Wave Propagation for Train-induced Vibrations: A Finite Infinite Element Approach

Contents......Page 14
Foreword......Page 8
Preface......Page 10
1.1 Ground-Borne Vibrations......Page 20
1.2 Analytical Approaches......Page 22
1.2.1 Classical theory of wave propagation......Page 23
1.2.2 Elastic medium subjected to moving loads......Page 29
1.2.2.1 Elastic unbounded body subjected to a moving point load......Page 31
1.2.2.2 Elastic half-space subjected to a moving line load......Page 33
1.2.2.3 Elastic half-space subjected to a moving point load......Page 34
1.2.3 Beam on elastic half-space subjected to moving loads......Page 35
1.2.4 Tunnel structure subjected to moving loads......Page 37
1.2.5 Load generation mechanism......Page 38
1.3 Field Measurement......Page 39
1.4 Empirical Prediction Models......Page 43
1.5 Numerical Simulation......Page 44
1.5.1 Two-dimensional modeling......Page 47
1.5.2 2.5-dimensional modeling......Page 48
1.6.1 Trenches......Page 51
1.6.2 Wave impeding block......Page 52
1.6.3 Floating slab track......Page 53
1.7 Evaluation Criteria of Vibration......Page 54
1.8 Concluding Remarks......Page 60
2.1 Introduction......Page 64
2.2.1 Equation of motion......Page 66
2.2.2 Triple Fourier transform......Page 68
2.3 Solution for the Soil Response......Page 69
2.3.1 Boundary conditions......Page 70
2.3.2 Steady state response in time domain......Page 72
2.4.1 General loading function of a moving train......Page 73
2.4.2 Distribution function f(z) of the loading......Page 74
2.4.2.2 A uniformly distributed wheel load......Page 75
2.4.2.3 An elastically distributed wheel load......Page 76
2.4.2.4 A sequence of wheel loads......Page 77
2.4.3 Interaction forces between wheels and rails......Page 78
2.5 Numerical Studies and Discussions......Page 79
2.5.1 Verification of the present approach......Page 80
2.5.2 Single moving point load......Page 82
2.5.3 A uniformly distributed moving wheel load......Page 89
2.5.4 An elastically distributed moving wheel load......Page 93
2.5.5 A sequence of moving wheel loads......Page 109
2.6 Concluding Remarks......Page 110
3.1 Introduction......Page 114
3.2 Formulation of the Problem......Page 116
3.3.1 Shape functions......Page 119
3.3.2 Element matrices......Page 123
3.3.3 Damping property of materials......Page 125
3.3.4 Method of numerical integration......Page 126
3.3.5 Selection of amplitude decay factor a......Page 127
3.3.6 Selection of wave number k......Page 130
3.4 Mesh Range and Element Size......Page 131
3.5 Mesh Expansion by Dynamic Condensation......Page 135
3.6 Numerical Examples......Page 139
3.7 Concluding Remarks......Page 143
4.1 Introduction......Page 144
4.2 Dynamic Stiffness and Compliance of Foundation......Page 145
4.3 Vibration of a Massless Rigid Strip Foundation......Page 147
4.3.1 Effect of bedrock depth (H/B)......Page 149
4.3.2 Effect of shear modulus ratio (G1/G2) of soil layers......Page 154
4.3.3 Effect of Poisson’s ratio......Page 157
4.3.4 Effect of material damping ratio......Page 161
4.4 Vibration of Rails and Ground under Harmonic Loads......Page 165
4.5 Applications to Practical Problems......Page 169
4.5.1 Problem 1: Uniform half-space......Page 172
4.5.2 Problem 2: Soil deposit resting on bedrock......Page 174
4.6 Concluding Remarks......Page 182
5.1 Introduction......Page 184
5.2 Considerations in Parametric Studies......Page 187
5.3 Vibration Isolation by Elastic Foundation......Page 191
5.3.1 Young’s modulus ratio (Es /Ee)......Page 193
5.3.3 Poisson’s ratios ( ne, n)......Page 197
5.3.4 Material damping ratio ( b)......Page 199
5.3.5 Normalized dimensions (T, E)......Page 201
5.3.6 Bedrock depth H......Page 204
5.4 Vibration Isolation by Open Trenches......Page 206
5.4.1 Distance L between the railway and open trench......Page 209
5.4.2 Depth D and width W of open trench......Page 210
5.5 Vibration Isolation by In-Filled Trenches......Page 211
5.5.1 Distance L between the railway and in-filled trench......Page 215
5.5.3 Shear modulus ratio (Gsb/Gss)......Page 216
5.5.4 Mass density ratio ( rb/ )......Page 219
5.5.5 Poisson’s ratios ( nb, ns)......Page 220
5.5.6 Depth D and width W of in-filled trench......Page 222
5.6 Effect of Frequencies of Traffic Loads......Page 224
5.7 Concluding Remarks......Page 225
6.1 Introduction......Page 226
6.2 Problem Formulation and Basic Assumptions......Page 230
