# Applicability of critical double impulse timing to corresponding sinusoidal wave

In the reference (Kojima and Takewaki 2015a), it was demonstrated that if the maximum value of the Fourier amplitude is selected as the key parameter for an adjustment (Section 1.2 in Chapter 1), the responses to the double impulse and the corresponding sinusoidal input exhibit a fairly good correspondence. In this section, it is investigated whether the critical timing derived from the double impulse is also an approximate critical timing of the sinusoidal input. Although the SDOF model with a fixed base is treated

*Figure 7.10* Schematic diagram for better understanding of effect of soil type on structure plastic deformation: (a) small input level (Case 2), (b) large input level (Case 3) (Kojima and Takewaki 2016).

here, it is applicable to the equivalent SDOF model by introducing the equivalent parameters, *V _{y}โ, d/, w{* etc.

Let *t _{0}^{c}* denote the critical timing of the double impulse and

*t*denote the general timing. In Kojima and Takewaki (2015a) and Chapter 2,

_{0}*t*has been derived as follows (CASE 3: large input level; plastic deformation after the first impulse).

_{0}โ

This relation is plotted in Figure 7.11. It can be understood that the critical timing is delayed due to plastic deformation as the input level increases.

Figure 7.12(a) shows the maximum deformation with respect to *tjt _{0}^{c}.* It can be observed that the critical time interval

*t*derived from the double impulse is a good approximate of that for the sinusoidal input. Figure 7.12(b) is the corresponding plot for the double impulse (Kojima and Takewaki 2015a).

_{0}^{c}*Figure 7.11* Interval time between first and second impulses with respect to input level (Kojima and Takewaki 2015a).

*Figure 7.12* Maximum deformation with respect to t_{0}/t_{0}^{c}: (a) sinusoidal input, (b) double impulse (Kojima and Takewaki 20 I 6).

# Toward better correspondence between double impulse and sinusoidal input

In Kojima and Takewaki (2015a) and Chapter 2, it was clarified that if the magnitude of the double impulse is adjusted so that the maximum values of the Fourier amplitudes of the double impulse and the corresponding sinusoidal input are the same (Section 1.2 in Chapter 1), the maximum elastic- plastic responses correspond well in the range of input level V/V,. < 3. However, in the range of V/V *>* 3, the maximum response of the double impulse becomes larger than that of the sinusoidal input. In order to seek a better correspondence over a wider range of input level, the amplitude of the sinusoidal input is amplified. This is because the effect of the sinusoidal input is rather small in the large input level V/V > 3 compared to the large influence of the first impulse in the double impulse. As in Section 7.4, although the SDOF model with a fixed base is treated here, it is applicable to the equivalent SDOF model by introducing the equivalent parameters, V/, *d;, (ยป{'* etc.

Figure 7.13(a) presents the plot of the coefficient *a* with respect to the timing of the double impulse for adjusting the maximum Fourier amplitudes of the double impulse and the sinusoidal input where the sinusoidal acceleration input is expressed as *u _{g}* (?) = Asin(;rt/t

_{0}) and the coefficient

*a*is defined by

*a = A/V.*Figures 7.13(b)โ(e) present the maximum normalized elastic-plastic deformations to the double impulse and the corresponding sinusoidal inputs amplified by 1.0, 1.1, 1.15, 1.2 from the original input with the same maximum Fourier amplitude as the double impulse. It can be found that the amplification 1.15 or 1.2 provide the best fitting for the input range of

*V/V*3.

_{y}># Applicability to recorded ground motions

Because the double impulse and the corresponding one-cycle sinusoidal wave have somewhat different natures from actual recorded ground motions, it seems important to investigate the applicability of the present theory to actual recorded pulse-type ground motions. As explained in the previous sections, the maximum deformation of the simplified SR model can be obtained from the equivalent SDOF model. Figure 7.7 enables the estimation of the maximum deformation of the equivalent SDOF model. Therefore, it is sufficient to investigate the response of the SDOF model. Moreover, since the consideration and reflection of ground conditions in actual recorded ground motions is complicated, an SDOF model with a fixed base is considered in this section.

Consider two pulse-type recorded ground motions, the Rinaldi station fault-normal (FN) component during the Northridge earthquake in 1994

*Figure 7.13* Better correspondence of maximum responses to double impulse and amplified sinusoidal inputs: (a) plot of coefficient *a* with respect to timing of double impulse; (b) amplification factor = 1.0; (c) amplification factor = I. I; (d) amplification factor = I. I 5; (e) amplification factor = 1.2 (Kojima andTakewaki 2016).

and the Kobe University NS component (almost fault-normal) during the Hyogoken-Nanbu (Kobe) earthquake in 1995. The input level of the equivalent double impulse for the Rinaldi station FN component is *V =* 1.64 [m/sec] and that for the Kobe University NS component is V = 0.677 [m/sec]. As in the previous chapter (Chapter 5), since the ground motions are fixed, the structural models are varied for the realization of the resonance, i.e., ctq or *d _{y}* in

*V*is varied. Figure 7.14 illustrates the modeling of the part of the recorded ground motion acceleration into a one-cycle sinusoidal input. Figure 7.15 shows the maximum amplitude of deformation for the recorded ground motions and the corresponding proposed one. As stated before, since the initial velocity V is determined in Figure 7.14,

_{y}= Wfdy*V*is changed here.

_{y}*Figure 7.14* Modeling of part of pulse-type recorded ground motion into corresponding one-cycle sinusoidal input: (a) Rinaldi station fault-normal component during the Northridge earthquake in I 994; (b) Kobe University NS component (almost fault-normal) during the Hyogoken-Nanbu (Kobe) earthquake in 1995 (Kojima and Takewaki 2016).

*Figure 7.15* Maximum amplitude of deformation for recorded ground motions and proposed one: (a) Rinaldi station fault-normal component; (b) Kobe University NS component (Kojima and Takewaki 2016).

Because w, is closely related to the resonance condition, *d _{y}* is changed principally. This procedure is similar to the well-known elastic-plastic response spectrum developed in 1960-1970. The solid line is obtained by changing

*V*for the specified V using the method for the double impulse and the dotted line is drawn by conducting the elastic-plastic time-history response analysis on each model with varied

_{y}*V*under the recorded ground motion. It can be observed that the result by the proposed method is a fairly good approximate of the recorded pulse-type ground motions.

_{y}