Introduction to Finite Elements in Engineering [With Access Code]

Cover......Page 1
PREFACE......Page 14
ABOUT THE AUTHORS......Page 17
Contents......Page 6
 1.2 Historical Background......Page 18
 1.4 Stresses and Equilibrium......Page 19
 1.5 Boundary Conditions......Page 21
 1.6 Strain-Displacement Relations......Page 22
 1.7 Stress-Strain Relations......Page 23
  Special Cases......Page 24
 1.8 Temperature Effects......Page 25
  Potential Energy,∏......Page 26
  Rayleigh-Ritz Method......Page 29
 1.10 Galerkin's Method......Page 31
 1.11 Saint Venant's Principle......Page 35
 1.13 Principle of Superposition......Page 36
 Historical References......Page 37
 Problems......Page 38
 2.1 Matrix Algebra......Page 45
  Matrix Multiplication......Page 46
  Differentiation and Integration......Page 47
  Identity Matrix......Page 48
  Matrix Inversion......Page 49
  Eigenvalues and Eigenvectors......Page 50
 2.2 Gaussian Elimination......Page 52
  General Algorithm for Gaussian Elimination......Page 54
  Symmetric Banded Matrices......Page 57
  Gaussian Elimination with Column Reduction......Page 59
  Skyline Solution......Page 61
 2.3 Conjugate Gradient Method for Equation Solving......Page 62
  Input Data/Output......Page 63
 Problems......Page 64
  Program Listings......Page 66
 3.1 Introduction......Page 68
  Element Division......Page 69
  Numbering Scheme......Page 70
 3.3 Shape Functions and Local Coordinates......Page 72
 3.4 The Potential-Energy Approach......Page 76
  Element Stiffness Matrix......Page 77
  Force Terms......Page 79
  Element Stiffness......Page 81
  Force Terms......Page 82
 3.6 Assembly of the Global Stiffness Matrix and Load Vector......Page 83
 3.7 Properties of K......Page 86
  Types of Boundary Conditions......Page 87
  Elimination Approach......Page 88
  Penalty Approach......Page 93
  Multipoint Constraints......Page 99
 3.9 Quadratic Shape Functions......Page 102
 3.10 Temperature Effects......Page 109
  Problem in Equilibrium......Page 113
  Two Elements with Same End Displacements......Page 114
  Input Data/Output......Page 115
 Problems......Page 116
  Program Listing......Page 128
 4.1 Introduction......Page 134
  Local and Global Coordinate Systems......Page 135
  Formulas for Calculating  l and m......Page 136
  Element Stiffness Matrix......Page 137
  Stress Calculations......Page 138
  Temperature Effects......Page 143
 4.3 Three-Dimensional Trusses......Page 146
  Assembly for Banded Solution......Page 148
  Skyline Assembly......Page 149
  Inclined Support in Three Dimensions—Line Constraint......Page 151
  Inclined Support in Three Dimensions—Plane Constraint......Page 152
  Symmetry and Antisymmetry......Page 153
  Input Data/Output......Page 155
 Problems......Page 156
  Program Listing......Page 164
 5.1 Introduction......Page 167
  Potential-Energy Approach......Page 168
  Galerkin Approach......Page 169
 5.2 Finite Element Formulation......Page 170
  Element Stiffness—Direct Approach......Page 174
 5.3 Load Vector......Page 175
 5.4 Boundary Considerations......Page 176
 5.5 Shear Force and Bending Moment......Page 177
 5.6 Beams on Elastic Supports......Page 179
 5.7 Plane Frames......Page 180
 5.8 Three-Dimensional Frames......Page 186
 5.9 Problem Modeling and Boundary Conditions......Page 190
  Input Data/Output......Page 191
 Problems......Page 193
  Program Listings......Page 200
 6.1 Introduction......Page 205
 6.2 Finite Element Modeling......Page 206
 6.3 Constant Strain Triangle (CST)......Page 208
  Isoparametric Representation......Page 209
  Element Stiffness......Page 215
  Force Terms......Page 216
  Galerkin Approach......Page 223
  Stress Calculations......Page 225
  Temperature Effects......Page 227
 6.4 Problem Modeling and Boundary Conditions......Page 229
  Patch Test......Page 232
 6.6 Orthotropic Materials......Page 233
  Temperature Effects......Page 237
  Input Data/Output......Page 239
 Problems......Page 242
  Program Listing......Page 255
 7.1 Introduction......Page 259
 7.2 Axisymmetric Formulation......Page 260
