BOOK
Introduction to Finite Elements in Engineering
Tirupathi R. Chandrupatla | Ashok D. Belegundu
(2014)
Additional Information
Book Details
Abstract
Introduction to Finite Engineering is ideal for senior undergraduate and first-year graduate students and also as a learning resource to practicing engineers.
This book provides an integrated approach to finite element methodologies. The development of finite element theory is combined with examples and exercises involving engineering applications. The steps used in the development of the theory are implemented in complete, self-contained computer programs. While the strategy and philosophy of the previous editions has been retained, the Fourth Edition has been updated and improved to include new material on additional topics.
Table of Contents
| Section Title | Page | Action | Price |
|---|---|---|---|
| Cover | Cover | ||
| PREFACE | 13 | ||
| ABOUT THE AUTHORS | 16 | ||
| Contents | 5 | ||
| 1 FUNDAMENTAL CONCEPTS | 17 | ||
| 1.1 Introduction | 17 | ||
| 1.2 Historical Background | 17 | ||
| 1.3 Outline of Presentation | 18 | ||
| 1.4 Stresses and Equilibrium | 18 | ||
| 1.5 Boundary Conditions | 20 | ||
| 1.6 Strain-Displacement Relations | 21 | ||
| 1.7 Stress-Strain Relations | 22 | ||
| Special Cases | 23 | ||
| 1.8 Temperature Effects | 24 | ||
| 1.9 Potential Energy and Equilibrium: The Rayleigh-Ritz Method | 25 | ||
| Potential Energy,& | 25 | ||
| Rayleigh-Ritz Method | 28 | ||
| 1.10 Galerkin's Method | 30 | ||
| 1.11 Saint Venant's Principle | 34 | ||
| 1.12 Von Mises Stress | 35 | ||
| 1.13 Principle of Superposition | 35 | ||
| 1.14 Computer Programs | 36 | ||
| 1.15 Conclusion | 36 | ||
| Historical References | 36 | ||
| Problems | 37 | ||
| 2 MATRIX ALGEBRA AND GAUSSIAN ELIMINATION | 44 | ||
| 2.1 Matrix Algebra | 44 | ||
| Row and Column Vectors | 45 | ||
| Addition and Subtraction | 45 | ||
| Multiplication by a Scalar | 45 | ||
| Matrix Multiplication | 45 | ||
| Transposition | 46 | ||
| Differentiation and Integration | 46 | ||
| Square Matrix | 47 | ||
| Diagonal Matrix | 47 | ||
| Identity Matrix | 47 | ||
| Symmetric Matrix | 48 | ||
| Upper Triangular Matrix | 48 | ||
| Determinant of a Matrix | 48 | ||
| Matrix Inversion | 48 | ||
| Eigenvalues and Eigenvectors | 49 | ||
| Positive Definite Matrix | 51 | ||
| Cholesky Decomposition | 51 | ||
| 2.2 Gaussian Elimination | 51 | ||
| General Algorithm for Gaussian Elimination | 53 | ||
| Symmetric Matrix | 56 | ||
| Symmetric Banded Matrices | 56 | ||
| Solution with Multiple Right Sides | 58 | ||
| Gaussian Elimination with Column Reduction | 58 | ||
| Skyline Solution | 60 | ||
| Frontal Solution | 61 | ||
| 2.3 Conjugate Gradient Method for Equation Solving | 61 | ||
| Conjugate Gradient Algorithm | 62 | ||
| Input Data/Output | 62 | ||
| Problems | 63 | ||
| Program Listings | 65 | ||
| 3 ONE-DIMENSIONAL PROBLEMS | 67 | ||
| 3.1 Introduction | 67 | ||
