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 |  موضوع: كتاب Finite Element Analysis - With Numeric and Symbolic MatLAB الأربعاء 27 مارس 2024, 1:55 am | |
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أخواني في الله
أحضرت لكم كتاب
Finite Element Analysis - With Numeric and Symbolic MatLAB
John E Akin
Rice University, USA
و المحتوى كما يلي :
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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