Fundamental-Finite-Element-Analysis-and-Applications

BOUNDARY VALUE PROBLEM CORRESPONDING TO A GIVEN FUNCTIONAL 683

This san be written as follows:

[ r" 2) r -2)Jt=o If If

8 x(x - If)U (x, dx - Jx=o x(x - If)U (x, dx

ff(- 21(OoUx )2+ 21(OoUy)2 + sm. (x)u)dx dy] = 0

A

Thus the variational functional is as follows:

L" L"leu) =
x(x - If)U (x, I-f) dx - x(x - If)U (x, - I-f ) dx
x=O 2 x=O 2

ff (-21(OoUx )2 + 21(OoUy )2 + .)

sm(x)u dxdy

'A

8.3 BOUNDARY VALUE PROBLEM CORRESPONDING
TO A GIVEN FUNCTiONAL

In this section we consider the situation opposite to that of Section B.2. Here yve assume
that a functional is known. Our goal then is to find the corresponding boundary value
problem. The procedure is useful in verifying that a given functional indeed is the correct
functional for the problem.

Since the minimum of the functional corresponds to the equivalent differential equation,
we start the process by setting the first.variation of the functional to zero. Mathematical ma-
nipulations involving integration by parts are then carried out to bring boundary conditions
into the picture. The process is illustrated through the following example.

Example JB.5 Axial Deformations In Chapter 2 we used the following potential en-
ergy functional for the nonuniform bar shown in Figure B.2:

Jor JorL L
= =II U - W 21 EA(x) (ddux)2dx - q(x)u(x)dx - Pu(L)

To show that this is a suitable functional for the governing equation for the axial deforma-

p
Figure B.Z. Tapered axially loaded bar

684 VARIATIONAL FORM FOR BOUNDARY VALUE PROBLEMS

tions, we take its first variation as follows:

r rL
Jo Joorr = 21: 0 [EA(x)(ddUx )2] dx - o[q(x)u(x)]dx - POu(L)

From the definition of variation

1 1L L
orr = EA(xd)u-o (d- U) dx - q(x)oudx - Pou(L)
o dx dx
0

Noting that order of differentiation and variation can be interchanged, we have

1orr = EA(xd)u-d-(dou-)dx -
LL
1o dx x
q(x) ou dx - Pou(L)

0

Integrating the first term by parts, we have

orr = [EA(xdd)-uoxu]oL - 1 1L L
[-d (EA(;Ii)d-Ud )] Sudx-« q(x)oudx-Pou(L)
odx x
0

At x = L, EA du/dx = P, the applied load, and thus at the this point the boundary term

cancels with the last term, and we have

Jor Jororr = -EA(O)-d-u-;(rO;-) ou(O) - L
[ddx (EdA(xU) d)x ] Sudx -
q(x) ou dx

At x = 0 we have a specified displacement. Requiring all assumed solutions to satisfy this

boundary condition means that ou(O) = O. Thus, with this restriction, the variation of the

potential energy functional can be written as follows:

orrFor to be zero (a necessary condition for the minimum), we have

1L [:x (EA(X)~:)+ q(X)] Sudx =0

Since the variation ou is arbitrary, the only way this integral can always be zero is if the
term inside the square brackets is zero. Thus

ddx (EA(x) ddUx ) + q(x) = 0

This is exactly the differential equation governing the axial deformation of a nonuniform
bar. Thus, as long as we 'restrict ourselves to solutions that satisfy the displacement bound-
ary condition, the given potential energy is the appropriate functional for the problem.

PROBLEMS 685

PROBLEMS

B.1 Compute the variation of the foliowing functionals:

(a) F[u,x] = .u02 + ~

(b) F[u', u, x] = L\~u/2 + u3 + x) dx
.0

(c) F[u, x] = u2u,2 + x3

(d) F[u",u',u,x] = J(-Il(Uu" +u'x)dx

B.2 Determine an equivalent variational form for the following boundary value problem:

0<x<1

with the boundary conditions u'(1) + 2u(l) = 1

u(O) = 1;

B.3 Consider the following boundary value problem: 7T/4 < x < 7T12

=u" sin(x) + u' cos(x) + u sin(x) 0;
=u(7T/4) 1 and U'(7TI2) = 2

Derive an equivalent variational functional for the problem. Note that the differential
equation can be written as follows:

~ [u' sin(x)] + u sin(x) = 0

dx
B.4 Consider the finite element solution of the following boundary value problem:

=<u" +x 0;' 7T/4 < x < 7T12
=u(7T/4) = 1 and U'(7TI2) + 2 0

Verify that the following is an appropriate functional for the problem: .

