A vehicle-bridge interaction vibration model considering bridge deck pavement

Abstract

The bridge pavement subjected to vehicle loads is the most vulnerable component in the highway bridge and overloaded vehicles can result in serious damage to the bridge pavement more easily. Therefore, it is necessary to consider the pavement in vehicle-bridge interaction (VBI) system. However, the bridge pavement is not considered in the traditional VBI system or the viscoelasticity of bridge pavement is ignored. In this study, a new vehicle-pavement-bridge interaction (VPBI) system is established. The viscoelastic asphalt pavement is simulated with the continuously and uniformly distributed spring-damper. The equivalent stiffness coefficient and equivalent damping coefficient of the pavement are determined through dynamic mechanical analysis (DMA) test and finite element method. The equations of motion for VPBI system can be derived using Lagrange equation and the modal superposition method. Then the crucial parameters such as displacement at mid-span of bridge, acceleration of vehicle body, tire contact force, and pavement deformation are obtained. The effects of vehicle velocity, bridge span, tire contact area, pavement stiffness coefficient, and pavement damping coefficient on VPBI responses are analyzed. The results show that (1) The pavement is compressive during vehicle passing bridge and its dynamic deformation fluctuates around initially static deformation of pavement. (2) Increasing the stiffness of pavement can rapidly reduce the deformation of pavement, while the pavement damping has the opposite effect. (3) The increase of bridge span can conduct exponentially growth of dynamic responses of VPBI system. The pavement deformation will increase by 7.9% if the bridge span is lengthened by 5 m. This study provides a reliability response analysis method for VPBI system.

Keywords

vehicle-bridge interaction viscoelastic pavement prony series spring-damper dynamic responses

Introduction

The vehicle bridge interaction (VBI) system model has been established by many researchers through the observation and investigation of the vibration of vehicle and bridge.¹ In the early VBI system, vehicles were generally simplified as a moving constant load² or a moving mass,³ and the dynamic responses of a bridge were discussed. Currently, more complicated and actual models are appeared and used to analyze VBI system. A three-mass VBI system based on semi-analytical approach was proposed by Judy P. Yang.⁴ The dynamics of a two-axle vehicle and a continuous elastic beam were studied to analyze the separation of the vehicle and the beam by Dan St˘ancioiu et al.⁵ The beam was supported by elastic bearing modelled as linear springs. The governing equations of the vehicle and the beam were derived and the results showed that most probably separation will occur if the beam has three linear elastic bearing. The VBI system with different boundary conditions was studied by Zhenhua Shi and Nasim Uddin.⁶ The vehicle was modelled as a damped sprung mass and the Euler–Bernoulli beam had three boundary conditions including both ends fixed, fixed simply supported, and one end fixed the other end free (cantilever). This study indicated that boundary conditions have obvious effect on the bridge frequency. The dynamic amplification factor of the bridge was investigated by Yingyan Li et al.⁷ via the VBI system. A quarter car with two degrees of freedom and a Euler–Bernoulli with T-shape webs were used and the influence of road roughness irregularities on the dynamic amplification. A two-mass vehicle model with single degree of freedom moving on a simply supported beam was proposed by Judy P. Yang and Bo-How Chen⁸ to extract frequencies more realistically. Most of the works mentioned above were based on the analytical or semi-analytical VBI system. In addition, the finite element method (FEM) is also a common technique for the analysis of VBI system.^9–12 Both the vehicle and bridge can be modelled and simulated by FEM. Although many achievements have been obtained in the research field of VBI system, the influence of pavement is always neglected or seldom considered directly in VBI system.

The pavement directly bears the loads from vehicles and environment and is important to the bridge maintenance and preservation.¹³ Leilei Chen et al.¹⁴ proposed a numerical multiscale analysis method to determine the critical cracking zone of steel bridge deck pavements. The full-scale bridge analysis, subsection analysis and sub-element analysis were utilized by the consideration of vehicle pavement interaction. Linbing Wang et al.¹³ proposed a coupled static-and-dynamics analysis to analyze the pavement structure stress state. An elastic layered model of the pavement was established by the dynamic module of ANSYS software and a half sine periodic loading was applied to bridge deck pavement. The results showed that the suggested thickness of the upper layer pavement is 3.5 cm–4.5 cm and this range for the lower layer is 5 cm–7 cm. Xuntao Wang et al.¹⁵ established a three-dimensional finite element steel bridge and its deck pavement by ANSYS software. The viscoelasticity of asphalt concrete is described by Burgers model and the parameters were converted into Prony series in the ANSYS software. The stress values of asphalt concrete deck pavement were calculated and analyzed with the consideration of the characteristic of asphalt concrete and the interlayer bonding condition. Niujing Ma¹⁶ proposed an eight-node solid-plate element for simulating the pavement and analyzed the local dynamic behavior of orthotropic steel bridge decks subjected to impact loads, wherein the impact loads considered as triangular loads. Its results indicated that the bridge deck pavement cannot be neglected in the mechanical analysis of bridge deck structure. Chenchen Zhang¹⁷ studied the mechanical performance of the steel-bridge deck pavement of the multitower suspension bridge under random traffic flow through the multiscale numerical approach and experimental program. It is found that the pavement layer can reduce the stress of the steel-box girder roof. However, most of the reported studies did not consider the effect of the interaction between pavement and vehicle-bridge. And the material of asphalt pavement is often considered as elastic properties rather than viscoelasticity in vehicle-bridge interaction. And the equation derivation models are fewer than FEM models for the pavement researches considering vehicle-bridge interaction.

To obtain dynamic responses of the pavement and understand mechanical behavior of the pavement in vehicle-bridge interaction, a vehicle-pavement-bridge interaction (VPBI) system model is proposed in this study. The vehicle model and bridge model are set as a quarter car and a simply supported Euler–Bernoulli beam, respectively. The viscoelastic asphalt pavement is regarded as continuously and uniformly distributed spring-dampers and its equivalent stiffness coefficient and damping coefficient are estimated by dynamic mechanical analysis (DMA) test and finite element method. The dynamic responses of VPBI system can be obtained by solving the derived motion equations of VPBI system. Furthermore, the effect of vehicle velocity, bridge span, tire contact area, pavement stiffness coefficient, and pavement damping coefficient on dynamic responses of VPBI system is also discussed quantitatively. The results of theoretical analysis provide practical guidelines for preventing serious damage to bridge pavements.

The motion equations of VPBI system

Vehicle-pavement-bridge interaction system model is depicted as Figure 1. The quarter vehicle model moves on a simply supported Euler–Bernoulli beam with constant speed v. The bridge damping property is neglected, and the beam cross section is constant. The quarter vehicle model has three degrees of freedom which are the vertical displacement y_i (i = 1,2,3) of the tire, the axle, and the vehicle body, respectively. $M_{i}$ (i = 1,2,3) are the mass of the tire, the axle, and the vehicle body, respectively. K₁ and C₁ are stiffness and damping of the tire. K₂ and C₂ are stiffness and damping of the suspension. The contact surface between the tire and the pavement is modeled as a rectangle with length b and width a. The mass of pavement is neglected. The pavement is modeled as continuously and uniformly distributed spring-dampers with the equivalent spring stiffness coefficient K_p and the equivalent damping coefficient C_p. In this model, it is assumed that the tire contacts with the pavement at all times and the spring-damper representing pavement is compressed only within the range of contact length b.

The motion equations of the VPBI system are given in the following derivation.

Figure 1.

The VPBI system.

