MATLAB.Codes.for.Finite.Element.Analysis
SOLID MECHANICS AND ITS APPLICATIONS Volume 157 Series editor. GML. GLADWELL Department of Civil Engineering University of Waterloo Waterloo, Ontario, Canada n2l 3gi Aims and scope of the series The fundamental questions arising in mechanics are: Why?, How?, and How much? The aim of this series is to provide lucid accounts written by authoritative researchers giving vision and insight in answering these questions on the subject of mechanics as it relates to solids The scope of the series covers the entire spectrum of solid mechanics. Thus it includes the foundation of mechanics; variational formulations; computational mechanics; statics, kinematics and dynamics of rigid and elastic bodies: vibrations of solids and structures; dynamical systems and chaos, the theories of elasticity, plasticity and viscoelasticity; composite materials; rods, beams, shells and membranes, structural control and stability; soils, rocks and geomechanics; fracture; tribology; experimental mechanics; biomechanics and machine design The median level of presentation is the first year graduate student. Some texts are monographs defining the current state of the field; others are accessible to final year undergraduates; but essentially the emphasis is on readability and clarity For other titles published in this series, go to www.springer.com/series/6557 MATLAB Codes for finite Element Analysis Solids and structures AM. Ferreira Universidade do porto Portugal S pringer AJ. M. Ferreira Universidade do porto Rua dr. roberto frias 4200-465 Porto Portugal erreira@ fe up. pt ISBN978-1-4020-9199-5 e-ISBN978-1-4020-9200-8 Library of Congress Control Number: 2008935506 All Rights reserved C 2009 Springer Science+Business Media B V No part of this work may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work Cover illustration: WMXDesign gmbh Printed on acid-free paper 987654321 springer. com reface This book intend to supply readers with some MaTLAB codes for finite element analysis of solids and structures After a short introduction to matlab. the book illustrates the finite element implementation of some problems by simple scripts and functions The following problems are discussed Discrete systems, such as springs and bars Beams and frames in bending in 2D and 3D Plane stress problems Pla tes in bending Free vibration of Timoshenko beams and Mindlin plates, including laminated composites Buckling of Timoshenko beams and Mindlin plates The book does not intends to give a deep insight into the finite element details ust the basic equations so that the user can modify the codes. The book was prepared for undergraduate science and engineering students, although it may be useful for graduate students The matlab codes of this book are included in the disk. Readers are welcomed to use them freely The author does not guarantee that the codes are error-free, although a ma jor ffort was taken to verify all of them. Users should use MaTLaB 7.0 or greater when running these codes Any suggestions or corrections are welcomed by an email to ferreira@fe up. pt Porto, Portugal Antonio ferreis 2008 Contents 1 Short introduction to matLaB 1.1 Introductio 1.2 Matr Operating with matrices 1.4 Statements 1.5 Matrix functions 3 1.6 Conditionals. if and switch 4 1.7 Loops: for and while 5 1. 8 Relatio 1.9 Scalar functions 1.10 Vector functions 1.11 Matrix functions 9 1.12 Submatrix 10 1.13 Logical indexing 12 1. 14 M-files, scripts and functions 13 1. 15 Graphics 14 1. 15.1 2D plots .14 1.15.2 3D plots 15 1.16 Linear algebra. 16 2 Discrete systems 2.1 Introduction 19 2.2 Springs and bars 19 2.3E uilibrium at nodes 20 2.4 Some basic steps 2.5 First problem and first MATLAB code .,,.21 2.6 New code using matlab structures 3 Analysis of bars 33 3.1a bar element 33 3.2 Numerical integration .36 Contents 3.3 An example of isoparametric bar 37 3.4 Problem 2, using MATLAB struct 41 3.5 Problem 3 44 4 Analysis of 2d trusses 4.1 Introduction 4.2 2D trusses 4.3 Stiffness matrix 52 4.4 Stresses at the element 53 4.5 First 2D truss problem .53 4.6 A second truss problem 8 4.7 An example of 2D truss with spring 5 Trusses in 3D space 5.1 Basic formulation ...69 5.2 A 3D truss problem 5.3 A second 3D truss example 73 6 Bernoulli beams .79 6.1 Introduction .79 6.2 Bernoulli beam problem 6.3 Bernoulli beam with spring 7 2D frames 89 7.1 Introduction 7.2 An example of 2d frame 7. 3 Another example of 2D frame 8 Analysis of 3D frames 103 1 Introduction 103 8.2 Stiffness matrix and vector of equivalent nodal forces 103 8.3 First 3D frame example 104 8.4 Second 3D frame example 108 9 Analysis of grids 113 9. 1 Introduction 11 9.2 A first grid example .116 9.3 A second grid example 119 10 Analysis of Timoshenko beams ....123 10.1 Introduction ..123 10.2 Formulation for static analysis 123 10.3 Free vibrations of Timoshenko beams 130 10.4 Buckling analysis of Timoshenko beams ...136 Contents 11 Plane stress ....143 11. 1 Introduction 143 11.2 Displacements, strains and stresses 143 11.3 Boundary conditions 144 11.4 Potential energy 145 11.5 Finite element discretization 145 11.6 Interpolation of displacements ..145 11.7 Element energy 146 11.8 Quadrilateral element Q4 147 11.9 Example: plate in traction 149 11.10 Example: beam in bending 152 12 Analysis of Mindlin plates 161 12.1I ..161 12.2 The Mindlin plate theory ..161 12.2.1 Strains 162 12.2.2 Stresses 163 12.3 Finite element discretization 163 12.4 Example: a square Mindlin plate in bending ....165 12.5 Free vibrations of Mindlin plates 182 12.6 Buckling analysis of Mindlin plates 192 13 Laminated plates 203 13.1 Introduction 203 13.2 Displacement field 203 13.3 Strains ..203 13.4 Strain-displacement matrix B 205 13.5 Stresses 205 13.6 Stiffness matrix .....207 13.7 Numerical example 208 13.8 Free vibrations of laminated plates 225 References 231 Index 233 Chapter 1 Short introduction to matLaB 1.1 Introduction MAtLab is a commercial software and a trademark of the math Works. Inc USA. It is an integrated programming system, including graphical interfaces and a large number of specialized toolboxes. MATLAB is getting increasingly popular in all fields of science and engineering This chapter will provide some basic notions needed for the understanding of the remainder of the book. a deeper study of matlab can be obtained from many MatLAB books and the very useful help of MATLAB 1.2 Matrices Matrices are the fundamental object of MATLAB and are particularly important in this book. Matrices can be created in MATLAB in many ways, the simplest one obtained by the commands >>A=[123;456;789] 4 7 258 Note the semi-colon at the end of each matrix line. We can also generate matrices by pre-defined functions, such as random matrices > rand (3) 0.81470.9134 0.2785 0.9058 0.6324 0.5469 0.1270 0.0975 0.9575 A.J.M. Ferreira, MATLAB Codes for Finite Element analysis 1 Solids and structures, Solid Mechanics and Its Applications 157. C Springer Science+Business Media B V. 2009