Vehicle-bridge coupled vibrations in different types of cable stayed bridges

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

Introduction

The study of coupled vibration of vehicle-bridge system could be dated back to the late 1990s. Many researchers used Railway Bridge as a typical example and derived the bridge-vehicle coupling equations to study the coupled vibration, through which the inertial force was usually neglected [ 1, 2]. Compared with the in situ test data, though the analysis results were pretty close, slight differences were still existed. In China, similarly, the study of the coupled vibration also started from the railway bridge system, Shen et al. [ 3], for example, studied the forced vibrations on bridge which based on a real engineering case of a suspended cable bridge. At the beginning of this century, Wang and Xu [ 4] published a monograph on the interactions between railways and bridge system, in which the knowledge of the coupled vibrations of railway bridge system was introduced and could be traced back to the 1950s. The previous studies, however, were not applicable to vehicle bridge system, because the loading conditions on highway bridge are relatively small compared to a railway bridge. Due to the rapid development of highway transportation, the study of the coupled vibration in the vehicle-bridge system is becoming much more important. Zibdeh [ 5], Sheng et al. [ 3] and Wang and Xu [ 4] have established coupled differential equations for vehicle-bridge system and provided numerical solutions based on the computational software. Foda and Abduljabbar [ 6] and Jiang et al. [ 7] have successfully solved the differential equations for vehicle bridge system at transient dynamic state that based on the ANSYS and APDL. The analysis results were compared with the measured in situ data which indicated that his method was very practical [ 8, 9].

Compared with the railway loading condition, highway traffic loadings have large variability, for example, the number of traffics, the model of the vehicle, the size of the vehicle, the speed, and the acceleration speed that all influence the loading conditions on highway bridges and pose great changes to the study of the response of coupled vibrations of the vehicle-bridge system [ 1, 2, 10, 11]. Of all the long span bridges in China, highway cable-stayed bridges take the dominate position, which has the larger spanning capacity and possess superior construction technology. There are sufficient research achievements based on wind vibration and seismic response, but the study of the influence of vehicle-bridge coupled vibration is usually neglected for the reason of the relatively small vehicle loads and impact factor. Among the existing references, the influence of vehicle-bridge coupled vibration is rare referred to. Due to the increasing resonance effect, load effect has multiplied at an astonishing rate. Therefore even if the vehicle static load is relatively low, the meaning of research on the effect of vehicle-bridge coupled vibration is pretty important.

The objective of this paper is to present a numerical analysis of vehicle-bridge coupled vibrations using ANSYS, and study the effects of the floating system and half floating system of cable stayed bridges on the vehicle-bridge coupled vibrations under traffic flows.

Numerical modeling for the response of coupled vibrations of vehicle-bridge system

Numerical modeling

The coupled vibration relationship is obtained by APDL program in ANSYS. The traffic condition, single vehicle and multi vehicles at constant speed, and uniform acceleration speed were incorporated into the model [ 4, 12]. Four steps were carried out in developing the model. The models of vehicles and bridge were set up separately in ANSYS without performing dynamic equations and iterative calculation. The presented method could be used to calculate dynamic response of all kinds of vehicle bridge models in different complex conditions. An example of single axis vehicle driving through the simply supported bridge in ANSYS was presented to show the coupled vibration. The model is shown in Fig. 1(a) and the parameters used in the model were tabulated in Table 1. ANSYS has numerous element types and powerful pre-processors and post-processors. Mass element (e.g., MASS21), spring-damper element (e.g., COMBIN14), beam element (e.g., BEAM3) are all widely used in modeling the vehicles and bridges independently. The model mentioned above and their element numbers are shown in Fig. 1(b).