The static equilibrium position is chosen as the reference position of the vehicle motion. The kinetic energy T and the potential energy U of the vehicle are described by equations (1) and (2), respectively

T = \frac{1}{2} M_{1} {\dot{y}}_{1}^{2} + \frac{1}{2} M_{2} {\dot{y}}_{2}^{2} + \frac{1}{2} M_{3} {\dot{y}}_{3}^{2}

(1)

U = \frac{1}{2} K_{1} {(y_{2} - y_{1})}^{2} + \frac{1}{2} K_{2} {(y_{3} - y_{2})}^{2}

(2)

And dissipation function R is expressed as¹⁸

R = \frac{1}{2} C_{1} {({\dot{y}}_{2} - {\dot{y}}_{1})}^{2} + \frac{1}{2} C_{2} {({\dot{y}}_{3} - {\dot{y}}_{2})}^{2}

(3)

The Lagrange equation is

\frac{d}{d t} (\frac{\partial L}{\partial {\dot{y}}_{i}}) - \frac{\partial L}{\partial y_{i}} + \frac{\partial R}{\partial {\dot{y}}_{i}} = Q_{i} i = 1,2,3, . . .

(4)

where

L = (T - U)

is Lagrangian.

y_{i}

{\dot{y}}_{i}

, and Q_i are the ith generalized displacement, generalized velocity, and the generalized force, respectively.

t

is time.

The number of total degrees of freedom is three. Substituting equations (1)–(3) into equation (4), the vehicle motion equations can be obtained as follows

M_{1} {\ddot{y}}_{1} + K_{1} (y_{1} - y_{2}) + C_{1} ({\dot{y}}_{1} - {\dot{y}}_{2}) = Q_{1}

(5)

M_{2} {\ddot{y}}_{2} + K_{1} (y_{2} - y_{1}) - K_{2} (y_{3} - y_{2}) + C_{1} ({\dot{y}}_{2} - {\dot{y}}_{1}) - C_{2} ({\dot{y}}_{3} - {\dot{y}}_{2}) = Q_{2}

(6)

M_{3} {\ddot{y}}_{3} + K_{2} (y_{3} - y_{2}) + C_{2} ({\dot{y}}_{3} - {\dot{y}}_{2}) = Q_{3}

(7)

where

Q_{1} = a b [K_{p} (y_{b} (x, t) |_{x = v t} - y_{1}) + C_{p} (\frac{\partial y_{b} (x, t)}{\partial t} |_{x = v t} - {\dot{y}}_{1})],

Q_{2} = 0,

Q_{3} = 0

The bridge motion equations are

ρ A \frac{\partial^{2} y_{b} (x, t)}{\partial t^{2}} + E I \frac{\partial^{4} y_{b} (x, t)}{\partial x^{4}} = F_{b} (t) [H (x - (v t - b)) - H (x - v t)] \cdot [H (t) - H (t - \frac{l + b}{v})]

(8)

where ρ

A

E I

are the density, the cross-section area, the bending stiffness of the bridge, respectively.

Here they are all constant. $H (Δ)$ represents the Heaviside function

H (Δ) = {\begin{cases} 0, Δ \leq 0 \\ 1, Δ > 0 \end{cases}

(9)

F_{b} (t)

is the force between the pavement and bridge.

F_{b} (t) = \frac{- (M_{1} + M_{2} + M_{3}) g + a b [K_{p} (y_{1} - y_{b} (x, t) |_{x = v t}) + C_{p} ({\dot{y}}_{1} - \frac{\partial y_{b} (x, t)}{\partial t} |_{x = v t})]}{b}

(10)

where

g

is the gravitational constant, and

a

b

are width and length of the rectangle contact area.

According to the Modal superposition method, $y_{b} (x, t)$ can be written as

y_{b} (x, t) = \sum_{i = 1}^{N} ϕ_{i} (x) q_{i} (t)

(11)

where

ϕ_{i} (x)

is the mode function,

q_{i} (t)

is the generalized coordinate of the beam and

N

is the total number of the modes.

For the simply supported beam, the mode functions are the sine function,¹ that is $ϕ_{i} (x) = \frac{\sin i π x}{l}$ . Therefore, equation (11) can be expressed as

y_{b} (x, t) = \sum_{i = 1}^{N} [\frac{\sin i π x}{l} q_{i} (t)]

(12)

Substituting equation (12) into equation (8), multiplying both sides of the equation by $ϕ_{j} (x)$ , and integrating over the length $l$ of the bridge, equation (13) is obtained¹

\begin{array}{l} \int_{0}^{l} ϕ_{j} (x) ρ A (\sum_{i = 1}^{N} ϕ_{i} (x) {\ddot{q}}_{i} (t)) d x + \int_{0}^{l} ϕ_{j} (x) E I (\sum_{i = 1}^{N} ϕ_{i}^{″ ″} (x) q_{i} (t)) d x \\ = \int_{0}^{l} ϕ_{j} (x) F_{b} (t) [H (x - (v t - b)) - H (x - v t)] \cdot [H (t) - H (t - \frac{l + b}{v})] d x \end{array}

(13)

Based on the orthogonality relation for the modal function, equation (13) can be simplified as

M_{b i} {\ddot{q}}_{i} + K_{b i} q_{i} = P_{i} (x, t)

(14)

The generalized mass $M_{b i}$ and the generalized stiffness $K_{b i}$ can be written as

M_{b i} = \int_{0}^{l} ρ A ϕ_{i} (x) ϕ_{i} (x) d x

(15)

K_{b i} = \int_{0}^{l} E I ϕ_{i}^{″} (x) ϕ_{i}^{″} (x) d x

(16)

where

P_{i} (x, t)

is the terms of right-hand side of equation (13).

When the vehicle moving on the bridge, the moving procedure can be divided into three components: the vehicle enters to the bridge, the vehicle is on the bridge and the vehicle leaves from the bridge. These procedures shown in Figure 2, and $P_{i} (x, t)$ can be reduced to

P_{i} (x, t) = {\begin{cases} \int_{0}^{v t} F_{b} (t) ϕ_{i} (x) d x, (0 < v t < b, 0 < x < v t) \\ \int_{v t - b}^{v t} F_{b} (t) ϕ_{i} (x) d x, (b < v t < l, v t - b < x < v t) \\ \int_{v t - b}^{l} F_{b} (t) ϕ_{i} (x) d x, (l < v t < l + b, v t - b < x < l) \end{cases}

(17)

Figure 2.

The moving procedure of the vehicle: (a) enter to the bridge, (b) on the bridge, and (c) leave from the bridge.

Considering the first three modes of the bridge, $F_{b} (t)$ is written as

F_{b} (t) = \frac{- (M_{1} + M_{2} + M_{3}) g + a b [K_{p} (y_{1} - \sum_{i = 1}^{3} ϕ_{i} (v t) q_{i} (t)) + C_{p} ({\dot{y}}_{1} - \sum_{i = 1}^{3} ϕ_{i} (v t) {\dot{q}}_{i} (t))]}{b}

(18)

Combining the equations (5)–(7) and equation (14), the motion equation for VPBI system is derived as a matrix form

[M_{s}] [\ddot{d}] + [K_{s}] [d] + [C_{s}] [\dot{d}] = [F_{s}]

(19)

where

\ddot{d}

d

, and

\dot{d}

are the generalized acceleration vector, the generalized displacement vector, and the generalized velocity vector, respectively

[\ddot{d}] = {[\begin{array}{c} {\ddot{y}}_{1} & {\ddot{y}}_{2} & {\ddot{y}}_{3} & \begin{array}{c} \begin{array}{c} {\ddot{q}}_{1} & {\ddot{q}}_{2} \end{array} & {\ddot{q}}_{3} \end{array} \end{array}]}^{T}