 7.3 Finite Element Modeling: Triangular Element......Page 262
  Potential Energy Approach......Page 265
  Rotating Flywheel......Page 266
  Surface Traction......Page 267
  Galerkin Approach......Page 269
  Stress Calculations......Page 272
  Cylinder Subjected to Internal Pressure......Page 273
  Press Fit on a Rigid Shaft......Page 274
  Press Fit on an Elastic Shaft......Page 275
  Belleville Spring......Page 276
  Thermal Stress Problem......Page 277
  Input Data/Output......Page 279
 Problems......Page 280
  Program Listing......Page 288
  Shape Functions......Page 290
  Element Stiffness Matrix......Page 293
 8.3 Numerical Integration......Page 296
  Stiffness Integration......Page 300
  Stress Calculations......Page 301
 8.4 Higher Order Elements......Page 303
  Nine-Node Quadrilateral......Page 304
  Eight-Node Quadrilateral......Page 306
  Six-Node Triangle......Page 307
  Integration on a Triangle—Symmetric Points......Page 308
  Integration on a Triangle—Degenerate Quadrilateral......Page 309
 8.5 Four-Node Quadrilateral for Axisymmetric Problems......Page 311
 8.7 Concluding Remarks and Convergence......Page 312
 8.8 References for Convergence......Page 314
  Input Data/Output......Page 315
 Problems......Page 317
  Program Listings......Page 325
 9.1 Introduction......Page 329
 9.2 Finite Element Formulation......Page 330
  Element Stiffness......Page 333
 9.3 Stress Calculations......Page 334
 9.4 Mesh Preparation......Page 335
 9.5 Hexahedral Elements and Higher Order Elements......Page 339
 9.6 Problem Modeling......Page 341
 9.7 Frontal Method for Finite Element Matrices......Page 343
  Connectivity and Prefront Routine......Page 344
  Elimination of Completed dof......Page 345
  Consideration of Multipoint Constraints......Page 346
  Input Data/Output......Page 347
 Problems......Page 349
  Program Listings......Page 353
 10.1 Introduction......Page 362
 10.2 Steady-State Heat Transfer......Page 363
  One-Dimensional Heat Conduction......Page 364
  One-Dimensional Heat Transfer in Thin Fins......Page 372
  Two-Dimensional Steady-State Heat Conduction......Page 376
  Two-Dimensional Fins......Page 386
 10.3 Torsion......Page 387
  Triangular Element......Page 389
  Galerkin Approach[sup(2)]......Page 390
  Potential Flow......Page 393
  Seepage......Page 395
  Electrical and Magnetic Field Problems......Page 396
  Fluid Flow in Ducts......Page 398
  Acoustics......Page 400
  One-Dimensional Acoustics......Page 401
  One-Dimensional Axial Vibrations......Page 403
  Two-Dimensional Acoustics......Page 405
  Input Data/Output......Page 406
 Problems......Page 408
  Program Listings......Page 419
 11.2 Formulation......Page 425
  Solid Body with Distributed Mass......Page 426
 11.3 Element Mass Matrices......Page 428
 11.4 Evaluation of Eigenvalues and Eigenvectors......Page 433
  Eigenvalue-Eigenvector Evaluation......Page 434
  Inverse Iteration Method......Page 437
  Generalized Jacobi Method......Page 440
  Bringing Generalized Problem to Standard Form......Page 444
  Tridiagonalization......Page 445
  Implicit Symmetric QR Step with Wilkinson Shift for Diagonalization[sup(2)]......Page 448
 11.5 Interfacing with Previous Finite Element Programs and a Program for Determining Critical Speeds of Shafts......Page 449
 11.6 Guyan Reduction......Page 450
 11.7 Rigid Body Modes......Page 453
  Input Data/Output......Page 455
 Problems......Page 457
  Program Listings......Page 463
  Region and Block Representation......Page 470
    Block Corner Nodes, Sides, and Subdivisions......Page 471
  Deformed Configuration and Mode Shape......Page 478
  Contour Plotting......Page 479
  Nodal Values from Known Constant Element Values for a Triangle......Page 480
  Least-Squares Fit for a Four-Noded Quadrilateral......Page 482
 12.4 Conclusion......Page 483
  Input Data/Output......Page 484
 Problems......Page 485
  Program Listings......Page 487
APPENDIX Proof of dA = det Jdξ dη......Page 500
BIBLIOGRAPHY......Page 503
ANSWERS TO SELECTED PROBLEMS......Page 507
 C......Page 509
 F......Page 510
 L......Page 511
 S......Page 512
 W......Page 513