Abstract

The dynamic interaction between a bridge and a moving train has been widely studied. However, there is a significant gap in our understanding of how the presence of isolation bearings influences the dynamic response, especially when an earthquake occurs. Here, we formulate a coupled model of a train-bridge-bearing system to examine the bearings’ dynamic effects on the system responses. In the analysis, the train is modeled as a moving oscillator, the bridge is a one span simply supported beam and one isolation bearing is installed under each support of the bridge. A mathematical model using fractional derivatives is used to capture the viscoelastic properties of the bearings. The vertical response is the focus of this investigation. Dynamic substructuring is used in modeling to efficiently capture the coupled dynamics of the entire system. Illustrative numerical simulations are carried out to examine the effects of the bearings. The results demonstrate that although the presence of bearings typically decreases the bridge seismic responses, there is a potential to increase the bridge response induced by the moving train.

References

References
1.
Stokes
,
S. G. G.
,
1849
, “
Discussion of a Differential Equation Relating to the Breaking of Railway Bridges
,”
Trans. Cambr. Phil. Soc.
,
8
(
5
), pp.
707
735
.
2.
Yang
,
Y.-B.
,
Yau
,
J.
,
Yao
,
Z.
, and
Wu
,
Y.
,
2004
,
Vehicle-Bridge Interaction Dynamics: With Applications to High-Speed Railways
,
World Scientific
,
River Edge, NJ
.
3.
Timoshenko
,
S.
,
1922
, “
Cv. on the Forced Vibrations of Bridges
,”
London, Edinburgh, Dublin Philos. Magazine J. Sci.
,
43
(
257
), pp.
1018
1019
. 10.1080/14786442208633953
4.
Jeffcott
,
H.
,
1929
, “
Vi. on the Vibration of Beams Under the Action of Moving Loads
,”
London, Edinburgh, Dublin Philos. Magazine J. Sci.
,
8
(
48
), pp.
66
97
. 10.1080/14786440708564857
5.
Prescott
,
J.
,
1934
, “
A Mathematical Treatise on Vibrations in Railway Bridges. by Ce Inglis. Pp. Xxv, 203, 21s. 1934.(cambridge)
,”
Math. Gazette
,
18
(
231
), pp.
329
330
. 10.2307/3605488
6.
Lowan
,
A. N.
,
1935
, “
Liv. on Transverse Oscillations of Beams Under the Action of Moving Variable Loads
,”
London, Edinburgh, Dublin Philos. Magazine J. Sci.
,
19
(
127
), pp.
708
715
. 10.1080/14786443508561407
7.
Bhatti
,
M.
,
Garg
,
V.
, and
Chu
,
K.
,
1985
, “
Dynamic Interaction Between Freight Train and Steel Bridge
,”
ASME J. Dyn. Syst. Meas. Control.
,
107
(
1
), pp.
60
66
. 10.1115/1.3140708
8.
Garg
,
V.
,
Chu
,
K.
, and
Wang
,
T.
,
1985
, “
A Study of Railway Bridge/Vehicle Interaction and Evaluation of Fatigue Life
,”
Earthquake Eng. Struct. Dynam.
,
13
(
6
), pp.
689
709
. 10.1002/eqe.4290130602
9.
Wang
,
T.-L.
,
Garg
,
V. K.
, and
Chu
,
K.-H.
,
1991
, “
Railway Bridge/Vehicle Interaction Studies With New Vehicle Model
,”
J. Struct. Eng.
,
117
(
7
), pp.
2099
2116
. 10.1061/(ASCE)0733-9445(1991)117:7(2099)
10.
Frỳba
,
L.
,
1996
,
Dynamics of Railway Bridges
,
Thomas Telford Publishing
,
Czech Republic
.
11.
Zhang
,
Q.-L.
,
Vrouwenvelder
,
A.
, and
Wardenier
,
J.
,
2001
, “
Numerical Simulation of Train-Bridge Interactive Dynamics
,”
Comput. Struct.
,
79
(
10
), pp.
1059
1075
. 10.1016/S0045-7949(00)00181-4
12.
Li
,
H.
,
Xia
,
H.
,
Soliman
,
M.
, and
Frangopol
,
D. M.
,
2015
, “
Bridge Stress Calculation Based on the Dynamic Response of Coupled Train-Bridge System
,”
Eng. Struct.
