Rayleigh Damping vs. Modal Damping Matrix Superposition for Steel Frames and Evaluation of Higher-Mode Contribution

Abstract

Rayleigh damping is usually adopted in seismic analysis of steel buildings to develop the damping matrix (C^R), where the initial stiffness (K_i) and the lumped mass (M_L) matrices, and the first two lateral vibration modes are often used. Additionally, it is a widespread practice to use M_L in the nonlinear dynamic equilibrium equation. Similarly, the contributions to the response of higher modes related to rotations and vertical displacement of joints (JR-VD) are generally ignored. In this study, some issues related to these practices are addressed, contrasting the results obtained through the use of C^R with those of a damping matrix (C^M) obtained with a more accurate model as that of modal damping matrix superposition (MDMS) procedure, and by comparing the responses derived from M_L with those of the consistent mass matrix (M_C). Three steel building models are used, which are considered low, medium and high height. If C^R is adopted, axial loads, bending moments, drifts, and interstory shears are underestimated, on average, by up to 42%, 23%, 22% and 20%, respectively, compared to the results obtained with the MDMS procedure; the horizontal damping forces are overestimated by up to 232%. The contribution of the JR-VD modes can be up 41% for axial loads and up to 20% for bending moments, interstory shears and drifts. If M_L and K_i are used, average overestimations of up to 80% are observed for axial loads with respect to those of M_C and the tangent stiffness matrix (K_t); bending moments and interstory shears, in contrast, can be underestimated by up to 24% and 16%, respectively. Hence, the C^M, K_t and M_C matrices should be used; the contributions of the JR-VD modes should not be overlooked; and if C^R is used, to increase the accuracy, mode 1 and a mode greater than the second must be considered to obtain the Rayleigh damping matrix. Failure to do so may result in non-conservative designs.