Bhatti-Fundamental FEA and Application Using MATLAB & Mathematica

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FUN AMEN L FINITEELE ENT ANALY IS

ND APPllC I NSWith Mathematica and MATLABComputations

M. ASGHAR BHATT~

~WILEY

JOHN WILEY 8t SONS, INC.

METU LIBRARY

Mathematica is a registeredtrademarkof WolframResearch,Inc.MATLAB is a registered trademarkof The MathWorks, Inc.ANSYS is a registered trademarkof ANSYS, Inc.ABAQUS is a registered trademarkof ABAQUS, Inc.This book is printed on acid-freepaper.eCopyright 2005 by John Wiley & Sims, Inc. All rights reserved.Published by John Wiley & Sons, Inc., Hoboken,New JerseyPublishedsimultaneouslyin Canada.

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Library ofCongress Cataloging-in-Publication Data

Bhatti, M. AsgharFundamental finiteelement analysis and applications: with Mathematica

and Matlab computations/ M. AsgharBhatti.p. cm.

Includes index.ISBN 0,471-64808-61. Structural analysis (Engineering) 2. Finite element method, J, Title.

TA646.B56 2005620' .001'51825-dc22

Printed in the United States of America1098765432r,~~; ~,:: Vr/\. ff (; ".+ TL'~'}j i'J\If,!~ ~J:'''(~

CONTENTS

CONTENTS OF THE BOOK WEB SITE

PREFACE

xi

xiii

1 FINITE ELEMENT METHOD: THE BIG PICTURE 11.1 Discretization and Element Equations / 2

1.1.1 Plane Truss Element / 41.1.2 Triangular Element for Two-Dimensional Heat Flow / 71.1.3 General Remarks on Finite Element Discretization / 141.1.4 Triangular Element for Two-Dimensional Stress Analysis / 16

1.2 Assembly of Element Equations -/ 211.3 Boundary Conditions and Nodal Solution / 36

1.3.1 Essential Boundary Conditions by Rearranging Equations / 371.3.2 Essential Boundary Conditions by Modifying Equations / 391.3.3 Approximate Treatment of Essential Boundary Conditions / 401.3.4 Computation of Reactions to Verify Overall Equilibrium / 41

1.4 Element Solutions and Model Validity / 491.4.1 Plane Truss Element / 491.4.2 Triangular Element for Two-Dimensional Heat Flow / 511.4.3 Triangular Element for Two-Dimensional Stress Analysis / 54

1.5 Solution ofLinear Equations / 581.5.1 Solution Using Choleski Decomposition / 581.5.2 ConjugateGradientMethod / 62

v

vi CONTENTS

1.6 Multipoint Constraints / 721.6.1 Solution Using Lagrange Multipliers / 751.6.2 Solution Using Penalty Function / 79

1.7 Units / 83

2 MATHEMATICAL FOUNDATION OF THEFINITE ELEMENT METHOD 982.1 Axial Deformation of Bars / 99

2.1.1 Differential Equation for Axial Deformations I 992.1.2 Exact Solutions of Some Axial Deformation Problems / 101

2.2 Axial Deformation of Bars Using Galerkin Method / 1042.2.1 Weak Form for Axial Deformations / 1052.2.2 Uniform Bar Subjected to Linearly Varying Axial Load / 1092.2.3 Tapered Bar Subjected to Linearly Varying Axial Load / 113

2.3 One-Dimensional BVJ;>Using .Galerkin Method / 115_... -,-

2.3.1 Overall Solution Procedure Using GalerkinMethod / 1152.3.2 Highet Order Boundary Value Problems / 119

2.4 Rayleigh-Ritz Method / 1282.4.1 Potential Energy for Axial Deformation of Bars / 1292.4.2 Overall Solution Procedure Using the Rayleigh-Ritz Method / 1302.4.3 Uniform Bar Subjected to Linearly Varying Axial Load I 1312.4.4 Tapered Bar Subjected to Linearly Varying Axial Load / 133

2.5 Comments on Galerkin and Rayleigh-Ritz Methods / 1352.5.1 Admissible Assumed S~lution / 1352.5.2 Solution Convergence-the Completeness Requirement / 1362.5.3 Galerkin versus Rayleigh-Ritz / 138

2.6 Finite Element Form of Assumed Solutions / 1382.6.1 LinearInterpolation Functions for Second-Order Problems / 1392.6.2 Lagrange Interpolation / 1422.6.3 Galerkin Weighting Functions in Finite Element Form / 1432.9.4 Hermite Interpolation for Fourth-Order Problems / 144

2.7 Finite Element Solution of Axial Deformation Problems / 1502.7.1 Two-Node Uniform Bar Element for Axial Deformations / 1502.7.2 Numerical Examples / 155

3 ONE-DIMENSIONAL BOUNDARYVALUE PROBLEM3.1 Selected Applications of 1D BVP / 174

3.1.1 Steady-State Heat Conduction / 1743.1.2 Heat Flow through Thin Fins / 175

173

CONTENTS

3.1.3 Viscous Fluid Flow between Parallel Plates-LubricationProblem / 176

3.1.4 Slider Bearing / 1773.1.5 Axial Deformation of Bars / 1783.1.6 Elastic Buckling of Long Slender Bars / 178

3.2 Finite Element Formulation for Second-Order ID BVP / 1803.2.1 Complete Solution Procedure / 186

3.3 Steady-State Heat Conduction / 1883.4 Steady-State Heat Conduction and Convection / 1903.5 Viscous Fluid Flow Between Parallel Plates / 1983.6 Elastic Buckling of Bars / 2023.7 Solution of Second-Order 1D BVP / 2083.8 A Closer Look at the Interelement DerivativeTerms / 214

4 TRUSSES, BEAMS, AND FRAMES 2224.1 Plane Trusses / 2234.2 Space Trusses / 2274.3 Temperature Changes and Initial Strains in Trusses / 2314.4 Spring Elements / 2334.5 Transverse Deformation of Beams / 236

4.5.1 Differential Equation for Beam Bending / 2364.5.2 Boundary Conditions for Beams / 2384.5.3 Shear Stressesin Beams / 2404.5.4 Potential Energy for Beam Bending / 2404.5.5 Transverse Deformation of a Uniform Beam / 2414.5.6 Transverse Deformation of a Tapered Beam Fixed at

Both Ends / 2424.6 Two-Node Beam Element / 244

4.6.1 Cubic Assumed Solution / 2454.6.2 Element Equations Using Rayleigh-Ritz Method / 246

4.7 Uniform Beams Subjected to Distributed Loads / 2594.8 Plane Frames / 2664.9 Space Frames / 279

4.9.1 Element Equations in Local Coordinate System / 2814.9.2 Local-to-Global Transformation / 2854.9.3 Element Solution / 289

4.10 Frames in Multistory Buildings / 293

vii

viii CONTENTS

5 TWO-DIMENSIONALELEMENTS5.1 Selected Applications of the 2D BVP / 313

5.1.1 Two-Dimensional Potential Flow / 3135.1.2 Steady-State Heat Flow / 3165.1.3 Bars Subjected to Torsion / 3175.1.4 Waveguidesin Electromagnetics / 319

5.2 Integration by Parts in Higher Dimensions / 3205.3 Finite Element Equations Using the Galerkin Method / 3255.4 Rectangular Finite Elements / 329

5.4.1 Four-Node Rectangular Element / 3295.4.2 Eight-Node Rectangular Element / 3465.4.3 Lagrange Interpolation for Rectangular Elements / 350

5.5 Triangular Finite Elements / 3575.5.1 Three-Node Triangular Element / 3585.5.2 Higher Order Triangular Elements / 371

311

6 MAPPED ELEMENTS 3816) Integration Using Change of Variables / 382

6.1.1 One-Dimensional Integrals / 3826.1.2 Two-Dimensional Area Integrals / 3836.1.3 Three-Dimensional VolumeIntegrals / 386

6.2 Mapping Quadrilaterals Using Interpolation Functions / 3876.2.1 Mapping Lines / 3876.2.2 Mapping Quadrilater~ Areas / 3926.2.3 Mapped Mesh Gene~ation / 405

6.3 Numerical Integration Using Gauss Quadrature / 4086.3.1 Gauss Quadrature for One-Dimensional Integrals / 4096.3.2 Gauss Quadrature for Area Integrals / 4146.3.3 Gauss Quadrature for VolumeIntegrals / 417

6.4 Finite Element Computations Involving Mapped Elements / 4206.4.1 Assumed Solution / 4216.4.2 Derivatives of the Assumed Solution / 4226.4.3 Evaluation of Area Integrals / 4286.4.4 Evaluation of Boundary Integrals / 436

6.5 Complete Mathematica and MATLABSolutions of 2D BVP InvolvingMapped Elements / 441

6.6 Triangular Elements by Collapsing Quadrilaterals / 4516.7 Infinite Elements / 452

6.7.1 One-DirnensionalBVP / 4526.7.2 Two-Dimensional BVP / 458

CONTENTS

7 ANALYSIS OF ELASTIC SOLIDS 4677.1 Fundamental Concepts in Elasticity / 467

7.1.1 Stresses / 4677.1.2 Stress Failure Criteria / 4727.1.3 Strains / 4757.1.4 ConstitutiveEquations / 4787.1.5 TemperatureEffects and Initial Strains / 480

7.2 GoverningDifferential Equations / 4807.2.1 Stress Equilibrium Equations / 4817.2.2 GoverningDifferential Equations in Terms of Displacements / 482

7.3 GeneralForm of Finite Element Equations / 4847.3.1 Potential Energy Functional / 4847.3.2 WeakForm / 4857.3.3 Finite Element Equations / 4867.3,4 Finite Element Equations in the Presence of Initial Strains / 489

7.4 Plane Stress and Plane Strain / 4907.4.1 Plane Stress Problem / 4927.4.2 Plane Strain Problem / 4937.4.3 Finite Element Equations / 4957.4.4 Three-Node Triangular Element / 4977.4.5 Mapped Quadrilateral Elements / 508