,
99
(
1
), pp.
334
345
. 10.1016/j.engstruct.2015.04.014
13.
Rocha
,
J. M.
,
Henriques
,
A. A.
, and
Calçada
,
R.
,
2016
, “
Probabilistic Assessment of the Train Running Safety on a Short-Span High-Speed Railway Bridge
,”
Struct. Infrastructure Eng.
,
12
(
1
), pp.
78
92
. 10.1080/15732479.2014.995106
14.
Truong
,
V. H.
,
Liu
,
J.
,
Meng
,
X.
,
Jiang
,
C.
, and
Nguyen
,
T. T.
,
2018
, “
Uncertainty Analysis on Vehicle-Bridge System With Correlative Interval Variables Based on Multidimensional Parallelepiped Model
,”
Int. J. Computat. Methods
,
15
(
05
), p.
1850030
. 10.1142/S0219876218500305
15.
Yang
,
Y.-B.
, and
Wu
,
Y.-S.
,
2001
, “
A Versatile Element for Analyzing Vehicle-Bridge Interaction Response
,”
Eng. Struct.
,
23
(
5
), pp.
452
469
. 10.1016/S0141-0296(00)00065-1
16.
Zhang
,
N.
, and
Xia
,
H.
,
2013
, “
Dynamic Analysis of Coupled Vehicle-Bridge System Based on Inter-System Iteration Method
,”
Comput. Struct.
,
114
(
1
), pp.
26
34
. 10.1016/j.compstruc.2012.10.007
17.
Yau
,
J.-D.
,
Wu
,
Y.-S.
, and
Yang
,
Y.-B.
,
2001
, “
Impact Response of Bridges With Elastic Bearings to Moving Loads
,”
J. Sound. Vib.
,
248
(
1
), pp.
9
30
. 10.1006/jsvi.2001.3688
18.
Song
,
M.-K.
,
Noh
,
H.-C.
, and
Choi
,
C.-K.
,
2003
, “
A New Three-Dimensional Finite Element Analysis Model of High-Speed Train–Bridge Interactions
,”
Eng. Struct.
,
25
(
13
), pp.
1611
1626
. 10.1016/S0141-0296(03)00133-0
19.
Zeng
,
Z.-P.
,
Zhao
,
Y.-G.
,
Xu
,
W.-T.
,
Yu
,
Z.-W.
,
Chen
,
L.-K.
, and
Lou
,
P.
,
2015
, “
Random Vibration Analysis of Train–Bridge Under Track Irregularities and Traveling Seismic Waves Using Train–Slab Track–Bridge Interaction Model
,”
J. Sound. Vib.
,
342
(
1
), pp.
22
43
. 10.1016/j.jsv.2015.01.004
20.
Alotta
,
G.
,
Di Paola
,
M.
, and
Pirrotta
,
A.
,
2014
, “
Fractional Tajimi–Kanai Model for Simulating Earthquake Ground Motion
,”
Bull. Earthquake Eng.
,
12
(
6
), pp.
2495
2506
. 10.1007/s10518-014-9615-z
21.
Li
,
H.
,
Gomez
,
D.
,
Dyke
,
S. J.
, and
Xu
,
Z.
,
2019
, “
Fractional Differential Equation Bearing Models for Base-Isolated Buildings: Framework Development
,”
J. Struct. Eng.
,
146
(
2
), p.
04019197
. 10.1061/(ASCE)ST.1943-541X.0002508
22.
Hwang
,
J.
, and
Hsu
,
T.
,
2001
, “
A Fractional Derivative Model to Include Effect of Ambient Temperature on HDR Bearings
,”
Eng. Struct.
,
23
(
5
), pp.
484
490
. 10.1016/S0141-0296(00)00063-8
23.
Yuan
,
Y.
,
Wei
,
W.
,
Tan
,
P.
,
Igarashi
,
A.
,
Zhu
,
H.
,
Iemura
,
H.
, and
Aoki
,
T.
,
2016
, “
A Rate-dependent Constitutive Model of High Damping Rubber Bearings: Modeling and Experimental Verification
,”
Earthquake Eng. Struct. Dynam.
,
45
(
11
), pp.
1875
1892
. 10.1002/eqe.2744
24.
Khajehsaeid
,
H.
