Fatigue reliability evaluation of aging prestressed concrete bridge accounting for stochastic traffic loading and resistance degradation

Abstract

Fatigue damage accumulation is a critical factor resulting in the failure of prestressed concrete (PC) bridges. The fatigue damage is usually caused by the coupled effect of cyclic vehicle loading and environmental corrosion. This study investigated probabilistic fatigue damage on aging PC bridges considering both stochastic traffic loading and corrosion. A stochastic traffic model was derived based on long-term monitoring data aiming to simulate fatigue stress spectra of critical rebar. The effect of cracks on the fatigue stress spectra was investigated in order to model the fatigue stress state more realistically. A three-stage traffic growth model was established based on traffic volume histories of three highways in China. A fatigue limit state function considering traffic growth and corrosion effect was deduced for fatigue reliability assessment of PC bridges. Numerical results show that the stress amplitude of rebar considering cracks is 1.53 times greater than the rebar with no-cracks, resulting in a decrease of fatigue life by 68 years. In addition, the three-stage traffic growth models lead to 25 years shorter fatigue life than the one considering a linear traffic growth model. Finally, the corrosion effect results in a fatigue life of 44 years. The numerical results provide a theoretical basis for fatigue life estimation and maintenance of aging PC bridges.

Keywords

crack corrosion fatigue reliability fatigue stress spectrum prestressed concrete bridge stochastic traffic load

Introduction

With the booming development in transportation market, the current highway freight volume has increased significantly. The continuous traffic growth may result in two critical issues on existing bridges, including fatigue damage of steel comments and cracking of concrete girders. The cracks make the rebar in the prestressed concrete (PC) bridge easy to be corroded, which accelerates the bridge fatigue damage. Therefore, fatigue damage of PC bridge structures with cracks deserves further investigation. In addition, it is urgent to evaluate the time-dependent fatigue reliability of PC bridges considering stochastic traffic loading and corrosion effect, where a theoretical basis can be provided for traffic management on highway bridges.

A practical fatigue stress spectrum is the precondition for fatigue life and reliability assessment of existing bridges. Extensive research efforts have been conducted on fatigue stress spectrum modeling using national standard fatigue vehicle load models (American Association of State Highway and Transportation Officials [AASHTO], 2012; Eurocode 1, 2003). Several researchers devoted to structural health monitoring (SHM) systems (Lee and Cho, 2016; Mao et al., 2019) and updated fatigue vehicle models (Chen et al., 2015; Wang et al., 2005). However, most researchers developed deterministic fatigue vehicle load models that cannot include the probabilistic characteristic of random traffic loads. In addition, SHM system is costly for extensive application. The combination of finite element method (FEM) and stochastic traffic load models was recently developed as an effective approach. Stress spectrum models were effectively developed considering various vehicle parameters, such as vehicle type, driving speed, vehicle configurations, and gross vehicle weight (GVW) (Lu et al., 2017, 2019a; Yan et al., 2017a; Zhu and Zhang, 2018). In addition, coupled load effects of stochastic traffic and wind (Han et al., 2020; Zhang et al., 2013) were investigated for fatigue evaluation of long-span bridges. In the aspect of PC bridges, girder cracks are common phenomenon, which impact the fatigue stress of rebar. However, fatigue evaluation of PC bridges considering practical traffic loads and crack effects is relatively insufficient.

In addition to load effects, degradation of bridge resistance due to corrosion is critical for fatigue issues of PC bridges. Garbatov et al. (2014) conducted fatigue strength experiments on corroded rebars and found a direct relationship between the initial corrosion surface roughness and the pit depth. Gehlen et al. (2016) demonstrated that the slope of the S-N curve was steeper in a corrosion fatigue environment for rebar. Lan et al. (2018) proposed a Weibull model of the C-S-N (Corrosion S-N) relationship for steel rebar, which was subsequently utilized to evaluate the influence of corrosion on the fatigue life. In summary, a large number of experimental studies have been conducted on corrosion fatigue issues. However, research effects on fatigue life assessments considering both stochastic traffic loads and corrosion effects are relatively insufficient.

