Finite Element Analysis

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1 Textbook of Finite Element Analysis P. Seshu

2 Textbook of Finite Element Analysis P. Seshu Professor Department of Mechanical Engineering Indian Institute of Technology Bombay Mumbai NEW DELHI 2012

3 TEXTBOOK OF FINITE ELEMENT ANALYSIS P. Seshu 2003 by PHI Learning Private Limited, New Delhi. All rights reserved. No part of this book may be reproduced in any form, by mimeograph or any other means, without permission in writing from the publisher. ISBN The export rights of this book are vested solely with the publisher. Tenth Printing L L January, 2012 Published by Asoke K. Ghosh, PHI Learning Private Limited, M-97, Connaught Circus, New Delhi and Printed by Mohan Makhijani at Rekha Printers Private Limited, New Delhi

4 Respectfully Dedicated to My Parents and Teachers

6 Contents Preface 1. Introduction Typical Application Examples Automotive Applications Manufacturing Process Simulation Electrical and Electronics Engineering Applications Aerospace Applications 14 Summary Finite Element Formulation Starting from Governing Differential Equations Weighted Residual Method Use of a Single Continuous Trial Function The General Weighted Residual (WR) Statement Weak (Variational) Form of the Weighted Residual Statement Comparison of Differential Equation, Weighted Residual and Weak Forms Piece-wise Continuous Trial Function Solution of the Weak Form One-dimensional Bar Finite Element One-dimensional Heat Transfer Element 57 Summary 61 Problems Finite Element Formulation Based on Stationarity of a Functional Introduction Functional and Differential Equation Forms Principle of Stationary Total Potential (PSTP) Rayleigh Ritz Method 75 v ix

7 vi Contents 3.4 Piece-wise Continuous Trial Functions Finite Element Method Bar Element Formulated from the Stationarity of a Functional One-dimensional Heat Transfer Element Based on the Stationarity of a Functional Meaning of Finite Element Equations 84 Summary 87 Problems One-dimensional Finite Element Analysis General Form of the Total Potential for 1-d Generic Form of Finite Element Equations The Linear Bar Finite Element The Quadratic Bar Element Determination of Shape Functions Element Matrices Beam Element Selection of Nodal d.o.f Determination of Shape Functions Element Matrices Frame Element One-dimensional Heat Transfer 132 Summary 138 Problems Two-dimensional Finite Element Analysis Introduction Dimensionality of a Problem Approximation of Geometry and Field Variable Simple Three-noded Triangular Element Four-noded Rectangular Element Six-noded Triangular Element Natural Coordinates and Coordinate Transformation Alternate Methods of Deriving Shape Functions Natural Coordinates Quadrilateral Elements Natural Coordinates Triangular Elements d Elements for Structural Mechanics Generic Relations Three-noded Triangular Element Four-noded Rectangular Element Compatibility of Displacements Four-node Quadrilateral Element Eight-node Quadrilateral Element Nine-node Quadrilateral Element Six-node Triangular Element 192

8 Contents vii 5.5 Numerical Integration Trapezoidal Rule Simpson s 1/3 Rule Newton Cotes Formula Gauss Quadrature Formula Gauss Quadrature in Two Dimensions Incorporation of Boundary Conditions Solution of Static Equilibrium Equations d Fluid Flow 220 Summary 225 Problems Dynamic Analysis Using Finite Elements Introduction Vibration Problems Equations of Motion Based on Weak Form Axial Vibration of a Rod Transverse Vibration of a Beam Equations of Motion Using Lagrange s Approach Formulation of Finite Element Equations Consistent Mass Matrices for Various Elements Consistent and Lumped Mass Matrices HRZ Lumping Scheme Form of Finite Element Equations for Vibration Problems Some Properties of Eigenpairs Solution of Eigenvalue Problems Transformation Based Methods Vector Iteration Methods Transient Vibration Analysis Modelling of Damping The Mode Superposition Scheme Direct Integration Methods Thermal Transients Unsteady Heat Transfer in a Pin-Fin 289 Summary 293 Problems Application Examples Finite Element Analysis of Crankshaft Torsional Vibrations Beam Element Model of Crankshaft Assembly Results and Discussion Dynamic Response Analysis Axisymmetric Finite Element Analysis of a Pressure Vessel Finite Element Formulation for Axisymmetric Loads Stress Analysis of a Pressure Vessel 305

