Response spectra of a three-dimensional frame building

As described in Linear analysis of a rod under dynamic loading, for structures with well-separated modes the TENP and the CQC methods reduce to the SRSS method, while the NRL and the ABS methods give similar results. Hence, for such structures, two summation rules would suffice, with ABS providing the more conservative results. However, when structures with closely spaced modes are analyzed, all five summation rules can yield very different results. This is even more apparent in three-dimensional problems. In the present example, the plane of the earthquake motion lies along the x-axis, so we expect that the structural response will be dominated by Frames 1 and 3 and will result in a significant base shear in the x-direction. All five methods are compared against a modal time history response using the same El Centro acceleration record in Table 1, where the base shear forces are summed up in the plane of each frame $S_i$, where i is the frame number. This comparison shows that the best approximation is generated by the CQC method. The other methods overestimate the shear in the y-direction, and some of them underestimate the base shear in the x-direction. The CQC method is generally recommended for asymmetrical three-dimensional problems with closely spaced structural modes. This method takes into account the sign of the mode shapes through cross-modal correlation factors and can correctly predict the response in directions perpendicular to the direction of excitation.