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577

Index

AAdmissible

KinematicallyDisplacement field . . . . 25

AiryStress Function Φ . 181, 313

Algol . . . . . . . . . . . . . . . . . . 433American

“American epsilon” 530, 533,534

AnalysisTensor . . . . . . . . . . . . . 516

AnalyticManipulations . . . . . . . . 433

AngleChange of . . . . . . . . . 12, 40

Anisotropy . . . . . . . . . . . . . . . 28Antiymmetric

Matrix . . . . . . . . . . . . . 515Approximate

SolutionEuler Column . . . . . . 545Plate buckling . . . . . . 335

ApproximationGood . . . . . . . . . . . . . . 530

AreaCross-sectional A . . 204, 205Effective Ae . . . 115, 235, 236

AsymmetricStructure . . . . . . . . . . . 538

AuxiliaryCondition . . . . 545, 548, 568

Axes of the cross-section . . . . 206Axial

Equilibrium . . . . . . . . . . 220Fiber

Strain εf . . . . . . . . . . 202Force N . . . . . . . . . . . . 202Stiffness . . . . . . . . 115, 122Strain ε . . . . . . 132, 261–263

AxisNeutral . . . . . . . . . . . . . 206

BBar

Deformation . . . . . . . . . 92Finite

Element . . . . . . . . . . . 433Base vector

Deformed gm . . . . . . . . . 11Undeformed ij . . . . . . . . . 8

Bauschinger effect . . . . . . . . . 80Beam

AxialStrain ε 94, 97, 105, 109

112, 132, 147Axis . . . . . . . . . . . . . . . 206Bending

Strain κ 94, 97, 105, 109113, 132

Bernoulli-Euler 94, 97, 109Fully nonlinear . . 95, 125

CurvatureStrain κ 94, 97, 105, 109

113, 132Curvature of . . . . . . . . . 131Fiber . . . . . . . . . . . . . . 200Fully nonlinear

Bernoulli-Euler . . . . . 95Curved Bernoulli-Euler 125

Rotation of . . . . . . . . . . 130Shear

Strain ϕ . . . . . . 105, 113Shear strain ϕ . . . . . . . . 95Timoshenko . . . 95, 104, 110

BehaviorPostbuckling . . . . . . . . . 342

BendingDeformation . . . . . . . . . 93Instability

Tubes . . . . . . . . . . . . 148Melosh

Element . . . . . . . . . . . 430Moment M . . . . . . . . . . 202Plates . . . . . . . . . . . . . . 159


579

E. Byskov, Elementary Continuum Mechanics for Everyone,Solid Mechanics and Its Applications 194, DOI: 10.1007/978-94-007-5766-0,Ó Springer Science+Business Media Dordrecht 2013

Index

Stiffness . . . . . . . . 115, 122Bernoulli-Euler

Beam . . . . . . . . . . 94, 97Fully nonlinear (curved) 125Fully nonlinear (straight) .

. . . . . . . . . . . . . . . . 95Linear . . . . . . . . . . . . 109Moderately nonlinear . 97

BifurcationBuckling . . . . . 281, 282, 299Load

Higher . . . . . . . . . . . . 301Biharmonic

Operator ∇4 (nabla) . . . 180Bilinear

Operator l11 . . . . . 162, 554Boundary

ConditionsKinematic . . . . . . . 15, 39Static . . . . . 18, 20, 48, 53

Kinematic . . . . . . . . . . . . 6Static . . . . . . . . . . . . . . 6

BrazierEffect . . . . . . . . . . . . . . 148Solution . . . . . . . . 154–157

BridgeQuebec . . . . . . . . . . . . . 368

Broken Pocket Calculator . . . 522Buckling

Bifurcation . . . 281, 282, 299Column

Model . . . . . . . . . . . . 291Fields u1, ε1, σ1 . . 300, 344Limit load . . . . . . . . . . . 281Load . . . . . . . . . . . 120, 318Mode u1 . . . . . . . . 300, 318Mode w(1) . . . . . . . . . . . 120Plate . . . . . . . . . . . . . . . 309Problem . . . . . . . . 300, 348Snap . . . . . . . . . . . . . . . 281

Load λs . . . . . . . . . . . 282Budiansky, B. . . . . . . . . . . . . 342Budiansky-Hutchinson

Notation . . . . . . . . . 25, 553Interpretation of . . . . 569

Bulk BModulus . . . . . . . . . . . . . 67

Byskov, E. . . . . . . . . . . . . . . 343Byskov, E. & Hansen, J.C. . . 343Byskov, E. & Hutchinson, J.W. 343Byskov, E., Damkilde, L. & Jensen,