[d] = {[\begin{array}{c} y_{1} & y_{2} & y_{3} & \begin{array}{c} \begin{array}{c} q_{1} & q_{2} \end{array} & q_{3} \end{array} \end{array}]}^{T}

(20)

[\dot{d}] = {[\begin{array}{c} {\dot{y}}_{1} & {\dot{y}}_{2} & {\dot{y}}_{3} & \begin{array}{c} \begin{array}{c} {\dot{q}}_{1} & {\dot{q}}_{2} \end{array} & {\dot{q}}_{3} \end{array} \end{array}]}^{T}

where

M_{s}

K_{s}

C_{s}

, and

F_{s}

are the mass matrix, the stiffness matrix, and the damping matrix, and the loading vector, respectively

[M_{s}] = d i a g [M_{1} M_{2} M_{3} \frac{ρ A l}{2} \frac{ρ A l}{2} \frac{ρ A l}{2}]

(21)

(22)

where

[k_{11}]

[k_{12}]

[c_{11}]

, and

[c_{12}]

are invariant in the whole procedure

[k_{11}] = [\begin{array}{c} a b K_{p} + K_{1} & - K_{1} & 0 \\ - K_{1} & K_{1} + K_{2} & - K_{2} \\ 0 & - K_{2} & K_{2} \end{array}] [k_{12}] = [\begin{array}{c} - a b K_{p} \sin (\frac{v t π}{l}) & - a b K_{p} \sin (\frac{2 v t π}{l}) & - a b K_{p} \sin (\frac{3 v t π}{l}) \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}]

(23)

[c_{11}] = [\begin{array}{c} a b C_{p} + C_{1} & - C_{1} & 0 \\ - C_{1} & C_{1} + C_{2} & - C_{2} \\ 0 & - C_{2} & C_{2} \end{array}] [c_{12}] = [\begin{array}{c} - a b C_{p} \sin (\frac{v t π}{l}) & - a b C_{p} \sin (\frac{2 v t π}{l}) & - a b C_{p} \sin (\frac{3 v t π}{l}) \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}]

(24)

But the other submatrices vary with the location of the vehicle.

When the vehicle enters the bridge, the variable submatrices are

[k_{21}] = [\begin{array}{c} l (\cos (\frac{π t v}{l}) - 1) a K_{p} & 0 & 0 \\ l (\cos (\frac{2 π t v}{l}) - 1) a K_{p} & 0 & 0 \\ \frac{l (\cos (\frac{3 π t v}{l}) - 1) a K_{p}}{3 π} & 0 & 0 \end{array}]

(25)

[k_{22}] = [\begin{array}{c} \frac{E I π^{4}}{2 l^{3}} - \frac{l (\cos (\frac{π t v}{l}) - 1) a K_{p} \sin (\frac{π t v}{l})}{π} & - \frac{l (\cos (\frac{π t v}{l}) - 1) a K_{p} \sin (\frac{2 π t v}{l})}{π} & - \frac{l (\cos (\frac{π t v}{l}) - 1) a K_{p} \sin (\frac{3 π t v}{l})}{π} \\ - \frac{l (\cos (\frac{2 π t v}{l}) - 1) a K_{p} \sin (\frac{π t v}{l})}{2 π} & \frac{8 E I π^{4}}{l^{3}} - \frac{l (\cos (\frac{2 π t v}{l}) - 1) a K_{p} \sin (\frac{2 π t v}{l})}{2 π} & - \frac{l (\cos (\frac{2 π t v}{l}) - 1) a K_{p} \sin (\frac{3 π t v}{l})}{2 π} \\ - \frac{l (\cos (\frac{3 π t v}{l}) - 1) a K_{p} \sin (\frac{π t v}{l})}{3 π} & - \frac{l (\cos (\frac{3 π t v}{l}) - 1) a K_{p} \sin (\frac{2 π t v}{l})}{3 π} & \frac{81 E I π^{4}}{2 l^{3}} - \frac{l (\cos (\frac{3 π t v}{l}) - 1) a K_{p} \sin (\frac{3 π t v}{l})}{3 π} \end{array}]

(26)

[c_{21}] = [\begin{array}{c} \frac{l (\cos (\frac{π t v}{l}) - 1) a C_{p}}{π} & 0 & 0 \\ \frac{l (\cos (\frac{2 π t v}{l}) - 1) a C_{p}}{2 π} & 0 & 0 \\ \frac{l (\cos (\frac{3 π t v}{l}) - 1) a C_{p}}{3 π} & 0 & 0 \end{array}]

(27)

[c_{22}] = [\begin{array}{c} - \frac{l (\cos (\frac{π t v}{l}) - 1) a C_{p} \sin (\frac{π t v}{l})}{π} & - \frac{l (\cos (\frac{π t v}{l}) - 1) a C_{p} \sin (\frac{2 π t v}{l})}{π} & - \frac{l (\cos (\frac{π t v}{l}) - 1) a C_{p} \sin (\frac{3 π t v}{l})}{π} \\ - \frac{l (\cos (\frac{2 π t v}{l}) - 1) a C_{p} \sin (\frac{π t v}{l})}{2 π} & - \frac{l (\cos (\frac{2 π t v}{l}) - 1) a C_{p} \sin (\frac{2 π t v}{l})}{2 π} & - \frac{l (\cos (\frac{2 π t v}{l}) - 1) a C_{p} \sin (\frac{3 π t v}{l})}{2 π} \\ - \frac{l (\cos (\frac{3 π t v}{l}) - 1) a C_{p} \sin (\frac{π t v}{l})}{3 π} & - \frac{l (\cos (\frac{3 π t v}{l}) - 1) a C_{p} \sin (\frac{2 π t v}{l})}{3 π} & - \frac{l (\cos (\frac{3 π t v}{l}) - 1) a C_{p} \sin (\frac{3 π t v}{l})}{3 π} \end{array}]

(28)

[F_{s}] = [\begin{array}{c} 0 \\ 0 \\ 0 \\ \frac{l (\cos (\frac{π t v}{l}) - 1) g (M 1 + M 2 + M 3)}{π b} \\ \frac{l (\cos (\frac{2 π t v}{l}) - 1) g (M 1 + M 2 + M 3)}{2 π b} \\ \frac{l (\cos (\frac{3 π t v}{l}) - 1) g (M 1 + M 2 + M 3)}{3 π b} \end{array}]

(29)

When the vehicle is on the bridge, the variable submatrices are

[k_{21}] = [\begin{array}{c} \frac{l (- \cos (\frac{π (- t v + b)}{l}) + \cos (\frac{π t v}{l})) a K_{p}}{π} & 0 & 0 \\ \frac{l (- \cos (\frac{2 π (- t v + b)}{l}) + \cos (\frac{2 π t v}{l})) a K_{p}}{2 π} & 0 & 0 \\ \frac{l (- \cos (\frac{3 π (- t v + b)}{l}) + \cos (\frac{3 π t v}{l})) a K_{p}}{3 π} & 0 & 0 \end{array}]

(30)

