Structural Engineering: Models and Methods for Statics, Instability and Inelasticity 3031235916, 9783031235917

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Lecture Notes in Applied and Computational Mechanics 100

Adnan Ibrahimbegovic Rosa-Adela Mejia-Nava

Structural Engineering Models and Methods for Statics, Instability and Inelasticity

Lecture Notes in Applied and Computational Mechanics Volume 100

Series Editors Peter Wriggers, Institut für Kontinuumsmechanik, Leibniz Universität Hannover, Hannover, Niedersachsen, Germany Peter Eberhard, Institute of Engineering and Computational Mechanics, University of Stuttgart, Stuttgart, Germany

This series aims to report new developments in applied and computational mechanics - quickly, informally and at a high level. This includes the fields of fluid, solid and structural mechanics, dynamics and control, and related disciplines. The applied methods can be of analytical, numerical and computational nature. The series scope includes monographs, professional books, selected contributions from specialized conferences or workshops, edited volumes, as well as outstanding advanced textbooks. Indexed by EI-Compendex, SCOPUS, Zentralblatt Math, Ulrich’s, Current Mathematical Publications, Mathematical Reviews and MetaPress.

Adnan Ibrahimbegovic · Rosa-Adela Mejia-Nava

Structural Engineering Models and Methods for Statics, Instability and Inelasticity

Adnan Ibrahimbegovic Laboratory Roberval Mecanique Universite de Technologie Compiegne Compiegne, France

Rosa-Adela Mejia-Nava Laboratory Roberval Mecanique Universite de Technologie Compiegne Compiegne, France

ISSN 1613-7736 ISSN 1860-0816 (electronic) Lecture Notes in Applied and Computational Mechanics ISBN 978-3-031-23591-7 ISBN 978-3-031-23592-4 (eBook) https://doi.org/10.1007/978-3-031-23592-4 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 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 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 Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

To our families

Preface

The roots of this book go back to my engineering education in ex-Yougoslavia, where during my studies till my master’s degree in 1984, I have learned from the most active professors in structural mechanics in different scientific centers throughout the country (Branislav Verbic and Ognjen Jokanovic from Sarajevo, Josip Dvornik from Zagreb, Peter Fajfar and Miran Saje from Ljubljana and Milos Kojic from Kragujevac). With the good fortune of winning a Fullbright Grant, I was able to complete my doctoral studies at the University of California at Berkeley, from 1986 to 1989. The UC Berkeley, in general, and Structural Engineering, Mechanics and Materials Division in particular, provided an excellent study and research environment, with the opportunities to exchange the ideas with some extraordinary talented people from all over the world. The high point of my education was during another couple of years as a post-doc, where I could strongly interact with my thesis mentors, Berkeley professors Edward L. Wilson and Robert L. Taylor, on very wide variety of topics, contributing to some of my most highly cited papers. The same good fortune was my subsequent research appointment from 1991 to 1994 at the Swiss Federal Institute of Technology in Lausanne at Structural and Continuum Mechanics Laboratory, directed by François Frey, who granted me complete freedom to carry on with further explorations. Further inspiration for the book contents came from my subsequent appointments in France, first in 1994 at the Compiègne University of Technology, where I was interacting with Jean-Louis Batoz and Gouri Dhatt, the authors of French books on linear structural mechanics, and then in 1999 at Ecole Normale Supérieure of Cachan, where I was in contact with Pierre Ladevèze, the author of French book on nonlinear structural mechanics. All these interactions at five different universities in four different countries, and many private discussions with colleagues and friends like Jürgen Bathe, Mike Crisfield and Ted Belytschko, have allowed me to enrich my understanding of structural mechanics in many different aspects, including a great diversity of labels that experts of structural mechanics attach to different models (e.g., shear deformable beam was Timoshenko beam for ones and Hancky beam for others). The diversity of labels and diversity of models are perhaps the most challenging aspect when writing a book on Structural Engineering, which we tried to handle in this work. vii

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The work on the book started in 2021, when I won my renewal in prestigious ‘Institut Universitaire de France’ (IUF), which allowed me to keep reduced teaching load and have more free time for research. This has coincided with my return to the Compiègne University of Technology with the position of Chair for Computational Mechanics, with the special role of teaching courses in Doctoral School and federating interdisciplinary research efforts. Here, the structural models have proved very helpful in constructing an accessible basis to experts in different domains of engineering science that were participating in such projects, in order to build best microscale representation (as replacement of atomistic models) in multi-scale models for various multi-physics phenomena we studied. That was the first motivation for writing this book. The second motivation comes from the intent to preserve the knowledge accumulated by long-term efforts of experts in structural mechanics (with many of them retired, or gone) and provide a priori selection of reduced model by means of corresponding kinematic hypotheses and constraints that reduce the cost compared to solid or continuum mechanics approach (e.g., see [175–177]). Such an approach is opposite of the currently active research on constructing such reduced model in a posteriori fashion, by using the statistical approach of data processing to provide the reduced model in terms of Proper Orthogonal Decomposition (POD) or Proper Generalized Decomposition (PGD), which seems to reduce the model construction skills to the application of artificial intelligence algorithms. I firmly believe that the knowledge of alternative methods with natural intelligence of engineers should also be brought to bear upon any completely successful solution to complex engineering problems, and that either approach should be tried in seeking such optimal results. In fact, we would like to illustrate with this book perhaps the best possible manner to combine the expertise in structural engineering with artificial intelligence skills. Namely, for models construction in terms of the best choice of hypotheses for given goal, one should have the expertise in structural engineering, and for models selection in terms of validation and verification, one should also include the adequate stochastic tools with statistical methods and probability (e.g., see [212–215]). Compiegne, France October 2022

Adnan Ibrahimbegovic Professor Classe Exceptionnelle

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Motivation and Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Main Topics Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Further Studies Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Summary of Main Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 5 11 12

2 Truss Model: General Theorems and Methods of Force, Displacement and Finite Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Truss Model—Strong Form and Weak Form . . . . . . . . . . . . . . . . . . . . 2.1.1 Strong or Differential Form: Analytic Solution . . . . . . . . . . . 2.1.2 Weak or Integral Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 General Theorems of Structural Mechanics on Truss Model . . . . . . 2.2.1 Principle of Virtual Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Principle of Complementary Virtual Work . . . . . . . . . . . . . . . 2.2.3 Principle of Minimum of Total Potential Energy . . . . . . . . . . 2.2.4 Applied General Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Castigliano’s Theorems, Force and Displacement Methods . . . . . . . 2.3.1 Castigliano’s Theorems—Stiffness and Flexibility . . . . . . . . 2.3.2 Force and Displacement Methods . . . . . . . . . . . . . . . . . . . . . . 2.4 Finite Element Method Implementation for Truss Model . . . . . . . . . 2.4.1 Local or Elementary Description . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Consistence of Finite Element Approximation . . . . . . . . . . . . 2.4.3 Equivalent Nodal External Load Vector . . . . . . . . . . . . . . . . . 2.4.4 Higher Order Finite Elements . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.5 Role of Numerical Integration . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.6 Finite Element Assembly Procedure . . . . . . . . . . . . . . . . . . . .

15 15 16 20 22 22 25 27 31 34 34 37 43 43 48 49 50 52 56

3 Beam Models: Refinement and Reduction . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Reduced Models of Solid Mechanics: Planar Beams of Euler, Timoshenko and Reissner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Euler-Bernoulli Planar Beam Model . . . . . . . . . . . . . . . . . . . .

59 59 59

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3.1.2 Solid Mechanics Versus Beam Model Accuracy for Planar Cantilever Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Timoshenko Planar Beam Model . . . . . . . . . . . . . . . . . . . . . . . 3.1.4 Brief on Reissner Planar Beam Model . . . . . . . . . . . . . . . . . . . 3.2 Beam Model Refinement and Reduction . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Method of Direct Stiffness Assembly for 3D Beam Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Beam Model Refinement: Flexibility Approach for Reduced Model in Deformation Space . . . . . . . . . . . . . . . 3.2.3 Beam Model Reduction: Joint Releases and Length Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Curved Shallow Beam and Non-locking FE Interpolations . . . . . . . . 3.3.1 Two-Dimensional Curved Shallow Beam: Linear Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Non-locking Finite Element Interpolation for Shallow Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Illustrative Numerical Examples and Closing Remarks . . . . 4 Plate Models: Validation and Verification . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Finite Elements for Analysis of Thick and Thin Plates . . . . . . . . . . . 4.1.1 Motivation: Timoshenko Beam Element Linked Interpolations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Reissner-Mindlin Plate Model and FE Discretization . . . . . . 4.1.3 Illustrative Numerical Examples and Closing Remarks . . . . 4.2 Discrete Kirchhoff Plate Element Extension with Incompatible Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Reissner-Mindlin Plate Model and Enhanced FE Interpolations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Illustrative Numerical Examples and Closing Remarks . . . . 4.3 Validation or Model Adaptivity for Thick or Thin Plates Based on Equilibrated Boundary Stress Resultants . . . . . . . . . . . . . . 4.3.1 Thick and Thin Plate Finite Element Models . . . . . . . . . . . . . 4.3.2 Model Adaptivity for Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Illustrative Numerical Examples and Closing Remarks . . . . 4.4 Verification or Discrete Approximation Adaptivity for Discrete Kirchhoff Plate Finite Element . . . . . . . . . . . . . . . . . . . . 4.4.1 Kirchhoff Plate Bending Model . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Kirchhoff Plate Finite Elements . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3 Error Estimates for Kirchhoff Plate Elements Based Upon Equilibrated Boundary Stress Resultants . . . . . . . . . . . 4.4.4 Implementation of Equilibrated Element Boundary Tractions Method For DKT Plate Element . . . . . . . . . . . . . . . 4.4.5 Examples on Error Indicators Comparison and Closing Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69 72 76 81 83 87 91 103 103 106 114 119 119 120 122 132 137 138 141 144 146 154 163 173 179 183 190 194 199

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5 Solids, Membranes and Shells with Drilling Rotations: Complex Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Solids with Drilling Rotations: Variational Formulation . . . . . . . . . . 5.1.1 Strong Form of the Boundary Value Problem . . . . . . . . . . . . . 5.1.2 Variational Formulation, Stability Analysis and Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.3 Alternative Variational Formulations, Extension to Nonlinear Kinematics and Closing Remarks . . . . . . . . . . . 5.2 Membranes with Drilling Rotations: Discrete Approximation . . . . . 5.2.1 Discrete Approximations with Quadratic and Cubic Displacement Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Illustrative Numerical Examples and Closing Remarks . . . . 5.3 Shells with Drilling Rotations: Linearized Kinematics . . . . . . . . . . . 5.3.1 Geometrically Linear Shallow Shell Theory . . . . . . . . . . . . . . 5.3.2 Incompatible Modes Based Finite Element Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Illustrative Numerical Examples and Closing Remarks . . . . 6 Large Displacements and Instability: Buckling Versus Nonlinear Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Large Displacements and Deformations in 1D Truss with Instabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Large Strain Measures for 1D Truss . . . . . . . . . . . . . . . . . . . . 6.1.2 Strong and Weak Forms for 1D Truss in Large Displacements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.3 Linear Elastic Behavior for 1D Truss in Large Displacements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.4 Finite Element Method for 1D Truss in Large Displacements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.5 Buckling, Nonlinear Instability and Detection Criteria . . . . . 6.2 Geometrically Nonlinear Curved Beam and Nonlinear Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Curved Reissner’s Beam: Nonlinear Kinematics . . . . . . . . . . 6.2.2 Finite Element Implementation for Curved Reissner’s Beam and Comment on Objectivity . . . . . . . . . . . . . . . . . . . . . 6.2.3 Control of Nonlinear Instability . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Buckling of (Heterogeneous) Euler’s Beam . . . . . . . . . . . . . . . . . . . . 6.3.1 Euler Instability Problem: Two Alternative Formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Analytic Solution Based on Strong Form . . . . . . . . . . . . . . . . 6.3.3 Numerical Solution Based on Reduced Models with Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Buckling Analysis of Complex Structures with Refined Models of Plates and Shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Buckling Problems for Plates and Shells . . . . . . . . . . . . . . . . .

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211 211 213 215 223 230 231 242 247 248 255 261 273 273 273 277 279 282 288 298 299 304 315 321 323 327 329 335 336

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6.4.2 Finite Element Shell Approximation Including Drilling Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.3 Illustrative Numerical Examples and Closing Remarks . . . . 6.5 Buckling Problems for Coupled Thermomechanical Extreme Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.1 Linear Thermoelasticity 1D . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.2 Linearized Instability for Thermoelastic Coupling . . . . . . . . 6.5.3 General Linear Eigenvalue Problem Solution Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.4 Thermomechanical Coupling Model: Illustrative and Validation Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.5 Brief on Instability for Thermomechanical Coupling in 1D Finite Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Inelasticity: Ultimate Load and Localized Failure . . . . . . . . . . . . . . . . . 7.1 Stress Resultants Finite Element Model for Reinforced-Concrete Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1 Plate Element for Reinforced-Concrete Slabs . . . . . . . . . . . . 7.1.2 Stress-Resultants Constitutive Model for Reinforced-Concrete Plates . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.3 Illustrative Numerical Examples and Closing Remarks . . . . 7.2 Stress Resultants Plasticity for Metallic Plates . . . . . . . . . . . . . . . . . . 7.2.1 Variational Formulation and Discrete Approximation for Metallic Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Stress Resultants Plasticity Formulation for Metallic Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.3 Illustrative Numerical Examples and Closing Remarks . . . . 7.3 Plasticity Criterion with Thermomechanical Coupling in Folded Plates and Non-smooth Shells . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Theoretical Formulation of Shell Model for Folded Plates and Non-smooth Shells . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2 Finite Element Implementation with Shell Element . . . . . . . 7.3.3 Stress Resultants Constitutive Model of Saint-Venant Plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.4 Thermomechanical Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.5 Operator Split Solution Procedure with Variable Time Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.6 Illustrative Numerical Examples and Closing Remarks . . . . 7.4 Stress Resultants Plasticity and Localized Failure of Reissner’s Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 Reissner’s Beam with Localized Elastoplastic Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.2 Stress Resultant Plasticity Discrete Approximations and Computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.3 Illustrative Numerical Examples and Closing Remarks . . . .

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8 Brief on Mulitscale, Dynamics and Probability . . . . . . . . . . . . . . . . . . . . 8.1 Mulitscale Approach to Quasi-brittle Fracture in Dynamics . . . . . . . 8.1.1 Geometrically Exact Shear Deformable Beam as Cohesive Link . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.2 Micro and Macro Constitutive Models for Dynamic Fracture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.3 Dynamics of Lattice Network and Time-Stepping Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.4 Illustrative Numerical Examples and Closing Remarks . . . . 8.2 Stochastic Upscaling, Size Effect and Damping Replacement . . . . . 8.2.1 Stochastic Upscaling in Localized Failure . . . . . . . . . . . . . . . 8.2.2 Probability-Based Size Effect in Ductile Failure . . . . . . . . . . 8.2.3 Damping Model Replacement of Rayleigh Damping . . . . . . 8.3 Reduced Stochastic Models for Euler Beam Dynamic Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Duffing Oscillator: Reduced Model for Euler Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2 Instability Studies in Dynamics Framework . . . . . . . . . . . . . . 8.3.3 Stochastic Solution to Euler Instability Problem . . . . . . . . . .

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References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517

Chapter 1

Introduction

Abstract We specify in this chapter our motivation for writing this book, the objectives and the contents of each chapter. We also give some remarks on further reading and chosen conventions for different notations.

1.1 Motivation and Objectives The Structural Engineering is perhaps the most classical scientific discipline, developed well before the computer-based simulations have become dominant in current engineering practice. These skills are used to develop the simplified models capable of providing the most efficient solution methods with sufficient accuracy to numerous problems from domains of Civil, Mechanical or Aerospace Engineering. The word ‘Structure’, which draws more than 4 billion entries in a Google search, has very broad meaning that is summarized in Wikipedia entry as ‘an arrangement and organization of interrelated elements in a material object or system’. Our main interest in present work pertains to the material structures, including man-made objects, such as buildings and machines or vehicles, and natural objects, such as biological organisms, rather than the abstract structures, including data structures in computer science and musical form. The main goal of this book is to help with the choice of model for any such material structure that is made by engineers in order to eliminate the unnecessary details by using certain simplifying hypotheses, and provide accompanying methods to accomplish the task on hands. Contrary to many classical works, the approach we follow relies heavily on the most advanced models developed by using the finite element method (e.g., see [175, 176]), with an ambition to present not only the basis of linear theory, but also the nonlinear theory dealing with structure instability and inelasticity. The nonlinear problems studied in this book are extremely difficult to solve other than by numerical modeling and the finite element method. For that reason, the first important role of a specialist in structural engineering is that of verification, testing the © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Ibrahimbegovic and R.-A. Mejia-Nava, Structural Engineering, Lecture Notes in Applied and Computational Mechanics 100, https://doi.org/10.1007/978-3-031-23592-4_1

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1 Introduction

quality of the discrete approximation and the numerical methods available in a given computer code. Here we seek to estimate the discretization error (e.g., see [255]) of the finite element model with respect to the exact solution of the relevant equations, and furnish an acceptable quality of the numerical results. We have provided the formal procedure for verification and its illustration on one of the most popular structural engineering models of DKT plate finite element (e.g., [36]). Perhaps even more important is the second role for a specialist in structural engineering, which is that of validation, where one seeks the right choice of the model with respect to the results of interest to be obtained. In other words, one seeks ’the right equations’ to be solved with respect to the modeling goals. There were many structural engineering specialists in early times, before intensive use of computers, who could choose the most appropriate ’beam model’ for just about any problem in structural engineering, for it was one of the few that can be solved by hand calculations. As a young student, I have been educated by many such experts, almost ending up to believe that the only thing one needs to know about structural engineering is the beam theory. At present times, this is naturally no longer the case, since many commercial software products currently used by engineers have a very broad library of elements. Thus, we have developed here the formal procedure for the goaloriented model validation and illustrated its performance for a plate model, where the modeling goal pertains to importance of including or not the shear deformation to choose between Reissner-Mindlin and Kirchhoff plate models. The final difficulty to handle by a specialist in structural engineering, given the great diversity of available models, is to make sure that all the most appropriate models for different structure components are indeed possible to connect within the corresponding finite element assembly procedure (e.g., see [175, 176]) in order to ensure the compatibility of the displacement field across the boundaries between adjacent elements. The latter is the basic convergence requirement of the finite element method (e.g., see [175, 176]), which can be rather difficult to achieve when modeling the structures. The case in point is the connection between structural mechanics elements (beams, plates and shells) and solid mechanics elements (membranes and bricks), where standard choice of displacement field interpolation violates such compatibility requirements. Namely, membranes or brick finite elements do not have rotational degrees of freedom, contrary to beams, plates and shells. Therefore, in this work we present the development of the solid mechanics elements with rotational degrees of freedom that can share the same discrete approximation with structural mechanics elements. The development of this kind is not a mere modification of discrete approximation, but a thorough treatment with a sound theoretical formulation based upon Hu-Washizu variational principle with independent rotation field, its corresponding regularization and finally the most appropriate finite element interpolation that can match those used for structural elements. The developments of this kind allow us to provide a unified discrete approximation of complex structural assemblies and greatly simplify the modeling task for structural engineers. This was the guiding principle in the work that we started many years ago with UC Berkeley Professors Ed Wilson and Bob Taylor, resulting in the most cited paper on membrane elements with drilling rotations [222]. This element has become one of the modeling tools

1.1 Motivation and Objectives

3

not only in well-known UC Berkeley finite element codes, like SAP of Ed Wilson [382] and FEAP of Bob Taylor [391], but also commercial codes, like ANSYS. Moreover, not only the membrane element, but those for beams, shells and solids with rotational degrees of freedom have been developed by the first author during his post-docs years at UC Berkeley for the first version of commercial computer code SAP2000 (URL: https://www.csiamerica.com/products/sap2000). Hence, this book can also be perceived as the theoretical manual for using this computer code by practicing engineers. Even higher level of problem complexity needs to treated by a specialist in structural engineering if there is an ambition to push the structure into nonlinear behavior regime. In this book we illustrate two such domains, the first of structure instability related to risk of structure failure brought by large displacement and rotations, and the second of inelasticity related to risk of accumulated damage leading to premature structure failure. Here again we seek the approach that is of direct interest to vast majority of problems to be treated by structural engineers. In particular, the instability problem that we discuss in detail pertains to so called linearized instability or so-called Euler buckling that assumes small pre-buckling displacement. We cast the approach proposed by Euler in terms of variational formulation where the key role is played by virtual von Karman deformation measure replacing the strong form approach of Euler with equilibrium equations posed at the deformed configuration. We show that such approach can allow us to formulate the linearized buckling problem for any type of structural models that can be discretized by finite elements. Hence, any such code can easily be adapted to become the solver for linearized instability problems. A simple illustration of this kind is available in SAP2000 commercial code, named P-delta effect, and mostly adapted to beam and frame structures. Here, we show how to generalize such an approach in a very consistent manner to any kind of structure, with the finite element model composed of truss, beam, membrane, plate or shell finite elements. We also present how such instability problems can further be generalized for load combinations, where the first load case can serve as pre-stressing state and the second load case can lead to instabilities. Such an application is used to provide an explanation of sadly famous collapse of the World Trade Center building, under combined influence of thermal loads due to fire and structure dead load. The final contribution concerns the various inelasticity models provided for structures, illustrated for beams and plates. In particular, we study the simple stressresultant plasticity models suitable for reinforced concrete, steel plates and masonry structures. Any such model is built by generalizing the corresponding solid mechanics plasticity models to a particular structure, thus building multi-scale model of plasticity for structures. What is interesting to see this way is that a particular mechanism at fine scale (the scale of the representative volume element of a particular material) can be quite simple, but the subsequent generalization to structures turns such criteria into multi-surface plasticity criteria that are certainly more complex, but also more efficient since the structure model brings reduced computational effort. We have also generalized such plasticity criteria for load combinations between mechanical and thermal loads in order to provide the same level of generality with the chapter on instability.

4

1 Introduction

One of the main goals of this book is to present all the ingredients for constructing numerical models of complex structures with nonlinear behavior, with typical tasks for structural engineering related to pre-mature failure due to instability or inelasticity. However, the book should also prove useful for those mostly interested in linear problems of mechanics, since a sure way to obtain a sound theoretical formulation of a linear problem is through the consistent linearization of a more general nonlinear problem. The case in point is the linear formulation of shallow shell structures given in [195]. The same point of view that all the problems of mechanics are nonlinear, and that any linear approximation (for example, Hooke’s law) has only limited range of applicability, has been adopted by the pioneers of nonlinear mechanics, such as Euler and Bernoulli. They boldly proceeded along such a path and managed to obtain some remarkable solutions of nonlinear problems in mechanics, such as the elastica or the Euler-Bernoulli beam. In fact, it is only the undeniable success of Cauchy’s linear theory for continuum mechanics which made us forget that the Euler-Bernoulli theory of beams was originally developed in a geometrically exact setting. The main reason for the success of Cauchy’s linear theory of mechanics lay in its ability to furnish a number of results of practical value in a situation where only analytic solutions were available. This restriction no longer applies, since numerical models are readily available in many commercial codes. Hence, this book should be useful in providing a thorough understanding on how to develop the superior performance models, which can be implemented in any such code in terms of user finite element. Given this final goal, we seek in this book to provide the readers with well-balanced developments regarding both the theoretical formulations of structural engineering models and finite element based numerical solution As such, this book differs from many works discussing mostly theoretical aspects with no discussion of numerical implementation issues (e.g., see [8] or [151]), and also those who only consider the finite element method detailed developments with no theoretical formulation (e.g., see [36] or [98]). Additional and not less important of our goals is to define the theoretical formulation and numerical implementation details with the greatest possible simplicity (simple, but not too simple). This is much contrary to the spirit of classical works on nonlinear mechanics of solids and structures (e.g., see [374] or [292]) using the general curvilinear coordinates and Christoffel symbols that scared away more than one inexperienced users. In fact, we have shown in many of our previous works that the most general development for space-curved membranes and shells can all be carried out by using local Cartesian coordinates that greatly simplify the resulting equations (e.g., see [173, 195]). Thus, we sincerely hope that potential readers of this book will have the same pleasure and ease in going through the developments presented herein.

1.2 Main Topics Outline

5

1.2 Main Topics Outline The main challenge in this work is to present the essential working knowledge for an expert in structural mechanics without producing overly large number of pages. This inevitably leads to reducing the number of topics and details one might need for different engineering applications. To achieve the goal we fixed, we have selected to present only the structural models that can finally be combined within a compatible finite element assembly, with pertinent aspects of theoretical formulation and finite element discretization. Hence, the complete overview of all research domains and historical aspects of different developments have not be included in this book, in order to keep the number of pages reasonable. We have chosen the traditional structure of presenting different structural engineering models, in terms of truss, beam, membrane, plate and shell. This has been presented in subsequent four chapters of the book that come after this introduction. For each one of them we are limited initially to linear theory, which allows defining the structural models with the highest clarity. The extension to nonlinear problems is delegated to remaining three chapters. The first one among them is dealing with instability brought about by large displacements and rotations, the next one is dealing with inelasticity brought about by the nonlinear material behavior of the materials used for building a particular structure and the last one concludes with the message on synergy between nonlinear mechanics and stochastics, which can provide the results way beyond traditional engineering tools. More precisely, in Chap. 2 we present the simple truss model, which is used as the platform to explain the main ideas in constructing the model ingredients and developing the solution methods that are later used for all other structural engineering problems presented in this book. The truss model is first used to illustrate the traditional core knowledge of structural engineering in terms of so-called general theorems that served well the generations of structural engineers to carry out simple hand calculations of structure displacements and internal forces by using traditional force and displacement methods, and successfully design a large number of structures. The same model is then used to present a more modern approach based upon the finite element method. In particular, we here present the basic ingredient of such a solution procedure including several important aspects of finite element technology: isoparametric elements, the parent element, numerical integration, the patch test and finite element assembly procedures. In Chap. 3, we study the beam theory that has served for long time as the ’bread and butter’ for a structural engineer. So what can possibly be stated new about such theory? Well, we try to find (many) points that might be new with respect to many classical works. First, we present the beam theory as the reduced model that can be obtained by imposing the kinematic constraints to solid mechanics formulation. The model reduction is carried out with respect to the given goal, which is here chosen as model capabilities to remove the strain components producing the shear and the change of thickness. We also give another reduced model interpretation with respect to the classical benchmark of a cantilever beam under free-end force that is

6

1 Introduction

solved both by the solid mechanics model and by structural mechanics models for a beam in order to illustrate why Timoshenko’s model might be better than EulerBernoulli’s beam model when the shear deformation contribution plays an important role. We then illustrate several possible derivations of the classical Euler-Bernoulli beam theory. This is done either by constructing directly the stiffness matrix by a sequence of elementary transformations or by first constructing the flexibility matrix and inverting it in the reduced space (with imposed supports to eliminate rigid body modes) where such flexibility matrix can be constructed. We also illustrate how to carry on with further model refinements and reduction by accounting for local constraints, like releases or hinges, or global constraints, like beam length invariance. Finally, we show how to obtain the same beam by linearizing the nonlinear formulation of Reissner’s beam [322]. In this manner we can provide a consistent linearized formulation for shallow Timoshenko beam, as presented in our previous work [191]. The model of this kind allows giving a very clear interpretation for so-called shear and membrane locking phenomena as inability of representing the fundamental deformation mode of pure bending, as well as providing different non-conventional finite interpolation that can handle these locking phenomena. The lesson learned here is that any structural finite element, in order to perform well and in robust way, cannot be reduced to standard isoparametric interpolation, for such elements are not capable of dealing with locking phenomena. In Chap. 4 we study the plate models as 2D extension of the beam models. Contrary to beams, here we don’t start with the classical theory of the so-called Kirchhoff plates as 2D generalization of the Euler beam, often referred to as a ‘thin’ plate model that does not account for shear deformation. Such model is not the best basis for constructing the discrete approximation, for its variational formulation requires the continuity of second derivatives of transverse displacement field which is not easy to achieve for distorted elements. Hence, we present an alternative model in terms of the Reissner-Mindlin plate as 2D generalization of the Timoshenko beam, yet referred to as a ‘thick’ plate model that can take into account the shear deformation. The variational formulation now requires the inter-element continuity of only the first derivatives of transverse displacement and (independent) rotations, which is much easier to handle. We present several finite element discrete approximations for the the Reissner-Mindlin plate model, which share the same order of accuracy. We also show how to recover the discrete approximation of the Kirchhoff plate model in terms of the discrete Kirchhoff plate finite element. This element is using non-conventional finite element interpolations enforcing the assumed shear strain values equal to zero along each element edge, which results with (assumed) shear deformation equal to zero throughout element domain. The discrete Kirchhoff plate element is wellknown, since it is used in many commercial computer codes. What is less known is the consistent displacement interpolations in terms of third order polynomials that show convergence from above in mesh refinement. Even less known is another plate element with incompatible modes, which allows including shear deformation within the same order of finite element displacement discrete approximation leading to discrete Reissner-Mindlin plate element, first presented in a paper dedicated to Ed Wilson on the occasion of his retirement [162]. This was an appropriate gift, since the

1.2 Main Topics Outline

7

challenge of developing the shear deformable plate element that has the same order of approximation as discrete Kirchhoff element for thin plates, was first brought by Ed Wilson to the first author during his post-doc years dedicated to development of the first version of SAP2000 computer code. This element removed the long-standing deficiency of conventional finite element approximations for Reissner-Mindlin plate element that reduced the order of approximation versus discrete Kirchhoff plate element, and produced illogical results that including the shear deformation would reduce the value of nodal displacement (which happened due to lower order of discrete approximation for the same number of element nodes). The plate models presented in this chapter are further used to illustrate typical goals of adaptivity procedure in structural engineering that pertain to model selection or validation, as well as to discrete approximation quality or verification. In particular, having developed the same order of discrete approximation for both Reissner-Mindlin and Kirchhoff plate elements, we can present the model adaptivity criterion for validation procedure, which reduces to simple choice of either including or not the contribution of shear deformation. This needs the Reissner-Mindlin plate elements with non-conventional finite element approximations that are presented in the first part of this chapter. Subsequently, we also present the procedure for discrete approximation error estimate or verification, which allows selecting the sufficiently refined finite element mesh in order to provide the result accuracy within the chosen tolerance. This is again illustrated on perhaps the most popular plate element in commercial codes, the discrete Kirchhoff triangular plate element. The verification procedure, which compares the accuracy to achieve with different orders of approximation, exploits the fifth order polynomial discrete approximation by triangular plate element of Argyris [12], This element was not much used as a computational tool, for it requires imposing the second order displacement derivatives or curvature as the boundary condition, which is not suitable for direct solution problem. However, the same element is nearly perfect for the verification procedure given that the discretization error is estimated independently in each element, with no difficulty related to boundary conditions. In Chap. 5 we present solid and membrane elements with rotational degrees of freedom, which are rather non-conventional choices with respect to the standard solid mechanics elements. Namely, the classical continuum based solid elements used for modeling of structural components at the material scales, or at the level of the representative volume element (e.g., see [176]), do not include the rotation field in stress computations, for it is only related to rigid body motion. Thus, it is rather difficult to achieve compatible connections between classical solid mechanics models with no rotations and already presented structural mechanics models with rotations. Hence, we propose different approach providing the membrane and solid finite elements with rotations (yet referred to as drilling rotations) that can easily be combined with structural mechanics elements (beams and plates) within a compatible finite element assembly. This task is not merely reduced to constructing non-conventional finite element interpolations, but also needs a fresh start for theoretical formulation of such problem in terms of regularized variational principle.

8

1 Introduction

More precisely, we have to provide a generalized theoretical formulation of solid mechanics by assuming the rotation field as independent. This also implies the need to keep the stress tensor non-symmetric, with not enforcing the moment equilibrium equation in the strong form. The solid mechanics formulations of this kind have already been proposed by Reissner [321] well before the finite element procedures have become dominant tools of engineering practice; hence, the numerical solution has not been studied. Much latter, the experts on computational mechanics took interest in reviving this procedure, and came up with the most suitable finite element approximations to accompany this non-conventional solid mechanics formulation. In particular, the choice of non-conventional finite element formulation along with regularized variational formulation [222] proved to be an optimal combination of theoretical formulation and finite element discrete approximation that provides an excellent performance of membrane element with drilling rotations. Such element can be used to prepare large-scale models of complex structures where one has to combine solid and structural mechanics elements (e.g., in-filled frames, stiffened shells, frame on soil foundation etc.), and has become an indispensable part of finite element libraries for several well-known computer codes. We finally show how to combine the proposed membrane element with drilling rotations with plate element in order to obtain a shell element. Such shell an element is developed consistently within the framework of shallow shell discrete approximation, which is proved to be a very powerful tool to handle various locking phenomena that plague the classical shell theory discrete approximation that can outperform most of the well-known shell elements in the so-called obstacle coarse proposed for shell problems [274]. In preparing the transition to large displacement, in this chapter we also briefly present a solid element with independent rotational field in a more general context of large rotations by following the variational formulation of Fraeils de Veubeuke [124]. This introduces the finite strain measure of Biot, which can be written as a product of finite rotations and finite deformation gradient. We show that the consistent linearization of such formulation recovers the Reissner formulation, which serves as the starting point of discrete approximation for solid elements with rotational degrees of freedom. The finite element interpolation can be written as a straightforward 3D extension of the non-conventional interpolation already explained for membrane with drills or as an alternative version based on a set of incompatible modes [225]. In Chap. 6, we present all the developments related to geometrically nonlinear approach in structural analysis and corresponding instability problems that can arise within the framework of large displacements and rotations. The main interest is to quantify the risk of pre-mature failure with a disproportional increase of displacement, strain and stress due to a small increase of external load, which is the intrinsic definition of instability. In particular, we first present the consistent formulation of large displacement and large rotation problems for structural engineering. Two models used for detailed illustration are geometrically nonlinear truss and geometrically nonlinear beam. We then point out the difference between the linearized instability or buckling with respect to nonlinear instability. The former is characterized by small pre-buckling displacements with moderate rotations resulting typically with bifurcation equilibrium point, whereas the latter are associated with large displacements

1.2 Main Topics Outline

9

and rotations when reaching the critical equilibrium state that leads to a limit point. We present three different criteria for identifying the critical equilibrium point, with some of them that allows computing the critical load and others also providing the critical mode of instability. All of them are using the tangent stiffness matrix which is obtained by the consistent linearization of nonlinear equilibrium equations. In the second part of this chapter we elaborate upon very powerful computational approach to computing the solution to linearized instability of complex structures within the finite element framework. The key role is played by so-called von Karman strain measure, which is obtained by simplifying the Green-Lagrange finite strain measure [176] with the hypothesis that the displacements and strains (but not rotations) remain small before reaching the critical bifurcation point. The von Karman virtual strain is used to provide the weak form of linearized instability problem that is further used as the basis for constructing the finite element discrete approximation. Thus, any structural engineering model that can be constructed by finite element method can be brought consistently within this framework in order to solve linearized instability for any complex structure. This is certainly more general result than the standard analytic solution that one introduces when teaching Euler beam buckling problem, and the only requirement is to be able to compute the solution numerically by using the finite element method. The classical problems of Euler beam buckling are used to illustrate the accuracy of the proposed numerical solution. We also develop the corresponding numerical models for more complex structures built of plates and shells. In the final part we briefly illustrate another original concept in constructing the solution to linearized instability problem under coupled thermomechanics conditions. Here, one load provides a pre-stressing effect and another load is used to estimate the risk of failure due to instability. The motivation for this approach is brought about the sadly famous disaster of World Trade Center building collapse under fire caused by the airplane attack. This clarifies that it was not the airplane crash that brought the building down, since other than local damage the building was able to sustain this impact. It was in fact the combined action of heat transfer under fire that weakened the structural material and dead load that resulted in building collapse. This is illustrated in a simplified manner for truss structure representing a crude model of Eiffel tower. In Chap. 7, we deal with another class of nonlinear problems in structural engineering that is brought about by the inelastic constitutive behavior of structural material. The main interest is in quantifying the risk of pre-mature structure failure due to the accumulation of internal material damage that reduced the load-bearing capacity from (linear) elastic behavior. Many books study this problem at the material scale of representative volume element (e.g., [175, 176]), or even finer scales of microstructure characterizing material heterogeneities (e.g., [56, 58]). Here, we do start with fine scales at the level of material, but then provide the inelastic constitutive model in terms of stress resultants, which is much more suitable for keeping the computational efficiency of structural engineering models. This procedure is much alike the classical homogenization that spans from microstructure heterogeneities to homogenized solution at the level of representative volume element for a particular material, for those who are familiar with homogenization. The main difference is in the fact

10

1 Introduction

that the representative volume element of a structure does not offer a homogeneous strain or stress field, but rather the variation that is in agreement with the particular kinematic hypothesis for a chosen structure model. We first study the inelastic behavior of the Reissner-Mindlin plate model, seeking to illustrate different plasticity criteria characterizing the inelastic behavior of typical structural materials, such as reinforced concrete, steel or masonry. This brings very diverse plasticity criteria that can quantify either ductile behavior typical of metallic materials, described by von Mises criterion and quasi-brittle behavior characterizing concrete and masonry behavior, which is described by St. Venant criterion. However, in each of these criteria development we follow the same path by identifying the governing mechanism of inelastic behavior at material scale, and applying the corresponding kinematic hypotheses of Reissner-Mindlin plate model in order to define corresponding plasticity criterion expressed directly in terms of stress resultants, bending moments and shear forces. In any such case, the resulting plasticity criterion becomes more complex, often turning into a multi-surface plasticity criterion. Moreover, for one of the criteria we developed in the framework of fire-resistance studies of masonry structures, we again seek to illustrate the effect of combined loading requiring to solve the time-evolution of non-stationary heat transfer. In the final part of this chapter we tackle the combined effect of large deformation and inelastic constitutive behavior, that jointly can push the structure into pre-mature failure. This development is illustrated on the Reissner beam model, which can be used to describe the large displacements and rotations, already briefly presented in Chap. 3 as the starting point of consistent linearization to obtain a shallow Timoshenko beam. We present the von Mises plasticity model for metallic materials that can handle both hardening and softening phase, which requires a multiplicative decomposition of deformation gradient. This kind of model is shown to provide very efficient computational approach to computing the ultimate limit load of steel frames. In Chap. 8, we turn to current applications in multiscale modeling and probability computations. We provide only a brief presentation of the useful synergy between the nonlinear mechanics and stochastic that can be used not only to build the model of interest for dealing with real-life heterogeneous structures, but also to develop solution procedures that reach way beyond traditional engineering tools. More precisely, we first present multiscale approach and corresponding constitutive model or Reissner’s beam used for quasi-brittle materials, as a geometrically nonlinear extension of St. Venant plasticity criterion, which is able to handle large displacements and rotations in localized failure of quasi-brittle materials. The development of this kind was demonstrated with a particular application of such plasticity criterion to constructing a discrete model for cohesive failure of microstructure of geomaterials, which was needed to study the risk of scabbing under airplane impact [223, 228]. Hence, it is essential to be able to account not only for heterogeneity, but also to large displacements and rotations (but small strain) that are characteristic of such application. We also present the multiscale approach that can handle both size and scale effects. The latter can bring a probability-based explanation to why the dominant failure modes in a large structure are not necessarily the same as in the small structure

1.3 Further Studies Recommendations

11

built of the same heterogeneous material. Also, the same approach explains how to replace the Rayleigh damping with the materials-based dissipation and recover characteristic exponential decay of vibration amplitude at the structure level due to the probability-based description of material heterogeneity. The final illustration is the use of reduced model in terms of the Duffing oscillator that can give a complete overall description of instability phenomena for geometrically exact (heterogeneous) Euler beam by solving corresponding stochastic differential equation obtained by model reduction. More details than this brief introduction can be found in our recent works [214, 215]. The main message of this chapter is on useful synergy of nonlinear mechanics and stochastics. This comes as an alternative to probability computation and statistics, often referred to as ‘artificial intelligence’, that are presently used to build statistical models in many different domains. In Structural Engineering domain, we believe, it is mechanics that should provide the model and stochastics that should provide the estimate for the model applicability to particular computational goals.

1.3 Further Studies Recommendations The book was first and foremost written for students in Structural Engineering in the broad domain of applications coming from Civil, Mechanical or Aerospace engineering. However, it also targets practising structural engineers and users of different software products for computational structural mechanics, as well as those in teaching and research seeking to further enhance their understanding of the theoretical formulations of the structural engineering models used in computer codes. What is presented in this book is roughly the current state of the art of the finite element modeling of structures, with the main difficulties and their solutions presented to the readers. The prerequisites for studying the book are kept to a minimum, and the good part of the presented discussion ought to be accessible to the experts coming from other fields, who are seeking the necessary background for interacting with researchers in Structural Engineering. Several courses which I have been teaching at Technical University Compiègne (UTC) and at École Normale Supérieure (ENS) in Paris, one of the ‘Grandes Écoles’ of the French system of higher education, pertain to the subjects studied in this book. Among them are both courses taught to the last year of Master of Engineering students and doctoral students in Graduate Program ‘Mechanics and Technology’, with the book content fully covering the course entitled ‘Integrity and instability of structures under extreme loading conditions’, where different methods are presented for dealing with ultimate load computations, geometric and material instabilities. The material for this book was also used in various short courses I gave in a number of Graduate Programs at different universities in Europe (Univ. Innsbruck Austria, TU Braunschweig Germany, TU Tampere Finland, NTNU Trondheim Norway, Univ. Luxembourg, Univ. Ljubljana Slovenia, TU Budapest Hungary), Latin

12

1 Introduction

America (Univ. Sao Pualo Brazil, IPN Mexico) and Asia (KAIST Korea, TU Istanbul Turkey, IIT Bombay, India). For further studies of the ideas in computational structural mechanics presented in this book, one can consult a number of different books each specialized in a particular domain. For example, for further studies of the physical aspects, one can start with the books of Hill [148], Prager [316], Lemaitre and Chaboche [262], Lubliner [270], François et al. [125], Maugin [284], Krajcinovic [248], Bornert et al. [56] or Brancherie et al. [58]. The theoretical formulation of solid mechanics, can further be elaborated by studying some of the classical works of Truesdell and Toupin [375], Truesdell and Noll [374], Green and Zerna [138], Sokolnikoff [349], Germain [131], as well as some later books on the subject, such as Chadwick [73], Gurtin [142], Ogden [306], Ladevèze [251], Duvaut [115] or Ibrahimbegovic [176]. Several more specialized works, as those of Bazant and Cedolin [41], Argyris et al. [11], Nguyen [300], Rougée [325], as well as more mathematically oriented works, such as Duvaut and Lions [116], Arnold [16], Goldstein [133], Marsden and Hughes [278], Antman [8]. For more detailed studies of the finite element methods, one can consult the books of Zienkiewicz and Taylor [394], Hughes [153], Bathe [27], Batoz and Dhatt [37], Dhatt and Touzot [110], Strang and Fix [360], Brezzi and Fortin [65], Johnson [239], Ciarlet [78], Washizu [379], Owen and Hinton [309], Crisfield [98], Simo and Hughes [339], Belytschko et al. [44], Wriggers [387], Laursen [259] or books by Ibrahimbegovic [175–178, 182, 204, 228]. Finally, if needed to further clarify some of the ideas from applied mathematics regarding the different methods presented herein, one can further explore the books of Strang [359], Luenberger [271], Dennis and Schnabel [109], Golub and Van Loan [135] or Ciarlet [80]. One can certainly consult other books on Structural Engineering to get an alternative point of view and complementary aspects with respect to what is covered in this book. This list is too long to be fully reproduced here, but one can mention the books that are written in the same spirit as this book, but typically with a much larger scope for they mix the aspects of solid mechanics and structural mechanics. For example, one can consult the books of Bathe [27], Crisfield [97, 98] or Belytschko et al. [44], which are written for nonlinear problem specialists. Contrary to those books, each close to 1000 pages, we have made a deliberate choice to limit exposure to a ‘reasonable’ size book, by leaving out the most advanced topics to specialists who should be able to read our numerous papers on the subject of 3D structural engineering models [161, 165, 167, 169–174, 179, 186, 192, 199, 202, 203, 205, 207–209, 214, 215, 217, 220, 221, 287, 299].

1.4 Summary of Main Notations We first lay down the general rules for distinguishing different notations employed in this book. We use Latin alphabet letters to denote the scalar fields in a boundary value problem in a 1D context, which is a frequently used vehicle for introducing new ideas at the beginning of each chapter. In 2D and 3D continuum settings, the

1.4 Summary of Main Notations

13

same fields are denoted by tensors bold face Latin letters, with a lower case letters reserved for vectors and an upper case for second order tensors. The only exception might be to accommodate long tradition in Structural Engineering by denoting the stress resultant vector (axial and shear forces) and bending moment in beams, with bold-face capital letters N and M. This notation convention is also modified for Greek letters, where the long tradition is respected by denoting several second order tensors by lower case Greek letters, such as σ and ε for the stress and the strain tensors, respectively. Higher-order tensors (fortunately, we do not have to go higher than order three of four) are denoted by calligraphic letters. In order to avoid the risk of possible confusion. Occasionally we also state the tensor equation in index notation, which refers explicitly to tensor components. The Greek letters are used for index notation in 2D case, when developing the variational formulations for membranes, plates and shells, with indices varying in set {1, 2}. In 3D case, we use Latin letters for indices varying in set {1, 2, 3}. The index notation is also useful when it comes to computations, since the value of any tensor has to be obtained component by component. Alternatively, the computed results are stored in a matrix, exploiting the possibility to reduce the number of stored tensor components for any symmetric tensor and thus increase computational efficiency. We use either bold-face letters to denote matrices, or the sans serif fonts for this matrix notation if the bold-face is already used in the same section for tensors. We also try to enforce the convention where the lower case letters are used to denote the vectors (one column matrices) and upper case to denote matrices. The index of each vector or matrix needed for finite element computations is denoted with either small letters in element local coordinate system or capital letters in structure global coordinate system. One final word on notation. Given a very large family of possibly interesting models in Structural Engineering, it seems to be next to impossible to propose a ‘purified’ notation, which does not take into account long tradition in different domains. The hard lesson is learned in one such work [167], where an attempt to enforce a ‘rigid’ notation convention led us to abandon some of the traditional symbols which made reading quite difficult. We sincerely hope that the potential readers of this book will be able to get used to different notation conventions that we used in trying to reach the best compromise between tradition and clarity, while trying to avoid a huge proliferation of different symbols. Already, keeping in mind the need for providing the computational approach for proposed models and methods, we have eliminated completely the use of curvilinear coordinates [294, 374] or manifolds [1, 278], which are not needed for curvilinear structural elements, as shown in [167, 170, 173, 199].

Chapter 2

Truss Model: General Theorems and Methods of Force, Displacement and Finite Elements

Abstract We first present the simple truss model, as the platform to explain the main ideas in constructing the model ingredients and developing the solution methods that are later used for all other structural engineering problems presented in this book. The truss model is also used to illustrate the traditional core knowledge of structural engineering in terms of so-called general theorems that served well the generations of structural engineers to carry out simple hand calculations of structure displacements and internal forces by using traditional force and displacement methods. The same model is then used to present a more modern approach based upon the finite element method, including several important aspects of finite element technology.

2.1 Truss Model—Strong Form and Weak Form We start with a study of the simplest model of an elastic bar that can be represented by a reduced model of one-dimensional linear elasticity referred to as Truss, which is obtained by using the following kinematic assumptions: 1. applied loading, linear elastic mechanical and geometric properties are such that the displacements and deformations remain small, which allows using so-called linearized kinematics or small displacement gradient theory. 2. truss-bar deformable solid body is homogeneous in transverse direction, with applied loading and boundary conditions that ensure displacement field variability only along bar axis (Bernoulli’s plane section law; see Fig. 2.1). 3. loading increases very slowly in time, so that inertia effects can be neglected and one only studies static equilibrium. This simple problem of structural mechanics is quite representative of any problem we solve in this book. Namely, we only seek three fields: displacement, strain and stress. We need three equations to compute these fields, which are kinematics, equilibrium and constitutive equations. In order to ensure the solution uniqueness, we also need the corresponding boundary conditions, which turns the bar into the structure. Here, this implies placing at last a single support which blocks the only rigid body mode in 1D truss-bar of translation along the bar; see Fig. 2.1. We will © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Ibrahimbegovic and R.-A. Mejia-Nava, Structural Engineering, Lecture Notes in Applied and Computational Mechanics 100, https://doi.org/10.1007/978-3-031-23592-4_2

15

16

2 Truss Model: General Theorems and Methods of Force …

Fig. 2.1 Problem: 1D truss bar and applied loading, resulting with uniform stress distribution within cross-section

present in this section two possible formulations for such problem: the strong form and the weak form, and discuss how to solve them.

2.1.1 Strong or Differential Form: Analytic Solution First, we present the strong form of this problem that allows defining displacement, strain and stress fields in terms of the analytic solution. By judicious choice of Cartesian coordinates with x axis placed along the bar, we can simplify the problem for Truss reduced model by turning all mechanics fields (displacement, strain and stress) into corresponding scalar fields depending upon a single parameter ’x’. Namely, by placing the coordinate system as indicated in Fig. 2.1, we can write: u(x, y, z) ≡ u(x), (x, y, z) ≡ (x) and σ (x, y, z) ≡ σ (x). Thus, the first description of this problem can then be posed as The Strong Form for Truss-Bar Model • Given: solicitations: distributed loading b(x), imposed displacement u, ¯ imposed traction t¯ geometric properties: length l, section (assumed uniform) A(x) ≡ A mechanical property: elasticity modulus (assumed constant) E(x) ≡ E • Find: displacement u(x), strain ε(x), stress σ (x) • Such that: kinematics, equilibrium and constitutive equations are valid ∀x ∈ [0, l], imposing boundary conditions (see Fig. 2.2) What is common among all the problems in mechanics, which we clearly see in the simple problem of truss-bar, is the need to define the corresponding equations to solve and compute three fields: displacement, stress and strain. When applying the static load and limited to linear elasticity, all the fields are defined only as the functions of x for all values from zero to l, with the coordinate system placed at the left end of the bar (see Fig. 2.2). Remark: When compared against a model of 3D elasticity (e.g., see [176]), the trussbar model is capable of significantly reducing the number of non-zero components of displacement vector and deformations and stresses: tensors, which all allow simplified representation with respect to 3D case

2.1 Truss Model—Strong Form and Weak Form

17

Fig. 2.2 1D model of truss-bar: initial configuration, loading and imposed boundary conditions

⎧ ⎫ ⎡ ⎤ ⎡ ⎤ εx x 0 0 σx x 0 0 ⎨ ux ⎬ u = 0 ; ε = ⎣ 0 0 0 ⎦ ; σ = ⎣ 0 0 0 ⎦ ↔ u x = u, εx x = ε, σx x = σ ⎩ ⎭ 0 0 00 0 00 (2.1) We continue further with the 1D truss-bar model. For computing three fields of displacement, strain and stress, we need three general equations: 1. Kinematics: providing the link between deformation ε(x) and displacement u(x), which is obtained by assuming infinitesimal deformation, based upon the linear theory with small displacement gradients du(x) ε(x) = dx

; with

du(x) d x 0. Hence, given that the virtual displacements are small displacements, we can finally present the total potential energy functional for any ’perturbed’ configuration (produced by superimposing the virtual displacement field on a given deformed configuration) in terms of the Taylor series, to conclude that the total potential energy will take a higher value ... Π (u i + δu i ) = Π (u i ) + δΠ (u i ) + δ 2 Π (u i ) +        =0

>0

(2.45)

≈0

This result can be interpreted as follows. In the structure is slightly perturbed from its equilibrium configuration (by applying a variation δu i ), it will have the ability to return to its deformed configuration spontaneously once this perturbation is gone. This implies that such equilibrium state is stable, under any kinematically admissible small perturbation. The stability of equilibrium state is a stronger result than the principle of virtual work can give. In later  we will study the structures which  2 chapters, Π < 0 and structures in the critical equilibmight have unstable behavior with δ   rium state with δ 2 Π = 0 where a small perturbation might lead to a disproportional change of structure response. We further quickly present the application of this theorem to 1D truss-bar structure, where we can carry out simplified computations given a single non-zero component of displacement, strain or stress. In particular, the total potential energy can now be written as l Π (u) = Π (u) + Π int

ext

(u) := 0

1 σ (x) ε(x) Ad x − 2   Eε(x)

du(x) dx

l

u(x) b(x) d x − u(l)t¯

0

(2.46) where we indicated that constitutive and kinematic equations apply point-wise with σ (x) = Eε(x) and ε(x) = du(x)/d x. Hence, Π is a functional, or a function of u(x), which is a function of x. We illustrate in Fig. 2.7 the strain energy for 1D case W (ε) = 21 εEε as the area of the triangle underneath the line representing the linear elasticity stress-strain response.

30

2 Truss Model: General Theorems and Methods of Force …

Fig. 2.7 Illustration of strain energy density W (ε) = 21 εEε and its complementary value W (σ ) = 21 σ E −1 σ for 1D case

The first variation of the internal and external energy can easily be computed to obtain l δΠ (u(x); δu(x)) :=

l δε(x) σ (x) Ad x −

0

δu(x) b(x) Ad x − δu(l)t¯ = 0

0

(2.47) and the second variation that always takes a positive value (given that δu(x) = cst. or else the boundary condition δu(0) would only allow zero value everywhere) l δ Π (δu(x)) :=

δu(x) E δu(x) Ad x > 0

2

(2.48)

0

Hence, we can again easily confirm the minimum of the total potential energy, with Π (u(x) + δu(x)) > Π (u(x)); ∀δu(x) which is kinematically admissible. Note on duality and complementary potential energy We briefly illustrate the notion of duality in the framework of 1D truss-bar model. Duality is another important notion that implies two opposite or complementary parts by which they integrate into a whole. We illustrate in Fig. 2.7 the complementary strain energy W ∗ (σ ), which when combined with strain energy amounts to the product of stress and strain σ ε. W (ε) + W ∗ (σ ) = σ ε ⇒ W ∗ (σ ) = σ ε − W (ε)

(2.49)

The notion of duality requires to clearly define the primal variables used to specify the structure equilibrium state (for example, strain ε in strain energy density W (ε)), versus the dual variable that can (easily) be recovered once the primal variables are known (for example, stress σ = Eε). The roles of primal and dual variables can be exchanged through Legendre transformation (e.g., see [176]). The dual approach allows developing another well-known variational principle in structural mechanics known as the Principle of Stationary Total Complementary Energy. For small deformation problems, this principle is only expressed in terms of stress variables and, most notably, under the subsidiary conditions of equilibrium,

2.2 General Theorems of Structural Mechanics on Truss Model

31

it can be regarded as a maximum principle. Furthermore, it can be obtained from the Principle of Minimum Total Potential Energy by means of a simple Legendre transformation, which requires the inversion of the constitutive relations to obtain ε = dW ∗ (σ )/dσ = E −1 σ . In addition, the maximum and minimum values of the respective functionals are the same. In other words, both principles play a dual role. It was only in 1953 that this dual role was mathematically demonstrated by the renowned mathematicians Courant and Hilbert. For 3D case, we can write the corresponding complementary energy functional as Π ∗ (σi j ) :=



W ∗ (σi j ) d V −

V

 n i σi j u¯ j d A

(2.50)

Au

where we indicate that complementary energy can be written as functions of stress only, with boundary term defined by the product n i σi j that represents the surface tractions t j exerted on Au . The corresponding variational equation results with the principle of complementary virtual work   δΠ σi j := ∗



 δσi j εi j d V −

V

δσi j n j u¯ j d A = 0

(2.51)

Au

2.2.4 Applied General Theorems We next present several brief developments of general theorems of structural mechanics which are easily deduced from fundamental theorems of energy and work, so we refer to them as applied. The typical use of these theorems is to provide quick computational results, especially for ’hand’ calculation. All these application theorems rely upon the linear elastic behavior of material described by Hooke’s law (with strain energy potential for hyperelastic case).

2.2.4.1

Clapeyron’s Theorem

For the case with imposed zero displacement at the Dirichlet boundary, we can recast PVW or PVD in (2.21) by choosing for virtual kinematics, the real deformations and displacements δεi j ≡ εi j and δu i ≡ u i . For such a case, the principle of virtual work results in the following equality 

 εi j σi j d V = V

 u i bi d V +

V

Aσ

u i t¯i d A

(2.52)

32

2 Truss Model: General Theorems and Methods of Force …

By comparing this result against the virtual work principle in (2.21), we see that we can express the strain energy using the external quantities in the following format Π int =

1 2

 u i bi d V +

1 2

V

 u i ti d A

(2.53)

A

which is known as Clapeyron’s theorem stating that the deformation energy is equal to the half-sum of the products of the external forces and corresponding displacements. Clapeyron’s theorems are the basis of the study of convergence and error in the finite element method. In 1D case for the truss-bar model, this theorem reduces to Π

int

1 = 2

l

1 u(x)b(x) d x + u(l) t¯ 2

(2.54)

0

which confirms the stored energy in a bar is equal to a half of the work of external forces.

2.2.4.2

Betti’s Reciprocity Theorem

Yet another theorem which is useful for computational purposes is Betti’s reciprocity theorem. Let (bi1 , ti1 ) and (bi2 , ti2 ) be two force systems applied successively to the same structure, producing respectively the displacement fields u i1 and u i2 . We recall that for each of these force systems Clapeyron’s theorem would apply, in the case we use one or the other system to replace the virtual field. By using Clapeyron’s theorem, we can then write for the first case when load 2 is taken to produce a virtual field   1 1 int u i2 bi1 d V + u i2 ti1 d A (2.55) Π = 2 2 V

A

and then for the second case when load 1 is taken to produce virtual field Π

int

1 = 2



1 u i1 bi2 d V + 2

V

 u i1 ti2 d A

(2.56)

A

Given the existence of strain energy potential, which implies that the left hand side remains the same in the last two equations, we can easily establish the following equality     u i1 bi2 d V + V

u i1 ti2 d A = A

u i2 bi1 d V + V

u i2 ti1 d A A

(2.57)

2.2 General Theorems of Structural Mechanics on Truss Model

33

This result is known as Betti’s reciprocity theorem (1872) stating that the sum of the products of the forces of the first load case by the associated displacements of the second is equal to the sum of the products of the forces of the second by the associated displacements of the first. !n 1D case for truss-bar model, Betti’s reciprocity theorem can be written as l

u 1 (x)b2 (x) d x + u 1 (l)t¯2 =

0

l

u 2 (x)b1 (x) d x + u 2 (l)t¯1

(2.58)

0

We often use this theorem in 1D case for quick computation of the given displacement at a chosen point of the truss-bar; for example, the end displacement at l due to the first applied load b1 (x) and t¯1 can be computed by selecting a unit force at the same end. Namely, choosing b2 (x) ≡ 0 and t¯2 = 1, results with u 2 (x) = ExA and u 2 (l) = ElA . We can then write the corresponding result that follows from (2.58) as u 1 (l) = =

l 0 l 0

u 2 (x)b1 (x) d x + u 2 (l)t¯1 (2.59) x b (x) d x EA 1

+

l ¯ t EA 1

which is the exact result we would obtain by integrating the weak form of the same problem. This observation follows from 1D case superconvergence property which confirms that linear approximation of weighting function delivers exact displacements at both ends of the bar (e.g., see [176]).

2.2.4.3

Maxwell’s Theorem

This is a special case of Betti’s reciprocity theorem where specific point values of displacement field for two different load cases are compared. To that end, a single concentrated unit force is applied in each load case; this can be done for any particular point and not only end points, by applying forces f 1 = 1 and f 2 = 1 e.g., see Fig. 2.8. From (2.57), from where we can easily conclude that u 1 | f2 =1 = u 2 | f1 =1

(2.60)

stating that the displacement at point i = 1 due to a unit force applied at point 2 is equal to the displacement at point i = 2 due to an applied unit force at point 1. This can be confirmed by computing the corresponding displacement fields due to the unit forces f 1 = 1 and f 2 = 1 in Fig. 2.8 to obtain the value in (2.60) equal to 2El A

34

2 Truss Model: General Theorems and Methods of Force …

Fig. 2.8 Imposed unit or zero forces f 1 and f 2 for two different load cases, to confirm Maxwell’s theorem

 u 1 (x) = u 2 (x) =

x ; x ∈ [0, l/2] EA l ; x ∈ [l/2, l] 2E A

x EA

(2.61)

; x ∈ [0, l]

Maxwell’s theorem, with its concentrated forces, is more reserved for structures than for solids. It is essentially used to demonstrate that the system of equations for the  force method is symmetric u i | f j = u j | fi . Similarly, the system for the displacement method is also shown to be symmetric.

2.2.4.4

Kirchhoff’s Theorem

Kirchhoff’s uniqueness theorem (1859): the solution of a (static) problem of (linear) elasticity is unique. This is easy to confirm by using the strain energy potential to provide the solution bounds in the case where the elastic properties are varying with E(x). The latter calls for homogenization (e.g. [56, 176]) with either replacing heterogeneous material by equivalent homogeneous material, or providing upper and lower bounds for strain energy. The uniqueness is granted by the total potential energy principle stating that any perturbed configuration can easily be handled with small perturbation restricted to small displacements and strains, and recovering the same deformed configuration once the perturbation is removed. Hence, we can provide the guarantee that the deformed configuration is in a stable equilibrium state. Reassuring.

2.3 Castigliano’s Theorems, Force and Displacement Methods 2.3.1 Castigliano’s Theorems—Stiffness and Flexibility 2.3.1.1

Castigliano’s First Theorem—Stiffness Matrix

This theorem plays the same role as the constitutive equation, but brought to the structure level, which allows defining so-called stiffness matrix. Namely, we choose a finite number of points to quantify the structure response. One often refers to this as a discrete approximation, which will further be elaborated upon later in this chapter. We will illustrate this theorem for a simple truss-bar presented in Fig. 2.9,

2.3 Castigliano’s Theorems, Force and Displacement Methods

35

Fig. 2.9 Imposed displacement components u 1 = 0 and u 2 = 1 for computing K 22

with respect to the end displacements u 1 and u 2 . We will assume that such a structure response can be presented with the displacement field in terms of a linear polynomial constructed from these end displacement values u(x) = (1 − x/l)u 1 + (x/l)u 2

(2.62)

For such displacement variation, we can compute the internal potential energy from (2.46) to obtain l Π

int

:= 0

1 du(x) du(x) 1 u2 − u1 u2 − u1 E Ad x = EA 2  dx  dx 2 l l

(2.63)

(u 2 −u 1 )/l

We have already explained the duality of the stress field with respect to strain in hyperelastic material, with a strain energy density that plays the role of the potential to obtain σ = dWdε(ε) . The first Castigliano theorem generalizes the same result to the structural level. In this particular case, we will thus define internal forces as partial derivatives of the internal energy with respect to chosen displacement components, which can be written as f 1 := f 2 :=

∂Π int ∂u 1 ∂Π int ∂u 2

= − ElA (u 2 − u 1 ) = ElA (u 2 − u 1 )

(2.64)

We can further restate the same result in matrix notation, by introducing the nodal displacements vector u = [u 1 , u 2 ]T and corresponding internal force vector f = [ f 1 , f 2 ]T . The Internal energy can then be written as a quadratic form in terms of this displacement vector, and provide internal forces as the potential  EA EA  ∂Π 1 T − l int l = Ku ; Π = u Ku ; K = f := − ElA ElA ∂u 2

(2.65)

where K is so-called stiffness matrix. The components of the stiffness matrix, K = [K ab ], with 1 ≤ a, b ≤ 2 will have clear physical interpretation as internal forces for unit value of corresponding displacement. Namely, K ab is the force that should be applied at node a in order to produce unit displacement at node b, while keeping all other displacements but the one at node b equal to zero. For clarity, in Fig. 2.9 we illustrate the imposed displacements for computing stiffness matrix component K 22

36

2 Truss Model: General Theorems and Methods of Force …

when applying at node 2 the force t¯ to produce unit displacement at the same node u 2 = 1, which results with given: u(0) ≡ u 1 = 0 ⇒ u(l) = ElA t¯ impose: u(l) ≡ u 2 = 1 ⇒ t¯ = K 22 =

EA l

(2.66)

which is precisely the result stated in the stiffness matrix above. The same result is obtained for stiffness component K 11 = ElA . Furthermore, since each column b of the stiffness matrix represents the (self-equilibrated) internal forces for unit displacement at node b and all other zero, it is easy to recover that K 12 = K 21 = − ElA . 2.3.1.2

Castigliano’s Second Theorem—Flexibility Matrix

The second part of Castigliano’s theorem relies upon the complementary energy to define the structure flexibility matrix. It has the same limitation as the complementary energy principles in that it requires eliminating the rigid body modes in order to be able to compute such a flexibility matrix. The structure in Fig. 2.9 can be used to illustrate this approach in the simplest possible setting. Namely, with only one end displacement free, to be denoted as u(l) = u, we can appeal to the result in (2.66) to easily obtain the connection between this displacement and the corresponding internal force N as u = ENlA . The complementary internal energy of the truss-bar can be written as a quadratic form in free-displacement, component u, which can be written as Π ∗,int := 21 u F11 u l = 21 NA E −1 NA Ad x 0 (2.67) l 1 N = 2N EA  F11

This allows us tp define the flexibility matrix, here with one single component, F11 = ElA . The complementary energy can be written as quadratic form in terms of internal axial force N . The second theorem of Castigliano then states that the partial derivative of the complementary energy with respect to a force applied at a particular node is equal to the displacement at that node in the direction of the applied load. This can be written as u :=

l ∂Π ∗,int = N ∂N EA 

(2.68)

F11

We note in passing the duality between flexibility and stiffness matrices. Namely, from the last expression, we obtain the stiffness matrix in reduced deformation space with a single support point removing the rigid body mode as the inverse of the

2.3 Castigliano’s Theorems, Force and Displacement Methods

37

flexibility matrix above, which leads to N :=

EA ∂Π int = u ∂u l 

(2.69)

d K 11

d where K 11 is the single component of the stiffness matrix in deformation space. Starting from this result, one can easily recover the stiffness matrix in full space defined in (2.65). Namely, by using equilibrium equations to connect internal force N and nodal forces f 1 and f 2 , as well as kinematic equation to connect displacement u in deformation space with nodal displacements u 1 and u 2 , we can obtain



f1 f2





         −1 f1 −1 E A u u1 = ⇒ = [−1 1] 1 N ; u = [−1 1] u2 f2 u2 1 1 l    K

(2.70) Finally, in closing this section, we would like to revisit example shown in Fig. 2.8 to give the flexibility matrix computed when two load cases with unit forces are applied on a statically determined beam, one in its middle and another at its end. We leave it to the readers to use the previous developments to confirm the result state below for the flexibility matrix that can be written as u = Ff ; u =

 l   u1 ; F = 2El A u2 2E A

l 2E A l EA

 ;f=

  f1 f2

(2.71)

One can note the symmetry of the flexibility matrix with F12 = F21 which follows from Maxwell’s theorem.

2.3.2 Force and Displacement Methods In this section, we briefly discuss the basic solution methods for structural engineering referred to Force and Displacement Methods, where one exploits, respectively, the flexibility and stiffness matrices derived previously by Castigliano’s theorems. The illustration is made with the presented truss-bar model, but the presented procedures can easily be generalized to any structural mechanics models, by using the corresponding flexibility and stiffness matrices.

2.3.2.1

Force Method

The simplest problem one can solve with the truss-bar model is the so-called statically determined problem with the minimum number of supports (here single support for

38

2 Truss Model: General Theorems and Methods of Force …

Fig. 2.10 Statically determined truss-bar loaded with distributed load and traction force (left) and a cut-out segment of the bar to access the internal force and stress distribution (right)

1D model), which eliminates the rigid body motion. One can isolate a segment of the bar from x and l, and easily obtain the stress field as already obtained by integration of strong form as defined in (2.11). Once the stress field is defined, we can further use the linear elastic constitutive equation to define the corresponding strain field, and then integrate kinematics equation to provide the displacement field. Given such ability to directly solve for stress, strain and displacement fields for the statically determined structure, we would like to build upon this kind of solution and reduce other problems with number of supports larger than minimum, which we refer to as statically undetermined structure. This is the approach followed by the Force Method used for finding the solution to displacement, strain and stress field of statically undetermined structure. One such statically determined structure is illustrated in Fig. 2.10, loaded with a uniformly distributed load b(x) = b and traction force at right (free) end t¯. As already stated by the label, for a statically determined structure we can access directly the internal force and stress field distribution by only using the static equilibrium equation. Namely, as illustrated in Fig. 2.10, from the equilibrium equation posed for the isolated segment spanning from x to the free end of the bar at l, we can write: N (x) = b(l − x) + t¯ ⇒ σ (x) :=

b(l − x) t¯ N (x) = + A A A

(2.72)

From this stress field, we can further compute the corresponding strain field by exploiting the linear elastic constitutive equation to obtain: ε(x) :=

σ (x) b(l − x) t¯ = + E EA EA

(2.73)

Finally, we can integrate the kinematics equation for this strain field to obtain the corresponding displacement field, given zero imposed displacement at the left (fixed) end of the bar u(0) = 0 x u(x) :=

ε(ξ )dξ = 0

b t¯ (lx − x 2 /2) + x + u(0)  EA EA 0

(2.74)

2.3 Castigliano’s Theorems, Force and Displacement Methods

39

Fig. 2.11 Statically undetermined truss-bar loaded with distributed load and traction force (left) and corresponding statically determined structure obtained by removing right support and replacing it with unknown force X 1 (right)

We note that the last expression allows computing the free-end displacement equal to t¯l bl 2 + (2.75) u(l) = 2E A EA We note in passing that we could easily obtain the reaction of the single support of statically determined structure from static equilibrium equation for the whole bar to obtain r := N (0) = bl + t¯ (2.76) We now consider statically undetermined structure in Fig. 2.11, which is created by adding another support at the right end of the statically determined truss-bar we have just studied. The static equilibrium equation no longer allows computing the supports’ reactions, nor to have direct access to stress field distribution. Thus, we use the Force Method to solve this problem, by first identifying the corresponding statically determined structure and adding an unknown force X 1 to such structure that should allow us to reproduce the same displacement, strain and stress fields as for the statically undetermined structure. With such an approach, we can then use the computational results for statically determined structure. Hence, it is logical to choose again the corresponding statically determined structure we have just studied, as illustrated in Fig. 2.11. We note in passing that the choice for statically determined structure is not unique, since we could have removed the support at the left end; however, the end results are the same. We can further use the superposition principle and compute the total displacement of the chosen statically determined structure in Fig. 2.11 under combined action of two loads, the first with uniformly distributed load b and traction t¯ denoted as u sd (x)|b,t¯ and the second the unknown force X 1 denoted as u sd (x)| X 1 , which allows recovering the displacement of the statically undetermined structure under the same external loads denoted as u su (x)|b,t¯. This can formally be written as u su (x)|b,t¯ := u sd (x)|b,t¯ + u sd (x)| X 1 =

X1 b t¯ (lx − x 2 /2) + x+ x (2.77) E A E A E A     

Of particular interest is the displacement at the free end of the statically determined structure, for which we know that it corresponds to imposed (e.g., zero) displacement of the support point of the statically undetermined structure. This can be used as the

40

2 Truss Model: General Theorems and Methods of Force …

condition for computing the correct value of unknown force X 1 . We can write this condition in terms of flexibility matrix resulting with l bl 2 bl t¯l X1 + + = 0 ⇒ X¯ 1 = − − t¯ EA 2E A EA 2 (2.78) We can finally replace the computed value of force X 1 into equation (2.77) to recover the final form of displacement field of statically undetermined structure F11 X 1 + u sd (l)|b,t¯ = u¯ su ⇔

u su (x)|b,t¯ := u sd (x)|b,t¯ + u sd (x)| X¯ 1 =

b bl (lx − x 2 /2) − x EA 2E A

(2.79)

The Force Method has been demonstrated here in the simplest possible case with one unknown imaginary force X 1 that has to be imposed when releasing one support in order to enforce the known displacement value from statically undetermined structure u¯ su . We can easily generalize this result to a more complex case where more than one such force should be used to release the sufficient number of supports in order to recover the corresponding statically determined structure. This can be written in terms of corresponding flexibility matrix F and the computed values of displacements at released support points on statically determined structure u¯ sd in order to provide the system for computing all unknown forces X = [X 1 , ..., X m ]T ; we can write: FX + u¯ sd = u¯ su ⇒ X = F−1 (u¯ su − u¯ sd )

2.3.2.2

(2.80)

Displacement Method

In this section, we briefly discuss the Displacement Method for structural engineering, where one now exploits the stiffness matrices derived previously by the first Castigliano’s theorem. We will briefly illustrate the main steps for the simplest problem of statically determined truss-bar with a single support, for a slightly more general case where the support is moved by known displacement u(0) = u. ¯ We note that such support displacement for 1D model will introduce the rigid body motion where the whole bar is also moved by such displacement. One can go back to integrating the strong form as defined in (2.11) to obtain the corresponding solution for the displacement field with x ε(ξ )dξ =

u(x) :=

b t¯ (lx − x 2 /2) + x + u¯ EA EA

(2.81)

0

We note in passing that the strain and stress fields will not change from the previously computed values given above. If we further go to discrete approximation where we only keep the endpoints of the truss-bar, and assume that the displacement field is interpolated as linear polynomial as indicated in (2.62), we can appeal to the principle

2.3 Castigliano’s Theorems, Force and Displacement Methods

41

of virtual work and write the corresponding set of equations that will express the static equilibrium as δu 1 ( f 1int − f 1ext ) = 0 (2.82) δu 2 ( f 2int − f 2ext ) = 0 where we can use already computed internal forces in the previous section, but with replacing u 1 with imposed support displacement u¯ to get ∂Π int ∂u 1 ∂Π int ∂u 2

f 1int := f 2int :=

= − ElA (u 2 − u) ¯ EA = l (u 2 − u) ¯

(2.83)

and where the external forces follow as the result of external virtual work computation with virtual displacement variation described by a linear polynomial the same as already used in (2.62) for the real displacement field: l f 1ext := (1 − x/l)b d x = 0

f 2ext

bl 2

(2.84)

l

:= (x/l)b d x + (l/l)t¯ = 0

bl 2

+ t¯

For the present case of linear elastic behavior of truss-bar, the result in (2.82) can be replaced by a system of equations featuring stiffness matrix K, displacement vector u and external force vector fext , which can be written as  Ku = f

ext

; K=

EA l − ElA

− ElA EA l





u1 ;u= u2

 ;f

ext

=



 bl 2 bl 2

+ t¯

(2.85)

We note that such a system with a singular matrix cannot be solved to provide a unique solution. However, the system can be reduced to a single equation given that the kinematically admissible virtual displacement field imposes that δu 1 ≡ δu(0) = 0, which reduces the problem to a reduced format with only the second equation that can easily be solved as bl 2 bl t¯l EA + t¯ ⇒ u 2 = u¯ + + u2 = l E A 2E A 2    K 22

(2.86)

f 2ext

We note that the displacements u 1 and u 2 at both ends of the bar are computed exactly, with the same end results as the one obtained by integrating the strong form, despite the fact that we used linear displacement variation in (2.62). This property holds for 1D problems and is often referred to as superconvergence (e.g., see [176]). The superconvergence property also applies in 1D case for strain and stress computations, only not at the end of the bar, but rather in the middle with

42

2 Truss Model: General Theorems and Methods of Force …

ε(l/2) = 2EblA + Et¯A σ (l/2) = 2blA + At¯

(2.87)

Having solved the reduced system for free nodal displacements, we can now proceed to compute the nodal reactions at nodes where displacements are imposed. In this particular case with non-zero imposed displacement at Dirichlet boundary u, ¯ we can not go back and release the support point with an imaginary displacement δu 1 which moves the complete truss-bar with an imposed rigid body motion and allows for recovering from the first equation in (2.82) the corresponding value of reaction r1     bl bl + + t¯ ⇒ r1 = bl + t¯ 0 = δu 1 −r1 + ( f 1int − f 1ext ) = δu 1 −r1 +  2 2   ∀  =0

(2.88) We now turn to the statically undetermined structure shown in Fig. 2.11 and construct the solution by using the Displacement Method. The equilibrium equations in (2.82) constructed from the principle of virtual work are still valid, as well as the external force vector written in (2.84). The only difference this time is that both end displacements are imposed (for example, non-zero values, further denoted as u¯ 1 and u¯ 2 , respectively). Hence, there is no need to solve the set of equations in (2.82) with known values of u¯ 1 and u¯ 2 , since internal force vector can directly be computed as f 1int := f 2int :=

∂Π int ∂u 1 ∂Π int ∂u 2

= − ElA (u¯ 2 − u¯ 1 ) = ElA (u¯ 2 − u¯ 1 )

(2.89)

Hence, it only remains to recover the corresponding reaction forces; this can be done by choosing appropriate virtual displacements to move one support at time with δu 1 and δu 2 , respectively. This can be written according to:   0 = δu 1 −r1 + ( f 1int − f 1ext ) = ⇒ r1 =     ∀ =0   0 = δu 2 −r2 + ( f 2int − f 2ext ) = ⇒ r2 =     ∀

bl 2

+

EA (u¯ 2 l

bl 2

+ t¯ −

− u¯ 1 )

EA (u¯ 2 l

− u¯ 1 )

(2.90)

=0

We note in closing this section that the displacement method proved to be more suitable for computer implementation, for it provided a unified approach known as the Finite Element Method which is discussed next.

2.4 Finite Element Method Implementation for Truss Model

43

2.4 Finite Element Method Implementation for Truss Model In many works, the finite element method is described merely as a convenient manner for constructing a discrete approximation for the solution of a boundary value problem, quite similar to the pioneering methods of Galerkin and Ritz. The important difference between the finite element method and these two historical predecessors, which is perhaps the main reason for the impressive success that the method has achieved, is in its ability to easily handle rather complex domains. Namely, unlike the methods of Ritz and Galerkin, the finite element method does not need the shape functions that are constructed globally within the domain, but only the element-based shape functions with non-zero values only within the element sub-domains connected to a particular node. With the finite element strategy for patching the discrete approximation from such element-wise contributions, we gain both simplicity and efficiency. Namely, the computation of integrals for stiffness matrix and equivalent nodal load vector can be performed element-wise, and thus limited to only those elements for which a particular shape function takes non-zero values. Moreover, the computations can be standardized for all the elements of the same kind, by referring them to their common parent element, and be carried out easily by means of numerical integration. The element-based approach to integral computations requires a special procedure to account for the contributions of different elements toward the global set of algebraic equations, which is known as the finite element assembly. All those aspects, which constitute the basis of a typical finite element computer program, are presented in detail in this section. Remark on history of Finite Element Method: The potential of the weak form for solving the complex problems in mechanics by local-support interpolation functions was already recognized in 1941 by R. Courant, which is often credited with the start of the era of the finite element method. Although the real application developments really started with the finite element pioneers coming form engineering community, such as R.W. Clough at Berkeley, who essentially reinvented the finite element method as a natural extension of matrix structural analysis and published the first work in 1956, as well as simultaneous effort by J. Argyris at the Imperial College in London, which independently led to another successful introduction of the method. Further important developments have continued with the work of O.C. Zienkiewicz group, with many fundamental contributions by B.M. Irons on isoparametric elements, numerical integration and patch test that became standard tools of many finite element computer codes.

2.4.1 Local or Elementary Description In order to construct the discrete approximation for the displacement field, one can use the global description by defining the global shape functions over the whole domain

44

2 Truss Model: General Theorems and Methods of Force …

u h (x)|Ω =

n node

Na (x)da

(2.91)

a=1

where any shape function Na (x), a = 1, 2, . . . , n node is a polynomial that takes unit value at node ’a’ and zero values at all other nodes. This implies that the parameters da are in fact nodal displacements. Such an approximation and a similar approximation for the virtual displacement field are then replaced into the weak form in (2.15), if we use Galerkin’s method, or into the total energy functional in (2.42), if we use the Ritz method. With the finite element method, we will replace the global description in (2.91) with a local or element-wise description. The latter implies first that we will account only for a portion of any global shape function Na (x), a = 1, 2, . . . , n node which corresponds to one element domain Ω e . That portion of the shape function will be denoted by Nae (·) (2.92) Nae (x) ≡ Na (x)|Ω e Moreover, in the local description we will introduce the change of coordinates, replacing ’x’ with the so-called natural coordinate ’ξ ’, which should center the element domain and reduce it to the fixed length (basically, changing the metric); see Table 2.1 The results in Table 2.1 provide the discrete approximations for real and virtual displacement fields in local description, which account for all the nodes of a particular element. This kind of approximation allows in return rewriting the discrete approximation of the weak form or Galerkin’s equation in (G1) by summing over all the elements in the inner loop. G(u h ; w h ) :=

elem

n

a

⇒

b

e wa (K ab db − f aext,e ) = 0

e=1

K ab db = f aext ; ∀wa = 0 (G1e)

b

Table 2.1 Global and local descriptions for 2-node truss-bar element domain # Ingredient Global description Local description (1) (2) (3) (4) (5) (6)

Domain Nodes Degrees of freedom Shape functions Real displacement field Virtual displacement field

[xa , xa+1 ] {a, a + 1} {da , da+1 } {Na (x), Na+1 (x)} u h (x) = Na (x) da +Na+1 (x)da+1 w h (x) = Na (x) wa +Na+1 (x)wa+1

[ξ1 , ξ2 ] {1, 2} {d1e , d2e } {N1e (ξ ), N2e (ξ )} u h (ξ ) = N1e (ξ )d1e +N2e (ξ )d2e u h (ξ ) = N1e (ξ )w1e +N2e (ξ )w2e

2.4 Finite Element Method Implementation for Truss Model

45

We note in passing that the product between the stiffness matrix and the nodal values of displacements can be redefined as internal force vector even for linear problems,  f aint,e = b K ab db ; thus, we can rewrite the last equation as G(u h ; w h ) :=

elem

n

a

⇒

b

wa ( fˆaint,e (db ) − f aext,e ) = 0

e=1

fˆaint,e (db )

= f aext ; ∀wa = 0 (G2e)

Note that for nonlinear problems, which we will also study in this book, the internal force vector is a nonlinear function of nodal displacements. The finite element method implies that the integrals defining the stiffness matrix and equivalent external load in (G1e) above, are computed by splitting the integration domain Ω into a number of element sub-domains Ω e to be integrated independently. Thus, as the result one obtains the element stiffness matrix Ke and equivalent nodal force vector fe , which will be assembled within the global arrays in order to provide the set of global equilibrium equations Kd = f to be solved for unknown nodal displacements d. This is carried out by the finite element assembly procedure, as explained at the end of this section. In principle, each element sub-domain Ω e can be different from others, even if they all belong to the same element type (for example, a 2-node truss-bar element). In order to avoid this kind of proliferation of different cases, we further standardize the computational task of computing element arrays by introducing the parent element as the (same) image of any real element corresponding to an interval of natural coordinate, ξ ∈ [ξ1 , ξ2 ]. We always choose the same interval for the truss-bar 2-node parent element with ξ1 = −1 and ξ2 = +1, which is imposed by requirements of numerical integration procedure used for element integrals computation as elaborated upon subsequently. Such change of coordinates between the real and the parent element has to be an affine transformation for a 2-node truss-bar element, which is represented by a linear polynomial x(ξ ) = c1 + c2 ξ

(2.93)

The coefficients c1 and c2 can be obtained by equating the values at the ends of the interval; for a real element placed between nodes ’a’ and ’a + 1’, we write x(−1) = xa x(+1) = xa+1



 =⇒

c1 = c2 =

xa +xa+1 2 e

le 2

; l = xa+1 − xa

(2.94)

Having defined the coordinate transformation, we can write the local description of the element shape function in (2.92) in terms of natural coordinate ξ according to Nae (ξ )

1 := Na (x(ξ )) = (1 + ξa ξ ) ; ξa = 2



−1 ; a = 1 +1 ; a = 2

(2.95)

46

2 Truss Model: General Theorems and Methods of Force …

With this result in hand, the affine transformation in (2.93) can also be rewritten: x(ξ )

Ωe

2

=

Nae (ξ ) xae

(2.96)

a=1

By choosing so-called isoparametric elements (see [27, 153, 390] or [176]), the shape functions for domain representation are also used in constructing the discrete approximations for real and virtual displacement fields u (ξ ) h

Ωe

w (ξ ) h

Ωe

=

2

Nae (ξ ) dae

(2.97)

Nae (ξ ) wae

(2.98)

a=1

=

2

a=1

where dae and wae are nodal values of real and virtual displacements, respectively. The discrete approximation of the infinitesimal deformation field can then readily be obtained from the displacement approximation in (2.97) resulting with 2 du h (ξ ) d Nae (ξ ) e  (ξ ) := = da dx dx a=1 h

1 d Nae de dξ d x(ξ )/dξ a 2

(−1)a e = da le a=1 =

(2.99)

where we used the result for the jacobian of affine transformation ‘d x/dξ = l e /2’, with l e as the length of the real truss-bar element. The same kind of approximation can also be constructed for the virtual deformation field with nodal values of virtual displacements wae replacing those of real displacement dae . We note that both real and virtual deformations remain constant in each element, which also implies (for homogeneous elastic material) that the stress approximation is constant. By exploiting these results, we can easily obtain element internal force for nonlinear elasticity.

2.4 Finite Element Method Implementation for Truss Model

47

fint,e = ( f aint,e (de )) ; 1 ≤ a ≤ 2  2

d Nbe (x) e d Na (x) Aσˆ ( db ) d x f ae (de ) = dx dx b=1 Ωe

1 = −1

1 = −1

2

d Nbe (ξ ) e d Nae (ξ ) Aσˆ ( db ) j (ξ )dξ dx dx b=1 2

(−1)a (−1)b e l e A σ ˆ ( db ) dξ e l le 2 b=1     h (ξ )

=

(−1)a Aσˆ (

2  b=1

(−1)b le

dbe )

(2.100)

The same result can be written in matrix notation, defining the internal force vector for a 2-node truss-bar element   −Aσ int,e = (2.101) f Aσ where we use stress resultant axial force for truss-bar with N = Aσ to simplify notation. In the same manner, we can obtain the element tangent stiffness matrix by replacing the previously defined finite element approximations into linearized form of (G2e) e ] ; 1 ≤ a, b ≤ 2 Ke = [K ab !  2 d Na (x(ξ )) ˆ d Nc (x(ξ )) e d Nb (x(ξ )) e AC dc dx K ab = dx dx dx c=1 e Ω ⎛ ⎞

1 = −1

⎜ 2 ⎟

d N e (ξ ) ⎟ d N e (ξ ) d Nae (ξ ) ˆ ⎜ c b e⎟ AC ⎜ d j (ξ )dξ c⎟ ⎜ dx d x d x ⎝ c=1 ⎠     h (ξ )

1 = −1

=

2 (−1)a ˆ (−1)c e AC dc he le c=1

(−1)a (−1)b le

ˆ AC(

2  c=1

(−1)c le

dce )

!

