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For long-span cable-stayed bridges, cables serve as one of the most important components to guarantee structural integrity. Forces of stay cables indicate not only the performance of cables themselves but also the overall condition of bridges. In order to help stakeholders to make maintenance decisions, an extreme cable force estimation method was proposed based on cable force measurements and traffic data from the weighing system. First, raw monitoring data were preprocessed based on a median filtering to obtain usable cable force signals. The multiresolution wavelet method was used to extract traffic-induced force component from mixed signals. Then, a Monte Carlo-based random vehicle model was developed using traffic data from the weighing system. Based on field temperature measurements and simulation of traffic-induced effects, extreme cable forces with respect to vehicle loads and temperature effects were predicted by extreme value theory. The Generalized Pareto Distribution (GPD) was adopted to establish the probability distribution models of the daily maximum cable force. Then, the extreme value within a return period of 100 years was determined and compared with the design loading demand. Finally, the effectiveness of the proposed method was validated through a cable-stayed bridge in China. As a result, the low-frequency varying component of cable force response had positive correlation with environmental temperatures, and the extreme value of the predicted cable force under prospective traffic volumes was within limit interval value according to the design code. The conclusions can be utilized by bridge owners to make maintenance decisions.

With rapid growth of infrastructure investment in recent years, more and more long-span cable-stayed bridges have been constructed over rivers and valleys [

For cable-stayed bridges, the stay cable is regarded as one of the most important components to ensure the allowable displacement and distribution of bending moments along the bridge deck with prestressing force [

To predict the performance of cables, the random vehicle model should be established to simulate traffic-induced cable force responses under prospective traffic volumes. Previously, the passing vehicles were only tracked and identified by monitoring camera system on bridges [

In this paper, the cable forces of cable-stayed bridges were discussed and the extreme value was predicted based on monitoring data and the developed random vehicle model. First, a median-filtering-based preprocessing was conducted to obtain usable cable force measurements. The multiresolution wavelet method was used to extract traffic-induced force. Meanwhile, the Monte Carlo based random vehicle model was developed using traffic data from bridge weighing system. Then, the extreme cable forces with respect to vehicle loads and temperature effects for the return period of 100 years were predicted by applying the Generalized Pareto Distribution (GPD) model. Finally, the reliability of the proposed method was validated by a case study of a long-span cable-stayed bridge, the third Nanjing Yangtze River Bridge in China.

At present, monitoring systems are widely installed in long-span cable-stayed bridges. The monitoring data, including cable force measurements from SHM system and traffic data from weighing system, can achieve continuous collection. However, further investigation is still needed to obtain valuable information from mass data. Considering the probable performance of stay cables within the return period (which can be regarded as an approximate frequency of occurrence [

The flowchart of the method.

At present, the recorded cable force data by SHM systems includes direct and indirect measurements. Compared with the common vibration-based measurements, the direct force measurements help to know the operational condition conveniently. The influence of conversion error was also eliminated to ensure data accuracy. However, direct measuring methods skip the transforming procedure such as fast Fourier transform (FFT), which can reduce signal noise simultaneously. Therefore, it is necessary to perform preprocessing to eliminate the influence of data noise before analyzing the direct cable force measurements. Figure

Preprocessing for monitoring data: (a) the raw signals; (b) the filtered signals.

To simplify, the monitoring cable forces in bridge service period are considered as the linear superposition of temperature effect and dead load-induced force and vehicle load-induced force, corresponding to slow-varying ingredient and dynamic ingredient in signal, respectively. Compared with traffic load, the wind effect to cable force is ignored in this paper for it does not significantly influence the results, which was also validated by existing research [

To separate signals of aforementioned two ingredients, the multiresolution wavelet method based on distinguished frequency bandwidths is adopted [

The multiresolution wavelet method transforms the signals to approximation and detail coefficients that depict the frequency distribution in the time and frequency domains, as shown in Figure ^{th} term of approximation, while Di denotes the ^{th} term of detail. Thus, the original signals can be separated. For instance, Figure

The schematic representation of wavelet method.

Separation results of signals.

Similar to other codes, the Chinese codes mention that the vehicle load should be modified based on the actual situation during the assessment [

Then, the establishment procedure for dynamic random vehicle model was developed in several stages, as shown in Figure

The establishment procedure of the random vehicle model.

