كتاب Vibration of Solids and Structures under Moving Loads

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Vibration of Solids and Structures under Moving Loads
LADISLAV FRYBA
Institute of Theoretical and Applied Mechanics,
Academy of Sciences of the Czech Republic,
Prague, Czech Republic

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Contents
Preface XIV
Symbols XIX
I Introduction
II One-dimensional solids
1 Simply supported beam subjected to a moving constant force 13
1.1 Formulation of the problem 13
1.2 Solution of the problem 16
1.3 Special cases 20
1.3.1 Static case 20
1.3.2 Case with no damping 20
1.3.3 Light damping 22
1.3.4 Critical damping 23
1.3.5 Supercritical damping 23
1.4 Application of the theory 24
1.4.1 The effect of speed 24
1.4.2 Application to bridges 25
1.4.3 Approximate solution of the effects of a moving mass 26
1.4.4 Experimental results 28
1.5 Additional bibliography 32
2 Moving harmonic force 33
2.1 Formulation and solution of the problem 33
2.2 The dynamic coefficient 35
2.3 Application of the theory 37
2.3.1 Comparison of theory with experiments 38
2.3.2 Critical speed and maximum dynamic coefficient for
bridges of various spans 41
2.4 Additional bibliography 43
VCONTENTS
3 Moving continuous load 44
3.1 Steady-state vibration 44
3.2 Arrival of a continuous load on a beam 48
3.3 Departure of a continuous load from a beam 50
3.4 Application of the theory 51
3.4.1 Approximate calculation of large-span railway bridges 52
3.4.2 Comparison of theory with experiments 53
3.4.3 The dynamic coefficient 55
3.4.4 Pipelines carrying moving liquid 56
3.5 Additional bibliography 56
4 Moving force arbitrarily varying in time 57
4.1 Force linearly increasing in time 58
4.2 Force exponentially varying in time 59
4.3 Moving impulses 60
4.4 Application of the theory 62
4.5 Additional bibliography 65
5 Beam stresses 66
5.1 Calculation by expansion in series 66
5.2 The integro-differential equation 69
5.2.1 Particular solution of the non-homogeneous integrodifferential equation 71
5.2.2 General solution of the homogeneous integro-differential
equation 75
5.2.3 Numerical example 77
5.3 Combined method 78
5.4 Application of the theory 81
5.5 Additional bibliography 82
6 Beams with various boundary conditions subjected
to a moving load 83
6.1 Generalized method of finite integral transformations 83
6.2 Motion of a force generally varying in time, along a beam 86
6.3 Motion of a force along a cantilever beam 91
6.4 Application of the theory 93
6.5 Additional bibliography 93
7 Massless beam subjected to a moving load 94
7.1 Formulation of the problem 94
7.2 Exact solution 95
7.3 Approximate solutions 98
7.3.1 The perturbation method 98
7.3.2 Method of successive approximations 100
7.3.3 Expansion in power series 101
VICONTENTS
7.4 Application of the theory 102
7.5 Additional bibliography 102
8 Beam subjected toa moving system with two degrees of freedom 103
8.1 Formulation of the problem 104
8.2 Solution of the problem 108
8.2.1 Dimensionless parameters 108
8.2.2 Reduction of the equations to the dimensionless form 112
8.2.3 Numerical solution 114
8.3 The effect of various parameters 115
8.3.1 The effect of speed 115
8.3.2 The effect of the frequency parameter of unsprung mass 117
8.3.3 The effect of the frequency parameter of sprung mass 118
8.3.4 The effect of variable stiffness of the elastic layer 118
8.3.5 The effect of the ratio between the weights of vehicle and
beam 119
8.3.6 The effect of the ratio between the weights of unsprung
and sprung parts of vehicle 119
8.3.7 The effect of beam damping 121
8.3.8 The effect of vehicle spring damping 121
8.3.9 The effect of initial conditions 121
8.3.10 The effect of other parameters 123
8.4 Application of the theory 125
8.4.1 Comparison of theory with experiments 125
8.4.2 Dynamic stresses in large-span railway bridges 128
8.5 Additional bibliography 128
9 Beam subjected to a moving two-axle system 129
9.1 Formulation of the problem 129
9.2 Solution of the problem 134
9.2.1 Dimensionless parameters 134
9.2.2 Reduction of the equations to the dimensionless form 137
