# Fatigue Life Evaluation of Parallel Steel-Wire Cables under the Combined Actions of Corrosion and Traffic Load

Authors: Sheng Yu; Cheng Yan; Chengyin Liu; Jinping Ou
Journal: Structural Control and Health Monitoring
 (2023)

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## Abstract

Fatigue and corrosion are the main reasons for the failure of stay cables. Traffic load also plays a significant role in cable fatigue. To study the fatigue life of stay cables under the combined action of traffic load and corrosion, this study originally deduced the fatigue probability model for steel wires, considering the stress range and average stress based on the nonlinear fatigue damage model. Then, the weight loss rate was introduced into the fatigue probability model for steel wires in order to obtain the fatigue life probability model for corroded steel wires. Subsequently, a Monte Carlo simulation (MCS) was conducted to predict the fatigue life of stay cables under different guaranteed probabilities. Finally, using a cable-stayed bridge on the Yangtze River in China as an example, the fatigue life of the stay cables in the bridge under the combined effects of different corrosive environments and measured traffic loads was evaluated. Results indicate that both the stress range and mean stress affect the fatigue life of the steel wire. The fatigue life of stay cables is determined by a few steel wires with shorter lives, and the stress redistribution accelerates the failure of the inner wire of the stay cables. The fatigue life of the stay cables on background engineering is much less than the design life of the bridge under the combined actions of a middle-level or high-level corrosive environment and traffic load.


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## Body

## 1. Introduction

As an important component of the cable-stayed bridge connecting the main tower and main girder, the cable plays the role of transmitting the dead load of the main girder and external dynamic load to the main tower. Because the stay cable is always under high stress and subjected to alternating loads, such as traffic loads, it is sensitive to fatigue and corrosion. In recent years, several bridge collapse accidents have been caused by the failure of stay cables [1]. In the statistics of nearly 40 cable-stayed bridges, the longest service life of the cables was 23 years, the shortest was only seven years, and the average service life was only 15 years. The actual service life of the cables was much lower than the design life of bridge [2].A stay cable is often composed of multiple parallel high-strength steel wires or steel strands, which can be regarded as a parallel system composed of multiple steel wires or strands. Some scholars have successively proposed fatigue life probability models of different steel strands or high-strength steel wires based on fatigue experiments of a single steel wire. Castillo et al. [3–6] proposed a multiparameter Weibull distribution model to describe the fatigue life of high-strength steel wires and strands under different stress ranges and verified the effect of the fatigue test sample length on the model parameter estimation. Ma et al. [7] established the S-N curve of an 1860-grade steel strand using the fatigue test data of this grade. Based on a three-parameter power function stress-life model, the conclusion was that the fatigue life obeys a two-parameter Weibull distribution under a specified stress range. Lan et al. [8] derived a multiparameter Weibull model characterizing the fatigue life of a single strand and estimated the parameters of the model based on the experimental fatigue data of 1860-grade steel strands. Lan et al. [9] also introduced a multiparameter Weibull model for the fatigue life of a single high-strength wire. Based on the parallel system theory, they established an S-N curve model of the stay cable with a wire breakage rate of 10%.When the protective system of the stay cable is damaged, the steel wire in the cable is directly exposed to the corrosive environment, which aggravates the corrosion of the high-strength steel wire in the stay cable [10–12]. The fatigue test data of corroded steel strands under different stress conditions in reference [13] showed that the stress state (stress range and mean stress) had a significant effect on the fatigue properties of the steel strands. Li et al. [14] established a time-dependent depth model of uniform and pitting corrosion based on accelerated corrosion experiments. The mechanical properties of corroded steel wires, including yield stress, ultimate stress, ultimate strain, and modulus of elasticity, are experimentally characterized, and the statistical models are established through regression analysis, and the deterioration models of high-strength steel wires were extended to assess the time-variant sectional area loss and remaining capacity of cables. Yao et al. [15] found that the degree of corrosion and corrosion rate of the steel strands or high-strength steel wires are related to the stress conditions under the same corrosion conditions. The corrosion rate under the alternating load was the highest, the corrosion rate under a constant load was the second highest, and the corrosion rate under no stress was the lowest. Yu et al. [16] obtained the fatigue life of steel strands with different degrees of corrosion through fatigue tests. The test data indicated that the fatigue life of steel strands decayed exponentially with an increase in the corrosion rate. Li et al. [17] evaluated the variation in the uniform and pitting corrosion depths of high-strength steel wires based on the accelerated corrosion experiments and established the extreme value distribution of the pit depth of corroding steel wires. Lan et al. [18] introduced the degree of corrosion as a covariate into the three-parameter Weibull model to characterize the fatigue life of steel strands. They constructed a multiparameter Weibull model that can quantitatively evaluate the effect of the degree of corrosion on the fatigue life of steel strands. The reference [19] established a multiparameter Weibull distribution model of the fatigue life of corroded high-strength parallel steel wires based on accelerated salt spray corrosion experiments. The effects of corrosion on the fatigue life of stay cables and fatigue life distribution of stay cables under different degrees of corrosion were further investigated using Monte Carlo simulations. Wang et al. [20] studied the fatigue damage evolution process of notched high-strength steel wires and corroded high-strength steel wires through experiments and numerical simulations. Under the combined action of an external load and corrosive environment, the corrosion of the steel wire in the stay cable intensifies, and the stay cable suffers from severe fatigue damage. The aforementioned fatigue life models of the corrosive steel wire and stay cable only consider the effect of the stress range and ignore the impact of the mean stress on the fatigue life of the steel wire and stay cable.With the development of current structural health monitoring technology, real-time monitoring of cable forces and external loads of cable-stayed bridges has become possible [21–24]. The fatigue assessment and life prediction of cables based on monitoring information have become one of the hot spots in the current in-service bridge condition assessment and life prediction [23, 25–28]. Currently, the fatigue assessment and life prediction methods of the stay cables can be generally classified into two categories: data-driven methods and finite element model-based methods. Data-driven methods use the data processing methods such as data statistics and data mining to directly assess the condition of the diagonal cable based on the monitoring responses [23, 28–31]. Unlike the data-driven method, the finite element model-based approach uses finite element analysis to indirectly obtain the tension force of cables under different loading conditions, and then perform condition assessment and life prediction of the stay cable using suitable fatigue life models [28, 32–34]. Currently, most of the fatigue assessment methods for the stay cables are proposed based on the linear Palmgren–Miner law. However, these proposed methods are hard to consider the effects of the sequence of traffic load action and the coupling effects of fatigue load and corrosion and other factors.In view of the research difficulty, this paper introduces an innovative method to characterize the fatigue life distribution of corroded high-strength wires in the stay cable, by considering the coupling effect of the external fatigue load and corrosion. The core idea of the method proposes a multiparameter Weibull distribution fatigue life model based the nonlinear Lemaitre fatigue damage evolution theory. Based on this model, the fatigue life of parallel steel wire cables under the combined actions of corrosion and traffic load of in-service stay cable are evaluated. Compared with the traditional evaluation method based on the linear Palmgren–Miner law, the proposed method in this paper can consider the effects of the sequence of traffic load action and the coupling effects of fatigue load and corrosion and other factors on the fatigue condition. The remainder of this paper is organized as follows: First, in Section2, the fatigue life model for steel-wire is proposed. Based on the nonlinear fatigue damage model, the fatigue probability model of the steel wire considering the stress range and mean stress was deduced, and then the parameters of the fatigue probability model were estimated based on the fatigue test data of the steel wire. Subsequently, in Section 3, the weight loss rate as a corrosion evaluation index is introduced into the fatigue life probability model for steel wires, and the fatigue life probability model for corroded steel wires is obtained. The parameters of the fatigue probability model for the corroded steel wires were estimated based on the fatigue test data. In Section 4, the corroded stay cable fatigue failure is simulated using a Monte Carlo simulation, and the fatigue life model for corroded cables under different guaranteed probabilities is presented. Section 5 considers a cable-stayed bridge on the Yangtze River in China as the background engineering and calculates the fatigue life of stay cables in three different corrosive environments. Finally, the conclusions drawn from this study are provided in Section 6.


