^{1}

^{1}

^{1}

^{2}

^{3}

^{1}

^{2}

^{3}

The random traffic flow model which considers parameters of all the vehicles passing through the bridge, including arrival time, vehicle speed, vehicle type, vehicle weight, and horizontal position as well as the bridge deck roughness, is input into the vehicle-bridge coupling vibration program. In this way, vehicle-bridge coupling vibration responses with considering the random traffic flow can be numerically simulated. Experimental test is used to validate the numerical simulation, and they had the consistent changing trends. This result proves the reliability of the vehicle-bridge coupling model in this paper. However, the computational process of this method is complicated and proposes high requirements for computer performance and resources. Therefore, this paper considers using a more advanced intelligent method to predict vibration responses of the long-span bridge. The PSO-BP (particle swarm optimization-back propagation) neural network model is proposed to predict vibration responses of the long-span bridge. Predicted values and real values at each point basically have the consistent changing trends, and the maximum error is less than 10%. Hence, it is feasible to predict vibration responses of the long-span bridge using the PSO-BP neural network model. In order to verify advantages of the predicting model, it is compared with the BP neural network model and GA-BP neural network model. The PSO-BP neural network model converges to the set critical error after it is iterated to the 226th generation, while the other two neural network models are not converged. In addition, the relative error of predicted values using PSO-BP neural network is only 2.71%, which is obviously less than the predicted results of other two neural network models. We can find that the PSO-BP neural network model proposed by the paper in predicting vibration responses is highly efficient and accurate.

With the progress of the era, transportation and automobile industries have achieved rapid development, while vehicle loads acting on bridge structures also are increased continuously. As a result, the traffic flow will be increased continuously, driving speeds of vehicles will be increased obviously, and a lot of heavy-load vehicles travel on highway. In some regions, the overload phenomena are serious, and vehicle dynamic loads become one of the main reasons for bridge deck damage, threatening the safe of long-span bridges [

However, when vehicle loads are considered, they compute vehicle loads according to bridge design specifications and neglected randomness of traffic flows. Therefore, the obtained conclusions had a great deviation from results of the experimental test. Computational results will further approach actual engineering situations, and engineering practice can be guided better by studied conclusions if random traffic flow conditions can be investigated fully; comprehensive statistical analysis can be conducted to parameters; and a random traffic flow simulation program based on tested data can be compiled on this basis and used for studying bridge structures. Yin and Deng [

At present, researches on random traffic flow loads acting on bridges are mainly focused on statistical analysis theories, depend too much on basic assumption about unchanging of vehicle type, vehicle weight, vehicle distance, and vehicle speed, and fail to fully study and consider random characteristics of each traffic flow parameters. Aiming at such situation, this paper tests bridge health and monitors traffic flow parameters including vehicle type, vehicle weight, and vehicle speed. Then, through mathematic statistical analysis, representative data of traffic situations is obtained. Simulation on random traffic flow is conducted, and a random vehicle flow program is compiled on this basis. A vehicle-bridge coupling model including random traffic flows is established, which can consider vehicle type, vehicle weight, vehicle lane, vehicle speed, and opposite driving functions. After that, response values of the bridge structure are extracted to be compared with tested values. In the testing, tested time is long, so there are many data parameters to be monitored. Size of data to be uploaded to the data center will be very huge. In view of the large bridge spans, monitoring with traditional wired sensor networks must be confronted with long wiring and high monitoring difficulty. Thanks to rapid development of big data analysis technology, microcomputer system, sensor technology, wireless communication technology, and low-power consumption embedding technology, it is possible to acquire and process data with a wireless sensor network. With increase of monitoring scope and monitoring points, the data throughout to be realized by the network shall be increasingly higher. Finally, massive data will be uploaded to the data center, belonging to the category of big data. The computational results are practical and reliable. The vehicle-bridge coupling vibration model with considering random traffic flows established in the paper is feasible.

This paper selected a long-span bridge with two towers as the studied object, as shown in Figure

Schematic diagram of long-span bridges.

According to size parameters in Figure

Local mesh model of long-span bridges.

Cross section

Cable tower

Stay cable

Connections between cable tower, pier, and girder are as follows: CP command of nodes in ANSYS was used to couple three translational freedom degrees and three rotational freedom degrees between two nodes. The cable towers and piers were processed by the complete consolidation, so freedom degrees in 6 directions were restrained, as shown in Figure

Boundary constraints of long-span bridges.

Modal vibration shapes of top 7 orders for long-span bridges.

First order

Second order

Third order

Fourth order

Fifth order

Sixth order

The paper will study vehicle-bridge coupling vibration responses. Therefore, bridge deck roughness should be input into the computational model. Bridge deck roughness refers to deviation levels of the bridge surface relative to a standard plane. According to a lot of experimental results, bridge deck roughness is an argotic and steady Gauss random process with a mean value of zero. Therefore, it could be simulated by different forms of trigonometric series. Within the spatial frequency scope of

Fitting expression of power spectrum density

Through comparing (

Sine wave functions corresponding to different intervals are overlaid, so a random bridge deck roughness model can be obtained.

Formula (

Bridge deck roughness of long-span bridges.

The paper researches vibration responses of bridges under random traffic flow, so random traffic flows need to be monitored. When vehicles run on a bridge, the vehicle speeds, vehicle types, vehicle distances, vehicle weights, and lanes would change randomly. All these parameters shall be taken into account. Monitoring time is long, so there are many data parameters to be monitored. Size of data to be uploaded to the data center will be very huge. In view of the large bridge spans, monitoring with traditionally wired sensor networks must be confronted with long wiring and high monitoring difficulty. Thanks to the rapid development of big data technology [

Application of the wireless sensor network.

