[영문] CONTENTS
PRINCIPAL NOTATION = xxi
1 INTRODUCTION TO FINITE ELEMENT METHOD = 1
1.1 Basic concept = 1
1.2 Historical backgrond = 1
1.3 General applicability of the method = 3
1.3.1 One-dimensional heat transfer = 3
1.3.2 One-dimensional fluid flow = 5
1.3.3 Solid bar under axial load = 5
1.4 Engineering applications of the finite element method = 6
1.5 General description of the finite element method = 8
1.6 Comparison of finite element method with other methods of analysis = 21
1.6.1 Derivation of the equation of motion for the vibration of a beam = 21
1.6.2 Exact analytical solution(separation of variables technique) = 23
1.6.3 Approximate analytical solution(Rayleigh's method) = 24
1.6.4 Approximate analytical solution(Galerkin method) = 26
1.6.5 Finite difference method of numerical solution = 28
1.6.6 Finite element method of numerical solution(displacement method) = 30
1.7 Finite element program packages = 32
References = 34
Problem = 36
2 SOLUTION OF FINITE ELEMENT EQUATIONS = 38
2.1 Introduction = 38
2.2 Solution of equilibrium problems = 40
2.2.1 Gaussian elimination method = 41
(ⅰ) Generalization of the method = 42
(ⅱ) Computer implementation of Gaussian elimination method(GAUSS) =43
2.2.2 Choleski method = 46
(ⅰ) Decomposition of［A］into lower and upper triangular matrices = 46
(ⅱ) Solution of equations = 47
(ⅲ) Choleski decomposition of symmetric matrices = 47
(ⅳ) Inverse of a symmetric matrix = 48
(ⅴ) Computer implementation of the Choleski method(DECØMP and SØLVE) = 49
2.2.3 Other methods = 52
2.3 Solution of eigenvalue problems = 52
2.3.1 Jacobi method = 54
(ⅰ) Method = 55
(ⅱ) Computer implementation of the Jacobi method(JACØBI) = 55
2.3.2 Power method =58
(ⅰ) Competing the largest eigenvalue by the power method = 58
(ⅱ) Computing the smallest eigenvalue by the power method = 60
(ⅲ) Computing intermediate eigenvalues = 60
2.3.3 Rayleigh-Ritz subspace iteration method = 63
(ⅰ) Algorithm = 63
(ⅱ) Computer implementation of subspace iteration method
2.3.4 Other methods = 73
2.4 Solution of propagation problems = 74
2.4.1 Numerical solution of Eq. (2.56) = 75
(ⅰ) Solution of a set of first order differential equations = 76
(ⅱ) Computer implementation of Runge-Kutta method (RUNGE) = 76
2.4.2 Numerical solution of Eq. (2.58) = 80
(ⅰ) Direct integration methods = 80
(ⅱ) Mode superposition methods = 82
(ⅲ) Solution of a general second order differential equation = 83
(ⅳ) Computer implementation of mode superposition method(MODAL) = 85
References = 89
Problems = 89
3 GENERAL PROCEDURE OF FINITE ELEMENT METHOD = 93
3.1 Discretization of the domain = 93
3.1.1 Basic element shapes = 93
3.1.2 Discretization process = 97
(ⅰ) Type of elements = 97
(ⅱ) Size of elements = 100
(ⅲ) Location of nodes = 102
(ⅳ) Number of elements = 102
(ⅴ) Simplifications afforded by the physical configuration of the body = 103
(ⅵ) Finite representation of infinite bodies = 103
(ⅶ) Node numbering scheme = 105
3.2 Interpolation polynomials = 107
3.2.1 Polynomial form of interpolation functions = 108
(ⅰ) Simplex, complex and multiplex elements = 110
(ⅱ) Interpolation polynomial in terms of nodal degrees of freedom = 111
3.2.2 Selection of the order of the interpolation polynomial = 112
3.2.3 Convergence requirements = 114
3.2.4 Linear interpolation polynomials in terms of global coordinates = 117
(ⅰ) One-dimensional simplex element = 117
(ⅱ) Two-dimensional simplex element = 119
(ⅲ) Three-dimensional simplex element = 121
(ⅳ) Interpolation polynomials for vector quantities = 123
3.2.5 Linear interpolation polynomials in terms of local coordinates = 126
(ⅰ) One-dimensional element = 128
(ⅱ) Two-dimensional(triangular) element = 130
(ⅲ) Three-dimensional(tetrahedron) element = 133
