Hutton - Fundamentals of Finite Element Analysis, страница 90

See alsoTime-dependent heat transfertwo-dimensional conduction withconvection, 235–261.See also Two-dimensionalconduction with convectionHermite polynomials, 214Higher-order isoparametricelements, 201Higher-order one-dimensionalelements, 170–173Higher-order rectangularelement, 186–187Higher-order tetrahedralelements, 190Higher-order triangular elements, 182Historical overview, 11–12Hooke’s law, 34h-refinement, 164Hydrostatic stress, 370Hutton: Fundamentals ofFinite Element AnalysisBack MatterIndex© The McGraw−HillCompanies, 2004IndexIIdentity matrix, 448Incomplete polynomial, 174–175Incompressible flow, 293Incompressible flow analysis.See Fluid mechanicsIncompressible viscousflow, 314–323Inertia coupling, 417Initial conditions, 280, 388, 391Integration step, 8Interelement boundaries, 145Interior nodes, 2Internal heat generation(two-dimensional heattransfer), 259–261Interpolation, 3Interpolation functions, 3, 163–221axisymmetric elements, 202–206brick element, 191–193C 0-continuity, 163compatibility, 165completeness, 166geometric isotropy.See Geometric isotropyhigher-order one-dimensionalelements, 170–173isoparametric formulation, 193–201mesh refinement, 164–165numerical integration (Gaussianquadrature), 206–213polynomial forms (geometricisotropy), 174–176polynomial forms (one-dimensionalelements), 166–173rectangular elements, 184–187tetrahedral element, 188–190three-dimensionalelements, 187–193triangular elements, 176–183Inverse of a matrix, 177, 451–454Inverse of the Jacobian matrix, 199Inviscid, 294Irrotational flow, 297Isoparametric element, 196Isoparametric formulation, 193–201Isoparametric formulation of planequadrilateral element, 347–356Isoparametric mapping, 196JJacobian, 350Jacobian matrix, 199, 200, 349KKnight, Charles E., 473LLagrangian approach, 417Lagrangian mechanics, 412Lagrange’s equations of motion, 412Lanczos method, 443Laplace’s equation, 298, 304Least squares, 132Legendre polynomials, 214Line elements, 131Line source, 259Linear elastic spring, 20Linear spring as finiteelement, 20–31Link element, 19Liquids, 295Load-deflection curve, 20Local coordinate system, 20Lower triangular matrix, 467LU decomposition, 467–470Lumped capacitance matrix, 278Lumped mass matrix, 407MMagnitude of gradient discontinuitiesat nodes, 145Many degree-of-freedomsystem, 398–402Mapped meshing, 374Mapping, 195Marching, 280, 285Mass, 437Mass matrix, 390Mass matrix for general element(equations of motion), 412–418Mass transport, 261–266Master degrees of freedom, 443Material property matrix, 459Matrix.

See also Matrix mathematicscapacitance, 278, 279coefficient, 452cofactor, 452column, 447conductance, 240–242consistent mass, 404damping, 428defined, 447diagonal, 419, 420, 448geometric mapping, 351identity, 448inverse of, 177, 451–453491Jacobian, 199, 200, 349lower triangular, 467lumped mass, 407mass, 390material property, 459modal, 419nodal acceleration, 390null, 448order, 447row, 447skew symmetric, 448square, 447stiffness. See Stiffness matrixsymmetric, 448system mass, 394upper triangular, 467zero, 448Matrix addition, 449Matrix inversion, 177, 451–453Matrix mathematics, 447–454addition/subtraction, 449algebraic operations, 449–450definitions, 447–448determinants, 450–451matrix partitioning, 454multiplication, 449–450scalar algebra, contrasted, 450Matrix multiplication, 449–450Matrix partitioning, 454Matrix subtraction, 449Maximum shear stress theory(MSST), 369Megapascal (MPa), 458Mesh, 4Mesh refinement, 164–165Meshing, 4, 374Mesh-refined models, 374Method of weighted residuals(MWR), 131–162application of Galerkin’s method tobeam element, 149–152application of Galerkin’s method tospar element, 148–149convergence, 137–138defined, 131–132Galerkin finite elementmethod, 140–148Galerkin’s weighted residualmethod, 133–139general concept, 132one-dimensional heatconduction, 152–158trial functions, 131, 138, 140variations, 132–133Hutton: Fundamentals ofFinite Element Analysis492Back MatterIndex© The McGraw−HillCompanies, 2004IndexMinimum potentialenergy, 44–47, 158Minor, 450Modal analysis, 387, 397Modal matrix, 419Modal superposition, 397, 399Mode superposition, 422–424Model definition step, 10Modulus of elasticity, 458Modulus of rigidity, 459Molten polymers, 315MPa, 458MSC/NASTRAN, 12, 18MSST, 369MWR.

