## 2 Establishment of the Active-Passive Single Pile Calculation Model
### Phase Description of the Additional Load Caused by an Adjacent Surcharge
#### 2.3.1 Elastic State  
The lateral pressure on the foundation caused by the surcharge is analyzed according to Figure 3. The surcharge is assumed to be a trapezoidally distributed load on a rectangular area, which can be thought of as a combination of the uniformly distributed loads and triangular distributed loads. According to the improved Boussinesq theory, the lateral additional load at point \(Q\) under the action of each load zone is calculated separately, and the total lateral additional load at this point can be obtained by the superposition. In the figure, \(p_{0}\) is the surcharge strength, and we assume that the surcharge width in the \(y\)-axis direction is \(a\), the total width of the bottom of the surcharge load in the \(x\)-axis direction is \(B\), the \(xoz\) plane is the symmetry plane in the \(y\)-axis direction, and the distance between the calculation point \(Q\) and the \(z\)-axis is \(e\).  
Horizontal additional pressure at point \(Q\) under a concentrated load micro-unit \(p_{0}dxdy\) in the uniformly loaded rectangular area can be written as  
\[d\sigma_{x}(z)=\frac{3z}{\pi}\frac{(x-e-b_{1})}{\left[\left(x-e-b_{1}\right)^ {2}+y^{2}+z^{2}\right]^{5/2}}p_{0}dxdy \tag{2}\]  
In addition, the horizontal additional stress at point \(Q\) under uniform load on the rectangular area can be obtained by integration.  
\[\sigma_{x1}(z)=\frac{p_{0}z}{\pi}\left\{\begin{array}{l}\frac{a(e+b_{1})}{ \left[\left(e+b_{1}\right)^{2}+z^{2}\right]\sqrt{\left(e+b_{1}\right)^{2}+a^{ 2}+z^{2}}}-\frac{1}{z}\text{arctan}\left[\frac{a(e+b_{1})}{z\sqrt{\left(e+b_{ 1}\right)^{2}+a^{2}+z^{2}}}\right]\\ +\frac{1}{z}\text{arctan}\left[\frac{a(e+b_{1}+b_{2})}{z\sqrt{\left(e+b_{1}+b _{2}\right)^{2}+a^{2}+z^{2}}}\right]+\left(\frac{a(-e-b_{1}-b_{2})}{\left[ \left(e+b_{1}+b_{2}\right)^{2}+z^{2}\right]\sqrt{\left(e+b_{1}+b_{2}\right)^{2 }+a^{2}+z^{2}}}\right)\end{array}\right\} \tag{3}\]  
The horizontal additional stress at point \(Q\) under right and left rectangular area triangle loads can be written as  
Figure 3: Diagram of the surcharge load.  
\[\sigma_{x2}(z)=\frac{p_{0}z}{\pi b_{1}}\left\{\begin{array}{l}\frac{-ab_{1}(e+b_{1 })}{\left[\left(e+b_{1}\right)^{2}+z^{2}\right]\sqrt{\left(e+b_{1}\right)^{2}+a ^{2}+z^{2}}}-\frac{e}{z}\arctan\left[\frac{a(e+b_{1})}{z\sqrt{\left(e+b_{1} \right)^{2}+a^{2}+z^{2}}}\right]\\ \\ +\ln\left[\frac{\left(e+b_{1}\right)^{2}+z^{2}}{e^{2}+z^{2}}\right]+\frac{e}{z }\arctan\left(\frac{ae}{z\sqrt{e^{2}+a^{2}+z^{2}}}\right)+\\ \\ 2\ln\left(\frac{a+\sqrt{e^{2}+a^{2}+z^{2}}}{a+\sqrt{\left(e+b_{1}\right)^{2}+a ^{2}+z^{2}}}\right)\end{array}\right\} \tag{4}\]  
\[\sigma_{x3}(z)=\frac{p_{0}z}{\pi b_{1}}\left\{\begin{array}{l}\frac{ab_{3}(e +b_{1}+b_{2})}{\left[\left(e+b_{1}+b_{2}\right)^{2}+z^{2}\right]\sqrt{\left(e +b_{1}+b_{2}\right)^{2}+a^{2}+z^{2}}}+\ln\left[\frac{\left(e+b_{1}+b_{2} \right)^{2}+z^{2}}{\left(e+B\right)^{2}+z^{2}}\right]\\ \\ +2\ln\left(\frac{a+\sqrt{\left(e+B\right)^{2}+a^{2}+z^{2}}}{a+\sqrt{\left(e+b_ {1}+b_{2}\right)^{2}+a^{2}+z^{2}}}\right)-\frac{e+B}{z}\arctan\left[\frac{a(e +b_{1}+b_{2})}{z\sqrt{\left(e+b_{1}+b_{2}\right)^{2}+a^{2}+z^{2}}}\right]\\ \\ +\frac{e+B}{z}\arctan\left(\frac{a(e+B)}{z\sqrt{\left(a+B\right)^{2}+a^{2}+z^ {2}}}\right)\end{array}\right\} \tag{5}\]  
The total horizontal additional stresses caused by the surcharge can be written as  
\[\sigma_{x}(z)=\sigma_{x1}(z)+\sigma_{x2}(z)+\sigma_{x3}(z) \tag{6}\]  
Finally, the passive line load set can be written as  
\[q(z)=\sigma_{x}(z)\cdot d \tag{7}\]