# Incorporating Setup Effects into the Reliability Analysis of Driven Piles
## 3 Estimation of the Reliability Index of Driven Piles
### Setup Effect in Reliability Evaluation of Driven Piles  
When the setup effect is incorporated into the driven pile design, the limit state function can be expressed as  
\[g=R_{0}+R_{\text{setup}}-Q_{\text{D}}-Q_{\text{L}}=0 \tag{5}\]  
In this work, the increase factor for the ultimate resistance is defined as the proportion of setup resistance to initial ultimate resistance, represented as \(M_{\text{setup}}\) by Equation (6).  
\[M_{\text{setup}}=\frac{R_{\text{setup}}}{R_{0}} \tag{6}\]  
Then, Equation (7) is derived using Equation (6) and LRFD method [29].  
\[\frac{\lambda_{\text{R}}R_{\text{n}}}{\lambda_{\text{QD}}Q_{\text{D}}+\lambda _{\text{QL}}Q_{\text{L}}}=\frac{\left(\lambda_{\text{R0}}+\lambda_{\text{ Setup}}M_{\text{setup}}\right)\text{FOS}(Q_{\text{D}}+Q_{\text{L}})}{ \lambda_{\text{QD}}Q_{\text{D}}+\lambda_{\text{QL}}Q_{\text{L}}}=\left( \lambda_{\text{R0}}+\lambda_{\text{Rsetup}}M_{\text{setup}}\right)\frac{ \text{FOS}(\rho+1)}{\lambda_{\text{QD}}\rho+\lambda_{\text{QL}}} \tag{7}\]  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Reference** & **Equation** & **Comments** \\ \hline [PERSON] and [PERSON] [25] & \(R_{\text{t}}=R_{0}\left(1+A\log\frac{t}{t_{0}}\right)\) & \(t_{0}=1.0\) and \(A=0.6\) in clay; \(t_{0}=0.5\) and \(A=0.2\) in sand; \(R_{\text{t}}\) is the predicted resistance at time \(t\) after driving; \(R_{0}\) is the measured resistance at time \(t_{0}\). \\ \hline [PERSON] et al. [3] & \(R_{\text{t}}=1.1R_{\text{EOD}}\mu\) & Values of \(\alpha\): average = 0.13, lower bound = 0.05, and upper bound = 0.18. \(R_{\text{EOD}}\) is the measured resistance at the EOD. \\ \hline [PERSON] et al. [26] & \(R_{\text{t}}=BR_{\text{EOD}}\mu\)\({}^{0.1}\) & Values of \(B\): lower bound = 1.025, and upper bound = 1.4. \\ \hline [PERSON] and [PERSON] [27] & \(R_{\text{t}}=R_{\text{u}}\left[0.2+0.8\left(\frac{t/T_{\text{gg}}}{1+t/T_{ \text{gg}}}\right)\right]\) & \(R_{\text{u}}\) is the ultimate resistance with 100\% of setup realized, \(T_{\text{gg}}\) is the time required to realize 50\% of pile setup. \\ \hline \hline \end{tabular}
\end{table}
Table 1: Empirical equation for predicting setup resistance of driven piles.  
where \(\rho=Q_{\rm D}/Q_{\rm L}\); FOS is the factor of safety.  
The computation formula for the reliability index of driven piles considering setup effects can be obtained by substituting Equation (7) into Equation (4) as follows:  
\[\beta=\frac{\ln\left[\left(\Lambda_{\rm R0}+\lambda_{\rm Rsetup}M_{\rm setup} \right)\frac{\rm FOS(\rho+1)}{\rm QQP}\sqrt{\frac{1+\rm COV_{\rm QD}^{2}+COV_{ \rm QL}^{2}}{1+\rm COV_{\rm R0}^{2}+COV_{\rm Setup}^{2}}}\right]}{\sqrt{\ln \left[\left(1+\rm COV_{\rm R0}^{2}+COV_{\rm Rsetup}^{2}\right)\left(1+\rm COV_ {\rm QD}^{2}+COV_{\rm QL}^{2}\right)\right]}} \tag{8}\]  
As there is an inescapable interplay between \(R_{0}\) and \(R_{\rm setup}\), the connection between \(R_{0}\) and \(R_{\rm setup}\) should be taken into account in reliability analysis. When considering the correlation between \(R_{0}\) and \(R_{\rm setup}\), the reliability index of driven piles is expressed as follows:  
\[\beta=\frac{\ln\left[\left(\Lambda_{\rm R0}+\lambda_{\rm Rsetup}M_{\rm setup} \right)\frac{\rm FOS(\rho+1)}{\rm QQP}\sqrt{\frac{1+\rm COV_{\rm QD}^{2}+COV_ {\rm QL}^{2}}{1+\rm COV_{\rm R0}^{2}+2\rho_{\rm R0,Rsetup}\rm COV_{\rm R0}\rm COV _{\rm Rsetup}+COV_{\rm Rsetup}^{2}}}\right]}{\sqrt{\ln\left[\left(1+\rm COV _{\rm R0}^{2}+2\rho_{\rm R0,Rsetup}\rm COV_{\rm R0}\rm COV_{\rm Rsetup}+COV_ {\rm Rsetup}^{2}\right)\right]}} \tag{9}\]  
where \(\rho_{\rm R0,Rsetup}\) is the correlation coefficient between \(R_{0}\) and \(R_{\rm setup}\).  
The relationship between failure probability and reliability index can be calculated with the following function:  
\[P_{f}=1-NORMDIST(\beta) \tag{10}\]