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+
+# CONTENTS
+
+# NOTATION
+
+# Chapter 1 INTRODUCTION 1
+
+1.1 The Finite Element Method 1   
+1.2 The Element Characteristic Matrix 7   
+1.3 Element Assembly and Solution for Unknowns 11   
+1.4 Summary of Finite Element History 14   
+1.5 Strain-Displacement Relations 15   
+1.6 Theory of Stress and Deformation 17   
+1.7 Stress-Strain-Temperature Relations 20   
+1.8 Warning: The Computed Answer May Be Wrong 24
+Problems 25
+
+# Chapter 2 THE STIFFNESS METHOD AND THE PLANE TRUSS 31
+
+2.1 Introduction 31   
+2.2 Structure Stiffness Equations 32   
+2.3 Properties of [K]. Solution for Unknowns 34   
+2.4 Element Stiffness Equations 36   
+2.5 Assembly of Elements. Plane Truss Example 38   
+2.6 Assembly Regarded as Satisfying Equilibrium 40   
+2.7 Assembly as Dictated by Node Numbers 41   
+2.8 Node Numbering That Exploits Matrix Sparsity 44   
+2.9 Automatic Assignment of Node Numbers 47   
+2.10 Displacement Boundary Conditions 48   
+2.11 Gauss Elimination Solution of Equations 53   
+2.12 Stress Computation. Support Reactions 55   
+2.13 Summary of Procedure 57
+Problems 59
+
+# Chapter 3 STATIONARY PRINCIPLES, THE RAYLEIGH-RITZ METHOD, AND INTERPOLATION 69
+
+3.1 Introduction 69   
+3.2 Principle of Stationary Potential Energy 70   
+3.3 Problems Having Many D.O.F. 73   
+3.4 Potential Energy of an Elastic Body 75
+
+<!-- source-page: 12 -->
+
+3.5 The Rayleigh-Ritz Method 78   
+3.6 Comments on the Rayleigh-Ritz Method Based on Assumed Displacement Fields 81   
+3.7 Stationary Principles and Governing Equations 83   
+3.8 A Piecewise Polynomial Field 88   
+3.9 - Finite Element Form of the Rayleigh-Ritz Method 90   
+3.10 Finite Element Formulations Derived from a Functional 93   
+3.11 Interpolation 95   
+3.12 Shape Functions for $C^0$ Elements 96   
+3.13 Shape Functions for $C^1$ Elements 99 Problems 101
+
+# Chapter 4 DISPLACEMENT-BASED ELEMENTS FOR STRUCTURAL MECHANICS
+
+109
+
+4.1 Formulas for Element Matrices [k] and $\{\mathbf{r}_e\}$ 109   
+4.2 Overview of Element Stiffness Matrices 113   
+4.3 Consistent Element Nodal Loads $\{\mathbf{r}_e\}$ 118   
+4.4 Equilibrium and Compatibility in the Solution 124   
+4.5 Convergence Requirements 126   
+4.6 The Patch Test 129   
+4.7 Stress Calculation 132   
+4.8 Other Formulation Methods 136 Problems 137
+
+# Chapter 5 STRAIGHT-SIDED TRIANGLES AND TETRAHEDRA
+
+147
+
+5.1 Natural Coordinates (Linear) 147   
+5.2 Natural Coordinates (Area and Volume) 149   
+5.3 Interpolation Fields for Plane Triangles 153   
+5.4 The Linear Triangle 154   
+5.5 The Quadratic Triangle 157   
+5.6 The Quadratic Tetrahedron 159 Problems 159
+
+# Chapter 6 THE ISOPARAMETRIC FORMULATION
+
+163
+
+6.1 Introduction 163   
+6.2 An Isoparametric Bar Element 164   
+6.3 Plane Bilinear Isoparametric Element 166   
+6.4 Summary of Gauss Quadrature 170   
