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Frontiers of Structural and Civil Engineering

ISSN 2095-2430

ISSN 2095-2449(Online)

CN 10-1023/X

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Front. Struct. Civ. Eng.    2016, Vol. 10 Issue (1) : 81-92    https://doi.org/10.1007/s11709-015-0306-x
RESEARCH ARTICLE
Vehicle-bridge coupled vibrations in different types of cable stayed bridges
Lingbo WANG1,*(),Peiwen JIANG2,Zhentao HUI3,Yinping MA1,Kai LIU4,Xin KANG5
1. Key Laboratory for Bridge and Tunnel of Shaanxi Province, Chang’an University, Xi’an 710064, China
2. Basic Construction Project Quality Supervision Station, Shaanxi Provincial Transport Department, Xi’an 710075, China
3. Yulin TianYuan Lu Ye Limited Company, 25 Shangjun Road, Yuyang District, Yulin 719000, China
4. School of Transportation Engineering, Hefei University of Technology, Hefei 230009, China
5. Department of Civil, Architectural, and Environmental Engineering, Missouri University of Science and Technology, Rolla, MO 65409, USA
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Abstract

Numerical analyses of the coupled vibrations of vehicle-bridge system and the effects of different types of cable stayed bridges on the coupled vibration responses have been presented in this paper using ANSYS. The bridge model and vehicle model were independently built which have no internal relationship in the ANSYS. The vehicle-bridge coupled vibration relationship was obtained by using the APDL program which subsequently imposed on the vehicle and bridge models during the numerical analysis. The proposed model was validated through a field measurements and literature data. The judging method, possibility, and criterion of the vehicle-bridge resonance (coupled vibrations) of cable stayed bridges (both the floating system and half floating system) under traffic flows were presented. The results indicated that the interval time between vehicles is the main influence factor on the resonance excitation frequency under the condition of equally spaced traffic flows. Compared to other types of cable stayed bridges, the floating bridge system has relatively high possibility to cause vehicle-bridge resonance.
Keywords vehicle-bridge coupled vibration      cable stayed bridge      resonances of vehicle-bridge system     
Corresponding Author(s): Lingbo WANG   
Online First Date: 19 November 2015    Issue Date: 19 January 2016
 Cite this article:   
Lingbo WANG,Peiwen JIANG,Zhentao HUI, et al. Vehicle-bridge coupled vibrations in different types of cable stayed bridges[J]. Front. Struct. Civ. Eng., 2016, 10(1): 81-92.
 URL:  
https://academic.hep.com.cn/fsce/EN/10.1007/s11709-015-0306-x
https://academic.hep.com.cn/fsce/EN/Y2016/V10/I1/81
bridge parameters vehicle parameters
span /m linear density /(kg·m−1) bending stiffness /(N·m2) suspension system mass (m1/kg) vehicle mass (m2/kg) spring stiffness (k/N·m−1) spring damping (c/kg·s−1)
16 9.36x103 2.05x1010 4.69x104 1.69x104 4.87x106 3.14x105
Tab.1  Model parameters for single axis vehicle-bridge system
Fig.1  (a) A single axis vehicle driving through the simply supported bridge Model in ANSYS; (b) vehicle and bridge element numbers in ANSYS
vehicle reaching moment FEA analysis vehicle leaving moment
1 Before vehicle load acting on the bridge, initial state U0(i), V0(i) and A0(i) were given or set static as its initial statevehicle model: free vibration under initial state;bridge model: static substep interval Δt, total time Δt,transient analysis Calculating results of bridge node: U1(i) = 0, V1(i) = 0, A1(i) = 0, i = 1−101;Calculating results of vehicle node: U1(i), numerical differentiation was used to obtain: V1(i) and A1(i)V1(i) = (U1(i)- U0(i))/ΔtA1(i) = (V1(i)- V0(i))/Δti = 201~202
2 Assume that there is no displacement at the No. 2 node for its short distance from the support.Interaction force was applied on bridge node No. 2: F = k ( U 1 ( 2 ) - U 1 ( 202 ) ) + c ( V 1 ( 2 ) - V 1 ( 202 ) ) + m 1 A 1 ( 2 ) + m 1 g substep interval Δt, total time 2Δt,transient analysis U2(i) were obtainedsimilarly:V2(i) = (U2(i)- U1(i))/ΔtA2(i) = (V2(i)- V1(i))/Δti = 1~101,201,202
