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

. Fatigue and corrosion are the main reasons for the failure of stay cables. Trafc load also plays a signifcant role in cable fatigue. To study the fatigue life of stay cables under the combined action of trafc 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. Ten, 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 diferent 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 efects of diferent corrosive environments and measured trafc loads was evaluated. Results indicate that both the stress range and mean stress afect the fatigue life of the steel wire. Te 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. Te 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 trafc load.


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 trafc 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.Te 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 highstrength 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 diferent steel strands or high-strength steel wires based on fatigue experiments of a single steel wire.Castillo et al. [3][4][5][6] proposed a multiparameter Weibull distribution model to describe the fatigue life of highstrength steel wires and strands under diferent stress ranges and verifed the efect 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 specifed 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 highstrength steel wire in the stay cable [10][11][12].Te fatigue test data of corroded steel strands under diferent stress conditions in reference [13] showed that the stress state (stress range and mean stress) had a signifcant efect 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.Te 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.Te 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 diferent degrees of corrosion through fatigue tests.Te 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.Tey constructed a multiparameter Weibull model that can quantitatively evaluate the efect of the degree of corrosion on the fatigue life of steel strands.Te 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.Te efects of corrosion on the fatigue life of stay cables and fatigue life distribution of stay cables under diferent 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 intensifes, and the stay cable sufers from severe fatigue damage.Te aforementioned fatigue life models of the corrosive steel wire and stay cable only consider the efect 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][22][23][24].Te 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][26][27][28]. Currently, the fatigue assessment and life prediction methods of the stay cables can be generally classifed into two categories: datadriven methods and fnite 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][29][30][31]. Unlike the data-driven method, the fnite element model-based approach uses fnite element analysis to indirectly obtain the tension force of cables under diferent loading conditions, and then perform condition assessment and life prediction of the stay cable using suitable fatigue life models [28,[32][33][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 efects of the sequence of trafc load action and the coupling efects of fatigue load and corrosion and other factors.
In view of the research difculty, 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 efect of the external fatigue load and corrosion.Te 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 trafc 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 efects of the sequence of trafc load action and the coupling efects of fatigue load and corrosion and other factors on the fatigue condition.Te remainder of this paper is organized as follows: First, in Section 2, 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.Te 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 diferent guaranteed probabilities is presented.Section 5 considers a cable-stayed bridge on the

Fatigue Life Model for Steel Wire
2.1.Fatigue Life Probabilistic Model for Steel Wire.Te threeparameter 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 diferent failure probability conditions.Tis model only considers the infuence of the stress range on the fatigue life of the steel wire and ignores its actual stress state.Te fatigue test results of the steel wire show that both the stress range and the mean stress afect its fatigue life [32,35,36].Terefore, it is critical to establish a steel wire fatigue model that considers the stress range and pretension.
Te nonlinear Lemaitre fatigue damage evolution model under a one-dimensional stress state can be expressed by the following equation (1) [37,38]: 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): where ∆σ and σ m represent the stress range and mean stress, respectively; B is the material parameter, and B � 2 β+2 × B. Equation ( 2) is integrated over one stress cycle to obtain equation (3): where D i 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.Te present incremental damage is not simply linearly superimposed on the previous damage but exhibits a signifcant nonlinear, which is obviously diferent from the linear fatigue damage model based on the Palmgren-Miner law.Under the specifed stress level, the fatigue life of the steel-wire can be expressed as 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) Under the specifed stress level, the fatigue life cumulative distribution function (CDF) of the steel wire can be expressed as [3] where β 0 and δ(∆σ, σ m ) are the shape and characteristic parameters of the Weibull distribution, respectively.We took the logarithm twice on both sides of the following equation (6): Comparing equation ( 7) with (5), we can assume that where K is a constant.Substituting equation ( 8) into equation (6) gives where K, β 0 , β, and α are unknown parameters that can be estimated using the test data of all the specimens.Tis formula is a Weibull distribution model that characterizes the fatigue life of steel wires under diferent stress levels.Te fatigue life of the steel wire follows a two-parameter Weibull distribution of the shape parameter β 0 and characteristic parameter 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 defned as Structural Control and Health Monitoring Equation ( 9) can be expressed as When the model parameters are known, the ft between the model and fatigue test data can be checked according to the Weibull probability diagram of the intermediate variable V.

