Introduction to Finit Elements in Engineering
Chandrupatla, T R
Introduction to Finit Elements in Engineering - 3rd - Delhi PHI Learning 2002 - 453p.
PREFACE XIII
ABOUT THE AUTHOR XVI
1 FUNDAMENTAL CONCEPTS 1
1.1 Introduction 1
1.2 Historical Background 1
1.3 Outline of Presentation 2
1.4 Stresses and Equilibrium 2
1.5 Boundary Conditions 4
1.6 Strain—Displacement Relations 5
1.7 Stress—Strain Relations 6
Special Cases, 7
1.8 Temperature Effects 8
1.9 Potential Energy and Equilibrium: The Rayleigh—Ritz Method 9
Potential Energy ß , 9
Rayleigh—Ritz Method, 12
1.10 Galerkin’s Method 14
1.11 Saint Venant’s Principle 18
1.12 Von Mises Stress 19
1.13 Principle of Superposition 19
1.14 Computer Programs 20
1.15 Conclusion 20
Historical References 20
Problems 21
2 MATRIX ALGEBRA AND GAUSSIAN ELIMINATION 28
2.1 Matrix Algebra 28
Row and Column Vectors, 29
Addition and Subtraction, 29
Multiplication by a Scalar, 29
Matrix Multiplication, 29
Transposition, 30
Differentiation and Integration, 30
Square Matrix, 31
Diagonal Matrix, 31
Identity Matrix, 31
Symmetric Matrix, 32
Upper Triangular Matrix, 32
Determinant of a Matrix, 32
Matrix Inversion, 32
Eigenvalues and Eigenvectors, 33
Positive Definite Matrix, 35
Cholesky Decomposition, 35
2.2 Gaussian Elimination 35
General Algorithm for Gaussian Elimination, 37
Symmetric Matrix, 40
Symmetric Banded Matrices, 40
Solution with Multiple Right Sides, 40
Gaussian Elimination with Column Reduction, 42
Skyline Solution, 44
Frontal Solution, 45
2.3 Conjugate Gradient Method for Equation Solving 45
Conjugate Gradient Algorithm, 46
Input Data/Output 46
Problems 47
Program Listings, 49
3 ONE-DIMENSIONAL PROBLEMS 51
3.1 Introduction 51
3.2 Finite Element Modeling 52
Element Division, 52
Numbering Scheme, 53
3.3 Shape Functions and Local Coordinates 55
3.4 The Potential-Energy Approach 59
Element Stiffness Matrix, 60
Force Terms, 62
3.5 The Galerkin Approach 64
Element Stiffness, 64
Force Terms, 65
3.6 Assembly of the Global Stiffness Matrix and Load Vector 66
3.7 Properties of K 69
3.8 The Finite Element Equations: Treatment
of Boundary Conditions 70
Types of Boundary Conditions, 70
Elimination Approach, 71
Penalty Approach, 76
Multipoint Constraints, 82
3.9 Quadratic Shape Functions 85
3.10 Temperature Effects 92
3.11 Problem Modeling and Boundary Conditions 96
Problem in Equilibrium, 96
Symmetry, 97
Two Elements with Same End Displacements, 97
Problem with a Closing Gap, 98
Input Data/Output, 98
Problems 99
Program Listing, 111
4 TRUSSES 117
4.1 Introduction 117
4.2 Plane Trusses 118
Local and Global Coordinate Systems, 118
Formulas for Calculating / and m, 119
Element Stiffness Matrix, 120
Stress Calculations, 121
Temperature Effects, 126
4.3 Three-Dimensional Trusses 129
4.4 Assembly of Global Stiffness Matrix for the Banded and Skyline
Solutions 131
Assembly for Banded Solution, 131
Skyline Assembly , 132
4.5 Problem Modeling and Boundary Conditions 134
Inclined Support in Two Dimensions, 134
Inclined Support in Three Dimensions–Line Constraint, 134
Inclined Support in Three Dimensions–Plane Constraint, 135
Symmetry and Antisymmetry , 136
Input Data/Output, 138
Problems 139
Program Listing, 147
5 BEAMS AND FRAMES 150
5.1 Introduction 150
Potential-Energy Approach, 151
Galerkin Approach, 152
5.2 Finite Element Formulation 153
Element Stiffness–Direct Approach, 157