The coupling between those two models was achieved by APDL, which could be applied at any time to the system. The method of developing vehicle model has utilized MASS21、COMBIN14 and BEAM4 elements, the following chart (Fig. 2) shows an example of a vehicle model, which is a three axis vehicle and 5 degree of freedoms. Vehicle load acting on each bridge node was activated by the order of vehicle driving through the bridge from No.1 to No.101 node. The time interval of vehicle load acting on each node is t = element length/vehicle velocity. The recursive calculation was conducted to figure out interaction force between vehicle and bridge at each node by using the approximate conditions, and the vehicle-bridge dynamic response at each node at the moment of vehicle passing by was finally obtained. Modeling process was shown in Table 2 in which ① and ② represented the node number and load step analysis, respectively. T_k represents the time that the vehicle load leaves node No. k which equals to the time that reaches the node No. k + 1. U_k(i), V_k(i) and A_k(i) represent the vertical displacement, vehicle velocity, and acceleration of the node No. k at that moment. FEA model was established in ANSYS based on the recurrence method in Table 2 and the dynamic response of vehicle-bridge coupled vibration was finally obtained. The analysis indicated that the dynamic response had significant differences than that of the constant vehicle load under high velocity conditions. When considering roughness of the bridge deck, it is only needed to replace the bridge node U_k(i) with U_k(i) + w(i) in which w(i) represents the bridge deck roughness at node No. i. When multiple vehicles are running on the bridge, independent vehicle models should be set up in ANSYS, and coupled analysis should be carried out between the vehicles and bridge. Similarly, models for bridges at different loading conditions could be achieved in the same environment under ANSYS, and could be solved by APDL.

Determination of the coordinates of wheels

MASS element, COMBIN element, BEAM element could be utilized to develop complex vehicle models in ANSYS. Figure 2 shows an example of a vehicle model, which is a three axis vehicle and has 5 degree of freedoms in which m_c is mass of the vehicle, I_c is moment at z axis, m_i、k_ui、c_ui、k_di and c_di (i = 1,2,3)were the mass of the wheels, stiffness, and damping ratio, respectively,b₁ and b₂ are the distance between centroid of the vehicle to the front and back axial,b₃and b₄ are the distance of the middle axial to the front and back axial, c₁ is the distance between driver to the front axial. Due to the uncertainly of vehicle axis range, bridge element length and vehicle velocity, the vehicle load could not be applied on bridge node step by step according to the method mentioned above under complex conditions. Thus it is important to determine the coordinates of wheels at every moment. The starting point M and ending point N of vehicles were used to define the direction of vehicle movement (M_x, M_y, M_z are the coordinates of point M, N_x, N_y, N_z are the coordinates for N), the coordinates of the wheels on each vehicle could be determined by the initial speed v₀, acceleration a, and running time t, then coordinates of wheels were obtained. Equation (1) shows the calculation results of wheels coordinates based on 3 axis vehicle shown in Fig. 2.

The starting point M and ending point N of vehicles were used to define the direction of vehicle movement (M_x, M_y, M_z are the coordinates of point M, N_x, N_y, N_z are the coordinates for N), the coordinates of the wheels on each vehicle could be determined by the initial speed v₀, acceleration a, and running time t.

(1) \begin{array}{l}{c}_{1x}=M_x+z\left(N_x-M_x\right)/l_c\hfill \\ c_{1y}=M_y+z\left(N_y-M_y\right)/l_c\hfill \\ c_{1z}=M_z+z\left(N_z-M_z\right)/l_c\hfill \\ c_{2x}=c_{1x}-b_4\left(N_x-M_x\right)/l_c\hfill \\ c_{2y}=c_{1y}-b_4\left(N_y-M_y\right)/l_c\hfill \\ c_{2z}=c_{1z}-b_4\left(N_z-M_z\right)/l_c\hfill \\ c_{3x}=c_{2x}-b_3\left(N_x-M_x\right)/l_c\hfill \\ c_{3y}=c_{2y}-b_3\left(N_y-M_y\right)/l_c\hfill \\ c_{3z}=c_{2z}-b_3\left(N_z-M_z\right)/l_c\hfill \end{array}\},

where, l_{\mathrm{c}}=\sqrt{{(M_x-N_x)}^{\mathrm{2}}+{(M_y-N_y)}^{\mathrm{2}}+{(M_z-N_z)}^{\mathrm{2}}} ; ; c_{ij} is wheel coordinates, I is wheel number, j is direction; refers to Fig. 2.

