Accuracy investigation by time-history response analysis to corresponding multi-cycle sinusoidal input
To check the accuracy in using the multiple impulse (Input Sequence 1) as a substituted version of the corresponding multi-cycle sinusoidal wave (representative of the long-duration ground motion input), the time-history response analysis of the undamped EPP SDOF model under the multi-cycle sinusoidal wave was conducted.
Figure 4.9 Response to Input Sequence 2 for V/Vy = 2: (a) displacement, (b) velocity, (c) restoring force, (d) force-deformation relation (Kojima and Takewaki 201 5c)
In this investigation, it is important to adjust the input level of the multiple impulse and the corresponding multi-cycle sinusoidal wave. This adjustment is conducted by using the equivalence criterion of the maximum Fourier amplitude (see Appendix 1) and a modification based on the response equivalence at some points with different input levels for better correspondence in the wider input velocity range (point fitting processing).
Figures 4.11(a) and (b) show the comparison of the ground displacement and velocity between the multiple impulse and the corresponding multicycle sinusoidal wave for the input level V/V = 3. In Figures 4.11(a) and (b), the structural parameters aq = 2л-(rad/s) (T, = 1.0s) and dy = 0.16(m) were used. The amplitude of the sinusoidal wave was amplified by 1.15 after both Fourier amplitudes of the sinusoidal wave, and the multiple impulse were adjusted (10 cycles). This amplification factor 1.15 was introduced based on the response equivalence at some points with different input levels (point fitting processing). This amplification coefficient was also introduced in the double impulse (Kojima and Takewaki 2016). It should be remarked that
Figure 4.10 Response to Input Sequence 2 for V/Vy = 3: (a) displacement, (b) velocity, (c) restoring force, (d) force-deformation relation (Kojima and Takewaki 20 I Sc).
the information on critical timing shown in Figure 4.5 was incorporated in Figure 4.12.
Figure 4.4 shows the comparison of the maximum plastic deformation Upldy and the residual displacement ujdy of the undamped EPP SDOF model under the multiple impulse and the corresponding multi-cycle sinusoidal wave with respect to the input velocity level. The multiple impulse provides a fairly good substitute of the multi-cycle sinusoidal wave in the evaluation of the maximum plastic deformation ujdy if the amplitudes of both inputs are adjusted. Although the residual displacement exhibits a rather good correspondence between the multiple impulse (Input Sequence 1) and the corresponding multi-cycle sinusoidal wave, the Input Sequence 2 shows the somewhat larger residual displacement. Flowever, since the Input Sequence 2 is used mainly for finding the critical timing, this discrepancy does not cause any problem.
Figure 4.12 presents the comparison of displacements to the multiple impulse (Input Sequence 1) and the corresponding sinusoidal wave for the
Figure 4.1 I Comparison of ground displacement and velocity between multiple impulse and corresponding multi-cycle sinusoidal wave for input level VIVy = 3 (Kojima andTakewaki 2015c).
Figure 4.12 Comparison of responses to multiple impulse and sinusoidal wave (phase lag has been adjusted): (a) VI Vy = 2, (b) VI Vy = 3 (Kojima and Takewaki 201 5c).
input velocity levels V/Vy = 2, 3 and the resonant impulse timing t0/t‘0 = 1.0. As pointed out above, since the steady state is very sensitive to the time increment in the time-history response analysis using an EPP model due to the plastic flow, the time increment was chosen as 1.0xl0-6[sec]. This time increment requires a lot of computational time and load. On the other hand, the closed-form expression derived in this chapter provides an extremely efficient and reliable tool for response evaluation. In Figure 4.12, the structural parameters a>= 2^[rad/sec] (T, = 1.0[sec]) and dy = 0.16[m] were used. The phase lag was adjusted for the comparison purpose. The ground displacement and velocity of the corresponding sinusoidal wave for V/Vy = 3 are shown in Figure 4.11. Although a slight difference exists in the first cycle, both responses show a fairly good correspondence in the steady state. If desired, the residual displacement can be evaluated from Figure 4.4(b). As is well known, the residual displacement is sensitive to the irregularity in the input in the case of the EPP system. This issue is beyond the scope of this chapter.
Proof of critical timing
Figure 4.13 illustrates the normalized plastic deformation amplitude upldy with respect to the timing of multiple impulse input for various input velocity levels V/Vy= 1,2, 3,4, 5 (Input Sequence 1). It can be understood that the critical timing t0 = to derived from the Input Sequence 2 provides the critical timing even under Input Sequence 1. Repetitive appearance of peaks with
Figure 4.13 Plastic deformation amplitude with respect to timing of multiple impulse input for various input levels (Input Sequence I).
the same amplitude indicates the existence of multiple solutions. However, the lowest (shortest) timing t0ltc0 = 1.0 may be meaningful from the viewpoint of occurrence possibility of such ground motion with long duration. The peak at the value of t0 larger than to (t0/to > 1.0) implies the action of the second impulse after the point with the zero restoring-force.