Introduction to Finite Element Methods [1st ed. 2024] 3658427418, 9783658427412

Citation preview

Dieter Dinkler Ursula Kowalsky

Introduction to Finite Element Methods

Introduction to Finite Element Methods

Dieter Dinkler · Ursula Kowalsky

Introduction to Finite Element Methods

Dieter Dinkler Institut für Statik und Dynamik Technische Universität Braunschweig Braunschweig, Germany

Ursula Kowalsky Institut für Statik und Dynamik Technische Universität Braunschweig Braunschweig, Germany

ISBN 978-3-658-42741-2 ISBN 978-3-658-42742-9  (eBook) https://doi.org/10.1007/978-3-658-42742-9 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2024 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer Vieweg imprint is published by the registered company Springer Fachmedien Wiesbaden GmbH, part of Springer Nature. The registered company address is: Abraham-Lincoln-Str. 46, 65189 Wiesbaden, Germany Paper in this product is recyclable.

Preface

The development and design of technical systems need a deep understanding of processes and phenomena to be described, if an analysis with respect to safety, production effort and resources is required. Nowadays, numerical investigations are at least of the same quality as experimental studies, but more efficient, if parametrical studies have to be accomplished or structures of high complexity have to be optimized. The Finite Element Method is one of the most important tools for engineers to numerically investigate and design structures and all kinds of components exposed to constant or time–varying external actions and excitations. Thus it is an essential need to teach students the fundamentals of the method and to introduce them to the possibilities of applications and the variety of different element formulations and their respective advantages. The textbook presented here gives an introduction to the Finite Element Method from an engineering point of view and offers various insights into the background and the details of the procedures to get finite elements of different characteristics, types of discretization, and the related methodological approaches. It is not the intention to present the most efficient types of elements, because the efficiency of elements depends on the criteria of evaluation, the kind of programming, and very often on their applications. Since the number of publications on the Finite Element Method extraordinarily increased over the last decades, the textbook is not able to discuss and to refer to all elements and all applications presented in the literature, but it is limited to heat conduction, membrane and bending structures of one and two dimensions. The extension to three dimensional structures is possible with little effort, if the presented methods are analogously extended to the third dimension. Furthermore, the representation does not deal with nonlinearities of different types and time–varying deformation behavior, since this would need further mathematical and mechanical background, and a deep insight into the phenomena of geometric nonlinear deformation behavior of structures, plastic deformation of materials and other physical processes, what is beyond the actual Finite Element Method.

VI The manuscript of this textbook has gradually evolved from lecture notes presented by the authors at the Technical University Braunschweig and Stuttgart University to students from the engineering faculties. Thus the textbook is addressed to students and engineers, who are dealing with structural mechanics and are interested to develop elements for their own applications. The authors like to remind of Hermann Ahrens and to gratefully acknowledge his longtime commitment in lecturing the fundamentals of the method and his support to develop this kind of presentation of the Finite Element Method. Finally we gratefully acknowledge Springer Vieweg publishers for the possibility to publish this monograph and for the great support until going to press. Dieter Dinkler and Ursula Kowalsky

VII

Glossary

Physical Description of Processes δWd = 0 δWσ = 0 ( )d ( )σ ( )e x, y, z s ( )n ( )t ξ, η, ζ λa , λb , λc p u ε σ E

Principle of virtual Displacements Principle of virtual Forces displacement stress external action physical cartesian coordinates boundary coordinate normal direction at the boundary tangential direction at the boundary normalized element coordinates area–coordinates within a triangle vector of external actions – loading, heat sources vector of field variables – displacements, temperature vector of strain variables vector of stress variables matrix of physical properties – elasticity, thermal conductivity

Matrix– and Vector–Symbols at the Element Level Ω φi v s f D E B K F S

matrix of shape functions shape functions element vector of discrete unknowns – nodal unknowns element vector of discrete stress variables – nodal stresses element vector of discretized actions – nodal loading, heat sources matrix of differentiation rules due to the kinematic conditions matrix of physical properties – elasticity, thermal conductivity element matrix of strains element matrix – stiffness, heat resistance element matrix – flexibility element matrix of subsequent stress analysis

Matrix– and Vector–Symbols at the System Level K v f

system matrix – stiffness, heat resistance system vector of discrete unknowns – nodal unknowns system vector of discretized actions – nodal loads, heat sources

VIII

Denomination of Elements

The denomination of the elements is necessary in order to distinguish the elements with respect to their properties. To avoid misunderstandings the textbook by hand uses the following abbreviations: K RM Θ D M HD HM Q T N-M RI SRI

Kirchhoff plate theory Reissner–Mindlin plate theory temperature–based formulation displacement–based formulation mixed formulation hybrid displacement formulation hybrid mixed formulation quadrilateral element triangular element element with N nodes and M degrees of freedom reduced integration selectively reduced integration

Heat Conduction Elements HC-ΘQ-4 HC-ΘT-3

temperature–based fomulation, bi–linear shape functions temperature–based fomulation, linear shape functions

Plane Stress Elements P-DQ-4 P-DQ-4-SRI P-DQ-9 P-DQ-9-SRI P-HMQ-4-5 P-HMQC-4-5 P-HMQ-8-13 P-DT-3 P-DT-6

displacement–based fomulation, bi–linear shape functions displacement–based fomulation, bi–linear shape functions displacement–based fomulation, bi–quadratic shape functions displacement–based fomulation, bi–quadratic shape functions hybrid–mixed fomulation, bi–linear shape functions, 5 stress variables hybrid–mixed fomulation, bi–linear shape functions, 5 stress variables due to Pian–Sumihara [80] hybrid–mixed fomulation, serendipity element, incomplete cubic shape functions for w, 13 stress variables displacement–based fomulation, linear shape functions displacement–based fomulation, quadratic shape functions

IX

Kirchhoff Plate Elements K-DQ-4-16 K-DQ-4-12 K-MQ-4-16 K-HMQ-4-16 K-DT-6-21 K-DT-3-18 K-MT-6-12 K-HDT-6-12 K-HDT-3-9 K-HMT-6-12 K-HMT-3-9 DKT-3-9

displacement–based formulation, bi–cubic shape functions displacement–based formulation, incomplete bi–cubic shape functions mixed formulation, bi–linear shape functions hybrid mixed formulation, bi–linear shape functions displacement–based formulation, 5th order shape functions displacement–based formulation, 5th order shape functions mixed formulation, cubic shape functions for w, λ–constraint due to the slope w,n hybrid–displacement formulation, cubic shape functions for w, w,n –constraint hybrid–displacement formulation, cubic shape functions for w, weak conformity hybrid–mixed formulation, cubic shape functions for w, w,n –constraint, quadratic shape functions for the stresses hybrid–mixed formulation, cubic shape functions for w, quadratic shape functions for the stresses, weak conformity displacement–based formulation, quadratic shape functions for the rotations, cubic shape functions for the deflection at the interface

Reissner–Mindlin Plate Elements RM-DQ-4 RM-DQ-4-SRI RM-DQ-4-HC RM-DQ-4-BHC RM-MQ-4-16 RM-HMQ-4-12 RM-HDQ-4-5

RM-DT-6-12 RM-DT-6-15

displacement–based formulation, bi–linear shape functions displacement–based formulation, bi–linear shape functions, selectively reduced integration displacement–based formulation, bi–linear shape functions, reduced integration and hour–glass stabilization displacement–based formulation, bi–linear shape functions, selectively reduced integration and hour–glass stabilization mixed formulation, bi–linear shape functions hybrid–mixed formulation, bi–linear shape functions, 12 stress variables hybrid–displacement formulation, bi–linear shape functions for the deflection and the rotations, 5 internal variables due to balanced quadratic shape functions for the deflection displacement–based formulation, quadratic shape functions for the deflection, linear shape functions for the rotations displacement–based formulation, cubic shape functions for the deflection, quadratic shape functions for the rotations

Table of contents

FOUNDATIONS

1

1

Introduction 3 1.1 Governing Equations and Approximate Solution . . . . . . . . . . . . . . . . . . . 7 1.2 General Aspects Concerning the Finite Element Method . . . . . . . . . . . 8 1.3 A Comparison of Exact Solution and Approximate Solution . . . . . . . . 11

2

Discretization of the Work Equation 2.1 Principle of Virtual Displacements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Principle of Virtual Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 The General Procedure to set up the Element Matrices . . . . . . . . . . . 2.4 Shape Functions and Convergence Criteria . . . . . . . . . . . . . . . . . . . . . . .

19 19 25 28 38

3

Structure and Solution of the System of Equations 3.1 Real and Virtual Nodal Displacements . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Equations of Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Structure of the System of Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Assembly of the Stiffness Matrix of the Entire System . . . . . . . . . . . . 3.5 The Storage and Solution of the System of Equations . . . . . . . . . . . . . 3.6 The Optimization of the Bandwidth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 The Fulfillment of Dirichlet Boundary Conditions . . . . . . . . . . . . . . . . .

51 52 52 53 55 58 59 61

4

Heat 4.1 4.2 4.3

63 63 69 77

5

Membrane Structures 81 5.1 Rectangular Elements Regarding Plane Stress Situation . . . . . . . . . . . 81 5.2 Rectangular Element Comprising Modified Shear Strains . . . . . . . . . . 96 5.3 Plane Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

6

Bending Structures 101 6.1 Element Matrices Regarding Euler–Bernoulli Beams . . . . . . . . . . . . . . . 101 6.2 Kirchhoff Plate Element with 16 DOF . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 6.3 Kirchhoff Plate Element with 12 DOF . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 6.4 The 12 DOF Element Employing a Weak Conformity . . . . . . . . . . . . . 129

Conduction Heat Conduction at One–Dimensional Description . . . . . . . . . . . . . . . . Heat Conduction Regarding Two Spatial Dimensions . . . . . . . . . . . . . . Example of Use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

XII

Table of contents

TRIANGULAR ELEMENTS

133

7

Triangular Elements – Description of Geometry 135 7.1 Local ξ–η–Coordinate System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 7.2 Description Employing Area Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . 140

8

Triangular Elements to Describe Heat Conduction 145 8.1 A Linear Approach Related to the ξ–η–Coordinate System . . . . . . . . 145 8.2 Example of Use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

9

Triangular Elements for Membrane Structures 153 9.1 Linear Shape Functions with Respect to ξ–η–Coordinates . . . . . . . . . 153 9.2 Description Employing Area Coordinates λa , λb , λc . . . . . . . . . . . . . . . . 158 9.3 Quadratic Approach Employing ξ–η–Coordinates . . . . . . . . . . . . . . . . . 159 9.4 Quadratic Shape Functions Using Area Coordinates . . . . . . . . . . . . . . . 166 9.5 A Comparison of Standard Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

10 Triangular Elements for Kirchhoff Plates 173 10.1 Choice of Shape Functions and Nodal Unknowns . . . . . . . . . . . . . . . . . 173 10.2 Complete Approach of the 5th Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 10.3 A Plate Element with 18 DOF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 10.4 A Comparison of Standard Plate Elements . . . . . . . . . . . . . . . . . . . . . . . 193 ISOPARAMETRIC ELEMENTS

197

11 Numerical Integration 199 11.1 Numerical Integration Using Gauss–Legendre Quadrature . . . . . . . . . . 200 11.2 Numerical Integration Applied to Membrane Elements . . . . . . . . . . . . 204 11.3 Triangular Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 12 Isoparametric Elements 215 12.1 Description of the Element Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 12.2 Isoparametric Elements Regarding Membranes . . . . . . . . . . . . . . . . . . . . 218 12.3 Quadrilateral Plate Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 HYBRID QUADRILATERAL ELEMENTS

229

13 Hybrid Finite Elements 231 13.1 Mixed Formulation of Governing Equations . . . . . . . . . . . . . . . . . . . . . . . 234 13.2 Mixed Formulation of Work Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 13.3 Hybrid Discretization of Work Equations . . . . . . . . . . . . . . . . . . . . . . . . . 242 14 Hybrid–Mixed Plane Stress Elements 249 14.1 Mixed Principle of Work for Plane Stress Structures . . . . . . . . . . . . . . 249 14.2 Work Equations of a Hybrid Plane Stress Element . . . . . . . . . . . . . . . . 252 14.3 Element Stiffness Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 14.4 Subsequent Stress Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

Table of contents 14.5 14.6 14.7 14.8

XIII

Linear Shape Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 Quadratic Shape Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 Linear Approaches with Coupling of Degrees of Freedom . . . . . . . . . . 264 Convergence Behavior of the Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . 268

15 Hybrid–Mixed Euler–Bernoulli Beam Elements 273 15.1 Mixed Formulation Employing Forces and Displacements . . . . . . . . . . 273 15.2 Work Equation in Hybrid Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 15.3 Element Stiffness Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 16 Hybrid–Mixed Kirchhoff Plate Elements 283 16.1 Mixed Principles of Work Concerning Kirchhoff Plates . . . . . . . . . . . . 283 16.2 Hybrid–Mixed Rectangular Plate Element . . . . . . . . . . . . . . . . . . . . . . . . 293 HYBRID TRIANGULAR PLATE ELEMENTS

299

17 Hybrid Triangular Plate Elements 301 17.1 Cubic Approach for Triangular Plate Elements . . . . . . . . . . . . . . . . . . . . 302 17.2 Hybrid–Displacement Elements Employing 10+3 DOF . . . . . . . . . . . . 306 17.3 Displacement–based Element with Weak Conformity . . . . . . . . . . . . . . 314 18 Hybrid–Mixed Triangular Plate Elements 317 18.1 Work Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317 18.2 Approaches Related to the Deflection and the Stresses . . . . . . . . . . . . 319 18.3 Element Stiffness Matrix and Load Vector . . . . . . . . . . . . . . . . . . . . . . . . 321 18.4 Fulfillment of the Continuity Conditions at the Interface . . . . . . . . . . 323 18.5 Subsequent Stress Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324 18.6 Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324 19 Discrete Kirchhoff –Theory Element 327 19.1 The Discrete Kirchhoff Triangular Element . . . . . . . . . . . . . . . . . . . . . . . 328 19.2 Stiffness Matrix, Load Vector and Stress Analysis . . . . . . . . . . . . . . . . . 336 19.3 Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 20 Benchmark Concerning Triangular Plate Elements 339 20.1 Square Plate Subjected to Distributed Loading . . . . . . . . . . . . . . . . . . . 339 20.2 Trapezoidal Plate Subjected to Distributed Loading . . . . . . . . . . . . . . . 343 SHEAR–DEFORMATION BEAM AND PLATE ELEMENTS

347

21 Timoshenko Beam Elements 349 21.1 Elements Employing Displacements as Primary Variables . . . . . . . . . . 350 21.2 Mixed Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354 21.3 Hybrid–Mixed Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356 21.4 Convergence Behavior Concerning the Beam Elements . . . . . . . . . . . . 357

XIV 22 Plate 22.1 22.2 22.3 22.4 22.5 22.6 22.7 22.8 22.9

Table of contents Elements Including Shear Deformations 359 Kirchhoff and Reissner–Mindlin Theories by Comparison . . . . . . . . . . 359 Governing Equations of the Reissner–Mindlin Theory . . . . . . . . . . . . . . 362 Weak Formulation of the Governing Equations . . . . . . . . . . . . . . . . . . . . 363 Quadrilateral Element Employing a Bi–linear Approach . . . . . . . . . . . . 365 Hybrid–Displacement Quadrilateral Element . . . . . . . . . . . . . . . . . . . . . . 368 Comparison of the Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373 Displacement–Based Triangular Element . . . . . . . . . . . . . . . . . . . . . . . . . 380 Mixed Quadrilateral Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387 Hybrid–Mixed Quadrilateral Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393

EVALUATION OF RESULTS 401 23 Error Estimation 403 23.1 The Least Squares Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404 23.2 Error Estimation by Applying the Principle of Virtual Work . . . . . . . . 407 23.3 Mesh–Adaptation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410 24 Quality of Elements 413 24.1 The Eigenvalue Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413 24.2 The Locking Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417 24.3 Improvement of Elements Suffering from Locking . . . . . . . . . . . . . . . . . 421 24.4 The Patch–Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425 REFERENCES

427

INDEX

437

FOUNDATIONS

1 Introduction

The detailed investigation of the phenomenology of structures and processes in engineering, medicine and environmental technology is important, if their safety has to be guaranteed with respect to limit loads and life cycle. In general experimental methods are well developed for prototypes as basis for production in series. If individual structures, large scale structures or systems as well as processes of higher complexity are to be investigated, experimental methods are too expensive in many cases. Here, computational methods are more efficient, if they are able to describe the behavior of structures with sufficient accuracy with respect to space and time. Basis of analytical and computational studies are models, which usually deal with partial differential equations or integral equations. The model equations have to be developed with respect to the level of accuracy, that is of interest. Engineering models, for which analytical solutions exist, often suffice for simplifying investigations. However, general processes that are to be investigated may be so complex, that a proper description has to include multiple dimensions in space with partially curved boundaries as well as the dimension in time, and should take into account different nonlinearities. Therefore, exact solutions are only possible for special cases. Hence, a general methodology must be available to get approximate solutions in order to investigate and to design engineering structures in a proper way. A universally applicable approximation method is the Finite Element Method (FEM). Originally developed for studying the deformation behavior of plate and membrane structures, nowadays FEM is applied in all fields of engineering, wherever the method is transferable. Since the Finite Element Method is a reliable numerical tool to solve the model equations, it has even become accepted in areas where investigations have been performed solely experimentally so far. Furthermore, due to its efficiency the FEM offers possibilities for parametric studies, the optimization of structures, and the consideration of uncertainties in terms of a probabilistic concept. This is the case, for example, regarding multi–physical processes, which exhibit strongly non–linear behavior on different spatial and temporal scales, or at investigations of structures with scattering material behavior and actions respectively. Thus modern commercial FEM–codes offer different elements to describe and to optimize physical as well as chemical processes in space and time domain. © The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2024 D. Dinkler und U. Kowalsky, Introduction to Finite Element Methods, https://doi.org/10.1007/978-3-658-42742-9_1

4

1 Introduction

As an example of a complex optimization process, the drive shaft of a water wheel, presented in Figure 1-1, is the result of the interaction of functionality, physical modeling, FEM–discretization and stiffness optimization.

Fig. 1-1 FEM–discretization of the drive shaft of a water wheel [51] The classical Finite Element Method has been developed over decades from 1956 on. The mathematical foundation goes back to Trefftz [98] and Courant [29]. With the development of the first computers the method became of interest for engineers. First pioneers of the method in the field of engineering were Clough [26, 99] at UC–Berkeley, Zienkiewicz [105, 106] at Swansea University and Argyris [4, 5] at Stuttgart University. Further textbooks for students have been presented by Bathe [12], Hughes [47], Onate [76, 77], and many others. In many industrial processes, the shape of products is developed by means of computer–aided design procedures (CAD), which employ B–splines to describe the geometry with higher order continuity in order to avoid kinks. The iso– geometric extension of the FEM, developed by Cotrell, Hughes and Basilevs [49, 31], employs non–uniform rational basis splines (NURBS) to approximate the deformation of structures, whereat the geometry is designed with the same kind of B-splines. This approach offers some advantages, if the deformation of structures, designed by means of CAD employing high continuity conditions, is of interest, where the geometry and the deformation field need the same order of approximation, and where the solution needs a high order of continuity as in the case of wave propagation as an example.

5 Recent developments of the FEM deal with an extension of the fundamental method to partly very sophisticated applications on multiple scales. Fundamentals of polygonal elements are discussed by Wachspress [101] and Sukumar and Tabarraei [94], and may be applied to structures of correspondent geometry as in the case of metallic micro–structures, if the physical and chemical behavior on this scale is of interest, cf. Figure 1-2-left. Nonetheless, the application to macroscopic structures is possible as well, if necessary, cf. 1-2–right.

Fig. 1-2 a) Metallic micro–structure with polygonal grains b) beam like structure [51] Applying the idea to the investigation of wood may lead to polygonal elements with a thin–walled geometry for the solid part and an internal polygonal element for the description of fluids, cf. Figure 1-3.

Fig. 1-3 Thin–walled polygonal elements to model wood on the meso–scale [51] A complete different approach to numerically investigate structures offers the Discrete Element Method. In contrast to the classical Finite Element Method, which is developed to investigate continua of different type, the Discrete Element Method deals with the dynamics of rigid particles, which interact with each other. The conditions of interaction can be described by means of springs, viscous dampers or other rheological elements.

6

1 Introduction

Originally the method is founded on the publications by Cundall [33, 34], who developed the method to describe the behavior of gravel like solids. The extension of the method offers the possibility to investigate the micro–structure of continua even down to the dynamics of molecules. The discretization of beam like structures with particles of different size and related sieve line as well as different physical and chemical properties offers the possibility to investigate the deformation behavior of structures with discontinuities down to the micro–structure, cf. Figure 1-4.

❄

❄ ✻

✻ Fig. 1-4 a) Discrete element mesh to describe the micro–structure of concrete b) Crack pattern due to four point loading [51] The publications on the development of the Finite Element Method deal with different approaches as basis of the discretization. The textbook at hand applies the Principle of virtual Work with respect to virtual displacements and virtual forces. Due to the variety of elements and applications, which can be hardly discussed sufficiently in a textbook, this book focuses on the basic methodology of the displacement–based formulation as well as mixed and hybrid formulations in order to explain the different approaches and the fulfillment of the governing equations. The method will be presented with respect to civil engineering structures, since it facilitates clear understanding of the derivation. Therefore we will limit ourselves to load–carrying and deformation behavior of one– and two–dimensional structures as well as to heat conduction in one and two dimensions. All results of examples, tests, and benchmarks have been computed by original investigations. Nevertheless, the correctness of numbers cannot be guaranteed.

1.1 Governing Equations and Approximate Solution

7

1.1 Governing Equations and Approximate Solution In structural mechanics there are two basic approaches available to describe the load–carrying and deformation behavior of structures: • Going back to Newton’s synthetic approach, the method of sections is also applied to derive the governing equations at the differential element. Therefore differential equations concerning the equilibrium and kinematics as well as algebraic equations to describe the material behavior can be derived regarding all engineering structures. Additionally, the boundary conditions regarding stress and deformation states must be considered. • Going back to Leibniz’s analytical approach, integration is performed over the entire structure without differentiating between boundary and domain or between different differential elements. This can comprise an integral statement concerning the work performed at deforming the structure for example, or the energy stored in the structure after the deformation process or similar. Both approaches are equivalent and are convertible into each other. The exact solution fulfills the governing equations in the domain as well as at the boundaries and at the intersections between different integration regions. Thus all other equations, to get an integral description of the problem, are fulfilled by implication. The exact solution is also named the strong solution. Concerning problems with more than one dimension in space and arbitrary boundaries, exact solutions generally turn out to be impossible. Thus for the analysis of complex structures, such as plates, membranes or shells, methods are required to approximately solve the governing equations. If the modeling equation is equivalent to the governing equations, the identified solution is an approximation of the exact solution. Hereby the equations applied to get the solution are still fulfilled exactly. However, the solution itself is only an approximation of the exact solution. Therefore, it no longer exactly fulfills the governing equations at the differential element. Hence the approximate solution is also named the weak solution. To get an approximate solution an approach for the primary variable is chosen, which depends on a normalized function describing the course of the variable in space, and a multiplier, which scales the solution. In the case of a displacement field u(x) the approach could be u(x) = φ(x) · uˆ .

8

1 Introduction

The function φ(x) is prescribed and is named the shape function, whereat polynomials are usually chosen. The multiplier uˆ is still unknown and is evaluated by means of the respective modeling equation. This idea, developed by the physicist Walter Ritz [88] to fulfill the Principle of Minimum Potential Energy, is the basis of all modern methods that apply integral work equations or principles of energy to compute an approximate solution. As an approximation to the exact solution, one may choose shape functions that are defined regarding the whole structure. However, a closed–form approach only makes sense if the stiffnesses are constantly distributed with respect to the domain and if the geometry of the structure is regular. The solutions concerning more general problems exhibiting arbitrary geometries as well as arbitrary boundaries, stiffnesses and external actions are limited employing an closed– form approach.

1.2 General Aspects Concerning the Finite Element Method At increasing complexity of geometry or of material and load–carrying behavior, the requirements with respect to the computational method increase, too. Regarding the optimization of processes and structures that presently proceeds in all engineering disciplines and that accompanies the description of the related systems by increasingly sophisticated models, it is important to have a computational tool at one’s disposal which is generally applicable. Due to the fact that the Finite Element Method is able to generalize the procedure for numerical investigations of arbitrary systems, the method has become more accepted than others. Thus the FEM has currently become the standard procedure concerning all kinds of structural analysis as limit load design and vibration analysis of structures as well as the analysis of temperature fields, investigation of transport processes in porous media and even the analysis of the material behavior at a microscopic level. The basic idea of FEM regarding structural mechanics may be described as follows: Instead of choosing a closed–form approach to describe the primary variables, the structure is virtually divided up into many finite elements. Concerning the single elements, shape functions are chosen to describe the course of the primary variables whereby the related multipliers, still to be determined, are dedicated to specified spatial positions called element nodes. These nodal unknowns are the degrees of freedom (DOF) of the finite model of the structure. Increasing the number of elements and

1.2 General Aspects Concerning the Finite Element Method

9

therefore also the number of DOF may not only result in an improved approximate solution but also the geometry may be taken into account in more detail, e.g. if curved boundaries are to be approximated by polygonal representation. An advantage of the FEM compared to methods dealing with a closed–form approach is the polynomial order needed to describe the physical fields inside the elements. The shape functions can be of lower order, thus, when regarding the entire structure, the primary variables are approximated by a polygonal shape or piecemeal parabolically, respectively. In general, the approximate solution improves the more elements that are applied. To ensure the convergence of the FE approximate solution against the exact solution, the approach has to fulfill the convergence criteria at mesh refinement, see Section 2.4.2, e.g. requirements related to the polynomial order of the shape functions and to the continuity of the physical fields to be determined at element intersections. Furthermore, choosing elements with identical approaches concerning the entire structure allows for the analysis of the stress–deformation behavior of a single element prior to instructing the computer to assemble the complete system schematically. The fundamental advantages of the Finite Element Method are: • applicability to arbitrary problems regarding the whole field of natural as well as engineering sciences; • no restrictions concerning the geometry of the structures to be analysed, also curvilinear boundaries are allowed; • linear as well as non–linear modeling equations can be solved approximately; • arbitrary external actions; • variable system characteristics; and the • schematic computational procedure enables a generalized implementation of different approximations into coding. When applying the FEM, the degrees of freedom are evaluated by means of work equations or principles of energy. Regarding the book at hand, the Principle of virtual Work is applied because in general, it is also valid for non–conservative systems. The description of the virtual work can be performed in the following different ways: • The Principle of virtual Displacements (PvD) results in a formulation, which applies the displacements as variables to describe the system behavior.

10

1 Introduction • The Principle of virtual Forces (PvF) yields a formulation, which applies forces or stresses as variables. • Mixed Principles of virtual Work (PvD and PvF) utilise displacements as well as stresses as variables to describe the system behavior.

Other approaches are possible, if the Principle of stationary values of Energy is applicable or if the governing equations are employed and solved approximately by means of the Method of weighted Residuals. Concerning the first part of the book at hand, the PvD provides the basis for the development of finite elements, which are commonly accepted in engineering practice. In addition, it is closely related to the displacement method applied in structural analysis. Concerning sizing, the displacement–based formulation has the disadvantage that the stresses are solely evaluated from a subsequent analysis, whereby the differentiation of the displacement approach yields a less smooth distribution. Applying the PvD, the sum of the external work due to external actions and of the internal work due to internal force variables vanishes at regarding the whole system. Dividing the complete structure into finite elements allows for the description of the virtual work of the entire system as the sum of the virtual work of the single elements. Regarding problems in practice and due to the relatively high amount of data which is generated, the Finite Element Method cannot be applied without using FE–programs. FE–programs vary only slightly at employing different types of elements, therefore they are transferable to diverse problems with just few modifications. Even to analyse arbitrary structures comprising different components, element formulations must be provided employing different types of elements. Such an element library could differentiate between: • the physics of the problem as bending, membrane behavior, heat conduction, fluid flow etc.; • the geometry of the element as quadrilaterals, rectangles, triangles, bars, etc.; and • the approximate approach to describe the displacements as linear, quadratic or higher order functions. Currently, in addition to the basic FE–programming systems, separate codes are common concerning pre–processing, what deals with the data generation and control, as well as post–processing, which is neccessary for the visual representation of the solution and for the interpretation of results.

1.3 A Comparison of Exact Solution and Approximate Solution

11

1.3 A Comparison of Exact Solution and Approximate Solution Assuming the exact solution to be unknown one may compute an approximate solution by applying the FEM by means of the Principle of virtual Displacements. Without referring to the related procedure in detail, the differing results regarding an approximate solution applying FEM and the exact solution should be presented concerning the bar referred to in Figure 1-5. H

case 1

EA p(x)

case 2

EA

x,u

units : x, u, l p(x) H EA

[m] [ N/m ] [N] [N]

l

Fig. 1-5 Bar – geometry and loading

1.3.1 Analytically Exact Solutions In order to describe the respective problem, the exact solution has to fulfill all governing equations regarding the domain to be investigated as well as the Dirichlet boundary condition for the displacements and the Neumann boundary condition for the first derivative of the deformation, which correlates with the longitudinal force. Regarding the bar depicted in Figure 1-5, this yields: a) Kinematics ε = u,x

in domain and

u(0) = 0

at the boundary,

(1.1)

b) Equilibrium N,x = 0

case 1 in domain,

N,x = −p (x)

case 2 in domain and

N (ℓ) = H

case 1 at the boundary,

N (ℓ) = 0

case 2 at the boundary.

(1.2)

c) Material equation – Hooke’s model N = EA · ε

in domain,

(1.3)

12

1 Introduction

The exact solutions follow to • case 1: external boundary action H H · x, EA H ε(x) = , EA N (x) = H . u(x) =

• case 2: constantly distributed external action p(x) p x2 · (ℓ · x − ) , EA 2 p ε(x) = · (ℓ − x) , EA N (x) = p · (ℓ − x) . u(x) =

To control the analytical solution, the solution is to be introduced into the governing Equations (1.1) to (1.3), and the boundary conditions are to be checked at positions x = 0 and x = ℓ respectively.

1.3.2 Approximate Solutions To start with, one may choose a single element to describe the entire structure, because the PvD does not make any statements concerning the number of elements needed. Then an approximate approach is chosen to describe the displacements in the domain, whereby its unknown multipliers are evaluated by applying the PvD, which is shown here with one nodal unknown: u(x) = φ(x) · uˆ . Here, u ˆ is the unknown multiplier and φ(x) is the given shape function, for which a linear run is feasible, see Figure 1-6.

x,u

Φ(x)

Fig. 1-6 Bar – linear approach to describe the displacement

1.3 A Comparison of Exact Solution and Approximate Solution

13

Even though the analytically exact solution is unknown concerning u(x) in general, the approximate solution may correspond to the strong solution in a particular case. With u(x) being an approximation, the equilibrium is integrally fulfilled for the entire structure applying the PvD, but it is only approximately fulfilled regarding the differential element. The other governing equations are still considered exactly, as they are introduced into the PvD equation. At the subsequent analysis stage, the strains ε(x) are evaluated from the displacements u(x) applying Equation (1.1). Further, the longitudinal forces N (x) are evaluated from ε(x) applying Equation (1.3). If u(x) is an approximation, ε(x) and N (x) are approximations, too.

Case 1: External Boundary Action H Regarding the entire bar, a computation applying the PvD and employing a linear approach for u(x) according to Figure 1-6 yields the solution: u(x) = φ(x) · u ˆ, φ(x) = x/l , u ˆ=

H ·l . EA

In this particular case the approximate solution regarding u(x) corresponds to the exact solution. ε(x) is evaluated from a subsequent analysis following Equation (1.1), and ε(x) equals the exact solution, too: ε = u,x = φ(x),x · u ˆ = 1/l · u ˆ=

H . EA

The evaluation of the longitudinal force following Equation (1.3) N = EA · ε = EA ·

H =H EA

also yields the exact solution. Introducing the longitudinal force N (x) into the equilibrium condition (1.2) and controlling the boundary condition confirms the PvD to provide the analytically exact solution in this particular case. With

p(x) = 0

it applies

with the boundary condition

N,x = 0 , N (ℓ) = H .

The Dirichlet boundary condition u(0) = 0 is already fulfilled by the approach. A computation employing several elements or subdomains also yields the exact solution regarding this external action.

14

1 Introduction

Case 2: Constantly Distributed External Actions p(x) Regarding the entire bar, a computation applying the PvD and employing a linear approach for u(x) according to Figure 1-6 yields the approximate solution of the displacement: u(x) = φ(x) · u ˆ, φ(x) = x/l , p · ℓ2 u ˆ= . 2EA Here, the approximate solution yields the exact boundary displacement, what may be explained by means of the Principle of virtual Forces, but can not be generalized. Regarding the domain, however, deviations exist with respect to the exact solution, see Figure 1-7. p . l2

u

2 EA

exact solution approximation

x

Fig. 1-7 Bar – approximation and exact solution regarding the displacement The strain ε(x) is evaluated by a subsequent analysis following the kinematic condition of Equation (1.1): ε = u,x = φ(x),x · u ˆ=

1 p·ℓ ·u ˆ= . ℓ 2EA

u(x) being an approximation results in ε(x) being an approximation, too. Only the mean value of the strains equals the strong solution, see Figure 1-8.

ε

p.l p.l

EA

2 EA

exact solution approximation

x

Fig. 1-8 Bar – approximation and exact solution regarding the strains The evaluation of the longitudinal force, following the material Equation (1.3), also yields an approximate solution N = EA · ε =

p·ℓ . 2

1.3 A Comparison of Exact Solution and Approximate Solution

15

Here, only the mean value of the approximation equals the exact solution, too, see Figure 1-9. N

p. l

p.l 2

exact solution approximation

x

Fig. 1-9 Bar – approximation and exact solution regarding the longitudinal force Although, in general, the courses of the state variables are only approximations of the strong solution, parts of the governing equations are exactly fulfilled. The kinematics following Equation (1.1) as well as the material behavior following Equation (1.3) are implicitly enclosed in the work equation and therefore are exactly fulfilled. The equilibrium conditions (1.2) are not exactly fulfilled, because the PvD is only a weak formulation of the equilibrium. Since the longitudinal forces are constant, inside the domain the equilibrium condition yields N,x + p(x) 6= 0 . |{z} 0

The Neumann boundary conditions are not exactly fulfilled either N (ℓ) =

p·ℓ 6= 0 . 2

When evaluating the results, the following is to be taken into account. Although the approximation regarding the distribution of the longitudinal forces seems to be relatively coarse compared to the displacements, the PvD optimally approximates the equilibrium and thus the longitudinal forces N (x), but not the displacements u(x).

1.3.3 The FE–Solution Regarding two Elements – Case 2 The bar is subdivided into two identical elements, whereby, regarding the displacements, a linear approach is employed for both, see Figure 1-10. This results in an approximation of the displacement comprising two degrees of freedom uˆ1 and u ˆ2 as well as the shape functions φ1 (x) and φ2 (x) u(x) = φ1 (x) · u ˆ1 + φ2 (x) · u ˆ2 . Although the kink at node 1 seems to pertube the solution, it is nonetheless admissible, what is rigorously proved later on. Compared to the approximation

16

1 Introduction

employing a linear approach and solely one element, the solution dealing with two degrees of freedom better approximates the exact solution but needs further discussion. Φ1(x)

Φ2(x) x,u 0

1

2

Fig. 1-10 Bar – approximation employing two degrees of freedom The principle statements concerning the nature of the approximation remain valid with regard to two elements. Regarding the nodes 1 and 2 the displacements turn out to be exact, whereat the deviations inside the elements become smaller in comparison to one element, cf. Figure 1-11. 2 3 p.l

8 EA

p . l2

u

2 EA

exact solution approximation

x

Fig. 1-11 Bar – approximation and exact solution regarding the displacements Performing the computation of the other state variables, inside the elements the kinematic conditions (1.1) and the material equation (1.3) are exactly satisfied again. This yields constant strains ε(x) as well as longitudinal forces N (x), whereat the analytical and the approximate solutions turn out to be identical at the midpoints of the elements, cf. Figure 1-12. Since the shape functions do not be differentiable, strains and stresses offer a jump at node 1. p.l N

3p . l 4

p.l 4

exact solution approximation

Fig. 1-12 Bar – approximation and exact solution of the longitudinal force

1.3 A Comparison of Exact Solution and Approximate Solution

17

Because of the weak formulation of the equilibrium by means of the Principle of virtual Displacements, the equilibrium condition and the Neumann boundary condition, given in Equation (1.2), are not exactly fulfilled, but the error becomes smaller: N,x + p(x) 6= 0 , |{z} 0

N (ℓ) =

p·ℓ 6= 0 . 4

The solution also offends against the intersection condition concerning the longitudinal force at node 1, where the jump is an indication of the error as well. Performing further elements, the approximate solution converges to the exact solution. The equilibrium as well as the Neumann boundary condition are only approximately fulfilled in general. In particular cases, the nodal displacements are identical to the analytical solution. This is only valid if the approximate approach corresponds to the solution of the homogeneous differential equation.

Remarks The comparison of the exact solution and the approximation leads to the following fundamental statements regarding the PvD. • The Principle of virtual Displacements satisfies the governing equations and the boundary conditions with different accuracy. • The equilibrium and the Neumann boundary conditions are satisfied in a weak sense. • The kinematics, the material equation, and the Dirichlet boundary conditions are satisfied exactly. • The accuracy of the approximation may be increased by increasing the number of elements. • The approximation of the displacement field leads to nodal displacements of high accuracy, whereby the stresses are approximated by lower order.

2 Discretization of the Work Equation

The main idea of FEM is to subdivide the whole structure into finite elements. Thereby it is possible to describe the virtual work of the entire structure as the sum of the virtual works performed at the single finite elements. This is required to develop the system of equations employing the degrees of freedom of the approach as unknowns, X δW = δWel = 0 . (2.1) This yields a two–part task to be worked on:

• Computation of work performed at each element; • Summing up the works performed at all elements to get the work performed on the entire structure. This section systematically describes the formal procedure at the element level. At first, the Principle of virtual Displacements (PvD) will be proven to be an equivalent formulation of the equilibrium conditions with regards to a bar, as an example. It is the basis for further derivations. As a second approach, the Principle of virtual Forces (PvF) will be applied to the bar in order to explain the equivalence of the principle to the kinematic conditions.

2.1 Principle of Virtual Displacements Regarding a self–contained system, the sum of internal as well as external works performed at the entire structure must be zero at the changeover from the undeformed to the deformed situation. X X W = (Wi + We ) = 0 . (2.2) The static case is described here, where deformations and thus also stresses and strains slowly increase simultaneously with loads from zero to the final value.

The bar is used to clearly and concisely represent the different parts of the work previously defined. Here, the work theorem according to Equation (2.2) is applied to a bar loaded by an external action p(x) as well as by an external boundary action H(ℓ) as shown in Figure 2-1. Due to the real loadings p(x) and H(ℓ), the deformation state characterized by u(x) and ε(x) arises as do the longitudinal forces N (x). © The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2024 D. Dinkler und U. Kowalsky, Introduction to Finite Element Methods, https://doi.org/10.1007/978-3-658-42742-9_2

20

2 Discretization of the Work Equation

If the Dirichlet boundary condition is fulfilled, internal work is performed on the strains by the longitudinal forces as well as external work is performed on the displacements by the external actions. As both internal stresses and strains are counteracting, the internal work is preceded by a negative sign Z Z X 1 1 1 W =− N · ε dx + p · u dx + [ H · u ]l = 0 . (2.3) 2 2 2 The deformations are generated by the forces itselves, therefore a 1/2 – coefficient in Equation (2.3) marks the active work, see also Figure 2-2.

2.1.1 Virtual Work In addition to the real external action, the system is loaded by a virtual – this means imagined and arbitrary small – external action δP now. This yields a total deformation u(x) + δu(x) associated with the strains ε(x) + δε(x) as well as with the normal forces N (x) + δN (x). p(x)

H

x l

δu(x) u(x)

δε(x) ε(x)

u

ε

Fig. 2-1 Real and virtual deformation state of a bar Regarding this overall state being achieved in two steps, the work theorem also has to be fulfilled: Z Z X 1 1 1 W =− N · ε dx + p · u dx + [ H · u ]x=ℓ 2Z 2 2 1 1 − δN · δε dx + [ δP · δu ]x=ℓ 2Z Z 2 − N · δε dx + p · δu dx + [ H · δu ]x=ℓ = 0 . (2.4)

2.1 Principle of Virtual Displacements

21

Referring to Equation (2.4), the 1/2 – coefficient marks the active work of the basic state regarded at first, and the additional virtual state added afterwards. Both parts of the work must be zero individually because of Equation (2.3). Thus the remaining virtual passive work, which the real stress state performs on the virtual deformation added subsequently, must be zero, too: Z Z δW = − N · δε dx + p · δu dx + [ H · δu ]l = 0 . (2.5) Figure 2-2 illustrates the meaning of the aforementioned parts of the work Equation (2.4), assuming a linear correlation between the forces and the related deformations. The area determined by the straight and the x–axis can be interpreted as the active work. The horizontally–lined triangles mark the active work of the basis state as well as the active work of the virtual state, the crosshatched rectangle marks the passive work according to Equation (2.5). N

δN δε

ε

Fig. 2-2 Active and passive work – PvD Referring to Equation (2.5) the virtual state must only fulfill kinematics. Thus, the virtual external action and the related virtual stress state, primarily introduced to enable a better understanding of the basic principle, both are not required any longer. Therefore, since only the virtual deformation state is required, this can be prescribed independently of any virtual load. The following may explain which assumptions ensure the Principle of virtual Displacements according to Equation (2.5) being equivalent to the equilibrium condition. Assuming that kinematics according to Equation (1.1) also apply to the virtual state δε = δu,x δu(0) = 0

in domain, at boundary,

Equation (2.5) turns to Z Z δW = − N · δu,x dx + p · δu dx + [ H · δu ]x=ℓ = 0 .

(2.6)

(2.7)

22

2 Discretization of the Work Equation

Integration by parts transforms Equation (2.7) into Z Z δW = N,x · δu dx − [ N · δu ]ℓ0 + p · δu dx + [ H · δu]x=ℓ Z = (N,x + p) · δu dx + [( H − N ) · δu]x=ℓ = 0 .

(2.8)

Referring to Equation (2.8) the differential equation related to the equilibrium is given within the brackets of the integral. The boundary term takes into account the boundary conditions concerning the real force N (ℓ) = H at position x = ℓ. The boundary condition concerning the virtual displacements δu(0) = 0 at position x = 0 is fulfilled a priori according to Equation (2.6). Thus, the following statement applies: The Principle of virtual Displacements according to Equation (2.5) is equivalent to the equilibrium conditions (1.2), if the virtual state fulfills kinematics (2.6) and (1.1) respectively.

2.1.2 Fulfillment of Governing Equations and Boundary Conditions In addition to the equilibrium condition, kinematics and material equations must be fulfilled concerning the real state as well, if the Principle of virtual Displacements shall be the basis of the analysis of the overall state. Hence, the internal forces N , applying Equation (2.7), may be described by the material equation (1.3) and kinematics (1.1). The Dirichlet boundary conditions have to be fulfilled a priori, since they are part of kinematics. Z Z δW = − N · δu,x dx + p · δu dx + [ H · δu ]l = 0 Z Z = − ε · EA · δu,x dx + p · δu dx + [ H · δu ]l = 0 Z Z = − u,x · EA · δu,x dx + p · δu dx + [ H · δu ]l = 0 . Now, all governing equations as well as all boundary conditions are taken into account, and the work equation describes the real problem completely and without any approximation. Since the work is solely described by real and virtual displacements, it is named displacement–based formulation. For the purpose of practical analyses it is convenient to multiply the equation by −1. Thus, the PvD according to the work equation (2.9) is applied to further considerations: Z Z − δW = u,x · EA · δu,x dx − p · δu dx − [ H · δu ]l = 0 . (2.9)

2.1 Principle of Virtual Displacements

23

However, it must be pointed out that the PvD integrally fulfills the equilibrium conditions concerning the entire structure. This is also known as a weak form of equilibrium. Kinematics and the material equations are still exactly fulfilled at every point of the structure, because these governing equations are taken into account implicitly by the PvD. Even the boundary conditions are fulfilled differently. While the Neumann boundary conditions of the real state, being equilibrium conditions, are fulfilled integrally by the work equation and named as natural boundary conditions, the Dirichlet boundary conditions of the real and of the virtual state must, in contrast, be fulfilled exactly by the displacement field. Therefore, they are named as substantial boundary conditions.

2.1.3 Fulfillment of the Conditions at the Element Intersections FE–analyses are applied where no analytical solution is known or can be found. If the unknown deformation state of a structure is evaluated by applying the work equation (2.9), approximations have to be chosen to describe the real and the virtual displacements respectively, which could match by chance the exact solution in special cases of geometry as well as of external actions. However, the approximations have to satisfy kinematics according to Equation (1.1) and Equation (2.6) in both cases. Thus the approximations for the real as well as the virtual displacements must be continuous as well as continuously differentiable regarding the entire domain. However, the virtual work regarding the entire structure according to Equation (2.9) is represented by a scalar quantity, which can also be described as the sum of the works performed at single integration domains, which are the elements: X δW = δWel = 0 . (2.10)

If the work equation is integrated element by element, it might also be convenient to approximate the displacement field within all elements by the same shape functions. Nevertheless, it must be ensured that, when integrating the work equation regarding one element at a time, all terms of work are considered and, moreover, the governing equations – including boundary and intersection conditions – must not get hurt. This can be achieved if the continuity of the approximations is ensured at the element intersections without any gaps of the displacements. As the shape functions are limited to single elements, kinematics might no longer be fulfilled at the element intersections, if the shape functions do only satisfy the continuity of displacements. Thus the strains may become discontinuous at intersections of neighboring elements.

24

2 Discretization of the Work Equation

By means of a simplifying example the fulfillment of kinematics and the equilibrium conditions at the intersections is explained as follows. Figure 2-3 represents a bar subdivided into two finite elements. A

B

1

C

2

H p(x) x l

l

Fig. 2-3 Subdivision into elements Figure 2-4 applies the method of sections to the elements considered, in order to visualize the internal forces and displacements at the respective nodes. B

N

1

B

2

B

N

A

N

A

H p

p A

1

uA1

C

B

B

2

B uB2

uB1

C uC2

Fig. 2-4 Elements being cut cleanly Following Equation (2.9), the work equation that applies the PvD to every single element is comprised of both internal and external work inside the domain as well as work performed at the element boundaries and yields Z Z −δWel = δu,x · EA · u,x dx − δu · p(x) dx − [ δWel−bound. ] (2.11) with the work performed at the element intersections, cf. Figure 2-4, [ δWel−bound. ](1) = [ δuB · NB − δuA · NA ](1) , [ δWel−bound. ](2) = [ δuC · H − δuB · NB ](2) .

(2.12)

When adding up the elements’ work to the system’s work, the work at the elements’ intersections is also added up. If the shape functions regarding the displacements are continuous at the element intersections, the following applies: uB(1) = uB(2)

and

δuB(1) = δuB(2) .

(2.13)

2.2 Principle of Virtual Forces

25

Furthermore, the internal force variables have to fulfill Equation (2.14) NB(1) = NB(2) ,

(2.14)

whereby the respective terms of internal work at the element intersections inside the structure compensate exactly for each other and can be ignored at the element level. In general the polynomial order of the shape functions regarding the displacements results in discontinuous longitudinal forces at the elements’ intersections and therefore Equation (2.14) might not be fulfilled exactly. However, if the work is disregarded at element intersections, Equation (2.14) is fulfilled implicitly in its weak form by applying the PvD to the entire system in the same way as all the other equilibrium conditions.

2.2 Principle of Virtual Forces In analogy to the Principle of virtual Displacements being a weak formulation of the equilibrium conditions, the Principle of virtual Forces is a weak formulation of kinematics. Both principles are applied to be the basis of the development of generalized hybrid finite elements, cf. Section 13. Thus in the following the equivalence of the Principle of virtual Forces and kinematics is shown in order to explain the duality of both principles. Here again, the work theorem is applied to the bar shown in Figure 2-5. The bar is loaded by a distributed external action p(x) and a prescribed displacement ue at the boundary. ue

p(x)

δP P

x l

δN = δP

Fig. 2-5 A bar loaded by a distributed external action and virtual load In contrast to the Principle of virtual Displacements the virtual load δP is applied to the bar at the position x = ℓ first. Thus the work performed by the virtual stress state on the related deformation yields Z X 1 1 W =− δN · δε dx + [ δP · δu ]x=ℓ = 0 . (2.15) 2 2

26

2 Discretization of the Work Equation

The deformations are generated by the forces itselves, therefore a 1/2 – coefficient in Equation (2.15) marks the active work, see also Figure 2-6.

Virtual Work In addition to the virtual action, the system is loaded by the real external action p(x), ue now. This yields a total deformation δu(x) + u(x) associated with the strains δε(x) + ε(x) as well as with the normal forces δN (x) + N (x). Regarding this overall state being achieved in two steps, the work theorem also has to be fulfilled: Z X 1 1 W =− δN · δε dx + [ δP · δu ]x=ℓ 2Z 2Z 1 1 1 − N · ε dx + p · u dx + [ N · ue ]x=0 2 2 2 Z − δN · ε dx + [ δP · u ]x=ℓ + [ δN · ue ]x=0 = 0 . (2.16) Referring to Equation (2.16), the 1/2 – coefficient marks the active work of the virtual state regarded at first, and the additional real state added afterwards. Both parts of the work must be zero individually because of Equation (2.15). Thus the remaining virtual passive work, which the virtual stress state performs on the real deformation added subsequently, must be zero, too: Z δW = − δN · ε dx + [ δP · u ]x=ℓ + [ δN · ue ]x=0 = 0 . (2.17) Figure 2-6 illustrates the meaning of the aforementioned parts of the work Equation (2.16), assuming a linear correlation between the forces and the related deformations. The area determined by the straight and the x–axis can be interpreted as the active work of the overall state. The horizontally–lined triangles mark the active work of the virtual state as well as the active work of the real state, the crosshatched rectangle marks the passive work according to Equation (2.17). δP

P u δu

Fig. 2-6 Active and passive work – PvF

2.2 Principle of Virtual Forces

27

Referring to Equation (2.17) the virtual state must only fulfill the equlibrium conditions. Thus, the virtual deformations and strains, primarily introduced to enable a better understanding of the basic principle, both are not required any longer. Therefore, since only the virtual stress state is required, this can be prescribed independently of any virtual deformation.

Weak Form of Kinematics The following may explain which assumptions ensure the Principle of virtual Forces according to Equation (2.17) to be equivalent to kinematics. Applying kinematics u,x − ε = 0 in domain, u − ue = 0 at boundary x = 0 , in a weak formulation Equations (2.18) turn to Z δW = δN · ( u,x − ε ) dx − [ δN · ( u − ue ) ]x=0 = 0 .

(2.18)

(2.19)

Referring to Equation (2.19) kinematics related to the real state is given within the brackets of the integral. The boundary condition concerning the real displacement u − ue = 0 at position x = 0 is considered in a weak sense as well. Integration by parts transforms Equation (2.19) into Z δW = − ( δN,x · u + δN · ε ) dx + [ δN · u]ℓ0 − [ δN · ( u − ue ) ]x=0 = 0 .

At x = 0, the boundary terms eliminate each other except for the prescribed displacement. At x = ℓ, the remaining bondary term belongs to the virtual load δN u|ℓ = δP u|ℓ . Furthermore, if the virtual state fulfills the equilibrium condition δN,x = 0, Z δW = − δN · ε dx + [ δN · ue ]x=0 + [ δP · u]x=ℓ = 0 (2.20)

yields the final statement. Comparing Equation (2.17) and Equation (2.20) it becomes clear, that the Principle of virtual Forces is a weak formulation of kinematics and thus may be applied to develop finite elements analogously to the Principle of virtual Displacements. Thus, the following sentence applies: The Principle of virtual Forces according to Equation (2.17) is equivalent to kinematics (2.18), if the virtual state fulfills the equilibrium conditions (1.2).

28

2 Discretization of the Work Equation

2.3 The General Procedure to set up the Element Matrices The weak form of the equilibrium condition described by the work equation applying the PvD (2.9) is valid for the entire structure. According to equation (2.10), the virtual work of the structure can be represented as the sum of the single finite elements’ work X δW = δWel = 0 .

Thereby the element’s work may either be formulated in analogy to the system’s work equation or may be reassembled at every single element, see Section 2.1.3. The benefit of the FEM is that every single element’s work can be described applying a pattern that is identical for all elements. In the following, all individual procedural steps are represented in detail up to the discretization of the work equation, as well as the evaluation of the internal force variables.

2.3.1 Solution Procedure According to Equation (2.9), the initial equation to compute an approximate solution is the weak form of the equilibrium regarding the entire structure Z Z − δW = u,x · EA · δu,x dx − p · δu dx − [ H · δu ]l = 0 . After the choice of shape functions and the integration of the work equation concerning the element domain, one gets the discrete form of the work equation, which depends on the still unknown nodal displacements. The discretization of the work equation takes place in three steps: 1. set up the work equation regarding a single element applying a displacement–based formulation, 2. choose shape functions regarding the displacements, and 3. introduce the shape functions into the work equation and perform integration. Furthermore the FE–procedure comprises: 4. sum up all elements’ work to get a description of the entire system and computation of the nodal displacements as presented in Section 3, and 5. subsequent stress analysis to evaluate the internal force variables element by element. The general procedure is shown by investigating a bar e.g. and is transferred to other problems as heat conduction, membranes, beams and plates in the following sections.

2.3 The General Procedure to set up the Element Matrices

29

Step One: Work Equation Regarding a Single Element Figure 2-3 shows the bar subdivided into two elements and loaded by a constantly distributed external action as well as by an external boundary action. Both elements are identical concerning length ℓ and elasticity EA. According to Section 2.1.3, work performed at the element boundaries does not have to be considered if the element intersection conditions are fulfilled. At element level, the work H δuc performed by the external boundary action H can be dropped, because it is considered at the system level. Therefore, when adding up the elements’ works to get the system’s work, only Z Z −δWel = δu,x · EA · u,x dx − δu · p(x) dx 6= 0 (2.21)

has to be taken into account, whereby this notation equals the work equation represented at system level. It must be pointed out that the element’s work given in Equation (2.21) does not include the work performed at the element intersections. Therefore, the work equation at element level cannot be directly used to evaluate the displacements.

Step Two: The Shape Functions If the strong solution is unknown, an approximate approach must be chosen to describe the displacements at element level. This approximation must fulfill the continuity conditions of the PvD concerning the displacements inside the related domain as well as the displacements at the beginning and at the end of the element. Here, a linear approach with two degrees of freedom will suffice to describe u(x). At first, the polynomial approach applies: u(x) = a0 · x0 + a1 · x1 = a0 + a1 · x .

(2.22)

u(x) = φA (x) · uA + φB (x) · uB .

(2.23)

The coefficients a0 , a1 are the degrees of freedom of the approach and x0 , x1 are general monomials of different order, whereat the coordinate 0 ≤ x ≤ ℓel is defined within each single element. Regarding this general polynomial approach, the degrees of freedom do not have any physically descriptive meaning. Furthermore, it might be difficult to fulfill the element’s intersection conditions concerning the displacements. Hence it makes more sense to employ the physically meaningful and easily interpretable nodal displacements uA and uB as the degrees of freedom:

The fundamental advantage of this descriptive approach is the fact that the continuity conditions at the element’s intersections as well as the Dirichlet

30

2 Discretization of the Work Equation

boundary conditions may be met more easily. The following will show how to transfer the general polynomial approach given in Equation (2.22) into the descriptive approach given in Equation (2.23), cf. Figure 2-7. uA

u(x)

uB

ΦA uA =

A

ΦB uB +

1

B x

1

ΦA = 1− xl

x ΦB = l

l

l

l

Fig. 2-7 Scaling of the approach with respect to the nodal displacements An evaluation of the approach in Equation (2.22) according to the element boundaries results in the two equations x = 0 : u(0) = a0 + a1 · 0 = uA , x = ℓ : u(ℓ) = a0 + a1 · ℓ = uB

(2.24)

yielding the solution a0 = u A , a1 = (uB − uA ) /ℓ . Thus, the approach according to Equation (2.22) leads to u(x) = uA + (uB − uA ) · x /ℓ , or merging in a more conventional way gives u(x) = (1 −

x x ) · uA + · uB , ℓ ℓ

(2.25)

where uA and uB are the nodal displacements and φA (x) and φB (x) are the related shape functions φA (x) = 1 −

x ℓ

and

φB (x) =

x . ℓ

These shape functions φi are also employed to describe the virtual displacements δu(x). Due to the fact that two degrees of freedom will need two conditional equations later on, the work equation is arranged according to two

2.3 The General Procedure to set up the Element Matrices

31

independent virtual displacement states. It is advantageous to choose the shape functions of the virtual displacements to be the same as the shape functions of the real displacements δuA (x) = φA (x) · δuA , δuB (x) = φB (x) · δuB , since this choice leads to a symmetrical system of equations and thus turns out to be efficient concerning the solution of the overall system of equations.

Step Three: Integration of the Element Work Following Equation (2.21), the real as well as the virtual displacements are to be differentiated with respect to x: u(x),x = φA (x),x · uA + φB (x),x · uB , δuA (x),x = φA (x),x · δuA ,

(2.26)

δuB (x),x = φB (x),x · δuB .

