# Summaries

The obtained results may be summarized as follows:

- 1. The multiple impulse input of equal time interval can be a good substitute for the long-duration earthquake ground motion which is expressed in terms of harmonic waves. A closed-form expression can be derived for the maximum response of an elastic-perfectly plastic (EPP) single-degree-of-freedom (SDOF) model under the critical multiple impulse input.
- 2. While the critical set of input velocity amplitude and input frequency (timing of impulse) has to be computed iteratively for the multi-cycle sinusoidal wave, it can be obtained directly without iteration for the multiple impulse input by introducing a modified version (only the timing between the first and second impulses is modified so that the second impulse is given at the zero restoring force). The resonance was proved by using energy investigation. It was made clear that the critical timing of the multiple impulses can be characterized by the time attaining the zero restoring force in the unloading process. This decomposition of input amplitude and input frequency overcame the difficulty in finding the resonant frequency without repetition. This is one of the most original contributions in this chapter.
- 3. Since only free-vibration is induced under such multiple impulse input, a simple energy approach plays an important role in the derivation of the closed-form expression for a complicated elastic-plastic critical response. The energy approach enabled the derivation of the maximum critical elastic-plastic response without resorting to solving the equation of motion. The maximum elastic-plastic response after each impulse can be obtained by equating the initial kinetic energy given by the initial velocity to the sum of hysteretic dissipation and elastic strain energies. The critical inelastic deformation and the corresponding critical input frequency can be captured by the substituted multiple impulse input depending on the input velocity level. This is the second one of the most original contributions in this chapter.
- 4. The validity and accuracy of the theory explained in this chapter were assured through the comparison with the time-history response to the corresponding equivalent multi-cycle sinusoidal input. It was found that if the adjustment of both inputs is made by using the equivalence of the maximum Fourier amplitude and a modification based on the response correspondence at some points with different input levels for better correspondence in a wider range of input level, the maximum elastic-plastic deformation to the multiple impulse exhibits a good correspondence with that to the multi-cycle sinusoidal wave.
- 5. While the conventional methods (Caughey 1960a, b, Iwan 1961, 1965a, b) are aimed at constructing an equivalent linear structural model or solving transcendental equations iteratively to an unchanged input (sinusoidal input) to enable the simple approximate computation of complicated elastic-plastic responses, the method explained in the present chapter is aimed at finding an equivalent input model for an unchanged exact elastic-plastic model. The most significant difference between two approaches is that while the conventional methods require the repetition both in the computation of equivalent model parameters or the solution of transcendental equations for one input frequency and the computation of the resonant frequency giving the maximum response, the approach explained in this chapter does not require any repetition in the computation of the critical input timing (resonant frequency) and the critical response. The approach explained in this chapter also enabled the computation of the steady-state response for an EPP model that cannot be treated by the conventional approaches.

The approach for an EPP model can be extended to a bilinear hysteretic model with some modifications (see Chapter 6).

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# A. Appendix 1: Multi-impulse and correspondingmulti-cycle sine wave with thesame frequency and samemaximum fourier amplitude

The velocity amplitude V of the multi-impulse is related to the maximum velocity of the corresponding multi-cycle sine wave with the same frequency (the period is twice the interval of the multi impulse) so that the maximum Fourier amplitudes of both inputs coincide. The detail is explained in this section.

The multi-impulse is expressed by

where V is the velocity amplitude of the multi-impulse and *S(t)* is the Dirac delta function. The Fourier transform of Eq. (4.A1) can be obtained as

Let *A _{h}* Т/, and

*w,= 2л/Т,*denote the acceleration amplitude, the period, and the circular frequency of the corresponding multi-cycle sine wave, respectively. The acceleration wave of the corresponding multi-cycle sine wave is expressed by

The time interval *t _{0}* of consecutive two impulses in the multi-impulse is related to the period T, of the corresponding multi-cycle sine wave by T; = 2

*1*Although the starting points of both inputs differ by

_{0}.*tj*2, the starting time of multi-cycle sine wave does not affect the Fourier amplitude. For this reason, the starting time of multi-cycle sine wave will be adjusted so that the responses of both inputs correspond well. In this section, the relation of the velocity amplitude V of the multi-impulse with the acceleration amplitude

*A,*of the corresponding multi-cycle sine wave is derived. The ratio

*a*of

_{M}^{h}(t_{0},N)*A,*to V is introduced by

The Fourier transform of * ^{u}g* in Eq.(4.A3) is computed by

From Eqs. (4. A2), (4. A5), the Fourier amplitudes of both inputs are expressed

by

The coefficient *a _{M}^{b}(t_{0},N)* can be derived from Eqs. (4.A4), (4.A6), (4.A7) and the equivalence of the maximum Fourier amplitude ILfco) =

ргнI •

N-l

In the numerator of Eq. (4.A8), max 0.5 + ^(-1)" *е~ ^{шпи>} +* 0.5

»=1

holds. The denominator of Eq. (4.A8) will be evaluated next.

Let us define the function *f _{M}^{b}(x,N)* given by

When we substitute x = *wt _{0}* in Eq. (4.A9), Eq. (4.A7) can be transformed into

Eq. (4.A9) means that once the maximum value of *f _{M}^{b}(x = wt_{0},N)* is obtained, the denominator of Eq. (4.A8) is evaluated. After some manipulation on the maximization of Eq. (4.A10), the following relation is derived.

where /_{Mmax}*(N) is the maximum value of *f _{M}^{b}(x,N)* for the variable x. Finally, for a sufficiently large number N(>20), the function

*f*becomes maximum at x =

_{M}^{b}(x,N)*к*and

*f*Then, Eq. (4.A11) is reduced to

_{M}^{b}(x = n,N) = N/(2л).

Eq. (4.A12) provides the relation between *V* and *A,* in terms of *t _{0}(=T,/2) *together with the relation between V and V;.