## 2 Method of Analysis
### Solutions of Building Response  
The relative position between the shallowly buried tunnel and the strip foundation building in the general case is shown in Figure 2. Two coordinate systems are established to facilitate analysis (Figure 3). The origin O\({}_{1}\) and O are set at the tunnel face and the left end of the building (the end of the building that is first passed by the tunnel face), respectively. The coordinates of the ground and the existing building are described as (\(x_{1}\), \(y_{1}\), \(z_{1}\)) and (\(x\), \(y\), \(z\)), respectively. A unit width \(B\) is taken along the short side of the existing building [13]. The building axis (parallel to the long side of the existing building) is aligned with \(x_{1}\)-axis at an angle \(\alpha\), and \(\alpha>0\) indicates the counterclockwise alignment. The relationship between the two coordinate systems can be obtained as follows:  
\[\left\{ \begin{array}{l}x_{1}=(y+s_{2})\cos\alpha\\ y_{1}=(y+s_{2})\sin\alpha+s_{1}\\ z_{1}=z\end{array} \right. \tag{5}\]  
where \(s_{1}\) is the distance between the origin O\({}_{1}\) and the intersection of the building axis and the \(y_{1}\)-axis; \(s_{2}\) is the distance between the origin O and the intersection of the building axis and the \(y_{1}\)-axis. Figure 1: Greenfield surface settlement above a tunnel axis. In the second stage, the existing building is simplified as an Euler-Bernoulli beam with the equivalent stiffness on the Pasternak model, as shown in Figure 4, which is able to consider the interaction between adjacent springs. The masonry building and the framed building with strip foundations are simplified as the pure bending beam and the shear bending beam in this work, respectively [39]. It is assumed that the building foundation is always in contact with the underlying soil, and there is no slippage on the soil-foundation interface. Additionally, the soil is assumed to be isotropic elastic material and the plastic behavior of the soil is not involved here. Then, the subgrade reaction can be expressed as  
\[p(y)=k[w(y)-w_{1}(x_{1},y_{1})]-G_{p}\frac{\text{d}^{2}[w(y)-w_{1}(x_{1},y_{1}) ]}{\text{d}y^{2}} \tag{6}\]  
where \(w(y)\) is the building settlement; \(w_{1}(x_{1},y_{1})\) is the greenfield surface settlement at the corresponding building location calculated by Equation (1); \(k\) and \(G_{p}\) denote the coefficient of the subgrade modulus [40] and the shear stiffness of the subgrade [41], respectively, which are given by  
\[k=\frac{0.65E_{s}}{B(1-v^{2})}\sqrt[2]{\frac{E_{s}B^{4}}{EI}} \tag{7}\]  
\[G_{p}=\frac{E_{s}t}{6(1+v)} \tag{8}\]  
where \(E_{s}\) is the elastic modulus of the soil; \(EI\) is the equivalent bending stiffness of the existing building; and \(t\) is the thickness of the elastic layer, with \(t=2.5B\), as suggested by  
Figure 3: Profile view of building axis parallel to tunnel axis. Figure 2: Plan view of tunnel and building. Xu [42]. Note that when \(G_{p}\) is equal to zero, the Pasternak model can be degenerated into the Winkler model. By imposing the greenfield surface settlement on the existing building, the governing differential equation for the building settlement is provided below [43]:  
\[EI\frac{\mathrm{d}^{4}w(y)}{\mathrm{d}y^{4}}-G\frac{\mathrm{d}^{2}w(y)}{\mathrm{ d}y^{2}}+kBw(y)=Bf(x_{1},y_{1}) \tag{9}\]  
where \(G\) represents the shear stiffness parameter, with \(G=G_{p}B\) for masonry buildings and \(G=G_{f}+G_{c1}+G_{p}B\) for framed buildings; \(G_{f}\) is the equivalent shear stiffness of the framed building; \(G_{c1}\) is the linear bending stiffness of the columns on the bottom story. The values of \(G_{f}\) and \(G_{c1}\) can be determined with reference to [PERSON] et al. [44]; \(f(x_{1},y_{1})\) denotes the additional load on the existing building and can be expressed as  
\[f(x_{1},y_{1})=q+kw_{1}(x_{1},y_{1})-G_{p}\frac{\mathrm{d}^{2}w_{1}(x_{1},y_{1 })}{\mathrm{d}y^{2}} \tag{10}\]  
where \(q\) is the equivalent vertical uniform pressure of the building acting on the subgrade.