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The Finite ElementMethod in Engineering

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The Finite ElementMethod in EngineeringFOURTH EDITIONSingiresu S. RaoProfessor and ChairmanDepartment of Mechanical and Aerospace EngineeringUniversity of Miami, Coral Gables, Florida, USAAmsterdam Boston Heidelberg London New YorkParis San Diego San Francisco Singapore Sydney OxfordTokyo

Elsevier Butterworth–Heinemann30 Corporate Drive, Suite 400, Burlington, MA 01803, USALinacre House, Jordan Hill, Oxford OX2 8DP, UKc 2005, Elsevier Inc. All rights reserved.Copyright No part of this publication may be reproduced, stored in a retrieval system, or transmittedin any form or by any means, electronic, mechanical, photocopying, recording, or otherwise,without the prior written permission of the publisher.Permissions may be sought directly from Elsevier’s Science & Technology RightsDepartment in Oxford, UK: phone: ( 44) 1865 843830, fax: ( 44) 1865 853333,e-mail: permissions@elsevier.com.uk. You may also complete your request on-linevia the Elsevier homepage (http://elsevier.com), by selecting “Customer Support”and then “Obtaining Permissions.”Recognizing the importance of preserving what has been written, Elsevier prints itsbooks on acid-free paper whenever possible.Library of Congress Cataloging-in-Publication DataAPPLICATION SUBMITTEDBritish Library Cataloguing-in-Publication DataA catalogue record for this book is available from the British Library.ISBN: 0-7506-7828-3For information on all Butterworth-Heinemann publicationsvisit our Web site at www.books.elsevier.com.04050607080910109Printed in the United States of America87654321

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CONTENTSxiiiPrefacePrincipal NotationxvINTRODUCTION11Overview of Finite Element Method1.1 Basic Concept1.2 Historical Background1.3 General Applicability of the Method1.4 Engineering Applications of the Finite Element Method1.5 General Description of the Finite Element Method1.6 Comparison of Finite Element Method with OtherMethods of Analysis1.7 Finite Element Program PackagesReferencesProblems3337101026434345BASIC PROCEDURE512Discretization of the Domain2.1 Introduction2.2 Basic Element Shapes2.3 Discretization Process2.4 Node Numbering Scheme2.5 Automatic Mesh polation Models3.1 Introduction3.2 Polynomial Form of Interpolation Functions3.3 Simplex, Complex, and Multiplex Elements3.4 Interpolation Polynomial in Terms of NodalDegrees of Freedom3.5 Selection of the Order of the Interpolation Polynomial3.6 Convergence Requirements3.7 Linear Interpolation Polynomials in Termsof Global Coordinates3.8 Interpolation Polynomials for Vector Quantities80808283vii8485869197

viiiCONTENTS3.9Linear Interpolation Polynomials in Termsof Local CoordinatesReferencesProblems45Higher Order and Isoparametric Elements4.1 Introduction4.2 Higher Order One-Dimensional Elements4.3 Higher Order Elements in Terms of Natural Coordinates4.4 Higher Order Elements in Terms of ClassicalInterpolation Polynomials4.5 One-Dimensional Elements Using ClassicalInterpolation Polynomials4.6 Two-Dimensional (Rectangular) Elements Using ClassicalInterpolation Polynomials4.7 Continuity Conditions4.8 Comparative Study of Elements4.9 Isoparametric Elements4.10 Numerical IntegrationReferencesProblemsDerivation of Element Matrices and Vectors5.1 Introduction5.2 Direct Approach5.3 Variational Approach5.4 Solution of Equilibrium Problems Using Variational(Rayleigh–Ritz) Method5.5 Solution of Eigenvalue Problems Using Variational(Rayleigh–Ritz) Method5.6 Solution of Propagation Problems Using Variational(Rayleigh–Ritz) Method5.7 Equivalence of Finite Element and Variational(Rayleigh–Ritz) Methods5.8 Derivation of Finite Element Equations UsingVariational (Rayleigh–Ritz) Approach5.9 Weighted Residual Approach5.10 Solution of Eigenvalue Problems Using WeightedResidual Method5.11 Solution of Propagation Problems Using WeightedResidual Method5.12 Derivation of Finite Element Equations Using WeightedResidual (Galerkin) Approach5.13 Derivation of Finite Element Equations Using WeightedResidual (Least Squares) 0187192193194198201202

