One-Dimensional Finite Elements: An Introduction To The Method 3662667576, 9783662667576

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Markus Merkel Andreas Öchsner

One-Dimensional Finite Elements An Introduction To The Method

One-Dimensional Finite Elements

Markus Merkel • Andreas Öchsner

One-Dimensional Finite Elements An Introduction To The Method

Markus Merkel Institute for Virtual Product Development Aalen University of Applied Sciences Aalen, Germany

Andreas Öchsner Hochschule Esslingen Esslingen, Germany

ISBN 978-3-662-66757-6 ISBN 978-3-662-66758-3 https://doi.org/10.1007/978-3-662-66758-3

(eBook)

This book is a translation of the original German edition “Eindimensionale Finite Elemente” by Merkel, Markus, published by Springer-Verlag GmbH, DE in 2010. The translation was done with the help of artificial intelligence (machine translation by the service DeepL.com). A subsequent human revision was done primarily in terms of content, so that the book will read stylistically differently from a conventional translation. Springer Nature works continuously to further the development of tools for the production of books and on the related technologies to support the authors. # Springer-Verlag GmbH Germany, part of Springer Nature 2023 This work is subject to copyright. All rights are reserved 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-Verlag GmbH, DE, part of Springer Nature. The registered company address is: Heidelberger Platz 3, 14197 Berlin, Germany

Dedicated to our wives

Preface

Preface to the First Edition The title of the book – One-Dimensional Finite Elements: An Introduction to the Method – stands for content and orientation. There is a great deal of literature available today on the subject of the finite element method. The different works reflect the diverse views and application possibilities. The basic idea of this introduction to the finite element method is based on the concept of explaining the complex method using one-dimensional elements. The aim is to introduce the many aspects of the finite element method and to provide the reader with a methodological understanding of important topics. Readers learn to understand the assumptions and derivations in various physical problems in structural mechanics and to critically evaluate the possibilities and limitations of the finite element method. Additional extensive mathematical description forms are omitted, arising only from the extended representation for two- or three-dimensional problems. Thus, the mathematical description remains largely simple and straightforward. The treatment of one-dimensional elements, however, is not a mere restriction to a simpler and clearer formal representation of the necessary equations. In structural engineering, there are numerous structures – for example bridges or high-transmission towers – which can usually be modelled using one-dimensional elements. Thus, this work also includes a ‘set of tools’ that also finds its application in practice. The focus on one-dimensional elements is new for a textbook and allows the treatment of a wide variety of fundamental and challenging physical problems in structural mechanics in a single textbook. This new concept thus allows the methodical understanding of important topics (e.g. plasticity or composites), which a prospective computational engineer encounters in professional practice, but which are rarely treated in this form at universities. Consequently, an easy access is possible, also into more advanced application areas of the finite element method. This book has been developed from a collection of lecture notes, which were distributed as written material for lectures, and training material for special courses on the finite element method. Typical questions of students and course participants are taken up, especially in the calculated examples and the advanced tasks.

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Preface

Prerequisites for a good understanding are basics in linear algebra, physics, materials science and strength of materials, as they are typically taught in the basic studies of a technical subject in the field of mechanical engineering. In the first chapters, the one-dimensional elements are introduced, which can be used to represent the basic load types tension/compression, torsion and bending. In each case, the differential equation and the basic equations from strength theory for kinematics, the constitutive relationship and the formation of equilibrium are derived. Subsequently, the finite elements are introduced with the usual definitions for force and displacement variables. Examples are used to clarify the basic procedure. For further tasks, short solutions are given in the appendix. In Chap. 6, questions are taken up independent of the type of loading and the associated element formulation. A general one-dimensional finite element, which can be built up from the combination of basic elements, the transformation of elements in the general threedimensional space and the numerical integration as an important tool in the implementation of the finite element method are dealt with. Chapter 7 presents the complete analysis of a complete structure. The overall stiffness relationship is created from the individual stiffness relationships of the basic elements, taking into account the connections to each other. With the boundary conditions, a reduced system is created from which the unknown quantities are determined. The procedure is presented using plane and generally three-dimensional structures as examples. Chapters 8, 9, 10, 11 and 12 deal with topics that are not part of the standard repertoire of a basic book. Chapter 8 introduces the beam element with a shear component. The basis is the Timoshenko beam. Chapter 9 introduces a finite element formulation for a special class of materials – composite materials. First, different description forms for direction-dependent material behavior are presented. The fiber composites are also briefly discussed. A composite element is demonstrated using the composite bar and the composite beam as examples. Chapters 10, 11 and 12 deal with nonlinearities. Chapter 10 briefly introduces the different types of nonlinearities. The case of nonlinear elasticity is examined in more detail. The problem is exemplified for bar elements. First, the principal finite element equation is derived considering the strain dependence. To solve the nonlinear system of equations, the direct iteration and the complete and modified Newton-Raphson iteration are derived and demonstrated by means of numerous examples. In Chap. 11, elasto-plastic behavior is considered, one of the frequently occurring forms of material nonlinearity. First, the continuum mechanical fundamentals for plasticity on the one-dimensional continuum bar are compiled. The yield condition, the flow rule, the hardening law and the elasto-plastic material modulus are introduced for uniaxial monotonic loading conditions. In the context of hardening, the description is limited to isotropic hardening. For the integration of the elasto-plastic constitutive law, the incremental predictor-corrector method is introduced in general and derived for the case of the fully

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implicit and the semi-implicit backward Euler algorithm. At crucial points, the difference between one- and three-dimensional descriptions is pointed out to ensure an easy transfer of the derived methods to general problems. With stability, a topic is taken up in Chap. 12 which is particularly taken into account in the design and dimensioning of lightweight components. The finite elements developed for this type of nonlinearity are used to solve the Euler buckling cases. In Chap. 13, an FE formulation for dynamic problems is presented. In addition to the stiffness matrices, mass matrices are also set up. Different assumptions about the distribution of the masses, whether continuous or concentrated, lead to different formulations. By way of example, the facts are discussed in the case of axial vibrations of the bar. For illustration, each chapter is deepened with both extensively calculated and commented examples as well as with further tasks – including short solutions. Each chapter concludes with an extensive bibliography.

Preface to the Second Edition The basic concept for the treatment of the finite element method with one-dimensional problems has been retained in the 2nd edition. Additionally, the stationary heat conduction and the principle of virtual work as a further method for the derivation of FE formulations have been included. Chapter 14 is supplemented with special elements for the modeling of singularities.

Preface to the Third Edition The basic concept for the treatment of the finite element method with one-dimensional problems has been retained in the 3rd edition. In addition, thermoelasticity has been included. Furthermore, numerous tasks with solutions have been added. Aalen, Germany Esslingen, Germany September 2019

Markus Merkel Andreas Öchsner

Acknowledgements

We would like to thank Springer-Verlag for their consideration regarding the orientation of the book and for the attractive layout of the book. Students and course participants are to be thanked for their contribution to the present form through critical scrutiny. Finally, thanks to our families for their understanding and patience during the writing of the book.