9 viii Contents Appendix A Suggested Mini-Project Topics Project 1: Thermal Analysis of a Pressure Vessel 309 Project 2: Structural Dynamic Analysis of a Pressure Vessel 310 Project 3: Dynamics of a Scooter Frame 312 Project 4: Automotive Chassis Dynamics 313 Project 5: Analysis of a Turbine Disk 316 Project 6: Dynamic Analysis of a Building 317 Project 7: Thermal Analysis of an IC Engine Cylinder 318 Project 8: Stress Concentration 319 Project 9: Dynamics of a Hard Disk Drive Read/Write Head Assembly 319 Appendix B Review of Preliminaries B1.1 Matrix Algebra 321 B1.2 Interpolation 322 Appendix C Typical Finite Element Program Index

10 Preface Many excellent books have been written on Finite Element Analysis, for example, books authored by R.D. Cook, D.S. Malkus and M.E. Plesha; J.N. Reddy; K.J. Bathe; O.C. Zienkiewicz, K. Morgan, et al. I have immensely benefitted from these books, both as a student and as a teacher. Most books, however, present finite element method (FEM) primarily as an extension of matrix methods of structural analysis (except, for example, the ones by J.N. Reddy and Huebner). Present-day applications of FEM, however, range from structures to bio-mechanics to electromagnetics. Thus the primary aim of the present book, based on several years of teaching the course at Indian Institute of Technology (IIT) Bombay, is to present FEM as a general tool to find approximate solutions to differential equations. This approach should give the student a better perspective on the technique and its wide range of applications. Finite element formulation, based on stationarity of a functional, is also discussed, and these two forms are used throughout the book. Over the past few years, several universities and engineering institutes in India have introduced a one-semester course on finite element method at the senior undergraduate level. The material presented in this book covers the syllabus of most such courses. Several workedout examples, drawn from the fields of structural mechanics, heat transfer and fluid flow, illustrate the important concepts. Some of the issues are in fact brought out through example problems. At the same time, some problems to investigate given at the end of each chapter encourage the student to think beyond what has been presented in the book. FEM is a technique (numerical tool), and the various nuances are best mastered by attempting challenging real-life problems. While teaching the course at IIT Bombay, I have successfully used two types of term projects (running through the semester) one where a near-real-life problem is modelled using a commercial finite element analysis software and the other where the student attempts to develop his own code and verify the same on simple textbook type problems. Some topics for the former are suggested in Appendix A and a sample code is given in Appendix C for isoparametric, nine-node quadrilateral element. I am indebted to IIT Bombay and my colleagues (particularly, Profs. C. Amarnath, Bharat Seth, and Kurien Issac) in the Mechanical Engineering Department for providing a congenial work environment. The financial support provided by the Quality Improvement ix

11 x Contents Preface Programme (QIP) at IIT Bombay for the preparation of the manuscript is gratefully acknowledged. My students Anurag Ganguli, Vignesh Raja, Gaurav Sharma, Kartik Srinivasan and V.K. Gupta did a splendid job by studiously going through the entire manuscript and offering their critical and incisive comments. I am also thankful to my student Pankaj Langote for assisting me in getting the diagrams drawn and to Mrs. Padma Amin for keying in the whole manuscript. I would like to thank my wife Uma and my little jewels Soumya and Saket for their continuous support and encouragement. Preparation of the manuscript took longer than I expected and I am grateful to my wife for helping me stay focussed till the very end. Finally, I wish to thank the Publishers, PHI Learning their sales, editorial and production team for patiently pursuing the scheduling of the manuscript and for its careful processing. P. Seshu

12 CHAPTER 1 Introduction Finite element analysis has now become an integral part of Computer Aided Engineering (CAE) and is being extensively used in the analysis and design of many complex real-life systems. While it started off as an extension of matrix methods of structural analysis and was initially perceived as a tool for structural analysis alone, its applications now range from structures to bio-mechanics to electromagnetic field problems. Simple linear static problems as well as highly complex nonlinear transient dynamic problems are effectively solved using the finite element method. The field of finite element analysis has matured and now rests on rigorous mathematical foundation. Many powerful commercial software packages are now available, enabling its widespread use in several industries. Classical analytical methods consider a differential element and develop the governing equations, usually in the form of partial differential equations. When applied to real-life problem situations, it is often difficult to obtain an exact solution to these equations in view of complex geometry and boundary conditions. The finite element method (FEM) can be viewed simply as a method of finding approximate solutions for partial differential equations or as a tool to transform partial differential equations into algebraic equations, which are then easily solved. Some of the key ideas used in finite element formulation are now summarised: Ø Ø Ø Since the solution for the field variable satisfying both the boundary conditions and the differential equation is unknown, we begin with an assumed trial solution. The trial solution is chosen such that the boundary conditions are satisfied. The trial solution assumed, in general, does not satisfy the differential equation exactly and leaves a domain residual defined as the error in satisfying the differential equation. In general, the domain residual varies from point to point within the domain and cannot be exactly reduced to zero everywhere. We can choose to make it vanish at select points within the domain, but we prefer to render the residual very small, in some measure, over the entire domain. Thus, the weighted sum of the domain residual computed over the entire domain is rendered zero. 1