2u(~)leu) = + l:2 Gu,2 +XU)dX

B.5 A functional is given as follows:

L'(1= (7fX))leu) -EAu'-? - u sm• - dx - -1~u(l)2 - -1u(O)
o 2 1 2 7T

Determine the corresponding boundary value problem.

686 VARIATIONAL FORMFOR BOUNDARY VALUE PROBLEMS
B.6 The potential energy for the problem of the torsion of a thin-walled section with
warping restraint can be written as follows:

JoII = (L(E-::JfrfJ1I2 + -Gfr1fJ/2 - t(x)rfJ) dx

where E is the modulus of elasticity, G is the shear modulus, J is the warping
IV

constant, Jo is the torsional constant, t is the thickness of the section, and rfJ is the
angle through which a cross section rotates. Determine the governing differential
equation and appropriate boundary conditions.

!

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!

INDEX

A Assembly of element equations, 21
ABAQUS applications, 641, 663 Assembly with p-modes,617
Axial deformation, 99
plane stress analysis, 671
steady-state heat flow, 666 bars, 99
truss analysis, 665 tapered bar, 102, 113, 158
ABAQUS execution procedure, 663 uniform bar, 101, 109
Admissible assumed solution, 108, 116, 122, using Galerkin method, 104,109, 113
using Rayleigh-Ritz method, 131,
135
Analysis of elastic solids, 467 133
Analytical solution for
B
axial deformations, 102, 103 Backward substitution, 59
beams, 242, 244 Bars subjected to torsion, 282, 317, 342
circular fin, 603 Basic conjugate gradient, 62
pressure vessel, 518
rotating disk, 524 algorithm, 64
ANSYS applications, 641, 642 method,62
plane stress analysis; 648 Beam bending, 236
pressurevessel, 523 Beam element, 244
rectangular shaft, 346 Bending moment, 236
rotating disk, 529 Body forces, 480
steady-state heat flow, 342, 651 Boundary conditions, 36
transient heat flow, 556 for beams, 238
truss analysis, 655 at infinity, 4S2
weld stresses, 530 Boundary element method, 1
ANSYS execution procedure 642 Boundary integral, 320, 438
Applied surface forces, 471 ' Boundary value problem (BVP), 115, 173,
Area coordinates, 372
Area integral, 428 311,586,604
Area of a triangle, 11
Aspect ratio, 404 C
Cantilever bracket, 32
Characteristic equation, 204

695

696 INDEX

Choleski decomposition, 58 linear element for ID BVP, 185
algorithm, 60 mapped quadrilateral elements for plane
method,58
stress/strain, 508
Circular fin, 599 nine-node rectangular element for 2D
Completeness requirement, 136
Computation ofreactions, 41 BVP, 354
Concentrated loads, 6 p-formulation element for ID BVP, 593
Conjugate gradient method, 62 p-formulation element for 2D BVP, 609
Constitutive equations, 19,478,492,494 plane frame element, 266
Continuity, 141 plane truss element, 5, 225, 232
Contour plot, 395 quadratic element for ID BVP, 186
Convection, 8 quarter-point element for crack-tip
Crack-tip singularity, 531
Culvert model, 406 singularity, 533
space frame element, 284
D space truss element, 229, 232
Degrees offreedom, 3, 139 torsional spring element, 234
Differential equation triangular element for 2D BVP, 361
triangularelement for heat flow, 11
for axial deformations, 99 triangular element for plane stress, 18,
for beam bending, 236
for linear elasticity, 483 498
Discretization, 2, 14 Element interfaces, 15
Displacement constraint, 502 Element shapes, 3
Dynamic equilibrium, 100, 238, 557 Element solution, 49
Elementary beam theory, 236
E ! Equilibrium check, 41
EBC (Essential Boundary Condition), 36, Equilibrium equations, 100,237,481
Equivalent stress, 55, 474
312 Equivalent variational form, 679
EBC approximate treatment, 40 Essential boundary condition (EBC), 36,
EBC with p-modes, 620
Eig command, 568 312
Eigensystem command, 568 EBC by modifying equations, 39
Eigenvalue-eigenvectors, 204, 354, 567 EBC by rearranging equations, 37
Eight-node rectangular element, 346 Euler buckling, 205
Elastic buckling of bars, 178, 202 Evaluation of
Elastic solids, 467 area integrals, 428
Elastic solids subjected to dynamic loads, boundary integrals, 436
Exact solution of differential equations, 101
557 Excessive curvature, 404
Electromagnetics, 319, 353 Extensional spring support, 240
Element, 2
Element with curved edges, 399 F
Element equations Field problems, 545
Finite difference method, 1
axial deformations, 153 Finite element
axial spring element, 234
beam element, 247 computations involving mapped elements,
eight-node quadrilateral element for 2D 420