This study investigated probabilistic fatigue damage on PC bridges considering stochastic traffic loading and corrosion. A stochastic traffic model was derived based on long-term monitoring data aiming to simulate fatigue stress spectra of critical rebar. Effect of cracks on the fatigue stress spectra was investigated in order to model the fatigue stress state more realistically. A three-stage traffic growth model was established based on traffic volume histories of three highways in China. A fatigue limit state function considering traffic growth and corrosion effect was deduced for fatigue reliability assessment of PC bridges. The numerical results provide a theoretical basis for fatigue life estimation and maintenance of PC bridges.

Theoretical basis

S-N curve considering fatigue strength degradation

In general, structural fatigue damage is investigated based on stress-life (S-N) curves and linear cumulative damage theory. Eurocode 3 (2005) is widely utilized due to the comprehensive classification of details in welded joints. The S-N curves recommended in Eurocode 3 can be written as follows:

\begin{matrix} Δ σ_{R}^{3} N_{R} = K_{C} & (\begin{matrix} K_{C} = 2 \times 10^{6} Δ σ_{C}^{3} & Δ σ_{R} \geq Δ σ_{D} \end{matrix}) \end{matrix}

(1a)

\begin{matrix} Δ σ_{R}^{5} N_{R} = K_{D} \end{matrix} (\begin{matrix} K_{D} = 5 \times 10^{6} Δ σ_{C}^{5} & Δ σ_{L} < Δ σ_{R} \leq Δ σ_{D} \end{matrix})

(1b)

where Δσ_R represents the stress amplitude; N_R represents the corresponding number of fatigue cycle; Δσ_C is the reference fatigue strength at N_R = 2 million cycles; Δσ_D is the fatigue limit for a constant amplitude stress range at N_R = 5 million cycles; Δσ_L represents the cut-off limit corresponding to a fatigue life of 10 million cycles. Eurocode 3 assumes that the fatigue damage is not considered for the stress cycle less than the cut-off limit. For instance, the welded joints is assumed no damage for the stress range Δσ_R≤Δσ_L.

Due to the truck overloading and environment corrosion, the phenomenon of concrete cracking and corrosion are quite common for existing PC bridges. Fatigue performance analysis and durability evaluation of corroded rebar has become a hot topic. Hahin (1994) developed a general corrosion fatigue model to investigate the effect of general corrosion and pitting corrosion. Zhang et al. (2012). proposed new S-N curves for artificially and naturally corroded steel bars. Lan et al. (2018) developed a corrosion S-N (C-S-N) relationship based on Weibull model and corrosion grades. This study utilized Hahin’s model, written as

N_{R} = K' Δ σ_{R}^{- m}

(2)

where K′ = K/K_f; K_f is the fatigue reduction factor, which is related to pitting (K_f = 1.2 + 11.54Rn); R represents the corrosion rate in inches/year (R = 0.0254 m/year); n is time in years; m is the fatigue exponent (m = 3.26).

As elaborated above, the S-N curves specified in Eurocode 3 and Hahin’s model provide a theoretical basis for the fatigue stress spectrum simulation of PC bridges under stochastic traffic loads.

Fatigue accumulation damage

The S-N curves are usually regressed based on fatigue test under constant loading. However, the fatigue stress of a bridge has the characteristics including variable-amplitude, low-stress, and high-cycle. Therefore, Palmgren-Miner rule (Miner, 1945) is used to model an equivalent constant-amplitude stress cycles based on the variable-amplitude stress cycles. The fatigue damage accumulation is evaluated by linear superposition of the equivalent fatigue damage. Based on the above assumption and the S-N curves in Eurocode 3, the cumulative fatigue damage of bridges under variable amplitude loading can be written as

D = \frac{N_{d} Δ σ_{re}^{m}}{K_{D}} = \sum_{S_{i} \geq σ_{D}} \frac{n_{i} S_{i}^{3}}{K_{C}} + \sum_{S_{i} < σ_{D}} \frac{n_{j} S_{j}^{3}}{K_{D}}

(3)

where n_i and n_j are extracted from the rain flow counting technique and represents the number of cycles of fatigue stress S_i and S_j, which are greater and less than Δσ_D, respectively. According to the equivalent principle of fatigue damage, the variable-amplitude stress is equivalent to the constant-amplitude stress △σ_re.

