The steadily growing traffic load has resulted in lots of bridge collapse events over the past decades, especially for short-to-medium span bridges. This study investigated probabilistic and dynamic traffic load effects on short-to-medium span bridges using practical heavy traffic data in China. Mathematical formulations for traffic-bridge coupled vibration and probabilistic extrapolation were derived. A framework for extrapolating probabilistic and dynamic traffic load effect was presented to conduct an efficient and accurate extrapolation. An equivalent dynamic wheel load model was demonstrated to be feasible for short-to-medium span bridges. Numerical studies of two types of simply-supported bridges were conducted based on site-specific traffic monitoring data. Numerical results show that the simulated samples and fitting lines follow a curve line in the Gumbel distribution coordinate system. It can be assumed that dynamic traffic load effects follow Gaussian distribution and the extreme value follows Gumbel distribution. The equivalent probabilistic amplification factor is smaller than the individual dynamic amplification factor, which might be due to the variability of individual samples. Eurocode 1 is the most conservative specification on vehicle load models, followed by the BS5400 specification. The D60-2015 specification in China and ASSHTO specification provide lower conservative traffic load models.

In recent decades, the highway freight volume has a significant increase leading by the steady expansion of transportation industry [

Several researchers indicated that the present design vehicle load model was underestimated compared to practical traffic loads. A study conducted by Han et al. [

Since the traffic is random in nature probabilistic analysis is essential to estimate the characteristic load effect on a bridge. Evaluating vehicle load effects on bridges using probabilistic extreme probability is one of the research hotspots in the field of bridge engineering [

With the development of weigh-in-motion (WIM) technology [

It is a common phenomenon that short-to-medium span bridges were collapsed or significantly damaged due to heavy traffic loads. In addition, the current freight traffic volume keeps a sustained growth tendency, which may evolve into a risk source for existing bridges. Thus, it is an urgent task to estimate and predict the traffic load effect on existing bridges with consideration of site-specific traffic loads rather than the design load. In this research area, most studies concentrate to the probabilistic traffic load effect modeling via static analysis. Several researchers developed dynamic analysis accounting for stochastic traffic loads. However, the probabilistic dynamic analysis for the bridges in extremely heavy traffic load area is still insufficient. Therefore, it is difficult to conduct a reasonable prediction of the maximum dynamic traffic load effect in a bridge lifetime.

This study presented a computational framework for extrapolating maximum dynamic traffic load effect on short-to-medium span bridges. A huge traffic data in the heavy traffic area was utilized to predict the maximum traffic load effect on short-to-medium span bridges. Two types of simply supported prestressed concrete bridges were selected to demonstrate the feasibility of the proposed framework. Parametric studies were conducted to investigate dynamic extrapolation effect. The numerical results were utilized for calibrating the vehicle load models in several design specifications.

The novelty of this study is the big traffic data collected in highway bridges in China, combined with the probabilistic analysis of the dynamic traffic load effects on short span bridges. Even though several researchers [

For short-to-medium span bridges, traffic dynamic effects are more significant than that of a long-span bridge. Thus, the dynamic effect should be considered in the numerical simulation. In general, the dynamic effect can be considered in a vehicle-bridge coupled vibration system. The equations of motion of the system can be established based on forces and displacement of the system, which are written by [

where, _{v}_{b}_{bv}_{vb}_{bg}_{vg}_{v}_{b}

The coupled equations of motion are differential equations with variable coefficients for parameters include displacements, velocities and accelerations. In general, the equation can be solved by utilizing a step-by-step integration method with the following procedures [

The above method can be used to calculate dynamic effects of a bridge under individual vehicle load. For multiple vehicle loads on the bridge, the estimation of dynamic effects is more complex. Zhou et al. [

where, _{a}_{j}_{j(t)} represents the equivalent load of the

By combining the equations of motion and the EDWL function, the traffic-bridge coupled vibration equation can be written as [

where, _{j}_{j}_{k}

In order to improve the computational efficiency, structural mode parameters are added to the motion equations. Considering the orthogonality of bridge modes, the equation of motion can be written as [

where,

where,

This equation can be solved by Newmark-

where,

In general, the vehicle load can be regarded as a stationary stochastic process according to the design specification and many research results [

where,

where,

For the topic of traffic load effect, the commonly used method is to divide the simulated data into groups, Therefore, according to probability function of the binomial distribution, the probability _{T}

where,

In the design reference period, a characteristic value _{k}

where, _{k}_{k}

In general, the traffic load effect on a bridge can be evaluated based on static influence lines of the bridge extracted from the finite element model. However, dynamic effects in the vehicle–bridge coupled system have no concern with the static influence lines. For the vehicle-bridge interaction analysis, there is a time consuming process to evaluate a large number of block maximum values for probabilistic modelling. Therefore, there should be a balance between the large number of dynamic simulations and the accurate probabilistic modeling. This study presented a comprehensive computational framework for a reasonable extrapolation of dynamic and probabilistic traffic load effects.

By combining the theories of traffic-bridge coupled vibration and probabilistic extreme theory, the maximum probabilistic and dynamic traffic load effect on the bridge can be evaluated. In order to improve the computational efficiency, the static influence line was utilized for the purpose of identifying the critical loading scenarios, which could be subsequently utilized for the dynamic analysis. A flow chart of the computational framework is shown in

As shown in

Compared with the traditional computational approach, the proposed computational framework has the following benefits. Firstly, the equivalent dynamic approach can provide efficient and considerable accurate dynamic solution for the traffic load effect on the bridge. Secondly, the stochastic traffic load model derived from the practical traffic monitoring data can provide real-time probabilistic parameters for the site-specific bridge. Finally, the traffic change due to traffic growth or overloading control can be reflected by the traffic monitoring data, where the change tendency of the maximum traffic load effect can be reflected. However, it is important to note that the accuracy of the probabilistic extrapolation mostly depends on the number of simulations. Meanwhile, more input traffic data will lead to a more realistic extrapolation.