3.3 Formulation of element characteristic matrices and vectors = 136
3.3.1 Direct approach = 137
(ⅰ) Bar element under axial load = 137
(ⅱ) Line element for heat flow = 138
(ⅲ) Line element for fluid flow = 140
(ⅳ) Line element for current flow = 141
(ⅴ) Triangular element under plane strain = 142
3.3.2 Variational approach = 144
(ⅰ) Specification of continuum problems = 145
(ⅱ) Approximate methods of solving continuum problems = 145
(ⅲ) Calculus of variations = 145
(ⅳ) Advantages of variational formulation = 150
(ⅴ) Solution of equilibrium problems using variational(Rayleigh-Ritz) method = 150
(ⅵ) Solution of eigenvalue problems using variational(Rayieigh-Ritz) method = 154
(ⅶ) Solution of propagation problems using variational(Rayleigh-Ritz) method = 155
(ⅷ) Equivalence of finite element method and variational(Rayleigh-Ritz) method = 155
(ⅸ) Derivation of finite element equations using variational(Rayleigh-Ritz) approach = 156
3.3.3 Weighted residual approach = 162
(ⅰ) Solution of equilibrium problems using weighted residual method = 163
(ⅱ) Solution of eigenvalue problems using weighted residual method = 167
(ⅲ) Solution of propagation problems using weighted residual method = 168
(ⅳ) Derivation of finite element equations using weighted residual(Galerkin) approach = 169
3.3.4 Coordinate transformation = 172
3.4 Assembly of element matrices and vectors and derivation of system equations = 173
3.4.1 Assemblage of element equations = 173
3.4.2 Computer implementation of the assembly procedure = 175
3.4.3 Incorporation of the boundary conditions = 184
3.4.4 Incorporation of boundary conditions in the computer program = 186
3.5 Solution of finite element(system) equations = 187
3.6 Computation of element resultants = 188
References = 188
Problems = 189
4 HIGHER ORDER AND ISOPARAMETRIC ELEMENT FORMULATIONS = 193
4.1 Introduction = 193
4.2 Higher order one-dimensional element = 194
4.2.1 Quadratic element = 194
4.2.2 Cubic element = 195
4.3 Higher order elements in terms of natural coordinates = 196
4.3.1 One-dimensional element = 196
4.3.2 Two-dimensional element (triangular element) = 198
4.3.3 Derivation of nodal interpolation functions = 200
4.3.4 Three-dimensional element(tetrahedron element) = 203
4.3.5 Two-dimensional element(qadrilateral element) = 205
4.3.6 Three-dimensional element(hexahedron elemet) = 209
4.4 Higher order elements in terms of classical interpolation polynomials = 213
4.4.1 Classical interpolation functions = 213
(ⅰ) Lagrange interpolation functions for n stations = 213
(ⅱ) General two-station interpolation functions = 215
(ⅲ) Zeroth order Hermite interpolation function = 216
(ⅳ) First order Hermite interpolation function = 218
(ⅴ) Second order Hermite interpolation function = 220
4.4.2 One-dimensional elements = 221
(ⅰ) Linear element = 221
(ⅱ) Quadratic element= 221
(ⅲ) Cubic element = 221
4.4.3 Two-dimensional elements : Rectangular elements = 222
(ⅰ) Using Lagrange interpolation polynomials = 222
(ⅱ) Using Hermite interpolation polynomials = 223
4.5 Continuity conditions = 225
4.6 Comparative study of elements = 227
4.7 Isoparametric elements = 228
4.7.1 Definitions = 228
4.7.2 Shape functions in coordinate transformation = 229
4.7.3 Curved-sided elements = 231
4.7.4 Derivation of element equations = 234
4.8 Numerical integration = 236
4.8.1 In one-dimension = 236
4.8.2 In two-dimensions = 238
(ⅰ) In rectangular regions = 238
(ⅱ) In triangular regions = 239
4.8.3 In three-dimensions = 240
(ⅰ) In rectangular prism type regions = 240
(ⅱ) In tetrahedral regions =240
References = 241
Problems = 242
5 SOLID AND STRUCTURAL MECHANICS = 245
5.1 Introduction = 246
5.2 Basic equations of solid mechanics = 246
5.2.1 introduction = 247
5.2.2 External equilibrium equations = 247