BVP, 447 discretization, 2, 14
eight-node rectangular element for 2D equations in the presence of initial strains,

BVP, 348 489
four-node quadrilateral element for 2D form of assumed solutions, 138
method, I
BVP, 443 solution of axial deformation problems,
four-node rectangular element for 2D
155
BVP, 332 steps, 2
infinite element for 1D BVP, 454 Finite element equations for 2D BVP, 326
infinite element for 2D BVP, 458

INDEX 697

;.. Hierarchical interpolation functions, 586
Fixed-end beam solution Higher-order boundary value problems,

trapezoidal loading, 262 119
uniform loading, 259 Higher-order triangular elements, 371
Flow around
a cylinder, 363 I
an object, 313 Inclined roller support, 73, 76
Forward elimination, 59 Incomplete Choleski preconditioning, 70
Four-node rectangular element, 329 Incorporating NBC for 1D BVP, 184
Fourth-order equation, 122. Infinite boundary, 452
Frames in multistory buildings, 293 Infinite elements, 452
Free-vibration analysis, 567 Initial strains in trusses, 231
Free-surface problem, 633 In-plane rigid floor system, 293
Functional, 684 Integration using change of variables, 382
Fundamental concepts in elasticity, 467 Integration by parts, 107
Integration by parts in higher dimensions,
G
Galerkin method, 104, 115, 135 320
Galerkin versus Rayleigh-Ritz, 138 Integration by parts in two dimensions, 322
Galerkin weighting function, 106 Integration over a triangle, 361
Galerkin weighting functions in the finite Interelement derivatives, 214
Internal hinge in a beam, 253
element form, 143 Internal modes, 606
Gauss's divergence theorem, 320 Interpolation functions
Gauss points and weights, 411
Gauss quadrature, 408 Hermite, 146,245
Lagrange, 142,350
for area integrals, 414 serendipity, 347, 399, 400
for one-dimensional integrals, 409 triangle, 359
for volume integrals, 417 Invalid mesh, 15
General formula for Hermite interpolation Irrotational fluid flow, 313
Isotropic material, 478
(2D BVP), 146
Generalized eigenvalue problem, 567 J
Generalized Hooke's law, 478
Global and local coordinates, 223, 266, 285 Jacobian, 384
Global equations, 21 Jacobi preconditioning, 68
Governing differential equations, 480 Joints in frames, 294
Green-Gauss theorem, 321
Groundwater flow, 633 L
Guidelines for mapped element shapes, Lagrange interpolation, 142
Lagrange interpolation for rectangular
403
elements, 350
H Lagrange multipliers, 75
Handling concentrated loads, 6 Least-squares weighted residual, 106
Handling essential boundary conditions, Least-squares weighting function, 106
Legendre polynomials, 588
37 Linear assumed solution, 151
Heat conduction through household iron, Linear interpolation functions for

189 second-order problems, 185
Heat flow LinearSolve command, 58
Linear solution, 185
differential equation, 8 Local to global transformation, 224, 228,
L shape, 8, 13, 337,445,449,624
through thin fins, 175, 190 266,285 .
Heat flux, 8
Heat loss, 194 M
Hermite interpolation, 144,245 Mapped elements, 381
Hermite interpolation for fourth-order Mapped mesh generation, 405
Mapped quadrilateral elements, 508
problems, 144
h-fonnulation, 586

