## 2 Computation Model
\begin{split}\prod=U-V&=\frac{1}{2}\int_{R}^{\infty}\int_{0}^{2 \pi}\int_{0}^{\infty}(\sigma_{rr}\varepsilon_{rr}+\sigma_{r\theta}\varepsilon _{\theta\theta}+\sigma=\varepsilon_{zz}+\tau_{r\theta}\gamma_{r\theta}+\tau_{ z}\gamma_{rz}+\tau_{\theta z}\gamma_{\theta z})\mathrm{d}r\mathrm{d}z\\ +\frac{1}{2}\int_{0}^{L}E_{p}I_{p}\bigg{(}\frac{\mathrm{d}^{2}w}{ \mathrm{d}z^{2}}\bigg{)}^{2}\mathrm{d}z+\frac{1}{2}\int_{L}^{\infty}\pi R^{2} G_{s}(\frac{\mathrm{d}w}{\mathrm{d}z})^{2}\mathrm{d}z-P_{0}w-M_{0}\frac{\mathrm{d}w}{ \mathrm{d}z}\end{split}  \tag{4}\]  
By substituting Equations (2) and (3) into Equation (4), the following can be obtained:  
\[ \begin{split}&\int_{0}^{\infty}\int_{R}^{\infty}\bigg{[}\pi( \lambda_{s}+3G_{s})w\frac{d\phi}{dr}\delta\bigg{(}w\frac{d\phi}{dr}\bigg{)}+2 \pi G_{s}\frac{dw}{dz}\phi\bigg{(}\frac{dw}{dz}\phi\bigg{)}\bigg{]}rdrdz\\ +\int_{0}^{L}E_{p}I_{p}\frac{d^{2}w}{dz^{2}}\bigg{(}\frac{d^{2}w}{ dz^{2}}\bigg{)}dz+\int_{L}^{\infty}\pi R^{2}G_{s}\frac{dw}{dz}\delta(\frac{dw}{dz})dz-P_{0} \delta w-M_{0}\delta\frac{dw}{dz}=0\end{split}  \tag{5}\]Based on the modified Vlasov foundation model, which accounts for soil continuity and varying properties with depth, this approach improves accuracy in predicting pile behavior in layered foundations. It offers better pile-soil interaction modeling and computational efficiency compared to traditional methods. For clarity, we use a pile in a three-layer foundation as an example:  
\[ \begin{split}&\sum_{i=1}^{3}\int_{H_{i-1}}^{H_{i}}(E_{p}I_{p}\frac {d^{4}w_{i}}{dz^{4}}+k_{i}w_{i}-2t_{i}\frac{d^{2}w_{i}}{dz^{2}})\delta w_{i}dz- \int_{L}^{\infty}(k_{3}w_{4}-2t_{4}\frac{d^{2}w_{4}}{dz^{2}})\delta w_{4}dz\\ &+\sum_{i=1}^{2}\biggl{[}\biggl{(}-E_{p}I_{p}\frac{d^{3}w_{i}}{dz^ {3}}+2t_{i}\frac{dw_{i}}{dz}\biggr{)}\delta w_{i}+\biggl{(}E_{p}I_{p}\frac{d^{3} w_{i+1}}{dz^{3}}-2t_{i+1}\frac{dw_{i+1}}{dz}\biggr{)}\delta w_{i+1}\biggr{]}_{z=H _{i}}\\ &-\biggl{[}E_{p}I_{p}\frac{d^{3}w_{1}}{dz^{3}}-2t_{1}\frac{dw_{1} }{dz}-P_{0}\biggr{]}\delta w_{1}\biggr{|}_{z=0}+2t_{4}\frac{dw_{4}}{dz}\delta w _{4}\biggr{|}_{z=\infty}\\ &+\biggl{[}-E_{p}I_{p}\frac{d^{2}w_{1}}{dz^{2}}+M_{0}\biggr{]} \delta\frac{dw_{1}}{dz}\biggr{|}_{z=0}+\biggl{[}\biggl{(}-E_{p}I_{p}\frac{d^{3 }w_{3}}{dz^{3}}+2t_{3}\frac{dw_{3}}{dz}\biggr{)}\delta w_{3}-2t_{4}\frac{dw_{4 }}{dz}\delta w_{4}\biggr{]}_{z=L}\\ &+\sum_{i=1}^{2}\left(E_{p}I_{p}\frac{d^{2}w_{i}}{dz^{2}}\frac{dw _{i}}{dz}-E_{p}I_{p}\frac{d^{2}w_{i+1}}{dz^{2}}\delta\frac{dw_{i+1}}{dz} \right)\biggr{|}_{z=H_{i}}+E_{p}I_{p}\frac{d^{2}w_{3}}{dz^{2}}\delta\frac{dw_{ 3}}{dz}\Bigr{|}_{z=L}\\ &+\int_{R}^{\infty}\biggl{[}\pi m_{s}(\frac{d^{2}\phi}{dr^{2}}r+ \frac{d\phi}{dr})-2\pi n_{s}\phi r\biggr{]}\delta\phi dr+\pi m_{s}\frac{d\phi }{dr}r\delta\phi\biggr{|}_{R}^{\infty}=0\end{split}  \tag{6}\]  
where  
\[ \begin{split}& t_{i}=\pi G_{si}\int_{R}^{\infty}\phi^{2}rdr;t_{ 4}=\pi G_{s_{3}}\int_{R}^{\infty}\phi^{2}rdr+\frac{1}{2}\pi G_{s_{3}}R^{2};(z \in[L,\infty));n_{s}=\sum_{i=1}^{3}G_{si}\biggl{[}\begin{matrix}H_{i}\\ H_{i-1}\end{matrix}(\frac{dw}{dz})^{2}dz;\\ & i=1,2,3;H_{0}=0;H_{3}=L;\text{m}_{s}=\sum_{i=1}^{3}\bigl{(} \lambda_{si}+3G_{si}\bigr{)}\int_{H_{i-1}}^{H_{i}}w_{i}^{2}dz;k_{i}=\pi( \lambda_{si}+3G_{si})\int_{R}^{\infty}\biggl{(}\frac{d\phi}{dr}\biggr{)}^{2} rdr.\end{split}  \tag{7}\]  
To facilitate analysis, dimensionless parameters are introduced:  
\[ \begin{split}& t_{i}^{*}=\frac{2a_{i}\xi_{i}\lambda^{2}}{(1+v_{si})};k_{i}^{*}=\frac{2(3-4v_{si})a_{i}\eta\lambda^{4}}{(1+v_{si})(1-2v_{si})};t_{4 }^{*}=\frac{2a_{i}\xi_{2}\lambda^{2}}{(1+v_{si})};\\ & P_{0}^{*}=\frac{4P_{0}L^{2}}{\pi R^{4}E_{p}};M_{0}^{*}=\frac{4M_ {0}L}{\pi R^{4}E_{p}};z^{*}=\frac{z}{L};w_{i}^{*}=\frac{w_{i}}{L}.\end{split}  \tag{8}\]  
where \(v_{si}\) is the Poisson's ratio of the \(i\)-th layer foundation, and \(\alpha_{i}=E_{si}/E_{p}\) is the elastic modulus ratio of the \(i\)-th layer foundation to the pile,  
Figure 2: Schematic diagram of pile–soil system displacement and soil stress around pile. (**a**) Displacement of pile–soil system.