The WIM data collected from a highway bridge in China was selected for probabilistic modeling of traffic loads. Illustration of the WIM system is shown in

All vehicles were classified into 6 types based on the vehicle configurations and axle characteristics. The proportions of all type of vehicles are shown in

Vehicle type | Vehicle configuration | Illustration | Proportion | |
---|---|---|---|---|

Slow lane (%) | Fast lane (%) | |||

V_{1} |
Light cars | 36.6 | 63.4 | |

V_{2} |
2-axle trucks | 84.5 | 15.5 | |

V_{3} |
3-axle trucks | 91.1 | 8.9 | |

V_{4} |
4-axle trucks | 96.5 | 3.5 | |

V_{5} |
5-axle trucks | 92.6 | 7.4 | |

V_{6} |
6-axle trucks | 98.1 | 1.9 |

As shown in

In order to study the probability distribution of axle weights and total weights of heavy trucks, the proportion of axle weights was assigned as a parameter.

Based on estimated probability distribution models, the stochastic traffic flow load model was established using Monte Carlo simulation.

Two typical simply-supported T-girder bridges with span-length of 20 and 40 m were selected as prototypes for the numerical simulation. The dimensions of the bridges are shown in _{b}

The simulated stochastic load model for dense traffic flow shown in

Since the EDWL approach was primarily developed by Chen et al. [

Subsequently, consider the critical loading vehicles passing on the bridge with a constant speed

Span length (m) | Maximum bending moment (kN.m) | Dynamic amplification factor | |||
---|---|---|---|---|---|

Static | |||||

641 | 715 | 905 | 1.11 | 1.41 | |

2917 | 3202 | 3701 | 1.09 | 1.26 |

Based on the above analysis, it can be inferred that a bridge with a shorter span-length and a worse RRC has a larger dynamic amplification factor. This conclusion is in accordance with the theoretical basis that a bridge with a shorter span-length has a higher stiffness, which leads to stronger vibrations. Therefore, the influence of dynamic effects on the probabilistic extrapolation of traffic load effects should be considered in the bridge with a shorter span length.

For the probabilistic evaluation, 10-year samples (

As shown in

Based on the established probability models, this study examined and compared the traffic load effect accounting for different national design specifications, such as AASHTO, Eurocode 1, BS5400, and D60-2015 in China. The referenced bending moment was considered as a return period of 1000 years and the RRC was considered as good.

Items | Standard values (kN.m) | Estimated return period (years) | ||
---|---|---|---|---|

Span length | ||||

Estimated values accounting for _{t} |
969 | 4070 | 1000 | 1000 |

AASHTO | 934 | 3454 | 909 | 645 |

Eurocode 1 | 1781 | 6500 | Infinite | Infinite |

BS5400 | 1284 | 5806 | Infinite | |

D60-2015 | 966 | 4214 | 980 | 1354 |

It is observed that Eurocode 1 has the highest standard value compared to the other design specifications and the actual value. In addition, the return period for the Eurocode 1 value is infinite, which can be treated as the most conservative design specification. Subsequently, the BS5400 value is much higher than the estimated value, ASSHTO value and D60-2015 value. The return period for the BS5400 value is approximately infinite. The AASHTO and the D60-2015 have the lowest values, which are close to the estimated values. For the D60-2015 code in China, the standard value for longer span bridges is conservative than short span bridges.

This study presented an efficient computational framework combining vehicle-bridge interaction analysis and probabilistic extrapolation, which provided reasonable characteristic dynamic traffic load effects on short-to-medium span bridges. A stochastic traffic load model can provide a reliable connection for transmitting probability distributions from site-specific traffic loads to the dynamic load effects on the bridge. Case studies of two simply supported bridges were conducted taking into account traffic monitoring data. Comparison of deterministic and probabilistic dynamic amplification factors shows advantages of the proposed computational framework. Based on the site-specific traffic data, the estimated bending moments of the two bridges were utilized to calibrate the characteristic values in several design specifications. The conclusions are summarized as follows:

(1) The stochastic traffic flow model contains the individual vehicle parameters and the probability characteristics estimated from the actual traffic data. In addition, the simulated traffic samples is appropriate for the vehicle-bridge interaction analysis in time domain, where the interspace between the deterministic simulation and probability extrapolation.

(2) The simulated samples follow a curve line in the Gumbel distribution coordinate system. With increase of the number of samples, the extreme value will follow a straight line. This phenomenon demonstrates that the dynamic traffic load effects follow Gaussian distribution, and the extreme values follow Gumbel distribution.

(3) The dynamic amplification factors for the 20 and 40 m girders with a good road roughness condition are 1.11 and 1.09, respectively. However, the equivalent probabilistic amplification factors are 1.07 and 1.05, respectively. It is an interesting phenomenon that the probabilistic dynamic factor is less than the individual dynamic factor, which might due to the variability of individual samples. In other words, equivalent probabilistic amplification factor contains a large number of simulations, while the individual dynamic amplification factor is just one of the data base. Therefore, the probabilistic amplification factor is more reliable.

(4) The vehicle load models in representative national design codes have a great deviation corresponding to the practical traffic data. The design vehicle load model in Eurocode 1 is the most conservative one, followed by the BS5400 specification. The D60-2015 specification in China provides an approximated load model for the practical traffic data. The traffic load model in ASSHTO specification provides the lowest traffic load value that is less conservative for the traffic load in China.