7.5 Planar Finite Element Models / 5177.5.1 Pressure Vessels / 5177.5.2 Rotating Disks and Flywheels / 5247.5.3 Residual Stresses Due to Welding / 5307.5.4 Crack Tip Singularity / 531

8 TRANSIENT PROBLEMS 5458.1 TransientField Problems / ,545

8.1.1 Finite Element Equations / 5468.1.2 TriangularElement / 5498.1.3 Transient Heat Flow / 551

8.2 Elastic Solids Subjected to Dynamic Loads / 5578.2.1 Finite Element Equations / 5598.2.2 Mass Matrices for Common Structural Elements / 5618.2.3 Free-VibrationAnalysis / 5678.2.4 Transient Response Examples / 573

lx

x CONTENTS

9 p-FORMULATION 5869.1 p-Formulation for Second-Order 1D BVP / 586

9.1.1 Assumed Solution Using Legendre Polynomials / 5879.1.2 Element Equations / 5919.1.3 Numerical Examples / 593

9.2 p-Formulation for Second-Order 2D BVP / 6049.2.1 p-Mode Assumed Solution / 6059.2.2 Finite Element Equations / 6089.2.3 Assembly of Element Equations / 6179.2.4 Incorporating Essential Boundary Conditions / 6209.2.5 Applications / 624

A USE OF COMMERCIAL FEA SOFTWARE 641A.1 ANSYS Applications / 642

A.1.1 General Steps / 643A.1.2 Truss Analysis / 648A.1.3 Steady-State Heat Flow / 651A.1.4 Plane Stress Analysis / 655

A.2 Optimizing Design Using ANSYS / 659A.2.1 General Steps / 659A.2.2 Heat Flow Example / 660

A.3 ABAQUSApplications / 663A.3.1 Execution Procedure / 663A.3.2 Truss Analysis / 66'5A.3.3 Steady-State Heat Flow / 666A.3.4 Plane Stress Analysis / 671

B VARIATIONAL FORM FOR BOUNDARYVALUE PROBLEMS 676B.1 Basic Concept of Variationof a Function / 676B.2 Derivation of Equivalent VariationalForm / 679B.3 Boundary ValueProblem Corresponding to a Given Functional / 683

BIBLIOGRAPHY 687

INDEX 695

CONTEN'TS OF THE BOOK WEB S~TE(www.wiley.com/go/bhatti)

ABAQUS Applications

AbaqusUse\AbaqusExecutionProcedure.pdfAbaqusUse\HeatFlowAbaqusUse\PlaneStressAbaqusUse\TmssAnalysis

ANSYS Applications

AnsysUse\AppendixA. AnsysUse\Chap5

AnsysUse\Chap7AnsysUse\Chap8AnsysUse\GeneralProcedure.pdf

Full Detail Text Examples

Full Detail Text Examples\ChaplExarnples.pdfFull Detail Text Examples\Chap2Examples.pdfFull Detail Text Examples\Chap3Examples.pdf .Full Detail Text Exarnples\Chap4Exarnples.pdfFull Detail Text Exarnples\Chap5Exarnples.pdfFull Detail Text Examples\Chap6Examples.pdfFull Detail Text Examples\Chap7Examples.pdf

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xii CONTENTS OF THE BOOKWEB SITE

Full Detail Text Examples\Chap8Examples.pdfFull Detail Text Examples\Chap9Examples.pdf

Mathematica Applications

MathematicaUse\MathChapl.nbMathematicaUse\MathChap2.nbMathematicaUse\MathChap3.nbMathematicaUse\MathChap4.nbMathematicaUse\MathChap5 .nbMathematicaUse\MathChap6.nbMathematicaUse\MathChap7.nbMathematicaUse\MathChap8.nbMathematicaUse\Mathematica Introduction.nb

MATLAB Applications

MatlabFiles\Chap IMatlabFiles\Chap2MatlabFiles\Chap3MatlabFiles\Chap4MatlabFiles\Chap5MatlabFiles\Chap6MatlabFiles\Chap7MatlabFiles\Chap8MatlabFiles\Common I

Sample Course Outlines, Lectures, and Examinations

Supplementary Material and Corrections

PREFACE

Large numbers of books have been written on the finite element method. However, effectiveteaching of the method using most existing books is a difficult task. The vast majority ofcurrent books present the finite element method as an extension of the conventional matrixstructural analysis methods. Using this approach, one can teach the mechanical aspects ofthe finite element method fairly well, but there are no satisfactory explanations for eventhe simplest theoretical questions. Why are rotational degrees of freedom defined for thebeam and plate elements but not for the plane stress and truss elements? What is wrongwith connecting corner nodes of a planar four-node element to the rnidside nodes of aneight-node element? The application of the method to nonstructural problems is possibleonly if one can interpret problem parameters in terms of their structural counterparts. Forexample, one can solve heat transfer problems because temperature can be interpreted asdisplacement in a structural problem.

More recently, several new textbooks on finite elements have appeared that emphasizethe mathematical basis of the finite element method. Using some of these books, the fi-nite element method can be presented as a method for .obtaining approximate solution ofordinary and partial differential equations. The choice of appropriate degrees of freedom,boundary conditions, trial solutions, etc., can now be fully explained with this theoreti-cal background. However, the vast majority of these books tend to be too theoretical anddo not present enough computational details and examples to be of value, especially toundergraduate and first-year graduate students in engineering.

The finite element coursesface one more hurdle. One needs to perform computationsin order to effectively learn the finite element techniques. However, typical finite elementcalculations are very long and tedious, especially those involving mapped elements. Infact, some of these calculations are essentially impossible to perform by hand. To alleviatethis situation, instructors generally rely on programs written in FORTRAN or some other

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xiv PREFACE

conventional programming language. In fact, there are several books available that includethese types of programs with them. However, realistically, in a typical one-semester course,most students cannot be expected to fully understand these programs. At best they use themas black boxes, which obviously does not help in learning the concepts.

In addition to traditional research-oriented students, effective finite element coursesmust also cater to the needs and expectations of practicing engineers and others interestedonly in the finite element applications. Knowing the theoretical details alone does not helpin creating appropriate models for practical, and often complex, engineering systems.

This book is intended to strike an appropriate balance among the theory, generality,and practical applications of the finite element method. The method is presented as a fairlystraightforward extension of the classical weighted residual and the Rayleigh-Ritz methodsfor approximate solution of differential equations. The theoretical details are presented inan informal style appealing to the reader's intuition rather than mathematical rigor. To makethe concepts clear, all computational details are fully explained and numerous examples areincluded showing all calculations. To overcome the tedious nature of calculations associ-ated with finite elements, extensive use of MATLAB and Mathematicd'' is made in theboole.All finite element procedures are implemented in the form of interactive Mathemat-ica notebooks and easy-to-follow MATLAB code. All necessary computations are readilyapparent from these implementations. Finally, to address the practical applications of thefinite element method, the book integrates a series of computer laboratories and projectsthat involve modeling and solution using commercial finite element software. Short tuto-rials and carefully chosen sample applications of ANSYS and ABAQUS are contained inthe book.

The book is organized in such a way that it can be used very effectively in a lecture/computer laboratory (lab) format. In over 20 years of teaching finite elements, using avariety of approaches, the author has found that presenting the material in a two-hourlecture and one-hour lab per week is i~eally suited for the first finite element course. Thelecture part develops suitable theoretical background while the lab portion gives studentsexperience in finite element modeling and actual applications. Both parts should be taughtin parallel. Of course, it takes time to develop the appropriate theoretical background inthe lecture part. The lab part, therefore, is ahead of the lectures and, in the initial stages,students are using the finite element software essentially as a black box. However, thisapproach has two main advantages. The first is that students have some time to get familiarwith the particular computer system and the finite element package being utilized. Thesecond, and more significant, advantage is that it raises students' curiosity in learning moreabout why things must be done in a certain way. During early labs students often encountererrors such as "negative pivot found" or "zero or negative Jacobian for element." When,during the lecture part, they find out mathematical reasons for such errors, it makes themappreciate the importance of learning theory in order to become better users of the finiteelement technology.

The author also feels strongly that the labs must utilize one of the several commerciallyavailable packages, instead of relying on simple home-grown programs. Use of commer-cial programs exposes students to at least one state-of-the-art finite element package withits built-in or associated pre- and postprocessors. Since the general procedures are verysimilar among different programs, it is relatively easy to learn a different package after this

PREFACE

exposure. Most commercial prol$nims also include analysis modules for linear and nonlin-ear static and dynamic analysis, buclding, fluid flow, optimization, and fatigue. Thus withthese packages students can be exposed to a variety of finite element applications, eventhough there generally is not enough time to develop theoretical details of all these topicsin one finite element course. With more applications, students also perceive the course asmore practical and seem to put more effort into learning.

TOPICS COVERED

The book covers the fundamental concepts and is designed for a first course on finite ele-ments suitable for upper division undergraduate students and first-year graduate students.It presents the finite element method as a tool to find approximate solution of differentialequations and thus can be used by students from a variety of disciplines. Applications cov-ered include heat flow, stress analysis, fluid flow, and analysis of structural frameworks.The material is presented in nine chapters and two appendixes as follows.

1. Finite Element Method: The Big Picture. This chapter presents an overview of thefinite element method. To give a clear idea of the solution process, the finite element equa-tions for a few simple elements (plane truss, heat flow, and plane stress) are presented in thischapter. A few general remarks on modeling and discretization are also included. Importantsteps of assembly, handling boundary conditions, and solutions for nodal unknowns and el-ement quantities are explained in detail in this chapter. These steps are fairly mechanical innature and do not require complex theoretical development. They are, however, central toactually obtaining a finite element solution for a given problem. The chapter includes briefdescriptions of both direct and iterative methods for solution oflinear systems of equations.Treatment of linear constraints through Lagrange multipliers and penalty functions is alsoincluded.

