## 1 Introduction
### Theoretical Analysis of the Control Bending Moment at the Bottom of the Column  
Considering various loads such as soil weight [22], water pressure, and surface vehicle loads [23; 24], we can simplify the distribution of these pressures on the side walls of the underground metro stations as the trapezoidal distributed load [25]. In addition, we can facilitate the control bending moment of the target-wall column as a mechanical problem in which the column is fixed-supported at the end and elastic-supported both in the middle and on the top under a trapezoidal distributed load with a specific gradient [26]. Based on the Code for metro design, the impermeability grade of the side wall of the metro station is P10, and the tolerated range of deformation is 0.2 mm, accordingly. Under this condition, the inner forces of side walls satisfy the linear superposition principle when the element is in the elastic stage [27]. Therefore, the trapezoidal distributed load can be decomposed into uniformly distributed load q and triangularly distributed load Q, as shown in Figure 8.  
Under the uniformly distributed load, elastic supports at the top and the middle can be removed and replaced by unknown forces X1 and X2, respectively, as shown in Figure 9. Thus, the statically indeterminate structure is simplified as a statically determinate structure with additional unknown forces X1 and X2 [28], and then the basic structure of force method under the uniformly distributed load is obtained (shown as Figure 9). Figure 8: Simplified calculation model of wall column. Figure 7: Mechanical model of wall column. where  
\(\delta_{\rm ii}=\int_{L}\left[\frac{1}{EA}\overline{F}_{\rm Ni}^{2}+\frac{K}{GA} \overline{F}_{\rm Ci}^{2}+\frac{1}{EI}\overline{M}_{\rm i}^{2}\right]\)\(ds\) indicates the virtual displacement of the \(i\)-th support of the system when \(X_{\rm i}=1\) is acting alone (\(i=1,\ 2\)). \(\delta_{\rm ii}=\delta_{\rm ii}=\int_{L}\left[\frac{1}{EA}\overline{F}_{\rm Ni }\overline{F}_{\rm Ni}+\frac{K}{GA}\overline{F}_{\rm Ci}\overline{F}_{\rm Ci} +\frac{1}{EI}\overline{M}_{\rm i}\overline{M}_{\rm i}\right]\)\(ds\) indicates the virtual displacement of the \(j\)-th support of the system when \(X_{\rm i}=1\) is acting alone (\(i=1,\ j=2\); or \(i=2,\ j=1\)). \(\Delta_{\rm iq}=\int_{L}\left[\frac{1}{EA}\overline{F}_{\rm Ni}\overline{F}_{ \rm Ni}\right]\)\(\rm{x}_{\rm{x}}+\frac{K}  
The deformation of the structure under the action of its weight is minimal and negligible, so it can be assumed that the system has no initial strain, that is, the initial normal strain, shear strain, and bending strain are all zero, shown as Equation (22). \[\epsilon_{0}=0,\gamma_{0}=0,\kappa_{0}=0 \tag{22}\]  
Then we can obtain \(\Delta_{10}=\Delta_{20}=0\). Then Equation (20) can be simplified as Equation (23). \[\left[ \begin{array}{cc}\delta_{11}\ \delta_{12}\ \Delta_{1q}\\ \delta_{21}\ \delta_{22}\ \Delta_{2q}\end{array} \right]\left\{ \begin{array}{cc}X_{1} \\ X_{2}\\ 1\end{array} \right\}=-\frac{1}{k_{eq}}\left\{ \begin{array}{cc}X_{1}\\ X_{2}\end{array} \right\} \tag{23}\]  
When the bending deformation is the main form of the wall column, we can ignore the influence of axial force and shear force, and conclude Equations (24) and (25). \[\delta_{11}=\frac{L^{3}}{3 EI},\ \delta_{22}=\frac{L^{3}}{24 EI},\ \delta_{12}= \delta_{21}=\frac{5L^{3}}{48 EI} \tag{24}\]  
\[\Delta_{1p}=-\frac{qL^{4}}{8 EI},\ \Delta_{2p}=-\frac{17 qL^{4}}{384 EI} \tag{25}\]  
where \(E\), \(I\), and \(L\) are the elastic modulus, the section moment of inertia, and the calculated length of the wall column, respectively. Let \(t=L^{3}/EI\), \(t_{b}=L_{b}{}^{3}/E_{b}I_{b}\), and \(\alpha=t_{b}/t\). Then, the rigidity \(k\) can be obtained, shown as Equation (26). \[k=1/\delta_{b}=192E_{b}I_{b}/L_{b}{}^{3}=192/\alpha t \tag{26}\]  
Substituting Equation (26) into Equation (23), we obtain the solutions of \(X_{1}\) and \(X_{2}\), respectively, shown as Equations (27) and (28).