Fatigue reliability evaluation of reinforced concrete bridges under stochastic traffic loading based on truck weight limits

Abstract

To investigate the fatigue reliability of reinforced concrete (RC) bridges under stochastic traffic loading and the effectiveness of limiting the truck load, this study presented an analysis method for evaluating the bridge fatigue reliability under the action of random traffic flows based on truck weight limits. Firstly, a simply RC T-beam bridge was selected as an example to illustrate the presented approach, and the traffic-induced stress spectrum of the tensile rebars in the critical girder of the bridge was obtained. Secondly, based on the collected traffic data from two specific areas, the corresponding random traffic flow was simulated respectively, and then calibrated using the weight limit referring to the current regulation and overload violation rates (VR) to exclude overweight trucks. Finally, based on the S–N curve from Eurocode 3 and Miner’s cumulative damage model, the effects of violation rate, average daily truck traffic (ADTT) and truck weight limit (TWL) on the fatigue reliability index of the bridge were studied. The results show that heavy duty trucks will induce serious cumulative fatigue damage, and degrade the durability of RC bridges, while limiting the truck weight can obviously slow down the reduction of the fatigue reliability index. Both the ADTT and the VR can significantly affect the fatigue reliability of the bridge. More specifically, the bridge fatigue reliability index shows a clear downward trend with the increase in the ADTT and the VR. Furthermore, the performance to satisfy the target fatigue reliability of the bridge was compared between TWL for each type of trucks and unified TWL for all the trucks, and then the unified TWL for all the trucks are determined to ensure the target reliability index of the RC bridge after 100 years’ service life.

Keywords

RC bridge truck weight limit fatigue reliability S–N curve violation rate average daily truck traffic

Introduction

With the intense growth of fierce competition of the global transportation market, the traffic volume and vehicle load on highway bridges have also been steadily increasing in the past few years (Yan et al., 2017a, 2017b). It is acknowledged that the longtime repeated action of the traffic load can induce serious cumulative fatigue damage, and degrade the durability of highway bridges (Cha et al., 2016).

Plenty of studies have indicated that the long-term effects of the traffic loading and overload vehicles have become a main trigger of bridge fatigue failure. Ma et al. (2018) developed the fatigue-loaded vehicle models to estimate the fatigue performance of steel box-girders of XiHouMen bridge based on the weigh-in-motion data. Using weigh-in-motion systems to collect the traffic information, Sacconi et al. (2021) presented a method for investigating fatigue life cycle cost of critical details in steel bridges. Lu et al. (2017) proposed a computational framework for probabilistic modeling of the bridge fatigue damage, and further developed a stochastic fatigue truck-load model for the fatigue reliability investigation of welded steel bridge decks. Yu et al. (2021) predicted the development of fatigue damage under traffic loading based on a new time-based fatigue crack growth model. Deng et al. (2021) investigated the fatigue performance of the novel composite orthotropic steel deck using ultra-high-performance concrete under dynamic vehicle loading using numerical simulations of a local girder segment from the Human bridge. With the development of machine learning approach, a new method on the prediction for the fatigue failure probability of bridges was proposed by the adopting the feedforward neural network and the Monte Carlo method (Yan et al., 2019).

Moreover, vehicular overloading may significantly decrease the bridge’ load-carrying capacity, and even shorten the bridge service life, resulting in the rapid increase in the strengthening cost of bridges. Therefore, it is highly desired to limit the traffic loading considering a good balance of the truck productivity, bridge safety, transportation efficiency, environmental sustainability, etc. Actually, extensive research effort has been devoted to studying vehicle weight limit for highway bridges. Wang et al. (2018) obtained the TWL for a typical simply-supported steel girder bridge under different traffic load cases based on the fatigue reliability theory and Miner’s cumulative damage model. Ghosn (2000) proposed a new method for determining vehicle load limit based on the structural reliability theory and the ultimate state equation of bridge bearing capacity, and a formula for calculating the vehicle load limit fulfilling the target reliability level in the AASHTO standard specification. Cohen et al. (1999) evaluated the impact of raising the truck load limit value the fatigue life of bridges. Yuan et al. (2017) analyzed the impacts of the bridge target reliability index and the regional-load characteristics on the limited truck mass, and made suggestions on the truck load limits for short- and medium-span highway bridges according to the standard of truck-load limitation in China. Gao et al. (2022) calculated the fatigue reliability of a prestressed concrete (PC) continuous beam bridge located in China based on the probabilistic density evolution method. Luo et al. (2021) deduced the fatigue limit state function (LSF) considering traffic growth and corrosion effect, and calculated the fatigue reliability of PC bridges. Meanwhile, the fatigue reliability assessment method of reinforced concrete (RC) bridges based on health monitoring data has been proposed (Bayane et al., 2021; Mankar et al., 2019). In addition, fatigue reliability studies of large-span bridges under combined vehicle and wind dynamic loads have been gradually carried out (Han et al., 2020; Jiang et al., 2020). It can be summarized that the reliability method has been widely used to study the traffic load limit on highway bridges based on the evaluation of bridge fatigue damage.