This chapter gives enough background to students so that they can quickly start usingavailable commercial finite element packages effectively. It plays an important role in thelecture/lab format advocated-for the first finite element course.

2. Mathematical Foundations of the Finite Element Method. From a mathematicalpoint of view the finite element method is a special form of the well-known Galerkinand Rayleigh-Ritz methods for finding approximate solutions of differential equations.The basic concepts are explained in this chapter with reference to the problem of axialdeformation of bars. The derivation of the governing differential equation is included forcompleteness. Approximate solutions using the classical form of Galerkin and Rayleigh-Ritz methods are presented. Finally, the methods are cast into the form that is suitablefor developing finite element equations. Lagrange and Hermitian interpolation functions,commonly employed in derivation of finite element equations, are presented in this chapter.

3. One-Dimensional Boundary Value Problem. A large humber of practical problemsare governed by a one-dimensional boundary value problem of the form

d ( dU(X))dx k(x)~ + p(x) u(x) + q(x) =0

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xvi PREFACE

Finite element formulation and solutions of selected applications that are governed by thedifferential equation of this form are presented in this chapter.

4. Trusses, Beams, and Frames. Many structural systems used in practice consist oflong slender members of various shapes used in trusses, beams, and frames. This chap-ter presents finite element equations for these elements. The chapter is important for civiland mechanical engineering students interested in structures. It also covers typical mod-eling techniques employed in framed structures, such as rigid end zones and rigid floordiaphragms. Those not interested in these applications can skip this chapter without anyloss in continuity.

5. Two-Dimensional Elements. In this chapter the basic finite element concepts are il--lustrated with reference to the following partial differential equation defined over an arbi-trary two-dimensional region:

The equation can easily be recognized as a generalization of the one-dimensional bound-ary value problem considered in Chapter 3. Steady-state heat flow, a variety of fluid flow,and the torsion of planar sections are some of the common engineering applications thatare governed by the differential equations that are special cases.of this general boundaryvalue problem. Solutions of these problems using rectangular and triangular elements arepresented in this chapter.

6. Mapped Elements. Quadrilateral elements and other elements that can have curvedsides are much more useful in accurately modeling arbitrary shapes. Successful develop-ment of these elements is based on the key concept of mapping. These concepts are dis-cussed in this chapter. Derivation of the Gaussian quadrature used to evaluate equations formapped elements is presented. Four-sand eight-node quadrilateral elements are presentedfor solution of two-dimensional boundary value problems. The chapter also includes pro-cedures for forming triangles by collapsing quadrilaterals and for developing the so-calledinfinite elements to handle far-field boundary conditions.

7. Analysis ofElastic Solids. The problem of determining stresses and strains in elasticsolids subjected to loading and temperature changes is considered in this chapter. Thefundamental concepts from elasticity are reviewed. Using these concepts, the governingdifferential equations in terms of stresses and displacements are derived followed by thegeneral form of finite element equations for analysis of elastic solids. Specific elementsfor analysis of plane stress and plane strain problems are presented in this chapter. Theso-called singularity elements, designed to capture a singular stress field near a crack tip,are discussed. This chapter is important for those interested in stress analysis. Those notinterested in these applications can skip this chapter without any loss in continuity.

8. Transient Problems. This chapter considers analysis of transient problems using fi-nite elements. Formulations for both the transient field problems and the structural dynam-ics problems are presented in this chapter.

9. p-Formulation. In conventional finite element formulation, each element is based ona specific set of interpolation functions. After choosing an element type, the only way to

PREFACE

obtain a better solution is to refihe the model. This formulation is called h-formulation,where h indicates the generic size of an element. An alternative formulation, called thep-formulation, is presented in this chapter. In this formulation, the elements are based oninterpolation functions that may involve very high order terms. The initial finite elementmodel is fairly coarse and is based primarily on geometric considerations. Refined solu-tions are obtained by increasing the order of the interpolation functions used in the formu-lation. Efficient interpolation functions have been developed so that higher order solutionscan be obtained in a hierarchical manner from the lower order solutions.

10. Appendix A: Use of Commercial FEA Software. This appendix introduces studentsto two commonly used commercial finite element programs, ANSYS and ABAQUS. Con-cise instructions for solution of structural frameworks, heat flow, and stress analysis prob-lems are given for both programs.

11. Appendix B: Variational Formfor Boundary Value Problems. The main body of thetext employs the Galerkin approach for solution of general boundary value problems andthe variational approach (using potential energy) for structural problems. The derivationof the variational functional requires familiarity with the calculus of variations. In the au-thor's experience, given that only limited time is available, most undergraduate studentshave difficulty fully comprehending this topic. For this reason, and since the derivationis not central to the finite element development, the material on developing variationalfunctionals is moved to this appendix. If desired, this material can be covered with thediscussion of the Rayleigh-Ritz method in Chapter 2.

To keep the book to a reasonable length and to make it suitable for a wider audience,important structural oriented topics, such as axisymmetric and three-dimensional elasticity,plates and shells, material and geometric nonlinearity, mixed and hybrid formulations, andcontact problems are not covered in this book. These topics are covered in detail in acompanion textbook by the author entitled Advanced Topics in Finite Element Analysis ofStructures: With Mathematico'" and lvIATLAB Computations, John Wiley, 2006.

UNIQUE FEATURES

(i) All key. ideas are introduced in chapters that emphasize the method as a way tofind approximate solution of boundary value problems. Thus the book can be usedeffectively for students from a variety of disciplines..

(ii) The "big picture" chapter gives readers an overview of all the mechanical detailsof the finite element method very quickly. This enables instructors to start usingcommercial finite element software early in the semester; thus allowing plentyof opportunity to bring practical modeling issues into the classroom. The authoris not aware of any other book that starts out in this manner. Few books thatactually try to do this do so by taldng discrete spring and bar elements. In myexperience this does not work very well because students do not see actual finiteelement applications. Also, this approach does not make sense to those who arenot interested in structural applications.

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xviii PREFACE

(iii) Chapters 2 and 3 introduce fundamental finite element concepts through one-dimensional examples. The axial deformation problem is used for a gentle intro-duction to the subject. This allows for parameters to be interpreted in physicalterms. The derivation of the governing equations and simple techniques for ob-taining exact solutions are included to help those who may not be familiar withthe structural terminology. Chapter 3 also includes solution of one-dimensionalboundary value problems without reference to any physical application for non-structural readers.

(iv) Chapter 4, on structural frameworks, is quite unique for books on finite elements.No current textbook that approaches finite elements from a differential equationpoint of view also has a complete coverage of structural frames, especially in threedimensions. In fact, even most books specifically devoted to structural analysis donot have as satisfactory a coverage of the subject as provided in this chapter.

(v) Chapters 5 and 6 are two important chapters that introduce key finite elementconcepts in the context of two-dimensional boundary value problems. To keepthe integration and differentiation issues from clouding the basic ideas, Chap-ter 5 starts with rectangular elements and presents complete examples using suchelements. The triangular elements are presented next. By the time the mappedelements are presented in Chapter 6, there are no real finite element-related con-cepts left. It is all just calculus. This clear distinction between the fundamentalconcepts and calculus-related issues gives instructors flexibility in presenting thematerial to students with a wide variety of mathematics background.

(vi) Chapter 9, on p-formulation, is unique. No other book geared toward the first fi-nite element course even mentions this important formulation. Several ideas pre-sented in this chapter are used in recent development of the so-called mesh lessmethods.

(vii) Mathematica and MATLAB ,implementations are included to show how calcula-tions can be organized using' a computer algebra system. These implementationsrequire only the very basic understanding of these systems. Detailed examplesare presented in Chapter 1 showing how to generate and assemble element equa-tions, reorganize matrices to account for boundary conditions, and then solve forprimary and secondary unknowns. These steps remain exactly the same for all im-plementations. Most of the other implementations are nothing more than elementmatrices written using Mathematica or MATLAB syntax.

(viii) Numerous numerical examples are included to clearly show all computations in-volved.

(ix) All chapters contain problems for homework assignment. Most chapters alsocontain problems suitable for computer labs and projects. The accompanyingweb site (www.wiley.com/go/bhatti) contains all text examples, MATLAB andMathematica functions, and ANSYS and ABAQUS files in electronic form. Tokeep the printed book to a reasonable length most examples skip some compu-tations. The web site contains full computational details of-these examples. Alsothe book generally alternates between showing examples done with Mathematicaand MATLAB. The web site contains implementations of all examples in bothMathematica and MATLAB.

PREFACE

tVPICAL COURSES

The book can be used to develop a number of courses suitable for different audiences.

First Finite Element Course for Engineering Students About 32 hours of lec-tures and 12 hours of labs (selected materials from indicated chapters):

Chapter l: Finite element procedure, discretization, element equations, assembly,boundary conditions, solution of primary unknowns and element quantities, reactions,solution validity (4 hr)Chapter 2: Weak form for approximate solution of differential equations, Galerkinmethod, approximate solutions using Rayleigh-Ritz method, comparison of Galerkinand Rayleigh-Ritz methods, Lagrange and Hermite interpolation, axial deformationelement using Rayleigh-Ritz and Galerkin methods (6 hr)Chapter 3: ID BVP, FEA solution ofBVP, ID BVP applications (3 hr)Chapter 4: Finite element for beam bending, beam applications, structural frames (3 hr)Chapter 5: Finite elements for 2D and 3D problems, linear triangular element forsecond-order 2D BVP, 2D fluid flow and torsion problems (4 hr)Chapter 6: 2D Lagrange and serendipity shape functions, mapped elements, evaluationof area integrals for 2D mapped elements, evaluation of line integrals for 2D mappedelements (4 hr)Chapter 7: Stresses and strains in solids, finite element analysis of elastic solids, CSTand isoparametric elements for plane elasticity (4 hr)Chapter 8: Transient problems (2 hr)Review, exams (2 hr)

About 12 hours of labs (some sections from the indicated chapters supplemented by docu-mentation of the chosen commercial software):

Appendix: Introduction to Mathematica and/or MATLAB (2 hr)Chapters 1 and 4: Software documentation, basic finite element procedure using com-mercial software, truss and frame problems (2 hr)Chapters 1 and 5: Software documentation, 2D mesh generation, heat flow problems(2 hr)Chapters 1 and 7: 2D, axisymmetric, and 3D stress analysis problems (2 hr)Chapter 8: Transient problems (2 hr)Software documentation: Constraints, design optimization (2 hr)

First Finite Element Course for Students Not Interested in Structural Appli-cations Skip Chapters 4 and 7. Spend more time on applications in Chapters 5 and 6.Introduce Chapter 9: p-Formulation. In the labs replace truss, frame, and stress analysisproblems with appropriate applications.