5.2.3 Equations of internal equilibrium = 247
5.2.4 Stress strain relations(Constitutive relations) = 249
(ⅰ) Three-dimensional case = 249
(ⅱ) Two-dimensional case(plane stress) = 250
(ⅲ) Two-dimensional case(plane strain) = 251
(ⅳ) One-dimensional case = 253
(ⅴ) Axisymmetric case = 253
5.2.5 Strain-displacement relations = 254
5.2.6 Boundary conditions = 256
5.2.7 Compatibility equations = 258
5.2.8 Stress-strain relations for anisotropic materials = 259
5.2.9 Formulations of solid and structural mechanics = 260
STATIC ANALYSIS
5.3 Formulation of equilibrium equations = 266
5.4 Analysis of trusses and frames = 271
5.4.1 Space truss element = 271
5.4.2 Space frame element = 279
(ⅰ) Axial displacements = 279
(ⅱ) Torsional displacements = 282
(ⅲ) Bending displacements in the plane xy = 284
(ⅳ) Bending displacements in the plane xz = 285
5.4.3 Planar frame element = 292
5.4.4 Beam element = 294
5.4.5 Computer program for frame analysis(FRAME) = 295
5.5 Analysis of plates = 303
5.5.1 Introduction = 303
5.5.2 Triangular membrane element = 303
5.5.3 Numerical results with membrane element = 311
(ⅰ) A plate under tension = 311
(ⅱ) Circular hole in a tension plate = 313
(ⅲ) Cantilevered box beam = 316
5.5.4 Computer program for plates under inplane loads(CST) = 317
5.5.5 Bending behaviour of plates = 323
5.5.6 Triangular plate bending element = 328
5.5.7 Numerical results with bending elements = 333
5.5.8 Analysis of three-dimensional structures using plate elements = 336
5.5.9 Computer program for the analysis of three-dimensional structures using plate elements(PLATE) = 340
5.6 Analysis of three-dimensional problems = 340
5.6.1 Introduction = 340
5.6.2 Tetrahedron element = 340
5.6.3 Hexahedron element = 343
5.6.4 Numerical results = 348
5.7 Analysis of solids of revolution = 348
5.7.1 Introduction = 348
5.7.2 Formulation of elemental equations for an axisymmetric ring element = 349
5.7.3 Numerical results = 353
5.7.4 Computer program(STRESS) = 354
DYNAMIC ANALYSIS
5.8 Dynamic equations of motion = 362
5.9 Consistent and lumped mass matrices = 365
5.10 Consistent mass matrices in global coordinate system = 366
5.10.1 Consistent mass matrix of a pin-jointed(space truss) element = 367
5.10.2 Consistent mass matrix f a frame element = 368
5.10.3 Consistent mass matrix of a triangular membrane element = 370
5.10.4 Consistent mass atrix of a triangular bending element = 371
5.10.5 Consistent mass matrix of a tetrahedron element = 372
5.11 Free vibration analysis = 373
5.12 Computer program for eigenvalne analysis of three-dimensional structures(PLATE) = 381
5.13 Condensation of the eigenvalue problem(eigenvalue economizer) = 395
(ⅰ) Natural frequencies of a square cantilever plate = 398
(ⅱ) Natural frequencies of a cantilevered box beam = 399
5.14 Dynamic response calculations using finite element method = 400
5.14.1 Uncoupling the equations of motion of an undamped system = 401
5.14.2 Uncoupling the equations of motion of a damped system = 402
5.14.3 Solution of a general second order differential equation = 403
5.15 Nonconservative stability and flutter problems = 410
References = 411
Problems = 412
6 HEAT TRANSFER = 418
6.1 Introduction = 418
6.2 Basic equations of heat transfer = 419
6.2.1 Energy balance equation = 419
6.2.2 Rate equations = 419
(ⅰ) For conduction = 419
(ⅱ) For convection = 420
(ⅲ) For radiation = 420
(ⅳ) Energy generated in a solid = 420
(ⅴ) Energy stored in a solid = 421
6.2.3 Governing differential equation for heat conduction in three-dimensional bodies = 421
6.2.4 Statement of the problem in differential equation form = 425
6.3 Derivation of finite element equations = 425
6.3.1 Variational approach = 425
6.3.2 Galerkin approach = 428
6.4 One-dimensional heat transfer = 431
