Ultimate 02% Yield Strength Strength Modulus of Sbeer Coefficient of
cr .. Cly Elasticity Modulus Thermal Expansion, Q Density. p E G
Material ksi MPa ksi MPa (ttr psi GPa) (Hf psi) 1O-6tF 1O-6rC IbJin? kglm3
Copper and its alloys CDA 145 copper,
hard 48 331(1') 44 303 16 110 6.1 9.9 17.8 0.323 8940 CDA 172 bery~ium
copper, hard 17S 1210(1') 240 965 19 131 7.1 9.4 17.0 0.298 8250 CDA 220 bronze,
hard 61 421(1') 54 372 17 117 6.4 10.2 18.4 0.318 8800 CDA 260 brass,
bard 76 524(1) 63 434 16 110 6.1 11.1 _ 20.0 0.308 8530
Magnesium alloy (8.5% AI) 55 380(1) 40 275 4.5 45 2.4 14.5 26.0 0.065 1800
Monel alloy 400 (Ni-Cu) Cold-worked 98 675(1') 85 580 26 180 7.7 13.9 0.319 8830 Annealed 80 550(1') 32 220 26 180 7.7 13.9 0.319 8830
Steel Structural
(ASTM-A36) 58 4OO(T) 36- 250 29 200 11.5 6.5 Ih7 0.284 7860 High-strength low-alloy
ASTM-A242 70 480(1) 50 345 • 29 200 11.5 6,5 11.7 0.284 7860 Quenched and tempered aHoy
ASTM-A514 120 825(1) 100 690 29 200 11.5 6.5 11.7 0.284 7860 Stainless, (302)
Cold-rolled 125 860(T) 75 520 28 190 10.6 9.6 17.3 0.286 7920 Annealed 90 620(T) 40 275 28 190 10.6 9.6 17.3 0.286 7920
Titanium alloy (6% AI, 4% V) 130 900(1) 120 825 16.5 114 6.2 5.3 9.5 0.161 4460
Concrete Medium strength 4.0 28(C) 3.5 25 5.5 10.0 0.084 2320 High strength 6.0 4O(C) 4.5 30 5.5· 10.0 0.084 2320
Granite 35 240(C) 10 69 4.0 7.0 0.100 2770 Glass, 98% silica 7 5O(Q 10 69 44.0 80.0 0.079 2190 Melamine 6 41(T) 2.0 13.4 17.0 30.0 0.042 1162 Nylon, molded 8 55(1) 0.3 2 45.0 81.0 0.040 1100 Polystyrene 48(1') 0.45 40.0 72.0 O.03S 1050 Rubbers
Natural 2 14(1) 90.0 l62.0 0.033 910 Neoprene 3.5 24(T) 0.045 1250
Timber, air dry, parallel to grain Douglas fir, construction
grade 7.2 50(C) 1.5 10.5 varies varies 0.019 525 Eastern spruce 5.4 37(C) 1.3 9 1.7- 3- 0.016 440 Southern pine,construction
grade 7.3 5O(C) 1.2 8.3 3.0 5-:4 0.022 610
The values given in !:he table are avenge mcclJaDica1 properties. Further verification may bt rICCCSSaI}' for fiDaI desi811 or analysis. For ductile maroials. !:he compressive strmgtb is lIOIIrl8Ily assumed 10 eqoallhe rensiIe streogth. Abbrrvii:aiMI: C. c:omprcssive·strength; T. 1l:8Si1e s:rength. F«- an explanation of the IIIlIIIbm associart:d wilh the aIuroimuns, cast irons, 'and stt.eIs. see ASM Metals Refe:renc:e Book. latest ed, AmeriGan Society for Melals, Metals Parle. Ohio 44073
Fourth Edition
THOMSON .. Australia Brazil Canada Mexico Singapore Spain United.Kingdom United States
A First Course in the Finite Element Method, 4e Daryl L. Logan
Copyright Q 2007 by Nelson a division of Thomson Canada Limited. Thomson Learning TM is a trademark used herein ~nder license.
ISBN: 81-315-0217-1
Printed & Bound in India by Rahul Print 0 Pack. DeIhi-20
ALL RIGHfS RESERVED. No part ofthis Publication may be reproduced, stored in a retrieval systems, transmitted, In any form or by any means-electronic, mechanical, photocopying. or otherwise without the prior permission in writing from original Publisher.
For sale in India, Pakistan, Bangladesh, Nepal and Sri Lanka only_ Circulation of this edition outside of these countries is STRICTLY PROHIBTED.
