## 2 Computation Model
(**b**) Stress diagram of soil around pile. \[\xi_{1}=\frac{1}{R^{2}}\int_{R}^{\infty}\phi^{2}rdr;\eta=\int_{R}^{\infty}\left( \frac{d\phi}{dr}\right)^{2}rdr;\xi_{2}=\frac{1}{R^{2}}\int_{R}^{\infty}\phi^{2} rdr+1 \tag{9}\]  
By substituting Equation (8) into Equation (6), the dimensionless displacement control equation for the pile in the i-th layer foundation can be obtained:  
\[i\frac{d^{4}w_{i}^{*}}{dz^{*4}}+k_{t}^{*}w_{i}^{*}-2t_{t}^{*}\frac{d^{2}w_{i}^ {*i}}{dz^{*2}}=0 \tag{10}\]  
According to the pile foundation constraint, the upper boundary condition is as follows (\(z^{*}=0\)):  
\[ \begin{split}&\frac{d^{3}w_{1}^{*}}{dz^{*3}}-2t_{1}^{*}\frac{dw_{1} ^{*}}{dz^{*}}-P_{0}^{*}=0\\ &\frac{d^{2}w_{1}^{*}}{dz^{*2}}+M_{0}^{*}=0\qquad\qquad\text{( Free)}\\ &\frac{dw_{1}^{*}}{dz^{*}}=0\qquad\qquad\qquad\qquad\text{(Fixed)} \end{split}  \tag{11}\]  
The deformation of the pile foundation at the interface between the \(i\)-th layer foundation and the \((i+1)\)-th layer foundation is continuous, then we have the following (\(z^{*}=H_{i}/L\), \(i=1,2\)):  
\[ \begin{split}& w_{i}^{*}=w_{i+1}^{*};\frac{dw_{i}^{*}}{dz^{*}}= \frac{dw_{i+1}^{*}}{dz^{*}};\frac{d^{2}w_{i}^{*}}{dz^{*2}}=\frac{d^{2}w_{i+1}^ {*}}{dz^{*2}};\\ &\frac{d^{3}w_{i}^{*}}{dz^{*3}}-2t_{i}^{*}\frac{dw_{i}^{*}}{dz^{* }}=\frac{d^{3}w_{i+1}^{*}}{dz^{*3}}-2t_{i+1}^{*}\frac{dw_{i+1}^{*}}{dz^{*}} \end{split}  \tag{12}\]  
Correspondingly, the lower boundary condition is as follows (\(z^{*}=1\)):  
\[w_{3}^{*}=0;\frac{dw_{3}^{*}}{dz^{*}}=0;\ \frac{d^{2}w_{3}^{*}}{dz^{*2}}=0;\frac{d ^{3}w_{3}^{*}}{dz^{*3}}-2t_{3}\frac{dw_{3}^{*}}{dz^{*}}-\sqrt{2k_{3}^{*}t_{4} ^{*}}w_{3}^{*}=0 \tag{13}\]  
The governing equation of the attenuation function can be obtained by the coefficient \(\delta\phi\) in Equation (6):  
\[\frac{d^{2}\phi}{dr^{2}}+\frac{1}{r}\frac{d\phi}{dr}-(\frac{\gamma}{R})^{2} \phi=0 \tag{14}\]  
where \(\left(\frac{\gamma}{R}\right)^{2}=\frac{2n_{s}}{m_{s}}\), and \(\gamma\) is the dimensionless attenuation coefficient. The boundary conditions are: \(\phi(R)=1\); \(\phi(\infty)=0\); the solution is as follows:  
\[\phi r=\frac{K_{0}\{(\gamma/R)r\}}{K_{0}(\gamma)} \tag{15}\]  
where \(K_{0}()\) is the second kind of zero-order Bessel function,  
\[\gamma=\frac{1}{\varphi}\sqrt{\frac{\sum\limits_{i=0}^{2}G_{si}\int_{H_{i}^{i +1}}^{H_{i+1}}{(\frac{dw_{i}^{*}}{dz^{*}})}^{2}dz^{*}+G_{s3}w_{3}^{*2}|_{z^{*} =1}\sqrt{k_{3}^{*}/8t_{4}^{*}}}} \tag{16}\]  
where \(H_{i}^{*}=H_{i}/L\), \(H_{i+1}^{*}=H_{i+1}/L\), \(H_{0}^{*}=0\), \(H_{3}^{*}=1\). Given that one of the primary focuses of this paper is the phenomenon of local pile-soil contact surface detachment in layered foundations, and to clearly demonstrate its impact on the mechanical behavior of piles, this study specifically considers the scenarios where certain soil layers are fully detached from the pile. Such detachment typically occurs due to insufficient lateral confinement provided by the surrounding soil or differential settlement under lateral loading conditions. By analyzing these extreme conditions, we aim to provide a comprehensive understanding of how pile-soil detachment affects lateral deformation, bending moments, and shear forces along the pile shaft. Based on Equation (10) and the specified boundary conditions, the displacement control equation for the pile at the pile-soil contact interface is derived by incorporating the interaction effects of degraded pile-soil contact. For brevity, the detailed derivation process is omitted, but the resulting equation forms the basis for subsequent parametric analysis. This approach allows us to quantify the extent to which local detachment influences the overall mechanical performance of piles in layered soil, offering practical insights for pile foundation design.