametric studies.
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Table 3. Non-dimensional critical loads (N¯ cr = N0 a2 /π 2 D) for isotropic square plates under biaxial compressions a/h
BCs SSSS
10
Present
CCCC SFSF
1.8247 4.5414 0.8701
Xiang [20] 1.8920 4.6203 0.8996 20
Error %
3.55
1.71
3.27
Present
1.9324 5.0862 0.9023
Xiang [20] 1.9719 5.1117 0.9213 Error %
2.00
0.49
2.06
Table 4. Effect of number of laminae on non-dimensional critical loads of laminated SSSS square FG-GPLRC plates under biaxial compressions NL
1
5
10
15
20
25
30
35
40
UD
7.0739
7.0739
7.0739
7.0739
7.0739
7.0739
7.0739
7.0739
7.0739
FG-V
7.0739
6.8103
6.7300
6.7000
6.6843
6.6747
6.6682
6.6635
6.6599
FG-O
7.0739
5.5728
5.0504
4.8778
4.7897
4.7377
4.7023
4.6775
4.6586
FG-X
7.0739
8.5093
8.9995
9.1530
9.2340
9.2796
9.3119
9.3335
9.3508
Fig. 4. Effect of number of laminae on non-dimensional critical loads of laminated SSSS square FG-GPLRC plates under biaxial compressions
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5.4 Effect of GPL Distribution Type and GPL Weight Fraction Effect of GPL distribution types on non-dimensional critical loads of laminated SSSS square FG-GPLRC plates under biaxial compressions with a/h = 10 are tabulated in Table 5 and displayed by the graph in Fig. 5. It is clearly seen that non-dimensional critical loads of the plates with FG-O pattern of GPL distribution have the lowest value and non-dimensional critical loads of the plate with the FG-X pattern of GPL have the highest value. Thus, to achieve the greater stiffness, the GPLs should be distributed close to the upper and lower surface of the plate. Moreover, it is obvious that the nondimensional critical loads increase as the GPLs weight fraction increases for all GPLs distribution patters. This proves that the GPLs weight fraction is higher, the stiffness of FG-GPLRC plates increases. Table 5. Effect of GPL distribution types on non-dimensional critical loads of laminated SSSS square FG-GPLRC plates under biaxial compressions ∗ (%) gGPL
0
0.2
0.4
0.6
0.8
1.0
UD
1.6364
2.7251
3.8131
4.9006
5.9875
7.0739
8.1597
FG-V
1.6364
2.6890
3.7056
4.7063
5.6981
6.6843
7.6665
FG-O
1.6364
2.2739
2.9053
3.5345
4.1625
4.7897
5.4163
FG-X
1.6364
3.1637
4.6848
6.2030
7.7193
9.2340
10.7473
1.2
Fig. 5. Effect of GPL distribution types on non-dimensional critical loads of laminated SSSS square FG-GPLRC plates under biaxial compressions
5.5 Effect of Boundary Conditions Effect of various BCs on non-dimensional critical loads of laminated square FG-GPLRC plates under in-plane biaxial compressions with a/h = 10 are tabulated in Table 6 and
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plotted in Fig. 6. As expected, from obtained results it can be seen that the CCCC plate has the highest non-dimensional critical loads compared to the plates with remaining BCs whereas the non-dimensional critical loads of the SFSF plate is the lowest. This is evident that the constraints of BCs play important role in the stability of FG-GPLRC plates. Table 6. Effect of various BCs on non-dimensional critical loads of laminated square UD-GPLRC plates under biaxial compressions ∗ (%) gGPL
0
0.2
0.4
SSSS
1.6364
2.7251
3.8131
4.9006
5.9875
7.0739
8.1597
CCCC
4.1950
6.9863
9.7766
12.5660
15.3544
18.1419
20.9284
CSCS
3.0568
5.0907
7.1237
9.1559
11.1874
13.2180
15.2478
CFCF
2.0793
3.4638
4.8486
6.2336
7.6188
9.0043
10.3900
SFSF
0.8055
1.3417
1.8779
2.4142
2.9505
3.4869
4.0233
0.6
0.8
1.0
1.2
Fig. 6. Effect of various BCs on non-dimensional critical loads of laminated square UD-GPLRC plates under biaxial compressions
5.6 Effect of Plate Length-To-Thickness Ratio Table 7 together with Fig. 7 illustrate the effect of length-to-thickness ratio (a/h) on nondimensional critical loads of laminated SSSS square FG-GPLRC plates under biaxial ∗ = 1%. From obtained results, it can be seen that compressions with input data: gGPL the a/h ratio increases as the non-dimensional critical loads increase for all types of laminated FG-GPLRC plates.
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Table 7. Effect of a/h ratio on non-dimensional critical loads of laminated SSSS square FGGPLRC plates under biaxial compressions a/h
5
UD
10
20
30
40
50
5.7805 7.0739
7.6988
7.8721
7.9417
7.9759
FG-V 5.5240 6.6843
7.2274
7.3735
7.4314
7.4597
FG-O 4.1019 4.7897
5.1048
5.1858
5.2171
5.2323
FG-X 7.2449 9.2340 10.2299 10.5198 10.6401 10.7003
Fig. 7. Effect of a/h ratio on non-dimensional critical loads of laminated SSSS square FG-GPLRC plates under biaxial compressions
5.7 Effect of Plate Aspect Ratio (b/a) Table 8 and Fig. 8 present the effect of b/a ratio on non-dimensional critical loads of laminated SSSS rectangular FG-GPLRC plates under biaxial compressions with ∗ = 1%. It is evident that the non-dimensional critical loads decrease as a/h = 10; gGPL aspect ratio increases for laminated UD-GPLRC plates and the remaining three types of laminated FG-GPL reinforced composite plates. However, the non-dimensional critical loads decrease quickly as the increase of the aspect ratio from 1 to 2.5, then decreases slowly.
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Table 8. Effect of b/a ratio on non-dimensional critical loads of laminated SSSS rectangular FG-GPLRC plates under biaxial compressions b/a
1
2
3
4
5
UD
7.0739 4.6905 4.2550 4.1020 4.0309
FG-V 6.6843 4.4314 4.0339 3.9015 3.8433 FG-O 4.7897 3.1311 2.8256 2.7183 2.6686 FG-X 9.2340 6.1950 5.6442 5.4506 5.3606
Fig. 8. Effect of b/a ratio on non-dimensional critical loads of laminated SSSS rectangular FGGPLRC plates under biaxial compressions
6 Conclusions Based on the pb2 Rayleigh-Ritz method and FSDT, buckling behaviors of FG-GPLRC plates are investigated in detail. The laminated FG-GPLRC plates are assumed to be perfectly bonded, in each lamina, the GPLs are assumed to be uniformly dispersed and GPLs weigh fraction varies across the thickness of the plate according to four different GPL distributions patterns. A Matlab code has been developed and used to validate the present solution against published results. Numerical results indicate that the GPL parameters (weight fraction, distribution pattern) significantly affect the critical loads of the FG-GPLRC plate. It is also seen that the FG-X distribution pattern of GPLs increases considerably the stability of the FG-GPLRC plates. Acknowledgements. This research is funded by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) (Grant No. 107.02-2018.322). The financial support is gratefully acknowledged.
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References 1. Coleman, J.N., Khan, U., et al.: Small but strong: a review of the mechanical properties of carbon nanotube–polymer composites. Carbon 44(9), 1624–1652 (2006) 2. Kim, M., Park, Y.B., et al.: Processing, characterization, and modeling of carbon nanotubereinforced multiscale composites. Compos. Sci. Technol. 69(3–4), 335–342 (2009) 3. Kanagaraj, S., Varanda, F.R., et al.: Mechanical properties of high density polyethylene/carbon nanotube composites. Compos. Sci. Technol. 67(15–16), 3071–3077 (2007) 4. Thostenson, E.T., Chou, T.-W.: On the elastic properties of carbon nanotube-based composites: modelling and characterization. J. Phys. D Appl. Phys. 36(5), 573 (2003) 5. Gibson, R.F.: A review of recent research on mechanics of multifunctional composite materials and structures. Compos. Struct. 92(12), 2793–2810 (2010) 6. Rafiee, M.A., Rafiee, J., et al.: Enhanced mechanical properties of nanocomposites at low graphene content. ACS Nano 3(12), 3884–3890 (2009) 7. Song, M., Yang, J., Kitipornchai, S.: Bending and buckling analyses of functionally graded polymer composite plates reinforced with graphene nanoplatelets. Compos. Part B Eng. 134, 106–113 (2018) 8. Song, M., Yang, J., et al.: Buckling and postbuckling of biaxially compressed functionally graded multilayer graphene nanoplatelet-reinforced polymer composite plates. Int. J. Mech. Sci. 131, 345–355 (2017) 9. Kitipornchai, S., Chen, D., Yang, J.: Free vibration and elastic buckling of functionally graded porous beams reinforced by graphene platelets. Mater. Des. 116, 656–665 (2017) 10. Yang, J., Wu, H., Kitipornchai, S.: Buckling and postbuckling of functionally graded multilayer graphene platelet-reinforced composite beams. Compos. Struct. 161, 111–118 (2017) 11. Shen, H.S., Xiang, Y., et al.: Buckling and postbuckling of functionally graded graphenereinforced composite laminated plates in thermal environments. Compos. Part B Eng. 119, 67–78 (2017) 12. Mirzaei, M., Kiani, Y.: Isogeometric thermal buckling analysis of temperature dependent FG graphene reinforced laminated plates using NURBS formulation. Compos. Struct. 180, 606–616 (2017) 13. Reddy, R.M.R., Karunasena, W., Lokuge, W.: Free vibration of functionally graded-GPL reinforced composite plates with different boundary conditions. Aerosp. Sci. Technol. 78, 147–156 (2018) 14. Guo, H., Cao, S., et al.: Vibration of laminated composite quadrilateral plates reinforced with graphene nanoplatelets using the element-free IMLS-Ritz method. Int. J. Mech. Sci. 142, 610–621 (2018) 15. Yang, J., Chen, D., Kitipornchai, S.: Buckling and free vibration analyses of functionally graded graphene reinforced porous nanocomposite plates based on Chebyshev-Ritz method. Compos. Struct. 193, 281–294 (2018) 16. Thai, C.H., Ferreira, A.J.M., et al.: Free vibration, buckling and bending analyses of multilayer functionally graded graphene nanoplatelets reinforced composite plates using the NURBS formulation. Compos. Struct. 220, 749–759 (2019) 17. Anamagh, M.R., Bediz, B.: Free vibration and buckling behavior of functionally graded porous plates reinforced by graphene platelets using spectral Chebyshev approach. Compos. Struct. 253, 112765 (2020)
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Design and Fabrication of Mecanum Wheel for Forklift Vehicle Thanh-Long Le1,2(B) , Dang Van Nghin3 , and Mach Aly1,2 1 Faculty of Mechanical Engineering, Ho Chi Minh City University of Technology (HCMUT),
268 Ly Thuong Kiet Street, District 10, Ho Chi Minh City, Vietnam {ltlong,aly.mach040920}@hcmut.edu.vn 2 Vietnam National University Ho Chi Minh City, Linh Trung Ward, Thu Duc City, Ho Chi Minh City, Vietnam 3 Van Lang University, Ho Chi Minh City, Vietnam [email protected]
Abstract. In this paper, some aspects of the Mecanum wheeled vehicles such as the structure introduction, wheel type’s classification, some remarkable advantages, and also the situation of research and development in over the world are proposed. The main content is that we present the methods of design profile of Mecanum rollers and the mechanical wheeled structure for omnidirectional forklift vehicles, the calculation for shape and parameter of the wheel components to get optimal dynamic capacity, the fabrication process of each component, and finally the full assembly of Mecanum wheel. We also implement some results and calculation methods for inspecting the limitation geometry of rollers to get the achievement for workspace requirements and loading capacity. At the end of this paper, we will compare the kinematics and pushing force between standard, Omni, and Mecanum wheel to show you an overview of this special type of wheel. Keywords: Omni-directional vehicle · Mecanum wheel · Forklift vehicle · Airtraxlift vehicle
1 Introduction Nowadays, the forklifts become an indispensable piece of equipment in manufacturing and warehousing. The orientation and movement of a forklift truck through the limited narrow space are a rather complex and difficult task. Therefore, some solutions to these problems are offered, one of them is to use the Mecanum wheel to turn the forklift into the “Omnidirectional vehicle”, which has capability of moving directly to the side. A forklift is a powered industrial truck used to lift and move materials over short distances. However, their working environment is relatively narrow, especially warehouses, where a large number of forklifts are needed to lift many heavy objects. Warehouses are often large for storing goods and for forklifts and trucks to move inside. But in reality, the warehouses are often very cramped because the quantity of goods is always in excess state. This makes the forklift trucks do not have enough space for steering or turning. Navigating and moving a forklift truck through a tight, confined space is quite © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 795–810, 2022. https://doi.org/10.1007/978-981-16-3239-6_61
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complicated and difficult. One of the solutions to this problem is the development of vehicles capable of direct sideways movement, known as an omnidirectional vehicle. These vehicles carry a flexible multidirectional wheel structure, they are applied to many fields such as industry, military, defense, daily life service, public health… The design of the Mecanum omnidirectional wheel allows the vehicle to operate and move in many different directions without using the traditional methods of turning the steering wheel of today’s cars. This type of wheel is often used for applications in robots where high maneuverability is required. Multidirectional wheels have been used in the robotics field, in industries, military defense, life service, and some other specialized fields for many years. With the concept of these omnidirectional wheels coming from companies that manufacture omnidirectional conveyor systems, the University of Western Australia design team developed and built the Mecanum wheeled robot model [1]. These types of flexible omnidirectional wheels are quite common in the robotics sector. This omnidirectional robot can move straight to the target while performing the rotation accurately. These omnidirectional wheels are also used in wheelchairs, airport pickup trucks, and many other applications [2]. A typical example of an organization with high experience in this field is the international aerospace company NASA, which has applied the special features of this wheel for research, testing, exploration, and exploration in hazardous and dangerous environments [3]. In addition, the trading company AIRTRAX also uses this wheel mechanism to design forklifts, vehicles that support overhead work, and flexible jobs. In general, the market today only commercializes small omnidirectional wheels with relatively high price. For example, AndyMark’s product has a diameter of 200 m and has a price of up to 3,000 USD, and the reload capacity is very low. With the design of forklifts weighing up to 1 ton and wheel diameters up to 600 mm, the market is not currently available. Therefore, the design and manufacture of large multidirectional Mecanum wheels with high load capacity are very interested today [4–10]. In the other hand, the kinematics and pushing force between standard, Omni, and Mecanum wheel to show you an overview of this special type of wheel are also investigated in this study.
2 Methodology The Mecanum wheel is based on a tireless wheel, with a series of rubberized external rollers obliquely attached to the whole circumference of its rim or hub, likes the Omni wheel type. But the difference from Omni type is these rollers typically each has an axis of rotation at 45° to the wheel plane and at 45° to the axle line. Each Mecanum wheel is an independent non-steering drive wheel with its own powertrain, and when spinning generates a propelling force perpendicular to the roller axle, which can be vectored into a longitudinal and a transverse component in relation to the vehicle. Then we need to fit up 4 Mecanum wheels into the corners of the rectangular frame in order to constitute the omnidirectional movement. This type of design is relatively more complicated than other Omni types because the rollers are arranged at a certain deflection angle, but the biggest advantage is that the load capacity of this wheel is significantly increased because the contact with the road surface is smoother and fewer vibrations.
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Fig. 1. The Mecanum wheel.
Classification: There are also some different types of Mecanum wheel structure. But in general, they have many common characteristics that each roller on them has an axis of rotation at 45° to the wheel plane and at 45° to the axle line, they also have drum profile, and all the rollers are laid on the periphery of the steady circular profile like normal wheels. Figure 2 shows that the rollers are laid with 2 solid hubs at the ends of them, these hubs are processed on a CNC machine. Moreover, the rollers are also designed based on two sides console form (Fig. 3). Especially there is another type that the hubs are processed by using the mechanical working method, which means they are pressed, and this type of Mecanum wheel is commercialized (Fig. 4).
Fig. 2. The rollers are assembled with 2 hubs at the ends.
Fig. 3. The rollers are laid as console form.
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Fig. 4. The product has been commercialized.
All of these models need to meet the requirements of the load, flexibility, superiority and application range of the multidirectional forklift (Fig. 1). In the context, the first model still has many problems in movement (the horizontal motion obstruction of two hubs) and the third one has been developed and commercialized, we decide to calculate and design the Mecanum wheel based on the console type (Fig. 3). In this paper, we would like to present the Mecanum roller profile design method and the wheel structure (Fig. 5).
Fig. 5. The 3D drawing of multidirectional forklift.
The Motion Characteristics: The movement of the omnidirectional vehicle with mecanum wheel is possible by controlling the four wheel independently that include
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speed and rotational. The direction of movement is navigated by rollers in mecanum wheel as depicted in Table 1. Table 1. The principle of omnidirectional vehicle movement using mecanum wheels. Direction of movement Wheel actuation Forward
All wheels forward same speed
Reverse
All wheels backward same speed
Right shift
Wheels 1, 4 forward; 2, 3 backward
Left shift
Wheels 2, 3 forward; 1, 4 backward
CW turn
Wheels 1, 3 forward; 2, 4 backward
CCW turn
Wheels 2, 4 forward; 1, 3 backward
Fig. 6. Omnidirectional vector mapping for Mecanum drive.
These rollers are having a unique shape with an axis of rotation at 45° to the plane of the wheel and 45° to a linethrough the center of the roller parallel to the axis of rotation
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of the wheel. From Fig. 6, to move a omnidirectional vehicle, we just need to choose the direction of the wheel’s rotation (forward, backward or inactive). The omnidirectional vehicle can not only move vertically and horizontally, it can also move diagonally. For example, if wheels 1, 4 forward and wheel 2, 3 are inactive, the vehicle will move upwards at an angle of 45° from the right direction. And if we keep wheels 1, 3 and 4 in the same state, let wheel 2 to move backwards, the car will move upwards at an angle less than 45° from the right direction. Developments and Implementations: The structure of this Mecanum wheel type includes many main components: Mecanum roller, roller shaft, hub and wing. So the developments for this project can be divided into 4 parts based on the wheel structure. Of which, the roller design is considered the most important part. Roller Design: The rollers (Fig. 6) are constructed based on the circular profile of the wheel when we bevel out a circular prism with a diameter based on the wheel’s outer diameter according to Doroftei [3]. The beveled plane is inclined at an angle γ = 45°.
Fig. 7. Principle of Mecanum roller profile shaping.
The Mecanum wheel is composed of multiple rollers arranged at a certain tilt angle with respect to the wheel’s axle; the tilt angle η is shown in the side projection of the wheel onto the Oxy plane (Fig. 7a). Figure 7b shows the projection of the wheel in the Oyz plane (Fig. 8). Let OX be an axis on the Oxy plane that is inclined at η angle with respect to the Ox axis. Then, OX becomes the main axis of the rollers. (Figure 9b shows the Mecanum wheel’s profile, and Fig. 9c shows a roller’s profile when situated on an inclined axis). From Fig. 9a and b, we got: y = X sin η
(1)
and z2 = R2 − y2
(2)
Combine (1) and (2): z2 = R2 – X2 (sin η)2 ⇒ X2 sin2 η + z2 = R2
(3)
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Fig. 8. The position of rollers.
(a)
(b)
(c)
Fig. 9. The wheel profile is defined on the coordinate axis.
So we can find out the roller profile satisfies the equation:
X2 R sin η
2 +
z2 =1 R2
(4)
(With R (mm) is the distance from the wheel’s center to the roller axle) Let L be the length of each roller (mm) Rwheel : The radius of the wheel (mm) RRim : The radius from the wheel’s center to the roller (mm) rrol : The maximum radius of the roller (mm) η: The roller’s inclination angle with respect the wheel’s main axle θ: Interior angle between each roller θt : The concealed angle between two adjacent rollers Let n be the number of rollers. From Fig. 10, we have: n(θ − θt ) = 2π → θ =
2π + θt n
(5)
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Fig. 10. Geometrical parameters of Mecanum Wheel.
That means the difference between θ and θt of n rollers exactly equal to 360° The relationship between parameters in the Mecanum wheels: Rwheel = RRim + 2rrol
(6)
Look at Fig. 10, when L is the real length of each roller, L’ will become the roller’s length in the wheel diameter plane. From Fig. 7, we have: L = 2R tan
θ θ = 2(RRim + rrol ) tan 2 2
(7)
(With RRim + rrol = R) Otherwise: L=
L sin η
(8)
From (5), we have the relationship: θ θt π = + 2 2 n
(9)
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From (4)–(6), we can deduce the formula of the roller length (Fig. 11): 2(RRim + rrol ) tan θ2t + πn L= sin η
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(10)
Fig. 11. Relationship between parameters.
With d is the roller radius along the diameter axle, the relationship is represented: d cos η + Rwheel − rrol = R2wheel − d2 sin4 η Squaring two sides: cos2 η + sin4 η d2 + 2d(Rwheel − rrol ) cos η + (rrol − 2Rwheel )rrol = 0 The roots of the quadratic equation above: −B cos η ± B2 cos2 η + ACrrol d= A
(11)
With: A = (cos2 η + sin4 η); B = 2(Rwheel − rrol ); C = (rrol − 2Rwheel )rrol With α (mm) is the minimum radius of the roller, in this design we choose rrol = 2α. We have the function r(X) depend on variable X (−L/2 ≤ X ≤ L/2): (12) r(X) = (Rwheel + 2rrol )2 − X2 sin2 η − (Rwheel − rrol ) The value of r(X) at X = L/2: 2 L L 2 = (Rwheel + 2rrol ) − r sin2 η − (Rwheel − rrol ) > α 2 2 Input parameters:
(13)
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The roller’s inclination angle with respect the wheel’s main axle η = 45° The number of roller pairs n = 6 The concealed angle between two adjacent rollers θt = 15° The value of maximum roller radius can be chosen in the within the range: [24; 49]. In this paper, we choose rrol = 45 mm
From the data above, we have the final calculation results: – – – –
Dwheel = 340 mm RRim = 80 mm rrol = 45 mm L = 240 mm
After calculating the dimensions of wheel, we replace the calculated values into the Eq. (13), L = 240 mm is still in the allowable range: 1 (2.170)2 − (2.22, 5 + 170 + 45)2 > L > 0 ≥ 309, 8387 > L > 0. sin 45◦ The Mecanum roller is manufactured by using a special processing method: using a computer to program the tool path in an elliptical orbit with geometrical parameters based on (1), it is processed on a CNC lathe machine to ensure about high processing precision. The material of the Mecanum wheel is POM plastic, roller surface is covered with a special material layer in order to ensure the loading-bearing capacity of the forklift truck. In this paper, the Mecanum wheel structure includes 6 roller pairs (Fig. 12) which are fitted on 6 roller axes.
Fig. 12. Mecanum roller.
Shaft Design: The Mecanum roller shaft is made of C45 steel material. This type of shaft has many steps, high inversion and symmetry requirements. To satisfy these requirements, it needs to be manufactured on a CNC lathe machine. Furthermore, the shaft is processed in the surface heat treatment method to ensure its reliability when it functions. This shaft is fitted with 4 radial ball bearings and a Mecanum roller pair (Fig. 13).
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Fig. 13. Mecanum roller shaft.
The shaft is lathed into 5 steps. The journal’s dimension is 45 mm. The dimension of the shaft, which is longest and jointed with the wing, is 35 mm. The shaft contacted with the roller is 25 mm. The remaining steps have the dimension is 30 mm for big ball-bearing and 20 mm respectively for the smaller one. So each roller shaft has 4 ball-bearing, all the them are chosen with the technical standard. The ends of the shaft are threaded with a M12-diameter, this part is used to fasten bolts and keep components from slipping. Hub Design: The hub is also manufactured on CNC milling machine to get high symmetry. It is milled into 6 beveled faces, each face is processed 4 screw holes and 2 locating pins. There is a hole in the center of the hub which is processed to fit the axle. Besides that, the edges and two sides of the hub are chamfered to reduce wheel weight (Fig. 14).
Fig. 14. The wheel hub.
The hub is designed so that its radius is smaller than the RRim radius (RRim = 80 mm is the distance from the wheel center to the roller) to avoid friction. In the report, we decided to select the pentagon hub with the distance from the center to the beveled surfaces is 55 mm. On each beveled surface, there are 4 holes for mounting wings, these 4 holes are arranged so that their center forms a square with 24 mm edge. The width of the hub is 50 mm, there is 1 large hole in the center and 6 small holes around to install the wheel on the forklift. The diameter of the big hole and the small one are 32 mm and 12 mm respectively (Fig. 15).
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Fig. 15. The dimensions of the hub.
Wing Design: The wing of the Mecanum roller is made of C45 steel material and manufactured by using the milling method on the CNC milling machine. Each wing includes 2 main details to assemble with other components. The first detail is processed with a slot to pin the latch of the Mecanum roller shaft. The second detail is the wing sole, it is processed 4 slots (Fig. 16) includes 2 screw holes to assemble with the first detail and 2 others with the hub. In addition, these details are also chamfered to reduce wheel weight.
Fig. 16. The wing of the Mecanum wheel.
The sole of the wing has 2 M5 holes to attach to the hub, the 2 other holes are covered by the wing. This sole has a square shaped 42 mm edge. Let h be the height from the bottom of the sole to the axis of the hole in the wing, we have h = RRim + rrol − 55 = 70 mm (55 mm is the distance from the wheel center to the beveled surface of the hub
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which was selected above). The big hole on the wing has diameter is 35 mm, there is a small slot on this hole for the latch of roller shaft (Fig. 17).
Fig. 17. The dimensions of the wing.
We also have another design for this component. By changing the single-wing above into the bracket (Fig. 15). This bracket is processed 2 holes on its sole to connect it to the hub by tightening the bolts, and it is also easier to manufacture than the single wing. It is cut by the CNC laser machine and pressed 2 ends down at an angle of 90° on the machine press. With this design, the roller no longer needs to be halved into a pair like above. But changing single-wing into the bracket has a disadvantage that is easily hindered at 2 ends when moving in an uneven environment (Fig. 18).
Fig. 18. Bracket designed for rollers.
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3 Results and Discussion Calculation Results: We have designed and fabricated the Mecanum wheel for forklift vehicle. Here are the parameter results: – – – – –
The diameter of Mecanum wheel D = 2Rwheel = 340 mm The length of the roller Lr = 240 mm The diameter of the roller drol = 90 mm The angle of each arc ϕ = 60° The number of the roller pairs n = 6
Design Results: The wings are fixed on the hub by locating pins to constitute the wheel rim. Then we insert 6 roller shafts into the hole in the centre of each wing. Finally, 6 roller pairs are fitted in order to complete the fully Mecanum wheel. Fabrication Results: Rollers fabrication is the most important part. We programmed the tool path on the CNC lathe machine to ensure the curvature profile and geometry parameters of the original design. Since its material is POM plastic, it is easier to process than metal. However, the loading-bearing capacity of plastic is not very good, so we covered the roller surface with a special material layer to reduce damage to rollers when the vehicle was carrying heavy loads. The shaft is easier to process because of its simple shape. It has many steps, there are spaces between them. To achieve high precision, we input the spacing parameters between the steps and the diameter of each step into the CNC lathe machine. Before being lathing, the shaft is tempered for high toughness and durability. In the hub fabrication, its large size and relatively heavy weight cause many difficulties for us. Firstly we created its simple shape by casting it in a hexagonal sand mold. Then the casted hub is milled into 6 bevel faces on the CNC milling machine. And finally, we lathed the edges and two sides to reduce its weight, and make its surface smoother and less angular. The wing or bracket can be processed by programing their parameters on the CNC laser machine, or cut, bended, and drilled by hand without the CNC machine. After fabricating the components, we fitted them together to complete the full assembly of Mecanum wheel for forklift truck. Comparison: Efficiency is not the right thing to compare due to the various advantages and disadvantages of each type of wheel. However, comparing speed and force can give a good comparison of the different types of wheels. Table 2 offers a quick and simple comparison (this assumes fictionless bearings and equal traction): The three columns are for Standard, Omni, and Mecanum 4 wheeled-Vehicles, respectively. The Omni vehicle’s wheels are mounted at 45°. All wheels same diameter. The first three rows are vehicle velocity: forward, strafe, and diagonal for a given wheel speed ω (radians/s). The second three rows are vehicle total pushing force: forward, strafe, and diagonal, for a given wheel torque τ. These last three rows assume frictionless Mecanum and Omni roller bearings and sufficient traction to support the floor reaction forces. The experiments showed that for the same wheel speeds, the Omni vehicle moved 41% faster than the Mecanum, and for the same wheel torque, the Mecanum vehicle had 41% more pushing force than the Omni.
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Table 2. Kinematics and pushing force comparison between Standard, Omni and Mecanum wheel. Standard Omni √ Kinematics Vƒ ω·r ω·r 2 √ Vr – ω·r 2 Vd – Force
Fƒ 4·τ/r Fr
–
Fd –
ω·r
Mecanum ω·r ω·r √ ω·r/ 2
√ 4·τ/(r 2) 4·τ/r √ 4·τ/(r 2) 4·τ/r √ 2·τ/r 2·τ 2/r
4 Conclusion The Mecanum wheel still has a lot of disadvantages that need to be improved, however, this type of wheel has great potential for development in the future. The strength of this wheel is the enhanced maneuverability of forklift truck that needs extreme maneuverability in narrow environment. But the roller material is not good enough for longlasting, so we need to research and find a better to replace it. In this paper, we have presented an overview of the Mecanum wheel, design direction, calculation methods, design and fabrication process, and final results. We have also analyzed the structure of different Mecanum wheel types to advance geometrical parameters for optimal dynamic capacity. The development directions such as design calculation, profile improvement, structure optimality, and simulation, attract the attention of many developers because of the geometrical complication as well as the structure of this wheel type. Acknowledgment. We acknowledge the support of time and facilities from Key Laboratory of Digital Control and System Engineering (DCSELab), Ho Chi Minh City University of Technology (HCMUT), VNU-HCM for this study.
References 1. McCandless, A.: Design and Construction of a Robot Vehicle Chassis. Faculty of Engineering, Computing and Mathematics, The University of Western Australia (2001) 2. Dehengre, N., Mogra, A., Verma, S., Gupta, A.: Design and Manufacturing of Mecanum wheel for Omnidirectional Robot. MPSTME, SVKM’s NMIMS (Deemed to be University), Shroff S. R. Rotary Institute of Chemical Technology, Bharuch, Gujarat, India (2018) 3. Doroftei, I.: Omnidirectional Mobile Robot – Design and Implementation. Technical University of Iasi, Romania (2007) 4. Kiran Kumar, N.A., Patil, V.M., Kunnur, S.: Design and Fabrication of Improved Mecanum Wheel by Drag Reduction Method. Department of Mechanical Engineering, Vivekanada College of Engineering & Technology, Puttur D.K., India (2019) 5. Hemanth, S., Pranavatheertha, K.G., Rajath, S.S., Vidish, S.: Design and Fabrication of Prototype Model of Mecanum Wheel Forklift – A Review. Department of Mechanical Engineering, Alva’s Institute of Engineering & Technology, Mijar (2019)
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6. Lanjewar, S., Sargaonkar, P.: Fabrication and Simulation of Mecanum Wheel for Automation. Department of Mechanical Engineering, Cummins College of Engineering, Nagpur, India (2016) 7. Krishnaraj, J., Sangeetha, K., Babu Tanneru, M.V., Harnadh Prasad, V.V.S., Vishnu Vardhan, M.: A Mecanum Wheel based Robot Platform for Warehouse Automation. MLR Institute of Technology, VNR Vignana Jyothi Institute of Engineering and Technology, Institute of Aeronautical Engineering, Vardhaman College of Engineering, Hyderabad, Telangana, India (2017) 8. Lengade, S., Shirodka, S.: Design and Analysis of Casted Mecanum Wheel. Mechanical Department from Jain College of Engineering (2017) 9. (Sid) Wang, S.L.: Motion Control and the Skidding of Mecanum-Wheel Vehicles. Department of Mechanical Engineering, North Carolina A&T State University, Greensboro, North Carolina, USA (2018) 10. Li, Y., Ge, S., Dai, S., Zhao, L., Yan, X., Zheng, Y., Shi, Y.: Kinematic Modeling of a Combined System of Multiple Mecanum Wheeled Robots with Velocity Compensation. Department of Mechanical Engineering, Stevens Institute of Technology, Hoboken, NJ 07030, USA, School of Mechatronic Engineering, China University of Mining and Technology, Xuzhou 221116, China, and School of Mechanical and Electrical Engineering, Xuzhou University of Technology, Xuzhou 221018, China (2019)
Muti Object Prediction and Optimization Process Parameters in Cooling Slope Using Taguchi-Grey Relational Analysis Anh Tuan Nguyen1,2(B) , Dang Giang Lai1 , and Van Luu Dao1 1 Le Quy Don Technical University, Hanoi, Vietnam 2 Air Defense - Air Force Technical School, Thanh Mai, Thanh Oai, Hanoi, Vietnam
Abstract. In the present work, the effect of processing parameters of cooling slope techniques of ADC 12 Aluminum alloy on its microstructural evolution has been studied in detail. Three important process parameters such as the pouring temperature (580 °C, 585 °C and 590 °C), slope length (300, 450 and 600 mm) and slope angle (30, 45 and 60°) were investigated in this study. The plan of experiments based on Taguchi’s was used for acquiring the data. Multi-object optimization parameters in cooling slope using Taguchi-Grey relation analysis. The results also indicated a new optimal value that has not been conducted in the experimental plan to improve the effectiveness of the experiment. The effect of processing parameters on the particle size and shape factor has been investigated by applying analysis of variance (ANOVA) for a grey relational grade. The resulting ANOVA shows that the slope angle (37.3%) has the greatest effect degree of sphericity and particle size followed by pouring temperature (35.3%) and slope length (27.3%). Keywords: Semi – solid · Cooling slope casting · Taguchi method · ANOVA · Grey relational analysis
1 Introduction Semi-solid processing originated from work by researchers at MIT in 1971 experimenting on the rheological behaviour of Sn-15Pb alloy. The semisolid slurry with a spheroidal microstructure of 0.4–0.6 weight fraction solid suspended in liquid had a very low value of flow resistance and that it would be possible to use this in developing new forming processes. The final product by SSM has been proved a lot of advantages compare with the conventional technique, such as lightweight potential resulting from enhanced mechanical properties, superior weldability and pressure tightness, near net shape casting, improve surface quality…[1]. The usual methods available for large scale production of semisolid billet with globular microstructure are mechanical stirring, electromagnetic stirring, etc. These suffer from drawbacks like complex design, high cost, structural inhomogeneity. The cooling slope is considered to be a simple but effective method because of its simple design and easy control of process parameters, low equipment and running costs, high production © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 811–822, 2022. https://doi.org/10.1007/978-981-16-3239-6_62
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efficiency and reduced inhomogeneity, but very effective in creating billet with globular microstructure [2]. One-object optimization method is a simple and effective optimization analysis method, so this method has been used by many authors for their research to optimize processing parameters for cooling slope. Das et al. [3] optimized the processing parameters of the cooling slope using the Taguchi method. The authors concluded that the length of the slope affects the α-Al (A356) morphology the most compared to the rest of the parameters (pouring temperature, angle and slope temperature). Vundavilli et al. [4] also used the Taguchi L9 method based on noise ratio and ANOVA analysis to evaluate the influence of the processing parameters of the slopes. The experiments indicated that the experimental value of hardness of the A356 alloy is very close to estimating value from Taguchi method. Gautam et al. [5] used surface response method to study the effects of pouring temperature, slope length and slope angle of the cooling slope on the average particle diameter and sphericity of the α-Al grain, the study found a regression model for these two output parameters and evaluate the influence of the processing parameters on the parameters. These studies often use one single object optimization because the algorithm is simple and often gives an explicit. However, most of the options actually have to choose not one but several criteria at the same time. The Taguchi-grey relational analysis does a good job of solving this problem. In this study, the cooling slope method is used to change the eutectic dendrite to a globular microstructure with small particle size to change the mechanical properties of ADC 12 alloy and study the effect of process parameters on degree of sphericity by taguchi-grey relational analysis using muti object optimization. ADC 12 has excellent castability with high fluidity and low shrinkage rate [6–8], so it is widely used in industry, especially in automotive and motorcycle piston casting. ADC 12 (Si 11.6%wt) has high strength and low ductility (about 1%) because it is a eutectic alloy with eutectic dendrite microstructure. Few investigators have attempted to modify the cast structure of ADC 12 Al alloy using the Gas Induced semi-solid technique (GISS), Strain-Induced metal activation (SIMA) and Mechanical Rotational Barrel (MRB) [9–11]. ADC 12 with globular microstructure increase the ductility of this alloy during the semi-solid forming process while remaining mechanical properties leads to an increase in the quality of finished parts such as the piston, increasing the reliability of the internal combustion engine.
2 Experimental Procedure 2.1 Materials Chemical composition of Aluminum ADC 12 alloy was determined by the Spectrolab machine in the Laboratory of Institute of Technology and presented in Table 1. It is worth noting that the silicon content is 11,6%. This proves that this is a eutectic alloy of Aluminum – Silicon and some alloys have other significant components such as Cu 2% (this metal was added to increase the strength and machinable [8]). The melting and solidification temperature of ADC12 alloy were determined through the DSC (Differ-
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ential Thermal Calorimetry) and shown in Fig. 1. The result shows that liquidus and solidus temperatures of ADC 12 are 573 °C and 520 °C respectively. Table 1. Chemical composition of ADC12 alloy Element
Si
Fe
Cu
Mn
Mg
Cr
Ni
Zn
Ti
Pb
Al
Wt%
11.58
0.63
2.09
0.17
0.081
0.023
0.055
0.77
0.048
0.056
84.5
T, oC 620
exo
600
571,7 oC
580 560 540 520 500 480 DSC, (mW/mg) -0.6
-1.2
-1.8
-2.4
-3.0
Fig. 1. DSC curve of ADC 12 aluminum alloy
2.2 Cooling Slope Casting Process The experimental setup of the cooling slope casting process is shown in Fig. 2. The cooling plate was adjusted to be inclined at 45°, 55°, 65° for the horizontal plane and was cooled by water circulation underneath. The low rate of water circulation was low and kept constant of 2l/min throughout the experiment. In once of the experiment, 1100 g aluminum alloy was put in a graphite crucible and melted by a Nabertherm electric resistance furnace and then cooling to pouring temperature. It is then poured onto the surface of the cooling slope plate made of stainless steel with a water-circulating cooling system, a slurry of alloy out of the plate collected into a stainless steel mold. Temperatures were monitored with a K – type thermocouple fixed at the top of the slope and one inserted in the mold to record the temperatures of the melt at entering and exit of the slope. Thermocouple accuracy and measuring equipment calibrated and verified at the Military Measurement Center.
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Fig. 2. Experimental setup of cooling slope
2.3 Analysis of Samples Sample obtained from the experiment and then cut by CNC Wire-Cutting Machine to size 1 × 1 × 1 cm (Fig. 3). The cut sections were subject to mechanical grinding, polishing and etched by 0.5% HF solution. The etched samples were investigated using a Carl Zeiss microscope. The images are analysed using ImageJ software to calculate average particle size and shape factor according to and Eq. (1) and Eq. (2).
Fig. 3. Sample and cut pattern
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n 4A
dave =
π
i=1
(1)
n
Where dave is the average particle size, A and n are the area of one particle and total particle in the microscopic, respectively. Primary particles’ morphology is a key factor in quantitative metallography. The sphericity of the particle is assessed through the shape factor. Shape factors (SF) are calculated according to the following Eq. (2) [2]. The sphericity factor or shape factor varies between 0 for objects having very elongated cross-sections and 1 for those having circular cross-sections. By using the average value of sphericity, it was that this parameter is responsive to small variation of the morphology of primary particles. SF =
4π A P2
(2)
where A is the total area of primary particles and P represents perimeter of liquid-primary particles interface. 2.4 Taguchi’s Experimental Design Taguchi’s method is a well-known and reliable tool for the design of a high-quality system. In this study, experiments were designed to apply the Taguchi’s methods L9 to establish the optimal processing conditions of the cooling slope casting process for the generation of the semi-solid feedstock of ADC12 alloy. The cooling slope casting process parameters involving the pouring temperature, slope angle and slope length are assigned to the first, second and third columns of the array, respectively. The process parameters and their levels are given in Table 2. Table 2. Cooling slope casting parameters and their values at 3 levels Parameters
Parameters designtion
Levels 1
2
3
Pouring temperature (°C)
A
580
590
600
Slope length (mm)
B
300
450
600
Slope angle (degrees) C
45
55
65
A three - level OA (L9 ) has been employed in the design of the experiments which includes 9 experiments (show in Table 3).
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C
Average Shape particle size factor
1
580
300 45 50.3
0.792
2
580
450 55 54.2
0.788
3
580
600 65 53.1
0.754
4
590
300 55 57.2
0.705
5
590
450 65 51.6
0.785
6
590
600 45 62.5
0.738
7
600
300 65 52.5
0.800
8
600
450 45 64.4
0.724
9
600
600 55 51.5
0.701
3 Grey Relational Analysis Normalize experiment data (pre-processing) in order to transfer initial measurement data to the same standard, which is comparable. After normalization, a dimensionless data is obtained, the value is in the range [0 1]. In this study, the normalized value of average particle size which is smaller-the-better [12] performance characteristic can be expressed as: xij =
max(yij ) − yij max(yij ) − min(yij )
(3)
where yij are original data. The normalized value of shape factor which is larger-the better performance characteristic can be expressed as: xij =
yij − min(yij ) max(yij ) − min(yij )
(4)
The normalized values of pouring temperature, slope length and slope angle are shown in Table 4. For a response j of experiment i, if the value x ij which has been processed by data preprocessing procedure is equal to 1 or nearer to 1 than the value for any other experiment, then the performance of experiment i is considered as the best for the response j. The reference sequence X 0 is defined as (x 01 , x 02 , …, x0j , …, x 0n ) = (1, 1, …, 1, …, 1), where x 0j is the reference value for jth response. Δij = x0j − xij is deviation sequence. Table 4 shows deviation sequences. The grey relational coefficient can be determined as: γ (x0j , xij ) =
(Δmin + ξ Δmax ) (Δij + ξ Δmax )
(5)
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for i = 1; 2; …; m and j = 1; 2; …; n and where y(x 0j , x ij ) is the grey relational coefficient between x ijand x 0j , x = − x Δ 0j ij ,Δmin = min Δij , i = 1, 2, ..., m; j = 1, 2, ..., n , Δmax = ij max Δij , i = 1, 2, ..., m; j = 1, 2, ..., n ,ξ is distinguishing coefficient, ξ ⊂ (0,1]. In this study the value of ξ was chosen to be 0.5. Distinguishing coefficient (ξ ) is the index for distinguishability. The smaller ξ , the higher its distinguishability. The quantity measurement equation in grey relational space is called the grey relational grade. The grey relational grade is a weighted sum of the grey relational coefficients and it is calculated using the formula: Γ (X0 , Xi ) =
n
wj γ (x0j , xij )
(6)
j=1
for i = 1; 2;…;m where nj=1 wjj = 1 Γ (X 0 , X i ) is the grey relational grade between comparability sequence X i and reference sequence X 0 . The weight of response jth is wj and depends on decision-makers’ judgement. The grey relational grade indicates the degree of similarity between the comparability sequence and the reference sequence. If an experiment gets the highest grey relational grade which means that experiment would be the best choice. Table 4. Results of grey relational analysis Runs Normalization
Deviation sequence Average particle size
Grey relational coefficient Shape factor
Average particle size
Grade
S/N
Average particle size
Shape factor
Shape factor
1
1.0000
0.9192 0.0000
0.0808 1.0000
0.8609 0.9304 −0.62628
2
0.7234
0.8788 0.2766
0.1212 0.6438
0.8049 0.7244 −2.80095
3
0.8014
0.5354 0.1986
0.4646 0.7157
0.5183 0.6170 −4.19387
4
0.5106
0.0404 0.4894
0.9596 0.5054
0.3426 0.4240 −7.45333
5
0.9078
0.8485 0.0922
0.1515 0.8443
0.7674 0.8059 −1.87463
6
0.1348
0.3737 0.8652
0.6263 0.3662
0.4439 0.4051 −7.84897
7
0.8440
1.0000 0.1560
0.0000 0.7622
1.0000 0.8811 −1.09968
8
0.0000
0.2323 1.0000
0.7677 0.3333
0.3944 0.3639 −8.78089
9
0.9149
0.0000 0.0851
1.0000 0.8545
0.3333 0.5939 −4.52516
Table 4 shows the grey relational coefficients and grade for each experiment. The highest grey relational grade is the order of 1. The experiment number 1, highlighted in bold, is the nearest optimum controllable parameters combination: pouring temperature of 580 °C (level 1), slope length of 300 mm (level 1), slope angle of 45 (level 1) – A1B1C1.
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The means of the grey relational grade for each level of controllable parameters were calculated from Table 4 and summarized in Table 5. The larger the grey relational grade, shown in bold in Table 5. From this calculation result, the best optimal value for the cooling slope system is not A1B1C1, the better optimal value must be chosen as A1B1C3. Therefore, the optimal values of the processing parameters are as follows: pouring temperature of 580 °C (level 1), slope length of 300 mm (level 1), slope angle of 65 (level 3) – A1B1C3. Table 5. Response table for grey relational grade. Parameters
Level 1
Level 2
Level 3
Rank [max/min]
Pouring temp
0.7573
0.5450
0.6130
1 (0.2123)
Slope length
0.7452
0.6314
0.5387
2 (0.2065)
Slope angle
0.5665
0.5808
0.7680
3 (0.2015)
Total mean value of the grey relational grade = 0.6384.
4 Results and Discussion 4.1 Microstructural Investigations Figure 4 indicates the results obtained in the 9 tests, each of which was performed at levels shown in Table 3 (standard orders). It shows various microstructural features such as rosettes and globular grains in different samples. As shown in Fig. 4, the experiment was successful in creating a globules microstructure. The microstructure of the eutectic aluminum-silicon billet usually takes the form of eutectic dendrite which is replaced by globules microstructure of α-Al, Silicon became plate-like which concentrated into slabs. It can be explained as follows, Grains nucleated on plate of slope along with the detached nuclei grow during movement from the top to the bottom of the cooling slope. Most of the nucleation that has occurred at the top of the cooling slope may be considered as a general source of nuclei. The evolution of the microstructure along the flow has also been attributed to the change in the flow pattern as the melt flows down the slope. The temperature of the molten metal dropped from 580 to 540 °C after flowing out of the slope as heat from molten metal was adsorbed by the cooling slope, and according to P.Das et al. [13], the temperature of the flowing molten metal on the slope surface could be assumed to be decreased linearly along the slope. As a result of the drop in temperature of molten metal along the slope as the melt flowed into the mold, the solid fraction increased in the last portion of slope, leading to an increase in the viscosity of the melt, or in other words, an increase in the Reynolds number, so the melt flow tends to be laminar in the lowest portion of the slope. Furthermore, the flow of the fluid down the slope causes force convection, which help to homogenize the temperature and concentration gradients of the molten metal, which would also lead to the suppression of dendritic growth, hence facilitating spheroid formation.
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Fig. 4. Micrographs of different samples (1–9) in standard order in Table 3
4.2 Analysis of Variance for Grey Relational Grade The analysis of variance (ANOVA) is a method which is used in this study to find out which controllable parameter significantly affects the performance characteristic. This is accomplished by separating the total variability of the grey relational grades, which is measured by the sum of the squared deviations from the total mean of the grey relational grade into contributions by each controllable parameter and the error. The percentage contribution by each of the process parameters in the total sum of the squared deviations was used to evaluate the importance of the controllable parameter change on the performance characteristic. The results of ANOVA for the grey relational grade values are shown in Table 6. The results indicate that the percentage contribution of the pouring temperature, slope length and slope angle are 35.3%, 27.3% and 37.3%, respectively. These three parameters significantly influenced the grey relational grade, the influence levels of these three parameters are quite similar. Thus, the semi-solid feedstock of ADC12 alloy is generated with optimum cooling slope casting parameters of pouring temperature 580 °C (level 1 – A1), slope length of 300 mm (level 1 – B1) and slope angle of 650 (level 3 – C3).
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Source
df
Seq SS
Adj SS
Adj MS
F-ratio
Contribution
Pouring temp
2
16.114
16.114
8.057
0.52
35.3%
Slope length
2
9.174
9.174
4.587
0.30
27.3%
Slope angle
2
18.426
18.426
9.213
0.60
37.3%
Residual error
2
30.805
30.805
15.402
Total
8
74.519
df - Degree of freedom, SS - Sum of square, MS - Mean square.
4.3 Predicting Optimal Value and Confirmation Experiments Using Taguchi method to calculate S/N ratio of the influencing parameters is shown in Table 7 for avarage particle size. Table 7. Computation of ANOVA for S/N ratio Level
Pouring temp (°C)
Slope length (mm)
Slope angle (degrees)
1
−34.40
−34.53
−35.38
2
−35.11
−35.04
−34.69
3
−34.94
−34.89
−34.39
Maximum (Max)
−34.40
−34.53
−34.39
Mean (m)
−34.82
−34.82
−34.82
The optimal value of S/N ratio of the cooling slope casting parameters was calculated by adding the average performance to the contribution of each parameter using formula [14]. S S S S = m + max − m + max − m + max −m N opt N A N B N C = − 34.82 + (−34.4 + 34.82) + (−34.53 + 34.82) + (−34.39 + 34.82) = − 33.68 Where m – the average value of S/N ratio. The predictive optimal average particle size value was determined by the formula [14]: Yopt = 10
(S/N ) opt − 20
= 10
− −33.68 20
= 48.33
The A1B1C3 is an optimal parameter combination of the cooling slope process obtained by grey relation analysis which has been conducted experimentally to confirm.
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The results of the confirmation test using the optimal controllable variables, average particle size and shape factor were improved from A1B1C1 (50.3 for average particle size, 0.792 for shape factor) to A1B1C3 (48.3 for average particle size, 0.82 for shape factor). The results showed the accuracy of the optimal calculation results.
5 Conclusions The research has optimized the processing parameters of cooling slope casting (a type of rheocasting) process to achieve the smallest particle size and the maximum particle sphere. The experiment is designed according to Taguchi’s L9 orthogonal array technique. The three parameters are pouring temperature, slope length and slope angle are important factors affecting output characteristics. The process variables have been considered at three different levels for designing the experiments. The conclusions emerging from the present work are listed below: The research was successful in creating a globular microstructure for an alloy of the ADC12 eutectic alloy with high globularity. ADC12 alloy is a eutectic alloy that is difficult to create a globular microstructure during semi-solid casting. Taguchi-grey relation analysis method has found a new optimal value based on multi-goal optimization, the new value is completely consistent with published studies. The new optimal value is a combination of cooling slope casting process parameters as, pouring temperature of 580 °C, slope length of 300 mm and slope angle of 65°. From the results of grey relation analysis combined with Taguchi analysis to evaluate the combined effects of the processing parameters on the output factors. The ANOVA results, the most significant influencing parameter on particle size is the slope angle, which accounts for 37.3% of the total effect, followed by pouring temperature (35.3%) and slope length (27.3%) respectively.
References 1. Winklhofer, J.: Semi-solid casting of aluminium from an industrial point of view. Solid State Phenom. 285, 24–30 (2019) 2. Nafisi, S., Ghomashchi, R.: Semi-solid processing of aluminum alloys. Springer (2016) 3. Das, P., Samanta, S.K., Das, R., Dutta, P.: Optimization of degree of sphericity of primary phase during cooling slope casting of A356 Al alloy: Taguchi method and regression analysis. Measurement 55, 605–615 (2014) 4. Vundavilli, P.R., Mantry, S., Mandal, A., Chakraborty, M.J.P.M.S.: A Taguchi optimization of cooling slope casting process parameters for production of semi-solid A356 alloy and A356-5TiB2 in-situ composite feedstock. 5, 232–241 (2014) 5. Gautam, S.K., Mandal, N., Roy, H., Lohar, A.K., Samanta, S.K., Sutradhar, G.: Optimization of processing parameters of cooling slope process for semi-solid casting of ADC 12 Al alloy. J. Brazil. Soc. Mech. Sci. Eng. 40(6), 1–15 (2018). https://doi.org/10.1007/s40430-018-1213-6 6. Zhao, H., Wang, F., Li, Y., Xia, W.J.: Experimental and numerical analysis of gas entrapment defects in plate ADC12 die castings. 209(9), 4537–4542 (2009) 7. Elliot, R.: Eutectic Soldification Processing: Crystalline and Glassy Alloys. Butterworths & Company, New York (1983)
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8. Barrirero, J.: Eutectic Modification of Al-Si Casting Alloys. Linköping University Electronic Press, Linköping (2019) 9. Janudom, S., Rattanochaikul, T., Burapa, R., Wisutmethangoon, S., Wannasin, J.: Feasibility of semi-solid die casting of ADC12 aluminum alloy. Trans. Nonferrous Metals Soc. China 20(9), 1756–1762 (2010) 10. Wang, Z., Ji, Z., Hu, M., Xu, H.: Evolution of the semi-solid microstructure of ADC12 alloy in a modified SIMA process. Mater. Character. 62(10), 925–930 (2011) 11. Hu, Z.-h., et al.: Microstructure evolution and mechanical properties of rheo-processed ADC12 alloy. 26(12), 3070–3080 (2016) 12. Tzeng, G.-H., Huang, J.-J.: Multiple Attribute Decision Making: Methods and Applications. CRC Press, Boca Raton (2011) 13. Das, P., Samanta, S.K., Bera, S., Dutta, P.: Microstructure evolution and rheological behavior of cooling slope processed Al-Si-Cu-Fe alloy slurry. Metallur. Mater. Trans. A 47(5), 2243– 2256 (2016). https://doi.org/10.1007/s11661-016-3356-3 14. Paskevich, A., Wies, T.: Verified software. theories, tools, and experiments. In: 9th International Conference, VSTTE 2017, Heidelberg, Germany, 22–23 July 2017, Revised Selected Papers. Springer (2017)
The Effect of Porosity on the Elastic Modulus and Strength of Pervious Concrete Viet-Hung Vu1 , Bao-Viet Tran2(B) , Viet-Hai Hoang2 , and Thi-Huong-Giang Nguyen2 1 University of Transport and Communications, Campus in Ho Chi Minh City, Ho Chi Minh
City, Vietnam 2 University of Transport and Communications, Hanoi, Vietnam
[email protected]
Abstract. Pervious/porous material is composed of a solid structure that is continuously arranged in an orderly or random way, forming a framework and between them exists empty spaces called pores filled with fluid (liquid, gas). One of the important applications of porous materials in the construction industry is high porous concrete that used to make surface pavement with natural water permeation capacity. This solution is called Sustainable Drainage Systems (SUDS). In this paper, a novel micromechanical model is developed to predict the relationship between the porosity and the strength of pervious concrete materials. Based on the three-phase composite sphere assemblage model with coated pore-concrete inclusions embedded in a fictitious effective medium, the strain, stress mean fields and the effective properties of material are constructed. Moreover, illustrative applications are reported by comparing the theoretical prediction with the previous experimental review to show the pertinence of model. According to the obtained results, it can be concluded that due to the additional information on the maximum porosity and elastic modulus of material, the new approximation model is better suited to experimental results than some of previous published analytical models, which demonstrates the effectiveness of this research. Keywords: Pervious concrete · Porosity · Elastic modulus · Strength
1 Introduction Pervious/porous material is composed of a solid structure that is continuously arranged in an orderly or random way, forming a framework and between them exists empty spaces called pores filled with fluid (liquid, gas). Porous materials are abundant in nature such as soil, stone, wood, etc. or artificial materials such as porcelain, metal, concrete, and plastic with high porosity to serve important practical applications such as energy management, damping, soundproofing, thermal insulation, water permeation, medical products, etc. One of the most important applications of porous materials in the construction industry is high porous concrete that used to make surface pavement with natural water permeation. This solution is called Sustainable Drainage Systems (SUDS). Contrary to traditional drainage, sustainable surface drainage is a solution to increase the natural water permeability on the covering surface. In Vietnam, a sustainable drainage system in general and © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 823–829, 2022. https://doi.org/10.1007/978-981-16-3239-6_63
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the mechanical, structural characteristics of porous materials constituting the system in particular have been researched and piloted for application through a number of scientific researches. However, pervious concrete or cement mortar is characterized by damage expressed through destructive strength values that tend to be lower than conventional materials. The two main characteristics of compressive strength and porosity of materials in general tend to be inversely proportional to each other: the higher the compressive strength, the lower the porosity, the less the ability to permeation performance and vice versa. The issue determining the relationship between these two characteristics, and further research to optimize this relationship are complex problems in science [1–4]. Therefore, developing a new analytical approximation formula based on material mechanics, multi-ratio approach, overcoming the disadvantages of previous studies and successful empirical verification is the main content of this research.
2 The Strain and Stress Fields Let us here consider situation concerning the macroscopically isotropic suspension of spherical pores (porosity p) in an infinite continuous concrete matrix having elastic raider C c (bulk modulus Kc , shear modulus μc ), tensile and compressive strength St0 and Sc0 . The porous concrete media has average mechanical parameters as C cp (Kcp , μcp ), St , Sc , respectively.
Fig. 1. Three-component problem of porous concrete
It is assumed that the loading applied to domain is defined by the tensor E representing the strain state at the macroscopic scale. The solution of the stress–strain fields linearly depends on E. = C cp : E = 3Kcp J + 2µcp K : E, (1) E = Scp : =
1 1 J+ K 3Kcp 2µcp
: .
(2)
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J = 1/31 ⊗ 1 and K = I − J, with I and 1 are respectively, the fourth-order and second-order (symmetric) identity tensors in an orthonormal frame. To determine average strain and stress field in concrete matrix, it is considered the solution of displacement fields in polar coordinates for a coated inclusion in a continuous matrix (Fig. 1) from [5]: (3) Ec = Ac : E = Asc J + Adc K : E, Ep = Ap : E = Asp J + Adp K : E, c = Bc : = Bcs J + Bcd K : .
(4)
Where Ap , Ac , Bc are respectively the strain and stress concentration tensor of pore and concrete matrix: Bcs =
Kc s d μc d Ac , Bc = A . Kcp μcp c
(5)
Because of isotropy and spherical symmetry of the structure, the four unknown coefficients in (3), (4) may be obtained from the solution of two elementary problems: the volumetric one with E = E01, concerning the spherical components (Asc , Asp ), and the simple shear one with (E = E0(e1 ⊗ e1 − e2 ⊗ e2 )) for (Adc , Adc ). Using the results in [5], it is given that: − 3Kcp + 4μcp μc s , Ac = (6) 3pKc μc − μcp − μc 3Kc − 4μcp 3Kcp + 4μcp (3Kc + 4μc ) 1 s . Ap = − (7) 4 3pKc μc − μcp − μc 3Kc − 4μcp While the particular expressions of scalars (Adc , Adc ) that depend only on the elastic moduli and the porosity are unfortunately too large to be given explicitly in this text.
3 Calculation of the Elastic Modulus of Porous Concrete Based on some results above relating the strain and stress fields, the self-consistence approximation is used to solve two elastic coefficients of porous concrete depending on the property of matrix concrete and porosity: pm (1 − p/pm )Kc Asc + (1 − pm )Kc Asc0 − Kcp = 0 pAsp + (1 − p)Asc + (1 − pm )Asc0
(8)
and pm (1 − p/pm )μc Adc + (1 − pm )μc Adc0 pAdp + (1 − p)Adc + (1 − pm )Adc0
− μcp = 0.
(9)
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While the elastic modulus and Poisson ratio could be calculated based on the classical relationships: E=
3K − 2μ 9Kμ ;ν = . 3K + μ 6K + 2μ
(10)
With Asc0 , Adc0 are respectively Asc , Adc at p = 0 and pm is maximum value of porosity. For elasticity-porosity relationship, new adaptive generalized self-consistent equations (Eqs. 8, 9) are proposed with a free parameter charactering by the maximum porosity (pm ). Then compared this prediction with experimental data in Miled et al. (2011) [6] and the Hashin-Strickman bound that coincide with the well-known MoriTanaka approximation in this geometric configuration [5]. To highlight the novelties of this work, the classical self-consistent estimate is added in Fig. 2 [5]. The good coherence shows effectiveness of this method at pm = 0, 74.
1
Ecp/Ec
HSB (MTA) 0.8
Model
0.6
Experimental data Self-consistent approximation
0.4 0.2 0 0
0.2
0.4
0.6
0.8
1
p Fig. 2. Relationship between elastic modulus and porosity
4 Calculation of the Compressive Strength of Porous Concrete To determine the compressive strength of porous concrete, it is assumed that the loading applied to domain is defined by hydrostatic stress = 1. The average stress in concrete matrix c is calculated: Kc s c = c 1 = Bc : = Bcs J + Bcd K : 1 = A (11) Kcp c ⇒ c =
Kc s A . Kcp c
(12)
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When c → St0 , it means that → St , given that: St =
Kcp 1 St0 . Kc Asc
(13)
Based on some experimental relationship between tensile and compressive strength of concrete material, it is proposed that an adaptive relationship as follows: St = λ(Sc )n .
(14)
where λ, n are adaptive coefficients, n is from 1/2 to ¾. Reporting (14) in (13), a semi-analytical formula was formed to calculate the compressive strength of porous concrete media: Kcp 1 1/n Sc = Sc0 . (15) Kc Asc It is necessary to note that, for strength-porosity relationship, we constructed new adaptive generalized self-consistent equations to predict the tensile strength and porosity relationship with a free parameter characterizing by the maximum porosity while the compressive strength and porosity relationship are estimated by adaptive equation of two free parameters characterizing by the maximum porosity and exponent coefficient. In order to verify the above results, some comparisons between the formula (15) with the experimental data of Zhong et al. (2016) [7] in Fig. 3, and Chen et al. (2013) [3] in Fig. 4 are performed. Table 1 presents parameters for each example. The analytical approximation of Li et al. (2018) [4] are also presented to show effectiveness of this method. Table 1. Parameters of model Ec (GPa) νc
Sc0 (MPa) pm
n
Figure 3 33.1
0.2
39
0.39 1/2
Figure 4 44.35
0.229 66
0.62 1/2
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Model Experimental data Li et al.
60
Sc (Mpa)
50 40 30 20 10 0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
p Fig. 3. Comparisons between the formula (15) with the experimental data of Zhong et al. (2016) [7]
70
Model Experimental data Li et al.
60
Sc (Mpa)
50 40 30 20 10 0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
p Fig. 4. Comparisons between the formula (15) with the experimental data of Chen et al. (2013) [3]
5 Conclusions Based on the construction of local stress and deformation fields and an empirical formula for the relationship between tensile and compressive strength of concrete, a theoretical
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formula for determining the tensile strength of porous concrete, which depends on the porosity and properties of substrate, was proposed. Due to the additional information on the maximum porosity and elastic modulus of the material, the new approximation model is better suited to previous experimental results than some of the published analytical models, which demonstrates the effectiveness of this research approach. However, further empirical researches are needed to expand the number of measurements for better verifying. Comparing the model with new numerical and experimental results that contain more information about the material structure and applying this model to other types of porous materials is the research direction that can be further developed. Acknowledgement. This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 107.02-2018.306.
References 1. Huang, B., Wu, H., Shu, X., Burdette, E.G.: Laboratory evaluation of permeability and strength of polymer-modified pervious concrete. Constr. Build. Mater. 24(5), 818–823 (2010) 2. Chen, X., Wu, S., Zhou, J.: Influence of porosity on compressive and tensile strength of cement mortar. Constr. Build. Mater. 40, 869–874 (2013) 3. Du, X., Jin, L., Ma, G.: Macroscopic effective mechanical properties of porous dry concrete. Cem. Concr. Res. 44, 87–96 (2013) 4. Li, D., Li, Z., Lv, C., Zhang, G., Yin, Y.: A predictive model of the effective tensile and compressive strengths of concrete considering porosity and pore size. Constr. Build. Mater. 170, 520–526 (2018) 5. Tran, B.V., Pham, D.C., Nguyen, T.H.G.: Equivalent-inclusion approach and effective medium approximations for elastic moduli of compound-inclusion composites. Arch. Appl. Mech. 85(12), 1983–1995 (2015). https://doi.org/10.1007/s00419-015-1031-6 6. Miled, K., Sab, K., Le Roy, R.: Effective elastic properties of porous materials: homogenization schemes vs experimental data. Mech. Res. Commun. 38(2), 131–135 (2011) 7. Zhong, R., Wille, K.: Compression response of normal and high strength pervious concrete. Constr. Build. Mater. 109, 177–187 (2016)
Predicting Capacity of Defected Pipe Under Bending Moment with Data-Driven Model Hieu C. Phan1(B) , Nang D. Bui1 , Tiep D. Pham1 , and Huan T. Duong2 1 Le Quy Don Technical University, 236 Hoang Quoc Viet, Hanoi, Vietnam
{phanchihieu,ducnangbui,phamductiep}@lqdtu.edu.vn
2 Vietnam National University of Agriculture, Trau Quy, Gia Lam, Hanoi, Vietnam
[email protected]
Abstract. Water mains which suffered from corrosive environment and various loads/effects. This leads to the simultaneous occurrence of the decrease of the pipe capacity and the appearance of significant bending moments within the pipe. Various investigation in literature focused on the stress of defected pipe due to burst pressure for oil and gas pipe under the high internal pressure, rather than the bended water mains. Primitive studies on this problem are at the observation step without providing an applicable model for practice. Since finding analytical solution for corrosion pipe is a challenging task because of the localized of the defects, the Finite Element Analysis (FEA) is an effective alternative. The critical drawback of FEA is that the problem needs to analysis separately with computational expense and required highly skilled experts. To ease these difficulties, a data-driven model is developed based on the database generated from FEAs and labeled by their results. The application of machine learning techniques improves the accuracy of the conventionally statistical regression models such as linear regression due to the flexible of the model and thus a trained data-driven model is a practical approach to solve the problem. Keywords: Defected pipe · Computational intelligence · Machine learning · Finite element analysis
1 Introduction Water mains are vital systems for human communities, however, these networks normally suffered from corrosive environment of surrounding soil and various loads/effects such as land sliding, non-uniform supports or temperature changing etc. Unlike pipeline used for transporting oil and gas which design mainly for internal pressure and a significant temperature change due to the difference between transported medium and environment [34], unexpected bending moment can be the major factor of failure for water main. Statistics in [12] shows that the critical failure mode of water mains dues to circular crack accounting for up to 56% of failure event in North America compared to this of 8% for longitudinal crack. This implies the most critical thread to water mains is the bending moment or the axial force (corresponding to circular crack) rather than © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 830–840, 2022. https://doi.org/10.1007/978-981-16-3239-6_64
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ring bending or internal pressure (corresponding to longitudinal crack). The bending moments may derive from various sources such as land sliding [29, 35], ununiform supports [2, 21], frost-induced ground deformations [36], strike-slip faults [38] etc. Along with circular crack, corrosion is another major threat to water mains with 28% of failures comes from such source of degradation [12] and the situation can be worse with the combination of such failure modes. Makar et al. [23] have conducted experiments and reported the closely relationship of corrosion and appearance of circular crack. The combination of defected pipe under bending moment thus has been gaining interest of researchers [20, 21, 27, 33, 40]. Roy et al. [33] have validated the FEA and experiment results of pipe under bending, axial load, internal pressure and temperature change. Liyanage and Dhar [21] investigated the defected pipe placed on springs and voids represented for soil support and erosion due to water leak, respectively. In a report of Liu et al. [20], the failure loci are provided with both experiment and Finite Element Analysis (FEA) simulation provides the insight behavior of pipelines with corrosion under combining loads. Mondal and Dhar in [27] have validated the results in [20] with discussion on its failure criteria and investigate the reduction of pipe internal pressure under the combination of loads (i.e. with bending and axial loads). Many attempts to modelized the capacity of defected pipe under bending moment are made and can be found in [1, 7, 8, 39, 40] with most of the study using FEA to obtain the failure moment. The requirement of simulation structure with FEA may lead to a cumbersome work with requirement expert level and computational cost. Besides, the nonlinear relationship of the inputs to the output variable(s) cannot be provided without repetition of FEA. In the case of combining structure capacity prediction with other algorithm (e.g. optimization or simulation-related reliability assessment), this drawback of the FEA approach surges and can be a critical problem for successfully finish such cumbersome work. Consequently, the data-driven models commonly applied based on the database generated from FEA or experiment as in [31] or [11]. The Random Forest Regression (RFR) is a machine learning model which has been successfully applied in predicting various interested variables with a medium-small database contained few hundreds of samples as in [17, 25, 41] and [31]. In this study, results of 208 FEA simulations are composed to be a database for a machine learning model, RFR. The following sections are organized such that: the second section discusses on the establish of a proper FE model, the third one presented the fundamental of the chosen data-driven model, the RFR. Discussion on the database and developed model for predicting pipe bending capacity will be in the fourth section and finally, conclusion of contribution and future works are at the last section.