Collection of random traffic flow loads using the wireless sensor network.

When the random traffic flow was obtained, we have prepared a program to use the random traffic flow. In the program, the vehicle lane, vehicle type, vehicle weight, vehicle speed, and roughness spectrum were considered. Finally, the prepared program was combined with the ANSYS software because ANSYS can output the command script file. Detailed methods, steps, and processes of numerical analysis on vehicle-bridge coupling vibration based on ANSYS are shown in Figure

Solution processes of vehicle-bridge coupling vibrations.

Vibration displacements and accelerations at cable towers and piers of the long-span bridge are computed, as shown in Figure

Vibration responses at different positions of long-span bridges.

Vibration displacements at cable tower

Vibration accelerations at cable tower

Vibration displacements at pier

Vibration accelerations at pier

Such complicated model of the long-span bridge is affected by many parameters, so its correctness should be verified by experimental test. As shown in Figure

Comparison of bridge vibration responses between experiment and simulation.

Vibration displacements at cable tower

Vibration displacements at pier

As mentioned, vibration responses of the long-span bridge are computed using finite element simulation. However, the computational process of this method is complicated and proposes high requirements for computer performance and resources. Therefore, the paper considers using a more advanced intelligent method to predict vibration responses of the long-span bridge. BP neural network model is a typical multilayer feed forward neural network. With a strong capacity of information classification and recognition, it can achieve approximation problem of any function [

BP neural work is widely applied in many fields and also achieves some effects [

Particle swarm optimization (PSO) algorithm is an optimization algorithm based on swarm intelligence theories and is also an effective global optimization algorithm [

Flow chart of PSO-BP neural network model.

The PSO-BP neural network model is used to predict vibration responses of the long-span bridge. Vibration responses obtained by numerical computation in Section

Training of PSO-BP neural network model.

Vibration displacements at cable tower

Vibration displacements at pier

Prediction of PSO-BP neural network model.

Vibration displacements at cable tower

Vibration displacements at pier

In order to verify advantages of the prediction model, we have extracted the predicting process of the vibration displacement at cable tower using the PSO-BP neural network model. Then, it is compared with the BP neural network model and GA-BP neural network model, as shown in Table

Comparison of predicted results for three kinds of neural network models.

Predicting algorithms | Training errors | Amount of iterations | RMS of predicted results (mm) | Real results (mm) | Relative errors (mm) | Predicting time |
---|---|---|---|---|---|---|

BPNN | 0.08 | 500 | 20.1 | 18.4 | 9.23 | 1.05 hours |

GA-BPNN | 0.016 | 398 | 19.2 | 18.4 | 4.34 | 0.70 hours |

PSO-BPNN | 0.016 | 226 | 18.9 | 18.4 | 2.71 | 0.50 hours |

Modal frequencies of the long-span bridge are 0.32 Hz, 0.39 Hz, 0.45 Hz, 0.72 Hz, 0.82 Hz, 0.91 Hz, and 1.07 Hz, respectively. Obviously, the frequencies are distributed densely, satisfying the feature of dense distribution of natural frequencies of large infrastructures. In addition, vibration shapes of the long-span bridge are not completely the simple torsion or bending vibration. Sometimes, vibration shapes are overlaid results of two vibration shapes.

In this paper, the experimental test was completed by the wireless sensor network technology. The tested time was too long, and parameters are too many, so that the collected data will be very huge, belonging to the category of big data. Then, experimental results and numerical simulation results are compared to perform the consistent changing trends. At some peaks, experimental values are slightly more than numerical simulation results. The reason is that boundary conditions of the numerical simulation are relatively ideal states and only consider effects of random traffic flow on the bridge, but they neglect practical wind excitation. In addition, material characteristics of numerical simulation can hardly be kept consistent with actual values. Wind speeds in experimental test are low, and excitation borne by the bridge is mainly generated from vehicles, so results between numerical simulation and experimental test do not have big errors. This result proves the reliability of the vehicle-bridge coupling model in the paper.

The PSO-BP neural network model is used to predict vibration responses of the long-span bridge. Predicted values and real values of the neural network at each position point basically have the consistent changing trends, only some peak values are different, and the maximum error is less than 10%. Hence, it is feasible to predict vibration responses of the long-span bridge using the PSO-BP neural network model.

In order to verify advantages of the prediction model, it is compared with the BP neural network model and GA-BP neural network model. The PSO-BP neural network model converges to the set critical error after it is iterated to the 226th generation, while the other two neural network models are not converged. The reason is that PSO can accelerate convergence of the neural network model. The training errors are 0.08 and 0.016, respectively, when the BP neural network reaches the maximum iteration 500, and GA-BP neural network reaches the 398th generation, but errors using the BP neural network are still more than the set critical error 0.016. In addition, the relative error of predicted values of PSO-BP neural network is only 2.71%, which is obviously less than the predicted results of other two neural network models. We can find that application of the PSO-BP neural network model proposed by the paper in predicting vibration responses is highly efficient and accurate.

The data used to support this study is currently under embargo while the research findings are commercialized. Requests for data, 6 months after publication of this article, will be considered by the corresponding author.

The authors declare that they have no conflicts of interest regarding this work.

This work is supported by the National Natural Science Foundation of China (51608137).