## 2 Establishment of the Active-Passive Single Pile Calculation Model
### Phase Description of the Additional Load Caused by an Adjacent Surcharge
#### 2.3.2 Elastoplastic State  
Calculation Model and Formulas  
When the soil around the pile is in an elastoplastic state, the earth pressure generated by the surcharge load can be calculated using an improved Ito theory: \(\theta_{1}=\frac{\pi}{8}+\frac{\varphi}{4},\theta_{2}=\frac{\pi}{4}-\frac{ \varphi}{2}\), \(\theta_{3}=\frac{\pi}{4}+\frac{\varphi}{2}\). Here, we assume that the wedge-shaped part of soil between piles (\(AEBB^{\prime}E^{\prime}A^{\prime}\)) reaches its ultimate strength under the influence of the soil lateral displacement and then enters the plastic state, while the rest of soil is still in the elastic state. Force analysis of micro-units in area \(AA^{\prime}E^{\prime}E\) and area \(EE^{\prime}B^{\prime}B\) (the shaded area in Figure 4) is shown in Figure 5a,b, respectively. The static equilibrium differential equation in the \(x\) direction is given by  
\[D_{2}d\sigma_{x}=2(\sigma_{n}\tan\varphi+c)dx \tag{8}\]  
\[\left[\sigma_{n}+(\sigma_{n}\tan\varphi+c)N_{\varphi}^{-1/2}-\sigma_{x}\right] dD-Dd\sigma_{x}=0 \tag{9}\]  
where \(\sigma_{n}=\sigma_{x}N_{\varphi}+2 cN_{\varphi}^{1/2}\), \(N_{\varphi}=\tan^{2}(\pi/4+\varphi/2)\), \(dD=2 dxN_{\varphi}^{1/2}\), \(c\) is the soil-cohesive force, and \(\varphi\) is the soil internal friction angle. These two differential equations can be solved separately to obtain  
\[\sigma_{1x}=\frac{C_{1}\exp\bigl{(}2 xN_{\varphi}\tan\varphi/D_{2}\bigr{)}-c \Bigl{(}2N_{\varphi}^{1/2}\tan\varphi+1\Bigr{)}}{N_{\varphi}\tan\varphi},\left(0 \leq x\leq\frac{D_{1}-D_{2}}{2}\tan\Bigl{(}\frac{\pi}{8}+\frac{\varphi}{4} \Bigr{)}\right) \tag{10}\]  
and  
\[\sigma_{2D}=\frac{(C_{2}D)^{N_{\varphi}^{1/2}\tan\varphi+N_{\varphi}-1}-c\Bigl{(} 2\tan\varphi+2N_{\varphi}^{1/2}+N_{\varphi}^{-1/2}\Bigr{)}}{N_{\varphi}^{1/2} \tan\varphi+N_{\varphi}-1},(D_{2}\leq x\leq D_{1}) \tag{11}\]  
In the above two formulas, \(\sigma_{1x}\) is the stress at any point \(x\) of zone \(AA^{\prime}E^{\prime}E\), \(\sigma_{2D}\) is the stress at any point \(D\) of zone \(EE^{\prime}B^{\prime}B\), \(C_{1}\) and \(C_{2}\) are the undetermined integration constants, \(D_{1}\) is the centerline spacing between piles, and \(D_{2}\) is the net distance between the piles. Figure 4: Calculation model using an improved Ito theory. Figure 5: Stress analysis of a micro-unit in the plastic zone. (**a**) Zone \(AA^{\prime}E^{\prime}E\); (**b**) zone \(EE^{\prime}B^{\prime}B\). Taking the horizontal stress at plane \(BB^{\prime}\) as the initial boundary condition, and substituting \(\sigma_{BB^{\prime}}(z)=\sigma_{x}(z)+K_{0}\gamma z\) into Equation (11) gives \(C_{2}\) so that \(\sigma_{2D}\) can be written as  
\[\sigma_{2D}=\frac{1}{G_{1}}\Bigg{\{}\left(\frac{D}{D_{1}}\right)^{G_{1}}( \sigma_{BB^{\prime}}(z)\cdot G_{1}+c\cdot G_{2})-c\cdot G_{2}\Bigg{\}} \tag{12}\]  
\(C_{1}\) can be obtained according to the stress continuity condition (\(\sigma_{2D}\big{|}_{D=D_{2}}=\sigma_{1x}\big{|}_{x=\frac{D_{1}-D_{2}}{2}\tan \left(\frac{\pi}{8}+\frac{\pi}{4}\right)}\)) of the common boundary interface \(EE^{\prime}\) in zone \(AA^{\prime}E^{\prime}E\) and \(EE^{\prime}B^{\prime}B\), and thus \(\sigma_{1x}\) can be written as  
\[ \begin{array}{rl}\sigma_{1x}=&\frac{\exp\left(2 xN_{\varphi} \tan\varphi/D_{2}\right)}{N_{\varphi}\tan\varphi}\cdot\Bigg{\{}\frac{c\left(2 N_{\varphi}^{1/2}\tan\varphi+1\right)}{\exp\left[(D_{1}-D_{2})G_{3}/D_{2} \right]}+\Bigg{[}\left(\frac{D_{2}}{D_{1}}\right)^{G_{1}}(\sigma_{BB^{\prime} }(z)\cdot G_{1}+c\cdot G_{2})-c\cdot G_{2}\Bigg{]}.\\ &\frac{N_{\varphi}\tan\varphi}{G_{1}\exp\left[(D_{1}-D_{2})G_{3}/D_{2} \right]}\Bigg{\}}-\frac{c\left(1+2N_{\varphi}^{1/2}\tan\varphi\right)}{N_{ \varphi}\tan\varphi}\\ &\mbox{where }G_{1}=N_{\varphi}^{1/2}\tan\varphi+N_{\varphi}-1,\ G_{2}=2\tan \varphi+2N_{\varphi}^{1/2}+N_{\varphi}^{-1/2},\mbox{ and }G_{1}=N_{\varphi}\tan\varphi\cdot\tan(\pi/8+\varphi/4). \end{array}