Solid Mechanics and Its Applications M. Reza Eslami Finite Elements Methods in Mechanics Solid Mechanics and Its Applications Volume 216 Series editors J. R. Barber, Ann Arbor, USA Anders Klarbring, Linköping, Sweden For furthervolumes: http://www.springer.com/series/6557 Aims and Scope of the Series The fundamental questions arising in mechanics are: Why?, How?, and How much? The aim of this series is to provide lucid accounts written by authoritative researchersgivingvisionandinsightinansweringthesequestionsonthesubjectof mechanics as it relates to solids. The scope of the series covers the entire spectrum of solid mechanics. Thus it includes the foundation of mechanics; variational formulations; computational mechanics;statics,kinematicsanddynamicsofrigidandelasticbodies:vibrations of solids and structures; dynamical systems and chaos; the theories of elasticity, plasticity and viscoelasticity; composite materials; rods, beams, shells and mem- branes; structural control and stability; soils, rocks and geomechanics; fracture; tribology; experimental mechanics; biomechanics and machine design. Themedianlevelofpresentationistothefirstyeargraduatestudent.Sometexts aremonographsdefiningthecurrentstateofthefield;othersareaccessibletofinal year undergraduates; but essentially the emphasis is on readability and clarity. M. Reza Eslami Finite Elements Methods in Mechanics 123 M.Reza Eslami Mechanical EngineeringDepartment AmirkabirUniversity ofTechnology Tehran Iran Additionalmaterial tothis bookcan bedownloaded from http://extras.springer.com ISSN 0925-0042 ISSN 2214-7764 (electronic) ISBN 978-3-319-08036-9 ISBN 978-3-319-08037-6 (eBook) DOI 10.1007/978-3-319-08037-6 Springer ChamHeidelberg New YorkDordrecht London LibraryofCongressControlNumber:2014941509 (cid:2)SpringerInternationalPublishingSwitzerland2014 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionor informationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodology now known or hereafter developed. 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While the advice and information in this book are believed to be true and accurate at the date of publication,neithertheauthorsnortheeditorsnorthepublishercanacceptanylegalresponsibilityfor anyerrorsoromissionsthatmaybemade.Thepublishermakesnowarranty,expressorimplied,with respecttothematerialcontainedherein. Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) This book is dedicated to my precious grandchildren: Arya, Roxan, Ava, and Lilia Preface TheauthorispleasedtopresentFiniteElementMethodinMechanics.Thisbookwill serve a wide range of readers, in particular, graduate students, Ph.D. candidates, professors, scientists, researchers in various industrial and government institutes, andengineers.Thus,thebookshouldbeconsiderednotonlyasagraduatetextbook, butalsoasareferencebooktothoseworkingorinterestedinareasoffiniteelement modellingofsolidmechanics,heatconduction,andfluidmechanics. The book is self-contained, so that the reader should not need to consult other sources while studying the topic. The necessary mathematical concepts and numericalmethodsarepresentedinthebookandthereadermayeasilyfollowthe subjectsbasedonthesebasictools.Itisexpected,however,thatthereadershould have some basic knowledge in the classical mechanics, theory of elasticity, and fluid mechanics. The book contains 17 chapters, where the chapters cover the finite element modeling of all major areas of mechanics. Chapter1 presents the history of development offinite element method, where the key references are given and the progress of this science is discussed. Chapter 2 is devoted to the basic mathematical concepts of finite element method. The method of calculus of variation is discussed and the distinction of boundary value problems versus the variational formulation is presented and several examples are given to make the reader familiar with the concepts of functionandfunctional.Thematerialthenfollowsintothediscussionoftraditional Ritz and Galerkin methods. Numerical examples show the powerful nature of these numerical techniques. IntroductiontothefiniteelementmethodisgiveninChap.3withthediscussion of elastic membrane. The subject of elastic membrane is selected because the height of membrane is approximated with finite element method. This gives a physical feeling for the finite element approximation to the reader. Sincealineartriangularelementisemployedtomodeltheelasticmembranein Chap. 3, Chap. 4 discuss the one-, two-, and three-dimensional elements with linearandhigherorderapproximation.Thediscussiongivesafeelingtothereader that there is no limitation in the type of elements and the order of approximation, geometrically and mathematically. The subparametric, isoparametric, and super- parametric elements are discussed and the natural coordinates are presented. vii viii Preface The finite element approximation of the field problems, harmonic and biharmonic, are given in Chap. 5. Chapter 6 deals with the finite element approximation of the heat conduction equations. One-, two-, and three-dimensional conduction in solids are discussed and the transient heat conduction problems are presented. Both variational and Galerkin techniques are presented. Up to this point, the reader learns how to obtain the element stiffness, capa- citance, and force matrices for one element. He questions how a solution domain with many number of elements should be modeled and solved to obtain the requireddomainunknowns.AcomprehensivetreatmentisgiveninChap.7togive a proper tool to the reader to write his own computer program. Many numerical examples are solved to show the numerical scheme, and proper algorithms are given. The assembly of global matrices, bandwidth calculation, the method to apply the boundary conditions, and the Gauss elimination method are presented. The method