6.4.1 Straight uniform fin analysis = 431
Computer program(HEATI) = 439
6.4.2 Tapered fin analysis = 441
6.4.3 Straight uniform fin analysis using quadratic elements = 445
6.5 Two-dimensional heat transfer = 448
Computer program(HEAT2) = 464
6.6 Axisymetric heat transfer = 468
Computer program(HEATAX) = 477
6.7 Three-dimensional heat transfer = 482
6.8 Unsteady state heat transfer problems = 487
6.8.1 Derivation of element capacitance matrices = 487
(ⅰ) For one-dimensional problems = 487
(ⅱ) For two-dimensional problems = 489
(ⅲ) For axisymetric problems = 489
(ⅳ) For three-dimensional problems = 490
6.8.2 Finite difference solution in the time domain = 493
6.9 Heat transfer problems with radiation = 495
Computer program(RADIAT) = 501
References = 504
Problems = 504
7 FLUID MECHANICS = 507
7.1 Introduction = 507
7.2 Basic equations of fluid mechanics = 508
7.2.1 Definitions = 508
7.2.2 Flow field = 508
7.2.3 Continuity equation = 509
7.2.4 Equations of motion or momentum equations = 510
(ⅰ) State of stress in a fluid = 510
(ⅱ) Relation between stress and rate of strain for Newtonian fluids = 511
(ⅲ) Equations of motion = 513
7.2.5 Energy equation = 515
7.2.6 State and viscosity equations = 517
7.2.7 Solution procedure = 517
7.2.8 Inviscid fluid flow = 517
7.2.9 Irrotational flow = 518
7.2.10 Velocity potential = 520
7.2.11 Stream function = 520
7.2.12 Bernoulli equation = 522
7.3 Inviscid incompressible flows = 523
7.3.1 Potential function formulation = 524
(ⅰ) Differential equation form = 524
(ⅱ) Variational form = 524
7.3.2 Stream function formulation = 533
(ⅰ) Differential equation form = 533
(ⅱ) Variational form = 533
7.3.3 Computer program(PHIFLO) = 535
7.4 Flow in porous media = 538
7.4.1 Governing equations = 539
7.4.2 Finite element solution = 540
7.4.3 Steady state unconfined flow through a dam = 532
7.4.4 Steady state flow towards a well = 543
7.5 Wave motion of a shallow basin = 544
7.5.1 Equation of motion = 544
7.5.2 Boundary and initial conditions = 546
7.5.3 Finite element solution of Eq. (7.133) using Galerkin approach = 546
7.5.4 Eigenvalue solution = 550
7.5.5 Solution of Eq. (7.161) by mode superposition method = 551
7.6 Incompressible viscous flow = 552
7.6.1 Statement of the problem = 552
7.6.2 Stream function formulation(using variational approach) = 552
7.6.3 Velocity-pressure formulation(using Galerkin approach) = 558
7.6.4 Stream function-vorticity formulation = 563
(ⅰ) Governing equations = 563
(ⅱ) Finite element solution(using variational approach) = 563
7.7 Flow of non-Newtonian fluids = 565
7.7.1 Governing equations = 565
(ⅰ) Flow curve characteristic = 565
(ⅱ) Equation of motion = 567
7.7.2 Finite element equations using Galerkin method = 567
7.7.3 Solution procedure = 568
References = 571
Problems = 572
8 ADDITIONAL APPLICATIONS AND GENERALIZATION OF THE FINITE ELEMENT METHOD = 573
8.1 Introduction = 573
8.2 Steady state field problems = 574
8.3 Transient field problems = 577
8.4 Space-time finite elements = 579
8.5 Solution of Poisson equation = 580
8.5.1 Derivation of the governing equation for the torsion problem = 580
8.5.2 Finite element solution = 582
8.5.3 Computer program(TORSON) = 588
8.6 Solution f Helmholtz equation = 590
8.7 Solution of Reynolds equation = 596
8.7.1 Hydrodynamic lubrication problem = 596
8.7.2 Finite element equations = 597
8.8 Least squares finite element approach = 601
8.8.1 Solution of a general linear partial differential equation = 601
8.8.2 Solution of unsteady gas dynamic equations = 605
8.9 Equilibrium, mixed and hybrid elements = 610
8.10 Miscellaneous applications = 611
References = 613
Problems = 617
APPENDIX A : GREEN-GAUSS THEOREM(Integration by parts in two and three dimensions) = 618