Finite Element Course for Practicing Engineers From the current book: Chapters1, 2, 6, and 7. From the companion advanced book: Chapters 1, 2, and 5 and selectedmaterial from Chapters 6, 7, and 8.

xix

xx PREFACE

Finite Element Modeling and Applications For a short course on finite elementmodeling or self-study, it is suggested to cover the first chapter in detail and then moveon to Appendix A for specific examples of using commercial finite element packages forsolution of practical problems.

ACKNOWLEDGMENTS

Most of the material presented in the book has become part of the standard finite elementliterature, and hence it is difficult to acknowledge contributions of specific individuals. Iam indebted to the pioneers in the field and the authors of all existing books and journalpapers on the subject. I have obviously benefited from their contributions and have used agood number of them in my over 20 years of teaching the subject.

I wrote the first draft of the book in early 1990. However, the printed version has prac-tically nothing in common with that first draft. Primarily as a result of questions from mystudents, I have had to make extensive revisions almost every year. Over the last coupleof years the process began to show signs of convergence andthe result is what you seenow. Thus I would like to acknowledge all direct and indirect contributions of my formerstudents. Their questions hopefully led me to explain things in ways that make sense tomost readers. (A note to future students and readers: Please keep the questions coming.)

I want to thank my former graduate student Ryan Vignes, who read through severaldrafts of the book and provided valuable feedback. Professors Jia Liu and Xiao Shaopingused early versions of the book when they taught finite elements. Their suggestions havehelped a great deat in improving the book. My colleagues Professors Ray P.S. Han, HosinDavid Lee, and Ralph Stephens have helped by sharing their teaching philosophy and bykeeping me in shape through heated games of badminton and tennis.

Finally, I would like to acknowledge the editorial staff of John Wiley for doing a greatjob in the production of the book. 1'/am especially indebted to Jim Harper, who, fromour first meeting in Seattle in 2003, has been in constant communication and has kept theprocess going smoothly. Contributions of senior production editor Bob Hilbert and editorialassistant Naomi Rothwell are gratefully acknowledged.

CHAPTER ONE5

FINITE ELEMENT METHOD:THE BIG PICTURE

Application of physical principles, such as mass balance, energy conservation, and equi-librium, naturally leads many engineering analysis situations into differential equations.Methods have been developed for obtaining exact solutions for various classes of differ-ential equations. However, these methods do not apply to many practical problems be-cause either their governing differential equations do not fall into these classes or theyinvolve complex geometries. Finding analytical solutions that also satisfy boundary condi-tions specified over arbitrary two- and three-dimensional regions becomes a very difficulttask. Numerical methods are therefore widely used for solution of practical problems in allbranches of engineering.

The finite element method is one of the numerical methods for obtaining approximatesolution of ordinary and partial differential equations. It is especially powerful when deal-ing with boundary conditions defined over complex geometries that are common in practi-cal applications. Other numerical methods such as finite difference and boundary elementmethods may be competitive or even superior to the finite element method for certainclasses of problems. However, because of its versatility in handling arbitrary domains andavailability of sophisticated commercial finite element software, over the last few decades,the finite element method has become the preferred method for solution of many practi-cal problems. Only the finite element method is considered in detail in this book. Readersinterested in other methods should consult appropriate references, Books by Zienkiewiczand Morgan [45], Celia and Gray [32], and Lapidus and Pinder [37] are particularly usefulfor those interested in a comparison of different methods.

The application of the finite element method to a given problem involves the followingsix steps:

2 FINITEELEMENTMETHOD:THE BIG PICTURE

1. Development of element equations2. Discretization of solution domain into a finite element mesh3. Assembly of element equations4. Introduction of boundary conditions5. Solution for nodal unknowns6. Computation of solution and related quantities over each element

The key idea of the finite element method is to discretize the solution domain into anumber of simpler domains called elements. An approximate solution is assumed overan element in terms of solutions at selected points called nodes. To give a clear idea ofthe overall finite element solution process, the finite element equations for a few simpleelements are presented in Section 1.1. Obviously at this stage it is not possible to givederivations of these equations. The derivations must wait until later chapters after we havedeveloped enough theoretical background. Few general remarks on discretization are alsomade in Section 1.1. More specific comments on modeling are presented in later chap-ters when discussing various applications. Important steps of assembly, handling boundaryconditions, and solutions for nodal unknowns and element quantities remain essentiallyunchanged for any finite element analysis. Thus these procedures are explained in detail inSections 1.2, 1.3, and 104. These steps are fairly mechanical in nature and do not requirecomplex theoretical development. They are, however, central to actually obtaining a finiteelement solution for a given problem. Therefore, it is important to fully master these stepsbefore proceeding to the remaining chapters in the book.

The finite element process results in a large system of equations that must be solved fordetermining nodal unknowns. Several methods are available for efficient solution of theselarge and relatively sparse systems of equations. A brief introduction to two commonlyemployed methods is given in Section 1.5. In some finite element modeling situations itbecomes necessary to introduce constraints in the finite element equations. Section 1.6presents examples of few such situations and discusses two different methods for handlingthese so-called multipoint constraints. A brief section on appropriate use of units in nu-merical calculations concludes this chapter.

1.1 DISCRETIZATION AND ELEMENT EQUATIONS

Each analysis situation that is described in terms of one or more differential equationsrequires an appropriate set of element equations. Even for the same system of governingequations, several elements with different shapes and characteristics may be available. Itis crucial to choose an appropriate element type for the application being considered. Aproper choice requires knowledge of all details of element formulation and a thoroughunderstanding of approximations introduced during its development.

A key step in the derivation of element equations is an assumption regarding the solutionof the goveming differential equation over an element. Several practical elements are avail-able that assume a simple linear solution. Other elements use more sophisticated functionsto describe solution over elements. The assumed element solutions are written in terms ofunknown solutions at selected points called nodes. The unknown solutions at the nodes are

DISCRETIZATION AND ELEMENT EQUATIONS

generally referred to as the nodal degrees offreedom, a terminology that dates back to theearly development of the method by structural engineers. The appropriate choice of nodaldegrees of freedom depends on the governing differential equation and will be discussedin the following chapters.

The geometry of an element depends on the type of the governing differential equation.For problems defined by one-dimensional ordinary differential equations, the elements arestraight or curved line elements. For problems governed by two-dimensional partial differ-ential equations the elements are usually of triangular or quadrilateral shape. The elementsides may be straight or curved. Elements with curved sides are useful for accurately mod-eling complex geometries common in applications such as shell structures and automobilebodies. Three-dimensional problems require tetrahedral or solid brick-shaped elements.Typical element shapes for one-, two-, andthree-dimensional (lD, 2D, and 3D) problemsare shown in Figure 1.1. The nodes on the elements are shown as dark circles.

Element equations express a relationship between the physical parameters in the gov-erning differential equations and the nodal degrees of freedom. Since the number of equa-tions for some of the elements can be very large, the element equations are almost alwayswritten using a matrix notation. The computations are organized in two phases. In the firstphase (the element derivation phase), the element matrices are developed for a typical ele-ment that is representative of all elements in the problem. Computations are performed ina symbolic form without using actual numerical values for a specific element. The goal isto develop general formulas for element matrices that can later be used for solution of anynumerical problem belonging to that class. In the second phase, the general formulas areused to write specific numerical matrices for each element.

One of the main reasons for the popularity of the finite element method is the wideavailability of general-purpose finite element analysis software. This software developmentis possible because general element equations can be programmed in such a way that,given nodal coordinates and other physical parameters for an element, the program returnsnumerical equations for that element. Commercial finite element programs contain a largelibrary of elements suitable for solution of a wide variety of practical problems.

3

ID Elements

2D Elements

3D Elements

Figure 1.1. Typical finite element shapes


To give a clear picture of the overall finite element solution procedure, the general fi-nite element equations for few commonly used elements are given below. The detailedderivations of these equations are presented in later chapters.

1.1.1 Plane Truss Element

Many structural systems used in practice consist of long slender shapes of various crosssections. Systems in which the shapes are arranged so that each member primarily resistsaxial forces are usually known as trusses. Common examples are roof trusses, bridge sup-ports, crane booms, and antenna towers. Figure 1.2 shows a transmission tower that canbe modeled effectively as a plane truss. For modeling purposes all members are consid-ered pin jointed. The loads are applied at the joints. The analysis problem is to find jointdisplacements, axial forces, and axial stresses in different members of the truss."

Clearly the basic element to analyze any plane truss structure is a two-node straight-line element oriented arbitrarily in a two-dimensional x-y plane, as shown in the Figure1.3. The element end nodal coordinates are indicated by (Xl' YI) and (x2' Y2). The elementaxis s runs from the first node of the element to the second node. The angle a defines theorientation of the element with respect to a global x-y coordinate system. Each node hastwo displacement degrees of freedom, u indicating displacement in the X direction and vindicating displacement in the y direction. The element can be subjected to loads only atits ends.

Using these elements, the finite element model of the transmission tower is as shownin Figure 1.4. The model consists of 16 nodes and 29 plane truss elements. The elementnumbers and node numbers are assigned arbitrarily for identification purposes.