## 2. Fatigue Life Model for Steel Wire

### 2.1. Fatigue Life Probabilistic Model for Steel Wire

The three-parameter power function model [8] mainly analyzes the stress range and fatigue life through the linear regression statistical method and obtains the empirical formula that is used to characterize the relationship between the fatigue life and stress range of the steel wire under different failure probability conditions. This model only considers the influence of the stress range on the fatigue life of the steel wire and ignores its actual stress state. The fatigue test results of the steel wire show that both the stress range and the mean stress affect its fatigue life [32, 35, 36]. Therefore, it is critical to establish a steel wire fatigue model that considers the stress range and pretension.The nonlinear Lemaitre fatigue damage evolution model under a one-dimensional stress state can be expressed by the following equation (1) [37, 38]:(1)D˙=σ2∣σ−σ¯∣βB¯1−Dα×σ˙,σ˙>0,0,σ˙≤0,where • is the McCauley bracket operator; σ and σ¯ represent the stress and mean stress, respectively; and α, β, and B¯ are the material parameters. Equation (1) is integrated over one stress cycle to obtain equation (2):(2)∂D∂N=∆σ+2×σm×∆σβ+3/2B1−Dα×β+3,where ∆σ and σm represent the stress range and mean stress, respectively; B is the material parameter, and B=2β+2×B¯. Equation (2) is integrated over one stress cycle to obtain equation (3):(3)Di+1=1−1−Di1+αi−1+αi×∆σ+2×σm×∆σβ+3/2B×β+31/1+αi,where Di represents fatigue damage after the ith stress cycle. From this equation, it can be seen that the damage induced by the current stress cycle is related to the stress cycle experienced and the previous damage state. The present incremental damage is not simply linearly superimposed on the previous damage but exhibits a significant nonlinear, which is obviously different from the linear fatigue damage model based on the Palmgren–Miner law. Under the specified stress level, the fatigue life of the steel-wire can be expressed as(4)N=B×β+31+α×∆σ+2×σm×∆σ−β+3/2,where N is the fatigue life; α is a function of the stress range ∆σ;α∆σ=a×∆σ+b. Taking the logarithm on both sides of the following equation (5):(5)logN=logB×β+3+log∆σ+2×σm×∆σ−β+3/21+α∆σ.Under the specified stress level, the fatigue life cumulative distribution function (CDF) of the steel wire can be expressed as [3](6)FN;∆σ,σm=1−exp−Nδ∆σ,σmβ0,where β0 and δ∆σ,σm are the shape and characteristic parameters of the Weibull distribution, respectively. We took the logarithm twice on both sides of the following equation (6):(7)logN=1β0×log1−ln1−F+logδ∆σ,σm.Comparing equation (7) with (5), we can assume that(8)δ∆σ,σm=K1+α∆σ×∆σ+2×σm×∆σ−β+3/2,where K is a constant. Substituting equation (8) into equation (6) gives(9)FN;∆σ,σm=1−exp−1+α∆σK×N×∆σ2+2×σm×∆σβ+3/2β0,where K, β0, β, and α are unknown parameters that can be estimated using the test data of all the specimens. This formula is a Weibull distribution model that characterizes the fatigue life of steel wires under different stress levels. The fatigue life of the steel wire follows a two-parameter Weibull distribution of the shape parameter β0 and characteristic parameter K/1+α∆σ×∆σ+2×σm×∆σ−β+3/2. Once all the unknown parameters have been estimated by the maximum likelihood estimation (MLE) or by the expectation-maximization (EM) [39] algorithm based on the fatigue test data, a variable denoted as V can be defined as(10)V=1+α∆σ×N×∆σ2+2×σm×∆σβ+3/2.Equation (9) can be expressed as(11)FV=1−exp−VKβ0.When the model parameters are known, the fit between the model and fatigue test data can be checked according to the Weibull probability diagram of the intermediate variableV.


### 2.2. Parameter’s Estimation of Fatigue Life Probabilistic Model for Steel Wire

The 1670-grade steel wire is widely used in cable-stayed bridges in China. Forty-four groups of samples with a diameter of 7 mm and a length of 250 mm were employed to study the fatigue properties of the 1670-grade steel wire [8]. The fatigue tests were performed using a servohydraulic universal testing machine (MTS Landmark). The stress ratio of the fatigue test was 0.5, the loading frequency was 10 Hz, and the stress range was set to four groups of 335, 418, 520, and 670 MPa. All test results are listed in Table 1. The unknown parameters in equation (9) can be estimated using the expectation-maximization (EM) algorithm based on the fatigue test data [39], and the solution details are shown in the Appendix A. The unknown parameter values for the fatigue life probability model proposed in this study are listed in Table 2. It is worth to note that the parameters for this model are estimated from a certain number of test samples of the 1670-grade steel wire. Parameters for more steel wire cases thus need of further investigation.Table 1
Fatigue test results of the 1670-grade steel wire.
Noσm (MPa)∆σ (MPa)N (thousands)1100567035210056703031005670404100567063510056703961005670527100567039810056707891005670351010056704811100567030127805202961378052014414780520107157805202401678052042317780520106187805205861978052038520780520250217805204622278052029723627418760246274189822562741810242662741867227627418867286274189632962741812083062741814833162741863032627418142533627418136334502.53352000∗35502.53352000∗36502.53352000∗37502.53352000∗38502.53352000∗39502.53352000∗40502.53352000∗41502.53352000∗42502.53352000∗43502.53352000∗44502.53352000∗∗indicates that the experimental data are truncated.Table 2
Estimated values of the unknown parameters.
ParametersKβ0βABEstimated value1.399 × 10282.5793.767−0.0013181.9Based on the test data in Table1 and the estimated values of the unknown parameters in Table 2, the Weibull plot of intermediate variable V for the 1670-grade steel wires is shown in Figure 1, where the estimated parameters are also given. The samples of intermediate variable V are nearly linear based on the estimated parameters, as shown in Figure 1. In addition, the standard Weibull distribution of the intermediate variable V can be accepted by the Kolmogorov–Smirnov (K-S) test at the 5% level of significance. The proposed Weibull model for the 1670-grade steel wire fits the fatigue data well.Figure 1
Weibull plot of variableV for 1670-grade steel wire.According to equation (9) and the estimated parameters of the model, the S-N curves of parallel steel wires with guarantee probabilities of 99.7%, 90%, and 50% are calculated as follows:(12)lgN99.7%=27.168−log101−0.0013×∆σ+181.91−3.3835×log10∆σ2+2×σm×∆σ,lgN90.0%=27.767−log101−0.0013×∆σ+181.91−3.3835×log10∆σ2+2×σm×∆σ,lgN50.0%=28.084−log101−0.0013×∆σ+181.91−3.3835×log10∆σ2+2×σm×∆σ.