+6.5 Computer Subroutines for the Bilinear Isoparametric Element 173   
+6.6 Quadratic Plane Elements 176
+
+<!-- source-page: 13 -->
+
+# CONTENTS
+
+xi
+
+6.7 Hexahedral (Solid) Isoparametric Elements 180
+
+6.8 Triangular Isoparametric Elements 182
+
+6.9 Consistent Element Nodal Loads $\{\mathbf{r}_e\}$ 185
+
+6.10 The Validity of Isoparametric Elements 186
+
+6.11 Appropriate Order of Quadrature 188
+
+6.12 Element and Mesh Instabilities 190
+
+6.13 Remarks on Stress Computation 194
+
+6.14 Examples. Effect of Element Geometry 196
+
+Problems 199
+
+# Chapter 7 COORDINATE TRANSFORMATION 209
+
+7.1 Introduction 209
+
+7.2 Transformation of Vectors 209
+
+7.3 Transformation of Stress, Strain, and Material Properties 211
+
+7.4 Transformation of Stiffness Matrices 213
+
+7.5 Examples: Transformation of Stiffness Matrices 214
+
+7.6 Inclined Support 216
+
+7.7 Joining Dissimilar Elements to One Another 218
+
+7.8 Rigid Links. Rigid Elements 220
+
+Problems 222
+
+# Chapter 8 TOPICS IN STRUCTURAL MECHANICS 228
+
+8.1 D.O.F. Within Elements. Condensation 228
+
+8.2 Condensation and Recovery Algorithms 231
+
+8.3 Parasitic Shear. Incompatible Elements 232
+
+8.4 Rotational D.O.F. in Plane Elements 236
+
+8.5 Assumed-Stress Hybrid Formulation 239
+
+8.6 A Plane Hybrid Triangle with Rotational D.O.F. 242
+
+8.7 User-Defined Elements. Elastic Kernel 244
+
+8.8 Higher Derivatives as Nodal D.O.F. 246
+
+8.9 Fracture Mechanics. Singularity Elements 247
+
+8.10 Elastic Foundations 250
+
+8.11 Media of Infinite Extent 252
+
+8.12 Finite Elements and Finite Differences 256
+
+8.13 Reanalysis Methods 256
+
+8.14 Substructuring 257
+
+8.15 Structural Symmetry 260
+
+8.16 Cyclic Symmetry 262
+
+Problems 263
+
+<!-- source-page: 14 -->
+
+# Chapter 9 CONSTRAINTS
+
+9.1 Constraints. Transformations 272   
+9.2 Lagrange Multipliers 275   
+9.3 Penalty Functions 276   
+9.4 Naturally Arising Penalty Formulations. Numerical Integration and Constraints 278   
+9.5 Constraint Counting 283   
+9.6 Additional Techniques for Incompressible Media 285
+Problems 288
+
+# Chapter 10 SOLIDS OF REVOLUTION
+
+10.1 Introduction 293
+
+10.2 Elasticity Relations for Axial Symmetry 294
+
+10.3 Finite Elements for Axial Symmetry 295
+
+10.4 Fourier Series 298
+
+10.5 Loads Without Axial Symmetry: Introduction 301
+
+10.6 Loads Without Axial Symmetry: Element Matrices 304
+
+10.7 Related Problems 307
+Problems 308
+
+# Chapter 11 BENDING OF FLAT PLATES
+
+11.1 Plate-Bending Theory 314   
+11.2 Finite Elements for Plates 319   
+11.3 Mindlin Plate Elements 323   
+11.4 A Triangular Discrete Kirchhoff Element 328   
+11.5 Boundary Conditions and Test Cases 332
+,Problems 335
+
+# Chapter 12 SHELLS
+
+12.1 Shell Geometry and Behavior. Shell Elements 340   
+12.2 Circular Arches and Arch Elements 343   
+12.3 Flat Elements for Shells 351   
+12.4 Shells of Revolution 352   
