# A Study on Dynamic Amplification Factor and Structure Parameter of Bridge Deck Pavement Based on Bridge Deck Pavement Roughness

Authors: Fei Han; Dan-hui Dan; Hu Wang
Journal: Advances in Civil Engineering
 (2018)

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## Abstract

In order to study the coupled influence of deck pavement roughness and velocity on dynamic amplification factor, a 2-DOF 1/4 vehicle model is employed to establish the vehicle-bridge-coupled vibration system. The random dynamic load of running vehicle simulated by software MATLAB is applied on bridge deck pavement (BDP) through ANSYS software. Besides, the influence of BDP parameters on control stress under static load and random vibration load is analyzed. The results show that if the surface of BDP is smooth, the dynamic magnification coefficient would first increase and then decrease with increasing of vehicle velocity and reach its maximum value whenv = 20 m/s; if the surface of BDP is rough, the maximal and minimum values of the dynamic amplification coefficient (DAC) occur, respectively, when the velocity reaches 10 m/s and 15 m/s. For a composite bridge deck with the cushion layer, the thickness of asphalt pavement should be not too thick or thin and better to be controlled for about 10 cm; with the increasing of cushion layer thickness, the control stress of deck pavement is all decreased and show similar change regularity under effect of different loads. In view of self-weight of structure, the thickness of the cushion layer is recommended to be controlled for about 4 cm.


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## Body

## 1. Introduction

The deck pavement of long-span bridges is usually susceptible to various destructions such as cracking, lamination, steel corrosion, and concrete degradation under vehicle loads. In the interaction of tire and bridge deck, the additional dynamic load will generate because of the roughness of BDPs. Therefore, the actual vehicle load applied on BDPs is of random load, the moving constant load cannot reflect the real dynamic behavior of BDPs. However, existing mechanical analysis of BDPs is concentrated on orthotropic plate pavement system under static vehicle load. Most literatures simplify the vehicle load as moving constant load when analyzing the dynamic behavior of BDPs, and the dynamic effect of vehicle load is considered using an empirical amplification factor.Dynamic vehicle load will bring bridges a greater effect on structural stress and deformation compared with the static vehicle load, and this response amplification phenomenon of stress and deformation caused by vehicle dynamic load is called the impact effect [1]. Dynamic load of vehicle consists of vehicle body weight and additional dynamic load caused by surface roughness of BDPs; the specific value of dead and additional load is defined as the dynamic amplification factor (DAF). In most existing studies, the vehicle-bridge-coupled vibration is taken into consideration ignoring the effect of BDPs [2–4]. DAFs play a vital role in the practice of bridge design and condition assessment. Accurate evaluation of DAFs will lead to safe and economical designs for new bridges and provide valuable information for condition assessment and management of existing bridges. However, the evaluation of a DAF is a rather complicated issue because of the sophisticated mechanism of the vehicle-bridge interaction (VBI) and a large number of parameters influencing DAFs, including the dynamic characteristics of both the bridge and the vehicle, road surface condition, and vehicle speed, where the road surface condition and vehicle speed are two critical index affecting DAFs significantly and have been widely investigated [5–9]. Though comprehensive studies of the DAF of highway bridges have been finished; due to the neglect of the influence of pavement layer, heavy maintenance cost of BDPs was paid during later maintenance and reinforce of the bridge. In China, BDPs were designed as an independent part isolated from the bridge girder, without considering the influence of deck pavement on structure [10]. However, BDPs are usually susceptible to cracking, lamination, steel corrosion, and concrete degradation, which accelerate the growth of damage; therefore, it is necessary to investigate the mechanical behavior of BDPs and analyze the effect of velocity and the roughness of BDPs on DAFs.Existing researches have analyzed the dynamic response of BDPs under vehicle dynamic load considering road roughness and vehicle velocity [11], and the variation rule of vehicle load caused by structural parameters of bridge and vehicle speed has been investigated as well [12, 13]. However, the effect of DAFs caused by the design parameter of BDPs has not been studied considering the effect of road roughness. Besides, the investigation of the relation between the roughness and DAF has not been considered yet. Based on this, in this paper, the vehicle random dynamic load has been first simulated by MATLAB based on International Roughness Index (IRI) to analyze the variation law of DAF under the effect of BDP roughness and velocity. Then, the vehicle random dynamic load is applied on the finite element model of a typical bridge to investigate the relation between design parameters of BDPs and pavement stress state.


