## 2 Theoretical Model
### Load Transfers between Pile and Soil
#### 2.2.1 The zone of the conical head  
The interaction between the conical head and soil is analyzed in this section. As depicted in Figure 5, the interaction force consists of a normal force vertical to the contact surface and friction parallel to the contact surface. The equilibrium equations can be acquired by decomposing the normal force and the friction along the coordinate direction.  
Horizontally:  
\[F_{N}(z^{**})\cos(\alpha)+F_{N}(z^{**})\tan(\varphi_{s})\sin\alpha=P_{s}(z^{**})k _{p} \tag{12}\]  
Vertically:  
\[F_{N}(z^{**})\sin(\alpha)-F_{N}(z^{**})\tan(\varphi_{s})\cos(\alpha)=F(z^{**}) \tag{13}\]  
where \(F_{N}(z^{**})\) is the normal stress; \(P_{s}(z^{**})\) is the vertical stress of the soft soil; \(F(z^{**})\) is the vertical component of the interaction stress; \(\varphi_{s}\) is the internal friction angle of the soil; and \(\alpha\) is the cone angle of the conical head.  
By associating Equation (12) with Equation (13), the following equation can be obtained:  
\[F(z^{**})=\frac{Mk_{p}}{C}P_{s}(z^{**}) \tag{14}\]  
where \(M\) and \(C\) represent the intermediate variables without physical significance, \(M=\sin(\alpha)-\tan(\varphi_{s})\)\(\cos(\alpha)\), \(C=\cos(\alpha)+\tan(\varphi_{s})\)\(\sin(\alpha)\).  
As shown in Figure 4, the surrounding soil within the conical head zone is divided into small pieces with thickness of \(\mathrm{dz}\). The vertical force equilibrium can be denoted as  
\[P_{s}(z^{**})A_{su}+F(z^{**})A_{b0}\mathrm{dz}^{**}=P_{s}(z^{**})A_{sd}+ \mathrm{d}P_{s}(z^{**})A_{sd} \tag{15}\]  
where \(A_{su}\) and \(A_{sd}\) respectively represent the upper and lower surface area of the soil element, \(Asu=Asd=\pi(re2-(r1-k1z^{**})2)\); \(A_{b0}\) denotes the perimeter of the soil element, \(A_{b0}=2\pi(r_{1}-k_{1}z^{**})\) and \(k1=tan(\alpha)\).  
Figure 5: Interaction between the conical head and the soil.  
Substituting Equation (14) into Equation (15) and integrating Equation (15), while introducing the stress continuity condition \(P\!s(z^{**}=0)\ =\ oo(z^{*}=h)\), the following formula thus can be obtained:  
\[P_{s}(z^{**})=[(r_{1}-r_{e})(r_{1}+r_{e})]^{\frac{D\!p}{C\!s_{1}}}[(r_{1}-r_{e} -k_{1}z^{**})(r_{1}+r_{e}-k_{1}z^{**})]\sigma_{o}(h) \tag{16}\]  
The sum of the vertical forces acting on the pile and the soil is equal to the gravity.  
\[m_{1}p_{p}(z^{**})+(1-m_{1})p_{s}(z^{**})=\gamma_{f}h+\gamma_{s}z^{**} \tag{17}\]  
where \(m_{1}\) is the ratio of the conical head area at any horizontal cross section to the whole cylinder area and \(m_{1}=(r_{1}-k_{1}z^{**})/r_{e}\)2; \(\gamma_{s}\) is the unit weight of the soft soil.  
Substituting Equation (16) into Equation (17), the stress formula of pile therein reads as follows:  
Footnote 2: The \(\gamma_{s}\) is the unit weight of the soft soil.  
\[\begin{array}{l}p_{p}(z^{**})=\frac{r_{e}^{2}}{(r_{1}-k_{1}z^{**})^{2}}[(l \eta_{f}+z^{**}\gamma_{s})\\ -[(r_{1}-r_{e})(r_{1}+r_{e})]^{\frac{D\!p}{C\!s_{1}}}[(r_{1}-r_{e}-k_{1}z^{**}) (r_{1}+r_{e}-k_{1}z^{**})](1-\frac{(r_{1}-k_{1}z^{**})^{2}}{r_{e}^{2}})\sigma_{ o}(h)]\end{array} \tag{18}\]  
In the conical head zone, the vertical stress on the pile and soft soil can be obtained from Equations (16) and (18). Therefore, the deformation of the soil and pile can be formulated as follows:  
\[\begin{array}{l}S_{s1}=\int\limits_{0}^{L_{0}}\frac{p_{s}(z^{**})}{E_{s}} \mathrm{d}z^{**}\\ \\ S_{p1}=\int\limits_{0}^{L_{0}}\frac{p_{p}(z^{**})}{E_{p}}\mathrm{d}z^{**} \end{array} \tag{19}\]