### 2.3. Stress Effects on the Fatigue Life Model

Figure2 shows the mean stress-stress range-fatigue life (M-S-N) surfaces based on equation (12), representing the relationship between the mean stress of the wire, stress range, and the logarithm of the fatigue life at a guaranteed probability of 50%. It can be seen that the fatigue life model proposed in this study can better describe the fatigue life distribution of steel wires under different stress levels compared to the model presented in reference [8]. The referenced model does not consider the effect of the mean stress on the fatigue life of steel wires (the model-based test results are listed in Table 1). When the mean stress of the steel wire was less than the mean stress corresponding to the experimental data, the fatigue life of the model proposed in this study was longer than that of the comparison model. When the mean stress of the steel wire was greater than the mean stress corresponding to the experimental data, the fatigue life of the model proposed in this study was less than that of the comparison model.Figure 2
M-S-N surfaces of a steel wire.The S-N curves of the steel wire are shown in Figure3, based on the estimated parameters of equation (9). From the figure, it can be seen that the experimental results exhibit a dispersive phenomenon, which may be attributed to the presence of possible damage. Therefore, the probabilistic model chosen to characterize the fatigue life of high-strength steel wires is reasonable in this paper. Under the corresponding stress level, all test data were well distributed on both sides of the corresponding S-N curve. The S-N curves of the steel-wires with different mean stress at a guaranteed probability of 50.0% are shown in Figure 3(a). Under a given mean stress value, the fatigue life of the parallel steel wire has a linear relationship with the stress range in a bilogarithmic coordinate system. The fatigue life of the steel wire decreased with an increase in the mean stress within the same stress range. The S-N curves of the steel-wires with different stress ranges at a guaranteed probability of 50.0% are shown in Figure 3(b). The fatigue life of the steel wire has a linear relationship with the mean stress in a bilogarithmic coordinate system under a given stress range. The fatigue life of the steel wire decreases with an increase in the stress range under the same mean stress.Figure 3
S-N curves of a steel wire. (a) Stress range-life curve. (b) Mean stress-life curve.
(a)(b)


## 2.1. Fatigue Life Probabilistic Model for Steel Wire

The three-parameter power function model [8] mainly analyzes the stress range and fatigue life through the linear regression statistical method and obtains the empirical formula that is used to characterize the relationship between the fatigue life and stress range of the steel wire under different failure probability conditions. This model only considers the influence of the stress range on the fatigue life of the steel wire and ignores its actual stress state. The fatigue test results of the steel wire show that both the stress range and the mean stress affect its fatigue life [32, 35, 36]. Therefore, it is critical to establish a steel wire fatigue model that considers the stress range and pretension.The nonlinear Lemaitre fatigue damage evolution model under a one-dimensional stress state can be expressed by the following equation (1) [37, 38]:(1)D˙=σ2∣σ−σ¯∣βB¯1−Dα×σ˙,σ˙>0,0,σ˙≤0,where • is the McCauley bracket operator; σ and σ¯ represent the stress and mean stress, respectively; and α, β, and B¯ are the material parameters. Equation (1) is integrated over one stress cycle to obtain equation (2):(2)∂D∂N=∆σ+2×σm×∆σβ+3/2B1−Dα×β+3,where ∆σ and σm represent the stress range and mean stress, respectively; B is the material parameter, and B=2β+2×B¯. Equation (2) is integrated over one stress cycle to obtain equation (3):(3)Di+1=1−1−Di1+αi−1+αi×∆σ+2×σm×∆σβ+3/2B×β+31/1+αi,where Di represents fatigue damage after the ith stress cycle. From this equation, it can be seen that the damage induced by the current stress cycle is related to the stress cycle experienced and the previous damage state. The present incremental damage is not simply linearly superimposed on the previous damage but exhibits a significant nonlinear, which is obviously different from the linear fatigue damage model based on the Palmgren–Miner law. Under the specified stress level, the fatigue life of the steel-wire can be expressed as(4)N=B×β+31+α×∆σ+2×σm×∆σ−β+3/2,where N is the fatigue life; α is a function of the stress range ∆σ;α∆σ=a×∆σ+b. Taking the logarithm on both sides of the following equation (5):(5)logN=logB×β+3+log∆σ+2×σm×∆σ−β+3/21+α∆σ.Under the specified stress level, the fatigue life cumulative distribution function (CDF) of the steel wire can be expressed as [3](6)FN;∆σ,σm=1−exp−Nδ∆σ,σmβ0,where β0 and δ∆σ,σm are the shape and characteristic parameters of the Weibull distribution, respectively. We took the logarithm twice on both sides of the following equation (6):(7)logN=1β0×log1−ln1−F+logδ∆σ,σm.Comparing equation (7) with (5), we can assume that(8)δ∆σ,σm=K1+α∆σ×∆σ+2×σm×∆σ−β+3/2,where K is a constant. Substituting equation (8) into equation (6) gives(9)FN;∆σ,σm=1−exp−1+α∆σK×N×∆σ2+2×σm×∆σβ+3/2β0,where K, β0, β, and α are unknown parameters that can be estimated using the test data of all the specimens. This formula is a Weibull distribution model that characterizes the fatigue life of steel wires under different stress levels. The fatigue life of the steel wire follows a two-parameter Weibull distribution of the shape parameter β0 and characteristic parameter K/1+α∆σ×∆σ+2×σm×∆σ−β+3/2. Once all the unknown parameters have been estimated by the maximum likelihood estimation (MLE) or by the expectation-maximization (EM) [39] algorithm based on the fatigue test data, a variable denoted as V can be defined as(10)V=1+α∆σ×N×∆σ2+2×σm×∆σβ+3/2.Equation (9) can be expressed as(11)FV=1−exp−VKβ0.When the model parameters are known, the fit between the model and fatigue test data can be checked according to the Weibull probability diagram of the intermediate variableV.


## 2.2. Parameter’s Estimation of Fatigue Life Probabilistic Model for Steel Wire

The 1670-grade steel wire is widely used in cable-stayed bridges in China. Forty-four groups of samples with a diameter of 7 mm and a length of 250 mm were employed to study the fatigue properties of the 1670-grade steel wire [8]. The fatigue tests were performed using a servohydraulic universal testing machine (MTS Landmark). The stress ratio of the fatigue test was 0.5, the loading frequency was 10 Hz, and the stress range was set to four groups of 335, 418, 520, and 670 MPa. All test results are listed in Table 1. The unknown parameters in equation (9) can be estimated using the expectation-maximization (EM) algorithm based on the fatigue test data [39], and the solution details are shown in the Appendix A. The unknown parameter values for the fatigue life probability model proposed in this study are listed in Table 2. It is worth to note that the parameters for this model are estimated from a certain number of test samples of the 1670-grade steel wire. Parameters for more steel wire cases thus need of further investigation.Table 1
Fatigue test results of the 1670-grade steel wire.
Noσm (MPa)∆σ (MPa)N (thousands)1100567035210056703031005670404100567063510056703961005670527100567039810056707891005670351010056704811100567030127805202961378052014414780520107157805202401678052042317780520106187805205861978052038520780520250217805204622278052029723627418760246274189822562741810242662741867227627418867286274189632962741812083062741814833162741863032627418142533627418136334502.53352000∗35502.53352000∗36502.53352000∗37502.53352000∗38502.53352000∗39502.53352000∗40502.53352000∗41502.53352000∗42502.53352000∗43502.53352000∗44502.53352000∗∗indicates that the experimental data are truncated.Table 2
Estimated values of the unknown parameters.
ParametersKβ0βABEstimated value1.399 × 10282.5793.767−0.0013181.9Based on the test data in Table1 and the estimated values of the unknown parameters in Table 2, the Weibull plot of intermediate variable V for the 1670-grade steel wires is shown in Figure 1, where the estimated parameters are also given. The samples of intermediate variable V are nearly linear based on the estimated parameters, as shown in Figure 1. In addition, the standard Weibull distribution of the intermediate variable V can be accepted by the Kolmogorov–Smirnov (K-S) test at the 5% level of significance. The proposed Weibull model for the 1670-grade steel wire fits the fatigue data well.Figure 1
Weibull plot of variableV for 1670-grade steel wire.According to equation (9) and the estimated parameters of the model, the S-N curves of parallel steel wires with guarantee probabilities of 99.7%, 90%, and 50% are calculated as follows:(12)lgN99.7%=27.168−log101−0.0013×∆σ+181.91−3.3835×log10∆σ2+2×σm×∆σ,lgN90.0%=27.767−log101−0.0013×∆σ+181.91−3.3835×log10∆σ2+2×σm×∆σ,lgN50.0%=28.084−log101−0.0013×∆σ+181.91−3.3835×log10∆σ2+2×σm×∆σ.