9.2.3 Reduction of the equations to a form suitable for numerical calculations 140
9.2.4 Numerical solution 142
9.3 The effect of various parameters 145
9.3.1 The effect of speed 145
9.3.2 The effect of the frequency parameter of unsprung mass 148
9.3.3 The effect of the frequency parameter of sprung mass 148
9.3.4 The effect of the ratio between the weights of vehicle and
beam 150
9.3.5 The effect of the ratio between the weights of unsprung
parts and whole vehicle 150
9.3.6 The effect of the depth of irregularities 150
9.3.7 The effect of the length of irregularities 151
VIICONTENTS
9.3,8 The effect of other parameters 152
9.4 Application of the theory 153
9.4.1 Comparison of theory with experiments 153
9.4.2 Dynamic stresses in short-span railway bridges 156
9.5 Additional bibliography 156
10 Beam subjected to a moving multi-axle system 157
10.1 Formulation of the problem 157
10.2 Solution of the problem 160
10.2.1 Dimensionless parameters 160
10.2.2 Reduction of the equations to the dimensionless form 162
10.2.3 Numerical solution 163
10.3 The effect of various parameters 165
10.3.1 The effect of speed 165
10.3.2 The effect of other parameters 169
10.4 Application of the theory 169
10.4.1 Comparison of theory with experiments 169
10.4.2 Dynamic stresses in medium-span railway bridges at high
speeds 172
10.5 Additional bibliography 172
11 Systems of prismatic bars subjected ta a moving load 173
11.1 Frame systems 175
11.1.1 Free vibration 176
11.1.2 Forced vibration 180
11.2 Continuous beams 183
11.2.1 Free vibration 184
11.2.2 Forced vibration 185
11.3 Trussed bridges 188
11.4 Application of the theory 189
11.4.1 Two-span continuous trussed bridge 189
11.4.2 Six-span continuous trussed bridge 190
11.5 Additional bibliography 191
12 Non-uniform beams and curved bars subjected to a moving
load 192
12.1 Straight non-uniform bar 192
12.1.1 The network method 192
12.1.2 Galerkin’s method 195
12.2 Curved bars 197
12.2.1 Normal-mode analysis 198
12.2.2 Circular arch of constant cross section 199
12.3 Application of the theory 205
12.4 Additional bibliography 205
VIIICONTENTS
13 Infinite beam on elastic foundation 206
13.1 Formulation of the problem 206
13.2 Poles of the function of a complex variable 209
13.2.1 Static case 211
13.2.2 Case with no damping 211
13.2.3 Light damping 211
13.2.4 Critical damping 212
13.2.5 Supercritical damping 213
13.3 Solution of various cases 213
13.3.1 Static case 216
13.3.2 Case with no damping 218
13.3.3 Light damping 219
13.3.4 Critical damping 219
13.3.5 Supercritical damping 222
13.4 Deflection, bending moment and shear force at the point of load
action 224
13.4.1 Static case 224
13.4.2 Case with no damping 224
13.4.3 Light damping 225
13.4.4 Critical damping 227
13.5 Application of the theory 228
13.5.1 The effect of moving mass 228
13.5.2 The effect of speed 229
13.5.3 The effect of foundation 230
13.6 Additional bibliography 230
14 String subjected to a moving load 231
14.1 String with a moving force 231
14.2 Motion of a mass along a massless string 233
14.3 String with ends suspended at unequal heights 239
14.3.1 Static deflection of string produced by dead weight and
by a concentrated force 241
14.3.2 Motion of a force along a string 241
14.4 Application of the theory 242
14.5 Additional bibliography 243
III Two-dimensional solids
15 Plates subjected to a moving load 247
15.1 Simply supported rectangular plate 249
15.1.1 Force variable in time moving parallel to x-axis 250
15.1.2 Motion of a constant force 251
15.1.3 Motion of a force along a straight line 251
15.2 Rectangular plate simply supported on opposite edges 253
15.2.1 Force variable in time moving parallel to x-axis under
arbitrary initial conditions 254
15.2.2 Motion of a constant force 256
15.3 Application of the theory 257
15.4 Additional bibliography 259
16 Infinite plate on elastic foundation 260
16.1 Steady state vibration 261
16.2 Solution in polar coordinates 262
16.2.1 Static solution 263
16.2.2 Approximate solution for subcritical speed 265
16.3 Application of the theory 266
16.4 Additional bibliography 266