CONTENTS67ixAssembly of Element Matrices and Vectorsand Derivation of System Equations6.1 Coordinate Transformation6.2 Assemblage of Element Equations6.3 Computer Implementation of the Assembly Procedure6.4 Incorporation of Boundary Conditions6.5 Incorporation of Boundary Conditions inthe Computer ProgramReferencesProblems221222223Numerical Solution of Finite Element Equations7.1 Introduction7.2 Solution of Equilibrium Problems7.3 Solution of Eigenvalue Problems7.4 Solution of Propagation Problems7.5 Parallel Processing in Finite Element 209209211215216APPLICATION TO SOLID MECHANICS PROBLEMS2778Basic Equations and Solution Procedure8.1 Introduction8.2 Basic Equations of Solid Mechanics8.3 Formulations of Solid and Structural Mechanics8.4 Formulation of Finite Element Equations (Static nalysis of Trusses, Beams, and Frames9.1 Introduction9.2 Space Truss Element9.3 Beam Element9.4 Space Frame Element9.5 Planar Frame Element9.6 Computer Program for Frame 34210 Analysis of Plates10.1 Introduction10.2 Triangular Membrane Element10.3 Numerical Results with Membrane Element10.4 Computer Program for Plates under Inplane Loads10.5 Bending Behavior of Plates10.6 Finite Element Analysis of Plate Bending357357357365371374378

xCONTENTS10.710.810.9Triangular Plate Bending ElementNumerical Results with Bending ElementsAnalysis of Three-Dimensional Structures UsingPlate Elements10.10 Computer Program for Three-Dimensional StructuresUsing Plate ElementsReferencesProblems37938338739139139211 Analysis of Three-Dimensional Problems11.1 Introduction11.2 Tetrahedron Element11.3 Hexahedron Element11.4 Analysis of Solids of 2 Dynamic Analysis12.1 Dynamic Equations of Motion12.2 Consistent and Lumped Mass Matrices12.3 Consistent Mass Matrices in Global Coordinate System12.4 Free Vibration Analysis12.5 Computer Program for Eigenvalue Analysis ofThree-Dimensional Structures12.6 Dynamic Response Using Finite Element Method12.7 Nonconservative Stability and Flutter Problems12.8 Substructures MethodReferencesProblems421421424425433APPLICATION TO HEAT TRANSFER PROBLEMS46513 Formulation and Solution Procedure13.1 Introduction13.2 Basic Equations of Heat Transfer13.3 Governing Equation for Three-Dimensional Bodies13.4 Statement of the Problem13.5 Derivation of Finite Element 014 One-Dimensional Problems14.1 Introduction14.2 Straight Uniform Fin Analysis14.3 Computer Program for One-Dimensional Problems14.4 Tapered Fin Analysis14.5 Analysis of Uniform Fins Using Quadratic Elements482482482488490493440444455455455458

CONTENTS14.6 Unsteady State Problems14.7 Heat Transfer Problems with Radiation14.8 Computer Program for Problems with RadiationReferencesProblemsxi49650250750850915 Two-Dimensional Problems15.1 Introduction15.2 Solution15.3 Computer Program15.4 Unsteady State ProblemsReferencesProblems51451451452752852852916 Three-Dimensional Problems16.1 Introduction16.2 Axisymmetric Problems16.3 Computer Program for Axisymmetric Problems16.4 Three-Dimensional Heat Transfer Problems16.5 Unsteady State APPLICATION TO FLUID MECHANICS PROBLEMS55517 Basic Equations of Fluid Mechanics17.1 Introduction17.2 Basic Characteristics of Fluids17.3 Methods of Describing the Motion of a Fluid17.4 Continuity Equation17.5 Equations of Motion or Momentum Equations17.6 Energy, State, and Viscosity Equations17.7 Solution Procedure17.8 Inviscid Fluid Flow17.9 Irrotational Flow17.10 Velocity Potential17.11 Stream Function17.12 Bernoulli 56756856957057257357418 Inviscid and Incompressible Flows18.1 Introduction18.2 Potential Function Formulation18.3 Finite Element Solution Using the Galerkin Approach575575576578