,
2018
, “
Application of Fractional Time Derivatives in Modeling the Finite Deformation Viscoelastic Behavior of Carbon-Black Filled Nr and Sbr
,”
Polym. Test.
,
68
(
1
), pp.
110
115
. 10.1016/j.polymertesting.2018.04.004
25.
Hwang
,
J.
, and
Wang
,
J.
,
1998
, “
Seismic Response Prediction of HDR Bearings Using Fractional Derivative Maxwell Model
,”
Eng. Struct.
,
20
(
9
), pp.
849
856
. 10.1016/S0141-0296(98)80005-9
26.
Markou
,
A. A.
, and
Manolis
,
G. D.
,
2016
, “
A Fractional Derivative Zener Model for the Numerical Simulation of Base Isolated Structures
,”
Bull. Earthquake Eng.
,
14
(
1
), pp.
283
295
. 10.1007/s10518-015-9801-7
27.
Markou
,
A. A.
,
Oliveto
,
N. D.
, and
Athanasiou
,
A.
,
2017
, “Modeling of High Damping Rubber Bearings,”
Dynamic Response of Infrastructure to Environmentally Induced Loads
,
Springer
,
Cham, Switzerland
, pp.
115
138
.
28.
Cortés
,
F.
, and
Elejabarrieta
,
M.
,
2007
, “
Viscoelastic Materials Characterisation Using the Seismic Response
,”
Mater. Des.
,
28
(
7
), pp.
2054
2062
. 10.1016/j.matdes.2006.05.032
29.
Lewandowski
,
R.
, and
Baum
,
M.
,
2015
, “
Dynamic Characteristics of Multilayered Beams With Viscoelastic Layers Described by the Fractional Zener Model
,”
Arch. Appl. Mech.
,
85
(
12
), pp.
1793
1814
. 10.1007/s00419-015-1019-2
30.
Tanabe
,
M.
,
Yamada
,
Y.
, and
Hajime
,
W.
,
1987
, “
Modal Method for Interaction of Train and Bridge
,”
Comput. Struct.
,
27
(
1
), pp.
119
127
. 10.1016/0045-7949(87)90187-8
31.
Xia
,
H.
,
Zhang
,
N.
, and
Guo
,
W.
,
2017
,
Dynamic Interaction of Train-Bridge Systems in High-Speed Railways: Theory and Applications
,
Beijing Jiaotong University Press and Springer-Verlag GmbH, Germany
.
32.
Podworna
,
M.
, and
Klasztorny
,
M.
,
2014
, “
Vertical Vibrations of Composite Bridge/Track Structure/High-Speed Train Systems. Part 2: Physical and Mathematical Modelling
,”
Bull. Polish Acad. Sci.: Tech. Sci.
,
62
(
1
), pp.
181
196
. 10.2478/bpasts-2014-0019
33.
Ticona Melo
,
L. R.
,
Ribeiro
,
D.
,
Calçada
,
R.
, and
Bittencourt
,
T. N.
,
2019
, “
Validation of a Vertical Train-Track–Bridge Dynamic Interaction Model Based on Limited Experimental Data
,”
Struct. Infrastruct. Eng.
,
16
(
1
), pp.
108
201
.
34.
Liu
,
H.
,
Yu
,
Z.
, and
Guo
,
W.
,
2019
, “
A Fast Modeling Technique for the Vertical Train-Track-Bridge Interactions
,”
Shock Vib.
,
2019
(
1
), pp.
1
15
.
35.
Gomez
,
D.
,
Dyke
,
S. J.
, and
Rietdyk
,
S.
,
2018
, “
Experimental Verification of a Substructure-based Model to Describe Pedestrian–Bridge Interaction
,”
J. Bridge Eng.
,
23
(
4
), pp.
1
14
. 10.1061/(ASCE)BE.1943-5592.0001204
36.
Gomez
,
D.
,
Dyke
,
S. J.
, and
Rietdyk
,
S.
,
2020
, “
Structured Uncertainty for a Pedestrian-Structure Interaction Model
,”
J. Sound. Vib.
,
474
(
1
), pp.
115237
. 10.1016/j.jsv.2020.115237
37.
Xia
,
H.
,
Zhang
,
N.
, and
De Roeck
,
G.
,
2003
, “
Dynamic Analysis of High Speed Railway Bridge Under Articulated Trains
,”
Comput. Struct.