INDEX = 621
PRINCIPAL NOTATION = xxi
1 INTRODUCTION TO FINITE ELEMENT METHOD = 1
1.1 Basic concept = 1
1.2 Historical backgrond = 1
1.3 General applicability of the method = 3
1.3.1 One-dimensional heat transfer = 3
1.3.2 One-dimensional fluid flow = 5
1.3.3 Solid bar under axial load = 5
1.4 Engineering applications of the finite element method = 6
1.5 General description of the finite element method = 8
1.6 Comparison of finite element method with other methods of analysis = 21
1.6.1 Derivation of the equation of motion for the vibration of a beam = 21
1.6.2 Exact analytical solution(separation of variables technique) = 23
1.6.3 Approximate analytical solution(Rayleigh's method) = 24
1.6.4 Approximate analytical solution(Galerkin method) = 26
1.6.5 Finite difference method of numerical solution = 28
1.6.6 Finite element method of numerical solution(displacement method) = 30
1.7 Finite element program packages = 32
References = 34
Problem = 36
2 SOLUTION OF FINITE ELEMENT EQUATIONS = 38
2.1 Introduction = 38
2.2 Solution of equilibrium problems = 40
2.2.1 Gaussian elimination method = 41
(ⅰ) Generalization of the method = 42
(ⅱ) Computer implementation of Gaussian elimination method(GAUSS) =43
2.2.2 Choleski method = 46
(ⅰ) Decomposition of［A］into lower and upper triangular matrices = 46
(ⅱ) Solution of equations = 47
(ⅲ) Choleski decomposition of symmetric matrices = 47
(ⅳ) Inverse of a symmetric matrix = 48
(ⅴ) Computer implementation of the Choleski method(DECØMP and SØLVE) = 49
2.2.3 Other methods = 52
2.3 Solution of eigenvalue problems = 52
2.3.1 Jacobi method = 54
(ⅰ) Method = 55
(ⅱ) Computer implementation of the Jacobi method(JACØBI) = 55
2.3.2 Power method =58
(ⅰ) Competing the largest eigenvalue by the power method = 58
(ⅱ) Computing the smallest eigenvalue by the power method = 60
(ⅲ) Computing intermediate eigenvalues = 60
2.3.3 Rayleigh-Ritz subspace iteration method = 63
(ⅰ) Algorithm = 63
(ⅱ) Computer implementation of subspace iteration method
2.3.4 Other methods = 73
2.4 Solution of propagation problems = 74
2.4.1 Numerical solution of Eq. (2.56) = 75
(ⅰ) Solution of a set of first order differential equations = 76
(ⅱ) Computer implementation of Runge-Kutta method (RUNGE) = 76
2.4.2 Numerical solution of Eq. (2.58) = 80
(ⅰ) Direct integration methods = 80
(ⅱ) Mode superposition methods = 82
(ⅲ) Solution of a general second order differential equation = 83
(ⅳ) Computer implementation of mode superposition method(MODAL) = 85
References = 89
Problems = 89
3 GENERAL PROCEDURE OF FINITE ELEMENT METHOD = 93
3.1 Discretization of the domain = 93
3.1.1 Basic element shapes = 93
3.1.2 Discretization process = 97
(ⅰ) Type of elements = 97
(ⅱ) Size of elements = 100
(ⅲ) Location of nodes = 102
(ⅳ) Number of elements = 102
(ⅴ) Simplifications afforded by the physical configuration of the body = 103
(ⅵ) Finite representation of infinite bodies = 103
(ⅶ) Node numbering scheme = 105
3.2 Interpolation polynomials = 107
3.2.1 Polynomial form of interpolation functions = 108
(ⅰ) Simplex, complex and multiplex elements = 110
(ⅱ) Interpolation polynomial in terms of nodal degrees of freedom = 111
3.2.2 Selection of the order of the interpolation polynomial = 112
3.2.3 Convergence requirements = 114
3.2.4 Linear interpolation polynomials in terms of global coordinates = 117
(ⅰ) One-dimensional simplex element = 117
(ⅱ) Two-dimensional simplex element = 119
(ⅲ) Three-dimensional simplex element = 121
(ⅳ) Interpolation polynomials for vector quantities = 123
3.2.5 Linear interpolation polynomials in terms of local coordinates = 126
(ⅰ) One-dimensional element = 128
(ⅱ) Two-dimensional(triangular) element = 130
(ⅲ) Three-dimensional(tetrahedron) element = 133