698 INDEX

Mapping plane stress problem, 58
plane truss, 51, 227
a curve, 389 plane truss element equations, 7
plane truss element results, 51
a quadraticcurve, 391 second-orderBVP using Galerkin

a straight line, 387 method,119
solution of 1D BVP, 187
Mapping lines, 387 solution of axial deformationproblems,

Mappingquadrilateral areas, 392 163
solution of bucklingproblem, 207
Mappingquadrilaterals, 392 space frames, 293
space truss, 231
Mappingquadrilaterals using interpolation tapered bar using Rayleigh-Ritz, 135
transient analysisof a plane frame, 576
functions, 392 transient analysisof a plane truss, 573
triangularelementfor 2D BVP,371
Mass matrices for common structural triangularelement for heat flow, 14
triangularelementfor plane stress, 21
elements, 561 triangularelementfor plane stress and

axial deformation element, 561 plane strain, 507
triangularelement for transient2D BVP,
beam element,564
557
frame element,565 Matrix notation, 11
Maximumshear-stressfailure, 472
plane stress/strainelement,566 Mesh compatibility, 15, 73
Midside node for a quadratic curve, 392
plane truss element,562 Minimumof potential energy, 154
Modal analysis, 567
space truss element,563 Mohr-Coulomb failure criterion, 474
Moment equilibrium,468
Master area, 393 Multipointconstraints,2, 72
Multistory building, 293
Master line, 388
N
Mathematica examples Natural boundarycondition (NBC), 36, 312
Naturalfrequency, 567
eight-nodequadrilateral elementfor plane NDSolve function, 556
Newmark's method, 561
stress and plane strain, 525 Nine-noderectangularelement, 352
Nodal degrees of freedom, 3, 139
1D BVP involvinginfinitedomain,454 Nodal solution, 36
Node, 2
third-orderequation, 125 Normal derivative, 312
Normal stress, 468
TM modesfor waveguides, 353 Notation for derivatives, 116
Notchedbeam, 16,510
MathematicalMATLAB implementations Numericalintegration, 408

beams, 262 o

eight-nodequadrilateralelementfor 2D Ode23command, 573
One-dimensional BVP (lD BVP), 173,586
BVP,447 One-dimensional BVP using Galerkin

eight-noderectangularelementfor 2D method,115
One-dimensional integrals,409
BVP,350 One-pointformula, 409

finiteelement assemblyprocedure,25

five-barplane truss assembly, 28 /

four-node quadrilateral elementfor 2D

BVP,442

four-nodequadrilateral elementfor plane

stress and plane strain, 516

four-noderectangularelement for 2D

BVP,342

fourth-orderBVP using Galerkinmethod,

125

heat flow, 53

heat flowelement results, 53

heat flow example assembly, 32

heat flow throughfins; 198

imposingessential boundaryconditions,

39

mapping areas, 405

mappinglines, 392

modal analysisof a plane frame, 570

modal analysisof a plane truss, 568

plane frames, 277

plane stress element results, 58

plane stress example assembly, 32

INDEX 699

Opfimizing design using ANSYS, 659 Rigid-body motion, 502
Orthogonality property, 588 Rigid diaphragm, 293
Orthotropic material, 478 Rigid element, 74
Overall solution procedure using Galerkin Rigid zone at beam-column connection, 294
Rotating disks and flywheels, 524
method,115 Rotational spring support, 240
Overall solution procedure using the
S
Rayleigh-Ritz method, 130 Second-order lD BVP, 173,586
Seepage through soil, 627
p Selected applications of 1D BVP, 174
Selected applications of the 2D BVP, 313
Penalty function, 79 Serendipity shape functions, 347, 399, 400
p-formulation, 586 Shape functions
Planar finite element models, 490, 517
Plane frame element, 266 Hermite, 146,245
Plane frames, 266 Lagrange, 142,350
Plane strain problem, 493 serendipity, 347, 399, 400
Plane stress analysis, 492 triangle, 359
Plane truss element, 4, 223 Shear force, 236
Plane trusses, 4, 223 Shear stresses in beams, 240
p~mode assumed solution, 605 Side modes, 606
p-mode parameters, 611 Six-node rectangular element, 351
p-modes,587 Skew in elements, 404
Potential energy Skew symmetry, 530
Slider bearing, 177
axial deformation, 129 Small displacement, 476
beam bending, 240 Solution for different right-hand sides, 59
elastic solids, 484 Solution of linear equations, 58
Potential function, 315 Solution of second-order 1D BVP, 208
Preconditioned conjugate gradient Space frames, 279
algorithm, 68 Space truss element, 227
method,66 Spring elements, 233
Preconditioning matrix, 66 Square duct, 28
Pressure vessels, 517 Steady-state heat conduction, 174, 188
Principal directions and principal stresses, Steady-state heat conduction and convection,

