# Reliability-Based Design of Driven Piles Considering Setup Effects
## 3 Estimation of the Resistance Factor of Driven Piles
### Considering Setup Effects in the LRFD Principle of Driven Piles  
In order to make the reliability-based design of driven piles conform to the LRFD framework, the principle of strength limit state function corresponding to setup effects is expanded as:  
\[g=R_{0}+R_{\text{setup}}-Q_{\text{D}}-Q_{\text{L}}=0 \tag{9}\]  
As the ultimate resistance of driven piles increases over time after pile driving, the setup effects further affect the resistance of driven piles. Incorporating the setup effect in the LRFD principle of driven piles is remarkable progress. For this, the LRFD standard can be re-expressed as:  
\[\phi\bigl{(}R_{0}+R_{\text{setup}}\bigr{)}\geq\gamma_{\text{QD}}Q_{\text{D}} +\gamma_{\text{QL}}Q_{\text{L}} \tag{10}\]  
The ratio of the setup resistance to the initial ultimate resistance of the driven piles is called the increase factor of the ultimate resistance of driven piles and denoted as \(M_{\text{setup}}\) by Equation (11). \[M_{\text{setup}}=\frac{R_{\text{setup}}}{R_{0}} \tag{11}\]  
Based on Equations (2) and (11), the increase factor \(M_{\text{setup}}\) is re-expressed by:  
\[M_{\text{setup}}=A\log\frac{t}{t_{0}} \tag{12}\]  
It can be seen from Equation (12) that \(M_{\text{setup}}\) is mainly determined by parameters \(A\) and \(\log(t/t_{0})\). The research in [PERSON] and [PERSON] [19] indicated the approximate range of \(A\) from 0.1 to 1 for driven piles in clay. Meanwhile, [PERSON] and [PERSON] [29] also defined the approximate range of \(A\) from 0.1 to 0.9 for driven piles in sand. Comparing the results about parameter \(A\) from [PERSON] and [PERSON] [19; 29], it is reasonable to accept the values of \(A\) from 0.1 to 1 in this study. Furthermore, time \(t\) after EOD for most cases occurs between 1 day and 100 days, for which \(\log(t/t_{0})\) with \(t_{0}=1\) is between 0 and 2. Therefore, compiling the analysis about parameters \(A\) and \(\log(t/t_{0})\), the increase factor \(M_{\text{setup}}\) is determined in the range of 0 to 2 for this study. Then, Equations (13) and (14) are obtained based on Equation (11) and the LRFD principle [28]:  
\[\lambda_{\text{R}}R_{\text{n}}=\lambda_{\text{RD}}R_{0}+\lambda_{\text{Rsetup }}R_{\text{setup}}=\bigl{(}\lambda_{\text{R0}}+\lambda_{\text{Rsetup}}M_{ \text{setup}}\bigr{)}R_{0} \tag{13}\]  
and  
\[\lambda_{\text{R}}R_{\text{n}}=\lambda_{\text{R}}\bigl{(}R_{0}+R_{\text{setup }}\bigr{)}=\lambda_{\text{R}}\bigl{(}1+M_{\text{setup}}\bigr{)}R_{0} \tag{14}\]  
so  
\[\lambda_{\text{R}}=\frac{\lambda_{\text{RQ}}+\lambda_{\text{Rsetup}}M_{ \text{setup}}}{1+M_{\text{setup}}} \tag{15}\]  
where \(\lambda_{\text{RQ}}\) is the bias factors for pile resistance at initial time \(t_{0}\). Substituting Equation (15) into (6), the computed formulas for the resistance factor of driven piles considering setup effects is as follows:\[\phi=\frac{\lambda_{\text{RQ}}+\lambda_{\text{Rsetup}}M_{\text{setup}}}{1+M_{ \text{setup}}}\times(\gamma_{\text{QD}}\rho+\gamma_{\text{QL}})\times\sqrt{\frac {1+\text{COV}_{\text{QD}}^{2}+\text{COV}_{\text{QL}}^{2}}{1+\text{COV}_{\text{ RQ}}^{2}+\text{COV}_{\text{Rsetup}}^{2}}} \tag{16}\]  
In Equation (16), the formation mechanisms of initial ultimate resistance and setup resistance indicate that the development of the total ultimate resistance of driven piles is the result of the synergistic effect of initial ultimate resistance and setup resistance. Therefore, with the interaction between the initial ultimate resistance and setup resistance of driven piles, the inevitable correlation that exists between \(R_{0}\) and \(R_{\text{setup}}\) should be seriously considered in reliability-based design.