2 Finite Element Model The FEA in this study is implemented on the Abaqus software which has the ability of simulating nonlinear behavior of the structure in the post-yielding state. A quarter pipe model with the defected on the top/crown and the symmetric constrains are applied as in Fig. 1. The x-symmetric and z-symmetric constrains are set along the longitudinal circular cuts of the pipe, respectively, to reduce the elements in the models and thus reduce the computation cost. The defection dimension including the depth, d; length, l;
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and width, warc , of the defect with sharp edge. Comparison between the difference of the sharp and smooth edges has been conducted in [6] with minor difference. However, the smooth edge required much more computational effort, and thus, the sharp edge is used. The beam-style Multiple Point Constrain, MPC, is used with the reference point located at the other end of the model. The bending moment on the model is applied on this reference point.
Fig. 1. Quarter model of defected pipe on Abaqus
Various material models are proposed as summarized in [6] and illustrated in Fig. 2. The non-linear true stress-strain curves based on experiment is ideal for exact simulation. However, the drawback is the increase of inputs for data-driven model break such curve into points or mathematically modelized them. The bilinear and elastic-perfectly plastic are preferable to present for the behavior of material due to the simplification and the bilinear model is chosen in this study to develop the database. Properties of material thus require the stress and strain of the yield and ultimate points including: yield strength, σy , ultimate strength, σu , and their corresponding strain, εy and εu . These parameters of metal material are collected from literature [10, 20, 22, 24, 30] and summarized in Table 1.
Fig. 2. Different material models [6]
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833
Table 1. Collected material properties Material
σy σu εy (MPa) (MPa) -
εu -
Reference
Cast Iron
161.9
218.64 0.0031 0.0067 [24]
Steel, X42
290
415
0.0014 0.1036 [20]
Steel, X60_1
452
542
0.0021 0.043
[10]
Steel, X60_2
414
600
0.0020 0.095
[10]
Steel, X65
464.5
563.8
0.0022 0.061
[30]
Steel, X70_1
508
667
0.0024 0.095
[22]
Steel, X70_2
523
701
0.0025 0.095
[22]
Steel, X80
524
685
0.0025 0.078
[22]
Various failure criteria are also used for defining the failure state of the pipe ([19, 20] and [27]). Liu et al. in [20] has defined that a pipe is considered to be failure if: (1) “The average von Mises equivalent stress across the full remaining thickness reaches the true ultimate tensile strength of the pipe material; or the von Mises equivalent stress at diametrically opposite from the corrosion defect reaches the yield strength of the pipe material; or the onset of local collapse or global instability/buckling.” However, the early failure of the pipe where the yield stress at the opposite point of the defect appear. The structures are considered to be failure due to the exceed of yield stress [27]. This leads to the over conservative prediction of pipe capacity. Consequently, [27] use the failure criteria as in [19] with the definition of pipe failure is: (2) “The average von Mises equivalent stress throughout the thickness reaches the true ultimate strength of the pipe material”. This study adjusts the failure criteria (2), the failure criteria to decide the ultimate bending capacity of pipe, M, is equal to the quantity of moment applied on pipe where: (3) “Failure occurred if any point in the structure reaches the ultimate stress”. Figure 3 illustrates the difference of material models and failure criteria are provided with the observation at the failure point in the ligament of the pipe. The control case is conducted with the inputs are: D = 203.2 mm, t = 82 mm, d = 4.1, l = 65.6 mm, warc = 65.6 mm, σy = 290 MPa, σu = 415 MPa, εy = 0.0014, εu = 0.1036 (steel, X42). The intact pipe capacity calculated based on classic theory is 109 kNm [27]. The moment capacity of pipe with the true/actual, bilinear and Elastic-perfectly plastic stress-stress curves (Mc_Tr , Mc_Bi , Mc_Ep ) are: 67, 93 and 98 kNm. Compared to the result in [27],
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Mc_Tr in this study is almost equal (~98 kNm versus 97 kNm, respectively) and provide a validation for the present FEA model. Analogously, the Mc_Ep , in this study compared to Liu et al. with failure criteria (1) have minor difference (67 kNm versus ~68 kNm, respectively). The simulated Mc_Bi with failure criteria (3) is within the range set by [Mc_Ep , Mc_Tr ] or it is higher than Mc_Ep and less than Mc_Tr .
Fig. 3. Comparative results at the failure point
3 Developing a Data-Driven Model with Random Forest Regression Unlike conventional mechanical models, data-driven model does not attempt to establishing and solving mechanical equilibrium equations. They rely heavily on a database labeled by the interested variable for each set of input. A learning process is conducted with the target to minimize the error between predicted and label or “actual” values. Once the model is developed, the effect of an input to the output is established. This process is fostered by the computer power and the redundancy of data available [28]. Technically, the database is splitted into train and test set for training and validating processes, respectively. The test set, as its name, used to test the performance of the developed model on the data that it has never “seen” or not appeared in the train set [14]. Various algorithms have been developed and can be used for developing a data-driven model. Some of them are: Artificial Neural Network, ANN (introduce by McCulloch and Pitts [26] and developed by many studies such as [5, 15, 18, 32]); Support vector machine, SVM ([3, 9, 37]); Random Forest, RF; Extreme Gradient Boosting (XGBoost) [13]; Adaptive neuro fuzzy inference system (ANFIS) [16] etc. In this study, the RF is chosen to develop the data-driven model based on samples generated from FEA. Random Forest (RF or Random Forest Regression, RFR) is an ensemble of the decision trees. Each tree in the forest established by the Classification and Regression Tree, CART, algorithm [4] with a random part of the database or a sub-database randomly selected. The CART algorithm minimizes the loss function which can be Entropy, Gini or errors of the tree [14] and the Mean Squared Error, MSE, is chosen in this study. The decision tree is repeatedly established to create a number of trees or a forest and a voting process is conducted to aggregate the final prediction.
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4 FEA Database and the Developed Model A wide range of materials and pipe dimension are used with different defects sizes are chosen as the generated input for the FEA. Pipe and defect dimension inputs are outer diameter, D, wall thickness, t, of pipe and depth, d, length, l, and width, warc , of defect. Because the bilinear material model is applied, inputs related to material properties include yield strength, σy , ultimate strength, σu , and their corresponding strain, εy and εu . The Abaqus Python script is applied to obtain a semi-automatic model generation of the database. This accelerate the data generating process and partly reduce the cumbersome work where a large number of simulations involved. Table 2 provides the list of inputs and some selected samples in the database. Table 2. Selective FEA database D (mm)
t (mm)
d (mm)
l (mm)
warc (mm)
σy (MPa)
σu (MPa)
εy -
εu -
M (kNm)
1
762
17.5
0
0
0
290
415
0.001376
0.1036
3860
2
762
17.5
0
0
0
452
542
0.002145
0.043
5040
3
762
17.5
0
0
0
414
600
0.001965
0.095
5430
…
…
…
…
…
…
…
…
…
…
…
198
914
25.4
0.5
100
50.05
464.5
563.8
0.002205
0.061
10800
199
914
25.4
0.5
100
50.05
523
701
0.002482
0.095
12000
200
914
25.4
0.5
100
50.05
524
685
0.002487
0.078
11880
Basic descriptive statistics of the generated database are given in Table 3 providing the mean, standard deviation, maximum and minimum values of each input. As mentioned in our previous study [11], the maximum and minimum value of each input in the database can be set as the boundaries for a valid prediction. The mean and standard deviation of bending moment capacity, M, in the database are 3566 and 4066 kNm, respectively. The minimum and maximum values of this variable are 45.4 kNm and 13580 kNm which established a wide range of possible pipe moment capacity. Table 3. Descriptive statistics of the FEA database D (mm)
t (mm)
d (mm)
l (mm)
warc (mm)
σy (MPa)
σu (MPa)
εy -
εu -
M (kNm)
mean
622.234
13.028
5.025
72.946
134.022
436.674
570.386
0.002117
0.090100
3566.439
std
338.915
5.837
4.025
93.367
164.046
85.050
103.314
0.000385
0.159162
4066.432
min
180.000
5.560
0.000
0.000
0.000
161.900
218.640
0.001376
0.006739
45.400
max
1219.000
25.400
14.000
500.000
718.168
524.000
701.000
0.003085
2.103600
13580.000
Further observation on the database is illustrated in Fig. 4 which presents the matrix of coefficients of correlation among input and output variables. In this figure, area of each
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square is equivalent to the Pearson correlation coefficient of variable in the horizontal and vertical axis. It is predictable that squares along the diagonal of this matrix are the largest indicating the self-correlation value at 1.0. The most significant relationships are the correlation of pipe diameter and wall thickness to the capacity of pipe. The dimensions of defect have less significant level. The depth, length and width of the defect have correlation coefficients with the descending order. It is predictable to observe the strong effect of d to the capacity of pipe. Also, the strong relationship between stress and strain of the material can be predictable. However, the stronger influence of defect length to defect wide is remarkable which indicate the more important level of l to warc .
Fig. 4. Correlation coefficient matrix among variables (larger squares implie larger value of coefficients)
Table 4. Hyperparameter of the RFR Hyperparameter
Value Description
bootstrap
True
criterion
MSE Use mean square error as the loss function
max_depth
None No limit for maximum depth of the trees
max_leaf_nodes
None No limit for leaf nodes
Use bootstrapping
min_samples_leaf 1
Minimum samples in a node
min_samples_split 2
Minimum samples for a split
n_estimators’
The number of trees in the forest
100
As earlier discussion, the database generated by Abaqus FEA and labeled with moment capacity of defected pipe is used to develop the RFR model. The database
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is splitted with the 0.8/0.2 ratio for train set and test set respectively. With 160-sample train set, the model is developed and then such model is validated with the 40-sample test set. The hyperparameters of the RFR are given in Table 4 with the bootstrapping used and number of trees in a forest is 100. Meanwhile, the minimum samples in a node and for a split are chosen at 1 and 2, respectively. Table 5 provides the evaluation metrics on both train and test set of the model after training process. The coefficient of determination, R2 , on both sub-databases are consistently higher than 0.99 indicating a well-matched of the model and the actual trend of the output. The Mean Absolute Error, MAE, at 66.93 kNm on train set and higher at 165.65 kNm on test set are minor compared to the mean value of (e.g. 3566.44 kNm) as in Table 2. The Mean Absolute Percentage Error, MAPE, reinforced this evaluation with quantity of this type of error are all less than 5% (0.021 on train set and 0.0486 on test set). The means of error between predicted and actual value of M are also closed to zero indicating a non-bias prediction along with the acceptable standard deviation of such error. Figure 5 illustrates the evaluation process visually with the closely following of the datapoints to the 1:1 line of the predicted and simulated axis. This implies that the trend (direction of the datapoints) of the model is satisfied and error (distance between datapoints and the 1:1 line) of such model are minor. Table 5. Evaluation metrics of the RFR model on train and test set Unit R2
–
MAE
(kNm)
MAPE
–
mean error (kNm) std error
–
Train set Test set 0.9991
0.9930
66.9364
165.6581
0.0210
0.0486
6.1003 −40.0941 125.2573
291.1223
Fig. 5. Predicted versus simulated (FEA) of pipe moment capacity
The feature importance for each input, which found by the average depth of each input appeared in decision trees [26], are showed in Fig. 6 with the firstly ranked position
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belongs to pipe diameter, D at 0.9. This is well-matched with the correlation matrix in Fig. 4. The other significant inputs are the wall thickness and the ultimate strength of material at 0.04 and 0.02, respectively. Unlike the relative positions obtained in Fig. 4, length of defect is the most important variable among defect dimension inputs. Meanwhile, the depth of the defect, d, conquered this position in Fig. 4. The strains of material are the most insignificant variables compared to other inputs with the important levels are less than 0.01.
Fig. 6. Feature importance of input variables
5 Conclusion This study has implemented an extensive simulation-based database for the defected pipe under the bending force which are commonly appeared in buried pipe as a major cause of failure. A validation is conducted to investigate the validation of the FEA model of this study. Besides, a brief comparison difference between material models, failure criteria are provided for observation. The chosen failure criteria (3) is observed to be between values of criteria (2) and criteria (1). Consequently, FEA models are repeatedly established based on bilinear material models with the incorporation between Abaqus and Python scripts to reduce the cumbersome work of data generating. A set of 208 labeled samples have been generated by FEA. The statistics analysis on this database provides the preliminary evaluation of input important levels. A RFR model then has been developed with appropriate performances such as R2 and MAPE are 0.993 and 0.0486, respectively. Further development where machine learning model combined with other advance techniques is promising with such high-quality model or the expansion of boundaries for the data-driven model can be conducted. Acknowledgements. This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number: 107.02-2020.04.
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26. McCulloch, W.S., Pitts, W.: A logical calculus of the ideas immanent in nervous activity. Bull. Math. Biophys. 5(4), 115–133 (1943) 27. Mondal, B.C., Dhar, A.S.: Burst pressure of corroded pipelines considering combined axial forces and bending moments. Eng. Struct. 186, 43–51 (2019) 28. Montáns, F.J., Chinesta, F., Gómez-Bombarelli, R., Kutz, J.N.: Data-driven modeling and learning in science and engineering. C. R. Méc. 347(11), 845–855 (2019) 29. Ni, P., Mangalathu, S., Yi, Y.: Fragility analysis of continuous pipelines subjected to transverse permanent ground deformation. Soils Found. 58(6), 1400–1413 (2018) 30. Oh, C.-K., Kim, Y.-J., Baek, J.-H., Kim, Y.-P., Kim, W.-S.: Ductile failure analysis of API X65 pipes with notch-type defects using a local fracture criterion. Int. J. Press. Vessels Pip. 84(8), 512–525 (2007) 31. Pham, T.D., Bui, N.D., Nguyen, T.T., Phan, H.C.: Predicting the reduction of embankment pressure on the surface of the soft ground reinforced by sand drain with random forest regression. In: IOP Conference Series: Materials Science and Engineering, pp. 072027. IOP Publishing (2020) 32. Ranzato, M.A., Huang, F.J., Boureau, Y.-L., LeCun, Y.: Unsupervised learning of invariant feature hierarchies with applications to object recognition. In: 2007 IEEE Conference on Computer Vision and Pattern Recognition, pp. 1–8. IEEE (2007) 33. Roy, S., Grigory, S., Smith, M., Kanninen, M., Anderson, M.: Numerical simulations of full-scale corroded pipe tests with combined loading (1997) 34. Sun, J., Shi, H., Jukes, P.: Upheaval buckling analysis of partially buried pipeline subjected to high pressure and high temperature. In: International Conference on Offshore Mechanics and Arctic Engineering, pp. 487–494 (2011) 35. Torii, A.J., Lopez, R.H.: Reliability analysis of water distribution networks using the adaptive response surface approach. J. Hydraul. Eng. 138(3), 227–236 (2012) 36. Trickey, S.A., Moore, I.D., Balkaya, M.: Parametric study of frost-induced bending moments in buried cast iron water pipes. Tunn. Undergr. Space Technol. 51, 291–300 (2016) 37. Vapnik, V.: Statistical Learning Theory. John Wiley&Sons. Inc., New York (1998) 38. Vazouras, P., Karamanos, S.A., Dakoulas, P.: Mechanical behavior of buried steel pipes crossing active strike-slip faults. Soil Dyn. Earthq. Eng. 41, 164–180 (2012) 39. Yu, W., Vargas, P.M., Karr, D.G.: Bending capacity analyses of corroded pipeline. J. Offshore Mech. Arct. Eng. 134(2) (2012) 40. Zheng, M., Luo, J., Zhao, X., Zhou, G., Li, H.: Modified expression for estimating the limit bending moment of local corroded pipeline. Int. J. Press. Vessels Pip. 81(9), 725–729 (2004) 41. Zhou, J., Shi, X., Du, K., Qiu, X., Li, X., Mitri, H.S.: Feasibility of random-forest approach for prediction of ground settlements induced by the construction of a shield-driven tunnel. Int. J. Geomech. 17(6), 04016129 (2017)
Image Recognition Using Unsupervised Learning Based Automatic Fuzzy Clustering Algorithm V. V. Tai1 and L. T. K. Ngoc2,3,4(B) 1 College of Natural Science, Can Tho University, Can Tho City, Vietnam
[email protected]
2 Faculty of Engineering, Van Lang University, Ho Chi Minh City, Vietnam
[email protected]
3 University of Science, Ho Chi Minh City, Vietnam 4 Vietnam National University, Ho Chi Minh City, Vietnam
Abstract. This article proposes a novel techniques for unsupervised learning in image recognition using automatic fuzzy clustering algorithm (AFCA) for discrete data. There are two main stages in order to recognize images in this study. First of all, new technique is shown to extract sixty four textural features from n images represented by a matrix n × 64. Afterwards, we use the proposed method based on Hausdorff distance to simultaneously determine the appropriate number of clusters. At the end of the unsupervised clustering process, discrete data belonging to the same cluster converge to the same position, which represents the cluster’s center. After determining number of cluster, we have probability of assigning objects to the established clusters. The simulation result built by Matlab program shows the effectiveness of the proposed method using the corrected rand, the partition entropy, and the partition coefficients index. The experimental outcomes illustrate that the proposed method is better than the existing ones as Fuzzy Cmean. As a result, we believe that the proposed method is filled with a potential possibility which can be applied in practical realization. Keywords: Automatic algorithm · Hausdorff distance · Image recognition · Fuzzy · Unsupervised clustering
1 Introduction In recent years, unsupervised learning techniques have made a great contribution to machine learning that is used to find the common natural cluster structure of an unlabeled data set. The conventional supervised learning methods utilized in many fields, containing as data analysis, require a supervisor to guide the machine, labeling inputs with the outputs we want the machine to learn from. Nevertheless, this labeling process is complex stage, especially when a number of data to label are being created up to the billions of items every day, such as data images on the Internet. As a result, unsupervised © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 841–856, 2022. https://doi.org/10.1007/978-981-16-3239-6_65
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learning techniques are required to solve this limitation. In unsupervised learning, unsupervised clustering is an important data mining technique, used to divide data points into groups. Such those objects in the same group will have a high degree of similarity while objects belonging to the other ones will have a high degree of dissimilarity. It has been widely used in many applications such as data analysis, pattern recognition and image processing [1, 2]. Unsupervised item recognition plays a vital role in many fields such as artificial intelligence, architecture, and engineering. Although it has been concerned by many scientists in many areas for a long time, it is quite challenging at present. To overcome these challenges, there are a great deal of approaches presented by [3, 4] to recognize items through some algorithms. However, these approaches are usually process compoundly, create inaccurate outcomes and take a huge amount of time to perform. Therefore, this paper shows a new method to image recognition using an automatic fuzzy clustering algorithm. There are two primary phases in order to recognize images in this study. To begin with a first phase, human eye is a discerning item like a combination of primary parts (color, texture, shape). Therefore, color extraction, texture extraction and shape extraction are three main approaches in image feature extraction technique. We can choose the method which is appropriate to our goal as each approach has its advantages and disadvantages. In this paper, we concentrate on the texture extraction method. In image recognition, there are two main approaches that attracted by many researchers: signal processing approach [5–7] and statistical approach [8–10]. To compare with the signal processing approach, statistical image analysis based on the grey level co-occurrence matrix (GLCM) is simple to carry out and takes less computational time. The experimental outcomes in Sect. 4 illustrate that the proposed algorithm is better than the existing ones about running time and accuracy. In fact, the proposed method gives a high clustering performance and is applied by many researchers. For example, Clausi [11] utilized GLCM and Fisher method to group natural textures, Bhogle and Patil [12] combined GLCM and Mahalabonis distance to detect oil spill. It is well-known that there are two major steps in texture-based image recognition. In the first step of recognition, we compute the GLCMs from each image. GLCM is a two-dimensional matrix of joint probabilities between pairs of pixels, separated by a distance d in a given direction θ. For example, Ayala [13] presented d = 2 and θ = {0, 90°, 180°, 270°} while Celebi and Alpkocak [10] claimed that using contrast with d = {1, 2, 3, 4} is the best. In the final step, we extract the features from GLCMs. Haralick [8] defined 14 statistical features from gray-level co-occurrence matrix for texture clustering, where contrast, correlation, homogeneity and energy are common features since they have strongly affects on clustering result. In GLCM, we will collect x × y × z typical variables for texture (TVT) if x, y and z are cases of distance d, direction θ and extracted features for each GLCM, respectively. In the second phase, we use the extracted features for clustering by the proposed method, called Automatic fuzzy clustering algorithm (AFCA). AFCA is a novel technique in unsupervised learning since it is built based on combining two techniques: the determination of the suitable number of cluster, and items belonging to the same cluster converge to the same position, which represents the cluster’s center in the end of the
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unsupervised clustering process. In the next step, we have the probability for assigning the items into the established clusters. As a consequent, the proposed method shows an outstanding effect rather than other ones through applicated experiments. In our knowledge, most technique used in unsupervised clustering are fuzzy c-mean clustering (FCM) [3, 4] to recognize the image. However, FCM technique requires prior k clusters so the choice of the number of cluster, k, is based on experience of researchers. This choice leads to no guarantee of the accuracy of the clustering results. To solve this drawback, we proposed AFCA which can be seen as a new method in unsupervised clustering. There are two main reasons why we believe that AFCA is the novel technique. First of all, we proposed the vector of weight λi where Π λi = 1, i = 1, …, k utilized to measure the difference of central clusters. In addition, we also build the objective function f(U) which is improved to optimize the initial matrix partition U. According to kind of objects, we have some common objects such as discrete elements [14] probability density functions [15–18], and interval data [19, 20]. Particularly, cluster analysis for discrete data (CAD) was studied first with a lot of announced results, which are related to both theory and application [14]. Furthermore, discrete data are becoming more and more popular in storing of data. Hence, we propose a novel technique for unsupervised clustering in image recognition using automatic fuzzy clustering algorithm (AFCA) for discrete data. Determining the criteria for evaluating the similarity between two objects and between two clusters in CAD, the commonly used distances in measuring the similarity between the two clusters are the min distance, the max distance, the average distance, and the Ward distance while the Euclidean distance (dE ) the Lp distance, and the city-block distance (dC ). They are the main criteria for evaluating the similarity among discrete data. In this paper, we use the Hausdorff distance (dH ) as primary measurement to evaluate the difference of data. The experiment results present that the proposed algorithm used dH is better than the other ones, where the corrected rand (CR), the partition entropy (PE), and the partition coefficients (PC) are indexes utilized to check effect of outcome clustering out. As a result, we believe that the proposed method is filled with a potential possibility which can be applied in practical realization. The remainder of the paper is organized as follows. In Sect. 2, some distance measurements are defined for multi-dimension cases. In addition, clustering evaluation criteria also defined in this section. The proposed technique contains image extraction method, the proposed algorithm and its convergence given in Sect. 3. Section 4 illustrates the proposed technique by two applications in image recognition and compares with existing methods based on PC and PE index. Section 5 is the conclusion of the paper.
2 Some Distance Measurements for Discrete Data 2.1 Measure Distance Between Points The Hausdorff Distance is commonly used in machine. In that field, a typical problem is that you are given an image and a model of what you want to match to. The goal is to find all the locations in the image which match the model. This is similar to the problem of matching protein motifs within protein sequences. This distance is different from some of the previously discussed measures, instead of forming a one-to-one mapping between the two, we allow a many-to-many correspondence in this case. Often times, it is easier
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to build a many-to-many correspondence, since if we wish to change one assignment, we no longer have a cascade of other assignments which now also need to be redone. Definition 2.1.1. Given two set X = {x 1 , x 2 , …, x n } and Y = {y1 , y2 , …, yn } in E 2 . The distance from X to Y is defined as. The Hausdorff distance dH (X , Y ) =
n
(hH (xi , yi ), hH (yi , xi ))
i=1
Where hH (X , Y ) =
n
max minxi − yi xi ∈X yi ∈Y
i=1
The Euclidean distance dE (X , Y ) =
n
1 2
(xi − yi )
2
i=1
The City – Block distance dC (X , Y ) =
n
|xi − yi |
i=1
2.2 Clustering Evaluation Criteria Definition 2.2.1. The corrected rand (CR) index [21] is utilized to evaluate the accuracy of the clustering result. CR index is defined as follows −1 R C ni nj n 2 i=1 j=1 i=1 2 j=1 2 −1 CR = C R C R nj ni nj n 1 ni + − 2 2 i=1 2 j=1 2 i=1 2 j=1 2 R C
nij 2
−
where nij represents the number of objects that are in clusters ui and vi ; ni indicates the number of objects in cluster ui ; nj indicates the number of objects in cluster vi ; and n is the total number of objects. CR is an external measure that can make the comparison between the partition produced by a clustering algorithm and the actual partition where “ground-truth” labeling is known. The closer the CR is to 1, the better the clustering result is.
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Definition 2.2.2. The partition coefficient and entropy are used to evaluate the quality of fuzzy clusters. They are given as follows PC =
k N 1 2 μji , N i=1 j=1
PE = −
k N 1 μji log(μji ), N i=1 h=1
where k and N are the number of clusters and objects, respectively. The PC index has the value in [1/k, 1]. The closer to unity the index the “crisper” the clustering is.
3 The Proposed Technique 3.1 Methods to Extract Image Data Image recognition which has many applications is a very interesting and challenging problem. It is the foundation for many applications in the fields of medicine, environment, security [4, 15, 16, 22]. In this study, we performed cluster analysis for images by the proposed algorithm. To identify the image, we must first extract data from the grayscale image. There are many methods of doing this as presented in [23]. Furthermore, we extracted the feature of the image based on the grey level co-occurrence matrix (GLCM) which is a popular and effective method today [24]. The GLCM is a common technique in statistical image analysis that is used to estimate image properties related to secondorder statistics. GLCM considers the relation between two neighbor pixels in one offset, as the second order texture, where the first pixel is called reference and the second one the neighbor pixel. The GLCM presents the information about intensities of pixels and their neighbors at fixed distance d and orientation θ . If we have an image with the size of M × N (M pixels in X - axis and N pixels in Y - axis) and G is the domain of grey level. Then, GLCM is a matrix P with the size of G × G. Each element p(i,j) presents the probability of the occurrence of intensity i and intensity j at fixed distance d and orientation θ. The formula to compute p(i,j) is presented by (1): pd θ (i, j) = {((x, y), (x , y )) ∈ M × N |d = (x, y), (x , y ), θ = ((x, y), (x , y )), f (x, y) = i, f (x , y ) = j}
d = (x, y), (x , y ),
pd θ (i, j) = (x, y), (x , y ) ∈ M × N
f (x, y) = i, f (x , y ) = j . θ = (x, y), (x , y ) (1) From GLCM, Haralick [8] defined 14 statistical measures that can be extracted. However, there are only 4 features consisting of contrast, correlation, homogeneity, and energy having strong affects on classification result. These features are presented in Table 1 (Fig. 1).
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Fig. 1. Illustrating how to calculate GLCM. Table 1. Formula to extract four features from GLCM.
3.2 The Proposed Algorithm Suppose we have N images X = {a1 , a2 ,…, an }. The Automatic Fuzzy Clustering Algorithm for discrete (AFCA) includes the two stages: In the first phase of the proposed algorithm, there are four steps of finding the number of suitable groups. The second phase in the AFCA consists of three steps which can determine the probability to assign each image to clusters. The steps of the AFCA are shown as follows. Step 1. Extract 64 features based on GLCM to data set I N × 64 where N is the number of images. Step 2. Initialize t = 0 and V (0) = X = {v1 , v2 ,…., vn } Step 3. Update the prototype images using (2): (t) (t) N f vi , vj (t+1) vj(t) , i = 1, . . . , N , vi = (2) N (t) (t) j=1 k=1 f vi , vk
Image Recognition Using Unsupervised Learning
where
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⎧ ⎨ exp −d v(t) , v(t) /σ if d v(t) , v(t) ≤ μαij (t), i j j (t) (t) i = f vi , vj ⎩ 0 if d v(t) , v(t) > μαij (t), i
j
with αij (0) = 1, αij (t) =
αij (t−1) , t (t−1) (t−1) 1+αij (t−1)f vi ,vj
≥ 1, 2 (0) (0) (0) (0) ⎛ 1 ⎞ d v − μ d vi , vj , σ = , v . μ= ⎛1⎞ i j 2 ⎝ 2 ⎠ i T BF . The maximum percent difference for conservative values are 4.6% at L b /h = 40 in case of T TF ≤ T BF , and 11.7% at L b /h = 10 in case of T TF > T BF .
(a) TTF ≤ TBF
(b) TTF > TBF
Fig. 7. Comparison of FEA and Proposed equation
4 Conclusions This study presents the elastic lateral-torsional buckling of steel H-beam under fire subjected to pure bending a finite element program, ABAQUS. The effect of non-uniform temperature distribution and length to height ratio were also considered. Based on the
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FEA results obtained, a proposed equation for the beam with non-uniform heating was developed. The theoretical results from the previous studies with uniform temperature distribution assumption by treating the beam temperature as the compression flange temperature and the average temperature in the steel beam were generated and compared. The results showed that the elastic critical buckling strengths of the steel beams using the equation from previous studies with the uniform heating assumption by treating the beam temperature as the compression flange temperature can give unconservative values. And the elastic critical buckling strengths of steel beams give over-conservative values when treating the beam temperature as the average temperature. The proposed equation provides conservative results and better agreement with FEA results. Acknowledgement. This work is supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (Ministry of Science and ICT) (No.2019R1F1A1060708).
References 1. Real, P.V., Franssen, J.M.: Numerical modeling of lateral-torsional buckling of steel I-Beams under fire conditions – comparison with Eurocode 3. J. Fire. Prot. Eng. 11(2), 112–128 (2001) 2. Real, P.V., Lopes, N., da Silva, L.S., Franssen, J.M.: Lateral - torsional buckling of stainless steel I-beams in case of fire. J. Constr. Steel Res. 64, 1302–1309 (2008) 3. Couto, C., Real, P.V., Lopes, N., Zhao, B.: Numerical investigation of the lateral–torsional buckling of beams with slender cross sections for the case of fire. Eng. Struct. 106, 410–421 (2016) 4. Varol, H., Cashell, K.A.: Numerical modelling of high strength steel beams at elevated temperature. Fire Saf. J. 89, 41–50 (2017) 5. Yin, Y.Z., Wang, Y.C.: Numerical simulations of the effects of non-uniform temperature distributions on lateral torsional buckling resistance of steel I-beams. J. Constr. Steel Res. 59, 1009–1033 (2003) 6. Wang, W.Y., Zhou, H., Zhou, Y., Xu, L.: A simplified approach for fire resistance design of high strength Q460 steel beams subjected to non-uniform temperature distribution. Fire Technol. 54(2), 437–460 (2017) 7. ABAQUS: Version 2017, Dassault Systèmes, RI, USA (2016) 8. Timoshenko, S., Gere, J.: Theory of elastic stability. McGraw-Hill Book Company, New York (1961) 9. American Institute of Steel Construction: Steel Construction Manual, 15th edn. AISC, USA (2017) 10. Eurocode 3 Part 1–2: Eurocode 3: Design of steel structures – Part 1–2: General rules– Structural fire design. European Committee for Standardisation, Brussels, Belgium (2005) 11. ISO-834: Fire resistance tests - Elements of building construction. International Organization for Standardization, Geneva, Switzerland (1999) 12. Eurocode 1 Part 1–2: Eurocode 1: Actions on structures – Part 1–2: General actions– Actions on structures exposed to fire. European Committee for Standardisation, Brussels, Belgium (2002) 13. European Convention for Constructional Steelwork (ECCS): Ultimate limit state calculation of sway frames with rigid joints. Publication No. 33, ECCS-Technical Committee No. 8. Brussels, Belgium (1984)
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14. Avery, P., Mahendran, M.: Distributed plasticity analysis of steel frame structures comprising non-compact sections. Eng. Struct. 22(8), 901–919 (2000) 15. Li, G.Q., Guo, S.X.: Experiment on restrained steel beams subjected to heating and cooling. J. Constr. Steel Res. 64, 268–274 (2008)
On the Thin-Walled Theory’s Application to Calculate the Semi-enclosed Core Structure of High-Rise Buildings Nguyen Tien Chuong and Doan Xuan Quy(B) Thuyloi University, Hanoi, Vietnam {chuongnguyentien,quydx}@tlu.edu.vn
Abstract. The shape of a high-rise building’s core structure is usually semienclosed; the coupling beams connecting the two boundary walls at the floors’ level play an essential role in the core’s working. The discrete-continuous model used to calculate semi-enclosed core structures based on the thin-walled theory proved useful and evaluated the overall working and structural components’ behavior in the core. However, this theory also has certain assumptions about the correlation between core boundary walls and coupling beams. The paper will investigate cases of core’s boundary walls corresponding to the several instances of the height of coupling beams according to the thin-walled theory and compare the results with those of the SAP2000 structural analysis program to evaluate and figure out the correct use of the calculation theory. The research results serve as a basis for the design of the core structures of high-rise buildings. Keywords: Semi-enclosed core · Coupling beam · Thin-walled theory - TWT · Boundary wall · Discrete-continuous model - DCM · SAP2000
1 Introduction The semi-enclosed core structure is commonly used in high-rise buildings to resist mainly horizontal loads. Calculations [1, 2] have shown the semi-closed core’s effective working compared to the open-section core of the same size due to the additional coupling beams connecting the two boundary walls at the level of the story. The core is called the thin-walled core, which has three dimensions whose values are different in degree. Let d be the thickness of the core; a is the size of a particular edge of the cross-section, l is the length, then the ratio between these dimensions satisfies the following conditions: d/a ≤ 0.1 and a/l ≤ 0.1 [1, 2]. The thin-walled theory (TWT) and the discrete-continuous model (DCM) [1–4] are used to analyze the performance of the core wall structure. This method is based on a closed core analysis (using TWT) with the coupling beams on each floor level is converted into a wall panel on the story height with an equivalent wall thickness (using DCM). Survey results by TWT and DCM [5, 6] clearly show the functional characteristics of © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 866–874, 2022. https://doi.org/10.1007/978-981-16-3239-6_67
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this type of structure in torsion – bending, and coupling beams play an essential role in reducing the rotation angle, the horizontal displacement, and warping stresses in the core. Vlasov’s TWT is constructed based on two assumptions [2, 3]: 1) when the core deforms the cross-sectional plane down to the plane perpendicular to the axis of the core, retaining the original shape of the section, and; 2) ignore the slip deformation in the medial face of the core. Besides, the wall’s thickness is small enough so that the normal stress is distributed over its thickness. Another assumption condition for using TWT and DCM in calculating a semienclosed core is that the bending rigidity in the boundary wall’s plane (Fig. 2) must be much larger than that of the connecting beams. Previous calculations [1, 2, 5–7] were performed with a certain length of the boundary walls. There is no investigation has been made on whether this change in length of the boundary walls will affect the calculation results according to TWT. The following paper will consider the calculation of a semi-enclosed core structure (one chamber) subjected to the lateral load taking into consideration the length of the core boundary walls to the analysis of the rotation angle of the core by TWT and DCM compared with those by SAP2000 [8]. The result is reached in this paper is the value of rotation angle, thereby evaluating TWT and DCM in calculating the semi-enclosed core structure of a high-rise building.
2 Working Characteristics and Deformation of the Semi-enclosed Core Structure Subjected to Torsion The characteristics of a semi-enclosed core subjected to torsion: opposite walls through openings have a vertical displacement in the opposite direction, rotating around the vertical axis in the same direction. These displacements of the walls at the two ends of the coupling beams make the beam be shear and bent. The shear forces at the ends of the connecting beam create additional vertical forces to the wall. The normal stresses around the core structure section are similar to those stresses in a closed core structure. Thus, the core’s torsional resistance increases, reducing the core structure’s torsion and warping and vertical stresses. TWT is applied to give an approximate analysis of the semi-enclosed core structure [1, 2, 7]. DCM relies on TWT to calculate a semi-enclosed core as an equivalent closed core from the original open core (without connecting beams). The equivalent core is a core in which a wall plate replaces the beam parts with a thickness of t1 , which is featured for the beam’s shear and bending. After calculating the integral for the equivalent closed core, we obtain the equivalent torsion inertia of the core structure with a semi-enclosed section [5]: J=
2 1 bi ti3 + n LLH L 3 + t1 t
In the formula (1): L is the length of the coupling beam;
(1)
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LLH is the summarized length of the opened core; is two times the area that is bounded by the middle line of the core section. The first part of Eq. (1) J1 is the torsion constant (the St.Venant torsion constant) of the original core wall with lengths of bi and thickness of ti of each segment of the core wall. The equivalent wall’s thickness under bending-shearing is determined by the equal deformation of the identical wall plate to the deformation due to the bending and shearing deformations of the coupling beam, calculated by the formula [1, 5]:
t1 = hLT
1 L2
G 12Ib E
+
1,2 Ab
(2)
In the formula (2): hLT is the height of the coupling beam; L is the length of the coupling beam; Ab is the section area of the coupling beam; Ib is the moment of inertia of the coupling beam. The rotation displacement due to torsion of the thin-walled core structure. Consider the case of core structure fixed at the bottom, free at the top, and subject to evenly distributed torque along with the height with the strength m(z) = m. Determine the rotation angle of the core due to the torsion by the following formula [5]: 1 z 2 mH4 1 (AshA + 1) 2 z (3) − θ(z) = − 1) − Ashαz + A (chαz E1 Iω A4 chA H 2 H In the formula (3): A = αH = H
GJ E1 Iω
(4)
In the formula (4): z is the height of the mentioned story; The modified elastic modulus: E1 = E(1−μ2 ); GJ is the corresponding shear torsional rigidity; E1 Iω is the warping rigidity of the core.
3 Investigation of the Behavior of the Semi-enclosed Core Wall Structure Subjected to Horizontal Load The building core wall structure has a diagram shown in Fig. 1 (both width and length are 20 m), subject to uniformly distributed horizontal wind pressure w = 0.5 KN/m2 . The building consists of 20 stories, with the height of each story is ht = 3.5 m; the total height of the building is H = 70 m. The core has a thickness t of 0.25 m; the size boundary wall h1 will be changed for investigation.
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Fig. 1. The story plan and the dimensions of the core
The coupling beams that connect the two boundary core walls from point A to F have a width equal to the wall thickness; the height ranges from 0.1ht to 0.3ht . The concrete material has the secant modulus of elasticity E of 2.50 × 107 KN/m2 , μ = 0.2. The next stage is to perform the analysis and calculate the core structure with a variety of h1 length by TWT and DCM, then compare the results calculated by SAP2000 to evaluate the two methods’ calculation results. It is necessary to determine the core characteristics without coupling beams (the originally opened core): the position of shear center e; the sectorial moment of inertia Iω ; the St.Venant torsion constant J1 . Then, determine the semi-enclosed core characteristics: the equivalent torsion inertia J and the equal wall thickness t1 , depending on the length of the boundary core wall h1 (Table 1). The core structure in SAP2000 is modeled along with the building plan (Fig. 1 and Fig. 2). Columns, beams are modeled small enough to not interfere with the core’s working; the floor’s thickness chosen to make the floor is strong enough to transfer the load to the core, which is automatically subdivided into small shells 1.5 m maximum. The core and connecting beam are modeled as a shell element of 0.175 × 0.25 m [6], which is small enough to ensure calculations accurately. When calculating by the TWT, the horizontal load w will be converted to a uniform force along with the height at the shear center wh = 20 × 0.5 = 10 KN/m, together with a torque value equal to the uniform force multiplied by the eccentricity of the load to the shear center (Table 1). When analyzing the structure, the ratio h1 /h is taken from 0 to 0.4 because of creating space enough to enter the doors (such as for lift or stairs). The torsion (rotation) angle of the core structure reaches its maximum value at the top, called the torsion displacement. The calculation result of the rotation angle is given
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h1
L
h1
The coupling beam
The boundary wall
ht
hLT h
a. 3D Model in SAP2000
b. Model and dimension for Walls and Coupling beams Fig. 2. Model of Structure in SAP2000
Table 1. Semi-enclosed core and torque parameters applied to the core when the core wall is calculated according to TWT and DCM Ratio h1 /h
e(m)
J1 (m4 )
Iω (m5 )
J (m4 ) corresponding to the case of hLT /ht 0.1
0.15
0.2
Torque value (KN.m/m)
0.3
0.000
1.11
0.0625
17.7
0.11
0.21
0.40
1.11
26.13
0.067
1.30
0.0667
25.4
0.13
0.29
0.57
1.60
27.97
0.167
1.56
0.0729
43.8
0.22
0.54
1.12
3.00
30.59
0.233
1.72
0.0771
62.4
0.35
0.95
1.95
4.78
32.20
0.333
1.92
0.0833
103.7
1.12
2.96
5.31
9.59
34.25
0.400
2.02
0.0875
143.1
3.69
7.48
10.42
13.71
35.24
in Table 2. Accordingly, the calculation results, according to TWT, will be compared simultaneously with those by SAP2000. The diagram in Fig. 3 shows the relationship between rotation angle and h1 /h ratio according to TWT compared with SAP 2000 results for some cases of the beam’s heights.
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Table 2. Summary of rotation angle value calculated by TWT and SAP2000 (at the top floor) Ratio Rotation- rad (TWT) corresponding to the h1 /h case of hLT /ht 0.1
0.15
0.2
0.3
Rotation - rad (SAP) corresponding to the case of hLT /ht 0.1
0.15
0.2
0.3
0.000 0.03340
0.01980 0.01149 0.00461 0.03330
0.02489 0.01978 0.01381
0.067 0.02748
0.01522 0.00856 0.00343 0.02726
0.01606 0.01099 0.00631
0.167 0.01815
0.00897 0.00487 0.00202 0.01801
0.00890 0.00536 0.00251
0.233 0.01216
0.00558 0.00302 0.00135 0.01214
0.00573 0.00336 0.00153
0.333 0.00473
0.00209 0.00125 0.00072 0.00472
0.00227 0.00134 0.00062
0.400 0.00170
0.00091 0.00067 0.00052 0.00176
0.00087 0.00054 0.00027
The diagram in Fig. 4 shows the summary of the relationship between the calculated rotation angles with the ratio of the boundary wall on the length of the boundary wall to the connecting beam heights. The values are arranged by increasing the beam heights in proportion to decreasing the rotation angle value. The suffix index (SAP and TWT) shows the method is used to calculate the core structure.
TWT
0.020 0.010
Rotation (Rad)
Rotation (Rad)
0.030 SAP2000
0.030
0.000
0.020 0.010 0.000 0.00 0.10 0.20 0.30 0.40 0.50
0.00 0.10 0.20 0.30 0.40 0.50
h1/h in case hLT = 0.15ht
h1/h in case hLT = 0.1ht 0.015
Rotaion (Rad)
Rotaion (Rad)
0.025 0.02 0.015 0.01 0.005 0 0.00 0.10 0.20 0.30 0.40 0.50 h1/h in case hLT = 0.2ht
0.010 0.005 0.000 0.00 0.10 0.20 0.30 0.40 0.50
h1/h in case hLT = 0.3ht
Fig. 3. Diagram of the relationship between the rotations with ratio h1 /h corresponding to connecting beam heights
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The investigated results show that the length of the boundary wall h1 significantly affects TWT and DCM’s accuracy when calculating the semi-enclosed core structure. However, depending on the height of the connecting beam, the level of influence is different. If the beam height is 0.1 times the story’s height, the boundary wall’s change does not affect the calculation results. According to TWT and according to SAP, the calculation has almost the same value (Fig. 3, Fig. 4). If the beam’s height is 0.15 of the story’s height, the TWT’s rotation result has a deviation from calculation according to SAP2000 if h1 /h is small. For an h1 /h ratio of 0.067 or more, the outputs computed using the two methods are identical (Fig. 3, Fig. 4). If the beam’s height equals 0.2 of the story’s height, TWT calculates well when h1 /h is higher than 0.167 (Fig. 3, Fig. 4). If the beam’s height equals 0.3 of the story height or more, TWT calculates well when h1 /h is higher than 0.2 (Fig. 3, Fig. 4). 0.040
Rotaion- Rad
0.035
LT0.1 SAP LT0.1 TWT LT0.15 SAP LT0.15 TWT LT0.2 SAP LT0.2 TWT LT0.3 SAP LT0.3 TWT
0.030 0.025 0.020 0.015 0.010 0.005 0.000 0.00
0.10
0.20
0.30
0.40
0.50
h1/h Fig. 4. Relationship diagram of rotation angle with ratio h1 /h with cases of connecting beams height
So when calculating by TWT, apart from the two original assumptions of Vlasov, it is necessary to pay attention to the third important factor: the length of the boundary wall of the core must be large enough for TWT to calculate correctly. This problem may because the connecting beam’s length is too long, then the beam’s behavior is not merely in the bending-shearing form. The conversion has overestimated the equivalent wall thickness, and leading to the calculation result of the rotation angle is much lower than those of SAP2000. When calculating the case of a small boundary wall, the reduction of equivalent wall thickness makes the results of rotation and displacement increase accordingly. That shows the above explanation is grounded. To summarize the application scope for the proper use of the Thin-walled theory and the Discrete-continuous model to calculate the semi-enclosed core wall structure is shown in Fig. 5. In this Figure, the vertical axis shows the ratio of hLT /ht ; the horizontal
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Fig. 5. The diagram between hLT /ht vs. h1 /h shows the thin-walled theory’s accuracy calculates the semi-enclosed core wall structure - the folding point is shown the difference of rotation results compared with those of SAP2000 is under 9%.
axis shows the ratio of h1 /h; the hidden line is to separate the two-zone by using TWT. The scope to use the TWT is outside the shade zone, which shows the results calculated by TWT are accurate.
4 Conclusion The paper has applied a discrete-continuous model based on thin-walled theory to calculate the semi-enclosed core structure of a high-rise building subjected to the horizontal loads; the calculation results compare with those of the software SAP2000. From the analysis results, we can propose the following conclusions: - The thin-walled theory and discrete-continuous model continuously apply calculations to semi-enclosed core structures, which estimate the torsional angle quite well, even though there are specific hypotheses to facilitate the analysis. - However, besides the two original assumptions of Vlasov, the TWT’s conditions to calculate the semi-enclosed core wall structure correctly is that the length of the boundary walls must be large enough to correspond to the case of beam heights. The paper’s survey results show that TWT calculates accurately when the ratio between the beam height to the height of the story (hLT /ht ) and the length ratio between the boundary walls to the core’s long edge (h1 /h) has a special relationship shown in Fig. 5.
References 1. Smith, B.S., Counll, A.: Tall Building Structures: Analysis and Design. Wiley (1991) 2. Chuong, N.T.: Structural analysis of high-rise buildings. Construction Publishing House, Hanoi (2015). (in Vietnamese) 3. V.Z. Vlasov, Thin-walled elastic beams, translated from Russian. Israel, 1961. 4. Taranath, B.S.: Reinforced Concrete Design of Tall Buildings. CRC Press, Newyork (2010)
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5. Chuong, N.T., Quy, D.X.: The effect of connecting beam stiffness to the torsional behavior of the core structure of high-rise buildings. In: The 14th National Scientific Conference on Solid Mechanics, Tran Dai Nghia University, Ho Chi Minh City, 19–20 July 2018, July 2018. (in Vietnamese). https://www.researchgate.net/publication/339550333_Anh_huong_cua_do_ cung_dam_noi_den_su_lam_viec_chiu_xoan_cua_ket_cau_loi_nha_cao_tang 6. Chuong, N.T., Quy, D.X.: Analysis of the working of the semi-enclosed core structure of high-rise buildings subjected to horizontal load. In: Annual Science Conference of Thuyloi University, Hanoi, November 2018. (in Vietnamese). https://www.researchgate.net/public ation/339738832_Phan_tich_su_lam_viec_cua_ket_cau_loi_nua_kin_nha_cao_tang_chiu_ tai_trong_ngang 7. Kheyroddin, A., Abdollahzadeh, D., Mastali, M.: Improvement of open and semi-open core wall system in tall buildings by closing of the core section in the last story. Int. J. Adv. Struct. Eng. (IJASE) 6(3), 1–12 (2014). https://doi.org/10.1007/s40091-014-0067-0 8. Wilson, E.L.: CSI analysis reference manual for SAP2000. Computers and Structures Inc, Berkely (2007)
Numerical Simulation of Full-Scale Square Concrete Filled Steel Tubular (CFST) Columns Under Seismic Loading Hao D. Phan1,2(B) , Ker-Chun Lin2,3 , and Hieu T. Phan4 1 Faculty of Civil Engineering, The University of Danang – University of Science and
Technology, Danang, Vietnam [email protected] 2 Department of Civil and Construction Engineering, National Taiwan University of Science and Technology, Taipei, Taiwan [email protected] 3 National Center for Research on Earthquake Engineering, Taipei, Taiwan 4 Department of Computer Science and Software Engineering, Miami University, Oxford, OH, USA [email protected]
Abstract. This paper presents numerical investigations on the seismic behavior of full-scale square concrete filled steel tubular (CFST) columns. The main objective is to understand the seismic behavior and evaluate the seismic performance of these composite columns under high levels of axial compression. Finite element analysis (FEA) models in ABAQUS software were used to simulate a series of columns subjected to axial compression and cyclic lateral loading. The CFST columns were modeled using eight-node reduced integration brick elements (C3D8R) for the infilled concrete with confinement effect, and four-node reduced integration shell elements (S4R) for the steel tube with consideration of steel-concrete interaction and steel wall’s buckling. The feasibility of the FEA models has been validated by published experimental results. The validated FEA model was further extended to conduct parametric studies with various parameters including width-to-thickness ratio (B/t), concrete strength, and axial compression level. The numerical analysis results reveal that with the same B/t and constituent materials, the higher the axial compression was, the lower the shear strength and the deformation capacity were. Also, the higher axial compression led to earlier local buckling of the steel tube, especially, in the case of the thinner steel wall (B/t of 41.7). The thicker steel wall (B/t of 20.8) resulted in higher strength and larger deformation capacity of the column. Increasing concrete material strength significantly improved the column’s shear strength for both thinner and thicker steel walls, but it led to significant development in deformation for the column having thicker steel walls. This study also reveals that only the square CFST columns with B/t of 20.8 using medium material strengths satisfy the seismic performance demand for the building columns in high seismic zones (ultimate interstory drift ratio (IDRu ) not less than 3% radian) under high axial compression (up to 55% of the nominal compression strength, P0 ).
© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 875–889, 2022. https://doi.org/10.1007/978-981-16-3239-6_68
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H. D. Phan et al. Keywords: Square CFST columns · Finite element analysis (FEA) model · Concrete confinement effect · Local buckling · Width-to-thickness ratio (B/t) · High axial compression · Seismic performance
1 Introduction Concrete filled steel tubular (CFST) columns are widely used in construction infrastructures such as building and bridge structures, especially for high-rise buildings in high seismic zones. It is due to the high performance of these composite members coming from an effective combination of constituent materials, namely steel and concrete. To have a good understanding of the seismic design of structures, many past studies were conducted both experimentally and analytically. In which, conducting experimental studies has brought significant benefits with a valuable test database on the mechanical behavior and load-bearing capacity (strength and ductility) of various structural members and systems. However, there were disadvantages of experimental studies including the relevant requirements of testing facilities, fabricated time, and cost of specimens, especially for testing the full-scale specimens [1]. Therefore, analytical studies have become an indispensable part to overcome the above limitations of the aforementioned method. This results in not only reducing research duration and cost but also increasing the range of comprehensive parametric studies to get a bigger database for further design applications in civil and construction engineering. For analytical studies, finite element analysis (FEA) modeling was widely used in recent decades with the support of some strong software packages such as ANSYS, ABAQUS, ATENA, LS-DYNA, etc. The mechanical behavior of CFST columns subjected to axial compression or combined axial and flexural loading was analytically investigated by many researchers. With CFST columns under axial compression, previous studies revealed that the cross-section shape and the width-(diameter)-to-thickness ratio (B/t or D/t) of steel tube significantly affected concrete confinement and then the column’s loading capacity [2–5]. In which, columns with circular sections demonstrated a better confinement effect than those with non-circular sections. Moreover, the concrete confinement effect has dramatically increased when applying axial compression only on the concrete core compared to other loading types [6–9]. To model the behavior of steel tube and infilled concrete, there were different models used in previous studies [10, 11]. With CFST columns under combined axial and flexural loading, the numerical analysis process was more complicated, especially in the case of cyclic loading. The mechanical behavior of CFST columns under combined axial and monotonic lateral loading was analytically studied by some researchers [12–15]. The numerical analysis conducted by Han et al. (2008) [12] showed that the axial compressive level affects the shear loading capacity of the steel tube. Namely, with the axial load ratio less than or equal to 0.2 the shear loading capacity of steel tube tends to increase when increasing the lateral displacement. Meanwhile, with the axial load ratio greater than or equal to 0.4 the shear loading capacity of steel tube reduces after reaching the peak load due to softening behavior. A nonlinear fiber element analysis method was developed by Liang (2008) [13] for predicting the P-M interaction diagrams of CFST columns subjected to
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axial load and biaxial bending. Moreover, an analytical investigation on circular CFST columns under combined axial and flexural loading, which was conducted by Moon et al. (2013) [15], indicated that the predictions of CFST column strength using current design codes are good just in the case of axial or bending demand. However, current provisions are conservative in predicting the CFST column strength in the case of combined loading. The seismic behavior of CFST columns under combined axial and cyclic lateral loading was also analytically studied in recent decades [16–21]. In which, accurate modeling the behavior of steel tube and infilled concrete components was a big challenge so far. Especially, the concrete stress-strain curve for modeling infilled concrete in CFST columns is significantly changed depending on some factors such as crosssection shape, B/t (D/t) value, materials’ strength, and loading condition. Research by Hajjar and Gourley (1996) [16] revealed that the uniaxial stress-strain curve for infilled concrete in rectangular or square CFST columns subjected to combined loading has a different shape in compressive behavior. It was shown that the lower the B/t value, the better the stress-strain curve with longer strain in the peak stress and higher stress in the post-peak portion. Zubydan and ElSabbagh (2011) [19] analytically investigated the local buckling effect of steel tube in rectangular CFST columns under combined loading. Their study results showed that neglecting the local buckling effect in modeling leads to overestimating the maximum and/or the post-peak capacity of the composite column according to the tube geometry and the constituent material properties. Another research conducted by Patel et al. (2014) [20] has developed cyclic stress-strain curves for steel tube and infilled concrete in rectangular CFST columns. Besides, Skalomenos et al. (2014) [21] numerically investigated the nonlinear behavior of square CFST columns subjected to constantly axial load and cyclically varying flexural loading. In which, some important factors influencing the behavior of CFST members, such as cyclic local buckling of steel tube, nonlinear behavior of confined concrete into tension and compression, cyclic softening and the interface action between steel tube and infilled concrete needed to be considered in modeling. Their model was also used for parameter identification of three previous well-known hysteresis models including the Bouc-Wen model, the Ramberg-Osgood model, and the Al-Bermani model. From past studies, some valuable models for both steel and concrete materials can be used to model the behavior of CFST columns under combined axial and flexural loading. However, to have an accurate confined concrete model used for infilled concrete is extremely difficult, especially, in the case of high and varying axial compression. Moreover, due to the lack of an experimental database, this research would conduct numerical investigations on the seismic behavior of full-scale square CFST columns with a high range of axial compression.
2 FEA Modeling Program 2.1 Modeling Procedure To construct a full CFST column model, four main steps including steel tube modeling, infilled concrete modeling, steel-concrete interaction modeling, boundary conditions, and loading application need to be conducted carefully to ensure that all the model parts are properly connected. In which, ABAQUS (2016) [22] would be used to model
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the column members by using eight-node reduced integration brick elements, C3D8R, for modeling the infilled concrete component, and four-node reduced integration shell elements, S4R, for modeling the steel tube component. The element types and meshing of steel tube and infilled concrete were chosen as shown in Fig. 1. In which, the mesh convergence study was conducted for selecting reasonable meshes. For meshing the steel tube, the finer mesh was done in the areas, which have a length equal to the column’s width, B, near the top- and bottom footings of the specimen. This leads to higher accuracy in the mechanical behavior of these large deformation areas that occurred due to local buckling of the steel tube.
Fig. 1. Steel tube and concrete core meshing
To define material properties of steel tube in ABAQUS, two models, namely (a) elastic and perfectly plastic model and (b) elastic and linear hardening plastic model as shown in Fig. 2, are reasonable to use. In the elastic stage, linear stress-strain behavior was defined based on the steel yield strength (f y ) and modulus of elasticity (E s ). In which, the value of f y was obtained from the tensile tests of standard specimens and then the value of E s was calculated from the test results, if not it was taken as 200 GPa with Poisson’s ratio (ν s ) of 0.3 in the modeling.
Fig. 2. Stress-strain models used for steel tubes
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To have a rational and accurate concrete model for infilled concrete in the square CFST columns, especially with full-scale ones, is a big challenge for the simulation process. Hence, the Concrete Damaged Plasticity (CDP) material model available in ABAQUS needs to be developed with consideration to the confinement effect for modeling in this study. Due to the previously confined concrete models [11, 20] that could not accurately simulate the seismic behavior of full-scale square CFST columns [1], new confined concrete models have been developed by the first author. The compressive behavior of infilled concrete depends on the axial load ratio, n = P/P0 , in which increasing n value leads to the reduction of both the confined strength of infilled concrete (fcc ) and the later confined strain at the peak stress (εcc2 ). The proposed confined concrete models according to different n values for the case of fc = 36.35 MPa are presented in Fig. 3. The relationship between the confined coefficient (k) and the axial load ratio n was constructed and expressed as in Eq. (1) as below. k = 1.84 − 0.5n, → 0.15 ≤ n ≤ 0.55
(1)
Fig. 3. Proposed confined concrete models
The *Contact pair option with surface-to-surface contact type was used to model steel-concrete interaction between the steel tube and infilled concrete portions. A pair of surfaces in this contact including master and slave surfaces requires to be defined. To reduce numerical errors, the slave surface should belong to a softer portion and has a finer mesh than the master one [22]. Therefore, the outer surfaces of the infilled concrete were set as master surfaces, whilst the inner surfaces of the steel tube were set as slave surfaces. The contact property between the master and slave surfaces was defined by normal behavior and tangential behavior. In which, the normal behavior is simulated by the “Hard” contact which allows for the separation of the two surfaces after contact.
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The tangential behavior between the two surfaces is simulated by the Coulomb friction model with a friction coefficient taken as 0.25 [23]. The full-scale square CFST column specimens tested in Phan and Lin (2020) [1] were fully fixed at the Top end and partially fixed at the Bottom end with releases of one degree of freedom (DOF) with moving along the axial axis (longitudinal direction) of the column and another DOF in the direction for applying cyclic lateral displacement load. In the experimental program, the fully-fixed column end was achieved using the column end connected to the top footing and then completely fixed to the cross beam of the MultiAxial Testing System (MATS). Meanwhile, the partially-fixed column end was achieved through the column end connected to the bottom footing and then completed fixed to the platen (shaking table) of the MATS, which can be applied by the axial compression and cyclic lateral displacement loading. Therefore, in modeling, the boundary conditions were applied in the two ends of the column specimen with six and four DOFs fixed at the Top end and Bottom end, respectively. 2.2 Numerical Validation The FEA model developed using new confined concrete models proposed in this study would be validated by the experimental data collected in [1]. The validation would be carried out in the case of full-scale square CFST columns with B/t of 41.7 having an actual yield strength of the steel tube, f ya = 351 MPa, and actual compressive strength of the infilled concrete, fca = 36.35 MPa. Some characteristics that need to be fixed in the validation process include the initial lateral stiffness of the composite columns, the value of column shear strength, shear strength degradation, and the local buckling of steel tube. The numerical validation indicates a good agreement between the analytical and experimental behavior of three specimens with various axial load levels, as shown in Fig. 4.
(a) CFST42-15C (n = 0.15)
(b) CFST42-35C (n = 0.35)
(c) CFST42-55C (n = 0.55)
Fig. 4. Comparisons of numerical results to experimental results by Phan and Lin (2020) [1]
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2.3 Parametric Studies To investigate the seismic behavior of square CFST columns under high axial compression, three axial load levels were chosen in the range from 0.35 to 0.55. Two other parameters B/t and fc would be chosen at two levels for each parameter. For all specimens, the column length was 3 m, in which the outer width and yield strength of steel tubes were 500 mm and 345 MPa, respectively. The details of all parameters investigated in this study are shown in Table 1. Herein, the symbol CFST42–345/35-45C illustrates a CFST column specimen with B/t = 41.7 (42), f y = 345 MPa and fc = 35 MPa subjected to a constant axial compressive load, P = 0.45P0 (P0 is the nominal compression strength of the CFST column). The validated FEA model was extended to conduct parametric studies for investigating the effects of the parameters mentioned above on the strength and deformation capacity of CFST columns. The numerical results and discussions would be presented in detail in the next section. Table 1. Matrix of CFST column specimens Specimen
B (mm)
t (mm)
B/t
f y (MPa)
fc (MPa)
n = P/P0
CFST42-345/35-35C
500
12
41.7
345
35
0.35
CFST42-345/35-45C
500
12
41.7
345
35
0.45
CFST42-345/35-55C
500
12
41.7
345
35
0.55
CFST42-345/70-35C
500
12
41.7
345
70
0.35
CFST42-345/70-45C
500
12
41.7
345
70
0.45
CFST42-345/70-55C
500
12
41.7
345
70
0.55
CFST21-345/35-35C
500
24
20.8
345
35
0.35
CFST21-345/35-45C
500
24
20.8
345
35
0.45
CFST21-345/35-55C
500
24
20.8
345
35
0.55
CFST21-345/70-35C
500
24
20.8
345
70
0.35
CFST21-345/70-45C
500
24
20.8
345
70
0.45
CFST21-345/70-55C
500
24
20.8
345
70
0.55
3 Numerical Results and Discussions 3.1 Numerical Results The numerical results including shear force vs. lateral displacement hysteresis loops, shear strength values, and ultimate interstory drift ratio (IDRu ) values of all specimens are presented below. In which, the hysteresis loops for shear force vs. lateral displacement were drawn for all twelve specimens, as shown in Figs. 5 and 6. The maximum and minimum values of shear strength, and the values of IDRu in positive and negative directions of all specimens were also calculated and presented as in Table 2. Note that
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IDRu was calculated at the post-peak point corresponding with a shear force equal to 80% of maximum or minimum shear force (V max or V min ) in the positive or negative direction of lateral displacement, respectively.
(a)
= 35 MPa, n = 0.35
(b)
= 35 MPa, n = 0.45
(c)
= 35 MPa, n = 0.55
(d)
= 70 MPa, n = 0.35
(e)
= 70 MPa, n = 0.45
(f)
= 70 MPa, n = 0.55
Fig. 5. Hysteresis loops of specimens with B/t = 41.7
(a)
(d)
= 35 MPa, n = 0.35
(b)
= 70 MPa, n = 0.35
(e)
= 35 MPa, n = 0.45
= 70 MPa, n = 0.45
(c)
(f)
= 35 MPa, n = 0.55
= 70 MPa, n = 0.55
Fig. 6. Hysteresis loops of specimens with B/t = 20.8
3.2 Effect of Axial Load Level The numerical results demonstrate that the axial load level has a significant effect on the seismic behavior of these composite columns such as shear strength and deformation capacity.
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Table 2. Numerical simulation results Specimen
V max,min (kN)
IDRu (%)
max
min
average
max
min
average
CFST42-345/35-35C
1221
−1229
1225
3.00+
3.00
3.00+
CFST42-345/35-45C
1095
−1094
1095
2.30
2.67
2.48
CFST42-345/35-55C
1011
−1020
1016
1.50
1.39
1.45
CFST42-345/70-35C
1551
−1548
1550
4.00+
4.00+
4.00+
CFST42-345/70-45C
1486
−1472
1479
2.68
1.97
2.33
CFST42-345/70-55C
1458
−1451
1455
2.00
1.92
1.96
CFST21-345/35-35C
1894
−1924
1909
4.00+
4.00+
4.00+
CFST21-345/35-45C
1869
−1884
1877
4.00+
4.00+
4.00+
CFST21-345/35-55C
1843
−1850
1847
3.00−
3.00+
3.00
CFST21-345/70-35C
2222
−2236
2229
5.00+
4.00+
4.50+
CFST21-345/70-45C
2220
−2229
2225
2.85
2.87
2.86
CFST21-345/70-55C
2210
−2209
2210
2.38
2.27
2.32
For shear strength evaluation, as can be seen in Fig. 7, with the same B/t, f y , and fc , the higher the axial compression, the smaller the column shear strength obtained. When increasing the axial compressive ratio from 0.35 to 0.55, the shear strength decreases more significantly in the case of higher B/t. Namely, with B/t of 41.7 and 20.8, the shear strength drops 17.1 and 3.2%, respectively, in the case of fc = 35 MPa. However, the shear strength drops just 6.1 and 0.9%, respectively, in the case of fc = 70 MPa. It reveals that using high strength concrete filled in the steel tube results in increasing the shear strength and keeping it more stable under higher axial compression in the case of lower B/t.
(a) Specimens with
= 35 MPa
(b) Specimens with
= 70 MPa
Fig. 7. Shear strength comparisons for the effect of axial load level
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For deformation capacity evaluation, the axial load level has also dramatically affected as analyzed in detail below. In general, the higher the axial compression, the smaller the deformation capacity was (see Fig. 8). Firstly, at the lowest axial load level, n = 0.35, the ultimate interstory drift ratio (IDRu ) of all specimens was more than 3% radian, especially, it was more than 4% radian in the cases of thicker steel tubes (B/t = 20.8) and higher strength concrete used (fc = 70 MPa). Secondly, at the middle axial load level, n = 0.45, the deformation capacity has a different changed trend compared with specimens in the case of lower axial compression. In which, the thicker steel tube still supports to increase IDRu of the column, however, using higher concrete strength leads to a reduction of column deformation capacity (only Specimen CFST21-345/3545C has IDRu larger than 3% radian). Thirdly, at the highest axial load level, n = 0.55, the changing trend of deformation capacity is pretty similar to the case of n = 0.45 (just Specimen CFST21-345/35-55C has IDRu of 3% radian). However, for specimens with thinner steel tubes (B/t = 41.7), using higher strength concrete results in an increase of IDRu value from 1.45% to 1.96% radian. In summary, the analysis reveals that only specimens with thicker steel tubes (B/t = 20.8) and using lower strength concrete (fc = 35 MPa) satisfies the deformation demand for CFST columns under high axial compression in the building located in the high seismic zones (IDRu larger than or equal to 3% radian).
(a) Specimens with
= 35 MPa
(b) Specimens with
= 70 MPa
Fig. 8. Deformation comparisons for the effect of axial load level
3.3 Effect of B/t Value The width-to-thickness ratio of the steel tube, B/t, is an important parameter that significantly affects the seismic performance of square CFST columns. It affects not only the column’s strength but also the deformation capacity. A general trend for the B/t effect shows that the lower the B/t of steel tube was, the higher the strength and deformation capacity of the CFST column obtained. As shown in Fig. 9, reducing B/t leads to an increase of shear strength for all cases of concrete strength and axial compressive level. When fc = 35 MPa (Fig. 9a), reducing
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B/t from 41.7 to 20.8 increases shear strength about 55.8, 71.4, and 81.8% for the axial compressive ratio of 0.35, 0.45, and 0.55, respectively. In the case of fc = 70 MPa (Fig. 9b), the corresponding shear strength increase is 43.8, 50.4, and 51.9% at the three axial load levels (n = 0.35, 0.45, and 0.55), respectively. It means that the higher axial compression has significantly affected to reduce the shear strength of square CFST columns when using lower strength concrete (35 MPa) filled in the thinner steel tube (B/t of 41.7). Moreover, the thicker steel tube (B/t of 20.8) keeps the shear strength more stable when increasing the axial load level from 0.35 to 0.55, especially, in the case of using higher strength concrete (70 MPa).