Introducing these approaches into the element’s work equation (2.21) leads to: Z Z −δWA = δuA · φA,x · EA · {φA,x · uA + φB,x · uB } dx − δuA · φA · p(x) dx , Z Z −δWB = δuB · φB,x · EA · {φA,x · uA + φB,x · uB } dx − δuB · φB · p(x) dx . Since the nodal values are independent of the integration, the element’s total work can be represented in a matrix notation:

− δWel = − δWA − δWB # " # Z " φA,x EA · φA,x φA,x EA · φB,x uA = [ δuA δuB ] · { dx · φB,x EA · φA,x φB,x EA · φB,x uB " # Z φA · p(x) − dx } . (2.27) φB · p(x) The element matrix is symmetrical only if the shape functions regarding the real and the virtual displacements are identical. We may start with p(x)

=p

= const. ,

φA (x),x = − 1/ℓ = const. , φB (x),x = + 1/ℓ = const. .

32

2 Discretization of the Work Equation

After analytical integration of the element’s work it follows:  " 1 # " #  2 − ℓ EA (− 1ℓ ) x − 1ℓ EA 1ℓ x uA p ( x − x2ℓ ) ℓ }0 · −δW = [ δuA δuB ] · { − 1 1 1 1 x2 uB EA (− ) x EA x p ( ) ℓ ℓ ℓ ℓ 2ℓ " # " # " # EA/ℓ −EA/ℓ uA pℓ/2 = [ δuA δuB ] · { · − }. (2.28) −EA/ℓ EA/ℓ uB pℓ/2 The integrated representation of the element matrix equals the stiffness matrix of the bar, as it has been established by the displacement method developed from structural analysis. Therefore the FEM’s element matrix may also be named as element stiffness matrix. The entries of the element matrix illustrate the nodal forces, which evolve from the real nodal displacements and which perform work on their corresponding virtual nodal displacements. Every single sum of a row represents the work of the element referring to a rigid body motion performed with respect to the corresponding virtual displacement state, wherein the nodal displacements uA and uB are the same size. Due to the fact that a rigid body motion must not produce any forces, the work and therefore every sum of a row regarding the element stiffness matrix is equal to zero. Just as the sum of a column describes the nodal forces’ work performed with respect to a virtual rigid body displacement.

Step Four: Change over to the Overall System Adding up the elements’ works to the system’s work provides the system’s stiffness matrix, the system’s load vector and the system’s vector of degrees of freedom. The set up of the system of equations to evaluate the nodal displacements is shown in Section 3.

Step Five: Subsequent Analysis of Stresses The unknown nodal degrees of freedom are evaluated by solving the overall system of equations. By applying a subsequent analysis regarding every element individually, the strains may be evaluated from the known u(x) by means of kinematics ε = u(x),x = φA (x),x · uA + φB (x),x · uB 1 1 = − · uA + · uB , ℓ ℓ

2.3 The General Procedure to set up the Element Matrices

33

and, furthermore, the longitudinal forces may be evaluated in terms of the material equation 1 1 N = EA · ε = EA · (− · uA + · uB ) . (2.29) ℓ ℓ When applying the matrix notation, the strain may be represented by     1 1 uA [ε]= − · uB ℓ ℓ as well as the longitudinal force  1 [ N ] = [ EA ] · [ ε ] = [ EA ] · − ℓ

1 ℓ

   uA · . uB

To evaluate the equations element by element, there is no need for integration or solving a system of equations. Due to the differentiation, the polynomial order of the courses to describe the distributions of strains and stresses is lower than the respective order for the description of the displacements.

2.3.2 Example Section 1.3 gives an approximate solution regarding the example referred to in Figure 1-5 and Figure 2-8, which is also compared to the analytical solution. H

case 1

EA p(x)

case 2

EA

x,u l

Fig. 2-8 Bar – geometry and loading According to the procedure presented in Section 2.3.1, the approximate solution is represented for one element in the following.

Loading Condition: Constantly Distributed External Action p(x) The discretized form of the PvD is given by " # " # " # EA/ℓ −EA/ℓ uA pℓ/2 [ δuA δuB ] · { · − } = 0. −EA/ℓ EA/ℓ uB pℓ/2

34

2 Discretization of the Work Equation

The boundary conditions uA = 0 and δua = 0, prescribed at x = 0, may be introduced by eliminating the first row and the first column of the system of equations. It remains       [ δuB ] · { EA/ℓ · uB − pℓ/2 } = 0 . Thus the displacement follows to     uB = pℓ2 /2 EA .

Applying the subsequent stress analysis, the longitudinal force is evaluated to     uA 1 1 [ N ] = [ EA ] · − · . uB ℓ ℓ Consideration of the boundary condition uA = 0 yields     1 EA [ N ] = [ EA ] · · [ uB ] = uB = [ pℓ/2 ] . ℓ ℓ

Loading condition: External Boundary Action The equivalent procedure may be applied to investigate the external boundary action H. Introducing the load vector   0 p= , H the displacement follows to [ uB ] = [ Hℓ/EA ] , and the longitudinal force gives [ N ] = [ H ].

2.3.3 Matrix Notation Transferring to matrix notation enables a more general formulation of the work equation. Therefore, symbols are introduced which are related to the matrices and vectors given in Section 2.3.1. Their specific contents are represented with regard to the bar. In the following, matrices are marked with capital bold letters, while vectors shall be identified by small bold letters.

2.3 The General Procedure to set up the Element Matrices

35

Step One: Work Equation Regarding a Single Element The work equation regarding the bar without considering any boundary terms Z Z −δWel = δu,x · EA · u,x dx − δu · p(x) dx can be generalized by applying the following notations: vector of displacements virtual displacements

: u : δu

= =

[ u(x) ] , [ δu(x) ] ,

operator matrix

:

D =

[

elasticity matrix load vector

: :

E = p =

[ EA ] , [ p(x) ] .

∂ ∂x

] = [ ∂x ] ,

The operator matrix D comprises the rule of differentiation that is to be applied to the shape functions, while the transposed operator matrix DT is applied to the function on the left hand side. The virtual displacements have to appear as transposed vectors regarding the work equation if both the displacements and the external actions are represented by applying vector notation. That is the only way the virtual work can be represented as having a scalar quantity. Introducing all matrix symbols into the work equation yields Z Z − δWel = δuT · DT · E · D · u dx − δuT · p dx . (2.30)

Step Two: Shape Functions The approach u(x) regarding the displacements of the bar u(x) = φA (x) · uA + φB (x) · uB

(2.31)

is transferred to matrix notation applying the matrix Ω, which incorporates all shape functions, and the nodal displacement vector v, which comprises the nodal degrees of freedom, yielding   uA u = [ φA φB ] · or uB u = Ω·v.

(2.32)

Hence the virtual displacements follow to δu = Ω · δv ,

(2.33)

whereby the virtual nodal displacements may be arbitrarily small but finite.

36

2 Discretization of the Work Equation

Step Three: Integration of the Element’s Work The description of the element’s work applying matrix notation may take place either following the detailed representation according to Section 2.3.1 or by applying Equation (2.30) and introducing the chosen symbols for the shape functions Z Z − δWel = [ δvT · ΩT · DT ] · E · [ D · Ω · v ] dx − [ δvT · ΩT ] · p dx . Since the nodal degrees of freedom v as well as the virtual nodal values δv are independent of x, they can be excluded from the integral of work Z Z − δWel = δvT · { [ ΩT · DT ] · E · [ D · Ω ] dx · v − ΩT · p dx } . (2.34) Moreover, the auxiliary matrix B= D·Ω

, BT = ( D · Ω )T = ΩT · DT

can be introduced to get a short form describing the field of strains. This yields Z Z − δWel = δvT · { BT · E · B dx · v − ΩT · p dx } | {z } | {z } K f

with the element stiffness matrix K, that is symmetrical if identical shape functions u and δu are employed in the same order, the element load vector f and the element displacement vector v. Assigning the element stiffness entries as well as the load vector to the respective nodes yields the following pattern clearly arranged: " # " # kAA kAB fA K= , f= . kBA kBB fB The matrix notation of the virtual work offers the big advantage of being applicable to all physical problems. Only the contents of the respective matrices and vectors as well as the integration region must be tentatively adjusted to the particular problem being investigated.

Step Four: Change over to the Overall System The work of the element is now defined by applying the matrix notation − δWelement = δvT { K v − f } .

2.3 The General Procedure to set up the Element Matrices

37

After adding up the works regarding all the elements to get the system’s work, and applying the same matrix symbols at the system level, the work equation follows to − δWsystem = δvT { K v − f } = 0 . Here, K is the stiffness matrix of the system, f is the load vector of the system and v is the displacement vector of the system. The procedure of adding up the work with regard to all elements is explained in more detail in Section 3. According to common practice with respect to the literature, no difference is made between element matrices and system matrices or between the respective vectors. In the following vectors will be characterized by small letters, matrices by capital letters. In addition to the elements’ works, the boundary conditions as well as the external boundary actions must be taken into account, but may be considered at the system’s level afterwards. This is advantegeous, because it does not perturbe the element by element procedure to get the system’s stiffness matrix and the load vector. After assembling the system of equations to compute the system’s displacement vector, v may be evaluated numerically by means of a standard equation solver, cf. Section 3.

Step Five: Subsequent Analysis of Stresses Applying the matrix notation the analysis of the longitudinal force follows from Equation (2.29)     uA ∂ [ φA φB ] . [ N ] = [ EA ] uB ∂x A symbol notation applies the vector of strains ǫ = [ ε(x) ] , and the vector of stresses σ = [ N (x) ] . Hereby, the computational rule regarding the strains follows to ǫ = D·u = D·Ω·v = B·v,

(2.35)

and regarding the stresses to σ = E·ǫ = E·D·Ω·v = E·B·v.

(2.36)

38

2 Discretization of the Work Equation

It is advantageous to multiply E and B a priori, what leads to σ = S·v,

(2.37)

whereat S is named the stress matrix. S incorporates the governing equations and the shape functions for the computation of the force variables within the element. Therefore, to get the longitudinal forces, it only has to be multiplied by the related nodal displacements v of the element, which must be extracted from the global solution vector. If special values of stresses are of interest the coordinates of the related positions ˜ Thus have to be taken into account for the computation of the stress matrix S. ˜·v σ ˜=S

(2.38)

no longer describes the polynomial approach of the stresses, but the specified values.

2.4 Shape Functions and Convergence Criteria The Finite Element Method substantially bases on the shape functions employed to approximately describe the displacement field inside the element’s domain. Usually these functions may be polynomials of different orders and are to be related to the dimensions of the element geometry. To improve the quality of the approximate solution and to converge against the exact solution – if it exists – either the order of the shape functions or the number of elements is to be increased. Thus, shape functions and convergence are strongly correlated to each other.

2.4.1 Shape Functions Polynomials may be selected by applying different representations according to different scaling factors. In principle, all polynomials are equivalent if they employ the same polynomial order and thus the same number of degrees of freedom. Thus, the transformation of scaling factors from one set of factors to another one is possible. An example is discussed in Section 2.3.1 and Equations (2.22) and (2.25). Nevertheless, according to FEM, representations are only appropriate when applying physically meaningful nodal degrees of freedom as scaling factors. The following groups of shape functions arise among others and may explain the different representations:

2.4 Shape Functions and Convergence Criteria

39

1. General Polynomials The most simplified representation of the displacement field consists of power terms of x, which are named monomials. Thus the complete approach is given by a general polynomial of specific order, e.g. a quadratic approximation of the displacement u(x) is given in Figure 2-9. u(x) = a0 x0 + a1 x1 + a2 x2

B

A l

x0

1

Φ 1(x)

x1

l

Φ 2 (x)

l2

Φ 3 (x)

2

x

Fig. 2-9 General polynomial of second order Although the displacement field is approximated, the generalized coordinates ai do not attach any physical meaning and are attributed to the origin of the coordinate x. In the following, these polynomials are characterized as general polynomials to make a clear difference to Lagrange and Hermite Polynomials. Nonetheless, the transformation of the scaling factors ai to scaling factors with a physical meaning could be done afterwards.

2. Legendre Polynomials Another type of approximation without any physical meaning of the scaling factors deals with Legendre Polynomials. Legendre Polynomials are shape functions employing the special feature 1 ℓ

Z

φi (x) · φj (x) dx =

2 · δij , 2i + 1

δij = {

1 f¨ ur i = j , 0 f¨ ur i = 6 j

which means, that they are orthogonal with respect to the integral. This leads to numerical advantages, when the virtual work equation has to be integrated. These polynomials, however, do not exhibit any physically meaningful degree of freedom. In this case the transformation of the scaling factors to scaling factors with a physical meaning can be done after the integration of the virtual work.

40

2 Discretization of the Work Equation

Regarding the Legendre Polynomials according to Figure 2-10, the following approach of the displacement field u(x) may be established: B

A

u(x) = a0 · φ0 + a1 · φ1 + a2 · φ2 + . . . ,

x l

l

φ0 (x) = 1

φ1 (x) =

1

1

x ℓ

1

1 x2 φ2 (x) = ( 3 2 − 1) 2 ℓ

Φ 1(x)

Φ 2 (x)

Φ 3 (x)

1 2

Fig. 2-10 Legendre Polynomials

3. Lagrange Polynomials If shape functions are chosen that employ nodal displacements at designated coordinates as scaling factors, so–called Lagrange Polynomials are obtained. In Figure 2-11 quadratic Lagrange Polynomials are depicted. C

A

u(x) = uA φ1 + uB φ2 + uC φ3

φ1 (x) = 1 − 3

φ2 (x) = −

x x2 +2 2 ℓ ℓ

l 2

l 2

Φ 1 (x)

1

x x2 +2 2 ℓ ℓ

φ3 (x) = +4

x x2 −4 2 ℓ ℓ

Fig. 2-11 Quadratic Lagrange Polynomials

B

1

1

Φ 2 (x)

Φ 3 (x)

2.4 Shape Functions and Convergence Criteria

41

The three scaling factors of the quadratic approach of the displacement field are the displacements uA , uB , uC at the element nodes A, B, C. The shape functions φ1 , φ2 , φ3 no longer keep single power terms, but functions, that describe the influence of the related nodal displacement within the element. Thus, Lagrange Polynomials are functions, which comprise the value of 1 in the node, that keeps the unknown displacement, and a value of 0 in all other nodes.

4. Hermite Polynomials Hermite Polynomials employ the nodal displacements as well as their higher derivatives at the domain’s boundaries and further nodes inside the domain as scaling factors. They are of advantage, if the intersection conditions between elements are described by means of higher derivatives of the displacement course. Applying the cubic Hermite Polynomials according to Figure 2-12, this means: u(x) = uA φ1 + u,x A φ2 + uB φ3 + u,x B φ4 . B

A x l 2

3

φ1 (x) = 1 − 3

x x +2 3 ℓ2 ℓ

φ2 (x) = x − 2

x2 x3 + 2 ℓ ℓ

φ3 (x) = 3

Φ1 (x)

1

1

Φ2 (x)

x2 x3 −2 3 2 ℓ ℓ

φ4 (x) = −

x2 x3 + 2 ℓ ℓ

1

1

Φ3 (x)

Φ4 (x)

Fig. 2-12 Cubic Hermite Polynomials Even higher order Hermite Polynomials may be employed if second or higher order derivatives are chosen as nodal degrees of freedom. Since the nodal degrees of freedom have different units, the Hermite Ploynomials have different units as well. Hermite Polynomials are normalised functions, which comprise the value of 1 with respect to the prevailing unknown, and a value of 0 with respect to all other unknowns.

42

2 Discretization of the Work Equation

2.4.2 Criteria of Convergence At mesh refinement the finite element approximation of the course of displacements needs to converge towards the analytical strong solution, assuming it exists. To ensure this convergence, the shape functions regarding the displacements must fulfill special conditions which have been partially brought up in Section 2.1.3 already. If these conditions – in the following named as convergence criteria – are fulfilled, the convergence is ensured towards the exact solution. The displacement–based formulation of the FEM applies the PvD, therefore the requirements with respect to the shape functions are correlated to the work equation developed in Section 2.1.

Criterion of at least Constant Strains At a minimum, the strains must be approximated constantly by the shape functions regarding the displacements inside the element. The same applies to the elongation ε regarding bars, or the curvature κ regarding beams. It is an important requirement, without which the computation of the virtual internal R work δWi = − δεT· σ dV , given here in the most general matrix notation, and of the stresses when applying a subsequent analysis σ = E · ε would not be possible. Concerning bars, the displacements must be approximated at least linearly to be able to describe constant strains and constant longitudinal forces within the element. This applies analogously, concerning the displacement fields at membrane structures with respect to coordinates x and y, and for spatial structures of three dimensions. Regarding the Euler–Bernoulli beam the curvature κ = w,xx is applied, since the effect of shear deformations is neglected. Therefore, at least quadratic shape functions are required to describe the deflections. This requirement also analogously applies for the course of the deflection w(x,y) regarding Kirchhoff ’s plate theory.

Criterion of Conformity The criterion of conformity requires continuity and moreover continuous differentiability of the shape functions within the element up to the strains, which are required to have a continuous course inside the element’s domain. Otherwise additional work must be taken into account when applying the PvD. At the element’s intersections the following conditions must be fulfilled and the same is valid for the system’s boundary conditions:

2.4 Shape Functions and Convergence Criteria

43

• When assembling the elements all substantial intersection conditions regarding real displacements as well as virtual displacements must be fulfilled. Regarding the bar, the continuity of the displacements is required. Only by this the work performed by the internal force variables cancels each other out at the element intersections. Note: Regarding the PvD, the conditions concerning the displacements are the substantial boundary conditions as well as intersection conditions. They have to be directly fulfilled by the shape functions. The boundary as well as intersection conditions concerning the force variables are fulfilled in their weak form which implicitly applies the PvD work equation as shown above. • If the boundary as well as the intersection conditions are fulfilled regarding the displacements only, which formally means the zero order derivative of the displacements, this is called a C0 –conformity of shape functions. Regarding a bar employing linear shape functions to describe u(x) and δu(x), this condition may be fulfilled by the displacements ( uA , uB ) and the virtual displacements ( δuA , δuB ), respectively. As a consequence the strains as well as the longitudinal forces may incorporate some gaps at the element intersections. • Regarding the Euler–Bernoulli beam the deflection and its first derivative are substantial boundary as well as intersection conditions. Therefore C1 –conformity of the shape functions is required, since one integration constant corresponding to w and one corresponding to w,x follow from kinematics. Analogous statements apply concerning the respective virtual displacement variables. Since the beam element has to fulfill the conditions at both element nodes A and B, the approach regarding w must comprise of at least four degrees of freedom, e.g. ( wA , w,xA , wB , w,xB ), which means at least a cubic approach. In contrast, bending moments as well as shear forces may exhibit discontinuities at the element intersections. • Regarding membrane and plate structures, equivalent demands are made on the approaches compared with the respective one–dimensional structure. Thus, membrane structures require C0 –conformity of shape functions, while plate structures require C1 –conformity. • Conformity of shape functions regarding shell structures is only achievable when employing high order shape functions. Therefore, if shape functions are applied, which are not fulfilling the criterion of conformity, the work performed at the element intersections has to be explicitly taken into account.

44

2 Discretization of the Work Equation

To conclude, regarding the shape functions and their respective derivatives at the element intersections, continuity is required up to the differential order that is lowered by one compared to the order which appears in the work equation concerning the element’s domain. There might be some elements which cannot fulfill all element intersection conditions a priori without increasing the polynomial order of the shape functions considerably. Thus, when applying the work equation, it might be more meaningful to explicitly consider the work performed by the internal force variables at the element intersections.

Criterion of Rigid–Body Displacement A rigid–body motion or rigid–body displacement does not describe a motion of a rigid body, but a motion without strains and stresses. This means, that an element may undergo a pure translation or/and rotation, cf. Figures 2-13 and 2-14, whereat strains and curvatures vanish. A

B

A*

B*

uA uB x

x=l

ε = u’X = 0

uA = uB

Fig. 2-13 Rigid–body displacement of a bar Since such displacement states may occur at any area inside a structure, depending on the situation, the shape functions must fulfil this criterion right from the start and must be able to describe these displacement fields a priori.

w

displacement and rotation

Fig. 2-14 Rigid–body displacement of a part of a beam structure The demand from the strains and curvatures not to occur in the case of a rigid–body displacement is generally fulfilled regarding plane structures such

2.4 Shape Functions and Convergence Criteria

45

as bars, membranes, beams or plates, if the shape functions are balanced on each other, cf. Section 2.4.1. In contrast, for curved structures like circular arcs and shell structures the rigid body criterion is hardly to fulfil, because of the special kinematic relations applying to curved structures. Regarding circular arcs, the strain ε additively follows from the first derivative of u with respect to the curved coordinate s and from the zero order derivative of w. If the same shape functions, for example linear courses, are employed to describe both displacements, the condition of rigid–body displacement without occuring strains cannot be fulfilled at every single point, see Figure 2-15. Instead a linear strain and thus a linear stress field occurs. exact solution

linear shape functions for u and w

πs , 2ℓ πs u(s) = uA · cos , 2ℓ w ε = u,s + = 0. R

s s w(s) = (1 − ) · wA + · wB , ℓ ℓ s s u(s) = (1 − ) · uA + · uB , ℓ ℓ 1 1 s ε = − · uA + · · wB 6= 0 . ℓ R ℓ

w(s) = wB · sin

A

uA s

A*

wA = 0

w u

uB = 0

R

wB = uA

wB

s=l B

B*

Fig. 2-15 Rigid–body displacement of a circular arc

Criterion of Coordinate Invariance Regarding two– and three–dimensional structures, the shape functions must be coordinate–invariant to avoid any influence of rotation of the coordinate system on the deformation state both of the element and of the system. This is achieved by ensuring the shape functions are equivalent with respect to all directions. The easiest way to check the coordinate–invariance of shape functions is to employ Pascale’s triangle, see Figure 2-16. Pascale’s triangle indicates the multiplicative combinations of power terms with respect to the x– and y– directions, which are possible in principle. If all products are taken into account up to a selected order, this is called a complete approach.

46

2 Discretization of the Work Equation

For example: • u(x, y) = a0 + a1 · x + a2 · y is a complete linear approach and • u(x, y) = a0 + a1 x + a2 y + a3 x2 + a4 xy + a5 y 2 is a complete quadratic approach.

1 y

x x2 x2y

x3 x4

x3y

y2

xy xy 2

x2y 2

y3 xy 3

y4

symmetric approach of second degree complete approach of second degree complete approach of fourth degree

Fig. 2-16 Pascale’s triangle Complete approaches are inherently coordinate–invariant. But to choose proper shape functions the element geometry and the number of possible nodes should be taken into account. Triangular elements need complete approaches, because they should have invariant properties, since they may have different positions in the x–y–plane. Furthermore, complete approaches match the number of possible nodes inside triangles, because three unknowns are correlated to the three corners and six unknowns are correlated to the corners and the midpoints of the edges. Rectangular as well as quadrilateral elements usually have four, eight or nine nodes, which do not match the number of the polynomials of complete approaches. Thus, symmetric approaches with respect to the x– and y–coordinates are chosen. Symmetric approaches arise if one–dimensional approaches of the same order with respect to the x– and y–directions are multiplied by each other.

2.4 Shape Functions and Convergence Criteria

47

They are invariant against rotation of the coordinate system by 90o and multiple thereof. Regarding two–dimensional plane structures, the approaches may be taken from the respective one–dimensional structure therefor. • Regarding membrane structures, it applies in analogy to bars: Employing a linear approach with respect to the x–direction and an equivalent linear approach with respect to 1 x y the y–direction, a bi–linear approach is xy generated to describe the displacements u(x, y) and v(x, y) that occupies 4 adjacent combinations in Pascale’s triangle. • Regarding plate structures, it applies in analogy to beams: The product of a cubic approach with respect to the x–direction and the equi1 x y valent cubic approach with respect to x2 xy y2 the y–direction generates a bi–cubic 3 2 2 approach to describe the deflection x x y xy y3 x3 y x2 y 2 xy 3 w(x, y), which occupies the adjacent x3 y 2 x2 y 3 combinations in Pascale’s triangle. This 3 3 x y leads to an approach employing 16 degrees of freedom.

2.4.3 Scaling Matrix Regarding two– or more–dimensional elements employing higher order approaches, difficulties often arise when trying to directly relate the shape functions to the physically meaningful degrees of freedom, especially when applying triangular elements. In such cases, a more general procedure, which is also well–suited to coding may be chosen to determine the shape functions. To approach the course of displacements u = [ u(x) ] inside the element, at first the general polynomial with the notation u=ψ·a (2.39) is chosen, where the vector ψ comprises the power terms of the approach corresponding to the coordinates x, y and a comprises the related scaling factors ai , cf. Section 2.3.1 and Equation (2.22). The transformation of the general polynomial into a physically meaningful one is divided into two steps. At first, the relationship has to be established between the scaling factors a = [ a1 a2 a3 . . . ] and the desired physically meaningful nodal degrees of freedom v = [ uA uB uC . . . ]. Here, it could be necessary to provide the derivatives of

48

2 Discretization of the Work Equation

the general polynomials with respect to the coordinates x, y, if derivatives of the displacements are selected to be nodal degrees of freedom, e.g. ϕ = w,x in the case of beams and plates. Applying the procedure for each physically meaningful nodal degree of freedom, the respective nodal coordinates are successively introduced into the general relation (2.39). This yields a system of equations which is comprised of the relations between the physically meaningful nodal degrees of freedom v and the scaling factors a of the general polynomial by numbers:     ˜A ψ uA   ˜    uB  =  ψ B ·a ... ... and in matrix notation

˜ · a. v=Ψ

(2.40)

˜ transforms the scaling factors a to the physically meaningThus, the matrix Ψ ˜ according to Equation (2.40) in a ful nodal degrees of freedom v. Reversing Ψ second step, one obtains ˜ −1 · v = G · v . a=Ψ

(2.41)

Thus the vector of scaling factors a regarding Equation (2.39) may be described directly as employing the nodal degrees of freedom. It follows that u = ψ ·G·v = Ω·v.

(2.42)

The matrix G is named as scaling matrix and scales the general polynomial according to Equation (2.39) with respect to the physically meaningful nodal degrees of freedom. The product ψ · G now describes the physically meaningful shape functions Ω. As an example, in the following Table 2.1, the scaling of a quadratic polynomial with the general coordinates [ a1 a2 a3 ] is represented with respect to the nodal displacements [ uA uB uC ] regarding a bar. The physically meaningful shape functions evaluated by this procedure turn out to be the quadratic Lagrange Polynomials according to Section 2.4.1.

2.4 Shape Functions and Convergence Criteria

49

Table 2.1 Relationship between the general polynomial and the physically meaningful shape functions direct approach

scaling procedure

u=Ω·v

u = ψ · a,

shape functions

general polynomial   ψ = 1 x x2

Ω = [ φA

φB

φA = 1 − 3

x ℓ

φC ] 2

+ 2 xℓ2

a= G·v

2

φB = − xℓ + 2 xℓ2

2

φC = +4 xℓ − 4 xℓ2

degrees of freedom v T = [ uA

uB

uC ]

aT = [ a0

a1

a2 ]

scaling matrix ˜ ·a v=Ψ      uA ψ(x = 0) 1  uB  =  ψ(x = ℓ)  · a =  1 uC ψ(x = ℓ/2) 1

 0 0 ℓ ℓ2  · a 2 ℓ/2 ℓ /4

˜ −1 · v = G · v a=Ψ   1 0 0 −1 ˜ 4/ℓ  G=Ψ =  −3/ℓ −1/ℓ 2 2 2/ℓ 2/ℓ −4/ℓ2 shape functions u=ψ·G·v

 1 0 0 4/ℓ  Ω = ψ · G = 1 x x2 ·  −3/ℓ −1/ℓ 2/ℓ2 2/ℓ2 −4/ℓ2 h i 2 2 2 = (1 − 3 x + 2 x2 ) (− x + 2 x2 ) (4 x − 4 x2 ) ℓ ℓ ℓ ℓ ℓ ℓ 





50

2 Discretization of the Work Equation

It is an essential fact that the integration of the element stiffness matrix and of the load vector can be simplified employing the scaling matrix and may be generally more efficiently established. Therefore, integration of the element stiffness matrix K and the load vector f may be performed by employing the general polynomial. After the integration procedure, the stiffness matrix as well as the load vector are multiplied by the scaling matrix, which now can be done numerically Z δvT · K · v = δvT · GT · ψ T · DT · E · D · ψ dA · G · v , (2.43) Z δvT · f = δvT · GT · ψ T · p dA . (2.44) Even if the scaling matrix appears to be a detour from computing the physically meaningful shape functions at first, it is often more advantageous as well as numerically more efficient to follow the procedure given in Equations (2.43) and (2.44), with regard to the programming of the stiffness matrix.

3 Structure and Solution of the System of Equations

In Section 2 the Principle of virtual Work is discretized at element level. Herewith, the virtual work is described by the virtual and the real nodal displacements, the stiffness matrix and the load vector. The work performed at a single element can be incorporated into the work of the entire system only if the conditions of transition at the element intersections are considered. These are, as shown in Section 2.1.3, the conditions of equilibrium and of deformation. The conditions of equilibrium are fulfilled weakly at system level by the work equations addressing the Principle of virtual Displacements. The conditions of deformation are fulfilled strongly for the virtual and the real displacements, for example, shown in Figure 3-1 for a tensile bar. p(x)

1

elements

3

2

1

2

nodes

4

3

u (x)

u1 δ u1 1

u2 δ u2 1

22

u4 δu4

u3 δ u3 2

33

3

possible qualitative final solution

unknown real nodal displacements virtual nodal displacements

4 final solution inside single elements

1

1

1

shape functions for real and virtual displacements inside single elements 1

1

1

Fig. 3-1 Shape functions for the course of displacements at a tensile bar © The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2024 D. Dinkler und U. Kowalsky, Introduction to Finite Element Methods, https://doi.org/10.1007/978-3-658-42742-9_3

52

3 Structure and Solution of the System of Equations

3.1 Real and Virtual Nodal Displacements The unknown real displacement field of the tensile bar is approximated element– wise with the shape functions and the related unknown nodal displacements. In order to satisfy the conditions of the work equation, shape functions and nodal displacements must directly fulfill kinematics inside the element and at the element intersections, cf. convergence criteria. Since the real displacements have to be continous at the element boundaries, the nodal displacements of two neighboring elements cannot be independent variables at the common node. Therefore, they will be assigned as nodal displacements of the corresponding node of the system. Thus, being the unknown multipliers of the shape functions, the nodal displacements ui are the unknowns of the given problem and completely determine the deformation state of the system. Since the Principle of virtual Work is applied for each element, the virtual displacements are also described for each element separately. As per Section 2.1.3, kinematics at the element intersections must also be fulfilled by the virtual displacements. This is only possible, if the virtual nodal displacements of the elements are the same at the common nodes of neighboring elements, and thus become virtual nodal displacements of the system. Hence the virtual work at the common nodes of neighboring elements can be summarised. Nonetheless, the polygon–shaped approximation of the course of the real as well as of the virtual displacements of the tensile bar, as shown in Figure 3-1, do not yet fulfill the displacement boundary condition. The boundary conditions will be considered later on in the global system of equations.

3.2 Equations of Condition The Principle of virtual Displacements is equivalent to the conditions of equilibrium, if the virtual displacements satisfy the kinematics, cf. Section 2.1.1. Since the virtual work performed at the bar is the summation of the virtual work performed on all chosen compatible virtual displacements, the virtual work performed on each single virtual displacement must be zero. This leads to a set of equations, which may be arranged in matrix–vector notation, employing the stiffness matrix, the load vector, and the vector of the real nodal displacements. Choosing the same sequence for the virtual and for the real displacements yields a symmetric stiffness matrix. Because the real nodal displacements are the only unknowns according to the PvD, they can be computed solving these equations, presuming the number of independent virtual displacements corresponds to the number of unknown nodal displacements.

3.3 Structure of the System of Equations

53

Each row of the system of equations comprises the virtual work of the real stress state on the respective virtual displacement, whereat the virtual nodal displacements occur as multipliers of the corresponding work equations. Thus the work equation can also be taken as the condition of equilibrium of the nodal forces acting in the direction of the virtual nodal displacements. Accordingly, the conditions of equilibrium do not need to be fulfilled strongly at all differential elements, but only approximately employing shape functions and nodal displacements.

3.3 Structure of the System of Equations The treatment of the equations of condition for the global unknowns is represented for the example of the tensile bar, given in Figure 3-1, and considering a discretization into three elements. At first, it is shown how to store the element matrices. Without considering the boundary conditions, the given example employs four unknowns, u1 to u4 . The four equations of condition, which describe the work performed by the real stress state on the related virtual displacements δu1 to δu4 , are now available for their determination. Assembling the entire system of equations element by element is particularly convenient for coding, and is universally accepted. Therefore, the schematic representation of the virtual work performed at a single element i, as shown in Figure 3-2, can be used. Due to the two element nodes A, B the element stiffness matrix K consists of 2 × 2 sub–matrices kAA , kAB , kBA , kBB and the element load vector f of two sub–vectors fA and fB , respectively. uA

δuA

i

kAA

uB

i

i

kAB

fA =K

δuB

i

kBA

i

i

kBB

= f

i

i

fB

Fig. 3-2 Stiffness matrix and load vector of the element i Since the virtual work of every single element is computed independently from each other, the summation of the virtual work of the elements to the total virtual work of the system has to satisfy the kinematic conditions of the Principle of

54

3 Structure and Solution of the System of Equations

virtual Displacements afterwards. This will be performed in two steps according to the real and the virtual shape functions. At first, the virtual work performed at every single element is not yet coupled with each other as schematically represented in Figure 3-3.

δ u1

u3 u4

u2 u3

u1 u2 K

1

f

1

f

2

f

3

δu 2 δu 2

K

2

δu 3 K

δu 3

3

δu 4

Fig. 3-3 The virtual work without considering the transition conditions The transition conditions of deformation are fulfilled for the final state at the element intersections, if the actual displacements ui at the common nodes of the neighboring elements are the same and therefore considered as the nodal displacements of the system: u2 is same for elements 1 and 2, u3 is same for elements 2 and 3. The virtual work thus exhibits the scheme shown in Figure 3-4. u1 u2 u3 u4 δ u1

K

1

f

1

f

2

f

3

δu 2 δu 2

K

2

δu 3 δu 3

K

3

δu 4

Fig. 3-4 Conditions of deformation for the actual nodal displacements are fulfilled

3.4 Assembly of the Stiffness Matrix of the Entire System

55

Due to the Principle of virtual Displacements the conditions of deformation must be fulfilled at every node concerning the virtual displacements δui as well: δu2 is same for elements 1 and 2, and δu3 is same for elements 2 and 3. Thus the virtual work performed on the same virtual displacements is to be summed up regarding the entire system, cf. Figure 3-5. This condition governs the overlap at the element stiffness matrices and the load vectors. u1 u2 δ u1 δu 2 δu 3

u3 K

u4

1

f1 K2

f2 K3

f3

δu 4

Fig. 3-5 Fulfilment of conditions of deformation for the virtual nodal displacements Now the virtual work fulfills all kinematic conditions at the element intersections. The columns of the stiffness matrix are to be multiplied by the unknown real nodal displacements. Each row is to be multiplied by the respective virtual displacement, whereupon the values of the variables δu1 to δu4 do not influence the results, since they can be excluded from the related equation.

3.4 Assembly of the Stiffness Matrix of the Entire System The work performed by the real stress state on all possible virtual displacements of the entire system is summarized in the global system of equations. Since the work performed on the entire system is equal to the sum of the work performed on the single elements, the work performed at every single element can be stored additively and sequentially into the respective rows and columns of the system’s matrix, corresponding to the respective real and virtual nodal displacements. The node–wise allocation of the work performed on an element is preserved in the global system of equations, even if more than one unknown exists per node. For the correct allocation of the work into the rows and columns of the global stiffness matrix, the node numbers of the entire system must be correlated to the node numbers of the element as well as to their sequence in the element.

56

3 Structure and Solution of the System of Equations

This information is summarized in the connectivity matrix. For the tensile bar in Figure 3-1, this yields the connectivity matrix in Table 3.1: Table 3.1 Connectivity matrix for the bar Figure 3-1 element

node B A 1 2 3

1 2 3

2 3 4

The Connectivity Matrix for 2–Dimensional Structures For 2–dimensional structures related to heat conduction, membranes and plates, the sequence of nodes is different at the element– and at the system–level respectively. Therefore the connectivity matrix is extremely important. Introducing a connectivity matrix corresponding to a generalized system with more than two nodes per element, several unknowns per node and arbitrary numbering of the total system nodes are possible. In the following, the problem is explained by means of the discretization of a rectangular area into four rectangular elements each employing four element nodes A, B, C, D. The partitioning of the rectangular area into four elements could yield the element– and node–numbers as given in Table 3.2. Table 3.2 Connectivity matrix for a plane structure

1 4

2

4 8

System with four elements, connectivity matrix element

6

5

3 7

3

2

1

9

1 2 3 4

node A node B node C node D 1 2 4 5

2 3 5 6

5 6 8 9

4 5 7 8

The element stiffness matrix K of a single element may split up into the sub– matrices klm , cf. Figure 3-6. The sub–matrices of the element matrix corresponding to the nodal unknowns of the element are square matrices of the size n×n, corresponding to the n real unknowns per node as well as to the n virtual unknowns. The real and the virtual unknowns may comprise only one nodal

3.4 Assembly of the Stiffness Matrix of the Entire System

57

unknown as in the case of heat conduction, cf. Section 4, two nodal unknowns as in the case of membranes, cf. Section 5, or four nodal unknowns as in the case of plates, cf. Section 6. Element matrix vA vB vC A

B i

D

C

vD

δvA

kAA

kAB

kAC

kAD

δvB

kBA

kBB

kBC

kBD

δvC

kCA

kCB

kCC

kCD

δvD

kDA

kDB

kDC

kDD

Fig. 3-6 Correlation of the node–numbers and the element stiffness matrix By means of the connectivity matrix, the 4 × 4 sub–matrices of the element matrix will be additively and sequentially stored into the entire system matrix of 9 × 9 sub–matrices. The storage is shown in Figure 3-7 for the element number 2. Because the sequence of the node numbers at the element level and the sequence of the node numbers at the system level do not coincide, the element matrix will not be stored as a whole, but its sub–matrices are to be placed carefully into the corresponding rows and columns depending on the node numbers at the system level. 1

2

3

4

5

6

2

2

2

2

2

2

2

2

7

8

9

1 2

2

2

2

2

2

2

2

2

k AA k AB

3

k BA k BB

k AD k AC k BD k BC

4

K

2

=

5

k DA k DB

6

k CA k CB

k DD k DC k CD k CC

7 8 9

Fig. 3-7 System matrix, wherein only element 2 is incorporated

58

3 Structure and Solution of the System of Equations

The procedure of building up the left hand side of the entire system of equations can be directly implemented into a finite element program, if the storage considers all elements of the connectivity matrix. It can be transferred to the system’s matrix and to the load vector as shown in Figure 3-8. 1

2

3

4

5

6

7

8

9

1

X

X

X

X

1 X

2

X

X

X

X

2 X

4

X

X

X

X

5

X

X

3

3

K=

+ + X X + +

4 X+

+ + + +

f=

5

X

+

6

6

+ + + +

7 8

7

+ + + +

8

+ +

9

9

X

element

1

element

2

+

element

3

element

4

Fig. 3-8 The assembly of the system matrix and the load vector

3.5 The Storage and Solution of the System of Equations The structure of the element matrix and its storage as given in Figure 3-7 and Figure 3-8 have some properties, which are of advantage regarding the required store and the solution of the system of equations. Since the element matrices are symmetric and are stored symmetrically the resulting system matrix is also symmetric. Thus, just the diagonal and half of the matrix have to be computed and stored, if computing time and storage capacity have to be reduced. Furthermore, the system matrix is filled with zero values outside the employed diagonals, which are of no influence on the solution process. Neglecting the zero diagonals reduces the required storage capacity as well as the computing time. The number of the non–zero diagonals depends on the connectivity of the node–numbers and the unknowns of the nodes, cf. Figure 3-8, and is named bandwidth (BW). The bandwidth of the system matrix is defined to the number of diagonals with non–zero elements in the upper triangle, including the main

3.6 The Optimization of the Bandwidth

59

diagonal. Thus the bandwidth can be determined by means of the connectivity matrix: BW = (the maximum difference of node numbers in an element + 1) × (number of unknowns per node). Because of the symmetry, only the upper– or lower–triangular matrix is needed. Since the components of the matrix lying outside the bandwidth have zero values, they do not have to be considered in the solution process. Thus the system matrix will be stored in such a manner that the main diagonal on the left hand side is stored in the first column of a rectangular matrix with BW as the number of columns, whereby only the components of the upper triangular matrix will be stored at all, as shown in Figure 3-9.

z 

⊗  ∗     ∗   ∗      

BW = 5 · ( nodal unknowns ) }| { ∗ ⊗ ∗ ∗ ∗ ∗

∗ ⊗ ∗ ∗

∗ ∗

⊗ ∗ ∗ ∗

∗ ∗ ∗ ∗ ⊗ ∗ ∗ ∗ ∗

∗ ∗

∗ ⊗ ∗ ∗

∗ ∗ ⊗ ∗

∗ ∗ ∗ ∗ ⊗ ∗



   −→   Utilisation of  ∗   symmetry and ∗   bandwidth  −→  ∗  ⊗

             

⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗

∗ ∗ ∗ ∗ ∗ ∗

∗ ∗ ∗ ∗

∗ ∗ ∗ ∗ ∗ ∗

 ∗ ∗     ∗   ∗       

Fig. 3-9 The storage of the symmetric system matrix Since a symmetric system of equations does not only reduce the storage capacity needed but also considerably reduces the computing time, the order of magnitude of the effort proportionally accounts for n · B W 2 , whereat n is the number of unknowns of the entire system.

3.6 The Optimization of the Bandwidth In a few cases the geometry of the structure allows a structured element mesh employing a node numbering, which can be developed manually. But very often an unstructured mesh or a mesh refinement is necessary, if strong gradients of the stress fields locally occur. Thus a situation may be quickly reached, wherein it no longer makes sense to manually partition a structure into finite elements including the allocation of element and node numbers.

60

3 Structure and Solution of the System of Equations

Dealing with more complex discretizations, preprocessing is needed wherein the FE–meshes can be generated automatically including the assembly of the connectivity matrix. In the literature, different algorithms for the development of unstructured meshes are offered as the Advancing–front method see for example Lo [63, 64], George [39], and L¨ohner [65, 66], and the Delauny triangulation, see Ruppert [89] and Shewchuk [90]. If the mesh development is finished, an optimization of the node numbering is able to rigorously reduce the numerical effort for solving the system of equations. These algorithms are able to reduce the bandwidth and the storage capacity by renumbering the system nodes with respect to minimal numerical effort, see Cuthill and McKee [32] and Liu and Sherman [62]. Here, the optimization of the node numbering may be part of the preprocessing, where it influences all kind of data with respect to the node numbers as the connectivity matrix, the boundary conditions, the loading etc. Though the disadvantage of an arbitrary node numbering is not evident at first, it will become quite clear when regarding the allocation of the global stiffness matrix, cf. Figure 3-10. The stiffness values of the element nodes are incorpo-

arbitrarily node numbering

optimized node numbering

Fig. 3-10 Distribution of matrix elements of the system matrix rated into the system matrix as related to their current system node numbers. Hence, arbitrary allocation of the node numbers results in the system matrix without any diagonal structure as shown in Figure 3-10–left. Since the computing time is related to n · BW 2 and the needed storage capacity with respect to n · BW a renumbering of the nodes is necessary, which may significantly

3.7 The Fulfillment of Dirichlet Boundary Conditions

61

decrease the storage capacity required as well as the computing time needed. The system matrix presented in Figure 3-10–right particular clearly shows the effects of optimization of node numbering with regard to the bandwidth.

3.7 The Fulfillment of Dirichlet Boundary Conditions In the Finite Element Method developed using the PvD, the Neumann boundary conditions do not have to be fulfilled separately. They are fulfilled weakly by the work equation, see Section 2.1.2. The Dirichlet boundary conditions are to be fulfilled strongly by presetting the related nodal unknowns to zero, as shown in Figure 3-1. In the given example, Figure 3-1, the first column with u1 = 0 and the first row with δu1 = 0 are affected. For the efficiency of the solution procedure, the reduction of the size of the system matrix as well as of the number of unknowns is not that essential. It is more important to preserve the structure of the FE–program as simple as possible. Hence, all nodes with prescribed boundary conditions are formally kept. Therefor two possibilities shall be looked at: 1. The values of the components of the respective rows and columns are set to zero which is similar to cancelling out the related rows and columns. Since this yields a singular matrix, the values on the related main diagonals are to be set to ’1’, whereby the related unknown will be formally computed to zero. 2. A fixed constraint can be considered to be a spring with high stiffness. The virtual work of the spring’s force can be introduced into the system matrix by adding the spring stiffness to the value of the respective main diagonal component. Inhomogeneous boundary conditions with a certain value of the unknown 6= 0, for example a preset displacement, can also be easily considered. 1. Before the related row and column are replaced by zero values, the related column is to be multiplied by the prescribed value of the unknown and is to be subtracted from the right–hand side. Afterwards, the addressed row and column are set to zero and the element at the main diagonal is set to be ’1’. The value of the corresponding element in the load vector is then replaced by the prescribed value. 2. The product of the prescribed displacement value and the spring stiffness is to be stored in the load vector. It should be noted that the boundary conditions does not affect the symmetry and the storage of the system’s matrix either.

4 Heat Conduction

Heat conduction is a quite another physical phenomenon in comparison to the bar of Section 2. Concerning the building practice, an analysis of temperature fields is required according to completely different tasks. Regarding buildings, thermal bridges are analysed as well as the efficiency of heat exchangers, the thermal curing of bulky concrete devices, and the heat conduction in the case of fire etc. Foundation ground freezing is applied to stabilise the soil if the ground is not sufficiently strong to perform a tunnel excavation or similar actions, or if partly–saturated soils are present. Regarding the FEM heat conduction gives a very comprehensive introduction into the discretization of multi–dimensional field equations. As such, the tasks to be worked on concern the type of description of the phenomenology as well as its introduction into the FE–algorithm. For convenience, the terminology will firstly be introduced regarding one spatial dimension, then it will be extended to two spatial dimensions and finally transferred to the FE–context. The units of the quantities describing the heat conduction are [K] = [K elvin] for the temperature, [W ] = [W att] for the power and [J] = [J oule] for the energy.

4.1 Heat Conduction at One–Dimensional Description A one–dimensional domain is given, that is heated at a boundary by an external temperature field Tb or inside the domain by a heat source qv . While the heating takes place, a part of the heat is conducted into and transported through the domain, whereby the temperature T [K] within the domain increases.

4.1.1 Governing Equations for One Dimension The heat transport is designated as heat flux q [W/m2 ] or as q [J/s m2 ] employing different units, which can be exchanged by means of W = J/s. The greater the difference in temperature regarding neighboring points in the domain, the greater the heat flux will occur. Thus in each point of the domain, Fourier’s model of heat conduction applies q(x) = −λ · T,x ,

(4.1)

that can be designated as the first governing equation. The negative sign captures the heat flux from hot to cold. λ [W/m K] or as [J/s m K] is the material parameter describing the thermal conductivity, cf. Figure 4-1. © The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2024 D. Dinkler und U. Kowalsky, Introduction to Finite Element Methods, https://doi.org/10.1007/978-3-658-42742-9_4

64

4 Heat Conduction z

dx/2

T

y

dx/2

dy

x

dT q (x)

dz

x

x

Fig. 4-1 Temperature gradient and heat flux If the amount of heat is not totally transported through the domain, a part of it may be stored inside the material. The stored energy ψ [ J/m3 ] depends on the temperature as well as on the material density ρ [kg/m3 ], and on the heat storage capacity or heat capacity c [W s/kg K] or c [J/kg K] respectively. This relation leads to the second governing equation to describe the stored heat or the stored energy ψ(x, t) = ρ · c · T . (4.2) At this, c describes the amount of heat that is required to warm up 1 kg of the material by 1 K. The stored energy depends on the position x and the time t. q-dq/2

ψ , qv

q+dq/2

0

x

dx

Fig. 4-2 Energy balance of stored energy and heat flux The third governing equation describes the balance of energy, cf. Figure 4-2. If one part of the heat is transported and one part is stored, the sum of the parts changed in space and in time needs to be balanced. Considering that the heat is not only transported from the boundary into the domain but also develops as qv (x, t) in the domain itself – for example due to chemical reactions as a result of hydration as the concrete hardens – it applies ψ˙ + q,x = qv .

(4.3)

The balance of energy confirms nothing else but the fact that the sum of heat storage and heat flux might equal the heat supplied to the system. Introducing the first and the second governing equation into the balance of energy gives (ρ · c · T )˙ − (λ · T,x ),x = qv .

4.1 Heat Conduction at One–Dimensional Description

65

In the following, only the stationary situation will be investigated, so the derivatives with respect to the time disappear. Thus it applies −(λ · T,x ),x = qv .

(4.4)

4.1.2 Boundary Conditions The governing equations comprise two derivatives with respect to the coordinate x, thus two boundary conditions need to be formulated. These are the Dirichlet boundary condition describing the temperature Tb and the Neumann boundary condition describing the heat flux qb T − Tb = 0 ,

q − qb = 0 .

(4.5)

Here, Tb and qb are the prescribed values at the boundary. In the case of temperature and heat flux being combined at the boundary, the Cauchy boundary condition needs to be fulfilled q = −h · (T∞ − T )

(4.6)

with h [W/K m2 ] being the coefficient of heat transfer. This includes the influence of thermal convection at the boundary, when T∞ is the outside temperature. Here, T and q are not prescribed but the result of the complete heat conduction process. If radiation at the surface of a structure is taken into account, the boundary condition deals with T 4 and needs algorithms to solve the nonlinearity. Thus, this type of boundary condition is not discussed here.

4.1.3 Weak Form of the Energy Balance Equation Unlike describing elastic structures, here the Principle of virtual Displacements can not be applied to develop an alternative to the governing equations. In complete analogy to the bar, a Method of weighted Residuals (WR) may be defined, which deals with a weak formulation of the describing differential equation. The differential equation describing the heat conduction − q,x + qv = 0

(4.7)

applies for each differential element. If the equation is multiplied by a weighting function, which could be interpreted as virtual temperature, it follows to δT {−q,x + qv } = 0 .

(4.8)

66

4 Heat Conduction

After integration over the entire domain, the weak form of the energy balance equation follows to Z δT {−q,x + qv } dV = 0 , (4.9)

which can be designated as weighted residual δR. Respective consideration of the boundary conditions dealing with virtual heat fluxes and the surface S gives Z Z δR = δT {−q,x + qv } dV + {[ δq (T − Tb ) ] + [ δT (q − qb ) ]} dS = 0 . (4.10) Integration by parts of the first term of the integral yields Z Z δR = {δT,x q + δT qv } dV + {[ δq (T − Tb ) ] + [ −δT qb ]} dS = 0 .

(4.11)

Taking into account Fourier’s model of heat conduction and presuming that the temperature (T − Tb ) and the virtual temperature δT exactly fulfill the boundary conditions, it follows Z Z −δR = {δT,xλ T,x − δT qv } dV + δT qb dS = 0 . (4.12)

This equation fully complies with the Principle of virtual Displacement regarding the bar, thus the weak form may be interpreted as Principle of virtual Temperature. If a Cauchy boundary condition is to be taken into account, the weak form is to be extended accordingly: Z Z −δR = {δT,x λ T,x − δT qv } dV + δT [ −h(T∞ − T ) ] dS = 0 . (4.13) The discretization, which is now enabled by applying the Finite Element Method, happens in complete analogy to the bar. However, here the unit of δR is [W ] and converting by [W = N m/s] may result in [N m/s] only if the unit of δT is 1.

4.1.4 Example of use The temperature field regarding a wall may be evaluated as follows. Within the domain heat production is neglected by assuming qV = 0. Considering the heat transfer between 0 l x the surface of the wall and the ambience results in the Cauchy boundary conT∞ 0 T0 T1 T∞ 1 ditions, which are taken into account as x=0

→

q0 = −h (T0 − T∞0 ) ,

x=ℓ

→

qℓ = −h (T∞1 − T1 ) .

h

h

4.1 Heat Conduction at One–Dimensional Description

67

The course of the temperature is to be found concerning the domain. Employing a cross section of 1 m2 and absence of heat conduction in y– and z–direction, the weak formulation may be simplified to Z −δR = {δT,x λ T,x − δT qV } dx + δT [ −h(T∞ − T ) ]10 = 0 , whereby the matrix notation is basis for the FE–description Z Z − δRel = δuT · DT · E · D · u dx − δuT · p dx . Hereby, the heat conductivity is described with E = [ λ ], and the differentiation of the temperature field with D = [ −∂x ], and the heat production within the domain with p = [ qv ]. The Cauchy boundary conditions are taken into account at the system’s level later on. Proper shape functions have to satisfy the convergence criteria, which lead to linear shape functions to describe the temperature field inside the element. Scaling the general polynomial T (x) = a0 + a1 · x to the nodal temperatures TA and TB leads to T (x) = TA · ( 1 −

x x ) + TB · ( ) , ℓ ℓ

cf. Figure 4-3, what can be transferred to the matrix notation of the temperaB

A x l

TA ( 1 - x / l )

1 1

TB ( x / l )

Fig. 4-3 Shape functions for the description of the temperature field ture field as [ T (x) ] = [ ( 1 − u = Ω·v,

x x ) ( )] · ℓ ℓ



TA TB



,

68

4 Heat Conduction

and also of the virtual temperature field δu = Ω · δv . Now, the integration of the weighted element residual can be performed analogously to Section 2.3.2. Indroducing the shape functions into the weighted residual gives Z Z − δRel = δvT · { [ ΩT · DT ] · E · [ D · Ω ] dx · v − [ ΩT ] · p dx } , what can be simplified by means of the auxiliary matrix B= D·Ω

, BT = ( D · Ω )T = ΩT · DT

as − δRel = δvT · {

Z

Z BT · E · B dx ·v − ΩT · p dx } . | {z } | {z } K

f

Employing the linear shape functions the residual can be written with element matrix K, element heat production vector f, and element vector of nodal temperatures v as " # " # " # λ/ℓ −λ/ℓ TA qV ℓ/2 − δRel = [ δTA δTB ] · { · − }. −λ/ℓ λ/ℓ TB qV ℓ/2 If just one element 0 ≤ x ≤ ℓ is taken into account, the element matrix equals the system matrix. Considering qV = 0 and Cauchy boundary conditions gives " # " # λ/ℓ −λ/ℓ T0 − δRsystem = [ δT0 δT1 ] · { · −λ/ℓ λ/ℓ T1 + δT0 [ −h ( T∞0 − T0 ) ] + δT1 [ −h ( T∞1 − T1 ) ] = 0 . Introducing the boundary conditions into the matrix notation yields " # " # " # λ/ℓ + h −λ/ℓ T0 h · T∞0 − δRsystem = [ δT0 δT1 ] · { · − } = 0. −λ/ℓ λ/ℓ + h T1 h · T∞1

4.2 Heat Conduction Regarding Two Spatial Dimensions Solving the equations towards T0 and T1 gives the unknown nodal temperatures T0 =

1 hℓ {(1 + ) T∞0 + T∞1 } , 2 + h ℓ/λ λ

T1 =

1 hℓ {(1 + ) T∞1 + T∞0 } 2 + h ℓ/λ λ

T∞ 0

T0

T1

h

T∞ 1

h

T T∞ 0

and therefore the linear distribution of the temperature inside the wall – see the figure on the right – T (x) =

69

T0

T1

0 - 0+

T∞ 1

1 - 1+

x

1 hℓ x hℓ x {[1 + (1 − )] T∞0 + [1 + · ] T∞1 } . 2 + h ℓ/λ λ ℓ λ ℓ

4.2 Heat Conduction Regarding Two Spatial Dimensions Generally, heat conduction is a three–dimensional phenomenon that may be reduced to a two– or a one–dimensional problem under specific conditions. Here the two–dimensional heat conduction shall be described up to the discretization of the weak form applying the FEM. Regarding two-dimensional heat conduction, all governing equations as well as the weak form need to be described in the x–y–subspace. Figure 4-4 shows the sign rule concerning the heat fluxes qx and qy respectively. x

y dq qx - 2 x

dx

dq qy - 2 y

qv , ψ dq qy + 2 y

dq qx + 2 x dz

dy

Fig. 4-4 Sign rule External heat sources can be point– or line–shaped as well as surface–orientated or volumetric. Regarding practical purposes, these could be e.g. welding anodes, solar radiation or hydration heat when concrete hardens.

70

4 Heat Conduction

4.2.1 Governing Equations and the Weak Form The governing equations – energy balance E and Fourier’s model of heat conduction F – are given below and, for comparison, the energy balance equation is given applying the method of weighted residuals as well as the weak form, which implicitly comprises all governing equations. The temperature field T (x, y) and the heat fluxes qx (x, y) and qy (x, y) are the related variables. The governing equations as well as the integral equations, which are basis for the discretization, are summarized including the Cauchy boundary conditions. Here it is postulated that heat conduction does not take place in z–direction. Thus the description can be reduced to x–y–plane. a) governing equations without heat storage (ψ˙ = 0) E : − qx,x − qy,y + qV = 0 F :

qx + λ · T,x

=0

qy + λ · T,y

=0

b) weighted residual Z Z δR = δT ( −qx ,x − qy ,y + qV ) dA + δT [−h(T∞ − T )] ds = 0

c) weak form Z Z Z − δR = λ ( δT,x T,x + δT,y T,y ) dA − δT qV dA − δT [−h (T∞ − T )] ds .

