Prestress applied on bridges affects the dynamic interaction between bridges and vehicles traveling over them. In this paper, the prestressed bridge is modeled as a beam subjected to eccentric prestress force at the two ends, and a half-vehicle model with four degrees-of-freedom is used to represent the vehicle passing the bridge. A new bridge–vehicle interaction model considering the effect of prestress with eccentricity is developed through the principle of virtual work. The correctness and accuracy of the model are validated with literature results. Based on the developed model, numerical simulations have been conducted using Newmark's β method to study the effects of vehicle speed, eccentricity and amplitude of the prestress, and presence of multiple vehicles. It is shown that prestress has an important effect on the maximum vertical acceleration of vehicles, which may provide a good index for detecting the change of prestress. It is also interesting to find that the later-entering vehicle on the prestressed bridge will largely reduce the maximum vertical acceleration of the vehicle ahead of it.

## Introduction

In recent decades, prestressed concrete has been extensively used in the construction of bridges for its unique advantages such as reduction in self-weight and high crack resistance. Based on the data from U.S. Department of Transportation [1], prestressed concrete bridges make up over 60% of the total bridge decking area for all new and replaced bridges built in U.S. in 2009. Furthermore, they make up 9% of all structurally deficient bridges in the U.S. Prestress loss is one of key reasons for the structural deficiency of prestressed bridges. Big loss of prestress would cause malfunction and even collapse of a prestressed bridge, which is a risk to the public safety. Therefore, it is essential to investigate the effect of prestress on dynamic responses of bridges and vehicles through bridge–vehicle interaction and detect the prestress loss.

Bridge–vehicle interaction is a classical topic, which has been studied by many researchers. Green and Cebon [2] investigated dynamic responses of highway bridges under heavy vehicle loads and good agreements are found for the measured dynamic bridge midspan displacement and the predicted bridge midspan displacement. Yang and Wu [3] developed a vehicle–bridge interaction element to capture the dynamic bridge and vehicle responses. Zhu and Law [4] studied the continuous bridge and vehicle interaction, in which the dynamics of the bridge deck under single and several vehicles moving in different lanes is analyzed using the orthotropic plate theory and modal superposition technique. Cai et al. [5] researched the dynamic response of the bridge caused by moving vehicles under different approach span conditions. Ahmari et al. [6] analyzed the effect of foundation settlement on dynamic load allowance of the bridge through the simulation of bridge–vehicle interaction.

Some literatures did take the prestress into account to some extent. Kocaturk and Simsek [7] utilized the Lagrange equations to solve the dynamic response of a simply supported beam subjected to an eccentric compressive force and a concentrated moving harmonic force, in which no interaction was presented. Lu and Law [8] proposed to use dynamic response sensitivity to detect the prestress force of a bridge deck, where the deck is represented by an Euler beam finite element model and the prestress force is simulated as the axial force in each beam element without eccentricity. Khang et al. [9] investigated transverse vibrations of prestressed continuous beams under the action of moving bodies by using the method of substructure, which neglected the eccentricity of the prestress. However, the true interaction between the prestressed bridges subjected to eccentric prestress force and vehicles traveling over them has not been considered in these works.

In the present study, a new bridge–vehicle interaction model with consideration of eccentric prestress is built through the principle of virtual work. With the built model, the dynamic responses of prestressed bridges and vehicles traveling over them can be solved using Newmark's β method. Then, a parametric study was performed to investigate the effects of vehicle speed, eccentricity and amplitude of the prestress, and presence of multiple vehicles on dynamic bridge vehicle interaction responses.

## Bridge Vehicle Interaction

### Equation of Motion for the Prestressed Bridge.

As shown in Fig. 1, a two-span continuous eccentrically prestressed bridge can be simplified as a continuous beam subjected to one axial force ($S$) and one initial moment ($Mo$) at the two ends.

Based on the modal superposition principle, dynamic deflection $w(x,t)$ of the beam can be described as
$w(x,t)=∑i=1NWi(x)qi(t)$
(1)

where $Wix$, $qi(t)$, and N are the ith mode shape function of the beam, the corresponding modal amplitude of the beam, and the selected number of mode shapes, respectively.

