The Finite Element - WordPress

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