(a) Specimens with
= 35 MPa
(b) Specimens with
= 70 MPa
Fig. 9. Shear strength comparisons for the effect of B/t value
The deformation comparisons in Fig. 10 were conducted between specimens in two levels of B/t, 41.7 and 20.8, using the steel tubes with f y of 345 MPa and filled with normal and high strength concrete, fc of 35 and 70 MPa. The comparison results show that there are some differences in the deformation responses when using normal and high strength concrete. In the case of fc = 35 MPa, reducing B/t value from 41.7 down to 20.8, the IDRu has a significant increase by 1.33, 1.63, and 2.07 times for the axial compressive ratio of 0.35, 0.45, and 0.55, respectively (Fig. 10a). It reveals that the higher the axial compression was, the bigger the increase of deformation capacity was when reducing B/t value in this case. Meanwhile, in the case of fc = 70 MPa, the IDRu has a light increase when reducing the B/t value from 41.7 down to 20.8. The IDRu increase is 1.09, 1.23, and 1.18 times for the axial compressive ratio of 0.35, 0.45, and 0.55, respectively (Fig. 10b).
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(a) Specimens with
= 35 MPa
(b) Specimens with
= 70 MPa
Fig. 10. Deformation comparisons for the effect of B/t value
3.4 Effect of Concrete Strength The compressive strength of the infilled concrete significantly contributed to developing the seismic performance of the square CFST columns in this study. The effect of concrete strength on the shear strength and deformation capacity of these composite columns is analyzed in detail in this subsection. Figure 11 presents shear strength comparisons of all specimens for the effect of concrete strength. The comparison results show that increasing the concrete strength from 35 to 70 MPa leads to an enhancement of column shear strength. In which, the larger increase happens in the case of CFST columns using thinner steel tubes (B/t = 41.7) and subjected to higher axial compression. Namely, the shear strength increase is 26.5, 35.1, and 43.2% for the case of the axial compressive ratio of 0.35, 0.45, and 0.55, respectively (see Fig. 11a). Meanwhile, the smaller increase happens in the case of CFST columns using thicker steel tubes (B/t = 20.8) with 16.7, 18.5, and 19.7% for the case of n = 0.35, 0.45, and 0.55, respectively (see Fig. 11b).
(a) Specimens with B/t = 41.7
(b) Specimens with B/t = 20.8
Fig. 11. Shear strength comparisons for the effect of concrete strength
Figure 12 shows deformation comparisons of all specimens for the effect of concrete strength. The comparison results show that increasing the concrete strength from 35 to
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70 MPa leads to both an increase and a decrease of the column’s deformation capacity depending on axial load level and B/t value. The deformation increase happens when B/t = 41.7, n = 0.35 and 0.55 and B/t = 20.8, n = 0.35, in which, the larger increase happens in the case of thinner steel tube used (Fig. 12a). Meanwhile, the deformation decrease happens in the cases of B/t = 41.7, n = 0.45 and B/t = 20.8, n = 0.45 and 0.55, in which, the larger decrease happens in the case of thicker steel tube used (Fig. 12b). Based on the comparisons, it also reveals that under higher axial compression (n = 0.45 and 0.55) the increase of concrete strength could not significantly enhance the deformation capacity of the column, especially, for the case of B/t = 20.8. As shown in Fig. 12, just specimens with thicker steel tubes and lower concrete strength have a good deformation capacity (with IDRu higher or equal to 3% radian) in different axial load levels.
(a) Specimens with B/t = 41.7
(b) Specimens with B/t = 20.8
Fig. 12. Deformation comparisons for the effect of concrete strength
4 Conclusions Based on the numerical analyses of seismic behavior of full-scale square CFST columns in this research, the following conclusions are drawn: • An FEA model was successfully developed in ABAQUS software for simulating CFST columns under seismic loading. In which, two stress-strain models for steel tubes, including the elastic and perfectly plastic model and elastic and linear hardening plastic model, were selected depending on B/t value. Especially, new confined concrete models were proposed for infilled concrete behavior, whose mechanical properties are significantly influenced by the axial load levels. • The FEA model was validated with experimental results obtained from recent companion research and further extended to conduct parametric studies for square CFST columns with some parameters such as axial load level, the width-to-thickness ratio of steel tube, and concrete strength. The numerical validation analysis shows a good agreement between numerical predictions and experimental results. Moreover, the
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numerical simulation demonstrates its effectiveness in time and cost expenditure compared to the experimental program. • The axial load level has a significant effect on the strength and deformation capacity of these CFST columns under seismic loading. The higher the axial compression applied, the lower the shear strength and IDRu obtained. In particular, there was a dramatic shear strength reduction in the case of using thinner steel tube and lower strength concrete. Moreover, a significant drop in IDRu also was found when increasing axial load up to higher levels, n of 0.45 and 0.55, for most cases of specimens. • The B/t value significantly affects the strength and deformation capacity of the CFST columns in this study. It reveals that the lower the B/t value was, the higher the shear strength and larger IDRu obtained. The highest increase in shear strength happened when using lower strength concrete and the highest axial load level. Meanwhile, the largest increase in IDRu happened when using lower strength concrete and higher axial load levels, n of 0.45 and 0.55. Additionally, the thicker steel tube (B/t = 20.8) helped to keep the shear strength more stable when increasing the axial load level. • Increasing concrete strength leads to the enhancement in shear strength of the CFST columns in all cases. However, it results in a decrease in IDRu for the cases of columns using thicker steel tubes under higher axial levels. The study results also reveal that only the composite columns with thicker steel tube (B/t = 20.8) and using lower strength concrete (fc = 35 MPa) have a good deformation capacity for all three levels of axial compression with their IDRu larger than or equal to 3% radian. Therefore, these columns satisfy the seismic performance demand of the building columns under high axial compression in high seismic zones.
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10. Tao, Z., Wang, Z.-B., Yu, Q.: Finite element modelling of concrete-filled steel stub columns under axial compression. J. Constr. Steel Res. 89, 121–131 (2013) 11. Thai, H.-T., et al.: Numerical modelling of concrete-filled steel box columns incorporating high strength materials. J. Constr. Steel Res. 102, 256–265 (2014) 12. Han, L.-H., Tao, Z., Yao, G.-H.: Behaviour of concrete-filled steel tubular members subjected to shear and constant axial compression. Thin-Walled Struct. 46(7–9), 765–780 (2008) 13. Liang, Q.Q.: Nonlinear analysis of short concrete-filled steel tubular beam–columns under axial load and biaxial bending. J. Constr. Steel Res. 64(3), 295–304 (2008) 14. Abdullah, J.A., Sumei, Z., Jiepeng, L.: Shear strength and behavior of tubed reinforced and steel reinforced concrete (TRC and TSRC) short columns. Thin-Walled Struct. 48(3), 191–199 (2010) 15. Moon, J., et al.: Strength of circular concrete-filled tubes with and without internal reinforcement under combined loading. J. Struct. Eng. 139(12), 04013012 (2013) 16. Hajjar, J.F., Gourley, B.C.: Representation of concrete-filled steel tube cross-section strength. J. Struct. Eng. 122(11), 1327–1336 (1996) 17. Inai, E., et al.: Behavior of concrete-filled steel tube beam columns. J. Struct. Eng. 130(2), 189–202 (2004) 18. Varma, A.H., et al.: Development and validation of fiber model for high-strength square concrete-filled steel tube beam-columns. ACI Struct. J. Am. Concr. Inst. 102(1), 73–84 (2005) 19. Zubydan, A.H., ElSabbagh, A.I.: Monotonic and cyclic behavior of concrete-filled steel-tube beam-columns considering local buckling effect. Thin-Walled Struct. 49(4), 465–481 (2011) 20. Patel, V.I., Liang, Q.Q., Hadi, M.N.: Numerical analysis of high-strength concrete-filled steel tubular slender beam-columns under cyclic loading. J. Constr. Steel Res. 92, 183–194 (2014) 21. Skalomenos, K.A., Hatzigeorgiou, G.D., Beskos, D.E.: Parameter identification of three hysteretic models for the simulation of the response of CFT columns to cyclic loading. Eng. Struct. 61, 44–60 (2014) 22. SIMULIA: Abaqus Analysis User’s and Abaqus/CAE User’s Guides (2016) 23. Ellobody, E.: Numerical modelling of fibre reinforced concrete-filled stainless steel tubular columns. Thin-Walled Struct. 63, 1–12 (2013)
Numerical Modeling of Shear Behavior of Reinforced Concrete Beams with Stirrups Corrosion: Finite Element Validation and Parametric Study Tan N. Nguyen(B) and Kien T. Nguyen Faculty of Building and Industrial Construction, National University of Civil Engineering, Hanoi, Vietnam [email protected]
Abstract. Stirrups in reinforced concrete (RC) beams are more vulnerable to corrosion than longitudinal steel rebars due to the thinner concrete cover. As a result, the stirrups corrosion can lead to a possibility of shear failure instead of bending failure for RC beams. While the shear behavior of RC beams with stirrup corrosion has been mainly studied using experimental investigations, the numerical methods on this subject have not been efficiently documented. In this study, finite element (FE) model is used to simulate the structural behavior of five RC beams with the dimensions of 130 × 260 × 2000 mm having different corrosion degrees of stirrups ranging from 0% to 30% calculated by the mass loss. To model the shear behavior of the tested beams, the existing constitutive models of the materials and the steel-concrete bond are used. The FE model validation is calibrated based on the experimental results (e.g. load-deflection relationship, crack pattern). A parametric study was then implemented to assess the effect of influencing parameters on the shear strength of corroded RC beams, such as concrete compressive strength, stirrup ratio, and steel-concrete bond strength. For the target corrosion degree, the results obtained in this study show that the decrease in concrete compressive strength causes the greatest reduction in ultimate load and deflection among the parameters analyzed. Moreover, as concrete compressive strength decreased or the stirrup ratio increased, the contribution of corroded stirrups to the shear strength of the beam increased followed by the rapid deterioration of shear strength and ductility in corroded RC beams. Keywords: Reinforced concrete · Corroded beams · Stirrups corrosion · Shear behavior · Nonlinear finite element
1 Introduction The corrosion of reinforcement is one of the most dominant deterioration mechanisms of reinforced concrete (RC) structures. It leads to complex distributions of strains and stresses, highly nonlinear, path-dependent behavior of corroded elements. The corrosion inflicts damages which induce a decrease in the performance as well as safety of RC © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 890–904, 2022. https://doi.org/10.1007/978-981-16-3239-6_69
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structures [1]. Most research works had investigated mainly the flexural behavior of corroded RC structures [2–4], but the effect of reinforcement corrosion on shear performance has not been given much attention. When corrosion degree is not high enough to cause cracking of concrete cover, stirrup corrosion has little effect on shear behavior. With the further increase in corrosion degree, the concrete cover begins to crack and even spall, and shear behavior will degrade significantly like flexural behavior. The degradation of shear behavior is directly related to the loss of cross-section area and deterioration of mechanical properties of corroded stirrups, damage of concrete crosssection caused by cover cracking and spalling, deterioration of bond strength between steel and concrete. Generally, steel stirrups have a small diameter and mechanical properties, which are much more sensitive to corrosion [5]. In fact, stirrups begin to be corroded earlier due to the thinner concrete cover and then have more severe corrosion than longitudinal steel rebars [6]. The cumulative probability of shear failure raises for corroded RC beams with increasing the corrosion degree of stirrups [7]. An experimental study performed by Rodriguez et al. [8] on corroded RC beams has shown that the corrosion of reinforcement had altered the failure mode from bending for non-corroded beams to shear for corroded beams. Therefore, it is necessary to design RC beams with a higher shear capacity to avoid such sudden failure. On the one hand, research works for assessing the effect of stirrups corrosion on the shear capacity of RC beams have been mainly performed using the experimental approaches that are based on both types, such as accelerated corrosion and natural corrosion. Rodriguez et al. [8] concluded that the pitting corrosion of the stirrups using accelerated corrosion test was the predominant factor in the reduction of the load-carrying capacity of corroded beams. Xia et al. [9] also studied the shear performance of reinforced concrete beams having corroded stirrups with different corrosion degrees and shown that the average width of concrete cracks due to corrosion has been identified to have the best correlation with the reduction of the shear capacity. Jeppsson et al. [10] investigated the effect of the loss of bond between longitudinal reinforcements and concrete on the shear capacity of RC beams. The obtained results of this study show that the shear capacity is moderately reduced with approximately 80% loss of bond. Regan et al. [11] identified also a similar result, in which the reduction in shear strength was determined as 14–33% for a 65–75% loss of the stirrup end anchorage. Meanwhile, Khan et al. [12] conducted an experimental study on the naturally corroded beams after 26 years of exposure and concluded that the stirrup corrosion (i.e. 30%–60% maximum loss of diameter) did not reduce the anchorage capacity despite the straight end anchorage of rebars and the appearance of large cracks due to corrosion. The stirrup corrosion caused a decrease in the ultimate deflection of deep beams. On the other hand, the finite element (FE) method has been widely used as a powerful tool for the analysis of corroded RC structures. Coronelli and Gambarova [13] studied the modeling of corroded RC beams. It stated that a critical aspect is an assessment of pitting corrosion in the FE model, which may induce brittle behavior in the steel rebars. Therefore, corrosion affects both the strength and the ductility of a structure. For assessing the serviceability of a corroded RC component, not only should concrete cover, steel rebar cross-section reduction be considered, but also the reduction of the
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concrete cross-section. A two-dimensional nonlinear FE model is developed in the study of Kallias et al. [14] to assess the structural performance of a series of RC beams damaged by varying corrosion degrees at different locations. This study shows that the loss of steel area and associated concrete damage/section loss (due to the accumulation of expansive corrosion products) are found to be the main causes of loss of strength and bending stiffness. The bond deterioration is responsible for changes in cracking patterns and widths. Consequently, modeling bond deterioration is highly significant for performance assessment at the serviceability limit state. The study of Saether et al. [15] had been conducted on how to use FE analysis to simulate the mechanical response of RC structures with corroding steel reinforcement. In this study, five tested beams with the dimensions of 130x260x2000 mm having different corrosion degrees of stirrups ranging from 0% to 30% were simulated using DIANA FEA software. It is possible to use the existing constitutive laws of the materials and the steel-concrete bond for modeling the structural behavior of the tested beams. The FE models were validated by calibrating on the experimental results, such as the relationship between load and mid-span deflection as well as the crack pattern (failure mode). Then, a parametric study has been realized to assess the effect of concrete compressive strength, stirrup ratio, and steel-concrete bond strength on the deterioration of shear capacity of RC beams.
2 Materials Law for Modeling Corroded RC Beams 2.1 Concrete Material Law The expansion of corrosion products induces the crack and spalling of concrete. Consequently, the concrete area that is degraded by corrosion damage-induced reduced strength compared to that of the undamaged concrete areas. The corrosion damage on the concrete cover is considered in the FE model by modifying the stress-strain relationship of the concrete, as suggested by Coronelli and Gambarova [13] as illustrated in Fig. 1. Where, GC and GF are compression and tension fracture energy of concrete, respectively. The deterioration of the concrete compressive strength can be described by Eq. (1) with fc,d being the compressive strength of
the corroded concrete, fc being the compressive strength of the non-corroded concrete, k’ being the coefficient related to rebar roughness and diameter, for the case of mediumdiameter ribbed rebars a value k’ = 0.1 has been proposed by Cape [16], ε0 being the strain at the compressive strength fc , and ε1 being the average smeared tensile strain in the transverse direction. Fig. 1. Constitutive law of concrete in fc,d = fc / 1 + k ε1 ε0 (1) compression and tension [1]
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The strain ε1 can be estimated by Eq. (2) with b0 being the section width in the state without corrosion crack, bf being the beam width expanded by corrosion cracking. ε1 = bf − b0 /b0 (2) bf − b0 = nbars wcr
(3)
Where, nbars is the number of rebars; and wcr is the total crack width at a given corrosion degree. The total crack width wcr can be determined as Eq. (4) proposed by Molina et al. [17]. wcr = 2(vrs − 1)Xd
(4)
Where, vrs is the ratio between the specific volumes of rust and steel that can be assumed to be 2. X d is the depth of the penetration attack that is determined by Eq. (5) proposed in the study of Val [5], with icorr (μA/cm2 ) being the corrosion current density in the steel rebar and t (years) being the duration of corrosion. Xd = 0.0116icorr t
(5)
2.2 Steel Reinforcement Law Previous studies reported that both strength and ductility of corroded reinforcement are affected mainly due to variability in steel cross-section loss over their lengths [2]. Because of the difficulty in implementing the actual variability of steel corrosion in the numerical model, an alternative approach is suggested by modeling the corroded steel rebar over a length based on average cross-section loss together with empirical coeffiFig. 2. Stress-strain relationship of the steel cients. The use of empirical coeffireinforcement [1] cients (whose values are smaller than 1) is to account for the reduction in strength and ductility of corroded rebar attributed to the irregular cross-section loss along the rebar length in addition to the reduction attributed to the average cross-section. Since the corrosion damage on the rebar is considered in the FE model by reducing the steel cross-sectional areas over the rebar length according to steel weight loss, the simplified bilinear constitutive stress-strain relationship of steel as illustrated in Fig. 2 is used without empirical coefficients, where the post-yield modulus is assumed to be 1% of its elastic modulus E s . Where, f y and f su are the yield tensile strength and ultimate tensile strength of steel. εy and εsu are the yield strain and maximum strain of steel, respectively.
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2.3 Constitutive Law of the Steel - Concrete Bond The two significant factors that have huge effects on the bond stress-slip relationship is the amount of steel corrosion and the confinement of the concrete. There is a consensus on its well-defined trend that the bond strength initially increased with the corrosion amount in the pre-cracking stage and then substantially decreased as the longitudinal corrosion cracking developed along with the steel reinforceFig. 3. Constitutive law of the deteriorated bond [1] ment [1]. However, bond failure in corroded rebars is mostly by splitting, for the commonly used concrete covers and stirrup amounts. Consequently, the parameters of the bond-stress relationship must be modified to reproduce such brittle behavior. Therefore, the residual bond stress-slip curve as proposed by Kallias et al. [14] is used herein for the deteriorated bond between steel and concrete, and bond stress-slip relation in CEB-FIP Model Code 1990 [18] is used to model the good bond as indicated in Fig. 3. (6) Umax,D = R[0.55 + 0.24(c/db )] + 0.191 Ast .fyt /Ss .db R = A1 + A2 mL
(7)
Where, Umax,D is the residual bond strength, which can be determined by Eq. (6), with c being the concrete cover, d b being the diameter of the longitudinal rebar, Ast being the cross-section area of the stirrup, f yt being the yield strength of the stirrup, S s being the stirrup spacing. R is the factor accountable for the residual contribution of concrete towards the bond strength as a function (Eq. (7)) of A1 = 0.861 and A2 = 0.014, which is related to the current density used in the accelerated corrosion test, and mL is the amount of steel weight loss in percentage. Equation (6) consists of two different terms: concrete and stirrup contributions to the bond strength are related to the first and second terms, respectively. The effectiveness of this equation is that the level of confinement can be varied with the changes in the stirrup spacing and concrete compressive strength for different specimens.
3 Validation of FE Models for Shear-Critical Corroded RC Beams 3.1 Presentation of the Tested Beam In this section, five beams with the dimensions of 130 × 260 × 2000 mm as illustrated in Fig. 4 from an experimental study conducted by Ye et al. [19] are used for modeling the
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shear behavior of RC beams subjected to stirrups corrosion. The tested beams were made of concrete having the compressive strengths at 28 days ranging from 27.5 to 32.2 MPa (cf. Table 1), with an average value of 30 MPa. The longitudinal reinforcements were constituted by two steel rebars with a nominal diameter of 18 mm in the bottom layer, two steel rebars with a nominal diameter of 12 mm in the top layer. The stirrups were used to be the plain steel rebar with a nominal diameter of 8 mm with a regular spacing of 150 mm. In fact, a non-corroded beam named L0LS was used as the control beam in order to compare with other corroded beams. The corroded beams named L1LS, L2LS, L3LS, and L5LS had the corroded stirrups in the shear span with different corrosion degrees ranging between 5%, 11%, 15%, and 27%, respectively. The corrosion degree is calculated by the mass loss divided to the initial mass of steel reinforcement. Table 1. Synthesis of test results Beam notation
Concrete compressive strength (MPa)
Diameter of tension steel rebars (mm)
Degree of corrosion
L0LS
27.45
18
0
0
91.9
Bending
L1LS
32.19
18
0.05
0
87.5
Bending
L2LS
31.91
18
0.11
0
86.9
Shear
L3LS
27.84
18
0.15
0
80.5
Shear
L5LS
30.85
18
0.27
0
59.2
Shear
nv
Ultimate force Failure P (kN) modes nx
Note: nv is the corrosion degree of stirrups; nx is the corrosion degree of longitudinal steel rebars
Fig. 4. Layout and cross-section of tested beams
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3.2 Modeling of the RC Corroded Beam with Stirrups Corrosion in Shear Span The concrete is modeled with an element mesh of 50 × 50 × 50 mm using a 20-node hexahedron solid element as illustrated in Fig. 5. For undamaged areas of all simulated beams, the compressive strength of concrete is assigned to be 30 MPa corresponding to the average value obtained from the compression test (cf. Table 1). Meanwhile, the residual compressive strength of damaged concrete in the target corrosion areas is calculated by Eq. (1) and reduced from 24 to 12 MPa when increasing the corrosion degree of stirrups from 5% to 27%. Moreover, the tensile strength of damaged concrete has remained the identical value as the undamaged concrete at 2.8 MPa. Lastly, the modulus of elasticity for damaged and undamaged concretes shown in Table 2 are calculated by the compressive strength using Eq. (2.1–15) in CEB-FIP Model Code 1990 [18]. In DIANA FEA software, the steel reinforcement is modeled as bond-slip reinforcements. A line-solid interface element has been used in order to simulate the influence of bond-slip behavior because it connects slip reinforcements to the continuum element in which the line element is located. The yield and ultimate tensile strengths of steel used are also summarized in Table 2 for three types of reinforcement diameter [19]. In the present simulation, the effect of steel corrosion is modeled by reducing the cross-section of the stirrups based on their corrosion degree. For non-corroded steel rebars (e.g. longitudinal reinforcements, stirrups in undamaged areas), the bond strength and slip parameters are selected based on CEB-FIP Model Code 1990 [18], such as the maximum bond strength τ max = 7.0 MPa, the residual bond strength τ f = 1.05 MPa, the slips S 1 = S 2 = 0.6 mm and S 3 = 2.5 mm, and the exponent coefficient α = 0.4. For corroded stirrups, in order to model the splitting failure mode of corroded plain steel, S 1 must be taken close to S 2 and the descending branch must be sharp, reducing the value of S 3 , and considering τ f close to zero. The deteriorated bond curve as illustrated in Fig. 3 is applied, then the residual bond strength is calculated by Eq. (6) in Sect. 2.3, ranging from 6.3 to 0.2 MPa when increasing the corrosion degree of stirrups from 5% to 27% in the beams L1LS to L5LS.
(a) Concrete mesh
(b) Reinforcement mesh
Fig. 5. Three-dimensional FE model of the tested corroded beams
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Table 2. Mechanical properties of materials for modeling tested RC beams Beam notation
L0LS
L1LS
L2LS
L3LS
L5LS
Corrosion degree of stirrup
0
0.05
0.11
0.15
0.27
Residual compressive strength of concrete (MPa)
30
24
21
15
12
Residual tensile strength of concrete (MPa)
2.8
2.8
2.8
2.8
2.8
Young’s modulus of concrete (MPa)
28300 27200 26300 24125 23000 ϕ8 mm
Yield tensile strength of steel (MPa)
Ultimate tensile strength of steel (MPa)
369.6
369.6
369.6
369.6
369.6
ϕ12 mm 443.1
443.1
443.1
443.1
443.1
ϕ18 mm 459.0
459.0
459.0
459.0
459.0
ϕ8 mm
550.3
550.3
550.3
550.3
550.3
ϕ12 mm 584.4
584.4
584.4
584.4
584.4
ϕ18 mm 607.4
607.4
607.4
607.4
607.4
Elastic modulus of steel (GPa)
210
210
210
210
210
Maximum bond strength (MPa)
7.0
6.3
5.6
3.5
0.2
3.3 Validation of FE Model
100
Load (kN)
Figures 6 and 7 illustrate the 80 comparison between the results of tested beams by experimental 60 and numerical approaches. It is L0LS FEM confirmed that the simulation by 40 L0LS EXP modifying the constitutive mate20 rials shown in Sect. 2 is capable of modeling the response of 0 corroded RC beams with reason0 10 20 30 40 able accuracy between the two Deflection at mid span (mm) approaches results for the load– deflection curves of the con- Fig. 6. Load-deflection curves of the control beam using trol beam L0LS and four cor- experiment and FEM roded beams L1LS, L2LS, L3LS, L5LS. FE model can predict the ultimate flexural strength of tested beams L0LS, L1LS with good accuracy. In the results from tested specimens, beams L0LS and L1LS failed in bending. Due to the increasing corrosion degree of stirrups in shear span of five tested beams, the cross-sectional areas of the stirrups decreased and the concrete cover cracked leading to a reduction of the effective cross-section. The shear strength of corroded RC beams decreased considerably with the increasing corrosion degree explain why the failure mode shifted from bending to shear failure in case of the beams L2LS, L3LS and L5LS as shown in Table 1. In comparison with bending failure, a shear failure is sudden and brittle, and the ultimate deformation is small while the maximum capacity decreases
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with the increasing corrosion degree of stirrups. Only 10% corrosion degree in stirrups was enough to transform the failure mode of the tested beams from ductile to extremely brittle. With the increase from 11% to 27% of corrosion degree in stirrups, the maximum capacities of the beams L2LS, L3LS and L5LS decreased from 4.8% to 35.6% of the control beam L0LS (cf. Table 1). The shear strength decreased rapidly after the peak applied load, and a small displacement increment caused a large decrease in shear strength. The beam L1LS with a 5% degree of corrosion in stirrups has the least reduction in load-carrying capacity, 87.5 kN in the test versus 86.4 kN in FEM. The ultimate load of the beam L2LS in the test was 86.9 kN compared with approximately 82.4 kN at the same deflection of FEM results. For the case of the beam L3LS, the reduction in maximum capacity was nearly 12% compared with the control beam. The failure load of the beam L3LS in FEM analysis was 73.6 kN, which is 8.6% smaller than the experimental result. For different degrees of corrosion in stirrups, the least capacity was observed in the beam L5LS with a nearly 35% reduction from the control beam, and there is a close to only 2.3% distinction between FEM and test. Thus, the difference of ultimate load at the failure between experimental and FEM results is less than 10% for all the tested beams. In general, the results of corroded beams in FEM simulation is shown the sensitive responses to the selection of the model of bond strength reduction with the corresponding parameters assigned in DIANA. The lack of a formulation for the bond strength in simulation for corroded bars with varying diameters in stirrups is a barrier against accurate estimation for the ductility of FEM simulated beams and tested beams. It can be seen by the differences in deflection at the failure stage of both methods. There was a large difference in the stiffness of the FEA model comparing with experimental beams, it can be ascribed to initial cracking in concrete before loading due to the corrosion of stirrups. Furthermore, the corrosion damage in accelerated corrosion tests has resulted in the variability of cracks forming around stirrups in corrosion-damaged areas which leads to the distinguished beam initial stiffness before the loading procedure which leads to the significant difference between two approaches in beam stiffness, especially in the elastic stresses stage. The results shown in Fig. 7 indicated that the corrosion of stirrups had little impact on the initial stiffness of all tested beams. Prior to cracking, the stirrup stress is low and corrosion of stirrups has little influence on shear behavior and cracking load. Once a beam cracks, the stirrup stress increases quickly with the increasing loading. With an increase in the corrosion degree, the cross-sectional area of stirrups decreases and the stirrup stress increases more quickly. Because of the decrease in cross-sectional areas, yielding strength and yielding strain of stirrups, the ultimate shear strength and the corresponding shear strain of concrete or ductility undergo an obvious decrease. Figure 8 shows the maps of concrete cracks due to loading for each tested beam, which are identified by the test and the FE model. FEM results can represent the crack pattern of the tested beams. For the control beam L0LS, there are only the flexural cracks that occurred mainly between two loading points and the target corrosion areas (cf. Fig. 4). These flexural cracks propagated gradually and became larger until the failure of concrete in compression area. For tested corroded beams, the flexural cracks preceded in the middle span, then the shear cracks occurred lately in a target corrosion
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80
80
60 L1LS FEM 40
L1LS EXP
Load (kN)
100
Load (kN)
100
20
899
60 L2LS FEM 40
L2LS EXP
20
0
0 0
10 20 30 Deflection at mid span (mm)
40
0
(a) Beam L1LS
10 20 30 Deflection at mid span (mm)
40
(b) Beam L2LS 100 80
60 L3LS FEM 40
L3LS EXP
20
Load (kN)
Load (kN)
80
60 L5LS FEM 40
L5LS EXP
20
0
0 0
10 20 30 Deflection at mid span (mm)
(c) Beam L3LS
40
0
10 20 30 Deflection at mid span (mm)
40
(d) Beam L5LS
Fig. 7. Load - deflection curves of corroded beams using experiment and FEM
area. The corroded beam was fractured at the location of the shear cracks. In addition, the number of flexural cracks reduced when increasing the corrosion degree of stirrups on the corroded beams.
(a)
(b)
(c)
(d)
(e)
Fig. 8. Comparison of the crack pattern on the tested beams using experiment and FEM: (a) L0LS, (b) L1LS, (c) L2LS, (d) L3LS and (e) L5LS
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4 Parametric Study As can be seen in Fig. 7, the experimental and numerical results of the tested beams show that the RC beam with a higher corrosion degree of stirrups leads to the shift of failure mode and the reduction in both stiffness and load-carrying capacity. By choosing the tested beam L2LS, the parametric study is performed to determine the influencing parameters which are dominant to the change of failure mode of degraded beams. In this section, the simulated beams have identical dimensions, material properties, corrosion degree, and its effect with the tested beam L2LS were used in order to evaluate the effects of corrosion-induced parameters on shear behavior and to identify the most decisive factor in degradation of the strength of a RC beam due to stirrups corrosion. Three simulation approaches have been considered, which the first case is the simulation to assess the effect of concrete compressive strength while the second and third ones take into account the impact of stirrup ratio and bond strength between stirrups and concrete, respectively. 4.1 Effect of Concrete Compressive Strength Figure 9 shows the numerical results of the load-deflection curves of the simulated beams having the corrosion degree of stirrups to be 11% and concrete compressive strength ranging from 20 MPa to 50 MPa (from C20 to C50 grade). It can be seen that both cracking load and ultimate shear strength of the beam are reduced with decreasing the concrete compressive strength. The load-carrying capacity of the simulated beam of C20 concrete is decreased by approximately 25% in comparison with that of the simulated beam of C30 concrete (62.8 kN versus 82.4 kN). In the case of small compressive strength, stirrups provide a greater contribution to the overall shear strength of the beam, resulting in a more obvious influence of stirrups corrosion on shear behavior. Once the beam cracks, the stirrup stress raises quickly with an increase in the loading, and the ultimate shear strain of concrete decreases. It is also found that the critical corrosion degree of stirrups, with which the beam begins to fail in shear instead of in bending, increases with an increase in the concrete strength. As an example, the failure of the simulated beams changes from shear mode to bending mode when the concrete compressive strength raises from C30 to C50 grade, with a slight increase in the ultimate load (e.g. 86 kN on the beam of C40 concrete versus 82.4 kN on the beam of C30 concrete). Therefore, it can be observed that the failure mode of the RC beams is not only dependent on the corrosion degree of stirrups but also depends on the compressive strength of concrete. Because of the dramatic change in both the failure mechanism and serviceability of the corroded beam, concrete strength should be received high attention in design RC structures that are in the corrosive environment. 4.2 Effect of Stirrup Ratio Figure 10 shows the numerical results of the load-deflection curves of the simulated beams made of stirrups with the nominal diameter ranging from d6 to d10 mm and the corrosion degree of 10%. When there is a change in the stirrup ratio, it can be seen
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100
Load (kN)
80 C20 FEM C30 FEM C40 FEM C50 FEM
60 40 20 0
0
5 10 15 Deflection at mid span (mm)
20
Fig. 9. Load - deflection curves with consideration the effect of concrete strength
that there is no difference in the load-carrying capacity and failure mechanism of the simulated beams. As illustrated in Fig. 11 for the stirrup stresses – deflection curve, in the failure stage of the analysis, the stress of stirrups in three simulated beams does not exceed the ultimate tensile strength of the steel which leads to the failure from concrete crushing in shear span instead of stirrup failure. The difference between the cases studied is the initial stiffness of the beam because of the difference of yield strength in steel material. Finally, the change of failure mode that is observed in the study of Ye et al. [19] is the result of a decrease in the compressive strength of the concrete in shear spans. Therefore, the increase of stirrup ratio, in this case, does not contribute any considerable effect on both load-carrying capacity and failure mode of the beam. And, in order to improve the shear strength of a beam, a larger contribution of concrete is preferred instead of a contribution of stirrups considering the adverse impact of stirrups corrosion.
Load (kN)
60
d6 d8 d10
40 20
Stirrup cauchy total stresses (N/mm2)
400
80
300
d6 d8
200
d10
100 0
0 0
5
10 15 Deflection at mid span (mm)
Fig. 10. Load-deflection curves with considering the effect of stirrup ratio
20
0
5 10 15 Deflection at mid span (mm)
Fig. 11. Stirrup stresses – deflection curve with considering the effect of stirrup ratio
20
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4.3 Effect of Bond Strength Load-deflection responses of the FE models with the deboned concrete-rebar interface are shown in Fig. 12. As seen in this figure, the ultimate loads of the models with the varying of loss in bond strength are relatively close to each other and ranged between 7% and 10% less than for the undamaged model. Moreover, even though the failure mode of the three cases is identical which is a shear failure, the load-deflection curve in the right side of the case of 90% loss of bond strength fail more steeply than the case of 50% loss of bond strength which indicates for the sudden failure of concrete without any bond with stirrups. 80
Load (kN)
60
Bond strength loss 50% Bond strength loss 90% Good bond strength
40 20 0 0
5 10 15 Deflection at mid-span (mm)
20
Fig. 12. Load - deflection curves with considering the effect of bond strength
5 Conclusions In this paper, the effects of the stirrups corrosion on the structural performance of five accelerated-corrosion RC beams have been simulated and discussed. FE analysis method with DIANA software has been provided to simulate the structural responses of the corroded beams when considering the corrosion of stirrups in the shear span. The validation of the FE model is presented by comparing the numerical and experimental results of the load-deflection curves. The main conclusions can be drawn as follows: • The FE model that was adopted in this paper provides a reasonable prediction of the structural behavior (e.g. load-carrying capacity, load-deflection curve) of the RC beams with stirrups corrosion in shear span by modeling the corrosion damage on the materials and the steel-concrete bond. However, there was a large difference in the stiffness of the FEA model comparing with experimental beams, it can be ascribed to initial cracking in concrete before loading due to the corrosion of stirrups. The data given from the experimental test should be sufficient in order to overcome some of
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the shortcomings of deterministic finite element modeling of corroded beams pointed out in this paper. • With a higher corrosion degree of stirrups in the shear span, the failure mode of a corroded beam can be shifted from bending to shear failure. As a result, with the increasing degree of corrosion in stirrups, the shear capacity of the beam decreases. For a target corrosion degree, the concrete compressive strength has the most significant effect on the shear behavior of the beam (e.g. load-carrying capacity, deflection, failure mode). • It is more rational to increase the concrete compressive strength instead of raising the stirrup ratio in order to improve the shear capacity and decrease the risk of shear failure of RC beams in aggressive environments, but the ratio should remain greater than the minimum requirement to avoid brittle failure of stirrups.
References 1. Lim, S., Akiyama, M., Frangopol, D.M.: Assessment of the structural performance of corrosion-affected RC members based on experimental study and probabilistic modeling. Eng. Struct. 127, 189–205 (2016) 2. Du, Y.G., Clark, L.A., Chan, A.H.C.: Residual capacity of corroded reinforcing bars. Mag. Concr. Res. 57(3), 135–147 (2005) 3. Dong, J., Zhao, Y., Wang, K., Win, J.: Crack propagation and flexural behaviour of RC beams under simultaneous sustained loading and steel corrosion. Constr. Build. Mater. 151, 208–219 (2017) 4. Soltani, M., Safiey, A., Brennan, A.: A state-of-the-art review of bending and shear behaviors of corrosion-damaged reinforced concrete beams. ACI Struct. J. 116(3), 53–64 (2019) 5. Val, D.V.: Deterioration of strength of RC beams due to corrosion and its influence on beam reliability. ASCE J. Struct. Eng. 133(9), 1297–1306 (2007) 6. Ye, Z., Zhang, W., Gu, X., Liu, X.: Experimental investigation on shear fatigue behavior of reinforced concrete beams with corroded stirrups. J. Bridge Eng. 24(2), 04018117 (2019) 7. Zhang, W., Ye, Z., Gu, X.: Effects of stirrup corrosion on shear behaviour of reinforced concrete beams. Struct. Infrastruct. Eng. 13(8), 1081–1092 (2017) 8. Rodriguez, J., Ortega, L.M., Casal, J.: Load carrying capacity of concrete structures with corroded reinforcement. Constr. Build. Mater. 11(4), 239–248 (1997) 9. Xia, J., Jin, W., Li, L.: Shear performance of reinforced concrete beams with corroded stirrups in chloride environment. Corros. Sci. 53, 1794–1805 (2011) 10. Jeppsson, J., Thelandersson, S.: Behavior of reinforced concrete beams with loss of bond at longitudinal reinforcement. J. Struct. Eng. 129(10), 1376–1383 (2003) 11. Regan, P.E., Kennedy Reid, I.L.: Shear strength of RC beams with defective stirrup anchorages. Mag. Concr. Res. 56(3), 159–166 (2004) 12. Khan, I., François, R., Castel, A.: Experimental and analytical study of corroded shear-critical reinforced concrete beams. Mater. Struct. 47(9), 1467–1481 (2013). https://doi.org/10.1617/ s11527-013-0129-y 13. Coronelli, D., Gambarova, P.: Structural assessment of corroded reinforced concrete beams: modeling guidelines. J. Struct. Eng. 130, 1214–1224 (2004) 14. Kallias, A.N., Rafiq, M.I.: Finite element investigation of the structural response of corroded RC beams. Eng. Struct. 32, 2984–2994 (2010)
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15. Sæther, I., Sand, B.: FEM simulation of reinforced concrete beams attacked by corrosion. ACI Struct. J. 39(2), 15–31 (2012) 16. Cape, M.: Residual service-life assessment of existing R/C structures. MS thesis, Chalmers Univ. of Technology (Goteborg, Sweden) and Milan Univ. of Technology (Italy, Erasmus Program) (1999) 17. Molina, F.J., Alonso, C., Andrade, C.: Cover cracking as a function of rebar corrosion: Part 2 - Numerical model. Mater. Struct. 26, 532–548 (1993). https://doi.org/10.1631/jzus.A13 00393 18. CEB-FIP Model Code: design code, p. 1993. Thomas Telford, London (1990) 19. Ye, Z., Zhang, W., Gu, X.: Deterioration of shear behaviour of corroded reinforced concrete beams. Eng. Struct. 168, 708–720 (2018)
Characterization of Stress Relaxation Behavior of Poscable-86 High-Strength Steel Wire Ngoc-Vinh Nguyen1 and Viet-Hung Truong2(B) 1 Department of Civil and Environmental Engineering, Sejong University, 98 Gunja-dong,
Gwangjin-gu, Seoul, Korea [email protected] 2 Faculty of Civil Engineering, Thuyloi University, 175 TaySon, DongDa, Hanoi, Vietnam [email protected]
Abstract. High-strength steels have been widely employed in public projects, especially in bridges and the cables supported structures since this material was designed as a key structural component in the structures to support high-tension loads in the cables. Generally, high-strength cables play an important role in the loading bearing capacity as well as the stiffness of structures and are usually designed to support long-term tension loads during the in-service state. The longterm tension loads can generate the stress relaxation/creep behavior in the cable components, resulting in the degradation of the tension force in the cables. This can cause the degradation of the initial stress as well as the degradation of the mechanical properties, for example, indentation hardness and yield strength, and finally the failure of whole structures. The degradation of mechanical properties is quite important in the case of the cable structures since these cables are the main components in the structures. Thus, this study reported the stress relaxation of Poscable-86 high-strength steel wire. For this purpose, three stress relaxation experiments were performed at three different initial stress relaxation levels, i.e. 0.5f u , 0.6f u , and 0.7f u . The results showed that stress relaxation behavior includes two main stages. The first stage was a transient relaxation occurring at a few hours of stress relaxation, while the second stage was a secondary relaxation exhibiting a stable rate of stress loss. At 0.5f u , the relaxation rate was limited within 1.5% in the period of 10 h, slightly increased from 1.37% to 1.96% at a testing time from 10 h to 500 h, and became stable at 1000 h. At 0.6f u , the relaxation rate quickly increased in the range of 0–1.5% at 10 h, slightly increased from 1.25% to 2.31% at the corresponding testing time from 10 h to 500 h, and finally became stable at 1000 h. The same behavior could be observed in the case of 0.7f u . Furthermore, relaxation rates of all stress levels were limited within 3% at a period of 1000 h, in which high-strength steel wire exhibited a more rapid transition to the secondary stage at a lower initial stress level. Another interesting feature of stress relaxation behavior is that the relaxation rate is strongly sensitive to the initial stress level. The long-term relaxation rate of the tested wire was investigated using a Fib model via the values of relaxation rates at 1000 h. The results indicated the relaxation rates of Poscable-86 wire at 0.7f u at 10 years, 20 years, and 50 years were 4.23%, 4.54%, and 4.98%, respectively. Similarly, at 0.5f u , the relaxation rates of 0.5f u are 3.34%, 3.57%, and 3.89% were calculated at 10 years, 20 years, and 50 years, respectively. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 905–914, 2022. https://doi.org/10.1007/978-981-16-3239-6_70
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N.-V. Nguyen and V.-H. Truong Keywords: Stress relaxation · Relaxation rate · Long term behavior · High-strength steel
1 Introduction High tensile strength cables have been attributed to appropriate material and popularly employed in public projects, especially in bridges and the cables supported structures since this material was designed as a key structural component in the structures to support high-tension loads in the cables [1–8]. Wang et al. have indicated that the tensionholding capability of high-strength steel wires/cables has a strong influence on not only the strength but also the stiffness of the whole structure because these high-strength wires/cables are mainly designed for supporting the high tension loading [9, 10]. Thus, these materials are the favorite option for the public projects generally designed to be in service for a long time, for example, 50 years or 100 years as pointed out by Wang et al. [11]. It is evident that the long-term characteristics of structural components, for example, high-strength cables/wires, strongly affect the performance of these public structures. Although these high-strength components were generally applied the tension force in the elastic state; however, the permanent deformation (or permanent strain) of the cable/wire structure was observed due to the influences of long-term rheologic properties. This permanent deformation may lead to the stress relaxation phenomenon in the cables/wires as well as the degradation of the stiffness structure. Therefore, it is very important to consider these rheologic properties of the cable/wire components in the practical designs of the whole structures [12], in which the determination of timedependent behavior and mechanical properties of high-strength cables/wires is the most important task [5, 13, 14]. Stress relaxation behavior that the tension stress in high-strength cables/wires can be observed to decrease characteristically under constant strain/temperature conditions [2, 15], is generally occurred in the pre-stressed concrete structures and long-span structures. Stress relaxation has been attributed to be strongly sensitive to the levels of applied stress in the cables as well as the environmental temperature conditions [15–18]. A study on the relaxation and creep behaviors of high-strength steel wire at several levels of temperature conditions was conducted by Papsdorf and Schwier [17] by performing the stress relaxation experiments and a collective survey in the literature. The research displayed that the stress relaxation of the tested cable strongly depended on the temperature conditions, in which the relaxation rate at an elevated temperature was higher than those at lower temperatures for both cases of initial stresses (i.e. 50% and 60% tensile strength) in the same testing period of 1000 h. A study of stress relaxation behavior in the in prestressing reinforcement was then conducted by Magura et al. [15], showing that the substantial stress relaxation losses mainly occurred beyond 1000 h, and the relaxation rates for 7-wire strand and single wire specimens were 2.0–6.1% and 4.2–9.0%, respectively. A new galvanized high-strength steel wire was successfully created by Yamaoka et al. [19]. The results of their research exhibited that the increase of Si content of the new steel wire led to the increase of the resistance to softening during the galvanizing process. Zhou and Li [20] and Fan [21] then performed the steady-state experiments on the prestressed steel strand at high temperatures.
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Li et al. [22] considered the deficiency and carried out secondary development for modeling wire strand/rope to obtain the dynamic tension of wire rope, while Wang et al. [23] established an innovative approach to simulate the stress relaxation behavior of the high-strength steel wire/cables using a commercial software by applying a simplified relation between stress relaxation and creep. By using this approach, the values of the relaxation rate of larger strands can be easily investigated and determined from the experimental data of only a single wire. Furthermore, the effects of the friction and contact of the wires in the strand could be also investigated. Zhao et al. [24] were then studied the long-term behaviors, including creep, stress relaxation, and fatigue of FRP tendon-anchor system, while the stress corrosion cracking fracture mechanism of high-carbon cable bolts was proposed by Wu et al. [25] using four-point bending experiments. However, to the best of the author’s knowledge, the stress relaxation behavior of a Poscable-86 high-strength wire under several initial stress levels has not been well established. Thus, the stress relaxation experiments on Poscable-86 high-strength steel wire were performed at a room-temperature and three initial stress levels, i.e. 50%, 60%, and 70% tensile strength to investigate the stress relaxation behavior of the tested high-strength steel.
2 Methodology The stress loss in the cable or relaxation rate (RR) can be determined based on the response of the loading history and the applied initial stress during the stress relaxation experiments as [3, 11, 15] RR =
σ0 − σ (t) , σ0
(1)
where σ (t) and σ0 are the stress at time t and the applied initial stress, respectively. The values of the relaxation rate for the long-term can be extrapolated based on the relaxation rate at periods of 100 h and 1000 h [26]. Therefore, the values of relaxation rates at 10 years, 20 years, and 50 years can be easily predicted by using the Fib model proposed by Switzerland (2013) as. RRLong-term = RR1000 (t/1000)K
(2)
where K is the factor determined as K = log[RR1000 /RR100 ], while RR100 and RR1000 are the experimental data at corresponding 100 h and 1000 h.
3 Experimental Procedure As previously mentioned, three stress relaxation experiments were performed at room temperature for three different initial stress levels to characterize the stress relaxation behavior of Poscable-86 high-strength steel wire. The investigated material has a chemical composition as C 0.91%, S 0.003%, Ni 0.01%, Cr 0.28%, Zn 0.31%, Mn 0.45%, Si 0.07%, and Fe balance. Figure 1 showed the machine and specimen for the stress relaxation experiment. Several standards suggested the upper bound of applied initial stress
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in the cables/wires of 70% tensile strength [27, 28], while the tension force is normally controlled within 50% tensile strength in the cable-supported structures [11]. Furthermore, the real tension state occasionally overcomes the level under additional loads or accidental conditions. Therefore, three different initial stress levels, such as 50% tensile strength, 60% tensile strength, and 70% tensile strength, with the same gradient interval were selected.
Fig. 1. Tension strength instrument 300 kN electromechanical for stress relaxation
As mentioned before, temperature and applied initial stress conditions strongly influence the stress relaxation behavior of the high-strength steel cable/wires. Wang et al. [11] investigated the stress relaxation behavior of several types of high-strength steel strands, indicating that the variation of the stress relaxation rate under the ambient temperature was significant and distinguishable. However, the main purpose of this investigation was to characterize the stress relaxation behavior of Poscable-86 high-strength steel, and thus the surrounding temperature was considered as a minor factor. The ISO standard [27] suggested that a suitable range of temperature during the stress relaxation experiments is from 15 ± 2 ◦ C to 25 ± 2 ◦ C. As a result, a room temperature of 20 ± 0.3 ◦ C was adopted and applied for all stress relaxation experiments in this study. Normally, the stress relaxation experiments were performed in the period of 100 h or 120 h to save the testing cost, and then the values of relaxation rate at 1000 h could be extrapolated via the experimental data. However, this method may not ensure a high accuracy of the prediction of the long-term behavior of the tested material. Thus, the testing time of 1000 h was designed for all stress relaxation experiments in this study to ensure the
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high accuracy in the estimation of the stress relaxation behavior and the determination of long-term relaxation rates.
4 Results and Discussion After performing the stress relaxation on Poscable-86 high-strength steel wire at three different initial stresses over a long duration of 1000 h, the relationship between the relaxation rate and testing time was well constructed and shown in Figs. 2, 3, and 4 for initial stress of 50% tensile strength, 60% tensile strength, 70% tensile strength, respectively. At the initial stress of 50% tensile strength, the results in Fig. 2 exhibited that stress relaxation behavior of tested material at room temperature included two main characteristic stages, i.e. the transient relaxation and secondary relaxation. The first stage normally occurs at the beginning of the stress relaxation testing (i.e. 2~3 h of testing for Poscable-86 high-strength steel wire). The results showed that the relaxation rate quickly increased from 0% to 1.2% during the transient stage with a high increasing speed as seen in Fig. 2. In contrast, the second stage, the secondary relaxation, occurs at the long duration of testing, exhibiting a low increasing speed of relaxation rate during the secondary stage. The same stress relaxation behavior of Poscable-86 high-strength steel at other initial stress levels could be observed in Fig. 3 and Fig. 4.
Fig. 2. Stress relaxation behavior of Poscable-86 high-strength steel at the initial stress of 50% tensile strength
To clearly understand the stress relaxation behavior, the comparison of the relaxation rate values at three testing durations, for example, 100 h, 500 h, and 1000 h, for three initial stress levels as illustrated in Fig. 5. It can be seen in Figs. 2, 3, 4, and 5 that the values of relaxation rates for all initial stress levels were limited to 2.25% at a testing time of 100 h, 2.5% at a testing time of 500 h, and 2.7% at a testing time of 1000 h. Another interesting feature from Fig. 5 is that stress relaxation strongly sensitive to the
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Fig. 3. Stress relaxation behavior of Poscable-86 high-strength steel at the initial stress of 60% tensile strength
Fig. 4. Stress relaxation behavior of Poscable-86 high-strength steel at the initial stress of 70% tensile strength
applied initial stress levels, in which the relaxation rate tends to increase with the further increase of initial stress. For example, the relaxation rate at 100 h seems to linearly increase from 1.7% to 2.1% when the initial stress increases from 50% tensile strength (0.5fu ) to 70% tensile strength (0.7fu ), respectively. At 1000 h, the relaxation rate also increases from 2.2% to 2.63% with increasing initial stress in the range of 0.5fu − 0.7fu . The present behavior of stress relaxation of Poscable-86 high-strength wire is in good
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Fig. 5. Stress relaxation of Poscable-86 high-strength steel at different testing durations
agreement with those reported for several types of high-strength steel cables/wires in the literature [14–16, 19] (Fig. 6).
Fig. 6. Long-term stress relaxation of Poscable-86 high-strength steel at different testing durations
The logarithmic expression was introduced to depict the relationship between relaxation rate and testing time as [27], RRi = Axln(t) + B, where i is initial stress levels, for example, 0.7fu , 0.6fu , and 0.5fu , and A and B are constant parameters estimated from the experiment data. The regression analysis was then conducted to estimate the fitting parameters A and B based on the experimental data in Figs. 2, 3, and 4. The
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results of the regression analysis were illustrated in Fig. 2, Fig. 3, and Fig. 4 for the stress relaxation behavior at the corresponding initial stress of 0.5fu , 0.6fu , and 0.7fu . The values of A and B were also listed in Table 1. It can be recognized that the logarithmic formula depicts well the experimental data of relaxation rate with a good value of R-square, and RR = 0.1832ln(t) + 0.8925, RR = 0.2239ln(t) + 0.9688, and RR = 0.2166ln(t) + 1.1336 were recommended to predict the values of stress relaxation at corresponding 0.5fu , 0.6fu , and 0.7fu (Table 2). Table 1. Estimation of constant parameters A and B Stress level A
R2
B
0.5fu
0.1832 ± 0.004
0.8925 ± 0.0025 0.985
0.6fu
0.2239 ± 0.002
0.9688 ± 0.001
0.994
0.7fu
0.2166 ± 0.0032 1.1336 ± 0.015
0.992
Table 2. Long-term stress relaxation behavior of Poscable-86 high-strength steel wire Stress level
R100
R1000
K
R10years
R20years
R50years
0.5fu
1.75
2.18
0.0954
3.34
3.57
3.89
0.6fu
2.00
2.47
0.0916
3.72
3.97
4.31
0.7fu
2.13
2.63
0.1014
4.23
4.54
4.98
It can be observed that the relaxation rates of high-strength steel wire at 0.5fu over a long-term period of 20 years and 50 years are 3.57% and 3.89%, respectively. At 0.6fu , the relaxation rates of high-strength steel wire at a long-term period of 20 years and 50 years are 3.97% and 4.31%, respectively. At the highest initial stress, the relaxation rates of 4.54% and 4.98% were well calculated at 20 years and 50 years. Another interesting feature of the long-term stress relaxation behavior is that there is no indication that the relaxation rates will approach the limiting values for all cases.
5 Conclusions In this study, the stress relaxation experiments were conducted on Poscable-86 highstrength steel wire to investigate the variation of relaxation rate at different applied initial stress levels. For this purpose, three stress relaxation experiments were performed at room temperature for three levels of initial stress, for example, 0.5fu , 0.6fu , and 0.7fu . The influence of initial stress on the stress relaxation behavior of Poscable-86 highstrength steel wire was investigated, and the long-term stress relaxation behavior of the tested material was also characterized. The experimental and analysis results support the following conclusions.
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1. Stress relaxation behavior of Poscable-86 high-strength steel wire included two main characteristic stages, including transient relaxation and secondary relaxation. 2. The first stage normally occurred at the beginning of the stress relaxation testing (i.e. 2~3 h of testing for Poscable-86 high-strength steel wire), while the secondary relaxation occurred at the long duration of testing. 3. A high increasing speed of relaxation rate could be observed at the transient relaxation, while a low increasing speed of relaxation rate during the secondary stage. 4. Stress relaxation strongly sensitive to the applied initial stress levels, in which the relaxation rate tends to increase with the further increase of initial stress. 5. RR = 0.1832ln(t) + 0.8925, RR = 0.2239ln(t) + 0.9688, and RR = 0.2166ln(t) + 1.1336 were recommended to predict the values of stress relaxation at corresponding 0.5fu , 0.6fu , and 0.7fu . 6. Relaxation rates of high-strength steel wire at 0.5fu over a long-term period of 20 years and 50 years are 3.57% and 3.89%, respectively.
References 1. Saitoh, M., Okada, A.: The role of string in hybrid string structure. Eng. Struct. 21, 756–769 (1999). https://doi.org/10.1016/S0141-0296(98)00029-7 2. Du, Y., Zhan Peng, J., Richard Liew, J.Y., Qiang, G.: Mechanical properties of high tensile steel cables at elevated temperatures. Constr. Build. Mater. 182, 52–65 (2018). https://doi. org/10.1016/j.conbuildmat.2018.06.012 3. Forn˚usek, J., Konvalinka, P., Sovják, R., Vítek, J.L.: Analysis of concrete cable-stayed creep, shrinkage and relaxation effects. Eng. Struct. 32, 829–842 (2010). https://doi.org/10.4028/ www.scientific.net/AMR.594-597.1498 4. Han, Q., Wang, L., Xu, J.: Test and numerical simulation of large angle wedge type of anchorage using transverse enhanced CFRP tendons for beam string structure. Constr. Build. Mater. 144, 225–237 (2017). https://doi.org/10.1016/j.conbuildmat.2017.03.150 5. Forn˚usek, J., Konvalinka, P., Sovják, R., Vítek, J.L.: Long-term behaviour of concrete structures reinforced with pre-stressed GFRP tendons. WIT Trans. Modelling Simul. 48, 535–545 (2009). https://doi.org/10.2495/CMEM090481 6. Chen, Z., Yu, Y., Wang, X., Wu, X., Liu, H.: Experimental research on bending performance of structural cable. Constr. Build. Mater. 96, 279–288 (2015). https://doi.org/10.1016/j.con buildmat.2015.08.026 7. Truong, V.H., Kim, S.E.: An efficient method of system reliability analysis of steel cablestayed bridges. Adv. Eng. Softw. 114, 295–311 (2017). https://doi.org/10.1016/j.advengsoft. 2017.07.011 8. Nguyen, N.-V., Vu, Q.-A., Kim, S.-E.: An experimental study on stress relaxation behaviour of high strength steel wire: microstructural evolution and degradation of mechanical properties. Constr. Build. Mater.. 261, 119926 (2020) 9. Wang, X., Yu, Y., Yan, X.: Theoretical model of symmetric wire breaks in semi-parallel wire cables. Tianjin DaxueXuebao (Ziran Kexue Yu Gongcheng Jishu Ban)/J. Tianjin Univ. Sci. Technol. (2016). https://doi.org/10.11784/tdxbz201605035 10. Au, F.T.K., Si, X.T.: Accurate time-dependent analysis of concrete bridges considering concrete creep, concrete shrinkage and cable relaxation. Eng. Struct. 33, 118–126 (2011). https:// doi.org/10.1016/j.engstruct.2010.09.024
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11. Wang, X., Chen, Z., Liu, H., Yu, Y.: Experimental study on stress relaxation properties of structural cables. Constr. Build. Mater. 175, 777–789 (2018). https://doi.org/10.1016/j.conbui ldmat.2018.04.224 12. Kmet, S., Mojdis, M.: Time-dependent analysis of cable domes using a modified dynamic relaxation method and creep theory. Comput. Struct. 125, 11–22 (2013). https://doi.org/10. 1016/j.compstruc.2013.04.019 13. Au, F.T.K., Si, X.T.: Time-dependent effects on dynamic properties of cable-stayed bridges. Struct. Eng. Mech. 41, 139–155 (2012). https://doi.org/10.12989/sem.2012.41.1.139 14. Enomoto, T., Harada, T., Ushijima, K., Khin, M.: Long-term relaxation characteristics of CFRP cables. Japan 21, 73–77 (2009) 15. Magura, D.D., Sozen, M.A., Seiss, C.P.: A study of stress relaxation in prestressing reinforcement. Struct. Res. Ser. 237, 13–57 (1964) 16. Abrams, C.R.C.M.S.: Behavior at high temperature of steel strand for pre-stressed concrete. Portl. Cem. Assoc. Res. Dev. Lab. 3, 8–19 (1961) 17. Papsdorf, W., Schwier, F.: Creep and relaxation of steel wire, particularly at slightly elevated temperatures. Stahl Un Eisen. 78 (1958) 18. Jevtic, D.: Relaxation and creep fatigue tests of steel wires for pre-stressed concrete. RILEM Bull. 4, 1–45 (1959) 19. Yamaoka, Y., Hamada, K., Tsubono, H., Kawakami, H., Oki, Y., Kawaguchi, Y.: Development of galvanized high-strength high-carbon steel wire. Trans. Iron Steel Inst. Japan. 26, 1059– 1064 (1986). https://doi.org/10.2355/isijinternational1966.26.1059 20. Zhou, S.C.J.H.T., Li, G.Q.: Experimental studies on the properties of steel strand at elevated temperatures. J. Sichuan Univ. 40, 106–110 (2008) 21. Fan, Z.L.J.: Experimental study on the pre-stressed steel strand at high temperature. Br. J. Psychiatry. 32, 50–63 (2002). https://doi.org/10.1192/bjp.112.483.211-a 22. Ai-ping, L., Chao-ping, J., Xue-mei, L.: Simulation of dynamic tension of wire rope based on ADAMS. Mod. Manuf. Eng. 1, 43–46 (2010) 23. Wang, X., Chen, Z., Yu, Y., Liu, H.: An innovative approach for numerical simulation of stress relaxation of structural cables. Int. J. Mech. Sci. 131–132, 971–981 (2017). https://doi. org/10.1016/j.ijmecsci.2017.08.011 24. Zhao, J., Mei, K., Wu, J.: Long-term mechanical properties of FRP tendon - anchor systems - a review. Constr. Build. Mater. 230, 117017 (2020). https://doi.org/10.1016/j.conbuildmat. 2019.117017 25. Wu, S., Li, J., Guo, J., Shi, G., Gu, Q., Lu, C.: Stress corrosion cracking fracture mechanism of cold-drawn high-carbon cable bolts. Mater. Sci. Eng. A. 769, 138479 (2020). https://doi. org/10.1016/j.msea.2019.138479 26. Switzerland, F.: Fib Model Code for Concrete Structures 2010. International Federation for Structural Concrete. Ernst Sohn Publ, House, Lausanne (2013) 27. EN ISO 15630–1. EN ISO 15630–1: Steel for the reinforcement and prestressing of concrete. Test methods - Part 1: Reinforcing bars, wire rod and wire. Ital. Board Stand. 248 (2010) 28. GB/T10120–2013. Metallic materials-Tensile stress relaxation- Method of test, Beijing China (2013)
Seismic Performance of Concrete Filled Steel Tubular (CFST) Columns with Variously Axial Compressive Loads Hao D. Phan1,2(B) and Ker-Chun Lin2,3 1 Faculty of Civil Engineering, The University of Danang – University of Science and
Technology, Danang, Vietnam [email protected] 2 Department of Civil and Construction Engineering, National Taiwan University of Science and Technology, Taipei, Taiwan 3 National Center for Research on Earthquake Engineering, Taipei, Taiwan [email protected]
Abstract. This paper presents a study on the seismic performance of square concrete filled steel tubular (CFST) columns under different axial compressive loads. A total of four full-scale specimens using the steel tube with a width-to-thickness ratio (B/t) of 42 was tested by combined axial compression and cyclic lateral loading. In which, three specimens were subjected to constant axial compression, P = 0.15, 0.35, and 0.55P0 , respectively, and one specimen was applied varied axial compression varying from 0.15 to 0.55P0 (P0 is nominal axial compression strength of the column) that is an assumption for designing exterior columns in a typical moment resistance frame (MRF) system. An identical loading protocol according to the AISC 341–16 code was adopted as the basis of cyclic lateral displacement loading for all four specimens. Test results reveal that there were significant differences in yielding and local buckling process in the steel tube, lateral (shear) strength, deformation capacity, and lateral stiffness degradation of the columns. For three CFST specimens with constant axial compression, the higher the axial compression was, the smaller the lateral strength and deformation capacity were. Another important finding in the specimen with varied axial compression, CFST42–15/55C, was a discrepancy in lateral strength and deformation capacity in positive and negative directions of cyclic lateral displacement. It was found that Specimen CFST42-55C possesses the lowest lateral strength and deformation capacity with the average ultimate interstory drift ratio (IDRu ) of only 1.29% radian. The study results also show that these square CFST columns satisfy the highly seismic requirement when the axial compression has not exceeded 0.35P0 with their average IDRu of more than 3% radian. To improve the seismic performance of these composite columns, reducing the B/t limit is one of the most effective ways. Keywords: CFST columns · Seismic performance · Axial compression · The ratio of B/t · Interstory drift ratio
© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 915–925, 2022. https://doi.org/10.1007/978-981-16-3239-6_71
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1 Introduction Concrete filled steel tubular (CFST) columns, recently, are dramatically used in buildings located in high seismic regions. Some related studies were conducted by structural engineering researchers. In which, the mechanical behavior of CFST columns having different cross-sections subjected to axial compression combined with monotonic or cyclic lateral loading is a significant issue. The seismic performance of CFST columns depends on some factors such as shape and dimension of the cross-section, width/diameter-tothickness ratio (B/t or D/t) of the steel tube, concrete- and steel strength, interaction between steel tube and concrete core, and axial compression level. The strength and energy dissipation capacity of the circular CFST columns significantly increased when using a thicker steel shell and shear studs (Boyd et al. 1995). However, another study revealed that using different bond and end loading conditions insignificantly affected the flexural strength and ductility of these composite columns (Fam et al. 2004). The use of high strength steel or low D/t in circular CFST columns increased the column ductility due to the concrete confinement effect, but using high strength concrete reduced the column ductility. The square CFST columns possess less ductility and moment strength than circular ones. Especially, for square CFST columns with higher B/t, the confinement effect was lower, and the local buckling phenomenon early occurred in the steel tubes resulting in the column ductility decrease. Also, a ductility reduction was found in the columns subjected to axially varied loading (Fujimoto et al. 2004, Inai et al. 2004). Other researches showed that stiffness and strength of square CFST columns were not significantly affected by loading type: monotonic or cyclic, however, the post-peak moment resistance reduced rapidly in the case of cyclic loading. The cyclic ductility decreased when increasing steel yield strength, B/t, and axial compression levels (Han et al. 2003, Varma et al. 2002a, 2002b, 2004 and 2005). The past studies revealed that few experimental data on the seismic behavior of full-scale square CFST columns under high axial compression. Hence, the purpose of this study is to investigate the seismic behavior of full-scale square CFST columns with different levels and types of axial compression. An experimental program using four full-scale specimens was conducted at the National Center for Research on Earthquake Engineering (NCREE), Taipei, Taiwan. In this paper, test observations and discussions on the seismic behaviors of these composite columns are presented in detail. The experimental results are also compared to the predicted results based on the AISC 360–16 specification.
2 Experimental Program 2.1 Specimen Design The matrix of CFST column specimens is designed as shown in Table 1. In which, the symbols CFST42-35C and CFST42–15/55C illustrated a specimen with the steel tube’s width-to-thickness ratio (B/t) of 42 subjected to constant compression of 0.35P0 (P0 is the column’s nominal axial compressive strength) and varied compression designed in a range from 0.15 to 0.55P0 combined with symmetrically cyclic lateral displacement loading, respectively. The 12-mm-thick steel plates are used to fabricate steel tubes with
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a hollow square section, including the outer width B = 500 mm and being rounded at corners with an inner and outer radius of 18 and 30 mm, respectively. The column length (L) equals 3000 mm and the two steel footings have their height of 595 mm plus a 5-mm aluminum plate for each end of the specimen. Table 1. Matrix of CFST column specimens Specimen
L (mm) B (mm) t (mm) B/t P/P0
CFST42-15C
3000
500
12
42 0.15
CFST42-35C
3000
500
12
42 0.35
CFST42-55C
3000
500
12
42 0.55
CFST42–15/55C 3000
500
12
42 0.15–0.55
*Notes: Actual steel grade f ya = 351 MPa; and actual concrete strength f ’ ca = 36.35 MPa
To fabricate a steel tube, the two steel plates were bent to make smooth corners and then connected each other by two welding lines using backing bars, which were located on the inner surface of the steel tube. The top and bottom footings were also fabricated using a combination of welded steel plates (12 and 20 mm of thickness) and welded to the steel tube to form a complete steel specimen. Figure 1 shows detailed dimensions of the specimen.
