^{1}

^{2}

^{3}

^{1}

^{1}

^{2}

^{3}

To study the vehicle load characteristics of bay bridges in China, especially truck loads, we performed a statistical analysis of the vehicle loads on Hangzhou Bay Bridge using more than 3 months of weigh-in-motion data from the site. The results showed that when all the vehicle samples were included in the statistical analysis, the histogram of the vehicles exhibited a multimodal distribution, which could not be fitted successfully by a familiar single probability distribution model. When the truck samples were analyzed, a characteristic multiple-peaked distribution with a main peak was obtained. The probability distribution of all vehicles was fitted using a weighting function with five normal distributions and the truck loads were modeled by a single normal distribution. The results demonstrated the good fits with the histogram. The histograms of different time periods were also analyzed. The results showed that the traffic mainly comprised two-axle small vehicles during the rush hours in the morning and the evening, and the histogram could be fitted approximately using three normal distribution functions. And the maximum value distributions of vehicles during the design life of the bay bridge were predicted by maximum value theory.

The rapid economic development of China is reflected by the construction of large-span bridges in recent decades. Over 60 bridges with spans of >400 m have been built [

Many previous studies have addressed vehicle loads. Studies of vehicle loads have been performed outside China for many years, but they have been continued as the traffic loads have increased. These studies only began recently in China, mainly because suitable weighing equipment was rarely available in the past and limited data were acquired. At present, newly built long-span bridges must be equipped with health monitoring systems to monitor the bridge situation [

Using statistical data for the current vehicle loads on National Expressway 110, Mei et al. [

These previous studies focused mainly on the vehicle loads of road bridges, including urban bridges. However, the vehicle load conditions of bay bridges are different from both of these types. The vehicle loads of busy road bridges mainly comprise multiaxle heavy trucks, whereas those of urban bridges mainly comprise cars, but the vehicle loads of bay bridges include both types. In the present study, the vehicle loads of Hangzhou Bay Bridge were analyzed. Based on WIM data obtained from Hangzhou Bay Bridge for over 3 months, the vehicle load histograms were determined and a probabilistic model of the truck loads was produced. In addition, the maximum value distribution was predicted for vehicles during the reference design period of bay bridges using the maximum value theory, thereby facilitating a better understanding of the vehicle load distributions of bay bridges. These data can provide a basis for the revision of specifications.

Hangzhou Bay Bridge is a cross-sea bridge that stretches over Hangzhou Bay in China. It officially entered use on May 1, 2008. Its main span is 325 m and its width is 33 m with six lanes of traffic in two directions. It extends for 36 km from Jiaxing City in Zhejiang Province in the north to Ningbo in the south (Figure

(a) Location of Hangzhou Bay Bridge. (b) Facade profile sketch map of the main towers of Hangzhou Bay Bridge.

The bridge employs a charging operational mode and the WIM system is embedded near the charging site at both entrances (two directions) to obtain information about vehicles, including the passing time, gross weight, axle weight, wheelbase, axle number, and speed. According to official statistics, there have been up to 19 million passing vehicles since its opening on May 1, 2010, that is, 23,000 per day. In this study, data in the paper were obtained from south entrance WIM system that recording three lane vehicles of the bridge during more than 3 months between September 2012 and January 2013. Before the analysis, the data were preprocessed to remove any obvious errors and meaningless data [

After preprocessing, there were 82 effective days and 931,774 valid data points. Initially, the mean values, standard deviations, and maximum values were calculated for the valid data, which showed that the mean values and coefficients of variation for the vehicle loads were relatively concentrated in specific periods, except the coefficients of variation (COV) which were higher on holidays such as New Year’s Day, thereby indicating that the vehicle loads were consistent in the short term; some samples are shown in Table

Statistical characteristics of the vehicle loads.