3.3 Formulation of element characteristic matrices and vectors = 136
3.3.1 Direct approach = 137
(ⅰ) Bar element under axial load = 137
(ⅱ) Line element for heat flow = 138
(ⅲ) Line element for fluid flow = 140
(ⅳ) Line element for current flow = 141
(ⅴ) Triangular element under plane strain = 142
3.3.2 Variational approach = 144
(ⅰ) Specification of continuum problems = 145
(ⅱ) Approximate methods of solving continuum problems = 145
(ⅲ) Calculus of variations = 145
(ⅳ) Advantages of variational formulation = 150
(ⅴ) Solution of equilibrium problems using variational(Rayleigh-Ritz) method = 150
(ⅵ) Solution of eigenvalue problems using variational(Rayieigh-Ritz) method = 154
(ⅶ) Solution of propagation problems using variational(Rayleigh-Ritz) method = 155
(ⅷ) Equivalence of finite element method and variational(Rayleigh-Ritz) method = 155
(ⅸ) Derivation of finite element equations using variational(Rayleigh-Ritz) approach = 156
3.3.3 Weighted residual approach = 162
(ⅰ) Solution of equilibrium problems using weighted residual method = 163
(ⅱ) Solution of eigenvalue problems using weighted residual method = 167
(ⅲ) Solution of propagation problems using weighted residual method = 168
(ⅳ) Derivation of finite element equations using weighted residual(Galerkin) approach = 169
3.3.4 Coordinate transformation = 172
3.4 Assembly of element matrices and vectors and derivation of system equations = 173
3.4.1 Assemblage of element equations = 173
3.4.2 Computer implementation of the assembly procedure = 175
3.4.3 Incorporation of the boundary conditions = 184
3.4.4 Incorporation of boundary conditions in the computer program = 186
3.5 Solution of finite element(system) equations = 187
3.6 Computation of element resultants = 188
References = 188
Problems = 189
4 HIGHER ORDER AND ISOPARAMETRIC ELEMENT FORMULATIONS = 193
4.1 Introduction = 193
4.2 Higher order one-dimensional element = 194
4.2.1 Quadratic element = 194
4.2.2 Cubic element = 195
4.3 Higher order elements in terms of natural coordinates = 196
4.3.1 One-dimensional element = 196
4.3.2 Two-dimensional element (triangular element) = 198
4.3.3 Derivation of nodal interpolation functions = 200
4.3.4 Three-dimensional element(tetrahedron element) = 203
4.3.5 Two-dimensional element(qadrilateral element) = 205
4.3.6 Three-dimensional element(hexahedron elemet) = 209
4.4 Higher order elements in terms of classical interpolation polynomials = 213
4.4.1 Classical interpolation functions = 213
(ⅰ) Lagrange interpolation functions for n stations = 213
(ⅱ) General two-station interpolation functions = 215
(ⅲ) Zeroth order Hermite interpolation function = 216
(ⅳ) First order Hermite interpolation function = 218
(ⅴ) Second order Hermite interpolation function = 220
4.4.2 One-dimensional elements = 221
(ⅰ) Linear element = 221
(ⅱ) Quadratic element= 221
(ⅲ) Cubic element = 221
4.4.3 Two-dimensional elements : Rectangular elements = 222
(ⅰ) Using Lagrange interpolation polynomials = 222
(ⅱ) Using Hermite interpolation polynomials = 223
4.5 Continuity conditions = 225
4.6 Comparative study of elements = 227
4.7 Isoparametric elements = 228
4.7.1 Definitions = 228
4.7.2 Shape functions in coordinate transformation = 229
4.7.3 Curved-sided elements = 231
4.7.4 Derivation of element equations = 234
4.8 Numerical integration = 236
4.8.1 In one-dimension = 236
4.8.2 In two-dimensions = 238
(ⅰ) In rectangular regions = 238
(ⅱ) In triangular regions = 239
4.8.3 In three-dimensions = 240
(ⅰ) In rectangular prism type regions = 240
(ⅱ) In tetrahedral regions =240
References = 241
Problems = 242
5 SOLID AND STRUCTURAL MECHANICS = 245
5.1 Introduction = 246
5.2 Basic equations of solid mechanics = 246
5.2.1 introduction = 247
5.2.2 External equilibrium equations = 247
