## 2 The Novel Equivalent Beam Model Calibration
### The Sub-Structure Approach (e-SBM)  
The equivalent beam model proposed by [PERSON] and [PERSON] [11] is based on the sub-structure approach, which uses an equivalent cantilever sandwich beam to model the lateral load-resisting system of a tall building starting from the stiffnesses of the different structural sub-systems (e-SBM). The equivalent sandwich beam is given by coupling one Timoshenko beam with bending stiffness, \(D_{0}\), and shear stiffness, \(S\), and one Euler-Bernoulli beam with bending stiffness, \(D_{I}\), assuming to have the same horizontal displacements at all levels. The parameter \(D_{I}\) is also referred to as \"local\" stiffness since it is obtained from each of the subsystem columns and beams. Moreover, the 3D behavior is also described using the Vlasov non-uniform torsion theory. In this Section, the main features of the model described in [11] that will be used to develop the proposed novel equivalent beam model are recalled. In the sub-structure approach, each lateral load-resisting sub-system of the building (i.e., wall, truss, frame, coupled shear wall) is replaced by a sandwich beam or a Timoshenko beam with suitable stiffness parameters. In this paper, a tall building with two sub-systems, frame and truss, will be considered. For these two cases, Table 1 in [11] gives the following stiffness values:\(k\)-th frame with \(n+1\) columns, having cross section area, \(A_{ci}\), and inertia, \(I_{ci}\), \(i=0,\ldots,n\), and \(n\) beams having cross section inertia, \(I_{bi}\), \(i=1,\ldots,n\): \[D_{lk} =\sum_{i=0}^{n}EI_{ci}\] (20) \[D_{0k} =\sum_{i=0}^{n}EA_{ci}c_{i}^{2}\] (21) \[S_{k} =\left(S_{b}^{-1}+S_{c}^{-1}\right)^{-1}\] (22) \[S_{b} =\sum_{i=1}^{n}\frac{12 EI_{bi}}{l_{i}h}\] (23) \[S_{c} =\sum_{i=0}^{n}\frac{12 EI_{ci}}{h^{2}},\] (24) where \(c_{i}\) is the distance of the \(i\)-th column from the frame axis, \(l_{i}\) is the span of the \(i\)-th beam (i.e., distance between columns \(i\) and \(i-1\)) and \(h\) is the constant inter-storey height. It is worth noting that these values do not depend on the number of floors. * \(k\)-th truss with columns and bracing having cross section area, \(A_{c}\) and \(A_{d}\), respectively: \[D_{0k} =\frac{1}{2}EA_{c}L^{2}\] (25) \[S_{k} =\frac{2 EthL^{2}A_{d}}{d^{3}}\] (26) where \(L\) is the span of the truss, \(h\) is the inter-storey height and \(d\) is the length of each of the two cross-bracing at each storey, with \(d=\sqrt{L^{2}+h^{2}}\). In this case, a pure Timoshenko beam is assumed, since \(D_{I}\) is null. In the general 3D case, the tall building equivalent sandwich beam parameters can be found by using the approach based on the deformation energy and can be collected into three matrices, \([D_{0}]\), \([D_{I}]\) and \([S]\). In the following, adopting the assumptions in [11], the reference system \(x,y,z\) is used with the beam axis in the \(x\) direction; \(v\) and \(w\) are the displacements of the beam axis in the \(y\) and \(z\) directions, while \(\psi\), \(\chi_{y}\), \(\chi_{z}\) are the rotations about the \(x\), \(y\), and \(z\) axis, respectively. Finally, \(\gamma_{y}\) and \(\gamma_{z}\) are the shear deformations, while  
\[\psi^{\prime}=\theta_{B}+\theta_{S}, \tag{27}\]  
is the torsional deformation with the two components, \(\theta_{B}\) and \(\theta_{S}\), due to bending and shear of the single sub-structures, respectively. In the approach proposed in [12; 13] the following equations hold  
\[v^{\prime} =\chi_{y}+\gamma_{y} \tag{28}\] \[w^{\prime} =\chi_{z}+\gamma_{z}\] (29) \[D_{0k} =\sum_{i=0}^{n}EA_{ci}c_{i}^{2}\]0  
It is worth noting that neglecting the shear deformation, i.e., \(\gamma_{y}=\gamma_{z}=\theta_{S}=0\) one obtains the standard Vlasov model with \(v^{\prime}=\chi_{y}\), \(w^{\prime}=\chi_{z}\) and \(\psi^{\prime}=\theta_{B}\). The values of \([S]\), \([D_{0}]\) and \([D_{I}]\) are obtained assuming that the deformation energy of the tall building equivalent sandwich beam is the same of the sum of the deformation energy of all the \(N\) sub-structures  
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