# Reliability-Based Design of Driven Piles Considering Setup Effects
## 3 Estimation of the Resistance Factor of Driven Piles  
### General LRFD Principle of Driven Piles  
In order to conduct reliability-based design of driven piles, the limit state equation should be established first. If the total ultimate resistance of driven piles (\(R\)) and the load effect (\(Q\)) are considered, the limit state equation can be expressed as:  
\[g=R-Q=0 \tag{3}\]  
Because the LFRD method can well explain the uncertainty in geotechnical engineering, this method is often used for the reliability-based design of driven piles. The basic formula for the limit state design by AASHTO is [28]:  
\[\phi R_{\text{n}}\geq\eta\sum\gamma_{i}Q_{i} \tag{4}\]  
where \(\phi\) is the resistance factor; \(R_{\text{n}}\) is the nominal resistance; \(\eta\) is the load modifier to account for effects of ductility, redundancy, and operational importance; \(Q_{i}\) is the load effect and \(\gamma_{i}\) is the load factor. The value of \(\eta\) generally is obtained as 1.  
The load effect \(Q\), herein, only includes the combination of the dead load \(Q_{\text{D}}\) and the live load \(Q_{\text{L}}\), and the LRFD criterion can be expressed as:  
\[\phi R_{n}\geq\gamma_{\text{QD}}Q_{\text{D}}+\gamma_{\text{QL}}Q_{\text{L}} \tag{5}\]  
Therefore, the resistance factor \(\phi\) can be estimated as [28]:  
\[\phi=\frac{\lambda_{\text{R}}(\gamma_{\text{QD}}\rho+\gamma_{\text{QL}}) \times\sqrt{\frac{1+\text{COV}_{\text{QD}}^{2}+\text{COV}_{\text{QL}}^{2}}{1+ \text{COV}_{\text{R}}^{2}}}}{(\lambda_{\text{QD}}\rho+\lambda_{\text{QL}}) \times\exp\biggl{\{}\beta\tau\sqrt{\ln\Bigl{[}\Bigl{(}1+\text{COV}_{\text{R}} ^{2}\Bigr{)}\Bigl{(}1+\text{COV}_{\text{QD}}^{2}+\text{COV}_{\text{QL}}^{2} \Bigr{)}\Bigr{]}\biggr{\}}\biggr{.}}} \tag{6}\]  
where \(\lambda_{\text{R}}\), \(\lambda_{\text{QD}}\), and \(\lambda_{\text{QL}}\) are the bias factors for resistance, dead load, and live load, respectively; \(\text{COV}_{\text{R}}\), \(\text{COV}_{\text{QD}}\), and \(\text{COV}_{\text{QL}}\) are the coefficients of variation (COVs) for resistance, dead load, and live load, respectively. The following relevant statistics for \(Q_{\text{D}}\) and \(Q_{\text{L}}\) are accepted for further study: \(\lambda_{\text{QD}}\) = 1.080, \(\lambda_{\text{QL}}\) = 1.150, \(\sigma_{\text{QD}}\) = 0.140, \(\sigma_{\text{QL}}\) = 0.207, \(\text{COV}_{\text{QD}}\) = 0.130, and \(\text{COV}_{\text{QL}}\) = 0.180 [28]. Meanwhile, the distributions of \(Q_{\text{D}}\) and \(Q_{\text{L}}\) are assumed to be lognormal.  
The corresponding factor of safety (FOS) in allowable stress design (ASD) can be calculated as follows:  
\[FOS=\frac{\gamma_{\text{QD}}\rho+\gamma_{QL}}{\phi(1+\rho)} \tag{7}\]Substituting Equation (6) for (7), the computed formulas for the factor of safety for driven piles are as follows:  
\[FOS=\frac{(\lambda_{\text{QD}}\rho+\lambda_{\text{QL}})\times\exp\biggl{\{}\beta _{\text{T}}\sqrt{\ln\Bigl{[}\Bigl{(}1+\text{COV}_{\text{R}}^{2}\Bigr{)}\Bigl{(} 1+\text{COV}_{\text{QD}}^{2}+\text{COV}_{\text{QL}}^{2}\Bigr{)}\Bigr{]}\Bigr{\}} \biggr{\}}}{\lambda_{\text{R}}(1+\rho)\times\sqrt{\frac{1+\text{COV}_{\text{QD }}^{2}+\text{COV}_{\text{QL}}^{2}}{1+\text{COV}_{\text{R}}^{2}}}} \tag{8}\]