# A Rapid Detection Method for Bridges Based on Impact Coefficient of Standard Bumping

Authors: Xiaolan Liu; Xianmin Zhang; Yadong Wang
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

The evaluation method of load transfer efficiency using falling weight deflectometer is unworkable in remote mountain areas and transportation difficult region. Therefore, a novation method of load transfer efficiency evaluation is proposed using the index of amplitude ratio. Finite element method is applied to study the influence of dowel bar parameters (diameter, length, spacing, and elastic modulus) and pavement structures parameters (thickness and modulus) on load transfer efficiency, frequency, and the ratio of amplitude. Results of finite element model show that the effects of dowel bar and pavement structure parameters on load transfer efficiency and the ratio of amplitude are similar. The load transfer efficiency, frequency, and the ratio of amplitude enhance with the increase of dowel bar diameter, length, and elastic modulus and the decrease of dowel bar spacing. The subgrade modulus has more significant influence on the load transfer efficiency, frequency, and the ratio of amplitude than other pavement parameters. Polynomial function method is utilized to established load transfer mode between deflection-based load transfer efficiency and the ratio of amplitude. The feasibility and reliability of new method is verified by static and dynamic load test. All results are helpful for the development of highway engineering and airport engineering.


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## Body

## 1. Introduction

Structural and functional defects, damage, and failure appear inordinately in bridges through the combined action of a moving vehicle and the environment. Thus, regular tests are necessary to guarantee the security and serviceability of bridges. Static load testing, as a main detection method of bridges, is backed by well-developed theory and technology; however, it is costly and time consuming and causes serious traffic congestion, which is hazardous when the traffic flow cannot be easily suspended to carry out load tests [1–3]. Thereby, dynamic load testing is now of considerable interest to bridge engineers, because it is faster and more efficient than its static equivalent [4, 5]. Moreover, it causes less traffic congestion by using moving vehicles as a continuous source of energy for the bridge [6, 7]. González et al. (2010) [8], He et al. (2014) [9], and Podworna and Klasztorny (2014) [10] found that the results of dynamic load testing are inevitably influenced by deck conditions, vehicle, and stiffness and form of bridges. Park et al. (2005) [11] researched the function of roughness on dynamic load testing through full-scale field tests. Ding et al. (2009) [12] explored the relationship between deck roughness and dynamic load coefficient through evolutionary spectral method. Liu et al. (2015) [13] and Jiang et al. (2011) [14] emphasized the influence of speed and deck roughness on dynamic load testing. Huang et al. (2011) [15] showed that fundamental frequency and impact coefficient have close correlation in dynamic load testing. Kim et al. (2007) [16] reported that bump height is a key factor of impact coefficient in dynamic load testing. Qi et al. [17] (2013) studied the effect factor of width–span ratio on dynamic load testing. Wang and Jiang (2016) [18] analyzed the effect factors of dynamic load testing, and they were able to obtain bridge health situation using the smallest vibration impact coefficient.However, existing dynamic load testing methods were not able to achieve the reliability and accuracy requirements of static load testing. Therefore, the main objective of our study is to test and analyze bridges displacement and strain with moving vehicle to obtain loading efficiency. As the influence of deck roughness on bridge dynamic tests must be considered, although difficult to quantitatively describe, a new method of standard bumping is proposed to desensitize the effect of deck roughness on vehicle–bridge interaction. Moreover, the software of MATLAB was used to solve coupled vibration equation caused by vehicle–bridge interaction, and the factors of vehicle–bridge coupled vibration were explored further based on dynamic load testing and standard bumping testing. The results of numerical simulation and experiments demonstrate that standard bumping tests effectively decreased the influence of deck roughness and avoid the difficulty of quantitative deck roughness, which in turn leads to the realization of rapid and accurate detection of load-carrying capacity of bridges.


## 2. Theoretical Analysis

### 2.1. Load Carrying Capacity Assessment of Bridges

Static load testing is the most direct and effective method to assess the load-carrying capacity of bridges, in which static loads are placed on bridges in the most unfavorable loading way to measure the actual stress, strain, and displacement of bridges [19]. The ratio between static load testing and theory calculating values is defined as the calibration coefficient of bridges to assess bridge load-carrying capacity, in which(1)η=SeSsHere, η is the calibration coefficient of bridges, Se states the measuring values in static load testing, and Ss represents the theory calculating values. When η is more than 1, bridges are seriously damaged and the load-carrying capacity of bridges would be insufficient. Otherwise, bridges are safe and have enough load-carrying capacity. The smaller the value of η is, the safer the bridge structure would be.As compared with static load testing, dynamic load testing not only surpasses disadvantages, but it also establishes the relationship of detection results in static and dynamic load testing: [20](2)1+μ=SdSewhere 1+μ is the impact coefficient of bridges and Sd represents the measuring values in dynamic load testing.It is known that the key in dynamic load testing is to obtain suitable impact coefficients to calculate static stress, strain, and displacement of bridges. Nonetheless, existing dynamic load testing methods show difficulty in deciding vehicle load to satisfy complete bridge evaluation and loading efficiency such as that when 1+μ is 1.5, a 20-T vehicle load in dynamic load testing would be equal to 30-T vehicle load in static load testing.Thus, we propose a standard bumping method in which a standard vehicle passes over bridges and bumps over a standard height wooden plank placed at the midspan position of the bridge; the bridge evaluation is completed by inserting the measured values of dynamic load testing into (2) and by taking the calculated static values from (2) into (1).


### 2.2. Vehicle–Bridge Interaction Model

In vehicle–bridge interaction model, vehicle loads and bridges are simplified as single-degree-of-freedom mass-spring system and one-dimensional structure, respectively, in which time (t) and distance along the bridge axis (x), instead of bridge width, are considered in vehicle–bridge interaction equation (Figure1).Figure 1
Vehicle–bridge interaction model.The factors affecting impact coefficient include vehicle speed, deck roughness, stiffness, and structural form of bridges. However, the effect of random deck roughness on impact coefficient of bridges is difficult to determine and test. Thus, vehicle–bridge coupling vibration partial differential equations, including the influence of deck roughness, are established based on D’ Alembert’s principle [21].(3)EI∂4y∂x4+m∂2y∂t2=Mg+kz-y-r+cz′-y′-y′v-r′vδx-vt(4)Md2zdt2+kz-y-r+cz′-y′-y′v-r′v=0Here, m, E, I,and y represent the unit length quality (kg/m), elastic modulus (Pa), bending inertia moments (m4), and deflection functions (m) of simply supported girder bridge, respectively. M, z, k, c, and v represent vehicle mass (kg), vehicle quality displacement relative to equilibrium position (m), vehicle suspension spring stiffness (N/m), spring damping (N•s/m), and speed (m/s), respectively. Lastly, r and δ are deck roughness (m) and Delta–Dirac function, respectively.According to the principle of modal superposition(5)y=∑n=1Nφnxqnt(6)φnx=sin⁡nπxLBy taking (5) and (6) into (3) and (4), the orthogonality of vibration mode function would be(7)∫0Lmφixφjxdx=0i≠j(8)∫0LEIφix′′φjx′′dx=0i≠jBoundary conditions: The beam displacement is zero at both ends, and the bending moment is equal to zero, (9)y0,t=∂2y0,t∂x2=yL,t=∂2yL,t∂x2=0Equations (3) and (4) can be simplified into(10)qit′′+ωi2qit=2mLMg+kz-∑n=1Nsin⁡nπxLqnt-rvt+cz′-∑n=1Nsin⁡nπxLqnt′-v∑n=1NnπL×sin⁡nπxLqnt-rvt′vsin⁡iπxL(11)Mz′′+kz-∑n=1Nsin⁡nπxLqnt-rvt+cz′-∑n=1Nsin⁡nπxLqnt′-v∑n=1NnπL∗sin⁡nπxLqnt-rvt′v=0Through the transformation, the complex problem solving of vehicle–bridge coupling vibration partial differential equations is converted into the problem solving of ordinary differential equations. Based on given initial conditions, which is qi(0)=0, qi(0)′=0, zi(0)=0, and zi0′=0, the fourth-order Runge–Kutta algorithm was used to solve the vehicle–bridge coupled vibration partial differential equations. Then, time series of qi(t), qi(t)′, zi(t), and zit′ were obtained by MATLAB program, and the numerical solutions of (3) and (4) were determined. Finally, the solution of vehicle–bridge coupling systems was completed.


