Singiresu s rao the finite element method in engineering, fifth edition elsevier butterworth heinemann (2011)

The Finite Element Method in Engineering This page intentionally left blank The Finite Element Method in Engineering Fifth Edition Singiresu S Rao Professor and Chairman Department of Mechanical and Aerospace Engineering University of Miami, Coral Gables, Florida, USA AMSTERDAM • BOSTON • HEIDELBERG • LONDON NEW YORK • OXFORD • PARIS • SAN DIEGO SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO Butterworth-Heinemann is an imprint of Elsevier Butterworth–Heinemann is an imprint of Elsevier 30 Corporate Drive, Suite 400, Burlington, MA 01803, USA The Boulevard, Langford Lane, Kidlington, Oxford, OX5 1GB, UK © 2011 Elsevier Inc All rights reserved No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our web site: www.elsevier.com/permissions This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein) Notices Knowledge and best practice in this field are constantly changing As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein MATLAB® is a trademark of The MathWorks, Inc and is used with permission The MathWorks does not warrant the accuracy of the text or exercises in this book This book’s use or discussion of MATLAB® software or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB® software Library of Congress Cataloging-in-Publication Data Application submitted British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library ISBN: 978-1-85617-661-3 For information on all Butterworth–Heinemann publications, visit our web site at: www.elsevierdirect.com Typeset by: diacriTech, Chennai, India Printed in the United States of America 10 11 12 13 14 10 This page intentionally left blank CONTENTS PREFACE xiii PART • Introduction CHAPTER Overview of Finite Element Method 1.1 Basic Concept 1.2 Historical Background 1.3 General Applicability of the Method 1.4 Engineering Applications of the Finite Element Method 1.5 General Description of the Finite Element Method 1.6 One-Dimensional Problems with Linear Interpolation Model 12 1.7 One-Dimensional Problems with Cubic Interpolation Model 24 1.8 Derivation of Finite Element Equations Using a Direct Approach 28 1.9 Commercial Finite Element Program Packages 40 1.10 Solutions Using Finite Element Software 40 PART • Basic Procedure CHAPTER Discretization of the Domain 53 2.1 Introduction 53 2.2 Basic Element Shapes 53 2.3 Discretization Process 56 2.4 Node Numbering Scheme 63 2.5 Automatic Mesh Generation 65 CHAPTER Interpolation Models 75 3.1 Introduction 75 3.2 Polynomial Form of Interpolation Functions 77 3.3 Simplex, Complex, and Multiplex Elements 78 3.4 Interpolation Polynomial in Terms of Nodal Degrees of Freedom 78 3.5 Selection of the Order of the Interpolation Polynomial 80 3.6 Convergence Requirements 82 3.7 Linear Interpolation Polynomials in Terms of Global Coordinates 85 3.8 Interpolation Polynomials for Vector Quantities 96 3.9 Linear Interpolation Polynomials in Terms of Local Coordinates 99 3.10 Integration of Functions of Natural Coordinates 108 3.11 Patch Test 109 CHAPTER Higher Order and Isoparametric Elements 119 4.1 Introduction 120 4.2 Higher Order One-Dimensional Elements 120 4.3 Higher Order Elements in Terms of Natural Coordinates 121 4.4 Higher Order Elements in Terms of Classical Interpolation Polynomials 130 vii CONTENTS viii 4.5 One-Dimensional Elements Using Classical Interpolation Polynomials 134 4.6 Two-Dimensional (Rectangular) Elements Using Classical Interpolation Polynomials 135 4.7 Continuity Conditions 137 4.8 Comparative Study of Elements 139 4.9 Isoparametric Elements 140 4.10 Numerical Integration 148 CHAPTER Derivation of Element Matrices and Vectors 157 5.1 Introduction 158 5.2 Variational Approach 158 5.3 Solution of Equilibrium Problems Using Variational (Rayleigh-Ritz) Method 163 5.4 Solution of Eigenvalue Problems Using Variational (Rayleigh-Ritz) Method 167 5.5 Solution of Propagation Problems Using Variational (Rayleigh-Ritz) Method 168 5.6 Equivalence of Finite Element and Variational (Rayleigh-Ritz) Methods 169 5.7 Derivation of Finite Element Equations Using Variational (Rayleigh-Ritz) Approach 169 5.8 Weighted Residual Approach 175 5.9 Solution of Eigenvalue Problems Using Weighted Residual Method 182 5.10 Solution of Propagation Problems Using Weighted Residual Method 183 5.11 Derivation of Finite Element Equations Using Weighted Residual (Galerkin) Approach 184 5.12 Derivation of Finite Element Equations Using Weighted Residual (Least Squares) Approach 187 5.13 Strong and Weak Form Formulations 189 CHAPTER Assembly of Element Matrices and Vectors and Derivation of System Equations 199 6.1 Coordinate Transformation 199 6.2 Assemblage of Element Equations 204 6.3 Incorporation of Boundary Conditions 211 6.4 Penalty Method 219 6.5 Multipoint Constraints—Penalty Method 223 6.6 Symmetry Conditions—Penalty Method 226 6.7 Rigid Elements 228 CHAPTER Numerical Solution of Finite Element Equations 241 7.1 Introduction 241 7.2 Solution of Equilibrium Problems 242 7.3 Solution of Eigenvalue Problems 251 7.4 Solution of Propagation Problems 262 7.5 Parallel Processing in Finite Element Analysis 268 PART • Application to Solid Mechanics Problems CHAPTER Basic Equations and Solution Procedure 277 8.1 Introduction 277 8.2 Basic Equations of Solid Mechanics 277 CONTENTS CHAPTER CHAPTER 10 CHAPTER 11 CHAPTER 12 PART 8.3 Formulations of Solid and Structural Mechanics 294 8.4 Formulation of Finite Element Equations (Static Analysis) 299 8.5 Nature of Finite Element Solutions 303 Analysis of Trusses, Beams, and Frames 311 9.1 Introduction 311 9.2 Space Truss Element 312 9.3 Beam Element 323 9.4 Space