| General note |
PREFACE XIII<br/><br/>ABOUT THE AUTHOR XVI<br/><br/>1 FUNDAMENTAL CONCEPTS 1<br/><br/>1.1 Introduction 1<br/><br/>1.2 Historical Background 1<br/><br/>1.3 Outline of Presentation 2<br/><br/>1.4 Stresses and Equilibrium 2<br/><br/>1.5 Boundary Conditions 4<br/><br/>1.6 Strain—Displacement Relations 5<br/><br/>1.7 Stress—Strain Relations 6<br/><br/>Special Cases, 7<br/><br/>1.8 Temperature Effects 8<br/><br/>1.9 Potential Energy and Equilibrium: The Rayleigh—Ritz Method 9<br/><br/>Potential Energy ß , 9<br/><br/>Rayleigh—Ritz Method, 12<br/><br/>1.10 Galerkin’s Method 14<br/><br/>1.11 Saint Venant’s Principle 18<br/><br/>1.12 Von Mises Stress 19<br/><br/>1.13 Principle of Superposition 19<br/><br/>1.14 Computer Programs 20<br/><br/>1.15 Conclusion 20<br/><br/>Historical References 20<br/><br/>Problems 21<br/><br/>2 MATRIX ALGEBRA AND GAUSSIAN ELIMINATION 28<br/><br/>2.1 Matrix Algebra 28<br/><br/>Row and Column Vectors, 29<br/><br/>Addition and Subtraction, 29<br/><br/>Multiplication by a Scalar, 29<br/><br/>Matrix Multiplication, 29<br/><br/>Transposition, 30<br/><br/>Differentiation and Integration, 30<br/><br/>Square Matrix, 31<br/><br/>Diagonal Matrix, 31<br/><br/>Identity Matrix, 31<br/><br/>Symmetric Matrix, 32<br/><br/>Upper Triangular Matrix, 32<br/><br/>Determinant of a Matrix, 32<br/><br/>Matrix Inversion, 32<br/><br/>Eigenvalues and Eigenvectors, 33<br/><br/>Positive Definite Matrix, 35<br/><br/>Cholesky Decomposition, 35<br/><br/>2.2 Gaussian Elimination 35<br/><br/>General Algorithm for Gaussian Elimination, 37<br/><br/>Symmetric Matrix, 40<br/><br/>Symmetric Banded Matrices, 40<br/><br/>Solution with Multiple Right Sides, 40<br/><br/>Gaussian Elimination with Column Reduction, 42<br/><br/>Skyline Solution, 44<br/><br/>Frontal Solution, 45<br/><br/>2.3 Conjugate Gradient Method for Equation Solving 45<br/><br/>Conjugate Gradient Algorithm, 46<br/><br/>Input Data/Output 46<br/><br/>Problems 47<br/><br/>Program Listings, 49<br/><br/>3 ONE-DIMENSIONAL PROBLEMS 51<br/><br/>3.1 Introduction 51<br/><br/>3.2 Finite Element Modeling 52<br/><br/>Element Division, 52<br/><br/>Numbering Scheme, 53<br/><br/>3.3 Shape Functions and Local Coordinates 55<br/><br/>3.4 The Potential-Energy Approach 59<br/><br/>Element Stiffness Matrix, 60<br/><br/>Force Terms, 62<br/><br/>3.5 The Galerkin Approach 64<br/><br/>Element Stiffness, 64<br/><br/>Force Terms, 65<br/><br/>3.6 Assembly of the Global Stiffness Matrix and Load Vector 66<br/><br/>3.7 Properties of K 69<br/><br/>3.8 The Finite Element Equations: Treatment<br/><br/>of Boundary Conditions 70<br/><br/>Types of Boundary Conditions, 70<br/><br/>Elimination Approach, 71<br/><br/>Penalty Approach, 76<br/><br/>Multipoint Constraints, 82<br/><br/>3.9 Quadratic Shape Functions 85<br/><br/>3.10 Temperature Effects 92<br/><br/>3.11 Problem Modeling and Boundary Conditions 96<br/><br/>Problem in Equilibrium, 96<br/><br/>Symmetry, 97<br/><br/>Two Elements with Same End Displacements, 97<br/><br/>Problem with a Closing Gap, 98<br/><br/>Input Data/Output, 98<br/><br/>Problems 99<br/><br/>Program Listing, 111<br/><br/>4 TRUSSES 117<br/><br/>4.1 Introduction 117<br/><br/>4.2 Plane Trusses 118<br/><br/>Local and Global Coordinate Systems, 118<br/><br/>Formulas for Calculating / and m, 119<br/><br/>Element Stiffness Matrix, 120<br/><br/>Stress Calculations, 121<br/><br/>Temperature Effects, 126<br/><br/>4.3 Three-Dimensional Trusses 129<br/><br/>4.4 Assembly of Global Stiffness Matrix for the Banded and Skyline<br/><br/>Solutions 131<br/><br/>Assembly for Banded Solution, 