[k_{22}] = [\begin{array}{c} \frac{E I π^{4}}{2 l^{3}} + \frac{l (\cos (\frac{π (- t v + b)}{l}) - \cos (\frac{π t v}{l})) a K_{p} \sin (\frac{π t v}{l})}{π} & \frac{l (\cos (\frac{π (- t v + b)}{l}) - \cos (\frac{π t v}{l})) a K_{p} \sin (\frac{2 π t v}{l})}{π} & \frac{l (\cos (\frac{π (- t v + b)}{l}) - \cos (\frac{π t v}{l})) a K_{p} \sin (\frac{3 π t v}{l})}{π} \\ \frac{l (\cos (\frac{2 π (- t v + b)}{l}) - \cos (\frac{2 π t v}{l})) a K_{p} \sin (\frac{π t v}{l})}{2 π} & \frac{8 E I π^{4}}{l^{3}} + \frac{l (\cos (\frac{2 π (- t v + b)}{l}) - \cos (\frac{2 π t v}{l})) a K_{p} \sin (\frac{2 π t v}{l})}{2 π} & \frac{l (\cos (\frac{2 π (- t v + b)}{l}) - \cos (\frac{2 π t v}{l})) a K_{p} \sin (\frac{3 π t v}{l})}{2 π} \\ \frac{l (\cos (\frac{3 π (- t v + b)}{l}) - \cos (\frac{3 π t v}{l})) a K_{p} \sin (\frac{π t v}{l})}{3 π} & \frac{l (\cos (\frac{3 π (- t v + b)}{l}) - \cos (\frac{3 π t v}{l})) a K_{p} \sin (\frac{2 π t v}{l})}{3 π} & \frac{81 E I π^{4}}{2 l^{3}} + \frac{l (\cos (\frac{3 π (- t v + b)}{l}) - \cos (\frac{3 π t v}{l})) a K_{p} \sin (\frac{3 π t v}{l})}{3 π} \end{array}]

(31)

[c_{21}] = [\begin{array}{c} \frac{l (- \cos (\frac{π (- t v + b)}{l}) + \cos (\frac{π t v}{l})) a C_{p}}{π} & 0 & 0 \\ \frac{l (- \cos (\frac{2 π (- t v + b)}{l}) + \cos (\frac{2 π t v}{l})) a C_{p}}{2 π} & 0 & 0 \\ \frac{l (- \cos (\frac{3 π (- t v + b)}{l}) + \cos (\frac{3 π t v}{l})) a C_{p}}{3 π} & 0 & 0 \end{array}]

(32)

[c_{22}] = [\begin{array}{c} \frac{l (\cos (\frac{π (- t v + b)}{l}) - \cos (\frac{π t v}{l})) a C_{p} \sin (\frac{π t v}{l})}{π} & \frac{l (\cos (\frac{π (- t v + b)}{l}) - \cos (\frac{π t v}{l})) a C_{p} \sin (\frac{2 π t v}{l})}{π} & \frac{l (\cos (\frac{π (- t v + b)}{l}) - \cos (\frac{π t v}{l})) a C_{p} \sin (\frac{3 π t v}{l})}{π} \\ \frac{l (\cos (\frac{2 π (- t v + b)}{l}) - \cos (\frac{2 π t v}{l})) a C_{p} \sin (\frac{π t v}{l})}{2 π} & \frac{l (\cos (\frac{2 π (- t v + b)}{l}) - \cos (\frac{2 π t v}{l})) a C_{p} \sin (\frac{2 π t v}{l})}{2 π} & \frac{l (\cos (\frac{2 π (- t v + b)}{l}) - \cos (\frac{2 π t v}{l})) a C_{p} \sin (\frac{3 π t v}{l})}{2 π} \\ \frac{l (\cos (\frac{3 π (- t v + b)}{l}) - \cos (\frac{3 π t v}{l})) a C_{p} \sin (\frac{π t v}{l})}{3 π} & \frac{l (\cos (\frac{3 π (- t v + b)}{l}) - \cos (\frac{3 π t v}{l})) a C_{p} \sin (\frac{2 π t v}{l})}{3 π} & \frac{l (\cos (\frac{3 π (- t v + b)}{l}) - \cos (\frac{3 π t v}{l})) a C_{p} \sin (\frac{3 π t v}{l})}{3 π} \end{array}]

(33)

[F_{s}] = [\begin{array}{c} 0 \\ 0 \\ 0 \\ \frac{l (- \cos (\frac{π (- t v + b)}{l}) + \cos (\frac{π t v}{l})) g (M 1 + M 2 + M 3)}{π b} \\ \frac{l (- \cos (\frac{2 π (- t v + b)}{l}) + \cos (\frac{2 π t v}{l})) g (M 1 + M 2 + M 3)}{2 π b} \\ \frac{l (- \cos (\frac{3 π (- t v + b)}{l}) + \cos (\frac{3 π t v}{l})) g (M 1 + M 2 + M 3)}{3 π b} \end{array}]

(34)

When the vehicle leaves from the bridge, the variable submatrices are

[k_{21}] = [\begin{array}{c} - \frac{l (\cos (\frac{π (- t v + b)}{l}) + 1) a K_{p}}{π} & 0 & 0 \\ - \frac{l (\cos (\frac{2 π (- t v + b)}{l}) - 1) a K_{p}}{2 π} & 0 & 0 \\ - \frac{l (\cos (\frac{3 π (- t v + b)}{l}) + 1) a K_{p}}{3 π} & 0 & 0 \end{array}]

(35)

[k_{22}] = [\begin{array}{c} \frac{E I π^{4}}{2 l^{3}} + \frac{l (\cos (\frac{π (- t v + b)}{l}) + 1) a K_{p} \sin (\frac{π t v}{l})}{π} & \frac{l (\cos (\frac{π (- t v + b)}{l}) + 1) a K_{p} \sin (\frac{2 π t v}{l})}{π} & \frac{l (\cos (\frac{π (- t v + b)}{l}) + 1) a K_{p} \sin (\frac{3 π t v}{l})}{π} \\ \frac{l (\cos (\frac{2 π (- t v + b)}{l}) - 1) a K_{p} \sin (\frac{π t v}{l})}{2 π} & \frac{8 E I π^{4}}{l^{3}} + \frac{l (\cos (\frac{2 π (- t v + b)}{l}) - 1) a K_{p} \sin (\frac{2 π t v}{l})}{2 π} & \frac{l (\cos (\frac{2 π (- t v + b)}{l}) - 1) a K_{p} \sin (\frac{3 π t v}{l})}{2 π} \\ \frac{l (\cos (\frac{3 π (- t v + b)}{l}) + 1) a K_{p} \sin (\frac{π t v}{l})}{3 π} & \frac{l (\cos (\frac{3 π (- t v + b)}{l}) + 1) a K_{p} \sin (\frac{2 π t v}{l})}{3 π} & \frac{81 E I π^{4}}{2 l^{3}} + \frac{l (\cos (\frac{3 π (- t v + b)}{l}) + 1) a K_{p} \sin (\frac{3 π t v}{l})}{3 π} \end{array}]

(36)

[c_{21}] = [\begin{array}{c} - \frac{l (\cos (\frac{π (- t v + b)}{l}) + 1) a C_{p}}{π} & 0 & 0 \\ - \frac{l (\cos (\frac{2 π (- t v + b)}{l}) - 1) a C_{p}}{2 π} & 0 & 0 \\ - \frac{l (\cos (\frac{3 π (- t v + b)}{l}) + 1) a C_{p}}{3 π} & 0 & 0 \end{array}]

(37)

[c_{22}] = [\begin{array}{c} \frac{l (\cos (\frac{π (- t v + b)}{l}) + 1) a C_{p} \sin (\frac{π t v}{l})}{π} & \frac{l (\cos (\frac{π (- t v + b)}{l}) + 1) a C_{p} \sin (\frac{2 π t v}{l})}{π} & \frac{l (\cos (\frac{π (- t v + b)}{l}) + 1) a C_{p} \sin (\frac{3 π t v}{l})}{π} \\ \frac{l (\cos (\frac{2 π (- t v + b)}{l}) - 1) a C_{p} \sin (\frac{π t v}{l})}{2 π} & \frac{l (\cos (\frac{2 π (- t v + b)}{l}) - 1) a C_{p} \sin (\frac{2 π t v}{l})}{2 π} & \frac{l (\cos (\frac{2 π (- t v + b)}{l}) - 1) a C_{p} \sin (\frac{3 π t v}{l})}{2 π} \\ \frac{l (\cos (\frac{3 π (- t v + b)}{l}) + 1) a C_{p} \sin (\frac{π t v}{l})}{3 π} & \frac{l (\cos (\frac{3 π (- t v + b)}{l}) + 1) a C_{p} \sin (\frac{2 π t v}{l})}{3 π} & \frac{l (\cos (\frac{3 π (- t v + b)}{l}) + 1) a C_{p} \sin (\frac{3 π t v}{l})}{3 π} \end{array}]