Frame Element 328 9.5 Characteristics of Stiffness Matrices 338 Analysis of Plates 355 10.1 Introduction 355 10.2 Triangular Membrane Element 356 10.3 Numerical Results with Membrane Element 367 10.4 Quadratic Triangle Element 369 10.5 Rectangular Plate Element (In-plane Forces) 372 10.6 Bending Behavior of Plates 376 10.7 Finite Element Analysis of Plates in Bending 379 10.8 Triangular Plate Bending Element 379 10.9 Numerical Results with Bending Elements 383 10.10 Analysis of Three-Dimensional Structures Using Plate Elements 386 Analysis of Three-Dimensional Problems 401 11.1 Introduction 401 11.2 Tetrahedron Element 401 11.3 Hexahedron Element 409 11.4 Analysis of Solids of Revolution 413 Dynamic Analysis 427 12.1 Dynamic Equations of Motion 427 12.2 Consistent and Lumped Mass Matrices 430 12.3 Consistent Mass Matrices in a Global Coordinate System 439 12.4 Free Vibration Analysis 440 12.5 Dynamic Response Using Finite Element Method 452 12.6 Nonconservative Stability and Flutter Problems 460 12.7 Substructures Method 461 • Application to Heat Transfer Problems CHAPTER 13 Formulation and Solution Procedure 473 13.1 Introduction 473 13.2 Basic Equations of Heat Transfer 473 13.3 Governing Equation for Three-Dimensional Bodies 475 13.4 Statement of the Problem 479 13.5 Derivation of Finite Element Equations 480 CHAPTER 14 One-Dimensional Problems 489 14.1 Introduction 489 14.2 Straight Uniform Fin Analysis 489 14.3 Convection Loss from End Surface of Fin 492 14.3 Tapered Fin Analysis 496 14.4 Analysis of Uniform Fins Using Quadratic Elements 499 ix PART ABAQUS and ANSYS Software and MATLAB®Programs for Finite Element Analysis EXAMPLE 23.5 (Continued ) nodes=[3 1; 4; 3; 6; 5; 8; 7; 10; 11 10 9; 10 11 12; 15 14 13; 14 15 16; 17 16 15; 16 17 18; 19 18 17; 18 19 20; 21 20 19; 20 21 22; 23 22 21; 22 23 24; 14 2; 14 16; 16 4; 16 18; 18 6; 18 20; 10 20 8; 20 10 22; 12 22 10; 22 12 24; 13 1; 13 15; 15 3; 15 17; 17 5; 17 19; 19 7; 19 21; 11 21 9; 21 11 23]; % ISTRES=array of element numbers in which stresses are to be found ISTRES=[10 20 30 40]; % -% input data for boundary conditions % -bcdof=[31 32 33 34 35 36 67 68 69 70 71 72]; % bedof = fixed dof numbers -% end of data [disp, STRESS]=CST3D (neltot, nelm, nemp, nels, nnel, ndof, nnode, sdof, effdof, topdof, edof, emodule, p oisson, tt, rho, nmode, NB, FF, CX, CY, CZ, gcoord, nodes, ISTRES, bcdof); % ************************ % OUTPUT:= fprintf (‘ Nodal displacements fprintf (‘Node X-disp Y-disp \n’); Z-disp\n’); for i=1:1:20 696 fprintf (‘%1d %6.8f %6.8f %6.8f\n’, i, disp (3* (i–1)+1), disp (3*(i–1)+2), disp (3* (i–1)+3)); end fprintf (‘ \n’); fprintf (‘ Stress in the structure fprintf (‘Element \n’); Principal Stress Principal Stress \n’); for IJK=1:1:NB; ii=ISTRES (IJK); %%% ISTRES is the array of element numbers in which stresses are to be found for jj=1:1:3; stressp (IJK, jj) = STRESS (jj, IJK); end A1=(STRESS (1, IJK) + STRESS (2, IJK))/2; A2=sqrt (((STRESS (1, IJK) −STRESS (2, IJK))/2) ^2+STRESS (3, IJK) ^2); boxbeam_stress (IJK, 1) = A1+A2; boxbeam_stress (IJK, 2) = A1−A2; boxbeam_stress; fprintf (‘ %1d % 6.8f % 6.8f\n’, ii, boxbeam_stress (IJK, 1), boxbeam_stress (IJK, 2)); end Nodal displacements Node X-disp Y-disp Z-disp −0.00143909 0.00009891 −0.01149128 −0.00142851 0.00005433 −0.01128080 −0.00135326 0.00009942 −0.00824795 −0.00133865 0.00003701 −0.00805697 CHAPTER 23 MATLAB Programs for Finite Element Analysis −0.00118348 0.00010738 −0.00530461 −0.00115996 0.00000364 −0.00514629 −0.00090429 0.00011121 −0.00280302 −0.00087101 −0.00003803 −0.00269616 −0.00051266 0.00009874 −0.00096246 10 −0.00047439 −0.00007322 −0.00091898 11 0.00137522 −0.00008099 −0.01131236 12 0.00136999 −0.00004809 −0.01109938 13 0.00131777 −0.00008645 −0.00820368 14 0.00130824 −0.00003279 −0.00801115 15 0.00115913 −0.00009939 −0.00526670 16 0.00114000 −0.00000073 −0.00510831 17 0.00088969 −0.00010641 −0.00276502 18 0.00086045 0.00003853 −0.00266030 19 0.00050770 −0.00009675 −0.00092457 20 0.00047142 0.00007256 −0.00089349 Element Principal Stress Principal Stress 10 −799.42490921 −1313.09508435 20 1304.82999848 794.41165622 30 1295.10699450 850.47289482 40 1394.79009474 897.81264906 Stress in the structure The results given by the program CST3D.m can be seen to be identical to the results shown in Table 10.3 697 23.6 TEMPERATURE DISTRIBUTION IN ONE-DIMENSIONAL FINS To find the temperature distribution in a 1D fin (details given in Section 14.2), a program called heat1.m is developed The program requires the following quantities: NN = number of nodes (input) NE = number of elements (input) NB = semibandwidth of the overall matrix GK (input) IEND = 0: means no heat convection from free end IEND = any nonzero integer: means that heat convection occurs from the free end (input) CC = thermal conductivity of the material, k (input) H = convection heat transfer coefficient, h (input) TINF = atmospheric temperature, T∞ (input) QD = strength of heat source, q (input) Q = boundary heat flux, q (input) NODE = array of size NE × 2; NODE(I, J) = global node number corresponding to J-th (right-hand side) end of element I (input) XC = array of size NN; XC(I) = x coordinate of node I (input) A = array of size NE; A(I) = area of cross-section of element I (input) PERI = array of size NE; PERI(I) = perimeter of element I (input) TS = array of size NN; TS(I) = prescribed value of temperature of node I (input) If the temperature of node I is not specified, then the value of TS(I) is to be given as 0.0 The program heat1.m requires the following subprograms: adjust.m, decomp.m, and solve.m The following example illustrates the use of the program heat1.f PART ABAQUS and ANSYS Software