xiiCONTENTS18.4 Stream Function Formulation18.5 Computer Program for Potential Function ApproachReferencesProblems58558858959019 Viscous and Non-Newtonian Flows19.1 Introduction19.2 Stream Function Formulation (Using Variational Approach)19.3 Velocity–Pressure Formulation (Using Galerkin Approach)19.4 Solution of Navier–Stokes Equations19.5 Stream Function–Vorticity Formulation19.6 Flow of Non-Newtonian Fluids19.7 Other 8613614616ADDITIONAL APPLICATIONS61920 Solution of Quasi-Harmonic Equations20.1 Introduction20.2 Finite Element Equations for Steady-State Problems20.3 Solution of Poisson’s Equation20.4 Computer Program for Torsion Analysis20.5 Transient Field 21 Solution of Helmholtz Equation21.1 Introduction21.2 Finite Element Solution21.3 Numerical ExamplesReferencesProblems64264264264464764822 Solution of Reynolds Equation22.1 Hydrodynamic Lubrication22.2 Finite Element Solution22.3 Numerical ix A Green–Gauss Theorem657Index659

PREFACEThe ﬁnite element method is a numerical method that can be used for the accuratesolution of complex engineering problems. The method was ﬁrst developed in 1956 forthe analysis of aircraft structural problems. Thereafter, within a decade, the potentialities of the method for the solution of diﬀerent types of applied science and engineeringproblems were recognized. Over the years, the ﬁnite element technique has been so wellestablished that today it is considered to be one of the best methods for solving a widevariety of practical problems eﬃciently. In fact, the method has become one of the activeresearch areas for applied mathematicians. One of the main reasons for the popularity ofthe method in diﬀerent ﬁelds of engineering is that once a general computer program iswritten, it can be used for the solution of any problem simply by changing the input data.The objective of this book is to introduce the various aspects of ﬁnite element methodas applied to engineering problems in a systematic manner. It is attempted to give detailsof development of each of the techniques and ideas from basic principles. New concepts areillustrated with simple examples wherever possible. Several Fortran computer programsare given with example applications to serve the following purposes:– to enable the student to understand the computer implementation of the theorydeveloped;– to solve speciﬁc problems;– to indicate procedure for the development of computer programs for solving anyother problem in the same area.The source codes of all the Fortran computer programs can be found at the Web sitefor the book, www.books.elsevier.com. Note that the computer programs are intended foruse by students in solving simple problems. Although the programs have been tested, nowarranty of any kind is implied as to their accuracy.After studying the material presented in the book, a reader will not only be able tounderstand the current literature of the ﬁnite element method but also be in a position todevelop short computer programs for the solution of engineering problems. In addition, thereader will be in a position to use the commercial software, such as ABAQUS, NASTRAN,and ANSYS, more intelligently.The book is divided into 22 chapters and an appendix. Chapter 1 gives an introductionand overview of the ﬁnite element method. The basic approach and the generality ofthe method are illustrated through simple examples. Chapters 2 through 7 describe thebasic ﬁnite element procedure and the solution of the resulting equations. The ﬁniteelement discretization and modeling, including considerations in selecting the numberand types of elements, is discussed in Chapter 2. The interpolation models in terms ofCartesian and natural coordinate systems are given in Chapter 3. Chapter 4 describes thehigher order and isoparametric elements. The use of Lagrange and Hermite polynomialsis also discussed in this chapter. The derivation of element characteristic matrices andvectors using direct, variational, and weighted residual approaches is given in Chapter 5.xiii