| 3.2 Finite Element Modeling | 68 | ||
| Element Division | 68 | ||
| Numbering Scheme | 69 | ||
| 3.3 Shape Functions and Local Coordinates | 71 | ||
| 3.4 The Potential-Energy Approach | 75 | ||
| Element Stiffness Matrix | 76 | ||
| Force Terms | 78 | ||
| 3.5 The Galerkin Approach | 80 | ||
| Element Stiffness | 80 | ||
| Force Terms | 81 | ||
| 3.6 Assembly of the Global Stiffness Matrix and Load Vector | 82 | ||
| 3.7 Properties of K | 85 | ||
| 3.8 The Finite Element Equations: Treatment of Boundary Conditions | 86 | ||
| Types of Boundary Conditions | 86 | ||
| Elimination Approach | 87 | ||
| Penalty Approach | 92 | ||
| Multipoint Constraints | 98 | ||
| 3.9 Quadratic Shape Functions | 101 | ||
| 3.10 Temperature Effects | 108 | ||
| 3.11 Problem Modeling and Boundary Conditions | 112 | ||
| Problem in Equilibrium | 112 | ||
| Symmetry | 113 | ||
| Two Elements with Same End Displacements | 113 | ||
| Problem with a Closing Gap | 114 | ||
| Input Data/Output | 114 | ||
| Problems | 115 | ||
| Program Listing | 127 | ||
| 4 TRUSSES | 133 | ||
| 4.1 Introduction | 133 | ||
| 4.2 Plane Trusses | 134 | ||
| Local and Global Coordinate Systems | 134 | ||
| Formulas for Calculating l and m | 135 | ||
| Element Stiffness Matrix | 136 | ||
| Stress Calculations | 137 | ||
| Temperature Effects | 142 | ||
| 4.3 Three-Dimensional Trusses | 145 | ||
| 4.4 Assembly of Global Stiffness Matrix for the Banded and Skyline Solutions | 147 | ||
| Assembly for Banded Solution | 147 | ||
| Skyline Assembly | 148 | ||
| 4.5 Problem Modeling and Boundary Conditions | 150 | ||
| Inclined Support in Two Dimensions | 150 | ||
| Inclined Support in Three Dimensions—Line Constraint | 150 | ||
| Inclined Support in Three Dimensions—Plane Constraint | 151 | ||
| Symmetry and Antisymmetry | 152 | ||
| Input Data/Output | 154 | ||
| Problems | 155 | ||
| Program Listing | 163 | ||
| 5 BEAMS AND FRAMES | 166 | ||
| 5.1 Introduction | 166 | ||
| Potential-Energy Approach | 167 | ||
| Galerkin Approach | 168 | ||
| 5.2 Finite Element Formulation | 169 | ||
| Element Stiffness—Direct Approach | 173 | ||
| 5.3 Load Vector | 174 | ||
| 5.4 Boundary Considerations | 175 | ||
| 5.5 Shear Force and Bending Moment | 176 | ||
| 5.6 Beams on Elastic Supports | 178 | ||
| 5.7 Plane Frames | 179 | ||
| 5.8 Three-Dimensional Frames | 185 | ||
| 5.9 Problem Modeling and Boundary Conditions | 189 | ||
| 5.10 Some Comments | 190 | ||
| Input Data/Output | 190 | ||
| Problems | 192 | ||
| Program Listings | 199 | ||
| 6 TWO-DIMENSIONAL PROBLEMS USING CONSTANT STRAIN TRIANGLES | 204 | ||
| 6.1 Introduction | 204 | ||
| 6.2 Finite Element Modeling | 205 | ||
| 6.3 Constant Strain Triangle (CST) | 207 | ||
| Isoparametric Representation | 208 | ||
| Potential-Energy Approach | 214 | ||
| Element Stiffness | 214 | ||
| Force Terms | 215 | ||
| Integration Formula on a Triangle | 222 | ||
| Galerkin Approach | 222 | ||
| Stress Calculations | 224 | ||
