# Critical earthquake response of an elastic–perfectly plastic SDOF model on compliant ground under double impulse

## Introduction

As in Chapters 2 and 5, the double impulse is introduced as a substitute of fling-step near-fault ground motion. It is well known that the ground beneath a structure influences the response of the structure during earthquake ground motions. It is therefore important and useful to include the effect of the ground on the responses of structures in the seismic-resistant design.

A closed-form expression for the elastic-plastic response of a structure on the compliant (flexible) ground by the “critical double impulse” is derived based on the expression for the corresponding structure with a fixed base. As in the case of the fixed-base model, only free-vibration appears under such double impulse, and the energy balance approach plays an important role in the derivation of the closed-form expression for a complicated elastic-plastic response on the compliant ground. It is remarkable that no iteration is needed in the derivation of the critical elastic-plastic response. It is shown via the closed-form expression that, in the case of a smaller input level of the double impulse to the structural strength, as the ground stiffness becomes larger, the maximum plastic deformation becomes larger. On the other hand, in the case of a larger input level of the double impulse to the structural strength, as the ground stiffness becomes smaller, the maximum plastic deformation becomes larger. The criticality and validity of the proposed theory are investigated through the comparison with the nonlinear time-history response analysis to the corresponding one-cycle sinusoidal input as a representative of the fling-step near-fault ground motion. The applicability of the proposed theory to actual recorded pulse-type ground motions is also discussed.

## Double impulse input

### Double impulse input

As explained in Kojima et al. (2015), Kojima and Takewaki (2015a, b), and Chapters 2 and 5, 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 a series of three sinusoidal wavelets. The fling step is caused by the permanent displacement of the ground induced by the fault dislocation, and the forward directivity effect is concerned with the relation of the movement of the rupture front with the site. The discussion in this chapter is intended to simplify typical nearfault ground motions by a double impulse (Kojima et al. 2015, Kojima and Takewaki 2015a). This is because the double impulse has a simple characteristic, and a straightforward expression of the response can be expected even for elastic-plastic responses based on an energy approach to free vibrations. Furthermore, the double impulse enables us to describe directly the critical timing of impulses (resonant frequency), which is not easy for the sinusoidal and other inputs without application of a repetitive procedure. It is remarkable to note that while most of the previous methods (Caughey 1960a, b, Roberts and Spanos 1990, Luco 2014) employ the equivalent linearization of the structural model with the input unchanged (see Figure 7.1(a) including an equivalent linear stiffness), the method proposed in Kojima and Takewaki (2015a, b) and in this chapter transforms the input into the double impulse with the structural model unchanged (see Figure 7.1(b)).

Figure 7.1 Comparison of present method with previous method: (a) previous method (equivalent linearization of structural model for unchanged input); (b) present method (transformation of input into double impulse for unchanged structural model) (Kojima andTakewaki 20 I 6).

Consider again a ground acceleration ug{t) as double impulse expressed

by

where V is the given initial velocity, S(t) is the Dirac delta function, and t0 is the time interval between two impulses. It is well understood that the double impulse is a good approximation of the corresponding sinusoidal wave even in the form of velocity and displacement. However, the correspondence in the response should be discussed carefully. This will be conducted in Section 7.5.

As shown in Chapters 2 and 5, the Fourier transform of iig(t) of the double impulse can be derived as

### Closed-form critical elastic-plastic response of SDOF system subjected to double impulse (summary of results in Chapter 2)

In Kojima and Takewaki (2015a) and Chapter 2, a closed-form expression for the critical response of an undamped elastic-perfectly plastic (EPP) SDOF system was derived for the double impulse (see Figure 7.2). The critical response plays a key role in the worst-case analysis (Drenick 1970, Takewaki 2002, 2007, Moustafa et al. 2010, Takewaki et al. 2012). Since this expression is used effectively in this chapter, the essence will be shown in this section.

Consider an undamped EPP SDOF system of mass m and stiffness k on the rigid ground. The yield deformation and force are denoted by d and f (see Figure 7.3). Let cot = sjklm, u, and f denote the undamped natural circular frequency, the displacement of the mass relative to the ground, and the restoring force of the model, respectively. The plastic deformation after the first impulse is expressed by wpl and that after the second impulse is denoted by upl.

Figure 7.2 SDOF model subjected to critical (resonant) double impulse.

As explained before, the impulse input changes the mass velocity by V instantaneously and the elastic-plastic response of the EPP SDOF system under the double impulse can be expressed by the continuation of free vibrations. Let итах] and um3Xl denote the maximum deformations after the first and second impulses, respectively, as shown in Figure 7.3. Those responses can be derived by an energy balance approach without solving the equation of motion directly. The kinetic energies given at the initial stage by the first impulse and at the subsequent time by the second impulse are transformed into the sum of the hysteretic energy and the strain energy corresponding to the yield deformation (see Figures 1.5, 1.7). The critical timing of the second impulse corresponds to the state of the system with a zero restoring force, and only the kinetic energy exists in this state as mechanical energies. By using this rule, the maximum deformation can be obtained in a simple manner (see Figure 1.7).

The maximum elastic-plastic response of the undamped EPP SDOF system under the critical double impulse can be classified into the three cases depending on the yielding stage. The parameter Vy(=a)1dy) denotes the input level of the velocity of the double impulse at which the undamped EPP SDOF system just attains the yield deformation after the first impulse. This parameter also indicates a strength parameter of the undamped EPP SDOF system. CASE 1 corresponds to the case of elastic response even after the second impulse and CASE 2 implies the case of plastic deformation only after the second impulse. Finally, CASE 3 presents the case of plastic deformation after the first impulse. Figure 7.3 shows the schematic diagram for these three cases.

Figure 7.3(a) shows the maximum deformation after the first impulse and that after the second impulse, respectively, for the elastic case (CASE 1) during the whole stage. The maximum deformations umaxi and um3Xl can be obtained as follows from the energy balance.

Figure 7.3 Prediction of maximum elastic-plastic deformation to double impulse based on energy approach: (a) CASE I: Elastic response; (b) CASE 2: Plastic response after second impulse; (c) CASE 3: Plastic response after first impulse (• : first impulse, A : second impulse) (Kojima and Takewaki 20 I 6).

Figure 7.4 Maximum normalized elastic-plastic deformation under double impulse with respect to input level (Kojima and Takewaki 201 5a).

A similar energy balance law enables rhe derivation of nmaxl and umax2 for CASEs 2 and 3 (Figures 7.3(b), (c)).

Figure 7.4 shows the maximum value of normalized elastic-plastic deformation under the double impulse with respect to input level.