An explicit limit on the input velocity level of the double impulse as a representative of the principal part of a near-fault ground motion has been derived for the overturning of a rigid block. The results may be summarized as follows.
1. The rocking vibration of a rigid block was formulated by using the conservation law of angular momentum and the conservation law of mechanical energy. The conservation law of angular momentum was used in determining the initial rotational velocity just after the first
Figure 9.11 Comparison of time-history responses 0(t)/a at overturning limit between double impulse and corresponding equivalent one-cycle sinusoidal wave magnified by coefficient about 1.53-1.54. (Nabeshima et al. 2016).
impulse and the rotational velocity change at the impact on the ground and the input of the second impulse. On the other hand, the conservation law of energy was used in obtaining the maximum rotational angles after the first impulse and that after the second impulse, which are needed for the computation of the overturning limit. This enabled us to avoid the computation of complicated nonlinear time-history responses.
- 2. The critical timing of the second impulse was characterized by the time of impact after the first impulse. It was clarified that the action of the second impulse just after the impact on ground corresponds to the critical timing.
- 3. The overturning of the rigid block can be characterized by the coincidence of the maximum rotational angle 02тлх after the second impulse with the limit value -« of rotation. This condition gives the critical velocity amplitude of the double impulse just inducing the overturning of the rigid block. Note that the area A2 in the restoring-force characteristic in the negative side can be obtained in closed form in terms of the velocity amplitude of the double impulse by using the conservation law of energy (Eq. (9.12)). In this process, the initial velocity of rotational angle was derived in closed form in Eq. (9.11) and the critical velocity amplitude limit of the double impulse can be obtained.
- 4. It was found from the explicit expression on the critical velocity amplitude limit of the double impulse that it is proportional to the square root of the size of the rigid block. This finding is important from the viewpoint of the structural design of monuments and tall buildings, i.e., as the structure becomes larger, it becomes more stable. This characteristic has been commonly understood and was confirmed here in a clear mathematical formulation.
- 5. Numerical examples including the comparison with the numerical simulation results by the Runge-Kutta method demonstrated the accuracy and reliability of the method explained in this chapter. However, the comparison of the response to the double impulse with that to the equivalent sinusoidal wave reminds us of the necessity to introduce a magnification coefficient (about 1.53-1.54 in this case). The introduction of two-step transformation of the magnitude of the equivalent sinusoidal waves (introduction of the one-cycle sinusoidal wave equivalent to the double impulse and magnification of the equivalent one-cycle sinusoidal wave) comes from the necessity of the equivalence of the maximum Fourier amplitude and from the viewpoint of connection to the previous work (Kojima and Takewaki 2016a).
- 6. The proposed magnified overturning limit exhibits a fairly good correspondence to the other available data for resonance (Makris and Kampas 2016, Dimitrakopoulos and Dejong 2012b).
Although only the critical input was dealt with in this chapter, it gives the lowest level of the limit input. If the input timing is not critical, it gives smaller responses (Taniguchi et al. 2017).
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Appendix 1: Verification of critical timing ofdouble impulse for various inputlevels
In order to verify the critical timing of the double impulse, time-history response analysis was conducted for various input levels. By using the relation between the areas At and A, in Figures 9.4 and 9.6 with the help of Eqs. (9.5), (9.6), (9.8), (9.10), (9.11), (9.12), and (9.15), the critical timing can be expressed by
In Eq. (9.A1), V is an arbitrary input velocity, V. is the critical overturning limit velocity obtained by Eq. (9.19), and a linear approximation sin(« - в) = (« - в) was used.
Figure 9.A1 shows the plot of dlmxJa with respect to tjt0c where t0c is given by Eq. (9.20) and t0 is an arbitrary timing of the second impulse. It can be confirmed that the assumption introduced in Section 9.3 (critical timing is just after the impact) is valid.
Figure 9.AI Verification of critical timing: plot of 02max/a with respect to t0/t0c (Nabeshima et al. 20 I 6).