K.J. . . . . . . . . . . . . 343

CC++ . . . . . . . . . . . . . . . . . . 433Calculator

Broken . . . . . . . . . . . . . 522Calculus

of variations . . . . . . . . . 521Cantilever

TimoshenkoBeam . . . . . . . . . . . . 122

CartesianCoordinate

System . . . . . . . . 8, 511Center

of Gravity . . . . . . . . . . . 206Change

of angle . . . . . . . . . . . 12, 41of length . . . . . . . . . . 11, 40

“Characteristic”Displacement . . 401, 544, 561

Christensen, C.D. . . . . . . . . . 343Circular

Cross-section . . 211, 226, 255Ring-shaped 216, 232, 260

FiniteElement . . . . . . . . . . . 449

ClassicalCritical load λc 120, 282, 299,

300Column

ElasticModel . . . . . . . . . . . . 383

Elastic-plastic . . . . . . . . 374Euler . . . . 119, 302, 360, 545Imperfect

Model . . . . . . . . . . . . 290Model . . . . . . . . . . . . . . 285

Elastic Shanley . . . . . 383Shanley plastic . . . . . . 380

PerfectModel . . . . . . . . . . . . 285


580

Index

Timoshenko . . . . . . . . . . 306Truss

Geometrically imperfect 369Geometrically perfect . 369

Vector . . . . . . . . . . . . . . 510Comma

Notation ( ),j . . . . . 9, 512Compatibility

Equations . . 14, 36, 247, 253Strain . . . . . . . . . . . . . . . 36Torsion . . . . . . . . . 247, 253

ComplementaryEnergy ΠC . . . . . . 480, 562

Modified ΠCM . . . . . . 479Strain

Energy function WC(εij) 58Conclusion

Shanley . . . . . . . . . . . . . 391Concrete . . . . . . . . . . . . . . . 77Condition

Auxiliary . . . . . 545, 548, 568Side . . . . . . . . . . . 545, 568

ConjugateQuantities . . . . . . . . . . . . 26Work . . . . . . . . . . . . . . 557

ConservativeLoad . . . . . . . . . . . . . . . 284

ConsistentTheory . . . . . . . . . . . . . 99

ConstantPostbuckling a . . . . 346, 349Postbuckling b . . . . 346, 351

ConstantsLame µ and λ . . . . . . . . . 65

ConstitutiveOperator . . . . . . . . . . . . 558

Linear H . . . . . . . . . . 558Relations . . . . . . . . . . . . . 60

Hyperelasticity . . . . . . . 28Linear hyperelasticity . . 28

ConventionSummation . 8, 511, 512, 514

CoordinateSystem

Cartesian . . . . . . . 8, 511Coordinates

Generalized

Functions . . . . . . . . . 519Vectors . . . . . . . . . . . 517

Transformation of . . . . . . 42Crawford, R.F. & Hedgepeth, J.M.

. . . . . . . . . . . . . . . . 343Criteria

Stability . . . . . . . . . . . . 281Criterion

Rigid-body . . . . 4, 449, 458Critical load

Classical λc . . . 120, 219, 300Cross-section

Circular . . . . . 211, 226, 255Ring-shaped 216, 236, 260

Elliptic . . . . . . . . . . . . . 257Equilateral

Triangle . . . . . . . . . . . 263I-shaped . . . . . . . . 214, 228Rectangular . . . 212, 225, 265Ring-shaped

Circular . . . . 216, 236, 260T-shaped . . . . . . . . . . . . 212

Cross-sectionalArea A . . . . . . . . . . . . . 204

Cross-sectional area A . . . . . . 205Curl

Curl v . . . . . . . . . . . . . 514Curvature

beam . . . . . . . . . . . . . . 131Geometric k0 and k . . . . 127Radii of ρ0 and ρ . . . . . . 127Strain κ 94, 122, 161, 202, 203

DDeficiency

MeloshElement . . . . . . . . . . . 430

DeformationBar . . . . . . . . . . . . . . . . 92Bending . . . . . . . . . . . . 93Shear . . . . . . . . . . . . . . 93Theory of plasticity . . . . 81

DeformedBase vector gm . . . . . . . . 11

DensityStrain


581

Index

Energy W (εij) . . . . . . . 66Energy W (γij) . . . . . . . 27

DerivativeGateaux . . . . . . . . 531, 558

DeterminateStatically . . . . . . . . . . . 116

DeviatorStrain εjk . . . . . . . . . . . . 67Stress σjk . . . . . . . . 67, 81

Displacement“Characteristic” 401, 544, 561Field

Kinematically admissible 25Generalized u . . . . . . . . 553Interpolation

Matrix [N ] . . . . . . . . . 415Vector

Element {v}j . . . . . . . 422Displacements

Infinitesimal . . . . . . . 35–60Large . . . . . . . . . . . . 5–31

Divergence . . . . . . . . . . . . . . 511Theorem . . . . . . . . . . . . . 21

Dot notation · . . . . . . . . . . . 555Dummy

Index . . . . . . . . . .. 8, 512

EEffect

Brazier . . . . . . . . . . . . . 148Effective

Area Ae . . 122, 234, 236, 239Shear

Stiffness GAe 122, 234, 236Stress σe . . . . . . . . . . . . 81

EigenvalueProblem . . . . . . . . . . . . 326

Eigenvalue problemLinear . . . . . . . . . . . . . . 120

Eigenvector . . . . . . . . . . . . . 326Elastic

TimoshenkoBeam . . . . . . . . . . . . 122

Elastic ShanleyModel

Column . . . . . . . . . . . 383

Elastica . . . . . . . . . . . . . . . . 134Elasticity

Hyperelasticity . . . . . . . . 28Modulus of E . . . . . . . . . 65Nonlinear . . . . . . . . . . . 74