Parameter's Estimation of Fatigue Life Probabilistic Model
for Steel Wire.Te 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].Te fatigue tests were performed using a servohydraulic universal testing machine (MTS Landmark).Te 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.Te 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. Te 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.
Based on the test data in Table 1 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.Te 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 signifcance.Te proposed Weibull model for the 1670-grade steel wire fts the fatigue data well.
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:

Stress Efects on the Fatigue Life Model.
Figure 2 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 diferent stress levels compared to the model presented in reference [8].Te referenced model does not consider the efect 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.Te S-N curves of the steel wire are shown in Figure 3, based on the estimated parameters of equation (9).From the fgure, it can be seen that the experimental results exhibit a dispersive phenomenon, which may be attributed to the presence of possible damage.Terefore, 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.Te S-N curves of the steel-wires with diferent 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.Te fatigue life of the steel wire decreased with an increase in the mean stress within the same stress range.Te S-N curves of the steel-wires with diferent stress ranges at a guaranteed probability of 50.0% are shown in Figure 3(b).Te fatigue life of the steel wire has a linear relationship with the mean stress in a bilogarithmic coordinate system under a given stress range.Te fatigue life of the steel wire decreases with an increase in the stress range under the same mean stress.

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 signifcantly afects 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 signifcant than that under no stress.Te corrosion of hightensile steel wire under alternating loading condition is the most signifcant.Terefore, it is meaningful to consider the coupling efect between corrosion and stress conditions to develop the fatigue life model for corroded steel wire.

4
Structural Control and Health Monitoring

Fatigue Life Probabilistic Model for Corroded Steel Wire.
Te 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 specifed stress range.In this study, the degree of corrosion was quantifed based on the weight loss rate w, that is, the average loss of the cross-section of the steel wire.From equation ( 9), the slope of the S-N curve of the steel wire is (β + 3)/2.Terefore, the slope of the S-N curve and parameter K in equation ( 9) can be expressed as where A, B, C, and D are the parameters that can be estimated using the test data.

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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: 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 defned as (17) Equation ( 15) can be expressed as When the model parameters are known, the ft between the model and fatigue test data can be checked according to the Weibull probability diagram of intermediate variable v.

Parameter Estimation of Fatigue Life Probabilistic Model
for Corroded Steel Wire.Tis 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 steel wires.Te corrosion test was carried out according to relevant regulation [40].Te 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 of the corrosion products on the samples.After the measurement of the weight loss, all samples were ready for fatigue tests.
Te fatigue tests were performed using a servohydraulic universal testing machine (MTS Landmark).Te stress ratio   Structural Control and Health Monitoring 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 3. Te unknown parameters in equation ( 16) can be estimated using the MLE based on the fatigue test data [39] 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 Figure 5.Each line in the fgure corresponds to a stress level.Te fatigue life of the parallel steel wire decreases with an increase in the degree of corrosion under the specifed stress level, indicating that the efect of corrosion on the fatigue performance of the steel wire is exceptionally signifcant.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 signifcantly from the theoretical model.Te 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.Terefore, 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%.

Steel Wire Time-Variant Corrosion Model.
Te exponential model is often used to calculate the uniform corrosion depth at diferent 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.Te uniform corrosion depth of each stage can be characterized by an exponential model, as shown in equation ( 19) [34].
where t is the corrosion time of the steel wire; t 0 is the time required for the galvanized layer on the surface of the steel wire to be completely corroded; C r1 and C r2 are the annual corrosion rates of the galvanized layer of the steel wire and the inner steelwire, respectively, and c 1 and c 2 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.Te mass loss of the steel wire at diferent 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 defnition of the weight loss rate of the steel wire mass, the relationship between the weight loss rate and uniform corrosion depth is as follows: where D Fe is the diameter of the steel wire, D Zn is the thickness of the zinc coating, and ρ Fe and ρ Zn 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.Te fatigue life model for the corroded stay cable can then be derived based on equation (21), as described in Section 4. 
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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 of m wires in parallel, and the external load remained constant.When i wires break, the stress of the remaining parallel wires is where ∆σ i and σ i m 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.Tis accelerated the fatigue fracture of the remaining parallel wires until the entire stay cable was destroyed.

