Theory of principal components analysis and applications to multistory frame buildings responding to seismic excitation

Described herein is a technique of multivariate statistical analysis applied to the post-processing of dynamic response data. The data may represent the linear or nonlinear response of structures, and may be obtained from computed simulations or from the measured response of instrumented structures. When applied to displacement response data, an ordered set of orthonormal mode shapes is obtained. The principal components analysis (PCA) mode shapes coincide with or are related to the elastic mode shapes for linear elastic systems, and depart from these shapes as nonlinear response becomes more prominent. The PCA modes provide an unambiguous and simple description of the ‘predominant’ mode of structures responding to earthquake ground motions, and thus improve the theoretical basis of nonlinear static procedures that use ‘equivalent’ single-degree-of-freedom (SDOF) systems for representing the response of structures subjected to damaging earthquake ground motions (e.g. the capacity spectrum and displacement coefficient methods). Where greater fidelity is desired, the most efficient representations are obtained by including as few PCA modes as are needed for the degree of precision desired. This paper presents the theory of PCA and illustrates its application to a 12-story frame building responding linearly and nonlinearly to earthquake ground motions. ‘Equivalent’ SDOF models of the structure are developed based on the PCA mode shapes, and these are applied to estimate the computed displacement histories.