Fig. 1. Dimensions of the specimen
Fig. 2. Set up a specimen on the MATS
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2.2 Material Properties The steel plates, SN 490B, having a nominal yield strength (f y ) of 345 MPa were used. The preliminary tension tests were conducted (ASTM E8–08 2008) to determine the steel mechanical properties. These include actual yield stress f ya = 351 MPa, ultimate stress f u = 518.4 MPa, yield strain εy = 0.00175, ultimate strain εu = 0.1542, and elastic module E s = 200 GPa. The concrete with a nominal compressive strength of 35 MPa was offered. To determine actual concrete compressive strength, f ca , fifteen concrete cylinders (D = 150, H = 300 mm) were made and cured in the non-water condition. In which, the first three concrete cylinders were tested at the age of 28 days. The other concrete cylinders were tested on the same day of the CFST column testing with three concrete cylinders per time. The average f ca was calculated as 36.35 MPa. 2.3 Test Procedure The Multi-Axial Testing System (MATS) machine in the NCREE was used to test these CFST specimens (Fig. 2). The test set up of a specimen on the MATS and the measuring instruments’s arrangement and set up were conducted on step by step as described below. Firstly, the specimen was lifted and put on the platen and moved to the central position. Then, the bottom footing and top footing of the specimen were fixed to the platen and the top crossbeam, respectively, combined with two aluminum plates to increase friction forces. Next, a set of thirty-four NDI makers was stuck in one surface (side) of the column to measure displacements of these points. In which, the NDI makers were divided into two vertical parallel lines (seventeen NDI markers per line) with a linespace of 400 mm, the dictances between NDI markers in each line were 150, 200, or 250 mm, depending the posistion of these markers along the column’s length (set up two NDI machines in both East and West). Then, all the strain gauges glued on the steel tube surfaces, which were designed to measure the longitudinal strain on the steel tube’s surfaces during the testing time, were connected to a computer for recording the strain value data. Next, two Tempos, name of the Linear Variable Differential Transformers (LVDTs), were mounted at the levels near to the bottom end and the top end of the column on the direction of cyclic lateral displacement loading to define the relative lateral displacements between two column’s ends. Meanwhile, the main computer was also connected to the MATS to record the loading and displacement data during the testing process. Finally, four cameras were set up in four sides to capture the global deformation of the column. All four CFST column specimens were tested under combined axial compression and cyclic lateral loading. The first three specimens were subjected to three different levels of constant axial compression: 0.15, 0.35, and 0.55P0 , respectively, combined with cyclic lateral loading (AISC 341–16). The last specimen subjected to varied axial compression designed varying from 0.15 to 0.55P0 , in which 0.35P0 was chosen as a started and middle value, combined with the same cyclic lateral loading protocol as mentioned above. The diagrams of the two loading protocols used in this study are presented in Fig. 3. In which, Fig. 3a shows the cyclic lateral loading protocol used for all specimens, meanwhile, Fig. 3b illustrates three levels of constant axial compression and the varied axial compression having the same changing rule with the lateral loading.
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Fig. 3. Proposed loading protocols
3 Test Results and Discussions 3.1 Test Results and Observations The test results (Table 2) and detailed discussions including test observations, lateral load vs. lateral displacement response, lateral strength, deformation capacity, and lateral stiffness degradation of these columns are presented below. Table 2. Test results of four specimens Specimen
(±) IDRmax H max or Qmax (kN)
Δu at 0.8Qmax (mm)
IDRu at 0.8Qmax (% radian)
±
Average ±
Average ±
Average
1347
147.5
4.92
+
2.0
1337
-
2.0
1357
CFST42-35C
+
1.5
1244
-
1.5
1249
CFST42-55C
+
1.0
1051
-
1.0
1041
1.5
1208
-
70.2
-
2.34
-
1.5
1334
-
90.0 +
-
3.00 +
-
CFST42-15C
CFST42–15/55C + -
153.9 141.0
1246
95.7
1046
40.5
5.13 4.70
90.9
3.19
38.7
1.35
86.1
3.03
2.87
36.9
1.29
1.23
In general, test observations reveal that significant deformation, outward buckling of the steel tube, and concrete and steel fractures always happened in the areas with a distance between B/4 and B/2 from each footing surface. Based on the local deformation of CFST column specimens, it can conclude that the plastic hinges in the column locate at the positions with a distance of B/4 from the footing surfaces. Test observations of Specimen CFST42-15C show that the steel tube started yielding in flanges and webs at the interstory drift ratio (IDR) of 1.5% and 2% radian, respectively. Then, local buckling of the steel tube developed in flanges and webs at 3% and 4% radian, respectively (Fig. 4a, b). Next, at the IDR of 5%, local buckling in the webs dramatically
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developed and then fractures at the steel tube corners and concrete crushing with a loud sound occurred near the two column’s ends. Finally, the specimen experienced the largest buckling and cracking fractures of the steel tube and the highest intensive crushing of the concrete core at 6% radian (Fig. 4c, d). From the test observations of Specimen CFST42-35C, steel yielding developed in flanges and webs at the IDR of 1.5% radian (Fig. 5a, b). The full yielding and the beginning of local buckling occurred in flanges after completion of 2%. Next, at the IDR of 3% radian, larger local buckling in flanges and buckling developing in webs happened, respectively (Fig. 5c, d). Finally, a reduction of the designed axial compressive loading occurred in the first cycle of 4% due to the axial shortening displacement of the specimen exceeded the maximum vertical displacement of the actuators in MATS. Hence, to evaluate seismic performance, it is concluded that this specimen failed after finishing one half of the first cycle at the IDR of 4% radian.
(a) Buckling in the top flange
(b) Buckling in the bottom end
(c) Fractures in the top end
(d) Fractures in the bottom end
Fig. 4. Deformation stages of Specimen CFST42-15C
(a) Yielding in the top flange
(c) Buckling in the top end
(b) Yielding in the bottom flange
(d) Buckling in the bottom end
Fig. 5. Deformation stages of Specimen CFST42-35C
Test observations of Specimen CFST42-55C show that, firstly, unlike the two previous specimens, yielding started in flanges and webs at the lower IDR of 0.5% and 0.75% radian, respectively. Then, the full yielding and local buckling developed in flanges and webs of the steel tube at the IDR of 1% radian during four cycles (Fig. 6a, b). Next, at 1.5%, the local buckling quickly developed in both flanges and webs due to the influence of high axial compression. Finally, at the beginning of 2%, the axial compressive force on the specimen was reduced suddenly due to the large axial shortening of the column with the same reason in Specimen CFST42-35C. Therefore, it is concluded that this specimen failed at the lowest loading level just after finishing two cycles at the IDR of 1.5% radian (Fig. 6c, d). From the test observations of Specimen CFST42–15/55C, the steel tube started yielding in its flanges and webs at the IDR of 0.75% and 1% radian, respectively. Then, the full steel yielding occurred in flanges and webs at the IDR of 1.5% radian (Fig. 7a, b). Next, the local buckling developed in flanges at around 2%. After that, the local buckling developed in both flanges and webs at the IDR of 3% radian (Fig. 7c, d). In
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(a) Yielding in the top flange
(c) Buckling in the top end
(a) Yielding in the top flange
(c) Buckling in the top end
(b) Yielding in the bottom flange
(d) Buckling in the bottom end
(b) Yielding in the bottom end
(d) Buckling in the bottom end
Fig. 6. Deformation stages of Specimen CFST42-55C
Fig. 7. Deformation stages of Specimen CFST42–15/55C
which, a notable finding is that buckling in webs was asymmetrical after half-cycle ‘−1’ and then the full buckling spread in entitle webs up to the cycle ‘2’ of this loading level. Finally, however, another response was found that going to the IDR of 4% radian, the axial compressive force gradually reduced on its expected increasing trend due to the same reason with Specimens CFST42-35C and CFST42-55C. It means that in the case of varied axial compression changed from 0.2 to 0.5P0 (corresponding to the IDR of 3% radian), the designed axial compressive force could be controlled to follow the varied axial compression loading protocol (Fig. 3b), and it could not when the axial compression was increasing to 0.55P0 . Hence, to evaluate seismic performance, it can be concluded that this specimen failed after finishing two cycles of the IDR of 3% radian. 3.2 Lateral Load vs. Lateral Displacement Curves The lateral load vs. lateral displacement (H-Δ) hysteresis loops are shown in Fig. 8. These hysteresis loops reveal that the axial compression level significantly affected the seismic behavior of CFST columns. For the lowest axial compression, Specimen CFST42-15C shows to be the best seismic resistance column compared to the three remains. Figure 8a illustrates that this column maintains a high lateral loading up to the IDR of more than 4% radian. Meanwhile, the Specimens CFST42-35C, CFST42-55C, and CFST42–15/55C just maintain the designed axial compression up to more than 3, 1.5, and 3% radian, respectively. Increasing the axial compression leads to early local buckling and then reduction of lateral strength of the column. Generally, the higher compression and larger lateral displacement, the more significant degradation of lateral strength. For instance, Specimen CFST42-55C was the worst seismic resistance column with the lowest loading and deformation capacities (Fig. 8c). Besides, the varied axial compression made an asymmetrical behavior of Specimen CFST42–15/55C. It is found that lateral load in the negative direction (lower axial compression) was larger than that in the positive direction. Figure 8d shows that at the IDR of 3% radian, higher lateral
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strength, and less strength degradation occurred in the negative direction compared to those in the positive direction.
Fig. 8. Lateral load vs. lateral displacement (H-) hysteresis loops
3.3 Lateral Strength and Deformation Capacity Test results are shown in Table 2, in which, Qmax and IDRmax denote the maximum lateral strength and the IDR corresponding to maximum lateral strength. The other two notations, Δu and IDRu denote the ultimate displacement and ultimate IDR, respectively. Herein, Δu is defined as the displacement corresponding to the point of 80% Qmax postpeak, and IDRu is calculated by Δu divided to L. For the first three specimens with constant axial compression, the values of Qmax , Δu , and IDRu were obtained by averaging two absolute values in the positive and negative directions. For the last specimen with varied axial compression, these values were separately calculated in each direction. Based on the data in Table 2 and Fig. 9, two main findings were found as follows. The first is, the higher the axial compression, the lower the lateral strength the column possesses. Compared to Specimen CFST42-15C, the lateral strength reduction of Specimen CFST42-35C was 7.5% and the largest drop was 22.3% coming from Specimen CFST42-55C. Meanwhile, Specimen CFST42–15/55C reduced about 10.3 and 1% of lateral strength in the positive (P/P0 = 0.425) and the negative (P/P0 = 0.275) lateral directions, respectively. Table 2 also shows that the higher the axial compression, the
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Fig. 9. Cyclic response envelopes of specimens
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Fig. 10. Lateral stiffness degradations of specimens
lower the lateral displacement level corresponding to Qmax of the column. The second is, deformation capacities of these CFST columns are different and significantly depend on the axial compression level. The higher the axial compression, the lower the deformation capacity the column possesses. From Table 2, it is clear that the highest IDRu is 4.92% radian obtained by Specimen CFST42-15C, then following by Specimen CFST42-35C and Specimen CFST42–15/55C in the negative side with IDRu of 3.03% and more than 3.00% radian, respectively. Meanwhile, Specimen CFST42–15/55C on the positive side has an IDRu of 2.34% radian. At the last, Specimen CFST42-55C showed the lowest deformation capacity compared to the three remaining specimens, which has an IDRu of 1.29% radian. In conclusion, it reveals that in the high seismic zones, the CFST columns subjected to the axial compression not exceeded 0.35P0 are safe (IDRu ≥ 3.0% radian). Moreover, based on the comparison of Qmax and QAISC at different levels of axial compression (Fig. 9), it reveals that all maximum lateral strengths of the CFST column obtained from the test results were higher than those calculated according to the AISC 360–16 specification. The ratio of Qmax /QAISC was ranged from 1.134 to 1.160, when the axial compression was applied with different levels. This means that the AISC 360–16 is reasonably conservative for predicting the loading capacity of square CFST columns investigated in this study. 3.4 Lateral Stiffness Degradation The lateral stiffness at the cyclic lateral loading level ith (S i ) of a column member was calculated by Eq. (1) according to the definition of the secant stiffness from the AISC 360–16 specification. n
j+ j=1 (Hi j+ j=1 (i
Si = n j+
j−
j+
j−
j−
− Hi ) j−
− i )
(1)
where Hi ,Hi and i ,i are peak lateral load and peak lateral displacement in the jth cycle for the positive and negative directions at the cyclic lateral loading level ith ,
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respectively; and n is the number of cycles tested at the corresponding cyclic lateral loading level ith . The comparison of lateral stiffness at different lateral displacement levels of four specimens are presented in Fig. 10. This shows that, at the initial IDR (0.375% radian), all specimens have the strongest lateral stiffness, ranging from 53.85 to 57.43 kN/mm. However, the larger the lateral displacement, the lower the lateral stiffness is; and the higher the axial compression, the faster the lateral stiffness degradation occurs. For instance, at the IDR of 1.5% radian, the lateral stiffness of Specimens CFST42-15C, CFST42-35C, and CFST42-55C is 29.44, 27.59, and 13.05 kN/mm, respectively, and it is 28.10 kN/mm for Specimen CFST42–15/55C. It also reveals that at smaller IDRs (0.375–1.5% radian), the lateral stiffness of Specimen CFST42–15/55C is completely higher than that of Specimen CFST42-35C, but at larger IDRs (2.0 and 3.0% radian) the lateral stiffness of Specimen CFST42-35C dominates.
4 Conclusions An experimental study was conducted to investigate the seismic performance of full-scale square CFST columns subjected to a combination of axial compression with different levels and types and cyclic lateral displacement loading. The conclusions can be drawn as below: • The higher the axial compression was, the earlier the yielding and local buckling of the steel tube occurred. As a result, it leads to a reduction of lateral strength and deformation capacity of these square CFST columns. It shows that increasing axial compression from 0.15 to 0.55P0 , the lateral strength reduces 22.3% and the IDRu decreases 3.8 times. • With the axial compression higher or equal to 0.425P0 , the IDRu is less than or equal to 2.34% radian, while with the axial compression lower or equal to 0.35P0 , the IDRu is larger or equal to 3.03% radian. It means that 0.35P0 is the upper limit value of axial compression for the CFST columns (with B/t of 42) in buildings located in high seismic zones (the required deformation capacity not less than 3% radian). • The higher the axial compression was, the stronger the initial lateral stiffness obtained. However, at higher levels of lateral displacement, the higher the axial compression was, the faster the lateral stiffness degradation occurred. Also, the varied axial compression negatively affects the seismic behavior of the column. • The comparison of column’s lateral strength between the test results and design code predictions according to AISC 360–16 specification at different axial compressive levels reveals that this code is reasonably conservative for calculating the loading capacity of square CFST columns. • To satisfy the deformation requirement of these composite columns under high axial compression conditions (exceeded 0.35P0 ), reducing B/t value is an effective way. This issue would be numerically studied in detail and presented in the next publication.
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References AISC 360-16 Committee on Specifications: Specification for structural steel building, American Institute of Steel Construction, Chicago (2016) AISC 341-16 Committee on Specifications: Seismic provisions for structural steel building, American Institute of Steel Construction, Chicago (2016) ASTM E8/E8M-08 Committee E28: Standard test methods for tension testing of metallic materials, American Association State Highway and Transportation Officials Standard, West Conshohocken (2008) Boyd, P.F., Cofer, W.F., Mclean, D.I.: Seismic performance of steel-encased concrete columns under flexural loading. ACI Struct. J. 92(3), 355–364 (1995) Fam, A., Qie, F.S., Rizkalla, S.: Concrete-filled steel tubes subjected to axial compression and lateral cyclic loads. J. Struct. Eng. 130(4), 631–640 (2004) Fujimoto, T., Mukai, A., Nishiyama, I., Sakino, K.: Behavior of eccentrically loaded concrete-filled steel tubular columns. J. Struct. Eng. 130(2), 203–212 (2004) Han, L.-H., Yang, Y.-F., Tao, Z.: Concrete-filled thin-walled steel SHS and RHS beam-columns subjected to cyclic loading. Thin-Walled Struct. 41(9), 801–833 (2003) Inai, E., Mukai, A., Kai, M., Tokinoya, H., Fukumoto, T., Mori, K.: Behavior of concrete-filled steel tube beam columns. J. Struct. Eng. 130(2), 189–202 (2004) Varma, A.H., Ricles, J.M., Sause, R., Lu, L.-W.: Experimental behavior of high strength square concrete-filled steel tube beam-columns. J. Struct. Eng. 128(3), 309–318 (2002) Varma, A.H., Ricles, J.M., Sause, R., Lu, L.-W.: Seismic behavior and modeling of high-strength composite concrete-filled steel tube (CFT) beam-columns. J. Constr. Steel Res. 58(5–8), 725– 758 (2002) Varma, A.H., Ricles, J.M., Sause, R., Lu, L.-W.: Seismic behavior and design of high-strength square concrete-filled steel tube beam columns. J. Struct. Eng. 130(2), 169–179 (2004) Varma, A.H., Sause, R., Ricles, J.M., Li, Q.: Development and validation of fiber model for highstrength square concrete-filled steel tube beam-columns. ACI Struct. J.-Am. Concr. Inst. 102(1), 73–84 (2005)
Damage Simulation Based on the Phase Field Method of Porous Concrete Material at Mesoscale Hoang-Quan Nguyen, Ba-Anh Le, and Bao-Viet Tran(B) University of Transport and Communications, Hanoi, Vietnam [email protected]
Abstract. In this research, a study on damage behavior of porous concrete was conducted. Based on the phase field theory and the generation process based upon Monte Carlo’s simulation method, we construct a numerical procedure to solve complex damage thermodynamic problems. The phase field variable obtained can be used to model crack behavior within porous concrete structure. Some factors that affect the results are discussed to make the predictions more accurate for the case of porous concrete material. Illustrations of applications are provided in examples to show the usefulness of the approach. Keywords: Damage modelling · Pervious concrete · Phase field method
1 Introduction Over many decades, determining the damage behavior of previous concrete structure has become a stimulating subject for numerous theoretical-experimental-numerical researches. To date, in order to predict this property, most of the research is usually based on two approaches: analytical model and numerical model. The numerical methods of characterizing damage in heterogeneous materials taking into account the effect of the microstructure of materials are interesting topics. The strength of concrete with high porosity is highly dependent on the distribution of voids and the ability to develop cracks in this material. The Finite Element Method followed by the Extended Finite Element Method (XFEM) or the phase-field method (PFM) (Francfort and Marigo 1998) was developed to allow for accurate timing and progressing of damage process in high porosity concrete materials. However, the complexity and its practical applicability at the Engineer level are the biggest drawbacks of the numerical method. So, in this paper, we are interested in construct a simple numerical approximation based on the phase field method within the framework of finite element method to model damage behavior of previous concrete structure. We recall from the literature in Sect. 2 some basics of procedure of discretization of the system of the governing equations at element level using the FEM for displacement and phase field variables. In Sect. 3, the Monte Carlo simulations are used to construct a generation procedure to generate 2D concrete structures at mesoscale with randomly circular aggregates of different fractions. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 926–934, 2022. https://doi.org/10.1007/978-981-16-3239-6_72
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In Sect. 4, the 2D structure are then meshed and the pervious concrete problem are elaborated and calculated in MATLAB for examples to show interests of the approximation. Some interesting conclusions will be put in the last section.
2 Finite Element Discretization from the Phase-Field Problem Framework Starting from works of Francfort and Marigo (Francfort and Marigo 1998), this method aims at constructing a diffusive phase field variable d (x) by minimizing the free energy that combines elastic bulk energy and crack surface energy, then this variable can be used to represent the crack network. The method alleviate the short coming of remeshing crack geometry by using a fixed mesh and a regularized description of crack dícontinuites. The phase field method is useful to model crack behavior of brittle material, however application of this method for concrete structure is still need to investigate (Li et al. 2019). In this method, assuming small strain, the regularized form of the energy describing the cracked structure can be written as follows: (1) E(u, d ) = W(u, d )d + gc γ (d )d
Where W is the elastic density energy, u(x) is the vector of displacements, gc is the fracture toughness and γ (d ) is the crack energy density, expressed by: γ (d , ∇d ) =
1 2 l2 d + ∇d · ∇d 2l 2
(2)
Where l is a regularization parameter related to the width of the smeared crack. Applying the principle of maximum dissipation and energy minimization, we obtain the set of coupled equation describing the mechanical problem and the phase-field problem on the domain associated with the structure, with boundary δ and outward nornal n: ⎧ gc 2 ⎪ 2(1 − d )H − d − l d = 0 in ⎪ ⎨ l (3) d (x) = 1 on Γ ⎪ ⎪ ⎩ ∇d (x) · n = 0 on ∂Ω ⎧ ⎪ ⎨ ∇ · σ (u, d ) = f in u(x) = u on ∂u (4) ⎪ ⎩ σ n =F on ∂F Where Γ represents the crack surface, H (t) is the history strain energy density function describing a dependence on history and possible loading – unloading. This function is expressed as follows:
H (x, t) = max Ψ + (x, τ ) (5) τ ∈[0, t]
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In (5), Ψ + is the tensile part of the elastic strain energy funtion serving to model the unilateral contact. It is given by: 2 λ 2 (6) Ψ + (ε) = Tr(ε)+ + μTr ε + 2 Where ε is the strain tensor and x± = (x ± |x|)/2 and ε± are tensile and compression parts of the strain tensor. In (4), σ = ∂W ∂ε is the second-order Cauchy stress tensor, f are the body forces, u, F are prescribed displacements and forces on the corresponding boundaries ∂Ωu and ∂Ω F , respectively. The constitutive law is given by: σ = (1 − d )2 + k λTrε+ 1 + 2με + + λTrε− 1 + 2με − . Where k is very small numerical parameter to avoid loss of stability in case of fully damaged elements. The problem described in Eq. (3) and (4) are solved by a standard FE procedure in a staggered scheme at each time step, i.e. the phase field problem and the mechanical problem are solved alternatively. The discretization of the system of the governing equation at element level using the FEM for displacement and phase-field variable can be expressed as follows: u = Nu δui u = Nu ui [ε(u)] = Bu ui [ε(δu)] = Bu δui ui d (u) = Nd (x)di ∇d = Bd (x)di δd (u) = Nd (x)δdi ∇δd (x) = Bd (x)δdi
(7)
Where ui , di are nodal displacements and nodal phase field at time tn+1 , respectively. Nu,d , Bu,d are vector of shape function and matrix of shape functions derivatives for displacement and phase field, respectively. Within the context of incremental scheme, the projection tensors defined at time n + 1 will be evaluated based on the result from previous step n as follows: (8) With R+ (ε n ) = 21 (sign(Trε n ) + 1); R− (ε n ) = 21 (sign(−Trε n ) + 1). Setting R± (ε n ) ≡ Rn± , P± (ε n ) ≡ Pn± where P± are the matrix forms associated . with the fourth –order tensors The finite element equation for the mechanical problem can be written as follows: {K1 (dn+1 , un ) + K2 (un )}un+1 = Fn+1 Where
K1 (dn+1 ) =
BTu
(9)
(1 − dn+1 )2 + k λRn+ [1]T [1] + 2μPn+ Bu d Ω
K2 =
BTu λRn+ [1]T [1] + 2μPn+ Bu d Ω
(10)
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Fn+1 =
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NuT fd +
NuT Fd
∂ΩF
The finite element equation for the phase field problem is given by:
where
Kd dn+1 = Fd
(11)
gc + 2Hn NdT Nd + gc lBTd Bd d Kd = l
(12)
and
Fd =
2NdT Hn d
(13)
Where Hn = H (un ) is computed from the previous load increment: + + (x) if Ψn+1 (x) > Ψn+ (x) Hn+1 (x) = Ψn+1 + (x) ≤ Ψn+ (x) Hn+1 (x) = Ψn+ (x) if Ψn+1
(14)
3 Arrangement of Aggregate Particle Based on Monte-Carlo Simulation At mesoscopic level, concrete could be represented as biphasic material: coarse aggregates and mortar matrix and an interfacial transition zone (ITZ) between them. The evaluation of the composite behavior of concrete at mesoscopic level requires the generation of a random aggregate structure in which the shape, size and distribution of coarse aggregate closely resemble real concrete in the statistical sense. The shape of aggregate particles depends on the aggregate types. In general, gravel aggregates have a rounded shape while crushed stone aggregates have an angular shape. In 2D numerical simulation, the aggregate shape could be simulated by a polygonal shape (Lopez et al. 2007) and elliptical or circular shape (Wang et al. 2015). The size distribution of concrete may be constructed based on an experimental sieving process. Alternatively, the grading of aggregate particle is designed by the Fuller curve which give an optimal density and strength of concrete mixture. The Fuller curve can be expressed as follows: D n (15) P(D) = 100 Dmax Where P(D) is the cumulative percentage passing a sieve with aperture diameter D. Dmax is the maximum size of aggregate and n is the exponent of the equations. Thus, for
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size, Di and Di+1 , then an interval Di , Di+1 defined by two sequentialsieve opening the area of aggregates within a grading segment Di , Di+1 can be calculated as: P(Di+1 ) − P(Di ) Aagg Di+1 − Di = Pagg A (16) P(Dmax ) − P(Dmin ) Where Aagg Di+1 − Di is the are of aggregate within the grading segment Di , Di+1 , Dmin is the minimum size of aggregate, Pagg is the area fraction of all aggregates and A is the total size of the concrete specimen. Regarding the simulation of the aggregate spatial distribution, the random sampling principle of Monte Carlo’s simulation method is used. This method is commonly called the take-and-place method. The random principle is applied by taking the aggregate sizes from a grading curve and placing each particle in the mortar matrix randomly so that intersection between aggregate and between aggregate and the boundary of the specimen is avoided. This method has been used by most researchers including (Bazant and Tabbara 1990), (Schlangen and Mier 1992).
4 Calculation of the Compression Strength of Porous Concrete The main purpose of this numerical example to demonstrate the potential of the phase field method to simulate the influence of porosity on the damage properties of porous concrete media. Figure 1 shows the geometry and boundary conditions for uniaxial tests. It consists of 50 mm x 50 mm square numerical specimens. The model is fixed at the bottom boundary and is subjected to a uniformly distributed displacement at the top boundary. In this study, the aggregate size distribution summarized in Table 1 is used. The aggregate particle whose size is greater than 5 mm is considered as coarse aggregate while fine aggregate together with cement matrix is treated as mortar. The interfacial transition zone between coarse aggregates and mortar matrix is neglected. Here, for the sake of simplicity, the coarse aggregate particles and pore media are geometrically represented by a circular shape. For normal strength concrete, the coarse aggregate represents around 40–50% the concrete volume. In this study, the area fraction of coarse aggregate is assumed to be equal to 40%. Coarse aggregates and mortar are described by linear elastic behavior. Young’s modulus is 70000 MPa for coarse aggregates and is 25000 MPa for mortar. Poisson’s ratio of both coarse aggregates and mortar is 0,2. Fracture energy is 0,06 N/mm for coarse aggregates and is 0,05 N/mm for mortar. The analyses are performed in plane stress condition and the out of plane thickness was unit. All analyses are ended at a displacement 0,05 mm. The computation is performed with monotonic displacement increments of 10–4 mm during 500 load increments. The length scale parameter is chosen as 0,35 mm. In this study, the domain does not contain pre-existing cracks, we cannot predict the crack nucleation and the crack pattern. Thus, to detect the crack nucleation, the domain is meshed by a regular triangular grid element whose characteristic size is about 0,15 mm. The total number of elements is 298921. Figure 2 represent random generation of aggregates resulted the Monte-Carlo Simulation inducted above. The results obtained in terms of final crack patterns are depicted
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in Fig. 3 (for the case of porosity being 10%). Their corresponding stress-displacement curves are plotted in Fig. 4 for several value of porosity. It is necessary to note that, in this exam, we verify possibility that we can use the Phase field method to model damage behavior of pervious concrete. In Fig. 3, macrocracks, starting from crack network, are predominantly perpendicular to the load direction. In Fig. 4, we receive interesting damage behavior curves of porous media with properties of pervious concrete. When porosity increase, tensile strength has corresponding smaller value. The important relationship between porosity and strength of pervious concrete could be obtained quickly with Phase field Method.
Fig. 1. Geometry of the specimen and boundary condition
Table 1. Sieve size parameter Sieve size (mm) Total percentage passing (%) 10
90%
5
15%
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Fig. 2. Sample before damage at porosity of 10%
Fig. 3. Predicted final crack pattern at porosity of 10%
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Fig. 4. Comparison of stress-displacement curves in uniaxial tension test for several porosity
5 Conclusions In this paper, we deal with the problem of modeling the damage and fracture behavior of porous concrete material. To do this, we construct a numerical approximation based on the phase field thermodynamic framework. Then, we are interested in only simple numerical tests composed of 2D configuration, circular aggregate, and no ITZ phase. In spite of this simplicity, some numerical results are shown to prove firstly that we can use Phase field method for further applications concerning the complex micro-structure of porous cement-based composite material. Acknowledgement. This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 107.02–2018.306.
Referencess Francfort, G., Marigo, J.: Revisiting brittle fracture as an energy minimization problem. J. Mech. Phys. Solids 46(8), 1319–1342 (1998) Li, X., Chu, D., Gao, Y., Liu, Z.: Numerical study on crack propagation in linear elastic multiphase composite materials using phase field method. Eng. Comput. 36, 307–333 (2019) Nguyen, T., Yvonnet, Y., Bornert, M., Chateau, C.C., Sab, K., Romani, R., Le Roy, B.: On the choice of parameters in the phase field method for simulating crack initiation with experimental validation. Int. J. Fract. 197(2), 213–226 (2016) Wang, X., Yang, Z., Yates, J., Jivkov, A., Zhang, C.: Monte carlo simulation of mesoscale fracture modelling of concrete with random aggregates and pores. Constr. Build. Mater. 15, 35–45 (2015) Lopez, C., Carol, I., Aguado, A.: Meso-structural study of concrete fracture using interface element i: numerical model and tensile behavior. Mater. Struct. 41, 583–599 (2007)
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Bazant, M. K. G. P.-C. Z.P., Tabbara, M.R.: Random particle model for facture of aggregate or fiber composites. J. Eng. Mech. 116, 1686–705 (1990) Schlangen, E., van Mier, J.: Simple lattice model for numerical simulation of fracture of concrete materials and structures. Mater. Struct. 25(156), 534–542 (1992)
Constitutive Response and Failure Mechanism of Porous Cement-Based Materials Under Triaxial Stress States Thang T. Nguyen1(B) , Lien V. Tran1 , Tung T. Pham1 , and Long N. Tran2 1 Faculty of Civil and Industrial Construction, National University of Civil Engineering, Hanoi,
Vietnam {Thangnt,lientv,tungpt}@nuce.edu.vn 2 Department of Civil Engineering, Vinh University, Vinh city, Nghe An, Vietnam [email protected]
Abstract. The constitutive response and failure mechanism of a highly porous cement-based material are generally controlled by the competition of the shear failure and pore collapse. These micromechanical processes are mainly dependent on the stress state. In this study, the pore-structure of the material is explicitly described by the discrete element modelling (DEM), the cohesive-frictional model and spherical particles are ultilised to model the mortar phase at the meso-scale. Numerical simulations of true triaxial tests are conducted on porous specimens with various stress paths. The mechanical data show that the transition, from brittle failure into cataclastic flow, is heavily dependent on the confining stress level. The insight into pore deformation and shear failure development is discussed. In addition, the octahedral section of yield surface of a typical highly porous material is presented in the paper. This feasible study demonstrates the potential of the DEM modelling in investigating the failure mechanism of the porous cement-based material under multi-axial tress state. Keywords: Porous cement-based material · Porous concrete · Triaxial · Constitutive behaviour · DEM
1 Introduction Recent years, light-weight materials such as porous concrete, porous rock are commonly used in building construction. The materials are recognised for being more environmental friendly and having advanced physical properties such as low cement ratio, less aggregate usage, very good thermal and sound insulation [1, 2]. Furthermore, different from the dense counterpart, the mechanical behaviour of a cement-based material with a high porous ratio is usually controlled by its micro-structure characteristics such as pore-structure, cement matrix and grain. Thus the material behaviour is shown to be more complicated. Further studies are needed to address many existing and potential challenges related to the material such as effects of varying density on its mechanical © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 935–944, 2022. https://doi.org/10.1007/978-981-16-3239-6_73
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properties or the understanding of microstructural differences between solid and porous cement-based materials. The numerical simulation of conventional cement-based has been developed for a long time using both continuum-based and discrete-based approaches. For example, numerous advanced numerical methods such as Finite Element Method (FEM) [3, 4], eXtended Finite Elements Method (XFEM) [5], Discrete Element Method (DEM) [6–9] or Hybrid Continuum-Discrete Element Method [10], the Smooth Particle Hydrodynamics (SPH) [11–13] have been developed and successfully applied to simulate cracking quasi- or quasi-brittle materials. However, since the porosity of porous cement-based materials is much higher than that of conventional counterpart, it is not reasonable to consider the material as a continuum or dense system. Thus, methods that can handle the discontinuous features of these materials, such as internal pore-size structure and fragmentation process, are generally suitable for the numerical simulations of this porous material. In this case, DEM shows to be a reasonable method to study the mechanical behaviour of highly porous cement-based materials [14–17]. To date, many experimental investigations on the mechanical behaviour of dense and low-porous cement-based materials have been made [18–21]. Nevertheless, the study of highly porous cementitious materials is still very limited, especially in the case of triaxial stress states. It may contribute to the fact that the material with a high porous ratio and complicated pore-structure is very sensitive to laboratory characterisations and thus challenging in keeping consistent on pore-structure for testing in many complicated stress states. In this sense, the paper aims to provide a numerical investigation on the constitutive response and fracture behaviour of a highly porous cementitious material with consistent pore-structure under various complicated stress states. For our DEM modelling, idealised porous cementitious specimens instead of realistic samples are considered in the study. The numerical results are then compared with the experimental data of similar realistic materials in the literature.
2 Porous Cement-Based Materials Under Triaxial Tests The aim of doing true triaxial tests for a porous cement-based material is to determine its mechanical behaviour under different 3D stress states which can represent the performance of the material in the field. Based on the test, the desired properties of the material can be obtained by either direct test or by calculation through theory. The triaxial test can be performed in either cylindrical or cube specimens as shown in Fig. 1. The cylindrical specimen is usually utilised in the conventional triaxial test, while the cubic specimen is used in the multi-axial test. In the conventional triaxial test, the confining stress is applied to all sides of the specimen. Then depending on the stress path, additional stresses will be applied on axial or horizontal directions. The two common types of tests have been considered are triaxial compression (σ1 > σ2 = σ3 ) and triaxial compression (σ1 = σ2 > σ3 ), where σ1 , σ2 , σ3 σ1 , σ 2 , σ3 are major, intermediate and minor principal stresses, respectively. This type of test helps to investigate the mechanical response and failure mechanism of the material. However, since the only axisymmetric stress states are simulated, the influence of the second principal stress is ignored in the conventional triaxial test. To remedy the shortcoming of the conventional triaxial test,
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Fig. 1. (a) Cylindrical specimen and (b) cube specimen for the triaxial test.
the multi-axial test or true triaxial test has been carried out in laboratory [19, 22–24] and in numerical modelling [25]. It is shown in the literature that the second principal stress can have a significant influence on the mechanical of behaviour porous cementbased materials [23, 24]. Under the multi-axial stress state, the stress-strain response of a sample can be changed owing to the variation of stress path, which can be controlled by an intermediate stress ratio or the Lode angle [22]. 2.1 DEM Simulation of Triaxial Tests A DEM specimen with the dimension of 50 × 50 × 100 (mm) is generated for the true triaxial test. The reasonable dimension could save the computational cost in the DEM simulation, while the effect of boundary condition is negligible [16]. The porous cement-based specimen has a porosity ratio of 68% corresponding to the material density of 800 kg/m3 . The pore structure of the material consists of both isolated air-pore and interconnected air-pore with an average radius of 2.61 mm. The loading is imposed through six frictionless facets in the DEM simulation (Fig. 2). The constitutive parameters for DEM modelling, which are pursued through a calibration and validation process in our previous publication [15], are utilised in this paper. The numerical modelling of the porous cement-based sample in this study includes hydrostatic compression test, conventional triaxial compression test, triaxial extension test and polyaxial test. To control the stress path, the stress ratio b and the Lode angle which are employed and defined as follows: σ2 − σ3 σ1 − σ3 √ 3 3 J3 cos(3θ ) = . 2 J 3/2 b=
(1) (2)
2
Where J2 is the second invariant and J3 is the third invariant stress which are expressed as follows: 1 (3) J2 = (σ1 − σ2 )2 + (σ2 − σ3 )2 + (σ1 − σ3 )2 6
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J3 =
1 (2σ1 − σ2 − σ3 )(2σ2 − σ1 − σ3 )(2σ3 − σ2 − σ1 ) 27
The relation between b and θ is expressed as follows: √ b 3 tan θ = 2−b
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The stress ratio b varies from 0 to 1. When b = 0 hence σ1 > σ2 = σ3 σ1 > σ2 = σ3 and θ = 0, the sample is under triaxial compression test. When b = 1 then σ1 = σ2 > σ3 and θ = π 3, the sample is under triaxial extension test. In other cases, the polyaxial test is conducted. The triaxial compression and extension tests help to build the meridian curves, while the polyaxial test assists in drawing the deviatoric shapes of yield and failure surfaces. In this study, the stress ratio b is set to 0, 0.2, 0.4, 0.6, 0.8, 1.0 respectively to create the yield in the deviatoric plane.
Time
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Fig. 2. DEM simulation of the triaxial test of porous cement-based materials
The mean stress (p) and differential/deviatoric stress (q) are defined as follows p=
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1 q = √ (σ1 − σ2 )2 + (σ2 − σ3 )2 + (σ1 − σ3 )2 2
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In the conventional triaxial test (σ1 > σ2 = σ3 ), the differiental stress is simply calculated as the following equation q = σ1 − σ2
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2.2 The Influence of Confining Stress The conventional triaxial test is conducted to study the failure behaviour of the porous materials under different confining stress levels. The isotropic pressure is first applied and the pressure is increased until it reaches the design value. After reaching the predefined lateral stress level, the major principal stress is increased by placing the vertical displacement through the top and bottom facets, while the confining stresses are kept constant to perform the test by DEM simulation. The stress-strain behaviour, failure pattern and pore deformation development of the specimen under three levels of confining stress, p = 0.1 Mpa, p = 4 Mpa and p = 8 Mpa, which corresponding to the low confining stress, average confining stress and high confining stress levels are illustrated in Fig. 3, Fig. 4 and Fig. 5, respectively. It is noted that Fig. 3a shows the fracture pattern of the specimen in the 2D format at the middle plane, while the development of pore deformation is fully demonstrated in 3D in Fig. 3b. The three typical points (1), (2) and (3) are selected for demonstrating the fracture pattern and pore deformation with respect to elastic stage, failure point and residual stage, respectively. A similar illustration is applied for average confining stress and high confining stress tests.
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Fig. 3. The stress-strain response, failure pattern (a) and the development of pore deformation of a porous cement-based material under low confining stress (p = 0.1 Mpa).
The deviatoric stress-axial strain relation in Fig. 3a indicates that the specimen experiences a brittle-like behaviour under the low confining stress level. In particular, the linear elastic behaviour is first presented and continues until the sample reaches its yield stress. The second stage is non-linear hardening behaviour, where the stress monotonically increases up to the failure stress. Beyond the failure stress, the softening behaviour is observed, in which the stress drops significantly to the residual level. The stress-strain response is shown to be a typical behaviour of brittle-like materials. The shear failure is dominant in the sample, in which the shear band is clearly seen once sample reaching its failure stress. These numerical observations strongly agree with experimental results of porous rocks, which failure under low confining stress [26]. Regarding the pore deformation, only air-pores sitting next to the shear band experiences little deformation during the hardening stage. The other pores reforms shortly as two parts of the sample slide from another during the softening and residual stages.
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Fig. 4. The stress-strain response, failure pattern (a) and the development of pore deformation of a porous cement-based material under average confining stress (p = 4.0 Mpa).
The mechanical behaviour of the sample at an average confining stress level (p = 4.0 Mpa) is shown in Fig. 4. The four typical stages including linear elastic stage, non-linear hardening stage, non-linear softening stage and residual stage are still well presented. However, in comparison to the previous test, the more ductile behaviour is generally observed. In particular, there is only a small drop in the differential stress after the sample reaches its ultimate strength and the stress gradually decreases to the residual stress level. It can be seen that lateral stress can have a profound influence on the failure response of the material. In particular, there is a transition from brittle behaviour into ductile behaviour following an increase of confining pressure. Consequently, the air-pores deform significantly during the test as can be seen in Fig. 4b. It is shown that the failure mode of the sample is a result of the competition between pore collapse and shear failure.
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Fig. 5. The stress-strain response, failure pattern (a) and the development of pore deformation of a porous cement-based material under high confining stress (p = 8.0 Mpa).
Figure 5 illustrates the mechanical response of the porous cemented-based material under a high confining stress level. Different from the two mentioned tests, the stressstrain curve of the sample shows the typical response of ductile behaviour. In particular, there is almost no softening stage and the residual stage is observed after the sample reaching its ultimate strength. In contrast with the failure pattern of the sample under low
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confining stress where the shear failure is dominant, the pore collapse is prevalent in case of high confining test. The pore deformation is significant and increases considerably after the sample reaches its failure stress. The above numerical DEM modelling investigates the effect of confining stress on the failure mechanism of porous cement-based materials. This suggests that the confining stress can have a profound influence on both stress-strain response and failure mode of the material. In particular, the sample shows the typical behaviour of brittle-like materials under low confining stress, whereas ductile behaviour is clearly seen in the test with a high level of lateral confining. Moreover, the brittle-ductile transition is taken within the sample under an average confining stress level. The numerical results are matched well with the existing experimental investigation about the porous cement-based materials such as pours rocks, porous sandstones or porous limestone [18–21]. For example, Scott and Nielsen [21] showed that porous sandstone experienced the brittle regime under low confining stress, whereas the material changed into ductile behaviour at a high confining stress level. The transition from brittle to ductile response was also observed in those laboratory tests at an average confining stress level. 2.3 Influence of Intermediate Principal Stress on Octahedral Plane The previous section has investigated the effect of lateral pressure on the mechanical behaviour of a porous cement-based material. Although the relationship between the stress state and the failure response of the sample has been well investigated, this only considers the symmetric problem (σ1 = σ2 ). Thus, the influence of the second principal stress or Lode angle is ignored. In reality, anisotropic stress states can take place in many practical applications such as in tunnel excavation [27–29]. Consequently, the influence of the second principal stress on the behaviour of the materials needs to be considered. For these above purposes, DEM numerical modelling of a polyaxial tress test is presented. The procedure of setting up the polyaxial is similar to the conventional triaxial test, the only difference is in the stress path which can be controlled by the stress ratio b and the Lode angle as mentioned in Eq. (1) and (2). Depend on the stress ratio b, triaxial compression (0.0 ≤ b < 0.5), triaxial extension (0.5 < b ≤ 1.0) or simple shear test (b = 0.5) are performed (see Fig. 6). In this study, the variation of the mean stress (p) is taken into account. Accordingly, the numerical results are plotted to study the effect of mean stress on the octahedral plane of the yield surface of the material. Figure 7 shows the octahedral plane of the material at four mean stress levels ranging from low mean stress level p = 0.5fc to high mean stress level p = 4fc , which fc = 2.5 Mpa is the yield stress of the sample under a uniaxial test. Accordingly, a considerable variation in the octahedral plane shape can be observed. In particular, at a low mean stress level, the shape is a rounded triangle with the vertice upward, which mean the third invariants of the deviatoric stress have a profound influence on the yield surface. As the mean stress increases to an average level, the octahedral section transforms into a relatively rounded shape. Consequently, the effect of the Lode angle is negligible at this stress level. Nonetheless, the octahedral section back to the rounded triangle again but with the vertice downward, suggesting the significant influence of the intermediate
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Fig. 6. Polyaxial test, stress path controlling and octahedral plane
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Fig. 7. Yield surface’s octahedral plane at different levels of mean stress.
principal stress at a high mean stress level. The DEM simulation results show to have a good agreement with experimental studies of porous rock in the literature [23].
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From these above investigations, it is suggested that not only the first and second stress invariants but also the third stress invariant J3 have to be considered in the function of the yield surface of the porous cement-based material.
3 Conclusion The discrete element modelling of conventional triaxial test and polyaxial test have been performed in the study to investigate the influence of confining stress on the constitutive response and failure mechanism of the material, and then the effect of the second principal stress on the octahedral plane of the yield surface. From the numerical investigations, some conclusions are made as follows: i.
The confining stress can have a profound influence on both stress-strain response and failure mode of the material. The material behaves as a brittle-like material under low confining stress, whereas the ductile behaviour is observed at a high level of lateral confining. Also, the brittle-ductile transition is taken within the material under an average confining stress level. ii. The shape of an octahedral section of porous cement-based material depends on mean stress level and the immediate principal stress. It is suggested that the third stress invariant must be presented in the function of the yield surface of the porous cement-based material. iii. Different classical failure criteria can be applied to model the yield and failure surface of porous cement-based materials. Each criterion is then utilized for a particular octahedral yield envelope. It is suggested that one single model is not enough to fit the variety of octahedral shapes for the given material.
Acknowledgements. This research is funded by Ministry of Education and Training for Science and Technology project under grant number: B2021-XDA-05.
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Development of Automatic Landing Control Algorithm for Fixed-Wing UAVs in Longitudinal Channel in Windy Conditions Hong Son Tran1(B) , Duc Cuong Nguyen2 , Thanh Phong Le3 , and Chung Van Nguyen4 1 The Faculty of Control Engineering, Le Quy Don Technical University, Hanoi, Vietnam 2 Vietnam Aerospace Associations, Hanoi, Vietnam 3 The Faculty of Control Engineering, Le Quy Don Technical University, Hanoi, Vietnam 4 Air Force Officers College, Nha Trang, Vietnam
Abstract. Nowadays, to develop the automatic landing system in windy conditions is still a significant challenge for scientists. The improvement of the automatic landing algorithm of fixed-wings UAVs will expand possible wind speed range that allows safe automatic landing. This paper presents the development of an automatic landing control algorithm for fixed-wing UAVs in a longitudinal channel in windy conditions based on the JAR-VLA standard [1, 2]. The simulation results using Simulink tools of MATLAB show that i) the proposed algorithm satisfies the landing requirements for fixed-wing UAVs in strong turbulence, and ii) the proposed algorithm can be used in developing an automatic landing controller of UAVs. Keywords: Fixed-Wings UAVs · Automatic landing · Windy condition · Improvement of PID controller
1 Introduction In today’s modern world, fixed-wing UAVs bring the benefits in different fields of life such as agriculture, scientific research, mapping, border patrol, and especially, promising leaps in military applications) [3–5]. The UAV flight consists of different phases: take off, climb, cruise, descent and landing. The landing phase is the most difficult in the UAV’s operational phases because any error occurring during this phase can lead to UAV accident. That is why the problems of automatic landing techniques have been developed and applied by many researchers. Over the last several decades, a number of methods including dynamic inversion, model predictive control, adaptive sliding mode control and backstepping control have been used to design UAV automatic landing system [6–9]. In addition, many authors have studied intelligent systems such as fuzzy logic, neural networks, or control optimization of the parameter [10, 11]. However, most of the study above does not mention constructing a predetermined wind model close to the reality of the landing process but only use the wind model available in Matlab © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 945–958, 2022. https://doi.org/10.1007/978-981-16-3239-6_74
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Simulink (The Dryden wind turbulence model [12]) for their investigating. The Dryden Wind Turbulence Model (Continuous) block uses the Dryden spectral representation to add turbulence to the aerospace model by passing band-limited white noise through appropriate forming filters. But we can not investigate controller in “extreme” points (the extreme point is when wind amplitude reaches a maximum) in this wind model. Therefore, in this study, we focus on the improving the UAV’s automatic landing control PID controller and evaluating its effectiveness in a predetermined wind model, which was developed by the authors. Predetermined wind model has been proposed based on the JAR-VLA wind standard, which is the standard for estimating ultra-light aircraft’s flight safety. Finally, simulation results are presented to demonstrate the effectiveness of the improved controller. 1.1 Effects of Turbulence on UAV’s Motion in the Longitudinal Channel As known, when describing the landing process of the UAV in the vertical plane with wind effects, it is necessary to calculate and distinguish between the air-speed and the ground speed and determine the value of the attack angle in the motion equation of UAV [13], which be shown in the system of differential Eqs. (1) ⎧ dVg ρ.Va2 ⎪ ⎪ ⎪ m dt = T cos α −CD . 2 .S − G sin θ ⎪ ⎪ ⎪ ⎪ mV d θ = T sin α + C + C z .ω . ba + C δe .δ . ρ.Va2 .S − G cos θ ⎪ g z e y y ⎪ L Va 2 ⎪ ⎨ dt 2 ba z α α . ρ.Va .S.b .ω . + m Jz ddtωz = mδz e .δe + m (1) z Va a z z 2 ⎪ ⎪ ⎪ dxo = Vg cos θ ⎪ dt ⎪ ⎪ dyo ⎪ ⎪ = Vg sinθ ⎪ ⎪ ⎩ ddtϑ = ωz dt Where: m- The mass of UAV; Vg - Ground Speed; Va - Air Speed; T - Propulsive Force; α- Angle of attack; θ - Flight path angle; ϑ- Pitch angle;CD - Drag coefficient;CL - Lift coefficient; ρ- Air density; S- Reference area (of the wing); δe - Elevator deflection angle; δe α z ωz - Pitch rate; ba - Reference length (mean aerodynamic chord); Cyz ;Cyδe ;m z ;mz ;mz Aerodynamic derivatives coefficient (z = ωz Vbaa -Pitch rate dimensionless); Jz - Moment of inertia of UAV (pitch channel). We will divide the wind speed vector into the vertical and horizontal components, which is showed in Fig. 1.
Fig. 1. Wind of arbitrary direction in the vertical plane and its vertical and horizontal components
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Relied on Fig. 1, we have the following expressions: Wx = W cos θw Wy = W sin θw
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− → Thus, if we have the wind speed ( W) and its direction related to the horizontal plane (θw ), we can completely determine the wind components in the vertical and horizontal directions. On the other hand, when considering the relationship between air-speed, ground speed and attack angle in the system of Eqs. (1), we will analyze in the ground reference system (Fig. 2).
Fig. 2. Relationship between speed components in the vertical plane
If we have already known the angle of the ground speed compared to the horizontal plane, we will divide the wind vector into two vertical and horizontal components as follows: Vgx = Vg cos θ (3) Vgy = Vg sin θ − → − → Similarly, after having the components of wind speed ( W) and ground speed (Vg ) − → analyzed, we can calculate the value of the air-speed (Va ). ⎧ ⎪ ⎪ ⎨ Vax = Vgx − Wx ; Vay = Vgy − Wy Va = V2ax + V2ay (4) ⎪ ⎪ ⎩ tgθ = Vay ; α = ϑ − θ a
Vax
a
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After that, we can complete the right part of the system of Eqs. (1) and we have a new system of motion Eq. (5) of UAV considering the wind turbulence effects. ⎧ dVg ρ.Va2 ⎪ ⎪ ⎪ m dt = T cos α −CD 2 .S − G sin θ ⎪ ⎪ 2 ⎪ ⎪ mVg d θ = T sin α + C + Cyz .ωz . ba + C δe .δe . ρ.Va .S − G cos θ ⎪ L L ⎪ dt V 2 a ⎪ ⎪ 2 ⎪ δe z d ωz ba α α . ρ.Va .S.b ⎪ ⎪ = m J .δ + m .ω . + m z e z a z z z ⎪ dt Va 2 ⎪ ⎪ dx ⎪ o ⎪ ⎪ ⎨ dydt = Vg cos θ o (5) dt = Vg sinθ ⎪ Vay dϑ ⎪ = ω ; α = ϑ − θ ; θ = arctg ⎪ z a a ⎪ dt Vax ⎪ ⎪ ⎪ ⎪ = V cos θ V gx g ⎪ ⎪ ⎪ ⎪ ⎪ Vgy = Vg sin θ ⎪ ⎪ ⎪ Vax = Vgx − Wx ; Vay = Vgy − Wy ⎪ ⎪ ⎪ ⎪ ⎩ V = V2 + V2 a
ax
ay
In fact, in the area with tropical climate [14], due to the sunlight impact on different surfaces, the surfaces convection air flow near to the ground are produced, which form wind spaces are almost circular figures indicated in Fig. 3.
Fig. 3. Model of convection wind in tropical climate [14]
The wind circles are assumed to be equal. If we combine the wind models exposed vertically and horizontally to cover the UAV’s landing space, we will create a predetermined wind space with the form in Fig. 4. The number of vertical and horizontal wind circles depends on turbulence scale.
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Fig. 4. Convection wind model using modified JAR-VLA model standard
Where: A- Beginning point of landing; B- Endpoint of landing (see 3). The author proposes to apply the modìied JA-VLA standard in a single wind circle model. JAR-VLA wind model can be used to very light aircraft according to European standards and it has the form: Wy =
2π(x0 − x0∗ ) W0 (1 − cos ) 2 L
(6)
Where: W0 - Wind amplitude (m/s); x0 − x0∗ - Range of flight when wind occurred (m); L- Turbulence scale (m). However, this is an asymmetric model and the mean value in the whole wind field is a non-zero because this standard was used for the cruise phase. Therefore, in order to match the wind model in this study wind of arbitrary direction in the vertical plane), we propose to modify the wind model JAR-VLA to symmetrical form with the following model: W = W0 cos
2π(x0 − x0∗ ) L
(7)
The modified JAR-VLA wind model has the following form: In the JAR-VLA wind model, the object’s motion will be perpendicular to the wind vector. So if we get the turbulence scale (L), the range of flight when wind occured (x0 − x0∗ ), the amplitude wind (W0 ) we will calculate the wind vector’s amplitude at any time, according to Eq. (7). The structure of a single wind circle model in which modified JAR-VLA model is shown in Fig. 6. Thus, when the UAV flies on a straight line through the center of the circular wind model, the form of wind model fits the modified standards.
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Fig. 5. Modified JAR-VLA wind model
Fig. 6. Structure of the proposed single wind circle model
2 Development of Automatic Landing Control Algorithm for Fixed Wing UAVs in Longitudinal Channel in Windy Conditions In this study, we consider only the UAV motion from the beginning point of landing (the UAV was aligned itself with the runway in the lateral direction) to a touchdown point in the runway. However, when we research the landing of UAV in real condition, we assume that the distance from the beginning point of landing to the touchdown point is about 1000 (m); Initial altitude is 100 (m). The touchdown point’s choice depends on the structural characteristics and the safety of the UAV in consideration. Besides, the center of gravity of the UAV is about 1 (m) from the ground. We also assume that the touchdown point is about 20 (m) from the head of the runway. We assume that the reference trajectory is a parabola. The motion process of the UAV is shown in Fig. 7. Thus, the construction of a reference trajectory is a parabola with the beginning point being at the landing preparation time, the endpoint being the touchdown point. And for the UAV to follow this reference trajectory, it is necessary to ensure that the flight path angle must follow a given programe in control algorithm. Both the trajectory of these two parameters are presented in Fig. 8.
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Fig. 7. The process of the UAV motion according to the reference trajectory
Fig. 8. UAV landing reference trajectory
From the above geometrical parameters, we have a system of equations describing the reference trajectory of UAV and the function θ (x) as follows: 11 11 h = 115600 x2 − 2890 x + 300 289 (8) 1 θ = 10 x−2 In this study, the authors propose to use the classical PID controller, which is a powerful controller, to compare with the suggested improved controller. The PID controller parameters (Kp , Kd , Ki ) were optimized using the tool “Signal Constraint” in the optimization tool “Simulink Design Optimization” of Matlab Simulink software. On the other hand, if we see Fig. 8, the PID controller is good when altitude error and flight path angle also reach zero. From there, we have the control law following the reference trajectory in the vertical plane shown in Eq. (9)
t δe = Kp h + Kd θ + Ki
hdt+δe_bb 0
Where:
(9)
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h = hcur − hdes - Altitude error (m); hcur - Current altitude (m); hdes - Desired altitude (m);
θ = θcur − θdes - Flight path angle error (rad); θcur - Current flight path angle (rad); θdes - Desired flight path angle (rad); Kp - Proportional gain (rad/m); Kd - Derivative gain (-); Ki - Integral gain (rad/m); δe_bb - Level flight elevator deflection angle (rad); Based on the control rule (9), we have a vertical channel closed loop diagram of the UAV during landing using the PID controller as follows:
Fig. 9. Block diagram of closed loop control of UAV in longitudinal channel
Although the optimal gains have been selected using the Signals Constraint tool in Matlab Simulink, the PID controller still faces the significant errors when the turbulence occurs. It is the process of finding the values of proportional, integral, and derivative gains of a PID controller to follow desired performance. In addition, during landing, we find that it needs a normal load factor signal to generate the UAV control signal for the UAV to deviation from its reference trajectory. On the other hand, we have built the reference landing trajectory of the UAV. From this reference trajectory, we have determined the desired altitude parameters (hdes ), the desired flight path angle (θdes ). Then we can determine the desired load factor parameter during the landing of the UAV by the expression: nydes = cosθdes . Thus, from the system of Eqs. (5), we need to add a current load factor to complete the program to improve the quality of the PID controller. Based on Eq. (2) of the system of Eqs. (5), we have the expression for calculating the current load factor of the UAV as follows: ny =
Vg .θ˙ + cosθ g
(10)
Thus, the control law following the reference trajectory when adding a load factor component has the following form:
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Where: ny = ny cur − ny des - Overload error;
t δe = Kp h + Kd θ + Ki
hdt+Kny ny + δe_bb
(11)
0
The gains of the improved PID controller were also selected by the same tools as classical PID. From the control rule (11), we have a block diagram of closed loop control of the UAV, which shown in Fig. 10 in the longitudinal channel.
Fig. 10. Improved vertical UAV vertical control loop diagram
Comment: The improvements noted in the new algorithm (11) are the addition of components from the reference trajectory according to normal load against deviation factor Kny nytt − nymm based on the current trajectory parameter, precisely determined by the modern sensor. Therefore, additional components against trajectory deviation help the UAV follow the reference trajectory with higher accuracy. And the improved control algorithm will still achieve high accuracy when the wind is impacted, which will be demonstrated in the simulation results.
3 Simulation Results and Conclusion Assuming the initial and ⎧ the final condition: ⎪ ⎨ h0 = 100 (m)
⎧ ⎪ ⎨ ht = 1 (m) Initial condition: x0 = −1000 (m) ; End condition: xt = 20 (m) ⎪ ⎪ ⎩ ⎩ va = 40 (m/s) vy = 0 (m/s) Classical PID parameters: Kp = 0,0078; Kd = −0,1911; Ki = 6,408e − 004. Improved PID parameters: Kp = 0,1634; Kd = −0,6959; Ki = 0,0035; Kny = 0,1333.
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Section No
Parameter
Value
Unit
Section No
Parameter
Value
Unit
−16.23
[-]
−2.2144
1/rad
−1.4798
1/rad
01
Length (l)
2707
Mm
06
02
Mass (m)
56.5
kg
07
03
Hight (h)
713
mm
08
mzω¯ z mδz c mαz
1.05
m2
9
CL
5.9123
1/rad
10
Cyδc
0.6126
1/rad
04 05
Wing area (S) Wingspan length (l a )
3000
mm
To evaluate the classical PID controller’s efficiency with optimized gains and improved PIDs, the authors propose investigating at times that the wind module reaches a maximum (upwind, downwind, headwind, tailwind) at different disturbance patterns. Case 1: L = 12 (m); W0 = 4 (m/s).
Fig. 11. Simulation results for upwind maximum case
Comment: In general, the improved PID controller’ control quality is better than that of the classic PID controller during the whole landing process, which is shown on the simulation results above. In all cases, the improved PID controller’s maximum altitude error is 0,05 (m) (Fig. 11, Fig. 12, Fig. 13 and Fig. 14) in the case where the touchdown point coincides with the rising wind velocity phase is the greatest, the different cases are below 0,04 (m) (Fig. 12, Fig. 13 and Fig. 14). In contrast, the classical PID controller altitude error varies widely under different circumstances. When the touchdown point coincides with the phase of the downwind and headwind is maximum, the altitude error reaches about 0,19 (m) (Fig. 11, Fig. 12, Fig. 13 and Fig. 14). The vertical speed of the two controllers did not have much difference in all cases. Through 4 cases, we see that in the case where the touchdown point coincides with the phase of the headwind speed is
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Fig. 12. Simulation results for downwind maximum case
Fig. 13. Simulation results for tailwind maximum case
Fig. 14. Simulation results for headwind maximum case
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the largest that significantly impacts the quality of the controller. To verify more clearly, we consider the next case with more incredible maximum wind speed in case 1 at the maximum headwind phase. Case 2: L = 12 (m); W0 = 5 (m/s).
Fig. 15. Simulation results for case headwind maximum
Comment: When the wind intensity continues to increase, we see that the improved PID controller’s control quality remains stable with a small altitude error and when it hits the ground about 0,05 (m). By contrast, for the classical PID controller, the altitude error is 0,22 (m) at the moment of touchdown. For the vertical speed, the classical PID controller and the improved PID controller are also ensured. Thus, with the same scale of disturbance, the change in wind amplitude leads to a large change of the controller in the whole operation process; To verify the disturbance scale’s effect, we consider the following case. Case 3: L = 14 (m); W0 = 6(m/s). Comment: When increasing both the turbulence scale and wind amplitude, we find that the difference between the improved PID controller and the classical PID controller is rising significantly. The altitude error tends to increase much more vital and at the time of the landing moment is 0,35 (m) of the classic PID controller compared to 0,05 (m) of the improved PID. The vertical speed parameters, in this case, are not too different and approximately 0,6–0,7 (m/s). Conclusion: In general, the wind turbublence scale and the maximum wind amplitude affect the UAV’s flight performance during the landing phase. However, although the optimization tool was used to select the gains in the controller at the moment of landing, the parameters were not guaranteed to reach the desired value when at a larger range of disturbances. The investigation results show that using an improved PID controller has significantly increased control quality under any wind in the vertical plane, especially
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Fig. 16. Simulation results for case headwind maximum
under the conditions of the disturbance scale and strong intensity wind based on modified JAR-VLA standard. In future work, the authors propose to study new control laws to apply in our research and expand the range of wind turbublence in the control process.
References 1. JAR-VLA: Joint Airworthiness Requirements for Very Light Aeroplanes (1990) 2. Dang, C.V.: Application of an adaptive control enhance flight safety for small-sized UAV in turbulence conditions. J. Mil. Sci. Technol. Spec. Issue (2016) 3. Herwitz, L., et al.: Imaging from an unmanned aerial vehicle: agricultural surveillance and decision support. Comput. Electron. Agric. 44(1), 49–61 (2004) 4. Nagai, T., Chen, R., Shibasaki, H., Kumagai, A.A. : Uavborne 3-D mapping system by multisensor integration. Geosci. Rem. Sens. IEEE Trans. 47(3), 701–708 (2009) 5. Girard, R., Howell, A.S., Hedrick, J.K.: Border patrol and surveillance missions using multiple unmanned air vehicles. Decis. Control 1, 620–625 (2004) 6. Shashiprakash, S., Padhi, R.: Automatic landing of unmanned aerial vehicles using dynamic inversión. In: Proceedings of the International Conference on Aerospace Science and Technology, (2008) 7. Mathisen, S.H., Gryte, K., Johansen, T., Fossen, T.I.: Non-linear model predictive control for longitudinal and lateral guidance of a small fixed-wing UAV in precision deep stall landing (2016) 8. Zheng, Z., Jin, Z., Sun, L., Zhu, M.: Adaptive sliding mode relative motion control for autonomous carrier landing of fixed-wing unmanned aerial vehicles. IEEE Access 5, 5556–5565 (2017) 9. Lungu, M.: Auto-landing of fixed wing unmanned aerial vehicles using the backstepping control. ISA Trans. (2019) 10. Nho, K., Agarwal, R.K.: Automatic landing system design using fuzzy logic. J. Guidance Control Dyn. 23(2), 298–304 (2000) 11. Ambati, P.R., Padhi, R.: Robust auto-landing of fixed-wing UAVs using neuro-adaptive design. Control Eng. Pract. (2016)
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12. Kargin, V.: Design of an autonoumous landing control algorithm for fixed wing UAV. Middle East Technical University (2007) - ng cua các khí cu bay tu, dô 13. Cuong, N.D.: Mô hình hóa mô phong chuyên dô . . . . ng (in Vietnamese) (2002) 14. .P.Dobpolencki. Dinamika polta v necpokono atmocfepe (in Russian) (1969) ij
ij
ij
An Assessment of Terrain Quality and Selection Model in Developing Landslide Susceptibility Map – A Case Study in Mountainous Areas of Quang Ngai Province, Vietnam Doan Viet Long1 , Nguyen Chi Cong1 , Nguyen Tien Cuong2 , Nguyen Quang Binh1 , and Vo Nguyen Duc Phuoc1(B) 1 University of Science and Technology, The University of Danang, Danang, Vietnam
{dvlong,nccong,nqbinh,vnducphuoc}@dut.udn.vn
2 Faculty of Vehicle and Energy Engineering, Phenikaa University, Hanoi 12116, Vietnam
[email protected] Abstract. Landslide is one of the most common natural disasters in mountainous area of Vietnam. Therefore, studying and developing a landslide susceptibility map would make a significant contribution to local authorities in taking initiative in landslide prevention and mitigation. The quality of input data and the choice of model building methods are two very important impacted factors to accuracy of produced maps. This study will focus on investigating the influences of terrain data, which is a significant causative factor on landslides, to select the most appropriate DEM model. The DEM will be combined with other impact factors to develop landslide susceptibility assessment by applying two landslide spatial analysis methods: Analytic Hierarchy Process (AHP) and Frequency Ratio (FR). Conducting the investigation with three free sources of Digital Elevation Models (DEMs) in mountainous areas of Quang Ngai province has shown that NasaDEM performs better than the other DEMs (TanDEM-X90 and STRM). For this purpose, a total of 339 landslide locations was collected in this area, and then randomly split into two parts to generate training (70%) and testing (30%) datasets for construction and validation of the map, respectively. In addition, seven affecting factors were selected, including slope, aspect, soil types, land use, distance to roads, distance to rivers, and rainfall for developing the models. Validation of the maps was done using two performances indexes namely Untainted Area Under the Curve (AUC) and Landslide Density (LD). The results show that that two methods are appropriate for producing landslide susceptibility maps. Meanwhile, analyzing using the FR would get better AUC (0.793 > 0.747) and LD index compared to the AHP. Keywords: Digital elevation map · Analytic hierarchy process · Frequency ratio · Area under the curve · Landslide density · Quang Ngai
1 Introduction In many pieces of research which aim to assessment and product construction of landslide risk maps, digital elevation models (DEMs) is one of the most important factors. DEMs © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 959–970, 2022. https://doi.org/10.1007/978-981-16-3239-6_75
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are concerned as an initial input to extract and estimate different model parameters, such as slope, aspect, curvature, elevation for landslide susceptivity [1–3]. These factors have leads significant impacts on the landslides and have been commonly used in many pieces of research. Recent review from Pourghasemi et al. [4] have indicated that slope, aspect, and elevation are in the top five impact factors which have been often applied in the most of landslide related research. Among these three factors, the slope is the most popular one with 94.2% usage. The review paper by Reichenbach et al. [5] conducted based on 565 papers related to the landslide susceptibility theme published from 1983 to 2016 also agreed with that finding. There are many landslides related variables which can be divided into 5 clusters including geological, hydrological, land cover, morphological and other. Among them, the morphological group which is producted from DEMs accounting for the highest proportion with 25.5%. Reichenbach et al. [5] also pointed out that elevation, slope, aspect, and curvature are in the top popularly used factors. There are many DEMs in different resolution have been utilized in landslide susceptibility map developing which can be classified into two major group consist of contour DEM [1, 3, 6] or satellite DEM [7–10]. There are many different high-quality satellite DEMs that have been developed and freely distributed to users. These data are considered as an alternative solution to study landslide susceptibility for the places or regions where the contour DEMs are not readily available. Several DEMs that have been widely used in landslide susceptibility researches include TanDEM-X DEM (12 m resolution and resampled to 30 m), ASTER DEM (30 m), SRTM DEM (30 m), NasaDEM (30 m). The influences of different DEM sources and resolutions to the accuracy of landslide susceptibility models have been investigated by several researchers, e.g., see [7, 8]. Brock et al. [7] applied different satellite DEM sources (including TanDEM-X DEM, ASTER DEM, SRTM DEM, and an interpolated DEM with 25 m resolution from a map with a scale of 1: 25.0000 to build a landslide susceptibility map based on machine learning methods. Results showed that the TanDEM-X (12 m) yielded the highest AUC while the Aster DEM provided the lowest accuracy. Chen et al. [8] has assessed the influence of different DEM resolutions (i.e. 30, 40, 50, 60, 70, 80, and 90 m) extracted from Aster DEM to the landslide susceptibility mapping. The results have indicated that the DEM with 70m resolution yielded the highest accuracy. It has been shown that the optimum DEM resolution depending on the characteristics of the study region and the landslide hazards [7]. In summary, the quality of a DEM has a great impact on the quality of the landslide susceptibility map. Hence, it is necessary to assess and select a suitable DEM before applying to project a landslide susceptibility model. This research would pay concentrate to evaluate recently free high resolution DEMs such as NasaDEM (2020), MERIT DEM (2018) and SRTM. Many mountainous regions in Vietnam are often affected by landslide hazards every year, especially, rainy and flood seasons. The fact that DEMs measuring projects in developing countries like Vietnam often take place in metropolitan areas. Whereas, the elevation data sources are often inadequate for the mountainous areas where landslides usually occur. Besides, in developing countries like Vietnam, the topography maps are subject to changes quite often due to the execution and operation of many kinds of activities (i.e. exploitation of natural resources, road constructions, and so on). Therefore, there are large bias and unreliable existing in topographic in mountainous area, which
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results in unpredictable uncertainty of landslide hazard projection. Hence, the assessment and application of satellite DEMs product which is often updated to reflect changes are the trends and alternative approaches for the regions lacking measured data [7, 8]. In many studies of landslide susceptibility, in addition to the DEM input, the modeling method is also a key. The models used for assessment of landslide risks are often categorized into 2 groups (i.e. quantitative and qualitative) or 3 groups (heuristic, statistical, and deterministic models) [4]. The heuristic and statistical models often yield good results for large study areas while the deterministic models require more data inputs and is often suitable for the local or small areas [4, 5]. [4] showed that the Analytic Hierarchy Process (AHP) method (one of the heuristic models) and Frequency Ratio (FR)method (one of the statistical models) are considered well-performed and often rated as top five of the most appropriate methods in the study of landslide susceptibility assessment and management. Based on the importance of DEMs and the landslide susceptibility model, this paper presents the analysis and assessment of different satellite DEM sources, including NasaDEM, TanDEM_X90, and SRTM. The most appropriate DEM is selected for landslide susceptibility models using the FR and AHP methods. The Performances indexes - Area Under the Curve (AUC) and Landslide Density (LD) criteria were used to evaluation of the performance of different models and select the most suitable model based on case studies of different mountainous areas in the Quang Ngai province, Vietnam.
