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 |  موضوع: كتاب Matrix and Finite Element Analyses of Structures السبت 29 يونيو 2024, 12:12 pm | |
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أخواني في الله
أحضرت لكم كتاب
Matrix and Finite Element Analyses of Structures
Madhujit Mukhopadhyay , Abdul Hamid Sheikh
و المحتوى كما يلي :
Contents
1 Basic Concepts of Structural Analysis . 1
1.1 Types of Structures 1
1.2 Objective of Structural Analysis . 1
1.3 Materials and Basic Assumptions 3
1.4 Loads . 3
1.5 General Methods of Analysis . 4
1.5.1 Equilibrium Conditions 4
1.5.2 Compatibility Conditions 5
1.6 Force–Displacement Relationship . 5
1.7 Statical Indeterminacy . 6
1.7.1 Plane Structure . 7
1.7.2 Space Structures 9
1.8 Kinematic Indeterminacy . 10
1.9 Two Approaches of Structural Analysis . 11
2 Energy Principles . 13
2.1 Introduction 13
2.2 Principle of Virtual Work . 13
2.3 Principle of Complementary Virtual Work 15
2.4 Principle of Minimum Potential Energy 15
2.5 Principle of Minimum Complementary Energy 16
2.6 Castigliano’s Theorems 16
2.7 Determination of Displacements . 18
References and Suggested Readings 19
3 Introduction to the Flexibility and Stiffness Matrix Methods 21
3.1 Introduction 21
3.2 The Flexibility Matrix Method 21
3.3 The Stiffness Method 33
3.4 Incorporation of Different Loading Conditions 50
3.5 Other Types of Loadings . 51
3.5.1 Treatment by the Flexibility Matrix Method 52
xixii Contents
3.5.2 Treatment by the Stiffness Method 54
3.6 Incorporation of Shear Deformation 58
3.7 Relation Between Flexibility and Stiffness Matrices 59
3.8 Equivalent Joint Loads . 61
3.9 Choice of the Method of Analysis . 62
References and Suggested Readings 73
4 Direct Stiffness Method 75
4.1 Introduction 75
4.2 Local and Global Coordinate System . 75
4.3 Transformation of Variables 76
4.3.1 Transformation of Member Coordinate Axes . 76
4.3.2 Transformation of Member Displacement
Matrix 78
4.3.3 Transformation of the Member Force Matrix . 79
4.3.4 Transformation of the Member Stiffness Matrix . 80
4.4 Transformation of the Stiffness Matrix of the Member
of a Truss 81
4.5 Transformation of the Stiffness Matrix of the Member
of a Rigid Frame 82
4.6 Transformation of the Stiffness Matrix of the Member
of a Grillage 85
4.7 Transformation of the Stiffness Matrix of the Member
of a Space Frame 87
4.8 Horizontally Circular Curved Beam Element 89
4.9 Overall Stiffness Matrix 91
4.10 Boundary Conditions 95
4.10.1 Boundary Conditions Corresponding to Skewed
Supports 95
4.11 Computation of Internal Forces 97
4.12 Computer Program for the Truss Analysis by the Direct
Stiffness Method 97
4.13 Computer Program for the Frame Analysis by Direct
Stiffness Method 103
4.14 Computer Program for the Grillage Analysis by the Direct
Stiffness Method 106
References and Suggested Readings 111
5 Substructure Technique for the Analysis of Structural Systems 113
5.1 Introduction 113
5.2 Basic Concepts . 114
5.3 Direct Stiffness Method Restated 115
5.4 The Substructure Technique 116
5.5 An Illustrative Example 119
5.6 Computer Program for the Truss Analysis
by the Substructure Technique 125Contents xiii
References and Suggested Readings 132
6 The Flexibility Matrix Method . 133
6.1 Introduction 133
6.2 Element Flexibility Matrix 133
6.3 Principle of Contragredience 134
6.4 The Equilibrium Matrix 136
6.5 Construction of the Flexibility Matrix of the Structure 138
6.6 Matrix Determination of the Displacement Vector 139
6.7 Determination of Member Forces 140
6.8 Procedure of the Analysis of Statically Indeterminate
Structures 141
6.9 Illustrated Examples . 141
6.10 Choice of the Released Structure 145
References and Suggested Readings 148