Considering that limits in the vehicle weight regulations have been changing continually throughout historic development of the highway system through the world (OBrien et al., 2012), and fatigue reliability evaluation of bridges is essential as it can avert fatigue-induced failures and guarantee uninterrupted and acceptable performance of bridges throughout their service life, a new analysis method on investigating the fatigue damage of RC bridges considering the effect of TWL in this study. Based on the S–N curve in Eurocode 3, the LSF under the traffic loading was formulated and the calculation framework of RC bridge’s fatigue reliability index was deduced and built. Besides, the effects of the traffic loading with different characteristics on the bridge were analyzed and compared. Furthermore, the parametric studies were carried out to investigate their effects on the bridge fatigue reliability index, and an analysis method was approached on determining the TWL of RC bridges based on the structural fatigue reliability. Compare with the existing methods on RC bridge’s fatigue damage evaluation, the analysis method approached can consider the realistic traffic models using the collected weigh-in-motion (WIM) data and the effects of important parameters such as VR, average daily truck traffic (ADTT) and TWL. Besides, the method can also evaluate the performance of TWL on ensuring the target fatigue reliability of the RC bridge after 100 years of service, which can be adopted as a supplement of the current TWL strategies considering under various traffic load conditions.

Theoretical basis

S–N curves considering fatigue details of bridge girder

It is acknowledged that the cumulative fatigue damage caused by the traffic loads can be evaluated through stress cycles in fatigue-prone details of bridges. In general, the tensile rebars in bridge girders are the key components of RC bridges, and the stress cycles in the rebars can be converted to fatigue damage by S–N curves. In the present study, the S–N curve of rebars is defined referring to Eurocode 3 European Standard EN 1991-2 (2003), which is expressed as follows:

{\begin{cases} Δ σ_{i}^{3} N_{i} = K_{C} Δ σ_{C}^{3} Δ σ_{i} \geq Δ σ_{D} \\ Δ σ_{i}^{5} N_{i} = K_{D} Δ σ_{C}^{5} Δ σ_{L} < Δ σ_{i} \leq Δ σ_{D} \end{cases}

(1)

Where K_C and K_D are the fatigue strength coefficients of stress amplitude not less than or not more than Δσ_D, respectively. Δσ_i and N_i are the i th stress range and the number of corresponding stress cycles, respectively. Δσ_D is the constant radiation fatigue limit. Δσ_L is the fatigue threshold. Δσ_C is the stress range at 2 million stress cycles when fatigue failure of the structural component occurs. More details of the parameters in the S–N curve referring to Eurocode 3 are presented in Table 1.

Table 1.

Parameters of S–N curve of steel bars in Eurocode 3.

Δσ_C (MPa)	Δσ_D (MPa)	Δσ_L (MPa)	K _C	K _D
80	59	32	1.02 × 10¹²	1.64 × 10¹⁴

Limit state function of fatigue reliability

The P-M rule was adopted to calculate the vehicle-induced cumulative fatigue damage on bridges, which has been widely used in bridge design (Fatemi and Yang, 1998). Assuming that N_i and D_i are the number of stress cycles and the cumulative fatigue damage under the stress level of Δσ_i, the cumulative fatigue damage can be written as follows:

D = \sum D_{i} = \sum \frac{n_{i}}{N_{i}}

(2)

where n_i is the actual number of stress cycle experienced at the stress level of Δσ_i. Fatigue failure will occur if D ≥ 1 occurs. After substituting equation (2) into equation (1), the cumulative fatigue damage of tensile rebars in RC bridge girder can be determined as follows:

D = \sum \frac{n_{i}}{N_{i}} = \sum \frac{N_{m} Δ σ_{m}^{3}}{K_{C}} + \sum \frac{N_{k} Δ σ_{k}^{5}}{K_{D}}

(3)

where Δσ_m is the mth stress range larger than Δσ_D. Δσ_k is the kth stress range less than Δσ_D. N_m and N_k are the number of cycles corresponding to Δσ_m and Δσ_k, respectively.

Since the variant-amplitude stress ranges in rebars embedded in girders, which derive from the traffic loading, will hardly exceed Δσ_D, it be transformed into constant-amplitude stress cycles using the P-M rule (Yan et al., 2017a). The equivalent stress range and the corresponding number of stress cycles are shown in equation (4):

{\begin{cases} Δ σ_{e q}^{5} = \frac{K_{D}}{\sum_{m} N_{m} + \sum_{k} N_{k}} (\sum_{m} \frac{N_{m} Δ σ_{m}^{3}}{K_{C}} + \sum_{k} \frac{N_{k} Δ σ_{k}^{5}}{K_{D}}) \\ N_{e q} = \sum_{m} N_{m} + \sum_{k} N_{k} \end{cases}

(4)

Based on equations (3) and (4), the cumulative fatigue damage of the bridge caused by the N_ad trucks can be formulated as:

D_{a d} = \frac{(\sum_{m} N_{m}^{a d} + \sum_{k} N_{k}^{a d}) Δ σ_{e q, a d}^{5}}{K_{D}} = \frac{N_{e q, a d} Δ σ_{e q, a d}^{5}}{K_{D}}

(5)

Where Δσ_eq,ad and N_eq,ad are the equivalent stress range of bars and the number of the corresponding stress cycles under the action of the N_ad trucks, respectively. And the average cumulative fatigue damage AD_ad caused by each truck can be obtained as:

A D_{a d} = \frac{D_{a d}}{N_{a d}} = \frac{1}{N_{a d}} (\frac{N_{e q, a d} Δ σ_{e q, a d}^{5}}{K_{D}})

(6)

It should be noted that AD_ad is affected obviously by the effect of N_ad and N_s (N_s is the number of Monte Carlo simulations) in the process of random traffic simulation. In addition, considering that K_D follows the lognormal distribution (Yan et al., 2017b), the sensitivity analysis of N_ad and N_s cannot be conducted to obtained the AD_ad using equation (6) directly under various N_ad and N_s due to the randomness of K_D. Thus, the average cumulative fatigue damage parameter (FDP) is defined to erase the influence of K_D on the sensitivity analysis of AD_ad, which can be written as follows:

F D P = A D_{a d} \cdot K_{D} = \frac{N_{e q, a d} Δ σ_{e q, a d}^{5}}{N_{a d}}

(7)

The ratio of FDP to K_D represents the average cumulative fatigue damage AD_ad caused by each of the N_ad trucks., and the subsequent sensitivity analysis of FDP under different N_ad and N_s will be studied in the chapter five.

