# 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  
The displacement, the distribution of the subgrade reaction, and the form of the \(p\)-\(y\) curve of the P-BF are similar to those of the PF, according to the \(p\)-\(y\) curve of the PF, recommended by the API specification, and modifying those of P-BF. Under the static force, the \(p\)-\(y\) curve expression in clay recommended by the API specification is expressed by (1). \[\left\{ \begin{array}{l}P=\frac{P_{ult}}{2}\left(\frac{y}{y_{50}}\right)^{1/3},\,y\leq 8y_{50}\\ P=P_{ult},y\leq 8y_{50}\\ y_{50}=2.5e_{50}d\end{array} \right. \tag{1}\]  
where \begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Model**} & \multicolumn{2}{c}{**Lateral Ultimate Soil**} & \multirow{2}{*}{**Improvement/\%**} \\  & **Resistance** \(P_{ult}\)/**KN/m** & \\ \hline \hline API & 1065 & - \\ PF (\(d\) = 6 m) & 1150 & - \\ P–BF (\(D/h\) = 18 m/12 m) & 3426 & 197.91 \\ P–BF (\(D/h\) = 24 m/6 m) & 5406 & 370.09 \\ P–BF (\(D/h\) = 24 m/12 m) & 5140 & 346.96 \\ P–BF (\(D/h\) = 24 m/18 m) & 4940 & 329.57 \\ P–BF (\(D/h\) = 30 m/12 m) & 6700 & 482.61 \\ \hline \hline \multicolumn{3}{c}{**(b)**} \\ \hline API & 1173 & - \\ PF (\(d\) = 6 m) & 1540 & - \\ P–BF (\(D/h\) = 18 m/12 m) & 3846 & 149.74 \\ P–BF (\(D/h\) = 24 m/6 m) & 5433 & 252.79 \\ P–BF (\(D/h\) = 24 m/12 m) & 6000 & 289.61 \\ P–BF (\(D/h\) = 24 m/18 m) & 5880 & 281.82 \\ P–BF (\(D/h\) = 30 m/12 m) & 8133 & 428.12 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: (a) The increment of \(P_{ult}\) at a depth of 2.25 m; (b) The increment of \(P_{ult}\) at a depth of 3.75 m.  
Figure 14: \(p\)-\(y\) curve of the pile–bucket foundation. \(P=\) actual lateral resistance, kN/m;  
\(P_{ult}=\) ultimate resistance, kN/m;  
\(y=\) actual lateral deflection, m;  
\(y_{50}=\) corresponding lateral deflection of the half lateral ultimate soil reaction force, m;  
\(\varepsilon_{50}=\) strain that occurs at one-half the maximum stress on laboratory unconsolidated undrained compression tests of undisturbed soil samples;  
\(d=\) pile diameter, m;  
\(P_{ult}\) resistance (in the API specification) under static load is calculated as  
\[\left\{ \begin{array}{l}P_{ult}=(3s_{u}+\gamma Z)d+Js_{u}Z,0<Z\leq Z_{r}\\ P_{ult}=9s_{u}d,Z>Z_{r}\\ Z_{r}=\frac{6s_{u}d}{7+Js_{u}}\end{array} \right. \tag{2}\]  
where \(S_{u}=\) undrained shear strength for undisturbed clay soil samples, kPa;  
\(\gamma=\) effective unit weight of soil, kN/m\({}^{3}\);  
\(J=\) dimensionless empirical constant with values ranging from 0.25 to 0.5 having been determined by field testing; a value of 0.5 is appropriate for normally consolidated clay;  
\(Z_{r}=\) depth below soil surface to bottom of reduced resistance zone, m.  
The point of first intersection of the two equations in Formula (2) is taken to be \(Z_{r}\). These empirical relationships may not apply where strength variations are erratic. In general, the minimum value of Zr should be about 2.5 pile diameters. The remaining parameters are consistent with the above. It can be obtained for (1) and (2) that \(P_{ult}\), \(y_{50}\), \(P\) and \(y\) are the key parameters for modifying the \(p\)-\(y\) curve of the P-BF. Combined with the results of ABAQUS, the \(p\)-\(y\) curve of the P-BF can be obtained by modifying the above parameters with reference to the method of large-diameter wingel pile \(p\)-\(y\) curve modification mentioned by [PERSON] [25] et al. For the \(p\)-\(y\) curve at different depths of the P-BF obtained by ABAQUS simulations, (\(d=6\) m \(h=12\) m), the ultimate resistance of the foundation is the platform section in the \(p\)-\(y\) curve. The ultimate resistance of the P-BF was calculated and is recorded in Table 8. The ultimate resistance of the P-BF is significantly increased with an increasing depth, and it is significantly higher than that of the PF. At the same time, a model was built to determine the relationship between the pile diameter and the reinforcement bucket diameter.