According to the principle of virtual work, the external virtual work $δWE$ is equal to the internal virtual work $δWI$
$δWE=δWI$
(2)
The virtual displacements $δqiWix$, i = 1,2,…,N are selected to be consistent with the assumed shape functions. The external virtual work is the sum of the works ($δWin$,$δWP$, $δWC$, $δWS,$ and $δWMo$) done by the inertia force ($m¯(∂2w/∂t2)$), the moving loads ($Fbint$), the damping forces ($-cbi(∂w/∂t)$), the prestress force ($S$), and the moment ($Mo$), which can be written as
$δWE=δWin+δWP+δWC+δWS+δWMo$
(3)
where
$δWin=-δqi∫0LWixm¯∂2w∂t2dx$

$δWP=δqi∫0L∑k=12Fbint(k)δx-xk̂tWi[xk̂t]dx$

$δWC=-δqi∫0Lcbi(∂w∂t)Wixdx$

$cbi=2m¯ωiζi$

$δWS=δqi∫0LS∂w∂xWi′xdx$

$δWMo=δqi[MoWi′0-MoWi′L]$
(4)
The internal virtual work performed by the bending moment is
$δWI=δqi∫0LEI∂2w∂x2Wi″xdx$
(5)

where $EI$ is flexural rigidity of the beam; $Wi″x$ denotes the second derivative of $Wix$ with respect to x.

Substituting Eq. (1) and Eqs. (3)(5) into Eq. (2) and cancelling $δqi$ at both sides give
$∑j=1Nq¨jMbij+∑j=1Nq˙jCbij+∑j=1Nqj(Kbij−KGij)=Wpi+(WMo)i$
(6)
where
$Mbij=∫0Lm¯WixWjxdx$

$Kbij=∫0LEIWi″xWj″xdx$

$KGij=∫0LSWi′xWj′xdx$

$Cbij=∫0LcbiWixWjxdx$

$Wpi=∑k=12Fbint(k)Wixk̂t$

$WMoi=MoWi′0-MoWi′L$
(7)
$qi˙$ and $q¨i$ denote the first and second derivative of $qi(t)$ with respect to time t.
Corresponding to the N independent virtual displacements $δqiWix$, i = 1, 2, …, N, there are N virtual work equations in the form of Eq. (6). Together they can be expressed in matrix form as
$MbQ¨+CbQ˙+Kb-KGQ=WbFbint+WMo$
(8)
where
$Q=q1t,…,qNtTFbint=Ft1;Ft2$

$Wb=W1(x1̂t)W1(x2̂t)⋮WN(x1̂t)WN(x2̂t)$

$WMo=Mo[Wi′0-Wi′L]⋮Mo[WN′0-WN′L]$
(9)

and $Mb$, $Kb$, $Cb$, $KG$ are the mass, stiffness, damping, and geometric stiffness matrices of the bridge, respectively with their (i, j)th element calculated in Eq. (7); $Q˙$, $Q¨$ are the first and second derivatives of $Q$ with respect to time t; $Ft1$, $Ft2$ are the bridge–vehicle interaction forces at the front and rear wheel locations shown in Eq. (17); $mf$, $mr$, $mc$, $s1$, and $s2$ are parameters of vehicle shown in Fig. 2; $g$ is the acceleration of gravity.

### Modal Analysis of the Prestressed Bridge.

For the free vibration of the beam, its vertical deflection can be expressed as
$wx,t=Wxeiωt$
(10)

where $ω$ is the natural frequency of the vibration and $i=-1$.

The mode shape function of the beam $Wx$ may be expressed in term of a series as
$W(x)=∑mAmφm(x)$
(11)

where $φmx$ is the assumed admissible function satisfying the boundary conditions of the beam and $Am$ is undetermined coefficient. The selection of $φmx$ follows the method proposed by Zhou [10]. Note that the moment ($Mo$) is taken as an acting force but not a boundary condition.

