Finite Element Analysis – With Numeric and Symbolic MatLAB

Finite Element Analysis – With Numeric and Symbolic MatLAB
اسم المؤلف
John E Akin
التاريخ
27 مارس 2024
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154
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Finite Element Analysis – With Numeric and Symbolic MatLAB
John E Akin
Rice University, USA
Contents
Preface v
About the Author vii
List of Examples xi
List of Matlab Scripts xvii
List of Useful Tables xxi

  1. Overview 1
  2. Calculus Review 33
  3. Terminology from Differential Equations 59
  4. Parametric Interpolation 69
  5. Numerical Integration 117
  6. Equivalent Integral Forms 145
  7. Matrix Procedures for Finite Elements 167
  8. Applications of One-Dimensional Lagrange
    Elements 205
  9. Truss Analysis 317
  10. Applications of One-Dimensional Hermite
    Elements 341
  11. Frame Analysis 383
    ixx Finite Element Analysis: With Numeric and Symbolic Matlab
  12. Scalar Fields and Thermal Analysis 405
  13. Elasticity 487
  14. Eigenanalysis 537
  15. Transient and Dynamic Solutions 597
    Index 62
    List of Examples
    Example 1.3-1 Boolean scatter of a column vector 8
    Example 1.3-2 Inverse of a 2 × 2 matrix 9
    Example 1.3-3 Matlab script to invert 3 × 3 matrix 10
    Example 1.3-4 Multiplication of a matrix by its inverse 10
    Example 1.3-5 Matlab script to solve a 2 × 2 linear system 11
    Example 2.2-1 Variable Jacobian of a L3, in unit
    coordinates
    37
    Example 2.2-2 Variable Jacobian of a L3, in natural
    coordinates
    38
    Example 2.2-3 Moment integral of interpolated line force,
    L2
    39
    Example 2.2-4 Jacobian matrix in cylindrical coordinates 40
    Example 2.2-5 Area of a rotated square by integration 41
    Example 2.2-6 Jacobian matrix of a quadrilateral element 41
    Example 2.2-7 Area calculation for the same quadrilateral 42
    Example 2.2-8 Geometric constants of an arbitrary triangle 43
    Example 2.3-1 Polar moment of inertia of rotated square 47
    Example 2.4-1 Line integral with integration by parts 49
    Example 2.4-2 Area integral with integration by parts
    (Greens theorem)
    49
    Example 2.4-3 Green’s theorem incorporates Neumann
    condition
    51
    Example 4.2-1 Interpolated value and slope using a
    four-noded line element, L4
    79
    xixii Finite Element Analysis: With Numeric and Symbolic Matlab
    Example 4.2-2 Change one value in Example 4.2-2 and
    graph with Matlab, L4
    82
    Example 4.2-3 Approximate and graph a circular arc using
    a four-noded line element, L4
    84
    Example 4.6-1 Exactly integrate the local area of a
    parametric triangle (T 3, or T 6, etc.)
    99
    Example 4.8-1 Shape of any edge of an eight-noded
    quadrilateral, Q8 or Q9
    104
    Example 4.9-1 Shape of any edge of an four-noded
    quadrilateral, Q4
    106
    Example 4.10-1 Integral of the solution over cubic line
    element, L4
    108
    Example 4.10-2 Moment of line pressure on linear line
    element, L2
    109
    Example 5.1-1 Numerical integration for length of a
    straight line, using L2 element
    121
    Example 5.1-2 Numerical integration for length of a
    straight line, using L4 element
    122
    Example 5.1-3 Numerical integration of interpolated
    quantity, using L4 element
    125
    Example 5.1-4 Numerical integration for moment of inertia
    of a line, using L2 element
    127
    Example 5.1-5 Three-point integration of the mass matrix
    of a L3 line element
    130
    Example 5.2-1 Four-point integration of local moment of
    inertia of a square, using Q4
    138
    Example 5.2-2 Area–Jacobian relation for straight sided
    triangles, L3
    139
    Example 5.2-3 Local polar moment of inertia of the unit
    triangle, L3
    139
    Example 6.2-1 Galerkin form matrices for first-order ODE 148
    Example 6.2-2 Above analytic matrices for linear line
    element, L2
    149
    Example 6.2-3 Assembly of two of the above matrices, L2 151
    Example 6.2-4 Insert numerical values & BC & solve
    above system, L2
    152List of Examples xiii
    Example 6.2-5 Repeat above solution for quadratic line
    element, L3
    154
    Example 6.5-1 Apply Euler Theorem to extract
    two-dimensional Poisson Eq. and NBC
    164
    Example 7.2-1 Assembly (scatter) of six column source
    vectors, L2
    176
    Example 7.2-2 Form connection lists for seven nodes
    connected in two ways, T 3, Q4
    177
    Example 7.3-1 Assembly (scatter) with a non-sequential
    connection list, L3
    181
    Example 7.7-1 Write the constraint equation for an
    inclined two-dimensional roller support
    190
    Example 7.7-2 Prepare sample data for Matlab truss
    analysis
    191
    Example 7.7-3 Write the constraint equation for two
    geared torsional shafts
    192
    Example 7-8.1 Factor a 5 × 5 symmetric matrix into
    upper and lower triangles
    194
    Example 7.9-1 Establish skyline storage for banded 6 × 6
    matrix
    199
    Example 7.10-1 Locate two terms in sparse matrix skyline
    format
    202
    Example 8.1-1 Element matrix integrals defined by ODE
    on a line
    211
    Example 8.1-2 Integrate internal source matrix for a
    quadratic line element, L3
    212
    Example 8.1-3 Form single element matrix equilibrium
    system for above ODE, L3
    213
    Example 8.1-4 Insert a Dirichlet BC and a secondary BC
    and solve matrix system, L3
    215
    Example 8.2-1 Chimney three layer wall one-dimensional
    temperatures and heat flux computed, L2
    221
    Example 8.2-2 Equilibrium of bar with mid-point load and
    support displacement, L2
    225
    Example 8.2-3 Exact solution hanging bar displacements
    and reactions, L3
    228xiv Finite Element Analysis: With Numeric and Symbolic Matlab
    Example 8.2-4 Thermal stress and reactions in fixed–fixed
    bar, L3
    230
    Example 8.2-5 Three layer chimney with convection BC
    temperatures, and heat flux, L2
    230
    Example 8.2-6 Planar wall with convection on one side 232
    Example 8.3-1 ODE source vector for linear internal source
    on a line, L3
    247
    Example 8.3-2 One element solution for linear internal
