# Convergence of critical impulse timing

In this section, it is investigated whether the response under the multiimpulse with the equal time interval *t _{0}^{c}* obtained in Section 6.3.4 converges to the steady state in which each impulse acts at the zero restoring-force point as shown in Figure 6.3. Another possibility is to act the multi-impulses at the zero-restoring-force timing. In this case, the time intervals of the multi-impulses are not constant at the beginning and may converge to a constant. In this section, the former analysis is made.

The closed-form expression for the time history response in the steady state can be derived (see Appendix 1). However, the transient response is complicated because the number of impulses for assuring convergence depends on the input velocity level and the post-yield stiffness ratio. The time-history response analysis is used to calculate the response under the multi-impulse with the time interval *t _{0}^{c} .* The parameters = 1.0 [sec],

*d*0.04 [m], Д

_{y}=*t =*1.0 x 10-

^{4}T, were used in the analysis. The parameter

*At*denotes the time increment used in the time-history response analysis. The response under the multi-impulse is calculated by adding ±V to the velocity of the mass at the impulse acting timing. Figures 6.9, 6.10, and 6.11 show

*Figure 6.9*** Response under multi-impulse with time interval t _{0}^{c}for **

*VIV*

_{y}**= 0.5 and**

*a =*

**tan(;r/8) = 0.414 (impulse timing is critical one): (a) displacement, (b) velocity, (c) restoring force, and (d) restoring force-deformation relation (circles indicate acting points of impulses) (Kojima and Takewaki 2017).**

*Figure 6.10* Response under multi-impulse with time interval t_{0}^{c} for *VIV _{y}* = 1.0 and

*a =*tan(/r/8) = 0.4 14 (impulse timing is critical one): (a) displacement, (b) velocity, (c) restoring force, and (d) restoring force-deformation relation (circles indicate acting points of impulses) (Kojima and Takewaki 2017).

the time histories of relative displacement, relative velocity, restoring force, and restoring force-deformation relation under the multi-impulse with the time interval *t _{0}^{c}* in the model with

*a*= tan (я/8) = 0.414 for

*V/V*0.5, 1.0, 1.5. This post-yield stiffness ratio was taken from the comparative past work (Iwan 1961). The time interval used in this section was obtained by using the assumption of the steady state. The circles in Figures 6.9,6.10, and

_{y}=6.11 indicate the acting points of impulses. It can be seen that the response converges to a state in which each impulse acts at the zero restoring force irrespective of the input velocity level, and the maximum deformation and the plastic deformation amplitude after convergence correspond to the closed-form expressions obtained in Section 6.3.1 and Section 6.3.2. In the model with *a =* tan (л/8) = 0.414, the input velocity levels *V/V _{y} =* 0.5, 1.0 correspond to CASE 1 in Section 6.3.1, and the acting points of impulses converge to the zero restoring force timing in the unloading process in Figures 6.9 and 6.10. From Figures 6.9 and 6.10, the required number of impulses for assuring convergence is about 25. On the other hand, the input

*Figure 6.1 I*** Response under multi-impulse with time interval t _{0}^{c} for **

*VIV*

_{y}=**1.5 and**

*a =*

**tan(n/8)**

*=*

**0.4 14 (impulse timing is critical one): (a) displacement, (b) velocity, (c) restoring force, and (d) restoring force-deformation relation (circles indicate acting points of impulses) (Kojima and Takewaki 201 7).**

velocity level V/V = 1.5 corresponds to CASE 2 in Section 6.3.2 and the acting points of impulses converge to the zero-restoring-force timing in the loading process in Figure 6.11. From Figure 6.11, CASE 2 requires over 100 impulses for convergence.

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

In order to check the accuracy of using the multi-impulse with the equal time interval as a substitute of the corresponding multi-cycle sinusoidal wave representing long-duration ground motions, the time-history response analysis of the bilinear hysteretic SDOF system under the corresponding multi-cycle sinusoidal wave is conducted.

In the present evaluation, it is important to adjust the input level of the multi-impulse and the corresponding multi-cycle sinusoidal wave based on the equivalence of the maximum Fourier amplitude (see Appendix 2 in this chapter and Appendix 1 in Chapter 4). It is noted that although two different multi-impulses with modification in the first impulse are treated in Chapter 4 and in this chapter, the resulting relation of the multi-impulse and the multi-cycle sinusoidal waves is the same. The period, the circular frequency, the acceleration amplitude, and the velocity amplitude of the corresponding sinusoidal wave are denoted by T_{;}, *w*, = *2n!T,, A _{b}* and V/ =

*A/w*respectively, and Т/ =

_{h }*2t*is used in this section. The number of cycles of the multi-cycle sinusoidal wave is a half of the number of impulses. In the derivation of the response under the multi-impulse, the steady state after a sufficient number of impulses is assumed as shown in Figures 6.9, 6.10, and 6.11. The relation between the input velocity level of the multi-impulse with the sufficient number of impulses (for example, over 20 impulses) and the acceleration amplitude of the corresponding multi-cycle sinusoidal wave with the sufficient number of cycles is expressed as follows (see Appendix 2 in this chapter and Appendix 1 in Chapter 4):

_{0}^{c}

The derivation of Eq. (6.21) was shown in Appendix 1 in Chapter 4 and in Appendix 2 of this chapter.

Figure 6.12 presents the comparison of the plastic deformation amplitude and the maximum deformation normalized by the yield deformation of the bilinear hysteretic SDOF system under the multi-impulse and the corresponding multi-cycle sinusoidal wave with respect to input velocity level. The response under the multi-impulse comes from the closed-form expressions derived in Sections 6.3.1 and 6.3.2, and the response under the corresponding multi-cycle sinusoidal wave is obtained by using the time-history response analysis. The structural and computational parameters T, = 1.0 [sec], *d _{y}* = 0.04 [m],

*At*= 1.0 x 10-

^{4}T, are used in the time-history response analysis, and the numbers of cycles used in the time-history response analysis are 100 cycles for « = tan (2л/180) = 0.035,500 cycles for

*a*= tan (л/8) = 0.414, and 1000 cycles for « = 0.9. These post-yield stiffness ratios were taken from the past comparable work (Iwan 1961).The multi-impulse providesa fairly good result for the multi-cycle sinusoidal wave in the evaluation of the maximum deformation and the plastic deformation amplitude if the maximum Fourier amplitude is adjusted. In order to relate the elastic-plastic responses under the multi-cycle sinusoidal wave to that under the multi-impulse, it is necessary to amplify the acceleration amplitude of the corresponding multi-cycle sinusoidal wave by 1.15 after both Fourier amplitudes of the sinusoidal wave and the multi-impulse are adjusted in the model with the elastic-perfectly plastic restoring force characteristics

*(a*= 0) (Kojima and Takewaki 2015c and

*Figure 6.12*** Comparison of plastic deformation and maximum deformation between critical multi impulse and corresponding multi-cycle sinusoidal wave: (a, b) ***a*** = 0.9, (c, d) ***a*** = tan ***(rrl***8) = 0.414, (e, f, g, h) ***a*** = tan (27T/I80) = 0.0349 ((g) and (h) are magnified ones of (e) and (f)) (Kojima andTakewaki 2017).**

Chapter 4 in this book). The maximum deformation under the multi-impulse is larger than that under the corresponding multi-cycle sinusoidal wave in *V/Vy* < -2 + *y/l/a* in CASE 1. On the other hand, the maximum deformation under the corresponding multi-cycle sinusoidal wave is larger than that under the multi-impulse in *V/V _{y} >* -2 +

*yj!a*in CASE 2.