# An Equivalent Non-Uniform Beam-Like Model for Dynamic Analysis of Multi-Storey Irregular Buildings
## 3 Applications
### Type 3: Building with Planar Asymmetry and Vertical Non-Uniform Distribution of Mass and Stiffness  
Type 3 is represented by a six-storey building, with decreasing size of the column cross-sections along the height and eccentricity between the CS and CM at each floor. The planar distribution of the cross section of the columns is reported for each floor in Figure 16a, while Figure 16b shows a scheme of the FEM model. Figure 14: Eight-storey benchmark. L’Aquila earthquake. Displacements time history along the (**a**) x and (**b**) y direction. Figure 13: Eight-storey benchmark. Santa Venerina earthquake. Displacements time history along the (**a**) x and (**b**) y direction. As it has been already foretold, for this building due to the contemporary presence of planar and vertical irregularities, both the frequencies and the modes of vibration were taken into account in the minimization procedure in order to reliably evaluate the stiffness correction coefficients. With the aim of proving the latter statement, the matrices MAC_Tp3A and MAC_Tp3B obtained, respectively including or neglecting the modal shapes in the objective function, are reported in the following:  
\[\text{MAC\_Tp3A}= \begin{array}{cc}&\begin{array}{ccc}\varphi_{1,FEM}& \varphi_{2,FEM}&\varphi_{3,FEM}\\ \varphi_{2,BL}\\ \varphi_{3,BL}\\ \varphi_{1,FEM}&\varphi_{2,FEM}&\varphi_{3,FEM}\\ \varphi_{1,BL}\\ \varphi_{2,BL}\\ \varphi_{3,BL}\\ \end{array}\end{array} , \tag{26}\]  
\[\text{MAC\_Tp3B}= \begin{array}{cc}&\begin{array}{ccc}\varphi_{1,BL}\\ \varphi_{2,BL}\\ \varphi_{3,BL}\\ \varphi_{3,BL}\\ \end{array}\end{array}  \tag{27}\]  
The observation of the matrices clearly shows that the addition of the modal shapes is necessary in order to obtain accurate results. The minimization of the described objective function provides for this building the following values of the stiffness correction coefficients: \(k_{x}=0.4459;\quad k_{y}=0.5142;\quad k_{c}=0.5224\). Table 4 and Figure 17 compare the modal properties, periods and modal shapes, respectively, for the conventional FEM and the proposed beam-like model. From the comparison of modes of vibration, either graphical through Figure 17 or numerical by means of the MAC_Tp3A matrix, it can be observed that the results for the two considered models show good agreement and confirm the reliability of the proposed approach. Figure 16: Six storey building benchmark (Model 2): (a) the column cross section plan and (b) the FEM model. Figure 15: Eight-storey benchmark. Maxima floor displacements in the (a) Santa Venerina and (b) L’Aquila earthquakes. For this irregular building also, the time histories of the displacement components of the considered control point, shown in Figures 18 and 19, and the maxima displacements at each floor reported in Figure 20, are sufficiently accurate. With the aim of investigating the influence of the number of Rayleigh-Ritz components on the seismic response of the beam-like model, some further numerical analyses were developed varying the number of modes of vibration considered in the discretization. Figure 21 shows the maxima displacements of the control point at each floor of the Type 3 building, subjected to the L'Aquila earthquake, evaluated with a variable number of shape functions. As it can be clearly observed, the increase in the number of adopted shape functions allows to reduce the differences with respect to the results obtained through the FEM model. \begin{table}
\begin{tabular}{c c c c} \hline Type 3 & Mode 1 & Mode 2 & Mode 3 \\ \hline FEM & 0.854 & 0.837 & 0.766 \\ Proposed beam-like model & 0.854 & 0.837 & 0.766 \\ \hline \end{tabular}
\end{table} 
Table 4: Modal Period comparison. Figure 17: Six-storey benchmark. Modal shapes comparison between the (**a**) FEM and (**b**) beam-like models. Figure 18: Six-storey benchmark. Santa Venerina earthquake. Displacements time history along the (**a**) x and (**b**) y direction. Figure 19: Six-storey benchmark. L’Aquila earthquake. Displacements time history along the (**a**) x and (**b**) y direction.