and MATLAB®Programs for Finite Element Analysis EXAMPLE 23.6 Find the temperature distribution in the 1D fin considered in Example 14.4 and shown in Figure 14.1 with two finite elements Solution A main program called main_heat1.m is created for solving the problem The listing of the program main_heat1.m and the nodal temperatures given by the program are shown below The results can be seen to agree with the results obtained in Example 14.4 with hand computations clc; clear all; % % Written by Singiresu S Rao % The Finite Element Method in Engineering % NN=3; NE=2; NB=2; IEND=1; CC=70; H=5; TINF=40; QD=0; Q=0; NODE=[1 2; 3] ; XC= [0; 2.5; 5]; A=[3.1416; 3.1416]; PERI=[6.2832; 6.2832]; 698 TS=[140; 0; 0]; PLOAD = HEAT1 (NN, NE, NB, IEND, CC, H, TINF, QD, Q, NODE, XC, A, PERI, TS); fprintf (‘%s\n’, ‘Node Temperature’) for i = 1:NN fprintf (‘%2.0f %15.4f\n’, i, PLOAD (i)) end Node Temperature 140.0000 80.4475 63.3226 23.7 TEMPERATURE DISTRIBUTION IN ONE-DIMENSIONAL FINS INCLUDING RADIATION HEAT TRANSFER To find the temperature distribution in a 1D fin including radiation heat transfer (details given in Section 14.6), a program called radiat.m is developed The program requires the following quantities: EPSIL = emissivity of the surface (input) EPS = a small number of the order of 10–6 for testing the convergence of the method (input) SIG = Stefan–Boltzmann constant = 5.7 × 10–8 W/m2-K4 (input) ITER = number of iterations used for obtaining convergence of the solution (output) The other quantities NN, NE, NB, IEND, CC, H, TINF, QD, NODE, P, PLOAD, XC, A, PERI, and TS have the same meaning as in the case of the subprogram heat1.m CHAPTER 23 MATLAB Programs for Finite Element Analysis The program radiat.m requires the following subprograms: adjust.m, decomp.m, and solve.m The following example illustrates the use of the program radiat.f EXAMPLE 23.7 Find the temperature distribution in the 1D fin considered in Example 14.13 using one finite element Solution A main program called main_radiat.m is created for solving the problem The listing of the program main_radiat.m and the nodal temperatures given by the program are given below It can be seen that the present results agree with those obtained in Example 14.13 using hand computations clear all; clc; close all; % % Written by Singiresu S Rao % The Finite Element Method in Engineering % fprintf(‘%s\n’, ‘Iteration Error Nodal Tempature’) P = zeros (2, 1); PLOAD = zeros (2, 1); GK = zeros (2, 2); EL = zeros (1, 1); PERI = zeros (1, 1); NN = 2; NE = 1; NB = 2; IEND = 0; CC = 70; H = 5; TINF = 40; QD = 0; Q=0; EPSIL = 0.1; EPS = 0.0001; SIG = 5.7e–8; NODE = [1, 2]; XC = [0;5]; 699 A = 3.1416; PERI = 6.2832; TS = [140;0]; PLOAD = radiat (NN, NE, NB, IEND, CC, H, TINF, QD, Q, EPSIL, EPS, SIG, NODE, XC, A, PERI, TS, P, PLOAD, GK,EL); Iteration Error Nodal Temperature 1.0000 140.0000 58.4783 0.0145 140.0000 52.2292 0.0002 140.0000 52.3106 0.0000 140.0000 52.3095 23.8 TWO-DIMENSIONAL HEAT TRANSFER ANALYSIS To find the solution of a 2D heat transfer problem, such as the temperature distribution in a plate (details given in Section 15.2), a program called heat2.m is developed using linear triangular elements The program requires the following quantities: NN = number of nodes (input) NE = number of triangular elements (input) NB = semibandwidth of the overall matrix (input) NODE = array of size NE × 3; NODE(I, J) = global node number corresponding to the J-th corner of element I (input) XC, YC = array of size NN, XC(I), YC(I) = x and y coordinates of node I (input) CC = thermal conductivity of the material, k (input) QD = array of size NE; QD(I) = value of q for element I (input) ICON = array of size NE; ICON = if element I lies on convection boundary and = otherwise (input) PART ABAQUS and ANSYS Software and MATLAB®Programs for Finite Element Analysis NCON = array of size NE × 2; NCON(I, J) = J-th node of element I that lies on convection boundary (input) Need not be given if ICON(I) = for all I Q = array of size NE; Q(I) = magnitude of heat flux for element I (input) TS = array of size NN; TS(I) = specified temperature for node I (input) If the temperature of node I is not specified, then the value of TS(I) is to be set equal to 0.0 H = array of size NE; H(I) = convective heat transfer coefficient for element I (input) TINF = array of size NE; TINF(I) = ambient temperature for element I (input) The program heat2.m requires the following subprograms: adjust.m, decomp.m, and solve.m The following example illustrates the use of the program heat2.f EXAMPLE 23.8 Find the temperature distribution in the square plate with uniform heat generation considered in Example 15.2 using triangular finite elements Solution A main program called main_heat2.m is created for solving the problem The listing of the program main_ heat2.m and the nodal temperatures given by the program are shown below clear all; clc; % % Written by Singiresu S Rao % The Finite Element Method in Engineering % NN = 9; NE = 8; 700 NB = 4; CC = 30; Node = [1, 4, 2, 5, 4, 7, 5, 8; 2, 2, 3, 3, 5, 5, 6, 6; 4, 5, 5, 6, 7, 8, 8, 9]’; XC = [0.0, 5.0, 10.0, 0.0, 5.0, 10.0, 0.0, 5.0, 10.0]’; YC = [0.0, 0.0, 0.0, 5.0, 5.0, 5.0, 10.0, 10.0, 10.0]’; QD = [100.0, 100.0, 100.0, 100.0, 100.0, 100.0, 100.0, 100.0]’; ICON = [0, 0, 0, 0, 0, 0, 0, 0]’; Q = [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]’; TS = [0.0, 0.0, 50.0, 0.0, 0.0, 50.0, 50.0, 50.0, 50.0]’; H = [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]’; TINF = [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]’; PLOAD = HEAT2(NN, NE, NB, NODE, XC, YC, CC, QD, ICON, Q, TS, H, TINF); fprintf(‘ Nodal No Nodal temperature\n’) for i=1:length (PLOAD) fprintf(‘%8.0f %20.4f\n’, i, PLOAD(i)) end Nodal No Nodal temperature 133.3333 119.4444 50.0000 119.4444 105.5556 50.0000 50.0000 50.0000 50.0000 The results can be seen to be identical to those obtained in Example 15.2 using hand computations CHAPTER 23 MATLAB Programs for Finite Element Analysis 23.9 CONFINED FLUID FLOW AROUND A CYLINDER USING POTENTIAL FUNCTION APPROACH To find the potential function distribution for confined inviscid and incompressible fluid flow around a cylinder (details given in Section 18.3), a program called phiflo.m is developed The potential function approach with linear triangular elements is used The program requires the following quantities: NN = number of nodes (input) NE = number of elements (input) NB = semibandwidth of the overall matrix GK (input) XC, YC = array of size NN; XC(I), YC(I) = x and y coordinates of node I (input) NODE = array of size NE × 3; NODE(I, J) = global node number corresponding to J-th corner of element I (input) GK = array of size NN × NB used to store the matrix ½KŠ ! ~ P = array of size NN used to store the vector P ~ Q = array of size NE; Q(I) = velocity of the fluid leaving the element I through one of its edges (input) A = array of size NE; A(I) = area of element I PS = array of size NN; PS(I) = specified value of ϕ at node I If the value of ϕ is not specified at node I, then the value of PS(I) is to be set equal to –1000.0 (input) ICON = array of size NE; ICON(I) = if element lies along the boundary on which the velocity is specified, and = otherwise (input) NCON = array of size NE × 2; NCON(I, J) = J-th node of element I that lies on the boundary on which the velocity is specified Need not be given if ICON(I) = for all I (input) PLOAD = array of size NN × used to store the final right-hand-side vector It represents the solution vector (nodal values of ϕ) upon return from the subroutine PHIFLO The program phiflo.m requires the following subprograms: adjust.m, decomp.m, and solve.m The following example illustrates the use of the program phiflo.f EXAMPLE 23.9 Find the potential function distribution in the confined flow around a cylinder considered in Example 18.4 using triangular finite elements Solution A main program called main_phiflo.m is created for solving the problem The listing of the program main_phiflo.m and the nodal values of potential function given by the program are shown below clear all; clc; close all; % % Written by Singiresu S Rao % The Finite Element Method in Engineering % A = zeros (13, 1) ; PLOAD = zeros (13, 1) ; P = zeros (13, 1) ; NN = 13; NE = 13; NB = 7; XC = [0.0, 5.0, 9.17, 12.0, 0.0, 5.0, 9.17, 12.0, 0.0, 5.0, 8.0, 9.17, 12.0] ; YC = [8.0, 8.0, 8.0, 8.0, 4.0, 4.0, 5.5, 5.5, 0.0, 0.0, 0.0, 2.83, 4.0] ; ICON = [1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0] ; (Continued ) 701 PART ABAQUS and ANSYS Software and MATLAB®Programs for Finite Element Analysis EXAMPLE 23.9 (Continued ) NODE = [5, 6, 6, 7, 7, 7, 5, 9, 10, 11, 6, 7, 7; 1, 1, 2, 2, 3, 4, 6, 6, 6, 6, 7, 12, 8; 6, 2, 7, 3, 4, 8, 9, 10, 11, 12, 12, 13, 13]’; NCON = [5, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0; 1, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0; 0 0 0 0 0 0 ]’; GK = zeros (NN, NB) ; for I = 1:NN PS (I) = –1000.0; end PS (4)=0.0; PS (8)=0.0; PS (13)=0.0; Q = [–1.0, 0.0, 0.0, 0.0, 0.0, 0.0, –1.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]; PLOAD = Phiflo (XC, YC, NODE, ICON, NCON, GK, A, PS, PLOAD, Q, P, NN, NE, NB); fprintf (‘%s\n’, ‘Node Potential function’) for i = 1:NN fprintf (‘%4.0f %15.4f\n’, i, PLOAD (i)) end Node 702 Potential function 14.9004 9.6754 4.4818 0.0000 15.0443 10.0107 4.7838 0.0000 15.2314 10 10.5237 11 8.4687 12 6.2288 13 0.0000 23.10 TORSION ANALYSIS OF SHAFTS To find the stress function distribution in a solid prismatic shaft subjected to a twisting moment (details given in Section 20.3), a program called torson.m is developed Linear triangular elements are used for modeling the cross-section of the shaft The program requires the following quantities: NN = number of nodes NE = number of elements NB = semibandwidth of the overall matrix GK XC, YC = array of size NN; XC(I), YC(I) = x and y coordinates of node I NFIX = number of nodes lying on the outer boundary (number of nodes at which ϕ = 0) NODE = array of size NE × 3; NODE(I, J ) = global node number corresponding to J-th corner of element I G = shear modulus of the material THETA = angle of twist in degrees per 100-cm length CHAPTER 23 MATLAB Programs for Finite Element Analysis IFIX = array of sixe NFIX; IFIX(I ) = I-th node number at which ϕ = A = array of size NE denoting the areas of elements; A(I) = area of I-th triangular element The program torson.m requires the following subprograms: adjust.m, decomp.m, and solve.m The following program illustrates the use of the program torson.f EXAMPLE 23.10 Find the stresses developed in prismatic shaft with a cm × cm cross-section that is subjected to a twist of degrees per meter length using triangular finite elements This example is the same as Example 20.1 Solution A main program called main_torson.m is created for solving the problem The listing of the program main_torson.m, the nodal values of stress function, and the shear stresses developed in the elements given by the program are shown below clear all; clc; close all; % —————————————————————————————————————————————————— % Written by Singiresu S Rao % The Finite Element Method in Engineering % —————————————————————————————————————————————————— NN = 6; NE = 4; NB = 5; NFIX = 3; G = 0.8e6; THETA = 2.0; NODE = [2 6;3 3;1 2 2]’; XC = [2 2]; YC = [2 1 0 0]; IFIX = [1 6]; A = zeros (4, 1); PLOAD = Torson (NN, NE, NB, NFIX, G, THETA, NODE, XC, YC, IFIX, A); fprintf (‘%s\n’, ‘Node Value of stress function’) for i = 1:NN fprintf (‘%4.0f %15.4f\n’, i, PLOAD (i)) end Node Value of stress function 0.0000 418.8482 0.0000 651.5416 465.3869 0.0000 PROBLEMS 23.1 Solve Problem 7.11 using the MATLAB program choleski.m 23.2 Modify the matrix and the right hand side vector considered in Problem 6.22 to incorporate the boundary conditions T5 = T6 = 50°C Use the MATLAB program adjust.m 23.3 Find the nodal displacements and element stresses in the truss considered in Problem 9.7 and Figure 9.18 using the MATLAB program truss3D.m 23.4 Find the stresses developed in the plate shown in Figure 10.16 using at least 10 CST elements Use the MATLAB program cst.m 23.5 Find the nodal displacements and element stresses developed in the box beam described in Section 10.3.3 and Figure 10.7 using the finite element model shown in Figure 10.8 Use the MATLAB program CST3D.m 23.6 Solve Problem 14.8 using the MATLAB program heat1.m 23.7 Solve Problem 14.8 by including radiation heat transfer from the lateral and end surfaces of the fin using the MATLAB program radiat.m Assume ε = 0.1 703 PART ABAQUS and ANSYS Software and MATLAB®Programs for Finite Element Analysis 23.8 Find the temperature distribution in the plate considered in Example 15.2 using 16 triangular finite elements Use the MATLAB program heat2,m 23.9 Determine the velocity distribution in the region ABCDEA shown in Figure 18.1 using at least 25 triangular elements Use the MATLAB program phiflo.m 23.10 Find the stress distribution in the elliptic shaft subjected to a torsional moment described in Problem 20.7 and Figure 20.3 using the MATLAB program torson.m Use at least 10 triangular elements in a quarter of the cross section 704 APPENDIX Green-Gauss Theorem (Integration by Parts in Two and Three Dimensions) In the derivation of finite element equations for two-dimensional problems, we need to evaluate integrals of the type ZZ ∂ϕ dx dy (A.1) ψ ∂x S where S is the area or region of integration, and C is its bounding curve We can integrate Eq (A.1) by parts, first with respect to x, using the basic relation Zxr Zxr u dv = − xl xr  v du + uv (A.2) xl xl to obtain ZZ S ∂ϕ ψ dx dy = − ∂x ZZ ∂ψ ϕ dx dy + ∂x S Zy2 xr  ðψϕÞ ⋅ dy xl (A.3) y = y1 where (xl, xr) and (y1, y2) denote the limits of integration for x and y as shown in Figure A.1 However, dy can be expressed as dy = ±dC ⋅ lx (A.4) where dC is an element of the boundary curve, lx is the cosine of the angle between the normal n and the x direction, and the plus and minus signs are applicable to the right-and left-hand-side boundary curves (see Figure A.1) Thus, the last term of Eq (A.3) can be expressed in integral form as Zy2 I  xr  ðψϕÞ ⋅ dy = ψ ϕ dClx (A.5) xl y1 C Thus, the integral of Eq (A.1) can be evaluated as ZZ I ZZ ∂ϕ ∂ψ dx dy = − ϕ dx dy + ψ ψ ϕlx dC ∂x ∂x S S The Finite Element Method in Engineering DOI: 10.1016/B978-1-85617-661-3.00034-9 © 2011 Elsevier Inc All rights reserved C (A.6) 705 APPENDIX Green-Gauss Theorem (Integration by Parts in Two and Three Dimensions) y xr n dy xI n dC dC S C y1 y2 x FIGURE A.1 Limits of Integration for x and y 706 Similarly, if the integral (A.1) contains the term ð∂ϕ/∂yÞ instead of ð∂ϕ/∂xÞ, it can be evaluated as ZZ I ZZ ∂ϕ ∂ψ ψ ψ ϕly dC dx dy = − ϕ dx dy + ∂y ∂y S S where ly is the cosine of the angle between the normal n and the y direction Equations (A.6) and (A.7) can be generalized to the case of three dimensions as ZZZ I ZZZ ∂ϕ ∂ψ dx dy dz = − ϕ dx dy dz + ψ ψ ϕlx dS ∂x ∂x V (A.7) C V (A.8) S where V is the volume or domain of integration and S is the surface bounding the domain V Expressions similar to Eq (A.8) can be written if the quantity ð∂ϕ/∂yÞ or ð∂ϕ/∂zÞ appears instead of ð∂ϕ/∂xÞ in the original integral INDEX Page numbers in italics indicate figures, tables and footnotes A ABAQUS fixed-fixed beam, 632 plate subjected to transverse load, 645 10-bar planar truss, 635 4-bar space truss, 638 25-bar space truss, 641 adjust.m, 686 Anisotropic materials, 285–287 stress-strain relations, 285–287 ANSYS analysis of a two-dimensional truss, 668–678 finite element discretization, 665–666 GUI layout, 664 heat transfer in a steel rod, 678–681 material properties, 666 mesh density control, 666 meshing methods, 666 stages in solution, 667–681 system of units, 667 terminology, 664–665 Area coordinates, 101, 102, 104, 115 Assembly of element equations, 204–211 Assembly procedure, 210 computer implementation, 250 Automatic mesh generation, 65–68 Axisymmetric heat transfer, 531 Axisymmetric problems, 531–536, 541 Axisymmetric ring element, 413–417, 423 B Bandwidth, 63–64 Bar element under axial load, 29 Basic concept, 3–4 Basic characteristics of fluids, 549–550 Basic element shapes, 53–55 Basic equations of fluid mechanics, 549–569 Basic equations of heat transfer, 473–475 Basic equations of solid mechanics, 277–309 Basic procedure, 51 Beam element, 24, 323–328, 338, 434 axial displacement, 324 axial strain, 324 axial stress, 323 deformation, 325 Beams, 311–352 Bending of plates, 376 Bernoulli equation, 564–566 Biharmonic operator, 592 Bingham fluid, 603 Boundary conditions computer implementation, 41 incorporation, 174, 211–219, 686–687 Boundary value problem, 9, 573 Box beam, 59, 59 C Calculus of variations, 159–162 Capacitance matrix, 502 Cauchy condition, 614 Central differences, 266 Characteristics of fluids, 549–550 Characteristics of stiffness matrices, 338–339 Checking the results of finite element analysis, 41 Choleski method, 245–250 choleski.m, 685 Circumference of a circle approximation, 4, bounds, circumscribed polygon, inscribed polygon, Classical interpolation functions, 130 