## 5 Concentrated vs. Consistent Mass  
### Comparison for Global Parameters  
Global response parameters, i.e., inter-story shears and lateral drifts, are estimated for each story, structural model, direction, strong motion, and seismic intensity, by considering that the mass distribution in the structure is given by the \(M_{L}\) matrix and are compared to the corresponding responses obtained for the \(M_{C}\) matrix. Before presenting the comparison, it is worth mentioning that, considering only the lateral periods of vibrations, the first and second periods are quite similar for the \(M_{C}\) and \(M_{L}\) matrices for the case of Model 1. The third period, in contrast, is 28% greater for \(M_{C}\). For Model 2, the first five periods are essentially the same for the \(M_{L}\) and \(M_{C}\) matrices, but for modes 6 to 9, they resulted to be larger for the case of \(M_{C}\), with the differences ranging between 5% and 16%. Similarly, for Model 3, the periods are quite similar for modes 1 through 11 for both types of matrices, but for modes 12 to 20 they are larger for \(M_{C}\); varying the differences between 6 and 16%. Results for inter-story shears are compared first. To get this purpose the \(R_{V}\) parameter is used which is defined as  
\[R_{V}=\frac{V_{ML}}{V_{MC}} \tag{4}\]  
where, for a given inter-story, building and strong motion, \(V_{ML}\) and \(V_{MC}\) are the inter-story shears corresponding to the lumped and consistent matrices, respectively. The values of the \(R_{V}\) ratio for the \(NS\) direction of Model 1 are presented in Figure 6a,c,e for seismic intensities of 0.2 g, 0.6 g and 0.8 g, which correspond to small deformations (elastic behavior), moderate yielding, and significant yielding, respectively. The \(R_{V}\) values for the other horizontal (\(EW\)) direction are given in Figure 6b,d,f. The term \(IS\) abbreviates the expression inter-story. It can be seen from such figures that, if the structure remains elastic (Figure 6a,b), the values of \(R_{V}\) are very close to unity implying that the inter-story shears are essentially the same regardless of the mass matrix model. For moderate and significant yielding, however, values smaller than unity can be seen in many cases implying that the magnitude of the shears is underestimated when the \(M_{L}\) is used. Values of about 0.75 can be seen for some individual seismic records implying underestimations of 25%. The underestimation of \(R_{V}\) seems to increase as one moves up the frame. The values of \(R_{V}\) smaller than unity are due to the fact that the higher- mode contribution, which increases as one moves up the model, is larger for the case of the \(M_{C}\) matrix model. Figure 5: Variation of damping with the modal frequency in Rayleigh Damping Model. Despite the strong motions were normalized according to the \(S_{avg}\) intensity measure, a considerable dispersion is observed, specifically for large seismic intensities and inter-stories 2 and 3, which reflects, as stated in traditional structural dynamics textbooks [44, 50, 51], the influence of the inherent variability of seismic records on the response of structures, in addition to that of higher mode contribution. Plots for the \(R_{V}\) parameter, similar to those shown in Figure 6, were also developed for other seismic intensities for the case of Model 1. Including those of Figure 6, a total of 10 plots were developed for this model. In the same manner, sets of plots of \(R_{V}\) were developed for Models 2 and 3, but they are shown either, only the mean values are presented in all cases; they are shown in Figure 7. Results illustrate that, although for Model 1 underestimations of up to 25% were observed for inter-story shears for some strong motions when \(M_{L}\) is used, on an average basis the underestimation is negligible (about 4%). Figure 6: Values of \(R_{V}\); Model 1; (**a**,**c**,**e**), \(S_{avg}\) = 0.2, 0.6, 0.8 g, respectively, \(NS\) direction; (**b**,**d**,**f**), \(S_{avg}\) = 0.2, 0.6, 0.8 g, respectively, \(EW\) direction.