(38)

[F_{s}] = [\begin{array}{c} 0 \\ 0 \\ 0 \\ - \frac{l (\cos (\frac{π (- t v + b)}{l}) + 1) g (M 1 + M 2 + M 3)}{π b} \\ - \frac{l (\cos (\frac{2 π (- t v + b)}{l}) - 1) g (M 1 + M 2 + M 3)}{2 π b} \\ - \frac{l (\cos (\frac{3 π (- t v + b)}{l}) + 1) g (M 1 + M 2 + M 3)}{3 π b} \end{array}]

(39)

Then the VPBI system responses can be obtained by solving equation (19) with the Newmark-β method.

The equivalent stiffness coefficient and equivalent damping coefficient of the pavement

In practice, asphalt mixtures with nonlinear viscoelastic properties are usually used for the highway bridges pavement. In order to simplify the calculation, the equivalent stiffness coefficient K_p and equivalent damping coefficient C_p are used to replace the nonlinear viscoelastic characteristic of the asphalt mixture. In this section, K_p and C_p can be obtained by analytical method and FEM.

The rigid plate-asphalt mixtures pavement model

A rigid plate-asphalt mixture pavement (RPAP) model is established and shown in Figure 3. The spring-dampers which simulate the asphalt mixture pavement are uniformly distributed under the rigid plate. The rigidity of the rigid plate is much greater than that of the asphalt mixture pavement. And the rigid plate width, length, and thickness are a, b, and h, respectively. P is the concentrated excitation force acting on the center of upper surface of the rigid plate.

Figure 3.

The RPAP model.

Derivation of the formulas for K_p and C_p

Similar to the energy-based method to identify the parameter of Pasternak foundation,¹⁹ the Lagrange equation is applied to the RPAP model. The kinetic energy, potential energy, and the dissipation function are expressed as follows

T = \int_{0}^{a} \int_{0}^{b} \frac{1}{2} \bar{m} {[\dot{w} (x, y, t)]}^{2} d x d y

(40)

U = \int_{0}^{a} \int_{0}^{b} \frac{1}{2} K_{p} {[w (x, y, t)]}^{2} d x d y

(41)

R = \int_{0}^{a} \int_{0}^{b} \frac{1}{2} C_{p} {[\dot{w} (x, y, t)]}^{2} d x d y

(42)

where

w (x, y, t)

is the deflection of the rigid plate.

\bar{m}

, K_p, and C_p are the areal density of the rigid plate, the stiffness coefficient, and damping coefficient of the asphalt mixture pavement, respectively.

Based on the assumption that the rigid plate cannot deform, the deflection of the rigid plate is only related to time $t$ and not to coordinates. Due to the concentrated excitation force acting on the center of upper surface of the rigid plate, the rigid plate vibrates only along vertical direction. This RPAP model is a single degree of freedom model. Therefore, equations (40)–(42) can be simplified

T = \frac{1}{2} m {[\dot{w} (t)]}^{2}

(43)

U = \frac{1}{2} a b K_{p} {[w (t)]}^{2}

(44)

R = \frac{1}{2} a b C_{p} {[\dot{w} (t)]}^{2}

(45)

where

m

is the mass of the rigid plate

m = a b \bar{m}

(46)

Substituting equations (43)–(45) into the Lagrange equation, the motion equation of RPAP model can be given as

m \ddot{w} + a b K_{p} w + a b C_{p} \dot{w} = P

(47)

where P is the concentrated excitation force acting on the rigid plate.

Further, the natural frequency $ω_{n}$ and the damping ratio $ζ$ are obtained

ω_{n} = \sqrt{\frac{a b K_{p}}{m}}

(48)

ζ = \frac{a b C_{p}}{2 \sqrt{a b K_{P} m}}

(49)

When the concentrated excitation force is an impact load, the motion of the RPAP system is a damped free vibration. In underdamped case, the frequency of the RPAP model is

ω_{d} = \frac{2 π}{τ_{d}} = ω_{n} \sqrt{1 - ζ^{2}}

(50)

where

τ_{d}

is the period of free-damped vibration.

In theory of vibration, the logarithmic decrement δ is expressed as

δ = \frac{1}{n} \ln \frac{w_{i}}{w_{i + n}} = ζ ω_{n} τ_{d}

(51)

where w_i+n represents the displacement amplitude after n periods have elapsed beginning with w_i.

Considering equations (50) and (51), the natural frequency ω_n and the damping ratio ζ can be written as follows

ω_{n} = \frac{2 π}{τ_{d} \sqrt{1 - ζ^{2}}}

(52)

ζ = \frac{\ln (w_{i} / w_{i + n})}{\sqrt{4 π^{2} n^{2} + {[\ln (w_{i} / w_{i + n})]}^{2}}}

(53)

From equations (48) and (49) and equations (52) and (53), the equivalent stiffness coefficient Kp and the equivalent damping coefficient Cp can be obtained

K_{p} = \frac{m [4 π^{2} n^{2} + {I n}^{2} (w_{i} / w_{i + n})]}{a b n τ_{d}^{2}}

(54)

C_{p} = \frac{2 m \ln (w_{i} / w_{i + n})}{a b n τ_{d}}

(55)

where τ_d, w_i, and w_i+n can be obtained by finite element method.

Simulation for the RPAP model

In this section, the FEM of RPAP model is established by ANSYS software, as shown in Figure 4. The rigid plate and the asphalt mixture pavement are simulated using SOLID185. Lateral displacement of the asphalt mixture pavement is constrained and only vertical displacement is considered. The RPAP model dimensions^13,20 are shown in Table 1. The four cases from one to four correspond to four type of contact area of tires²⁰ and Case0 is point contact acted as a control. The material parameters of the rigid plate¹⁹ are shown in Table 2.

Figure 4.

The finite element model of the RPAP model.

Table 1.

The dimensions of rigid plate and asphalt mixture pavement.

	Width a (m)	Length b (m)	Thickness h (m)
Case1	0.2003	0.391	0.1
Case2	0.2328	0.46	0.1
Case3	0.3459	0.615	0.1
Case4	0.4083	0.744	0.1
Case0	Point contact

Table 2.

The material parameters of the rigid plate.

Modulus of elasticity $(MPa)$	Poisson’s ratio	Density $(kg / m^{3})$
3×10⁷	0.2	2500

Based on the small strain theory, the constitutive equation of isotropic viscoelastic asphalt mixture pavement is²¹

σ = \int_{0}^{t} 2 G (t - τ) \frac{d e}{d τ} d τ + I \int_{0}^{t} K (t - τ) \frac{d Δ}{d τ} d τ

(56)

where σ, e, and Δ are the Cauchy stress, the deviatoric part of the strain and the volumetric part of the strain, respectively. G(t) and K(t) are shear and bulk relaxation kernel function. I, t, and τ are unit tensor, current and past time, respectively.

Prony series can be expressed by the following formula²¹

G (t) = G_{0} [α_{\infty} + \sum_{i}^{n} α_{i} \exp (- \frac{t}{τ_{i}})]

(57)

where G₀ is the instantaneous shear modulus.

