1. Introduction
Cable-stayed bridges (CSBs) show excellent merits in terms of their beautiful appearance, great spanning capacity, and fast construction. Hence, CSBs are extensively applied in many states as elements of railway and highway transportation networks [
1] Recently, seismic activity has become more frequent, and many CSBs have been damaged by earthquakes. The seismic behavior of CSBs under earthquake loading and extreme earthquake events has been a significant issue and has been widely studied by researchers from different perspectives [
2]. The increasing deterioration of transoceanic CSBs due to aging and chloride-induced corrosion has led to a decrease in their seismic performance with increasing service life [
3].
Because of uncertainties associated with structures, ground motions, and the corrosion of materials, the seismic performance of corrosive bridges should be investigated using probabilistic analysis methods [
4,
5]. In the probabilistic analysis approach, the seismic reliability analysis (SRA) of structures is the major methodology to evaluate the probability that the structural seismic need does not surpass the related structural ability [
6]. The SRA of long-span CSBs with multiple components and multiple failure models has always been a great challenge to scholars and engineers. The life-cycle SRA of bridge systems considering the time-dependent characteristics of materials degradation is a crucial and challenging research topic.
At present, most researchers focus on the life-cycle seismic reliability of bridge systems on small-span bridges, for instance, continuous girder bridges and rigid frame bridges. Initially, a system reliability analysis method was developed by Estes and Frangopol, including both ultimate and serviceability limit situations, in order to decrease the life-cycle cost of a deteriorating structure, providing great merits, including a proper evaluation of the assumed risk of failure. Then, Biondini et al. [
7] examined the time evolution of the uncertainty roles related to various coefficients defining the probabilistic structural performance of two CSBs in Italy. This research put forward a universal methodology for the time-varying reliability exploration of concrete structures that suffer diffusive attacks. Ghosh et al. [
8] developed time-dependent seismic fragility curves of multi-span continuous steel girder bridges, accounting for variation in ground motion and corrosion parameters. They illustrated the impact of aging on not only components but also on system reliability.
Li et al. [
9] researched different ground motions in space, and the time dependence and seismic fragility of reinforced concrete bridges corroded by chloride. In probabilistic finite element modeling of different life-cycle time nodes, the time-varying characteristics of chloride corrosion and the non-determinacy associated with the structural, material, and corrosion coefficients of reinforced concrete bridges are considered. After the presentation of a copula technique at the bridge system level by Song et al. [
10], time-variant seismic fragility curves for corroded bridges at the system level were developed, which took realistic time-ranging dependence among component seismic needs into consideration. On the basis of mechanisms of material metamorphism and incremental dynamic changes, the time-evolving seismic needs of parts were acquired in the form of boundary possibility distributions. The classical time-invariant structural design criteria and methodologies need to be revised to account for the proper modeling of a structural system over its entire life-cycle by taking into account the effects of deterioration processes, time-variant loadings, and maintenance and repair interventions, among others [
3]. Ang et al. [
11] proposed that due to uncertainties in material and geometrical properties in the physical models of the deterioration processes; in the mechanical and environmental stressors; and in modeling loads, resistances and load effects, a measure of the time-variant structural performance is realistically possible only in probabilistic terms. In addition, the illustration of concepts and approaches is made on both single bridges and bridge networks. To evaluate the life-cycle performance of a malignant bridge built with FRC piers, Pang et al. [
12] proposed a probabilistic methodology. The role of the corrosive environment in the seismic performance of RC bridges was assessed under uncertainty through the seismic resilience framework.
Following up, Biondini et al. [
13] put forward that for progressive deterioration, the time-variant properties of structures are mostly modeled explicitly and continuously in the time horizon of interest. Given the many uncertainties involved in the prediction, time-variant reliability analysis is often used to consider detrimental effects due to progressive deterioration. Xin et al. [
14] put forward an integrated optimization scheme based on reliability, taking into account the life-cycle cost. Complex and non-linear problems can be conducted using the powerful capacity of Artificial Neural Networks (ANNs), which are carried out to forecast the performance of asphalt pavement on the basis of the training information chosen from the long-term pavement performance plan. Another time-dependent seismic fragility assessment framework was proposed by Li et al. [
15], which takes into account the variable association of random structural coefficients for aging highway bridges subject to non-uniform corrosion attacks induced by chloride.