4.2.2 Matrix Representation of the Weak Form The element matrix as well as the element load vector are derived regarding the weak form, applying the matrix representation according to Equation (2.34) Z Z ΩT · p dA } (4.14) − δR = δvT · { ΩT · DT · E · D · Ω dA · v − A A {z } | {z } | element matrix element load vector or regarding the abbreviated form: Z Z − δR = δvT · { BT · E · B dA · v − ΩT · p dA } . A

(4.15)

A

Due to their clear arrangement, the element matrices are not derived from the weak form, but, applying the matrix representation, are developed according to the bar. Therefore, the matrix symbols need to be linked to the context described by the governing equations.

4.2 Heat Conduction Regarding Two Spatial Dimensions

71

4.2.3 Heat Conduction Matrix E and Operator Matrix D Fourier’s model of heat conduction describes the relation between the heat flux and the derivatives of the temperature field. A value corresponding to the strains regarding elasticity is possible but uncommon. Thus the heat conduction is directly represented by q and T . Employing the governing equations defined in Section 4.2.1 it applies "

σ = E · D· u, # " # " #   qx +λ −∂x = · · T . qy +λ −∂y

(4.16)

The vector u now comprises the temperature field T , which is described by employing the shape functions and the nodal temperatures.

4.2.4 Linear Shape Functions Regarding the Temperature The choice of the shape functions regarding the temperature follows from the convergence criteria. The number of nodes employing nodal degrees of freedom with respect to the temperature and their placing inside the element complies with the element geometry, since all nodes need to be considered equally, see Figure 4-5. ly

lx

A

D

B x

y

C

Fig. 4-5 Geometry and coordinate system regarding the rectangular element The convergence criteria show that a linear symmetric approach regarding T (x, y) and δT (x, y) is sufficient. In the following this simplest possible approach is chosen. Regarding the mathematical formal derivation, the temperature field is approximated by symmetric generalized polynomials of the first order according to the x–y–coordinates employing the coefficients ai being initially unknown: T (x, y) = a0 + a1 x + a2 y + a3 xy . (4.17) The symmetry of the approach with respect to the coordinates may be controlled by applying Pascale’s triangle according to Figure 2-16.

72

4 Heat Conduction

It is possible to transform the general polynomial into the physically meaningful approach, however it is not shown here. A product approach employing linear, physically meaningful shape functions and the corresponding nodal degrees of freedom, well–known from the one–dimensonal heat conduction problem, achieves this purpose more directly. The bi–linear shape functions are illustrated in Figure 4-6, whereby the nodal temperatures TA , TB , TC , TD are the correlated physically meaningful unkowns. 1 A 1 B

1 D 1 C

Fig. 4-6 Linear shape functions regarding rectangular elements Therefore, the approach regarding the temperature fields T and δT is given with T (x, y) = φA · TA + φB · TB + φC · TC + φD · TD , δT (x, y) = φA · δTA + φB · δTB + φC · δTC + φD · δTD .

(4.18)

If the primary nodal unknowns of the element are arranged in the vector v by nodes   v T = TA TB TC TD , (4.19)

the shape functions are to be merged according to the matrix Ω by the sequence of the primary nodal unknowns regarding the temperature field T Ω=



φA

φB

φC

φD



.

(4.20)

Table 4.1 explicitly represents the shape functions and their derivatives with respect to the coordinates.

4.2 Heat Conduction Regarding Two Spatial Dimensions

73

Table 4.1 Bi–linear shape functions regarding rectangular elements node i

shape function φi

A

(1 −

B

x lx

derivative φi,x

y x ) · (1 − l ) lx y

· (1 −

C

x lx

D

(1 −

·

− l1 · (1 − x

y ) ly

1 lx

y ly

· (1 − 1 lx

y x )· l lx y

·

y ) ly

derivative φi,y (1 −

y ) ly

x lx

y ly

− l1 · x

y ly

x ) · (− l1 ) lx y

· (− l1 ) y

x lx

(1 −

·

1 ly

x ) · l1 lx y

Thus the approach regarding the temperature follows to u = Ω·v,

(4.21)

and analogously to the virtual temperature δu = Ω · δv .

(4.22)

4.2.5 Differentiation of the Temperature Field The auxiliary matrix B= D·Ω

(4.23)

incorporates the differentiated shape functions according to the differentiation rule D of Equation (4.16) and should be introduced into the weighted residual. For the chosen bi–linear shape functions matrix B reads as follows " # −φA,x −φB,x −φC,x −φD,x B= . (4.24) −φA,y −φB,y −φC,y −φD,y The derivatives of the shape functions applied at B are given in Table 4.1.

4.2.6 Element Matrix Applying the matrices B and E the element matrix may be computed according to Equation (4.15) Z Z ℓy

K=

0

ℓx

0

BT · E · B dx dy .

(4.25)

The integrand comprises a quadratic form resulting in a 4 × 4–matrix. The multiplication scheme is given in Figure 4-7. The columns of the integrand are

74

4 Heat Conduction

assigned to the real nodal temperatures and the rows to the virtual nodal temperatures respectively. The symmetry of the integrand’s matrix entries follows from its quadratic form and the choice of identical shape functions for the real and virtual temperatures. Therefore the element matrix is symmetric, too.

E

*

O

O

*

BT

B

BT E B

Fig. 4-7 Multiplication scheme concerning the element matrix To get a clear view, the integrand is multiplied beforehand employing the following scheme and is integrated afterwards: BT · E · B = λ · I .

(4.26)

The integration matrix I is given below whereupon the differentiated shape functions are given in Table 4.1. The products applied regarding the matrix I are constant according to the respective direction, if both shape functions are partially differentiated with respect to this direction, and quadratic according to the other direction since two linear functions are multiplied by each other.   φA,x φA,x φA,x φB,x φA,x φC,x φA,x φD,x  + φA,y φA,y + φA,y φB,y + φA,y φC,y + φA,y φD,y       φ  φ φ φ φ φ φ φ B,x A,x B,x B,x B,x C,x B,x D,x   +φ  B,y φA,y + φB,y φB,y + φB,y φC,y + φB,y φD,y     I=   φC,x φA,x φC,x φB,x φC,x φC,x φC,x φD,x     + φC,y φA,y + φC,y φB,y + φC,y φC,y + φC,y φD,y       φD,x φA,x φD,x φB,x φD,x φC,x φD,x φD,x  + φD,y φA,y + φD,y φB,y + φD,y φC,y + φD,y φD,y

According to the Equations (4.25) and (4.26), the element matrix K follows after the integration of I and is given by the representation below. As already explained regarding bars, the sums of single columns as well as rows of the

4.2 Heat Conduction Regarding Two Spatial Dimensions

75

element matrix disappear, since a constant temperature field without heat flux needs to be represented.  ℓ  ℓ ℓ ℓy ℓx 2 ℓy + 2 ℓℓx −2 ℓy + ℓℓx − ℓy − ℓℓx − 2 ℓx ℓy  y x y x y  x   ℓy ℓy ℓy ℓy   ℓx ℓx ℓx −2 ℓ + ℓ 2 ℓ + 2 ℓ −2 ℓ − ℓ − ℓℓx   ℓ x y x y x y x y  λ  K= ·  ℓy ℓy ℓy 6  ℓy ℓx ℓx ℓx  − 2 2 + 2 −2 +  − ℓ − ℓℓx  ℓx ℓy ℓx ℓy ℓx ℓy  x y    ℓy ℓy ℓy ℓy ℓx ℓx ℓx ℓx − 2 − − −2 + 2 + 2 ℓ ℓ ℓ ℓ ℓ ℓ ℓ ℓ x

y

x

y

x

y

x

y

4.2.7 Element Vector of Thermal Action Due to Equation 4.15 the vector of thermal action follows Z f= ΩT · p dA A

and can be evaluated, if the spatial course of the external action is given. Here, a constantly distributed heat source qV inside the element is assumed. Applying   p = qV = const. p can be excluded from the integral Z f= ΩT dA · p .

(4.27)

A

Employing the matrix of shape functions  φA φ  B ΩT =   φC φD

    

and the shape functions according to Section 4.2.4 the integration gives the value ℓx ℓy /4 regarding all shape functions:     fA 1  f  ℓ ℓ  1     B   x y  f = · =  · qV .  fC  4  1  fD

1

A physically meaningful interpretation of the element vector of thermal action is possible if the integral of the heat source qV is interpreted as four nodal sources of equal size.

76

4 Heat Conduction

4.2.8 Subsequent Flux Analysis The temperature field is known from the computation of the nodal temperatures of the entire system, as shown in Section 3. If the heat fluxes are of interest, a subsequent flux analysis additionally needs to be performed. Therefore, Fourier’s model of heat conduction qx + λ · T,x = 0 , qy + λ · T,y = 0 is evaluated element by element, applying the temperature field now known. Since the nodal temperatures are assigned to the shape functions in the related element, the respective temperature field may be represented as a course of function. However, the courses of function of the heat fluxes are not usually of interest, but discrete values at characteristic points are, where element nodes as well as other positions may be preferred. This can be achieved by introducing the respective coordinates into the course of function. The matrix, representing the heat fluxes at the chosen nodes depending on the nodal temperatures, is ˜ as known from one–dimensional structures. The course designated as matrix S, of the heat flux inside the element is formally given here by σ, which requires the matrix representation defined in Section 2.3.2 to: σ = E · ǫ,

σ = E · D· u, σ = E·D·Ω·v = E·B·v = S·v.

(4.28)

If the heat conduction matrix E is constant, the course of function of the heat flux σ already follows with the course of function regarding B. The special contents of the matrices required for the evaluation are known from the derivation of the element matrices. Here the stress matrix S describes the heat fluxes normalized to the nodal temperatures. Employing the shape fuctions given in Section 4.2.4 yields " # φA,x φB,x φC,x φD,x S = E · B = −λ · , φA,y φB,y φC,y φD,y whereat the derivatives of the shape functions are included in Table 4.1. The stress matrix S reveals the course of the heat fluxes inside the element, constantly related to the respective direction and linearly perpendicular to it.

4.3 Example of Use

77

The nodal values σ ˜ of the heat fluxes may be evaluated after the nodal coor˜ follows dinates have been introduced into B. Thus the discrete stress matrix S ˜ with the discrete auxiliary matrix B: ˜ ·v, σ ˜ =E·B ˜ σ ˜ = S·v.

(4.29)

In detail the heat fluxes according to the element nodes may be evaluated to

    

σ(x = 0 , y = 0 ) σ(x = ℓx , y = 0 ) σ(x = ℓx , y = ℓy ) σ(x = 0 , y = ℓy )

˜ ·v, σ ˜=S   S(x = 0 , y = 0 )      S(x = ℓx , y = 0 ) =   S(x = ℓx , y = ℓy ) S(x = 0 , y = ℓy )

and with the chosen bi–linear shape functions to  1 1  "  − ℓx ℓx #   qx  −1    ℓy    q   y  A   " 1  −1 #    ℓx ℓx   q  x      − ℓ1  qy   y  B  " #    = −λ ·     qx        q y   − ℓ1  C   " y #       qx      qy D  −1 ℓy



  ·v,   

1 ℓy

1 ℓy 1 ℓx 1 ℓy

− ℓ1

1 ℓx

− ℓ1

x

x

1 ℓy

                ·            

TA TB TC TD



  .  

4.3 Example of Use In the following, the heat conduction inside a wall is simulated employing rectangular elements as explained above. Figure 4-8 shows the geometry of a horizontal cut in the region of the entrance of a building, whereat the door is assumed to be wooden. Symmetry conditions are applied at the left and at the right edge of the structure and are realized by means of qx = 0.

78

4 Heat Conduction

A constant temperature of 220 C inside the building and of 60 C outside the building is assumed as boundary conditions for the temperature field inside the wall. Cauchy boundary conditions are not taken into account. The different materials are indicated by different shadings. λ1 = 0. 08 W/mK is valid for brickwork and λ2 = 0. 12 W/mK is valid for wood. The discretization is performed by means of rectangular elements with bi–linear shape functions for the temperature. The dimensions [ m ] and FE–discretization chosen for the investigation are represented in Figure 4-8. 0.50

1.25 0.125

22

0.50 A

λ 2 = 0.12

o

6

B

0.125

o

λ 1 = 0.08

0.50

0.25 y x

0.75

0.25

0.75

Fig. 4-8 2–dimensional heat conduction – initial configuration The heat conduction process results in a stationary temperature field shown in Figure 4-9. The temperature field is linearly distributed perpendicular to the wall in regions, where an almost undisturbed heat conduction exists – cross sections A and B. Near the outer corners the boundary conditions dominate the temperature field inside the wall, what leads to thermal bridges. The heat fluxes qx [ W/m2 ] and qy [ W/m2 ] are visualized in Figures 4-10 and 4-11. Due to the bi–linear approximation of the temperature field, the heat fluxes are constant inside the elements in the direction of the heat flux, and thus cause the gaps at the intersection of the elements. It is obvious, that the heat flux is larger in the region of the larger heat conduction coefficient as well as in the direction of the shorter transport distance. Disturbancies occur at the inner and outer corners of the wall, where the heat flux is following the direction of the lowest resistance. It has to be mentioned, that singularities occur at the inner corners of the wall, which are indicated by high gradients of the heat fluxes.

4.3 Example of Use

79

22. 0 o

C

6. 0 Fig. 4-9 2–dimensional heat conduction – temperature T (x, y)

6. 3 W/m2 0. 0 Fig. 4-10 2–dimensional heat conduction – heat flux qx

0. 0 W/m2 −15. 4 Fig. 4-11 2–dimensional heat conduction – heat flux qy

5 Membrane Structures

Membrane structures carry their loading along the main plane structure, so the equilibrium conditions also may be addressed at this structural level. Thus, instead of 3–dimensional coordinates and volume integrals, all governing equations as well as the work equations may be represented in the 2–dimensional x–y–coordinate system, what may be equivalent to an integration with respect to the third coordinate.

5.1 Rectangular Elements Regarding Plane Stress Situation Regarding plane stress situations, the stresses rectangular to the main plane structure are identical to zero, so the respective strains may be determined in a subsequent analysis if required. Figure 5-1 shows the sign rule concerning the displacements u(x, y) and v(x, y) as well as the stresses σxx , σyy and σxy with respect to the x–y–coordinate system. dx x,u px

y,v

dy py

dy

σxx

px py

dx

σxy σyx σyy

Fig. 5-1 Sign rule

5.1.1 Governing Equations and Work Equations In the following, the governing equations – kinematics K, equilibrium conditions E, material equations M – and for comparison the weak form of equilibrium applying the PvD and moreover the work equations are presented wherein the last ones implicitly comprise all governing equations. External actions are the distributed loadings px , py [ N/m3 ]. Line–shaped actions as well as point–shaped loads are possible, too, but will not be considered here. The variables employed by the governing equations are the displacement © The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2024 D. Dinkler und U. Kowalsky, Introduction to Finite Element Methods, https://doi.org/10.1007/978-3-658-42742-9_5

82

5 Membrane Structures

fields u(x, y), v(x, y) [ m ], the stresses σxx (x, y), σyy (x, y), σxy (x, y), [ N/m2 ] as well as Green’s strains εxx (x, y), εyy (x, y), εxy (x, y) [ 1 ]. a) Governing equations K:

E:

M:

εxx − u,x εyy − v,y εxy − 21 ( u,y + v,x ) σxx ,x + σyx ,y + px σyy ,y + σxy ,x + py σxy − σyx σxx − σyy − σxy −

E · (εxx + νεyy ) 1−ν 2 E · (εyy + νεxx ) 1−ν 2 E · εxy 1+ν

=0 =0 =0 =0 =0 =0 =0 =0 =0

b) Principle of virtual Displacements Z δW = (−δεxx σxx − δεyy σyy − 2 δεxy σxy + δu px + δv py ) dV = 0

c) Work equations: Instead of the integration with respect to the volume the integration with respect to the z–coordinate usually is conducted a priori. Assuming the thickness t to 1 m and employing the loadings px , py [ N/m2 ] this leads to: Z Z E 1 −δWu = {δu, (u, + ν v, ) + (1 − ν) δu, (u, + v, )} dA − δu px dA , x x y y y x 1 − ν2 2 Z Z E 1 −δWv = {δv, (v, + ν u, ) + (1 − ν) δv, (u, + v, )} dA − δv py dA . y y x x y x 1 − ν2 2

5.1.2 Matrix Notation of the Work Equations To get a clear arrangement of the element matrices the matrix notation developed regarding the bar has to be linked with respect to the relations formulated in the governing equations. The derivation of the element stiffness matrix and of the element load vector fits with the work equation given in the matrix notation according to Equation (2.34): Z Z −δW = δvT · { ΩT · DT · E · D · Ω dA · v − ΩT · p dA } . (5.1) A A | {z } | {z } element stiffness matrix element load vector

5.1 Rectangular Elements Regarding Plane Stress Situation Or rather, in shortened representation, it follows Z Z −δW = δvT · { BT · E · B dA · v − ΩT · p dA } . A

83

(5.2)

A

5.1.3 Elasticity Matrix E and Operator Matrix D Hooke‘s model describes a linear–elastic material behavior by the elasticity matrix E. Regarding a plane stress situation, the material equations follow to:



σ =E·ε ,  

1 σxx E    ν σ =  yy   1 − ν2 σxy

ν 1

 

 εxx     ·  εyy  . 1 2εxy 2 (1 − ν)

(5.3)

Applying the governing equations from Section 5.1.1 kinematics may be represented by matrix notation, which is applied to virtual strains analogously:



εxx   εyy 2εxy

ε = D·u ,    " # ∂x u    ∂y  · . = v ∂y ∂x

(5.4)

The displacement vector u includes the displacements u(x, y) and v(x, y), which can be described by means of shape functions and nodal degrees of freedom.

5.1.4 Bi–linear Shape Functions Regarding Displacements Usually it is non–trivial to choose the shape functions concerning the displacements. The order of the approach follows from the convergence criteria, while the nodal degrees of freedom and their positions within the element follow from the element geometry, since all nodes must be handled equally, see Figure 5-2. ly

lx

A

D

B x, u

y, v

C

Fig. 5-2 Geometry and coordinate system regarding the rectangular element

84

5 Membrane Structures

The convergence criteria yield a symmetrically linear approach. These most simple shape functions are chosen below. At the mathematically formal derivation, the displacement fields are given as symmetric polynomials of the first order with respect to the x–y–coordinates employing the so far unknown coefficients ai and bi : u(x, y) = a0 + a1 · x + a2 · y + a3 · xy , v(x, y) = b0 + b1 · x + b2 · y + b3 · xy .

(5.5)

The symmetry of shape functions may be checked with Pascale’s triangle as shown in Figure 2-16. It should be noted that the general polynomials may be transformed into the physically meaningful approach according to Section 2.4.3. At this point, a product approach, derived from the linear shape functions well– known from the bar and being physically meaningful, achieves the goal faster. Thus, the displacements u in x–direction and v in y–direction are approximated by the unknown nodal displacements at nodes A to D and the bi–linear shape functions φA (x, y), . . . which are correlated to the unknowns. Therefore, the physically meaningful approach regarding the displacement fields u(x, y) and v(x, y) yields: u(x, y) = φA · uA + φB · uB + φC · uC + φD · uD , v(x, y) = φA · vA + φB · vB + φC · vC + φD · vD .

(5.6)

If the nodal degrees of freedom of the respective element are arranged in the vector v by nodes,   (5.7) vT = uA vA uB vB uC vC uD vD ,

the shape functions need to be arranged in the matrix Ω in the same order as the nodal degrees of freedom in both displacement fields u " # φA φB φC φD Ω= . (5.8) φA φB φC φD Hence the approach regarding the displacements yields u = Ω·v.

(5.9)

Analogously to the description of the real displacements, the same approach is also employed regarding the virtual displacements δu = Ω · δv .

(5.10)

5.1 Rectangular Elements Regarding Plane Stress Situation

85

The bi–linear shape functions of the displacement fields are depicted in Figure 5-3. All shape functions φA , . . . are normalized to 1 at the nodes, linear in the x– and y–direction and quadratic along the diagonal. Table 5.1 presents the 1 A 1 B

1 D 1 C

Fig. 5-3 Bi–linear shape functions regarding displacements at rectangular elements shape functions and their derivatives with respect to the coordinates. Table 5.1 Bi–linear shape functions regarding rectangular elements node i

shape function φi

A

(1 −

B

x lx

derivative φi,x

y x ) · (1 − l ) lx y

· (1 −

C

x lx

D

(1 −

·

y ) ly

y ly

y x )· l lx y

− l1 · (1 − x

1 lx

· (1 − 1 lx

·

y ) ly

y ly

− l1 · x

y ) ly

y ly

derivative φi,y (1 − x lx

x ) · (− l1 ) lx y

· (− l1 ) y

x lx

(1 −

·

1 ly

x ) · l1 lx y

5.1.5 Differentiation of Displacement Fields The work equation employs the auxiliary matrix B defined according to the rule of differentiation D, see Equation (5.4), B = D· Ω,

(5.11)

86

5 Membrane Structures

that describes the strain field. Regarding the complete element, it follows   φA,x φB,x φC,x φD,x   φA,y φB,y φC,y φD,y  . B= (5.12) φA,y φA,x φB,y φB,x φC,y φC,x φD,y φD,x The differentiated shape functions are listed in Table 5.1.

5.1.6 Element Stiffness Matrix Applying the matrices B and E, the element stiffness matrix can be computed according to Equation (5.2) Z ℓy Z ℓx K= BT · E · B dx dy . (5.13) 0

0

The integrand employs a quadratic form that yields an 8 × 8-matrix. The multiplication scheme is represented in Figure 5-4. While the columns of the inteE

* * * *

0 0

B 0 0

*

BT

B TE

B TE B

Fig. 5-4 Multiplication–scheme regarding the stiffness matrix grand are linked to the real nodal degrees of freedom, the rows are linked to the respective virtual nodal displacements. The symmetry of the integrand as well as of the element stiffness matrix follows from the quadratic form of the integrand and from the identical choice of shape functions regarding real and virtual displacements.

5.1 Rectangular Elements Regarding Plane Stress Situation

87

For the sake of a clear view, the integrand is split into three parts according to the terms occurring in the elasticity matrix BT · E · B =

E νE E · I1 + · I2 + · I3 2 2 1−ν 1−ν 2(1 + ν)

(5.14)

with φA,x φA,x  φA,y φA,y   φB,x φA,x   φB,y φA,y I1 =   φC,x φA,x   φC,y φA,y   φD,x φA,x φD,y φA,y 

φA,x φA,y  φA,y φA,x   φB,x φA,y   φB,y φA,x  I2 =  φC,x φA,y   φC,y φA,x   φD,x φA,y φD,y φA,x 



     I3 =      

φA,y φA,y φA,x φA,y φB,y φA,y φB,x φA,y φC,y φA,y φC,x φA,y φD,y φA,y φD,x φA,y

φA,y φA,x φA,x φA,x φB,y φA,x φB,x φA,x φC,y φA,x φC,x φA,x φD,y φA,x φD,x φA,x

φA,x φB,x

φA,x φC,x φA,y φB,y

φB,x φB,x

φB,x φC,x φB,y φB,y

φC,x φB,x

φB,x φD,x φB,y φC,y

φC,x φC,x φC,y φB,y

φD,x φB,x

φC,x φD,x φC,y φC,y

φD,x φC,x φD,y φB,y

φD,x φD,x φD,y φC,y

φA,x φB,y φA,y φB,x φB,x φB,y

φC,y φC,x

 φA,y φD,x φA,x φD,x  φB,y φD,x  φB,x φD,x   φC,y φD,x φC,x φD,x  φD,y φD,x

φC,y φD,x φD,x φC,y

φD,y φC,x

φA,y φB,x φA,x φB,x φB,y φB,x φB,x φB,x φC,y φB,x φC,x φB,x φD,y φB,x φD,x φB,x

φA,y φD,y φA,x φD,y φB,y φD,y φB,x φD,y φC,y φD,y φC,x φD,y φD,y φD,y φD,x φD,y

φB,y φD,x φC,x φC,y

φD,x φB,y φD,y φB,x

φD,y φD,x

φB,x φC,y

φC,x φB,y φC,y φB,x

 φA,x φD,y   φB,x φD,y    φC,x φD,y    φD,x φD,y

φA,y φD,x

φB,y φC,x

φA,y φC,y φA,x φC,y φB,y φC,y φB,x φC,y φC,y φC,y φC,x φC,y φD,y φC,y φD,x φC,y

φA,y φC,x φA,x φC,x φB,y φC,x φB,x φC,x φC,y φC,x φC,x φC,x φD,y φC,x φD,x φC,x



φA,y φD,y    φB,y φD,y    φC,y φD,y  

φD,y φD,y

φA,x φC,y φA,y φC,x

φB,y φB,x

φA,y φB,y φA,x φB,y φB,y φB,y φB,x φB,y φC,y φB,y φC,x φB,y φD,y φB,y φD,x φB,y

φA,x φD,x φA,y φC,y

φD,x φD,x

The differentiated shape functions are given in Table 5.1. Because of the bi– linear shape functions regarding the displacements, quadratic functions at maximum appear in the integrand. According to Equations (5.13) and (5.14), the element stiffness matrix follows from the sum of the three sub–matrices after integrating: K = K1 + K2 + K3 . In the following the matrices K1 , K2 , K3 are given by numbers.

88

5 Membrane Structures 

ℓ

ℓ

2 ℓy

ℓ

−2 ℓy

ℓy ℓx

− ℓy

x x x    ℓx ℓx 2ℓ − ℓℓx −2 ℓℓx  ℓy y y y   ℓy ℓy ℓy ℓy  2ℓ −ℓ  −2 ℓx ℓx x x    ℓx ℓx ℓx 2ℓ −2 ℓ − ℓℓx  ℓy y y y E  K1 =  ℓy ℓy ℓy 6(1 − ν 2 )  ℓy 2ℓ −2 ℓ  − ℓx ℓx x x    ℓx − ℓℓx −2 ℓℓx 2 ℓℓx  ℓy y y y    ℓy ℓy ℓy ℓy  ℓ −ℓ −2 ℓ 2ℓ x x x  x  ℓx ℓx ℓx −2 ℓ −ℓ 2 ℓℓx ℓ y



1

 1     1 νE  K2 = 2 4(1 − ν )    −1   4 ℓℓx

y    3    ℓx  2  ℓy    −3  E  K3 = 24(1 + ν)   −2 ℓx  ℓy    −3     −4 ℓx  ℓy  3

y

1

−1 −1

−1 −1 1

1

−1

−1

3

1 1 −1

1

2 ℓℓx y

1

−3 −2 ℓℓx y

−1 −1

−3 −4 ℓℓx y

                          

      1     −1  1

1

1

−1

1

−1

−1

y

−1

1

−1



y



3



  ℓ 2 ℓy  x    ℓx ℓx ℓx 3 4ℓ −3 −4 ℓ −3 −2 ℓ 3   y y y  ℓy ℓy ℓy ℓy  −4 ℓ −3 4 ℓ 3 2ℓ 3 −2 ℓ  x x x x    −3 −4 ℓℓx 3 4 ℓℓx 3 2 ℓℓx −3   y y y  ℓy ℓy ℓy ℓy  −2 ℓ −3 2 ℓ 3 4ℓ 3 −4 ℓ  x x x x    ℓx ℓx ℓx −3 −2 ℓ 3 2ℓ 3 4ℓ −3   y y y  ℓ ℓ ℓ ℓ 2 ℓy 3 −2 ℓy −3 −4 ℓy −3 4 ℓy ℓ 4 ℓy x

x

3

ℓ −4 ℓy x

x

−3

ℓ −2 ℓy x

x

−3

x

5.1 Rectangular Elements Regarding Plane Stress Situation

89

5.1.7 Element Load Vector The computation of the integral f=

Z

A

ΩT · p dA

is possible, if the distribution of the external action is known. At first, only a special case is discussed regarding an external action constantly distributed inside the related element. With " # px p= = const. py p may be extracted from the integral: Z f = ΩT dA · p .

(5.15)

A

The matrix of shape functions 

φA

   φB   ΩT =   φC     φD



φA     φB     φC    φD

and the shape functions φi are given according to Section 5.1.4. The integration provides the value ℓx ℓy /4 concerning all shape functions. Thereby the element load vector follows   1  1     fA 1  " #   f  ℓ ℓ   px 1  B  x y  · f = · . =  4   fC  py 1   1   fD 1  1 An interpretation of the load vector is possible by interpreting the integral of the constantly distributed external action px , or rather py , as four equally sized nodal actions.

90

5 Membrane Structures

A variable external action can be taken into account if it is linearly approximated, similar to the description of the displacements. Thus, the external action may be approached by employing the matrix of shape functions already known as well as the nodal ordinates of the external action which yields p =Ω·p ˜ with Ω and

:

p ˜ :

(5.16)

matrix of shape functions external action ordinates at the nodes.

A representation in detail gives: px = φA · pxA + φB · pxB + φC · pxC + φD · pxD , py = φA · pyA + φB · pyB + φC · pyC + φD · pyD .

Thus it follows f =

Z

ΩT · p dA =

Z

ΩT · Ω · p ˜ dA =

Z

ΩT · Ω dA · p ˜.

(5.17)

The procedure of computing the integral is similar to the procedure concerning the element stiffness matrix and yields a symmetric 8 × 8–matrix, see Equation (5.18)     4 2 1 2 p    xA   4 2 1 2  pyA           2   pxB  4 2 1         2 4 2 1   pyB  ℓx · ℓy     f= (5.18) . · pxC  36  2 4 2  1         pyC   1 2 4 2          p   2  xD 1 2 4     pyD 2 1 2 4

5.1.8 Subsequent Stress Analysis After the calculation of the nodal displacements of the entire system, presented in Section 3, the deformation state of the structure is well–known and the problem is all but solved. However, the state of stress must be known for dimensioning of a structure. At derivation of the work equation applying the PvD, the stress variables are described by applying the material equations to the respective strains. In turn,

5.1 Rectangular Elements Regarding Plane Stress Situation

91

these strains are described by kinematics relating to the respective derivatives of the displacements. Correspondingly, in the subsequent stress analysis, the stresses may now be evaluated vice versa from the related nodal displacements. As the nodal displacements are connected to the related shape functions in each element, the displacements and thus also the stresses may not only be indicated as nodal values, but also the courses of the function may be evaluated in the whole element. For dimensioning, however, the courses of the stresses are not essentially necessary in each element, as discrete values at exclusive positions are sufficient. This is achieved by introducing the coordinates of the selected positions into the respective course of the function. The matrix, which evaluates the stress variables at the selected coordinates dependent on the respective ˜ sometimes simply S. nodal displacements, is called stress matrix S, Applying the matrix notation defined in Section 2.3.2, the computation of the stresses σ regarding the respective element follows to σ = E · ǫ, σ = E · D· u,

(5.19)

σ = E·D·Ω·v = E·B·v = S·v.

If the elasticity matrix E is constant, the courses of the stresses σ are already determined with the course of the function of B. The particular contents of the matrices required for the computation of stresses are known from the derivation of the element matrices. By means of the elasticity matrix E and the matrix of the differentiated shape functions B, the stress matrix S can be determined. Employing the shape functions, cf. Section 5.1.4, the following applies:   φ νφA,y φB,x νφB,y φC,x νφC,y φD,x νφD,y E  A,x νφA,x φA,y νφB,x φB,y νφC,x φC,y νφD,x φD,y  . S =E·B= 1 − ν2 eφA,y eφA,x eφB,y eφB,x eφC,y eφC,x eφD,y eφD,x

As such, the abbreviation e = (1 − ν)/2 is used here. Table 5.1 comprises the derivatives of the shape functions. It can be seen from the stress matrix S that the course of the function inside the element is constant in the respective direction of differentiation, and is linear perpendicular to it. Assuming that Poisson’s ratio is not zero, additional linearly–constant terms have to be superposed. Therefore, the course of the function is generally bi–linear, unlike as at the bar.

92

5 Membrane Structures

The nodal values of the stresses σ ˜ are obtained by inserting the coordinates of ˜ the stress matrix S ˜ yields: the nodes into B. Applying the discrete matrix B, ˜ ·v, σ ˜ =E·B

    

˜ ·v, σ ˜=S   S(x = 0 , y = 0 ) σ(x = 0 , y = 0 )   S(x = ℓx , y = 0 ) σ(x = ℓx , y = 0 )   =  S(x = ℓx , y = ℓy ) σ(x = ℓx , y = ℓy )   S(x = 0 , y = ℓy ) σ(x = 0 , y = ℓy )

                         

σxx σyy σxy σxx σyy σxy σxx σyy σxy σxx σyy σxy



−ℓ1 − ℓν x

y

 ν −    ℓx  e  −    ℓy      1  − A    ℓx   ν  −    ℓx        E  B   = 1 − ν2                C                D   −ℓe

− ℓ1

y

y

1 ℓx ν ℓx

− ℓe

x

1 ℓx ν ℓx − ℓe y

y

e ℓx

y

− ℓe

y

− ℓ1 y



ν ℓy 1 ℓy

y

− ℓ1

ν ℓy 1 ℓy e ℓy

− ℓν

− ℓν

− ℓν y − ℓ1 y

  ·v,  

e ℓx

x

− ℓe



e ℓy 1 ν ℓx ℓy ν 1 ℓx ℓy e e ℓy ℓx

− ℓ1

1 ℓx ν ℓx

− ℓ1 x − ℓν x e ℓy

e ℓx

x

− ℓν x

− ℓe

x

ν ℓy 1 ℓy − ℓe x

                  ·                  

  u  v A    u  v B   .  u  v C    u  v D

5.1.9 Example of Use The cantilever according to Figure 5-5 serves as an example of how to evaluate the quality of the element. It is loaded by a constantly distributed external action in the y–direction at the free right–hand side boundary. Assuming Poisson’s ratio is zero, the load–bearing behavior is equivalent to the behavior of a cantilever arm loaded at the free end. Local disturbances may occur at the position of the load application, if it does not correspond to the shear stress distribution regarding an undisturbed area. Due to kinematic constraints, disturbances in

5.1 Rectangular Elements Regarding Plane Stress Situation

93

shear stress behavior are also present at the clamping cross section. x = l/2

l 3

x, u

B

PY

y, v l l = 15 m, t = 1 m, E = 100 000 N/m2, ν = 0, PY = 2 N/m2

Fig. 5-5 Cantilever – geometry and loading In the following, cf. Figure 5-6, the stresses σxx and σxy are depicted regarding a mesh employing 24 × 8 elements without taking into account the antisymmetry. The jumps at the element intersections are clearly visible, following from the subsequent stress analysis executed element by element. σxx 180. 0

−180. 0 σxy 24. 0

−7. 8 Fig. 5-6 Cantilever – Distribution of stresses σxx and σxy The approximation of the shear stresses that shows a change of sign at the element intersections in the x–direction yet is particularly poor. An improvement of these results employing the same number of elements as well as keeping the order of shape functions unchanged may only be possible when applying a special procedure e. g. according to Section 5.2.

94

5 Membrane Structures

The jumps in the distribution of stresses are depicted for the undisturbed area in the center of the structure at x = 7. 50 m regarding meshes differently refined, see Figure 5-7, where only the upper half of the cantilever is represented. The convergence of the normal stresses σxx is quite good, whereat the discretization error can be evaluated directly from the stress jumps. The shear stresses converge very slowly and show particularly large jumps with changing signs in some parts, so the results appear altogether questionable. 0.0

σxx

50.0 6 384

0.0 384

6 24

σxy

20.0 24

6

6

neutral axis x = 7.5 ε x = 7.5 + ε

ε

0

Fig. 5-7 Cantilever – distribution of stresses at position x = 7. 50 m In Figure 5-8 the convergence behavior of the vertical displacement at the free boundary and of the stresses regarding position x = 7. 50 m is depicted dependent on mesh refinement. The values of displacements and of stresses are normalized with respect to the values regarding 384 elements, which is here assumed to be the reference values. The discretization employs n = 3 elements for the entire cantilever and n > 3 elements for the upper half of the cantilever. The relatively good convergence of the displacements is characteristic applying the FE–formulation employing displacements as primary variables. The reason of this phenomenon is the bi–linear shape function for the displacements, which describes the smooth deformation of the cantilever quite well, compare the results of the bar in Section 1.3. To represent the convergence behavior of the stresses, the averaged values of the stresses are shown at the element intersections. Although the averaged values are taken into account, the relatively poor convergence of the shear stresses is obvious. The shape functions do not only approximate the parabolic shear stress distribution in a poor way regarding the cross section of the structure, but also result in oscillations inside an element.

5.1 Rectangular Elements Regarding Plane Stress Situation

σxx σe

95

x = 7,5 m

1,0 0,5

3

σxy σe

6

24

96

384

no. of elements

24

96

384

no. of elements

24

96

384

no. of elements

x = 7,5 m

1,0 0,5

3

v ve

6

x = 15,0 m

1,0 0,5

3

6

Fig. 5-8 Cantilever – convergence behavior of stresses and displacements Overall, the distribution of stresses regarding the cantilever may only be poorly approached when employing a rectangular plane stress element with bi–linear shape functions, even if a relatively fine discretization is applied. Reason for this is the poor approximation of the shear stiffness inside the element, which is clearly visible in this example. An improvement of the element behavior is explained in Section 5.2.

96

5 Membrane Structures

5.2 Rectangular Element Comprising Modified Shear Strains The results of the rectangular element employing a bi–linear approach are not that sufficient, although the displacements of the cantilever slab may be approximated quite well, see Figure 5-5. The reason for the relatively bad courses of the stresses is the poor approximation of the shear stiffness in the element. This leads to the large oscillations in the course of the shear stresses beyond the edges of the elements, which can be seen in the example. If non–physical oscillations occur in the distribution of stresses or of displacements, this indicates numerical effects which may be motivated by the choice of shape functions. Here, the shape functions are already determined by the convergence criteria, so an improvement of the results is possible by increasing the order of the shape functions, cf. the results in Section 9.4 employing quadratic shape functions. When employing linear shape functions, an improvement is possible only if the characteristics of the stiffness matrix are analysed in more detail. Compressive as well as tensile stresses are well described by the element already, if the lower order of approximation compared to that regarding the displacements is ignored. In contrast, the shear stresses are very poorly approximated - depending on the displacement field. In Figure 5-9 the case of pure shear deformation is shown, whereby the shear strains and thus the shear stresses are constant inside the element and therefore may be represented exactly by adequately employing linear shape functions. εyx A

B x

εxy

εxy

uA = uB = −uC = −uD ,

y D

C

εyx

vA = −vB = −vC = vD .

Fig. 5-9 Pure shear deformation Due to the definition of shear strain, 2 εxy = u,y + v,x , multiplicative shape functions are sufficient for u(x, y), if they are constant in the x–direction, and for v(x, y), if they are constant in the y–direction, in order to correctly describe the shear strains inside the element.

5.2 Rectangular Element Comprising Modified Shear Strains

97

The second case, as given in Figure 5-10, describes a pure bending deformation of the element whereby the strains in the x– as well as in the y–direction are exactly represented, but additional shear deformations γAD and γBC occur, showing changing signs and being non-physical. They arise because of the linear shape functions that are employed, which cannot describe the element curvature at pure bending. The drawback results in activation of shear stiffness in the case of pure bending and therefore the respective bending deformation converges more slowly against the exact solution. bending described by linear functions

pure bending A

B

A

B

x

x

εxy

y

D

C

D ∆u

uA = −uB = uC = −uD

and

εxy

y C ∆u

vA = vB = vC = vD = 0 .

Fig. 5-10 Deficient description of bending deformation

5.2.1 Selectively Reduced Integration An elimination of this defect is possible, if the work term Z E 2 · δεxy · · 2 · εxy dA 2(1 + ν) is integrated comprising a constant δεxy as well as εxy according to Figure 5-9, and, referring to this, the stiffness matrix is modified. In this way, pure shear deformation can be described but the non–physical shear deformation at pure bending is not, see Figure 5-10. This procedure is called selectively reduced integration, since only the constant terms of the work integral are evaluated and all higher order ones are ignored, what is equivalent to a constant shear strain field. The results obtained from this modified element are surprisingly good, and do not show any oscillations in the shear stress distribution. This description of the shear state corresponds to Figure 5-9. In the following, the resulting stiffness matrix K3 is given. When compared to K3 as derived in Section 5.1.6, the procedure presented here can be interpreted as averaging.

98

5 Membrane Structures

The entries are constant except for the quotients of lengths and the sign.  ℓx  ℓx 1 −1 − ℓℓx −1 − ℓℓx 1 ℓy ℓy y y    ℓy ℓy ℓy ℓy   1 1 − ℓ −1 − ℓ −1 ℓ   ℓx x x x     ℓx  ℓ ℓ ℓ x x x   1 −1 − −1 − 1  ℓy  ℓy ℓy ℓy    ℓy ℓy   −1 − ℓy −1 ℓy  1 1 − ℓx ℓx ℓx ℓx  3·E   . K3 =  24(1 + ν)  ℓx ℓx  − ℓx −1 − ℓx 1 1 −1   ℓy  ℓy ℓy ℓy     ℓ ℓ ℓ ℓ y y y y  −1 − −1 ℓ 1 1 −ℓ   ℓx ℓx x x     ℓx  ℓx ℓx ℓx −  −1 − 1 1 −1  ℓy  ℓy ℓy ℓy   ℓy ℓy ℓy ℓy 1 1 − −1 − −1 ℓ ℓ ℓ ℓ x

x

x

x

The stress matrix applied at subsequent analysis is given below regarding constant shear stresses. This corresponds to an averaging of the shear stresses according to Section 5.1.8. As such, e = (1 − ν)/2/2 is applied.  1 1 ν  −ℓ − ℓν ℓx ℓy x y  ν ν 1  − − 1      ℓx ℓy ℓx ℓy  σxx  e  e e e   − − e −e e −ℓe   σyy     ℓ ℓ ℓ ℓ ℓ ℓ ℓ y x y x y x y x            1  1 ν ν u  σxy A  −  − ℓx ℓy ℓy     ℓx     σxx   ν   v A  ν 1 1   −    −   σ    ℓx    ℓx ℓy ℓy  yy    e   u  e e e e e e e   − −    − −  σxy   ℓy ℓx ℓy ℓx ℓy ℓx ℓy ℓx   v B  E  B =   ·  .    σ  1 − ν 2   u  ν 1 ν 1 xx       − − ℓy ℓx ℓy ℓx      v    σyy       C 1 ν 1 ν        − −  σ     u  ℓy ℓx ℓy ℓx  xy C   e    e e e e    − − e −e e −ℓ  v D  σxx   ℓy ℓx ℓy ℓx ℓy ℓx ℓy x      σyy   ν 1 1 ν        − − ℓy ℓx ℓx ℓy     σxy D ν  − ℓ1 − ℓν ℓ1    ℓ y

−ℓe − ℓe y

x

x

−ℓe

y

e ℓx

e e ℓy ℓx

x

e ℓy

y

−ℓe

x

5.2 Rectangular Element Comprising Modified Shear Strains

99

5.2.2 Example of Use Once again the cantilever, as given in Figure 5-5, serves as a test to investigate the quality of the element. The increase in quality becomes particularly clear when compared to the results employing the unmodified shape functions. Even in case of 3 elements the error of the tip displacement achieves only 4 %. In the following Figures 5-11, the stresses σxx and σxy are depicted applying a mesh with 24 × 8 elements, without taking into account the antiysmmetry. Due to the constant shear strains inside the element, the shear stresses are also constant inside each element. As a result, the real stress distribution can now be approximated much better, whereat the averaged stresses at the element intersections are very close to the exact values. Local disturbances at the position of the load application as well as at the clamping cross section are preserved because of the reasons mentioned above. σxx 180. 0

−180. 0 σxy 15. 0

2. 5 Fig. 5-11 Cantilever – distributions of stresses σxx and σxy , employing selectively reduced integration

Remarks Even if the selectively reduced integration allows for very good results here, the results may be poor at other applications, cf. Section 24.2 and 24.4. Furthermore, if the complete stiffness matrix would be integrated by a reduced integration scheme, displacement fields, not comprising any strain energy, may occur, which are named zero–enery modes. These displacement fields may be recognized by an oscillation of the displacements in terms of an hourglass and therefore are called hourglass modes.

100

5 Membrane Structures

5.3 Plane Strain The load–carrying behavior of a 3–dimensional structure, where strains does not occur with respect to the direction of thickness, is called plane strain. Compared to plane stress, it is a characteristic of plane strain that stresses σzz exist due to restraints in the direction of thickness. Plane strain is applicable to many problems in geotechnics. Here many 3–dimensional situations as tunnel constructions, pit excavations or continuous footing may be investigated by employing 2–dimensional modeling, if the load–carrying behavior can be assumed to be invariant in the third direction. The governing equations regarding plane strain can be derived from those of plane stress by substituting E/(1 − ν 2 ) in place of E and ν/(1 − ν) in place of ν with respect to all previous derivations at Section 5.1. However, it is formally more elegant, and also more advantageous concerning the description of nonlinear material behavior, to modify the elasticity matrix E, the operator matrix D as well as the vectors of the stresses σ and the strains ǫ. The stress σzz and the strain εzz are to be considered additionally regarding the direction of thickness, whereby εzz is set here to zero explicitly. Thus, the material equations and kinematics can be applied as given below: σ = E·ε,          

σxx σyy σzz σxy





  E    =  (1 + ν)(1 − 2ν)  ǫ = D· u,

εxx εyy εzz 2εxy





    =  



∂x

∂y

 

1−ν ν ν ν 1−ν ν ν ν 1−ν

  ∂y  u  . · v 

(1−2ν)/2

    ·  

εxx εyy εzz 2εxy



   , (5.20) 

(5.21)

∂x

Except for the fact that the matrix B is comprised of an additional row employing zero entries, there are no significant differences in the further derivations, therefore these are not shown here.

6 Bending Structures

Thin walled structures, which are loaded perpendicular to their main spatial dimensions, carry their loading by bending. Modeling of bending structures usually requires kinematic assumptions, which yield either the Euler–Bernoulli beam theory at investigating 1–dimensional structures or the Kirchhoff theory of plates for 2–dimensional structures. Investigating the bending behavior of thin bending structures, shear deformations may be neglected, what requires special finite elements, dealing with C1 –conformity at the element intersections.

6.1 Element Matrices Regarding Euler–Bernoulli Beams The kinematic assumption that governs the Euler–Bernoulli beam theory is the Bernoulli hypothesis neglecting the shear deformations by specifying εxz = 0. Thus the shear force does not perform any work and the deflection w being the only independent displacement variable. Employing the beam element described below, the definition of signs applies as it is represented in Figure 6-1. p

z

Me

x

V

e

p

ϕ

z

z,w

M - dM 2 Q-

dQ 2

x

M + dM 2 dQ Q+ 2

Fig. 6-1 Euler–Bernoulli beam – sign definition

6.1.1 Governing Equations and Work Equations a) Governing equations K:

E:

ϕ,x − κ

=0

w,x − ϕ

=0

Q,x + pz

=0

M,x − Q

=0

or :

w,xx − κ = 0

or :

M,xx + pz = 0

M : M − EI · (−κ) = 0 © The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2024 D. Dinkler und U. Kowalsky, Introduction to Finite Element Methods, https://doi.org/10.1007/978-3-658-42742-9_6

102

6 Bending Structures

b) Principle of virtual Displacements Z δWd = {δκ M + δw pz } dx = 0 c) Work equation − δWd =

Z

δw,xx · EI · w,xx dx −

Z

δw · pz dx = 0 .

6.1.2 Matrix Notation of the Work Equation In order to derive the element stiffness matrix and the load vector regarding Euler–Bernoulli beams, matrix notation is applied according to Equation (2.34) T

−δWd = δv {

Z

0

ℓ

T

Z

T

Ω · D · E · D · Ω dx · v −

ℓ

0

ΩT · p dx } ,

(6.1)

or in a short form T

− δWd = δv {

Z

0

ℓ

T

B · E · B dx · v −

Z

0

ℓ

ΩT · p dx } .

(6.2)

6.1.3 Elasticity Matrix E and Operator Matrix D Concerning Euler–Bernoulli beams, only the bending moment M does perform work as a stress variable σ. The negative curvature −κ is the corresponding strain variable here, since the positive moment and the positive curvature are defined opposed to each other. The bending stiffness EI is the only element of the elasticity matrix E here, M = EI · (−κ) , σ = E · ǫ.

(6.3)

The differentiation rule, necessary to compute the curvature κ from the deflection w, is given by the operator matrix D. Applying matrix notation, the vectors of strains ε and of displacements u, which only comprise the deflection w(x), yield −κ = −w,xx = −d2x w ǫ = D ·u.

→

D = [−d2x ] , (6.4)

6.1 Element Matrices Regarding Euler–Bernoulli Beams

103

6.1.4 Shape Functions to Describe the Deflection According to the convergence criteria, the deflection w(x) is to be described by a polynomial approach of the 3rd order at minimum, thus the mathematical formulation yields w(x) = a0 + a1 · x + a2 · x2 + a3 · x3 .

(6.5)

To get a physically meaningful approach, it makes sense not only to use the displacements wi , but also the rotations ϕ|i = w,x |i as degrees of freedom at the element nodes, since, regarding convergence criteria, the requirement of C1 –conformity may be easily fulfilled hereby w(x) = φ1 · wA + φ2 · ϕA + φ3 · wB + φ4 · ϕB .

(6.6)

The related shape functions φi , represented in Figure 6-2, are the Hermite– Polynomials of the 3rd order. The courses of the functions are given as well as the respective second derivatives. It is to be emphasized that the approach is chosen to describe the deflection w(x), therefore the rotations are nodal displacements variables only. i

φi (x)

B

A

φi (x),xx

x l

x2

1

1−3

2

x − 2 xℓ +

ℓ2

x3 ℓ3

− 6 ℓ12 + 12 ℓx3

x3 ℓ2

− 4 1ℓ + 6 ℓx2

+2

2

2

3

6 ℓ12 − 12 ℓx3

2

x3 ℓ2

− 2 1ℓ + 6 ℓx2

3

3 xℓ2 − 2 xℓ3

4

− xℓ +

1

1

Applying the nodal displacements ϕA

Φ2 (x)

1

Fig. 6-2 Cubic Hermite–Polynomials

vT = [ wA

Φ1 (x)

1

wB

ϕB ]

Φ3 (x)

Φ4 (x)

104

6 Bending Structures

and the shape functions merged into the matrix of shape functions Ω = [ φ1

φ2

φ3

φ4 ]

the approach to describe the deflection follows to u=Ω·v

(6.7)

and in analogy, regarding the virtual deflection, to δu = Ω · δv .

(6.8)

6.1.5 Differentiation of Shape Functions The work equation requires the shape functions Ω being differentiated according to the differentiation rule D, yielding the matrix B of curvatures: B = D · Ω = [ −φ1,xx − φ2,xx − φ3,xx − φ4,xx ] .

(6.9)

The entries φi,xx given in B are listed in Figure 6-2.

6.1.6 Element Stiffness Matrix The matrices E and B, required for the computation of the stiffness matrix Z ℓ K= BT · E · B dx , (6.10) 0

are already known. The matrix product, being  φ1,xx φ1,xx φ1,xx φ2,xx   φ2,xx φ1,xx φ2,xx φ2,xx BT · E · B = EI ·    φ3,xx φ1,xx φ3,xx φ2,xx φ4,xx φ1,xx

φ4,xx φ2,xx

part of the integrand, yields  φ1,xx φ3,xx φ1,xx φ4,xx  φ2,xx φ3,xx φ2,xx φ4,xx  .  φ3,xx φ3,xx φ3,xx φ4,xx  φ4,xx φ3,xx

φ4,xx φ4,xx

The second derivatives of the shape functions are linear according to Figure 6-2. The products may be analytically integrated, what results in:   12 6 6 − 12 3 2 3 2 ℓ ℓ ℓ ℓ    6  4 6 2   − ℓ2  ℓ2  ℓ ℓ . K = EI ·  (6.11)  12 6 12 6   − 3  − − 2 3 2  ℓ ℓ ℓ ℓ    6 2 6 4 − ℓ2 ℓ ℓ2 ℓ

6.1 Element Matrices Regarding Euler–Bernoulli Beams

105

Compared to the displacement method, the matrix K corresponds to the exact stiffness matrix of Euler–Bernoulli beams clamped at both ends. Therefore, the entries of the first and of the third row can be interpreted as shear forces at the ends of the beam, which perform work on the respective virtual nodal deflections, and the entries of the second and of the fourth row as bending moments, which perform work on the respective virtual nodal rotations. If the nodal displacement variables are ordered by nodes, the element stiffness matrix may be clearly arranged applying sub–matrices, which explicitly point out the allocation of the stiffness entries to the element nodes " # kAA kAB . K= kBA kBB

6.1.7 Element Load Vector Regarding constantly distributed external actions p = [ p ] = constant in the element, p can be extracted from the integral, so that the element load vector is determined by the integral of the shape functions Z ℓ Z ℓ T f= Ω · p dx = ΩT dx · p . (6.12) 0

0

The integration of the matrix Ω of shape functions gives  ℓ  f=

"

fA fB

#

   =  

2 ℓ2 12 ℓ 2 ℓ2 − 12

    · p.  

(6.13)

The load vector corresponds to the supporting forces and moments of the beam clamped at both ends and loaded by a constantly distributed external action. A varying action p(x) may be approximated by a linear course of the function inside the element. Applying the matrix Ωp h x x i Ωp = 1 − = [ φA φB ] , ℓ ℓ

employing linear shape functions according to Section 2.3.1 and the related ˜ = [ pA pB ], it applies nodal values p p = Ωp · p ˜ = φA · pA + φB · pB .

(6.14)

106

6 Bending Structures

The integral to be computed to get the load vector Z ℓ f= ΩT · Ωp dx · p ˜ 0

comprises the shape functions ΩT , employing polynomials of the third order to describe the virtual displacements, as well as the linear shape functions Ωp that approximates the course of external action.

6.1.8 Subsequent Stress Analysis To evaluate the stress resultants, the formal procedure of the stress analysis, already known from the bar as well as from the membrane element, is applicable σ = E · ǫ, σ = E · D· u,

(6.15)

σ = E·D·Ω·v = E·B·v = S·v. If the elasticity matrix E is constant, the course of stesses σ follows the course ˜ of the stresses are obtained by of curvatures given by B. The nodal values σ ˜ can introducing the coordinates of the nodes into B. Then, the stress matrix S ˜ be evaluated from the matrix of discrete curvatures B, which yields ˜ ·v, ˜ =E·B σ ˜ ·v. ˜ =S σ

Regarding Euler–Bernoulli beams, the material model is defined over the bending moments. The stress matrix follows with E = [ EI ]

and

B = [ −φ1,xx − φ2,xx − φ3,xx − φ4,xx ] ,

comprising a linear course of the bending moment inside the element h i 4 6x 6 12x 2 6x σ = EI · ℓ62 − 12x ·v. − − + − ℓ3 ℓ ℓ2 ℓ2 ℓ3 ℓ ℓ2

Evaluation at the element nodes gives the nodal moments MA and MB " # " # " 6 4 2 # − ℓ62 σ(x = 0) MA ℓ2 ℓ ℓ ˜= σ = = EI · ·v. 6 4 σ(x = ℓ) MB − ℓ62 − 2ℓ − ℓ2 ℓ The shear forces are evaluated applying the equilibrium condition h i 6 12 6 Q = M,x = dx M = dx σ = EI · − 12 ·v. − − 3 2 3 2 ℓ ℓ ℓ ℓ

6.1 Element Matrices Regarding Euler–Bernoulli Beams

107

Thus the shear forces are constantly approximated inside an element, but allocated both element nodes. The stress matrix regarding the shear forces as well as the bending moments can be summarised and ordered by nodes     12 − 12 − ℓ62 − ℓ62 3 3 QA ℓ ℓ        6 4 6 2  −  MA   ℓ2 2 ℓ ℓ ℓ    = EI ·  ·v. (6.16)    12  6 12 6  QB   − ℓ3 − ℓ2 − 3 2  ℓ ℓ     6 4 MB − ℓ62 − 2ℓ − 2 ℓ ℓ In this formulation, the stress matrix is identical to the element stiffness matrix according to Equation (6.11), except of the signs.