(−1)b l e dξ le 2

(2.102)

48

2 Truss Model: General Theorems and Methods of Force …

The element tangent stiffness matrix for a 2-node truss-bar element can also be written in matrix notation according to CA K = e l



e

1 −1 −1 1

 ; (if linear elastic: C = E)

(2.103)

The element stiffness matrix keeps the same form in the case of linear elasticity with tangent modulus C in the last expression being replaced by Young’s modulus E, which defines the element stiffness matrix that is obtained from (G1e).

2.4.2 Consistence of Finite Element Approximation The discrete approximation for the displacement field, constructed by the finite element method, should remain consistent and approach the exact solution, when the number of elements increases and the length of each element becomes smaller. The conditions which guarantee the consistence of the finite element approximation can be specified for each finite element or its shape function, according to c1) Nae ∈ C 1 (Ω e ) c2) Na ∈ C (Ω) ; Na = o

n( elem

Nae

e=1

c3)

n en

Nae = 1

(2.104)

a=1

where n en and n elem are, respectively, the number of nodes for each element and the total number of elements chosen in the mesh. The first condition ensures that the computation of the internal force vector or the tangent stiffness matrix for each element, defined by integrals in (2.100) and (2.102), can be carried out with no need to further subdivide the domain of integration Ω e in order to guarantee the unique expression for integrand. The second condition imposes that the finite-element-based discrete approximation will provide a continuous displacement field approximation from one element to another, allowing only for displacement derivative (or strain) discontinuities. For such a case, the chosen finite elementstrategy of subdividing the total domain of integration into element  sub-domains Ω (·) = e Ω e remains very well suitable.2 Finally, the third condition in (2.104) is imposed to ensure that the finite element discrete approximation of displacement field is capable of representing the rigid body modes. The latter 2

The second condition presents the minimum required regularity, since relaxing this requirement by allowing a discontinuous displacement approximation over finite element boundaries could produce  the integrals which are not well defined, such as Ω δ 2 d x, where δ(·) is the Dirac function, defined  according to Ω δ(x − xd )g(x) d x = g(xd ).

2.4 Finite Element Method Implementation for Truss Model

49

pertains in 1D case to pure translation, where all the nodal values of displacement field would remain constant (say, equal to c0 ), resulting with u

rb

=

2

Nae (ξ ) dae =⇒ dae = c0 ; ∀a ∈ {1, 2}

a=1 2

= 

a=1

! Nae (ξ )  =1

c0

(2.105)



2.4.3 Equivalent Nodal External Load Vector If trying to standardize the description of the external loading by the finite element approximation, we use again the same element shape functions. Therefore, for a 2node truss-bar element, we will allow at most a linear variation of distributed external loading, which leads to the following equivalent nodal external load vector fe = ( f ae )  2

Na (x)( Nb (x)bb ) d x f ae = b=1

Ωe

=

2 

1

Nae (ξ )Nbe (ξ ) j (ξ )dξ bb

b=1 −1

=

 2 le 1;a = b (1 + δab )bb ; δab = 0 ; a = b 6 b=1

The same result can also be written in matrix notation as   l e 2b1 + b2 fe = 6 b1 + 2b2

(2.106)

(2.107)

By using Taylor’s formula for representing the true variation of distributed external load, we can show that the proposed finite element approximation for external load will be of the order of O((l e )2 ), which is comparable with the displacement approximation provided by 2-node finite elements. If this kind of approximation is not acceptable, we ought to employ the finite elements with higher order displacement approximations (quadratic, cubic etc.), which requires introducing truss-bar elements with more than 2 nodes.

50

2 Truss Model: General Theorems and Methods of Force …

ξ1 = −1

ξ2 = 0

xe1

ξ3 = +1

xe2

xe3

x(ξ) N1e (ξ) N2e (ξ) N3e (ξ)

le /4

le /2

N1e (ξ) =

1 ξ(ξ 2

le /4 − 1)

N2e (ξ) = (1 + ξ)(1 − ξ) N1e (ξ) =

1 ξ(ξ 2

+ 1)

Fig. 2.12 Isoparametric truss-bar element with 3 nodes, its parent element and its shape functions

2.4.4 Higher Order Finite Elements The local description, already presented for the 2-node truss-bar element, can easily be extended to higher order isoparametric elements. The domain of the parent element for any such higher order element still remains the same interval in natural coordinate ξ ∈ [−1, +1], but with n en > 2 element nodes located within. The change of coordinates between ξ and x is no longer linear, but rather a higher order polynomial which can be written: x(ξ ) =

n en

Nae (ξ )xae ; n en ≥ 2

(2.108)

a=1

where xae , a = 1, 2, . . . , n en are real element nodal coordinates and Nae (ξ ) are the corresponding shape functions. The definition of any such shape function Nae (ξ ) remains the same: it takes a unit value at node a and zero value at all other element nodes (but not equal to zero in-between the nodes). The shape functions of this kind can easily be constructed by using the Lagrange polynomials (e.g., see Zienkiewicz and Taylor [390], p. 119, Hughes [153], p. 126). For a higher order element with n en nodes, we can construct the shape functions as the Lagrange polynomials of order n en − 1 Nae (ξ ) :=

n en ) b=1,b=a

(ξ − ξb ) (ξ − ξ1 ) . . . (ξ − ξa−1 )(ξ − ξa+1 ) . . . (ξ − ξn en ) = (ξa − ξb ) (ξa − ξ1 ) . . . (ξa − ξa−1 )(ξa − ξa+1 ) . . . (ξa − ξn en )

(2.109) For example, as illustrated in Fig. 2.12, the shape functions for a 3-node isoparametric element are quadratic polynomials. The jacobian of the coordinate transformation for a higher order element in (2.108) is not a constant, but a polynomial of the order n en − 2 n en d Nae (ξ ) e d x(ξ ) (2.110) j (ξ ) := = xa dξ dξ a=1

2.4 Finite Element Method Implementation for Truss Model

51

The same consistency conditions of the finite element discrete approximation as those established in (2.104) will remain valid for higher order elements. These conditions are easy to verify for each particular element of this Lagrangian family. For example, one can readily confirm the capability of a 3-node isoparametric n en e element Na = 1. The in Fig. 2.12 to represent the rigid body modes by verifying that a=1 displacement field continuity between neighboring elements is also apparent, since they share the same nodal value of the displacement. Finally, the condition on strain field continuity within an element, which calls for computation of shape functions derivatives, will require that the coordinate transformation x(ξ ) of a higher order isoparametric element remains bijective. By the application of the chain rule, we can readily confirm that such a computational requirement will ask for a positive value of the jacobian throughout the element domain j (ξ ) > 0, ∀ξ ∈ [−1, +1], ∂ Nae 1 ∂ Nae = ; Nea (x) ∈ C 1 (Ω e ) ∂x ∂ξ j (ξ ) For a 3-node isoparametric element, this kind of requirement will limit the acceptable position of the center node within the inner half of the element length; see Fig. 2.12 for illustration j (ξ ) := l e /2 + ξ(x1 − 2x2 + x3 ) > 0; ∀ξ ∈ [−1, +1] ; l e = (x2e − x1e ) ξ = +1 =⇒ x2 < (x1 + x3 )/2 + l e /4 (2.111) ξ = −1 =⇒ x2 > (x1 + x3 )/2 − l e /4 The isoparametric element employs the same shape functions as those in (2.108), for constructing the displacement field approximation; u h Ωe

=

n en

Nae (ξ )dae

(2.112)

a=1

where dae are displacement nodal values. With these results in hand, we can easily obtain the discrete approximation for the deformation field  (ξ ) h

Ωe

=

n en

Bae (ξ )dae

(2.113)

a=1

where Bae (ξ ) are defined with Bae (ξ ) :=

1 d Nae (ξ ) d Na (x(ξ )) = dx j (ξ ) dξ

(2.114)

The element internal force vector for an isoparametric element with n en nodes can then be written:

52

2 Truss Model: General Theorems and Methods of Force …

fint,e = ( f aint,e ) ; 1 ≤ a ≤ n en 1 int,e fa = Bae (ξ )Aσˆ ( h (ξ )) j (ξ )dξ

(2.115)

−1

and its tangent stiffness matrix reduces to e ] ; 1 ≤ a, b ≤ n en Ke = [K ab 1  e ˆ h (ξ ))Bbe (ξ ) j (ξ )dξ K ab = Bae (ξ )AC(

(2.116)

−1

Finally, by using the higher order isoparametric elements, we can also increase the precision of the element external load vector representation, which can be written: fe = ( f ae ) ; 1 ≤ a ≤ n en 1 n en

e fa = Nae (ξ )( Nbe (ξ ) bb ) j (ξ )dξ −1

(2.117)

b=1

2.4.5 Role of Numerical Integration We could see in the previous section that the higher order elements provide higher order approximations for displacement, strain and stress fields than a simple 2-node truss-bar element. However, the higher order approximations also make the computations of element arrays (element stiffness or internal force vector) quite laborious. For that reason, the computation of the integrals for element arrays components are best carried out by using numerical integration (e.g., see Irons [235]). The main idea of numerical integration is rather simple. First, instead of computing the element integrals exactly, we will approximate the integrand by a polynomial function.3 This kind of polynomial, denoted as g(ξ ), will take the same value as the integrand at the chosen points in the integration domain of the parent element ξl , l = 1, 2.., n in , referred to as the abscissas of numerical integration. We will then integrate analytically the polynomial integrand g(ξ ), and write the final result accordingly 1 g(ξ ) dξ = −1

n in

wl g(ξl )

(2.118)

l=1

where wl are the weights of numerical integration. 3

Recall that the approximation of external load components is also written in terms of polynomials.

2.4 Finite Element Method Implementation for Truss Model

53

Among several possibilities to carry out numerical integration (e.g., see Hughes [153], p.132, Zienkiewicz and Taylor [390], p. 121 or Bathe [27], p. 462), the optimal choice is the Gauss quadrature rule, since it is capable of providing the most accurate result with the smallest number of integration points. More precisely, for Gauss quadrature with n in integration points we consider both the abscissas and weights as free parameters to choose, which let us to uniquely define an approximating polynomial of order 2n in − 1, which is then integrated exactly. For example, one Gauss quadrature point will integrate exactly any linear polynomial g(ξ ) = c0 + c1 ξ 1 −1

(c0 + c1 ξ ) dξ = 2c0 ≡ 2 g(0)   

(2.119)

g(ξ )

The comparison of the two results in the last expression readily shows that 1 point Gauss quadrature rule should use the abscissa of the matching point in the center of the parent element and the corresponding integration weight equal to 2, n in = 1 =⇒ ξ1 = 0 ; w1 = 2

(2.120)

For higher order approximation polynomials, the abscissas and weights of the numerical integration can also be obtained by comparison with the analytic results obtained for a generic polynomial integrand. For example, with 2 Gauss points n in = 2, we can integrate exactly any third order polynomial g(ξ ) = c0 + c1 ξ + c2 ξ 2 + c3 ξ 3 , where c0 , c1 , c2 , c3 are constants 1 −1

(c0 + c1 ξ + c2 ξ 2 + c3 ξ 3 ) dξ =    g(ξ )

2

wl g(ξl )

(2.121)

l=1

In order to ensure invariance with respect to element node numbering, we impose −ξ1 = ξ2 ≡ ξ and w1 = w2 ≡ w, leading to a reduced form of the last result 2 2c0 + c2 = 2w(c0 + c2 ξ 2 ) 3

(2.122)

√ The last equality is verified if and only if w = 1 and ξ = 3/3. Hence, the abscissas and weights of 2-point Gauss quadrature, integrating exactly cubic polynomials, are given as √ (2.123) n in = 2 =⇒ w1 = w2 = 1 ; ξ2 = −ξ1 = 1/ 3 The same kind of analysis can be carried out for polynomials of even higher order, with 3-point Gauss quadrature used for integrating the polynomials of order 5, 4-point rule for a polynomial of order 7 etc. (see Zienkiewicz and Taylor [390]).

54

2 Truss Model: General Theorems and Methods of Force …

The Gauss quadrature can be used in the computation of internal force vector components according to fint,e = ( f aint,e ) ; 1 ≤ a ≤ n en n in

f aint,e = Ba (ξl )Aσˆ ( h (ξl )) j (ξl )wl

(2.124)

l=1

This result illustrates quite well the important role of numerical integration in reducing the necessary information on stress field variation to Gauss points only. This reduction is especially important for nonlinear constitutive models. Moreover, only the numerical value of any other field at Gauss integration points is finally needed, and the finite element residual computation is thus reduced to algebraic operations that are perfectly suitable for computer implementation. The same cost and data reduction applies to the computation of the element tangent stiffness matrix, where we only need the Gauss point values of elastic tangent modulus e ] ; 1 ≤ a, b ≤ n en Ke = [K ab n in

e ˆ h (ξl ))Bb (ξl ) j (ξl )wl K ab = Ba (ξl )AC(

(2.125)

l=1

What is the minimum number of Gauss points to use for computing the components of element residual vector or tangent stiffness matrix, defined by integrals in (2.124) and (2.125), respectively? The answer is easy to provide for 1D problems: the minimum  +1 Gauss quadrature rule must be capable of integrating exactly −1 j (ξ ) dξ . With this choice of numerical integration, we can ensure the representation of constant deformation and constant stress mode in each element.4 A more demanding criterion pertains to a number of Gauss integration points required to ensure the best possible order of convergence for deformation field with respect to the chosen element shape functions. In 1D, this criterion leads to the so-called reduced integration rule, with the number of Gauss points being equal to the number of element nodes reduced by 1. Hence, the reduced integration implies using 1 point for a 2-node truss-bar element, 2 points for a 3-node element etc. The reduced integration is the best rule to apply in 1D, since those Gauss quadrature points coincide with the points of higher order accuracy for strain and stress fields (see Zienkiewicz and Taylor [390], p. 348). This interesting property was initially pointed out by Barlow [25], and for that reason, the reduced integration points are yet referred to as Barlow points.

4

Verifying that the element can represent a constant stress state is referred to as the ’patch test’ (see [390], [153], or [27]), which guarantees the finite element method convergence in the limit where all elements become sufficiently small that the stress state in each element approaches a constant.

2.4 Finite Element Method Implementation for Truss Model

55

With 1 Gauss point, reduced integration rule for a 2-node truss-bar element, we can obtain the element internal force vector f aint,e = 2Bae (0)Aσˆ ( h (0))

le 2

= (−1)a σˆ ( h (0))

(2.126)

as well as its tangent stiffness matrix e

e ˆ h (0))Bbe (0) l K ab = 2Bae (0)AC( 2 (−1)a (−1)b ˆ h = C( (0)) le

(2.127)

These two numerical results of optimal accuracy, turn out to be the same as the analytic results. However, the reduced integration rule is not sufficiently accurate for integrating the external nodal load vector for the same element. Namely, for a linear load variation in a 2-node truss-bar element, 1 point Gauss quadrature rule will lead to a wrong result: f aext,e

=

1

2Na (0)(

l=1

2

Nb (0) bb )

b=1

le = (b1 + b2 ) 4

le 2 (2.128)

A simple way to improve this result is by increasing the order of Gauss quadrature rule to 2, which will provide the exact result for external load vector 2 2   e f aext,e = [Na ( ξ1 )( Nb (ξ1 ) bb ) w1 +Na ( ξ2 )( Nb (ξ2 ) bb ) w2 ] l2   b=1  b=1 √ −1/ 3

=

le 6

2 

1

√ 1/ 3

(1 + δab )bb

b=1

(2.129) This difference between quadrature rule for computing external load vector versus the reduced integration rule used for internal force computation, does not pose any practical problem. On the contrary, we can thus obtain the optimal results for both ingredients of the final set of algebraic equations to be solved. It is important, however, to use the same integration rule for computing the element internal force vector and its tangent stiffness matrix, especially for inelastic constitutive models employing the internal variables, in order to ensure the consistency between the stress and tangent modulus computations.

56

2 Truss Model: General Theorems and Methods of Force …

2.4.6 Finite Element Assembly Procedure The main advantage of the local (or element) description based upon the parent element pertains to the size reduction of element arrays and the complete standardization of the computational procedure for all finite elements of the same kind. The main advantage becomes the main inconvenience when the results for element arrays should be placed within the structural assembly, since the information of the elements positions is not available from the local description. The main role of the finite element assembly procedure is to reestablish this connectivity and carry out the summation of local (element) arrays with n en size to provide the set of n eqs (global) structural equations to be solved. For that reason, for each element we define the connectivity matrix Le , with size n en × n eqs , which provides the relationship between the nodal values of virtual displacement for a particular element we and the vector w containing the nodal values of virtual displacement for the chosen finite element mesh. we = Le w ; we = [w1e , w2e , . . . , wne en ]T ; w = [w1 , w2 , . . . , wn eqs ]T

(2.130)

The entries in the connectivity matrix depend on the position of a particular element within the finite element mesh. In general, the connectivity matrix Le is just a Boolean matrix, with each component equal to 0 or 1. The latter is reserved for all the components that correspond to a particular node to which the chosen truss-bar element is attached. In order to illustrate how the entries in the connectivity matrix are chosen, we present in Fig. 2.13 a simple example of the finite element mesh for a truss structure composed of either 2-node or 3-node elements. The connectivity matrices for the first mesh with 2-node elements can be written:         10000 01000 00100 00010 1 2 3 4 L = ;L = ;L = ;L = 01000 00100 00010 00000 (2.131) whereas the connectivity matrices for the mesh with 3-node elements are written: ⎡

1000 L1 = ⎣0 1 0 0 0010

⎤ ⎡ 0 0010 0⎦ ; L2 = ⎣0 0 0 1 0 0000

⎤ 0 0⎦ 0

(2.132)

We note in passing that in each of two cases, the entries of the connectivity matrix for node 5 is equal to 0, since that node is fixed by the support (and the corresponding virtual displacement is equal to 0). By exploiting the connectivity matrices connecting the nodal values of virtual displacement of every single element to those of the structure, we can rewrite the discretized weak form (or Galerkin equation) as

2.4 Finite Element Method Implementation for Truss Model

1

2

mesh a) 1

2

mesh b)

1

3 1 (2)

4

1

2 2 3

2 (1)

2

(2)

1

a)LM (2, 4)

5 1 (4)

57

3

2

1 2 b)LM (3, 2) 1 2 3

2 3 4 3 4 0 3 4 0

Fig. 2.13 Finite element mesh composed of: a 2-node elements b 3-node elements

0=

n elem

w

eT

(f

int,e

−f

ext,e

e=1

)=

n elem

(Le w)T (fint,e − fext,e )

(2.133)

e=1

= wT (

n elem

Le T fint,e −

e=1

n elem

Le T fext,e )

e=1

The last result implies that the assembly of element contributions to internal and external force vectors for the chosen finite element mesh should be taken into account after multiplication with the transpose of the element connectivity matrix

fint =

n elem e=1

n elem

Le T fint,e =: A fint,e e=1

; (2.134)

fext =

n elem e=1

n elem

Le T fext,e =: A fext,e e=1

The linearized form of this system, or an equivalent linear problem governed by a set of linear algebraic equations, will require the assembly of the tangent stiffness matrix. By taking into account that each element connectivity matrix for connecting the nodal values of real displacement at local and structural level remains the same as the one written for virtual displacement, we obtain

K=

n elem e=1

n elem

Le T Ke Le =: A Ke

(2.135)

e=1

A more efficient manner to carry out the finite element assembly procedure, first proposed by Wilson [382], pertains to reducing the size of the connectivity matrix from its full format n en × n eqs . For each element e ∈ [1, n elem ], we will write instead a single vector which contains only the node numbers for non-zero entries of the particular element connectivity matrix. As shown in Fig. 2.13, all these vectors can

58

2 Truss Model: General Theorems and Methods of Force …

be arranged in a single matrix, denoted as L M. The finite element assembly can then be performed according to ∀e ∈ [1, n elem ] & ∀a, b ∈ [1, n en ] =⇒ f ext (L M(a, e)) ←− f ext (L M(a, e)) + f aext,e ; f int (L M(a, e)) ←− f int (L M(a, e)) + f aint,e ; e K (L M(a, e), L M(b, e)) ←− K (L M(a, e), L M(b, e)) + K ab

(2.136)

It is clear that the efficiency of the finite element assembly is thus improved significantly by operating only upon the contributing terms with non-zero components of each connectivity matrix. The same kind of finite element assembly can easily be extended to 2D or 3D problems, with the only difference regarding more unknowns per node (e.g., Hughes [121], p. 92 or Zienkiewiecz and Taylor [390], p. 15 or Bathe [27], p. 28).

Chapter 3

Beam Models: Refinement and Reduction

Abstract We here study the beam theory that has served for long time as the ’bread and butter’ to structural engineers. We first present the beam theory as the reduced model that can be obtained by imposing the kinematic constraints to solid mechanics formulation. Thus, we illustrate why Timoshenko’s model might be better than EulerBernoulli’s beam model when the shear deformation contribution plays an important role. We also illustrate several alternative derivations of the classical Euler-Bernoulli beam theory with a sequence of elementary transformations, along with further model refinements and reduction by accounting for local constraints, like releases or hinges, or global constraints, like beam length invariance. Finally, we briefly explain geometrically exact Reissner’s beam theory, which can provide a consistent linearized formulation for shallow Timoshenko?s beam. This model is finally used to explain different manners to construct non-conventional finite interpolation that can handle the locking phenomena.

3.1 Reduced Models of Solid Mechanics: Planar Beams of Euler, Timoshenko and Reissner 3.1.1 Euler-Bernoulli Planar Beam Model The reduced model of solid mechanics referred to as a ’beam’ can represent in a sufficiently accurate manner the deformation of an elongated deformable solid body, with one dimension (the beam length) that is much bigger than the other two dimensions (placed in the beam cross-section). With the rule of thumb, we take this ratio to be equal to ten for a ’slender beam’ represented by the Euler-Bernoulli model, but this also depends on the constitutive behavior that can favor the contribution of shear deformation. The beam is certainly the structural model the most frequently used by engineers. For simplicity, in this section we consider the planar beam placed in 2D plane (x, y), with the straight beam axis (the line along the length) aligned with coordinate x. This beam model is defined by its geometry (including length l measured along x axis and the cross-section with area A and moment of inertia I ); the material property as Young’s modulus E; and a distributed loading p y showed © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Ibrahimbegovic and R.-A. Mejia-Nava, Structural Engineering, Lecture Notes in Applied and Computational Mechanics 100, https://doi.org/10.1007/978-3-031-23592-4_3

59

60

3 Beam Models: Refinement and Reduction

Fig. 3.1 Beam model: geometry (with x placed along the beam axis and y and z in principal axes of inertia of the cross-section) and applied loading

in Fig. 3.1. We study here bending behavior of the beam which is characterized only by transverse (vertical) displacement in y direction. The longitudinal displacement under load px in x direction is computed by using the truss-bar model equations. The (reduced) model of the Euler-Bernoulli beam is based upon the following two hypotheses: 1. plane cross-section, which is perpendicular to the beam axis in the initial configuration (depicted as the straight line for 2D case in Fig. 3.1), is assumed to remain perpendicular to the beam axis in the deformed configuration; 2. there is no change of beam thickness (distance measured along cross-section) between initial and deformed configurations. We can easily see that under these assumptions the 3D solid mechanics model can be reduced to 1D model of beam placed along x axis with only one non-zero strain component εx x = 0, as well as only one non-zero stress component σx x = 0 when using a simplified beam constitutive model. Namely, by hypothesis 1, we can obtain that the beam shear strain remains equal to zero, γx y := 2εx y = 0 and by hypothesis 2, we can obtain that the through-the-thickness strain remains equal to zero  yy = 0. We present the corresponding developments for kinematics, constitutive and equilibrium equations for a reduced model of the Euler-Bernoulli beam. Kinematics: we consider the beam-bending motion for a point G(x, 0), located at the intersection of the beam axis and cross-section at point x. If v(x) represents the vertical displacement of a point G(x, 0), then by the Euler-Bernoulli beam hypotheses we can obtain the displacement of any arbitrary point P(x, y) which belongs to the same cross-section. Displacement of point P can be expressed according to the beam deformation illustrated in Fig. 3.2 with two components u p (x, y) and v p (x, y). ≈θ

≈1

      u (x, y) = −y sin(θ ); v p (x, y) + y = v(x) + y cos(θ ) p

(3.1)

In agreement with the hypothesis of small displacement gradients, this can be reinterpreted in particular as small rotations, θ = ddvx  1 ⇒ cos θ ∼ = 1 and sin θ ∼ =θ which results with; u p (x, y) = −yθ (x) = −y

dv(x) ; v p (x, y) = v(x) dx

(3.2)

3.1 Reduced Models of Solid Mechanics: Planar…

61

Fig. 3.2 Euler-Bernoulli beam: a displacement; b normal stresses; c forces acting on the section length d x

It follows that the beam axis does not change the length in deformed configuration, dx ∼ or more simply, it does not deform under bending with ds = cos d x. θ = Having defined the displacement field in agreement with the beam reduced model constraints, we can further simply use linearized kinematics equations from solid mechanics to define the relationship between strain field and displacement field. In p geometrically linear theory, the axial strain εx x is defined as the first derivative of p axial displacement u (x) εxpx =

d 2 v(x) du p (x) = −y dx dx2

⇒

εxpx = yκ(x)

(3.3)

where κ(x) is (infinitesimal) curvature. The latter is further referred to as the generalized bending strain measure for the Euler-Bernoulli beam model κ(x) = −

d 2 v(x) dx2

(3.4)

We can easily check that all other strain components remain equal to zero with: p ε yy =

dv p dv dv dv p du p = 0 and γxpy = + =− + =0 dy dy dx dx dx

(3.5)

Constitutive equation: we next use the linear elastic constitutive law to define relationship between stress field and strain field. With constitutive law, we can describe particular behavior of a material. Here we will consider the simplest case of linear elasticity, where the stress-strain relationship is described by 1D version of Hooke’s

62

3 Beam Models: Refinement and Reduction

law. This constitutive model remains valid for many materials in a small strain regime (which occurs well before inelastic behavior or fracture). By combining Hooke’s law and expression (3.4), we can further obtain the stress at point P(x, y) σxpx (x, y) = Eεxpx (x, y) = E yκ(x)

(3.6)

where E is Young’s modulus. By combining the last two expressions, we can conclude that the stresses are linearly distributed across the cross-section in beam bending. Hence, in this case, one can define the generalized stress measure at each point x of the reduced beam model in terms of bending moment M(x). For illustration, we p refer to Fig. 3.2b depicting normal stress σx x acting on the infinitesimal area d A, which produces resultant force F. Such a force acts on distance y from the beam axis, thus producing the bending moment M that can be obtained by integrating over the beam cross-section. The end result is the beam model version of the linear elastic constitutive law  M(x) =

I

   p yσx x d A = Eκ y 2 d A = E I κ(x)

A

(3.7)

A

where we used the result in (3.6) to introduce the moment of inertia I . We note that for the beam bending behavior the linear elastic constitutive law can be expressed directly in terms of stress resultant bending moment M(x) and corresponding generalized bending strain as beam curvature κ(x) M(x) = E I κ

(3.8)

By introducing (3.8) into (3.6), we can obtain Navier’s formula which confirms the linear stress distribution within the cross-section in pure bending σxpx (x, y) =

y M(x) I

(3.9)

Remark on defining the beam neutral axis: We previously found out that the beam axis length in pure bending remains unchanged in the deformed configuration. Thus one can obtain the beam axis position within any cross-section by imposing the condition of zero stress resultant normal force with N (x) = 0  N (x) := A

Sy

    p σx x (x, y) d A = Eκ(x) y d A = Eκ(x) S y = 0 ⇒ S y = 0 (3.10)    A

=0

In other words, when x axis passes through geometric center G, static moment Sy is zero. This axis is called the neutral axis of the beam with corresponding value of

3.1 Reduced Models of Solid Mechanics: Planar…

63

stress σx x (x, 0) = 0 in pure bending. The beam neutral axis is a geometric place of G points. Equilbrium equation: the final set of equations to define are equilibrium equations, providing the balance between external loads and internal beam stress resultants. We consider an infinitesimal segment CD of the beam of length d x, isolated by imaginary cuts in the deformed configuration as illustrated in Fig. 3.2c. The beam is externally loaded with transversal distributed load p y (x) which generates internal forces in terms of shear forces V (x) and bending moments M(x), acting at both ends of cut, placed at x and at x + d x. Positive orientation of shear force on the right side of the cut is taken in y direction and clockwise rotational direction for bending moment M(x + d x), while on the left side internal forces have opposite directions in order to maintain equilibrium. One can first state the force equilibrium equation taking into account all the forces acting in y direction 

Fy := −V (x) + V (x + d x) + p y (x)d x = 0

(3.11)

By employing Taylor’s formula for V (x + d x) term and keeping only the first order terms in this expansion, we can further obtain V (x + d x) ≈ V (x) +

d V (x) dx dx

(3.12)

By combining the previous two results, the force equilibrium can finally be reduced to d V (x) + p y (x) = 0 (3.13) dx We next provide the moment equilibrium equation that can be written by taking into account all moments around z axis acting at point D (see Fig. 3.2c) 

dx Mz D := M(x) − M(x + d x) + V (x)d x − p y (x)d x =0  2 

(3.14)

≈0

where higher order term p y (x)d x d2x can be neglected. Furthermore, by expanding M(x + d x) with the use of Taylor’s formula and by keeping only the first order term, we can write d M(x) dx (3.15) M(x + d x) ≈ M(x) + dx By combining the previous two results, we can finally obtain a reduced moment equilibrium equation d M(x) − V (x) = 0 (3.16) dx

64

3 Beam Models: Refinement and Reduction

Strong Form: We can combine the local equations of kinematics, equilibrium and constitutive equations into a single equation in terms of transverse displacement of the beam, which is referred to as the differential or the strong form of the EulerBernoulli beam in bending. Namely, we first replace the kinematics result defining the beam curvature κ in (3.4) into the beam constitutive equation in (3.8) to express moment directly in terms of the transverse displacement field κ=−

d 2 v(x) dx2

⇒

M(x) = E I κ = −E I

d 2 v(x) dx2

(3.17)

We can also combine the result for moments equilibrium in (3.16) d M(x) − V (x) = 0 dx

⇒

V (x) =

d M(x) dx

(3.18)

with the force equilibrium in (3.13) to obtain a single equilibrium equation for the bending moment d V (x) + p y (x) = 0 dx

⇒

d 2 M(x) + p y (x) = 0 dx2

(3.19)

Finally, putting together the results in (3.17) and (3.19), as well as assuming that E I = cst.), we arrive at the strong or the differential form of the Euler-Bernoulli beam model d2 dx2

 d 2 v(x) −E I + p y (x) = 0 dx2

⇒

EI

d 4 v(x) − p y (x) = 0 dx4

(3.20)

which represents beam equilibrium expressed only as a function of transverse displacement v(x). It is clear that Eq. (3.20) requires v(x) to be four times differentiable in order to match the given variation of distributed loading p y (x) at any point x within the beam domain. The fourth-order differential equation must be integrated four times in order to obtain the strong form solution for transverse displacement v(x) which allows calculating all other quantities. In order to obtain a unique solution to the strong form, we need to impose four boundary conditions. In particular, this can be done by imposing two boundary conditions at each end of the beam, with the highest order smaller than 2m = 4 in order not to clash with the equilibrium equation that has to be valid at any point. For example, we can impose the boundary conditions upon: d 2v d 3v dv , , (3.21) v, dx dx2 dx3 The physical interpretation of different conditions at either the Dirichlet Γ D or the Neumann Γ N boundary can be as follows:

3.1 Reduced Models of Solid Mechanics: Planar…

(i) (ii) (iii) (iv)

65

displacement v|Γ D = v, rotation of the section (slope axis) θ |Γ D = θ (θ = 2 bending moment M|Γ N = M (M = −E I dd xv2 ), 3 shear force V |Γ N = V (V = ddMx = −E I dd xv3 ).

dv ), dx

Weak Form: The final step is to provide the integral or the weak form, which can also be obtained directly from a physical interpretation as the principle of virtual work. The latter introduces the notion of (infinitesimal) virtual displacement δv(x), which is in agreement with kinematic hypotheses of the Euler-Bernoulli beam and also equal to zero on imposed displacement boundary. We can then obtain the virtual curvature δκ = −d 2 (δv(x))/d x 2 which is virtual work conjugate to real bending moment M(x). The virtual work principle states that the Euler-Bernoulli beam in equilibrium will produce an equal amount of internal and external virtual work, which allows writing: l

l M(x)δκ(x)d x =

0

p y (x)δv(x)d x + δTBC

(3.22)

0

where δTBC is the virtual work of external boundary tractions in agreement with natural boundary conditions imposed on bending moment or shear force at the boundary. To ensure the solution uniqueness, the result in (3.22) must be accompanied by essential boundary conditions on imposed displacement and rotations. Given sufficient stress smoothness, we can integrate by parts the weak form equation in (3.22) two times, and thus recover the strong form and the natural or static boundary conditions as the corresponding Euler-Lagrange equations, by following the same procedure as presented for solid mechanics in [176]. Direct application of the beam element described above can only be valid in calculations for continuous straight beams resting on several supports and placed along x axis. For any other case, we need to use a combination of axial and bending responses. In general, the structures modeled by beam elements, such as frames, portal frames or crutch bridges, deform both under axial force and bending moment. These two types of beam deformation can be superposed and remain uncoupled if the axis passes through point G of the cross-section. In such a case we can describe beam behavior with a more general constitutive law N = E Aε

M = EIκ

⇒



 N EA 0 ε = M 0 EI κ

(3.23)

The corresponding result for the strong form can be written as EA

d 4 v(x) − p y (x) = 0 dx4 dv , N , V, M. + boundary conditions on u, v, dx

d 2 u(x) + px (x) = 0 dx2

and

EI

(3.24)

66

3 Beam Models: Refinement and Reduction

Table 3.1 Stresses, elastic constants and strains in different formulations Model Stress Elastic constant Strain Elasticity 1D Elasticity 3D Truss-Bar (axial) Beam

σ σi j N M

E Ci jkl EA EI

ε εi j ε κ

Finally, the weak form of such a beam model can be written as l

l (N δε + Mδκ)d x =

0

( px δu + p y δv)d x + δTBC

(3.25)

0

+ boundary conditions on u, v, θ. Remark on constitutive equation format: With respect to the solid mechanics problems in 1D and 3D elasticity, the beam and truss-bar structural elements introduce the generalized strain measures, which are work-conjugate to stress resultants. The two are connected by corresponding elastic constants in a constitutive model that can be obtained in agreement with a particular theory, which is summarized in Table 3.1. The formulation of the generalized deformation bending measures and stress resultants are derived from the kinematic hypotheses and the integration over the cross-section, which jointly allow reducing the 3D model to the corresponding 1D counterpart. Remark on shear stress computation: The Euler-Bernoulli beam formulation constrains the shear strain to zero by the chosen kinematic hypothesis. However, this does not imply that shear stress is equal to zero, since in stating the force equilibrium the shear force is a necessary ingredient. The shear occurs due to the moment variation = V (x). We postulate that shear force is produced by tangential stresses τ with d M(x) dx that occur in the cross-section and longitudinally between fibres, with τ = τx y = τ yx . Clearly, these stresses cannot be computed from strain since γ (= γx y ) = 0. We can argue that shear stiffness in the Euler-Bernoulli beam is infinite and shear strain γ remains equal to zero γ = Gτ , which implies that the shear modulus should take an infinite value. We can further estimate the shear force from equilibrium equation V (x) = d M(x) dx and obtain the corresponding values of shear stress τ . The values obtained in this manner are approximate, yet they are sufficiently accurate in the case of slender beams, because the shear strain has indeed a rather low value. Let us consider a case of the beam with a rectangular cross-section (b × h, where is h > b). The shear stress τ can be exteriorized by making a longitudinal cut L-L of the beam depicted in Fig. 3.3. The forces acting on the upper part above the cut acting over area (b d x), must satisfy equilibrium, which leads us to

3.1 Reduced Models of Solid Mechanics: Planar…

67

Fig. 3.3 Shear stress estimation by equilibrium







Fx := −

σd A + A

(σ + dσ )d A − τ b d x = 0

(3.26)

A

With further reduction of the equilibrium equation in (3.26) above, we can obtain  dσ d A − τ b d x = 0

(3.27)

A

We can recast the last result in terms of stress resultants by using the equilibrium equation in (3.18) dM = V ⇒ d M = V dx (3.28) dx and normal stress distribution equation in (3.9) σ =

My I

⇒

dσ =

y dM I

⇒

dσ =

y V dx I

(3.29)

Thus, it follows that the first term in (3.27) can be recast as  A

S(y)

      V y V V dx d A = dσ d A = yd A d x = S(y)d x I I I A

(3.30)

A

where S(y) is a static moment of A with respect to z axis. Finally, the previous result can be replaced into (3.27) to obtain the expression for shear stress τ dependence on shear force V S(y) V S(y)d x = τ (y)bd x ⇒ τ (y) = (3.31) I Ib This is known as a formula of Zhuravskii, derived in 1855. Here, we assumed that tangential stresses τ were uniformly distributed in the cross section over the beam width b, which is sufficiently precise if b < h. To obtain a specific variation of shear stress in direction of thickness of the beam, we chose the rectangular cross-section,

68

3 Beam Models: Refinement and Reduction

Fig. 3.4 Beam with rectangular section: a cross-section; b stress σ diagram; c shear stress τ diagram; d deformation of the cross-section d x under shear stresses τ

shown in Fig. 3.4, with I =

bh 3 , 12

A = (

h h − y)b, S(y) = A (y + ) 2 2

(3.32)

Replacing this result into Eq. (3.31), we obtain a parabolic distribution of shear stresses τ (y) =

2y 3V [1 − ( )2 ] 2bh h

⇒

τmax = τ | y=0 =

V 3V = 1, 5 2bh A

(3.33)

Remark on section warping: If we take a finite value of shear modulus G ≡ μ, it follows from the result in (3.33) that shear strain can be defined as γ (y) =

τ (y) G

(3.34)

The shear strain is not a constant over the cross-section, therefore the section should warp under the action of shear stresses as depicted in Fig. (3.4)d. The (infinitesimal) vertical displacement dv due to the shear force V is calculated by the average value of shear strain γm : (3.35) dv = γm d x which is defined as γm =

V G(k A)

(3.36)

where k A is called reduced shear area and k is a coefficient depending on the shape of the beam cross-section that takes into account the effect of warping. For the rectangular cross-section k = 5/6. By combining the results in (3.35) and (3.36), we can calculate displacement v(x) due to shearing by integrating dv(x) V (x) = dx G(k A)

(3.37)

3.1 Reduced Models of Solid Mechanics: Planar…

69

Fig. 3.5 Cantilever beam with rectangular section

The results obtained on this manner are not so inaccurate for very slender beams. Remark on alternative approach to shear stress computation: Equation (3.31) can also be established from the first equilibrium equation of 2D elasticity (with bx = 0) ∂τx y ∂σx x + =0 ∂x ∂y

(3.38)

and these two approaches are equivalent.

3.1.2 Solid Mechanics Versus Beam Model Accuracy for Planar Cantilever Beam The objective of this section is to quantify the difference between the Euler-Bernoulli planar beam model and the 2D solid mechanics elasticity model. We study here a cantilever beam with a rectangular cross-section shown in Fig. 3.5a. We choose that b = 1 is significantly smaller than h = 2c in order to reduce to planar beam domain 2D elasticity problem in plane stress. As the beam structure is clamped on the left end (x = 0) and loaded with a concentrated vertical force P on the right end (x = l). We will apply the same to 2D elasticity model boundary conditions in the agreement with the stress distribution obtained for the Euler-Bernoulli beam. We first provide the analytic solution for the Euler-Bernoulli beam model. With the diagrams of the internal forces, moments M and shear V , given in Fig. 3.5b, we can easily compute the analytic solution for the beam deflection by Mohr’s analogy [241]. The normal and tangential stresses in arbitrary section have linear and parabolic distribution according to the results in Fig. 3.4b, c. The transverse displacement v(x) can be obtained by integrating the strong form in (3.20) four times with zero load term p y = 0;  d2 d 2 v(x) −E I =0 (3.39) dx2 dx2 The general solution of displacement field v(x) is a cubic polynomial v(x) =

1 c1 x 3 c2 x 2 ( + + c3 x + c4 ) EI 6 2

(3.40)

70

3 Beam Models: Refinement and Reduction

with 4 integration constants c1 , c2 , c3 and c4 , which can be computed by imposing the boundary conditions v(0) = 0, θ (0) = 0, V (l) = P, M(l) = 0 V (l) = −c1 = P M(l) = −c1 x − c2 = 0 2

1 c1 x ( + c2 x + c3 ) = 0 EI 2 c2 x 2 1 c1 x 3 ( + + c3 x + c4 ) = 0 v(0) = EI 6 2

θ (0) = −

⇒

c1 = −P

⇒

c2 = −c1 l = Pl

⇒

c3 = 0

⇒

c4 = 0

By introducing the computed values of integration constants into general solution (3.40) we obtain P 2 x (3l − x) (3.41) v(x) = 6E I where the maximum displacement v(x) can be calculated for x = l vmax (l) :=

Pl 3 Pl 3 2c3 = ). ; (with I = 3 3E I 2Ec 3

(3.42)

We next compute the analytic solution by using the 2D elasticity model of solid mechanics. The analytic solution is obtained for the 2D domain, imposing the boundary conditions as depicted in Fig. 3.6: (i) kinematic conditions: v(0, 0) = 0 u(0, c) = 0 u(0, −c) = 0 (ii) static conditions (with I = tx (x, ±c) = t y (x, ±c); tx (l, y) = 0; tx (0, y) =

−Ply ; I

(3.43)

2c3 ): 3

t y (x, ±c) = 0; P 2 (c − y 2 ); t y (l, y) = 2I P t y (0, y) = − (c2 − y 2 ); 2I

∀x ∈ (0, l) ∀y ∈ [−c, c] ∀y ∈ [−c, c]

The minimum number of supports is chosen with the sole purpose of eliminating the rigid body motion. We note that kinematic boundary conditions chosen for that elasticity problem are not exactly identical to those imposed in the beam model problem, since the cross-section x = 0 does not remain plane. However, the static boundary conditions are the same, since the imposed surface tractions represent beam model stress with shear stress distribution obtained by the Zhuravskii formula. One can easily check that the following stress field makes it possible to satisfy all equilibrium equations and thus represent the exact solution for the imposed stress boundary conditions in this elasticity problem:

3.1 Reduced Models of Solid Mechanics: Planar…

71

Fig. 3.6 Beam model in 2D elasticity

σx x = −

P x¯ y P(l − x)y P 2 =− ; σ yy = 0; τx y = (c − y 2 ) I I 2I

(3.44)

with x¯ = l − x. Even though these two models share the same stress distribution, the strain and displacement fields will be different. Namely, by introducing these stresses into 2D Hooke’s law for the state of plane stress, we get x x = −

P x¯ y P x¯ y P ;  yy = ν ; γx y = (c2 − y 2 ) EI EI 2G I

(3.45)

E with G ≡ μ = 2(1+ν) . Finally, from the kinematic equations in 2D elasticity we can obtain the displacements by integration (see [371] for details) resulting with

P [3y(x¯ 2 − l 2 ) − (2 + ν)y(y 2 − c2 )] 6E I P {(x¯ 3 − l 3 ) − [3l 2 + (4 + 5ν)c2 ](x¯ − l) + 3ν x¯ y 2 } v(x, y) = 6E I

u(x, y) =

We can see that the displacements computed by 2D elasticity are different from those obtained by the Euler-Bernoulli beam. In particular, they depend on Poisson’s ratio ν, which is completely absent in the beam theory. The analytic solution for v(x, 0) obtained by 2D elasticity does not match the solution for the Euler-Bernoulli beam model in (3.41). Moreover, the cross-section for 2D elasticity model does not remain plane since u(x, y) is a cubic polynomial function (with y 3 ) representing the warping of the cross-sections under the effect of shear. Finally, the change of thickness is present ∀x = cst., since v(x, y) varies with y and thus ε yy = 0. In order to make more meaningful comparison with the beam theory, we introduce ν = 0 into v(x, y) and consider only the displacement along the beam axis (y = 0); thus we obtain P P 2 x (3l − x) + x (3.46) v(x, 0) = 6E I GA

72

3 Beam Models: Refinement and Reduction

with G = E2 and A = 2c. Compared to (3.41), there is one more term, which clearly takes into account the shear strain contribution. This term can be recovered by the Timoshenko beam model that we study next.

3.1.3 Timoshenko Planar Beam Model We assume a rectilinear prismatic beam with a constant cross-section which is placed along x axis. In Timoshenko’s model, the following kinematic hypotheses are made: 1. beam cross-sections initially perpendicular to beam axis remain straight in the deformed configuration, but no longer perpendicular to the beam axis in the deformed configuration; 2. there is no change of thickness in the transverse direction with respect to beam axis. Unlike the Euler-Bernoulli beam model, the straight cross-section in the deformed configuration does not have to remain normal to the beam axis, but it can rotate independently (see Fig. 3.7a). The rotation of the cross-section becomes an unknown field θ (x). The rotational angle γ (x), measured between the cross-section and the axis which remains orthogonal to the beam gives rise to the shear stress. The stress state remains plane with the non-zero components σx and τx y . Kinematics: We study here bending behavior, which is here characterized by two fields: the transverse displacement v(x) at the point which belongs to the beam axis and the cross-section rotation angle θ (x). The beam behavior for a motion u(x) along x axis, is again governed by the truss-bar equations. The displacement of an arbitrary point P in beam bending motion can be expressed as illustrated in Fig. 3.7a: ≈θ

   u (x, y) = −y sin(θ ) p

≈1

   v (x, y) + y = v(x) + y cos(θ ) p

(3.47)

By using the hypothesis of small displacement gradients, where sin(θ ) ∼ = θ and cos(θ ) ∼ = 1 for θ  1, we can further obtain u p (x, y) = −yθ (x) v p (x, y) = v(x)

(3.48)

We use again the linearized kinematics that allows writing the strain-displacement equations ∂θ (x) ∂u p (x, y) = −y ∂x ∂x p (x, y) ∂v p =0 ε yy (x, y) = ∂y ∂v p (x, y) ∂u p (x, y) dv(x) γxpy (x, y) = + = − θ (x) ∂x ∂y dx εxpx (x, y) =

3.1 Reduced Models of Solid Mechanics: Planar…

73

Fig. 3.7 Timoshenko’s beam: a displacements; b average (and true) warpage; c normal stresses σ ; average (and true) shear stresses τ

The kinematic equations can be used to define the curvature κ and shear strain γ defined as dv(x) dθ (x) γ (x) = − θ (x) (3.49) κ(x) = − dx dx Constitutive equations: For constitutive equations, we start again with Hooke’s law of 2D elasticity. Hooke’s law can further be recast in terms of stress resultants, moment M and shear force V , connecting them with the curvature κ and the shear strain γ as generalized deformation measures for Timoshenko’s beam model. Thus, we can obtain  dθ (x) p p σx x = Eεx x ⇒ M(x) = yσxpx d A = −E I = E I κ(x) (3.50) dx A



 τxpy = Gγxpy

⇒

V (x) =

τxpy d A = G A

dv(x) − θ (x) = G Aγ (x) dx

A

(3.51) Equilibrium: We can reuse previously developed equilibrium equations in the case of the Euler-Bernoulli beam model (3.13) and (3.16), for they remain valid given that both beam models have the same stress resultants d V (x) d M(x) + p y (x) = 0 − V (x) = 0 dx dx

(3.52)

74

3 Beam Models: Refinement and Reduction

Strong Form: By introducing internal forces (3.49) and (3.50) into the equilibrium equations (3.52), we obtain the strong form in terms of two coupled differential equations with two unknown finds v(x) and θ (x). We note that the strong form of equilibrium equations are now second-order differential equations (2m = 2), which can be written: d (G Aγ (x)) + p y (x) = 0 dx d (E I κ(x)) − G Aγ (x)) = 0 dx

d 2 v(x) dθ + py = 0 − dx2 dx (3.53)  dv(x) d 2θ EI 2 + GA −θ =0 dx dx 

⇒ ⇒

GA

In order to obtain the unique solution, we need to impose two boundary conditions at each end of the beam, which can be related to either the values of kinematics variables v(x) and θ (x), or stress resultants M(x) and V (x). We note that the boundary conditions now concern only the first derivatives of state variables, and will not place any constraint on the second derivative featuring in the strong form v, θ (essential conditions), V, M (natural conditions).

(3.54)

Weak Form: Weak or integral form can be developed in rather similar manner as done in (3.22) for the Euler-Bernoulli beam model by employing the principle of virtual work. This time an additional term of internal virtual work is included which pertains to the shear forces V and conjugate virtual shear strain δγ l

l (Mδκ + V δγ )d x =

0

p y δvd x + δTBC

(3.55)

0

with the essential boundary conditions imposed v = v¯ and θ = θ¯ . Combined axial and bending response: the longitudinal deformation governed by truss-bar equations can be included within Timoshenko’s beam model. We can rewrite constitutive law for such combined behavior in matrix form ⎡ ⎤ ⎡ ⎤⎡ ⎤ N EA 0 0 ε N = E Aε V = G Aγ M = E I κ ⇒ ⎣ V ⎦ = ⎣ 0 G A 0 ⎦ ⎣γ ⎦ (3.56) M 0 0 EI κ We can also update the principle of virtual work with an additional term which pertains to the virtual work of axial force N and virtual strain δε

3.1 Reduced Models of Solid Mechanics: Planar…

l

75

l (N δε + V δγ + Mδκ)d x =

0

( px δu + p y δv)d x + δTBC

(3.57)

0

together with boundary conditions imposed on u, v, θ . Let us make the same comments on the main features for Timoshenko’s beam model. With including axial deformation, we can count three strain measures in the Timoshenko beam model, dilatation ε, shear γ and curvature κ. The shear stress is constant over the cross-section; but the cross-section does not have any warping. It turns out that the cross-section will be on average in the true warping as illustrated in Fig. 3.7b. Furthermore, over the cross-section τ (= τx y ) = cst., as shown in Fig. 3.7c, and again equal the average of the true shear stress distribution. We recall that the true shear stress has a parabolic distribution in a rectangular cross-section, and that it should cancel on the upper and lower beam fibbers. Finally, the curvature κ for Timoshenko’s beam does not represent the curvature of the deformed beam axis, contrary to the case of the Euler-Bernoulli beam, but rather the change of crosssection rotation. Application to the cantilever beam: Returning to the cantilever beam problem, illustrated in Fig. 3.6, we first calculated θ (x) by integrating (3.49), with M = −P(l − x) and the boundary condition θ (0) = 0, where we get dθ (x) M(x) = −E I dx

 ⇒

P(l − x) Px  x dx = l− + c1 (3.58) EI EI 2

θ (x) =

θ (x = 0) = c1 = 0

⇒

θ (x) =

x Px  l− EI 2

(3.59)

With the last result on hand we can further integrate (3.49) to find displacement v(x) with imposed shear force V (x) = P and boundary condition v(0) = 0, hence  V (x) = G A v(x) =

dv(x) − θ (x) dx





 ⇒

v(x) =

P

1 x  x + l− dx GA EI 2 (3.60)

P 2 P x (3l − x) + x + c1 v(x = 0) = c1 = 0 6E I GA ⇒

v(x) =

P P 2 x (3l − x) + x 6E I GA

(3.61)

(3.62)

Now, we can match the result obtained for 2D elasticity (with ν = 0), since the Timoshenko beam model includes the effect of shear.

76

3 Beam Models: Refinement and Reduction

Remarks on beam theory inconsistencies: Despite this improved result for the cantilever beam by Timoshenko’s beam, one can find several deficiencies in both beam theories. (1) the (kinematic) assumption of non-deformable cross-section implies ε yy = εzz = 0; by introducing these values into Hooke’s law for 3D solid mechanics, we get E E (1 − ν)εx x σ yy = σzz = νεx x (1 + ν)(1 − −2ν) (1 + ν)(1 − −2ν) (3.63) resulting in typical values of the plane strain case! Here σx x = Eεx x and σ yy = 0 and σzz = 0, unless ν = 0, which is extremely rare for any real material. However, this hypothesis is limited to the beam kinematics, made to simplify expression; thanks to it v(x, y) ≡ v(x). On the other hand, the static assumption σ yy ∼ = 0 and σzz ∼ = 0, introduced in Hooke’s 3D law, provides ε yy + εzz = 2νεx x , then σx x = Eεx x exactly, as it should be. σx x =

(2) in any case there is a contradiction between these two hypotheses! However, we can demonstrate that the resulting error is more than one order of magnitude smaller than the exact magnitudes, as long as the beam is sufficiently slender (l > 10h). (3) in Timoshenko’s beam model, shear stress τ is constant over the beam crosssection, which is in severe contradiction with the boundary conditions on the top and bottom fibbers! This is correct again; since the kinematic assumption is only approximate (the section remains flat on average through the warping), resulting τ are also approximate. Just know it! More refined theories have been developed (e.g., see [175–178, 182, 204, 228]), but the resulting computational cost of a more complex versus beam-reduced model is often not justified for the required accuracy.

3.1.4 Brief on Reissner Planar Beam Model We assume a beam of length l and constant cross-section A along the length. The rectilinear beam is positioned along x axis. In Reissner’s beam model, the following kinematic hypotheses are made: 1. straight cross-sections remain straight in the deformed configuration, but not perpendicular to the beam axis in the deformed configuration; 2. no change in thickness with transverse normal stress that is negligible; 3. the deformed beam configuration remains unknown and not close to initial configuration, for displacements and rotations can be large.

3.1 Reduced Models of Solid Mechanics: Planar…

77

Fig. 3.8 Reissner’s beam: a displacements; b average (and true) warping; c normal stresses σ ; average (and true) shear stresses τ

Also, the straight cross-section in the deformed configuration doesn’t have to remain orthogonal to the beam axis, but it rotates, and this time with possibly large rotation. In a general sense, Reissner’s beam model can be understood as the counterpart of Timoshenko’s beam model in the finite strain theory, where obviously small strain approximations are not valid. Hence, we can no longer take into account that initial and deformed beam configurations are the same as in small displacement gradient theory, but we should make a distinction between them. Here, we will consider shear and bending of the Reissner beam subjected to the large displacement and rotation regime (see Fig. 3.8). Kinematics: We study here planar straight beam, where the beam axis in initial configuration is placed along x and cross-section along y. The beam axis in deformed configuration is obtained with transverse displacement v(x) and axial displacement u(x) and the beam cross-section is placed by rotation denoted as θ (x). Displacement of any arbitrary point P which belongs to the beam cross-section can be expressed by intrinsic parametrization via generalized coordinates u(x), v(x) and θ (x). According to the kinematic hypothesis illustrated in Fig. 3.8, we can write u p (x, y) = u(x) − y sin θ (x); v p (x, y) = −y + v(x) + y cos θ (x)

(3.64)

We note that in expression (3.64) the displacements consist of the parts related to both translation and rotation of the cross-section. In order to develop a strain-displacement relationship we should first introduce deformation gradient F, which is defined as identity I plus displacement gradient (e.g., see [176]):

78

3 Beam Models: Refinement and Reduction

 ∂u p (x,y) F := I +

∂u p (x,y) ∂x ∂y ∂v p (x,y) ∂v p (x,y) ∂x ∂y

 =

1+

du(x) − y dθ(x) cos θ (x) dx dx dv(x) dθ(x) − y sin θ (x) dx dx

− sin θ (x) (3.65) cos θ (x)

The deformation gradient is a two-point tensor or a linear transformation which maps a vector from the initial to the deformed configuration. The infinitesimal strain measure ε can no longer be employed, since the kinematic hypotheses mentioned previously are not valid in large displacement and rotation gradients. The finite strain setting inherently consists of the rigid body motion which can pertain to both translation and rotation, and the stretch which measures deformation. Hence, we would like to separate the rigid body motion from pure deformations, which are the actual source of stresses. This can be done by using the polar decomposition (e.g., see [176]) of deformation gradient F, which provides multiplicative split between the stretch U providing an intrinsic strain measure for large strains, and the rigid body rotation R, (3.66) F = RU ⇒ U = RT F In Reissner’s beam formulation, the large rotation tensor R is orthogonal tensor that can be represented in terms of rotation angle θ (x), which gives us direct manner to compute the stretch tensor U according to: R=

cos θ (x) − sin θ (x) sin θ (x) cos θ (x)

;

) cos θ (x) + dv(x) sin θ (x) − y dθ(x) 0 (1 + du(x) dx dx dx U= du(x) dv(x) −(1 + d x ) sin θ (x) + d x cos θ (x) 1

(3.67)

The stretch tensor U is but one of many finite strain measures that we can introduce in geometrically nonlinear setting (e.g., see [175, 176], each one adapted to a particular problem. Here, the most suitable strain measure is the so-called Biot strain measure H, which can be related to generalized strains for Reissner’s beam [322] defined as ⎡

⎤ Σ K       ⎢ ⎥ ⎢ 1 + du(x) cos θ (x) + dv(x) sin θ (x) − 1 −y dθ (x) 0⎥ ⎢ ⎥ dx dx dx ⎥ H := U − I = ⎢ Γ ⎢ ⎥     ⎢ ⎥ ⎣ ⎦ dv(x) du(x) sin θ (x) + cos θ (x) 0 − 1+ dx dx (3.68) Hence, we can extract three strain components for Reissner beam geometrically exact model which are yet called rotated strain measures for flexible beam

3.1 Reduced Models of Solid Mechanics: Planar…

 dv(x) du(x) cos θ (x) + Σ(x) = 1 + sin θ (x) − 1 dx dx  dv(x) du(x) sin θ (x) + cos θ (x) Γ (x) = − 1 + dx dx dθ (x) K (x) = dx

79

(3.69)

It can easily be verified that the rotated strain measure components for the Reissner beam can be reduced to the Timoshenko beam generalized strains by consistent linearization assuming small displacement gradients and small rotations; we thus obtain Σ(x) ≈ ε(x) := du(x) |1 | du(x) dx dx dv(x) ⇒ (3.70) Γ (x) ≈ γ (x) := − θ (x) | dv(x) |  1 dx dx |θ (x)|  1 K (x) ≈ κ(x) := dθ(x) dx Stress tensors: In geometrically nonlinear theory under applied loads the deformed configuration changes significantly, so that we can no longer use the initial configuration as a sufficiently accurate replacement in order to express the Cauchy (or true) stress σ , which is acting in the deformed configuration. Thus, the Reissner beam poses a more difficult problem than the Timoshenko or the Euler-Bernoulli beam, because the deformed configuration is unknown and so is the stress. The Lagrangian formulation (e.g., see [176]) comes to the rescue by reducing the number of unknowns with expressing the stress (replacement) in the initial configuration, which is well known for a given structure. Like for large strain measures, there is a lack of uniqueness in the true stress representation; in fact, as many different kinds of stress can be introduced to match each different large strain measure. Here, we will only be interested in stress work-conjugate to rotated strain already introduced in Reissner’s beam theory development. The stress representation of this kind can be derived by starting with corresponding replacement of the true or Cauchy stress in strong form of equilibrium equations, which is known (e.g., see [176]) as the first Piola-Kirchhoff stress P; the latter is defined from true stress σ with P = J σ F−T

J = det|F|

(3.71)

where J is Jacobian or determinant of the deformation gradient F. The stress tensor P is also featuring in the Cauchy principle to produce the traction vector acting in deformed configuration, but operating on the unit normal in the initial configuration (e.g., see [176]). For Reissner’s beam placed along x axis (as shown in Fig. 3.8), the corresponding traction vector acting on the cross-section in the deformed configuration will have only components [P11 P21 ]T . Given the Reissner beam hypothesis where the cross-section is placed in a deformed configuration by rotation tensor R, we can provide the corresponding replacement of the first Piola-Kirchhoff stress P by Biot stress T, which is work-conjugate to the Biot strain measure H, simply by rotation. Namely, by comparing two different manners to write the Cauchy stress vector with the stress components acting in the cross-section of Reissner’s beam

80

3 Beam Models: Refinement and Reduction

with unit normal e1 in the initial configuration, we can obtain

T e1 = R P e1

⇔

T



 cos θ (x) sin θ (x) P11 T11 = T21 − sin θ (x) cos θ (x) P21

(3.72)

where [T11 T21 ]T are the traction components represented over initial configuration. Constitutive behavior: We can further define the constitutive equation of linear elasticity for Reissner’s beam, by assuming Hook’s law for work-conjugate pairs of rotated strain measures and corresponding stress resultants of Biot stress. The latter can be used to define the internal forces very much in the spirit of Timoshenko’s beam theory, which counts axial force N , shear force V and bending moment M; namely, we can write 

 T11 d A =

N (x) := A

A

 V (x) :=

EΣ(x) d A = E A Σ(x) 

T21 d A = A



M(x) := −

GΓ (x) d A = G A Γ (x) A



yT11 d A = − A

(3.73)

y E K (x) d A = E I K (x) A

Equilibrium: By introducing Biot stress T representation into equilibrium equations (3.13) and (3.16) instead of Cauchy stress σ , we can rewrite equilibrium equations for Reissner’s beam model with respect to the initial configuration. First, for external distributed load px and p y , we obtain the force equilibrium equations with N for normal force and V for shear force  d N p (3.74) R + x =0 p V dx y which should be accompanied by the moment equilibrium equation for bending moment M with distributed external moment m z dϕ dM N − × RN + m z = 0 ; N = V dx dx

(3.75)

We note that all loads are parameterized with respect to coordinates in the initial configuration with px (x), p y (x), m z (x). Strong form: By introducing expression for internal forces in (3.73) into the equilibrium equations in (3.74) and (3.75), we obtain three coupled nonlinear differential equations. The first two are force equilibrium equations written in terms of three unknowns u(x), v(x) and θ (x). We note that these three equations remain tightly coupled, with both displacement components that appear in all of them. It is only the linearized form of these equations that allows uncoupling the bending and axial

3.2 Beam Model Refinement and Reduction

81

deformation modes and to recover the strong form of Timoshenko’s beam model in bending already stated in (3.53). The unique solution to these equations require to impose the boundary conditions, both on primary variables u, v, and θ for the essential conditions and on dual variables N , V and M for the natural boundary conditions. However, the exact solution is very difficult to find, except for a few academic problems. One of them is pure bending of a cantilever beam with constant ¯ which leads to a linear properties (E I = cst.) under free-end bending moment M, ¯ variation of rotation field along the beam and constant bending strain κ = EMI ; thus, any deformed configuration is a part of a circle (e.g., see [170, 287]): κ(x) :=

dθ (x) M¯ M¯ x = = cst. ; θ (0) = 0 ⇒ θ (x) = dx EI EI

(3.76)

Weak form: For most other problems to be solved with Reissner’s beam model, one can only find the numerical solution to the weak or integral form of these equations. The latter can be obtained by the principle of virtual work featuring the resultants of Biot stress and variation of rotated strain measure (e.g., see [191, 221]). We can write l

l (N (x) δΣ(x) + V (x) δΓ (x) + M(x)δ K (x)) d x =

0

( px δu(x) + p y δv(x) 0

+ m z δθ (x)) d x + δTBC (3.77) where δu, δv and δθ have to satisfy homogeneous essential boundary conditions on the Dirichlet boundary. It remains rather difficult to construct non-locking finite element interpolations for geometrically nonlinear problems, except for geometrically nonlinear Reissner beam where optimal finite element approximation can deliver the exact solution for pure bending problem (e.g., see [170, 287]); see Fig. 3.9 for illustrative result computed with bending moment value that produces deformed shape as half a circle. More on this in later chapters.

3.2 Beam Model Refinement and Reduction In the previous section, we studied the reduced beam model that transforms the solid mechanics problem into the structural mechanics problem by using either the EulerBernoulli or the Timoshenko beam model. Here, we study the discrete approximation, which is constructed by using the finite element method [26, 153, 390] to transform the weak form of structural mechanics problem into discrete system analysis. In a mathematical sense, we switch from a set of differential equations to a set of algebraic equations within the framework of static analysis [159]. The basic advantage of such an approximation procedure is its ability of constructing the discrete model as an assembly of structure constituents (or finite elements) no matter what complexity

82 Fig. 3.9 Deformed shape of a cantilever with constant properties in pure bending motion under free-end moment M¯ = E I π computed with Reissner’s beam model, resulting with θ(l) = π

3 Beam Models: Refinement and Reduction

DISPLACEMENT 6 0.00E+00 2.62E-01 5.24E-01 7.85E-01 1.05E+00 1.31E+00 1.57E+00 1.83E+00 2.09E+00 2.36E+00 2.62E+00 2.88E+00 3.14E+00

Time = 1.00E-01

of the structural system might be [160]. Such an approach is used for computing the structure stiffness matrix, as the crucial result of the finite element technology. The assembly procedure of finite element stiffness matrices is presented in a clear manner by using the commands of the computer program CAL [383]. Here, a detailed illustration concerns the class of problems that deserve special attention by structural engineers pertaining to 3D frame analysis and enhancements of the basic beam element used for modeling different model refinements. More precisely, in this section, we first present a method for the systematic construction of the stiffness matrix for an arbitrary beam finite element in 3D space. The stiffness matrix of the basic beam model is constructed by applying a series of elementary transformations. The procedure of this kind is capable of including a number of model refinements (addition of shear deformation, variable cross-section etc.) that are not easily accessible to the standard displacement-based finite element method. We also show that such a procedure still applies even if we need to modify the beam element stiffness matrix in order to accommodate different constraints, such as point constraints in terms of joint releases (or hinges) for moments or shear forces (e.g., see [285]). We show that the reduction at the element (local) level provides a more effective approach than the alternative one at the structure (global) level having initially increased by one the nodal degrees of freedom for each new release. We note in passing that the proposed approach applies to geometrically nonlinear framework [207] and it can also be extended to materially nonlinear case [67]. Finally, we elaborate upon the reduction method imposing the length-invariant global constraints to frame structure elements, where each element can have an arbitrary position in space. This allows recovering 3D version of technical deformation theory [241] imposed on 3D beam finite elements. Several numerical examples are used to illustrate the performance of the proposed procedures. The computations are carried out by a modified version of computer code CAL [383].

3.2 Beam Model Refinement and Reduction

83

3.2.1 Method of Direct Stiffness Assembly for 3D Beam Elements Static analysis of a discrete model which employs the assembly procedure (e.g., see [176]) inevitably reduces to the solution of a system of linear algebraic equations, which are written in matrix notation as Kd = f

(3.78)

where K is the global stiffness matrix, d the displacement vector and f the external force vector. By using the so-called direct stiffness method, the global stiffness matrix K is obtained as the assembly of the element stiffness matrices Ke as follows K=



Le T Ke Le ; de = Le d ; ; f = Le, T fe

(3.79)

e

This result for the global stiffness matrix K represents a symbolic summation of elementary stiffness matrices Ke which are placed in the corresponding slots as determined by each element connectivity matrix Le . More precisely, for a 2-node beam element, with 3 displacements and 3 rotations at each node, the vector collecting element nodal degrees of freedom de = [de1 , de2 ]T has in total 12 components, which implies that each connectivity matrix Le is rectangular matrix of size (12 × n eq ), where n eq is the total number of equations corresponding to free nodal displacements and rotations gathered in structure displacement vector d. In assembly procedure (3.79) local and global displacements are defined in the same coordinate system (here the Cartesian coordinate system), so that each connectivity matrix is reduced to the Boolean matrix with entries either one or zero, i.e. only identifying the correspondence between local and global displacements. Thus, as already explained for the truss-bar model, one can be operating only on non-zero terms of each connectivity matrix in order to make the finite element assembly procedure (much) more efficient. Therefore, we will only focus on the challenge of how to produce the beam element stiffness matrix Ke that must finally be defined in the global coordinate system.

3.2.1.1

3D Beam Element Stiffness Matrix

We first derive the element stiffness matrix K p for a beam element in a local coordinate system, with the beam axis aligned with the local x-axis. In such a case, for the EulerBernoulli beam we construct discrete approximation by employing the so-called Hermite polynomials to interpolate the real and virtual transverse displacement fields from nodal values p = [ p1 , p2 , . . . , p12 ]T shown in Fig. 3.10 u 2 (x) = H1 (x) p2 + H2 (x) p6 + H3 (x) p8 + H4 (x) p12 u 3 (x) = H1 (x) p3 + H2 (x) p5 + H3 (x) p9 + H4 (x) p11

(3.80)

84

3 Beam Models: Refinement and Reduction

p11

p5 p2 p4

p1 p3

p8

u2 (x) u1 (x)

p7

u3 (x)

p6

p10

p9 p12

Fig. 3.10 Nodal degrees of freedom in local coordinate system p = [ p1 , p2 , ...., p12 ]T for 2-node beam element with 6 degrees-of-freedom (dof) per node, 3 displacements and 3 rotations

where Hermite polynomials H1 (x), ..., H4 (x) can be written as H1 (x) = 1 − 3(x/l)2 + 2(x/l)3 ; H2 (x) = (x/l)3 − 2(x/l)2 + x/l ; H3 (x) = −2(x/l)3 + 3(x/l)2 ; H4 (x) = (x/l)3 − (x/l)2

(3.81)

The Hermite polynomials of this kind are the exact solution to the Euler-Bernoulli beam element exposed to nodal loads (corresponding to a row or column in the element stiffness matrix) producing only one generalized displacement equal to one and all the others equal to zero. Moreover, due to the property of superconvergence (e.g., see [176]), the exact values of the generalized nodal displacements will also be obtained for a more general load case, as long as the equivalent nodal loads are also computed from external virtual work with virtual displacements interpolated with the same Hermite polynomials. For axial displacement and torsional rotation of the beam, we use linear polynomials that are exactly the same as those used for the truss-bar element studied in the previous chapter. u 1 (x) = N1 (x) p1 + N2 (x) p7 (3.82) θ1 (x) = N1 (x) p4 + N2 (x) p10 where we have N1 (x) = 1 − x/l ; N2 (x) = x/l

(3.83)

We note in passing that such a difference in the order of polynomials used for transverse displacement versus axial displacement can lead to locking phenomena for a more general constitutive model where axial and bending deformations are coupled. Such a locking problem can be solved by increasing the axial displacement variation to quadratic (e.g., see [72]). The components of the stiffness matrix can be computed by replacing the chosen finite element approximations with linear or Hermite polynomials into the virtual work of internal forces and integrating along the beam length; a partial list of the corresponding computations resulting in non-zero values can be written as

3.2 Beam Model Refinement and Reduction

K p (1, 1) = K p (1, 7)

 l

l 0

d N1 dx

K p (10, 10) = K p (2, 2) = K p (2, 8) = K p (3, 3) = K p (3, 9) = ...

d N1 dx

l 0 l 0 l 0 l 0

E A ddNx1 d x ; K p (7, 7) =

E A ddNx2 d x ; K p (4, 4) = l 0

85

d N2 G J ddNx2 dx

l 0

l 0

d N2 dx

d N1 G J ddNx1 dx

d x ; K p (4, 10) =

2 d 2 H1 (x) E I d dHx12(x) dx2

E A ddNx2 d x

l 0

d x ; K p (2, 6) =

d N1 G J ddNx2 dx

l 0

d H1 (x) E I d dHx32(x) dx2

d x ; K p (2, 12) =

2 d 2 H1 (x) E I d dHx12(x) dx2

d x ; K p (3, 5) =

2 d 2 H1 (x) E I d dHx32(x) dx2

d x ; K p (3, 11) =

2

2

dx

l 0

dx

2 d 2 H1 (x) E I d dHx22(x) dx2

l 0

d H1 (x) E I d dHx42(x) dx2 2

2

2 d 2 H1 (x) E I d dHx22(x) dx2

l 0

dx

(3.84)

dx

dx

2 d 2 H1 (x) E I d dHx42(x) dx2

dx

The end result of these integrations stemming from the principle of virtual work can be written in matrix notation as (3.85) fp = Kpp where f p is the vector of equivalent nodal forces (stress resultants) conjugate to the generalized nodal displacements p shown in Fig. 3.10, and K p is the corresponding element stiffness matrix in local coordinate system. We recall here the componentwise solution for the stiffness matrix K p for a 3D beam element with length l, constant cross-section properties (A, I, J = cst.) and homogeneous material (E, G = cst.) that can be written as: ⎤ ⎡ EA 0 0 0 0 0 − ElA 0 0 0 0 0 l 12E Iz Iz 6E I ⎥ ⎢ 0 0 0 6El 2Iz 0 − 12E 0 0 0 ⎥ ⎢ l3 l2 l2 ⎥ ⎢ 12E I y 6E I y 12E I y 6E I y 0 − 0 0 0 − 0 − 0 ⎥ ⎢ 3 2 3 2 l l l l ⎥ ⎢ GJ GJ 0 0 0 0 0 − 0 0 ⎥ ⎢ l l ⎥ ⎢ 4E I y 6E I y 2E I y ⎥ ⎢ 0 0 0 0 0 l l2 l ⎥ ⎢ 4E Iz 6E Iz 2E Iz ⎥ ⎢ 0 − l2 0 0 0 l l ⎥ ⎢ Kp = ⎢ EA ⎥ 0 0 0 0 0 ⎥ ⎢ l 12E Iz 6E Iz ⎥ ⎢ 0 0 0 − 3 2 ⎥ ⎢ l l ⎥ ⎢ 12E I y 6E I y sym. 0 0 ⎥ ⎢ l3 l2 ⎥ ⎢ GJ 0 0 ⎥ ⎢ l ⎥ ⎢ 4E I y ⎣ 0 ⎦ l 4E Iz l

(3.86) In order to carry out the structural assembly in (3.79), the beam finite element stiffness matrix in (3.86) has to further be transformed into global coordinate system

86

3 Beam Models: Refinement and Reduction

de11

Fig. 3.11 Nodal degrees of freedom in global coordinate system de = (d1e , d2e , ...., e )T for 2-node beam d12 element (6 dof per node, with 3 displacements and 3 rotations)

de8 e e de9 d7 d10

y x z

de5

de12

Y

de2 de6

de3 de1

de4

X Z

with axes X, Y i Z (see Fig. 3.11). This is a usual element transformation from local to global coordinate system that is carried out by using an orthogonal rotation matrix T (e.g., see [32],[176]) that gathers local base vectors (gi , i = 1, 2, 3) components representation within global coordinate system T  T = g1 g2 g3

(3.87)

Namely, we first establish the relationship between the nodal generalized displacements p defined in a local coordinate system x, y, z and the nodal generalized displacements d defined in the global coordinate system X, Y, Z by using transformation matrix T pd p = T pd de , T pd = diag[T, T, T, T] (3.88) where T pd is a 12 × 12 block diagonal matrix whose 3 × 3 blocks represent the inclinations of local coordinate system with respect to the global one [32]. We note in passing that the same kind of transformation connects the two sets of stress resultants expressed in global and local coordinate systems with fe = TT f p , which can be easily obtained from equilibrium equations. Thereafter, the principle of virtual work can also be expressed in the global coordinate system, and for any arbitrary virtual displacement we can finally obtain fe = Ke de , Ke = TTpd K p T pd

(3.89)

Since the element stiffness matrix Ke for beam element nodal degrees of freedom d is now derived in the global coordinate system X,Y,Z, it can directly be assembled into the structural stiffness matrix K, where each element contribution in equation (3.79) is used to overwrite the previous value of the global stiffness matrix; this can be written as e = 1, 2, . . . n el T (3.90) K ← K + Le Ke Le e

3.2 Beam Model Refinement and Reduction

87

The result for the element stiffness matrix obtained in this section remains valid only for a beam with uniform geometrical and mechanical properties. The question of model refinement of this basic beam model (adding shear or internal constraints) is studied next. This method can be used combined with Euler’s beam model presented here; an alternative that accounts for shear deformation would require Timoshenko’s beam model with non-conventional interpolations to be studied in the next section (see also [164]).

3.2.2 Beam Model Refinement: Flexibility Approach for Reduced Model in Deformation Space We show here that the beam element stiffness matrix for a more general case with variable cross-section and material properties also accounting for shear deformation can be obtained at no extra cost nor additional degrees of freedom by using the flexibility matrix approach. The main idea is to start with the reduced model pertaining to the deformation space where rigid body modes for a 2-node beam element in 3D setting are removed by the minimum number of six supports; see Fig. 3.12. In such a deformation space, we can first define the flexibility matrix, which is the inverse of the corresponding stiffness matrix. The flexibility matrix components can be computed exactly by complementary virtual work or Mohr’s analogy [241]. This can be done at no extra cost for any variation in mechanical or geometric beam properties and furthermore accounting for shear deformation. Namely, we can extend the computations of the flexibility matrix components already presented for truss-bar in (2.67) to account for all internal forces in a beam with l  Fi j = 0

Ni (x) N j (x) Vi (x) V j (x) y Mi (x) y M j (x) + + E(x)A(x) A(x) G(x)A(x) A(x) E(x)I (x) I (x)

dx

(3.91) where internal forces Ni , Vi , M j and N j , V j , M j correspond to unit forces applied at nodes i and j, respectively. The stiffness is then computed as the inverse of such a flexibility matrix. Finally, the stiffness matrix in reduced deformation space is then mapped into the full 3D form by employing the corresponding transformation between two different sets of generalized displacements. Although this procedure can be used for a general beam with variable properties, we will briefly illustrate it on a straight 3D beam element with constant properties in Fig. 3.12. First, we define the minimum of six supports removing six rigid body modes in a 2-node beam element. A natural choice restraining three displacements at one end, two displacements at another and one rotation in the local coordinate system (see Fig. 3.12) for which x axis is directed along the beam length while y and z axes are placed in cross-section plane perpendicular to the beam axis.

88

3 Beam Models: Refinement and Reduction

v1

v2 y

x

z

v5 v6

v3

v4 G, E, A, Iy, Iz, J

Fig. 3.12 Pure deformation degrees of freedom for 2-node beam element in the local coordinate system with the minimum number of supports

The generalized nodal displacements in reduced deformation space for a 2-node beam element are defined according to the following order (see Fig. 3.12): v1 and v2 are rotations around y axis at both element ends, v3 and v4 are the corresponding rotations around z axis, v5 is the rotation around x at the element free-end and finally v6 represents the the extension at the same element end. The displacements v1 to v6 are six generalized (independent) beam displacements induced by deformation, which are separated from six degrees of freedom corresponding to the rigid body modes. The rigid body displacements are restrained with the minimum number of six beam element supports (see Fig. 3.12). The idea of distinguishing between generalized deformation modes and rigid body modes is similar to the ’natural approach’ presented in [9] for large displacements and small strains, but the choice of deformation modes in our case is different from the one made in [9]. With the flexibility approach, we start by first computing the beam flexibility matrix for the unknown displacements described in Fig. 3.12. The computation of the entries of the flexibility matrix Fv for a straight or a curved spatial beam with variable properties can again exactly be performed by employing some established methods, such as Mohr’s analogy or the method of conjugate structure (e.g., see [201]). We can thus obtain the geometrically exact results of this kind (exact within the framework of small displacement gradient theory). For the simplest case of a beam with constant properties and without shear, we find the 6 × 6 flexibility matrix Fv ⎡

l l 3E I 6E I l l 6E I 3E I

⎢ ⎢ ⎢ 0 Fv = ⎢ ⎢ 0 ⎢ ⎣ 0 0

0 0 0 0

0 0

0 0

l l 3E I 6E I l l 6E I 3E I

0 0

0 0

0 0 0 0 l GJ

0

0 0 0 0 0

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(3.92)

l EA

The corresponding form of the stiffness matrix in reduced space is obtained by inverting this flexibility matrix to get Kv = F−1 v

(3.93)

3.2 Beam Model Refinement and Reduction

89

⎡ 4E I

where

y 2E I y l l 2E I y 4E I y l l

⎢ ⎢ ⎢ ⎢ Kv = ⎢ 0 ⎢ 0 ⎢ ⎣ 0 0

0 0 0 0

0 0

0 0

4E Iz 2E Iz l l 2E Iz 4E Iz l l

0 0

0 0

0 0 0 0 GJ l

0

⎤

0 0 0 0 0

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(3.94)

EA l

The stiffness matrix Kv in equation (3.94) is given for a straight beam with constant mechanical and geometrical properties. The first four rows of this matrix, for our choice of generalized displacements (deformations), can be obtained directly from the equations of Takabeya (see [241]), whereas the 5th and 6th equation represent the well-known torsional and axial beam stiffness. For a curved beam with arbitrarily variable properties Kv will in general be a fully populated matrix. One can use this kind of stiffness matrix to restate the discrete form virtual work principle in deformation space, resulting in the corresponding relation between generalized displacements and the work-conjugate forces in deformation space that can be written as (3.95) fv = Kv v The same result can be restated in full space (with both deformation and rigid body modes) in order to recover the results we wrote in (3.85). Namely, given the finite element approximation in (3.80) one can develop mapping between displacements v1 to v6 and displacements p1 to p12 show in Figs. 3.10 and 3.12 to obtain the corresponding transformation v = Tvp p , v = [v1 , . . . , v6 ]T , p = [ p1 , . . . , p12 ]T

(3.96)

For the choice of deformation modes made here, the transformation matrix Tvp takes the following form ⎡

Tvp

0 ⎢ 0 ⎢ ⎢ 0 =⎢ ⎢ 0 ⎢ ⎣ 0 −1

0 − 1l 0 − 1l 1 0 l 1 0 l 0 0 0 0

0 0 0 0 −1 0

1 0 0 0 0 0

0 0 1 0 0 0

0 0 0 0 0 1

0 0 − 1l − 1l 0 0

1 l 1 l

0 0 0 0

0 0 0 0 1 0

0 1 0 0 0 0

⎤ 0 0⎥ ⎥ 0⎥ ⎥ 1⎥ ⎥ 0⎦ 0

(3.97)

If we finally apply the principle of virtual work (e.g., see [32]) on the previously shown beam, constructing discrete approximation for virtual displacements (denoted with hat) in the same fashion as for real displacements (Figs. 3.10 and 3.12), then we can write  pT f p =  vT fv (3.98)

90

3 Beam Models: Refinement and Reduction

where f p is a set of generalized forces which are work-conjugate to generalized displacements described in Fig. 3.10. Consecutively exploiting the Eqs. (3.96), (3.95) and again (3.96) we wind up with   T  pT f p − Tvp Kv Tvp p = 0

(3.99)

Since virtual displacements are mutually independent (which is equivalent to the fundamental lemma of variational calculus in discrete formulation [147]), the Eq. (3.99) results with T Kv Tvp (3.100) f p = K p p , K p = Tvp which confirms the explicit format of the stiffness matrix K p already written in (3.86). We can now include the shear deformation contribution to Euler-Bernoulli beam stiffness, by exploiting the flexibility approach to modify the element stiffness and replace the result in (3.94). For a 3D beam element with deformation degrees of freedom shown in Fig. 3.12, we first start by recomputing the components of the flexibility matrix Fv to include displacements due to the shear, which leads to: ⎡

l + kz G1 Al 3E I y ⎢− l + 1 ⎢ 6E I y k z G Al

⎢ ⎢ Fv = ⎢ ⎢ ⎢ ⎣

0 0 0 0

− 6El I y + kz G1 Al 0 0 1 l + kz G Al 0 0 3E I y l 1 l + − + k y G1 Al 0 3E Iz k y G Al 6E Iz 0 − 6El Iz + k y G1 Al 3El Iz + k y G1 Al 0 0 0 0 0 0

0 0 0 0 l GJ

0

0 0 0 0 0 l EA

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(3.101) By inverting this matrix we get a new stiffness matrix in deformation space Kv that includes the shear deformation contribution. Finally, exploiting the Eqs. (3.97) and (3.100) we can determine the element stiffness matrix K p with shear deformation included, which is written as ⎤ ⎡ Kx 0 0 0 0 0 −K x 0 0 0 0 0 ⎢ 0 0 K y2 ⎥ K y1 0 0 0 K y2 0 −K y1 0 ⎥ ⎢ ⎢ K z1 0 −K z2 0 0 0 −K z1 0 −K z2 0 ⎥ ⎥ ⎢ ⎢ T 0 0 0 0 0 −T 0 0 ⎥ ⎥ ⎢ ⎢ 0 0 K z2 0 K z4 0 ⎥ K z3 0 ⎥ ⎢ ⎢ 0 0 K y4 ⎥ K y3 0 −K y2 0 ⎥ Kep = ⎢ ⎢ Kx 0 0 0 0 0 ⎥ ⎥ ⎢ ⎢ 0 0 0 −K y2 ⎥ K y1 ⎥ ⎢ ⎢ sym. K z1 0 K z2 0 ⎥ ⎥ ⎢ ⎢ T 0 0 ⎥ ⎥ ⎢ ⎣ 0 ⎦ K z3 K y3 (3.102)

3.2 Beam Model Refinement and Reduction

91

where we write explicitly Kx =

EA ; l (1+0.25φ y )4E Iz K y3 = (1+φ y )l T = Gl J ;

K y1 = ; K y4 = K z1 =

12E Iz ; (1+φ y )l 3 (1−−0.5φ y )4E Iz (1+φ y )l 12E I y ; (1+φz )l 3

6E Iz (1+φ y )l 2 12E Iz ; k y G Al 2

K y2 = ; φy =

; (3.103)

...

and note that the remaining coefficients are easily obtained by subscript interchange. We mention that k y A is the effective shear for transverse shear deformation in y direction (and k z A is analogous for z direction). Finally, if the beam is not placed along x axis, we ought to transform again this stiffness matrix into the global coordinate system in order to obtain the beam element stiffness matrix Ke , which will be ready for the finite element assembly procedure. It is clear that the proposed approach can easily include or exclude the shear deformation contribution will not change the number of degrees of freedom. With the development in this section, we presented how to compute the element stiffness matrix Ke through a series of elementary transformations which enhance the stiffness matrix in the deformation space (the one with all rigid body modes restrained). This facilitates the construction of Ke matrix as well as the understanding of the principles of that construction. Additionally, this approach enables us to form a stiffness matrix for a special beam element that enforce constraints of joint release or length invariance, as shown next.

3.2.3 Beam Model Reduction: Joint Releases and Length Invariance Here, we present an additional transformation of the beam element stiffness matrix, which is imposed by further constraints that can allow for model reduction. Two illustrative cases are considered: joint releases or hinges and length invariance constraint.