## 2. Dynamic Model of Vehicle Vibration

The most typical vehicle vibration model used for pavement mechanics analysis is the 2-DOF single-wheel vehicle model [14]. As shown in Figure 1, the total mass of bodywork, spring, and suspension is denoted by ms; mt represents tire mass; cs and ks represent damping and stiffness of spring; and ct and kt represent the damping and stiffness of tires, respectively. Other vehicle vibration models, such as 3-DOF single-wheel and 4-DOF vehicle vibration models, can be seen from Figures 2 and 3, respectively [15]. Though more complicated model such as biaxial 5-DOF and four-axle 10-DOF vibration models [16, 17] are raised by relevant study, 2-DOF single-wheel vehicle model is powerful and persuasive when simulating vehicle dynamic load [18]. Without loss of generality, the 2-DOF single-wheel vehicle model is employed here to analyze DAF. Vehicle system parameters of standard truck used in this paper can be seen from Table 1 [19].Figure 1
Single-wheel double-degree of freedom vehicle mode.Figure 2
Three-mass vehicle mode.Figure 3
Four-degree of freedom vehicle model.Table 1
Vehicle system parameters.
Parameter
Value
m
s
kg
9000
m
t
kg
1000
k
s
N
⋅
m
−
1
480 × 103
c
s
N
⋅
s
⋅
m
−
1
14 × 103
k
t
N
⋅
m
−
1
1900 × 103
c
t
N
⋅
s
⋅
m
−
1
3 × 103According to the coordinate system shown in Figure1, it is supposed that the vertically downward direction is positive, and the unknown displacement vector can be represented as U=−xs,−xtT. Vehicle random dynamic load can be calculated by the following equation:(1)MU¨+CU˙+KU=F¯,where M, U, K are the mass matrix, damping matrix, and stiffness matrix of the vehicle model, respectively, and F is the vertical load of vehicle applied on the bridge deck. Bridges and vehicles are connected by the interaction of tire and deck pavements. The force vector F can be solved by the following equation:(2)F=kiΔi+ciΔ˙i,where ki and ci the  stiffness and damping of the ith tire and Δi is the  vertical contact displacement between the ith tire and pavement surface,(3)Δi=ui−−ri,where ui is the vertical displacement of the ith tire measured from the equilibrium position and ri is the shape function of pavement roughness xg.


## 3. Generation of Deck Pavement Roughness

Erratic fluctuation and pit caused by local damages of pavement are called pavement roughness, which is widely evaluated by International Roughness Index (IRI). According to Paterson’s research, there are six different pavement roughness degrees, where the IRI of theoretical smooth longitudinal profile is 1; the IRI of moderate roughness is about 6, and the IRI of highly rough surfaces is 10 [20]. IRI is a dimensionless number which is customarily described by m/km. In China, deck pavement smoothness is often evaluated by the bump accumulated value BI or variance. Both of the two indexes have good linear correlation with IRI. To simulate a pavement roughness excitation, there are four existing simulated methods for the time-domain model: white noise filtration method, superposition of harmonic (namely, trigonometric series superposition method), AR (autoregression), and AMAR method based on the discrete time sequence and discrete sampling by PSD [21]. In this paper, superposition of the harmonic method is chosen because the algorithmic mechanism is clear, without too much consideration of the discretization degree of the model, and the resulting sample results are continuous. Denote that Gdω is the pavement surface power spectral density expressed in angular frequency, and xg is the stochastic response of deck pavements, which can be expressed as follows:(4)xg=∑k=1MAkcosωkt+ϕk,where M is a positive integral and ϕk is a random variable distributed in [0, 2π]. By substituting ω=2πvn, (4) can be converted to the frequency domain:(5)xg=∑k=1M3.26K0Δnn0nkcos2πvnkt+ϕk,where Δn=nm−n1/M, nm=2, and n1=0.1 and v is the velocity of vehicle. It will be more efficient to use fast Fourier transform (FFT) when simulation [22, 23]. Take M=400, n=1, v=20m/s, when IRI=2; the stochastic response of deck pavement xg is simulated as shown in Figure 4.Figure 4
Pavement simulation with roughness when IRI = 2.After gettingxg, the dynamics differential equation of the 2-DOF vehicle vibration model in Figure 1 can be expressed as follows:(6)msx¨s=−ksxs−xt−csx¨s−x¨t,mtx¨t=ksxs−xt+csx¨s−x¨t−ktxt−xg−ctx¨s−x¨g,where xs and xt are the vertical vibration displacement of suspension and tire, respectively, and xg is the pavement surface roughness excitation displacement. Ft is the random load of vehicle applied on deck pavement,(7)Ft=Fd+G,where Fd is the additional dynamic load of tire and G=ms+mtg, then(8)Fd=ktxt−xg+ctx˙t−x˙g.F
d is shown in Figure 5, and the DAF μ=Fd/G.Figure 5
Additional dynamic load of vehicle when IRI = 2.