Elastic-plasticColumn . . . . . . . . . . . . . 374

ElementDisplacement vector {v}j 422Isoparametric . . . . . . . . 459Load

Vector {r}j . . . . . . . . 420Vector at system level {R}j

. . . . . . . . . . . . . . . . 421Stiffness

Matrix [k]j . 409, 416, 417Matrix at system level [K]j

. . . . . . . . . . . . . . . . 419Transformation

Matrix [T ]j . . . . . . . . 418Elimination

InternalNodes . . . . . . . . . . . . 437

StressField Parameters . . . . 488

EllipticCross-section . . . . . . . . . 257

Elongation“Fiber” . . . . . . . . . . . . . . 11

EnergyComplementary ΠC . 58, 480,

562Density

Strain W (εij) . . . . . . . 66Strain W (γij) . . . . . . . 27

ModifiedPotential ΠPM . . 460, 548

Potential ΠP . 28, 30, 55, 56,521, 558

Minimum of . . . . . . . . 560Stationarity of . . . . . . 559

EngineeringStrain e . . . . . . . . . . . . . 132

epsilon“American epsilon” 530, 533,

534Equations


582

Index

Compatibility 14, 36, 247, 253Equilibrium . . . . . . . . 15, 48Finite element . . . . . . . . 421

EquilibriumAxial . . . . . . . . . . . . . . 220Equations . . . . . . . . . 15, 48Internal . . . . . . . . . . . 15, 48Moment . . . . . . . . . . . . 222Transverse . . . . . . . . . . . 220

EulerColumn . . . 119, 302, 360, 545

Approximate solution . 545Load . . . . . . . . 120, 373, 383

Expansionof λ . . . . . . . . . . . . . . . 346Theorem . . . . . . . . . . . . 320

ExperimentShanley . . . . . . . . . . . . . 380

FFEM . . . . . . . . . . . . . . . . . . 395

IntroductoryExample . . . . . . . . . . 397

FiberBeam . . . . . . . . . . . . . . 200Elongation γ . . . . . . . . . . 11Stress

Linear over the cross-section. . . . . . . . . . . . . . . . 210

Fields . . . . . . . . . . . . . . . . . . 510Buckling u1, ε1, σ1 300, 344Postbuckling u11, u111, ε11,

ε111, σ11, σ111 . . . . 346Finite

ElementBar . . . . . . . . . . . . . . 433Circular . . . . . . . . . . . 449Equations . . . . . . . . . 421Hybrid . . . . 479, 483, 489Isoparametric . . . . . . . 498Melosh . . . . . . . . . . . 426Method . . . . . . . . . . . 395Notation . . . . . . . . . . 326Plate . . . . . . . . . . . . . 425Stress hybrid . . . . . . . 479Torsion . . . . . . . . . . . 499

GGateaux

Derivative . . . . . . . 531, 558Generalized

CoordinatesFunctions . . . . . . . . . 5Vectors . . . . . . . . . . . 517

Displacement u . . . . . . . 553Hooke’s Law . . . . . . . . . . 27Load T . . . . . . . . . . . . . 555Strain . . 26, 62, 97, 105, 109,

113, 132, 134, 147, 161,184, 276, 398, 556, 559

Beams: ε and κ . . 97, 109,147

Beams: ε, ϕ and κ 105, 113

FirstMoment Sz . . . . . . 203, 204Order

Moment Sz . . . . . . . . 203Problem . . . . . . . . . . 348

Fitch, J.R. . . . . . . . . . . . . . . 343Force

Axial N . . . . . . . . . . . . 202Membrane Nαβ . . . . . . .Vector

. . . . . . . . . 417Formula

Grashof’s . . . . . . . . . . . 224Navier’s . . . . . . . . . . . . 210

Fortran . . . . . . . . . . . . . . . . . 433Frame

Roorda’s . . . . . . . . . . . . 329Free

Index . . . . . . . . . . . . . . 514Function

StressTorsion T . . . . . . . . . 245

Functional Π . . 521, 522, 529, 530Torsion . . . . . . . . . . . . . 499

Functional ΠVariation δΠ of . . . . . . . 529

FunctionsGeneralized

Coordinates . . . . . . . . 519

74


Nodal {q}

583

19

Index

Plates: εαβ and καβ . 161,184, 276

Strain εj . . . . . . . . . . . . . 62Stress . . 26, 62, 100, 105, 109,

113, 134, 147, 162, 184,276, 398, 557, 559

Beams: N and M 100, 109,147

Beams: N , V and M . 113Beams: N , V and M . 105Plates: Nαβ and Mαβ 162,

184, 276Stress σj . . . . . . . . . . . . . 62

GeometricCurvature k0 and k . . . . 127Matrix KG

mn . . . . . 325, 328Stiffness

Matrix KGmn . . . 325, 328

Vector . . . . . . . . . . . . . . 510Good

Approximation . . . . . . . 530Grashof’s formula . . . . . . . . . 224Gravity

Center of . . . . . . . . . . . . 206Greek

Index . . . . . . . . . . 160, 514

HHardening

Isotropic . . . . . . . . . . . .Kinematic . . . . . . . . . . .Nonlinear . . . . . . . . . . . 375Strain . . . . . . . . . . . . . .