The LSF in this study is defined as follows:

Z = g_{t} (X) = D_{Δ} - D_{t} (x) = D_{Δ} - 365 \cdot A D T T \cdot F D P \cdot t / K_{D}

(8)

where X is a random variable presenting the truck weights in this study. t is bridge service time. g_t(X) is the fatigue function of bridge structure in the tth year. D_Δ is critical cumulative fatigue damage index. D_t(x) is the cumulative fatigue damage of RC bridges.

Truck weight limit method for RC bridges

Violation rate

Driven by the goal of profit maximization, not all drivers will strictly adhere to the vehicle weight regulations, thus a quantitative of overweight vehicles may cross bridges illegally. It is found that violation rates (VR) exceeding to 5% are common, and large VR will probably result in a significant increase in the traffic loading (Asantey and Bartlett, 2005). In this study, the violation rate was defined as the proportion of the number of trucks with gross vehicle weight exceeding the weight limit that cross the bridge to the total number of overweight trucks in the traffic flow (Asantey and Bartlett, 2005), which can be written as follows:

I_{V R} = \frac{N_{V R_{p}}}{N_{V R_{a}}}

(9)

Where

N_{V R_{p}}

is the number of overweight trucks that violate the weight regulation and pass the bridge.

N_{V R_{a}}

is the total number of overweight trucks in the traffic flow.

Regulations of truck weight limit

Plenty of research effort has been devoted to studying the unified vehicle weight limit for all types of vehicles, while a few researchers attempt to study the effectiveness of limiting gross vehicle weights for each type of vehicles. To validate the necessity of TWL for each type of trucks, simply-supported girders with span of 15, 20, 25 and 30 m were selected as an illustration, and the truck models loading on the girders were plotted in Table 2 (Deng et al., 2020).

Table 2.

The information of truck models.

Truck type	Axle load split ratio and mean axle spacing of each truck
2-axle
3-axle
4-axle
5-axle
6-axle

Based on truck models in Table 2, the weight of each truck was assumed to be 300 kN. Using the bending moment influence line in the mid-span of each girder, the bending moment curve was obtained by moving each truck model across each girder step-by-step, which was showed in Figure 1.

Figure 1.

The bending moment curves in the mid-span of girders. (a) 15 m. (b) 20 m. (c) 25 m. (d) 30 m.

From Figure 1, it can be observed that multiple peaks occur in the curves due to the superposition of multiple axle load effects. Moreover, for a given truck load, the maximum bending moment in the mid-span increases with the span length and decreases with the number of truck axles. It can be demonstrated that TWL for each type of trucks may be more reasonable, as trucks with different axles, even with the same weight, will cause totally different load effects on bridges. Thus, it is necessary to evaluate the effectiveness of limiting the weight for each type of truck. Referring to the Regulations for the administration of highway with over limit transport vehicles in China Communications Press Co., Ltd. (2018), the gross TWL for each type of trucks in this study is listed in Table 3.

Table 3.

Gross truck weight limit referring to regulations in China.

Truck type	Truck weight limit (kg)
2-axle	18,000
3-axle	25,000
4-axle	31,000
5-axle	43,000
6-axle	46,000

Calculation framework of bridge fatigue reliability

In general, the main procedures of analyzing the RC bridge fatigue reliability based on the TWLs are as follows: (1) Obtain the bending moment influence line for the mid-span of the critical girder of the RC bridge. (2) Simulate the random traffic flow under the effect of TWL based on the collected traffic information. (3) Calculate the tensile stress ranges and the number of corresponding stress cycles for the rebars under the traffic loading based on the S–N curve and P-M rule. (4) Evaluate the fatigue reliability of the bridge considering the effect of VR and ADTT. The specific process is shown in Figure 2.

Figure 2.

Bridge fatigue reliability calculation process.

Case study

Bridge details

To illustrate the proposed analysis method for evaluating the RC bridge fatigue reliability, a typical simply-supported RC T-beam bridge with a standard span of 20 m was taken as an example (Bi, 2016). The expected service life of the bridge is 100 years. C40 concrete and HRB335 steel were used for the main girder. The cross section of the RC bridge is shown in Figure 3. The tensile rebars are fatigue critical components at the bottom of the cross section, and the tensile stress of the rebar will have a risk of fatigue damage when the concrete of the girder cracks. In addition, η_ij represents the reaction on beam _i when a unit load acts on beam _j , and the calculated values of η_ij are listed in Table 4, and then the transverse load influence line of each girder can be calculated by η_ij. When arranging vehicle loads, the wheel pressures can be applied as concentrated forces on the transverse load influence line to obtain the transverse load distribution coefficient of each girder, and then the bending moment influence line of the girder can be calculated.

Figure 3.

Size of T-beam bridge and reinforcement of cross section (unit: cm).

Table 4.

The calculated values of η_ij.

i	η_i1	η_i2	η_i3	η_i4	η_i5
1	0.5705	0.3853	0.2	0.01473	−0.1705
2	0.3853	0.2926	0.2	0.1074	0.01473
3	0.2	0.2	0.2	0.2	0.2

Traffic loading

To compare the difference in fatigue damage of the RC bridge caused by truck traffic loading in different situations, the truck load spectra showing characteristic of heavy and light truck loading respectively was obtained from the existing research (Liang et al., 2011; Yang, 2018), which is shown in Tables 5 and 6, respectively. Besides, the distribution properties of the gross truck weight and the proportion of each truck are summarized in Tables 7 and 8. The distribution properties of the headway distance of trucks were listed in Table 9. In addition, the truck flow was only loaded in the slow lane of the bridge for fatigue consideration as suggested (Deng and Yan, 2018), and the impact factor adopted for analyzing the bridge’s fatigue damage was taken as half of the impact factor value calculated for strength design (Zhang, 2014).

