# Spatial Deformation Calculation and Parameter Analysis of Pile-Anchor Retaining Structure
## 3 Calculation of Retaining Structure Deformation  
### Calculation of Strain Energy of Support Structure  
In this study, the total potential energy of the support structure system is referred to as the resistance potential energy, while the potential energy generated by the soil on the support structure is referred to as the active potential energy. Specifically, the resistance potential energy comprises the bending strain energy of the crown beam, the bending strain energy of the waist beam, the bending strain energy of the support pile, and the elastic strain energy of the anchor cable. On the other hand, the active potential energy is determined by the effect of active and passive soil pressures on the support structure system. The expression for each strain energy is presented below.  
Crown beam bending strain energy, \(W_{g}\):  
\[W_{g}=\frac{1}{2}\int\limits_{0}^{l}E_{\mathrm{g}}I_{\mathrm{c}}\phi^{\prime \prime}{}^{2}\mathrm{d}x=\frac{\pi^{4}E_{g}I_{\mathrm{g}}}{4L^{3}}\delta_{0}{}^ {2}=T_{gb}\delta_{0}{}^{2} \tag{7}\]  
The setting heights of anchor cables and waist beams are \(h_{i}=\xi_{i}H\). If there are \(M\) rows of waist beams in total, the strain energy, \(W_{yi}\), of the waist beam at the \(i\)-th row is:  
\[W_{yi}=\frac{1}{2}\int\limits_{0}^{l}E_{y}I_{y}\delta_{y}{}^{\prime\prime}{}^{ 2}\mathrm{d}x=\frac{\pi^{4}E_{y}I_{y}}{4L^{3}}\delta_{0}{}^{2}((n_{1}\xi_{i}{} ^{2}H^{2}+n_{2}\xi_{i}H+n_{3})=T_{wbi}\delta_{0}{}^{2} \tag{8}\]  
Assuming that a total of \(n\) piles is set in the retaining structure and the pile spacing is \(s\), the bending strain energy, \(W_{z}\), of all piles is expressed as:  
\[W_{zj}=\sum\limits_{j=1}^{n}\frac{1}{2}\int\limits_{0}^{H}E_{z}I_{z}u(x,z)^{ \prime\prime}{}^{2}\mathrm{d}z=\frac{E_{z}I_{z}\delta_{0}^{2}n_{1}^{2}HL}{s}=T _{gb}\delta_{0}{}^{2} \tag{9}\]  
Generally, the anchor cable is set at the same position as the waist beam. Assuming that the number of anchor cables and piles is the same, there are \(n\) anchor cables in a row with the same anchor cable spacing of \(s\), and the stiffness \(K_{m}=EA/L\), where \(E\), \(A\), and \(L\) are the elastic modulus, cross-sectional area, and length of the anchor cable, respectively. The elastic strain energy of the anchor cable, \(W_{m}\), is given by:  
\[W_{m}=\sum\limits_{j=1}^{n}\frac{1}{2}K_{m}\Delta^{2}=T_{mi}\delta_{0}^{2} \tag{10}\]  
In summary, the total resistance potential energy, \(W_{D}\), of the foundation pit retaining structure is given by Equation (11):  
\[W_{D}=W_{g}+W_{y}+W_{z}+W_{m} \tag{11}\]  
The expression of active soil pressure potential energy, \(W_{pa}\), is given by Equation (12):  
\[W_{pa}=-\int\limits_{0}^{l}\int\limits_{0}^{h}\left[(q+\gamma z)ka-2c\sqrt{ka} \right]\cdot[\frac{\delta\mathrm{max}}{2}\sin\frac{\pi x}{L}(1+\cos\frac{\pi z }{H})]dzdx=-T_{pa}\delta_{0} \tag{12}\]  
The expression of passive soil pressure potential energy, \(W_{pp}\), is given by Equation (13):  
\[W_{pp}=\int\limits_{0}^{l}\int\limits_{0}^{h}m(z-h)\cdot[\delta\mathrm{max} \sin\frac{\pi x}{L}(1+\cos\frac{\pi z}{H})]^{2}dzdx=T_{pp}\delta_{0} \tag{13}\]  
The expression of the active potential energy received by the retaining structure is given by Equation (14):  
\[W_{Z}=W_{pa}+W_{pp} \tag{14}\]