## 2.3. Stress Effects on the Fatigue Life Model

Figure2 shows the mean stress-stress range-fatigue life (M-S-N) surfaces based on equation (12), representing the relationship between the mean stress of the wire, stress range, and the logarithm of the fatigue life at a guaranteed probability of 50%. It can be seen that the fatigue life model proposed in this study can better describe the fatigue life distribution of steel wires under different stress levels compared to the model presented in reference [8]. The referenced model does not consider the effect of the mean stress on the fatigue life of steel wires (the model-based test results are listed in Table 1). When the mean stress of the steel wire was less than the mean stress corresponding to the experimental data, the fatigue life of the model proposed in this study was longer than that of the comparison model. When the mean stress of the steel wire was greater than the mean stress corresponding to the experimental data, the fatigue life of the model proposed in this study was less than that of the comparison model.Figure 2
M-S-N surfaces of a steel wire.The S-N curves of the steel wire are shown in Figure3, based on the estimated parameters of equation (9). From the figure, it can be seen that the experimental results exhibit a dispersive phenomenon, which may be attributed to the presence of possible damage. Therefore, the probabilistic model chosen to characterize the fatigue life of high-strength steel wires is reasonable in this paper. Under the corresponding stress level, all test data were well distributed on both sides of the corresponding S-N curve. The S-N curves of the steel-wires with different mean stress at a guaranteed probability of 50.0% are shown in Figure 3(a). Under a given mean stress value, the fatigue life of the parallel steel wire has a linear relationship with the stress range in a bilogarithmic coordinate system. The fatigue life of the steel wire decreased with an increase in the mean stress within the same stress range. The S-N curves of the steel-wires with different stress ranges at a guaranteed probability of 50.0% are shown in Figure 3(b). The fatigue life of the steel wire has a linear relationship with the mean stress in a bilogarithmic coordinate system under a given stress range. The fatigue life of the steel wire decreases with an increase in the stress range under the same mean stress.Figure 3
S-N curves of a steel wire. (a) Stress range-life curve. (b) Mean stress-life curve.
(a)(b)


## 3. Fatigue Life Model for Corroded Steel Wire

Yao et al. [13, 15] concluded that the coupling of the stress state (stress amplitude and pretension), and the corrosive environment significantly affects the fatigue performance of high-tensile steel wires. Under the same corrosive environment, the corrosion of high-tensile steel wire under alternating and constant tensile stress conditions is more significant than that under no stress. The corrosion of high-tensile steel wire under alternating loading condition is the most significant. Therefore, it is meaningful to consider the coupling effect between corrosion and stress conditions to develop the fatigue life model for corroded steel wire.


### 3.1. Fatigue Life Probabilistic Model for Corroded Steel Wire

The fatigue test results in reference [16] indicate that the slope of the S-N curves in a bi-logarithmic coordinate for a corroded wire rebar is a linear function of its corrosion degree, and the slope of the S-N curve increases with an increase in the degree of corrosion. Simultaneously, the fatigue life of the corroded steel wire decreased exponentially with an increase in the corrosion rate under a specified stress range. In this study, the degree of corrosion was quantified based on the weight loss rate w, that is, the average loss of the cross-section of the steelwire. From equation (9), the slope of the S-N curve of the steel wire is β+3/2. Therefore, the slope of the S-N curve and parameter K in equation (9) can be expressed as(13)β+32=C+D×w,(14)K=A×exp−B×w,where A, B, C, and D are the parameters that can be estimated using the test data.Substituting equations (13) and (14) into equation (9), we obtain the CDF and probability density function (PDF) of the fatigue lives N for the corroded steel wire as follows:(15)FN;∆σ,σm,w=1−exp−1+α∆σA×exp−B×w×N×∆σ2+2×σm×∆σC+D×wβ0,(16)fN;∆σ,σm=β0×1+α∆σA×exp−B×w×∆σ2+2×σm×∆σC+D×wβ0×Nβ0−1×exp−1+α∆σA×exp−B×w×N×∆σ2+2×σm×∆σC+D×wβ0.Once all the unknown parameters have been estimated by the maximum likelihood estimation (MLE) or by the expectation-maximization (EM) algorithm based on the censored data, a variable denoted can be defined as(17)v=1+α∆σexp−B×w×N×∆σ2+2×σm×∆σC+D×w.Equation (15) can be expressed as(18)Fv=1−exp−vKβ0.When the model parameters are known, the fit between the model and fatigue test data can be checked according to the Weibull probability diagram of intermediate variablev.


### 3.2. Parameter Estimation of Fatigue Life Probabilistic Model for Corroded Steel Wire

This study analyzes the fatigue test data of corroded steel wires obtained from the reference [19]. Ninety-nine groups of samples with a diameter of 7 mm and length of 250 mm were employed to study the fatigue properties of the corroded 1670-grade steelwires. The corrosion test was carried out according to relevant regulation [40]. The experiment started with adding some glacial acetic acid to 5 ± 0.5% sodium chloride solution to drop the solution pH value to about 3, then making the solution acidic and forming a sour salt spray accordingly. During the corrosion test, the temperature in the chamber should keep at 35 ± 2°C, and the corrosion time is designed as follows: 1, 7, 10, 30, and 60 days, with a total of 5 test cases and 99 groups of samples. When each group of samples reached the corrosion time, they were immediately removed from the salt spray chamber and put into the cleaning solution composed of sodium chloride to clean off the corrosion products on the samples. After the measurement of the weight loss, all samples were ready for fatigue tests.The fatigue tests were performed using a servohydraulic universal testing machine (MTS Landmark). The stress ratio of the fatigue test was 0.5, the loading frequency was 10 Hz, and the stress range was set to four groups of 335, 418, 520, and 670 MPa. All test results are listed in Table3. The unknown parameters in equation (16) can be estimated using the MLE based on the fatigue test data [39], and the solution details are shown in Appendix B. The unknown parameter values for the fatigue life probability model proposed in this study are listed in Table 4.Table 3
Fatigue test results of the corroded 1670-grade steel wire.
Now (%)σm (MPa)∆σ (MPa)N10.25502.53354,304,06720.23502.53353,995,56230.24502.53354,368,98340.18502.53354,213,84750.196274181,014,32060.26274181,205,97470.226274181,105,68780.356274181,048,18890.34627418826,526100.24780520232,963110.21780520246,119120.2780520256,961130.22780520235,852140.32780520166,893150.32780520325,609160.2100567051,257170.18100567050,600180.21100567055,595190.36100567051,388200.32100567030,283210.25100567037,063221.74502.53353,086,541231.76502.53353,167,280241.82502.53352,926,584251.8627418704,511261.91627418826,526271.87627418776,680281.98627418725,609291.75627418776,680301.88780520185,381311.82780520194,257321.97780520215,145331.81780520199,445342.14780520169,445351.78100567037,917361.79100567038,453371.79100567040,558381.62100567040,847391.82100567029,901402.06100567033,705411.75502.53352,453,711421.73502.53352,469,835431.87502.53352,542,617441.83502.53352,529,487451.96627418607,373461.88627418641,078471.97627418685,124481.96627418631,859492.10780520167,765501.89780520185,080511.86780520173,118521.91780520169,237531.99100567035,845541.88100567037,842551.91100567038,438562.06100567039,533572.35100567031,625582.05100567031,523592.44100567030,228602.43502.53351,381,860612.04502.53351,218,463622.39502.53351,293,784632.53502.53351,325,683642.37627418554,096652.37627418508,510662.61627418540,250672.57627418489,166682.21780520132,792692.31780520154,328702.32780520125,827712.27780520140,952722.8678052065,080732.6780520119,342742.8378052097,209752.49100567034,160762.36100567037,720772.27100567037,284782.4100567035,232793.04100567024,565802.89100567032,293812.63100567024,553823.32502.5335875,665833.53502.5335783,086843.12502.5335805,852853.35627418406,845863.44627418412,214872.97627418381,393883.27627418391,646893.11780520102,127903.5278052097,727913.51780520119,37692378052073,713933.3578052090,641943.34100567029,436953.51100567028,496963.62100567031,740973.33100567029,825983.48100567025,664993.23100567029,137Table 4
Estimated values of the unknown parameters for steel wire considering corrosion effects.
Parametersβ0ABCDabEstimated value3.9093.796 × 10293.796126.880−3.131−0.007985.349Based on the test data in Table3 and the estimated values of the unknown parameters in Table 4, the Weibull plot of intermediate variable v for the corroded 1670-grade steel wires is shown in Figure 4. The samples of intermediate variable v are nearly linear based on the estimated parameters, as shown in Figure 4. In addition, the standard Weibull distribution of intermediate variable v can be accepted by the K–S test at a 5% level of significance. The proposed Weibull model for the corroded 1670-grade steel wire fits the fatigue data well.Figure 4
Weibull plot of variablev for the corroded 1670-grade steel wire.Based on the estimated model parameters, the relationship between the fatigue life of the corroded steel wire and the degree of corrosion under a guaranteed probability of 50.0% is shown in Figure5. Each line in the figure corresponds to a stress level. The fatigue life of the parallel steel wire decreases with an increase in the degree of corrosion under the specified stress level, indicating that the effect of corrosion on the fatigue performance of the steel wire is exceptionally significant. When the corrosion rate exceeded 1.5%, all experimental samples were evenly distributed on both sides of the curve, and when the corrosion rate was lower than 0.5%, the experimental fatigue samples of each group deviated significantly from the theoretical model. The main reason is that, in the early stage of corrosion, the damage to the quality of the steel wire mainly comes from the galvanized layer on the surface of the steel wire, and the fatigue properties of the steel wire did not change substantially. Therefore, the fatigue assessment of a steel wire can completely rely on the wire fatigue S-N curve in the noncorrosion state when the steel wire corrosion rate is less than 1.5%.Figure 5
Fatigue life versus corrosion degree for steel wire.