3.2.3.1

Beam Element Reduction for Joint Releases (Hinges)

The developments in this section are carried out again in the most general setting of global coordinates and illustrated on the beam element shown in Fig. 3.11. We want to include at the second node of this element a ’hinge’, or more precisely a spherical joint release which precludes the possibility to resist any of three bending moment components in that particular node, The corresponding weak form discrete approximation in Eq. (3.89) can then be restated as follows

fed,n 0



Kenn Kenz = Kezn Kezz



dn dz

(3.104)

92

3 Beam Models: Refinement and Reduction

where the vector dn = [d1 , . . . , d9 ]T represents the local independent beam displacements (with fed,n as corresponding work-conjugate forces), whereas dz = [d10 , d11 , d12 ]T is the vector of dependent generalized displacements (here the hinge rotations) that are to be eliminated. We show here that the procedure for elimination of dz is identical to static condensation [32] or [384]. e e Knn dn ,  Knn = Kenn − Kenz Kezz−1 Kezn fd,n = 

(3.105)

and can be implemented as a partial decomposition of Ke matrix employing Gaussian elimination, which is described in [384]. The proposed method to modify the element stiffness matrix in the presence of joint releases can be applied to any of the three versions of the stiffness matrix discussed in the previous section, i.e. Kv , K p or Kd . For example, the joint release described by a spatial hinge for Kd has a completely identical effect on the modification of K p matrix, given in (3.85), since the total rotation vector is simply decomposed into another component set. For a joint release of the local hinge type at the element end, one needs to modify Kv matrix by eliminating degrees of freedom v2 (hinge around local y-axis) and v4 (hinge around local z-axis) from the Eq. (3.95). For a joint release pertaining to shear force at an element end, which modifies the matrix K p , it is necessary to eliminate the generalized displacements p8 and p9 from the Eq. (3.100). Following the same principle, in the matrix K p we will obtain the joint release for the axial force by eliminating the displacement p7 , whereas we will impose the joint release for the torsional moment by eliminating the displacement p10 . The structural analysis of frames with joint releases can again be performed by using the previously described assembly procedure. However, we have to employ the element stiffness matrices modified for the presence of joint releases. Symbolically, it can be written  e  Le T  (3.106) K Le K= e

The proposed approach eliminates the dependent displacements at the beam element (local) level and uses only the reduced element stiffness in the finite element assembly procedure, as indicated in (3.106). We will show this to be a more effective analysis approach for larger structural systems compared to an alternative reduction of the structure stiffness matrix that accounts for releases. More precisely, an alternative to the previous method for analysis of frames with joint releases implies the elimination of dependent generalized displacements at the global level, i.e. by modifying the structure stiffness matrix K. In this approach, the presence of releases at a particular structure node imposes two sets of (global) generalized displacements with different values at that node. It is understood that this pertains only to the generalized displacements influenced by the release, for example, two sets of rotations for the spatial hinge connection (see the illustrative example given later in this section). This approach considers every beam element as standard (without the reduction of

3.2 Beam Model Refinement and Reduction

93

element stiffness due to joint release) and the assembly is carried out as indicated in (3.79). However, all beam elements that are connected to a node with release have to pick up the correct set of global generalized displacements at that node. If we denote the basic generalized displacements of the joints with rn , and the additional set of generalized displacements stemming from the presence of joint releases with rz , then the system equilibrium equation can be written

fr rn Knn Knz = 0 Kzn Kzz rz

(3.107)

where fr represents the vector of generalized nodal forces. The procedure of static condensation can now be applied at the level of the entire structure Knn rn ,  Knn = Knn − Knz K−1 fr =  zz Kzn

(3.108)

The global stiffness matrix  Knn is identical to the global stiffness matrix in (3.106) obtained by the assembly procedure of the modified stiffness matrices; hence the resulting nodal displacements are the same. The condensation of the stiffness matrix at the global level is achieved with the triangular decomposition phase of the Gauss elimination method [176]. Since this is the most costly part of the solution procedure (proportional to n 3 , where n is a number of equations to be operated upon), the gain of efficiency is not negligible if the stiffness matrix reduction is carried out at the local rather than the global level. Illustrative Example for Joint Releases in Beam Element We here present the results of several numerical simulations, which were carried out by the beam elements with releases described herein and implemented in the computer program CAL [383]. In order to better understand the procedure, the list of commands, which is used in CAL-program [383] to solve this problem, is briefly explained at the end of this section. We first analyze a simple frame built with two elements (see Fig. 3.13 for its properties), which has a moment release (hinge) placed at the elements’ connection. Then we consider a 3D frame with a larger number of degrees of freedom. Joint Release for Moments Here, two approaches were compared: the global approach, in which the components of the rotation vector left and right from the hinge in the given node were treated as independent degrees of freedom, and the local approach, in which we have modified the element stiffness matrix for the presence of joint release in one of the beams. For both models, the global displacements are marked in Fig. 3.14. In the first analysis, the global displacements 7, 8 and 9 were eliminated at the structural level, and the total number of operations required to perform global stiffness reduction equals 540. On the other hand, the element reduction costs 980 operations, which is much larger than the global reduction as well as the solution of the system without any reduction (equal to 669 exactly). We can conclude that a difference between both the local and the global reduction does not play a significant role for small structural systems.

94

3 Beam Models: Refinement and Reduction

l = 10 E = 30000 G = 12000 J = 0.001 Iy = 0.003 Iz = 0.003 F1 = 10 F2 = 30 F3 = 20

F1

Z

F3 F2

l

Y X

Fig. 3.13 Simple two-element frame: geometrical and mechanical properties and applied loading

6 3 1 4

6

9

3

8 7

2

1

5

(a)

2

5

4

(b)

Fig. 3.14 Two-element frame models with moment release: a degrees of freedom for global approach; b degrees of freedom for local approach

Next, CAL commands for solving the problem in Fig. 3.14a are given as follows: C FRAME WITH MOMENT (GLOBAL) RELEASES C C NODAL COORDINATES LOAD XYZ R=3 C=3 10 0 0 10 10 0 0 10 0 C FORM ELEMENT STIFFNESS MATRICES C (Ki(12*12)) FRAME3 K1 T1 E=30000 A=0.16 J=0.001 I=0.003,0.003 G=12000 N=1,2 P=1,0 FRAME3 K2 T2 E=30000 A=0.16 J=0.001 I=0.003,0.003 G=12000 N=2,3 P=1,0 C LOCAL - GLOBAL DISPLACEMENTS LOADI IDT R=2 C=12 0 0 0 0 0 0 1 2 3 4 5 6 1 2 3 7 8 9 0 0 0 0 0 0 TRAN IDT ID C ASSEMBLE GLOBAL STIFFNESS MATRIX ZERO KK R=9 C=9

ADDK KK K1 ID N=1 ADDK KK K2 ID N=2 C CONDENSATION OF GLOBAL STIFFNESS MATRIX LOADI IDKT R=1 C=9 4 5 6 7 8 9 1 2 3 TRAN IDKT IDK ZERO KKP R=9 C=9 ADDK KKP KK IDK N=1 ZERO A R=9 C=1 SOLVE KKP A EQ=3 DUPSM KKP KKC R=6 C=6 L=4,4 C LOAD VECTOR LOAD RT R=1 C=6 30 20 -10 0 0 0 TRAN RT R C SOLVE SYSTEM OF EQUILIBRIUM EQUATIONS SOLVE KKC R C DISPLACEMENTS P R RETURN

3.2 Beam Model Refinement and Reduction

95

The computed results we obtained for frame with moment (global) release shown in Fig. 3.14a are given as follows: LOAD XYZ R=3 C=3 ARRAY NAME = XYZ NUMBER OF ROWS = 3 NUMBER OF COLUMNS = 3 FRAME3 K1 T1 E=30000 A=0.16 J=0.001 I=0.003,0.003 G=12000 N=1,2 P=1,0 FRAME3 K2 T2 E=30000 A=0.16 J=0.001 I=0.003,0.003 G=12000 N=2,3 P=1,0 LOADI IDT R=2 C=12 ARRAY NAME = IDT NUMBER OF ROWS = 2 NUMBER OF COLUMNS = 12 TRAN IDT ID ZERO KK R=9 C=9 ADDK KK K1 ID N=1 ADDK KK K2 ID N=2 LOADI IDKT R=1 C=9 ARRAY NAME = IDKT NUMBER OF ROWS =1 NUMBER OF COLUMNS = 9 TRAN IDKT IDK ZERO KKP R=9 C=9 ADDK KKP KK IDK N=1

ZERO A R=9 C=1 SOLVE KKP A EQ=3 TOTAL SOLUTION OF DUPSM KKP KKC R=6 LOAD RT R=1 C=6 ARRAY NAME = RT NUMBER OF COLUMNS TRAN RT R SOLVE KKC R TOTAL SOLUTION OF P R

Ax = B C=6 L=4,4 NUMBER OF ROWS = 1 = 6

Ax = B

COL# = 1 ROW 1 .62465E-01 ROW 2 .41643E-01 ROW 3 -18.519 ROW 4 -2.7778 ROW 5 .00000 ROW 6 -.93697E-02 RETURN

Next, we display CAL code commands for solving the same problem but with local releases as shown in Fig. 3.14b. C FRAME WITH MOMENT (LOCAL) RELEASES C C NODAL COORDINATES LOAD XYZ R=3 C=3 10 0 0 10 10 0 0 10 0 C FORM ELEMENT STIFFNESS MATRICES C (Ki(12*12)) FRAME3 K1 T1 E=30000 A=0.16 J=0.001 I=0.003,0.003 G=12000 N=1,2 P=1,0 FRAME3 K2 T2 E=30000 A=0.16 J=0.001 I=0.003,0.003 G=12000 N=2,3 P=1,0 C STATIC CONDENSATION FOR C BEAM ELEMENT 2 LOADI IDKT R=1 C=12 4 5 6 1 2 3 7 8 9 10 11 12 TRAN IDKT IDK ZERO A R=12 C=1 ZERO K2P R=12 C=12

ADDK K2P K2 IDK N=1 SOLVE K2P A EQ=3 C LOCAL - GLOBAL DISPLACEMENTS LOADI IDT R=2 C=12 0 0 0 0 0 0 1 2 3 4 5 6 0 0 0 1 2 3 0 0 0 0 0 0 TRAN IDT ID C ASSEMBLE GLOBAL STIFFNESS MATRIX ZERO KK R=6 C=6 ADDK KK K1 ID N=1 ADDK KK K2P ID N=2 C LOAD VECTOR LOAD RT R=1 C=6 30 20 -10 0 0 0 TRAN RT R C SOLVE SYSTEM OF EQUILIBRIUM EQUATIONS SOLVE KK R C DISPLACEMENTS P R RETURN

The corresponding results for frame with moment (local) release as defined in Fig. 3.14b are now given as: LOAD XYZ R=3 C=3 ARRAY NAME = XYZ NUMBER OF ROWS = 3 NUMBER OF COLUMNS = 3 FRAME3 K1 T1 E=30000 A=0.16 J=0.001 I=0.003,0.003 G=12000 N=1,2 P=1,0 FRAME3 K2 T2 E=30000 A=0.16 J=0.001 I=0.003,0.003 G=12000 N=2,3 P=1,0 LOADI IDKT R=1 C=12 ARRAY NAME = IDKT NUMBER OF ROWS = 1 NUMBER OF COLUMNS = 12 TRAN IDKT IDK ZERO A R=12 C=1 ZERO K2P R=12 C=12 ADDK K2P K2 IDK N=1 SOLVE K2P A EQ=3

TOTAL SOLUTION OF Ax = B LOADI IDT R=2 C=12 ARRAY NAME = IDT NUMBER OF ROWS = 2 NUMBER OF COLUMNS = 12 TRAN IDT ID ZERO KK R=6 C=6 ADDK KK K1 ID N=1 ADDK KK K2P ID N=2 LOAD RT R=1 C=6 ARRAY NAME = RT NUMBER OF ROWS = 1 NUMBER OF COLUMNS = 6 TRAN RT R SOLVE KK R TOTAL SOLUTION OF Ax = B

96

3 Beam Models: Refinement and Reduction

Fig. 3.15 3D frame with 288 DOFs with 144 DOFs to be released by reduction

P R COL# = ROW 1 ROW 2

1 .62465E-01 .41643E-01

ROW 3 -18.519 ROW 4 -2.7778 ROW 5 .00000 ROW 6 -.93697E-02 RETURN

One can see that for both cases of joint releases for all three moment components, the corresponding displacement components calculated from model a and model b are identical. Example for 3D Frame In the next example, we analyze a 3D only with respect to a number of operations needed to solve the problem, hence the geometrical and mechanical properties are of no interest. The frame consists of continuous beams interconnected with beams that have spatial hinges at both ends (see Fig. 3.15). This example has a much higher number of degrees of freedom compared to the previous example, and more importantly, many degrees of freedom have to be eliminated in order to account for hinge releases. Here we have the total number of unknown displacements equal to 288, which, however, can be reduced to 144. The cost of solving the system without any reduction equals approximately 16 · 106 , with the cost of global reduction slightly below. However, the element reduction costs only ∼ 2 · 106 operations, which is an order of magnitude smaller. Appendix: Selected Commands of Program CAL LOAD M1 R=? C=? load matrix M1 of real numbers with ’R=’ number of rows and ’C=’ number of columns LOADI M1 R=? C=? load array M1 of integer numbers ZERO M1 R=? C=? load array M1 with all entries equal to zero P M1 list array named M1 TRAN M1 M2 transpose matrix M1 to form matrix M2 DUPSM M1 M2 R=? C=? L=L1,L2 duplicates submatrix M2 from location M1(L1,L2) M2 is NR x NC SOLVE M1 M2 EQ=? solve symmetric system of equations M1 x = M2; EQ = number of equations to be decomposed FRAME3 Ki Ti E=? A=? I=I3,I2 J=? G=? N=Ni,Nj P=P1,P2 forms a (12$\,\times\,$12) spatial beam element stiffness matrix; E is Young’s modulus, A is cross-sectional area, I=I2,I3 are second moments of inertia around axes 2 and 3, J is torsional moment of inertia, N=Ni,Nj are node numbers of element start and end used to extract coordinates from the previously defined coordinate matrix XYZ, P=P1,P2 are also defined by XYZ and are used to specify the direction of z axis in local coordinate system; the degrees of freedom are 3 translations and 3 rotations at the element start and 3 translations and 3 rotations at the element end

3.2 Beam Model Refinement and Reduction

97

ADDK K Ki ID N=? assembly od element stiffness matrices Ki into the global stiffness matrix K; ID is the connectivity matrix, N specifies the column number in ID corresponding to the considered element

3.2.3.2

Beam Element Reduction: Invariant Length Constraint

The Displacement Method and subsequent finite element variant have replaced and eliminated many classical numerical methods from engineering practice when it comes to frame structures analysis. With an increase in computational power, the solution of the resulting set of algebraic equations (with the typical size for frame structures) is not a major problem, which led to the displacement-based frame models implemented in the vast number of computer programs. Here, we would like to revisit this method in order to recover the ’hand-calculation’ predecessor in terms of Technical Displacement Method where the frame element length is enforced to remain invariant. Rather than providing the connection to this classical approach, we use this development to illustrate imposing global constraints in beam model reduction. We will carry out this development for the 2D case with planar beams, where the degrees of freedom (dof) for the standard Displacement Method for each node are two displacements and one rotation. However, this is no longer the case in Technical Displacement Method which assumes axially rigid beams and thus further reduces the number of independent displacements. The proposed approach of the Technical Displacement Method can also improve the system conditioning when the axial deformation is significantly smaller than the flexural one, where one should rather exclude it completely. This is the main motivation for further elaborating upon the appropriate implementation of such a reduced model within the standard framework of the finite element method. The Technical Displacement Method can easily be implemented for frame structures where each and every beam element is aligned with one of the global axes. Here, the only modification needed in Technical Displacement Method is reducing the size of the element stiffness matrix to (4 × 4) and identifying the local (element) displacement components with corresponding global (structure) displacements. We leave this to the reader as an exercise to try out, and we further look into a more interesting case where the frame structure will have inclined beam elements, such as shown in an illustrative example in Fig. 3.16. In this case the element stiffness matrix for the Technical Displacement Method Ke,tdm is still of size (4 × 4) corresponding to local nodal displacements and rotations ue,tdm of size (4 × 1). One thus needs the element transformation matrix Te,tdm of size (4 × 6) to connect the nodal displacements and rotations in the local frame with those in the global frame denoted as de ; the latter can be written as

98

3 Beam Models: Refinement and Reduction

⎤ u1 ⎥ v1 − sin α cos α 0 0 00 ⎢ ⎢ v1 ⎥ ⎥ ⎢ ⎢ ⎢ ⎥ θ1 ⎥ 0 0 1 0 0 0 ⎥ ⎢ θ1 ⎥ ⎥ ue,tdm = Te,tdm de ⇔ ⎢ =⎢ ⎥ ⎣ v2 ⎦ ⎣ 0 0 − sin α cos α 0 0 ⎦ ⎢ ⎢ u2 ⎥ ⎣ θ2 v2 ⎦ 0 0 0 0 01 θ2 (3.109) We can then carry out with the finite element assembly procedure to compute the global stiffness matrix K of size (n eq × n eq ) by using the connectivity matrix for each particular element Le of size (6 × n eq ), which can be written as ⎡

⎤tdm

K=

⎡



Le T Te T Ke Te Le

⎤

⎡

(3.110)

e

where n eq = 3n − r is number of equations, with n number of nodes and r number of supports. However, the system to be solved in Technical Displacement Method is not simply Kd = f, since we have to impose at this stage the global constraints that enforce the length invariance for each beam element in this structure stiffness matrix. We proceed as follows. First, we split the total number of unknowns to those gathering rotations θ and displacements u and reorder unknowns to write: d = (θ, u)T

(3.111)

For illustration on global constraint enforcement, let us consider a 2D system with n nodes, e elements, r point supports and c free nodal rotations. Thus vector d will have m components including unknown rotations θ with c components and nodal displacement u with 2n − r components. For Technical Displacement Method Ke are (4 × 4) matrices, Te are (4 × 6) transformation matrices and Le are (6 × m) connectivity matrices, where m = c + 2n − r (3.112) We note that n eq = m for Displacement Method since we only separated the count for displacements from the one for rotations. At this point, we will enforce the length invariance constraint of each beam element. Namely, due to the assumption of axial rigidity, the total number of unknown independent displacements uid is reduced to: n id = 2n − r − e

(3.113)

since each element will reduce the number of independent displacement components by one. Hence, we can write the reduced set of global equilibrium conditions and unknown displacements: Ktdm dtdm = ftdm ; dtdm = [θ, uid ]T

(3.114)

3.2 Beam Model Refinement and Reduction

99

where uid gathers the unknown independent displacements at the structural level. How do we find these independent displacements? The most direct way is by exploiting the basic theorem of rigid body (linearized) kinematics that the projections of nodal displacements on the beam axis have to be the same in order to enforce length invariance. We thus obtain e homogeneous linear equations with 2n − r unknown projections on the global axes: Pu = 0 (3.115) We can further split unknowns in this equation into independent displacements uid to be picked up and remaining dependent displacements udd to be eliminated, along with the corresponding split of projection matrix P as u=

udd uid

  ; P := Pdd Pid

(3.116)

where one has to ensure that Pdd is a regular matrix when picking up independent displacements. With such a split, one can then obtain from (3.115) above how to express dependent displacements udd in terms of independent displacements uid leading to displacement vector reduction that can be written as ¯ id u = Gu

(3.117)

¯ can be obtained column by column by successively setting The reduction matrix G one entry in uid to 1 and all others to 0. We note that the global displacement vector is reduced with no reduction on rotations, which implies that the complete set of unknowns is transformed with

I 0 d = G dtdm ; G= (3.118) ¯ 0G The assembly procedure for the global stiffness matrix for the Technical Displacement Method can now be defined according to: Ktdm = GT

 

 Le T Te T Ke Te Le G = GT KG

(3.119)

e

The corresponding equivalent nodal force vector is redefined with: ftdm = GT f

(3.120)

Once we solve the system for unknown rotations and independent displacement to obtain dtdm , we can recover the internal nodal forces for each element by accounting for reduction with: (3.121) fe = Ke Te Le G dtdm

100

3 Beam Models: Refinement and Reduction

Fig. 3.16 Geometry and loads of 2D frame: (left) with n = 6 nodes, e = 6 beam elements, r = 4 supports and c = 4 free rotations, resulting with m := c + 2n − r = 12 equations for Displacement Method and (right) imposing the beam length constraints resulting with n id := 2n − r − e = 2 independent displacements Technical Displacement Method

Fig. 3.17 Independent displacement patterns 1 (left) and 2 (right) for Technical Displacement Method

Example of 2D Frame with Length-Invariant Beam Model: In this example we take a 2D frame shown in Fig. 3.16, defining the geometry and loads, and the number of unknown for the Displacement Method on the left versus a number of unknowns for the Technical Displacement Method including two independent displacements on the right. For the results presented further, we take the value of Young’s modulus as E = 3 · 107 k N /m 2 . In Fig. 3.17 we show the displacement pattern that accommodates the length invariance constraint for each of two selected independent displacements for the Technical Displacement Method In contrast to the nodal displacement and rotation vector for Displacement Method d with 12 unknown components T  d = θ3 θ4 θ5 θ6 u 3 v3 u 4 v4 u 5 v5 u 6 v6

(3.122)

3.2 Beam Model Refinement and Reduction

101

Fig. 3.18 Diagrams computed by Technical Displacement Method for bending moment (left) and normal force (right)

By exploiting the length invariance constraint in Technical Displacement Method, we reduce the number of unknowns to only 6, with the corresponding nodal displacement and rotation vector dtdm that can be written as T  dtdm = θ3 θ4 θ5 θ6 u 4 v6

(3.123)

By applying the proposed procedure further, we arrive at different results for the reduction arrays that are given as follows:  T udd = u 3 v3 v4 u 5 v5 v6 ⎡ 1 0 −1 0 0 0 ⎢ 0 1 0 0 0 0 ⎢ ⎢ 0 0 0 1 0 0 P=⎢ ⎢ 0 0 0 1 0 0 ⎢ ⎣ 0 0 0 0 1 0 0.8 0.6 0 0 −0.8 −0.6  T  T uid uid 1 = 1 0 2 = 0 1

T 1 0 1 0 0 4/3 0 0 ¯ G = 0 0 0 0 1 −4/3 1 0

0 0 0 0 −1 0

⎤ 0 0 ⎥ ⎥ 0 ⎥ ⎥ −1 ⎥ ⎥ 0 ⎦ 0

(3.124)

Finally, in Fig. 3.18 we present the resulting diagrams for bending moment and normal force.

102

3.2.3.3

3 Beam Models: Refinement and Reduction

Closing Remarks

In this work, we have addressed several issues pertinent to constructing model refinements in materially and geometrically linear structural analysis of spatial frames. In terms of such model refinement, we derived the element stiffness matrix starting from the reduced model pertaining to the deformation space (with rigid modes suppressed), where we first define the corresponding flexibility matrix, the inverse of the stiffness matrix. The proposed approach enables us to easily account for shear deformation in a more direct manner than the standard alternative based on the displacement-based Timoshenko’s beam element, which requires non-conventional interpolations and the increase of computational cost (with additional degrees of freedom) to deliver the same quality of results. The merit of the proposed flexibility approach can further become clear when deriving the stiffness matrices of beam elements with varying cross-section or Young’s modulus, where standard cubic interpolation functions would produce rather poor results. However, the flexibility approach is based on the principle of virtual forces which assumes that there exists an equilibrium between the external and internal sets of infinitesimal forces before the application of real loads/displacements. In the case of nonlinear analysis, this principle cannot be used because the linear relationship between external and internal forces no longer holds. On the other hand, Timoshenko’s beam elements can be extended for material nonlinearity, but the additive decomposition of total rotations is not in accord with the multiplicative nature of large three-dimensional rotations. We have also addressed the corresponding model reduction issues with the element stiffness matrix in the presence of joint releases (or hinges) for moments and shear force. This is an alternative approach to doubling the corresponding nodal degrees of freedom for every release and it is applicable for nonlinear problems. The number of independent displacements is reduced compared to the standard beam element, thus yielding a system with a smaller number of equations. In the case of a structural system with many members, the element reduction is much cheaper than solving the system without condensation or with condensation at the global level. Hence we consider it to be computationally more effective. Additional model reduction pertaining strictly to linear analysis is discussed to provide the systematic computer implementation of the technical displacement method with no change of element length. A detailed illustration is provided for the implementation of the technical displacement method for frames with inclined members by exploiting the fundamental theorem from rigid body kinematics. Global reduction of the beam model which introduces length-invariance results in a system with fewer unknown displacements, and it could also be interesting for improving the system conditioning when the axial deformation remains very small. This is obtained by means of model reduction providing a more effective approach than the alternative one in which the global number of degrees of freedom has to be increased by one for each new release. Such an approach also applies to geometrically nonlinear framework [207] and it can be extended for materially nonlinear case [67, 314] including thermomechanical coupling [298].

3.3 Curved Shallow Beam and Non-locking FE Interpolations

103

3.3 Curved Shallow Beam and Non-locking FE Interpolations We discuss the finite element analysis of planar deformations of elastic, initially curved beams with both linear and nonlinear kinematics, taking into account the shear deformation. Since this problem is of great practical interest, there exists an abundant list of accomplishments on the subject, too long to review in this work. A book of Crisfield [97] summarizes most of the approaches proposed up to date. In this section we focus on several novelties when constructing a reduced model for the beam, which can be stated as: (i) Giving a new formulation for a curved shallow beam with linear kinematics, which accounts for shear deformation. We arrive at the present formulation by the consistent linearization of the beam theory of Reissner [322]. In this respect, our derivation is different than many others already available (e.g., see [47, 358, 368] or [97]) and end result is more suitable for constructing the finite element approximations. (ii) The finite element approximation cannot be chosen as the standard isoparametric element for its exhibits inability to represent pure bending mode, or ’locking’. The presented beam gives a very clear insight into the causes of shear and membrane locking and allows exploring different directions for constructing non-locking interpolations when considering shallow shell formulation (e.g., see [197]). We also demonstrate that in a linear analysis of beams all proposed locking cures, which we have encountered in the literature, yield the same result. It is not so in nonlinear analysis, nor for shells. (iii) Finally, we also briefly illustrate that the finite element model for Reissner’s beam that accounts for initially curved configuration can increase the accuracy compared with an equivalent, but initially straight beam element of Simo et al. [340]. An outline of this section is as follows. First, we give a new formulation of shallow curved beams derived from the consistent linearization of Reissner’s beam theory. We then discuss different possibilities for constructing enhanced finite element interpolations which eliminate the shear and membrane locking. An illustrative numerical example and closing remarks are stated at the end.

3.3.1 Two-Dimensional Curved Shallow Beam: Linear Kinematics In this section, we give a detailed formulation for a two-dimensional curved shallow beam. We try to merge two prominent assumptions: one of Marguerre [277] on curved beam being shallow and one of Timoshenko [369] on accounting for shear deformation. The formulation for merging these two assumptions presented herein is different

104

3 Beam Models: Refinement and Reduction

a2 g2 e2

x

ϕ

g1

α

a1

α+ψ

v u

y = f (x) L

e1

l Fig. 3.19 Curvilinear beam: initial configuration (solid line) and deformed configuration (dotted line), and moving orthogonal frame (ai = gi ) attached to beam cross-section

from many other attempts (e.g. Belytschko et al. [47], Tessler and Spiridgliozzi [368] or Stolarski and Chiang [358]), who all seem somewhat ad-hoc. Namely, we arrive at the proposed formulation by linearizing the finite strain planar beam theory formulation of Reissner [322]. We note that the three-dimensional curved finite strain beam theory given by Ibrahimbegovic [170] can be used to extend our considerations to the 3D case. We note in passing that the proposed formulation is also more suitable for finite element implementation given that displacement components remain uncoupled, so that one can handle shear and membrane locking separately. First, we follow Ibrahimbegovic et al. [197] in considering that the shallow beam configuration is derived from the straight beam by an isometric transformation. If (e1 , e2 ) are basis vectors for a fixed orthogonal reference frame (see Fig. 3.19), than the orthogonal basis for shallow beam (g1 , g2 ) is derived by

g1T g2T

=

cos α sin α − sin α cos α



e1T e2T

(3.125)

We further recall the generalized deformation measures proposed by Reissner [322]. For that purpose, it is convenient to consider a moving orthogonal frame (see [170, 335]), which is obtained by rotating the frame (g1 , g2 ) so that a new position of g1 , further denoted as a1 , remains orthogonal to the cross-section in the deformed configuration and the new position of g2 , which we denote as a2 ), remains in the crosssection plane. We note that (a1 , a2 ) remains orthogonal frame, since Reissner’s beam cross-section is non-deformable and only moved by a 2D large rotation defined by angle ψ (see Fig. 3.19). Thus, we can further write:

a1T a2T



cos ψ sin ψ = − sin ψ cos ψ



g1T g2T



cos ψ sin ψ = − sin ψ cos ψ



cos α sin α − sin α cos α

e1T e2T (3.126)



3.3 Curved Shallow Beam and Non-locking FE Interpolations

105

The axial and shear strain measures, ε and γ , are then obtained as   dϕ ε x +u = − a1 ; ϕ = γ f (x) + v ds

(3.127)

where u and v are displacement components in fixed reference frame (see Fig. 3.19). Following Ibrahimbegovic et al. [197], we linearize the expressions in (3.125) by using an assumption that the beam is shallow  ds =

1+

df dx

2

 dx ≈ dx ⇒

df dx

2 1

(3.128)

which means that the metric of the straight line can be used rather than the metric of the curve. In other words, we can use d d (•) ≈ (•) ds dx

(3.129)

By the same token, we can write cos α ≈ 1 ; sin α ≈

df dx

(3.130)

which ensures that g1 and g2 remain the ’unit’ vectors in the framework of the linear approximation in (3.128). Furthermore, by assuming the small displacement gradient theory, we can linearize the transformation for the vector basis a1 , a2 in (3.126). Namely, by using the orthogonal matrix linearization result of Argyris [10],



!



cos ψ sin ψ !! 10 0ψ ≈ + − sin ψ cos ψ !ψ→0 01 −ψ 0

(3.131)

and the result in (3.130), we can write the linearized form for a ’unit’ vector a1 in (3.126) that becomes  1 − ψ ddxf (3.132) a1 ≈ d f +ψ dx Finally, by substituting the results in (3.129) and (3.132) into (3.127), we obtain the linearized axial and shear strain measures for the curved shallow beam as     du 1 + du 1 − ψ ddxf ε + ψ ddxf dx dx ≈ df (3.133) − df = dv γ + dv −ψ +ψ dx dx dx dx We note that such a result is different from earlier attempts given in [47, 97], who suggested coupling of transverse displacement derivative through shallow geometry

106

3 Beam Models: Refinement and Reduction

rather than rotation that we obtain herein. However, our proposed result not only comes out by consistent linearization, but it is also more suitable for constructing non-locking finite element interpolations as shown further on. The linearized measure for the curvature (Reissner [322]) coincides with the one for Timoshenko’s beam dψ dψ ≈ (3.134) κ= ds dx In matrix notation, the complete set of strain  and displacements d are given as " # " #  T = ε; γ ; κ ; uT = u; v; ψ

(3.135)

where ε, γ and κ are defined by (3.133) and (3.134). The stress resultants N , V and M are defined in (3.136) below with simple linear elastic constitutive equations only, since our main interest is in the locking phenomena; hence, we assume " # r = C ; r = N ; V ; M ; C = diag(E A; G A; E I )

(3.136)

Restricting ourselves, for simplicity, to homogeneous Dirichlet boundary conditions, we can easily obtain the total energy potential as the sum of internal energy, written as a quadratic form in Reissner’s strain measures, and external force energy, which is given as l (u, v, ψ) = 0

1 T  C d x − 2  r

l uT p d x

(3.137)

0

Note that in (3.137) above, all the integrals are taken over projected (straight) length of the beam (see Fig. 3.19). In the finite element implementation, every beam element can have its own reference system, thus extending the validity of the assumption on shallow beams.

3.3.2 Non-locking Finite Element Interpolation for Shallow Beam We discuss the selected interpolation schemes for the shallow beam element proposed to remove the shear and membrane locking, which plague behavior of such an element. The list of possible cures for locking is a long one: selective/reduced integration (e.g., see Hughes et al. [157], Stolarski and Belytschko [356]); assumed strain method (e.g., see Hughes and Tezduyar [158], Bathe and Dvorkin [29], de Ville de Goyet and Frey [105]); enhanced displacement method (e.g., see Tessler and Dong [366], Tessler and Spiridgliozzi [368]); mode-decomposition method (e.g., see

3.3 Curved Shallow Beam and Non-locking FE Interpolations projected configuration reference configuration v1

ψh

v2

vh

u2

u1 ψ1

107

ψ2

h0

l

Fig. 3.20 Shallow beam 2-node finite element: initial and projected configurations

Belytschko et al. [47], Stolarski and Chiang [358]) and mixed methods with independent force interpolation (e.g., see Malkus and Hughes [276] and Noor and Peters [302]). Subsequently, we will review a number of proposals for enhanced finite element interpolations that can handle shear and membrane locking, in order to pick up the simplest acceptable solution that leads to the most efficient implementation.

3.3.2.1

Shear Locking

Within the discrete formulation, the curvature κ h and the shear strain γ h are given as dψ h dv h ; γh = − ψh (3.138) κh = dx dx First, we consider a simple 2-node shallow beam element in Fig. 3.20, where the local frame defining projection plane is placed in the center of the element. Note that for the consideration of shear locking the actual shape (curved shallow element) is not important, since in shallow beam theory there is no coupling with f (x) for shear. In fact, the shear and bending strain measures are identical to those of straight Timoshenko’s beam element aligned with x-axis, since we use the shallow beam approximation. Thus, the reference configuration is defined with simple isoparametric mapping (e.g., see [176]) x h (ξ ) = N1 (ξ )x1 + N2 (ξ )x2 ; l = x2 − x1 ; ξ =

2x l

(3.139)

where Na (ξ ) are standard linear polynomials in natural coordinate ξ ∈ [−1, +1] that can be written as 1 1 (3.140) N1 (ξ ) = (1 − ξ ), N2 (ξ ) = (1 + ξ ) 2 2 Suppose that we want to keep the simplest choice and use the same isoparametric approximation for rotation field ψ h and displacement field v h ψ h (ξ ) = N1 (ξ )ψ1 + N2 (ξ )ψ2

(3.141)

108

3 Beam Models: Refinement and Reduction

v h (ξ ) = N1 (ξ )v1 + N2 (ξ )v2

(3.142)

Adopting these isoparametric interpolations, we can compute the curvature κ h and the shear strain γ h from (3.138), which produces 1 (ψ2 − ψ1 ) l

(3.143)

1 1 ξ (v2 − v1 ) − (ψ1 + ψ2 ) + (ψ1 − ψ2 ) l 2 2

(3.144)

κ h (ξ ) = and γ h (ξ ) =

Assume that the pure bending deformation (yet called Kirchhoff’s mode) needs to be enforced on such a 2-node shallow beam element. We would thus need to confirm the ability to represent constant bending moment, and constant (zero) shear in the element. (3.145) γ h (ξ ) ≡ 0 ⇔ γ h (ξ ) = 0, ∀ξ However, in view of result in (3.144), enforcing a constant (zero) value of γ h with the constraint in (3.145) above imposes to eliminate the term linear in ξ leading to ψ2 − ψ1 = 0 ⇒ κ h =

1 (ψ2 − ψ1 ) = 0 l

(3.146)

which, in view of (3.143), implies that the only possible solution is zero curvature. In mathematical terms (e.g., see [61]), the constraint of the type (3.145) has placed such a severe limitation on the choice of the available functions (piece-wise linear and continuous), that the only available one as a minimizer of (3.137) is the function which renders the curvature to be identically equal to zero. We are herein looking for a possible solution to the locking problem within the proposed space of functions, in order to facilitate an easier and more consistent extension to nonlinear analysis. The proposed methodology is to use projection to filter out a troublesome term (see Brezzi and Bathe [61]) and facilitate enforcement of the constraint in (3.145) within the proposed interpolations. Here, we further apply some possible remedies available in the literature: Selective/reduced integration If the shear strain is evaluated by a single point Gauss quadrature (placed at ξ = 0), we can see that the last term in (3.144) is directly eliminated to give ! 1 1 γ h !r =0 = (v2 − v1 ) − (ψ1 + ψ2 ) l 2

(3.147)

which enables to capture the pure bending response. The reduced integration is certainly simple, if not the simplest remedy to shear locking (which also works for nonlinear kinematics).

3.3 Curved Shallow Beam and Non-locking FE Interpolations

109

Assumed shear strain method We can directly assume a constant shear strain that can eliminate locking by keeping only the constant term γ h = γ¯ h :=

1 1 (v2 − v1 ) − (ψ1 + ψ2 ) l 2

(3.148)

The expression (3.148) can be directly substituted in (3.137) when computing the shear contribution to stiffness. This method essentially shrinks the space of γ h by leaving out the terms responsible for locking. Enhanced displacement method In this method, we enhance the displacement field interpolation so that Kirchhoff’s mode (with pure bending—vanishing shear) is attainable. The enhancement of the displacement field is performed, so that the constraint in (3.145) is directly enforced. We follow the approach in [226] to augment the lateral displacement v h interpolation in a hierarchical manner v h (ξ ) = N1 (ξ )v1 + N2 (ξ )v2 + N3 (ξ )Δv

(3.149)

N3 (ξ ) = 1 − ξ 2

(3.150)

where

and leave the rotation interpolation as before in (3.141). The discrete interpolation for curvature (3.143) remains the same since it depends on rotations only. The shear strain, however, changes into

1 1 1 4 h (3.151) γ = (v2 − v1 ) − (ψ1 + ψ2 ) + ξ (ψ1 − ψ2 ) − Δv3 l 2 2 l so that a proper value of Δv3 can now cancel the term with linear variation in ξ . Linked interopolations If we further want to enforce the constraint in (3.145), we need to set the term in square brackets in (3.151) equal to zero, to get the final form of the hierarchical interpolation for v h as l v h (ξ ) = N1 (r )v1 + N2 (r )v2 + N3 (ξ ) (ψ1 − ψ2 ) 8

(3.152)

which is capable of eliminating the shear locking. This is often called linked interpolation, which is similar in concept to the one used for the Euler-Bernoulli beam. Mode decomposition method We follow Belytschko et al. [47] to get a pure bending mode (Kirchhoff’s mode) by integrating the expression for the curvature in (3.138) vbh (x) =

1 (ψ2 − ψ1 ) x 2 + ψ1 x 2l

(3.153)

110

3 Beam Models: Refinement and Reduction

where the constants of integration are selected so that the nodal rotations for bending mode coincide with the total rotations. In other words, the relationship between nodal displacements in bending mode db and the total displacements dt can be written ⎞ ψ1 ⎜ ψ2 ⎟ ⎟ db = Pb dt , dt = ⎜ ⎝ v1 ⎠ , v2 ⎛

⎡

1 0 ⎢ 0 1 Pb = ⎢ ⎣ 0 0 l/2 l/2

0 0 0 0

⎤ 0 0⎥ ⎥ 0⎦ 0

(3.154)

We next assume that the bending mode nodal displacements db are additive to shear mode nodal displacements ds to give the total displacements, which can be written as (3.155) ds = dt − db , ds = Ps dt , Ps = I4 − Pb By direct multiplication in (3.154) and (3.155), it can be shown that Pb and Ps are projection operators, which leads to Pb Pb = Pb ; Ps Ps = Ps ⇒ Ps Pb = Pb Ps = 0

(3.156)

The shear strain which avoids locking is then computed from the shear mode only by

+ * 1 1 γˆ h = bT ds ; bT = − (1 − ξ ); − (1 + ξ ); −1/l; 1/l 2 2

(3.157)

or, alternatively, by using (3.155), we get γˆ h = bT Ps dt =

1 1 (v2 − v1 ) − (ψ1 + ψ2 ) l 2

(3.158)

In conclusion, we note that for the simple beam element, which we have just considered, all the procedures to cure shear locking yield identical result for the shear strain. This is not surprising, since the non-locking shear strain as a constant should be unique. This is at variance with the case when the same methods are applied to shells (see Ibrahimbegovic and Frey [195]). Another obvious remedy to the locking problem is to enlarge the space of functions over which we seek the solution. We consider, for example, a 3-node curved shallow beam element in Fig. 3.21. The element reference configuration is defined with x h (ξ ) = N1 (ξ )x1 + N2 (ξ )x2 + N3 (ξ )x3

(3.159)

where, N I (ξ ) are the quadratic Lagrange polynomials (e.g., see [176]) N1 (ξ ) =

 1 2 ξ −ξ , 2

  N2 (ξ ) = 1 − ξ 2 ,

N3 (ξ ) =

 1 2 ξ +ξ 2

(3.160)

3.3 Curved Shallow Beam and Non-locking FE Interpolations projected configuration reference configuration v1 u1 ψ1

111

v2

v3

u2 ψ2

u3 ψ3 l

Fig. 3.21 Shallow beam 3-node finite element: initial and projected configurations

The jacobian of this transformation from Cartesian to natural coordinates is no longer constant, but a linear function in ξ except for the central position of the middle node. j (ξ ) =

x1 + x3 l l + ξ (x1 − 2x2 + x3 ) ⇒ j = , if x2 = 2 2 2

(3.161)

For simplicity, we will consider such a regular element with a constant jacobian, which preserves quadratic interpolations for field variables. Suppose that we want to use the isoparametric approximation for rotation field ψ h and displacement field v h ψ h (ξ ) = N1 (ξ )ψ1 + N2 (ξ )ψ2 + N3 (xi)ψ3

(3.162)

v h (ξ ) = N1 (ξ )v1 + N2 (ξ )v2 + N3 (ξ )v3

(3.163)

Adopting the isoparametric interpolations above, we can compute the curvature κ h and the shear strain γ h defined in (3.138), which produces κh =

1 2 (ψ3 − ψ1 ) + ξ (ψ1 − 2ψ2 + ψ3 ) l l

(3.164)

and

2 1 1 (v3 − v1 ) − ψ2 + ξ (v1 − 2v2 + v3 ) − (ψ3 − ψ1 ) l l 2

1 (3.165) −ξ 2 (ψ1 − 2ψ2 + ψ3 ) 2

γh =

Enforcing Kirchhoff’s mode constraint in (3.165) will cause the term dependent on ξ 2 to vanish, which then, by the virtue of (3.164), causes that linearly varying term for curvature to vanish as well. Thus in this case, Kirchhoff’s constraint is reducing the order of approximation for curvature to constant over an element. Hence, the locking persists and again impairs the rate of convergence with respect to the optimal rate. If we want to avoid locking, we should consider, for example, enhanced displacement interpolation for lateral displacement

112

3 Beam Models: Refinement and Reduction

v h (ξ ) = N1 (ξ )v1 + N2 (ξ )v2 + N3 (ξ )v3 + N4 (ξ )Δv4 ;   N4 (ξ ) = ξ 1 − ξ 2 where Δv4 = −

l (ψ1 − 2ψ2 + ψ3 ) 12

(3.166)

(3.167)

so that the new interpolation for shear strain becomes

1 1 = (v3 − v1 ) − (ψ1 + 4ψ2 + ψ3 ) l 6

2 1 +ξ (v1 − 2v2 + v3 ) − (ψ3 − ψ1 ) l 2

γ

h

(3.168)

which eliminates the trouble-making term varying with ξ 2 . By comparing (3.165) and (3.168), we note that reduced (2-point) Gauss quadrature (with abscissas at ξ = ± √13 ) will yield identical results for both of them, and thus also be able to cure shear locking.

3.3.2.2

Membrane Locking

The membrane locking phenomenon stems from an inability to capture a state of pure bending with in-extensional deformation, i.e. to satisfy ε h (ξ ) ≡ 0 ⇔ ε h (ξ ) = 0, ∀ξ

(3.169)

which appears instead of the constraint in (3.145) responsible for shear locking. However, as opposed to the shear strain in (3.138), the membrane strain depends on the beam geometry interpolation. Namely, the discrete approximation for the membrane strain is given as εh =

dfh du h + ψh dx dx

(3.170)

Therefore, the result of enforcing the constraint in (3.169) will depend on the chosen interpolation for the geometry of the curved shallow beam f h (x h ). Let us consider the simplest case with a 2-node beam element presented in Fig. 3.20, where we use the co-rotational type formulation (e.g., see [98]) to position the local coordinate system to coincide with the center of the beam element. Such a choice will exhibit the membrane locking phenomena. The beam true configuration will be interpolated with isoparametric shape functions, with the nodal distance h 0 being the same at either node, which allows writing

3.3 Curved Shallow Beam and Non-locking FE Interpolations

f h (ξ ) = N1 (ξ ) (−h 0 ) + N2 (r ) (+h 0 ) = r h 0 ,

113

2 h0 dfh = dx l

(3.171)

We use rotation interpolation as in (3.141), which yields curvature interpolation as in (3.143). The displacement component u h is given by u h (ξ ) = N1 (ξ )u 1 + N2 (ξ )u 2

(3.172)

The discrete approximation for the axial strain can then be defined as ε h (ξ ) =

1 h0 h0 (u 2 − u 1 ) + (ψ1 + ψ2 ) + ξ (ψ2 − ψ1 ) l l l

(3.173)

If the Kirchhoff mode constraint (3.169) of vanishing membrane strain needs to be satisfied for any value of ξ , then we have to eliminate the term varying linearly in ξ , which again imposes ψ2 − ψ1 = 0 ⇒ κ h =

1 (ψ2 − ψ1 ) = 0 l

(3.174)

and the only possible solution leads to zero curvature. Possible solutions to this problem are entirely parallel to the shear locking problem and lead to the final satisfying solution that can be written as ε h = ε¯ h =

1 h0 (u 2 − u 1 ) + (ψ1 + ψ2 ) l l

(3.175)

which can be used to compute the axial beam stiffness in (3.137) free of membrane locking. We finally consider the membrane locking problem for a 3-node shallow beam element shown in Fig. 3.21. The beam element geometry interpolation is now given by quadratic polynomial   f h (ξ ) = 1 − ξ 2 h 0 ,

4h 0 dfh =− ξ dx l

(3.176)

We use the same rotation interpolation as in (3.162), which yields curvature interpolation as in (3.164). Let us immediately consider the interpolation for displacement component u h which can eliminate locking. The discrete approximation for the axial strain then becomes u h (ξ ) = N1 (ξ )u 1 + N2 (ξ )u 2 + N3 (ξ )u 3 + N4 (ξ )Δu 4 + N5 (ξ )Δu 5 ;   (3.177) N5 (ξ ) = ξ 2 1 − ξ 2 If the hierarchical displacement components are chosen as

114

3 Beam Models: Refinement and Reduction

Fig. 3.22 Timoshenko cantilever beam: geometrical and mechanical properties and applied loading

Δu 4 =

F

h0 h0 (ψ1 − ψ3 ) ; Δu 5 = − (ψ1 − 2ψ2 + ψ3 ) 3 4

l - variable E = 30000 G = 12000 b = 0.4 h = 0.6 k = 5/6 F = 100

(3.178)

then the membrane strain interpolation takes a locking-free form

1 2h 0 (u 3 − u 1 ) + (ψ1 − ψ3 ) l 3l

2 h0 +ξ (u 1 − 2u 2 + u 3 ) − (ψ1 + 2ψ2 + ψ3 ) l l

ε h (ξ ) =

(3.179)

Note again that reduced 2-point Gauss quadrature will yield the same results as the presented locking-free interpolation.

3.3.3 Illustrative Numerical Examples and Closing Remarks Here, we present the results of several numerical simulations in order to illustrate the performance of the proposed finite element discretization.

3.3.3.1

Shear Locking in Timoshenko Cantilever Beam

Here, we consider a 2D beam of length l clamped at one end and free at the other (see Fig. 3.22). We will compare the calculated vertical displacements for different values of element length against the exact deflection of the cantilever tip computed analytically and equal to Fl Fl 3 + (3.180) v(l) = 3E I kG A where the first term is the free-end displacement portion corresponding to the bending of the Euler-Bernoulli beam and the second is the shear force contribution to free-end displacement. The numerical results are computed for very coarse beam discretization by using the simplest possible mode with one element mesh for each case.

3.3 Curved Shallow Beam and Non-locking FE Interpolations

115

Shear locking for different interpolations 1.2

Relative deflection v/vexact

1

0.8

0.6 Equal−order Enhanced Linked Euler−Bernoulli Exact

0.4

0.2

0

0

1

2

3

4

5

6

7

8

9

10

Slenderness ratio l/h

Fig. 3.23 Locking problem for different Timoshenko’s beam formulations

The evaluated results are displayed in Fig. 3.23. From the diagram, one can see that the element with equal-order interpolation obtained by using linear shape functions and the isoparametric finite element approach yields acceptable results only for very thick structural elements. Otherwise, the element exhibits shear locking, being unable to adequately represent the deformation pattern in bending-dominated problems. The enhanced interpolation for the displacement field with one additional term can only partially solve the problem, resulting in roughly 20 % stiffer response for slender elements. For comparison, the Euler-Bernoulli beam element captures the relevant displacement in problems dominated by bending strain; however, it predicts poor results in shear dominated-deformation mode. Finally, the beam element with linked interpolation successfully combines the Euler-Bernoulli model for slender elements with the advantages of the Timoshenko model for thick beams yielding accurate displacements regardless of l/ h aspect ratio. We will further extend these considerations again in the next section on finite element implementation of Timoshenko’s shallow beam element.

3.3.3.2

Shallow Arch Under Point Load

One illustrative numerical example is presented in this section to demonstrate a very good non-locking performance of the proposed finite element formulation. All numerical computations are performed with the computer program FELINA (see

116

3 Beam Models: Refinement and Reduction

E = 2 × 107 b/d = 0.2/1.

P

P = 10 l = 40

h

h=1 l Fig. 3.24 Shallow arch: geometrical and mechanical properties and applied load Table 3.2 Shallow arch: computed results for tip displacement versus number of elements No. Elem. Vertical Displacement 5 10 20 Exact

0.04516 0.04830 0.04833 0.04834

[127]). The chosen example presents the results for the linear analysis of the shallow arch shown in Fig. 3.24. This example is adapted from the work in [105], where the analytical solution is reported as well. The numerical results are obtained for several meshes and presented in Table 3.2. We can see that even rather coarse mesh can achieve an excellent quality of computed results

3.3.3.3

Circular Cantilever Under End Force

This example presents the nonlinear analysis of a cantilever with the reference configuration in the form of a half-circle, loaded with the concentrated force at the free end (see Fig. 3.25). Here, we would like to point out that the curved geometry’s better representation allows for further improving the quality of the computed results. The numerical results are obtained for different meshes of quadratic elements and presented in Table 3.3, along with the exact solution reported by [328]. In order to display enhanced accuracy which results from taking into account curved geometry, the same computations are repeated using the straight beam quadratic elements, and accordingly, a polygonal approximation of the reference configuration. Those results are also presented in Table 3.3.

3.3 Curved Shallow Beam and Non-locking FE Interpolations

117 P = 0.641713 E = 7200 G = 3600 A = 1010 I = 0.5 R = 50

R P

Fig. 3.25 Circular cantilever beam Table 3.3 Circular cantilever under end force No. Elem. Curved Elem. Vertical disp. Horizontal disp. 1 2 5 10 20 Exact

3.3.3.4

112.804 116.269 116.570 116.579 116.580 116.580

18.015 18.968 19.094 19.097 19.097 19.097

Straight Elem. Vertical disp. Horizontal displ. 44.293 90.882 111.983 115.412 116.287 116.580

12.281 13.423 17.906 18.788 19.019 19.097

Closing Remarks

We have presented a new shallow beam element for geometrically linear analysis of planar deformation of beams, which accounts for the effect of shear. The element formulation is obtained by the consistent linearization of the nonlinear beam theory of Reissner [322]. The consistency of the derivation and the end result are at variance with previous mostly ad-hoc attempts of the direct merging of Marguerre’s and Timoshenko’s assumptions to produce a similar shallow beam theory. The proposed shallow beam formulation is more suitable for finite element implementation. Namely, in the axial strain measure, only the rotation field is coupled with the longitudinal displacements through the shallow beam geometry. This provides that the membrane and shear locking can be studied separately, and simplifies the choice for non-locking finite element interpolations. With this formulation, we are provided with a very revealing interpretation of shear and membrane locking phenomena. In linear analysis, all available locking cures are applicable and lead to the same result. However, not all considered remedies for locking are pertinent to geometrically nonlinear analysis, since membrane and shear locking effects, in this case, are coupled. We have used the reduced integration which is the cheapest one computationally.

118

Structural Engineering

The finite element implementation of the beam theory of Reissner is addressed in a different manner than in the earlier works, with respect to considering the curved reference geometry and non-locking finite element interpolations for all field variables. It is shown that by taking into account the curved geometry (as opposed to straight beam elements), we can further improve the accuracy of the computed results.

Chapter 4

Plate Models: Validation and Verification

Abstract We here study the plate models as 2D extension of the beam models. We don’t start with the classical theory of the so-called Kirchhoff plates as 2D generalization of the Euler beam, often referred to as a ‘thin’ plate model that does not account for shear deformation. Such model requires the continuity of second derivatives of transverse displacement field that is not easy to achieve for distorted elements. Hence, we present an alternative model in terms of the Reissner-Mindlin plate as 2D generalization of the Timoshenko beam, yet referred to as a ‘thick’ plate model that can take into account the shear deformation. The variational formulation now requires the inter-element continuity of only the first derivatives of transverse displacement and (independent) rotations, but again lead to locking phenomena in finite element discrete approximations. We present several Reissner-Mindlin plate elements with non-conventional interpolations that can handle locking. We also show how to recover the discrete approximation of the Kirchhoff plate model in terms of the discrete Kirchhoff plate finite element. The plate models presented in this chapter are further used to illustrate typical goals of adaptivity procedure in structural engineering that pertain to model selection or validation, as well as to discrete approximation quality or verification.

4.1 Finite Elements for Analysis of Thick and Thin Plates The early approach to the finite element analysis of plates relied mostly on the Kirchhoff thin plate theory. A plate finite element based on the Kirchhoff plate theory, featuring the second derivatives of the transverse displacement in the weak form, will impose that the displacement and its derivatives are continuous across the element boundaries. This required quite an ingenious procedure for constructing the interpolation of this kind with a limited number of possible choices (for illustrations of pioneering works, see [83] for quadrilateral or [12] for triangular plate element). Therefore, many others (e.g. see [30, 64, 158, 367, 395]) have later turned towards the theory of plates proposed by Reissner [319] and Mindlin [290] as a starting point of the finite element discretization in order to reduce the continuity requirements on the displacement interpolation. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Ibrahimbegovic and R.-A. Mejia-Nava, Structural Engineering, Lecture Notes in Applied and Computational Mechanics 100, https://doi.org/10.1007/978-3-031-23592-4_4

119

120

4 Plate Models: Validation and Verification

Another very important reason for favoring the Reissner-Mindlin versus the Kirchhoff plate theory is that we normally get a more reliable representation of the threedimensional solution. Namely, the Kirchhoff plate model does not take the shear deformations into account, so it should not be used in any analysis of thick plates. Moreover, Babuška and Scapolia [20] pointed out the inaccuracy of the Kirchhoff plate model which occurs in the analysis of skewed plates, even when they are thin. For all that reasons, we here focus only upon the Reissner-Mindlin plate model. In the finite element implementation of the Reissner-Mindlin plates, however, one is faced with the problem of incapability of the discrete approximation to capture the limit behavior of the thin plate model, i.e. the Kirchhoff constraint with vanishing shear. Typically, a very stiff response is obtained and this phenomenon is referred to as shear locking. The first completely successful solution to the shear locking problem emerged as a clever ad-hoc engineering approach of the ’assumed strain method’ (e.g. see [158] or [30]), which considers a direct interpolation for shear strain not necessarily consistent with the interpolation schemes for the displacements and the rotations. Subsequently, the assumed strain method is put into the proper variational framework in [338]. The relationship between the assumed strain method and the mixed method is discussed in [391] and the convergence study of several Reissner-Mindlin plate elements which employ an assumed shear strain interpolation is performed in [64]. However, in the assumed strain method, the underlying displacement interpolation of the order which is consistent with the assumed shear strain field is not clearly identified and it is not clear what is the proper definition of the consistent loading and the consistent mass. In this section, we present two quadrilateral plate elements with explicitly defined hierarchical displacement and rotation interpolations which appear consistent with the assumed strain field. The corresponding shear strain interpolation is capable of reproducing the constant shear strain state and the Kirchhoff constraint in the limit case of a thin plate. As a byproduct of this work, we arrive at a consistent displacement interpolation for the well-known discrete Kirchhoff plate element [38]. The section outline is as follows. First, the motivation for the present methodology is briefly explained on the related problem of Timoshenko’s beam. We then discuss the Reissner-Mindlin plate elements and provide a new derivation for the discrete Kirchhoff plate element. Numerical evaluations of the proposed plate elements are presented at the end, along with some closing remarks.

4.1.1 Motivation: Timoshenko Beam Element Linked Interpolations 4.1.1.1

Beam Element with Quadratic Displacement Field

We first consider a two-node Timoshenko beam element presented in Fig. 4.1. The reference configuration is determined with x(ξ ) = N1 (ξ )x1 + N2 (ξ )x2

(4.1)

4.1 Finite Elements for Analysis of Thick and Thin Plates Fig. 4.1 Timoshenko beam element

w1 θ1

6 r

121

 6 w 6 Δw3 A b

θA

Δθ3



where N1 (ξ ) =

l

1 1 (1 − ξ ), N2 (ξ ) = (1 + ξ ) 2 2

w2 θ2

r

-

(4.2)

In order to construct the displacement and rotation interpolations free of shear locking, the ‘Kirchhoff mode’ [366] must be attainable. Therefore the displacement interpolation should be a polynomial of one degree higher order than the polynomial which interpolates the rotation field. Hence, if the assumed rotation field is linear θ (ξ ) = N1 (ξ )θ1 + N2 (ξ )θ2

(4.3)

then the displacement field must be quadratic w(ξ ) = N1 (ξ )w1 + N2 (ξ )w2 + N3 (ξ ), Δw3

(4.4)

N3 (ξ ) = 1 − ξ 2

(4.5)

where and Δw3 is hierarchical mid-side displacement. Using the displacement (4.4) and the rotation interpolation (4.3), we obtain the discrete approximation for the curvature κ(ξ ) :=

1 dθ = (θ2 − θ1 ) ; l = x2 − x1 dx l

(4.6)

and the shear strain   1 dw 1 1 4 − θ = (w2 − w1 ) − (θ1 + θ2 ) + ξ (θ1 − θ2 ) − Δw3 (4.7) γ (ξ ) := dx l 2 2 l If we use the isoparametric displacement interpolation (with Δw3 = 0), then, considering (4.7) above, it is impossible to have a constant shear strain unless θ2 = θ1 . However, by virtue of (4.6) this will lead to an overly stiff bending response, i.e. shear locking phenomenon. In order to remove the shear locking deficiency, we constrain the term enclosed in brackets to zero, which allows replacing (4.7) with a hierarchical displacement interpolation l w(ξ ) = N1 (ξ )w1 + N2 (ξ )w2 + N3 (ξ ) (θ1 − θ2 ) 8

(4.8)

122

4 Plate Models: Validation and Verification

4.1.1.2

Beam Element with Cubic Displacement Field

To get a beam element with a cubic displacement field, we proceed from the interpolations (4.3) and (4.8) extending them in a natural way to get the hierarchical rotation interpolation in the form of quadratic polynomial θ (ξ ) = N1 (ξ )θ1 + N2 (ξ )θ2 + N3 (ξ ) Δθ3

(4.9)

and the hierarchical displacement interpolation in the form of cubic polynomial l w(ξ ) = N1 (ξ )w1 + N2 (ξ )w2 + N3 (ξ ) (θ1 − θ2 ) + N4 (ξ )α l Δθ3 8

(4.10)

where N4 (ξ ) = ξ(1 − ξ 2 )

(4.11)

Additional rotation interpolation parameter Δθ3 is relative to the rotation field given by (4.3) and α in (4.10) is yet undetermined parameter. If we impose again the same constraint on constant shear strain along the beam 1 1 dw − θ = (w2 − w1 ) + 2α(1 − 3ξ 2 ) Δθ3 − (θ1 + θ2 ) − (1 − ξ 2 ) Δθ3 dx l 2 (4.12) we get the corresponding value of α = 1/6. Therefore, the hierarchical displacement interpolation (4.10) becomes

γ (ξ ) :=

l l w(ξ ) = N1 (ξ )w1 + N2 (ξ )w2 + N3 (ξ ) (θ1 − θ2 ) + N4 (ξ ) Δθ3 8 6

(4.13)

It is easy to check that the interpolation (4.13) for w is the exact solution for the adjoint differential equation of the Timoshenko beam. Therefore, when using this interpolation, the exact solution at the nodal points is obtained for an arbitrary loading (see [391]). It seems that this fact was not recognized in [366], although it was manifested in the numerical examples presented therein. The presented interpolation schemes are next extended to the Reissner-Mindlin plate elements. This is considered in the next section.

4.1.2 Reissner-Mindlin Plate Model and FE Discretization The variational formulation for the Reissner-Mindlin plate theory (e.g. see [153]) can be given directly by generalizing the hypotheses on cross-section for Timoshenko’s beam to director vector for plates. Namely, we assume that the so-called director vector, which is initially orthogonal to the mid-surface of the Reissner-Mindlin plate, will remain non-extensible and placed in the new configuration by a (small) rotation

4.1 Finite Elements for Analysis of Thick and Thin Plates

123

vector θ independently on vertical displacement w of the plate mid-surface. By placing the plate mid-surface in the plane (x1 , x2 ), with the director vector oriented along the unit vector e3 , we can express the displacement at any point p at vertical distance x3 from the mid-surface by using the corresponding displacement vector u at the mid-surface; moreover, by assuming small rotations; we can write: u p (x1 , x2 , x3 ) = u(x1 , x2 ) + θ(x1 , x2 ) × (x3 e3 )

(4.14)

In plates, we will ignore the so-called ‘drilling’ rotation component θ3 = 0 (see next chapter for membranes and shells including drilling rotations) and recover the component form of the expression above that reads: p

u 1 (x1 , x2 , x3 ) = u 1 (x1 , x2 ) + x3 θ2 (x1 , x2 ) p u 2 (x1 , x2 , x3 ) = u 2 (x1 , x2 ) − x2 θ1 (x1 , x2 ) p u 3 (x1 , x2 , x3 ) = u 3 (x1 , x2 ) ≡ w(x1 , x2 )

(4.15)

We can easily recover the Kirchhoff plate theory (equivalent to the Euler-Bernoulli beam theory), by assuming that the director vector remains perpendicular to the deformed mid-surface, which allows expressing in-plane displacement components in terms of derivatives of the transverse displacement w 1 ,x 2 ) u 1 (x1 , x2 , x3 ) = u 1 (x1 , x2 ) − x3 ∂w(x ∂ x1 θ1 = ∂∂w p ∂w(x1 ,x2 ) x2 ⇒ u 2 (x1 , x2 , x3 ) = u 2 (x1 , x2 ) − x2 ∂ x 2 θ2 = − ∂∂w p x1 u 3 (x1 , x2 , x3 ) = w(x1 , x2 )

p

(4.16)

The Kirchhoff plate theory requires providing very special finite element interpolation schemes, which was the topic of very active research in the early days of finite element developments resulting in not many successful solutions (e.g. the most wellknown are TUBA triangular plate element of Argyris [12] and refined quadrilateral plate element of Clough and Felippa [83]). Given difficulties in finding a successful discrete approximation for Kirchhoff’s plates, and also an encouraging early success of using reduced integration to alleviate locking phenomena in the ReissnerMindlin’s plates [157, 396], most of the subsequent research turned towards the latter as the starting point of discrete approximation, seeking different manners how to improve the model performance and completely eliminate locking. This is the path we followed in this chapter, resulting in plate finite elements among currently the best performers. Given 3D displacement field resulting from the kinematic hypothesis of ReissnerMindlin’s plates, we can carry on with applying the standard expressions for infinitesimal strains from solid mechanics (e.g. see [176]) to obtain the corresponding results that can be expressed in terms of generalized strain measures for plates κ and γ as

124

4 Plate Models: Validation and Verification p

i j =

1 2



∂u j ∂u i + ∂x j ∂ xi



αβ (x1 , x2 , x3 ) = αβ (x1 , x2 ) + x3 καβ (x1 , x2 ) p 1 ,x 2 ) 2 13 (x1 , x2 , x3 ) ≡ γ1 = ∂w(x + θ2 (x1 , x2 ) ∂ x1 ⇒ (4.17) p ∂w(x1 ,x2 ) 2 23 (x1 , x2 , x3 ) ≡ γ2 = ∂ x2 − θ1 (x1 , x2 ) p 1 ,x 2 ) =0 33 (x1 , x2 , x3 ) = ∂w(x ∂ x3

where we introduced index notation with Greek letters varying between 1 and 2, i.e. α, β ∈ {1, 2}. Keeping the same index notation to denote rotation-vector components θα around axes xα and vertical displacement field as w(xα ), we can write the weak form for Reissner-Mindlin’s plate model according to ˆ θˆα ) := G(w, θα ; w,



 (κˆ αβ m αβ + γˆα qα ) 

dΩ −

Ω

d x1 d x2

wˆ p dΩ = 0 , α, β ∈ {1, 2} Ω

(4.18) where κˆ αβ are virtual curvature tensor components, which can be obtained from the director virtual rotations, in the same manner as the real curvature strains bending strains καβ are obtained from the real rotation vector; however, it is even more convenient to use a slightly modified convention with καβ

1 = 2



∂ θ˜α ∂ θ˜β + ∂ xβ ∂ xα

(4.19)

where the director rotation θ˜ is related to the right-hand-rule rotation vector θ via an alternating tensor eαβ   0 +1 θ˜α = eαβ θβ , eαβ = (4.20) −1 0 In (4.18), γα are shear strain components that can now be written with modified rotation convention ∂w − θ˜α (4.21) γα = ∂ xα The stress resultants m αβ and qα in (4.18) above are obtained from the constitutive equations for the Reissner-Mindlin plates, which can be written by adapting plane-stress elasticity tensor to plates m αβ =

t3 Cαβγ δ κγ δ ; qα = cμt γα 12

(4.22)

For simplicity, in (4.22) we assumed linear elastic constitutive equations, along with uniform loading p and the Dirichlet boundary conditions in (4.18). However, the discussion to follow on how to construct non-locking finite element interpolation also applies to more general constitutive equations and other kinds of boundary conditions.

4.1 Finite Elements for Analysis of Thick and Thin Plates

125

The discrete formulation which corresponds to (4.18) is written in matrix notation h

G(w h , θ h ; wˆ h , θˆ ) :=



 κˆ h T C B κ h dΩ +

Ωh

 γˆ h T C S γ h dΩ −

Ωh

wˆ h f dΩ = 0 Ωh

(4.23) The mapping of generalized strain measures which is introduced in the discrete formulation (4.23), i.e. καβ → κ h = [−∂θ2 /∂ x1 ; ∂θ1 /∂ x2 ; ∂θ1 /∂ x1 − ∂θ2 /∂ x2 ]T

(4.24)

γα → γ h = [γ1 ; γ2 ]T

(4.25)

and

determines the form of the constitutive matrices C B and C S . For example, in the case of an isotropic linear elastic plate, we will have ⎤   1ν 0 Et 10 ⎣ν 1 0 ⎦ ; C S = Etc CB = 12(1 − ν 2 ) 0 0 1−ν 2(1 + ν) 0 1 2 ⎡

3

(4.26)

where t is the plate thickness, E is Young’s modulus and ν is Poisson’s ratio. The shear correction factor c is usually set to 5/6. Superscript h in (4.23) is the mesh parameter which is usually used to denote the quantities in the discrete approximation. To simplify notation in the subsequent discussion of the discrete formulation, we drop superscript h.

4.1.2.1

Plate Element with Quadratic Displacement Interpolation

We consider a four-node quadrilateral plate element in Fig. 4.2, which is the first candidate to provide non-locking finite element interpolations by following the rules for constructing the quadratic linked interpolation for Timoshenko’s beam element. The reference configuration of the element is defined by the bilinear mapping x(ξ, η) =

4 

N I (ξ, η)x I

(4.27)

I =1

where x = [x1 ; x2 ]T is the vector of local coordinates, x I are nodal values of that vector and N I (ξ, η) are standard bilinear shape functions (e.g., see [153]) in natural coordinates 1 (4.28) N I (ξ, η) = (1 + ξ I ξ )(1 + η I η) ; I = 1, 2, 3, 4 4

126

4 Plate Models: Validation and Verification

Fig. 4.2 Plate element with quadratic displacement interpolation

The natural coordinates (ξ, η) are defined on interval {−1, 1}, covering the domain of parent element depicted in Fig. 4.2. The displacement and the rotation interpolations are constructed by generalizing the Timoshenko beam interpolations (4.8) and (4.3) to get  h 4  θ1 h = θ (ξ, η) = N I (ξ, η) θ I θ2h

(4.29)

I =1

for the discrete approximation of the rotation field in terms of nodal rotations θ I and w h (ξ, η) =

4 

N I (ξ, η)w I +

I =1

8  L=5

N L (ξ, η)

lJ K T n (θ J − θ K ) 8 JK

(4.30)

for the hierarchical displacement interpolation, where l J K and n J K are, respectively, the length and the outward unit normal vector for the plate element edge between corner nodes J and K (see Fig. 4.2), defined by l J K = ((x K 1 − x J 1 )2 + (x K 2 − x J 2 )2 )(1/2) ; n J K =

  cos α J K sin α J K

(4.31)

The corner nodes J and K can be defined by a FORTRAN-like expression L = 5, 6, 7, 8 ; J = L − 4 ; K = mod(L , 4) + 1

(4.32)

In equation (4.30) above, we also use Serendipity shape functions [390] N L (ξ, η) =

1 (1 − ξ 2 )(1 + η L η) ; L = 5, 7 2

(4.33)

N L (ξ, η) =

1 (1 − η2 )(1 + ξ L r ξ ) ; L = 6, 8 2

(4.34)

4.1 Finite Elements for Analysis of Thick and Thin Plates

127

The displacement interpolation (4.30) can be rewritten in a more compact form as w (ξ, η) = h

4 

N I (ξ, η)w I +

I =1

where

4 

ˆ I (ξ, η)θ I N

(4.35)

I =1

ˆ I (ξ, η) = N M (ξ, η) l I J nTIJ − N L (ξ, η) l I K nTIK N 8 8

(4.36)

with indices in (4.36) above being defined as I = 1, 2, 3, 4; J = mod(I, 4) + 1 K = I − 1 + 4 ∗ int (1/I ); L = K + 4; M = I + 4

(4.37)

The curvature vector is determined by substituting (4.29) into (4.24) to get ⎡

⎤ wI κ h (ξ, η) = B I (ξ, η)u I ; u I = ⎣θ1 I ⎦ θ2 I I =1 4 

(4.38)

where B I has the form ⎡

⎤ 0 0 0 0 ∂ N I (ξ, η)/∂ x2 ∂ N I (ξ, η)/∂ x1 ⎦ B I (ξ, η) = ⎣ −∂ N I (ξ, η)/∂ x1 0 −∂ N I (ξ, η)/∂ x2

(4.39)

For the Timoshenko beam element hierarchical displacement interpolation (4.8) and (4.13) are sufficient to alleviate shear locking. However, in the plate element constructed with a similar hierarchical interpolation, the shear locking is still present. Some remedies to such a shear locking are either using adjusted parameters [367] or explicitly enforcing the constant shear strain [164]. The latter is similar to the assumed shear strain method for alleviating the shear locking (see, e.g. [158] or [30]), which provides a bilinear distribution for the assumed shear strain in the form   4  γ1 = γ (ξ, η) = N I (ξ, η)γ I γ2

(4.40)

I =1

where nodal parameters γ I are computed to be consistent with the constant shear strain distribution along each edge. For a typical node I we get γI =

   1 1 1 1 1 n w + n w − n + n IJ K IK J IJ I K wI lI J lI K lI J t TIJ n I K l I K  1 1 1 T T T T + n I J n I K θ K − n I K n I J θ J + (n I J n I K − n I K n I J )θ I 2 2 2

(4.41)

128

4 Plate Models: Validation and Verification

where indices are again defined by (4.37). The detailed derivation of this equation is given later in this section. We note that the proposed shear strain interpolation (4.40) corresponds to the plate element presented in [158]. For undistorted configuration, this element performs very similarly to the one presented in [164]. After applying the constraint (4.41) for each edge, we obtain ⎡

⎤ wI Bˆ I (ξ, η)u I ; u I = ⎣θ1I ⎦ γ = θ2I I =1 4 

(4.42)

Having defined the matrix notation in (4.38) and (4.42), the element stiffness matrix can be obtained from (4.23) as  ˆ TI (ξ, η) C S Bˆ J (ξ, η)] dΩ , I, J = 1, 2, 3, 4 K I J = [BTI (ξ, η) C B B J (ξ, η) + B Ωe

(4.43) By using the matrix notation in (4.35), the consistent load vector can be written as fI =

 

ˆ I (ξ, η) N I (ξ, η); N

T

f dΩ , I = 1, 2, 3, 4

(4.44)

Ωe

and the consistent mass matrix, needed in dynamic analysis, is defined by  MI J =

[A I (ξ, η)]T ρt [A J (ξ, η)] dΩ, I, J = 1, 2, 3, 4

(4.45)

Ωe

where ρ is the mass density and  AI =

4.1.2.2

 0 N I (ξ, η) ˆ I (ξ, η) N I (ξ, η) ; N I (ξ, η) = N I (ξ, η)I N

(4.46)

Plate Element with Cubic Displacement Interpolation

We refer to Fig. 4.3 to note that the element reference configuration is again defined by a bilinear mapping in (4.27) from its parent element. The displacement and rotation interpolations are provided by generalizing the Timoshenko beam cubic interpolations (4.9) and (4.13) to plates. More precisely, the rotation is interpolated as an incomplete quadratic polynomial  h 4 8   θ1 h = θ (ξ, η) = N (ξ, η) θ + N L (ξ, η)n J K Δθ J K I I θ2h I =1

L=5

(4.47)

4.1 Finite Elements for Analysis of Thick and Thin Plates

129

Fig. 4.3 Plate element with cubic displacement interpolation

where Δθ J K is a hierarchical rotation (or edge degree of freedom) for the element edge between corner nodes J and K , while N I (ξ, η), I = 1, . . . , 4 are bilinear and N L (ξ, η), L = 5, . . . , 8 are Serendipity shape function given by (4.28), (4.33) and (4.34). We note that a unique unit normal vector n J K must be selected at the global (structure) level for each side between corner nodes J and K , so that Δθ J K is properly shared between adjacent plate elements. The displacement interpolation is w h (ξ, η) =

4 

N I (ξ, η)w I +

I =1

8 

N L (ξ, η)

L=5

lJ K T n (θ J − θ K ) 8 JK (4.48)

8 

lJ K Δθ J K M L (ξ, η) + 6 L=5 where M L (ξ, η) =

1 (1 + η L η)ξ(1 − ξ 2 ) ; L = 5, 7 2

(4.49)

M L (ξ, η) =

1 (1 + ξ L ξ )η(1 − η2 ) ; L = 6, 8 2

(4.50)

The indices in Eqs. (4.47) and (4.38) above are again determined by expressions in (4.32). The assumed shear strain interpolation has again the bilinear form (4.40) and the nodal shear interpolation parameters are now given by γ I = γ ∗I +

1 T tI J nI K



2 2 n I J Δθ I K − n I K Δθ I J 3 3

 (4.51)

where γ ∗I are given by Eq. (4.41). The indices in (4.51) are defined by (4.37).

130

4 Plate Models: Validation and Verification

Having defined the preceding interpolation schemes for the cubic element, the element stiffness matrix, the consistent mass matrix and the consistent load vector can be obtained in the same manner as for the quadratic element. See Eqs. (4.43), (4.44) and (4.45). In this case, however, the total number of element degrees of freedom is increased by four mid-side hierarchical rotations.

4.1.2.3

Assumed Shear Interpolations

We first consider the plate element with quadratic displacement interpolation. The shear interpolation nodal parameters are determined as follows. We consider a typical node I , with adjacent nodes J and K , according to indices in Eq. (4.37). The shear strain along edge I J , γt I J , is constant and equal to γt I J =

1 1 (w J − w I ) − nTIJ (θ J + θ I ) lI J 2

(4.52)

Similarly, the shear strain along edge I K , γt I K , is a constant equal to γt I K =

1 1 (w I − w K ) − nTIK (θ I + θ K ) lI K 2

(4.53)

We next impose that projection of shear interpolation nodal parameter γ I on the edges I J and I K should be equal, respectively, to γt I J and γt I K , which can be written as (4.54) t TIJ γ I = γt I J and t TIK γ I = γt I K

(4.55)

This gives us two equations to solve for two unknown components of γ I . The solution can be written as γI =

1 t TIJ n I K

  n I K γt I J − n I J γt I K

(4.56)

After we substitute the values of γt I J and γt I K from (4.52) and (4.53), respectively, in Eq. (4.56) above, we recover the expression for γ I given by (4.41). For the plate element with cubic displacement interpolation, the only difference is in the expression for γt I J and γt I K . In this case, we have 2 γt I J = γt∗I J − Δθ I J 3

(4.57)

4.1 Finite Elements for Analysis of Thick and Thin Plates

131

where γt I J denotes the constant hear strain from equation (4.52), and 2 γt I K = γt∗I K − Δθ I K 3

(4.58)

where γt I K denotes the constant hear strain from Eq. (I.2). Using the new values for γt I J and γt I K in Eq. (4.52), we recover Eq. (4.51).

4.1.2.4

Discrete Kirchhoff Plate Element

In this section, we give a new derivation for the well-known discrete Kirchhoff plate element (e.g. see [38]). First, we briefly explain the main idea by considering the Timoshenko beam element with cubic displacement interpolation in (4.13) and quadratic interpolation for the independent rotation field in (4.9). If we now impose that the shear strain in (4.12) must be equal to zero, we get a condition for evaluating the hierarchical rotation Δθ3 =

3 3 (w2 − w1 ) − (θ1 + θ2 ) 2l 4

(4.59)

It can easily be verified that such a value of hierarchical rotation parameter would allow recovering the Hermite polynomials from cubic displacement interpolation in (4.13), and thus would lead us to the Euler-Bernoulli displacement interpolations. We now turn to plates by imposing the same constraint of vanishing shear strain along every edge of the plate element with cubic displacement interpolation. For a typical element edge between corner nodes J and K , we get Δθ J K =

3 3 (w K − w J ) − nTJ K (θ J + θ K ) 2l J K 4

(4.60)

Utilizing (4.60) to eliminate mid-side hierarchical rotations in the ReissnerMindlin plate element, from (4.47) we get a new interpolation for the rotation field  3 3 T θ= N I (ξ, η)θ I + N L (ξ, η) n J K (w K − w J ) − n J K n J K (θ J + θ K ) 2l J K 4 I =1 L=5 (4.61) with indices in (4.61) varying according to (4.32). Note that in Eq. (4.61) N I (ξ, η), I = 1, . . . , 4 are bilinear and N L (ξ, η), L = 5, . . . , 8 are Serendipity shape functions defined by (4.28), (4.33) and (4.34), which give an 8-node Serendipity element interpolation, but written in hierarchical form. The expression for the rotation field interpolation of Batoz and Tahar [38] is recovered, if one uses the classical form of the shape functions in the 8-node Serendipity element (see [390]). Therefore, the interpolation for the rotation field (4.61) is precisely the one for the discrete Kirchhoff plate element. This expression can be used as proposed in [227] to derive a unified formulation for both triangular and quadrilateral discrete Kirchhoff plate elements. The 4 

8 



132

4 Plate Models: Validation and Verification

triangular discrete Kirchhoff plate element [34] is recovered by simply degenerating the quadrilateral. Similarly, we can also use the constraint (4.60) to eliminate mid-side rotations in the displacement interpolation (4.48) of the Reissner-Mindlin plate element and obtain a consistent displacement interpolation for the discrete Kirchhoff plate element w=

4 

N I (ξ, η)w I +

I =1 8 



8  L=5

N L (ξ, η)

lJ K T n (θ J − θ K ) 8 JK

1 lJ K T (w K − w J ) − n (θ J + θ K ) M L (ξ, η) + 4 8 JK L=5



(4.62)

where M L (ξ, η), L = 5, . . . , 8 are hierarchical cubic shape functions defined by (4.49) and (4.50). The interpolations for the displacement (4.62) and the rotations (4.61) can be used in computing the consistent loading and consistent mass matrix for the discrete Kirchhoff plate element, in the same way as defined in Eqs. (4.44) and (4.45). Note that the chosen displacement and rotations interpolations yield vanishing shear stain along every element side. Thus, following the procedure presented in the previous section, we obtain the assumed shear strain interpolation with all the nodal shear interpolation parameters γ I being equal to zero, and the contribution of the shear strain to the element stiffness matrix in (4.43) drops out. Therefore, the discrete Kirchhoff plate element fits in a consistent way within a more general framework of Reissner-Mindlin plate elements discussed herein.

4.1.3 Illustrative Numerical Examples and Closing Remarks Several numerical examples are solved with different plate elements we presented. The plate element with quadratic displacement interpolation is denoted P Q2 and the element with cubic displacement interpolation is denoted P Q3. We have solved the problems for both thick and thin plates to demonstrate that the presented elements do not exhibit shear locking. All computations are performed by using the computer program F E L I N A [127]. 4.1.3.1

Uniform Loading on Simply Supported Square Plate

The test problem of a simply supported square plate under uniform loading (see Fig. 4.4) is used to point out the difference in using consistent loading defined in (4.44) versus lumped loading which is obtained if in (4.44) we drop the displacement dependence on nodal interpolation parameters for the rotations. Most of the assumed strain-based elements, including the discrete Kirchhoff, use what we call lumped loading. The same example is used to compare our elements with the dis-

4.1 Finite Elements for Analysis of Thick and Thin Plates

133

CL CL

6 HS

HS : w = 0, θn = 0 CL : θt = 0

a/2

? 

a/2

HS

-

Fig. 4.4 Uniform loading on a simply supported square plate

crete Kirchhoff element and, for that reason, only so-called hard simply supported boundary conditions are used with w = 0 and θn = 0. The plate is made of linear elastic isotropic material, with Young’s modulus E = 10.92 and Poisson’s ratio ν = 0.3. The side length a = 10 and two values for the plate thickness t = 0.1 (thin plate) and t = 1 (thick plate) are selected. An analytic solution for a thin square plate is given in [372] and it can easily be corrected to account for shear deformation and thus also recover the solution for a thick square plate. The numerical results are obtained for both plate elements presented herein by modeling one quadrant using uniform finite element meshes (see Fig. 4.4). The results are presented in Table 4.1 for the thin plate and in Table 4.2 for the thick plate along with the results obtained by the discrete Kirchhoff element [38]. First, we want to comment on using consistent versus lumped loading. The results obtained by the element P Q2 using lumped loading converge from below in all three ‘norms’: the center displacement, the center bending moment and the energy. Actually, for lumped loading plate element P Q2 is equivalent to T 1 plate element [158]. For consistent loading, however, P Q2 element converges from above in center displacement, center bending moment and energy. Since the convergence form above is very significant in practical engineering applications, we think that it is an important improvement of T1 class of elements. The same difference in convergence tendencies for consistent and lumped loading is present for the element P Q3 if we consider the energy and the center displacement. Center bending moment, however, is converging from above for both loading vectors, lumped and consistent, and has the same values as those obtained by the discrete Kirchhoff element. Moreover, the overall difference of elements P Q3 and D K Q appears to be insignificant, thus indicating the equivalence of these two elements in the thin plate limit. The result presented in Table 4.2, showing convergence in center displacement, center bending moment and energy, indicates that both P Q2 and P Q3 can be used

134

4 Plate Models: Validation and Verification

Table 4.1 Uniform loading on thin square plate Element PQ2 PQ3 Mesh/load Lumped Consistent Lumped Center displacement 1×1 31,915 2×2 39,712 4×4 40,436 8×8 40,593 16 × 16 40,631 ‘Exact’—thick 40,644 ‘Exact’—thin 40,623 Center bending moment 1×1 3.316 2×2 4.763 4×4 4.790 8×8 4.789 16 × 16 4.789 ‘Exact’ 4.793 Energy 1×1 99,732 2×2 182,118 4×4 205,026 8×8 210,948 16 × 16 212,440 ‘Exact’—thick 212,939 ‘Exact’—thin 212,814

Consistent

DKQ Lumped

Consistent

53,172 43,835 41,411 40,834 40,692 40,644 40,623

37,874 40,478 40,621 40,640 40,643 40,644 40,623

56,393 44,481 41,590 40,880 40,703 40,644 40,623

37,847 40,456 40,600 40,619 40,622 40,644 40,623

56,366 44,459 41,569 40,859 40,682 40,644 40,623

5.527 5.153 4.877 4.810 4.794 4.793

6.031 5.010 4.839 4.801 4.792 4.793

6.994 5.342 4.923 4.822 4.797 4.793

6.031 5.010 4.839 4.801 4.792 4.793

6.994 5.342 4.923 4.822 4.797 4.793

276,887 227,711 216,461 213,807 213,516 212,939 212,814

118,356 187,798 206,564 211,340 212,539 212,939 212,814

282,322 232,288 217,923 214,195 213,254 212,939 212,814

118,273 187,684 206,441 211,215 212,414 212,939 212,814

282,239 232,174 217,800 214,070 213,128 212,939 212,814

successfully in an analysis of thick plates. The discrete Kirchhoff element, however, is naturally converging again to the ‘thin plate solution’, thus introducing significant errors in the computed quantities.

4.1.3.2

Point Load on Simply Supported Square Plate

The test problem of a point load in the center of the simply supported square plate is used to illustrate the performance of the plate elements presented herein in the presence of singularity. Namely, the exact solution for the displacement under the point load, according to the Reissner-Mindlin plate theory, is infinite. The solution for the thin plate limit is finite and it is one used for comparison. We want to demonstrate that for a reasonably fine mesh in the thin plate limit our elements do not exhibit a deteriorated performance. The computations are repeated for the square plate in the previous example (E = 10.92, ν = 0.3, a = 10, t = 0.1 in Fig. 4.4), with a point load P = 1 applied in the

4.1 Finite Elements for Analysis of Thick and Thin Plates Table 4.2 Uniform loading on thick square plate Element P Q2 P Q3 Mesh/load Lumped Consistent Lumped Center displacement 1×1 34.566 2×2 41.902 4×4 42.545 8×8 42.684 16 × 16 42.717 ‘Exact’ 42.728 Center bending moment 1×1 3.316 2×2 4.763 4×4 4.790 8×8 4.789 16 × 16 4.789 ‘Exact’ 4.793 Energy 1×1 108.020 2×2 193.429 4×4 217.166 8×8 223.302 16 × 16 224.849 ‘Exact’ 225.365

135

Consistent

DK Q Lumped

Consistent

55.825 46.025 43.521 42.924 42.777 42.728

42.526 42.675 42.731 42.730 42.729 42.728

59.044 46.678 43.701 42.970 42.789 42.728

37.847 40.456 40.600 40.619 40.622 42.728

56.366 44.459 41.569 40.859 40.682 42.728

5.527 5.153 4.877 4.810 4.794 4.793

6.031 5.010 4.839 4.801 4.792 4.793

6.994 5.342 4.923 4.822 4.797 4.793

6.031 5.010 4.839 4.801 4.792 4.793

6.994 5.342 4.923 4.822 4.797 4.793

285.174 239.023 228.601 226.161 225.564 225.365

126.643 199.104 218.703 223.694 224.947 225.365

290.609 243.594 230.062 226.549 225.662 225.365

118.273 187.684 206.441 211.215 212.414 225.365

282.239 232.174 217.800 214.070 213.128 225.365

Table 4.3 Point load on thin square plate Mesh/elem. P Q2

P Q3

DK Q

Center displacement 1×1 2×2 4×4 8×8 16 × 16 ‘Exact’

1515 1271 1195 1172 1165 1160

1514 1269 1194 1170 1163 1160

1277 1152 1156 1160 1162 1160

center. The results are reported in Table 4.3, along with the results obtained by using the discrete Kirchhoff plate element. In Table 4.3, we can see that the displacement singularity does not influence accuracy for a reasonably fine mesh (from practical standpoint). This is in sharp contrast to the singularity present in the Kirchhoff plate model for the rhombic plate studied in the next example.

