# Seismic Performance Evaluation of Highway Bridges under Scour and Chloride Ion Corrosion
## 2 Earthquake and Scour Combination in the Failure Probability of Bridge
### Application for a Practical Project
#### 2.2.6 Solution of Structural Failure Probability and Reliability Index  
When the distribution function of seismic effect and the distribution of the sectional resistance of the pile foundation under different scour depths were identified, the failure probability and reliability index of the pile foundation could be precisely calculated by means of direct integration. The failure probability is as follows.  
Figure 16: Cumulative probability density function fitting for the seismic bending moment of the pile foundation in the transverse direction under different scour depths.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Average Value \(\mathbf{\eta_{\text{R}}}\)/(kN\(\cdot\)m)** & **Standard Deviation \(\sigma_{\text{R}}\)/(kN\(\cdot\)m)** & **Probability Distribution** \\ \hline
33,548.8 & 4743.8 & Lognormal distribution \\ \hline \hline \end{tabular}
\end{table}
Table 3: Moment resistance value of the pile foundation section.  
\[P_{\text{f}}=P(S>R)=\int_{-\infty}^{+\infty}F_{\text{R}}(x)f_{\text{S}}(x)dx=\int_{ -\infty}^{+\infty}[1-F_{\text{S}}(x)]f_{\text{R}}(x)dx \tag{9}\]  
where \(F_{\text{S}}(x)\) is the distribution function of the seismic effect; \(f_{\text{R}}(x)\) is the function of resistance probability density.  
The solution of the above integral makes the data on failure probability and the reliability index of pile foundation elements under seismic action in the condition of different scour depths accessible, which were non-linearly fitted, respectively, to determine the failure probability of pile foundations suffering from seismic action under different scour conditions, as shown in Figures 17 and 18. It can be observed that the failure probability climbed with the increase of scour depth while the reliability index declined, both indicating that the seismic performance of the pile foundation will weaken due to scour action.  
The fitted nonlinear relationship between the scour depth and reliability index is  
\[\beta(H)=5.5094-0.29362H-0.06214H^{2}+0.01315H^{3}-6.98177E-4H^{4} \tag{10}\]  
The nonlinear relationship between the scour depth and failure probability is  
\[P_{\text{f}}(H)=1-\Phi(\beta(H)) \tag{11}\]  
where \(\beta(H)\), \(P_{\text{f}}(H)\) are the reliability index and the function of failure probability, respectively, with respect to the scour depth \(H\); \(\Phi(\cdot)\) is the function of standard normal distribution.  
Figure 17: Relationship between scour depth and failure probability of the seismic action of the pile foundation.  
Figure 18: Relationship between scour depth and reliability index of the seismic action of the pile foundation.  
The joint probability density between the scour depth and seismic action is calculated and the average failure probability under the joint probability density is solved as  
\[\overline{P_{\rm f}}=\int_{0}^{H_{max}}f_{\rm H}(H)P_{\rm f}(H)dH \tag{12}\]  
where \(f_{\rm H}(H)\) is the function of probability density related to scour depth \(H\); \(P_{\rm f}(H)\) is the failure probability corresponding to scour depth \(H\).  
The integration of the above equation gives the expression of average failure probability as  
\[\overline{P_{\rm f}}=9.79\times 10^{-5} \tag{13}\]  
The failure probability is converted into the reliability indicator as  
\[\overline{\beta}\ =\ 3.7244 \tag{14}\]