As discussed above, the life-cycle SRA of bridges is an exciting and critical topic for the safety evaluation of structures. It is still an open challenge affected by multiple uncertainties, especially for complex non-linear systems, such as long-span CSBs. In addition, in the past, there have been few and limited life-cycle SRAs of CSBs, taking into account the double uncertainties of structural and ground motions. On the other hand, during the long servicing period, CSBs affected by the corrosion environment and inevitably suffer dynamic loading action, such as earthquake events. Furthermore, due to the influence of chloride ion-induced corrosion, the stiffness and strength of structural components will be degraded at different service times, and the corresponding seismic resistance performance of structural components will also be weakened. At the same time, as a high-order statically indeterminate structure, the failure of a particular component of the CSB does not mean the collapse of the overall structure, so the life-cycle SRA of the structure system takes into account structural materials degradation has always been a hot and challenging research issue. Therefore, this study aims at promoting practical applications of the SRA theory by developing an efficient life-cycle framework for seismic reliability analysis of complicated bridges taking into account material corrosion and double uncertainties of bridge and ground motions. Based on the OpenSees software, the SRA of a railway CSB was researched. The current research provides a direct framework for the discussion of the probabilistic seismic performance of CSB to consult the seismic probability design.
The manuscript is organized below.
Section 2 sums up the fundamental theory and methodology of seismic reliability analysis, such as the multiplier dimensional reduction methods (MDRM), the maximum entropy approach based on the fractional moments (FM-MEM), and the product of the conditional marginal (PCM) approach.
Section 3 introduces the basic theory of the principal component time-varying model. The finite element model is based on OpenSees, the time-dependent models of principal components.
Section 4 summarizes the failure modes of vulnerable components. The probability density function (PDF) of structural seismic response and failure possibility of components and systems at a different servicing time can be found in
Section 5. Some critical conclusions are summarized in the final section.
6. Conclusions
In this study, considering material corrosion and degradation, the sample of non-linear time history analysis of the CSB was determined at the different servicing times with uncertainties of the structure and ground motions, and then based on the OpenSees batch program to carry out a large number of numerical calculations to obtain the time-dependent non-linear seismic response and PDF of seismic response via the MDRM and FM-MEM. Next, the time-varying failure probabilities of each component and the association coefficients between different failure modes of each component were acquired. In the end, the life-cycle failure possibility of the system is acquired with the PCM approach. The following conclusions were drawn from this investigation.
(1) In general, the failure probabilities of the three components in the longitudinal and transverse directions under different ground motions have prominent discrete characteristics, which indicates that the randomness of the ground motions has a significant influence on the failure probabilities of structural. Comparing and analyzing the failure probability of the three components at each servicing time in two directions, the failure probability of the bearings does not significantly change much with the servicing time. In contrast, the failure probability of piers and towers at the different service times increased with the servicing time. In detail, the mean values of failure probability of pier at 20 years, 45 years, 70 years, 90 years, and 100 years increased by 51.94%, 95.76%, 111.78%, 138.48%, and 156.25% in the longitudinal direction compared to the initial time, respectively. In the transverse direction, the failure probability of piers has increased by 230.85% at servicing 100 years compared to the initial time. Due to the failure possibility of the tower, the influence of the corrosive environment is also very significant.
(2) Under 30 ground motion excitations, the correlation coefficient divergence of the components at different servicing time is small, the correlation coefficient curves almost overlap, and the average value also fluctuates around 0.50. Whether in the longitudinal or transverse direction, the correlation coefficients of the components gradually decrease with increasing service time. concerning specific values, the mean value of the association coefficient of the bearing-pier in a longitudinal direction from 0.4617 at the initial time reduces to 0.4501 at servicing 100 years, with a reduction rate of 2.52%. The average value of the component correlation coefficients is not sensitive to changes in servicing time, whether it is longitudinal or transverse direction.
(3) Comparing and analyzing the failure possibility of the series, parallel, and hybrid systems at each servicing time in the longitudinal and transverse direction, the failure probability of the series system gradually increases from the initial probability value of 0.5635 to 0.7654, 0.8497, 0.8975, 0.9040 and 0.9119 corresponding to 20, 45, 70, 90 and 100 years in the longitudinal direction, respectively. For the series system, the failure probability in the transverse direction increases gradually from 0.5533 at the initial time to 0.9441 at 100 years of service. For the parallel and hybrid systems, the system failure probability in the longitudinal direction increases from 0.0031 and 0.0562 at the initial time to 0.0431 and 0.1239 at 100 years of service, respectively. For the parallel and hybrid systems, the failure probability values increase from 0.0035 and 0.0560 at the initial time to 0.0433 and 0.1380, respectively, at 100 years. According to the three system models, the system failure possibility of the CSB in a corrosive environment increases significantly both in the longitudinal and transverse directions.
The fatigue characteristics of reinforced concrete structures are not considered in this study, and in-depth research on this aspect will be conducted in future work.