### 3.3. Steel Wire Time-Variant Corrosion Model

The exponential model is often used to calculate the uniform corrosion depth at different service stages in a corrosive environment. For galvanized steel wires, corrosion can be divided into two stages: corrosion of the surface galvanized layer and corrosion of the inner steel wire. The uniform corrosion depth of each stage can be characterized by an exponential model, as shown in equation (19) [34].(19)du,Znt=Cr1×tγ1t≤t0,du,Fet=Cr2×t−t0γ2t>t0,where t is the corrosion time of the steel wire; t0 is the time required for the galvanized layer on the surface of the steel wire to be completely corroded; Cr1 and Cr2 are the annual corrosion rates of the galvanized layer of the steel wire and the inner steelwire, respectively, and γ1 and γ2 are the model exponent constants.Under corrosive environmental conditions, the mass loss of the steel wire equals the mass of the uncorroded steel wire minus the mass of the steel wire after corrosion. The mass loss of the steel wire at different corrosion stages can be calculated by multiplying the uniform corrosion depth by the exposed area of the steel wire and metal density. According to the definition of the weight loss rate of the steel wire mass, the relationship between the weight loss rate and uniform corrosion depth is as follows:(20)wt=DFe+2×DZn2−DFe+2×DZn−2×du,Znt2ρFe/ρZn−1×DFe2+DFe+2×DZn2t≤t0,wt=ρZn×DFe+2×DZn2−DFe2+ρFe×DFe2−DFe2−DFe−2×du,Fet2ρFe×DFe2+ρZn×DFe+2×DZn2−DFe2t>t0,where DFe is the diameter of the steel wire, DZn is the thickness of the zinc coating, and ρFe and ρZn are the densities of the steel wire and zinc coating, respectively.Substituting equation (20) into equation (15), the CDF of the corroded steel wire fatigue life can be obtained by considering time-varying corrosion. The fatigue life model for the corroded stay cable can then be derived based on equation (21), as described in Section 4.(21)FN;∆σ,σm,wt=1−exp−1+α∆σA×exp−B×wt×N×∆σ2+2×σm×∆σC+D×wtβ0t>t0.


## 3.1. Fatigue Life Probabilistic Model for Corroded Steel Wire

The fatigue test results in reference [16] indicate that the slope of the S-N curves in a bi-logarithmic coordinate for a corroded wire rebar is a linear function of its corrosion degree, and the slope of the S-N curve increases with an increase in the degree of corrosion. Simultaneously, the fatigue life of the corroded steel wire decreased exponentially with an increase in the corrosion rate under a specified stress range. In this study, the degree of corrosion was quantified based on the weight loss rate w, that is, the average loss of the cross-section of the steelwire. From equation (9), the slope of the S-N curve of the steel wire is β+3/2. Therefore, the slope of the S-N curve and parameter K in equation (9) can be expressed as(13)β+32=C+D×w,(14)K=A×exp−B×w,where A, B, C, and D are the parameters that can be estimated using the test data.Substituting equations (13) and (14) into equation (9), we obtain the CDF and probability density function (PDF) of the fatigue lives N for the corroded steel wire as follows:(15)FN;∆σ,σm,w=1−exp−1+α∆σA×exp−B×w×N×∆σ2+2×σm×∆σC+D×wβ0,(16)fN;∆σ,σm=β0×1+α∆σA×exp−B×w×∆σ2+2×σm×∆σC+D×wβ0×Nβ0−1×exp−1+α∆σA×exp−B×w×N×∆σ2+2×σm×∆σC+D×wβ0.Once all the unknown parameters have been estimated by the maximum likelihood estimation (MLE) or by the expectation-maximization (EM) algorithm based on the censored data, a variable denoted can be defined as(17)v=1+α∆σexp−B×w×N×∆σ2+2×σm×∆σC+D×w.Equation (15) can be expressed as(18)Fv=1−exp−vKβ0.When the model parameters are known, the fit between the model and fatigue test data can be checked according to the Weibull probability diagram of intermediate variablev.