3 Vehicle model: vertical displacement U2 (3) was applied on bottom node of vehicle model based on consistency in deformation between vehicle wheel and bridgeInteraction force was applied on bridge node No. 2: F = k ( U 2 ( 3 ) - U 2 ( 202 ) ) + c ( V 2 ( 3 ) - V 2 ( 202 ) ) + m 1 A 2 ( 3 ) + m 1 g substep interval Δt, total time 3Δt,transient analysis U3(i) were obtainedsimilarly:V3(i) = (U3(i)- U2(i))/ΔtA3(i) = (V3(i)- V2(i))/Δti = 1~101,201,202
j Similarly, vertical displacement Uj-1(j) was applied on bottom node of the vehicle modelInteraction force was applied on bridge node No. i: F = k ( U j − 1 ( j ) - U j − 1 ( 202 ) ) + c ( V j − 1 ( j ) - V j − 1 ( 202 ) ) + m 1 A j − 1 ( j ) + m 1 g substep interval Δt,total time i*Δt,transient analysis Ui(i) were obtainedsimilarly:Vj(i) = (Uj(i)- Uj-1(i))/ΔtAj(i) = (Vj(i)- Vj-1(i))/Δti = 1~101,201,202
Tab.2  Recurrence method of vehicle-bridge coupled vibration
Fig.2  Vehicle model with 3 axis and 5 degree freedom (Note: mc is mass of the vehicle, Ic is Moment at z axis, mi、kui、cui、kdi and cdi (i = 1,2,3)were the mass of the wheel, stiffness and damping ratio, respectively, b2 and b1 are the distance between centroid of the vehicle to the front and back axial, b4and b3 are the distance of the middle axial to the front and back axial, c1 is the distance between driver to the front axial)
Fig.3  (a) Equivalent displacement transformation; (b) equivalent load transformation
Fig.4  Flow chart of the response of the coupled vibration of the vehicle-bridge system
Fig.5  Comparisons between the calculated results (a) and published results (b)
floating system half floating system (fixed support) half floating system (none fixed support) rigid system
first mode frequency 0.191 0.603 0.191 0.617
second mode frequency 0.604 0.673 0.605 0.704
third mode frequency 0.825 0.825 0.938 1.269
Tab.3  Frequency comparisons of different types of cable stayed bridges
Fig.6  The finite element model of the bridge
Fig.7  Comparison of vehicle position at different time periods
Fig.8  First mode shape of floating system (basal frequency fb = 0.191)
Fig.9  First mode shape of half floating system (single horizontal restraint, basal frequency fb = 0.603)
Fig.10  First mode shape of half floating system (none horizontal restraint, basal frequency fb = 0.191)
Fig.11  First mode shape of rigid system (basal frequency fb = 0.617)
Fig.12  First mode shape of vehicle (basal frequency fv = 0.198)
conditions vehicle arrangements structure system vehicle velocity/(km/h) vehicle basal frequency/Hz interval time of vehicle/s frequency of traffic flow/Hz bridge frequency /Hz judging results of the presented method
condition 1 single vehicle floating system 60,120 0.198 / / 0.191 none resonance
condition 2 single vehicle half floating system(horizontal restraint) 60,120 0.198 / / 0.603 none resonance
condition 3 single vehicle half floating system(none horizontal restraint) 60,120 0.198 / / 0.191 none resonance
condition 4 single vehicle rigid system 60,120 0.198 / / 0.617 none resonance
condition 5 10 vechileequally spaced floating system 60,120 0.198 5s 0.2 0.191 resonance
condition 6 10 vechileequally spaced half floating system(horizontal restraint) 60,120 0.198 5s 0.2 0.603 none resonance
condition 7 10 vechileequally spaced half floating system(none horizontal restraint) 60,120 0.198 5s 0.2 0.191 resonance
condition 8 10 vechileequally spaced rigid system 60,120 0.198 5s 0.2 0.617 none resonance
condition 9 10 vechileequally spaced floating system 120 0.198 2.5s,10s 0.4,0.1 0.191 none resonance/resonance weakened
condition 10 10 vechilerandom spaced floating system 60 0.198 Random 2-7s / 0.191 none resonance
Tab.4  Calculation for conditions of vehicle bridge resonance
Fig.13  Time displacement response of condition 1
Fig.14  Time displacement response of condition 2
Fig.15  Time displacement response of condition 3
Fig.16  Time displacement response of condition 4
Fig.17  Time displacement response of condition 5
Fig.18  Time displacement response of condition 6
Fig.19  Time displacement response of condition 7
Fig.20  Time displacement response of condition 8
Fig.21  Time displacement response of condition 9
Fig.22  Time displacement response of condition 10
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