Hooke’s lawGeneralized . . . . . . . . . . . 27

Hutchinson, J.W. . . . . . . . . . 342Hybrid

FiniteElement . . . . 479, 483, 489

Hyperelasticity . . . . . . . . . . . . 27Linear . . . . . . . . . . . . 27, 28Potential energy . . . . . . . 29

II1 . . . . . . . . . . . . . . . . . . . . . 55

Trace of σij . . . . . . . . . . . 55I2 . . . . . . . . . . . . . . . . . . . . . 55

Quadratic invariant of σij 55I3 . . . . . . . . . . . . . . . . . . . . . 55

Determinant of σij . . . . . . 55Il’yushin

Theory . . . . . . . . . . . . . 374Imperfect

ColumnModel . . . . . . . . . . . . 290

ImperfectionSensitivity . . . . . . . 342, 357

IncrementalTheory of plasticity . . . . 80

IndexDummy . . . . . . . . . . 512Free . . . . . . . . . . . . . . . 514Greek . . . . . . . . . . 160, 514Notation . . . . . . . . 511, 512Repeated . . . . . . . . . . . . .

Lower-case . . . . . 512, 514Roman . . . . . . . . . . . . . 512Summation . . . . . . . . . . 512

InertiaMoment I of . . . . . . . . . 205

InfinitesimalDisplacements . . . . . . 35–60Rotation ωij . . . . . . . 14, 35Strain eij . . . . . . . . . 13, 35Stress σmn . . . . . . . . . . . 47Theory

Equilibrium . . . . . . . . . 48Kinematic boundary condi-

tions . . . . . . . . . . . 39Kinematics . . . . . . . . . 35Linear elasticity . . . . . . 63Nonlinear Constitutive Mod-

els . . . . . . . . . . . . . 74Plasticity . . . . . . . . . . 75Static boundary conditions

. . . . . . . . . . . . . . . . 53Initial

PostbucklingBehavior . . . . . . 343, 345

State . . . . . . . . . . . . . . . .Young’s Modulus E or E0

Inner

8584

79

780

8

8,


584

Index

Product . . . . . .Instability

Tubes . . . . . . . . . . . . . . 148Integration

Reduced . . . . . . . . 183, 462Interaction

Between modes . . . . . . . 343Mode

Imperfections . . . . . . . 366Internal

Equilibrium . . . . . . . . 18, 48Mismatch . . . . 449, 457, 459Nodes

Elimination . . . . . . . . 437Interpretation

Budiansky-Hutchinson Nota-tion . . . . . . . . . . . . 569

ElementMatrix [k] . . . . . . . . . 417

of strains . . . . . . . . . . . . . 39of stresses . . . . . . . . . . . . 48System

Stiffness matrix [K] . . 410Invariants

Strain J1, J2, J3 . . . . . . . 46Stress I1, I2, I3 . . . . . . . . 55

I-shapedCross-section . . . . . 214, 228

IsoparametricElement . . . . . . . . . . . . 459Finite

Element . . . . . . . . . . . 498Isotropic

Hardening . . . . . . . . . . .Isotropy . . . . . . . . . . . . . . 28, 64

JJ1 . . . . . . . . . . . . . . . . . . . . . 46

Trace of εij . . . . . . . . . . . 46J2 . . . . . . . . . . . . . . . . . . . . . 46

Quadratic invariant of εij . 46J3 . . . . . . . . . . . . . . . . . . . . . 46

Determinant of εij . . . . . . 46

KKinematic

Boundary . . . . . . . . . . . .Conditions . . . . . . . 15, 39

Hardening . . . . . . . . . . .Kinematically

AdmissibleDisplacement field . . . . 25

Kinematically moderately nonlinear,3-D . . . . . . . . . . . . 33

KinematicsInfinitesimal

Theory . . . . . . . . . . . . 35Large displacements . . . . .

Kirchhoff-Love plate theory . . 159Koiter, W.T. . . . . . . . . 342, 345Koiter, W.T. & Kuiken, G.D.C. 343Kronecker

delta δij . . . . . . . . . . . . 513Kronecker delta δij . . . . .

LLagrange

Multiplier . 44, 183, 459, 480,545, 547, 548, 568

Strain γij . . . . .Lame constants µ and λ . . . . . 65Large

Displacements . . . . . .Large displacements

Boundary conditionsKinematic . . . . . . . . . . 15Static . . . . . . . . . . . . . 20

Constitutive relations . . . . 27Equilibrium

Internal . . . . . . . . . . . . 18Internal equilibrium . . . . . 18Kinematic boundary condi-

tions . . . . . . . . . . . 15Kinematics . . . . . . . . . . .Static boundary conditions 20Statics . . . . . . . . . . . . . . 15

LawTresca’s . . . . . . . . . . . . .

9, 518, 555

85

6

84

7

8, 524

9, 132, 553

5–31

7

83


585

Index

von Mises’ . . . . . . . . . . .Laying

Pipelines . . . . . . . . . . . . 149Length

Change of . . . . . . . . . 11, 40of the line element . . . . . 129

Limit loadBuckling . . . . . . . . . . . . 281

Line elementLength of . . . . . . . . . . . 129

LinearBernoulli-Euler

Beam . . . . . . . . . . . . 109Constitutive

Operator H . . . . . . . . 558Eigenvalue problem 120, 300Elastic

Plate . . . . . . . . . 175, 273Hyperelasticity . . . . . . . . 27Operator l1 . . . . . . 162, 556Prebuckling . . . . . . 298, 341Timoshenko

Beam . . . . . . . . . . . . 110Load

Buckling . . . . . . . . 120, 318Classical Critical load λc 282Conservative . . . . . . . . . 284Critical

Classical λc . . . . . . . . 282Euler . . . . . . . 120, 373, 383Generalized T . . . . . . . . 555Non-conservative . . . . . . 284Reduced