    source and two Dirichlet BCs, L3
    249
    Example 8.3-3 Linear solution for internal source, one
    Dirichlet, one Neumann BC, L3
    251
    Example 8.3-4 Two elements, internal source, one Dirichlet,
    one Neumann BC, L3
    253
    Example 8.3-5 One L2 ODE reaction for linear internal
    source and two Dirichlet BCs, L2
    253
    Example 8.3-6 Line conduction, convection matrices, two
    EBCs, temperature solution, L3
    256
    Example 8.3-7 Line conduction, line convection matrices,
    EBC, zero NBC, L3
    259
    Example 8.3-8 Line conduction, convection matrices, EBC,
    non-zero NBC, L3
    261
    Example 8.4-1 Five element convecting fin temperature,
    heat flux, convection loss, L2
    265
    Example 8.4-2 Five element one-dimensional convecting fin
    check of heat flux balance, L2
    267
    Example 8.4-3 Five element one-dimensional convecting fin
    temperatures with base heat flux input, L2
    269
    Example 8.4-4 Structural pile displacements analogous to
    convecting fin temperature
    270
    Example 8.6-1 Tapered conical shaft in torsion matrices by
    numerical integration
    287
    Example 8.6-2 Variable coefficient ODE gives
    non-symmetric matrices, Matlab solution
    293
    Example 8.6-3 Variable coefficient ODE element reactions
    for flux recovery
    297
    Example 8.7-1 Symbolic integration of linear taper axial bar
    stiffness matrix, L3
    302
    Example 8.7-2 Thermal stress (initial strain) in taper axial
    bar with fixed ends, L3
    304List of Examples xv
    Example 9.1-1 Two-bar truss displacements and reactions,
    for point force, L2
    323
    Example 9.1-2 Two-bar truss, element axial displacements
    and reactions, L2
    326
    Example 9.1-3 Two-bar truss displacements and reactions,
    due to temperature change, L2
    327
    Example 9.1-4 Three-bar truss with inclined roller and
    point load, L2
    329
    Example 9.2-1 Sample data for a space truss, L2 333
    Example 10.1-1 Least squares finite element Hermite form
    for second-order ODE, L2C1
    342
    Example 10.6-1 Quintic beam center deflection and
    reactions for end settlement, L3C1
    361
    Example 10.6-2 Fixed–fixed quintic beam, triangular load;
    deflections and reactions, L3C1
    363
    Example 10.6-3 Fixed–fixed two cubic beams, triangular
    load; deflections, reactions, L2C1
    365
    Example 10.6-4 Two-span continuous beam with line load,
    moment and shear, L2C1
    366
    Example 11.2-1 Pin–pin planar frame with line load, node
    deflections, reactions, F 2C1
    391
    Example 11.2-2 Same frame member results graphed, F 3C1 392
    Example 11.2-3 Plane frame rotated member stiffness
    matrix, F 2C1
    394
    Example 11.2-4 Recover inclined member system reactions,
    F 2C1
    395
    Example 11.2-5 Verify above plane reactions using statics 395
    Example 12.8-1 Jacobian matrix of two-dimensional
    quadrilateral, Q4
    424
    Example 12.8-2 Pressure gradient in two-dimensional
    quadrilateral with pressure data, Q4
    425
    Example 12.8-3 Pressure gradient in two-dimensional
    rectangle with pressure data, R4
    427
    Example 12.8-4 Pressure gradient along edge in
    Ex 12.8-2, L2
    428
    Example 12.8-5 Pressure gradient at centroid parallel to
    edge, Q4
    429
    Example 12.13-1 Temperatures in square with internal heat
    generation, T 3
    465xvi Finite Element Analysis: With Numeric and Symbolic Matlab
    Example 12.13-2 Heat flux in square with internal heat
    generation, T 3
    468
    Example 12.13-3 Square temperature with two edge fluxes
    and two edge temperatures, L3
    469
    Example 12.13-4 Graph approximate diagonal temperature
    in above example
    472
    Example 14.3-1 Natural frequency of a bar with distributed
    and point masses, L2
    544
    Example 14.4-1 Effect of tension on string natural
    frequency, L3
    548
    Example 14.4-2 Matlab script for string vibration modes
    and frequencies, L3
    548
    Example 14.4-3 First two frequencies of fixed-free elastic
    bar, L3
    555
    Example 14.5-1 Matlab script for torsional vibrations of a
    fixed-free shaft, L3
    557
    Example 14.8-1 Buckling load for a two-bar truss, L2 569
    Example 14.8-2 Buckling load for a fixed–pinned
    beam-column, L3C1
    570
    Example 14.8-3 Buckling of a fixed–pinned beam-column
    with spring support, L3C1
    571
    Example 14.8-4 Matlab script for buckling of fixed–pinned
    beam-column, L3C1
    572
    Example 14.9-1 Matlab script for buckling of fixed–pinned
    tensioned beam-column, L3C1
    580
    Example 14.12-1 Matlab script to find principal stresses for
    three-dimensional stress tensor
    587
    Example 14.12-2 Find maximum shear stress for
    three-dimensional stress tensor
    588
    Example 15.4-1 Transient solution of a symmetric
    conducting square, T 3
    608
    Example 15.5-1 Estimate the critical time step size for
    transient solution, L2
    613List of Matlab Scripts
    Example 1.3-3 Inversion of 3 by 3 matrix 10
    Example 1.3-5 Solve a linear 2 by 2 matrix system 11
    Figure 4.2-2 Symbolic derivations of the quadratic line
    interpolation functions
    74
    Figure 4.2-6 Constant and linear results are included in
    quadratic interpolation
    78
    Figure 4.2-7 Placing interpolation functions in a script for
    a library of one-dimensional elements
    79
    Figure 4.2-11 A Matlab script to graph a cubic element
    (L4 C0)
    84
    Figure 4.2-13 Script to plot a single curved parametric
    element
    86
    Figure 4.3-1 Symbolic Lagrange quadratic line
    interpolation in natural coordinates
    87
    Figure 4.4-1 First two C1 Hermite line interpolations and
    physical derivatives
    89
    Figure 4.4-2 Symbolic derivation of the cubic C1 line
    element interpolation
    90
    Figure 4.5-2 Symbolic derivation of four-noded Lagrangian
    quadrilateral interpolations
    92
    Figure 4.5-4 Top of the Lagrange quadrilaterals script 94
    Figure 4.6-3 Symbolic derivation for a Lagrangian
    quadratic triangle
    98
    xviixviii Finite Element Analysis: With Numeric and Symbolic Matlab