600570540

480

42010001b

10001b300

180

o

300 180 96 o 6096 180 300in

Figure 1.2. Transmission tower

yDISCRETIZATION AND ELEMENT EQUATIONS 5

Nodal dof End loads

Figure 1.3. Plane truss element

x

Element numbers

Figure 1.4. Planetruss element model of the transmission tower

Using procedures discussed in later chapters, it can be shown that the finite elementequations for a plane truss-dement are as follows:

IslnsIn;

-lslns-In;

-1;-Is Ins

z2sIslns

where E = elastic modulus of the material (Young's modulus), A = area of cross section ofthe element, L =length of the element, and Is. Ins are the direction cosines of the elementaxis (line from element node 1 to 2). Here, Is is the cosine of angle a between the elementaxis and the x axis (measured 'counterclockwise) and Ins is the cosine of angle between theelement axis and the y axis. In terms of element nodal coordinates,


In the element equations the left-hand-side coefficient matrix is usually called the stiffnessmatrix and the right-hand-side vector as the nodal load vector. Note that once the elementend coordinates, material property, cross-sectional area, and element loading are specified,the only unknowns in the element equations are the nodal displacements.

It is important to recognize that the element equations refer to an isolated element, Wecannot solve for the nodal degrees of freedom for the entire structure by simply solvingthe equations for one element. We must consider contributions of all elements, loads, andsupport conditions before solving for the nodal unknowns. These procedures are discussedin detail in later sections of this chapter.

Example 1.1 Write finiteelement equations for element number 14 in the finite elementmodel of the transmission tower shown in Figure 1.4. The tower is made of steel (!i..=-29 x 106Ib/in2) angle sections. The area of cross section of element 14 is 1.73 in2 .

The element is connected between nodes 7 and 9. We can choosee~as th~ firstnode of the element. Choosing node 7 as the first node establishes the element s axis asgoing from node 7 toward 9. The origin of the global x-y coordinate system can be placedat any convenient location. Choosing the centerline of the tower as the origin, the nodalcoordinates for the element 14 are as follows:

First node (node 7) = (-60, 420) in;Second node (node 9) = (-180, 480) in;

XI =-60;x2 =-180;

YI = 420Y2 = 480

Using these coordinates, the element length and the direction cosines can easily be calcu-lated as follows:

Element length:

Element direction cosines:-------

L =~(X2 -xll + (Y2 - YI)2 =60-{5 in;' _ x2 - XI _ 2. 112 =Y2 - YI = _l_Is - -L- - - -{5' s L -{5

From the given material and section properties,

E =29000000Ib/in2; E: =373945. lb/inUsing these values, the element stiffness matrix (the left-hand side of the element equa-tions) can easily be written as follows:

[

z2k = EA Isl~s

L -I;-lm,

Isms112;

-Isms-112;

-I;-Isms

12sIsms

-Isms] [299156.-m; _ -149578.Isms - "':299156.112; 149578.

-149578.74789.

149578.-74789.

-299156.149578.299156.'

-149578.

149578.]-74789.

-149578.74789.

The right-hand-side vector of element equations represents applied loads at the elementends. There are no loads applied at node 7. The applied load of 1000 lb at node 9 is sharedby elements 14, 16,23, and 24. The portion taken by element 14 cannot be determined


without detailed analysis of the tower, which is exactly what we are attempting to do in thefirst place. Fortunately, to proceed with the analysis, it is not necessary to know the portionof the load resisted by different elements meeting at a common node. As will become clearin the next section, in which we consider the assembly of element equations, our goal is togenerate a global system of equations applicable to the entire structure. As far as the entirestructure is concerned, node 9 has an applied load of 1000 lb in the -y direction. Thus, it isimmaterial how we assign nodal loads to the elements as long as the total load at the nodeis equal to the applied load. Keeping this in mind, when computing element equations, wecan simply ignore concentrated loads applied at the nodes and apply them directly to theglobal equations at the start of the assembly process. Details of this process are presentedin a following section.

~Assuming nod'!JJQ.ads are tQ.~dedgU:~:Jly~t.Q..!h.g12Qe1.~q1!,gJjQ!1~~,!h~.1injj:e el~entequ~!~ons!2E..c::!.~ment 14 ar:...f9JIRWJ/;,.

7

[

299156. -149578. -299156.-149578. 74789. 149578.-299156. 149578. 299156.

149578. -74789. -149578.

149578.] [U7] [0]-74789. v7 _ 0

-149578. u9 ' - 0 '74789. v9 0

~ MathematicafMATLAB Implementation :n..l on the Book Web Site:Plane truss element equations

1.1.2 Triangular Element for Two-Dimensional Heat Flow

Consider the problem of finding steady-state temperature distribution in long chimneylikestructures. Assuming no temperature gradient in the longitudinal direction, we can talce aunit slice of such a structure and model it as a two-dimensional problem to determine thetemperature T(x, y). Using conservation of energy on a differential volume, the followinggoverning differential equation can easily be established.:

_. a (aT) a ( aT)ax kx ax + ay ky ay + Q=0

where kx and kyare thermal conductivities in the x and y directions and Q(x, y) is specifiedheat generation per unit volume. Typical units for k are W/m- C or Btu/hr ft OF and thosefor Qare W1m3 or Btu/hr . ft3. The possible boundaryconditions are as follows:

(i) Known temperature along a boundary:

T =To specified

(ii) Specified heat flux along a boundary:

8 FINITEELEMENTMETHOD: THE BIG PICTURE

y (m)

0.03

0.015 qo

oTo

o

n

0.03 0.06x (m)

n

Figure 1.5. Heat flow through an L-shaped solid: solution domain and unit normals

where nx and ny are the x and Y, components of the outer unit normal vector to theboundary (see Figure 1.5 for an example):

Inl =~n; + n; =1On an insulated boundary or across a line of symmetry there is no heat flow andthus qo = O. The sign convention for heat flow is that heat flowing into a body ispositive and that flowing out of the body is negative.

(iii) Heat loss due to convection along a boundary:

st (aT aT)-k an == - kx ax nx + ky ay ny =h(T - Too)

where h is the convection coefficient, T is the unknown temperature at the bound-ary, and Too is the known temperature of the surrounding fluid. Typical units for hare W/m2 C and Btulhr ft2 "P,

As a specific example, consider two-dimensional heat flow over an L-shaped bodyshown in Figure 1.5. The thermal conductivity in both directions is the same, kx = ky =

DISCRETIZATION AND ELEMENTEQUATIONS

45 Wlm . C. The bottom is maintained at a temperature of To = 110C. Convection heatloss takes place on the top where the ambient air temperature is 20C and the convectionheat transfer coefficient is h = 55 W/m2 C. The right side is insulated. The left side issubjected to heat flux at a uniform rate of qo = 8000 W/m2. Heat is generated in the bodyat a rate of Q =5 X 106 W1m3 .

Substituting the given data into the governing differential equation and the boundaryconditions, we see that the temperature distribution over this body must satisfy the follow-ing conditions:

9

Over the entire L-shaped region

On the left side (lix = -1, ny = 0)

On the bottom of the region

On the right side (nx =1,ny =0)

On the horizontal portions ofthe top side (nx =0, ny =1)

On the vertical portion of thetop side (nx =1, l1y =0)

45 (a2 ; + a2 ; ) + 5 X 106 =0ax ay

_ (45 aT (-1) = 8000 => aT = 8000 along x =0ax ax 45

T =110 along y =0

aT =0 along x =0.06ax

( aT ) ei 55- 45 ay (1) =55(T - 20) => ay =- 45 (T - 20)

( aT ) et 55- 45 ax (1) =55(T - 20) => ax =- 45 (T - 20)

Clearly there is little hope of finding a simple function T(x, y) that satisfies all these re-quirements. We must resort to various numerical techniques. In the finite element method,the domain is discretized into a collection of elements, each one of them being of a simplegeometry, such as a triangle, a rectangle, or a quadrilateral.

A triangular element for solution of steady-state heat flow over two-dimensional bod-ies is shown in Figure 1.6. The element can be used for finding temperature distribution

y

------x

Figure 1.6. Triangular element for heat flow

10 FINITE ELEMENTMETHOD: THE BIG PICTURE

over any two-dimensional body subjected to conduction and convection. The element isdefined by three nodes with nodal coordinates indicated by (xI' YI)' (Xz' Yz), and (x3'Y3)'The starting node of the triangle is arbitrary, but we must move counterclockwise aroundthe triangle to define the other two nodes. The nodal degrees of freedom are the unknowntemperatures at each node Tp Tz' and 13.

For the truss model considered in the previous section, the structure was discrete to startwith, and thus there was only one possibility for a finite element model. This is not the casefor the two-dimensional regions. There are many possibilities in which a two-dimensionaldomain can be discretized using triangular elements. One must decide on the number ofelements and their arrangement. In general, the accuracy of the solution improves as thenumber of elements is increased. The computational effort, however, increases rapidly aswell. Concentrating more elements in regions where rapid changes in solution are expectedproduces finite element discretizations that give excellent results with reasonable com-putational effort. Some general remarks on constructing good finite element meshes arepresented in a following section. For the L-shaped solid a very coarse finite element dis-cretization is as shown in Figure 1.7 for illustration. To get results that are meaningful froman actual design point of view, a much finer mesh, one with perhaps 100 to 200 elements,would be required.

