Study On Post-earthquake Residual Displacements Of SDOF Systems With Tangent-stiffness Proportional Damping Model

The residual displacement is an important performance indicator for the evaluation of post-earthquake repairability.Reasonable analysis and accurate estimation of residual displacements are of great significance for the seismic damage control and performance assessment,of which the single-degree-of-freedom(SDOF)system analysis is the basis.At present,the research related to residual displacements mainly focuses on SDOF systems.Note that the constant damping coefficient(CDC)assumption is generally adopted in nonlinear time history analysis(NTHA).However,the theoretical analysis and experimental test have demonstrated that the CDC assumption does not apply to the modeling of inherent damping energy dissipation mechanism,which can lead to unrealistic analysis results.Although the tangent-stiffness proportional damping(TSPD)has been verified to be more appropriate for modeling of structural inherent damping,the issue of negative damping,which is irreconcilable with physical reality,will be inevitably introduced when the structure exhibits post-yield softening behavior.In addition,the current research has paid insufficient effort on the residual displacement estimation and repairability evaluation,particularly for the normalization of residual displacements,the seismic design method considering both maximum inelastic and residual displacements,and the estimation of residual displacements of structures subjected to mainshock-aftershock sequences.Furthermore,the current codes for seismic design of bridges and buildings are still not perfect enough in considering the repairability and lack the specific calculation method of residual displacements.To solve the aforementioned problems,this dissertation conducts the study on damping models in NTHA and post-earthquake residual displacements of SDOF systems.The main research contents and conclusions are as follows:(1)The TSPD model is revised to avoid the conceptual issue of negative damping,and then the validity of the revised model is experimentally verified with the help of a shaking table test of a steel frame structure.With the combination of revised TSPD and mass proportional damping(MPD),the more versatile tangent-stiffness Rayleigh damping(TSPRD)assumption is developed,which can be applicable to modeling of the energy dissipation mechanism of structural inherent damping,supplemental damping devices,and structures outfitted with supplement damping devices.By applying the TSPRD model to NTHA of SDOF systems,the numerical iteration formula is theoretically derived,and the general numerical solution scheme for nonlinear dynamic equilibrium equation is developed for various viscous damping models.(2)Using the proposed TSPRD model and the numerical solution scheme for nonlinear dynamic equilibrium equation,the significance of the choice of damping models to post-earthquake residual displacements is examined with different combinations of mass proportional and tangent-stiffness proportional damping terms in TSPRD model.The results indicate that the viscous damping models have a significant effect on residual displacements,depending on natural periods,initial damping ratios and stiffness degradation.The residual displacement response spectra,developed based on the statistical results of NTHA using CDC assumption,will underestimate the residual displacement demands.The results for TSPD model are statistically organized to analyze the influence of structural parameters on residual displacements,which provides a theoretical basis for controlling(mitigating)post-earthquake residual displacements.(3)Based on the statistical results of NTHA of SDOF systems with TSPD model,the correlation between residual displacements and peak inelastic as well as elastic spectral displacements is examined.The two displacement quantities are utilized for the normalization of residual displacements,and the effects of ground motion and structural characteristics on the two definitions of residual displacement ratios are investigated.The constant-strength residual displacement ratio response spectra are developed for both two normalizations to estimate the residual displacements of conventional structures(without supplemental damping devices).The results reveal that the residual-to-peak-inelastic displacement ratios have the advantages of small dispersion,samples without any outliers,and relatively symmetric distribution,which benefits from the strong correlation between residual and peak inelastic displacements.Moreover,the residual-to-peak-inelastic displacement ratios are almost independent of site conditions,significant duration,natural periods and strength reduction factors(only for relatively high levels).From the viewpoint of simplicity and convenience of engineering application,the residual-to-peak-inelastic displacement ratios are more appropriate for residual displacement estimation.(4)The NTHA and nonlinear static analysis(NSA)are utilized to calculate the maximum inelastic displacements,then the NTHA results are regarded as a benchmark to revise Jacobsen’s equivalent damping ratio derived from the equal energy dissipation principle.The ratios of residual displacements to maximum displacements estimated from NSA using the revised equivalent damping ratio are statistically analyzed,and a probability-based procedure for residual displacement estimation using NSA is developed.The approach that allows estimating the residual displacements from maximum inelastic displacements emphasizes that the post-earthquake reparability assessment should be based on the premise of no collapse,which can not only calculate the maximum displacement consistent with NTHA results in the mean sense,but also accurately estimate the residual displacements under different exceedance probabilities.(5)The effect of aftershock on residual displacements is examined by considering mainshock only and mainshock-aftershock sequences.The statistical analysis is performed for the maximum value of residual displacements of the mainshock and mainshock-aftershock sequence to investigate the impact of ground motion and structural characteristics on maximum residual displacements,and the constant-ductility residual displacement ratio response spectra are developed for mainshock-aftershock sequences.It shows that the aftershock will reduce the dispersion of residual displacement ratios,whereas it can increase their mean values.The aftershock intensity,natural periods,displacement ductility and post-yield stiffness ratios are the critical factors that influence the residual displacements,while the frequency content and significant duration have negligible impact.The developed response spectra for mainshock-aftershock sequences adequately consider the effect of aftershock,and are more safe and reliable than those for the single earthquake ground motion.