Theory of matrix structural analysis

Cover......Page 1
Title Page......Page 4
Copyright Page......Page 5
Preface......Page 6
Contents......Page 8
1.1 Introduction......Page 14
1.2 Design Iterations......Page 16
1.3 Methods of Analysis......Page 20
1.4 Areas of Structural Analysis......Page 22
CHAPTER 2 BASIC EQUATIONS OF ELASTICITY......Page 23
5.11 Triangular Plates in Bending I......Page 24
2.2 Stress-Strain Equations......Page 25
2.3 Stress-Strain Equations for Initial Strains......Page 33
2.4 Equations of Equilibrium......Page 34
2.5 Compatibility Equations......Page 36
3.1 Introduction......Page 38
3.2 Work and Complementary Work; Strain Energy and Complementary Strain Energy......Page 40
3.3 Green's Identity......Page 45
3.4 Energy Theorems Based on the Principle of Virtual Work......Page 47
3.5 Energy Theorems Based on the Principle of Complementary Virtual Work......Page 51
3.6 Clapeyron's Theorem......Page 55
3.7 Betti's Theorem......Page 56
3.9 Summary of Energy Theorems and Definitions......Page 57
PROBLEMS......Page 59
4.1 Structural Idealization......Page 62
4.2 Energy Equivaience......Page 66
4.3 Structural Elements......Page 69
5.1 Methods of Determining Element Force-displacement Relationships......Page 74
5.2 Determination of Element Stiffness Properties by the Unit-displacement Theorem......Page 75
5.3 Application of Castigliano's Theorem (Part 1) to Derive Stiffness Properties......Page 79
5.4 Transformation of Coordinate Axes: A Matrices......Page 80
5.5 Pin jointed Bar Elements......Page 82
5.6 Beam Elements......Page 83
5.7 Triangular Plate Elements (]n-plane Forces)......Page 96
5.8 Rectangular Plate Elements (In-plane Forces)......Page 102
5.9 Quadrilateral Plate Elements (In-plane Forces)......Page 115
5.10 Tetrahedron Elements......Page 120
5.12 Rectangular Plates in Bending......Page 128
5.13 Method for Improving Stiffness Matrices......Page 135
PROBLEMS......Page 141
6.1 Matrix Formulation of the Displacement Analysis......Page 142
6.2 Elimination of the Rigid-body Degrees of Freedom: Choice of Reactions......Page 150
6.3 Derivation of the Transformation Matrix V from Equilibrium Equations......Page 152
6.4 Derivation of the Transformation Matrix T from Kinematics......Page 156
6.5 Condensation of Stiffness Matrices......Page 160
6.6 Derivation of Stiffness Matrices from Flexibility......Page 161
6.7 Stiffness Matrix for Constant-shear-flow Panels......Page 163
6.8 Stiffness Matrix for Linearly Varying Axial-force Members......Page 166
6.9 Analysis of a Pin jointed Truss by the Displacement Method......Page 168
6.10 Analysis of a Cantilever Beam by the Displacement Method......Page 172
6.11 Equivalent Concentrated Forces......Page 174
PROBLEMS......Page 176
7.1 Methods of Determining Element Displacement-force Relationships......Page 179
7.2 Inversion of the Force-displacement Equations: Flexibility Properties of P in jointed Bars and Beam Elements......Page 180
7.3 Determination of Element Flexibility Properties by the Unit-load Theorem......Page 184
7.4 Application of Castigliano's Theorem (Part II) to Derive Flexibility Properties......Page 186
7.6 Pin jointed Bar Elements......Page 187
7.7 Beam Elements......Page 188
7.8 Triangular Plate Elements (In-plane Forces)......Page 190
7.9 Rectangular Plate Elements (In-plane Forces)......Page 194
7.10 Tetrahedron Elements......Page 197
7.13 Rectangular Plates in Bending......Page 201
PROBLEMS......Page 205
8.1 Matrix Formulation of the Unit-load Theorem for External-force Systems......Page 206
8.2 Matrix Formulation of the Unit-load Theorem for Internal-force Systems: Self-equilibrating Force Systems......Page 210
8.3 Matrix Formulation of the Force Analysis: Jordanian Elimination Technique......Page 213
8.4 Matrix Force Analysis of a Pin jointed Truss......Page 219
8.5 Matrix Force Analysis of a Cantilever Beam......Page 232
8.6 Comparison of the Force and Displacement Methods......Page 239