Every effort has been made to trace ownership of all copyright material and to secure perm,ission from copyright holders. In the event of any question arising as to the use of any material, we will be pleased to make the necessary corrections in futUre printings.
1 Introduction 1
1.4 General Steps of the Finite Eleme~t Method 7
I.S Applications of the Finite Element Method 15
1.6 Advantages of the Finite Element Method 19
1.7 Computer Programs for the Finite Element ~ethod 23
References 24
Problems 27
Introduction 28
2.1 Definition of the Stiffness Matrix 28
2.2 Derivation of the Stiffness Matrix for a Spring Element 29
2.3 Example of a Spring Assemblage 34
2.4 Assembling the Total Stiffness Matrix by Superposition
" '\ (Direct Stiffness Method) 37 .
2.5 Boundary Conditions 39
2.6 ~otential Energy Approach to Derive Spring Element Equations 52
iv A Contents
Introduction 65
3.1 Derivation of the Stiffness Matrix for a Bar Element in Local Coordinates 66
3.2 Selecting Approximation Functions for Displacements 72
3.3 Transformation of Vectors in Two Dimensions 75
3.4 Global Stiffness Matrix 78
3.5 Computation of Stress for a Bar in the x-y Plane 82
3.6 Solution of a Plane Truss 84
3.7 Transformation Matrix and Stiffness Matrix for a Bar in Three-Dimensional Space 92
3.8 Use of Symmetry in Structure 100
3.9 Inclined) or Skewed, Supports 103
3.10 Potential Energy Approach to Derive Bar Element Equations 109
3.11 Comparison of Finite Element Solution to Exact Solution for Bar 120
3.12 Galerkin's Residual Method and ItS Use to Derive the One-Dimensional Bar Element Equations 124
3.13 Other Residual Methods and Their Application to a One-Dimensional Bar Problem 127
References 132
Problems 132
Introduction 151
4.1 Beam Stiffness 152
4.2 Example of Assemblage of Beam Stiffness Matrices 161 4.3 Examples of Beam Analysis Using the Direct Stiffness Method 163
4.4 Distributed Loading 175
4.5 Comparison of the Finite Element Solution to the Exact Solution for a Beam 188
4.6 Beam Element with Nodal Hinge 194
. :
References 203
Problems 204
Introduction 214 5.1 Two-Dimensional Arbitrarily Oriented Beam Element 214
5.2 Rigid Plane Frame Examples 218
5.3 Inclined or Skewed Supports-Frame Element 237
5.4 Grid Equations 238
5.6 Concept of Substructure Analysis 269
References 275
Problems 275
6 Development of the Plane Stress and Plane Strain Stiffness Equations 304
Introduction 304
6.1 Basic Concepts of Plane Stress and Plane Strain 305 6.2 Derivation of the Constant-Strain Triangular Element
Stiffness Matrix and Equations 310
6.3 Treatment of Body and Surface Forces 324 '6.4 Explicit Expression for the Constant-Strain Triangle Stiffness Matrix 329 6.5 Finite Element Solution of a Plane Stress Problem 331
ReferenCes 342
Problems 343
7 Practical Considerations in Modeling; Interpreting Results; and Examples of Plane Stress/Strain Analysis
Introduction 350
7.1 Finite Element Modeling 350 7.2 Equilibrium and Compatibility of Finite Element Results 363
350
7.3 Convergence of Solution 367
7.4 Interpretation of Stresses 368 7.5 Static Condensation 369 7.6 Flowchart for the Solution of Plane Stress/Strain Problems 374 7.7 Computer Program Assisted Step-by-Step Solution, Other Models,
and Results for Plane Stress/Strain Problems 374
References 381
Problems 382
Introduction 398
8.1 Derivation of the Linear-Strain Triangular Element Stiffness Matrix and Eq"\lations 398
8.2 Example LST Stiffness Determination 403
8.3 Comparison of Elements 406
References 409
Problems 409
9.1 Derivation of the Stiffness Matrix 412
9.2 Solution of an Axisymmetric Pressure Vessel 422 9.3 Applications of Axisymmetric Elements 428 References 433
Problems 434
Introduction 443 10.1 Isoparametric Formulation of the Bar Element Stiffness Matrix 444
10.2 Rectangular Plane Stress Element 449
10.3 lsoparametric Fonnulation of the Plane Element Stiffness Matrix 452
10.4 Gaussian and Newton-Cotes Quadratufe (Numerical Integration) 463
10.5 Evaluation' of the Stiffness Matrix and Stress Matrix by Gaussian Qua~ture 469
Contents • vU
References 484 Problems 484
Introduction 490
11.1 Three-Dimensional Stress and Strain 490 11.2 Tetrahedral Element 493
11.3 Isoparametric Formulation 501