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Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Overview of the Finite Element Method . . . . . . . . . . . . . . . . . . . . . 1.2 Fundamentals of Model Building . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 4

2

Motivation for the Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . 2.1 Procedures Motivated by Engineering Judgement . . . . . . . . . . . . . . . 2.1.1 The Matrix Stiffness Method . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Transition to the Continuum . . . . . . . . . . . . . . . . . . . . . . . 2.2 Integral Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 The Weighted Residuals Method . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Procedures Based on the Inner Product . . . . . . . . . . . . . . . . 2.3.2 Procedure Based on the Weak Formulation . . . . . . . . . . . . . 2.3.3 Procedure Based on the Inverse Formulation . . . . . . . . . . . . 2.4 Example Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 5 6 10 16 18 19 22 24 25 31

3

Bar Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Basic Description of the Tension Bar . . . . . . . . . . . . . . . . . . . . . . . 3.2 The Finite Element Tension Bar . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Derivation Via Potential . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Derivation Via Castigliano’s Theorem . . . . . . . . . . . . . . . . 3.2.3 Derivation Via the Principle of Weighted Residuals . . . . . . . 3.2.4 Derivation Via the Principle of Virtual Work . . . . . . . . . . . 3.3 Example Problems and Further Tasks . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Example Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Advanced Tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33 33 36 39 41 42 45 47 47 52 52

4

Analogies to the Tension Bar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Basic Descriptions of the Torsion Bar . . . . . . . . . . . . . . . . . . . . . . . 4.2 The Finite Element Torsion Bar . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4.3 Basic Descriptions of the Temperature Bar . . . . . . . . . . . . . . . . . . . 4.4 The Finite Element Temperature Bar . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Thermoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Examples and Additional Tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

60 62 64 65 70

5

Bending Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introductory Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Basic Description of the Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Material Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.4 Differential Equation of the Bending Line . . . . . . . . . . . . . . 5.2.5 Analytical Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 The Finite Element Plane Bending Beam . . . . . . . . . . . . . . . . . . . . . 5.3.1 Derivation via Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Derivation via the Principle of Weighted Residuals . . . . . . . 5.3.3 Derivation via the Principle of Virtual Work . . . . . . . . . . . . 5.3.4 Notes on the Derivation of the Shape Functions . . . . . . . . . 5.4 The Finite Element Bending Beam with Two Deformation Planes . . . 5.5 Determination of Equivalent Nodal Loads . . . . . . . . . . . . . . . . . . . . 5.6 Example Problems and Further Tasks . . . . . . . . . . . . . . . . . . . . . . . 5.6.1 Example Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.2 Further Tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71 71 73 73 78 80 82 83 86 92 95 99 101 104 106 111 111 121 122

6

General 1D Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Superimposition on the General 1D Element . . . . . . . . . . . . . . . . . . 6.1.1 Example 1: Bar in Tension and Torsion . . . . . . . . . . . . . . . 6.1.2 Example 2: In-Plane Beam with Tensile Component . . . . . . 6.2 Coordinate Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Plane Supporting Structures . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 General Three-Dimensional Structures . . . . . . . . . . . . . . . . 6.3 Numerical Integration of a Finite Element . . . . . . . . . . . . . . . . . . . . 6.4 Interpolation Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Unit Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Further Tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

125 125 127 128 130 131 133 136 139 141 143 143

7

Plane and Spatial Framestructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Structure of the Total Stiffness Relationship . . . . . . . . . . . . . . . . . . 7.2 Solving the System Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

145 145 149

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7.3 7.4

Postprocessing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Examples in the Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 Plane Structure with Two Bars . . . . . . . . . . . . . . . . . . . . . . 7.4.2 Plane Structure: Beam and Bar . . . . . . . . . . . . . . . . . . . . . . 7.5 Examples in Three Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Continuing Tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

150 151 151 155 161 172 173

Beams with Shear Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Introductory Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Basic Description of the Beam with Shear Contribution . . . . . . . . . . 8.2.1 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.2 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.3 Law of Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.4 Differential Equations of the Bending Line . . . . . . . . . . . . . 8.2.5 Analytical Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 The Finite Element of Plane Bending Beams with Shear Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Derivation via Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2 Derivation via Castigliano’s Theorem . . . . . . . . . . . . . . . . . 8.3.3 Derivation via the Principle of Weighted Residuals . . . . . . . 8.3.4 Derivation via the Principle of Virtual Work . . . . . . . . . . . . 8.3.5 Linear Interpolation Functions for the Deflection and Displacement Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.6 Higher Interpolation Functions for the Beam with Shear Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Example Problems and Further Tasks . . . . . . . . . . . . . . . . . . . . . . . 8.4.1 Example Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.2 Advanced Tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

175 175 179 181 181 182 183 184

213 219 219 229 230

Composite Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Composite Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Anisotropic Material Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.1 Special Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.2 Engineering Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.3 Transformation Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.4 Plane Stress States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Introduction to Micromechanics of Fibre Composites . . . . . . . . . . . . 9.4 Multilayer Composite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.1 A Layer in the Compound . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.2 The Multi-layered Composite . . . . . . . . . . . . . . . . . . . . . . .

233 233 234 236 238 241 243 248 250 250 253

8

9

186 189 193 194 199 202

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Contents

9.5

A Finite Element Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.1 The Composite Bar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.2 The Composite Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6 Example Problems and Further Tasks . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

254 255 256 257 258

10

Nonlinear Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Introductory Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Element Stiffness Matrix for Strain-Dependent Elasticity . . . . . . . . . 10.3 Solution of the Non-Linear System of Equations . . . . . . . . . . . . . . . 10.3.1 Direct Iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.2 Complete Newton-Raphson Method . . . . . . . . . . . . . . . . . . 10.3.3 Modified Newton-Raphson Method . . . . . . . . . . . . . . . . . . 10.3.4 Convergence Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Example Problems and Further Tasks . . . . . . . . . . . . . . . . . . . . . . . 10.4.1 Example Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

259 259 260 267 268 271 283 285 287 287 298

11

Plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Fundamentals of Continuum Mechanics . . . . . . . . . . . . . . . . . . . . . 11.1.1 Flow Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.2 Flow Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.3 Consolidation Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.4 Elasto-Plastic Material Module . . . . . . . . . . . . . . . . . . . . . 11.2 Integration of the Material Equations . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Derivation of the Full Implicit Backward Euler Algorithm . . . . . . . . 11.3.1 Mathematical Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.2 Interpretation as a Convex Optimization Problem . . . . . . . . 11.4 Derivation of the Semi-implicit Backward Euler Algorithm . . . . . . . 11.5 Example Problems and Further Tasks . . . . . . . . . . . . . . . . . . . . . . . 11.5.1 Example Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5.2 Advanced Tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

299 299 300 302 303 304 306 312 312 317 321 323 323 346 348

12

Stability (Buckling) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 Stability in the Bar/Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Large Deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3 Stiffness Matrices for Large Deformations . . . . . . . . . . . . . . . . . . . . 12.3.1 Bar with Large Deformations . . . . . . . . . . . . . . . . . . . . . . . 12.3.2 Beams with Large Deformations . . . . . . . . . . . . . . . . . . . . 12.4 Examples of Buckling: The Four Euler Buckling Cases . . . . . . . . . . 12.4.1 Analytical Solution to the Euler Buckling Cases . . . . . . . . .

