Hutton - Fundamentals of Finite Element Analysis, страница 89

(c) Are your results in accordwith classic geartooth theory?E9.3 A flat plate of thickness 25 mm is loaded as shown; the material has modulusof elasticity E = 150 GPa and Poisson’s ratio 0.3. Determine the maximumdeflection, maximum stress, and the reaction forces assuming a state of planestress.E9.4 Repeat Problem E9.3 if the thickness varies from 25 mm at the left end to 15 mmat the right.E9.5 A thin, 0.5 thickness, steel plate is subjected to the loading shown. Determinethe maximum displacement and the stress distribution in the plate.

UseE = 30(10 6 ) and Poisson’s ratio 0.3.Hutton: Fundamentals ofFinite Element AnalysisBack MatterAppendix E: Problems forComputer Solution© The McGraw−HillCompanies, 2004E.4 Chapter 910,000 N3m9mProblem E9.350 lb/in.15 in.40 in.150 lbProblem E9.5E9.6 A uniform thin plate subjected to a uniform tensile stress as shown has a centralrectangular opening. Use the finite element method to determine the stressconcentration factor arising from the cutout. Use the material properties of steel.Would your results change if you use the material properties of aluminum? Why?Why not?6 in.3 in.␴0␴00.5in.Problem E9.6E9.7 The figure shows a common situation in mechanical design. A fillet radius isused to smooth the transition between sections having different dimensions.Use the finite element method to determine the stress concentration factor arisingfrom the fillet radius at the section change.

Material thickness is 0.25 and E =15(106) psi. How do you model the moment loading?485Hutton: Fundamentals ofFinite Element Analysis486Back MatterAPPENDIX EAppendix E: Problems forComputer Solution© The McGraw−HillCompanies, 2004Problems for Computer Solution58 mm57 mmR ⫽ 7.5 mm40 mm25 mm4000 N · m4000 N · mProblem E9.7E9.8 The gusset plate shown is attached at the upper left via a 1.5 cm diameter rivetand held free at the lower left (model as a pin connection).

The plate is loadedas shown. Assuming that the rivet is rigid, compute the stress distribution aroundthe circumference of the rivet. Also determine the maximum deflection. Thegusset has thickness 14 mm, modulus of elasticity 207 GPa, and Poisson’sratio 0.28.2.5 cm5 cm40 cm8000 N45 cmProblem E9.8E9.9 Noncircular shaft sections are often used for quick-change couplings. The figureshows a hexagonal cross section used for such a purpose. The shaft length is 6and subjected to a net torque of 2800 in.-lb.

If the material is steel, compute thetotal angle of twist. (Note: It is highly likely that your FE software will have noelement directly applicable to this problem. Analogy may be required.)1 in.Problem E9.9Hutton: Fundamentals ofFinite Element AnalysisBack MatterAppendix E: Problems forComputer Solution© The McGraw−HillCompanies, 2004E.5 Chapter 10E.5 CHAPTER 10For each truss of Problems E3.1–E3.7 and E4.1–4.7, determine the lowestfive natural frequencies and mode shapes. How do these vary with pin jointversus rigid frame assumptions? (Note that, where a material is not specified,the instructor will provide the density value.)E10.8 Use the modal analysis capability of your finite element software to determinethe natural frequencies and mode shapes of the cantilevered beam shown. Usemesh refinement to observe convergence of the frequencies.

Compare withpublished values in many standard vibration texts. What do the higherfrequencies represent? How many frequencies can you calculate?E10.1–10.7E, IzLProblem E10.8487Hutton: Fundamentals ofFinite Element AnalysisBack MatterIndex© The McGraw−HillCompanies, 2004INDEXAAbsolute viscosity, 294Active zone, 468Adjoint, 452Admissible functions, 132Air, 294ALGOR, 12Amplitude, 389, 391Amplitude ratio, 396ANSYS, 12Applicationsfluid mechanics, 293–326.See also Fluid mechanicsGalerkin’s method (beamelement), 149–152Galerkin’s method (sparelement), 148–149heat transfer, 222–292.See also Heat transfersolid mechanics, 327–386.See also Solid mechanicsArea coordinates, 179–181Aspect ratio, 194Assembly of global stiffnessmatrix, 61–67Associative, 449Automeshing, 374–375Automeshing software, 374Axial strain, 357Axial stress, 113, 120Axisymmetric elements, 202–206Axisymmetric heat transfer, 271–276Axisymmetric problems, 202Axisymmetric stress analysis, 356–364BBack substitution, 469Backward difference method, 283–284Backward sweep, 469Bandwidth, 319Bar element, 19, 31–38Bar element consistent massmatrix, 402–407Bar element mass matrix(two-dimensional trussstructures), 434–441488Beam cross sections, 92Beam elements, 407–412Beam theory.