References 508 Problems 509
Introduction 514
12.1 Basic ~ncep~ of Plate Bending 514 12.2 Derivation of a Plate Bending Element Stiffness Matrix
and Equations 519 12.3 Some Plate EJemep.t Numerical Compa.tjsons 523 12.4 Computer Solution for a Plate Bending Problem 524 References 528
Problems 529·
13 Heat Transfer and Mass Transpor't S34
Introduction ·534 13.1 Derivation of the Basic Differenti3.l Equation 535 13.2 Heat Transfer'with Convection 538
13.3 Typical Units; Thermal Conductivities, K; and Heat-Transfer Codficients,h 539
13.4 One-Dimensional Finite Element Formulation Using a Variational Method S40
13.5 Two-Dimensional Finite Element FonnuJation 555 13.6 Line or Point sOurces . 564 13.7 Three-Dimensional Heat Transfer Finite Element FormUlation 566 13.8 One-Dimensional Heat Transfer with Mass Transport 569 ,
viii ... Contents
13.9 Finite Element Fonnulation of Reat Transfer with Mass Transport by Galerkin's Method 569
13JO Flowchart and Examples ofa Heat-Transfer Program 574
References 577
Problems 577
References 612 Problems 613
15 Thermal Stress 617
15.1 Fonnulation of the Thermal Stress Problem 'and Examples 617
Reference 640
Problems 641
16 Structural Dynamics and T'me-D~pen~ent Heat Transfer 647
Introduction 647 16.1 Dynamics of a Spring-Mass System 647
16.2 Direct Derivation of the Bar Element Equations 649 16'.3 Nwnerica1 Integratio~ in Time 653
16.4 Nat~l Frequencies of a One-Dimensional Bar 665
16.5 Time-Dependent One-Dimensional Bar Analysis 669
16.6 Beam Element Mass Matrices and Natural Frequ~cies 674
16.7 Truss, Plane Frame, Plane Stress/Strain, Axisymmetric, and Solid 'Element Mass Matrices: 681
16.8 Time-Dependent Heat T~~f~ 686
Contents A ix
References 702
Problems 702
Introduction 708 A.l Definition of a Matrix 708
A.2 Matrix Operations 709 .
A.3 Cofactor or Adjoint Method to Determine the Inver:se of a Matrix 716.
A.4 Inverse of a Matrix by Row Reduction 718
References 720
Problems 720
Appendix B Methods for Solution of Simultaneous Linear Equations 722
Introduction 722
B.2 Uniqueness, Nonuniqueness, and Nonexistence of Solution 723
B.3 Methods for Solving Lin~r Algebraic Equat.ions 724
B.4 Banded-5ymmetric Matrices, Bandwidth, Skyline, and Wavefront Methods 735'
References 741
Problems 742
IntfOduction 744
C.3 Stress/S~in Relationships 748
Reference' 751
Problems 152
References 758
Appendix F Properties of Structural Steel and Aluminum Shapes 759
Answers to Selected Problems 773
Index 799
The purpose of this fourth edition is again to provide a simple, basic approach to the finite element method that can be understood by both undergraduate and graduate students without the usual prerequisites (such as structural analysis) .required by most available texts in this area. The book is written primarily as a 'basic learning tool for the undergraduate student in civil and mechanical engineering whose main interest is in stress analysis and heat transfer. Howeyer, the concepts are presented in sufficiently simple form so that the book serves as a valuable learning aid for students with other backgrounds, as well as for practicing engineers. The text is geared toward .those who want to apply the finite element method to solve practical physical problems.
General principles,are presented for each topic, followed by traditional applica­ tions of these principles, which are in tum followed by computer applications where relevant. This approach is taken to· illustrate concepts used for computer analysis of large-scale problems.
The book proceeds from basic to advanced topics and can be suitably used in a two-course sequence. Topics include basic treatments of (1) simple springs and bars, leading to two· and three-dimensional truss analysis; (2) beam bending, leading to plane frame and grid analysis and space frame analysis; (3) elementary plane stress/strain elements> leading to more advanced plane stress/strain elements; (4) axisymmetric stress; (5) isoparametric formulation of the finite element method; (6) three-dimensional stress; (7) plate bending; (8) heat transfer and fluid mass transport; (9) basic fluid mechanics; (10) t~ennal stress; and (11) time.dependent stress and heat transfer.