5.3 Load Vector 158
5.4 Boundary Considerations 159
5.5 Shear Force and Bending Moment 160
5.6 Beams on Elastic Supports 162
5.7 Plane Frames 163
5.8 Three-Dimensional Frames 169
5.9 Problem Modeling and Boundary Conditions 173
5.10 Some Comments 174
Input Data/Output, 174
Problems 176
Program Listings, 183
6 TWO-DIMENSIONAL PROBLEMS USING CONSTANT STRAIN TRIANGLES 188
6.1 Introduction 188
6.2 Finite Element Modeling 189
6.3 Constant Strain Triangle (CST) 191
Isoparametric Representation, 192
Potential-Energy Approach, 198
Element Stiffness, 198
Force Terms, 199
Integration Formula on a Triangle, 206
Galerkin Approach, 206
Stress Calculations, 208
Temperature Effects, 210
6.4 Problem Modeling and Boundary Conditions 212
Some General Comments on Dividing into Elements, 215
6.5 Patch Test and Convergence 215
Patch Test, 215
6.6 Orthotropic Materials 216
Temperature Effects, 220
Input Data/Output, 222
Problems 225
Program Listing, 238
7 AXISYMMETRIC SOLIDS SUBJECTED TO AXISYMMETRIC LOADING 242
7.1 Introduction 242
7.2 Axisymmetric Formulation 243
7.3 Finite Element Modeling: Triangular Element 245
Potential-Energy Approach, 248
Body Force Term, 249
Rotating Flywheel, 249
Surface Traction, 250
Galerkin Approach, 252
Stress Calculations, 255
Temperature Effects, 256
7.4 Problem Modeling and Boundary Conditions 256
Cylinder Subjected to Internal Pressure, 256
Infinite Cylinder, 257
Press Fit on a Rigid Shaft, 257
Press Fit on an Elastic Shaft, 258
Belleville Spring, 259
Thermal Stress Problem, 260
Input Data/Output, 262
Problems 263
Program Listing, 271
8 TWO-DIMENSIONAL ISOPARAMETRIC ELEMENTS
AND NUMERICAL INTEGRATION 273
8.1 Introduction 273
8.2 The Four-Node Quadrilateral 273
Shape Functions, 273
Element Stiffness Matrix, 276
Element Force Vectors, 279
8.3 Numerical Integration 279
Two-Dimensional Integrals, 283
Stiffness Integration, 283
Stress Calculations, 284
8.4 Higher Order Elements 286
Nine-Node Quadrilateral, 287
Eight-Node Quadrilateral, 289
Six-Node Triangle, 290
Integration on a Triangle–Symmetric Points, 291
Integration on a Triangle–Degenerate Quadrilateral, 292
8.5 Four-Node Quadrilateral for Axisymmetric Problems 294
8.6 Conjugate Gradient Implementation of the Quadrilateral Element 295
8.7 Concluding Remarks and Convergence 295
8.8 References for Convergence 297
Input Data/Output, 298
Problems 300
Program Listings, 308
9 THREE-DIMENSIONAL PROBLEMS IN STRESS ANALYSIS 312
9.1 Introduction 312
9.2 Finite Element Formulation 313
Element Stiffness, 316
Force Terms, 317
9.3 Stress Calculations 317
9.4 Mesh Preparation 318
9.5 Hexahedral Elements and Higher Order Elements 322
9.6 Problem Modeling 324
9.7 Frontal Method for Finite Element Matrices 326
Connectivity and Prefront Routine, 327
Element Assembly and Consideration of Specified dof, 328
Elimination of Completed dof, 328
Backsubstitution, 329
Consideration of Multipoint Constraints, 329
Input Data/Output, 330
Problems 332
Program Listings, 336
10 SCALAR FIELD PROBLEMS 345
10.1 Introduction 345
10.2 Steady State Heat Transfer 346
One-Dimensional Heat Conduction, 347
One-Dimensional Heat Transfer in Thin Fins, 355
Two-Dimensional Steady-State Heat Conduction, 359
Two-Dimensional Fins, 369
Preprocessing for Program Heat2D, 370
10.3 Torsion 370
Triangular Element, 372
Galerkin Approach, 373
10.4 Potential Flow, Seepage, Electric and Magnetic Fields,
