# Design of stiffness and strength for specified velocity and period of triple impulse and specified response ductility

As in the case of the double impulse as a substitute for the near-fault fling- step input, it may be meaningful to present a procedure (flowchart) for the design of stiffness and strength for the specified velocity and period of the near-fault forward-directivity input and the response ductility. This design concept is based on the philosophy that if we focus on the worst resonant case, the safety for other nonresonant cases is guaranteed (see Takewaki 2002). This fact will be explained in the following section.

Since Figures 3.4 and 3.7 are nondimensional ones, they can be used for such design. Figure 3.14 shows the procedure (flowchart) for the design of stiffness and strength. Let us present one example: Figure 3.14 Flowchart for design of stiffness and strength (Kojima and Takewaki 2015b).

[Specified conditions] V=2.00(m/s), fo=0.500(s), umJdy=4.00, m = 4.00x 106(kg)

[Design results] V/Vy= 1.70, Vj=1.18(m/s), T,=1.00(s), ^.=0.188(01), k= 1.58 x 108(N/m),fy = 2.97 x 107(N)

Figure 3.4 provides V/V=1.70 for the specified ductility umaJdy=4.0. Then, V =1.18(m/s) is derived from the specified condition V=2.00(m/s) and V/Vy= 1.70. In the next step, T,=1.00(s) is found from Figure 3.7 for V/Vy= 1.70 and fo=0.5(s). In this model, t/y=0.188(m) is determined from the definition V = w^dy and T,(=2^/ai,)=1.00(s). Finally the stiffness k = 1.58 x 108(N/m) is obtained from k = w^m and the yield force fy = 2.97 x 107(N) is derived from fy = kdy.

# Approximate prediction of response ductility for specified design of stiffness and strength and specified velocity and period of triple impulse

Until Section 3.5, only the critical set of velocity and period of near-fault ground motion input and the corresponding critical response have been treated for a specified design of stiffness and strength. On the other hand, in Figure 3.15 Schematic diagram of prediction method of response ductility for specified design of stiffness and strength and specified velocity and period of near-fault ground motion (Kojima and Takewaki 201 5b).

Section 3.6, the design flowchart of stiffness and strength for the specified velocity and period of the near-fault ground motion input and specified response ductility was presented. In this section, an approximate method is explained for predicting the response ductility for the specified design of stiffness and strength and the specified velocity and period of near-fault ground motion input. If a more exact response is desired, the time-history response analysis for an arbitrary timing of impulses and an arbitrary input level can be done.

Figure 3.15 shows a schematic diagram for approximately predicting (only predicting the upper bound) the response ductility /r = umJdy for a specified design of stiffness and strength and a specified velocity and period of the near-fault ground motion input using the corresponding critical response. Generally, the specified set of velocity and period of the near-fault ground motion input is not the critical set for a given structure. In such a case, consider the critical set (point C) of velocity and period of the input corresponding to the specified set of velocity. Let цА and //c denote the response ductilities corresponding to point A and C. From Appendix 1, fiA < //c can be shown directly. This enables an approximate prediction of response ductility (only upper bound) for a specified design of stiffness and strength and a specified velocity and period of near-fault ground motion input.

# Comparison between maximum response to double impulse and that to triple impulse

Since it may be meaningful to show the comparison between the maximum response to the double impulse explained in Chapter 2 and that to the triple impulse, it is presented in this section. Figure 3.16 Comparison between maximum response to double impulse and that to triple impulse (Kojima et al. 2019).

Figure 3.16 shows such comparison. It can be understood, while the maximum deformation to the double impulse is larger than that to the triple impulse in the range of 0.5 < VIV < 1.5, it reverses in the range of V/V > 1.5. This phenomenon may be connected to the relation between the magnitudes of the first and second impulses. When the magnitude of the first impulse is the same as the second impulse in the double impulse, the maximum deformation after the second impulse is rather small because of the large deformation in the reverse direction after the first impulse. In the case of the triple impulse, the smaller velocity amplitude 0.5 V may relax this effect. In Figure 3.16, the maximum deformation of the corresponding elastic model is also plotted for reference. It can be confirmed from the energy balance law that the maximum deformations to the double impulse and the triple impulse are the same.