Dynamic Reliability of Continuous Rigid-Frame Bridges under Stochastic Moving Vehicle Loads

Engineering Research Center of Catastrophic Prophylaxis and Treatment of Road & Traffic Safety of Ministry of Education, Changsha University of Science & Technology, Changsha 410114, China Industry Key Laboratory of Traffic Infrastructure Security Risk Management, Changsha University of Science and Technology, Changsha 410114, China School of Civil Engineering, Hunan University of Technology, Zhuzhou 412007, China


Introduction
In the design phase of a bridge, the structural safety is usually evaluated considering design traffic loads, which were formulated according to large traffic monitoring data during the past decades [1,2]. However, the current or future vehicle load might be beyond the design value with the rapid development of transportation industry. As a result, numerous bridges were collapsed or badly damaged due to truck overloading [3][4][5]. In case that growing traffic load is becoming more and more critical for in-service bridges, the bridge safety under actual traffic loads deserves investigation. e traffic load effect on a bridge is stochastic due to the random nature of traffic loads [6]. In addition, a moving vehicle leads to dynamic effects motivated by the road roughness and the bridge deflection. In view of the above reasons, the structural dynamic reliability theory is appropriate for the safety evaluation of existing bridges under actual stochastic traffic loads. In addition, the design vehicle load model can be calibrated with the target reliability index.
In general, the dynamic effect induced by a moving vehicle can be evaluated in a vehicle-bridge interaction system [7,8]. A great number of literatures focus on dynamic traffic load effects on both short-span and long-span bridges. Chen and Wu [9] investigated dynamic effects of a cable-stayed bridge under stochastic vehicle loads and investigated influence of the sparse and dense states of traffic loads on the probabilistic characteristics of bridge load effect. Lu et al. [10] evaluated lifetime deflection of a suspension bridge considering dynamic and growing traffic loads. Zhou and Chen [8] investigated probabilistic characteristics of the dynamic response of a cable-stayed bridge under wind load and stochastic vehicle load. Lu et al. [11] investigated the first-passage probability of a cable-stayed bridge under stochastic vehicle loads. Obrien et al. [12] investigated the dynamic increment of extreme traffic loading on short-span bridges. Jiang et al. [13] investigated the fatigue damage of cables of cable-stayed bridge under combined effect of traffic and wind. Li et al. [14] investigated the safety of suspenders of Tsing Ma Bridge under traffic loads However, there are relatively few studies on probabilistic dynamic traffic load effects on long-span continuous rigid-frame bridges. e unique feature for the continuous rigid-frame bridge is the high-rise piers, which are usually space-flexible, affecting the dynamic behaviour of the bridge.
us, the probabilistic dynamic traffic load effects on continuous rigid-frame bridges deserve more investigation.
Most studies treated the traffic load effect on a bridge as a random variable. In practice, the load effect on a bridge is a time-varying random process [15]. e commonly used theoretical basis for the probabilistic dynamic analysis of bridges under vehicle load is the first-passage criterion [16,17]. e first-passage failure probability is usually estimated based on Rice's level-crossing principle [18]. However, applications of the first-passage theory are mostly concentrated in seismic engineering, while the application to traffic load analysis is relatively insufficient. e bottleneck is that the root mean square of traffic load effects is usually evaluated in frequency domain, while the vehicle-bridge coupled vibration is mostly analysed in time domain [19]. erefore, the key point is how to extract the probability parameters from the dynamic load effect.
is study evaluated probabilistic dynamic traffic load effects on rigid-frame bridges under highway traffic monitoring loads. e site-specific traffic monitoring data were utilized to establish stochastic traffic models. e dynamic effect was considered in a vehicle-bridge coupled vibration model, and the probability estimation was conducted based on the first-passage criterion of the girder deflection. e prototype bridge is a continuous rigid-frame bridge with mid-span length of 200 m and pier height of 182 m. It is demonstrated that the dynamic traffic load effect follows Gaussian distribution and thus can be treated as a stationary random process. Parametric studies were conducted, accounting for the road roughness condition, the bridge span length, and the traffic growth ratio.