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Figure 6 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.Te 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 N ∆σ 1 , the steel wire with the shortest fatigue life will break, and then the stress of the stay cable will be redistributed: Repeat the previous process until the fatigue limit of the stay cable is reached, and the stay cable will fail.
According to the CDF the corroded steel wire in equation ( 21), the fatigue life of the steel wire under diferent stress levels has the same failure probability: 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 12 Structural Control and Health Monitoring When the second wire breaks, the number of load cycles N (2) experienced by the stay cable is By analogy, when the ith steel wire breaks, the number of load cycles N i experienced by the stay cable is

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Substituting equation (24) into equation ( 26): Te 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 diferent corrosion states was calculated based on the Monte Carlo method.Te calculation steps are as follows: (1) According to the PDF of the fatigue life of a single steel wire under a specifed stress level and under diferent corrosion states, 253 groups of samples were randomly generated to simulate the fatigue life of each steel wire in the stay cable.(2) Te samples were arranged from small to large, then the fatigue life of the stay cables under diferent broken wire conditions was calculated according to equation (A.2).
Te previous process can be repeated to obtain the fatigue life distribution samples of stay cables under diferent corrosion states.Te number of fatigue life simulation samples of the corroded stay cables was set to 10,000 for each degree of corrosion.Te 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).
According to the simulation results under diferent corrosion conditions, the corrosion-stress range fatigue life (C-S-N) surfaces for the corroded steel wire and corroded stay cable are shown in Figure 7. Te 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%.Te C-S-N surfaces of the parallel steel wire are parallel to the C-S-N surface of the stay cable under diferent 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%.Tis 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.

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 Figure 8. Tere 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.Te cable numbers are shown in Figure 9. Te number of steel wires in a stay cable increased with the length of the stay cable, ranging from 109 to 253.Te longest stay cable, A18, comprises 253 steel wires of grade 1670 with a diameter of 7 mm.ABAQUS was used to establish the fnite element model, as shown in Figure 10.Te stay cable is simulated using a truss element that only bears tension, and the cable force is applied by defning the initial strain.To consider the infuence of the sag efect 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.Te main girder was simulated using a fshbone beam model.Cross-section windresistant and vertical supports at the pylon location were simulated using spring elements with a specifed stifness.Te established fnite element model has been updated with in-site measurements to ensure its efectiveness, and more details on the model updating can be found in reference [42].

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Weibull process were used to simulate the vehicle arrival process and stochastic trafc process of each lane, respectively.Te Markov chain Monte Carlo (MCMC) method generated the vehicle sequence of each lane at each moment.Finally, a trafc fow stochastic process model was established and verifed.More details on the trafc fow model can be found in reference [42,43].
Based on the fnite element model, the cable force infuence line of any stay cable and the cable force-time history of each stay cable can be obtained by combining the infuence line of each stay cable and the trafc fow stochastic process model.Tis study uses A05, J05, A18, and J18 cables as examples.Te 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 trafc load and dead load, the stress of the A05 cable is between 415.2-511.4MPa, the stress of the J05 cable is between 415.7-484.8MPa, the stress of the A18 cable is between 426.5-457.9MPa, and the stress of the J18 cable is between 490.0-518.7 MPa.As shown in the fgure, compared with the other stay cables, the force of stay cables A18 and J18 at the end of the side span will be signifcantly relaxed under the action of the trafc load on the bridge.
To count the stress range, average stress, and corresponding cycle times of the cable under the combined action of the trafc load and dead load, the rain fow counting method was used to count the stress time history, and the stress spectrum of the cable was obtained, as shown in Figure 12.It can be seen from the fgure 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 trafc load is much less than the stress of the cables under the dead load.