The finite element equations for a triangular element for two-dimensional steady-stateheat flow are derived in Chapter 5. The equations are based on the assumption of linear

y Element numbers0.03

0.0250.02

0.0150.01

0.0050

x0 0.01 .0.02 0.03 0.04 0.05 0.06

y Node numbers0.03

0.0250.02

0.015 210.01

0.005 200 1 6 11 16 19

x0 0.01 0.02 0.03 0.04 0.05 0.06

Figure 1.7. Triangular element mesh for heat flow through an L-shaped solid


temperature distribution over the element. In terms of nodal temperatures, the temperaturedistribution over a typical element is written as follows:

where

The quantities Ni , i = 1, 2, 3, are known as interpolation or shape functions. The.superscriptT over N indicates matrix transpose. The vector d is the vector of nodal unknowns. Theterms b l , cI ' ... depend on element coordinates and are defined as follows:

11

CI=X3-X2;

II =XiY3 - x3Y2 ;C2 =xI -X3;

12 =X3YI -XIY3;

b3 =YI - Y2C3 = X2 - X j

13 = X IY2 - X2YIThe area of the triangle A can be computed from the following equation:

where det indicates determinant of the matrix.

A note on the notation employed for vectors and matrices in this book is in order here.As an easy-to-remember convention, all vectors are considered column vectors and aredenoted by boldface italic characters. When an expression needs a row vector, a super-script T is used to indicate that it is the transpose of a column vector. Matrices are alsodenoted by boldface italic characters. The numbers of rows and columns in a matrixshould be carefully noted in the initial definition. Remember that, for matrix multipli-cation to make sense, the number of columns in the first matrix should be equal to thenumber of rows in the second matrix. Since large column vectors occupy lot of space ona page, occasionally vector elements may be displayed in arow to save space. However,for matrix operations, they are still treated as column vectors.

As shown in Chapter 5, the finite element equations for this element are as follows:

12 FINITE ELEMENTMETHOD:THE BIG PICTURE

where kx =heat conduction coefficient in the x direction, ky =heat conduction coefficientin the y direction, and Q = heat generated per unit volume over the element. The matrixJe" and the vector r" take into account any specified heat loss due to convection along oneor more sides of the element. If the convection heat loss is specified along side 1 of theelement, then we have

Convection along side 1: [2 1 0)k = hL12 1 2 0 ."6 '000 - hTooL12 [ 11)r" - --2- .owhere h = convection heat flow coefficient, Too = temperature of the air or other fluidsurrounding the body, and L12 = length of side 1 of the element. For convection heat flowalong sides 2 or 3, the matrices are as follows:

J. =hL23[~ 0 n r" =hT~~3 0)Convection along side 2: e" ' 6 20 1, ~hI,,[~ 0 ~); - hTooL31[~)Convection along side 3: e" 6 0 r,,---?-I' 1 0 - 1

where L23 and L31 are lengths of sides 2 and 3 of the element. The vector rq is due topossible heat flux q applied along one or more sides of the element:

Applied flux along side 1:



r, ~ q~" UJr,~ qi'[!j

r,~ qi' mIf convection or heat flux is specified on more than one side of an element, appropriatematrices are written for each side and then added together. For an insulated boundary q = 0,and hence insulated boundaries do not contribute anything to the element equations.


As mentioned in the previous section, we cannot solve for nodal temperatures by simplysolving the equations for one eiement. We must consider contributions of all elements andspecified boundary conditions before solving for the nodal unknowns. These proceduresare discussed in detail in later sections in this chapter.

Example 1.2 Write finite element equations for element number 20 in the finite elementmodel of the heat flow through the L-shaped solid shown in Figure 1.7.

The element is situated between nodes 4, 10, and 5. We can choose any of the threenodes as the first node of the element and define the other two by moving counterclockwisearound the element. Choosing node 4 as the first node establishes line 4-10 as the first sideof the element, line 10-5 as the second side, and line 5-4 as the third side. The origin ofthe global x-y coordinate system can be placed at any convenient location. Choosing node1 as the origin, the coordinates of the element end nodes are as follows:

13

Node 1 (global node 4) = (O., 0.0225) m;Node 2 (global node 10) = (O.015, 0.03) m;

Node 3 (global node 5) = (O., 0.03) m;

XI =0.;x2 =0.015;x3 =0.;

YI =0.0225Y2 = 0.03Y3 =0.03

Using these coordinates, the constants bi' ci, and I, and the element area can easily becomputed as follows:

b, =0.;cI = -0.015;II =0.00045;

b2 =0.0075;c2 =0.;12 =0.;

b3 =-0.0075c3 =0.01513 =-0.0003375

Element Area.= 0.00005625

From the given data the thermal conductivities and heat generated over the solid are asfollows:

Q= 5000000

Substituting these numerical values into the element equation expressions, the matrices lckand rQ can easily be written as follows:

There is an applied heat flux on side 3 (line 5-4) of the element. The length of this side ofthe element is 0.0075 m and Withq = 8000 (a positive value since heat is flowing into thebody) the rq vector for the element is as follows:

Heat flux on side 3 with coordinates ({O., 0.0225) (O., 0.03)),L =0.0075; q =8000

(45.

lck = O.-45. (

93.75]"a = 93.75

93.75


[30.)rq = a ,30.The side 2 of the element is subjected to heat loss by convection. The convection termgenerates a matrix khand a vector rho Substituting the numerical values into the formulas,these contributions are as follows:

Convection on side 2 with coordinates ((0.015, 0.03) (0.,0.03}),L =0.015; h =55; Too =20

kh=[~ ~.275 ~.1375);rh=[8.~5)'a 0.1375 0.275 8.25

Adding matrices kk and k h and vectors rQ , rq , and rh , the complete element equations areas follows:

[45.

O.-45.

O. -45. )[T4 ) [123.75)11.525 -11.1125 TIO = 102.

-11.1125 56.525 Ts 132. MathematicalMATLAB Implementation 1.2 on the Book Web Site:Triangular element for heat flow

1.1.3 General Remarks on Finite Element DiscretizationThe accuracy of a finite element analysis depends on the number of elements used inthe model and the arrangement of elements. In general, the accuracy of the solution im-proves as the number of elements is' increased. The computational effort, however, in-creases rapidly as well. Concentrating more elements in regions where rapid changes insolution are expected produces finite element discretizations that give excellent resultswith reasonable computational effort. Some general remarks on constructing good finiteelement meshes follow.

1. Physical Geometry of the Domain. Enough elements must be used to model thephysical domain as accurately as possible. For example, when a curved domain is to bediscretized by using elements with straight edges, one must use a reasonably large numberof elements; otherwise there will be a large discrepancy in the actual geometry and the dis-cretized geometry used in the model. Figure 1.8 illustrates error in the approximation of acurved boundary for a two-dimensional domain discretized using triangular elements. Us-ing more elements along the boundary will obviously reduce this discrepancy. If available,a better option is to use elements that allow curved sides.

2. Desired Accuracy. Generally, using more elements produces more accurate results.3. Element Formulation. Some element formulations produce more accurate results

than others, and thus formulation employed in a particular element influences the num-ber of elements needed in the model for a desired accuracy.

DISCRETIZATION AND ELEMENTEQUATIONS

Actual boundary

Figure 1.8. Discrepancy in the actual physical boundary and the triangular element model geometry

15

Valid meshx

Invalid mesh

Figure 1.9. Valid and invalid mesh for four-node elements

4. Special Solution Characteristics. Regions over which the solution changes rapidlygenerally require a large number of elements to accurately capture high solution gradients.A good modeling practice is to start with a relatively coarse mesh to get an idea of thesolution and then proceed with more refined models. The results from the coarse model areused to guide the mesh refinement process.

5. Available Computational Resources. Models with more elements require more com-putational resources in terms of memory, disk space, and computer processor.

6. Element Interfaces. J;:~ements are joined together at nodes (typically shown as darkcircles on the finite element meshes). The solutions at these nodes are the primary variablesin the finite element procedure. For reasons that will become clear after studying the nextfew chapters, it is important to create meshes in which the adjacent elements are alwaysconnected from comer to comer. Figure 1.9 shows an example of a valid and an invalidmesh when empioying four-node quadrilateral elements. The reason why the three-elementmesh on the right is invalid is because node 4 that forms a comer of elements 2 and 3 isnot attached to one of the four comers of element 1.

7. Symmetry. For many practical problems, solution domains and boundary conditionsare symmetric, and hence one can expect symmetry in the solution as well. It is impor-tant to recognize such symmetry and to model only the symmetric portion of the solutiondomain that gives information for the entire model. One common situation is illustratedin the modeling of a notched-beam problem in the following section. Besides the obviousadvantage of reducing the model size, by taking advantage of symmetry, one is guaran-teed to obtain a symmetric solution for the problem. Due to the numerical nature of the

16 FINITEELEMENT METHOD: THE BIG PICTURE

Figure 1.10. Unsymmetrical finite element mesh for a symmetric notched beam

501b/in2

Figure 1.11. Notched beam

finite element method and the unique characteristics of elements employed, modeling theentire symmetric region may in fact produce results that are not symmetric. As a simpleillustration, consider the triangular element mesh shown in Figure 1.10 that models theentire notched beam of Figure 1.11. The actual solution should be symmetric with respectto the centerline of the beam. However, the computed finite element solution will not beentirely symmetric because the arrangement of the triangular elements in the model is notsymmetric with respect to the midplane.

A general rule of thumb to follow in a finite element analysis is to start with a fairlycoarse mesh. The number and arrangement of elements should be just enough to get a goodapproximation of the geometry, loading, and other physical characteristics of the problem.From the results of this coarse model, select regions in which the solution is changingrapidly for further refinement. To see solution convergence, select one or more criticalpoints in the model and monitor the solution at these points as the number of elements(or the total number of degrees of freedom) in the model is increased. Initially, when themeshes are relatively coarse, there should be significant change in the solution at thesepoints from one mesh to the other. The solution should begin to stabilize after the numberof elements used in the model has reached a reasonable level.

1.1.4 Triangular Element for Two-Dimensional Stress Analysis

As a final example of the element equations, consider the problem of finding stresses in thenotched beam of rectangular cross section shown in Figure 1.11. The beam is 4 in thick inthe direction perpendicular to the plane of paper and is made of concrete with modulus ofelasticity E =3 x 1061b/in2 and Poisson's ratio v =0.2.