,
81
(
26–27
), pp.
2467
2478
. 10.1016/S0045-7949(03)00309-2
38.
Yau
,
J.
,
Martínez-Rodrigo
,
M. D.
, and
Doménech
,
A.
,
2019
, “
An Equivalent Additional Damping Approach to Assess Vehicle-Bridge Interaction for Train-Induced Vibration of Short-Span Railway Bridges
,”
Eng. Struct.
,
188
(
1
), pp.
469
479
. 10.1016/j.engstruct.2019.01.144
39.
Shabana
,
A. A.
,
Zaazaa
,
K. E.
, and
Sugiyama
,
H.
,
2007
,
Railroad Vehicle Dynamics: A Computational Approach
,
CRC Press
,
Boca Raton, FL
.
40.
Pritz
,
T.
,
1996
, “
Analysis of Four-Parameter Fractional Derivative Model of Real Solid Materials
,”
J. Sound. Vib.
,
195
(
1
), pp.
103
115
. 10.1006/jsvi.1996.0406
41.
Podlubny
,
I.
,
1998
,
Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications
,
Academic Press
,
San Diego, CA
.
42.
Monje
,
C. A.
,
Chen
,
Y.
,
Vinagre
,
B. M.
,
Xue
,
D.
, and
Feliu-Batlle
,
V.
,
2010
,
Fractional-order Systems and Controls: Fundamentals and Applications
,
Springer Science & Business Media
,
New York
.
43.
Irvine
,
T.
,
2012
, “
Bending Frequencies of Beams, Rods, and Pipes
,”
Compare
,
500
(
s50
), pp.
31
38
.
44.
MATLAB
,
2018
. version 9.4.0 (R2018a). The MathWorks Inc., Natick, Massachusetts.
45.
Museros
,
P.
,
Moliner
,
E.
, and
Martínez-Rodrigo
,
M. D.
,
2013
, “
Free Vibrations of Simply-Supported Beam Bridges Under Moving Loads: Maximum Resonance, Cancellation and Resonant Vertical Acceleration
,”
J. Sound. Vib.
,
332
(
2
), pp.
326
345
. 10.1016/j.jsv.2012.08.008
46.
Liu
,
K.
,
De Roeck
,
G.
, and
Lombaert
,
G.
,
2009
, “
The Effect of Dynamic Train–Bridge Interaction on the Bridge Response During a Train Passage
,”
J. Sound. Vib.
,
325
(
1–2
), pp.
240
251
. 10.1016/j.jsv.2009.03.021
47.
Doménech
,
A.
,
Museros
,
P.
, and
Martínez-Rodrigo
,
M.
,
2014
, “
Influence of the Vehicle Model on the Prediction of the Maximum Bending Response of Simply-Supported Bridges Under High-Speed Railway Traffic
,”
Eng. Struct.
,
72
(
1
), pp.
123
139
. 10.1016/j.engstruct.2014.04.037
48.
Cantero
,
D.
,
Arvidsson
,
T.
,
OBrien
,
E.
, and
Karoumi
,
R.
,
2016
, “
Train–Track–Bridge Modelling and Review of Parameters
,”
Struct. Infrastruct. Eng.
,
12
(
9
), pp.
1051
1064
. 10.1080/15732479.2015.1076854
49.
Cortés
,
F.
, and
Sarría
,
I.
,
2015
, “
Dynamic Analysis of Three-Layer Sandwich Beams With Thick Viscoelastic Damping Core for Finite Element Applications
,”
Shock Vib.
,
2015
(
1
), pp.
1
9
. 10.1155/2015/736256
50.
Li
,
X.
,
Zhang
,
Z.
, and
Zhang
,
X.
,
2016
, “
Using Elastic Bridge Bearings to Reduce Train-induced Ground Vibrations: An Experimental and Numerical Study
,”
Soil Dynam. Earthquake Eng.
,
85
, pp.
78
90
. 10.1016/j.soildyn.2016.03.013
51.
Zhao
,
C.
, and
Ping
,
W.
,
2018
, “
Effect of Elastic Rubber Mats on the Reduction of Vibration and Noise in High-Speed Elevated Railway Systems
,”
Proc. Inst. Mech. Eng., Part F: J. Rail Rapid Transit
,
232
(
6
), pp.
1837
1851
. 10.1177/0954409717752201
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