Table 5.

Truck models of heavy traffic loading.

No.	Axle-1		Axle-2		Axle-3		Axle-4		Axle-5		Axle-6
No.	AWR	AS(1-2)	AWR	AS(2-3)	AWR	AS(3-4)	AWR	AS(4-5)	AWR	AS(5-6)	AWR
A	0.2	3.3	0.8	—	—	—	—	—	—	—	—
B	0.14	4.0	0.42	1.5	0.44	—	—	—	—	—	—
C	0.07	3.5	0.34	7.0	0.27	1.5	0.32	—	—	—	—
D	0.08	3.0	0.21	1.65	0.22	7.5	0.21	1.5	0.28	—	—
E	0.075	3.0	0.18	1.5	0.19	5.5	0.135	1.5	0.21	1.5	0.21

Note. AWR: axle weight ratio; AS: axle spacing (unit: m).

Table 6.

Truck models of light traffic loading.

No.	Axle-1		Axle-2		Axle-3		Axle-4		Axle-5		Axle-6
No.	AWR	AS(1-2)	AWR	AS(2-3)	AWR	AS(3-4)	AWR	AS(4-5)	AWR	AS(5-6)	AWR
A	0.35	4.0	0.65	—	—	—	—	—	—	—	—
B	0.16	3.5	0.42	1.4	0.42	—	—	—	—	—	—
C	0.12	2.0	0.32	5.6	0.28	1.4	0.28	—	—	—	—
D	0.1	3.0	0.3	6.1	0.2	1.4	0.2	1.4	0.2	—	—
E	0.06	3.0	0.2	1.4	0.2	6.5	0.18	1.4	0.18	1.4	0.18

Note. AWR: axle weight ratio; AS: axle spacing (unit: m).

Table 7.

Weight distribution types and parameters for trucks in heavy traffic loading.

No.	Truck type	Distribution type	Parameter statistics	Truck weight (ton)	Ratio
A	2-axle	Normal distribution	μ = 1.452, σ = 0.358	1.2 ∼ 25	0.25
B	3-axle		μ₁ = 14.342, σ₁ = 4.149 μ₂ = 36.328, σ₂ = 6.749	9 ∼ 85	0.19
C	4-axle	Log-normal distribution	p = 0.480	9 ∼ 79	0.16
D	5-axle		μ₁ = 19.166, σ₁ = 4.935 μ₂ = 125.006, σ₂ = 9.968	10 ∼ 133	0.19
E	6-axle		p = 0.202	10 ∼ 143	0.21

Table 8.

Weight distribution types and parameters for trucks in light traffic loading.

No.	Truck type	Distribution type	Parameter statistics	Truck weight (ton)	Ratio
A	2-axle	Normal distribution	μ = 19.99, σ = 3.72	10 ∼ 31	0.67
B	3-axle		μ₁ = 27.22, σ₁ = 13.97	5 ∼ 65	0.11
B	3-axle		μ₂ = 12.07, σ₂ = 2.07 p = 0.56	5 ∼ 65	0.11
C	4-axle	Log-normal distribution	μ₁ = 37.74, σ₁ = 18.49	8 ∼ 80	0.08
C	4-axle		μ₂ = 14.13, σ₂ = 2.51 p = 0.54	8 ∼ 80	0.08
D	5-axle		μ₁ = 53.78, σ₁ = 21.85	10 ∼ 105	0.07
D	5-axle		μ₂ = 17.90, σ₂ = 2.32 p = 0.58	10 ∼ 105	0.07
E	6-axle		μ₁ = 62.71, σ₁ = 25.53	10 ∼ 115	0.07
E	6-axle		μ₂ = 22.87, σ₂ = 6.39 p = 0.52	10 ∼ 115	0.07

Table 9.

The distribution properties of the headway distance of trucks.

Truck traffic loading	Distribution type	Parameter statistics
Heavy loading	Log-normal distribution	μ = 2.873, σ = 0.6201
Light loading	Log-normal distribution	μ = 4.027, σ = 0.697

Based on the collected traffic information, the random traffic flow showing characteristic of heavy and light truck loading was modeled using Monte-Carlo method respectively, and the corresponding scatterplot of the truck weight distribution within 1 day by assuming that the ADTT is 10,000, which is shown in Figure 4.

Figure 4.

Vehicle weight distribution of random traffic. (a) Heavy loading. (b) Light loading.

As shown in Figure 4, the truck weights with heavy loading are mainly concentrated in the two ranges of 10–50 ton and 100–140 ton. By contrast, the majority of truck weights in random traffic flow of light loading is less than 30 ton, while a few trucks weigh more than 100 ton.

Analysis method of bridge fatigue reliability

Sensitivity analysis of FDP

In this study, the sensitivity analysis of FDP under different N_s and N_ad is carried out, and a series of FDP values corresponding to N_s and N_ad are calculated respectively. Among them, N_s is selected as 100, 1000 and 10,000 respectively, and N_ad is selected as 25, 50, 100, 150, 200, 250 and 300 respectively, and the results are shown in Figure 5.

Figure 5.

FDPs under different N_s and N_ad. (a) Heavy truck traffic loading. (b) Light truck traffic loading.

Figure 5 shows that the variation of the FDPs will become insignificant when N_ad is more than 200. To keep a good balance of the calculation efficiency and the accuracy of the results, N_ad =200 and N_s = 100 were adopted in the calculation of FDPs.