and Fluid Flow in Ducts 376
Potential Flow, 376
Seepage, 378
Electrical and Magnetic Field Problems, 379
Fluid Flow in Ducts, 381
Acoustics, 383
Boundary Conditions, 384
One-Dimensional Acoustics, 384
One-Dimensional Axial Vibrations, 386
Two-Dimensional Acoustics, 388
10.5 Conclusion 389
Input Data/Output, 389
Problems 391
Program Listings, 402
11 DYNAMIC CONSIDERATIONS 408
11.1 Introduction 408
11.2 Formulation 408
Solid Body with Distributed Mass, 409
11.3 Element Mass Matrices 411
11.4 Evaluation of Eigenvalues and Eigenvectors 416
Properties of Eigenvectors, 417
Eigenvalue—Eigenvector Evaluation, 417
Inverse Iteration Method , 420
Generalized Jacobi Method, 423
Tridiagonalization and Implicit Shift Approach, 427
Bringing Generalized Problem to Standard Form, 427
Tridiagonalization, 428
Implicit Symmetric QR Step with Wilkinson Shift
for Diagonalization, 431
11.5 Interfacing with Previous Finite Element Programs and a Program
for Determining Critical Speeds of Shafts 432
11.6 Guyan Reduction 433
11.7 Rigid Body Modes 436
11.8 Conclusion 438
Input Data/Output, 438
Problems 440
Program Listings, 446
12 PREPROCESSING AND POSTPROCESSING 453
12.1 Introduction 453
12.2 Mesh Generation 453
Region and Block Representation, 453
Block Corner Nodes, Sides, and Subdivisions, 454
12.3 Postprocessing 461
Deformed Configuration and Mode Shape, 461
Contour Plotting, 462
Nodal Values from Known Constant Element Values
for a Triangle, 463
Least-Squares Fit for a Four-Noded Quadrilateral, 465
12.4 Conclusion 466
Input Data/Output, 467
Problems 468
Program Listings, 470
APPENDIX 483
BIBLIOGRAPHY 486
ANSWERS TO SELECTED PROBLEMS 490
INDEX 492
Resources
Show resources for
9788120321069
620.00151825 CHA-I
Introduction to Finit Elements in Engineering - 3rd - Delhi PHI Learning 2002 - 453p.
PREFACE XIII
ABOUT THE AUTHOR XVI
1 FUNDAMENTAL CONCEPTS 1
1.1 Introduction 1
1.2 Historical Background 1
1.3 Outline of Presentation 2
1.4 Stresses and Equilibrium 2
1.5 Boundary Conditions 4
1.6 Strain—Displacement Relations 5
1.7 Stress—Strain Relations 6
Special Cases, 7
1.8 Temperature Effects 8
1.9 Potential Energy and Equilibrium: The Rayleigh—Ritz Method 9
Potential Energy ß , 9
Rayleigh—Ritz Method, 12
1.10 Galerkin’s Method 14
1.11 Saint Venant’s Principle 18
1.12 Von Mises Stress 19
1.13 Principle of Superposition 19
1.14 Computer Programs 20
1.15 Conclusion 20
Historical References 20
Problems 21
2 MATRIX ALGEBRA AND GAUSSIAN ELIMINATION 28
2.1 Matrix Algebra 28
Row and Column Vectors, 29
Addition and Subtraction, 29
Multiplication by a Scalar, 29
Matrix Multiplication, 29
Transposition, 30
Differentiation and Integration, 30
Square Matrix, 31
Diagonal Matrix, 31
Identity Matrix, 31
Symmetric Matrix, 32
Upper Triangular Matrix, 32
Determinant of a Matrix, 32
Matrix Inversion, 32
Eigenvalues and Eigenvectors, 33
Positive Definite Matrix, 35
Cholesky Decomposition, 35
2.2 Gaussian Elimination 35
General Algorithm for Gaussian Elimination, 37
Symmetric Matrix, 40
Symmetric Banded Matrices, 40
Solution with Multiple Right Sides, 40
Gaussian Elimination with Column Reduction, 42
Skyline Solution, 44
Frontal Solution, 45
2.3 Conjugate Gradient Method for Equation Solving 45
Conjugate Gradient Algorithm, 46
Input Data/Output 46
Problems 47
Program Listings, 49
3 ONE-DIMENSIONAL PROBLEMS 51
3.1 Introduction 51
3.2 Finite Element Modeling 52