6.1.9 Example of Use The single span beam with a length of l = 8 m and a bending stiffness of EI = 10,000 kN m2 is analysed with regard to the loading and boundary conditions as given in Figure 6-3. 1

1

2

2

3

3

4

4

5

x

EI = 10,000 kNm2

z

l = 8.0 m 2.0

2.0

2.0

2.0

lc 1

p = 10 kN/m

lc 2

P = 16 kN

lc 3

w1 = 0.04 m

Fig. 6-3 Single span beam – geometry and loading The discretization is chosen employing four elements and five nodes. The results concerning the loading conditions two and three represented in Figure 6-4 are identical to the exact analytical solution, since the approach for the deflection is the solution of the homogeneous differential equation, and the particular solution part disappears due to the external actions selected here. The results

108

6 Bending Structures

concerning loading condition one are characteristic for the FEM, being only an approximate solution. Regarding the selected discretization, the quality of the approximation can be identified from the jumps in the courses of the M – and Q–courses and from the fulfillment of the boundary condition for the bending moment. The jumps in the M –course disappear at the element intersections, since the error resulting from the particular solution due to a constantly distributed loading is equal at the respective left and right hand side of a node. The error present at the left hand side support ∆M = 3. 33 kN m cannot be clearly seen due to the measuring unit used. The shear force is determined from the M –course and therefore is also an approximation. 2

1 1

w1 w2 w3 lc 1: M Q

lc 2: M Q

lc 3: M Q

4

3 2

3

5 4

20 mm

−76.67 = M5 −40.00 = Q 4

−24.00 = M5 −11.00 = Q 4

−18.75 = M5 −2.34 = Q 4

Fig. 6-4 Single span beam – state variables regarding loading conditions 1 to 3

6.2 Kirchhoff Plate Element with 16 DOF

109

6.2 Kirchhoff Plate Element with 16 DOF The Kirchhoff theory of plates [55] is basis for the investigation of the deformation behavior of thin plates. Applying the Kirchhoff theory of plates, the deformations caused by shear forces are assumed to be negligible, comparable to the Euler–Bernoulli beam theory. This can be realized by the kinematic constraints, that straight lines perpendicular to the mid-surface remain perpendicular to the mid-surface and straight after the deformation, and the thickness does not change during the deformation. As a consequence of the Kirchhoff hypothesis, effective shear forces must be defined and prescribed on the free boundaries of the plate. However, this is not essential regarding the displacement–based formulation investigated here, since the Neumann boundary conditions are only fulfilled weakly by the PvD. Employing the rectangular plate element described below, the definition of signs applies as it is represented in Figure 6-5. p x y

p

z,w

dx dy

units : p [ N/m2 ] w [m] q [ N/m ] m [ Nm/m ]

dx

dy m xy m yy

m xx qx

m yx qy

Fig. 6-5 Stress resultants of plates – definition of signs In the following subsections, as a first approach, the rectangular element developed by Bogner, Fox, Schmit [20] will be described in detail.

6.2.1 Governing Equations and Work Equations a) Governing Equations K : w,x − ϕx w,y − ϕy ϕx ,x − κxx ϕy ,y − κyy (ϕx ,y + ϕy ,x ) − 2κxy

=0 =0 =0 =0 =0 or : w,xx − κxx = 0 w,yy − κyy = 0 w,xy − κxy = 0

110 E:

6 Bending Structures qx ,x + qy ,y + p mxx ,x + mxy ,y − qx myy ,y + mxy ,x − qy

=0 =0 =0 or : mxx ,xx + 2mxy ,xy + myy ,yy + p = 0

M : mxx − B (−κxx − νκyy ) = 0 myy − B (−κyy − νκxx ) = 0 mxy − B (1 − ν) (−κxy ) = 0 with plate bending stiffness

B=

E t3 12 (1 − ν 2 )

b) Principle of virtual Displacements Z − δWd = { δκxx mxx + 2δκxy mxy + δκyy myy + δw p } dA = 0 c) Work equation Z − δWd = B { δw,xx (w,xx + νw,yy ) + δw,yy (w,yy + νw,xx ) Z + δw,xy 2(1 − ν) w,xy } dA − δw p dA .

6.2.2 Matrix Notation of the Work Equation To derive the element stiffness matrix as well as the load vector regarding Kirchhoff ’s plate theory, once again the general work equation is applied in the matrix notation Z Z − δW = δvT · { ΩT · DT · E · D · Ω dA · v − ΩT · p dA } (6.17) A

A

or in a short form

− δW = δvT · {

Z

A

BT · E · B dA · v −

Z

A

ΩT · p dA } .

(6.18)

The matrix symbols are filled with physical content, if the governing equations are interpreted in a corresponding manner.

6.2.3 Elasticity Matrix E and Operator Matrix D According to Kirchhoff ’s plate theory, only bending and twisting moments, employed as stress variables σ, perform work on the respective virtual curvatures δκ, employed as generalized strain variables δε.

6.2 Kirchhoff Plate Element with 16 DOF

111

The material equations link stresses σ and strains ǫ to each other:



σ = E ·ε,  

1 mxx    ν  myy  = B ·  mxy

ν 1

 

 −κxx     ·  −κyy  . 1 −2κxy (1 − ν) 2

(6.19)

The differentiation rule D to compute the strains ǫ from the deflection is defined by the kinematic conditions



−κxx   −κyy −2κxy

ε = D · u,    −∂xx     =  −∂yy  · [ w ] . −2∂xy

(6.20)

The only independent state variable is the deflection surface w(x, y), which is approximated by shape functions and nodal displacement variables w(x, y) = Ω(x, y) · v .

(6.21)

6.2.4 Shape Functions to Describe the Deflection The choice of the shape functions to describe the deformation behavior of plates is more difficult than regarding membrane structures because the convergence criteria require a C1 –conformity analogous to that of beams. This means that the deflection must be continuous and continuously differentiable parallel and rectangular to common element intersections. In analogy to the beam this requires at least cubic polynomials with respect to the x– and y–coordinate employing deflections and derivatives of the deflection surface as nodal displacement variables at the element boundaries. The shape functions assigned to the nodal displacement variables can be developed as a product approach employing Hermite–Polynomials that have been already introduced regarding bending at Euler–Bernoulli beams. The combination of the respective four shape functions of the beam concerning the x– and y–directions as a product approach yields 16 shape functions in total to describe the deflection. If the nodal displacement variables vnode = [ w w,x w,y w,xy ]node are chosen as degrees of freedom in the four corners of the rectangular element,

112

6 Bending Structures

the element intersection conditions can be fulfilled. Overall, the symmetric approach employs 16 degrees of freedom, ordered by the nodes here vT = { [ [ [ [

w w w w

w,x w,x w,x w,x

w,y w,y w,y w,y

w,xy w,xy w,xy w,xy

]A ]B ]C ]D } .

(6.22)

The shape functions assigned to the displacement variables of a single node are depicted in Figure 6-6.

w

w,x

w,y

w,xy

Fig. 6-6 Shape functions regarding displacement variables related to node D The shape functions related to the displacement variables of the other nodes can be obtained analogously, if those are combined with each other appropriately. The order of the nodal displacement variables also defines the order of the shape functions in the vector Ω ={[ [ [ [

φ1 (x) · φ1 (y) φ3 (x) · φ1 (y) φ3 (x) · φ3 (y) φ1 (x) · φ3 (y)

φ2 (x) · φ1 (y) φ4 (x) · φ1 (y) φ4 (x) · φ3 (y) φ2 (x) · φ3 (y)

φ1 (x) · φ2 (y) φ3 (x) · φ2 (y) φ3 (x) · φ4 (y) φ1 (x) · φ4 (y)

φ2 (x) · φ2 (y) φ4 (x) · φ2 (y) φ4 (x) · φ4 (y) φ2 (x) · φ4 (y)

]A ]B ]C ]D }

or written in a short form related to the products of the shape functions Ω = [ φA1 φB1 φC1 φD1

φA2 φB2 φC2 φD2

φA3 φB3 φC3 φD3

φA4 φB4 φC4 φD4

].

(6.23)

6.2 Kirchhoff Plate Element with 16 DOF

113

Thereby the deflection w(x, y) follows in a short form to u = [ w(x, y) ] = Ω(x, y) · v .

(6.24)

If the more formal way is chosen, that is also correct and applies the scaling matrix, at first the symmetric general polynomial approach is chosen: w(x, y) = a1 + a2 x + a3 y + a4 x2 + a5 xy + a6 y 2 + a7 x3 + a8 x2 y + a9 xy 2 + a10 y 3 + a11 x3 y + a12 x2 y 2 + a13 xy 3 + a14 x3 y 2 + a15 x2 y 3 + a16 x3 y 3 .

(6.25)

The symmetry of this approach can be proved by Pascale’s triangle, cf. Figure 2-16. The transformation of the general polynomial into an approach employing physically meaningful degrees of freedom is possible according to Section 2.4.3, when the scaling is related to the nodal displacement variables according to Equation (6.22). With the general polynomial [ w(x, y) ] = ψ · a the physically meaningful degrees of freedom of Equation (6.22) can be represented by ˜ · a, v=Ψ ˜ may be computed by introducing the coordiwhereat the 16 × 16–matrix ψ nates of the related nodes either into the general polynomial itself or into the respective derivatives. Now, the scaling matrix G is obtained from the solution of the system of equations with respect to a −1

˜ a=Ψ

·v = G·v.

Hereby, the general polynomial ψ can be transformed into the approach Ω employing physically meaningful nodal displacement variables Ω = ψ · G. The scaling matrix G is given below.

                          



1 0 0

0 1 0

0 0 1

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0



x y

x y

x y

x y

x y

x y

x y

x y

x y

x y

x y

x y

x y

x y

x y

x y

              3 2 3 1   − − 0 0 − 0 0 0 0 0 0 0 0 0 0 a1 2 lx lx lx  lx2     a2   0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0       a3   3  3 0 − l2 0 0 0 0 0 0 0 0 0 0 − l1 0   − ly2  ly2 y y a4    2  1   0 0 − l23 l12 0 0 0 0 0 0 0 0 0 0  a5  2 lx   lx3  x x    3 a6  0 − l32 − l2 0 0 − l1 0 0 0 0 0 0 0 0     0 2 lx x x x     a7   3 2 3 1   0 −l 0 0 0 0 0 0 0 0 0 0 −l   0 − ly2 ly2 a8  y y = 2   1 2 1    a9   l 3 0 0 0 0 0 0 0 0 0 0 − 0 0 2 3 2  ly ly ly  y    a10  2 1   0  0 0 0 − l23 l12 0 0 0 0 0 0 0 0  3 2   lx lx  x x a11       9 6 6 4 9 3 6 2 9 3 3 1 9 6 3 2  a12    l2 l2 l l2 l2 l l l − l2 l2 l l2 − l2 l l l l2 l2 − l l2 − l2 l l l − l2 l2 − l l2 l2 l l l   x y x y x y x y x y x y   xy xy xy xy x y x y x y x y x y x y a13     2 1 2 1    a14   0  0 0 0 0 0 0 0 0 0 0 − 0 ly3 ly2 ly3 ly2       a15  6  3 4 2 6 3 4 2 6 3 2 1 6 3 2 1 − 3 2 − 2 2 − 3 − 2  − − − − − − a16  lx ly lx ly lx ly lx ly lx3 ly2 lx2 ly2 lx3 ly lx2 ly lx3 ly2 lx2 ly2 lx3 ly lx2 ly lx3 ly2 lx2 ly2 lx3 ly lx2 ly     6  − l2 l3 − l 4l3 − l23l2 − l 2l2 l26l3 − l 2l3 l23l2 − l 1l2 − l26l3 l 2l3 l23l2 − l 1l2 l26l3 l 4l3 − l23l2 − l 2l2  x y x y x y x y x y x y  xy xy xy xy xy x y x y x y x y x y   4 2 2 1 − l34l3 l22l3 − l32l2 l21l2 l34l3 − l22l3 − l32l2 l21l2 − l34l3 − l22l3 l32l2 l21l2 l3 l3 l2 l3 l3 l2 l2 l2



  w  w,x    w,y    w,xy A    w   w,x    w,y    w,xy B    w   w,x     w,y    w,xy C   w    w,x     w,y  w,xy D

114 6 Bending Structures

6.2 Kirchhoff Plate Element with 16 DOF

115

6.2.5 Differentiation of Shape Functions The differentiation rule D is applied to the shape functions Ω regarding the deflection surface and yields the matrix B = D ·Ω.

(6.26)

The physical meanings of the components of B turn out to be the curvatures with respect to the related shape functions. To differentiate the entries of the matrix of shape functions, the matrix scheme may be applied, given in Figure 6-7. Here, the matrix B is arranged with respect to the nodes of the element. Thus, each sub–matrix has the same structure but with respect to the shape functions related to the node. node A

node B

node C

node D

Ω D B

Fig. 6-7 Multiplication scheme B = D · Ω The submatrix BA comprising the shape functions φA1 , φA2 , φA3 , φA4 applies for the node A, e.g., as follows φA1

φA2

φA3

φA4

−φA3 ,xx

−φA4 ,xx

−∂xx

−φA1 ,xx −φA2 ,xx

−2∂xy

−2φA1 ,xy −2φA2 ,xy −2φA3 ,xy −2φA4 ,xy

−∂yy

−φA1 ,yy −φA2 ,yy

−φA3 ,yy

−φA4 ,yy

.

The differentiation of the shape functions with respect to the coordinates may be seen more clearly applying product notation 

 −φ1 (x),xx φ1 (y) −φ2 (x),xx φ1 (y) −φ1 (x),xx φ2 (y) −φ2 (x),xx φ2 (y)   BA = −φ1 (x) φ1 (y),yy −φ2 (x) φ1 (y),yy −φ1 (x) φ2 (y),yy −φ2 (x) φ2 (y),yy  . −2φ1 (x),x φ1 (y),y −2φ2 (x),x φ1 (y),y −2φ1 (x),x φ2 (y),y −2φ2 (x),x φ2 (y),y A

116

6 Bending Structures

6.2.6 Element Stiffness Matrix The element stiffness matrix follows from the integration of K=

Z

0

lxZ ly 0

BT · E · B dx dy .

(6.27)

The integrand may be computed applying the scheme according to Figure 6-8 and is assigned to the corresponding element nodes.

A

B

C

D

0 E ** ** 0 0 0

B

*

A

B

C

D BT

BT E B

Fig. 6-8 Multiplication scheme BT · E · B The 16 × 16 stiffness matrix may be also computed and clearly arranged by applying the 4 × 4 submatrices kij that are assigned to the nodes 

  K= 

kAA kBA kCA kDA

kAB kBB kCB kDB

kAC kBC kCC kDC

kAD kBD kCD kDD



  . 

All submatrices kAA , kAB , . . . are similarly arranged; only the shape functions applied to build them are different.

6.2 Kirchhoff Plate Element with 16 DOF The submatrix kAA , e.g., follows to Z kAA = BTA · E · BA dA

117

A

=B

Z

IAA1 dA + νB A

Z

IAA2

A

Z dA + 2B(1 − ν) IAA3 dA , A

whereat the multipliers of the integrals result from the elasticity matrix E. In detail this yields   φA1 ,xx φA1 ,xx φA1 ,xx φA2 ,xx φA1 ,xx φA3 ,xx φA1 ,xx φA4 ,xx  +φA1 ,yy φA1 ,yy +φA1 ,yy φA2 ,yy +φA1 ,yy φA3 ,yy +φA1 ,yy φA4 ,yy     φ , φ , φA2 ,xx φA2 ,xx φA2 ,xx φA3 ,xx φA2 ,xx φA4 ,xx  A2 xx A1 xx    +φ , φ , +φ , φ , +φ , φ , +φ , φ ,  A2 yy A1 yy A2 yy A2 yy A2 yy A3 yy A2 yy A4 yy   IAA1 =  ,  φA3 ,xx φA1 ,xx φA3 ,xx φA2 ,xx φA3 ,xx φA3 ,xx φA3 ,xx φA4 ,xx     +φA3 ,yy φA1 ,yy +φA3 ,yy φA2 ,yy +φA3 ,yy φA3 ,yy +φA3 ,yy φA4 ,yy     φ , φ , φA4 ,xx φA2 ,xx φA4 ,xx φA3 ,xx φA4 ,xx φA4 ,xx  A4 xx A1 xx +φA4 ,yy φA1 ,yy +φA4 ,yy φA2 ,yy +φA4 ,yy φA3 ,yy +φA4 ,yy φA4 ,yy 

IAA2

φA1 ,xx φA1 ,yy φA1 ,xx φA2 ,yy  +φA1 ,yy φA1 ,xx +φA1 ,yy φA2 ,xx   φ , φ , φA2 ,xx φA2 ,yy A2 xx A1 yy   +φ , φ , +φ , φ , A2 yy A1 xx A2 yy A2 xx  =  φA3 ,xx φA1 ,yy φA3 ,xx φA2 ,yy   +φA3 ,yy φA1 ,xx +φA3 ,yy φA2 ,xx   φ , φ , φA4 ,xx φA2 ,yy A4 xx A1 yy +φA4 ,yy φA1 ,xx +φA4 ,yy φA2 ,xx 

  IAA3 =  

 φA1 ,xx φA3 ,yy φA1 ,xx φA4 ,yy +φA1 ,yy φA3 ,xx +φA1 ,yy φA4 ,xx   φA2 ,xx φA3 ,yy φA2 ,xx φA4 ,yy   +φA2 ,yy φA3 ,xx +φA2 ,yy φA4 ,xx   , φA3 ,xx φA3 ,yy φA3 ,xx φA4 ,yy   +φA3 ,yy φA3 ,xx +φA3 ,yy φA4 ,xx   φA4 ,xx φA3 ,yy φA4 ,xx φA4 ,yy  +φA4 ,yy φA3 ,xx +φA4 ,yy φA4 ,xx 

φA1 ,xy φA1 ,xy

φA1 ,xy φA2 ,xy

φA1 ,xy φA3 ,xy

φA1 ,xy φA4 ,xy

φA2 ,xy φA1 ,xy

φA2 ,xy φA2 ,xy

φA2 ,xy φA3 ,xy

φA3 ,xy φA1 ,xy

φA3 ,xy φA2 ,xy

φA3 ,xy φA3 ,xy

φA4 ,xy φA1 ,xy

φA4 ,xy φA2 ,xy

φA4 ,xy φA3 ,xy

φA2 ,xy φA4 ,xy   . φA3 ,xy φA4 ,xy 

Thus the stiffness matrix follows with three parts as well

φA4 ,xy φA4 ,xy

kAA = kAA1 + kAA2 + kAA3 , where k1 is assigned to the integral I1 , k2 to the integral I2 and k3 to the integral I3 .

118

6 Bending Structures

The representation of the integrals may be clarified by introducing the coefficients α = ℓy /ℓ3x , β = ℓx /ℓ3y and γ = 1/ℓxℓy , which are related to the differentiation of the shape functions and to the integration with respect to the element area. The numbers of the entries of the submatrices resulting from integration are listed below for the submatrix kAA . All submatrices are symmetric. Therefore, only the upper triangle matrix is represented. The rows and the columns of the matrices have to be multiplied by the specified coefficients. These coefficients show that the units of the different virtual as well as real nodal displacement variables are different. It is applied

kAA1α =

kAA1β =

kAA2 =

kAA3

B·α · 6300

B·β · 6300

ν ·B·γ · 6300

1 28080

ℓx 14040 9360

ℓy 3960 1980 720

ℓx ℓy 1980 1320 360 240

1 28080

ℓx 3960 720

ℓy 14040 1980 9360

ℓx ℓy 1980 360 1320 240

1 18144

2 (1 − ν) · B · γ = · 6300

ℓx 9072 2016

1 9072

ℓy 9072 7686 2016

ℓx 756 1008

ℓx ℓy 1386 1008 1008 224 ℓy 756 63 1008

·1 ·ℓx , ·ℓy ·ℓx ℓy

·1 ·ℓx , ·ℓy ·ℓx ℓy

·1 ·ℓx , ·ℓy ·ℓx ℓy ℓx ℓy 63 84 84 112

·1 ·ℓx . ·ℓy ·ℓx ℓy

6.2 Kirchhoff Plate Element with 16 DOF

119

6.2.7 Element Load Vector Assuming a constantly distributed external action p = [ p ] the element load vector follows in matrix notation to Z Z f= ΩT · p dA = ΩT dA · p . (6.28) A

A

Integration yields

f T = [ 36 6ℓx 36 −6ℓx 36 −6ℓx 36

6ℓx

6ℓy 6ℓy −6ℓy

ℓx ℓy −ℓx ℓy ℓx ℓy

−6ℓy

−ℓx ℓy ] ·

(6.29)

ℓx ℓy 144

·p.

Under arbitrarily distributed external surface actions, the loading can be approximated employing bi–linear shape functions and nodal values, in analogy to a membrane element. With p = Ωp · p ˜ describing the course of the surface action regarding the values p ˜ at the element nodes it applies Z Z f= ΩT · p dA = ΩT · Ωp dA · p ˜ . (6.30) A

A

The integrand is comprised of a 16 × 4 matrix, which is multiplied by the vector of nodal action values after integration. Usually this procedure is avoided, since it hardly influence the results, if small elements are needed for a sufficient approximation of the deformation behavior.

6.2.8 Subsequent Stress Analysis The bending and the twisting moments are computed from the material equations introduced at Section 6.2.1. The elasticity matrix is given by   1 ν   E=B· ν 1  1 (1 − ν) 2

and the matrix of curvatures by  −φA1 ,xx −φA2 ,xx −φA3 ,xx  −φA2 ,yy −φA3 ,yy B =   −φA1 ,yy −2φA1 ,xy −2φA2 ,xy −2φA3 ,xy

  −φA4 ,xx " # " # " #   −φA4 ,yy  . B C D −2φA4 ,xy

120

6 Bending Structures

In this way, the stress matrix Sm = E · B and thus the courses of the plate’s moments may be computed with 



−φA1 ,xx − νφA1 ,yy

 σ = E · B · v = B ·   −νφA1 ,xx − φA1 ,yy −(1 − ν)φA1 ,xy

A2

A3

A4

 

"# "# "# B

C

D



 ·v.

The stress analysis concerning the moments is performed at the corner nodes of the element yielding the nodal moments, which can be evaluated by applying ˜m explicitly given on the next page. the stress matrix S ˜m · v . σ ˜=S The shear forces may be computed analogously to the procedure presented for the Euler–Bernoulli beam by means of the equilibrium conditions   " # " # " # mxx qx mxx ,x + mxy ,y ∂x 0 ∂y   = = ·  myy  = Dq · σ . qy myy ,y + mxy ,x 0 ∂y ∂x mxy Employing the shape functions regarding the deflection it explicitly applies " # "" #   # −φA1 ,xxx − φA1 ,yyx qx =B· ·v qy −φA1 ,yyy − φA1 ,xxy A2 A3 A4 B C D or in matrix notation q = Dq · σ = Dq · ( E · B ) · v = Sq · v . The stress analysis concerning the shear forces is performed at the corner nodes of the element as well. It applies ˜q · v , q ˜=S ˜ q is the stress matrix related to the shear forces at the element nodes. whereat S It should be mentioned, that the shear forces may also be computed by means of the equilibrium conditions and the first derivatives of the moments, if the moments are described with a bi–linear approximation with respect to the corner values.

                   



mxx myy mxy mxx myy mxy mxx myy mxy mxx myy mxy

6 + 6ν ℓ2x ℓ2y

4 ℓx

4ν ℓy

0

0

0

  6ν 6 4ν 4  ℓ2 + ℓ2 ℓx ℓy y  x     0 0 0      −6 −2 0   ℓ2x ℓx   A    6ν 2ν  − 2 − 0   ℓx ℓx      0  0 0  B  =B·    0 0 0        0  0 0  C       0 0 0      − 6ν D 0 − 2ν  ℓ2y ℓy   6  − 2 0 − ℓ2  ℓy y



0 0

0 0 0

0

0

0

− ℓ2 0

− ℓ62

0 y

0

0

0

0

0

0

0

0

0

0

y

y

0 − 2ν 0 ℓ

y

− 6ν ℓ2

0

0

0

0

0

0

f

0

0

4 ℓy

0

x

− 4ν ℓ

0

6ν + ℓ62 ℓ2x y

4ν ℓy

0

0

x

0

0

0

− ℓ4

0

0

0

6 + 6ν ℓ2x ℓ2y

x

2ν ℓx

2 ℓx

0

0

− 6ν ℓ2

0 f

− ℓ62

0 x

y

0

0

2ν ℓx

2 ℓx

x

− ℓ62

− 6ν ℓ2x

y

0

0

0

0

0

0

0

0

0

0

y

0

0

0

f

− 4ν − ℓ4 0 ℓ 0 x

x

0

2 ℓy

2ν ℓy

0

0

0

− ℓ4 − 4ν 0 ℓ

0

0

0

0

0

0

0

6ν + ℓ62 ℓ2x y

6 + 6ν ℓ2x ℓ2y

0

y

− ℓ62

− 6ν ℓ2

0

0

0

x

y

x

0

0

4ν ℓx

4 ℓx

6µ 6 + ℓ2 ℓ2x y 6ν + ℓ62 ℓ2x y

0

x

− 2ν ℓ

− ℓ2

0

0

0

0

0

0

0

x

− 6ν ℓ2

− ℓ62

0

0

0

0

− ℓ62

y

− 6ν ℓ2

y



  0    0    0     0    0   ·v.  0    0     0    0     0  

0

0 f

− ℓ4 y

− 4ν ℓ

0

0

0

0

0

0

0

2 ℓy

2ν ℓy

˜m The moments, regarding the element nodes A to D, are evaluated applying the following stress matrix S including the abbreviation f = −(1 − ν)

6.2 Kirchhoff Plate Element with 16 DOF 121

              



qx qy

qx qy

qx qy

qx qy

6 6 − 12 − ℓ3 ℓ2 ℓ2

0

y

4 ℓy

y

 x y x  12  − 3 0 62− 62 4   ℓx ℓy ℓx  ℓy      − 12 −6 0 0 A  ℓ3x ℓ2x       0  0 − ℓ62 − ℓ2  x x  B    = B ·  0  0 0 0     C     0 0 0 0     D  0 − 62 0 − ℓ2 ℓy  y  12 6 − ℓ3 0 − ℓ2 0

 0

x

− ℓ62

0 0 0

0 0

y

− 12 ℓ3

y

− ℓ62

0

0

y

− 12 ℓ3

12 6 6 − ℓ3x ℓ2y ℓ2x

0

12 ℓ3x

y

0

0

− ℓ62

0

0

0

0

y

− ℓ2

6 6 − −4 ℓ2x ℓ2y ℓx

0

4 ℓy

2 ℓx

− ℓ62 x

0

0

0

− ℓ62 y

0

0

0

− ℓ62 x

12 ℓ3x

0

0

12 ℓ3y

12 6 6 − ℓ3x ℓ2y ℓ2x

12 ℓ3y

0

0

0

y

y

− ℓ4

0

2 ℓy

0

0

0 2 ℓx

0 − ℓ62

x

6 6 − −4 ℓ2x ℓ2y ℓx

0

− ℓ62

0

0

0

x

0

0

0

x

y

− ℓ62

0

12 ℓ3y

0

6 6 − 12 − ℓ3x ℓ2y ℓ2x

0

− 12 − ℓ62 ℓ3

0

0

12 ℓ3y

0

0 y

x

6 6 − ℓ2x ℓ2y

0

− ℓ62

0

0

0

− ℓ62

˜q The shear forces, regarding the element nodes, are evaluated applying the related stress matrix S

4 ℓx



  0     0    0    ·v. 0     − ℓ2  x   − ℓ4  y 

2 ℓy

122 6 Bending Structures

6.2 Kirchhoff Plate Element with 16 DOF

123

6.2.9 Example of Use As an example of use the square plate loaded by a constantly distributed external action according to Figure 6-9 is investigated. The boundaries at their respective opposite sides are clamped or simply supported. Thus, only a quarter of the plate may be discretized, if symmetry conditions are considered. The discretization employing 2 × 2 elements is depicted in Figure 6-9. x y

GE

clamped constantly distributed loading: p = 1 kN/m2 7 2 E = 3.0 . 10 kN/m

E

T

hinged

M

ly

t = 0.1 m G

ν = 0.2 l x = 10.0 m l y = 10.0 m

lx

Fig. 6-9 Quadratic plate – geometry and loading

Representation of Results The course of the deflection and the moments is depicted in Figure 6-10 regarding the 16 DOF element at selected sections applying a discretization of 2 × 2 and 4 × 4 elements respectively. The deflection gives only little changes due to mesh refinement, which indicates that the chosen approach may approximate the deformation behavior very well. The course of the bending moments is comprised of linear parts as well as of cubic parts superposed because of lateral curvatures. The respective linear part is represented only, which still shows significant changes in the case of mesh refinement due to the second derivatives of the deflection surface. The quality of the approximate solution of the respective discretization may be identified from the jumps of the bending moments at the element intersections as well as from the fulfillment of the boundary condition mxx = 0 at position G. The twisting moment is directly computed employing the nodal displacement variable w,xy , thus it is approximated as well as the deflection surface itself. The course of the twisting moment is parabolic along the element boundaries and given here as a polygonal line.

124

6 Bending Structures section G − M:

0

G

M

G

M

5 w [mm] 0

+

1.0

+

m xx [kNm/m]

section M − E:

0

M

E

5 w [mm] M 0 +

E

3.0 m yy [kNm/m] 0

M +

1.0

E

m xx [kNm/m]

section G − GE:

G

GE

0 − 1.0

m xy [kNm/m]

Fig. 6-10 Square plate – state variables regarding different discretizations

6.2 Kirchhoff Plate Element with 16 DOF

125

Investigation of Convergence Table 6.1 represents the convergence behavior of the most important state variables regarding the 16 DOF element, cf. Figure 6-11 and the comparison with other elements is given in Section 10.4. As shown in Section 6.1.9, the deflection Table 6.1 16 DOF plate element – convergence behavior of state variables element pattern

1×1

2×2

4×4

8×8

16 × 16

0. 74627

0. 73603

0. 73616

0. 73618

0. 73618

÷

−1. 516

−1. 536

−1. 537

−1. 537

wM

[cm]

mxyT

[kNm/m]

mxxM

[kNm/m]

myyM

[kNm/m]

5. 040

3. 481

3. 236

3. 183

3. 170

myyE

[kNm/m]

qxG

[kN/m]

−4. 664

−6. 194

−6. 760

−6. 925

−6. 969

qyE

[kN/m]

1. 87

3. 15

4. 06

4. 59

4. 87

2. 814

1. 87

2. 219

1. 63

2. 164

1. 93

2. 156

2. 16

2. 154

2. 29

exhibit only little errors, even when employing a small number of elements. Due to the chosen shape functions the convergence behavior concerning the twisting moment mxy is just as effective, because mxy may be directly computed from the twisting w,xy , which is a nodal degree of freedom, but not from the derivative of the deflection surface. The bending moments converge more slowly against the exact solution, since the second derivative of the deflection surface causes a roughening of the course of the corresponding state variable. The shear forces converge even less effectively, since they are computed by applying the third derivative of the deflection surface. Figure 6-11 shows the development of the state variables z with respect to mesh refinement, whereat the reference variables zexact are chosen from the 16 × 16 mesh. z z exact myy M mxx M 1.0 0.5

w mxy T myy E

4

16

64

elements

16 36

100

324

unknowns

1

Fig. 6-11 Square plate – convergence behavior regarding the state variables

126

6 Bending Structures

6.3 Kirchhoff Plate Element with 12 DOF Due to the twistings w,xy being nodal displacement variables, the approach is no longer coordinate–invariant. To get a coordinate–invariant approach, nodal displacement variables w,xx and w,yy would also be required to uniquely transform the curvatures. As an alternative to the 16 DOF element, an approach may be chosen without comprising any twisting w,xy , which consequently employs only 12 nodal displacement variables, [ w w,x w,y ] at each node. Although this approach is not conform, it still converges to the correct solution, cf. Section 6.2.9. A direct derivation of the approach comprising 12 nodal displacement variables is not possible, but it may be derived by means of a general polynomial comprising 12 degrees of freedom ai . 10 degrees of freedom are assigned to a complete cubic approach according to Pascale’s triangle and employing 10 power terms. In addition, the power terms x y 3 and x3 y are related to the remaining two degrees of freedom. This approach originally was developed by Adini and Clough [1] and Melosh [71]. It comprises w(x, y) = a1 + a2 x + a3 y + a4 x2 + a5 xy + a6 y 2 + a7 x3 + a8 x2 y + a9 xy 2 + a10 y 3 + a11 x3 y + a13 xy 3 , or u = ψ 12 · a12 .

(6.31)

The scaling matrix of the 12 DOF element can now be derived from the approach of the 16 DOF element, as follows. According to Equation (6.25), the degrees of freedom a12 , a14 , a15 , a16 are assigned to the power terms, which are no longer taken into account. Thus the corresponding columns of the scaling ˜ related to the 16 DOF element must be eliminated. Moreover, the matrix Ψ fourth, eighth, twelfth and sixteenth row is to be eliminated, since the twisting nodal displacement variables are omitted. It applies ˜ 12 · a12 v12 = Ψ and subsequently −1

˜ 12 · v12 = G12 · v12 . a12 = Ψ In this way, the shape functions employing physically meaningful nodal degrees of freedom may be computed, applying the scaling matrix, to Ω12 = ψ 12 · G12 . All further steps to compute the element matrices are the same with regard to the 16 DOF element.

                  



a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a13

                           =                           



x

0 1 0 x

1 2 lx 2 lx ly

0 0

2 3 lx

3 2l lx y

3 lx ly2

2 ly3

x y

− l 2l3

x

x

x y

0

1 ly2

2 lx ly

2 lx ly3

2 3l lx y

0

x y

− l 3l2

x y

− l23

0

0 − l23l

0

− l2 y

0 − l 1l2

− l32l − l21l x y x y

0

y

− l32 y

x y

1 lx ly

− l 1l − l1 − l1

0 0 0 3 2 lx

0 0 1 0

− l32 − l2

1 0 0 x

0

− l21l x y

0

1 lx ly2

0

0

x y

0

0

0

1 lx

0

0 0 0

0 − l 2l

1 lx ly

1 2 lx

0

0

− l1

0 0 0

0

x y

− l 2l3

0

1 lx ly2

0

0

x y

y

2 lx ly3

2 3l lx y

− l23

x y

x y

0

3 ly2

1 lx ly

0

0 0 0

x y

0

1 ly2

1 lx ly

0

0

− l1 y

0

0

0 0 0

0 − l 1l2

1 2l lx y

0

0

x y

0

0

1 ly

0

0 0 0

0 − l23l − l 2l

0

0

0

0

0 0 0

0 − l 1l − l 3l2

x y

− l 1l

0

0

0

0

0 0 0

− l32l l21l x y x y

0

3 lx ly2

3 2l lx y

0

0

x y

− l 1l

0

0 0 0                             ·                        



















w w,x w,y w w,x w,y w w,x w,y w w,x w,y



D

C



B



A





                  



6.3 Kirchhoff Plate Element with 12 DOF 127

128

6 Bending Structures

The shape functions regarding the 12 DOF element are depicted in Figure 6-12. The breach of the element intersection conditions is obvious, since the shape functions related to the rotations become linear in one direction, and thus exhibit a kink at the intersection to neighboring elements.

Fig. 6-12 Shape functions of the 12 DOF element

Example of Use For comparison the following Table 6.2 shows the convergence behavior of the 12 DOF element. The deflection converges significantly less effectively in comparison to the 16 DOF element, since the clamping and the element intersection conditions with respect to the rotations cannot be exactly described, which yields a more flexible structural behavior. In contrast, the moments as well as the shear forces are relatively well approximated, because the omitted twisting degrees of freedom w,xy does not influence the bending moments mxx , myy and the shear forces qx , qy that much. It should be mentioned that the twisting moment mxy now has to be computed at the element level with poorer convergence, since the corresponding curvature w,xy is no nodal degree of freedom as it is in the case of the 16 DOF element. Table 6.2 12 DOF element – convergence behavior of state variables element pattern

1×1

wM

[cm]

0. 9702

mxyT

[kNm/m]

mxxM

[kNm/m]

3. 107

myyM

[kNm/m]

myyE

[kNm/m]

qxG

[kN/m]

qyE

[kN/m]

2×2

4×4

8×8

16 × 16

0. 8017

0. 7527

0. 7403

0. 7372

−1. 751 −1. 524

−1. 570 −1. 553

−1. 546 −1. 542

−1. 539 −1. 538

6. 443

3. 777

3. 308

3. 201

3. 175

−6. 064

−6. 721

−6. 899

−6. 960

−6. 977

2. 80

3. 75

4. 36

4. 74

4. 94

1. 14

2. 385

1. 60

2. 207

1. 92

2. 167

2. 16

2. 157

2. 29

6.4 The 12 DOF Element Employing a Weak Conformity

129

6.4 The 12 DOF Element Employing a Weak Conformity The plate element with 12 DOF employs the deflection w and the rotations w,x and w,y as nodal unknowns and hence does not have the difficulties recognized in plate elements with 16 DOF. Nonetheless, the element does not satisfy the conditions of conformity at the intersections of neighboring elements. Instead of introducing additional variables at the intersections to satisfy the conditions, the work equation may be extended by the work of the stress variables at the respective intersection. This leads to a weak formulation of the conformity with respect to the slope w,n and improves the properties of the element remarkably.

6.4.1 Stiffness matrix The work equation of the Kirchhoff theory of plates is already given explicitly in Section 6.1 with Z −δW = B {δw,xx · (w,xx + νw,yy ) + δw,yy · (νw,xx + w,yy ) Z 1−ν +2 δw,xy · w,xy } dxdy − δw · p dxdy . (6.32) 2 Because of the non–conforming approach the work equation must be completed by the work performed by the moment on the conjugated rotation along the boundary. The nodal unknowns w w,x w,y , transformed to the boundary coordinates, are available on each boundary as w w,s w,n , with s being the coordinate in the direction of the boundary and n normal to it. The DOF w and w,s suffice to fulfill the element boundary conditions for w and w,s as a 3rd order Hermite–polynomial. The two remaining nodal DOF w,n suffice to describe the slope w,n as a linear polynomial along the boundary and thus may reflect the corresponding stiffness. Hereby the virtual work with respect to Equation (6.32) satisfies the continuity condition at the intersection up to a linear course of w,n but neglect the quadratic part, which is necessary for the cubic approach, see Figure 6-13. quadratic part w,n A

linear part

Fig. 6-13 Course of w,n at the element boundary

w,n B

130

6 Bending Structures

In order to satisfy the continuity with respect to the rotations w,n , the work equation has to be extended by additional terms at the element boundaries, which take into account the work perfomed by the bending moments mn on the corresponding rotations. The bending moments are replaced by the material equations, thus leading to a pure displacement–based formulation with Z Z − δWel.b. = δmn · (w,n quad − w,n lin )} ds + mn · (δw,n quad − δw,n lin )} ds Z = B · (δw,nn + ν δw,ss ) · (w,n quad − w,n lin )} ds Z + B · (w,nn + ν w,ss ) · (δw,n quad − δw,n lin )} ds . Applying the extended work equation, the continuity conditions for w,n and δw,n can be fulfilled weakly at the system level, too.

6.4.2 Example of Use The example deals with the square plate with clamped and hinged boundaries which is already investigated in Section 6.2.9 with respect to a constantly distributed loading, cf. Figure 6-14. x y

GE

clamped constantly distributed loading: p = 1 kN/m2 7 2 E = 3.0 . 10 kN/m

E

T

hinged

M

ly

t = 0.1 m G

ν = 0.2 l x = 10.0 m l y = 10.0 m

lx

Fig. 6-14 Quadratic plate – system and loading, see Section 6.2.9 The deflection at the centre of the plate is given in Table 6.3 concerning the element with 12 DOF, taking into account boundary integrals of work, and applying it to different meshes. The results obtained by the modifications are excellent, cf. Figure 6-15, where the results are compared to the different rectangular elements with 16 DOF and 12 DOF respectively. The deflection converges almost as well as for the element with 16 DOF.

6.4 The 12 DOF Element Employing a Weak Conformity

131

The convergence of the bending moments is comparably good. This is confirmed for the bending moments mxx and myy at the centre of the plate M , see Table 6.3. Since the twisting w,xy is no longer a nodal degree of freedom, the twisting moment has to be computed inside the elements by means of the differentiated shape functions. Thus the twisting moment mxyT comprises the two values of the neighboring elements. Table 6.3 12 DOF element – convergence of state variables 1×1

2×2

4×4

8×8

16 × 16

wM

[cm]

0. 7692

0. 7449

0. 7408

0. 7376

0. 7365

mxyT

[kNm/m]

mxxM

[kNm/m]

2. 564

−1. 648 −1. 740

−1. 723 −1. 587

−1. 566 −1. 556

−1. 543 −1. 542

myyM

[kNm/m]

5. 128

3. 204

3. 181

3. 170

3. 167

myyE

[kNm/m]

qxG

[kN/m]

−4. 808

−6. 036

−6. 705

−6. 910

−6. 965

qyE

[kN/m]

2. 24

3. 27

4. 16

4. 65

4. 91

normalized centre deflection

element pattern

0. 96

1. 869

0. 46

2. 097

0. 96

2. 140

1. 22

2. 150

1. 36

1.3

1.2

16 DOF element 12 DOF element 12 DOF element, weak conformity

1.1

1.0 1

4

16 64 number of elements

256

Fig. 6-15 Quadratic plate – convergence of the centre deflection wM

TRIANGULAR ELEMENTS

7 Triangular Elements – Description of Geometry

A disadvantage of rectangular elements is undeniably the restricted field of application, which, according to Section 4, 5 and 6, is limited to rectangular geometries. Furthermore the mesh refinement, often required, is very inefficent if it covers the whole domain. An element geometry allowing the discretization of arbitrary geometries as well as an efficent mesh refinement is the triangle. Triangular elements seem to have the ideal geometry for the FEM, but the derivation of the element matrices is slightly more difficult when compared to rectangular elements. The rectangular elements previously studied are described in the orthogonal x–y–coordinate system, so that the edges of the elements are parallel to the direction of the coordinates. Therefore, the choice of shape functions and the integration of the element matrices as well as the load vectors turn out to be relatively straight forward. This only holds for right–angled triangles if the coordinate system forms the right angle. In contrast, an arbitrary triangle would have to be divided into multiple integration domains when integrated in x– and y–coordinates, which is very time–consuming. The definition of shape functions related to triangular geometries in x–y–coordinates would become very complex, too, because in general, distinctions between different cases would be necessary. Thus when employing triangular elements, it is more advantageous to apply a local element coordinate system. Since the work equation Z Z −δWel = δvT · { ΩT · DT · E · D · Ω dxdy · v − ΩT · p dxdy} (7.1) is related to the x–y–coordinates it turns out, that applying a local coordinate system needs a transformation of the coordinates, a transformation of the derivatives with respect to the coordinates, and a transformation of the integration rules.

7.1 Local ξ–η–Coordinate System In addition to the global x–y–coordinate system, in which the structure is located and in which the governing equations as well as the weak formulation are described, local ξ–η–coordinates are defined with respect to two of the element edges, see Figure 7-1. Each of the three element corner nodes is equivalent to the others and may be chosen as the origin of the local coordinate system. © The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2024 D. Dinkler und U. Kowalsky, Introduction to Finite Element Methods, https://doi.org/10.1007/978-3-658-42742-9_7

136

7 Triangular Elements – Description of Geometry

Because of the coordinate invariance, this must not influence the results. The ξ–η–coordinates are normalised to the length of the element edges ℓAB and ℓAC , so that the conditions 0≤ξ≤1

and

0≤η≤1

are fulfilled. xc

xa A

ya

η yc

xb

x

( ξ = 0; η = 0 ) ξ

C ( 0;1 )

yb

B ( 1;0 )

y

Fig. 7-1 The coordinate system for triangular elements With regard to the element domain, the shape functions and the integration limits can be clearly formulated by applying local coordinates.

7.1.1 Transformation of Coordinates The transformation from one coordinate system into the other can be performed by means of the corner coordinates, see Figure 7-1, applying x = xa + (xb − xa ) · ξ + (xc − xa ) · η , y = ya + (yb − ya ) · ξ + (yc − ya ) · η . Thus it might be useful to employ the following abbreviations concerning the coordinate differences of the related element nodes aa = xc − xb , ab = xa − xc , with aa + ab + ac = 0 , ac = xb − xa and

ba = yb − yc , bb = yc − ya , bc = ya − yb

with

ba + bb + bc = 0 .

7.1 Local ξ–η–Coordinate System

137

Hence the matrix notation of the transformation of coordinates gives " # " # " # " # x xa ac −ab ξ + = · . y ya −bc bb η

(7.2)

The inverse representation is needed with respect to the re–transformation. The matrix notation yields " # " # " # " # ξ b b ab x xa 1 = · ·{ − } (7.3) det η b c ac y ya applying det = ac · bb − ab · bc .

7.1.2 Transformation of Partial Derivatives The work equation comprises the partial derivatives of the displacement field u(x, y) with respect to the coordinates x and y. If the displacement field is described in relation to local coordinates ξ and η, the derivatives of the displacement field with respect to the coordinates ξ and η need to be transformed in accordance with the chain rule ∂u ∂u ∂ξ ∂u ∂η = · + · , ∂x ∂ξ ∂x ∂η ∂x ∂u ∂u ∂ξ ∂u ∂η = · + · . ∂y ∂ξ ∂y ∂η ∂y The derivatives of the local coordinates with respect to the global coordinates follow Equation (7.3), e.g. ∂ξ ∂ 1 bb = { · [ bb (x − xa ) + ab (y − ya ) ]} = . ∂x ∂x det det Concerning all derivatives, the matrix notation yields    ∂ξ ∂η    " ∂u ∂u bb 1 ∂x ∂x ∂x ∂ξ  =  = · ∂u ∂ξ ∂η ∂u det ab ∂y

∂y

∂y

The inverse representation gives   " 

∂u ∂ξ ∂u ∂η

=

∂η

ac −ab

−bc bb

# 

∂u ∂x ∂u ∂y



.

bc ac

# 

∂u ∂ξ ∂u ∂η



.

(7.4)

(7.5)

138

7 Triangular Elements – Description of Geometry

The coefficient matrix of the inverse representation is named Jacobian matrix J. The determinant of the Jacobian matrix gives det J = ac · bb − ab · bc . Thus J=

"

ac −ab

holds as well as −1

J

1 = det J

"

−bc bb bb ab

#

bc ac

(7.6) #

.

7.1.3 The Integration of the Element Area To prepare the element stiffness matrix, at integration of the work performed in the respective element, the differential domain dA = dx · dy needs to be described by means of an equivalent expression in dξ and dη. Using vector calculus, an area can be computed by the vector product of the vectors spanning the domain. As shown in Figure 7-2, the differential vectors in the direction of the local coordinates are ξ = 0, η = 0

ξ=0



η=0

A

A

η η=1

ξ=1

ξ

  (xb − xa ) dξ dvξ =  (yb − ya ) dξ  =  0    (xc − xa ) dη dvη =  (yc − ya ) dη  =  0

 ac dξ −bc dξ  , 0  −ab dη bb dη  . 0

Fig. 7-2 Description of the triangle’s area The vector product provides a third vector whereby its norm is related to the value of the area     ex ey ez 0 . 0 dvξ × dvη = det  ac dξ −bc dξ 0  =  −ab dη bb dη 0 [ ac · bb − ab · bc ] dξ dη

7.1 Local ξ–η–Coordinate System

139

The component in z–direction gives the area of the differential parallelogram spanned by the coordinates dξ and dη, cf. Figure 7-2, dA = [ ac · bb − ab · bc ] dξ dη = det J dξ dη .

(7.7)

At integration of the work related to the domain of the triangle, the integration limits need to be adapted to the triangular geometry. Because of the crooked boundaries, the upper integration limit is variable in the ξ–direction, Z Z Z 1 Z 1−η Integrand dx dy = Integrand · det J dξ dη . y

0

x

0

Thus the area of the triangular domain can be evaluated Z 1 Z 1−η Z 1 1 A∆ = det J dξ dη = (1−η) det J dη = [ η−η 2 /2 ]10 det J = det J . 2 0 0 0 When regarding integrals employing polynomials of an arbitrary order, the solution can be analytically obtained by Z 1 Z 1−η i! k! ξ i · η k dξdη = . (7.8) (i + k + 2)! 0 0

7.1.4 Linear Shape Functions and Nodal Degrees of Freedom The linear shape functions employed to describe the displacement field are chosen as shown in Figure 7-3. Each shape function describes a plane tilted 1

1 B

A

A

B

A

B 1

C

C

C

Fig. 7-3 Linear shape functions regarding triangular elements over one edge of the element wherein its value is equal to one regarding the opposite node u = [ u(x, y) ] = Ω · v   uA   = [ φA φB φC ]  uB  , uC

140

7 Triangular Elements – Description of Geometry

with the nodal degrees of freedom v T = [ uA uB uC ] and the shape functions φa = 1 − ξ − η,

φb = ξ ,

φc = η .

Such, all shape functions are equivalent regarding the corner nodes A, B and C, see Figure 7-3.

7.2 Description Employing Area Coordinates As an alternative, the stiffness matrix can be derived employing area coordinates. In comparison to the local ξ–η–coordinates, area coordinates have the advantage that the shape functions are clearly recognized ab initio as being symmetric concerning all element nodes and thus the element stiffness matrix will be. The position of an arbitrary point P within a triangular element can be described by means of the triangles formed by the lines connecting P with the element corners, cf. Figure 7-4. xp

x

A ( λ a = 1, λ b = 0, λ c = 0 )

yp C ( 0, 0, 1 )

Ab Ac P Aa B ( 0, 1, 0 )

y

Fig. 7-4 The definition of area coordinates The triangle is subdivided into the three triangles Aa , Ab and Ac . The indices correspond to the opposite corner points. The arbitrary position of P within the triangle is set by means of the area coordinates λa =

Aa , A△

λb =

Ab , A△

λc =

Ac , A△

(7.9)

7.2 Description Employing Area Coordinates

141

whereby, because of Aa + Ab + Ac = A△ , it holds additionally λa + λb + λc = 1 .

(7.10)

The area coordinates fulfill the constraint (7.10), so only two independent coordinates act here as well. The area coordinates λa , λb and λc are normalized and thus are independent of the size of the element as well as of its position in the x–y–coordinate system.

7.2.1 Transformation between Cartesian and Area Coordinates The coordinates x and y of an arbitrary point P (x, y) inside the triangle can be described by means of area coordinates as follows: x = xa · λa + xb · λb + xc · λc , y = ya · λa + yb · λb + yc · λc .

(7.11)

This can be controlled very easily with respect to the corner nodes regarding Figure 7-4, as the value for the related λ has to equal 1, and the two others must equal 0. At the midpoints of the element edges, two values for λ equal 1/2, and the third one equals 0. Concerning the centre of the triangle, all coordinates equal 1/3. Taking into account the constraint (7.10) with respect to λi the transformation instruction applies in the matrix notation       1 1 1 1 λa  x  =  xa xb xc  ·  λb  . (7.12) y ya yb yc λc The re–transformation tion    λa 1      λb  = det λc

may be performed by applying the inverse representaxb yc − xc yb xc ya − xa yc xa yb − xb ya

yb − yc yc − ya ya − yb

   xc − xb 1    xa − xc  ·  x  . xb − xa y

The determinant of the coefficient matrix equals twice the triangular area, cf. Section 7.1.3 det

1 xa ya

1 xb yb

1 xc yc

= (xb yc − xc yb ) − (xa yc − xc ya ) + (xa yb − xb ya ) = (xb − xa )(yc − ya ) − (xc − xa )(yb − ya ) = ac · bb − ab · bc = 2A△ .

142

7 Triangular Elements – Description of Geometry

The re–transformation can be shortened to    xb yc − xc yb ba λa 1     xc ya − xa yc bb  λb  = 2A∆ xa yb − xb ya bc λc

   aa 1    ab  ·  x  ac y

(7.13)

by employing the abbreviations defined above. Subsequently, the re–transformation is required to perform the transformation of the derivatives.

7.2.2 Transformation of Partial Derivatives The differentiation of a displacemente field u(x, y), defined with respect to the Cartesian coordinates x and y, can be performed by applying the chain rule, if the shape functions for the description of the displacement are given employing area coordinates. Therefore it holds that ∂u ∂u ∂λa ∂u ∂λb ∂u ∂λc = · + · + · , ∂x ∂λa ∂x ∂λb ∂x ∂λc ∂x (7.14) ∂u ∂u ∂λa ∂u ∂λb ∂u ∂λc = · + · + · . ∂y ∂λa ∂y ∂λb ∂y ∂λc ∂y The derivatives of the area coordinates can be computed with respect to the Cartesian coordinates regarding Equation (7.13) ∂λa ∂x ∂λb ∂x ∂λc ∂x ∂λa ∂y ∂λb ∂y ∂λc ∂y

= = = = = =

1 (yb − yc ) 2A△ 1 (yc − ya ) 2A△ 1 (ya − yb ) 2A△ 1 (xc − xb ) 2A△ 1 (xa − xc ) 2A△ 1 (xb − xa ) 2A△

= = = = = =

1 2A△ 1 2A△ 1 2A△ 1 2A△ 1 2A△ 1 2A△

ba , bb , bc , aa , ab , ac .

Thus it follows from Equation (7.14)   ∂u 1 ∂u ∂u ∂u = ba + bb + bc , ∂x 2A△ ∂λa ∂λb ∂λc   ∂u 1 ∂u ∂u ∂u = aa + ab + ac . ∂y 2A△ ∂λa ∂λb ∂λc

(7.15)

7.2 Description Employing Area Coordinates

143

A Jacobian matrix can be formally given, as with ξ–η–coordinates, as a 2 × 3– matrix, but this is of no further relevance.

7.2.3 Integration with Respect to Area Coordinates The integration of the work with respect to the area coordinates has to take into account, that the three coordinates λa , λb , λc must satisfy the condition (7.10). Thus, the integration has to be performed with respect to two area coordinates only. Analogously to Figure 7-2 the differential area dx dy = detJ dξ dη may also be transformed to the area coordinates, if dξ is replaced by dλa and dη is replaced by dλb . Z Z Z 1 Z 1−λb Integrand dx dy = Integrand · det J dλa dλb . y

0

x

0

Hence, the area of the triangular domain can be evaluated with Z 1 Z 1−λb Z 1 A∆ = det J dλa dλb = (1 − λb ) det J dλb 0

0

0

1 = [ λb − λ2b /2 ]10 det J = det J . 2

The integration of the individual terms is performed by applying the integration rule related to the product terms of the area coordinates ZZ i! j! k! λia · λjb · λkc dλa dλb = . (7.16) (i + j + k + 2)! It should be noted that λc is dependent on λa and λb for example, therefore the integration has to be performed with respect to two area coordinates only.

7.2.4 Shape Functions Employing Area Coordinates Considering the same nodal displacements in the same order as in Section 7.1.2, the general shape of the matrix comprising the shape functions remains the same, see Figure 7-3. The representation of the shape functions employing area coordinates and their respective derivatives are symmetric according to the area coordinates: φA = λa ,

φB = λb ,

φC = λc .

The comparison with the shape functions depicted in Figure 7-3 shows that they are identical.

144

7 Triangular Elements – Description of Geometry

Shape functions of higer order The development of complete shape functions of higher order employing area coordinates happens differently from the procedure concerning ξ–η–coordinates, whereby there is no representation comparable to Pascale’s triangle. However, the number of the independent shape functions remains the same as in Pascale’s triangle. Thus a complete approach of n–th order includes all (n + 1) · (n + 2)/2 possible products of n–th order of the area coordinates. As such, a linear approach employs three independent functions that depend linearly on the area coordinates λa , λb , λc . A quadratic approach comprises six independent quadratic shape functions λ2a , λ2b , λ2c , λa · λb , λa · λc , λb · λc . A scaling to the corner and the mid points of the edges leads to, cf. Figure 7-5, φA = λa (2λa −1) , φB = λb (2λb −1) , φC = λc (2λc −1) , φD = 4 λa λb ,

φE = 4 λb λc ,

φF = 4 λa λc .

1 1

A

C

1 B 1 1/4 1 D

1/4 1

F

1/4

E

Fig. 7-5 Quadratic shape functions A cubic approach comprises ten independent cubic shape functions λ3a , λ3b , λ3c , λ2a · λb , λ2a · λc , λ2b · λa , λ2b · λc , λ2c · λa , λ2c · λb , λa · λb · λc . Shape functions of higher orders can be developed appropriately.

8 Triangular Elements to Describe Heat Conduction

The basic idea when performing the element matrix as well as the load vector is still the weak formulation of the energy balance equation applying the matrix notation Z Z T T T −δRel = δv · { Ω · D · E · D · Ω dxdy · v − ΩT · p dxdy} . (8.1) At first, the element matrices regarding elements with a linear temperature approach are derived, describing constant heat fluxes inside the element. Elements employing a quadratic or even higher polynomial approach may be derived but are not looked at here. Concerning linear shape functions, all convergence criteria are fulfilled. The derivation of the element matrices does not essentially change compared with rectangular elements, thus the presentation is focused on the application of the local element coordinate system.

8.1 A Linear Approach Related to the ξ–η–Coordinate System The approach to describe the temperature inside the element domain can be chosen in relation to the local coordinate system, since all necessary transformations are known between the coordinate systems. The convergence criteria need to be fulfilled and checked as with any other type of element. With regard to heat conduction, at least Co –conform shape functions are to be employed, which have to be complete shape functions to fulfill the criterion of coordinate invariance in the case of triangular elements.

8.1.1 Linear Shape Functions and Nodal Unknowns The linear shape functions employed to describe the temperature field are chosen as shown in Figure 8-1. Such, all shape functions are equivalent concerning the corner nodes A, B and C. Each linear function describes a plane tilted over one edge of the element wherein its value is equal to one regarding the opposite node u = [T ] = Ω·v with the nodal degrees of freedom v T = [ TA TB TC ] . © The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2024 D. Dinkler und U. Kowalsky, Introduction to Finite Element Methods, https://doi.org/10.1007/978-3-658-42742-9_8

146

8 Triangular Elements to Describe Heat Conduction

The shape functions chosen with Ω and their respective derivatives are: φA = 1 − ξ − η φA ,ξ = −1 φA ,η = −1

φB = ξ φB ,ξ = 1 φB ,η = 0

φC = η φC ,ξ = 0 φC ,η = 1 1

1 A

B

A

B

A

B 1

C

C

C

Fig. 8-1 Linear shape functions regarding triangular elements Hence it applies explicitly to 

T (x, y)



=



φA

φB

φC

  TA    TB  . TC

8.1.2 Differentiation of the Temperature Field In the weak form of the balance equation, the differentiation rule related to the computation of the heat fluxes from the temperatures is comprised of the derivatives with respect to the x–y–coordinates. If the shape functions are described in local ξ–η–coordinates, the derivatives need to be transformed with respect to the local coordinates. Thus the operator matrix D is transformed according to Section 7.1.2 to  ∂ξ  " # " # ∂η ∂ + ∂ ξ η −∂x bb ∂ξ + bc ∂η 1 ∂x ∂x =− D= = −  ∂ξ , (8.2) ∂η det J ab ∂ξ + ac ∂η −∂y ∂ξ + ∂η ∂y

∂y

whereby the negative sign is positioned in front of the matrix for convenience here. Thus the matrix B, comprising the derivatives of the shape functions, yields B = D·Ω 1 =− · det J

""

bb φA ,ξ + bc φA ,η ab φA ,ξ + ac φA ,η

#

A

"

#

B

"

#

C

#

8.1 A Linear Approach Related to the ξ–η–Coordinate System =− =−

1 · det J 1 · det J

 "

−bb − bc −ab − ac



ba

bb

bc

aa

ab

ac



A

#

bb ab



B



bc ac



C

.