G_{0} = G_{\infty} + \sum_{i = 1}^{n} G_{i}

, of which G_∞ is the equilibrium shear modulus and G_i is the shear modulus.

α_{i} = G_{i} / G_{0}

and

τ_{i}

are relative modulus and relaxation time of each Prony series term.

α_∞ can be calculated when t is zero. equation (57) can be written as

G_{0} = G_{0} [α_{\infty} + \sum_{i}^{n} α_{i}]

(58)

Then

α_{\infty} + \sum_{i}^{n} α_{i} = 1

(59)

That is

α_{\infty} = 1 - \sum_{i}^{n} α_{i}

(60)

Therefore, the equation (57) for Prony series can be determined if the parameters G_∞, G_i and $τ_{i}$ are obtained. Considering the relation $G = E / [2 (1 + μ)]$ , only the parameters E_∞, E_i, and $τ_{i}$ need to be determined.

In this paper, the DMA test is implemented to determine material property of the asphalt mixture as shown in Figure 5. Under the reference temperature 20°C, the storage modulus is obtained at each frequency. And the tested data of storage modulus can be fitted by the formula as follows²²

E^{'} (ω) = E_{\infty} + \sum_{i = 1}^{n} E_{i} \frac{ω^{2} τ_{i}^{2}}{1 + ω^{2} τ_{i}^{2}}

(61)

where E^′ is the storage modulus, ω is the corresponding frequency, E_∞ is the equilibrium modulus, and E_i is the elastic modulus.

Figure 5.

The DMA test.

Considering the relaxation time

τ_{i}

within 10⁻⁷ s–10^3.5 s and the time interval of 10^0.5 s, E_∞, 22 groups of E_i, and

τ_{i}

are fitted by equation (61). Then, using equations (58)–(60) and the relation

G = E / [2 (1 + μ)]

, the parameters of equation (57) are calculated. The material parameters of asphalt mixture and the parameters of Prony series are listed in Table 3 and Table 4. Figure 6 shows the storage modulus, respectively, obtained by DMA test and by equation (61). From Figure 6, the test data of storage modulus and the calculated storage modulus are fitted well. The error calculated by equation (62) is only 0.086%

e r r o r = \frac{\sqrt{\sum_{i = 1}^{870} \frac{{(E_{t e s t, i}^{'} - E_{f i t t e d, i}^{'})}^{2}}{E_{t e s t, i}^{'}}}}{870}

(62)

where

E_{t e s t, i}^{'}

is the ith test value of storage modulus and

E_{fitted, i}^{'}

is the ith calculated value of storage modulus by equation (61) and 870 is total number of test data.

Table 3.

The material parameters of asphalt mixture.

Instantaneous modulus $G_{0} (MPa)$	Poisson’s ratio $μ$	Density $ρ_{a} (kg / m^{3})$
10,346.8778	0.25	2470

Table 4.

The relative modulus and relaxation time.

$i$	$α_{i}^{G}$	$τ_{i}^{G}$
1	0.0005974	10⁻⁷
2	0	10^−6.5
3	0	10⁻⁶
4	0	10^−5.5
5	0	10⁻⁵
6	0.047869326	10^−4.5
7	0	10⁻⁴
8	0.083400593	10^−3.5
9	0.058920447	10⁻³
10	0.116541204	10^−2.5
11	0.12273351	10⁻²
12	0.142564731	10^−1.5
13	0.130305626	10⁻¹
14	0.109674072	10^−0.5
15	0.075683226	1
16	0.049587757	10^0.5
17	0.026455055	10
18	0.016553892	10^1.5
19	0.006578082	10²
20	0.006379565	10^2.5
21	0	10³
22	0.003917694	10^3.5

Figure 6.

Storage modulus comparison of testing and fitting.

The material parameters of asphalt mixture and the relative modulus and relaxation time (listed in Table 3 and Table 4) are input into RPAP model by APDL command in Appendix A, then viscoelastic property can be assigned to those elements of the asphalt mixture pavement. For excitation, a load P is acted on the RPAP model and lasts 0.01 s to simulate the impact load, as shown in Figure 7. Then the RPAP model does damping free vibration. Because the rigid plate model is not ideal rigid plate, the average vertical displacements of node A and node B (shown in Figure 4) of the rigid plate are used to describe dynamic response of the RPAP model, shown in Figure 8.

Figure 7.

The load-time curve of load P

Figure 8.

Dynamic response of the RPAP model

From Figure 8, it is illustrated that the vibration of the RPAP model decays over the time due to the asphalt mixture viscoelasticity. After 0.11s, the RPAP model vibrates freely with damping. The parameters τ_d, w_i, and w_i+n is obtained in this stage and they are substituted into equation (54) and (55) to calculate K_p and C_p. For case 1 to case 4, four groups of K_p and C_p are shown in Figure 9.

Figure 9.

K_p and C_p under different cases.

It shows that there are some differences for K_p and C_p due to the dimensional difference of rigid plate (i.e., dimensional difference of tire contact surface). Therefore, K_p and C_p are calculated considering the practical dimension of tire contact surface when K_p and C_p are used to calculate responses of VPBI system.

Model validation

Dynamic responses of the VPBI system are compared with the results of VBI system calculated by M. F. GREEN²³ to verify the validity of model in this paper. The VBI system is shown in Figure 10, which is a quarter vehicle model with 2-DOF and the vehicle model moves on a simply supported bridge at a constant velocity. From Figure 1 and Figure 10, there are some differences between the VPBI system and the VBI system. Such as the pavement and the tire contact area are considered in the VPBI system, but not considered in the VBI system. In addition, the modal damping ratio of bridge is 0.02 in VBI system although it has no effect on model validity.¹ Except these differences, other components of these two systems are same. The parameters of the bridge, the vehicle^20,23,24 and the pavement are given in Table 5 and Table 6. When the velocity is 25m/s, substituting the vehicle, bridge and pavement parameters into equation (19), the VPBI system responses can be obtained. The vertical displacement responses at the bridge mid-span of VBI and VPBI are shown in Figure 11.

Figure 10.

The VBI system of M. F. GREEN²³.

Table 5.

Parameters of the beam and the pavement.

Symbol	name	Value	unit
EI	Bending stiffness of bridge	127500×10⁶	N∙m²
m _b	Mass of unit length of bridge	12,000	kg/m
l	Span length of bridge	40	m
K _p	Stiffness coefficient of pavement (case1)	1.813×10¹²	Pa/m
C _p	Damping coefficient of pavement (case1)	9.436×10⁴	Pa∙s/m

Table 6.

Parameters of the vehicle.

Symbol	Name	Value	unit
$M_{1}$	Tire mass	98	$kg$
$M_{2}$	Unsprung mass	4000	$kg$
$M_{3}$	Sprung mass	36,000	$kg$
$K_{1}$	Tire stiffness	7.2×10⁷	$N / m$
$K_{2}$	Suspension stiffness	1.8×10⁷	$N / m$
$C_{1}$	Tire damping	14.4×10⁴	$N \cdot s / m$
$C_{2}$	Suspension damping	14.4×10⁴	$N \cdot s / m$
a	Contact width of the tire	0.2003	m
$b$	Contact length of the tire	0.391	m
$g$	Acceleration of gravity	9.81	${m / s}^{2}$

From Figure 11, two lines are very close and have similar variation trend. The vertical displacement of the VPBI system is little smaller than those in Reference [23]. From the enlarged partial view, it can be seen that the maximum two lines are about 0.0041 m and 0.0042 m, respectively. Therefore, VPBI system model is correct.

Figure 11.

Vertical displacement at the bridge mid-span calculated by VPBI and VBI.