Since the beam thickness is small as compared to the other dimensions, it is reasonableto consider the analysis as a plane stress situation in which the stress changes in the thick-ness direction are ignored. Furthermore, we recognize that the loading and the geometryare symmetric with respect to the plane passing through the midspan. Thus the displace-ments must be symmetric and the points on the plane passing through the midspan do notexperience any displacement in the horizontal direction. Taking advantage of these simpli-

DISCRETIZATION AND ELEMENTEQUATIONS 17

x

xso

so

40

4030

3020

20

Element numbers

10

10

y121086420

0

y121086420

0

Figure 1.12. Finite element model of the notched beam

fications, we need to construct a two-dimensional plane stress finite element model of onlyhalf of the beam. As an illustration, a coarse finite element model of the right half of thebeam using triangular elements is shown in Figure 1.12. All nodes on the right end are fixedagainst displacement because of the given boundary condition. The left end of the modelis on the symmetry plane, and thus nodes on the left end cannot displace in the horizontaldirection. Once again, in an actual stress analysis a much finer finite element mesh willbe needed to get accurate values of stresses and displacements. Even in the coarse modelnotice that relatively small elements are employed in the notched region where high stressgradients are expected.

A typical triangular element for the solution of the two-dimensional stress analysis prob-lem is shown in Figure 1.13. The element is defined by three nodes with nodal coordinatesindicated by (XI' YI)' (x2' Y2)' and (x3' Y3)' The starting node of the triangleis arbitrary, butwe must move counterclockwise around the triangle to define the other two nodes. Thenodal degrees of freedom are the displacements in the X and Y directions, indicated byu and v. On one or more sides of the element, uniformly distributed load in the normaldirection qn and that in the tangential direction qr can be specified.

The element is based on the assumption of linear displacements over the element. Interms of nodal degrees of freedom, the displacements over an element can be written asfollows:

u(x, y) = NI ul + N2u2 + N3u3vex, y) =NI VI + N2v2 + N3v3

ul

~JVI

( u(x, y) ) =(NI 0 N2 0 N3 u2 =NTdvex, y) 0 NI 0 N2 0 v2

u3v3


y

-------------x

Figure 1.13. Plane stress triangular element

where the Ni , i = 1, 2, 3, are the same linear triangle interpolation functions as those usedfor the heat flow element:

Cj=X3-XZ;

II = XZY3 - X3Yz;Cz =XI -X3;

I z =X3Yl - XlY3;C3 = Xz -Xl

I 3 =XlYz - XZYI

The element area A can be computed as follows:

Using these assumed displacements, the element strains can be written as follows:

o bz 0 b3o Cz 0

Cz bz c3

where Ex and

DISCRETIZATION AND ELEMENT EQl'iATIONS

where C is the appropriate constitutive matrix. For homogeneous, isotropic, and elasticmaterials under plane stress conditions,

19

(1 v

C- E v 1-1- V200 ,~,]

where E = Young's modulus and v = Poisson's ratio.The finite element 'equations for this element are derived in Chapter 7 and are as follows:

kd=rq

where k is the element stiffness matrix given by

where h = element thickness. The vector rq represents equivalent nodal load due to anyapplied distributed loads along one or more sides of an element. For a uniformly distributedload on side 1 of the element with components q" and q/ in the normal and tangentialdirections of the surface, the equivalent load vector is as follows:

T hLI2rq =T(l1xqll - l1yq/ l1yq" + nxq/ I1xq" - nyq/ l1yq" + I1xq/ 0 0)

where LI2 =~ (x2 - xJ + (Y2 - YI)2 is the length of side 1 and I1x and l1y are the compo-nents of the unit normal to side. Note that rq is a 6 x 1 column vector. It is written as a rowto save space. The components of the unit normal to the side can be computed as follows:

n =-Y2- Y l.x L

12'

x -x11 =__2__1Y L I2

A pressure component is considered positive if it is along the positive direction of thenormal or tangent to the side. As shown in Figure 1.13, while moving counterclockwisearound the element, the positive normal vector points in the outward direction. The positivetangent vector is 900 counterclockwise from the positive normal vector.

For a uniformly distributed load on side 2,

hLrT = ~(O 0 n q - 11 qq 2 X" Y t

where L23 =~ (x3 - x2)2 + (Y3 - Y2)2 is the length of side 2 andn = Y3 -Y2.

x L23 '.;1:"3 -x211 =----Y , L

23.

For a uniformly distributed load on side 3,

T hL31rq =-2-( nxq" - l1yq/ l1yq" + I1xqt 0 0 I1xq" - l1yqt l1yq" + I1xq/ )


n = YI -Y3.x ~l'

If loads are specified on more than one side of an element, appropriate vectors are.writtenfor each side and then added together. As mentioned with the plane truss element, any con-centrated applied load at a node is added directly to the global equations during assembly.This will be illustrated in a later section. Furthermore, we cannot solve for nodal displace-ments by simply solving the equations for one element. We must consider contributionsof all elements and specified boundary conditions before solving for the nodal unknowns.These procedures are discussed in detail in later sections in this chapter.

Example 1.3 Write finite element equations for element number 2 in the finite elementmodel of the notched beam shown in Figure 1.12.

The element is connected between nodes 4, 7, and 11. We can choose any of the threenodes as the first node of the element and define the other two by moving counterclockwisearound the element. Choosing node 4 as the first node establishes line 4-7 as the first sideof the element, line 7-11 as the second side, and line 11-4 as the third side. The origin ofthe global x-y coordinate system can be placed at any convenient location. Choosing theorigin as shown in Figure 1.12, the coordinates of the element end nodes are as follows:

Node 1 (global node 4) =(0., 12.)in;Node 2 (global node 7) = (5., 9.66667)in; .

Node 3 (global node 11) =(6., 12.) in;,I

Xl =0.;x2 =5.;x3 =6.;

YI =12.Y2 = 9.66667Y3 = 12.

Using these coordinates, the constants in the B matrix can easily be computed as follows:

b l =-2.33333;ci =1.;

Substituting the given data, we have

b2 =0.;c2 = -6.;

b3 = 2.33333

ASSEMBLY OF ELEMENT EQUATIONS

Thus the element stiffness matrix. is

21

2.60913-0.625-1.07143

1.25-1.5377

~0.625

-0.6251.418652.5

-2.67857-1.875

1.25992

-1.071432.56.42857o

-5.35714-2.5

1.25-2.67857o

16.0714-1.25-13.3929

-1.5377, -1.875-5.35714-1.25

6.894843.125

-0.6251.25992

-2.5-13.3929

3.12512.1329

There is an applied load in the negative outer normal direction on side 3 (nodes (11,4}) ofthe element. The equivalent nodal load vector rq for the element is computed as follows:

Specified load components: qn =-50; qt =0End nodal coordinates: (6., l2.) (0., l2.}), giving side length L = 6Components of unit normal to the side: Hx = 0.; Hy = 1.Using these values, we get r~ =(0. -600. 0 0 O. -600.)

Thus the complete element equations are as follows:

2.60913 -0.625 -1.07143 1.25 -1.5377 -0.625 u4 O.-0.625 1.41865 2.5 -2.67857 -1.875 1.25992 v4 -600.

106 -1.07143 2.5 6.42857 0 -5.35714-2.5 u7 O.

1.25 -2.67857 0 16.0714 -1.25 -13.3929 v7 O.-1.5377 -1.875 -5.35714 -1.25 6.89484 3.125 ull O.-0.625 1.25992 -2.5 -13.3929 3.125 12.1329 vll -600.

Mathematica/MATLARImplementation :R..3 on the Book Web Site:Triangular element for plane stress

1.2 ASSEMBLY OF ELEMENT EQUATIONS

The finite element discretization divides a solution domain or structure into simple ele-ments. For each element the finite element equations can be written by substituting nu-merical values into the formulas for appropriate element type. In the assembly process, wemust put the split-up solution domain back together before proceeding with the solution.The key concept in the assembly process is that at a node common between several ele-mentsthe nodal solution is the same for all elements sharing this node. Thus contributionsto that degree of freedom from all adjacent elements must be added together.

To illustrate the assembly process, consider the lO-node and 8-element finite elementmesh for the heat flow problem shown in Figure 1.14. The nodal degrees of freedom arethe temperatures at the nodes. Since there are 10 nodes, the global system of equations is10 x 10. Thus we start the assembly process by initializing a 10 x 10 system of equations


[111 201 301)[T!) (11)201 222 232 Tz = 12301 232 333 Ts 13

Now we consider the assembly of element 1 into the global system. This element con-tributes to the nodal degrees of freedom (1,2,5). Since the element involves degrees offreedom Tl' Tz' and Ts, these equations will be added to global equations 1, 2, and 5, re-spectively. Written in expanded form, the first equation of this element is

lIlT! + 20lTz + 30lTs = 11

. Expanding it to include al1lO degrees offreedom in the model, we have

ASSEMBLY OF ELEMENTEQUATIONS 23

In the global system this is the first equation, and therefore the equation can be insertedinto the global system as follows:

111 201 0 0 301 0 0 0 0 0 t; 110 0 0 0 0 0 0 0 0 0 T2 00 0 0 0 0 0 0 0 0 0 T3 00 0 0 0 0 0 0 0 0 0 T4 00 .0 0 0 0 0 0 0 0 0 T5 00 0 0 0 0 0 0 0 0 0 T6 00 0 0 0 0 0 0 0 0 0 T? 00 0 0 0 0 0 0 0 0 0 Ts 00 0 0 0 0 0 0 0 0 0 T9 0O' 0 0 0 0 0 0 0 0 0 TIO 0

The other two equations for the element are expanded in a similar manner.