## 4. The Relationship betweenμ, v, and IRI

To investigate the effect ofv and IRI on μ, let IRI be 0.5 (nearly absolutely smooth), 2 (smoothness is well), 4 (slightly rough), 6 (medium rough), 8 (highly rough), and 10 (very poor condition), the response curves of μ for different IRIs when v varies from 5 m/s to 35 m/s are shown in Figure 6.Figure 6
The relationship betweenv and μ. (a) IRI = 0.5, (b) IRI = 2, (c) IRI = 4, (d) IRI = 6, (e) IRI = 8, and (f) IRI = 10.
(a)
(b)
(c)
(d)
(e)
(f)When IRI = 0.5, it can be seen from Figure6(a) that with velocity increasing, μ increases gradually at first and then decreases. When v = 20 m/s, μ reaches its maximum value and increases more than doubling compared to v = 5 m/s. Therefore, if deck pavement is very well, the additional dynamic load of vehicle is not ever-increasing but diminishing when v > 20 m/s with speedup; If IRI = 2, the variation curve of μ increases rapidly at first from v = 5 m/s to v = 10 m/s and gets its peak value at v = 10 m/s. Then with speedup, μ began to decrease quickly till v = 15 m/s, and as v exceeds 15 m/s, the curve increases as a whole. As IRI = 4, 6, 8, 10, respectively, we can see some common characteristics clearly from Figures 6(c) and 6(f) that there are two obvious summits and a trough of the line and the maximum will be achieved when v = 10 m/s and minimum when v = 15 m/s. These manners could be explained from the definition equation of additional dynamic load Fd, which is determined by the displacement xg and its derivate x˙g. Because xg and x˙g are the trigonometric functions of velocity v which can be seen from (5), μv must superposed by several trigonometric functions. All of these explains why μ increases and decreases alternately with the variation of v. In fact, the DAF μ fluctuates continuously as v changes.Based on analysis above, we can conclude that deck pavement roughness greatly affects the dynamic amplification factor, and under the same IRI, the response curve ofμ exists two obvious summits and a trough of the line and differs from that of ideal smooth pavement which first increases and then decreases. When a heavy vehicle running over the bridge whose smoothness is general or poor, the velocity should be better controlled at 15 m/s, and once the speed reaches 10 m/s or 25 m/s, the impact effect caused by vehicle vibration would significantly influence the driving comfortableness and service life of deck pavement.If the velocity is constant and IRI is a variation, it can be seen from Figure7 that when IRI < 4, the influence of v on μ for different v is similar, while once IRI > 4, the character of response curves for different v differs significantly.Figure 7
The relationship between IRI andμ.


## 5. The Effect of Deck Pavement Structure on Pavement Stress

In existing researches, many literatures study the influence of deck pavement thickness on control stresses only under dead load or moving constant load of vehicle, regardless of the function of additional dynamic load caused by deck pavement roughness. In this section, the influence of different types of load on control stress response will be considered in order to further study the influence law of pavement structure on control stress.


### 5.1. Thickness of Asphalt Pavement

Taking a simple supported bridge with a span of 20 m and width of 12.6 m as an example, the section of the BDP is shown in Figure8. Then, the response of control stresses under different load when asphalt layer thickness changes from 4 cm to 16 cm is investigated by software ANSYS. In this study, σz and σx represent the longitudinal and transverse tensile stress of asphalt pavement (AP), respectively; σ1 represents the first principle stress of the AP layer (tensile); τxy represents the transverse shear stress in the contact surface between the AP and WP layers; τyz represents the longitudinal shear stress in the contact surface between the AP and WP layers; and σy represents normal tensile stress in the contact surface. These six stresses are collectively referred to as the control stress of the BDP, while τxy, τyz, and σy are interlayer contact stresses.Figure 8
The section of bridge deck pavement.The four response curves in figures below, respectively, stand for vehicle dead load fixed in middle span (dead load), moving load without considering additional dynamic load caused by roughness (moving constant load), vehicle random dynamic load when IRI = 4 andv = 10 m/s (I = 4, v = 10), and vehicle random dynamic load when IRI = 4 and v = 20 m/s (I = 4, v = 20).It can be seen from Figures9–14 that (1) the response regularities of extreme values of σx and σ1 are similar, and both increase linearly with asphalt pavement thickness increasing. Furthermore, the response of random dynamic load is greater than that of dead load and moving constant load. The maximum value of stress response is the curve “I = 4, v = 20” and the response of dead load is no different than moving constant load; (2) the response curve of τxy is escalating first and then descending. The maximum value will be achieved when the thickness is 6 cm. The effect of dead load and moving constant load on τxy is similar and less than random dynamic load; (3) the extreme response of σy and σz is similar under different loads: descending rapidly with thickness increase and later escalating, the minimum value is achieved when thickness is 6 cm; (4) the variation regularity of the response curve of τyz is relatively complicate, and the minimum value will be achieved when thickness is 10 cm.Figure 9
Response curve ofσx.Figure 10
Response curve ofσy.Figure 11
Response curve ofσz.Figure 12
Response curve ofσ1Figure 13
Response curve ofτxy.Figure 14
Response curve ofτyz.Through comprehensive analysis we can conclude that if asphalt pavement is too thin, the extreme response ofσy and σz would be too large; increase thickness properly will contribute to diminish the response, but asphalt layer should not be too thick because it will be bad for the response of σx and σ1; under the effect of stochastic dynamic load, τyz and τxy only reduce by 7.5% and 6.6%, respectively, when thickness increases from 10 cm to 16 cm. Besides, the increasing of thickness could not only augment the self-weight of the bridge but lead to unnecessary waste economically. So, for the composite deck pavement structure, the thickness of asphalt should be better controlled for about 10 cm.


