## 3 Analysis
### Calculation of Single Pile Effective Compaction Range  
The theory of cylindrical cavity expansion can be used to examine the squeezing impact of compacted piles. Assuming that the geometric shape, constraint, and load of the soil are symmetrical with respect to the central axis of the pile, the compaction range of the soil can be determined by applying cylindrical cavity expansion theory to the plane axis symmetry problem. The soil stress balance equation and the Mohr-Coulomb failure condition are shown in Equations (1) and (2):  
\[\frac{d\sigma_{\mathrm{r}}}{dr}+\frac{\sigma_{\mathrm{r}}-\sigma_{\mathrm{ \varphi}}}{r}=0 \tag{1}\]  
\[\sigma_{\mathrm{r}}-\sigma_{\mathrm{\varphi}}=(\sigma_{\mathrm{r}}+\sigma_{ \mathrm{\varphi}})\sin\varphi+2c\cdot\cos\varphi \tag{2}\]  
From the stress balance Equation (1) and the Mohr-Coulomb failure condition (2), the differential equation is as follows:  
\[\frac{d\sigma_{\mathrm{r}}}{dr}=\frac{2\sin\varphi}{1+\sin\varphi}\cdot\frac{ \sigma_{\mathrm{r}}}{r}+\frac{2c\cdot\cos\varphi}{1+\sin\varphi}\frac{1}{r}=0 \tag{3}\]  
\[\sigma_{\mathrm{r}}=(p_{\mathrm{u}}+c\cdot c\cdot cq\varphi)(\frac{R_{\mathrm{ u}}}{r})^{\frac{2\sin\varphi}{1+\sin\varphi}}-c\cdot c\cdot cq\varphi \tag{4}\]  
Before bored pile construction, it is assumed that the soil within the depth range possesses isotropic effective stress q. Based on the radial displacement of the Lame solution and boundary conditions, the volume strain in the plastic zone is introduced to produce \(R_{\mathrm{p}}\)[21]:  
\[I_{\mathrm{r}}=\frac{E}{2(1+v)(c+qlg\varphi)} \tag{5}\]  
\[I_{\mathrm{rr}}=I_{\mathrm{r}}/(1+I_{\mathrm{r}}\Delta\sec\varphi) \tag{6}\]  
\[R_{\mathrm{p}}=R_{\mathrm{u}}\sqrt{I_{\mathrm{rr}}\sec\varphi} \tag{7}\]  
The compaction effect of the bored piles increased dramatically as the center distance of the pile decreased. When the soil density between piles reaches a particular threshold, the possibility of collapse is eliminated. The effective compaction range of a bored pile is defined as the range in which the soil collapsibility between the heaps is completely eliminated. An effective compaction coefficient was introduced to determine the effective compaction range of bored piles. \[R_{\mathrm{Y}}=\partial R_{\mathrm{p}} \tag{8}\] \begin{table}
\begin{tabular}{c c c c c} \hline
**Center Distance of Piles** & **1.0 m** & **1.25 m** & **1.5 m** & **T2** \\ \hline Range value & 0–0.013 & 0–0.014 & 0.041–0.098 & 0.057–0.084 \\ Average value & 0.04 & 0.07 & 0.062 & 0.072 \\ \hline \end{tabular}
\end{table} 
Table 4: Collapsibility coefficient of soil between piles. \begin{table}
\begin{tabular}{c c c c c} \hline
**Center Distance of Piles** & **1.0 m** & **1.25 m** & **1.5 m** & **T3** \\ \hline Range value & 0–0.004 & 0–0.007 & 0.07–0.075 & 0.009–0.074 \\ Average value & 0.002 & 0.003 & 0.035 & 0.048 \\ \hline \end{tabular}
\end{table} 
Table 5: Self-weight collapsibility coefficient of soil between piles. where \(E\)s is the pile side soil compression modulus, \(c\) is the cohesion of soil close to the pile side, \(\varphi\) is the pile side soil internal friction angle, \(Ir\) is the pile side soil stiffness coefficient, \(\gamma\) represents the pile side soil bulk density, \(Irr\) is the pile side soil corrected stiffness coefficient, \(R_{\rm u}\) is the diameter of the pile hole after compaction treatment, \(R_{\rm P}\) is the pile side soil radius of the plastic zone, and \(R_{\rm Y}\) is the effective compaction range. \(E=E_{\rm S}\times(1-2\mu^{2}/(1-\mu))\); \(\mu\) is the pile side soil Poisson's ratio; \(\Delta=(e_{1}-e_{2})/(1+e_{1})\), \(e_{1}\) and \(e_{2}\) are weighted average values of the pore ratio of the soil around the pile before and after compaction treatment. According to the geological prospecting data, the characteristics of soil in the treatment depth range are essentially identical. By putting the average values of measured foundation soil parameters in the depth range into Equation (7), the average plastic radius of foundation soil RY is evaluated to be 2.9D. [PERSON] et al.