MultiPhysicsVault/.raw/AFirstCourseInTheFiniteElementMethod/AFirstCourseInTheFiniteElementMethod_083.md

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Free convection, 538, 540

Fringe carpet, 369

Functional, defined, 12

# G

Galerkin’s method, 12–13, 124–127, 131, 201–203

bar element formulation, 125–127

beam element equations, 201–203

general formulation, 124–125

one-dimensional bar element equations, 124–127, 131

residual method, 124–127, 131 use of, 12–13

Gauss-Jordan method, 718–720

Gauss-Seidel iteration, 733–735

Gaussian elimination, 726–733

Gaussian quadrature, 463–466, 469–475

element stresses, evaluation of, 473–475

one-point, 463–464

sti¤ness matrix, evaluation of, 469–473

three-point, 465–466

two-point formula, 464–465

Global equations, 13–14, 34, 70, 161–163, 320–322, 601

assemblage of, 13–14

bar element, 70

beam element, 161–163

constant-strain triangular (CST) element, 320–322

fluid flow, 601

spring element, 34

Global sti¤ness matrix, 36, 78–81. See also Total sti¤ness matrix

bar element, 78–81

inverse, 80

spring assembly, 36

transverse, 80

Gradient/potential relationship, 599, 607

Grid, defined, 238

Grid equations, 214, 238–255

determination of, 238–255

introduction to, 214

open sections, 241

polar moment of inertia, 240

torsional constant, 240–241, 242

# H

h method of refinement, 355–356

Harmonic motion, simple, 649

Heat flux, 542, 546

Heat flux/temperature gradient relationship, 542, 556–557

Heat transfer, 534–593, 686–693

coe‰cients, 539–540

convection, 538–539, 540

di¤erential equations, 535–538

element conduction matrix, 542–546, 557–558

finite element formulation, 540–555, 555–564, 566–568, 569–574

flowchart for, 574

Galerkin’s method, 569–574

heat conduction, one-dimensional, 535–537

heat conduction, two-dimensional, 537–538

heat flux/temperature gradient relationship, 542, 556–557

heat-transfer coe‰cients, 539–540

introduction to, 534–535

line sources, 564–566

mass transport, 569–574

nodal temperature, 546

numerical time integration, 687–683

one-dimensional, 540–555, 569

point sources, 564–566

program, examples of, 574–576

temperature function, 541, 556

temperature gradient/temperature relationships, 542, 556–557

thermal conductivities, 539–540

three-dimensional, 566–568

time-dependent, 686–693

two-dimensional, 555–564, 574–567

units of, 539–540

variational method, 540–555

Hermite cubic interpolation function, 155–156

Heterosis element, 523

Hooke’s law, 11, 67

# I

Identity matrix, 712

Inclined supports, 103–109, 237

frame equations, 237

truss equations, 103–109

Infinite medium, 361

Infinite stress, 360–361

Integration, see Numerical Integration

Interpolation functions, 32, 74. See also Approximation functions

Intrinsic coordinate system, 444

Inverse, defined, 80

Inverse of a matrix, 712, 716–718, 718–720

adjoint method, 718

cofactor method, 716–717

defined, 712

Gauss-Jordan method, 718–720

row reduction, 718–720

Isoparametric formulation, 443–489, 501–508

bar element sti¤ness matrix, 444–449

defined, 444, 483

element stresses, evaluation of, 473–475

Gaussian quadrature, 463–466, 469–475

intrinsic coordinate system, 444

introduction to, 443

linear hexahedral element, 501–504

natural coordinate system, 444

Newton-Cotes quadrature, 467–469

numerical integration, 463–469

plane element sti¤ness matrix, 452–462

plane stress element, 449–452

quadratic hexahedral element, 504–508

shape functions, higher-order, 475–484

sti¤ness matrix, evaluation of, 469–473

stress analysis, 501–508

transformation mapping, 444

#

Jacobian function, 447

Joint force, see Nodal force

# K

Kirchho¤ assumptions, 515–517

# L

LaGrange interpolation, 482

Least squares method, 130

Line elements, defined, 304

Line sources, 564–566

Linear elements, 9

Linear-elastic bar element, see Bar elements; Truss equations

Linear hexahedral element, 501–504

Linear-strain triangle (LST) equations, 398–411

CSTelements,comparisonof,406–408

defined, 398, 401

derivation of, 389–403

displacement function, 399–401

element type, selection of, 399

introduction to, 398

Pascal triangle, 400

quadratic-strain triangle (QST) element, 400

sti¤ness, determination of, 403–406

sti¤ness matrix, 398–403

strain/displacement relationships, 401–402

stress/strain relationships, 401–402

Load replacement, 177–178

Local sti¤ness matrix, 34

Longitudinal wave velocity, 670

LST, see Linear-strain triangle (LST) equations

Lumped-mass matrix, 651, 682

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# M

Mass matrix, 650–653, 674–681, 681–685

axisymmetric element, 684–685

bar element, 650–653

beam element, 674–681

consistent-mass, 651–653, 682–985

lumped-mass, 651, 682

natural frequencies and, 674–681

plane frame element, 682–683

plane stress/strain element, 683–684

tetrahedral (solid) element, 685

truss element, 681–682

Mass transport, 569–574

Galerkin’s method, 569–574

heat transfer and, 569–574

mass flow rate, 569

Matrix, 4–6, 11, 28–29, 29–34, 36, 37–39, 66–72, 78–81, 92–100,

216, 259–260, 304–305, 309,

310–324, 329–331, 519–523,

542–546, 557–558, 620–622,

650–653, 647–681, 681–685,

708–721. See also Matrix algebra;

Mass matrix; Sti¤ness matrix

algebra, 708–721

column, 4, 708

consistent-mass, 651–653

constant-strain triangular (CST)