xivPREFACEThe assembly of element characteristic matrices and vectors and the derivation of systemequations, including the various methods of incorporating the boundary conditions, areindicated in Chapter 6. The solutions of ﬁnite element equations arising in equilibrium,eigenvalue, and propagation (transient or unsteady) problems, along with their computerimplementation, are brieﬂy outlined in Chapter 7.The application of the ﬁnite element method to solid and structural mechanics problems is considered in Chapters 8 through 12. The basic equations of solidmechanics — namely, the internal and external equilibrium equations, stress–strain relations, strain–displacement relations and compatibility conditions — are summarized inChapter 8. The analysis of trusses, beams, and frames is the topic of Chapter 9. Thedevelopment of inplane and bending plate elements is discussed in Chapter 10. The analysis of axisymmetric and three-dimensional solid bodies is considered in Chapter 11. Thedynamic analysis, including the free and forced vibration, of solid and structural mechanicsproblems is outlined in Chapter 12.Chapters 13 through 16 are devoted to heat transfer applications. The basic equationsof conduction, convection, and radiation heat transfer are summarized and the ﬁniteelement equations are formulated in Chapter 13. The solutions of one-, two-, and threedimensional heat transfer problems are discussed in Chapters 14–16, respectively. Boththe steady state and transient problems are considered. The application of the ﬁniteelement method to ﬂuid mechanics problems is discussed in Chapters 17–19. Chapter 17gives a brief outline of the basic equations of ﬂuid mechanics. The analysis of inviscidincompressible ﬂows is considered in Chapter 18. The solution of incompressible viscousﬂows as well as non-Newtonian ﬂuid ﬂows is considered in Chapter 19. Chapters 20–22present additional applications of the ﬁnite element method. In particular, Chapters 20–22discuss the solution of quasi-harmonic (Poisson), Helmholtz, and Reynolds equations,respectively. Finally, Green–Gauss theorem, which deals with integration by parts in twoand three dimensions, is given in Appendix A.This book is based on the author’s experience in teaching the course to engineeringstudents during the past several years. A basic knowledge of matrix theory is requiredin understanding the various topics presented in the book. More than enough materialis included for a ﬁrst course at the senior or graduate level. Diﬀerent parts of the bookcan be covered depending on the background of students and also on the emphasis tobe given on speciﬁc areas, such as solid mechanics, heat transfer, and ﬂuid mechanics.The student can be assigned a term project in which he/she is required to either modifysome of the established elements or develop new ﬁnite elements, and use them for thesolution of a problem of his/her choice. The material of the book is also useful for selfstudy by practicing engineers who would like to learn the method and/or use the computerprograms given for solving practical problems.I express my appreciation to the students who took my courses on the ﬁnite elementmethod and helped me improve the presentation of the material. Finally, I thank my wifeKamala for her tolerance and understanding while preparing the manuscript.MiamiMay 2004S. S. Raosrao@miami.edu

PRINCIPAL NOTATIONaax , ay , azAA(e)Ai (Aj )b BccvC1 , C2 , . . .[C]D[D]EE (e)Eiif1 (x), f2 (x), . . .FgGGijh(0)Hoi (x)(j)HkiiI i, (I)I (e)IzzJ j (J)[J]length of a rectangular elementcomponents of acceleration along x, y, z directions of a ﬂuidarea of cross section of a one-dimensional element; area of a triangular(plate) elementcross-sectional area of one-dimensional element ecross-sectional area of a tapered one-dimensional element at node i(j)width of a rectangular elementbody force vector in a ﬂuid {Bx , By , Bz }Tspeciﬁc heatspeciﬁc heat at constant volumeconstantscompliance matrix; damping matrixﬂexural rigidity of a plateelasticity matrix (matrix relating stresses and strains)Young’s modulus; total number of elementsYoung’s modulus of element eYoung’s modulus in a plane deﬁned by axis ifunctions of xshear force in a beamacceleration due to gravityshear modulusshear modulus in plane ijconvection heat transfer coeﬃcientLagrange polynomial associated with node ijth order Hermite polynomial( 1)1/2functional to be extremized;potential energy;area moment of inertia of a beamunit vector parallel to x(X) axiscontribution of element e to the functional Iarea moment of inertia of a cross section about z axispolar moment of inertia of a cross sectionunit vector parallel to y(Y ) axisJacobian matrixxv