Element Division, 52
Numbering Scheme, 53
3.3 Shape Functions and Local Coordinates 55
3.4 The Potential-Energy Approach 59
Element Stiffness Matrix, 60
Force Terms, 62
3.5 The Galerkin Approach 64
Element Stiffness, 64
Force Terms, 65
3.6 Assembly of the Global Stiffness Matrix and Load Vector 66
3.7 Properties of K 69
3.8 The Finite Element Equations: Treatment
of Boundary Conditions 70
Types of Boundary Conditions, 70
Elimination Approach, 71
Penalty Approach, 76
Multipoint Constraints, 82
3.9 Quadratic Shape Functions 85
3.10 Temperature Effects 92
3.11 Problem Modeling and Boundary Conditions 96
Problem in Equilibrium, 96
Symmetry, 97
Two Elements with Same End Displacements, 97
Problem with a Closing Gap, 98
Input Data/Output, 98
Problems 99
Program Listing, 111
4 TRUSSES 117
4.1 Introduction 117
4.2 Plane Trusses 118
Local and Global Coordinate Systems, 118
Formulas for Calculating / and m, 119
Element Stiffness Matrix, 120
Stress Calculations, 121
Temperature Effects, 126
4.3 Three-Dimensional Trusses 129
4.4 Assembly of Global Stiffness Matrix for the Banded and Skyline
Solutions 131
Assembly for Banded Solution, 131
Skyline Assembly , 132
4.5 Problem Modeling and Boundary Conditions 134
Inclined Support in Two Dimensions, 134
Inclined Support in Three Dimensions–Line Constraint, 134
Inclined Support in Three Dimensions–Plane Constraint, 135
Symmetry and Antisymmetry , 136
Input Data/Output, 138
Problems 139
Program Listing, 147
5 BEAMS AND FRAMES 150
5.1 Introduction 150
Potential-Energy Approach, 151
Galerkin Approach, 152
5.2 Finite Element Formulation 153
Element Stiffness–Direct Approach, 157
5.3 Load Vector 158
5.4 Boundary Considerations 159
5.5 Shear Force and Bending Moment 160
5.6 Beams on Elastic Supports 162
5.7 Plane Frames 163
5.8 Three-Dimensional Frames 169
5.9 Problem Modeling and Boundary Conditions 173
5.10 Some Comments 174
Input Data/Output, 174
Problems 176
Program Listings, 183
6 TWO-DIMENSIONAL PROBLEMS USING CONSTANT STRAIN TRIANGLES 188
6.1 Introduction 188
6.2 Finite Element Modeling 189
6.3 Constant Strain Triangle (CST) 191
Isoparametric Representation, 192
Potential-Energy Approach, 198
Element Stiffness, 198
Force Terms, 199
Integration Formula on a Triangle, 206
Galerkin Approach, 206
Stress Calculations, 208
Temperature Effects, 210
6.4 Problem Modeling and Boundary Conditions 212
Some General Comments on Dividing into Elements, 215
6.5 Patch Test and Convergence 215
Patch Test, 215
6.6 Orthotropic Materials 216
Temperature Effects, 220
Input Data/Output, 222
Problems 225
Program Listing, 238
7 AXISYMMETRIC SOLIDS SUBJECTED TO AXISYMMETRIC LOADING 242
7.1 Introduction 242
7.2 Axisymmetric Formulation 243
7.3 Finite Element Modeling: Triangular Element 245
Potential-Energy Approach, 248
Body Force Term, 249
Rotating Flywheel, 249
Surface Traction, 250
Galerkin Approach, 252
Stress Calculations, 255
Temperature Effects, 256
7.4 Problem Modeling and Boundary Conditions 256
Cylinder Subjected to Internal Pressure, 256
Infinite Cylinder, 257
Press Fit on a Rigid Shaft, 257
Press Fit on an Elastic Shaft, 258
Belleville Spring, 259
Thermal Stress Problem, 260
Input Data/Output, 262
Problems 263
Program Listing, 271
8 TWO-DIMENSIONAL ISOPARAMETRIC ELEMENTS
AND NUMERICAL INTEGRATION 273
8.1 Introduction 273
8.2 The Four-Node Quadrilateral 273
Shape Functions, 273
