Response spectrum analysis - RSA

Response spectrum analysis (RSA) [e.g. Rosenblueth, 1951; Chopra, 2001; EN 1998-1, 2004], is a linear elastic static - (pseudo)dynamic statistical analysis method which provide the peak values of response quantities, such as forces and deformations, of a structure under seismic excitation. It is called (pseudo)dynamic because the peak response can be estimated directly from the response spectrum of the ground motion by carrying out static analysis rather than time-history dynamic analysis. In this context, the time - acceleration history imposed to the supports of the structure is replaced by the equivalent static forces, which are distributed to all free DOFs of the structure and they represent the contribution from each natural mode of vibration. These equivalent forces are derived per mode as the product of two quantities: (1) the modal inertia force distribution (thus Eigenvalue analysis is needed), and (2) the pseudo-acceleration response per structural mode (obtained from the response spectrum). For each meaningful mode a static analysis is conducted, and then every final peak response quantity is derived by the superposition of the quantities corresponding to each analysis.

It is noted that it is not possible to determine the exact peak value, because, in general, the modal response quantities attain their peaks at different time instants. Approximations are introduced by implementing one of the modal combination (statistical) rules, such as the absolute sum (ABSSUM), square-root-of-sum-of-squares (SRSS) and the complete quadratic combination (CQC). CQC suggested when periods are closely spaced, with cross-correlation between mode shapes. SRSS can be used when periods differ by more than 10%, whilst ABSSUM offers an upper limit of response. This procedure is repeated for each desired seismic direction by using different or same response spectra. RSA is typically considered as useful for design decision-making because it relates structural type-selection to dynamic performance. Structures of shorter period experience greater acceleration, whereas those of longer period experience greater displacement. For design purposes, it is usually requested that two or three seismic loading directions (EX, EY, EZ) are to be considered simultaneously, together with the gravity static loads (G+Q) of structure.

Users are asked to provide as input the response spectrum and the seismic loading combination(s) for which the RSA will output the results. The response spectrum may be defined directly or may be calculated from a given accelerogram. This spectrum is employed for both the two horizontal (EX, EY) and the vertical (EZ) seismic directions. Alternatively, different response spectrum factors between horizontal and vertical directions may be defined in the loading combination module. Further, the modal combination rule (ABSSUM, SRSS, CQC) should be determined, as well as which modes are to be combined, as a function of the target cumulative effective modal mass.

Finally, for each case of loading combination (G+Q±E), users are asked to define the factor of static gravity loading (f_G+Q) and the factor of the seismic loading (f_E). Seismic loading directions may be combined linearly (E = ±EX±EY±EZ) with different factors per direction (f_EX, f_EY, f_EZ) or by the SRSS rule (E = ± ). It should be noted that the gravity loads have an explicitly defined algebraic sign, while for the seismic loadings both signs for every direction are taken into account. Consequently, the results of RSA loading combinations in terms of any response quantity are presented as envelopes.