To realize extreme cable force estimation in the field of statistics for the return period, it is required to estimate the extreme cable forces from current records [

Step 1: determination of subsample

Based on abundant monitoring data, subsample

Step 2: maximum excess quantity calculation

Then, the maximum excess quantity for each subsample

Step 3: GPD fitting and parameter estimation

Assuming that it is independent, each maximum excess quantity can be fitted by the GPD model with 2 parameters, which can be expressed as follows:

where

The GPD fitting is conducted with data to estimate the shape parameter

Step 4: extreme value estimation

With estimated parameters, the POT model described by the GPD is established to estimate maximum excess quantity. Therefore, the GPD-based extreme value of data series

where

The monitoring data in this paper were collected from an SHM system and a weighing system on a highway cable-stayed bridge crossing Yangtze River in Nanjing, China. Several typical cables were analyzed to select the critical position. Usually, cables designed with maximum force or the longest length were emphatically studied. The traffic data from weighing system was used as a probabilistic database for random simulation. In addition, all the sensors on the bridge are brand new in early years so that their measurements can be utilized for analysis.

The third Nanjing Yangtze River Bridge was built in 2005, which connects Nanjing City and its Pukou District. It is a cable-stayed bridge with a main span of 648 m and a tower height of 215 m and has bidirectional 6 lanes. The main girder is supported by 168 stay cables, each consisting of 109 to 241 wires with a diameter of 7 mm. The appearance of the bridge is shown in Figure

The third Nanjing Yangtze River Bridge.

The bridge weighing system is set at the bridge toll stations (both south and north banks). For each vehicle passing the bridge, the system will record its pass time, vehicle class, total weight, axle weight, axle distance, and driving lane. The recorded traffic data are used to establish random vehicle models combined with other parameters, such as speed and acceleration.

For a practical cable-stayed bridge, the longest cables near the mid-span are usually designed to bear the maximum force. Figure ^{2}. The wires are galvanized and have an ultimate strength of 1670 MPa.

The designed cable forces and cable length of completed bridge.

The structure of cable NJ21.

In Figure

The subsignal of monitoring data after wavelet separation.

The obvious linear relationship can be observed between

The relationship between separated cable force and temperature.

Extreme value analysis was then conducted for temperature data with the GPD model. The threshold

The PDF of the GPD for temperature excess quantity.

The two parameters of the GPD processing were then substituted into equation (

The weighing system data of the third Nanjing Yangtze River Bridge from January 2006 to April 2009 were extracted as a sample to analyze the vehicle characteristics. According to different vehicle axle distribution, the data were divided into 12 types, as shown in Table

Vehicle types and associated information.

Vehicle type | Graphic illustration | Vehicle axle | Number | Wheelbase (m) | |
---|---|---|---|---|---|

I | 2 + 2 | 5.47 | 335101 | 2.92 | |

II | 2 + 4 | 54.07 | 3313643 | 4.862 | |

III | 2 + 8 | 3.30 | 201965 | 3.987 + 1.377 | |

IV | 2 + 2 + 4 | 3.70 | 226936 | 1.811 + 5.331 | |

V | 2 + 2 + 8 | 2.87 | 176074 | 1.791 + 4.216 + 1.361 | |

VI | 2 + 4 + 4 | 1.33 | 81205 | 3.525 + 5.810 | |

VII | 2 + 4 + 8 | 11.43 | 700783 | 3.525 + 5.5897 + 1.312 | |

VIII | 2 + 4 + 12 | 12.54 | 768784 | 3.525 + 6.811 + 1.313 + 1.313 | |

IX | 2 + 8 + 8 | 0.56 | 34156 | 3.233 + 1.376 + 5.867 + 1.312 | |

X | 2 + 8 + 12 | 3.33 | 203808 | 3.233 + 1.376 + 6.811 + 1.313 + 1.313 | |

XI | 2 + 2+4 + 8 | 0.04 | 2290 | 1.89 + 2.416 + 5.897 + 1.312 | |

XII | 2 + 2+4 + 12 | 1.37 | 83668 | 1.89 + 2.416 + 6.811 + 1.313 + 1.313 |

Based on the associated information of vehicle types, the mixture normal distribution (MND), mixture lognormal distribution (MLD), and EM algorithm were applied to establish the statistical models of vehicle weight of different vehicle types, and the t-distribution was used to establish the statistical models of vehicle speed. The vehicle arrival time intervals were determined according to Poisson process theory. Figure

Examples of statistical models: (a) vehicle weight for type III; (b) vehicle speed for type III; (c) the time interval.

In order to verify the reliability of the random vehicle model developed in this study, the simulated vehicle flow was acted on structural influence lines to calculate the time history of vehicle-induced responses, which was then compared with the monitoring cable forces. Similarly, the cable NJ21 was selected to be investigated. The cable force influence line for cable NJ21 and the layout of lanes are shown in Figure

The random vehicle loading procedure: (a) cable force influence line for NJ21; (b) layout of lanes.