of solution of the transient and dynamic finite element equations are thenpresented.Thecentraldifferencemethod,theHouboltMethod,theNewmark Method, and the Wilson-h method are presented. At this stage, the reader learns how to write his own computer problem. Now, he should learn how different problemsofmechanicsareformulatedbythefiniteelementapproximation.These techniques are discussed in the following chapters. Chapter 8 deals with the finite element approximation of beams. Static beam deflectionequation,basedontheEulerbeamtheory,ispresentedandtheGalerkin and variational formulations are obtained. The axial, torsional, and lateral vibra- tions of beams are modeled. Finally, the vibrations of Timoshenko beam are presented. Chapters9and10presentthefiniteelementformulationsofelasticityproblems based on Galerkin and variational formulations. Torsion of prismatic bars and rods are given in Chap. 11 and quasistatic thermoelasticity theory is discussed in Chap. 12. Chapter 13 is devoted to the finite element solution of viscous fluid mechanic problems. Derivation of the Navier-Stokes equations is presented and the finite element formulation of the two-dimensional fluid flow based on the velocity components and pressure are derived. In the following section, the vorticity transportmodeloftheNavier-Stokesequationsareobtainedandthefiniteelement formulations are derived. The method of solution of the resulting nonlinear finite element equation is presented. Chapter 14 presents one-dimensional higher order elements. The local natural coordinate for the quadratic and cubic elements are derived and the Jacobian matrixisobtained.Todescribetheapplication,fieldproblemforone-dimensional case is discussed and the element of the matrices are calculated. The chapter is completed with a discussion of layerwise theory for composite beams, where one-dimensional higher order element is used to discuss the problem. The higher order element in two dimension in discussed in Chap. 15. The triangular element with quadratic and cubic shape functions are given in terms of the area element and the Jacobian matrix is calculated. The quadratic element is Preface ix employed to obtain the element matrices for a two-dimensional field problem. In the following, the quadrilateral element is discussed and its application to the field problem is presented. Chapter 16 presents the linear coupled thermoelasticity problems, and their method of solution by finite element method. This chapter is unique in the literature offinite element analysis of solid elastic continuum. The most general form of the three-dimensional classical coupled thermoelasticity equations are considered and the finite element formulations are presented. ComputerprogramsforthreedifferenttypesofproblemsaregiveninChap.17. The first program is related to the elastic membrane problem, where Poisson’s equation is solved. This program may be used for any other application of Poisson’s equation, such as the steady-state heat conduction, torsion of prismatic bars, inviscid incompressible fluid flow problems, and the pressure in porous media. The second computer program handles two-dimensional elasticity problems, and the third computer program presents three-dimensional transient heat conduction problems. The programs are written in C++ environment. At the end of all the chapters, except Chap. 1, there are a number of problems for students to solve. Also, at the end of each chapter, there is a list of relevant references. The book was prepared over some 40 years of teaching the graduate finite element course. During this long period of time, the results of classwork assignmentsandstudentresearcharecarefullygatheredandputintothisvolumeof work. The author takes this opportunity to thank all his students who made possible to provide this piece of work. October 2013 M. Reza Eslami Contents 1 Introduction and History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2 Mathematical Foundations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Statement of Extremum Principle. . . . . . . . . . . . . . . . . . . . 8 2.3 Method of Calculus of Variation . . . . . . . . . . . . . . . . . . . . 9 2.4 Function of One Variable, Euler Equation. . . . . . . . . . . . . . 10 2.5 Higher Order Derivatives. . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.6 Minimization of Functions of Several Variables. . . . . . . . . . 14 2.7 Cantilever Beam. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.8 Approximate Techniques. . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.8.1 A: Weighted Residual Methods . . . . . . . . . . . . . . . 21 2.8.2 B: Stationary Functional Method . . . . . . . . . . . . . . 23 2.9 Further Notes on the Ritz and Galerkin Methods . . . . . . . . . 23 2.10 Application of the Ritz Method . . . . . . . . . . . . . . . . . . . . . 26 2.10.1 Non-homogeneous Boundary Conditions. . . . . . . . . 28 2.11 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Further Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3 Finite Element of Elastic Membrane. . . . . . . . . . . . . . . . . . . . . . 35 3.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.2 Poisson’s Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.2.1 Physical Examples . . . . . . . . . . . . . . . . . . . . . . . . 36 3.3 Weightless Elastic Membrane (Method I) . . . . . . . . . . . . . . 38 3.4 Membrane Analysis (Method II). . . . . . . . . . . . . . . . . . . . . 39 3.5 Strain Energy of Elastic Membrane. . . . . . . . . . . . . . . . . . . 41 3.6 Application of Calculus of Variation. . . . . . . . . . . . . . . . . . 43 3.7 Introduction to the Finite Element Method. . . . . . . . . . . . . . 44 3.7.1 The Elastic Membrane. . . . . . . . . . . . . . . . . . . . . . 45 3.7.2 Boundary Value Problem. . . . . . . . . . . . . . . . . . . . 45 xi