### 2.3. Deck Roughness

Deck roughness refers to the deviation of the ideal flat surface, and it is a zero mean and ergodic stationary Gauss random process, as described by roughness power spectral density: [22](12)Gxn=Gxn′nn′-wwhere,n and n′ represent spatial frequency and benchmark spatial frequency (Hz·s/m) and Gx(n) and Gxn′ state power spectral densities ofnand n′ (10−6m2(m−1)−1), which are shown in Table 1. w is frequency index, which descripts the trend of power spectral density that is usually 2.Table 1
Deck roughness coefficientGxn′ of different levels.
Level Gx(n′) (10−6m2(m-1)-1) Level Gx(n′) (10−6m2(m-1)-1) A 16 E 4,096 B 64 F 16,384 C 256 G 65,536 D 1,024 H 262,144Due to isolation effects of vehicle suspension system, effective spatial frequency interval (i.e., (nu, nl)) has significant effects on vehicle vibration and is calculated using (13). Equation (12) is solved by trigonometric series method as shown in (14) and (15).(13)f=vnHere,f states vehicle vibration frequency, and v represents vehicle speed.(14)rx=∑k=1NAnkcos⁡2πnkx+ϕk(15)Ank=4GxnkΔnHere, nk is a discrete value in effective spatial frequency interval, and nk=nl+(k-0.5)×Δn. Φk states random phase angle in uniform distribution of 0-2π (rad). Ank represents trigonometric function amplitude of nk. Δn is the resolution of effective spatial frequency, Δn=(nu-nl)/N.


## 2.1. Load Carrying Capacity Assessment of Bridges

Static load testing is the most direct and effective method to assess the load-carrying capacity of bridges, in which static loads are placed on bridges in the most unfavorable loading way to measure the actual stress, strain, and displacement of bridges [19]. The ratio between static load testing and theory calculating values is defined as the calibration coefficient of bridges to assess bridge load-carrying capacity, in which(1)η=SeSsHere, η is the calibration coefficient of bridges, Se states the measuring values in static load testing, and Ss represents the theory calculating values. When η is more than 1, bridges are seriously damaged and the load-carrying capacity of bridges would be insufficient. Otherwise, bridges are safe and have enough load-carrying capacity. The smaller the value of η is, the safer the bridge structure would be.As compared with static load testing, dynamic load testing not only surpasses disadvantages, but it also establishes the relationship of detection results in static and dynamic load testing: [20](2)1+μ=SdSewhere 1+μ is the impact coefficient of bridges and Sd represents the measuring values in dynamic load testing.It is known that the key in dynamic load testing is to obtain suitable impact coefficients to calculate static stress, strain, and displacement of bridges. Nonetheless, existing dynamic load testing methods show difficulty in deciding vehicle load to satisfy complete bridge evaluation and loading efficiency such as that when 1+μ is 1.5, a 20-T vehicle load in dynamic load testing would be equal to 30-T vehicle load in static load testing.Thus, we propose a standard bumping method in which a standard vehicle passes over bridges and bumps over a standard height wooden plank placed at the midspan position of the bridge; the bridge evaluation is completed by inserting the measured values of dynamic load testing into (2) and by taking the calculated static values from (2) into (1).


## 2.2. Vehicle–Bridge Interaction Model

In vehicle–bridge interaction model, vehicle loads and bridges are simplified as single-degree-of-freedom mass-spring system and one-dimensional structure, respectively, in which time (t) and distance along the bridge axis (x), instead of bridge width, are considered in vehicle–bridge interaction equation (Figure1).Figure 1
Vehicle–bridge interaction model.The factors affecting impact coefficient include vehicle speed, deck roughness, stiffness, and structural form of bridges. However, the effect of random deck roughness on impact coefficient of bridges is difficult to determine and test. Thus, vehicle–bridge coupling vibration partial differential equations, including the influence of deck roughness, are established based on D’ Alembert’s principle [21].(3)EI∂4y∂x4+m∂2y∂t2=Mg+kz-y-r+cz′-y′-y′v-r′vδx-vt(4)Md2zdt2+kz-y-r+cz′-y′-y′v-r′v=0Here, m, E, I,and y represent the unit length quality (kg/m), elastic modulus (Pa), bending inertia moments (m4), and deflection functions (m) of simply supported girder bridge, respectively. M, z, k, c, and v represent vehicle mass (kg), vehicle quality displacement relative to equilibrium position (m), vehicle suspension spring stiffness (N/m), spring damping (N•s/m), and speed (m/s), respectively. Lastly, r and δ are deck roughness (m) and Delta–Dirac function, respectively.According to the principle of modal superposition(5)y=∑n=1Nφnxqnt(6)φnx=sin⁡nπxLBy taking (5) and (6) into (3) and (4), the orthogonality of vibration mode function would be(7)∫0Lmφixφjxdx=0i≠j(8)∫0LEIφix′′φjx′′dx=0i≠jBoundary conditions: The beam displacement is zero at both ends, and the bending moment is equal to zero, (9)y0,t=∂2y0,t∂x2=yL,t=∂2yL,t∂x2=0Equations (3) and (4) can be simplified into(10)qit′′+ωi2qit=2mLMg+kz-∑n=1Nsin⁡nπxLqnt-rvt+cz′-∑n=1Nsin⁡nπxLqnt′-v∑n=1NnπL×sin⁡nπxLqnt-rvt′vsin⁡iπxL(11)Mz′′+kz-∑n=1Nsin⁡nπxLqnt-rvt+cz′-∑n=1Nsin⁡nπxLqnt′-v∑n=1NnπL∗sin⁡nπxLqnt-rvt′v=0Through the transformation, the complex problem solving of vehicle–bridge coupling vibration partial differential equations is converted into the problem solving of ordinary differential equations. Based on given initial conditions, which is qi(0)=0, qi(0)′=0, zi(0)=0, and zi0′=0, the fourth-order Runge–Kutta algorithm was used to solve the vehicle–bridge coupled vibration partial differential equations. Then, time series of qi(t), qi(t)′, zi(t), and zit′ were obtained by MATLAB program, and the numerical solutions of (3) and (4) were determined. Finally, the solution of vehicle–bridge coupling systems was completed.


## 2.3. Deck Roughness

Deck roughness refers to the deviation of the ideal flat surface, and it is a zero mean and ergodic stationary Gauss random process, as described by roughness power spectral density: [22](12)Gxn=Gxn′nn′-wwhere,n and n′ represent spatial frequency and benchmark spatial frequency (Hz·s/m) and Gx(n) and Gxn′ state power spectral densities ofnand n′ (10−6m2(m−1)−1), which are shown in Table 1. w is frequency index, which descripts the trend of power spectral density that is usually 2.Table 1
Deck roughness coefficientGxn′ of different levels.
Level Gx(n′) (10−6m2(m-1)-1) Level Gx(n′) (10−6m2(m-1)-1) A 16 E 4,096 B 64 F 16,384 C 256 G 65,536 D 1,024 H 262,144Due to isolation effects of vehicle suspension system, effective spatial frequency interval (i.e., (nu, nl)) has significant effects on vehicle vibration and is calculated using (13). Equation (12) is solved by trigonometric series method as shown in (14) and (15).(13)f=vnHere,f states vehicle vibration frequency, and v represents vehicle speed.(14)rx=∑k=1NAnkcos⁡2πnkx+ϕk(15)Ank=4GxnkΔnHere, nk is a discrete value in effective spatial frequency interval, and nk=nl+(k-0.5)×Δn. Φk states random phase angle in uniform distribution of 0-2π (rad). Ank represents trigonometric function amplitude of nk. Δn is the resolution of effective spatial frequency, Δn=(nu-nl)/N.