xvikk x , ky , kzkr , kθ , k z k (K) [k(e) ](e)[K (e) ] [Kij ][K] [Kij ][K] [K] ijll(e)lx , ly , lzlox , mox , noxlij , mij , nijLL1 , L2L1 , L2 , L3L1 , L2 , L3 , L4 mMMx , My , MxyMz[m(e) ][M (e) ][M ]][M nNi[N ]pP cPPi (Pj )Px , Py , Pzp (e)(e) (e))p b (PbPRINCIPAL NOTATIONthermal conductivitythermal conductivities along x, y, z axesthermal conductivities along r, θ, z axesunit vector parallel to z(Z) axisstiﬀness matrix of element e in local coordinate systemstiﬀness matrix of element e in global coordinate systemstiﬀness (characteristic) matrix of complete body after incorporationof boundary conditionsstiﬀness (characteristic) matrix of complete body beforeincorporation of boundary conditionslength of one-dimensional elementlength of the one-dimensional element edirection cosines of a linedirection cosines of x axisdirection cosines of a bar element with nodes i and jtotal length of a bar or ﬁn; Lagrangiannatural coordinates of a line elementnatural coordinates of a triangular elementnatural coordinates of a tetrahedron elementdistance between two nodesmass of beam per unit lengthbending moment in a beam; total number of degrees of freedomin a bodybending moments in a platetorque acting about z axis on a prismatic shaftmass matrix of element e in local coordinate systemmass matrix of element e in global coordinate systemmass matrix of complete body after incorporation ofboundary conditionsmass matrix of complete body before incorporation ofboundary conditionsnormal directioninterpolation function associated with the ith nodal degree of freedommatrix of shape (nodal interpolation) functionsdistributed load on a beam or plate; ﬂuid pressureperimeter of a ﬁnvector of concentrated nodal forcesperimeter of a tapered ﬁn at node i(j)external concentrated loads parallel to x, y, z axesload vector of element e in local coordinate systemload vector due to body forces of element e in local (global)coordinate system

PRINCIPAL NOTATION p i (Pi(e)(e)) s )p s (P(e)(e) (e) {P (e) }Pi {Pi }P {P }P iqq̇qxQiQx , QyQx , Qy , Qz (e) ) q (e) (Q Q Qjr, sr, s, tr, θ, z(ri , si , ti )RSS1 , S2S (e)(e)(e)S1 , S2tTTiT0T (e)TiT (e)load vector due to initial strains of element e in local (global)coordinate systemload vector due to surface forces of element e in local (global)coordinate systemvector of nodal forces (characteristic vector) of element e in globalcoordinate systemvector of nodal forces of body after incorporation ofboundary conditionsvector of nodal forces of body before incorporation ofboundary conditionsrate of heat ﬂowrate of heat generation per unit volumerate of heat ﬂow in x directionmass ﬂow rate of ﬂuid across section ivertical shear forces in a plateexternal concentrated moments parallel to x, y, z axesvector of nodal displacements (ﬁeld variables) of element e in local(global) coordinate systemvector of nodal displacements of body before incorporation ofboundary conditionsmode shape corresponding to the frequency ωjnatural coordinates of a quadrilateral elementnatural coordinates of a hexahedron elementradial, tangential, and axial directionsvalues of (r, s, t) at node iradius of curvature of a deﬂected beam;residual;region of integration;dissipation functionsurface of a bodypart of surface of a bodysurface of element epart of surface of element etime; thickness of a plate elementtemperature;temperature change;kinetic energy of an elastic bodytemperature at node itemperature at the root of ﬁnsurrounding temperaturetemperature at node i of element evector of nodal temperatures of element exvii

xviii T uu, v, w UV VwWWiWp (e)Wx(xc , yc )(xi , yi , zi )(Xi , Yi , Zi )ααiδεiiεijε(e) εε 0θ[λ(e) ](t)ηjμννijππcπpπRπ (e)ρσiiPRINCIPAL NOTATIONvector of nodal temperatures of the body before incorporation ofboundary conditionsﬂow velocity along x direction; axial displacementcomponents of displacement parallel to x, y, z axes; components ofvelocity along x, y, z directions in a ﬂuid (Chapter 17)vector of displacements {u, v, w}Tvolume of a bodyvelocity vector {u, v, w}T (Chapter 17)transverse deﬂection of a beamamplitude of vibration of a beamvalue of W at node iwork done by external forcesvector of nodal displacements of element ex coordinate;axial directioncoordinates of the centroid of a triangular element(x, y, z) coordinates of node iglobal coordinates (X, Y, Z) of node icoeﬃcient of thermal expansionith generalized coordinatevariation operatornormal strain parallel to ith axisshear strain in ij planestrain in element estrain vector {εxx , εyy , εzz , εxy , εyz , εzx }T for athree-dimensional body; {εrr , εθθ , εzz , εrz }T for an axisymmetric bodyinitial strain vectortorsional displacement or twistcoordinate transformation matrix of element ejth generalized coordinatedynamic viscosityPoisson’s ratioPoisson’s ratio in plane ijpotential energy of a beam;strain energy of a solid bodycomplementary energy of an elastic bodypotential energy of an elastic bodyReissner energy of an elastic bodystrain energy of element edensity of a solid or ﬂuidnormal stress parallel to ith axis