element, 304–305, 310–324, 329–331

constitutive, 309, 522

curvature, 521–522

defined, 4, 708–709

element conduction, 542–546, 557–558

element sti¤ness, 11

global nodal displacement, 36

global nodal force, 36

global sti¤ness, 36, 78–81

identity, 712

local sti¤ness, 34

lumped-mass, 651

mass, 650–653, 647–681, 681–685

moment, 521–522

notation for, 4–6

orthogonal, 713–714

quadratic form, 716

rectangular, 4, 708

row, 708

singular, 718

square, 708

sti¤ness, 28–29, 29–34, 66–72, 92–100, 519–523, 650–653

sti¤ness influence coe‰cients, 5

stress/strain, 309

symmetric, 712

system sti¤ness, 36

thermal strain, 620–622

three dimensions, for bars in, 92–100

total sti¤ness, 36, 37–39

transformation (rotation), 92–100, 216, 259–260

unit, 712

Matrix algebra, 708–721

addition of matrices, 710

adjoint method, 718

cofactor method, 716–717

definitions of, 708–709

di¤erentiation’s, 714–715

Gauss-Jordan method, 718–720

identity matrix, 721

integrating, 715–716

inverse of, 712, 716–718, 718–720

multiplication by a scalar, 709

multiplication of matrices, 710–711

operations, 709–716

orthogonal matrix, 713–714

row reduction, 718–720

symmetric matrices, 712

transpose, 711–712

unit matrix, 712

Maximum distortion energy theory, 341–342

Mindlin plate theory, 523, 526

Minimum potential energy, principle of, 52–53, 57–59, 111

finite element equations, 111

spring element equations, 52–53, 57–59

Modeling, 350–397

adaptive refinement, 355

aspect ratio (AR), 351, 352–353

checking, 362

compatibility of results, 363–367

computer program assisted step-bystep solutions, 374–380

concentrated loads, 360–361

connecting (mixing) elements, 361–362

convergence of solution, 367–368

discontinuities, natural subdivisions at, 354, 357

equilibrium of results, 363–367

finite element, 350–363

flowcharts, 374

general considerations, 351

h method of refinement, 355–356

infinite medium, 361

infinite stress, 360–361

introduction to, 350

natural subdivisions, 354, 357

p method of refinement, 358–359

point loads, 360–361

postprocessor results, 362–363

refinement, 355–356, 358–359

static condensation, 369–373

stresses, interpretation of, 368–369

symmetry, 351–354, 355–356

transition triangles, 359–360

Modes, natural, 666, 668

Modulus of elasticity, 748

Moment matrix, 521–522

#

Natural convection, 538, 540

Natural coordinate system, 444, 447

Jacobian function, 447 use of, 444

Natural frequencies, 649, 665–669, 674–681

amplitude, 649

bar element, one-dimensional, 665–669

beam element, 674–681

circular, 649

mass matrices, 674–681

modes, 666, 668

rule of thumb for, 668

Natural subdivisions at discontinuities, 354, 357

Newmark’s method of numerical integration, 659–663

Newton-Cotes quadrature, 467–469

intervals, 467

numerical integration, 467–469

Nodal displacements, 34, 36, 70, 322

bar element, 70

constant-strain triangular (CST) element, 322

global matrix, 36

spring element, 34

Nodal forces, 178–182, 232–233, 752–754

e¤ective, 232–233

e¤ective global, 181–182

equivalent, 178–180, 752–754

load displacement, beams, 178–182

rigid plane frames, 232–233

Nodal hinge, beam elements, 194–199

Nodal potentials, 601

Nodal temperature, 546

Nodes, 29, 152, 370

actual, 370

condensed out, 370

defined, 29

sign conventions for beams, 152

Nonexistence of solution, 724

Nonuniqueness of solution, 723–724

Numerical comparisons, plate bending element, 523–524

Numerical integration, 463–469, 653–665, 687–693

central di¤erence method, 653, 654–659

direct integration, 653

dynamic systems, 653–665

explicit, 689

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flowcharts for, 656, 661

Gaussian quadrature, 463–466, 469–475

heat-transfer, 687–693

Newmark’s method, 659–663

Newton-Cotes quadrature, 467–469

Simpson one-third rule, 463, 467

time, 653–665, 687–693

trapezoid rule, 463, 467–468, 687

Wilson’s method, 664–665

# O

One-dimensional elements, 124–127, 127–131, 540–555, 569, 598–601, 665–669, 669–674