| Temperature Effects | 226 | ||
| 6.4 Problem Modeling and Boundary Conditions | 228 | ||
| Some General Comments on Dividing into Elements | 231 | ||
| 6.5 Patch Test and Convergence | 231 | ||
| Patch Test | 231 | ||
| 6.6 Orthotropic Materials | 232 | ||
| Temperature Effects | 236 | ||
| Input Data/Output | 238 | ||
| Problems | 241 | ||
| Program Listing | 254 | ||
| 7 AXISYMMETRIC SOLIDS SUBJECTED TO AXISYMMETRIC LOADING | 258 | ||
| 7.1 Introduction | 258 | ||
| 7.2 Axisymmetric Formulation | 259 | ||
| 7.3 Finite Element Modeling: Triangular Element | 261 | ||
| Potential Energy Approach | 264 | ||
| Body Force Term | 265 | ||
| Rotating Flywheel | 265 | ||
| Surface Traction | 266 | ||
| Galerkin Approach | 268 | ||
| Stress Calculations | 271 | ||
| Temperature Effects | 272 | ||
| 7.4 Problem Modeling and Boundary Conditions | 272 | ||
| Cylinder Subjected to Internal Pressure | 272 | ||
| Infinite Cylinder | 273 | ||
| Press Fit on a Rigid Shaft | 273 | ||
| Press Fit on an Elastic Shaft | 274 | ||
| Belleville Spring | 275 | ||
| Thermal Stress Problem | 276 | ||
| Input Data/Output | 278 | ||
| Problems | 279 | ||
| Program Listing | 287 | ||
| 8 TWO-DIMENSIONAL ISOPARAMETRIC ELEMENTS AND NUMERICAL INTEGRATION | 289 | ||
| 8.1 Introduction | 289 | ||
| 8.2 The Four-Node Quadrilateral | 289 | ||
| Shape Functions | 289 | ||
| Element Stiffness Matrix | 292 | ||
| Element Force Vectors | 295 | ||
| 8.3 Numerical Integration | 295 | ||
| Two-Dimensional Integrals | 299 | ||
| Stiffness Integration | 299 | ||
| Stress Calculations | 300 | ||
| 8.4 Higher Order Elements | 302 | ||
| Nine-Node Quadrilateral | 303 | ||
| Eight-Node Quadrilateral | 305 | ||
| Six-Node Triangle | 306 | ||
| Integration on a Triangle—Symmetric Points | 307 | ||
| Integration on a Triangle—Degenerate Quadrilateral | 308 | ||
| 8.5 Four-Node Quadrilateral for Axisymmetric Problems | 310 | ||
| 8.6 Conjugate Gradient Implementation of the Quadrilateral Element | 311 | ||
| 8.7 Concluding Remarks and Convergence | 311 | ||
| 8.8 References for Convergence | 313 | ||
| Input Data/Output | 314 | ||
| Problems | 316 | ||
| Program Listings | 324 | ||
| 9 THREE-DIMENSIONAL PROBLEMS IN STRESS ANALYSIS | 328 | ||
| 9.1 Introduction | 328 | ||
| 9.2 Finite Element Formulation | 329 | ||
| Element Stiffness | 332 | ||
| Force Terms | 333 | ||
| 9.3 Stress Calculations | 333 | ||
| 9.4 Mesh Preparation | 334 | ||
| 9.5 Hexahedral Elements and Higher Order Elements | 338 | ||
| 9.6 Problem Modeling | 340 | ||
| 9.7 Frontal Method for Finite Element Matrices | 342 | ||
| Connectivity and Prefront Routine | 343 | ||
| Element Assembly and Consideration of Specified dof | 344 | ||
| Elimination of Completed dof | 344 | ||
| Backsubstitution | 345 | ||
| Consideration of Multipoint Constraints | 345 | ||
| Input Data/Output | 346 | ||
| Problems | 348 | ||
| Program Listings | 352 | ||
| 10 SCALAR FIELD PROBLEMS | 361 | ||