7 Elements of Elasticity 149
7.1 Introduction 149
7.2 Some Notations and Relations in the Theory of Elasticity . 149
7.2.1 Surface and Body Forces 149
7.2.2 Components of Stresses . 150
7.2.3 Components of Strain . 151
7.2.4 Stress–Strain Relationship . 151
7.3 Two-Dimensional Problems 152
7.3.1 Plane Stress 152
7.3.2 Plane Strain 153
7.3.3 Differential Equations of Equilibrium 154
7.4 Bending of Thin Plates . 156
7.4.1 Basic Assumptions . 156
7.4.2 Deformation of the Plate . 156
7.4.3 Strain–Displacement Relationship 157
7.4.4 Stress–Strain Relationship . 157
7.4.5 Equilibrium Equations 159
7.4.6 Differential Equation for Deflection . 160
7.4.7 Shearing Forces 161
7.5 Boundary Conditions 161
7.5.1 Simply Supported Edge . 161
7.5.2 Clamped Edge . 162
7.5.3 Free Edge . 162
7.5.4 Elastically Supported Edge . 163
7.5.5 Edge Having Elastic Rotational Restraint 164
7.6 Concluding Remarks 164
References and Suggested Readings 165xiv Contents
8 Introduction to the Finite Element Method 167
8.1 Introduction 167
8.2 The Finite Element Method . 167
8.3 Brief History of the Development of the Finite Element
Method 169
8.4 Basic Steps in the Finite Element Method for the Solution
of Static Problems . 170
8.5 Advantages and Disadvantages of the Finite Element
Method 173
References and Suggested Readings 174
9 Finite Element Analysis of Plane Elasticity Problems . 177
9.1 Introduction 177
9.2 Three-Noded Triangular Element 177
9.2.1 Displacement Function 178
9.2.2 Displacement Function Expressed in Terms
of Nodal Displacements . 179
9.2.3 Strain–Nodal Parameter Relationship 180
9.2.4 Stress–Strain Relationship . 181
9.2.5 Derivation of the Element Stiffness Matrix . 182
9.2.6 Determination of Element Stresses 183
9.3 Criteria for the Choice of the Displacement Function . 184
9.4 Polynomial Displacement Functions . 185
9.5 Verification of the Convergence Criteria
of the Displacement Function of 3-Noded Triangular
Element . 185
9.6 Number of Terms in a Polynomial . 186
9.7 Four-Noded Rectangular Element . 187
9.7.1 Displacement Function 188
9.7.2 Displacement Function in Terms of Nodal
Displacements . 189
9.7.3 Strain-Nodal Displacement Relationship . 190
9.7.4 Stress–Strain Relationship . 190
9.7.5 Derivation of the Element Stiffness Matrix . 191
9.7.6 Evaluation of Element Stresses . 191
9.8 A Note on the Rectangular Element 191
9.9 A Note on Element Stresses 192
9.10 Computer Program for the Plane Stress Analysis Using
Three–Noded Triangular Element . 192
Bibliography 196
10 Isoparametric and Other Element Representations
and Numerical Integrations 197
10.1 Introduction 197
10.2 Shape Function or Interpolation Function . 197
10.3 Determination of Shape Functions . 198Contents xv
10.3.1 Linear 2-D Element . 198
10.3.2 Quadratic 2-D Element 200
10.4 Plane Stress Isoparametric Linear Element 201
10.4.1 Displacement Function in Terms of Nodal
Parameters . 201
10.4.2 Strain-Nodal Parameter Relationship 201
10.4.3 Evaluation of [B] Matrix . 202
10.4.4 Element Stiffness Matrix 204
10.4.5 Convergence of Isoparametric Elements . 204
10.4.6 Concept of Isoparametric Element 206
10.5 Numerical Integration 206
10.5.1 Gaussian Quadrature Formula 207
10.5.2 Gaussian Integration of Two Variables . 207
10.6 Lagrangian Interpolation . 210
10.7 Natural Coordinates and Higher Order Triangular
Elements . 211
10.7.1 One-Dimensional Element . 211
10.7.2 Higher Order Triangular Elements 212
10.8 The Quadratic Triangle for the Plane Stress Problem . 214
10.9 Numerical Integration of Area Coordinates . 216
10.10 Triangular Isoparametric Elements for the Analysis
of Plane Stress Problems . 216
10.11 Allman’s Triangular Plane Stress Element 218
10.12 Computer Program for the Solution of Plane Stress