131<br/><br/>Skyline Assembly , 132<br/><br/>4.5 Problem Modeling and Boundary Conditions 134<br/><br/>Inclined Support in Two Dimensions, 134<br/><br/>Inclined Support in Three Dimensions–Line Constraint, 134<br/><br/>Inclined Support in Three Dimensions–Plane Constraint, 135<br/><br/>Symmetry and Antisymmetry , 136<br/><br/>Input Data/Output, 138<br/><br/>Problems 139<br/><br/>Program Listing, 147<br/><br/>5 BEAMS AND FRAMES 150<br/><br/>5.1 Introduction 150<br/><br/>Potential-Energy Approach, 151<br/><br/>Galerkin Approach, 152<br/><br/>5.2 Finite Element Formulation 153<br/><br/>Element Stiffness–Direct Approach, 157<br/><br/>5.3 Load Vector 158<br/><br/>5.4 Boundary Considerations 159<br/><br/>5.5 Shear Force and Bending Moment 160<br/><br/>5.6 Beams on Elastic Supports 162<br/><br/>5.7 Plane Frames 163<br/><br/>5.8 Three-Dimensional Frames 169<br/><br/>5.9 Problem Modeling and Boundary Conditions 173<br/><br/>5.10 Some Comments 174<br/><br/>Input Data/Output, 174<br/><br/>Problems 176<br/><br/>Program Listings, 183<br/><br/>6 TWO-DIMENSIONAL PROBLEMS USING CONSTANT STRAIN TRIANGLES 188<br/><br/>6.1 Introduction 188<br/><br/>6.2 Finite Element Modeling 189<br/><br/>6.3 Constant Strain Triangle (CST) 191<br/><br/>Isoparametric Representation, 192<br/><br/>Potential-Energy Approach, 198<br/><br/>Element Stiffness, 198<br/><br/>Force Terms, 199<br/><br/>Integration Formula on a Triangle, 206<br/><br/>Galerkin Approach, 206<br/><br/>Stress Calculations, 208<br/><br/>Temperature Effects, 210<br/><br/>6.4 Problem Modeling and Boundary Conditions 212<br/><br/>Some General Comments on Dividing into Elements, 215<br/><br/>6.5 Patch Test and Convergence 215<br/><br/>Patch Test, 215<br/><br/>6.6 Orthotropic Materials 216<br/><br/>Temperature Effects, 220<br/><br/>Input Data/Output, 222<br/><br/>Problems 225<br/><br/>Program Listing, 238<br/><br/>7 AXISYMMETRIC SOLIDS SUBJECTED TO AXISYMMETRIC LOADING 242<br/><br/>7.1 Introduction 242<br/><br/>7.2 Axisymmetric Formulation 243<br/><br/>7.3 Finite Element Modeling: Triangular Element 245<br/><br/>Potential-Energy Approach, 248<br/><br/>Body Force Term, 249<br/><br/>Rotating Flywheel, 249<br/><br/>Surface Traction, 250<br/><br/>Galerkin Approach, 252<br/><br/>Stress Calculations, 255<br/><br/>Temperature Effects, 256<br/><br/>7.4 Problem Modeling and Boundary Conditions 256<br/><br/>Cylinder Subjected to Internal Pressure, 256<br/><br/>Infinite Cylinder, 257<br/><br/>Press Fit on a Rigid Shaft, 257<br/><br/>Press Fit on an Elastic Shaft, 258<br/><br/>Belleville Spring, 259<br/><br/>Thermal Stress Problem, 260<br/><br/>Input Data/Output, 262<br/><br/>Problems 263<br/><br/>Program Listing, 271<br/><br/>8 TWO-DIMENSIONAL ISOPARAMETRIC ELEMENTS<br/><br/>AND NUMERICAL INTEGRATION 273<br/><br/>8.1 Introduction 273<br/><br/>8.2 The Four-Node Quadrilateral 273<br/><br/>Shape Functions, 273<br/><br/>Element Stiffness Matrix, 276<br/><br/>Element Force Vectors, 279<br/><br/>8.3 Numerical Integration 279<br/><br/>Two-Dimensional Integrals, 283<br/><br/>Stiffness Integration, 283<br/><br/>Stress Calculations, 284<br/><br/>8.4 Higher Order Elements 286<br/><br/>Nine-Node Quadrilateral, 287<br/><br/>Eight-Node Quadrilateral, 289<br/><br/>Six-Node Triangle, 290<br/><br/>Integration on a Triangle–Symmetric Points, 291<br/><br/>Integration on a Triangle–Degenerate Quadrilateral, 292<br/><br/>8.5 Four-Node Quadrilateral for Axisymmetric Problems 294<br/><br/>8.6 Conjugate Gradient Implementation of the Quadrilateral Element 295<br/><br/>8.7 Concluding Remarks and Convergence 295<br/><br/>8.8 