### 5.2. Design Principle

Concrete cushion is paved above girder to protect and level the bridge deck. To analyze its effect on pavement stress, suppose that the thickness of the cushion layer changes within the range of 3 cm to 10 cm.Through analysis we can find that overall increasing cushion thickness can help to decrease pavement stresses to some extent; the extreme responses ofσy, σx and σ1 all descend linearly under different loads while the stress response of dead load and moving constant load is less than the random dynamic load; response curves of τxy, τyz, and σz are reducing greatly when cushion thickness increases from 3 cm to 4 cm and then descends slowly as thickness continues increasing. It follows that the response trend of each stress is relatively similar under action of different loads. The relative reduction of most stress extreme values is biggest when cushion thickness increases from 3 cm to 4 cm. Besides, allowing for the effect of self-weight of bridge caused by increasing cushion layer thickness, the thickness of the cushion layer should be better controlled for about 4 cm.


### 5.3. Elastic Modulus of Deck Pavement Materials

The elastic modulus of common asphalt materials ranges from 1000 MPa to 2000 MPa; through analysis we can find that the extreme values of control stresses all increase in different degrees; the extreme response curves ofσx, σ1, τxy, and τyz have similar variation trend, and the maximum response is caused by the random dynamic load; the relative variation of extreme values of each stress caused by increment of elastic modulus is very close.Enhancing elastic modulus of the cushion layer can diminish pavement stress response to some extent, but the relative reduction of each stress is less than 2% for every 2 GPa, and the elastic modulus of cushion increases. It is because that the local effect on the deck pavement layer is obvious under vehicle loads, and the variation of elastic modulus of cushion can seldom improve the stress state of the upper layer.


## 5.1. Thickness of Asphalt Pavement

Taking a simple supported bridge with a span of 20 m and width of 12.6 m as an example, the section of the BDP is shown in Figure8. Then, the response of control stresses under different load when asphalt layer thickness changes from 4 cm to 16 cm is investigated by software ANSYS. In this study, σz and σx represent the longitudinal and transverse tensile stress of asphalt pavement (AP), respectively; σ1 represents the first principle stress of the AP layer (tensile); τxy represents the transverse shear stress in the contact surface between the AP and WP layers; τyz represents the longitudinal shear stress in the contact surface between the AP and WP layers; and σy represents normal tensile stress in the contact surface. These six stresses are collectively referred to as the control stress of the BDP, while τxy, τyz, and σy are interlayer contact stresses.Figure 8
The section of bridge deck pavement.The four response curves in figures below, respectively, stand for vehicle dead load fixed in middle span (dead load), moving load without considering additional dynamic load caused by roughness (moving constant load), vehicle random dynamic load when IRI = 4 andv = 10 m/s (I = 4, v = 10), and vehicle random dynamic load when IRI = 4 and v = 20 m/s (I = 4, v = 20).It can be seen from Figures9–14 that (1) the response regularities of extreme values of σx and σ1 are similar, and both increase linearly with asphalt pavement thickness increasing. Furthermore, the response of random dynamic load is greater than that of dead load and moving constant load. The maximum value of stress response is the curve “I = 4, v = 20” and the response of dead load is no different than moving constant load; (2) the response curve of τxy is escalating first and then descending. The maximum value will be achieved when the thickness is 6 cm. The effect of dead load and moving constant load on τxy is similar and less than random dynamic load; (3) the extreme response of σy and σz is similar under different loads: descending rapidly with thickness increase and later escalating, the minimum value is achieved when thickness is 6 cm; (4) the variation regularity of the response curve of τyz is relatively complicate, and the minimum value will be achieved when thickness is 10 cm.Figure 9
Response curve ofσx.Figure 10
Response curve ofσy.Figure 11
Response curve ofσz.Figure 12
Response curve ofσ1Figure 13
Response curve ofτxy.Figure 14
Response curve ofτyz.Through comprehensive analysis we can conclude that if asphalt pavement is too thin, the extreme response ofσy and σz would be too large; increase thickness properly will contribute to diminish the response, but asphalt layer should not be too thick because it will be bad for the response of σx and σ1; under the effect of stochastic dynamic load, τyz and τxy only reduce by 7.5% and 6.6%, respectively, when thickness increases from 10 cm to 16 cm. Besides, the increasing of thickness could not only augment the self-weight of the bridge but lead to unnecessary waste economically. So, for the composite deck pavement structure, the thickness of asphalt should be better controlled for about 10 cm.


