# Risk-Cost Optimized Maintenance Strategy for Steel Bridge Subjected to Deterioration
## 2 Formulation of Maintenance Strategy  
The bridge system consists of many components (e.g., deck, girder, etc.). A component can fail in many modes (namely, limit state). This paper focused on a steel-concrete composite bridge with the configuration shown in Figure 1, which is a common bridge type [30]. The failure of this bridge system is logically expressed in Figure 2. The system was modeled as a series system because maintenance needs to be performed when there is a violation of the limit state for any failure modes [28]. It is worth noting that bridge decks are usually overdesigned [31,32]. Thus, their flexure, shear deflection, and fatigue failure modes were not considered. Additionally, deterioration is normally inspected on bridge decks and girders for this bridge type [2]. Hence, the failure of bridge piers was not considered in this study. Figure 1: Configuration of the bridge in the working example (Unit: mm). (**a**) Girder system, (**b**) Cross-section of girder. The maintenance strategy for the bridge was developed by combining the concepts of system reliability and optimization. The justification for the maintenance strategy was that the maintenance will only be carried out when the probability of failure determined by any failure modes is greater than the acceptable limit, and the probability of whole bridge system failure is kept below the acceptable failure mode [16]. Considering this justification, the merit of developing this maintenance strategy was to minimize, if not eliminate, the number of maintenance actions without affecting the safety of the bridge. The problem can then be expressed mathematically as follows [17]:  
\[\textit{Minmize Risk}=\sum_{i=1}^{N_{r}}\sum_{j=1}^{N_{c}}\sum_{k=1}^{N_{m}}p _{fs,s}(t_{i})\times C_{jk}(t_{i})\]  
\[\textit{Subject to }p_{f}(t_{j})\leq P_{f,a}\]  
\[p_{fs,s}(t_{j})\leq P_{fs,sa}\]  
\[0\leq t_{j}\leq t_{L} \tag{1}\]  
where \(t_{i}\) is the maintenance time sequence with \(i\) referring to the number of times, \(j\) to structural component, and \(i\) to failure mode. \(N_{r}\) is the number of maintenance actions, \(N_{c}\) is the number of components, and \(N_{m}\) is the number of failure modes for each component. \(C_{jk}\) is the cost (including interest rate) of failure for the \(j\)th component due to the \(k\)th failure mode. \(p_{f}\) and \(P_{f,a}\) are the probability and acceptable probability of each failure mode. \(p_{fs,s}\) and \(P_{fs,sa}\) are the probability of failure and the acceptable probability of failure for the bridge system. Both \(P_{f,a}\) and \(P_{fs,sa}\) were determined as 0.011% based on AS 5100.1 [33]. \(t_{L}\) is the lifetime of the structure. The optimization (Equation (1)) finds the number of maintenances \(N_{r}\) and the time of each maintenance, i.e., \(t_{i}\). The most influential component within the system can also be determined at each maintenance time through this optimization. To be specific, the probabilities of failure of different components were determined and ranked (shown later in Equation (5)) in this optimization process. The component with the highest probability of failure was treated as the critical component, as bridge structure was considered a series system in this study. The maintenance action was only performed on the critical component and the cost of failure related to the repair for this component. This is shown schematically in Figure 3. Furthermore, the failure mode with the highest failure probability was repaired for the critical component. Figure 2: Schematic diagram of the bridge system. The probability of bridge failure \(p_{f}\) for each failure mode in Equation (1) can be determined as follows:  
\[p_{f}(t)=P[G(S,R,t)\leq 0]=P[R(t)\leq S(t)] \tag{2}\]  
where \(P\) denotes the probability of an event. \(S(t)\) is the bridge response and \(R(t)\) is the structural resistance. \(G\) is the limit state function, which is \(G(S,R,t)=R(t)-S(t)\). Equation (2) represents a typical down-crossing problem, which can be determined using first passage probability theory [29].