Coaxial cable, 197 Collocation method, 175–176 Commercial packages, 40 Commutative, 160 Comparative study of elements, 139–140 Comparison of finite element method with other methods, 41, 169 Compatibility, 304, 144–145 Compatibility equations, 291–293, 295, 303 Compatible element, 83 Complementary energy, 297–298 Complete element, 83 Complex element, 78 Compliance matrix, 286 Composite wall, 50, 511, 514 Conduction, 474 Confined flow, 572 Conforming element, 83, 384, 384 Constant strain triangle, 228 Consistent load vector, 314, 325, 360–362, 404–409 Consistent mass matrix bar element, 430–431 beam element, 434 in global coordinate system, 439–440 planar frame element, 436 planar truss element, 432–433 space frame element, 434–435 space truss element, 431–434 tetrahedron element, 438–439 triangular bending element, 437–438 triangular membrane element, 436–437 Constitutive relations, 280–285 Continuity conditions, 137–139 Continuity equation, 551 Continuum problems approximate methods, 159 specification, 158–159 Convection, 474 Convection heat loss from end, 492–496 Convective boundary condition, 531 Convergence requirements, 82–85, 170–175 Coordinate transformation, 199–204 Critically damped case, 455 CST element, 229, 362, 691–693 cst.m, 691, 692 CST3D.m, 694 Cubic element, 120–125, 129–130, 134–135 Cubic interpolation model, 24–28, 120, 122, 124, 129 Cubic model, 78, 337 Current flow, 33–40 Curved-sided elements, 143–144, 143 Cylindrical coordinate system, 476 Cr continuity, 83 C0 continuity, 137–138 C1 continuity, 137–139 D Damped system, 453–455 Damped wave equation, 622 Damping matrix, 454 Darcy law, 587 Decomposition of a matrix, 245–246 Diffusion equation, 139, 140 Direct approach, 28, 158 bar under axial load, 29 current flow, 33–40 fluid flow, 32–33 heat flow, 30–31 Direct integration method, 265–266 Directional constraint, 228, 228 Dirichlet condition, 477, 613 Discretization of domain, 53–73 Displacement-force method, 294, 295 Displacement method, 294 707 INDEX Discretization process, 56–62 Dissipation function, 428, 556, 557 Drilling machine, 65, 65, 347 Dynamic analysis, 427–469 Dynamic equations of motion, 427–430 Dynamic response, 452–460 E 708 Eigenvalue analysis, 251 Eigenvalue problem, 167–168, 182–183, 251–254 Electrical network, 34 Electric resistor element, 33–40 Electric potential, 613 Electrostatic field, 614 Element capacitance matrix, 502 Element characteristic matrix, 34, 36, 205, 579, 586 Element characteristic vector, 205 Element damping matrix, 205, 211, 586 Element equations, derivation, 145–147 Element load vector, 18, 205, 207, 326 due to body forces, 302 due to initial strains, 302 due to surface forces, 302 Element mass matrix, 430, 438 Element matrices assembly, 199–240 derivation, 157–197 direct approach, 158 variational approach, 158 Element shapes, 53–55 Element stiffness matrix, 26, 205, 303, 412 Element vectors, derivation, 157–197 Energy balance equation, 474, 475, 556 Energy equation, 556 Energy generated in a solid, 474 Energy stored in a solid, 475 Engineering applications, 9, 10, 454–455 Equations of motion, 430, 452, 453–455, 552–556, 560 in fluid flow, 555–556 Equilibrium equations, 11, 294, 295, 299, 302, 304, 339, 616 external, 278–279 internal, 279–280 Equilibrium problem, 175, 242–251, 241 Essential boundary condition, 164, 168 Euler equation, 161, 556 Eulerian method, 550, 550 Euler-Lagrange equation, 161, 162 Exact analytical solution, 278 Expansion theorem, 453 F Field problem, 241, 613 Field variable, 29, 30, 82 Fighter aircraft, Finite difference method, 266 Finite difference solution, 505–507 Finite element, 3, 4, 9, 53, 78, 624 Finite element equations, 28–40, 169–174, 184–189, 241–274, 299–303, 480–484, 604–605, 615 Finite element method basic procedure, 169 comparison with other methods, 41, 169 displacement method, 294 engineering applications, 9, 10 general applicability, 7–9 general description, 9–12 historical background, 4–6 overview, 3–50 program packages, 40 Fin, temperature distribution, 492, 495, 697–699 Fixed-fixed beam, 44, 350 Flexural rigidity, 378 Flow curve characteristic, 602–603 Flow field, 550 Flow through nozzle, 565 Fluid film lubrication, 614 Fluid flow, one-dimensional, Fluid mechanics problems basic equations, 549–569 continuity equation, 551–552 energy, state, and viscosity equations, 556 inviscid and incompressible flow, 571 momentum equations, 552–556 viscous and non-Newtonian flow, 591–609 Fluids, 549–550 Flutter problem, 460–461 Forced boundary condition, 162 Force method, 295 Fourier equation, 8, 476 Frame element, 240, 328–338, 434–436 Frames, 311–352 Free boundary condition, 161 Free vibration analysis, 440–452 Functional, 18, 21, 22, 82, 160–162 initial conditions, 477–479 spherical coordinates, 477 Helical spring, 59 Heat transfer problems axisymmetric problems, 531–536, 541 basic equations, 473–475 boundary conditions, 477–479 cylindrical coordinates, 476 finite element equations, 480–484 governing equation, 476 initial conditions, 477–479 one-dimensional problems, 7–8, 489–515 spherical coordinates, 477 three-dimensional problems, 531–545 two-dimensional problems, 517–530 Hermite interpolation formula, 130 Hermite polynomials first-order, 133–134 zeroth-order, 132–133 Hexahedron element, 55, 409–413, 422 Higher order element natural coordinates, 121–130 one-dimensional, 120–121 three-dimensional, 129–130 two-dimensional, 125–128 Hillbert matrix, 273 Historical background, 4–6 h-method, 76, 85 Homogeneous solution, 455–456 Hyperosculatory interpolation formula, 130 I Galerkin method, 178–180, 573, 604–605 Gaussian elimination method, 243–245 Gauss integration, 148 one dimension, 148 two dimensions, 