Distribution of FDPs

It is well-known that the lognormal distribution is widely-used in engineering calculations, and Chi-square test was adopted to determine the statistical properties of FDP. The interval number of histograms with specific samples, the corresponding degree of freedom and the threshold value were determined according to the Sturges’ rule (Deng et al., 2021) The results in this study are listed in Table 10, which show that FDP can follow the lognormal distribution well. Furthermore, the statistical properties of FDP are listed in Table 11.

Table 10.

Values of Chi-square test for FDP using lognormal distribution.

Truck traffic loading	I _VR	Interval number of histogram	Degree of freedom	Log-normal	Threshold value
Heavy loading	0	7	4	7.94	13.28
	0.2	8	5	3.93	15.09
	0.4	7	4	3.04	13.28
	0.6	8	5	6.05	15.09
	0.8	9	6	4.85	21.67
	1	9	6	2.28	21.67
Light loading	0	8	5	4.78	15.09
	0.2	6	3	8.25	11.34
	0.4	7	4	8.67	13.28
	0.6	7	4	1.76	13.28
	0.8	7	4	7.11	13.28
	1	8	5	2.91	15.09

Table 11.

Statistical properties of FDPs.

Truck traffic loading	I _VR	Mean (MPa⁵)	Coefficient of variation (COV)
Heavy loading	0	3.68 × 10⁴	0.21
	0.2	1.91 × 10⁶	0.32
	0.4	3.24 × 10⁶	0.18
	0.6	4.72 × 10⁶	0.15
	0.8	5.44 × 10⁶	0.13
	1	7.39 × 10⁶	0.11
Light loading	0	6.10 × 10³	0.25
	0.2	1.20 × 10⁵	0.95
	0.4	1.60 × 10⁵	0.38
	0.6	2.09 × 10⁵	0.42
	0.8	2.79 × 10⁵	0.29
	1	4.23 × 10⁵	0.29

Function of bridge fatigue reliability index

In the present study, the distribution type and the statistical properties of D_Δ and K_D in equation (8) are listed in Table 12 (Yan et al., 2017b).

Table 12.

Statistics of random variables.

Variable	Distribution type	Mean	SD	COV
D _Δ	Log-normal distribution	1	0.3	0.3
K _D	Log-normal distribution	1.64 × 10¹⁴	0.56 × 10¹³	0.0341

It should be noted that all the random variables in equation (8) follow the lognormal distribution, and the probability density function can be written as follows

f_{X} (x) = \frac{1}{x σ_{\ln x} \sqrt{2 π}} \exp (- \frac{{(\ln (x) - μ_{\ln x})}^{2}}{2 σ_{\ln x}^{2}})

(10)

Where X is a random variable. The expected value and the variance of X are written as follows:

{\begin{cases} E (x) = \int_{- \infty}^{+ \infty} x f (x) d x = \int_{- \infty}^{+ \infty} \frac{1}{σ_{\ln x} \sqrt{2 π}} \exp (- \frac{{(\ln (x) - μ_{\ln x})}^{2}}{2 σ_{\ln x}^{2}}) d x \\ D (x) = E (x^{2}) - {[E (x)]}^{2} \end{cases}

(11)

Where the function of E(x²) is shown as:

E (x^{2}) = \int_{- \infty}^{+ \infty} x^{2} f (x) d x = \int_{- \infty}^{+ \infty} \frac{x}{σ_{\ln x} \sqrt{2 π}} \exp (- \frac{{(\ln (x) - μ_{\ln x})}^{2}}{2 σ_{\ln x}^{2}}) d x

(12)

As it is known that X ∼ logn(μ_x, σ_x²), ln(X) ∼ N (

μ_{\ln x}

σ_{\ln x}^{2}

), thus, μ_x, σ_x² and δ_x are defined as follows:

{\begin{cases} μ_{x} = E (x) = e^{μ_{\ln x} + \frac{σ_{\ln x}^{2}}{2}} \\ σ_{x}^{2} = D (x) = e^{2 μ_{\ln x} + σ_{\ln x}^{2}} (e^{σ_{\ln x}^{2}} - 1) \\ δ_{x} = \frac{σ_{x}}{μ_{x}} \end{cases}

(13)

Then, $μ_{\ln x}$ and $σ_{\ln x}^{2}$ can be deduced as follows:

{\begin{cases} μ_{\ln x} = \ln (\frac{μ_{x}}{\sqrt{1 + δ_{x}^{2}}}) \\ σ_{\ln x} = \ln (\sqrt{1 + δ_{x}^{2}}) \end{cases}

(14)

Based on equation (8), the fatigue failure probability of the bridge is given as follows:

p_{f} = P [g_{t} (X) < 0] = P [D_{Δ} - D_{t} (x) < 0] = P [\ln \frac{D_{Δ}}{D_{t} (x)} < \ln 1] = P [\ln \frac{D_{Δ}}{D_{t} (x)} < 0]

(15)

It can be noted that the variables in equation (8) are positive, thus equation (15) can be simplified as follows:

\begin{array}{l} p_{f} = P [\ln \frac{D_{Δ}}{365 \cdot A D T T \cdot F D P \cdot t / K_{D}} < 0] \\ = P [\ln D_{Δ} - \ln (365 \cdot A D T T \cdot t) - \ln (F D P) + \ln (K_{D}) < 0] \end{array}

(16)

Furthermore, g_t (X) can be written as follows:

Z = \ln D_{Δ} - \ln (365 \cdot A D T T \cdot t) - \ln (F D P) + \ln (K_{D})

(17)

