INFLUENCE OF VISCOUS DAMPING MODEL ON TIME-HISTORY RESPONSE ANALYSIS OF ELASTIC-PLASTIC SYSTEMS (PART 1): COMPARISON OF DAMPING MODELS

<p> In time-history analysis of building structures, damping is commonly modeled by classical viscous damping models among which Rayleigh damping is favored due to its computational efficiency and ability to assign near-constant damping ratios over a wide range of frequencies. However, researchers recognize that Rayleigh damping can produce unrealistically large damping forces in elastic-plastic analysis, and, in turn, lead to underestimation of computed response. Now that ever sophisticated nonlinear analysis with high-fidelity hysteresis models is at our disposal, more attention should be directed towards damping models.</p><p> This paper describes a fundamental study on how the time-history response of elastic-plastic systems can be affected by the selection of damping models. Comparison is made between 13 damping models listed in Table 1, including massor stiffness-proportional models, Rayleigh model, Caughey series, modal-damping proportional model, and modifications of these models. Some models require step-by-step update of the damping matrix. Models 4, 7 and 13 require step-by-step eigenvalue analysis, which by nature is computationally expensive. In the post-elastic state, the damping matrix may become nonclassical, remain classical, or always retain the original damping ratios. Fig. 2 compares the models for a partially-yielded state of a 5-DOF system, plotting the effective modal damping ratio defined in Eq. (4) against the updated eigenfrequencies. Summation was taken over all modes for the Caughey series (Model 9) and modal-damping proportional models (Models 12 and 13). The Rayleigh model (Model 5) led to large damping ratios, while the modified Rayleigh models, Models 7 and 8, suggested by Charney and Hall, respectively, retained the damping ratios near the original target value. Although computationally expensive, Model 13 retains the damping ratio under any state, and may therefore be viewed as an exact model.</p><p> Time-history response was compared through a 5-DOF system with bilinear kinematic hysteresis (see Fig. 3) subjected to a strong earthquake ground motion. Integration over time was conducted using the central difference method, expressed by equations (10) to (12), and a time increment of 0.002 s. The resulting story drift distribution varied substantially in both elastic analysis (see Fig. 5b) and elastic-plastic analysis (see Fig. 6). Mass- or stiffness-proportional models or their variations (Models 1 to 4) resulted in large deviation from Model 13. The Rayleigh model (Model 5) resulted in somewhat smaller response than Model 13. Fig. 7 illustrates, for the third-story response, how the damping force from Model 5 was substantially greater than Model 13 when the system yielded. Fig. 9 illustrates that the damping force in Model 5 was consistently dictated by the first mode response although the normalized effective modal mass (defined in Eq. (6)) for the first mode fluctuated from the original 0.85 to a minimum 0.2 and maximum 1.0. Fig. 11 correlates damping energy (computed using Eq. (13)) and damping ratio averaged over all modes and time. Most models overestimated the damping energy while Models 7 and 8 matched Model 13 very closely.</p><p> Time-history analysis was repeated for a 20-DOF system (see Fig. 12). Fig. 13 shows that, as in the 5-DOF system, the modified Rayleigh models (Models 7 and 8) yielded very similar results to Model 13. Fig. 14 shows that at least 5 of 20 terms must be taken for Model 13 to yield a result as accurate as Models 7 or 8.</p>