See Flexure elementsBending stress, 113, 120Blending functions. See InterpolationfunctionsBody forceaxisymmetric stressanalysis, 362–363equilibrium equations, 461plane stress, 379Book, overview, 16–17Boundary conditionsaxisymmetric heat transfer, 275defined, 1one-dimensional conduction withconvection, 213stream function, 300–304torsion, 377truss structures, 67–68two-dimensional conduction withconvection, 240–253Boundary value problems, 1Boyle’s law, 293Brick element, 191–193CC 0-continuity, 163C1-continuity, 163C n-continuity, 163Calculus of variations, 45Capacitance matrix, 278, 279Castigliano’s first theorem, 40–44Central difference method, 284–285Chain rule of differentiation, 272, 274Characteristic equation, 395Circular frequency, 389Coefficient matrix, 452Cofactor matrix, 452Cofactors, 451Column matrix, 447Column vector, 447Commutative, 449Compatibility, 165Compatibility conditions, 24Complete polynomial, 174Complete structure, 21Completeness, 166Compressible flow, 293Compressible flow analysis, 295Computer softwareALGOR, 12ANSYS, 12aspect ratio, 194automeshing software, 374conductance matrix, 240COSMOS/M, 12damping, 432FEPC software, 473–475fluid elements, 323indication of failure, 371pressure on transverse faceof beam, 152problems for computersolution, 476–487reaction equations, 241structural weight, 394–395three-dimensional heat transfer, 270transient dynamic response, 434Conditionally stable, 434Conductance matrix, 240–242Conformable for multiplication, 449Conservative force, 45Consistent capacitance matrix, 278Consistent mass matrix, 404, 414Constant acceleration method, 432Constant parameter mapping, 196Constant strain triangle(CST), 179, 330–333Constitutive equations, 458Constraint equation, 25Continuity equation, 295, 296Convection, 227Convective inertia, 315Convergencecompatibility, 165displacement of tapered cylinder, 4–6isoparametric quadrilateralelement, 355mesh refinement, 164–165MWR solution, 137–138structural dynamics, 442Hutton: Fundamentals ofFinite Element AnalysisBack MatterIndex© The McGraw−HillCompanies, 2004IndexCOSMOS/M, 12Coupling, 417Cramer’s rule, 350, 463–465Creeping flow, 315Critical damping coefficient, 426Critically damped, 426, 427CST, 179, 330–333Curved-boundary domain, 4Cyclic frequency, 392DDamped natural circular frequency, 427Damping, 424–432critical damping coefficient, 426matrix, 428over/underdamped, 426, 427physical forms, 424ratio, 426Rayleigh, 430, 432software packages, 432structural, 428Damping matrix, 428Damping ratio, 426Dashpot, 425Deflected beam element, 92Degrees of freedomcalculating, 3dynamics, 443many degrees-of-freedomsystem, 398–402master, 443N degrees-of-freedom system, 402two degrees-of-freedom system, 395DET, 369–371Determinant, 450–451Diagonal matrix, 419, 420, 448Differential equation, 388, 390Differential equation theory, 8Dirac delta, 260Direct assembly of global stiffnessmatrix, 61–67Direct stiffness method, 53, 63Direction cosines, 61Displacement, 6, 12Displacement method, 12Distortion energy theory(DET), 369–371Distributed loads, workequivalence, 106–114Dot notation, 295Double subscript notation(shearing stresses), 459nDouble-dot notation, 404Dynamic analysis.

See StructuraldynamicsDynamic degrees of freedom, 443489Equivalent viscous dampingcoefficient, 428Euler’s method, 280Exterior nodes, 2EEigenvalue problem, 397Eigenvector, 402Eight-node brickelement, 191–193, 367Eight-node rectangular element, 186Elastic bar element, 31–38Elastic coupling, 417Elastic failure theory, 369–371Element capacitance matrix, 278Element conductance matrix, 242Element coordinate system, 20Element damping matrix, 428Element displacement locationvector, 66Element free-body diagrams, 55Element load vector, 102–106Element stiffness matrix, 21–22Element transformation, 58–61Elementary beam theory, 91–94Elementary strength of materialstheory, 150Element-node connectivity table, 66Elements (matrix), 447Element-to-system displacementcorrespondence, 104Energy dissipation, 424.See also DampingEquationcharacteristics, 395compatibility, 461–462constitutive, 458constraint, 26continuity, 295, 296equilibrium, 460–461frequency, 395, 417Laplace’s, 298Navier-Stokes, 315nodal equilibrium, 53–58one-dimensional wave, 403Equations of elasticity, 455–462compatibility equations, 461–462equilibrium equations, 460–461strain-displacementrelations, 455–458stress-strain relations, 458–460Equations of motion, 412–418Equipotential lines, 304Equivalent stress, 370FFailure theories, 369–371FEA.