Additional features include how to handle inclined or skewed supports, beam element with nodal hinge, beam element arbitrarily located in space, and the concept of substructure analysis.
xii ~ Preface
The direct approach, the principle of minimwn potential energy, and Galerkin's residual method are introduced at various stages, as required. to deve10p the equations needed for analysis.
Appendices provide material on the following' tOpics; (A) basic matrix algebra used throughout the text, (B) solution methods for simultaneous equations, (C) basic theory of elasticity, (D) equivalent nodal forces, (E) the principle of virtual work, and (F) properties of structural steel and aluminum shapes.
More than 90 examples apPear throughout the text. These example~ are solved "longhand" to illustrate the concepts. More than 450 end-of-chapter problems are provided to reinforc,e concepts. Answers to many problems are included in the back of the book. Those end-of-chapter problems to be solved using a computer program are marked with a computer symbol.
New features of this edition inc1uCle additional information on modeling, inter­ preting results, and comparing finite element solutions with analytical solutions. In addition, general descriptions of and detailed examples to illustrate specific methods of weighted residuals (collocation, least squares) subdomain} and Galerkin's method) are included. The Timoshenko beam stiffness matrix has ~n added to the text, along with an example comparing the solution of the Timoshenko beam ~esults with the c1assic Euler-Bernoulli beam stiffness matrix results. Also, the h and p convergence methods and shear locking are described. Over 150 new problems for solution have been included, and additional design-type problems have been added to chapters 3) 4, 5, 7, 11, and 12. New real world applications from industry have also been added. For convenience, taples of common structural steel and aluminum shapes have been added as an appendix. This edition deHberately leaves out consideration of special­ purpose computer programs and suggests that instructors choose a program they are familiar with.
Following is an outline of suggested topics for a first course (approximately 44 lectures, 50 minutes each) in which this textbook is used. '
Appendix A Appendix B Chapter 1 Chapter 2
Topic
Chapter 3, Sections 3.1-3.11 Exam I Chapter 4, Sections 4.1-4.6 Chapter 5, Sections 5.1-5.3, 5.5 Chapter 6 Chapter 7 Exam 2 Chapter 9 Chapter 10 Chapter 11 Chapter 131 Sections 13.1-13.7 Exam 3
Number of Lectures
2 3 5 1 4 4 4 3 I 2 4 3 5 1
Preface ... xiii
This outline can be used in a one-semester course for undergraduate and graduate students in civil and mechanical engineering. (If a total stress analysis emphasis is desir::d, Chapter 13 can be replaced, for instance, with material from Chapters 8 and 12, or parts of Chapters 15 and 16. The r~t of the text can be finished in a second semester course with additional material provided by the instructor.
I express my deepest appreciation to the staff at Thomson Publishing Company, especially Bill Stenquist and Chris Carson, Publishers; Kamilah Reid Burrell and Hilda Gowans, Developmental Editors; and to Rose Kernan of RPK Editorial Services, for their assistance in producing this new ectition.
I am grateful to Dr. Ted Belytschko for his excellent teaching of the finite ele-­ ment method, whi~h aided me in writing this text. I want to thank Dr. Joseph Rends for providing analytical solutions to structural dynamics problems for comparison to finite element solutions in Chapter 16.1. Also, I want to thank the many students who used the notes that developed into this text. I am especially grateful to Ron CenfeteHi, Barry Davignon, Konstantinos Kariotis, Koward Koswara, Hidajat Harintho. Hari Salemganesan, Joe Keswari, Yanping Lu, and Khailan Zhang for checking and solv­ ing problems in the first two editions of the text and for the suggestions of my students at the university on ways to make the topics in iNs book easier to understand.
I thank my present students, Mark Blair and Mark Guard of th.e University of Wisconsin-Platteville (UWP) for contributing three-dimensional models from the finite elemen~ course as shown in Figures 11-1 a and 11-1 b, respectively. Thank you also to UWP graduate students, Angela Moe, David Walgrave, and Bruce Figi for con­ tributions of Figures 7-19, 7-23, and 7-24, respectively, and to graduate student William Gobeli for creating the results for Table 11-2 and for Figure 7-21. Also, special thanks to Andrew Heckman, an alum of UWP and Design Engineer at Sea­ graves Fire Apparatus for permission to use Figure 11-10 and to Mr. Yousif Omer. Structural Engineer at John Deere Dubuque Works for allowing pennission to use Figure 1-10.