2 Methodology 2.1 Study Area Quang Ngai province which located in the central region of Vietnam (see Fig. 1) (14°46 55 ÷ 15°10 5 N, 108°13 48 ÷ 108°46 37 ) was selected as the case study for the research. This study focuses on three mountainous areas covers an area of 1.352 km2 . The west part of the study area is adjacent to Central Highland provinces on the Truong Son mountain range with the highest altitude at 1694 m. The eastern of the study area is next to the lowland area of Quang Ngai Province [11]. This region is highly vulnerable to landslides as many landslide incidents happened recently [11]. In this area, landslides usually happen from September to December along with the appearances of tropical storms and tropical monsoon, accounting for accounting for over 70% rainfall of 2500 mm annually) [11]. The methodology of this paper is as follows: (i) DEM analysis (NasaDEM, TanDEM_X90, and SRTM) and selecting the best DEM; (ii) Database preparation for landside susceptibility assessment, including: landslide inventory and landslide causal factors; (iii) developing landslide models (AHP, FR) to assess landslide susceptibility; (iv) validating and producing the landslide susceptibility map. Detailed information of the methodology can be seen in flow chart of Fig. 2.
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Fig. 1. Study area and recorded landslides
2.2 Data Collection a. Landslide inventory Landslide inventory map of this area was produced by mapping the landslide sites which were collected from many sources, including: satellite images, published reports and field surveys [11]. A total of 339 sallow landslide sites were collected in this area, they were then randomly splitting in two parts to generate training (70% landslide sites) and testing (30% landslide sites) datasets, see Fig. 1. b. Landslide affecting factors Beside the three impact factors: slope, aspect and elevation which were extracted from selected DEM, this paper also uses other conditioning factors, including: distance to road, distance to stream, soil type, land cover. The data was collected from local authorities. Rainfall is considered the triggering factor that lead to landslide in this area. Instead of using average annual rainfall like most of previous research [3, 12], the study of [11, 13] showed that 3-day antecedent rainfall is the most significant trigger to landslide. Thus, we use the 3-day antecedent rainfall (50% frequency_P50) which was created from Regional Frequency Analysis (RFA) method for assessment. Detailed information of these factors is shown in Table 1.
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Fig. 2. Diagram of methodology
Table 1. Thematic data layer of the study area No Landslide causal factors
Landslide causal factors
1
Slope (degree)
(1) 0–10; (2) 10–20; (3) 20–30; (4) 30–40; (5) 40–50; (6) 50–60;(7) > 60
2
Aspect
(1) 0–45; (2) 145–175; (3) 175–415; (4) 415 – 565; (5) 565 – 720; (6) 720 – 900; (7) 900 – 1135; (8) > 1135
3
Elevation (m)
(1) 0–60; (2) 60–120; (3) 120–180; (4) 280– 240; (5) > 240
4
Distance to road (m)
(1) 0–60; (2) 60–120; (3) 120–180; (4) 280 – 240; (5) > 240
5
Distance to stream (m)
(1) Others; (2) Hapli Humic Acrisols; (3) Hapli Ferralic Acrisols;
6
Soil type (group)
(3) Epi Lithi Humic Acrisols ARC/INFO; (4) Epi Lithi Ferralic Acrisols
7
Land cover
(1) Residental area; (2) Agriculture; (3) Forest; (4) Bush; (5) Timber
8
Rainfall (3day-P50) (mm) (1) 0–410; (2) 410–430; (3) 430–455; (4) 455–480; (5) > 480
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2.3 DEM Analysis The accuracy of these mentioned DEMs are evaluated based on vertical error which are compared to available spot heigh in the research area. The vertical errors are calculated by subtracting reference elevation from TanDEM-x 90, SRTM and NasaDEM elevation following the equation. The accuracy then is analyzed using Mean Error (ME), Mean Absolute error (MAE), Root Mean Square Error (RMSE), standard deviation (STD) and skewness. hi = hi,GDEM − hi,ref
(1)
where hi is the vertical height error of pixel i, hi,GDEM is elevation of a GDEM pixel, and hi, ref is the elevation of a reference DEM pixel. To establish that evaluation, all elevation datasets must be in the same vertical datum. While SRTM and Nasa DEM represent height of pixel based on EGM 1996 datum, TanDEM elevation are accorded to ellipsoid WGS 1984. Therefore, TanDEM-X 90 value are transferred to EGM 1996 using Vdatum Version 4.1 software from NOAA (https:// vdatum.noaa.gov/). Meanwhile, Vietnam local quasi Geoid have correlation with EGM 2008, which can represent by equation: ξ¯ ∗ = ξ¯ + D0 = ξ¯ + 0.890 m
(2)
2.4 Landside Susceptibility Model a. Analytic Hierarchy Process (AHP) Analytic Hierarchy Process (AHP) [14] method has been used widely to develop an appropriate landslides susceptibility map. According to the method, an estimation of landslide causative factors contribution would be risen by pair-wise comparison matrix. Based on researches carried out by, a range was suggested for comparison of criteria which includes numerical values from 1 to 9. Based on researches carried out by Saaty and Saaty et al. [14, 15], a range was suggested for comparison of criteria which includes numerical values from 1 to 9 according to the importance of the factor. In which, the value of 9 shows that the factor is extremely strong importance of a criterion compared to another. Also, the Consistency Ration (CR) and Consistency Index (CI) would be calculated to estimate the consistency of the AHP process [14]. Note that CR value must be less than 0,1. If not the pairwise table is considered inconsistent [14]. CI =
λmax − n n−1
(3)
CI RI
(4)
CR =
b. Frequency Ratio (FR) FR is a quantiative model of statistical method for landslide investigation. This model is developed based on the relationship between the landslide inventory map
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and each landslide causal factor [16, 17]. Each affecting factor will be divided into many classes. The value of FRij in each class is calculated according to the following formula 5 [1]: FRij =
N (L ∩ Cij )/N (L) N (Cij )/N (C)
(5)
Where: C ij is the j-th class of affecting factor C i (i = 1, 2,…, n). n is the number of affecting factors, n = 8 in this study. N (L ∩ Cij ): is the number of landslide pixels in C ij . N(L) is the total number of landslide pixels. N(C ij ) is the total number of pixels in the class C ij . N(C) is the total number of pixels in the study area. The frequency ratios FRij were summed to calculate the landslide susceptibility index (LSI): FR = FRij (6)
2.5 Accuracy Assessment The accuracy of the FR model and the AHP model in this study was evaluated based on the success rate curve and predictive curves. The success rate curve is created from the training dataset (237 landslide points) and the prediction rate curve is calculated from the testing dataset (102 landslide points). The success rate curve shows the relationship between the cumulative percentage of the area of the susceptibility classes (x-axis) and the cumulative percentage of observed landslide occurrences in different landslide susceptibility classes (y-axis) [3, 17]. The area under the rate curve (AUC) indicates the prediction accuracy of the susceptibility map. An AUC value in the range of 0.90−1.00 is considered as an indicator of excellent model quality while a value in the range of 0.80−0.90 indicates good, fair (0.70−0.80), poor (0.60−0.70), and fail (0.50−0.60) [12, 18]. Also, landslide density (LD) was also used to evaluate the effectiveness of the model. LD is constructed by calculating the ratio of pixels with occurred landslides over the ratio of non-occurred landslide pixels for each classified susceptible zone in a diagram. The landslide susceptibility map will perform well if most of landslide sites occurre on very high and high classes [17].
3 Result and Discussion 3.1 DEM Selection Evaluation results of NasaDEM and TanDEM X-90 and STRM represent that NasaDEM is the most accurate elevation model comparted to TanDEM X-90 and SRTM. While TanDEM and SRTM represent mean of errors are higher than zero, the figure for
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NasaDEM represents a minus value. Mean absolute errors of Nasadem only 6.167 as the near haft of MAE of SRTM. The same figure is also seen as RMSE satistical index. So, reproduce of SRTM, NasaDEM is used to develop landslide susceptibility in the mountainous area of Quang Ngai province. Table 2. Comparison of three DEM models DEM TanDEM X-90 SRTM NasaDEM
ME (m)
MAE (m)
RMSE (m)
STD (m)
Skewness −0.617
1.126
7.494
10.333
10.271
7.396
11.081
14.3964
12.351
0.019
−0.987
6.167
7.238
8.581
−0.254
Fig. 3. Comparison of three DEM models
3.2 Landslide Susceptibility Mapping The landslide susceptibility map was produced by multi criteria decision based on Geographic Information System (based on QGIS 3.12) to project landslide susceptibility indexes (LSI). In FR model, LSI value is obtained by the FR value which was defined by formula 6. Each pixel of the landslide susceptibility map will get a corresponding FR value, the calculation is supported by the Raster Calculator tool in the ArcGIS software. In the case of AHP model, LSI value was calculated in SAGA toolkit product with the LSI range from 1.536 to 7.471. The higher the LSI values, the greater of landslide susceptibility. Based on percentages of area, the LSI values were classified into five
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susceptibility classes: very high (10%), high (10%), moderate (20%), low (20%), very low (40%) [1]. The landslide susceptibility maps that obtained from AHP method and the FR method were illustrated in Fig. 4.a and Fig. 4.b. These maps can be very useful for natural disasters management and for land use planning.
Fig. 4. : Landslide susceptibility mapping based on: (a) AHP method, (b) FR method.
3.3 Landslide Susceptibility Assessment Landslide susceptibility models were evaluated by AUC value using training and testing dataset and LD value using the testing dataset. The AUC values are shown in Fig. 5 and Fig. 6 indicates that the FR model giving better performance than the AHP model in both the success rate curve graph and predictive rate curve graph. In success rate curve graph (Fig. 5), the FR model (AUC = 0.775) outperforms the AHP model (AUC = 0.732). The same results happen in the prediction rate curve graph (Fig. 6) when the FR model (AUC = 0.793) performs better than the AHP model (AUC = 0.743). In both the FR model and the AHP model, the AUC value of testing cases is better than the training cases. With the range of AUC value from 0.732 to 0.793, both landslide models have performed well for landslide susceptibility assessment. Figure 7 indicates the certainty of landslide susceptibility models following the AHP method and the FR method for this study area. The results represent that most landslide susceptibilities fall into very high and high categories. In a very high class, the LD value of the FR model (65%) is moderately higher than the APH model (47%). In contrast, the LD value of the AHP model is higher than the FR model in other classes. This result once again confirms that the FR model outperforms the AHP model. The results from the two model can be used effectively in prediction of landslide in area as fundamental for landscape planning and disaster management in the area.
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Fig. 5. Success rate curve of the AHP and FR Fig. 6. Prediction rate curve of the AHP and methods FR methods
Fig. 7. Landslide density graph of the AHP and FR methods.
4 Conclusion The DEMs is an important data in the assessment of landslide susceptibility, the optimal DEM data depends not only on the study area but also on the characteristics of the landslide. For areas where measured elevation data is in poor quality, satellited-based DEMs are potential resource for research. Currently, there are many sources of these DEMs that are freely available to users with different build-time characteristics, computation methods, and resolutions. This study has performed the evaluation of satellite DEM data NasaDEM, TanDEM_X90 and SRTM according to spot heigh data. The results show that NasaDEM has the highest accuracy. This DEM data will be analyzed to create slope, aspect, elevation maps in combination with other factors such as soil type, land use, rainfall, distance to road and distance to river to create the landslide conditioning factors set. These influencing factors data combined with the 339-landslide points data were collected to create the landslide susceptibility model input parameter set. The two models: FR models represent the quantitative method and AHP represents the qualitative method were selected. The results of the AUC and LD assessment show that the FR model has higher accuracy and is suitable for building landslide susceptibility maps for mountainous districts of Quang Ngai province.
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Acknowledgments. [Doan Viet Long, VINIF.2020.TS.135] was funded by Vingroup Joint Stock Company and supported by the Domestic Master/ PhD Scholarship Programme of Vingroup Innovation Foundation (VINIF), Vingroup Big Data Institute (VINBIGDATA).
References 1. Bui, D.T., Pradhan, B., Lofman, O., Revhaug, I., Dick, O.B.: Landslide Susceptibility assessment at Hoa Binh Province of Vietnam using frequency ratio model. Adv. Biomed. Eng. 6, 476–484 (2012) 2. Dikshit, A., Sarkar, R., Pradhan, B., Jena, R., Drukpa, D., Alamri, A.M.: Temporal probability assessment and its use in landslide susceptibility mapping for eastern Bhutan. Water 12(1), 267 (2020) 3. Pham, B.T., Tien Bui, D., Indra, P., Dholakia, M.: Landslide susceptibility assessment at a part of Uttarakhand Himalaya, India using GIS–based statistical approach of frequency ratio method. Int. J. Eng. Res. Technol. 4(11), 338–344 (2015) 4. Pourghasemi, H.R., Teimoori Yansari, Z., Panagos, P., Pradhan, B.: Analysis and evaluation of landslide susceptibility: a review on articles published during 2005–2016 (periods of 2005– 2012 and 2013–2016). Arab. J. Geosci. 11(9), 1–12 (2018). https://doi.org/10.1007/s12517018-3531-5 5. Reichenbach, P., Rossi, M., Malamud, B.D., Mihir, M., Guzzetti, F.: A review of statisticallybased landslide susceptibility models. Earth-Sci. Rev. 180, 60–91 (2018) 6. Althuwaynee, O.F., Pradhan, B.:Ensemble of data-driven EBF model with knowledge based AHP model for slope failure assessment in GIS using cluster pattern inventory. In: FIG Congress Engaging the Challenges–Enhancing the Relevance Kuala Lumpur, Malaysia, pp. 16–21 (2014) 7. Brock, J., Schratz, P., Petschko, H., Muenchow, J., Micu, M., Brenning, A.: “The performance of landslide susceptibility models critically depends on the quality of digital elevations models. Geomatics. Nat. Hazards Risk 11(1), 1075–1092 (2020) 8. Chen, Z., Ye, F., Fu, W., Ke, Y., Hong, H.:The influence of DEM spatial resolution on landslide susceptibility mapping in the Baxie River basin, NW China. Nat. Hazards, 1–25 (2020) 9. Das, G., Lepcha, K.: Application of logistic regression (LR) and frequency ratio (FR) models for landslide susceptibility mapping in Relli Khola river basin of Darjeeling Himalaya, India. SN Appl. Sci. 1(11), 1–22 (2019). https://doi.org/10.1007/s42452-019-1499-8 10. Pham, B.T., et al.: A novel hybrid approach of landslide susceptibility modelling using rotation forest ensemble and different base classifiers. Geocarto Int. 35(12), 1267–1292 (2020) 11. Chi Cong Nguyen, T., Vo, D.P., Long, D.V., Binh, N.Q.: Assessment of the effects of rainfall frequency on landslide susceptibility mapping using AHP method: a case study for a mountainous region in central Vietnam. J. Crit. Rev. 7(10) (2020) 12. Vakhshoori, V., Zare, M.: Landslide susceptibility mapping by comparing weight of evidence, fuzzy logic, and frequency ratio methods, Geomatics. Nat. Hazards Risk 7(5), 1731–1752 (2016) 13. Cong, N.C., Binh, N.Q., Phuoc, V.N.D.: Landslide susceptibility mapping by combining the analytical hierarchy process and regional frequency analysis methods: a case study for Quangngai Province (Vietnam). In: International Conference on Asian and Pacific Coasts, pp. 1327–1334 (2019) 14. Saaty, R.W.: The analytic hierarchy process—what it is and how it is used. Math. Model. 9(3–5), 161–176 (1987)
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15. Saaty, T.L., Vargas, L.G.: Prediction, projection, and forecasting: applications of the analytic hierarchy process in economics, finance, politics, games, and sports. Kluwer Academic Pub, (1991) 16. Bonham-Carter, G.F.: Geographic information systems for geoscientists-modeling with GIS. Comput. methods Geosci. 13, 398 (1994) 17. Pradhan, B., Seeni, M.I., Kalantar, B.: Performance evaluation and sensitivity analysis of expert-based, statistical, machine learning, and hybrid models for producing landslide susceptibility maps. In: Laser scanning applications in landslide assessment, Springer, pp. 193–232 (2017) . https://doi.org/10.1007/978-3-319-55342-9_11. 18. Hasanat, M.H.A., Ramachandram, D., Mandava, R.: Bayesian belief network learning algorithms for modeling contextual relationships in natural imagery: a comparative study. Artif. Intell. Rev. 34(4), 291–308 (2010)
Analysis of the Impact of Ponds Locations in Flood Cutting from a Developing City in Vietnam Quang Binh Nguyen, Duc Phuoc Vo(B) , Hoang Long Dang, Hung Thinh Nguyen, and Ngoc Duong Vo Faculty of Water Resources Engineering, University of Science and Technology, The University of Danang, 54 Nguyen Luong Bang, Lienchieu, Da Nang, Vietnam [email protected]
Abstract. With the expansion of imperviousness due to urbanization, flooding frequency and intensity are more likely to increase in many cities. The application of ponds and lakes has seemed like one of a sustainable alternative solution to abate flow urban and developing areas. The fact that design parameters and management procedures of the stormwater conveying system are strongly dependent on local conditions, which resulting in difficulty in choosing the most appropriate plan for pond and lakes design. Thus, this research compares performances of a drainage system by simulation in the Stormwater Management Model (SWMM) under the climate conditions of Quang Ngai city, located on the Central Coast of Vietnam. Results represent that multi-ponds scenarios are more efficiencies than single points scenarios in reducing risks of flooding. Compare to single pond scenarios, water levels of the multi-ponds system increase slightly in upstream while reducing significantly in downstream areas. The study provides an insight into selecting and locating of ponds and lakes in the planning of stormwater management systems. Keywords: Urban flooding · Low impact development · Flood reduction · Stormwater management
1 Introduction Urban flooding Urbanization in recent years leads to an increase in flooding problems in many cities. In terms of climate change and urbanization, flooding and inundation are becoming significant in many cities. The problem is likely significant in tropical storm areas where rainfall intensity increases during the landing of storms and tropical monsoons due to influence from climate change [1, 2]. The projection of many RCM and GCM model have confirmed increases in extreme precipitation [3]. In addition to this, reduce of infiltration rate result from urbanization would enhance surface runoff and reduce time concentrate, increase risks of flooding [2, 4, 5]. There are two approaches to reduce the vulnerability of flooding, which are enlarging conveying system capacity while the other improving integrated water cycle management. The most popular method to improving capacity of conveying system is enlargement of conduit size and developing new conveying system for routing stormwater to © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 971–980, 2022. https://doi.org/10.1007/978-981-16-3239-6_76
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receive waters [6, 7]. The system is designed to collected runoff generated from storm water which over initial abstraction. In the case of flow generation is over than conveying system capacity, flood happen. On the other hand, integrate water cycle management problems to conventional system are more likely use near natural techniques to enhance storage of stormwater in the system and reduce time concertation of flow generation [8, 9]. Consequently, these measures considerably make contribution to reduce stress to existing conventional problems and improving in groundwater recharging to water cycle [1]. Near natural integrate water level can be categorized into two classes, infiltrationbased measures and detention measures [1]. Some of measure such as sediment basin, water tanks, infiltration trend are widely used successfully in stormwater management in many places. Ponds and constructed wetland are considered place an important role in water sensitive urban design [10]. Even though the measure has been application in many places effectively [1, 10], mechanism of multi ponds and constructed wetland in tropical storm climate condition is remain unclear, which lead to hesitate from local authorities in application the measures on the fields. Pond and constructed wetland take responsibility in reduce both of process to reduce stormwater runoff [11]. On the one hand, storm water storage in these objects as long as possible to reduce total amount of the runoff through majorly through evapotranspiration and from infiltration. and recharging water to underground [12]. Detention storage of pond and constructed wetland can impact on hydrology and hydraulic procedure which resulting in reducing of peak flow and prolong duration of flood [1, 12]. The problem is the impact mechanism of pond and constricted wetland have been carryout in many palaces which majorly less impact by tropical monsoon. However, the fact that this research evaluates capability of pond in the term of extremely rainfall which may over 300 mm per day or even over 500 mm in 3 days cause by influences of tropical monsoon and storm. Recording in recent year have confirm the severe flooding happened in Ho Chi Minh City (annually), Ha Noi Capital (2008, 2019) and Danang city which majorly result from extreme rainfall resulting from tropical monsoon and storm. Understanding effectiveness of these measures under the condition of tropical monsoon could make contribution to reduce risks of flooding in the region. The effectiveness of ponds and constructed wetland strongly depend on size, location, and design of these pond. Due to urbanization, it is difficult to find large enough places for implementing large pond and constructed wetland. Meanwhile proportion of multiple design constructed wetland is not fully understanding for tropical monsoon. This research aims to highlight impact mechanism of multiple-pond allocation in stormwater management system in a tropical monsoon. This research would investigation effectiveness of ponds and constructed wetland as alternatives measure to reduce the impact of flooding. This measure is expected to reduce flood susceptibility in the term of increase extreme precipitation in climate change. A case study considered in this paper is the city of Quang Ngai which suffers very much from inundation and is already inundated under 30% return periods of rainfall.
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2 Methodology and Materials 2.1 Study Area In this research, we focus on highlight performance extremely rainfall condition of Quang Ngai city 180°48’E and 15°08’N. The city is in right band of Tra khuc river which located in Central Coast Region of Vietnam. Recently, the city is facing with many severe inundations due to significant urbanization and increasing in extremely rainfall even though surface elevation much higher than average flood water level. Flooding in the region can be classify into two categories. One the one hand, there is flooding event related to fluvial flooding when water level in Tra Khuc river rising over riverbank. On the other hand, the city is inundated due to stormwater convey discharge unable to release large amount of runoff generation cause by extremely rainfall. Recently, urban flooding is more likely to increase due to replacement of pervious landcover such as agriculture area and ponds into residential areas. The study area has typical features of the climate on the south-central coast of Viet Nam. The rainy season usually occurs from September to December, accounting for 80% of the annual rainfall. Specifically, most of the heavy rains occur in October and November together with monsoon and tropical storm and become the main reason for flooding in the city. According to the statistics of the southern central hydro-meteorological station, a heavy rainfall in 2016 occurred from December 14 to December 15 resulting in flooded many areas of Quang Ngai city, especially in Southern part of area. Initial assesses claim that the event is result from insufficient in design of convey in system and impact of urbanization in the city. Quang Ngai’s drainage system is used for both wastewater and storm water, thereafter, discharge to three main destinations which are Tra Khuc river, Bau Giang river and Bau Lang canal. The city catchment is divided into three separately sub-catchments that their drainage system works independently. To meet scope of this study, the author selected the southern urban area, located along the Southbank of Tra Khuc River to implement the research (Fig. 1). Drainage characteristics of the southern urban area, the water does not escape to the river but the whole country concentrates on Bau Ca pumping station, there are 2 units (and 1 backup group) direct discharge pump to Tra Khuc river. The study area is a mixed basin, including parts of high urbanization and low urbanization areas and agricultural production. The urban area including the old core zone of Quang Ngai city and the newly formed urban areas in recent years, this area has a very large proportion of waterproof land. The centre of the study area still has a large area of agricultural production. However, this area can be replaced by new residential areas under pressure of urbanization.
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Fig. 1. Study area
2.2 Scenarios In order to achieve the objective of the research, we compared performances of the system under two scenarios of convey system. On the one hand, the conventional conveying system account for existing stormwater management system in the city, which did not represent participation of integrated stormwater management strategies. On the other hand, an integrated multi-pond would be allocated to assessment ability of the system as potential measure to reduce risk of flooding. We suggested for establishing an addition ponds in lowland agriculture area near To Hien Thanh Streets to collect and detent stormwater from upstream areas (Fig. 2). The pond is expected to reduce stress at downstream conveying system. However, estimating most appropriate design criteria is likely difficult in completed stormwater management system in the city. Proposed area scenarios of the pond are promoted in the table below (Table 1). Table 1. Scenarios of developing additional pond Pond
Scenario 1 Scenario 2 Scenario 3
Area (m2 ) 21200
35200
81800
Rainfall data used for the process based on analysing return period from 1976– 2019 hourly rainfall data according to regulation from designing standard for drainage water system. The effective area of addition pond area is investigating following 10 min recorded rainfall even (corresponding to return frequency of 30%) from 14th to 15th in December 2016. Also, scenarios also are investigated under high return frequencies P = 5%, 10% và 20% (see in Table 2) follow above 2016 rainfall pattern (see Fig. 3). Drainage system data are provided from flooding and storm control agency. The drainage represents the newest update for the year of 2017, which appropriate for simulating in 2017 and 2017 flooding event. Only main conduits with is more than 800 mm in diameter are used for simulating of the model (Fig. 4).
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Fig. 2. Proposed location for pond
Table 2. Maximum rainfall for different scenarios (total rainfall max in 6 h period) Frequency
P = 5% P = 10% P = 20% P = 30%
Rainfall (mm) 126.11
109.01
91.18
81.8
Fig. 3. Observed 10-min rainfall, 14th to 15th in December 2016
2.3 Methodology The capacity of the drainage system in the south of the city of Quang Ngai was simulated by using the Stormwater Management Model (SWMM). SWMM is a 1-D model that is widely used and accepted by engineers and researchers. Recently, the model shows its capacities in evaluating the impact of climate change [4], investigating low impact development, or assessing the stormwater quality. The fact that monitoring systems have not been set up for measuring water level, flow as well as a discharge for the city. So, simulation is more likely appropriate to simulate the performance of the conveying systems in reducing flood. So, the Storm Water Management Model (SWMM) is selected to implement the convey system in this research.
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Fig. 4. Simulation model
The model is calibrated by using flood marks recorded on the wall of many houses in Quang Ngai Province caused by a flood event in December 2016. There are for flood marked were measured to get water elevation for the hydraulic calibration model. Because the observation data in the research area is inadequate for calibrating the model. Thus, the hydrology model was set up based on the author’s experiences, and similarly revised and tested are at Da Nang and Quy Nhon [13, 14]. The error after calibration with water depth in four point in the model are shown in Table 3 below. Table 3. Error at four calibration notes in research area Note
Location
Simulated water depth (m)
Observation water depth (m)
Error (m)
112
Phan Boi Chau – Nguyen Tu Tan junction
2.23
2.47
0.24
200
Phan Boi Chau
2.61
2.83
0.22
J218
Phan Dinh Phung – Hung Vuong junction
1.86
2.07
0.21
J221
Quang Trung – Hung Vuong junction
2.24
2.40
0.16
Simulation results would be investigated in typical notes before and after ponds and outfall of the system to investigate capacity in urban flood cutting off the system.
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3 Result and Discussion Simulation result have represented that, the current conveying system is unable to cope with increase in runoff generation following increase of rainfall according to different frequencies scenarios.
Fig. 5. Water depth after detention point according to variation of rainfall frequencies scenarios (a: P = 30%, b: P = 20%, c: P = 10%, and d: P = 5%) according to four scenarios of stormwater conveying system (current system (red line) and integrated scenarios additional ponds with surface of 2.12 ha (orange line), 3.52 ha (green line), and 8.18 ha (blue line)
Simulation results of water depth and total inflow at before and after additional pond represent efficiently in cutting down of flood of the ponds in the system. Developing scenarios of an additional pond reduce significant water depth in the system, which may save the volume of storage for a potential increase in flow generation in the terms of climate change and urbanization. However, the effectiveness of pond scenarios is different according to rainfall frequencies. The ponds are more likely to capture a large proportion of flow in low frequencies (P30% and P20%). The figure for larger frequencies reduces significantly for larger rainfall at P10% and P5% (Fig. 5). Simulation results also represent the influences differently of the additional pond to the duration of the flow in the system. In the case of rainfall equal to P30%, the peak of flow tends to reach the peak later than current scenarios, while the figures represent a contrasting figure in the rest of the rainfall value. Also, flow scenarios in 2 are more suitable as the position of detention ponds with reducing the concentration-time of the system (Fig. 6). The pond is stated that it would take part in effectively reduce the flowrate and water depth at the outfall. Following an increase of area, from scenario 1 to scenario 3, the total discharge and inundation depth of the entire basin also decrease (Fig. 7), respectively. At the outlet of the basin, the effects of scenario 2 (regulating lake area
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Fig. 6. Flow in conveying system after detention point according to variation of rainfall frequencies scenarios (a: P = 30%, b: P = 20%, c: P = 10%, and d: P = 5%) according to four scenarios of stormwater conveying system (current system (red line) and integrated scenarios additional ponds with surface of 2.12 ha (orange line), 3.52 ha (green line), and 8.18 ha (blue line))
Fig. 7. Water depth at outfall according to variation of rainfall frequencies scenarios (a: P = 30%, b: P = 20%, c: P = 10%, and d: P = 5%) according to four scenarios of stormwater conveying system (current system (red line) and integrated scenarios additional ponds with surface of 2.12 ha (orange line), 3.52 ha (green line), and 8.18 ha (blue line)
of 35200 m2 ) and scenario Three (regulating lake area of 81800 m2 ) correspond to all four rainfall frequencies nearly similar, while the pond surface area of scenario 3 is 2.32
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times higher than that of scenario 1. Thus, with the catchment characteristics of Quang Ngai city, when building a regulating lake, it is recommended to build with an area of about 35200 m2 will maximize the regulatory efficiency of the lake, and at the same time reduce land acquisition of the city.
Fig. 8. Correlation of flow between current scenarios of integrated scenarios additional ponds with surface of 2.12 ha (blue triangle), 3.52 ha (green diamond), and 8.18 ha (purple square)
Figure 8 represent that the simulated flow value increases with the frequency of precipitation (the number of points and values above the symmetric line) and decreases with the scenario of increasing the area of the additional pond. The flood is more likely to reach the peak earlier to current area corresponds to 4 frequencies of rainfall.
4 Conclusion This research has successfully evaluated the capability of multipound in reduce flooding in the city of Quang Ngai. Using the Stormwater management model, the research has investigated three scenarios under four scenarios of rainfall accounting for respectively return periods 30%, 20%, 10%, and 5%. The pond has play important role in reducing the risk of flooding in the city. However, it also confirms that the effect of multi-pond will reach a limitation where the increase of surface area would not lead to efficiencies in reduce flooding.
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References 1. Eckart, K., McPhee, Z., Bolisetti, T.: Performance and implementation of low impact development - a review. Sci. Total Environ. 607–608, 413–432 (2017) 2. Yazdanfar, Z., Sharma, A.: Urban drainage system planning and design–challenges with climate change and urbanization: a review. Water Sci. Technol. 72(2), 165–179 (2015) 3. Tran, T., Neefjes, K.: Viet Nam Special Report on Managing the Risks of Extreme Events and Disasters to Advance Climate Change Adaptation (SREX Viet Nam). The Ministry of Natural Resources and Environment of Vietnam (2015) 4. Huong, H.T.L., Pathirana, A.: Urbanization and climate change impacts on future urban flooding in Can Tho city, Vietnam. Hydrol. Earth Syst. Sci. 17(1), 379–394 (2013) 5. Baek, S.S., et al.: Optimizing low impact development (LID) for stormwater runoff treatment in urban area, Korea: experimental and modeling approach. Water Res. 86, 122–131 (2015) 6. Bell, C.D., McMillan, S.K., Clinton, S.M., Jefferson, A.J.: Hydrologic response to stormwater control measures in urban watersheds. J. Hydrol. 541, 1488–1500 (2016) 7. Burns, M.J., Fletcher, T.D., Walsh, C.J., Ladson, A.R., Hatt, B.E.: Hydrologic shortcomings of conventional urban stormwater management and opportunities for reform. Landsc. Urban Plan. 105(3), 230–240 (2012) 8. Clary, J., et al.: Integration of low-impact development into the international stormwater BMP database. J. Irrig. Drain. Eng. 137(3), 190–198 (2011) 9. Elliott, A., Trowsdale, S.: A review of models for low impact urban stormwater drainage. Environ. Model. Softw. 22(3), 394–405 (2007) 10. Roy, A.H., et al.: Impediments and solutions to sustainable, watershed-scale urban stormwater management: lessons from Australia and the United States. Environ. Manage. 42(2), 344–359 (2008) 11. Ashley, R., Garvin, S., Pasche, E., Vassilopoulos, A., Zevenbergen, C.: Advances in Urban Flood Management, CRC Press (2007) 12. Clar, M.L., A.S.o.C.E.L.I.D. Committee, Lichten, K.: Low Impact Development Technology: Design Methods and Case Studies. American Society of Civil Engineers (2015) 13. Vo, D.P.. Bhuiyan, M.A., Vo, N.D.: Climate Change Impact on Urban Flooding in Quy Nhon City, Vietnam. Computer-Aided Civil and Infrastructure Engineering (2019) - dánh ´ ,ng dung mô hình MIKE URBAN dê ´ giá hê. thông 14. Bình, N.Q., Phu,o´,c, T.L., Hùng. N.T.: U . ,, , ´ ´ ˜ ´ thoát nuoc mua cua quâ.n Câm Lê., thành phô Ðà Na˘ ng. Ky yêu hô.i thao khoa ho.c "Công ´ phát triên bên ` vu˜,ng lân ` thu´, 2 – ATCESD 2016, 2016. Ða.i ´ hu,o´,ng d-ên nghê. Xây du.,ng tiên tiên ho.c Bách khoa Ðà Na˜˘ ng ij
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Turning Electronic and Optical Properties of Monolayer Janus Sn-Dichalcogenides By Biaxial Strain Vuong Van Thanh1(B) , Nguyen Thuy Dung1 , Le Xuan Bach1 , Do Van Truong1 , and Nguyen Tuan Hung2 1 School of Mechanical Engineering, Hanoi University of Science and Technology, Hanoi,
Vietnam [email protected] 2 Frontier Research Institute for Interdisciplinary Sciences, Tohoku University, Sendai 980-8578, Japan
Abstract. We investigate mechanical, electronic, and optical properties of monolayer Janus SnSTe and SnSeTe with 1T structure under biaxial strain based on first-principles calculations. We find that both SnSTe and SnSeTe are metallics for the unstrain case. SnSTe and SnSeTe become semiconductors at the biaxial strains of 6% and 4%, respectively. The biaxial strain effectively modulates the optical absorption of the monolayer Sn-dichalcogenides. Besides, the theoretical strengths of SnSTe and SnSeTe are also investigated. Our obtained results are helpful for applications in nano-electromechanical, optoelectronic, and photocatalytic devices based on the monolayer SnSTe and SnSeTe. Keywords: Theoretical strength · First-principles · Optical absorption · Transition metal dichalcogenides
1 Introduction Two-dimensional (2D) Janus transition metal dichalcogenides (TMDs) have been demonstrated many potential applications in the field of nanoelectronic, optoelectronic, and photocatalytic devices [1–3]. The structure of the monolayer Janus MoSSe was synthesized [4, 5] by replacing the S layer on the monolayer MoS2 with the Se layer. The monolayer Janus MoSSe has many unique properties compared with the 2D conventional TMDs because of the structural symmetry-breaking of MoSSe. For example, the Janus MoSSe has a low carrier recombination rate and high lattice thermal conductivity, making it could be a candidate material for photocatalytic and thermoelectric applications [6]. The electronic properties of the 2D Janus TMDs are calculated by first-principles calculations, in which the 2D Janus TMDs show the metallic or semiconductor for the 1T and 1H structures, respectively [7–9]. Using the density functional theory (DFT) calculations, we also investigated the mechanical, electronic, and optical properties of the Janus TMDs [10], in which the optical absorption coefficient of the Janus TMDs increases by © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 981–989, 2022. https://doi.org/10.1007/978-981-16-3239-6_77
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applying the tensile strain. Although the 2D Janus TMDs have been mentioned in many studies [7–10], these studies mainly focused on the 2H structure. Recently, another structure of the 2D Janus TMDs is the 1T structure that has been investigated [11]. For example, Gue et al., [12] showed the electronic and thermal properties of the monolayer ZrSSe with 1T structure. At room temperature, the thermal conductance of ZrSSe is 33.6 WK−1 , which is smaller than that of Zr (47.8 WK−1 ) [12]. Moreover, ZrSSe has a larger thermoelectric performance than that of ZrS2 , making it becomes a good candidate for n-type thermoelectric devices. By using the firstprinciples calculations, carrier mobility, and piezoelectric of the monolayer 1T-SnSSe were investigated [12]. In addition, Nguyen et al. [13] showed that the electronic and optical properties of Janus SnSSe are turned by applying external strain or electric field. Based on the successful preparation of the monolayer Janus TMDs and the extraordinary properties of the Janus TMDs with the 1T structure, we investigate the mechanical, electronic, and optical properties of the monolayer SnSTe and SnSeTe in 1T structure under the strain by using the first-principles calculations. We demonstrate that the mechanical properties of the Janus structures exhibit strong anisotropic behavior under the tensile strain. The bandgap of SnSTe and SnSeTe can be controlled under the biaxial strain. Furthermore, the effect of the biaxial strain on the optical properties of SnSTe and SnSeTe is also investigated in this study.
2 Methodology In the present paper, we use the DFT calculations for the simulations by using the Quantum ESPRESSO [14]. The generalized gradient approximation (GGA) of Perdew-BurkeErnzerhof (PBE) functional [15] is performed for the exchange-correlation energy. A (16 × 16 × 1) Monkhorst-Pack k-mesh grid in the Brillouin zone is chosen for all models [16]. The calculations are converged with an cut-off energy of 60Ry. In Fig. 1, we illustrate the crystal structure of the monolayer Janus material, in which the periodic boundary condition is applied for three directions. A vacuum region with a thickness of 30 Å is set to reduce the interaction between layers. The atomic structures are fully relaxed until the forces and the stresses are smaller than 5 × 10–4 Ry/a.u. and 5 × 10–2 GPa, respectively, based on the Broyden-Fletcher-Goldfarb-Shanno (BFGS) minimization method [17–20]. The elastic constants C ij of SnSTe and SnSeTe are determined by using Thermo-pw code [21]. The calculated stresses for the monolayer Janus material are determined by multiplying the product of the stresses of unit-cell (units of N/m2 ) and the thickness of unit-cell (∼30 Å [22]). The Young’s modulus, E, and the Poisson’s ratio, ν, of SnSTe and SnSeTe are calculated by [23–25]. 2 2 C11 − C12 C12 E = , and v = (1) C11 C11 To determine the theoretical strength σ and theoretical strain ε of the Janus SnSTe and SnSeTe, a tensile strain is applied with an increment of 2% per step. The tensile strain is expressed by ε = (L- L 0 )/L 0 , where L and L 0 are the deformed and initial equilibrium lattice constants, respectively.
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Fig. 1. Crystal structure of monolayer Janus Sn-dichalcogenides: (a) top view and (b) front view. (c) The first Brillouin Zone.
The absorption coefficient α(ω) and the electron energy loss spectrum L(ω) of SnSTe and SnSeTe are calculated as, [26] √ 1/2 2ω α(ω) = ε12 (ω) + ε22 (ω) − ε1 (ω) (2) c and L(ω) = −Im(ε−1 (ω))
(3)
respectively, where ε1 is the real part and ε2 is the imaginary part. c and ω are the speed of light and the angular frequency, respectively.
3 Results and Discussion First, the Janus structures are optimized by the DFT calculation, in which the optimized parameters are listed in Table 1. The optimized lattice parameters of SnSTe and SnSeTe are 3.89Å and 3.98Å, respectively, which are consistent with the earlier theoretical study [27]. The lattice constant of SnSTe ranges between that of SnS2 (a = 3.70Å) and SnTe2 (a = 4.12Å) [27]. The bucking height of SnSeTe (3.43Å) is larger than that of SnSTe (3.33Å). Interestingly, the calculated bucking height of SnSTe is the same with that of MoSTe (3.33Å) [10]. The obtained bond length of Sn-Se and Sn-Te is 2.78Å and 2.96 Å, respectively, which is longer than that of Zr-Se (2.71Å) [12], and Sn-Se (2.705Å) [13].
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Table 1. Lattice parameters a (Å), bucking heights h (Å), elastic constants C ij (N/m), Young’s modulus E (N/m), Poisson’s ratios ν, σ xx (N/m) (εxx ), σ yy (N/m) (εxx ), and σ bia (N/m) (εbia ) of the Janus monolayers. Materials a
h
C 11
C 22
C 12
C 66
E
v
σ xx (εxx ) σ yy (εyy ) σ bia (εbia )
SnSTe
3.89 3.33 39.12 39.12 23.76 7.68 24.68 0.60 1.54 (0.10)
4.57 (0.22)
4.08 (0.08)
SnSeTe
3.98 3.43 41.85 41.85 24.74 8.55 27.22 0.59 2.65 (0.13)
4.85 (0.32)
3.50 (0.08)
Next, we examine the dynamic and mechanical stabilities of the monolayer SnSTe and SnSeTe. In Fig. 2, we display the phonon spectra of SnSTe and SnSeTe with 1T phase. We can see that SnSTe and SnSeTe demonstrate very small negative frequencies (∼3 cm−1 ) around the -point, indicating a structural instability. It is noted that the bending of the 2D plane leads to negative frequencies [28]. Nevertheless, the Janus TMDs are fabricated on the substrate material [4], therefore, instability for bending can be avoided experimentally. With three atoms in the unit cell, there are 3 acoustic modes and 6 optical phonon modes, as illustrated in Fig. 2. For SnSeTe, the A1 and E’ modes are 210 cm−1 and 131 cm−1 , respectively, which are consistent with the previous report [27]. We examine the mechanical stability of SnSTe and SnSeTe based on the elastic constants C ij . The calculated results show that C 11 > |C 12 |> 0 and C 66 > 0 for two Janus SnSTe and SnSeTe, as shown in Table 1, which satisfy the Born stability criteria [29]. Therefore, Janus SnSTe and SnSeTe are stable in mechanics. The Young’s modulus and Poisson’s ratio of SnSTe and SnSeTe are calculated by Eq. (1). Our calculated results show that the Janus SnSeTe (E = 27.22 N/m) is stiffer than SnSTe (E = 24.68). On the other hand, the Young’s modulus of SnSTe and SnSeTe are lower than that of SnSSe (57.5 N/m) [12], MoSeTe (92.15 N/m) and MoSTe (114.47 N/m) [10], indicating that Janus SnSTe and SnSeTe exhibit excellent mechanical flexibility.
Fig. 2. Phonon spectra of (a) SnSTe and (b) SnSeTe at the equilibrium state.
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Besides the mechanical parameters of the Janus monolayer such as the Young’s modulus and Poisson’s ratios, the theoretical strength and theoretical strain are also important parameters, because they are used to evaluate the stability and the life of devices. In Figs. 3(a) and (b), we show the calculated stress-strain curves of SnSTe and SnSeTe along with the x, y, and biaxial directions, respectively. As shown in Figs. 3(a) and (b), we can see that both the Janus monolayers show anisotropic behavior under the tensile strain. We note that MoS2 [30, 31] and the Janus MoXY and WXY (X/Y = S, Se, and Te) [10] also have anisotropic behavior. The theoretical strength of the Janus structures along with the zigzag direction is higher than that along with the armchair and biaxial directions, as shown in Figs. 3(a) and (b). The ideal strengths of SnSTe and SnSeTe in the zigzag direction are 4.57 N/m and 4.85 N/m, respectively, which is smaller than that of the monolayer MoSTe (7.32 N/m) and WSTe (8.68 N/m) [10]. The theoretical strength and theoretical strain of the Janus structures are summarized in Table 1. It notes that the mechanical properties of the atomic layer materials can be experimentally conducted for graphene and MoS2 [32–34].
Fig. 3. Calculated stress-strain curves of (a) SnSTe and (b) SnSeTe under uniaxial x, y, and biaxial tensile strain.
In Figs. 4 (a) and (b), we demonstrate the band structures of SnSSe and SnSeTe at the unstrain case εxx = εyy = 0) and the biaxial strain case, respectively. It can be seen that the band structures of the Janus monolayers are significantly tuned by the strain, which is close to the earlier data [10]. At the unstrained case, the obtained results indicate that both the Janus SnSTe and SnSeTe are metallics. SnSTe and SnSeTe become semiconductor at εbia = 0.06 and 0.04, respectively. We note that the bandgap of graphene was also opened by strain [35]. The bandgaps of both the Janus structures increase with increasing of the biaxial strain up to ~0.1 and then the bandgaps decrease, as shown in Fig. 5. The calculated maximum bandgaps of SnSTe and SnSeTe under the biaxial strain are 0.383 eV and 0.35 eV at εbia = 0.1 and εbia = 0.08, respectively. In Figs. 6 (a) and (b), we plot the absorption coefficient of the Janus monolayers for several biaxial strain values along with the in-plane (α // (ω) direction, respectively. The optical absorption α // (ω) is calculated from Eq. (2). The calculated results indicate
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Fig. 4. Band structures of SnSTe (a) and SnSeTe (b) under the different biaxial strain values. The Fermi levels are set to be zero.
Fig. 5. Bandgap of SnSTe and SnSeTe under the biaxial tensile strain.
that the biaxial strain has significantly improved the optical absorption of SnSTe and SnSeTe, as shown in Figs. 6 (a) and (b). At the unstrain case (εbia = 0), the maximum absorption coefficients of SnSTe and SnSeTe are 45 × 104 cm−1 and 50.82 × 104 cm−1 at 4.80 eV and 4.79 eV, respectively, in the range of the middle ultraviolet lights. In the photon energy range from 4 to 5 eV, the optical absorption of SnSTe and SnSeTe changes dramatically with the photon energy. As shown in Fig. 6 (a), for the Janus SnSTe, α // (ω) increases from 40 × 104 cm−1 (εbia = 0) to 47.8 × 104 cm−1 (εbia = 0.10). In
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Fig. 6. Optical absorption of (a) SnSTe and (b) SnSeTe for the several values of the biaxial strain along the in-plane (α// (ω)).
Figs. 7 (a) and (b), we show the electron energy loss spectrum L(ω) as a function of the biaxial strain εbia . We can see that L(ω) depends strongly on the εbia in the middle ultraviolet. Our obtained results demonstrate that the optical properties of SnSTe and SnSeTe can be improved by the biaxial tensile strain, which is a significant property for their applications in the optoelectronic and electro-mechanical devices.
Fig. 7. Electron energy loss spectrum L(ε bia ) of (a) SnSTe and (b) SnSeTe for the several values of the biaxial strain along the in-plane (α// (ω)).
4 Conclusions By using the first-principles calculations, we investigate the mechanical, electronic, and optical properties of the monolayer Janus SnSTe and SnSeTe with the 1T structure under the biaxial strain. We show that the mechanical properties of the monolayer Janus Sn-dichalcogenides are strongly anisotropic. Without strain, both SnSTe and SnSeTe are metallics. When applying the biaxial strain, SnSTe and SnSeTe become indirect
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semiconductors at the biaxial strain of 6% and 4%, respectively. The maximum values of the bandgaps of SnSTe and SnSeTe are achieved at the biaxial strain of 10% and 8%, respectively. In addition, the absorption coefficients of SnSTe and SnSeTe are increased by applying the biaxial strain for the middle ultraviolet lights. Our obtained results could be helpful for design and manufacture the optoelectronic devices based on the Janus Sn-dichalcogenides.
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19. Goldfarb, D.: A family of variable-metric methods derived by variational means. Math. Comput. 24(109), 23–26 (1970) 20. Shanno, D.F.: Conditioning of quasi-newton methods for function minimization. Math. Comput. 24(111), 647–656 (1970) 21. Dal Corso, A.: Elastic constants of beryllium: a first-principles investigation, J. Phys. Condens. Matter. 28(7), 075401 (2016) 22. Peng, Q., Ji, W., De, S.: Mechanical properties of the hexagonal boron nitride monolayer: Ab initio study. Comput. Mater. Sci. 56, 11–17 (2012) 23. Hung, N.T., Nugraha, A.R., Saito, R.: Two-dimensional MoS2 electromechanical actuators. J. Phys. Appl. Phys. 51(7), 075306 (2018) 24. Van Thanh, V., Hung, N.T., et al.: Charge-induced electromechanical actuation of Mo-and W-dichalcogenide monolayers. RSC Adv. 8(67), 38667–38672 (2018) 25. Van Thanh, V., Hung, N.T., et al.: Charge-induced electromechanical actuation of two dimensional hexagonal and pentagonal materials. Phys. Chem. Chem. Phys. 21(40), 22377–22384 (2019) 26. Li, X., Zhao, J., Yang, J.: Semihydrogenated BN sheet: a promising visible-light driven photocatalyst for water splitting. Sci. Rep. 3, 1858 (2013) 27. Zhang, X., Cui, Y., Sun, L., Li, M., Du, J., Huang, Y.: Stabilities, and electronic and piezoelectric properties of two-dimensional in dichalcogenide derived Janus monolayers. J. Mater. Chem. C 7(42), 13203–13210 (2019) 28. Cheng, Y., Zhu, Z., Tahir, M., Schwingenschlögl, U.: Spin-orbit-induced spin splittings in polar transition metal dichalcogenide monolayers. EPL (Europhy. Lett.) 102 (5) 57001 (2013) 29. Mouhat, F., Coudert, F.: Necessary and sufficient elastic stability conditions in various crystal systems. Phys. Rev. 90 224104 (2014) 30. Li, T.: Reply to Comment on ideal strength and phonon instability in single-layer MoS2. Phys. Rev. 90 167402 (2014) 31. Cooper, R.C., Kysar, J.W., Marianetti, C.A.: Comment on ideal strength and phonon instability in single-layer MoS2. Phys. Rev. 90 167401 (2014) 32. Mazilova, T., Wanderka, N., Sadanov, E., Mikhailovskij, I.: Measurement of the ideal strength of graphene nanosheets. Low Temp. Phys. 44(9), 925–929 (2018) 33. Bertolazzi, S., Brivio, J., Kis, A.: Stretching and breaking of ultrathin MoS2 , ACS Nano. 5(12), 9703–9709 (2011) 34. Castellanos-Gomez, A., Poot, M., Steele, G.A., van der Zant, H.S., Agraït, N., Rubio Bollinger, G.: Elastic properties of freely suspended MoS2 nanosheets, Adv. Mater. 24 (6), 772–775 (2012) 35. Gui, G., Li, J., Zhong, J.: Band structure engineering of graphene by strain: first-principles calculations. Phys. Rev. 78(7), 075435 (2008)
A Meshfree Method Based on Integrated Radial Basis Functions for 2D Hyperelastic Bodies Thai Van Vu1,2 , Nha Thanh Nguyen1,2(B) , Minh Ngoc Nguyen1,2 , Thien Tich Truong1,2 , and Tinh Quoc Bui3 1 Department of Engineering Mechanics, Faculty of Applied Science, Ho Chi Minh City University of Technology (HCMUT), 268 Ly Thuong Kiet Street, District 10, Ho Chi Minh City, Vietnam [email protected] 2 Vietnam National University Ho Chi Minh City, Linh Trung Ward, Thu Duc District, Ho Chi Minh City, Vietnam 3 Department of Civil and Environmental Engineering, Tokyo Institute of Technology, 2-12-1-W8-22 Meguro-ku, Ookayama 152-8552, Tokyo, Japan [email protected]
Abstract. This paper presents a meshfree method based on integrated radial basis functions (iRBF) for hyperelastic bodies with compressible and nearlyincompressible behavior. The neo-Hookean model is used for hyperelastic material and the nonlinear elastic behavior of 2D bodies are simulated under finite deformation state with total Lagrange formulation. In the approximation of field nodes, the present iRBF shape functions are constructed through integration. With the property of damping out or at least containing an inherent inaccuracy, the iRBF produces a greatly improved approximation of its derivatives. Moreover, the meshfree numerical approach shows its advantage to analyze large deformation problems by the feature of “free of mesh”, especially it does not suffer from the volumetric locking. A number of numerical examples are given to compare the results of the proposed method with the reference solutions given by other methods. Keywords: Integrated radial basis function · Meshfree method · Hyperelasticity
1 Introduction Hyperelastic materials are special elastic material for which the stress is derived by the strain energy density function that determined by the current state of deformation. One of the attractive properties of these rubber-like materials is their ability to have large strains under small loads and retains initial configuration after unloading. Moreover, hyperelastic materials have lightweight and good form-ability so they are widely used in various engineering applications such as shock-absorbing matters in transport vehicles, sport devices and buildings protection from earthquakes. There are various forms of strain energy potentials to model the nonlinear stress-strain relationship of such materials including Neo-Hookean, Mooney-Rivlin, Yeoh, Ogden and so on. Because these © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 990–1003, 2022. https://doi.org/10.1007/978-981-16-3239-6_78
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materials mainly work in large strain condition so the finite deformation analyses need to be considered to warrant their toughness and durability. Generally, hyperelastic problems are often considered in highly non-linear state and it is infeasible to find out the analytic solutions for these problems. In practice, experiments are usually adopted to verify the behavior of hyperelastic structures but their costs are high and it takes too much time to do a lot of tests for obtaining an optimal design. For several decades, together with the rapidly developing of computer and numerical methods, the finite element methods (FEM) are very strong and popular method in computational engineering. FEM has been used as the most powerful method in various types of structural analysis problems with most of material families, including non-linear elastic problems [1–3]. However, FEM and other mesh-based methods also have disadvantages because of the existence of the mesh of elements. Especially in geometrical non-linear problems, when large deformation cannot be passed over, the elements can be distorted and they cannot give good approximated results. To maintain the convex shape of the whole system of elements, an appropriate re-meshing algorithm need to be applied, of course it is time consuming. In order to overcome the drawbacks of mesh-based methods, several meshless or meshfree approaches have been developed, the main purpose is to remove the depending on mesh of finite element models. In meshfree methods, there is no finite element required for the domain but a system of scattered nodes is used for the approximation. The most advantage of meshfree approach is that field nodes can be removed, added or changed position easily in each computation step, it is useful in problems that the domain changing occurs continuedly. There are some studies using meshless methods for finite deformation problems have been proposed in recent years. In 2000, the reproducing kernel particle method (RKPM) was presented by Li et al. in [4] for large deformation problem of thin shell structures. Five years later, the meshless local Petrov-Galerkin (MLPG) method was proposed by Han et al. [5] for solving nonlinear problems with large deformations and rotations. In 2007, Hu et al. applied MLPG method for large deformation contact analysis of elastomers [6]. After that, a meshless local Kriging method had been presented for large deformation analyses by Gu et al. [7]. In 2013, the MLPG was further applied for dynamic impact analysis of hyperelastic bodies by the group of Hu [8]. Recently, Khosrowpour et al. presented a strong-form meshfree method for stress analysis of hyperelastic materials [9]. To improve the accuracy for the approximation of the derivative of a function, a meshless approach based on integrated radial basis functions (iRBF) has been proposed by Mai et al. [10, 11]. In contrast to approximation methods based on conventional differentiated radial basis function, these authors suggested that the approximation procedure starts with the highest-order derivative of the original function. The lower-order derivative and the original function itself are then obtained by integration. This approach has been shown advantages of the accuracy and stability of the numerical solution for various engineering problems. Moreover, the iRBF shape function also possesses Kronecker delta property for which boundary conditions can be easily enforced similar to the finite element method. In 2016, Ho et al. applied meshfree method based on iRBF for dual yield design [12]. This year, Ho et al. also presented meshfree method based on iRBF for quasilower bound shakedown analysis [13]. In this study, the iRBF based meshless method
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is proposed for large deformation analysis of hyperelastic 2D problems. Neo-Hookean model is used for hyperelastic behavior, both compressible and nearly-incompressible models are considered in computation. Standard Newton-Raphson algorithm is applied for these geometrical and material non-linear problems. The paper is constructed as follows. After this introduction part, Sect. 2 shows the constitutive laws for hyperelastic material models. The iRBF-based meshfree method is presented in Sect. 3 and the Sect. 4 is intended for numerical examples that are verified with reference solution. Finally, main conclusions and remarks about the presented method are discussed in the last part.
2 Constitutive Equations of Hyperelastic Material Consider a general solid of the hyperelastic material that is subjected to external forces and displacements so that its geometry is changed. The strain energy density function ψ exists naturally and it can be constructed by right Cauchy-Green deformation tensor C. Stress can be obtained from the first-order derivative of the strain energy density function with respect to the Lagrangian strain. The deformation gradient tensor at the current configuration of solid is defined as Fij =
∂xi ∂ui = + δij ∂Xj ∂Xj
(1)
The right Cauchy-Green deformation tensor C and Lagrangian strain are given below C = FT F; E =
1 (C − I) 2
(2)
where I is identity matrix. As mentioned above, the second Piola-Kirchhoff stress S can be derived from the first-order derivative of the strain energy density function. S=
∂ψ 2∂ψ = ∂E ∂C
(3)
The Cauchy stress (real stress) can be obtained by relationship as follows σ =
1 FSFT J
(4)
where J is the determinant of deformation gradient tensor, describes the volume change between current and initial configuration. J = det(F)
(5)
In this paper, the Neo-Hookean model is adopted. [14], the strain energy density function is given as ψ=
κ μ (J − 1)2 + [I1 − 3 − 2 ln(J )] 2 2
(6)
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where κ and μ are the bulk and shear modulus, respectively. I 1 is the first invariant of the right Cauchy-Green deformation tensor. From Eq. (3) and (6), the nonlinear stress-strain relation for the compressible Neo-Hookean model can be written as (7) S = κ J 2 − J C−1 + μ(I − C−1 ) In the hyperelasticity, the constitutive tensor D is a function of deformation and it is achieved by differentiating the second Piola-Kirchhoff stress S. D=
∂S ∂E
(8)
3 IRBF-Based Meshfree Method for Hyperelasticity 3.1 Integrated Radial Basis Function Approximation The approximation based on integrated radial basis functions is performed in contrast to other methods like polynomial basis functions and radial basis functions due to the fact that it is started with the highest-order derivative of the original function. Consider a problem domain, which is presented by a set of arbitrarily distributed nodes. The approximation of functions u(xi ) in a local support domain of the problem domain is conducted by all nodes in it. To reduce the error of derivative, this paper uses the secondorder derivative of the original function as a started point. The first-order derivative and the original function are obtained by integration as follows k (x) = u,ij
N
hc (x)wc = Q2 (x)a
(9)
c=1
u,ik (x) =
k u,ij (x)dxj =
hc (x)wc dxj =
c=1
uk (x) =
N
u,ik (x)dxi =
¨ N c=1
hc (x)wc dxj dxi =
N
Hc (x)wc + I1 =
c=1 N
Hc (x)wc + I1 xi + I2 =
c=1
N +p1
Q1 (x)a (10)
c=1
N +p2
Q0 (x)a
(11)
c=1
where N is the number of nodes in the local support domain, p1 and p2 denote the number of integration constants (p2 = 2p1 ), hc is the radial basis functions. hc , H¯ c and Hc are formulated in [10] clearly. a is a vector of unknow coefficients. Q2 , Q1 and Q0 are vectors which has (N + p2 ) columns Q2 = [h1 , ........, hN , 0, ..., 0]
(12)
Q1 = H¯ 1 , ..., H¯ N , ..., H¯ N +p1 , 0, ..., 0
(13)
Q0 = [H1 , ......, HN , ......., HN +p2 ]
(14)
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In this study, multiquadric (MQ) function is employed due to accuracy and easy integration. (15) hc = (x − xc )2 + (y − yc )2 + ε where ε is a positive value which is chosen based on the minimum distance from the interested node to neighbors in the local support domain. Collocating Eq. (11) at the set of scattered nodes in the local support domain leads to u = QT a
(16)
u = [u1 , u2 , . . . , uN ]T
(17)
⎤ ... ... ... ... ... QT = ⎣ H1 (xc ) ... HN (xc ) ... HN +p2 (xc ) ⎦ ... ... ... ... ...
(18)
a = Q−1 T u
(19)
where
⎡
From Eq. (16)
Substitute Eq. (19) into Eqs. (9), (10), (11) k (x) = Q2 (x)Q−1 u,ij T u = ,ij u
(20)
u,ik (x) = Q1 (x)Q−1 T u = ,i u
(21)
uk (x) = Q0 (x)Q−1 T u = u
(22)
where is the vector of the shape functions, ,i and ,ij denote the vectors of first-order and second-order derivative, respectively. It should be noticed that the integration constants appear naturally in iRBF method. It leads to QT matrix not square, usually is ill-conditioned. Some additional constraints can be applied to achieve the square system. 3.2 The Weak Form and Meshfree Method In this section, the weak form of hyperelasticity is obtained by mean of the principle of minimum potential energy. Given a material solid of hyperelasticity, the continuum domain is in static equilibrium, under external forces t* on boundary t , displacement u on boundary u , body force b, as shown in Fig. 1. The weak form can be written as ¯ ¯ ¯ ∗d = 0 S : Ed − ubd − ut (23)
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Fig. 1. Boundary conditions of hyperelastic model
in which, the first term denotes the strain energy and the others present work done by body and traction forces where u¯ is similar to virtual displacement in the principle of vitual work. Tensor S is the second Piolar-Kirchhoff stress tensor and E¯ is defined as follows E¯ = Bd¯
(24)
where vector d¯ is the variation of nodal displacements and B is the nonlinear displacement-strain matrix d¯ = ⎡ B3×2N
⎢ =⎣
N
i u¯ i
(25)
⎤ F21 1,1 ... F11 N ,1 F21 N ,1 F11 1,1 ⎥ F12 1,2 F22 1,2 ... F12 N ,2 F22 N ,2 ⎦ F11 1,2 + F12 1,1 F21 1,2 + F22 1,1 ... F11 N ,2 + F12 N ,1 F21 N ,2 + F22 N ,1 i=1
(26) in which, N is the number of nodes in the local support domain. i,x1 and i,x2 are the derivatives of iRBF shape function at node i with respect to x1 , x2 coordinate in the current configuration. Substituting Eq. (24) into Eq. (23), the discrete form of weak form can be written as ⎫ ⎧ N ⎬ ⎨ T T T BT Sd − i bd + i t ∗ d = d¯ f int − d¯ f ext = 0 (27) u¯ Ti d¯ ⎭ ⎩ i=1
In this paper, the Newton-Raphson method is used to solve Eq. (27). The vector of residuals of Eq. (27) is defined as the difference between f int and f ext R = d¯ (f ext − f int ) T
(28)
Apply the linearization into the strain energy, Eq. (23) can be written as ¯ ¯ ¯ ¯ ∗d = 0 E : D : E + S : E d − ubd − ut
(29)
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or T d¯ KT d = R
where KT =
BT DB + GT MG d
(30)
(31)
⎤ 1,1 0 ... N ,1 0 ⎢ 1,2 0 ... N ,1 0 ⎥ ⎥ G4×2N = ⎢ ⎣ 0 1,1 ... 0 N ,1 ⎦ 0 1,2 ... 0 N ,2 ⎤ ⎡ S11 S12 0 0 ⎢ S12 S22 0 0 ⎥ ⎥ M4×4 = ⎢ ⎣ 0 0 S11 S12 ⎦ 0 0 S12 S22 ⎡
(32)
(33)
4 Numerical Examples To estimate convergence of the meshfree method based on iRBF approximation, two numerical examples of compressible and nearly-incompressible behaviors are presented. The results then are compared with [14] for checking the accuracy and stability of this approach. The Neo-Hookean model is used for the numerical tests, the material parameters are chosen similar to examples in [14]. The shear modulus is given as 80.194 N/mm2 , the bulk modulus is assumed as 120.291 N/mm2 in the compressible case and 400889.806 N/mm2 for nearly-incompressible case in this section. In these examples, the standard Newton-Raphson algorithm is used. The force residual, i.e. difference between internal force and external force, is used to verify the convergence of the non-linear solution. With a pre-defined tolerance value of 10–6 , usually 3 to 4 iterations are required to reach convergence in each load step. 4.1 Inhomogeneous Compression Problem In this example, the inhomogeneous compression problem is studied. This problem is given first to evaluate the convergence and accuracy of the developed approach for a regular domain. Considering a rectangular 2D solid which has the essential boundary at the top edge (horizontal displacement) and the bottom edge (vertical displacement) are set to be zero. The geometry and distributed external force f of the solid are shown in Fig. 2. In this case, only half of the solid is studied due to the symmetry. For the purpose of evaluate the convergence, some various values of distributed force are used for compressible (f = 50; 100; 150 and 200 N/mm2 ) and nearly-incompressible behaviors (f = 100; 150; 200 and 250 N/mm2 ). The percentages of compression at
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Fig. 2. Inhomogeneous compression problem: model and boundary conditions
the interested point N versus to the total number nodes of compressible and nearlyincompressible regimes are shown in Figs. 3 and 4, respectively. The charts show the good convergence of the proposed iRBF for non-linear analysis. Moreover, in iRBF approach, the number of nodes used for computation is smaller than that in VDQ (45 × 45) [14]. In others words, iRBF can give good results with less degrees of freedom than VDQ.
Fig. 3. Percent of compression at point N for various values of distributed force in the compressible inhomogeneous compression problem
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Fig. 4. Percent of compression at point N for various values of distributed force in the nearlyincompressible inhomogeneous compression problem
The deformed configurations of the model under f = 200 N/mm2 and f = 250 N/mm2 are shown in Figs. 5 and 6 in the cases of compressible and nearly-incompressible behaviors, respectively. The color indicates the equivalent Cauchy stress [15] distribution on the model. 2 (34) σeqv = σ12 + σ22 − σ1 σ2 + 3τ12 As seen in the two figures, the percent compression at point N is 59.96% (compressible case) and 44.68% (nearly-incompressible case). These obtained results match well with the reference solutions reported in [14] in which the percentages of compression at that point are 59.69% and 45.09% for the two cases, respectively. It is clearly to see that the results obtained by the proposed method are in excellent agreement with the reference values reported in [14]. In addition, it is worth notice that the VDQ method used a mesh that is fine at the edges of the solid while the meshfree method based on integrated radial basis functions takes regular nodes in this case.
