Response Spectrum Method for integrated and differential spatial seismic ground motions

The Response Spectrum Method (RSM) is a widely known method for solving linear dynamic problem in earthquake engineering. However, it is often believed that the RSM can only be applied to the integrated (or averaged over the base of the building) seismic ground motion, when a ground base moves like a rigid body. In this paper we show how to apply spectral technique not only to the integrated six-component seismic motion that is included three translational and three rotational components, but also to the differential seismic motion, when each support point of structure makes individual spatial motion. The RSM is a quasistatic modal method, in which the modal seismic forces are defined as static loads that depend on spectral accelerations. To use the RSM for the differential seismic motion it is necessary to solve two key problems: firstly, correctly define the relative motion of a structure about the moving points of the ground and, secondly, determine the spatial distribution of spectral accelerations. These problems are solved by introducing an influence matrix (it relates the generalized coordinates of the support points and the generalized coordinates of the structure) and a matrix of spatial variations of intensity. The use of these matrices allows to obtain the equations of relative differential motion which are similar to the equations of the integrated seismic ground motion. Therefore, it's easy to apply the RSM to them. This article presents a theory of spectral technique for the general cases of the integrated and differential ground motions. To illustrate the application of the RSM, we consider a simple spatial structure (a rigid plate with shifted flexural center on four steel columns under two-component seismic action, the m-file with solution is attached in the Appendix) for the integrated and differential ground motions.