Simplifications of CQC method and CCQC method

Abstract

The response-spectrum mode superposition method is widely used for seismic response analyses of linear systems. In using this method, the complete quadratic combination (CQC) is adopted for classically damped linear systems and the complex complete quadratic combination (CCQC) formula is adopted for non-classically damped linear systems. However, in both cases, the calculation of seismic response analyses is very time consuming. In this paper, the variation of the modal correlation coefficients of displacement, velocity and displacement-velocity with frequency and damping ratios of two modes of interest are studied, Moreover, the calculation errors generated by using CQC and square-root-of-the-sum-of-the-squares (SRSS) methods (or CCQC and CSRSS methods) for different damping combinations are compared. In these analyses, some boundary lines for classically and non-classically damped systems are plotted to distinguish the allowed minimum frequency ratio at given geometric mean of the damping ratios of both modes if their relativity is neglected. Furthermore, the simplified method, which is a special mode quadratic combination method considering only relativity of adjacent modes in CQC method and named simplified CQC or partial quadratic combination (PQC) method for classically damped linear system, is proposed to improve computational efficiency, and the criterion for determination of how many correlated modes should be adopted is proposed. Similarly, the simplified CCQC or complex partial quadratic combination (CPQC) method for the non-classically damped linear system and the corresponding criterion are also deduced. Finally, a numerical example is given to illustrate the applicability, computational accuracy and efficiency of the PQC and CPQC methods.