5.2.3 Equations of internal equilibrium = 247
5.2.4 Stress strain relations(Constitutive relations) = 249
(ⅰ) Three-dimensional case = 249
(ⅱ) Two-dimensional case(plane stress) = 250
(ⅲ) Two-dimensional case(plane strain) = 251
(ⅳ) One-dimensional case = 253
(ⅴ) Axisymmetric case = 253
5.2.5 Strain-displacement relations = 254
5.2.6 Boundary conditions = 256
5.2.7 Compatibility equations = 258
5.2.8 Stress-strain relations for anisotropic materials = 259
5.2.9 Formulations of solid and structural mechanics = 260
STATIC ANALYSIS
5.3 Formulation of equilibrium equations = 266
5.4 Analysis of trusses and frames = 271
5.4.1 Space truss element = 271
5.4.2 Space frame element = 279
(ⅰ) Axial displacements = 279
(ⅱ) Torsional displacements = 282
(ⅲ) Bending displacements in the plane xy = 284
(ⅳ) Bending displacements in the plane xz = 285
5.4.3 Planar frame element = 292
5.4.4 Beam element = 294
5.4.5 Computer program for frame analysis(FRAME) = 295
5.5 Analysis of plates = 303
5.5.1 Introduction = 303
5.5.2 Triangular membrane element = 303
5.5.3 Numerical results with membrane element = 311
(ⅰ) A plate under tension = 311
(ⅱ) Circular hole in a tension plate = 313
(ⅲ) Cantilevered box beam = 316
5.5.4 Computer program for plates under inplane loads(CST) = 317
5.5.5 Bending behaviour of plates = 323
5.5.6 Triangular plate bending element = 328
5.5.7 Numerical results with bending elements = 333
5.5.8 Analysis of three-dimensional structures using plate elements = 336
5.5.9 Computer program for the analysis of three-dimensional structures using plate elements(PLATE) = 340
5.6 Analysis of three-dimensional problems = 340
5.6.1 Introduction = 340
5.6.2 Tetrahedron element = 340
5.6.3 Hexahedron element = 343
5.6.4 Numerical results = 348
5.7 Analysis of solids of revolution = 348
5.7.1 Introduction = 348
5.7.2 Formulation of elemental equations for an axisymmetric ring element = 349
5.7.3 Numerical results = 353
5.7.4 Computer program(STRESS) = 354
DYNAMIC ANALYSIS
5.8 Dynamic equations of motion = 362
5.9 Consistent and lumped mass matrices = 365
5.10 Consistent mass matrices in global coordinate system = 366
5.10.1 Consistent mass matrix of a pin-jointed(space truss) element = 367
5.10.2 Consistent mass matrix f a frame element = 368
5.10.3 Consistent mass matrix of a triangular membrane element = 370
5.10.4 Consistent mass atrix of a triangular bending element = 371
5.10.5 Consistent mass matrix of a tetrahedron element = 372
5.11 Free vibration analysis = 373
5.12 Computer program for eigenvalne analysis of three-dimensional structures(PLATE) = 381
5.13 Condensation of the eigenvalue problem(eigenvalue economizer) = 395
(ⅰ) Natural frequencies of a square cantilever plate = 398
(ⅱ) Natural frequencies of a cantilevered box beam = 399
5.14 Dynamic response calculations using finite element method = 400
5.14.1 Uncoupling the equations of motion of an undamped system = 401
5.14.2 Uncoupling the equations of motion of a damped system = 402
5.14.3 Solution of a general second order differential equation = 403
5.15 Nonconservative stability and flutter problems = 410
References = 411
Problems = 412
6 HEAT TRANSFER = 418
6.1 Introduction = 418
6.2 Basic equations of heat transfer = 419
6.2.1 Energy balance equation = 419
6.2.2 Rate equations = 419
(ⅰ) For conduction = 419
(ⅱ) For convection = 420
(ⅲ) For radiation = 420
(ⅳ) Energy generated in a solid = 420
(ⅴ) Energy stored in a solid = 421
6.2.3 Governing differential equation for heat conduction in three-dimensional bodies = 421
6.2.4 Statement of the problem in differential equation form = 425
6.3 Derivation of finite element equations = 425
6.3.1 Variational approach = 425
6.3.2 Galerkin approach = 428
6.4 One-dimensional heat transfer = 431