In summary, the fatigue damage accumulation formula for a bridge rebar under variable-amplitude loading has been deduced. The formulation can provide a foundation for the subsequent fatigue stress spectrum analysis of the bridge under stochastic traffic loading.

Simulation of fatigue stress spectrum

Stochastic traffic flow model based on WIM system

Compared with the fatigue load models recommended by BS 5400 (1980), AASHTO (2012), Eurocode 1 (2003) and JTG D64-2015 (2015), stochastic traffic flow models based on big data analytics and statistics have been widely applied for a real traffic condition (Cui et al., 2018; Lu et al., 2017, 2019b). Therefore, a statistical analysis of vehicle type, vehicle lane, vehicle weight, and vehicle speed collected from the WIM system on the Yangtze River Bridge was conducted here for the stochastic traffic flow model, which are shown in Figure 1. For detailed statistics, pleased refer to author’s previous papers (Yan et al., 2017a, 2017b).

Figure 1.

Statistical analysis of traffic flow parameters: (a) occupation ratios for vehicle type, (b) occupation ratios for six-axle trucks in traffic lane, (c) probability distribution of the Gross vehicle weight (GVW), and (d) probability distribution of the vehicle speed.

Based on the statistical analysis of traffic flow parameters in Figure 1, the Monte Carlo Sampling (MCS) method was used to establish the stochastic traffic flow model. The simulated stochastic traffic flow during 1 h is shown in Figure 2.

Figure 2.

Stochastic traffic flow during 1 h.

Based on the probability distribution characteristics of the related vehicle parameters, stochastic traffic flow load models were simulated. Subsequent fatigue stress spectrum is simulated on this basis. The influence of changes in traffic flow on the fatigue stress of bridges is studied based on parametric analysis.

Simulation of the fatigue stress spectrum

In this study, a 25 m-span T-shaped bridge shown in Figure 3 was used here for the fatigue assessment. Related parameters of the bridge can be refer to Yan et al. (2017a) . The formula for cumulative fatigue damage repesented in equation (3) indicates that fatigue damage decreases exponentially with increasing stress amplitude. Therefore, it is sensitive to accurately calculate the stress amplitude of rebars. The response surface function (Yan et al., 2017a) is utilized to simulate the fatigue stress spectrum considering vehicle parameters and road roughness condition (RRC). For an average RRC and six-axle trucks, the response surface of the fatigue stress spectrum is shown in Figure 4.

Figure 3.

Cross-sectional dimensions of bridge midspan (unit: meters).

Figure 4.

Response surface of fatigue stress spectrum for average RRC under fatigue loading effect six-axle trucks.

A linear interpolation response surface was fitted for each vehicle type and RRCs. Subsequently, the stochastic traffic samples were assumed running on the response surface to compute the stress time-history. The equivalent stress cycles were computed by the rain-flow counting approach and the equivalent S-N curves. The fatigue stress spectrum considering an average RRC is shown in Figure 5. The fatigue stress spectrum simulated by the GMM provides relatively accurate data for subsequent parameter analysis of the fatigue assessment.

Figure 5.

Fatigue stress spectrum considering an average RRC.

In order to consider the concrete cracking effect, the previous stress spectrum should be updated. Previous experimental results (Feng et al., 2006; Han et al., 2014) showed that cracking induced stress changes in rebars and prestress tendons are not the same. The stress amplitudes of rebars with good bonding properties increase faster than those of prestress tendons. The stress amplitude ratios for rebars and prestress tendons follow a three-phase growth model, where the second phase is critical for the fatigue life. As suggested by Han et al. (2014), a mean stress amplitude ratio of 0.65 in the second phase is used in the present study to simulate the fatigue stress spectrum of cracked bridge girders, as shown in Figure 6.