+12.5 Isoparametric General Shell Elements 358
+
+Problems 362
+
+# Chapter 13 FINITE ELEMENTS IN DYNAMICS AND VIBRATIONS
+
+13.1 Introduction 367   
+13.2 Dynamic Equations. Mass and Damping Matrices 368   
+13.3 Mass Matrices, Consistent and Diagonal 370
+
+<!-- source-page: 15 -->
+
+13.4 Damping 376   
+13.5 Natural Frequencies and Mode Shapes 378   
+13.6 Time-History Analysis. Modal Methods 381   
+13.7 Mass Condensation. Guyan Reduction 387   
+13.8 Component Mode Synthesis 391   
+13.9 Time-History Analysis. Direct Integration Methods 395   
+13.10 Explicit Direct Integration Methods 397   
+13.11 Implicit Direct Integration Methods 405   
+13.12 Other Implicit and Explicit Methods. Mixed Methods 407   
+13.13 Stability Analysis. Accuracy of Direct Integration Methods 410   
+13.14 Concluding Remarks on Time-History Analysis 417
+Problems 418
+
+# Chapter 14 STRESS STIFFENING AND BUCKLING 429
+
+14.1 Introduction 429   
+14.2 Stress Stiffness Matrices for Beams and Bars 432   
+14.3 Stress Stiffness Matrix of a Plate Element 435   
+14.4 A General Formulation for $[k_{\sigma}]$ 437   
+14.5 Bifurcation Buckling 441   
+14.6 Remarks on $[\mathbf{K}_{\sigma}]$ and Its Uses 444   
+14.7 Remarks on Buckling and Buckling Analysis 446
+Problems 448
+
+# Chapter 15 WEIGHTED RESIDUAL METHODS 455
+
+15.1 Introduction 455   
+15.2 Some Weighted Residual Methods 455   
+15.3 Example Solutions 458   
+15.4 Galerkin Finite Element Method 461   
+15.5 Integration by Parts 466   
+15.6 Two-Dimensional Problems 468   
+Problems 470
+
+# Chapter 16 HEAT CONDUCTION AND SELECTED FLUID PROBLEMS 474
+
+16.1 Introduction to Heat Conduction Problems 474   
+16.2 A One-Dimensional Example 475   
+16.3 Heat Conduction in a Plane 477   
+16.4 General Solids and Solids of Revolution 479   
+16.5 Finite Element Formulation 480   
+16.6 Thermal Transients 484   
+16.7 Related Problems. Fluid Flow 486
+
+<!-- source-page: 16 -->
+
+16.8 Fluid Vibration and Waves, Pressure Formulation 488
+
+16.9 Fluid-Structure Interaction 491
+
+Problems 495
+
+# Chapter 17 AN INTRODUCTION TO SOME NONLINEAR PROBLEMS
+
+501
+
+17.1 Introduction 501
+
+17.2 Some Solution Methods 502
+
+17.3 One-Dimensional Elastic-Plastic Analysis 510
+
+17.4 Small-Strain Plasticity Relations 515
+
+17.5 Elastic-Plastic Analysis Procedures 519
+
+17.6 Nonlinear Dynamic Problems 522
+
+17.7 A Problem Having Geometric Nonlinearity 529
+
+17.8 Other Nonlinear Problems 532
+
+Problems 533
+
+# Chapter 18 NUMERICAL ERRORS AND CONVERGENCE 542
+
+18.1 Introduction. Error Classification 542
+
+18.2 Ill-Conditioning 543
+
+18.3 The Condition Number 546
+
+18.4 Diagonal Decay Error Tests 550
+
+18.5 Residuals 552
+
+18.6 Discretization Error: Analysis 553
+
+18.7 Discretization Error: Estimation and Extrapolation 558
+
+18.8 Tests of Element Quality 563
+
+18.9 Concluding Remarks 566