136

4 Plate Models: Validation and Verification

Fig. 4.5 Uniform loading on simply supported rhombic plate

Table 4.4 Uniform load on rhombic plate Mesh/elem. T1

P Q2

DK Q

Center displacement 2×2 4×4 8×8 16 × 16 ‘Exact’

0.04627 0.04271 0.03971 0.04206 0.04455

0.20804 0.08303 0.05533 0.04835 0.04455

4.1.3.3

0.02780 0.03918 0.03899 0.04187 0.04455

Uniform Loading on Rhombic Plate

Babuška and Scapolia [20] have pointed out the inaccuracies of the Kirchhoff plate model in the analysis of simply supported rhombic plate (Fig. 4.5). The reasons for inaccuracies are explained by the influence of the displacement singularity and hard simple support boundary conditions (w = 0, θn = 0). Reissner-Mindlin’s plate model with soft simple support boundary conditions (w = 0), which is used herein, renders the results much closer to the three-dimensional solution. The rhombic plate model selected in this example is made of linear elastic material with Young’s modulus E = 10 × 106 , Poisson’s ratio ν = 0.3, plate side a = 100 and plate thickness t = 1. The solution for the center displacement under unit uniform load q = 1, obtained by Morley [291], is used for comparison with numerical results. The numerical results are obtained using the plate element P Q2 and consistent loading. They are also compared to the results obtained by using the T 1 plate element [158] and D K Q element [38], which both use what we call lumped loading. We observe in Table 4.4 that the element P Q2 gives consistently better results than T 1 element and that the consistent loading again is beneficial for accuracy. The accuracy of DKQ element has significantly deteriorated.

4.1.3.4

Closing Remarks

We have presented a couple of plate elements which employ a hierarchical displacement interpolation and an assumed shear strain, and which are versatile in an analysis of both thick and thin plates. We have also demonstrated how the well-known discrete Kirchhoff plate element fits within the proposed framework.

4.2 Discrete Kirchhoff Plate Element Extension with Incompatible Modes

137

The explicitly defined hierarchical displacement interpolation that allows constructing the consistent loading, which is in contrast with the commonly used low order (lumped) loading approximation for the assumed strain elements. Using the consistent loading for the elements, in the presented examples we have observed the convergence form above, which is rather significant in practical applications. We did not perform any dynamic analysis, but it is reasonable to expect that the consistent mass matrix, defined herein, will also have beneficial effects on the accuracy. Since all finite element interpolations are explicitly defined, the extension to inelastic analysis is straightforward (as shown later in a chapter on inelasticity).

4.2 Discrete Kirchhoff Plate Element Extension with Incompatible Modes A couple of quadrilateral plate elements have been presented in the previous section, following the approach first proposed in [164] The elements are based on the plate theory proposed by Reissner [319] and Mindlin [290], a hierarchical displacement interpolation and an assumed shear strain field [158]. Moreover, we have also demonstrated how the well-known discrete Kirchhoff plate element [38] fits consistently within the proposed framework. The quadrilateral thick plate element proposed in previous section, which is equivalent to the discrete Kirchhoff quadrilateral, has totally 16 degrees of freedom: 12 nodal degrees of freedom and 4 edge degrees of freedom (normal mid-sides rotations). Introducing edge degrees of freedom may be desirable to facilitate easier construction of a complete interpolation polynomial, which has proven superior accuracy in many applications (e.g. see [378]). However, the presence of the edge degrees of freedom is somewhat undesirable from the standpoint of a standard computer program architecture, which normally carries degrees of freedom associated with the nodal points only. Therefore, on the basis of the thick plate element [164] with twelve nodal and four edge degrees of freedom, we construct an element with degrees of freedom associated with the nodal points only. In that process, the mid-side normal rotations are identified as the incompatible modes [225] interpolation parameters and via use of the static condensation procedure [384] they are eliminated at the element level. It is shown here that such an element still keeps all the desirable properties of its compatible predecessor. The successful addition of incompatible modes is performed by following the incompatible mode methodology proposed in the work of Ibrahimbegovic and Wilson [225]. The presented element can be used as an equivalent to the discrete Kirchhoff plate element in a thin plate analysis. However, as opposed to the discrete Kirchhoff, this element can also successfully be applied to the analysis of thick plates. The triangular plate element can be obtained by degenerating the proposed quadrilateral, as it was done with the discrete Kirchhoff element in [227].

138

4 Plate Models: Validation and Verification

4.2.1 Reissner-Mindlin Plate Model and Enhanced FE Interpolations The variational formulation for the Reissner-Mindlin plate theory is given in (4.18), where for simplicity, we assumed linear elastic constitutive equations and the Dirichlet boundary value problem with the homogeneous boundary conditions. However, the discussion to follow applies to other kinds of boundary conditions (as shown later in this chapter), as well as to more general constitutive equations (as shown in Chap. 7). The discrete formulation which corresponds to (4.18) is h G(w , θ ; wˆ , θˆ ) := h

h



  κ

h

hT

h

Ωh

  γ

C κ dΩ + B

hT

C γ dΩ − S

Ωh

w h f dΩ = 0

h

Ωh

(4.63) where the generalized strain measures introduced in the discrete formulation are written in matrix notation as κi j → κ h = [−∂θ2 /∂ x1 ; ∂θ1 /∂ x2 ; ∂θ1 /∂ x1 − ∂θ2 /∂ x2 ]T

(4.64)

γi → γ h = [γ1 ; γ2 ]T

(4.65)

and

In (4.63) above, we indicated that the stress-resultant moments and shear are computed from the strong form of the constitutive equations, with matrices C B and C S . For example, in the case of the isotropic linear elastic plate ⎤ ⎡   1ν 0 3 Et Etc 10 B S ⎦ ⎣ ν1 0 ; C = C = 12(1 − ν 2 ) 0 0 1−ν 2(1 + ν) 0 1 2

(4.66)

where t is the plate thickness, E is Young’s modulus and ν is Poisson’s ratio. The shear correction factor c is usually set to 5/6. Superscript h in (4.63) is the mesh parameter which is used to denote the quantities in the discrete approximation. 4.2.1.1

Finite Element Interpolation

We further consider a four-node quadrilateral plate element in Fig. 4.6 in order to construct enhanced finite element interpolations. The reference configuration of the element is defined by the bilinear mapping xh (ξ, η) |Ω e =

4  I =1

N I (ξ, η)x I

(4.67)

4.2 Discrete Kirchhoff Plate Element Extension with Incompatible Modes

139

Fig. 4.6 DKQ plate element with incompatible modes

where xh = [x1 ; x2 ]T is the discrete approximation for the position vector in natural coordinates, x I are the nodal values of that vector, and N I (ξ, η) are standard bilinear shape functions (e.g., see [153, 176, 390]) N I (ξ, η) =

1 (1 + ξ I ξ )(1 + η I η) ; I = 1, 2, 3, 4 4

(4.68)

The natural coordinates (ξ, η) are defined on interval {−1, 1}, which spans the domain of the paren element shown in Fig. 4.6. The curvature vector discrete approximation κ h is constructed by superposing the compatible and incompatible contribution as κ h (ξ, η) |Ω e =

4 

B I (ξ, η)θ I +

I =1

8  L=5

  θ  B L (ξ, η)n J K α Le ; θ I = 1 I θ2 I

where B I has the form   0 ∂ N I (ξ, η)/∂ x2 ∂ N I (ξ, η)/∂ x1 B I (ξ, η) = −∂ N I (ξ, η)/∂ x1 0 −∂ N I ξ, η)/∂ x2

(4.69)

(4.70)

The outward unit normal vector n J K for the plate element edge between corner nodes J and K (see Fig. 4.6) is given by nJ K

  cos φ J K = sin φ J K

(4.71)

where the corner nodes J and K can be defined by a FORTRAN-like expression L = 5, 6, 7, 8 ; J = L − 4 ; K = mod(L , 4) + 1

(4.72)

140

4 Plate Models: Validation and Verification

The strain-displacement matrix B L (ξ, η), L = 5, 6, 7, 8, has the same form as the matrix B I in (4.70), but the Serendipity shape functions [390] are used instead of the bilinear ones, where N L (ξ, η) =

1 (1 − ξ 2 )(1 + η L η) ; L = 5, 7 2

(4.73)

N L (ξ, η) =

1 (1 − η2 )(1 + ξ L ξ ) ; L = 6, 8 2

(4.74)

The strain-displacement matrix  B L (ξ, η) in equation (4.69) is produced by purifying B L of constant curvature states, in order to satisfy the patch test [176] requirements. Namely, we use the modification [225] 1  B L (ξ, η) = B L (ξ, η) − e Ω

 B L (ξ, η) dΩ

(4.75)

Ωe

In equation (4.69), α Le , L = 5, 6, 7, 8, are incompatible modes interpolation parameters, which correspond to the mid-sides hierarchical rotations. Namely, the curvature-rotation matrix in (4.69) is consistent with a hierarchical rotation interpolation 4 8   N I (ξ, η) θ I + N L (ξ, η)n J K α Le (4.76) θ h |Ω e = θ e = I =1

L=5

Assumed shear strain field is bilinear over an element, i.e. γ |Ω e = γ e =

4  I =1

N I (ξ, η)γ I +

4  I =1

N I (ξ, η)

2 3t TIJ n I K

[n I J α eM − n I K α Le ] (4.77)

where nodal parameters γ I are computed to be consistent with the constant shear strain distribution along each edge. For a typical node I we get    1 1 1 1 1 nI J wK + nI K w J − nI J + nI K wI γI = T lI J lI K lI J tI J nI K l I K  1 1 1 T T T T + n I J n I K θ K − n I K n I J θ J + (n I J n I K − n I K n I J )θ I 2 2 2

(4.78)

In Eqs. (4.77) and (4.78), indices are defined by I = 1, 2, 3, 4; J = mod(I, 4) + 1; K = I − 1 + 4 ∗ int (1/I ); L = K + 4; M = I + 4

(4.79)

and l I J is the length of the plate element edge between corner nodes I and J (see Fig. 4.6), i.e.

4.2 Discrete Kirchhoff Plate Element Extension with Incompatible Modes

l I J = ((x I 1 − x J 1 )2 + (x I 2 − x J 2 )2 )(1/2)

141

(4.80)

The detailed derivation of Eq. (4.78) is given in the previous section. Having defined the curvature (4.69) and the shear strain interpolation (4.77), we can compute the element stiffness matrix from the equation (4.63). It is initially of the size 16 × 16. After the static condensation is performed to eliminate four incompatible modes parameters α e , we obtain the standard format with 12 × 12 size for the element stiffness matrix. In the basic variational structure of the incompatible modes formulation [225], there are no element loads associated with the incompatible modes parameters. For simplicity, we can use the bilinear interpolation (lumped form) for loading vector, defined by  fI =

N I (ξ, η) p dΩ , I = 1, 2, 3, 4

(4.81)

Ωe

4.2.2 Illustrative Numerical Examples and Closing Remarks Several numerical examples are solved by using the plate element presented herein. In the results to follow, the plate element is denoted as P Q I . All numerical computations are performed by using 2 × 2 Gauss quadrature (e.g. see [176, 390]). The bending moments are first computed at the Gauss quadrature points to obtain their optimal accuracy [25]. They are then projected to the nodes by using the least square fit over an element to the bilinear smoothed field [390].

4.2.2.1

Uniform Loading on Simply Supported Square Plate

The test problem of a simply supported square plate under uniform loading (see Fig. 4.6) is used to compare our element with the discrete Kirchhoff plate element [38]. The plate is made of linear elastic isotropic material, with Young’s modulus E = 10.92 and Poisson’s ratio ν = 0.3. The side length a = 10 and two values for the plate thickness t = 0.1 (thin plate) and t = 1 (thick plate) are selected. An analytic solution for the thin plate is given in [372] and it can be corrected to account for shear deformation to get the solution for the thick plate. Both so-called hard (w = 0 and θn = 0) and soft (w = 0) simply supported boundary conditions are used. In the element P Q I representation of the hard simple support boundary condition is disturbed by the inability to set the corresponding incompatible modes equal to zero (recall that they can be identified with hierarchical mid-sides rotations). However, this does not influence significantly the accuracy of the computed results. The numerical results are obtained by modeling one quadrant of the plate using uniform finite element meshes (see Fig. 4.7). The results are presented in Table 4.5

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4 Plate Models: Validation and Verification

Fig. 4.7 Uniform loading on a simply supported square plate Table 4.5 Uniform loading on thin square plate Result Center displacement Support Hard Soft Mesh/elem. 1×1 2×2 4×4 8×8 16 × 16 32 × 32 ‘Exact’

PQI 46026 42512 41115 40761 40673 40651 40644

DK Q 37847 40456 40600 40619 40622 40623 40644

PQI 48708 44613 42273 41395 41060 40961 –

DK Q 37903 40458 40600 40619 40622 40623 –

Center moment Hard PQI 8.129 5.597 4.979 4.836 4.800 4.792 4.793

DK Q 6.031 5.010 4.839 4.801 4.792 4.789 4.793

Soft PQI 8.959 5.659 5.081 4.892 4.835 4.819 –

DK Q 6.126 5.006 4.839 4.801 4.792 4.789 –

for the thin plate and in Table 4.6 for the thick plate along with the results obtained by the discrete Kirchhoff element [38]. Considering the results in Table 4.5, we can state that the overall convergence rates for elements P Q I and D K Q appear to be similar. The results presented in Table 4.6, showing convergence in center displacement and center bending moment, indicate that P Q I can be used successfully in an analysis of thick plates. The discrete Kirchhoff element, however, is naturally converging again to the ‘thin plate solution’, thus introducing significant errors in the computed quantities.

4.2.2.2

Uniform Loading on Simply Supported Circular Plate

The test problem of a uniform loading on the simply supported circular plate is used to illustrate the performance of the plate element P Q I in an arbitrarily distorted

4.2 Discrete Kirchhoff Plate Element Extension with Incompatible Modes Table 4.6 Uniform loading on thin square plate Result Center displacement Support Hard Soft Mesh/elem. 1×1 2×2 4×4 8×8 16 × 16 32 × 32 ‘Exact’

PQI 48.682 44.709 43.225 42.852 42.759 42.736 42.728

DK Q 37.847 40.456 40.600 40.619 40.622 40.623 42.728

PQI 51.687 47.743 46.122 46.002 46.103 46.150 –

DK Q 37.903 40.458 40.600 40.619 40.622 40.623 –

143

Center moment Hard PQI 8.115 5.591 4.979 4.836 4.800 4.792 4.793

DK Q 6.031 5.010 4.839 4.801 4.792 4.789 4.793

Soft PQI 8.914 5.808 5.238 5.117 5.099 5.096 –

DK Q 6.126 5.006 4.839 4.801 4.792 4.789 –

Fig. 4.8 Uniform loading on simply supported circular plate

Table 4.7 Uniform loading on circular plate Result Center displacement No. elem./thick. t = 1 t = 0.1 3 12 48 192 ‘Exact’

41.029 41.544 41.608 41.610 41.599

39,293 39,775 39,841 39,842 39,832

Center bending moment t =1 t = 0.1 5.209 5.178 5.159 5.157 5.156

5.233 5.180 5.158 5.157 5.156

element configuration. The original method of incompatible modes [386] did not converge in such a case. The computations are performed on a quarter of the circular plate of radius R = 5 (see Fig. 4.8), using the same properties as in the previous example (E = 10.92, ν = 0.3, t = 0.1 and 1 and q = 1). The accuracy of the results, which are reported in Table 4.7, show that the presented plate element, based on the modified method of incompatible modes [6], faces no difficulties in an arbitrarily distorted element configuration.

144

4.2.2.3

4 Plate Models: Validation and Verification

Closing Remarks

In this section, we presented a methodology which allows shearing deformations to be incorporated into the standard D K Q and D K T plate bending formulations and implemented it in a plate element denoted as P Q I . In constructing the finite element interpolation scheme for the plate element P Q I , a fundamental role is played by the assumed shear strain interpolation and a set of incompatible modes used to enrich the bending strain interpolation. In order to provide a satisfying performance for the method of incompatible modes, we rely on the methodology presented in collaboration with Ed Wilson [225]. The plate element P Q I shares with the D K Q the same number of degrees of freedom (incompatible modes are eliminated at the element level) and the same discrete approximation accuracy. Thus, in an analysis of very thin plates, the two lead to the same results. However, as opposed to the D K Q element, the element P Q I can also be used successfully in an analysis of thick plates.

4.3 Validation or Model Adaptivity for Thick or Thin Plates Based on Equilibrated Boundary Stress Resultants Analysis considering a complex plate structure (with most general shape, loading and boundary conditions) is one of the most frequently encountered task in structural engineering practice. A problem of selecting the most suitable computational model for a particular plate structure between thick and thin plates is the topic of this section. If successfully solved, it can lead to an efficient and accurate plate analysis, which is of great practical interest. The plate structures are often combined with frame and other skeletal structures, for which one can develop by far the most efficient finite element analysis by exploiting the 1D model and the superconvergence properties of the corresponding finite element method, which implies the smallest error in energy norm (e.g. see [153, 176, 390]). Given that the superconvergence property of the finite element solution does not carry over to 2D/3D problems, we seek to construct the best possible solution for the plate problem by appealing to validation (the best choice for a model) and verification (the best choice for a discrete approximation), both of them are defined again in the energy norm. Thus, finding the solution to the above-mentioned problems would clearly have a very high practical value. Here, we start by studies of validation or adaptive modeling in structural analysis that has the goal to produce the most suitable computational model for a particular structural component that can be represented by a plate. The computational model suitability is measured in terms of two errors, which normally occur in finite element modeling: the discretization error and the modeling error. Since adaptive modeling relies on both discretization and modeling error, in order to drive the adaptive process of computational modeling, the best possible estimate for both errors is of crucial importance for its success. The discretization error comes as a consequence of a cho-

4.3 Validation or Model Adaptivity for Thick …

145

sen finite element discretization of a particular solid/structural mathematical model. In simple words: it measures how close the discretized finite element solution is to the exact solution of governing equations of the underlying mathematical model. The modeling error is related to the suitability of the mathematical model itself. In plate problems, it arises because we use a simplified 2D model, i.e. the Kirchhoff plate or the Reissner-Mindlin plate, as the corresponding approximation of a 3D solid. Among other reasons, such simplifications are of interest due to the computational savings (through a dimensional reduction) and higher computational robustness of 2D plate models with respect to 3D model in the case of plate-like domains, with one dimension (thickness) much smaller than the other two (mid-surface). In order to produce an efficient computational model, we thus have to control both the discretization and the modeling error. The discretization error is controlled by suitable mesh grading of the domain, which is usually called mesh refinement or h-adaptivity. Many error estimators and corresponding mesh adaptivity procedures are available (see e.g. [21, 305] for continuum model and [260, 397] for plates). On the other hand, not many procedures are suitable for modeling error control. It has been shown previously e.g. by [304, 307, 353, 354] or [355] that modeling error for 2D/3D linear elasticity can be controlled by so-called equilibrated boundary traction approach to error estimation. Our departure point is the assumption that a family of hierarchical plate models is provided, and the adaptive analysis starts with the lowest order model. From a posteriori computations that include the higher model in the hierarchy, we can obtain a suitable model error indicator. The regions where a more refined model should be used can thus be identified. In this section, we focus upon the development of model adaptivity procedure for plates. We assume that mesh and model adaptivity can be treated separately. Ideally, the model adaptivity procedure should start from the final output of the mesh adaptivity, which would distribute evenly the discretization error throughout the mesh. The presented procedure can be seen as an application of the equilibrated boundary traction approach to error estimation to model adaptivity for plates. It can select automatically which particular model, chosen from a set of available hierarchical plate models, is the most suitable for any finite element of the chosen mesh. We address in detail a particular case of two low-order models, i.e. those of Kirchhoff that includes no shear and of Reissner-Mindlin that includes shear strain. We note that the procedure developed for these two models (we will call it KRM adaptivity procedure) has a very similar form to the procedure that would include higher hierarchical models, i.e. so-called (1, 1, 2), (3, 3, 4) or ‘zig-zag’ plate models that take into account through-the-thickness stretching and nonlinear distribution of displacements through the plate thickness (e.g. see [22] or [60]). The chosen elements are quadrilaterals; i.e. discrete Kirchhoff quadrilateral (DKQ) and the corresponding Reissner-Mindlin quadrilateral (RMQ). They have the ability to provide approximately the same order of the discretization error for bending moments, leaving the only difference in the shear part of the error norm, which justifies uncoupling the model and mesh adaptivity procedures. It is obvious that the main reason for using the model adaptivity procedure that chooses between the DKQ and the RMQ plate elements is not so much an increase

146

4 Plate Models: Validation and Verification

in efficiency. Namely, the DKQ element allows only slight computational savings with respect to the RMQ (by skipping the shear deformation contribution to stiffness matrix), and one may simply use the RMQ throughout the mesh. However, there are several aspects to such a procedure that deserve attention: (i) the proposed KRM procedure explained in this rather simple setting (choosing to compute the element stiffness with or without shear contribution) can easily be generalized to more complex models including all higher-order plate models and thus achieve much more significant computational savings; (ii) the DKQ plate element is among the most robust and most often used finite element for plates in many commercial codes (e.g. SAP2000 [385]) and mesh adaptivity performed with the DKQ element is easier to handle than using the RMQ element (e.g. see [260]); (iii) from the theoretical point of view, the proposed KRM procedure can be used to locate regions in the plate where shear deformation should be accounted for. The outline of this section is as follows. First, we recall the plate finite elements already presented in this chapter and used for the proposed validation procedure (see also [162, 164]). These elements have an exceptional feature of sharing the same order of discrete approximation with non-conventional finite element interpolations for displacement and bending strains for thick and thin plates. We also briefly comment on the possible manners for choosing an optimal finite element mesh applicable to any of these plate models. We next discuss how to test any of the plate elements of the chosen mesh in order to choose between the Kirchhoff plate and the Reissner-Mindlin plate models. Finally, we present a number of numerical examples illustrating the proposed validation procedure satisfying performance, as well as closing remarks.

4.3.1 Thick and Thin Plate Finite Element Models In this section, we briefly recall the plate finite elements that will further be used in the model adaptivity procedure.

4.3.1.1

Theoretical Formulation

We model a plate as a 2D body occupying a domain Ω in the (x1 , x2 ) plane. The weak form of the boundary value problem for the Reissner-Mindlin plate model is given in (4.18) as ˆ θˆα ) := G(w, θα ; w,



 (κˆ αβ m αβ + γˆα qα ) dΩ −

Ω

wˆ p dΩ = 0 , α, β ∈ {1, 2} Ω

(4.82)

4.3 Validation or Model Adaptivity for Thick …

147

where the generalized strain measures are used to obtain the stress resultants with a set of linear elastic constitutive equations t3 Cαβγ δ κγ δ ; qα = cμt γα 12 1 ∂ θ˜α ∂ θ˜β ∂w καβ = + − θ˜α ; γα = 2 ∂ xβ ∂ xα ∂ xα   ˜θα = eαβ θβ , eαβ = 0 +1 −1 0

m αβ =

(4.83)

It is important to note that in the Reissner-Mindlin plate model, the generalized strains are computed as the first derivatives of the state variables, displacement w and rotations θα , resulting in the standard continuity requirement of discrete approximations (e.g. see [176]), but also with non-zero shear strains that require special non-locking interpolations. On the other hand, the Kirchhoff plate model eliminates the locking phenomena from the start by removing completely the contribution of shear strains by enforcing the constraints γα = 0, which introduces a new definition of the curvature tensor components ∂w ∂ 2w ⇒ κ˜ αβ = (4.84) γα = 0 ⇒ θ˜α = ∂ xα ∂ xα ∂ xβ We can thus restate the variational formulation in (4.82) as  G(w; w) ˆ := Ω

(κˆ˜ αβ m αβ ) dΩ −

 Ω

wˆ p dΩ = 0; κˆ˜ αβ =

∂ 2 wˆ ∂ xα ∂ xβ

(4.85)

where we changed the generalized strain measures for bending but kept the same linear elastic constitutive equations as in (4.83). We will next focus upon the model adaptivity for plates, and propose a procedure that can be used in the context of discrete approximation (the mesh adaptivity will be discussed in the next section). Therefore, we will use the most general framework for a discrete approximation that can accommodate both Reissner-Mindlin’s plates (with shear) and Kirchhoff’s plates (without shear). The key feature of the proposed procedure is to construct both plate models to share the same order of discrete approximation, which can help to eliminate the influence of discretization error. We note in passing that providing such a discrete approximation in plates is certainly more complicated than for beam models, where the flexibility approach (see Chap. 3) allows us to add shear deformation contribution to Euler-Bernoulli beams. The discrete approximation, which corresponds to the most general format of variational formulation in (4.82), can be written as

148

4 Plate Models: Validation and Verification



h

G(w h , θ h ; wˆ h , θˆ ) :=

 κˆ h T C B κ h dΩ +

Ωh

 γˆ h T C S γ h dΩ −

Ωh

wˆ h f dΩ = 0 Ωh

(4.86)

where the following mappings are defined κihj → κ h = [−∂θ2h /∂ x1 ; ∂θ1h /∂ x2 ; ∂θ1h /∂ x1 − ∂θ2h /∂ x2 ]T

CiBjkl

(4.87)

γih → γ h = [∂w h /∂ x1 + θ2h ; ∂w h /∂ x2 − θ1h ]T

(4.88)

⎤ ⎡   1ν 0 3 Et 10 ⎣ν 1 0 ⎦ ; CiSj → C S = cEt → C B = 12(1 − ν 2 ) 0 0 1−ν 2(1 + ν) 0 1 2

(4.89)

In (4.89) above, t is plate thickness, E is Young’s modulus and ν is Poisson’s ratio. The shear correction factor c is usually set to 5/6 (see [27] or [36]). Superscript h in (4.86) is the mesh parameter, which is usually used to denote the quantities in the discrete approximation. In what follows we will restrict ourselves to quadrilateral plate elements. The discretized Ω h is represented by a finite element mesh of plate elements, n el domain h e Ω = e=1 Ω , where n el is the number of elements in the mesh. The geometry of an element is defined by the bilinear mapping ξ ∈  → xh ∈ Ω e xh (ξ )|Ω e =

4 

N I (ξ ) x I ; x I = [x1I ; x2I ]T ; ξ = [ξ, η]

(4.90)

I =1

where  ∈ [−1, +1] × [−1, +1], x I are coordinates of a finite element node I, and N I (ξ, η) =

4.3.1.2

1 (1 + ξ I ξ )(1 + η I η) 4

I 1 2 3 4 ξ I -1 +1 +1 -1 η I -1 -1 +1 +1

(4.91)

Reissner-Mindlin Quadrilateral (RMQ) Plate Element

Interpolation of the rotation field for plate element in Fig. 4.9 is based on quadratic polynomials (see [162] for details)  h 4 8   θ1 h e = = θ (ξ )| N (ξ ) θ + N L (ξ )n J K Δθ J K Ω I I θ2h I =1

L=5

(4.92)

4.3 Validation or Model Adaptivity for Thick …

149

Fig. 4.9 Plate element with cubic displacement interpolation

where N L (ξ ) = 21 (1 − ξ )2 (1 + η J η); L = 5, 7 N L (ξ ) = 21 (1 − η)2 (1 + ξ J ξ ); L = 6, 8

L 5678 J 1234 K2341

(4.93)

and 1/2  l J K = (x1K − x1J )2 ) + (x2K − x2J )2

n J K = [cos α J K ; sin α J K ]T ;

Location of rotations θ I and Δθ J K is illustrated in Fig. 4.9. Displacement field is interpolated by cubic polynomials w h (ξ )|Ω e =

4  I =1

N I (ξ )w I +

8 

N L (ξ )

L=5

8  lJ K T lJ K n J K (θ J − θ K ) + Δθ J K M L (ξ ) 8 6 L=5 (4.94)

M L (ξ ) = 21 η J (1 − ξ )2 ξ(1 + η J η); L = 5, 7 M L (ξ ) = 21 ξ J (1 − η)2 η(1 + ξ J ξ ); L = 6, 8

(4.95)

Interpolation of bending strains follows from (4.87) and (4.92) κ h (ξ )|Ω e =

4  I =1

B I (ξ )θ I +

8  L=5

B˜ L n J K Δθ J K

(4.96)

150

4 Plate Models: Validation and Verification

⎡

0

B I = ⎣ N I ,x2 N I ,x1

⎤ −N I ,x1 0 ⎦; −N I ,x2

⎡

0

B˜ L = ⎣ N L ,x2 N L ,x1

⎤ −N L ,x1 0 ⎦ −N L ,x2

∂ξ

j where notation (·),xi = ∂(·) ; ξ1 = ξ ; ξ2 = η is used. We further choose a bilinear ∂ξ j ∂ xi distribution for the assumed shear strain in the form



 4  γ1h h e = γ (ξ )| = N I (ξ )γ I Ω γ2h

(4.97)

I =1

where nodal shear strains γ I are consistent with the constant shear strain distribution along each edge. For a node I we have γI =

1 T tI J nI K



1 lI K

nI J wK +

1 lI J

 nI K w J −

1 lI K

nI J +

1 lI J

 nI K wI

1 1 1 + n I J nTIK θ K − n I K nTIJ θ J + (n I J nTIK − n I K nTIJ )θ I 2 2 2  2 2 + n I J Δθ I K − n I K Δθ I J ; 3 3

I 1234 J 4123 K2341

(4.98)

Notation for strains can further be simplified by using κ h (ξ )|Ω e =

4 

 B I u I , γ h (ξ )|Ω e

I =1

⎛

⎞ wI = GI uI , uI = ⎝ θ I ⎠ Δθ J K I =1 4 

ˆ I follows from (4.96) and G I from (4.97) and (4.98). where B The element stiffness matrix can be computed on the node-to-node basis as  KeI J = Ωe

T  B I (ξ ) C B  B J (ξ ) dΩ +

 GTI (ξ ) C S G J (ξ ) dΩ

(4.99)

Ωe

Both bending and shear strains have been taken into account. The presented plate element has 8 nodes and 16 dofs, 3 per each vertex node and 1 hierarchical rotation Δθ I J at each mid-side node. Remark: Reissner-Mindlin plate with only 3 dofs per node and incompatible internal dofs can be developed by using the method of incompatible modes and by eliminating the relative rotations at the element level (see [164]).

4.3 Validation or Model Adaptivity for Thick …

4.3.1.3

151

Discrete Kirchhoff Quadrilateral (DKQ) Plate Element

On the basis of the Reissner-Mindlin quadrilateral of the preceding section one can derive the DKQ element. If we impose a constraint that the shear strain vanishes along all element edges (see [162] for details), we can express the hierarchical rotations as Δθ J K =

3 3 (w K − w J ) − n J K · (θ J + θ K ) 2l J K 4

(4.100)

The discrete approximation for the rotation field in (4.92) can then be rewritten as θ h (ξ )|Ω e =

4 

N I (ξ )θ I

I =1

+

8 

 N L (ξ )

L=5

3 3 n J K (w K − w J ) − n J K nTJ K (θ J + θ K ) 2l J K 4



(4.101) and the explicit form of the discrete approximation of the displacement field in (4.95) now reads w h (ξ )|Ω e =

4 

N I (ξ )w I +

I =1

8 

N L (ξ )

L=5

8 

lJ K T n (θ J − θ K ) 8 JK



1 lJ K T n (θ J + θ K ) M L (ξ ) (w K − w J ) − + 4 8 JK L=5

 (4.102)

The displacement and rotation finite element interpolations (4.101) and (4.102) will make nodal shear strains (4.98) equal to zero. Due to (4.97), the shear strain discrete approximation is zero throughout the whole element domain. The element stiffness matrix and load vector for such a plate element can be computed on the node-to-node basis as   T T K eI J = B¯ I (ξ ) C B B¯ J (ξ ) dΩ, f I =  (4.103) N I (ξ ) f dΩ; Ωe

where w h |Ω e =

Ωe

4  I =1

T  N I u I , κ h |Ω e =

4  I =1

B¯ I u I , u I =



wI θI



with B¯ I that follows from (4.96) and (4.100) and  N I that follows from (4.102). Note, that this 4-node quadrilateral element has only 3 dofs per node.

152

4.3.1.4

4 Plate Models: Validation and Verification

Mesh Adaptivity to Reduce Discretization Error

In order to obtain the best possible result on model adaptivity, one should first make sure that the mesh density and discretization error are sufficiently reduced. Namely, one can control the mesh density by refinement driven by the mesh adaptivity and the estimates of the discretization error. Since our main interest is the plate models, we will adapt our procedure to stress resultants, by using a posteriori discretization error that can provide the estimates in the energy norm of the difference between the ‘true’ and the finite element (FE) stress resultants. The ‘true’ ones are obviously not known, but one can produce their replacements by improving the computed values of FE stress resultants. This can be achieved by smoothing the FE stress resultants across the whole mesh, looking for the best possible fit of smoothed solution in the least square sense to the FE solution. The computation of this kind can be carried out independently for each stress resultant component. h . The FE Let us illustrate the procedure for bending moment component m αβ h results are available at each Gauss point of each plate element m αβ (ξ G )|Ω e = CiBjkl κklh (ξ G )|Ω e . The corresponding smooth field can be obtained over each element by using the standard bilinear shape functions N I (ξ ) and unknown nodal values a I , m ∗αβ (ξ )|Ω e =

4 

N I (ξ )a I

(4.104)

I =1

The least squares fit results in el 1 2 e=1

n



h [m ∗αβ (ξ ) − m αβ (ξ )]2 dΩ → min

(4.105)

Ωe

The last two equations can be combined to provide the minimization problem, which leads to a set of equations for computing the unknown parameters a J el el Ane=1 [M Ie J ]a J = Ane=1 beI

(4.106)

with  M Ie J =

N I (ξ )N J (ξ ) dΩ = Ωe

beI

nint 

N I (ξ G )N J (ξ G ) j (ξ G )wG

(4.107)

N I (ξ G ) m ihj (ξ G ) j (ξ G )wG

(4.108)

G=1

 N I (ξ ) m ihj

= Ωe

dΩ =

nint  G=1

4.3 Validation or Model Adaptivity for Thick …

153

el where Ane=1 denotes the finite element assembly procedure, nint is the number of element integration points, wG are the weights of integration points, and j is the transformation Jacobian  → Ω e . Different proposals are made in literature in order to reduce the computational cost of this global problem for computing the smooth approximation. For example, one can use it only at the level on a single element (e.g. see [150]), or at the level of a patch of elements surrounding a particular node (e.g. see [397]), or yet at the global level but using only diagonal form of matrix M to make the computation more efficient (e.g. see [391]). The discretization error can be estimated in the energy norm. For the DKQ plate element, this can be based on the difference between the improved moments m∗ and the moments m = C B κ h obtained from the finite element solution  −1 2 (4.109) e = (m∗ − m)T C B (m∗ − m)dΩ

Ωh

For the RMQ plate element, other strategies can be applied that also take into account the influence of boundary layers, see e.g. [260]. Integral over the whole domain (4.109) can be decomposed into element integrals and the total error estimator can be expressed as the sum of element error estimators e2 =



ee 2

(4.110)

e

This information can be used to obtain each element’s contribution to the total discretization error, as well as to estimate the need to refine the mesh. In order to construct optimal mesh with uniform distribution of discretization error, one needs to relate element size with the discretization error. Typical element size h e can be defined as  h e = Ae /π where Ae is element area. From the a priori analysis of discretization error we get (see [390]) the following relation ee  = Ch ep

(4.111)

where p stands for the polynomial order of used interpolation and C is a sizeindependent constant. The aim is to construct a mesh with Ne elements and a uniform distribution of discretization error with its total value e ≈ T O L. Following (4.110) the element discretization error should then be T OL ee  ≈ √ Ne

(4.112)

154

4 Plate Models: Validation and Verification

Desired element size can thus be determined from (4.111) as h˜ e =



T OL √ C Ne

(1/ p) (4.113)

Since the estimate of discretization error ee  for an element of size h e is known, the value of a constant can be deduced from it C = ee / h ep

(4.114)

The estimated new size of the element is then h˜ e = h e



T OL √ ee  Ne

(1/ p) (4.115)

Based on the obtained estimate of element size distribution the new mesh is automatically generated. Mesh refinement is repeated until the desired accuracy is met.

4.3.2 Model Adaptivity for Plates To locate regions of the domain where the chosen plate model (here, the thin plate model, the simplest among available hierarchical models) no longer performs well, one has to provide an estimate of the model performance. Ideally, such an estimate for the chosen model should follow from the comparison with the best possible mathematical model, which provides the closest solution to 2D/3D solid model. Computing the solution with the best model estimate is often not feasible, since it remains prohibitively expensive or simply inaccessible. Thus, in order to provide the practical model error estimates, we further compare the chosen model results against the same type of results obtained by a plate model which is known to deliver better performance than the chosen plate model. Such model will be referred to as the enhanced model. Yet further modification is needed in order to increase computational efficiency. In principle, two global computations would be required to obtain the best results for comparison between two different plate models: the chosen against the enhanced model. However, the computational cost would have not been justified for merely providing the model error estimates, for we could simply keep the results of the enhanced model. Hence, we seek to reduce the computational cost with the enhanced model that is needed only to provide an estimate of the true stress field. This is made possible by extracting a portion of the domain, typically one finite element based on the enhanced model, and reducing the computational cost by processing one finite element at a time. For such computations, we apply along element boundary the traction loads representative of the true stress state, compute the local enhanced

4.3 Validation or Model Adaptivity for Thick …

155

solution and compare it element-wise against the solution provided by the chosen plate model. The proposed procedure efficiency should not be made at the expense of accuracy, given that one still needs the best possible estimate of true stress state. Here, we can construct the corresponding approximation by post-processing of the FE solution obtained with the chosen model in order to provide the best possible approximation of the true stress state. This is done independently with the FE solution for any stress component, which is discontinuous between the elements, by enforcing it to be continuous across the element boundaries as a logical approximation to the true solution (assuming no discontinuity in plate thickness, material or loading). In accordance with the best-possible approximation of the true stress state, the edge loading (or boundary traction) for each finite element of the mesh is computed. That loading is further used in setting the local (element-wise) Neumann problems based on the enhanced model. Comparison of two plate models can thus be carried out in terms of the solution energy norm computed by global computation for one and a set of local computations for other. In the case of plate problem, the procedure described above implies the need to construct the solution for a hierarchical family of plate models under boundary tractions, the same as done for 3D solid model (e.g. see [21]). In order to construct the boundary stress resultants (which are plate counterparts of solid boundary traction), we adapt the procedure outlined for 3D elasticity (see [253, 304, 353]). The local problem that needs to be solved (to get a model error indicator/estimator) deals with a single plate element, which is a floating structure, loaded on the surface and along its boundary with self-equilibrated tractions. One can refer to such model adaptivity procedure for plates as computing enhanced model solution under self-equilibrated boundary stress resultants.

4.3.2.1

Construction of Equilibrated Boundary Tractions for 2D/3D Solid

The idea of equilibrated boundary traction (edge loading) comes from the observation that the finite element equilibrium equation for a single element can be written as f int,e − f ext,e = re

(4.116)

where f ext,e , re and f int,e are nodal values of external, residual and internal force, respectively. The internal force is proportional to the stiffness matrix Ke and the nodal displacements de , so that (4.116) can be rewritten as: Ke de = f ext,e + re

(4.117)

The last result indicates that the residual force can be seen as the traction exerted by the surrounding elements across the element boundary.

156

4 Plate Models: Validation and Verification

The next step is to obtain such boundary traction forces that can replace the residual forces and are continuous across element boundaries. We thus seek the boundary traction tΓe acting on an edge Γ (Γ ⊂ ∂Ω e , where ∂Ω e denotes element boundary), which will replace equivalent (in energy-norm) action of residual nodal forces and reflect the continuity of the stress field. We can write 

reI · vˆ hI =

 Γ

Γ

I

tΓe · vˆ Γh ds



tΓe + tΓe = 0

(4.118)

(4.119)

 N I vˆ hI are element virtual displacements, vˆ Γh =  vh |Γ are edge virtual where  vh = h displacements, and vˆ I are nodal virtual displacements. Elements e and e share the same edge Γ . We note that the condition in (4.119) follows from tΓe = σ ne and ne = −ne , where ne is normal to the edge Γ and σ is a continuous approximation for the true stress tensor. We further express the nodal value of the element residual force reI in (4.118) as the sum (4.120) reI = reI,Γ1 + reI,Γ2 where Γ1 and Γ2 are two edges of an element e that have in common node I . We can see reI,Γ as residual forces at node I of an element e due to the boundary traction applied on edge Γ , i.e. reI,Γ =

∂ WΓe (tΓe ) ; ∂ vˆ hI

 WΓe (tΓe ) =

Γ

tΓe · vˆ Γh ds

(4.121)

By collecting Eqs. (4.120) and (4.119) for all elements in the mesh (e = 1, . . . , n el ), all element edges (Γ = 1, . . . n Γ ) and all element nodes (I = 1, . . . n en ), we get a global system of equations for the unknowns reI,Γ . Note that the number of unknowns in such a system is reduced if there are regions of the discretized domain where boundary traction forces are prescribed. In order to illustrate the procedure for computation of reI,Γ , we consider a patch of four 2D elements surrounding node I of the FE mesh (see Fig. 4.10). Edges and elements are numbered in the counter-clockwise manner. Computation indicated in equation (4.120) in this case leads to 1 1 reI 1 = reI,Γ + reI,Γ 4 1 2 2 reI 2 = reI,Γ + reI,Γ 1 2 3 3 reI 3 = reI,Γ + reI,Γ 2 3 4 4 reI 4 = reI,Γ + reI,Γ 3 4

4.3 Validation or Model Adaptivity for Thick …

157 Γ3

Fig. 4.10 A patch of four elements e4

e3

eI

Γ4

e1

Γ2

e2

Γ1

The demand for continuity leads to 1 2 + reI,Γ =0 reI,Γ 1 1 2 3 + reI,Γ =0 reI,Γ 2 2 3 4 + reI,Γ =0 reI,Γ 3 3 4 1 reI,Γ + reI,Γ =0 4 4

A combination of the last two results on nodal residual forces leads to the following set of equations ⎡ e1 ⎤ ⎡ ⎤ ⎡ e1 ⎤ r I,Γ1 rI +1 0 0 −1 e2 ⎥ ⎢ reI 2 ⎥ ⎢ −1 +1 0 0 ⎥ ⎢ r I,Γ2 ⎥ ⎢ e ⎥=⎢ ⎥⎢ (4.122) ⎢ ⎣ r 3 ⎦ ⎣ 0 −1 +1 0 ⎦ ⎣ re3 ⎥ I I,Γ3 ⎦ 4 0 0 −1 +1 reI 4 reI,Γ 4 k with reI,Γ as unknowns. Since all the unknowns refer only to node I of the FE mesh, k the system is independent of other nodes. One thus obtains a local problem for each patch of elements around a single node. Thus, the global system needs not to be solved, but only a number of independent patch-wise smaller systems of equations. By solving such a local system (4.122) for each node of the FE mesh, we can obtain k for all element nodes (I = 1, . . . n en ). the corresponding value of residual force reI,Γ We note that the data for setting the local problems are provided by all elements of the mesh (ek = e = 1, . . . , n el ) and all element edges (Γ = 1, . . . n Γ ). With reI,Γ known, we next can obtain the equivalent element boundary traction, with the procedure carried out independently in each element. Let us consider an element edge between element nodes I and J , further denoted as Γ . We introduce discrete approximation for k-th component of the boundary traction along this edge tΓe as  e (4.123) tΓ k = pkI ψkI + pkJ ψkJ

158

4 Plate Models: Validation and Verification

  where ψkI are shape functions (yet to be chosen) and pkI are nodal values of tΓe k . The variation of virtual displacements along the same edge Γ can be written as  h I J vˆmI + Nk,m vˆmJ  vΓ k = Nk,m

(4.124)

I are the where vˆmI is related to the m-th degree of freedom at node I , and Nk,m I corresponding shape functions. In other words, the shape function Nk,m interpolates nodal value vˆmI for the k-th component of virtual displacements. We can now insert the above parameterizations into (4.121) and express residuals at nodes I and J with respect to the boundary traction on edge Γ (that spans between I and J ). The result can be written in a compact form as:

reI,Γ reJ,Γ

! =

MΓI I MΓI J MΓJ I MΓJ J

!

pΓI pΓJ

!

 IJ = ; MΓ,nm

I Nm,n ψmJ ds

(4.125)

Γ

#T " where pΓI = pkI , k = 1, . . . n dim , and n dim is vector dimension. By solving (4.125) for every element edge Γ , we obtain parameters pΓI I J , which define element boundary traction tΓe through (4.123). A convenient choice suggested in [3] or [304] for shape functions ψkJ and ψkI makes MΓI J and MΓJ I equal to zero and matrix M becomes block diagonal. With such a choice, (4.125) simplifies to (4.126) reI,Γ = MΓI I pΓI This choice reduces bookkeeping and decreases the computational cost. However, our experience is not as favorable, for this discrete approximation of the boundary traction does not represent well the stress field for plates. We also note that the specific choice of parametrization (4.126) is not necessary, since local patch-wise computation can be performed, see (4.122).

4.3.2.2

Regularization of Local System in (4.122)

The solution of the local system of equations (4.122) is not unique unless we impose an adequate additional regularization. The latter can be derived in accordance with the estimate of the true stress state with the continuous boundary traction across the element interfaces. However, the true stress state is not known, but only its (discontinuous) element-wise approximations σ eF E . Continuous stresses at the edge Γ denoted as σ 0 |Γ follow from the smooth stress recovery which may be coupled with a previous mesh adaptivity procedure. Boundary traction forces resulting from e such stresses σ 0 |Γ are obtained by the use of the Cauchy principle t 0,e Γ = σ 0 |Γ n . It is now possible to calculate the effect of this edge loading on neighboring nodes by using (4.121)

4.3 Validation or Model Adaptivity for Thick …

re,0 I,Γ =

159

∂ WΓe (t 0,e Γ )

(4.127)

∂ vˆ hI

We want reI,Γ to be as close as possible to this result in the least square sense. Thus, we obtain the corresponding value by solving patch-wise the following constrained minimization problem: L{reI,Γ ; λΓ ; σ e } =

%2 1   $ e,0 r I,Γ − reI,Γ 2 e∈P Γ I   + σ Ie (reI − reI,Γ ) Γ

e∈P I

+



λΓI (reI,Γ



+ reI,Γ )

(4.128)

Γ

where we introduced the constraints (4.119) and (4.120) by means of the Lagrange multipliers σ Ie and λΓI to obtain the corresponding Lagrangian. Note again that the constrained minimization problem is solved over the patch of elements P I around node I of the FE mesh. The corresponding Kuhn-Tucker optimality conditions are then obtained by computing the partial derivates of this Lagrangian with respect to unknowns reI,Γ to obtain: e Γ reI,Γ − re,0 I,Γ − σ I + λ I = 0

(4.129)

The same equation also holds for the residual force on the neighboring element reI,Γ : reI,Γ − reI,Γ,0 − σ Ie + λΓI = 0 (4.130) If we sum up the last two equations and take into account the continuity condition, enforcing reI,Γ + reI,Γ = 0, we can express the Lagrange multiplier λΓI as λΓI =

1 e e ,0 (σ I + σ Ie + re,0 I,Γ + r I,Γ ) 2

(4.131)

The unknowns reI,Γ can then be expressed in terms of the multiplier σ Ie reI,Γ =

1 e e ,0 (σ I − σ Ie + re,0 I,Γ − r I,Γ ) 2

(4.132)

The condition reI = reI,Γ1 + reI,Γ2 can now be rewritten as reI =

1 e,0 1 1 1 ,0 2 ,0 − reI,Γ ) + (re,0 − reI,Γ ) + (σ Ie − σ Ie1 + σ Ie − σ Ie2 ) (r 1 2 2 I,Γ1 2 I,Γ2 2

(4.133)

160

4 Plate Models: Validation and Verification

By exploiting the above results, the local system for the patch of four elements can be rewritten as ⎡ e1 ⎤ ⎡ ⎤ ⎡ e1 ⎤ σI r˜ I 2 −1 0 −1 e ⎢ r˜ 2 ⎥ 1 ⎢ −1 2 −1 0 ⎥ ⎢ σ e2 ⎥ ⎢ eI 3 ⎥ = ⎢ ⎥ ⎢ Ie3 ⎥ (4.134) ⎣ r˜ I ⎦ 2 ⎣ 0 −1 2 −1 ⎦ ⎣ σ I ⎦ e4 −1 0 −1 2 r˜ I σ Ie4 where we introduced the notation r˜ eI = reI −



e,0 < re,0 I,Γ >; < r I,Γ >=

Γ ∈∂Ωe,I

1 e,0 (r I,Γ − reI,Γ,0 ) 2

(4.135)

where < re,0 I,Γ > represents averaged boundary traction on Γ (evaluated from FE solution) ‘projected’ to node I . To summarize, the element-wise boundary tractions are computed from FE solution (based on chosen plate model) in two steps: (i) by solving the patch-wise problems in (4.134) and (4.132) to obtain reI,Γ ; (ii) by solving element-wise problems (4.125) to compute nodal values pkI of boundary tractions (4.123). 4.3.2.3

Equilibrated Stress Resultants for the DKQ Element

In order to find equilibrated stress resultants for the DKQ element, we start with interpolations for displacement and rotations. The variation of displacement along the edge Γ , which spans between element nodes I and J , is defined by: wΓh = w I ϕ1 + w J ϕ2 +

lI J lI J n I J · ( θ I − θ J ) ϕ3 + Δθ I J ϕ4 4 3

(4.136)

where parameter Δθ I J can be expressed in terms of nodal displacements and rotations of neighboring nodes as Δθ = 2l3I J (w J − w I ) − 43 n I J · ( θ I + θ J ). Variation of rotations vector along the same edge can then be written as θ Γh = θ I ϕ1 + θ J ϕ2 + n I J Δθ I J ϕ3

(4.137)

The functions ϕi are ϕ1 = (1 − ξ )/2 ϕ2 = (1 + ξ )/2 ϕ3 = (ξ 2 − 1)/2 ϕ4 = ξ(ξ 2 − 1)/2

(4.138)

4.3 Validation or Model Adaptivity for Thick …

161

In accordance with (4.136), (4.137) and notation used in (4.124), one can write the interpolation functions on the edge Γ (between nodes I and J ) according to I = ϕ1 − ϕ4 /2 Nw,w

J Nw,w = ϕ2 + ϕ4 /2

I = −l n x (−ϕ3 + ϕ4 )/4 Nw,θ x

J Nw,θ = −l n x (+ϕ3 + ϕ4 )/4 x

I Nw,θ = −l n y (−ϕ3 + ϕ4 )/4 y

J Nw,θ = −l n y (+ϕ3 + ϕ4 )/4 y

NθIx ,w = −(3n x /2l) ϕ3

NθJx ,w = +(3n x /2l) ϕ3

NθIx ,θx = ϕ1 − 3n 2x ϕ3 /4

NθJx ,θx = ϕ2 − 3n 2x ϕ3 /4

NθIx ,θ y = −3n x n y ϕ3 /4

NθJx ,θ y = −3n x n y ϕ3 /4

NθIy ,w = −(3n y /2l) ϕ3

NθJy ,w = +(3n y /2l) ϕ3

NθIy ,θx = −3n x n y ϕ3 /4

NθJy ,θx = −3n x n y ϕ3 /4

NθIy ,θ y = ϕ1 − 3n 2y ϕ3 /4

NθJy ,θ y = ϕ2 − 3n 2y ϕ3 /4

(4.139)

(4.140)

(4.141)

where l = l I J , n I J = [n x , n y ], θx = θ1 and θ y = θ2 . Note, that with respect to (4.124) and notation in previous section one has: k ∈ [w, θx , θ y ], m ∈ [w, θx , θ y ], (vˆΓh )w = h h wˆ Γh , (vˆΓh )θx = θˆ1 Γ , (vˆΓh )θ y = θˆ2 Γ . The boundary stress resultants for the DKQ plate element at the edge Γ will have three components " # t = q, m x , m y (4.142) where q is the shear force along the edge defined in direction of x = x3 , and m x , m y are the moments defined in the direction of x = x1 and y = x2 , respectively. We choose to parameterize them as q = q I ϕ1 + q J ϕ2 m x = m x I ϕ1 + m x J ϕ2 m y = m y I ϕ1 + m y J ϕ2

(4.143)

With respect to the notation introduced in (4.123), we note that ψkI = ϕ1 ψkJ = ϕ2

(4.144)

and pqI = q I

pqJ = q J

pmI x = m x I

pmJ x = m x J

pmI y = m y I

pmJ x = m y J

162

4 Plate Models: Validation and Verification

To get the parameters of (4.143) from (4.125) we have to evaluate integrals & nodal IJ I = Γ Nm,n ψmJ ds, which have explicit forms as M Γ,nm

IJ M Γ,nm

⎛ I ⎞ J J I I J +1 Nw,w ψw Nθx ,w ψθx Nθ y ,w ψθ y l ⎜ NI ψJ NI ψJ NI ψJ ⎟ = ⎝ w,θx w θx ,θx θx θ y ,θx θ y ⎠ dξ 2 I Nw,θ ψwJ NθIx ,θ y ψθJx NθIy ,θ y ψθJy −1 y

(4.145)

Remark on results validation: The so-obtained DKQ boundary stress resultants are exact replacements of the nodal residuals of the DKQ solution. If used for elementwise computation with the DKQ elements, they should produce exactly the same results as the original global computation. This fact can be used to check the correctness of the boundary stress resultants computation. Remark on computations for RMQ plate element: Equilibrated stress resultants for the RMQ element can be obtained in a very similar way as for the DKQ element. I need to be found in accordance with the interpolations introduced in Functions Nm,n the previous section and used in (4.145).

4.3.2.4

Model Error Indicator for the DKQ Element

A model error indicator for the DKQ element is obtained by comparing its results with the ones obtained by an element based on a more refined plate model (e.g. RMQ). As shown above, the later results can be obtained by element-wise computation with the DKQ boundary stress resultants applied as external edge loading. Thus, a Neumann-type problem has to be solved for each element of the mesh. An element with only the Neuman boundary conditions is essentially a floating structure, and one thus ought to eliminate the rigid body modes to get a unique result. This can be done simply by using the element geometry, as shown in [312]. A rigid body displacement deR of an element is a linear combination of all its rigid body modes (4.146) deR = De α where α is a vector of amplitudes that correspond to rigid body modes, and De is a matrix containing the rigid body modes of the element (arranged column-wise). Since the element stiffness matrix Ke is singular due to the rigid body modes contribution, T one can form a modified nonsingular stiffness matrix Ke by adding a product De De to Ke T (4.147) Ke = Ke + De De

The inversion of Ke is possible and solution for element nodal displacements/ rotations de can be obtained, with a risk of the rigid-body ‘pollution’. Since the matrix De De T projects onto the space spanned by the rigid body modes, we have de = de + deR

(4.148)

4.3 Validation or Model Adaptivity for Thick …

163

where de are real nodal displacements/rotations. As we are interested only in stress resultants, such a pollution is not critical due to the following property Ke deR = 0. Having defined the stress resultants for the RMQ, the RMQ deformation energy can be calculated, by taking into account the corresponding bending and shear deformations E

R M, e

e =

m

R M, T

C

B −1

e m

RM

dΩ +

Ω

=

e E BR M,

T

q R M, C S

−1 R M

q

dΩ

Ω

+

e E SR M,

(4.149)

where m R M = [m 11 , m 22 , m 12 ]T and q R M = [q1 , q2 ]T . Since DKQ and RMQ plate elements predict very similar values for moments e and the DKQ assumes zero shear strains, the shear deformation energy E SR M, of RMQ can be used to determine how well the DKQ plate model performs in a given situation. If shear strain contribution is low enough, the DKQ plate element could be considered as a sufficiently good model. Otherwise, it should be replaced by a more suitable element based on the Reissner-Mindlin model (i.e. by the RMQ plate element). A fairly reasonable choice for a model error indicator ηeM we use herein is therefore the ratio of the energy norm of the shear difference between the RMQ and the DKQ element to the total energy of the RMQ element. Due to the zero shear strain assumption in the DKQ, and similar prediction for moments of both elements, one can define the model error indicator as ,e

ηeM = E SR M /E R M,

e

(4.150)

This indicator is used below in all numerical examples.

4.3.3 Illustrative Numerical Examples and Closing Remarks We present in this section the results of numerical examples, which were chosen to illustrate performance of the proposed approach to adaptive modeling of plates. We do not elaborate on the mesh adaptivity for controlling the discretization error, since this has been extensively covered in the literature (e.g., see [256, 305, 390]). However, the departure point for the present model adaptivity procedure should be an optimally adapted FE mesh, with the discretization error uniformly distributed and placed within a predefined limit. We note that the plate elements in such an automatically generated mesh of quadrilaterals, which takes into account predefined element size distribution, can be quite distorted. For this reason, we also present examples where the influence of mesh distortion on the chosen model error indicator is studied.

164

4.3.3.1

4 Plate Models: Validation and Verification

Sensitivity of Model Error Indicator on Mesh Distortion: Simply Supported Square Plate

The problem of simply supported (SSSS) square plate under uniform loading is used to test the sensitivity of model error indicator on mesh distortion; as well as to compare those effects for thin and thick plate situations. The material behavior is linear elastic and isotropic, with Young’s modulus E = 10.92 and Poisson’s ratio ν = 0.3. The side length is a = 10. Two values for the plate thickness t = 0.1 (thin) and t = 1 (thick) are selected. Due to the symmetry, only one-quarter of the plate (the lower left one) is chosen for the numerical analysis. The model error indicator (4.150) is computed for each DKQ element of the mesh. This value is then compared with the corresponding analytical (reference) value, which can be for this simple problem obtained by using an analytic solution for Mindlin plates, see e.g. [59, 261]. In Fig. 4.11, we show the comparison of two structured (regular) meshes with the same number of elements for the case of a thick plate. The meshes differ in discretization error, which is lower for the second mesh that is more refined towards the outer edges. It can be observed that the computed model error indicator is very close to the reference one, except for the corner element. This is due to the fact that the equilibration is not possible to carry out at the corner node. We thus obtain as a consequence that the estimated boundary stress resultants for the corner element are not as good as for the inner elements. To estimate the effect of mesh distortion, the model error indicator (4.150) is computed (for the case of a thick plate) with two distorted meshes. The first mesh is obtained by adaptive mesh refinement with 5% tolerance in discretization error. The second one is derived from the mesh shown in Fig. 4.11 by a parametrical displacement of the nodes. The results are presented in Fig. 4.12. Again, the comparison is made with the analytical (reference) solution. We note that the performance of the model error indicator is slightly better for the parametrical distortion of the mesh. One may conclude from the results shown in Fig. 4.12 that the model error indicator is slightly (but not severely) influenced by mesh distortion. We repeat the above-mentioned comparison also for the case of thin plates. The results are shown in Figs. 4.13 and 4.14 (notice the change of scale). The model error indicator shows that the DKQ is sufficiently good (almost) for the whole mesh. Comparison with the analytic solution shows that the sensitivity of the model error indicator is in the same range as for the case of a thick plate.

4.3.3.2

Square Plate with Two Simply Supported and Two Free Edges

The second problem we study is a square plate with two opposite edges simply supported and two other edges free (SFSF). It has been chosen to test the ability of the proposed procedure to detect boundary layer effect, which is most prominent along the free edges, see [14]. The width of the boundary layer is proportional to the thickness of the plate, and it is most pronounced for the transverse shear component, which varies as 1/t. The geometry and material parameters are the same as in the

4.3 Validation or Model Adaptivity for Thick …

165 100.0 90.0 80.0 70.0 60.0 50.0 40.0 30.0 20.0 10.0 0.0 [%]

(a)

(b)

100.0 90.0 80.0 70.0 60.0 50.0 40.0 30.0 20.0 10.0 0.0 [%] (a)

(b)

Fig. 4.11 Reference value of model error indicator (a) and its estimated value (b) for thick SSSS plate for rectangular meshes

first example. Due to the symmetry, only one-quarter of the plate (the lower left one) is used in the analysis; the boundary conditions are imposed as follows: the left edge in simply supported, the lower one is free, and the right and the upper ones take into account symmetry. Since the boundary layer effect is intrinsic to the Reissner-Mindlin theory, the DKQ can not detect it. In the present procedure, the equilibrated boundary stress resultants are derived from the DKQ solution and subsequently applied to the RMQ in order to evaluate model error indicator. It is therefore interesting to see whether this procedure can effectively detect the boundary layer. The result in Fig. 4.15 indicates that the procedure is able to detect a region with increased shear deformation energy (along the left edge) but is not very successful to detect the boundary layer (along the lower free edge), which is clearly observable in the reference (analytical) solution.

166

4 Plate Models: Validation and Verification 100.0 90.0 80.0 70.0 60.0 50.0 40.0 30.0 20.0 10.0 0.0 [%] (a)

(b) 100.0 90.0 80.0 70.0 60.0 50.0 40.0 30.0 20.0 10.0 0.0 [%]

(a)

(b)

Fig. 4.12 Reference value of model error indicator (a) and its estimated value (b) for thick SSSS plate for distorted meshes

Comparison with the analytical solution shows (see Figs. 4.15, 4.16, 4.17 and 4.18) that the sensitivity of the model error indicator is of the same range as in the case of SSSS plate.

4.3.3.3

Adaptive Analysis of L-Shaped Plate

Adaptive model analysis of an L-shaped plate under uniform unit pressure is considered in this example. Long sides (a = 10) of the plate are clamped and all other sides (b = 5) are free. The plate is made of linear elastic isotropic material, with Young’s modulus E = 10.92 and Poisson’s ratio ν = 0.3. The thickness of the plate is equal to 1. A non distorted mesh was chosen for this example in order to avoid the effect of element distortion on the model error indicators.

4.3 Validation or Model Adaptivity for Thick …

167 5.0 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 [%]

(a)

(b)

5.0 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 [%] (a)

(b)

Fig. 4.13 Reference value of model error indicator (a) and its estimated value (b) for thin SSSS plate for rectangular meshes

The computation started with the analysis of the plate by using DKQ elements. The model error indicator (4.150) was further computed and the result is shown in Fig. 4.19. The indicator identified several regions where the model error was high. The limit of 15% model error was chosen to determine the elements of the mesh, where the RMQ should be used instead of the DKQ. Those elements are colored dark grey in Figs. 4.20, 4.21 and 4.22b. Finally, the computation with both the DKQ (light grey in Figs. 4.20, 4.21 and 4.22b) and the RMQ elements (dark grey in Figs. 4.20, 4.21 and 4.22b), used at different regions of the plate, was performed. Kinematic coupling of elements based on different models was considered, i.e. the hierarchical rotation Δθ of the RMQ was restrained on edges where the RMQ elements were adjacent to the DKQ elements in order to account for zero transverse shear modelling along those edges.

168

4 Plate Models: Validation and Verification 5.0 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 [%] (a)

(b)

5.0 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 [%] (a)

(b)

Fig. 4.14 Reference value of model error indicator (a) and its estimated value (b) for thin SSSS plate for distorted meshes

The comparison of stress resultants evaluated with the DKQ elements (Figs. 4.20, 4.21 and 4.22a), with the RMQ elements (Figs. 4.20, 4.21 and 4.22c) and with both the DKQ and the RMQ elements (Figs. 4.20, 4.21 and 4.22d) is shown. It can be seen that the RMQ based computation and the mixed DKQ-RMQ based computation produces almost identical results. On the other hand, the results are not equal to those obtained by using only DKQ elements. Therefore, the comparison of these different results (and especially close agreement between the DKQ-RMQ and the RMQ computations), clearly indicate that the adaptive procedure of this kind performs successfully. One can hope for the same trend when applying the same procedure to higher order plate models, where significant computational savings could also be obtained in the process.

4.3 Validation or Model Adaptivity for Thick …

169 100.0 90.0 80.0 70.0 60.0 50.0 40.0 30.0 20.0 10.0 0.0 [%]

(a)

(b) 100.0 90.0 80.0 70.0 60.0 50.0 40.0 30.0 20.0 10.0 0.0 [%]

(a)

(b)

Fig. 4.15 Reference value of model error indicator (a) and its estimated value (b) for thick SFSF plate for rectangular meshes

4.3.3.4

Adaptive Analysis of Morley’s Skew Plate

Adaptive model analysis of Morley’s 30◦ skew plate (see [291]) with thickness t = 1, side length a = 10, simple supports on all sides, and unit uniform pressure is performed in this example. The plate is built of linear elastic isotropic material, with Young’s modulus E = 10.92 and Poisson’s ratio ν = 0.3. The most interesting feature of the solution to this problem concerns two singular points at the two obtuse corners of the plate, which strongly influence the quality of the computed results (e.g. see [164]). The chosen FE mesh is made more dense near the sides of the plate where singularities are expected. The computation started with the analysis by using DKQ elements. The model error indicator (4.150) was further computed, see Fig. 4.23. It can be seen from Fig. 4.23 that the error was increased near the sides and in the vicinity of the obtuse corners. Again, the limit of 15% model error was chosen to determine the elements

170

4 Plate Models: Validation and Verification 100.0 90.0 80.0 70.0 60.0 50.0 40.0 30.0 20.0 10.0 0.0 [%] (a)

(b)

100.0 90.0 80.0 70.0 60.0 50.0 40.0 30.0 20.0 10.0 0.0 [%] (a)

(b)

Fig. 4.16 Reference value of model error indicator (a) and its estimated value (b) for thick SFSF plate for distorted meshes

of the mesh, where the RMQ should be used instead of the DKQ; all such elements are colored dark grey in (b) part of Figs. 4.24 and 4.25. Finally, the computation with both the DKQ (light grey in Figs. 4.24b and 4.25b) and the RMQ elements (dark grey in Figs. 4.24 and 4.25b), used at different regions of the plate, was performed. It can be seen in Fig. 4.24 that in this case, the difference in results for moments between all three computational models is negligible. The difference however still shows in the results for the shear force, which are presented in Fig. 4.26.

4.3.3.5

Closing Remarks

We have presented a detailed development for the plate model adaptivity procedure capable of selecting automatically the best suitable choice between the two plate

4.3 Validation or Model Adaptivity for Thick …

171 5.0 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 [%]

(a)

(b)

5.0 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 [%] (a)

(b)

Fig. 4.17 Reference value of model error indicator (a) and its estimated value (b) for thin SFSF plate for rectangular meshes

models, the first for a thin Kirchhoff plate and the second for a thick ReissnerMindlin plate. The model adaptivity computation is carried out independently for each element, and it starts with the optimal finite element mesh selected by a mesh adaptivity procedure in order to achieve the same (acceptable) discretization errors throughout the mesh. We illustrated the proposed approach by using two low-order plate finite element models, one represented with discrete Kirchhoff quadrilateral (DKQ) and the other with the Reissner-Mindlin quadrilateral (RMQ); but the same procedure would carry over to higher-order plate models that capture more accurately local 3D effects. It is the key advantage of the presented approach to the plate model adaptivity to remain uncoupled with respect to the mesh adaptivity procedure employed to reduce the discretization error, and thus can be combined with any already available mesh adaptivity procedure to ensure optimal overall accuracy. This advantage

172

4 Plate Models: Validation and Verification 5.0 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 [%] (a)

(b)

5.0 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 [%] (a)

(b)

Fig. 4.18 Reference value of model error indicator (a) and its estimated value (b) for thin SFSF plate for distorted meshes

stems from the proposed special choice of the finite element interpolations for both RMQ and DKQ plate elements, which results with the same quality of the discrete approximation for bending moments. Numerical examples illustrate clearly that the proposed procedure is capable of capturing any significant contribution of shear deformations. The case in point concerns the shear layers which typically occur for different kinds of boundary conditions.

4.4 Verification or Discrete Approximation Adaptivity for Discrete …

173 100 90 80 70 60 50 40 30 20 10 0 [%]

Fig. 4.19 Model error indicator

4.4 Verification or Discrete Approximation Adaptivity for Discrete Kirchhoff Plate Finite Element There exist numerous applications in aerospace, civil and mechanical engineering, which call for the thin plate finite element (FE) modeling. Thus, it is certainly of great practical interest to provide the finite element discrete approximation error estimates that can guide the optimal mesh choice with respect to a required level of solution accuracy. This is further reinforced in this section by the main focus upon perhaps most frequently used plate finite elements, referred to as the Discrete Kirchhoff (DK) plate elements (e.g. [33, 34, 38] or [51]). These elements are widely popular for their capabilities to deliver higher-order approximations for thin plate structures, even for finite elements with a small number of nodes. The plate elements of this kind are part of many commercial FE codes, either as a 3-node triangle (DKT) or as a 4-node quadrilateral (DKQ). The DK elements are the result of a long line of developments on Kirchhoff plate elements requiring C 1 continuity of transverse displacement interpolation. Early developments using cubic polynomials were only partially successful, since they resulted with very complex interpolation schemes, not easy to generalize to any other than particular triangular in [84] or quadrilateral element in [82]. Only partial success was also met by interpolation scheme for Kirchhoff plates in form of the complete fifth-order polynomial for triangular element proposed in [12]. Such an interpolation employs the nodal interpolation parameters that include not only displacements and

174

4 Plate Models: Validation and Verification 11.70

10.16

8.61

7.07

5.52

(a)

(b)

3.98

2.43

0.89

-0.66

-2.20

-3.75 (c)

(d)

Fig. 4.20 Stress resultant m x x a DKQ model, c RMQ model and d DKQ-RMQ model. The result presented in (d) is obtained with a mesh shown in (b), where dark areas represent RMQ and light areas DKQ model

rotations (or the first derivatives), but also the curvatures (or the second derivatives). The curvatures as kinematic nodal unknowns can overconstrain the curvatures in the constitutive equations. One such example pertains to setting the curvatures to zero in corner points of the skew plate with Neumann boundary, which can cause the element to lock. For these reasons, the Kirchhoff plate elements were abandoned and preference was given to the Reissner-Mindlin (RM) thick plate elements, using only C 0 interpolations for displacement and independent rotations. The interpolations of this kind are very easy to generate for different numbers of element nodes. However, they reduce the order of approximation with respect to Kirchhoff plate elements with the same number of nodes, as already shown earlier for Timoshenko’s vs. EulerBernoulli’s beams. More importantly, the Reissner-Mindlin plate elements turned to be very sensitive to shear locking phenomena, and required special interpolation schemes. For a summary of a vast number of remedies, we refer to [391] or [153]. The DK plate element was introduced within the RM plate theory framework, trying to deal with shear locking by completely removing the shear deformation. This was done by using simple linear interpolations for both displacement and rotations, tying them by imposing the Kirchhoff constraint at each element edge (e.g. see [33,

4.4 Verification or Discrete Approximation Adaptivity for Discrete …

175 3.52

3.03

2.55

2.06

1.57

(a)

(b)

1.08

0.60

0.11

-0.38

-0.87

-1.36 (c)

(d)

Fig. 4.21 Stress resultant m x y a DKQ model, c RMQ model and d DKQ-RMQ model. The result presented in (d) is obtained with a mesh shown in (b), where dark areas represent RMQ and light areas DKQ model

34] or [51]). The fully consistent mixed formulation of Discrete Kirchhoff Quadrilateral (DKQ) plate elements is proposed in [164], along with a non-conventional displacement field interpolation in terms of cubic polynomial. The same approach is extended here to Discrete Kirchhoff Triangular (DKT) plate elements. It was shown in [162] that the same mixed variational framework and modified non-conventional interpolations, properly enhanced with incompatible modes, can be used to develop a thick plate equivalent to DKQ plate element, which also provides a bi-linear approximation for bending moments in 4-node element. The latter was exploited in [55] to construct the plate model adaptivity procedure, which can effectively indicate the elements for which the shear energy is not negligible. The previous works on the theoretical foundation of Kirchhoff or Reissner-Mindlin plate theories (e.g. see [4] or [57]) also make use of this kind of mixed variational formulation in order to provide the plate models justification with respect to three-dimensional elasticity. We note in passing that these results, which showed the importance of taking into account a quadratic variation in transverse displacement in the through-the-thickness direction, can eventually be used to complete the model adaptivity procedure [55] with estimates pertinent to the change of thickness. These results are also in agree-

176

4 Plate Models: Validation and Verification 15.59

12.96

10.34

7.71

5.08

(a)

2.45

(b)

-0.18

-2.80

-5.43

-8.06

-10.69 (c)

(d)

Fig. 4.22 Stress resultant gx a DKQ model, c RMQ model and d DKQ-RMQ model. The result presented in (d) is obtained with a mesh shown in (b), where dark areas represent RMQ and light areas DKQ model 100.00 90.00 80.00 70.00 60.00 50.00 40.00 30.00 20.00 10.00 0.00 [%]

Fig. 4.23 Model error indicator

4.4 Verification or Discrete Approximation Adaptivity for Discrete …

177 0.36 0.24 0.13 0.01

(a)

(b)

-0.10 -0.22 -0.33 -0.45 -0.56 -0.68 -0.79

(c)

(d)

Fig. 4.24 Stress resultant m x x computed with: a DKQ model, c RMQ model and d DKQ-RMQ model. The result presented in (d) is obtained with a mesh shown in (b), where dark areas represent RMQ and light areas DKQ model 0.43 0.37 0.32 0.27 (a)

(b)

0.22 0.16 0.11 0.06 0.01 -0.04 -0.10

(c)

(d)

Fig. 4.25 Stress resultant m x y computed with: a DKQ model, c RMQ model and d DKQ-RMQ model. The result presented in (d) is obtained with a mesh shown in (b), where dark areas represent RMQ and light areas DKQ model

ment with our previous works on geometrically nonlinear shell models featuring the same quadratic variation in the through-the-thickness direction that can accommodate directly 3D constitutive laws (e.g. see [60]). However, how to make the best choice for the finite element discretization or construct the optimal finite element mesh with DK plate elements is not fully understood, unless we provide the discretization error estimates for guiding the mesh refinement. That is the main goal of this section. The discretization by FE method provides an approximate solution to a given boundary value problem ([176, 391]). Estimates of the discretization error of the FE solution are of interest for the adaptive mesh refinement (e.g. [3, 253, 352]). The results on discretization-error-estimates for the biharmonic operator typical of Kirchhoff plates are available (e.g. see [81]) only for the conventional Kirchhoff

178

4 Plate Models: Validation and Verification 1.01 0.81 0.61 0.40 (a)

(b)

0.20 0.00 -0.20 -0.40 -0.61 -0.81 -1.01

(c)

(d)

Fig. 4.26 Stress resultant qx computed with: a DKQ model, c RMQ model and d DKQ-RMQ model. The result presented in (d) is obtained with a mesh shown in (b), where dark areas represent RMQ and light areas DKQ model

elements [84] or [12], which are hardly used nowadays. Only limited success is provided for non-conventional Kirchhoff plate elements, for the simplest choice with constant approximation in each element known as Morley triangle (e.g. [101, 152, 267]). For a more refined plate element with non-conventional interpolations like Discrete Kirchhoff, the most appropriate error estimates are established only later [54]. In constructing FE discretization error estimates, the preference is given to a posteriori version that starts from the computed solution to the FE-produced equilibrium equations. One can consider two types of methods: (i) explicit estimates that bound the displacement discretization error with an unknown global constant, multiplied with the norm of residual of equilibrium equations (e.g. [253]) or (ii) implicit estimates that rely upon the subsequent post-processing of the FE solution in order to obtain the enhanced displacement and stress of increased accuracy and thus provide the corresponding error estimates by comparison against FE solution, with no need for constants estimate (e.g. [23]). The latter is the favorite choice for solid mechanics problems, with the two main types of procedures for constructing enhanced solution by using either the superconvergent patch recovery (SPR) or the equilibrated boundary stresses (EqR). The SPR enhanced solution (e.g. see [389, 397]) is the least square best fit by a smooth stress interpolation of the FE results computed in superconvergence points (roughly, the reduced numerical integration points of a particular element). The same implementation is kept for both continuum and DK plate elements, with either stresses or bending moments being processed, one component at a time. The EqR enhanced solution (e.g. see [3, 43, 253, 253, 352]) makes use of the equilibrated element boundary traction. Such a solution can be obtained with higher-order FE approximations to provide the solution to a local Neumann boundary value problem in each single element (e.g. [2, 66, 354]). The main thrust in this section is directed towards the direct application of the EqR procedure for constructing a posteriori error estimates for the Discrete Kirch-

4.4 Verification or Discrete Approximation Adaptivity for Discrete …

179

hoff plate elements. The SPR procedure is used only for comparison, showing that its performance need not always be optimal for the elements with higher order interpolations, such as the Discrete Kirchhoff plate element. The first novelty concerns explicit definition of the finite element interpolation with cubic displacement field for the Discrete Kirchhoff Triangular (DKT) element. We also adapt to DKT element the EqR approach that was initially derived for solid mechanics finite elements with only C 0 interpolations. Among several alternatives for constructing the enhanced test space, the main novelty concerns using the conforming Kirchhoff plate element of [12], which has not been often used for computations by the engineering community for it requires handling the delicate issue of imposing the boundary conditions on curvatures. The outline of this section is as follows. We first briefly summarize the governing equations of the Kirchhoff plate theory, its strong and weak forms. We then present two triangular Kirchhoff plate finite elements employed as the main ingredients of the proposed approach, the Discrete Kirchhoff triangular (DKT) and Argyris TUBA plate elements. All the governing equations of residual based local and global error indicators are presented in detail, along with their numerical implementation in EqR method for DKT plate element. Numerical examples are presented at the end together with closing remarks.

4.4.1 Kirchhoff Plate Bending Model The main governing equations of the Kirchhoff plate model are summarized here (for details, see e.g. [318]), in order to define the solid theoretical basis for the developments to follow.

4.4.1.1

Model Main Ingredients

Consider an elastic plate occupying the domain Ω (see Fig. 4.27). Its middle surface is placed in x y plane, with"the middle-plane boundary Γ p = Γ N ∪ Γ D , Γ N ∩ Γ D = ∅, # t  z. At a boundary point we define local coordinate , and the plate thickness t, −t 2 2 system (s, n) introducing two unit vectors, first the exterior normal at the plate boundary n = [n x , n y ]T and tangent vector defined as s = [−n y , n x ]T . Coordinates s and n change, respectively, along the boundary and in the direction of the normal. The plate is loaded by distributed area loading f : Ω → R, line boundary moment in s direction m s : Γ N → R and line boundary force in z direction specifying the effective shear q¯e f : Γ N → R, which is precisely defined in (4.160). The Kirchhoff plate model is built upon the main kinematic constraint of throughthe-thickness fibre that remains perpendicular to the plate middle surface, which allows to define the fibre rotations θ = [θx , θ y ]T as the derivatives of the plate transverse displacement w : Ω ∪ Γ p → R:

180

4 Plate Models: Validation and Verification

Fig. 4.27 Notation for plate problem

∇ K w − θ = 0,

∇K =



∂ , − ∂∂x ∂y

T (4.151)

The above constraint sets the transverse shear strains to zero. Thus, the only generalized deformations are middle-plane curvatures κ = [κx x , κ yy , 2κx y ]T , defined as the second derivatives of w   ∂θ y ∂θ y ∂θx ∂ 2w ∂ 2w ∂θx ∂ 2w , κ yy = , κx y = = − /2 =− = κx x = ∂x2 ∂x ∂ y2 ∂y ∂ x∂ y ∂x ∂y (4.152) The curvatures relate to the bending moments m = [m x x , m yy , m x y ]T through stress-resultant constitutive equations: m=D 

"

1ν0

#T " #T " , ν10 , 00 

1−ν 2

#T  Et 3 κ; D= 12(1 − ν 2 )

(4.153)

CB

where E is Young’s modulus and ν is Poisson’s ratio. According to the Kirchhoff hypothesis and the corresponding zero value of shear strains, we cannot use similar constitutive equations for shear. Rather, the shear forces have to be computed directly from corresponding moment equilibrium equations with  qx = −

∂m x y ∂m x x + ∂x ∂y



 , qy = −

∂m x y ∂m yy + ∂y ∂x

 (4.154)

The last two results can be inserted into force equilibrium equation, in order to provide a single equilibrium equation in terms of moments as ∂m 2yy ∂q y ∂ 2m x y ∂qx ∂m 2x x + + f = 0 =⇒ + + 2 = f ∂x ∂y ∂x2 ∂ x∂ y ∂ y2

(4.155)

4.4 Verification or Discrete Approximation Adaptivity for Discrete …

181

The following transformations applied at the plate boundary allow us to define the boundary rotations θn =

∂w ∂w ∂w ∂w ∂w ∂w = −n y = nx + nx , −θs = + ny ∂s ∂ x ∂ y ∂n ∂ x ∂y 







−θ y

−θ y

θx

(4.156)

θx

along with the boundary moment components m nn = n 2x m x x + n 2y m yy + 2n x n y m x y

(4.157)

  m ns = −n x n y m x x + n x n y m yy + n 2x − n 2y m x y

(4.158)

and the boundary shear forces  qn = n x q x + n y q y = −

∂m ns ∂m nn + ∂n ∂s

 (4.159)

The Kirchhoff hypothesis imposes the effective shear force, combining the shear with twisting moments derivative along the mid-plane boundary, as the only appropriate way for imposing the Neumann boundary conditions qe f = qn −

∂m ns ∂m n on Γ p , q¯e f = q¯ − on Γ N ∂s ∂s

(4.160)

In general, the boundary can be split into the Neumann Γ N , where we impose the normal moment and effective shear force (m¯ nn , q¯e f ), and the Dirichlet boundary Γ D , where we impose displacement and its normal derivative w, ¯ ∂∂nw¯ ). We can also have a mixed case or Navier boundary Γ M , where we impose the displacement and normal moment (w, ¯ m¯ nn ). 4.4.1.2

Strong and Weak Forms

By combining the above defined kinematic, constitutive, and equilibrium equations into a single equation, we define the strong form of the boundary value problem for the Kirchhoff plate. It can be stated as: Given: distributed load f in Ω, imposed boundary shear and moments q¯e f , m¯ s and imposed displacement and rotation w, ¯ θ¯s , Find: w, such that ΔΔw = f /D ; Δ = ∂∂x 2 + ∂∂y 2 in Ω w = w¯ ; − ∂w = θ¯s on Γ D ∂n w = w¯ ; m nn (w) = −m¯ s on Γ M qe f (w) = q¯e f ; m nn (w) = −m¯ s on Γ N 2

2

(4.161)

182

4 Plate Models: Validation and Verification

and that for any corner point P on Γ N , −m ns | P− + m ns | P+ = −m n | P− + m n | P+ . The corresponding variational formulation, considering the weak form of plate equilibrium equation along with strong form of kinematics and constitutive equations, can be written as a(w, v) = l(v), w ∈ V, ∀v ∈ V0

(4.162)

where (we assume for clarity in what follows that the integral over Γ M,h is zero)  κ T (v) C B κ(w) dΩ

a(w, v) = Ω

 v f dΩ +

l(v) = Ω

 ΓI J

⎛ ⎝



(4.163) ⎞

(v q¯e f + θs m s ) ds + [vm n ] IJ ⎠

(4.164)

ΓI J

 Here, I, J ∈ G N , Γ I J ⊂ Γ N , and Γ I J = Γ N is the counter-clockwise sum of boundary sections. The trial and the test spaces in (4.162) are defined as: * ) ∂w = θ¯s on Γ D | w = w¯ on Γ M V = w ∈ H2 (Ω) | w = w, ¯ − ∂n * ) ∂v = θs = 0 on Γ D | v = 0 on Γ M V0 = v ∈ H2 (Ω) | v = 0, − ∂n

(4.165) (4.166)

where H2 is the second order Sobolev space (e.g. [3]). For future use, let us rewrite the weak form (4.162) as a(u, v) = l(v), u ∈ V , ∀v ∈ V 0 , θ y = − ∂w ]T and v = [v, φx = where u = [w, θx = ∂w ∂y ∂x and the test spaces (4.165), (4.166) are rewritten as ) V =

= − ∂∂vx ]T . The trial

w ∈ H2 (Ω) , θ y = − ∂w θ = ∂w ∂y ∂x " 1 #2 | x w = w, ¯ − ∂w = θs = θ¯s on Γ D ; w = w¯ on Γ M θ ∈ H (Ω) ∂n )

V0 =

∂v , φy ∂y

(4.167)

v ∈ H2 (Ω) , φ y = − ∂∂vx φ = ∂v ∂y " 1 #2 | x ∂v v = 0, − ∂n = φs = 0 on Γ D ; v = 0 on Γ M φ ∈ H (Ω)

where H1 is the first order Sobolev space.

* (4.168)

* (4.169)

4.4 Verification or Discrete Approximation Adaptivity for Discrete …

183

4.4.2 Kirchhoff Plate Finite Elements The FE approximation wh to the solution w of the variational form (4.162) is sought in Vh ⊂ V. The subspace Vh consists of element-wise polynomial functions of order p over the FE partition Ωh , each satisfying the Dirichlet boundary conditions on Γ D,h and on Γ M,h (e.g. see [176, 391]). The difficulty in designing FE for the Kirchhoff plate problem lies in the fact that the natural variational space (4.165) for the biharmonic strong form is the second order Sobolev space H2 . Thus, C 1 continuity is imposed on wh , which requires higher-order polynomials over FE domain Ωh,e ⊂ Ωh . The plate element of Argyris [12] delivers this kind of approximation, but it uses the second derivatives of displacement or curvatures, which renders imposing the Dirichlet boundary conditions quite difficult. In order to overcome this difficulty for lower-order polynomials, non-standard FE approximations for the Discrete Kirchhoff plate element are needed, but they no longer provide continuous moments across the element boundaries.