Problem Using Isoparametric Element . 222
References and Suggested Readings 234
11 Finite Element Analysis of Plate Bending Problems . 235
11.1 Introduction 235
11.2 Beam Element 235
11.2.1 Displacement Function 236
11.2.2 Displacement Function in Terms of Nodal
Displacements . 237
11.2.3 Strain (Curvature)–Nodal Parameter
Relationship . 238
11.2.4 Stress (Moment)–Strain (Curvature)
Relationship . 238
11.2.5 Derivation of the Element Stiffness Matrix . 239
11.2.6 Determination of Equivalent Loading
on the Beam . 240
11.3 Rectangular Plate Bending Element 241
11.3.1 Displacement Function 242
11.3.2 Displacement Function Expressed in Terms
of Nodal Displacements . 244
11.3.3 Strain–Nodal Parameter Relationship 245xvi Contents
11.3.4 Stress (Moment)–Strain (Curvature)
Relationship . 246
11.3.5 Derivation of the Element Stiffness Matrix . 247
11.4 Parallelogram Element of Plate Bending 249
11.4.1 Displacement Function 251
11.4.2 Displacement Function in Terms of Nodal
Displacements . 251
11.4.3 Curvature–Nodal Parameter Relationship 252
11.4.4 Moment–Curvature Relationship 253
11.4.5 Element Stiffness Matrix 254
11.5 Hermitian Polynomial Interpolation 255
11.6 A Conforming Plate Bending Element 256
11.7 Isoparametric Plate Bending Element 257
11.7.1 Displacement Function 257
11.7.2 Strain–Nodal Displacement Relationship . 259
11.7.3 Stress–Strain Relationship . 260
11.7.4 Element Stiffness Matrix 260
11.7.5 Reduced Integration Technique . 261
11.8 Smoothed Stresses 262
11.9 Triangular Plate Bending Elements 263
11.10 DKT Element . 264
11.10.1 Constraint Equations 265
11.10.2 Transformation Matrix 266
11.10.3 Element Stiffness Matrix 269
11.11 The Patch Test 271
11.11.1 The Patch Test for the Plane Stress Element 273
11.11.2 The Patch Test for Plate Bending Elements . 273
11.12 Horizontally Curved Isoparametric Beam . 276
11.12.1 Displacement Function in Terms of Nodal
Parameters . 276
11.12.2 Stress–Strain Relations 278
11.12.3 Strain–Displacement Relationship 279
11.12.4 Element Stiffness Matrix 280
11.13 Nonuniform Straight Beam Element . 280
11.14 Computer Program for Isoparametric Quadratic Bending
Element . 282
References and Suggested Readings 285
12 Finite Element Analysis of Shells . 287
12.1 Introduction 287
12.2 Flat Shell Element . 287
12.2.1 Transformation of the Stiffness Matrix
and Assembly 289
12.3 Shell of Revolution 292
12.4 General Shell Finite Element of Triangular Shape 295Contents xvii
12.4.1 Derivation of the Stiffness Matrix . 296
12.4.2 Consistent Load Vector 302
12.4.3 Condensation of Stiffness Matrix . 303
12.5 Isoparametric General Shell Element . 304
12.5.1 Geometry of the Shell Element . 305
12.5.2 Displacement Field . 306
12.5.3 Strains Inside the Element . 308
12.5.4 Stress–Strain Relationship . 310
12.5.5 Stiffness Matrix of the Shell Element 311
12.6 Vertically Curved Beam Element 312
12.7 Computer Program for the Finite Element Analysis
of Shallow Shells of General Shape Using Triangular
Element . 314
References and Suggested Readings 317
13 Semi-analytical and Spline Finite Strip Method of Analysis
of Plate Bending 319
13.1 Introduction 319
13.2 Beam Function . 320
13.3 Model of the Plate . 324
13.4 The Displacement Function . 325
13.5 Curvature-Nodal Parameter Relationship . 326
13.6 Moment—Curvature Relationship . 327
13.7 Strip Stiffness Matrix 327
13.8 Loading Matrix . 329
13.9 Force Displacement Relationship 330
13.10 Spline Finite Strip Method of Analysis of Plate Bending 331
13.10.1 The Spline Function 332
13.10.2 Displacement Functions . 333
13.10.3 Strain–Displacement Relationship 334
13.10.4 Stiffness Matrix 335
13.10.5 The Loading Matrix . 336