Modulus . . . . . . 373, 386Tangent

Modulus . . . 373, 376, 381Vector . . . . . . . . . . 420, 421

Locking . . . . . . . . . 449, 457, 459Membrane . . . . . . . . . . . 183

LowerBound of shear stiffness . 237

LunchesNo

Free . . . . . . . . . . . . . 440

MManipulations

Analytic . . . . . . . . . . . . 433Material

Anisotropic . . . . . . . . . . . 28Compliance matrix [C] . . 178Isotropic . . . . . . . . . . . . . 28Stiffness

Matrix [D(ξ)]j . . . . . . 416Stiffness matrix [D] . . . . 176

MatrixAntiymmetric . . . . . . . . 515Displacement

Interpolation . . . . . . . 415Element

Stiffness . . . . 409, 416, 419Transformation . . . . . 418

GeometricStiffness KG

mn . . 325, 328Material

Stiffness . . . . . . . . . . . 416Stiffness . . 325, 328, 410, 420

Geometric KGmn . 325, 328

StrainDistribution . . . . . . . . 416

Symmetric . . . . . . . . . . . 515System

Stiffness [K] . . . . . . . . 418Two-dimensional . . . . . . 510

maxima . . . . . . . . . . . . . . . . . 433Program . . . . . . . . . . . . 442

MeanStrain ε . . . . . . . . . . . . . . 67Stress σ . . . . . . . . .

MeloshElement

Bending . . . . . . . . . . . 430Deficiency . . . . . . . . . 430

FiniteElement . . . . . . . . . . . 426

MembraneForce Nαβ . . . . . . . . . . .Locking . . . . . . . . . . . . . 183Strain εαβ . . . . . . .

MethodFinite

82

67, 82

74

74, 161


586

Index

Element . . . . . . . . . . . 395Mindlin

PlateTheory . . . . . . . . . . . 170

MinimumRayleigh Quotient Λ[φ] . 324

MismatchInternal . . . . . . 449, 457, 459

ModeBuckling . . . . . . . . . . . . 318Buckling E

Buckling w(1) . . . . . . . 120Interaction . . . . . . . . . . 343

Imperfections . . . . . . . 366Mode Interaction

Imperfections . . . . . . . . 366Model

ColumnBuckling . . . . . . . . . . 285Elastic Shanley . . . . . 383Imperfect . . . . . . . . . . 290Perfect . . . . . . . . . . . 285Plastic Shanley . . . . . 381

Moderate displacements . . . . . 33Moderately Nonlinear

Bernoulli-EulerBeam . . . . . . . . . . . . . 97

Moderately nonlinear, 3-D . . . . 33Modified

ComplementaryEnergy ΠCM . . . . . . . 479

Potential energy ΠPM . . 548Potential

Energy ΠPM . . . . . . . 460Modulus

Bulk B . . . . . . . . . . . . . . 67Shear G . . . . . . . . . . . . . 65Tangent ET . . . . . . . 81, 376Young’s

Initial E or E0 . . . . . . . 81Modulus of elasticity E . . . . . . 65Møllmann, H. & Goltermann, P. 343Moment

Bending M . . . . . . . . . . 202Equilibrium . . . . . . . . . . 222First Sz . . . . . . . . . . . . 204First-Order . . . . . . . . . . 203

Inertia I . . . . . . . . . . . . 205Second-order . . . . . . . . . 205Static Sz . . . . . . . . . . . . 204Static S . . . . . . . . . . . . 205Torsional MT . . . . 243, 248Twisting MT . . . . . 243, 248Zeroth Order . . . . . . . . . 203

Moment of inertia Izz . . . . . . 205Multiplier

Lagrange 44, 45, 183, 459, 480,545, 547, 548, 568

MuPAD . . . . . . . . . . . . . . . . 433

Nnabla (∇4), biharmonic operator .

. . . . . . . . . . . . . . . . 180Navier’s formula . . . . . . . . . . 210Neutral axis . . . . . . . . . . . . . 206No

FreeLunches . . . . . . . . . . . 440

NodalForce

Vector {q} . . . . . . . . . 417Nodes

Elimination . . . . . . . . . . 437Internal

Elimination . . . . . . . . 437Non-conservative

Load . . . . . . . . . . . . . . . 284Nonlinear

Elasticity . . . . . . . . . . . . 74Hardening . . . . . . . . . . . 375Kinematically moderate, 3-D

. . . . . . . . . . . . . . . . 33Plates . . . . . . . . . . . . . . 160Prebuckling . . . . . . 296, 343Timoshenko

Beam . . . . . . . . . . . . 104Notation

Budiansky-Hutchinson 25, 553Comma ( ),j . . . . . . . 9, 512Dot · . . . . . . . . . . . . . . 555Finite element . . . . . . . . 326Index . . . . . . . . . . 511, 512

ν


587

Index

Value of . . . . . . . . . . . . . 66

OOperator

Biharmonic ∇4 (nabla) . 180Bilinear l11 . . . . . . 162, 554Linear l1 . . . . . . . . 162, 554Quadratic l2 . . . . . 162, 554

Operators . . . . . . . . . . . . . . . 510Orthogonality

Condition on buckling modes. . . . . . . . . . . . . . . . 301

Ovalization . . . . . . . . . . . . . . 148Overbar ¯ . . . . . . . . . . . . . . 509

PPascal . . . . . . . . . . . . . . . . . . 433Peek, R. & Kheyrkhahan, M. . 343Perfect

ColumnModel . . . . . . . . . . . . 285

PermutationSymbol

Three-dimensional case eijk

. . . . . . . . . . . . . . . . 513Two-dimensional case eαβ .