## 3. Impact Coefficient Analysis of Standard Bumping

### 3.1. Factors Influencing Impact Coefficient

The physical parameters of simply supported girder bridges are as follows: flexural rigidity isEI=0.75e10 Pa•m4, actual span isL=13 m, and unit mass ism=14,700 kg/m. The physical parameters of vehicles are as follows: mass isM=24,000 kg; suspension spring rigidity isk=2.8e6 N/m; spring damp isc=24,000 N•s/m; and vehicle speed isv=2m/s,v=4m/s,v=6m/s,v=8m/s,v=10m/s,v=20m/s,v=30m/s, andv=40m/s. Midspan displacement is 1.43 mm with stationary vehicle load placed on the middle of bridges. The results of vehicle passing bridges at a uniform speed are shown in Figure 2 and Table 2.Table 2
Maximum dynamic displacement and impact coefficient of midspan without deck roughness level.
Vehicle speed (m/s) Maximum dynamic displacement (mm) Impact coefficient 2 1.45 1.01 4 1.46 1.02 6 1.47 1.03 8 1.49 1.04 10 1.50 1.05 20 1.55 1.08 30 1.66 1.16 40 1.69 1.18Figure 2
(a) Midspan displacement of simply supported girder bridges with 2-10m/s. (b) Midspan displacement of simply supported girder bridges with 10-40m/s.
(a) (b)Figure2 and Table 2 show that impact coefficient moderately increased with vehicle speed growth and that growth margin is minimal at speed of less than 10 m/s. When vehicle speed is less than 20 m/s, the impact coefficient variation of unit vehicle speed is 0.38%, and when vehicle speed is between 20 m/s and 30 m/s, the impact coefficient variation of unit vehicle speed is 0.80%; furthermore, when vehicle speed is between 30 m/s and 40 m/s, the impact coefficient variation of unit vehicle speed is 0.20%. Therefore, the influence of vehicle speed on the impact coefficient is negligible when the speed is less than 10 m/s.The physical parameters of simply supported girder bridges and vehicles are as aforementioned, and deck roughness levels are assigned as A, B, C, D, E, and F. The results of vehicle passing bridges at the speed of 4 m/s are presented in Figure3 and Table 3.Table 3
Maximum dynamic displacement and impact coefficient of midspan with different deck roughness levels.
Deck roughness level Maximum dynamic displacement (mm) Impact coefficient A 1.53 1.07 B 1.73 1.21 C 2.03 1.42 D 3.32 2.32 E 6.06 4.23 F 9.59 6.71Figure 3
Midspan displacement of simply supported girder bridges with different deck roughness levels.Figure3 and Table 3 illustrate that deck roughness level significantly affected impact coefficient, which gradually increased with deck roughness levels ranging from A to F. Deck roughness levels A, B, C, and D are common in actual bridges, and impact coefficient moderately varies with deck roughness from A to D.Equations (3) and (4) state that both vehicle speed and deck roughness influence vehicle–bridge interaction. Vehicle speed is a quantitative value, whereas deck roughness is random and difficult to be accurately measured. Hence, statistical method is used to analyze the influence of deck roughness on vehicle–bridge interaction. The physical parameters of simply supported girder bridges and vehicles are as aforementioned and vehicle speed isv=2m/s,v=4m/s,v=6m/s,v=8m/s, andv=10m/s. One-hundred deck roughness samples of deck roughness levels from A to D were randomly generated to calculate the impact coefficient of the midspan position. The results of vehicle passing bridges at uniform speed are shown in Figure 4 and Table 4.Table 4
Impact coefficients with different deck roughness levels (without standard bumping).
Vehicle speed(m/s) Absolute smooth Deck roughness level of A Deck roughness level of B Deck roughness level of C Deck roughness level of D Mean Variance Mean Variance Mean Variance Mean Variance 2 1.01 1.05 0.0005 1.09 0.0013 1.21 0.0069 1.43 0.0245 4 1.02 1.09 0.0015 1.18 0.0054 1.42 0.0246 1.95 0.1012 6 1.03 1.09 0.0021 1.20 0.0061 1.46 0.0312 1.94 0.1047 8 1.04 1.10 0.0027 1.19 0.0053 1.41 0.0263 1.92 0.0936 10 1.05 1.10 0.0033 1.18 0.0111 1.43 0.0472 1.85 0.1349Figure 4
Probability density curve of impact coefficient at the speed of 10 m/s without standard bumping.Figure4 and Table 4 show that mean and variance of impact coefficient gradually increased with deck roughness levels ranging from A to D. Moreover, impact coefficient became more discrete as deck roughness level was reduced. Therefore, deck roughness has a significant and random effect on impact coefficient.As fundamental frequency is a function of bending stiffness (EI), span (l), and mass of unit length (m), the relationship between fundamental frequency and impact coefficient can reflect the dynamic characteristics of vehicle–bridge interaction vibration. Taking three hollow slab bridges and four T beam bridges (physical parameters of bridges shown in Table 5) as examples, the physical parameters of vehicles are as aforementioned and vehicle speed isv=4m/s. The results with absolute smooth deck are shown in Table 6.Table 5
Physical parameters and fundamental frequency of bridges.
Code Section form E I (pa·m4) m (kg/m) l (m) f (Hz) 1 hollow slab 1.84E+10 14,000 19.0 4.99 2 hollow slab 2.78E+10 17000 17.4 6.63 3 hollow slab 1.11E+10 13,000 13.0 8.59 4 T beam 1.49E+10 9,000 20.3 4.90 5 T beam 1.42E+10 8,000 17.8 6.61 6 T beam 1.20E+10 8,000 16.3 7.24 7 T beam 7.99E+09 7,000 10.3 15.82Table 6
Physical parameters and impact coefficient of bridges.
Code f(Hz) Dynamic displacement (mm) Static displacement (mm) Impact coefficient 1 4.99 1.856 1.827 1.016 2 6.63 0.941 0.929 1.013 3 8.59 0.983 0.970 1.013 4 4.90 2.789 2.751 1.014 5 6.61 1.969 1.946 1.012 6 7.24 1.789 1.768 1.012 7 15.82 0.675 0.670 1.007In Tables5 and 6, the maximum impact coefficient of seven bridges is equal to 1.015; low speed is beneficial in reducing the influence of deck roughness on impact coefficient with different section form, bending stiffness, span, and mass of unit length. Impact coefficients were reduced with the increase in fundamental frequency of the same section form bridges; however, the correlation of impact coefficient and fundamental frequency are not evident for different section form bridges. Thereby, it is necessary to analyze the variation of impact coefficient with different section form bridges.


