## 2 Theoretical Methods
### Soil Arching Analysis  
Based on the above research, the trapezoidal limited \(c\)-\(\varphi\) soil slip wedge and the stresses acting on the horizontal soil strip at depth \(z\) are shown in Figure 2. The soil slip wedge is surrounded by the red lines delimiting the ground surface ab, the existing side wall segment bc, the slip surface segment cd, and the retaining wall segment ad. The external forces acting on the horizontal soil strip within the slip wedge are the horizontal normal stress \(\sigma_{\rm wh}\) and shear stress \(\tau_{\rm w}\) near the retaining wall and the existing side wall; the vertical stress \(\sigma_{\rm s}\) and shear stress \(\tau_{\rm s}\) near the slip surface cd; and the vertical stress \(\sigma_{\rm v}\) and \((\sigma_{\rm v}+d\sigma_{\rm v})\) and shear stress \(\tau_{\rm x}\) and \((\tau_{\rm x}+d\tau_{\rm x})\) above and below the horizontal soil strip, respectively. Based on the Mohr stress circle analysis as shown in Figure 3, the stresses at any distance \(x\) from the retaining wall within the slip wedge are deduced as:  
\[\left\{ \begin{array}{l}\sigma_{h}=\left(\frac{1+\cos 2\theta\sin\varphi}{1+ \sin\varphi}\right)(\sigma_{1}+c\cot\varphi)-c\cot\varphi\\ \sigma_{\rm v}=\left(\frac{1-\cos 2\theta\sin\varphi}{1+\sin\varphi} \right)(\sigma_{1}+c\cot\varphi)-c\cot\varphi\\ \tau_{x}=\frac{\sin 2\theta\sin\varphi}{1+\sin\varphi}(\sigma_{1}+c\cot \varphi)\end{array} \right., \tag{4}\]  
in which \(\theta\) corresponds to \(x\). Equation (4) corresponds to the stress state at the wall surface (points A, B, C) when \(\theta=\theta_{\rm w}\), whereas it corresponds to the stress state at soil slip surface (point D) when \(\theta=\theta_{\rm s}\). The ranges of \(\theta\) for soil strips AB and CD are \([\theta_{\rm w},\pi-\theta_{\rm w}]\) and \([\theta_{\rm w},\pi-\theta_{\rm s}]\), respectively. In order to satisfy the shear stress integral of the horizontal soil strip to be zero within the slip wedge, i.e., \(\int_{-B_{2}/2}^{B_{2}/2}\tau(x)dx=0\), the stress arch of the horizontal soil strip must be bilaterally symmetric, in which \(B_{2}\) is the soil width at depth \(z\). \(B_{\rm z}=B\) when \(0\leq z\leq(h\text{-}\text{Btan}\beta)\), and the stress arch is identical everywhere. However, \(B_{\rm z}=(h\text{-}z)\cot\varphi\) when \((h\text{-}\text{Btan}\beta)<z\leq h\), and the arc length decreases as the burial depth increases. The stress state of the horizontal soil strip, in the active state, is like that of an inverted arch. In order to satisfy the self-support of the inverted arch, and the traits unchanged after compressing or stretching [17; 48; 49], the symmetric catenary curve is proposed to describe the stress arc of limited soil in active state and its mathematical expression is:  
\[y=\frac{a}{2}\bigg{[}\exp\left(\frac{x}{a}\right)+\exp\left(-\frac{x}{a} \right)-\exp\left(\frac{B_{\rm z}}{2a}\right)-\exp\left(-\frac{B_{\rm z}}{2a} \right)\bigg{]}\left(-\frac{B_{\rm z}}{2}\leq x\leq\frac{B_{\rm z}}{2}\right), \tag{5}\]  
where \(B_{\rm z}\) is the horizontal soil strip width at depth \(z\), \(a\) is the undetermined coefficient, which indicates the size of the catenary opening and is a positive value in the active state, and \(x\) is the horizontal distance from the vertex. When \(x=-B_{\rm z}/2\), \(y^{\prime}\) (\(x=-B_{\rm z}/2\)) = \(-\cot\theta_{\rm w}\), and thus:  
\[a=\frac{B_{z}}{2\ln\left(\cot\theta_{w}+\sqrt{\cot^{2}\theta_{w}+1}\right)}, \tag{6}\]  
Figure 5: Comparison of slip surface with centrifuge model tests. \[\cot\theta=-\frac{1}{2}\Big{[}\exp\left(\frac{x}{a}\right)-\exp\left(-\frac{x}{a} \right)\Big{]}. \tag{7}\]  
The stress values at any position on the horizontal soil strip within the slip wedge can be obtained by substituting Equation (7) into Equation (4), which is represented by a symmetric catenary arch curve.