## 3.2. Parameter Estimation of Fatigue Life Probabilistic Model for Corroded Steel Wire

This study analyzes the fatigue test data of corroded steel wires obtained from the reference [19]. Ninety-nine groups of samples with a diameter of 7 mm and length of 250 mm were employed to study the fatigue properties of the corroded 1670-grade steelwires. The corrosion test was carried out according to relevant regulation [40]. The experiment started with adding some glacial acetic acid to 5 ± 0.5% sodium chloride solution to drop the solution pH value to about 3, then making the solution acidic and forming a sour salt spray accordingly. During the corrosion test, the temperature in the chamber should keep at 35 ± 2°C, and the corrosion time is designed as follows: 1, 7, 10, 30, and 60 days, with a total of 5 test cases and 99 groups of samples. When each group of samples reached the corrosion time, they were immediately removed from the salt spray chamber and put into the cleaning solution composed of sodium chloride to clean off the corrosion products on the samples. After the measurement of the weight loss, all samples were ready for fatigue tests.The fatigue tests were performed using a servohydraulic universal testing machine (MTS Landmark). The stress ratio of the fatigue test was 0.5, the loading frequency was 10 Hz, and the stress range was set to four groups of 335, 418, 520, and 670 MPa. All test results are listed in Table3. The unknown parameters in equation (16) can be estimated using the MLE based on the fatigue test data [39], and the solution details are shown in Appendix B. The unknown parameter values for the fatigue life probability model proposed in this study are listed in Table 4.Table 3
Fatigue test results of the corroded 1670-grade steel wire.
Now (%)σm (MPa)∆σ (MPa)N10.25502.53354,304,06720.23502.53353,995,56230.24502.53354,368,98340.18502.53354,213,84750.196274181,014,32060.26274181,205,97470.226274181,105,68780.356274181,048,18890.34627418826,526100.24780520232,963110.21780520246,119120.2780520256,961130.22780520235,852140.32780520166,893150.32780520325,609160.2100567051,257170.18100567050,600180.21100567055,595190.36100567051,388200.32100567030,283210.25100567037,063221.74502.53353,086,541231.76502.53353,167,280241.82502.53352,926,584251.8627418704,511261.91627418826,526271.87627418776,680281.98627418725,609291.75627418776,680301.88780520185,381311.82780520194,257321.97780520215,145331.81780520199,445342.14780520169,445351.78100567037,917361.79100567038,453371.79100567040,558381.62100567040,847391.82100567029,901402.06100567033,705411.75502.53352,453,711421.73502.53352,469,835431.87502.53352,542,617441.83502.53352,529,487451.96627418607,373461.88627418641,078471.97627418685,124481.96627418631,859492.10780520167,765501.89780520185,080511.86780520173,118521.91780520169,237531.99100567035,845541.88100567037,842551.91100567038,438562.06100567039,533572.35100567031,625582.05100567031,523592.44100567030,228602.43502.53351,381,860612.04502.53351,218,463622.39502.53351,293,784632.53502.53351,325,683642.37627418554,096652.37627418508,510662.61627418540,250672.57627418489,166682.21780520132,792692.31780520154,328702.32780520125,827712.27780520140,952722.8678052065,080732.6780520119,342742.8378052097,209752.49100567034,160762.36100567037,720772.27100567037,284782.4100567035,232793.04100567024,565802.89100567032,293812.63100567024,553823.32502.5335875,665833.53502.5335783,086843.12502.5335805,852853.35627418406,845863.44627418412,214872.97627418381,393883.27627418391,646893.11780520102,127903.5278052097,727913.51780520119,37692378052073,713933.3578052090,641943.34100567029,436953.51100567028,496963.62100567031,740973.33100567029,825983.48100567025,664993.23100567029,137Table 4
Estimated values of the unknown parameters for steel wire considering corrosion effects.
Parametersβ0ABCDabEstimated value3.9093.796 × 10293.796126.880−3.131−0.007985.349Based on the test data in Table3 and the estimated values of the unknown parameters in Table 4, the Weibull plot of intermediate variable v for the corroded 1670-grade steel wires is shown in Figure 4. The samples of intermediate variable v are nearly linear based on the estimated parameters, as shown in Figure 4. In addition, the standard Weibull distribution of intermediate variable v can be accepted by the K–S test at a 5% level of significance. The proposed Weibull model for the corroded 1670-grade steel wire fits the fatigue data well.Figure 4
Weibull plot of variablev for the corroded 1670-grade steel wire.Based on the estimated model parameters, the relationship between the fatigue life of the corroded steel wire and the degree of corrosion under a guaranteed probability of 50.0% is shown in Figure5. Each line in the figure corresponds to a stress level. The fatigue life of the parallel steel wire decreases with an increase in the degree of corrosion under the specified stress level, indicating that the effect of corrosion on the fatigue performance of the steel wire is exceptionally significant. When the corrosion rate exceeded 1.5%, all experimental samples were evenly distributed on both sides of the curve, and when the corrosion rate was lower than 0.5%, the experimental fatigue samples of each group deviated significantly from the theoretical model. The main reason is that, in the early stage of corrosion, the damage to the quality of the steel wire mainly comes from the galvanized layer on the surface of the steel wire, and the fatigue properties of the steel wire did not change substantially. Therefore, the fatigue assessment of a steel wire can completely rely on the wire fatigue S-N curve in the noncorrosion state when the steel wire corrosion rate is less than 1.5%.Figure 5
Fatigue life versus corrosion degree for steel wire.


## 3.3. Steel Wire Time-Variant Corrosion Model

The exponential model is often used to calculate the uniform corrosion depth at different service stages in a corrosive environment. For galvanized steel wires, corrosion can be divided into two stages: corrosion of the surface galvanized layer and corrosion of the inner steel wire. The uniform corrosion depth of each stage can be characterized by an exponential model, as shown in equation (19) [34].(19)du,Znt=Cr1×tγ1t≤t0,du,Fet=Cr2×t−t0γ2t>t0,where t is the corrosion time of the steel wire; t0 is the time required for the galvanized layer on the surface of the steel wire to be completely corroded; Cr1 and Cr2 are the annual corrosion rates of the galvanized layer of the steel wire and the inner steelwire, respectively, and γ1 and γ2 are the model exponent constants.Under corrosive environmental conditions, the mass loss of the steel wire equals the mass of the uncorroded steel wire minus the mass of the steel wire after corrosion. The mass loss of the steel wire at different corrosion stages can be calculated by multiplying the uniform corrosion depth by the exposed area of the steel wire and metal density. According to the definition of the weight loss rate of the steel wire mass, the relationship between the weight loss rate and uniform corrosion depth is as follows:(20)wt=DFe+2×DZn2−DFe+2×DZn−2×du,Znt2ρFe/ρZn−1×DFe2+DFe+2×DZn2t≤t0,wt=ρZn×DFe+2×DZn2−DFe2+ρFe×DFe2−DFe2−DFe−2×du,Fet2ρFe×DFe2+ρZn×DFe+2×DZn2−DFe2t>t0,where DFe is the diameter of the steel wire, DZn is the thickness of the zinc coating, and ρFe and ρZn are the densities of the steel wire and zinc coating, respectively.Substituting equation (20) into equation (15), the CDF of the corroded steel wire fatigue life can be obtained by considering time-varying corrosion. The fatigue life model for the corroded stay cable can then be derived based on equation (21), as described in Section 4.(21)FN;∆σ,σm,wt=1−exp−1+α∆σA×exp−B×wt×N×∆σ2+2×σm×∆σC+D×wtβ0t>t0.


## 4. Fatigue Life Prediction Approach for Corroded Stay Cables

A parallel wire stay cable is composed of multiple parallel steel wires, which can be regarded as a parallel system composed of numerous steel wires. A single steel wire can be considered as the basic unit of a stay cable. When the parallel wires in the stay cable break, the stress state of the parallel wires in the stay cable will be redistributed. It was noted that the stay cable was composed ofm wires in parallel, and the external load remained constant. When i wires break, the stress of the remaining parallel wires is(22)∆σi=mm−i×∆σ,σmi=mm−i×σm,where ∆σi and σmi are the wire stress range and mean stress after i wires break, respectively. As the number of cycles increased, the parallel wires inside the stay cable gradually broke, which in turn led to an increase in the stress level of the unbroken wires. This accelerated the fatigue fracture of the remaining parallel wires until the entire stay cable was destroyed.Figure6 shows the relationships between the fatigue life of cable and stress range under constant load, and the bold black lines are the evolution process of fatigue damage of the stay cable. The lower right corner letter of fatigue life N represents the number of broken wires in the cable, and the upper right corner represents its corresponding stress range. When the number of external load actions reaches N1∆σ, the steel wire with the shortest fatigue life will break, and then the stress of the stay cable will be redistributed: ∆σi=m/m−i×∆σ, σm1=m/m−1×∆σm. Repeat the previous process until the fatigue limit of the stay cable is reached, and the stay cable will fail.Figure 6
Relationship between fatigue life and stress range.According to the CDF the corroded steel wire in equation (21), the fatigue life of the steel wire under different stress levels has the same failure probability:(23)FNi∆σ,σm;∆σ,σm;wt=FNi∆σi−1,σmi−1;∆σi−1,σmi−1;wt,where the letter in the lower right corner of fatigue life N indicates the number of broken wires in the cable, and the upper right corner indicates the stress level. Substituting equation (21) into equation (23) yields(24)Ni∆σi−1,σmi−1=1+a×∆σ+b1+a×∆σi−1+b×∆σ2+2×∆σ×σm∆σi−12+2×∆σi−1×σmi−1β+3/2×Ni∆σ,σm.When the second wire breaks, the number of load cyclesN2 experienced by the stay cable is(25)N2T=N1∆σ,σm+N2∆σ1,σm1−N1∆σ1,σm1.By analogy, when theith steel wire breaks, the number of load cycles Ni experienced by the stay cable is(26)NiT=N1∆σ,σm+∑j=2iNj∆σj−1,σmj−1−Nj−1∆σj−1,σmj−1.Substituting equation (24) into equation (26):(27)NiT=N1∆σ,σm+∑j=2iNj∆σ,σm−Nj−1∆σ,σm×1+a×∆σ+b1+a×∆σj−1+b×mm−j+1−β+3,where i=2,3,⋯,m.The fatigue life of a single corroded steel wire was used to deduce the fatigue life model of a stay cable composed of multiple corroded parallel steel-wires. Taking a cable consisting of 253 corroded parallel wires as an example, the fatigue life distribution of the stay cable under different corrosion states was calculated based on the Monte Carlo method. The calculation steps are as follows:(1)
According to the PDF of the fatigue life of a single steel wire under a specified stress level and under different corrosion states, 253 groups of samples were randomly generated to simulate the fatigue life of each steel wire in the stay cable.(2)
The samples were arranged from small to large, then the fatigue life of the stay cables under different broken wire conditions was calculated according to equation (A.2).The previous process can be repeated to obtain the fatigue life distribution samples of stay cables under different corrosion states. The number of fatigue life simulation samples of the corroded stay cables was set to 10,000 for each degree of corrosion. The number of cycles when the wire breakage rate reached 10% was considered as the fatigue life of the stay cable [19].Based on the previous calculation steps, the fatigue life of the corroded stay cable at guaranteed probabilities of 99.7%, 90%, and 50% is shown in equation (28).(28)lgN99.7%=29.200+log10exp−126.880×wt−log101−0.008×∆σ+5.3487−3.8522−3.131×wt×log10∆σ2+2×σm×∆σ,lgN90.0%=29.2393+log10exp−126.880×wt−log101−0.008×∆σ+5.3487−3.8522−3.131×wt×log10∆σ2+2×σm×∆σ,lgN50.0%=29.2706+log10exp−126.880×w−log101−0.008×∆σ+5.3487−3.8522−3.131×wt×log10∆σ2+2×σm×∆σ.According to the simulation results under different corrosion conditions, the corrosion-stress range fatigue life (C-S-N) surfaces for the corroded steel wire and corroded stay cable are shown in Figure7. The C-S-N surfaces from top to bottom were steel wires with a guaranteed probability of 50%, steel wires with a guaranteed probability of 90%, stay cables with a guaranteed probability of 50%, and steel wires with a guaranteed probability of 99.7%. The C-S-N surfaces of the parallel steel wire are parallel to the C-S-N surface of the stay cable under different guarantee probabilities. Under the same stress level and corrosion conditions, the median fatigue life of the stay cables is smaller than that of a single wire, with a guaranteed probability of 90%. This is mainly because the fatigue life of stay cables is determined by a few steel-wires with shorter lives, and stress redistribution accelerates the failure of the inner wire of the stay cable.Figure 7
C-S-N surfaces for the corroded steel-wire and corroded stay cable.