. . . . . . . . . . . . 182, 514Piola-Kirchhoff stress tij . . . . . 17Pipelines

Laying . . . . . . . . . . . . . 149Plane

Strain . . . . . . . . . . . . . . . 72Stress . . . . . . . . . . . . . . . 73

Plasticity . . . . . . . . . . . . . . . . 74Deformation theory of . . . 82Incremental theory of . . . . 81Multi-Axial States . . . . . . 82One-Dimensional Case . . . 75Perfect . . . . . . . . . . . . . . 76Rigid, perfect . . . . . . . . . 76Total theory of . . . . . . . . 82

PlateBending . . . . . . . . . . . . 159Buckling . . . . . . . . . . . . 309Curvature strain καβ . . . 161

ExternalVirtual work . . . . . . . 163

FiniteElement . . . . . . . . . . . 425

GeneralizedStrains εαβ, καβ . 161, 184,

276Stresses Nαβ and Mαβ 162,

184,276Internal

Virtual work . . . . . . . 162Kirchhoff-Love . . . . . . . . 159Linear

Elastic . . . . . . . . 175, 273Membrane

Strain εαβ . . . . . . . . . 161Mindlin

Theory . . . . . . . . . . . 170Nonlinear . . . . . . . . . . . 160Shearing . . . . . . . . . . . . 159Stretching . . . . . . . . . . . 159Thick . . . . . . . . . . . . . . . 72Thin . . . . . . . . . . . . . . . . 73

159.........namraKnovPL/I . . . . . . . . . . . . . . . . . . . 433Pocket Calculator

Broken . . . . . . . . . . . . . 522Poisson’s ratio ν . . . . . . . . . . . 65Postbuckling

Behavior . . 341, 343, 345, 346Initial . . . . . . . . 343, 345

Constant a . . . . . . 346, 349Constant b . . . . . . 346, 351Fields u11, u111, ε11, ε111, σ11,

σ111 . . . . . . . . . . . . 346Neutral . . . . . . . . . . . . . 342Nonlinear

Prebuckling . . . . . . . . 343Problem . . . . . . . . . . . . 349Stable . . . . . . . . . . . . . . 342Unstable . . . . . . . . . . . . 342

“Potato” . . . . . . . . . . . . . . . . . . 6Potential

Energy ΠP 28, 30, 55, 56, 524,558

Hyperelasticity . . . . . . . 29Minimum of . . . . . . . . 560


588

Index

Modified ΠPM . . 460, 548Energy ΠP

Stationarity of . . . . . . 559Potential Π . . . . . . . . . 523, 557

Variation δΠ of . . . 529, 557Prebuckling

Linear . . . . . . . . . . 298, 341Nonlinear . . . . . . . 296, 343

PrincipalStrains . . . . . . . . . . . . . . 44Stresses . . . . . . . . . . . . . . 54

PrincipleMinimum Potential Energy .

. . . . . . . . . . . . . . . . 560Stationary Potential Energy .

. . . . . . . . . . . . . . . . 559Variational . . . . . . . . . . 521Virtual

Displacements . 21, 48, 555Forces . . . . . . . . 27, 56, 57Work . . . . . . . . . . . . . . 21

ProblemBuckling . . . . . . . . 300, 348First-order . . . . . . . . . . 348First-order postbuckling . 349Linear eigenvalue . . . . . . 300Postbuckling . . . . . . . . . 349Second

Order . . . . . . . . . . . . 349Second-order . . . . . . . . . 349Third

Order . . . . . . . . . . . . 350Third-order . . . . . . . . . . 350Third-order postbuckling 350

ProcedureRayleigh-Ritz . . . . . . . . 324

ProductInner . . . . . . . . . 9, 518, 555Scalar . . . . . . . . . . . . 9, 518

Programmaxima . . . . . . . . . . . . . 442

PythagorasTheorem of . . . . . . . . . . . 96

QQuadratic

Operator l2 . . . . . . 162, 554Quadratic operator l2 . . . . . . 162Quadrilateral

HybridFinite Element . . . . . . 489

QuantitiesConjugate . . . . . . . . . . . . 26

Quebec Bridge . . . . . . . . . . . 368Quotient

Rayleigh Λ[φ] . . . . . . . . 321

RRadii of curvature ρ0 and ρ . . 127Rayleigh

Quotient Λ[φ] . . . . . . . . 321Rayleigh Quotient Λ[φ]

Minimum . . . . . . . . . . . 324Stationary . . . . . . . . . . . 322

Rayleigh-Ritz Procedure . . . . 324Application of . . . . 329, 335

Rearrangementof Strain and Stress Compo-

nents . . . . . . . . . . . 61Rectangular

Cross-section . . 210, 225, 265Reduced

Integration . . . . . . 183, 462Modulus

Load . . . . . . 373, 379, 383Relations

Constitutive . . . . . . . . . . 61Reloading . . . . . . . . . . . . . . . . 80Repeated

