# Optimization of Uplift Piles for a Base Plate Considering Local Anti-Floating Stability
## 2 Optimization Method
### Design Variable and Sensitivity Coefficient  
In the BESO method, the design variable affects the change of the objective function. Therefore, it is necessary to use an appropriate design variable to minimize the objective function.  
The pulling force \(F_{pi}\) is mainly provided by the lateral friction resistance of the pile side and the pile weight. Under the assumption that friction resistance varies linearly with the pile length, \(F_{pi}\) can be expressed as follows:  
\[F_{pi} =2\pi r^{2}L_{i}\rho g+\lambda q_{sk}\pi DL_{i} \tag{2}\]  
where \(r\) is the radius of the pile, \(\rho\) is the material density, \(D\) is the pile diameter, \(L_{i}\) is the pile length, \(\lambda\) is the reduction coefficient, and \(q_{sk}\) is the standard value of the limited lateral resistance of the pile. According to Equation (2), under the conditions of the same soil layer and the pile diameter, \(F_{pi}\) can be expressed as a single-valued function of \(L_{i}\), namely  
\[F_{pi}=f_{1}(L_{i}) \tag{3}\]  
According to Figure 2, under the specific site condition, the pulling force \(F_{pi}\) and earth pressure \(p_{s}\) change with the different design parameters of uplift piles among the multiple forces. Furthermore, \(p_{s}\) varies with \(\sum_{i=1}^{n}F_{pi}\). Therefore, \(F_{pi}\) is the only variable that affects the external force of the base plate element, namely  
\[f=\sum_{i=1}^{n}f_{2}\big{(}F_{pi}\big{)} \tag{4}\]  
where \(f\) is the external force vector acting on the base plate element. The numerical calculation is a process that the internal force and external force of the base plate elements are gradually balanced, and the residual force tends to zero, namely  
\[R=f-f^{\text{int}}=0 \tag{5}\]  
where R is the residual force vector. Combining with Equations (1) and (3)-(5), we obtain  
\[C=\int\sum_{i=1}^{n}f(L_{i})du \tag{6}\]  
It is inferred from Equation (6) that \(L_{i}\) affects the strain energy \(C\). Therefore, when other parameters are kept unchanged, \(L_{i}\) can be used as the design variable of the BESO method to reduce the deformation degree of the base plate.  
In the BESO method, the sensitivity coefficient \(\alpha\) needs to be introduced, which controls the variation in the design variable (i.e., \(L_{i}\) in this paper). In order to minimize the deformation of the base plate, which is subjected to buoyancy, each uplift pile under the base plate is required to generate approximately the same vertical displacements. If the vertical displacement of some piles is large, it is implied that the anti-floating force provided by these piles is relatively insufficient (i.e., \(F_{pi}\) is relatively insufficient). On the contrary, if the vertical displacement of some piles is much smaller than other piles, the \(F_{pi}\) of these piles should be reduced. Therefore, the vertical displacement of the single pile can be used as the indicator to determine whether the \(F_{pi}\) provided by this pile is reasonable. At the same time, Equation (2) shows that \(F_{pi}\) is only related to \(L_{i}\), and \(L_{i}\) is the design variable in this paper. Therefore, the vertical displacement of the single pile can be used as the indicator to determine whether the pile length \(L_{i}\) is reasonable, that is, as the sensitivity coefficient to control the variation in \(L_{i}\). Furthermore, according to the assumption that the pile is a rigid body, the displacement of the pile end can be regarded as the displacement of the pile and can be used as the sensitivity coefficient \(\alpha\).