Modal Analysis of Generally Damped Linear Structures Subjected to Seismic Excitations; Technical rept

摘要

Motivated by the need for a systematic approach for seismic evaluation and design of civil engineering structures with supplemental damping, a general modal analysis method, in which over-damped modes are taken into account, is developed and described in this report. This general modal analysis method deals with a unified formulation used to evaluate most structural response quantities of interest, such as displacements, velocity, inter-story drifts, story shear, damping forces and absolute accelerations etc. In addition, a novel general real-valued transformation matrix is established, which can be utilized to decouple the equations of motion of a generally damped structure in terms of real-valued modal coordinates. Non-singularity of this matrix and other properties related to this transformation, such as modal responses to initial conditions, modal energy distribution, modal effective masses and modal truncation etc., are discussed in details to explain the dynamic nature of the generally damped structural system. Furthermore, on the basis of the general modal response history analysis and the white noise input assumption as well as the theory of random vibration, two general modal combination rules for the response spectrum analysis, GCQC and GSRSS are formulated to handle non-classical damping and over-damped modes. To enable the new rules applicable to the practical earthquake engineering, a conversion procedure to construct an over-damped mode response spectrum compatible with the given 5% standard design response spectrum is established. The adequacy of this conversion procedure is also validated. Examples are given to demonstrate the application of the modal analysis method, to assess the accuracy of the new modal combination rules, and to show that over-damped modes may develop in structures with supplemental damping which can provide significant response contributions to certain response parameters.