Modelling Inelastic Response in Direct Displacement-Based Design

The direct displacement-based design (DDBD) method requires the definition of equivalent viscous damping to accurately predict the peak nonlinear response. Equivalent viscous damping is usually specified as the sum of a viscous and hysteretic component, where the former is assumed to be constant, and the latter depends on the ductility and hysteresis model.

The characterisation of viscous damping in time history analysis is discussed. Although it has been more common in the past to use a constant damping coefficient for single-degree-of-freedom time history analyses, it is contended that tangent-stiffness proportional damping is a more realistic assumption for inelastic systems. Analyses are reported showing the difference in peak displacement response of single-degree-of-freedom systems with various hysteretic characteristics analysed with 5% damping ratio applied as either a constant damping coefficient or tangent-stiffness proportional damping. The difference is found to be significant, and dependent on hysteresis rule, ductility level and period. The relationship between the level of elastic viscous damping assumed in time-history analysis, and the value adopted in DDBD is investigated. It is shown that the difference in characteristic stiffness between time-history analysis (i.e. the initial stiffness) and displacement based design (the secant stiffness to maximum response) requires a modification to the elastic viscous damping added to the hysteretic damping in DDBD.

Numerical analyses are carried out to study the combination of hysteretic and viscous energy dissipation in nonlinear analysis. Expressions are calibrated that describe the ductility and period dependence of the equivalent viscous damping, for a range of hysteresis and damping models. It is found that simple equations are able to provide accurate values of equivalent viscous damping for both analytical research, and practical design applications of DDBD.