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Fig. 5. Equivalent Cauchy stress and deformed configuration for f = 200 N/mm2 in the compressible inhomogeneous compression problem (colors indicate stress (N/mm2 ))
Fig. 6. Equivalent Cauchy stress and deformed configuration for f = 250 N/mm2 in the nearlyincompressible inhomogeneous compression problem (colors indicate stress (N/mm2 ))
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4.2 Curved Beam Problem In the next numerical example, an irregular domain is considered to evaluate the effectiveness of the developed method for non-linear problem. Assumed a curved beam subjected the distributed shearing force f at the end and clamped at the other end. The geometry, boundary conditions and a nodal distribution are shown in Fig. 7. Similar to the previous example, the convergence of the proposed method is investigated by using four various values of the shearing force (f = 0.2, 0.3, 0.4 and 0.5 N/mm2 ) for compressible behavior. The convergence solutions for the displacement at point N with various shearing forces are plotted in Fig. 8 and compared with those of VDQ [14]. As is observed from the graph, the present method can provide an acceptable solution for the irregular domain. Plots in Fig. 9 show the vertical displacement distribution of the beam with four values of distributed force f. Finally, the deformation of curved beam for four values of shearing force mentioned above is plotted in Fig. 10 in which the colors indicate the equivalent Cauchy stress.
Fig. 7. Curved beam problem: model and boundary conditions
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Fig. 8. Vertical displacement at point N for various values of shearing force in the compressible curved beam problem
Fig. 9. Vertical displacement distribution in the compressible curved beam problem
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Fig. 10. Equivalent Cauchy stress and deformed configuration for various values of shearing force in the compressible curved beam problem (colors indicate the equivalent stress (N/mm2 ))
5 Conclusion In this paper, a meshfree method based on integrated radial basis functions is proposed for analyzing the compressible and nearly-incompressible behaviors of hyperelastic domains under finite deformation state. By using the iRBF approach for approximating the derivative of a function in a local support domain, the developed method has shown its good convergence for finite deformation analysis in hyperelastic media as expected. The validity of the proposed method is shown through comparing the obtained numerical results with reference solutions. The iRBF based meshless method is promised for advanced analysis such as contact, crack propagation and damage problems in hyperelastic materials. Acknowledgment. This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 107.02–2019.327. The authors are gratefully acknowledged.
References 1. Ramabathiran, A.A., Gopalakrishnan, S.: Automatic finite element formulation and assembly of hyperelastic higher order structural models. Appl. Math. Model. 38, 2867–2883 (2014)
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2. Nomoto, A., Yasutaka, H., Oketani, S., Matsuda, A.: 2-dimensional homogenization FEM analysis of hyperelastic foamed rubber. Procedia Eng. 147, 431–436 (2016) 3. Angoshtari, A., Shojaei, M.F., Yavari, A.: Compatible-strain mixed finite element methods for 2d compressible nonlinear elasticity. Comput. Meth. Appl. Mech. Eng. 313, 596–631 (2017) 4. Li, S., Hao, W., Liu, W.K.: Numerical simulations of large deformation of thin shell structures. Comput. Mech. 25, 102–116 (2000) 5. Han, Z.D., Rajendran, A.M., Atluri, S.N.: Meshless local Petrov-Galerkin (MLPG) approaches for solving nonlinear problems with large deformations and rotations. CMES 10, 1–12 (2005) 6. Hu, D., Long, S., Han, X., Li, G.: A meshless local petrov-galerkin method for large deformation contact analysis of elastomers. Eng. Anal. Boundary Elem. 31, 657–666 (2007) 7. Gu, Y., Wang, Q., Lam, K.: A meshless local kriging method for large deformation analyses. Comput. Methods Appl. Mech. Eng. 196, 1673–1684 (2007) 8. Hu, D., Sun, Z., Liang, C., Han, X.: A mesh-free algorithm for dynamic impact analysis of hyperelasticity. Acta Mech. Solida Sin. 26(4), 362–372 (2013). https://doi.org/10.1016/ S0894-9166(13)60033-6 9. Khosrowpour, E., Hematiyan, M., Hajhashemkhani, M.: A strong-form meshfree method for stress analysis of hyperelastic materials. Eng. Anal. Boundary Elem. 109, 32–42 (2019) 10. Nam, M.-D., Thanh, T.-C.: Approximation of functions and its derivative using radial basis function networks. Appl. Math. Model. 27, 1997–2220 (2003) 11. Nam, M.-D., Thanh, T.-C.: Solving biharmonic problem with scattered-point discretization using indirect radial-basis-function networks. Eng. Anal. Boundary Elem. 30, 77 (2006) 12. Ho, P.L.H., Le, C.V., Tran-Cong, T.: Displacement and equilibrium mesh-free formulation based on integrated radial basis functions for dual yield design. Eng. Anal. Bound. Elem. 71, 92–10 (2016) 13. Ho, P.L.H., Le, C.V.: A stabilized iRBF mesh-free method for quasi-lower bound shakedown analysis of structures. Comput. Struct. 228, 106157 (2020) 14. Hassani, R., Ansari, R., Rouhi, H.: Large deformation analysis of 2D hyperelastic bodies based on the compressible nonlinear elasticity: A numerical variational method. Int. J. Non-Linear Mech. 116, 39–54 (2019) 15. Pascon, J.P.: Large deformation analysis of plane-stress hyperelasticity problems via triangular membrane finite elements. Int. J. Adv. Struct. Eng. 11, 331–350 (2019)
Structural Damage Identification of Plates Using Two-Stage Approach Combining Modal Strain Energy Method and Genetic Algorithm Thanh-Cao Le1,2,3 and Duc-Duy Ho1,2(B) 1 Faculty of Civil Engineering, Ho Chi Minh City University of Technology (HCMUT),
Ho Chi Minh City, Vietnam [email protected], [email protected] 2 Vietnam National University Ho Chi Minh City, Ho Chi Minh City, Vietnam 3 Faculty of Civil Engineering, Nha Trang University, Nha Trang, Vietnam
Abstract. In this study, a two-stage approach combining modal strain energy method and genetic algorithm (GA) to identify the location and the extent of damage in plate-like structures is developed. In the first stage, a criteria based on the change in modal strain energy namely Modal Strain Energy Damage Index (MSEDI) is utilized to determine the damage’s location. The modal strain energy is determined by using the modal analysis of plate-like structures in both states, before and after the occurrence of damages. In the second stage, the GA is employed to minimize the objective function with the variables relating to the vector of thickness reduction of the potential damaged elements, which are the result of the previous stage. The objective function is also based on modal strain energy. The effectiveness of the proposed method is analyzed and evaluated by numerical simulations for a plate with various damaged scenarios. The results show that the proposed method has the capability of exactly identifying the occurrence, the location and the severity of damages in plate-like structures. Keywords: Damage identification · Genetic algorithm · Modal strain energy · Plate · Vibration
1 Introduction The occurrence of damages is inevitable during service life of structures. If the damages are not detected timely, they will cause catastrophic incidents for the safety of not only self-structures but also the humans and society. One of the promising ways to guarantee the structural safety and integrity is to enact Structural Health Monitoring (SHM) in a regular periodic manner and to detect critical damage in its early stage. Structural health monitoring is defined as the process of implementing a damage identification and health evaluation strategy for engineering structures. In recent years, SHM has been playing a very important role for the sustainable performance of civil structures. Recently, the development of the vibration-based damage detection methods has been attracting the interest from many researchers. In addition, vibration-based damage identification © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 1004–1017, 2022. https://doi.org/10.1007/978-981-16-3239-6_79
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technologies have been widely used in mechanical, aerospace, and civil applications. In particular, the Modal Strain Energy (MSE) method has been proven to be one of the highly effective methods for damage detection in structures [1]. Stubbs et al. (1995) firstly applied MSE method to identify the damage in beam structures [2]. Then, Cornwell et al. (1999) have expanded MSE for the plate structures [3]. In both studies, a damage index is proposed on the basis of the comparison of modal strain energy before and after damage. Hu and Wu (2008) verified MSE method using experimental vibration results to detect surface cracks in thin, isotropic aluminum plate with the free boundary condition [4]. Le and Ho (2015) have developed MSE method for thin plate with different boundary conditions [5]. Fu et al. (2016) established a twostep procedure using a combination of MSE and response sensitivity analysis to identify damage in the plate structures using isotropic homogeneous material [6]. In the field of structural engineering, optimization algorithms have been applied for model updating to solve the problem of structural optimization and damage identification. Dinh et al. (2018) presented an effective two-step approach based on modal strain energy change and Jaya algorithm for damage assessment in plate-like structures [7]. Samir et al. (2019) have introduced a two-step approach for damage assessment in beam-like structures using normalized Modal Strain Energy Indicator (nMSEDI) and the Teaching-Learning-Based Optimization (TLBO) [8]. Among optimization tools, the genetic algorithm (GA) is used very popularly in optimization problems. This algorithm is a stochastic search algorithm based on principles of natural competition between individuals to find the individual with the best features. Then, the algorithm is applied in the optimization problem to find the variable that gives the best value of the objective function. Friswell et al. (1998) applied GA to identify both the position and extent of damage on the cantilever beam and cantilever plate structure. In this study, the objective functions used are frequency and vibration mode shapes [9]. Chou and Ghaboussi (2001) used GA to detect the existence, the location and th extent of damage for a plane structure. Parameters used in this study include measured and computed static displacements at few degrees of freedom on actual structures [10]. Marano et al. (2011) have proposed modified real-coded genetic algorithm to identify the unknown parameters (the mass, the stiffness, and the damping coefficients) of large structural systems subject to the dynamic loads [11]. From the previous studies, the damage identification in the plate structures has been successfully applied, including both the location and the extent of the damage. However, these studies use nodal displacements of mode shape vectors to calculate the strain energy values of the elements. From there, the damage indices are derived from these values. This only makes sense in theory, but not feasible for practical implementation because we need to collect too much nodal displacement components. It is also impossible to determine the rotational displacement components when the plate structure vibrates. In this study, the MSE method is improved to reduce modal data while ensuring the accuracy of the damage identification results. This improved method uses only the vertical displacement component of the nodes in the plate structures. In practice, the vertical displacement component on a real plates can easily be acquired through the measurement and analysis of acceleration signal at the corresponding nodes. A two-step damage identification
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procedure using MSE combined with genetic algorithm is proposed to identify both the location and extent of damage in the plate structures.
2 Modal Strain Energy and Genetic Algorithm-Based Damage Identification Approach 2.1 Finite Element Method for Free Vibration Analysis of Plate Structures The mode shape data of the plate structure with both the intact state and damaged state are input of MSE method. In this study, the finite element (FE) method was used to analyze the free vibration of plate structure. Four-node isometric quadrilateral element, denoted Q4, is used to model the plate structure. In Cartesian coordinate system, Q4 element has four nodes with the node coordinates shown in Fig. 1. When establishing the stiffness matrix and the mass matrix of the element, there is occurrence of the surface integrals. In order to approximate these quantities, Cartesian coordinate system is converted to a natural coordinate system of ξ η. Standard element in the natural system is a square with the coordinates of 4 nodes (−1, −1), (1, −1), (1,1), (−1,1), respectively. In a natural coordinate system, the surface integral willbe calculated based on Gaussian points. Each node has 5 displacement components ui vi wi θxi θyi , where (ui vi wi ) are three straight displacement components in the x, y and z directions and θxi θyi are two rotational components around the x and y axis, respectively.
Node 4th (x4;y4) Node 1st (x1;y1)
Node 3rd (x3;y3)
y Node 2nd (x2;y2)
x
z Fig. 1. Four-node isometric quadrilateral element in Cartesian coordinate system
The equation of motion for undamped free vibration of a reactangular plate structure can be expressed as: M d¨ + Kd = 0
(1)
in which d and d¨ are the displacement vectors and acceleration vectors of all nodes in whole plate, respectively; K and M are the global stiffness and mass matrices, respectively. These matrixes are assembled from the element matrices. The numerical integration method is used to determine stiffness and mass matrices of elements:
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I=
1 1 −1 −1
f (ξ, η)d ξ d η ≈
1 n −1 1
wi f (ξi , η)d η ≈
n
wj
1
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n
n n wi f ξi , ηj ≈ wi wj f ξi , ηj
1
i=1 j=1
(2)
For 4-node element, the membrane stiffness and flexural stiffness are determined using the numerical integration with 2 × 2 Gauss points. Meanwhile, the shear stiffnes of the element is determined using the numerical integration with 1 Gauss point. The coordinates of Gauss points and the respective weight coefficients are shown in Fig. 2. In the case of using 1 Gauss point, the node coordinate is (0,0) and the weight coefficient is 2. η
N4(-1;1) W4(-0,577;0,577)
x
N3(1;1) W3(0,577;0,577)
x
ζ
W1(-0,577;-0,577)
x
W2(0,577;-0,577)
x
N1(-1;-1)
N2(1;-1)
Fig. 2. Standard element in the natural system
Equation (1) is solved using the eigenvalue method; The solutions of the equation are the vibration frequencies (eigenvalues) and the mode shape vectors (eigenvectors), respectively. 2.2 Modal Strain Energy Method A thin plate with undamped free vibrations is divided into sub-regions as shown in Fig. 3. For a particular mode shape φk (x, y), the modal strain energy of the sub-region (i, j) is determined as follows [3]: 2 2 2 2 2 Dij yj xi+2 ∂ 2 φk ∂ φk ∂ φk ∂ φk + 2(1 − v) MSEk,ij = + 2v dxdy 2 yj xi ∂ 2x ∂ 2x ∂ 2y ∂x∂y (3) where Dij is the flexural stiffness of the sub-region (i, j); ν is the coefficient of Poisson. Total strain energy of the plate during elastic deformation is given by: MSEk =
Ny Nx i=1 j=1
M SEk,ij
(4)
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x
Element (i,j)
Element (Nx,Ny)
Lywith Ny elements
Ly
yi+1 yi
y xi xi+1 Lx with Nx elements
O
Lx
Fig. 3. Schematic of plate structure
The fractional strain energy the sub-region (i, j) is determined as follows: N
Fk,ij
Ny
x MSEk,ij = ; Fk,ij = 1 MSEk
(5)
i=1 j=1
Considering m mode shapes, the damage index of the sub-region (i, j) is defined as: m
βij =
k=1 m
∗ Fk,ij
(6) Fk,ij
k=1
where, the symbol “*” denotes damaged state of the structure. The damage index after normalization is determined as follows: MSEDIij =
βij − β¯ij σij
(7)
where β¯ij , σij are mean and standard deviation of the damage indices, respectively. MSEDIij values are used to locate the position of damaged elements in the plate structure. In Eq. (3), there are the occurrences of the second derivatives of the mode shape to two variables x and y in the modal strain energy formula. These values are calculated using the Center Differential Method (CDM) [5].
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2.3 Genetic Algorithm Damage in a structure will cause changes in the geometric or material properties of the structure, for example, a structural crack (stiffness change), bridge pillar silting (boundary condition change), counterweight balancing loss (mass change) or looseness in a bolted joint (connectivity change). For plate structure, a damage is assumed by reducing the stiffness of individual elements. The damage ratio or the stiffness reduction ratio of the jth element is described by the following equation: (8) kjs = 1 − αj kjo where: kjo , kjs are the stiffness of the jth element in the undamaged and damaged state, αj is the damage ratio of the jth element. Then, the damage ratio of the whole plate is represented by a vector as follows: α = {α1 α2 ... αk }
(9)
where k is the number of damaged elements in the plate. The vector α in Eq. (9) indicates the extent of damage of the elements needed to be identified. In this study, the objective function based on the modal strain energy value of the structure is shown as follows: m MSE d − MSE s (α s ) s i i (10) OF α = min αs MSEid i=1
where α s is the variable corresponding to the stiffness reduction of the elements; MSEid is the total modal strain energy of the ith mode shape corresponding to the damaged state and MSEis (α s ) is the total modal strain energy of the ith mode shape corresponding to the assumed damaged level. In this study, the genetic algorithm is used as an optimization tool. The goal of the algorithm is to find the variable α s that gives the minimum value of the objective function in Eq. (10). Any vector α s giving a smaller objective function value is considered as more accurate damage identification. 2.4 Flow Chart Based on the theory of the modal strain energy method and genetic algorithm, a twostage damage identification approach for plate-like structures is proposed in this study. In the first stage, the modal strain energy method is used to locate the damaged element. Then, genetic algorithm is deployed to identify the extend of damage in the second stage. The flow chart of the proposed approach is shown in Fig. 4.
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Intact plate structure
Damage plate structure
Mode shape Φh (acquired via simulation) Calculate MSEh
Mode shape Φd (acquired via measurement) Calculate MSEd
Locate damage elements
i=1
Assign αi a ramdon value Form objective function Update αi value for the next generation
Update stiffness matrix Ki based on αi
Calculate objective function value
Apply genetic algorithm in ith generation
i=i+1
No
Calculate mode shape Φ(αi) according to Ki
Stop condition
Yes
Conclude damage ratio of plate based on αi value
Fig. 4. Flow chart of two-stage damage identification approach
3 Numerical Verification 3.1 Properties of Plate An aluminum plate is 300 × 200 × 4 mm in size, four edges are fixed, as depicted in Fig. 5. The material parameters of the plate are given by: Young’s modulus E = 68.9 GPa, Poisson’s ratio ν = 0.33, and mass density ρ = 2710 kg/m3 . The FE model of the plate is discretized into a mesh 30 × 20 of four-node isometric quadrilateral elements. The parameters of the damage scenario are based on the research of Kumar et al. (2016) [12]. For the purpose of demonstrating the robustness of the proposed method, a scenario
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in which element 410th is assumed with stiffness reduction equals to 10% is considerd. The capability of the proposed method in identifying different levels of damage is also investigated.
Fig. 5. Mesh 30 × 20 of the plate and location of damaged element
3.2 Modal Analysis MATLAB software is used to simulate and analyze the free vibration for the target plate at both intact state and damaged state. Four-node isometric quadrilateral element is used to simulate plate following the Mindlin theory. Tables 1 and 2 show the first five frequencies and mode shapes for both states. It can be seen that for the intact case, the obtained frequencies are in excellent agreement with the corresponding results from the study [12] and calculated by SAP2000 software. The difference of the natural frequency values is very small, about 0.1 ~ 0.8%. 3.3 Identification of the Damage’s Location In the first stage, MSEDI indicators are used to locate the damage. The damage threshold has been chosen to be 50% to eliminate noise effects [5]. The key advantage in this study compared to previous studies is that in Eq. (3), the MSE values are calculated using only the vertical displacement component of every node. The damage scenario is investigated and evaluated the identification accuracy corresponding to some combination of mode shapes. MSEDI indicators are calculated with three cases: using only one mode; using
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1
2
3
4
5
MATLAB
661.9 1020.7 1629.4 1632.8 1955.4
Kumar [12] 660.3 1018.8 1615.9 1624.4 1947.2 SAP2000
661.4 1019.2 1619.3 1624.6 1945.7
a combination of the first three modes and using a combination of the first five modes. The damage location identification results are shown in Table 3. Table 2. The first five mode shapes in the intact state. Mode
MATLAB
SAP2000
1 Y(m)
X(m)
2
3
4
5
The damage location identification results of element 410th in plate structure show that: mode 1 and mode 2 have the best location identification results; mode 5 can successfully locate the actual damaged element with negligible false alarms; mode 3 can also locate element 410th but false alarms are significantly, whereas mode 4 can not locate the damaged element. Thus, the ability to detect the damage location of every mode is not the same. Some modes have the capability of damage location identification with high accuracy. Meanwhile, some modes show the low accuracy and a few modes fail to locate damage. However, the combination of three or five modes shows the high accuracy for
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Table 3. The capability of localization damaged element using different modes. Mode
Damage index chart
Identified damaged element
The capability of localization 410th
1
410
Yes
2
410
Yes
3
301, 302, 329, 330, 410
Yes
4
11, 20, 41, 50
No
5
315, 410
Yes
1~3
410
Yes
1~5
410
Yes
damage localization. This is a very important result, ensuring the proposed approach stable and reliable, regardless of the used mode shapes or the different locations of the damaged elements. 3.4 Identification of the Damage’s Extent In the second stage, the genetic algorithm are used to determine the extent of damage. The parameters of the genetic algorithm are given by: the range of the variable is based on range of the extent of damage, α s ∈ [0 ; 1]; The population size is 200 individuals; the minimum number of generations equals to 1. The algorithm will stop when one of two conditions is met: (1) the number of iterations equals to 100; (2) the objective function value has reached a certain pre-defined value, called fitness value. In this study, the
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fitness value equals to 1.0E-4. In order to evaluate the effectiveness of the damage extent identification algorithm, when the stopping condition is met, the following two values are suggested to evaluate: (1) the number of iterations had performed when the algorithm stopped; (2) the error in the damage extent identification when the stopping condition is satisfied. The damage extent identification results and the efficiency of proposed method are listed in Tables 4, 5, 6, 7, 8 and 9. Table 4. The damage extent identification results using mode 1 Number of iterations
1
Estimated damage extent (410th ) 0.084
2
3
0.084
0.100
Objective function value
8.3E-04 8.3E-04 2.5E-05
Error (%)
16.35
16.35
0.49
Table 5. The damage extent identification results using mode 2 Number of iterations
1
Estimated damage extent (410th ) 0.099 Objective function value
1.4E-05
Error (%)
0.91
Table 6. The damage extent identification results using mode 3 Number of iterations
10
72
Estimated damage extent (301th ) 0.286 Estimated damage extent (302th ) 0.111
1
0.073
0.073
0.650
0.653
Estimated damage extent (329th ) 0.220
0.167
0.171
Estimated damage extent (330th ) 0.018 Estimated damage extent (410th ) 0.187
0.158
0.158
0.424
0.425
Objective function value
8.3E-04 8.3E-04 2.5E-05
Error (%)
87.22
324.26
325.38
The analytical results in Tables 4, 5, 6, 7, 8 and 9 show: Mode 1 and mode 2 accurately identify the damage extent of element 410th after just three and one iterations, respectively. Even though the objective function value has been reached the fitness value, mode 3 and mode 5 are not able to detect the damage extent. Thus, when using only one mode, some modes may not be able to determine the damage extent. Meanwhile, when using the combination of the first three modes, the algorithm gives the accurate damage
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Table 7. The damage extent identification results using mode 5 Number of iterations
1
Estimated damage extent (315th ) 0.181 Estimated damage extent (410th ) 0.380
2
3
0.213
0.211
0.479
0.479
Objective function value
7.8E-05 1.6E-05 3.8E-07
Error (%)
279.90
378.72
378.70
Table 8. The damage extent identification results using a combination of the first three modes Number of iterations
1
3
5
6
7
Estimated damage extent (410th )
0.097
0.100
0.100
0.100
0.100
Objective function value
8.8E-02
3.0E-03
2.8E-03
1.3E-04
1.4E-05
Error (%)
2.99
0.10
0.09
0.00
0.00
Table 9. The damage extent identification results using a combination of the first five modes Number of iterations
1
3
5
7
8
Estimated damage extent (410th )
0.105
0.101
0.100
0.100
0.100
Objective function value
7.0E-02
1.1E-02
2.7E-03
7.4E-04
2.0E-05
Error (%)
4.88
0.80
0.19
0.05
0.00
extent identification results after 6 iterations and after eight iterations with the case of using the first five modes. The convergence investigation results of the algorithm according to the damage extent of the element 410th are shown in Table 10. The algorithm uses the first three mode shapes and the damage threshold is 50%. The results show that the proposed algorithm can identify exactly many diffenrent damage extents. In case of the damage ratio as small as 3%, the algorithm still gives accurate identification result after 12 iterations. When the damage ratio is as low as 1%, the algorithm can not identify correctly the damage extent after 30 iterations. However, in practice, damage ratio of less than 1% has almost no effect on the performance of the structure and can be neglected.
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Real damage extent (410th )
50%
10%
5%
3%
1%
Estimated damage extent (410th )
50.0%
10.0%
5.0%
3.0%
2.8%
Error (%)
0.00%
0.00%
0.03%
0.01%
182.00%
Objective function value
7.3E-05
1.9E-05
9.3E-05
1.2E-05
9.7E-02
Number of iterations
25
13
13
12
30
4 Conclusions In this study, an effective two-stage approach based on modal strain energy method and genetic algorithm was proposed for damage identification in plate-like structures. The proposed approach has the capability of exactly identifying the occurrence, the location and the severity of damages in plate-like structures. In the first stage, the potential damage locations are detected by normalized modal strain energy-based damage index (MSEDI), and then genetic algorithm is utilized in the second stage to estimate the extent of these damages. A numerical example was carried out to demonstrate the capability of the proposed approach. Based on the obtained results, some conclusions can be withdrawn as follows: (1) The MSEDI provides a high accuracy for locating the actual sites of damage. The genetic algorithm optimization with the objective function based on the modal strain energy value is proven to be an efficient tool for solving damage quantification problem. (2) Despite of using only vertical displacement component of the mode shape, proposed method still shows superior accuracy. This advantage plays a very important role in reducing the nodal displacement data and reflects the potential applicability of the method. (3) The study shows that the combination of the first few modes is sufficient to accurately identify the occurrence, the location and the extent of damage in next stage after only a few iterations. Despite this, the application of the proposed method in real conditions requires further investigation for dealing with more complex damage scenarios, the effect of modelling errors and environmental conditions.
Acknowledgements. This research is funded by Ho Chi Minh City University of Technology (HCMUT), VNU-HCM, under grant number BK-SDH-2021–1880698. We would like to thank Ho Chi Minh City University of Technology (HCMUT), VNU-HCM for the support of time and facilities for this study.
References 1. Fan, W., Qiao, P.: Vibration-based damage identification methods: a review and comparative study. Struct. Health Monit. 10, 83–111 (2011)
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2. Stubbs, N., Kim, J.T., Farrar, C.R.: Field verification of a nondestructive damage localization and severity estimation algorithm. In: Proceedings of 13th International Modal Analysis Conference, vol. 1, pp. 210–218 (1995) 3. Cornwell, P., Doebling, S.W., Farrar, C.R.: Application of the strain energy damage detection method to plate-like structures. J. Sound Vib. 224, 359–374 (1999) 4. Hu, H.W., Wu, C.B.: Nondestructive damage detection of two dimensional plate structures using modal strain energy method. J. Mech. 24, 319–332 (2008) 5. Le, T.C., Ho, D.D.: Damage detection in plate-like structures using modal strain energy-based approach. J. Constr. 6, 100–105 (2015) 6. Fu, Y.Z., Liu, J.K., Wei, Z.T., Lu, Z.R.: A two-step approach for damage identification in plates. J. Vib. Control 22(13), 3018–3031 (2016) 7. Dinh, C.D., Vo, D.T., Ho, H.V., Nguyen, T.T.: Damage assessment in plate-like structures using a two-stage method based on modal strain energy change and Jaya algorithm. Inverse Prob. Sci. Eng. 27(2), 166–189 (2018) 8. Samir, K., Magd, A.W., Boutchicha, D., Tawfiq, K.: Structural health monitoring using modal strain energy damage indicator coupled with teaching-learning-based optimization algorithm and isogoemetric analysis. J. Sound Vib. 448, 230–246 (2019) 9. Friswell, M.I., Penny, J.E.T., Garvey, S.D.: A combined genetic and eigensensitivity algorithm for the location of damage in structures. Comput. Struct. 69(5), 547–556 (1998) 10. Chou, J.-H., Ghaboussi, J.: Genetic algorithm in structural damage detection. Comput. Struct. 79(14), 1335–1353 (2001) 11. Marano, G.C., Quaranta, G., Monti, G.: Modified genetic algorithm for the dynamic identification of structural systems using incomplete measurements. Comput.-Aided Civ. Infrastruct. Eng. 26(2), 92–110 (2011) 12. Kumar, K.A., and Reddy, D.M.: Application of frequency response curvature method for damage detection in beam and plate like structures. In: IOP Conference Series: Materials Science and Engineering, vol. 149, pp. 1–11 (2016)
Auxetic Structure Design: A Multi-material Topology Optimization with Energy-Based Homogenization Approach T. M. Tran1 , Q. H. Nguyen1 , T. T. Truong2,3 , T. N. Nguyen2,3 , and N. M. Nguyen2,3(B) 1 Department of Computational Engineering, Vietnamese German University, Binh Duong,
Vietnam [email protected] 2 Department of Engineering Mechanics, Faculty of Applied Science, Ho Chi Minh City University of Technology, 268 Ly Thuong Kiet Street, District 10, Ho Chi Minh City, Vietnam [email protected] 3 Vietnam National University Ho Chi Minh City, Linh Trung Ward, Thu Duc District, Ho Chi Minh City, Vietnam
Abstract. In this research, the auxetic structure is designed by employing a multimaterial topology optimization approach to form the unit cells that reproduce negative Poisson’s ratio property. In detail, the multi-phase topology problem is divided into a series of binary phase topology optimization sub-problems, which are partially solved by homogenizing the material constitutive parameters concerning element mutual energies. With constraints on individual material volume fraction, this numerical framework allows extremizing the objective function constructed by homogenized Poisson’s ratio. Several two-dimensional numerical examples are studied to illustrate the performance of this algorithm. Besides, benchmark auxetic designs are verified through finite element simulations. Keywords: Topology optimization · Homogenization · Multi-phase · Auxetic material
1 Introduction Topology optimization of structures is nowadays a well-developed field with many different approaches and a wealth of applications. Several advanced techniques, such as level-set [1, 2], phase-field [3, 4], homogenization [5], or more recently, the direct multiscale topology optimization [6–8], are developed to design the optimum shape and size of engineering structures under different constraints. Among them, homogenization-based topology optimization, introduced in the early eighties, is one of the earliest methods. It became well-known in its simplified version, Solid Isotropic Material with Penalization (SIMP), which preserves only the assumption of material density and forgets about proper composite materials with optimal microstructures. Thomsen [9] was the first work where the material compositions are controlled by employing the SIMP method in © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 1018–1032, 2022. https://doi.org/10.1007/978-981-16-3239-6_80
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a utilized linear interpolation scheme to design multi-material structure through topology optimization. Various interpolation schemes to model multiple materials are later investigated, and Bendsøe and Sigmund [10] summarized the rules of multi-material interpolation under the density-based framework. Recently, Gaynor et al. [11], implemented both this scheme and a new alternative combinatorial SIMP method in association with minimum feature sizes, related to the manufacturing process resolution, to design multi-material compliant mechanisms. Results demonstrate that the combinatorial SIMP method performed better control over feature sizes and the sizes of every single material. Watts and Tortorelli [12] proposed improvement by implementing a smooth thresholding scheme to specify the volume fraction of each material. This advanced scheme eliminates the nonlinearity issues seen Bendsøe and Sigmund’s material interpolation scheme when going beyond three materials. Even though topology optimization methods for multi-material are widely studied, their applications on designing auxetic materials - artificial materials with negative Poisson’s ratio - are still limited. To author’s best knowledge, Vogiatzis et al. [13], is the first report of proposing an optimized structure for multi-phase auxetic metamaterials employing reconciled level set-based scheme. By representing each distinct material with a single level set function and adapting the resulted level set functions with the Merriman–Bence–Osher operator, they have successfully achieved novel designs for two-dimensional (2D) multi-material, and exceptionally innovative three-dimensional (3D) multi-metamaterials where a new assembly of multiple microstructures with a different Poisson’s ratio in each direction can be generated. In a similar motivation of implementing newly advanced topology optimization method for inventing multimaterial auxetic structure, this paper proposes a robust two-dimensional computational framework. We follow the concept of Xia et al. [14], where an equivalent energy-based homogenization approach employing average stress and strain theorems is adapted to predict the effective material properties. Besides, the multi-phase topology optimization problem is conducted using the alternating active phase algorithm (AAPA) [15]. In particular, the problem is split into a series of binary phase topology optimization sub-problems; the binary phase solutions are later coupled based on the block coordinate descent method (see more details in Tavakoli et al. [15]). Formerly, the AAPA includes two loops: one inner loop for the binary-phase subproblems and one outer loop for the multi-phase problem. However, it is found that the convergence of the inner loop is only loosely checked, and thus, the inner loop is removed in this research. The second modification is made following a suggestion of [16]: The target phase is not updated after each sub-problem; instead, it is updated by using a weighted sum of the results of all sub-problems in which it is involved. The third modification is the enforcement that the softer phase is not distributed into elements which are dominated by stiffer phases. The enforcement is done simply by using passive elements. The organization of this paper is as follows: homogenization theory and the prediction of material effective properties are briefly reviewed in the next Sect. 2. Section 3 presents an algorithm of multi-phase topology optimization problem. Section 4 discusses the optimization model, while Sect. 5 gives several numerical experiments using the proposed algorithm. Finally, this work is summarized in Sect. 6.
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2 Homogenization with Periodic Boundary Conditions Homogenization is a method of averaging to overcome the computational-cost difficulty in the analysis of the boundary value problems consisting of media with a large number of heterogeneities. One crucial assumption of this technique is that the microstructure is spatially periodic, which means the characteristic functions representing the physical quantity of the heterogeneous medium must have the following property: (x + NY) = (x). x is a position vector of the material point, N is a diagonal matrix of integer numbers, and Y is a constant vector determining the repeated pattern of the media. In the theory of homogenization, the Y-periodic is assumed to be infinitesimal compared with dimensions of the domain. Hence, with the high level of heterogeneity inside the material, the characteristic functions will vary rapidly within a tiny neighbourhood of a point x. This fact inspires the reason to claim that all quantities have two explicit dependencies: one on the macroscopic or global level x, and the other on the microscopic or local level y = ηx . In this assumption, η is a ratio between the period and the characteristic size of the structure equals to η 1. By using the chain rule, derivatives of any function g(x, y) with respect to x is written as ∂g ∂g 1 ∂g = + . ∂xi ∂xi η ∂yi
(1)
Let us consider a linear elastic deformation of a material body , constructed with a periodic cellular microstructure Y. The whole system is exerted by body forces f and traction t. The smooth boundary includes d ( where displacements are prescribed) and t (the traction boundary) as in Fig. 1. t Γt y
f x
Ω Γd
Fig. 1. Illustration of a macro-micro scale
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Following the asymptotic homogenization discussed by Guedes [17], the macro displacement field u can be approximately expanded: (x, y) = u0 (x) + ηu1 (x, y) + η2 u2 (x, y) + ...,
(2)
where u0 is continuum level displacements, and u1 , u2 , etcetera are perturbations of the displacement field due to the microstructure. The small strain deformation ε is written in index notation as: ∂uj ∂u0j ∂u1j 1 ∂u0i ∂u1i 1 ∂ui = + + O(ηn ). (3) εij (u) = + + + 2 ∂xj ∂xi 2 ∂xj ∂xi ∂yj ∂yi − ε ij
Neglecting the higher-order terms of O(ηn ), the average macroscopic strain tensor and the fluctuating strain tensor εij can be defined from Eq. (3) εij (u) = ε¯ ij + εij∗ ,
ε¯ ij =
∂u0j 1 ∂u0i , + 2 ∂xj ∂xi
εij∗ =
∂u1j 1 ∂u1i . + 2 ∂yj ∂yi
(4)
The accuracy of the analysis may be improved by including the higher-order terms, but it also needs additional calculations ([18]). Similarly, the virtual displacement v and the virtual strain ε(v) is also asymptotically expanded, and then, the virtual strain components are assigned after ignoring the higher-order terms O(ηn ) ∂v0j ∂v1j 1 ∂v0i 1 ∂v1i 0 1 0 1 εij (v) = εij + εij , εij = + + , εij = . (5) 2 ∂xj ∂xi 2 ∂yj ∂yi By substituting the components of linear strain and virtual strain deformation Eq. (4), Eq. (5) into the standard weak form of the equilibrium equation, we obtain the following expansion ∗ ∫ εij0 (v) + εij1 (v) Cijkl ε¯ kl (u) + εkl (6) (u) d = ∫ vi fi d + ∫ vi ti d .
t
Here, v is an arbitrary test function belonging to Sobolev space; hence, it may be kept to vary only on the macro or micro scale. If v is chosen to vary independently on x, Eq. (6) becomes ∗ ∫ εij1 (v)Cijkl ε¯ kl (u) + εkl (7) (u) d = 0
and if v is chosen to vary independently on y, Eq. (6) can be simplified to ∗ ∫ εij0 (v)Cijkl ε¯ kl (u) + εkl (u) d = ∫ vi fi d + ∫ vi ti d .
t
(8)
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For the smooth enough Y-periodic function and assuming that η limits to zero, Eqs. (7) and (8) are rewritten as 1 ∗ ∫ ∫ εij1 (v)Cijkl ε¯ kl (u) + εkl (9a) (u) dY d = 0 |Y | Y 1 ∗ ∫ Cijkl ε¯ kl (u) + εkl ∫ εij0 (v) (9b) (u) dY d = ∫ vi fi d + ∫ vi ti d , |Y | Y t where |Y | stands for the area (or volume in the three-dimensional region) of the periodic −
cell. Due to the linear elastic problem, the arbitrary ε kl (u) can be represented as a linear −kl
combination of four (2D) (or nine (3D)) independent unit test strain fields ε pq 10 00 01 00 11 22 12 21 ε¯ pq =− =− =− =− , ε¯ pq , ε¯ pq , ε¯ pq . 00 01 00 10 The Eqs. (9a) and (10) are therefore simplified to kl ∗kl ∫ εij1 (v)Cijpq ε¯ pq − εpq (u) dY = 0,
(10)
(11a)
Y
H ∫ εij0 (v)Cijkl d = ∫ vi fi d + ∫ vi ti d .
(11b)
t
kl (u) is determined from Eq. 12, the homogenized stiffness tensor C H Once εpq ijkl defined by the mean of the integral over the periodic pattern Y , is obtained: H Cijkl =
1 kl ∗kl ∫ Cijpq ε¯ pq − εpq (u) dY . |Y | Y
(12)
According to Sigmund [19], the average elasticity tensor in Eq. (14) can be expressed in an equivalent form in terms of element mutual energies to implement effective existing algorithms used in the topology optimization H Cijkl =
1 ij ∗ij kl ∗kl ∫ Cpqrs ε¯ pq − εpq (u) ε¯ rs − εrs (u) dY . |Y | Y
(13)
In finite element analysis, the periodic pattern Y is discretized into N elements so that Eq. (17) is approximated by 1 ij ue ke uekl , |Y | N
H Cijkl =
(14)
e=1
−kl
with uekl is element displacement corresponding to the unit test strain fields ε , and ke is the element stiffness matrix.
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Under the assumption of the periodic structure, the periodic boundary conditions must be applied to each repeated base cell Y to imply that these patterns have the same deformation mode and there is no separation or overlap between the neighbouring. As stated by Michel [20], this condition on the boundary of representative domain Y is: uk = ε¯ kl yl + uk∗ ,
(15)
−
with uk is the displacement field of the base cell, ε kl is the global strain computed via −kl
unit test strain fields ε pq , and uk is the periodic fluctuation of displacement field on the boundary surfaces. In general, uk is unknown; thus, (15) cannot be directly imposed on the boundaries. Xia et al. [21] proposed a more explicit form of (15), considering the displacements at a pair of nodes on the opposite boundaries of the base cell uki+ = ε¯ kl yli+ + uk∗ ,
(16a)
uki− = ε¯ kl yli− + uk∗ ,
(16b)
here the index "i + " and "i − " denote the pair of opposite boundaries normal to the i th direction. The periodic fluctuation term uk can be omitted through the difference between the above two displacements uki+ − uki− = ε¯ kl yli+ − yli− = ε¯ kl yli , (17) −
with a specified ε kl , yli is unchanging for any given base cell. Thus, the right-hand side of (17) becomes constant. This unified periodic form of boundary conditions meets the necessities of periodicity and continuity for both displacement and stress fields and adapts easier to finite element analysis.
3 Alternating Active Phase Algorithm (AAPA) of Topology Optimization The single material topology optimization problem can be considered as a binary phase problem in which one phase is a solid material, and the other is voided region. For multimaterial topology optimization, nph kinds of materials are involved, in which voided region is also considered as a phase. The problem is to find the optimal distribution of nph kinds of materials that lead to the minimized value of objective function, e.g. compliance. Mathematically, the problem can be expressed in the following form min J (α, U (α))
(18)
subject to : R(M (α), U (α) = 0 in
(19)
α
In Eq. (21), J is the objective function, and α is the vector field that denotes the material distribution, i.e. α = {α1 , α2 , ..., αnph }. Equation (22) is the constraint in the
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form of a partial differential equation (PDE), e.g. the equilibrium equation in case of mechanical elasticity. U (α) is the solution of the PDE corresponding to the material distribution α. M (α) denotes the function that interpolates the material properties from local volume fraction data. The summation of volume fraction at every point x ∈ should be equal to unity nph
αi = 1.
(20)
i=1
A global volume constraint is also usually imposed on the total volume of each material inside ∫ αi dx = vi ||, i = 1, 2, ..., nph ,
(21)
where 0 ≤ vi ≤ 1 are user-defined indicating the global volume fraction of
parameters, nph vi = 1. each material phase. Obviously, i=1 Initially proposed by Tavakoli and Mohseni [15], the Alternating active phase algorithm AAPA aims to generalize the procedure for binary phase problem into multimaterial topology optimization problems. The key idea of the algorithm is to split a multi-phase problem into a series of binary phase sub-problems, in which a standard procedure for binary phase topology optimization can be applied. An nph -phases probn (n −1)
lem is decomposed into ph 2ph binary sub-problems. At a particular time, only one binary sub-problem is solved, i.e. only two phases are considered as active and are updated, while the rest nph − 2 phases are kept unchanged. Two nested loops implement the algorithm. An outer loop sequentially selects the two active phases, and an inner loop is used to solve the binary phase problem. Denoting the two active phases as phase a and phase b, their total volume fraction in each element can be calculated as
nph
αi = 1 − αj (22) j=1,j={a,b}
i=a,b
Following the single material algorithm, where the material region is updated, here the phase a, namely the target phase, is renewed. Then the density of phase b is computed by
αb = αi − αa (23) i=a,b
The advantage of AAPA is its simplicity. Once a single material algorithm is validated, it is straightforward to extend to multi-material topology optimization by combination with AAPA. However, it is found in the MATLAB code provided by [15] that the inner loop is redundant. The inner loop is stopped after some user-specified number of iterations, without any convergence check. Convergence is required for the global problem (multimaterial problem), not the local problem (binary phase sub-problem). Therefore, in the current research, the inner loop is removed, i.e. only one iteration is done for each binary phase sub-problem.
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The second modification is on how the target phase is updated, following the discussion presented by Cui and Chen [16]. By the original AAPA [15], the target phase a is immediately updated the sub-binary problem in which it is involved is solved. Each target phase a (a = 1, 2, ..., nph − 1) is involved in (nph − 1) binary sub-problems. Therefore within one iteration of the outer loop, the target phase a is updated (nph − 1) times. Since the sub-problems are sequentially solved, the results of the previous subproblem may affect the subsequent ones. In the current research, the results of all the binary sub-problems are stored. Finally, a weighted average is taken as the values of the target phase a, for any a = 1, 2, ..., nph − 1.
nph
i=1,i=a ki αai
αa (updated ) =
nph
j=1,j=a kj
,
(24)
where αai denotes the value of αa calculated in the binary sub-problem between phase a and phase i. The weighted coefficient ki is calculated by vi ki = nph
(25)
j=1,j=a vj
The voided phase is then updated by αnph = 1 −
nph −1 i=1
αi .
(26)
It is noticed that by using AAPA, one point generally contains all the material phases. In 3D printing, it is preferred that one point should be either one solid phase or voided phase. Therefore, a modification is proposed in this work. If a stronger phase already dominates an arbitrary point, i.e. the local volume fraction of the one in that point is higher than 0.75, the weaker phases are not allowed to be distributed to that point. This constraint is done simply by using passive elements.
4 Optimization Model Following [14], the objective function is defined by H H H min J (α) = C1122 . − β iter * C1111 + C2222 αe
(27)
in which design variables αe = {α1 , α2 , ..., αnph } are the density values of nph phases at H are the components of the homogenized elastic tensor, computed each elemente. Cijkl by Eq. (16). Parameter β ∈ [0, 1] is a fixed number chosen by the user, and iter is the iteration. With the objective function in Eq. (30), the optimizer tends to maximize the H H . After a few iterations, the and C2222 horizontal and vertical stiffness moduli (i.e. C1111 H optimizer then tends to minimize C1122 ; and thus the Poisson’s ratio is reduced, even to a negative value. When β < 1 is selected, the second term becomes smaller as the H is aggressively forced to decrease, in number of iteration increases, hence the term C1122 order to minimize objective functionJ . However, large change after each iteration may
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happen with a small value of β, which makes it harder to get convergence. In this paper, β = 0.8 is adopted. For a two-dimensional plane stress problem, the Poisson’s ratio ν is determined by ν=
H C1122 H C1111
(28)
The following constraints are defined in the optimization problem K · U = F.
Ne e=1
αi,e = vi ||, i = 1, 2, ..., nph ,
(29) (30)
in which Eq. (32) is the equilibrium, while Eq. (33) expresses the volume constraint on each phase. The power-law SIMP model (Solid isotropic material with penalization) [22] is employed s Ei,0 ) . (31) E αi,e = αi,e Here, Ei,0 is the modulus of phase i and s is the penalty power in the SIMP model. It should be noted that the voided region is also considered as one material phase, whose modulus is chosen to be very small to avoid the singularity. The modulus of element e due to the existence of multi-materials is calculated by Ee =
n−1 Ph i=1
Ei,e .
(32)
5 Numerical Examples 5.1 Single-material Auxetic Patterns The proposed algorithm is first applied to a unit cell of a single material, i.e. a two-phase problem with one solid phase and one voided phase. The initial design is chosen as depicted in Fig. 2. The centre of the domain is determined; namely, point O. A circle whose centre is point O and radius is 1/6 of the domain side is defined. Zero density (voided region) is then assigned to all the elements inside that circle. The other elements are assigned by a density value equal to the volume fraction of the solid phase. Material properties of the solid phase are given by Young’s modulus E = 1 Mpa and Poisson’s ratio ν = 0.3. The unit cell is uniformly discretized into 100 × 100 quadrilateral elements. Figure 3 and Fig. 4. presents the auxetic patterns of the single material with volume fraction being 0.3 and 0.5, respectively.
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Fig. 2. Initial pattern of auxentical single material
Fig. 3. Auxetic pattern of the single material with 30% volume fraction of the solid phase, displayed by one unit cell and 3 × 3 cells. The homogenized Poisson’s ratio is ν = −0.6604
Fig. 4. Auxetic pattern of the single material with 50% volume fraction of the solid phase, being displayed by one unit cell and 3 × 3 cells. The homogenized Poisson’s ratio is ν = −0.6358
5.2 Multi-material Auxetic Patterns In this example, without loss of generality, a three-phase problem is investigated, including two solid phases and one voided phase. The initial design for the three-phase problem is chosen similarly to that in the single material problem, as depicted in Fig. 5. Instead of being voided, the circular region is now not filled (density = 1) by the weak solid phase. All the rest of the elements are assigned by a density equal to the volume fraction of the strong solid phase. The Young’s modulus of the strong solid phase is E = 5 MPa, and
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that of the weak solid phase is E = 1 MPa. Both solid materials have Poisson’s ratio of value ν = 0.3. Similar to the single-material example, here, the design domain (the unit cell) is also uniformly discretized into 100 × 100 quadrilateral elements. Three cases of volume fraction are considered: [0.1, 0.4, 0.5] (i.e. 10% of strong solid phase, 40% of weak solid phase and 50% voided region), [0.25, 0.25, 0.5], and [0.4, 0.1, 0.5]. Familiar auxetic patterns are found in all numerical results. Especially, the patterns obtained for single material are close to those reported in kinds of literature [13, 23]. The homogenized Poisson’s ratio is calculated by Eq. (31), showing negative value as expected. The pattern presented in Fig. 8. is studied by finite element analysis (FEA), which exhibits the characteristic response of the auxetic structure, such that when the structure is stretched along a vertical direction, the horizontal displacement is also positive. FEA results of other patterns display similar behaviour. Therefore, they are not mentioned for the sake of brevity.
Fig. 5. Initial pattern of auxentical multi-material
Fig. 6. Multi-material auxetic pattern with 10% of the strong solid phase (red), 40% of the weak solid phase (blue), and 50% of the voided region, being displayed by one unit cell and 3 × 3 cells. The homogenized Poisson’s ratio is ν = −0.4780.
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Fig. 7. Multi-material auxetic pattern with 25% of the strong solid phase (red), 25% of the weak solid phase (blue), and 50% of the voided region, being displayed by one unit cell and 3 × 3 cells. The homogenized Poisson’s ratio is ν = −0.5193.
Fig. 8. Multi-material auxetic pattern with 40% of the strong solid phase (red), 10% of the weak solid phase (blue), and 50% of the voided region, being displayed by one unit cell and 3 × 3 cells. The homogenized Poisson’s ratio is ν = −0.6503.
5.3 Benchmark Verification In order to demonstrate the auxetic characteristic of micro-structured materials, the above-optimized result in Fig. 8 is imported into the engineering software ANSYS for simulating the deformation of the NPR (Negative Poisson’s Ratio) structure. We stretch the proposed structure along the y-direction (1E-03 mm), and the extended depth is much less than the length (1 mm) and width (1 mm) to maintain the plane stress condition to some extent. The finite element mesh and its displacement field in the x-direction are shown in Fig. 9. Meanwhile, the initial configuration is also plotted in the black line. Based on the results in the simulation, the maximum displacement in the left and right boundaries are equal to 0.682E-03 mm and 0.679E-03 mm, respectively. One can easily see the confirmation of auxetic feature that the structure expands along the width direction when stretched in the height direction.
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Fig. 9. NPR finite element mesh and displacement field in ANSYS.
6 Conclusions This paper proposes a numerical scheme to determine multi-material auxetic patterns by SIMP-based topology optimization. The material properties are homogenized, and the structure is assumed to be formed by repeating unit cells. The design domain is then defined in one unit cell with periodic boundary conditions. Using the Alternating active phase algorithm (AAPA), the multi-material topology optimization is decomposed into a series of binary phase problems, which are sequentially solved in every iteration.
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There is no unique solution for auxetic patterns. As it is observed in numerical examples, the patterns may differ concerning volume fraction of solid phases. Therefore, in practice, the volume fraction of each solid phase has to be clearly defined. A constraint may also be added into the problem, e.g. material cost should not exceed a specified value.
References 1. Allaire, G., Jouve, F., Toader, A.-M.: A level-set method for shape optimization. C.R. Math. 334, 1125–1130 (2002) 2. Wang, M.Y., Wang, X., Guo, D.: A level set method for structural topology optimization. Comput. Methods Appl. Mech. Eng. 192, 227–246 (2003) 3. Bourdin, B., Chambolle, A.: Design-dependent loads in topology optimization. . ESAIM: Control Optim. Calc. Var. 9, 19–48 (2003) 4. Takezawa, A., Nishiwaki, S., Kitamura, M.: Shape and topology optimization based on the phase-field method and sensitivity analysis. J. Comput. Phys. 229, 2697–2718 (2010) 5. Bendsøe, M.P., Kikuchi, N.: Generating optimal topologies in structural design using a homogenization method. Comput. Methods Appl. Mech. Eng. 71, 197–224 (1988) 6. Hoang, V.-N., Tran, P., Nguyen, N.-L., Hackl, K., Nguyen-Xuan, H.: Adaptive concurrent topology optimization of coated structures with non-periodic infill for additive manufacturing. Comput. Aided Des. 129, 102918 (2020) 7. Hoang, V.-N., Tran, P., Vu, V.-T., Nguyen-Xuan, H.: Design of lattice structures with direct multiscale topology optimization. Compos. Struct. 252, 112718 (2020) 8. Hoang, V.-N., Nguyen-Xuan, H.: Extruded-geometric-component-based 3d topology optimization. Comput. Methods Appl. Mech. Eng. 371, 113293 (2020) 9. Thomsen, J.: Topology optimization of structures composed of one or two materials. Struct. Optim. 5, 108–115 (1992) 10. Bendsøe, M.P., Sigmund, O.: Material interpolation schemes in topology optimization. Arch. Appl. Mech. (Ingenieur Archiv) 69, 635–654 (1999) 11. Gaynor, A.T., Meisel, N.A., Williams, C.B., Guest, J.K.: Multiple-material topology optimization of compliant mechanisms created via polyjet three-dimensional printing. J. Manuf. Sci. Eng. 136, 1–11 (2014) 12. Watts, S., Tortorelli, D.A.: An n -material thresholding method for improving integerness of solutions in topology optimization. Int. J. Numer. Meth. Eng. 108, 1498–1524 (2016) 13. Vogiatzis, P., Chen, S., Wang, X., Li, T., Wang, L.: Topology optimization of multi-material negative Poisson’s ratio metamaterials using a reconciled level set method. Comput. Aided Des. 83, 15–32 (2017) 14. Xia, L., Breitkopf, P.: Design of materials using topology optimization and energy-based homogenization approach in Matlab. Struct. Multidiscip. Optim. 52(6), 1229–1241 (2015). https://doi.org/10.1007/s00158-015-1294-0 15. Tavakoli, R., Mohseni, S.M.: Alternating active-phase algorithm for multi-material topology optimization problems: a 115-line MATLAB implementation. Struct. Multidiscip. Optim. 49, 621–642 (2014) 16. Cui, M., Chen, H.: An improved alternating active-phase algorithm for multi-material topology optimization problems. Appl. Mech. Mater. 635–637, 105–111 (2014) 17. Guedes, J., Kikuchi, N.: Preprocessing and postprocessing for materials based on the homogenization method with adaptive finite element methods. Comput. Methods Appl. Mech. Eng. 83, 143–198 (1990) 18. Bourgat, J.F.: (1979) Numerical experiments of the homogenization method. Comput. Methods Appl. Sci. Eng. I, 330–356 (1977)
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19. Sigmund, O.: Materials with prescribed constitutive parameters: An inverse homogenization problem. Int. J. Solids Struct. 31, 2313–2329 (1994) 20. Michel, J.-C., Moulinec, H., Suquet, P.: Effective properties of composite materials with periodic microstructure: a computational approach. Comput. Methods Appl. Mech. Eng. 172, 109–143 (1999) 21. Xia, Z., Zhang, Y., Ellyin, F.: A unified periodical boundary conditions for representative volume elements of composites and applications. Int. J. Solids Struct. 40, 1907–1921 (2003) 22. Sigmund, O.: Morphology-based black and white filters for topology optimization. Struct. Multidiscip. Optim. 33, 401–424 (2007) 23. Gao, J., Xue, H., Gao, L., Luo, Z.: Topology optimization for auxetic metamaterials based on isogeometric analysis. Comput. Methods Appl. Mech. Eng. 352, 211–236 (2019)
Fuzzy Structural Identification of Bar-Type Structures Using Differential Evolution Ba-Duan Nguyen and Hoang-Anh Pham(B) Department of Structural Mechanics, National University of Civil Engineering, Hanoi, Vietnam {duannb,anhph2}@nuce.edu.vn
Abstract. In this paper, a non-probabilistic procedure is introduced to identify the member stiffness of bar-type structures considering uncertainties. The approach is a combination of fuzzy set theory, finite element model updating methodology, and differential-based optimization technique. Uncertainties associated with the measured outputs, as well as the unknown stiffness parameters, are described as fuzzy quantities. The theoretical fuzzy model outputs are predicted based on local sensitivity indices. The fuzzy identification is formulated as a constrained optimization problem at each α-cut level, with the objective function being the difference between the bounds of measured outputs and those of the predicted model outputs. A modified differential evolution algorithm (DE-NNC) is suggested to determine the bounds of the stiffness parameters corresponding to each α-level are. Two simulation examples, including a 2D truss and a 2D frame, are investigated to show the effectiveness and accuracy of the presented fuzzy structural identification procedure. It is further demonstrated that DE-NNC is superior to some popular meta-heuristics in updating the member stiffness of the structures. Keywords: Structural identification · Fuzzy model updating · DE-NNC · Bar-type structure
1 Introduction In practice, there always exist differences and discrepancies between a structure in its actual condition and its initial reference state, especially when damage occurs in its members. The damage can be due to, for example, the defect and crack in structural elements. Damages in structural members often lead to the reduction of structural stiffness and load caring capacity of the structure. Therefore, the structural health needs to be properly assessed during the operation of a structure to provide an appropriate strengthening/repairing plan. Mostly, quantitative and qualitative damage detections are possible with the utilization of a mathematical model (e.g., a finite-element-model) and the measured data from dynamic non-destructive-testing, such as natural frequencies and mode shapes [1]. The parameter values of the model will be determined through an optimal correlation between the measured data and the model output. Then, the damage can be quantified and localized by comparing the updated parameter values with their initial © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 1033–1051, 2022. https://doi.org/10.1007/978-981-16-3239-6_81
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values. Thus, the focal issue is the identification of the unknown parameter values for the model from the experimental data [2]. Unfortunately, experimental data are contaminated with noises, which may affect the correctness of the identified results. Therefore, uncertainty quantification methods have been suggested to be used in the identification procedure [3]. The identification of the unknown parameter values is usually formulated as an optimization problem with the objective function being the difference between the theoretical model outputs and the measured data, and the design variables being the model parameter values. The main issues in solving such kinds of inverse problems are: 1) the objective function is often non-smooth and multi-modal, which is very challenging to minimize; 2) the evaluation of the objective function can be costly. Conventional gradient-based techniques are efficient in terms of convergence rate, but they are easily trapped in local minima. In recent years, the utilization of meta-heuristics (MHs) to solve the inverse problem has gained extensive research interest. For example, some popular MHs implemented successfully in the structural identification include genetic algorithm (GA) [4–6], particle swarm optimization (PSO) [6–9], artificial bee colony [10, 11], ant colony optimization [12], teaching-learning-based optimization (TLBO) [13], cuckoo search [14], Jaya algorithm [8, 15, 16], or hybrid technique [17]. However, MHs are in general computationally expensive, since they often require a huge number of objective function evaluations to attain a well acceptable solution. This paper, therefore, suggests an alternative MH that is useful to reduce the computation burden in solving the structural identification problem. The method, named Differential Evolution with Nearest Neighbor Comparison (DE-NNC), is integrated with the fuzzy finite element (FE) model updating based on the α-cut strategy [18] to update the stiffness of bar-type structures considering fuzzy uncertainties. First, Sect. 2 briefly presents the fuzzy FE model updating procedure. An interval analysis based on the local sensitivity index is suggested to predict the theoreticalssss fuzzy model outputs effectively. Then, in Sect. 3, DE-NNC is presented as the optimizer to determine the bounds of the stiffness parameters. In Sect. 4, the proposed procedure is investigated by identifying the member stiffness of two structures, including a plane truss and a plane frame. The efficiency of DE-NNC is highlighted in comparison with some popular MHs. The accuracy of the identified results is also discussed in Sect. 4. Finally, Sect. 5 gives some conclusions.
2 Fuzzy Fe Model Updating 2.1 FE Model Updating Procedure The FE model updating procedure for the structural identification problem of a structure is described in Fig. 1. In the beginning, the FE model of the structure is built based on the initial assumption about the unknown model parameters. These parameters will be updated according to the experimental measurements of the actual behavior of the structure (Fig. 1). The determination of suitable parameter values is usually done by solving an optimization problem with the objective function being the error between the predicted model behavior and the measured data. Let θ = {θ1 , θ2 , ..., θm } be a vector containing
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Uncertain structure
FE modelling
Experimental measurement
Predicting behavior of the model
Measured data
Yes
No
Optimal parameter values
Updating parameter values Fig. 1. Flow chat of FE model updating
the unknown parameter values of the model that will be determined through the updating procedure. The error between the theoretical behavior of the model and the measured data is determined by δ(θ ) = u∗ − umodel (θ )
(1)
where umodel is the theoretical behavior of the model; and u∗ is the corresponding measured data. The optimization problem of finding the model parameter vector is given as follows: Minimize δ(θ ) θj min ≤ θj ≤ θjmax , j = 1, . . . , m
(2)
where θj min and θjmax are the lower and upper bounds of the j-th model parameter θj , respectively.
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2.2 Fuzzy Concept of Uncertainty The fuzzy set theory, which was introduced by Zadeh [19] in 1965, can be used to describe the uncertain quantities, like structural parameters and experimental data, due to insufficient, incomplete, and vague information. Unlike the crisp set, where an element either belongs or does not belong to the set, in a fuzzy set, the belongingness of an element to the set is measured by the grade of membership. A fuzzy set x˜ is defined as [19]: x˜ = {(x, μx˜ (x))|x ∈ X , μx˜ (x) ∈ [0, 1]}
(3)
where X is a set, and μx˜ is called the membership function of x˜ . Corresponding to each member x ∈ X , the value μx˜ (x) defines the level of belongingness of x to x˜ . If μx˜ (x) = 0, x does not belong to x˜ ; if μx˜ (x) = 1, x definitely belongs to x˜ ; and if 0 < μx˜ (x) < 1, x has a certainly degree of belongingness to x˜ . The fuzzy set commonly used in engineering practice is the triangular fuzzy set, which has the membership function given as: μx˜ (x) = 0, x ≤ xL ; xU < x x − xL μx˜ (x) = , xL < x ≤ xM xM − xL μx˜ (x) =
(4)
xU − x , xM < x ≤ xU xU − xM
In practice, the analysis of a problem with fuzzy inputs is often carried out by using the α-cut approach. In the α-cut approach, the membership function of fuzzy inputs is discretized into a finite number of levels. The α-cut of the fuzzy input x˜ at an α-level is defined as: Xα = {x ∈ X , μx˜ (x) ≥ α}
(5)
Hence, for each α-level, the interval Xα = [xαL , xαU ] is obtained, where xαL , xαU are the lower and upper bounds of Xα . By this way, the fuzzy analysis is transformed to a number of interval analyses at finite sublevels [20]. 2.3 Updating Fuzzy Model Parameters 2.3.1 Objective Function and Constraints In the fuzzy FE model updating, the inputs are described by fuzzy quantities, including unknown model parameters and the measured data. The aim of the updating procedure is to determine the membership functions of m fuzzy model parameters θ = {θ˜1 , θ˜2 , . . . , θ˜j , . . . , θ˜m } so that the error between the membership function of the model responses and that of the measured data is minimized [21]. By adopting the α-cut approach as suggested in Ref. [18], the objective function for this optimization problem is formulated as: δα (θ αL , θ αU ) =
n i=1
wL,i
∗ 2 ∗ 2 model (θ , θ model (θ , θ uαL,i − uαL,i uαU ,i − uαU αL αU ) αL αU ) ,i + wU ,i ∗ ∗ uαL,i uαU ,i
(6)
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where θ αL and θ αU are, respectively, the vectors of lower bounds and upper bounds of ∗ ∗ and uαU the model parameters corresponding to the α-level; uαL,i ,i are, respectively, the model lower and upper bounds of the i-th measured response corresponding to the α-level; uαL,i model and uαU ,i are, respectively, the lower and upper bounds of the i-th model response corresponding to the α-level; wL,i and wU ,i are the weight factors. The optimization problem has 2m variables being the lower bounds and the upper bounds of m model parameters at each α-level. In case the measured responses are described by triangular fuzzy numbers, the updating at the first α-level (α = 1) is simply a deterministic optimization with m variables. To ensure the updated parameters physically realistic and their membership functions convex, the values of the parameter bounds should be constrained such that [18]: k−1 k ≤ θjαL θj min ≤ θjαL k−1 k θjαU ≤ θjαU ≤ θj max
(7)
k k and θjαU are, respectively, the lower and upper bounds of the updating where θjαL parameter θj at the k-th α-level; θj min and θj max are the lower and upper limits of the j-th k−1 k−1 parameter, respectively; θjαL and θjαU are, respectively, the obtained lower and upper bounds of the j-th parameter at the previous (k-1)-th α-level. The fuzzy updating procedure is performed by minimizing the objective function of Eq. (6) with the constraints in Eq. (7) from high α-level to low α-level to obtain a discrete membership function of the updating parameters. In the evaluation of Eq. (6), model and the upper bound an interval analysis is necessary to compute the lower bound uαL,i model of the i-th model response. In the next subsection, an efficient interval analysis uαU ,i based on local sensitivity indices of the response with respect to the model parameters is proposed to determine these bounds.
2.3.2 Interval Model Output Quantification The local sensitivity index of a model output uimodel with respect to the model parameter θj is computed as sij1 =
uimodel (θj + θj ) − uimodel (θj ) , j = 1, . . . , m θj
(8)
where θj is a small variation of the model parameter θj , and it is taken as 0.001*θj in this study. It is notable that the local sensitivity indices are estimated only one time after obtaining the optimal parameter values at the first α-level (α = 1). Therefore, the estimation of m local sensitivity indices requires m structural analysis. It is assumed that the model response uimodel varies monotonically with respect to the model parameters. This assumption is usually valid for structural modal characteristics like the natural frequency. Therefore, local sensitivity indices are sufficient to derive the parameter combinations that result in the extremes of the response. The extremes of the model and umodel , are determined for each α-level as follows. If s1 > 0, θ k response, uαL,i αU ,i ij jαL model , and θ k model . Conversely, if s1 < 0, is use to compute uαL,i is taken to compute u jαU αU ,i ij k model , and θ k is taken to compute umodel . θjαU is use to compute uαL,i jαL αU ,i
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3 Differential Evolution with Nearest Neighbor Comparison In this study, a modified differential evolution algorithm is suggested to perform the fuzzy FE model updating. Differential evolution (DE) is a direct search method invented by Storn and Price [22] for solving global optimization problems. The DE algorithm has a simple structure with few control parameters. DE has become one of the most efficient methods, which is suitable for various design problems [23]. DE has been introduced to structural system identification [24–26], and damage detection [27–31]. 3.1 Basic of Differential Evolution DE belongs to the class of population-based optimization algorithms. Suppose that it is necessary to minimize the objective function: (9) u = f (x) : Rm → R, x = xj , xj ∈ [xj min , xjmax ], j = 1, ..., m where xj min and xjmax are the lower and upper bounds of the variable xj , respectively. The initial population consists of NP individuals, which are vectors xk (0), k = 1, ..., NP, of randomly generated elements: xk,j (0) = xj min + rand [0, 1] · (xjmax − xj min ), j = 1, ..., m
(10)
where rand [0, 1] is a uniformly distributed random number in the range [0, 1]. In the (t + 1)th generation, corresponding to each vector xk (t) in the population of the t th generation, a mutant vector is created through the mutation operation as follows: (11) y = xr1 (t) + F · xr2 (t) − xr3 (t) where r1 , r2 , r3 are randomly chosen integers satisfying 1 ≤ r1 = r2 = r3 = k ≤ NP; F is a scaling factor, 0 < F < 1. Vector y is combined with vector xk (t) to produce a trial vector z. The element of z are defined by:
yj when (rand [0, 1] ≤ Cr) or (r = j) (12) zj = xk,j (t) when (rand [0, 1] > Cr) and (r = j) In Eq. (12), Cr is the crossover rate (Cr ∈ [0, 1]), and r is a random integer in [1,m]. The trial vector z is compared with xk (t): if z has better value of objective function than xk (t) does, z will be selected as an individual of the population in the (t +1)th generation, i.e. xk (t + 1) = z, otherwise xk (t + 1) = xk (t). 3.2 Nearest Neighbor Comparison Method In DE, many trial vectors will not survive in the selection. The evaluation of the objective function for such trial vectors is to be useless and should be avoided. A simple method, named Nearest Neighbor Comparison (NNC), has been recently proposed for this purpose. NNC is a technique that can judge a solution candidate without determining its objective function value. This technique was proposed by Pham [32] and used in a number of improved DE algorithms to solve different engineering optimization problems
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[33–36], as well as structural health monitoring [37]. The implementation of NNC is as follows. First, in the population of t th generation, we find the vector xc (t) which is closest to the trial vector z by using the Euclidean distance measure: ⎛ ⎞2 m xj − zj ⎝ ⎠ (13) d (x, z) = max xk,j − min xk,j j=1
k
k
where d (x, z) is the Euclidean distance between two vectors x and z; max xk,j and k
min xk,j are, respectively, the maximum and minimum values of the jth variable in the k
current population. Then, xc (t) is compared with xk (t). If xc (t) is worse than xk (t), the trial vector z is likely worse than xk (t), and it will be omitted. Otherwise, the objective function value of z is calculated for comparison. By this way, unpromising trial vectors can be skipped, and computation cost is reduced [32]. The DE with NNC method, named DE-NNC, is supposed to be able to save the computational cost considerably in solving the fuzzy structural identification problem. The demonstration is presented in the numerical examples in the next section.
