# Application to recorded ground motions

To demonstrate the applicability of the explained evaluation method, it was applied to a recorded ground motion shown in Chapter 1 (Rinaldi Station FN during the Northridge earthquake in 1994). Figure 3.17 shows the accelerogram of the Rinaldi station fault-normal component. The main part of this accelerogram is modeled by the 1.5-cycle sinusoidal wave as shown in Figure 3.17. The maximum velocity and the period of the corresponding 1.5-cycle sinusoidal wave are denoted by and Tp. To investigate the effect of variation of the parameter of the sinusoidal wave, Vp = 1.05[m/s] and Tp = 0.8,0.85,0.9,0.95[s] were adopted for the Rinaldi station fault-normal component. Although the critical triple impulse was determined for the Figure 3.17 Rinaldi station fault-normal component and corresponding 1.5-cycle sinusoidal wave (Kojima et al. 2019).

structural parameter V, the structural parameter V is selected for the input velocity level of the actual recorded ground motion to maximize the elastic- plastic response under the actual recorded ground motion because the recorded ground motion is fixed (the input velocity level is fixed). This treatment is similar to the case in Section 2.7 (Chapter 2).

Figure 3.18 shows the comparison of the elastic-plastic response under the critical triple impulse and that under the Rinaldi station fault-normal component. The critical elastic-plastic response under the Rinaldi station fault-normal component was obtained by the time-history response analysis. The ordinate axes in Figure 3.18(a) and Figure 3.18(b) present the maximum deformation and the maximum amplitude (positive + negative) of deformation. The abscissa is the normalized input velocity level V/V . The input velocity level V of the triple impulse as a substitute for the Rinaldi station fault-normal component is V = 1.687[m/s]. Figures 18(a) and (b) demonstrates that the response under the critical triple impulse well simulates the critical elastic-plastic response under the Rinaldi station fault-normal component regardless of some variation of the parameter of the sinusoidal wave.

# Summaries

The obtained results may be summarized as follows:

1. The triple impulse input can be a simplified version of the forward- directivity near-fault ground motion, and a closed-form expression can be derived for the critical response of an undamped elastic-per- fectly plastic (EPP) single-degree-of-freedom (SDOF) model under this triple impulse input. Figure 3.18 Comparison of elastic-plastic response under critical triple impulse and that under Rinaldi station fault-normal component, (a) maximum deformation, (b) maximum deformation amplitude (Kojima et al. 2019).

2. Since only free-vibration is induced under such triple impulse input, the energy approach plays an important role in the derivation of the closed-form expression of a complicated critical elastic-plastic response. In this process, the input of impulse is expressed by the instantaneous change of velocity of the mass. The maximum elastic- plastic response after impulse can be obtained by equating the initial kinetic energy given by the initial velocity to the sum of hysteretic and elastic strain energies. It was shown that the maximum inelastic deformation can occur after either the second or third impulse depending on the input level.

• 3. The validity and accuracy of the theory explained in this chapter were investigated through the comparison with the time-history response to the corresponding three wavelets of sinusoidal input as a representative of the forward-directivity near-fault ground motion. It was found that if the level of the triple impulse is adjusted so that its maximum Fourier amplitude coincides with that of the corresponding three wavelets of sinusoidal input, the maximum elastic-plastic deformation to the triple impulse exhibits a very good correspondence with that to the three wavelets of sinusoidal waves. This good correspondence is extremely different from the case of the double impulse. The reason may result from the fact that since the velocity level of the second impulse is double of that of the first impulse, the response after the second impulse becomes critical in the broader range of input level in the triple impulse different from the case of the double impulse.
• 4. While the resonant equivalent frequency has to be computed for a specified input level by changing the excitation frequency parametrically for the sinusoidal input, no iteration is required in the method for the triple impulse. This is because the resonant equivalent frequency (resonance can be proved by using energy investigation) can be obtained directly without resorting to the repetitive procedure (the timing of the second impulse can be characterized as the time corresponding to the zero restoring force). In the triple impulse, the analysis can be conducted without determining the input frequency (timing of impulses) before the second impulse. While a simple and clear concept of critical input was defined in the case of double impulse (Kojima and Takewaki 2015a), the criticality can be used only before the third impulse in the present triple impulse. This is because the timing of the third impulse, determined already for the first and second impulses, decreases the maximum deformation нтзх2 after the second impulse and may increase the maximum deformation nmax3 after the third impulse.
• 5. Only critical response (upper bound) is captured by the method, and the critical resonant frequency can be obtained automatically for the increasing input level of the triple impulse. Once the frequency and amplitude of the critical triple impulse are computed, the corresponding three wavelets of sinusoidal motion as a representative of the forward-directivity motion can be identified by the equivalence of the maximum Fourier amplitude.
• 6. Using the relations among response ductility, input velocity, and input period, a flowchart was presented for the design of stiffness and strength for the specified velocity and period of the near-fault ground motion input and the specified response ductility. It was demonstrated that this flowchart can provide a useful result for such design.
• 7. An approximate method of prediction of response ductility (only prediction of upper bound) using the corresponding critical response can be developed for a specified design of stiffness and strength and a specified velocity and period of near-fault ground motion input.
• 8. The comparison between the maximum response to the double impulse and that to the triple impulse is meaningful. While the maximum deformation to the double impulse is larger than that to the triple impulse in the range of 0.5 < V/Vy < 1.5, it reverses in the range of V/Vy > 1.5. This phenomenon may be connected to the relation between the magnitudes of the first and second impulses.
• 9. To demonstrate the applicability of the expression for the maximum response to the triple impulse, the maximum response to the critical triple impulse was compared with the critical response under the Rinaldi station FN component during the Northridge earthquake in 1994. The explained method for the triple impulse provides the critical response under the recorded near-fault fault-normal earthquake ground motion with reasonable accuracy.