## 2 Method of Analysis
### Solutions of Building Response
Since it is difficult to obtain an analytical solution of Equation (10), the finite difference method is used herein to solve Equation (10) numerically. As shown in Figure 5, the beam is divided into \(n\) elements of length \(l=L/n\), and \(L\) is the beam length. For the purpose of the differential operation, extra two virtual nodes are added at the two ends of the beam. Regarding the particular node \(i\), the following central difference forms can be deduced:  
\[\left(\frac{\mathrm{d}w}{\mathrm{d}y}\right)_{i}=\frac{-(w_{i+2}-w_{i-2})+8(w_ {i+1}-w_{i-1})}{12l} \tag{11}\]  
\[\left(\frac{\mathrm{d}^{2}w}{\mathrm{d}y^{2}}\right)_{i}=\frac{-(w_{i+2}+w_{i -2})+16(w_{i+1}+w_{i-1})-30w_{i}}{12l^{2}} \tag{12}\]  
\[p(y)=k[w(y)-w_{1}(x_{1},y_{1})]-G_{p}\frac{\text{d}^{2}[w(y)-w_{1}(x_{1},y_{1}) ]}{\text{d}y^{2}} \tag{6}\]0  
\[p(y)=k[w(y)-w_{1}(x_{1},y_{1})]-G_{p}\frac{\text{d}^{2}[w(y)-w_{1}(x_{1},y_{1}) ]}{\text{d}y^{2}} \tag{6}\]1  
where \(w_{i-2}\), \(w_{i-1}\), \(w_{i}\), \(w_{i+1}\), and \(w_{i+2}\) are the beam settlements at the nodes \(i-2\), \(i-1\), \(i\), \(i+1\), and \(i+2\), respectively. \[p(y)=k[w(y)-w_{1}(x_{1},y_{1})]-G_{p}\frac{\text{d}^{2}[w(y)-w_{1}(x_{1},y_{1}) ]}{\text{d}y^{2}} \tag{6}\]2 Soil–building interaction model. where \(\{W\}\) = \(\{w_{0}\), \(w_{1}\), \(w_{2}\), \(\ldots\), \(w_{n}\}^{\rm T}\) and \(\{F\}\) = \(\{f_{0}\), \(f_{1}\), \(f_{2}\), \(\ldots\), \(f_{n}\}^{\rm T}\) represent the beam settlement vector and the additional load vector, respectively; and \([K]\) is defined as the displacement stiffness matrix of the beam, which can be expressed as  
\[[K]=\left[ \begin{array}{cccccccccccc}\frac{30}{7}a_{1}+\frac{15}{7}a_{2}+a_{3 }&-\frac{37}{7}a_{1}-\frac{2}{7}a_{2}&\frac{16}{7}a_{1}+\frac{1}{7}a_{2}&&&\\ \frac{15}{7}a_{1}+a_{2}&-\frac{7}{7}a_{1}+a_{3}&\frac{1}{7}a_{1}+a_{2}&a_{1}&&& \\ a_{1}&a_{2}&a_{3}&a_{2}&a_{1}&&&\\ &a_{1}&a_{2}&a_{3}&a_{2}&a_{1}&&&\\ &&\ddots&\ddots&\ddots&\ddots&\ddots&\\ &&&a_{1}&a_{2}&a_{3}&a_{2}&a_{1}&\\ &&&&&&a_{1}&a_{2}&a_{3}&a_{2}&a_{1}\\ &&&&&&&a_{1}&\frac{1}{7}a_{1}+a_{2}&-\frac{9}{7}a_{1}+a_{3}&\frac{15}{7}a_{1}+a_ {2}\\ &&&&&&&\frac{16}{7}a_{1}+\frac{1}{7}a_{2}&-\frac{37}{7}a_{1}-\frac{2}{7}a_{2}& \frac{30}{7}a_{1}+\frac{15}{7}a_{2}+a_{3}\end{array} \right] \tag{27}\]  
Similar to Equation (15), Equations (17)-(19) can be further written in the matrix form as the following equations:  
\[\{\theta\}=[K_{\theta}]\{W\} \tag{28}\]  
\[\{M\}=[K_{M}]\{W\} \tag{29}\]  
\[\{Q\}=[K_{Q}]\{W\} \tag{30}\]  
where \(\{\theta\}\) = \(\{\theta_{0}\), \(\theta_{1}\), \(\theta_{2}\), \(\ldots\), \(\theta_{n}\}^{\rm T}\), \(\{M\}\) = \(\{M_{0}\), \(M_{1}\), \(M_{2}\), \(\ldots\), \(M_{n}\}^{\rm T}\), and \(\{Q\}\) = \(\{Q_{0}\), \(Q_{1}\), \(Q_{2}\), \(\ldots\), \(Q_{n}\}^{\rm T}\) denote the vectors of the rotation angle, bending moment, and shear force of the beam, respectively; \([K_{\theta}]\), \([K_{M}]\), and \([K_{Q}]\) represent the stiffness matrices of the rotation angle, bending moment, and shear force of the beam, respectively, which can be expressed as follows:  
\[[K_{\theta}]=\frac{1}{12l}\left[ \begin{array}{cccccccccccc}-\frac{90}{7}& \frac{96}{7}&&&&&&&&&\\ -\frac{41}{7}&-\frac{7}{7}&\frac{57}{7}&-1&&&&&&\\ 1&-8&0&8&-1&&&\\ &1&-8&0&8&-1&&&\\ &&\ddots&\ddots&\ddots&\ddots&\ddots&\\ &&1&-8&0&8&-1\\ &&&&&1&-8&0&8&-1\\ &&&&&1&-\frac{57}{7}&\frac{9}{7}&\frac{41}{7}\\ &&&&&&&\frac{6}{7}&-\frac{96}{7}&\frac{90}{7}\end{array} \right] \tag{31}\]  
\[[K_{M}]=-\frac{EI}{12l^{2}}\left[ \begin{array}{cccccccccccc}0&0&0&&&&&&&\\ \frac{97}{7}&-\frac{20l}{7}&\frac{111}{2}&-1&&&&&&\\ -1&16&-30&16&-1&&&\\ &-1&16&-30&16&-1&&&\\ &&\ddots&\ddots&\ddots&\ddots&\ddots&\\ &&&&&-1&16&-30&16&-1\\ &&&&&-1&116&-30&16&-1\\ &&&&-1&11&7&-\frac{201}{7}&\frac{97}{7}\\ &&&&&&&0&0&0\end{array} \right] \tag{32}\]\[\left[K_{Q}\right]=-\frac{EI}{2l^{3}}\left[