Element Stiffness Matrix, 276
Element Force Vectors, 279
8.3 Numerical Integration 279
Two-Dimensional Integrals, 283
Stiffness Integration, 283
Stress Calculations, 284
8.4 Higher Order Elements 286
Nine-Node Quadrilateral, 287
Eight-Node Quadrilateral, 289
Six-Node Triangle, 290
Integration on a Triangle–Symmetric Points, 291
Integration on a Triangle–Degenerate Quadrilateral, 292
8.5 Four-Node Quadrilateral for Axisymmetric Problems 294
8.6 Conjugate Gradient Implementation of the Quadrilateral Element 295
8.7 Concluding Remarks and Convergence 295
8.8 References for Convergence 297
Input Data/Output, 298
Problems 300
Program Listings, 308
9 THREE-DIMENSIONAL PROBLEMS IN STRESS ANALYSIS 312
9.1 Introduction 312
9.2 Finite Element Formulation 313
Element Stiffness, 316
Force Terms, 317
9.3 Stress Calculations 317
9.4 Mesh Preparation 318
9.5 Hexahedral Elements and Higher Order Elements 322
9.6 Problem Modeling 324
9.7 Frontal Method for Finite Element Matrices 326
Connectivity and Prefront Routine, 327
Element Assembly and Consideration of Specified dof, 328
Elimination of Completed dof, 328
Backsubstitution, 329
Consideration of Multipoint Constraints, 329
Input Data/Output, 330
Problems 332
Program Listings, 336
10 SCALAR FIELD PROBLEMS 345
10.1 Introduction 345
10.2 Steady State Heat Transfer 346
One-Dimensional Heat Conduction, 347
One-Dimensional Heat Transfer in Thin Fins, 355
Two-Dimensional Steady-State Heat Conduction, 359
Two-Dimensional Fins, 369
Preprocessing for Program Heat2D, 370
10.3 Torsion 370
Triangular Element, 372
Galerkin Approach, 373
10.4 Potential Flow, Seepage, Electric and Magnetic Fields,
and Fluid Flow in Ducts 376
Potential Flow, 376
Seepage, 378
Electrical and Magnetic Field Problems, 379
Fluid Flow in Ducts, 381
Acoustics, 383
Boundary Conditions, 384
One-Dimensional Acoustics, 384
One-Dimensional Axial Vibrations, 386
Two-Dimensional Acoustics, 388
10.5 Conclusion 389
Input Data/Output, 389
Problems 391
Program Listings, 402
11 DYNAMIC CONSIDERATIONS 408
11.1 Introduction 408
11.2 Formulation 408
Solid Body with Distributed Mass, 409
11.3 Element Mass Matrices 411
11.4 Evaluation of Eigenvalues and Eigenvectors 416
Properties of Eigenvectors, 417
Eigenvalue—Eigenvector Evaluation, 417
Inverse Iteration Method , 420
Generalized Jacobi Method, 423
Tridiagonalization and Implicit Shift Approach, 427
Bringing Generalized Problem to Standard Form, 427
Tridiagonalization, 428
Implicit Symmetric QR Step with Wilkinson Shift
for Diagonalization, 431
11.5 Interfacing with Previous Finite Element Programs and a Program
for Determining Critical Speeds of Shafts 432
11.6 Guyan Reduction 433
11.7 Rigid Body Modes 436
11.8 Conclusion 438
Input Data/Output, 438
Problems 440
Program Listings, 446
12 PREPROCESSING AND POSTPROCESSING 453
12.1 Introduction 453
12.2 Mesh Generation 453
Region and Block Representation, 453
Block Corner Nodes, Sides, and Subdivisions, 454
12.3 Postprocessing 461
Deformed Configuration and Mode Shape, 461
Contour Plotting, 462
Nodal Values from Known Constant Element Values
for a Triangle, 463
Least-Squares Fit for a Four-Noded Quadrilateral, 465
12.4 Conclusion 466
Input Data/Output, 467
Problems 468
Program Listings, 470
APPENDIX 483
BIBLIOGRAPHY 486
ANSWERS TO SELECTED PROBLEMS 490
INDEX 492
Resources
Show resources for
9788120321069
620.00151825 CHA-I