### 3.2. Validation of Impact Coefficient Based on Standard Bumping

Approximate closed-form solutions were determined for the vehicle and bridge based on the following practical assumptions:(1) the vehicle remains in contact with the bridge at all times, as it is being driven and during bumping; (2) the effect of the wooden plank on deck roughness is negligible; (3) the spring–mass system executes vertical free fall with a certain initial velocity after leaving the wooden plank; and (4) the gravitational potential energy of mass is completely converted into kinetic energy in the process of bumping, with no loss of energy. The standard bump model is shown in Figure 5. As external force applied to a simply supported bridge directly determines its displacement, velocity, and acceleration with respect to the equilibrium position, the movement of mass in spring–mass system is the focus of analysis. Taking the equilibrium position of spring–mass system in its initial state as the origin of the coordinate system and the under orientation as the positive direction, the velocity and displacement of the mass after bumping were determined as z(i+1)′=z(i)′+2×9.8×h andz(i+1) =z(i), respectively, where h is the bumping height and z′(i) andz(i) are the initial velocity and displacement of mass before bumping, respectively.Figure 5
Standard bumping model.The physical parameters of simply supported girder bridges and vehicles are as aforementioned. The deck roughness levels are A, B, C, and D; vehicle speed isv=2m/s,v=4m/s, v=6m/s,v=8m/s, andv=10m/s, and bumping height is 5 cm, 10 cm, and 15 cm. The results of 5 cm bumping height are shown in Figure 6 and Table 7, and the results of 15 cm bumping height are shown in Table 8.Table 7
Impact coefficients with different deck roughness levels (with 5 cm bumping height).
Vehicle speed(m/s) Absolute smooth Deck roughness level of A Deck roughness level of B Deck roughness level of C Deck roughness level of D Mean Variance Mean Variance Mean Variance Mean Variance 2 2.32 2.33 0.0011 2.32 0.0040 2.33 0.0240 2.38 0.0575 4 2.33 2.33 0.0046 2.37 0.0157 2.32 0.0569 2.51 0.1779 6 2.32 2.32 0.0049 2.33 0.0188 2.35 0.0969 2.66 0.2658 8 2.26 2.26 0.0056 2.26 0.0220 2.35 0.0883 2.54 0.2211 10 2.26 2.27 0.0051 2.28 0.0171 2.31 0.820 2.59 0.2101Table 8
Impact coefficients with different deck roughness levels (with 15 cm bumping height).
Vehicle speed(m/s) Absolute smooth Deck roughness level of A Deck roughness level of B Deck roughness level of C Deck roughness level of D Mean Variance Mean Variance Mean Variance Mean Variance 2 3.29 3.30 0.0011 3.30 0.0045 3.29 0.0159 3.30 0.0566 4 3.31 3.31 0.0053 3.32 0.0209 3.32 0.0679 3.35 0.1932 6 3.29 3.29 0.0053 3.28 0.0197 3.31 0.0632 3.46 0.3103 8 3.22 3.22 0.0075 3.21 0.0186 3.21 0.1130 3.43 0.2943 10 3.22 3.23 0.0042 3.20 0.0222 3.26 0.0928 3.37 0.2464Figure 6
Probability density curve of impact coefficient at the speed of 10 m/s with 5-cm bumping height.Comparing Figures4 and 6, the standard bumping test reduced the discreteness of impact coefficient, which can effectively reduce the influence of deck roughness on impact coefficient. Table 7 demonstrates that the relative variation ratio of impact coefficient, which is the ratio between the differences of maximum and minimum values and average values of impact coefficient, is <4.78% at speeds of ≤10 m/s and for bump height of 5 cm with relatively smooth decks (levels A and B). However, the relative variation ratio of impact coefficient is <3.68% at speeds ≤10 m/s for bump height of 15 cm with rougher decks (levels C and D) in Table 8. Thus, lower bump height can reduce the influence of deck roughness levels A and B on impact coefficient, whereas higher bump height can reduce the influence of deck roughness levels C and D on impact coefficient at the speed of less than 10 m/s.To further verify the reliability and accuracy of standard bumping, two pieces of reinforced rectangular concrete section beams were made with C25 concrete (32.5 ordinary Portland cement, with density of 2,500 kg/m3 and cement, sand, stone, and water quality ratio of 1:1.44:2.67:0.5), HRB335 steel, and HPB235 stirrup. As shown in Table9 and Figures 7-8, the calculation spans of 1# and 2# simply supported girder bridges were both 3,300 mm. The bending inertia moments of 1# and 2# simply supported girder bridges were 4.32 e6mm4 and 9.36 e6mm4, respectively. The elastic moduli of 1# and 2# test beams were 2.52 e10Pa and 1.89 e10Pa, respectively, with a 30-kg concrete block and dial indicator applied in the bottom of midspan beam.Table 9
Parameter of two test beams of simply supported girder bridge.
Beam code Calculationspan (mm) Sectionsize (mm) Main reinforcement diameter (mm) Main reinforcement quantity Stirrup diameter (mm) Stirrup spacing (mm) 1# 3,500 300×80 12 3 6 120 2# 3,500 600×80 12 5 6 130Figure 7
Section diagram of (a) 1# test beam and (b) 2# test beam.
(a) (b)Figure 8
Preparation of (a) 1# test beam and (b) 2# test beam.
(a) (b)In Figure9, the two test beams were tested by DHDAS-5907 bridge modal apparatus and analysis system. Three speed sensors, which were placed at 1/4, 1/2, and 3/4 span of the test beams, were used to collect vibration speed signal for 6-8 mins. The fundamental frequencies of 1# and 2# test beams were determined as 7.36 HZ (as shown in Figure 10) and 6.38 HZ through spectrum analysis of vibration speed signal. The section sizes of 1# and 2# test beams were 0.080 × 0.3 m2 and 0.080 × 0.6 m2, but the calculation spans of 1# and 2# test beams were both 3.3m. The elastic modulus of 1# and 2# test beams were 2.52 e10Pa and 1.89 e10Pa. The Poisson ratios of 1# and 2# test beams were both 0.2. The fundamental frequencies of 1# and 2# test beams were 7.51 Hz (as shown in Figure 11) and 6.75 Hz as measured by Block Lanczos modal method with errors of 2.04% and 5.80%, respectively, compared with measured frequency. Thus, the measurements verify that the two beam models are appropriate.Figure 9
Laboratory test of standard bumping.
(a) (b)Figure 10
Measured frequency of 1# test beam.Figure 11
Calculated frequency of 1# test beam.The physical parameters of vehicles are as follows: front axle and rear axle mass were 10 kg and 45 kg, respectively; distance between front axle and rear axle was 1.25 m; spring stiffness and tire stiffness of front axle were 5 KN/m and 12 KN/m, respectively. Physical parameters of 1# test beam are as aforementioned. The static displacement was measured as 2.02 mm by static load testing. The dynamic displacement of measured value and calculated value were 4.94 mm and 4.77 mm, respectively; and impact coefficients were 2.45 and 2.36, with 3.67% error, thus proving that standard bumping method can exactly simulate vehicle–bridge interaction.


## 3.1. Factors Influencing Impact Coefficient

The physical parameters of simply supported girder bridges are as follows: flexural rigidity isEI=0.75e10 Pa•m4, actual span isL=13 m, and unit mass ism=14,700 kg/m. The physical parameters of vehicles are as follows: mass isM=24,000 kg; suspension spring rigidity isk=2.8e6 N/m; spring damp isc=24,000 N•s/m; and vehicle speed isv=2m/s,v=4m/s,v=6m/s,v=8m/s,v=10m/s,v=20m/s,v=30m/s, andv=40m/s. Midspan displacement is 1.43 mm with stationary vehicle load placed on the middle of bridges. The results of vehicle passing bridges at a uniform speed are shown in Figure 2 and Table 2.Table 2
Maximum dynamic displacement and impact coefficient of midspan without deck roughness level.
Vehicle speed (m/s) Maximum dynamic displacement (mm) Impact coefficient 2 1.45 1.01 4 1.46 1.02 6 1.47 1.03 8 1.49 1.04 10 1.50 1.05 20 1.55 1.08 30 1.66 1.16 40 1.69 1.18Figure 2
(a) Midspan displacement of simply supported girder bridges with 2-10m/s. (b) Midspan displacement of simply supported girder bridges with 10-40m/s.
(a) (b)Figure2 and Table 2 show that impact coefficient moderately increased with vehicle speed growth and that growth margin is minimal at speed of less than 10 m/s. When vehicle speed is less than 20 m/s, the impact coefficient variation of unit vehicle speed is 0.38%, and when vehicle speed is between 20 m/s and 30 m/s, the impact coefficient variation of unit vehicle speed is 0.80%; furthermore, when vehicle speed is between 30 m/s and 40 m/s, the impact coefficient variation of unit vehicle speed is 0.20%. Therefore, the influence of vehicle speed on the impact coefficient is negligible when the speed is less than 10 m/s.The physical parameters of simply supported girder bridges and vehicles are as aforementioned, and deck roughness levels are assigned as A, B, C, D, E, and F. The results of vehicle passing bridges at the speed of 4 m/s are presented in Figure3 and Table 3.Table 3
Maximum dynamic displacement and impact coefficient of midspan with different deck roughness levels.
Deck roughness level Maximum dynamic displacement (mm) Impact coefficient A 1.53 1.07 B 1.73 1.21 C 2.03 1.42 D 3.32 2.32 E 6.06 4.23 F 9.59 6.71Figure 3
Midspan displacement of simply supported girder bridges with different deck roughness levels.Figure3 and Table 3 illustrate that deck roughness level significantly affected impact coefficient, which gradually increased with deck roughness levels ranging from A to F. Deck roughness levels A, B, C, and D are common in actual bridges, and impact coefficient moderately varies with deck roughness from A to D.Equations (3) and (4) state that both vehicle speed and deck roughness influence vehicle–bridge interaction. Vehicle speed is a quantitative value, whereas deck roughness is random and difficult to be accurately measured. Hence, statistical method is used to analyze the influence of deck roughness on vehicle–bridge interaction. The physical parameters of simply supported girder bridges and vehicles are as aforementioned and vehicle speed isv=2m/s,v=4m/s,v=6m/s,v=8m/s, andv=10m/s. One-hundred deck roughness samples of deck roughness levels from A to D were randomly generated to calculate the impact coefficient of the midspan position. The results of vehicle passing bridges at uniform speed are shown in Figure 4 and Table 4.Table 4
Impact coefficients with different deck roughness levels (without standard bumping).
Vehicle speed(m/s) Absolute smooth Deck roughness level of A Deck roughness level of B Deck roughness level of C Deck roughness level of D Mean Variance Mean Variance Mean Variance Mean Variance 2 1.01 1.05 0.0005 1.09 0.0013 1.21 0.0069 1.43 0.0245 4 1.02 1.09 0.0015 1.18 0.0054 1.42 0.0246 1.95 0.1012 6 1.03 1.09 0.0021 1.20 0.0061 1.46 0.0312 1.94 0.1047 8 1.04 1.10 0.0027 1.19 0.0053 1.41 0.0263 1.92 0.0936 10 1.05 1.10 0.0033 1.18 0.0111 1.43 0.0472 1.85 0.1349Figure 4
Probability density curve of impact coefficient at the speed of 10 m/s without standard bumping.Figure4 and Table 4 show that mean and variance of impact coefficient gradually increased with deck roughness levels ranging from A to D. Moreover, impact coefficient became more discrete as deck roughness level was reduced. Therefore, deck roughness has a significant and random effect on impact coefficient.As fundamental frequency is a function of bending stiffness (EI), span (l), and mass of unit length (m), the relationship between fundamental frequency and impact coefficient can reflect the dynamic characteristics of vehicle–bridge interaction vibration. Taking three hollow slab bridges and four T beam bridges (physical parameters of bridges shown in Table 5) as examples, the physical parameters of vehicles are as aforementioned and vehicle speed isv=4m/s. The results with absolute smooth deck are shown in Table 6.Table 5
Physical parameters and fundamental frequency of bridges.
Code Section form E I (pa·m4) m (kg/m) l (m) f (Hz) 1 hollow slab 1.84E+10 14,000 19.0 4.99 2 hollow slab 2.78E+10 17000 17.4 6.63 3 hollow slab 1.11E+10 13,000 13.0 8.59 4 T beam 1.49E+10 9,000 20.3 4.90 5 T beam 1.42E+10 8,000 17.8 6.61 6 T beam 1.20E+10 8,000 16.3 7.24 7 T beam 7.99E+09 7,000 10.3 15.82Table 6
Physical parameters and impact coefficient of bridges.
Code f(Hz) Dynamic displacement (mm) Static displacement (mm) Impact coefficient 1 4.99 1.856 1.827 1.016 2 6.63 0.941 0.929 1.013 3 8.59 0.983 0.970 1.013 4 4.90 2.789 2.751 1.014 5 6.61 1.969 1.946 1.012 6 7.24 1.789 1.768 1.012 7 15.82 0.675 0.670 1.007In Tables5 and 6, the maximum impact coefficient of seven bridges is equal to 1.015; low speed is beneficial in reducing the influence of deck roughness on impact coefficient with different section form, bending stiffness, span, and mass of unit length. Impact coefficients were reduced with the increase in fundamental frequency of the same section form bridges; however, the correlation of impact coefficient and fundamental frequency are not evident for different section form bridges. Thereby, it is necessary to analyze the variation of impact coefficient with different section form bridges.