    Figure 4.6-4 Top of a script to access Lagrange triangle
    interpolations
    99
    Figure 5.1-2 Portion of the unit coordinate line
    quadrature data
    119
    Figure 5.1-3 Portion of the natural coordinate tabulated
    data
    120
    Figure 5.1-4 Numerical integration of a solution result on
    a line element
    124
    Figure 5.1-5 Numerical line integration using tables and
    the element library
    125
    Figure 5.1-7 Planar curve segment length by numerical
    integration
    135
    Figure 5.2-2 Creating quadrilateral integration rule from
    the one-dimensional rule
    136
    Figure 5.2-4 Selected triangular quadrature data values 137
    Figure 6.2-2 FEA solution of u′ + au = F, u(0) = 0 157
    Figure 7.2-1 Calculation of system equation (DOF)
    numbers for an element
    170
    Figure 7.3-1 Assembling element square and column array
    into the system equations
    180
    Figure 7.4-1 Matrix partitions using vector subscripts 184
    Figure 7.9-3 Calculating the column height for each
    equation
    198
    Figure 7.9-4 Extracting the system skyline from the
    element connection list
    200
    Figure 7.10-1 Locating a full matrix term in the skyline
    vector
    201
    Figure 8.2-1 Element assembly loop: One-dimensional
    linear, or, quadratic, or cubic
    218
    Figure 8.5-2 Assign general control numbers and logic
    flags, allocate arrays
    273
    Figure 8.5-3 Set the coefficient data, build element arrays,
    assemble into system arrays
    275
    Figure 8.5-4 Enforce EBC, solve the system, and recover
    the reaction
    275
    Figure 8.5-5 Post-processing the results at selected points 277
    Figure 8.6-1 Loop to automate the integration of the
    element matrices
    280List of Matlab Scripts xix
    Figure 8.6-3 Numerical integration of matrices for
    tapered bar line elements
    283
    Figure 8.6-4 Post-processing numerically integrated
    tapered axial bar
    287
    Figure 8.6-5 Post-processing a tapered torsional shaft 288
    Figure 8.6-6 Partial post-processing for a tapered shaft
    in torsion
    289
    Figure 8.6-10 Post-processing integrals for a
    hydrodynamic bearing
    295
    Figure 8.6-11 Interpolating variable coefficients for
    numerical integration
    296
    Figure 8.7-1 Symbolic solution of ODE with non-zero
    EBC and NBC
    301
    Figure 8.7-3 Symbolic stiffness matrix for a quadratic
    tapered axial bar
    304
    Figure 10.5-1 Symbolic integration to form the
    rectangular line-load conversion matrix
    356
    Figure 10.5-2 Symbolically computing the resultant
    source vector from line-loads
    357
    Figure 10.5-3 Symbolic solution of a quintic fixed–fixed
    beam with a triangular line load
    357
    Figure 10.6-4 Partitioning the displacements into three
    sets
    362
    Figure 10.8-2(a) Beam sketch, controls, basic data, and
    memory allocation
    371
    Figure 10.8-2(b) Assemble beam matrices, solve for
    displacements, and recover reactions
    372
    Figure 11.1-3 Combining axial and bending arrays to
    form a frame member stiffness
    388
    Figure 11.2-1 Recovering the frame member global and
    local reactions
    390
    Figure 13.6-1 Constitutive arrays for solid elasticity
    analysis
    501
    Figure 13.14-1 Strain–displacement matrix for
    plane–stress or –strain
    511
    Figure 13.14-2 Constitutive arrays for plane–stress
    analysis
    511
    Figure 13.14-3 Constitutive arrays for plane–strain
    analysis
    513xx Finite Element Analysis: With Numeric and Symbolic Matlab
    Figure 13.16-2 Strain–displacement matrix for
    axisymmetric stress model
    522
    Figure 13.16-3 Constitutive arrays for axisymmetric stress
    analysis
    523
    Figure 14.4-1 Tensioned string eigenvalue–eigenvector
    calculations
    550
    Figure 14.4-2 Torsional frequencies for a shaft with
    end-point inertia
    551
    Figure 14.6-2 Natural frequencies of a cantilever with a
    transverse spring
    559
    Figure 14.8-6 Linear buckling load and mode shape for a
    fixed–pinned beam
    577
    Figure 14.9-2 Frequencies of beam-column with axial
    load
    581
    Figure 14.12-1 Computing ductile material failure criteria 588
    Figure 15.3-2 Computing the transient node
    temperatures
    605
    Figure 15.4-2(a) Preparing for a transient integration of the
    finite element matrices
    607
    Figure 15.4-2(b) Time stepping the independent DOF and
    recovering the reactions
    608
    List of Useful Tables
    Table 1.7-1 Alternate interpretations of spring networks 28
    Table 2.5-1 Exact physical integrals for constant
    Jacobian elements
    52
    Table 3.3-1 Boundary condition classes for even-order
    partial differential equations
    63
    Table 3.3-2 Example one-dimensional boundary
    conditions
    63
    Table 4.10-1 Interpolation column and matrix integrals for
    one-dimensional constant Jacobian
    107
    Table 4.10-2 Asymmetric constant Jacobian line element
    integrals
    108
    Table 5.1-1 Abscissas and weights for Gaussian
    Quadrature in Unit Coordinates
    118
    Table 7.2-1 Relating local and system equation numbers 174
    Table 7.2-2 Equation numbers for truss element 21 175
    Table 12.10-1 Interpolation integrals for straight-edged
    triangles
    432
    Table 12.10-2 Diffusion integrals for isotropic straight-edged
    triangles
    433
    Table 12.10-3 Diffusion integrals for orthotropic
    straight-edged triangles
    434
    xxixxii Finite Element Analysis: With Numeric and Symbolic Matlab
    Table 12.10-4 Interpolation integrals for rectangular
    elements
    434
    Table 12.10-5 Diffusion integrals for orthotropic rectangles 435
    Table 12.15-1 Brick elements selected inputs, node 11
    result, gradient vector components
    478
    Table 14.8-1 Interpretation of the buckling load factor 567
    Index
    A
    abs min, 576
    absolute maximum shear stress, 586
    absolute temperature, 264
    acceleration update, 494, 509, 615
    acoustical pressure, 541
    acoustical vibration, 537
    adaptive mesh, 563
    addpath, 200
    adjoint, 63, 67
    advection matrix, 205, 209, 211, 413,
    415