147

 (8.3)

The matrix B turns out to be constant. Because the heat flux is also computed by means of B, the heat flux is also constantly distributed inside the element. Considering the chosen abbreviations for the coordinate differences, B may be recognized now to be symmetric with respect to all element nodes. It is remarkable that the order of the shape functions is reduced by one relating to the whole element, not only in the respective direction like it does concerning rectangular elements employing symmetric shape functions.

8.1.3 Element Matrix The element matrix K follows analogously to Section 4 Z K= BT · E · B dA A 1

=

Z

0

Z

1−η

0

BT · E · B · det J dξ dη .

In addition to the matrix of heat conductivity E " # λ E= , λ B is also constant here. When applying det J = 2A△ , the whole integrand turns out to be constant. Thus, in this case, the element matrix is performed by matrix multiplication BT · E · B analogously to the multiplication scheme given in Section 4.2 and finally by multiplying the product by the factor det J ·

1 1 = 2 A△ · = A△ . 2 2

Comparing to Section 4.2, the element matrix comprises three rows and three columns, that are related to the virtual as well as to the real nodal temperatures. This yields   b a · b a + aa · aa b a · b b + aa · ab b a · b c + aa · ac λ K= ·  b b · b a + ab · aa b b · b b + ab · ab b b · b c + ab · ac  . 4A△ b c · b a + ac · aa b c · b b + ac · ab b c · b c + ac · ac

148

8 Triangular Elements to Describe Heat Conduction

8.1.4 Vector of Thermal Action At first, the vector of thermal action is computed for the special case of element oriented, constantly distributed external energy input, cf. Section 4.2. With p = [ qV ] = constant , this yields f=

Z

A

ΩT · p dA =

Z

A

ΩT dA · p .

In the case of linear shape functions, the integral can be solved independently of the chosen coordinate system. The external energy input is therefore equally distributed to the three nodes by 1/3 each, because the shape functions are equivalent     f =  

A△ 3 A△ 3 A△ 3

   · [ qV ] .  

A fluctuating external energy input may be described and taken into account at performing the integration by means of the matrix of shape functions and the nodal values of qV .

8.1.5 Subsequent Flux Analysis Employing the chosen linear shape functions for the description of the temperature field, the matrix of the derivatives B is constant in terms of each entry, resulting in constant heat fluxes. Therefore, the heat fluxes qx , qy are evaluated only once for the whole element and can be assigned, for example, to the centre of the element. At first, this holds in general " # qx σ= = E·B·v = S·v. qy The stress matrix S follows, regarding the chosen linear shape functions to " # ba bb bc λ S=E·B=− · . 2F△ aa ab ac

8.2 Example of Use

149

8.2 Example of Use The following example is already investigated applying rectangular elements in Section 4.3. Figure 8-2 shows the geometry and the dimensions [ m ] of a horizontal cut in the region of the entrance of a building. A constant temperature of 220 C within the building and 60 C outside the building are assumed as Dirichlet boundary conditions for the temperature field inside the wall. Cauchy boundary conditions are not taken into account. The different materials are indicated by different gray scale values. λ1 = 0. 08 W/mK is valid for brickwork and λ2 = 0. 12 W/mK is valid for wood. The discretization with triangular elements is represented in Figure 8-2 as well. 0.50

1.25 0.125 λ 2 = 0.12

22 o

0.50 A

B

0.125

6o λ 1 = 0.08

0.50

0.25 y x

0.75

0.25

0.75

Fig. 8-2 Brickwork – Geometry and heat conduction coefficients λ [ W/mK ] The results for linear shape functions, cf. Figures 8-3, 8-4 and 8-5, and for quadratic shape functions, cf. Figures 8-6, 8-7 and 8-8, are plotted inside the elements, whereat the heat fluxes are discontinous at the element intersections. Figures 8-9, 8-10 and 8-11 represent the results after averaging them at the nodes of neighboring elements. The heat conduction process results in a stationary temperature field. The temperature field is nearly linear in regions, where an almost linear heat conduction without disturbance exists – see cross sections A and B. Thermal bridges develop close to the outer corners, where the boundary conditions dominate the temperature field inside the wall in two directions. The heat fluxes qx [ W/m2 ] and qy [ W/m2 ] are correlated to the gradients of the temperature field. Thus the heat flux is larger in the direction of the shorter transport distance, cf. qy close to the cross sections A and B, and larger in the region of the larger heat conduction coefficient – right part of the structure.

150

8 Triangular Elements to Describe Heat Conduction

22. 0 o

C

6. 0 Fig. 8-3 Brickwork, linear shape functions – temperature T (x, y)

5. 8 W/m2 0. 0 Fig. 8-4 Brickwork, linear shape functions – heat flux qx (x, y)

−0. 7 W/m2 −15. 4 Fig. 8-5 Brickwork, linear shape functions – heat flux qy (x, y)

8.2 Example of Use

151

22. 0 o

C

6. 0 Fig. 8-6 Brickwork, quadratic shape functions – temperature T (x, y)

8. 4 W/m2 −1. 2 Fig. 8-7 Brickwork, quadratic shape functions – heat flux qx (x, y)

0. 2 W/m2 −17. 2 Fig. 8-8 Brickwork, quadratic shape functions – heat flux qy (x, y)

152

8 Triangular Elements to Describe Heat Conduction

22. 0 o

C

6. 0 Fig. 8-9 Brickwork, quadratic shape functions – temperature T (x, y)

8. 4 W/m2 −1. 2 Fig. 8-10 Brickwork, quadratic shape functions – averaged heat flux qx (x, y)

0. 2 W/m2 −17. 2 Fig. 8-11 Brickwork, quadratic shape functions – averaged heat flux qy (x, y)

9 Triangular Elements for Membrane Structures

In this section different approaches concerning the choice of the applied coordinate system are represented when employing triangular elements for the description of plane stress linear elasticity. The procedure to derive triangular elements for the description of linear elasticity is almost the same as for the description of stationary heat conduction. Only the transformation of the partial derivatives has to be extended to a vector notation regarding the primary variables. Again, basis of the development of the element stiffness matrix and the load vector is the matrix notation of the work equation Z Z −δWel = δvT · { ΩT · DT · E · D · Ω dxdy · v − ΩT · p dxdy} . (9.1) The description of the geometry, the transformation of coordinates, and the transformation of derivatives follow Section 7 for the ξ–η–coordinates as well as for the area coordinates λa , λb , λc . Thus the following sections describe triangular element formulations for both coordinate systems and for linear as well as quadratic shape functions. Basis of the development of the stiffness matrix and the load vector is the choice of the approximation of the displacement fields, in matrix notation u = Ω·v, which have to be applied to the shape functions and the coordinate system.

9.1 Linear Shape Functions with Respect to ξ–η–Coordinates The approach to describe the displacements u(x, y), v(x, y) inside the element domain is chosen in relation to the local coordinate system. The convergence criteria need to be fulfilled and checked as with any other type of element. With regard to membrane elements, at least Co −conform shape functions are to be employed, which have to be complete shape functions to fulfill the criterion of coordinate invariance in the case of triangular elements.

9.1.1 Shape Functions and Nodal Unknowns The linear shape functions employed to describe the displacements are chosen as shown in Figure 9-1. Such, all shape functions are equivalent concerning the © The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2024 D. Dinkler und U. Kowalsky, Introduction to Finite Element Methods, https://doi.org/10.1007/978-3-658-42742-9_9

154

9 Triangular Elements for Membrane Structures

corner nodes A, B and C. Each linear function describes a plane tilted over one edge of the element wherein its value is equal to one regarding the opposite node. As represented in Section 8.1.1, the shape functions and their respective derivatives are given with: φA = 1 − ξ − η φA ,ξ = −1 φA ,η = −1

φB = ξ φB ,ξ = 1 φB ,η = 0

φC = η φC ,ξ = 0 φC ,η = 1 1

1 A

B

A

B

A

B 1

C

C

C

Fig. 9-1 Linear shape functions regarding triangular elements According to the plane stress situation, cf. Section 5.1, the displacements u(x, y) and v(x, y) have to be approximated, which yields the nodal unknowns v of linear shape functions vT = [ uA vA uB vB uC vC ] . Hence the matrix notation of the displacement fields applies explicitly to

"

u v

#

=

"

φA

φB φA



φC φB

φC

 #       

uA vA uB vB uC vC



    .    

9.1.2 Differentiation of Displacement Fields In the work equation, the differentiation rules related to the computation of the strains from the displacements are comprised of the derivatives with respect to the x–y–coordinates. Because the shape functions are described in local ξ–η– coordinates, their derivatives need to be transformed with respect to the local coordinates. Thus the operator matrix D of the plane stress is transformed

9.1 Linear Shape Functions with Respect to ξ–η–Coordinates according to Section 7.1.1 to    ∂x       D= ∂y  =     ∂y ∂x =



1   det J

∂ξ ∂ ∂x ξ

∂ξ ∂ ∂y ξ

+

+

∂η ∂ ∂x η

∂η ∂ ∂y η

∂ξ ∂ ∂y ξ ∂ξ ∂ ∂x ξ

+ +

∂η ∂ ∂y η ∂η ∂ ∂x η

    



bb ∂ξ + bc ∂η ab ∂ξ + ac ∂η

155

 ab ∂ξ + ac ∂η  . bb ∂ξ + bc ∂η

(9.2)

Applying the differentiation rules to the shape functions yields the matrix B, which is needed to compute the stiffness matrix, B = D·Ω



1  ·  det J



bb φA ,ξ + bc φA ,η











      ab φA ,ξ + ac φA ,η       ab φA ,ξ + ac φA ,η bb φA ,ξ + bc φA ,η A B C        −bb − bc bb bc 1        −ab − ac   ab  ac   = ·   det J −ab − ac −bb − bc A ab b b B ac b c C   ba bb bc   1 = · aa ab ac  (9.3)  . det J aa b a ab b b ac b c =

The matrix B turns out to be constant. Because the strains and therefore also the stresses are computed by means of B, they are constant inside the element, too. Considering the chosen abbreviations for the coordinate differences, B may be recognized now to be symmetric with respect to all element nodes.

9.1.3 Element Stiffness Matrix The matrix formulation of the element stiffness matrix K follows analogously to Section 5 Z Z 1 Z 1−η K= BT · E · B dA = BT · E · B · det J dξ dη . A

0

0

156

9 Triangular Elements for Membrane Structures

In addition to the elasticity matrix E, here, B is also constant. When applying det J = 2A△ , the whole integrand turns out to be constant. Thus in this case, the element stiffness matrix is performed by matrix multiplication BT · E · B analogously to the multiplication scheme given in Section 5.1 and finally by multiplying the product by the factor det J ·

1 1 = 2 A△ · = A△ . 2 2

The element stiffness matrix may be summed up by three parts, which can be assigned to the three different components of the elasticity matrix, as shown in Section 5.1. With K = K1 + K2 + K3 this yields

E K1 = 4A△ (1 − ν 2 )

νE K2 = 4A△ (1 − ν 2 )



ba · ba

   bb · ba ·    bc · ba 

 aa · b a   ·  ab · b a   ac · b a 

   E K3 = ·  8A△ (1 + ν)   

aa · aa b a · aa ab · aa b b · aa ac · aa b c · aa

aa · aa ab · aa ac · aa b a · aa b b · aa b c · aa aa · b a ba · ba ab · b a bb · ba ac · b a bc · ba

ba · bb bb · bb bc · bb

aa · b b ab · b b ac · b b aa · ab b a · ab ab · ab b b · ab ac · ab b c · ab

aa · ab ab · ab ac · ab b a · ab b b · ab b c · ab aa · b b ba · bb ab · b b bb · bb ac · b b bc · bb

ba · bc bb · bc bc · bc

aa · b c ab · b c ac · b c aa · ac b a · ac ab · ac b b · ac ac · ac b c · ac



aa · ac    , ab · ac    ac · ac b a · ac



aa · b c ba · bc ab · b c bb · bc ac · b c bc · bc



  b b · ac  ,   b c · ac     .   

For the same reasons as regarding rectangular elements, the sums over the row’s components as well as over the column’s components of the element stiffness matrix must turn out to be identical to zero.

9.1 Linear Shape Functions with Respect to ξ–η–Coordinates

157

9.1.4 Load Vector At first, the load vector is computed for the special case of constantly distributed external actions, cf. Section 5.1. With " # px p= = constant , py this yields f=

Z

A

ΩT · p dA =

Z

A

ΩT dA · p .

In the case of linear shape functions, the integral can be solved independently of the chosen coordinate system. The load is therefore equally distributed to the three nodes by 1/3 each, because the shape functions are equivalent        f =     

F△ 3 F△ 3 F△ 3

F△ 3 F△ 3 F△ 3

   " #   px · .  py    

A fluctuating external action may be described and taken into account analogously to Section 5.1.7, when performing the integration by means of the matrix of shape functions and the nodal values of the loading.

9.1.5 Subsequent Stress Analysis Concerning the chosen linear shape functions for the description of the displacements, the matrix of strains B is constant in terms of each element, resulting in constant stresses. Therefore, the stresses are evaluated only once for the whole element and can be assigned, for example, to the centre of the element. At first, this holds in general σ = E·B·v = S·v. The stress matrix S follows, employing the chosen linear shape functions and the abbreviation e = (1 − ν)/2, to   ba νaa bb νab bc νac E ·  νba aa νbb ab νbc ac  . S= E·B= 2A△ (1 − ν 2 ) eaa eba eab ebb eac ebc

158

9 Triangular Elements for Membrane Structures

9.2 Description Employing Area Coordinates λa , λb , λc As an alternative, the stiffness matrix can be derived employing area coordinates. In comparison to the local ξ–η–coordinates, area coordinates have the advantage that the shape functions and thus the element stiffness matrix are clearly recognized ab initio as being symmetric concerning all element nodes.

9.2.1 Linear Shape Functions Employing Area Coordinates The shape functions employing area coordinates are symmetric according to the area coordinates, cf. Figure 9-2, and the comparison with the shape functions depicted in Figure 9-1 shows that they are of the same shape. φA φA ,λa φA ,λb φA ,λc

= λa =1 =0 =0

φB φB ,λa φB ,λb φB ,λc

= λb =0 =1 =0

φC φC ,λa φC ,λb φC ,λc

= λc =0 =0 =1

1

1 B

A

A

B

A

B 1

C

C

C

Fig. 9-2 Linear shape functions regarding area coordinates Analogously to Section 9.1.2 and taking into account Section 7.2.2, the operator matrix D results in   P c ∂λi ∂   λ i   i=a ∂x ∂   c  x    P ∂λi    ∂  D= ∂y  =  ∂y λi  i=a   ∂y ∂x c c   P P ∂λi ∂λi ∂ ∂ ∂y λi ∂x λi i=a

=

1 2A△

i=a

 P c b ·∂  i=a i λi      c  P ai ·∂λi i=a

c P

i=a c P i=a



ai ·∂λi bi ·∂λi

   .   

9.3 Quadratic Approach Employing ξ–η–Coordinates

159

Therefore the derivative of the shape functions can be represented by   ba bb bc 1   aa ab ac  . B=D·Ω= · 2A△ aa b a ab b b ac b c

The matrix B is constant and of course can be derived identically to Section 9.1.2. Hence the element stiffness matrix K, as well as the element load vector p and the stress matrix S, are also performed identically to the matrices presented in Sections 9.1.3, 9.1.4 and 9.1.5. By employing area coordinates, the integration is performed with respect to the chosen notation as shown explicitly in Sections 9.1.3 and 9.1.4. The formulation employing area coordinates offers the advantage of shape functions being clearly arranged as well as being equivalent for all nodal displacements.

9.3 Quadratic Approach Employing ξ–η–Coordinates The previously chosen linear approach to describe the displacements has the disadvantage of only constantly approximating the stresses inside an element. A quadratic approach to describe the displacements allows for a linear distribution of strains and stresses. The complete quadratic approach is comprised of six DOF regarding triangles and, when related to u(x, y) for example, the mathematical representation holds that u(x,y) = a0 + a1 x + a2 y + a3 x2 + a4 xy + a5 y 2 , which can be checked by means of Pascale’s triangle, see Section 2.4.2. The displacements of the corner nodes A to C and of the midpoints of the edges D to F are chosen to represent the DOF of the physically meaningful approach, cf. Figure 9-3. x A ( ξ = 0, η = 0 ) F ( 0, 0.5 ) C ( 0, 1 )

D ( 0.5, 0 )

η E ( 0.5, 0.5 )

y

B ( 1, 0 )

ξ

Fig. 9-3 Nodal coordinates concerning the 6–nodes triangular element

160

9 Triangular Elements for Membrane Structures

9.3.1 Quadratic Shape Functions At first, the local ξ–η–coordinate system is also chosen for the 6–nodes element, cf. Figure 9-3. The shape functions are chosen according to the physically meaningful degrees of freedom and are depicted in Figure 9-4. φA = (1−ξ−η) · (1−2ξ−2η) φA ,ξ = −3 + 4ξ + 4η φA ,η = −3 + 4ξ + 4η

φB = ξ (2ξ − 1) φB ,ξ = 4ξ − 1 φB ,η = 0

φC = η (2η−1) φC ,ξ = 0 φC ,η = 4η − 1

1 1

A

C

1 B 1 1/4

1/4

1

1

D

F

1/4

E

φD = 4ξ (1 − ξ − η) φD ,ξ = 4 − 8ξ − 4η φD ,η = −4ξ

φE = 4ξη φE ,ξ = 4η φE ,η = 4ξ

φF = 4η (1 − ξ − η) φF ,ξ = −4η φF ,η = 4 − 4ξ − 8η

Fig. 9-4 Quadratic shape functions concerning triangular elements The approach of the displacements and analogously of the virtual displacements is given with u=

"

u(ξ, η) v(ξ, η)

#

= Ω·v,

whereat the nodal displacments v as well as the matrix of shape functions Ω

9.3 Quadratic Approach Employing ξ–η–Coordinates

161

need to be extended to the 6 nodes A to F according to Figure 9-3 # " φA φB φC φD φE φF , Ω= φA φB φC φD φE φF   vT = uA vA uB vB uC vC uD vD uE vE uF vF .

The transformation, the partial derivatives and the notes related to the integration given in Section 9.1 are also valid here and are independent of the shape functions. Thus the operator matrix D remains unchanged when compared to Section 9.1.2. Because the number and particularly the courses of the shape functions φi in Ω are changed, the differentiation rule D leads to different derivatives regarding the matrix B:   bb φA ,ξ bb φB ,ξ ...  + bc φA ,η + bc φB ,η ...        a φ , a φ , . . . 1 b A ξ b B ξ   B=D·Ω= · . det J  + ac φA ,η + ac φB ,η . . .       ab φA ,ξ bb φA ,ξ ab φB ,ξ bb φB ,ξ . . .  + ac φA ,η

+ bc φA ,η

+ ac φB ,η

+ bc φB ,η

...

The matrix B is represented here only concerning the nodes A and B, for to the nodes C to F the procedure has to be performed analogously. The derivatives of the shape functions are represented explicitly in Figure 9-4. Since the matrix B comprises linear functions, the subsequent stress analysis yields linear strains and stresses inside the element domain.

9.3.2 Element Stiffness Matrix Regarding the integral to perform the element stiffness matrix Z 1 Z 1−η Z 1 Z 1−η K= BT · E · B det J dξdη = BT · E · B · (2A△ ) dξdη , 0

0

0

0

the integrand BT · E · B is not constant but is of the second order in parts, since the differentiation rules lower the order of the shape functions all over by ’one’. The multiplication of the matrices equals the scheme given in Figure 5-4 in Section 5.1.6 and leads to a stiffness matrix of the size 12 × 12.

162

9 Triangular Elements for Membrane Structures

As an example the integration is shown concerning the entry K[10,7] of the element stiffness matrix:

B=

1 2A△



...  · ... ...

bb φD ,ξ + bc φD ,η 0 ab φD ,ξ + ac φD ,η 

... E  ... E·B = ·  2A△ (1 − ν 2 ) ... BT =

1 2A△



  · 

 ...  ...  ...

7th column ,

 ... ...   1−ν (a φ , + a φ , ) . . . b D ξ c D η 2 bb φD ,ξ + bc φD ,η ν(bb φD ,ξ + bc φD ,η )

.. .. . . 0 ab φE ,ξ + ac φE ,η .. .. . .

.. . bb φE ,ξ + bc φE ,η .. .

    

7th column ,

10th row .

The product of the 10th row and the 7th column gives Z 1Z 1−η  1 E K[10,7] = · ν(bb φD ,ξ + bc φD ,η )(ab φE ,ξ + ac φE ,η ) 2A△ 2A△ (1 − ν 2 ) 0 0  1−ν + (ab φD ,ξ + ac φD ,η )(bb φE ,ξ + bc φE ,η ) · 2A△ dξdη . 2

When employing the derivatives of the shape functions given in Figure 9-4, it follows Z 1Z 1−η E K[10,7] = { ν [ bb (4 − 8ξ − 4η) + bc (−4ξ) ] [ ab 4η + ac 4ξ ] 2A△ (1−ν 2 ) 0 0 +

1−ν [ ab (4 − 8ξ − 4η) + ac (−4ξ) ] [ bb 4η + bc 4ξ ] } dξdη . 2

Concerning integrals employing polynomials of an arbitrary order, the solution can be analytically obtained by applying Z 1Z 1−η i! k! ξ i · η k dξdη = . (i + k + 2)! 0 0 Again, the element stiffness matrix K is divided according to the three different parts of the elasticity matrix K = K1 + K2 + K3 .

E · 4A△ (1 − ν 2 )

 ba ba   0 aa aa    1   − bb ba  0 bb bb  3    1 0 − 3 ab aa 0 ab ab . . . symmetric. . .    1   − bc ba 0 − 1 bc bb 0  bc bc 3  3    1 1 0 − 3 ac aa 0 − 3 ac ab 0 ac ac      4  8/3(bbbb + 4  bb ba  0 b b 0 0 0 a b 3  3  b b +b b ) b a a a     8/3(a a + b b 4 4   0 0 0 0 0 3 ab aa 3 aa ab   a a +a a ) b a a a     8/3(b b + c c 4 4 8   0 0 b b 0 b b 0 b b 0 3 c b 3 b c 3 a c   bc bb +bb bb )     8/3(ac ac + 4 4 8   0 0 0 a a 0 a a 0 a a 0 c b b c a c 3 3 3   a a +a a ) c b b b    4  8/3(b b + c c 4 8 8  bc ba  0 0 0 3 ba bc 0 0 0 3 bb bc 3 ba bb  3  b b +b b ) c a a a     8/3(a a + c c 4 4 8 8 0 a a 0 0 0 a a 0 a a 0 a a 0 3 c a 3 a c 3 b c 3 a b ac aa +aa aa )



K1 =

9.3 Quadratic Approach Employing ξ–η–Coordinates 163

νE · 4A△ (1 − ν 2 )

0  0  aa b a  1  0 − 3 b b aa 0   1 − ab b b 0 . . . symmetric. . .  3 ab b a 0  1 1  0 − 3 b c aa 0 − 3 b c ab 0   1 ac b c 0  − 3 ac ba 0 − 31 ac bb 0  4 4  0 b a 0 b a 0 0 0 b a a b 3 3   4 4/3(a a ba + 4  ab b a 0 0 0 0 0 3 aa b b  3 a b +a b b c bc )   4/3(aa bc 4  0 0 0 0 43 bb ac 0 0 3 b c ab  +ac ba )   4/3(aabc 4/3(aa ba + 4  0 0 0 43 ab bc 0 0 0 3 ac b b  +a b ) ab bb +acbc ) c a   4/3(ab bc 4/3(ab ba 4  0 0 0 0 43 ba ac 0 0 3 b c aa  +a b ) +aa bb ) c b   4 4/3(ab bc 4/3(aabb 4 0 0 0 3 aa b c 0 0 0 3 ac b a +ac bb ) +ab ba )



K2 =

4/3(aa ba + 0 ab bb )+ac bc )

0

                               



164 9 Triangular Elements for Membrane Structures

aa aa

bc bc

. . . symmetric. . .

0

4 4 3 b a ac 3 b a b c

0

4/3(ab bc +ac bb )

8 3 bb bc

4/3(ab ba +aa bb )

8 3 ba bb

4/3(aa ba + (bc bc + ab bb +ac bc ) bc ba +ba ba )

8/3(ab ab + ab aa +aa aa ) 4/3(aa ba + 8/3(bb bb + 0 0 ab ba +acbc ) bb ba +ba ba ) 4/3(ac ba 8/3(ac ac + 4 4 8 3 ab ac 3 ab b c 3 aa ac +aa bc ) ac ab +ab ab ) 4/3(a b 4/3(aa ba + 8/3(bcbc + a c 4 4 8 3 b b ac 3 b b b c 3 ba bc +ac ba ) ab bb +ac bc ) bc bb +bb bb ) 4/3(acbb 4/3(aa bb 8/3(acac + 4 4 8 8 3 aa ac 3 aa b c 3 ab ac 3 aa ab +ab bc ) +ab ba ) ac aa +aa aa )

b c ac

ac ac

E · 8A△ (1 + ν)

  b a aa ba ba   1  − 3 ab aa − 13 ab ba ab ab   1  − 3 bb aa − 13 bb ba bb ab bb bb   1 1 1  − 3 ac aa − 3 ac ba − 3 ac ab − 31 ac bb   1  − 3 bc aa − 13 bc ba − 31 bc ab − 31 bc bb    4  3 ab aa 43 ab ba 43 aa ab 43 aa bb    4  3 bb aa 43 ba ba 43 ba ab 43 ba bb    4 4  0 0 3 ac ab 3 ac b b    4 4  0 0 3 b c ab 3 bc bb    4  3 ac aa 43 ac ba 0 0    4 4 0 0 3 b c aa 3 bc ba



K3 =

                                    



9.3 Quadratic Approach Employing ξ–η–Coordinates 165

166

9 Triangular Elements for Membrane Structures

9.4 Quadratic Shape Functions Using Area Coordinates Following Section 7.2, every point of the triangle is defined by the area coordinates λa , λb and λc . In particular, Figure 9-5 shows the values related to the six nodes of the element. x

A ( λ a = 1, λ b = 0, λ c = 0 ) F ( 0.5, 0, 0.5 ) D ( 0.5, 0.5, 0 ) C ( 0, 0, 1) E ( 0, 0.5, 0.5 )

y

B ( 0, 1, 0 )

Fig. 9-5 The area coordinates of a 6–node triangular element Employing area coordinates the quadratic shape functions to describe the displacements given in Figure 9-4 and their respective derivatives follow to φA = λa (2λa −1) , φA,λa = 4λa −1 , φA,λb = 0 ,

φA,λc = 0 ,

φB = λb (2λb −1) ,

φB,λa = 0 ,

φB,λb = 4λb −1 , φB,λc = 0 ,

φC = λc (2λc −1) ,

φC,λa = 0 ,

φC,λb = 0 ,

φC,λc = 4λc −1 ,

φD = 4 λa λb ,

φD,λa = 4 λb ,

φD,λb = 4 λa ,

φD,λc = 0 ,

φE = 4 λb λc ,

φE,λa = 0 ,

φE,λb = 4 λc ,

φE,λc = 4 λb ,

φF = 4 λa λc ,

φF,λc = 4 λc ,

φF,λb = 0 ,

φF,λa = 4 λa .

Here again, the shape functions and their derivatives turn out to be symmetric according to the respective area coordinate.

9.4.1 Stiffness Matrix Following Section 9.2.1, the operator matrix D can be put up to  P  c bi ·∂λi  i=a    c   P 1   a ·∂ D= i λ i  .  2A△  i=a  c c  P  P ai ·∂λi bi ·∂λi i=a

i=a

9.4 Quadratic Shape Functions Using Area Coordinates

167

In this way, the differentiation of the shape functions yields B= D·Ω  b (4λa −1) bb (4λb −1) bc (4λc −1)  a  1  = aa (4λa −1) ab (4λb −1) ac (4λc −1) 2A△   aa (4λa −1) ba (4λa −1) ab (4λb −1) bb (4λb −1) ac (4λc −1) bc (4λc −1) 4ba λb +4bb λa

4bb λc +4bcλb 4aa λb +4ab λa

4aa λb +4ab λa

4ba λb +4bb λa

4ba λc +4bc λa 4ab λc +4ac λb

4ab λc +4ac λb

4bb λc +4bc λb

4aa λc +4ac λa



    4aa λc  . +4ac λa    4ba λc  +4bc λa

When applying the matrix B, the integration of the element stiffness matrix can be performed again following this repeated scheme: ZZ K= BT · E · B · (2A△ ) dλa dλb .

The integration of the individual terms follows the integration rule related to the product terms of the area coordinates as shown in Section 7.2.4 ZZ i! j! k! λia · λjb · λkc dλa dλb = . (i + j + k + 2)! Thus the evaluation of the entry K[10,7] regarding the stiffness matrix yields ZZ E K[10,7] = [ ν (4ba λb + 4bb λa )(4ab λc + 4ac λb ) 2A△ (1−ν 2 ) 1−ν + (4aa λb + 4ab λa )(4bb λc + 4bc λb ) ] · dλa dλb 2 =

=

νE 1 2 1 1 (16ba ab + 16ba ac + 16bb ab + 16bb ac ) 2A△ (1−ν 2 ) 24 24 24 24 E 1 2 1 1 + (16aa bb + 16aa bc + 16ab bb + 16ab bc ) 4A△ (1+ν) 24 24 24 24 νE E (aa bc + ac ba ) + (aa bc + ac ba ) . 2 3A△ (1−ν ) 6A△ (1+ν)

Of course, the stiffness matrix K equals the stiffness matrix employing the ξ–η–coordinates derived in Section 9.3.1.

168

9 Triangular Elements for Membrane Structures

9.4.2 Load Vector Here, the computation of the load vector is performed by employing area coordinates and considering the special case of a constantly distributed external action Z Z ZZ f = ΩT · p dA = ΩT dA · p = 2A△ ΩT dλa dλb · p . Because of the nodal symmetry, the integration of the shape functions is to be performed for only one corner node and one midpoint node of the respective edge. For node A, this holds   ZZ ZZ 2 1 2 2A△ φA dλa dλb = 2A△ (2λa − λa ) dλa dλb = 2A△ 2 · − = 0, 24 6 and for node D, it yields   ZZ ZZ 1 A△ 2A△ φD dλa dλb = 2A△ (4λa λb ) dλa dλb = 2A△ 4 · = . 24 3 Thus the corner nodes A, B and C do not take any amount of the external action defined above, and only the midpoints of the edges D, E and F take A△ /3 each.

9.4.3 Subsequent Stress Analysis Employing quadratic shape functions to describe the displacements in a triangular element results in linearly distributed strains, because the previously computed matrix B fluctuates linearly. Therefore, the stresses in the element are also linearly distributed. It gives sense to only compute the stresses related to the corner nodes A, B and C to be able to describe a linear course in the ˜ of the corner nodes is given by element. Thus the stress matrix S ˜ =E·B ˜. S The coordinates related to the nodes A, B and C need to be introduced into the matrix B, as given in Section 9.4.1. Applying the abbreviation e = (1 − ν)/2, the stress matrix explicitly follows to

                   



σxxc σyyc σxyc

σxxb σyyb σxyb

σxxa σyya σxya



3ba   3νb   a     3eaa         −ba   E   =  −νba  2A△ (1 − ν 2 )    −eaa         −ba       −νba −eaa



−νaa −aa −eba

−νaa −aa −eba

3νaa 3aa 3eba

−bb −νbb −eab

3bb 3νbb 3eab

−bb −νbb −eab

−νab −ab −ebb

3νab 3ab 3ebb

−νab −ab −ebb

3bc 3νbc −3eac

−bc −νbc −eac

−bc −νbc −eac

3νac 3ac −3ebc

−νac −ac −ebc

−νac −ac −ebc

0 0 0

4ba 4νba 4eaa

4bb 4νbb 4eab

0 0 0

4νaa 4aa 4eba

4νab 4ab 4ebb

4bb 4νbb 4eab

4bc 4νbc 4eac

0 0 0

4νab 4ab 4ebb

4νac 4ac 4ebc

0 0 0

4ba 4νba 4eaa

0 0 0

4bc 4νbc 4eac

 4νac  4ac   4ebc     0    0 ·v .  0     4νaa    4aa  4eba

9.4 Quadratic Shape Functions Using Area Coordinates 169

170

9 Triangular Elements for Membrane Structures

9.5 A Comparison of Standard Elements The cantilever–like structure depicted in Figure 9-6 is investigated employing linear and quadratic shape functions respectively related each to triangular and rectangular elements. Because of the antisymmetry of the problem, u = 0 is to be predefined with respect to the center line. Therefore, only the upper half of the structure needs to be discretized. The external action at the right end of the structure is defined as a constantly distributed boundary action. The following element types are to be compared: 1. The triangular element employing a linear approach, 2. the triangular element employing a quadratic approach, 3. the rectangular element employing a linear approach but performing selectively reduced integration – SRI), 4. the rectangular element employing a quadratic approach related to 9 nodes, which is introduced in Section 11.2.1. x = l/2

l 3

x, u

B

PY

y, v l l = 15 m, t = 1 m, E = 100 000 N/m2, ν = 0, PY = 2 N/m2

Fig. 9-6 Cantilever – geometry and loading Investigations are performed applying two different discretizations regarding each element type. With regard to the coarser mesh, the upper half of the structure is discretized by means of 39 nodes (78 unknowns), which yields either two rows with 12 rectangular elements each employing a linear approach, or one row with six rectangular elements employing a quadratic approach, see Figure 9-7. Concerning a discretization employing triangular elements, the respective rectangular element is divided by a diagonal rising from left to right. Regarding the more refined mesh with 250 unknowns, the number of elements in the x– and y–directions are each doubled. When evaluating the accuracy of the results, both the number of unknowns of the system of equations and the

9.5 A Comparison of Standard Elements

171

bandwidth should be taken into account. As stated previously in Section 3, the computing time needed to solve the system of equations increases depending on the square of the bandwidth. 1

4

7

10

13

16

19

21

24

27

30

34

37

2

5

8

11

14

17

20

22

25

28

31

35

38

3

6

9

12

15

18

21

23

26

29

32

36

39

Fig. 9-7 Cantilever – meshes employing triangular and rectangular elements The vertical displacement at the end of the cantilever is normalized by the external action as well as by Young’s modulus, and is compared in Table 9.1 with respect to the different element types. It is obvious that higher order shape functions lead to better results when requiring approximately the same numerical effort. The selectively reduced integration (SRI) also leads to better results than expected. Table 9.1 Cantilever – displacement vB for different element types and meshes Elements

Unknowns

Bandwidth vB · E/P

1. Triangle linear approach

78 250

8 12

94. 9 109. 2

2. Triangle quadratic approach

78 250

14 22

114. 8 115. 0

3. a Rectangle bi–linear approach

78 250

10 14

111. 2 114. 0

3. b Rectangle bi–linear approach – SRI

78 250

10 14

114. 5 114. 9

4. Rectangle quadratic approach

78 250

18 26

114. 9 115. 0

172

9 Triangular Elements for Membrane Structures

The improvement of the results by employing quadratic shape functions as well as by performing selectively reduced integration is especially obvious with respect to the stresses. In the following, the stresses to the left as well as to the right of the cut in x = ℓ/2 are displayed for ℓ = 15 m, Py = 2 N/m2 and for the more refined discretization. The results regarding triangular elements employing the linear as well as the quadratic approach are depicted on the left, while the results regarding rectangular elements employing the linear approach as well as the quadratic approach combined with the selectively reduced integration scheme are depicted on the right.

σxx

σxy

Fig. 9-8 Cantilever – stress distributions for different element types at x = 7. 50 m

10 Triangular Elements for Kirchhoff Plates

The governing equations as well as the work equation concerning the Principle of virtual Displacements are given in Section 6.2 regarding Kirchhoff ’s plate theory. They are independent of the element formulation, so that they can be adopted unchanged when considering triangular elements. The coordinate system for the description of the triangular elements for plates can be taken to be the same as the one introduced in Section 7. Because of the requirements necessitated by C1 –conformity and the demand of coordinate invariance, it becomes much more difficult to choose shape functions and nodal unknowns concerning the triangular elements for Kirchhoff ’s plate theory when compared to the membrane elements.

10.1 Choice of Shape Functions and Nodal Unknowns The required order n of the shape functions for the description of the deflection surface w depends on the convergence criteria to be fulfilled, cf. Section 2.4.2: • Constant deformations – curvature or twisting – require at least quadratic shape functions for the bending surface w(x, y). • Among others the coordinate invariance requires all comparable element nodes to employ the same degrees of freedom. Therefore the DOF of an element node must be clearly transformable. Thus in addition to [ w ], two first order derivatives [ w,x w,y ] and possibly three second order derivatives [ w,xx w,yy w,xy ] must be available at each element node. • C1 –conformity requests the deflection inside the element domain to be described by a complete polynomial of nth–order. The chosen bending surface w(x, y) must result in a continuous polynomial of nth–order at the element intersections to all neighboring elements, and the same must hold for the slopes w,s (s) and w,n (s) with polynomials of the (n − 1)th– order along the local intersection coordinate s. This means that at all element intersections, (n + 1) nodal degrees of freedom are needed for the description of w(s) and w,s (s) and additional n independent degrees of freedom for the description of w,n are to be defined. Thus each element edge requires (2n + 1) independent DOF. If n DOF are chosen for each element corner, an additional DOF is needed in the centre of each edge. © The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2024 D. Dinkler und U. Kowalsky, Introduction to Finite Element Methods, https://doi.org/10.1007/978-3-658-42742-9_10

174

10 Triangular Elements for Kirchhoff Plates

The C1 –conformity is achievable, but requires high effort and provides conforming elements with restricted applications. Thus, in the following, various shape functions with ascending polynomial order are investigated, whereat linear shape functions are not able to describe constant curvatures, since these are 2nd order derivatives of the bending surface w(x, y). A complete quadratic approach (n = 2) employs 6 degrees of freedom and has been applied first by Morley [73]. As shown in Figure 10-1, there are 3 DOF per element edge, since [ w ] is chosen in the corner nodes and the slope [ w,n ] is considered in the centre of the edges. Although 2 · 2 + 1 = 5 DOF are formally needed at each edge, the solution converges towards the correct value in the case of uniform meshes. Because of this restriction, the element cannot be generally applied. The convergence is poor because the curvature is only constantly approached inside the element domain. n w A

s w,n

w,n F C w

D E w,n

B w

Fig. 10-1 Triangular plate element with non–conforming quadratic shape functions Complete cubic shape functions (n = 3) employ 10 degrees of freedom. Therefore, 2 · 3 + 1 = 7 DOF are required at each element edge. As shown in Figure 10-2, the DOF [ w w,x w,y ] are chosen in the corner nodes and [ w ] in the triangle’s centre of gravity, thus [ w ] is approximated as a cubic polynomial on the element boundary, and subsequently w,s is approached quadratically. w,n is, at most, linearly describable, concerning the two remaining degrees of freedom per edge. Thus a unique transformation of the slope is not possible. w w,x w,y A

M C w w,x w,y

B w w,x w,y

Fig. 10-2 Triangular plate element with non–conforming cubic shape functions

10.1 Choice of Shape Functions and Nodal Unknowns

175

Complete shape functions of the 4th order employ 15 degrees of freedom. Therefore, 2 · 4 + 1 = 9 DOF are required at each element edge. If [ w w,x w,y ] are chosen in the corner nodes and [ w w,n ] in the centres of the edges, as shown in Figure 10-3, a maximum of 8 degrees of freedom is obtained. Thus w could be described as a polynomial of the 4th order along the element boundary and thus w,s would subsequently be approached as a 3rd order polynomial. The nodal degrees of freedom w,n relating to the corners and to the centres of the edges, enable, at most, a description of the slope w,n with a quadratic polynomial. n w w, x w, y

s

A w w, n

w w, n F

D E

C w w, x w, y

w w, n

B w w, x w, y

Fig. 10-3 Triangular plate element with non–conforming 4th order shape functions Complete shape functions of the 5th order employ 21 degrees of freedom. Thus at each element edge 2 · 5 + 1 = 11 nodal unknowns are needed to satisfy the compatibility conditions. If [ w w,x w,y w,xx w,xy w,yy ] are chosen as physically meaningful unknowns in the corner nodes as well as [ w,n ] in the centres of the edges, 13 DOF are obtained per element boundary and therefore all convergence criteria can be satisfied. n A

v={

s

[ w w,x w,y w,xx w,xy w,yy ]A [ w w,x w,y w,xx w,xy w,yy ]B

F C

[ w w,x w,y w,xx w,xy w,yy ]C

D

[ w,n ]d

E

[ w,n ]e

[ w,n ]f

}

B

Fig. 10-4 Triangular plate element with conforming 5th order shape functions Despite of being a valuable approximation of the bending surface, the possibilities of an application are limited. At first, the displacement boundary conditions can not be easily fulfilled in all cases when considering the 2nd order derivatives of w. Further, the element will not fulfill the moment equilibrium at the

176

10 Triangular Elements for Kirchhoff Plates

element intersections, if the plate bending stiffness changes between neighboring elements. Nevertheless, triangular elements with complete shape functions of the 5th order are introduced in detail to explain the general procedure for coping with triangular elements. Shape functions of an even higher order can be chosen, but are not efficient as the effort needed to compute the stiffness matrix increases sharply and, moreover, the corresponding choice of unknowns creates further difficulties. Because of the difficulties in the fulfillment of the C1 –conformity concerning Kirchhoff ’s plate theory, triangular plate elements have been developed, which go beyond the standard displacement–based formulation of the Principle of virtual Displacements, cf. Sections 17, 18, 19.

10.2 Complete Approach of the 5th Order The triangular plate element with a complete 5th order approach was originally investigated by Bosshard [21] and Argyris, Fried and Scharpf [5]. The complete 5th order approach employs 21 degrees of freedom, cf. Figure 10-4. Therefore, the bending surface w(x, y) may be described as a complete polynomial of the 5th order depending on the coordinate s along the element edge. The course of w(s) is defined as a Hermite Polynomial of the 5th order with the degrees of freedom [ w w,s w,ss ], transformed to the boundary coordinates of both corner nodes of the respective edge. Thus w,s is automatically approached as a polynomial of the 4th order. Subsequently, the slope w,n is also a combination of the slopes w,x and w,y . Therefore, it is described as a polynomial of 4th order in s, too. Regarding w,n and w,ns in the corner nodes and w,n in the centres of the edges, this polynomial is also complete. Thus the requirements of the element intersection conditions are fulfilled for w, w,s and w,n and therefore the conformity criterion for Kirchhoff ’s plate theory is satisfied as well. The additional unknowns w,nn take care of the unique transformation of the curvatures in the corners.

10.2.1 The Process to Assemble the Element Stiffness Matrix In order to develop the element stiffness matrix, it is advantageous to utilize the symmetry of the area coordinates, cf. Section 7.2. Additionally, it is difficult to directly assign shape functions according to every individual unknown in the case of a 5th order polynomial, as it has been done in Section 6.2 for the respective rectangular element. Hence the general polynomial is chosen here as a starting point for the formulation, which is also numerically much more

10.2 Complete Approach of the 5th Order

177

efficient. The deflection in the element is approached to w = ψ · a.

(10.1)

The vector ψ contains the monomials of the chosen general polynomial in area coordinates. The vector a contains the coefficients a1 −a21 of the polynomial. At first the relationship between the coefficients ai and the physically meaningful nodal unknowns v is to be formulated. This results in ˜ · a. v=Ψ

(10.2)

˜ defines the scaling matrix G. Thus The inverse matrix of Ψ a= G·v

(10.3)

is obtained. The vector of the physically meaningful shape functions Ω is now obtained by employing a in Equation (10.1) w = ψ ·G·v = Ω·v.

(10.4)

Nevertheless, the computation of the element stiffness matrix, following Section 10.2.5, applies the general polynomial here, since this procedure has substantial numerical advantages.

10.2.2 General Polynomial Employing Area Coordinates The general polynomial approach for the deflection is w (λa , λb , λc ) = ψ · a .

(10.5)

For an approach of the 5th order, and following the definition of area coordinates given in Section 7.2, this yields ψ = [ λ5a λ5b λ5c

λ4a λb λ4b λc λ4c λa

λ4a λc λ4b λa λ4c λb

λ3a λ2b λ3b λ2c λ3c λ2a

λ3a λb λc λ3b λa λc λ3c λa λb

λ3a λ2c λ3b λ2a λ3c λ2b

λ2b λc λ2a λ2c λa λ2b λ2a λb λ2c

(10.6) ].

To derive the scaling matrix, the derivatives of the deflection surface must be evaluated with respect to the area coordinates.

178

10 Triangular Elements for Kirchhoff Plates

This results in w,λa

= [ 5λ4a 0 0

4λ3a λb 0 λ4c

4λ3a λc λ4b 0

3λ2a λ2b 3λ2a λb λc 3λ2a λ2c 2λ2b λc λa 0 λ3b λc 2λ3b λa λ2c λ2b 3 3 2λc λa λc λb 0 2λa λb λ2c ] · a ,

w,λa λa = [ 20λ3a 12λ2a λb 12λ2a λc 6λa λ2b 6λa λb λc 6λa λ2c 0 0 0 0 0 2λ3b 0 0 0 2λ3c 0 0

2λ2b λc 0 2λb λ2c

] · a,

w,λa λb = [

0 0 0

4λ3a 0 0

0 4λ3b 0

6λ2a λb 0 0

3λ2a λc 3λ2b λc λ3c

0 4λb λc λa 2 6λb λa 2λ2c λb 0 2λa λ2c ] · a ,

w,λa λc = [

0 0 0

0 0 4λ3c

4λ3a 0 0

0 0 6λ2c λa

3λ2a λb λ3b 3λ2c λb

6λ2a λc 2λ2b λa 0 2λc λ2b 0 4λa λb λc ] · a ,

0 5λ4b 0

λ4a 4λ3b λc 0

w,λb

=[

w,λb λb = [

0 0 0 2λ3a 0 0 3 2 2 20λb 12λb λc 12λb λa 6λb λ2c 6λb λa λc 6λb λ2a 0 0 0 0 0 2λ3c

w,λb λc = [

0 0 0

w,λc

0 0 0 λ4b 5λ4c 4λ3c λa

=[

w,λc λc = [

0 2λ3a λb λ3a λc 0 2λb λc λ2a 4λ3b λa 3λ2b λ2c 3λ2b λa λc 3λ2b λ2a 2λ2c λa λb λ4c 0 λ3c λa 2λ3c λb λ2a λ2c ] · a,

0 4λ3b 0

0 0 4λ3c λ4a 0 4λ3c λb

0 6λ2b λc 0

λ3a 3λ2b λa 3λ2c λa

2λc λ2a 2λ2c λa 0

] · a,

0 2λb λ2a 0 4λc λa λb 6λ2c λb 2λ2a λc ] · a ,

0 λ3a λb 2λ3a λc λ2b λ2a 2λ3b λc λ3b λa 0 2λc λa λ2b 2 2 2 2 2 3λc λa 3λc λa λb 3λc λb 2λ2a λb λc ] · a ,

0 0 0 0 0 2λ3a 3 0 0 0 2λb 0 0 20λ3c 12λ2c λa 12λ2c λb 6λc λ2a 6λc λa λb 6λc λ2b

0 2λa λ2b 2λ2a λb

] · a.

10.2 Complete Approach of the 5th Order

179

10.2.3 Derivatives with Respect to x–y–Coordinates The transformation of the 1st order derivatives of the deflection with respect to the area coordinates λi to the respective derivatives with respect to coordinates x and y follows Section 7.2.2. Analogously the complete transformation of the 1st as well as the 2nd order derivatives is provided by the following scheme in matrix notation    ∂λb ∂λa ∂λc ∂w 0 0 0 ∂x ∂x ∂x   ∂x     ∂λb  ∂w   ∂λa ∂λc 0 0 0  ∂y   ∂y ∂y ∂y     2   2  ∂ w   ∂ λa ∂ 2 λb ∂ 2 λc ( ∂λa )2 ( ∂λb )2 ( ∂λc )2  ∂x2  =  ∂x2 ∂x2 ∂x2 ∂x ∂x ∂x     2   2 2 2  ∂ w   ∂ λa ∂ λb ∂ λc ( ∂λa )2 ( ∂λb )2 ( ∂λc )2  ∂y2   ∂y2 ∂y 2 ∂y 2 ∂y ∂y ∂y     ∂2w   2 2 2 ∂x∂y

∂ λa ∂x∂y

∂ λb ∂x∂y

∂ λc ∂x∂y

∂λa ∂λa ∂x ∂y

∂λb ∂λb ∂x ∂y

∂λc ∂λc ∂x ∂y



0

0

0

0

0

0

a 2 ∂λ ∂x

∂λb ∂x

a ∂λc 2 ∂λ ∂x ∂x

2

∂λb ∂λc ∂x ∂x

a 2 ∂λ ∂y

∂λb ∂y

a ∂λc 2 ∂λ ∂y ∂y

2

∂λb ∂λc ∂y ∂y

∂λa ∂λb ∂λa ∂λb + ∂y ∂x ∂x ∂y

∂λa ∂λc ∂λa ∂λc + ∂y ∂x ∂x ∂y

∂λb ∂λc ∂λb ∂λc + ∂y ∂x ∂x ∂y

                   ·                   

∂w ∂λa ∂w ∂λb ∂w ∂λc ∂2w ∂λ2a ∂2w ∂λ2b ∂2w ∂λ2c ∂2w ∂λa λb ∂2w ∂λa λc ∂2w ∂λb λc



             .            

The first and the second row describe the transformation of the 1st order derivatives as developed in Section 7.2.2. The rows three, four and five describe the transformation of the 2nd order derivatives, whereat the 2nd order derivatives of the area coordinates with respect to the x–y–coordinates occur, which depend on the element geometry. In the case of a linear transformation as taken into account at hand and given in Section 7.2.1 the 2nd order derivatives of the area coordinates disappear and lead to a simplification of the scheme as follows. After introducing the abbreviations for the derivatives of the area coordinates,

180

10 Triangular Elements for Kirchhoff Plates

following Section 7.2.2, this yields        

w,x w,y w,xx w,yy w,xy



   = 1 ·  2A∆  



 w,λa  w,   λb      w,λc  ba bb bc   a a a  w,   a b c  λa λa     2 2 2  ba bb bc 2 aa ab 2 aa ac 2 ab ac  w,λb λb . (10.7)       a2a a2b a2c 2 ba bb 2 ba bc 2 bb bc    w,λc λc    ba aa bb ab bc ac ba ab +aa bb ba ac +aa bc bb ac +ab bc  w,λa λb     w,   λa λc  w,λb λc

10.2.4 Scaling Matrix ˜ is separately performed row by row The computation of the scaling matrix Ψ for each nodal unknown, as the corresponding nodal coordinates are introduced into the approach and its derivatives respectively, following Section 10.2.2. Such, the deflection in node A is computed as follows w|A = w (λa = 1, λb = 0, λc = 0) = [ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] · a. Accordingly, the slope of the deflection surface in x–direction, see Equation (10.7), follows to w,x =

1 · [ ba · w,λa + bb · w,λb + bc · w,λc ] , 2A∆

and, introducing the derivatives of the approach, is evaluated for node A ap-

10.2 Complete Approach of the 5th Order

181

plying the coordinates (λa = 1, λb = 0, λc = 0), to ba 2A∆ bb + 2A∆ bc + 2A∆ 1 = 2A∆

w,x|A =

·[ 5

0

0

00 00 0 00 00 00 0 00 00 00 ]·a

·[ 0

1

0

00 00 0 00 00 00 0 00 00 00 ]·a

·[ 0

0

1

00 00 0 00 00 00 0 00 00 00 ]·a

· [ 5ba bb

bc

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] · a.

This procedure is transferred to the other unknowns with respect to the corner nodes, which addresses the first 18 DOF of the vector v. The scaling of the approach with respect to the slopes w,n in the centres of the edges is more time–consuming, as the derivatives are defined with respect to the global x–y–coordinates, but these must be transformed with respect to the local element boundary coordinates. To identically define the unknowns w,n in neighboring elements at assembling the system, the direction of the element boundary coordinates n, s should be defined globally. The positive direction of the normal with respect to the edge of the triangular element considered, will be defined such that it encloses an angle of 0 ≤ β < π with the x–axis, as shown in Figure 10-5. The angle between the normal and the x–axis is explicitly evaluable when considering the differences in the coordinates of the respective corner nodes. x A

F

D

βf C

βd

E βe

B

tan( βd −

π yb − ya bc )= =− 2 xb − xa ac

tan( βe −

π yc − yb ba )= =− 2 xc − xb aa

tan( βf −

π ya − yc bb )= =− 2 xa − xc ab

y

Fig. 10-5 The definiton of the positive normal direction The transformation of the slopes w,x and w,y into the slopes w,s and w,n can be performed with respect to the element boundary, as shown in Figure 10-6,

182

10 Triangular Elements for Kirchhoff Plates

by π ) · w,x + sin( β − 2 π w,n = − sin( β − ) · w,x + cos( β − 2 w,s = + cos( β −

π ) · w,y , 2 π ) · w,y . 2

x w, n D

w,y

s

w,s

s

(β − π) 2 td

td y

nd

w,x

nd

Fig. 10-6 Definition of w,n and w,s according to w,x and w,y ˜ By means of the derivatives w,x and w,y the last three rows of the matrix Ψ can be computed as follows. Due to the transformation of coordinates the slope w,n|D satisfies w,n|D = π 1 )· [ ba · w,λa + bb · w,λb + bc · w,λc ](λa = 21 , λb = 12 , λc =0) 2 2A∆ π 1 + cos( β − ) · [ aa · w,λa + ab · w,λb + ac · w,λc ](λa = 21 , λb = 12 , λc =0) 2 2A∆ − sin( β −

and may be evaluated analogously to w,x|A . The same transformation leads to the slopes w,n|E and w,n|F . ˜ The entire Ψ–matrix employs 21 rows and columns and links the coefficients of the general polynomial with the physically meaningful nodal degrees of freedom vT = { [ w w,x [ w w,x [ w w,x [ [ w,n ]D

w,y w,xx w,y w,xx w,y w,xx [ w,n ]E

w,xy w,yy w,xy w,yy w,xy w,yy [ w,n ]F

]A ]B ]C ] }.

(10.8)

10.2 Complete Approach of the 5th Order

183

˜ is structured as follows The asymmetric matrix Ψ  ˜ Ψ11 0 0  ˜ Ψ22 0  0 ˜ =  Ψ  0 ˜ 0 Ψ33  ←

˜n Ψ

→



  .  

