# Optimization of Uplift Piles for a Base Plate Considering Local Anti-Floating Stability
## 3 Verification Analysis
### Variation of Total Strain Energy and Maximum Differential Deformation  
By conducting the BESO program and the numerical calculation, the pile lengths of Case 1 and Case 2 are optimized, respectively. Figure 6a,b are the varied history of the maximum differential deformation and the total strain energy of the base plate, respectively. When n is the number of iterations, \(\Delta u\) is the maximum differential deformation, and C is the total strain energy. Initially, the total volume of pile elements in both cases is V\({}_{0}\). For Case 1, the residual volume of pile elements is 0.9 V\({}_{0}\) after 12 iterations (i.e., 10% total volume of the pile elements is reduced). After 21 iterations, the residual volume is 0.8 V\({}_{0}\) (i.e., 20% total volume of the pile elements is reduced). The total strain energy C of Case 1 gradually decreases and becomes stable as the iterations increase. Eventually, the iterative process converges. The maximum differential deformation \(\Delta u\) oscillates in the initial iterations, then decreases rapidly and eventually stable as the iterations increase. For Case 2, the residual volume of pile elements is 0.9 V\({}_{0}\) after 7 iterations and 0.8 V\({}_{0}\) after 16 iterations. However, for Case 2, the variation characteristics of C and \(\Delta u\) are different from those in Case 1. The value of C and \(\Delta u\) are small and oscillate as the iterations increase. It is indicated that although the BESO process reduces the total volume of the pile elements in Case 2, C and \(\Delta u\) do not decrease significantly with the iteration process and remain oscillating at a low level. Figure 7 shows the vertical deformation contour of the base plate with different residual pile elements volumes in the two cases. Figure 7 has been standardized; the average deformation value of the base plate is used as the origin of the ordinate in each residual volume condition. For Case 1, when it is not optimized (i.e., residual volume is V\({}_{0}\)), the vertical deformation of the base plate is large in the middle and small in the periphery. As the iterations increase and the residual volume decreases, the maximum \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Material Properties** & **Pile** & **Base Plate** & **Soil** & **Interface Element** \\ \hline Elastic modulus (GPa) & 25 & 25 & 0.1 & / \\ Poisson’s ratio & 0.2 & 0.2 & 0.3 & / \\ Friction angle (\({}^{\circ}\)) & / & / & 20 & 20 \\ Cohesion (kPa) & / & / & 30 & 30 \\ Tangential stiffness (kN/m) & / & / & / & \(1\times 10^{5}\) \\ Normal stiffness (kN/m) & / & / & / & \(1\times 10^{5}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of numerical model. Figure 6: The varied history: (a) maximum differential deformation; (b) total strain energy. deformation in the middle gradually decreases, and minimum deformation in the periphery gradually increases, and the base plate gradually becomes flat. However, the positions of the maximum and minimum vertical deformation are basically fixed and do not change with the iterative process. Under the three residual volume conditions, the maximum differential deformation \(\Delta u\) is 16.09 mm, 8.56 mm, and 3.90 mm, respectively. The total strain energy C is 1.26 \(\times\) 10\({}^{4}\) J, 9.12 \(\times\) 10\({}^{3}\) J, and 8.08 \(\times\) 10\({}^{3}\) J, respectively. It indicates that the BESO process can significantly reduce the deformation degree of the base plate in that case by optimizing the distribution of pile lengths. However, for Case 2, the base plates are relatively flat under the three residual volume conditions; the positions of the maximum and minimum vertical deformation will change with the iteration process. It confirms that the deformation of the base plate maintains a relatively flat level and oscillates during the iteration process in that case. In general, under the same residual volume conditions, the total strain energy C and maximum differential deformation \(\Delta u\) of the base plate in Case 2 are smaller than that in Case 1.