Traffic-Bridge Interaction Model.
A vehicle passing on a bridge leads to the vibration of the bridge due to the road surface roughness [20]. Meanwhile, the vibration of the bridge impacts the vibration of the vehicle. us, the vehicle-bridge interaction is a coupled vibration system. Figure 1 shows the plane motion model of two-axle vehicle, where x represents the road roughness and C and K represent the damping and stiffness parameters, respectively. e vehicle model has four degrees of freedom including vertical motion of two axles and the motion and rotation of the vehicle body.
Based on the simplified model as shown in Figure 1, the differential equation of motion of the vehicle-bridge coupled system can be written as [21] M where M v and M b represent the mass matrix of the vehicle and that of the bridge, respectively; C v and C b represent the damping matrix of the vehicle and that of the bridge, respectively; K v and K b represent the stiffness matrix of the vehicle and that of the bridge, respectively; u v and u b represent the displacement vector of the vehicle and that of the bridge, respectively; F vb and F bv represent the interaction forces between the vehicle and the bridge; F vg and F bg represent the weight of the vehicle and that of the bridge, respectively. In order to consider the multiple-vehicle effect on the bridge response, this study utilized an equivalent dynamic wheel load (EDWL) approach proposed by Chen and Cai [22]. Based on the EDWL approach, the force of the jth vehicle on a bridge can be simplified as a time-varying force, which can be written as where R j represents wheel weight ratio of the jth vehicle; G j represents the GVW of the jth vehicle; x j and d j represent longitudinal and lateral positions on the bridge, respectively; h k and a k represent the kth-order vertical and torsional bending modes of the bridge; n and n v , respectively, represent modal orders of bridges and the number of vehicles on the bridges; K i vl and C i vl represent stiffness and damping matrixes of the vehicle, respectively; Y i vl and _ Y i vl , respectively, represent the vertical displacement and the velocity of the ith axle; EDWL j (t) represents the equivalent dynamic axis load of the jth vehicle at time t. e effectiveness of EDWL approach has been verified by Wu and Chen [23] and Lu et al. [24] for long-span cablestayed bridges and suspension bridges, respectively. is study develops the EDWL approach in probability analysis and reliability evaluation for continuous rigid-frame bridges.

First-Passage Criterion.
e first-passage criterion is usually utilized for Rice's level-crossing probability analysis of a random process. e principle of Rice's level-crossing is shown in Figure 2, where t is time, x is a random process, that is, the traffic load effect in the present study, and v(x) is the level-crossing rate fitted to the histograms of the number of crossings. us, the critical content of the level-crossing theory is to count the number of crossings for the random process.
Suggest that the traffic load effect can be assumed as a Gaussian random process, which will be demonstrated in the case study. For a random process x(t), Rice [25] provided the number of crossings for a level b during time T, which is written as where is a joint probabilistic density function for is function is difficult to estimate for a nonstationary process. Fortunately, the traffic load effect on a bridge can be treated as a stationary random process, as mentioned by researchers [26,27]. On this basis, the level- For a Gaussian stationary random process with the mean value of zero, the level-crossing probability can be simplified further as where v b is a constant for level b and σ x is the root mean square value. In addition, the number of level-crossings of stochastic traffic load effects can be assumed as Poisson erefore, the probability of exceeding a threshold can be written as Note that the accuracy of the estimation mostly depends on the fitting to the curves as shown in Figure 2. In addition, a higher threshold b will result in a more reliable estimation. In general, the mean value of the traffic load effect on a bridge is not zero, which is different from seismic load effects. erefore, this study developed the above procedures to a nonzero mean value process, which is rewritten as where m x is the mean value of the traffic load effect. e density of traffic loads changes with time, where the traffic will be congested and free-flowing for the day and night, respectively. erefore, the daily vehicle flow was divided into sparse and dense states based on the density of stochastic vehicle model. e probability can be treated as the superposition of the two types of level-crossing probability, which is written as where v b1 and v b2 represent the level-crossing rates for the congested and free-flowing vehicle flows, respectively; a 1 and a 2 represent the proportions of the congested and freeflowing vehicle flows, respectively. ere is a relationship between the structural reliability and the first-passage probability. On this basis, the dynamic reliability can be written as where β is the reliability index, Φ() is the cumulative distribution function (CDF) of the standard normal distribution, and Φ −1 () is the corresponding inverse function.

