Design of stiffness and strength for specified velocity and period of double impulse and specified response ductility
It may be useful to provide a procedure (or flowchart) for the design of stiffness and strength for the specified velocity and period of the near-fault ground motion input and response ductility. This design concept comes from the design philosophy that, if we focus on the worst case of resonance, the safety for other nonresonant cases is guaranteed (see Takewaki 2002).
Nondimensional figures, Figures 2.4 and 2.5, can be used for such design. Figure 2.12 illustrates the procedure (flowchart) for the design of stiffness and strength. One example is presented here:
[Specified conditions] V=2(m/s) (velocity of double impulse), t0=0.5(s) (interval of the double impulse and half the period of the corresponding sine wave), um3X/dy=4.0 (ductility), m = 4.0 x 106(kg)
[Design results] V/Vy=2.5, V,=0.80(m/s), T,=0.74(s), dv=0.094(m),
k = 2.9x 108(N/m), fy = 2.7 x 107(N)
Figure 2.4 provides V/V,.=2.5 for the specified ductility um3XJdy=4.0. Then V,,=0.80(m/s) is derived from the specified condition V=2(m/s) and V/V,=2.5. In the next step, T,=0.74(s) is found from Figure 2.5 for V7V,=2.5 and fo=0.5(s). In this model, c/v=0.094(m) is determined from the definition
Figure 2.12 Flowchart for design of stiffness and strength.
V, = Wdy and rhe condition T,(=2;r/a>1)=0.74(s). Finally the stiffness k = 2.9 x 108(N/m) is obtained from k = w^m and the yield strength fy = 2.7 x 107(N) is derived from fy = kdy.
It may be useful to remind that, while most of the previous researches on near-fault ground motions are aimed at disclosing the response characteristics of elastic or elastic-plastic structures with arbitrary stiffness and strength parameters and require tremendous amount of numerical task for clarification, the present chapter focused on the critical response (worst resonant response) and enabled the drastic reduction of computational task. It should be remarked that even the nonresonant case can be treated explicitly by using the energy balance law (see Appendix 1 and Kojima and Takewaki 2016a).
Application to recorded ground motions
To demonstrate the applicability of the explained evaluation method, it was applied to two recorded ground motions shown in Chapter 1 (Rinaldi Station FN (Northridge 1994), Kobe Univ. NS (Hyogoken-Nanbu 1995)). Figure 2.13 shows the comparison between the sum of the maximum displacements in both directions and the corresponding value computed by the time-history response analysis. It should be noted that, since the ground motions are specified, the structural parameters, шъ dy, were selected so as for each model to be resonant to the ground motions (refer to Section 7.6 and Kojima and Takewaki 2016a,b for the detailed procedure). It can be observed that the explained method exhibits fairly good performance.
Figure 2.13 Comparison between sum of maximum displacements in both directions to double impulse and corresponding value computed by time- history response analysis to recorded ground motions, (a) Rinaldi Station FN (Northridge 1994), (b) Kobe Univ. NS (Hyogoken-Nanbu 1995) (Kojima and Takewaki 2016a).
The obtained results may be summarized as follows:
- 1. The double impulse input can be a good substitute of the fling-step (fault-parallel) near-fault ground motion and a closed-form expression can be derived for the maximum response of an undamped elas- tic-perfectly plastic (EPP) single-degree-of-freedom (SDOF) model under the critical double impulse input.
- 2. Since only free-vibration is induced under such double impulse input, the energy approach plays an important role in the derivation of the closed-form expression for a complicated elastic-plastic response. In other words, the energy approach enables the derivation of the maximum elastic-plastic seismic response without resorting to solving the differential equation directly. In this process, the input of impulse is expressed by the instantaneous change of velocity of the mass. The maximum elastic-plastic response after impulse can be obtained through the energy balance law by equating the initial kinetic energy computed by the initial velocity to the sum of hysteretic and elastic strain energies. The maximum inelastic deformation can occur either after the first impulse or after the second impulse depending on the input level.
- 3. The validity and accuracy of the theory explained in this chapter were investigated through the comparison with the response analysis result to the corresponding one-cycle sinusoidal input as a substitute of the main part of the fling-step near-fault ground motion. If the level of the double impulse is adjusted so as for its maximum Fourier amplitude to coincide with that of the corresponding one-cycle sinusoidal wave, the maximum elastic-plastic deformation to the double impulse exhibits a good correspondence with that to the one-cycle sinusoidal wave. 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.
- 4. While the resonant equivalent frequency has to be computed for a specified input level by changing the excitation frequency parametrically in dealing with the sinusoidal input, no iteration is required in the method for the double impulse. This is because the resonant equivalent frequency can be obtained directly without repetitive procedure. The resonance was proved by using energy investigation and it was made clear that the timing of the second impulse can be characterized by the time corresponding to the zero restoring force in the unloading process.
- 5. Only critical response (upper bound) was captured by the method and it was shown that the critical resonant frequency can be obtained automatically for the increasing input level of the double impulse. Once the frequency and amplitude of the critical double impulse are computed, the corresponding one-cycle sinusoidal motion can be identified.
- 6. Using the newly derived nondimensional relations among response ductility, input velocity, and input period, a flowchart was presented for the design of stiffness and strength for the specified velocity and period of the near-fault ground motion input and the specified response ductility. This flowchart can provide a useful result for such design.