## Results
### Calibration of The e-TBM  
The last step of the proposed procedure is the calibration of the mechanical parameters of the 3D Timoshenko beam model, e-TBM, using the optimization procedure presented in Section 2.3. Using the target response parameters of the e-SBM computed in Section 3.3. In particular, the results in Table 2 suggested using five natural frequencies together with the two static displacements in the \(x\) and \(y\) directions at the beam free end. The function in Equation (53) becomes:  
\[ \begin{split} f\big{(}A,A_{s,x}.A_{s,y},I_{x},I_{y},I_{t}\big{)}& =w_{1}\Bigg{|}1-\frac{\omega_{1}^{e-TBM}}{\omega_{1}^{e-SBM}} \Bigg{|}+w_{2}\Bigg{|}1-\frac{\omega_{2}^{e-TBM}}{\omega_{2}^{e-SBM}}\Bigg{|} +w_{3}\Bigg{|}1-\frac{\omega_{3}^{e-TBM}}{\omega_{3}^{e-SBM}}\Bigg{|}+\\ &+w_{4}\Bigg{|}1-\frac{\omega_{4}^{e-TBM}}{\omega_{4}^{e-SBM}} \Bigg{|}+w_{5}\Bigg{|}1-\frac{\omega_{5}^{e-TBM}}{\omega_{5}^{e-SBM}}\Bigg{|} +w_{6}\Bigg{|}1-\frac{u^{e-TBM}}{u^{e-SBM}}\Bigg{|}+w_{7}\Bigg{|}1-\frac{v^{ e-TBM}}{v^{e-SBM}}\Bigg{|},\end{split}  \tag{90}\]  
where the e-TBM response parameters are evaluated using a numerical model and the assumed design variables are  
* cross-section area, \(A\);
* shear area in \(x\) and \(y\) directions, \(A_{s,x}\) and \(A_{s,y}\);
* second inertia moments around \(x\) and \(y\), \(I_{x}\) and \(I_{y}\);
* torsional stiffness, \(I_{t}\). and the assumed weights, to make the convergence easier and to drive the solution to better approximate the first three natural frequencies and the two static displacements, are summarized in Table 3.  
Assuming the following data  
* \(L=179.4\) m (building height);
* \(M=2.792\times 10^{7}\) kg (total mass of the building);
* number of elements of the FE model equal to the number of floors, \(N_{FE}=46\);
* \(E=2.1\times 10^{11}\) N/mm\({}^{2}\) (Young's modulus);
* \(G=8.75\times 10^{10}\) N/m\({}^{2}\) (Shear modulus). and using genetic algorithms to minimize the function in (90), the design variables result as follows: \(A=36.7125\) m\({}^{2}\), \(A_{s,x}=0.1628\) m\({}^{2}\), \(A_{s,y}=0.1652\) m\({}^{2}\), \(I_{x}=168.1814\) m\({}^{4}\), \(I_{y}=173.9372\) m\({}^{4}\), \(I_{t}=6.5456\) m\({}^{4}\). Table 4 compares the response of the obtained e-TBM with the e-SBM and the detailed FE model. The calibration makes the e-TBM response almost identical to the e-SBM. This is confirmed by the relative difference between the e-TBM and FE model response (eF), which is very close to the corresponding values in the last column in Table 2. It is worth noting that the minimum of Equation (90) is very close to zero, i.e., 0.1194, and a sensitivity analysis suggests that a global minimum is found. Indeed, Figure 3 shows the sensitivity \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**k** & **1** & **2** & **3** & **4** & **5** & **6** & **7** \\ \hline \(w_{k}\) & 1.00 & 1.00 & 0.67 & 0.33 & 0.33 & 1.00 & 1.00 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Weights used in Equation (90). analysis for couples of design variables, keeping the remaining four at the optimal values. In particular Figure 3a-c show the variation of the objective function with the second inertia moments, \(I_{x}\) and \(I_{y}\), the shear areas, \(A_{s,x}\) and \(A_{s,y}\), the cross-section area and torsional stiffness, \(A\) and \(I_{t}\), respectively. The red dot in the three sub-figures indicates the value of the function in (90) at the optimal design parameters. Another interesting feature of this case study is that the weights used in Equation (90) seem to have no effect given the perfect match of the chosen response parameters. The ability of the e-TBM to describe the tall building dynamic structural response can be also verified with time domain seismic analysis. To this aim, spectrum-compatible acceleration time histories are generated using the method proposed in [18] and the corresponding response time series is computed by direct integration using [PERSON]'s method.