Equivalent viscous damping equations for direct displacement-based design

Student: Carlos Blandon
Supervisor: Prof. M.J.N. Priestley


Estimation of the equivalent viscous damping factor (EVDF) is an important step in the methodology of the direct displacement based design. The dynamic response of the substitute structure is characterized by an effective stiffness and an equivalent viscous damping, simplifying considerably the dynamic problem and making this approach very desirable for design purposes. However, errors in the estimation of these parameters characteristics lead to consequent errors in the ductility demand of the designed elements.

The most used procedure to estimate the equivalent viscous damping is the Jacobsen's approach, which estimates this factor based on the ratio between the elastic stored energy and the dissipated energy by a given hysteretic model. However this approximation assumes complete loops under a sinusoidal excitation (steady-state harmonic response). The real situation during an earthquake is different given that the complete loops are not formed in each cycle and the system is subjected to a random excitation.

Evidence from past investigations has shown that the Jacobsen approach can estimate, for some hysteretic models, the EVDF with the same accuracy as more elaborated techniques such as Gulkan's approach. This technique balances the input energy from the earthquake and the energy necessary to bring the system to rest using viscous damping. However, there are also investigations that indicate that Jacobsen's approach overestimates the value of the damping for some hysteretic models and for some earthquake characteristics such us pulses. Thereafter it is needed to carry out additional analyses to find the limitations of this approach.

A comparison of the displacements obtained from a non-linear time-history analysis and a spectral design was carried out for a specific single degree of freedom system (SDOF), in order to evaluate how accurate can these displacement be estimated using Jacobsen's approach. This procedure was repeated for six different hysteretic rules which covered a wide range of energy dissipation: a thin and a fat Takeda model, a bilinear model with high post yielding stiffness, an elastic perfectly plastic model, a Ramberg Osgood type model and a ring spring model. Six records were used, one synthetic, adjusted to a EC8 type target spectrum and five artificial records adjusted to Caltrans design spectra for soil type C (PGA=0.7g). The SDOF model was designed for five different levels of ductility (2 to 6) and effective periods from 0.5 to 4 s in steps of 0.5s. In iterative methodology was carried out until the displacements from the non-linear time-history analysis matched the design displacement obtained from the damped response spectra. In order to separate the effects of elastic-viscous and hysteretic damping the designs and analyses were carried without elastic viscous damping.

It was found that in general the Jacobsen approach overestimates the values of the equivalent viscous damping with a few exceptions; As a consequence, modified equations were proposed in order to estimate the EVDF.

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