471 175, 190
Principal moment of inertia, 279 Strain(s),475
Principle of virtual displacements, 108, 486 Strain-displacement, 476
Properties of variation, 678 Strain energy

Q axial, 129
bending, 241
Quadratic form, 62 elastic solids, 484
Quadrilaterals Strain due 'to truss fabrication error, 231
Stream function, 314
with curved sides, 399 Stress(es),467
with straight sides, 392 on an inclined plane, 469
Quarter-point elements, 532 Stress components; 469
Stress equilibrium equations, 481
R Stress failure criteria, 472
Rayleigh-Ritz method, 128, 130, 135 Stress tensor, 4"68
Reactions, 41 Strong and weak forms, 105
Real constants, 644 Symmetry boundary conditions
Rectangular finite elements, 329 beams, 250
Removing rows for boundary condition, 38 eigenvalue problem, 354
Residual stresses due to welding, 530
Restrictions on mapping

of areas, 394
oflines, 390
Rigid-body modes for plane triangle, 75

700 INDEX

Symmetry boundary conditions (continued) Triangular element for plane stress/strain,
fluid flow, 314, 363 497,566
frames, 277, 291
general remarks, 15 Triangular element for two dimensional heat
heat flow; 28 flow, 7
plane stress/strain, 502, 510, 518,525,
530 Triangular element for two dimensional
torsion, 342, 367 stress analysis, 16

T Triangular elements by collapsing
quadrilaterals, 451
Taper in elements, 404
Truss analysis, 4, 223, 227
Tapered bar, 102, 133 Truss supporting rigid plate, 80
Trusses, beams, and frames, 222
Tapered beam fixed at both ends, 242 Two-dimensional area integrals, 383
Two-dimensional BVP, 311
Temperature changes in trusses, 231 Two-dimensional potential flow, 313
Two-node beam element, 244
Temperature effects and initial strains, 231 . Two-node linear element, 185
Two-node uniform bar element for axial
Thermal and initial strains, 480
deformations, 150
Thermal stresses, 502 Two-point formula, 409

Thermal stresses in trusses, 231

Third-order equation, 125

Three-dimensional volume integrals, 417

Three-node method, 286 U
Uniform bar, 109
Three-node quadratic element, 185 Uniform beam, 259
Units, 83
Three-node triangular element, 358, 497, Use of commercial FEA software, 641

549,566

Three-point formula, 410

TMmodes, 353 V
Valid mesh, 15
Torsion of a C shape, 367 Variation as total derivative, 677
Variation of a function, 676
Torsion of a rectangular shaft, 342 Variation of functionals, 677
Vector notation, 11
Torsional constant, 280 Vibration modes, 567
Viscous fluid flow between parallel plates,
Transformation matrix, 49, 225, 229, 268,
176,198
285 von Mises failure criterion, 473
von Mises stress, 474
Transient field problems, 545

Transient heat flow, 551 !
Transient problems, 545

Transient response examples, 573

Transition elements, 516

Transition region, 73

Transverse deformation of beams, 236 W

Transverse deformation of a uniform beam, Waveguides in electromagnetics, 319, 353

241 Weak form, 105,181,325 .

Tresca criterion, 472 Weak form for axial deformations, 105

Triangle shape functions, 11 Weighted residual, 105

Triangular element, 358, 497, 549, 566· Weld bead, 530