Computational Procedures.
Based on the derived formulations of the vehicle-bridge coupled vibration system and the first-passage criterion, a computational framework is essential to combine the components as a system. is study presented a comprehensive framework for dynamic and probabilistic analysis for traffic load effects on rigid-frame bridges based on traffic monitoring data. Figure 3 shows the flow chart of the procedures. As shown in Figure 3, the framework consists of three main procedures: the stochastic traffic modelling, the trafficbridge interaction analysis, and the probability evaluation. Detailed illustrations of the procedures are shown as follows. e first step is the stochastic traffic modelling. Initially, the screened traffic monitoring data should be collected from site-specific highways. Subsequently, all vehicles are classified as 6 groups according to the configuration associated with the number of axles. On this basis, the statistical analysis can be conducted for probability modelling of the gross vehicle weight (GVW), the driving lanes, and the vehicle spacing. Finally, the stochastic traffic load model can be simulated utilizing the Monte Carlo simulation (MCS) approach. e second step is the traffic-bridge interaction analysis. e critical step is to model the dynamic parameters of the vehicle model and the modal parameters of the bridge. Usually, a 2D vehicle model as shown in Figure 1 can be utilized for a relatively precise analysis. e dynamic characteristics of the bridge are considered as mode shapes and natural frequencies. Since the long-span rigid-frame bridge is more complex than a conventional-girder bridge, the numbers of mode shapes and frequencies should be considered as more as possible. Subsequently, the EDWL approach is utilized to convert the moving stochastic traffic flow into time-varying forces. Finally, the time history of traffic load effects can be simulated based on the pseudodynamic analysis. e third step is the probability evaluation. is step is conducted based on Rice's level-crossing theory. Based on the time histories evaluated from the traffic-bridge interaction analysis, the number of level-crossings can be counted. e number of crossings is formed as histograms, where the level-crossing rate can be fitted accurately as shown in Figure 2(b). Subsequently, the maximum traffic load effect can be extrapolated with consideration of a return period. Finally, the first-passage probability and reliability index can be evaluated based on the assumption of Poisson distribution.

Probabilistic Modelling of WIM Measurements.
Due to the truck overloading behaviour, the GVW usually follows a multimodal distribution. In this regard, this study utilized a Gaussian mixture model (GMM), which is a superposition of several Gaussian models. In practice, the GMM method is widely used in the application of data clustering [28], but it is relatively insufficient in the vehicle weight modelling. In general, a GMM can be represented as where w is the GVW, a i is a weight coefficient, g is a Gaussian distribution function, and M is the number of Gaussian functions. e Gaussian is written as where μ i represents mean value of the ith Gaussian distribution function and σ i represents standard deviation of the ith Gaussian distribution function. e parameters of GMMs are usually estimated based on the maximum likelihood optimization.
is study utilized the Expectation-Maximization (EM) algorithm to evaluate these parameters. e number of components is estimated based on the Akaike information criterion (AIC) [29]. e authors have conducted extensive analyses on the probability modelling of traffic parameters as shown in the doctoral dissertation [30]. Due to the limit of article, only the probability distributions for 6-axle trucks are provided. Figure 3 plots the fitted probability distribution function (PDF) of the axle weight and the GVW, respectively. In Figure 4, W 64 represents the second axle load of six-axle vehicle.
It is observed that the axle weight and the GVW follow bimodal distribution. In addition, the GMMs have ideally fitted to the empirical histograms. In other words, the GMM has captured the probability behaviour of truck overloading. Note that the threshold weight for 6-axle trucks is 550 kN according to the traffic law in China. It can be observed that half of the 6-axle trucks are overloaded, which is indeed a risk for existing bridges.

