## Results
### Torsional Stiffness, Natural Frequencies and Static Displacements of The e-SBM  
The torsional stiffness can be computed using Equation (46)  
\[\begin{split} D_{t}&=\sum_{h=1}^{f}\left((S_{i})_{ y}(z_{i}-z_{CR})^{2}+(S_{i})_{z}(y_{i}-y_{CR})^{2}\right)=\\ &=2\cdot\left(S^{ifx}+S^{ify}\right)\cdot\left(2.15^{2}+6.45^{2 }+10.75^{2}\right)+2\cdot\left(S^{efy}+S^{efy}\right)\cdot 15.05^{2}=1.041\cdot 10^{ 13}\,\mathrm{N}\,\mathrm{m}^{2}.\end{split} \tag{81}\]  
The solution of the e-SBM is obtained by first computing the stiffness parameters solving Equation (31), and secondly, evaluating the natural frequencies solving the problem  
Figure 2: Bracing on the external frame in \(x\) direction.  
in (42). Table 2 summarizes the obtained results in terms of natural periods and compares these values with those computed using a detailed SAP2000 FE model.  
The first two modes are in the \(y\) and \(x\) direction (Figure 1), while the third mode shape is torsional. For these three modes, the error from the detailed FE model solution is below 11%. The torsional model has a satisfactory estimate given the inclusion of the warping contribution in the e-SBM. These results suggested that the sub-structure approach for the considered case study can be effectively used as a reference for the first five modal frequencies that will be used as quantities of interest in Equation (53). It is worth noting that, in this case, the effect of the bracing system on the external frames is significant: neglecting this term in the sub-structures would result in a first natural period rising to 9.135 s, more than double the actual value.  
Beside modal frequencies, the novel calibration procedure described in Section 2.3 requires knowledge of the static response of the e-SBM. In this case study the response displacements to uniformly distributed loads over the height, \(z\), of the building are used. In particular, the displacements at the top floor in the \(x\) and \(y\) directions, \(u\) and \(v\), obtained, respectively, for a static load in the \(x\) and \(y\) directions are computed using the modal analysis results, in terms of \(nm\) natural periods, \(T_{k}\), and modal shapes, \(\phi_{k}(z)\), \(k=1,\ldots,nm\).  
When a beam of length \(H\) is affected by a load \(F(z,t)\), the \(k\)-th elementary oscillator is ruled by the following equation  
\[\ddot{q}_{k}+2v_{k}\omega_{k}q_{k}+\omega_{k}^{2}q_{k}=\frac{f_{k}(t)}{m_{k}} \tag{82}\]  
with  
\[\omega_{k}=\frac{2\pi}{T_{k}},\quad m_{k}=\int_{0}^{H}\mu(z)\phi_{k}(z)^{2}dz \tag{83}\]  
and  
\[f_{k}(t)=\int_{0}^{H}F(z,t)\phi_{k}(z)dz. \tag{84}\]  
In this case study, a pseudo-static uniformly distributed load is considered, than can be represented by  
\[F(z,t)=F_{0}\sin\biggl{(}\frac{2\pi}{T}t\biggr{)}\quad\text{with}\;T\gg T_{k}. \tag{85}\]  
In this case, the steady-state response is given by  
\[v(z)=\sum_{k=1}^{\infty}q_{k}\phi_{k}(z) \tag{86}\]  
with  
\[q_{k}=\frac{f_{k}}{m_{k}\,\omega_{k}}=\frac{F_{0}\int_{0}^{H}\phi_{k}(z)dz}{ \omega_{k}\,\mu\int_{0}^{H}\phi_{k}(z)^{2}dz}, \tag{87}\]  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Mode** & \(\mathbf{T}^{e-SBM}\) & \(\mathbf{T}^{FEM}\) & **Error** \\  & [s] & [s] & [\%] \\ \hline
1 & 4.491 & 4.832 & 7.596 \\
2 & 4.457 & 4.779 & 7.222 \\
3 & 1.142 & 1.017 & \(-\)10.946 \\
4 & 1.119 & 0.989 & \(-\)11.618 \\
5 & 1.118 & 0.984 & \(-\)11.987 \\
6 & 0.624 & 0.464 & \(-\)25.604 \\
7 & 0.576 & 0.462 & \(-\)19.810 \\
8 & 0.537 & 0.307 & \(-\)42.761 \\
9 & 0.526 & 0.169 & \(-\)67.799 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Natural periods between the e-SBM and FE models.  
where the summation is stopped after a finite number of terms. In this case study, the first \(nm=6\) modes and \(F_{0}=100\) kN/m are assumed giving the following top floor, \(z=H\), static displacements  
\[u =0.467\,\text{m} \tag{88}\] \[v =0.478\,\text{m}. \tag{89}\]  
It is worth noting that these displacements differ less than 3.5% from the corresponding values computed using the detailed FE model (\(u_{FEM}=0.484\) m, \(\
u_{FEM}=0.493\) m).