## 4 Discussion and Perspectives  
The work presented in this paper proposed a novel two-step approach to calibrate a reduced-order 3D Timoshenko beam model to accurately describe the structural response of complex tall buildings affected by stochastic loads without the need for time-consuming and expensive detailed finite element models. The novelty of the paper consists of merging two equivalent beam models proposed in the literature in order to enhance the advantages and to overcome the drawbacks of each of the two. The first step consisted of computing the mechanical characteristics of an equivalent sandwich beam model using the sub-structures approach, first proposed by [PERSON] and [PERSON], and evaluating both the static and modal response (natural frequencies and mode shapes). These response parameters were used in the second step to calibrate the mechanical features of the Timoshenko beam model recently proposed by these Authors, which is able to describe the coupled bending, shear and torsional behavior. The calibration was obtained by a suitable optimization genetic algorithm that can be driven to better describe the response parameters of interest using appropriate weights in the function to be minimized. The proposed procedure improves the equivalent sandwich beam model, whose dynamic response solution needs specific parameters depending on the mode shape of interest. On the contrary, once the equivalent Timoshenko beam is calibrated, an exact closed-form solution is obtained for all the modal response. A tall building with a square plan but different stiffness to horizontal loads in two directions was used as a case study. The main results, validated using a detailed finite element model, can be summarized as follows:  
1. the equivalent sandwich beam modal response differs by less than 12% from the detailed numerical model in the first five natural frequencies;
2. the modal response of the calibrated equivalent Timoshenko beam perfectly matches the corresponding sandwich beam model and therefore has the same accuracy in describing the actual tall building response;
3. seismic response maximum and average displacements estimated using direct time integration differ less than 4% from the detailed model;
4. computational time ratio for 10 s of seismic dynamic analysis between the detailed numerical model and the reduced-order equivalent Timoshenko beam model is higher than 300. The obtained results confirm the effectiveness of the proposed procedure to calibrate equivalent Timoshenko beam models without the need for preliminary time-consuming complex numerical finite element models to describe the tall building structural response. This is of paramount importance when performing structural optimization strategies for the global response in the context of reliability analysis. The main immediate perspective of this work is the validation of the proposed novel calibration procedure for complex irregular tall buildings (non-symmetric). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Seismic Direction** & **Response** & \multicolumn{2}{c}{\(\boldsymbol{\mu_{max}}\)} & \multicolumn{2}{c}{\(\boldsymbol{\mu_{ave}}\)} & \multicolumn{2}{c}{\(\boldsymbol{\mu_{\sigma}}\)} \\  & **FE Model** & **e-TBM** & **FE Model** & **e-TBM** & **FE Model** & **e-TBM** \\ \hline \(x\) & \(u(H)\) & 0.318 & 0.320 & 0.000 & 0.000 & 0.105 & 0.105 \\ \(x\) & \(V_{x}\) & 27410 & 34622 & \(-14.38\) & \(-12.4\) & 6687.9 & 6704.3 \\ \(y\) & \(\varphi(H)\) & 0.318 & 0.317 & 0.000 & 0.001 & 0.106 & 0.107 \\ \(y\) & \(V_{y}\) & 27549 & 31598 & \(-14.34\) & \(-13.8\) & 6835.5 & 6748.9 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Response statistics for the e-TBM and the FE model top displacements (\(u(H)\) and \(v(H)\)) and base shear (\(V_{x}\) and \(V_{y}\)). **Author Contributions:** Conceptualization, F.C., S.F., V.G. and M.G. ; methodology, F.C., V.G. and M.G. ; software, F.C. and S.F. ; validation, F.C., S.F., V.G. and M.G. ; data curation, F.C. and S.F. ; writing--original draft preparation, F.C. and M.G.