# Adaptive Optimization Method for Piled Raft Foundations Based on Variable Pile Spacing
## 6 Reduction in the Cost of the Pile Group  
### Translation of the Excess Bearing Capacity of the Piles into Economic Savings  
As is shown in Table 5, the maximum pile top stresses are all below the UBC after optimization design, which means the piles still have excess bearing capacity (EBC). This part tries to translate EBC into the reduction of the cost of the pile group. In order to keep enough safety stock for the pile groups, some constraints are given as follows:  
1. Max-Stress at the pile top is below the UBC whether before or after the optimization design;
2. After the optimization design, the maximum pile top stress of the pile group shall not be greater than the initial design calculation result or half of the ultimate bearing capacity of a single pile bearing capacity (short for (1/2)UBC), which is commonly used in the Chinese code [31]. During this process, it is assumed that the load is only taken by piles: \[P_{total}=\sum_{i=1}^{N}P_{i}=\sum_{i=1}^{N}\sigma_{i}A_{P}\] (14) where \(P_{total}\) is the load on top of the raft, \(P_{i}\) is the load supported by pile \(i\), \(N\) is the number  
with that in the initial design, the pile spacing of the pile group increases obviously. The range of the pile spacing after AOM-ES in Case 1 is 3.7 m to 5.5 m, in Case 2 3.3 m to 5.7 m, and in Case 3 3.5 m to 5.9 m.  
Compared with Figure 11a,c,e, Figure 12 presents the 3 D view of the raft settlements in each case after AOM-ES, which shows that the raft settlements are more uniform than that in the initial design. Table 7 gives the detailed comparison of the corresponding data. The minimum and maximum settlements of the raft in the three cases are all increased, which are affected by the reduction of the number of piles, while these increases did not affect the decrease of the differential settlements (the optimization rates are still no less than 80%). For the same reason, fewer piles are used in the models after AOM-ES, and the minimum and maximum pile top stresses are all increased, which indicates that the bearing capacity of the piles are more utilized than that in the initial design and first optimization design (by AOM), without pile top stress excess in the corresponding UBC.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Model** & **C1-117-I** & **C2-117-I** & **C3-211-I** \\ \hline \(\sigma_{target}\) (MPa) & 4.85 & 5.22 & 7.95 \\ \(\sigma_{max,o}\) (MPa) & 3.91 & 4.3 & 6.52 \\ \(F_{extm}\) & 19\% & 17\% & 18\% \\ \(N_{new}\) & 94 & 97 & 173 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Comparison of the three cases during AOM-ES.  
Figure 11: File arrangements of initial designs and AOM-ES: (**a**) initial design and AOM-ES (C1-94-O) of Case 1; (**b**) initial design and AOM-ES (C2-97-O) of Case 2; (**c**) initial design and AOM-ES (C3-173-O) of Case 3.  
Figure 12: _Cont._  
As mentioned in the previous part, the bearing capacity of the soil is neglected. However, the load shared by the soil actually increases as the number of piles reduces. Therefore, the load shared by the piles in models after AOM-ES is always less than that in the optimization design only by AOM, as follows:  
\[N\sigma_{i}A_{P}\geq N_{new}\sigma_{i,new}A_{P} \tag{18}\]  
As a result, \(\sigma_{i,new}\) will not exceed \(\sigma_{target}\) because of the existence of the soil and the influence of the pile-soil interaction. Finally, the volume of the pile group in C1-94-O is reduced by 19%, in C2-97-O by 17%, and in C3-173-O by 18% compared with the initial design, which means the costs of the pile groups are reduced by the same percent without sacrificing too many bearing capacity stock of the pile group.