13.11 Computer Program for the Spline Finite Strip Method
of Analysis of Plates in Bending . 337
References and Suggested Readings 341
14 Dynamic and Instability Analyses by the Finite Element
Method 343
14.1 Introduction 343
14.2 Dynamic Analysis . 343
14.2.1 Torsional Vibration of Shafts . 343
14.2.2 An Example . 346
14.2.3 Flexural Vibration of Beams . 347
14.2.4 In-Plane Vibration of Plates 348
14.2.5 Flexural Vibration of Plates 349
14.3 Elastic Instability Analysis . 351xviii Contents
14.3.1 Column Instability Analysis 351
14.3.2 Plate Instability Analysis 356
References and Suggested Readings 360
15 The Finite Difference Method for the Analysis of Beams
and Plates 361
15.1 Introduction 361
15.2 Finite Difference Representation of Derivatives 362
15.2.1 First Derivative . 362
15.2.2 Second Derivative 362
15.2.3 Third Derivative 363
15.2.4 Fourth Derivative . 363
15.3 Errors in the Finite Difference Expressions 364
15.4 Equivalent Concentrated Load 365
15.5 Boundary Conditions for Beam Bending 366
15.5.1 Simple Support . 366
15.5.2 Fixed End . 366
15.5.3 Free End 367
15.6 A Statically Determinate Static Problem 367
15.7 A Statically Indeterminate Static Problem . 370
15.8 Free Vibration of Beams . 371
15.9 Buckling of Columns 373
15.10 Finite Difference Representation of the Plate Equation 375
15.10.1 A Plate Example . 377
References and Suggested Readings 380
16 Adaptive Finite Element Analysis . 381
16.1 Introduction 381
16.2 The Adaptive Finite Element Technique 381
16.3 Superconvergent Patch Recovery Technique . 382
16.4 Example of Verification of SPR . 385
16.5 Error Estimation 386
16.6 ZZ Error Estimator 389
16.7 ZZ—Refinement Framework . 391
16.8 Adaptive Mesh Generation . 392
16.8.1 Mesh Generation Based on Mapping 394
16.8.2 Delaunay Triangulation Method 394
16.8.3 Domain Decomposition Method (Quadtree) 394
16.8.4 Advancing Front Technique 395
References and Suggested Readings 403
17 Geometrical Nonlinear Finite Element Analysis 405
17.1 Introduction 405
17.2 Nonlinear Equation Solving Procedures 405
17.2.1 Direct Iteration Method 406
17.2.2 Newton–Raphson Method . 407Contents xix
17.2.3 Modified Newton–Raphson Method . 408
17.2.4 Incremental Techniques . 409
17.3 Formulation of the Geometric Nonlinear Problem 410
17.3.1 Equilibrium Equations 411
17.3.2 Incremental Equilibrium Equation 412
17.4 Large Deflection Analysis of Plates in b-notation 413
17.5 Large Deflection Analysis of Plates in n-notation 418
17.6 Example of a Pin-Jointed bar . 422
17.7 Computer Program for Geometrically Nonlinear Analysis
of Plates . 425
References and Suggested Reading . 429
18 Finite Element Method of Analysis of Stiffened Plates 431
18.1 Introduction 431
18.2 Modeling the Plate and the Stiffener . 431
18.3 Rectangular Stiffened Plate Bending Element . 432
18.3.1 Stiffness Matrix of the Stiffener Element . 433
18.4 Isoparametric Stiffened Plate Bending Element 436
18.4.1 Stiffness Matrix of Arbitrarily-Oriented
Eccentric Stiffener 436
References and Suggested Readings 441
19 Selected Topics 443
19.1 Rayleigh–Ritz Method . 443
19.2 An Example 444
19.3 Rayleigh–Ritz Finite Element Method 445
19.4 Weighted Residual Methods 447
19.5 Galerkin Method 449
19.5.1 An Example of Galerkin Method 450
19.6 Galerkin Finite Element Method . 453
19.7 Torsional Stiffness of Prismatic Beam Element 455
19.8 Torsion of Noncircular Sections . 458
19.9 Axi-symmetrical Element 461
19.10 Three-Dimensional Elements . 463
19.10.1 Linear Element (8 Nodes) (Fig. 19.8a) . 463
19.10.2 Quadratic Element (20 Nodes) (Fig. 19.8b) . 464
19.10.3 Cubic Element (32 Nodes) (Fig. 19.8c) 464
19.10.4 A16 Noded Solid (Fig. 19.9a) 465
19.10.5 A24 Noded Solid (Fig. 19.9b) 466
References and Suggested Readings 466