(10.9)

˜ ii are rectangular matrices of the size The sub–matrices at the main diagonal Ψ ˜ ii –matrices are arranged for each 6 × 7. The factors positioned behind the Ψ respective row to be multiplied by that factor:   1 0 0 0 0 0 0   1  5b bb bc 0 0 0 0   a  · 2A∆   1  ·  5aa  2A a a 0 0 0 0 b c ∆   ˜ Ψ11 =   1  20b2 2 2  · 8ba bb 8ba bc 2bb 2bb bc 2bc 0  4A2  a   1∆    20aa ba 4(aa bb +ab ba ) 4(aa bc +ac ba ) 2ab bb ab bc +ac bb 2ac bc 0  · 4A2   ∆ 1 2 2 2 · 20aa 8aa ab 8aa ac 2ab 2ab ac 2ac 0 4A2∆   1 0 0 0 0 0 0  1   5b bc ba 0 0 0 0   · 2A∆  b  1  ·   2A  5ab a a 0 0 0 0 c a ∆   ˜ 22 =  Ψ  1   20b2 2 2 · 8bb bc 8ba bb 2bc 2ba bc 2ba 0  4A2  b  1∆     20ab bb 4(ac bb +ab bc ) 4(ab ba +aa bb ) 2ac bc ac ba +aa bc 2aa ba 0  · 4A2   ∆ 1 20a2b 8ab ac 8aa ab 2a2c 2aa ac 2a2a 0 · 4A2 ∆   1 0 0 0 0 0 0   1  5b ba bb 0 0 0 0    · 2A∆ c   1  ·  5ac  2A a a 0 0 0 0 a b ∆   ˜ Ψ33 =   1  20b2 2 2  8ba bc 8bb bc 2ba 2ba bb 2bb 0  · 4A2  c   1∆    20ac bc 4(ac ba +aa bc ) 4(ac bb +ab bc ) 2aa ba ab ba +aa bb 2ab bb 0  · 4A2   ∆ 1 20a2c 8aa ac 8ab ac 2a2a 2aa ab 2a2b 0 · 4A2 ∆

184

10 Triangular Elements for Kirchhoff Plates

˜ n –matrix is completely taken and has 3 rows and 21 columns. The inThe Ψ troduction of the abbreviations π π 1 ) + aa · cos(βd − )} · , 2 2 32A∆ π π 1 b = {−bb · sin(βe − ) + ab · cos(βe − )} · , 2 2 32A∆ π π 1 c = {−bc · sin(βf − ) + ac · cos(βf − )} · , 2 2 32A∆

a = {−ba · sin(βd −

and consideration of the derivatives given in Section 10.2.2 yields the rows 19, 20 and 21:  5a 4a + b c 3a + 2b c 0 c ˜n =  0 0 0 0 0 0 Ψ  0 5a b c + 4a 0 b 2c + 3a 0 5b c 5b 4b + c 0 0

a + 4b 0 c a 3b + 2c a 0 0 0

2a + 3b 0 0 a 0 0

0 5c 5c

0 b + 4c b

 0 0  2b + 3c 0  . 0 b

0 a 4c + a

0 0 0 a 3c + 2a b

To get the scaling matrix, the inversion −1

˜ G=Ψ

can be computed numerically for each element in the FE–program when applying its respective coordinates. Here, the scaling matrix is also explicitly specified with   G11 0 0 0  0 G22 0 0       0  0 G 0 33 G= .  ← g7 →     ← g →  ←

14

g21

→

Although the vectors gi are computed as rows 7, 14, 21 of G they are embedded in the scaling matrix as rows 19, 20, 21 for simplicity. By employing the

10.2 Complete Approach of the 5th Order

185

abbreviations a, b and c the vectors gi are determined to   g7 · c    g14 · a  = g21 · b   

−5(11a + 5b + 5c) 0 −5(11a + 5c + 5b)

−ac (16a + 9b + 4c) + 5cab 0 ab (16a + 9c + 4b) − 5bac

cab ac −a2c (1. 5a + b) bc ac (3a + 2b)−c(ab bc + ac bb ) 0 0 bab ac −a2b (1. 5a + c) bb ab (3a + 2c)−b(ab bc + ac bb ) −5(11b + 5a + 5c) −5(11b + 5c + 5a) 0

ac (16b + 9a + 4c) − 5caa −aa (16b + 9c + 4a) + 5aac 0

caa ac −a2c (1. 5b + a) bc ac (3b + 2a)−c(aa bc + ac ba ) aaa ac −a2a (1. 5b + c) ba aa (3b + 2c)−a(acba + aa bc ) 0 0 0 −5(11c + 5b + 5a) −5(11c + 5a + 5b)

0 aa (16c + 9b + 4a) − 5aab −ab (16c + 9a + 4b) + 5baa

0

0 ba aa (3c + 2b)−a(abba + aa bb ) bb ab (3c + 2a)−b(abba + aa bb )

aaa ab −a2a (1. 5c + b) baa ab −a2b (1. 5c + a) 1 0 0

0 1 0

bc (16a + 9b + 4c) − 5cbb 0 −bb (16a + 9c + 4b) + 5bbc cbb bc − b2c (1. 5a + b) 0 bbb bc − b2b (1. 5a + c) −bc (16b + 9a + 4c) + 5cba ba (16b + 9c + 4a) − 5abc 0 cbc ba − b2c (1. 5b + a) aba bc − b2a (1. 5b + c) 0 0 −ba (16c + 9b + 4a) + 5abb bb (16c + 9a + 4b) − 5bba 0 aba bb − b2a (1. 5c + b) bbb ba − b2b (1. 5c + a) 0 0 1



 .

186

10 Triangular Elements for Kirchhoff Plates

The sub–matrices at the main–diagonal are computed as 6 × 6 matrices

G11

G22

G33



1 0 5 ac 5 −ab 10 4ac 20 4(ac − ab ) 10 −4ab

0 0 −bc 0 bb 0 −4bc 0,5a2c −4(bc − bb ) −ab ac 4bb 0,5a2b

0 0 0 −bc ac ab b c + ac b b −bb ab

0 0 0 0,5b2c −bb bc 0,5b2b





1 0 0 0 5 aa −ba 0 5 −ac bc 0 10 4aa −4ba 0,5a2a 20 4(aa − ac ) −4(ba − bc ) −ac aa 10 −4ac 4bc 0,5a2c

0 0 0 −ba aa ac b a + aa b c −bc ac

0 0 0 0,5b2a −bc ba 0,5b2c





1 0 0 0 5 ab −bb 0 5 −aa ba 0 10 4ab −4bb 0,5a2b 20 4(ab − aa ) −4(bb − ba ) −aa ab 10 −4aa 4ba 0,5a2a

0 0 0 −bb ab aa b b + ab b a −ba aa

0 0 0 0,5b2b −ba bb 0,5b2a



    =    

    =    

    =    

    ,    

    ,    

    .    

10.2.5 Element Stiffness Matrix The element stiffness matrix K is computed with Z T K = G · ψ T · DT · E · D · ψ dA · G = GT · Kgen · G .

(10.10)

Here, Kgen incorporates the coefficients of the general polynomial and will be transformed into the element stiffness matrix K related to the physically meaningful nodal unknowns by applying the scaling matrix G. Thus integration is performed with regard to the general polynomial. This results in simplified programs and therefore in a substantial reduction in numerical cost. The computation of the integrand is demonstrated for two different possibilities.

10.2 Complete Approach of the 5th Order

187

1st Approach Here, the B–matrix is not evaluated with the shape functions Ω but with the general polynomial given with respect to the area coordinates B = D·ψ.

(10.11)

The differentiation with respect to x and y can be replaced by the differentiation with respect to the area coordinates λi following Equation (7.14). The multiplication and the integration procedure involved when computing the element stiffness matrix Kgen employing the general polynomial described by area coordinates are performed here, along the lines of Section 7.2, yielding Z Z Kgen = BT · E · B · 2A∆ dλa dλb . Figure 10-7 shows the related multiplication scheme.

E 3

21

3

21

3

3

GT

21

21

21

21

K =

21

dA .

. BT

21

21

B

3

G

Fig. 10-7 Matrix scheme for the computation of K The method of explicit integration to get the stiffness matrix is not performed here, since the second approach is more efficient and clearly arranged.

2nd Approach Here, to get the element stiffness matrix, the matrix operations in the integrand are performed in advance with D and E, as per Section 6.2, DT · E · D =

1−ν (10.12) xy ∂∂yx } . 2 The multiplication yields a 1×1–matrix. For example, xx ∂ describes the partial double differentiation with respect to x as related to the shape function in front of it, and ∂xx describes the application of partial double differentiation B · { xx ∂∂xx +

yy ∂∂yy

+ νxx ∂∂yy + νyy ∂∂xx + 4

188

10 Triangular Elements for Kirchhoff Plates

with respect to x as related to the shape function behind it. When applying this representation, it is possible to compute each term in the stiffness matrix separately by applying the related differential operator. Such, the entry in the 8th row and the 10th column of the stiffness matrix Kgen can be performed, for example, by applying the 8th shape function from Equation (10.6) in front of the differential operator and by applying the 10th shape function from Equation (10.6) behind the differential operator: 8th function : 10th function : Kgen [ 8, 10 ] =

Z Z

λ5b λ4b λa

λ5b · DT · E · D · λ4b · λa · 2A∆ dλa dλb .

The differentiation with respect to x and y must be replaced by the differentiation with respect to the area coordinates. To get the entries of the stiffness matrix, the integration with respect to the area coordinates can be performed analogously to Section 7.2. The final stiffness matrix K, as it is related to the degrees of freedom, is transformed numerically using the scaling matrix.

10.2.6 Element Load Vector The integration of the virtual external work is performed here concerning a constantly distributed action. For the integration the general polynomial is applied for convenience, which gives Z Z Z Z f = ΩT · p dA = ΩT dA · p = GT ψ T · 2A∆ dλa dλb · p . (10.13) A

A

After the integration, multiplication by the scaling matrix G is performed with f = GT · fgen . The load vector fgen , which is not scaled yet, is integrated analytically T fgen = 2A∆ · { [

1 1 2 1 2 2 1 6·7 5·6·7 5·6·7 4·5·6·7 4·5·6·7 4·5·6·7 3·4·5·6·7

]

[

1 1 1 2 1 2 2 6·7 5·6·7 5·6·7 4·5·6·7 4·5·6·7 4·5·6·7 3·4·5·6·7

]

[

1 1 1 2 1 2 2 6·7 5·6·7 5·6·7 4·5·6·7 4·5·6·7 4·5·6·7 3·4·5·6·7

] }·p .

The computation of the element load vector concerning linearly varying actions is not commonly accepted and hence is not shown here.

10.3 A Plate Element with 18 DOF

189

10.2.7 Subsequent Analysis for Stress Resultants The instruction to compute the stresses, cf. Equation (2.39), σ =E·D·Ω·v = E·B·v is given here considering the general polynomial related to area coordinates, σ (λi ) = E · D · ψ (λi ) · G · v , whereby σ stores the moments. At first, similar to the computation of the element stiffness matrix, the differential operator D · ψ (λi ) is to be performed if the moments’ magnitudes should be computed at any arbitrary position in the element. The distribution of the moments’ magnitudes inside the element and along its edges is described cubically. If the magnitudes are to be computed at the corner nodes, the nodal unknowns of the curvatures can be transformed according to the material equations in order to directly compute the values. The computation of the shear forces is performed similarly to that in Section 6.2.8, employing the 3rd order derivative of the deflection surface. At first the matrix notation gives q = Dq · σ = Dq · (E · D · ψ (λi ) · G · v) = Sq · v , with Sq representing the stress matrix for shear forces. Explicitly introducing the shape functions for the deflections yields " # " # ψ (λi ) ,xxx + ψ (λi ) ,yyx qx = −B · ·G·v. qy ψ (λ ) ,yyy + ψ (λ ) ,xxy i

i

10.3 A Plate Element with 18 DOF By eliminating the unknowns w,n related to the centres of the edges, one gets an element which is substantially easier to implement into a program than the element with 21 DOF. The elimination reduces the number of independent unknowns, but the approach of 5th order remains unchanged. Very early, Butlin and Ford [25], Cowper [30] and Bell [16] have independently developed elements without the unknowns w,n .

190

10 Triangular Elements for Kirchhoff Plates

The starting point of this derivation is the 21 DOF element previously described with the unknowns T v21 ={ [ [ [ [

w w,x w w,x w w,x [ w,n ]D

w,y w,xx w,y w,xx w,y w,xx [ w,n ]E

w,xy w,yy w,xy w,yy w,xy w,yy [ w,n ]F

]A ]B ]C ] }.

Concerning the 18 DOF element, the slopes in the normal directions w,n at the centres of the edges are described as a linear combination of the nodal unknowns at the corner points. Therefore, the nodal unknowns T v18 ={ [ w [ w [ w

w,x w,x w,x

w,y w,y w,y

w,xx w,xx w,xx

w,xy w,xy w,xy

w,yy w,yy w,yy

]A ]B ]C

}

suffice. The elimination is possible only if the remaining unknowns are able to consistently fulfill the element intersection conditions along the element edges, i.e. without a gap in w and in w,n . The 5th order approach of the 21 DOF element fulfills all element intersection conditions, i.e. for the slope w,n as a polynomial of the 4th order employing 5 DOF at each edge. Eliminating w,n at the centres of the edges, 4 DOF of the quality [ w,n w,ns ] are left at the element corners, and a Hermite Polynomial of the 3rd order can be defined at each edge relating to w,n . Therefor, the slopes w,n at the centres of the edges are described by a linear combination of the nodal unknowns [ w,x w,y w,xx w,yy w,xy ] at the corner points, which have to be transformed to the edge coordinates n, s. Thus the element intersection conditions for [ w w,s ], described as a polynomial of the 5th order, are still fulfilled exactly. However, w,n is only described with a polynomial of the 3rd order instead of the 4th order, what causes a small loss in accuracy but does not provide any other disadvantage. At first, the general polynomial gives w,n = a0 + a1 · s + a2 · s2 + a3 · s3 , which is adapted to the nodal unknowns [ w,n w,ns ] at the corner points of the edges. In this way, the slopes at the corner points w,s and w,n are described by means of w,x and w,y , cf. Equation (10.8). Similarly w,ns may be computed. In general, the relationship between the nodal unknowns of the elements with 18 DOF and 21 DOF is defined by v21 = T · v18

10.3 A Plate Element with 18 DOF

191

with the transformation matrix T=

"

I Tn

#

.

The first 18 rows describe the identities of the nodal unknowns at the element corners. The relationship between the slopes at the centres of the edges and the nodal unknowns at the corners is established only by the last three rows. Applying the transformation matrix T, the transformation rule to get the element with 18 DOF from the element with 21 DOF yields stiffness matrix: K18 = TT · K21 · T , stress matrix:

S18 = S21 · T ,

f18 = TT · f21 .

load vector:

Concerning the FE–program, it is convenient to multiply, once and in advance, the scaling matrix G21 of the element with 21 DOF by the transformation matrix T from the right. The product G18 can be assumed to be the scaling matrix of the element with 18 DOF: 

G18

    = G21 · T =     

G11 0 0 ← ← ←

0 G22 0 g7 g14 g21

0 0 G33 → → →



    .    

It is given explicitly here and is employed after the integration of the general polynomial instead of the scaling matrix G21 with the main diagonal sub– matrices G11 , G22 , G33 as given in Section 10.2.4. The vectors gi comprise 18 columns and are summarised into a matrix as follows. Here ℓa , ℓb , ℓc mean the lengths of the respective element side and β holds for the angle between the side and the x–axis, see Figure 10-5. 

 g7 · c    g14 · a  = g21 · b

192

10 Triangular Elements for Kirchhoff Plates



−5(11a + 5b + 5c)     0   −5(11a + 5c + 5b)

−ac (16a + 9b + 4c) +5cab − sin β/2 0 ab (16a + 9c + 4b) −5bac − sin β/2

bc (16a + 9b + 4c) −5cbb + cos β/2 0 −bb (16a + 9c + 4b) +5bbc + cos β/2

cab ac −a2c (1,5a + b) + sin β cos β lc /8 0 bab ac −a2b (1,5a + c) − sin β cos β lb /8

bc ac (3a + 2b) − cab bc −cac bb −(1−2 sin2 β)lc /8 0 −bab bc + bb ab (3a + 2c) −bac bb +(1−2 sin2 β)lb /8

cbb bc −b2c (1,5a + b) − sin β cos β lc /8 0 bbb bc −b2b (1,5a + c) + sin β cos β lb /8

−5(11b + 5a + 5c)

ac (16b + 9a + 4c) −5caa − sin β/2 −aa (16b + 9c + 4a) +5aac − sin β/2 0

−bc (16b + 9a + 4c) +5cba + cos β/2 ba (16b + 9c + 4a) −5abc + cos β/2 0

−5(11b + 5c + 5a) 0

caa ac− a2c (1,5b + a) bc ac (3b + 2a) − caa bc − sin β cos β lc /8 −cac ba +(1−2 sin2 β)lc /8 aaa ac− a2a (1,5b + c) ba aa (3b + 2c) − aac ba + sin β cos β la /8 −aaa bc −(1−2 sin2 β)la /8 0 0 0 −5(11c + 5b + 5a) −5(11c + 5a + 5b) 0 2 aaa ab −aa (1,5c + b) − sin β cos β la /8 baa ab −a2b (1,5c + a) + sin β cos β lb /8

cbc ba −b2c (1,5b + a) + sin β cos β lc /8 aba bc −b2a (1,5b + c) − sin β cos β la /8 0

0 aa (16c + 9b + 4a) −5aab − sin β/2 −ab (16c + 9a + 4b) +5baa − sin β/2

0 −ba (16c + 9b + 4a) +5abb + cos β/2 bb (16c + 9a + 4b) −5bba + cos β/2

0 ba aa (3c + 2b) − aab ba −aaa bb +(1−2 sin2 β)la /8 bb ab (3c + 2a) − bab ba −baa bb −(1−2 sin2 β)lb /8

0 2 aba bb −ba (1,5c + b) + sin β cos β la /8 bbb ba −b2b (1,5c + a) − sin β cos β lb /8



   .  

10.4 A Comparison of Standard Plate Elements

193

10.4 A Comparison of Standard Plate Elements In the following, a square plate is investigated which is loaded by a constantly distributed loading p = 1. 0 N/m2 . The support is partially defined by clamping and bearing respectively, as shown in Figure 10-8, where only one fourth of the plate is discretized due to its symmetry. The investigation of convergence, cf. Figures 10-10, 10-11 and 10-12, is limited to a few standard elements. x y

GE

clamped constantly distributed loading: p = 1 kN/m2 7 2 E = 3.0 . 10 kN/m

E

T

hinged

M

ly

t = 0.1 m G

ν = 0.2 l x = 10.0 m l y = 10.0 m

lx

Fig. 10-8 Square plate – geometry and loading The element meshes are depicted in Figure 10-9, whereat two triangles cover the same area as one rectangular element.

Fig. 10-9 Square plate – meshes for rectangular and triangular elements Since the solution employing the element comprised of 18 DOF only deviates insignificantly from the solution employing the element with 21 DOF, just one solution is presented here. Concerning the general polynomial the difference between the rectangular elements comprised of 12 DOF and the rectangular element comprised of 16 DOF is explained in Section 6.2.

194

10 Triangular Elements for Kirchhoff Plates

The quality of an element may be evaluated by means of the convergence and the numerical effort to get the solution. The computing time required to solve the system of equations is to be regarded as well as the bandwidth and the number of unknowns. Additionally, the computational cost must be taken into account, which is afforded to generate the stiffness matrix. The discretizations for rectangular and triangular elements are given in Table 10.1. Table 10.1 Square plate – mesh size, unknowns and bandwidth 12 elements unknowns bandwidth

1 12 12

4 27 15

16 48 21

16 64 243 33

1 16 16

4 36 20

16 100 28

△18 64 324 44

2 24 18

8 54 24

32 150 36

128 486 60

The following figures show the convergence behavior of state variables, which indicate the quality of the elements. It is obvious, that due to the higher polynomial degree the triangular element with 18 DOF yields superior results, whereat the quality of the moments is of the same order as of the deflection. Nonetheless, the rectangular elements converge to the right results as well but very slowly regarding the moments. As mentioned above the elements with 16 DOF respectively 18 DOF employ curvatures as nodal DOF, what restricts their application to plates without stiffeners and without discontinuities in stiffness distribution. % error of wM ∆ 18

* 16 * 12 - weak conformity * 12 5

0

20

100

500

dof

Fig. 10-10 Square plate – the convergence behavior of the deflection wM

10.4 A Comparison of Standard Plate Elements

195

% error of mxxM

40

∆ 18

* 16 30

* 12 - weak conformity * 12

20 10

0

20

100

500

dof

-10

-20

Fig. 10-11 Square plate – the convergence behavior of the moment mxxM % error in m yyE

0

20

100

500

dof

-5

-10

-15 ∆ 18 -20

-25

* 16 * 12 - weak conformity * 12

-30

Fig. 10-12 Square plate – the convergence behavior of the moment myyE

ISOPARAMETRIC ELEMENTS

11 Numerical Integration

When computing the element stiffness matrix and the load vector, an important part is the solution of integrals like Z Z Z f (x) dx , f (x,y) dA and f (x,y,z) dV . x

A

V

The integrand g(x) · h(x) or other products of functions are also included along with the integrand f (x). Concerning approaches of lower order, analytical integration can still be reasonably managed, but for approaches of higher order, for example those shown in Section 10, the effort increases enormously. For different problems as described in Section 12 the analytical integration is not possible, and thus numerical integration must be performed. Therefor different methods can be applied. Besides the Newton–Cotes method, the Gauss–Legendre quadrature is also used prevalently, since it offers the best possible solution for minimal effort. The derivation of the Gauss–Legendre quadrature will not be shown in detail here but its application in the FEM context. Basis for the numerical integration scheme is the application of a normalized element coordinate system, as has been described previously for the triangular elements in Section 7. xA

x0

xB

l/2

ξ = -1

l/2

ξ= 0

ξ = +1 ξ

A

B

Fig. 11-1 Global and local coordinate systems Considering one dimension, the transformation of coordinates may be described by means of Figure 11-1 with x = x0 +

ℓ ·ξ, 2

where x is the global coordinate, x0 is the center of the element, ℓ is the length of the element and −1. 0 < ξ < + 1. 0 is the local normalized coordinate. Thus all integrals of the work equations can be transformed from the global © The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2024 D. Dinkler und U. Kowalsky, Introduction to Finite Element Methods, https://doi.org/10.1007/978-3-658-42742-9_11

200

11 Numerical Integration

coordinates to the local one, if x is exchanged by ξ. Assuming f (x) and g(ξ) are normalized shape functions to approach the displacement field, the integration can be changed to local coordinates with x=x Z B

ξ=1 Z

f (x) dx =

x=xA

g(ξ)

ℓ dξ , 2

ξ=−1

where the integration with respect to local coordinates can be separated from the element geometry, if the shape functions are described in local coordinates, too. In the following, the rules for numerical integration with respect to cartesian coordinates of rectangular elements and to area coordinates of triangular elements are explained.

11.1 Numerical Integration Using Gauss–Legendre Quadrature In general, the representation in normalized element coordinates allows for the following integral to be solved numerically. In the case of continuous polynomials the integration can be performed exactly as the sum of the weighted function values at the given supporting points ξi , which yields Z1

f (ξ) dξ =

n X i=1

−1

f (ξi ) · wi .

(11.1)

Here, each function value f (ξi ) is to be multiplied by a fixed weighting factor wi . As such, it is fundamental that the integration is performed with respect to the normalized coordinate ξ, which can be transformed to any arbitrary real length. As an example, the positions of the origin of coordinates and supporting points are shown in Fig. 11-2. f (ξ ) f (ξ i )

−1

i

f ( ξ i+1)

0

i +1

+1

ξ

Fig. 11-2 Gauss–Legendre quadrature – supporting points of a function At numerical integration, the weighting factors wi may define the domain of influence with respect to the corresponding supporting points. Related to the

11.1 Numerical Integration Using Gauss–Legendre Quadrature

201

supporting points ranging from n = 1 to n = 6, the coordinates of the supporting points as well as the weighting factors are given in Table 11.1. The number of supporting points n depends on the order of the function to be integrated. Considering n supporting points, the integration is exact up to an order of m = 2n − 1. Table 11.1 One dimension – supporting points ξi and weighting factors wi n

m = 2n − 1

ξi

wi

1

1

2

3

3

5

4

7

±0. 3399 8104 3585 0. 6521 4515 4863 ±0. 8611 3631 1594 0. 3478 5484 5137

5

9

0. 0 0. 5688 8888 8889 ±0. 5384 6931 0106 0. 4786 2867 0499 ±0. 9061 7984 5939 0. 2369 2688 5056

6

11

±0. 2386 1918 6083 0. 4679 1393 4573 ±0. 6612 0938 6466 0. 3607 6157 3048 ±0. 9324 6951 4203 0. 1713 2449 2379

0. 0 √ ±1/ 3

2. 0

p 0. 0 ± 3/5

8/9 5/9

1. 0

An analogous procedure can be passed if the integration of a product of multiple functions is performed, e.g. the product of virtual and real shape functions: Z1

−1

f (ξ) · g(ξ) dξ =

n X i=1

f (ξi ) · g(ξi ) · wi .

This is true, since the product of two functions gives a new function, which has to be integrated numerically by the same integration rule.

Example Analytical Integration: Z+1 Z+1 1 1 2 f (ξ) · g(ξ) dξ = (1 − ξ) · (ξ 2 ) dξ = [ ξ 3 − ξ 4 ]+1 . −1 = 3 4 3

−1

−1

202

11 Numerical Integration

√ Numerical integration with n = 2, ξi = ±1/ 3, wi = 1. 0: Z+1 1 1 1 1 2 (1 − ξ) · (ξ 2 ) dξ = 1 · [ (− √ )2 − (− √ )3 ] + 1 · [ (+ √ )2 − (+ √ )3 ] = . 3 3 3 3 3

−1

Multi–dimensional integration Similarly, integrals can be solved regarding coordinates in two directions. The positions of the supporting points are shown in Figure 11-3 for n = 3 in ξ– and η–directions. Here the summation is to be performed in both directions Z1 Z1

f (ξ, η) dξ dη =

n X n X i=1 j=1

−1−1

f (ξi , ηj ) · wi · wj .

(11.2)

1

1

1

3/5

1

3/5

ξ

3/5

3/5

η

Fig. 11-3 Rectangular domains – supporting points regarding a square at n = 3 Similarly, the numerical integration can be extended to volume integrals: Z1 Z1 Z1

f (ξ, η, ζ) dξ dη dζ =

n X n X n X i=1 j=1 k=1

−1−1−1

f (ξi , ηj , ζk ) · wi · wj · wk .

(11.3)

Triangles Concerning the triangular elements, the integration is done using a single summation regarding the supporting points and the weighting factors, as shown in Table 11.2 Z1 1−η Z n 1 X f (ξ, η) dξ dη = · f (ξi , ηi ) · wi . (11.4) 2 i=1 0

0

11.1 Numerical Integration Using Gauss–Legendre Quadrature

203

Table 11.2 Triangular domains – supporting points ξi and weighting factors wi exact till the power m

position of the supporting points

supporting points ξ η λa λb λc

weightingfactors wi

A

1

a

C

c

a

C

B

b

A b

3

a c

C

d

B

A e

5

1 3

1 3

1 3

1

a

1 2

1 2

0

1 3

b

0

1 2

c

1 2

0

1 2 1 2

1 3 1 3

a

1 3

1 3

1 3

− 27 48

b

0. 6

0. 2

0. 2

c

0. 2

0. 6

0. 2

d

0. 2

0. 2

0. 6

25 48 25 48 25 48

a

1 3

1 3

1 3

0. 225 000 000 0

b c

α1 β1

β1 α1

β1 β1

0. 132 394 152 7

d

β1

β1

α1

e

α2

β2

β2

f

β2

α2

β2

g

β2

β2

α2

B A

2

a

c

C g

a

d f

b

B

α1 = 0. 059 715 871 7 , β1 α2 = 0. 797 426 985 3 , β2

0. 470 142 064 1 0. 101 286 507 3

0. 125 939 180 5

204

11 Numerical Integration

11.2 Numerical Integration Applied to Membrane Elements To perform numerical integration, the element coordinate system, cf. Figure 5-2, Section 5.1.4, is to be transformed into a normalized coordinate system as given in Figure 11-4. Concerning the transformation between the global x–y– coordinate system and the local ξ–η–coordinate system, the respective rule is defined as x = x0 + lx · ξ and (11.5) y = y0 + ly · η . xo

x

2 lx A (−1;−1)

yo

2 ly

B(1;−1)

ξ

C ( ξ = 1; η = 1)

D (−1;1) η y

Fig. 11-4 Local coordinates related to a rectangular element Starting with the Principle of virtual Displacements the matrix formulation in global coordinates is given with Z Z −δWd = δvT ΩT · DT · E · D · Ω dA v − δvT ΩT · p dA . (11.6) A

A

The transformation to local coordinates has to consider the following steps: • Shape functions with respect to local coordinates. • Derivatives of shape functions with respect to local coordinates. • Integration of the virtual work. So far, the transformation of the coordinates has been formally prepared. To perform the differentation of the shape functions with respect to the ξ–η– coordinates as well as the integration of the element work, the shape functions

11.2 Numerical Integration Applied to Membrane Elements

205

must also be given in local coordinates. In the following, symmetric linear and quadratic shape functions and its derivatives are given with respect to local coordinates.

11.2.1 Shape Functions Employing Local Coordinates The linear shape functions are given in local coordinates as already covered in Section 5.1.4: φA = 14 (1 − ξ)(1 − η) φB = 14 (1 + ξ)(1 − η) φC = 14 (1 + ξ)(1 + η) φD = 14 (1 − ξ)(1 + η)

φA,ξ = − 14 (1 − η) 1 (1 − η) 4 φC,ξ = 14 (1 + η) φD,ξ = − 14 (1 + η)

φB,ξ =

A

1 4 D

φA,η = − 14 (1 − ξ) φB,η = − 14 (1 + ξ) φC,η = φD,η =

1 (1 + ξ) 4 1 (1 − ξ) 4

1 2 1 B

η

ξ C

Fig. 11-5 Linear shape function φB of the rectangular membrane element The symmetric quadratic approach is a product of one of three quadratic shape functions both in the x– and in the y–direction. Apart from the displacements at the corners, the nine DOF require physically meaningful displacements at the centres of the edges and at the centre point, see the nodes A...I in Figure 11-6. The Lagrange Polynomials assigned to the nodal displacements vT = [ uA uB . . . uH uI | vA vB . . . vH vI ] are subsequently given for the element employing quadratic shape functions. The vector of the shape functions φ = [ φA φB φC φD φE φF φG φH φI ] incorporates the following shape functions in local coordinates

206

11 Numerical Integration φA =

1 ξ (1 − ξ) · η 4

· (1 − η)

φA,η φB = − 14 ξ (1 + ξ) · η

· (1 − η)

φB,ξ φB,η

φC =

1 ξ (1 + ξ) · η 4

· (1 + η)

φC,ξ φC,η

φD = − 14 ξ (1 − ξ) · η

· (1 + η)

φD,ξ φD,η

φE = − 12

(1 − ξ 2 ) · η · (1 − η)

φE,ξ φE,η

φF =

1 ξ (1 + ξ)(1 − η 2 ) 2

φF,ξ φF,η

φG =

1 2

(1 − ξ 2 ) · η · (1 + η)

φG,ξ φG,η

φH = − 12 ξ (1 − ξ)(1 − η 2 )

φH,ξ φH,η

φI = (1 − ξ 2 )(1 − η 2 )

1 (1 − 2ξ) · η · (1 − η) 4 = 14 ξ (1 − ξ)(1 − 2η) = − 14 (1 + 2ξ) · η · (1 − η) = − 14 ξ (1 + ξ)(1 − 2η) = 14 (1 + 2ξ) · η · (1 + η) = 14 ξ (1 + ξ)(1 + 2η) = − 14 (1 − 2ξ) · η · (1 + η) = − 14 ξ (1 − ξ)(1 + 2η) = − 12 (−2ξ) · η · (1 − η) = − 12 (1 − ξ 2 )(1 − 2η) = 12 (1 + 2ξ)(1 − η 2 ) = 12 ξ (1 + ξ)(−2η) = 12 (−2ξ) · η · (1 + η) = 12 (1 − ξ 2 )(1 + 2η) = − 12 (1 − 2ξ)(1 − η 2 ) = − 12 ξ (1 − ξ)(−2η)

φA,ξ =

φI,ξ = − 2 ξ (1 − η 2 )

φI,η = (1 − ξ 2 )(−2η) . A E

H D

G

η

1

I

B

F C

ξ

Fig. 11-6 Quadratic shape function φB of the rectangular membrane element

11.2 Numerical Integration Applied to Membrane Elements

207

11.2.2 Derivatives with respect to local coordinates The transformation of the partial derivatives with respect to the global coordinates into partial derivatives with respect to the local coordinates is done accordingly to Section 11.1 with Equation (11.7). " #  ∂x ∂y  " # " # " # ∂ξ ∂x ℓx 0 ∂x ∂ξ ∂ξ · = = · . (11.7) ∂y ∂x ∂η ∂y 0 ℓy ∂y ∂η ∂η | {z } J

Here it holds

det J =

∂x ∂ξ

·

∂y ∂η

−

∂y ∂ξ

·

∂x ∂η

= ℓx · ℓy .

The inverse representation yields " # " # " 1  −1  ∂x ∂ξ = J · = ℓx ∂y ∂η 0

0 1 ℓy

# " ·

∂ξ ∂η

#

.

(11.8)

11.2.3 Element Stiffness Matrix and Load Vector The operator matrix D, as defined in Section 5.1.3, can be processed to build the derivatives with respect to local ξ–η–coordinates. Therefor it is advantageous to represent the operator matrix D as a product of two auxiliary matrices, to enable a decoupling between the derivatives with respect to x and y       ∂x 0 ∂x 0 1 0 0 0        ∂y 0  D =  0 ∂y  =  0 0 0 1  ·  .  0 ∂x  ∂y ∂x 0 1 1 0 0 ∂y Next, the derivatives with respect to x–y–coordinates will be replaced by the derivatives with respect to ξ–η–coordinates, using Equation (11.8)     ∂ξ 0 " # 1 0 0 0  ∂  J−1 0    η 0  D= 0 0 0 1 · · (11.9) . −1  0 ∂ξ  0 J 0 1 1 0 0 ∂η

Thereby, differentiation of the shape functions can be conducted in sub–steps, if the nodal unknowns are not listed in the order of the nodes as before, but in the order of the displacements. With vT = [ uA uB uC uD | vA vB vC vD ]

208

11 Numerical Integration

the matrix of linear shape functions results in " # φ 0 Ω= with φ = [ φA φB φC φD ] 0 φ and thus 

1  B=D·Ω=  0 0

0 0 0 0 1 1



0  1 · 0

"

J−1 0

0 J−1

or, briefly,

#



  · 

φ,ξ φ,η 0 0

0 0 φ,ξ φ,η

B = D · Ω = B1 · B2 · B3 .

    

(11.10)

(11.11)

The elasticity matrix E, given in Section 5.1.3, and the matrix B, subdivided into three parts, yield the element stiffness matrix with respect to local coordinates Z Z 1Z 1 K= BT · E · B · dA = BT · E · B · det J dξdη . (11.12) A

−1

−1

The integral may be transformed by means of Equation (11.11) to K=

Z

1

−1

Z

1 −1

BT3 BT2 BT1 · E · B1 B2 B3 · det J dξ dη .

(11.13)

B2 , B1 and E as well as det J = ℓx ·ℓy were discussed in Section 11.2.2. All the terms are constant inside an element and hence they have the same numerical value at each supporting point of the numerical integration. Thus the matrix multiplication ˆ = BT2 · BT1 · E · B1 · B2 · det J E can be performed in advance, what yields the element related constant matrix  ℓy  0 0 ν ℓ x   1−ν  0 1−ν · ℓx 0  E   2 ℓy 2 ˆ · (11.14) E= . 1−ν 1−ν ℓy  1 − ν2  0 · 0   2 2 ℓx ℓx ν 0 0 ℓ y

11.2 Numerical Integration Applied to Membrane Elements

209

It simplifies the integrand in Equation (11.13), which gives Z 1Z 1 ˆ · B3 dξ dη . K= BT3 · E −1

(11.15)

−1

The matrix B3 is comprised of the shape functions, differentiated to the local coordinates. For a linear approach this results in  φA,ξ φB,ξ φC,ξ φD,ξ 0   φA,η φB,η φC,η φD,η 0 B3 =   0 φA,ξ φB,ξ φC,ξ φD,ξ  0 φA,η φB,η φC,η φD,η

with respect 

  .  

The sequence of the nodal unknowns yields a clear partition of the matrix B3 and the stiffness matrix K into 4 sub–matrices. Table 11.3 illustrates the arrangement of the integrand of Equation (11.15) employing linear shape functions. To compute the element stiffness matrix by numerical integration, the matrix ˆ · B3 is to be performed considering the numerical values multiplication BT3 · E ˆ of the shape functions at the supporting points. Thereby, the zero entries in E will be omitted immediately. When employing bi–linear shape functions, n = 2 supporting points are needed ˆ is constant and the matrix B3 is still linear in each direction. Since the matrix E in one direction because of the partial differentiation in the other direction, then the integrand in Equation (11.15) will employ power terms at a maximum order of m = 2 in each direction. The selectively reduced integration of the stiffness matrix, given in Section 5.2, is performed by integrating the terms multiplied by eˆ22 , eˆ23 , eˆ32 , eˆ33 using a lower order of integration. If numerical integration is to be performed using 2 × 2 supporting points for example, then the terms defined above will be evaluated using only a 1–point integration. The 1–point integration matches a constant distribution of shear stresses. Employing bi–quadratic shape functions to approach the displacements, 9 nodes are available per element, so that B3 is to be extended to the shape functions φA until φI . In this case n = 3 supporting points are needed to numerically integrate the stiffness matrix, since the integrand in Equation (11.15) incorporates power terms at a maximum order of m = 4 in each direction. In this case, a reduced integration with respect to each direction does not resolve the issue. Here, the shape functions for the shear terms are to be chosen according to the shear deformations being linearly distributed inside the element.

φA,η φB,η φC,η φD,η 0 0 0 0

φA,ξ φB,ξ φC,ξ φD,ξ 0 0 0 0

φA,η φB,η φC,η φD,η

φA,ξ φB,ξ φC,ξ φD,ξ

eˆ14 0 0 eˆ44

0 0 0 0

0 eˆ23 eˆ33 0

0 0 0 0

ˆ E

BT3

0 eˆ22 eˆ32 0

eˆ11 0 0 eˆ41

eˆ11 φA,ξ eˆ22 φA,η eˆ32 φA,η eˆ41 φA,ξ

φA,ξ φA,η 0 0

φC,ξ φC,η 0 0

ˆ · B3 )δv u (BT3 · E

ˆ · B3 )δu u (BT3 · E

eˆ11 φB,ξ . . . eˆ22 φB,η . . . eˆ32 φB,η . . . eˆ41 φB,ξ . . .

φB,ξ φB,η 0 0

ˆ · B3 BT3 · E

0 0 φC,ξ φC,η

ˆ · B3 )δv v (BT3 · E

ˆ · B3 )δu v (BT3 · E

eˆ14 φB,η . . . eˆ23 φB,ξ . . . eˆ33 φB,ξ . . . eˆ44 φB,η . . .

φB,ξ φB,η

φA,ξ φA,η eˆ14 φA,η eˆ23 φA,ξ eˆ33 φA,ξ eˆ44 φA,η

0 0

0 0

ˆ · B3 E

φD,ξ φD,η 0 0

B3

φD,ξ φD,η

0 0

210 11 Numerical Integration

Table 11.3 Matrix scheme of the integrand to compute the stiffness matrix

11.2 Numerical Integration Applied to Membrane Elements

211

11.2.4 Element Load Vector A constantly distributed external action yields an integral in local coordinates Z Z Z 1Z 1 f = ΩT · p dA = ΩT dA · p = ΩT (ξ, η) · det J dξdη · p . −1

−1

In the case of a rectangular element, the determinant of the Jacobian matrix is constant, thus only the shape functions are to be integrated numerically. Since these functions employ lower order terms compared to the terms to be integrated when evaluating the stiffness matrix, less supporting points are to be considered at numerical integration.

11.2.5 The Subsequent Stress Analysis Although numerical integration is not required for the computation of the stress matrix, it is given here for completeness. The preparation shows some similarities compared to the computation of the stiffness matrix. The stress matrix yields, see Section 5.1.8, ˜ ·v=S ˜·v σ ˜ =E·B or, after transformation, ˜ =E·B ˜ = E · B1 · B2 · B ˜3 = E ˜ ·B ˜3 . S ˜ is also computed once in advance. Here, E  1 0  ℓx E  ν ˜= 0 · E 1 − ν 2  ℓx 0 1−ν 2ℓ y

0

ν ℓy 1 ℓy

1−ν 2ℓx

0

0



  . 

˜ will be immediately omitted in the program. At the The zero entries in E ˜ ·B ˜ 3 , the fact that the sequence of unknowns is multiplication of the term E ˜ 3 should be taken care of. However, fixed to uA , uB , uC . . . , vA , vB , vC . . . in B the sequence uA , vA , uB , vB , . . . should be applied in the program code, cf. the procedure to compute the element stiffness matrix. In the case of the corresponding local coordinates being given, the stresses can be evaluated either at the element nodes or at the supporting points of the Gauss–Legendre quadrature. It can be shown that the stresses are best approximated at the supporting points. This can be explained by the fact that the integral of the internal work δε·σ is also evaluated at the supporting points at the numerical integration.

212

11 Numerical Integration

11.3 Triangular Elements Triangular elements for membranes applying local coordinates are explained in Section 9 and for plates in Section 10. The shape functions as well as the integration are represented in both local ξ–η–coordinates and surface coordinates λa , λb , λc , but integration is performed analytically so far. To switch to numerical integration, only the supporting points need to be taken from Section 11.1. It should be mentioned that in the case of triangular elements, the directions of integration are not distinguished from each other, instead a simple single–summation is performed over all the supporting points.

11.3.1 Triangular Elements for Membranes Considering a linear approach for the displacements and therefore constant strains, the numerical integration of the element stiffness matrix requires only a single supporting point, since the function to be integrated is constant. Employing a quadratic approach for the displacements and thus linear strains three supporting points are to be considered, since the integrand is quadratic. Similarly to Section 11.2.3, the following notation is applied at integration. For triangular elements the inverse of the Jacobian matrix is represented by   1 bb bc −1 · J = ab ac det J with det J = ac · bb − ab · bc being constant within the element. Thus the ˆ multiplication of the matrices E, B1 , B2 yields the following E–matrix due to Equation (11.14) ˆ = BT2 · BT1 · E · B1 · B2 · det J , E which may be coded directly. As an abbreviation, e = (1 − ν)/2 is used. ˆ= E 

1 E · · 1 − ν 2 det J b2b + e a2b

b b b c + e ac ab

(e + ν) bb ab e bc ab + ν ac bb

  b b b c + e ac ab e a2c + b2c e bb ac + ν bc ab (e + ν) ac bc  ·  (e + ν) bb ab e bb ac + ν bc ab e b2b + a2b e b b b c + ac ab  e bc ab + ν ac bb (e + ν) ac bc e bb bc + ac ab a2c + e b2c



   .  

11.3 Triangular Elements

213

When assembling B3 , three shape functions for the linear approach and six shape functions for the quadratic approach are to be considered for u and v respectively. Therefore, the integrand of the stiffness matrix ˆ · B3 BT3 · E is comprised of the same terms as in Section 11.2.3. At the numerical integration the factor 1/2 in Equation (11.4) is to be taken into account. The load vector is computed analogously to the case of rectangular elements. ˜ and B3 are to be To perform the subsequent stress analysis the matrices, E adapted. Thus the stress matrix, analogously taken from Section 9.4.3, yields ˜ = E · B1 · B2 · B ˜ =E·B ˜3 = E ˜ ·B ˜3 , S with ˜ = E · B1 · B2 . E

˜ is computed for the whole element to be evaluated Here, E  bb bc ν ab ν ac  E 1 ˜= ν bc ab ac E · ·  ν bb 1 − ν 2 det J  1−ν 1−ν 1−ν 1−ν ab ac bb bc 2 2 2 2



 . 

(11.16)

11.3.2 Triangular Elements for Plates Triangular plate elements employing 18 or 21 DOF are derived in Section 10. Because of the employment of the general polynomial the numerical integration of the virtual work may be focussed on the integrals Z Z Kgen = ψ T DT E D ψ dA and fgen = ψ T p dA . The stiffness matrix regarding the general polynomial and the corresponding load vector have to be transformed to the physically meaningful nodal unknows by means of the scaling matrices G18 and G21 , which are not affected by the numerical integration. The computation of the scaling matrix and the determination of the globally positive direction of the normal vector with respect to each edge of the element is presented explicitly in Sections 10.2.4 and 10.3. The matrix multiplications K = GT 18 · Kgen · G18

as well as

f = GT 18 · fgen

214

11 Numerical Integration

are performed after the numerical integration of the general polynomial has been executed. With constantly distributed external actions, the integration can also be performed explicitly, see Section 10.2.6. The following should be taken care of at numerical integration: A complete approach of the 5th order is presented for the element employing 21 DOF. Since the general polynomial is differentiated twice, it leads to an integrand of the element stiffness matrix which comprises a polynomial of 6th order. For an exact integration of the element stiffness matrix, 12 supporting points are to be considered together with the corresponding weighting factors, see the figure on the right. The coordinates of the supporting points are given at 15 places behind the decimal point in Table 11.2. The integration is performed, as it is the rule for triangular elements, using a single–summation over all supporting points.

A

2 10

12 1

11 7 C

5 3

4 9 8

6 B

The stress analysis is performed similarly to the other elements. The bending– and twisting–moments at the corner nodes are computed formally using the stress matrix, whereat in this case the stress matrix comprises a direct correlation of the moments and the curvature DOF at the corner nodes.

12 Isoparametric Elements

In Section 11.2, to map a rectangular element to a unit square, a constant transformation rule is applied element by element. Now the same (= iso) shape functions φ, which are chosen to describe the physical behavior of an element, may also be used to describe the element geometry. This idea is first published by Taig [95] and later on introduced by Irons and Zienkiewicz [52] as isoparametric element concept. rectangle

quadrilateral

x

x (1;−1) linear functions

(−1;−1)

ξ

y (−1;1)

η

y

(1;1)

η xa quadr. y functions a

ξ η

ξ (−1;1)

(1;1)

x

y

(1;−1)

(−1;−1)

yh yd

A

xe

xb

E

B

D

F

I

H G

x

ξ C

η y

Fig. 12-1 Geometry variations employing isoparametric elements The most significant advantage of an isoparametric element is its adaptability to any existing geometry, e.g. angular or curvilinear boundaries, see Figure 12-1. Sometimes super– or sub–parametric elements are also employed. With the former ones, the order of the shape functions used to describe the element geometry is higher than the order used to describe physics and with the latter ones, the reverse is true. An example of a sub–parametric element is the membrane element employing 9 DOF and thus quadratic shape functions to describe the displacements u(x, y) and v(x, y) while describing the geometry by only employing linear shape functions. © The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2024 D. Dinkler und U. Kowalsky, Introduction to Finite Element Methods, https://doi.org/10.1007/978-3-658-42742-9_12

216

12 Isoparametric Elements

Generally, isoparametric elements considering both four or nine nodes are to be applied carefully. Since the typically used multiplicative approach of the displacements is not coordinate–invariant, the results become increasingly inferior as the geometry deviates more and more from a rectangle. Hence the mesh needs to be inspected, for example, the nodal angles are to be looked at. This could be evaluated by 90o − ε ≤ ϕ ≤ 90o + ε, when ε is a measurement of tolerance, e.g. of 10o . In principle the idea of an isoparametric approach can also be applied to triangular elements. A triangular element, arbitrarily lying in the x–y–plane, see Section 7.1, may be interpreted as a triangular isoparametric element. In this case, the description of the geometry is also performed using a linear transformation from local ξ–η–coordinates to global x–y–coordinates. Thereby the employment of complete shape functions is advantageous, because this automatically results in a coordinate–invariant approach.

12.1 Description of the Element Geometry The additional step from a rectangular element as given in Section 11.2 to an isoparametric quadrilateral element consists of performing the changed transformation rule between the ξ–η–coordinates and the x–y–coordinates. Inside an element, each arbitrary pair of coordinates (x, y) is described now by shape functions related to scaled coordinates x = φ(ξ, η) · x ˜, y = φ(ξ, η) · y ˜.

(12.1)

Thereby, the shape functions are the same as before φ(ξ, η) = [ φA

φB

φC

φD

... ]

and x ˜ as well as y ˜ represent the vectors of the coordinates of the element nodes     xa ya  xb   yb       xc    x ˜= y ˜ =  yc  . ,  xd   yd      .. .. . .

12.1 Description of the Element Geometry

217

Since the coordinates x and y now depend on ξ and η, the transformation of the partial derivatives is performed by " #  ∂x ∂y  " # ∂ξ ∂x ∂ξ ∂ξ · = . ∂y ∂x ∂η ∂y ∂η ∂η | {z } J

The matrix entries of the Jacobian matrix J may be computed applying Equations (12.1): ∂x = φ,ξ · x ˜, ∂ξ ∂x = φ,η · x ˜, ∂η

∂y = φ,ξ · y ˜, ∂ξ ∂y = φ,η · y ˜. ∂η

In general the Jacobian matrix follows with  φA,ξ xa  + φB,ξ xb " #   + ... φ,ξ · x ˜ φ,ξ · y ˜  J= =  φ,η · x ˜ φ,η · y ˜ φA,η xa   + φB,η xb

φA,ξ ya + φB,ξ yb + ...

    , φA,η ya   + φB,η yb  + ...

+ ...

which yields the inverse of the Jacobian matrix " # " i11 i12 i22 1 −1 J= , J = · det J i21 i22 −i21

with and



−i12 i11

(12.2)

#

det J = i11 · i22 − i12 · i21 i11 = i21 =

P k

P k

φk,ξ · xk ,

i12 =

φk,η · xk ,

i22 =

P k

P k

φk,ξ · yk , φk,η · yk ,

where k runs over all nodes and shape functions inside the element respectively. In principle, the Jacobian matrix which is still dependent on ξ and η is not constant. Hence, J−1 and det J cannot be evaluated for the entire element but only for each separate coordinate pair ξi and ηj . Thus the integration cannot be performed analytically any longer, and numerical integration needs to be applied, e.g. the Gauss–Legendre quadrature. Normally, the number of supporting points is chosen as in the case of a rectangular element.

218

12 Isoparametric Elements

12.2 Isoparametric Elements Regarding Membranes The procedure of how to develop isoparametric elements related to membranes shall now be demonstrated, whereby the respective rectangular element has already been prepared for numerical integration, cf. Section 11.2.3. The extension to non–rectangular elements needs the following supplement.

12.2.1 The Stiffness Matrix and the Stress Matrix Following Section 11.2.3 the element stiffness matrix of a rectangular element can be computed with Equation (12.3). Z 1Z 1 K= BT3 BT2 BT1 · E · B1 B2 B3 · det J dξ dη . (12.3) −1 −1

Applying the transformation of coordinates as represented in Section 12.1 yields the stiffness matrix of an quadrilateral isoparametric element. Hence, performing the integration to get the element stiffness matrix, the matrices J,

ˆ det J , J−1 , E

need to be computed at every supporting point, since the transformation of the derivatives may vary inside the element. This gives   1 ν   1−ν 1−ν   E 2 2 T  . E 1 = B1 · E · B1 = ·  1−ν 1−ν 1 − ν2    2 2 ν

1

ˆ at each supporting point. ApIt is followed by the numerical evaluation of E plying " # J−1 0 B2 = 0 J−1 yields ˆ = BT2 · E1 · B2 · det J . E

ˆ is now completely filled, since B2 is not a In contrast to Section 11.2.3, E ˆ can be computed analytically and can be evaluated for diagonal matrix. E each supporting point.

12.2 Isoparametric Elements Regarding Membranes

219

The multiplication of the matrices yields ˆ= E 

E 1 · · 2 1 − ν det J i222 + e i221

−i22 i12 − e i11 i21 −(e + ν)i22 i21

e i12 i21 + νi11 i22



   −i22 i12 − e i11 i21 e i211 + i212 e i22 i11 + ν i12 i21 −(e + ν)i11 i12    · . 2 2  −(e + ν)i22 i21 e i22 i11 + ν i12 i21 e i22 + i21 −e i22 i12 − i11 i21    e i12 i21 + νi11 i22 −(e + ν)i11 i12 −e i22 i12 − i11 i21 i211 + ei212

As an abbreviation, e = (1 − ν)/2 is used.

Similarly, at the computation of the stress matrix, only part of the matrix ˜ can be performed independently of the coordinates, at multiplications to get E which the stresses are to be evaluated. Again, the stresses may be determined at both the supporting points or at the element nodes, cf. Section 11.2.3. Thus the stress matrix, analogously taken from Section 11.2.3, yields ˜ =E·B ˜ = E · B1 · B2 · B ˜3 = E ˜ ·B ˜3 . S ˜ is computed for each point to be evaluated E  i22 −i12 ˜ = E · 1 · ν i −ν i12 E 22  1 − ν 2 det J 1−ν 1−ν − 2 i21 i 2 11

−ν i21 −i21

1−ν i 2 22

(12.4)

ν i11 i11 −

1−ν i 2 12



 .

(12.5)

12.2.2 The Selectively Reduced Integration Detrimental load–carrying behavior concerning bending of the elements employing linear shape functions has already been discussed in Sections 5.2, and a procedure using modified shear deformations has been suggested as a way out. For isoparametric membrane elements, the modification of the element stiffness matrix is possible by means of selectively reduced integration, too. When employing a linear approach for the displacements, this can be achieved if the terms, resulting from the entries (1,2), (1,3), (2,1), (2,2), (2,3), (2,4), (3,1), ˆ (3,2), (3,3), (3,4), (4,2) and (4,3) of the E–matrix, are integrated numerically applying a single supporting point. It means that the parts of the integrand with respect to the terms u,y and v,x must be described to be constant inside the element. The results given in Section 12.2.3 illustrate the improvements concerning accuracy.

220

12 Isoparametric Elements

By employing quadratic shape functions for the displacements, reduced integration can be performed by applying two supporting points in each direction, concerning the parts of the integrand relating to the terms u,y and v,x . This procedure provides very good results although it cannot be compared directly to that regarding linear shape functions. Nevertheless, in this case, integration should correspond to a linear distribution of u,y and v,x respectively. Thereby the integrands would arise quadratic at maximum, which does not correlate with integration when applying two supporting points, whereby third order polynomials may also be exactly integrated. Hence, at the integration procedure chosen here, unwanted parts of a higher order are also integrated. Nevertheless, the results are excellent, see Section 12.2.3. As an alternative, the terms u,y and v,x could be described directly using linear shape functions, so that the higher order terms disappear. Therefore, the numerical integration can be performed applying the well–known scheme and the order of integration with respect to the other parts of the integral. The reduced integration does not need to be considered at the subsequent stress analysis, since correct nodal displacements are automatically followed by the evaluation of correct stresses.

12.2.3 Comparison of Standard Membrane Elements The quality of isoparametric quadrilateral elements can be checked, if the results are compared to the results obtained from triangular elements, which are comprised of complete shape functions, cf. Section 9. A benchmark often given in the literature, is the cantilever first analysed by Cook [28], shown in Figure 12-2. c x b

B y

D

1N

a

C

A

E = 1.0 N/mm 2 ν = 0.33 a = 44 mm b = 16 mm c = 48 mm

Fig. 12-2 Cook’s cantilever – geometry and loading

12.2 Isoparametric Elements Regarding Membranes

221

In the following, the cantilever is investigated by employing 4– and 9–node quadrilateral elements as well as 3– und 6–node triangular elements. The meshes related to triangular elements develop from the division of the quadrilaterals related to the smaller diagonal. The results definitely are of inferior quality if the quadrilaterals are divided along the longer diagonal. The following Table 12.1 shows the convergence of the vertical end displacement in the centre line of the cantilever regarding different types of elements. Employing the same number of unknowns, the results obtained from the quadratic approach are definetely of a superior quality compared to those obtained from the linear approach. Applying the selectively reduced integration of the quadrilateral elements as presented in Section 12.2.2 improves the results substantially, especially regarding the coarse element mesh. Table 12.1 Cook’s cantilever – vertical displacement vD El.

mesh

1 2 3 4 5 6

iso - 4 iso - 4 - SRI iso - 9 iso - 9 - SRI tri - 3 tri - 6

3×3

5×5

9×9

17 × 17

33 × 33

65 × 65

11. 845 20. 043 19. 644 22. 626 11. 991 18. 358

18. 299 22. 648 23. 289 23. 653 18. 283 23. 301

22. 079 23. 569 23. 839 23. 885 22. 022 23. 856

23. 430 23. 846 23. 925 23. 937 23. 411 23. 935

23. 817 23. 927 23. 949 23. 954 23. 815 23. 951

23. 924 23. 953 23. 960 23. 962 23. 924 23. 961

Tables 12.2 and 12.3 present the convergence of the stresses σxx at the positions A and B. The convergence behavior already noted regarding the displacements is more clearly detected here. The best convergence can be observed for the 9–node element by applying selectively reduced integration, cf. Section 12.2.2. Table 12.2 Cook’s cantilever – compressive stresses σxx · (−1) at position A 9×9

17 × 17

33 × 33

65 × 65

El

left

right

left

right

left

right

left

right

1 2 3 4 5 6

0. 1907 0. 1497 0. 1349 0. 1546 0. 0987 0. 1207

0. 0947 0. 1619 0. 1438 0. 1412 0. 0960 0. 1435

0. 1676 0. 1411 0. 1298 0. 1354 0. 1126 0. 1263

0. 1149 0. 1468 0. 1333 0. 1328 0. 1154 0. 1331

0. 1499 0. 1353 0. 1289 0. 1302 0. 1204 0. 1281

0. 1228 0. 1382 0. 1300 0. 1299 0. 1230 0. 1299

0. 1397 0. 1321 0. 1287 0. 1291 0. 1245 0. 1285

0. 1261 0. 1335 0. 1291 0. 1290 0. 1261 0. 1290

222

12 Isoparametric Elements

Table 12.3 Cook’s cantilever – tensile stresses σxx at position B 9×9

17 × 17

33 × 33

65 × 65

El.

left

right

left

right

left

right

left

right

1 2 3 4 5 6

0. 1610 0. 1779 0. 1808 0. 1819 0. 1619 0. 1790

0. 1852 0. 1609 0. 1894 0. 1921 0. 1452 0. 1848

0. 1759 0. 1805 0. 1830 0. 1833 0. 1759 0. 1828

0. 1917 0. 1736 0. 1846 0. 1844 0. 1694 0. 1839

0. 1808 0. 1816 0. 1832 0. 1833 0. 1808 0. 1831

0. 1893 0. 1786 0. 1835 0. 1834 0. 1779 0. 1834

0. 1824 0. 1825 0. 1832 0. 1832 0. 1823 0. 1832

0. 1867 0. 1810 0. 1832 0. 1832 0. 1810 0. 1832

As known from rectangular elements the jump of the stresses at the intersection of neighboring elements is an indication of the discretization error and the convergence behavior. As expected, the convergence of stresses develops poorly compared to the convergence of displacements. Because of the non–rectangular element geometry, which normally yields poor shear stress approximations, the distributions of the other stresses are also badly influenced. Furthermore, the fulfillment of the Neumann boundary conditions needs a combination of all stresses, what includes a transformation of the equilibrium conditions at the boundary. The course of the stresses σxx at the cross–section A–B is shown in Figure 12-3 for the mesh sized with 65 × 65 nodes. Due to the oblique boundaries at the upper and the lower edge of the cantilever the course is not linear but slightly curved. B

48 x

44

16

B y

- 0.2

0.0

A

- 0.1

0.1

0.2

A

sxx

Fig. 12-3 Cook’s cantilever – stress σxx at section A–B regarding 65 × 65 nodes

12.3 Quadrilateral Plate Elements

223

Figure 12-4 gives the distributions of the stresses σxx , σyy , σxy over the whole structure. Stress σxx

−0. 20

Stress σyy

Stress σxy

N/mm2 + 0. 20

Fig. 12-4 Cook’s cantilever – the stresses regarding 65 × 65 nodes

12.3 Quadrilateral Plate Elements An isoparametric approach equivalent to the rectangular plate element employing 16 DOF is subject to restrictions, so that other types of elements should be chosen in order to enable arbitrary mesh geometries, cf. Figure 12-5. The plate element employing 12 DOF, cf. Section 6.3, does not, in fact, behave conform but can be applied as a sub–parametric element. A y

x

(-1,-1)

z, w

D

(1,-1)

(-1,1)

η

B

ξ

w (1,1)

local coordinates (ξ, η)

C

Fig. 12-5 Local coordinates of a quadrilateral plate element

224

12 Isoparametric Elements

At both the development of the stiffness matrix and the scaling matrix, and in contrast to the membrane elements, the 1st as well as the 2nd order derivatives of the deflection need to be transformed because of the curvature. The 1st and the 2nd order derivatives with respect to the local coordinates ξ, η are transformed into the respective derivatives with respect to the global coordinates x and y altogether by matrix operation, cf. the following scheme 

∂w  ∂ξ   ∂w    ∂η  2  ∂ w   ∂ξ 2   ∂2w    ∂η 2  2  ∂ w ∂ξ∂η



 ∂x   ∂w ∂y   ∂ξ  ∂ξ ∂x      ∂x  ∂y ∂w         ∂y ∂η   ∂η    2   ∂2w 2   ∂ x ∂ y ∂x ∂y ∂x ∂y  2 2 = ( ) ( ) 2( )    ∂ξ 2 ∂ξ 2 ∂ξ ∂ξ ∂ξ ∂ξ   ∂x2    2   ∂2x 2  ∂ w ∂ y ∂x 2 ∂y 2 ∂x ∂y    ( ) ( ) 2( )   2 2   ∂y 2 ∂η ∂η ∂η ∂η ∂η   ∂η    2 2  ∂ x ∂ y ∂x ∂x ∂y ∂y ∂x ∂y ∂y ∂x   ∂ 2 w ( ) ( ) ( + ) ∂ξ∂η ∂ξ∂η ∂ξ ∂η ∂ξ ∂η ∂ξ ∂η ∂ξ ∂η ∂x∂y



        .        