Stochastic Traffic Flow Simulation.
With the probability models established based on the weigh-in-motion (WIM) data, this study utilized MCS to establish different types of stochastic traffic flows. e detailed procedures are summarized in Figure 5, where N ADT is the total number of daily traffic. e programming package MATLAB was utilized to translate the traffic as a matrix that will be called in the subsequent vehicle-bridge interaction analysis. e preliminary statistical analytical results of the actual traffic monitoring data indicate that the vehicle spacing between two following vehicles follows the Weibull distribution for congested traffic, while it follows Gamma distribution for free-flowing state. Figure 6 plots the sample of a free-flowing stochastic traffic flow, where V1 denotes the lightweight cars and busses, and V2∼V6 denote the 2-axle to 6-axle trucks, respectively. Details of the configuration of the vehicle types can be found in [28]. e benefit of the stochastic traffic flow is the probability distribution in accordance with the site-specific traffic monitoring data. In addition, the stochastic traffic model can be updated with the consideration of the traffic volume growth and the limit of truck overloading.  Figure 8. e material of the box-girder is C60 concrete. e pier is a composite structure with concrete filled steel tube. e internal and external concrete grades are C80 and C50, respectively. e finite element model is shown in Figure 9. All of the elements were simulated with beam elements. e boundary condition of the bridge piers is the constraint of all degrees of freedom at the bottom elements. e top nodes of mid-span piers are connected with the closest girder nodes in rigid connection. e top nodes of side-span piers are connected with the closet nodes with the compressive only connection. e top 50 vertical modes of vibration were extracted for the following vehicle-bridge interaction analysis.

Influence of RRC on Probability Distribution.
Since the road roughness condition (RRC) has a significant influence on the vehicle-bridge interaction system, this study selected "good" and "poor" RRCs for the comparative study. e RRCs were simulated by an inverse Fourier transformation function and corresponding power spectral density. e coefficients for "good" and "poor" RRCs are 32 × 10 −6 m 3 / cycle and 512 × 10 −6 m 3 /cycle, respectively. Figure 10  Shock and Vibration 5 pseudodynamic approach. Figure 11 plots the vertical deflection histories of the critical point in the mid-span of the bridge accounting for both good and poor RRCs, respectively. It is observed from Figure 10 that the poor RRC results in more deviations compared with the good RRC. e maximum deflection is located at 162 s, where several heavy trucks load on the mid-span point simultaneously. e maximum deflections for good and poor RRCs are −0.196 m and −0.246 m, respectively. us, it is demonstrated that the RRC has a significant influence on the maximum load effect.
Since the basis for extreme value extrapolation is the assumption of Gaussian distribution, the probability distribution should be fitted. Figure 12 plots the histograms of the deflections and the fitted PDF and CDF curves of Gaussian distribution. It is obvious that both good and poor RRCs follow Gaussian distribution. In addition, the shape of histograms for the poor RRC is more flat than that of the good RRC. e hypothesis testing was conducted to check the fitting to Gaussian distribution. e fitting variances are 0.0028 and 0.0019 for the good and poor RRCs, respectively. It is confirmed that the samples do not refuse the Gaussian distribution. In order to capture the influence of the two types of RRCs on the probabilistic load effect, the histograms of numbers of level-crossings were counted as shown in Figure 13.
As shown in Figure 13, Rice's level-crossings were fitted to the top 30% samples. It is observed that the poor RRC leads to larger number of level-crossings compared to the good RRC. Based on the above analysis, it can be concluded that the RRC can affect the probabilistic characteristics of traffic load effects, and thus a poor RRC will amplify the maximum traffic load effect.

Influence of Bridge Span Length on the Dynamic
Deflection. In order to investigate the influence of bridge span length on the deflection, both shorter-span and longerspan continuous rigid-frame bridges were selected for comparison study.
e main span lengths of the three bridges are 140 m, 200 m, and 248 m, respectively. In order to make the comparison of general significance, the same traffic flow as shown in Figure 6 was utilized for the three bridges. e bridge parameters and the maximum deflections are shown in Table 1.
As observed from Table 1, the maximum deflection increases with increase of the bridge main span length.

Shock and Vibration
However, the ratio between the good and poor RRC effects decreases with the increase of the bridge span length. is phenomenon can be explained by the theoretical basis that the bridge with a longer span length has a lower frequency, which will weaken the vehicle-bridge interaction. erefore, the influence of the RRC on the dynamic effect is weakened with the increase of the bridge span length.

