Critical response of a single-degree-of-freedom system with bilinear hysteresis and viscous damping under triple impulse

A triple impulse is used as a mathematical model of forward-directivity inputs, which are fault-normal components of near-fault earthquake ground motions. In this paper, two types of the triple impulse input are employed. One is the triple impulse with the same time interval. The critical triple impulse of this sequence is defined as the triple impulse with the critical time intervals maximizing the displacement response, and the critical time intervals are necessary to be captured by changing the time intervals. The other is the triple impulse, with second and third impulses acting at zero-restoring force points. The elastic–plastic responses of single-degree-of-freedom (SDOF) systems with bilinear hysteresis and viscous damping under the two types of triple impulses are obtained by time-history response analysis and compared. Furthermore, approximate expressions are derived for the responses of the damped bilinear hysteretic SDOF system under the triple impulse (the latter triple impulse stated above), with impulses acting at the zero-restoring force timing. In response derivation, a quadratic function approximation of the damping force–displacement relationship and an energy balance approach are employed. The validity of the triple impulse as the model of the forward-directivity input and the accuracy of approximate expressions to the triple impulse (the latter triple impulse stated above) are checked by comparing them with responses under the equivalent three wavelets of sinusoidal waves and the Ricker wavelet.