Abstract
After Parkfield earthquake (1966), San Fernando earthquake (1971), Northridge earthquake (1994) and Hyogoken-Nanbu earthquake (1995), many aspects of near-fault ground motions have been made clear and the effects of near-fault ground motions on structural response have been investigated. The fling-step (parallel to the fault plane) and forward-directivity (perpendicular to the fault plane) inputs have been characterized by two or three sinusoidal wavelets. For such near-fault ground motions, many analyses have been conducted from various viewpoints. However, as far as a forced base input is used, both a free-vibration term and a forced-vibration term arise inevitably and the closed-form expression of the elastic-plastic response may be difficult. In order to overcome this difficulty, the double impulse has been introduced by some of the present authors as a good substitute for the near-fault ground motion and the closed-form expression has been derived for the undamped elastic-plastic response and linearly damped elastic-plastic response of a structure under the critical double impulse. Furthermore, this approach has been extended to other various vibration models, e.g. soil-structure interaction problems, dynamic collapse problems, repeated ground motion problems, overturning rocking problems of rigid blocks.
The double impulse input is introduced here again as a substitute for the fling-step near-fault ground motion and some closed-form expressions are derived for the elastic-plastic response of a structure with nonlinear viscous damping under the ‘critical double impulse’. It is shown that, since only the free vibration appears under such double impulse, the energy approach enables the derivation of the closed-form expression of a complicated elastic-plastic response with nonlinear viscous damping. It is also shown that the critical timing of the second impulse is the time with the zero restoring force in the case where the input velocity is small. On the other hand, the critical timing of the second impulse is the time with the maximum velocity in the case where the input velocity is large. The quadratic-function or elliptical-function approximation for the damping force-deformation relationship is introduced. The combination of the timings of the structural yielding and the damper relieving is considered in detail and the closed-form expressions are derived for all the combinational cases.
The validity of the proposed theory using the quadratic-function or elliptical-function approximation and the assumption of the critical impulse timing has been investigated through the comparison with the critical elastic-plastic response under double impulse using the time history response analysis. The validity of the proposed closed-form solution has also been demonstrated through the comparison with the response analysis to the corresponding one-cycle sinusoidal input as a representative of the fling-step near-fault ground motion. It has been demonstrated that the maximum response to the critical double impulse and the response to the corresponding one-cycle sinusoidal input coincide fairly well. This supports the validity of the proposed theory.