Lower-caseIndex . . . . . . . . 512, 514

Repeated index . . . . . . . . . . . . . 8Restriction

Thermodynamic . . . . . . . 69Right-hand side

VectorSystem {R} . . . . . . . . 421

RigidPlasticity . . . . . . . . . . . . 76

Rigid-bodyCriterion . . . . . . . 4, 449, 458

Ring-shaped


589

Index

Cross-section . . 216, 232, 260Roman

Index . . . . . . . . . . . . . . 510Roorda’s frame . . . . . . . . . . . 331Rotation

beam . . . . . . . . . . . . . . 130Infinitesimal ωij . . . . . 14, 35

RowVector . . . . . . . . . . . . . . 510

SScalar

Product . . . . . . . . . . 9, 518Scalar product . . . . . . . . . . . . . 9Second

OrderMoment . . . . . . . . . . . 205Problem . . . . . . . . . . 349

SelfStrain . . . . . . . . . . 449, 459

SensitivityImperfection . . . . . 342, 355

Shanley . . . . . . . . . . . . . . . . 373Conclusion . . . . . . . . . . 341Experiment . . . . . . . . . . 380Model

Column plastic . . . . . . 381Shear

Deformation . . . . . . . . . 393Effective

Stiffness GAe 122, 234, 236Modulus G . . . . . . . . . . . 65Stiffness . . . . . . . . . . . . 234

Lower bound of . . . . . 237Shearing

Plates . . . . . . . . . . . . . . 159Side

Condition . . . . . . . 545, 568Snap Buckling . . . . . . . . . . . 281Snap buckling

Load λs . . . . . . . . . . . . 282Snap-through . . . . . . . . . . . . 282Softening

Strain . . . . . . . . . . . . . . . 80Solution

Approximate

Euler Column . . . . . . 545Plate buckling . . . . . . 337

Brazier . . . . . . . . . 154–156Special

Strain and stress states . . 72Stability

Criteria . . . . . . . . . . . . . 281State

Initial . . . . . . . . . . . . . . . . 7Virgin . . . . . . . . . . . . . . . . 7

StatesSpecial . . . . . . . . . . . . . . 72

StaticBoundary . . . . . . . . . . . . . 6

Conditions . . 18, 20, 48, 53Moment Sz . . . . . . . . . . 204Moment S . . . . . . . . . . . 205

Statically determinate . . . . . . 116Statics . . . . . . . . . . . . . . . . . . 15

Large displacements . . . . . 15Stationary

Rayleigh Quotient Λ[φ] . 323Steel . . . . . . . . . . . . . . . . . . . 76Stiffness

Axial . . . . . . . . . . 115, 122Bending . . . . . . . . 115, 122Effective

Shear . . . . . . . . . . . . 122Geometric

Matrix KGmn . . . 327, 330

Material . . . . . . . . . . . . 416Matrix 325, 328, 409, 410, 416,

419–421Geometric KG

mn . 327, 330System [K] . . . . 410, 420

Shear . . . . . . . 122, 234, 237System

Matrix [K] . . . . . . . . . 410Strain

Axial ε . . . . 84, 132, 202, 203Fiber εf . . . . . . . . . . 202

Bending κ . . . . . . . . . . . . 84Compatibility . . . . . . . . . 36Curvature κ 84, 132, 202, 203Curvature καβ . . . . . . . . 161Deviator εjk . . . . . . . . . . 67Deviatoric part εjk . . . . . 67


590

Index

DistributionMatrix [B] . . . . . . . . . 416

EnergyComplementary WC(εij) 58Density W (εij) . . . . . . 66Density W (γij) . . . . . . 27Function W (γij) . . . . . 27Function W (εij) . . . 57, 66

Engineering e . . . . . . . . 134Generalized . 26, 62,132, 134,

398,554, 557Generalized εj . . . . . . . . . 62Hardening . . . . . . . . . . . . 79Infinitesimal eij . . . . . 13, 35Interpretation of . . . . . . . 39Invariants J1, J2, J3 . . . . . 48Lagrange γij . . . . 9, 134, 553Mean ε . . . . . . . . . . . . . . 67Membrane εαβ . . . . . . . . 161

Membrane εαβ . . . . . . . 74Plane . . . . . . . . . . . . . . . 72Principal . . . . . . . . . . . . . 44Self . . . . . . . . . . . . 449, 458Shear ϕ . . . . . . . . . . . . . . 95Softening . . . . . . . . . . . . . 80Transformation of . . . . . . 41Two-dimensional . . . . . . . 72

StressDeviator σjk . . . . . . . . . . 67Deviatoric part σjk . . 67, 82Effective σe . . . . . . . . . . . 82Field Parameters

Elimination of . . . . . . 488Function

Airy Φ . . . . . . . . 281, 313Torsion T . . . . . . . . . 245

Generalized . 26, 62, 134, 398,555,557

Generalized σj . . . . . . . . . 62Hybrid

Finite element . . . . . . 483Infinitesimal σmn . . . . . . . 47Interpretation of . . . . . . . 48Invariants I1, I2, I3 . . . . . 55Mean σ . . . . . . . . . . . 67, 82Piola-Kirchhoff stress tij . 17Plane . . . . . . . . . . . . . . . 73

Principal . . . . . . . . . . . . . 54Transformation of . . . . . . 53Two-dimensional . . . . . . . 72

Stress hybridFinite

Element . . . . . . . . . . . 479Stretch λ . . . . . . . . . . . . . . . 126Stretching

Plates . . . . . . . . . . . . . . 159Structure

Asymmetric . . . . . . . . . . 358Symmetric . . . . . . . . . . . 357Too

Flexible . . . . . . . . . . . 563Stiff . . . . . . . . . . . . . . 560

SummationConvention . . 8, 511, 512, 514Index . . . . . . . . . . . . . . 512

SurfaceTractions τm . . . . . . . . . . 20Yield . . . . . . . . . . . . . . . 82

SymbolPermutation

Three-dimensional case eijk

. . . . . . . . . . . . . . . . 513Two-dimensional case eαβ .