## 3.2. Validation of Impact Coefficient Based on Standard Bumping

Approximate closed-form solutions were determined for the vehicle and bridge based on the following practical assumptions:(1) the vehicle remains in contact with the bridge at all times, as it is being driven and during bumping; (2) the effect of the wooden plank on deck roughness is negligible; (3) the spring–mass system executes vertical free fall with a certain initial velocity after leaving the wooden plank; and (4) the gravitational potential energy of mass is completely converted into kinetic energy in the process of bumping, with no loss of energy. The standard bump model is shown in Figure 5. As external force applied to a simply supported bridge directly determines its displacement, velocity, and acceleration with respect to the equilibrium position, the movement of mass in spring–mass system is the focus of analysis. Taking the equilibrium position of spring–mass system in its initial state as the origin of the coordinate system and the under orientation as the positive direction, the velocity and displacement of the mass after bumping were determined as z(i+1)′=z(i)′+2×9.8×h andz(i+1) =z(i), respectively, where h is the bumping height and z′(i) andz(i) are the initial velocity and displacement of mass before bumping, respectively.Figure 5
Standard bumping model.The physical parameters of simply supported girder bridges and vehicles are as aforementioned. The deck roughness levels are A, B, C, and D; vehicle speed isv=2m/s,v=4m/s, v=6m/s,v=8m/s, andv=10m/s, and bumping height is 5 cm, 10 cm, and 15 cm. The results of 5 cm bumping height are shown in Figure 6 and Table 7, and the results of 15 cm bumping height are shown in Table 8.Table 7
Impact coefficients with different deck roughness levels (with 5 cm bumping height).
Vehicle speed(m/s) Absolute smooth Deck roughness level of A Deck roughness level of B Deck roughness level of C Deck roughness level of D Mean Variance Mean Variance Mean Variance Mean Variance 2 2.32 2.33 0.0011 2.32 0.0040 2.33 0.0240 2.38 0.0575 4 2.33 2.33 0.0046 2.37 0.0157 2.32 0.0569 2.51 0.1779 6 2.32 2.32 0.0049 2.33 0.0188 2.35 0.0969 2.66 0.2658 8 2.26 2.26 0.0056 2.26 0.0220 2.35 0.0883 2.54 0.2211 10 2.26 2.27 0.0051 2.28 0.0171 2.31 0.820 2.59 0.2101Table 8
Impact coefficients with different deck roughness levels (with 15 cm bumping height).
Vehicle speed(m/s) Absolute smooth Deck roughness level of A Deck roughness level of B Deck roughness level of C Deck roughness level of D Mean Variance Mean Variance Mean Variance Mean Variance 2 3.29 3.30 0.0011 3.30 0.0045 3.29 0.0159 3.30 0.0566 4 3.31 3.31 0.0053 3.32 0.0209 3.32 0.0679 3.35 0.1932 6 3.29 3.29 0.0053 3.28 0.0197 3.31 0.0632 3.46 0.3103 8 3.22 3.22 0.0075 3.21 0.0186 3.21 0.1130 3.43 0.2943 10 3.22 3.23 0.0042 3.20 0.0222 3.26 0.0928 3.37 0.2464Figure 6
Probability density curve of impact coefficient at the speed of 10 m/s with 5-cm bumping height.Comparing Figures4 and 6, the standard bumping test reduced the discreteness of impact coefficient, which can effectively reduce the influence of deck roughness on impact coefficient. Table 7 demonstrates that the relative variation ratio of impact coefficient, which is the ratio between the differences of maximum and minimum values and average values of impact coefficient, is <4.78% at speeds of ≤10 m/s and for bump height of 5 cm with relatively smooth decks (levels A and B). However, the relative variation ratio of impact coefficient is <3.68% at speeds ≤10 m/s for bump height of 15 cm with rougher decks (levels C and D) in Table 8. Thus, lower bump height can reduce the influence of deck roughness levels A and B on impact coefficient, whereas higher bump height can reduce the influence of deck roughness levels C and D on impact coefficient at the speed of less than 10 m/s.To further verify the reliability and accuracy of standard bumping, two pieces of reinforced rectangular concrete section beams were made with C25 concrete (32.5 ordinary Portland cement, with density of 2,500 kg/m3 and cement, sand, stone, and water quality ratio of 1:1.44:2.67:0.5), HRB335 steel, and HPB235 stirrup. As shown in Table9 and Figures 7-8, the calculation spans of 1# and 2# simply supported girder bridges were both 3,300 mm. The bending inertia moments of 1# and 2# simply supported girder bridges were 4.32 e6mm4 and 9.36 e6mm4, respectively. The elastic moduli of 1# and 2# test beams were 2.52 e10Pa and 1.89 e10Pa, respectively, with a 30-kg concrete block and dial indicator applied in the bottom of midspan beam.Table 9
Parameter of two test beams of simply supported girder bridge.
Beam code Calculationspan (mm) Sectionsize (mm) Main reinforcement diameter (mm) Main reinforcement quantity Stirrup diameter (mm) Stirrup spacing (mm) 1# 3,500 300×80 12 3 6 120 2# 3,500 600×80 12 5 6 130Figure 7
Section diagram of (a) 1# test beam and (b) 2# test beam.
(a) (b)Figure 8
Preparation of (a) 1# test beam and (b) 2# test beam.
(a) (b)In Figure9, the two test beams were tested by DHDAS-5907 bridge modal apparatus and analysis system. Three speed sensors, which were placed at 1/4, 1/2, and 3/4 span of the test beams, were used to collect vibration speed signal for 6-8 mins. The fundamental frequencies of 1# and 2# test beams were determined as 7.36 HZ (as shown in Figure 10) and 6.38 HZ through spectrum analysis of vibration speed signal. The section sizes of 1# and 2# test beams were 0.080 × 0.3 m2 and 0.080 × 0.6 m2, but the calculation spans of 1# and 2# test beams were both 3.3m. The elastic modulus of 1# and 2# test beams were 2.52 e10Pa and 1.89 e10Pa. The Poisson ratios of 1# and 2# test beams were both 0.2. The fundamental frequencies of 1# and 2# test beams were 7.51 Hz (as shown in Figure 11) and 6.75 Hz as measured by Block Lanczos modal method with errors of 2.04% and 5.80%, respectively, compared with measured frequency. Thus, the measurements verify that the two beam models are appropriate.Figure 9
Laboratory test of standard bumping.
(a) (b)Figure 10
Measured frequency of 1# test beam.Figure 11
Calculated frequency of 1# test beam.The physical parameters of vehicles are as follows: front axle and rear axle mass were 10 kg and 45 kg, respectively; distance between front axle and rear axle was 1.25 m; spring stiffness and tire stiffness of front axle were 5 KN/m and 12 KN/m, respectively. Physical parameters of 1# test beam are as aforementioned. The static displacement was measured as 2.02 mm by static load testing. The dynamic displacement of measured value and calculated value were 4.94 mm and 4.77 mm, respectively; and impact coefficients were 2.45 and 2.36, with 3.67% error, thus proving that standard bumping method can exactly simulate vehicle–bridge interaction.


