# Study on the Disturbance of Existing Subway Tunnels by Foundation Sloping Excavation
## 3 The Foundations of Theory Calculation
### Model of Foundation Sloping  
According to the previous analysis process, the magnitude and distribution width of the unloading load after the sloping excavation of the foundation pit must be determined first. There are usually two forms of sloping excavation: excavating the middle soil and reserving the core soil on both sides or excavating on both sides and reserving core soil the middle, as shown in Figure 4. In this figure, \(H\) is the excavation depth of the pit, (m); \(L\) is the excavation width of the pit, (m); \(\theta\) is the angle of the foot of the slope of the sloping excavation, (\({}^{\circ}\)); \(b\) is the excavation width of the sloping excavation body, (m); \(b_{1}\) is the width of the bottom of the pit of the sloping excavation body, \(b_{1}=b+H\mathrm{cot}\theta\), \(b_{2}=b-H\mathrm{cot}\theta\), (m). The numbers 1, 2 represent the order of excavation. Figure 4: Load calculation schematic diagram of the proposed sloping excavation. Figure 3: Simplified presentation of the Mindlin calculation. The vertical unloading loads of the two sloping excavation methods can be calculated from the density and volume of the excavated soil. The action position is still assumed to be at the bottom of the foundation pit. However, there is no research on the range and width of unloading load after sloping excavation. For the method of excavating the middle soil and reserving the core soil on both sides, the soil in the middle of the pit is excavated first. In this paper, the width of the unexcavated soil body's load distribution is the top edge of the trapezoid, the bottom edge of the trapezoid and the middle line of the trapezoid, respectively. The corresponding unloading distribution widths are \(l_{a1}=L-2b\), \(l_{a2}=L-2b\)\(-2H\cot\theta\), \(l_{a3}=L-2b-H\cot\theta\). For Figure 4a, the unloading load magnitude for the three load distribution width cases are formulated as follows:  
\[q_{a1}=\frac{\gamma H\alpha(2L-4b-2H\cot\theta)}{2(L-2b)} \tag{3}\]  
\[q_{a2}=\frac{\gamma H\alpha(2L-4b-2H\cot\theta)}{2(L-2b-2H\cot\theta)} \tag{4}\]  
\[q_{a3}=\frac{\gamma H\alpha(2L-4b-2H\cot\theta)}{2(L-2b-2H\cot\theta/2)} \tag{5}\]  
where \(\gamma\) is the soil weight, (N/m\({}^{3}\)) and \(\alpha\) is the residual stress coefficient. For the three load distribution cases using the method of excavating soil on both sides and reserving the middle core soil in Figure 4b, the proposed unloading load distribution widths are \(l_{b1}=b\), \(l_{b2}=b_{2}\), \(l_{b3}=(b+b_{2})/2\). The size of the unloading load is determined as follows:  
\[q_{b1}=\frac{\gamma H\alpha(2b-H\cot\theta)}{2b} \tag{6}\]  
\[q_{b2}=\frac{\gamma H\alpha(2b-H\cot\theta)}{2(b-H\cot\theta)} \tag{7}\]  
\[q_{b3}=\frac{\gamma H\alpha(2b-H\cot\theta)}{2(b-H\cot\theta/2)} \tag{8}\]  
For the method of excavating the middle soil and reserving the core soil on both sides, Figure 5 shows the calculation diagram of the lateral unloading load. For the excavation method of reserving core soil on both sides, the unloading effect on the sides is basically very small because the enclosure structure is applied on both sides and there are retained soil bodies on both sides. Therefore, our study assumes that there is no consideration of unloading effects on sides 2 and 3. After the excavation of the middle soil, the free face will be formed at sides 4 and 5. Considering the restraint of the deformation of the underground diaphragm wall by the sloping soil on both sides, the width of the unloading surface is assumed as \(l=(L-2b-2 Hcot\theta)\) in this study for sides 4 and 5. The unloading load magnitude of the lateral sides is calculated as  
\[\sigma_{\text{pt,5}}=\gamma HK_{0}a \tag{9}\]  
Figure 4:0 Lateral unloading load calculation diagram. where \(a\) is the unloading discount factor considering the role of the support structure, which generally takes the value of 0.1-0.3. When considering the most adverse effect, it takes a value of 0.5.