149 three dimensions, 150 Gauss integration rectangular region, 149 tetrahedral region, 151 triangular region, 150 Geometric boundary condition, 212 Generalized Hooke’s law, 285 Global coordinates, 85–96, 334, 359, 439–440 Global stiffness matrix, 332–336 Green-Gauss theorem, 706 Idealization of aircraft wing, 58 Incompressible flow, 571–589 Infinite body, 61–62 Infinite element, 63 Initial conditions, 477–479 Initial strain, 281 Intergration by parts, 186, 623, 706 Interpolation function, 75, 77–78, 96, 127, 130–134 Interpolation model, 75–117 Interpolation polynomial local coordinates, 99–107 selection of the order, 80–82 vector quantities, 96–99 Inverse of a matrix, 247–250 Inviscid fluid flow, 559–560 in a tube, 21 Irrotational flow, 560–561 of ideal fluids, 614 Isoparametric element, 76, 119–156 Isotropic materials, 280–285 H J Hamilton’s principle, 298–299 Harmonic operator, 592 heat1.m, 697 heat2.m, 699 Heat conduction, 480 Heat flow equation boundary conditions, 477–479 Cartesian coordinates, 475 cylindrical coordinates, 476 Jacobian, 143 Jacobian matrix, 126, 147 Jacobi method, 254–256 G L Lagrange equations, 428 Lagrange interpolation functions, 130–131 Lagrangian, 298, 550 Lagrangian approach, 550 INDEX Laminar flow, 550 Laplace equation, 8, 615 Least squares method, 180–182 Linear element, 55, 76, 126–128, 127, 134 Linear interpolation polynomials, 85–96, 99–107 Linear model, 77 Linear triangle element, 150 Line element, 54, 56, 66 for heat flow, 30–31 Load vector, 302, 317 Local coordinates one-dimensional element, 100–101 three-dimensional element, 104–107 two-dimensional element, 101–104 Location of nodes, 60 Lower triangular matrix, 245–246 LST, 371 Lumped mass matrix bar element, 430–431 beam element, 434 in global coordinate system, 439–440 planar frame element, 436 planar truss element, 434 space frame element, 434–435 space truss element, 431–434 M Magnetostatics, 614 Mapping of elements, 142 Mass matrix beam element, 434 planar frame element, 436 space frame element, 434–435 tetrehedron element, 438–439 triangular bending element, 437–438 triangular membrane element, 436–437 MATLAB, 41, 683 MATLAB programs adjust.m, 686 choleski.m, 684, 685 cst.m, 691 CST3D.m, 694 heat1.m, 697 heat2.m, 699 phiflow.m, 701 radiat.m, 698 torson.m, 702 truss3D.m, 687 MATLAB program for analysis of plates under in-plane loads, 691–693 analysis of plates (three-dimensional structures), 694–697 analysis of space trusses, 687–691 confined fluid flow around a cylinder, 701–702 incorporation of boundary conditions, 686–687 solution of linear simultaneous equations, 684–686 temperature distribution in onedimensional fins, 697–698 temperature distribution in fins (with radiation), 698–699 torsion analysis of shafts, 702–703 two-dimensional heat transfer, 699–700 Matrix inversion, 42 Maxwell-Betti reciprocity theorem, 14, 338–339 Maxwell’s theorem, 14 Membrane element, 367–369 Mesh refinement, 83, 303 Milling machine structure, 3, Mode superposition method, 267–268 Momentum equations, 552–556, 592 M-orthogonalization of modes, 449–452 Multiplex element, 78 Multipoint constraint, 212, 223–226, 229 N NASTRAN, 40 Natural boundary conditions, 161, 212, 596, 599 Natural coordinates higher-order elements, 121–130 integration, 108–109 one-dimensional element, 100–101 three-dimensional element, 101–104 two-dimensional element, 125–126 Nature of finite element solutions, 303 Navier–Stokes equations, 555, 598–600 Network of pipes, 32, 32 Neumann condition, 477, 573, 585, 614 Newmark method, 266–267 Newtonian fluid, 550, 552–554, 567, 602, 603 Nodal interpolation functions, 121–122, 136, 138, 325 Node numbering scheme, 63–65, 64, 93 Nonconforming element, 384 Nonconservative problems, 460–461 Non-Newtonian fluid, 550, 567, 602–607 Number of elements, 60, 76, 137, 205, 401 Numerical integration one-dimension, 148–149 three-dimensions, 150–151 two-dimensions, 149–150 Numerical solution, 241–274 O Octree method, 66 One-dimensional element, 54, 54, 77, 85–88, 100–101, 108, 120–124, 134–135, 140, 231 One-dimensional fluid flow, One-dimensional heat transfer, 7–8, 489, 499–502 Order of polynomial, 138, 148 Orthogonalization of modes, 449–452 Orthotropic materials, 286 Osculatory interpolation formula, 130 Overdamped case, 455 Overview, 3–50 P Parallel processing, 268–269 Partial differential equations elliptic, 614 hyperbolic, 614 parabolic, 614 Particular integral, 455–456 Pascal pyramid, 81 Pascal tetrahedron, 81 Pascal triangle, 81 Patch test, 109–111 Path line, 550 Penalty method multipoint constraints, 223–226 single point constraints, 223 symmetry conditions, 226–228 phiflo.m, 701 Pipe element, 32–33 Planar frame element, 337–338, 337, 436 Planar truss, 206, 311 Plane strain, 146, 282, 291, 413 Plane stress, 146, 281, 291, 357 Plate inplane loads, 355, 386, 691–693 transverse loads, 355 Plate bending element, 379–383 Plate fin, 238, 514, 515 Plate under tension, 367–369 Plate with a hole, 59, 60, 61 Plates, 231–232, 355–399, 691–693 Poisson equation, 8, 604, 615 Polynomial approximation, 76 Polynomial form, 77–78 Post-processing, 41 Potential energy, 159, 190, 220, 223, 227, 295–297, 299 Potential function, 8, 561, 701–702 Potential function formulation, 573 Power method, 254, 256–261 Prandtl’s stress function, 616 p-refinement, 85 Pre-processing, 40, 663 Primary nodes, 76 Principal stresses, 293–294 Principle of minimum complementary energy, 297–298 Principle of minimum potential energy, 190, 295–299 Principle of stationary Reissner energy, 298 Program packages, 40 Propagation problems, 168–169, 183–184, 242, 262–268 Q Quadratic element, 120–122, 124, 128–129, 134, 499–502 Quadratic model, 121, 499, 503 Quadrilateral