# Accuracy check by time-history response analysis to one-cycle sinusoidal wave

To investigate the accuracy of using the double impulse as a substitute for the one-cycle sinusoidal wave in representing the fling-step near-fault ground motions, time-history response analysis of the EPP SDOF system with viscous damping under the one-cycle sinusoidal wave was conducted. The maximum velocity *V _{p}* of the corresponding one-cycle sinusoidal wave is adjusted so that the maximum Fourier amplitude of the one-cycle sinusoidal wave is equal to that of the double impulse (Kojima and Takewaki 2015a, 2016a, b). The adjustment procedure can be found in Section 1.2 in Chapter 1. Eq. (5.2) is used as an acceleration waveform of the one-cycle sinusoidal wave. The period

*T*of the one-cycle sinusoidal wave is

_{p}*T*where

_{p}= 2t_{0}^{c},*t*is calculated by time-history response analysis, as shown in Figure 5.10.

_{0}^{c}Figure 5.13 shows a comparison of the maximum deformation of the EPP SDOF system with viscous damping under the critical double impulse with that under the corresponding one-cycle sinusoidal wave for the damping ratios *h* = 0, 0.02, 0.05, 0.1, 0.2, 0.5. The maximum deformation of the undamped EPP SDOF system under the critical double impulse is in good agreement with that under the one-cycle wave in the range of the input velocity level *V/V _{y} <* 3. As the damping ratio increases, the maximum deformation under the critical double impulse corresponds well to that under the one-cycle sinusoidal wave in a wider range of input velocity level. This is because the maximum deformation after the first impulse (governing the maximum deformation in the input velocity range

*V/V*3) exhibits better correspondence with that under the first half-cycle of the corresponding one-cycle sinusoidal wave as the damping ratio increases. This result clearly indicates that the adjustment procedure of the input level of the double impulse and the corresponding one-cycle sinusoidal wave based on the

_{y}>*Figure 5.13* Comparison of maximum elastic-plastic deformation *u _{m3x}ld_{y}* of model with viscous damping under critical double impulse (quadratic-function approximation) with that under equivalent one-cycle sine wave: (a)

*h =*0, (b)

*h =*0.02, (c)

*h =*0.05, (d)

*h = 0.*1, (e)

*h =*0.2, (f)

*h =*0.5 (Kojima et al. 201 7, 201 8).

equivalence of the maximum Fourier amplitude is appropriate for the elastic-plastic system with viscous damping.

# Applicability of proposed theory to actual recorded ground motion

The applicability of the theory explained in this chapter to actual recorded ground motions is investigated through the comparison of the critical elastic-plastic response under the near-fault ground motion with that under the critical double impulse. The Rinaldi station fault-normal component during the Northridge earthquake in 1994 and the Kobe University NS component (almost fault-normal) during the Hyogoken-Nanbu (Kobe) earthquake in 1995 were used as the near-fault ground motions. The accelerograms of these two ground motions are shown in Figure 5.14. Although these are the fault-normal ground motions, these are represented by the double impulse in this section. The main part of the recorded ground motion acceleration is modeled as a one-cycle sinusoidal wave, as shown in Figure 5.14, and the one-cycle sinusoidal wave is substituted by the double impulse by using the method shown in Sections 5.2 and 1.2 (Eq. (1.16)).

Although the critical double impulse was determined for a given structural parameter V,. in Section 5.6, the structural parameter was selected to approximately maximize the response for a given input velocity V of the actual recorded ground motion in this section. This procedure is similar to the elastic-plastic response spectrum (changing the strength parameter), which was developed in 1960-1970 (Veletsos et al. 1965). In this section, a method for evaluating the critical elastic-plastic response under the

*Figure 5.14* Recorded near-fault ground motion and corresponding one-cycle sine wave: (a) Rinaldi Station FN comp. (Northridge 1994), (b) Kobe Univ. NS comp. (Hyogoken-Nanbu I 99S) (Kojima et al. 20 I 7, 2018).

*Figure 5.15* Comparison of maximum elastic-plastic deformation (double amplitude) of model with viscous damping *(h **=* 0.05) under critical double impulse (quadratic-function approximation) and recorded ground motions: (a) Rinaldi Sta. FN comp., (b) Kobe Univ. NS comp. (Kojima et al. 2018).

near-fault ground motion that is used in the literature (Kojima and Takewaki 2016b) was employed. The input velocity level of the Rinaldi station fault- normal component is V = 1.64[m/s] and that of the Kobe University NS component is V = 0.677[m/s] by the method in the literature (Kojima and Takewaki 2016b).

Figures 5.15(a), (b) show a comparison of the critical elastic-plastic response of the EPP SDOF system with viscous damping under the Rinaldi station fault-normal component and the Kobe University NS component with the proposed closed-form expression of the elastic-plastic response under the critical double impulse. The ordinate presents the maximum amplitude of deformation (the sum of *u _{maxi}* and и

_{тах2}), and the abscissa is the input velocity level

*V/V*. In comparison with the undamped case shown in Figures 2.13(a), (b) in Chapter 2 (Kojima and Takewaki 2016a, b), the elastic-plastic response under the critical double impulse corresponds well to the critical elastic-plastic response under the actual recorded ground motion in a wide range of the input velocity level, owing to the existence of viscous damping.