4.4.2.1

Conforming Kirchhoff Triangle of Argyris

The starting point in the development of the discrete approximation is the displacement-type weak form in (4.162) requiring C 1 -continuity of displacement field. This can be achieved by using the conforming Kirchhoff triangle of [12] with discrete approximation of the displacement field wh,e = wh|e in terms of complete 5th order polynomial. The latter counts 21 terms, and thus the Argyris element has 21 degrees of freedom, counting among them: w I , w I,x , w I,y , w I,x x , w I,yy , w I,x y at any vertex node (I = 1, 2, 3), and w J,n at mid-side node (J = 4, 5, 6); see Fig. 4.28 for illustration. The trial and the test subspaces for such plate finite elements can be written as: , + VhA RG = wh ∈ V | wh|e ∈ P5 (e) ∀e ∈ Ch , + A RG V0,h = vh ∈ V0 | vh|e ∈ P5 (e) ∀e ∈ Ch

(4.170) (4.171)

where P5 (e) is the space of polynomials of degree at most 5 on triangle e, and Ch represents the collection of all the elements in the mesh. The discrete approximation for the displacement field is wh,e (ζ ) = ω(ζ )p, where  ζ = [ζ1 , ζ2 , ζ3 ] are area coordinates ( 3I =1 ζ I = 1), [391], p is vector of 21 nonzero constants and ω is transposed vector of 21 polynomials that define complete 2-dimensional quintic interpolation. Those polynomials are defined in [12] as ω = [ζ15 , ζ25 , ζ35 , ζ1 ζ24 , ζ2 ζ34 , ζ3 ζ14 , ζ2 ζ14 , ζ3 ζ24 , ζ1 ζ34 , ζ13 ζ22 , ζ23 ζ32 , ζ33 ζ12 , ζ23 ζ12 , ζ33 ζ22 , ζ13 ζ32 , ζ13 ζ2 ζ3 , ζ1 ζ23 ζ3 , ζ1 ζ2 ζ33 , ζ12 ζ22 ζ3 , ζ12 ζ2 ζ32 , ζ1 ζ22 ζ32 ]

(4.172)

184

4 Plate Models: Validation and Verification

The constants p relate to the natural degrees of freedom of the element wρ wρ = [w1 , w 1 ,12 , w 1 ,31 , w 1 ,122 , w 1 ,312 , w 1 ,232 , w2 , w 2 ,23 , w 2 ,12 , w 2 ,122 , w 2 ,312 , w 2 ,232 ,

(4.173)

w3 , w 3 ,31 , w 3 ,23 , w 3 ,122 , w 3 ,312 , w 3 ,232 w 1−2 ,n , w 2−3 ,n , w 3−1 ,n ]T as p = Awρ . In wρ , the derivatives are formed with respect to the natural coordinates. Hence, w K , I J is derivative of wh,e with respect to s I J ∈ [0, L I J ] that runs from vertex node I towards vertex node J (L I J is the length of that edge), evaluated at vertex node K . Similarly, w K , I J 2 is 2nd the derivative of wh,e with respect to that parameter, evaluated at vertex node K . The above derivatives are defined as ∂wh,e ∂wh,e ∂ 2 wh,e ∂wh,e ∂wh,e ∂ 2 wh,e − , wh,e , I J 2 = −2 + 2 ∂ζ J ∂ζ I ∂ζ I ∂ζ J ∂ζ J ∂ζ I2 (4.174) Furthermore, w I −J ,n in (4.173) denotes derivative of wh,e with respect to the normal to the edge (defined by vertex nodes I and J ) evaluated at the mid-side node wh,e , I J =

     ∂wh,e ∂wh,e ∂wh,e ∂wh,e ∂wh,e − + − (4.175) w I −J ,n = 2 + μI J ∂ζ K ∂ζ I ∂ζ J ∂ζ J ∂ζ I |ζ I −J 1−2 2−3 3−1 vertex nodes notation mid-side node notation, Fig. 4.28 4 5 6 Here, K = I = J is the 3rd vertex node, μ I J = 2xTIJ (x K − (x I + x J ) /2) /L 2I J , x I J = x J − x I , x K = [x K , y K ]T , and ζ I −J is value of ζ at the mid-side node. The matrix A (with constant entries) can be obtained by evaluating wh,e (ζ ) = ω(ζ )p and derivatives (4.174) at vertex nodes, and (4.175) at mid-side nodes 1 2 3 Node " 1 41 # " 51 1 # " 1 6 1 # ζ [1, 0, 0] [0, 1, 0] [0, 0, 1] 2 , 2 , 0 0, 2 , 2 2 , 0, 2

(4.176)

and equating the results to wρ . Once A is known, we can transform nodal ζ -derivatives to nodal x, y -derivatives. The following transformations are valid for any point of the triangle (see [12]): ⎡

⎤ ⎡ ⎤  wh,e ,12 x12 y12  ⎣ wh,e ,23 ⎦ = ⎣ x23 y23 ⎦ wh,e ,x wh,e , y wh,e ,31 x31 y31

(4.177)

4.4 Verification or Discrete Approximation Adaptivity for Discrete …

185

⎡

⎤ ⎡ 2 2 ⎤⎡ ⎤ wh,e ,x x w h,e ,122 x12 y12 x12 y12 2 2 ⎣ w h,e ,232 ⎦ = ⎣ x23 y23 x23 y23 ⎦⎣ wh,e , yy ⎦ 2 2 w h,e ,312 x y x31 y31 2wh,e ,x y 

 31 31

  

κρ

(4.178)

κh

T−1

where x I J = x J − x I and y I J = y J − y I . With the transformations (4.177) and (4.178) in hand, one can provide relation wρ = XT ue , where ue are element degrees of freedom indicated in Fig. 4.28 ue = [w1 , w 1 ,x , w 1 , y , w 1 ,x x , w 1 ,x y , w 1 , yy ,

(4.179)

w2 , w 2 ,x , w 2 , y , w 2 ,x x , w 2 ,x y , w 2 , yy , w3 , w 3 ,x , w 3 , y , w 3 ,x x , w 3 ,x y , w 3 , yy w 1−2 ,n , w 2−3 ,n , w 3−1 ,n ]T and XT is a matrix of constant entries. It follows from the above that wh,e (ζ ) = ω(ζ )AXT ue , where ω(ζ )AXT = ω(ζ )X is transposed vector of shape functions. The approximation of curvature vector κ h in (4.178) can now written as ⎤ ω,122 κ h = Tκ ρ = T⎣ ω,232 ⎦AXT ue = Tωρ Xue  



ω,312 ωρ   X ⎡

(4.180)

ωρ

∂ω ∂ω where ω, I J 2 is obtained as, see (4.174), ω, I J 2 = ∂ω − 2 ∂ζ + ∂ω . ∂ζ J2 ∂ζ I2 I ∂ζ J The same type interpolation as defined for trial functions wh,e is also used for test functions vh,e , with appropriate nodal values ve and corresponding constraint of kinematic admissibility. By further replacing these interpolations into the bilinear form in (4.162), we can obtain the element stiffness matrix Ke :           a(wh,e , vh,e ) = κ hT vh,e mh wh,e dΩ = κ ρT vh,e Gκ ρ wh,e dΩ 2

Ωe,h



Ωe,h

ωρT Gωρ dΩue = ve,T Ke ue

= ve,T

2

(4.181)

Ωe,h

where G = TT C B T. The last integral in (4.181) is obtained by analytical integration. The element load vector follows from using the above interpolation in the linear form (4.164)

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4 Plate Models: Validation and Verification

Fig. 4.28 Degrees of freedom of Argyris plate triangular element

3

6 5 w1 1 w1,x , w1,y w1,xx , w1,xy , w1,yy

⎡ ⎢ l(vh,e ) = ve,T XT ⎣



Ωe,h

=v

ω T f dΩ +

 ΓI J

⎛ ⎝



 T ω qef

w5,n

4 2

⎞⎤  " #J ⎥ T − ω,n m s ds + ω T m n I ⎠⎦

ΓI J

e,T e

f

(4.182)

% $ % $ ∂ω ∂ω ∂ω ∂ω ∂ω + μ , see (4.175). The intewhere ω,n = ∂ω | = 2 − + − Γ I J I J ∂n ∂ζ K ∂ζ I ∂ζ J ∂ζ J ∂ζ I grals in (4.182) are obtained in closed form. For more details for this element, we refer to [12]. The Kirchhoff triangular plate element of Argyris ensures the continuity of bending moments across the element boundaries. Under such conditions, one can provide the error estimates and confirm the convergence for the plate discretization of this kind (e.g. see [81], p. 296), resulting with the error estimate for k-th order polynomial approximations that reads:  w − wh 2,Ω ≤ C h k−1 |w|k+1,Ω

(4.183)

where C is a constant independent of characteristic element size h. In other words, for the regular solution w ∈ H 6 (Ω), the error energy norm in the displacement field computed by the Argyris plate element is O(h 4 ). The Argyris element provides the continuity of moments, which is also the key requirement for the enhanced solution employed in error estimates (e.g. see [63] or [19]) However, the continuity of moments also enforces continuity of curvatures, which requires imposing the curvature nodal values. The latter turned into the main weakness of the Argyris element that kept it from becoming the standard engineering tool for plates, and the Discrete Kirchhoff element finally earned the place of favorite. However, we show here that the Argyris plate can perfectly serve in tandem with the DKT plate element for constructing discretization error estimates by solving the local (element-wise) Neumann problems, as explained next.

4.4 Verification or Discrete Approximation Adaptivity for Discrete …

4.4.2.2

187

Discrete Kirchhoff Triangle

In order to explain how non-standard DKT interpolations are designed, we can start with the mixed variational formulation (e.g. [190]), replacing as a starting point the weak form for Kirchhoff plate in (4.167). The main goal is to reduce the displacement continuity requirements by introducing an independent shear strain field γ , along with the corresponding weak form of the Kirchhoff hypothesis. We can write this mixed variational formulation as follows: & aγ (u, γ , q; v) = l(v); aγ (u, γ ; v) = {κ T (v) m(u) + [γ (v) − γ ∗ ]T q} dΩ Ω & ∗,T {q [γ (u) − γ ]T }; dΩ = 0 Ω

(4.184) where u = [w, θx , θ y ]T and v = [v, φx , φ y ]T are kinematics state variables and + θ y ; ∂w − θx ]T and γ (v) = [ ∂∂vx + φ y ; ∂v − φx ]T are their variations, γ (u) = [ ∂w ∂x ∂y ∂y displacement-based shear strains and their variations, while q are shear forces. By assuming the shear force is defined independently in each element, we can recover (e.g. see [190]) the variational consistency of the assumed shear strain formulation by enforcing the validity of the variational equation (4.184) in each element. Such a mixed variational formulation is the basis for the Discrete Kirchhoff plate element approximation. The DKT plate element provides cubic interpolation of the displacement field and quadratic interpolation of the rotation field, along with the zero value of assumed shear strain. Yet, DKT finite element requires only standard degrees of freedom, with (w, θx , θ y ) defined at each corner node. For clarity on how to construct the finite element interpolations of this kind, let us first consider each edge of DKT plate element as a 2-node planar beam finite element of length L. We define the coordinates s B = ξ B L2 + L2 ∈ [0, L], ξ B ∈ [−1, +1]. Along each edge, we choose the following interpolations for displacement (cubic) and rotation (quadratic): whB =

2  I =1

w I N IB +

4 

w I N IB , θhB =

I =3

2 

θ I N IB + θ3 N3B

(4.185)

I =1

where N IB are hierarchical Lagrangian functions N1B = (1 − ξ B )/2, N2B = (1 +  2  2 ξ B )/2, N3B = (1 − ξ B ), N4B = ξ B (1 − ξ B ). At the final stage, only two nodal displacements, w1 , w2 , and two nodal rotations, θ1 , θ2 , will remain as acceptable parameters. The other parameters in (4.185) are condensed out by enforcing the normality hypothesis of the Kirchhoff theory, i.e. by setting to zero the constant, the ∂w B linear, and the quadratic terms in ∂s Bh − θhB = 0. This results with: w3 =

L L (θ1 − θ2 ), w4 = 8 4



 w2 − w1 6 1 − (θ1 + θ2 ) , θ3 = w4 L 2 L

(4.186)

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4 Plate Models: Validation and Verification

By replacing (4.186) into (4.185), we finally recover a four-parameter interpolation along ξ B . Hence, with this interpolation the DKT will have the zero transverse shear imposed along each edge, which for assumed strain interpolation (see [190]) further implies zero shear strains everywhere within the element domain. This result is directly applicable to the DKT element by recognizing that θhB = nT θ h,e = θn,h,e and that the Kirchhoff normality constraint implies ∂w∂sh,e − θn,h,e = 0 for each edge of element e. Thus, we obtain the following DKT approximations of displacement and rotations represented by the sum of nodal, I = 1, 2, 3, and edge contributions, I J = 12, 23, 31 wh|e = wh,e =

3 

wI N I +

I =1

θ h|e = θ h,e =

3 

  w3,I J N I J + w4,I J M I J

(4.187)

IJ

θ I NI +

I =1



θ 3,I J N I J

IJ

Here, w3,I J , w4,I J and θ 3,I J are in analogy with parameters in (4.186) LIJ T L I J w J − wI 1 n I J (θ I − θ J ), w4,I J = ( − nTIJ (θ I + θ J )) 8 4 LIJ 2 6 (4.188) = w4,I J n I J LIJ

w3,I J = θ 3,I J

where n I J = [n x,I J , n y,I J ]T is the outward unit normal of edge between nodes I and J , and L I J is the length of that edge. The functions in (4.187) may be defined in area coordinates ζ I , [391], as N I = ζ I , N I J = 4ζ I ζ J , and M I J = 4ζ I ζ J (ζ J − ζ I ). The above DKT interpolations can also be presented in the matrix notation wh,e =

3 

Nw,I u I , θ h,e =

I =1

3 

Nθ,I u I , κ h =

3 

I =1

Bκ,I u I

(4.189)

I =1

where vector u I = [w I , θx I , θ y I ]T collects the degrees of freedom of node I . We will also use notation u¯ e = [u1T , u2T , u3T ]T . Explicit forms of Nw,I , Nθ,I and Bκ,I can be obtained from (4.187), (4.188) and (4.152). T T ] , By using the so defined DKT interpolations for both trial, uh,e = [wh,e , θ h,e T T and test functions, vh,e = [vh,e , φ h,e ] , in the weak form (4.162), we obtain  ae (uh,e , vh,e ) =

κ hT (vh,e ) mh (uh,e ) dΩ =

I,J =1

Ωh,e



KeI J

=

T Bκ,I C B Bκ,J dΩ Ωh,e

3 

v TI KeI J u J

(4.190)

4.4 Verification or Discrete Approximation Adaptivity for Discrete …

189

where v I = [v I , φx I , φ y I ]T , and [KeI J ] is the element stiffness matrix. The element T  consistent load vector f e = f Ie,T follows from  le (vh,e ) =

vh,e

⎡ ⎤   J ⎣ (vh,e q¯e f + φs,h,e m s ) ds + [vh,e m n ] I ⎦ f dΩ + ΓI J

Ωh,e

ΓI J

(4.191) =

3 

v TI f Ie =

I =1

3 

e e v TI (f ef,I + ft,I + fc,I )

I =1

#T " where φs,h,e = sTI J φ h,e , s I J = −n y,I J , n x,I J , edge Γ I J ⊂ Γ N ,h , and  f ef,I =



e f Nw,I dΩ, ft,I =

(q¯e f Nw,I + m s sTI J Nθ,I )ds

ΓI J Γ IJ

Ωh,e

e = [m n | I + − m n | I − , 0, 0]T , ∀ node I ∈ Γ N ,h fc,I

(4.192)

The trial and the test spaces for a plate problem, discretized by the DKT finite element, can be written as: ⎧ ⎫ ⎨ wh ∈ U | wh|e ∈ P3 (e) ∀e ∈ Ch ⎬ (4.193) VhD K T = θ h ∈ U | θ h|e ∈ [P2 (e)]2 ∀e ∈ Ch ⎩ T ⎭ n I J ∇ K wh|e − θ h|e |Γ I J = 0 ∀Γ I J ∈ e ∧ ∀e ∈ Ch DK T V0,h

⎧ ⎫ ⎨ vh ∈ U0 | vh|e ∈ P3 (e) ∀e ∈ Ch ⎬ = φ h ∈ U0 | φ h|e ∈ [P2 (e)]2 ∀e ∈ Ch ⎩ T ⎭ n I J ∇ K vh|e − φ h|e |Γ I J = 0 ∀Γ I J ∈ e ∧ ∀e ∈ Ch

(4.194)

where P3 (e) and P2 (e) are the spaces of polynomials of degrees at most 3 and 2, respectively, on triangle e with edges Γ I J , I J = 12, 23, 31, and nodes I = 1, 2, 3. The Ch represents the collection of all the elements of the mesh, and 4

3

)



2

¯ on Γ D ∧ Γ M , U = θ ∈ H1 (Ω) U = w ∈ H1 (Ω) | w = w 3

4

)



2

U0 = v ∈ H (Ω) | v = 0 on Γ D ∧ Γ M , U0 = φ ∈ H (Ω) 1

1

| sT θ = θ¯s on Γ D

*

(4.195) * | s φ = 0 on Γ D T

(4.196) Thus, (4.168) and (4.169) are for the DKT replaced with ) V =

w ∈ H1 (Ω) " #2 | w = w, ¯ sT θ = θs = θ¯s on Γ D ;w = w¯ on Γ M θ ∈ H1 (Ω)

* (4.197)

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4 Plate Models: Validation and Verification

and ) V0 =

v ∈ H1 (Ω) " #2 | v = 0, sT φ = φs = 0 on Γ D ; v = 0 on Γ M φ ∈ H1 (Ω)

* (4.198)

4.4.3 Error Estimates for Kirchhoff Plate Elements Based Upon Equilibrated Boundary Stress Resultants In this section, we define the discretization error and present the computational procedure based on the equilibrated element boundary stress resultants (EqR), which can be used for any Kirchhoff plate element. The procedure follows the path proposed earlier for classical elasticity problems in [253] or [354].

4.4.3.1

Global Versus Local Residuals and Equilibrated Edge Tractions

The discretization error is a continuous field defined as the difference between the exact solution u and the FE solution uh : e = u − uh

(4.199)

Given that both exact and FE solutions satisfy the weak form of the plate equilibrium equations, by linearity property for both linear form l(·) and bilinear form a(·, ·), we can easily show the orthogonality in the energy-norm of the true error with respect to any test function chosen in discrete approximation space; thus, we can write: * a(u, vh ) = l(vh ) (4.200) ⇒ a(e, vh ) = 0 ; ∀vh ∈ V 0,h ⊂ V 0 a(uh , vh ) = l(vh ) This confirms the best approximation property (e.g. [391] or [176]), which implies that the finite element method delivers the best approximate solution in the chosen discrete space measured in terms of the energy norm, defined by the corresponding bilinear form a(·, ·). In order to quantify the error, we would need to provide the exact solution, which is not available, in general. Thus, our main task here reduces to providing instead a sufficient quality replacement of the exact solution, or rather its variation v ∈ V 0 . Given such a replacement, we can define the corresponding error estimates in terms of the global residual equation: a(e, v) := a(u, v) − a(uh , v) = l(v) − a(uh , v) ; ∀v ∈ V 0

(4.201)

4.4 Verification or Discrete Approximation Adaptivity for Discrete …

191

The computational effort can significantly be reduced if the global residual in (4.201) is decomposed into contributions from all individual elements in a particular finite element mesh ⎤ ⎡  " #   T ⎣ a(e, v) = ge (ve ) − ae (uh,e , ve ) + ve te,Γ ds ⎦ ∀v, ve ∈ V 0 e

Γ

e

Γ

(4.202) where ge (ve ) = Ωh,e v f dΩ and te,Γ are the element boundary tractions on each edge Γ in a particular finite element e. We will use the notation te,Γ = [qe f,Γ , m s,Γ , m n,Γ ]T , where m s,Γ = −m nn |Γ and m n,Γ = m ns |Γ . Since the sum of local residuals in (4.202) should match the global residual in (4.201), the total virtual work of element boundary traction forces over all edges must remain equal to zero. Given the continuity of test functions across any edge, this further implies that, for any enhanced solution, the element boundary tractions must remain continuous across the interior edge between two adjacent elements: &



te,Γ + te ,Γ = 0 for e and e sharing edge Γ

(4.203)

The enforcement of condition in (4.203) above requires the special procedure, which is discussed subsequently for DKT plate element. Note that the continuity requirement concerns the ‘moment traction vector’ (as generalization oh the Cauchy principle [176]) with components being moments m nn and m ns . This can also be recast in terms of only corner nodal moments m ns ) and the effective shear force qe f across all the interior edges. This implies that the moment m ss , and the shear force qn need not be continuous across the interior edges. Given such a choice of element boundary tractions, we can reduce the computation of discretization error ϑ e ∈ V in (4.201) to local (element-wise) residual computations to be carried out independently in each element e ae (ϑ e , ve ) = ge (ve ) − ae (uh,e , ve ) +

 Γ

veT te,Γ ds ∀ve ∈ V

(4.204)

Γ

We note that any element edge placed on the Dirichlet boundary Γ ⊂ Γ D,h is also considered as an interior edge, where we need to compute the equilibrated element boundary tractions. Only the element edges placed on the Neumann boundary Γ ⊂ Γ N ,h are left out of this computations, and the corresponding values of element boundary tractions are set equal to the imposed traction vales: #T " te,Γ = q¯e f , m s , m n for Γ ⊂ Γ N ,h

(4.205)

This condition should ensure the accuracy of the computed error with respect to boundary conditions.

192

4 Plate Models: Validation and Verification

Another condition that must be imposed upon the element boundary tractions te,Γ concerns the element equilibrium. In other words, for ve = [1, 0, 0]T , ve = [x, 0, −1]T , ve = [y, 1, 0]T , the edge traction forces should exactly equilibrate the applied external loading. For ve = vh,e , the Galerkin orthogonality is recovered on the left-hand side in (4.204), with ae (ϑ e , vh,e ) = 0. This further leads to an additional condition on element boundary traction, which can be written as: 

T vh,e te,Γ I J ds = ae (uh,e , vh,e ) − ge (vh,e ) ⇔

ΓI J Γ IJ

   T e v TI reI,Γ I J + v TJ reJ,Γ I J = vL RL ΓI J

(4.206)

L

Here, ReL denotes the nodal value at the FE node L of element residual, obtained as ReL = KeL J u J − f ef,L , where KeL J and f ef,L are the stiffness matrix and the external nodal load vector, respectively. The component reI,Γ I J may be regarded as a projection of ReI onto the edge Γ I J of a particular element e. Similarly, reJ,Γ I J can be viewed as projection of ReJ onto edge Γ I J of element e. The procedure to construct the element boundary tractions satisfying (4.206), as well as constraints (4.203) and (4.205), must be specified for each particular element; in the next section we discuss the specific choice for DKT plate element.

4.4.3.2

Enhanced Test Space, Local Neumann Problem and Error Estimates

ue − ue,h , where 5 ue ∈ V In seeking the error estimate, we can define the error ϑ e = 5 is an enhanced approximation of the exact solution ue ; this allows to restate the result in (4.204) as:  ue , ve ) = ge (ve ) + veT te,Γ ds ∀ve ∈ V (4.207) ae (5 Γ

Γ

One can then solve (4.207) instead of (4.204), to obtain an enhanced solution. This is a local (element-wise) Neumann problem. For simple elements, one can provide the analytic solution to this problem. However, for plates elements, this can lead to a prohibitive computational cost (e.g. see [48]). A more efficient alternative is to solve this local Neumann problem with a higher-order approximation of test space, i.e. vh,e → vh + ,e : ae (uh + ,e , vh + ,e ) = ge (vh + ,e ) +

 Γ

Γ

vhT+ ,e te,Γ ds ∀vh + ,e ∈ V h + ⊂ V

(4.208)

4.4 Verification or Discrete Approximation Adaptivity for Discrete …

193

The result (4.208) can be written explicitly as 

 κ hT+ (vh + ,e ) mh + (uh + ,e ) dΩ = Ωh,e

+





ΓI J

vh + f dΩ Ωh,e

  vh + qe f,Γ + φs,h + m s,Γ ds + [vh + m n,Γ ] IJ

ΓI J

where uh + ,e = [wh + , θ hT+ ]T ∈ V h + , vh + ,e = [vh + , φ hT+ ]T , and [qe f,Γ , m s,Γ , m n,Γ ]T = [q e f , m s , m n ]T for Γ I J ⊂ Γ N ,h . The local Neumann problem in (4.209) above results with a set of linear algebraic equations to be solved independently for each single element e e u¯ + = f+e K+

(4.209)

The main difficulty in solving this system pertains to the singularity of the element e , which occurs due to pure Neumann boundary and presence of stiffness matrix K+ rigid body modes. Thus, a special procedure need to be used in order to solve this system of equations. Namely, we first collect three rigid body modes of plate finite element into a matrix D, which allows for any nodal displacements/rotation vector due to the rigid body motion, to write u¯ eR = Dα, where α is a set of rigid body mode amplitudes. By using further the observation that the rigid body motion of the plate e Dα. We can now finite element produces zero force, we can write: 0 = Ke+ u¯ eR = K+ ensure the solvability of the system in (4.209) by regularization, where we replace e + DDT , which is no longer a singular K with a new system matrix defined as K+ matrix. Hence, we can thus obtain the unique solution e e + DDT ) u+ = f+e (K+

(4.210)

Finally, having solved this local Neumann problem, the rigid body modes can be e e e e to get the final nodal displacements u¯ + = u+ − DDT  u+ . purged from the solution u+ We note in passing that the bending moments are not affected by the rigid body modes e e or u¯ + . and energy norm can be computed either with  u+ e With enhanced solution u¯ + in hand, local error estimates can be computed as ϑ e = uh + ,e − uh,e

(4.211)

The corresponding energy norm of the local error indicator can then be readily obtained as  "    #T "    # κ h + uh + ,e − κ h uh,e eh,e 2E = C B κ h + uh + ,e − κ h uh,e dΩ Ωh,e

(4.212)

194

4 Plate Models: Validation and Verification

It has been noted (e.g. [354]) that the same results can be obtained directly from the local Neumann problem by using the error field as the test function ee 2E ≡ a(ee , e) = l(v − vh ) − a(uh , v − vh ) ; ∀vh ∈ V 0h ; ∀v ∈ V 0

(4.213)

When the computations are performed for each element of the mesh, the upper bound of the corresponding global error indicators can be estimated simply by using the Cauchy-Schwarz inequality e2E ≤ eh 2E =



eh,e 2E

(4.214)

e

4.4.4 Implementation of Equilibrated Element Boundary Tractions Method For DKT Plate Element The equilibrated element boundary resultants method is outlined in the previous Sect. 4.4.3 for any Kirchhoff plate element. We will provide in this section the implementation details specific to the DKT plate element.

4.4.4.1

Element Boundary Stress Resultants

To find the relationship between the equilibrated tractions te,Γ I J and the nodal edge projections reI,Γ I J and reJ,Γ I J , the first part of Eq. (4.206) is used 

  T vh,e te,Γ I J ds = v TI reI,Γ I J + v TJ reJ,Γ I J

(4.215)

ΓI J

For DKT plate element, (4.215) above can be written explicitly as 



 vh qe f,Γ I J + sTI J φ h m s,Γ I J ds + [vh m n,Γ I J ] IJ = v TI reI,Γ I J + v TJ reJ,Γ I J

(4.216)

ΓI J

where the DKT interpolations of the transverse displacement and the rotations along the edge Γ I J follow from (4.187) and (4.188) as LIJ T LIJ n (φ − φ J ) ϕ3 − Δφ I J ϕ4 4 IJ I 3 φ h = φ I ϕ1 + φ J ϕ2 + 2n I J Δφ I J ϕ3 vh = v I ϕ1 + v J ϕ2 +

(4.217)

4.4 Verification or Discrete Approximation Adaptivity for Discrete …

195

" #T Here, L I J is the length of the edge, n I J = n x , n y is the edge outward nor#T " mal, s I J = −n y , n x , and Δφ I J = 2 L3 I J (v J − v I ) − 43 nTIJ (φ I + φ J ). The shape functions in (4.216) are: ϕ1 = (1 − ξ )/2, ϕ2 = (1 + ξ )/2, ϕ3 = (1 − ξ 2 )/2, ϕ4 = ξ(1 − ξ 2 )/2, and ξ ∈ [−1, 1]. In order to evaluate the integral in (4.216), we need to provide distribution of qe f,Γ I J and m s,Γ I J along the edge. The functions ψ I and ψ J will be used for this purpose, both for qe f,Γ I J and m s,Γ I J . By denoting nodal parameters as qeI f,Γ I J , qeJf,Γ I J , J I m s,Γ , and m s,Γ , the corresponding interpolations of qe f,Γ I J and m s,Γ I J read: IJ IJ qe f,Γ I J = ψ I qeI f,Γ I J + ψ J qeJf,Γ I J ,

I J m s,Γ I J = ψ I m s,Γ + ψ J m s,Γ IJ IJ

(4.218)

J I and m n,Γ . The For m n,Γ I J , we will need only the nodal values denoted as m n,Γ IJ IJ relationship between the nodal parameters of equilibrated tractions and these edge projections follows from (4.216) in a form of six linear equations



reI,Γ I J reJ,Γ I J



 = Aq

qeI f,Γ I J qeJf,Γ I J



 + Am s

I m s,Γ IJ J m s,Γ IJ



 + Am n

I m n,Γ IJ J m n,Γ IJ

 (4.219)

where 

T 100000 A = (−n y A + n x A ), A = (4.220) 000100 ⎡ I ⎤  k k   Nk,w a a I ⎦ k k I I I J ⎣ N , aI J = ψ J ds, k ∈ [q, m x , m y ] A = k k k,θx aJ I aJ J I Nk,θ ΓI J y ms

mx

my

mn

and I = ϕ1 − ϕ4 /2, Nq,w I Nq,θ x I Nq,θ y

= −L I J n x (−ϕ3 + ϕ4 )/4, = −L I J n y (−ϕ3 + ϕ4 )/4,

NmI x ,w = 3n x /L I J ϕ3 , NmI x ,θx NmI x ,θ y

= ϕ1 +

3n 2x ϕ3 /2,

= 3n x n y ϕ3 /2,

J Nq,w = ϕ2 + ϕ4 /2 J Nq,θ x J Nq,θ y

= −L I J n x (ϕ3 + ϕ4 )/4 = −L I J n y (ϕ3 + ϕ4 )/4

NmJ x ,w = −3n x /L I J ϕ3 NmJ x ,θx NmJ x ,θ y

(4.221)

= ϕ2 +

3n 2x ϕ3 /2

= 3n x n y ϕ3 /2

(4.222)

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4 Plate Models: Validation and Verification

NmI y ,w = 3n y /L I J ϕ3 ,

NmJ y ,w = −3n y /L I J ϕ3

NmI y ,θx = 3n x n y ϕ3 /2,

NmJ y ,θx = 3n x n y ϕ3 /2

NmI y ,θ y = ϕ1 + 3n 2y ϕ3 /2,

NmJ y ,θ y = ϕ2 + 3n 2y ϕ3 /2

(4.223)

If one chooses ψ I = ϕ1 ,

ψ J = ϕ2

(4.224)

the following relationship between the nodal parameters of equilibrated edge tractions and the edge projections is obtained from (4.219)-(4.223)  12  m m mx mx n x (2r I,Γ + 3r J,Γ ) + n y (2r I,Γy I J + 3r J,Γy I J ) (4.225) 2 IJ IJ LIJ 2 m m mx mx = (n y (−2r I,Γ + r J,Γ ) + n x (2r I,Γy I J − r J,Γy I J )) IJ IJ LIJ 1 m m q mx mx =− (n x (r I,Γ + 7r J,Γ ) + n y (r I,Γy I J + 7r J,Γy I J )) − r I,Γ I J IJ IJ LIJ

qeI f,Γ I J = − I m s,Γ IJ I m n,Γ IJ

 12  m m mx mx n x (3r I,Γ + 2r J,Γ ) + n y (3r I,Γy I J + 2r J,Γy I J ) (4.226) 2 IJ IJ LIJ 2 m m mx mx = (n y (r I,Γ − 2r J,Γ ) + n x (−r I,Γy I J + 2r J,Γy I J )) IJ IJ LIJ 1 m m q mx mx =− (n x (7r I,Γ + r J,Γ ) + n y (7r I,Γy I J + r J,Γy I J )) + r J,Γ I J IJ IJ LIJ

qeJf,Γ I J = J m s,Γ IJ J m n,Γ IJ

m

q

mx The notation reI,Γ I J = [r I,Γ I J , r I,Γ , r I,Γy I J ]T has been used above. By inserting IJ (4.225)–(4.226) into (4.218), the equilibrated edge tractions te,Γ I J are given as functions of projections reI,Γ I J and reJ,Γ I J for the chosen interpolation (4.224).

4.4.4.2

Construction of Equilibrated Boundary Tractions

We now elaborate on the construction of nodal edge projections. Starting from the second part of (4.206), we obtain the following relation at the FE node I reI,Γ1 + reI,Γ2 = ReI

(4.227)

where Γ1 and Γ2 are two edges of the element e that meet at the element node I . Furthermore, the condition (4.203) requires that edge projections are continuous across the edge; at node K of the FE mesh, it should hold that



reK ,Γ + reK ,Γ = 0 for e and e sharing edge Γ

(4.228)

4.4 Verification or Discrete Approximation Adaptivity for Discrete …

197 Γ3

Fig. 4.29 A patch of four elements surrounding node K e4

e3

dK

Γ4 e1

Γ2 e2

Γ1

From (4.227) and (4.228), one can get a set of linear algebraic equations. Each sets of this kind is formed for the patch of elements attached to a particular node of the FE mesh. For illustration, let us consider a patch of four elements P K that surround node K of the mesh (see Fig. 4.29). Equations (4.227) results in this case with ReK1 = r Ke1,Γ4 + r Ke1,Γ1 ReK2 ReK3 ReK4

= = =

r Ke2,Γ1 r Ke3,Γ2 r Ke4,Γ3

(4.229)

+ r Ke2,Γ2 + r Ke3,Γ3 + r Ke4,Γ4

Moreover, the continuity condition (4.228) reads r Ke1,Γ1 + r Ke2,Γ1 = 0 r Ke2,Γ2 r Ke3,Γ3 r Ke4,Γ4

+ r Ke3,Γ2 + r Ke4,Γ3 + r Ke1,Γ4

(4.230)

=0 =0 =0

The last result can be written as the set of equations with reKk ,Γk as unknowns ⎡

⎤ ⎡ ReK1 +1 ⎢ Re2 ⎥ ⎢ −1 ⎢ eK3 ⎥ = ⎢ ⎣ RK ⎦ ⎣ 0 ReK4 0

0 +1 −1 0

0 0 +1 −1

⎤ ⎡ e1 ⎤ r K ,Γ1 −1 ⎢ reK2,Γ ⎥ 0 ⎥ 2 ⎥ ⎥⎢ 0 ⎦ ⎣ reK3,Γ3 ⎦ +1 reK4,Γ4

(4.231)

The solution to system (4.231) is not unique. This is the consequence of many possible choices for splitting ReK into two parts. To obtain the unique solution of (4.231 ), some kind of regularization has to be used. The following idea can be exploited for the regularization of the system in (4.231): search projections reI,Γ that should the closest possible in the least squares sense to some known projections r˜ eI,Γ .

198

4 Plate Models: Validation and Verification

We choose r˜ eI,Γ satisfying 

T ˜ vh,e te,Γ I J ds = v TI r˜ eI,Γ I J + v TJ r˜ eJ,Γ I J

(4.232)

ΓI J

where t˜e,Γ are assumed to be known edge tractions that are continuous across interior edges, and equal to prescribed loading on plate boundary #T " t˜e,Γ = q¯e f , m s , m n for Γ ⊂ Γ N ,h

(4.233)

By minimizing function, defined over a patch P K that surrounds node K of the FE mesh   1   e 2 r K ,Γ K I − r˜ eK ,Γ K I −→ min JK reK ,Γ K I = 2 e∈P Γ K

q

(4.234)

KI

m

y x the projections reK ,Γ = [r K ,Γ , r Km,Γ , r K ,Γ ]T can be obtained and further used in (4.226). Simple averaging can be used to get t˜e,Γ in (4.232)

te,Γ +  te ,Γ )/2 for e and e sharing edge Γ t˜e,Γ = (

(4.235)

#T " The computation of  te,Γ = q˜e f,Γ , m 5s,Γ , m 5n,Γ goes as follows. The moments over an entire DKT domain are first obtained as m 5=

3 

#T " m 5 I NI , m 5= m 5x x , m 5yy , m 5x y

(4.236)

I =1

where N I are the Lagrange interpolation functions. Nodal values m 5 I are computed by using the least squares fit of m 5 (ξgp ) to mh (ξgp ), where ξgp are isoparametric coordinates of integration points. At an element edge Γ the transformations (4.157) 5ns |Γ , m 5s,Γ = −5 m nn |Γ , are used to get the twisting and the normal moment, m 5n,Γ = m respectively. The shear force along the same edge is q˜Γ = q˜ x n x + q˜ y n y , where q˜ x = ∂m 5 ∂m 5 ∂m 5 5x x + ∂ yx y ) and q˜ y = −( ∂ yyy + ∂ xx y ). Finally, the effective shear force at the −( ∂ m ∂x m 5n . With these results,  te,Γ in (4.235) edge under consideration is q˜e f,Γ = q˜Γ − ∂∂s becomes fully defined.

4.4.4.3

Chosen Enhancements of Test Space

With an already well-performing finite element, such as DKT plate, there are not that many optimal solutions for constructing the enhanced test space in (4.208). Some among those initially proposed (e.g. [48]) can lead to quite a laborious procedure.

4.4 Verification or Discrete Approximation Adaptivity for Discrete …

199

Fig. 4.30 Subdivision schemes for the DKT element wI θx,I ; θy,I

wI θx,I ; θy,I

SDKT

CDKT

Here, we evaluate three simpler solutions for constructing the enhanced space. Two of them are based upon the most straightforward approach with subdividing a single DKT into a number of smaller DKT elements (e.g. [253]), and the third is the favorite one proposed in terms of the Argyris plate element. The simplest enhancement of the DKT test space is the subdivision of DKT into 3 triangular elements, as shown in Fig. 4.30. The resulting element, which is further referred to as the SDKT, has an additional node in the center and 3 additional degrees of freedom. Besides the center node, the SDKT shares all the nodes with the original DKT element. Therefore, the SDKT interpolation will exactly match the e DKT interpolation on the edges (vh+,e = vh,e on Γ ). It follows that R e+ I = R I , which renders the equilibration procedure unnecessary. The procedure using the SDTK to compute error indicators will be called the EqR-SDTK. Another subdivision scheme, called the CDKT, uses the SDKT as the starting point. It further subdivides each of 3 triangular elements into 2 by using the symmetry lines between the centre node and the mid-side nodes (see Fig. 4.30). Thus, the CDKT element has in total 7 nodes and 21 degrees of freedom. The procedure that uses the SDTK to compute error indicators will be called the EqR-CDTK. The final and the best choice for constructing an enhanced test space of the DKT is by using a conforming plate element presented in Sect. 4.4.2.1. This plate element, first proposed in [12] for computing the solution rather than for constructing error estimates, will further be called Argyris element. This procedure will be called the EqR-ARGY.

4.4.5 Examples on Error Indicators Comparison and Closing Remarks In this section, we carry out a number of numerical simulations in order to provide the illustration of error indicators performance with different enhanced spaces: EqRSDTK , EqR-CDTK and EqR-ARGY. The results obtained by using the standard smoothing procedure for bending moment computed by DKT plate element over supercovergence patch (e.g. see [352, 391] or [253]) and further referred to as SPR, are also used for comparison.

200

4 Plate Models: Validation and Verification

For any such comparison of different results, we need the exact solution and well-defined ‘true’ error. For the exact solution, we take the strong form solutions, if available (e.g. [59, 363]). Otherwise, the exact solution replacement was obtained by using a very fine mesh with Argyris elements (with curvature values left free). The results are often compared in terms of the effectivity index of the proposed error indicator, which is defined as: eh  E (4.237) Θ= e E The second role of numerical examples is to illustrate various aspects of our strategy for mesh adaptivity. For an adaptive change of the mesh, we will be using ∗ the local error indicator. At a single element level, these are denoted as: e h,e  E = ηe . ∗2 ∗2 Their sum over all the elements in the mesh will be denoted as: η = e ηe . We ∗ , defined as: will also make use of the relative local error indicator ηe,r ∗ ηe,r

eh,e  E = , nh,e 2E = nh,e  E

 κ hT mh ds

(4.238)

Ωe

as well as the relative global error indicator ηr∗ , defined as: ηr∗ =

eh  E , nh  E

nh 2E =



nh,e 2E = nh 2

(4.239)

e

The main goal of our adaptive strategy is to generate a mesh where the local element error roughly remains constant over the whole domain, i.e. equal in each element (within the given tolerance) to a prescribed target value. Given such a target value η¯ e , the desired element size h¯ for the error indicator of order p can be obtained as: (4.240) ηe∗ = Ch p ; η¯ e = C h¯ p ⇒ h¯ = h(η¯ e /ηe∗ )1/ p where we assumed that a constant C is independent of element size h. The computed target values of element size h¯ is stored into a scalar field that is used to guide remeshing in a subsequent step of this adaptive procedure. The software remacle09a [132] was employed to generate meshes in all different steps.

4.4.5.1

Clamped Square Plate Under Uniform Loading

We consider a clamped square plate of side length a = 10 and thickness t = 0.01 under uniform loading f = 1. The material data is: E = 10.92 1010 , ν = 0.3. The closed form analytical expression for the moments, given in [363], is taken as the reference solution (200 terms in series solution were used). The reference solution has no singularities. In Fig. 4.31 we present the curves that show the dependency of the global error indicator η∗ on the number of elements in a structured mesh. The

4.4 Verification or Discrete Approximation Adaptivity for Discrete …

201

Fig. 4.31 Clamped square plate under uniform loading—global energy error indicator

latter is expressed as 1/ h, where h 2 = Ωh /ne, where ne is the number of elements in the mesh. The EqR-SDKT indicator curve is below the “true error”. This indicator thus underestimates the error. The curves of other indicators are larger than the “true error”. The effectivity index (Fig. 4.32) is always close to 1.3 for the SPR. The effectivity indices of the EqR-CDKT and the EqR-ARGY are somewhat larger than 1.3. However, It is important to note the advantage for EqR-ARGY over EqR-CDKT, since both have the same number of 21 unknowns in the local Neumann problem. Another advantage is offered by EqR-ARGY with respect to all other error indi∗ . Namely, cators if we look at a comparison of the relative local error indicators ηe,r as shown in Fig. 4.33, the local error in the corners is quite highly underestimated by both SPR and simple EqR-SDKT error indicators.

4.4.5.2

Morley’s Skew Plate Under Uniform Loading

The well-known benchmark of Morley’s 30◦ skew plate [291] is analyzed in this example. The chosen values for plate thickness t = 1 and side length a = 10. The plate is resting on simple supports on all sides, under uniformly distributed loading f = 1. The linear elastic material properties are: Young’s modulus E = 10.92 and Poisson’s ratio ν = 0.3. The ‘exact’ solution is provided in [291] and is confirmed by computation with Argyris element on regular mesh of 100 × 100 elements with side h = 0.1. The solution is characterized by two singularity points for moments at the obtuse corners of the plate. The influence of the singularities spreads strongly influencing the quality of computed results (e.g. see [164]). In Fig. 4.34, we show the global error indicator η∗ versus the number of elements computed with a uniform mesh. We first note that the EqR-SDKT exhibits erratic behavior, while SPR underestimates the error. The other two indicators, EqR-

202

4 Plate Models: Validation and Verification

Fig. 4.32 Clamped square plate under uniform loading—effectivity index for the global energy error indicator

ARGY and EqR-CDKT, are rather on the safe side by overestimating the error, with equivalent performance for finer meshes and some advantage for EqR-ARGY for coarse meshes. The convergence is not monotonic. The effectivity index curves are presented in Fig. 4.35. The comparison of the relative local error indicators, given in Fig. 4.36, clearly shows that the EqR-SDKT is completely incapable of capturing the singularities at the obtuse corners, and even the SPR procedure severely underestimates the error. We can also see in this figure that both EqR-ARGY and EqR-CDKT have no difficulties in capturing the singularities.

4.4.5.3

Uniformly Loaded Clamped Square Plate

In this example, we consider the adaptive mesh refinement in the square plate already considered in the previous example, where the first FE solution was obtained with a uniform (original) mesh. Here, the problem is then recomputed, with a new mesh generated according to a distribution of the local error indicators ηe∗ . It is required that the number of the elements of the new mesh would approximately match the number of the elements of the original mesh. Thus, the average element size can be corrected accordingly. The p = 1.1 is used in (4.240), which is in accordance with the convergence trends shown in Fig. 4.31 for η¯ e = 0.05. The “true error” is computed by using the reference solution from [363] in the same way as in example 4.4.5.1. The results of the computations with the above-derived error indicator procedures are summarized in Table 4.8, which shows new meshes, the distributions of the element error ηe = ηe∗ , and their histograms. The histograms of the local element error are relatively narrow,

4.4 Verification or Discrete Approximation Adaptivity for Discrete … (a)

(m)

(b)

(c)

203

70

60

50 (d)

(e)

40

30

20

10

0

Fig. 4.33 Clamped square plate under uniform loading— comparison of relative local error indi∗ in [%] on the mesh (m): a True error, b SPR, c EqR-SDKT, d EqR-CDKT, e EqR-ARGY cator ηe,r

except for the original mesh, as expected. The histogram of the second row is not just a single peak. This is due to the limited capability of the mesh generation algorithm, and due to the lack of ability of the error estimate in (4.240) to take into account the effect of element distortion. The global results do not firmly indicate a superiority of any of the applied procedures, and the major difference pertains to the resulting structure of generated meshes. Thus, one has to look again into the distribution of the local error indicators clearly showing the difference between the original mesh and the adapted meshes for different choices of enhanced test space. We note again that EqR-SDKT and SPR cannot fully capture the true need for mesh refinement at the plate corners, which can be obtained correctly with EqR-ARGY and EqR-CDKT.

204

4 Plate Models: Validation and Verification

True error SPR indicator EqR - SDKT EqR - CDKT EqR - ARGY

0.80

log(η ∗ )

0.30

−0.20

−0.70

−1.20 −0.75

−0.50

−0.25

0.00 0.25 log(1/h)

0.50

0.75

1.00

Fig. 4.34 Morley’s skew plate under uniform loading—global energy error indicator 7.0 True error SPR indicator EqR - SDKT EqR - CDKT EqR - ARGY

6.0 5.0

θ

4.0 3.0 2.0 1.0 0.0 −0.75

−0.50

−0.25

0.00 0.25 log(1/h)

0.50

0.75

1.00

Fig. 4.35 Morley’s skew plate under uniform loading—effectivity index for the global energy error indicator

4.4.5.4

Uniformly Loaded L-Shaped Plate

In this example, we study an L-shaped plate resting on simple supports under uniformly distributed loading f = 1. We choose the plate thickness t = 0.01 and side length a = 10. The linear elastic material parameters are: Young’s modulus E = 10.92 109 and Poisson’s ratio ν = 0.3. The exact solution exhibits a singularity in stress resultant components at the obtuse corner. The corresponding singularities in m x x and m yy are both governed by the term r λ−2 , whereas the singularity in m x y has governing term r λ−3 , where r is the radial distance to the singularity point and λ is the exponent which depends

4.4 Verification or Discrete Approximation Adaptivity for Discrete … (a)

(m)

(b)

(c)

205

100 90 80 70 (d)

(e)

60 50 40 30 20 10 0

Fig. 4.36 Morley’s skew plate under uniform loading—comparison of relative local error indicators ∗ in [%] on the mesh (m): a True error, b SPR, c EqR-SDKT, d EqR-CDKT, e EqR-ARGY ηe,r

on the opening angle α, with λ = π/α (e.g. see [261]). In our case, α = 3π/2, thus λ = 2/3. The first solution was obtained on the regular structured mesh shown in Fig. 4.37. The problem was further reanalyzed in six iterations, with the mesh refinement based upon one of three different strategies. (i) The first and most straightforward mesh refinement strategy was a uniform refinement of a structured mesh. This strategy does not take into account any kind of information about discretization errors.

206

4 Plate Models: Validation and Verification

Fig. 4.37 L-shaped plate under uniform loading—initial mesh

(ii) The second mesh refinement strategy, which belongs to the same category, was a uniform refinement of an unstructured mesh. For that purpose, the initial structured mesh from Fig. 4.37 was replaced by an unstructured mesh with approximately the same number of elements. (iii) The third strategy is adaptive meshing based upon different error indicators defined previously. The ‘true’ error was computed by using the reference solution obtained with a refined mesh of Argyris elements, each with element side h = 0.1. At each iteration, the distribution of the error indicator ηe = ηe∗ was computed. Based on the estimated new element-size density (4.240), a new mesh was generated with p = 1 and η¯ e = 0.05. Comparison of the results convergence is shown in Fig. 4.38. The convergence of the adaptive meshing (iii) is considerably faster than uniform mesh refinements (i) and (ii). The difference between applied error indicators in the framework of (iii) disappears after the first two iterations. This is due to the limitation of the minimum allowable element size in the meshing algorithm. If that limitation had not existed, the element size at the singular points would have tended to zero. Table 4.9 compares results of different error indicators after the third iteration. The comparison reveals that the distribution of the local error indicator is considerably more uniform for (iii) than for (i), the latter shown in the first row. This is precisely one of the primary goals of adaptive meshing, making the local error indicators approximately equal for all elements in the mesh. Another important confirmation of the intrinsic value of mesh refinement strategy is that even though the average of local error indicators is smaller for the uniformly refined mesh (see the histogram in the first row of Table 4.9), the total energy norm of the error is still larger (see Fig. 4.38).

4.4.5.5

Computational Cost for Different EqR Mesh Adaptivity Procedures

Here, we briefly summarize the important information regarding the computational cost for the proposed EqR mesh adaptivity procedures and the comparison strategy

4.4 Verification or Discrete Approximation Adaptivity for Discrete …

207

Table 4.8 Adaptive meshing of a square plate by using different procedures Mesh

Local element error ηe 6. × 10-7

Original mesh

5.03 × 10-5

1. × 10-4

Histogram of local element error ηe 300 200 100

1156 elements, ηr : 6.2% Mesh based on true error

0

−6

−5

−4

log10 ηe 300 200 100

1171 elements, ηr : 5.4%

Mesh based on SPR

0

−6

−5

−4

log10 ηe 300 200 100

1170 elements, ηr : 5.7% Mesh based on EqR-SDKT

0

−6

−5

−4

log10 ηe 300 200 100

1168 elements, ηr : 5.5% Mesh based on EqR-CDKT

0

−6

−5

−4

log10 ηe 300 200 100

1158 elements, ηr : 5.4% Mesh based on EqR-Argyris

0

−6

−5

−4

log10 ηe 300 200 100

1261 elements, ηr : 5.4%

0

−6

−5

log10 ηe

−4

208

4 Plate Models: Validation and Verification

Table 4.9 Adaptive meshing of L shaped plate by using different procedures Mesh after 3rd refinement step Uniformly refined mesh 1716 elements, ηr : 18.1% Mesh based on true error 1559 elements, ηr : 5.8%

Mesh based on SPR

1568 elements, ηr : 6.5% Mesh based on EqR-SDKT 1575 elements, ηr : 6.0% Mesh based on EqR-CDKT 1574 elements, ηr : 5.9% Mesh based on EqR-Argyris 1557 elements, ηr : 5.6%

Local element error ηe 5. × 10-6

1.25 × 10-4

2.5 × 10-4

Histogram of local element error ηe 600 500 400 300 200 100 0

−5

−4

−3

log10 ηe 600 500 400 300 200 100 0

−5

−4

−3

log10 ηe 600 500 400 300 200 100 0

−5

−4

−3

log10 ηe 600 500 400 300 200 100 0

−5

−4

−3

log10 ηe 600 500 400 300 200 100 0

−5

−4

−3

log10 ηe 600 500 400 300 200 100 0

−5

−4

log10 ηe

−3

4.4 Verification or Discrete Approximation Adaptivity for Discrete … −0.3

(a) (b)

−0.5

(c)

) ref nh ref nh

nh

log(

209

(d)

−0.7

(e)

−0.9

(f) (g)

−1.1 −1.3 −1.5 2.00

2.50

3.00

3.50

4.00

4.50

log(1/h)

Fig. 4.38 L-shaped plate under uniform loading—convergence for: a Uniform mesh refinement of structured mesh, b Uniform mesh refinement of unstructured mesh, c Adaptive meshing based on true error, d Adaptive meshing based on SPR, e Adaptive meshing based on EqR-ARGY, (f) Adaptive meshing based on EqR-SDKT, g Adaptive meshing based on EqR-CDKT

SPR. The computational effort of the SPR is proportional to the total number of nodes n nodes in a mesh, since it relies on patch-wise computations for each node. The number of computational operations for the SPR N S P R is thus equal to: N S P R = C S P R ; n nodes

(4.241)

The proportionality constant C S P R is directly related to the order of local interpolation and the number of element superconvergent points. When linear interpolation is used, and 3 superconvergent points are taken into account for each element, C S P R ≈ 10. The computational cost for EqR procedure pertains to its three main steps: (i) equilibration of nodal element residuals, (ii) computation of edge generalized forces from the edge projections, and (iii) construction and solution of local (element) problems. Thus, the total number of operations is equal to: N Eq R = C(i) n nodes + C(ii) n edges + C(iii) n elements where C(i) , C(ii) and C(iii) are constants that are independent of n nodes (they are, however, proportional to the number of element degrees of freedom n do f ). The number of elements and the total number of element edges are denoted by n elements and n edges , respectively. Both are directly proportional to n nodes n elements = K 1 n nodes , n edges = K 2 n elements = K 2 K 1 n nodes = K 3 n nodes

210

Structural Engineering

The above constants are approximately K 1 ≈ 2, and K 2 ≈ 3/2 for the triangular unstructured mesh. The computational effort of the EqR method can thus be estimated as N Eq R = C Eq R n nodes , C Eq R = n do f (c(i) + c(ii) K 2 + c(iii) K 1 K 2 )

(4.242)

The constant c(i) is proportional to the number of edges meeting at each node, while constants c(ii) and c(iii) are proportional to the log(n do f ). In the case of unstructured triangular mesh and the DKT element, C Eq R ≈ 50. From (4.241) and (4.242), we can see that the computational effort for both the SPR and the EqR is finally proportional to the number of nodes n nodes . The ratio C Eq R /C S P R defines the computational effort of the EqR method compared to the SPR. For the DKT elements, the computational effort of the EqR is approximately 5 times higher than the computational effort of the SPR.

4.4.5.6

Closing Remarks

The main thrust of this section is directed towards a question of the choice of the error estimates in discrete approximation and adaptive mesh refinement for the Discrete Kirchhoff Triangular (DKT) plate bending finite element. The pertinent error indicators are constructed by using the equilibrated residual method adapted to suit this particular element, which is one of the favorite choices in engineering analysis of plates. Therefore, we believe that the proposed method to provide the DKT error indicators is of great practical interest. Regarding the different possibilities for constructing an enhanced test space needed for equilibrated element residual method, the best was proved to be the one constructed by using the Argyris plate element. The Argyris element discrete approximation is the complete fifth-order polynomial, which can provide the corresponding enhancement of the third order polynomial used by DKT plate element for transverse displacement field. The main difficulty in using the Argyris plate element for engineering analysis, concerning imposed nodal values of curvatures, does not occur when solving the local Neumann problem for constructing the corresponding enhanced space by the Argyris element. The superiority of the Argyris element-based construction of error indicators by equilibrated element residual method was also shown with respect to the classical technique of superconvergent patch recovery, especially in terms of its to properly provide the local error indicators. This improved result quality does impose a somewhat higher computational cost than the one typical of superconvergent patch recovery. Moreover, the computational cost of the proposed error indicator solving 21 equations in each local Neumann problem is still quite acceptable (and lower than the cost of earlier proposals; e.g. [48]).

Chapter 5

Solids, Membranes and Shells with Drilling Rotations: Complex Structures

Abstract We here present solid and membrane elements with rotational degrees of freedom, which are rather non-conventional choices with respect to the standard solid mechanics elements that do not include the rotation field in stress computations, for it is only related to rigid body motion. The models are this kind are needed for dealing with complex structures and achieve compatible connections between classical solid mechanics models with no rotations and already presented structural mechanics models with rotations. This task is not merely reduced to constructing nonconventional finite element interpolations, but also needs a fresh start for theoretical formulation of such problem in terms of regularized variational principle. We also show how to combine the proposed membrane element with drilling rotations with plate element in order to obtain a shell element. Such shell an element is developed consistently within the framework of shallow shell discrete approximation, which is proved to be a very powerful tool to handle various locking phenomena that plague the classical shell theory discrete approximation that can outperform most of the well-known shell elements. In preparing the transition to large displacement, we also briefly present a solid element with independent rotational field in a more general context of large rotations.

5.1 Solids with Drilling Rotations: Variational Formulation In the modeling of complex structural systems, typically encountered in civil, mechanical or aerospace engineering practice, one can face some serious difficulties pertaining to the compatibility of models used for different system components. In short, the difficulties are caused by connecting the parts of the system which are described by the classical continuum-type of formulation with the remaining parts of the system which are described by the oriented-media formulations with so-called Cosserat continuum [90], where one defines an independent rotation field. Such a formulation was already used for the analysis of beams (e.g. classical Euler-Bernoulli’s or Timoshenko’s beam), plates (Kirchhoff’s or Reissner-Mindlin’s plate) and can also be used for constructing shell theories (e.g. see [294]). However, for modeling complex structures, we also need to define 2D membranes in plane stress [222, 227] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Ibrahimbegovic and R.-A. Mejia-Nava, Structural Engineering, Lecture Notes in Applied and Computational Mechanics 100, https://doi.org/10.1007/978-3-031-23592-4_5

211

212

5 Solids, Membranes and Shells with Drilling Rotations: Complex Structures

and the 3D solids [196, 203] that fit together within the same framework with the classical continuum elements. Some practical examples where the connection of two media is needed are infilled frames and the connected shear walls, where the membrane and beam elements interact, folded plates or cellular structures (box-girders), where the plate and membrane elements interact, as well as the connections of typical frame structures with the foundation models in the studies of structure-foundation interaction, where the beam element interacts with the membrane for two-dimensional or with the solid elements for three-dimensional analysis. In short, the main difficulty concerns that the kinematics of the classical continuum is described in a different manner than the kinematics of the oriented medium, the latter being enriched with the presence of the so-called directors and independent rotation field. The difference encountered in the continuum formulation carries over to the discrete case, i.e. to the finite element implementations, as well. Namely, if one uses a standard formulation of membrane elements (e.g., see [26, 153]) together with the standard formulation of beam and plate elements [26, 153], the compatibility of two kinds of elements is impossible to establish. Of course, one must fulfill the compatibility requirement in order to ensure the uniform convergence [26, 153]. Rightly so, the violation of the compatibility requirement is declared to be the ‘variational crime’ in the mathematics community [360]. In the linear analysis the violation of the compatibility requirement [386] might cause some serious errors, as demonstrated in [32]. In the nonlinear analysis, the violation of the compatibility leads to even larger problems. The problem caused by the lack of compatibility is not appreciated enough in the engineering community, since comparative studies in solving practical problems are seldom done. In many commercial finite element codes, nothing is done to circumvent this problem. In some, however, the special transition elements (e.g., see [26], p. 250) are used. The transition elements are based on clever, but empirical tricks of the finite element technology and, as such, they are not a satisfying solution for modeling the connections of two media, especially since such places are typically exposed to very higher stress gradients. The main idea we elaborate upon here is to unify the modeling of different structural subassemblies via consistent use of oriented media-type formulation. In other words, we try to solve the incompatibility problems at the continuum level, rather than constructing some empirical interpolations of transition elements. This approach is in sharp contrast to a series of early attempts carried out in constructing a membrane element with rotational degrees of freedom (e.g., see [5, 50, 71, 88, 237, 275]), which utilize some clever tricks of the finite element technology to obtain satisfying finite element performance. Here, the membrane and the three-dimensional continuum are built starting with the variational formulation for a continuum with independent rotation fields. Such an approach provides a sound theoretical basis for constructing continuum elements with rotational degrees of freedom, which are fully compatible with the Cosserat continuum that was used as starting point for constructing the models for beams, plates and shells (e.g. [294]).

5.1 Solids with Drilling Rotations: Variational Formulation

213

The first paper in which this kind of formulation is discussed appears to be written by Reissner [321]. The same results were independently obtained by Naghdi [293]. Reissner’s formulation was then revisited in a paper by Hughes and Brezzi [154] on membrane problems and the addition of so-called ‘drilling’ degrees of freedom, i.e. the rotational degrees of freedom perpendicular to the membrane plane. In that paper, a modification of Reissner’s variational principle is suggested in order to preserve the stability of the discrete formulation and its finite element implementation. Here, we further reconsider the formulation of Hughes and Brezzi [154] within the context of the three-dimensional continuum. The proposed model of the continuum with an independent rotation field provides the same richness in the structure of kinematics as the oriented continuum. Different possibilities for constructing a variational principle governing this formulation are discussed in order to provide the starting point for the discrete formulation and the finite element implementation. The details of finite element implementation are discussed in the next section. At the end of these introductory remarks, it should be stated that the formulation set in this chapter provides a unified modeling approach to different kinds of finite elements that allows placing them in a single computer code having six degrees of freedom per node, three translations and three rotations. This is especially advantageous from the standpoint of standard architecture of a finite element code for structural engineering, which typically carries six degrees of freedom per node. This is of special interest for shell elements, presented in the last section of this chapter. The classical shell problem formulation (e.g., see [294]) considers locally defined five degrees of freedom (drilling rotation is absent), which needs to be mapped into the six global degrees of freedom for the shell of arbitrary geometry, taking care if possible, to avoid any singularities in that mapping [336, 337]. In the present formulation, we introduce the sixth degree of freedom at the local level, i.e. getting it ’for free’, since the neither the total number of unknown nodal parameters for the shell of arbitrary geometry nor the computational effort is increased. In addition, the presence of the sixth degree of freedom is especially beneficial for a nearly co-planar shell geometry, where it prevents ill-conditioning or singularity from which it suffers the classical shell formulation in numerical computations [336, 337]. No ad-hoc devices to cure this singularity (e.g., see [153], p. 404) are needed in the work presented herein.

5.1.1 Strong Form of the Boundary Value Problem For the model problem studied in this section, we choose a three-dimensional continuum with linearized kinematics (small displacement gradient theory). We will also take linear elastic constitutive response. However, there is nothing in the basic theory to prevent a general form of the constitutive equations (e.g. nonlinear elasticity, plasticity etc.), apart from the assumption of the existence of the strain energy function. In addition, since the inertial effect is not pertinent to the considerations to follow, we limit ourselves to statics problems.

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5 Solids, Membranes and Shells with Drilling Rotations: Complex Structures

We consider the standard boundary value problem first. We will write all local equations of the strong form for the 3D problem in elasticity by using index notation, with Latin letters denoting the indices. Standard summation convention on repeated indices is applied throughout, with indices varying in the set {1, 2, 3}. Let Ω ⊂ R3 be the region occupied by the body. Then, ∀ x = (xi ) ∈ Ω, it holds σi j, j + f i = 0, i j =

1 (u i, j + u j,i ) ≡ symmu i, j 2

(5.1) (5.2)

and σi j = Ci jkl kl

(5.3)

where (5.1)–(5.3) are, respectively, the equilibrium equations, the definition of the strain tensor as the symmetric part of the deformation gradient and the constitutive equations. Next to index notation, we also use abbreviation with ’comma’ denoting the partial derivatives, such as σi j, j ≡ ∂σi j /∂ x j . For a nonlinear elastic material, the constitutive equations as in (5.3) can still be considered, but only applied in incremental form, i.e. with elasticity tensor relating the increments of stress and strain. We assume the positive definiteness of the elasticity tensor, i.e. (5.4) i j Ci jkl kl > 0 , ∀ i j =  ji = 0 as a provision for the solution’s existence and uniqueness, provided a sufficient number of supports for removing the rigid body modes. In addition, as a consequence of the assumed existence of the strain energy function, the symmetry of the constitutive modulus follows as well with Ci jkl = Ckli j . For simplicity, let us consider the Dirichlet boundary value problem with homogeneous boundary conditions. Thus, it holds that displacements are zero at each point of the domain boundary (5.5) u i |Γ = 0 ; ∀x ∈ ∂Ω ≡ Γ Different boundary conditions present no difficulties for the considerations to follow and they can be handled in a standard way (e.g., see [153, 176, 379]). At this point, we depart from the standard boundary value problem of the classical theory of continuum mechanics by taking the stress tensor σi j as non-symmetric. In the standard boundary value problem, the symmetry of the stress tensor is implicit both in the equilibrium equations (5.1) and in the constitutive equations (5.3). This implies that besides the strain tensor i j , defined by (5.2) as the symmetric part of the deformation gradient, we have to consider the skew-symmetric part of the displacement gradient representing the skew-symmetric tensor of infinitesimal rotation ψi j . The basic kinematic split of the displacement gradient into symmetric and skew-symmetric parts is defined

5.1 Solids with Drilling Rotations: Variational Formulation

u i, j = i j + ψi j

215

(5.6)

In other words, the Euclidean decomposition of the second order tensors is employed for the displacement gradient, where we further define the infinitesimal strain i j as the symmetric part and the infinitesimal rotation ψi j as the skewsymmetric part u i, j = symmu i, j + skewu i, j i j = symmu i, j := 21 (u i, j + u j,i ) (5.7) ψi j = skewu i, j := 21 (u i, j − u j,i ) In the same way, we split non-symmetric stress tensor σi j into its symmetric and skew-symmetric parts, σi j = symmσi j + skewσi j symmσi j := 21 (σi j + σ ji ) (5.8) skewσi j := 21 (σi j − σ ji ) where the symmetric part is work-conjugate to the infinitesimal strain and the skewsymmetric part is work-conjugate to the infinitesimal rotation. The strong form of the boundary value problem with non-symmetric stress tensor can then be written as σi j, j + f i = 0 (5.9) skewσi j = 0

(5.10)

ψi j = skewu i, j

(5.11)

symmσi j = Ci jkl symmu k,l

(5.12)

and

where Eqs. (5.9)–(5.12) stand for, respectively, the balance of linear and angular momentum, the definition of the independent infinitesimal rotation field and the constitutive equations. Note that in (5.12) we have directly introduced the definition of the strain tensor in (5.2), thus reducing the number of unknowns. Our goal is construct a variational equation with the Euler-Lagrange equations corresponding to Eqs. (5.9)–(5.12). This is presented in the next section.

5.1.2 Variational Formulation, Stability Analysis and Regularization Variational formulation of the boundary value problem defined by (5.9)–(5.12) with the boundary conditions (5.5) is given by Reissner [321] in the form

216

5 Solids, Membranes and Shells with Drilling Rotations: Complex Structures

 Π (u¯ i , ¯i j , σ¯ i j ) = − 21  +

symm σ¯ i j Ci−1 ¯ kl dΩ jkl symm σ 

Ω

σ¯ i j ¯i j dΩ − Ω

u¯ i f i dΩ

(5.13)

Ω

and via the use of the kinematic relationship (5.6) it can be transformed into the equivalent form [154]  Π (u¯ i , ψ¯ i j , σ¯ i j ) = − 21 symm σ¯ i j Ci−1 ¯ kl dΩ jkl symm σ   Ω + σ¯ i j (u¯ i, j − ψ¯ i j )dΩ − u¯ i f i dΩ Ω

(5.14)

Ω

where σ¯ i j ∈ S, u¯ i ∈ U and ψ¯ i j ∈ W are trial variables. The variational statement (5.14) requires that the stress σ¯ i j and the infinitesimal rotation tensor ψ¯ i j both belong to the space of square integrable functions over the region Ω, further denoted as L 2 (Ω). It holds that (5.15) S = {σ¯ i j | σ¯ i j ∈ L 2 (Ω)} and

W = {ψ¯ i j | ψ¯ i j ∈ L 2 (Ω); symm ψ¯ i j = 0}

(5.16)

The space of trial displacements, however, must be more regular and belong to the function space with derivatives in L 2 (Ω), i.e. to the Sobolev space H 1 (Ω). In addition, the space of trial displacements must satisfy the zero boundary conditions on ∂Ω. Thus it belongs to the subset of H 1 (Ω) denoted as H01 (Ω) U = {u¯ i | u¯ i ∈ H01 (Ω)}

(5.17)

In the finite element context, that means that the displacements must be continuous across the element boundaries. The rotations and the stresses need not. By taking the variations of (5.14) with respect to trial variables, we get 

 u¯ i, j σi j dΩ − Ω

u¯ i f i dΩ = 0

(5.18)

Ω

 −

ψ¯ i j σi j dΩ = 0

(5.19)

Ω

 − Ω

symm σ¯ i j Ci−1 jkl symmσkl dΩ

 +

σ¯ i j (u i, j − ψi j )dΩ = 0

(5.20)

Ω

Using the Gauss divergence theorem for the Dirichlet boundary value problem with the homogeneous boundary conditions, we get

5.1 Solids with Drilling Rotations: Variational Formulation

217



 u¯ i, j σi j dΩ = − Ω

u¯ i σi j, j dΩ

(5.21)

Ω

If we introduce the Euclidean decomposition of the displacement gradient and use the result in (5.21) above, the Euler-Lagrange equations (5.18)–(5.20) can be rewritten as  (5.22) − u¯ i (σi j, j + f i )dΩ = 0 Ω



ψ¯ i j skewσi j dΩ = 0

(5.23)

symm σ¯ i j (Ci−1 jkl symmσkl − symmu i, j )dΩ

(5.24)

− Ω

 − Ω

 σ¯ i j (skewu i, j − ψi j )dΩ = 0

+

(5.25)

Ω

The Euler-Lagrange equations (5.22) and (5.23) will allow recovering the linear and angular momentum balance equations, written in (5.9) and (5.10), respectively. Similarly, the Euler-Lagrange equations in (5.25) and (5.24) can be used to recover the constitutive equations (5.12) and the definition of the infinitesimal rotation field (5.11). Note that we obtained equation (5.23) from (5.19) simply by using the wellknown result [142, 176] that the scalar product of the symmetric and the skewsymmetric tensor is zero. The variational equation (5.13) is the starting point in the stability analysis performed in the next section. However, before we proceed in that direction, we want to present another form of the variational principle [154, 321], which can also serve as a starting point for constructing the discrete approximation. Namely, a Hu-Washizu [379] type of functional can be formed by introducing the definition of the infinitesimal strain tensor (5.2) as an additional Euler-Lagrange equation. The appropriate functional is given as  Π (u¯ i , ψ¯ i j , ¯i j , symm σ¯ i j , skewσ¯ i j ) = 21 symm σ¯ i j Ci−1 jkl symm σ¯ kl dΩ   Ω  + symm σ¯ i j (symm u¯ i, j − ¯i j )dΩ + skewσ¯ i j (skewu¯ i, j − ψ¯ i j )dΩ − u¯ i f i dΩ Ω

Ω

Ω

(5.26)

The variational formulation (5.26) is useful in providing the sound theoretical basis [225] for the method of incompatible modes [386], thus eliminating the adhoc correction introduced previously [362]. The incompatible modes interpolation is helpful in alleviating different locking tendencies which can occur within the

218

5 Solids, Membranes and Shells with Drilling Rotations: Complex Structures

H

H d6

 n H

H
HH H HH H HH H H H H H H HH H H H H HH H

d≡n

6

d = director n = normal

welded

clamped

Fig. 5.1 Boundary conditions for Cosserat rod ψ1 = 0

s H H HH HH H HHψ2 s s H H HH HH H HH s welded

ψ1 = 0

s

=0

ψ 2 = 0

s

s

ψ = rotation

s cl a m p e d

Fig. 5.2 Boundary conditions for solid with independent rotation field

present formulation, e.g. locking for nearly incompressible materials, Poisson’s ratio stiffening etc. We will elaborate on using the incompatible modes in the second part of this section on finite element interpolation. In passing, we want to point out the common features of the present formulation of the continuum with the independent rotation fields to the oriented (or directed) media formulation of Cosserat continuum [294]. First, within the context of the discrete approximation, the independent rotation field can be brought into the direct correspondence with the rotation of the directors in the Cosserat continuum, thus providing a compatible interpolation at the contact of the finite elements of these two kinds. Second, the richness of the structure of the kinematics allows capturing the subtle differences in the boundary conditions. For example, for the Cosserat rod [7] the difference between the welded and clamped end is presented in Fig. 5.1. If we choose the hierarchical interpolation [222] for the solid with independent rotation field in the four-node membrane element, then the corresponding boundary conditions to those in Fig. 5.1 are given in Fig. 5.2

5.1 Solids with Drilling Rotations: Variational Formulation

5.1.2.1

219

Stability Analysis and Regularization

The variational problem described by Eqs. (5.18)–(5.20) can be put in the operator form by introducing two bilinear forms  a(σ¯ , σ ) = −

symm σ¯ i j Ci−1 jkl symmσi j dΩ,

Ω

(5.27)



b(σ¯ , {u, ψ}) =

σ¯ i j (u i, j − ψi j )dΩ

(5.28)

Ω



and a linear form ¯ = f (u)

u¯ i f i dΩ

(5.29)

Ω

to get

and

a(σ¯ , σ ) + b(σ¯ , {u, ψ}) = 0 ; ∀σ¯ i j ∈ S

(5.30)

¯ = f (u) ¯ ψ}) ¯ ; ∀u¯ i ∈ U, ψ¯ i j ∈ W b(σ , {u,

(5.31)

The symmetry of the constitutive modulus C implies that the bilinear form a(·, ·) is symmetric. We define the norm induced by Poincare’s inequality  u

U2 =

u i, j  = 2

u i, j u i, j dΩ ; ∀u i ∈ U

(5.32)

ψi j ψi j dΩ ; ∀ψi j ∈ W

(5.33)

σi j σi j dΩ ; ∀σi j ∈ S

(5.34)

Ω

and L 2 (Ω) norms  ψ

2W =

ψi j  = 2

Ω 

 σ 2S = σi j 2 = Ω

From (5.32) and (5.33) we also define  {u, ψ} 2V = u U2 +  ψ 2W ; ∀ u i ∈ U, ψi j ∈ W

(5.35)

Problem defined by Eqs. (5.30) and (5.31) belongs to the class of mixed methods. Its existence and uniqueness are established by the conditions identified by Babuška [18] and Brezzi [62] in the form

220

5 Solids, Membranes and Shells with Drilling Rotations: Complex Structures

(i)

| a(σ¯ , σ ) |≤ c1  σ¯  S  σ  S ; ∀σ¯ i j , σi j ∈ S | b(σ¯ , {u, ψ}) |≤ c2  σ¯  S  {u, ψ} V ; ∀σ¯ i j ∈ S, ∀{u i , ψi j } ∈ V

(ii)

(5.36) (5.37)

| a(σ¯ , σ¯ ) |≥ α  σ¯ 2S ; ∀σ¯ i j ∈ K

(5.38)

K = {σ¯ i j ∈ S | b(σ¯ , {u, ψ}) = 0 ; ∀{u i j , ψi j } ∈ V }

(5.39)

| b(σ¯ , {u, ψ}) | ≥ β  {u, ψ} V ; ∀{u i , ψi j } ∈ V  σ¯  S

(5.40)

where

and (iii) sup |σ¯ ∈S

In the continuous case, it can be shown [154] that all the conditions (i) to (iii) are satisfied. The key observation is the property of the constrained set K which guarantees that the skew-symmetric part of the stress tensor is zero  K = {σ¯ i j ∈ S | skew σ¯ i j = 0,

σ¯ i j u i, j dΩ = 0 ; ∀u i ∈ U }

(5.41)

Ω

Let us now consider the discrete approximation. We denote U h , W h and S h to be, respectively, the finite-dimensional subspaces of U , W and S, constructed by the finite element method. The superscript h is used to denote and distinguish the variables in the discrete approximation. The discrete formulation of the problem defined in (5.30) and (5.31) now becomes

and

a(σ¯ h , σ h ) + b(σ¯ h , {uh , ψ h }) = 0 ; ∀σ¯ ihj ∈ S h

(5.42)

b(σ h , {u¯ h , ψ¯ h }) = f (u¯ h ) ; ∀u¯ ih ∈ U h , ψ¯ ihj ∈ W h

(5.43)

The existence and the uniqueness of the discrete mixed problem above is guaranteed by the conditions which are analogous to (5.36)–(5.40), which can be written as (i h ) (5.44) | a(σ¯ h , σ h ) |≤ c1  σ¯ h  S  σ h  S ; ∀σ¯ ihj , σihj ∈ S h | b(σ¯ h , {uh , ψ h }) |≤ c2  σ¯ h  S  {uh , ψ h } V ; ∀σ¯ ihj ∈ S h , ∀{u ih , ψihj } ∈ V h (5.45) (ii h ) (5.46) | a(σ¯ h , σ¯ h ) |≥ α  σ¯ h 2S ; ∀ψ¯ ihj ∈ K h

5.1 Solids with Drilling Rotations: Variational Formulation

221

where K h = {σ¯ ihj ∈ S h | b(σ¯ h , {uh , ψ h }) = 0 ; ∀{u ihj , ψihj } ∈ V h }

(5.47)

and (iii h ) sup |σ¯ h ∈S h

| b(σ¯ h , {uh , ψ h }) | ≥ β  {uh , ψ h } hV ; ∀{u ih , ψihj } ∈ V h  σ¯ h  S

(5.48)

We want to establish that the conditions (i h ) to (iii h ) are satisfied for an arbitrary choice of finite element interpolation. The simplest choice that seems natural to start with (e.g. see [154, 155]) is the same order continuous polynomials for the displacements and the rotations, together with the discontinuous stress which can then be eliminated at the element level. To establish Babuška-Brezzi condition (iii h ), we choose the stress tensor given as

and

symm σ¯ ihj = symm u¯ i,h j

(5.49)

skew σ¯ ihj = −ck ψ¯ ihj

(5.50)

where ck is the constant in Korn’s inequality (see [141], pp. 38–39). For the Dirichlet boundary value problem under consideration ck = 1/2. For a more complicated choice for the boundary conditions, the estimate of ck by using Korn’s inequality becomes more involved [122]. If we now try to establish the condition (iii h ) for the same choice of stress as in (5.49) and (5.50), we get only | a(σ¯ h , σ¯ h ) |≤ c1  symm σ¯ h 2S ; ∀σ¯ ihj ∈ K h ,

(5.51)

We recall that in the discrete approximation the discrete set K h is given by K h = { σ¯ ihj ∈ S h |

 Ωh

skew σ¯ ihj ψihj dΩ = 0,

∀ u ih ∈ U h , ∀ψihj ∈ W h }

 Ωh

σ¯ ihj u i,h j dΩ = 0 ;

(5.52)

Therefore, the space of continuous polynomials for ψ¯ ihj is much smaller than the space of the discontinuous polynomials for σ¯ ihj and it does not provide the control against the occurrence of some non-trivial skew-symmetric stress members. This is in sharp contrast with the continuous problem; see Eq. (5.41). In order to establish the condition (iii h ), Hughes and Brezzi [154] have suggested the regularization of the functional (5.14) by adding an extra term

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5 Solids, Membranes and Shells with Drilling Rotations: Complex Structures

Πγ (u¯ i , ψ¯ i j , σ¯ i j ) := Π (u¯ i , ψ¯ i j , σ¯ i j ) −

1 2



γ −1 skew σ¯ i j skew σ¯ i j dΩ

(5.53)

Ω

This regularization is reflected by the change of the bilinear form a(·, ·) into  aγ (σ¯ , σ ) := a(σ¯ , σ ) −

γ −1 skew σ¯ i j skewσi j dΩ

(5.54)

Ω

and the condition (ii h ) can now be established for any member σ¯ ihj ∈ S h and not only those σihj ∈ K h . It holds that | aγ (σ¯ h , σ¯ h ) | :=| a(σ¯ h , σ¯ h ) | +γ −1  skew σ¯ h 2 ≥ α  symm σ¯ h 2 +γ −1  skew σ¯ h 2 = min{α, γ −1 }  σ¯ h  S ; ∀σ¯ i j ∈ S h

(5.55)

The regularized variational formulation of Reissner takes the form Πγ (u¯ i , ψ¯ i j , σ¯ i j ) = − 21 − 21

 Ω



symm σ¯ i j Ci−1 ¯ kl dΩ + jkl symm σ 

Ω

γ

−1

skew σ¯ i j skew σ¯ i j dΩ −



σ¯ i j (u¯ i, j − ψ¯ i j )dΩ

Ω

u¯ i f i dΩ Ω

(5.56) Taking the variations with respect to the trial variables and using (5.21) again, we get the Euler-Lagrange equations in the form  −

u¯ i (σi j, j + f i )dΩ = 0

(5.57)

ψ¯ i j skewσi j dΩ = 0

(5.58)

Ω

 − Ω

 − symm σ¯ i j (Ci−1 jkl symmσkl − symmu i, j )dΩ Ω  + σ¯ i j (skewu i, j − ψi j − γ −1 skewσi j )dΩ = 0

(5.59)

Ω

By (5.58) we still have skewσi j = 0, so that the Euler-Lagrange equations are the same as in given by (5.22)–(5.25). However, we also get an additional relationship from (5.59)2 above (5.60) skewσi j = γ (skewu i, j − ψi j ) which can eventually be used to eliminate skewσi j in the variational equations.

5.1 Solids with Drilling Rotations: Variational Formulation

223

5.1.3 Alternative Variational Formulations, Extension to Nonlinear Kinematics and Closing Remarks By using the Euler-Lagrange equations, we can eliminate chosen trial variables in the variational formulation (5.56) to get various alternative forms useful in applications. First, we eliminate the symmetric part of the stress tensor, symmσi j , via the use of the constitutive equations (5.59)1 to get Mixed-type variational formulation Πγ (u¯ i , ψ¯ i j , skew σ¯ i j ) =



1 2

 Ω

symm u¯ i, j Ci jkl symm u¯ k,l dΩ

+ skew σ¯ i j (skew u¯ i, j − ψ¯ i j )dΩ Ω  − 21 γ −1 skew σ¯ i j skew σ¯ i j dΩ − u¯ i f i dΩ Ω

(5.61)

Ω

By taking variations with respect to different trial variables, we find 

 symm u¯ i, j Ci jkl symmu k,l dΩ + Ω

 −

 skew u¯ i, j skewσi j dΩ −

Ω

u¯ i f i dΩ = 0 Ω

ψ¯ i j skewσi j dΩ = 0

(5.62) (5.63)

Ω



skew σ¯ i j (skewu i, j − ψi j − γ −1 skewσi j )dΩ = 0

(5.64)

Ω

Equations (5.62)–(5.64) serve as the starting point of the mixed-type discrete formulation considered in the second part of this chapter. As opposed to the Reissner’s variational formulation in the regularized variational formulation (5.56) we can also eliminate the skew-symmetric part of the stress tensor, skewσi j , via use of equation (5.60). If we do that starting with the variational formulation (5.61), we get Displacement-type variational formulation Πγ (u¯ i , ψ¯ i j ) =

1 2



Ω 1

symm u¯ i, j Ci jkl symm u¯ k,l dΩ

γ (skew u¯ i, j − ψ¯ i j )(skewu i, j − ψi j )dΩ 2 Ω − u¯ i f i dΩ +

(5.65)

Ω

We call the variational formulation (5.65) displacement-type just for distinction from other alternatives. Strictly speaking, kinematic variables are used, both the displacement u i and the rotations ψi j .

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5 Solids, Membranes and Shells with Drilling Rotations: Complex Structures

By taking the variations with respect to the trial variables in (5.65), we get 

 symm u¯ i, j Ci jkl symmu k,l dΩ +

Ω

 γ skew u¯ i, j (skewu i, j − ψi j )dΩ −

Ω

 −

u¯ i f i dΩ = 0 Ω

γ ψ¯ i j (skewu i, j − ψi j )dΩ = 0

(5.66) (5.67)

Ω

The variational equations (5.66) and (5.67) serve as a starting point for the displacement-type discrete formulation considered in the second part of this chapter. Finally, if one intends to incorporate the incompatible modes [225], the starting point is provided by regularizing the Hu-Washizu-type variational formulation given by (5.26). An optimal choice for the regularization parameter γ can be given for the isotropic elasticity where i j Ci jkl kl (5.68) 2μ = min |i j =0 i j i j and μ = E/2(1 + ν) is the shear modulus. For the case of the Dirichlet boundary value problem with homogeneous boundary conditions, where the constant in Korn’s inequality ck = 1/2, it is recommended in [154] to take optimal choice for γ = μ. However, the stability of the discrete formulation is guaranteed for any positive value of γ . Moreover, the computations performed in [222] show that the formulation is not very sensitive to the choice of γ even for other kinds of boundary conditions if we use the optimal discrete approximation with hierarchical displacement interpolations.

5.1.3.1

Equivalent Variational Formulations for Nonlinear Kinematics

In this section, we discuss the variational formulations for the continuum with an independent rotation field in geometrically nonlinear theory. The starting point in our considerations can be provided by the classical potential energy principle for finite displacements proposed by Fraeijs de Veubeke (e.g., see [124, 166]) in the following format   W (H(u)) dΩ − u · f dΩ , (5.69) Π (u) = Ω

Ω

where W (H(u)) is a stored energy given as a function of a chosen finite strain measure proposed by Biot [52], separating displacements and rotations. Throughout this section an index-free tensor notation is utilized, such as found in modern expositions on finite elasticity (e.g. see [176]), with index notation supplied only to clarify the adopted convention. So, for example, the second integral in the last expression represents the external work for the Dirichlet boundary value problem we are considering

5.1 Solids with Drilling Rotations: Variational Formulation

225

herein, which can now be written u · f = u i fi ,

(5.70)

denoting the inner product of displacement vector u and body force f, where the standard summation convention on repeated indices is implied. The finite strain measure H, often called Biot strain (see [52]), can be the most explicitly defined via the polar decomposition theorem (e.g., see [176]). Namely, if the deformation is a vector field ϕ, which is, without loss of generality for our purposes, specified with respect to the Euclidean coordinate system with the base vectors ei , i.e. if (5.71) xϕ = ϕ(x) ; x = xi ei ; ϕ = ϕi ei , then the deformation gradient can be written as F = ∇ϕ ; F =

∂ϕi ei ⊗ e j , ∂x j

(5.72)

where ⊗ denotes the tensor product. Introducing the displacement vector field: u = ϕ(x) − x, the deformation gradient can be rewritten in terms of the displacement gradient ∇u as F = ∇ϕ = I + ∇u ; ∇u =

∂u i ei ⊗ e j ; I = δi j ei ⊗ e j . ∂x j

(5.73)

The polar decomposition theorem states that the deformation gradient can be factored in a unique way into the orthogonal tensor R and symmetric positive-definite tensor, called a right stretch tensor, U F = RU ,

(5.74)

where U describes the deformation, while R describes rotation. The Biot strain tensor is defined with H =U−I. (5.75) By using the results in (5.73)–(5.75), the polar decomposition theorem can be rewritten as I + ∇u = R(H + I) . (5.76) By eliminating the rotation field in (5.76) via orthogonality of R, we get a functional relationship between H and u 1 H + H2 = ∇u + (∇u)T + (∇u)T ∇u , 2

(5.77)

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5 Solids, Membranes and Shells with Drilling Rotations: Complex Structures

which is needed in (5.69). We note that the polar decomposition in (5.76) is recovered as the Euler-Lagrange equation of a new variational formulation rather than having it as a subsidiary condition of the variational formulation in (5.69). The Lagrange multiplier procedure (e.g. see [176]) can be used to impose that condition in the form  P · [(I + ∇u) − R(I + H)] dΩ ,

PF =

(5.78)

Ω

where P is the Lagrange multiplier and · denotes a natural inner product of two second-order tensors (e.g., see [176]). For clarity, if A and B are two second order-tensors, then their inner product is A · B = trace(AT B) = Ai j Bi j .

