# Risk-Cost Optimized Maintenance Strategy for Steel Bridge Subjected to Deterioration
## 4 Failure Mode  
As mentioned previously, a performance criterion should be established to assess the risk of failure. In the structural reliability theory, this criterion can be expressed in the form of limit state function. The limit state functions for a bridge girder and deck under different performance criteria (failure modes) are presented in this section. _Flexural failure (Girder)._ The limit state function can be written as follows for flexure failure. \[G(M_{R},M_{S},t)=M_{R}(t)-M_{S} \tag{10}\]  
where \(M_{R}(t)\) is the flexural resistance of steel section at time \(t\) and \(M_{S}\) is the flexural moment due to the load effect, which is treated as a deterministic parameter in this research. Moreover, \(M_{R}(t)\) and \(M_{S}\) are resistance and load effect at the mid-span. \(M_{R}(t)\) is computed based on Australian Standard (AS 5100.6 2017):  
\[M_{R}(t)=\sigma_{ye}(t)Z_{e}(t) \tag{11}\]  
where \(Z_{e}(t)\) is the effective section modulus of mid-girder of the span, and \(\sigma_{ye}(t)\) is the yield strength of the girder at mid-span at time \(t\). The bending moment \(M_{S}\) is case-specific. The value is presented in Section 7 for the working example. _Shear Failure (Girder)._ The limit state function can be written as follows for shear failure. \[G(V_{R},V_{S},t)=V_{R}(t)-V_{S} \tag{12}\]where \(V_{R}(t)\) is the shear resistance of girder at time \(t\). \(V_{S}\) is the shear force due to load effects. \(V_{R}(t)\) is calculated as follows (AS 5100.6 2017):  
\[V_{R}(t)=0.6\sigma_{ye}(t)A_{w}(t) \tag{13}\]  
where \(A_{w}(t)\) is the steel web section area at time \(t\). The shear force \(V_{S}\) is also case-specific with values shown in Section 7. Deflection Failure (Girder)For deflection failure, the limit state function is calculated as follows:  
\[G(\Delta_{R},\Delta_{S},t)=\Delta_{R}-\Delta_{S}(t) \tag{14}\]  
where \(\Delta_{R}\) is the allowable deflection under live load. According to Australian Standard (AS5100.2 2017), maximum mid-span deflection under live load should not exceed 1/640 of the span length. \(\Delta_{S}\) is the deflection at the middle of the span due to live load effect at time \(t\). \(\Delta_{S}\) is case-specific, which can be calculated based on the elastic modulus of steel, the second moment of inertia of girder cross-section, corrosion rate, and time. Fatigue Failure (Girder)For fatigue failure, the limit state function is calculated as follows:  
\[G(D,\Delta,t)=D(t)-\Delta \tag{15}\]  
where \(\Delta\) is the threshold that is equal to 1, \(D(t)\) is the Miner's damage accumulation index at time \(t\), which can be determined as follows [37]:  
\[D(t)=\frac{num(t)}{A(t)}E\Big{[}S_{s}(t)^{B(t)}\Big{]}-\Delta \tag{16}\]  
where \(A(t)\) and \(B(t)\) are the fatigue strength coefficient and fatigue strength exponent. \(S_{s}\) is the stress range. The detailed explanations of \(A\), \(B\) and \(S_{s}\) are given in Li et al. (2020). \(num(t)\) is the total number of fatigue stress cycles applied during time \(t\), which can be determined as follows [37]:  
\[num(t)=f_{load}t \tag{17}\]  
where \(f_{load}\) is the load frequency in the lifetime and \(t\) is time in years. \(E[S_{s}(t)^{B(t)}]\) is the expected value of \(S_{s}{}^{B}\) at time \(t\), which can be calculated as follows [37]:  
\[E[S_{s}(t)^{B(t)}]=(\sqrt{2}S_{so}(t))^{B(t)}\Gamma(\frac{B(t)}{2}+1) \tag{18}\]  
where \(S_{so}(t)\) is a statistical parameter at time \(t\), which can be determined as follows:  
\[S_{so}(t)=\sqrt{\frac{\pi}{2}}E[S_{s}(t)] \tag{19}\]  
where \(E[S_{s}(t)]\) is the mean stress effect at time \(t\), which can be either a normal stress effect or a shear stress effect. The mean stress effect can be determined based on the maximum stress and minimum stress in the largest load cycle as follows [36]:  
\[M_{R}(t)=\sigma_{ye}(t)Z_{e}(t) \tag{11}\]0  
where \(\sigma_{max}(t)\) and \(\sigma_{min}(t)\) are the maximum and minimum stress on the girders.