Figure 6.

Fatigue stress spectrum considering cracking.

It is observed from Figure 6 that the GMM fits the stress histogram very well. Compared the Figure 5 with Figure 6, it is observed that the high stress cycles moves to the right side and the proportion of high-stress amplitude increased. This phenomenon can be explained by that the stress amplitude in the rebar is 1.53 times higher than that in prestressed tendon, mainly due to different bonding properties. Since fatigue damage increases exponentially with increasing stress amplitude, increased fatigue stress in rebar due to cracking cannot be ignored. The simulated stress spectrum in Figure 6 was used in the subsequent fatigue life assessment of the bridge with cracking.

Fatigue reliability assessment considering multiple influencing factors

Limite state functions

According to the fatigue damage accumulation theory, fatigue failure is defined as a fatigue damage factor. The random variable in the fatigue limit state function involves the stochastic traffic variables, S-N curves, and corrosion effects. Therefore, the fatigue limit state function for the rebar in the bridge girder under stochastic traffic load is written as (Yan et al., 2017a)

g_{n} (X) = D_{Δ} - D_{n} (X) = D_{Δ} - \frac{365 \cdot Δ σ_{re}^{m} \cdot N_{d} \cdot e}{K} \sum_{n = 1}^{T} n

(4)

where D_△ represents the critical fatigue damage, D_n(X) is the cumulative fatigue damage in the n-th year, △σ_re is the equivalent stress, m is the fatigue exponent, N_d represents the number of the daily cycles, and e is the distribution coefficient of wheel tracks in the lateral direction.

It is worth noting that m and K are depending on the S-N curves. For the S-N curve recommended by Eurocode 3, m is defined as 5, and K is defined as K_D, becased the stress spectrum is mostly less than Δσ_D accounting for the majority. Based on the S-N proposed by Hahin (1994), m = 3.26, and K = K_D/1.2 + 11.54 Rn. The fatigue damage function can be written as

g_{n} (X) = D_{Δ} - D_{n} (X) = D_{Δ} - \frac{365 \cdot Δ σ_{re}^{5} \cdot N_{d} \cdot e}{K_{D}} \sum_{n = 1}^{T} n

(5a)

\begin{matrix} g_{n} (X) = D_{Δ} - D_{n} (X) = D_{Δ} \\ - \frac{365 \cdot Δ σ_{re}^{3.26} \cdot N_{d} \cdot e (1.2 + 11.54 Rn)}{K_{D}} \sum_{n = 1}^{T} n \end{matrix}

(5b)

The fatigue damage function presented in equation (5) is mainly influenced by four random variables: critical fatigue damage D_△, equivalent stress △σ_re, distribution coefficient of the wheel tracks in the lateral direction e, and number of the daily cycles N_d. Among them, D_△ is assumed to obey a log-normal distribution with a mean value of 1 and standard deviation of 0.3 (Wirsching, 1984). The distribution coefficient of the wheel tracks in the lateral direction e obeys a normal distribution with a mean value of 0.78 and standard deviation of 0.078 (Yan et al., 2017a). The fatigue strength coefficient K_D follows a lognormal distribution with a mean value and standard deviation of 1.64 × 10¹⁴ and 0.56 × 10¹³, respectively (Yan et al., 2017a). The probability distributions of △σ_re and N_d refer to the simulated fatigue stress spectrum and statistical analysis above.

In practical engineering problems, the target reliability index β_target, which is the minimum design safety level, is widely adopted to evaluate the structure safety. The target reliability index of the fatigue limit state of railway bridges in China is specified between 2.3 and 3.5 (TB 10091-2017, 2017). Helmerich et al. (2007) calibrated the standards of the European institute of steel construction, and suggested that the target reliability index for the fatigue life of steel structures should be in the range of 2.0 and 3.5. The AASHTO (2012) LRFD code stipulates a target reliability index of bridge structures of 3.5. In ACI 318-2002 (2002), the target reliability index is in the range of 2.5–4.0, depending on the type of structure. In summary, there are significant differences in the target reliability index in various specifications, methods, and engineering fields. In this study, the target reliability index of the PC beam bridge is assumed as 2 and the corresponding failure probability is 2.3%. The developed fatigue limit state function and the probability distribution of the related random variables will be utilized for fatigue reliability evolution of PC bridges.

