# Accuracy investigation by time-history response analysis to corresponding three wavelets of sinusoidal waves

To conduct the accuracy investigation of using the triple impulse as a substitute for the corresponding three wavelets of sinusoidal waves (representative of the forward-directivity input), the time-history response analysis of the undamped EPP SDOF model under the three wavelets of sinusoidal waves was conducted.

In the evaluation procedure, it is important to adjust the input level of the triple impulse and the corresponding three wavelets of sinusoidal waves based on the equivalence of the maximum Fourier amplitude. The period of the corresponding three wavelets of sinusoidal waves is twice of the time interval of the triple impulse (2*1 _{0}).* The adjustment procedure is shown in Appendix 3. Figure 3.8 shows one example of that correspondence for the input level

*V/V*3. Figures 3.9(a) and (b) illustrate the comparison of the ground displacement and velocity between the triple impulse and the corresponding three wavelets of sinusoidal waves for the input level V/V, = 3. Only in Figures 3.8 and 3.9(a), (b) are = 2^(rad/s) (T, = 1 .Osj and

_{y}=*d*= 0.16(m) used.

_{y}Figure 3.10 presents the comparison of the ductility (maximum normalized deformation) of the undamped EPP SDOF model under the triple impulse and the corresponding three wavelets of sinusoidal waves with respect to the input level. It can be found that the triple impulse is a good substitute for the three wavelets of sinusoidal waves in the evaluation of the maximum deformation if the maximum Fourier amplitude is adjusted

*Figure 3.8* Adjustment of input level of triple impulse and corresponding three wavelets of sinusoidal waves based on Fourier amplitude equivalence (Kojima andTakewaki 2015b).

*Figure 3.9* Comparison of ground displacement and velocity between triple impulse and corresponding three wavelets of sinusoidal waves: (a) displacement, (b) velocity (Kojima andTakewaki 201 5b).

*Figure 3.10* Comparison of ductility of elastic-plastic structure to triple impulse and corresponding three wavelets of sinusoidal waves (Kojima and Takewaki 201 5b).

appropriately. It should be remarked that, while the correspondence of the maximum deformation between the double impulse and the one-cycle sinusoidal input has some accuracy problem in the large input level (V/V > 3), the correspondence of the maximum deformation between the triple impulse and the three wavelets of sinusoidal waves is extremely good in a wide range of input level.

Figure 3.11 shows the comparison of the earthquake input energies by the triple impulse and the corresponding three wavelets of sinusoidal waves. An extremely good correspondence can also be observed in the input energy, and this is quite different from the case of the double impulse (the double impulse caused slightly larger input energy compared to the one-cycle sine wave). As stated before, the ratio 0.5 of the amplitude of the first impulse to the second impulse may be the main reason for this good correspondence of the maximum responses. This supports the validity of the triple impulse as a substitute for the forward-directivity near-fault ground motion.

Figure 3.12 illustrates the comparison of response time histories (normalized deformation and restoring-force) under the triple impulse and those under the corresponding three wavelets of sinusoidal waves. The parameters w, = 2^(rad/s) (T, = 1.0s), *d _{y}* = 0.16(m) are used here. While a rather good correspondence can be seen in general, the amplitude of deformation after the third impulse exhibits a slightly different value resulting from the difference in the applied timing of the third impulse. At the same time, a difference in phase can be observed both in the deformation and restoring-force. This may also result from the difference in the applied timing of the third impulse.

*Figure 3.1 I* Comparison of earthquake input energy by triple impulse and corresponding three wavelets of sinusoidal waves (Kojima andTakewaki 2015b).

*Figure 3.12* Comparison of response time history to triple impulse with that to corresponding three wavelets of sinusoidal waves: (a) normalized deformation, (b) normalized restoring-force (Kojima and Takewaki 2015b).

*Figure 3.13* Comparison of restoring-force characteristic under triple impulse with that under corresponding three wavelets of sinusoidal waves (Kojima and Takewaki 201 5b).

The difference in the amount of energy input at the third impulse seems to influence the later response.

Figure 3.13 presents the comparison of the restoring-force characteristic under the triple impulse and that under the corresponding three wavelets of sinusoidal waves. The parameters a», = 2^(rad/s) (T, = 1.0s), *d* = 0.16(m) are also used here. It can be observed from Figure 3.12 that, while the maximum deformations after the first and second impulses exhibit a rather good correspondence, the deformation response after the third impulse exhibits a non-negligible difference. Flowever, since the deformation response after the third impulse does not affect the maximum deformation in an overall time range in a larger input level of CASE 3 and CASE 4, this difference may not be significant.