## 5 Concentrated vs. Consistent Mass
### Comparison for Local Parameters  
The axial loads and flexural moments acting on the members of the models with the lumped mass matrix are now compared with the corresponding ones from the consistent mass matrix. For axial loads, the comparison is made for interior and exterior columns, whereas for the case of flexural moments, in addition to columns, a comparison is also made for interior and exterior beams, from the base up to the top. The ratios \(R_{A}\) and \(R_{B}\) given by Equations (5) and (6) are used to make the comparison for axial loads and flexural moments, respectively. \[R_{A}=\frac{A_{ML}}{A_{MC}} \tag{5}\]  
\[R_{B}=\frac{B_{ML}}{B_{MC}} \tag{6}\]  
In the earlier equations, the symbols \(A\) and \(B\) stand for axial and bending, respectively, so, similar to the \(R_{V}\) ratio, \(A_{ML}\) for example, represents the axial load demands on the frames when \(M_{L}\) is used. Since the results are quite similar for both horizontal directions, the discussion is mainly centered on those of the \(NS\) direction. In addition, only the mean values are presented. The axial loads are compared first. The \(R_{A}\) mean values for exterior columns can be seen in Figure 8a,c,e, for Models 1, 2 and 3, respectively, while the corresponding mean values for the interior ones are given in Figure 8b,d,f. It can be seen that, for exterior columns, unlike shears and displacements, the \(R_{A}\) mean values are in many cases significantly larger than unity implying that the axial loads on these structural members can be considerably overestimated if the \(M_{L}\) model is used. Values of up to 1.6 can be seen for inter-story 2 of Model 1 implying, on an average basis overestimations of up to 60%. Even for elastic structural behavior (\(Sa_{\text{avg}}=0.2\) g) \(R_{A}\) mean values of about 1.3 can be observed. Overestimation larger than 100% (not shown) occurred for individual seismic records. Excepting the last inter-story of the 3- and 9-story models, the values increase as \(Sa_{\text{avg}}\) increases; however, they tend to decrease with the building height, so for the tallest model the maximum level of overestimation is observed to be about 5%. There are many considerable differences in the responses for interior columns with respect to those of exterior columns. The axial loads on interior columns of the 3-story model, the individual, and average underestimations are up to 44% and 30%, respectively. The corresponding values for the interior columns of the 9- and 20-story models are about 18% and 8%. One of the reasons for this is that, for a given floor, the interaction between beams and exterior columns is expected to be very different from that of the interiors, since the former are connected to columns only at one end, while the latter are connected at both ends. Figure 8: _Cont._The mean values of \(R_{B}\) for the \(NS\) direction, for all columns and models, are shown in Figure 9. It is observed that for exterior columns (Figure 9a,c,e) the flexural moments may be considerably underestimated when \(M_{L}\) is used. The amounts of underestimation increase as \(Sa_{avg}\) and the story number increase; in contrast, they tend to decrease as the model becomes taller. The maximum values are about 0.65, 0.68 and 0.80, indicating average underestimations of 35%, 32% and 20%, for Model 1, 2 and 3, respectively. Considering individual seismic records, the corresponding maximum levels of underestimation are about 66%, 58% and 42%. As for axial loads, the underestimation of flexural moments on interior columns is less significant than that of the exteriors; for Models 1 and 2 the maximum individual and average underestimations are around 21% and 11%, respectively; for Model 3, on the other hand, the flexural moments are essentially the same for \(M_{L}\) and \(M_{C}\). The mean values of \(R_{B}\) for exterior and exterior beams were also calculated, but the results are not presented. It is worth mentioning, however, that the level of underestimation is much smaller than that of interior columns.