Date | Vehicle number | Mean value/ton | Standard deviation/ton | Coefficient of variation | Maximum value/ton |
---|---|---|---|---|---|

20120916 | 7408 | 11.29345 | 9.787491 | 0.866652 | 65.5 |

20120926 | 7399 | 13.15604 | 10.88008 | 0.827002 | 72.4 |

20121102 | 14627 | 7.913687 | 10.33094 | 1.305452 | 72.7 |

20121103 | 11351 | 8.199344 | 10.54972 | 1.286654 | 73.5 |

20121106 | 13263 | 8.948771 | 11.30286 | 1.263063 | 75.1 |

20121107 | 13641 | 8.587805 | 10.70217 | 1.246205 | 80.95 |

20121110 | 10426 | 8.58123 | 11.2873 | 1.315347 | 124.4 |

20121111 | 11694 | 7.499042 | 9.344368 | 1.246075 | 71.3 |

20121112 | 11633 | 8.413505 | 9.937082 | 1.181087 | 71.5 |

20121114 | 14232 | 7.807947 | 9.695331 | 1.241726 | 89.2 |

20121127 | 12080 | 8.341387 | 10.46041 | 1.254038 | 103.5 |

20121128 | 11771 | 8.037461 | 9.97701 | 1.241314 | 102.3 |

20130101 | 10459 | 6.523783 | 9.956516 | 1.526187 | 70.4 |

20130102 | 10835 | 6.878782 | 10.37906 | 1.508851 | 69.9 |

Note that the vehicle number of Table

Figures

Variability in the mean values and maximum values.

Variability in the coefficients of variation and vehicle numbers.

The vehicle loads of bridges comprised vehicle loads of various type; thus there was a multimodal distribution, which could not be described by familiar probability distributions. If we suppose that

Based on (

Note that the forms of the cumulative probability distribution and probability density function may differ between (

Histogram of the vehicle loads.

One normal distribution fitted.

Two normal distributions fitted.

Three normal distributions fitted.

Four normal distributions fitted.

Five normal distributions fitted.

The analysis of the index of fit (

The proportion of two-axle vehicles was 75.4%, which included two-axle small cars, two-axle minivans, and two-axle coaches. The proportion of three-, four-, five-, and six- or more axle vehicles was 7.5%, 5.8%, 4.5%, and 6.8%, respectively. The vehicle types that comprised two-axle vehicles were more frequent than others. The two-axle vehicles were characterized mainly by a multimodal distribution, which was similar to that shown in Figure

Probability distribution parameters for the vehicle loads.

Two-axle | Three-axle | Four-axle | Five-axle | Six- or more axle | |
---|---|---|---|---|---|

Mean value | 1.54 | 14.55 | 19.5 | 26.56 | 31.23 |

Standard deviation | 1.01 | 5.92 | 7.89 | 11.75 | 11.84 |

Truck load models are always used for the design and performance evaluations of bridges [

Proportions of different vehicles in terms of different load intensities.

Number of axles | [5, 10 |
[10–15 |
[15–20 |
[20–25 |
>25 |
---|---|---|---|---|---|

2 | 87.1% | 57.6% | 29.7% | 11.6% | 2.3% |

3 | 10.8% | 22.5% | 28.2% | 25.7% | 6.4% |

4 | 2.1% | 11.7% | 19.3% | 29.6% | 17.1% |

5 | 0 | 5.1% | 10.4% | 15.6% | 25.8% |

6 | 0 | 3.1% | 12.4% | 17.6% | 48.5% |

According to the data collected, the numbers of five- and six-axle trucks were 42,116 and 63,424, respectively, and the probability histogram is shown in Figure

Histogram of five-axle (or more) trucks.

Normal distribution fitting.

Log-normal distribution fitting.

Weibull distribution fitting.

To determine the vehicle loads during the rush hour periods in the morning and the evening, the probability characteristics in different time intervals were analyzed. The traffic jam conditions on Hangzhou Bay Bridge during the morning and the evening rush hours, and the actual lane loads, were also compared based on the mean speed of vehicles. In general, the rush hour period in the morning occurred from 7:00 am to 9:00 am and the evening period lasted from 5:00 pm to 7:00 pm. Given the length of Hangzhou Bay Bridge, the morning rush hour period extended from 6:00 am to 9:00 am. Figure

Proportions of vehicles in the morning and the evening rush hour periods.

