# Risk-Cost Optimized Maintenance Strategy for Steel Bridge Subjected to Deterioration
## 1 Introduction
and mechanical properties (e.g., yield strength, S-N curve, etc.). It is thus vital to consider both degradation impacts in risk-cost optimization of the deteriorated bridge [23]. Most existing works have only considered the corrosion-induced changes in sectional properties, as the cases in [PERSON] et al. [24], [PERSON] et al. [25], and [PERSON] et al. [26]. Nonetheless, [PERSON] et al. [27] suggested that there can be a 20.9% loss in yield strength of the steel corroed in seawater conditions with a 60% reduction in weight loss. [PERSON] et al. [23] tested the mechanical properties of corroded steel and the results suggested that there can be a 6.1% loss in yield strength when corrosion loss increases to 1.36 mm. Furthermore, [PERSON] and [PERSON] [28] suggested that ignoring the reduction in fatigue strength of steel due to corrosion can result in a 57.5 times probability of underestimation of failure. This underestimation indicates that the degradation in mechanical properties of corroded steel should not be ignored in risk-cost optimization and maintenance strategy development. Thirdly, few studies have used the first passage probability in failure prediction and risk-cost optimization of deteriorated bridges. In detail, bridge deterioration, whatever the causes, is a time-by-time process of accumulation. The autocorrelation of deterioration at each point of time is high [28; 29]. To consider this autocorrelation, the first passage probability theory can be used for the failure prediction of deteriorated bridges [29] and the subsequent risk-cost optimization works. Mathematically, first passage probability theory determines the probability of failure of each failure mode by estimating the mean outcrossing rate (up- or down-crossing rate) of a stochastic process (e.g., flexure moment, shear force, deflection, fatigue damage, and chloride concentration) from a threshold and obtaining a solution to the so-called \"first passage probability\" [17]. In calculating the mean outcrossing rate, the autocorrelation of stochastic processes within time is determined [17]. Recent studies have used the first passage probability on failure prediction of bridges, e.g., the prediction probability of fatigue failure carried out by [PERSON] and [PERSON] [28], but they did not go one step further for risk-cost optimization and maintenance plan development. This paper intends to fill the research gaps mentioned above and develop a bridge maintenance strategy based on risk-cost optimization. Five common failure modes are considered for the deteriorated bridge: flexure, shear, deflection, fatigue failure for the girder, and chloride attack for the concrete deck. The load effect of each failure mode is modeled as a stochastic process to consider the deterioration. The degradation effect of corrosion on the mechanical properties of bridge steel is considered in the modeling. The first passage probability theory and system reliability analysis theory are used for the time-dependent reliability of bridges considering multiple failure modes of different bridge components. A risk-cost optimization program is then applied to the bridge to develop an optimum maintenance plan based on the reliability analysis outcome. A working example is conducted to demonstrate the application of the proposed maintenance strategy on an existing bridge. The significance of the research is in advancing the knowledge in the failure prediction and maintenance of deteriorated structures.