## 2 Theoretical Methods
### Slip Surface
According to Figure 3 and Equation (1), the slip surface angle \(\beta\) of \(c\)-\(\varphi\) soil at depth \(z\) of horizontal soil strip CD is  
\[\beta_{z}=\frac{\pi}{4}+\frac{\varphi-\delta}{2}+\frac{1}{2}\mathrm{arcsin} \frac{\sin\delta[(\gamma z+q)-c\cos\varphi+c_{w}\cot\delta(1+\sin\varphi)]}{ \sin\varphi(\gamma z+q+c\cot\varphi)}, \tag{2}\]  
Figure 3: Mohr stress circle analysis. Figure 2: Stress analysis of horizontal soil strip within the slip wedge. thus, \(\beta_{\rm z}\) is not affected by \(B\), but increases with increasing \(\gamma\), \(q\), and \(\varphi\). \(\beta_{\rm z}\) is a function of \(\varphi\) and \(\delta\) and the slip surface is a plane when \(c_{\rm w}=c\cot\varphi/\cot\delta\). \(\beta_{\rm z}\) is equal to (\(\pi/4+\varphi/2\)) when \(\delta=0\), which is consistent with Rankine's slip surface angle, and the soil arching effect disappears. Furthermore, \(\beta_{\rm z}\) is \(\varphi/2\) and the slip surface coincide with the retaining wall when \(\delta=\varphi\). The relationship between \(c_{\rm w}\) and \(\delta\) are discussed to get clear the soil-wall interaction mechanism. Here we assume that \(c_{\rm w}\) and \(\delta\) are independent from each other; thus, the influence of \(c_{\rm w}\) and \(\delta\) on \(\beta\) is shown in Figure 4a,b, respectively, in which \(\gamma=19\) kN/m\({}^{3}\), \(q=0\), \(\varphi=37^{\circ}\), \(\delta=25^{\circ}\), and \(c=30\) kPa, then \(c\times\cot\varphi/\cot\delta=18.564\) kPa. When \(\beta=90^{\circ}\), \(c_{\rm w}=23.118\) kPa. Comparison of Figure 4a,b, it shows that the influence of \(\delta\) on the curvature of the slip surface can offset the effect of \(c_{\rm w}\) and \(\beta\) independents of \(z\). Then the slip surface is a plane, such as \(c_{\rm w}=18.564\) kPa in Figure 4a and \(\delta=25^{\circ}\) in Figure 4b. For the limit equilibrium method of a horizontal soil strip, the horizontal force is not disturbed when the burial depth \(z\) is deeper than the wall depth \(h\). After this, the slip surface needs to pass through the wall toe. Furthermore, when \(c_{\rm w}\cot\delta=c\cot\varphi\), the double-fold line slip surface for limited soil is a plane passing through the wall toe and intersecting the side wall of the existing underground structure, then till the ground surface along the existing wall. The inclination angle of the double-fold line slip surface for the limited \(c\)-\(\varphi\) soil can be simplified as:  
\[\beta=\left\{ \begin{array}{cc}\frac{\pi}{2}&;(0\leq z\leq(h-B\tan\beta))\\ \frac{\pi}{4}+\frac{\varphi-\delta}{2}+\frac{1}{2}\arcsin\frac{\sin\delta}{ \sin\varphi};((h-B\tan\beta)<z\leq h)\end{array} \right.. \tag{3}\]  
Figure 5 shows the comparison of the double-fold line slip surface from Equation (3) with the results obtained from centrifuge model tests [6]. The results of this study are consistent with those obtained from centrifuge model tests. Therefore, by setting \(c_{\rm w}=c\times\tan\delta/\tan\varphi\), the soil-wall interaction can be determined by \(\delta\) and the calculation of active earth pressure distribution based on the slip surface is simplified. \(c_{\rm w}=c\times\tan\delta/\tan\varphi\) is assumed in the following calculation procedure. Figure 4: Influence of (**a**) \(c_{\rm w}\) and (**b**) \(\delta\) on the slip surface.