# Adaptive Optimization Method for Piled Raft Foundations Based on Variable Pile Spacing
## 4 Illustrative Example  
### PRF Model and Material  
The initial designs of the PRFs used in this paper are designed by the traditional design method (\"identical\" piles with the same length, pile diameter, and constant pile spacing). Both of the PRFs in Case 1 and Case 2 consist of 117 concrete piles arranged in a 9 \(\times\) 13 pattern and connected with a raft (40 m \(\times\) 58 m) with a thickness of 1.8 m. The only difference is that the load in Case 1 is a uniform load and in Case 2 is a non-uniform load. Case 3 consists of 211 concrete piles and is connected with an irregular raft (see Figure 6) with the same thickness of 1.8 m, and the load is a non-uniform load. The length (37 m) and diameter (1.5 m) of the piles in the three models are the same, and the length and diameter are located with approximately the same center-to-center spacing equal to 3 diameters. In order to reduce the effect of boundary conditions, the sizes of the PRFs in the horizontal direction are approximately 3-4 times greater than the raft size; the detailed dimensions are shown in Figure 7. The detailed material parameters of the raft, pile, and bedrock are listed  
Figure 5: Flowchart of AOM.  
in Table 1. The ground of the models contains the same soils, which has seven soil layers and one bedrock layer; the detailed parameters are reported in Table 2. The solid element is used to model the soil and piled raft foundations. Considering the influence of pile-soil interaction, the soil around the pile is adopted with a more precise unit size than other soil to show the difference between the deformation characteristics of the soil around the pile and other soil. As a result, the maximum element sizes in the vertical and horizontal direction are 0.7 m and 6 m, respectively. There are about 2.4 \(\times\) 10\({}^{5}\)-4 \(\times\) 10\({}^{5}\) nodes and 3.9 \(\times\) 10\({}^{5}\)-6.4 \(\times\) 10\({}^{5}\) elements.  
Figure 6: Detailed dimensions of the irregular raft in Case 3.  
Figure 7: _Cont._  
In this paper, the Duncan-Chang (DC) model [33; 34] used in this paper is an incremental nonlinear (hyperbolic) stress-dependent model, in which the tangent modulus and Poisson's ratio are expressed in an incremental relationship, and the relationship between strain and stress is nonlinear. During the analysis, the initial principal stress (\(\sigma_{1}\) and \(\sigma_{3}\)) of the soil is obtained through a finite element calculation, and then the initial elastic modulus of the soil can be obtained by inputting it into the DC constitutive model, and then the external load is gradually applied. After the calculation of each load step, the three principal stresses of each soil element are output, and then the adjusted soil elastic modulus and Poisson's ratio are obtained from the DC model and used as the soil parameters for the next calculation. Equation (13) is used to obtain \(\sigma_{1}\) and \(\sigma_{3}\), where z is the distance from the ground to the center of soil element, \(\gamma\) is the gravity per unit volume of the soil, and \(K_{0}\) is the static lateral pressure coefficient.  
\[\left\{\begin{aligned} &\sigma_{1}=\gamma z\\ &\sigma_{3}=K_{0}\gamma z\end{aligned}\right.\quad\text{ where, }K_{0}=0.95-\sin\varphi \tag{13}\]