Rayleigh's method is then used to determine the natural frequencies and mode shapes of the prestressed beam. The maximum potential and kinetic energies of the prestressed beam over a vibration cycle can be expressed as
$ESo=∫0L12EI∂2Wx∂x22-12S∂w∂x2dx$

$EKo=∫0L12m¯ω2Wx2dx$
(12)
Substituting Eq. (11) into Eq. (12) and taking the first derivation of Rayleigh's quotient with respect to each coefficient $Am$ would lead to the eigenvalue equations
$K-ω2MA=0$
(13)
where
$A=A1,A2,….AmTKij=∫0L[EIφi″xφj″x-Sφi′xφj′x]dxMij=∫0Lm¯φixφjxdxi=1,2,…,m;j=1,2,…,m$
(14)

Natural frequencies $ω$ and coefficients $Am$ can be determined from Eq. (13). Then the mode shape functions of the beam $Wx$ can be determined through Eq. (11).

### Vehicle Model.

A half vehicle model shown in Fig. 2 is adopted in present study to simulate the vehicle traveling over the bridge. It has four degrees-of-freedom, corresponding to the vertical displacement of vehicular body ($zc$), rotation of vehicular body about the transverse axis ($θc$), and vertical displacements of the front wheel ($zf$) and rear wheel ($zr$).

Applying the Lagrange method, the equations of motion for the vehicle model can be derived and expressed as follows:
$MvZ¨+CvZ˙+KvZ=Fvint$
(15)
where
$Z=ZcθcZfZrMv=mc0000Ic0000mf0000mr$

$Fvint=00kt1w1+ct1w1˙kt2w2+ct2w2˙$

$Cv=cs1+cs2s2cs2-s1cs1-cs1-cs2s2cs2-s1cs1s12cs1+s22cs2s1cs1-s2cs2-cs1s1cs1cs1+ct10-cs2-s2cs20cs2+ct2$

$Kv=ks1+ks2s2ks2-s1ks1-ks1-ks2s2ks2-s1ks1s12ks1+s22ks2s1ks1-s2ks2-ks1s1ks1ks1+kt10-ks2-s2ks20ks2+kt2$
(16)
$Z˙$, $Z¨$ are the first and second time derivatives of $Z$; $mc$, $Ic$, $mf$, $mr$ are half of vehicular body mass, half of vehicular body lateral mass moment of inertia, the mass of a front and rear wheels, respectively; $ks1$, $ks2$, $cs1$, $cs2$ are the stiffness and damping coefficients of the front and rear suspensions, respectively; $kt1$, $kt2$, $ct1$, $ct2$ are the stiffness and damping coefficients of the front and rear tires, respectively; $s1$, $s2$ are the distance of gravity center of the vehicular body from the front and rear axles, respectively; $w1$, $w2$ are the deflection of the bridge at the location of the front and rear wheels, respectively; $w1˙$, $w2˙$ are the time derivative of $w1$, $w2$. Note that $Z$ is measured from the static equilibrium position of the vehicle, resulting in no gravity terms in Eq. (17).

### Bridge–Vehicle Interaction Force.

The interaction forces between the bridge and vehicle shown in Fig. 2 can be described as follows:
$Ft1=kt1(Zf-w1)+ct1(Zf˙-w1˙)-mf+mcs2s1+s2gFt2=kt2(Zr-w2)+ct2(Zr˙-w2˙)-(mr+mcs1s1+s2)g$
(17)

## Numerical Implementation

The dynamic responses of the prestressed bridge and vehicle can be calculated from Eqs. (8), (15) and (17) using Newmark's β method through an iterative procedure. The implementation procedure is shown in Fig. 3, in which $v$ represents the speed of the vehicle, $Δt$ is the time-step, and error representing the difference between the results of two consecutive iterations is defined as follows:
$error=w1(x,t)j-w1(x,t)j-1w1(x,t)j2+w2(x,t)j-w2(x,t)j-1w2(x,t)j2$
(18)

where $w1(x,t)j-1$, $w1(x,t)j$ are the deflection of the bridge at the front wheel location in the ($j-1$)th iteration and ($j$)th iteration, respectively; $w2(x,t)j-1$, $w2(x,t)j$ are the deflection of the bridge at the rear wheel location in the ($j-1$)th iteration and the ($j$)th iteration, respectively.

## Validation

The published results in Ref. [3] have been used to validate the effectiveness and accuracy of the proposed method in capturing the dynamic responses of the bridge and vehicle.

As can be seen from Figs. 46, good agreements have been achieved between the results obtained from present study and the literature results.

