# 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 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 # 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 # 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 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 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 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 # 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

[ ]	Rectangular or square matrix.
{ }, [ ], [ ]	Column, row, and diagonal matrices.
[ ]T	Matrix, transpose.
[ ]-1, [ ]-T	Matrix inverse and inverse transpose; that is, ([ ]-1)T ≡ ([ ]T)-1.
|| ||	Norm of a matrix or a vector.
.	Time differentiation; for example, i = du/dt, ii = d2u/dt2.
,	Partial differentiation if the following subscript(s) is literal; for example, w,x = ∂w/∂x, w,xy = ∂2w/∂x ∂y.
{ ∂Π/∂a}	Represents [ ∂Π/∂a1 ∂Π/∂a2 · · · ∂Π/∂an]T, where Π is a scalar function of a1, a2, . . . , an.

LATIN SYMBOLS

A	Area or cross-sectional area.
[A]	Relates {d} to {a}; {d} = [A]{a}.
{a}	Generalized coordinates.
B	Bulk modulus, $B = E/(3 - 6\nu)$ .
[B]	Spatial derivative(s) of the field variable(s) are [B]{d}.
$C^{m}$	Field continuity of degree m (Section 3.11).
[C]	Damping matrix. Constraint matrix.
d.o.f.	Degree(s) of freedom.
D	Displacement. Flexural rigidity of a plate or shell.
{D}, {d}	Nodal d.o.f. of structure and element, respectively.
E	Modulus of elasticity.
[E]	Matrix of elastic stiffnesses (Section 1.7).
{F}	Body forces per unit volume.
G	Shear modulus.
I	Moment of inertia of cross-sectional area.
[I]	Unit matrix (also called identity matrix).
J	Determinant of [J] (called the Jacobian).
[J]	The Jacobian matrix.
k	Spring stiffness. Thermal conductivity.
[K], [k]	Structure and element conventional stiffness matrices.
$[K_{\sigma}]$ , $[k_{\sigma}]$	Structure and element stress stiffness matrices.
L, $L_{T}$	Length of element, length of structure.
$\ell$ , m, n	Direction cosines.
$n_{\text{eq}}$	Number of equations.
[M], [m]	Structure and element mass matrices.

[N], [N]	Shape (or basis, or interpolation) functions.
O	Order; for example, $O(h^{2}) =$ a term of order $h^{2}$ .
[0], {0}	Null matrix, null vector.
{P}	Externally applied concentrated loads on structure nodes.
q	Distributed load (surface or line).
{R}	Total load on structure nodes; {R} = {P} + $\Sigma$ {re}.
{re}	Loads applied to nodes by element, for example, by temperature change or distributed load (Eq. 4.1-6).
S, Se	Surface, element surface.
T	Temperature.
t	Thickness. Time.
[T]	Transformation matrix.
U, U0	Strain energy, strain energy per unit volume.
u, v, w	Displacements, for example, in directions x, y, z.
{u}	Vector of displacements; {u} = [u v w]T.
V, Ve	Volume, element volume.
x, y, z	Cartesian coordinates.

GREEK SYMBOLS

$\alpha$	Coefficient of thermal expansion, penalty number.
$[\Gamma]$	Jacobian inverse; $[\Gamma] = [J]^{-1}$ .
$\{ \epsilon \}, \{ \epsilon_0 \}$	Strains, initial strains.
$[\kappa], \{ \kappa \}$	Matrix of thermal conductivities, vector of curvatures.
$\lambda$	Eigenvalue. Lagrange multiplier.
$\nu$	Poisson's ratio of an isotropic material.
$\xi, \eta, \zeta$	Isoparametric coordinates.
$\xi_1, \xi_2, \xi_3$	Area coordinates.
$\Pi$	A functional; for example, $\Pi_p =$ potential energy.
$\rho$	Mass density.
$\{ \sigma \}, \{ \sigma_0 \}$	Stresses, initial stresses.
$\phi$	A dependent variable. Meridian angle of a shell.
$\{ \Phi \}$	Surface tractions.
$\omega$	Circular frequency in radians per second.