13 2 Introduction Ø Ø The accuracy of the assumed trial solution can be improved by taking additional, higher order terms, but the computations become tedious and do not readily render themselves for automation. Also, for complex real-life problems, choosing a single continuous trial function valid over the entire domain satisfying the boundary conditions is not a trivial task. We therefore prefer to discretise the domain into several segments (called finite elements) and use several piece-wise continuous trial functions, each valid within a segment (finite element). Trial functions used in each segment (finite element) are known as element level shape functions. These are defined in the form of interpolation functions used to interpolate the value of the field variable at an interior point within the element from its value at certain key points (called the nodes) in the element. Typical elements commonly used in finite element analysis are shown in Figure 1.1. w 2 v 2 j 2 u 2 q 2 y 2 v 1 w 1 j 1 q 1 u 1 y 1 (a) General frame element (Six d.o.f. Per node) (b) Common 2-d elements

14 Introduction 3 y W j v q u y W j v q u (d) Plate and shell elements (e) Common 3-d elements Fig. 1.1 Typical finite elements Ø Ø With these shape functions, the weighted sum of the domain residual is computed for each element and then summed up over all the elements to obtain the weighted sum for the entire domain. For all elements using the same shape functions, the computations will be identical and, thus, for each type of element we have element level characteristic matrices. These characteristic matrices for several types of elements are derived a priori and programmed into a finite element software such as ANSYS, NASTRAN, IDEAS, etc. The user can choose to discretise (model) his domain with a variety of different finite elements. The computer program sets up the characteristic matrices for each element and then sums them all up for the entire finite element mesh to set up and solve the system level equations. The basic steps of finite element analysis, as outlined above, are quite generic and can be applied to any problem be it from the field of structural mechanics or heat transfer, or fluid flow or electromagnetic fields, given the appropriate differential equation and boundary conditions. In view of the similarity in the form of governing differential equations, the finite element formulation for a particular type of differential equation can be used to solve a class of problems. For example, a differential equation of the type 2 d f AC 2 dx + q = 0

15 4 Introduction describes axial deformation of a rod when we use the connotation that f represents the axial deformation, q represents the load, and A, C stand for cross-sectional area and Young s modulus, respectively. The same equation, when interpreted with the connotation that f stands for temperature, q represents internal heat source and A, C stand for cross-sectional area and coefficient of thermal conductivity, respectively will be the governing equation for onedimensional heat conduction. Thus, a finite element formulation developed for the above differential equation can be readily used to solve either of the physical problems. Sometimes, the governing equations are more readily available in the form of minimization of a functional. For example, in problems of structural mechanics, the equilibrium configuration is the one that minimizes the total potential of the system. Finite element formulation can be developed readily for a problem described by a functional, rather than a differential equation. When both the forms are available for a given problem, the differential equation and functional forms are equivalent and can be derived from each other. The finite element method essentially grew up as a tool for structural mechanics problems, as an extension of the matrix methods of structural analysis. While such an approach towards the study of finite element formulation enables easy visualisation in the form of lumped springs, masses, etc., the approach outlined above highlights the generic nature of the method, applicable for a variety of problems belonging to widely varying physical domains. It is felt that this approach gives a proper perspective on the entire field of finite element analysis. In the chapters that follow, we elaborate on the various basic steps outlined above for one- and twodimensional, static and dynamic problems. We now present several examples of application of finite element analysis to real-life problems, to give an overview of the capabilities of the method. Our application examples are drawn from the fields of structural mechanics, aerospace, manufacturing processes, electromagnetics, etc. 1.1 Typical Application Examples Automotive Applications In a vehicle having monocoque construction, the body itself is connected to the suspension. Therefore, the body panels are subjected to road loads. Hence, stresses and strains in these body panels are of interest. Figure 1.2 shows a FE mesh of a floor panel from the rear end of the vehicle. Provision for spare wheel as well as the various depressions used as stiffeners can be seen in the figure. A total of about 13,000 quadrilateral and triangular shell elements have been used to perform modal analysis, torsional stiffness analysis, and service load analysis. The same finite element mesh is also used for crash analysis using LS-DYNA software. An automotive engine cylinder block experiences severe pressures and temperature gradients and other transient loads. It is essential to predict accurately the stresses and the vibration levels for further correlation with noise predictions. Figure 1.3 shows a typical finite element (shell element) model of a four cylinder in-line diesel engine cylinder block. Such a model is used to predict the system natural frequencies and mode shapes, response to combustion gas pressure, etc.

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Finite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras. Lecture - 36

Finite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras Lecture - 36 In last class, we have derived element equations for two d elasticity problems