Further, the VPBI system is compared with the VBI system under the case without tire contact area, shown in Figure 12. The parameters of these two systems are listed in Table 5 and Table 6. And the dynamic responses of them are shown in Figure 13.

Figure 12.

The VPBI (a) and VBI (b) with point contact.

Figure 13.

Dynamic responses comparison of VPBI and VBI: (a) displacement at mid-span of bridge, (b) acceleration at mid‐span of bridge, (c) displacement of vehicle body, (d) acceleration of vehicle body.

From Figure 13, it illustrates that dynamic response curves of the VPBI system and VBI system are very similar. Use a L-2 norm of the dynamic response difference value between VPBI system and VBI system to describe the effect of the pavement. And then the L-2 norm is 0.0005, 0.2193, 0.0012, and 0.3806 corresponding to Figure 13 (a) to (d) in turn. It indicates that the pavement has more effect on acceleration than displacement, especially on the acceleration of vehicle body.

Analysis of the dynamic responses

In this section, the dynamic responses of VPBI system (shown in Figure 1) are analyzed. The dynamic responses include displacement at bridge mid-span, acceleration of vehicle body, tire contact force, and pavement deformation under the tire. The tire contact force is determined by

F_{c o n} = (M_{1} + M_{2} + M_{3}) g + M_{1} {\ddot{y}}_{1} + K_{1} (y_{1} - y_{2}) + C_{1} ({\dot{y}}_{1} - {\dot{y}}_{2})

(63)

where M₁, M₂, and M₃ are the mass of tire, vehicle axle, and vehicle body, respectively.

\ddot{y}

₁,

\dot{y}

₁, and y₁ are the acceleration, velocity, and displacement of tire.

\dot{y}

₂ and y₂ are the velocity and displacement of vehicle axle. K₁ and C₁ are tire stiffness and tire damping. g is the gravitational constant.

The deformation of pavement y_p is defined by the following expression, that is

y_{p} = | {y_{b} |}_{x = v t} | - | y_{1} | + Δ_{s t}

(64)

where y_b|_{x = vt} and y₁ are the bridge displacement and tire displacement at the location x = vt, respectively. ∆_st is the initially static deformation of pavement under the action of the vehicle’s weight, here is 2.7704×10⁻⁶ m.

Effect of vehicle velocity

When the vehicle velocity is 40 km/h, 80 km/h, 120 km/h, respectively, and other parameters are the same as Table 5 and Table 6, the dynamic response curves of VPBI system are given in Figure 14. The variation trends of the maximum dynamic responses are also shown in Figure 14, as the vehicle velocity increasing from 10 km/h to 120 km/h with an interval of 10 km/h.

Figure 14.

Effect of vehicle velocity: (a) displacement at mid‐span of bridge, (b) variation trend of the maximum displacement, (c) acceleration of vehicle body, (d) variation trend of the maximum acceleration, (e) tire contact force, (f) variation trend of the maximum tire contact force, (g) deformation of pavement, (h) variation trend of the maximum pavement deformation.

From Figure 14, as the velocity increases, the VPBI responses have similar variation pattern, except that the fluctuation amplitude increases and the number of the fluctuation peaks decreases. The curves of ${\ddot{y}}_{3}$ and F_con are almost identical, except for the different values. This shows a strong correlation between ${\ddot{y}}_{3}$ and F_con. From Figure 14 (g), the pavement deformation curve resembles a sine wave. When the vehicle enters the bridge, acts on the mid-span, and leaves the bridge, the deformation is about static deformation. When the vehicle moves near the quarter span, the deformation is maximal. When the vehicle moves around three quarters of the span, the deformation is minimal. Furthermore, the maximum responses have an increasing trend, especially the maximum of the acceleration of vehicle body and the tire contact force. They increase almost linearly. The slopes are 0.00401 and 147.7881 by linear fitting, respectively. From Figure 14 (b), when the velocity is 90 km/h, the maximum of y_b is dropped, and then gradually increases. Figure 14 (h) shows that the maximum pavement deformation has similar trend to Figure 14 (b) and the maximum deformation is dropped at 80 km/h.

Effect of bridge span

Only the bridge span is changed while other parameters of vehicle, pavement, and bridge are the same as Table 5 and Table 6. When the vehicle velocity is 90 km/h, dynamic responses of VPBI are shown in Figure 15 within the span range in 20 m–40 m.

Figure 15.

Effect of bridge span: (a) displacement at mid‐span of bridge, (b) variation trend of the maximum displacement, (c) acceleration of vehicle body, (d) variation trend of the maximum acceleration, (e) tire contact force, (f) variation trend of the maximum tire contact force, (g) deformation of pavement, (h) variation trend of the maximum pavement deformation.

It can be seen that the bridge span has a great influence on the dynamic responses of the VPBI system. From Figure 15 (a) and (b), with the increase of the bridge span, the vertical displacement at mid-span of the bridge increases significantly. The maximum displacement when the span is 40m is about 8 times of the maximum displacement when the bridge span is 20m. From Figure 15 (c), (e), and (g), the fluctuation of the curves increases greatly with the increase of the bridge span. Figure 15 (b), (d), (f), and (h) show that the maximum of these responses varies exponentially as the bridge span increases. These exponential expressions can be given by fitting as follows

{| y_{b} |}_{\max} = 6.6778 \times 10^{- 4} e^{\frac{l}{18.8572}} - 0.00143

(65)

{| {\ddot{y}}_{3} |}_{\max} = 4.6825 \times 10^{- 4} e^{\frac{l}{6.1891}} + 0.02768

(66)

{| F_{c o n} |}_{\max} = 7.2099 e^{\frac{l}{5.4590}} + 394455.43

(67)

{| y_{p} |}_{\max} = 4.0558 \times 10^{- 8} e^{\frac{l}{13.4171}} + 2.6799 \times 10^{- 6}

(68)

where l is the bridge span. R-square (COD) of fitting is 0.99933, 0.99943, 0.99915, and 0.99988 corresponding to equations (65)–(68), respectively.

These equations can be used in studies to estimate the maximum responses of a VPBI system when the bridge span is known.

Effect of tire contact area

Considering five contact cases in Table 1 and parameters of the vehicle, pavement, and bridge in Table 5 and Table 6, dynamic responses of VPBI are shown in Figure 16 when the vehicle velocity is 90 km/h. The tire contact area from case 1 to case 4 is increased.

Figure 16.

Effect of tire contact area: (a) displacement at mid‐span of bridge, (b) variation trend of the maximum displacement, (c) acceleration of vehicle body, (d) variation trend of the maximum acceleration, (e) tire contact force, (f) variation trend of the maximum tire contact force, (g) deformation of pavement, (h) variation trend of the maximum pavement deformation.

From Figure 16, the tire contact area has little effect on VPBI responses. In Figure 16 (b) and (h), ${| y_{b} |}_{\max}$ and ${| y_{p} |}_{\max}$ gradually decrease from case 0 to case 4, and the relative difference between case 0 and case 4 is only 0.06% and 0.44%, respectively. From Figure 16 (d) and (f), the maximum of ${\ddot{y}}_{3}$ and F_con increases first and then decreases. The maximum relative differences between case 1 and case 4 are 0.15% and 0.004%, respectively.

Effect of stiffness coefficient of pavement

The stiffness coefficient K_p=1.813×10¹² $\frac{P a}{m}$ calculated is used as a reference value K_p0. K_p1 and K_p2 are considered as that K_p0 is increased and decreased by 10 times, respectively. It means that the stiffness coefficient is K_p1=1.813×10¹¹ $\frac{P a}{m}$ , K_p0=1.813×10¹² $\frac{P a}{m}$ , and K_p2=1.813×10¹³ $\frac{P a}{m}$ , while other parameters are the same as Table 5 and Table 6. When the vehicle velocity is 90 km/h, the dynamic responses of VPBI are shown in Figure 17.