## 4. Impact Coefficient Formula

### 4.1. Vehicle Model

To make the results a more legitimate representative of actual bridge vibrations, ANSYS 10.0 simulation software was used to create vehicle–bridge interaction model in which the mass of suspension system was assigned to the front and rear axles of the vehicle in a mass ratio of 1:4 [23]. The testing vehicle was modeled as a three-dimensional mass–spring–damper system with seven degrees of freedom (DOFs), as shown in Table 10 and Figure 12. The tandems at the drive and rear axles of the test vehicle were lumped as single wheel for simplicity. Therefore, this model consisted of five rigid masses that represented the truck and its four wheels. The vehicle was assigned three DOFs, which correspond to the vertical displacement at the mass center, pitching rotation, and rolling rotation. Each wheel was modeled as a lumped mass with vertical displacement only. The vehicle was represented by beam elements (MPC184). The suspension and nonsuspension systems and the inertia moments of pitching and rolling rotations were represented by mass elements (Mass21). The spring and damper were represented by a spring–damper element (Combine14).Table 10
Parameters of the test vehicle.
Measurement Parameter Overall mass (kg) M0=30,000 Mass of suspension above (kg) m c z=27,000 Mass of front axle (kg) m Q Z=600 Mass of rear axle (kg) m H Z=2,400 Mass moment of inertia of the X axis (kg·m2) J x=1.0e6 Mass moment of inertia of the Y axis (kg·m2) J y=3.6e5 Suspension stiffness of front axle (N/m) K x Q Z=0.8e6 Suspension stiffness of rear axle (N/m) K x H Z=2.0e6 Tire stiffness of front axle (N/m) K L Q Z=1.2e6 Tire stiffness of rear axle (N/m) K L H Z=4.8e6 Suspension damping of front axle (N·s/m) C x Q Z=5.0e3 Suspension damping of rear axle (N·s/m) C x H Z=2.0e4 Tire damping of front axle (N·s/m) C L Q Z=6.0e3 Tire damping of rear axle (N·s/m) C L H Z=2.4e4 Distance between center of gravity and front axle (m) L Q Z=3.6 Distance between center of gravity and rear axle (m) L H Z=0.9 Distance between center of gravity and left and right wheels L y=0.9Figure 12
Vehicle model with seven DOFs.


### 4.2. Bridge Model

The reinforced concrete structure of the whole bridge was represented by Solid65 elements. Solid65 elements are also called reinforced concrete entity elements and they are used to simulate the unreinforced or reinforcement of the three-dimensional entity structure. Finally, the method of transient dynamic analysis was used to solve vehicle–bridge interaction model.


### 4.3. Impact Coefficient Formula

Finite element models of five simply supported hollow-slab bridges and five T-beam bridges were created in ANSYS 10.0 based on the design shown in Table11. The fundamental frequencies of each bridge were determined through modal test (Figure 13). The displacements of the front and rear axles and the total structure were statically determined in a bumping test wherein a 30-T test vehicle passed over a 15 cm wooden plank in the midspan of the test bridge (Table 12). Due to limitation in space, only the results of the simply supported hollow-slab bridge were presented in our study.Table 11
Parameters and analysis of simply supported hollow-slab bridges by static load testing.
Bridge name Width(m) Span(m) Frequency(Hz) Front axle displacement(mm) Rear axle displacement (mm) Total displacement (mm) Luanhe Bridge 10.0 19.0 5.11 0.32 1.83 2.15 Tatou Bridge 12.0 17.4 6.13 0.15 0.93 1.08 Cichang Bridge 12.4 13.0 6.77 0.15 1.43 1.58 Xiazhuang Bridge 7.5 10.0 10.44 0.04 1.18 1.22 Xieliangkou Bridge 10.1 13.0 8.26 0.11 0.97 1.08Table 12
Summary of impact coefficients of 15 cm bumping height.
Frequency 5.11HZ  (Luanhe Bridge) 6.13HZ  (Tatou Bridge) 6.77HZ  (Cichang Bridge) 8.26HZ  (Xieliangkou Bridge) 10.44HZ  (Xiazhuang Bridge) 2 m/s 3.54 3.28 3.01 2.87 2.76 4 m/s 3.55 3.41 3.05 2.81 2.78 6 m/s 3.57 3.29 3.09 2.81 2.74 8 m/s 3.60 3.33 3.06 2.78 2.68 10 m/s 3.57 3.31 3.07 2.77 2.67 Change 1.77% 3.98% 2.53% 3.73% 4.04%Figure 13
Fundamental frequency and mode shape of Luanhe Bridge.Tables11 and 12 show that vehicle speeds that are less than 10 m/s have minimal effect on impact coefficient. The relative variation ratio of the impact coefficient, or the ratio between the difference and mean of the maximum and minimum impact coefficients, was <5.0% for each bridge and thus can be neglected. However, fundamental frequency of bridges has a significant influence on impact coefficient; and impact coefficients decline with the increase in fundamental frequency, as observed in the Ontario Highway Bridge Design Code (1982) but not in the General Code for Design of Highway Bridges and Culverts (JTG D60-2004). Thus, vehicle–bridge interaction vibration was able to affect impact coefficient but not JTG D60-2004.The relationship between impact coefficient and fundamental frequency is better captured by a logarithmic function than a polynomial one. The fitting formula for the impact coefficient of a simply supported hollow-slab bridge is shown in (16). Impact coefficient fitting formulas of different bridge forms and bumping heights are listed in Table 13.(16)1+μ=-1.224ln⁡f+5.497As double-vehicle loading is used in bridge testing to verify the lateral connection integrity and as single-vehicle loading concentrates the load distribution on a wide deck, five simply supported hollow-slab bridges were used to determine the impact coefficients for double-vehicle loading (Table 14). Parameters of vehicle and bridge are as aforementioned, and double vehicles passed over a bumping height of 15 cm side by side at the speed of 4 m/s with a spacing of 1 m.Table 13
Impact coefficient formulas of different bridge forms and bumping heights.
Simply supported bridge form Bumping height(cm) Impact coefficient formula Application scope (Hz) Correlation coefficient(R2) Hollow-slab 5 1+u=-0.616ln⁡(f)+3.235 5 ≤f ≤ 11 0.9681 10 1+u=-0.894ln⁡(f)+4.313 0.9721 15 1+u=-1.224ln⁡(f)+5.497 0.9122 T-beam 5 1+u=-0.657ln⁡(f)+3.379 5 ≤f ≤ 16 0.9661 10 1+u=-0.748ln⁡(f)+4.092 0.9362 15 1+u=-0.616ln⁡(f)+3.235 0.9681Table 14
Impact coefficients for single-vehicle and double-vehicle loading with 15 cm bumping height.
Bridge Impact coefficient(single vehicle) Impact coefficient(double vehicle) Impact coefficientrelative variation ratio Luanhe Bridge 3.55 3.68 3.66% Tatou Bridge 3.41 3.57 4.69% Cichang Bridge 3.05 3.14 2.95% Xiazhuang Bridge 2.78 2.87 3.24% Xieliangkou Bridge 2.81 2.88 2.49%Table14 illustrates that, for all the simply supported hollow-slab bridges, the double-vehicle impact coefficient is only marginally greater than that of single-vehicle, with a variation of <5.0%. Therefore, the single-vehicle midspan impact coefficient formulas can also be used for double-vehicle cases.