Effect of traffic growth on fatigue reliability

In the previous study (Yan et al., 2017a), the effect of traffic growth on the fatigue reliability of PC bridges was investigated by assuming that traffic increases linearly at a rate of 1%–3% per year. In practice, traffic growth ratio is not constant in every year. The traffic volume of a highway tends to become saturated when the local economy develop to a stable stage. A three-stage theory of traffic growth in volume developed by Kuang (2010) was utilized in the present study for fatigue life assessment of PC bridges during service period. The three-stage theory divides the growth process of traffic volume into the formation stage in a straight line, the rapid growth stage in a curve and the stable stage in a horizontal line. The second stage is the dominant stage involving fast growth. In general, the second stage can be described as a growth curve.

y = e^{(b_{0} + b_{1} t)}

(5)

where y represents the traffic volume; t is the time in years; b₀ and b₁ are model parameters related to the initial traffic volume, upper limit of traffic flow, and location of the highway.

Kuang (2010) selected three representative highways in the eastern and western regions of China and concluded that the regional characteristic parameters of highways are b₁ = 0.147 and 0.117 for eastern and western regions of China, respectively. The highway traffic volume in eastern China generally reaches the maximum in about 15 years, and the duration is about 10 years. The upper limit of two-way four-lane traffic flow was considered as 5.5 × 10⁴ in accordance with the literatures.

Based on the research foundation of the above paper, this study assumes b₁ = 0.147 for the eastern region, and assumes the maximum traffic volume 5.5 × 10⁴ in the 15th year. Based on the growth curve equation of the highway, the traffic growth curve from the 5th to the 15th year was obtained. Finally, the first stage is approximately described as linear growth. With the known WIM monitoring data, that is, the initial daily traffic volume is 6037, the traffic volume growth curve for the period from 0 to 5 years can be obtained. Therefore, the traffic growth curve is shown in Figure 7.

Figure 7.

Three-stage growth curve of traffic volume.

It can be observed from Figure 7 that the traffic volume increases linearly from 6037 to 1.27 × 10⁴ in the first 5 years (initial stage). The traffic volume increases from 1.27 × 10⁴ to an upper limit of 5.5 × 10⁴ in the subsequent 10 years (rapid growth stage). The traffic volume keeps constant in the subsequent stabilization stage. The practical traffic growth will be utilized for fatigue life assessment of PC bridges.

Applying the three-stage growth curve of the traffic shown in Figure 7 and fatigue damage function described by equation (5a), the lifetime reliability of the bridge is plotted in Figure 8.

Figure 8.

Lifetime fatigue reliability accounting for traffic growth.

It is observed from Figure 8 that traffic growth has a greater impact on the fatigue life of PC bridges. More specifically, the fatigue reliability index of the PC bridge decreases from 4.7 to 2 in 80 years of service without considering traffic growth. The fatigue life of the PC bridge decreases to 44 years with consideration of a linear growth rate of 3%. However, the fatigue life sharply decreases to 19 years with consideration of the three-stage growth model. The two different traffic growth models resulted in different fatigue lives of the PC bridges, and it is demonstrated that the traffic growth model is an important factor for fatigue life prediction of existing bridges. In general, the three-stage traffic growth model is more realistic to simulate the predict the current traffic growth trends in China. Therefore, this study provides a practical traffic model to predict fatigue life for existing bridges.

Effect of cracks on the fatigue life of PC beam bridges

Duo to the phenomenon of bridge aging and truck overloading, cracks are widely observed in PC bridges. Based on the simulated of stress spectra, it is observed that the fatigue stress in rebar increases by 1.53 times with considering of cracks. Thus, it is essential to conduct fatigue life assessments of PC bridge with cracks. Based on the fitted stress spectrum and the fatigue damage function in equation (5a), the lifetime fatigue reliability of the PC bridge with cracks was evaluated via Monte Carlo simulation. Figure 9 plots the reliability index impacted by cracks.