As it is known that $\ln D_{Δ}$ ∼ $N (μ_{\ln D_{Δ}}, σ_{\ln D_{Δ}}^{2})$ , $\ln F D P$ ∼ $N (μ_{\ln F D P}, σ_{\ln F D P}^{2})$ , $\ln K_{D}$ ∼ $N (μ_{\ln K_{D}}, σ_{\ln K_{D}}^{2})$ , and ADTT and t values are treated as constants each time, μ_Z and σ_Z can be deduced as follows:

{\begin{cases} μ_{Z} = μ_{\ln D_{Δ}} - \ln (365 \cdot A D T T \cdot t) - μ_{\ln F D P} + μ_{\ln K_{D}} \\ σ_{Z} = \sqrt{σ_{\ln D_{Δ}}^{2} + σ_{\ln F D P}^{2} + σ_{\ln K_{D}}^{2}} \end{cases}

(18)

Moreover, the reliability index can be defined as follows:

β = \frac{μ_{Z}}{σ_{Z}} = \frac{μ_{\ln D_{Δ}} + μ_{\ln K_{D}} - \ln (365 \cdot A D T T \cdot t) - μ_{\ln F D P}}{\sqrt{σ_{\ln D_{Δ}}^{2} + σ_{\ln F D P}^{2} + σ_{\ln K_{D}}^{2}}}

(19)

Using equation (14) to transform $μ_{\ln υ}$ and $σ_{\ln υ}$ to $μ_{υ}$ and $σ_{υ}$ (υ= D_Δ, K_D, FDP) respectively, β can be written as follows:

β = \frac{κ_{D_{Δ}} + κ_{K_{D}} - (κ_{F D P} + \ln (t \cdot 365 \cdot A D T T))}{\sqrt{ξ_{D_{Δ}}^{2} + ξ_{K_{D}}^{2} + ξ_{F D P}^{2}}}

(20)

Where

ξ_{υ}

and

κ_{υ}

are given as follows:

{\begin{cases} ξ_{υ} = \sqrt{\ln (1 + δ_{υ}^{2})} \\ κ_{υ} = \ln (μ_{υ}) - ξ_{υ}^{2} \end{cases}

(21)

Where μ_υ and δ_υ are the mean value and the coefficient variation of υ.

Parameter analysis of bridge fatigue reliability

The random truck traffic flow was modeled based on the traffic statistic information. Using the TWLs in Table 3 to limit the truck loading, curves of the fatigue reliability index of the bridge changing with the bridge service time under different ADTTs and VRs are shown in Figures 6 and 7.

Figure 6.

Variation of bridge fatigue reliability index with change in bridge service time and VR under the heavy truck traffic loading. (a) ADTT = 4000. (b) ADTT = 6000. (c) ADTT = 8000. (d) ADTT = 10,000.

Figure 7.

Variation of bridge fatigue reliability index with change in bridge service time and VR under the light truck traffic loading. (a) ADTT = 4000. (b) ADTT = 6000. (c) ADTT = 8000. (d) ADTT = 10,000.

From Figures 6 and 7, it can be found that the heavy truck loading on the bridge can significantly decrease the reliability index comparing with the light truck loading. Besides, as trucks with weights exceeding the TWLs were all removed in the traffic flow when VR = 0 is adopted, the reliability index will remain a higher level. While the reliability index of the bridge will decrease rapidly with the increase of the VR, especially for the bridge under the heavy truck loading. To be more specific, compared with the light traffic loading, the VR will have more significant effect on the fatigue reliability index under the action of the heavy traffic loading.

In order to further analyze the influence of the AADT and VR on the fatigue reliability index of the bridge considering the heavy traffic loading, Figure 8 shows the reliability index changing with the ADTTs, where I_VR = 0.4 and I_VR = 0.8 were adopted respectively.

Figure 8.

The influence of the ADTT on the fatigue reliability index under the heavy truck traffic loading. (a) I_VR = 0.4. (b) I_VR = 0.8.

For the RC bridge under consideration in this study, the target reliability index for evaluation of bridge fatigue life is taken as 2.0 (Luo et al., 2021). Based on Figure 8, ADTT required for the reliability index to decrease to 2.0 at the end of 100 years of service is about 500, in which I_VR =0.4. while the fatigue life will decrease to 81 years, in which ADTT =500 and I_VR = 0.8. When I_VR =0.8 is adopted, the bridge fatigue life will decrease from 81 years to 40 years with ADTT increasing from 500 to 1000. It can be deduced that VR must be strictly controlled to improve the fatigue life of RC bridges.

To compare the effectiveness of the unified TWL for all the trucks and that of TWLs for corresponding type of trucks, the random traffic flow of the heavy traffic loading was simulated with TWL = 55 ton referring to the current specifications in China (Deng et al., 2017) for all the trucks and I_VR = 0. Using Chi-square test to determine the statistical properties of FDP under the condition considered, and the statistical properties of lognormal distribution for FDP are listed in Table 13. Furthermore, the effect of different TWL strategies on the bridge fatigue reliability index at the end of 100 years’ service life is shown in Figure 9.

Table 13.

Values of Chi-square test and the statistical parameters for FDP.

Results of chi-square test				Statistical parameters of log-normal
Interval number of histogram	Degree of freedom	Log-normal	Threshold value	Mean (MPa⁵)	COV
7	4	7.96	13.28	1.03 × 10⁵	0.20

Figure 9.

Bridge fatigue reliability index using different TWL strategies.

It can be seen from Figure 9 that TWLs in Table 3 exhibits better performance in improving the bridge fatigue reliability index by contrast to the truck limit strategy of TWL = 55 ton. Besides, when the ADTT exceeds 20,000, the reliability index will decrease to the target reliability index. If ADTT and VR can be strictly controlled within a proper range, both TWL strategies can satisfy the expected service time.