IV Three-dimensional solids
17 Elastic space with a moving force 269
17.1 Quasi-stationary motion of a force in elastic space 270
17.1.1 Subsonic speed 273
17.1.2 Transonic speed 276
17.1.3 Supersonic speed 279
17.2 Stresses in elastic space 281
17.3 Application of the theory 284
17.4 Additional bibliography 284
18 Force moving on elastic half-space 285
18.1 Motion of a concentrated force on elastic half-space 286
18.1.1 Subsonic speed 288
18.1.2 Transonic speed 291
18.1.3 Supersonic speed 292
18.2 Motion of a line load on elastic half-space 293
18.2.1 Subsonic speed 295
18.2.2 Transonic speed 297
18.2.3 Supersonic speed 301
18.3 Motion of a force on elastic half-plane 303
18.4 Application of the theory 303
18.5 Additional bibliography 305
V Special problems
19 Load motion at variable speed 309
19.1 Motion of a concentrated force 310
19.2 Arrival of a continuous load 314
19.3 The effect of inertial mass of a load moving at variable speed 316
19.3.1 Galerkin’s method 318
19.3.2 The perturbation method 320
19.3.3 Series expansion 322
19.4 Application of the theory 323
19.5 Additional bibliography 324
20 Beams subjected to an axial force and a moving load 325
20.1 Beams subjected to a static axial force under a moving load 328
20.1.1 Moving concentrated force 328
20.1.2 Arrival of a continuous load on a beam 329
20.1.3 The effect of mass of a moving continuous load 330
20.2 Suspended beams 333
20.3 Application of the theory 337
20.4 Additional bibliography 338
21 Longitudinal vibration of bars subjected to a moving load 339
21.1 Moving load 341
21.2 Bending and longitudinal vibrations of bars 341
21.3 Application of the theory 342
21.4 Additional bibliography 342
22 Thin-walled beams subjected to a moving load 343
22.1 Beam section with vertical axis of symmetry 345
22.1.1 Vertical constant force 347
22.1.2 Vertical harmonic force 348
22.1.3 Horizontal force and torsion moment 349
22.2 Beam section with two axes of symmetry 352
22.2.1 Horizontal force 353
22.2.2 Torsion moment 353
22.3 Application of the theory 354
22.3.1 The effect of moving mass of a vehicle 355
22.3.2 The torsional effect of counterweights 355
22.3.3 Lateral impacts 356
22.4 Additional bibliography 356
23 The effect of shear and rotatory inertia 357
23.1 Simply supported beam 361
XICONTENTS
23.1.1 Timoshenko beam 362
23.1.2 Shear beam 364
23.1.3 Rayleigh beam 365
23.2 Infinite beam on elastic foundation 366
23.2.1 Timoshenko beam 369
23.2.2 Shear beam 380
23.2.3 Rayleigh beam 380
23.2.*4 Bernoulli-Euier beam 381
23.3 Application of the theory 382
23.4 Additional bibliography 382
24 Finite beams subjected to a force moving at high speed 383
24.1 Finite beams 384
24.2 Simply supported beam 385
24.3 Application of the theory 387
24.4 Additional bibliography 387
25 Non-elastic properties of materials 388
25.1 Viscoelastic beam subjected to a moving load 388
25.1.1 Perfectly elastic Hooke solid 391
25.1.2 Kelvin solid 391
25.1.3 Maxwell solid 394
25.1.4 Standard linear solid 395
25.2 Rigid-plastic beam subjected to a moving load 397
25.2.1 Perfectly rigid beam with plastic hinge 398
25.2.2 Moving force 405
25.2.3 Massless beam 408
25.3 Application of the theory 408
25.4 Additional bibliography 409
26 Moving random loads 410
26.1 General theory 411
26.1.1 Correlation analysis 412
26.1.2 Spectral density analysis 413
26.2 Moving random force 414
26.3 Moving random continuous load 418
26.4 Infinite beam on random elastic foundation traversed by a random force 421
26.5, Application of the theory 426
26.6 Additional bibliography 427
XIICONTENTS
VI Appendix and bibliography
27 Tables of integral transformations 431
27.1 Laplace-Carson integral transformation 432
27.2 Fourier sine finite integral transformation 446
27.3 Fourier cosine finite integral transformation 447
27.4 Fourier complex integral transformation 448
27.5 Fourier sine integral transformation 449
27.6 Fourier cosine integral transformation 450
27.7 Hankel integral transformation 451
Bibliography 452
Supplementary bibliography 467
Author index 478
Subject index

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