## 5. Fatigue Life Evaluation for Stay Cables of a Real Bridge

### 5.1. Background Engineering and Finite Element Model

Background engineering was a steel box girder cable-stayed bridge with a main span of 460 m and girder width of 38.8 m on the Yangtze River in China, as shown in Figure8. There were 72 pairs of stay cables in the bridge, and each stay cable was made of the 1670-grade steel-wire with a diameter of 7 mm. The cable numbers are shown in Figure 9. The number of steel wires in a stay cable increased with the length of the stay cable, ranging from 109 to 253. The longest stay cable, A18, comprises 253 steel wires of grade 1670 with a diameter of 7 mm.Figure 8
The cable-stayed bridge on the Yangtze River.Figure 9
Number of stay cables.ABAQUS was used to establish the finite element model, as shown in Figure10. The stay cable is simulated using a truss element that only bears tension, and the cable force is applied by defining the initial strain. To consider the influence of the sag effect of the stay cables, according to Ernst [41], a calculation formula for the equivalent elastic modulus was proposed to correct the elastic modulus of each stay cable. The main girder was simulated using a fishbone beam model. Cross-section wind-resistant and vertical supports at the pylon location were simulated using spring elements with a specified stiffness. The established finite element model has been updated with in-site measurements to ensure its effectiveness, and more details on the model updating can be found in reference [42].Figure 10
Multiscale finite element model.


### 5.2. Time History Analysis of Cable Force

The author of this study established the vehicle fatigue load spectrum of a specific section of the bridge through statistical analysis of the traffic flow information monitored by the background engineering bridge weigh-in-motion (WIM) system from the third quarter of 2011 to the fourth quarter of 2015. Then, the Weibull stochastic process and filtering composite Weibull process were used to simulate the vehicle arrival process and stochastic traffic process of each lane, respectively. The Markov chain Monte Carlo (MCMC) method generated the vehicle sequence of each lane at each moment. Finally, a traffic flow stochastic process model was established and verified. More details on the traffic flow model can be found in reference [42, 43].Based on the finite element model, the cable force influence line of any stay cable and the cable force–time history of each stay cable can be obtained by combining the influence line of each stay cable and the traffic flow stochastic process model. This study usesA05, J05, A18, and J18 cables as examples. The time histories of the tension force for those cables in a day are shown in Figure 11. It is noted that the dead load cable forces were measured in bridge loading experiments before opening. Under the combined action of the traffic load and dead load, the stress of the A05 cable is between 415.2–511.4 MPa, the stress of the J05 cable is between 415.7–484.8 MPa, the stress of the A18 cable is between 426.5–457.9 MPa, and the stress of the J18 cable is between 490.0–518.7 MPa. As shown in the figure, compared with the other stay cables, the force of stay cables A18 and J18 at the end of the side span will be significantly relaxed under the action of the traffic load on the bridge.Figure 11
Time history of tension force for stay cables: (a)A05 stay cable, (b) A18 stay cable, (c) J05 stay cable, and (d) J18 stay cable.
(a)(b)(c)(d)To count the stress range, average stress, and corresponding cycle times of the cable under the combined action of the traffic load and dead load, the rain flow counting method was used to count the stress time history, and the stress spectrum of the cable was obtained, as shown in Figure12. It can be seen from the figure that most of the stress range of the cables is less than 20 MPa and the average stress is more than 440 MPa, indicating that the stress of the cables under the traffic load is much less than the stress of the cables under the dead load.Figure 12
Stress spectrum of stay cables in a day: (a)A05 stay cable, (b) A18 stay cable, (c) J05 stay cable, and (d) J18 stay cable.
(a)(b)(c)(d)


### 5.3. Fatigue Life Analysis of Corroded Stay Cables

Determining each cable’s corrosive environment grade and corrosion level for the detail bridge is a challenge relying on technologies and methods selected. Jiang et al. [34] set three values for the annual corrosion rate of a high-strength steel wire: 15, 30, and 45 in equation (19), corresponding to the low, medium, and high corrosion environment, respectively. Accordingly, the parameters representing the corrosion development trend are set as 0.45, 0.60, and 0.75 in equation (19), respectively, based on the experiment and simulation results of steel wire corrosion [44, 45]. The testing data from Shimen Bridge, Yonghe Bridge, and Williamsburg Bridge were used to verified the correctness of the model. The uniform corrosion rate model parameters of steel wires under three different corrosion environments are listed in Table 5. Therefore, this paper directly selects the previous corrosive types to evaluate the fatigue life of stay cables.Table 5
Values of the parameters in equation (19).
Corrosive environment gradeCr (μm/year)γt0Low level150.455.2Middle level300.605.2High level450.755.2The fatigue life of the cable can be obtained by substituting the stress spectrum of the cable into the fatigue life model of the corroded stay cable, as shown in Figure13. The figure shows the fatigue life for each corrosive environment under the current traffic load for each stay cable at a 50% guarantee probability and 10% wire breakage rate. Under the combined actions of a low-corrosive environment and traffic load, the fatigue life of the stay cables is between 210.5 and 389.1 years, which is more than the design life of 100 years for cable-stayed bridges. Under the combined actions of a moderately corrosive environment and traffic load, the fatigue life of the stay cables was between 33.0–44.0 years. Under the combined actions of a highly corrosive environment and traffic load, the fatigue life of the stayed cables was between 14.5–17.5 years, which is less than the design life of cable-stayed bridges. It can be seen from the figure that in the three corrosive environments, the shortest fatigue life is not the J18 cable with the largest stress, but the A05 and J05 cables. Therefore, more attention should be paid to the fatigue damage of cables A05 and J05 during actual operation. It can be found that the fatigue life of the stay cables varies greatly under the three corrosion conditions, indicating that corrosion has a significant effect on the fatigue life of the stay cables.Figure 13
Fatigue life of the stay cables in different corrosion environments: (a) Low-corrosive environment. (b) Moderately corrosive environment. (c) Highly corrosive environment.
(a)(b)(c)