Employing a linear approach to describe the coordinate transformation x(ξ, η) and y(ξ, η), the second derivatives with respect to ξ and η disappear. The matrix of coefficients equals the Jacobian matrix. The inverse of the Jacobian matrix is to be introduced into the integrand of the Principle of virtual Work, similar to the way by which membrane elements were handled. Additionally, the Jacobian matrix is needed to compute the scaling matrix and to perform the subsequent stress analysis.

12.3.1 The 16 DOF Plate Element The approach employing 16 DOF is not complete regarding 4–node Kirchoff plate elements, cf. Section 6.2. Hence, it must inevitably run into problems with respect to the isoparametric concept. It must also be noted that the shape functions are related to the physically meaningful unknowns, the nodal rotations w,x and w,y , and partially the curvature w,xy . It is essential that the rotations w,x and w,y are transformed consistently from local to global coordinates, but the curvature w,xy is not, as the curvatures w,xx and w,yy are missing. Thus w,xy remains solely an element variable, which is coupled with the respective local coordinate system. The curvature w,xy bears the same meaning for two neighboring elements only if the local coordinates pass from one element to the next without a kink at the intersection line, see Figure 12-6. Therefore, the plate element employing 16 DOF is only satisfac-

12.3 Quadrilateral Plate Elements

225

torily applicable, if regular element meshes are generated without any kinks at the mesh lines.

regular mesh lines − possible

irregular mesh lines − not possible

Fig. 12-6 Sub–parametric plate elements – meshes employing quadrilaterals When performing numerical integration of the elment matrices, the following needs to be considered. The deflection of the element employing 16 DOF is described by means of a bi–cubic approach. Although the shape functions are double–differentiated in one direction, the integrand is comprised of a polynomial of the 6th order with respect to the other direction. Therefore, numerical integration is to be performed considering four supporting points in each direction. Because the origin of the coordinate system now lies in the centre of the related element, the Hermite Polynomials of Section 6.2 need to be transformed to the local coordinate system. This yields φ1 (ξ) = (2 − 3ξ + ξ 3 )/4 2

3

φ2 (ξ) = (1 − ξ − ξ + ξ ) · ℓx /8 φ3 (ξ) = (2 + 3ξ − ξ 3 )/4 2

3

φ4 (ξ) = (−1 − ξ + ξ + ξ ) · ℓx /8

φ1 (η) = 2 − 3η + η 3 )/4 φ2 (η) = (1 − η − η 2 + η 3 ) · ℓy /8 φ3 (η) = (2 + 3η − η 3 )/4

φ4 (η) = (−1 − η + η 2 + η 3 ) · ℓy /8 .

12.3.2 The 12 DOF Plate Element The rectangular plate element employing 12 DOF exhibits the deflection w and the rotations w,x and w,y at the corner nodes as unknowns, and thus does not have the difficulties mentioned above. The choice of the shape functions for the element with 12 DOF is possible by means of a general polynomial. Ten DOF ai are assigned to a complete cubic

226

12 Isoparametric Elements

approach as per Pascal’s triangle. Additionally the exponents ξ · η 3 and ξ 3 · η are chosen for the remaining two DOF. This is done similar to Equation (6.25) w(x,y) = a1 + a2 ξ + a3 η + a4 ξ 2 + a5 ξη + a6 η 2 + a7 ξ 3 + a8 ξ 2 η + a9 ξη 2 + a10 η 3 + a11 ξ 3 η + a13 ξη 3 .

(12.6)

In a first step the stiffness matrix and the load vector should be integrated concerning the general polynomial. Four supporting points are needed in each direction, because of the cubic order of the approach. The scaling of the coefficients of the general polynomial to the physically meaningful unknowns is performed afterwards by means of the multiplication of the stiffness matrix as well as of the load vector by the scaling matrix G12 . The scaling matrix for the element with 12 DOF can be derived for the isoparametric quadrilateral element with [ w(x,y) ] = ψ 12 · a12

˜ 12 · a12 v12 = Ψ

and ˜ −1 a12 = Ψ 12 · v12 = G12 · v12 . Thus the approach for the element with 12 DOF is established by the scaling matrix developed on the next pages. Since the continuity conditions of the rectangular element with 12 DOF are not fulfilled as discussed in Section 6.3, it is obvious, that the continuity conditions at the intersections of the isoparametric element are not satisfied as well. Nonetheless, the element converges against the correct solution.

Fig. 12-7 Shape functions for the element with 12 DOF

12.3 Quadrilateral Plate Elements

227

The scaling matrix G12 with respect to the nodal degrees of freedom   vT = [ w w,x w,y ]A [ w w,x w,y ]B [ w w,x w,y ]C [ w w,x w,y ]D

may be developed by means of the Jacobian matrix and the following procedure. Taking into account a quadrilateral element as represented in Figure 12-5 the element geometry is described by the differences of the coordinates a1 = 0. 25 · (−xA + xB + xC − xD ) a2 = 0. 25 · (+xA − xB + xC − xD ) a3 = 0. 25 · (−xA − xB + xC + xD ) b1 = 0. 25 · (−yA + yB + yC − yD ) b2 = 0. 25 · (+yA − yB + yC − yD ) b3 = 0. 25 · (−yA − yB + yC + yD ) . Therewith, considering the linearly fluctuating geometry, the Jacobian matrix of the element " 4 # " # Σi=1 φi ,ξ xi Σ4i=1 φi ,ξ yi a1 + a2 · η b 1 + b 2 · η J= = Σ4i=1 φi ,η xi Σ4i=1 φi ,η yi a3 + a2 · ξ b 3 + b 2 · ξ

depends just on the quantities ai , bi . The inversion of the Jacobian matrix yields " # " # i11 i12 b3 + b2 · ξ −b1 − b2 · η 1 −1 J = = · det J i21 i22 −a3 − a2 · ξ a1 + a2 · η whereat the determinant is given with det J = (a3 + a2 · ξ) · (b1 + b2 · η) − (b1 + b2 · η) · (a3 + a2 · ξ) . For simplicity the entries of the inverse are named by ikl , cf. Section 12.1. Since the Jacobian matrix is developed with respect to the transformation of derivatives  ∂    ∂  i11 i12 ∂x  =  ∂ξ  ∂ ∂ i21 i22 ∂y ∂η

it can be applied to describe the nodal degrees of freedom w,x and w,y by means ˜ of the unknowns a12 of the general polynomial (12.6). The complete Ψ–matrix is represented on the next page, whereat the rows have to be evaluated with respect to the local coordinates of the respective node.

˜ 12 Ψ

             =            



1 2i11 2i21

1 −2i11 −2i21

1 1 1 0 i11 i12 0 i21 i22

1 −1 1 0 i11 i12 0 i21 i22

−1 i11 − i12 i21 − i22

1 i11 + i12 i21 + i22 1 2i12 2i22

1 2i12 2i22

−1 1 −i11 + i12 −2i12 −i21 + i22 −2i22

1 2i11 2i21

1 1 −1 0 i11 i12 0 i21 i22

−1 2i11 + i12 2i21 + i22

−1 i11 + 2i12 i21 + 2i22

1 2i11 + i12 2i21 + i22

−1 3i12 3i22

−1 3i12 3i22

1 3i12 3i22

1 1 i11 + 2i12 +3i12 i21 + 2i22 3i22

−1 1 −1 3i11 −2i11 + i12 i11 − 2i12 3i21 −2i21 + i22 i21 − 2i22

1 3i11 3i21

1 −1 1 3i11 −2i11 + i12 i11 − 2i12 3i21 −2i21 + i22 i21 − 2i22

−1 3i11 3i21

 ,

−1 3i11 − i12 3i21 − i22

1 3i11 + i12 3i21 + i22

−1 i11 − 3i12 i21 − 3i22

1 i11 + 3i12 i21 + 3i22

−1 −1 −3i11 + i12 −i11 + 3i12 −3i21 + i22 −i21 + 3i22

1 1 −3i11 − i12 −i11 − 3i12 −3i21 − i22 −i21 − 3i22

[ w w,x w,y ]A [ w w,x w,y ]B [ w w,x w,y ]C [ w w,x w,y ]D

1 1 1 −2i11 −i11 − i12 −2i12 −2i21 −i21 − i22 −2i22



1 −1 −1 0 i11 i12 0 i21 i22

vT =

˜ 12 · a , v=Ψ

node D

     node A         node B         node C     



228 12 Isoparametric Elements

HYBRID QUADRILATERAL ELEMENTS

13 Hybrid Finite Elements

At finite element formulations employing the PvD the equilibrium condition is satisfied in a weak sense, whereas the other governing equations are fulfilled exactly by introducing these equations into the work equation of the PvD, cf. Section 2.1. As a consequence, the displacements persist as primary variables, whereas the stresses or stress resultants, which are the more important state variables for the structural design, are only computed by the subsequent stress analysis. Since the displacements have to be differentiated with respect to kinematics, this procedure yields stress distributions, which are generally more roughened compared to the approach of the displacements. More general approaches deal with displacement and stress variables, whereat the Principle of virtual Displacements and the Principle of virtual Forces are applied to fulfill the equilibrium and the condition of deformation in a weak sense. This idea leads to the Mixed Finite Element Method and the Hybrid Finite Element Method, which are applied in the following sections to develop finite elements for membranes and plates. The most general formulation is established by Hu and Washizu [44, 102] and thus known as the Hu–Washizu Principle. Generalized variational principles are applied to finite elements very early by Pian and Tong [80], Greene et al. [41] and McLay [67].

Mixed Finite Element Method The Mixed Finite Element Method deals with the equilibrium applying the PvD incorporating the stresses and kinematics applying the PvF incorporating displacements u and strains ε. Replacing the strains ε at the PvF by the material equation yields the condition of deformation in weak form, described by displacement and stress variables respectively. The integral representations of both principles PvD and PvF to discretize the equilibrium and the condition of deformation is named mixed formulation of virtual work. The idea of Mixed Principles of Work is originally developed as variation of a potential and first published by Hellinger [43]. Prange [82] describes an equivalent formulation for frame structures transferring the force–based method and the displacement–based method into each other by a Legendre transformation. Later on Reissner [87] develops the today’s common formulation. In literature, the mixed formulation often is named as Hellinger–Reissner Principle. The mixed formulation of the finite element method is of advantage, if nonlinearities have to be taken into account, cf. Kr¨oplin and Dinkler [58, 59, 60], but is restricted to structures without structural bifurcations. © The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2024 D. Dinkler und U. Kowalsky, Introduction to Finite Element Methods, https://doi.org/10.1007/978-3-658-42742-9_13

232

13 Hybrid Finite Elements

Applying the mixed formulation of the virtual work the displacements and the stresses are employed both as primary variables. Thus shape functions are to be chosen for the displacement variables as well as for the stress variables. Thereby it is advantageous that the stress variables are directly computed and no subsequent analysis is needed. Moreover, in particular regarding bending problems, approaches of less polynomial order can be chosen concerning the displacements likewise fulfilling the convergence criteria. A drawback might result from a higher number of degrees of freedom at every node.

Hybrid Finite Element Method The disadvantages of mixed elements can be avoided, if the stress variables are still defined inside the element, but are eliminated at the element level after the discretization. This procedure leads to an element dealing with displacement variables at the system level. Pian and Tong [80] and Pian and Sumihara [81] have been the first to propose this method as well as to apply it for the development of plane stress elements. Noor and Peters [74, 75] develop hybrid elements to investigate the nonlinear behavior of shells. The following is substantial to the procedure: • Parts of the shape functions as well as of the related degrees of freedom are defined inside the element. This part of the approach does not fulfill all conditions of continuity at the element interfaces. • Solely at the element interfaces additional degrees of freedom are defined. Thereby further conditions of continuity can be fulfilled either weakly or exactly related to the entire interface length. y z

x

interface

A a

B d

element domain

b

D

element boundary

c C

Fig. 13-1 Element domain and interface A 2–dimensional element is taken as an example to clarify the partioning of the element into the element domain, wherein the virtual work is discretized employing continuous shape functions, and the element interface, whereat the conditions of continuity are to be fulfilled, cf. Figure 13-1.

233 The classification into element domain and element interface results in multiple possibilities at formulating and discretizing the work equations. Displacement as well as stress fields are defined relating to the element domain and to the element interfaces, therewith the conditions of equilibrium and of deformation are to be fulfilled. Due to the different parts of the element, which are adressed as domain and as interfaces, the formulation is named as hybrid element method. In total three different hybrid formulations are defined in literature, cf. Figure 13-2: 1. HMM The hybrid–mixed model, whereby conditions of equilibrium as well as of deformation are weakly fulfilled in the element domain. The hybrid–mixed model comprises displacement as well as stress variables as variables inside the element. 2. HSM The hybrid–stress model, whereby the conditions of equilibrium are exactly fulfilled inside the element domain. This is equivalent to the force method in structural analysis. 3. HDM The hybrid–displacement model, whereby the conditions of deformation are exactly fulfilled inside the element domain. This is equivalent to the displacement method in structural analysis. The denotation goes by the degrees of freedom defined inside the element domain. Concerning the element interfaces normally displacement variables are defined as primary variables of the system. Finite Element Methods ✥ ❵❵❵ ✥✥✥ ❵❵❵ ✥✥✥ ❵ ✥ ❵ hybrid elements

mixed elements

displacement elements

hybrid–mixed model

equilibrium and def. condition weakly fulfilled elem. DOF: displacement and stress variables

hybrid–stress model

equilibrium exactly, def. condition weakly fulfilled elem. DOF: displacement and stress variables

hybrid–displacement model

equilibrium weakly, kinematics exactly fulfilled elem. DOF: displacement variables

Fig. 13-2 Overview of the different finite element methods

234

13 Hybrid Finite Elements

At first the hybrid–mixed model HMM shall be illustrated by means of examples of a bar and a rectangular plane stress structure. These both examples of use demonstrate the principles of the procedure. Due to C0 –conformity of the formulations the conditions at the interface are fulfilled already by the approach. Thus advantages at the choice of the shape functions to describe the behavior inside the element domain as well as at the interfaces exist in comparison to the model employing only displacements as primary variables, which may even increase at investigating nonlinear load–carrying behavior. Global system variables are the displacement variables at the element interfaces. Regarding bars and plane stress structures, these are the displacements at the nodes of the system. Further applications are Euler–Bernoulli beams and Kirchhoff plates, whereby C1 –conformity is to be ensured, that is, concerning mixed models with approaches related to w and M , strongly fulfilled for w and weakly fulfilled for ϕ. Concerning hybrid models additional arrangements are to be set up to fulfill all conditions at the interface. Hereinafter the principle procedure at deriving either mixed or hybrid–mixed formulations is developed relating to the example of the bar.

13.1 Mixed Formulation of Governing Equations In Section 2.1 the PvD is derived from the work equation and is shown to be equivalent to the differential equation of equilibrium. In the succeeding sections this procedure is also chosen to derive the mixed principle of work applying the PvD and the PvF to the bar depicted in Figure 13-3. p(x)

He c, EA

x, u

l

Fig. 13-3 Bar loaded by constantly distributed and concentrated external action The governing equations are given already in Section 1.3.1, whereby hereinafter the condition of equilibrium is extended by a tangential bedding force Fc . The material equations describe the bedding force with the bedding modulus c and the impressed strains εT from heating by T0 with the coefficient of thermal expansion αT .

13.1 Mixed Formulation of Governing Equations

235

a) Kinematics u,x − ε = 0

in the domain,

u − ue = 0

(13.1)

at fixed boundary: ue = 0 .

b) Equilibrium −fc + N,x + p(x) = 0 e

N −H =0

in the domain,

(13.2) e

at free boundary: H = 0 .

c) Material Equations fc − c · u = 0 ,

(13.3)

εel − N/EA = 0 ,

(13.4)

εT − αT · T0 = 0 .

(13.5)

Condition of Deformation Applying the formulation employing the displacements as primary variables, the material equations as well as the governing equations related to kinematics are introduced into the PvD and hence into the condition of equilibrium. Regarding the mixed formulation only the material equation related to bedding is introduced into the condition of equilibrium. The condition, which enforces the equality of strains due to heating and elasticity and due to kinematics, is named condition of deformation: ε = εel + εT .

(13.6)

Introducing the kinematics, Equation (13.1), and the material equations (13.4, 13.5) yield the condition of deformation in the domain, which is employed to connect the displacements with the stress variables, and the displacement boundary conditions: u,x −

1 · N − αT · T0 = 0 EA

and

u − ue = 0 .

(13.7)

Equilibrium (13.2) and the condition of deformation (13.7) are summarized applying the matrix vector notation: " # " # " # " # −c ∂x u p 0 · + = . (13.8) 1 N −αT · T0 0 ∂x − EA

236

13 Hybrid Finite Elements

13.2 Mixed Formulation of Work Equations To derive the finite element method in mixed formulation Equation (13.8) as well as Equation (13.2) and Equation (13.6) are formulated equivalently employing the principle of virtual work.

The Principle of Virtual Displacements – PvD The Principle of virtual Displacements represents a weak formulation of the conditions of equilibrium. Thereby the real forces perform work on the virtual displacements Z ℓ (−δu · Fc − δε · N + δu · p ) dx + [ δu · H e ]bound. = 0 . 0

If the virtual strains fulfill kinematics δu,x − δε = 0, integration by parts of the second term yields Z ℓ δu · (−Fc + N,x + p ) dx − [ δu · (N − H e ) ]bound. = 0 . 0

It becomes clear, that the PvD is equivalent to the conditions of equilibrium in the domain and at the boundary. Introducing the material equation related to the bedding force into the PvD yields the formulation, which bases the discretization of the finite element method Z ℓ Z ℓ Z ℓ − δu · c · u dx − δu,x · N dx + δu · p dx + [ δu · H e ]bound. = 0 . (13.9) 0

0

0

This formulation of the PvD completely corresponds to the displacement–based finite element formulation. Only by employing the Principle of virtual Forces to formulate the condition of deformation in a weak representation a completely independent method arises.

The Principle of Virtual Forces – PvF Along the lines of the Principle of virtual Displacements the Principle of virtual Forces is a weak representation of the conditions of deformation related to the domain as well as to the boundary, cf. Section 2.2. Regarding the PvF the virtual longitudinal force performs work on the real strains and displacements Z ℓ − δN · (u,x − ε) dx + [ δN · (u − ue ) ]bound. = 0 . 0

13.2 Mixed Formulation of Work Equations

237

The internal work terms get a negative sign here, since the internal virtual force variables and the strains act against each other, cf. Equation (13.9). Replacing the real strains by the material equation, yields in a first step Z ℓ N − δN · (u,x − − αT · T0 ) dx + [ δN · (u − ue ) ]bound. = 0 . EA 0 | {z } −ε

Regarding the integral of the internal work the condition of deformation (13.7) can be recognized now, including the displacement boundary condition in weak form. The request to exactly fulfill the displacement boundary condition by the approach itself yields the work equation of the PvF in the following representation, which is applied for the discretization: Z ℓ N − δN · (u,x − − αT · T0 ) dx = 0 . (13.10) EA 0 Concerning the work equations of the PvD and of the PvF, the displacements and the forces are independent variables. Thus for both independent approaches can be chosen in the context of finite element methods. Nonetheless, the boundary conditions at the boundary of the structure and the interface conditions between two elements are fulfilled in a different manner.

13.2.1 Shape Functions, Element Matrix, Load Vector Applying the convergence criteria analogously to Section 2.4.2 to mixed displacement and force variables formulation yields linear shape functions to describe u(x) and constant shape functions to describe N (x), following the criterion of at least constant strains. The same is valid for the virtual displacements as well as for the virtual forces. The criterion of conformity requires linear shape functions to describe u(x). To approach N (x) constant shape functions are adequate, since, concerning N , system boundary conditions as well as element interface conditions are weakly fulfilled by the work equation. Thus an element is derived, which incorporates different degrees of freedom relating to the element nodes and induces special indexing and control procedures at programming plane stress elements. In general linear shape functions are chosen to describe the longitudinal forces N (x), too. This might be advantageous at investigating geometric as well as physical nonlinearities. Choosing linear shape functions to describe the displacements and the forces, already known from the formulation solely employing displacements as primary

238

13 Hybrid Finite Elements

variables, cf. Section 2.3.2 and Figure 13-4, u(x) and N (x) may be scaled to nodal values as follows u(x) = φA (x) · uA + φB (x) · uB , N (x) = φA (x) · NA + φB (x) · NB . x

φA

=1-x/l

x

φB

=x/l

Fig. 13-4 Linear shape functions Concerning the virtual states δu, δN , the same shape functions are chosen δu(x) = φA (x) · δuA + φB (x) · δuB , δN (x) = φA (x) · δNA + φB (x) · δNB . Employing linear shape functions to describe N (x) according to the nodal degrees of freedom NA und NB , the conditions at the element interfaces Ni = Ni+1 result in the fact, that concentrated external loadings must not act at element interface nodes but are only reasonable at system boundaries.

13.2.2 Matrix Notation of the Work Equations To derive a general matrix representation the notation following Section 2.3.3 is applied, whereby the contents of the matrices related to bars may be adopted, too. To distinguish the discrete displacement variables from the stress variables, the indices ( )s relating to stresses or stress resultants and ( )v relating to displacements are introduced. Applying the matrix notation of Table 13.1 the work equations (13.20) can be generally represented relating to an arbitrary element in mixed formulation. At element level the work performed at the element interfaces as well as the work performed at the system boundaries are not taken into account. Concerning the PvD it follows Z Z Z − δWd = δvT · { ΩTv C Ωv dx · v + (ΩTv DT ) Ωs dx · s − ΩTv p dx}. (13.11) Accordingly, the work performed on virtual stresses PvK is formulated to Z Z Z − δWσ = δsT ·{ ΩTs (D Ωv ) dx·v− ΩTs E−1 Ωs dx·s− ΩTs ǫT dx}. (13.12)

13.2 Mixed Formulation of Work Equations

displacements

in general

related to the bar

u = Ωv · v

u = Ωv = vT = δvT =

δu = Ωv · δv stresses

σ = Ωs · s

δσ = Ωs · δs kinematics strains material equation strains due to heating bedding external action

239

ǫ =D·u ǫ = ǫel + ǫT ǫel = E−1 · σ

σ Ωs sT δsT

= = = =

D = ǫ = E−1 = ǫT = C = p =

[ u(x) ] [ φA φB ] [ uA uB ] [ δuA δuB ] [ N (x) ] [ φA φB ] [ NA NB ] [ δNA δNB ] [ ∂x ] [ǫ]  1  EA

[ αT · T0 ] [c] [ px ]

Table 13.1 Matrix notation of the governing equations of bars Along the lines of the displacement–based formulation the following abbreviations may be introduced Bv = D · Ωv ,

BTv = (D · Ωv )T = ΩTv · DT , and may be applied in the same meaning. The work equations (13.11) and (13.12) may be summarized applying the matrix notation, line by line extracting the virtual displacements and the virtual stresses: # " # Z " T # Z " T Ωv p  T T Ωv C Ωv ΩTv DT Ωs v − δW = δv δs { dx · − dx} . T T T −1 Ω Ωs D Ωv −Ωs E Ωs s s ǫT | {z } | {z } element matrix load vector Since the heat conduction is not included as part of the work equations, the strains εT are considered as external action in the load vector.

240

13 Hybrid Finite Elements

Element Matrix Concerning a generally applicable notation the representation arranging stresses behind displacements, is convenient. However, applying the notation to special element formulations, the arrangement with respect to nodal degrees of freedom is more reasonable. Therewith the element matrix related to the bar and employing linear shape functions follows to:     c·ℓ 1 c·ℓ 1 δuA : uA − − 2 6 2  3        1 ℓ 1 ℓ    δNA :  N  − 2 − 3EA 2 − 6EA   A   · . (13.13)  c·ℓ    1 c·ℓ 1     δuB :  6 u B 2 3 2        1 ℓ 1 ℓ − 2 − 6EA 2 − 3EA δNB : NB In contrast to the displacement–based formulation the matrix cannot be understood as a stiffness matrix.

Load Vector Regarding element by element constantly distributed external actions p as well as constant heating T0 the load vector results from the integral of external work   ℓ/2 0 " # " #  0 ℓ/2  fA p   = . (13.14) ·  ℓ/2 0  fB εT 0 ℓ/2 Linearly varying actions may be considered employing linear shape functions and related nodal values.

13.2.3 Test In order to evaluate the quality of the mixed formulation the following test is performed. Figure 13-5 shows the bar as given in Section 1.3.2 loaded by constantly distributed external actions. p(x)

EA = 6 kN l = 5m p(x) = 1 N/m

EA

x,u l 1

2

3

4

5

Fig. 13-5 Bar – geometry, loading, and discretization with four elements

13.2 Mixed Formulation of Work Equations

241

Employing linear shape functions for displacements and stresses the discretizations with four, five, six, and seven elements yield the study of convergence represented in Table 13.2, whereat the boundary condition of the force N |ℓ is fulfilled in a weak sense. Table 13.2 Bar – convergence of displacements and stresses number of nodes

1

2

3

4

5

6

7

8

N

[ kN ]

5. 00

3. 75

2. 50

1. 25

0. 00

u

[m]

0. 00

0. 95

1. 56

1. 99

2. 083

N

[ kN ]

4. 90

4. 10

2. 90

2. 10

0. 90

0. 10

u

[m]

0. 00

0. 77

1. 34

1. 76

2. 02

2. 083

N

[ kN ]

5. 00

4. 16

3. 33

2. 50

1. 66

0. 83

0. 00

u

[m]

0. 00

0. 65

1. 15

1. 58

1. 85

2. 04

2. 083

N

[ kN ]

4. 95

4. 34

3. 52

2. 91

2. 09

1. 48

0. 66

0. 05

u

[m]

0. 00

0. 56

1. 02

1. 41

1. 71

1. 92

2. 05

2. 083

The displacements at the nodes become partly the same values as at employing the displacement-based element, whereby the displacements by chance partly match the exact values. The stresses approach the linear course of the exact solution quite well, depending on the number of elements. The stresses match the exact values in the case of an equal number of elements but oscillate around the exact solution in the case of an uneven number of elements, since the condition of deformation Z N δN (u,x − ) dx = 0 EA is fulfilled in a weak sense. This indicates some inconsistencies at the element formulation. Nonetheless the element converges against the exact solution. Oscillations may be avoided, if the approaches for the displacements and the stresses are adapted to each other. This means, that a linear approach of the displacements needs a constant approach for the stresses, and a quadratic approach of the displacements needs a linear approach of the stresses. In these cases the results have the same quality as at employing the displacement–based elements. Nonetheless, if the order of the shape functions for N is lower than the

242

13 Hybrid Finite Elements

order of the shape functions for the displacements, the number of element nodes does not coincide with the number of stress variables. This means, that the stresses can be dealt with as internal variables as present in the displacement– based formulation. The disadvantages concerning the bar element in the mixed formulation may be summarized as follows: • The mixed formulation has got twice as much unknowns in comparison to the displacement–based element, without any improvement of the results. • If N is chosen as nodal variable, an external nodal action is impossible, since this would effect a jump with respect to the Force N . • If stresses are nodal degrees of freedom, due to the equilibrium conditions two elements may be connected at a node at most, because more elements would generate different stresses at the node with respect to each element. Thus trusses can not be analysed applying the mixed element. • If the approach of the stresses is of lower order than the approach of the displacements, the degrees of freedom related to the stresses are only element variables, without any connection to neighboring elements. In this case the element is comparable to the displacement–based formulation. In order to overcome the disadvantages of the mixed formulation of the work equations, the hybrid formulation has been developed as discussed in the following sections.

13.3 Hybrid Discretization of Work Equations The procedure to discretize the work equations employing hybrid elements is developed regarding the example of the bar, see Figure 13-6. HeA

p(x)

HeB

B

A x

Fig. 13-6 Element of a bar loaded by distributed and nodal external actions The Principle of virtual work in mixed displacement–force formulation is given with PvD and PvF, see Section 13.2. Applying the work equation related to the PvD (13.9) the equilibrium is weakly fulfilled in the domain as well as at

13.3 Hybrid Discretization of Work Equations

243

the system boundary. The interface condition related to the forces is fulfilled strongly or weakly depending on the approach. Applying the work equation related to the PvF (13.10) the conditions of deformation are weakly fulfilled with respect to the domain but the boundary as well as the interface conditions are strongly fulfilled concerning the displacements. Applying hybrid elements both types of work equations are discretized in the domain employing shape functions without fulfillment of the element interface conditions. The element interface conditions related to kinematics and equilibrium are separately looked at.

13.3.1 Work Equation at Element Level Regarding hybrid elements the element domain and the interface are to be distinguished. Applying the method of sections the segmentation to element domain and interface is possible, thus differing conditions may be formulated for both parts. Figure 13-7 shows the bar element with the element domain a − b and the interfaces A and B, which are identical to the system nodes, where the transition to the neighboring elements is to be ensured. HeA A

HeB

p(x) Na

a

u E(x), NE(x)

b

Nb

B

Fig. 13-7 State variables at the cut cleanly bar element The cut cleanly of the interfaces, cf. Figure 13-7, yields: • the element domain ( )E between a and b, • the element boundaries a and b, • the interfaces A and B, • the cut cleanly longitudinal forces Na , Nb .

At employing hybrid–mixed elements the work related to the element domain is described by longitudinal forces and displacements. However, at the element interfaces – these are the system nodes A and B – only displacements are defined. Relating to the element the work equations may be specified according to the indices ( )E for the element domain, for the element boundaries a, b and for the e e interfaces A, B. The work performed by the external actions HA , HB is taken into account, when the systems’ equations are assembled.

244

13 Hybrid Finite Elements

Applying the PvD to the element domain including element boundaries yields Z δWd = {−δεE · NE − δuE · c · uE + δuE · p(x)} dx |domain + [−δua · Na + δub · Nb ]el.bound. = 0 .

(13.15)

At the element intersection A between the elements j − 1, j and at the element intersection B between the elements j, j + 1 the PvD describes the work performed by the forces Na , Nb on the virtual displacements δuA , δuB of the intersections, which are independent of δua , δub so far: h i e δWd, A = δuA · −Nbj−1 + Naj + HA = 0, (13.16) h i e δWd, B = δuB · −Nbj + Naj+1 + HB = 0. (13.17) Equations (13.16) and (13.17) fulfill the equlibrium at the system level in a weak sense, if the forces Na , Nb are replaced by the variables at the element domain of the neighboring elements employing Equation (13.15). Due to the elimination of the forces Na , Nb no system variables exist, which are related to longitudinal forces at the interfaces. Thus, at the element level, introducing kinematics into the PvD yields Z − δWd = {δu,x E · NE + δuE · c · uE − δuE · p(x)} dx 6= 0 , (13.18) Applying the PvF kinematics is given in a weak formulation with: Z 1 δWσ = {−δNE · εE + δNE · · NE + δNE · αT · T0 } dx |domain EA

− δNa · [ua − uA ]trans.A−a + δNb · [ub − uB ]trans.B−b = 0 . (13.19)

e

Gaps ∆u are possible between element boundary and interface, but are not taken into account here. Strong fulfillment of the kinematic conditions at the interface related to the real as well as to the virtual displacements between element boundaries and interfaces, uA = ua ,

uB = ub ,

δuA = δua ,

δuB = δub ,

yields in a first step Z 1 − δWσ = {δNE · εE − δNE · · NE − δNE · αT · T0 } dx = 0 EA

and introducing kinematics into the work equation succeeds in Z 1 − δWσ = {δNE · u,x E − δNE · · NE − δNE · αT · T0 } dx = 0 . (13.20) EA

13.3 Hybrid Discretization of Work Equations

245

Applying this formulation of the virtual work, the element interface conditions related to the longitudinal forces can only be fulfilled weakly, since the longitudinal forces are defined solely at the element level and not at the interfaces, cf. Equations (13.16) and (13.17). Thus no direct coupling to the neighboring elements is present concerning N and δN , hence the PvF is to be arranged at the element level only.

13.3.2 Stiffness Matrix and Load Vector of the Hybrid Element The discretization of the work equation is related to the element domain, employing linear shape functions as for the mixed formulation, cf. Section 13.2. Applying matrix notation the integrals of the work equations related to the domain (13.18) and (13.20) can be given as follows: # " # Z " # Z " ΩT C Ω ΩTv DT Ωs δv : v ΩTv p v v dx − dx (13.21) ΩTs D Ωv −ΩTs E−1 Ωs δsE : sE ΩTs εT with the degrees of freedom, slightly changed in comparison to Section 13.2,

and

v T = [ uA uB ]

at system level

sTE

at element level.

= [ Na Nb ]

The integrals related to Equation (13.21) are directly comparable with the mixed formulation corresponding to Equation (13.13). Thereby the element matrix and the load vector are already derived, see Equation (13.14). At element level the PvF is valid Z −δWσ = δsTE {ΩTs D Ωv · v − ΩTs E−1 Ωs · sE − ΩTs εT } dx = 0 . (13.22) Applying Equation (13.22) the longitudinal forces sE can be described depending on the nodal displacements v Z Z T −1 −1 sE = { Ωs E Ωs dx } · { [ ΩTs D Ωv · v − ΩTs εT ] dx } . (13.23) In this form the equation can be used to eliminate sE from the equation applying the PvD. Out of it follows a representation, only incorporating the degrees of freedom at the system level v: Z  T − δWd = δv ΩTv C Ωv dx Z Z Z  + ΩTv DT Ωs dx · ΩTs E−1 Ωs dx }−1 · { [ ΩTs D Ωv · v − ΩTs εT ] dx Z − δvT ΩTv p dx . (13.24)

246

13 Hybrid Finite Elements

According to programming it might be advantageous to apply matrix notation to Equation (13.21) and to the elimination procedure subject to Equation (13.23), whereat the virtual variables are extracted again: δv : δsE :

"

Hvv Hsv

Hvs −F

# " ·

v sE

# " −

fp fT

#

dx

6= 0 at element level, (13.25) = 0 at element level.

The following abbreviations are introduced: Hvv = Hvs = F= fp = fT =

Z

ΩTv C Ωv dx ,

Z

ΩTs E−1 Ωs dx ,

Z

ΩTs εT dx .

Z

ΩTv DT Ωs dx = HTsv ,

Z

ΩTv p dx ,

Following the integration of the virtual work terms the computation of sE is numerically performed solving sE = F−1 [ Hsv · v − fT ] .

(13.26)

Introducing sE into the PvD yields −δWd = δvT { [ Hvv + HTsv F−1 Hsv ] · v − fp − HTsv F−1 fT } 6= 0 .

(13.27)

The expression in the square brackets of Equation (13.27) corresponds to the stiffness matrix of the displacement–based method.

Linear shape functions Applying linear shape functions, cf. Section 13.2, and employing the nodal degrees of freedom

and

v T = [ uA uB ]

at system level,

sTE

at element level,

= [ Na Nb ]

13.3 Hybrid Discretization of Work Equations

247

which satisfy the continuity of the displacements but not of the stresses, gives the element matrix according to Equation (13.21) HTsv

Hvv δuA : δuB : δNa : δNb :



cℓ 3 cℓ 6

      −1  2  − 12

cℓ 6 cℓ 3

− 12

− 12

1 2 1 2

ℓ − 3EA

ℓ − 6EA

1 2

ℓ − 6EA

1 2

ℓ − 3EA

−F

Hsv





uA



        uB     .  ·      Na     Nb

(13.28)

The elimination of the longitudinal forces sE at the element level is performed following Equations (13.25) and (13.26) with " # " # " # " # " # 1 1 − 12 12 Na uA 1 εT ℓ ℓ 3 6 − · 1 1 · + · = . (13.29) EA 2 Nb uB 1 − 12 12 6 3 Solving the equations (13.29) with respect to the stresses gives in a first step " # " # " # " # " # Na 4 −2 − 12 12 uA 1 εT ℓ EA = · ·{ · − } ℓ 2 Nb −2 4 uB 1 − 21 12 and after multiplication the final result " # " # " # " # Na −1 1 uA 1 EA = · · − EA εT . ℓ Nb −1 1 uB 1

(13.30)

Thus comparable to the element employing displacements only as primary variables and linear shape functions, it follows Na = Nb . Hence, concerning N , a constant approach is possible, leading to the same result here. Introducing Equation (13.30) into the PvD, the stiffness matrix follows, according to Equation (13.27), to " # " # " # 1 1 1 −1 uA EA T 3 6 }· δv ·K·v = [ δuA δuB ]·{c·ℓ . (13.31) 1 1 + ℓ −1 1 uB 6

3

In this case, the stiffness matrix of the hybrid–mixed method is identical by chance to the matrix according to displacement–based elements, cf. Section 2.3.1, and thus leads to the same results applying the element to structural analysis. Nonetheless, the hybrid–mixed method gives more freedom at the choice of shape functions for displacements and stresses inside the element.

14 Hybrid–Mixed Plane Stress Elements

The advantages of the hybrid–mixed formulation become more obvious at transferring the concept to plane stress structures. The freedom at the choice of shape functions and at the fulfillment of conditions of continuity is visible only now. Once again Figure 14-1 distinguishes between element domain and interface. At the element domain the shape functions to describe the displacements and the stresses can be chosen. So far, they do not have to fulfill the conditions at the element interfaces, due to the fact that the conditions of continuity only have to be fulfilled at the interfaces themselves. y z

x

interface

A a

B d

element domain

b

D

element boundary

c C

Fig. 14-1 Hybrid–mixed plane stress element

14.1 Mixed Principle of Work for Plane Stress Structures Succeeding, the governing equations as well as the work equations concerning PvD and PvF are prepared by analogy with the bar. To discretize the work equations a rectangular element employing linear shape functions is chosen. The matrix notation comprises the same symbols as already given in Section 13.3.

14.1.1 Governing Equations The governing equations are already represented in matrix notation, cf. Section 5.1. Thus, incorporating the matrix symbols into kinematics, it follows ǫ = Du · u , © The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2024 D. Dinkler und U. Kowalsky, Introduction to Finite Element Methods, https://doi.org/10.1007/978-3-658-42742-9_14

250

14 Hybrid–Mixed Plane Stress Elements   εxx ∂x    ε =  yy   0 2εxy ∂y 

 0   u  ∂y  · . v ∂x

(14.1)

The conditions of equilibrium can be given with

Dσ · σ + p = 0 ,   " # " # σxx ∂x 0 ∂y px   ·  σyy  + = 0, (14.2) 0 ∂y ∂x py σxy as well as the material equations according to the plane stress state with σ = E · ǫ, 

  σxx 1 E     σyy  =  ν 1 − ν2 σxy

ν 1

and in inverse representation

 

(14.3)

 

(14.4)

 εxx     ·  εyy  , 1 2εxy 2 (1 − ν)

ǫ = E−1 · σ ,   1 εxx 1     εyy  =  −ν E 2εxy 

−ν 1

 σxx     ·  σyy  . 2(1 + ν) σxy

Expressions concerning heating or bedding, as given for the bar, are not considered here. By analogy with the bar and to eliminate the strains, the material equation (14.4) is introduced into kinematics (14.1), what leads to the condition of deformation: Du · u − E−1 · σ = 0 . (14.5) Here, the differential operator Du correlates with the differential operator Dσ , since stresses and strains are energetically conjugated variables. Summarizing the governing equations, the conditions of equilibrium as well as of deformation may lead to the matrix formulation " # " # " # 0 Dσ u p · = . (14.6) Du E−1 σ 0 In addition to the governing equations, which are to be considered in the domain, the respective boundary conditions are to be taken into account. However they are not especially addressed here, cf. Section 5.

14.1 Mixed Principle of Work for Plane Stress Structures

251

14.1.2 Work Equations in Mixed Formulation By analogy with the bar the governing equations are formulated equivalently applying the Principle of virtual Work now.

The Principle of Virtual Displacements The Principle of virtual Displacements is equivalent to the conditions of equilibrium (14.2), if the virtual displacements fulfill kinematics. As a start, the following is valid Z δWd = {−δεxx · σxx − δεyy · σyy − 2δεxy · σxy + δu · px + δv · py } dA Z A + {δun σn + δus σt } ds = 0 . s

At the boundary ds the indices n and t indicate the normal and the tangential direction respectively. Replacing the virtual strains by kinematics yields Z δWd = {−δu,x · σxx − δv,y · σyy − (δu,y + δv,x ) · σxy + δu · px + δv · py } dA Z A + {δun σn + δut σt } ds = 0 s

or in matrix notation Z Z Z T T T − δWd = ( δu · D ) · σ dA − δu · p dA − δuT · σ e ds = 0 . A

A

(14.7)

s

At each boundary, the stress vector σ e only comprises the tangential and the normal component, which are related to the displacements un and ut . The brackets in Equation (14.7) imply to which state variable the differential operator D is to be applied.

The Principle of Virtual Forces The Principle of virtual Forces is equivalent to the condition of deformation. After have been introducing the material equations the weak form of the condition of deformation results in Z 1 ν ν 1 δWσ = {− δσxx · (u,x − σxx + σyy ) − δσyy · (v,y + σxx − σyy ) E E E E A 2(1 + ν) − δσxy · (u,y + v,x − σxy } dA E Z + {δσn · ( un − uen ) + δσt · ( ut − uet )} ds = 0 . s

252

14 Hybrid–Mixed Plane Stress Elements

Requiring the displacement boundary conditions as well as the displacement continuity conditions to be fulfilled concerning the real displacements by the shape functions with un − uen = 0 and ut − uet = 0, yields δWσ =

Z

1 ν ν 1 σxx + σyy ) − δσyy · (v,y + σxx − σyy ) E E E E A 2(1 + ν) − δσxy · (u,y + v,x − σxy } dA = 0 . E {− δσxx · (u,x −

Thus the conditions of deformation are fulfilled weakly with the PvF in the domain and are fulfilled strongly at the boundary. It follows in matrix notation − δWσ =

Z

A

δσ T · ( D · u − E−1 · σ ) dA = 0 .

(14.8)

Both principles of virtual work may be summarized by analogy with Equation (14.6) by matrix formulation # " # 0 DT u δu δσ · · dA D −E−1 σ A " # " # Z Z  T   T  p σe T T = δu δσ · dA + δu δσ · ds . 0 0 A s Z



T

T



"

(14.9)

14.2 Work Equations of a Hybrid Plane Stress Element Regarding hybrid elements the element domain is distinguished from the element boundary and the interface. In the mixed formulation the virtual work is arranged according to the Principle of virtual Work given by the PvD and the PvF, see Section 14.1 – without contour integrals. In addition the virtual work at the element boundaries and at the interfaces is to be described, in a first step without fulfillment of the conditions of continuity between the elements. Figure 14-2 identifies the stress variables σn , σt , that perform work at the element boundary b–c and at the interface B–C, as an example. With the PvD and the PvF the displacements and the stressess are identified to be the variables of description in the element domain and at the interfaces. Hence, concerning the following discretization, different procedures are possible at choosing adequate shape functions and at fulfilling the element continuity conditions.

14.2 Work Equations of a Hybrid Plane Stress Element

y z

x

253

A a B b

d D c

σt

σt

σn

σn C

Fig. 14-2 Normal and tangential stresses at element boundary and interface

14.2.1 Work Equations at Element Level At element level the PvD comprises the same expressions as the mixed formulation of the Principle of virtual Work following Equation (14.7). In addition the work is to be considered, that the stresses perform on the virtual displacements at the element boundaries as well as at the interfaces. Subsequently the work equations are formulated for a rectangular element to particularly illustrate the conditions of continuity related to the displacements between the element boundaries and the respective interfaces. The Principle of virtual Displacements yields Z δWd = {−δεxx · σxx − δεyy · σyy − 2δεxy · σxy + δu · px + δv · py } dA A Z   + δun,a/b − δun,A/B · σn ds |transitionA/B−a/b Z   + δun,b/c − δun,B/C · σn ds |transitionB/C−b/c Z   − δun,d/c − δun,D/C · σn ds |transitionD/C−d/c Z   − δun,a/d − δun,A/D · σn ds |transitionA/D−a/d Z   + δut,a/b − δut,A/B · σt ds |transitionA/B−a/b Z   + δut,b/c − δut,B/C · σt ds |transitionB/C−b/c Z   − δut,d/c − δut,D/C · σt ds |transitionD/C−d/c Z   − δut,a/d − δut,A/D · σt ds |transitionA/D−a/d . (14.10)

254

14 Hybrid–Mixed Plane Stress Elements

Concerning the Principle of virtual Forces, in addition to the mixed principle of work, the work performed by the virtual stresses on the real displacements at the element boundaries and at the interfaces is to be considered, since the related courses do not have to be chosen identically: Z 1 ν δWσxx = −δσxx · {εxx − · σxx + · σyy } dA E E A Z   − δσn · un,a/d − un,A/D ds |transitionA/D−a/d Z   − δσn · un,b/c − un,B/C ds |transitionB/C−b/c , (14.11) δWσyy =

δWσxy =

Z

ν 1 −δσyy · {εyy + · σxx − · σyy } dA E E A Z   − δσn · un,a/b − un,A/B ds |transitionA/B−a/b Z   − δσn · un,d/c − un,D/C ds |transitionD/C−d/c ,

Z

A

2(1 + ν) −δσxy · {2εxy + · σxy } dA E Z   − δσt · ut,a/b − ut,A/B ds |transitionA/B−a/b Z   − δσt · ut,b/c − ut,B/C ds |transitionB/C−b/c Z   − δσt · ut,d/c − ut,D/C ds |transitionD/C−d/c Z   − δσt · ut,a/d − ut,A/D ds |transitionA/D−a/d .

(14.12)

(14.13)

14.2.2 Conditions of Deformation at the Element Interface The condtions of continuity related to the displacements between two elements are fulfilled weakly by applying the contour integrals (14.11), (14.12) and (14.13). Thus, the approaches to describe the displacements may differ concerning the element boundary and the interface. The polynomial order of the shape functions with respect to the displacements at the element boundary is to be chosen equally or higher compared to the polynomial order at the interface, in order to be able to transfer sufficient information from the element domain to the interface. Tolerating small gaps between element boundary and interface and strongly fulfilling the respective conditions of continuity related

14.2 Work Equations of a Hybrid Plane Stress Element

255

to the real displacements un,interf ace = un,element boundary ,

ut,interf ace = ut,element boundary ,

results in disappearing of all boundary terms from PvF in Equations (14.11), (14.12) and (14.13). At choosing the same conditions related to the virtual displacements δun,interf ace = δun,element boundary ,

δut,interf ace = δut,element boundary ,

the contour integrals with respect to PvD are omitted, too, cf. Equation (14.10). After having introduced kinematics into the work equations concerning real as well as virtual strains, the work performed at element level follows to Z δWd = {−δu,x · σxx − δv,y · σyy − (δu,y + δv,x ) · σxy δWσxx = δWσyy = δWσxy =

Z

Z

Z

+ δu · px + δv · py } dA |domain 6= 0 ,

(14.14)

− δσxx · {u,x −

1 ν · σxx + · σyy } dA |domain = 0 , E E

(14.15)

− δσyy · {v,y +

ν 1 · σxx − · σyy } dA |domain = 0 , E E

(14.16)

− δσxy · {(u,y + v,x ) −

2 (1 + ν ) · σxy } dA |domain = 0 . (14.17) E

Thereby it is essential that the displacements are defined with respect to the element domain and to the interface and that the stresses are only defined with respect to the element domain.

14.2.3 Stress Conditions at the Element Interface The element continuity conditions related to the stresses are only to be fulfilled weakly, since the stresses are only defined in the element domain as wells as at the section between element boundary and interface but are not defined at the interface itself. Presuming the equality of the virtual displacements at the interface between two elements j and j + 1 yields omitting of the respective work integrals at system level. With Z δun · [ −σn |jb−c + σn |j+1 a−d ] ds = 0 , Z δut · [ −σt |jb−c + σt |j+1 a−d ] ds = 0

256

14 Hybrid–Mixed Plane Stress Elements

the equilibrium is fulfilled weakly at the interface between the neighboring elements j and j + 1 applying the PvD at system level. The same is valid concerning the respective work expressions at the element boundary and the interface. Thus, they do not have to be considered any further. When concentrated or line–shaped external loadings occur, the respective external work expressions have to be additionally taken into account at the interface. Hence, the real as well as the virtual stresses are not directly coupled to the neighboring elements and the PvF is employed at element level only.

14.3 Element Stiffness Matrix Developing the element stiffness matrix, the approach for the displacements may be chosen as a general polynomial or as shape functions dealing with physically meaningful degrees of freedom u = Ωv · v ,

or

u = Ψu · au = Ψu · G · v ,

whereat G represents the scaling matrix. The approach for the sresses is chosen as a general polynomial with respect to the element coordinates σ = Ψs · as . To start with, the work equation representing the PvD comprises the degrees of freedom related to the system level v as well as the degrees of freedom as at element level Z Z  −δWd = δvT (ΩTv DT ) Ψs dA · as − ΩTv p dA . (14.18) Applying the PvF, Equations (14.15) to (14.17), the following is valid just as well Z −δWσ = δaTs { ΨTs (D Ωv ) · v − ΨTs E−1 Ψs · as } dA = 0 . (14.19) Thus the degrees of freedom as may be computed from the nodal displacements v solving Z Z as = { ΨTs E−1 Ψs dA }−1 · { ΨTs (D Ωv ) dA} · v . (14.20) The solution can be applied to eliminate as from the PvD equation.

14.4 Subsequent Stress Analysis

257

Concerning programming it is convenient to formulate the Equations (14.18) and (14.19) as well as the subsequently following elimination process according to Equation (14.20) in matrix notation. Extracting the virtual variables yields   −δW = δvT δaTs · # " # Z " # Z " 0 (ΩTv DT ) Ψs v ΩTv p { dA − dA} (14.21) ΨTs (D Ωv ) −ΨTs E−1 Ψs as 0 comprising the degrees of freedom v at system level and as at element level corresponding to the approaches according to Sections 14.5, 14.6 and 14.7. The integrals according to Equation (14.21) may be numerically evaluated following Gauss. Thus the degrees of freedom as can be numerically eliminated. By analogy with the bar, " # " # " # δv : 0 HTsv v fp 6= 0 at element level, · − (14.22) δas : Hsv −F as 0 = 0 at element level is valid. The computation of as can be performed now with as = F−1 · Hsv · v . Introducing as into the conditions of equilibrium yields  −δWd = δv · [ 0 + HTsv · F−1 · Hsv ] · v − fp .

(14.23)

(14.24)

The expression given in square brackets at Equation (14.24) corresponds to the stiffness matrix of the displacement–based method.

14.4 Subsequent Stress Analysis Alternative A The calculation of the stresses related to the element domain can be performed by evaluating the governing equations equivalently to the procedure at applying displacement–based elements, cf. Section 12. This procedure is advantageous, since the approaches related to the displacements may be directly evaluated applying kinematics. Thus as a start the following is valid: σ = E · ǫ,

σ = E · D· u, σ = E·D·Ω·v = E·B·v = S·v.

(14.25)

258

14 Hybrid–Mixed Plane Stress Elements

The general formulation of the stress matrix S is performed by analogy with Section 12.2.2, and the evaluation is conducted preferably at the corner nodes of the element or the supporting points of the numerical integration with ˜ =E·B ˜ = E · B1 · B2 · B ˜3 = E ˜ ·B ˜3 . S ˜ is computed for each point to be evaluated with E  i22 −i12 −ν i21 E 1  ˜ ν i22 −ν i12 −i21 E= · · 1 − ν 2 det J 1−ν 1−ν 1−ν − 2 i21 i i 2 11 2 22

(14.26)

ν i11 i11 −

1−ν i 2 12



  . (14.27)

Concerning the element employing linear approaches to describe the displace˜ with the columns 2 and 3 of matrix E ˜ should ments the product of matrix B be evaluated in the centre of the element and with the columns 1 and 4 in the corner nodes of the element respectively. Thus physically non–desirable oscillations in the courses of stresses may be avoided, whereby this procedure is comparable to applying the selectively reduced integration according, cf. Section 5.

Alternative B The calculation of the stresses, which is performed here by employing the shape functions to describe the stresses and the PvF according to   σxx  σyy  = σ = Ψs · as and as = F−1 · Hsv · v , (14.28) σxy

yields more accurate results and is more efficient on top, if, after the discretization of the work equation concerning the PvF, the product F−1 · Hsv is stored element by element and is availabe for the subsequent stress analysis. After the computation of as the stresses may be evaluated by means of the general polynomials Ψs .

14.5 Linear Shape Functions The shape functions to describe the geometry of the quadrilateral element and to describe the displacements in the element domain may be taken from the displacement–based formulation, cf. Sections 5.1 and 12.2. Applying linear shape functions the element coordinates and element nodes are used as described in Figure 14-3. The following approach employ bi–linear shape functions to describe the displacements and three balanced polynomials with five general

14.5 Linear Shape Functions

259

degrees of freedom to describe the stresses yielding an element, which is named P-HMQ-4-5. y z

x

A

local coordinates

(ξ, η)

(-1,-1)

a (-1,1)

B d

ξ

η

b (1,-1)

D c

(1,1)

C

Fig. 14-3 Element nodes and coordinates – linear shape functions

Linear Approaches Concerning Geometry and Displacements At applying isoparametric elements the shape functions to describe the geometry are given as product expressions. Regarding bi–linear approaches employing the shape functions φ(ξ, η)i , this yields 1 (1 + ξi · ξ)(1 + ηi · η) xi , 4 1 y = Σi (1 + ξi · ξ)(1 + ηi · η) yi . 4 x = Σi

Thereby xi , yi , ξi and ηi are the global and the local coordinates of the respective element nodes. By complete analogy the approach related to the displacements is chosen to 1 (1 + ξi · ξ)(1 + ηi · η) ui , 4 1 v(ξ, η) = Σi (1 + ξi · ξ)(1 + ηi · η) vi . 4 u(ξ, η) = Σi

Thus, concerning the real displacements, it follows u = Ωv · v with uT = [ u v ] and the nodal displacements vT = [ uA vA uB vB uC vC uD vD ] as well as with the same related to the virtual displacements δu = Ωv · δv .

260

14 Hybrid–Mixed Plane Stress Elements

Linear Approaches Concerning the Stresses The element continuity conditions are only fulfilled weakly by the stresses. Therefore, the approaches related to the stresses inside the element domain may be chosen such, so as to adapt best their courses to the distribution of the displacements and strains respectively. In general, the number of degrees of freedom of the approaches to describe the stresses must be equal or larger than the number of degreees of freedom to describe the strains, which yields a regular stiffness matrix. Mathematical evidence related to this statement can be found at Babuska [6, 7], Brezzi [22] and Ladyzhenskaya [61] and is denoted as LBB– Condition. Thereby, as a mechanical interpretation, each possible displacement field can be connected to a specific strain-related stiffness. The bi–linear approaches to describe the displacements u(ξ, η) and v(ξ, η) incorporate four nodal displacement variables each. Thereby, in two dimensions, three rigid body motions – two translations and a rotation – may be described as well as five displacement fields, which are available to describe the strains εxx εyy εxy . Therefor an approach which balances stresses and strains should incorporate at least five degrees of freedom related to the stresses. At choosing the approaches it should be claimed that the courses of the strain εxx = u(x, y),x and thus also of the stress σxx are constant in x–direction. Concerning the strain εyy = v(x, y),y and the shear strain 2εxy = u(x, y),y + v(x, y),x the corresponding facts are valid. This is reached by employing the following non– scaled approaches σxx = a1 + a2 · η

constant in ξ–direction,

σyy = a3 + a4 · ξ

constant in η–direction,

σxy = a5

constant.