351 351 353 355 356 358 360 360

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12.4.2 Finite Element Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5 Further Tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

362 363 363

13

Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1 Fundamentals of Linear Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 The Mass Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3 Modal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4 Forced Vibrations, Periodic Loads . . . . . . . . . . . . . . . . . . . . . . . . . 13.5 Direct Integration Methods, Transient Analyses . . . . . . . . . . . . . . . . 13.5.1 Integration According to Newmark . . . . . . . . . . . . . . . . . . . 13.5.2 Central Difference Procedure . . . . . . . . . . . . . . . . . . . . . . . 13.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.6.1 Definition of Mass and Stiffness Matrices . . . . . . . . . . . . . . 13.6.2 Axial Vibrations in the Tension Bar . . . . . . . . . . . . . . . . . . 13.7 Supplementary Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

365 365 368 368 371 372 373 374 376 376 381 396 397

14

Special Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1 Elastic Foundation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2 Stress Singularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3 Infinite Extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4 Supplementary Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

399 399 402 407 412 413

Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

415

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

461

Formula Symbols and Abbreviations

Latin Formula Characters (Capital Letters) A B C Celpl D D E Eelpl Epl E F F G I K K KT L L1 Lni Lk M M N N Q Q_ Q

Area, cross-sectional area Matrix with derivatives of the shape functions Substance matrix, damping matrix Elasto-plastic matrix Diameter Substance matrix Modulus of elasticity Elasto-plastic modulus Plastic modulus Average modulus Flow condition, force Column matrix of the external load Shear modulus Area moment of inertia Compression modulus, stress intensity factor Overall stiffness matrix Tangent stiffness matrix Element length Differential operator 1st order Lagrange polynomial Buckling length Moment Mass matrix Interpolation function Row matrix of the interpolation functions, N = [N1 N2 . . . Nn] Plastic potential Heat flow Fabric matrix, plane case, shear force

xix

xx

R S S T T V W X Y Z

Formula Symbols and Abbreviations

Radius Bar force Compliance matrix Torsional moment Transformation matrix Volume Weight function Global spatial coordinate Global spatial coordinate Global spatial coordinate

Latin Formula Characters (Lower Case) a b c d h e f g h i j k ks ke m mt m n q q_ q r r t tij ux uy uz

Geometric dimension Geometric dimension, width Integration constant Geometric dimension Geometric dimension, height Unit vector Function Function, acceleration due to gravity Function of the solidification change Increment number, variable Iteration index, variable Modulus of subgrade reaction, spring stiffness, yield stress Thrust correction factor Element stiffness matrix Number of elements, slope polynomial degree mass Continuously distributed torsional moment per length Residual function Number of nodes, variable, state Line load, order of integration, modal coordinates Heat flux density Inner variables matrix Function of flow direction, radius, residuum Flow direction vector Time, geometric dimension Component of the transformation matrix Displacement in x direction Displacement in y direction Displacement in z direction

Formula Symbols and Abbreviations

u v x y z

Column matrix of node displacements Argument vector (Newtonian method) Spatial coordinate Spatial coordinate Spatial coordinate

Greek Formula Characters (Capital Letters) Γ Λ Π Π Πext Πint Φ Ω Θ

Boundary Parameter (Timoshenko bar) Energy Complementary energy Potential of external loads Elastic strain energy Modal matrix Space, volume Temperature

Greek Formula Characters (Lower Case) α β γ δ ε εij ε εpl eff εth κ λ dλ ν ξ π σ ρ σ trial n þ1 σ ij σ τ η

Coefficient of thermal expansion, constant, angle Angle, constant Shear strain Virtual Strain Strain tensor Column matrix of the strain Effective plastic strain Thermal strain Inner variable (plasticity), curvature (beam bending) Eigenvalue, thermal conductivity Consistency parameter Transverse contraction number (Poisson’s ratio) Unit coordinate (-1 ≤ ξ ≤ 1) Volume-specific work, volume-specific energy Stress, normal stress Density Trial stress state Stress tensor Column matrix of stress Shear stress Coordinate (-1 ≤ η ≤ 1)

xxi

xxii

ζ ψ ϕ φ ω

Formula Symbols and Abbreviations

Coordinate (-1 ≤ ζ ≤ 1) Phase angle Angle of rotation, torsion Angle of rotation, torsion Natural frequency

Indices, Superscript ⋯V ⋯e ⋯el ⋯ext ⋯geo ⋯glo ⋯init ⋯k ⋯lo ⋯pl ⋯red ⋯th ⋯trial

Composite Element Elastic External size Geometric Global Initial (initial yield point) k-th layer Local Plastic Reduces Thermal Trial state (back projection)

Indices, Subscript ⋯Im ⋯Re ⋯b ⋯c ⋯eff ⋯f ⋯k ⋯l ⋯m ⋯p ⋯s ⋯t ⋯w

Imaginary part of an imaginary number Real part of an imaginary number Bend Compression, damping Effective value Fibrous composite Buckling Lamina Matrix in compound, inertia Node value Shear Torsion, tension Wall

Mathematical Symbols (⋯)T j ⋯j k⋯ k 

Transposed Amount Norm Dyadic product

Formula Symbols and Abbreviations

sgn IR

Sign function Set of real numbers

Abbreviations 1D 2D CAD FE FEM inc SI

One-dimensional Two-dimensional Computer-aided design Finite elements Finite element method Increment number International System of Units

xxiii

1

Introduction

Abstract

In this first chapter, the content and orientation of this book are classified in many respects. First of all, the development of the finite element method is briefly discussed, and various points of view are taken into account.

1.1

Overview of the Finite Element Method

In terms of time, the roots of the finite element method lie in the middle of the last century. Thus, this method is a relatively young tool compared to other tools and aids in the dimensioning and design of components. The development of the finite element method started in the 1950s. Researchers and users from very different disciplines have contributed ideas and made the method a universal tool, which today cannot be imagined without in research, development and engineering applications. In the beginning, the focus was rather on fundamental questions, such as questions regarding the solvability in principle. From today’s point of view, only rudimentary resources were available with regard to the program-technical implementation. Preprocessing consisted of punching cards, which were fed in batches to a calculating machine. Errors in programming were promptly acknowledged by flashing lamps. As computer development progressed, the programming environment became more convenient and algorithms could be tested and optimized on sophisticated examples. From the point of view of engineering applications, the problems to be analyzed by means of the finite element method were limited to simple examples. The computer capacities only allowed a very rough modeling. Today, many fundamental questions have been clarified, the focus of the questions is rather on the application side. Finite element program packages are available in a large