PROBLEMS......Page 242
9.1 Substructure Analysis by the Matrix Displacement Method......Page 244
9.2 Substructure Displacement Analysis of a Two-Bay Truss......Page 254
9.3 Substructure Analysis by the Matrix Force Method......Page 259
9.4 Substructure Force Analysis of a Two-bay Truss......Page 270
PROBLEMS......Page 276
10.1 Formulation of the Dynamical Problems......Page 277
10.2 Principle of Virtual Work in Dynamics of Elastic Systems......Page 279
10.3 Hamilton's Principle......Page 280
10.5 Equations of Motion and Equilibrium......Page 282
10.6 Static and Dynamic Displacements in a Uniform Bar......Page 286
10.7 Equivalent Masses in Matrix Analysis......Page 291
10.8 Frequency-dependent Mass and Stiffness Matrices for Bar Elements......Page 294
10.9 Frequency-dependent Mass and Stiffness Matrices for Beam Elements......Page 297
PROBLEMS......Page 300
11.1 Equivalent Mass Matrices in Datum Coordinate System......Page 301
11.2 Equivalent Mass Matrix for an Assembled Structure......Page 303
11.3 Condensed Mass Matrix......Page 304
11.5 Uniform Beam......Page 305
11.6 Triangular Plate with Translational Displacements......Page 310
11.7 Rectangular Plate with Translational Displacements......Page 312
11.8 Solid Tetrahedron......Page 313
11.9 Solid Parallelepiped......Page 314
11.10 Triangular Plate with Bending Displacements......Page 315
11.11 Rectangular Plate with Bending Displacements......Page 318
PROBLEMS......Page 322
12.1 Vibration Analysis Based on Stiffness......Page 323
12.2 Properties of the Eigemnodes: Orthogonality Relations......Page 328
12.3 Vibration Analysis Based on Flexibility......Page 331
12.4 Vibration of Damped Structural Systems......Page 333
12.5 Critical Damping......Page 334
12.6 Longitudinal Vibrations of an Unconstrained Bar......Page 335
12.7 Longitudinal Vibrations of a Constrained Bar......Page 340
12.8 Transverse Vibrations of a Fuselage-Wing Combination......Page 341
12.9 Determination of Vibration Frequencies from the Quadratic Matrix Equation......Page 349
PROBLEMS......Page 352
CHAPTER 13 DYNAMIC RESPONSE OF ELASTIC SYSTEMS......Page 354
13.1 Response of a Single-degree-of-freedom System: Duhamel's Integrals......Page 355
13.2 Dynamic Response of an Unconstrained (Free) Structure......Page 358
13.3 Response Resulting from Impulsive Forces......Page 362
13.5 Steady-state Harmonic Motion......Page 363
13.7 Dynamic Response to Forced Displacements: Response to Earthquakes......Page 364
13.8 Determination of Frequencies and Modes of Unconstrained (Free) Structures Using Experimental Data for the Constrained Structures......Page 370
13.9 Dynamic Response of Structural Systems with Damping......Page 372
13.10 Damping Matrix Proportional to Mass......Page 374
13.11 Damping Matrix Proportional to Stiffness......Page 376
13.12 Matrix C Proportional to Critical Damping......Page 377
13.13 Orthonormalization of the Modal Matrix p......Page 379
13.14 Dynamic Response of an Elastic Rocket Subjected to Pulse Loading......Page 380
13.15 Response Due to Forced Displacement at One End of a Uniform Bar......Page 384
PROBLEMS......Page 386
CHAPTER 14 STRUCTURAL SYNTHESIS......Page 388
14.1 Mathematical Formulation of the Optimization Problem......Page 389
14.2 Structural Optimization......Page 393
CHAPTER 15 NONLINEAR STRUCTURAL ANALYSIS......Page 396
15.1 Matrix Displacement Analysis for Large Deflections......Page 397
15.2 Geometrical Stillness for Bar Elements......Page 399
15.3 Geometrical Stiffness for Beam Elements......Page 401
15.4 Matrix Force Analysis for Large Deflections......Page 405
15.5 Inelastic Analysis and Creep......Page 408
15.6 Stability Analysis of a Simple Truss......Page 409
15.7 Stability Analysis of a Column......Page 413
15.8 Influence of a Constant Axial Force on Transverse Vibrations of Beams......Page 416
PROBLEMS......Page 419
APPENDIX A MATRIX ALGEBRA......Page 422
APPENDIX B BIBLIOGRAPHY......Page 458
INDEX......Page 478