## 6 Objective 1: Local Ductility, Dynamic Analysis
Unlike what observed for beams, the mean values of \(\mu_{L\phi}\) for columns tend to decrease with the story number. For both, beams and columns, the mean values tend to increase with the seismic intensity and are larger for the \(NS\) that for the \(EW\) direction; the uncertainty in the estimation is moderate in most of the cases. The statistics for the 10-level building (Table 4) resemble those of the 3-level building in the sense that the mean values of \(\mu_{L\phi}\) are much larger for beams than for columns and that the uncertainty in the estimation is, in general, moderate. However, unlike the 3-level building, the mean values tend to decrease with the story number for beams, and as for the 3-story building the \(\mu_{L\phi}\) mean values tend to decrease with the story number for columns; no yielding occurs in columns in most of the cases for low intensities of the strong motions (\(S_{a}\) = 0.2 g and 0.4 g). The only additional observation that can be made is that, for the larger strong motion intensities, the maximum mean values of the bending local ductility demands are observed to be larger for the 10-level building. Local ductility values associated to axial elongations (\(\mu_{L\delta}\)) were also calculated. It is worth to mention that since axial deformations of beams are negligible in comparison with those of bending, \(\mu_{L\delta}\) is not calculated for beams. In other words, plasticization of beams (formation of plastic hinges) is essentially produced by bending. In addition, it was observed that, due to their location, the axial loads at interior columns are smaller than that of exterior columns in such a way that plasticization of interior columns are essentially due to the action of bending moments too. Thus, that the \(\mu_{L\delta}\) parameter is calculated only for exterior columns  
Similar to the case of \(\mu_{L\phi}\), for a given story and building, the \(\mu_{L\delta}\) values are averaged (but in this case only over the two exterior columns) and graphs for individual members are developed; only the fundamental statistics for the 3-level building are presented (Table 5), however. The main observations that can be made are that the mean values of \(\mu_{L\delta}\) are comparable to those of \(\mu_{L\phi}\) for columns but much smaller than those of \(\mu_{L\phi}\) for beams. However, unlike the case of \(\mu_{L\phi}\) for beams or columns, the maximum values occur for the upper story. The previous discussion clearly indicates that the ductility demands associated to column axial deformations of steel framed structures are much smaller than those associated to bending of beams; for example, for the 3-level building and the deformation state close to collapse (\(S_{a}\) = 1.2 g), the average ductility demand associated to bending of beams ranges from 7.8 to 11.86, while that associated to axial elongation on columns ranges from 1.00 to 3.57. Thus, even though tensional ductility demands are considered as an important issue in some research reports [9] and steel structural members may have a high ductility capacity in tension, these types of ductility demands are less relevant for the type of the structural system under consideration.