## 2 The Novel Equivalent Beam Model Calibration  
### The Timoshenko Beam with De Saint-Venant Torsional Behavior (e-TBM)  
The equivalent beam model presented in [8] to estimate the three-dimensional response of asymmetric tall buildings was based on the Timoshenko beam with coupled De Saint-Venant torsional behavior (e-TBM). As it is well known, the Timoshenko beam model enhances the Euler-Bernoulli beam model by considering the shear stiffness, the cross-section rotation inertia and the possibility to have deformation where the beam axis can also be not normal to the cross section. In this Section the ruling equations are briefly recalled and additional details can be found in [8]. Assuming a reference system where \(z\) is the beam axis, \(x\) and \(y\) are the cross-section principal axis, the problem in the planar case, \(y-z\), can be described by the following equations:* the balance equations \[\frac{dM_{x}}{dz} =V_{y}(z)-m_{x}(z)\] (1) \[\frac{dV_{y}}{dz} =-q_{y}(z),\] (2) where \(q_{y}(z)\) and \(m_{x}(z)\), \(V_{y}(z)\) and \(M_{x}(z)\) are the distributed load and moment, the shear and the bending moment, respectively. * the constitutive equations \[M_{x} =EI_{x}\,\frac{d\theta_{x}}{dz}\] (3) \[V_{y} =GK_{y}\,\gamma_{y},\] (4) where \(d\theta_{x}/dz\), \(\gamma_{y}\), \(EI_{x}\) and \(GK_{y}\) are curvature, shear deformation, bending stiffness and shear stiffness, respectively. * the compatibility equation, assumed as in [PERSON] (1921) \[\frac{du_{y}}{dz}=\gamma_{y}-\theta_{x}.\] (5) Assuming constant stifness along the beam axis, the following equations are obtained \[GK_{y}\left(\frac{d^{2}u_{y}}{dz^{2}}+\frac{d\theta_{x}}{dz} \right)+q_{y}=0\] (6) \[EI_{x}\frac{d^{2}\theta_{x}}{dz^{2}}-GK_{y}\left(\frac{du_{y}}{dz} +\theta_{x}\right)+m_{x}=0,\] (7) which can be extended to the equations of motion assuming inertial forces and moments per unit length and D'Alembert's principle \[GK_{y}\left(\frac{\partial^{2}u_{y}}{\partial t^{2}}+\frac{ \partial\theta_{x}}{\partial z}\right)+q_{y}-\rho A\frac{\partial^{2}u_{y}}{ \partial t^{2}}=0\] (8) \[EI_{x}\frac{\partial^{2}\theta_{x}}{\partial z^{2}}-GK_{y}\left( \frac{\partial u_{y}}{\partial z}+\theta_{x}\right)+m_{x}-\rho I_{x}\frac{ \partial^{2}\theta_{x}}{\partial t^{2}}=0,\] (9) where \(\rho\) is the mass density, assumed to be uniform over both the beam cross-section area and beam length. The previous planar model can be easily extended to the 3D case writing the equations of motion in the \(x\)-\(z\) plane and describing the torsional deformation by the classical [PERSON]'s model where, assuming the torsional moment, \(M_{t}\), stiffness, \(GJ_{t}\), and distributed moment, \(m_{z}\), and using the constitutive equation  
\[M_{t}=GJ_{t}\frac{d\phi}{dz}, \tag{10}\]  
and the balance equation  
\[\frac{dV_{y}}{dz} =-q_{y}(z),\]0  
one obtains  
\[\frac{dV_{y}}{dz} =-q_{y}(z),\]1Finally, assuming uniform mass density \(\rho\) and using D'Alembert's principle, the equation of motion becomes  
\[\frac{dV_{y}}{dz} =-q_{y}(z),\]2  
where \(I_{0}=\int_{\Omega}\bigl{(}x^{2}+y^{2}\bigr{)}d\Omega\) is the polar moment of inertia and \(\Omega\) is the cross section surface. The equations ruling the 3D dynamic problem can be obtained by coupling the Timoshenko beam model in the \(x\) and \(y\) directions with the torsional model in (13) and assuming a reference system with the origin in the cross-section centroid  
\[\frac{dV_{y}}{dz} =-q_{y}(z),\]3 \[\frac{dV_{y}}{dz} =-q_{y}(z),\]4 (15) \[\frac{dV_{y}}{dz} =-q_{y}(z),\]5 (16) \[\frac{dV_{y}}{dz} =-q_{y}(z),\]6 (17) \[\frac{dV_{y}}{dz} =-q_{y}(z),\]7 (18) \[\frac{dV_{y}}{dz} =-q_{y}(z),\]8  
where  
\[\frac{dV_{y}}{dz} =-q_{y}(z),\]9 \[M_{x} =EI_{x}\,\frac{d\theta_{x}}{dz}\]0 \[M_{x} =EI_{x}\,\frac{d\theta_{x}}{dz}\]1  
with \(S_{x}^{M}\), \(S_{y}^{M}\), \(I_{x}^{M}\), \(I_{y}^{M}\) and \(I_{xy}^{M}\) being the classical first and second inertia moments and \(x_{S}\) and \(y_{S}\) are the coordinates of the shear centre.