Figure 9.

Lifetime fatigue reliability accounting of PC bridge with cracking.

It is observed from Figure 9 that cracks lead to a distinct decrease in the lifetime reliability index. More specifically, the lifetime reliability index of the bridge without cracks decreased from 4.7 to 1.7, and it reduces to the target lifetime reliability index in the 78th year. However, the lifetime reliability index for the bridge with cracks decreased from 3.4 to 0.7 and decreases to the target reliability index in the 10th year. The decrease rate is greater in the early stages of cracking. The numerical results show that cracks in PC bridges greatly shorten the fatigue life by nearly 68 years. Therefore, the effect of cracks on the stress amplitude of rebar should be considered in the fatigue assessment of PC bridges. The numerical result also provides a theoretical basis for maintenance of PC bridges with cracks.

Effect of corrosion on fatigue life of PC bridges

Previous investigation efforts (Zhang and Yuan, 2014) showed that the fatigue life of bridges under the combined effects of vehicle loads and corrosion is much greater than individual effects. Therefore, this study combined the probability distribution of parameters in equation (5b) to consider the corrosion effect. The lifetime reliability of the PC bridge considering corrosion is shown in Figure 10.

Figure 10.

Lifetime reliability accounting for corrosion.

It is observed that corrosion effect leads to a sharp decrease of the fatigue life greatly, based on the fatigue damage function of equation (5b). Specifically, when corrosion is taken into account, the fatigue reliability index of the PC bridge reaches the target reliability index in 44th year. Regardless of corrosion, the PC bridge can operate normally within 100 years of service. In conclusion, corrosion effect is a significant factor for fatigue life evaluation of aging PC bridges.

Conclusions

This study investigated fatigue life and fatigue reliability of aging PC bridges considering the effects of vehicle loading and resistance degradation. The simulated fatigue stress spectrum and fatigue life assessments are more practical due to the stochastic traffic load model the three-stage traffic growth model based on highway traffic monitoring data in China. In addition, fatigue stress in rebar and prestress tendon were compared with considering of concrete cracking. The conclusions are summarized as follows:

Due to differences in bonding properties, the concrete crack induce fatigue stress redistribution in rebar and prestress tendon are not the same. The fatigue stress in rebars is 1.53 times greater than that of prestress tendons. The fatigue stress induced by crack leads to a decrease in fatigue life by 68 years.

The adopted three-stage traffic growth model is more practical than the linear growth model, and thus can provide a reliable fatigue life for existing bridges. The fatigue life curve of the bridge based on the three-stage growth model decreased faster compared to that of linear traffic growth model. The three-stage growth model leads to the fatigue life decrease by 25 years.

An updated S-N curve considering the corrosion effect was utilized to evaluate the fatigue life of the PC bridge. The fatigue life is only 44 years for a PC bridge considering corrosion effect.

Further studies can be conducted on the following aspects. The three-stage traffic growth model should be verified with long-term traffic monitoring data. Numerical simulation methods of fatigue stress considering crack effect should be verified with experimental data. A fatigue-corrosion test should be concluded to investigate the S-N curve of corroded rebar in PC bridges.

Footnotes

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) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This study was supported by the National Natural Science Foundation of China (Grant No. 51608189); the Natural Science of Foundation of Hunan Province of China (Grant No. 2019JJ50130; Grant No. 2020JJ5140; Grant No. 2020JJ5143), the Key Laboratory of Bridge Engineering Safety Control by the Department of Education (Changsha University of Science & Technology) (Grant No. 19KB02); the Industry Key Laboratory of Traffic Infrastructure Security Risk Management (Changsha University of Science & Technology) (Grant No. 16KE01), Scientific Research Project of Hunan Provincial Department of Education (Grant No. 20C0639). The opinions, findings, and conclusions expressed in this study are those of the authors and do not necessarily represent the views of the sponsors.

ORCID iD

Yuan Luo

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