Ductility Demand Distribution Patterns

In the conventional DBD in which the ductility is assumed uniform over the height, the effective stiffness would not change with the ductility. If the ductility is distributed according to the ductile design of braces over the height for example based on the elastic modal vibration of the structure, the brace characteristics will interfere with the lateral displacement. The increase in the resulted ductility in comparison with the uniform distribution, will cause reduction in the effective mass and results in an increase in the effective period due to the rise in effective displacement and thus reduce the resulted effective stiffness of the substitute SDOF structure. In the presented DBD method, the lateral displaced shape of the structure is modified using multi-modal, polynomial and exponential distributions of ductility over the height of the structure in order to take into account higher mode effects and combined shear and flexural lateral deformations. Higher mode effect cause considerable changes in the dynamic response of large period or flexible structures such as tall buildings. Besides, the ductile behavior of the building also results in considerable increases in the system period. This issue has been discussed in the parametric study. In low rise buildings the shear behavior often governs the response and in medium rise buildings a combined shear and flexural deformation is normally expected.

For modal distribution of ductility it is assumed that the distribution is approximately conformed to the some first mode shapes which have a more than 98 percents of the system mass. For most of the structures, the first three modes of a cantilever with known equations may be assumed or alternatively an elastic modal analysis of the structure can be performed and then a mode combination procedure followed. However in order to avoid high strain demand in members in the structural design of the braced or wall buildings, the participation of higher modes must be limited. This issue is generally
considered in the capacity design of the structure so that the mass portion of the first mode does not decrease to values less than 70 percents or alternatively reduce the number of modes that own the 98 percents of the system mass. In the presented study, as an alternative approach and in order to conform to the capacity design criteria, the effect of higher modes on the ductility demand distribution in the stories below the effective height have been neglected,

I f H

i=1

The second selected pattern is an exponential function with parameter a as follows, ф =,, 1 – EXP(- ah/H)

ФлЕХР] ^max’ 1 – EXP(-a)

The third function is a simple polynomial function with parameter b as,

b

The comparison of various ductility patterns using the mentioned functions has been plotted in fig. 3. The final displaced shape is obtained by multiplying the initial profile obtained from the mechanism models, introduced in the previous section, by ductility demand distribution. The maximum ductility capacity of the each story (defined as the ratio of maximum displacement capacity and yield displacement of the story) must also be determined. This parameter defines the limit state or the performance point. For concentric and chevron braced frames this is governed by the capacity of the brace to beam connection and for eccentric braces the plastic rotation capacity of the link beam defines the ductility. Some modifications for the effects of strain hardening and cumulative damage can also be assessed in the DBD method that has been discussed in the next section.