(5.79)

That Lagrange multiplier P is actually the non-symmetric Piola-Kirchhoff stress tensor, which follows directly from a result on energy conjugate pairs (e.g., see [176]). The non-symmetric Piola-Kirchhoff tensor P is related to the Cauchy (true) stress tensor σ via Piola transform P = J σ F−T ; J = detF .

(5.80)

If one is to introduce the pull-back of P performed with the rotation part R of the deformation gradient, we obtain the new stress tensor T = R T P = J R T σ F−T ,

(5.81)

which is also non-symmetric. By using the properties of the inner product and result in (5.81), we can rewrite the result in (5.78) in the form  (5.82) P F = T · [R T (I + ∇u) − (I + H)] dΩ . Ω

The weak form of the polar decomposition in (5.82) can be added to the variational formulation in (5.69) to get a general Hu-Washizu type variational formulation which is valid for geometrically nonlinear theory 

 {W (H) − T · H + T · [R T (I + ∇u) − I)]} dΩ −

Π (u, R, T, H) = Ω

u · f dΩ . (5.83) Ω

The associated variational equations and the Euler-Lagrange equations can be obtained by taking the directional derivative (see [176]) in the direction of virtual

5.1 Solids with Drilling Rotations: Variational Formulation

227

displacements, virtual rotations, virtual stresses and virtual strains, in order to recover, respectively, force equilibrium equation, moment equilibrium equation, kinematics and constitutive equation. The work of Fraeijs de Veubeke [124] and most of the works related to it (e.g. see [246, 323]) have concentrated on establishing the validity of the complementary energy principle in the regime of large displacements. Therein, the crucial assumption is that the complementary energy Σ(symmT) exists and that it can be recovered by the Legendre transform − H · symmT + W (H) = −Σ(symmT) .

(5.84)

where we have introduced the Euclidean decomposition for a second order tensor into its symmetric and skew-symmetric part for the Biot stress tensor T with T = symmT + skewT ; symmT :=

1 1 (T + TT ) ; skewT := (T − TT ) . (5.85) 2 2

By introducing the transformation in (5.84) into the variational formulation in (5.83), we can eliminate the strain field H to get the three-field variational formulation 

 {−Σ(symmT) + T · [R T (I + ∇u) − I]} dΩ −

Π (u, R, T) = Ω

u · f dΩ . Ω

(5.86) The variational equations and Euler-Lagrange equations of linear and angular momentum balance remain preserved, while the definition of strains and the constitutive equations give rise to a new variational equation, which connects directly the stresses with the displacement gradient and rotations In the work of Fraeijs de Veubeke, an additional step is taken to eliminate the displacement field via picking up the stress field which satisfies linear momentum balance equations, so that in the final form of the complementary energy principle only stress and rotation fields are retained. We do not follow this procedure to devise the variational principles, because the absence of the displacement field can be undesirable in the solution to a practical problem. In the remaining part of this section, we want to return again to geometrically linear theory. For that purpose, we simply use the consistent linearization of the preceding results around the reference configuration, which is assumed free of initial stress and initial deformation. For example, by using the results [10, 205], we can linearize an orthogonal tensor to get the corresponding infinitesimal rotation tensor , which is skew-symmetric lin.

R −→ I +  ; T = − .

(5.87)

From the finite strain measure H we recover a standard definition of infinitesimal strain as the symmetric part of displacement gradient

228

5 Solids, Membranes and Shells with Drilling Rotations: Complex Structures lin.

H −→  =

1 (∇u + (∇u)T ) , 2

(5.88)

and from the result in (5.87) we recover the definition of infinitesimal rotation field  as the skew-symmetric part of the deformation gradient =

1 (∇u − (∇u)T ) . 2

(5.89)

The stress tensor is assumed to remain non-symmetric lin.

T −→ σ ; σ T = σ .

(5.90)

If these results are introduced into the variational formulation in (5.86), and higher order terms are neglected, we get the final product of the consistent linearization that can be written in tensor notation as   Π (u, , σ ) = {−Σ(symmσ ) + σ · (∇u − )} dΩ − u · f dΩ . (5.91) Ω

Ω

It is pleasing to recover the variational principles (5.91), which appeared in earlier developments in geometrically linear theory (see [154, 222]), for this provides the basis for proposing the regularized forms of the variational principles that we discussed in the previous section. In closing this section, we give the results obtained in nonlinear computations of a cantilever under end moment [166]. This is an illustrative test problem for large rotation analysis and ability of the element to carry the concentrated moment loading. According to the classical Euler formula, a cantilever beam with Young’s modulus E = 1200, Poisson’s ratio ν = 0, length l = 10 and a square cross-section of unit area, under a bending moment M = 10π will bend in the form of half-a-circle. Figure 5.3 displays the computed response for a model with a very coarse mesh of five solid elements with rotational degrees of freedom, each of which of the same size in the reference configuration. This type of formulation can further be extended to the so-called micropolar continuum defined in early works by Cosserat brothers [90], and implemented both in geometrically linear [136] and nonlinear framework [137]. An illustrative example is given from our recent work [137], with large displacements and rotations in the corkscrew motion of a T-shape bottle opener; see Fig. 5.4.

5.1.3.2

Closing Remarks

We have presented a general methodology for the development of a three-dimensional continuum element with an independent rotation field. The present considerations can be easily specialized to two-dimensional cases. This enables the continuum

5.1 Solids with Drilling Rotations: Variational Formulation

229

Fig. 5.3 Deformed configuration of the cantilever beam under end moment

Fig. 5.4 Deformed configuration of T-shaped cantilever beam obtained for different load steps in corkscrew motion

230

5 Solids, Membranes and Shells with Drilling Rotations: Complex Structures

elements (e.g. membrane, brick etc.) to be fully compatible with the beams, plates and shells. It also provides a unified treatment for all subassemblies of the arbitrarily complex structural systems which occur frequently in civil, mechanical or aerospace engineering practice. In addition, the development of the shell element with six degrees of freedom can be accomplished based on the presented considerations. The methodology discussed herein is in sharp contrast to the widespread use of ad-hoc engineering approaches, which are used in attempts to solve this problem. The sound theoretical basis in this work is provided by the variational principle of Reissner [321] and its regularized form of Hughes and Brezzi [154]. We briefly illustrated that such variational formulation can be obtained by the consistent linearization of the variational formulation of a three-dimensional continuum with an independent rotation field proposed in [124, 166] for large displacements and rotations.

5.2 Membranes with Drilling Rotations: Discrete Approximation In this section, we discuss the discrete formulation of the variational problem for the three-dimensional continuum with independent rotation fields presented in the previous section. We assume the plane stress condition holds so that the corresponding formulation simplifies accordingly. Such a problem is of great interest for the finite element analysis of shells using the elements with six degrees of freedom per node. Namely, a versatile tool for the analysis of shells can be obtained by superposing the membrane elements discussed in the foregoing and the plate elements discussed by Ibrahimbegovic [164]. However, the membrane problem of this kind is of interest in its own right when, for example, one considers structures like in-filled frames, box girders, folded plates etc. There has been been a lot of interest in constructing the membrane elements with rotational degrees of freedom, for their ability to easily construct the finite element model of a complex structure. The initial efforts have followed a couple of independently proposed approaches of Allman [5] and Bergan and Fellipa [50]. See, for example, [6, 71, 88, 237] or [275]. However, most of the proposed formulations were based on some clever tricks of finite element technology, rather than a sound variational foundation. This has the undesirable consequences of spurious modes occurrence, need of adjusted parameters, lack of physical interpretation for the rotation field etc. The sound variational framework for this class of problems is proposed later by Hughes and Brezzi [154]. It relies on the regularized variational formulation initially given by Reissner [321] (and similar work given in [293]) for the continuum with an independent rotation field. Follow up work by Hughes et al. [155] have also proposed several membrane elements with equal order interpolations for the displacement and the independent rotation fields. However, those results remain highly sensitive to the choice of regularization parameter (see [154]. Hence, the final truly satisfying solution needed special interpolations, which are first proposed in [222]). This is the path we follow in the developments presented further.

5.2 Membranes with Drilling Rotations: Discrete Approximation

231

Namely, here we take the same regularized variational formulation of the continuum with independent rotation field as a starting point of the discrete formulation, but construct rather different finite element subspaces. A special hierarchical displacement interpolation is used to enhance the element performance. An additional very important motivation for introducing the displacement interpolation of this kind is in providing the benefits of full compatibility with the typical beam and plate finite elements. For example, the membrane element with quadratic displacement variation is compatible with the Timoshenko beam and the Reissner-Mindlin plate element (see [164]). We want to emphasize that the membrane elements presented herein are free of any spurious modes and do not need any adjusted parameters. Moreover, the rotation field corresponds to the continuum-mechanics definition of the rotations (the skew-symmetric part of the displacement gradient) and therefore it brings a new quality in the analysis of membranes connected to beams, plates and shells. A couple of new membrane elements with rotational degrees of freedom are presented in this section. M Q2 is a quadrilateral membrane element with quadratic displacement interpolation and M Q3 is a quadrilateral with cubic displacement interpolation. Mid-side tangential displacements are used as the incompatible modes interpolation parameters to improve the performance of the elements in the distorted configuration. Both membrane elements are based on a sound variational foundation and both exhibit an excellent performance over a set of problems. An outline of this section is as follows. First, we give the discrete formulation for membrane problems. Next, we consider, respectively, the membrane elements with quadratic and cubic displacement interpolations. For both elements, the displacement interpolations are enriched with the set of incompatible modes, which are constructed by using the principles of the modified method of incompatible modes of Ibrahimbegovic and Wilson [225]. Finally, we present the evaluation of the elements performance, using the set of problems in elastostatics, including an illustrative problem from standard engineering practice. At the end of the section, we state some closing remarks.

5.2.1 Discrete Approximations with Quadratic and Cubic Displacement Fields For the presentation of the discrete formulation, we make the transition from index notation to the matrix notation suitable for manipulation with discrete quantities. In that respect, we assume that the (x1 , x2 ) plane of the local coordinate system coincides with the membrane plane. For arbitrarily placed membrane, we need to transform the resulting matrices from a local to the global coordinate system using the standard coordinates transformation (e.g., see [153]). The symmetric part of the displacement gradient, denoted as symm u i, j is mapped into the vector

232

5 Solids, Membranes and Shells with Drilling Rotations: Complex Structures

⎤ ∂u 1 /∂ x1 ⎦ → symm∇uh = ⎣ ∂u 2 /∂ x2 ∂u 1 /∂ x2 + ∂u 2 /∂ x1 ⎡

symmu i, j

(5.92)

so that the final result for the scalar product of work-conjugate stress and strain tensors remains preserved in the discrete formulation. In equation (5.92) and in the foregoing, we use superscript h (finite element mesh parameter) to denote discrete approximations. Each skew-symmetric tensor is represented via its axial vector (e.g., see [142]), which has only one non-zero component perpendicular to the membrane plane. Thus skewu i, j → skew∇uh =

1 (∂u 2 /∂ x1 − ∂u 1 /∂ x2 ) 2

(5.93)

ψi j → ψ h

(5.94)

skewσi j → skewσ h

(5.95)

Using (5.92)–(5.95) we can rewrite the discrete form of the mixed-type variational formulation  Πγ (uˆ h , ψˆ h , skew σˆ h ) = 21 (symm∇ uˆ h )T C(symm∇ uˆ h ) dΩ Ωh    h h + skew σˆ (skew∇ uˆ − ψˆ h )dΩ − 21 γ −1 skew σˆ h skew σˆ h dΩ − uˆ h T fdΩ Ωh

Ωh

Ωh

(5.96) The corresponding variational equations can easily be obtained by computing directional derivatives with respect to state variables, in order to obtain 

 (symm∇ uˆ h )T C(symm∇uh ) dΩ +

0= Ωh

 0=−

 skewσ h (skew∇ uˆ h )dΩ −

Ωh

uˆ h T fdΩ

(5.97)

Ωh

skewσ h (ψˆ h )dΩ

(5.98)

Ωh

and  0=

 skew σˆ (skew∇u − ψ )dΩ − h

Ωh

h

h

γ −1 skew σˆ h skewσ h dΩ

(5.99)

Ωh

The discrete approximation for the displacement-type variational formulation can be written as

5.2 Membranes with Drilling Rotations: Discrete Approximation

233

 Πγ (uˆ h , ψˆ h ) = 21 (symm∇ uˆ h )T C(symm∇ uˆ h ) dΩ Ωh   + 21 γ (skew∇ uˆ h − ψˆ h )(skew∇ uˆ h − ψˆ h )dΩ − uˆ h T fdΩ Ωh

(5.100)

Ωh

This reduces the corresponding number of variational equations, which can be written as  0 = (symm∇ uˆ h )T C(symm∇uh ) dΩ   Ωh (5.101) + γ (skew∇ uˆ h )(skew∇uh − ψ h )dΩ − uˆ h T fdΩ Ωh

Ωh



and 0=−

γ ψˆ h (skew∇uh − ψ h )dΩ

(5.102)

Ωh

Within the framework of the discrete approximation, it is possible to establish the equivalence of the mixed-type formulation (5.96) and the displacement-type formulation (5.100) for corresponding choices of the finite-dimensional subspaces for displacements, infinitesimal rotations and the skew-symmetric part of the stress tensor. It is referred to as the ’equivalence theorem’ of Malkus and Hughes [276]. In our previous work (see [222]) the equivalence of the two formulations was slightly disturbed by a bubble function hierarchical interpolation. We have observed therein that the mixed-type formulation was slightly advantageous. Here, we do not use bubble function interpolation (for it offers only an insignificant enhancement), but the incompatible modes. The pertinent consideration of the modified method of incompatible modes for standard isoparametric elements is given by Ibrahimbegovic and Wilson [225]. In this case, the equivalence of two formulations, mixed-type and displacement-type, can be established again. For that reason, in the foregoing, we limit our consideration to the mixed-type variational formulation.

5.2.1.1

Membrane with Quadratic Displacement Interpolation

We consider a 4-node quadrilateral element with degrees of freedom as shown in Fig. 5.5. However, with a trivial modification (see [227]) we can degenerate this quadrilateral into the 3-node triangular element, so that the same considerations apply. The element geometry is defined by a bilinear map, i.e. xh (ξ, η) |Ω e =

4  I =1

N I (ξ, η)x I

(5.103)

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5 Solids, Membranes and Shells with Drilling Rotations: Complex Structures

where we use bilinear shape functions (e.g., see [176]) N I (ξ, η) =

1 (1 + ξ I ξ )(1 + η I η) 4

(5.104)

We next present the chosen interpolations. The independent rotation field is interpolated as a standard bilinear field over each element. Accordingly ψ h (ξ, η) |Ω e =

4 

N I (ξ, η)ψ I

(5.105)

I =1

The in-plane displacement approximation is derived from the 8-node parent element (see Fig. 5.5) interpolation written in hierarchical form

u 1h u 2h

= uh (ξ, η) |Ω e

4  I =1

N I (ξ, η)u I +

8 

N I (ξ, η)Δu I

(5.106)

I =5

where Δu I are hierarchical displacements, i.e. displacements relative to 4-node isoparametric mapping. In (5.106) we employ Serendipity shape functions defined by (see [176]) 1 (5.107) N I (ξ, η) = (1 − ξ 2 )(1 + η I η) ; I = 5, 7 2 N I (ξ, η) =

1 (1 + ξ I ξ )(1 − η2 ) ; I = 6, 8 2

(5.108)

Hierarchical mid-side displacements Δu I , I = 5, 6, 7, 8 are rotated in local (n, t) coordinate system (see Fig. 5.5) by

s2 B t12 n12 BM α12 B B u2 B6 u1 Bs  1 3s   ψ 1   4s 

4r

6 b7 r

1r

-

x1

Fig. 5.5 Membrane element with quadratic displacement interpolation

3 ξ

8b

x2 6 x3

η

6 b-

b

5

r

2

5.2 Membranes with Drilling Rotations: Discrete Approximation

235

Δu n = nT Δu = Δu 1 cos α + Δu 2 sin α Δu t = t T Δu = −Δu 1 sin α + Δu 2 cos α

(5.109)

where n is an outward unit normal vector and t is a unit tangent vector of the corresponding element side. The tangential components of the mid-side displacements Δu t are retained as the interpolation parameters for the incompatible modes (see [386]). The straindisplacement matrix for this part of the interpolation is made orthogonal to the constant stress field using the procedure introduced in [225], so that the convergence in the spirit of the patch test (see [176]) is ensured. The mid-side displacement component Δu n perpendicular to the element sides are eliminated in terms of the nodal rotations ψ I , to get the non-conventional displacement interpolation uh (ξ, η) |Ω e = +

8  I =5

4  I =1

N I (ξ, η)u I +

8  I =5

N I (ξ, η) l J8K (ψ K − ψ J )n J K (5.110)

N I (ξ, η) Δu t I t J K

where l J K is the length of the element side associated with the corner nodes J and K, i.e. (5.111) l J K = ((x K 1 − x J 1 )2 + (x K 2 − x J 2 )2 )(1/2) and FORTRAN-like definition of adjacent corner nodes is I = 5, 6, 7, 8 ; J = I − 4 ; K = mod(J, 4) + 1

(5.112)

Finally, the skew-symmetric stress field is chosen to be constant over each element, i.e. (5.113) skewσ h (ξ, η) |Ω e = σ0 which closes the description of chosen interpolation fields. We further define the matrix notation for the infinitesimal strains symm∇u (ξ, η) | = h

4 

Ωe

(B I (ξ, η)u I + G I (ξ, η)ψ I ) +

I =1

8 

R M (ξ, η)Δu t M

M=5

(5.114) where u I , ψ I and Δu t M are, respectively, the nodal values of the displacement, the rotation field and mid-side incompatible displacement parameters. The B I matrix in (5.114) has the standard form BI =

0 ∂ N I /∂ x2 ∂ N I /∂ x1 0 ∂ N I /∂ x2 ∂ N I /∂ x1

(5.115)

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5 Solids, Membranes and Shells with Drilling Rotations: Complex Structures

The part of the displacement interpolation associated with the rotation defines GI =

1 [l I J B L n I J − l I K B M n I K ] 8

(5.116)

and the displacement interpolation of the tangential incompatible mid-side displacement gives rise to (5.117) RM = BM tI K In Eqs. (5.115)–(5.117) above, a FORTRAN-like definition of indices is I = 1, 2, 3, 4; J = I − 1 + 4 int (1/I ); K = mod(I, 4) + 1; L = J + 4; M = I + 4

(5.118) We impose the requirement that the strains associated with the incompatible modes be orthogonal to the constant stress field. For the incompatible modes, this will ensure convergence of the analysis in the spirit of the patch test (see [176]). The modification (see [225]) fits into the framework of well-known B-bar methods (e.g., see [153, 176]) and reduces to changing strain-displacement matrix R into =R− 1 R Ωe

 R dΩ

(5.119)

Ωe

The strain-displacement matrix associated with rotational degrees of freedom is purified of constant strain modes. That amounts to the similar modification =G− 1 G Ωe

 G dΩ

(5.120)

Ωe

We briefly discuss the motivation for the modification (5.120). Since the rotation field ψ h in our formulation corresponds to the continuum-mechanics rotations, it follows from the chosen interpolation (5.105) that so do the nodal values of that field ψ I . We refer to Fig. 5.6 for the pure stretch of a distorted element and the pure infinitesimal rigid body rotation. In a pure stretch, the skew-symmetric part of the displacement gradient must be zero, i.e. for that strain state the rotation contribution to the strain must vanish. That is precisely what we achieve with the modification (5.120). Rigid body infinitesimal rotation requires that all nodal values of the rotation field are to be the same. Under that condition, the displacement interpolation (5.110) reduces to the standard isoparametric interpolation which can represent rigid body rotations without difficulties. Furthermore, we introduce the matrix notation for the infinitesimal rotation fields as (skew∇uh − ψ h ) |Ω e =

4  I =1

(b I u I + g I ψ I ) +

8  M=5

r M Δu t M

(5.121)

5.2 Membranes with Drilling Rotations: Discrete Approximation

sh Xh Xh Xh Xh hhh Xh XX h hc s

c s B s  c B B B B Bc  sB  s Bc

(c ( s   ((((  (  ( ( s( ( Pure stretch

237

Infinitesimal rigid body rotation

Fig. 5.6 Displacement patterns of the quadrilateral membrane

where

1 < −∂ N I /∂ x2 ; ∂ N I /∂ x1 > 2

(5.122)

1 [l I J b L n I J − l I K b M n I K ] − N I 8

(5.123)

bI = and gI = while

r M = bM tI K

(5.124)

with indices I, J, K , L , M, in Eqs. (5.122)–(5.124), again defined by Eq. (5.118). Having defined the matrix notation (5.114) and (5.121) we proceed with recasting the discrete approximation of the mixed-type formulation into matrix form. The first term in equation (5.97) gives rise to the element stiffness matrix  K=

R] T C[B; G; R] dΩ [B; G;

(5.125)

Ωe

The second term in (5.97) combined with (5.98) is denoted as  h= < b; g; r >T dΩ

(5.126)

Ωe

and the last term in (5.97) defines the loading vector f. With this notation at hand, the discrete mixed-type formulation can be rewritten as

T

K h h −γ −1 Ω e



a σ0



⎛ ⎞

 u f = ; a=⎝ ψ ⎠ 0 Δu

(5.127)

Since the skew-symmetric part of the stress is interpolated independently in each element, the corresponding part of the stiffness matrix (5.127), may be eliminated at

238

5 Solids, Membranes and Shells with Drilling Rotations: Complex Structures

the element level to yield a rank-one matrix update to the element stiffness. Equation (5.126) can be rewritten as =f; K = K + γ hhT Ka Ωe

(5.128)

is then used to Static condensation (see [384]) on the element stiffness matrix K eliminate the relative tangential displacements Δu at the element level, so that the remaining element global degrees of freedom are corner nodes displacements u and corner nodes drilling rotations ψ, i.e. ⎛

⎞

 u ⎝ ψ ⎠ → u ψ Δu 5.2.1.2

(5.129)

Membrane with Cubic Displacement Interpolation

We wish to extend the displacement interpolation scheme of the membrane element of the previous section to the incomplete cubic field. For that purpose, we include additional interpolation parameters, four mid-side hierarchical rotations Δψ J K . Namely, the rotation interpolation is extended to quadratic interpolation in the form ψ h (ξ, η) |Ω e =

4 

N I (ξ, η)ψ I +

I =1

8 

N I (ξ, η) Δψ J K

(5.130)

I =5

where N I (ξ, η), I = 1, ..., 8 are bilinear and Serendipity shape functions defined by (5.104), (5.107) and (5.108). However, the element geometry is still defined by the bilinear mapping defined in (5.103) (see Fig. 5.7). Therefore, the hierarchical displacement interpolation can be obtained by following the same consideration which takes us from (5.106) to (5.110). Thus uh (ξ, η) |Ω e = +

8  I =5

4  I =1

N I (ξ, η)u I +

N I (ξ, η) Δu t I t J K +

8  I =5

8  I =5

N I (ξ, η) l J8K (ψ K − ψ J )n J K (5.131)

M I (ξ, η) l J6K Δψ J K n J K

where the shape functions M I (r, s) related to the hierarchical mid-sides rotations are given as 1 M I (ξ, η) = sign(η I ) ξ(1 − ξ 2 )(1 + η I η) ; I = 5, 7 2

(5.132)

5.2 Membranes with Drilling Rotations: Discrete Approximation

s2 B t BM 12 n12 B c α12 c B u2 B 6 u1 BBs  1 3s   ψ1 c  c    Δψ41 4 s x2 6 x3

239 4r

η

6 b7

3 ξ

8b 1r

r

b 6b

5

r

2

-

x1

Fig. 5.7 Membrane element with cubic displacement interpolation

and

1 M I (ξ, η) = −sign(ξ I ) η(1 − η2 )(1 + ξ I ξ ) ; I = 6, 8 2

(5.133)

The indices in Eq. (5.131) are again defined by (5.112). Note that the displacement interpolation (5.131) can still represent pure stretch if the strain displacement matrix for the rotational degrees of freedom is purified of the constant strain modes in the manner of (5.120). Infinitesimal rigid body rotation can be represented as well. In that case, all corner rotations are the same, and all mid-side hierarchical rotations are equal to zero so that the remaining isoparametric displacement interpolation represents the rigid body rotation. The skew-symmetric stress field is interpolated by the polynomial given in global coordinates, which is discontinuous over the element boundaries. Namely ⎞ σ0 skewσ h |Ω e =< 1; x1 ; x2 > ⎝ σ1 ⎠ = qT σ σ2 ⎛

(5.134)

Having defined the interpolations for the displacements (5.131), the rotations (5.130) and the skew-symmetric part of the stress field (5.134), we can follow the same procedure as given in the previous section by (5.121)–(5.128) to define the element stiffness matrix. The infinitesimal strain field is computed by symm∇uh (ξ, η) |Ω e =

4  I =1

(B I u I + G I ψ I ) +

8 

(P M Δψ I K + R M Δu t M )

M=5

(5.135)

240

5 Solids, Membranes and Shells with Drilling Rotations: Complex Structures

where PM =

lI K BM nI K 6

(5.136)

and the remaining matrices and the indices in (5.135) are the same as defined in (5.114) and (5.118), respectively. Similarly, the matrix notation for an infinitesimal rotation field is an extension of (5.121) (skew∇uh − ψ h ) |Ω e =

4 

(b I u I + g I ψ I ) +

I =1

where pM =

8 

( p M Δψ I K + r M Δu t M )

M=5

(5.137) lI K bM nI K − N M 6

(5.138)

The element stiffness for the membrane with cubic displacement field is  K=

T C[B; G; dΩ [B; G; P; R] P; R]

(5.139)

Ωe

and the mid-side The strain-displacement matrix for corner nodes rotations, G, incompatible tangential displacements, R, are orthogonalized against the constant strain states, as given in (5.120) and (5.119), respectively. The strain-displacement matrix for hierarchical mid-side rotations, P, is orthogonalized against both the constant strain states and the pure bending modes. The generalized Gram-Schmidt procedure is used for that purpose, which is briefly discussed in the subsequent section. This orthogonalization is crucial for passing the basic patch test (constant strain), and the higher order patch test (pure bending). We also define  (5.140) HT = q < b; g; p; r > dΩ Ωe



and A=

γ −1 qqT dΩ

(5.141)

Ωe

and, with the preceding matrix notation, the matrix form of the mixed-type variational formulation is ⎛ ⎞ u



 ⎜ ψ ⎟ a f K HT ⎟ = ; a=⎜ (5.142) ⎝ Δψ ⎠ H −A σ 0 Δu

5.2 Membranes with Drilling Rotations: Discrete Approximation

241

The skew-symmetric stress interpolation parameters σ may be eliminated at the element level to get =f; K = K + HA−1 HT (5.143) Ka Finally, the static condensation (see [384]) is used to eliminate the relative tangential displacements Δu at the element level. The final set of element degrees of freedom, which is obtained after static condensation of tangential mid-side displacements, consists of four corner nodes displacement vectors u, four corner nodes rotations ψ and four mid-side hierarchical rotations Δψ, which we can indicate as ⎛

⎞ ⎛ ⎞ u u ⎜ ψ ⎟ ⎜ ⎟ ⎝ ψ ⎠ ⎝ Δψ ⎠ → Δψ Δu 5.2.1.3

(5.144)

Generalized Gram-Schmidt Procedure

The strain-displacement matrix P determines the contribution of the mid-sides rotations to the total strain, i.e. symm∇u(Δψ) = P Δψ

(5.145)

We wish to purify the matrix P of a selected set of modes c c = E c

(5.146)

For example, for a membrane element placed in (x1 , x2 ) plane of a local coordinate system, the selected set of modes of constant strain and the pure bending modes can be such that ⎡ ⎤ 1 0 0 ⎢ 0 1 0⎥ ⎢ ⎥ ⎥ (5.147) E=⎢ ⎢ 0 0 1⎥ ⎣ x2 0 0 ⎦ 0 x1 0 If we want to make two sets of modes in (5.145) and (5.146) orthogonal in L 2 norm, that means that we need  ET P dΩ = 0 (5.148) Ωe

242

5 Solids, Membranes and Shells with Drilling Rotations: Complex Structures

where P is a purified strain-displacement matrix, which can be constructed as 

P = P − ET−1

ET P dΩ

(5.149)

Ωe



where T=

ET E dΩ

(5.150)

Ωe

In the case the matrix P is to be purified only against the constant strain modes, then it follows from (5.147) that E is a 3 × 3 identity matrix. i.e. E=I

(5.151)

T = ΩeI

(5.152)

and, by virtue of (5.150), we get

Therefore, equation (5.149) reduces to 1 P=P− e Ω

 P dΩ

(5.153)

Ωe

which is precisely the modification of the form (5.120) used for the matrix G.

5.2.2 Illustrative Numerical Examples and Closing Remarks Several numerical examples are presented to demonstrate the accuracy of the membrane elements presented herein. Both membrane elements with quadratic and cubic displacement fields are evaluated. In the results to follow they are denoted as M Q2 and M Q3, respectively. All numerical computations are performed by the computer program F E L I N A (see [127]).

5.2.2.1

The Patch Test

First, two examples are the fundamental tests of the element performance; the patch tests. Both a constant strain and a higher-order patch test are performed. First, the patch test is performed in its most stringent version on a one-element test (see [176]). A skewed element is fixed with a minimum number of constraints and exposed to uniform tension (see Fig. 5.8). All rotational degrees of freedom are left unconstrained. Both M Q2 and M Q3 pass this patch test and allow recovering constant strain and constant stress values.

5.2 Membranes with Drilling Rotations: Discrete Approximation

243

r F    F r-

br bbr

Fig. 5.8 Single element path test: uniform tension

gg

g

regular mesh

load case 1: couple

-



load case 2: moment

   

JJ JJ



distorted mesh

Fig. 5.9 Simple beam in pure bending

Some of the other membrane elements with rotational degrees of freedom (e.g. see [5] or [237]) do not pass the one-element patch test due to the presence of a spurious mode.

5.2.2.2

A Simple Beam: The Higher Order Patch Test

A simple beam with a length equal to 10 and height equal to 1 is subjected to pure bending. The beam is of unit thickness. It is made of a homogeneous elastic material with Young’s modulus E = 100 and Poisson’s ratio ν = 0. The beam is modeled by one row of six membrane elements with drilling degrees of freedom as shown in Fig. 5.9. No drilling degree of freedom is restrained; only a minimum number of restraints is imposed. Two load cases are considered. The first load case is a unit couple applied at the free ends, which represents consistent loading for a higher-order patch test (see [176]). When a regular mesh is used, the solution is exact. See Table 5.1. For the membrane element with cubic displacement interpolation, M Q3, the purification of the straindisplacement matrix against pure bending modes is crucial for the satisfaction of the higher order patch test. However, for an undistorted element configuration, it makes M Q3 to perform in the same way as M Q2. For a distorted mesh (see Fig. 5.9) the accuracy for both membrane elements is still very good. The membrane element M Q3 is now slightly superior to M Q2. The second load case is a new test proposed in [222]. The loading is again a unit moment, but this time it is applied as a concentrated moment at the drilling degrees of freedom at both ends. The computed value for vertical displacement is again exact,

244

5 Solids, Membranes and Shells with Drilling Rotations: Complex Structures

Table 5.1 A simple beam displacements Element Mesh Load M Q2

Regular Distor. Regular Distor. Regular Distor. Regular Distor.

M Q3

Couple Couple Moment Moment Couple Couple Moment Moment

g

Center displ.

End rot.

1.5000 1.1742 1.5001 1.1739 1.5000 1.2625 1.5003 1.4160

0.6000 0.5036 0.6533 0.5654 0.6000 0.6172 0.7056 0.6843

6

gg

regular mesh

B

BB 







distorted mesh

Fig. 5.10 Short cantilever beam

while the end rotation is slightly different than the exact solution. The difference from the exact solution, for a regular mesh, is due to the fact that a single concentrated moment at a drilling degree of freedom is not a consistent loading, which follows from the displacement interpolations (5.110) and (5.131). The membrane element M Q3 again performs slightly better than M Q2 when the distorted finite element mesh is used. The results of the analysis can be compared with the exact solution, obtained by the beam theory, given as 1.5 for the beam central deflection and 0.6 for end rotation.

5.2.2.3

Short Cantilever Beam

An additional test of the membrane elements is performed to evaluate their performance in a distorted configuration as well as the accuracy of the stress recovery. For that purpose, the shear-loaded cantilever beam (see Fig. 5.10 is selected as a test problem for the membrane elements, which is used by many authors (e.g., see [6, 50, 155, 274]). The elasticity solution (e.g., see [371]) for the tip displacement is u2 =

(4 + 5ν)Pl Pl 3 + = 0.3553 3E I 2Eh

5.2 Membranes with Drilling Rotations: Discrete Approximation Table 5.2 Tip displacement of short cantilever beam Mesh Allman [6] MacNeal Frey [126] [274] 4×1 8×2 16 × 4 4 × 1a a Irregular

0.3026 0.3394 0.3512 –

0.3409 – – 0.2978

a Irregular

M Q2

M Q3

0.3491 0.3524 0.3545 0.3181

0.3491 0.3524 0.3545 0.3212

mesh after [274]

Table 5.3 Selected stress of short cantilever Mesh Allman [6] 4×1 8×2 16 × 4 4 × 1a

0.3283 0.3460 0.3529 –

245

52.7 58.4 59.7 –

Frey [126]

M Q2

60.0 61.3 60.8 –

60.41 59.93 60.00 55.25

mesh after [274]

for the properties selected as: end shear P = 40, Young’s modulus E = 30,000, Poisson’s ratio ν = 0.25, length l = 48, height h = 12 and unit thickness, which gives I = 144. The finite element solution is obtained for a coarse mesh of four square elements, for both a regular and the distorted one, and also for finer meshes constructed by bisection of the regular mesh. For regular meshes, M Q2 and M Q3 give the same results. However, for the distorted mesh, M Q3 is slightly superior. Moreover, all the results obtained are superior to the results available in the literature; see Table 5.2. Stress computation for the membrane elements is first performed at the optimal accuracy points (see [25]) which correspond to 2 × 2 Gauss quadrature points, and then projected to the nodes by using a bilinear assumed stress distribution (see [176]) and finally averaged. The computed results for the maximum normal stress in the Sect. 12 units away from the fixed end are reported in Table 5.3 along with the results of [6], and [126]. The results converge to the analytical value of 60 and are somewhat better than those already available in the literature.

5.2.2.4

Beam-to-Shell Connection

A simple example of a square plate supported by a single column with a circular cross-section, is used to illustrate the performance of both membrane elements in a practical example of a beam-to-shell connection. The model consists of a 400 × 400 square plate of unit thickness, connected to a column placed in the center of the plate perpendicularly to the plate. The column length is l = 300 and its St. Venant torsional rigidity is J = 790.55. Both the plate and the column are made of isotropic elastic material with Young’s modulus E = 2 × 103 and Poisson’s ratio ν = 0.3. The finite element mesh is shown in Fig. 5.11.

246

5 Solids, Membranes and Shells with Drilling Rotations: Complex Structures 1 

1 6

1 -

1 ? 200



200



finite element mesh



200



200

external loads

Fig. 5.11 Slab supported by a central column Table 5.4 Tip rotation of the column Load Frey [126] Forces Moments

0.39473 0.39473

M Q2

M Q3

0.39466 0.39466

0.39466 0.39466

Two load cases, given in terms of the concentrated forces and concentrated moments, are used. See Fig. 5.11. Both load cases produce beam torsion with the resultant torque of 800. The analytic solution for the beam tip rotation of 0.39466 is computed by the St. Venant torsion theory. This example is taken from [126], used therein to evaluate the performance of J E T 2 shell element, with the results given in Table 5.4, along with the results obtained by M Q2 and M Q3. Note that both M Q2 and M Q3 are capable to reproduce the exact solution, i.e. they provide full compatibility in a beam-to-shell connection.

5.2.2.5

Closing Remarks

We have presented two membrane elements with rotational degrees of freedom. Both elements exhibit excellent performance in a set of standard problems for either regular or distorted finite element mesh. The sound variational foundation, the chosen hierarchical displacement interpolations and the modified set of incompatible modes are the main ingredients of the successful solution when constructing the membrane elements of this kind. It is important to note that the membrane elements presented herein are free of spurious modes, adjusted parameters and capable of robust performance when modeling beam-to-shell connections. Therefore, they can be of great value to the solution of practical engineering problems, e.g. in-filled frames, folded plates, frame-foundations systems and frame-to-shell connections. This work was extended to nonlinear kinematics, constructing the geometrically nonlinear membrane elements starting with both displacement-based [196] and mixed

5.3 Shells with Drilling Rotations: Linearized Kinematics

247

variational formulations [163]. We have also shown how to further increase the accuracy of these elements by using the geometrically nonlinear method of incompatible modes [192]. Finally, we note that the membrane element with drilling rotations is also valuable for the analysis of shells, especially for the nearly co-planar configurations. In order to demonstrate that, we have constructed a shell element with quadratic displacement interpolation, discussed in the next section, and proved it has a very satisfying performance in the set of standard test problems.

5.3 Shells with Drilling Rotations: Linearized Kinematics The development shell theory with drilling rotations first started in [168] with a consistent formulation of the geometrically nonlinear theory with special emphasis on problems of parameterizing the finite rotations. In [194] we have discussed the details of its numerical implementation with non-standard interpolations based on the geometrically nonlinear method of incompatible modes [192]. In this section, we will build on these developments and discuss the geometrically linear shell theory with drilling rotations, which can be obtained from its geometrically nonlinear counterpart by using the consistent linearization (e.g., see [278]) applied at the shell reference configuration. We actually go a step further in that direction by assuming the shell reference configuration to be shallow. Consequently, we obtain a geometrically linear shallow shell theory to be considered in the foregoing. More specifically, the presented shell theory merges the Reissner [319] - Mindlin [290] kinematic hypotheses with Marguerre’s assumption on shallow reference geometry (see [277]), and, in addition, it provides three-rotation parameters due to the presence of the drilling rotations (e.g., see [197]) While some excellent shell elements with drilling rotations based on Kirchhoff’s shell theory already exist (e.g., see [237, 361] and [222]), the shell elements with Reissner-Mindlin’s kinematics and drilling rotations are not widely reported in the literature. This development aims to close that gap. However, in addition to providing a shell element useful in the analysis of thick shells and complex problems of shell intersections and connections with beams and stiffeners, other noteworthy features, we believe, are: (i) We have clarified the issue of a proper form of the shell theory which attempts to merge the Reissner-Mindlin kinematic hypotheses with Marguerre’s assumption on shallow reference geometry. The final form of the equations for membrane strains is different from some earlier proposals given in the works on curved beam and shell elements of this kind (e.g., see [47] or [97], p. 237). The geometrically linear shallow shell theory discussed herein is obtained by the consistent linearization of the geometrically nonlinear theory at the reference configuration. The same final result is obtained by departing either from geometrically nonlinear shell theory [168] or from the variational principle of Fraeijs de Veubeke

248

5 Solids, Membranes and Shells with Drilling Rotations: Complex Structures

[124] for the geometrically nonlinear theory of three-dimensional continuum with independent rotation field. The presented derivation provides a sound justification for the proposal of Tessler [365] on the proper way to combine the Reissner-Mindlin kinematics and Marguerre’s assumption on shallow shell configuration. (ii) By assuming the shell reference configuration to be shallow, we simplify the task of constructing the finite element interpolations which are capable of eliminating the shear and membrane locking phenomena (e.g. see [47]). The proposed finite element interpolations are based on the method of incompatible modes (see [225]). We have already shown, in a somewhat simpler setting of two-dimensional plane problems (see [225]), that the incompatible-mode-based interpolations can deliver essentially the same high accuracy and mesh distortion insensitivity as hybrid-stress interpolations of Pian-Sumihara (see [315]). The present work further extends these findings. On the set of numerical examples, we demonstrate that the same high accuracy is achieved with our shell elements as with what appears to be an optimally tuned shell element in [337], based on the hybrid-stress interpolations of stress resultant and stress couple components. This result is even more significant since it is obtained with no special effort made in modeling the shell reference configuration as accurately as in [337], and in the presence of drilling rotations, which normally introduces a stiffening effect. The outline of this section is as follows. First, we derive the governing equations of the geometrically linear shallow shell theory with drilling rotations. Next, we discussed the choice of incompatible-mode-based interpolations, along with their relationship with assumed strain interpolations. Finally, the shell element performance is illustrated for the set of standard test problems and closing remarks are given.

5.3.1 Geometrically Linear Shallow Shell Theory 5.3.1.1

From Geometrically Nonlinear Shell Theory

In [168], we have presented a regularized variational principle for the geometrically nonlinear shell theory with drilling rotations in the form Πγ (ϕ, θ ) =

1 2

 A

αβ

αβ

{α · C N β + α · Cαβ γ β + α · C Q β

˜ αβ κβ } dA − Πext , α, β ∈ [1, 2] +κα · C M

(5.154)

where, for compactness, we used direct notation with strain measures denoted as α = ϕ,α − aα

(5.155)

5.3 Shells with Drilling Rotations: Linearized Kinematics

and

κα = ωα = T(θ ) θ,α ,

where T(θ ) =

θ − sinθ sinθ 1 − cosθ

+ θ ⊗θ . I+ 2 θ θ θ3

249

(5.156)

(5.157)

In (5.154)–(5.157), ϕ is the position vector in the deformed configuration, and θ is the rotation vector (with three rotation parameters) used to parameterize finite rotations. The finite rotations, (θ ), describe the current position of the local Cartesian basis ai as a rotated position of the corresponding Cartesian basis gi in the shell reference configuration, i.e. ai = gi (5.158) = 0 ei . As indicated in (5.158), a convenient way to construct the reference Cartesian basis gi is by rotating the global Euclidean basis ei with an orthogonal matrix 0 , which is a known function defined over the reference configuration. The orthogonal matrix of finite rotations  is related to the rotation vector θ via (e.g. see [133], p. 164) 1 − cosθ sinθ

+ θ ⊗θ . (5.159)  = cosθ I + θ θ2 We next proceed with linearizing the expressions above in order to obtain geometrically linear theory. For (θ → 0), from the expression in (5.159), it follows immediately that the infinitesimal rotations around reference configuration are described by a skew-symmetric matrix , since ⎡

⎤ 0 θ3 −θ2 (θ ) → I + ; = ⎣−θ3 0 θ1 ⎦ θ2 −θ1 0 lin

(5.160)

Given that the matrix T in (5.157), upon linearization around the reference configuration, reduces to the identity matrix, the linearized bending strain measure can directly be obtained from (5.156) as lin

ωα → θ,α = κ lin .

(5.161)

We specialize the present derivation of linear shell theory one step further, by assuming the shell reference configuration to be shallow. The main reason for introducing this assumption is that it simplifies the construction of the locking-free interpolations. No significant limitations are introduced with this assumption since at the later stage, in the finite element implementation, the reference configuration is approximated as shallow over each shell finite element. Thus, we can also converge to deep-shell solutions.

250

5 Solids, Membranes and Shells with Drilling Rotations: Complex Structures

Fig. 5.12 Shallow shell reference configuration

DD D

g2 , s2

x3 = f (x1 , x2 )

e3

6

-

e1 , x 1

g1 , s1

e2 , x 2

Utilizing Marguerre’s assumption, the pertinent equations can further be simplified. First, in this case (see [197]), the metric of the plate can be used rather than the metric of the shell, and integration in (5.154) can be performed over the projected plane rather than over the true shell domain. Moreover, the partial derivatives with respect to surface coordinate in (5.155) and (5.156) can be approximated with the partial derivative with respect to the plane coordinates xα which are used to parameterize the shell projected configuration. (see Fig. 5.12). If x3 = f (x1 , x2 ) represents the functional form of the shallow shell reference geometry (see Fig. 5.12), then, similarly to (5.160), the linearized form of 0 can be written as ⎡ ⎤ 0 0 f ,1 lin 0 f ,2 ⎦ 0 → I + 0 ; 0 = ⎣ 0 (5.162) − f ,1 − f ,2 0 where, within the order of approximation for shallow shells, we were able to use θ01 ≈ f ,2  1 and θ02 ≈ − f ,1  1. The linearized form of the position vector in (5.155) is given as ⎡

⎤ x1 + u 1 ϕ → ⎣x2 + u 2 ⎦ f + u3 lin

(5.163)

Hence, using the results given in (5.160), (5.162) and (5.163), from (5.155) and (5.158) the linearized form of the membrane and shear strain measures reduces to ⎤ ⎡ ⎤ 1 + θ2 f ,1 1 + u 1,1 = ⎣ u 2,1 ⎦ − ⎣θ3 − θ1 f ,1 ⎦ u 3,1 + f ,1 −θ2 + f ,1 ⎡

1lin

(5.164)

⎡

and 2lin

⎤ ⎡ ⎤ u 1,2 −θ3 + θ2 f ,2 = ⎣ 1 + u 2,2 ⎦ − ⎣ 1 − θ1 f ,2 ⎦ u 3,2 + f ,2 θ1 + f ,2

(5.165)

If the sign convention for the rotation vector components is chosen as it is commonly done for the Reissner-Mindlin plate theory (e.g., see [153], p. 311) with

5.3 Shells with Drilling Rotations: Linearized Kinematics

θα = −eαβ θ˜β , θ3 = ψ , eαβ =

251



0 1 −1 0

(5.166)

then rearranging the components of the bending strain in the same way, lin lin = eαβ κβγ , κ˜ αγ

(5.167)

preserves the form of the expression for the bending strain measures as in (5.161) lin = θ˜α,β . κ˜ αβ

(5.168)

On the other hand, the membrane and shear strains in (5.164) and (5.165) now become ⎡ ⎤ u 1,1 + θ˜1 f ,1 (5.169) 1lin = ⎣u 2,1 + θ˜2 f ,1 − ψ ⎦ u 3,1 − θ˜1 and

⎡

2lin

⎤ u 1,2 + θ˜1 f ,2 + ψ = ⎣ u 2,2 + θ˜2 f ,2 ⎦ u 3,2 − θ˜2

(5.170)

Finally, following the corresponding developments in geometrically nonlinear theory (see [168]), we split the strain measures into their symmetric and skewsymmetric parts. The bending strain measure reduces to lin = κ˜ (αβ)

1 (θ˜α,β + θ˜β,α ) , 2

(5.171)

while the shear strain measures can be written as γαlin = u 3,α − θ˜α .

(5.172)

Note that both the bending and shear strain measures are those of the classical Reissner-Mindlin plate theory (e.g., see [153], p. 314). Hence, we can directly benefit from a rich experience accumulated in searching for an optimal finite element interpolation for the Reissner-Mindlin plates. The expression for the membrane strains, which follows from (5.169) and (5.170), is given as 1 1 lin = (u α,β + u β,α ) + (θ˜α f ,β + θ˜β f ,α ) . (5.173) (αβ) 2 2 The expression above differs from the corresponding expression usually encountered in the literature (e.g., see [97], p. 237), for having the rotation field θ˜α (rather than the derivatives of the lateral displacement field) coupled with the membrane strains in the warped element configuration. For this form of membrane strains, the problems

252

5 Solids, Membranes and Shells with Drilling Rotations: Complex Structures

of shear and membrane locking appear uncoupled, which simplifies the construction of non-locking finite element interpolations. The skew-symmetric part of the membrane strain takes the form lin = [12]

5.3.1.2

1 1 (u 2,1 − u 1,2 ) − ψ + (θ˜2 f ,1 + θ˜1 f ,2 ) . 2 2

(5.174)

From Three-Dimensional Continuum with Independent Rotation Field

In the remainder of this section, we want to show that the same equations of geometrically linear shallow shell theory as those in (5.171)–(5.174) can be derived if we start with the variational principle of Fraeijs de Veubeke [124]. This variational principle, which is proposed for the geometrically nonlinear theory of three-dimensional continuum with an independent rotation field, can be derived from  {W (H) + P · (I + ∇u − (I + H))}dB − Πext ,

Π (u, , H, P) =

(5.175)

B

where W (H) is the strain energy density given in terms of Biot strain tensor H, which is defined via polar factorization of the deformation gradient (e.g., see [79, 176]) I + ∇u = (I + H) .

(5.176)

The second term in (5.175) is just the weak form of the polar factorization, with the Lagrange multiplier, P, being the non-symmetric Piola-Kirchhoff stress tensor. In addition,  in (5.175) is the independent rotation field, and u is the displacement field. If we use the Legendre transformation to define the complementary energy V (T) in terms of the Biot stress T = T P, i.e. using −V (T) = W (H) − P · H = W (H) − T · H ,

(5.177)

the variational principle in [124] can be rewritten as  {−V (T) + P · (I + ∇u) − P · }dB − Πext .

Π (u, , T) =

(5.178)

B

As elaborated upon in [246], the transformation in (5.177) can be unambiguously defined for so-called semi-linear material, where the complementary energy is assumed to take a quadratic form. Moreover, due to the symmetry of H, only the symmetric part of the Biot stress contributes to the complementary energy. It was shown in [197] that linearizing the variational principle in (5.178) we recover the

5.3 Shells with Drilling Rotations: Linearized Kinematics

253

principle of Reissner [321] for the geometrically linear theory of three-dimensional continuum with independent rotations. We next assume that the variational principle in (5.178) applies in the shallow shell reference configuration B = {A × [−t/2, t/2]}. Following Ibrahimbegovic et al. [197], we assume that the shallow shell configuration is derived from the projected (plate) configuration A˜ by a point-wise isometric transformation, whose deformation gradient is therefore an orthogonal tensor 0 = ∂xss /∂x. Under this kind of transformation, the displacement gradient in shallow shell configuration ∇uss can be written as ∂u i ∂u i ∂ x ss = ss k =⇒ ∇uss = ∇u0T . ∂x j ∂ xk ∂ x j

(5.179)

Similarly, recalling that P is a two-point tensor, we have Pss = P0T . Therefore, the three-dimensional shallow shell approximation of the variational equation in (5.178) becomes  {−V (T) + P · 0 + P · ∇u − P · 0 }dζ d A˜ − Πext . Π (u, , T) = ˜ A×[−t/2,t/2]

(5.180) We next proceed with linearizing the variational principle in (5.180) above. Upon linearization, the stress tensor still remains non-symmetric lin

P → σ ; σ = σ T .

(5.181)

However, as noted already, the complementary energy depends only on the symmetric part of the stress tensor V (σ ) =

1 1 symm{σ } · C−1 symm{σ } ; symm{σ } = (σ + σ T ) . 2 2

(5.182)

By using the approximation of orthogonal matrices  and 0 given, respectively, in (5.160) and (5.162), we can find that lin

0 − 0 → − −

0 ,

(5.183)

which leads to the linearized form of the variational principle  Π (u, θ , σ ) =

{−V (σ ) + σ · ∇u − σ · ( +

0 )}dζ d A˜ − Πext .

˜ A×[−t/2,t/2]

(5.184) We next regularize the variational principle for the geometrically linear theory of continuum with an independent rotation field in the same way as it was done

254

5 Solids, Membranes and Shells with Drilling Rotations: Complex Structures

in [154]. The regularization reduces to adding a quadratic term in skew-symmetric stress component Πγ (u, θ , σ ) = Π (u, θ , σ ) −

1 2



{skew(σ ) · γ −1 skew(σ )}dζ d A˜ ,

˜ A×[−t/2,t/2]

(5.185) where γ is a regularization parameter with a recommended value equal to shear modulus (see [154]). Finally, we want to reduce this variational principle to the stress resultant form. Following Reissner [320] we introduce some usual simplifying assumptions regarding the stress tensor components which are appropriate for the shallow shell theory on hand. Namely, we use σ33 = 0 σ13 = σ31 = q1 /t (5.186) σ23 = σ32 = q2 /t , where qα are shear stress resultants. Once this assumption on shear stress distribution has been made, the regularization in (5.185) will affect only the in-plane shear stress components, which is an exact counterpart of the corresponding work in geometrically nonlinear shell theory (see [168]). By virtue of non-symmetry of σ , the stress resultants n αβ and m˜ αβ are also nonsymmetric. However, we will disregard the skew-symmetric part m˜ [αβ] , and assume the distributions of the remaining stress components as σ(αβ) = 1t n (αβ) + σ[αβ] =

1 n t [αβ]

.

12ζ t3

m˜ (αβ)

(5.187)

The rotation field is assumed constant through the thickness, and the notation convention for its components defined in (5.166) is invoked again. The displacement vector components are given in accordance with the Reissner-Mindlin kinematic hypotheses u α (xα , ζ ) = u α (xα ) − ζ θ˜α (xα ) (5.188) u 3 (xα , ζ ) = u 3 (xα ) . In (5.188) above, we have assumed that the infinitesimal rotation field, which is obtained by linearizing the finite rotation field featuring in the polar factorization, is equal to the rotation field appearing in the assumed displacement distribution. Using the assumptions on the stress and displacement fields in (5.186)–(5.188), we obtain a stress resultant form of the variational principle. If its Euler-Lagrange equations are used in order to eliminate the symmetric parts of stress resultants, we get mixed-type stress resultant variational principle. The variational principle of this kind represents a shallow shell counterpart of the three-dimensional principle used in [222]. It takes the form

5.3 Shells with Drilling Rotations: Linearized Kinematics

r

x3

@

b projected configuration A˜e r warped configuration Ae

b HH  HH   HH  ˜e b A  Hb b "h b " 6 r Ae b r " b " r I b " b " b" b

6 x1

255

x2

Fig. 5.13 A quadrilateral shallow-shell element

 Πγ (u, θ , n [αβ] ) = { 21 [(αβ) n (αβ) + κ˜ (αβ) m˜ (αβ) + γα qα ] + [αβ] n [αβ] A˜

− 21 n [αβ] γ −1 n [αβ] }dζ d A˜ − Πext ,

(5.189)

where the strain measures have the identical forms as those given in (5.171)–(5.174). Alternatively, by using the corresponding Euler-Lagrange equation to eliminate n [12] , we can get a displacement-type variational principle which is based on kinematic variables only. That kind of principle represents a linearized version of the principle in (5.185). Hence, starting either with a geometrically nonlinear shell or with a geometrically nonlinear three-dimensional continuum, the final form of the governing principle for geometrically linear analysis of shallow shells turns out to be the same.

5.3.2 Incompatible Modes Based Finite Element Approximation In the finite element approximation of the governing variational formulation discussed in the previous section, we approximate  N eltruee shell configuration via an assemA . In general, each of the elements bly of shell finite elements, i.e. A ≈ Ah = e=1 represents a non-planar surface shown in Fig. 5.13. The shell element surface Ae is assumed to be shallow with respect to the local coordinate system adopted for this element (see Fig. 5.13), so that the integration can be performed over its projected domain A˜ e . The projected configuration A˜ e of each element belongs to the plane determined with two lines connecting the mid-points of the opposite element sides. With this construction, the maximum distances between the warped and projected configuration appear at the nodal points, and they all have the same value, which is here denoted as h. The stiffness matrix and the load vector of each element are first computed in its local coordinate system and subsequently transformed into the global coordinate

256

5 Solids, Membranes and Shells with Drilling Rotations: Complex Structures

system used for the whole shell structure. This transformation is performed on a node-to-node basis, using the rotation matrices (e.g., see [153], p. 376). By choosing the local coordinate system for each element we can quite considerably extend the applicability of the shallow shell theory which was considered in the previous section. Moreover, as argued in [229], since the local coordinate system is Cartesian we can obtain convergence to a deep shell solution. Using the shallow shell approximation provides a very clear insight into the membrane and shear locking phenomena. Moreover, uncoupling the membrane and bending actions in a shallow shell element considerably simplifies the task of constructing non-locking interpolations.

5.3.2.1

Bending Deformation

We first consider the plate bending part of the problem. The plate (in-plane) rotation components are interpolated in the isoparametric manner (e.g., see [390]). Namely, if the element projected configuration is interpolated as   xh (ξ, η)

A˜ e

=

4 

N I (ξ, η) x I ,

(5.190)

I =1

then the rotation interpolation is also given as   ˜θ h (ξ, η) 

A˜ e

=

4 

N I (ξ, η) θ˜I ,

(5.191)

I =1

where θ˜I are nodal values of the rotation field and N I (ξ, η) =

1 (1 + ξ I ξ )(1 + ηη I ) 4

(5.192)

are the standard bilinear shape functions, with (ξ, η) ∈ [−1, 1] being the natural coordinates and ξ I and η I their nodal values equal to −1 or +1. In contrast with isoparametric interpolation of in-plane rotation components, the bending strain measures (curvatures) are enriched with a set of incompatible modes ([192]. The enhanced curvature modes are formally constructed as the gradients of an incompatible rotation field, i.e.   κ˜ (ξ, η) h

A˜ e

=

4  I =1

B B I (ξ, η) θ˜I +

2  J =1

B J (ξ, η) α κJ , G

(5.193)

5.3 Shells with Drilling Rotations: Linearized Kinematics

where

BB I = LB NI ; GB J = LB M J M1 =  1 − ξ 2 ; M2= 1 − η2 ∂ 0 ∂ L B = ∂ x1 ∂ ∂∂x2 0 ∂ x2 ∂ x1

257

(5.194)

The incompatible bending mode interpolation needs to be modified in order to meet the patch test requirements (e.g. see [364]). This modification is performed by using the methodology suggested in [225] to get B = GB − 1 G A˜ e



G B d A˜ .

(5.195)

A˜ e

An alternative way to construct incompatible bending deformation modes, as given in [343], is to first construct an enhanced curvature field in the isoparametric space and then transform that field into the physical space using the constant Jacobian transformation matrix evaluated at the center of the element. We will elaborate upon this methodology in the next section, for it provides a convenient explanation for constructing an assumed shear strain interpolation.

5.3.2.2

Shear Deformation

In constructing a shear strain interpolation, one of the main concerns is elimination of the shear locking phenomena which plague the behavior of the Reissner-Mindlin elements (e.g., see [158]). In order to briefly recapitulate the shear locking problem, we consider the case when we use an isoparametric interpolation of the lateral displacement field, given as   w (ξ, η) h

A˜ e

=

4 

N I (ξ, η) w I ,

(5.196)

I =1

along with the in-plane rotation isoparametric interpolation in (5.191). For a regular (rectangular) element, this results in the shear strain interpolation of the form



γ˜ξ 3 b − aθ1 + (dw − cθ1 )η − bθ1 ξ − dθ1 ξ η =: γ˜ = w γ˜η3 cw − aθ2 + (dw − bθ2 )ξ − cθ2 η − dθ2 ξ η

(5.197)

where the coefficients bw , ..., dθ are known combinations of the nodal values of the lateral displacement w I and the in-plane rotations θ˜I . The interpolation in (5.197) follows directly upon substituting the interpolations in (5.191) and (5.196) into the shear strain expressions in (5.172), i.e.

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5 Solids, Membranes and Shells with Drilling Rotations: Complex Structures

  γ  h

A˜ e

=

4  I =1

BS I

θ˜I wI

; BS I



 ∂  −1 0 ∂ x1 = LS N I ; LS = 0 −1 ∂∂x2

(5.198)

A potential shear locking problem can be seen in the shear strain interpolations in (5.198). Namely, due to the presence of the last two terms, vanishing shear or pure bending deformation (or Kirchhoff mode, see [158]) can not be attained without imposing some constraints on the element rotation field (bθ1 = cθ2 = dθ1 = dθ2 = 0), which reduces the space of bending strain approximations. One possibility is to construct a non-locking shear strain interpolation via enhanced strain method [343]. Namely, the shear strain interpolation is simply enriched by the set of incompatible modes chosen in such a way as to cancel the extra-terms in (5.198). The chosen incompatible shear strain modes are   inc  γ˜ 

A˜ e

⎡

ξ ⎢ 0 = GS αγ ; GS = ⎢ ⎣ξ η 0

⎤ 0 η⎥ ⎥. 0⎦ ξη

(5.199)

For an isotropic plate, with the matrix of shear moduli proportional to the identity, we can go a step further and a priori eliminate the incompatible mode parameters α γ to obtain an assumed shear strain interpolation B¯ S ⎡



⎢ B¯ S = B S − G S ⎣

A˜ e

⎤−1 ⎥ GTS G S d A˜ ⎦



GTS B S d A˜ .

(5.200)

A˜ e

It can easily be shown by direct computations that the standard form of the stiffness matrix obtained with the assumed shear strain in (5.200) is in fact identical to the corresponding form obtained from the static condensation (see [384]) of α γ parameters. For a regular element, the resulting enhanced strain interpolation is of the same form as the assumed shear strain interpolation proposed in [30]



bw − aθ1 + (dw − cθ1 )η γ˜ξ 3 = γ˜ = γ˜η3 cw − aθ2 + (dw − bθ2 )ξ

1 (1 + η)γ˜ξA3 + 21 (1 − η)γ˜ξC3 = 12 B D (1 + ξ )γ˜η3 + 21 (1 − ξ )γ˜η3 2

(5.201)

D are the corresponding mid-side values. where γ˜ξA3 , ..., γ˜η3 Hence, this equivalence can be used at the outset to eliminate the computational effort involved in computing (5.200). It is interesting to note that, even for a skew shell element, these two methodologies, assumed strain and enhanced strain method, are nearly equivalent when it comes to

5.3 Shells with Drilling Rotations: Linearized Kinematics

259

constructing the shear strain interpolations. As proposed in [30], the assumed shear strain interpolation for a skew element can be constructed by using the relationship between the natural strain components ˜i j and the physical strain components i j given as (5.202)  = ˜i j gi ⊗ g j = i j ei ⊗ e j , where gi are the contravariant base vectors in natural coordinate system, while ei ≡ ei are base vectors of element local Cartesian system. Using isoparametric mapping in (5.190), we can first compute the covariant base vectors gi as ⎡ ⎤ ⎡ ⎤

F F ∂x ⎣ 11 ⎦ ∂x ⎣ 12 ⎦ F11 F21 = F21 ; g2 = = F22 ; g3 = e3 ; F = g1 = F12 F22 ∂ξ ∂η 0 0

(5.203)

from where the contravariant base vectors directly follow (e.g., see [278]) as gi =



G−1 0T (g1 · g1 ) (g2 · g1 ) gi ; G = . 0 1 (g1 · g2 ) (g2 · g2 )

(5.204)

However, noting that detG = (detF)2 , we can further simplify the expression in [30] to get explicit forms for the contravariant base vectors ⎡ ⎡ ⎤ ⎤ F22 −F21 1 1 ⎣−F12 ⎦ ; g2 = ⎣ F11 ⎦ ; g3 = e3 . g1 = detF detF 0 0

(5.205)

Taking into account symmetry of the strain tensor and denoting γα3 = 2α3 , from the general relationship in (5.202) we can get the shear strain interpolation for a distorted element in the form



γ13 (e1 · g1 ) (e2 · g1 ) γ˜ξ 3 = γ23 (e1 · g2 ) (e2 · g2 ) γ˜η3



(5.206) F22 −F12 γ˜ξ 3 1 , = detF −F21 F11 γ˜η3 where γξ 3 and γη3 are the corresponding interpolation in the natural coordinates defined in (5.201). The enhanced strain method uses the same transformation for the distorted elements as in (5.206) (see [343]), with only difference being a center point evaluation of the terms enclosed in the brackets when transforming the interpolation part related to the incompatible modes (see [225]). As shown in a numerical example, the resulting difference in numerical results obtained with two formulations is marginal.

260

5.3.2.3

5 Solids, Membranes and Shells with Drilling Rotations: Complex Structures

Membrane Deformation

Finally, the chosen interpolations for the membrane part of the problem are considered. Out-of-plane rotation field is interpolated in the same manner as the in-plane rotations in (5.191) via   ψ h (ξ, η)

A˜ e

=

4 

N I (ξ, η) ψ I .

(5.207)

I =1

When interpolating the membrane strains, however, we make use of the incompatible mode method again. Namely, the symmetric membrane strain components are interpolated as    (ξ, η) h

A˜ e

=

4 

B M I (ξ, η) u I +

I =1

2 

M J (ξ, η) α J . G

(5.208)

J =1

Similarly to the incompatible bending modes, the incompatible membrane modes are formally computed as the gradient of the incompatible displacement field with BM I = LM N I ; GM J = LM M J M1 = 1− ξ 2 ; M2 = 1 − η2 ∂ 0 ∂ L M = ∂ x1 ∂ ∂∂x2 . 0 ∂ x2 ∂ x1

(5.209)

M is produced with the same kind of modification as given in (5.195). The matrix G The skew-symmetric part of the membrane strains is given similarly to (5.208) as  

4  1  ∂ NI

=

;−

∂ NI  uI − N I ψI ∂ x1



2  1  ∂ MJ

∂ MJ   αJ . 2 ∂ x2 2 ∂ x2 ∂ x1 A˜ e I =1 J =1 (5.210) The interpolations given in (5.207)–(5.210) are valid for a flat shell element. However, an element configuration is in general warped (see Fig. 5.13), so that the additional terms in membrane strain expression appear as given in (5.173). Merely using isoparametric interpolations for a warped shell element, introduces a severe membrane locking effect (e.g., see [47]), i.e. the inability of the element to represent a pure bending deformation mode without imposing additional constraints on the rotation field. A simplified model problem exhibiting the same kind of difficulties is given by a 2-node Timoshenko-Marguerre beam element (see [191]). In trying to eliminate the undesirable membrane locking effect in a warped element configuration, we use the mode decomposition method (e.g., see [105, 357] or [47]. Namely, we make an additive decomposition of the total displacement field into the part which corresponds to the pure bending deformation state (or the Kirchhoff h [12] (ξ, η)

+

;−

5.3 Shells with Drilling Rotations: Linearized Kinematics

261

mode), and the remaining part, which is then used to compute the contribution of the membrane deformation to the element stiffness matrix. We want to use a simple procedure; Therefore, in identifying the Kirchhoff mode we consider each side of the element independently. Along the element edge I J , the Kirchhoff mode u KIJ can be computed as (see [47])  u KIJ

=−

f ,ξ θ (ξ ) dξ = −hθ I − hθ J .

(5.211)

IJ

Subtracting this nodal pattern from the total nodal displacement field, we will get the final nodal displacement vector for a warped shell element which, on the node-to-node basis, can be written as uI −

u IK





uI h0 . = [I; A] ˜ ; A = 0h θI

(5.212)

Introducing the transformation above in the element stiffness matrix computation, we will produce the coupling of its membrane and plate part. This coupling will increase the element accuracy, without entailing the membrane locking phenomena.

5.3.3 Illustrative Numerical Examples and Closing Remarks The present element is tested over a number of standard benchmark problems proposed in [274] and [47]. The element performance is compared versus some of the top-performing elements in this category, such as a 4-node shell element based on DKQ bending formulation (e.g., see [237, 361] and [197]), a 4-node hybrid-stress element based on Pian-Sumihara interpolations for stress resultants proposed in [337], as well as with higher order elements, such as a 9-node shell element based on uniform reduced integration and stabilization as in [47]. In the examples to follow the corresponding results are cited as [361] (DKQ-4), [337] (Hybrid-4) and [47] (URI-9).

5.3.3.1

Patch Test

First, we have verified that the shell element discussed herein passes the patch test (e.g., see [364]). Both a single element patch and the regular patch of arbitrarily distorted elements (see Fig. 5.14) have been used to check the ability of the element to capture homogeneous deformation states: extension, shear, bending and twisting.

262

5 Solids, Membranes and Shells with Drilling Rotations: Complex Structures

One-element test

Patch test

Fig. 5.14 Different versions of the patch test

g

6

gg

regular mesh

B

BB 







distorted mesh

Fig. 5.15 Short cantilever beam Table 5.5 Tip displacement of short cantilever Mesh Present Displ. (%) 4 × 1a 4×1 8×2 16 × 4 Exact a Irregular

5.3.3.2

0.3344 0.3491 0.3524 0.3543 0.3553

94.12 98.25 99.18 99.72 100.00

MacNeal [275] Displ.

(%)

0.3150 0.3404 – – 0.3553

88.67 95.81 – – 100.00

mesh after MacNeal and Harder [275]

Short Cantilever Beam

A short cantilever beam under free-end shear force (see Fig. 5.15) is often used to test the performance of the membrane elements (e.g., see [275]). The exact elasticity solution for the tip displacement is equal to 0.3553, for the properties selected as: end shear force P = 40, Young’s modulus E = 30,000, Poisson’s ratio ν = 0.25, length l = 48, height h = 12 and unit thickness. The finite element solution is obtained for a coarse mesh of four square elements, and for finer meshes constructed by bisection. In order to demonstrate mesh distortion insensitivity, the same result is computed for an arbitrarily distorted mesh used in [274]. All the results are presented in Table 5.5.

5.3.3.3

Rhombic Plate

Test problem proposed in Morley [291] for the skew plate under uniformly distributed loading (see Fig. 5.16) is often used to evaluate the performance of the plate and shell elements.

5.3 Shells with Drilling Rotations: Linearized Kinematics

263

SS            SS     a                        SS                  o 30     

SS : w = 0



a

SS

-

Fig. 5.16 Uniform loading on simply supported rhombic plate Table 5.6 Center displacement of rhombic plate Mesh Present-ANS Displ. (%) 4×4 8×8 16 × 16 Exact

0.04212 0.04224 0.04374 0.04455

94.55 94.81 98.18 100.00

Present-ENS Displ.

(%)

0.04218 0.04262 0.04424 0.04455

94.68 95.67 99.30 100.00

The plate model characteristics are: Young’s modulus E = 10 × 106 , Poisson’s ratio ν = 0.3, plate side a = 100 and plate thickness t = 1. The solution for the center displacement under unit uniform load q = 1 equals to 0.04455 (see [291]). The computed numerical results are presented in Table 5.6. As presented in Fig. 5.17, the shell element which uses the DKQ bending formulation is strongly influenced by the moment singularity in the obtuse corners, and its performance deteriorates. However, the present element performs extremely well in this example, both when the assumed shear interpolation is used (Present-ANS) and enhanced shear strain interpolation (Present-ENS). The difference between the two of them is marginal. Since using the assumed shear strain increases the computational efficiency with respect to enhanced shear strain, only the former option is used in the examples to follow. The same test problem is used in order to demonstrate the accuracy of the prediction of bending moments. For that purpose, the bending moment distribution along the shorter diagonal of the skew plate (moment components along the diagonal and perpendicular to it) are computed with a uniform mesh of 16 × 16 elements. The results are plotted in Fig. 5.18, together with the analytical results given in [291]. Considering that the mesh is uniform rather than biased to favor the singularity, the accuracy of computed bending moments is quite satisfying. Moreover, the opposite sign of the moment components in the vicinity of the obtuse vertex is predicted correctly even for a coarse mesh. This is in sharp contrast with a number of thin plate elements which, as stated in [153], p. 347, yield pathological results for this problem in that the moments with the same signs are obtained.

264

5 Solids, Membranes and Shells with Drilling Rotations: Complex Structures

Fig. 5.17 Center displacements computed for the rhombic plate

Fig. 5.18 Bending moments computed for the rhombic plate

5.3.3.4

Twisted Beam

The twisted cantilever beam shown in Fig. 5.19 is one of the standard test problems proposed in [274]. Two load cases are considered: a unit free-end shear force in the thickness direction, and a unit shear force in the width direction. The characteristics of the beam are: Young’s modulus E = 29 × 106 , Poisson’s ratio ν = 0.22, beam length l = 12, width h = 1.1 and depth t = 0.32. The total angle of twist from the built-in to the free end is 90o . The exact solutions for two load cases are equal to 0.005424 and 0.001754, respectively (see [274]). This is an excellent test for assessing the element performance when the reference configuration is non-planar (warped). Any shell element which does not account properly for the warped reference configuration fails in this test. The present element has no difficulties in that respect; see Table 5.7.

5.3.3.5

Scordelis-Lo Barrel Vault

Another standard test problem for testing the performance of a shell element is the barrel vault supported by the end diaphragm and subjected to self-weight dead load (see Fig. 5.20). The characteristics of the shell are: Young’s modulus E = 432 ×

5.3 Shells with Drilling Rotations: Linearized Kinematics

265

Fig. 5.19 Twisted cantilever beam

x3 6

  +

clamped

-

x2

x1

6 -

Table 5.7 End displacement of twisted beam Mesh Load Case 1 Displ. (%) 6×1 12 × 2 24 × 4 Exact

0.005390 0.005405 0.005411 0.005424

Fig. 5.20 Scordelis-Lo Barrel vault

99.37 99.65 99.76 100.00

Load Case 2 Displ.

(%)

0.001759 0.001754 0.001751 0.001754

100.29 100.00 99.94 100.00

       6   free  *   R   l  x @  63 I  x2 @  φ @ x1 @

106 , Poisson’s ratio ν = 0, cylindrical shell radius R = 25, central angle φ = 40o , cylinder length l = 50 and the thickness t = 0.25. Self weight dead load is vertical uniformly distributed loading q = 90. Due to the symmetry, only a quarter of the vault is modeled by a finite element mesh of uniform shell elements. The numerical results (see Table 5.8) for the vertical displacement in the center of the free boundary, obtained by different finite element meshes can be compared with the reference value of 0.3024 suggested in [274]. Shell element’s ability to capture complex membrane states of stress is the dominant factor for the rate of convergence in this problem. The present element faces no difficulties converging at the same rate as other top performers. See Fig. 5.21.

266

5 Solids, Membranes and Shells with Drilling Rotations: Complex Structures

Table 5.8 Max. displacement of Scordelis-Lo Barrel vault Mesh Present Displ. 2×2 4×4 8×8 16 × 16 Exact

0.4190 0.3166 0.3039 0.3016 0.3024

(%) 138.56 104.70 100.50 99.74 100.00

Fig. 5.21 Maximum deflection in Scordelis-Lo Barrel vault

Table 5.9 Max. displacement of pinched cylinder Mesh Present Displ. 4×4 8×8 16 × 16 Exact

5.3.3.6

0.02194 0.02381 0.02439 0.02439

(%) 89.95 97.62 100.00 100.00

Pinched Cylindrical Shell Without Diaphragm

A cylindrical tube of finite length is pinched with a pair of forces placed along its diameter, in the middle of the length. The ends of the cylinder are left free. The cylinder characteristics are: Young’s modulus E = 10.5 × 106 , Poisson’s ratio ν = 0.3125, radius of the cylinder R = 4.953, length l = 10.35 and thickness t = 0.01548. Each of the forces P = 0.1. One quadrant of the shell is modeled with a mesh of uniform elements, and the results obtained are compared versus the reference value of 0.02439 (e.g., see [361]). They are all presented in Table 5.9. The present test is used to evaluate the ability of the shell element to deliver a non-locking performance in a nearly in-extensional membrane state. The rate of

5.3 Shells with Drilling Rotations: Linearized Kinematics

267

Fig. 5.22 Maximum deflection in pinched cylinder

Table 5.10 Max. displacement of pinched cylinder with diaphragm Mesh Present Mesh Present Displ. (%) Displ. 4×4 8×8 16 × 16 Exact

× 10−5

0.6760 1.3444 × 10−5 1.7050 × 10−5 1.8249 × 10−5

a Non-uniform

37.00 73.67 93.43 100.00

4 × 4a 8 × 8a 16 × 16a Exact

1.4654 × 10−5

1.7601 × 10−5 1.8309 × 10−5 1.8249 × 10−5

(%) 80.30 96.45 100.33 100.00

mesh

convergence in this problem indeed appears quite satisfying. As opposed to DKQ-4 element which converges to the somewhat bigger displacement, the present element recovers the exact solution; See Fig. 5.22.

5.3.3.7

Pinched Cylinder with End Diaphragm

A pinched cylinder with end diaphragms (e.g., see [47]) is a very severe test for a shell element performance. A cylindrical shell, very much like the one in the previous example except for the rigid diaphragms being placed at both ends, is pinched in the middle of the length with a pair of unit forces along its diameter. The characteristics of the shell are: Young’s modulus E = 3 × 106 , Poisson’s ratio ν = 0.3, radius of the cylinder R = 300., length l = 600 and thickness t = 3. We have again modeled one quadrant of the shell with a mesh of uniform elements. The obtained results are presented in Table 5.10, and compared with the analytical value of 1.82488 × 10−5 (see [47]). The presence of the rigid diaphragm changes the deformation pattern with respect to the previous cylindrical shell test, producing a very significant displacement variation over a very narrow zone around the loaded points. While fairly poor results (see Table 5.10) are obtained for uniform meshes, significantly better results are obtained for non-uniform meshes with the same number of elements, but refined around the loaded point. (In each non-uniform mesh grid spacing is selected so that 75% of all

268

5 Solids, Membranes and Shells with Drilling Rotations: Complex Structures

Fig. 5.23 Maximum deflection in the pinched cylinder with diaphragm

Fig. 5.24 Hemispherical shell with hole

x3

6 free symm.

x1

symm.

free

@ @

x2

elements were grouped in the inner third of the cylinder around the loaded point.) This observation indicates the significance of an adaptive mesh refinement as part of a robust solution strategy. For a mesh of uniform elements, the elements with higher order displacement interpolations, capable of spanning high gradients over a single element, typically perform better in this example. See Fig. 5.23.

5.3.3.8

Hemispherical Shell with 18o Hole

In this test problem (see [274]) we consider a hemispherical shell with an open hole at the top in the form of 18o spherical cap (see Fig. 5.24), under two pairs of pinching forces (each P = 2), one applying loading inward and one outward at the hemispherical edge. The characteristics of the hemispherical shell are: Young’s modulus E = 68.25 × 106 , Poisson’s ratio ν = 0.3, radius R = 10, thickness t = 0.04. Only a quarter of the shell is modeled with a structured mesh. The results obtained in the analysis are given in Table 5.11, and compared with the reference value equal to 0.093, as quoted in [337].

5.3 Shells with Drilling Rotations: Linearized Kinematics

269

Table 5.11 Max. displacement in hemispherical shell with hole Mesh Present Displ. (%) 2×2 4×4 8×8 16 × 16 Exact

0.08577 0.09399 0.09315 0.09309 0.09300

92.23 101.06 100.16 100.10 100.00

Fig. 5.25 Maximum deflection in the hemispherical shell with hole

Despite the fact that very little attention is paid to correct modeling of the shell reference configuration due to the shallow shell approximation, the results obtained herein exhibit the same very high accuracy as the geometrically exact hybrid shell element in [337]. See Fig. 5.25. In all the examples presented up to now, full 2 × 2 Gauss integration is used on all terms which contribute to the element stiffness matrix. In this example, however, we found it necessary to use a reduced one-point integration on the skew-symmetric membrane strain contribution to the element stiffness matrix. It is important to note that the reduced integration of this kind produces no rank deficiency of the element stiffness matrix in global coordinates and that one-point reduced integration can be safely used whenever the local-global transformation is truly three-dimensional.

5.3.3.9

Full Hemispherical Shell

A variation of the last problem, the full hemispherical shell (e.g., see [47]), is also very often used to test a shell element’s performance. The chosen shell characteristics (Young’s modulus E = 68.25 × 106 , Poisson’s ratio ν = 0.3, radius R = 10, thickness t = 0.04) and the loading (P = 2) are exactly the same as in the previous example, but there is no more hole on the top. The reference solution for the maximum displacement is only slightly different from the previous case; it is equal to 0.0924 (e.g., see [47]),

270

5 Solids, Membranes and Shells with Drilling Rotations: Complex Structures

Fig. 5.26 Full hemispherical shell

6 x3

symm.

x1

Table 5.12 Max. displacement in full hemispherical shell Mesh Present Displ. 3×1 3×4 3 × 16 3 × 64 Exact

0.08062 0.09278 0.09201 0.09193 0.09240

symm.

free

@ @

x2

(%) 87.25 100.41 99.58 99.49 100.00

A mesh of three elements for a quarter of the shell is constructed first (see Fig. 5.26). Finer meshes are then obtained by subdividing each of the three patches on the spherical surface. Each shell element in the mesh is distorted and for the same number of nodes per side fewer elements are used than in the previous example. Hence, this is a more demanding test than the previous one. The accuracy of the results obtained, however, is not reduced significantly. See Table 5.12. In this particular example, the presented element outperforms all the elements we have considered in the comparison. See Fig. 5.27. The reason for that is perhaps in significant participation of the rigid body rotations about the normals to the shell surface in the final displacement pattern. The presence of the drilling rotations in a shell element seems to be advantageous for a situation of this kind. On the other hand, as pointed out by [47], some elements with 5 degrees of freedom per node experience difficulties in a test of this kind due to the spurious straining in rotation about the normal to the shell surface.

5.3.3.10

Closing Remarks

We have presented a 4-node shell element for geometrically linear analysis of shell structures. As opposed to the vast majority of shell elements, the present shell element possesses three rotation parameters, i.e. drilling rotations are included, which makes it a very versatile tool in the analysis of practical engineering problems.

5.3 Shells with Drilling Rotations: Linearized Kinematics

271

Fig. 5.27 Maximum deflection in full hemispherical shell

The most important features of the presented developments can be stated as follows: (i) The element is based on a shallow shell theory; Consequently no high precision in modeling shell reference configuration should be expected for the coarse meshes. Such a limited effort in modeling the reference configuration is in sharp contrast with the recent works (e.g. see [130, 337]) which place an extra effort to account for the true shell geometry. However, it seems that the convergence rates are not impaired with this modeling approach, at least when one considers low-order shell elements such as the present 4-node element. Essentially the same results of very high accuracy are obtained over the complete set of standard benchmark problems as compared with the geometrically exact 4-node shell element in [337]. (ii) The crucial reason for a very satisfying present element performance is in the choice of interpolations based on the set of incompatible modes. As already observed in a somewhat simpler model problem of plane elasticity (see [225]), the incompatible mode based interpolation can deliver essentially the same high-level element performance as the hybrid stress method. The results presented here extend the same findings to the linear analysis of shells. (iv) One of the most useful features of the present element is its robust performance in arbitrarily distorted configurations, i.e. high mesh distortion insensitivity. As such, the element can be considered a very suitable candidate for an adaptive finite element refinement strategy, which we already discussed in the chapter on plates. The usefulness of this mesh adaptivity concept is merely demonstrated in the demanding test problem of the pinched cylinder with a diaphragm. (iv) For extending the present consideration to the geometrically nonlinear framework, we refer to our previous works in [168, 173, 192]. An extension to materially nonlinear analysis can also be readily accomplished, as already demonstrated in somewhat simpler setting of Reissner-Mindlin plates in [190, 193], flat shells in [187] or membrane shells in [167, 171, 199].

Chapter 6

Large Displacements and Instability: Buckling Versus Nonlinear Instability

Abstract We here present the geometrically nonlinear approach to structural analysis and corresponding instability problems that can arise within the framework of large displacements and rotations. The main interest is to quantify the risk of premature failure with a disproportional increase of displacement, strain and stress due to a small increase of external load, which is the intrinsic definition of instability. Two models used for detailed illustration are geometrically nonlinear truss and geometrically nonlinear beam. We also point out the difference between the linearized instability or buckling, characterized by small pre-buckling displacements, versus the nonlinear instability, associated with large displacements and rotations at critical equilibrium point. We present three different criteria for identifying the critical equilibrium point. We finally present a very powerful computational approach to computing the solution to linearized instability of any complex structures within the finite element framework. We point out the key role played by so-called von Karman strain measure, based upon the hypothesis that the displacements and strains (but not rotations) remain small before reaching the critical bifurcation point. Finally, we briefly illustrate another original concept in constructing the solution to linearized instability problem under coupled thermomechanics conditions.