    advection-diffusion equation, 407
    algebraic system, 46
    algorithm constant, 599
    analogies, 440
    analytic inverse matrix, 8
    angle of rotation, 346
    angle of twist, 438
    analytic matrix inverse, 10
    angular velocity, 272
    anisotropic material, 406, 408, 499
    anti-symmetric mode, 546, 563
    anti-symmetry, 306, 369, 432
    application library, 284, 329
    applied torque, 410, 440
    apply mpc type 2.m, 192
    approximate contours, 442
    area coordinates, 29
    artificial hip, 398
    assembly, 3, 167, 176, 181, 211, 257,
    365, 505, 509
    assembly example, 466
    assembly of elements, 149
    assembly of springs, 492
    assembly symbol, 505
    assumed form, 308
    automatic mesh generator, 274, 516
    automation, 271, 373
    average acceleration method, 616
    average mass matrix, 541, 555, 584,
    611
    axial bar, 226
    axial compression, 354
    axial displacement, 490
    axial force, 385, 490
    axial load, 345
    axial stiffness, 188, 384
    axial strain, 227, 281, 283
    axial stress, 227, 281
    axial vibration, 555
    axisymmetric analysis, 520
    axisymmetric fields, 474
    axisymmetric solid, 487
    axisymmetric stress, 497, 501, 521
    B B
    axisym elastic.m, 521
    B matrix elastic.m, 507
    B planar elastic.m, 510
    backward difference method, 600
    backward-substitution, 196, 600
    bar, 225, 236, 245, 319, 488, 497
    baracentric coordinates, 29
    bar member, 383
    beam, 488
    beam bending, 63
    beam column, 345
    627628 Finite Element Analysis: With Numeric and Symbolic Matlab
    beam element, 88, 341
    beam line load resultant, 625
    beam on an elastic foundation
    (BOEF), 345, 401
    beam theory, curved, 514
    beam vibration, 557
    beam with axial load, 578
    beam-column vibration, 341, 383, 581
    beam stiffness matrix, 380
    beam thermal moment, 381
    bearing pressure, 289, 292
    boundary region, 13
    bending moment, 346
    bending stiffness, 384, 386, 401, 559
    bending stiffness matrix, 351
    bi-linear quadrilateral, 112, 424
    Biggs exact time history, 618
    binary file, 606
    body force, 494, 504
    Boolean array, 31, 479
    Boolean matrix, 9, 13, 147, 168, 210
    bottom hole assembly (BHA), 556
    boundary condition flag, 389
    boundary displacements, 505
    boundary flux, 255
    boundary integral, 50, 413
    boundary interpolation, 3, 503
    boundary matrices, 413
    boundary property, 62
    boundary segment, 3, 411, 414, 505,
    526
    boundary source, 471
    boundary value problem, 64
    buckled mode shape, 568, 576
    buckled shape, 542
    buckling factor, 318, 358, 385, 542,
    565
    buckling load factor (BLF), 566
    C
    calculus review, 33
    cantilever beam, 488, 515, 558
    capacitance matrix, 414
    capacity matrix, 598, 608
    carpet plot, 442, 607
    cassic beam, 378, 400
    Castigliano’s theorem, 514
    catastrophic failure, 565, 568
    Cauchy condition, 62, 262
    centrifugal acceleration, 300
    centripetal force, 578
    centroid, 284
    change of variables, 14
    characteristic equation, 555
    chimney, 221, 239
    circular arc, 84, 131, 133
    color control integer, 445
    circular shaft, 218, 488
    circumferential stress, 514, 518
    classes of boundary conditions, 61
    coefficient of thermal expansion
    (CTE), 245, 352, 499
    collocation method, 159
    color scalar result.m, 445
    column buckling, 566, 578–579
    column heights, 196
    column matrix, 64
    column vector, 5
    complex number, 539
    compliance matrix, 499
    compressive yield stress, 586
    concentration, 410
    conduction, 28
    conditionally stable, 600
    conduction matrix, 211, 257, 310,
    337, 434, 466, 598
    conformable matrices, 6
    conical shaft, 286
    connection list, 13, 19, 23, 168, 170,
    217, 266, 279, 282, 373, 414
    connectivity list, 189, 372
    consistent mass matrix, 143, 541, 546,
    554, 557, 561, 583, 611, 613
    consistent units, 222
    constant determinant, 37, 423
    constant Jacobian, 53, 139
    constant source, 213, 471
    constitutive law, 488
    constitutive matrix, 523
    constitutive relation, 499
    constraint equation, 181, 192Index 629
    continuity level, 70
    contour result on mesh.m, 446
    control integers, 271, 273
    control numbers, 273
    count EBC MPC flags.m, 192
    count MPC eqs.m, 189
    convection coefficient, 62, 205, 216,
    239, 241, 265, 268, 310
    convection condition, 4
    convection loss, 265
    convection matrix, 257, 413, 432, 434
    coordinates, 3
    corresponding PDE, 163
    Crank–Nicolson method, 600, 608,
    612
    critical dampening, 614, 622
    critical time step, 612
    cross product, 54
    cubic bar, 624
    cubic beam element, 353, 365, 571,
    578, 625
    cubic interpolation, 76, 79–81, 122
    cubic polynomial, 343
    curvature, 70
    curve length, 428
    curve tangent, 35
    curved beam theory, 513–514
    curved elements, 412, 435
    Cuthill-McGee algorithm, 198
    cyclic permutation, 45, 54
    cyclic symmetry, 436–437, 526
    D
    damping matrix, 509, 614
    Darcy’s Law, 409
    DC circuit, 28
    DC current, 25
    decrease element size, 612
    deflection, 364
    degenerate quadrilaterals, 53
    degree of freedom numbers, 203
    degrees of freedom (DOF), 167, 169,
    173, 182, 210, 350, 400, 490, 502,
    538
    dependent variable, 59
    derivative of a matrix, 5
    det, 8
    determinant, 541
    deformation, 15
    diag, 539, 576
    diagonally dominant, 21
    diagonal mass matrix, 541–542, 583,
    611
    diagonal matrix, 5
    diff, 300
    differential area, 41
    differential equations, 59
    differential geometry, 37
    differential length, 50
    differential operator, 412
    differential volume, 36, 55, 480
    diffusion, 205
    diffusion coefficients, 410
    diffusion matrix, 209, 211, 413, 434
    diffusion matrix integrals, 433
    dimensional homogeneity, 159
    Dirac Delta distribution, 217, 346
    direct assembly, 179
    direct time integration, 598, 614
    direction angles, 573
    direction cosines, 164, 318
    directional derivative, 428
    Dirichlet boundary condition, 146,