Appendix A: Fixed-End Forces . 467
Appendix B . 469
Index . 471
Index
A
Adaptive finite element, 381–389, 391, 392,
394–397, 399, 400, 402, 403
Adaptive mesh generation, 389, 392
Advancing front technique, 395, 397
Area coordinate, 264, 265
Assembly of elements, 76, 96
Axisymmetric element, 461
B
Background mesh, 395
Banded matrix, 113
Basic functions
Beam function, 320–329, 331
Body forces, 149, 154
Boundary conditions, 161–164, 320,
331–333, 361, 366, 369, 370
Buckling, 351–354, 359
C
Castigliano’s theorems, 16–19, 182
Characteristics functions, 320
Compatibility conditions, 5
Complementary energy, 16, 17
Concentric stiffener, 432
Consistent load, 302
Consistent mass matrix, 345, 347, 348
Constant strain triangle, 177
Continuum, 167, 168, 170–172, 174
Convergence criteria, 184, 186, 189, 198,
204, 206, 222
D
Damping matrix, 345
Delaunay triangulation, 394
Displacements, determination, 18
DKT element, 264
Domain decomposition, 394
Dynamic problems, 343
E
Eccentric stiffener, 432
Eigenvalues, 319, 346, 354
Equilibrium conditions, 4, 154, 159
Equilibrium matrix, 136
Error estimation, 386
Error in finite difference method, 364
F
Finite difference method, 362–373,
375–377
Finite element, definitions, 167–169
Finite strip, 319–331
Flat shell, 287, 295
Free vibration, 343, 345
Frontal solution, 113
G
Galerkin finite element, 453
Galerkin method, 449
Gauss quadrature, 207, 208, 222
Geometric stiffness matrix, 413, 417
H
h-adaptivity, 389, 413, 417
Half band width, 95, 106
Hermitian polynomial, 255, 256
Higher order element, 200, 211–213
Horizontally curved beam, 276, 277, 279,
280
Hybrid element, 169
I
Interpolation function, 197, 198
Isoparametric element, 197–199, 201–208,
210–223, 232
Isoparametric general shell element,
304–312
J
Jacobian, 202, 204, 206, 217, 307, 309
K
Kinematic indeterminacy, 10
L
Lack of fit, 52
Lagrangian interpolation, 210
Linear triangle, plane stress
Loads, 3, 4
Lumped mass matrix, 348, 351
M
Mesh, 167, 168, 174
N
Natural coordinates, 211
Newton-Cotes formula, 207
Non-linear problems, 405–418
Numerical integration, 204–206, 216, 217,
222
O
Orthogonality relationship, 324, 329, 445
Orthotropic plates, 159, 222, 246, 260
P
p-adaptivity, 389
Pascal triangle, 185
Patch test, 271
Plane strain, 153, 154, 177, 183
Plane stress, 152, 154, 177, 178, 180, 181,
183, 185, 186, 188, 190, 192
Plate bending, 156–161
Potential energy, 15, 16
Prestrain, 52
Principle of contragredience, 134, 136
Principle of virtual work, 13–15, 239, 327,
440
R
Rayleigh-Ritz finite element, 445
Rayleigh-Ritz method, 443
Rectangular element, plane stress, 187–191
Rectangular element, plate bending,
241–243, 245–247, 249
Reduced integration, 260–262
Rotational restraints, elastic, 323, 324
Rotation matrix, 78, 88, 303
S
Semianalytical methods, 319, 331
Settlement of supports, 51–54
Shape functions, 198, 200, 201, 205
Shear deformation, 59, 109, 257, 260
Shells, 287–314
Shells of revolution, 292–294
Skew element, 249
Skew support, 95
Smoothed stress, 261
Space frame, 9, 10
Spline finite strip, 331–336, 338
Statical indeterminacy, 6, 7, 9–11
Stiffened plate, 431–441
Stiffener element, 433–436
Structural analysis, objective, 1
Substructure techniques, 113–125, 127–130
Superconvergent patch recovery, 382
Surface force, 149
T
Three-dimensional element, 463
Transfinite interpolation, 394
Transformation matrix, 79, 81, 84–86, 88,
93, 96, 103, 106, 291, 293, 298, 299
Triangular element, bending, 263
Triangular element, plane stress, 218–221
V
Vertically curved beam, 312, 314
Vibration, 170, 173
Virtual work, 13–15Index 473
W
Weighted residual method, 447
Z
zz-error estimator, 389
zz-refinement, 391
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