6.4.1 Straight uniform fin analysis = 431
Computer program(HEATI) = 439
6.4.2 Tapered fin analysis = 441
6.4.3 Straight uniform fin analysis using quadratic elements = 445
6.5 Two-dimensional heat transfer = 448
Computer program(HEAT2) = 464
6.6 Axisymetric heat transfer = 468
Computer program(HEATAX) = 477
6.7 Three-dimensional heat transfer = 482
6.8 Unsteady state heat transfer problems = 487
6.8.1 Derivation of element capacitance matrices = 487
(ⅰ) For one-dimensional problems = 487
(ⅱ) For two-dimensional problems = 489
(ⅲ) For axisymetric problems = 489
(ⅳ) For three-dimensional problems = 490
6.8.2 Finite difference solution in the time domain = 493
6.9 Heat transfer problems with radiation = 495
Computer program(RADIAT) = 501
References = 504
Problems = 504
7 FLUID MECHANICS = 507
7.1 Introduction = 507
7.2 Basic equations of fluid mechanics = 508
7.2.1 Definitions = 508
7.2.2 Flow field = 508
7.2.3 Continuity equation = 509
7.2.4 Equations of motion or momentum equations = 510
(ⅰ) State of stress in a fluid = 510
(ⅱ) Relation between stress and rate of strain for Newtonian fluids = 511
(ⅲ) Equations of motion = 513
7.2.5 Energy equation = 515
7.2.6 State and viscosity equations = 517
7.2.7 Solution procedure = 517
7.2.8 Inviscid fluid flow = 517
7.2.9 Irrotational flow = 518
7.2.10 Velocity potential = 520
7.2.11 Stream function = 520
7.2.12 Bernoulli equation = 522
7.3 Inviscid incompressible flows = 523
7.3.1 Potential function formulation = 524
(ⅰ) Differential equation form = 524
(ⅱ) Variational form = 524
7.3.2 Stream function formulation = 533
(ⅰ) Differential equation form = 533
(ⅱ) Variational form = 533
7.3.3 Computer program(PHIFLO) = 535
7.4 Flow in porous media = 538
7.4.1 Governing equations = 539
7.4.2 Finite element solution = 540
7.4.3 Steady state unconfined flow through a dam = 532
7.4.4 Steady state flow towards a well = 543
7.5 Wave motion of a shallow basin = 544
7.5.1 Equation of motion = 544
7.5.2 Boundary and initial conditions = 546
7.5.3 Finite element solution of Eq. (7.133) using Galerkin approach = 546
7.5.4 Eigenvalue solution = 550
7.5.5 Solution of Eq. (7.161) by mode superposition method = 551
7.6 Incompressible viscous flow = 552
7.6.1 Statement of the problem = 552
7.6.2 Stream function formulation(using variational approach) = 552
7.6.3 Velocity-pressure formulation(using Galerkin approach) = 558
7.6.4 Stream function-vorticity formulation = 563
(ⅰ) Governing equations = 563
(ⅱ) Finite element solution(using variational approach) = 563
7.7 Flow of non-Newtonian fluids = 565
7.7.1 Governing equations = 565
(ⅰ) Flow curve characteristic = 565
(ⅱ) Equation of motion = 567
7.7.2 Finite element equations using Galerkin method = 567
7.7.3 Solution procedure = 568
References = 571
Problems = 572
8 ADDITIONAL APPLICATIONS AND GENERALIZATION OF THE FINITE ELEMENT METHOD = 573
8.1 Introduction = 573
8.2 Steady state field problems = 574
8.3 Transient field problems = 577
8.4 Space-time finite elements = 579
8.5 Solution of Poisson equation = 580
8.5.1 Derivation of the governing equation for the torsion problem = 580
8.5.2 Finite element solution = 582
8.5.3 Computer program(TORSON) = 588
8.6 Solution f Helmholtz equation = 590
8.7 Solution of Reynolds equation = 596
8.7.1 Hydrodynamic lubrication problem = 596
8.7.2 Finite element equations = 597
8.8 Least squares finite element approach = 601
8.8.1 Solution of a general linear partial differential equation = 601
8.8.2 Solution of unsteady gas dynamic equations = 605
8.9 Equilibrium, mixed and hybrid elements = 610
8.10 Miscellaneous applications = 611
References = 613
Problems = 617
APPENDIX A : GREEN-GAUSS THEOREM(Integration by parts in two and three dimensions) = 618
INDEX = 621