# Accuracy investigation by time-history response analysis to corresponding one-cycle sinusoidal input

In order to investigate the accuracy of using the double impulse as a substitute of the corresponding one-cycle sinusoidal wave (representative of the fling-step input), the time-history response analysis of the undamped EPP

SDOF model under the corresponding one-cycle sinusoidal wave was conducted.

In the evaluation procedure, it is important to adjust the input level of the double impulse and the corresponding one-cycle sinusoidal wave based on the equivalence of the maximum Fourier amplitude as noted in Chapter 1 (Section 1.2). The period of the corresponding one-cycle sinusoidal wave is twice that of the time interval of the double impulse (2f_{0}). Another criterion on the adjustment may be possible from different viewpoints. Figure 2.6 shows one example of the Fourier amplitudes for the input level *V/V _{y} =* 3. Figures 2.7(a) and (b) illustrate the comparison of the ground displacement and velocity between the double impulse and the corresponding one-cycle sinusoidal wave for the input level V7V,, = 3. In Figures 2.6, 2.7(a), (b), w, = 2tf(rad/s) (T, = 1.0s) and

*d*0.16(m) are used.

_{y}=Figure 2.8 presents the comparison of the ductility (maximum normalized deformation) of the undamped EPP SDOF model under the double impulse and the corresponding one-cycle sinusoidal wave with respect to the input level. It can be seen that the double impulse provides a fairly good substitute of the one-cycle sinusoidal wave in the evaluation of the maximum deformation if the maximum Fourier amplitude is adjusted appropriately. Although some discrepancy is observed in the large deformation range *(V/V _{y} >* 3), that response range (relatively large plastic deformation) is out of interest in the earthquake structural engineering. The reason for such discrepancy is due to the difference of effect after the first impulse. The modification of the adjustment of the input level between the double impulse and the corresponding one-cycle sine wave can be made in order to guarantee the response correspondence in a broader input range (Chapter 7 and Kojima and Takewaki 2016b).

*Figure 2.6* Adjustment of input level of double impulse and corresponding one- cycle sinusoidal wave based on Fourier amplitude equivalence (Kojima and Takewaki 201 5).

*Figure 2.1* Comparison of ground displacement and velocity between double impulse and corresponding one-cycle sinusoidal wave: (a) Displacement, (b) Velocity (Kojima and Takewaki 2015).

*Figure 2.8* Comparison of ductility of elastic-plastic structure under double impulse and corresponding one-cycle sinusoidal wave (Kojima and Takewaki 20 I 5).

*Figure 2.9* Comparison of earthquake input energy by double impulse and corresponding one-cycle sinusoidal wave (Kojima andTakewaki 2015).

Figure 2.9 presents rhe comparison of the earthquake input energies by the double impulse and the corresponding one-cycle sinusoidal wave. Although a good correspondence can be observed at a lower input level, the double impulse tends to provide a slightly larger value in the larger input level. This property can be understood from the time-history responses shown in Figures 2.10 and 2.11, i.e., a rather clear difference in deformation after the first impulse (the double impulse has a large impact after the first impulse). As stated above, the modification of the adjustment of the input level of the double impulse and the corresponding one-cycle sine wave is possible in order to guarantee the response correspondence in a broader input range (Chapter 7 and Kojima and Takewaki 2016b).

Figure 2.10 illustrates the comparison of response time histories (normalized deformation and restoring-force) under the double impulse and those under the corresponding one-cycle sinusoidal wave. The structural parameters *w*j = 2л-(rad/s) (T, = 1.0s), *d =* 0.16(m) were also used here. While rather good correspondence can be seen in the restoring-force (restoring- force is not sensitive to input variation due to yielding phenomenon), the maximum deformation after the first impulse exhibits a rather larger value in the double impulse. The difference in the initial condition may affect these response discrepancies. Flowever, it is noteworthy that the maximum deformation after the second impulse demonstrates rather good correspondence. This may result from the fact that the effect of the initial condition becomes smaller in this stage and the discrepancy in larger input level in Figure 2.8. Figure 2.11 shows the comparison of the restoring-force characteristic under the double impulse and that under the corresponding one-cycle sinusoidal wave. The structural parameters a», = 2zr(rad/s) (T, = 1.0s), *d _{y}* = 0.16(m) were also used here. As seen in Figure 2.10, while the maximum deformation after the first impulse exhibits a rather larger value under the double

*Figure 2.10* Comparison of response time history under double impulse with that under corresponding one-cycle sinusoidal wave: (a) Normalized deformation, (b) Restoring-force (Kojima and Takewaki 2015).

*Figure 2.11* Comparison of restoring-force characteristic under double impulse with that under corresponding one-cycle sinusoidal wave (Kojima and Takewaki 201 5).

impulse compared to that under the corresponding one-cycle sinusoidal wave, the maximum deformation after the second impulse demonstrates a rather good correspondence.