## 4.1. Vehicle Model

To make the results a more legitimate representative of actual bridge vibrations, ANSYS 10.0 simulation software was used to create vehicle–bridge interaction model in which the mass of suspension system was assigned to the front and rear axles of the vehicle in a mass ratio of 1:4 [23]. The testing vehicle was modeled as a three-dimensional mass–spring–damper system with seven degrees of freedom (DOFs), as shown in Table 10 and Figure 12. The tandems at the drive and rear axles of the test vehicle were lumped as single wheel for simplicity. Therefore, this model consisted of five rigid masses that represented the truck and its four wheels. The vehicle was assigned three DOFs, which correspond to the vertical displacement at the mass center, pitching rotation, and rolling rotation. Each wheel was modeled as a lumped mass with vertical displacement only. The vehicle was represented by beam elements (MPC184). The suspension and nonsuspension systems and the inertia moments of pitching and rolling rotations were represented by mass elements (Mass21). The spring and damper were represented by a spring–damper element (Combine14).Table 10
Parameters of the test vehicle.
Measurement Parameter Overall mass (kg) M0=30,000 Mass of suspension above (kg) m c z=27,000 Mass of front axle (kg) m Q Z=600 Mass of rear axle (kg) m H Z=2,400 Mass moment of inertia of the X axis (kg·m2) J x=1.0e6 Mass moment of inertia of the Y axis (kg·m2) J y=3.6e5 Suspension stiffness of front axle (N/m) K x Q Z=0.8e6 Suspension stiffness of rear axle (N/m) K x H Z=2.0e6 Tire stiffness of front axle (N/m) K L Q Z=1.2e6 Tire stiffness of rear axle (N/m) K L H Z=4.8e6 Suspension damping of front axle (N·s/m) C x Q Z=5.0e3 Suspension damping of rear axle (N·s/m) C x H Z=2.0e4 Tire damping of front axle (N·s/m) C L Q Z=6.0e3 Tire damping of rear axle (N·s/m) C L H Z=2.4e4 Distance between center of gravity and front axle (m) L Q Z=3.6 Distance between center of gravity and rear axle (m) L H Z=0.9 Distance between center of gravity and left and right wheels L y=0.9Figure 12
Vehicle model with seven DOFs.


## 4.2. Bridge Model

The reinforced concrete structure of the whole bridge was represented by Solid65 elements. Solid65 elements are also called reinforced concrete entity elements and they are used to simulate the unreinforced or reinforcement of the three-dimensional entity structure. Finally, the method of transient dynamic analysis was used to solve vehicle–bridge interaction model.


## 4.3. Impact Coefficient Formula

Finite element models of five simply supported hollow-slab bridges and five T-beam bridges were created in ANSYS 10.0 based on the design shown in Table11. The fundamental frequencies of each bridge were determined through modal test (Figure 13). The displacements of the front and rear axles and the total structure were statically determined in a bumping test wherein a 30-T test vehicle passed over a 15 cm wooden plank in the midspan of the test bridge (Table 12). Due to limitation in space, only the results of the simply supported hollow-slab bridge were presented in our study.Table 11
Parameters and analysis of simply supported hollow-slab bridges by static load testing.
Bridge name Width(m) Span(m) Frequency(Hz) Front axle displacement(mm) Rear axle displacement (mm) Total displacement (mm) Luanhe Bridge 10.0 19.0 5.11 0.32 1.83 2.15 Tatou Bridge 12.0 17.4 6.13 0.15 0.93 1.08 Cichang Bridge 12.4 13.0 6.77 0.15 1.43 1.58 Xiazhuang Bridge 7.5 10.0 10.44 0.04 1.18 1.22 Xieliangkou Bridge 10.1 13.0 8.26 0.11 0.97 1.08Table 12
Summary of impact coefficients of 15 cm bumping height.
Frequency 5.11HZ  (Luanhe Bridge) 6.13HZ  (Tatou Bridge) 6.77HZ  (Cichang Bridge) 8.26HZ  (Xieliangkou Bridge) 10.44HZ  (Xiazhuang Bridge) 2 m/s 3.54 3.28 3.01 2.87 2.76 4 m/s 3.55 3.41 3.05 2.81 2.78 6 m/s 3.57 3.29 3.09 2.81 2.74 8 m/s 3.60 3.33 3.06 2.78 2.68 10 m/s 3.57 3.31 3.07 2.77 2.67 Change 1.77% 3.98% 2.53% 3.73% 4.04%Figure 13
Fundamental frequency and mode shape of Luanhe Bridge.Tables11 and 12 show that vehicle speeds that are less than 10 m/s have minimal effect on impact coefficient. The relative variation ratio of the impact coefficient, or the ratio between the difference and mean of the maximum and minimum impact coefficients, was <5.0% for each bridge and thus can be neglected. However, fundamental frequency of bridges has a significant influence on impact coefficient; and impact coefficients decline with the increase in fundamental frequency, as observed in the Ontario Highway Bridge Design Code (1982) but not in the General Code for Design of Highway Bridges and Culverts (JTG D60-2004). Thus, vehicle–bridge interaction vibration was able to affect impact coefficient but not JTG D60-2004.The relationship between impact coefficient and fundamental frequency is better captured by a logarithmic function than a polynomial one. The fitting formula for the impact coefficient of a simply supported hollow-slab bridge is shown in (16). Impact coefficient fitting formulas of different bridge forms and bumping heights are listed in Table 13.(16)1+μ=-1.224ln⁡f+5.497As double-vehicle loading is used in bridge testing to verify the lateral connection integrity and as single-vehicle loading concentrates the load distribution on a wide deck, five simply supported hollow-slab bridges were used to determine the impact coefficients for double-vehicle loading (Table 14). Parameters of vehicle and bridge are as aforementioned, and double vehicles passed over a bumping height of 15 cm side by side at the speed of 4 m/s with a spacing of 1 m.Table 13
Impact coefficient formulas of different bridge forms and bumping heights.
Simply supported bridge form Bumping height(cm) Impact coefficient formula Application scope (Hz) Correlation coefficient(R2) Hollow-slab 5 1+u=-0.616ln⁡(f)+3.235 5 ≤f ≤ 11 0.9681 10 1+u=-0.894ln⁡(f)+4.313 0.9721 15 1+u=-1.224ln⁡(f)+5.497 0.9122 T-beam 5 1+u=-0.657ln⁡(f)+3.379 5 ≤f ≤ 16 0.9661 10 1+u=-0.748ln⁡(f)+4.092 0.9362 15 1+u=-0.616ln⁡(f)+3.235 0.9681Table 14
Impact coefficients for single-vehicle and double-vehicle loading with 15 cm bumping height.
Bridge Impact coefficient(single vehicle) Impact coefficient(double vehicle) Impact coefficientrelative variation ratio Luanhe Bridge 3.55 3.68 3.66% Tatou Bridge 3.41 3.57 4.69% Cichang Bridge 3.05 3.14 2.95% Xiazhuang Bridge 2.78 2.87 3.24% Xieliangkou Bridge 2.81 2.88 2.49%Table14 illustrates that, for all the simply supported hollow-slab bridges, the double-vehicle impact coefficient is only marginally greater than that of single-vehicle, with a variation of <5.0%. Therefore, the single-vehicle midspan impact coefficient formulas can also be used for double-vehicle cases.


## 5. Engineering Application

### 5.1. Test Bridge Description

The test bridge was built in 2000 in the town of Xiong in Baoding City, Hebei Province, China. It is a reinforced concrete, simply supported, and core-slabbed bridge with 4 × 12.5 m spans, thus giving a total length of 50 m (Figure14). The lateral structure of the bridge comprises nine hollow slabs that were 0.90 × 1.20 m with fissure connections. The hinge joints and deck pavement were made from C40 and C30 waterproof concrete, respectively. The surface is hot-mix bituminous concrete. During testing, the deck slab, as viewed from the surface, was integral and was free from cracks, pits, spalling, subsidence, or any other defects. Likewise, as viewed from the bottom, the deck slab was free from cracks, seepage, hinged-joint exfoliation, etc. The fixed support was in uniform compression, and the spherical pedestal was free from defects. Therefore, the whole bridge structure was in a satisfactory mechanical state with a deck roughness level of B.Figure 14
Overview of the tested bridge and tested vehicle.