To evaluate the accuracy of simulation results, the actual average hourly traffic volumes of March 2009 were used as an input of the established random vehicle model. Meanwhile, cable force monitoring data for the same period were extracted and separated. A comparison for the calculated result and the monitoring data of cable NJ21 was conducted during the period from 11:00 to 12:00 on March 6, 2009, as shown in Figure

The comparison of simulation and monitoring data.

Comparison of calculated cables forces with monitoring cable forces.

Maximum value | Minimum value | Average value | |
---|---|---|---|

Calculated cable force (kN) | 91.6294 | −21.6761 | 6.7399 |

Monitoring data (kN) | 88.3274 | −19.0815 | 6.5012 |

Moreover, the Kullback–Leibler (KL) divergence was introduced to quantitatively evaluate the simulation. The calculated KL divergence should be a positive value, and the more its value is close to zero, the more accurate the simulation method used is. Assuming that

The average daily traffic of the third Nanjing Yangtze River Bridge in the analysis period was about 27,000. To predict the extreme value of vehicle-induced responses, prospective traffic volume should be analyzed. In this respect, influencing factors including population, industry, and business development need to be considered. In this study, the traffic volume of another long-span bridge (Jiangyin Bridge) located at Yangtze River near the case bridge was investigated to analyze prospective traffic, for the operational situation of this bridge was close to the saturation level. The average daily traffic of Jiangyin Bridge was about 70,000 vehicles, while its ADTT (average daily truck traffic) was about 11,000. The traffic jam occurs in Jiangyin Bridge when the daily traffic exceeded 90,000. When the daily traffic volume reached the peak value of 130,000 vehicles on a holiday, a severe traffic jam happened. Mean while, the corresponding ADTT was only 5,000 in the same period. This traffic condition was almost the operational limitation of Jiangyin Bridge. Since ADTT is an important parameter that influences the vehicle-induced cable force, the passenger cars dominating traffic volume are not suitable to calculate response extreme value. As these two bridges are located in adjacent regions of the middle-lower Yangtze plains, the traffic data of Jiangyin Bridge was used as a sample to determine the prospective average daily traffic volume of the third Nanjing Yangtze River Bridge. The result was estimated to be 80,000 with the ADTT of 15,000.

Then, the received prospective traffic volume was taken as an input parameter of random vehicle model to calculate prospective vehicle-induced responses of cable NJ21. In order to consider various adverse conditions that may appear in service periods, vehicle-induced daily maximum cable force of cable NJ21 was simulated for a long period of five years; see Figure

Five-year vehicle-induced daily simulated maximum cable forces.

The PDF of the GPD for cable force excess quantity.

The values of the two parameters of the GPD were substituted into equation (

To illustrate the predicted extreme value of cable forces in prospective operational conditions, Figure

Variation of the predicted extreme value with return periods: (a) the vehicle-induced response for NJ21 and temperature; (b) the total cable force.

The extreme value of total cable force in prospective operational conditions with the return period of 100 years can be calculated by equation (

This paper proposed a practical approach to analyze and predict the performance of stay cables of long-span cable-stayed bridges based on cable force monitoring data and traffic data. After raw data preprocessing, a random vehicle model was developed and applied to predict the prospective cable force with extreme value theory. To validate the reliability of the proposed method, measurements from the SHM system and the weighing system of the third Nanjing Yangtze River Bridge were involved for a case study. According to the results of the study, the following conclusions can be drawn:

The cable forces for a cable-stayed bridge are considered as the combination of different action effects. The wavelet transform method is suitable to separate the monitoring cable force induced by different effects.

The multiparameter vehicle modelling method was developed to generate Monte Carlo random traffic flow based on the collected bridge weighing system data and then calculated vehicle-induced response. The vehicles can be divided into specific several types according to the numbers of axles and wheels.

As a supplement to the normal distribution, the GPD-based extreme value method was utilized to predict extreme cable force in a specific return period. In the application, the estimated extreme cable force in prospective operational conditions with the return period of 100 years was 4239.04 kN.

The proposed analysis and estimation methods were applied to the third Nanjing Yangtze River Bridge. Based on the actual cable force monitoring data and the bridge weighing system data, the extreme cable force in prospective traffic operational condition for cable NJ21 was obtained. The method provided reliable estimates for the cable forces in specific return periods with limited data.

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

The research was supported by the Natural Science Foundation of Jiangsu Province (no. BK20181278), Transportation Science Research Project in Jiangsu (no. 2019Z02), the Postgraduate Research & Practice Innovation Program of Jiangsu Province (no. KYCX19_0099), and the Academician Special Science Research Project of CCCC (no. YSZX-03-2020-01-B).