## 5.1. Background Engineering and Finite Element Model

Background engineering was a steel box girder cable-stayed bridge with a main span of 460 m and girder width of 38.8 m on the Yangtze River in China, as shown in Figure8. There were 72 pairs of stay cables in the bridge, and each stay cable was made of the 1670-grade steel-wire with a diameter of 7 mm. The cable numbers are shown in Figure 9. The number of steel wires in a stay cable increased with the length of the stay cable, ranging from 109 to 253. The longest stay cable, A18, comprises 253 steel wires of grade 1670 with a diameter of 7 mm.Figure 8
The cable-stayed bridge on the Yangtze River.Figure 9
Number of stay cables.ABAQUS was used to establish the finite element model, as shown in Figure10. The stay cable is simulated using a truss element that only bears tension, and the cable force is applied by defining the initial strain. To consider the influence of the sag effect of the stay cables, according to Ernst [41], a calculation formula for the equivalent elastic modulus was proposed to correct the elastic modulus of each stay cable. The main girder was simulated using a fishbone beam model. Cross-section wind-resistant and vertical supports at the pylon location were simulated using spring elements with a specified stiffness. The established finite element model has been updated with in-site measurements to ensure its effectiveness, and more details on the model updating can be found in reference [42].Figure 10
Multiscale finite element model.


## 5.2. Time History Analysis of Cable Force

The author of this study established the vehicle fatigue load spectrum of a specific section of the bridge through statistical analysis of the traffic flow information monitored by the background engineering bridge weigh-in-motion (WIM) system from the third quarter of 2011 to the fourth quarter of 2015. Then, the Weibull stochastic process and filtering composite Weibull process were used to simulate the vehicle arrival process and stochastic traffic process of each lane, respectively. The Markov chain Monte Carlo (MCMC) method generated the vehicle sequence of each lane at each moment. Finally, a traffic flow stochastic process model was established and verified. More details on the traffic flow model can be found in reference [42, 43].Based on the finite element model, the cable force influence line of any stay cable and the cable force–time history of each stay cable can be obtained by combining the influence line of each stay cable and the traffic flow stochastic process model. This study usesA05, J05, A18, and J18 cables as examples. The time histories of the tension force for those cables in a day are shown in Figure 11. It is noted that the dead load cable forces were measured in bridge loading experiments before opening. Under the combined action of the traffic load and dead load, the stress of the A05 cable is between 415.2–511.4 MPa, the stress of the J05 cable is between 415.7–484.8 MPa, the stress of the A18 cable is between 426.5–457.9 MPa, and the stress of the J18 cable is between 490.0–518.7 MPa. As shown in the figure, compared with the other stay cables, the force of stay cables A18 and J18 at the end of the side span will be significantly relaxed under the action of the traffic load on the bridge.Figure 11
Time history of tension force for stay cables: (a)A05 stay cable, (b) A18 stay cable, (c) J05 stay cable, and (d) J18 stay cable.
(a)(b)(c)(d)To count the stress range, average stress, and corresponding cycle times of the cable under the combined action of the traffic load and dead load, the rain flow counting method was used to count the stress time history, and the stress spectrum of the cable was obtained, as shown in Figure12. It can be seen from the figure that most of the stress range of the cables is less than 20 MPa and the average stress is more than 440 MPa, indicating that the stress of the cables under the traffic load is much less than the stress of the cables under the dead load.Figure 12
Stress spectrum of stay cables in a day: (a)A05 stay cable, (b) A18 stay cable, (c) J05 stay cable, and (d) J18 stay cable.
(a)(b)(c)(d)


## 5.3. Fatigue Life Analysis of Corroded Stay Cables

Determining each cable’s corrosive environment grade and corrosion level for the detail bridge is a challenge relying on technologies and methods selected. Jiang et al. [34] set three values for the annual corrosion rate of a high-strength steel wire: 15, 30, and 45 in equation (19), corresponding to the low, medium, and high corrosion environment, respectively. Accordingly, the parameters representing the corrosion development trend are set as 0.45, 0.60, and 0.75 in equation (19), respectively, based on the experiment and simulation results of steel wire corrosion [44, 45]. The testing data from Shimen Bridge, Yonghe Bridge, and Williamsburg Bridge were used to verified the correctness of the model. The uniform corrosion rate model parameters of steel wires under three different corrosion environments are listed in Table 5. Therefore, this paper directly selects the previous corrosive types to evaluate the fatigue life of stay cables.Table 5
Values of the parameters in equation (19).
Corrosive environment gradeCr (μm/year)γt0Low level150.455.2Middle level300.605.2High level450.755.2The fatigue life of the cable can be obtained by substituting the stress spectrum of the cable into the fatigue life model of the corroded stay cable, as shown in Figure13. The figure shows the fatigue life for each corrosive environment under the current traffic load for each stay cable at a 50% guarantee probability and 10% wire breakage rate. Under the combined actions of a low-corrosive environment and traffic load, the fatigue life of the stay cables is between 210.5 and 389.1 years, which is more than the design life of 100 years for cable-stayed bridges. Under the combined actions of a moderately corrosive environment and traffic load, the fatigue life of the stay cables was between 33.0–44.0 years. Under the combined actions of a highly corrosive environment and traffic load, the fatigue life of the stayed cables was between 14.5–17.5 years, which is less than the design life of cable-stayed bridges. It can be seen from the figure that in the three corrosive environments, the shortest fatigue life is not the J18 cable with the largest stress, but the A05 and J05 cables. Therefore, more attention should be paid to the fatigue damage of cables A05 and J05 during actual operation. It can be found that the fatigue life of the stay cables varies greatly under the three corrosion conditions, indicating that corrosion has a significant effect on the fatigue life of the stay cables.Figure 13
Fatigue life of the stay cables in different corrosion environments: (a) Low-corrosive environment. (b) Moderately corrosive environment. (c) Highly corrosive environment.
(a)(b)(c)


## 6. Conclusions

An innovative fatigue life evaluation method of parallel steel wire stay cables under the combined actions of corrosion and the traffic load was proposed. The method draws its strength from the development of fatigue probability model for steel wires, based on the nonlinear Lemaitre fatigue damage evolution theory. Taking a cable-stayed bridge on the Yangtze River in China as an example, the fatigue life of the stay cables under the three corrosion conditions and traffic loads was calculated. The conclusions are summarized as follows:(1)
Both the uncorroded steel wire fatigue life model and corroded steel wire fatigue life model proposed in this study, fit the fatigue test data well. Both the stress range and mean stress affect the fatigue life of the wire. Under a given mean stress value, there is a linear relationship between the fatigue life of the steel wire and the stress range in a bilogarithmic coordinate system, and the fatigue life decreases with an increase in the stress range. Under a given stress range, there is a linear relationship between the wire fatigue life and mean stress in a bilogarithmic coordinate system, and the fatigue life decreases with an increase in the mean stress.(2)
The C-S-N surface of the stay cable is parallel to the C-S-N surface of the parallel steel-wires under different guarantee probabilities. Under the same stress level and corrosion conditions, the median fatigue life of the stay cables is less than the fatigue life of a single wire with a guarantee probability of 90%. The fatigue life of the stay cables is determined by a few steel-wires with shorter lives, and stress redistribution accelerates the failure of the inner wire of the stay cable.(3)
Under the combined actions of a highly corrosive or moderately corrosive environment and traffic load, the fatigue life of the cable is less than the 100-year design life of the cable-stayed bridge. In a highly corrosive environment, the shortest fatigue life of the cable is 14.5 years, meaning more attention must be paid to the anticorrosion of the cable during its service period. The cables with the shortest fatigue lives were theA05 and J05 cables in the three corrosive environments.(4)
The fatigue load of the cable in this study only considers the traffic load. More consideration could be given to the influence of temperature and wind load on cable fatigue. Meanwhile, it should be noted that the weight loss rate may not be able to specifically represent the degree of corrosion on stayed cables. A more accurate and reliable corrosion index deserves of further investigation in the future.



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*Source: 5806751-2023-02-16.xml*