6.1 Large Displacements and Deformations in 1D Truss with Instabilities 6.1.1 Large Strain Measures for 1D Truss We recall that replacing the coordinates in deformed by those in the initial configuration within the framework of linearized kinematics (or small displacement gradient theory) is the key result which allows simplifying the development of either strong or weak forms of the equilibrium equations. This implicitly implies that the two configurations are identical when computing the derivatives and integrals of the true stress acting in the deformed configuration, computing them with respect to the coordinates chosen in the initial configuration. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Ibrahimbegovic and R.-A. Mejia-Nava, Structural Engineering, Lecture Notes in Applied and Computational Mechanics 100, https://doi.org/10.1007/978-3-031-23592-4_6

273

274

6 Large Displacements and Instability: Buckling Versus Nonlinear Instability

For the geometrically nonlinear theory with large displacements and large displacement gradients, this kind of simplification can no longer be justified. In other words, with large displacements and deformations, we are obliged to choose the configuration with which we are working on the development of the theoretical formulation and numerical solution of a boundary value problem. In that sense, we can choose between the Lagrangian formulation considering that all the unknown variables are functions of coordinates xi in the initial configuration, and the Eulerian ϕ formulation where all the variables depend upon the coordinates xi in the deformed configuration. In principle, the Eulerian formulation is well suitable for problems of fluid mechanics where the only configuration of interest is the current deformed configuration (for example, the problems of flooding in a city area) and where the constitutive behavior does not depend on the deformation trajectory (for Newtonian fluids, for example, the Cauchy stress is directly proportional to the spatial velocity gradient; see Duvaut [115]). The Lagrangian formulation is more suitable for solid mechanics, since it uses the configuration we should know the best - the initial configuration of the solid body. Moreover, this kind of formulation requires taking into account of the complete deformation trajectory leading to a particular deformed configuration, which allows defining the corresponding evolution of the internal variables and the resulting value of stress for solid materials with inelastic behavior (for example, large strain plasticity and damage models). We elaborate further on these ideas for the 1D problem of an elastic truss-bar in the ¯ = [0, l], large displacement regime. In the initial configuration of the bar, denoted  we suppose that the stress-free position of each particle is described by its position vector, with the only non-zero component x ¯ ;  ¯ = [0, l] x ∈

(6.1)

We can describe the motion of the solid as the following transformation: ¯ × [0, T ] → R ϕ:

(6.2)

where t ∈ [0, T ] is the pseudo-time loading parameter. For a fixed value of pseudotime ‘t¯’, we can describe the deformation of the solid body by specifying the new position of each particle: (6.3) x ϕ = ϕt (x) The one-dimensional case allows us to introduce the displacement field u t (x), and describing the motion according to: x ϕ := ϕt (x) = x + u t (x)

(6.4)

The assembly of particles in the new position will constitute the deformed or current configuration of the solid body: ¯ ϕt = [0, ltϕ ] ; ltϕ = ϕt (l) 

(6.5)

6.1 Large Displacements and Deformations in 1D Truss with Instabilities ϕt (x)

r

r x

-

dx

t¯ϕ

R

b -t

-

ltϕ

l

xϕ

dxϕ

-b(x )dx ϕ



275

σ(xϕ )

ϕ

σ(xϕ + dxϕ )

xϕ - dxϕ

Fig. 6.1 Initial and deformed configurations of 1D truss in large displacement regime

The imposed zero displacement on the Dirichlet boundary for the choice of the reference frame in Fig. 6.1 implies that both the initial and deformed configurations include the origin of x-axis, with Γu := {0}: u t (0) = 0 (ϕt (0) = 0)

(6.6)

ϕ The Neumann boundary condition, corresponding to the imposed traction t¯t at the right end of the bar, can be written for the same case as: ϕ

ϕ

ϕ

σt (lt ) = t¯t ϕ

(6.7)

where σt (·) is the true or Cauchy stress. The boundary condition in (6.7) is written in spatial or Eulerian description, assuming that we are working with the deformed configuration and that all the variables are expressed as functions of coordinates x ϕ . The Neumann boundary will change constantly for such a case according to: ϕ Γσ = {lt = ϕt (l)}. Choosing the material or Lagrangian description, considering that all the variables are expressed as the function of the coordinate x in the initial configuration will allow us to impose the Neumann boundary condition at Γσ = {l}. However, this kind of choice will change the corresponding boundary condition, by requiring the appropriate representation of stress according to: σ (ϕ(x)) = P(x). Remark on the choice of formulation: other than two basic theoretical formulations of mechanics at large displacements, Eulerian and Lagrangian, many other choices are possible. Namely, within the framework of the incremental/iterative analysis used for a nonlinear problem of this kind, we can move the reference frame in each increment or in each iteration and thus obtain so-called updated Lagrangian formulation (e.g. see Bathe [27]). This motion of the reference frame at each increment or iteration can also be performed by a rotation, in which case we obtain the co-rotational formulation (e.g. see Crisfield [98]). The main advantage of any such formulation is that the moving frame would allow exploiting the results from the small strain theory pertinent to the tangent operator or internal force vector, which could simplify their computations for nonlinear theory (ideally, by only using a simple transformation, such as the rotation). However, fairly vague hypothesis on small strains with large (or moderate) rotations which would allow for this kind of simplification, as well as a large variety of the potential choices for the moving frame, make any such formulation rather difficult to develop in a very rigorous manner. There are basically only a few exceptions to this rule, such as a 2-node truss-bar element in 1D, a 3-node CST element in 2D or a 4-node tetrahedral element in 3D.

276

6 Large Displacements and Instability: Buckling Versus Nonlinear Instability

Each of these elements provides a constant approximation for the strain and rotation fields throughout the element domain and allows for an arbitrary placement of the moving reference frame separating large rotations from the small strains. Another type of theoretical formulation of interest is so-called arbitrary Eulerian-Lagrangian formulation (e.g. see Belytschko et al. [44]), which is suitable for the class of interaction problems where two interacting fields have very different nature of their motion properties (such as in fluid-structure interaction, with fluid motion which is handled by Eulerian and solid motion by Lagrangian formulation). 1D case: In the one-dimensional setting, the deformation gradient F takes a diagonal form: ⎡ ⎤ λ(x) 0 0 1 0⎦ (6.8) dxϕ = F dx ; F = ⎣ 0 0 01 The only component of deformation gradient for 1D case, which happens to be different from zero or one, is called stretch λ. With the corresponding choice of the reference frame, the stretch can be written: d x ϕ = λt (x) d x ; λt (x) :=

∂u t (x) dϕt (x) =1+ dx ∂x

(6.9)

For a homogeneous strain field, the stretch is independent of x, and it can be computed as the ratio of the deformed and the initial length of the 1D truss: ϕ

λt (x) ≡ λt = lt /l

(6.10)

It thus follows that no deformation is produced with the stretch equal to one, λt = 1. Moreover, with both initial and deformed length being positive, the stretch always remains positive. In other words, we can not reduce the deformed length d x ϕ = λt (x) d x to zero, nor can we make it infinitely long: 0 < λt (x) < ∞ ; ∀(x, t) ∈ [0, l] × [0, T ]

(6.11)

The stretch remains a useful deformation measure for 3D large deformations, even though it is not as easy to compute as in 1D case. Other deformation measures using the stretch can also be employed in a 3D case. In fact, one can provide an infinite number of large deformation measures, which can all be expressed as a monotonically increasing function of the stretch f (λ), verifying the following conditions: f (λ) : (0, ∞) → R ; f (1) = 0 & f  (1) = 1 & f  (λ) > 0 ; ∀λ

(6.12)

The first of the conditions in (6.12) above will ensure that zero deformation is obtained for a unit stretch. The second condition allows recovering from the proposed form the infinitesimal strain measure for the case of small displacement gradients. The final condition on monotonically increasing form of f (·) ensures that the infinite

6.1 Large Displacements and Deformations in 1D Truss with Instabilities

277

strain will be produced by the infinite stretch. One such family of the functions f (·), which all verify the conditions imposed in (6.12), is proposed by Doyle and Ericksen [113] and also by Hill [149]:  f (λ) =

1 (λm m

− 1) ; m = 0 lnλ ; m = 0

(6.13)

It is easy to see that the Green-Lagrange deformation is only a particular member of this family of large deformation measures, obtained for m = 2, which can be written in the 1D case as: E t (x) =

1 {[λt (x)]2 − 1} 2

(6.14)

One exceptional member of this family, which is a frequently used for experimental measurements, is obtained for m = 0 in the form of the natural or logarithmic large deformation measure: (6.15) t (x) = ln[λt (x)] The vast diversity of large deformation measures may seem inconvenient in forcing us to define an equally large number of the corresponding work-conjugate stress tensors, and identify the matching pair for each of them. However, such a large diversity of work-conjugate couples of stress and strain tensors in large deformations regime also provides an important advantage for easing the task of constructing the most appropriate constitutive model governing the material behavior for different deformation modes and various strain regimes, from very small to very large strains.

6.1.2 Strong and Weak Forms for 1D Truss in Large Displacements In the geometrically nonlinear case where the displacements are no longer small enough to ignore the difference between the initial and final deformed configurations, the equilibrium equations are no longer represented in a unique manner. There exist many different possibilities for expressing equilibrium in the large deformation problems, each one using a particular representation of the true stress tensor. The most frequently used stress tensors are presented subsequently, and their relationship with the Cauchy or true stress is explained. It is interesting to write the corresponding form of the equilibrium equations in the 1D setting. In this case, the only component of the deformation gradient which can change is the stretch λt (x), and any stress tensor has only one non-trivial component, with σ ϕ (x ϕ ) for Cauchy stress and P(x) for the first Piola-Kirchhoff stress. It is easy to check from the corresponding 1D form of (??) that these two stress components will always have the same numerical value:

278

6 Large Displacements and Instability: Buckling Versus Nonlinear Instability

⎡

λ0 F = ⎣0 1 00

⎤ 0 0 ⎦ ; Pt = J (σ ϕ ◦ ϕ)F−T =⇒ P(x) = σ ϕ (x ϕ ) 1

(6.16)

We can obtain the same conclusion from the Cauchy principle, by exploiting the 1D version of Nanson’s formula, which leads to: ntϕ ◦ ϕ t = Jt Ft−T nt =⇒ n ϕt (ϕt (x)) ≡ n t (x) = 1 ϕ ϕ tt := σt (x ϕ ) n t (x ϕ ) = Pt (x) n t (x) =⇒ σt (x ϕ ) ◦ φt (x) = Pt (x)  

 

(6.17)

=1

=1

The last result and the chain rule application with ϕ

∂σ (x ϕ ) ∂ϕt (x) ∂ Pt (x) = t ϕ ∂x ∂x  ∂ x

λt (x)

allow us to write the equilibrium equation of an infinitesimal segment in the deformed configuration, d x ϕ = λ d x, according to: ⎤

⎡ 0=

ϕ ⎢ ∂σ (ϕt (x)) λt (x) ⎣ t ϕ ∂x

⎥ + btϕ (x ϕ ) ⎦  

bt (x)/λt (x)

∂ Pt (x) + bt (x) = ∂x

(6.18)

We finally provide a brief summary of all these results in the 1D setting, which can further illustrate the relations between different work-conjugate stress-strain pairs for large deformation case. The 1D case allows representation of the deformation gradient by the identity matrix, apart from a single diagonal component expressing the stretch as the ratio between the deformed and the initial length of an infinitesimal element, λ = d x ϕ /d x. We start by expressing the internal virtual work by using the intrinsic form featuring the Cauchy stress: ϕ

G int (λ; w ) := ϕ lt

dw ϕ ϕ ϕ σ dx dxϕ

(6.19)

and then rewrite this result in several alternative forms, which introduce different stresses as:

6.1 Large Displacements and Deformations in 1D Truss with Instabilities

G int (λ; w) :=

 l

=

 l

=

 l

dw dx 

279

σP d x

d d [F =λ ]|=0

dw 1 λσ d x x λ 

d

(6.20)

τ

d d [lnλ ]|=0

dw λ x

d

d 1 2 d [E  = 2 (λ −1)]|=0

1 σ dx λ 

S

With such a large variety of possible choices for stress and strain tensors available for large strain problem formulation, the question arises which one should we favor? Often an important criterion for choosing a particular stress-strain couple concerns the corresponding constitutive model formulation, which should provide the most reliable representation of particular material behavior. These issues are studied in detail in the next section.

6.1.3 Linear Elastic Behavior for 1D Truss in Large Displacements All different possibilities for expressing the internal virtual work indicated in (6.20) are only different material representations of the same work, which is defined in terms of the Cauchy (or true) stress and infinitesimal deformation. In the case when the displacement gradient is small, all different possibilities for finite strain measure ought to reduce to this infinitesimal strain and all different stresses will be the same as the Cauchy stress. We can therefore conclude that any such material description of an elastic constitutive law for large deformations should reduce to Hook’s law for small deformation case. The constitutive model corresponding to Saint-Venant– Kirchhoff material takes this hypothesis one step further by postulating that Hook’s law also remains valid for large deformations, defined in terms of the Green-Lagrange strain measure. Thus, we can easily compute the corresponding values of the workconjugate second Piola-Kirchhoff stress, as well as confirm the model capability to recover Hook’s law for the limit case of small deformations:      du 1 1 du 2 σ =C + ⇔ σ = C 

 S = CE =⇒ lim λ→1,(du/d x) 1 λ dx 2 dx du/d x

(6.21) We can easily draw the same conclusion regarding Hook’s law validity for the limit case of the small deformations with many other large deformation measures. In other words, if the material behavior starts as linear elastic for the small deformation case, we should recover from different material models the same stress representation and the same elasticity tensor.

280

6 Large Displacements and Instability: Buckling Versus Nonlinear Instability

It is not possible to write the universal form of constitutive law in a large deformation regime. For example, if we wish to express the Saint-Venant-Kirchhoff constitutive model, defined in (6.21) as Hook’s law for Green-Lagrange strain and the second Piola-Kirchhoff stress, by using another work-conjugate couple of stress and strain tensors, it is no longer possible to keep the elasticity tensor with constant entries, as previously done in [31]. This can easily be shown for 1D case by expressing the Saint-Venant-Kirchhoff material law in terms of the first Piola-Kirchhoff stress and the stretch used as the conjugate large deformation measure. Thus, we can easily obtain a new format of constitutive law in nonlinear elasticity and the corresponding elastic tangent modulus: S = C 21 (λ2 − 1) & S = λ−1 P =⇒ P = C 21 (λ2 − 1)λ 1 =⇒ ddλP = C (3λ2 − 1) 2  

(6.22)

CF F

Similarly, we can obtain the appropriate form of the constitutive relationship for any other particular choice of the large deformation measure that we would like to use, resulting with a wide variety of possible stress-strain relations to use for the same material model. The unique form of constitutive relation, which contains any such relation as a special case, can be written for a hyperelastic material model in terms of the strain energy potential. For the Saint-Venant-Kirchhoff material model, the strain energy for the 1D case can be written:   2 S = ∂( 1∂ψ(·) 1 2 1 2 2 (λ −1)) (λ − 1) =⇒ ψ(λ) = C ∂ψ(·) 2 2 P = ∂λ

(6.23)

We have great difficulty in using the Saint-Venant-Kirchhoff constitutive model, pertaining to material instability in large compressive deformation case, which precludes that a very large strain be accompanied by a very large value of the true or Cauchy stress. A clear illustration of this difficulty can be provided for a simple1D case, where the strain energy can be written according to (6.23). We can thus easily compute the resulting values of the stress which accompany very large tensile and compressive strains, produced by an infinite and zero value of stretch, respectively: λ → 1 ⇒ P → 0 ; ψ → 0 λ → ∞ ⇒ P → ∞ ; ψ → ∞ λ → 0 ⇒ P → 0 ; ψ → C/8

(6.24)

In the 1D case, where the first Piola-Kirchhoff stress P and the true stress σ share the same numerical value, the last result would imply that producing an infinite compressive deformation and reducing the deformed bar length to zero, the SaintVenant-Kirchhoff material would require the zero value of the true stress:

6.1 Large Displacements and Deformations in 1D Truss with Instabilities

281

1 λ → 0 ⇔ (u → −l) ⇒ σ ≡ P := C λ(λ2 − 1) → 0! 2

(6.25)

The last result represents a paradox that can not be justified for any real material. One can explain what went wrong by computing the second variation of the strain energy, or the elastic tangent modulus for the present 1D framework, which quickly reveals that there is the minimum value of compressive stress with the corresponding stretch value beyond which the material instability will occur:  C 1 dP = (3λ2 − 1) = 0 ⇒ λ = 1/3 ⇒ Pmin = − √ C dλ 2 3 3

(6.26)

Such result of the Saint-Venant-Kirchhoff model will not allow decreasing the stress beyond the minimum value, which is clearly not justified in elasticity. An alternative point of view for explaining the Saint-Venant–Kirchhoff model deficiency in representing very large compressive strains can be provided through the loss of convexity of the strain energy. The latter occurs because of the presence of the inflextion point corresponding to the value of stretch in (6.26), which in turn precludes any large value of compressive stress (see Fig. 6.2). Remark on alternative linear elastic model: One can choose alternative strain measures to define the linear elastic constitutive model in large strain characterized by Young’s modulus, here denoted as C. For example, one can use logarithmic strain connected to Kirchhoff stress τ = C lnλ, or the stretch with the first PiolaKirchhoff stress P = Cλ, as a couple of alternative possibilities for formulating the virtual work of internal forces in (6.20). All these reduce to Hooke’s law for small strain, but do not define the same behavior in large strain, and should be transformed according to the rules in (6.20). One can also define 1D incompressible (nonlinear) elastic behavior (e.g.  see [161]) with strain energy density given as  W (λ) = r μαrr λαr + 2λ−0.5αr − 3 , where μr and αr are constitutive coefficients, Remark on space-curved membranes: one can take these considerations to spacecurved membranes, keeping many of the concepts introduced here but adapted to the 2D case, especially with using the principal stretches (the principal values of either ˜ ψ/E 6

P/E 6

λ

1

1 √ 1/ 3

λ

Fig. 6.2 Saint-Venant–Kirchhoff constitutive model: strain energy and the first Piola-Kirchhoff stress as a function of stretch

282

6 Large Displacements and Instability: Buckling Versus Nonlinear Instability

left or right stretch tensor; see [176]). In some of our previous works, we presented the rubber-like membranes using 2D incompressible constitutive law [199] and elastoplastic membranes with large elastic and large plastic strains using multiplicative decomposition of deformation gradient [167]. Remark on polyconvexity: This kind of approach for establishing the well-posed form of the strain energy can be generalized to the 3D case in terms of the polyconvexity conditions (see Ball [24]), which should provide the guarantees that the large strains remain accompanied by large stresses.

6.1.4 Finite Element Method for 1D Truss in Large Displacements In order to illustrate all the main steps in constructing the finite element approximation of large displacement elasticity with Saint-Venant–Kirchhoff material model, we choose the simplest problem of 1D elasticity with the truss-bar element. More precisely, we study the large displacements of an elastic bar (see Fig. 6.3) of length l, the cross-section A, Young’s modulus E, built-in on the right end and loaded on the left end by a traction force t¯i along with a uniformly distributed loading b. Without loss of generality, we consider that the bar is straight and its motion is restricted to the plane x1 -x2 . By placing x1 axis along the initial configuration of the bar and by taking into account that there remains only one non-zero stress component in the bar, we can write the discrete approximation of the weak form of the boundary value problem for the 1D case given in (6.20), to obtain: G(u i ; wi ) :=

l 0

with:

Γ11 S11 d x −

Γ11 =

dϕ1 dw1 dx dx

+

l

wi bi d x 0 dϕ2 dw2 dx dx

S11 = C1111 E 11 ; E 11 

− wi (l)t¯i = 0 (6.27)

1 1 2 2 2 = [( dϕ ) + ( dϕ ) − 1] dx 2 dx

E

where S11 is the second Piola-Kirchhoff stress, E 11 is the Green-Lagrange deformation and Γ11 is its variation. This formulation is used as the starting point for constructing the discrete approximation of the solution by Galerkin’s method. Such an approximation belongs to a

Fig. 6.3 Elastic truss-bar in large displacements: initial and deformed configurations for 1D model

x2

6t¯ i -

x1 ≡ x

d1

XXX XXX 6 d2 bi , E, AX =X 1X XX X l

6.1 Large Displacements and Deformations in 1D Truss with Instabilities

H Na (x)  H  1.0 H  H Hs s s a−1

⎧ ⎪ (x − xa )/la−1 ; xa−1 ≤ x ≤ xa ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

(xa+1 − x)/la ; xa ≤ x ≤ xa+1 0 ; otherwise

a+1

a la−1

Na (x) =

283

la

Fig. 6.4 Piece-wise linear discrete approximation of the displacement field

finite-dimensional space, which we choose as the appropriate sub-space of the true, infinite-dimensional solution space. The same kind of finite dimensional sub-space is chosen for constructing the discrete approximation of the weighting functions or h virtual displacements, with w h ∈ Vi,0 . Here, the subscript ‘0’ indicates that the virtual displacements must remain equal to zero on the Dirichlet boundary, where the real displacement takes the imposed value. The required regularity of this finite-dimensional approximation space depends, in general, upon the choice of the strain measure and the constitutive law. For the choice made in (6.27), with the GreenLagrange deformation and the Saint-Venant–Kirchhoff material model, we can admit the simplest discrete approximation for real and virtual displacement fields which are linear in each element (see Fig. 6.4). The linear approximation of the displacements implies that their derivatives remain constant in each element: ∂ϕ1h ∂x

=

n 

d Na (x) dx

(xa + da,1 )

d Na (x) dx

da,2

a=1

∂ϕ2h ∂x

=

n 

⎧ 1 ⎨ la−1 ; xa−1 ≤ x ≤ xa d Na = −1 =⇒ ; xa ≤ x ≤ xa+1 ⎩ la dx 0 ; otherwise

a=1

It thus follows that the discrete approximations of the Green-Lagrange strain and the second Piola-Kirchhoff stress will also be constant in each element. By introducing these approximations into the weak form in (6.27), we can obtain the final result in terms of the Galerkin equation expressing the principle of virtual work (or the weak form of equilibrium) within the framework of the chosen discrete approximation: n  2  wa,i ra,i = 0 (6.28) a=1 i=1

284

6 Large Displacements and Instability: Buckling Versus Nonlinear Instability

ra,i :=

eiT 

l −

l  d Na dx

0 0

0

 n  d Nb

d Na dx

b=1

(xb + dx ∂ Nb d d x b,2

db,1 )

 h S11 dx





int f a,i (d)

Na bi d x − Na (l)t¯i

0

 S11 = E



ext f a,i

n n 1   [ 2 a=1 b=1

 d Na

(xa + dx d Na d d x a,2

da,1 )

 T  d Nb

(xb + dx d Nb d d x b,2

db,1 )

 − 1]

By considering arbitrary non-zero nodal values of virtual displacements wa,i , where a = 1, 2, ..., n and i = 1, n dm , we can obtain from Galerkin’s equation a set of nonlinear algebraic equations: int ext (d) = f a,i ra,i (d) = 0 =⇒ f a,i

(6.29)

This set of equations governing the equilibrium of an elastic bar in large displacements can also be written in matrix notation according to: ⎡

fint (d) = fext

⎡ int ⎤ ⎡ ext ⎤ ⎤ f 1,1 f 1,1 d1,1 ⎢ d1,2 ⎥ ⎢ f int ⎥ ⎢ f ext ⎥ ⎢ ⎢ 1,2 ⎥ ⎢ 1,2 ⎥ ⎥ ⎢. ⎥ ⎢ ⎢ ⎥ ⎥ ⎢ ⎥ int ⎢ . ⎥ ext ⎢ . ⎥ ⎢ ⎢ ⎢ ⎥ ⎥ ⎥ ; d = ⎢. ⎥ ; f = ⎢. ⎥ ; f = ⎢. ⎥ ⎢. ⎥ ⎢. ⎢. ⎥ ⎥ ⎢ ⎢ int ⎥ ⎢ ext ⎥ ⎥ ⎣ dn,1 ⎦ ⎣ f n,1 ⎦ ⎣ f n,1 ⎦ ext int dn,2 f n,2 f n,2

(6.30)

We show subsequently how to use the finite element method in order to construct the set of nonlinear equations in (6.30) more efficiently. To that end, we abandon the global point of view of Galerkin’s method, in favor of the local point of view for each finite element. We can thus obtain an important benefit of processing in a unified manner all the finite elements of the same type, such as all truss-bar elements with the same number of nodes. For a 2-node bar element in large displacements (see Fig. 6.5), the local description of this kind is defined in Table 6.1. The initial configuration of a 2-node bar element is reconstructed from its parent element according to: x(ξ )

= le

2  a=1

Nae (ξ )xa ; Nae =

1 (1 + ξa ξ ), ξa = 2



−1 , a = 1 1, a = 2

(6.31)

6.1 Large Displacements and Deformations in 1D Truss with Instabilities 1

s

2 ...

s

1

2 ...

s

a

s

s

a+1 e

s

@ 1. @

Na (x)

s

285

nnode

s

nelem

@ 1. @

Na+1 (x)

Fig. 6.5 Finite element model of a truss-bar in large displacements composed of 2-node truss-bar elements Table 6.1 Local description of a 2-node truss-bar element Ingredient Global description Local description Domain Nodes d.o.f. Shape functions Interpolations

[xa , xa+1 ] {a, a + 1} {da,1 , da,2 , da+1,1 , da+1,2 } {Na (x), Na+1 (x)} u ih (x) = Na (x)da,i +Na+1 (x)da+1,i

[ξ1 , ξ2 ] (1, 2) e , de , de , de } {d1,1 1,2 2,1 2,2 {N1 (ξ ), N2 (ξ )} e u ih (ξ ) = N1e (ξ )d1,i e e +N2 (ξ )d2,i

For an isoparametric element, the same shape functions Nae (ξ ) are used for constructing the discrete approximation of the displacement field. The isoparametric elements will thus allow us to construct any deformed configuration very easily: ⎧ ⎪ ⎪ ⎨ ϕ1h (ξ ) ⎪ ⎪ ⎩ ϕ2h (ξ )

= le

= le

2  a=1 2  a=1

e Nae (ξ ) (xa + da,1 )

(6.32) e Nae (ξ ) da,2

In the Lagrangian formulation framework, we ought to construct the derivatives of the displacement components with respect to coordinate x in the initial configuration. For any isoparametric element the derivative computation ought to be carried out by exploiting the chain rule. For the chosen 2-node bar element this results with: j (ξ )

= x,ξ = le

=⇒

∂ Na ∂x

=

x2 −x1 2

∂ Na 1 ∂ξ j (ξ )

=

le 2

=

ξa 2 2 le

 =

(6.33)

−1/l e , a = 1 1/l e , a = 2

We can thus easily confirm that the resulting discrete approximation of derivatives is constant within each element: dϕ1h ∂ N1 ∂ N1 ∂ N2 dϕ2h ∂ N2 = = (x1 + d1,1 ) + (x2 + d2,1 ) ; d1,2 + d2,2 (6.34) dx ∂x ∂x dx ∂x ∂x 







−1/l e

1/l e

−1/l e

1/l e

286

6 Large Displacements and Instability: Buckling Versus Nonlinear Instability

By taking into account that l e = x2e − x1e and 0 = y2e − y1e , we can rewrite the last result in matrix notation:      dϕ1 2 dϕ2 2 2 1 + = e 2 xe T Hde + e 2 de,T Hde + 1 (6.35) dx dx (l ) (l ) le where we denote: ⎤ ⎡ e ⎤ ⎡ ⎤ d1,1 x1 1 0 −1 0 ⎢ y1 ⎥ ⎢ e ⎥ ⎢ 0 1 0 −1 ⎥ e ⎥ ; de = ⎢ d1,2 ⎥ ⎢ ⎥ xe = ⎢ e ⎦ ; H =⎣ ⎣ x2 ⎦ ⎣ d2,1 −1 0 1 0 ⎦ e y2 0 −1 0 1 d2,2 ⎡

(6.36)

With these results in hand, we can write the corresponding discrete approximation of the Green-Lagrange strain for a 2-node bar element according to:

=

h E 11 le

1 xe T He de (l e )2

+

1 de T He de 2(l e )2

(6.37)

For an elastic bar whose constitutive behavior is governed by the Saint-Venant– Kirchhoff material model, we can then obtain the corresponding finite element approximation of the second Piola-Kirchhoff stress by multiplying the last expression with Young’s modulus:

h S11 le

= E( (l e1)2 xe T Hde +

1 de T Hde ) 2(l e )2

(6.38)

The finite element approximation of the derivatives of virtual displacement field can be constructed in the same manner as the one for real displacement field in (6.35). We can further obtain the corresponding finite element approximation of the virtual Green-Lagrange strain: ⎤ e w1,1 ⎢ we ⎥ 1,2 ⎥ ; we = ⎢ e ⎦ ⎣ w2,1 e w2,2 ⎡

Γ11 =

1 (xe T (l e )2

+ de T )Hwe

(6.39)

It is interesting to note that the last result can also be obtained as the directional derivative of the Green-Lagrange strain approximation in (6.37) in the direction of virtual displacement. With these results in hand, we can write a particular 2-node bar element contribution to the discrete approximation of the virtual work according to:

6.1 Large Displacements and Deformations in 1D Truss with Instabilities

we T fint,e (de ) =



287

Γ11 S11 d x

le



1 HT (xe + de )S11 d x e 2 l e (l )  

const.

1 1 = we T HT e (xe + de )S11 e dx l l e l  

= we T

(6.40)

=1

⇒

fint,e = HT

1 e h (x + de )S11 le

where the second Piola-Kirchhoff stress approximation given in (6.38). The same isoparametric finite element approximations are employed in constructing the linearized weak form, which is the basis of Newton’s iterative solution procedure. For a 2-node bar element, this results with: Lin[re (de )] = re (de ) + Ke (de ) ue ;

Kme

Ke = Kme + Kge

;

1 E 1 S11 = H e (xe + de ) e e (xe T + de T )H ; Kge = e H l l l l

(6.41)

T

where Ke is the consistent tangent stiffness matrix for such element. We note that the tangent stiffness consists of two parts; the first Kme is referred to as the material stiffness and the second Kge is the geometric stiffness matrix which is characteristic of large displacement problems. Even though the derivation of the element stiffness matrix and its residual force vector is carried out for a 2-node truss-bar element placed along x1 axis of the global coordinate system, the final results will also apply to an arbitrary position of the element. Namely, any constant value of strain or stress obtained from 2-node element approximation, will remain invariant under reference frame rotation by an arbitrary angle α; hence, for general orientation of the 2-node truss-bar element, with cos α = (x2 − x1 )/l e and sin α = (y2 − y1 )/l e ), we can still write: ⎡

fint,e

⎤ −(x2 − x1 ) − (d2,1 − d1,1 ) 1 ⎢ −(y2 − y1 ) − (d2,2 − d1,2 ) ⎥ ⎥ S11 = e⎢ l ⎣ (x2 − x1 ) + (d2,1 − d1,1 ) ⎦ (y2 − y1 ) + (d2,2 − d1,2 )

(6.42)

Remark on higher order finite element interpolations: it is possible (e.g. see Ibrahimbegovic [161]) to generalize the finite element approximation presented in this section to a bar element with n en > 2 nodes and a higher order interpolation; to that end, we employ the shape functions in terms of the Lagrange polynomi-

288

6 Large Displacements and Instability: Buckling Versus Nonlinear Instability

als of order n en − 1, the same as defined in (2.109). The elements of this kind can have an arbitrary curvilinear axis, parameterized by the arc-length s. The derivative computation of the element shape functions can be carried out by using the chain rule: " d N e (ξ ) d Nae (ξ ) dy 2 = j (ξ1 ) dξa ; j (ξ ) = ( ddξx )2 + ( dξ ) ; ds n n en en  d Nae (ξ ) e dy  d Nae (ξ ) e dx = xa ; dξ = ya ; dξ dξ dξ a=1

dx(ξ ) ds dϕ(ξ ) ds

= le

=

n en  a=1 n en  a=1

le

a=1

Bae xae

du(ξ ) ds

;

Bae (xae

+

dae )

=

a=1

le

;

n en 

Bae

Bae dae ;

= diag[

(6.43)

d Nae (ξ ) d Nae (ξ ) , ds ] ds

The only non-zero components of the second Piola-Kirchhoff stress and the internal force vector can then be computed (see [161]) according to: ) = E 21 [ dϕds(ξ ) dϕ(ξ − 1] ds T

S11 (ξ ) le

fint,e = [faint,e ] ; faint,e =

n en −1  l=1

l) Bae dϕ(ξ S11 (ξl ) j (ξl )wl ds

(6.44)

We indicated in the last expression that the internal force vector for this higher order element is computed by numerical integration by using n en − 1 Gauss quadrature points with corresponding choice of abscissas ξl and weights wl . It is easy to verify that for n en = 2, the last result will be the same as the one in (6.42). We can carry on the same way to compute the tangent stiffness matrix with both material and geometric parts: Ke := Kme + Kge ; e e Kme = [Km,ab ] ; Km,ab =

Kge =

e [Kg,ab ]

;

e Kg,ab

=

n en −1 

l=1 n en −1  l=1

l) Bae dϕ(ξ E dϕds(ξl ) Be,T b j (ξl )wl ds T

(6.45)

Bae Be,T b S11 (ξl ) j (ξl )wl

6.1.5 Buckling, Nonlinear Instability and Detection Criteria 6.1.5.1

Linear Instability or Buckling

Euler’s buckling: is the classical example of (linear) geometric instability. We make a brief digression to briefly illustrate this linear instability with the classical Euler beam model submitted to a compressive axial force P and a small perturbation in terms of transverse displacement v(x) (see Fig. 6.6). We can judge the stability of such an equilibrium state with respect to the perturbation effects. Namely, if

6.1 Large Displacements and Deformations in 1D Truss with Instabilities

ϕ(x) = (x; v(x))

6 e2 -P - s e1

e 3 ≡ eϕ 3



c s

x

c s -

289

deformed configuration initial configuration

sP l

j

Fig. 6.6 Euler’s buckling: initial and deformed configuration of beam under critical force in the deformed configuration produced by perturbation

the beam response remains small and returns to the original configuration once the perturbation is removed, then the equilibrium state is stable. On the other hand, if the displacement increases in a disproportional manner with respect to such a perturbation, the equilibrium state is unstable. Somewhere in-between these two is placed the so-called critical equilibrium state, which marks the transition from stable to unstable equilibrium states (or vice versa). For the critical equilibrium state, the beam will remain in equilibrium under the critical value of axial force P = Pcr , in the perturbed configuration produced with a transverse displacement v(x). Therefore, we can establish the equilibrium equation in this deformed configuration according to: M(x) = −Pv(x)

(6.46)

where M(x) is the bending moment in the beam and P is the compressive force acting at both ends. The last equation, establishing equilibrium in the deformed configuration, is the only nonlinear equation that we use in defining Euler’s (linear) buckling problem. Namely, given the crucial hypothesis that the displacement v(x) remains small before reaching critical equilibrium state, we will use linear kinematics equation to compute 2 . Moreover, we will consider the curvature of the beam according to: κ(x) = − d dv(x) x2 linear elastic constitutive equation in terms of Hook’s law: σ (x, y) = E(x, y), along with the classical hypothesis for Euler’s beam on plane sections that remain plane leading to a linear variation of strain through the beam thickness (x, y) = yκ(x). The last two equations jointly lead to the constitutive equation in terms of momentcurvature relation, which can be written: M(x) = E I

d 2 v(x) ; M(x) = dx2



−yσ (x, y) d A ; I =

A

y2 d A A

where E is Young’s modulus and I is the moment of inertia of the beam. By exploiting this equation we can express the equilibrium equation directly in terms of the displacement field, whose analytic solution provides the well-known result for Euler buckling load (see Fig. 6.6), which can be written:

290

6 Large Displacements and Instability: Buckling Versus Nonlinear Instability

⎫ 4 2  0 = E I d dv(x) + P d dv(x) ;⎬ x4 x2 vcr (x) = sin πlx v(0) = v(l) = 0 ; =⇒ Pcr = π 2 E I /l 2 ⎭ 2 d 2 v(l) = E I = 0; E I d dv(0) 2 2 x dx

(6.47)

The critical equilibrium point of this kind is referred to as bifurcation, with a typical symmetry-breaking post-bifurcation response. In order to obtain the result for Euler buckling, we combined a linear kinematics equation and linear constitutive equation with a nonlinear equilibrium equation, with the latter being established in the deformed configuration. These are typical ingredients of what we refer to as the linear instability problem, where the displacements, strains and stresses remain small before reaching the critical equilibrium point.

6.1.5.2

Nonlinear Instability

Next, we develop a more general framework for the study of instability problems than the one used for solving the Euler buckling problem, where we assume that the displacements, rotations and strains can be (very) large before arriving to the critical equilibrium point, and moreover we admit nonlinear inelastic constitutive behavior. We will call those the problems of nonlinear instability. The increase of complexity of nonlinear instability problems is such, that it is practically impossible to obtain the analytic solution (of the strong form) for any such problem, comparable to the one we have constructed for linear instability problem of Euler buckling. Therefore, we can only obtain the corresponding solution of the weak form by using the finite element method. We will first address the nonlinear instability problems where displacement can be large when arriving at the critical equilibrium point, but the constitutive behavior remains elastic. We will introduce this class of problems by means of a simple example, considering a shallow truss composed of 2 truss-bar elements with 2-nodes, which is loaded by a vertical force at the apex (see Fig. 6.7). By assuming that the elastic constitutive behavior of each bar is described by Saint-Venant–Kirchhoff material model, and by taking into account the symmetry, we can write the weak form of the equilibrium equations according to:

−f

c

E, A, l

6f, v c ` ` `l````` `c h b

6

fcr1

-

2h

h

−v

fcr2

Fig. 6.7 Shallow truss and its force-displacement diagram with two critical equilibrium states

6.1 Large Displacements and Deformations in 1D Truss with Instabilities

291

h

G

ext

=G

int

1   ⇔ w f = w 2 AS11 ( y2 − y1 +v) ; S11 = E l12 (hv + 21 v 2 ) l 



(6.48)

f int

=⇒ f =

2E A (h l3

+ v)(hv + 21 v 2 )

In the last expression, we took into account that the second Piola-Kirchhoff stress is constant in each bar and can be written explicitly in terms of vertical displacement v. It is easy to see that maintaining equilibrium will require zero value of vertical external force for the displacement values v = −h and v = −2h. These two equilibrium states correspond, respectively, to the truss equilibrium state where both bars are horizontal and the equilibrium state where bars have been moved through to the opposite side so that the deformed length of each bar becomes the same as in the initial configuration, which implies zero internal force. However, even if they share the same (zero) value of external force, these two equilibrium states are not the same: the first is unstable since any small force increase will lead to a very large increase in displacement, whereas the second is stable with a small force increase accompanied by a proportionally small displacement increase. It thus follows that in-between these two states we will eventually find a critical equilibrium state, which is the first to bring disproportionately large displacement increase accompanying a small force increase. This condition can be written in the inverse form, clearly showing a very large response increase for a small force perturbation, which can be recast as the equivalent condition stating that the zero value of tangent stiffness corresponds to a critical equilibrium state d f int dv → ∞ & f = f int =⇒ =: K = 0 df dv

(6.49)

In fact, in this example, we can compute two critical equilibrium states, which correspond to: 0 = K :=

d f int 2E A 2E A 1 = 3 (h + v)2 + 3 (hv + v 2 ) =⇒ dv l l 2

√ √ 3 3 vcr 1 = −h(1 − 3/3) =⇒ f cr 1 = − 2ElAh 3 9 √ √ 3 3 vcr 2 = −h(1 + 3/3) =⇒ f cr 2 = 2ElAh 3 9

(6.50)

(6.51)

The first critical state concerns passing from stable to unstable equilibrium states, whereas the second critical state marks return to stable equilibrium states.

6.1.5.3

Detection Criterion for Critical Equilibrium State

On the basis of the discussion in the previous section, we can provide the first general criterion for detecting a critical equilibrium state in problems of practical interest

292

6 Large Displacements and Instability: Buckling Versus Nonlinear Instability

with a large number of equilibrium equations. Namely, we could check when the tangent stiffness matrix becomes singular, with its determinant taking zero value at the critical state of equilibrium: int ˆ cr ) = ∂f (dcr ) ˆ cr )] = 0 ; K(d det[K(d ∂d

(6.52)

Although theoretically correct, the criterion for detection of equilibrium state in terms of zero determinant is not practical to use, for two reasons. First, the cost of computing the determinant of n × n matrix is prohibitively high (of the order of n!) and second, a rapid increase in determinant values in the neighborhood of the critical point can lead to significant convergence difficulties of Newton’s iterative method for computing the corresponding displacement dcr resulting with the zero value of the determinant. We are thus prompted to seek yet other detection criteria for critical equilibrium state and the more suitable choice for computation of critical equilibrium points. Detection criterion for instability based upon a variation of total potential energy is an alternative criterion for detecting a critical equilibrium state. The critical point is reached for the equilibrium state where the second variation of the total potential energy becomes zero. For a hyperelastic constitutive model, it is easy to establish the direct connection of this energy criterion with the one previously proposed based upon the singularity of the tangent stiffness matrix. Namely, for a geometrically nonlinear problem with the total potential energy Π (ϕ), we can obtain the first and the second variations and establish their connection with the equilibrium equation and the tangent stiffness matrix, respectively: d [Π (ϕ ε ]ε=0 = G(ϕ; w) = w T (ˆfint (d) − fext ) = 0 Dw Π (ϕ) := dε d2 d T Dw [Dw Π (ϕ)] := dε 2 [Π (ϕ ε )]ε=0 = dε [G(ϕ ε ; w)]ε=0 = w Kw

(6.53)

With these results in hand, we can express the total potential energy in the neighborhood of the chosen equilibrium state ϕ, for any new equilibrium state which is produced by a small, kinematically admissible perturbation w. To that end, we can use the Taylor series representation to write: Π (ϕ + w) ≈ Π (ϕ) + Dw Π (ϕ) + Dw [Dw Π (ϕ)] ≈ Π (ϕ) + G(ϕ; w) +w T Kw  

(6.54)

=0

With the first variation which is equal to zero (since it represents equilibrium equation of an equilibrium state), the difference in total potential energy of these adjacent equilibrium states will be controlled by the second variation. We can thus conclude that the given equilibrium state is stable, if any other equilibrium state in its neighborhood, which is produced by a kinematically admissible perturbation, will impose an increase in energy. For such a case, removing the perturbation will

6.1 Large Displacements and Deformations in 1D Truss with Instabilities

293

allow returning to the original equilibrium state before perturbation. We can also conclude that any stable equilibrium state will have the positive definite tangent stiffness matrix, which is written: Π (ϕ + w) > Π (ϕ) ; ∀w ∈ V0 =⇒ Dw [Dw Π (ϕ] := w T Kw > 0 ⇔ K positive definite

(6.55)

The equilibrium state is considered unstable if there exists a small, kinematically admissible perturbation that will reduce the total potential energy. The tangent stiffness matrix of the unstable equilibrium state is negative definite: Π (ϕ + w) < Π (ϕ) ; ∃w ∈ V0 =⇒ Dw [Dw Π (ϕ] := w T Kw < 0 ⇔ K negative definite

(6.56)

The principle of minimum potential energy only applies to stable equilibrium states, which guarantees the return to the original equilibrium state following any perturbation. The same principle does not apply to unstable equilibrium states, where a small perturbation will produce the state with smaller energy which would no longer allow recovering the original equilibrium state after perturbation removal. In fact, the unstable equilibrium state perturbation is very likely not to remain limited to small displacements, but rather to lead to large displacements and strains with a great risk of subsequent structural failure. Needless to say, the risk of unstable, or even critical equilibrium states, should not in general be acceptable. The critical equilibrium state indicates passing from stable to unstable states (or vice versa), with the second variation of its total potential energy equal to zero. The tangent stiffness matrix of the critical equilibrium state is a singular matrix, for which we can write: Π (ϕ + w) = Π (ϕ) ; ∀w ∈ V0 =⇒ Dw [Dw Π (ϕ] := w T Kw = 0 ⇔ detK = 0 ; K singular matrix

(6.57)

A graphical illustration of these results deduced from the detection criterion based upon the total potential energy variations is given in Fig. 6.8, for the case where the potential energy potential is defined by gravity field.

d t d

Π(ϕ + w)

d t d

Π(ϕ) i)

ii)

Π(ϕ) Π(ϕ + w)

d t d

Π(ϕ) ≈ Π(ϕ + w)

iii)

Fig. 6.8 Total potential energy of equilibrium state: (i) stable - Π (ϕ) → min., (ii) unstable Π (ϕ) → max., (iii) critical Dw [Dw Π (ϕcr )] = 0.

294

6 Large Displacements and Instability: Buckling Versus Nonlinear Instability

Detection criterion based on zero eigenvalues is another detection criterion for instability, which can be used to verify if the tangent stiffness is a singular matrix. Namely, by choosing for the virtual displacement vector in (6.57) the instability mode ψ with zero eigenvalue, we can further write: w = ψ =⇒ 0 = [K − 

λ I]ψ = Kψ

(6.58)

=0

The main advantage of such a criterion is in providing not only the indication of the critical equilibrium state, but also on the kind of perturbation which reveals instability, in terms of the corresponding eigenvector ψ. We will further illustrate the main advantage of this detection criterion, which can indicate the type of instability mode by computing the eigenvector of the tangent stiffness at the critical equilibrium point associated with zero eigenvalue. For that reason, we will go back to the example of a simple truss structure, which consists of two bars with linear hyperelastic constitutive behavior described by the Saint-Venant-Kirchoff model. We will accept this time any form of the initial configuration, with either shallow or deep truss decided by the corresponding values of b and h (see Fig. 6.7). We use again for each bar a 2-node isoparametric finite element, which allows us to construct the tangent stiffness matrix by the finite element assembly procedure: element 1:  T  b + u EA b + u := + = + h + v l3 h + v E 1 2 1 2 = l 2 (bu + hv + 2 u + 2 v ) 

K S

(1)

(1)

Km(1)

Kg(1)

S (1) A l



10 01

 ;

element 2:  T  −b + u E A −b + u + l3 h+v h+v = lE2 (−bu + hv + 21 u 2 + 21 v 2 ) 

K(2) := Km(2) + Kg(2) = S (2)

S (2) A l



10 01

 ;

which results with: K=K

(1)

+K

(2)

2E A := 3 l



   2E A 1 2 1 2 10 u 2 + b2 u(h + v) + 3 (hv + u + v ) u(h + v) (h + v)2 01 l 2 2

The symmetry of the structure imposes that the horizontal displacement will remain equal to zero under applied vertical force at the apex of the truss, until reaching the critical equilibrium point; with this result (u cr = 0), the tangent stiffness matrix becomes a diagonal matrix and the solution to eigenvalue problem Kcr ψ = 0 for critical mode computation can be obtained in closed form. We can have two cases: the first, where the perturbation of the equilibrium configuration brought by instability mode will push down the truss further in the same direction ψ T = (0, 1); we thus recover the previously presented result on nonlinear instability of this truss with a very large value of vertical displacement vcr at the critical equilibrium point:

6.1 Large Displacements and Deformations in 1D Truss with Instabilities

 0 = Kcr ψ := =⇒

2E A l3

2 )0 b2 + (hvcr + 21 vcr 2 ) 0 (h + vcr )2 + (hv + 21 vcr

295

  0 1

u cr = 0 √ vcr = −h(1 ± 3/3)

This kind of critical point provides the maximum load level which can be carried by the truss when both bars are compressed, and it is referred to as the limit load point. In the second case, the perturbation is produced by a horizontal displacement with instability mode ψ T = (1, 0), leading to the bifurcation point, where lateral motion, either to the left or to the right, breaks the symmetry of the structure:  0 = Kcr ψ := =⇒

2E A l3

2 )0 b2 + (hvcr + 21 vcr 2 ) 0 (h + vcr )2 + (hvcr + 21 vcr

  1 0

u cr = 0 √ vcr = −h ± h 2 − 2b2

We note that the shallow truss with h 2 − 2b2 < 0, can not have the bifurcation point. The latter will be typical, however, of a deep truss with h  b (which guarantees that h 2 − 2b2 > 0), where the last expression will provide the solution for vertical displacement vcr at bifurcation point, leading to phenomena very much equivalent to the Euler buckling. In conclusion, the last detection criterion for nonlinear instability, which is based upon the computation of the eigenvectors of tangent stiffness matrix that reveals the type of instability, will also remain applicable to the linear instability problems and provide an equivalent result to Euler’s buckling solution. Moreover, the same conclusion remains valid for much more complex structures [220] than the simple truss used herein for illustration.

6.1.5.4

Direct Computation of Critical Equilibrium Points

Next, we discuss the solution methods for the boundary value problem in presence of geometric instabilities, which provides the direct computation of the critical equilibrium points (e.g. see [179]). This method provides the complementary information to usual solution methods used in the presence of instabilities, such as arc-length, which seeks to achieve stabilization by a number of different proposals (see Riks [324], Batoz and Dhatt [35] or Crisfield [96]) Two methods are often used together, especially when the direct computations of the critical points are carried out by Newton’s iterative procedure where we need the best possible initial guess to ensure convergence. By using these two methods combined, we can trace the complete forcedisplacement diagram, detect the critical points, identifies their true nature between limit and bifurcation point, and explore any post-bifurcation path. We postpone this to the next section, and focus only on direct computations of critical equilibrium points.

296

6 Large Displacements and Instability: Buckling Versus Nonlinear Instability

The proposed procedure for direct computation of critical points seeks to detect the critical equilibrium states for nonlinear instability phenomena and converge the iterative procedure directly to this state, with no need to visit any other equilibrium state. The final result of this computation leads to the displacement vector and critical load parameter value for the critical equilibrium state, accompanied by the supplementary information which can reveal if we have detected a limit or a bifurcation point. The direct computation is carried out by Newton’s iterative procedure. Therefore, it is very important to have a very good initial guess in order to ensure the convergence of this computation. For that reason, we rely upon the arc-length method in order to get closer to the critical point and to provide the best iterative guess for initial values which will be placed within the radius of convergence of Newton’s method. Practically, as soon as the determinant of tangent stiffness will change-the-sign within the current step, this will indicate the presence of the critical point. We can then take the displacement and load parameter from the beginning of that step for the initial values of our direct computation. Thus, besides the equilibrium equations for the critical state, the direct computation needs an additional condition, which can be written in two different manners:  d2 det[Kcr ] = 0 [Π (ϕ )] = 0 ⇔ (6.59) cr,ε Kcr ψ = 0 dε2 ε=0 where Kcr = ∂ ˆfint (dcr )/∂d is the tangent stiffness matrix at the critical equilibrium state. The final choice we will make depends upon what kind of information about the critical state we would like to provide. By choosing the first condition, we can obtain the formulation where we only compute the critical value of load parameter and corresponding displacements, by solving the following augmented system: 

ˆfint (dcr ) − λcr f0 det[Kcr ]

 = 0 → (dcr , λcr )

(6.60)

This is, in general, a well-posed problem with n + 1 nonlinear algebraic equations with the same number of unknowns, including the displacement at the critical state and the corresponding load parameter (dcr , λcr ). By using Newton’s iterative procedure for solving this problem, at each iteration we perform the consistent linearization of these equations, obtain the increments for critical displacements and load parameters and carry out the corresponding updates: (i) = 1, 2, . . .  (i)  (i)   (i)  Δdcr λcr f0 − fint (d(i) ] K tr [K−1 ∂K cr ) ∂d = −1 Δλ(i) 0 f0ext cr cr (i) (i+1) (i) d(i+1) = d(i) = λ(i) cr cr + Δdcr ; λcr cr + Δλcr

(6.61)

6.1 Large Displacements and Deformations in 1D Truss with Instabilities

297

The iterative procedure will continue until the convergence is reached within the specified tolerance. One disadvantage of this method concerns the lack of information about the computed equilibrium point. In order to obtain this kind of information, we need an alternative formulation of the extended system, where the critical mode ψ which renders the tangent stiffness matrix singular is also computed. This kind of formulation leads to a set of ‘2n + 1’ nonlinear algebraic equations with the same number of unknowns: ⎡ ⎤ ˆfint (dcr ) − λcr f0 ) ⎣ ⎦ = 0 → (dcr , λcr , ψ) (6.62) Kcr ψ l(ψ) The last of equations in (6.62) above represents a complementary condition which is chosen in agreement with the kind of instability we would like to detect, between a limit and a bifurcation point. The precise form of the complementary condition to impose can be obtained from the linearized form of equilibrium equations at the critical point, by scalar-multiplication with the critical mode ψ and by taking into account the symmetry of the tangent stiffness matrix to obtain the following result: =0



  0 = ψ {ˆfint (dcr ) − λcr f0 +Kcr Δd − f0ext Δλ} = ΔdT Kcr ψ −ψ T f0ext Δλ  

T

(6.63)

=0

The last result can be verified for a critical equilibrium state pertaining to a limit point where the external load has reached the maximum value with no further increase, which implies Δλ = 0. In such a case, one complementary condition that can be used in (6.62) will serve to normalize the computed critical mode with: l(ψ) := ψ  −1 = 0

(6.64)

However, the result in (6.63) above can also be verified for the case where the external load can still increase, but the critical mode will push the structure in the direction that is orthogonal to the applied external load direction resulting with ψ T f0ext = 0. We will thus find a bifurcation point, if the complementary equation in (6.62) is written: (6.65) l(ψ) := ψ T f0ext = 0 The direct computation of the critical point with the corresponding information on the type of critical mode, can also be carried out by Newton’s iterative procedure, where at each iteration we obtain the consistently linearized form of our system, compute the increments in nodal displacements, load parameter and critical mode, and carry out the corresponding updates:

298

6 Large Displacements and Instability: Buckling Versus Nonlinear Instability

(i) = 1, 2, . . . ⎤ ⎤(i) ⎡ (i) ⎤ ⎡ (i) ⎡ λcr f0 − fint (d(i) Δdcr K ∂(Kψ)/∂d 0 cr ) (i) (i) ⎦ ⎣ 0 K ∂l/∂ψ ⎦ ⎣Δψ (i) ⎦ = ⎣ −Kcr ψ (i) (i) −f0 ∂(Kψ)/∂λ 0 Δλcr −l(ψ ) cr

(6.66)

(i+1) (i) (i) (i+1 (i) (i) d(i+1) = d(i) = ψ (i) cr + Δψ cr cr cr + Δdcr ; λcr ) = λcr + Δλcr ; ψ cr

The iterations are stopped when the specified tolerance is reached. The main difficulty in applying this solution procedure is related to the choice of initial values for unknowns in the iterative procedure. Namely, we can again use the displacements and load parameter value provided by the arc-length solution procedure for the beginning of the step where the presence of the critical point was noticed (by the change of the sign of determinant of the tangent stiffness matrix), but there is no indication, in general, what to use for the first iterative guess of the critical bifurcation mode ψ (1) . A poor guess of this kind can significantly slow down or even prevent the iterative procedure from converging. After passing through a critical bifurcation point, we can explore different postbifurcation responses, where the computation along each path can be started with a small perturbation of the critical equilibrium state in the direction of the computed critical mode with: dcr + εψ, where              SS     a         SS                 =      o 30     

SS : w = 0





a

SS

-

Fig. 7.15 Simply supported skew plate

The converged value of s ≡ s j+1 is than used to update the displacement for the next iterative step (7.134) u(i+1) = u(i) + s Δu(i) Note that the solution of Eq. (7.133) requires the consecutive evaluation of the residual, but not of the tangent stiffness matrix.

7.2.3 Illustrative Numerical Examples and Closing Remarks Several numerical examples are solved in order to demonstrate the performance of the presented element in the elastoplastic analysis. A couple of singularity-dominated elastic problems are solved as well, showing that the element does not suffer a deteriorated performance in such a case. It is very important to establish such a desirable property, since the accuracy of inelastic response can be significantly degraded because of an unsatisfying performance in the initial elastic steps. All computations are performed with the computer program F E L I N A (see [127]). 7.2.3.1 Uniform Loading on Simply Supported Skew Plate The skew plate shown in Fig. 7.15 is often used as a benchmark problem in the elastic analysis of plates, although its usefulness has been questioned in [20] on the grounds of a strong influence of the moment singularity in obtuse corner. The selected characteristics for the plate are: the thickness t = 1, the side length a = 100, Young’s modulus E = 107 and Poisson’s ratio ν = 0.3. Some otherwise very versatile plate elements, for example, the discrete Kirchhoff element (D K Q) and its analogous element P Q3 that includes shear deformation (see [164]), exhibit a degraded performance if a uniform mesh is used. However, the presented element exhibits an excellent performance, not influenced by singularity. It also compares favorably to T 1 plate element (see [158]).

7.2 Stress Resultants Plasticity for Metallic Plates Table 7.4 Uniform loading on skew plate Center displacement Mesh/Elem. Present DK Q 4×4 8×8 16 × 16 ‘Exact’

0.04212 0.04224 0.04374 0.04455

0.08303 0.05533 0.04835 0.04455

Fig. 7.16 Simply supported square plate

407

P Q3

T1

0.08311 0.05542 0.04844 0.04455

0.03918 0.03899 0.04187 0.04455

CL CL

6 HS

HS : w = 0, θn = 0 CL : θt = 0

a/2

? 

a/2

HS

-

The solutions for the center displacement under the uniform loading q = 1, are obtained for different plate elements and presented in Table 7.4. The reference value used for comparison is the numerical result of Morley [291]. 7.2.3.2 Point Load on Simply Supported Square Plate The second test problem is a simply supported (hard supports) square plate loaded by a point load in the center. See Fig. 7.16. The plate thickness is t = 0.1, side length a = 10, and material characteristics are chosen to model elastic-perfectly plastic plate with E = 10.92, ν = 0.3, σ y = 1000 and H = 0. For the unit value of the point load, the response is elastic. Therefore, this is used as an additional test of singularity dominated elastic response, since the displacement under the point load for the Reissner-Mindlin plate becomes infinite. However, even for a rather fine uniform mesh, as well as for a more refined mesh used in the elastoplastic analysis, the accuracy of the results presented in Table 7.5 is not influenced by the singularity and converges to the finite solution value for Kirchhoff’s plate (see [372]). We next performed an elastoplastic analysis by further incrementing the load until the limit state is reached. The computations are performed with a uniform mesh of 16 × 16 elements and the refined mesh shown in Fig. 7.17. The refined mesh is

408 Table 7.5 Unit point load on square plate Mesh 2×2 4×4 8×8 16 × 16 32 × 32 Refineda ‘Exact’ a

7 Inelasticity: Ultimate Load and Localized Failure

Center displ. ×10−4 0.1170 0.1160 0.1161 0.1162 0.1162 0.1161 0.1160

Refined mesh in Fig. 7.17

Fig. 7.17 Refined mesh for square plate used in elastoplastic analysis under point load

constructed by using the results of elastoplastic analysis obtained on 16 × 16 mesh. The refinement is performed in the zones which are plastified in that analysis. The load-displacement diagram is shown in Fig. 7.18 along with the analytical solution for the upper bound given in [240], p. 529. The spreading of the plastic zones is shown in Fig. 7.19 for different values of center point load. We note that the resultant formulation of plasticity exhibits significant smearing of the yield lines. Nevertheless, the obtained numerical results are in good agreement with the analytic solution. Although the difference in the numbers of degrees of freedom for two meshes is fairly large, the difference in results obtained is not very pronounced. This indicates that the results of acceptable accuracy can be obtained without refining the mesh extensively. In the examples to follow, we use only uniform meshes, and, indeed, obtain the results of very satisfying accuracy.

7.2 Stress Resultants Plasticity for Metallic Plates

409

Fig. 7.18 Point load—center displacement diagram for square plate

Fig. 7.19 Spreading of the plastic zones for the square plate under point load

7.2.3.3 Uniform Loading on Simply Supported Square Plate The same simply supported square plate (see Fig. 7.16) used in the previous example (a = 10, t = 0.1, E = 10.92, ν = 0.3, σ y = 1000, H = 0) is now loaded with the uniform loading. An elastoplastic analysis is performed using a uniform 16 × 16 mesh. The load-displacement diagram is presented in Fig. 7.20. The computed solution is bracketed between the lower and upper-bound analytic solutions presented in [270], pp. 377–378. The spreading of the plastic zones, presented in Fig. 7.21, is again notable.

410

7 Inelasticity: Ultimate Load and Localized Failure

Fig. 7.20 Uniform loading—center displacement diagram for square plate

Fig. 7.21 Spreading of the plastic zones for the square plate under uniform loading

7.2.3.4 Uniform Loading on Simply Supported Circular Plate This test problem is used to illustrate the performance of the element in an arbitrarily distorted configuration. The original method of incompatible modes of Wilson et al. [386] did not converge in such a case. However, this element suffers no difficulties, since when constructing the element interpolations we followed the methodology of the modified method of incompatible modes in [225]. The computations are performed on a quarter of circular plate (see Fig. 7.22), with selected characteristics: radius R = 5, thickness t = 0.1, Young’s modulus E = 10.32, Poisson’s ration ν = 0.3, yield stress σ y = 1000 and hardening modulus H =0.

7.2 Stress Resultants Plasticity for Metallic Plates Fig. 7.22 Simply supported circular plate

411

6 SS R

XXX X C

?L C L 

C C C

C

R

SS : w = 0 CL : θt = 0

-

Fig. 7.23 Uniform loading—center displacement diagram for circular plate

The numerical solution of governing differential equation for thin plate, presented in [270] p. 379, is used for comparison with numerical results. An excellent agreement is obtained (see Fig. 7.23), since the plate is rather thin and the contribution of shear force is negligible. The spreading of the plastic zones is presented in Fig. 7.24. Radial spreading reflects the one-dimensional nature of this problem. 7.2.3.5 Closing Remarks The presented formulation for stress resultant plasticity in Reissner-Mindlin plate is incorporated into a very accurate but computationally inexpensive four-node plate element. The presented plate element appears to be optimally tuned for a four-node element, which enables a very efficient implementation in nonlinear analysis. The efficiency of the proposed formulation is further reinforced with the implementation of the elastoplastic constitutive model directly in terms of stress resultants

412

7 Inelasticity: Ultimate Load and Localized Failure

Fig. 7.24 Spreading of the plastic zones for circular plate under uniform loading

(which eliminates costly through-the-thickness-integration), and with the split of plastic flow computations. The main application of the proposed formulation is to limit analysis of plates, both thick and thin. We can consider a general quadratic form of the yield criterion and the linear elastic compliance (which is not necessarily isotropic). Therefore, a wide range of material models for metal plates can be accommodated.

7.3 Plasticity Criterion with Thermomechanical Coupling in Folded Plates and Non-smooth Shells The shell structures have been considered as ‘primadonna’ of the engineering practice and have attracted a vast interest in the computational mechanics research community. A number of issues related to the choice of the best possible theoretical basis for computational shell models, choice of locking-free finite element interpolations and treatment of large rotation parameters for shells have been debated and to a large extent settled (e.g. see a review in [173]). The latter does not imply the end of research activity on shell models. As the matter of fact, a renewed interest is fueled by a wide variety of industrial applications involving shell structures, which are often faced with multiphysics and multiscale problems. One such industrial application is addressed in this work. The motivation for the work presented herein is the development of predictive models capable of describing the inelastic behavior of cellular structures, build either of folded plates and/or non smooth shells, under sustained long-term effect of high temperature. The model of this kind should initially supplement and then eventually

7.3 Plasticity Criterion with Thermomechanical Coupling …

413

replace the standard testing procedure for evaluating the fire resistance of the cellular structures, built of clay or concrete hollow blocks. The model should be able to account for a number of complex phenomena of heat conduction and radiation, as well as the inelastic behavior of the material with thermomechanical coupling. While the heat transfer problems appear to be firmly under control for solid mechanics (e.g. see [13, 86, 263, 334, 350] or [186]) including the pertinent aspects of thermomechanical coupling, the present problem of structural mechanics gives rise to a number of novel issues (e.g. see [53] or [333]), both in terms of describing pertinent heat transfer phenomena and of accounting for thermomechanical coupling. In particular, the main contributions presented in this section are the following: (i) The modeling of cellular structures of this kind built of hollow units use a modified version of the shell element of Ibrahimbegovic (e.g. see [168, 195]). This allows for a detailed modeling a single hollow unit of a cellular structure, where the bending action in one shell element is coupled with the membrane action of its neighbor. The coupling of this kind is provided by the added nonstandard feature of the shell element pertaining to so-called drilling degrees of freedom [195, 222]. (ii) From the standpoint of the modeling of the whole structure the shell element is placed at the ‘micro-scale’, in the sense that very many of them are used to build a model of a single brick unit. In trying to construct a predictive model for the whole structure, where at such ‘macro-scale’ we consider a large number of units and therefore an even larger number of shell elements, one must represent only fairly simple failure mechanisms at the level of a single shell element. In this work, the latter is selected in terms of Saint-Venant [327] (or Rankine-like) failure criterion which can be calibrated with respect to brittle-like materials used for the structures of this kind. Moreover, in the spirit of the shell model, the Saint-Venant failure criterion is recast in terms of stress resultants in the format of multi-surface plasticity (e.g. see [342]). We note in passing that a predictive model of this kind can also be constructed directly incorporating experimental results obtained on a specimen corresponding to a particular shell element (e.g. traction, compression or bending test performed on a plate-like piece of clay). (iii) Sustained high temperatures impose that the thermomechanical coupling also be considered for this problem, and not only with respect to the usual concern of modifying the mechanical properties with respect to temperature. The additional concern in this work is the modification that one has to make for heat conduction and radiation problem related to spreading of failure zone, where two (or more) adjacent cells connect upon the complete damage of the shell barriers in-between. (iv) Operator split solution procedure for solving the coupled thermomechanical problem is developed in order to reduce the computational cost and avoid working with full-size non-symmetric format ([334] or [13]). In this work we take this development a step further by providing the operator split solution procedure where the time-integration schemes for mechanics and heat transfer do not necessarily need to use the same time step. In that way, one can choose

414

7 Inelasticity: Ultimate Load and Localized Failure

the optimal value of the time step for each sub-problem in order to adapt the obtained result accuracy to the time scale of the evolution process for either thermal or mechanics part. The choice which is made for the case of practical interest for this work of long-term fire resistance, is to combine a large time step for thermal sub-problem with a number of smaller time steps for mechanical sub-problem. The outline of this section is as follows. First, we briefly present the theoretical formulation for the shell model which can be employed for folded plates and non-smooth shells and summarize the finite element implementation details. Next, thermomechanical coupling and the chosen constitutive model of Saint-Venant plasticity are addressed along with the operator split solution procedure for thermomechanical coupling employing different time steps for thermal and mechanical parts. Several illustrative numerical examples and conclusions are presented at the end.

7.3.1 Theoretical Formulation of Shell Model for Folded Plates and Non-smooth Shells In modeling folded plates or non-smooth shells, one can not rely on the classical shell model (e.g. see [294] or [183]). The main difficulty in that sense is the lack of compatibility between the displacement-degrees-of-freedom-only, which one uses to describe the membrane deformation field, and the displacement combined with rotations, which are necessary to describe the bending deformations. Such difficulty is resolved by replacing the classical shell model with the shell model with socalled drilling rotations (e.g. see [139, 168, 194, 195]). By including rotational component around normal to the shell, one can construct the shell model which remains fully compatible in coupling the membrane and bending modes in shell intersections, as well as in shell connections with beams or stiffeners. From the standpoint of the finite element implementation the most convenient format of the shell theory with drilling rotations is the one for shallow shells (see [195]). The latter combines the kinematics hypothesis appropriate for a particular shell model, such as the Kirchhoff hypothesis where the shear deformation is neglected or the ReissnerMindlin hypothesis including constant shear deformation, with the hypothesis of Marguerre on shallow shell geometry. This approach allows us to work with the projected form of the shallow shell. More precisely, one can replace the coordinates on the shell surface by the coordinates on the projected plane (see Fig. 7.25), which significantly simplifies the computation of derivatives and integrals featuring in the weak form of the problem. With a hypothesis of this kind, one can show (e.g. see [195]) that the bending and shear deformation components are identical to those of a plate. For the case of Reissner-Mindlin hypothesis, these strain components can be written as

7.3 Plasticity Criterion with Thermomechanical Coupling …

415

Fig. 7.25 Shallow shell: initial and reference configurations

κ(αβ) = for bending deformation and

 1  θα,β +  θβ,α 2

θα γα = u 3,α − 

(7.135)

(7.136)

for shear. In the case of the Kirchhoff plate, this shear deformation vanishes and one recovers a modified form of the bending strains computed as the second derivatives of the out-of-plane displacement component. In the subsequent development this option will not be used. Rather, if one needs to impose the vanishing of the shear strain, this will be done by the discrete Kirchhoff technology (e.g. see [164]); which first makes the shear strain equal to zero in a number of chosen points within a shell element and then constructs from those points an assumed shear strain thus enforcing the shear strain to vanish everywhere. Therefore, for either case of thick or thin shells, θα will be chosen rotation parameters, which relate to the rotation vector components according to  0 −1  θα = eαβ θβ ; ψ = θ3 ; eαβ = (7.137) 1 0 where ψ is the drilling rotation and eαβ is the alternator tensor. The shallow shell coupling appears only at the level of the membrane strains, which can be written as ε(αβ) =

 1  1 u α,β + u β,α +  θα f ,β +  θβ f ,α 2 2

(7.138)

where u α are in-plane displacement components and f ,α are the partial derivatives of the function f (x1 , x2 ) describing the shallow shell mid-surface with respect to the coordinates selected in the projected plane. The coupling which occurs through membrane strain expression between the in-plane displacement and the corresponding rotation components, is thus defined by the shallow shell form.

416

7 Inelasticity: Ultimate Load and Localized Failure

Another particular feature of the shell theory of this kind (see [195]) is the presence of skew-symmetric membrane strain component, which also couples the drilling rotation with remaining in-plane displacement and rotation components according to   1 1 u 2,1 + u 1,2 − ψ +  θ2 f ,1 +  (7.139) θ1 f ,1 ε[12] = 2 2 The variational formulation of the shell problem with drilling rotation ought to be regularized (e.g. see [154] or [222]) in order to provide the basis for any convenient choice of finite element interpolation. The regularized form can be written in terms of the Hu-Washizu principle (e.g. see [392–394]) featuring displacements, rotations and stress tensor components. In accordance with the usual shell hypothesis, one assumes a linear through-the-thickness variation of the virtual in-plane displacement field along with the constant variation of the out-of-plane virtual displacement, leading to θα∗ (xα ); u ∗3 (xα , ζ ) = u ∗3 (xα ) u ∗α (xα , ζ ) = u ∗α (xα ) − ζ 

(7.140)

The weak form of the shallow shell equilibrium equation can then be written as ˆ θˆ˜ ) := θ , n [αβ] ; u, 0 = G u,θ (u, 

 

∗ ∗ ∗ [ε(αβ) n (αβ) +  κ(αβ) m (αβ) + γ(αβ) qα ]

(7.141)

A ∗ + ε[αβ] n [αβ] d A − G ext (·)

θ , n [αβ] ; n [αβ] ˆ ) := 0 = G n [·] (u, 

  1 ε[αβ] − n [αβ] d A γ

(7.142)

A

In 7.141 and 7.142 we defined the stress resultants of membrane and shear forces and bending moments according to t/2 n (αβ) =

t/2 σ(αβ) dζ ; qα =

−t/2

t/2 σα3 dζ ; m (αβ) =

−t/2

ζ σ(αβ) dζ

(7.143)

−t/2

along with the skew-symmetric membrane force n [12] . The particular choice of the constitutive equations, as those addressed later on among different consistent forms (e.g. see [91] or [254]), will relate these stress resultants to the strain measures defined in (7.135) to (7.138). The regularized form of the variational formulation allows us to eliminate the shew-symmetric membrane force and recover an alternative format which employs only kinematic variables—displacements and rotations—which are a very convenient starting point for the finite element implementation as described next.

7.3 Plasticity Criterion with Thermomechanical Coupling …

417

Fig. 7.26 Isoparametric element—warped and plane configurations

7.3.2 Finite Element Implementation with Shell Element In the finite element implementation, the shallow shell approximation is furnished within the framework of a single element—we refer to the corresponding choice for a 4-node quadrilateral element in Fig. 7.26. This allows us to extend the validity of the shallow shell approximation and also ensures the convergence towards the deep shell solutions (e.g. see [229]). The projected configuration is parameterized with respect to the parent element by making use of the isoparametric mapping (e.g. see [392–394]) 4 

N I (ξ, η) x I ; xh (ξ, η) Ae =

N I (ξ, η) =

I =1

1 (1 + ξ I ξ )(1 + η I η) 4

(7.144)

where N I (ξ, η) are bilinear shape functions, ξ I and η I are nodal values of natural coordinates (equal to ±1) and x I are nodal values of physical coordinates of the shell element nodes. The initial (warped) configuration of the shell quadrilateral element is also parameterized by the natural coordinates, by using the isoparametric representation of the shallow shell surface f (ξ, η)| Ae =

4 

N I (ξ, η) h I

(7.145)

I =1

where h I = ±h all have the same absolute value for the particular choice of the local coordinate system for each shell quadrilateral element. In keeping with the spirit of the isoparametric interpolations, the displacement and rotation fields are approximated by using the same shape functions

418

7 Inelasticity: Ultimate Load and Localized Failure 4

! u i −→ uh (ξ, η) Ae = N I (ξ, η) u I I =1

4

!  θα −→ θ h (ξ, η) Ae = N I (ξ, η)  θI

ψ −→ ψ h (ξ, η) Ae =

I =1 4 !

(7.146)

N I (ξ, η) ψ I

I =1

where u I ,  θ I and ψ I are the corresponding nodal values of displacements and rotations. The same type of interpolation is chosen for virtual as well as incremental displacements and rotations. In sharp contrast with the choice made for interpolating displacement and rotation, none of the finite element interpolations for strain fields is isoparametric. The latter is needed in order to avoid the locking phenomena (i.e. the inability of the shell element to represent fundamental deformation modes) and to achieve the optimal element performance. In particular, the bending field is computed by exploiting the method of incompatible modes (e.g. [225]), which can be written as 4 2  

 J (ξ, η) α J G B I (ξ, η)  θI +  καβ −→ κ h (ξ, η) Ae = I =1

(7.147)

J =1

where strain displacement operators can be written as ⎡∂N

I

⎢ ∂ x1 BI = ⎣ 0

∂ NI ∂ x2

0 ∂ NI ∂ x2 ∂ NI ∂ x1

⎤

⎡∂M

J

⎥ ⎢ ∂ x1 ⎦ ; GJ = ⎣ 0

∂ MJ ∂ x2

0 ∂ MJ ∂ x2 ∂ MJ ∂ x1

⎤ ⎥ ⎦;

M1 (ξ ) = 1 − ξ 2 M2 (η) = 1 − η2

(7.148)

The following modified form of G J assures orthogonality of incompatible bending modes with respect to the constant bending moment field and the satisfaction of the patch test (e.g. see [390]):  J (ξ, η) = G J (ξ, η) − 1e G A

 G J (ξ, η) d A A

(7.149)

e

The shear strain finite element interpolation is constructed by using the method of assumed strains (e.g. see [30] or [337]). Namely, the shear strain approximation is chosen not as the one consistent with those used for displacements and rotations. Rather only mid-point values of consistent shear strains are selected and the shear field is constructed such that the pure bending modes remain well represented with,

7.3 Plasticity Criterion with Thermomechanical Coupling …

419

γξ 3 = P1 (ξ )γξA3 + P2 (ξ )γξC3  γη3 =  P1 (ξ ) = where:

B P1 (η)γη3

+

1 (1 − ξ ); 2

(7.150a)

D P2 (η)γη3

P2 (ξ ) =

(7.150b) 1 (1 + ξ ) 2

γξA3 = (w,ξ −  θ1 ) A ; γξB3 = (w,ξ −  θ2 ) B

etc . . .

(7.150c)

(7.151)

are the mid-values of the shear strain in the natural frame. The transformation between the natural and physical coordinates is carried out by the standard form of the Jacobian matrix,    (e1 · g1 ) (e1 · g2 )  γξ 3 γ13 = (7.152) γ23 γη3 (e2 · g1 ) (e2 · g2 )  where ⎞ ⎛ ⎛ ⎞ F22 F21 1 1 ⎝−F12 ⎠ ; g2 = ⎝ F11 ⎠ ; g3 = e3 g1 = det F det F 0 0 ⎛ ⎞ ⎛ ⎞ F F ∂x ⎝ 11 ⎠ ∂x ⎝ 12 ⎠ g1 := = F21 ; g2 := = F22 ; g3 = e3 ∂ξ ∂η 0 0

(7.153a)

(7.153b)

For the considerations to follow, we will assume that the shear strain vanishes everywhere, which implies working with thin shells. A simple manner is by setting the mid-point values of the shear strain parameters equal to zero, which allows us to recover the vanishing shear strain in a shell element in terms of the discrete Kirchhoff approximation. The latter implies (e.g. see [164]) that a non-standard displacement approximation can be written for the shell element of this kind. The membrane strain interpolation is also handled by applying both the incompatibles modes and the assumed strain method. More precisely, the first part of the expression in (7.138) is treated by the standard isoparametric approximations enriched by a set of incompatible modes, very similar to those used for bending. The discrete approximation of this term can thus be written as 4 2  

 J (ξ, η) β J B I (ξ, η) u I + G u (α,β) −→ ε h (ξ, η) Ae = I =1

(7.154)

J =1

where β J are membrane incompatible mode parameters. We also construct a special interpolation for the second ‘Marguerre’ term. The latter is computed in accordance with the chosen displacement and rotation interpolations only for the mid-side values of membrane strains, such as

420

7 Inelasticity: Ultimate Load and Localized Failure

θ y1 + θ y2 2 θ y3 + θ y4 C C  εξ ξ = (− f ,ξ θ y ) = ζ etc . . . 2  εξAξ = (− f ,ξ θ y ) A = ζ

(7.155a) (7.155b)

The interpolation of the Marguerre component of the membrane strain in the natural frame is done according to the following assumed field ⎞ ⎡ ⎤ P1 (η)εξAξ + P2 (η)εξCξ εξ ξ B B ⎦ ⎝εηη ⎠ = ⎣ P2 (ξ )εηη + P1 (ξ )εηη C A B B εξ η P1 (η)εξ η + P2 (ξ )εξ η + P2 (η)εξ η + P1 (ξ )εξ η ⎛

(7.156)

Finally, we transform these approximations to the physical frame and combine the end result with the interpolation of the first part in (7.155) to obtain the final form of discrete approximation for membrane strains. The corresponding discrete approximations of stress resultants are obtained simply by multiplying the presented approximations of strains by the chosen constitutive coefficients. One such choice is described next.

7.3.3 Stress Resultants Constitutive Model of Saint-Venant Plasticity In order to specify the constitutive equations for the mechanics part of the shell problem, we have to select the kind of material model which is the most suitable for our applications. In that respect, we note that in the last section and many previous works (e.g. see [341] or [190], among others) one can find the generalization of the Von Mises plasticity criterion, which is the most appropriate for ductile metals that can easily be extended to thermomechanics framework [185] to tackle numerous applications in (hot) metal forming processes. In this section we consider a constitutive model which is geared toward describing behavior of brittle materials in terms of a generalized plasticity model based on the Saint-Venant [327] (or Rankinelike) yield criterion. The key idea of this model is to limit the elastic domain by the principal elastic strain values, which agrees with the experimentally observed behavior for brittle materials, where the Rankine criterion [176] related to principal stress is often used. Namely, for this kind of material, the failure is mainly driven by extensions (positive strains) leading to cracking in the direction perpendicular to the principal tension stress or yet parallel to the principal compressive stress (see [176]). Accordingly, the elastic domain is defined in terms of principal values by using a multi-surface plasticity criterion of the mechanical strain (see [176] for more details). This criterion can be recast in the stress resultant space, leading to a multi-surface yield criterion which consists of four surfaces intersecting in a non-smooth fashion of the form:

7.3 Plasticity Criterion with Thermomechanical Coupling …

421

Φ1 =



K + 4μ K − 2μ 3 3  n αβ + m  n αβ + m αβ I − αβ I I − (σ y (θ ) − q(θ )) 2μ 2μ (7.157a)

Φ2 =



K + 4μ K − 2μ 3 3  n αβ + m  n αβ + m αβ I I − αβ I − (σ y (θ ) − q(θ )) 2μ 2μ (7.157b)

Φ3 =



K + 4μ K − 2μ 3 3  n αβ − m  n αβ − m αβ I − αβ I I − (σ y (θ ) − q(θ )) 2μ 2μ (7.157c)

Φ4 =



K + 4μ K − 2μ 3 3  n αβ − m  n αβ − m αβ I I − αβ I − (σ y (θ ) − q(θ )) 2μ 2μ (7.157d)

Here K is bulk modulus, μ shear modulus and |·| I /I I denotes first/second principal values of the symmetric tensor of stress resultants and couples, which are written in a normalized format defined with n αβ t 6m αβ = 2 t

 n αβ =

(7.158a)

m αβ

(7.158b)

In regard to the subsequent thermomechanical modification of the given model, the temperature dependence is assumed both for the elastic limit σ y (θ ) and for the variable which controls the evolution of the elastic domain q(θ ). The latter is typically related to stress softening branch which eventually drives the stress to zero (see [181]). A special provision is taken (see [176]) to incorporate different behavior and corresponding fracture energies in compression and in tension. The plasticity model of this kind also features the standard additive split of generalized strain measures for both membrane and bending components with p

p

e e + εαβ ; καβ = καβ + καβ εαβ = εαβ

(7.159)

The free energy can then be written in terms of the elastic strain components according to e e , καβ , ξ) = Ψ (εαβ

3  1 e  e t  e εαβ t Cαβγ δ εγe δ + καβ ) Cαβγ δ κγe δ +H(ξ, εαβ 2 12

(7.160)

αβγ δ is the fourth-order elasticity tensor modified for plane stress case and where C ξ is the internal variable which controls hardening/softening response.

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7 Inelasticity: Ultimate Load and Localized Failure

Further developments of this plasticity model ingredients follow in the footsteps of the works in [176, 342]. In particular, the evolution equations for plastic components of the generalized strain measures are obtained by the Koiter rule for multi-surface plasticity (e.g. see [176]), which can be written as p

ε˙ αβ =

 j∈Jact

γ˙ j

 ∂Φ j ∂Φ j p ; κ˙ αβ = γ˙ j ∂n αβ ∂m αβ j∈J

(7.161)

act

with j ∈ Jact denoting the active yield surfaces and γ˙ j denoting the corresponding plastic multipliers. In the determination of Jact , we have to pay attention to the case in which    0 tCαβγ δ ε˙ γ δ ∂Φ2 ∂Φ2 ≤0 (7.162) · 3 t ∂ n αβ ∂ m αβ κ ˙γ δ 0 C αβγ δ 12 and γ˙ 2 > 0 at the same time (the same problem exists for i = 4, see [87] for details). The plastic multipliers are computed from the consistency condition of the given plastic state and consistent linearization provides elastoplastic tangent moduli. The time integration of the constitutive response is carried out by the backward Euler scheme and the return mapping algorithm. Namely, by starting with known p p values of internal variables at time tn , denoted as e pn = (εαβ,n , καβ,n ) and ξn , we seek to recover their values at time tn+1 . By first assuming the elastic trial step, where all the values of internal variables at time tn remain frozen, we carry out the computation of the trial values for stress resultants and couples as well as the corresponding yield criterion values with, n trial αβ,n+1 = leading to :

∂Ψ e,trial ∂εαβ,n+1

; m trial αβ,n+1 =

∂Ψ e,trial ∂καβ,n+1

trial ; qn+1 =

∂Ψ trial ∂ξn+1

trial trial trial i (n trial =Φ Φi,n+1 αβ,n+1 , m αβ,n+1 , qn+1 ); i ∈ [1 . . . 4]

(7.163)

(7.164)

The elastic trial step is admissible only if trial ≤ 0; ∀i ∈ [1, 2, 3, 4] Φi,n+1

leading to:

p

p

p

p

εαβ,n+1 = εαβ,n ; καβ,n+1 = καβ,n ; ξn+1 = ξn

(7.165)

(7.166)

If any of the yield surfaces is active, as indicated by a positive trial value, we have to correct the elastic trial step values by computing the corresponding plastic flow: p

p

p

p

trial > 0 ⇒ εαβ,n+1 = εαβ,n ; καβ,n+1 = καβ,n ; ξn+1 = ξn ∃i, Φi,n+1

(7.167)

7.3 Plasticity Criterion with Thermomechanical Coupling …

423

Quite a subtle point of the present plasticity model is that only a single yield surface being active will not necessarily imply that only a single Lagrange multiplier is non-zero, i.e. trial trial > 0 and Φ2,n+1 ≤ 0  γ1,n+1 > 0 and γ2,n+1 = 0 Φ1,n+1

(7.168)

trial trial > 0 and Φ3,n+1 ≤ 0  γ3,n+1 > 0 and γ4,n+1 = 0 Φ3,n+1

(7.169)

or,

By taking into account these and other possibilities to obtain the number of unknown Lagrange multipliers, we arrive at the complete set of nonlinear equations p p with εαβ,n+1 , καβ,n+1 , ξn+1 and λi,n+1 = Δtγi,n+1 as the unknowns, rn+1 (·) = 0

(7.170)

By the consistent linearization of the last expression and the systematic application of the static condensation (see [198]) we can obtain the elastoplastic tangent modulus needed for the incremental iterative solution procedure carried out by Newton’s method. At typical iteration (i), we get ep, (i)

(i+1) (i) (i) ] = rn+1 + Drn+1 ⇒ Cn+1 Lin[rn+1

(7.171)

With this result in hand, we can write the consistent tangent matrix for the mechanics part of the problem according to Kuu, (i) =



T ep, (i) Cn+1 B d A B 

(7.172)

A

where B is the strain-displacement matrix which contains all the corresponding submatrices for membrane and bending strain fields, as defined in (7.154) and (7.147), respectively.

7.3.4 Thermomechanical Coupling In this section, we set to complete the development of the variational formulation for the chosen shell model. The first ingredient we need should provide the basis for taking into account the thermomechanical coupling that is capable of dealing with the kind of applications we are targeting herein. In particular, we would like to furnish a heat transfer formulation which is fully compatible with stress resultant formulation of mechanics part of the shell problem as described in the previous sections.