4 Numerical Examples In this section, the proposed procedure is applied to identify the member stiffness of two structures, including a 2D truss and a plane frame. Simulated fuzzy natural frequencies are utilized to update the actual condition of the member stiffness. To show the advantage of DE-NNC, some popular MHs, including PSO [38], TLBO [39], Jaya [40], and classical DE, are also implemented for comparison purpose. The parameter setting for the considered MHs is given as follows: – All MHs: the population size is 30; the maximum iteration is 250. – DE and DE-NNC: the scaling factor and crossover rate are 0.7 and 0.9, respectively. – PSO: the inertia weight 0.5, cognitive learning rate 1.0, and social learning rate are used. – TLBO and Jaya: no parameter setting. The termination condition for all algorithms is the objective function value less than 10–4 . 4.1 2D Truss An example of a 2D truss structure taken from [41] is considered. The layout of the truss is shown in Fig. 2. The cross-section areas of all bars are 0.0025 m2 . Material density and Young’s modulus are 7.850 kg/m3 and 200 GPa, respectively. In the structure, the element 2 and 9 are damaged, which leads to the reduction of element stiffness of 50% and 25%, respectively. It is assumed that the damages do not
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alter the weight of the bars. The stiffness parameters to be identified are denoted as sj , j = 1, . . . , 9, where sj = 1 corresponds to the stiffness of the bar to be unchanged, and sj = 0 corresponds to the zeros stiffness of the bar. The allowable range for the stiffness parameter is [0, 1.5].
3 2
1
1
3
2
4m
6
5 4
5 7
4
4m
9 8
3m
6
4m
Fig. 2. Layout of the 2D truss
The responses of the damaged structure used for the identification procedure include the first nine natural frequencies. The fuzziness of ±5% is added to the simulated data to obtain fuzzy frequencies. The membership function of the fuzzy frequencies is described by the triangular form. Table 1 gives the nominal value, the lower bound, and the upper bound of the fuzzy frequencies. Table 1. Natural frequencies (Hz) of the damaged 2D truss structure Mode
Lower bound α=0
Nominal value α=1 35.026
Upper bound α=0
1
33.2744
36.777
2
62.9642
66.278
69.592
3
96.5503
101.632
106.713
4
178.802
188.213
197.623
5
243.1738
255.972
268.771
6
318.0206
334.759
351.496
7
323.6144
340.647
357.679
8
340.7676
358.703
376.638
9
414.3065
436.112
457.918
First, the deterministic updating corresponding to α = 1 is conducted to determine the nominal values of the stiffness parameters. Using the implemented MHs, the obtained results are listed in Table 2. including the best solution, together with the best value (f best ), the mean value (f mean ), the worst value (f worst ), and the standard deviation (std)
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of the objective function. The average number of function evaluations (NFE) is also given in Table 2. These results are derived from 20 independent runs of each MH. The statistical results of the stiffness parameter values obtained by DE-NNC are given in Table 3. and Fig. 3 depicts the average convergent curves of the implemented MHs. As seen from the obtained results, the stiffness parameter values are identified with high accuracy. On average, DE-NNC requires fewer function evaluations and converges better than PSO, TLBO, and DE. Comparing with Jaya, DE-NNC is a little bit slower and needs a slightly higher number of function calls. With a very small deviation of objective function values in 20 runs, DE-NNC is also as stable as TLBO, Jaya, and DE. PSO appears to be the most unstable algorithm. Table 2. Statistical results for the 2D truss by different MHs Parameter
PSO
TLBO
Jaya
DE
DE-NNC
s1
1.0010
0.9993
0.9992
1.0009
1.0000
s2
0.5004
0.5001
0.5000
0.4999
0.4998
s3
1.0010
1.0003
1.0002
1.0004
1.0003
s4
0.9976
1.0009
1.0006
0.9992
1.0002
s5
0.9993
1.0000
1.0002
0.9998
1.0000
s6
0.9986
0.9999
1.0009
0.9992
1.0001
s7
1.0009
0.9996
0.9993
1.0002
0.9999
s8
1.0010
0.9997
0.9995
1.0005
1.0001
s9
0.7500
0.7499
0.7500
0.7501
0.7502
fbest
9.6066e-05
7.3752e-05
6.6793e-05
4.2781e-05
6.2478e-05
fmean
0.0042
9.3171e-05
8.8044e-05
8.7360e-05
8.7457e-05
fworst
0.0403
9.9763e-05
9.8726e-05
9.9968e-05
9.7403e-05
std
0.0113
8.5216e-06
9.2901e-06
1.2902e-05
8.7252e-06
NFE
6208
9924
1979
5104
2263
Table 3. Updated nominal values of the stiffness parameters for the 2D truss by DE-NNC Parameter s1
s2
s3
s4
s5
s6
s7
s8
s9
Exact
1
0.5
1
1
1
1
1
1
0.75
Min
0.9988 0.4996 0.9993 0.9983 0.9993 0.9986 0.9991 0.9993 0.7495
Max
1.0007 0.5003 1.0005 1.0009 1.0010 1.0010 1.0008 1.0007 0.7507
Mean
1.0000 0.5000 1.0000 0.9997 0.9999 1.0002 1.0001 1.0001 0.7500
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PSO TLBO Jaya DE DE-NNC
Objective value
0.08
0.06
0.04
0.02
0 2000
4000
6000 NFE
8000
10000
12000
1
1
0.8
0.8
0.8
0.6 0.4 0.2 0 0.9
0.4
0.95
1
1.05
1.1
0 0.4
1.15
0.6 0.4 0.2
0.45
0.5
0.55 s
1
0.6
0 0.9
0.65
1 0.8
0.6 0.4 0.2
0.2 0 0.8
Membership level
1 0.8
0.4
0.9
1
1.1 s
1.2
0 0.9
1.3
0.95
1
1.05
1.1
0 0.8
1.15
0.8
0.8 Membership level
0.8
Membership level
1
0.6 0.4 0.2
0.95
1
1.05 s
7
0.9
1 s
1
0.2
1.1
1.15
0 0.9
1.15
1.1
1.2
1.3
0.8
0.85
0.4
s5
0.4
1.1
0.2
4
0.6
1.05
0.6
1
0 0.9
1 s3
1
0.6
0.95
2
0.8 Membership level
Membership level
0.6
0.2
s
Membership level
Membership level
1
Membership level
Membership level
Fig. 3. Convergence of the best objective value for the 2D truss
6
0.6 0.4 0.2
0.95
1
1.05 s
8
1.1
1.15
0 0.65
0.7
0.75 s
9
Fig. 4. Membership functions of updated stiffness parameters for 2D truss
0.9
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By applying the proposed fuzzy identification procedure with five membership levels, α = 0.8, 0.6, 0.4, 0.2, and 0.0, the membership functions of the updated stiffness parameters are computed and presented in Fig. 4. Then, the updated fuzzy stiffness parameters are used to estimate the membership functions of the fuzzy natural frequencies of the updated model. The fuzzy uncertainty propagation based on the local sensitivity indices is carried out at all membership function levels. The obtained results are shown in Fig. 5. Excellent agreements between the updated and the actual (denoted “measured”) fuzzy membership functions are observed for all frequencies. The results confirm the accuracy of the proposed fuzzy identification procedure in capturing the uncertain model parameters. 1
1 measured updated
0.4 0.2
35 f1 [Hz]
36
66 f2 [Hz]
68
185 190 f4 [Hz]
195
0.4
245
250
255 f [Hz]
260
265
0.2
340 f7 [Hz]
320
350
360
0.6 0.4
0 340
330 340 f6 [Hz]
350
360
1 measured updated
0.6 0.4 0.2
0.2
330
108
0.4
0.8 Membership level
Membership level
0.4
106
0.6
0 310
270
measured updated
0.8
0.6
104
0.2
1 measured updated
102 f3 [Hz]
measured updated
5
1 0.8
100
0.8
0.6
0 240
200
98
1
measured updated
0.2
180
0.4
0 96
70
Membership level
Membership level
Membership level
0.2
Membership level
64
0.8
0.4
0.6
0.2
1 measured updated
0.6
0 320
0.4
0 62
37
1 0.8
0 175
0.6
0.2
34
measured updated
0.8 Membership level
0.6
0 33
1 measured updated
0.8 Membership level
Membership level
0.8
350
360 f8 [Hz]
370
380
Fig. 5. Fuzzy frequencies of 2D truss
0 410
420
430 440 f9 [Hz]
450
460
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4.2 2D Frame The second example is a 2D frame structure taken from Ref. [42] (Fig. 6). The FE model consists of ten nodes and nine beam elements. In this example, the damage is assumed to occur in the elements and results in the reduction of the moment of inertia. Tables 4 and 5 give the structural input data for both the initial model and the damaged model. The “measured” (simulated in this study) natural frequencies are used to identify the stiffness parameters. Table 6 lists the first twelve natural frequencies of the damaged model, including the nominal value, the lower bound, and the upper bound. These data correspond to an assumed maximum error of ±5% of the natural frequencies. The membership function of the fuzzy frequencies is again described by the triangular form. The allowable range for the stiffness parameter is [0, 2.5]. 10 9
7
0.5m
9
7
8
4
5
5
3
2
8
2
3
4
6
0.5m
6
1
1m
1 0.25m 0.25m 0.25m 0.25m
Fig. 6. Layout of the 2D frame
The nominal values of the stiffness parameters corresponding to α = 1 are first computed by using DE-NNC and the other implemented MHs. The statistical results from 20 independent runs are listed in Tables 7 and 8 As seen in Table 8 the means of the stiffness parameter values obtained by DE-NNC are much closed to the exact values. The variation of the updated results in 20 runs is very small. These mean values of the updated parameters are utilized to analyze the local sensitivity of the natural frequencies. Moreover, with only 2708 function evaluations on average, DE-NNC is the best algorithm in terms of computational cost. The convergence of the objective values depicted
Fuzzy Structural Identification of Bar-Type Structures
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Table 4. Element properties of the 2D frame model Element No
Density (105 kg/m3 )
Area (1e-4 m2 )
Moment of inertial (1e-6 m4 )
sj
Initial model
Damaged model
1
0.5384
7.4283
369.52
369.52
1
2
1.0510
5.7103
5.2872
5.2872
1
3
1.4010
6.8425
7.1377
7.1377
1
4
1.0770
9.2854
527.88
527.88
1
5
1.0510
5.7103
5.2872
5.2872
1
6
1.4010
4.5682
6.6090
6.6090
1
7
5.0450
0.1487
2.5943
1.9457
0.75
8
10.090
0.1487
3.7598
2.8198
0.75
9
0.0
0.0857
0.7138
0.4640
0.65
Table 5. Nodal mass data of the 2D frame model Node Mass (m) Rotational mass (kgm2 ) 1
0.0
0.0
2
0.0
0.0
3
180.0
42.0
4
165.0
42.0
5
0.0
0.0
6
0.0
0.0
7
14.0
1.8
8
39.0
4.2
9
360.0
37.5
10
12.0
1.0
in Fig. 7 shows that DE-NNC is as fast as Jaya, and it converges better than the other algorithms. PSO and TLBO appear to be unstable and easily trapped in a local minimum. Using five membership levels, α = 0.8, 0.6, 0.4, 0.2, and 0, the membership functions of the updated stiffness parameters are computed and presented in Fig. 8. The fuzzy membership functions of the natural frequencies of the updated frame are depicted in Fig. 9, together with the exact (“measured”) ones. As seen in the figures, there are good agreements between the updated and the exact fuzzy membership functions for all frequencies, except for the 8th frequency. The difference between the two data may be explained by the fact that there exists no updated parameter interval that can exactly describe the corresponding exact bounds. It is notable, however, that the errors at different
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B.-D. Nguyen and H.-A. Pham Table 6. Natural frequencies (Hz) of the damaged 2D frame model Mode
Lower bound α=0
Nominal value α=1
Upper bound α=0
1
20.076
21.133
22.190
2
22.716
23.912
25.107
3
26.070
27.442
28.814
4
33.773
35.551
37.328
5
34.430
36.243
38.055
6
37.548
39.524
41.501
7
40.151
42.264
44.377
8
44.633
46.983
49.332
9
47.962
50.486
53.010
10
49.726
52.343
54.960
11
99.607
104.85
110.092
12
107.939
113.62
119.301
Table 7. Statistical results for the 2D frame by different MHs Parameter
PSO
TLBO
Jaya
DE
DE-NNC
s1
1.0007
0.9999
1.0004
1.0004
0.9999
s2
1.0001
1.0000
0.9993
1.0000
1.0002
s3
1.0010
1.0005
1.0008
1.0004
0.9995
s4
0.9991
0.9994
0.9852
0.9809
0.9948
s5
0.9615
0.9797
0.9927
0.9842
1.0010
s6
1.0010
0.9998
0.9996
1.0005
1.0007
s7
0.7583
0.7542
0.7517
0.7532
0.7499
s8
0.7491
0.7502
0.7499
0.7498
0.7500
s9
0.6499
0.6500
0.6501
0.6501
0.6500
fbest
1.7618e-04
8.8434e-05
4.7917e-05
6.8663e-05
6.5600e-05
fmean
0.0061
0.0035
8.8429e-05
8.8511e-05
8.6963e-05
fworst
0.0267
0.0338
9.9871e-05
9.9838e-05
9.7642e-05
std
0.0063
0.0104
1.2598e-05
8.8109e-06
9.1065e-06
NFE
7530
13611
3470
5718
2708
levels are relatively small. The maximum errors of the lower and upper bounds of the 8th frequency are 3.5% and 1.7%, respectively.
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Table 8. Updated nominal values of the stiffness parameters for the 2D frame by DE-NNC Parameter s1
s2
s3
s4
s5
s6
s7
s8
s9
Exact
1
1
1
1
1
1
0.75
0.75
0.65
Min
0.9999 0.9991 0.9993 0.9969 0.9874 0.9979 0.7499 0.7493 0.6499
Max
1.0008 1.0009 1.0010 1.0005 1.0010 1.0008 0.7550 0.7507 0.6501
Mean
1.0003 0.9999 1.0004 0.9995 0.9975 0.9996 0.7521 0.7501 0.6501
PSO TLBO Jaya DE DE-NNC
0.05
Objective value
0.04
0.03
0.02
0.01
0 2000
4000
6000
8000 NFE
10000
12000
14000
Fig. 7. Convergence of the best objective value for the 2D frame
Overall, the results from the considered examples show the effectiveness and computational efficiency of the proposed procedure in identifying the fuzzy member stiffness of the structures.
B.-D. Nguyen and H.-A. Pham 1
1
0.8
0.8
0.6 0.4 0.2
0.6 0.4 0.2
1
1.05
1.1 s1
1.15
1.2
0.4
1
1.05 s2
1.1
0 0.8
1.15
1
1
0.8
0.8
0.4
0 0.8
Membership level
1
0.6
0.6 0.4 0.2
1
1.2
1.4 s4
1.6
1.8
1
1.5 s5
2
0
2.5
0.8 Membership level
0.8 Membership level
1
0 0.66
0.6 0.4 0.2
0.68
0.7
0.72
0.74
s7
0
0.5
1
1.5
2
2.5
0.6 0.4 0.2
0 0.7
0.76
1.05
s6
1
0.2
1
0.4
1
0.4
0.95
0.6
0.8 0.6
0.9
0.2
0 0.5
2
0.85
s3
0.8
0.2
Membership level
0.6
0.2
0 0.95
1.25
Membership level
Membership level
0 0.95
Membership level
1 0.8
Membership level
Membership level
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0.8
0.9 s
1
0 0.55
1.1
0.6
0.65 s
8
0.7
0.75
9
Fig. 8. Membership functions of updated stiffness parameters for 2D frame 1
1 measured updated
0.4 0.2
0.6 0.4 0.2
20
20.5
21 f [Hz]
21.5
22
23
24 f [Hz]
1
0.6 0.4
35
0.6 0.4
36
37
0 34
38
35
36
1
37
38
0 37
39
43
44
45
41
42
measured updated
0.8
0.6 0.4
0 44
40
1 measured updated
0.2
0.2
7
39 6
Membership level
Membership level
0.4
f [Hz]
38
f [Hz]
0.8
0.6
29
0.4
1
measured updated
28.5
0.6
5
0.8
28
measured updated
f [Hz]
4
42
27.5 f [Hz]
0.2
f [Hz]
41
27
0.8
0.2
34
26.5
1
measured updated Membership level
Membership level
Membership level
0 26
26
3
0.8
0.2
Membership level
25
1 measured updated
0.8
0 40
0.4
2
1
0 33
0.6
0.2
0 22
22.5
measured updated
0.8 Membership level
0.6
0 19.5
1
measured updated
0.8 Membership level
Membership level
0.8
0.6 0.4 0.2
45
46
47 f [Hz]
48
49
50
8
Fig. 9. Fuzzy frequencies of 2D frame
0 46
48
50 f [Hz] 9
52
54
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5 Conclusion The fuzzy structural identification for bar-type structures has been considered in this paper. A fuzzy structural identification procedure has been established by adopting an improved differential evolution algorithm (DE-NNC) to perform the fuzzy FE model updating at different fuzzy membership levels. Furthermore, a fuzzy quantification approach based on the local sensitivity index has been developed. The established approach has been applied to identify the element stiffness of two structures, including a 2D truss and a 2D frame considering the fuzziness in the stiffness parameters and the natural frequencies. Simulated fuzzy natural frequencies were used as measured data, and the fuzzy membership functions of the uncertain stiffness parameters of the structures were estimated. It has been shown that the uncertain stiffness parameters of the structures could be quantified in terms of fuzzy quantities with relatively high accuracy. Moreover, DE-NNC implemented as the optimizer for the proposed procedure has been shown to be superior to some popular MHs in terms of computational efficiency. Future study is planned to validate the method with a physical test. Acknowledgement. This study is supported by National University of Civil Engineering (NUCE) project 73–2020/KHXD.
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Application of Artificial Intelligence for Structural Optimization Tran-Hieu Nguyen(B) and Anh-Tuan Vu National University of Civil Engineering, Hanoi, Vietnam [email protected]
Abstract. The conventional structural design is frequently implemented using the “trial-and-error” method in which the final result strongly depends on the designers’ experience. The design process can be significantly improved by adopting some Artificial Intelligence (AI) technologies. The paper presents an AI-based approach that combines the Differential Evolution optimization algorithm and Artificial Neural Network for finding the optimal structural solution. In more detail, an artificial neural network model is built for structural safety classification. The whole model is trained by a number of data points that are generated using the Latin Hypercube Sampling method. The trained model is then used to completely eliminate the unnecessary finite element analyses during the optimization process. By using AI techniques, the computation cost could be significantly reduced. The efficiency of the proposed approach is tested in the 47-bar planar tower with discrete variables. The numerical results show that the proposed approach is accurate, robust, and faster than traditional optimization. Keywords: Optimization · Artificial Intelligence · Differential evolution · Machine learning · Artificial neural network
1 Introduction Up to now, the world has experienced four industrial revolutions. The first industrial revolution happened in the period from the 1700s to the 1800s with the invention of steam-powered engines. The second industrial revolution started in 1870 with the use of electricity in factories. In the late 1960s, the rise of computer and computer networks is the motivation of the third industrial revolutions. In the past few decades, a fourth industrial revolution (also known as Industry 4.0) has emerged. The term “Industry 4.0” was firstly presented in a report of the German government in 2011. The main components of Industry 4.0 include Big Data, Internet of Things, Additive Manufacturing, Cloud Computing. A large amount of data produced in the digitalization era due to the huge number of cheaper digital cameras as well as precise sensors leads to the demand for data mining tools. Artificial Intelligence (AI) has been proved to be able to analyze big data quickly and accurately. Therefore, there has been a growing interest in the application of AI in all engineering domains, including the field of Architecture, Engineering, and Construction (AEC). © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 1052–1064, 2022. https://doi.org/10.1007/978-981-16-3239-6_82
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The application of AI in the AEC field is the research topic of previous review articles [1–4]. The findings of these studies indicated that some sub-domains that commonly use AI technologies include structural health monitoring, damage detection, structural reliability, performance evaluation, project management, structural optimization. In particular, the possibility of applying AI in the field of structural optimization has been studied early. Since 1986, Stanislaw Jozwiak applied AI to improve the effectiveness of structural optimization programs [5, 6]. In 1992, Genetic Algorithm was used to optimize truss structures [7, 8]. Since then, many evolutionary algorithms have been successfully applied for solving optimization problems. Some famous algorithms can be listed here such as Genetic Programming (EP) [9], Evolution Strategy (ES) [10], Differential Evolution (DE) [11, 12]. All the above-mentioned algorithms are designed based on the natural selection mechanism, and they are categorized into Evolutionary Computation, a subfield of AI. In comparison with the gradient-based optimization algorithm, the evolutionary algorithms have capable of seeking the optimal results over the large searching space but a huge number of structural analyses must be performed to evaluate the fitness of candidates. In some cases where the structures are complex or the nonlinear inelastic analyses are required, the computation time of each structural analysis is very time-consuming. Consequently, the total optimization time becomes too large. Shortening the optimization time by reducing the number of unnecessary function evaluations is the goal of many previous studies. Obviously, it can be seen that many offspring produced during the evolutionary optimization process are worse than their parents and they will be absolutely neglected at the selection phase. Performing the structural analysis for such candidates is useless. Therefore, some techniques for comparing the child and the corresponding parent have been proposed, such as estimated comparison [13, 14], nearest neighbor comparison [15, 16], multi-comparison technique [17, 18]. Recently, due to the rapid development of machine learning, many hybrid approaches combining a machine learning technique and an evolutionary algorithm have been introduced. For example, SVC-DE was developed by coupling Support Vector Machine classification and Differential Evolution optimization [19]. DE-kNN, introduced by Y. Liu and F. Sun, use the k-Nearest Neighbors classifier for saving the fitness function calls [20]. In these approaches, the precision of the classifier plays a key role. In 2012, Alex Krizhevsky and his colleagues developed a deep neural network (DNN), now known as AlexNet, to classify images. This model has achieved impressive accuracy in the ImageNet Large Scale Visual Recognition Challenge. In this study, some novel techniques were firstly introduced including ReLU activation function, dropout, as well as GPU-based training [21]. Following this success, many high accurate classification models were designed based on DNN such as GoogLeNet [22], VGG [23], ResNet [24], in which the numbers of hidden layers increase more and more. Until now, DNNs seem to be the most powerful classification model.
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Because of the above reasons, this study proposes a novel approach that utilizes a DNN-based classifier for omitting unpromising candidates. The efficiency of the proposed approach has been proved through an illustrative example of a planar truss. The rest of the article is organized as follows: Sect. 2 describes the optimization problem in detail; Sect. 3 focuses on developing a DNN model for structural safety assessment; the structural optimization process and the obtained result are presented in Sect. 4; conclusions are pointed out in Sect. 5.
2 Problem Description In this article, a planar 47-bar power line tower as displayed in Fig. 1 is chosen for optimization. This problem has been previously studied by Lee et al. [25], Kaveh and Mahdavi [26]. The tower consists of 22 nodes and 47 members which are categorized into 27 groups as presented in Table 1. Members in the same group are assigned the same cross-section. The members’ cross-sections are selected from a list of 64 available profiles in which the area values are listed in Table 2. All members are made of steel which has the material properties as follows: the modulus of elasticity E = 30,000 ksi (206.842 kN/mm2 ); the material density ρ = 0.3 lb/in3 (8303.97 kg/m3 ). The tower is subjected to one of three independently load cases as follows: (LC1) a vertical load of 14 kips along the negative y-direction and a lateral load of 6.0 kips along the positive x-direction acting at two nodes 17 and 22; (LC2) a vertical load of 14 kips along the negative y-direction and a lateral load of 6.0 kips along the positive x-direction acting at nodes 17; and (LC3) a vertical load of 14 kips along the negative y-direction and a lateral load of 6.0 kips along the positive x-direction acting at nodes 22. All members in the tower are designed according to the stress and buckling constraints. The allowable tensile and compressive stresses are 20 ksi (103.421 MPa) and −15 ksi (137.895 MPa), respectively. The allowable buckling stress of the ith member can be calculated by the following equation: i =− σcr
KEAi Li
(1)
where: K is the buckling constant that is fixed to 3.96 in this study; E is the modulus of elasticity of the used material; Ai and L i are the cross-sectional area and the length of ith member, respectively.
Application of Artificial Intelligence for Structural Optimization
Fig. 1. 47-bar planar tower
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T.-H. Nguyen and A.-T. Vu Table 1. Description of member groups in the 47-planar tower.
Design variables
Members
Design variables
Members
Design variable
Members
Design variable
Members
A(1)
1, 3
A(8)
13, 14
A(15)
27
A(22)
38
A(2)
2, 4
A(9)
15, 16
A(16)
28
A(23)
39, 40
A(3)
5, 6
A(10)
17, 18
A(17)
29, 30
A(24)
41, 42
A(4)
7
A(11)
19, 20
A(18)
31, 32
A(25)
43
A(5)
8, 9
A(12)
21, 22
A(19)
33
A(26)
44, 45
A(6)
10
A(13)
23, 24
A(20)
34, 35
A(27)
46, 47
A(7)
11, 12
A(14)
25, 26
A(21)
36, 37
Table 2. List of available cross-sectional areas. No.
Areas (in2 )
No.
Areas (in2 )
No.
1
0.111
17
1.563
33
2
0.141
18
1.620
34
3
0.196
19
1.800
4
0.250
20
5
0.307
21
6
0.391
7 8
Areas (in2 )
No.
Areas (in2 )
3.840
49
11.500
3.870
50
13.500
35
3.880
51
13.900
1.990
36
4.180
52
14.200
2.130
37
4.220
53
15.500
22
2.380
38
4.490
54
16.000
0.442
23
2.620
39
4.590
55
16.900
0.563
24
2.630
40
4.800
56
18.800
9
0.602
25
2.880
41
4.970
57
19.900
10
0.766
26
2.930
42
5.120
58
22.000
11
0.785
27
3.090
43
5.740
59
22.900
12
0.994
28
3.130
44
7.220
60
24.500
13
1.000
29
3.380
45
7.970
61
26.500
14
1.228
30
3.470
46
8.530
62
28.000
15
1.266
31
3.550
47
9.300
63
30.000
16
1.457
32
3.630
48
10.850
64
33.500
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3 Building the Deep Neural Network Classification Model In this section, the procedure for building the DNN-based structural safety assessment is described. The machine learning-based structural safety assessment has been studied in some recent articles. In 2018, a machine learning framework for assessing post-earthquake structural safety of damaged buildings was developed by Y. Zang et al. [27]. Fire collapse assessment of steel frames using machine learning was conducted by F. Fu [28]. V.-H. Truong et al. developed a Gradient Tree Boosting based method for the safety evaluation of steel trusses [29]. Moreover, the performances of five machine learning models in both regression and classification of the ultimate load-carrying capacity of steel frames were compared in [30]. All the aforementioned studies have the same concept in which the training data for machine learning models have been collected using finite element method (FEM) simulations. The structural safety classification model used in this article was built similarly. Firstly, the cross-sectional areas of tower members were randomly generated in the range from 0.1 to 33.5 in2 (0.6–216.13 cm2 ) using Latin Hypercube Sampling (LHS) technique. These values were also considered as the input data. In this case, the tower contains 47 members which are arranged into 27 groups. It means that the input data is a vector A = [A(1) , A(2) ,…, A(27) ] where 0.1 in2 ≤ A(i) ≤ 33.5 in2 . Subsequently, the structures were analysed using the FEM open-source code PyNiteFEA [31] and the obtained internal forces were used to check the design constraints. In case that all design constraints are satisfied, the structure was considered safe and the output was assigned the label “0”. In contrast, the structure was unsafe and the output label was “1”. Two datasets were generated separately. The training dataset which contains 1000 samples was used to train the model. The testing dataset with 100 samples was used to evaluate the accuracy of the predictive model. After collecting the training dataset and the testing dataset, the classification model was built based on a deep neural network with the (27-100-100-100-2) architecture as presented in Fig. 2. The DNN model was developed using the machine learning library Scikit-learn [32]. ReLU was used as the activation function in hidden layer neurons. The network was trained for 1,000 epochs. Other parameters of the network were set as follows: the optimization algorithm was Adam, the batch size equaled to 2, and the loss function was log-loss. Figure 3(a) shows the confusion matrix of the trained model on the training dataset. When examined on the testing dataset, the obtained confusion matrix is presented in Fig. 3(b). The values of different evaluation metrics of the classification model are presented in Table 3.
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A(1) A(2)
A(26)
. . .
. . 100 neurons . .
. . 100 neurons . .
. . 100 neurons . .
Label “0” Label “1”
A(27)
Input layer
Hidden layers
Output layer
Fig. 2. Neural network architecture for the structural safety classification of the 47-bar tower
(b) Testing dataset
(a) Training dataset
Fig. 3. Confusion matrices of the classification model
Table 3. Performance of the classification model. Precision
Recall
F1-score
Accuracy
Label “0”
0.50
0.45
0.47
0.71
Label “1”
0.78
0.82
0.80
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4 Structural Optimization Using Differential Evolution The optimization problem of the 47-bar tower can be stated as follows: Find A = [A(1) , A(2) ,…, A(27) ] where A(i) is selected from Table 1 to minimize: W (A) = ρ subject to :
n
Ai li
(2)
i=1 i i σt ≤ 20 ksi σ ≤ min −15 ksi, σ i c
(3)
cr
The DE algorithm was used for solving this problem. The original DE procedure consists of four basic operations including initialization, mutation, crossover, and selection [33]. However, the procedure used in this study was modified by embedding the machine learning classification model into the optimization process. In more detail, the process is split into two stages. Stage I is aimed at searching for new solutions in a new region. In this stage, four basic operations are performed sequentially and every produced solution is exactly evaluated using the FEA. In Stage II, the solution produced by the mutation and crossover operations will be preliminarily assessed using the classification model. If the predicted label is “0”, the constraints of the current solution will be evaluated by performing the FEA. On the contrary, the solution which is lighter than and equals to its parent solution will be analysed. Otherwise, the heavier solution will be rejected without conducting the structural analysis. By this way, the number of FEAs is reduced and the computation time can be shorted. The flowchart of the proposed approach is presented in Fig. 4. Using the proposed approach for solving the discrete optimization problem of the 47-bar tower. The DE optimization parameters were taken based on the recommendation in [34] as follows: the mutation factor F = 0.8; the crossover probability Cr = 0.9; the population size NP = 50. Two parameters max_iter1 and max_iter2 should be carefully chosen to ensure the balance between the exploration and the ability to reduce function evaluations. After a parameter sensitive analysis, the values of two parameters were set as follows: max_iter1 = 100 and max_iter2 = 400. The optimization was independently performed 30 times. A typical convergence history of the optimization process is plotted in Fig. 5. The best result obtained from 30 independent runs are presented in Table 4 in comparison with literature. It can be seen that the proposed approach found the lightest solution with the lowest number of structural analyses. Remarkably, the number of FEAs reduces by approximately 20% when compared with the original DE algorithm as shown in Fig. 6.
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Stage I
Initial population Constraint Evaluation using FEA Mutation Crossover N
Constraints Evaluation using FEA Selection i > max_iter1 Y
Stage II
Mutation Crossover Assessment using Classification Model N
Predicted Label = “0”
Predicted Label = “1” N
Wchild > Wparent
Y
Constraint Evaluation using FEA Selection i > max_iter2 Y End
Fig. 4. Flowchart of the proposed approach
Elimination
Application of Artificial Intelligence for Structural Optimization
Fig. 5. Convergence history of the optimization process
Fig. 6. Number of FEAs during the optimization process
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T.-H. Nguyen and A.-T. Vu Table 4. Optimization results
Design variables
Lee et al. [25]
Kaveh and Mahdavi [26]
Present work
A(1)
3.84
3.84
3.84
A(2)
3.38
3.38
3.38
A(3)
0.766
0.785
0.766
A(4)
0.141
0.196
0.111
A(5)
0.785
0.994
0.785
A(6)
1.99
1.8
1.99
A(7)
2.13
2.13
2.13
A(8)
1.228
1.228
1.228
A(9)
1.563
1.563
1.563
A(10)
2.13
2.13
2.13
A(11)
0.111
0.111
0.111
A(12)
0.111
0.111
0.111
A(13)
1.8
1.8
1.8
A(14)
1.8
1.8
1.8
A(15)
1.457
1.563
1.457
A(16)
0.442
0.442
0.563
A(17)
3.63
3.630
3.63
A(18)
1.457
1.457
1.457
A(19)
0.442
0.307
0.25
A(20)
3.63
3.09
3.09
A(21)
1.457
1.266
1.228
A(22)
0.196
0.307
0.391
A(23)
3.84
3.84
3.84
A(24)
1.563
1.563
1.563
A(25)
0.196
0.111
0.111
A(26)
4.59
4.59
4.59
A(27)
1.457
1.457
1.457
Weight (lb)
2396.8
2386.0
2374.095
Constraint violation value
0.9994
0.9913
0.9965
Number of FEAs
45,557
25,000
19,843
5 Conclusions This paper discovers the potential of applying AI technologies in structural optimization. In more detail, the Differential Evolution algorithm is used to search the optimal solution
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over the design space meanwhile the classification based on Artificial Neural Network serves as a coarse filter for removing poor solutions during the optimization process. In the illustrative example, the proposed approach achieves good performance in comparison with other optimization algorithms from the literature. Additionally, the use of the machine learning classification technique helps to save about 20% of the required finite element analyses. Acknowledgment. This research is supported by the National University of Civil Engineering, Vietnam (NUCE) under grant number 77-2020/KHXD. Furthermore, the first author gratefully acknowledges the financial support of the Domestic Ph.D. Scholarship Programme of Vingroup Innovation Foundation (VINIF).
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15. Pham, H.A.: Reduction of function evaluation in differential evolution using nearest neighbor comparison. Vietnam J. Comput. Sci. 2(2), 121–131 (2015) 16. Pham, H.-A.: Truss optimization with frequency constraints using enhanced differential evolution based on adaptive directional mutation and nearest neighbor comparison. Adv. Eng. Softw. 102, 142–154 (2016) 17. Truong, V.-H., Kim, S.-E.: Reliability-based design optimization of nonlinear inelastic trusses using improved differential evolution algorithm. Adv. Eng. Softw. 121, 59–74 (2018) 18. Ha, M.-H., Vu, Q.-V., Truong, V.-H.: Optimization of nonlinear inelastic steel frames considering panel zones. Adv. Eng. Softw. 142, 102771 (2020) 19. Lu, X., Tang, K., Yao, X.: Classification-assisted differential evolution for computationally expensive problems. In: Proceedings of the 2011 IEEE Congress of Evolutionary Computation (CEC). IEEE (2011) 20. Liu, Y., Sun, F.: A fast differential evolution algorithm using k-Nearest Neighbour predictor. Expert Syst. Appl. 38(4), 4254–4258 (2011) 21. Krizhevsky, A., Sutskever, I., Hinton, G.E.: Imagenet classification with deep convolutional neural networks. Adv. Neural Inf. Process. Syst. (2012) 22. Szegedy, C., et al.: Going deeper with convolutions. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (2015) 23. Simonyan, K., Zisserman, A.: Very deep convolutional networks for large-scale image recognition. arXiv:1409.1556 (2014) 24. Kaiming, H.E., et al.: Deep residual learning for image recognition. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (2016) 25. Lee, K.S., Han, S.W., Geem, Z.W.: Discrete size and discrete-continuous configuration optimization methods for truss structures using the harmony search algorithm. Iran Univ. Sci. Technol. 1(1), 107–126 (2011) 26. Kaveh, A., Mahdavi, V.R.: Colliding bodies optimization method for optimum discrete design of truss structures. Comput. Struct. 139, 43–53 (2014) 27. Zhang, Y., et al.: A machine learning framework for assessing post-earthquake structural safety. Struct. Saf. 72, 1–16 (2018) 28. Fu, F.: Fire induced progressive collapse potential assessment of steel framed buildings using machine learning. J. Construct. Steel Res. 166, 105918 (2020) 29. Truong, V.-H., et al.: A robust method for safety evaluation of steel trusses using Gradient Tree Boosting algorithm. Adv. Eng. Softw. 147, 102825 (2020) 30. Kim, S.-E., et al.: Comparison of machine learning algorithms for regression and classification of ultimate load-carrying capacity of steel frames. Steel Compos. Struct. 37(2), 193–209 (2020) 31. PyNiteFEA. https://pypi.org/project/PyNiteFEA/. Accessed 04 Nov 2020 32. Scikit-learn. https://scikit-learn.org/. Accessed 04 Nov 2020 33. Storn, R., Price, K.: Differential evolution - a simple and efficient heuristic for global optimization over continuous spaces. J. Glob. Optim. 11(4), 341–359 (1997) 34. Krempser, E., et al.: Differential evolution assisted by surrogate models for structural optimization problems. In: Proceedings of the International Conference on Computational Structures Technology, vol. 49. Civil-Comp Press (2012)
A Sliding Mode Controller for Force Control of Magnetorheological Haptic Joysticks Diep B. Tri1,3 , Le D. Hiep2 , Vu V. Bo3 , Nguyen T. Nien3 , Duc -Dai Mai1 , and Nguyen Q. Hung2(B) 1 Faculty of Civil Engineering, HCMC University of Technology and Education,
Ho Chi Minh City, Vietnam {tridb.ncs,daimd}@hcmute.edu.vn, [email protected] 2 Faculty of Engineering, Vietnamese-German University, Binh Duong, Viet Nam [email protected] 3 Industrial University of Ho Chi Minh City, Ho Chi Minh, Viet Nam {vuvanbo,nguyenthinien}@iuh.edu.vn
Abstract. In this research, a sliding mode controller is proposed and implemented in force control of a 3D haptic joystick featuring two bidirectional magnetorheological actuators (BMRA) and one linear magnetorheological brake (LMRB). After a review of haptic joystick development featuring magnetorheological fluid (MRF) and chattering reduction approaches in sliding mode controllers (SMC), the configuration and a prototype of the 3D haptic joystick are presented with both simulated and experimental performance characteristics. A sliding mode controller is then proposed to control feedback force of the joystick. The proposed SMC is implemented for the prototype of the 3D haptic joystick. Experimental results on feedback force of the joystick with the proposed SMC are then obtained and compared with PID controller with discussions. Keywords: MR fluid · MRB · Haptic joystick · Bidirectional MR actuator · Linear MRB
1 Introduction Currently, smart material attracts a lot of researchers on researching its behavior and haptic application. More haptic systems are investigated such as SMA haptic, pneumatic haptic, servo haptic, or MRF haptic system. High power and compact actuator are critical requirements of haptic systems. Therefore, MRF actuator get a lot of attention from numerous researcher because of fast response, easy control, and more compact [1–3]. However, in previous works, magneto-rheological brakes (MRBs) were used as actuators which exhibited several problems such as uncontrollable torque (the friction torque at zero-field) and the passive force feedback (the feedback force can only be felt by the operator when a motion of the operation handle is performed). To remedy these drawbacks, recently researchers have focused on development of bidirectional magnetorheological actuator (BMRA) for haptic systems. Nguyen P B et al. developed a BMRA © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 1065–1077, 2022. https://doi.org/10.1007/978-981-16-3239-6_83
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driven by a DC servo motor for employing in 2D haptic joysticks [4, 5]. In that research, conventionally, the expiation coils are wound on a bobbin (nonmagnetic) attached to a cylindrical housing. This configuration possesses several disadvantages such as high disc thickness, magnetic bottle-neck problem, and manufacturing and maintenance difficulties. To solve these problems, more recently, Nguyen Q H et al. [6, 7] developed a new BMRA in which the coils were placed directly on side housings. In addition, to avoid contamination of the coil with the MRF, a thin-wall configuration is used to separate the MRF and the coils. After that, the BMRA developed by Nguyen et al. was implemented in a 2D-haptic joystick shown in Fig. 1a [8]. Experimental works showed that a desired sinusoidal feedback force can be considerably achieved. In practical applications, 3D feed force are often required which leads to the necessity of 3D haptic joysticks. In this works, the BMRA in [6] and LMRB in [3] is employed to design 3D haptic system. The optimal results of each actuator are used to design and manufacture prototypes. The experimental results of each actuator are derived mathematical model of each actuator via system identification procedure. From desired force Fx, Fy, Fz the required torque of each BMRA and force of LMRB is calculated via the kinematic model. Traditional controller PID [9] and sliding mode controller (SMC) [10] are derived to control the desired force. In summary, the contributions of this paper are: i) 3D haptic system using BMRAs and LMRB is for the first time presented and demonstrated, ii) Kinematic model of haptic system is derived for controlling feedback force, iii) System identification results for both BMRA and LMRB are given, iv) the comparison between PID controller and SMC is provided.
2 The Proposed 3D Haptic Joystick In this work, BMRA in [6] and LMRB in [3] are employed to design 3D haptic system given in Fig. 1b. In this works, the 2D haptic system in [8] is developed to 3D haptic system by using two BMRA for reflecting force in x and y direction, and 1 LMRB for z direction. The BMRA configuration is given in Fig. 2. Two coils are located directly in both housings. Two discs are attached in two input shafts rotating counter-clockwise and encounter clockwise, respectively. The output shaft is connected with housing that can generate both pull and resistance torque depending on triggering coils. Several bearings are installed between shaft and housing to ensure the rotation of shaft round housing. Besides, lip-seals are employed to prevent MRF leaking. The magnetic field through MR duct is analyzed by using ANSYS software. After that, generated torque can be determined as [6]. The optimization of BMRA is conducted with the objective is to minimize the mass of BMRA with a maximum required torque. First-order optimization algorithm integrated the ANSYS optimization toolbox is implemented to obtain the optimal results, given in Table 1. The configuration of LMRB is demonstrated in Fig. 3. Two coils are wounded directly in outer housing of LMRB. LMRB in this work using shear mode work which can generate high force and compact structure. However, in LMRB, the disadvantage is off-state force caused by friction of two o-ring can not be eliminated. After analyzing the magnetic field through MR duct, the generated force is determined as [11]. The optimization
A Sliding Mode Controller for Force Control of Magnetorheological Haptic Joysticks
a) 2D haptic in [8]
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b) Proposal 3D haptic
Fig. 1. a) 2D haptic in [8]. b) Proposal 3D haptic.
problem of LMRB is to determine the minimum off-state force with required maximum generated force. The optimal solution of LMRB is also provided in Table 2. After simulation and optimization, the BMRA and LMRB prototypes are designed and manufactured. It is noted that all components of the prototypes are machined by CNC machines to ensure the accuracy dimensions. The coils are wound for BMRA and LMRB with sufficient stretch of wires. The magnetic test is also the next step before assembling all components of LRMB and BMRA. After that, MR gaps of each prototype are filled fully by MRF.
Fig. 2. The BMRA configuration
Fig. 3. The LMRB configuration
Figure 4 shows the experimental 3D-haptic system. It comprises two BMRAs in x and y direction which is driven by two corresponding Geared Servo Motor (Gearing ratio is 10:1) via the bevel gear system to rotate the inner and outer shafts of the BMRAs in opposite directions at 60 rpm. The BMRA output shaft is connected with the gimbal mechanism. On the handle of the gimbal, an LMRB is attached. In this research, the
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D. B. Tri et al. Table 1. Optimal results of the BMRAs.
Design variable (mm)
Optimized performance characteristics
Coils: wc1 = wc2= 4.25; hc1 = 7.4, hc2 = 6.2; Rci1 = 29, Rci2 = 47.5; No. of turns: n1 = 95, n2 = 80 Discs: Ri = 20, Ro = 56.5, t d = 4.0 Housing: R = 59.4, t o = 2, t h = 3.2, L = 24.3 Thin-wall: 0.6 MRF gap: 0.8
Max. output torque: 4.98 Nm General mass: 2.1 kg Power applied to the coil: 24.0 W Resistance of coils (): Rc1 = 1.68, Rc2 = 2.16
Table 2. Optimal results of the LMRB. Design variable (mm)
Optimized performance characteristics
Coil size: hcl = 7.5, wcl = 11.3, cr = 2.85, cl = 4.9 No. of coil turns = 230 Housing: L po = 4.0, L pi = 8.0, L l = 38.7; t w = 0.5, t o = 2.1, R = 14.8 Shaft radius: Rsl = 4.5 MRF gap: 0.5
Off-state torque: 5.0 N Max. brake force: 24.95 Nm General mass: 0.2 kg Coil Power: 14.5 W Resistance of coils: 2*1.15 = 2.3 Ω
operation range of handle is limited from −60° to 60° in both x and y direction. Two encoders are employed to determine the angular position of handle in comparison with the original position. The 3D force (OptoForce) is attached at the end of the handle to measure actual force Fx, Fy, Fz. The PCI Card NI-6008 is used to collect data from 3D Force and encoder. Afterward, the errors between actual force and desired force are determined. From the updated position of handle and errors of force, the required current is calculated and provided to BMRAs and LMRB via current amplifier. The supplied current is limited from 0 to 2.5A to ensure the safety of coils. To control the system, firstly, mathematic models of the BMRAs and the LMRB are identified experimentally. In the off-state condition, despite no current provides to any coils of the BMRA, there exists a small output torque called the off-state torque. This can be explained by the different friction torque of two inputs. It is necessary to offset to zero before identifying mathematical model of BMRA. From experimental results with each step input current, the output torque is employed to identify the mathematical model by MATLAB software. Step response and system identification results of BMRA are given in Fig. 5. This can be seen that the difference between the measured torque and identity torque is a good agreement. The system model of BMRA is determined as: ..
.
a T +b T +T = u(I )
(1)
Where u(I ) = 0.01025 − 0.53308I − 1.638528I 2 + 0.42608I 3 , a = 1 26590, b = 2452 26590. T is desired torque (Nm) and I is applied current (A).
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Fig. 4. Experimental system to test the 3D haptic joystick
Similarly, the step response and system identification results of LMRB is given in Fig. 6. The system model of LMRB is provided as: ..
.
a F +b F +F = u(I )
(2)
Where u(I ) = 5.01899 + 9.757391I + 1.28363I 2 − 0.796I 3 , a = 1 649.5, b = 60.69 649.5. F is desired force (N) and I is applied current (A). 6
2.5A 1.25A
1.75A 0.75A
1
1.5A 0.5A
2.5A 1.25A
0
4
Torque (Nm)
Torque (Nm)
5
2A 1A
3
2
2A 1A
1.75A 0.75A
1.5A 0.5A
-1
-2
-3
-4
1
-5
0 0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
0.0
0.5
1.0
a) disc1 activated
1.5
2.0
Time(s)
Time(s)
b) disc 2 activated
Fig. 5. Step response of the BMRAs
2.5
3.0
3.5
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2.5A 1.5A
2A 1A
1.75A 0.75A
0.5A
Force (N)
25
20
15
10
5 0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
Time (s)
Fig. 6. Step response of the LMRB
3 Controller Design for the 3D Haptic Joystick 3.1 Analysis of the 3D Haptic System Kinematic diagram of the joystick are shown in Fig. 7. By applying virtual work principle, the following relation can be obtained Tx δφx + Ty δφy + Fb δl + Fxp δxP + Fyp δyp + Fzp δzp = 0
Fig. 7. Kinematic schematics of the joystick
(3)
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Equation (3) can be rewritten as following T T [Tx Ty Fb ] δφx δφy δl + [Fxp Fyp Fzp ] δxp δyp δzp = 0
(4)
Here, T x and T y are the output torque of the BMRA of the X-shaft (BMRA_x) and the Y-shaft (BMRA_y) respectively, F b is the LMRB braking force, δφ x is the virtual angular displacement of the BMRA_x shaft, δφ y is the virtual angular displacement of the BMRA_y shaft, δl is the virtual displacement of the LMRB shaft, F xp , F yp and F zp are the force applied to the operation knob respectively in X, Y and Z direction, δx p , δyp and δzp are respectively virtual displacement of the operation knob in X, Y and Z direction, which can be determined by T δrp = δxp δyp δzp = δrp,xy + δrp,l
(5)
Where δrp,xy is the displacement regarding to the virtual displacements δφ x and δφ y , δrp,l is the displacement relating to the virtual displacements δl. These displacements can be calculated by δrp,xy = [Rδφx Rδφy − I ]rp δrp,l =
(6)
x y z T rp p p p δl δl = l l l l
(7)
In the above, Rδφx and Rδφy are respectively the rotation matrices of the virtual displacement δφ x and δφ y . For very small value of δφ x and δφ y , we can get ⎡
Rδφx Rδφy
⎤ 1 0 δφy ⎣0 1 −δφx ⎦ −δφy δφx 1
(8)
From Eqs. (5), (6), (7) and (8), we can derive the following relation ⎡ ⎡ ⎤ ⎤ ⎡ ⎤⎡ δφ ⎤ δxp δφx x 0 zp xp /l ⎢ ⎥ ⎢ ⎥ ⎥ ⎢ ⎣ δyp ⎦ = ⎣ −zp 0 yp /l ⎦⎣ δφy ⎦ = J ⎣ δφy ⎦ δzp
yp
−xp zp /l
δl
(9)
δl
Plug Eq. (9) into (4), we have [Tx Ty − Fb ][δφx δφx dl]T + [Fxp Fyp Fzp ]J [δφx δφx δl]T = 0
(10)
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The, the actuating torque/force can be derived from the acting force at the operating knob by T T Tx Ty Fb = −J T Fxp Fyp Fzp
(11)
The applied force in the local frame fixed to the handle are related to the force in the global frame by
Fxp Fyp Fzp
T
T = R Fx2 Fy2 Fz2
(12)
Place Eq. (11) into (12), the following equation can be obtained T T Tx Ty Fb = −J T R Fx2 Fy2 Fz2
(13)
Noteworthily, the normal feedback force is always equal to the damping force and decoupled from the tangent forces. Then Eq. (13) can be rewritten by
Tx Fx2 T (14) = −[J R]2x2 ; Fb = Fz2 Ty Fy2 where –[J T R]2x2 is the principal 2x2 sub-matrix of the –[J T R]. 3.2 Close Loop Control of the Feedback Force In this work, closed loop controller is designed to obtain desired feedback force to operator. Figure 8 shows flow chart to achieve a required reflection force at an arbitrary position of the operation knob, determined by T rP = xp , yp , zp = R[0, 0, l]T
(15)
where R is the total rotation matrix (resulting from a series rotation a bout a fixed frame). In the experiment, the rotation matrix R is updated at every sample time (Δt = 0.01s) by the following R(t + t) = R(t)R(t)
(16)
where, the rotation matrix R(Δt) is approximately estimated by ⎡
R(t) = Rd φx Rd φy
⎤ 1 0 d φy ⎣0 1 −d φx ⎦ −d φy d φx 1
(17)
where dφ x and dφ y are angular displacement of the handle in the interval time (Δt), measured by encoders. As the knob position is determined, the matrix J is also calculated by Eq. (9).
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From the error of the feedback force (EF ), the corresponding error of the actuating torque/force of the BMRAs and the LMRB (ET )can be evaluated by Eq. (14). This error is then fed to an appropriate controller for applied currents to the coils of the BMRA_x, BMRA_y and the LMRB (I x , I y , I z ).
Fig. 8. Closed loop controllers for desired forces
3.3 Design of PID Controller for the Feedback Force In the PID controller, the required current is generally determined as follows: . I (t) = kp e(t) + ki e(t) + kd e(t)
(18)
where kp , ki and kd are the proportional, integral, and derivative gains, respectively.e is the error between input and output. Tuning parameters as follows: BMRA_x Kp = 10 Ki = 0.00006 Kd = 0.00008 BMRA_y Kp = 9 Ki = 0.00007 Kd = 0.00007 LMRB
Kp = 9 Ki = 0.00007 Kd = 0.0002
3.4 Design of SMC Controller for the Feedback Force General Form x˙ 1 = x2 x˙ 2 =
u x1 bx2 − − + a a a
(19) (20)
determined from system [x1 x2 ] is state vector. u is control input. a, b are parameters identification. In this work a = 1 26590, b = 2452 26590.
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comprises disturbance and uncertainty of the system e = xd − x
(21)
e is the error between input and output. The sliding surface can be defined as: s = ce + e˙ , c > 0 as hurwitz condition
(22)
s˙ = c˙e + e¨
(23)
s˙ = −ksign(s), k > 0
(24)
Control effort can be determined as:
bx2d x1d u = a −ksign(s) − c˙e + x˙ 2 + + a a
(25)
Prove the stability of the system: The Lyapunov function can be determined as: V =
1 2 s 2
(26)
.
V = s˙s = s(c˙e + e¨ )
(27)
= s(c˙e + (˙x2d − x˙ 2 ))
(28)
⎛ a −ksign(s) − c˙e + x˙ 2 + x˙a1d + ⎜ ⎜ ⎜ = s⎜ ⎝c˙e + ⎝ a ⎛
= s c˙e +
−ksign(s) − c˙e + x˙ 2 +
bx2d a
bx2d x˙ 1d + a a
⎞⎞ −
−
⎟⎟ x1d bx2d ⎟ − + − x˙ 2 ⎟ ⎠⎠ a a
bx2d x1d − + − x˙ 2 a a
= s(−ksign(s) + ) < 0 when k > ||
(29) (30) (31)
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Tuning parameters as follows: BMRA_x c = 0.0015 k = 14 BMRA_y c = 0.0014 k = 197 c = 0.0017 k = 10
LMRB
4 Results and Discussions Figure 9 and 10 present the results of controlling feedback force of PID and SMC at 3 Hz of sine function, respectively. It can be seen that SMC well follows the desired force with a small tracking error of 8% which is smaller than PID. The tracking force of PID oscillates continuously around the desired force caused by chattering continuously of current of each actuator. This is easy to understand that in the system with more disturbance, noise, and uncertainty, PID cannot deal with all these drawbacks. However, the input current of the SMC is smoother than PID controller. Besides, the actuator force control by SMC tracks well with the desired force. It is also explained that SMC can deal with disturbance, uncertainty, and the variation of the system. In both controllers, 40 Desired Fx Measured Fx
3
Desired Fy Desired Fz Measured Fy Measured Fz
Current for LMRB
2
Current(A)
Force (N)
20
0
1
-20
0
1
2
3
4
0
5
0
Time (s)
2
3
4
5
Time(s)
a) Tracking Force Fx, Fy, . Fz
b) Required current for LMRB.
5
5
Current provides for first side BMRA Current provides for second side BMRA
4
Current provides for first side BMRA Current provides for second side BMRA
4
3
3
2
Current (A)
Current (A)
1
1 0
2 1 0
-1
-1
-2
-2 -3
-3 0
1
2
3
4
5
Time (s)
c ) Required current of the BMRA_x
0
1
2
3
4
5
Time (s)
d) Required current of the BMRA_y
Fig. 9. Experimental results of the feedback forces using PID controller
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30
3.0
Desired Fy Desired Fz Measured Fy Measured Fz
Current provides for LMRB 2.5
Current (A)
Force (N)
20
10
0
2.0
1.5
-10
1.0
-20
0.5
-30 0
1
2
3
4
0.0
5
0
1
2
Time (s)
a) Tracking Force Fx, Fy, Fz.
4
5
b) Required current for LMRB.
3
3
Current provides for first side BMRA Current provides for second side BMRA
2
Current provides for first side BMRA Current provides for second side BMRA
2
1
Current (A)
Current (A)
3
Time (s)
0
1
0
-1
-1
-2
-2
-3
-3 0
1
2
3
4
5
Time (s)
c ) Required current of the BMRA_x
0
1
2
3
4
5
Time (s)
d) Required current of the BMRA_y
Fig. 10. Experimental results of the feedback forces using SMC controller
it can be seen that the actual force Fz cannot track the required force in which required force smaller than 5.3N caused by off-state force of LMRB.
5 Conclusions In this study, a 3D Haptic system is constructed using MR actuator. Firstly, BMRA and LMRB are analyzed and optimized to have optimal geometry dimensions. These prototypes are designed and manufactured. The mathematical model of each actuator is provided by using the system identification of MATLAB. The kinematic equation of the 3D system is derived to determine the required torque of each BMRA. PID controller and SMC are designed and implemented for the real system. The results showed that the SMC is more suitable than PID in force control of the haptic system. The limitation of this work is LMRB which has a large off-state force. In future work, the novel active LMRB can be proposed to get rid of off-state force. Moreover, other robust controllers are also taken into consideration. Acknowledgment. This work was supported by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant no. 107.01–2018.335.
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References 1. Senkal, D., Gurocak, H.: Spherical brake with MR fluid as multi degree of freedom actuator for haptics. J. Intell. Mater. Syst. Struct. 20(18), 2149–2160 (2009) 2. Nguyen, Q.H., Choi, S.B., Lee, Y.S., Han, M.S.: Optimal design of a new 3D haptic gripper for telemanipulation, featuring magnetorheological fluid brakes. Smart Mater. Struct. 22(1), 015009 (2012) 3. Bui, D.Q., Hoang, V.L., Le, H.D., Nguyen, H.Q.: Design and evaluation of a shear-mode MR damper for suspension system of front-loading washing machines. In: Nguyen-Xuan, H., Phung-Van, P., Rabczuk, T. (eds.) ACOME 2017. LNME, pp. 1061–1072. Springer, Singapore (2018). https://doi.org/10.1007/978-981-10-7149-2_74 4. Nguyen, P.B., Choi, S.B.: A Bi-directional magneto-rheological brake for medical haptic system: optimal design and experimental investigation. Adv. Sci. Lett. 13(1), 165–172 (2012) 5. Nguyen, P.B., Choi, S.B.: Accurate torque control of a bi-directional magneto-rheological actuator considering hysteresis and friction effects. Smart Mater. Struct. 22(5), 055002 (2013) 6. Diep, B.T., Vo, V.C., Nguyen, Q.H.: Development of a new magnetorheological actuator for force feedback application. Int. J. Electron. Electr. Eng. 5(4), 280–283 (2017) 7. Nguyen, Q.H., Diep, B.T., Vo, V.C., Choi, S.B.: Design and simulation of a new bidirectional actuator for haptic systems featuring MR fluid. In: Proceedings of SPIE 10164, Active and Passive Smart Structures and Integrated Systems (2017). https://doi.org/10.1117/12.2259752 8. Diep, T.B., Le, H.D., Van Vo, C., Nguyen, H.Q.: Performance evaluation of a 2D-Haptic joystick featuring bidirectional magneto-rheological actuators. In: Nguyen-Xuan, H., PhungVan, P., Rabczuk, T. (eds.) ACOME 2017. LNME, pp. 1051–1059. Springer, Singapore (2018). https://doi.org/10.1007/978-981-10-7149-2_73 9. Truong, T.D., Nguyen, V.Q., Diep, B.T., Le, D.H., Le, D.T., Nguyen, Q.H.: Speed control of rotary shaft at different loading torque using MR clutch. In: Proceedings SPIE 10967, Active and Passive Smart Structures and Integrated System XIII, vol. 109671N (2019) 10. Nguyen, Q.H., Choi, S.B.: A new method for speed control of a DC motor using magnetorheological clutch. In: Active and Passive Smart Structures and Integrated Systems, vol. 9057, p. 90572T (2014) 11. Nguyen, Q.H., Han, Y.H., Choi, S.B., Wereley, N.M.: Geometry optimization of MR valves constrained in a specific volume using the finite element method. Smart Mater. Struct. 16, 2242–2252 (2007)
Experimental and Numerical Investigations on Flexural Behavior of Retrofitted Reinforced Concrete Beams with Geopolymer Concrete Composites Do Van Trinh(B) and Khong Trong Toan Faculty of Civil Engineering, Ho Chi Minh City University of Technology (HUTECH), Ho Chi Minh City, Vietnam {dv.trinh,kt.toan}@hutech.edu.vn
Abstract. Existing multi-storey buildings in earthquake prone regions of India are vulnerable to severe damage under earthquakes. So there is an urgent need for retrofitting of deficient and damaged buildings. Portland cement (PC) production is under critical review due to high amount of carbon dioxide gas released to the atmosphere and Portland cement is also one among the most energy-intensive construction material So retrofitting of existing deficient building using eco-friendly material which higher structural performance than the original building is essential Geopolymer concrete (GPC), a non-Portland cement binder based on alkaline activation of industrial wastes such as fly ash and GGBS. The application Geopolymer concrete in retrofitting can reduce the production of cement and can create an ecofriendly construction atmosphere to an extent. So the feasibility of geopolymer concrete in retrofitting of damaged structures is required to be studied. Non-linear finite element analysis of beams by 3D modeling of concrete (solid65 element) and discrete modeling of reinforcement (Link 8 element) was carried out using ANSYS software. The load deflection characteristics, failure modes and crack patterns are obtained from the experimental and analytical studies. Keywords: Geopolymer concrete (GPC) · Finite element modelling · Fly ash · Ground Granulated Blast Furnace Slag (GGBS) · Retrofitting
1 Introduction Concrete is a versatile material having number of desirable properties, so widely preferred in construction industry. But a having major disadvantage that under severe vibrations such as earthquakes it gets cracked and concrete has poor modification and repair quality. Therefore, some repair or strengthening must be performed so that damaged element must stand for further useful life. Many civil structures are no longer considered safe due to increased load specifications in the design codes. Such structures must be strengthened in order to maintain their serviceability.
© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 1078–1088, 2022. https://doi.org/10.1007/978-981-16-3239-6_84
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The various strengthening techniques that are currently available includes steel plate bonding, polymer injection followed by concrete jacketing, use of advanced composite materials like Glass Fibre Reinforced Polymer(GFRP), Carbon Fibre Reinforced Polymer(CFRP), mats and Ferrocement (Amarnath, C., et al., 2008) etc. [1]. But all the above mentioned techniques are very sensitive to increase in temperature. The task of retrofitting of existing structure must ensure that the strength and stiffness of overall structure is resorted to original design requirements for long life and also the material used for retrofitting should produce less greenhouse gases, because the cement is proving to be ecologically hazardous material due to its inherent high internal energy content besides occurrence of emission of large quantities of carbon dioxide during its production. The production of cement means the production of pollution because of the emission of CO2 during its production. To produce an environmental friendly concrete, it is necessary to replace the cement with ecofriendly material. In this respect, the new technology Geopolymer concrete (GPC) is a promising technique which uses industrial waste such as fly ash (FA) and ground granulated blast furnace slag (GGBS) [2]. A huge amount of fly ash (FA) is generated in thermal power plants, causing several disposal-related problems; the total utilization of FA is only about 50% of produced. Disposal of fly ash is a growing problem as only 15% of fly ash is currently used for high value addition applications like concrete and building blocks, the remaining being used for landfills. Geopolymer technology can be appropriate process technology utilize all classes and grades of FA and therefore there is a great potential for reducing stockpiles of waste FA materials [3]. GGBS is a by-product from the blast furnaces used to make iron. GGBS is also used as a binder component in Geopolymer concrete. It has been reported that geopolymer material does not suffer from alkali-aggregate reaction even in the presence of high alkalinity and possesses excellent fire resistant. Geopolymer is used as the binder, instead of cementitious material, to produce concrete. The manufacture of geopolymer concrete is carried out using the usual concrete technology methods.
2 Research Significance One of the most important applications of geopolymeris in construction industry. However, the suitability of GPC to various structural components is yet to be established by large number of experimental studies. Various experimental works have been done to find out the suitability of geopolymer concrete to replace the Portland cement concrete (PCC) and based on these results the geopolymer concrete application can create an environmental friendly construction industry. Since the majority of existing buildings are deficient for resisting earthquakes, so nowadays retrofitting of the building has become a major issue. The recent earthquakes in India occurred during 2001 in Gujarat, damaged many buildings that were seismically deficient. This calls for techniques those are environmental friendly, technically sound and economically feasible to upgrade deficient structures. Various retrofitting techniques available are steel plate bonding, polymer injection followed by concrete jacketing, use of advanced composite materials like Glass Fibre Reinforced Polymer (GFRP), Carbon Fibre Reinforced Polymer (CFRP), and Ferrocement etc. But since these above mentioned methods are just strengthening the existing structure without removing the
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damaged portion many disadvantages exist, like their behavior changes as temperature varies, and also the materials tensile strength and stiffness not match with the concrete properties. So in order to overcome these drawbacks, a new attempt was made i.e. retrofitting of damaged portion using geopolymer concrete. In this study the retrofitting is done after the removal of damaged concrete and is rebuilt with geopolymer concrete. The experimental and analytical studies on control beams and retrofitted beams having two different dimensions one with 100 × 150 × 1500 mm having 3.35% reinforcement and another with 100 × 200 × 1500 mm with 1.28% reinforcement were carried out. Also the GPC was made with different concentration of sodium hydroxide solution i.e. GPC made from 3M and 5M alkaline solution. The behaviour of reinforced GPC (RGPC) beams 100 × 150 × 1500 mm retrofitted with 3M (RGPC3M) and 5M (RGPC5M) GPC was also carried out.
3 Experimental Work Ordinary Portland cement conforming to IS 12269, fine aggregates, coarse aggregates and portable water were used for the control specimen RC specimens. While for GPC beams and for retrofitting Ground Granulated Blast Furnace Slag (GGBS) as per IS 3812, Fly Ash (FA) conforming to IS 12089, fine aggregates (river sand) and coarse aggregates (crushed granite stones-12 mm size) as per IS 2386 were used [5]. Catalytic liquid system (CLS) is an alkaline activator solution (AAS) for GPC. It is a combination of solutions of alkali silicates and hydroxides, besides distilled water. The role of AAS is to activate the geopolymeric source materials (containing Si and Al) such as fly ash and GGBS [6].
(a)
(b)
Fig. 1. Reinforcement and dimension details of (a) 100 × 200 × 1500 mm RPPC beam (with 1.28% reinforcement) and (b) 100 × 150 × 1500 mm RPPC and RGPC beam (with 3.35% reinforcement)
The beam specimens include 4 numbers of 100 mm wide and 200 mm deep beams and 3 numbers of 100 mm wide and 150 mm deep beams. They are 1500 mm in length and simply supported over an effective span of 1350 mm with M30 grade concrete. The beams of 100 × 200 mm cross section is having 1.28% reinforcement with 2#8 mmφ compression reinforcement and 2#10 mmφ tension reinforcement, the compression reinforcement is provided only at the end portion of beams, not at the mid-span. Also 8 mmφ
Experimental and Numerical Investigations on Flexural Behavior
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stirrups are provided at 125 mm c/c. The clear cover of the beam was 20 mm as shown in Fig. 1(a). Of the 4 RPPC beams 2 beams are retrofitted with 5M GPC and other 2 beams are retrofitted with 3M GPC. The beams with 100 × 150 mm cross section were designed as doubly reinforced sections with total reinforcement 3.35%. All the beams are reinforced with two numbers of 16 mm diameter rods at the bottom and two numbers of 8 mm diameter rods at top of the beam. Apart from it, 8 mm diameter stirrups are provided at 100 mm c/c spacing. The clear cover of the beam is 25 mm as shown in Fig. 1(b). Table 1 is of the 4 specimens (3.35%) two beams are reinforced PPC (RPPC) beam retrofitted with GPC one with 5M and the other one with 3M; other 2 beams are reinforced GPC (RGPC) beam retrofitted with geopolymer concrete one with 5M and other with 3M. Table 1. Details of specimens Specimen ID
Type
Details
RC5M-1
RPPC3
3.35% reinforcement, PPC beam retrofitted with 5M GPC
RC5M-2
RPPC1
1.28% reinforcement, PPC beam retrofitted with 5M GPC
RC5M-3
RPPC1
1.28% reinforcement, PPC beam retrofitted with 5M GPC
RC3M-1
RPPC2
1.28% reinforcement, PPC beam retrofitted with 3M GPC
RC3M-2
RPPC2
1.28% reinforcement, PPC beam retrofitted with 3M GPC
RC3M-3
RPPC4
3.35% reinforcement, PPC beam retrofitted with 3M GPC
GPC5M
RGPC1
3.35% reinforcement, PPC beam retrofitted with 5M GPC
GPC3M
RGPC2
3.35% reinforcement, PPC beam retrofitted with 5M GPC
RPPC1-C
RPPC1-C
3.35% reinforcement, PPC control beam
RPPC2-C
RPPC2-C
1.28% reinforcement, PPC control beam
RPPC-C
RPPC-C
3.35% reinforcement, GPC control beam
Concrete is removed from the damaged portion of the beam. The damaged portions of concrete were cut and loose materials were removed and blower was used to clean the surface. The removed portion of the damaged beam is about 300 mm at the mid-span for all specimens. Prior to casting, the inner walls of mould were coated with lubricating oil to prevent adhesion with the hardening concrete. GPC was mixed in a pan mixer machine of 60 kg capacity for about three minutes. At the end of this mixing, the alkaline activator solution (AAS) was added to the dry materials. Then mixing was continued for another four minutes till a uniform consistency was achieved. Immediately after mixing, the fresh concrete was casted in three layers of equal thickness and each layer was thoroughly compacted. Specimens were demoulded after 24 h. The retrofitted beams were cured with subsequently air cured under ambient conditions in the laboratory for a period up to 28 days after casting. The Fig. 2 shows the steps involved for the preparation of retrofitted specimen.