bar analysis, 665–669, 669–674

bar element equations, 124–127

bar element problems, 127–131

fluid flow, 598–601

heat-transfer problems, 540–555, 569

mass transport, 569

natural frequencies, 665–669

time-dependent, 669–674

Open sections, 241

Orthogonal matrix, 713–714

#

p method of refinement, 358–359

Parasitic shear, 342

Pascal triangle, 400

Penalty formulation, 331

Penalty method, 50–52

Period of vibration, 649

Pipes, fluid flow in, 596–598

Plane element, 452–463, 682–684

body forces, 460

consistent-mass matrix, 683–684

displacement functions, 455–456

equations, 459–460

isoparametric formulation, 452–463

mass matrices, 682–684

quadrilateral element, 684

selection of, 453–455

sti¤ness matrix, 452–463

strain/displacement relationships, 456–459

stress/strain relationships, 456–459, 683–684

surface forces, 460

Plane frames, 218–236, 682–683

element, 682–683

mass matrices, 682–683

rigid, 218–236

Plane strain, 305–309, 374–380, 683–684

concept of, 305–309

consistent-mass matrix, 683–684

defined, 305

flowchart for, 374

program assisted step-by-step solutions, 374–380

Plane stress, 305–309, 331–342, 374–380, 449–452, 683–684

concept of, 305–309

consistent-mass matrix, 683–684

defined, 305

discretization, 331–332

displacement functions, 450–451

element, 449–452

finite element solution of, 331–342

flowchart for, 374

isoparametric formulation, 449–452

maximum distortion energy theory, 341–342

principal angle, 307

program assisted step-by-step solutions, 374–380

rectangular element, 449–452

sti¤ness matrix assemblage for, 332–341

von Mises (von Mises-Hencky) theory, 341–342

Plane truss, solution of, 84–92

Plate bending element, 514–533

computer solution for, 524–528

concept of, 514–518

deformation of, 514–515

displacement function, 519–521

equations, 519–523

geometry of, 514–515

heterosis element, 523

introduction to, 514

Kirchho¤ assumptions, 515–517

Mindlin plate theory, 523, 526

numerical comparisons, 523–524

potential energy, 518

rigidity of, 517

selection of, 519

sti¤ness matrix, 519–523

strain/displacement relationships, 521–522

stress/strain relationships, 517–518, 521–522

Point loads, 360–361

Point sources, 564–566

Polar moment of inertia, 240

Porous medium, fluid flow in, 594–596

Potential energy approach, 52–60, 109–120, 199–201, 518

admissible variation, 55

bar element equations, 109–120

beam element equations, 199–201

minimum potential energy, principle of, 52–53, 57–59, 111

plate bending element, 518

spring element equations, 52–60

stationary value, 54

total potential energy, 53, 518

truss equations, 109–120

variation, 55

Potential function, 589

Pressure vessel, axisymmetric, solution of, 422–428

Primary unknowns, defined, 14

Principal angle, 307

Principal stresses, 307

#

Q8 element, 480

Q9 element, 482

Quadratic elements, 9

Quadratic form, 716

Quadratic hexahedral element, 504–508

Quadratic-strain triangle (QST) element, 400

Quadrilateral element consistent-mass matrix, 684