## Parametric Study Results

To investigate the effect of parameters such as vehicle speed, eccentricity and amplitude of prestress, and presence of multiple passing vehicles on dynamic responses of the prestressed bridge and the vehicle, numerical simulations have been conducted using the bridge and vehicle model shown in Figs. 1 and 2, respectively. The bridge has the following properties: $L=18+18=36 m$, $EI=3.2448×109N·m2$, $m¯=2052 kg/m$, $S=3.113×103kN$, $Mo=1.058×103kN·m$, where $EI$ is the flexural rigidity of the bridge and $m¯$ is the mass per unit length of the bridge. The parameters for the vehicle in Marchesiello et al. [11] are used in the simulations with some of them modified to accommodate the half vehicle model and listed as follows: $mc=8500kg$, $Ic=4.5×104kg·m2$, $mf=300kg$, $mr=500kg$, $ks1=1.16×105N/m$, $ks2=3.73×105N/m$, $kt1=7.85×105N/m$, $kt2=1.57×106N/m$, $cs1=2.5×104N·s/m$, $cs2=3.5×104N·s/m$, $ct1=100N·s/m$, $ct2=200N·s/m$.

Based on the developed prestressed bridge vehicle interaction model, a program was coded in Matlab to compute the dynamic bridge and vehicle responses using Newmark's β method with γ = 1/2 and β = 1/2, indicating a constant average acceleration over a time-step with unconditional stability. The first ten mode shapes of the bridge were used in the calculation, and a time-step of 5 × 10−5 s was selected. The tolerance error between two consecutive iterations was set to 0.01. To focus on the prestress effect, the surface of the bridge pavement was assumed to be smooth, indicating no road surface roughness. Meanwhile, to keep the problem traceable, the vehicle was assumed to enter and exit to the bridge smoothly, no jump in or jump out.

### Effect of the Prestress Force.

Varying percentages of prestress force ($S$) from 0% to 140% have been applied to the two-span continuous bridge shown in Fig. 1 to study its effect, with other properties including the eccentricity of the prestress kept constant.

Figure 7(a) shows the time histories of absolute displacements of the bridge at the middle of the first span for 0%, 50%, 80%, and 100% of prestress force. Here, the absolute displacement means that it was measured from the horizontal line through the supports, and for all the other displacements appearing in this paper, they were relative displacements measured from the static equilibrium position of the bridge.

Time histories of the vertical accelerations of the vehicle traveling over the bridge with different percentages of prestress force have been plotted in Fig. 7(b).

In Fig. 7(c), absolute displacement of the bridge at the front wheel location when the vehicle is crossing the bridge is plotted. Figure 7(d) shows the maximum vertical acceleration of the vehicle under different levels of prestress.

### Effect of the Eccentricity.

In the last section, the effect of the prestress force has been studied with the eccentricity kept constant. Actually, the eccentricity is also an essential component of the prestress in practical situations. Therefore, there is a need to investigate its effect.

First, keeping the initial moment constant ($Mo=S·e)$, three cases are studied: (2.0S, 0.5e), (1.0S, 1.0e), and (0.5S, 2.0e), where S represents the amount of prestress force and e is its eccentricity. The comparison of initial deflection of the bridge, time histories of the displacement at the middle of the first span, and vertical acceleration of the vehicle are plotted in Figs. 8(a)8(c), respectively.

Figure 8(d) shows the comparison of maximum vertical acceleration of the vehicle for two series of cases: one case with e fixed but S changing from 0% to 140% and the other with S fixed but e changing from 0% to 140%.

### Effect of Multiple Vehicles.

In practice, there are often multiple vehicles traveling over the bridge. Thus, it is important to know the performance of the bridge under multiple moving vehicles. Consider the bridge with a modest traffic flow, a 10.0 m following distance is assumed to include multiple vehicles on the bridge. Figure 9(a) shows the model of multiple vehicles passing the bridge; Fig. 9(b) shows time histories of midspan displacement of the bridge. The maximum vehicle vertical accelerations for different cases are summarized in Table 1.