| 10.1 Introduction | 361 | ||
| 10.2 Steady-State Heat Transfer | 362 | ||
| One-Dimensional Heat Conduction | 363 | ||
| One-Dimensional Heat Transfer in Thin Fins | 371 | ||
| Two-Dimensional Steady-State Heat Conduction | 375 | ||
| Two-Dimensional Fins | 385 | ||
| Preprocessing for Program HEAT2D | 386 | ||
| 10.3 Torsion | 386 | ||
| Triangular Element | 388 | ||
| Galerkin Approach[sup(2)] | 389 | ||
| 10.4 Potential Flow, Seepage, Electric and Magnetic Fields, and Fluid Flow in Ducts | 392 | ||
| Potential Flow | 392 | ||
| Seepage | 394 | ||
| Electrical and Magnetic Field Problems | 395 | ||
| Fluid Flow in Ducts | 397 | ||
| Acoustics | 399 | ||
| Boundary Conditions | 400 | ||
| One-Dimensional Acoustics | 400 | ||
| One-Dimensional Axial Vibrations | 402 | ||
| Two-Dimensional Acoustics | 404 | ||
| 10.5 Conclusion | 405 | ||
| Input Data/Output | 405 | ||
| Problems | 407 | ||
| Program Listings | 418 | ||
| 11 DYNAMIC CONSIDERATIONS | 424 | ||
| 11.1 Introduction | 424 | ||
| 11.2 Formulation | 424 | ||
| Solid Body with Distributed Mass | 425 | ||
| 11.3 Element Mass Matrices | 427 | ||
| 11.4 Evaluation of Eigenvalues and Eigenvectors | 432 | ||
| Properties of Eigenvectors | 433 | ||
| Eigenvalue-Eigenvector Evaluation | 433 | ||
| Inverse Iteration Method | 436 | ||
| Generalized Jacobi Method | 439 | ||
| Tridiagonalization and Implicit Shift Approach | 443 | ||
| Bringing Generalized Problem to Standard Form | 443 | ||
| Tridiagonalization | 444 | ||
| Implicit Symmetric QR Step with Wilkinson Shift for Diagonalization[sup(2)] | 447 | ||
| 11.5 Interfacing with Previous Finite Element Programs and a Program for Determining Critical Speeds of Shafts | 448 | ||
| 11.6 Guyan Reduction | 449 | ||
| 11.7 Rigid Body Modes | 452 | ||
| 11.8 Conclusion | 454 | ||
| Input Data/Output | 454 | ||
| Problems | 456 | ||
| Program Listings | 462 | ||
| 12 PREPROCESSING AND POSTPROCESSING | 469 | ||
| 12.1 Introduction | 469 | ||
| 12.2 Mesh Generation | 469 | ||
| Region and Block Representation | 469 | ||
| Block Corner Nodes, Sides, and Subdivisions | 470 | ||
| 12.3 Postprocessing | 477 | ||
| Deformed Configuration and Mode Shape | 477 | ||
| Contour Plotting | 478 | ||
| Nodal Values from Known Constant Element Values for a Triangle | 479 | ||
| Least-Squares Fit for a Four-Noded Quadrilateral | 481 | ||
| 12.4 Conclusion | 482 | ||
| Input Data/Output | 483 | ||
| Problems | 484 | ||
| Program Listings | 486 | ||
| APPENDIX Proof of dA = det Jd& | 499 | ||
| BIBLIOGRAPHY | 502 | ||
| ANSWERS TO SELECTED PROBLEMS | 506 | ||
| INDEX | 508 | ||
| A | 508 | ||
| B | 508 | ||
| C | 508 | ||
| D | 509 | ||
| E | 509 | ||
| F | 509 | ||
| G | 510 | ||
| H | 510 | ||
| I | 510 | ||
| J | 510 | ||
| K | 510 | ||
| L | 510 | ||
| M | 511 | ||
| N | 511 | ||
| O | 511 | ||
| P | 511 | ||
| Q | 511 | ||
| R | 511 | ||
| S | 511 | ||
| T | 512 | ||
| V | 512 | ||
| W | 512 |