# R

Refinement, 355–356, 358–359

adaptive, 355

h method, 355–356

p method, 358–359

Reflective (mirror) symmetry, 100–103

Rigid plane frames, 218–236

defined, 218

examples of, 218–236

Row reduction, 718–720

#

Serendipity element, 481

Shape functions, 32, 155–156, 475–484

beam element, 155–156

defined, 32

higher-order, 475–484

isoparametric formulation, 475–484

LaGrange element, 482

Q8 element, 480

Q9 element, 482

serendipity element, 481

Shear locking, 342

Sign conventions, beams, 152, 256–257

Simultaneous linear equations, 722–743

banded-symmetric method, 735–741

Cramer’s rule, 724–725

Gauss-Seidel iteration, 733–735

Gaussian elimination, 726–733

general form of, 722–723

introduction to, 722

inversion of coe‰cient matrix, 726

methods for solving, 724–735

nonexistence of solution, 724

nonuniqueness of solution, 723–724

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Simultaneous linear equations (Continued )   
skyline method, 735–741   
uniqueness of solution, 723   
wavefront method, 735–741   
Sizing of elements, 355–356, 358–359   
Skew, defined, 370–371   
Skewed supports, 103–109, 237   
frame equations, 237   
truss equations, 103–109   
Skyline method, 735–741   
Smoothing process, 369   
Solid bodies, fluid flow around, 596–598   
Solid element, see Tetrahedral element   
Spring elements, 29–34, 34–37, 52–60 assemblage of, 34–37   
compatibility requirement, 35   
continuity requirement, 35   
degrees of freedom, 29   
displacement function, 31–32   
element type, 30–31   
equations, 52–60   
global equation for, 34   
nodal displacements, 34   
nodes, 29   
potential energy approach, 52–60   
spring constant, 29   
sti¤ness matrix for, 29–34   
Spring-mass system, 647–649 amplitude, 649   
dynamics of, 647–649   
harmonic motion, simple, 649   
natural circular frequency, 649   
period of vibration, 649   
Static condensation, 369–373   
concept of, 369–373   
condensed load vector, 370   
condensed out nodes, 370   
condensed sti¤ness matrix, 370   
directional sti¤ness bias, 371   
skew, 370–371   
Stationary value, 54   
Sti¤ness equations, 304–349   
constant-strain triangular (CST)   
element, 304–305, 310–324,   
324–329, 329–331   
explicit expression, 329–331   
finite element solution, 331–342   
introduction to, 304–305   
maximum distortion energy theory, 341–342   
plane strain, 305–309   
plane stress, 305–309, 331–342   
von Mises (von Mises-Hencky) theory, 341–342   
Sti¤ness influence coe‰cients, 5   
Sti¤ness matrix, 28–29, 29–34, 36, 66–72, 92–100, 153–158, 158–161, 161–163, 304–305,

310–324, 332–341, 369–373,

402–403, 403–406, 419–422,

423–428, 444–449, 451–452,

452–463, 469–473, 497–500,

519–523, 599–601, 608, 735–741

axisymmetric element, 419–422, 423–428

banded-symmetric method, 735–741

bar element, 66–72, 444–449

beam equations, 153–158, 158–161, 161–163

beams, examples of assemblage of, 161–163

bending deformations, 153–158

body forces, 419–420, 448

condensed, 370

constant-strain triangular (CST)