+
+Problems 566
+
+# Chapter 19 MODELING, PROGRAMS, AND PROGRAMMING 573
+
+19.1 Modeling 573
+
+19.2 Programming and Programs 584
+
+# Appendix A MATRICES: SELECTED DEFINITIONS AND MANIPULATIONS
+
+589
+
+# Appendix B SIMULTANEOUS ALGEBRAIC EQUATIONS 592
+
+B.1 Introduction 592
+
+B.2 Solution of Simultaneous Linear Algebraic Equations by Gauss Elimination 593
+
+<!-- source-page: 17 -->
+
+CONTENTS
+
+XV
+
+# Appendix C EIGENVALUES AND EIGENVECTORS 598
+
+C.1 The Eigenproblem 598
+
+C.2 The Standard Eigenproblem 598
+
+C.3 The General Eigenproblem 599
+
+C.4 Remarks on Special Forms 602
+
+C.5 Solution Algorithms 603
+
+# REFERENCES 605
+
+# INDEX 623
+
+<!-- source-page: 18 -->
+
+<!-- source-page: 19 -->
+
+# NOTATION
+
+What follows is a list of principal symbols. Less frequently used symbols, and symbols that have different meanings in different contexts, are defined where they are used. Matrices and vectors are denoted by boldface type.
+
+MATHEMATICAL SYMBOLS 
+
+<table><tr><td>[ ]</td><td>Rectangular or square matrix.</td></tr><tr><td>{ }, [ ], [ ]</td><td>Column, row, and diagonal matrices.</td></tr><tr><td>[ ]T</td><td>Matrix, transpose.</td></tr><tr><td>[ ]-1, [ ]-T</td><td>Matrix inverse and inverse transpose; that is, ([ ]-1)T ≡ ([ ]T)-1.</td></tr><tr><td>|| ||</td><td>Norm of a matrix or a vector.</td></tr><tr><td>.</td><td>Time differentiation; for example, i = du/dt, ii = d2u/dt2.</td></tr><tr><td>,</td><td>Partial differentiation if the following subscript(s) is literal; for example, w,x = ∂w/∂x, w,xy = ∂2w/∂x ∂y.</td></tr><tr><td>{ ∂Π/∂a}</td><td>Represents [ ∂Π/∂a1 ∂Π/∂a2 · · · ∂Π/∂an]T, where Π is a scalar function of a1, a2, . . . , an.</td></tr></table>
+
+LATIN SYMBOLS
+
+<table><tr><td>A</td><td>Area or cross-sectional area.</td></tr><tr><td>[A]</td><td>Relates {d} to {a}; {d} = [A]{a}.</td></tr><tr><td>{a}</td><td>Generalized coordinates.</td></tr><tr><td>B</td><td>Bulk modulus,  $B = E/(3 - 6\nu)$ .</td></tr><tr><td>[B]</td><td>Spatial derivative(s) of the field variable(s) are [B]{d}.</td></tr><tr><td> $C^{m}$ </td><td>Field continuity of degree m (Section 3.11).</td></tr><tr><td>[C]</td><td>Damping matrix. Constraint matrix.</td></tr><tr><td>d.o.f.</td><td>Degree(s) of freedom.</td></tr><tr><td>D</td><td>Displacement. Flexural rigidity of a plate or shell.</td></tr><tr><td>{D}, {d}</td><td>Nodal d.o.f. of structure and element, respectively.</td></tr><tr><td>E</td><td>Modulus of elasticity.</td></tr><tr><td>[E]</td><td>Matrix of elastic stiffnesses (Section 1.7).</td></tr><tr><td>{F}</td><td>Body forces per unit volume.</td></tr><tr><td>G</td><td>Shear modulus.</td></tr><tr><td>I</td><td>Moment of inertia of cross-sectional area.</td></tr><tr><td>[I]</td><td>Unit matrix (also called identity matrix).</td></tr><tr><td>J</td><td>Determinant of [J] (called the Jacobian).</td></tr><tr><td>[J]</td><td>The Jacobian matrix.