### 5.2. Instrumentation

The main testing instruments used were BDI Wireless Structural Testing System (STS-WiFi), sensors for acceleration displacement and strain, wireless data transmission nodes and base station, a Dynamic Habitual Displacement Analysis system (3817N), and a magnet support. Acceleration sensors provided information on the vibration of bridges, fundamental frequency, and mode shape obtained from modal analysis. Displacement sensors were applied to collect the signal of time-displacement relationship, and strain sensors were used to monitor the time–strain relationship.


### 5.3. Procedure and Analysis

Three methods of testing were used in our study. In modal testing, acceleration sensors placed at 1/6, 1/3, and 1/2 span of the test bridge collected the vibration signals produced by moving vehicles for 360–480 seconds. The frequency and mode shape of the bridge were then determined by vibrational signal analysis. In static load testing, a 30-T two-axle vehicle was stationed at the midspan of the bridge for 30 seconds, perpendicular to the bridge axis, with displacement and strain sensors placed at the midspan. In standard bumping testing, the same vehicle passed over a 10-cm tall wooden plank at the midspan of bridges, with a speed of 5 m/s.Frequency spectrum analysis was applied to the acceleration-sensor vibration signal. The fundamental frequency of the test bridge is 8.67Hz in Figure15. Taking 8.67Hz into the formula of 1+u=-0.894ln(f)+4.313, the calculated impact coefficient is 2.38.Figure 15
Fundamental frequency of the tested bridge.Figures16 and 17 show static strain and displacement and dynamic strain and displacement, in which tensile strain is defined as the positive direction and compression as the negative direction. Impact coefficients were calculated by static and dynamic values and are listed in Table 15.Table 15
Impact coefficients of static load testing and standard bumping testing.
Static load testing Standard bumping testing Measured impact coefficients Calculated impact coefficients Error Strain 32.64με 74.86με 2.29 2.38 3.9% Displacement 1.61 mm 3.67 mm 2.28 2.38 4.5%Figure 16
(a) Tensile strain in the midspan of the test bridge in static loading testing. (b) Displacement in the midspan of the test bridge in static loading testing.
(a) (b)Figure 17
(a) Tensile strain in the midspan of the test bridge in standard bumping testing. (b) Displacement in the midspan of the test bridge in standard bumping testing.
(a) (b)Table15 shows that the errors of measured and calculated impact coefficients are less than 5% in both strain and displacement, which implicates the precision of impact coefficient formulas in this study.Connecting theory displacement of 2.53 mm and static displacement of 1.61 mm, the calculated calibration coefficient of bridges was 0.64 with static load testing. By taking displacement of standard bumping testing into (2) to solve for the static displacement of 1.54 mm, the calculated calibration coefficient of bridges was 0.61 with standard bumping testing, as shown in Table 16.Table 16
Calibration coefficient of static load testing and standard bumping testing.
Measured displacement (mm) Theory displacement (mm) Calibration coefficient Error Static load testing 1.61 2.53 0.64 4.7% Standard bumping testing 3.67 2.53 0.61Table16 presents the 4.7% error of calibration coefficient in static load testing and standard bumping testing, which verifies the precision and feasibility of standard bumping testing.


## 5.1. Test Bridge Description

The test bridge was built in 2000 in the town of Xiong in Baoding City, Hebei Province, China. It is a reinforced concrete, simply supported, and core-slabbed bridge with 4 × 12.5 m spans, thus giving a total length of 50 m (Figure14). The lateral structure of the bridge comprises nine hollow slabs that were 0.90 × 1.20 m with fissure connections. The hinge joints and deck pavement were made from C40 and C30 waterproof concrete, respectively. The surface is hot-mix bituminous concrete. During testing, the deck slab, as viewed from the surface, was integral and was free from cracks, pits, spalling, subsidence, or any other defects. Likewise, as viewed from the bottom, the deck slab was free from cracks, seepage, hinged-joint exfoliation, etc. The fixed support was in uniform compression, and the spherical pedestal was free from defects. Therefore, the whole bridge structure was in a satisfactory mechanical state with a deck roughness level of B.Figure 14
Overview of the tested bridge and tested vehicle.


## 5.2. Instrumentation

The main testing instruments used were BDI Wireless Structural Testing System (STS-WiFi), sensors for acceleration displacement and strain, wireless data transmission nodes and base station, a Dynamic Habitual Displacement Analysis system (3817N), and a magnet support. Acceleration sensors provided information on the vibration of bridges, fundamental frequency, and mode shape obtained from modal analysis. Displacement sensors were applied to collect the signal of time-displacement relationship, and strain sensors were used to monitor the time–strain relationship.


## 5.3. Procedure and Analysis

Three methods of testing were used in our study. In modal testing, acceleration sensors placed at 1/6, 1/3, and 1/2 span of the test bridge collected the vibration signals produced by moving vehicles for 360–480 seconds. The frequency and mode shape of the bridge were then determined by vibrational signal analysis. In static load testing, a 30-T two-axle vehicle was stationed at the midspan of the bridge for 30 seconds, perpendicular to the bridge axis, with displacement and strain sensors placed at the midspan. In standard bumping testing, the same vehicle passed over a 10-cm tall wooden plank at the midspan of bridges, with a speed of 5 m/s.Frequency spectrum analysis was applied to the acceleration-sensor vibration signal. The fundamental frequency of the test bridge is 8.67Hz in Figure15. Taking 8.67Hz into the formula of 1+u=-0.894ln(f)+4.313, the calculated impact coefficient is 2.38.Figure 15
Fundamental frequency of the tested bridge.Figures16 and 17 show static strain and displacement and dynamic strain and displacement, in which tensile strain is defined as the positive direction and compression as the negative direction. Impact coefficients were calculated by static and dynamic values and are listed in Table 15.Table 15
Impact coefficients of static load testing and standard bumping testing.
Static load testing Standard bumping testing Measured impact coefficients Calculated impact coefficients Error Strain 32.64με 74.86με 2.29 2.38 3.9% Displacement 1.61 mm 3.67 mm 2.28 2.38 4.5%Figure 16
(a) Tensile strain in the midspan of the test bridge in static loading testing. (b) Displacement in the midspan of the test bridge in static loading testing.
(a) (b)Figure 17
(a) Tensile strain in the midspan of the test bridge in standard bumping testing. (b) Displacement in the midspan of the test bridge in standard bumping testing.
(a) (b)Table15 shows that the errors of measured and calculated impact coefficients are less than 5% in both strain and displacement, which implicates the precision of impact coefficient formulas in this study.Connecting theory displacement of 2.53 mm and static displacement of 1.61 mm, the calculated calibration coefficient of bridges was 0.64 with static load testing. By taking displacement of standard bumping testing into (2) to solve for the static displacement of 1.54 mm, the calculated calibration coefficient of bridges was 0.61 with standard bumping testing, as shown in Table 16.Table 16
Calibration coefficient of static load testing and standard bumping testing.
Measured displacement (mm) Theory displacement (mm) Calibration coefficient Error Static load testing 1.61 2.53 0.64 4.7% Standard bumping testing 3.67 2.53 0.61Table16 presents the 4.7% error of calibration coefficient in static load testing and standard bumping testing, which verifies the precision and feasibility of standard bumping testing.


## 6. Conclusions

A new method of standard bumping testing was proposed to assess the load-carrying capacity of bridges based on impact coefficient. First, vehicle speed that is less than 10 m/s had no significant effect on impact coefficient, and deck roughness had a slight influence on impact coefficient with reasonable vehicle speeds and bumping heights; moreover, fundamental frequency, which has close relationship with stiffness and form of bridges, did have a significant effect on impact coefficient in standard bumping testing. Second, impact coefficient fitting formulas, or the relationship between impact coefficient and fundamental frequency, were established for simply supported bridges of different section forms, bumping heights, and fundamental frequency by numerical simulation. Third, impact coefficients produced by double vehicles were only marginally greater than those produced by single vehicles, with error of less than 5%, which implies that midspan impact coefficient formulas developed for single vehicles could also be applied to double vehicles. Finally, the accuracy of impact coefficient formulas was verified through laboratory and field tests, and the precision and feasibility of standard bumping testing were validated through field testing, which offers a new method for the load-carrying capacity assessment of bridges by calibration coefficient.



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