Vertical loading and deflection transformation at the intersect between wheels and bridge element points

Once the coordinates of the wheels are determined, which may not be coincided with the bridge element points, the vertical loading and deflections generated underneath the wheel have to be transformed. The equivalent load on the beam element could be calculated based on Eq. (2), the result is shown in Eq. (3), at which the W^e is the work from outside of the element, \bar{p}(x) is the loading function, v(x) is the displacement function, N(x) is the element shape array, and q^e is the element point displacement array, as shown in Fig. 3(a).

(2) W^e={\displaystyle {\int }_l\overline{p}(x)\cdot v(x)dx=\left[{\displaystyle {\int }_l\overline{p}(x)\cdot N(x)dx}\right]}\cdot q^e=[R_A{M}_A{R}_B{M}_B]\cdot q^e,

(3) \begin{array}{l}{R}_A=-\left(P{b}^2/L^3\right)/\left(3a+b\right)\hfill \\ R_B=-\left(P{a}^2/L^3\right)/\left(a+3b\right)\hfill \\ M_A=-Pa{b}^2/L^2\hfill \\ M_B=P{a}^{2}b/L^2\hfill \end{array}\}.

The displacement field (Fig. 3(b)) of the pure bending beam is calculated by Eq. (4), in which the v(x) is the displacement function, {\theta }_{\mathrm{1}} , {\theta }_{\mathrm{2}} and v_{\mathrm{2}} are the angle of rotations at both side of the beam and the deflection, l is the length of the beam element.

(4) \begin{array}{l}v(x)=(\mathrm{1}-\mathrm{3}{\xi }^{\mathrm{2}}+\mathrm{2}{\xi }^{\mathrm{3}})v_{\mathrm{1}}+l(\xi -\mathrm{2}{\xi }^{\mathrm{2}}+{\xi }^{\mathrm{3}}){\theta }_{\mathrm{1}}+(\mathrm{3}{\xi }^{\mathrm{2}}-\mathrm{2}{\xi }^{\mathrm{3}})v_{\mathrm{2}}+l({\xi }^{\mathrm{3}}-{\xi }^{\mathrm{2}}){\theta }_{\mathrm{2}}\hfill \\ \xi =x/l\hfill \end{array}\}.

Model validation

The proposed model was validated by comparison with literature data and the In situ test data. The parameters of the vehicle and the beam were determined based on literature data from Shen et al. [ 3], and the models were developed based on the flow chart in Fig. 4. The numerical analysis results were compared with the published data [ 3] as shown in Fig. 5. The model predicted deflections were pretty close to those published data under constant vehicle moving speed condition (as indicated in the Fig. 5).

By comparing the calculated results and the tested data, it showed that the analysis exhibited the similar trend with those measured from the published results from literature, which strongly supported that the analysis in ANSYS has great accuracy in predicting the vehicle-bridge coupled vibration responses. This method could also be extended to other kinds of bridge under complex traffic loading conditions, which on the other hand indicates that it has great potential in engineering practices.

The vehicle bridge resonance

According to the structure system, cable stayed bridges can be divided into four types: floating system, half floating system, pylons constraint system, and rigid-frame system. This paper researched the possibilities and criterions of the vehicle bridge resonance in those different types of system mentioned above. For the load effect caused by individual vehicle is relatively small compared to the designing load of the bridges, even high impact factor has little influence on the security of bridges. The time for vehicle passing through the bridge is limited which suggests that resonance effect could not be persistent. Considering the above factors, this paper studies the possibilities and criterions of the vehicle-bridge coupled vibration based on equally spaced traffic flow, which has maximum disadvantage over the response of the resonance effect.

