## 3 Results  
### Processing Method for Strain Data of the Retaining Structure  
The curvature distribution of the pile body was used to calculate the lateral displacement of the retaining structure. The distribution of the pile body's curvature (\(\omega\)) along the depth (\(z\)) of the pile body was as follows, according to the elastic beam theory:  
\[\omega(z)=\frac{\epsilon_{A}-\epsilon_{B}}{D} \tag{3}\]  
Figure 11: Test group. (**a**) Single-layer strutted retaining structure; (**b**) double-layer strutted retaining structure. Figure 10: Test arrangement. (**a**) U-shape sheet pile wall interlocking connection; (**b**) sheet pile wall arrangement and sand filling. In this equation, \(\varepsilon_{A}\) is the strain value of side \(A\) of the sheet pile; \(\varepsilon_{B}\) is the strain value on side \(B\); and \(D\) is the thickness of the sheet pile wall (m). The lateral displacement calculation method introduced the finite integral method (FIM) [23]. The principle is based on the known values of the derivative function (\(\delta^{\prime}\)), which are used to perform numerical analysis within the defined domain \([0,\mathrm{h}]\) to obtain the corresponding function values \(\delta\). We conducted a numerical analysis of the curvature distribution of the U-shape sheet piles based on this principle. The U-shape sheet pile's height range \([0,\mathrm{h}]\) was divided into even segments each with a length of \(c\). As shown in Figure 12, the design divided the sheet piles into 18 segments along the axial direction, yielding \(c=1.8\) m/18 = 0.1 m.  
This article used the first-order derivative function value (\(\delta^{\prime}\)) of displacement at nodes within the interval of depth \(z\in[ic,\)\((i+2)\)\(c]\) for the U-shape sheet pile to perform quadratic parabolic interpolation, where \(i\) = (0, 2, 4, 6, 8, 10, 12, 14, 16), and after integration, we obtained:  
\[\left\{ \begin{array}{l}\delta_{i+1}=\frac{c}{12}\left(5\delta^{\prime}_{i}+ 8\delta^{\prime}_{i+1}-\delta^{\prime}_{i+2}\right)+\delta_{i}\\ \delta_{i+2}=\frac{c}{3}\left(\delta^{\prime}_{i}+4\delta^{\prime}_{i+1}+ \delta^{\prime}_{i+2}\right)+\delta_{i}\end{array} \right. \tag{4}\]  
Based on Equation (4), the relationship between the lateral displacement value \(\left\{\delta\right\}\) and the first-order derivative function value \(\left\{\delta^{\prime}\right\}\) of displacement could be obtained as follows:  
\[\left\{\delta\right\}=\frac{c}{12}\left[N\right]\left\{\delta^{\prime}\right\} +\delta_{0}\left\{I\right\} \tag{5}\]  
In this equation, \(\left\{I\right\}\) is the unit column vector of 19 elements; \(\left[N\right]\) is a square matrix of size 19; and \(\delta_{0}\) is the lateral displacement of the pile top (m). \[\left[N\right]= \begin{bmatrix}0&0&0&0&0\\ 5&8&-1&0&0\\ 4&16&4&0&0&\ldots\\ 4&16&9&8&-1\\ 4&16&8&16&4\\ &&\vdots&&&\end{bmatrix}  \tag{6}\]  
Similarly, according to Equation (5), the relationship between the first-order derivative value \(\left\{\delta^{\prime}\right\}\) and the second-order derivative value of lateral displacement \(\left\{\delta^{\prime}\right\}\) could be obtained as follows:  
\[\left\{\delta^{\prime}\right\}=\frac{c}{12}\left[N\right]\left\{\delta^{ \prime\prime}\right\}+\delta^{\prime}_{0}\left\{I\right\} \tag{7}\]  
By solving Equations (5) and (7) simultaneously, we established a relationship between the lateral displacement value \(\left\{\delta\right\}\) and the second-order derivative value of lateral displacement \(\left\{\delta^{\prime\prime}\right\}\). According to the theory of small deflection, the curvature was treated as  
Figure 12: Discretization diagram of the FIM for retaining structures. (\(\omega=-\delta^{\prime\prime}\)).