# Springer-Verlag GmbH Germany, part of Springer Nature 2023 M. Merkel, A. Öchsner, One-Dimensional Finite Elements, https://doi.org/10.1007/978-3-662-66758-3_1

1

2

1

Introduction

variety and are applied in quite different ways. On the one hand, there are program packages which are mainly used in teaching. The aim is to demonstrate the systematic approach. Source codes are also available for such programs. On the other hand, there are commercial program packages that have been developed to the utmost in terms of both program technology and content. Program modules specially adapted to a computer platform or computer architecture (parallel computers) are very efficient and enable the processing of very extensive problems. With regard to the content, the authors dare to say that there is no physical discipline for which no finite element program exists. With regard to the further development of the finite element method, the focus today is on cooperation and integration with other development tools, such as the interface to design. The two classic disciplines of calculation and design are increasingly growing together and have already partially merged under a common user interface. In addition to stand-alone finite element software packages, solutions integrated into a CAD system are also available on the market. From the users’ point of view, the focus is on FE-compatible problem preprocessing and postprocessing of his/her special problem. The time-consuming process steps of geometry preparation should not mean any significant additional effort for the application of the finite element method. Calculation results should be able to be seamlessly integrated into the respective process chain. In terms of application areas, there are no limits to the use of the finite element method. The focus in mechanical and manufacturing engineering and in vehicle development is certainly the dimensioning and design of components, subsystems or complete machines. The use of the finite element method or simulation tools in general in product development is often seen as a competing tool to experimentation or testing. The authors see here rather an ideal supplement. Individual test benches or entire test scenarios can be optimized in advance using finite element simulation. In turn, test results help to create more precise simulation models.

1.2

Fundamentals of Model Building

The initial situation for the use of the finite element method is a model of a physical or technical problem (see Table 1.1). The complete description of the problem includes • • • •

the geometry to describe the domain, the field equations in the domain, the boundary conditions and the initial conditions in time-dependent problems.

In the context of this book, only one-dimensional problems are treated. The basic procedure is similar for two- and three-dimensional problems. However, the mathematical scope is much more complex.

1.2

Fundamentals of Model Building

3

Table 1.1 Physical problem areas in the context of differential equations Problem Heat conduction Pipe flow Viscous flow Elastic rods Elastic torsion Electrostatics

Field size u(x) Temperature Θ Pressure p Velocity vx

Coefficients a Heat conduction λ Pipe resistance 1/R Viscosity ν

Displacement u Angle φ Electr. Potential Φ

Stiffness EA Stiffness GIp Dielectricity

f Heat sources qw

Pressure gradient dp dx

Axial forces f Torsional moments mt Charge density ρ

Adapted from [1]

Usually, the problem situations can be described by means of differential equations. Here, the second-order differential equations are in the foreground. For example, the differential equations of a certain class of physical problems can be generally described with -

duðxÞ d - f = 0: a dx dx

ð1:1Þ

Depending on the physical problem, the variable u(x) and the parameters a and f are given different meanings. In order for a problem to be completely described, it is necessary to specify corresponding boundary conditions in addition to the differential equation. The local boundary conditions (BCs) can generally be divided into three groups: • Boundary condition of first type or Dirichlet boundary condition (also called essential, fundamental, geometric or kinematic BC): A BC of the first kind exists if the BC is expressed in terms of quantities in which the differential equation is formulated. • Boundary condition of second type or Neumann boundary condition (also called natural or static BC): A BC of the second type exists if the BC specifies the derivation in the direction of the normal of Γ. • Boundary condition of third kind or Cauchy boundary condition (also called mixed or Robin boundary condition):

4

1

Table 1.2 Different boundary conditions of a differential equation

Differential equation L{u(x)} = b

Dirichlet u

Introduction

Neumann

Cauchy

du dx

αu þ β du dx

Defines a weighted sum of Dirichlet and Neumann boundary condition at the boundary. These three types of boundary conditions are also summarized in Table 1.2. Note that one speaks of homogeneous boundary conditions if the corresponding variables on the boundary are zero. In this book, the finite element method is examined from the perspective of mathematics, physics, or engineering applications. From the mathematical point of view, the finite element method is a suitable tool to solve partial differential equations. From the physics point of view, the finite element method can be used to solve a variety of physical problems. The areas range from electrostatics to diffusion problems and elasticity theory. Engineers use the finite element method for the design and dimensioning of products and processes. With respect to the physical problem areas, only elastomechanical problems are discussed here. Within elastostatics • the tension rod, • the torsion bar and • the bending beam without and with shear contribution are treated. Vibrations of bars and beams are taken up as dynamic problems. This book is already the third English edition [2, 3] and based on three German editions [4–6].

References 1. Reddy JN (2006) An introduction to the finite element method. McGraw-Hill, Singapore 2. Öchsner A, Merkel M (2013) One-dimensional finite elements: an introduction to the FE method, 1. Aufl. Springer, Berlin 3. Öchsner A, Merkel M (2018) One-dimensional finite elements: an introduction to the FE method, 2. Aufl. Springer, Berlin 4. Merkel M, Öchsner A (2010) Eindimensionale Finite Elemente: Ein Einstieg in die Methode. Springer, Berlin 5. Merkel M, Öchsner A (2014) Eindimensionale Finite Elemente: Ein Einstieg in die Methode. Springer, Berlin 6. Merkel M, Öchsner A (2020) Eindimensionale Finite Elemente: Ein Einstieg in die Methode. Springer, Berlin

2

Motivation for the Finite Element Method

Abstract

Access to the finite element method can come from different motivations. Essentially, three paths can be identified: • a rather descriptive way, which has its roots in the engineering way of working, • a physical or • mathematically motivated approach. Depending on the point of view, different formulations result, which, however, lead to the common principal equation of the finite element method. The forms of description are presented in detail starting from • the matrix methods, • the physically based working and energy principles and • the principle of weighted residuals. The finite element method is used to solve various physical problems. Only finite element formulations for structural mechanics are considered here.

2.1

Procedures Motivated by Engineering Judgement

In elastostatics, for the analysis of complex structures, the matrix methods can be considered as a starting point for applications of the finite element method [1, 3]. Although this book focuses on one-dimensional finite elements, the basic facts are first presented on a plane structure for better clarity (see Fig. 2.1). A three-dimensional representation would be

# Springer-Verlag GmbH Germany, part of Springer Nature 2023 M. Merkel, A. Öchsner, One-Dimensional Finite Elements, https://doi.org/10.1007/978-3-662-66758-3_2

5

6

2 Motivation for the Finite Element Method

Fig. 2.1 Plane structure from substructures. (Adapted from [4])

too complex. The mathematical description will be given later using one-dimensional examples. The structure consists of several substructures I, II, III and IV. The substructures are called elements. The elements are coupled to each other at nodes 2, 3, 4 and 5. The overall structure is supported at nodes 1 and 6, and an external load engages at node 4. Wanted are • the displacements and reaction forces at each intermediate node and • the support reactions as a result of the applied load. The matrix methods are suitable for solving this problem. In the matrix methods, a distinction is made between the force methods (static methods), which are based on a direct determination of the statically indeterminate forces, and the displacement methods (kinematic methods), which consider the displacements as unknown quantities. Both methods can be used to determine the quantities sought. The decisive advantage of the displacement method is that it is not necessary to distinguish between statically determined and statically indetermined problems in the application. Therefore, this method will also be used in the following.