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7 Inelasticity: Ultimate Load and Localized Failure

To start, we recall the classical 3D strong form of the energy balance equation [176] for the case where the mechanics part contribution is ignored (e.g. no straining or a rigid conductor) (7.173) c θ˙ = −qα,α + r with c heat capacity, θ˙ rate of change in temperature, qα heat flux components and r heat source. This strong form of energy balance must be completed by specifying a set of initial condition (7.174) θ = θ0 in Ω as well as the boundary conditions, θ = θ on ∂Ωθ

(7.175a)

qα n α = q n on ∂Ωq

(7.175b)

To develop a shell-like formulation, we split the representation of the shell 3D domain into a product of 2D mid-surface A and thickness t. In analogy with mechanics part, we further assume a linear variation of the weighting temperature field, θ ∗ (xα , ζ ) = ϑ ∗ (xα ) + ζ ϕ ∗ (xα )

(7.176)

where ϑ ∗ is the mid-surface temperature and ϕ ∗ is the through-the-thickness gradient. The weak form of the energy balance equation (7.173) in a shell-like domain can then be written as, 0 = G θ (ϑ, ϕ) :=

    ∗  t3 ∗ ct ϑ ∗ ϑ˙ + c ϕ ∗ ϕ˙ d A − r α + ϕ ∗ p3 d A ϑ,α pα + ϕ,α 12 A

   ∗ t ∗ +  ∗ t ∗ − + ϑ + 2 ϕ qn + ϑ − 2 ϕ qn d A

A

A

(7.177) In (7.177) above we denote the resultant heat fluxes by t

t

+ 2 pα (xα ) =

+ 2 qα dζ ; rα (xα ) =

t − 2

qα ζ dζ

(7.178)

t − 2

and by qn− and qn+ the heat fluxes on the upper and lower surface, respectively. The resultant heat fluxes can be related to temperature and temperature gradient through a set of constitutive relations. For example, the choice which is made in this work to assure the compatibility with the stress resultant variations leads to the following form

7.3 Plasticity Criterion with Thermomechanical Coupling …

425

pα (xα ) = −ktϑ,α ; p3 = ϕe3

(7.179a)

3

rα (xα ) = − kt12 ϕ,α

(7.179b)

where k is the thermal conductivity coefficient. The finite element implementation of the thermal part of the problem is carried out in the standard manner by using the isoparametric interpolations. Namely, for the chosen 4-node shell element we have: 4 4  

N I (ξ, η) ϑ I ; ϕ −→ ϕ h (ξ, η) Ae = N I (ξ, η) ϕ I ϑ −→ ϑ h (ξ, η) Ae = I =1

I =1

(7.180) where ϑ I and ϕ I are nodal values of temperature and temperature gradients. By choosing the same kind of interpolations for weighting temperature, we can reduce the weak form for the thermal part in (7.177) to a set of algebraic equations, which can be written as   (7.181) f θ = 0; f θ = f Iθ f Iθ =

  t3 ct N I ϑ I,n+1Δt−ϑ I,n + c 12 N I ϕ I,n+1Δt−ϕ I,n d A − N I,α ( pα + rα + p3 ) d A A

In (7.181) above, we have applied the backward Euler method to integrate the temperature evolution over a given time step Δt = tn+1 − tn . In the absence of any thermomechanical coupling and no radiative exchange, this thermal problem remains linear and the consistent linearization of (7.181) leads to the exact result. In the case, we take into account the thermomechanical coupling for the shell model under consideration, the constitutive model should change. The starting point in defining the constitutive response can be selected in terms of the Helmholtz free energy which is now defined as 3  1 e  e t  e εαβ t Cαβγ δ εγe δ + καβ , ϑ, ϕ α ) Cαβγ δ κγe δ +H(ξ, εαβ 2 12 t3  ρc  2 tϑ + ϕ 2 − 2θr e f 12 3  e  e t αβγ δ δγ δ ϕ αβγ δ δγ δ ϑ + καβ − εαβ αC tα C 12 (7.182) where α is the thermal expansion coefficient. In the last expression, the free energy is written by adding upon the mechanics part in the first term a more general temperature-dependent form of the hardening potential in the second term, followed by the thermal potential in the third term and the thermomechanical coupling effect in the fourth term. With this choice of free energy, one obtains a modified form of the stress resultant constitutive equations which contains the temperature-dependent term according to: e e , καβ , ξ, ϑ, ϕ α ) = Ψ (εαβ

426

7 Inelasticity: Ultimate Load and Localized Failure

n αβ =

∂Ψ meca αβγ δ δγ δ ϑ + tα C e = n αβ ∂εαβ

(7.183a)

m αβ =

∂Ψ t3  meca α Cαβγ δ δγ δ ϕ = m + αβ e ∂καβ 12

(7.183b)

For the case of thermomechanical coupling, we only replace the stress resultant in (7.183) above in the weak form of the equilibrium equations for the mechanical part. Moreover, with the free energy (7.182), one obtains the constitutive equations defining entropy with both average and gradient component with, η=−

∂Ψ ∂Ψ ;ς =− ∂ϑ ∂ϕ

(7.184)

The weak form of the thermal balance equation will not change if we neglect the e e t3  αβγ δ δγ δ  0 and κ˙ αβ structural heating terms (˙εαβ tα C α Cαβγ δ δγ δ  0) and the plas12 tic dissipation. The former is done because the time variation of elastic deformations is very slow and the latter is done because we deal herein with brittle materials which have very small inelastic dissipation. However, even when simplifying the problem this way, the thermomechanical coupling still remains present through boundary conditions. Namely, the classical format of radiative heat exchange is exploited herein, implying that the resultant flux over the surface of each hollow cell ought to be equal to zero, which can be written as 

n sur f ± ± ←− qn,i −σ qn,i

4  j A j Fi j (θ ± j )

(7.185)

j=1 ± represent the outgoing flux, whereas the last term represents the conwhere qn,i tribution of incoming fluxes dependent upon the Stefan-Boltzmann constant σ , the surface emittance  j , the area of the surface A j , the relative shape factor of each pair of surfaces Fi j and the surface temperature to the power four. We note that each such equation is nonlinear in temperature terms (or in terms of the mid-surface and temperature gradient) and it requires several iterations at the global level to converge. In addition, the modification of such an equation in (7.185) has to be done each time the fracture of the barriers will change the hollow cell configuration. We refer to [87] for a more detailed discussion of this issue.

7.3.5 Operator Split Solution Procedure with Variable Time Steps The set of equations governing the semi-discretized problem of thermomechanical coupling of shells consists of the nonlinear algebraic equations expressing mechanics

7.3 Plasticity Criterion with Thermomechanical Coupling …

427

equilibrium equations along with differential equations describing the heat flow. These equations are accompanied by the evolutions equations of internal variables (plastic strain and hardening variable), which are defined and solved at the local level, at each Gauss numerical integration point. The problem can thus formally be written as  ru (du , dθ , ε p (θ ), κ p (θ ), ξ(θ )) =0 (7.186) r := Mθθ d˙θ − rθ (dθ , ε p (θ ), κ p (θ ), ξ(θ )) ! ⎞ i ε˙ p − i γ˙ i ∂Φ ∂n ! ⎝κ˙ p − i γ˙ i ∂Φi ⎠ = 0; Φi ≤ 0; ∀ GNP ∂m ! i ξ˙ − i γ˙ i ∂Φ ∂q ⎛

(7.187)

The dependence on real time is therefore present only in the heat transfer part of problem if we take a rate-independent constitutive response. However, since all the equations are, in general, tightly coupled, the same time parameter should be employed throughout. The system is thus first recast in the form where only algebraic equations will appear, which is done by integrating the evolution equations for internal variables as well as the heat transfer equation by using the backward Euler scheme. The problem thus reduces to a set of nonlinear algebraic equations, defining the central problem of computational plasticity (e.g. see [176]), which can be written as:

p p given: dun , dθn and (εn , κn , ξn ) GNP ; ∀GNP

p p find: dun+1 , dθn+1 and (εn+1 , κn+1 , ξn+1 ) GNP such that:   p p ru (dun+1 , dθn+1 , εn+1 , κn+1 , ξn+1 rn+1 := =0 (7.188) dθ −dθ p p Mθθ n+1Δt n − rθn+1 (dθn+1 , εn+1 , κn+1 , ξn+1 ) ! ⎞ p p i εn+1 − εn − i λi ∂Φ ∂n ! p p i ⎠ ⎝κn+1 − κn − i λi ∂Φ = 0; Φi ≤ 0; ∀ GNP ∂m ! i ∂Φ i ξn+1 − ξn − i λ ∂q ⎛

(7.189)

It is important to note that the thermomechanical coupling effect in the discretized heat transfer equation is implemented between temperature on one side and internal variables on the other side, not only through the plastic heating effect, but also through the corresponding value of plastic deformation which induces modification of the cell assembly and determines the heat radiation conditions. Therefore, even when we neglect the plastic heating, which appears to be justified for the brittle models of this kind where no large ductile deformations occur, the coupling still persists through a special re-meshing procedure which one has to provide. By reducing the coupling effect to a minimum, the linearized form of the system of algebraic equations to be solved can be written as

428

7 Inelasticity: Ultimate Load and Localized Failure

 uu K 0

Kuθ 1 θθ M + Kθθ Δt

(k)

 u (k+1) Δd · = −r(k) n+1 Δdθ n+1 n+1

(7.190)

These equations are solved for the converged values of internal variables, which would ensure the plastic admissibility of the stress state Φi ≤ 0. An alternative strategy to the proposed simultaneous solution, which we have also tested, is to employ a sequential solution procedure. In this case, the heat transfer problem is treated apart from the mechanics equilibrium, which also includes the solution to evolution equations for internal variables. More precisely, the problem split consists of the thermal phase at fixed configuration, followed by the mechanical equilibrium phase at fixed temperature. As suggested in [334] or [13], such an isothermal split leads to only a conditionally stable solution scheme, and it should be replaced by an isentropic split where entropy rather than the temperature is kept constant in the mechanical step. However, for the brittle materials for interest to present work, where we neglect structural heating, two kinds of split problems remain the same; see [244]. The last modification which was introduced in this sequential procedure concerns the idea of using variable time steps for different solution phases. Namely, the heat transfer problem is solved by using larger time steps than the mechanics problem, which needs smaller steps to converge. Moreover, in order to increase the robustness of the algorithm, we have tested the adaptive time-step procedure for the mechanical phase. The step size control is chosen on the basis of the evolution of the hardening/softening variable ξ , which is quite representative of the extent of the development of the plastic process. In accordance with the suggestion in et al. [380], we take the value of indicator R, defined with R = max GNP

(Δξn+1 )GNP Δξ n+1

(7.191)

as the guide on the time-step refinement. In this expression, Δξ n+1 is a target value and we aim to keep the ratio R close to 1. Consequently, we choose: • if R > 1.25 the solution is rejected and the new time-step size is Δt  =

0.85 Δt R

(7.192)

• else if R ≤ 1.25 the solution is accepted and the next time increment evolves as: – if R ≤ 0.5 then Δt  = 1.5Δt ; – if 0.5 < R ≤ 0.8 then Δt  = 1.25Δt ; – if 0.8 < R ≤ 1.25 then Δt  = Δt/R.

7.3 Plasticity Criterion with Thermomechanical Coupling …

429

7.3.6 Illustrative Numerical Examples and Closing Remarks Here we show several numerical examples in order to illustrate the predictive capabilities of the presented model. The chosen examples include both a simple problem for which we also have an analytic solution as well as a more complex problem of practical engineering interest. 7.3.6.1 Circular Ring Heating In this example, we consider a circular ring with an average radius R and initial temperature θ0 . With respect to the chosen reference frame and cylindrical coordinates (r, φ, z), we applied to all points satisfying φ = 0 a temperature θ1 (see Fig. 7.27). For convenience, we recast this problem using dimensionless expressions and write, ϕ=

ϕ 2π

θ=

θ − θ1 θ0 − θ1

Fo =

at R2

(7.193)

where a is the thermal diffusivity [m2 s−1 ] and Fo the Fourier number. The analytic solution of the heat equation for the middle fiber at r = R can be written as: θ(ϕ, Fo ) =

∞  n=0

  4 2n+1 sin (2n + 1)π ϕ e(− 2 Fo ) (2n + 1)π

(7.194)

In particular, at the point ϕ = 1/2, the dimensionless temperature evolves according to: ∞  (−1)n (− 2n+1 Fo ) θ (Fo ) = (7.195) e 2 2n + 1 n=0

Fig. 7.27 Circular ring transient problem

t  R

430

7 Inelasticity: Ultimate Load and Localized Failure

Fig. 7.28 Circular ring—mesh

3

1

1.2

Dimensionless temperature

Fig. 7.29 Temperature evolution

2

1 Exact Numerical

0.8

0.6

0.4

0.2

0

0

2

4

6

8

10

Fourier number

Coupled thermomechanical analysis has been performed limited to the elastic regime, by using the finite element model constructed with the proposed shell element. Considering the symmetry, the model is constructed for only half of the cylinder by using a 6-element mesh (see Fig. 7.28). Figure 7.30 shows the evolution of the bending moment at the opposite point ϕ = 1/2. We note the peak value is reached during the transient heat transfer phase, which is followed by a decrease to zero limit in the subsequent steady state phase. Figure 7.29 shows the comparison between the exact solution and the numerical results which pertain to the evolution of the dimensionless temperature for the same point. The two results are practically the same. However, this is not the case for other fibers placed at r = R. The reason for that is the following: in the transient phase, the temperature distribution along the r coordinates could be described using Bessel’s functions of the first kind. In the steady state, the limit distribution is a logarithmic one. Thus, it is only as the thickness of the element decreases, such a limit expression becomes closer to a linear distribution as assumed for the shell element. It follows that the strategy we developed is the most suitable for thermal analysis of thin shells.

7.3 Plasticity Criterion with Thermomechanical Coupling … Fig. 7.30 Bending moment

431

14

Bending moment

12 10 8 6 4 2 0 0

1

2

3

4

5

6

Fourier number

7.3.6.2 Cellular Structures We next turn to a typical practical engineering problem which is of interest to us. Considering cellular structures like hollow bricks, the evaluation of their fire resistance leads to a coupled thermomechanical problem. Moreover, the geometry of such a structure leads naturally to the choice of flat shell elements for constructing the finite element model. Concerning heat transfer, the large variation of temperature requires that one should take into account the radiative heat exchanges. The application we present here consists of a flue block. The latter consists of a hollow structure made of clay, which is vertically assembled in order to form a vertical stack, whose role is to evacuate smoke. The stack is submitted to an important temperature increase at the inner face, which may lead to the appearance of cracks and a loss of efficiency in the smoke evacuation. In order to study the cracking of the flue block, we carry out a thermomechanical analysis employing non-linear mechanical model. By taking into account the symmetries of the structure, we analyze only a quarter of the model and employ a mesh of 16 shell elements and 5 radiative solid elements (see Fig. 7.31). On the inner face, boundary conditions of the third kind are applied with a driven convection temperature which increases from 0 to 1000◦ in ten minutes. Figure 7.32 shows the comparison of the corresponding average temperature evolution at four different positions, with decreasing values from inside to outside. We see that the temperature gradient between the inner face and the outer face is still very large even after 10 min. This gradient is at the origin of thermally induced stresses which can lead to flue cracking and eventually failure. Since we are interested in eliminating the risk of cracking, we look into the thermal strain field and induced stresses. In Fig. 7.33, we show the stress evolution in the horizontal direction for the outer face. Since the inner face is heated, this part is in tension. Moreover, a supplementary global bending effect of the block is produced by two-dimensional nature of the problem and a non-uniform distribution of the temperature on the inner face. We see from Fig. 7.33 that we are able to capture very well both the peak stress level and the delay effect before this peak is reached. The latter is a major factor in trying to estimate the block quality.

432

7 Inelasticity: Ultimate Load and Localized Failure

Fig. 7.31 Flue block—mesh Fig. 7.32 Flue block—temperatures evolution from inside to outside

Validation temperatures evolution 800 Model Experiments

temperature (°C)

600

400

200

0

0

200

400

600 t (s)

800

1000

7.3 Plasticity Criterion with Thermomechanical Coupling … Fig. 7.33 Flue block—reactions evolution

433 Validation stress evolution

8 Experiments Model

Stress (MPa)

6

4

2

0 0

200

400

600

800

1000

t (s)

Fig. 7.34 Unit assembly in a brick wall and periodic BC

joint brick

vertical periodicities horizontal periodicity

7.3.6.3 Thermomechanical Analysis of Hollow Brick Wall In this example, we consider a thermomechanical coupling in the cellular units placed with a brick wall. We assume that the geometry and the loading allow for exploiting the periodicity conditions. This implies that the analysis can be carried out on a single cellular unit isolated from the whole structure at the level of interface with neighboring units, by applying the corresponding boundary conditions which assure periodicity. More precisely, for the typical unit assembly in a brick wall (see Fig. 7.34) with only partial overlapping of successive layers, the same periodicity conditions are enforced only over half of the brick. Therefore, the domain which is retained in the analysis corresponds to one typical unit of the size 570 × 200 × 200 mm3 . This domain also includes half of the vertical and horizontal joints with a thickness equal to 10 mm.

434

7 Inelasticity: Ultimate Load and Localized Failure

Fig. 7.35 FE mesh for a cellular unit

The chosen finite element mesh (see Fig. 7.35) consists of three vertical layers of flat shell elements which brings the total number of these elements to 384 for the entire brick. Another subtlety of the model is the choice which is made for the representation of the interface joints. This one is model by elastic solid elements covering the cells of the brick placed only at the top. However, the latter does not introduce any nonsymmetry in the problem, considering the periodicity in the boundary conditions. Both mechanical and thermal loading is applied in this case. The mechanical loading is supposed to represent the dead load on the brick chosen as a compressive loading of 1.3 MPa, which is introduced directly at the level of each element as the initial compressive loading in the bricks, remaining constant afterwards. The thermal loading is then applied, in terms of the uniform temperature field applied only at the brick facet exposed to fire. The time evolution of this temperature field is given as θ (t) = θ0 + 345 log(8t + 1)

(7.196)

where θ0 is the initial temperature and t the time in minutes. The mechanical and thermal properties of the brick material are chosen as given in Table 7.6. The properties of the interface are given in Table 7.7. First, the results are presented in terms of the temperature field. Figure 7.37a shows the evolution of the temperature in three different cells (see Fig. 7.36 for locating). The experimental results are provided by thermocouples inside the cells. Therefore, we compare these values with the temperatures of the two surfaces on both sides of each cell obtained by the finite element analysis. The comparison shows we are able to capture the temperature evolutions even far from the exposed face of the wall. This result is confirmed by Fig. 7.37b which shows a temperature profile 48 min after the beginning of heating. The key point in order to obtain such a good result is the introduction of radiative exchanges in the heat transfer model.

7.3 Plasticity Criterion with Thermomechanical Coupling …

435

Table 7.6 Mechanical and thermal properties of the brick Density 1.870 Heat capacity 836 J kg−1 K−1 Conductivity (parallel to flakes) 0.55 W m−1 K−1 Conductivity (perpendicular to flakes) 0.35 W m−1 K−1 Thermal expansion coefficient at θr e f 7 × 10−6 K−1 Young modulus 12 GPa Poisson ratio 0.2 σ y at θr e f 14.5 MPa Fracture energy 80 J m−2

Table 7.7 Mechanical and thermal properties of the interface Density 2.100 Heat capacity 950 J kg−1 K−1 Conductivity 1.15 W m−1 K−1 Thermal expansion coefficient 1 × 10−5 K−1 Young modulus 15 GPa Poisson ratio 0.25

a b

c d

e f

g h

i j

k l

mn

o p

q r

s t

exposed face

Fig. 7.36 Brick—cells facets locating 1000 Expériments Models

f g

j k

500

800

Temperature (°C)

1000

Temperature (°C)

Experiments Models

b c

600

400

200

0

0

100

200

0

0

t (mn)

Fig. 7.37 Brick—temperature evolutions and profile after 48 mn

50

100

Distance (mm)

150

200

436

7 Inelasticity: Ultimate Load and Localized Failure

Model Experiments

25

0,5 20 0,4

d (mm)

Vertical reaction on a line (MN)

30 0,6

0,3

1st line 2nd line 3rd line 4th line

0,2 0,1

0

0

20

40

60

t (mn)

80

15

10

5

100

0

0

100

50

t (mn)

Fig. 7.38 Brick—total vertical reactions for the first lines and horizontal displacement

From the mechanical point of view, Fig. 7.38a shows the evolution of the sum of vertical reactions at selected nodes. Each curve corresponds to a line of nodes parallel to the exposed face of the wall and positive values are for compression (with prestressed initial value due to mechanical constant loading). Figure 7.38b shows the comparison on the horizontal displacement of a wall built with ten rows of bricks. This bending is due to the temperature gradient through the wall. We show that the stiffness provided by the analysis is quite correct even if the displacement is slightly over-estimated by the absence of mechanical boundary conditions. Global analysis on the entire wall (e.g. without periodic boundary conditions) has been made in order to improve this result. 7.3.6.4 Closing Remarks We presented the model for thermomechanical coupling in folded plates or nonsmooth shells, which can be used for analysis of fire-resistance of cellular structures. The need to deal with cellular structure, or folded plates and non-smooth shells, will impose that the shell model also carries the so-called drilling rotation components, which can account for the coupling between bending and membrane deformations of two neighboring shell elements along a non-smooth joint edge. The limit load computation of any such structure should include a corresponding model of inelastic behavior. In that respect, the constitutive model proposed herein in terms of the Saint-Venant (or Rankine-like) multi-surface plasticity criteria is not only quite adequate for the application on hand, but it is also very representative of other stress resultant plasticity models one can conceive for describing the inelastic behavior of shells. The problem of thermomechanical coupling, whose solutions confirms the fire resistance of the shell, along with a number of model modifications which are typical of shells (such as the choice of not only mid-surface temperature but also temperature gradients, or choice of non-locking interpolations between temperatures and temperature gradients on one side and the membrane and bending strains on another), is a worthy illustration of very subtle modeling issues one has

7.4 Stress Resultants Plasticity and Localized Failure of Reissner’s Beam

437

to face when dealing with multi-physics problems involving shell structures. The problems of this kind are very likely to continue as the subject of ‘shell research’ in the future.

7.4 Stress Resultants Plasticity and Localized Failure of Reissner’s Beam The model capable of predicting the complete failure (collapse) of a frame structure is very important in the limit load design. A typical application is a push-over analysis used in earthquake engineering, the nonlinear static analysis of a building structure subjected to an equivalent static load that is pushing a structure towards its limit capacity. Traditional approach is based upon the concept of a ‘plastic hinge’, where one assumes the load bearing capacity of a particular beam cross-section to be limited to plastic moment M y . The limit load analysis can then be performed as event-to-event strategy and introduce corresponding releases into the structure, and further transform the stiffness matrix as described in [285] (or see the previous discussion in Chap. 3). A more consistent approach was initially developed in work [114] in the small displacement framework. Namely, contrary to the classical approach, we presented in [114] a rigorous development of stress resultant plasticity that can take into account plastic hardening phenomena until ultimate section resistance followed by softening until complete failure of the particular section reducing the corresponding stress resultant component to zero. For improved prediction of final failure mechanisms, it is also necessary to include geometric non-linearities. This is presented in this section. The truly large kinematics is needed for frame structures made of steel and other ductile materials that can handle large displacements and deformations of a structure during the limit load analysis. Thus, one needs a geometrically-exact beam with non-linear kinematics and non-linear constitutive behavior that should be capable of following the response of a structure to the complete failure (collapse). In this work, we propose elasto-plastic beam element in a geometrically non-linear regime [191] that can handle softening response until complete failure with corresponding stress resultant reduced to zero. In the formulation of the proposed beam element we use, as the starting point, the previous works [191, 340]. The proposed beam element includes non-linear kinematics and non-linear constitutive response. The constitutive behavior is defined as plasticity with linear hardening that includes interaction between axial force, shear force and bending moment. The evolution equations for internal variables are developed in rate form, imposing the need to employ a numerical time integration scheme, here chosen as the backward Euler scheme. Subsequently, when the beam model has reached the ultimate capacity of a cross-section, we activate one of three failure modes in either bending moment, shear or axial force, with a non-linear softening response. For simplicity, we keep failure modes uncoupled assuming that only one

438

7 Inelasticity: Ultimate Load and Localized Failure

softening failure mechanism can be activated at a time. The softening phenomena lead to corresponding localized plastic deformations, with either curvature, shear or axial strain reducing to a narrow domain (or a ‘hinge point’) with high strain gradients, which requires special interpolation (e.g. see [176]). In discrete approximation, the failure mode is represented by a field discontinuity and placed within the framework of incompatible-mode method, see [192]. The outline of the section is as follows. First, we present the main ingredients of the geometrically exact beam with elasto-plastic constitutive response. The interactions between axial force, shear force and bending moment are taken into account in the hardening regime, whereas failure modes in softening are handled separately by introducing corresponding kinematic enhancement in terms of ‘discontinuity’ or ‘jump’ in the displacement field or the rotational field depending upon the activated failure mode. The enhancement is included as an incompatible mode in the geometrically non-linear framework. The details FEM implementation, along with the results of illustrative numerical simulations, followed by concluding remarks.

7.4.1 Reissner’s Beam with Localized Elastoplastic Behavior Here, we give a detailed formulation of the two-dimensional beam in the framework of large displacement and large elasto-plastic strains. The formulation of Reissner’s beam [322] kinematics equations employs rotated strain measure. The linearization of these strain measures allows us to recover the infinitesimal strains for Timoshenko’s beam [191]. The plastic strains corresponding to stress resultant follow from the yield criterion introducing the interaction between axial force, shear force and bending moment. These constitutive equations can also be expressed in rate form [340]. The solution method is based on the finite element method, and the consistent linearization of the weak form of equilibrium equations provides a tangent stiffness matrix, for both material and geometric parts. Providing the beam element with the embedded discontinuity within the framework of a large displacement is needed for modeling softening phase. The softening can also be produced by the failure process in the connections [233]; hence, we model separately the failure modes in bending, in shearing or in axial extension. The latter requires a multiplicative decomposition of the deformation gradient into regular and singular parts, which corresponds to the additive decomposition of the displacement gradient, as already observed in [60]. Here, the weak form of equilibrium equations is recast within the framework of incompatible modes [192], which allows handling of the embedded discontinuity computation at the element level. 7.4.1.1 Geometrically Non-linear Kinematics at Localized Failure The initial beam configuration of the length L is described by it’s neutral axis position s and corresponding cross-section A. The beam volume is generated by sweeping the cross-section A along beam domain s ∈ [0, L]. Without loss of generality, we

7.4 Stress Resultants Plasticity and Localized Failure of Reissner’s Beam

439

Fig. 7.39 Beam kinematics: initial configuration (thick solid line) and deformed configuration (thin dashed line) with large displacements u and v of beam axis and large rotation ψ of beam cross-section

present the developments for an initially straight beam aligned with x axis, which allows simple parameterization with s ≡ x; see Fig. 7.39. Let the beam domain Ω be an open bounded set with a piece-wise smooth boundary such that ∂Ω = ∂Ωu ∪ ∂Ωσ and ∅ = ∂Ωu ∩ ∂Ωσ . The intrinsic parametrization of the shear-flexible Reissner’s beam can be done with a set of generalized coordinates further denoted at = (u t , vt , ψt )T , at ∈ U, which describe the motion of beam axis position and cross-section orientation in deformed configuration with respect to change of pseudo-time parameter t. The position vector ϕ t ∈ V in Euclidean space R2 can be presented by following transformation f : U ⊂ R3 → V ⊂ R2 , and thus the position vector in deformed configuration can be written   − sin ψt x + ut +y (7.197) φ t = ϕ t + ya2,t = vt cos ψt where x and y are coordinate in reference configuration, u and v are displacement components in the global coordinate system, ζ is the coordinate along the normal to the beam axis in the reference configuration (here ζ ≡ y) and ψ is cross-section rotation. The corresponding form of the deformation gradient Ft := ∇φ t will contain the contributions from both displacement and rotation fields, which can be written as: Ft =

 dψt 0 −y d x cos ψt − sin ψt + dvt t 0 −y dψ sin ψt cos ψt dx dx     

 1+ 

du t dx

I+∇ut

(7.198)

I+∇ψt

In the elastic regime, the finite strain measures remain the same as already stated in (6.72) for material and in (6.73) for spatial description. For plasticity, we have to account for irreversible plastic deformation, which requires precisely describing

440

7 Inelasticity: Ultimate Load and Localized Failure

the loading program with pseudo-time parameter t and obtaining the corresponding evolution of internal variables for plasticity by providing their rate equations. We next present a modified form of the deformation gradient at localized plastic failure, which is typical of softening phenomena with stress decrease for increasing deformation that reduces the size of the plastic zone (e.g. see [176, 181]). The failure modes in axial, shear and bending can be represented respectively by jumps in displacement components u, v and in rotation ψ, which splits any of these fields ¯ Namely, we can into a regular part (·) and an enhanced part (·) that occurs at point x. write: & u(x, t) = u(x, t) + Hx (x) u(t) 0 , x < x¯ v(x, t) = v(x, t) + Hx (x) v(t) ; Hx¯ (x) = 1 , x ≥ x¯ ψ(x, t) = ψ(x, t) + Hx (x) ψ(t)

(7.199)

where we represented the jump with the Heaviside function Hx¯ that captures the corresponding displacement or rotation jumps. The derivative computations can be written by introducing the Dirac delta function δx¯ (x) resulting in the additive decomposition of displacements and rotation gradients as: ∂u(x,t) = ∂u(x,t) + δx (x) u(t) ∂x ∂x ∂v(x) ∂v(x,t) = + δx (x) v(t) ; ∂x ∂x ∂ψ(x) ∂ψ(x,t) = ∂ x + δx (x) ψ(t) ∂x

& δx¯ (x) =

∞ , x = x¯ 1 , x = x¯

(7.200)

In the finite deformation framework (e.g. see [191]), such an additive split of displacement and rotation gradients in (7.199) above will allow us to recover the multiplicative decomposition of deformation gradient for both contributions from displacements and rotation fields: ' (  '  −1 ( F = [I + ∇u] I + δx ∇u (I + ∇u)−1 + I + ∇ψ I + δx ∇ψ I + ∇ψ             Fu

Fu

Fψ

Fψ

(7.201) We will seek the final development in stress resultants that act in a cross-section of Reissner’s beam. Thus, we perform the polar decomposition of the deformation ¯ which allows gradient and use only the regular part of rotation resulting in F = RU, us to further define the rotated strain measure H as:  cos ψ¯ − sin ψ¯ T ¯ ¯ H = U − I; U = R F ; R = (7.202) sin ψ¯ cos ψ¯ ¯ is the rotation tensor where U is the stretch tensor, I is the identity tensor and R adapted to Reissner’s beam kinematics. With the results in (7.201) and (7.202), we can obtain the corresponding additive decomposition of the stretch tensor:

7.4 Stress Resultants Plasticity and Localized Failure of Reissner’s Beam

¯ T ∇u + R ¯ T ∇ψ ¯ T (I + ∇u) + δx R ¯ T (I + ∇ψ) + δx R U=R             ¯u U

¯¯ U u

¯ψ U

441

(7.203)

¯¯ U ψ

where the first part is a regular strain field that can be written explicitly as: Uu =

    dψ 1 + ddux cos ψ¯ + ddvx sin ψ¯ 0 −y d x 0 ; U = ψ − 1 + ddux sin ψ¯ + ddvx cos ψ¯ 1 0 1

(7.204)

and the second part is an enhanced strain that is needed to properly handle the localized failure phenomena:   (δx u) cos ψ¯ + (δx v) sin ψ¯ 0 −y(δ ψ) 0 x ; Uψ = Uu = −(δx u) sin ψ¯ + (δx v) cos ψ¯ 0 0 0 

(7.205)

¯¯ as plastic stretch ¯ equivalent to elastic stretch and U We will further consider U which quantifies the localized plastic deformation. Remark on equilibrium equations: We note that all the modification one has to introduce to nonlinear kinematics in order to describe plastic localized failure does not affect the equilibrium equations, which remain the same as those already defined for Reissner’s beam model in (6.76) and (6.77) for strong form, or in (6.92) for weak form, with both expressed in stress resultants of the Biot stress already defined in (6.78). The only thing that changes is how to compute the stress tensor for Reissner’s beam for finite deformation plasticity, which is discussed next. 7.4.1.2

Constitutive Model of Finite Deformation Plasticity and Rate Equations With these results on deformation gradient and corresponding strain measures in hand, we can construct the constitutive model for finite deformation plasticity with localized failure by using the Biot stress T as the work-conjugate pair of stretch tensor. For the elastic case, one can express the strain energy density potential for any point x ∈ [0, L] of Reissner’s beam with cross-section A as: 

 Ψ (U) d A := A

1¯ F·PdA ≡ 2

A

 A

1 ¯ ¯T U ·  R P dA 2

(7.206)

T

We note that only non-zero Biot stress components that are of interest for computations with Reissner’s beam model in stress resultant form are related to corresponding non-zero components of the first Piola-Kichhoff stress P are defined by using the Cauchy principle (e.g. see [176]) to define the stress vector t1 according to: ¯ T Pe1 =⇒ t1 = t1 := Te1 ≡ R



 11 T 11 T P ¯ = R T 21 P 21

(7.207)

442

7 Inelasticity: Ultimate Load and Localized Failure

For localized plasticity, we have to modify the free-energy potential and introduce two novelties. First, we have to ‘regularize’ the rotation field by introducing a chosen ˜ ‘influence’ G(x) that can represent the effect of the localized plasticity process at x¯ and corresponding energy transfer to the rest of the structure, which allows us to write: ˜ + δx¯ (x)U(t) (7.208) U(x, t) = U(x, t) + G(x)U(t) This implies that for localized plasticity the strain energy in (7.206) above should be split between a regular part for elastic response Ψ e and a singular part for plastic softening contribution Ξ p . We further illustrate the particular case with plastic hinge when localized failure occurs at x¯ in beam bending with u = v = 0 (for more general case, see [232]) and introduce the local support2 for influence function denoted as L˜ that surrounds the failure point x¯ , which allows us to simplify the result in (7.208) above to ψ ψ ˜ U(x, t) = U(x, t) + G(x)U (t) +δx¯ (x)U (t) (7.209)    ˜ U(x,t)

Such an additive split in localized bending failure requires the additive split of strain energy ˜ ξ p ) := Ψ (U,

1 ˜T ˜ +δx¯ 1 ξ p K¯ ξ p ; ∀x ∈ (x¯ − L/2, ˜ ˜ U · CU x¯ + L/2) 2 2      ˜ Ψ e (U)

(7.210)

Ξ p (ξ p )

where C is the elasticity tensor, K¯ is the corresponding softening modulus and ξ p is the equivalent plastic bending strain as an internal variable that describes the plastic flow in softening for localized failure at x¯ due to bending moment. This failure mode evolution is governed by the yield criterion that defines a constraint on plastically admissible Biot stress, which can be written in terms of stress-resultant traction at the ¯ and stress-like variable q conjugate to an internal variable: discontinuity, tm = M(x) Φ(tm , q) := [|tm | − (Mu − q)]x¯ ≤ 0; q := −

dΞ p (ξ p ) = − K¯ ξ p ; dξ p

(7.211)

where Mu ultimate value of the bending moment that triggers softening failure. We note that softening leads to reduction of elastic domain (with K¯ < 0 and ξ p ≥ 0) until the final failure reducing Reissner’s beam bending moment bearing capacity to p zero at ξu = Mu /| K¯ |. Second, we have to use the non-local version of the second principle of thermodynamics (e.g. see [176, 181]) stating that the plastic dissipation is always either positive (in plastic regime) or zero (in elastic regime). The non-local interpretation 2 In embedded discontinuity discrete approximation we place localization point x¯ at the element Gauss point and limit the influence function support to one element only with G˜ e (x) as the derivative of the corresponding shape function (e.g. see [176, 181, 216]).

7.4 Stress Resultants Plasticity and Localized Failure of Reissner’s Beam

443

of dissipation is applied by integrating the dissipation rate over the complete volume (V = L × A) of Reissner’s beam, which can be written as: )   & ˙ − d Ψ (U, ξx¯p ) dA dx T·U dt L A )    & dΨ e ˙ dA dx + t ψ˙ T− ·U = m dU x¯

  0≤

Dp d A dx = L

A

L− L˜ A

  &

− L˜



A

(7.212)

) dΨ e ˙ ψ · U dA dx ˜ dU

d p p − Ξ (ξ ) dt

x¯

where we used the well-known result regarding the Dirac delta function (e.g. see [351]) to conclude  L ψ ˙ δx¯ (x) {T · U d A d x = M(x) ¯ ψ (7.213)    tm 0 A    ˙ ψ M(x)

For elastic process, where the yield function value is negative (Φ(·) < 0) and internal variable remains frozen (with ξ˙ p = 0), the plastic dissipation is equal to zero. Thus, we can obtain from the first term in (7.212) the set of local constitutive equations for computing the Biot stress (at current value of ψ) T := T :=

dΨ e dU dΨ e ˜ dU

 ˜ ˜ = CU; ∀x ∈ (0, x¯ − L/2) (x¯ + L/2, L); ˜ ∀x ∈ (x¯ − L/2, ˜ ˜ = CU; x¯ + L/2)

(7.214)

For the plastic process corresponding to zero value of the yield criterion (Φ(·) = 0), we have to compute the plastic flow, while assuming that the constitutive equation remains valid with the rate of change of internal variable that is much faster than the one for stress. Provided we enforce the weak form of equilibrium equation within the domain of influence of localized failure ˙ h(·) := tm ψ −

˜ x+ ¯ L/2

˜ G(x)

˜ x− ¯ L/2

 A



˙ψ T · U d A dx = 0 

˙ M(x)ψ

(7.215)



from (7.212) above we can obtain the only contribution to plastic dissipation is reduced to discontinuity

444

7 Inelasticity: Ultimate Load and Localized Failure

0 < Dx¯ = [q ξ˙ p ]x¯ p

(7.216)

Given a clear definition of localized plastic dissipation, we can obtain all the remaining results by using the maximum principle of plastic dissipation (e.g. [176, 269]) stating that the plastically admissible values of stress are those that maximize plastic dissipation, or otherwise: q=

min [−Dx¯ (q ∗ )]x¯ p

(7.217)

Φ(tm ,q)=0

The principle of maximum plastic dissipation applied to the case where only discontinuity remains active can be written by introducing the Lagrange multiplier in the form: γ˙ (x, t) = δx¯ (x)γ˙x¯ (t) (7.218) which further restricts the corresponding plastic Lagrangian to the discontinuity according to: L L (·) := p

max min L p (q, γ˙x¯ ); γ˙

p −Dx¯ (·)

+

∀(tm ,q)

γ˙ Φ(·)

(7.219)

0

= [−q ξ˙ p + γ˙x¯ Φ(·)]x¯ The Kuhn-Tucker optimality condition then reduces to ∂L p ∂Φ = 0 ⇒ ξ˙ p = γ˙x¯ ∂q ∂q 

(7.220)

=1

accompanied by loading-unloading conditions, if one includes both elastic and plastic process ∂L p = 0 ⇒ Φ(·) ≤ 0; γ˙x¯ ≥ 0; γ˙x¯ Φ(·) = 0 (7.221) ∂ γ˙x¯ This concludes the development of all local governing equations. In discrete approximation, we can solve all governing equations at each Gauss point of finite element mesh, to enforce the local constraint of plastic admissibility of stress. For the plastic step, this requires to use time integration scheme to compute the evolution of the internal variable corresponding to rotation discontinuity. We note that the model parameter for softening plasticity such as softening modulus K¯ < 0 is not easy to obtain, and one should rather use fracture energy G f . We can compute the fracture energy G f in a pure bending test (with M(x, t¯) = cst.), as the equivalent energy of external forces applied to a Reissner’s beam with constitutive behavior of softening plasticity that is needed to reduce the effective stress-resultant to zero, with

7.4 Stress Resultants Plasticity and Localized Failure of Reissner’s Beam

445

p

ξu

   Mu − qu = Mu − | K¯ | Mu /| K¯ | = 0   

(7.222)

qu

This computation is performed by integrating the total dissipation as defined from the Clausius-Duhem inequality in (7.212) for a given proportional loading program, which starts with zero applied traction at time t0 , peaks to a maximum value at time tu and goes down to zero value again as the result of softening at time t f ; we can thus write: Mu ξ˙ p

  ⎤ ⎡ t f  t f  t f  d ⎣ Mu γ˙ d x ⎦ dt = M κ˙ d x dt − δx¯ Ξ (ξ p ) d x dt dt tu t0 L tu L L          p

Mu ξu

Gf

(7.223)

p p −(1/2)ξu | K¯ |ξu

* *t ˜ d x dt = 0. For the softening plasticwhere we took into account that t0 f dtd V Ψ e (U) ity model with linear softening modulus K¯ = cst. < 0, the zero value of the effective p stress corresponds to the maximum value of the rotation discontinuity ξu = Mu /| K¯ |; from (7.223) above, we can further obtain that: Mu

Mu 1 Mu ¯ Mu 1 Mu2 = Gf + |K | =⇒ | K¯ | = 2 | K¯ | 2Gf | K¯ | | K¯ |

(7.224)

The last result shows that the fracture energy is equal to the area of the triangle below the softening part of the response, which starts at the peak value of resistance (where M = Mu ) and which ends at the final failure state with zero effective stress (where M = 0). It is also possible to obtain the corresponding expression for G f with any other form of post-peak softening response (e.g. see [176]).

7.4.2 Stress Resultant Plasticity Discrete Approximations and Computations Here we will provide the stress resultant format of the proposed localized plasticity model for Reissner’s beam, its discrete approximation and a brief outline of the computational procedure. 7.4.2.1 Stress Resultant Plasticity First, we recast the stress equations of this plasticity model in stress resultant form suitable for Reissner’s beam model. The latter imposes the kinematic constraint on rigid cross-section moved by large rotation, which results in the only non-zero strain

446

7 Inelasticity: Ultimate Load and Localized Failure

components of the rotated strain tensor H11 = Σ − y K and H21 = Γ expressed in terms of strain measures proposed by Reissner [322]. For finite deformation plasticity with softening in beam bending, these strain measures are regularized by influence ˜ which can be written in matrix notation as function G,  dv du cos ψ + sin ψ − 1 Σ˜ = 1 + dx dx    

du Γ˜ = − 1 + dx 



Σ

dv sin ψ + cos ψ dx  

(7.225)

Γ

dψ ˜ dψ + K˜ = G d x dx   K

K

We note in passing that all the developments to follow also hold for Kirchhoff’s beam, which enforces constraint Γ = 0 (e.g. see [231]). Here, we carry on with Reissner’s beam model with the result in (7.225) that can be rewritten in matrix notation:     Σ˜ = "T h a − a1 + G˜ K e3 (7.226) where ⎞ ⎡ Σ˜ cos ψ Σ˜ = ⎝ Γ˜ ⎠ ; " = ⎣ sin ψ 0 K˜ ⎛ d uˆ ⎛ ⎞ u  dx ⎜ ˆ a = ⎝ v ⎠ ; b aˆ = ⎝ dd vx d ψˆ ψ ⎛

dx

⎤ − sin ψ 0 cos ψ 0 ⎦ ; a1 = "e1 0 1 ⎞ ⎞ ⎛ ⎛ ⎞ ⎛ ⎞ 1 + ddux 1 0   ⎟ dv ⎠ ⎝ ⎝ ⎠ ⎝ 0 0⎠ ; h ; e a = = ; e = ⎠ 1 3 dx dψ 0 1 dx

(7.227) By using further the same matrix notation for the virtual strains (denoted with ˆ we can write the weak form of equilibrium equations in stress superposed (a)), resultants with:    ˆ T (a) d" r d x − aˆ T pext d x = 0 ˆ := ˆ bT (a)" + ψh G a¯ (a, ψ; a) dψ L L (7.228)  T ˆ ˆ ˆ G e¯¯ (a, ψ; ψ) := ψ tm − ψ G˜ e M d x = 0; ∀e with Φ = 0 ψ

Le

7.4 Stress Resultants Plasticity and Localized Failure of Reissner’s Beam

447

where virtual displacement and rotation field derivatives for the regular part are written in (7.227). In (7.228) above, N , V and M denote stress resultants expressed in terms of the Biot stress, which can be gathered in the vector of internal forces and written in matrix notation:    T 11 21 r = (N , V, M) ; N = T d A ; V = T d A ; M = − yT 11 d A (7.229) A

A

A

If behavior remains elastic, the internal forces are computed from the current values of strain measures (at frozen jump) by multiplying them with Reissner’s beam constitutive matrix r = CΣ˜ ; C = diag (C N , C V , C M )

(7.230)

where C N = E A, C V = G A and C M = E I are respectively elastic stiffness of the beam cross-section for axial force, shear force and bending moment. On the opposite, one first has to carry out the local (element-wise) computation of plastic flow and converge at the local level to provide the plastically admissible value of the stress that satisfies Φ(·) = 0. Subsequently, we keep the current values of discontinuity parameters frozen and then carry out one sweep of iterative procedure at the global (structure) level. If Newton’s iterative procedure is used, we need to carry out consistent linearization to obtain the tangent operators that provide a quadratic convergence rate. The latter also requires that we first discretize the problem and then linearize, but in the 1D case, these two procedures can be inverted (e.g. see [176]). Thus, we can perform the consistent linearization of the weak form in (7.228)1 to obtain the following contributions: '

( ˆ |t=0 G a¯ (a + tΔa, ψ + tΔψ; a)      b Δa dx ψˆ DmK + DgK Δψ L   d" ˆ T ˆ T ˜ + b (a)" + ψh (a) Ce3 GΔψ dx = 0 dψ

ˆ + d L [G a¯ (·)] = G a¯ (a, ψ; a) dt    = G a¯ (·) + b aˆ

(7.231)

L

where DmK and DgK are defined in (7.232) below ' "  C "T = T h a d" dψ 

DmK

d"T dψ

 ( h a ; DgK =



0 T

r T d" dψ

d" r dψ  2 hT a ddψ"2 r

 (7.232)

7.4.2.2 Finite Element Discrete Approximation The numerical implementation of the softening plasticity model with displacement discontinuity can nicely fit within the framework of the incompatible mode method

448

7 Inelasticity: Ultimate Load and Localized Failure

[225]. We will illustrate the pertinent details of numerical implementation for the simplest finite element model using a 2-node isoparametric Reissner’s beam elements. In order to allow for rotation discontinuity representation, without modifying the displacement values at the nodes, we need an additional parameter α e (which can further be treated in the same manner as the incompatible mode parameter): x(ξ )|Ω e =

2 

Na (ξ )xa ; Na (ξ ) =

a=1

u h (x, t)|Ω e =

2 

1 (1 + ξa ξ ); 2

Na (x)da,1 (t);

a=1

v h (ξ, t)|Ω e =

2 

(7.233)

Na (ξ )da,2 (t);

a=1

ψ h (ξ, t)|Ω e =

2 

Na (ξ )da,3 (t) + M e (ξ )α e (t);

a=1

& M e (ξ ) =

− 21 (1 + ξ ), si ξ ∈ [−1, 0] 1 (1 − ξ ), si ξ ∈ [0, 1] 2

The proposed interpolation will place the rotation discontinuity in the center of this element. The corresponding derivatives approximation defines the bending strain measure, which can then be written:  dψ h (x, t) (−1)a |L e = Ba (x) da,3 (t) + G e (x) α e (t); Bae (ξ ) = ;       dx Le a=1 2

d Na dx

dM dx

d N2 (x) ; δ0 (x) := G e (x) = G˜ e (x) + δ0 (x); G˜ e (x) = − dx

&

0 , ξ = 0 ∞ ,ξ = 0 (7.234) where δ0 is the Dirac function centered within the element. In order to ensure the method convergence in the spirit of the patch test, we will make sure that the incompatible mode variation remains orthogonal to the constant stress in each element σ e ; such a work-conjugate couple should thus satisfy: 

 e

M  cst. L e

G dx = 0 ⇔ e

Le

1 − e dx = − L    G˜ e

 δ0 d x

(7.235)

Le

For the present 1D case, with displacement discontinuity placed in the center of the element, the patch test condition in (7.235) is automatically satisfied leaving G e intact. We can thus keep exactly the same approximation for the virtual strain field as the one used for real strains

7.4 Stress Resultants Plasticity and Localized Failure of Reissner’s Beam

449

2  d ψˆ h (x, t) B (x) dˆ (t) + G e (x) αˆ e (t) |Ω e =  a  a,3    dx a=1 d Na dx

(7.236)

dM dx

Both of these approximations are constructed with a single Gauss point (n int = 1) placed at the center of the element with a weighting factor equal to 2 (e.g. see [176]). Such approximation gives the constant values for stress resultant r := (N h , V,h M h )T , and can be written in component form as: ˜ h (0, t) N h (0, t) = E A Σ h V h (0, t) = G A Γ˜ (0, t)  2   h e e M (0, t) = E I Ba (0)ψa (t) + G˜ (0)α (t)

(7.237)

a=1

With these results in hand, we can construct the discrete approximation for the residual vector for a 2-node Reissner’s beam element, first with the standard part of element’s internal force dˆ T fint,e =

n int =1 l=1

d"h h ˆ T (ah )

r }|ξ =0 j (0)  { bT (aˆ h )"h + ψh w  dψ    2 L/2

(7.238)

¯T dˆ T B

and then for the extra terms for the element in which plastic localized failure occurs αˆ h e = αˆ tm + αˆ

n int =1 l=1

{G˜ e M h }|ξ =0 j (0)  w 

(7.239)

2

L/2

7.4.2.3 Computational Procedure for Localized Plasticity The stress resultant field approximation can be obtained from the regular part of the real strain field in (7.234) with no contribution from the singular part representing the plastic strain; we can thus write: M (x, t)| L e = E I h

- 2 

. ˜e

Ba (x) da,3 (t) + G (x) α (t) e

(7.240)

a=1

By exploiting the approximations for virtual strains and stress, defined in (7.236) and (7.240) respectively, the weak form of equilibrium equations can be recast in the format typical of incompatible mode method:

450

7 Inelasticity: Ultimate Load and Localized Failure n elem

A (fint,e − fext,e ) = 0; fint,e =



e,T

B

e=1

r(d, α e )d x

Le



h = 0 ∀e ∈ [1, n elem ]; h = e

e

G˜ e,T M(d, α e )d x + tm (α)

(7.241)

Le

where tm = M|x¯ is the traction stress-resultant acting at discontinuity. This traction, which indirectly depends upon the value of parameter α e , enters the corresponding plasticity criterion: (7.242) Φ(tm , q) := |tm | − (Mu − q) ≤ 0 where q is the variable defining the current plasticity threshold depending upon the rotation discontinuity. We will solve this problem by using the operator split solution procedure for finding the solution to equations in (7.241) and (7.242) in an equivalent manner to the one already proposed for the standard plasticity model. Namely, we will treat separately the local phase from the global phase of computation and solve the former (defined in 7.241) for total displacement field (and incompatible mode parameter) and then the latter (defined in 7.242) for the corresponding localized plastic flow. There remains, however, one subtle point to be addressed concerning the role of parameter α e , which contributes to both local and global phases of computations. By starting with the local phase of computations, we will assume to be given the best iterative value of displacement de,(i) n+1 , for which we can further perform: Local computation: for element e with rotation discontinuity: e Given: de,(i) n+1 , αn , h = tn+1 − tn e Find: αn+1 , such that Φn+1 ≤ 0

(7.243)

Between two possibilities for the corresponding value of yield function at time tn+1 (Φn+1 ≤ 0), we will first test for the elastic trial state produced with a zero value trial of the plastic multiplier associated with discontinuity γ˙¯¯n+1 = 0: trial γ¯¯ n+1 := γ˙¯¯n+1 h = 0 e = αe αn+1

) ⇒

⎧ ⎨ ⎩

trial tm,n+1 =−

* Le

  2 ! e (i) dx Ba da,n+1 + G˜ e αne G˜ T E I a=1

trial trial Φn+1 = |tm,n+1 | − (Mu − qn )

(7.244) If the elastic trial state does not produce a positive value of the yield function, trial ≤ 0, we conclude that the step is indeed elastic and accept the trial values as Φn+1 the final ones, with no modification of the plastic strain computed in the previous step. In such a case the incompatible mode parameter αne will also remain intact, and the only change of the traction force can come from a displacement increment; we can thus write:

7.4 Stress Resultants Plasticity and Localized Failure of Reissner’s Beam

Lin[tm,n+1 ]αe := tm,n+1 + Kd Δdn+1 ; Kd := K αe

∂ t¯m := =0 ∂α e

∂ t¯ = E I BT ∂d

451

(7.245)

trial On the other hand, if the trial value of the yield function is positive, Φn+1 > 0, the elastic trial step is no longer considered plastically admissible. We are thus sure that the current step is plastic, and that we have to modify the elastic strain and the incompatible mode parameter αne to re-establish the plastic admissibility of the driving traction at discontinuity. Such a computation is carried out in the manner which allows exploiting the results obtained for the trial elastic state. Namely, we can first express the traction at discontinuity:

˜e e t¯m,n+1 = E I (Bd(i) n+1 + G αn+1 )  1 EI e (i) − αe ) = E I Bdn+1 − e αn − e (αn+1 L L   n    γ¯¯ sign(t¯ ) trial t¯m,n+1

trial = t¯m,n+1 −

n+1

(7.246)

n+1

EI ¯ γ¯n+1 sign(t¯n+1 ) Le

and then use the last result to write the final, plastically admissible value of the yield function in terms of the corresponding modification of its trial value: 0 = Φ(t¯m,n+1 , qn+1 ) := |tm,n+1 | − (Mu − qn+1 ) EI trial | − (Mu − qn ) +(qn+1 − qn ) − e γ¯¯ n+1 = |t¯m,n+1 L   

(7.247)

trial Φn+1

For a linear softening law, where q = − K¯ α e with a constant value of K¯ , we can express the last term in (7.247) above in terms of γ¯¯ n+1 : qn+1 − qn = − K¯ γ¯¯ n+1

(7.248)

We can thus obtain the final value of the plastic multiplier in this step, which is proportional to the computed trial value of the yield function, and carry out the corresponding update of the incompatible mode parameter: γ¯¯ n+1 =

trial φn+1

( EL eI

+ K¯¯ )

e trial = αne + γ¯¯ n+1 sign(t¯m,n+1 ) ; αn+1

(7.249)

452

7 Inelasticity: Ultimate Load and Localized Failure

For a nonlinear softening law, the result equivalent to (7.249) ought to be obtained iteratively, according to: ( j) = 1, 2, . . . E I ¯ ( j) e,( j) + [q(α ˆ n+1 ) − q(α ˆ ne )] > 0 (≈ tol.) γ¯ L e n+1 ( j) ( j) ( j) ( j) (7.250) =⇒ Lin[φn+1 ] := φn+1 + Dγ¯¯ φn+1 Δγ¯¯n+1 ( j)

trial − as long as: φn+1 := φn+1

( j) ( j) ( j+1) ( j) e,( j+1) e,( j) trial ) γ¯¯n+1 = γ¯¯n+1 + Δγ¯¯ n+1 ; αn+1 = αn+1 + Δγ¯¯ n+1 sign(tm,n+1

( j) ←− ( j + 1) In a plastic step, the modification of the discontinuity traction is produced by a change of incompatible mode parameter α e ; hence, the corresponding value of the elasto-plastic tangent modulus can be written: K α :=

∂ t¯ ∂ t¯ trial ); Kd := = K¯¯ sign(tm,n+1 =0 ∂α ∂d

(7.251)

Having converged with local computation to the final value of incompatible mode e (for either elastic or plastic step), we turn back to the global compuparameter αn+1 tation in order to check the convergence of the displacement field; the latter can be written: Global computation: equilibrium equations (i) e e e Given: d(i) n+1 , αn+1 , with h n+1 (dn+1 , αn+1 ) = 0 n elem

e e,ext as long as:  A (ˆfe,int (d(i) ) > 0 (≈ tol.) n+1 , αn+1 ) − f e=1 ( ' n elem e e,ext , α ) − f ) =⇒ 0 = Lin A (ˆfe,int (d(i) n+1 n+1 e=1 ( ' n elem n elem e e,ext ˆ e,(i) Δd(i) , α ) − f ) + A := A (ˆfe,int (d(i) K n+1 n+1 n+1 n+1 e=1

(7.252)

e=1

(i) (i) d(i+1) n+1 = dn+1 + Δdn+1

We assumed in (7.252) above that Newton’s iterative procedure can be employed to provide an improved guess on the best iterative value of displacement field; consequently, the contribution of a typical element to the linearized form of the system in (7.252) is defined as: e e,ext e e,ext Lin[fe,int (d(i) ] = fe,int (d(i) n+1 , αn+1 ) − f n+1 , αn+1 ) − f e,(i) e,(i) e,(i) + Kn+1 Δde,(i) n+1 + Fn+1 Δα n+1 (i) e,(i),T e e e e Lin[h en+1 (d(i) n+1 , αn+1 )] = h n+1 (dn+1 , αn+1 ) +Fn+1 Δdn+1   

+

=0 (i),e Hn+1 Δα en+1

+ Kd Δden+1 + Kα Δα en+1

(7.253)

7.4 Stress Resultants Plasticity and Localized Failure of Reissner’s Beam

453

where the element matrices can be written explicitly:  e = Kn+1

B¯ e,T Cn+1 B¯ e d x

Ωe



e Fn+1

=

˜ e dx B¯ e,T Cn+1 G

(7.254)

Ωe



Hen+1 =

˜ e,T Cn+1 G ˆ e dx G

Ωe

The right choice of matrices Kd and K α in (7.253) above is made according to (7.245) for elastic or to (7.251) for plastic step. It is indicated in (7.253) that the residual at discontinuity is equal to zero, since we have obtained the converged e . This allows to carry out the static value of the incompatible model parameter α¯ n+1 condensation and thus reduce the tangent stiffness matrix to the standard form with the size corresponding to the nodal displacement vector only: e,(i) e,(i) e,int,(i) e,T e,ext e e − Fn+1 (Hn+1 + K α )−1 (Fn+1 + Kd )] Δdn+1 = fn+1 − fn+1 [Kn+1   

(7.255)

e Kˆ n+1

We can thus proceed with the standard finite element assembly procedure to account for each element’s contribution to global equilibrium equations, solve this system and carry out the corresponding displacements update: (i) (i+1) (i) (i) ext int Δd(i) Kˆ n+1 n+1 = fn+1 − fn+1 =⇒ dn+1 = dn+1 + Δdn+1

(7.256)

With this result in hand, we can restart, if needed, another sweep of the operator split procedure.

7.4.3 Illustrative Numerical Examples and Closing Remarks A couple of numerical examples are presented in this section to illustrate the performance of the proposed finite element formulation. All numerical computations are performed with a research version of the computer program FEAP [390]. 7.4.3.1 Straight Cantilever Under Imposed End Rotation In this example, we present a nonlinear plastic response including both hardening and softening for a cantilever beam under a free-end bending load. The geometric properties of the cross-section correspond to standard hot rolled profile IPE 200 and material properties take values for steel class S235. More precisely, The chosen properties of the cantilever are: Young’s modulus: E = 2 × 104 kN/cm2 ; Hardening

454

7 Inelasticity: Ultimate Load and Localized Failure

Fig. 7.40 Cantilever beam plastic failure with hardening and softening: deformed shapes (left), force-displacement diagram (right)

modulus: K = 0.05E; Moment of inertia: I = 1940 cm4 ; Area of the cross-section: A = 28.5 cm2 ; Yield bending moment: M y = 3100 kN cm; and ultimate bending moment: Mu = 3800 kN cm. The mesh for the initially straight cantilever beam model is constructed with Reissner’s beam model using either 2, 4 or 8 elements. Each analysis is performed under imposed end rotation ψ = π and the corresponding bending moment is obtained as a reaction. The analysis represents the elasto-plastic response that goes into both hardening and softening when entering the localized failure response phase. The failure is localized in the middle of the cantilever, where one element is weakened (see Fig. 7.40 for deformed shape). Response diagrams, computed for three different mesh gradings with 2, 4 or 8 elements, show the mesh indifference of the proposed formulation (see Fig. 7.40). 7.4.3.2 Push-Over Analysis of a Symmetric Frame In this example, we present the results of a push-over analysis of a symmetric steel frame. The frame geometry is given in Fig. 7.41. The material properties for all frame members are equal (Young’s modulus: E = 2 × 104 kN/cm2 ; Hardening modulus: K = 0.05E; Moment of inertia: I = 1940 cm4 ; Area of the crosssection: A = 28.5 cm2 ; Yield bending moment: M y = 3100 kN cm; Ultimate bending moment: Mu = 3800 kN cm; Yield shear force: Vy = 355 kN; Ultimate shear force: Vu = 400 kN, Fracture energies: G f,M = 550 and G f,V = 450), except the fact that the cross-section properties of the columns are 10% stronger than the crosssection properties of the beams. The elements which connect beams to columns are 10% weaker than cross-section properties of beams; these elements are chosen to simulate the behavior of connections in the global analysis of the steel frame structure. The vertical load was applied to all beam members. This load is kept constant throughout pushover analysis in order to simulate the dead load effect. The lateral loading is applied in terms of an imposed incremental displacement u top at the upper corner (point A, see Fig. 7.41). In Fig. 7.41b, we present the deformed configuration of the steel frame and the corresponding distribution of the bending moments.

7.4 Stress Resultants Plasticity and Localized Failure of Reissner’s Beam utop A

qv

qv

qv

qv

qv

qv

qv

qv=14kN/m

qv

qv

qv

qv

455

Bending moment Hc=3,0m

Hc=3,0m

Hc=3,0m

Hc=3,0m Lb=5,0m

Lb=5,0m

Lb=5,0m

Frame geometry and loading

Deformed shape and bending moment distribution

Fig. 7.41 Multi-storey frame: structure geometry and loading (left) and deformed shape and bending moment distribution (right)

Fig. 7.42 Multi-storey frame: localized hinges at failure (left) and force-displacement diagram (right)

The position of activated plastic hinges in the final stage of failure and the computed softening response in terms of the force-displacement diagram are both shown in Fig. 7.42. 7.4.3.3 Closing Remarks The presented geometrically non-linear planar beam model provides the ability to account for both bending and shear failure. The proposed constitutive model contains both coupled plasticity with isotropic hardening and non-linear law for softening with three different failure mechanisms. The hardening response providing the interaction between bending moment, shear force and axial force can be calibrated against damage to beams or columns in a steel frame. The softening response can be activated to model the failure mode in the connections with different failure mechanisms. Which of these mechanisms will be activated depends on interplay and stress redistribution during the limit load analysis. By using the proposed beam element we can perform an ultimate limit analysis of any frame planar steel structure, including different failure mechanisms. The geometrically non-linear approach allows the ulti-

456

7 Inelasticity: Ultimate Load and Localized Failure

mate limit analysis with large displacement without any need for correction of the proposed property as suggested in [114]. This advantage is very important in a steel frame structure because of the large ductility of steel. The results for all numerical examples illustrate an excellent performance of the proposed beam element.

Chapter 8

Brief on Mulitscale, Dynamics and Probability

Abstract We here turn to current applications in multiscale modeling and probability computations. We provide only a brief presentation of the useful synergy between the nonlinear mechanics and stochastic that can be used not only to build the model of interest for dealing with real-life heterogeneous structures, but also to develop solution procedures that reach way beyond traditional engineering tools. Hence, we first present multiscale approach and corresponding constitutive model or Reissner’s beam used for quasi-brittle materials, which was needed to study the risk of scabbing under airplane impact. We also present the multiscale approach that can handle size effect with a probability-based explanation to why the dominant failure modes in a large structure are not necessarily the same as in the small structure built of the same heterogeneous material. The same approach explains how to replace the Rayleigh damping with the materials-based dissipation and recover characteristic exponential decay of vibration amplitude at the structure level due to the probability-based description of material heterogeneity. The final illustration is the use of reduced model in terms of the Duffing oscillator that can give a complete overall description of instability phenomena for geometrically exact (heterogeneous) Euler beam by solving corresponding stochastic differential equation obtained by model reduction.

8.1 Mulitscale Approach to Quasi-brittle Fracture in Dynamics The characteristic inelastic responses of heterogeneous brittle materials, such as concrete, ceramics or metallic powder, are very difficult to interpret without appealing to their microstructure. The standard test procedures to obtain the mechanical properties of brittle materials, such as a simple traction or compression test or yet a 3-point bending test, show a significant dispersion of experimental results. The latter concerns not only local properties, such as crack spacing and characteristic profile, but also global response represented with the resulting force-displacement diagram. In order to develop a clear interpretation of these experimental results and predictive models for the heterogeneous brittle material behavior, we abandon the usual framework of continuum mechanics in favor of so-called discrete models © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Ibrahimbegovic and R.-A. Mejia-Nava, Structural Engineering, Lecture Notes in Applied and Computational Mechanics 100, https://doi.org/10.1007/978-3-031-23592-4_8

457

458

8 Brief on Mulitscale, Dynamics and Probability

which start from the smallest computationally acceptable scale of micromechanics, where a heterogenous material is considered as an assembly of elementary particles held together by cohesive forces. This kind of approach, pioneered by Cundall in the late 70’s (e.g. see [100]), has mainly been restricted to granular soils (e.g. see [70] for an extensive review), where the material is considered as an assembly of (spherical) particles whose interaction is represented by frictional contact models of Mohr-Coulomb type. Later, these models were extended to other heterogeneous brittle materials, introducing a number of modifications concerning the shape of interaction particles, the presence of cohesive forces and the possible mechanisms representing the fracture. Some of the representative works of this kind are those of Kuhn and Hermann [250], d’Addetta et al. [102], Schlangen and Garboczi [329] or Chang et al. [75] who all use the Euler-Bernoulli beam lattice network to represent cohesive forces, then works of Camacho and Ortiz [69] or Ortiz and Pandolfi [308] who developed contact elements for modeling cohesive forces in 2D and 3D fracture problems or yet the work of Belytschko and Organ [45] who include the cohesive forces within the framework of the element-free Galerkin method. With respect to these works the main novelties introduced herein concern the following: (i) The construction of the beam lattice network representing the cohesive forces, for which we employ the geometrically nonlinear beam model of Reissner (e.g. [322]). The beam model of this kind, which includes the shear deformation, allows us to easily represent all the fracture mechanisms leading to either crack opening displacement or crack sliding displacement, the latter not being available for Euler-Bernoulli beam lattice. Moreover, the ability of the present model to handle large rotations is important for correctly representing the behavior of a heavily damaged zone as well as the displacement patterns of an assembly of several connected particles which split from the main structure. The constitutive model for cohesive forces obtained from analysis of the interface between the neighboring particles can be implemented within the framework of reduced integration of Reissner’s beam (e.g. see [191]). (ii) Using 1D discrete interface models of Delaplace et al. ([107]) where the mechanical properties are considered as random variables, which are here extended to structures with an arbitrary crack orientation. It is shown that the analysis of this kind can be brought to bear upon the identification of intrinsic material properties, such as internal length parameter for fracture, and the development of irreversible damage phenomena. Furthermore, the results obtained by this kind of models pertaining to instability producing localization problems can be interpreted through the appropriate studies of response fluctuation and statistical distribution of avalanches (e.g. see [145, 310] or [108]). (iii) The constantly increasing demand to generalize the heterogeneous models of this kind in order to make them applicable in the analysis of structures requires establishing the link between the micromechanics models of this kind with the standard phenomenological models developed at the macroscale. This work can be carried out without major difficulties (e.g. see Chang et al. [75]) only for

8.1 Mulitscale Approach to Quasi-brittle Fracture in Dynamics

459

elastic response. The fracture process can no longer be represented in detail at the macroscale, where crack patterns are ‘smeared’ and artificial characteristic length parameters must be introduced. The potential direction in constructing a macroscale model that is shortly explored in this work concerns the models of beam lattice networks constructed at the microscale, where both elastic and inelastic fracture responses can be developed in a systematic manner. The outline of the section is as follows. First, we present the governing equations of the geometrically nonlinear beam model of Reissner used as a component of the beam lattice network. The constitutive models for fracture are then discussed both at the microscale and macroscale, along with the final model for dynamic fracture composed of the beam lattice network and the time-stepping schemes. Finally, we present the results of a couple of illustrative numerical computations and concluding remarks.

8.1.1 Geometrically Exact Shear Deformable Beam as Cohesive Link In order to provide a reliable description of a complex heterogeneous structure, we make use of Voronoi cells, with each cell represented as a convex polygon. The Voronoi polygon construction can either be obtained from a random tessellation of the plane domain occupied by the structure or as a more or less reliable scanner-like representation of the observed heterogeneities of a structural assembly to be tested. In either case, one starts by choosing a set of points in the plane and then assigning to each point the part of the plane domain which is closer to it than to any other of the chosen points. For the given set of points Pi , i = 1, ..., n, the plane domain Ω will be covered by non-overlapping polygons Ω = ∪Ωi where each polygon Ωi associated with point Pi is defined as: Ωi = {P such that d(P, Pi ) ≤ d(P, P j ), ∀ j = i}

(8.1)

where d(·, ·) denotes the usual distance in the Euclidean space (see Fig. 8.1). It follows from (8.1) that each side of a given polygon splits the distance d(Pi , P j ) = li j = l e in half. Hence, the interface between two adjacent Voronoi cells coincides with the center of the beam cohesive link (see Fig. 8.1), which is an important observation to be exploited later. The lattice network representing the heterogeneous structure is constructed by connecting all the points in neighboring Voronoi polygons by 2-node straight beam elements, which leads to Delaunay triangularization (e.g. see [128]) where the beam elements are placed along the sides of each triangle. In the rest of this section we briefly describe the main features of interest for developments to follow of the chosen geometrically nonlinear model of the shear deformable beam. For a more elaborate description we refer to [8, 206, 322].

460

8 Brief on Mulitscale, Dynamics and Probability

Fig. 8.1 a Voronoi polygon microstructure representation on representative volume element domain Ω; b two neighboring Voronoi cells connected by cohesive links

Ωi

Y

y Pi

Ωj x

Pj

lij /2 lij /2

(a)

X

(b)

Fig. 8.2 Initial and deformed configurations of shear deformable geometrically nonlinear Reissner’s beam: initial configuration (dashed line) and deformed configuration (solid line)

In the local coordinate system (see Fig. 8.1) we describe the initial and deformed configurations of the beam as (x e1 + y e2 ) → φ = (x + u) e1 + v e2 +y a2 ;    ϕ



a2 = Λe2 ;

cos ψ − sin ψ Λ= sin ψ cos ψ

(8.2)



where u and v are displacement components of the beam axis and a2 is the unit vector indicating the position of the beam cross-section in the deformed configuration (see Fig. 8.2). According to the kinematic hypothesis of Reissner [322], the plane section remains plane, but not necessarily perpendicular to the beam axis, if we also want to take the shear into account. The corresponding strain measures for dilatation and shear indicated in Fig. 8.2 can then be written as   ε 1 + u  − cos ψ ⇔ ε = ϕ  − a1 = v  − sin ψ γ

(8.3)

8.1 Mulitscale Approach to Quasi-brittle Fracture in Dynamics

461

where (·) = ∂(·)/∂ x. We note in passing that if the beam displacements and rotations remain small, such that cos ψ ≈ 1 and sin ψ ≈ ψ, we obtain from (8.3) above the standard format of the strain measures for Timoshenko’s beam (see e.g. [191]). For large displacements and rotations, there exist many alternative forms for expressing the strain measures (e.g. see [278] or [8]) with the Biot strains as the most suitable for this kind of beam theory (e.g. see [191])  ∂φ ∂φ ; = [φ  ; a2 ]; H = Λ F − I2 , F = ∂x ∂y 

T



where

10 I2 = 01

(8.4)



The Biot strain tensor in (8.4) above can also be expressed in terms of components H11 (x, y) = Σ(x) − y K (x) H21 (x, y) = Γ (x)

(8.5)

where Σ(x) = a1 · φ  − e1 · e1 ; K (x) = a2 · a1 ; Γ (x) = a2 · φ  − e2 · e1 The Biot stress tensor components (e.g. [52, 191]) are employed as work-conjugate to the Biot strain in (8.5) above. For example, for a linear elastic behavior of the beam, we have T11 (x, y) = E(x, y)H11 (x, y); T21 (x, y) = G(x, y)H21 (x, y)

(8.6)

where E and G are, respectively, Young’s and shear moduli. In following the standard developments for beam theories, we can further define the stress resultants N and V and couple M according to



T11 dA; V =

N= Ae

T21 dA; M = −

Ae

yT11 dA

(8.7)

Ae

where Ae is the area of the beam cross-section. If the cross-section is symmetric and the local reference frame is placed in the center of gravity, we can rewrite the elastic constitutive equations in (8.6) directly in terms of the stress resultants with N = E Ae Σ; V = G Ae Γ ; M = E I e K where I e =

 Ae

y 2 dA is the section moment of inertia.

(8.8)

462

8 Brief on Mulitscale, Dynamics and Probability

With this kind of work-conjugate pairs, we can write the total potential energy of the elastic beam as ⎡ ⎤

1 ⎣ e e e e e Π = (8.9) Σ (C A )Σ +Γ (G A )Γ +K (E I )K ⎦ d x − Πext          2 Le

N

V

M

With the main goal in mind of developing the discrete model of heterogeneous structure employing the beam lattice network, we can use the finite element discretization of the presented beam model to provide the corresponding representation of the lattice network component. In particular, by assuming the simplest choice of finite element approximations, we assume that each strain component in (8.5) remains constant along the beam, which allows us to express them as functions of the beam end displacements and rotations according to (e.g. see [191])   u2 − u1 v2 − v1 ψ1 + ψ2 ψ1 + ψ2 ) 1+ ) + sin( −1 Σ = cos( 2 Le 2 Le   ψ1 + ψ2 u2 − u1 v2 − v1 ψ1 + ψ2 + cos( (8.10) ) 1+ ) Γ = − sin( 2 Le 2 Le ψ2 − ψ1 K = Le One can thus easily show that the beam length change Δs can be expressed with Δs = [(1 + Σ)2 + Γ 2 ]1/2 L e

(8.11)

which is illustrated in Fig. 8.3. We note that the strain measures in (8.10) can also be obtained by using the reduced, one-point Gauss integration rule (see [191]) along with the standard isoparametric interpolations. We also note that the reduced Gauss integration rule makes use of the cross-section placed at the mid-spam of the beam, which can be exploited to accommodate within the same framework the developments of the beam element arrays for more elaborate constitutive models allowing for inelastic behavior as shown next.

8.1.2 Micro and Macro Constitutive Models for Dynamic Fracture In this section, we present the constitutive model for brittle fracture implemented within the framework of the beam lattice network. The model is first developed at the microscale, where each beam should provide representation of only two elementary fracture modes, measured by either the beam length change or by relative rotations of end cross-sections. The mechanical properties of the beam components in the

8.1 Mulitscale Approach to Quasi-brittle Fracture in Dynamics

463

Fig. 8.3 Graphic interpretation of the Biot strain measure components (axial strain Σ, shear strain Γ ) and cross-section rotation ψ

Fig. 8.4 Two neighboring Voronoi cells with Reissner’s beam as the cohesive link

lattice network are considered to be random variables with the Gaussian density distribution. The alternative form of the constitutive model is also developed at the mesoscale where the mechanical properties are average values of those from the microscale and are therefore considered as deterministic (the constitutive model of this kind can be obtained by integrating through the thickness).

8.1.2.1

Microscale Constitutive Model

The main hypothesis in constructing the microscale constitutive model is that the size of each Voronoi cell corresponds to the representative size of heterogeneities. In other words, each cell is the characteristic of a single grain. Like grains, the Voronoi cells are kept together by cohesive forces. These cohesive forces are represented by the beam lattice network, which is constructed so that the beams connect the center of gravity of each pair of neighboring cells. This is contrary to the arrangement shown in Fig. 8.1b, where Pi and P j are placed in the center of gravity of their cells (see Fig. 8.4). This kind of choice allows us to consider that the elastic response of each beam is given by a diagonal matrix, as already indicated in (8.8), with the corresponding values of area and moment of inertia for the beam cross-section

464

8 Brief on Mulitscale, Dynamics and Probability

computed from the length h i j = h e of the common size of the neighboring cells taken as the fictitious thickness, to obtain Ae = h e ,

Ie =

(h e )3 12

(8.12)

The chosen elastic constitutive response of each beam representing cohesive forces is limited to the elastic domain. Outside that domain the fracture is assumed, with all the cohesive forces and moments being reduced instantaneously to zero. The fracture criterion is set in the strain space. It takes into account two possible modes of fracture: first, the traction-induced separation between the neighboring cells is limited to [(1 + Σ)2 + Γ 2 ]1/2 =

Δs ≤ εicrj Le

(8.13)

and second, the relative bending deformation is restricted to    ψ j − ψi   ≤ θ cr |K | =  ij Le 

(8.14)

The elastic limits εicrj and θicrj are supposed to be random variables with Gaussian distribution. For each one of them, we thus have to define the mean value and the standard deviation. The mean value of the fracture deformation εicrj controls the global constitutive behavior in traction where the rupture of all the cohesive links or cracks are perpendicular to the loading direction and can thus be deduced from the corresponding peak value of the traction-displacement diagram in a simple tension test. The mean value of the relative rotation θicrj is related to the case of compressive loading where cracks might appear parallel to the loading direction due to the Poisson’s ratio effect, where so-created isolated beam components can further buckle and break. The chosen rotation limit will thus control post-peak softening constitutive behavior. The final ingredient of the constitutive model at the microscale takes into account the interaction of the Voronoi cells which are not initially the neighbors and thus not connected by a cohesive beam components, but can come in contact in the later stage of the deformation with the extensive spreading of the fracture zone. The penalty like frictionless contact model is used for such a purpose, where the cell interpenetration is allowed with the contact force proportional to the overlapping area applied in the direction connecting the centers of gravity for two cells (see Fig. 8.5).

8.1.2.2

Macroscale Constitutive Model

In mesoscale constitutive modeling we assume that each Voronoi cell is a collection of elementary particles of a sufficiently large size which allows us to take the average

8.1 Mulitscale Approach to Quasi-brittle Fracture in Dynamics Fig. 8.5 Contact of Voronoi cells which are not initially neighbors

465

(i)

lij (j)

α ij

and no longer consider the mechanical properties as the random variables. For the particular choice of the beam lattice network the constitutive model of this kind can be constructed by integration through the thickness of the beam. The beam crosssection to be considered for this integration is chosen to coincide with the common side separating two neighboring Voronoi cells. In order to remain consistent with the one-point Gauss quadrature used to compute the beam element stiffness matrix and residual vector, we ought to assure that the considered cross-section is placed in the middle of the beam spam, which implies that original points Pi and P j are kept as the nodes of the beam lattice network. This also implies that the beam reference frame no longer passes through the center of gravity of the section. The elastic response of the beam will thus introduce the coupling of axial and bending strains as ⎡

⎤ ⎡ ⎤⎡ ⎤ N E Ae 0 E S e Σ ⎣ V ⎦ = ⎣ 0 G Ae 0 ⎦ ⎣ Γ ⎦ M K E Se 0 E I e

(8.15)

where, for the section of unit width, we can write

h e+ A =

h e+ dy = h ,

e

e

h e−

y dy =

1 e+ 2 [(h ) − (h e− )2 ], 2

h e−

h e+ Ie =

S = e

y 2 dy =

1 [(h e+ )3 − (h e− )3 ] 12

(8.16)

h e−

The inelastic response of the beam is also computed by integrating through the thickness. However, this time around, contrary to analytic result in (8.16), numerical integration is performed which requires that the chosen inelastic constitutive response (and the tangent modulus) be obtained for all the values of the natural coordinate η in a through-the-thickness direction in accordance with the chosen numerical integration rule. In particular, for any chosen value ηl , l = 1, ..., n int , we can compute the relative normal and tangential displacements by making use of the results in (8.5) and (8.10) as δn (ηl ) = (Σ + y(ηl )K ) L e (8.17) δt (ηl ) = Γ L e

466

8 Brief on Mulitscale, Dynamics and Probability

t (η l)

Fig. 8.6 Tractiondisplacement damage behavior for macroscale constitutive model

tf

Gf

δ(η l)

We can then follow the idea of Jeng and Shah [236], which is also used in [45], [69] or [308], to combine the crack opening displacement δn (ηl ) and crack sliding displacement δt (ηl ) into effective opening displacement δ(ηl ) =



β 2 [δt (ηl )]2 + [δn (ηl )]2

(8.18)

where parameter β can be varied with respect to the default unit value in order to assign different weights to each component of crack displacement. The mixed mode cohesive law can thus be constructed to relate the effective displacement δ(ηl ) and the effective traction  (8.19) t (ηl ) = β −2 [tt (ηl )]2 + [tn (ηl )]2 In constructing this kind of traction-displacement law we ought to take into account its compatibility with the elastic response in (8.15) and the irreversible nature of the fracture process. The former is achieved by setting the fracture traction value t f and the latter is described through the loading/unloading conditions δ˙ = γ˙

∂φ , ∂t

γ˙ ≥ 0, φ ≤ 0, γ˙ φ = 0

(8.20)

where φ(·) is the fracture criterion given as φ(t, q) = t − [t f − q(ξ )]

(8.21)

In (8.21) above, we denote by q the stress-like variable conjugate to the internal variable ξ which describes the softening behavior (see Fig. 8.6). The tractiondisplacement constitutive model can be written in the rate-form within the standard format of damage models (e.g. see [176]) according to  t˙(ηl ) =

˙ φ 0) there exists a solution of (8.105), which is a stochastic process almost surely continuous and unique. Let us now consider process X(t) that verifies (8.105), as well as a function V (t; x)((t; x) ∈ [t0 , T ] × Rn ), which allows partial derivative until order 2 in x and order 1 in t. The differential formula of Ito gives us (denoting a = σ σ T ). ⎡ ⎤ n n n  2   V (X (t)) ∂ V (X (t)) 1 ∂ ⎦ dt d V (X (t)) = ⎣ bi (X (t)) + ai j (t, X (t)) ∂ xi 2 i=1 j=1 ∂ xi ∂ x j i=1 +

k n  

σri (X (t))

i=1 r =1

∂ V (X (t)) dWr (t) ∂ xi (8.110)

or yet in integral form:

t V (X (t)) = V (X (t0 )) +

L V (X (u))du + t0

k

n   i=1 r =1 t0

t

σri (X (u))

∂ V (X (u)) dWr (u) ∂ xi

(8.111) where L is the generator of the Markov process X (t), which is defined so that for any function V (t; x)((t; x) ∈ [t0 , T ] × Rn ) we have L V (X (t)) =

n  i=1

∂ V (X (t))   ∂ 2 V (X (t)) + ai j (X (t)) ∂ xi ∂ xi ∂ x j i=1 j=1 n

bi (X (t))

n

(8.112)

Duffing oscillator equation: we will further specialize our development for the case where the system function corresponds to Duffing oscillator, where f (x(t), ˙ x(t)) = δ x(t) ˙ + αx 3 (t) + βx(t), with δ > 0 and α > 0. We will thus have /

x(t) ¨ + δ x(t) ˙ + αx(t)3 + βx(t) = η W˙ (t) , η > 0 (x(0), ˙ x(0)) = (x˙0 , x0 ) a.s.

(8.113)

Here, with the damping term being always positive, the energy of the system that is not loaded by any external loads will diminish until finally becoming equal to zero. For α = β = δ = 1 and (x(0), ˙ x(0)) = (0, 1), we have represented in Fig. 8.33 deterministic behavior of the oscillator (with no stochastic loads), as well as one possible trajectory of the same oscillator loaded by white noise. The computations are performed by using the Euler time integration scheme when applying deterministic loads and by using the Euler-Maruyama scheme when applying a white noise loading. For Duffing oscillator, by denoting y(t) = x(t), ˙ we will further have (x(t), y(t) as the bidimensional Markov process verifying the following SDE:

512

8 Brief on Mulitscale, Dynamics and Probability

Fig. 8.33 Response of Duffing oscillator under deterministic load (left) and white noise (right)

 d

x(t) y(t)



 y(t) 0 # dt + dW (t) , η > 0 η − δy(t) + αx(t)3 + βx(t)       

=

"

σ

b

(8.114)

for which the generator takes the form: L=y

# ∂ " ∂ η2 ∂ 2 − δy + αx 3 + βx + ∂x ∂y 2 ∂ y2

(8.115)

The stationary law of the above diffusion process X(t), if it exists, is given by: L∗ p = 0

(8.116)

where L * is the adjoint operator of L and p is probability density function. Existence of stationary law: The first important question concerning Duffing oscillator concerns the existence of a stationary law for the bidimensional process. In order to prove the existence of a stationary law, we will exploit the theorem [245], which can be written as: Theorem 3 If coefficients b and σ in eq. (8.114) verify conditions in (8.108) and (8.108) in an open ball of radius R > 0, if there exists a real-valued function V ∈ C 2 (R n ), that satisfies: V (x) ≥ 0

(8.117)

sup L V (x) = −A R → −∞ when R → ∞

(8.118)

and |x|>R

8.3 Reduced Stochastic Models for Euler Beam Dynamic Instability

513

where L is the operator defined in (8.112), and if X(t) is regular, we then conclude for SDE in (8.105) the solution existence for the stationary law; for proof see [215]. The density of the probability distribution of the stationary law is given next. Computation of stationary law: Here we will further be interested in computing the stationary law for Duffing oscillator system. To that end, we will exploit the FokkerPlanck equation. Let us consider the SDE in (8.114). If coefficients b and σ verify conditions (8.108) and (8.109), we know that the SDE will allow a unique solution for all random variables X (t) . We know that the transition probability P(s, y; t, d x) = P(X (t) ∈ d x|X (s) = y). (or conditional probability that X (t) belongs to d x given that X (s) = y, with 0 ≥ s ≥ t) is a solution of the Fokker-Planck partial differential equation. By supposing that P(s, y; t, d x) is sufficiently regular to allow defining the probability density function P(s, y; t, d x) = p(s, y; t, d x) d x P(s, y; t, dx) = p(s, y; t, x)dx, we have: " " # # n n n ∂ p(s, y, t, x)  ∂ b j (x) p(s, y, t, x) 1   ∂ 2 ai j (x) p(s, y, t, x) + − =0 ∂t ∂x j 2 i=1 j=1 ∂ xi ∂ x j j=1 (8.119) We note now the stationary law as Ps (d x) = ps (x)d x. By definition, this stationary law verifies (0 ≤ s ≤ t), the following equation:

+∞ ps (x) =

p(s, y, t, x) ps (y)dy

(8.120)

−∞

By multiplying (8.119) by ps (y) and integrating, we obtain the reduced FokkerPlanck equation: " # # " n n n  ∂ b j (x) ps (x) 1   ∂ 2 ai j (x) ps (x) − =0 ∂x j 2 i=1 j=1 ∂ xi ∂ x j j=1

(8.121)

The solution of this PDE gives the stationary law as a stationary solution X (t) of SDE in (8.114), if such stationary solution exists. For Duffing equation, we have already shown that (8.114) admits a solution for stationary law. The corresponding reduced Fokker-Planck can be written as: " " # # η2 ∂ 2 ps (x, y) ∂ ps (x, y) ∂ − δy + αx 3 + βx ps (x, y) + − y =0 ∂x ∂y 2 ∂ y2

(8.122)

This equation admits an analytic solution (see [68] for more details on solution 2 4 2 method) leading to ps (x, y) = Ce−δ(y +αx /2+βx ) (where C is a normalization constant. If we are only interested in stationary law, that we denote (with a slight abuse of notation) as ps (x), we obtain

514

8 Brief on Mulitscale, Dynamics and Probability

Fig. 8.34 Stationary solution for system in (8.114), with exact (red line) and numerical (histogram) with parameters values δ = α = η = β = 1 (left) versus δ = α = η = 1 and β = −2 (right)

Fig. 8.35 Exact solution for stationary law for system in (8.114) for parameters δ = α = η = β = 1 (left) versus δ = α = η = 1 and β = −2 (right)

+∞ ps (x) =

ps (x, y) dy = C  e−δ(αx

4

/2+βx 2 )

(8.123)

−∞

In Fig. 8.34 we have represented the stationary solution for position x for the considered system, obtained by numerical simulation using the Monte-Carlo method, as well as the exact solution obtained for different values of system parameters. In Fig. 8.35, we have represented the exact stationary law of the couple position/velocity (x, y = x) ˙ for different values of parameters.

8.3.3.2

Closing Remarks

In conclusion, we have presented the stochastic solution to the Euler beam instability problem, by making use of a reduced model constructed in terms of the Duffing oscillator. We have used the same kind of procedure to solve this instability problem in the stochastic framework, using the white noise stochastic process used as excitation

8.3 Reduced Stochastic Models for Euler Beam Dynamic Instability

515

replacing fast harmonic loads. We showed that we are able to construct the corresponding probability distribution by solving the Fokker-Planck equations. This is shown both for homogeneous beam material and for beam material heterogeneities described in terms of a fast oscillating stochastic process, which is typical of the time evolution of internal variables describing plasticity and damage. The additional computational cost of the stochastic framework is to a large extent compensated by an overall estimate of instability load for heterogeneous materials. All these developments, which have been carried out for conservative loads, can easily be extended to non-conservative loads by following the works in [99, 129, 144, 279, 287], but with an adequate selection of damping phenomena (e.g. [214]). We only change the type of perturbation force, with an impulse replacing harmonic force in order to detect the instability in a geometrically exact setting, which is shown to solve the Bolotin paradox stating that an addition of damping reduces the critical buckling load (see [287]).

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