See Finite elementmethod (FEM)FEA software. See ComputersoftwareFEM. See Finite elementmethod (FEM)FEPCIP, 473FEPCOP, 473Ferris wheel, 297Field, 1Field problems, 1Field variables, 1Fillet radius, 485Finite difference methodbackward differencemethod, 283–284central difference method, 284–285finite element method,compared, 7–10forward difference method, 280key parameter, 285time step, 279, 285what is it, 279Finite element, 2, 12Finite element analysis (FEA).See Finite element method (FEM)Finite element formulationaxisymmetric heat transfer, 273–276axisymmetric stressanalysis, 359–360general three-dimensional stressanalysis, 365–368one-dimensional conduction withconvection, 227–230plane stress, 330–333stream function, 299–300torsion, 378two-dimensional conduction withconvection, 236–240Finite element method (FEM)basic premise, 19defined, 1exact solutions, compared, 4–7examples, 12–15finite difference method,compared, 7–10Hutton: Fundamentals ofFinite Element Analysis490Back MatterIndex© The McGraw−HillCompanies, 2004IndexFinite element method—Cont.historical overview, 11–12how does it work, 1–4objective, 164postprocessing, 11preprocessing step, 10solution phase, 10–11Finite Element Method Primerfor Mechanical Design,A (Knight), 473Finite element method software.See Computer softwareFinite Element Personal Computer(FEPC) program, 473First derivative, 279First theorem of Castigliano, 40–44Flexibility method, 12, 52Flexure element stiffnessmatrix, 98–101Flexure element with axialloading, 114–120Flexure elements, 91–130element load vector, 102–106elementary beam theory, 91–94flexure element stiffnessmatrix, 98–101flexure element with axialloading, 114–120general three-dimensional beamelement, 120–124stress stiffening, 1142-D beam (flexure element), 94–98work equivalence (distributedloads), 106–114Flexure formula, 150Flow net, 304Flow with inertia, 321–323Fluid, 293Fluid mechanics, 293–326continuity equation, 295, 296incompressible viscousflow, 314–323incompressible/compressibleflow, 293Laplace’s equation, 298literature, 323rotational/irrotational flow, 296–297software packages, 323Stokes flow, 315–321stream function, 298–304velocity potential function, 304–314viscosity, 293–295viscous flow with inertia, 321–323Fluid viscosity, 293–295Forced convection, 227Forced response, 393Forced vibration, 392–393Forcing frequency, 393Forcing functions, 452Formal equilibrium approach, 53Forward difference scheme, 280Forward sweep, 469Fourier’s lawaxisymmetric heat transfer, 274one-dimensional conduction withconvection, 228three-dimensional conduction withconvection, 267two-dimensional conduction withconvection, 237–240Fourier’s law of heat conduction, 153Four-node quadrilateral element, 195Four-node rectangular element,184–185Four-node tetrahedral element,188–190Free meshing, 374Free vibration, 389Frequency equation, 395, 417Friction force, 45Frontal solution method, 470–472Functionadmissible, 132forcing, 452potential, 304Prandtl’s stress, 376, 377stream, 298–304Fundamental frequency, 396GGalerkin finite elementmethod, 140–148, 285Galerkin’s weighted residualmethod, 133–139Garbage in, garbage out, 10Gases, 295Gauss elimination, 465–467Gauss points, 207Gaussian quadrature, 206–213Gauss-Jordan reduction, 453Gauss-Legendre quadrature, 206General structural damping, 427–432General three-dimensional beamelement, 120–124Generalized displacements, 420Generalized forces, 422Geometric interpolationfunctions, 195Geometric isotropybrick element, 192complete polynomial, 174h-refinement, 176incomplete polynomial, 174mathematical function, 174rectangular element, 184triangular element, 178two-dimensional conduction withconvection, 240Geometric mapping matrix, 351Global capacitance matrix, 279Global coordinate system, 21Global damping matrix, 428Global displacement notation, 54Global stiffness matrix, 58, 61–67Green-Gauss theorem, 238Green’s theorem in the plane, 238Guyan reduction, 442HHalf-symmetry model, 254Harmonic oscillator, 387–393, 412Harmonic response, 417Harmonic response using modesuperposition, 422–424Heat transfer, 222–292axisymmetric, 271–276mass transport, with, 261–266one dimensional conduction withconvection, 227–235one-dimensional conduction(quadratic element), 222–227three-dimensional, 267–271time-dependent, 277–285.