2.1.1

The Matrix Stiffness Method

The primary sub-objective is to establish the stiffness relationship for the overall structure from Fig. 2.1. Following the description for a linear spring with Force = Stiffness × Change of Length, the following stiffness relation is to be established:

2.1

Procedures Motivated by Engineering Judgement

7

F = Ku:

ð2:1Þ

Here F and u is a column matrix, K is a square matrix. All nodal forces are summarized in F and all nodal displacements are summarized in u. The matrix K represents the stiffness matrix of the complete structure. For the problem, a single element is identified as the basic building block. A single element is characterized by being coupled to other elements via nodes. Displacements and forces are introduced at each node. In solving the overall problem, it is necessary that • the compatibility and • the equilibrium are fulfilled. In the matrix displacement method, one introduces the nodal displacements as significant unknowns. The displacement vector at a node is defined as jointly valid for all elements connected at this node. Thus, the compatibility of the overall structure is satisfied a priori. A Single Element Forces and displacements are introduced at the single element for each node (see Fig. 2.2). For an unambiguous representation, the node forces and the node displacements are provided with the index p in order to emphasize that they are quantities that are defined at nodes. The vectors of the nodal displacements up and nodal forces Fp generally consist of several components for the respective coordinates. An additional index e indicates to which element the quantities refer.1 Thus, the nodal forces according to Fig. 2.2 result in Fei =

F ix F iy

,

Fej =

,

uj =

F jx F jy

,

Fem =

,

um =

F mx

,

F my

ð2:2Þ

and the nodal displacements to ui =

uix uiy

ujx ujy

umx umy

:

ð2:3Þ

If all nodal forces and nodal displacements at an element are combined to a vector, the following results

1

The additional index e is omitted for the displacements, because in the displacement method the nodal displacements are identical for each connected element.

8

2 Motivation for the Finite Element Method

Fig. 2.2 Single element (e) with displacements and forces

Fi Entire nodal

force vector Fep

=

ð2:4Þ

Fj Fm

for the forces at all nodes as well as

entire nodal displacement vector up =

ui uj

ð2:5Þ

um for the displacements at all nodes. With the vectors for the nodal forces and displacements, the stiffness relation for a single element can be given as: Fep = ke up ,

ð2:6Þ

Fer = kers us ðr, s = i, j, mÞ:

ð2:7Þ

respectively per node:

The single stiffness matrix ke links the nodal forces with the nodal displacements. In the present example, the single stiffness relationship is formally as follows F ix F iy F jx F jy F mx F my

=

k eii k eji k emi

k eij k ejj kemj

keim kejm kemm

uix uiy ujx ujy umx umy

ð2:8Þ

2.1

Procedures Motivated by Engineering Judgement

9

Fig. 2.3 Equilibrium at node 4 to the problem from Fig. 2.1

For the further course it is assumed that the single stiffness matrices of elements I, II, III and IV are known. For one-dimensional elements, the single stiffness relations for different types of loading will be explicitly derived in the next chapters. The Overall Stiffness The equilibrium of each single element is satisfied by the single stiffness relation in (2.6). The overall equilibrium is ensured by placing each node in equilibrium. As an example, the equilibrium is set up for node 4 in Fig. 2.3. With F 4x F 4y

Fext 4 =

ð2:9Þ

applies: Fext 4 =

IV Fe4 = FIII 4 þ F4 :

ð2:10Þ

e

If the nodal forces are replaced by the nodal displacements via the individual stiffness relationships, the following follows III III IV IV IV Fext 4 = k43 u3 þ k44 þ k44 u4 þ k45 u5 þ k46 u6 :

ð2:11Þ

If one sets up the equilibrium at each node accordingly and writes all relations in the form of a matrix equation, one obtains the total stiffness relation F=Ku

ð2:12Þ

with keij ,

K= e

or rather written to in detail

ð2:13Þ

10

2 Motivation for the Finite Element Method R1 0 0 F ext 4 0 R6

=

k I11 k I21 k I31 0 0 0

k I12 k I22 þkII22 k I32 0 k II52 0

kI13 kI23 k I33 þk III 33 kIII 43 0 0

0 0 k III 34 IV kIII 44 þk 44 IV k 54 k IV 64

0 k II25 0 kIV 45 II k55 þk IV 55 kIV 65

0 0 0 k IV 46 k IV 56 k IV 66

0 u2 u3 u4 u5 0

:

ð2:14Þ

This equation is also called the principal equation of the finite element method. On the lefthand side is the vector of the external loads (imposed loads or support reactions) and on the right-hand side the vector of all nodal displacements. Both are coupled with each other via the total stiffness matrix K. The elements of the total stiffness matrix result from the addition of the corresponding elements of the individual stiffness matrices according to (2.13). In the displacement vector, the support conditions u1 = 0 and u6 = 0 are already taken into account. From the matrix equations two to five the unknown nodal displacements u2, u3, u4 and u5 can be determined. If these are known, the unknown support reactions R1 and R6 are obtained by substituting them into the matrix equations one and six. The matrix displacement method is exact as long as the individual stiffness matrices can be set up and as long as the different elements are coupled with each other in defined nodes. This is the case, for example, for trusses or frame structures within the theories valid for these. With the method presented so far, the nodal displacements and forces can be determined as a function of the external loads. For the strength analysis of a single element, the distortion and stress state inside the element is decisive. Usually, the displacement course is described via the nodal displacements up and approximation functions. The strain field can then be determined via the kinematic relationship and from this the stress field via the material law.

2.1.2

Transition to the Continuum

The starting point is a continuum with boundary Γ (see Fig. 2.4). In the previous section, the matrix displacement method was discussed on an hinged structure. In contrast, in the continuum the imaginary discretized finite elements are connected at an infinite number of nodes. However, in real applications of the matrix displacement method, only finitely many nodes can be considered. Thus, of the required conditions for compatibility and equilibrium, both cannot be satisfied exactly at the same time. Either the compatibility or the equilibrium is only fulfilled on average. In principle, the procedure can be represented using the force method or the displacement method. In the following, only the displacement method will be considered further. Here the

2.1

Procedures Motivated by Engineering Judgement

11

Fig. 2.4 Continuum with boundary Γ, boundary conditions and loads

• the compatibility within the approximation degree exactly and • the equilibrium in the mean are fulfilled. This Results in the Following Procedure • The continuum is discretized. For two-dimensional problems, it is divided by imaginary lines and for three-dimensional problems by imaginary surfaces into subareas, so-called finite elements. • The force flow from element to neighboring element occurs exclusively via the discrete nodes. Within the displacement method, the displacements at the nodes are introduced as fundamental unknowns. • Within an element, the displacement state is described using a nodal approach. The displacement field is approximated by nodal displacements and shape functions. • Within an element, the strain state is determined from the displacement state via the kinematic relationship and the stress state is further determined via the material law. This describes the displacement, strain and stress field as a function of the nodal displacements. • By means of the principle of virtual work, statically equivalent resulting nodal forces are assigned to the stresses prevailing along the imaginary element edges. • From the requirement of equilibrium at each individual node, the overall equilibrium emerges. The total stiffness relationship can be formulated. After considering the kinematic boundary conditions, a reduced system of equations is obtained from the total stiffness relation. The unknown nodal displacements are calculated via equation solution. In a subsequent step, the post-processing, the displacement field is determined element by element from the now known nodal displacements by means of shape functions and from this the strain field via the kinematic relationship and further the stress field via the material law.

