Preface ........................................................ xi
Acknowledgments .............................................. xiii
1 Introduction ................................................. 1
1.1 Structural analysis and design .......................... 1
1.2 Structural idealisation ................................. 2
1.3 Structural members and elements ......................... 2
1.4 Structural systems ...................................... 6
1.5 Types of loads .......................................... 8
1.6 Supports for structures ................................ 10
2 Statics of structures: Equilibrium and support reactions .... 13
2.1 Introduction ........................................... 13
2.2 Coordinate systems ..................................... 13
2.3 Force .................................................. 15
2.4 Moment of a force ...................................... 16
2.5 Resultant force and moment ............................. 19
2.6 Reactions .............................................. 25
2.7 Free-body diagram ...................................... 25
2.8 Equilibrium equations for planar structures ............ 28
2.9 External statical determinacy and stability ............ 29
2.9.1 Internally stable structures .................... 30
2.9.2 Internally unstable structures .................. 31
2.10 Determination of reactions ............................. 36
2.11 Equilibrium and reactions in three-dimensional
structures ............................................. 40
Problems .................................................... 43
3 Internal actions of beams and frames ........................ 55
3.1 Introduction ........................................... 55
3.2 Internal actions at a cross-section .................... 55
3.3 Sign convention of internal actions .................... 57
3.4 Determination of internal actions and statical
determinacy ............................................ 60
3.5 Axial force, shear force and bending moment diagrams ... 64
Problems ............................................... 75
4 Statically determinate trusses .............................. 83
4.1 Introduction ........................................... 83
4.2 Assumptions for truss analysis ......................... 84
4.3 Sign convention and notation ........................... 85
4.4 An introduction to the method of joints ................ 86
4.5 Method of joints in matrix form ........................ 92
4.6 Method of sections .................................... 100
4.7 Statical indeterminacy and stability of trusses ....... 105
4.8 Deformation of trusses ................................ 111
4.9 Trusses with loaded members ........................... 115
4.10 Space trusses ......................................... 118
Problems ................................................... 127
5 Euler-Bernoulli beam model ................................. 135
5.1 Introduction .......................................... 135
5.2 Equilibrium of a small length of beam ................. 135
5.3 Kinematic (or strain-displacement) equations .......... 137
5.3.1 Axial deformations and displacements ........... 137
5.3.2 Bending (flexural) deformations and
displacements .................................. 139
5.3.3 Combining axial and flexural deformations ...... 141
5.4 Constitutive equations ................................ 141
5.5 Method of double integration .......................... 149
5.6 Governing differential equations (as a function of
displacements) ........................................ 152
5.6.1 Boundary conditions for the axial
displacement ................................... 154
5.6.2 Boundary conditions for the vertical
displacement ................................... 154
5.7 Relationship between bending moment, shear force and
member loading ........................................ 163
Problems ................................................... 176
6 Slope-deflection methods ................................... 183
6.1 Introduction .......................................... 183
6.2 Method of double integration with step functions ...... 184
6.3 Moment-area method .................................... 186
6.4 Conjugate beam method ................................. 195
6.5 The slope-deflection equations ........................ 204
6.5.1 Sign convention for support moments and
rotations ...................................... 204
6.5.2 Rotation at support А, θА ...................... 205
6.5.3 Rotation at support В, θB ...................... 206
6.5.4 Fixed-end moments caused by applied loads ...... 206
6.5.5 Support settlement Δ ........................... 207
6.5.6 Slope-deflection equations ..................... 208
6.5.7 Frames without sidesway ........................ 213
6.5.8 Frames with sidesway ........................... 217
Problems ....................................... 222
7 Work-energy methods ........................................ 229
7.1 Strain energy ......................................... 229
7.1.1 Axially loaded members ......................... 230
7.1.2 Beams in bending ............................... 230
7.2 The work theorem ...................................... 233
7.3 Virtual work .......................................... 236
7.4 Virtual work applied to trusses ....................... 236
7.4.1 Principle of virtual forces .................... 236
7.4.2 Principle of virtual displacements ............. 240
7.4.3 Transfer coefficients .......................... 241
7.5 Virtual work applied to beams and frames .............. 242
7.5.1 Principle of virtual forces .................... 243
7.5.2 Principle of virtual displacements ............. 247
7.6 Castigliano's theorem ................................. 250
7.6.1 Application to trusses ......................... 251
7.6.2 Application to beams and frames ................ 255
Problems ................................................... 258
8 The force method ........................................... 263
8.1 Introduction .......................................... 263
8.2 The force method applied to trusses ................... 264
8.2.1 Determination of member forces in an n-fold