. . . . . . . . . . . . 182, 514Symmetric

Matrix . . . . . . . . . . . . . 515Structure . . . . . . . . . . . 357

SystemRight-hand

Vector {R} . . . . . . . . 421Stiffness

Matrix [K] . . 410, 418, 420

T“Taking”

Variations . . . . . . . 531, 538Tangent

Modulus ET . . . . . . 81, 376Load . . . . . . 373, 376, 384

TensorAnalysis . . . . . . . . . . . . 516

TheoremDivergence . . . . . . . . . . . 21


591

Index

Expansion . . . . . . . . . . . 321Pythagoras . . . . . . . . . . . 96

TheoryConsistent . . . . . . . . . . . . 99Il’yushin . . . . . . . . . . . . 374

ThermodynamicRestriction . . . . . . . . . . . 69

ThickPlate . . . . . . . . . . . . . . . . 72

ThinPlate . . . . . . . . . . . . . . . . 73

ThirdOrder

Problem . . . . . . . . . . 350Thompson, G.M.T & Hunt, G.W.

. . . . . . . . . . . . . . . . 343Tilde ˜ . . . . . . . . . . . . . . . . 509Timoshenko

Beam . . . . . . . . 95, 104, 110Cantilever . . . . . 122, 234Elastic . . . . . . . . . . . . 122Linear . . . . . . . . . . . . 110Nonlinear . . . . . . . . . 104

Column . . . . . . . . . . . . . 106Too

FlexibleStructure . . . . . . . . . . 563

StiffStructure . . . . . . . . . . 560

Torque MT . . . . . . . . . . 243, 248Torsion . . . . . . . . . . . . . 241–269

Compatibility . . . . 247, 253Finite

Element . . . . . . . . . . . 499Functional . . . . . . . . . . . 499Warping . . . . . . . . 242, 254

TorsionalMoment MT . . . . . 243, 248

TotalTheory of plasticity . . . . . 82

TractionsSurface τm . . . . . . . . . . . 20

TransformationMatrix [T ]j . . . . . . . . . . 418of coordinates . . . . . . . . . 42of strain . . . . . . . . . . . . . 41of stress . . . . . . . . . . . . . 53

TransverseEquilibrium . . . . . . . . . . 220

Tresca’s “Law” . . . . . . . . . . . . 83Triangular

Cross-sectionEquilateral . . . . . . . . . 263

Truss columnGeometrically imperfect . 369Geometrically perfect . . . 369

T-shapedCross-section . . . . . . . . . 212

TubeBending

Instability . . . . . . . . . 148Instability . . . . . . . . . . . 148

Tvergaard, V. . . . . . . . . . . . . 343Twisting

Moment MT . . . . . 243, 248Two-dimensional

Matrix . . . . . . . . . . . . . 510Strain and stress states . . 72

UUnloading . . . . . . . . . . . . . . . 80

VValue

of ν . . . . . . . . . . . . . . . . . 66van der Neut, A. . . . . . . . . . . 343Variation . . . . . . . . . . . . . . . . 22

δΠ of a functional Π . . . 529δΠ of a potential Π 529, 557

VariationalPrinciple . . . . . . . . . . . . 521

Finite degree system . 532Functional . . . . . . . . . 522Infinitely many degrees of

freedom . . . . . . . . . 538Lagrange Multiplier . . 545Lagrange Multiplier η 568Potential energy ΠP . 533,

535Variations . . . . . . . . . . . . . . . 520

Calculus of . . . . . . . . . . 521“Taking” . . . . . . . . 531, 538


592

Index

VectorBase

Deformed gm . . . . . . . . 11Undeformed ij . . . . . . . . 8

Column . . . . . . . . . . . . . 510Displacement

Element {v}j . . . . . . . 422Element

Load . . . . . . . . . 420, 421“Geometric” . . . . . . . . . . 510Right-hand side . . . . . . . 421Row . . . . . . . . . . . . . . . 510

VectorsGeneralized

Coordinates . . . . . . . . 517Virgin

State . . . . . . . . . . . . . . . . . 7Virtual

DisplacementsPrinciple of 21, 48, 165, 555

ForcesPrinciple of . . . . 27, 56, 57

WorkPlates . . . . . . . . 162, 163Principle of . . . . . . . . . 21

159...yroehtetalpnamraKnovvon Mises’ “Law” . . . . . . . . . . 82

WWarping (torsion) . . . . . 240, 254Wood . . . . . . . . . . . . . . . . . . . 78Work

Conjugate . . . . . . . . . . . 557

YYield

Stress σY

Initial . . . . . . . . . . . . . 81Stress σy

Subsequent . . . . . . . . . 81Surface . . . . . . . . . . . . . . 82

Young’s Modulus E . . . . . . . . 65Initial E or E0 . . . . . . . . 81

ZZeroth

OrderMoment A . . . . . 203, 204


593

bibliography - springer978-94-007-5766-0/1.pdf · bibliography arfken, ... k. j. (1996). finite...

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