See Method of weightedresiduals (MWR)NN degree-of-freedom system, 402NASTRAN, 12Natural circular frequency, 389Natural convection, 227Natural coordinates, 185, 186Natural frequency, 392Natural modes of vibration, 387, 443Navier-Stokes equations, 315Net force, 21Neutral surface, 92Newmark method, 432–434Newton’s law of viscosity, 294Newton’s second lawbar elements, 402linear spring, 22multiple degrees-of-freedomsystems, 394simple harmonic oscillator, 388No slip condition, 294Nodal acceleration matrix, 390Nodal displacement correspondencetable, 62Nodal displacements, 21Nodal equilibrium equations, 53–58Nodal free-body diagrams, 55Nodal load positive convention, 102Node, 2Noncircular shaft sections, 486Nonconservative force, 45Nonhomogeneous boundarycondition, 29Normal strain, 455–456Normalized coordinates, 185Null matrix, 448Numerical integration (Gaussianquadrature), 206–213OOctahedral shear stress theory(OSST), 371One dimensional conduction withconvection, 227–235One-dimensional conduction(quadratic element), 222–227One-dimensional heatconduction, 152–158One-dimensional wave equation, 403Order (matrix), 447Orthogonality, 418Orthogonality of principalmodes, 418–421Orthonormal, 419OSST, 371Overdamped, 426, 427Overview of book, 16–17PParent element, 195Partitioning (matrix), 454Pascal pyramid, 175Pascal triangle, 174, 175Period of oscillation, 392Phase angle, 389, 391Plane quadrilateral element, 347–356Plane strain (rectangularelement), 342–347Plane stress, 328–342assumptions, 328distributed loads/body face, 335–342finite element formulation(CST), 330–333stiffness matrix evaluation, 333–335Plate bending, 372–373Point collocation, 132Poisson’s ratio, 458Polynomial formsgeometric isotropy, 174–176one-dimensional elements, 166–173Polynomial trial functions, 138Postmultiplier, 449Postprocessing, 11Potential function, 304Practical considerationssolid mechanics, 372–375structural dynamics, 424–443Prandtl’s stress function, 376, 377p-refinement, 164Premultiplier, 449Preprocessing, 10Principal planes, 369Principal stresses, 369Principle of conservation ofenergy, 153Principle of conservation of mass, 295Principle of minimum potentialenergy, 44–47, 158Product (matrices), 448Pure rotation, 296QQ/2, 254Q/4, 253Quadrilateral element, 194–196Quarter-symmetry model, 253Quick-change coupling, 486RRadial strain, 357Ratioamplitude, 396aspect, 194damping, 426Poisson’s, 458Rayleigh damping, 430, 432Reaction equationsstream function, 303two-dimensional conduction withconvection, 241, 254Rectangular elements, 184–187Rectangular parallelopiped (brickelement), 191–193Recurrence relation, 8Reduced eigenvalue problem, 443Refined finite element mesh, 4Residual error, 132Resonance, 393Resonant frequency, 393Right-hand rule, 238Rotational flow, 297Row matrix, 447Row vector, 447SSampling points, 207Second-order differential equation, 388Serendipity coordinates, 185Shape functions.

See InterpolationfunctionsShear modulus, 459Shear strain, 455–456Simple cantilever truss, 51–52Simple harmonicoscillator, 387–393, 412Hutton: Fundamentals ofFinite Element AnalysisBack MatterIndex© The McGraw−HillCompanies, 2004IndexSingular, 22, 452Six-node quadratic triangularelement, 319Six-node triangular element, 181–182Skew symmetric matrix, 448Software packages. See ComputersoftwareSolid mechanics, 327–386.See also Stressautomeshing, 374–375axisymmetric stressanalysis, 356–364DET, 369–371failure theories, 369–371general three-dimensional stresselements, 364–368isoparametric formulation of planequadrilateral element, 347–356MSST, 369OSST, 371plane strain (rectangularelement), 342–347plane stress, 328–342practical considerations, 372–375strain/stress computation, 368–372torsion, 375–382Solution, 476–487Solution convergence.

See ConvergenceSolution phase, 10–11Solution techniques for linear algebraicequations, 463–472Cramer’s method, 463–465frontal solution, 470–472Gauss elimination, 465–467LU decomposition, 467–470Spar element, 19spar element stiffness matrix, 121Sparse, 470Spring constant, 20Spring rate, 20Spring stiffness, 20Spring-mass system, 394–402Square matrix, 447Static condensation, 442, 454Stiffness matrixelement, 21–22flexure element, 98–101global, 58, 61–67spar element, 121system, 25xy plane flexure, 121xz plane bending, 121Stiffness method, 52Stokes flow, 315–321Strain, 4Strain energy, 38–39Strain energy density, 39Strain energy per unit volume, 39Strain/stress computation, 368–372Stream function, 298–304Streamlines, 298Stressequivalent, 370hydrostatic, 370plane, 338–342principal, 369von Mises, 370Stress analysis. See Solid mechanicsStress stiffening, 114Stress vector, 460Structural damping, 428Structural dynamics, 387–446bar element mass matrix(two-dimensional trussstructures), 434–441bar-element-consistent massmatrix, 402–407beam elements, 407–412energy dissipation (structuraldamping), 424–432.See also Dampingharmonic response using modesuperposition, 422–424mass matrix for general element(equations of motion), 412–418multiple degrees-of-freedomsystems, 394–402Newmark method, 432–434orthogonality of principalmodes, 418–421practical considerations, 442–443simple harmonic oscillator, 387–393transient dynamicresponse, 432–434Subspace iteration, 443Superposition procedure, 25Symmetric matrix, 22, 448Symmetryincomplete polynomial, 174–175matrix, 448simplification of mathematics ofsolution, 252stiffness matrix, 22two-dimensional conduction withconvection, 253–254System mass matrix, 394System stiffness matrix, 25System viscous damping matrix, 428493TTapered cylinder, 4–6Ten-node tetrahedral element, 188Tensile stress, 113Tetrahedral element, 188–190Theory of continuum mechanics, 461Theory of thin plates, 372–373Thin curved plate structures, 373Three-dimensional elements, 187–193Three-dimensional heattransfer, 267–271Three-dimensional stresselements, 364–368Three-dimensional trusses, 79–83Three-node triangularelement, 178–179Time step, 279, 285Time-dependent heat transfer, 277–285capacitance matrix, 278, 279finite difference method, 279–285Torque, 377–378Torsion, 375–382Torsional finite element notation, 122Total potential energy, 45Transient dynamic response, 432–434Transient effects.