    205
    Dirichlet conditions, 2, 61
    discontinuous flux, 370, 469
    discontinuous source, 243
    disk, 557
    disp, 547, 558
    displacement components, 487, 495
    displacement derivatives, 495
    displacement field, 490
    displacement gradients, 487, 495
    displacement transformation, 320
    displacement vector, 318, 491, 495,
    502–503, 510, 517
    displacements, 227, 282, 305, 317,
    326, 487
    distorted elements, 38
    distortional energy criterion, 586
    distributed transverse load, 351630 Finite Element Analysis: With Numeric and Symbolic Matlab
    division by zero, 501
    do-loop, 179
    DOF numbering, 169
    dot product, 491
    drill string, 219, 556
    ductile material, 587
    dynamic solutions, 613
    E
    earthquake, 617
    EBC code, 3, 146, 158, 182, 229, 231,
    233
    EBC location symbol, 442
    eddy currents, 493
    edge based elements, 503
    edge interpolation, 428
    eig, 538, 547, 558, 586, 588
    eig.m, 549
    eigenproblem, 66, 537, 541
    eigenvalue, 541, 554, 572, 612
    eigenvector, 541
    eigs, 538, 584
    electric field intensity, 503
    el qp xyz fluxes.txt, 445
    el shape n local deriv.m, 77, 97
    elastic foundation, 297, 345, 375
    elastic modulus, 270, 284, 499
    elastic stiffness matrix, 507
    elastic support, 558
    elasticity matrix, 487, 500, 512
    electrical conductor, 606
    electrical engineering, 503
    electrical network, 25
    electrical resistance, 606
    electromagnetics, 503
    electrostatics, 405, 410
    element axes, 387
    element displacements, 505
    element domain, 31
    element length, 386
    element loop, 274, 276–277, 282, 287
    element mass matrix, 509
    element measure, 427
    element properties, 217
    element reactions, 299, 389, 493
    element type, 3, 278, 373
    elliptic equation, 163
    elliptical differential equation, 2
    elliptical PDE, 145
    emissivity, 264
    energy minimization, 17
    enforce EBC, 183, 185, 276, 344, 363,
    366
    enforce MPC equations.m, 189
    enforce NBC, 344
    enforcing EBC, 168
    engineering shear strains, 496
    equation of equilibrium, 220
    equation of motion, 556
    equilibrium equations, 15, 319, 325,
    349, 353, 491
    equivalent integral form, 2, 205
    equivalent stress, 586
    essential boundary condition (EBC),
    2–3, 17, 67, 146, 158, 168, 182, 185,
    208, 229, 231, 233, 319, 405, 492
    Euler’s Theorem, 145
    even order equations, 60, 62, 67
    exact integrals, 37, 51–52, 432
    exercises, 114, 313
    exterior corner, 439, 447
    external couple, 385
    external forces, 493
    external impact force, 617
    extreme eigenvalues, 540
    F
    Factor of Safety (FOS), 568, 577
    factorization, 8, 195, 600
    failure criteria, 346, 586
    failure criterion, 487, 512
    fake convection, 264
    fake material, 442
    Fick’s Law, 410
    field 2d types.m, 442
    field analysis, 405
    fillet, 439
    film thickness, 291
    fin, 265, 267, 269, 412
    finite differences in time, 599Index 631
    finite elements in time, 599
    fire brick, 225
    first-order ODE, 60, 148
    fixed joint, 389
    fixed–pinned beam, 571
    fixed–fixed beam, 363
    fixed–pinned column, 577–578
    flat plate, 488
    flexural stiffness, 345
    flux components, 409
    flux vector, 430
    for-loop, 179
    force vector, 491
    force-displacement curve, 497
    Fortran, 177, 374
    forward difference (Euler) method,
    600
    forward-substitution, 196, 204
    foundation matrix, 257
    foundation modulus, 270
    foundation pressure, 376
    foundation stiffness, 217, 401
    foundation stiffness matrix, 351
    four-noded tetrahedron, 413
    Fourier’s Law, 65, 225, 409, 430, 469,
    474
    Fourier number, 611
    fourth-order ODE, 345
    Fox–Goodwin method, 616
    frame member, 383
    fread, 289, 293
    free joint, 389
    free unknowns, 541
    frequency range, 614, 622
    functions library, 507, 510
    G
    Galerkin method, 145, 157–158, 165,
    205, 256, 348
    Galerkin-in-time method, 600
    Galileo, 488
    gap, 375
    gather, 171, 175, 276, 282, 422, 469
    Gauss points, 251
    Gaussian quadrature, 117
    gear, 192
    gen trap history.m, 606
    generalized mass matrix, 209, 352,
    541, 583
    generalized trapezoidal integration,
    599, 606
    geometric Jacobian, 422
    geometric stiffness matrix, 351, 542,
    567, 571, 575, 578, 625
    get element index.m, 374
    get and add pt mass.m, 543
    get and add pt stiff.m, 543
    get constraint eqs.m, 189
    get element index.m, 170, 287, 319,
    373
    get mesh elements.m, 278, 373
    get mesh nodes.m, 276, 373
    get mesh properties.m, 278, 374
    get point sources.m, 276, 373
    get quadrature rule.m, 280–281
    global axes, 387
    global constant, 540
    geometry mapping, 32
    governing matrix form, 147
    governing matrix system, 546
    gradient operator, 309, 337, 408
    gradient vector, 474
    graph L3 C1 moment.m, 374
    graph L3 C1 result.m, 374
    graph L3 C1 shear.m, 374
    gravity load, 337
    Green’s theorem, 48, 50, 56, 157, 481
    H
    half symmetry, 442, 515
    handbook solution, 365, 544
    hanging bar, 228, 230, 232
    heat conduction, 62
    heat convection coefficient, 263
    heat flow, 242, 267, 269, 409, 468
    heat flux vector, 225, 241, 409, 430
    heat generation rate, 216, 242, 465,
    606
    heat loss, 268
    heat source, 467632 Finite Element Analysis: With Numeric and Symbolic Matlab
    heat transfer, 216, 221, 405
    Helmholtz equation, 405, 407, 538,
    545
    Hermite elements, 341
    Hermite interpolation, 70, 86, 341,
    350, 546
    Hermite polynomials, 347
    Hermite 1D C1 library.m, 89, 90, 343
    hexahedra, 135
    hidden result surface.m, 446
    highest derivative, 60
    Hilber–Hughes–Taylor method, 616
    hinge, 389
    homogeneous solution, 60, 347
    Hooke’s law, 488, 499, 510, 521, 523
    hoop strain, 520
    hoop stress, 520
    hydrostatic pressure, 501
    hyperbolic cosine, 267
    hyperbolic equation, 545
    hydrodynamic lubrication, 289, 291
    I
    independent displacements, 21
    implied loop, 180
    impossible temperatures, 611
