# Risk-Cost Optimized Maintenance Strategy for Steel Bridge Subjected to Deterioration
## 1 Introduction  
Bridges are critical infrastructures that play an important role in transport systems, and the deterioration of bridges has become a global issue [1]. Deterioration is the change to the mechanical properties or the geometric properties of bridge components that affect the bridge's structural performance. There are many causes of deterioration, including corrosion, fatigue, overloading, etc. [2]. Deterioration can lead to capacity loss, remaining life reduction, and bridge failure [1, 3]. [PERSON] and [PERSON] [4] pointed out that 5% of bridges have collapsed worldwide from 1807 to 2007 because of deterioration and lack of maintenance, for example, the Silver Bridge in 1967 [5], Lowe's Motor Speedway in 2000 [6], and the Minneapolis Interstate 35 W Bridge in 2007 [7]. These failures indicate that an effective maintenance strategy is essential for a deteriorated bridge to avoid catastrophic failure. To make maintenance strategy effective, it is essential to minimize the total inspection, repair, and expected failure cost and limit the probability of failure of bridges. Different studies have been carried out to develop the maintenance strategy for bridges, using either the deterministic method or reliability theory. For the deterministic method, [PERSON] and [PERSON] [8] and [PERSON] et al. [9] used condition rating and the rating deterioration model for life cycle performance prediction and maintenance of bridges. [PERSON] et al. [10] and [PERSON] and [PERSON] [11] developed a bridge maintenance strategy through the life cycle cost analysis and discounted cash flow method. [PERSON] [12] incorporated reliability theory and life cycle cost in bridge maintenance. [PERSON] and [PERSON] [13], [PERSON] and [PERSON] [14] and [PERSON] and [PERSON] [15] further developed a maintenance strategy through risk-cost optimization. In this optimization, the deterioration of the bridge is modeled as a stochastic process. Time-dependent reliability theory is used to predict the probability of failure of thebridge. An optimization algorithm is then applied to determine the maintenance strategies that ensure the safe operation of the bridge and minimize the total risk [16]. Risk-cost optimization considers the uncertainties in bridge deterioration (e.g., corrosion rate, fatigue damage). Therefore, it can give a more reliable maintenance strategy compared with the deterministic method [17]. Although risk-cost optimization has been used for maintenance strategy development of deteriorated bridges, there are three major research gaps. Firstly, a bridge contains many components (e.g., girder, decks, etc. ), and each component has multiple failure modes (e.g., flexure, deflection, and fatigue) [2]. However, most studies only consider the failure of a single component in the risk-cost optimization of bridges. Moreover, [PERSON] [2] suggested that bridges can be more vulnerable to fatigue failure than other failure modes. Very limited studies have considered fatigue failure mode in the risk-cost optimization. For example, [PERSON] and [PERSON] [18], [PERSON] and [PERSON] [19], and [PERSON] [20] considered the failure modes (e.g., chloride penetration) only for bridge decks in their risk-cost optimization. The failure modes of other components (e.g., fatigue failure of the bridge girder) have not been considered. [PERSON] and [PERSON] [21] and [PERSON] et al. [22] considered the girders' flexure, shear, and deflection failure modes in the risk-cost optimization. Fatigue failure of girders and the failure modes of decks have not been studied. Based on the system reliability theory, the bridge's probability of failure can be underestimated if any failure modes are not considered. This underestimation can lead to the inefficiency of the developed maintenance strategy [17]. Secondly, corrosion is a common cause of deterioration for steel components (e.g., girder) of the bridge. Corrosion can degrade both sectional properties (e.g., area, effective section modulus, etc.)