Histogram and curve fitting results for vehicle loads in the morning and the evening rush hour periods.

Proportions of vehicles with different axle types.

Different load distributions of two-axle vehicles.

The embedded WIM system was located close to a charging station near south entrance; thus the speed of most of the vehicles was relatively low, that is, about 20 km/h in about 90% of the cases. Therefore, we could not analyze the actual speed and traffic jam condition for vehicles based on the speed measured at the Hangzhou Bay Bridge WIM site.

The analysis requires the solution of the maximum problem when the maximum truck load is needed during the reference design period for a bridge. A previous study [

Based on the Poisson process and the load characteristics, the following hypothesis can be proposed.

The truck loads during the bridge service life are completely random and the time interval of an event is very small relative to the overall bridge life.

The truck loads are also random and statistically time independent during each interval

The probability of each event during a time interval is an incremental process and the possibility of the reoccurrence of the event in a time interval is negligible.

According to the Poisson process, the truck load in each time interval can be considered independent and the intensity distributions are all the same. And in order to simplify the calculation, each time interval is supposed to be equal during the bridge service life and there is no overlap between each interval

According to the above hypothesis of an independent identical distribution, the following equation is obtained:

Suppose that the cumulative probability distribution of the truck load in each time interval is

Its probability density is

In the uniform standard of structural reliability design employed in Chinese highway engineering (GB50068-2001) [

Based on more than 3 months of data and the probability distribution, using (

Figures

Probability density in 1 week.

Probability density in 2 weeks.

During the extrapolation process, a problem that needed to be considered is the accuracy of the probability distribution in the tail fitted to the data collected. The tail reflects the distribution of heavy trucks; thus the error would be amplified when extrapolating using (

According to (

Figure

Maximum value distribution of the truck loads during the reference design period.

In this study, a statistical analysis of the vehicle loads on Hangzhou Bay Bridge was performed based on WIM data collected for more than 3 months. The following conclusions can be made.

The statistical analysis of all the vehicle samples showed that the mean values of the vehicle loads were relatively concentrated and the coefficients of variation were not high. The vehicle load comprised different types of vehicles, thus it was obviously characterized by a multimodal distribution. The fitting results were best using five normal functions, probably because the vehicles that passed over Hangzhou Bay Bridge mainly belonged to five types. And the occurrence frequencies of other types of vehicles were low and the dispersion of the gross weight was high.

The probabilistic characteristics of trucks with five axles (or more) were analyzed independently and the results showed that a normal distribution fitted well to the peak and tail. The mean value was 29.38 tons and the coefficient of variation was 37.2%. Thus, a normal distribution can be used to describe the probability distribution of the truck load.

A statistical analysis of the vehicle loads in the morning and the evening rush hour periods was also performed, where the results showed that the vehicles mainly comprised small cars in these periods, which could be fitted approximately using three normal distribution functions.

Only the truck load samples were used to analyze the maximum value of the truck load, rather than all the vehicle samples. The tail fitting of the histogram agreed well with the probability distribution of the truck load. Because sufficient data were available, the inverse method was used to determine the value of the basic time interval by referring to previous research. The results showed that the mean value of the maximum truck load in 100 years would be up to 65.0 tons with a coefficient of variation of 16.3%.

The results and conclusions presented in the paper are of authors’ and do not necessarily reflect the view of the sponsors.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This study is jointly funded by Basic Institute Scientific Research Fund (Grant no. 2012A02), the National Natural Science Fund of China (NSFC) (Grant no. 51308510), and Open Fund of State Key Laboratory Breeding Base of Mountain Bridge and Tunnel Engineering (Grant nos. CQSLBF-Y14-15 and CQSLBF-Y15-5).