element, 54, 109, 109, 125–128, 143 Quadtree method, 66 Quasi-harmonic equation, 613–627 R radiat.m, 698–699 Radiation, 474, 507–511, 698–699 Radiation heat transfer coefficient, 508 Rate equation, 474–475 Rayleigh-Ritz method, 163–169 Rayleigh-Ritz subspace iteration, 261–262 Rayleigh’s method, 163–169 709 INDEX Rectangular element bending loads, 377, 384 inplane loads, 372–376 Reissner energy, 298 Reynolds equation, 33 Reynolds number, 33 Rigid element, 33 Ring element, 413–417, 541 Rotational flow, 561 r-refinement, 85 Runge-Kutta method, 264 S 710 Secondary nodes, 99 Second-order differential equation, 453, 455–460 Seepage flow, Semi-bandwidth, 684 Shape functions, 77, 86, 96, 124, 127, 142–143, 146, 373, 496 Shape of elements, 53–55, 665 Simplex, 78 Simplex element one-dimensional, 85–88, 95 three-dimensional, 91–95, 105 two-dimensional, 88–90, 95, 101 Size of elements, 59–60 Software applications, 41 Software for finite element analysis, 629 Solid bar under axial load, Solid mechanics basic equations, 277–294 boundary conditions, 290–291 compatibility equations, 291–293 constitutive relations, 280–285 equilibrium equations, 278–280 formulations, 294–299 strain-displacement relations, 287–290 stress-strain relations, 285–287 three-dimensional problems, 278 Solids of revolution, 413–421 Solution of differential equations, 263–265, 601 Solution of finite element equations, 241–274 Space frame element global stiffness matrix, 332–336 transformation matrix, 333–334 Space-time finite element, 624 Space truss element, 239, 312–323, 431–434 Spherical coordinate system, 477 Spring element, 29–30 Springs in combination, 34 Stability problems, 460–461 Standard eigenvalue problem, 252–254 State equation, 505 State of stress in fluid, 552 Static analysis, 299–303 Static condensation, 136, 461 Stationary value, 160–163 Steady-state field problem, 614 Steady-state problem, 491, 615 Stefan-Boltzmann constant, 474, 507 Stepped bar, 29, 29 Stiffened plates and shells, 231–232 Stiffness matrix, 11, 24, 145, 205, 229, 314, 330, 331–332, 338, 382, 437 Straight uniform fin analysis, 489–492 Strain displacement relations, 287–290, 403, 410–412, 415 Strain energy density, 13, 14 Stream function, 562–564 formulation, 584–586, 592–596 vorticity formulation, 600–602 Streamline, 550, 562, 572, 596–595 Stress analysis of beams, 25 Stress analysis of stepped bar, 12 Stress concentration, 60, 369 Stress function, 292–291, 616–617 Stress-rate of strain relations, 553–554 Stress-strain relations, 280–287, 294, 357, 378, 410–412 Strong form formulation, 189–191 Subdomain collocation method, 176–178 Subparametric element, 141 Subspace iteration method, 261–262 Substructures method, 461 Superparametric element, 141 Symmetric geometry, 61 Symmetry conditions, 61, 226–228, 385 System equations, 37, 39, 211, 268, 461, 502, 599, 615, 620 T Tapered fin analysis, 496–499 Temperature distribution in a fin, 18, 492–494, 501 Tentative solution, 160 Tesselation method, 66, 67 Tetrahedral coordinates, 105, 129 Tetrahedron element, 93, 106–107, 116, 401–409, 404–405, 422, 466, 539 consistent load vector, 404–409 stiffness matrix, 403 Thermal diffusivity, 476, 479 Thermal strain, 281, 404 Thermal stress, 473, 681 Three-bay frame, 63, 63 Three-dimensional elements, 54, 139, 401, 440 Three-dimensional heat transfer, 536–541 Three-dimensional plate element, 386 Three-dimensional problems, 401–426, 531–545 Three-dimensional structures, 386–388, 694–697 Time domain, finite difference solution, 505–507 torson.m, 702 Torsion, prismatic shafts, 616, 702–703 Total derivative, 551 Total solution, 456 Transformation for vertical element, 336 Transformation matrix, 200, 333–334, 336, 358–360 Transient field problem, 622–624 Transient problem, 9, 269 Triangular coordinates, 124, 519 Triangular element, 54, 66, 78, 112, 138, 139 Triangular bending element, 379 Triangular membrane element, 356–367, 364, 397 Triangular ring element, 425, 541 Truss, 311–352 truss3D.m, 687–688 Truss element, 311 Turbulent flow, 550, 557 Two-bar truss, 202–201, 344, 444–445, 464 Two-dimensional elements, 54, 60, 111, 138 Two-dimensional heat transfer, 517–530, 699–701 Two-station interpolation functions, 131–132 Types of elements, 56, 57–58, 205, 303 U Uncoupling of equations of motion, 453–455 Undamped system, 453–454 Underground pipe, 72 Uniform fin analysis, 489–492 quadratic elements, 499–502 Unsteady heat transfer, 502 Unsteady state problems, 502–507, 526 axisymmetric problems, 541 three-dimensional problems, 541–542 Upper triangular matrix, 245, 247 V Variational approach, 158–163, 480–482, 592–596, 601–602 Variational formulation, 163–164, 295–299 Variational operator, 160 Vector quantities, 96–99 Velocity potential, 561–562, 571 Velocity-pressure formulation, 596–598 Vibration analysis, 440–452 Viscosity equation, 557 Viscous flow, 550 Volume coordinates, 98, 105, 105 Vorticity, 561, 600 W Weak form formulation, 189–191 Weighted residual approach, 158–159, 175–182 Weighted residual methods, 5, 175–184 Z Zeroth-order interpolation function, 132–133 ... considerations in selecting the number and types of elements, is discussed in Chapter The interpolation models in terms of Cartesian and natural coordinate systems are given in Chapter Chapter discusses the. .. stress analysis, use the stress-strain and strain-displacement relations of a beam Step 1: Idealize finite elements By assuming the two fixed ends of the beam as the end nodes and introducing an additional... system of springs connected in series as shown in Figure 1.12(a) The analysis of the system (to find the nodal displacements under a prescribed set of loads) can be conducted using the finite element