Approach on determining the TWL of RC bridges

Since the unified TWL for all the trucks has been widely adopted in practice (Deng et al., 2017, 2018), the effect of the unified TWL on the fatigue reliability of RC bridges was further studied. To be more specific, five different unified TWLs of all trucks, namely, 20, 30, 40, 50 and 55 ton were adopted respectively with the VR of 0.2 in this study. The Chi-square test results in Table 14 shows that the TWL_S follow lognormal distribution and the corresponding statistical properties are listed in Table 15.

Table 14.

Values of Chi-square test of FDPs for different TWLs.

TWL (ton)	Interval number of histogram	Degree of freedom	Log-normal	Threshold value
20	7	4	7.85	13.28
30	8	5	3.68	15.09
40	6	3	1.68	11.34
50	8	5	2.33	15.09
55	8	5	5.41	15.09

Table 15.

Statistical properties of FDPs for different TWLs.

TWL (ton)	Mean (MPa⁵)	Coefficient of variation (COV)
20	1.16 × 10⁶	0.09
30	1.23 × 10⁶	0.11
40	1.59 × 10⁶	0.19
50	1.87 × 10⁶	0.22
55	2.02 × 10⁶	0.21

Using the TWLs to limit the truck loading, curves of the fatigue reliability index of the bridge changing with the ADTTs at the end of 100 years’ service life are shown in Figure 10.

Figure 10.

Variation of β with change in ADTT under the effect of TWL.

As shown in Figure 10, ADTT required for β of the bridge girder to decrease to the target fatigue reliability index of 2.0 is around 1940, where TWL = 20 ton and I_VR = 0.2. In contrast, if TWL = 40 ton, the ADTT required for the β of the bridge girder to decrease to the target reliability index is about 1340. Likewise, if ADTT remains around 1500 vehicles, the TWL may have to be controlled below 40 ton to ensure the bridge fatigue reliability reach the target reliability index at the end of 100 years’ service life.

Moreover, to determine the reasonable TWL for the RC bridge to achieve the target fatigue reliability index at the end of 100 years of service, the variation of fatigue reliability index with change in TWL was obtained, where I_VR, and ADTT were taken as 0.2 and 1500 respectively. The results are shown in Figure 11. As expected, the fatigue reliability index decreases with the increase of the TWL. Specifically, a TWL of 36 ton is needed for the bridge girder to achieve the target fatigue reliability index of 2.0 after 100 years of service.

Figure 11.

Variation of β with change in TWL.

In the present study, though the analysis method has been approached to evaluate the RC bridge fatigue reliability under the effect of TWLs based on a specific RC bridge and the assumed traffic data, it can be applied on different RC bridges and traffic load conditions. The key procedures can be summarized and further developed as follows: (1) Extract the influence line (bending moment, stress or strain) of the key bridge cross sections from theoretical analysis, finite element method or field bridge test, and calculate the stress time history of the bridge’s fatigue failure detail such as tensile rebars based on the traffic data under the given TWLs. (2) Obtain the stress ranges and the number of corresponding stress cycles for the failure detail based on the S–N curve, and calculate the FDP using equation (7) and obtain its statistical properties under the given VR, ADTT or any other influence factors. (3) Calculate the fatigue reliability index considering the effect of influence factors based on the P-M rule and the statistical properties of FDP, and evaluate the fatigue damage comparing with the target reliability index through the 100 years’ service life of the RC bridge under the given TWLs.

Summary and conclusions

Based on the method proposed, effects of the heavy traffic loading and light loading on the bridge were analyzed and compared. Besides, parametric studies were also conducted to investigate the effect of ADTT and VR on the bridge fatigue reliability index. Furthermore, the effectiveness of the unified TWL for all the trucks and that of TWLs for corresponding type of trucks were studied. Based on the results of this study, the following conclusions can be drawn:

1. The FDP was defined to illustrate the fatigue reliability of the RC bridge based on the S–N curve and P-M rule referring to Eurocode 3. Meanwhile, FDPs with specific samples were deduced to follow the lognormal distribution and can be estimated reasonably according to Chi-square test results.

2. Based on the RC bridge in this study, the fatigue reliability index will decrease significantly with the increase of ADTT and VR. Besides, heavy traffic loading will probably lead to more serious fatigue damage as the fatigue damage are proportional to the fifth power of the stress range according to Eurocode 3.

3. Using TWL for each type of trucks may show better performance in slowing down the trend of the development of bridge fatigue damage. And, it was found that both the TWL strategies can ensure the safety of the bridge safety well if VR can be limited very strictly.

4. Based on the target reliability index, the fatigue life of the RC bridge can be calculated under different ADTT and VR, and this method can be used to predict the service time of RC bridges with considered traffic loading. Furthermore, the value of TWL for all the trucks can also be determined to ensure the target reliability index of the RC bridge after 100 years’ service life.

To develop the method on evaluating the vehicle-induced bridge fatigue damage, the nonlinear effect derived from the rebar corrosion, concrete cracking and excessive deformation should be further considered corresponding with realistic bridge-vehicle interaction. Moreover, machine learning approaches can also be adopted to develop the evaluation method more practical and accurate in future studies.

Footnotes

Acknowledgements

The authors acknowledge the support of Southeast University and constructive comments from tutors, in addition, good conditions for scientific research are provided in the peaceful era.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iDs

Tao Wang

Dong-Dong Zhao

Shi-Yu Wu

References

Asantey

SBA

Bartlett

(2005) Impact of posted load limits on highway bridge reliability. Journal of Bridge Engineering 10(3): 321–330.

