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 # 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 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 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 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 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)

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)

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)

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

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)

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)

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.)