Investigated in this report is the basic premise that the roof displacement of a multistory building can be determined from the deformation of an SDF system. For this purpose, the response of both systems is determined rigorously by nonlinear response history analysis, without introducing any of the approximations underlying the simplified methods for estimating the deformation of an SDF system (see, e.g., FEMA-273 or ATC-40 guidelines). The statistics of the SDF-system estimate of roof displacement are presented for a variety of building frames and six SAC buildings subjected to ground motion ensembles.

Two sets of structural systems and ground motions are considered. The first set is generic one-bay frames of six different heights: 3, 6, 9, 12, 15, and 18 stories designed for ductility factor μ= 1, 1.5, 2, 4, and 6 subjected to 20 large-magnitude, small-distance records. The second set is six “SAC” buildings—9- and 20-story model buildings designed according to Los Angeles, Seattle, and Boston codes—subjected to 20 ground motion records representing 2% probability of exceedance in 50 years.

Presented are the statistics of two roof-displacement, ur, ratios, (ur*)SDF = (ur)SDF ÷ (ur)NL-RHA and (ur*)MPA =(ur)MPA ÷ (ur)NL-RHA, where the subscripts NLRHA, MPA, and SDF denote the exact peak value determined by nonlinear RHA, approximate value from modal pushover analyses (MPA), and the SDF-system estimate. The data presented include histograms of the 20 values, range of values, median value, and dispersion measure.

These data for generic frames indicate that the first-“mode” SDF system overestimates the median roof displacement for systems subjected to large ductility demand μ, but underestimates for small μ, The bias and dispersion tend to increase for longer-period systems for every value of μ. Similar data for SAC buildings demonstrate that the bias and dispersion on the SDF estimate of roof displacement increases when P-delta effects (due to gravity loads) are included. The SDF estimate of roof displacement due to individual ground motions can be alarmingly small (as low as 0.312 to 0.817 of the “exact” value for the six SAC buildings) or surprisingly large (as large as 1.45 to 2.15 of the “exact” value for Seattle and Los Angeles buildings), especially when P-delta effects are included. The situation is worse than indicated by these data because they do not include several cases where the first-“mode” SDF system collapsed but the building as a whole did not. This large discrepancy arises because for individual ground motions the SDF system may underestimate or overestimate the yielding-induced permanent drift in the “exact” response determined by nonlinear RHA.

While this discrepancy is not improved significantly by including higher “mode” contributions, the MPA procedure has the advantage of reducing the dispersion in the roof displacement and the underestimation of the median roof displacement for elastic or nearly elastic cases at the expense of increasing slightly the overestimate of roof displacement of buildings responding far into the inelastic range.


Civil and Environmental Engineering

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