Bridge model and vehicle model

The bridge in this study has a length of 670m, with two big towers, which has auxiliary piers at the side span. The upper part of the bridge has a dimension of (50+ 120+ 330+ 120+ 50) m. The diamond type of concrete bridge pylon is 135 m high, and the distance between stiffening girder and pylon top is 100m. On the basis of the structural features of the floating system and half floating system, boundary conditions are set up at the linking sites between towers and deck based on the different structural types such as floating system, half floating system (including single fixed support and none fixed support) and rigid system. Figure 6 shows a finite element model of this cable stayed bridge. The vehicle model has 3 axes and 5 degrees of freedom, the parameters are shown in Table 1.

Methods for judging the vehicle-bridge resonance

The calculation results show that the possibility of causing vehicle-bridge resonance by disorder traffic flow is low. In this section the study focuses on the driving condition of same type and equally spaced vehicle models. When the bridge covered with equally spaced traffic flow, the interval of vehicles on bridge at the time t₀ is same with that of the vehicles at the time t₀ + t as shown in Fig. 7. Thus interval time t is the cycle of the load effect caused by the equally spaced traffic flow, and excitation frequency f_v can be calculated by Eq. (5).

(5) f_v=v/s_v,

where v is vehicle speed; s_v is longitudinal separation of the vehicles.

The relationship between excitation frequency f_{\mathrm{v}} and basal frequency f_b takes the dominant position when judging the resonance. The first three mode frequency of vertical vibration is shown in Table 3, the first mode shape of all types of cable stayed bridges mentioned above are shown in Fig. 8 to Fig. 11 and the mode shape of vehicle is shown in Fig. 12.

Judgment of the vehicle bridge resonance

With the vehicle bridge model above as an example, analysis results of the time displacement response for all conditions are shown in Figs. 13−22 and Table 4 shows the computation parameters.

The basal frequencies of floating system and half floating system (none fixed support) are usually low because the stiffening girder doesn’t have any longitudinal constraint, thus may causing vehicle bridge resonance under the equally spaced traffic flow. The displacement of pylon is larger than that of the stiffening girder in the corresponding mode shape. So the pylon produces significant response of resonance than the girder in this situation. But the resonance phenomenon would not occur under the traffic flow of same type within the security scope when considering the cable stayed bridges of rigid system and half floating system with fixed support for their relatively high basal frequency [ 10, 13]. Even if the frequency of traffic flow approaches structure basal frequency, the resonance also would not occur because the first two mode shape is symmetry and antis-symmetry state and the corresponding frequencies are pretty close to each other which mode superposition couldn’t be applied. Excitation frequency of the bridge is mainly decided by the interval time of vehicles under the conditions of the equally spaced traffic flow. Deck roughness of the bridge has effect on the impact factors but they do not change the states of the resonance. The structure resonance would be weakened or disappeared under conditions of single vehicle or unequally spaced traffic flow. The basal frequency of vehicle is close to bridge basal frequency in all conditions, but the response of vibration and resonance make great differences in these conditions, thus conclusions can be drawn that vehicle mode has little influence on resonance judgment. The reducing of the basal frequency that leads to the decreases of the impact factor according to the current criterion, however, the cable stayed bridges (especially the bridges that doesn’t have any fixed supports) do not follow this rule when considering the response of resonance.

Conclusions

A numerical analysis of the response of vehicle-bridge coupled vibrations is presented in this paper. The results showed that the interval time between vehicles is the main influence factor which contributes to the resonance excitation frequency under the condition of equally spaced traffic flows. Based on the cable stayed bridge system and conditions of equally spaced traffic flows, a vehicle-bridge resonance judging method was presented in this paper. The model results of the coupled vibration agreed well with the actual condition which directly supported that the proposed model is a robust tool for studying the vehicle-bridge coupled vibrations. Finally, in order to prevent structural resonance in the cable stayed bridges, traffic signs of deceleration should be set up at different locations which can help reducing the resonance frequency.

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