## 2 Theoretical Methods
### Slip Surface  
For soil of limited width, there are two kinds of horizontal soil strips between the retaining wall and the existing underground structure. AB is the horizontal soil strip from the retaining wall to the side wall of existing underground structure, and CD is the horizontal soil strip from the retaining wall to the soil slip surface. AB and CD with an extremely small thickness \(dz\) are shown in Figure 2, which are behind the retaining wall at an arbitrary burial depth \(z\). AB disappears for semi-infinite soil or when the slip surface does not intersect the existing structure. The width of AB is \(B\) and the width of CD depends on the slip surface shape and its position. And they can be expressed by the superposition of the inclined angle \(\beta\) of the slip surface. The parameter \(t\) in Figure 2 indicates the ratio of limited soil width. \(t\) equals to \(B\)tan\(\beta\)/\(h\) when the slip surface is a plane, in which \(\beta\) is the slip surface angle with respect to the horizontal, and \(h\) is the depth of retaining wall. \(\beta_{z}\) equals to \(\beta\) when the slip surface is a plane. However, \(\beta_{z}\) changes with the burial depth \(z\) when the slip surface is curved. In this case, \(t\) needs to be calculated using the integral of cot\(\beta_{z}\). This study mainly focuses on \(0\leq t\leq 1\). As the distance \(x\) of an arbitrary point E on the horizontal soil strip AB or CD from the retaining wall increases, the angle between the principal stress direction and the horizontal direction \(\theta\) gradually increases from \(\theta_{\rm w}\) at the retaining wall surface (point A and C) to \(90^{\circ}\). Then it gradually decreases to \(\theta_{\rm w}\) at the existing side wall at point B because of the same soil-wall friction angle with the retaining wall. When \(x\) tends to \(B\), \(\theta\) tends to \(\theta_{\rm s}\) for horizontal soil strip CD. \(\theta_{\rm s}\) equals to \(\theta_{\rm w}\) to make the integral of the horizontal shear stress to be 0. The analysis of any point E on the horizontal soil strip AB and CD using the Mohr stress circle was implemented, as shown in Figure 3. The stresses \(\sigma\) and \(\tau\) also transform gradually from A to B and from C to D, respectively. There is an angle of \(\theta_{\rm\beta}\) between the principal stress direction and the vertical of slip surface at the active state. \(\theta_{\rm\beta}\) equals to (\(\pi\)/4+\(\varphi\)/2) according to the Mohr stress circle analysis. From Figure 2 we can obtain that \(\theta_{\rm s}-\theta_{\rm\beta}=\pi\)/2 - \(\beta_{\rm z}\). \(\sigma_{1}\) (\(\sigma_{1}=\gamma z+q\)) and \(\sigma_{3}\) (\(\sigma_{3}=K_{\rm a}\sigma_{1}\)) are separately the major and the minor principal stress in Figures 2 and 3, where \(\gamma\) is the soil saturated unit weight, \(q\) is the ground surface overload, \(K_{\rm a}\) is the coefficient of the active earth pressure, and \(\varphi\) is the soil internal friction angle. The deflection angle of the minor principal stress direction \(\theta_{\rm w}\) at depth \(z\) of the \(c\)-\(\varphi\) soil is deduced from Figure 3 as:  
\[\theta_{w}=\frac{\pi}{2}+\frac{\delta}{2}-\frac{1}{2}\mathrm{arcsin}\frac{ \sin\delta[\gamma z+q-c\cos\varphi+c_{w}\cot\delta(1+\sin\varphi)]}{\sin\varphi (\gamma z+q+c\cot\varphi)}, \tag{1}\]  
where \(c\) is the soil cohesion, \(c_{\rm w}\) is the soil-wall cohesion, and \(\delta\) is the soil-wall friction angle. When \(\delta=0\), \(\theta_{\rm w}\) is equal to \(\pi\)/2, and the soil arching effect disappears. In general, if \(c_{\rm w}\)cot\(\delta=c\)cot\(\varphi\), then \(\theta_{\rm w}=[\pi+\delta-\mathrm{arcsin}(\sin\delta/\mathrm{sin}\varphi)]/2\), which is consistent with [PERSON]'s [PERSON] (1966) results.