</td></tr><tr><td>k</td><td>Spring stiffness. Thermal conductivity.</td></tr><tr><td>[K], [k]</td><td>Structure and element conventional stiffness matrices.</td></tr><tr><td> $[K_{\sigma}]$ ,  $[k_{\sigma}]$ </td><td>Structure and element stress stiffness matrices.</td></tr><tr><td>L,  $L_{T}$ </td><td>Length of element, length of structure.</td></tr><tr><td> $\ell$ , m, n</td><td>Direction cosines.</td></tr><tr><td> $n_{\text{eq}}$ </td><td>Number of equations.</td></tr><tr><td>[M], [m]</td><td>Structure and element mass matrices.</td></tr></table>
+
+<!-- source-page: 20 -->
+
+<table><tr><td>[N], [N]</td><td>Shape (or basis, or interpolation) functions.</td></tr><tr><td>O</td><td>Order; for example,  $O(h^{2}) =$  a term of order  $h^{2}$ .</td></tr><tr><td>[0], {0}</td><td>Null matrix, null vector.</td></tr><tr><td>{P}</td><td>Externally applied concentrated loads on structure nodes.</td></tr><tr><td>q</td><td>Distributed load (surface or line).</td></tr><tr><td>{R}</td><td>Total load on structure nodes; {R} = {P} +  $\Sigma$  {re}.</td></tr><tr><td>{re}</td><td>Loads applied to nodes by element, for example, by temperature change or distributed load (Eq. 4.1-6).</td></tr><tr><td>S, Se</td><td>Surface, element surface.</td></tr><tr><td>T</td><td>Temperature.</td></tr><tr><td>t</td><td>Thickness. Time.</td></tr><tr><td>[T]</td><td>Transformation matrix.</td></tr><tr><td>U, U0</td><td>Strain energy, strain energy per unit volume.</td></tr><tr><td>u, v, w</td><td>Displacements, for example, in directions x, y, z.</td></tr><tr><td>{u}</td><td>Vector of displacements; {u} = [u v w]T.</td></tr><tr><td>V, Ve</td><td>Volume, element volume.</td></tr><tr><td>x, y, z</td><td>Cartesian coordinates.</td></tr></table>
+
+GREEK SYMBOLS 
+
+<table><tr><td> $\alpha$ </td><td>Coefficient of thermal expansion, penalty number.</td></tr><tr><td> $[\Gamma]$ </td><td>Jacobian inverse;  $[\Gamma] = [J]^{-1}$ .</td></tr><tr><td> $\{ \epsilon \}, \{ \epsilon_0 \}$ </td><td>Strains, initial strains.</td></tr><tr><td> $[\kappa], \{ \kappa \}$ </td><td>Matrix of thermal conductivities, vector of curvatures.</td></tr><tr><td> $\lambda$ </td><td>Eigenvalue. Lagrange multiplier.</td></tr><tr><td> $\nu$ </td><td>Poisson&#x27;s ratio of an isotropic material.</td></tr><tr><td> $\xi, \eta, \zeta$ </td><td>Isoparametric coordinates.</td></tr><tr><td> $\xi_1, \xi_2, \xi_3$ </td><td>Area coordinates.</td></tr><tr><td> $\Pi$ </td><td>A functional; for example,  $\Pi_p =$  potential energy.</td></tr><tr><td> $\rho$ </td><td>Mass density.</td></tr><tr><td> $\{ \sigma \}, \{ \sigma_0 \}$ </td><td>Stresses, initial stresses.</td></tr><tr><td> $\phi$ </td><td>A dependent variable. Meridian angle of a shell.</td></tr><tr><td> $\{ \Phi \}$ </td><td>Surface tractions.</td></tr><tr><td> $\omega$ </td><td>Circular frequency in radians per second.</td></tr></table>