## 1 Introduction
### Calculation of Additional Settlement of Lower Pile Toe  
The geometric distribution of the self-balanced upper and lower pile elements is shown in Figure 1. The pile element's vertical extra displacement caused by the shear act of other unit loads can be described as Equation (1). \[\{w_{\rm s}\}=\frac{D}{E_{\rm s}}[I_{b}]\{\tau\} \tag{1}\]where \(w_{s}\) is the pile element's extra displacement; \(D\) is the test pile's diameter; \(E_{s}\) is the pile's elastic modulus; \(\tau\) is the shear stress of the pile; and \(I_{b}\) is the influence factor of additional stress on the load in the semi-infinite plane. For the \(k\) micro-units of the upper section pile, the lower pile's extra displacement \(w_{bs}\) end caused by the load of the pile body can be described as Equation (2). \[\{w_{bs}\}=\frac{D}{E_{s}}\sum_{j=1}^{k}\left(I_{bj}\vvet{\tau_{j}}\right) \tag{2}\]  
where \(\vvet{\tau_{j}}\) are the shear stress vector functions of any unit \(j\) in the upper section pile, and the upper pile is negative and the lower pile is positive. \(I_{bj}\) is the additional settlement influencing factor of the lower pile toe caused by the shear stress of unit \(j\). \[I_{bj}=\pi\int\limits_{(j-1)\Delta L}^{j\Delta L}I_{p}dh \tag{3}\]  
According to [PERSON]'s solution for Equation (3), \(I_{p}\) is the effect factor of vertical displacement, \(\Delta L\) is the length of unit \(j\), \(\Delta L=L/n\), and \(h\) is the burial depth of unit \(j\). \[I_{p}=\frac{(1+v_{s})}{8\pi(1-v_{s})}\Bigg{\{}\frac{{z_{1}}^{2}}{{R_{1}}^{3}}+ \frac{(3-4v_{s})}{R_{1}}+\frac{(5-12v_{s}+8v_{s}{}^{2})}{R_{2}}+\frac{(3-4v_{ s})z^{2}-2 hz+2h^{2}}{{R_{2}}^{3}}+\frac{6 hz^{2}(z-h)}{{R_{2}}^{5}}\Bigg{\}} \tag{4}\]  
where \(v_{s}\) is the Poisson's ratio of each unit corresponding to the soil layer; \(z\) and \(z_{1}\) are the calculated lengths; and \(R_{1}\) and \(R_{2}\) are the calculation coefficients, \({R_{1}}^{2}={z_{1}}^{2}+D^{2}/4\), \({R_{2}}^{2}={z}^{2}+D^{2}/4\). Substituting Equation (4) into (3) and (2). Figure 1: The geometry of the self-balanced pile. \[I_{bj}=\frac{(1+v_{s})}{8\pi(1+v_{s})}\bigg{\{}\frac{z_{1}}{R_{1}}-4(1+v_{s})\ln(z _{1}+R_{1})+8(1-2v_{s}+v_{s}{}^{2})\ln(z+R_{2})+\frac{2L^{2}z/r^{2}-4L-(3-4v_{s}) z}{R_{2}}+2\frac{\left(L^{2}-L^{2}z^{3}/r^{2}\right)}{R_{2}^{3}}\bigg{\}} \tag{5}\]  
where \(r\) is the test pile's radius. Equation (5) is substituted for Equation (2). The settlement \(w_{bs}\) at the lower pile toe caused by the side shear stress of \(n\) units in the pile body can be calculated as Equation (6). \[w_{bs}=\frac{\sum\limits_{j=0}^{k}I_{bj}K_{zj}w_{j}}{E_{s}U\Delta L/D+\sum \limits_{j=0}^{k}I_{bj}K_{z,j}} \tag{6}\]  
where \(U\) is the test pile's perimeter, \(K_{zj}\) is connected to the shear stress vector functions, which is the pile-shear soil's stiffness vector corresponding to the unit \(\tau_{j}\). [PERSON] et al. researched the negative friction resistance of the side caused by water immersion in expansive soil and considered that the pile-soil interface model proposed by [PERSON] [40] (as demonstrated in Figure 2) can adequately reflect the negative frictional resistance characteristics [41]. [PERSON] et al. considered that the nonlinear behavior of a single friction pile is described here by a nonlinear model which was proposed by [PERSON] [42]. Although the pile side resistance to friction is negative in the self-balanced upper pile, the pile-soil interaction is the same as in the positive friction resistance model. The Kraft model can conveniently take into account the stratification of soil and the nonlinear deformation of soil around the pile, and also consider the continuity of soil around the pile. In this study, it is presumed that the relationship between the upper and lower piles' side friction resistance follows Kraft's form, and the pile's side friction is written as Equation (7).