## Discussion

First, the effect of the prestress force on the dynamic bridge and vehicle responses needs to be analyzed. From Fig. 7(a), one observes that four lines share very similar trends with nearly the same amplitude of variability, which implies the prestress has little effect on the bridge relative displacement. As can be seen in Fig. 7(b), similar trends with different amplitudes appeared for the vehicles passing the prestressed bridges (50%, 80%, and 100%), while the trend for the vehicle passing the nonprestressed bridge is very different. The reason for that may be seen from Fig. 7(c) that shows the displacement of the bridge at the front wheel location when the vehicle is passing the bridge. As shown in Fig. 7(c), the trace of the front wheel passing the nonprestressed bridge consists of two nearly symmetric downward bumps. In contrast, the traces of the front wheel passing the prestressed bridges are composed of two asymmetric upward bumps. Thus, a conclusion can be drawn from Fig. 7(c) that the prestress force greatly changes the performance of the bridge under moving vehicles.

Moreover, to some extent, Fig. 7(c) may be used to explain the results shown in Fig. 7(d). Figure 7(d) shows the maximum vertical acceleration of the vehicle under different levels of prestress. With the increase in the percentage of prestress from 0% to 140%, the maximum vertical acceleration of the vehicle will decrease first and then increase, and specially increases almost linearly when the percentage of prestress increases from 50% to 140%, corresponding to the increasing height of the bumps in Fig. 7(c) from 50% to 100%. For the same travelling distance and speed, the higher bump as shown in Fig. 7(c) is, the larger acceleration as shown in Fig. 7(d) will be, which consists with the common sense of life that stiffer slope leads to stronger vibration of the vehicle. Meanwhile, the speed of the vehicle plays an important role in the variation of its maximum acceleration. Generally, the higher the speed is, the larger the maximum acceleration is. Furthermore, the higher speed may increase the slope of the linear line section from 50% to 140% of prestress force, indicating that the vehicle becomes more sensitive to the change of the bridge prestress when the speed increases. This is easy to understand because high-speed vehicle possesses stronger vibration and also more sensitive.

Second, Fig. 8 proves that eccentricity as a component of the prestress of the bridge has its own effect. As can be seen from Figs. 8(a)8(c), three lines representing three cases are very close to each other, indicating little difference among three cases. Note that these three cases share the same initial moment. As shown in Fig. 8(d), as long as having the same initial moment, the corresponding cases in two series nearly possess the same amount of maximum vehicle acceleration. Thus, a general conclusion can be obtained that the change of eccentricity has the same effect as the change of prestress and the initial moment is the controlling factor of bridge–vehicle interaction response. In other words, it is only the initial moment that matters but not its constitution. Another thing that needs to mention is the nonlinearity of the lines in Fig. 8(d). As discussed in the last paragraph for Figs. 7(c) and 7(d), higher bump causes larger acceleration; thus, the acceleration for 0% prestress is higher than the ones for 30% and 50%, preventing the existence of linear lines. However, there may exist a specific percentage of prestress that produce a minimum acceleration of even closing to 0, which enables the passenger to enjoy the most ride comfort.

Finally, the case of multiple vehicles passing the bridge was investigated. As can be seen from Fig. 9(b), there is no big difference between the amplitude of bridge response due to single vehicle and these due to multiple vehicles. It is interesting to find that the number of peaks of the curve in Fig. 9(b) is equal to the number of vehicles. One observation obtained from Table 1 is that the later-entering vehicle will significantly decrease the maximum vertical acceleration of the vehicle ahead of it, which mainly attributes to the suppression of the initial curvature of the prestressed bridge due to the later-entered vehicles.

## Conclusions

In this paper, a new bridge–vehicle model with consideration of eccentric prestress effect has been developed through the principle of virtual work to address the prestressed continuous bridges and vehicle interaction. The correctness and accuracy of the current model are validated with the existing published results. Based on the developed model, numerical simulations have been conducted to study effects of vehicle speed, eccentricity and amplitude of prestress, and presence of multiple vehicles.

Several conclusions were reached, which are as follows:

1. (1)

Prestress has an important effect on the maximum vertical acceleration of vehicles, which may provide a good index for detecting the change of prestress.

2. (2)

Eccentricity is an essential component of the prestress of the bridge and the initial moment induced is the controlling factor of the bridge–vehicle interaction response.

3. (3)

The later-entered vehicle on the prestressed bridge will largely reduce the maximum vertical acceleration of the vehicles ahead of it.

It is anticipated that these findings can be helpful to the design and prestress loss identification of the prestressed bridges in future.

## Funding Data

• Federal Highway Administration (Grant No. UsDOT MPC 394).

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