    improper mesh, 611
    improper time step, 611
    inclined member, 387
    inclined roller, 193, 329, 330
    incomplete polynomial, 424
    incompressible material, 499, 501
    incorrect interpolation functions, 102
    independent variable, 59
    infinite gradient, 562
    inflow heat flux, 469
    initial condition, 598, 608
    initial strain work, 236, 245, 304, 499,
    508
    initial stress matrix, 542
    initial stress stiffness, 575
    inner product, 63
    instability, 385
    insufficient memory, 369
    insulation, 225
    integral form, 145
    integral of a matrix, 5
    integration by parts, 48, 157, 206,
    348, 545
    integration loop, 136, 279, 282
    integration points, 518
    intensity, 517
    inter-element continuity, 60, 101, 341,
    348
    interior boundary curve, 441
    internal force, 492
    internal heat generation, 608
    internal nodes, 307
    interpolation functions, 61, 309, 337,
    546
    interpolation integrals, 432
    interpolation matrix, 283
    inverse Jacobian matrix, 37, 46–47,
    480
    inverse matrix, 364
    invertible map, 43
    inviscid fluid, 410
    Iron’s Theorem, 540, 612
    isoparametric element, 96, 243, 425
    isotropic material, 408, 470, 500
    iterative solution, 264
    ivert 3 by 3.m, 10
    J
    Jacobian determinant, 36, 81
    Jacobian inverse, 81
    Jacobian matrix, 35–36, 42, 44, 80,
    124, 126, 131, 421, 480
    jumps, 358
    K
    Kelvin, 407
    kinematically unstable, 325
    kinetic energy, 509
    Kirchhoff’s law, 27
    L
    L-shaped membrane, 560–561
    Lagrange interpolation, 70–71
    Lagrange quadrilaterals.m, 93–94Index 633
    Lagrangian 1D library.m, 77
    Lagrangian triangles.m, 97
    Laplace equation, 405
    largest eigenvalue, 540
    laser beam, 611
    least square fit, 612
    least squares method, 159, 165, 342
    line element integrals, 47, 107–108
    line load resultant, 242, 358
    linear acceleration method, 616
    linear algebraic equations, 167
    linear bar, 623
    linear buckling, 568
    linear elastic spring, 15
    linear interpolation, 75, 78, 149
    linear matrix system, 7, 600
    linear spring, 490
    linear tetrahedron, 100, 113
    linear triangle, 44, 93–94, 115
    load case, 567
    load per unit length, 217
    load transfer matrix, 354, 386
    local derivatives, 73
    local stiffness, 402
    logic flags, 273
    long bone, 397
    lubrication, 289
    M
    magnetic field intensity, 494, 503
    magnetic vector potential, 503
    magnetostatics, 405
    mass damping, 356, 614
    mass density, 272, 352, 407, 509, 545,
    556
    mass matrix, 130, 212, 244, 356, 402,
    578
    massless spring, 542
    material axes, 4
    material failure, 587
    material interface, 503
    material properties, 282
    Matlab backslash, 7
    Matlab colon, 177
    Matlab logo, 560
    Matlab single quote, 6
    Matlab symbolic, 300
    matrix addition, 6, 8
    matrix equation of motion, 509
    matrix equations of equilibrium, 208,
    492, 508
    matrix equilibrium equations, 203
    matrix factorization, 204, 599
    matrix inverse, 7
    matrix multiplication, 54, 479
    matrix notation, 4
    matrix partition, 182, 260
    matrix system, 61
    matrix transpose, 54
    maximum principal stress, 589
    maximum shear stress, 284, 288, 445,
    517, 587
    measure, 77, 99
    mechanical strain, 239, 276
    mechanical work, 490–491, 504
    mechanics of materials, 513
    member end forces, 389
    member rotation matrix, 574
    member weight, 317
    membrane analogy, 439
    membrane stiffness matrix, 561
    membrane tension, 560
    membrane thickness, 560
    membrane vibration, 560
    Membrane vibration.m, 565
    memory allocation, 274
    mesh, 146
    mesh at shock surface, 612
    mesh connections, 274
    mesh control, 562
    mesh coordinates, 274
    mesh generator, 4
    mesh refinement, 253
    method of moments, 160
    methods of weighted residuals
    (MWRs), 157
    minimum total potential energy
    (MTPE), 14, 489
    microwave oven, 537
    minimum state, 491
    mirror plane, 306, 369634 Finite Element Analysis: With Numeric and Symbolic Matlab
    mixed boundary condition, 62, 262,
    411, 414
    mixed condition, 4
    mode shape, 545–546, 549
    mode shape surface.m, 565
    modulus of elasticity, 384
    Mohr’s circle, 496
    moment diagram, 359, 376
    moment of inertia, 128, 129, 557
    msh bc xyz.txt, 3, 191, 277, 373
    msh ebc.txt, 3, 191
    msh load pt.txt, 191, 276, 373
    msh mass pt.txt, 543
    msh mpc.txt, 191
    msh properties.txt, 4, 191, 278, 374
    msh stiff pt.txt, 543
    msh typ nodes.txt, 3, 191, 278
    multiple span beam, 370
    multiple-step method, 600
    multipoint constraints (MPC), 168,
    186–187, 192, 203, 323, 329, 389,
    437
    N
    natural boundary condition (NatBC),
    4, 62, 261, 263, 406, 481
    natural boundary condition matrix,
    214
    natural coordinates, 42–43, 85, 110,
    141
    natural frequency, 542, 548, 578
    necessary and sufficient convergence,
    307
    Neumann boundary condition, 146,
    205
    Neumann condition, 4, 62, 164
    neutral state, 491
    Newmark Beta method, 615
    Newton’s Laws, 232, 234, 329, 364,
    493, 556
    Newton’s third law, 23
    node based elements, 503
    node reaction.txt, 445
    node results.txt, 445
    non-circular shaft, 405, 438, 488
    non-dimensional area, 99
    non-essential boundary condition
    (NBC), 4, 67, 146, 158, 259, 261
    non-flat surface, 422
    non-Fourier heat transfer, 611
    non-overlapping elements, 209
    normal boundary gradient, 164
    normal derivative, 62
    normal flux, 411
    normal gradient, 406
    normal heat flux, 431, 471
    normal slope, 306
    normal strain, 496, 499
    normal stress, 498, 518
    normal vector, 50, 65, 411
    normalized eigenvector, 541
    number of boundary segments, 505
    number of degrees of freedom, 493
    number of element equations, 502
    number of integration points, 425
    number of mesh nodes, 506
    number of nodes per element, 502
    number of quadrature points, 131,
    136–137, 285
    number of segment nodes, 505
    number of spatial dimensions, 493