12

2 Motivation for the Finite Element Method

Fig. 2.5 Discretization of a plane surface

Comments on the Individual Steps • Discretization Discretization divides the entire continuum into elements. A partial area is in contact with one or more neighboring elements. In two dimensions the contact areas are lines, in three dimensions surfaces. Figure 2.5 shows the discretization for a plane case. Graphically, the discretization can be interpreted as follows: Individual points do not change their geometric position in the continuum. The relationship to the neighboring points does change. While in the continuum every point interacts with its neighbouring point, in the imaginary discretized continuum this is only true within one element. If two points lie in different elements, they are not directly connected. • Nodes and displacements The information flow between individual elements is only through the nodes. In the displacement method, displacements are introduced as significant unknowns at the nodes (see Fig. 2.6). The displacements are identical for each element adjacent to the node. Forces only flow over the nodes, no forces flow over the element edges, even if the element edges coincide geometrically. • Approximation of the displacement course A common way to describe the displacement field ue(x) inside an element is to approximate the field by the displacements at the nodes and so-called shape functions (see Fig. 2.7): ue ðxÞ = NðxÞup :

ð2:15Þ

The discretization must not lead to holes in the continuum. To ensure compatibility between individual elements, a suitable description of the displacement field must be

2.1

Procedures Motivated by Engineering Judgement

13

Fig. 2.6 Nodes with displacements

Fig. 2.7 Approximation of the displacement course in the element

chosen. The choice of the shape functions has a decisive influence on the quality of the approximation and is discussed in detail in Sect. 6.4. • Strain and stress fields From the displacement field ue(x) one arrives via the kinematic relation εe ðxÞ = L1 ue ðxÞ

ð2:16Þ

to the strain field. Here L1 is a first order differential operator.2 The stresses in the interior of an element can be determined via the material law: σ e ðxÞ = Dεe ðxÞ = DL1 N ðxÞup = DBðxÞup :

ð2:17Þ

The expression L1N(x) contains the derivatives of the shape functions. Usually, a new matrix with the designation B is introduced for this purpose (Fig. 2.8). • Principle of virtual work, single stiffness matrices

2

d In the one-dimensional case, the differential operator simplifies to the derivative dx .

14

2 Motivation for the Finite Element Method

Fig. 2.8 Displacements, strains and stresses in the element

Fig. 2.9 Principle of virtual work on an element

While in the continuum any point can interact with its neighboring point, in the discretized structure this is only possible within an element. A direct exchange across the element boundaries is not provided. With the principle of virtual work, a suitable tool is available to assign statically equivalent nodal forces to the stresses along the imaginary element boundaries (see Fig. 2.9). For this purpose, the nodal forces are combined to a vector Fep . The virtual displacements δup perform with the nodal forces the external virtual work δΠext, the virtual strains δε perform with the stresses σ e in the interior the internal work δΠint: δΠ ext = F ep δΠ int =

Ω

T

δup ,

ðσ e ÞT δεe dΩ:

ð2:18Þ

According to the principle of virtual work: δΠext = δΠint : Transposing the equation F ep

T

δup =

Ω

ðσ e ÞT δεe dΩ

and using (2.16) and (2.17) respectively, it follows that

ð2:19Þ

2.1

Procedures Motivated by Engineering Judgement T

δup Fep = δup

T

BT D B dΩup :

15

ð2:20Þ

Ω

From this one gets the single stiffness relation Fep = ke up

ð2:21Þ

with the element stiffness matrix ke =

BT D BdΩ:

ð2:22Þ

Ω

• Total stiffness relationship The total stiffness relationship F=Ku

ð2:23Þ

is obtained from the total equilibrium. This is achieved by establishing the equilibrium at each node. The unknown quantities cannot yet be obtained from the total stiffness relation. In the context of solving the equation, the system matrix is not regular. Only after at least the rigid body motion (displacement and rotation) have been taken out of the total system, a reduced system is obtained Fred = K red ured p ,

ð2:24Þ

which can be solved. A description of the equation solution can be found in Sect. 7.2 and in Appendix A.1.5. • Determination of the element-related field quantities After solving the equation, the nodal displacements are known. Thus, the displacement, strains and stress curve inside can be determined for each element. In addition, the support reactions can be determined.

16

2.2

2 Motivation for the Finite Element Method

Integral Principles

The finite element method is often derived using so-called energy principles. Therefore, a concise summary of some important principles is provided in the context of this chapter. The total potential or the total potential energy of a system can generally be defined as the Π = Πint þ Πext

ð2:25Þ

where Πint represents the elastic distortion energy (strain energy3) and Πext the potential of the external loads. The elastic distortion energy – or work of the internal forces – results by means of the column matrix of stresses and strains generally for linear-elastic material behavior to: Πint =

1 2

σT εdΩ:

ð2:26Þ

Ω

The potential of the external loads – which corresponds to the negative work of the external loads – can be written for the column matrix of the external loads F and the displacements u as Πext = - FT u:

ð2:27Þ

Principle of Virtual Work The principle of virtual work includes the principle of virtual displacements4 and the principle of virtual forces. The principle of virtual displacements states that when a body is in equilibrium, for any compatible, small, virtual displacements that satisfy the geometric boundary conditions, the total internal virtual work is equal to the total external virtual work: σ T δεdΩ = FT δu: Ω

Accordingly, the principle of virtual forces results in:

3 4

The strain energy is also often split into the volume and deviatoric strain energy. Also called the principle of virtual displacements.