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].Te testing data from Shimen Bridge, Yonghe Bridge, and Williamsburg Bridge were used to verifed the correctness of the model.Te uniform corrosion rate model parameters of steel wires under three diferent corrosion environments are listed in Table 5. Terefore, this paper directly selects the previous corrosive types to evaluate the fatigue life of stay cables.
Te 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 Figure 13.Te fgure shows the fatigue life for each corrosive environment under the current trafc 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 trafc 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 trafc load, the fatigue life of the stay cables was between 33.0-44.0years.Under the combined actions of a highly corrosive environment and trafc 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 fgure 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.Terefore, 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 signifcant efect on the fatigue life of the stay cables.

Conclusions
An innovative fatigue life evaluation method of parallel steel wire stay cables under the combined actions of corrosion and the trafc load was proposed.Te 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 trafc loads was calculated.Te 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, ft the fatigue test data well.Both the stress range and mean stress afect 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) Te 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%.Te 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 trafc 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.Te cables with the shortest fatigue lives were the A05 and J05 cables in the three corrosive environments.(4) Te fatigue load of the cable in this study only considers the trafc load.More consideration could be given to the infuence 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.

A. Parameter Estimation of Fatigue Model for Steel Wire
Te parameters in equation ( 9) can be estimated based on the experimental data in Table 1.Derivation of equation ( 9), the PDF of the fatigue life of steel wire can be obtained as follows: Assume that the fatigue test contains r stress levels σ i m (i � 1, 2, . . ., r), each stress level contains l i stress range ∆σ ij (j � 1, 2, . . ., l i ), and each stress contains n ij the experimental data.Te MLE method is used to estimate the parameters, and the expression of the likelihood function is as follows: Taking the partial derivative of the likelihood function equation (A.2) yields the following nonlinear equations: Te unknown parameters in equation ( 9) can be obtained by solving the previous equations.

B. Parameter Estimation of Fatigue Model for Corroded Steel Wire
Te parameters in equation ( 16) can be estimated based on the experimental data in Table 3. Assume that the fatigue test contains r corrosion degree w i (i � 1, 2, . . .r), each corrosion degree w i contains l i stress level σ

(B.3)
Te unknown parameters in equation ( 16) can be obtained by solving the previous equations.

Figure 3 :
Figure 3: S-N curves of a steel wire.(a) Stress range-life curve.(b) Mean stress-life curve.

Figure 4 :
Figure 4: Weibull plot of variable v for the corroded 1670-grade steel wire.

Figure 6 :
Figure 6: Relationship between fatigue life and stress range.

Figure 5 :
Figure 5: Fatigue life versus corrosion degree for steel wire.
Force.Te author of this study established the vehicle fatigue load spectrum of a specifc section of the bridge through statistical analysis of the trafc fow information monitored by the background engineering bridge weigh-in-motion (WIM) system from the third quarter of 2011 to the fourth quarter of 2015.Ten, the Weibull stochastic process and fltering

Figure 9 :
Figure 9: Number of stay cables.

Figure 7 :
Figure 7: C-S-N surfaces for the corroded steel-wire and corroded stay cable.

Figure 8 :
Figure 8: Te cable-stayed bridge on the Yangtze River.

Figure 13 :
Figure 13: Fatigue life of the stay cables in diferent corrosion environments: (a) Low-corrosive environment.(b) Moderately corrosive environment.(c) Highly corrosive environment.

Table 1 :
Fatigue test results of the 1670-grade steel wire.
*indicates that the experimental data are truncated.StructuralControl and Health Monitoring

Table 2 :
Estimated values of the unknown parameters.
, and the solution details are shown in Appendix B. Te unknown parameter values for the fatigue life probability model proposed in this study are listed in Table4.Based on the test data in Table3and the estimated values of the unknown parameters in Table4, the Weibull plot of intermediate variable v for the corroded 1670-grade steel wires is shown in Figure4.Te samples of intermediate variable v are nearly linear based on the estimated parameters, as shown in Figure4.In addition, the standard Weibull distribution of intermediate variable v can be accepted by the K-S test at a 5% level of signifcance.Te proposed Weibull model for the corroded 1670-grade steel wire fts the fatigue data well.

Table 3 :
Fatigue test results of the corroded 1670-grade steel wire.

Table 4 :
Estimated values of the unknown parameters for steel wire considering corrosion efects.