    number of strains, 497
    number of system equations, 506
    number of unknowns per node, 423,
    502
    numbering of the displacements, 502
    numerical integration, 117, 121–122,
    127, 130, 141, 282, 285, 287, 431
    numerical integration, 279
    numerical manipulations, 185
    O
    one-eighth symmetry, 465
    one-step method, 599
    operator matrix, 412, 431
    ordinary differential equation, 598
    orthogonal functions, 146
    orthogonal matrix, 390
    orthotropic diffusion, 435
    orthotropic material, 49, 408, 430, 474Index 635
    orthotropic properties, 406
    orthotropic strains, 521
    oscillating results, 211
    oscillations, 415
    P
    packed flag, 332, 358
    packed integer code, 323
    padded array, 374
    padded connection list, 170
    padded list, 170
    parametric coordinates, 78
    parametric derivatives, 35, 73, 421
    parametric space, 14, 31
    parametric transformation, 41
    particular solution, 60
    partitioned B matrix, 507
    partitioned interpolation, 502
    partitioned matrix, 250
    partitioned stiffness, 360
    patch test, 307
    penalty method, 186
    penalty number, 189
    permeability, 410
    Petrov–Galerkin method, 211
    physical area, 43, 45, 417, 427
    physical derivative, 35, 46, 78
    physical gradient, 79, 423
    physical length, 77, 88, 121–122
    physical space, 31
    physical space dimension, 412
    pile, 270
    pin support, 325, 327, 331
    planar elasticity, 510
    planar frame, 341, 383
    Planar Frame.m, 388
    planar truss, 317, 330, 337
    Planar Truss.m, 318, 329
    plane frame, 383, 402
    plane of symmetry, 433
    plane–strain, 488, 495, 497
    plane–stress, 487, 497, 510
    plane–stress script, 513
    plane–stress vibration, 583
    point couples, 358
    point inertia, 557
    point load, 324, 386, 494
    point mass, 542–543, 556
    point matrices, 310
    point moment, 348
    point spring, 543
    point stiffness, 543
    Poisson’s equation, 51, 164, 405,
    438
    Poisson’s ratio, 499
    polar coordinates, 41
    polar moment of inertia, 47, 139, 218,
    284, 288, 556
    polynomial degree, 285
    polynomial interpolation, 70
    porous media, 405, 409
    positive definite, 499
    post-buckling, 568
    post-processing, 185, 225, 267,
    276–277, 281–282, 468, 510
    potential energy, 491
    potential flow, 409
    pressure gradient, 115, 289, 426, 429
    principal directions, 406
    principal normal stresses, 519
    principal stresses, 586
    principle axes, 383
    principle inertia axes, 397
    properties list, 374
    propped cantilever, 558
    pseudo-element, 188, 189
    Q
    Q16, 93
    Q25, 93
    Q9, 93
    qp rule Gauss.m, 119
    qp rule nat quad.m, 135–136
    qp rule unit Gauss.m, 119, 286
    qp rule unit tri.m, 137
    quadratic bar, 623
    quadratic element, 154, 515
    quadratic interpolation, 71, 74, 76,
    78, 130
    quadratic line element, 71636 Finite Element Analysis: With Numeric and Symbolic Matlab
    quadratic tetrahedron, 113
    quadratic triangle, 583
    quadrature loop, 277, 282, 286
    quadrature point, 586
    quadrature point locations, 136
    quadrature weights, 10
    quadratures, 117, 119
    quadrilateral element, 91, 373
    quadrilateral quadrature, 135
    quarter-symmetry, 436
    quintic beam element, 354, 571, 625
    quintic interpolation, 76, 88
    R
    radial acceleration, 272
    radial displacement, 520
    radial stress, 514
    radiation, 264
    radius of gyration, 568
    rate of heat generation, 441
    rational functions, 425
    reaction force, 516
    reaction vector, 209, 467
    reactions, 168, 182–183, 185, 215,
    221, 227, 238, 259, 267, 276, 284,
    299, 306, 324, 361, 364, 411, 468,
    472, 490
    reactions sum, 468, 472
    real, 87, 539, 547, 558
    rectangular element, 427
    rectangular element integrals, 434
    rectangular interpolation matrix, 502
    rectangular matrix, 248, 386, 422
    rectangular transfer matrix, 354
    reduced integration, 501
    reentrant corner, 439, 446
    repeated freedoms, 439
    residual error, 146, 159, 342
    resultant forces, 386
    results graph, 443
    result on const y.m, 446
    result surface plot.m, 446
    Reynolds 1D Lub.m, 291
    Reynolds’ Equation, 289
    right angle triangle, 465
    righthand side (RHS), 21
    rigid body motion, 347, 538
    rigid body rotation, 496, 514
    rigid body translation, 516
    rigid link, 188
    Robin condition (RBC), 62, 164, 262
    roller support, 329, 331
    rotating bar, 272, 300
    rotational inertia, 556
    rotational pendulum, 556
    rotational spring, 559
    rotational transformation, 496
    row matrix, 6
    rows in B, 412
    rubber, 499
    Runge–Kutta integration, 599
    S
    salar interpolation, 12
    satter, 19
    scalar field problem, 407, 598
    scalar product, 491
    scalar result surface.m, 445
    scaled diagonal mass, 583, 598, 614
    scatter, 167, 171, 175, 180, 263, 413,
    466
    second derivative, 349, 412
    second-order tensor, 496, 586
    second-moments of inertia, 383
    second-order ODE, 60, 205
    seepage, 405
    Seiche motion, 538
    self-adjoint operator, 64
    Serendipity interpolation, 70
    Serendipity quadrilaterals, 101
    settlement, 227–228, 270
    seven bar truss, 331
    shaft, 192, 219, 410
    shape change, 496
    shape functions, 72
    sharp transients, 598
    shear diagram, 359, 376
    shear force, 359
    shear modulus, 218, 284, 499, 556
    shear strain, 284, 499Index 637
    shear strain tensor, 496
    shear stress components, 284, 288,
    409–410, 439, 498
    shells, 487
    simple harmonic motion (SHM), 545
    simplex element, 93
    simplify, 304
    single step methods, 600, 617
    singular matrix, 209, 215
    singular point, 65, 439, 561
    singularity element, 562
    singularity points, 66
    skyline storage mode, 196
    slenderness ratio, 568
    slope continuity, 343, 364, 546
    small deflections, 349
    smallest eigenvalue, 540
    soap film, 440
    soil, 270, 406
    solid elasticity, 501
    solid element, 488
    solid stress, 497
    SolidWorks simulation, 514
    solution bounds, 488
    solution domain, 526