## 5.2. Design Principle

Concrete cushion is paved above girder to protect and level the bridge deck. To analyze its effect on pavement stress, suppose that the thickness of the cushion layer changes within the range of 3 cm to 10 cm.Through analysis we can find that overall increasing cushion thickness can help to decrease pavement stresses to some extent; the extreme responses ofσy, σx and σ1 all descend linearly under different loads while the stress response of dead load and moving constant load is less than the random dynamic load; response curves of τxy, τyz, and σz are reducing greatly when cushion thickness increases from 3 cm to 4 cm and then descends slowly as thickness continues increasing. It follows that the response trend of each stress is relatively similar under action of different loads. The relative reduction of most stress extreme values is biggest when cushion thickness increases from 3 cm to 4 cm. Besides, allowing for the effect of self-weight of bridge caused by increasing cushion layer thickness, the thickness of the cushion layer should be better controlled for about 4 cm.


## 5.3. Elastic Modulus of Deck Pavement Materials

The elastic modulus of common asphalt materials ranges from 1000 MPa to 2000 MPa; through analysis we can find that the extreme values of control stresses all increase in different degrees; the extreme response curves ofσx, σ1, τxy, and τyz have similar variation trend, and the maximum response is caused by the random dynamic load; the relative variation of extreme values of each stress caused by increment of elastic modulus is very close.Enhancing elastic modulus of the cushion layer can diminish pavement stress response to some extent, but the relative reduction of each stress is less than 2% for every 2 GPa, and the elastic modulus of cushion increases. It is because that the local effect on the deck pavement layer is obvious under vehicle loads, and the variation of elastic modulus of cushion can seldom improve the stress state of the upper layer.


## 6. Conclusion and Discussion

Existing researches of vehicle-bridge vibration only consider the interaction of vehicle and bridge and seldom take deck pavement into consideration. Meanwhile, the overloaded vehicles’ additional dynamic load caused by pavement roughness degree is also neglected by designers during bridge’s operating process. So, accurately analyzing the dynamic amplification factor caused by IRI andv is meaningful and necessary. In first part, this study investigated the relationship between μ with IRI and v, while the second part analyzed the influence of different types of load on control stress response and further studied the influence regularity of pavement structure on control stress so as to study the influence factor and change law of dynamic load deeply and to further guide the design of pavement structure safely and reliably. The main results are shown as follows:(1)
The roughness of BDP greatly affects the dynamic amplification factor, and under the same IRI, the response curve ofμ exists between two obvious summits and a trough of the line and differs from that of ideal smooth pavement which first increases and then decreases. When a heavy vehicle running over the bridge whose smoothness is general or poor, the velocity should be better controlled at 15 m/s, and once the speed reaches 10 m/s or 25 m/s, the impact effect caused by vehicle vibration would significantly influence the driving comfortableness and service life of deck pavement(2)
The increasing of thickness could not only augment the self-weight of bridge but lead to unnecessary waste economically. The thickness of asphalt should be better controlled about 10 cm.(3)
Increasing cushion thickness can help to decrease pavement stresses to some extent, and allowing for the effect of self-weight of bridge caused by increasing the cushion layer thickness, the thickness of the cushion layer should be better controlled about 4 cm.(4)
Enhancing elastic modulus of the cushion layer can diminish pavement stress response to some extent, but the relative reduction of each stress is less than 2% for every 2 GPa, and the elastic modulus of cushion increases. Thus, it is uneconomical to decrease pavement stress by using high-grade concrete cushion.



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*Source: 9810461-2018-04-01.xml*