indeterminate truss ............................ 264
8.2.2 Determination of joint displacements ........... 276
8.3 The force method applied to beams and frames .......... 279
8.3.1 Determination of internal actions .............. 279
8.3.2 Flexibility coefficients and transfer
functions ...................................... 286
8.3.3 Deformations of statically indeterminate
beams and frames ............................... 291
Problems ....................................... 293
9 Moment distribution ........................................ 299
9.1 Introduction .......................................... 299
9.2 Basic concepts ........................................ 300
9.3 Continuous beams ...................................... 302
9.3.1 Basic approach ................................. 302
9.3.2 Modification for an end span with a pinned
support ........................................ 307
9.4 Frames without sidesway ............................... 313
9.5 Frames with sidesway .................................. 315
Problems .............................................. 326
10 Truss analysis using the stiffness method .................. 331
10.1 Overview of the stiffness method ...................... 331
10.2 Sign convention, notation, coordinate systems and
degrees of freedom .................................... 331
10.2.1 Sign convention and notation ................... 331
10.2.2 Local and global coordinate systems ............ 331
10.2.3 Degrees of freedom of the structure ............ 333
10.3 Derivation of the stiffness matrix in local
coordinates ........................................... 333
10.4 Transformation between local and global coordinate
systems ............................................... 338
10.4.1 Transformation matrix for vectors .............. 338
10.4.2 Transformation matrix for the truss element .... 342
10.5 Truss element in global coordinates ................... 345
10.6 Assembling ............................................ 347
10.7 Solution procedure .................................... 351
10.8 Calculation of internal actions ....................... 352
10.9 Nodal coordinates ..................................... 356
10.10 Space truss .......................................... 362
Problems ................................................... 365
11 Beam analysis using the stiffness method ................... 369
11.1 The beam element ...................................... 369
11.2 Derivation of the stiffness matrix .................... 371
11.3 Beam element in global coordinates .................... 374
11.4 Assembling of the stiffness elements .................. 375
11.5 Member loads .......................................... 375
11.6 Solution procedure and post-processing ................ 378
Problems ................................................... 392
12 Frame analysis using the stiffness method .................. 397
12.1 The frame element ..................................... 397
12.2 Derivation of the element stiffness matrix ............ 397
12.3 Transformation between local and global coordinate
systems ............................................... 400
12.3.1 Transformation matrix for vectors .............. 400
12.3.2 Transformation matrix for the frame element .... 401
12.4 Frame element in global coordinates ................... 403
12.5 Member loads .......................................... 403
12.6 Assembling, solution and post-processing .............. 405
Problems ................................................... 420
13 Introduction to the finite element method .................. 425
13.1 Introduction .......................................... 425
13.2 Euler-Bernoulli beam model ............................ 425
13.2.1 Kinematic model ................................ 426
13.2.2 Weak form ...................................... 428
13.2.3 Finite element formulation ..................... 430
13.2.4 Solution procedure ............................. 436
13.2.5 Post-processing ................................ 437
13.2.6 Remarks on the consistency requirements for
finite elements ................................ 437
13.3 Timoshenko beam model ................................. 445
13.3.1 Kinematic model ................................ 445
13.3.2 Finite element formulation ..................... 447
Problems ....................................... 457
14 Introduction to the structural stability of columns ........ 459
14.1 Introduction .......................................... 459
14.2 Assumptions ........................................... 459
14.3 Critical load from equilibrium ........................ 462
14.4 Critical load from potential energy ................... 465
14.5 Buckling of an elastic column ......................... 469
14.6 Effective buckling length ............................. 479
14.7 Buckling stresses ..................................... 480
14.8 Imperfections in columns .............................. 485
Problems ................................................... 487
15 Introduction to nonlinear analysis ......................... 489
15.1 Introduction .......................................... 489
15.2 Nonlinear material properties ......................... 489
15.3 Illustrative examples ................................. 492
15.3.1 Axially loaded members ......................... 492
15.3.2 Beams in bending ............................... 494
15.4 Nonlinear analysis using the Newton-Raphson method .... 502
15.4.1 Overview of the Newton-Raphson method .......... 502
15.4.2 Cross-sectional analysis using the Newton-
Rapbson method ................................. 504
15.5 Finite element analysis using the Newton-Rapbson
method ................................................ 516
Problems ................................................... 527
Appendix A: Properties of plane sections ...................... 529
Appendix B: Fixed- end moments ................................ 543
Appendix C: Matrix algebra .................................... 545
Index ......................................................... 557
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