element, 304–305, 310–324

defined, 28–29

Euler-Bernouli theory, based on, 153–158

evaluation of, 469–473

fluid flow, 599–601, 608

Gaussian quadrature, 469–473

isoparametric formulation, 444–449, 469–473

linear-strain triangle (LST) element, 402–403, 403–406

local, 34

plane element, 452–463

plane stress element, 451–452

plane stress problem, assemblage of for, 332–341

plate bending element, 519–523

skyline method, 735–741

spring element, 29–34

static condensation, 369–373

superposition, assemblage by, 332–341, 423–428

surface forces, 420–421, 448–449

tetrahedral element, 497–500

threedimensions, forbarsin,92–100

Timoshenko theory, based on, 158–161

total (global), 36, 37–39, 332–341

transition matrix and, 92–100

transverse shear deformations, 158–161

wavefront method, 735–741

Sti¤ness method, 7, 28–64

boundary conditions, 34, 39–52

direct, 37–39

introduction to, 28–64

minimum potential energy, principle of, 52–53, 57–59

penalty method, 50–52

potential energy approach, 52–60

spring constant, 29

spring elements, 29–34, 34–37, 52–60

sti¤ness matrix, 28–29, 29–34, 36

superposition, 37–39

total potential energy, 53

total sti¤ness matrix, 37–39

use of, 7

Strain, 306–309. See also Plane strain

normal, 308

shear, 308

two-dimensional state of, 306–309

Strain/displacement relationships, 11, 33, 69, 156–157, 315–320, 401–402, 417–419, 446–447, 451, 456–459, 490–493, 496–497, 521–522, 746–748

axisymmetric element, 417–419

bar element, 69

beam element, 156–157

condition of compatibility, 748

constant-strain triangular (CST) element, 315–320

deformation, 33

elasticity theory, 746–748

Hooke’s law, 11, 67

isoparametric formulation, 446–447, 456–459

linear-strain triangle (LST) elements, 401–402

plane element, linear, 456–459

plane stress element, 451

plate bending element, 521–522

spring element, 33

stress analysis, 490–493

tetrahedral element, 496–497

Stress, 82–83, 306–309, 341–342, 360–361, 368–369, 473–475. See also Plane stress; Thermal stress

computation of for a bar element, 82–83

Coulomb-Mohr theory, 342

e¤ective, 341

equivalent, 341

evaluation of, 473–475

fringe carpet, 369

Gaussian quadrature, 473–475

infinite, 360–361

interpretation of, 368–369

maximum distortion energy theory, 341–342

principal, 307

smoothing process, 369

two-dimensional state of, 306–309

von Mises (von Mises-Hencky) theory, 341–342

Stress analysis, 490–513

isoparametric formulation, 501–508

linear hexahedral element, 501–504

quadratic hexahedral element, 504–508

strain/displacement relationships, 490–493

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stress/strain relationships, 490–493

tetrahedral element, 493–500

three-dimensional, 490–513

Stress/strain relationships, 11, 14, 33, 69, 156–157, 315–320, 401–402, 417–419, 446–447, 451, 456–459, 490–493, 496–497, 517–518, 521–522, 748–751