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(b)
Fig. 2. Retrofitting procedure (a) Damaged concrete is removed and molds are placed (b) Damaged portion is retrofitted with GPC.
A photograph of test setup is shown in Fig. 3. The beam specimen was mounted in a UTM of 100 tons capacity. The beam specimen was simply supported on reaction blocks mounted on a sturdy box section of length 1700 mm placed on the bottom plate of the UTM. The beam was simply supported over a span of 1350 mm, which is considered as the effective span. The load was applied on two points each 225 mm symmetrically away from centre of the beam towards the support through a load spreader. Two dial gauges of 0.01 mm least count were used for measuring deflections under the loading points. Another dial gauge of same least count was used at mid span for measuring the respective deflection. All the specimens were white washed in order to facilitate marking of cracks. The load was applied in increments of 2.5KN until the first crack was obtained. Later, the load was applied in increments in 5KN. The deflection at mid span and under the load points were measured using dial gauges. The behaviour of the beam was observed carefully. All the measurements including deflections, strain values and crack widths were recorded at regular intervals of load until the beam failed. The failure mode of the beams was also recorded.
Fig. 3. Test setup
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The retrofitted beam specimens were tested under two-point loading until failure. As the load increased, beam started to deflect and flexural cracks developed along the span of the beams. All the beams failed in the same pattern due to yielding of the tensile steel (primary tension failure) followed by crushing of concrete at the compression face (secondary compression failure). Figure 4 shows the comparison of load deflection characteristics of control RPPC and retrofitted RPPC beams. Thus from the comparison the retrofitted beam attains 80% strength of the control beam for RPPC beams. Comparison of load deflection characteristics of retrofitted RGPC and control RGPC is shown in Fig. 5. The load-deflection behavior of retrofitted RGPC beams are similar to control RGPC beams. It was observed that retrofitted RGPC beams perform better than retrofitted RPPC beams. Based on the analysis of results, Geopolymer concrete can be used for strengthening of reinforced concrete structures and structural components. Table 2 shows the ultimate load and ultimate deflection of RPPC beams with two different % of reinforcement, before and after retrofitting with GPC. It also lists the ultimate load and deflection of RGPC control and retrofitted beam with RGPC (Fig. 6).
Fig. 4. Comparison of mid-span load deformation characteristics of RPPC controlled and retrofitted beam (having 3.35% reinforcement)
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Fig. 5. Comparison of mid-span load deformation characteristics of RGPC controlled and retrofitted beam (having 3.35% reinforcement)
Fig. 6. Comparison of mid-span load deformation characteristics of RGPC retrofitted beam with 3M and 5M GPC. Table 2. Comparicon of retrofitted and controlled RPPC and RGPC beams Grid
Percent of reinforcement %
Ultimate load kN
Ultimate deflection mm
RC5M-1
3.35
87.1
16.65
RC3M-3
3.35
80
13.54
RPPC1-C
3.35
104.9
8.53
RC5M-2
1.28
65.55
14.66 (continued)
Experimental and Numerical Investigations on Flexural Behavior
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Table 2. (continued) Grid
Percent of reinforcement %
Ultimate load kN
Ultimate deflection mm
RC3M-1
1.28
60.15
18
RPPC2-C
1.28
80
10.82
GPC5M-1
3.35
104.84
12.59
GPC3M-1
3.35
78.05
12
RGPC-C
3.35
104.39
11.6
4 Finite Element Modeling For analytical study on flexural behaviour of reinforced beams, finite element method was adopted by using ANSYS. Solid 65 elements were used to model PPC concrete as well as GPC. The input parameters like modulus of elasticity and Poisson’s ratio for PPC and GPC are different. The rebar capability of this model was not considered. All reinforcements were modeled using Link 8- 3D spar element. Flexural behaviour of control beam and retrofitted beam was studied using ANSYS. The Fig. 7 shows ANSYS modelling of control beam. Similarly, the Fig. 8 shows ANSYS modeling of RPPC beam retrofitted with geopolymer concrete.
Fig. 7. ANSYS modeling of the Controlled beam (100 × 150 × 1500 mm)
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Fig. 8. ANSYS modeling of the Retrofitted beam (100 × 150 × 1500 mm)
5 Comparison Experimental and Analytical Results The analytical and experimental results showed good agreement. The load deformation characteristics of RPPC beam retrofitted with geopolymer concrete in experiment and ANSYS is compared as shown in Fig. 9.
Fig. 9. Comparison of load – deformation characteristics of Retrofitted RPPC beam between experimental and analytical
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The flexural cracks initiated in the pure bending zone as expected. As the load increased existing cracks propagated and the new cracks are developed along the span. The flexural cracks gave way to inclined cracks due to the effect of shear force. The spacing of cracks varied along the span. The total number of the flexural cracks developed was almost same for all the beams, with small variation of 2%to 3%. In case of retrofitted beams initially only flexural cracks develops then to shear cracks, due to good bond no cracks developed at the interface of GPC and PPC. A few cracks appeared in the flexural zones of the concrete at initial loads. Subsequently the cracks widened as the load increased. The cracks at the midspan opened widely near failure. At failure loads, the beams deflected significantly. The failure pattern of the beam specimens was found to be similar. The beams failed initially by yielding of the tensile steel followed by the crushing of concrete in the compression face. Since the retrofitted beams initially subjected to ultimate load, the flexure cracks appear early for retrofitted beams compared to control beams. The comparison of analytical and experimental crack patterns of retrofitted RPPC beam is shown from Fig. 10.
(a)
(b)
Fig. 10. Comparison of crack patterns of Retrofitted beam in (a) Analytical and (b) Experimental
6 Conclusion Based on the experimental and analytical investigation following conclusions are made: The load deflection characteristics at the mid span of the RPPC beams and RGPC beams were found almost similar. Also, the RGPC beams showed slightly more deflection at the same load than the RPPC beams. The ultimate strength of GPC retrofitted RPPC beam of 100 × 150 × 1500 mm is nearly 80% of the ultimate strength of control beam. The flexural capacity of retrofitted beams increase with increase in tensile reinforcement. The crack patterns of the retrofitted beams and control beams are similar. The crack first occurs at the tensile side i.e. flexural crack. Then flexural crack give way to inclined shear crack. The 3D ANSYS modeling is able to properly stimulate the nonlinear behaviour of retrofitted RPPC beam with GPC and control RPPC beam with reinforcements. The load deflection behaviour of retrofitted and control RPPC in ANSYS shows good agreement with experimental results. The failure mode and crack patterns of control and retrofitted RPPC beams predicted by ANSYS are in good agreement with experimental results.
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The ultimate deflection value for retrofitted beams are higher compared to control specimen; this is because the retrofitted specimen has yielded reinforcement. Based on the analysis of results, Geopolymer concrete can be used for strengthening of damaged RC structure.
References 1. Amarnath, C., Menon D., Sengupta, A.K.: Handbook on Seismic Retrofit of Buildings (2008) 2. Menna, C., et al.: Use of geopolymers for composite external reinforcement of RC members. Composites: part B 45, 1667–1676 (2013) 3. Pacheco-Torgal, F., Abdollahnejad, Z., Miraldo, S., Baklouti, S., Ding, Y.: An overview on the potential of geopolymers for concrete infrastructure rehabilitation. Constr. Build. Mater. 36, 1053–1058 (2012) 4. Ronagh, H.R., Eslami, A.: Flexural retrofitting of RC buildings using GFRP/CFRP- A comparative study. Composites: part-B 46, 188–196 (2013) 5. IS456: Indian code of practice for plain and reinforced concrete, Bureau of Indian standards, New Delhi (2000) 6. Rangan,. B.V., Hardjito, D.: Development and properties of low calcium fly ash based geopolymer concrete. Research report GC-1, Faculty of Engineering, Curtin University of Technology, Perth, Australia (2005)
A Study of Fluid-Structure Interaction of Unsteady Flow in the Blood Vessel Using Finite Element Method S. T. Ha(B) , T. D. Nguyen, V. C. Vu, M. H. Nguyen, and M. D. Nguyen Department of Mechanical Engineering, Le Quy Don Technical University, Hanoi, Vietnam [email protected]
Abstract. The paper presents a numerical simulation for fluid-structure interaction (FSI) of unsteady blood flow interacting with the vessel wall. The present work aims to provide a simple approach for very large deformation of the wall. The implementation of the method is straightforward and can be employed for a large scale problem with a cheap computational cost. The classical finite element discretization is adopted both for fluid and solid domains on tetrahedral elements. The monolithic scheme is used for the strong coupling of fluid and structure to satisfy kinematic and dynamic equilibrium conditions at the interface. The Navier-Stokes equations of an incompressible flow are solved by using the integrated method based on the Arbitrary Lagrangian-Eulerian (ALE) formula for the moving grid, and the total Lagrangian formulation is used for the non-linear hyper-elastic material of the vessel wall with the Mooney-Rivlin material model adopted as a constitutive equation for the solid domain. The numerical solutions for the FSI of blood vessel problem are quite similar to the experimental data. After validation the code, two problems of the blood vessel walls are considered: The carotid bifurcation and the aortic valve problems. The simulation results can be used for predicting the risk of cardiovascular diseases. Keywords: Fluid-structure interaction · Unsteady flow · Blood vessel · Finite element method
1 Introduction Hemodynamic characteristics in the blood vessel wall play an essential role in science and technology, and it is closely related to the progression of cardiovascular diseases. Primarily, transcatheter aortic valve implantation (TAVI) has been carried out as an alternative to patients with severe aortic stenosis, who are at high risk for surgical therapy. The biomechanical environment of TAVI is closely related to the interaction of the motion of the aorta as well as leaflets with the aortic hemodynamics with unsteady blood flow [1]. FSI occurs when fluid flow creates a deformation of the structure. This deformation, in turn, changes the boundary conditions of the fluid flow. Numerical simulations of fluidstructure interaction of blood vessels such as aortic bifurcation, carotid artery and aortic valve have received much attention in the last several decades. An accurate simulation of © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 1089–1101, 2022. https://doi.org/10.1007/978-981-16-3239-6_85
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the FSI problems plays an important part in the diagnosis and treatment of cardiovascular diseases. Recently, there has been a substantial improvement in the numerical method for simulating FSI problems and it can be employed to simulation the blood flow interacting with vessel problem [2–6]. The computational simulations for the blood flow in the artery have been performed by many investigators with different approaches. It is found that FSI is a robust method to investigate the mechanisms of the blood vessel and predict early atherosclerosis. However, FSI simulation for the large deformation of a vessel with unsteady flow remains a challenging problem. The ALE framework is the most common approach for FSI simulation. However, it may less performance or problematic for a large displacement of the wall such as the aortic valve problem. In order to treat this difficulty, some techniques are required to smooth the grid for fluid domain [1]. In addition, the IBM method is also a good alternative method to solve the large displacement of the wall problem [7]. In the IBM method, the fluid domain is discretized on a regular cartesian grid on which the solid body is free to move independently to fluid mesh. The advantage of IBM versus ALE is that there is no need for a re-meshing procedure for fluid mesh that may require parallel computation. Recently, the accuracy of the simulation is much improved by using a high order of element technique such as the polygonal finite element [6, 8, 9]. Since the ratio of blood to the vessel is closed to one, the added mass effect is strong and leads to a big challenging of the convergence of the FSI coupling problem. In this case, the explicit coupling (weak coupling) or some traditional strong coupling methods such as fixed-point approaches could not solve the problem [2]. For the material behavior of the wall, most previous researches used a linear model of the stress-strain relation. This model may not correctly express the real material of the blood vessel problem in the human aorta. In this work, a non-linear behavior of the Mooney-Rivlin model is employed for the FSI simulation of unsteady fluid flow in the vessel wall. The unstructured grid is used with the finite element discretization for both fluid and solid domains to easily generate the grids for a very complex simulation domain. Although there are many techniques and other more accurate finite elements that can be used to model the considered FSI problem, the classical finite element method (FEM) for triangular/tetrahedral was adopted in this work because of the straightforward implementation code and good performance for the large scale problem which requite parallel computation. The rest of the paper is organized as follows: Sect. 2 gives a brief description of the governing equation of fluid and structure and corresponding FEM formula for the FSI problem. Section 3 details the numerical solution of the two typically problem in blood vessel. Lastly, some conclusions are drawn in Sect. 4.
2 Governing Equations and the Finite Element Method 2.1 Governing Equations The blood flow in the vessel wall is assumed as an incompressible flow of a Newtonian fluid, and the governing equations are the incompressible Navier-Stokes equations which can be written as follows in the arbitrary Lagrangian-Eulerian (ALE) framework [2]: ∇ · v =0 in f m f ρf ∂v ∂t + (v − v ) · ∇v = ∇ · σ
in f
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where ρf , v, vm and σf denote the fluid density, the fluid velocity, the grid velocity, and the fluid stress tensor, respectively. Fluid domain and the boundary are denoted by f , f . The corresponding constitutive equations for fluid flow in Eq. (1) are written as follows: σf = −pI + τ, τ = μ[∇v + (∇v)T ]
(2)
where p, μ, τ and I indicate the pressure, the fluid dynamic viscosity, the shear stress tensor, and the second-order identity tensor, respectively. The boundary conditions are described as follows: f
v = v on v , f f σf · nf = t on t
(3) f
where nf denotes the outward unit normal vector to the fluid boundary, and v and f f t are the boundaries on which the velocity ( v) and traction ( t ) are imposed on the Dirichlet and Neumann boundary conditions, respectively. The governing equation for solid domain in the Lagrangian framework is writte as follows: ρs
∂ 2u = ∇ · σs in s ∂t 2
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with boundary conditions: f
on u , u=u s σs · ns = t on ts
(5)
where ρs , u and σs denote the solid density, the displacement of solid, and the solid stress tensor, respectively. The solid domain is denoted by s with a boundary s . The constitutive equation of the solid domain is written as follows [10] σs = J−1 FSFT ; T = SFT ; S = C : E,
(6)
where C denotes a fourth-order tensor representing the material behavior. T and S are the first and second Piola-Kirchhoff stress tensor, respectively, and F and J denote the deformation gradient tensor and its Jacobian. The fourth-order tensor C in this work is nonlinear behavior using the Mooney-Rivlin model. The elastic strain energy ψ for the Mooney-Rivlin model is written by [11]: 2 2 ψ = c10 I 1 − 3 + c01 I 2 − 3 + c20 I 1 − 3 + c11 I 1 − 3 I 2 − 3 + c02 I 2 − 3 + 3 2 2 3 k + c30 I 1 − 3 + c21 I 1 − 3 I 2 − 3 + c12 I 1 − 3 I 2 − 3 + c03 I 2 − 3 + (J − 1)2 (7) 2
where I 1 , I 2 and J are the invariants of deformation tensor described detail in Ref. [2]. Let f /s be the interface between the fluid/structure domains (F/S interface). For non-slip condition, both the velocity and traction of fluid domain are equilibrium with those of the solid at the F/S interface. The two conditions are described by the following formula: f /s v = ∂u ∂t on σf · nf +σs · ns = 0 on f / s
(8)
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2.2 Finite Element Formulation for FSI Coupling The integrated formulation in which the pressure and velocity of fluid flow are solved simultaneously in one system is adopted in this work. The pressure/velocity variable is linearly/quadratically interpolated in a finite element, where the pressure variable is allocated on the vertices and the velocity variables are on both vertices and mid-nodes. Finite element formulation of the governing equation for unsteady incompressible fluid flow is written as follows [2]: Find v ∈ H1h (), p ∈ L2h (), such that q∇ · vd =0 f (9) m f w · ρf ∂v d = w · (σf · nf ) d ∂t + (v − v ) · ∇v+∇w:σ f
f
for all admissible functions w ∈ Vh , q ∈ Ph , where
f Vh = w|w ∈ H1h (), w = 0 on v , Ph = q|q ∈ L2h ()
(10)
In a solid domain, the displacement variable is quadratically interpolated in an unstructured finite element. Its formulations for the solid domain are written as follows [2]: Find u ∈ H 1h (), such that
2 s∂ u w · ρ0 2 + ∇w : T d = w · (T · n0s ) d (11) ∂t s0
0s
for all admissible functions w ∈ V h , where
Vh = w|w ∈ Hh1 (), w = 0 on us .
(12)
The strong coupling of fluid and structure equations is accomplished based on the ALE framework in present work. In this work, we use the monolithic algorithm [3] for the strong coupling of FSI problem. For this method, the velocity variables are shared at the fluid-structure interface so that the kinematic constraint is satisfied automatically by sharing the velocity field at the interface. Summation of fluid and structure equations at the interface cancel out the stress terms of both sides, satisfying the equilibrium constraint. The total Lagrangian formulation is used for the non-linear hyper-elastic material of the vessel wall with the Mooney-Rivlin material model adopted as a constitutive equation for the solid domain as described in Ref [2]. Therefore, the system of FEM discretization for the coupling FSI problem is shown as following: f s G v f + H f pf + G v s = P f + P s , (13) f T f H v =0 f
where G and Hf are the stiffness matrices of velocity and pressure variables of fluid flow are constructed by FEM descretization, respectively. The details of these matrices and force vectors Pf , Ps are discussed in [3].
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3 Results and Discussions In this section, some FSI problems are studied by using the monolithic FEM described in Sect. 2. The program was written in the FORTRAN language on a single-core machine of a desktop. The mesh smoothing is required for each time step because of the large deformation of the wall. A Laplace equation was solved with a Drichlet boundary condition at the interface for smoothing mesh [2], and a remeshing procedure was used whenever the element shape is too distorted. In all simulation problems, P2/P1 tetrahedral elements were employed. Each element consists of four vertices and six mid-nodes, and 34/30 degree of freedom (DOF) per element for fluid/solid domain in this 3D case. The size of element for fluid domain is chosen similar to that of solid domain. From our independent grid test, it was found that only two layers of element (P2P1) for the solid wall is fine enough to get an accurate solution. 3.1 Validation Firstly, the code was validated by simulation of a simple case. A pressure wave propagation of incompressible fluid in a 3D flexible tube as reported in the previous works [2] was adopted in this section. The schematic is shown in Fig. 1, where the tube has a length of L = 5.0 cm, a diameter of D = 1.0 cm and a wall thickness of δ = 0.1 cm. Fluid density and dynamic viscosity are ρf = 1.0 g/cm3 and μ = 0.03 Poise. The Young’s modulus of E = 3 × 105 Pa and a Poisson ratio of ν = 0.3 and the density of ρs = 1.0 g/cm3 were employed for the tube wall. The tube is fixed in all directions at the two ends. Both fluid and solid are initially at rest. At the inlet, a pressure of pin = 10 mmHg is applied in the first short period of 3 × 10–3 s and then set back to zero later. A P2/P1 tetrahedral mesh is used for both fluid and solid domains as shown in Fig. 2 in which the numbers of the nodes for fluid and solid domains are 55,045 and 32,320, respectively. The detail descriptions of the problem was listed in Ref. [2]. Figure 3 plots the evolution of movement (axial and radiation displacement) at the centre point of the inner tube wall (point M in Fig. 2) for a quantitative comparison. It can be seen that the present results agree well with those obtained by Eken and Sahin [12].
Fig. 1. Schematic of flow in a straight flexible tube [2]
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Fig. 2. Grid of fluid and solid domains
a) Axial displacement
b) Radial displacement
Fig. 3. Comparison of the movement of point M obtained from present work and results by Eken and Sahin [12]
3.2 Blood Flow in a Carotid Bifurcation The first application simulation of the present method is blood flow interacting with a blood vessel in a carotid bifurcation, which includes a common carotid artery (CCA) branched into the internal and external carotid arteries (ICA and ECA, respectively). The lumen boundary of the carotid bifurcation is constructed by a 3D reconstruction from 2D cross-sectional ultrasound images described by Yeom et al. [13]. The blood vessel is then obtained by adding a tube of the constant thickness of 1 mm on the blood/vessel interface. A P2P1 tetrahedral grid is used both for fluid and solid domains as shown in Fig. 4. The problem consists of 121,358 nodes and 103,121 element with total of 375,396 DOF. The average of element length is 0.5 mm and there are two layers of P2P1 element of the wall. From our experience, this size of element is fine enough for the blood vessel problem. Boundary conditions for inlet and outlet are shown in Fig. 5 for pulsatile pressure and velocity, respectively. A time-step size of 5 × 10–5 is used f for the whole time in this simulation. The vessel density is set of ρ0 = 1366 kg/m3 at the reference configuration and the material parameters in Eq. (7) are given as follows:
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c10 = 0.07, c20 = 3.2, c21 = 0.0716 MPa and other cij set to zero. Bulk modulus is k = 0.2 MPa as suggested in Ref [11]. Numerical results are examined at four different instants: t1 = 0.1T, t2 = 0.15T, t3 = 0.32T, and t4 = 0.75T. The deformations of the blood/vessel interface at contraction phase (t1 = 0.1T) and expansion phase (t3 = 0.32T) are shown in Fig. 6. It can be seen that the numerical solutions are quite similar to the results from experimental data in Refs [13]. The contours of velocity magnitude on the intersection of the plane A with ECA and ICA near the bifurcation are shown in Fig. 7. The velocity magnitude in the interaction with ECA is a little higher than that with ICA, and the highest value is located close to the inner boundary of the branch. Blood velocity reaches the maximum value (~0.8 m/s) at the peak flow-rate (t2 = 0.15T), and then decreased to ~0.2 m/s at t4 = 0.75T. Accordingly, there is a rapid increase in wall shear stress (WSS) on the blood-vessel interface at the peak flow-rate as shown in Fig. 8.
Blood domain
Vessel wall domain
Fig. 4. Grid used for blood domain and vessel wall domain
Pressure at inlet [14]
Flow rate at outlet [13]
Fig. 5. Boundary conditions for simulating of blood flow in a carotid bifurcation [14]
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Fig. 6. The vessel wall deformation obtained by numerical solution
Fig. 7. The contours of velocity magnitude of ECA and ICA near the branch
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Fig. 8. WSS distribution on the blood vessel interface
WSS is then decreased as the flow-rate decreases until t4 = 0.75T. It also shows that WSS of ICA is a little higher than that of ECA. Moreover, the maximum value of WSS is located around the branch of carotid artery in all instants. Von-Mises stress (VMS) distribution in Fig. 9 shows that the highest value of VMS is located around the branch. It also shows that a larger deformation of blood vessel at the expansion phase leads to a higher VMS distribution. 3.3 Blood Flow Through the Aortic Valve The second simulation is for a pulsatile blood flow interacting with an aortic valve. The aortic domain and the valve are chosen from the model in [15]. Figure 10 shows the geometry of the valve inside an aortic artery and the corresponding tetrahedral meshes which are generated by using ANSA/ICEM software. The size of element is chosen similar to the problem in Sect. 3.1 with the two layers of element were adopted for the valve. The number of nodes and element are 150,429 and 128,189 with total of 467,306 DOF. The boundary condition at the inlet is a pulsatile flow rate as shown in Fig. 11 for one period of T = 1.1 s. At the end of systole ( s ∼ 0.35T), the direction of the inflow is inverted to create a backflow that is physiologically consistent and helps the valve close. And then, the diastole starts when the valve is closed and the inflow is set to zero. A zero-pressure is set at the outlet of the aorta [1]. The blood density and dynamic viscosity are given ρ f = 1.05 g/cm3 and μ = 0.004 Pa.s as provided in [16]. The density of the aortic valve at the reference configuration is set of ρ0s = 1.0 g/cm3 . The stress-strain relation of the aortic valves is assumed as the Mooney-Rivlin model with the material parameters are given in Refs. [17, 18]. The time step size t was set to 5 × 10–5 s on the systole phase and is 10–4 s on the diastole phase. The numerical results are examined at the two different instants: t1 = 0.18T and t2 = 0.6T for the maximum and the zero flow rates, respectively. The deformations of the valve at the two instants are illustrated in Fig. 12. The valve is fully opened at the systolic phase (Fig. 12-a), where the flow rate reaches the
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maximum value (~25 l/min). The valve is closed when the flow rate is zero at the diastolic phase (Fig. 12-b). According to the flow through the valve, there is a rapid increase in WSS on the inner wall of the valve at the peak flow-rate as shown in Fig. 13. WSS is then decreased as the flow-rate goes to zero when the valve is closed (t2 = 0.6T). Von-Mises stress distribution in Fig. 14 shows that the highest value is located around the commissure of the valve. It also shows that a larger deformation of the valve at the systole leads to a higher VMS distribution.
Fig. 9. Von-Mises Stress contours
Fig. 10. The geometry and mesh generation of valve Fig. 11. Flow rate at the inlet of aortic inside an aortic artery [15] valve [1]
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Fig. 12. Deformation of aortic valve at systolic phase (a) and diastolic phase (b)
Fig. 13. WSS distribution on the aortic valve at systolic phase (a) and diastolic phase (b)
Fig. 14. Von-Mises stress contours of aortic valve at systolic phase (a) and diastolic phase (b)
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4 Conclusion In this paper, we solve the FSI problem of unsteady blood flow in the vessel wall. The two special vessel walls are considered named as: the carotid bifurcation and the aortic valve. The monolithic is used for the FSI coupling of an incompressible fluid with a large displacement of the solid wall. The Mooney-Rivlin model is adopted for the material of the blood vessel to cache the real material behavior. A P2P1 tetrahedral grid is employed for finite element discretization of the 3D domain for both fluid and solid parts. The numerical solution obtained by the present approach are quite similar to the experimental data. The simulation results for both problems show that the maximum of wall shear stress (WSS) is appeared at the peak-flow rate. In this phase, the Von-Mises stress also reaches to the maximum values.
References 1. Jeannette, H.S., Johan, J., Niclas, J., Johan, H.: 3D fluid-structure interaction simulation of aortic valves using a unified continuum ALE FEM model. Front. Physiol. 9, 363 (2018) 2. Ha, S.T., Choi, H.G.: Investigation on the effect of density ratio on the convergence behavior of partitioned method for fluid–structure interaction simulation. J. Fluids Struct. 96, 103050 (2020) 3. Ha, S.T., Ngo, L.C., Saeed, M., Jeon, B.J., Choi, H.G.: A comparative study between partitioned and monolithic methods for the problems with 3D fluid–structure interaction of blood vessels. J. Mech. Sci. Technol. 31, 281–287 (2017) 4. Kang, S., Choi, H.G., Yoo, J.Y.: Investigation of fluid–structure interactions using a velocitylinked P2/P1 finite element method and the generalized-α method. Internat. J. Numer. Methods Eng. 90, 1529–1548 (2012) 5. Ha, S.T., Choi, H.G.: Simulation of the motion of a carotid artery interacting with blood flow by using a partitioned semi-implicit algorithm. Korean Soc. Comput. Fluids Eng. (2019) 6. Vu, T.H., Phung, V.P., Nguyen, X.H., Wahab, M.A.: A polytree-based adaptive polygonal finite element method for topology optimization of fluid-submerged breakwater interaction. Comput. Math. Appl. 76(5), 1198–1218 (2018) 7. Yao, J., Liu, G.R., Narmoneva, D.A., Hinton, R.B., Zhang, Z.Q.: Immersed smoothed finite element method for fluid–structure interaction simulation of aortic valves. Comput. Mech. 50(6), 789–804 (2012) 8. Vu, T.H., Le, T.C., Nguyen, X.H., Abdel, M.A.: An equal-order mixed polygonal finite element for two-dimensional incompressible stokes flows. Eur. J. Mech.-B/Fluids 79, 92–108 (2020) 9. Vu, T.H., Le, T.C., Nguyen, X.H., Abdel, M.A.: A high-order mixed polygonal finite element for incompressible Stokes flow analysis. Comput. Methods Appl. Mech. Eng. 356, 175–198 (2019) 10. Holzapfel, G.A.: Nonlinear solid mechanics: a continuum approach for engineering science. Meccanica 37, 489–490 (2002) 11. Nobuko, K., Joji, A., Chen, X., Hisada, T.: Multiphysics simulation of blood flow and LDL transport in a porohyperelastic arterial wall model. J. Biomech. Eng. 129(3), 374–385 (2007) 12. Eken, A., Sahin, M.: A parallel monolithic algorithm for the numerical simulation of largescale fluid structure interaction problems. Int. J. Numer. Methods Fluids 80, 687–714 (2016) 13. Yeom, E., Nam, K.H., Jin, C., Paeng, D.G., Lee, S.J.: 3D reconstruction of a carotid bifurcation from 2D transversal ultrasound images. Ultrasonics 54(8), 2184–2192 (2014)
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14. Jingliang, D., Zhonghua, S., Kiao, I., Jiyuan, T.: Fluid–structure interaction analysis of the left coronary artery with variable angulation. Comput. Methods Biomech. Biomed. Engin. 18(14), 1500–1508 (2015) 15. Ha, S.T.: Development of a new geometric multi-grid finite element method and a semiimplicit partitioned algorithm for fluid-structure interaction simulation. Ph.D. Dissertation, Seoultech (2019) 16. Ryo, T., et al.: Fluid–structure interaction analysis of a patient-specific right coronary artery with physiological velocity and pressure waveforms. Commun. Numer. Meth. Eng. 25, 565– 580 (2009) 17. Einstein, D.R., Reinhall, P., Nicosia, M., Cochran, R.P., Kunzelman, K.: Dynamic finite element implementation of nonlinear, anisotropic hyperelastic biological membranes. Comput. Methods Biomech. Biomed. Eng. 6(1), 33–44 (2003) 18. Ranga, A., Mongrain, R., Biadilah, Y., Cartier, R.: A compliant dynamic FEA model of the aortic valve. In: 12th IFToMM World Congress, Besançon, France, pp. 1–6 (2007)
Stable Working Condition and Critical Driving Voltage of the Electrothermal V-Shaped Actuator Kien Trung Hoang and Phuc Hong Pham(B) Hanoi University of Science and Technology (HUST), 1-Daicoviet, Hanoi, Vietnam [email protected]
Abstract. This work proposes one model of the electrothermal V-shaped actuator (EVA) and applies finite differential method to find out accurately the maximum temperature of a silicon V-beam to avoid an overheating phenomenon, which may damage to beam structure. Furthermore, value of critical voltages as U m , U n is determined corresponding to beam-buckling condition and limitation temperature of the beam. In order to guaranty safe working condition and lifetime of EVA, the critical voltage U m should be lower than U n , and can be considered as the safe condition while design V-shaped actuator. The influence of V-beam dimensions such as beam length L, beam width w and ratio L/w are also examined to figure out dimensional domain satisfying the stable working condition of the EVA system. Keywords: Electrothermal V-shaped actuator · Safe working condition · Critical driving voltage · Finite differential equation
1 Introduction Micro electrothermal actuators (ETA) play a vital role in category of MEMS actuators thanks to some great features such as a large output force, low driving voltage, simple structure and batch fabrication. In general, ETA can be classified into three main types: V-shaped, U-shaped and Z-shaped actuator. Comparing to other structures as U-shaped and Z-shaped, V-shaped actuator provides both larger force and displacement [1, 2]. Therefore, the electrothermal V-shaped actuator (EVA) is widely preferred to driving various devices like: micro gripper [3], nano material testing device [4], safe thermal device [5] or linear motor [6], etc. Recent researches of EVA focus on some topics as improving heat transfer model [7], optimizing dimensions [8], enhancing working quality [9], developing [10] or applying new fabrication process and material [11]. These works provided theoretical models with higher accuracy or methods in order to improve working quality of EVA basing on dimensional optimization and applying new machining technology. The stable working condition and critical driving voltage of EVA are also attracting topics for researchers. In [12], the authors established a nonlinear relation between thermal expansion force and stress in order to determine maximal temperature and limitation © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 1102–1112, 2022. https://doi.org/10.1007/978-981-16-3239-6_86
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voltage. This result could be used for optimizing and designing beam’s dimension aims to avoid plastic deformation. The stability of EVA fabricated by surface micromachining with thin structure in nano-scale has been achieved by selecting reasonable incline angle of beam as shown in [13]. The stable displacement against driving frequency as well as the relation between maximum driving voltage and incline angle of beam to overcome a V-beam buckling phenomenon, which was mentioned in [14]. The above publications pointed out the limited temperature and voltage, critical driving frequency, and buckling condition of EVA’s beam. However, the relation between mechanical stability and thermal deformation of EVA has not been explained clearly. In other words, the relation between the critical voltages in cases of beam-buckling problem and thermal deformation caused by high temperature on the beam should be computed and compared particularly. This paper focuses on establishing finite differential model, aims to calculate more accurately the temperature distribution along the V-beam due to considering the dependence of material properties on temperature. Besides, the effect of beam’s dimension (such as beam length L, beam with w and inclined angle θ ) on a driving voltage to avoid the beam-buckling phenomenon as well as a damaging temperature of silicon beam are also computed and examined.
2 Theoretical Model of EVA 2.1 Configuration of EVA A typical configuration of the electrothermal V-shaped actuator (EVA) is shown in Fig. 1. The actuator consists of a shuttle ➀ at the center, V-beam system ➁ at the both sides and fixed electrodes ➂. The shuttle is suspended by symmetric V-beams. It can move up and down in Y-direction while EVA is working. The other end of V-beams is connected to the fixed electrodes.
O θ
Y
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w
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h ga
ws Substrate
SiO2 layer
Device layer
Fig. 1. Configuration of EVA
Here: L, h and w are the length, the thickness and the width of a single beam, respectively; θ is an incline angle of beam in X-direction; ga is the air space between the beam and substrate; L s and ws are the length and the width of the shuttle.
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When applying a voltage on two fixed electrodes, an electric current transmits through the thin beams and generates heat emission. The increase of temperature on the beams causes thermal expansion and pushes the shuttle moving in Y-direction. If voltage drops to zero, temperature of beam reduces towards room temperature, and the beams will shrink and pull the shuttle backward. 2.2 Heat Transfer Model Heat transfer in a thin beam is complex process. For simple, there are some assumptions proposed as following: heat transfer is one-way and along the length of beam (x-direction); radiation, convection and losing heat from the shuttle to substrate through air gap ga are ignored. Considering the heat transmission in a V-beam with the total length of 2L, the equation of heat energy balance for a tiny element beam in x-direction with Δx-length is expressed as: qi+1 + qi−1 + qsi + qei = qsti
(1)
Where, qi+1 and qi-1 are the heat energy transmitting from (i + 1)th and (i – 1)th elements to ith element, respectively; qsi is the heat energy transmitting from ith element of beam to substrate through air gap; qei is the heat energy in ith element generated by electrical current; qsti is the internal energy storage within ith element. They are expressed as following [14]: j+1
j
− Ti .w · h · x t
(2)
qi+1 = k · w · h.
Ti+1 − Ti x
(3)
qi−1 = k · w · h.
Ti−1 − Ti x
(4)
qst i = C · d .
Ti
qls i = ka · S. qe i =
Ti − T0 .w · x ga
U2 .w · h · x 4ρ · L2
(5) (6)
Where k, C and d are the thermal conductivity coefficient, the specific heat and the density of single crystal silicon, respectively; k a is the thermal conductivitycoefficient of air; Δx is the tiny section of V-beam; U is the driving voltage; S = 0.6265 wh + 1.1188 is the shape factor of the beam (determined by simulation); ρ = ρ0 .[1 + λ.(T − T0 )] is the resistivity of silicon; λ is the temperature coefficient of resistivity; T 0 is room temperature; Δt is the time increment. Considering at steady state of heat transfer process (when qsti = 0), by substituting Eqs. (2)–(6) into Eq. (1) and simplifying, we have: Ti−1 − (2 +
ka ka U2 S S · x2 ) · Ti + Ti+1 + · x2 · T0 + · x2 = 0 · · k h · ga k h · ga 4 · k · ρ · L2
(7)
Stable Working Condition and Critical Driving Voltage
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According to Fig. 1, the first and n-th elements of a V-beam are connected to two fixed electrodes ➂ on both sides. These electrodes anchored to the substrate with larger area and leading better heat emission when comparing to the elements at the middle of beam. Therefore, it can be assumed that the temperature of the first and n-th elements is almost equal to the temperature of fixed electrodes (i.e. temperature T 0 ). The boundary condition for solving Eq. (7) is: T1 = Tn = T0
(8)
Where, T 1 and T n is the temperature of elements i = 1 and i = n, respectively. One V-beam with length 2L will be divided into n elements and we have n equations similar to Eq. (7). Each equation indicates the heat transfer state of one element (i.e. temperature of element ith ). By solving these equations with boundary condition (8), the temperature of all beam elements T i (i = 1, 2…, n) are determined. 2.3 Thermal Expansion Force The expansion of a single beam is calculated by differential equation: 1 α.(Ti − T0 ).x 2 n
L =
(9)
i=1
Where, α is thermal expansion coefficient of silicon. Thermal expansion force along the single beam is computed by [14]: Fb = E.A.
L L
(10)
Where, E = 169 GPa is a Young’s modulus of silicon; A is the cross-section area of beam.
3 Critical Voltages of Mechanical and Thermal Stability 3.1 Critical Voltage for Beam-Buckling Condition When driving voltage increase, a buckling phenomenon of thin beam may be occurred and leading the mechanical instability of EVA (i.e. when the thermal expansion force F b is larger than critical force F cr for axial stability of the beam). A model for determining beam-buckling condition is illustrated as in Fig. 2. Here N is a reaction force generated by load on the shuttle F load . According to [14], the stable condition along the single beam is expressed as: 4π 2 · E · I (A · L2 − 12I ) · sin2 θ · cos2 θ ≤ Fb . 1 − (11) L2 A · L2 · sin2 θ + 12I · cos2 θ Where, I =
h·w3 12
is the inertial moment of beam’s cross-section area.
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K. T. Hoang and P. H. Pham
Fload Buckling N
N
Initial position Fig. 2. Model for determining beam-buckling condition
From (9), (10) and (11) we have:
n (A · L2 − 12I ) · sin2 θ · cos2 θ E · A 4π 2 · E · I 1− . α · (T − T ).x ≤ (12) i 0 2L L2 A · L2 · sin2 θ + 12I · cos2 θ i=1 Finally, Eq. (12) can be written: 2 ⎛ ⎞ L 2 θ · cos2 θ n − 1 . sin w 2π 2 .w ⎝1 − ⎠ α.(Ti − T0 ).x ≤ L 2 L 2 2 3 w i=1 w . sin θ + cos θ
(13)
The buckling condition (13) is strongly depended on the temperature and the dimensional parameters of beam such as ratio L/w, w and angle θ. On the other hand, the temperature of elements in (7) depends on both driving voltage and the dimensional parameters of beam. By substituting result of T i inferred from Eq. (7) into (13), a driving voltage U m can be found when using the sign “ =” in formula (13). This voltage is also named as a critical voltage U m of EVA in order to satisfy a mechanical stability condition. In this case, the function fsolve of MATLAB is utilized for computing U m . The material properties of single crystal silicon are listed in Table 1 and Table 2 as following. Table 1. Material properties of silicon d (kg/m3 )
E (MPa)
k a (W/m.K)
C p (J/kg.K)
ρ 0 ( .m)
λ (1/K)
2330
1.69 × 105
0.0257
712
230 × 10–6
1.0 × 10–3
Here, the influence of ratio L/w and beam width w to the critical voltage U m is examined with the optimal incline angle of beam θ = 2° (explained in [14]). The relation of critical voltage U m to the ratio L/w and the beam width w is graphed in Fig. 3. It is clear that the U m is proportional with the ratio L/w and the beam width w.
Stable Working Condition and Critical Driving Voltage
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Table 2. Material parameters depend on temperature [15] T (K)
300
400
500
600
700
800
900
1000
1100
1200
1300
k (W/m.K)
156
105
80
64
52
43
36
31
28
26
25
α (10–6 /K)
2.62
3.25
3.61
3.84
4.02
4.15
4.18
4.26
4.32
4.38
4.44
Fig. 3. The relation between critical voltage Um with ratio L/w and w
3.2 Critical Voltage for Thermal Stability Condition Generally, while increasing beam temperature will lead to reducing strength of material. In this case, the safe temperature of EVA (i.e. limitation temperature) is chosen about 1200 °C and has to be lower than the melting point of silicon at 1414 °C [15]. The temperature of each element i determined from Eq. (7) is various and always exists a maximum temperature T max at the middle of beam length. Thus, the thermal stability condition relating to critical temperature is limited by upper bound value: Tmax ≤ 1200 ◦ C
(14)
Considering an example of silicon V-shaped actuator with the dimensions are listed in Table 3 and the material properties as mentioned above. The relation between maximum temperature T max on the beam and driving voltage U is indicated in Fig. 4. As can be seen in Fig. 4, the maximum temperature calculated from Eq. (7) is nearly proportional to driving voltage and reaches the critical temperature of 1200 °C at the driving voltage of about 33 V. Whereas, the maximum temperature obtained by simulation is a nonlinear proportion to driving voltage and reaches the critical temperature at
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K. T. Hoang and P. H. Pham Table 3. Geometric dimensions of the EVA model
L (μm)
h (μm)
w (μm)
ga (μm)
θ (0 )
Beam number
L s (μm)
W s (μm)
750
30
6
4
2
10
300
50
1800 1600
Max. Temp. Tmax (0C)
1400 1200 1000 800 600
Tmax of calculation
400
Tmax of simulation
200
Critical temp.
0 10
15
20
25
30
35
40
Driving voltage U (Volt) Fig. 4. Relation between maximum temperature and driving voltage
the driving voltage of about 31.5 V. For the dimensions listed in Table 3, the limitation driving voltages are 33 V and 31.5 V corresponding to calculation and simulation. These values are called as critical voltage U n of thermal stability condition. The temperature of beam elements and maximum temperature depend on the material properties, the dimensional parameters of beam and the driving voltage. With a set of beam dimension as ratio L/w and w, a value of critical voltage U n is always determined to satisfy the sign “ =” in the formula (14). As a result, the relation of the critical voltage U n with the ratio L/w and beam width w is indicated in Fig. 5.
Stable Working Condition and Critical Driving Voltage
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Fig. 5. The Relation between Un with ratio L/w and w
4 Safe Working Condition of the V-Shaped Actuator As mentioned at the previous section, both of U m and U n are effected by the dimensional parameters of V-beam as the beam length L (or ratio L/w) and the beam width w. When comparing voltages at the same value of dimension (i.e. ratio L/w and w), there are three situations: U m > U n , U m = U n or U m < U n . In case of a larger load applying to the EVA, it needs a higher driving voltage to balance this load and may be larger than critical voltages U m or U n . If the driving voltage overcomes U n , the thin beam of EVA may be failure because of melting phenomenon and could not restore. In case of the applied voltage is larger than U m but lower than U n , the V-beam of EVA is only buckled/deformed and it could be restored if reducing the voltage. It is evident that the EVA structure will be protected safety when mechanical instability occurs before melting phenomenon caused by overheating. In other words, the critical voltage U m should be lower than U n , and this can be seemed as the safe condition while design EVA. The comparison of U m and U n according to ratio L/w at different values of w is shown as in Fig. 6 (with optimal incline angle θ = 2°). In the range of typical beam width w changing from 4 to 7 μm, the voltage U m will be larger than U n if the ratio L/w is lower than 100, and U m will be lower than U n if the ratio L/w is larger than 125. As a suggestion, the safe condition can be obtained if the EVA dimension is selected so as to the ratio L/w is larger than 125. Besides, while the ratio L/w changing between 100 and 125, the voltage U m and U n are approximate at some values of ratio (L/w)cr and wcr . These values may be called as “critical” dimensions of EVA structure.
K. T. Hoang and P. H. Pham w = 4μm
37
Um and Un (V)
32
32
Um Un
27 22
22
50 60 70 80 90 100 110 120 130 140 150
50 60 70 80 90 100 110 120 130 140 150
Ratio L/w
Ratio L/w
w = 6μm
42
w = 7μm
47 42
Um Un
27 22 17
Um and Un (V)
Um and Un (V)
27
12
12
32
Um Un
17
17
37
w = 5μm
37
Um and Un (V)
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37
Um Un
32 27 22 17 12
12 50 60 70 80 90 100 110 120 130 140 150
50 60 70 80 90 100 110 120 130 140 150
Ratio L/w
Ratio L/w
Fig. 6. The comparison of Um and Un according to ratio L/w at various values of beam width w
The critical dimensions can be determined by taking the intersection points of both curved surfaces U m (L/w, w) and U n (L/w, w) as in Figs. 3 and 5. The relation between critical ratio (L/w)cr and the beam width wcr at the boundary of dimensional domain (U m = U n ) satisfying the safe working condition is indicated in Fig. 7. For example, considering two EVA structures with various values of the beam length and beam width as following: EVA_1 with L 1 = 750 μm, w1 = 6 μm; EVA_2 with L 2 = 450 μm, w2 = 4 μm. For the EVA_1, the ratio L 1 /w1 is 125 and larger than the critical ratio (L/w)cr = 105.3 at value w = 6 μm as shown in Fig. 7. Therefore, the dimensions of EVA_1 satisfy the safe working condition described above. With the EVA_2, the ratio L 2 /w2 is only 112.5 and lower than the critical ratio of 121.5 at the value w = 4 μm (Fig. 7). In this case, we can conclude that the EVA_2 structure is not satisfying the safe working condition.
Stable Working Condition and Critical Driving Voltage
1111
140 130
Um =Un (6; 125)
Ratio L/w
120 110
Domain of Um < Un
(4; 112.5)
100 90
Domain of Um > Un
80 70 3
4
5
6 7 Beam width w (μm)
8
9
10
Fig. 7. Domain of dimension satisfying EVA’s safe working condition
5 Conclusion This paper established the heat transfer model and formula of thermal expansion force basing on finite differential method to calculate more precisely the temperature distribution on a V-beam. A new approach was proposed to determine the EVA’s critical voltages in case of occurring the beam-buckling and the overheating phenomena. Moreover, this research also figured out the relation between critical ratio L/w (i.e. ratio of the beam length per the beam width) and the beam width so that satisfying the stability condition of EVA while working (i.e. safe condition U m < U n ). Finally, aims to get the stable working of EVA, the driving voltage U and critical voltages U m , U n have to satisfy: U ≤ U m < U n. These results are valuable suggestion/direction not only using for design of the electrothermal V-shaped actuators, but also for other kinds of micro actuator, as well as to avoid the instability of the system while working and extend lifetime of micro devices. Acknowledgement. This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant number “107.01-2019.05”.
References 1. Potekhina, A., Wang, C.: Review of electrothermal actuators and applications. Actuators 8(4), 69 (2019) 2. Zhang, Z., Yu, Y., Liu, X., Zhang, X.: A comparison model of V- and Z-shaped electrothermal microactuators. In: IEEE International Conference on Mechatronics and Automation ICMA, pp. 1025–1030 (2015) 3. Shivhare, P., Uma, G., Umapathy, M.: Design enhancement of a chevron electrothermally actuated microgripper for improved gripping performance. Microsyst. Technol. 22(11), 2623– 2631 (2016). https://doi.org/10.1007/s00542-015-2561-0
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4. Pantano, M.F., Pugno, N.M.: Design of a bent beam electrothermal actuator for in situ tensile testing of ceramic nanostructures. J. Eur. Ceram. Soc. 34(11), 2767–2773 (2014) 5. Zhao, Y.-L., Hu, T.-J., Li, X.-Y., Jiang, Z.-D., Ren, W., Bai, Y.-W.: Design and characterization of a large displacement electro-thermal actuator for a new kind of safety-and-arming device. Energy Harvest. Syst. 2, 143–148 (2015) 6. Maloney, J.M., Schreiber, D.S., DeVoe, D.L.: Large-force electrothermal linear micromotors. J. Micromech. Microeng. 14(2), 226–234 (2004) 7. Zhang, Z., Yu, Y., Liu, X., Zhang, X.: Dynamic modelling and analysis of V- and Z-shaped electrothermal microactuators. Microsyst. Technol. 23(8), 3775–3789 (2016). https://doi.org/ 10.1007/s00542-016-3180-0 8. Suen, M.S., Hsieh, J.C., Liu, K.C., Lin, D.T.W.: Optimal design of the electrothermal V-beam microactuator based on GA for stress concentration analysis. In: IMECS 2011 - International MultiConference of Engineers and Computer Scientists, vol. 2, pp. 1264–1268 (2011) 9. Hu, T., Zhao, Y., Li, X., Zhao, Y., Bai, Y.: Design and fabrication of an electro-thermal linear motor with large output force and displacement. In: Proceedings of IEEE Sensors (2017) 10. Nguyen, D.T., Hoang, K.T., Pham, P.H.: Larger displacement of silicon electrothermal Vshaped actuator using surface sputtering process. Microsyst. Technol. 27(5), 1985–1991 (2020). https://doi.org/10.1007/s00542-020-04985-5 11. Fogel, O., Winter, S., Benjamin, E., Krylov, S., Kotler, Z., Zalevsky, Z.: 3D printing of functional metallic microstructures and its implementation in electrothermal actuators. Addit. Manuf. 21, 307–311 (2018) 12. Enikov, E.T., Kedar, S.S., Lazarov, K.V.: Analytical and experimental analysis of folded beam and V-shaped thermal microactuators. In: Proceedings of the SEM X International Congress Costa Mesa, CA, USA, vol. 1 (2004) 13. Zhu, Y., Corigliano, A., Espinosa, H.D.: A thermal actuator for nanoscale in situ microscopy testing: Design and characterization. J. Micromechanics Microengineering 16(2), 242–253 (2006) 14. Hoang, K.T., Nguyen, D.T., Pham, P.H.: Impact of design parameters on working stability of the electrothermal V-shaped actuator. Microsyst. Technol. 26(5), 1479–1487 (2020). https:// doi.org/10.1007/s00542-019-04682-y 15. Hull, R.: Properties of Crystalline Silicon. INSPEC, London (1999)
Electromechanical Properties of Monolayer Sn-Dichalcogenides Le Xuan Bach1 , Vuong Van Thanh1(B) , Hoang Van Bao1 , Do Van Truong1 , and Nguyen Tuan Hung2 1 School of Mechanical Engineering, Hanoi University of Science and Technology, Hanoi,
Vietnam [email protected] 2 Frontier Research Institute for Interdisciplinary Sciences, Tohoku University, Sendai 980-8578, Japan
Abstract. We demonstrate electromechanical properties of monolayer SnX2 (X = Se, Te) with 1T structure as a function of charge doping using first-principles methods. We indicate that the monolayer SnSe2 shows a semiconductor-metal transition for the case of heavy electron doping, while SnTe2 retains the metallic properties under both electron and hole dopings. The actuation strain of SnX2 in the case of electron doping is significantly higher than those of hole doping. In addition, we also compute the ideal strength and ideal strain of the monolayer SnX2 under charge doping. Keywords: Artificial muscles · Charge doping · First-principles · Electromechanical
1 Introduction Artificial muscles are materials that can respond like a natural muscle to generate strain and force when stimulated [1–3]. There are many actuation materials, which have been examined to replace the natural muscle such as polymer nanofiber films [4], shapememory alloys [5], dielectric elastomers [6], and carbon nanotube bundles [7]. Recently, two-dimensional (2D) materials show great potential for the artificial muscle thanks to their unique physical and chemical properties [8–11]. Acerce et al. [12] observed that 2D nanosheet MoS2 generates mechanical stresses of about 17 MPa with an actuation strain ~0.8% at low voltage values ranging from −0.3 to 0.3 V. In the previous theoretical study, Thanh et al. [13] investigated the electromechanical properties of another 2D MoS2 family and indicated that the monolayers 1H-WTe2 and 1T-WS2 show the best actuation performances among MX2 (M = Mo, W; X = S, Se, and Te) compounds. Very recently, another 2D material as monolayer SnSe2 has been successfully synthesized [14, 15]. Qu et al. [16] explored the electronic and mechanical properties of the monolayer SnX2 by using the density functional theory (DFT) calculations and pointed out that under strain, a semiconductor-metal transition occurs on SnX2 with the 2H structure, while the change of the bandgap appears in SnX2 with the 1T structure. Zeng et al. [17] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 1113–1119, 2022. https://doi.org/10.1007/978-981-16-3239-6_87
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reported the gate-induced 2D superconductivity in the 1T-SnSe2 structure. Moreover, the magnetism of SnX2 (X = S, Se) was studied [18], in which there is a nonmagnetic to ferromagnetic ground state transition at the critical hole density (~1014 cm−2 ). Up to date, almost all researches of the monolayer SnX2 have mainly concentrated on the synthesis, mechanical, electronic, and magnetism properties [14–18], while their electromechanical properties are still lacking. In this paper, by using DFT methodology, we explore the electromechanical properties of the monolayer SnX2 (X = Se, Te) with the 1T structure (1T- SnX2 ) under the charge doping. We show that the semiconductor-metal transition happens in 1T-SnSe2 under the heavy electron doping, while 1T-SnTe2 retains metallic properties under charge doping. SnSe2 displays a reversible strain ranging from 0.5% to 1.8%, making them suitable for actuation applications. Moreover, the stress-strain relations of the monolayer SnX2 under the different charge doping levels are investigated as well in this study.
2 Computational Methods Our calculations are performed through the DFT calculations by using the Quantum ESPRESSO package [19] with the energy cut-off of 60 Ry. A k-mesh Monkhorst– Pack of 16 × 16 × 1 is used in all the calculations [21]. The exchange-correlation energy is evaluated through the general gradient approximation (GGA) and the Perdew– Burke–Ernzerhof (PBE) function [20]. In Fig. 1, we display the atomic structure of the monolayer SnX2 . A vacuum of 30 Å is set to prevent the interaction between layers. The atomic structures of the monolayer SnX2 are relaxed until the atom forces and the stresses are smaller than 5 × 10–4 Ry/a.u. and 5 × 10–2 GPa, respectively, based on the Broyden–Fletcher–Goldfarb–Shanno (BFGS) minimization method [22–25].
Fig. 1. Atomic structure of SnX2 . (a) and (b) are the top and front view of 1T-SnX2 . (c) The Brillouin zone.
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To investigate the electromechanical properties of the monolayer SnX2 , the atom structures are relaxed at the values of charge doping between −0.1 and +0.1 e/atom. To determine the stress-strain curves of the monolayer SnX2 , we apply an axial tensile strain to the structures in the axial direction, in which the tensile strain is calculated by ε = (L − L 0 )/L 0 , where L and L 0 are the lengths of the unit cells for deformed and undeformed structures, respectively.
3 Results and Discussion Our optimized lattice constants of the monolayer SnX2 are listed in Table 1, which are close to the earlier studies [16, 26, 27], indicating that our calculations are consistent. The buckling height (h, see Fig. 1(b)) of SnSe2 (3.24 Å) is smaller than that of SnTe2 (3.60 Å) and MoSe2 (3.34 Å) [28]. The bond lengths of Sn-Se and Sn-Te atoms are 2.74 Å and 2.98 Å, respectively. Our calculated bond lengths are consistent with the calculated results by a theoretical study [16]. Table 1. Lattice constant a0 (Å), buckling height h (Å), and bond lengths of Sn-X and X-X atoms (Å) of monolayer SnX2 . Materials a
h
Sn-X X-X
SnSe2
3.83 3.24 2.74
3.92
SnTe2
4.11 3.60 2.98
4.31
Fig. 2. Band structures of SnSe2 at the several values of (a) electron and (b) hole dopings.
To investigate the electronic properties of SnX2 under charge injection, we calculate the band structures of SnX2 for the neutral (q = 0) and charge doping cases (q = 0). In Figs. 2 (a) and (b), we demonstrate the band structures of SnSe2 under the electron and hole dopings, respectively. The obtained results revealed that SnSe2 is an indirect
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Fig. 3. Band structure of SnTe2 at the several values of (a) electron and (b) hole dopings.
semiconductor with a bandgap of 0.71 eV at q = 0, which is close to the previously calculated result (0.8 eV) [28, 29]. We note that, the 1H-MoS2 monolayer is direct-gap (1.65 eV) semiconductor [30], while the monolayer 1T-ZeS2 is an indirect-gap (1.1 eV) semiconductor [31]. For the electron doping case, the energy bands of monolayer SnSe2 shift downward compared with the energy bands in the neutral case. At q = −0.04 e/atom, the monolayer SnSe2 becomes metallic. In the hole doping case (q < 0), the band structures of SnSe2 are almost unchanged with the increase of q, as illustrated in Fig. 2 (b). On the other hand, the monolayer 1T-SnTe2 is metallic at q = 0 and q = 0, as illustrated in Fig. 3 (a) and (b). With the stable metallic, 1T-SnTe2 shows potential for the artificial muscle that can work with a low applied voltage. Acerce et al. [12] indicated that artificial muscle using the metallic 1T-MoS2 works with very low voltage ranging from −0.3 V to 0.3 V. In Fig. 4, we demonstrate the relationship between the actuator strain εq and charge doping q of the monolayer SnX2 , in which q ranged from −0.1 to +0.1 e/atom. At q = 0, we get εq = 0. For the both electron and hole doping cases, εq of SnX2 is non-linear function of q. For the electron doping case (q < 0), εq of the monolayer SnX2 increases with decreasing of q. At q = −0.1 e/atom, the actuator strains of monolayer SnSe2 reaches to 1.8%, eight times higher than maximum strain (0.21%) for graphene-based actuator [11]. For the hole doping case (q > 0), SnTe2 shows the highest compression strain (εq = −0.9%) at q = 0.06 e/atom, while SnSe2 shows the highest compression strain (εq = −0.7%) at q = 0.05 e/atom and the expansion strain about (εq = 0.5%) at q = +0.1 e/atom. We finally determine the ideal strength σ i and the ideal strain εi of the monolayer SnX2 at the several charge doping levels. In Figs. 5 (a) and (b), we demonstrate the stress-strain relations of SnSe2 and SnTe2 for several values of the charge dopings, respectively. The obtained results indicate that SnX2 shows anisotropic behavior. We note that the experiment can be conducted to the 2D materials by using a four-point apparatus [32]. At q = 0, σ i (εi ) of SnSe2 is 4.76 N.m−1 (0.15) and 5.08 N.m−1 (0.32) along the x and y directions, respectively. In addition, σ i of SnSe2 is higher than that of SnTe2 for both x and y directions, as illustrated in Figs. 5 (a) and (b). For the hole doping
Electromechanical Properties of Monolayer Sn-Dichalcogenides
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Fig. 4. Relationship between the actuator strain and charge doping levels of monolayer SnX2 .
Fig. 5. Engineering stress of monolayer SnX2 under the charge doping.
case, the ideal stress is decreased by increasing |q|, in which the highest reduction of σ i is found in SnSe2 along the armchair direction, as shown in Fig. 5 (a). In contrast, for the electron doping case, σ i of SnSe2 and SnTe2 is increased by 3.8% and 13.4% along the y-direction, respectively, as illustrated in Fig. 5 (b).
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4 Conclusions Based on first-principles calculations, we have investigated the electromechanical properties of the monolayers SnSe2 and SnTe2 with 1T structure under the charge doping. Our calculated results show that the monolayer SnSe2 has a semiconductor-metal transition at q = −0.04 e/atom. In the case of the heavy electron doping, the actuator strain of SnSe2 and SnTe2 is achieved up to 1.8% and 1.6%, respectively. Moreover, the ideal strength of SnSe2 and SnTe2 is enhanced by 3.8% and 13.4% in the zigzag direction for the electron doping, respectively. The obtained results provide useful information to the practice of the monolayer SnX2 -based artificial muscles.
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Author Index
A Aly, Mach, 795 Anh, Le Thi Ngoc, 301 B Bach, Le Xuan, 981, 1113 Bich, Dao Huy, 489, 506 Binh, Chu Thanh, 271, 761 Binh, Nguyen Quang, 959 Bo, Vu V., 1065 Bui, Duy Q., 125 Bui, Nang D., 830 Bui, Thi Loan, 731 Bui, Tinh Quoc, 990 C Chan, Do Quang, 743 Chinh, Vu Dan, 431 Cho, Sang-Rai, 416 Chu, Thanh Binh, 780 Chung, Nguyen Thai, 169 Chuong, Nguyen Tien, 866 Cong, Nguyen Chi, 959 Cuong, Nguyen Tien, 959 D Dang, Hoang Long, 971 Dang, Hung X., 229 Dang, Ngoc Thanh, 149 Dang, Xuan Hung, 780 Dao, Van Luu, 811 Dien, Nguyen Phong, 393 Dinh, Tran Binh, 458
Do, Quang Thang, 416 Doan, Le Xuan, 489, 506 Doan, Tran Ngoc, 316 Dong, Dang Thuy, 256 Du, Nguyen Trong, 393 Duc, Do Minh, 691 Duc, Ngo Trong, 458 Duc, Vu Minh, 240 Dung, Nguyen Anh, 203, 476 Dung, Nguyen Thuy, 981 Duong, Huan T., 830 Duong, Tham Hong, 716 Duyen, Do Q., 54 G Gan, Buntara S., 87 H Ha, Le Thi, 100, 112 Ha, S. T., 1089 Hai, Le Thanh, 761 Hau, Pham Hien, 431 Hien, Ho Thi, 287 Hiep, Le D., 1065 Hieu, Do H. M., 54 Ho, Duc-Duy, 1004 Ho, Ky-Thanh, 63, 537 Hoa, Le Kha, 743 Hoai, Bui Thi Thu, 743 Hoan, Pham Van, 743 Hoang, Kien Trung, 1102 Hoang, Viet-Hai, 823 Hoang, Vuong L., 125
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 N. Tien Khiem et al. (Eds.): Modern Mechanics and Applications, LNME, pp. 1121–1123, 2022. https://doi.org/10.1007/978-981-16-3239-6
1122 Huan, Duong Thanh, 271, 287 Hung, Dang Xuan, 72 Hung, Nguyen Ba, 362 Hung, Nguyen Duy, 755 Hung, Nguyen Q., 54, 1065 Hung, Nguyen Tuan, 981, 1113 Hung, Tran Quang, 691 Hung, Tran Van, 316 Hung, Vo Duy, 446, 584 Huong, Truong Q., 229 Huynh, Trieu Nhat, 158, 606, 617, 631, 641 J Jeon, Myungjun, 546 K Khiem, Nguyen Tien, 13 Kien, Nguyen Dinh, 301 Kien, Nguyen Trung, 755 L La, Ngoc-Tuan, 537 Lai, Dang Giang, 811 Lam, Hoang Trong, 584 Le, Ba-Anh, 926 Le, Thanh Phong, 945 Le, Thanh-Cao, 1004 Le, Thanh-Long, 795 Le, Vu Dan Thanh, 149 Lee, Myung Jin, 857 Lien, Pham Thi Ba, 13 Lim, Ocktaeck, 362 Lim, Seung Hyeon, 546 Lin, Ker-Chun, 875, 915 Long, Doan Viet, 959 Long, Phan Thanh, 663 M Mai, Dai D., 125 Mai, Duc -Dai, 1065 Mai, Thi Loan, 546 Minh, Le Duy, 573 Minh, Tran Quang, 240, 256 N Nam, Vu Hoai, 256 Nghi, Nguyen Ba, 573 Ngo, Duc Chinh, 219 Ngo, Quoc-Huy, 63, 537 Ngoc, L. T. K., 841 Ngoc, Nguyen Linh, 596 Ngon, Dang Thien, 382 Nguyen, Anh Tuan, 811 Nguyen, Anh Tuan, 149, 653 Nguyen, Ba-Duan, 1033
Author Index Nguyen, Duc Cuong, 945 Nguyen, Hoang-Quan, 926 Nguyen, Hung Q., 125 Nguyen, Hung Thinh, 971 Nguyen, Huu-Anh-Tuan, 560 Nguyen, Khac-Tuan, 63, 537 Nguyen, Kien T., 890 Nguyen, Lieu B., 673 Nguyen, M. D., 1089 Nguyen, M. H., 1089 Nguyen, Minh Ngoc, 990 Nguyen, N. M., 1018 Nguyen, Ngoc Tan, 219, 331 Nguyen, Ngoc-Vinh, 905 Nguyen, Nha Thanh, 990 Nguyen, Q. H., 1018 Nguyen, Quang Binh, 971 Nguyen, T. D., 1089 Nguyen, Tan N., 890, 1018 Nguyen, Thac Quang, 857 Nguyen, Thang T., 935 Nguyen, Thanh Xuan, 137 Nguyen, The Hung, 1 Nguyen, Thi-Huong-Giang, 823 Nguyen, Tran-Hieu, 1052 Nguyen, Trung-Kien, 707 Nguyen, Van Dinh, 416 Nguyen, Van-Du, 537 Nguyen, Viet Duc, 331 Nguyen, Viet Tuan, 158 Nguyen, Xuan Tung, 857 Nguyen, Y. Quoc, 158, 606, 617 Nguyen-Duy, Thao, 522 Nguyen-Thi-Kim, Loan, 522 Nguyen-Xuan, H., 189, 673 Nguyen-Xuan, Toan, 522 Nien, Nguyen T., 1065 Ninh, Vu Thi An, 301 P Park, Jong Sup, 857 Pham, Hoang-Anh, 229, 1033 Pham, Phuc Hong, 1102 Pham, Tiep D., 830 Pham, Tung T., 935 Phan, Hao D., 875, 915 Phan, Hieu C., 830 Phan, Hieu T., 875 Phong, Le Trung, 203 Phu, Khuc Van, 506 Phung, Binh Van, 149 Phung-Van, P., 189 Phuoc, Vo Nguyen Duc, 959 Phuong, Nguyen Thi, 240
Author Index Q Quan, Tang Ha Minh, 382 Quang, Duong Van, 316 Quoc Nguyen, Y, 631, 641 Quoc, Tran Huu, 271, 287, 347 Quy, Doan Xuan, 866 S Sang, Nguyen Vinh, 476 Son, Nguyen Hai, 373 T Tai, Nguyen P., 54 Tai, V. V., 841 Thai, Chien H., 189, 673 Thai, N. Canh, 401 Thang, N. Ngoc, 401 Thang, Nguyen Cao, 573 Thang, Nguyen Tat, 458 Thang, Nguyen V., 54 Thanh, Pham Ngoc, 26 Thinh, Tran Ich, 26 Thu, Duong Thi Ngoc, 169 Tinh, Nguyen Ngoc, 596 Toan, Khong Trong, 1078 Tran, Bao-Viet, 823, 926 Tran, Dai Hao, 780 Tran, Danh Thanh, 716 Tran, Dinh Khoi, 653 Tran, Dong, 331 Tran, Hong Son, 945 Tran, Lien V., 935 Tran, Long N., 935 Tran, Long Tuan, 137 Tran, Minh Tu, 780 Tran, T. M., 1018 Tran, Van Dang, 331 Tran-Van, Duc, 522
1123 Tri, Diep B., 1065 Trinh, Do Van, 1078 Truong, Huong Quy, 72 Truong, T. T., 1018 Truong, Thien Tich, 990 Truong, Viet-Hung, 905 Tu, Tran Minh, 72, 347, 691, 761 Tuan, Nguyen Minh, 13 V Van Bao, Hoang, 1113 Van Dang, Nguyen, 169 Van Lang, Tran, 301 Van Lien, Tran, 458 Van Long, Nguyen, 761 Van My, Nguyen, 446 Van Nghin, Dang, 795 Van Nguyen, Chung, 945 Van Nguyen, Tung, 617 Van Phu, Khuc, 489 Van Tham, Vu, 271, 347 Van Thanh, Vuong, 981, 1113 Van Toan, T., 401 Van Truong, Do, 981, 1113 Van Xuan, N., 401 Viet, La Duc, 573 Vinh, Ngo C., 54 Vo, Duc Phuoc, 971 Vo, Huan NguyenPhu, 716 Vo, Ngoc Duong, 971 Vu, Anh-Tuan, 1052 Vu, Thai Van, 990 Vu, V. C., 1089 Vu, Viet-Hung, 823 Y Yoon, Hyeon Kyu, 546