Triple impulse input

As pointed out in the paper (Kojima and Takewaki 2015a) and Chapter 1 in this book, it is well accepted that the fling-step input (fault-parallel) of the near-fault ground motion can be represented by a one-cycle sinusoidal wave and the forward-directivity input (fault-normal) of the near-fault ground motion can be expressed by three wavelets of sinusoidal input (see Figure 3.1(a), (b)). The latter one is also called “a 1.5-cycle sinusoidal wave” and is similar to the well-known Ricker wavelet (Mavroeidis and Papageorgiou 2003). In this chapter, it is intended to simplify the latter typical near-fault ground motion by the triple impulse. This is because the triple impulse has a simple characteristic, and a straightforward expression on the maximum response can be expected even for elastic-plastic responses based on a simple energy balance approach to free vibrations as in the double impulse in Chapter 2. Furthermore, the triple impulse enables us to describe directly the critical timing of impulses (resonant frequency), which is not possible for the sinusoidal and other inputs without a repetitive procedure.

Consider a ground motion acceleration ii_g (t) in terms of the triple impulse, as shown in Figure 3.1(b), expressed by

where 0.5 V is the given initial velocity of the first impulse, 8[t) is the Dirac delta function and t₀ is the time interval among the consecutive two of three impulses. As shown in Figure 3.1(b), the velocity amplitudes 0.5 V, -V, 0.5 V are given so as to attain zero velocity and displacement at the end of input.

Figure 3.1 Simplification of ground motion (acceleration, velocity, displacement): (a) fling-step input and double impulse; (b) forward-directivity input and triple impulse (Kojima and Takewaki 2015a, b).

The comparison with the corresponding three wavelets of sinusoidal waves as a representative of the forward-directivity input of the near-fault ground motion (Mavroeidis and Papageorgiou 2003, Kalkan and Kunnath 2006) is also plotted in Figure 3.1(b). The corresponding velocity and displacement of such triple impulse and three wavelets of sinusoidal waves are shown in Figure 3.1(b). The Fourier transform of the triple impulse input u_g(t) can be derived as

SDOF system

Consider an undamped elastic-perfectly plastic (EPP) single-degree-of- freedom (SDOF) system of mass m and stiffness k. The yield deformation and the yield force are denoted by d_y and f_y, respectively. Let co_{ = lklm, u, and f denote the undamped natural circular frequency, the displacement (deformation) of the mass relative to the ground, and the restoring force of the system, respectively. The time derivative is denoted by an over-dot as in Chapter 2.

Maximum elastic-plastic deformation of SDOF system to triple impulse

The elastic-plastic response of the undamped EPP SDOF system to the triple impulse can be expressed by the continuation of free-vibrations with a sudden change of velocity of mass at the impulse acting points. The maximum deformations after the first, second, and third impulses are denoted by n_maxl, и_тах2, and и_тах3, respectively, as shown in Figure 3.2. The maximum deformations M_maxi, w_max2, and и_тах3 are the absolute value. The input of each impulse results in the instantaneous change of velocity of the structural mass. Such response can be derived by the combination of a simple energy balance approach and the solution of equations of motion for free vibration. It should be remarked that while the solution of equations of motion was not necessary for the double impulse input, this is required in the triple impulse input. This is because a complicated situation exists at the input of the third impulse. The kinetic energy given at the initial stage (the time of action of the first impulse), that at the time of action of the second impulse, and the kinetic energy plus the elastic strain energy at the time of action of the third impulse are transformed into the sum of the hysteretic dissipation

Figure 3.2 Prediction of maximum elastic-plastic deformation under triple impulse based on energy approach: (a) (b) (c) CASE I: Elastic response, (d) (e) (f) CASE 2: Plastic response after the third impulse, (g) (h) (i) CASE 3: Plastic response after the second impulse, (j) (к) (I) CASE 4: Plastic response after the first impulse (• : first impulse, ▲ : second impulse, ■: third impulse) (Kojima and Takewaki 2015b).

energy and rhe elastic strain energy corresponding to the yield deformation. By using this rule and incorporating the information from the equations of motion, the maximum deformation can be obtained in a simple manner. It should be noted that while a simple and clear concept of critical input was defined in the case of double impulse (Kojima and Takewaki 2015a and Chapter 2 in this book), the criticality can be used only before the third impulse in the present triple impulse. This is because the timing of action of the third impulse, determined already for the first and second impulses, decreases the maximum deformation и_тах2 after the second impulse and may increase the maximum deformation и_тах3 after the third impulse in the larger input level where the system goes into the yielding stage after the second impulse. However, it is shown that this treatment of determination of timing provides the true criticality in an input level of practical interest.

Although stated in Introduction in this chapter, it should be emphasized again that while the resonant equivalent frequency can be computed for a specified input level by changing the excitation frequency in a parametric or mathematical programming-oriented manner in dealing with the sinusoidal input (Caughey 1960, Liu 2000, Moustafa et al. 2010), no iteration is required in the method for the triple impulse. In the analysis for the triple impulse, the resonant equivalent frequency (resonance can be proved by using energy investigation: see Appendix 1) can be obtained directly without repetition (the timing of the second impulse can be characterized as the time with zero restoring force). It should be reminded again that the resonance is defined before the third impulse.

Only critical response (upper bound for fixed velocity amplitude and variable impulse interval) is captured by the method explained in this chapter, and the critical resonant frequency can be obtained automatically for the increasing input level of the triple impulse. One of the original points in this chapter is the introduction of the concept of “critical excitation” in the elastic-plastic response (Drenick 1970, Abbas and Manohar 2002, Takewaki 2007, Moustafa et al. 2010, Kojima and Takewaki 2015a). Once the frequency and amplitude of the critical triple impulse are computed, the corresponding three wavelets of sinusoidal waves as a representative of the forward-directivity motion can be identified.

Let us explain the procedure for evaluating и_тах1, и_тах2, and и_тах3. Note that the maximum deformations и_тах1, w_max2, and и_тах3 are the absolute value. The plastic deformation after the first impulse is expressed by u_pl, that after the second impulse is described by и_рЪ and that after the third impulse is denoted by u_pi. Four cases can be categorized depending on the yielding stage.

CASE 1: Elastic response during all response stages (и_тах3 is the largest) CASE 2: Yielding after the third impulse (и_тах3 is the largest)

CASE 3: Yielding after the second impulse (и_тах2 or w_max3 is the largest)

• 3-1: The third impulse acts in the unloading stage.
• 3-2: The third impulse acts in the yielding (loading) stage.

CASE 4: Yielding after the first impulse (и_тах2 is the largest)

In comparison with the case of the double impulse, the triple impulse brings difficulties in deriving the critical timing in a general case. This is because the timing of three impulses is fixed by one parameter t₀ and many complicated situations arise. In this chapter, a case is treated where the critical timing is defined only before the input of the third impulse (the critical time interval is obtained from the zero restoring-force timing of the second impulse, after the first impulse). This means that, if the third impulse does not exist, that timing gives the maximum value of и_тах2. Since the amplitude of the first impulse is half of the second impulse in the triple impulse, it usually occurs that the maximum response after the second impulse is larger than that after the first impulse.