Figure 17.

Effect of stiffness coefficient: (a) displacement at mid‐span of bridge, (b) variation trend of the maximum displacement, (c) acceleration of vehicle body, (d) variation trend of the maximum acceleration, (e) tire contact force, (f) variation trend of the maximum tire contact force, (g) deformation of pavement, (h) variation trend of the maximum pavement deformation.

It can be known from Figure 17 (a), (c), (e), and (g) that the curve of y_b,

{\ddot{y}}_{3}

and F_con is insensitive to variation of stiffness coefficient but y_p is affected obviously. For the maximum of y_b,

{\ddot{y}}_{3}

, F_con, and y_p, they decrease as the increase of stiffness coefficient except F_con. Moreover, the relative differences of VPBI maximum responses are calculated by equation (69) and are listed in Table 7

Relative difference = \frac{| K_{p i} r e s p o n s e - K_{p 0} r e s p o n s e |}{K_{p 0} r e s p o n s e} \times 100 %, (i = 1,2)

(69)

where K_Pi response and K_P0 response is VPBI maximum responses when K_P is equal to K_Pi and K_P0, respectively.

Table 7.

The relative differences of VPBI maximum responses (K_p=K_p1, K_p2).

	K_P1	K_P2
${\| y_{b} \|}_{\max}$	0.018%	0.002%
${\| {\ddot{y}}_{3} \|}_{\max}$	0.006%	0.003%
${\| F_{c o n} \|}_{\max}$	0.001‰	0.0002‰
${\| y_{P} \|}_{\max}$	14.19%	0.17%

From Table 7, it indicates that the increase of stiffness coefficient can decrease pavement deformation, but the variation tends to be stable as increasing stiffness coefficient.

Effect of damping coefficient of pavement

Like the analysis about stiffness coefficient, the damping coefficient C_p=9.436×10⁴ $\frac{P a \cdot s}{m}$ calculated is used as a reference value C_p0. C_p1 and C_p2 are considered as that C_p0 is increased and decreased by 10 times, respectively, that is, C_p1=9.436×10³ $\frac{P a s}{m}$ , C_p0=9.436×10⁴ $\frac{P a s}{m},$ and C_p2=9.436×10⁵. Other parameters are the same as Table 5 and Table 6. Dynamic responses of VPBI are shown in Figure 18.

Figure 18.

Effect of damping coefficient: (a) displacement at mid‐span of bridge, (b) variation trend of the maximum displacement, (c) acceleration of vehicle body, (d) variation trend of the maximum acceleration, (e) tire contact force, (f) variation trend of the maximum tire contact force, (g) deformation of pavement, (h) variation trend of the maximum pavement deformation.

From Figure 18 (a), (c), (e), and (g), these response curves are almost overlapped when damping coefficient is changed. And the effect of damping coefficient on the maximum response of y_b,

{\ddot{y}}_{3}

F_con and y_p is shown in Table 8. The result shows that damping coefficient has slight effect on responses of VPBI, especially for y_b,

{\ddot{y}}_{3}

and F_con, and the effect on them can be almost neglected. The pavement deformation is easier to be affected than y_b,

{\ddot{y}}_{3}

and F_con. As damping coefficient increasing, the pavement deformation increases slightly. The reason is that pavement damping dissipates energy by deformation of itself. More energy is dissipated with a large damping coefficient, resulting in greater deformation of pavement.

Table 8.

The relative differences of VPBI maximum responses (C_p=C_p1, C_p2).

	C_P1	—	C_P2
${\| y_{b} \|}_{\max}$	Less than 0.0005‰	Less than 0.0005‰	Less than 0.0005‰
${\| {\ddot{y}}_{3} \|}_{\max}$	Less than 0.0005‰	Less than 0.0005‰	Less than 0.0005‰
${\| F_{c o n} \|}_{\max}$	Less than 0.0005‰	Less than 0.0005‰	Less than 0.0005‰
${\| y_{P} \|}_{\max}$	0.06‰	—	0.1%

$Relative difference = \frac{| C_{p i} r e s p o n s e - C_{p 0} r e s p o n s e |}{C_{p 0} r e s p o n s e} \times 100 %, (i = 1,2)$ Where, C_pi response and C_p0 response is VPBI maximum responses when C_p is equal to C_pi and C_p0, respectively.

Conclusions

In this study, a novel VPBI system model and its motion equations were proposed. Particularly, the pavement was modeled by continuously and uniformly distributed spring-dampers. The equivalent stiffness coefficient and damping coefficient of pavement are determined by DMA test and FEM based on a RPAP model. And the responses of VPBI system and VBI system were also compared. Furthermore, the effect of vehicle velocity, bridge span, tire contact area, pavement stiffness coefficient, and pavement damping coefficient on the VPBI system was analyzed. The major conclusions are listed as follows:

(1) The pavement has slight effect on the responses of vehicle and bridge in VBI system.

(2) During the vehicle passing through the bridge, pavement is compressive and its deformation first increases, then decreases, and finally increases again. The deformation fluctuates around the initially static deformation of pavement in the form of a sine wave.

(3) Responses of VPBI system increase with the increment of vehicle velocity. The maximum acceleration of vehicle body and the maximum tire contact force increase linearly. The maximum displacement of the bridge and the maximum deformation of the pavement tend to decrease first and then increase within higher vehicle velocity.

(4) Bridge span has significant effect on dynamic responses of VPBI system. The maximum response increases exponentially with the increment of bridge span.

(5) The increase of tire contact area can reduce dynamic responses of VPBI system. Moreover, the effect of tire contact area on pavement deformation is the most obvious.

(6) The pavement stiffness coefficient has slight effect on bridge displacement, acceleration of vehicle body and tire contact force. However, it has significant effect on pavement deformation. The increase of stiffness coefficient can reduce pavement deformation to a certain extent.

(7) The pavement damping coefficient has almost no effect on bridge displacement, acceleration of vehicle body, and tire contact force. However, a larger damping coefficient can slightly increase the pavement deformation.

All the findings above can serve as a practical guide for bridge health monitoring. In the future, the VPBI system in this study can be expanded to three-dimension system or vehicle-continuous bridge interaction system. However, in practical engineering, asphalt mixtures, the vehicle parameters, the number of the vehicles, the bridge types, etc., all have great uncertainty. In the future, perhaps it is possible to use probabilistic approach to deal with the VPBI model uncertainty to find the rules and improve the robustness of the proposed model. In addition, in-lab model experiments and real bridge experiments will be carried out to verify and optimize the VPBI model in the future work.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (NSFC) (11972238, 11902206, 11872255), Natural Science Foundation of Hebei Province, China (A2022210007, E2022210029, A2021210009) and 2022 Hebei Provincial Department of Education Project of China (ZD2022019). The financial aid is gratefully acknowledged.

ORCID iD

Yuhang Chen

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A vehicle-bridge interaction vibration model considering bridge deck pavement

Abstract

Keywords

Introduction

The motion equations of VPBI system

The equivalent stiffness coefficient and equivalent damping coefficient of the pavement

The rigid plate-asphalt mixtures pavement model

Derivation of the formulas for K p and C p

Simulation for the RPAP model

Model validation

Analysis of the dynamic responses

Effect of vehicle velocity

Effect of bridge span

Effect of tire contact area

Effect of stiffness coefficient of pavement

Effect of damping coefficient of pavement

Conclusions

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

Appendix

References

Derivation of the formulas for K_p and C_p