ð2:28Þ

2.2

Integral Principles

17

a

b

F

σ

¯ Π

¯ = udF dΠ

¯ π

dπ¯ = εdσ

Π

dΠ = Fdu

π

dπ = σdε u

ε

Fig. 2.10 For the definition of deformation energy and deformation supplementary energy: (a) external (b) internal

δσ T εdΩ = δFT u:

ð2:29Þ

Ω

The Principle of the Minimum of the Total Potential According to this principle, the total potential in the equilibrium position assumes an extreme value: Π = Πint þ Πext = minimum:

ð2:30Þ

The Principle of Castigliano Castigliano’s first theorem states that the partial derivative of the complementary deformation energy (deformation complementary energy, compare Fig. 2.10a) after an external force Fi results in the displacement of the point of application of the force in the direction of this force. Correspondingly, the partial derivation of the complementary deformation energy after an external moment Mi results in the rotation at the point of application of the moment in the direction of this moment: ∂Πint = ui , ∂F i

ð2:31Þ

∂Πint = φi : ∂M i

ð2:32Þ

Castigliano’s second theorem states that the partial derivative of the strain energy (compare Fig. 2.10a) after a displacement ui gives the force Fi in the direction of the displacement at

18

2 Motivation for the Finite Element Method

the point under consideration. An analogous relationship also applies to the rotation and the moment:

2.3

∂Πint = Fi , ∂ui

ð2:33Þ

∂Πint = Mi: ∂φi

ð2:34Þ

The Weighted Residuals Method

The starting point of the weighted residuals method is the differential equation describing the physical problem. In the one-dimensional case, such a physical problem in a domain Ω can be generally described by the differential equation L u0 ð x Þ = b

ð x 2 ΩÞ

ð2:35Þ

and described by the boundary conditions given on the boundary Γ. The differential equation is also called the strong form of the problem, since at each point x of the domain the problem is described exactly. In (2.35), L{. . .} represents an arbitrary differential operator, which can take the following forms, for example:

Lf. . . g =

Lf. . . g =

d2 f. . .g, dx2

ð2:36Þ

Lf. . . g =

d4 f. . .g, dx4

ð2:37Þ

d d4 f. . .g þ f. . .g þ f. . .g: dx dx4

ð2:38Þ

Furthermore, b in (2.35) represents a given function, and in the case of b = 0 one speaks of a homogeneous differential equation: L{u0(x)} = 0. The exact or true solution of the problem, u0(x), satisfies the differential equation and the geometric and static boundary conditions prescribed on the boundary Γ at every point in the domain x 2 Ω. Since the exact solution to most engineering problems cannot generally be computed, the goal of the following derivations is to obtain as good an approximate solution as possible

2.3

The Weighted Residuals Method

19

uðxÞ ≈ u0 ðxÞ

ð2:39Þ

to be determined. For the approximate solution in (2.39), an approach of the form n

uðxÞ = α0 þ

αk φk ðxÞ

ð2:40Þ

k=1

where α0 has to satisfy the non-homogeneous boundary conditions, φk(x) represents a set of linearly independent approximation functions and αk are the free values of the approximation approach, which are determined by the respective approximation method in such a way that the exact solution u0 is approximated as best as possible by the approximation solution u.

2.3.1

Procedures Based on the Inner Product

Substituting the approximation for u0 into the differential equation (2.35), we obtain a local error, called the residual r: r = LfuðxÞg - b ≠ 0:

ð2:41Þ

In the weighted residuals method, this error is weighted with a weight function W(x) and integrated over the entire domain Ω so that the error just vanishes in the mean: WrdΩ = Ω

W ðLfuðxÞg - bÞdΩ ! = 0:

ð2:42Þ

Ω

This formulation is also called the inner product. Note that the weight or test function W(x) allows the error in the range Ω to be weighted differently. However, the total error must just become zero on average, that is, integrated over the range. The structure of the weight function is usually assumed to be similar to that of the approximation function. n

W ð xÞ =

βk ψ k ðxÞ,

ð2:43Þ

k=1

where βk are arbitrary coefficients and ψ k(x) are linearly independent ansatz functions. Depending on the choice of the number of summands k and the functions ψ k(x), the approach (2.43) includes the class of methods with the same ansatz functions for the approximate solution and the weight function (φk(k) = ψ k(x)) and the class of methods

20

2 Motivation for the Finite Element Method

where the ansatz functions are chosen differently (φk(k) ≠ ψ k(x)). Depending on the choice of the weight function, the following classical methods can be distinguished [2, 5]: Point Collocation Procedure: ψ k(x) = δ(x - xk) The point collocation method makes use of the property of the delta function. The error r is supposed to just exactly vanish at the n freely selectable points x1, x2, ⋯, xn, with xk 2 Ω, the so-called collocation points, and thus the approximate solution exactly satisfies the differential equation in the collocation points. The weight function can thus be written as n

W ðxÞ = β1 δðx - x1 Þ þ ⋯ þ βn δðx - xn Þ = ψ1

β k δ ð x - xk Þ

ð2:44Þ

k=1

ψn

where the delta function is defined as follows: δðx - xk Þ =

for x ≠ xk for x = xk

0 1

:

ð2:45Þ

Substituting this approach into the inner product given by (2.42) and noting the property of the delta function, xk þε

1

δðx - xk Þdx =

δðx - xk Þdx = 1, xk - ε

-1

xk þε

1

f ðxÞδðx - xk Þdx = -1

ð2:46Þ

f ðxÞδðx - xk Þdx = f ðxk Þ,

ð2:47Þ

xk - ε

results in n linear independent equations for the calculation of the free values αk: rðx1 Þ = Lfuðx1 Þg - b = 0,

ð2:48Þ

r ðx2 Þ = Lfuðx2 Þg - b = 0,

ð2:49Þ

r ðxn Þ = L fuðxn Þg - b = 0:

ð2:50Þ

Note that the approximation approach must satisfy all boundary conditions, that is, the essential and natural boundary conditions. Due to the property of the delta function, ΩrW(δ)dΩ = r = 0, no integral, i.e. no integration via the inner product, has to be calculated with the point collocation method. This saves integration – for example, compared to the Galerkin method – and the approximate solution is obtained more quickly.

2.3

The Weighted Residuals Method

21

A disadvantage is, however, that the collocation points are freely selectable. These can therefore also be chosen unfavourably. Subdomain Collocation Method: ψ k(x) = 1 in Ωk and Zero Otherwise This method is also a collocation method, but instead of requiring that the error vanish at certain points, here one requires that the integral of the error over different domains, the subdomains, become zero:

Ωi

rdΩi = 0

for a subregion Ωi :

ð2:51Þ

This method can be used, for example, to derive the finite difference method. Procedure of the Minimum of the Error Squares: ψ k(x) = ∂r/∂αk With the method of least squares, one optimizes the mean square error ðLfuðxÞg - bÞ2 dΩ = minimum,

ð2:52Þ

Ω

respectively d dαk

Ω

ðLfuðxÞg - bÞ2 dΩ = 0,

ð2:53Þ

Ω

dðLfuðxÞg - bÞ ðLfuðxÞg - bÞdΩ = 0: dαk

ð2:54Þ

Petrov-Galerkin Procedure: ψ k(x) ≠ φk(x) Under this name, all methods are summarized in which the ansatz functions of the weight function and the approximate solution are different. Thus, for example, the subdomain collocation method can be assigned to this group. Galerkin Procedure: ψ k(x) = φk(x) The basic idea of the Galerkin or Bubnov-Galerkin method is to choose the same ansatz functions for the approximation approach and the weight function approach. Thus, for this method, the weight function results in:

22

2 Motivation for the Finite Element Method n

W ð xÞ =

βk φk ðxÞ:

ð2:55Þ

k=1

Since the same ansatz functions φk(x) were chosen for u(x) and W(x) and the coefficients βk are arbitrary, the function W(x) can be written as a variation of u(x) (with δα0 = 0): n

W ðxÞ = δuðxÞ = δα1 φ1 ðxÞ þ . . . þ δαn φn ðxÞ =

δαk . φk ðxÞ:

ð2:56Þ

k=1

The variations can be virtual quantities, such as virtual displacements or velocities. Inserting this approach into the inner product according to (2.42), yields for a linear operator L{. . .} a set of n linearly independent equations for the determination of the n unknown free values αk: ðLfuðxÞg - bÞ . φ1 ðxÞdΩ = 0,

ð2:57Þ

ðLfuðxÞg - bÞ . φ2 ðxÞdΩ = 0,

ð2:58Þ

ðL fuðxÞg - bÞ . φn ðxÞdΩ = 0:

ð2:59Þ

Ω

Ω

Ω

Conclusion on Inner Product Based Procedures These formulations require that the ansatz functions – which are assumed to be defined over the entire domain Ω – satisfy all boundary conditions, i.e. the essential and natural ones. This requirement, as well as the differentiability of the ansatz functions demanded by the L -operator, often lead to a difficult finding of suitable functions in practical applications. In addition, generally asymmetric coefficient matrices arise (if the L operator is symmetric, then the coefficient matrix of the Galerkin method is also symmetric).

2.3.2

Procedure Based on the Weak Formulation

For the derivation of a further class of approximation methods, the inner product is partially integrated as often as the derivation of u(x) and W(x) has the same order, and one arrives at the so-called weak formulation. In this formulation, the differentiability requirement for the solution function is lowered, but the requirement for the weight function is increased. If one uses the idea of the Galerkin method, i.e. the same ansatz functions for the approximation

2.3

The Weighted Residuals Method

23

approach and the weight function, the requirement for the differentiability of the ansatz functions is reduced overall. For a 2nd or 4th order differential operator, that is. L2 fuðxÞgW ðxÞdΩ,

ð2:60Þ

L4 fuðxÞgW ðxÞdΩ,

ð2:61Þ

Ω

Ω

once partial integration of (2.60) yields the weak form L1 fW ðxÞgL1 fuðxÞgdΩ = ½W ðxÞL1 fuðxÞg]Γ ,

ð2:62Þ

Ω

respectively, twice partial integration is the weak form of (2.61): L2 fW ðxÞgL2 fuðxÞgdΩ = ½L1 fW ðxÞgL2 fuðxÞg - W ðxÞL3 fuðxÞg]Γ :

ð2:63Þ

Ω

For the derivation of the finite element method one proceeds to domain-wise defined ansatz functions. For such a domain, i.e. a finite element with Ωe < Ω and a local element coordinate xe, for example, the weak formulation of (2.62) results in: L2 fW ðxe ÞgL2 fuðxe ÞgdΩe = ½L1 fW ðxe ÞgL2 fuðxe Þg - W ðxe ÞL3 fuðxe Þg]Γe : Ω

ð2:64Þ

e

Since the weak formulation contains the natural boundary conditions – see also the example problem 2.2 – it can be demanded in the following that the approach5 for u(x) only has to fulfil the essential boundary conditions. According to the Galerkin method, it is required for the derivation of the finite element main equation that the same approach functions are chosen for the approximation and weight function. In the context of the finite element method, the nodal values uk are chosen for the free values αk and the ansatz functions φk(x) are referred to as shape or ansatz functions Nk(x). Thus, the following representations result for the approximate solution and the weight function:

The index “e” for the element coordinate is omitted in the following – if it does not impair understanding. 5

24

2 Motivation for the Finite Element Method n

uðxÞ = N 1 ðxÞu1 þ N 2 ðxÞu2 þ ⋯N n ðxÞun =

N k ðxÞuk ,

ð2:65Þ

k=1 n

W ðxÞ = δu1 N 1 ðxÞ þ δu2 N 2 ðxÞ þ ⋯δun N n ðxÞ =

δuk N k ðxÞ,

ð2:66Þ

k=1

where n represents the number of nodes per element. Importantly, this approach minimizes the error at the nodes whose locations are defined by the user. This is a clear difference to the classical Galerkin method based on the inner product, which has just found the points with r = 0 itself. For further derivation of the principal finite element equation, the approaches (2.65) and (2.66) are written in matrix form and inserted into the weak form. For the further details of the derivation, we refer here to the remarks in Chaps. 3 and 5. In the context of the finite element method, the so-called Ritz’s method is also often addressed. The classical method considers the total potential Π of a system. In this total potential, an approximation approach of the form (2.40) is used, which, however, is defined for the entire domain Ω in the Ritz method. The approximate functions φk must satisfy the geometric but not the static boundary conditions.6 By deriving the potential according to the unknown free values αk, i.e. determining the extremum of Π, a system of equations is obtained for determining the k free values, the so-called Ritz coefficients. In general, however, it is difficult to find ansatz functions with unknown free values that satisfy all geometric constraints of the problem. If, however, the classical Ritz method is modified in such a way that only the domain Ωe of a finite element is considered, and if an approximation according to (2.65) is used, the finite element method is also obtained here.

2.3.3

Procedure Based on the Inverse Formulation

Finally, it should be noted here that for the derivation of a further class of approximation methods, the inner product can be partially integrated as often as necessary until the derivations of u(x) have been completely transferred to W(x). This leads to the so-called inverse formulation. Depending on the choice of the weight function, one obtains the following methods: • Choice of W so that L(W) = 0 but L(u) ≠ 0. • Method: Boundary element method (boundary integral equation of 1st kind). • Use of a so-called fundamental solution W = W*, i.e. a solution that satisfies the equation L(W*) = (-)δ(ξ). 6

Since the static boundary conditions are implicitly contained in the total potential, the initial functions do not have to fulfill them. However, if the initial functions also fulfill the static boundary conditions, a more accurate approximation can be expected.

2.4

Example Problems

25

• Method: Boundary element method (boundary integral equation of 2nd kind). The coefficient matrix of the corresponding system of equations is full and not symmetrical. Crucial for the application of the method is the knowledge of a fundamental solution for the L -operator (in the theory of elasticity such an analytical solution is known by the Kelvin solution – single load in full space). • The same ansatz functions for approximation approach and weight function approach. • Procedure: Trefftz’s method. • The same ansatz functions for approximation approach and weight function and it holds L(u) = L(W ) = 0. • Procedure: Variant of Trefftz’s method.

2.4

Example Problems

Example 2.1: Galerkin Method Based on the Inner Product Since the term Galerkin method is frequently used in the context of the finite element method, the original Galerkin method will be illustrated below in the context of this example. For this purpose, one considers the differential equation defined in the domain 0