## 6 One vs. More than One Element per Member
### Local Response Parameters, Two Elements per Beam (2E)  
To compare the axial loads on the columns of Models _1E_ with those of Model _2E_, the \(R_{A1}\)_ ratio, defined by Equation (8), is used. The numerator and denominator in such an equation have the same meaning as those of Equation (7), but now axial loads in columns are analyzed instead.  
\[R_{A1}=\frac{A_{1}}{A_{2}} \tag{8}\]  
The average of the \(R_{A1}\) values for the exterior columns as well as for the interiors are presented in Figure 11 for all the models. Only the results for the _NS_ direction are presented. Results from the figure indicate that, unlike the case of shears and drifts, axial loads on columns may be significantly overestimated when one element per beam is considered.  
Figure 10: Mean values of \(R_{V1}\); (**a,b**),_NS-EW_ direction, respectively, Model 1; (**c,d**),_NS-EW_ direction, respectively, Model 2; (**e,f**),_NS-EW_ direction, respectively, Model 3.  
For a given model the magnitude of the overestimation increases as the story number and the seismic intensity increase, but it decreases as the model becomes taller. The average overestimations can be up to 95%, 55% and 5%, for Models 1, 2 and 3, respectively; the corresponding maximum overestimations considering individual seismic records can be up to 140%, 86%, and 13%. Even for the case of elastic behavior, significant overestimations can be seen for the case of the 3-, and 9-story models.  
The \(R_{B1}\) ratio given by Equation (9) is used to compare the seismic responses in terms of flexural moments. Similar to the \(R_{V1}\) and \(R_{A1}\) parameters, \(B_{1}\) and \(B_{2}\) represent the flexural moments at columns for the 1E and \(2E\) structural representations.  
\[R_{B1}=\frac{B_{1}}{B_{2}} \tag{9}\]  
The \(R_{B1}\) averages for exterior columns as well as for the interiors are shown in Figure 12. As for the \(R_{A1}\) parameter, the results are similar for the two horizontal directions; consequently, only the values associated to the \(NS\) direction are presented. The mean values of \(R_{B1}\) resemble those of \(R_{A1}\) in the sense that, for the case of significant  
Figure 11: Mean values of \(R_{A1}\), \(NS\) direction: (**a,c,e**), exterior columns of Models 1-2-3, respectively; (**b,d,f**), interior columns of Models 1-2-3, respectively.  
structural deformations, the flexural moments of exterior columns may be significantly overestimated when no intermediate node is used, with the level of overestimation decreasing with the height of the model. The maximum overestimations (on average) are 48%, 32%, 11%, and for Models 1, 2 and 3, respectively; however, considering individual seismic records, the corresponding maximum overestimations are 68%, 59%, and 19%. For the lowest structural deformations, on the other hand, the flexural moments can be underestimated; on an average and individual basis, the maximum overestimations can be up to 12%, 24%, 29% and 5%, 10%, 14%, respectively. It is observed that the overestimation is much smaller for the case of interior columns.  
The \(R_{B1}\) averages for interior beams, as well as for the interiors, are presented in Figure 13 for all the models. It is seen that for elastic behavior (smallest values of \(S_{avg}\)), the flexural moments at both, exterior and interior beams are underestimated; the level of underestimation increases with the story number; the observed amounts of individual and average underestimation are about 38%, 60%, 45% and 19%, 35%, 22% for the Models 1, 2 and 3, respectively. For intermediate or large deformations, on the other hand, the values are essentially equal to unity in almost all cases, which  
Figure 12: Mean values of \(R_{B1}\), \(NS\): (**a**,**c**,**e**), exterior columns of Models 1, 2 and 3, respectively; (**b**,**d**,**f**), interior columns of Models 1, 2 and 3, respectively.  
moments in the beams without intermediate nodes are very similar to those resulting from the consideration of an intermediate node.