References for Convergence 297<br/><br/>Input Data/Output, 298<br/><br/>Problems 300<br/><br/>Program Listings, 308<br/><br/>9 THREE-DIMENSIONAL PROBLEMS IN STRESS ANALYSIS 312<br/><br/>9.1 Introduction 312<br/><br/>9.2 Finite Element Formulation 313<br/><br/>Element Stiffness, 316<br/><br/>Force Terms, 317<br/><br/>9.3 Stress Calculations 317<br/><br/>9.4 Mesh Preparation 318<br/><br/>9.5 Hexahedral Elements and Higher Order Elements 322<br/><br/>9.6 Problem Modeling 324<br/><br/>9.7 Frontal Method for Finite Element Matrices 326<br/><br/>Connectivity and Prefront Routine, 327<br/><br/>Element Assembly and Consideration of Specified dof, 328<br/><br/>Elimination of Completed dof, 328<br/><br/>Backsubstitution, 329<br/><br/>Consideration of Multipoint Constraints, 329<br/><br/>Input Data/Output, 330<br/><br/>Problems 332<br/><br/>Program Listings, 336<br/><br/>10 SCALAR FIELD PROBLEMS 345<br/><br/>10.1 Introduction 345<br/><br/>10.2 Steady State Heat Transfer 346<br/><br/>One-Dimensional Heat Conduction, 347<br/><br/>One-Dimensional Heat Transfer in Thin Fins, 355<br/><br/>Two-Dimensional Steady-State Heat Conduction, 359<br/><br/>Two-Dimensional Fins, 369<br/><br/>Preprocessing for Program Heat2D, 370<br/><br/>10.3 Torsion 370<br/><br/>Triangular Element, 372<br/><br/>Galerkin Approach, 373<br/><br/>10.4 Potential Flow, Seepage, Electric and Magnetic Fields,<br/><br/>and Fluid Flow in Ducts 376<br/><br/>Potential Flow, 376<br/><br/>Seepage, 378<br/><br/>Electrical and Magnetic Field Problems, 379<br/><br/>Fluid Flow in Ducts, 381<br/><br/>Acoustics, 383<br/><br/>Boundary Conditions, 384<br/><br/>One-Dimensional Acoustics, 384<br/><br/>One-Dimensional Axial Vibrations, 386<br/><br/>Two-Dimensional Acoustics, 388<br/><br/>10.5 Conclusion 389<br/><br/>Input Data/Output, 389<br/><br/>Problems 391<br/><br/>Program Listings, 402<br/><br/>11 DYNAMIC CONSIDERATIONS 408<br/><br/>11.1 Introduction 408<br/><br/>11.2 Formulation 408<br/><br/>Solid Body with Distributed Mass, 409<br/><br/>11.3 Element Mass Matrices 411<br/><br/>11.4 Evaluation of Eigenvalues and Eigenvectors 416<br/><br/>Properties of Eigenvectors, 417<br/><br/>Eigenvalue—Eigenvector Evaluation, 417<br/><br/>Inverse Iteration Method , 420<br/><br/>Generalized Jacobi Method, 423<br/><br/>Tridiagonalization and Implicit Shift Approach, 427<br/><br/>Bringing Generalized Problem to Standard Form, 427<br/><br/>Tridiagonalization, 428<br/><br/>Implicit Symmetric QR Step with Wilkinson Shift<br/><br/>for Diagonalization, 431<br/><br/>11.5 Interfacing with Previous Finite Element Programs and a Program<br/><br/>for Determining Critical Speeds of Shafts 432<br/><br/>11.6 Guyan Reduction 433<br/><br/>11.7 Rigid Body Modes 436<br/><br/>11.8 Conclusion 438<br/><br/>Input Data/Output, 438<br/><br/>Problems 440<br/><br/>Program Listings, 446<br/><br/>12 PREPROCESSING AND POSTPROCESSING 453<br/><br/>12.1 Introduction 453<br/><br/>12.2 Mesh Generation 453<br/><br/>Region and Block Representation, 453<br/><br/>Block Corner Nodes, Sides, and Subdivisions, 454<br/><br/>12.3 Postprocessing 461<br/><br/>Deformed Configuration and Mode Shape, 461<br/><br/>Contour Plotting, 462<br/><br/>Nodal Values from Known Constant Element Values<br/><br/>for a Triangle, 463<br/><br/>Least-Squares Fit for a Four-Noded Quadrilateral, 465<br/><br/>12.4 Conclusion 466<br/><br/>Input Data/Output, 467<br/><br/>Problems 468<br/><br/>Program Listings, 470<br/><br/>APPENDIX 483<br/><br/>BIBLIOGRAPHY 486<br/><br/>ANSWERS TO SELECTED PROBLEMS 490<br/><br/>INDEX 492<br/><br/>Resources<br/>Show resources for |