axisymmetric element, 417–419

bar element, 69

beam element, 156–157

constant-strain triangular (CST) element, 315–320

constitutive law, 11

deformation, 33

elasticity theory, 748–751

isoparametric formulation, 446–447, 456–459

linear-strain triangle (LST) elements, 401–402

modulus of elasticity, 748

plane element, linear, 456–459

plane stress element, 451

plate bending element, 517–518, 521–522

solving for, 14

spring element, 33

stress analysis, 490–493

tetrahedral element, 496–497

Structural dynamics, see Dynamics

Structural steel, properties of, 759–772

Structures, 100–103, 214–303

frame equations, 214–237

grid equations, 238–255

rigid plane frames, 218–236

substructure analysis, 269–275

symmetry in, 100–103

Subdivisions, natural, 354, 357

Subdomain method, 129–130

Subparametric formulation, 483–484

Substructure analysis, 269–275

Superposition, 37–39, 332–341, 423–428. See also Direct sti¤ness method

axisymmetric element, assemblage for by, 423–428

plane stress problem, assemblage for by, 332–341

total (global) sti¤ness matrix, assemblage by, 37–39, 332–341

Surface forces, 326–329, 420–421, 448–449, 460, 498

axisymmetric elements, 420–421

bar element, 448–449

natural coordinate system, 448–449

plane element, 460

tetrahedral element, 498

treatment of, 326–329

Symmetry, 100–103, 351–354, 355–356

axial, 100

finite element modeling, 351–354, 355–356

reflective (mirror), 100–103, 351

structures, use of in, 100–103

Symmetric matrix, 712

System sti¤ness matrix, see Total sti¤ness matrix

#

Temperature, 541–542, 546, 556, 574–576

distribution, examples of, 574–576

function, 541, 556

gradients, 542, 546

nodal, 546

Temperature gradient/temperature relationships, 542, 556–557

Tetrahedral element, 493–500, 685

body forces, 497–498

consistent-mass matrix, 685

displacement functions, 494–496

equations, 497–498

selection of, 493–494

sti¤ness matrix, 497–500

strain/displacement relationships, 496–497

stress/strain relationships, 496–497

surface forces, 498

Thermal conductivities, 539–540

Thermal strain matrix, 620–622

Thermal stress, 617–646

coe‰cient of thermal expansion, 618

formulation of, 617–640

introduction to, 617

thermal strain matrix, 620–622

Three-dimensional elements, 490–513, 566–568

heat-transfer problems, 566–568

space, 92–100

sti¤ness matrix for a bar, 94–100

stress analysis, 490–513

tetrahedral element, 493–500

transformation matrix for a bar, 92–94

Time, numerical integration in, 653–665, 687–689

Time-dependent, 649–653, 669–674, 686–693

bar analysis, one-dimensional, 669–674

heat transfer, 686–693

longitudinal wave velocity, 670

numerical time integration, 687–693

stress analysis, 649–653

structural dynamics, 649–653, 669–674

Timoshenko theory, 158–161

Torsional constant, 240–241, 242

Total equations, see Global equations

Total potential energy, defined, 53

Total sti¤ness matrix, 36, 37–39, 162. See also Global sti¤ness matrix beam element, 162

direct sti¤ness method, assembly by, 37–39

spring assembly, 36

superposition, assembly by, 37–39

Transformation mapping, 444

Transformation (rotation) matrix, 92–100, 216, 259–260, 713

Transition triangles, 359–360

Transpose of a matrix, 711

Transverse, defined, 80

Transverse shear deformations, 158–161

Trapezoid rule, 467–468, 687

Truss equations, 65–149, 681–682. See also Bar elements

approximation functions, 72–74

bar elements, 67–72, 92–100, 109–120, 120–124, 124–127, 127–131

boundary conditions, 103–109

collocation method, 129

consistent-mass matrix, 682

displacements, 72–74

exact solution, 120–124

finite element solution, 120–124

Galerkin’s residual method, 124–127, 131

global sti¤ness matrix, 78–81

inclined supports, 103–109

introduction to, 65

least squares method, 130

local coordinates for, 66–72

lumped-mass matrix, 682

mass matrices, 681–682

plane truss, solution of, 84–92

potential energy approach, 109–120

residual methods, 124–127, 127–131

skewed supports, 103–109

sti¤ness matrix, 66–72, 92–100

strain/displacement relationships, 69

stress, computation of for a bar element, 82–83

stress/strain relationships, 69

subdomain method, 129–130

symmetry, use of in structures, 100–103

transformation (rotation) matrix, 92–100

vectors, transformation of in two dimensions, 75–77

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Two dimensional elements, 75–77, 214–218, 304–349, 555–564, 574–576, 606–610 beam elements, arbitrarily oriented, 214–218 flowchart for heat-transfer process fluid flow, 606–610 heat-transfer problems, 555–564 plane stress and strain equations, 304–349 temperature distribution, 574–576 vectors, transformation of in, 75–77

U Uniqueness of solution, 723 Unit matrix, 712

V Variation, defined, 55 Variational methods, 52, 540–555 Vectors, 75–77, 370

condensed load, 370 transformation of in two dimensions, 75–77 Velocity, 602, 670 fluid flow 602 longitudinal wave, 670 Velocity/gradient relationship, 599, 607 Virtual work, principle of, 755–758 compatible displacements, 755 D’Alembert’s principle, 755–756 Volumetric flow rates, 602 Von Mises (von Mises-Hencky) theory, 341–342