Bayane

Pai

SGS

Smith

IFC

, et al. (2021) Model-based interpretation of measurements for fatigue evaluation of existing reinforced concrete bridges. Journal of Bridge Engineering 26(8): 04021054.

(2016) Study of Vehicle Weight Limit of Reinforced Concrete Highway Bridges Based on Fatigue Strength. Changsha, China: Hunan University. (in Chinese).

Cha

Liu

Prakash

, et al. (2016) Effect of local damage caused by overweight trucks on the durability of steel bridges. Journal of Performance of Constructed Facilities 30(1): 04014183.

China Communications Press Co., Ltd. (2018) Compilation of Policies, Regulations and Standards for the Governance of Oversize and Overloading Transportation of Highway. Beijing, China: China Communications Press Co., Ltd.

Cohen

Al-Dakkak

, et al. (1999) Bridge fatigue load modeling for truck weight limit changes. In: Avent

Alawady

(eds), Structural Engineering In the 21st Century. New Orleans, LA: 1999 New Orleans Structures Congress, pp. 508–511.

Deng

, et al. (2017) Vehicle weight limit analysis method for reinforced concrete bridges based on fatigue life. China Journal of Highway and Transport 30(4): 72–78. (in Chinese).

Deng

Yan

(2018) Vehicle weight limits and overload permit checking considering the cumulative fatigue damage of bridges. Journal of Bridge Engineering 23(7): 04018045.

Deng

Wang

, et al. (2020) Impact of vehicle axle load limit on reliability and strengthening cost of reinforced concrete bridges. China Journal of Highway and Transport 33: 92–100. (in Chinese).

10.

Deng

Zou

Wang

, et al. (2021) Fatigue performance evaluation for composite OSD using UHPC under dynamic vehicle loading. Engineering Structures 232: 111831.

11.

European Standard EN 1991-2 (2003) Eurocode 1: Actions On Structures-Part 2: Traffic Loads on Bridges. Maastricht, Netherlands: European Union. Incl.

12.

Fatemi

Yang

(1998) Cumulative fatigue damage and life prediction theories: a survey of the state of the art for homogeneous materials. International Journal of Fatigue 20(1): 9–34.

13.

Gao

Nie

(2022) Stochastic harmonic function-based traffic load simulation method for fatigue damage assessment of concrete bridges. Probabilistic Engineering Mechanics 69: 103308.

14.

Ghosn

(2000) Development of truck weight regulations using bridge reliability model. Journal of Bridge Engineering 5(4): 293–303.

15.

Han

Cai

, et al. (2020) Fatigue reliability assessment of long-span steel-truss suspension bridges under the combined action of random traffic and wind loads. Journal of Bridge Engineering 25(3): 04020003.

16.

Jiang

Cai

, et al. (2020) Fatigue analysis of stay cables on the long-span bridges under combined action of traffic and wind. Engineering Structures 207: 110212.

17.

Liang

Dong

Zhao

, et al. (2011) Research on load standard of highway bridges suitable for heavy load traffic. Highway; 2011(3): 30–36. (in Chinese).

18.

Noori

Liu

(2017) Fatigue reliability assessment of welded steel bridge decks under stochastic truck loads via machine learning. Journal of Bridge Engineering 22(1): 04016105.

19.

Luo

Zheng

Zhang

, et al. (2021) Fatigue reliability evaluation of aging prestressed concrete bridge accounting for stochastic traffic loading and resistance degradation. Advances in Structural Engineering 24(13): 3021–3029.

20.

Wang

, et al. (2018) Vehicle models for fatigue loading on steel box-girder bridges based on weigh-in-motion data. Structure and Infrastructure Engineering 14(6): 701–713.

21.

Mankar

Bayane

Sørensen

, et al. (2019) Probabilistic reliability framework for assessment of concrete fatigue of existing RC bridge deck slabs using data from monitoring. Engineering Structures 201: 109788.

22.

Obrien

Daly

O’Connor

, et al. (2012) Increasing truck weight limits: implications for bridges. Procedia - Social and Behavioral Sciences 48: 2071–2080.

23.

Sacconi

Venanzi

Ierimonti

, et al. (2021) Fatigue reliability assessment and life-cycle cost analysis of roadway bridges equipped with weigh-in-motion systems. Structure and Infrastructure Engineering 2(1): 25.

24.

Wang

Deng

, et al. (2018) Truck weight limit for simply supported steel girder bridges based on bridge fatigue reliability. Journal of Aerospace Engineering 31(6): 04018079.

25.

Yan

Luo

, et al. (2017a) Fatigue stress spectra and reliability evaluation of short- to medium-span bridges under stochastic and dynamic traffic loads. Journal of Bridge Engineering 22(12): 04017102.

26.

Yan

Luo

Yuan

, et al. (2017b) Lifetime fatigue reliability evaluation of short to medium span bridges under site-specific stochastic truck loading. Advances in Mechanical Engineering 9(3): 168781401769504.

27.

Yan

Deng

Zhang

, et al. (2019) Probabilistic machine learning approach to bridge fatigue failure analysis due to vehicular overloading. Engineering Structures 193: 91–99.

28.

Yang

(2018) Research on Limited Load of Bridge Vehicles in Typical Cities in Hefei. Hefei, China: Hefei University of Technology. (in Chinese).

29.

Kurian

Zhang

, et al. (2021) Fatigue damage prognosis of steel bridges under traffic loading using a time-based crack growth method. Engineering Structures 237: 112162.

30.

Yuan

Huang

Han

, et al. (2017) Reliability based research on truck-load limit of medium-small-span bridges. Engineering Mechanics 34(8): 161–170. (in Chinese).

31.

Zhang

(2014) Study of Highway Bridges Vehicle Load. Beijing, China: China Communication Press.