# A Numerical Analysis on Lateral Resistance of Pile-Bucket Foundation for Offshore Wind Turbines
## 5 Analysis on _p-y_ Curve
### Modification of \(p\)-\(y\) Curves for P-BF
#### 5.3.3 Modified \(P/P_{ult}\)-\(y/y_{50}\) Curves  
The dimensionless \(p\)-\(y\) curve (\(P/P_{ult}\)-\(y/y_{50}\)) is adopted in API to reflect the \(p\)-\(y\) curve of PF under a lateral load. This curve form can reflect the relationship between soil reaction and lateral displacement of PF. Modifying \(P/P_{ult}\)-\(y/y_{50}\) according to the formula given in API is an effective way to quantitatively analyze the \(p\)-\(y\) curve of P-BF. The \(p\)-\(y\) values of P-BF obtained by the finite element simulation are calculated and plotted in a dimensionless way as \(P/P_{ult}\)-\(y/y_{50}\) curves (see Figure 18). It can be found that the \(P/P_{ult}\)-\(y/y_{50}\) of the bucket of P-BF is basically the same as the recommended curve of API, but there are also some differences. Before reaching the lateral ultimate soil reaction, the results of the finite element simulation and API curves are typical power function distributions. However, when the API curve reaches the lateral ultimate soil reaction, the \(y/y_{50}\) value is significantly  
Figure 17: Modified displacement of the pile.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Depth** & & & \\ \hline \multicolumn{2}{c}{**Z/m**} & \multicolumn{2}{c}{**2.25**} & \multicolumn{2}{c}{**3.75**} & \multicolumn{2}{c}{**5.25**} \\ \hline \(d\)/m & \multicolumn{2}{c}{**D/m**} & \multicolumn{2}{c}{**y\({}_{50}\)/m**} & \\ \hline
3 & 18 & 0.07920 & 0.10430 & 0.09713 \\
4 & 18 & 0.07559 & 0.09622 & 0.10450 \\
5 & 18 & 0.08069 & 0.09894 & 0.10996 \\  & 18 & 0.08572 & 0.09659 & 0.11423 \\
6 & 24 & 0.09077 & 0.11274 &higher than the results of the finite element simulation, which leads to a great gap between the secant slope of the recommended curve in the API code and the results of the finite element simulation. Nevertheless, the curves of the current code can still be applied to the P-BF after being modified.  
If we transpose Formula (1) and rewrite the coefficient, we can obtain:  
\[P/P_{ult}=a(y/y_{50})^{\frac{1}{b}} \tag{9}\]  
When the eigenvalue of the pile displacement \(y_{50}\) is half of the ultimate lateral soil reaction, the corresponding lateral pile displacement is as follows. That is, \(y=y_{50}\), \(P=1/2\)\(P_{ult}\), \(a=0.5\), which then yields  
\[P/P_{ult}=0.5(y/y_{50})^{\frac{1}{b}} \tag{10}\]  
Applying a logarithm on both sides of Equation (10) gives  
\[\ln(P/P_{ult})=\ln 0.5+\frac{1}{b}\ln(y/y_{50}) \tag{11}\]  
Based on the ordinal \((y/y_{50}\), \(P/P_{ult})\) data to calculate [\(\ln\) (\(y/y_{50}\), \(\ln\) (\(P/P_{ult}\))], we use the numerical approximation method to obtain the undetermined coefficient \(b\). Thus, the \(p\)-\(y\) curve formula, which is suitable for P-BF, is obtained:  
\[\left\{\begin{array}{l}P=\frac{P_{ult}}{2}\Big{(}\frac{y}{y_{50}}\Big{)}^{1 /2},\ y\leq 3.99y_{50}\\ P=P_{ult},y>3.99y_{50}\end{array}\right. \tag{12}\]  
The other parameters are the same as above.  
The \(P/P_{ult}\)-\(y/y_{50}\) curve is calculated through the revised \(p\)-\(y\) curve formula, and the \(p\)-\(y\) curve at the depths of 2.25 and 3.75 m of the P-BF (taking the four groups of models with a diameter of 18 m as examples) is compared with the results of the ABAQUS finite element simulation, as shown in Figures 18 and 19. It can be seen from the graph that the modified \(p\)-\(y\) curve has satisfactory results. It can basically reflect the relationship between the lateral soil reaction of P-BF and the lateral displacement of the pile under lateral load.  
Figure 18: \(P/P_{ult}\)-\(y/y_{50}\) curve.  
Figure 19: Modified \(p\)-\(y\) curve. (**a**) Depth of 2.25 m. (**b**) Depth of 3.75 m. (**c**) Depth of 5.25 m.