PRINCIPAL NOTATIONσijσ (e) στφφx , φy , φz φ φ̄ΦΦx , Φ y , Φ zΦiΦ i(e)Φi (e)Φ Φ Φ ψωωjωxω̃iΩsuperscript earrow over a symbol (X) T ([ ]T )Xdot over asymbol (ẋ)shear stress in ij planestress in element estress vector {σxx , σyy , σzz , σxy , σyz , σzx }T for athree-dimensional body; {σrr , σθθ , σzz , σrz }T for an axisymmetric bodyshear stress in a ﬂuidﬁeld variable;axial displacement;potential function in ﬂuid ﬂowbody force per unit volume parallel to x, y, z axesvector valued ﬁeld variable with components u, v, and wvector of prescribed body forcesdissipation function for a ﬂuidsurface (distributed) forces parallel to x, y, z axesith ﬁeld variableprescribed value of φivalue of the ﬁeld variable φ at node i of element evector of nodal values of the ﬁeld variable of element evector of nodal values of the ﬁeld variables of complete body afterincorporation of boundary conditionsvector of nodal values of the ﬁeld variables of complete body beforeincorporation of boundary conditionsstream function in ﬂuid ﬂowfrequency of vibrationjth natural frequency of a bodyrate of rotation of ﬂuid about x axisapproximate value of ith natural frequencybody force potential in ﬂuid ﬂowelement e X 1 X2column vector X . . ])transpose of X([dxderivative with respect to time ẋ dtxix

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INTRODUCTION

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1OVERVIEW OF FINITE ELEMENTMETHOD1.1 BASIC CONCEPTThe basic idea in the ﬁnite element method is to ﬁnd the solution of a complicated problemby replacing it by a simpler one. Since the actual problem is replaced by a simpler onein ﬁnding the solution, we will be able to ﬁnd only an approximate solution rather thanthe exact solution. The existing mathematical tools will not be suﬃcient to ﬁnd the exactsolution (and sometimes, even an approximate solution) of most of the practical problems.Thus, in the absence of any other convenient method to ﬁnd even the approximate solutionof a given problem, we have to prefer the ﬁnite element method. Moreover, in the ﬁniteelement method, it will often be possible to improve or reﬁne the approximate solution byspending more computational eﬀort.In the ﬁnite element method, the solution region is considered as built up of manysmall, interconnected subregions called ﬁnite elements. As an example of how a ﬁniteelement model might be used to represent a complex geometrical shape, consider themilling machine structure shown in Figure 1.1(a). Since it is very diﬃcult to ﬁnd theexact response (like stresses and displacements) of the machine under any speciﬁed cutting(loading) condition, this structure is approximated as composed of several pieces as shownin Figure 1.1(b) in the ﬁnite element method. In each piece or element, a convenientapproximate solution is assumed and the conditions of overall equilibrium of the structureare derived. The satisfaction of these conditions will yield an approximate solution for thedisplacements and stresses. Figure 1.2 shows the ﬁnite element idealization of a ﬁghteraircraft.1.2 HISTORICAL BACKGROUNDAlthough the name of the ﬁnite element method was given recently, the concept datesback for several centuries. For example, ancient mathematicians found the circumferenceof a circle by approximating it by the perimeter of a polygon as shown in Figure 1.3.In terms of the present-day notation, each side of the polygon can be called a“ﬁnite element.” By considering the approximating polygon inscribed or circumscribed,one can obtain a lower bound S (l) or an upper bound S (u) for the true circumference S.Furthermore, as the number of sides of the polygon is increased, the approximate values3

4OVERVIEW OF FINITE ELEMENT METHODFigure 1.1. Representation of a Milling Machine Structure by Finite Elements.Figure 1.2. Finite Element Mesh of a Fighter Aircraft (Reprinted with Permission from AnametLaboratories, Inc.).