Element GlobalEquation EquationNumber Number Equation

2 2 201Tj + 222T2 + OT3 + OT4 + 232Ts + OT6 + OT?+ OTs + OT9 + OTIO = 12

3 5 301Tj + 232T2 + OT3 + OT4 + 333Ts + OT6 + OT?+ OTs + OT9 + OTIO = 13

Placing these equations in the second and fifth rows, the global equations after assemblyof element 1 are as follows:

222 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0 0

232_0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 0

The above procedure of reordering and expanding element equations is quite tedious. For-tunately, it is not necessary to formally carry out these steps in detail. The appropriatelocations of the entries in the global equations can be determined simply by taking the listof degrees of freedom to which the element is contributing. This list is called the locationvector. For element 1 the location vector is as follows: .


For assembling into a global vector (right-hand side), the entries in the location vectordirectly indicate the locations where the corresponding element quantity will contribute:

[;1]Locations for element 1 contributions to a global vector:To determine the global locations of the entries in an element matrix (left-hand-side coef-ficient matrix), we simply take all combinations of the indices in the location vector. Forthe first row, the locations have the row index of 1 and column indices are 1,2, and 5. Forthe second row, the row index is 2 and the column indices are 1,2, and 5. For the third row,the row index is 5 and the column indices are 1,2, and 5. Thus the locations in the globalmatrix where the corresponding element quantities are added are as follows:

Locations for element 1 contributions to the global matrix: [[1, 1] [1,2][2, 1] [2,2][5, 1] [5,2]

[1, 5])[2,5][5,5]

This indicates that 111 from the element matrix goes to the location [1, 1] in the globalmatrix, 201 into the [1,2], etc. Clearly, this gives us exactly the same global matrix afterassembly of this element as before.

Each element in a finite element model is processed in exactly the same manner. As afurther illustration, consider assembly of element 2. Assume that the equations for element2 are as follows:

[77 80 90 )[T2 ) [21)80 ;' 88 100 T6 = 2290 100 99 Ts 23

The location vector and the locations to which this element contributes in the global matrixare as follows:

[ 2~ )Element 2 location vector:Locations for element 2 contributions to a global vector:

Locations for element 2 contributions to a global matrix:

[~) /[[2, 2] [2,6] [2,5])

[6,2] [6,6] [6,5][5,2] [5,6] [5,5]

This indicates that 77 from the element matrix is added to the location [2, 2] in the globalmatrix, 80 into the [2,6], etc. Thus the global equations after assembly of this element are

ASSEMBLY OF ELEMENT EQUATIONS 25

as follows.

111 201 0 0 301 0 0 0 0 0 t, 11201 222 + 77 0 0 232+ 90 80 0 0 0 0 Tz 12+ 21

0 0 0 0 0 0 0 0 0 0 T3 00 0 0 0 0 0 0 0 0 0 T4 0

301 232+ 90 0 0 333 + 99 100 0 0 0 0 Ts 13 +230 80 0 0 100 88 0 0 0 0 T6 220 0 0 0 0 0 0 0 0 0 T7 00 0 0 0 0 0 0 0 0 0 Ts 00 0 0 0 0 0 0 0 0 0 T9 00 0 0 0 0 0 0 0 0 0 TIO 0

The equations for the remaining six elements can be assembled in exactly the same manner.The following examples,involving plane truss, heat flow, and plane stress elements, furtherillustrate the assembly procedure. ~

~ MathematicalMATLAB Implementation 1.4 on the Book Web Site:Finite element assembly procedure

Example 1.4 Five-Bar Truss Write element equations and assemble them to formglobal equations for the five-bar plane truss shown in Figure 1.15. The area of cross sectionfor elements 1 and 2 is 40 em", for elements 3 and 4 is 30 crrr', and for element 5 is 20 cnr'.The first four elements are made of a material with E = 200 GPa and the last one withE =70 GPa. The applied load P = 150 kN.

Each node in the model has two- displacement degrees of freedom. They are identified bythe letters u and v with a subscript indicating the corresponding node number and are shownin Figure 1.16, Without considering the specified zero displacements at the supports, themodel has a total of eight degrees of freedom. Thus the global equations will be a systemof eight equations in eight unknowns.

\5 l 30----4-----::;;0

Figure 1.15. Five-bar plane truss

4

3

2

o

o

p

2 3 4

.. \-;;,


Figure 1.16. Five-bar plane truss finite element model

The next step is to get finite element equations for each element in the model. We simplyneed to substitute the appropriate numerical values into the plane truss element equations.The concentrated nodal load is added directly into the global equations at the start of as-sembly. Since the load is acting downward and the displacements are assumed positivealong the positive coordinates, the load at node 2 is (0, -ISO kN). Since the displacementsare usually small, it is convenient to use newton-millimeters. The displacements will come~ITIillimeters and the stresses in megapascals. .

For each element we substitute the numericar-aata into the plane truss stiffness matrixand assemble them into the global equations using the assembly procedure discussed ear-lier. The complete computations are as follows. All numerical values are in newtons andmillimeters.

The specified nodal loads are as follows:

Node dof Value

2 u2 0112 -150000

The global equations at the start of the element assembly process are,I

t

0 0 0 0 0 0 0 0 u j 00 0 0 0 0 0 0 0 v j 00 0 0 0 0 0 0 0 u2 00 0 0 0 0 0 0 0 112 -150000:=:0 0 0 0 0 0 0 0 u3 00 0 0 0 0 0 0 0 113 00 0 0 0 0 0 0 0 u4 00 0 0 0 0 0 0 0 114 0

The equations for element 1 are as follows:E :=: 200000; A :=: 4000Element Node Global Node Number x Y

1 1 0 02 2 1500. 3500.

x j :=: 0; Yj :=: 0; x2 :=: 1500.; Y2 :=: 3500.L :=:~ (x2 - X j)2 + (Y2 - Yj)2 :=: 3807.89

ASSEMBLY OF ELEMENT EQUATIONS

Direction cosines' I = Xz - xI ,;, 0393919' m = Yz - YI =0 919145. s L . 's L .

Substituting into the truss element equations, we get

[

32600.2 76067.2 -32600.2 -76067.2] [Uj '] [0.]76067.2 177490. -76067.2 -177490. vI _ O.

-32600.2 -76067.2 32600.2 76067.2 Uz - O.-76067.2 -177490. 76067.2 177490. v2 O.

The element contributes to [1,2, 3,4) global degrees of freedom:

Locations for element contributions to a global vector: [ ~][

[1, 1] [1,2] [1,3] [1,4]]d 1 b 1 . [2, 1] [2,2] [2, 3] [2,4]an to ago a matnx: [3, 1] [3,2] [3,3] [3,4]

[4,1] [4,2] [4,3] [4,4]Adding element equationsinto appropriate locations, we have

27

32600.276067.2

-32600.2-76067.2

oooo

76067.2177490.-76067.2

-177490.oooo

-32600.2-76067.2

32600.276067.2

oo

_ 0o

-76067.2-177490.

76067.2177490.

oooo

o 0 0 0 ujo 0 0 0 VI0000 u2o 0 0 0 v20000 u3o 0 0 0 v3o 0 0 0 u4o 0 0 0 v4

ooo

-150000.oooo

The equations for element 2 are as follows:E =200000;A =4000Element Node Global Node Number x Y

3500.5000

1 2 1500.2 4 5000

x j = 1500.;YI= 3500.; x2 =5000;Yi =5000

L =~ (x2 - xji + (Y2 - y1i =3807.89Direction cosines: Is = X2 ~ X j =0.919145; ms = Y2 ~ YI. =0.393919Substituting into the truss element equations, we get

[

177490. 76067.276067.2 32600.2

-177490. -76067.2-76067.2 -32600.2

-177490.-76067.2177490.76067.2

-76067.2] [U2] [0.]-32600.2 v2 _ O.

76067.2 u4 - O.32600.2 v4 O.


The element contributes to (3, 4, 7, 8} degrees of freedom:

Locations '0' element contributions to a global vector: [!][[3.3] [3,4] [3,7] [3.81]

. [4,3] [4,4] [4,7] [4,8]and to a global matnx: [7, 3] [7,4] [7,7] [7,8]

[8,3] [8,4] [8,7] [8,8]Adding element equations into appropriate locations, we have

32600.2 76067.2 -32600.2 -76067.2 0 0 0 0 ul 076067.2 177490. -76067.2 -177490. 0 0 0 0 VI 0

-32600.2 -76067.2 210090. 152134. 0 0 -177490. -76067.2 u2 0-76067.2 -177490. 152134. 210090. 0 0 -76067.2 -32600.2 v2 -150000.

0 0 0 0 0 0 0 0 u3 00 0 0 0 0 0 0 0 v3 00 0 -177490. -76067.2 0 0 177490. 76067.2 u4 00 0 -76067.2 -32600.2 0 0 76067.2 32600.2 v4 0

The remaining elements can be processed in exactly the same manner. After assemblingall elements, the global equations for the model are as follows:

32600.2 76067.2 -32600.2 -76067.2 0 0 0 0"I 0

76067.2 297490. -76067.2 -177490. 0 -120000 0 0 VI 0-32600.2 -76067.2 243089. 119136. -32998.3 32998.3 -177490. -76067.2

"2 0-76067.2 -177490. 119136. . 243089. 32998.3 -32998.3 -76067.2 -32600.2 v2 -150000.

0 0 -32998.3 32998.3 1.~i998. -32998.3 -120000 0"3 0

0 -120000 32998.3 -32998.3 -32998.3 152998. 0 0 v3 00 0 -177490. -76067.2 -120000 0 297490. 76067.2

"4 00 0 -76067.2 -32600.2 0 0 76067.2 32600.2 v4 0

MathematicalMATLAB Implementation 1.5 on the Book Web Site:Five-bar plane truss assembly

Example 1.5 Heat Flow through a Square Duct The cross section of a 20 X 20-cmduct made of concrete walls 20 ern thick is shown in Figure 1.17. The inside surface ofthe duct is maintained at a temperature of 300'C due to hot gases flowing from a furnace.On the outside the duct is exposed to air with an ambient temp

Bhatti-Fundamental FEA and Application Using MATLAB & Mathematica

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