Modal analysis of arbitrarily damped three-dimensional linear structures subjected to seismic excitations
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Modal analysis is a powerful approach that is used to analyze the responses of a structure under dynamic loadings. This approach allows the equation of motion to be decoupled in the modal coordinate space, and subsequently used to evaluate the dynamic response of a structure in the modal coordinate system, which significantly simplifies and accelerates the response calculation. Past research has shown that the modal analysis approach is applicable in earthquake engineering, resulting in its widespread use. For example, the seismic design and analysis of structures with added damping devices is based on the modal analysis concept, in which the motion within a plane and the assumption of classical damping are usually made. However, three-dimensional (3-D) structures with complex geometric shapes enhanced with added damping devices may be highly non-classically damped, possess over-damped modes, and exhibit significant out-of-plane motions. These uncertainties may affect the accuracy of the modal analysis approach in practice. This dissertation presents a theoretical framework for the seismic analysis of arbitrarily damped 3-D linear structures. First, a complex modal analysis-based approach is developed to analyze seismic responses to multi-directional excitations. This approach is formulated in a 3-D manner and allows the eigenvalues to be real, which correspond to over-damped modes. As a result, the responses resulting from the over-damped modes and the out-of-plane coupled motions can be properly considered. Several useful modal properties are identified and their mathematical proofs are provided. Next, a new modal combination rule is developed to calculate the peak response of arbitrarily damped 3-D linear structures when the seismic inputs are given in terms of response spectra. This modal combination rule is based on the theory of stationary random vibration and the existence of the principal axes of ground motions. In this rule, the correlations among two perpendicular excitation components and between modes are considered. Finally, an over-damped mode response spectrum that accounts for the peak modal response resulting from the over-damped modes is proposed.