# Risk-Cost Optimized Maintenance Strategy for Steel Bridge Subjected to Deterioration
## 7 Working Example
For shear failure, \(V(t)\) was determined by Equations (13), (23) and (24) based on the geometry \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Symbol** & **Parameter** & **Statics Distribution** & **Mean** & **COV** & **Resource** \\ \hline \(C_{0}\) & Surface chloride content & & 0.8\% & 0.3 & \\ \hline \(D_{f}\) & Diffusion coefficient & & 31.5 & 0.2 & [38] \\ \hline \(K\) & Model parameter of corrosion & & 0.0802 mm & 0.15 & [41] \\ \hline \(n\) & Model parameter of corrosion & & 0.593 & 0.01 & \\ \hline \(b_{f}\) & Flange width at \(t=0\) & & 229 mm & 0.02 & \\ \hline \(d_{f}\) & Flange thickness at \(t=0\) & & 19.6 mm & 0.02 & \\ \hline \(b_{w}\) & Web length at \(t=0\) & & 572 mm & 0.02 & \\ \hline \(d_{w}\) & Web thickness at \(t=0\) & & 11.9 mm & 0.02 & \\ \hline \(c\) & Cover thickness of slab & & 30 mm & 0.02 & \\ reinforcement & & & 80 per day & 0.05 & Department of Railway \\ \hline \(f\) & Train frequency & & 210 GPa & 0.02 & Transportation \\ \hline \(E\) & Elastic modulus of girder steel & & & \\ \hline \(\sigma_{\text{swap}}\) & Yield strength of steel at \(t=0\) & & 320 MPa & 0.02 & \\ \hline \(q\) & Imposed load on girder & & 0.92 kN/m\({}^{2}\) & - & \\ \hline \(P_{1}\) & Axle load for a motor car & & 118 kN & - & \\ \hline \(P_{2}\) & Axle load for a trailer car & & 108.5 kN & - & \\ \hline \(l\) & Total length of the girder & & 12 m & - & \\ \hline \(\text{C}_{0}\) & Critical threshold chloride & & 0.05\% & - & [38] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Values of basic variables. Figure 5: Axle load system on each of the four tracks [2]. of web and yield strength of the girder summarized in Table 1. \(V_{s}\) was determined by Equation (29). For deflection failure, \(\Delta_{S}(t)\) was determined by Equations (23) and (30), and \(\Delta_{R}\) was determined as 0.019 m (\(l/640\)). For fatigue failure \(D(t)\), \(n(t)\) were determined by Equation (17) based on the load frequency. \(A(t)\) and \(B(t)\) were determined by Equations (25)-(27). \(E[S(t)]\) was determined by Equations (18)-(20), and (26) based on parameters in the corrosion model, sectional geometries, the maximum and minimum bending moment for girder. For chloride attack \(C(t)\) was determined based on Equation (22). The statistics of \(M(t),\ V(t),\ \Delta_{S}(t),D(t),C(t)\) were obtained using Monte Carlo simulation. A sample size of 1,000,000 was found to achieve a reasonable accuracy (convergence) in simulating each parameter [2]. With variables given in Table 1 and the above preparations, the probability of failure of different failure modes was determined by [PERSON] and [PERSON] [29] over time. The probability of failure of each failure mode was the same for the two girders as they were subjected to identical loading conditions. The corrosion effect on sectional properties and mechanical properties at any point of time was related to the corrosion effect in previous times [37]. Therefore, there was a high autocorrelation among the corrosion effects at each point in time, and autocorrelation coefficient \(\rho\) was selected as 0.9. It can be seen from Figure 6 that probability of failure for fatigue was the highest at each time point. This indicated that fatigue failure is the most critical failure mode for the bridges, and more care should be taken to protect the girder from fatigue damage. The changes in probability of failure for different bridge components (i.e., girders and deck) and the entire bridge system were then determined based on Equations (2)-(5), with the results shown in Figure 7.  
Figure 6: Probability of failure for different failure modes. Figure 7: System probability of failure of the bridge and its components. Development of maintenance strategiesA risk-cost program was used to minimize the risk based on Figure 4 and a developed MATLAB code. The program determined the number and period of maintenance that minimized the total risk of steel structures. As inputs for the program, the lifetime of steel structures \(t_{L}\) was 60 years.