Therewith the approach to describe the stresses is given by aTs

σ = Ψs · as

and = [ a1 a2 a3 a4 a5 ]. By analogy the approach to describe the virtual stresses might be chosen. There is a small discrepancy in assuming the strains in direction of the coordinates x, y to be constant, but inside the element to be constantly arranged in the direction of the loacal coordinates ξ, η. This drawback may be avoided at employing isoparametric quadrilateral elements including an additional transformation. However, no substantial improvement concerning the quality of the results is achieved. A comparison with respect to other elements is represented in Section 14.8 by means of two benchmarks.

14.6 Quadratic Shape Functions

261

14.6 Quadratic Shape Functions At deriving the formulation employing the displacements as primary variables it has turned out, that the convergence behavior of quadratic approaches is significant better in comparison to linear approaches, cf. Section 9.5. Thus it seems to be natural to employ equivalent approaches to derive hybrid elements, too. Applying incomplete approaches of third order to describe the displacements and approaches with 13 degrees of freedom to describe the stresses yields an element, which is named P-HMQ-8-13. y z

x

local coordinates (ξ, η)

A

E

(-1,-1) a

H

(0,-1)

(-1,0)

ξ

η

d

D

g (0,1)

G

B

e

h (-1,1)

b (1,-1) f

c

(1,0)

F

(1,1)

C

Fig. 14-4 Element nodes and coordinates – quadratic approaches Concerning non–rectangular quadrilateral elements, the shape functions to describe the geometry are still given with 1 (1 + ξi · ξ)(1 + ηi · η) xi , 4 1 y = Σi (1 + ξi · ξ)(1 + ηi · η) yi . 4 x = Σi

Hereby xi , yi , ξi and ηi are the coordinates of the respective element nodes, cf. Figure 14-4.

Bi–Quadratic Approaches Concerning the Displacements To describe the displacements u, v incomplete cubic approaches are chosen, that employ eight degrees of freedom each and that are related to the corners as well as to the midpoints of the edges. Since the element midpoint is neglected, the approaches do not comprise all parts of a symmetrical approach, because the term ξ 2 · η 2 is neglected. This fact turns out to be no drawback, but avoids unphysical oscillations of the solution inside the element domain. Employing

262

14 Hybrid–Mixed Plane Stress Elements

the approach for the displacements u(ξ, η) u(ξ, η) = a1 + a2 ξ + a3 η + a4 ξ 2 + a5 ξη + a6 η 2 + a7 ξ 2 η + a8 ξη 2 yields the matrix notation u(ξ, η) = Ψ8 · au . The scaling of u(ξ, η) with respect to the nodal displacements vu T = [ uA uB uC uD uE uF uG uH ] results in 

       ˜8 =  Ψ       

1 −1 −1 1 1 −1 1 1 1 1 −1 1 1 0 −1 1 1 0 1 0 1 1 −1 0

1 1 1 −1 1 1 1 −1 0 0 1 0 0 0 1 0

1 1 1 1 1 0 1 0

 −1 −1  −1 1   1 1    1 −1   0 0    0 0   0 0   0 0

˜ −1 with the inverse representation G8 = Ψ 8   −0,25 −0,25 −0,25 −0,25 0,50 0,50 0,50 0,50    0 0 0 0 0 0,50 0 −0,50     0 0 0 0 −0,50 0 0,50 0       0,25  0,25 0,25 0,25 −0,50 0 −0,50 0 −1 ˜  . Ψ8 =  0,25 −0,25 0 0 0 0   0,25 −0,25     0,25  0,25 0,25 0,25 0 −0,50 0 −0,50    −0,25 −0,25 0,25 0,25 0,50 0 −0,50 0    −0,25 0,25 0,25 −0,25 0 −0,50 0 0,50 Choosing an equivalent description for the displacement v(ξ, η) yields the matrix notation concerning quadratic approaches for uT = [ u(ξ, η) v(ξ, η) ] u = Ψ16 · au,v .

14.6 Quadratic Shape Functions

263

Scaling au,v with respect to the nodal displacements vT = [ uA vA uB vB uC vC uD vD uE vE uF vF uG vG uH vH ] yields the physically meaningful shape functions Ωv u = Ψ16 · G16 · v , = Ωv · v . The approach to describe the virtual displacements is equivalently chosen.

Quadratic Approaches Concerning the Stresses The incomplete cubic approach to describe the displacements u(ξ, η) and v(ξ, η) comprises 2 × 8 nodal displacements. Therewith, in two dimensions, three rigid body motions – two translations and one rotation – may be described as well as 13 displacement fields, that are available for the description of the strains [ εxx εyy εxy ]. Therefore, an approach to describe the stresses and being balanced to the strains comprises 13 degrees of freedom as well. By analogy with the linear approach and with respect to the strains, one may request on εxx = u(x, y),x and thus also on the stresses σxx being linear in x–direction but quadratic in y–direction. Correspondingly, requests are to be made with respect to εyy = v(x, y),y and, regarding the shear strains, to 2·εxy = u(x, y),y + v(x, y),x . These demands are fulfilled employing the following non– scaled approaches σxx = a1 + a2 · ξ + a3 · ξη + a4 · η + a5 · η 2

linear in ξ–direction,

σyy = a6 + a7 · η + a8 · ηξ + a9 · ξ + a10 · ξ

linear in η–direction,

σxy = a11 + a12 · ξ + a13 · η

2

bi–linear.

Hence the approach to describe the stresses is given with σ = Ψs · as and aTs = [ a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 ], an analogous approach is chosen to describe the virtual stresses. Equivalent to the linear approach, small discrepancies arise at assuming the courses of strains being linear in direction of the coordinates x, y, but being addressed in direction of the coordinates ξ, η inside the element. Introducing a modification at the approaches related to the stresses, these deviations may be considered, cf. Section 14.7. The benchmarks represented in Section 14.8 illustrate the high quality of the element.

264

14 Hybrid–Mixed Plane Stress Elements

14.7 Linear Approaches with Coupling of Degrees of Freedom The approaches to describe the stresses are chosen with respect to the local ξ–η– coordinate system so far, cf. Sections 14.5 and 14.6. In the case of geometrically deformed elements this may result in undesirable oscillations as well as bad convergence. These drawbacks may be partially eliminated by a modification of the approaches, that takes into account the variation of the components of the Jacobian matrix with respect to the element coordinates. Figure 14-5 depicts the angles ϕξ and ϕη of the local ξ–η–axes with respect to the global x–y–coordinates as well as the modification of the element geometry along the coordinates. x

xo

1

A

B

ϕξ

4 yo

2

ξ

ϕη

D 3

η

C

y

Fig. 14-5 Isoparametric quadrilateral element The element geometry is described by the differences of the coordinates a1 = 0. 25 · (−xA + xB + xC − xD ) a2 = 0. 25 · (+xA − xB + xC − xD ) a3 = 0. 25 · (−xA − xB + xC + xD ) b1 = 0. 25 · (−yA + yB + yC − yD ) b2 = 0. 25 · (+yA − yB + yC − yD ) b3 = 0. 25 · (−yA − yB + yC + yD ) . Therewith, considering the linearly fluctuating geometry, the Jacobian matrix

14.7 Linear Approaches with Coupling of Degrees of Freedom of the element may be computed by means of ai , bi " 4 # " a1 + a2 · η Σi=1 φi ,ξ xi Σ4i=1 φi ,ξ yi J= = Σ4i=1 φi ,η xi Σ4i=1 φi ,η yi a3 + a2 · ξ

b1 + b2 · η b3 + b2 · ξ

265

#

.

Concerning the origin of the local coordinate system, this yields det J|0 = det J|ξ=0, η=0 = b3 · a1 − b1 · a3 . The angles between the coordinate axes are given with cos ϕη = 2 · b3 /ℓη , sin ϕη = −2 · a3 /ℓη , cos ϕξ = 2 · a1 /ℓξ , sin ϕξ = 2 · b1 /ℓξ ,

employing the differences of the coordinates of the midpoints of the edges p ℓξ = 2 · (a1 )2 + (b1 )2 , p ℓη = 2 · (a3 )2 + (b3 )2 .

14.7.1 Approach Following Pian and Sumihara By analogy with Section 14.5, Pian and Sumihara [81] develop a 4–node element comprising bi–linear approaches to describe the displacements as well as linear approaches employing five degrees of freedom βi to describe the stresses. Deviating from Section 14.5 the approaches to describe the stresses are not independent from each other, but are coupled by the degrees of freedom β4 , β5 . This element is named P-HMQC-4-5, because of the coupling of the shape functions. The coefficients ai , bi , which are employed in the approaches   β1      β2  σxx 1. 0 0 0 a1 · a1 · η a3 · a3 · ξ     σyy  =  0 1. 0 0  b1 · b1 · η b3 · b3 · ξ ·  β3  ,  σxy 0 0 1. 0 a1 · b1 · η a3 · b3 · ξ β4  β5

take into account the geometry of the element. The approach is deviated by employing additional internal quadratic polynomials concerning the displacements, see [81]. The dependencies on units, that exist in the approaches concerning the degrees of freedom β4 , β5 , could be eliminated by a convenient scaling procedure. Employing these approaches yields computational results of high quality, which is proven by comparison to the quadratic approach presented in Section 14.6, cf. the study of convergence at Section 14.8.

266

14 Hybrid–Mixed Plane Stress Elements

14.7.2 Approach with Transformation of Coordinates The coupled linear approaches comprising five degrees of freedom may be deviated completely different by means of a transformation of the stresses between the global x–y coordinate system and the local ξ–η coordinates. Figure 14-6 illustrates the transformation with two sectional views. x

σηη

x

σηξ

σyy

σyx

1

5

5

6

σηη

ϕξ

σxy

σξξ σξη

σxx

σηξ

σξξ

σξη

6

ξ η

3

ϕη

ϕξ ξ

y

η

ϕη

y

Fig. 14-6 Coordinate transformation from x–y to ξ–η The lengths of the edges, that are required for the transformation procedure, are identified from the corner nodes of the sections. The coordinates of the midpoint of the edges 1 and 3 are given with Figure 14-5. Thus the lengths ℓ13 = ℓη , ℓ15 = ℓη sin ϕη = −2 · a3 , ℓ35 = ℓη cos ϕη = 2 · b3 , may be computed directly, since the coordinates of the corner node 5 are determined by the midpoints of the edges 1 and 3. The coordinates of the node 6 may be determined with respect to corner node 5 by ∆y56 b1 = , ∆x56 a1 ∆x56 −a3 tan ϕη = = . b3 − ∆y56 b3 tan ϕξ =

Introducing ∆x56 = −a1 · a3 · b3 / det J0 , ∆y56 = −b1 · a3 · b3 / det J0

14.7 Linear Approaches with Coupling of Degrees of Freedom

267

the still missing lengths follow to p ℓ56 = (∆x56 )2 + (∆y56 )2 = −a3 · b3 · ℓξ / det J0 , ℓ36 = (b3 − ∆y56 )/ cos ϕη = a1 · b3 · ℓη / det J0 , ℓ16 = ℓ13 − ℓ36 = −b1 · a3 · ℓη / det J0 . In a first step the conditions of equilibrium related to the x– and the y–direction are assembled following Figure 14-6, and they are rearranged to transform the stresses now. Thus matrix notation yields   ℓ36 ℓ56   cos ϕξ sin ϕη − ℓℓ36 sin ϕη − ℓℓ56 cos ϕξ  σξξ  σxx ℓ ℓ 35 35 35  35  ℓ56 ℓ16 ℓ56  σ   ℓ16 sin ϕξ   cos ϕη cos ϕη sin ϕξ   yy   ℓ15   σηη  ℓ15 ℓ15 ℓ15   =  ℓ36  . ℓ ℓ ℓ 56 36 56  σxy    σξη  cos ϕη − ℓ sin ϕξ  ℓ35  ℓ35 sin ϕξ − ℓ35 cos ϕη  35 ℓ16 ℓ56 σyx σηξ cos ϕξ − ℓ56 sin ϕη − ℓ16 sin ϕη cos ϕξ ℓ15

ℓ15

ℓ15

ℓ15

Replacing the trigonometric functions as well as the length of the edges by the differences of the coordinates and taking into account σxy = σyx results in 

 σxx 1    σyy  = det J0 σxy



ℓ

ℓ

a1 a1 ℓη

a3 a3 ℓ ξ

a1 a3

b3 b3 ℓ

b1 b3

η ℓξ

ξ ℓη

  ·  b1 b1 ℓ ξ  ℓ a1 b1 ℓη

η

ℓ

a3 b 3 ℓ ξ

η

ξ

a1 b 3

a1 a3



  b1 b3    b 1 a3

σξξ σηη σξη σηξ



  . 

In a second step and following Section 14.5 the linear approach to describe the local stresses is chosen to σξη = σηξ employing the 5 degrees of freedom βi     

σξξ σηη σξη σηξ





1     0 = 0  0

0 1 0 0

0 0 1 1

and is applied to transform the stresses 



σxx 1    σyy  = det J0 σxy

    

ℓ a1 a1 ℓη ξ ℓ b1 b1 ℓη ξ ℓ a1 b1 ℓη ξ

ℓ a3 a3 ℓ ξ η ℓ b3 b3 ℓ ξ η ℓ a3 b 3 ℓ ξ η

η 0 0 0





0   ξ   0   0

β1 β2 β3 β4 β5

ℓ 2a1 a3 a1 a1 ℓη ξ ℓ 2b1 b3 b1 b1 ℓη ξ ℓ a1 b3 + b1 a3 a1 b1 ℓη ξ

     

·η ·η ·η

(14.29)

ℓ a3 a3 ℓ ξ η ℓ b3 b3 ℓ ξ η ℓ a3 b 3 ℓ ξ η

 ·ξ    · ξ    ·ξ

β1 β2 β3 β4 β5



  .  

268

14 Hybrid–Mixed Plane Stress Elements

Transforming the degrees of freedom βi with ℓ ℓ β˜1 = β1 ℓη /det J0 , β˜2 = β2 ℓξ /det J0 , β˜3 = β3 /det J0 , η

ξ

β˜4 =

ℓ β4 ℓη /det J0 , ξ

β˜5 =

ℓ β5 ℓξ /det J0 η

the approach can be represented by 





σxx a1 a1     σyy  =  b1 b1 σxy a1 b 1

a3 a3 b3 b3 a3 b 3

2a1 a3 2b1 b3 a1 b 3 + b 1 a3

 ˜ β1 a3 a3 · ξ  β˜2   ˜ b3 b3 · ξ  ·   β3  β˜4 a3 b 3 · ξ β˜5 

a1 a1 · η b1 b1 · η a1 b 1 · η



  .  

The appproach presented here is equivalent to the approach developed by Pian und Sumihara following a completely different path. It can be transferred applying a scaling procedure to the degrees of freedom β˜1 , β˜2 , β˜3 . The advantage of the formulation presented here is the possibility to apply it to higher order polynomials by analogy with Equation (14.29).

14.8 Convergence Behavior of the Elements The hybrid–mixed plane stress elements partially yield very good results. Due to the special adaptation of the courses of stresses to the strain fields the physically non–reasonable zero energy modes are avoided.

Example of Use 1 As a first example of use a benchmark is investigated at first published by Pian and Sumihara, cf. Figure 14-7. P2 2

2

A

1

1

4

P1

B

C

P2

2

y, v x, u

D 1

1

2

3

3

P1

P2

E = 1500 N/m 2 , ν = 0.25, P1 = 1000 N, P2 = 150 N, l x = 10 m, l y = 2 m

Fig. 14-7 Cantilever published by Pian and Sumihara [81] – geometry and loading

14.8 Convergence Behavior of the Elements

269

The cantilever is examplarily investigated employing linear approaches to describe the displacements and relating linear approaches to describe the stresses according to Sections 14.5 and 14.7 as well as employing quadratic approaches according to Section 14.6. Due to the complex element geometry and the chosen approaches the different element formulations approximate the displacements as well as the stresses with different accuracy. Applying the finite element mesh according to Figure 14-7 the subsequent Table 14.1 comprises the displacement results related to the loading case P1 = 1000 N . The three hybrid element formulations are compared to the analytical solution, that is evaluated with respect to a linearly distributed loading in y–direction p1 (y) = 2000 · ( y − 1 ) N/m. Table 14.1 Loading case bending P1 – comparison of different elements vA reference solution P-HMQ-4-5 – Section 14.5 P-HMQ-8-13 – Section 14.6 P-HMQC-4-5 – Section 14.7

0. 0000 −0. 3847 −0. 00825 −0. 552

vB

vC

vD

25. 000 21. 508 24. 847 25. 538

100. 000 76. 818 99. 991 94. 012

100. 000 77. 542 100. 026 96. 184

The stresses σxx related to P1 are represented in Figure 14-8. The results illustrate the influence of the element geometry on the approximation of the stresses. The quadratic approaches P-HMQ-8-13 yield the best results as expected. The stresses evaluated from the element P-HMQC-4-5 published by Pian and Sumihara yield good results although an approach of low order is employed. In contrast the bi–linear approach P-HMQ-4-5 is not adequate. N/m2

σxx

-3000

P1 = 1000 N analytical solution P-HMQ-4-5 P-HMQ-8-13 P-HMQC-4-5

-2000 x

Fig. 14-8 Stress σxx at the upper boundary of the cantilever for P1

270

14 Hybrid–Mixed Plane Stress Elements

The second loading case is more difficult to be described by the given elements, since the stresses σxx develop linearly along the x–axis, whereat the stresses σxy are quadratically distributed along the y–axis. Due to the parabolic distribution of the stresses σxy in y–direction the load is transferred to concentrated nodal actions according to the PvD. The subsequent Table 14.2 comprises the displacement results concerning the loading case P2 = 150 N . As in loading case P1 the quadratic approach gives the best results, whereat the linear approach, according to Section 14.5, is hardly acceptable. Table 14.2 Loading case bending P2 – comparison of different elements

reference solution P-HMQ-4-5 – Section 14.5 P-HMQ-8-13 – Section 14.6 P-HMQC-4-5 – Section 14.7

vA

vB

vC

vD

0. 0000 −0. 509 −0. 0251 −0. 809

32. 600 27. 765 32. 472 32. 138

102. 750 81. 714 102. 439 97. 533

102. 750 82. 023 102. 421 98. 188

Applying the mesh according to Figure 14-7 the stresses σxx are represented in Figure 14-9 related to the loading case P2 . As in loading case P1 the results illustrate the strong influence of the element geometry on the approximation of the stresses. The quadratic approach yields the best results and describes the stresses σxx to be linearly distributed within the element without a larger gap at the interface of neighboring elements. The stresses evaluated from the element published by Pian and Sumihara yield good results, although an approach of low order is employed. In contrast the bi–linear approach is not adequate, since it leads to a stepwise approximation with larger gaps at the interfaces of neighboring elements. N/m2

σxx

-4000

P2 = 150 N reference solution P-HMQ-4-5 P-HMQ-8-13 P-HMQC-4-5

0 x

Fig. 14-9 Stress σxx at the upper boundary of the cantilever for P2

14.8 Convergence Behavior of the Elements

271

Example of Use 2 c x b

B

D

y

1N

C E = 1.0 N/mm 2 ν = 0.33 a = 44 mm b = 16 mm c = 48 mm

a

As a further example of use Cook’s cantilever [28] is chosen, since comparable results are available. Without completely representing the study of convergence according to Section 12.2.3, the results for different element formulations are compared in Table 14.3 applying meshes with 3 × 3 and 5 × 5 nodes. Concerning the hybrid elements the respective subsequent stress analysis is performed each with alternative B.

A

Table 14.3 Cook’s cantilever – vertical displacements in D, stresses in A and B nodes

3×3

5×5

5×5

v|D

v|D

σxx |B

mm

mm

element 1 2 3 4 5 6 7 8 9

iso - 4 iso - 4 - SRI iso - 9 iso - 9 - SRI tri - 3 tri - 6 hybrid, 14.5 hybrid, 14.6 hybrid, 14.7

11. 845 20. 038 19. 644 22. 626 11. 991 18. 358 17. 756 21. 063 21. 128

18. 299 22. 648 23. 289 23. 653 18. 283 23. 301 21. 935 23. 411 23. 021

5×5 2

N/mm

σxx |A

N/mm2

left

right

left

right

0. 1319 0. 1775 0. 1635 0. 1596 0. 1323 0. 1567 0. 1909 0. 1985 0. 1783

0. 1474 0. 1305 0. 2066 0. 2220 0. 0887 0. 1878 0. 1660 0. 1565 0. 1612

−0. 2005 −0. 1491 −0. 1350 −0. 1738 −0. 0776 −0. 1086 −0. 1375 −0. 1153 −0. 1127

−0. 0474 −0. 1854 −0. 1757 −0. 1808 −0. 0482 −0. 1766 −0. 1936 −0. 1570 −0. 1451

Regarding the hybrid–mixed element formulations and due to the especially chosen approaches to describe the stresses the displacements are approximated very well even at applying few elements. The stresses σxx |B = 0,1832 N/mm2 and σxx |A = −0,1290 N/mm2, evaluated according to Section 12.2.3 and chosen as a reference here, are also considerably better approximated in comparison to other element formulations employing an approach of a comparable order. Due to the excellent behavior of convergence the hybrid elements employing linear approaches according to Pian and Sumihara as well as those employing quadratic approaches according to Section 14.6 might be preferred compared to other element formulations.

15 Hybrid–Mixed Euler–Bernoulli Beam Elements

The derivation of a hybrid–mixed element formulation for the Euler–Bernoulli beam takes place by analogy with the bar.

15.1 Mixed Formulation Employing Forces and Displacements Concerning FEM formulations that employ the displacements as primary variables the choice of the approach to describe the deflection has been discussed already with respect to C1 –conformity. Applying the mixed formulation this specific characteristic can be avoided by incorporating w and M as primary variables, whereat the definition of signs is given in Figure 15-1. p

z

Me

x

V

e

p

ϕ

z

z,w

M-

dM 2

dQ Q2

x

dM M+ 2 dQ Q+ 2

Fig. 15-1 Euler–Bernoulli beam – coordinates and definition of signs

15.1.1 Governing Equations The governing equations related to beams according to the Euler–Bernoulli theory have been represented already at Section 6.1, whereat the description dealing with differential equations of second order is used: a) Kinematics in the domain at the boundary

w,xx − κ = 0 , ϕ − ϕe = 0 , w − we = 0 .

(15.1)

M,xx + pz = 0 , Q−Ve = 0, M + Me = 0 .

(15.2)

b) Equilibrium in the domain at the boundary

© The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2024 D. Dinkler und U. Kowalsky, Introduction to Finite Element Methods, https://doi.org/10.1007/978-3-658-42742-9_15

274

15 Hybrid–Mixed Euler–Bernoulli Beam Elements

c) The material equation EI · κ + M = 0

(15.3)

connects the bending moment to the curvature. Regarding Eq. (15.1) and replacing the curvature by the bending moment incorporating the material equation (15.3) yields the condition of deformation 1 · M = 0. (15.4) EI The equations (15.2) and (15.4) may be given in matrix notation with # " # " # " 0 ∂x2 w pz = 0. (15.5) · + 1 0 M ∂x2 EI w,xx +

15.1.2 Work Equations The governing equations as well as the related boundary conditions may be formulated equivalently applying the Principle of virtual Work.

The Principle of Virtual Displacements – PvD The Principle of virtual Displacements describes the virtual work that the bending moment and the external actions pz (x), V e , M e perform on a virtual state of deformation. Z −δWd = {δκ · M + δw · pz } dx + [ δw · V e + δϕ · M e ]bound. = 0 .

The PvD is equivalent to the conditions of equilibrium, if the virtual displacements strongly fulfill kinematics. This becomes obvious when describing the virtual curvature by the deflection line, Z {δw,xx · M + δw · pz } dx + [ δw · V e + δϕ · M e ]bound. = 0 . (15.6) Considering Eq. (15.6) the boundary conditions related to the force variables are naturally fulfilled. Integrating by parts of the first expression and taking into account δw,x = δϕ yields Z {−δw,x · M,x + δw · pz } dx + [δw · V e + δϕ · (M e + M )]bound. = 0 . (15.7)

Hereby the boundary condition concerning the bending moment occurs as an substantial condition. Integrating by parts once again results in Z {δw ·M,xx +δw ·pz } dx+[ δw ·(V e −M,x )+δϕ·(M e +M ) ]bound. = 0 . (15.8)

15.1 Mixed Formulation Employing Forces and Displacements

275

Applying the integral formulation the conditions of equilibrium are fulfilled weakly with respect to the domain as well as to the boundary. The boundary conditions related to the force variables both are fulfilled exactly by the contour expressions, if corresponding variables are available and the contour expressions disappear.

The Principle of virtual Forces – PvF The Principle of virtual Forces describes kinematics of the real state of deformation in a weak formulation, if the virtual forces strongly fulfill the conditions of equilibrium. As a start Z −δWσ = δM · (−w,xx + κ) dx − [ δQ · (w − we ) + δM · (ϕ − ϕe ) ]bound. = 0

is valid. Replacing the curvature by the material equation and thus by the bending moment yields Z M δM · (−w,xx − ) dx − [ δQ · (w − we ) + δM · (ϕ − ϕe ) ]bound. = 0 . (15.9) EI Concerning the domain the condition of deformation is weakly fulfilled by the integral. The contour expressions take into account the displacement boundary conditions. Integrating the first expression in the integral by parts and introducing w,x = ϕ yields Z M {δM,x · w,x − δM } dx − [ δQ · (w − we ) − δM · ϕe ]bound. = 0 . (15.10) EI A second integration by parts of the same term and replacing δM,x = δQ results in Z M {−δM,xx · w − δM } dx + [ δQ · we + δM · ϕe ) ]bound. = 0 . (15.11) EI

In Eq. (15.9) the contour expressions comprise the boundary conditions related to the displacement variables w and ϕ. After integration by parts the boundary condition related to the rotation can only be fulfilled weakly applying Eq. (15.10). In contrast the boundary condition related to the deflection may be fulfilled strongly with w − we = 0. Repeated integration by parts yields weak fulfillment of both boundary conditions applying Eq. (15.11).

Remarks With both principles of work, applying the Equations (15.6), (15.7) and (15.8) as well as (15.9), (15.10) and (15.11), different but equivalent formulations are availabe, which may be chosen as a basis concerning the discretization.

276

15 Hybrid–Mixed Euler–Bernoulli Beam Elements

Applying Equations (15.6) and (15.9) requires quadratic approaches to describe w and δw as well as at least constant approaches to describe M and δM . Since the boundary conditions related to w and δw and also those related to w,x and δw,x are to be fulfilled, even cubic approaches must be employed with respect to w and δw. In comparison to the displacement–based formulation this line of action yields no advantages and, on top, it also requires cubic approaches to describe the bending moments, since otherwise the system matrix would become singular, cf. the conditions according to Babuˆska [6] and Brezzi [22]. Choosing Equations (15.8) and (15.11) at least constant approaches are required to describe w and δw, at least cubic approaches are to be employed to describe M and δM . Here the boundary conditions related to the displacement variables are to be fulfilled naturally with the PvF and are to be fulfilled strongly related to M and M,x as well as to δM and δM,x . If only first order derivatives of the state variables occur, which means that Equations (15.7) and (15.10) are applied, it is adequate to employ linear approaches for all state variables. Therewith the boundary conditions related to w and δw as well as related to M and δM can be fulfilled strongly, which is simply possible according to the approaches for w and M . Thus it remains Z {δw,x · M,x − δw · pz }dx − [ δw · V e ]bound. = 0 , (15.12) Z M }dx + [ δM · ϕe ]bound. = 0 . (15.13) {δM,x · w,x − δM EI The boundary conditions related to ϕ and Q are fulfilled naturally according to the work equations.

15.1.3 Boundary and Element Interface Conditions Applying the PvD according to the formulation given with (15.12), the boundary conditions related to δw and M are fulfilled strongly. In contrast the conditions related to Q = M,x as well as to δϕ = δw,x are fulfilled weakly. At the element interfaces the virtual displacements of neighboring element boundaries need to be continuous, thus δwl = δwr = δw is fulfilled strongly by the approaches. The continuity condition related to the shear force Q = M,x is fulfilled naturally, if the vertical equilibrium at the internal node M,x l − M,x r = V e , here additionally comprising a concentrated external action, is formulated applying the PvD δwl · M,x l − δwr · M,x r = δw · V e .

15.2 Work Equation in Hybrid Formulation

277

Both terms, occuring at the left hand side, are implicitly incorporated in the work equation. The external work δw · V e is taken into account by the load vector. Applying the PvF according to Eq. (15.13) the boundary conditions related to δM and w are fulfilled strongly and the conditions related to δQ = δM,x as well as to ϕ = w,x are fulfilled weakly. At a simply supported system boundary the contour expression vanishes, since δM = 0 is valid. Regarding a clamped system boundary a rotation of the support ϕe is to be considered with δM · ϕe in the load vector. Concerning a fixed clamping it follows ϕe = 0. At the element interfaces the virtual bending moments are continuous. Here, the condition δMl = δMr = δM is fulfilled strongly, in contrast, the continuity condition related to the rotation w,x l − w,x r = ϕl − ϕr = ∆ϕe is fulfilled naturally. ∆ϕe corresponds to a sharp kink at the internal node here, which is to be considered in the load vector by −δM · ∆ϕe , comparable to the procedure at the system boundary. Numerical results are represented in Section 21.2 and Section 21.3 with an extension to shear deformations by means of simplified structures.

15.2 Work Equation in Hybrid Formulation The derivation of the hybrid–mixed formulation for Euler–Bernoulli beams applies the Equations (15.12) and (15.13). The development aims at getting an element employing the variables w, M at element level, whereby bending moments are to be numerically eliminated in a second step. Besides the work performed by distributed loadings p(x), the work equations related to the system level also comprise the work performed by concentrated actions V e as well as by single bending moments M e , that act at system nodes. Additionally bedding of the beam is taken into account to illustrate the differences of the formulation compared to elements employing only displacements as primary variables. In comparison to bars and plane stress structures a significant difference arises from the choice of the element as well as of the system variables. Concerning the beam w und M are defined at element level. At the system level w and in addition, to ensure C1 –conformity, the rotation ϕ are introduced as system variables, and the virtual rotation δϕ is introduced correspondingly. Thus va-

278

15 Hybrid–Mixed Euler–Bernoulli Beam Elements

riables are defined at the interfaces, which are not available at the element domain. At Figure 15-2 a beam element is represented together with the interfaces A and B, that are identical with the system nodes. Possible gaps ∆we between element boundaries and interfaces are not considered. Vei

Vei+1

pz

A i

b

a wi ϕi

Ma Qa

B i+1

wE ME

Mb Qb

wi+1 ϕ i+1

Fig. 15-2 Hybrid–mixed beam element – element domain and interface Cutting cleanly of the element domain including the element boundaries a and b from the interfaces A and B yields the bending moments Ma and Mb as well as the shear forces Qa = (M,x )a and Qb = (M,x )b . The bending moments are defined only related to the element domain and to the cut, but not related to the interface as primary variables. As kinematic variable the deflection wE is chosen. The nodal displacements and the nodal rotations are defined at the system level to ensure the element continuity conditions with C1 –conformity. Applying the PvD by analogy with Section 13.1 the equilibrium of forces is formulated concerning the element. In addition the equilibrium of moments at the interfaces – which means the system nodes i and i + 1 – is to be ensured by employing the virtual rotations. Z δWd = {δκE · ME − δwE · c · wE + δwE · pz } dx + Ma · (−δϕA + δw,x |a ) − Mb · (−δϕB + δw,x |b ) + Qa · (δwA − δwa ) − Qb · (δwB − δwb ) . Applying the PvF at element level the conditions of deformation are fullfilled yielding Z 1 · ME } dx δWσ = {δME · κE + δME · EI + δMa · (−ϕA + w,x |a ) − δMb · (−ϕB + w,x |b ) + δQa · (wA − wa ) − δQb · (wB − wb ) = 0 . If the continuity of the real displacements as well as of the virtual displacements

15.3 Element Stiffness Matrix

279

is required with wA = wa ,

wB = wb ,

δwA = δwa ,

δwB = δwb ,

the corresponding contour expressions of the principles of work disappear. Due to the independence of the virtual displacements δw from the virtual rotations δϕ the virtual work performed on δw and δϕ may be separately looked at. Thus it remains Z δw : δWw = {δκE · ME − δwE · c · wE + δwE · pz } dx + {−(δw,x )b · Mb + (δw,x )a · Ma }el.bound. ,

δϕ : δME :

δWϕ = {−δϕA · Ma + δϕB · Mb }interf ace , Z 1 · ME } dx δWM = {δME · κE + δME · EI + {−δMb · (w,x )b + δMa · (w,x )a }el.bound. + {−δMa · ϕA + δMb · ϕB }interf ace = 0 .

Applying kinematics κE = wE ,,xx and accordingly δκE = δwE ,xx as well as integration by parts of the respective work expressions yields the work equations, that are introduced into the discretization further on Z δw : − δWw = {δwE,x · ME,x + δwE · c · wE − δwE · pz } dx , (15.14) δϕ :

δME :

− δWϕ = − {−δϕA · Ma + δϕB · Mb }interf ace , Z 1 − δWM = {δME,x · wE ,x − δME · · ME } dx EI

− {−δMa · ϕA + δMb · ϕB }interf ace = 0 .

(15.15)

(15.16)

As at the procedure concerning bars and plane stress structures the virtual work related to the PvF disappears at element level, thus Equation (15.16) can be taken to eliminate the bending moments ME .

15.3 Element Stiffness Matrix Concerning the element domain, approaches are chosen to describe wE and ME , which are given in matrix notation by [ wE ] = Ωv · w

und wT = [ wa

[ M E ] = Ωs · s

und s

T

= [ Ma

wb ] , Mb ] .

280

15 Hybrid–Mixed Euler–Bernoulli Beam Elements

Along the lines of Section 13.2 and after the discretization of the virtual work performed at the element domain the matrix notation of Equations (15.14), (15.15) and (15.16) gives " R T R T T # " # "R T # δw : Ωv C Ωv dx Ωv D · D Ωs dx w Ωv p dx · − . R T T R δs : Ωs D · D Ωv dx − ΩTs E−1 Ωs dx s 0

Since the continuity conditions wA = wa , wB = wb , δwA = δwa , δwB = δwb are fulfilled, the deflections as well as the rotations at the interfaces are defined as nodal displacement variables vT = [ wA

wB

ϕA

ϕB ] .

Thereby the work performed by the nodal rotations can be introduced into the work equation applying matrix notation. Employing linear shape functions to describe wE and ME as well as the rotations ϕA , ϕB only being defined at the interfaces and integrating the work performed at the element domain yields HTsv

Hvv δwA :



cℓ 3 cℓ 6

  δwB :    δϕA :     δϕB :    1 δMa :   ℓ  δMb : − 1ℓ

cℓ 6 cℓ 3

1 ℓ − 1ℓ

+1

− 1ℓ

ℓ − 3EI

+1

1 ℓ

ℓ − 6EI

−1

− 1ℓ



         −1     ℓ  − 6EI   ℓ − 3EI 1 ℓ

−F

Hsv

wA



  wB    ϕA    .  ϕB    Ma   =0  =0 Mb

(15.17)

By analogy with the bar again, the calculation of the element stress resultants s takes place applying the deformation condition s = F−1 · Hsv · v.

(15.18)

After introduction the stress resultants into the PvD the element stiffness matrix and the load vector follow from eliminating the bending moments  −δWd = δvT [ Hvv + HTsv · F−1 · Hsv ] · v − fp . (15.19) Replacing

−1

F

· Hsv =

"

6

−6

4ℓ

2ℓ

−6

6

−2ℓ

−4ℓ

#

·

EI ℓ2

(15.20)

15.3 Element Stiffness Matrix

281

yields the part of the stiffness matrix without bedding   12 −12 6ℓ 6ℓ    −12 12 −6ℓ −6ℓ  EI   T −1 Hsv · F · Hsv =  · 3 . 2 2   6ℓ −6ℓ 4ℓ 2ℓ  ℓ  6ℓ

−6ℓ

2ℓ2

(15.21)

4ℓ2

Assuming constantly distributed loading p(x) = const. as well as concentrated e e at the interfaces the load vector yields and Mie , Mi+1 actions Vie , Vi+1 

  fp =   

pℓ/2 + Vie e pℓ/2 + Vi+1 e Mi e Mi+1



  .  

(15.22)

Equation (15.21) corresponds to the stiffness matrix of the displacement method, though not sorted with respect to the nodes yet. In addition to the element matrix given by Equation (15.21) and considering Equation (15.19), the work performed due to bedding Hvv related to Equation (15.17) is to be summed up. Differences to the formulation employing displacements as primary variables become obvious only when R taking into account bedding, since at the hybrid–mixed model the integral wE · c · δwE dx is evaluated employing linear approaches, whereby cubic approaches are to be employed to describe wE at the displacement–based formulation. The different approaches regarding the hybrid–mixed model and the formulation employing displacements as primary variables also result in different formulations concerning the load vector Equation (15.22). Identical results may be only achieved in special cases. Numerical results are represented in Section 21.2 and Section 21.3 with an extension to shear deformations by means of simplified structures.

16 Hybrid–Mixed Kirchhoff Plate Elements

The formulation of the hybrid–mixed quadrilateral plate elements is derived in analogy to the Euler–Bernoulli beam. Again, the element is split into the element domain with the element boundaries and the interfaces, which are identical with the system grid lines being cut cleanly.

y

x

z,w

A ϕn(A,B)

ϕn(D,A) a

B

p b

d D

qn

mt mn ϕn(C,D)

mt

c

mt qn

qn

mn

qn mt ϕn(B,C)

C

Fig. 16-1 Hybrid plate element – element domain and interfaces

16.1 Mixed Principles of Work Concerning Kirchhoff Plates To develop the mixed plate element Kirchhoff ’s plate theory is chosen, to illustrate the analogy with the mixed Euler–Bernoulli beam element.

16.1.1 Governing Equations The governing equations are represented at full length in Section 6.2. Figure 162 shows the differential element and the related stress resultants at the positive cutting line. By analogy with the Euler–Bernoulli beam the conditions of equilibrium are summarized by a differential equation of 2nd order mxx ,xx + myy ,yy + 2mxy ,xy + p = 0 . © The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2024 D. Dinkler und U. Kowalsky, Introduction to Finite Element Methods, https://doi.org/10.1007/978-3-658-42742-9_16

(16.1)

284

16 Hybrid–Mixed Kirchhoff Plate Elements p x y

p

z,w

dx dy

dy

dx

m xy m yy

m xx qx

m yx qy

Fig. 16-2 Kirchhoff ’s plate theory – coordinates and definition of signs Thus, applying matrix notation, it follows   mxx   [ ∂xx ∂yy 2∂xy ] ·  myy  + [ pz ] = [ 0 ]. mxy

(16.2)

Again, the condition of equilibrium (16.2) may be represented with the matrix symbols already known −DT · σ + pz = 0 . (16.3) Kinematics is described as a 2nd order system, too: ǫ =D·u    −κxx −∂xx      −κyy  =  −∂yy  · [ w ] . −2κxy −2∂xy 

(16.4)

The material equation is applied in its inverse representation, thereby the flexibility F = 1/B(1 − ν 2 ) = 12/Et3 is introduced as a material parameter. In matrix notation this yields ǫ = E−1 · σ, 

  −κxx F     −κyy  =  −νF −2κxy

−νF F

 

 mxx     ·  myy  . 2(1 + ν)F mxy

(16.5)

16.1 Mixed Principles of Work Concerning Kirchhoff Plates

285

Incoporating the material equation into kinematics results in the deformation condition D · u − E−1 · σ = 0 , 

  −∂xx F     −∂yy  · [ w ] −  −νF −2∂xy

−νF F

 

   mxx 0       ·  myy  =  0  . (16.6) 2(1 + ν)F mxy 0

The matrix symbols of the mixed formulation are determined from the governing equations, hence the work equations may be described correspondingly.

16.1.2 Work Equations Here again, the governing equations are formulated equivalently to the Principle of virtual Work.

The Principle of Virtual Displacements – PvD Concerning Kirchhoff ’s plate theory the Principle of virtual Displacements comprises the internal work performed by the moments and the external work performed by distributed external actions pz (x, y) Z δWd = {mxx · δκxx + myy · δκyy + 2 · mxy · δκxy + pz · δw} dx = 0 . (16.7) Introducing kinematics to replace the virtual curvature yields Z δWd = {mxx · δw,xx + myy · δw,yy + 2 · mxy · δw,xy + pz · δw} dx = 0 . (16.8) Choosing a notation by analogy with the Euler–Bernoulli beam in Section 15.1.2, which only comprises first order derivatives of the deflection w and the moments, the work performed by the moments on the curvature is to be integrated by parts once. Thereby the first as well as the second expression is to be integrated by parts with respect to x and y respectively. The third expression is processed correspondingly, but split up one half each related to x and y. The contour expressions, resulting from integration, become line integrals now Z Z δWd = − δw,x · mxx ,x dA + δw,x · mx dy|x = konst. ZA Zy − δw,y · myy ,y dA + δw,y · my dx|y = konst. A

x

286

16 Hybrid–Mixed Kirchhoff Plate Elements − − +

Z

ZA ZA A

δw,x · mxy ,y dA + δw,y · mxy ,x dA + δw · p dA = 0 .

Z

Zx y

δw,x · mx dx|y = konst. δw,y · my dy|x = konst. (16.9)

At a system boundary, the contour expressions concerning Equation (16.9) may be interpreted as follows, furthermore employing the coordinate describing the contour ds and the indices t, n to indicate the tangential and the normal direction respectively. The contour integral Z δWbound. = δw,n · mn ds (16.10) s

disappears, if, regarding a clamped boundary, the virtual rotations and, at a simply supported or free boundary, the conjugated real moments are equal to zero. Regarding a flexible clamping the contour integral is to be evaluated, since the clamping moments can be described by means of a corresponding material equation. If the element continuity conditions related to the moments are strongly fulfilled with mn l − mn r = 0 , the element continuity conditions related to the virtual slopes are weakly fulfilled with mn ( δw,n l − δw,n r ) = 0 . Thus the respective expressions of the work equation also disappear at the element interface. The contour integral Z δWbound. = δw,s · mt ds (16.11) s

disappears regarding a boundary supported by δw,s = 0. At a free boundary the boundary conditions related to the twisting moment as well as to Kirchhoff ’s effective shear force are fulfilled weakly, if the contour integral is evaluated. Starting at Equation (16.11) the contour integral may be interpreted as follows. A second integration by parts initially yields Z δWbound. = [−δw · mt ,s + (δw · mt ),s ] ds . s

16.1 Mixed Principles of Work Concerning Kirchhoff Plates

287

The first term at the integral corresponds to the part of the work performed by Kirchhoff ’s effective shear force, that is not captured by the shear force yet. The last term may be integrated following the respective boundary. This yields Z δWbound. = −δw · mt ,s ds + δw · mt |corner . s

The term comprising the work performed at the corner disappears, since either the twisting moment at the free corner or the virtual displacement at the supported corner is equal to zero. The remaining integrals are summarized to Kirchhoff ’s effective shear force qKn employing the shear forces qx , qy and qn implicitly incorporated in the work equation. Thus, the element continuity conditions related to the effective shear forces following the contour coordinate s of the element are also fulfilled weakly at the system level with Z δw[ (qn + mt ,s )l − (qn + mt ,s )r ] ds = 0 . s

With Equation (16.9) the work equation related to the PvD follows to Z − δWd = {δw,x · mxx ,x + δw,y · myy ,y + δw,x · mxy ,y + δw,y · mxy ,x } dA ZA Z − δw,s · mt ds − δw · p dA = 0 (16.12) s

A

forming the basis of the discretization. Thus, besides the equilibrium in the domain, the boundary conditions qKn = 0 at the free boundary as well as mt = 0 at the free corner are fulfilled naturally. The virtual work performed by line–shaped external actions in s–direction as well as work performed by concentrated actions may be additively taken into account summing up the external work at the interfaces in the load vector. Regarding the PvD, the boundary conditions related to the bending moments as well as to the virtual displacements are substantial ones and therefore are to be fulfilled by the approaches.

The Principle of Virtual Forces – PvF The PvF describes the deformation conditions in the domain as well as at the boundary in a weak form by Z δWσ = {− δmxx [ w,xx + F · ( mxx − ν myy ) ] − δmyy [ w,yy + F · ( ν mxx − myy ) ]

288

16 Hybrid–Mixed Kirchhoff Plate Elements − δmxy [ 2w,xy + 2(1 + ν)F · mxy ] } dA Z Z + δqKn ( w − we ) ds + δmn ( ϕn − ϕen ) ds = 0 .

Integrating the terms related to the domain by parts again, yields Z δWσ = {δmxx ,x w,x − δmxx · F · (mxx − ν myy ) + δmyy ,y w,y − δmyy · F · ( ν mxx − myy )

+

Z

+ (δmxy ,y w,x + δmxy ,x w,y ) − δmxy · 2(1 + ν)F · mxy } dA Z Z δqKn ( w − we ) ds + δmn (−w,n + ϕn − ϕen ) ds − δmt w,s ds = 0 .

The boundary conditions concerning the real deflection w − we = 0 are substantial and are to be fulfilled by the approach describing w. Hence, the first contour integral disappears. If the deformation conditions related to the slopes ϕn = w,n are fulfilled naturally – which means by the work equation – , the second contour integral is reduced to the expression δmn · ϕen . Thereby impressed kinks at the element interfaces or impressed rotations at the plate boundaries may be realized at incorporating the expression in the load vector. Together with the part of Kirchhoff ’s effective shear force that is related to the twisting moment, the third contour integral describes the work performed by the virtual corner force δmt on the deflection w at a non–supported corner of the plate, which has to disappear to δmt |corner = 0 : Z (δmt ,s · w + δmt · w,s ) ds = [ δmt · w ]|corner = 0 . Due to the first part being implictly considered already by the first contour integral, the third contour integral of the work equation is to be evaluated explicitly. Thus the work equation related to the PvF yields Z δWσ = {δmxx ,x w,x − δmxx · F · ( mxx − ν myy ) + δmyy ,y w,y − δmyy · F · ( ν mxx − myy )

−

Z

+ (δmxy ,y w,x + δmxy ,x w,y ) − δmxy · 2(1 + ν) · F · mxy } dA Z δmn ϕen ds − δmt w,s ds = 0 . (16.13)

Comparing the contour integrals to the corresponding terms being derived at the PvD, the symmetry as well as the duality of both of the principles of work become obvious.

16.1 Mixed Principles of Work Concerning Kirchhoff Plates

289

Matrix Notation of PvD and PvF To build the subsequently following element matrix it is convenient to represent the work expressions related to the PvD and to the PvF in matrix notation. Without considering the load vector it is valid Z   δw δmxx δmyy δmxy · −δW = A

     

x∂

x∂

∂x y ∂ ∂y (x ∂ ∂y + y ∂ ∂x )∗

∂x −F νF

   ∂y (x ∂ ∂y + y ∂ ∂x )∗ w      mxx  νF ·    m  dA . (16.14) −F   yy  −2(1 + ν)F mxy

y∂

At the positions indicated by ∗ and besides the domain integrals given here, the contour integrals related to Eqns. (16.13) and (16.12) are to be evaluated.

16.1.3 Element Matrix and Load Vector Due to the integration by parts the PvD and the PvF only comprise first order derivatives of the variables of description, thus linear approaches to describe [ w mx my mxy ] may be chosen at discretization. The shape functions related to the virtual as well as to the real state variables exhibit the same courses as at plane stress structures: u = Ωv · v

σ = Ωs · s

with

u

= [ w(x, y) ] ,

Ωv = [ φA φB φC φD ] ,

with

vT = [ wA wB wC wD ] ,   mxx (x, y)   σ =  myy (x, y)  , mxy (x, y)  φ φ φ φ  A B C D  Ωs =  φA φB φC φD

 φA φB φC φD

 , 

sT = [ mxxA mxxB mxxC mxxD | myyA myyB myyC myyD | mxyA mxyB mxyC mxyD ] .

290

16 Hybrid–Mixed Kirchhoff Plate Elements

Applying the approach for the deflection and the moments to Equation (16.14) directly leads to the element matrix. Therefore, corresponding to Equation (16.14), the discretized representation of the virtual work is given summarizing the terms to be integrated for the element matrix with φT = [ φA φB φC φD ]:   0 φT,x φ,x φT,y φ,y (φT,x φ,y + φT,y φ,x )∗  Z    φT,x φ,x −F φT φ νF φT φ 0  dA .    T T T φ,y φ,y νF φ φ −F φ φ 0 A  T T T ∗ (φ,x φ,y + φ,y φ,x ) 0 0 −2(1 + ν)F φ φ

The domain integrals may be integrated analogously to the plane stress element according to Section 14. Moreover, besides the domain integrals, the contour integrals are to be evaluated, which are indicated by ( )∗ . Although the contour integral is only to be evaluated at the free boundary, regarding the computation in practice, it may be more convenient to evaluate the integral in general related to each element and to each element boundary. This results in an all– purpose element matrix, that can be applied to all different possibilities of support. Disregarding the free boundary the expressions cancel out each other at assembling the elements to the entire structure. Assembling the nodal variables in the vector z with zT = [ wA mxxA myyA mxyA | wB mxxB myyB mxyB | wC mxxC myyC mxyC | wD mxxD myyD mxyD ] yields the element matrix A, which is given on the next page. Thus the Principle of virtual Work is discretized on the element level as −δW = δzT {A · z − f } .

Load Vector Due to the linear approaches the computation of the load vector takes place by analogy with the plane stress elements according to Section 14 and thus is taken over. At the PvD distributed external actions are taken into account by Z δWd = δvT ΩT · p dA . (16.15) A

At the PvF no curvatures are considered arising from non–uniform heating αT ∆T /d, however, by analogy with plane stress structures, they may be prepared in x– as well as in y–direction.

0

  A1 − F91  3  A2  − F92   1 3  0  2 −1   0 − A31    − A1 − F 1  3 18  A2 F2  −  6 18   0− 1 0 2  A=  0 − A61    − A1 − F 1  6 36   − A2 − F 2 6 36   1  −2 + 0 0   A1  0  6  A1 F1  − 6 18    − A32 − F182  0 − 12 0



0

A2 3 2 −F 18 F1 − 18

0

A2 3 2 −F 36 1 −F 36

0

A2 3 2 −F 18 1 −F 18

0 1 2

1 2

3 −F 18

0

0

0+

3 −F 36

0

0

A1 6 − A62 1 +0 2

0

A1 6 − A32 0 + 12

0

A1 3 A2 3 − 12 + 1

0

F2 = −ν ·

3 −F 18 − 12 + 0

0

0

0+

− F93

lx ly , Et3 /12

− F91

Abbreviations: F1 =

0

0

0

A2 3 − F182 − F181

0

− F363

0

0

+0

− F183

0

0

− F93 0 − 12

A1 3 A2 6

0

1 2

−1

0−

1 2

A1 3 A2 3

0

0

0

A2 3 2 − F181 − F 18 F2 F1 − 18 − 18

− A31

0

− F92 − F91 0

A1 =

ly , lx

1 2

3 −F 18

0

0

0+

− F93 0 A1 3 A2 3 − 12 + 1

0

0

− F92 − F91

lx , ly

− F91

A2 =

. . . symmetric. . .

lx ly , Et3 /12

− F91

F3 = 2(1 + ν) ·

A2 1 3 2 − F361 − F362 − F362 − F361

− A61

0

A1 6 − F181 − F182

0

− F92 − F91

− F91

lx ly , Et3 /12

− F93

                                         



16.1 Mixed Principles of Work Concerning Kirchhoff Plates 291

292

16 Hybrid–Mixed Kirchhoff Plate Elements

16.1.4 Convergence Behavior Concerning the Mixed Element As an example to test the formulation, again the square plate is investigated according to Section 6.2, cf. Figure 16-3. By analogy with Section 6.2 and x y

GE

clamped constantly distributed loading: p = 1 kN/m2 7 2 E = 3.0 . 10 kN/m

E

T

ly

t = 0.1 m M

hinged

G

ν = 0.2 l x = 10.0 m l y = 10.0 m

lx

Fig. 16-3 Square plate – system and external actions concerning different variables of state, the convergence behavior is summarized in the Table 16.1. In comparison to displacement–based elements an inferior convergence can be observed regarding the deflection and a partially better convergence regarding the moments. Table 16.1 Convergence Behavior of mixed plate elements mesh wM mxxM myyM myyE mxyT

[ cm ] [ kN m/m ] [ kN m/m ] [ kN m/m ] [ kN m/m ]

1×1

0. 9524 4. 167 6. 548 −5. 952 ÷

2×2

0. 8364 2. 627 4. 100 −6. 731 −1. 388

4×4

0. 7640 2. 262 3. 367 −6. 913 −1. 633

8×8

0. 7435 2. 181 3. 216 −6. 966 −1. 548

16 × 16

0. 7381 2. 161 3. 178 −6. 979 −1. 539

Although the element converges against the accurate solution, the mixed plate element comprises some properties, which may limit the applications. • Investigating oblique plates leads to problems with the fomulation of the boundary conditions with respect to the moment mn at hinged or free boundaries, since a transformation of the moment is needed there. • The computation of stiffened plates is impossible, if the stiffeners have to be discretized as well. In these cases the bending moment can not be a nodal degree of freedom.

16.2 Hybrid–Mixed Rectangular Plate Element

293

16.2 Hybrid–Mixed Rectangular Plate Element The mixed formulation dealing with the variables [ w mxx myy mxy ] concerning the element domain and already known from Section 16.1, is adopted. At the interfaces the deflection related to the corner nodes are defined as degrees of freedom of the system. To link the corner nodes, linear approaches are chosen to describe the deflection. To ensure the C1 –conformity and by analogy with the Euler–Bernoulli beam additional rotations with respect to the interfaces are defined. The following degrees of freedom are possible: • at the nodes of the system A, B, C, D : w, ϕx , ϕy → 12 degrees of freedom This formulation is comparable to the displacement–based formulation comprising 12 DOF, cf. Section 6.3. In contrast to the Euler–Bernoulli beam different approaches can be chosen here to describe the degrees of freedom at the interfaces: • w, ϕx , ϕy each empploying a linear course along the interface. The shape functions are only defined at the interface, see Figure 16-4. • ϕn , ϕt employing a constant or linear course along the interface between the nodes of the system. ϕn and ϕt are not defined according to the corners. x

y

A

z

D

interface ’I’

a

d

element domain ’E’

1

B b 1

c C 1 1

Fig. 16-4 Linear approaches concerning w, ϕx , ϕy at the interfaces Subsequently, the general procedure at computing the element matrices is shown. The element exactly fulfills the element continuity conditions related to w. The element continuity conditions concerning w,n and w,s are ensured by the rotations ϕx and ϕy at the interfaces. In the following section this element is referred to as K-HM-4-16.

294

16 Hybrid–Mixed Kirchhoff Plate Elements

16.2.1 Work Equations The formulation of the equilibrium applying the PvD takes place, corresponding to the mixed principle of work presented in Section 16.1, considering kinematics κxx = w,xx ,

κyy = w,yy ,

κxy = w,xy

in the element domain.

Cutting cleanly the interfaces yields the shear forces and the moments at the element boundaries E and at the interfaces I according to Figure 16-1. Incorporating the work performed at the transition between element boundary and interface results in: Z δWd = {δw,xx · mxx + δw,yy · myy + 2δw,xy · mxy + δw · p} dA AZ Z − mn · (δw,n E − δϕn I ) ds − mt · (δw,s E − δw,s I ) ds Z + qn · (δwE − δwI ) ds . Enforcing continuity of real as well as virtual displacements wE = wI and δwE = δwI and, by analogy with the beam, integrating the internal work by parts, yields the virtual work performed on the virtual deflection surface Z − δWd,w = {δw,x · mxx,x + δw,y · myy,y + δw,y · mxy,x + δw,x · myx,y } dA A Z Z − δw · p dA + mt · δw,s ds (16.16) A

I

as well as performed on the virtual rotations at the interfaces Z δWd,ϕ = − mn · δϕn ds .

(16.17)

I

The formulation of the deformation conditions takes place by analogy with Section 16.1 applying the PvF, whereby the work δmn ϕn is to be taken into account. Z 12 − δWσ,mxx = {δmxx ,x w,x − δmxx 3 (mxx − ν · myy )} dA Et A Z − δmn ϕn ds = 0 , (16.18) I Z 12 − δWσ,myy = {δmyy ,y w,y − δmyy (myy − ν · mxx )} dA Et3 A Z − δmn ϕn ds = 0 , (16.19) I

16.2 Hybrid–Mixed Rectangular Plate Element − δWσ,mxy =

295

Z

12 {δmxy ,y w,x + δmxy ,x w,y − 2δmxy 3 (1 + ν) · mxy }E dA Et A Z − δmt w,s ds = 0 . (16.20) I

All work expressions related to the interfaces are >0

at the positive cut