Thank you also to the reviewers of the fourth edition: Raghu· B. Agan'val, San Jose State University; H. N. Hashemi, Northeastern University; Anf Masud, University of Illinois-Chicago; S. D. Rajan, Arizona State University; Keith E. Rouch, University of Kentucky; Richard Sayles, University of Maine; Ramin Sedagbati, Concordia University, who made significant suggestions to make the book even more complete.
Finally. very special thanks to my wife Diane for her many sacrifices during the development of this fourth edition.
English Sym~ols .
A
!l
matrix relating strains to nodal displacements or relating temperature gradient to nodal temperatures
specific heat of a mate~ matrix relating stresses to nodal displacements direction cosine in two dimensions direction cosines in three dimensions element and structure nodal displacement matrix, both in global
coordinates local.coordina~ element nodal displacement matrix bending rigidity of a plate matrix relating stresses to strains operator matrix given by Eq. (10.3.16) exponential function modulus of elasticity global.coordinate nodal force matrix
local.coordinate element nodal force matrix body force matrix heat transfer force matrix heat flux force matrix
xvi ... Notation
h i,j,m I
M'
p
heat source force matrix surface force matrix global-coordinate structure force matrix condensed force matrix global nodal forces equivalent force matrix temperature gradient matrix or hydraulic gradient matrix shear modulus heat-transfer (or convection) coefficient nodes of a triangular element principal moment of inertia Jacobian matrix spring stiffness global-coordinate element stiffness or conduction matrix condensed stiffness matrix, and conduction part of the stiffness matrix
in heat-trarisfer problems . local-coordinate element stiffness matrix convective part of the stiffness matrix in heat-transfer problems g1obal-coordinate structure stiffness matrix th~rmal conductivities (or penneabilities, for fluid mechanics) in the x
and y directions, respectively length of a bar or beam element maximum difference in node numbers in an element general moment expression moments in a plate locaJ mass matrix local nodal moments global mass matrix matrix uSed to relate displacements to generalized coordinates for a
linear-strain triangle formu1ation matrix used to relate strains to generalized coordinates for a linear-
strain triangle fonnu1ation bandwidth of a structure number of degrees of freedom per node shape (interpolation or basis) function matrix shape functions surface pressure (or nodal heads in fluid mechanics) radial and axial (1ongitudinal) pressures, respectively concentrated load concentrated local force matrix
q
Rb Ri::cJ~.jy S,I,Z'
I Ii u,v,w U AU v V w W
Xj,Yi,Zj
X Xb,Yb' Zb
Notation A xvii
heat flow (flux) per urnt area or distributed loading on a plate rate of heat flow heat flow per unit area on a boundary surface heat source generated per unit volume or internal fluid source line or point heat sOurce transverse shear line loads on a plate radial, circumferential, and axial coordinates, respectively residual ill Galerkin's integral- body force in the radial direction nodal reactions in x and y directions, reSpectively natural coordinates attached to isoparametric element surface area thickness of a plane element or a plate element nodal temperatures of a triangular element
temperature function free-stream temperature displacement, force, and s~ess transformation matrix surface traction matrix in the i direction displacement functions in the x, y, and z directions, respectively strain energy change in stored energy velocity of fluid flow shear force in a beam distributed loading on a beam or along an edge of a plane element work nodal coordinates in the x, y, and Z directions, respectively local element coordinate axes structure global or reference coordinate axes b9dy force matrix body forces in the x and y dire~tions, respectively body force in longitudinal direction (axisymmetric case) or. in the z
direction (three-dimensional case)
Greek Symbols
(X coefficient of therMal expansion Cf.;,Pi' Yi'c5[ used to express the shape functions defined by Eq. (6.2.10) and Eqs.
(I 1.2.5)-( 11.2.8} ~ "\ spring or bar deformation e normal strain
xviii ... Notation
llT thennal strain matrix ICXI ICy, ICxy curvatures in plate bending v Poisson's ratio ¢i nodal angle of rotation or slope in a beam element 1!h functional for heat-transfer problem 7lp total potential energy P mass density of a material Pili weight density of a material co angular velocity and natural circular frequency n potential energy of forces ¢ fluid head or potential, or rotation or slope in a beam (J normal stress fET thennal stress matrix t shear stress and period of vibration (J angle between the x axis and the local i axis for two-dimensional
problems…