W Wavefront method, 735–741 Weighted residuals, methods of, 12–13, 124–127, 127–131, 201–203

bar element equations, 124–127, 127–131 beam element equations, 201–203 collocation method, 129 Galerkin’s method, 12–13, 124–127, 131, 201–203 introduction to, 12–13 least squares method, 130 one-dimensional problems, 127–131 subdomain method, 129–130 Wilson’s (Wilson-Theta) method of numerical integration, 664–665 Work methods, 12, 52–53, 57–59, 176–177, 755–758 Castigliano’s theorem, 12 introduction to, 12 minimum potential energy, principle of, 52–53, 57–59 virtual work, principle of, 755–758 work-equivalence, 176–177

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Cross-sectional diagram of a mechanical device with internal components and color-coded heat flow (no text or labels)
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Fuel injector—The turbine engine fuel injector is part of a turbine engine used in road transport vehicles designed by an engineering firm. Shown is the steady-state heat transfer analysis performed in ALGOR to determine the temperature distribution from convection loads applied to the inner shaft and the outside surface of the entire assembly. Brick elements (not shown) were used in the model. (Courtesy of ALGOR, Inc.)

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Color-coded 3D thermal or stress simulation visualization of a cylindrical mechanical component (no text or symbols)
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Housing model—The housing model made of ASTM A-572, grade 50 steel, is the rear-axle housing of a mining truck. A finite element analysis of the housing was necessary to determine why the housing failed in the field. The stress analysis performed using brick elements with torsional loads applied showed that the area around the padeye (shown in red color) was subjected to critical stresses, validating the visual inspection of the damaged part. The analysis was performed by a structural engineer working for the mining company. (Courtesy of ALGOR, Inc.)

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3D finite element mesh model of a mechanical component with color-coded stress or flow visualization (no text or symbols)
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Cylinder head—The cylinder head model made of stainless steel AISI 410, is part of a prototype diesel engine that would provide reduced heat rejection and increased power density. Shown is the ALGOR steady-state heat transfer analysis (using brick elements) revealing the high temperatures of 1500 degrees F in red color at the interface between the two exhaust ports. These temperatures were then fed into the linear stress analyzer to obtain the thermal stresses ranging from 85 ksi to 200 ksi. The linear stress analysis confirmed the behavior that the engineers saw in the initial prototype tests. The highest thermal stresses coincided with the part of the cylinder head that had been leaking in the preliminary prototypes. (Courtesy of ALGOR, Inc.)

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3D simulation of a mechanical assembly with blue and yellow components, no visible text or symbols
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Subsoiler—The 12-row subsoiler used in agricultural equipment was designed to prepare 10 inch wide seed beds spaced 40 inches apart as commonly used in cotton production. One of these load conditions was simulating the shanks of the subsoiler pulling through 18 inches of hardpan soil. The ALGOR linear static stress analysis program was used to optimize the thickness, shape, and material of the frame, hitch and hinge components to reduce high stresses. The stress shown is the von Mises stress plot when the load is simulating the shanks pulling through approximately 18 inches of soil. From these results the designers can determine the parts that need to be made of stronger steel alloys. (Courtesy of ALGOR, Inc.)

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3D model of a pink structural frame with visible supports and components (no text or symbols)
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Truck frame—Th e tru ck fra m e s h own is a fi n ite e l e m e nt m od e l m ad e of b ric k e l e m e nts. Th e stee l fra m e was d esig n ed to retrofit a t r u c k wi t h a n e l ect ri c m oto r wi t h batte ri es . (Co u rtesy of Tr u eG ri d 8.)

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3D CAD model of a mechanical component with colored internal sections and a coordinate axis indicator (no text or symbols on the model itself)
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Bearing housing—The steel bearing housing model is used to support one end of reel spool in the paper industry. A finite element model was created to study the deflection and stress in the bearing housing. The model consisted of beam elements to model the journal inside of the bearing, brick elements to model the bearings (multi-colored inside of the green colored bearing housing), bearing housing, and rail (orange color), universal joints to connect the journal to the bearing surface, surface contact pairs to represent the bearing-to-housing interface and housingto-rail interface. The model was created in Algor using FEMPRO. (Compliments of UW—Platteville students, Jason Fencl and David Stertz.)