    solution energy, 32
    solution gradient, 16, 24
    solution integral, 32
    sort, 539, 584
    sound system, 537
    source discontinuity, 205, 243
    source rate per unit volume, 407
    source vector, 209, 413, 432, 434
    sources sum, 468
    space frame, 384, 396
    Space Truss.m, 333
    space truss, 332
    space-time finite elements, 599
    sparse storage, 196
    spatial coordinates, 421
    spatial derivatives, 13
    spatial interpolation, 147, 209
    specific weight, 323
    SQ12, 102
    SQ8, 102, 110
    spring stiffness, 490
    spring stiffness matrix, 16
    spring-mass system, 542, 618
    springs in series, 492
    standard output files, 445
    statically indeterminate, 370
    stationary point, 491
    stationary state, 490–491
    steady state, 415, 482, 600, 607, 621
    steel, 306, 323, 331, 339
    Stefan–Boltzmann constant, 264
    stiffness matrix, 283, 285, 302,
    319–320, 516, 541, 554
    straight sided triangle, 139, 141
    straight triangle integrals, 432, 434
    straight triangles, 139
    strain components, 487, 497, 506
    strain energy density, 490, 492, 494,
    498, 528
    strain matrix, 506
    strain-displacement matrix, 506
    strain-displacement relation, 487
    strain-energy density, 498
    strain-stress relation, 527
    stress analysis, 487
    stress averaging, 517
    stress components, 487, 497
    stress function integral, 410, 430, 438,
    440, 446
    stress intensity, 586–587
    stress recovery, 282
    stress stiffening, 542
    stress tensor, 586
    stress-free state, 499
    stress-strain law, 487
    stress-strain relation, 528, 568
    string tension, 548
    String vib 2 L3.m, 549
    string vibration, 545
    strong form, 60, 158, 205
    structural buckling, 565
    structural damping, 614
    structural instability, 565
    structural stiffness matrix, 542
    sub-parametric, 247
    subset, 168, 500, 503, 526
    sum of integrals, 147638 Finite Element Analysis: With Numeric and Symbolic Matlab
    sum to unity, 75
    summary, 53, 67, 140, 165, 203, 309,
    336, 378, 400, 479, 526, 621
    summary and notation, 110
    support movement, 359
    surface area, 55, 493
    surface normal, 55
    surface stress vector, 439
    surface tangent, 35, 55
    surface traction, 493, 504
    switch, 79
    symbolic derivation, 87–88, 90, 92
    symbolic Matlab, 1
    symbolic solutions, 300
    symmetric integrals, 107
    symmetric matrix, 21
    symmetric mode, 546, 563
    symmetry, 306, 432
    symmetry restraint, 516
    system equation number, 371
    system equations, 168
    system equilibrium, 490
    system equilibrium matrices, 172
    system matrices, 414
    system reactions, 23
    T
    tangent vector, 429
    Tapered Axial Bar.m, 284
    tapered bar, 281, 302
    tapered shaft, 281, 287
    temperature, 239, 242, 265, 267–270,
    304, 322, 385, 409, 472
    temperature change, 352
    temporal integration, 599
    tensile yield stress, 586
    tension, 545
    tensioned-beam, 346
    tetrahedra, 94
    thermal analogy, 440
    thermal bending moment, 352
    thermal conductivity, 83, 222, 265,
    409, 441
    thermal expansion, 384
    thermal load, 236, 305, 310, 322, 327,
    337, 386, 529
    thermal moment, 402
    thermal shock, 611
    thermal strains, 245, 304, 499–500
    thermal stress, 245
    thin solid, 510
    thin-walled members, 446
    three-point rule, 140, 142
    time dependent EBC, 606
    time dependent loads, 614
    time dependent reactions, 606
    time history, 598, 606
    time history graph, 607
    time oscillations, 612
    time step size, 540, 599
    Tong’s Theorem, 347
    torque, 219, 284, 287
    torsion, 192, 409, 410
    torsion control integer, 445
    torsional constant, 438
    Torsional Vib BHA L3.m, 557
    torsional shaft, 28, 218, 284, 396, 555
    torsional stiffness matrix, 219, 438,
    556
    torsional vibration, 547, 556, 558
    total potential energy, 490, 491
    transformation matrix, 322, 389
    transformed material property, 431
    transient analysis, 597
    transient history, 610
    transient matrix system, 415, 481, 621
    transpose of a product, 6, 110
    transverse displacement, 345, 560
    transverse moment, 348
    transverse shear, 346, 354, 383
    transverse shear force, 348
    transverse spring, 572
    triangle matrix, 204, 373
    triangle quadrature, 139
    triangular matrix, 5
    triangular quadrature rule selection,
    141
    triple matrix product, 16
    truss buckling, 193, 573
    truss element stiffness, 321, 337Index 639
    truss member, 318
    turbine blade, 578
    twist angle, 219, 284
    two-bar truss, 323, 339
    two force member, 317
    two-node beam, 402
    two-point rule, 118
    U
    U-clamp, 513
    uniaxial tension test, 587
    union, 65, 309, 337, 526
    unique results, 106
    unique solution, 61
    unit coordinates, 80, 94, 123
    unit normal vector, 164
    unit triangle, 139
    unsymmetric integrals, 108
    unsymmetric matrix, 209
    V
    validation, 488, 544
    variable coefficient, 281, 302
    variable Jacobian, 38
    variable source, 242, 281
    variable thickness, 512, 514
    variational calculus, 489
    variational form, 162
    vector elements, 503
    vector interpolation, 13, 502
    vector subscript, 170, 177, 210, 274,
    281, 360, 361, 374, 384–385, 540,
    547, 558
    velocity potential, 65, 405, 410,
    509
    velocity update, 615
    velocity vector, 409
    vertical pile, 217
    vibration, 538
    viscous fluid, 493
    voltage, 25
    Voigt notation, 497
    Voigt stress notation, 586
    volume change, 286, 496
    volume integral, 53
    volume of revolution, 520
    volumetric rate of heat generation,
    476
    von Mises effective stress, 514
    von Mises stress, 586
    W
    warp function, 438
    wave equation, 545
    wave propagation, 614
    waveguide, 537
    weak boundary condition, 4
    weak form, 61, 158, 206
    weighted residuals, 157, 165
    Wilson method, 616
    Winkler foundation, 347
    wood, 406
    Y
    yield stress, 284, 568, 587
    Z
    zero eigenvalue, 538
    zero Jacobian, 38
    zeros, 276, 373, 554

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