Probability Estimation.
It was concluded by many researchers that dynamic response of bridges under stochastic vehicle flow can be assumed as stationary random process [31]. In order to investigate the influence of random process samples on the characteristics of stationary random processes, the mean value and the correlation coefficient in random process were analysed. Figure 14 It is observed that the mean value and standard value of the initial position are greatly affected by 7 seconds. In fact, the mean value and correlation coefficient tend to be constants with the increase of the samplings number.
erefore, if the sample is large enough, this process can be assumed as stationary random process. e random process under stochastic vehicle loads can be represented by the load effects, which was calculated by a large sample of vehicle flows. e mean value and root mean square deflection of the continuous rigid-frame bridge were obtained by the same method. Figure 15 shows the distribution of mean value and root mean square deflection along the girders.
As observed in Figure 15, the maximum mean value and the root mean square deflection of a continuous rigid-frame  erefore, this study selected the mid-span point as representation. e following investigation will focus on the deflection of the critical point under stochastic traffic loads.

First-Passage Reliability Evaluation.
In general, the firstpassage reliability analysis can be conducted based on probabilistic model associated with maximum extrapolation. e upper threshold deflection of a continuous rigid-frame bridge is a � L/500 � 0.4 m according to designed codes in China. Figure 16 shows the time-varying reliability index based on the first-passage criterion of the maximum girder deflection.
As shown in Figure 16, the initial reliability index of the bridge is 6.45. However, the reliability index decreases significantly with the time and the proportion of dense traffic flow. With consideration of the proportion of dense traffic flow as 1.2%, 2.4%, and 3.6%, the corresponding reliability indexes of the bridge in 100 years are 5.76, 5.62, and 5.60, respectively. erefore, it is concluded that the proportion of dense vehicle flow has significant influence on the first-passage probability of the bridge deflection. us, the control of dense vehicle    flow is essential for ensuring the bridge safety in service period.

Conclusions
is study developed the vehicle-bridge coupled vibration model to the dynamic reliability analysis of long-span bridges. A simplified analytical method was presented for evaluating probabilistic dynamic load effect induced by stochastic vehicle loads. e effectiveness of the proposed computational framework was demonstrated in the case study of a continuous rigid-frame bridge. Influence of the RRC, the bridge span length, and the traffic growth on the bridge deflection was investigated. e conclusions are summarised as follows: (1) e mean value and the root mean square deflections induced by stochastic traffic loads are constants, which are uncorrelated with time. erefore, the stochastic traffic-bridge coupled vibration is demonstrated as a stationary Gaussian random process, which can be treated as a long time history. (2) e level-crossing is able to capture the probability characteristics of the dynamic traffic load effect and thus provides a connection between the dynamic effect and the probability model. (3) e RRC will impact the probabilistic characteristics of traffic load effects. A poor RRC leads to larger number of level-crossings compared to a good RRC, and thus a poor RRC will amplify the maximum traffic load effect. (4) e influence of the RRC on the dynamic effect is weakened with the increase of the bridge span length. is phenomenon can be explained by the theoretical basis that the bridge with a longer span length has a lower frequency, which will weaken the vehicle-bridge interaction.
(5) With consideration of the proportion of dense traffic flow as 1.2%, 2.4%, and 3.6%, the corresponding reliability indexes of the bridge in 100 years are 5.76, 5.62, and 5.60, respectively. e proportion of dense vehicle flow has significant influence on the first-passage probability of the bridge deflection. us, the control of dense vehicle flow is essential for ensuring the bridge safety in service period.
Even though the proposed framework is verified for a rigidframe bridge, it can be applied for more types of bridges. However, more studies are necessary to improve the computational efficiency and accuracy. A more refined traffic-bridge interaction analysis approach instead of the simplified EDWL approach is critical to make the simulation more reasonable. In addition, the simulated traffic load effect needs to be compared with structural health monitoring data.
Data Availability e original traffic monitoring data are available from the authors if necessary.