# An Equivalent Non-Uniform Beam-Like Model for Dynamic Analysis of Multi-Storey Irregular Buildings
## 3 Applications
In order to make an appropriate comparison between the two models, the in-plane rigid floor, no live loads and inextensible concrete columns were considered in the FEM model. Aiming to simulate representative seismic inputs that may occur on the Italian peninsula, linear dynamic analyses were performed by considering two real seismic records [34] in the \(x\) and \(y\) directions that occurred in Santa Venerina (2018) and L'Aquila (2009), plotted, respectively, in Figure 5a,b. Some characteristic data of the two records (PGA, PGV, PGD, distance for epicenter) are reported in Table 1. In all the time-histories, constant modal damping ratios \(\xi_{j}=0.05\), for all considered modes, were assumed. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Record**} & \multirow{2}{*}{**State**} & \multirow{2}{*}{**Date**} & \multirow{2}{*}{**Mw**} & \multirow{2}{*}{**Dist.** & **PGA** & **PCV** & **PGD** & **Lat.** & **Long** \\  & & & & & (Km) & (cm/s) & (cm/s) & (cm) & (C) & (C) \\ \hline Santa Venerina & Italy & 2018-12-26 & 02:19:17 & 4.9 & 4.5 & -547.932 (N) & 37.128 (N) & 5.696 (N) & 37.644 & 15.116 \\ \hline L’Aquila & Italy & 2009-04-06 & 01:32:40 & 6.1 & 4.9 & 644.246 (E) & -42.720 (N) & 6.789 (E) & 42.342 & 13.380 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Records adopted in numerical simulation. Figure 4: Plan archetypes: (**a**) Type 1 low- or mid-rise and (**b**) Plan B mid- or high-rise buildings. Figure 5: Accelerograms of (**a**) Santa Venerina (2018) and (**b**) L’Aquila (2009). Aiming to numerically compare, for each considered building, the modal shapes that have been obtained by means of the beam-like and the FEM models, a consistency indicator is needed. The Modal Assurance Criterion (MAC) is here used calculating the following matrix elements:  
\[MAC_{ij}=\frac{\left(\boldsymbol{\varphi}_{i,BL}\cdot\boldsymbol{\varphi}_{j, FEM}^{T}\right)^{2}}{\left(\boldsymbol{\varphi}_{i,BL}\cdot\boldsymbol{\varphi}_{i,BL}^{T} \right)\left(\boldsymbol{\varphi}_{j,FEM}\cdot\boldsymbol{\varphi}_{j,FEM}^{T }\right)}, \tag{23}\]  
where each term of the MAC matrix is bounded between 0 and 1. Taking into account the orthogonality properties of the modes of vibration, a perfect coincidence between the modal shapes in the two models would provide a diagonal unitary matrix. Therefore, values close to 1 in the main diagonal of the matrix indicate a good correspondence of the modal shapes, while low values indicate that the modes are not consistent [35]. For the Type 1 building analyzed in this section, the matrix MAC_Tp1 assume the following expression:  
\[\text{MAC\_Tp1}= \begin{array}{cccc}&\varphi_{1,FEM}&\varphi_{2,FEM}& \varphi_{3,FEM}\\ \varphi_{1,BL}&0.9958&0.0002&0\\ \varphi_{2,BL}&0.0001&0.7971&0.513\\ \varphi_{3,BL}&0&0.0006&0.8702\end{array}  \tag{24}\]  
As it can be observed in Figure 7, and confirmed by the MAC_Tp1 matrix, there is satisfactory agreement between the results obtained for the beam-like model, for all the modal properties related to the first two translational modes and to the third torsional one. Figure 8 reports the displacement time histories of the control point in both the \(x\) and \(y\) directions, calculated by means of the beam-like and FEM models during the considered seismic excitation. In particular Figure 8a,b report, respectively, the \(x\) and \(y\) displacements due to the Santa Venerina earthquake loading. The correspondent results for the L'Aquila earthquake are reported in Figure 9.  
Once again, it must be remarked how the proposed approach is able to correctly reproduce the time history of a chosen point on the building. Finally, as it can be observed in Figure 10, also the maxima displacements of the same corner located at each floor, calculated by means of the beam-like model, give reliable results. Figure 8: Four-storey benchmark. Santa Venerina earthquake. Displacements time history along the (**a**) x and (**b**) y direction.