6.3 Scheme for Generating Finite/Infinite Element Mesh......Page 231
6.4 Parametric Studies for Open Trenches......Page 233
6.4.1 Normalized distance L from the structure......Page 235
6.4.2 Normalized depth D and width W of trench......Page 236
6.5.1 Normalized distance L from the structure......Page 237
6.5.2 Normalized depth D and width W of trench......Page 238
6.5.3 Impedance ratio of in-filled trench......Page 239
6.5.4 Poisson’s ratios ( nb, ns)......Page 241
6.6 Effect of Frequencies and Soil Conditions......Page 242
6.6.1 Soil with no bedrock......Page 244
6.6.2 Soil with bedrock......Page 246
6.7 Concluding Remarks......Page 248
7.1 Introduction......Page 250
7.2 Formulation of the Problem and Basic Assumptions......Page 253
7.3 Procedure of Derivation for Finite/Infinite Elements......Page 255
7.4 Wave Numbers for the Case with Moving Loads......Page 258
7.5 Shape Functions of Infinite Element......Page 263
7.6 Wave Propagation Properties for Different Vehicle Speeds......Page 264
7.7 Selection of Element Size and Mesh Range......Page 270
7.8 Selection of Wave Number k......Page 272
7.9 Selection of Amplitude Decay Factor a of Displacement......Page 273
7.10 Verification of the Present Approach......Page 274
7.10.1 Responses in frequency domain for moving loads at sub-, trans- and super-critical speeds......Page 275
7.10.2 Responses in frequency domain for moving loads with self oscillation......Page 277
7.10.3 Effectiveness and accuracy of condensation procedure......Page 278
7.10.4 Responses in time domain for sub-critical speed case......Page 281
7.11 Case Study......Page 285
7.12 Concluding Remarks......Page 290
8.1 Introduction......Page 296
8.2 Measurement of Vibration Attenuation for Soils......Page 298
8.3 Problem Description and Element Meshes......Page 299
8.4.2 Effect of Poisson’s ratio......Page 308
8.4.3 Effect of damping ratio with no self oscillation......Page 311
8.4.4 Effect of damping ratio for different oscillation frequencies......Page 314
8.5 Parametric Study for Single Soil Layer Overlying a Bedrock......Page 321
8.5.1 Effect of stratum depth for a quasi-static moving load......Page 322
8.5.2 Effect of stratum depth for a moving load with self oscillation......Page 328
8.5.3 Effect of self oscillation frequency......Page 334
8.5.4 Effect of load-moving speed......Page 338
8.6 Parametric Study for Multi Soil Layers......Page 342
8.6.1 Effect of soil layers for a quasi-static moving load......Page 343
8.6.2 Effect of soil layers for a moving load with self oscillation......Page 346
8.6.3 Effect of load-moving speed for multi-layered soils......Page 350
8.7 Concluding Remarks......Page 355
9.1 Introduction......Page 358
9.2 Major Considerations in Parametric Studies......Page 360
9.3.1 Moving loads with no self oscillation......Page 363
9.3.1.1 Effect of load-moving speed......Page 364
9.3.1.2 Effect of trench depth......Page 370
9.3.1.3 Effect of trench width......Page 372
9.3.2 Moving loads with self oscillation......Page 374
9.4.1.1 Effect of load-moving speed......Page 379
9.4.1.2 Effect of trench depth......Page 383
9.4.1.3 Effect of trench width......Page 387
9.4.1.4 Effect of shear wave speed of trenches......Page 388
9.4.2 Moving loads with self oscillation......Page 394
9.5.1.1 Effect of load-moving speed......Page 398
9.5.1.2 Effect of depth of WIB......Page 406
9.5.1.3 Effect of thickness of WIB......Page 407
9.5.1.4 Effect of shear wave speed of WIB......Page 409
9.5.2 Moving loads with self oscillation......Page 411
9.6 Comparison and Discussion......Page 417
9.7 Concluding Remarks......Page 423
10.1 Introduction......Page 426
10.2 Problem Formulation and Basic Assumptions......Page 428
10.3 Formulation of 2.5D Finite/Infinite Element Method......Page 431
10.4 Verification of the Present Approach......Page 433
10.5 Numerical Modeling and Related Considerations......Page 437
10.6.1 Effect of number of carriages......Page 443
10.6.2 Effect of load-moving speed......Page 445
10.6.3 Effect of bedrock depth H......Page 452
10.6.4 Effect of damping ratio......Page 454
10.6.5 Effect of tunnel lining thickness......Page 459
10.6.6 Effect of tunnel depth......Page 463
10.7 Concluding Remarks......Page 467
Appendix Steady-State Response in Finite Integrals by Eason (1965)......Page 470
Bibliography......Page 472
Author Index......Page 484
Subject Index......Page 488
Untitled......Page 194