HISTORICAL BACKGROUND5Figure 1.3. Lower and Upper Bounds to the Circumference of a Circle.converge to the true value. These characteristics, as will be seen later, will hold true inany general ﬁnite element application. In recent times, an approach similar to the ﬁniteelement method, involving the use of piecewise continuous functions deﬁned over triangular regions, was ﬁrst suggested by Courant [1.1] in 1943 in the literature of appliedmathematics.The basic ideas of the ﬁnite element method as known today were presented in thepapers of Turner, Clough, Martin, and Topp [1.2] and Argyris and Kelsey [1.3]. The nameﬁnite element was coined by Clough [1.4]. Reference [1.2] presents the application of simpleﬁnite elements (pin-jointed bar and triangular plate with inplane loads) for the analysis ofaircraft structure and is considered as one of the key contributions in the development ofthe ﬁnite element method. The digital computer provided a rapid means of performing themany calculations involved in the ﬁnite element analysis and made the method practicallyviable. Along with the development of high-speed digital computers, the application of theﬁnite element method also progressed at a very impressive rate. The book by Przemieniecki[1.33] presents the ﬁnite element method as applied to the solution of stress analysisproblems. Zienkiewicz and Cheung [1.5] presented the broad interpretation of the methodand its applicability to any general ﬁeld problem. With this broad interpretation of theﬁnite element method, it has been found that the ﬁnite element equations can also bederived by using a weighted residual method such as Galerkin method or the least squaresapproach. This led to widespread interest among applied mathematicians in applying theﬁnite element method for the solution of linear and nonlinear diﬀerential equations. Overthe years, several papers, conference proceedings, and books have been published on thismethod.A brief history of the beginning of the ﬁnite element method was presented byGupta and Meek [1.6]. Books that deal with the basic theory, mathematical foundations,mechanical design, structural, ﬂuid ﬂow, heat transfer, electromagnetics and manufacturing applications, and computer programming aspects are given at the end of thechapter [1.10–1.32]. With all the progress, today the ﬁnite element method is considered one of the well-established and convenient analysis tools by engineers and appliedscientists.

6OVERVIEW OF FINITE ELEMENT METHODFigure 1.4.Example 1.1 The circumference of a circle (S) is approximated by the perimeters ofinscribed and circumscribed n-sided polygons as shown in Figure 1.3. Prove the following:lim S (l) Sn andlim S (u) Sn where S (l) and S (u) denote the perimeters of the inscribed and circumscribed polygons,respectively.Solution If the radius of the circle is R, each side of the inscribed and the circumscribedpolygon can be expressed as (Figure 1.4)r 2R sinπ,ns 2R tanπn(E1 )Thus, the perimeters of the inscribed and circumscribed polygons are given byS (l) nr 2nR sinπ,nS (u) ns 2nR tanπn(E2 )

GENERAL APPLICABILITY OF THE METHOD7which can be rewritten asπ sin 2πR π n ,n S (l)As n ,π tan 2πR π n n S (u)(E3 )π 0, and hencenS (l) 2πR S,S (u) 2πR S(E4 )1.3 GENERAL APPLICABILITY OF THE METHODAlthough the method has been extensively used in the ﬁeld of structural mechanics, ithas been successfully applied to solve several other types of engineering problems, suchas heat conduction, ﬂuid dynamics, seepage ﬂow, and electric and magnetic ﬁelds. Theseapplications prompted mathematicians to use this technique for the solution of complicated boundary value and other problems. In fact, it has been established that the methodcan be used for the numerical soluti

1 Overview of Finite Element Method 3 1.1 Basic Concept 3 1.2 Historical Background 3 1.3 General Applicability of the Method 7 1.4 Engineering Applications of the Finite Element Method 10 1.5 General Description of the Finite Element Method 10 1.6 Comparison of Finite Element Method with Other Methods of Analysis

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