# Critical earthquake response of an elastic–perfectly plastic SDOF model with viscous damping under double impulse

## Introduction

In Chapter 1, the earthquake energy balance law was introduced for a damped single-degree-of-freedom (SDOF) model (Eq. (1.28) in Section 1.3.2). It was pointed out there that the energy dissipated by the viscous damping has to be evaluated in terms of the maximum displacement in an appropriate manner even if approximate in order to obtain the maximum displacement from the energy balance law. In this chapter, the critical earthquake response of an elastic-perfectly plastic (EPP) SDOF model with viscous damping under the double impulse is derived in an explicit manner. The quadratic-function approximation for the damping force-deformation relationship is introduced and enables the application of the earthquake energy balance law to the damped model. The validity and accuracy of the proposed theory are investigated through comparison of the results of the nonlinear time-history response analysis to the corresponding one-cycle sinusoidal input and actual recorded ground motions.

## Modeling of near-fault ground motion with double impulse

In this chapter, the double impulse is used as in Chapters 1 and 2. The critical time interval in the double impulse for the undamped elastic-perfectly plastic (EPP) SDOF system, which corresponds to a half of the resonant period in the one-cycle sinusoidal wave, can be obtained directly. In contrast, the resonant frequency of the one-cycle sinusoidal wave has to be computed for a specified input level by changing the input frequency parametrically and transforming the structural model to an equivalent model or solving complicated transcendental equations in the conventional approach (Caughey 1960, Iwan 1961).

A ground acceleration *ii _{g}* (t) in terms of the double impulse is expressed by

where V is the velocity amplitude of the double impulse, *S(t)* is the Dirac delta function, and *t _{0}* is the time interval of two impulses. As shown in Chapter 1, the acceleration M

_{g}

^{sw}of the corresponding one-cycle sine wave is expressed by

where *A _{p}* is the acceleration amplitude and

*w*is the circular frequency. The Fourier transform of Eq. (5.1) leads to

_{p}

On the other hand, the Fourier transform of Eq. (5.2) was presented in Section 1.2 (Eq. (1.5)).

## Elastic–perfectly plastic SDOF model with viscous damping

Consider a damped EPP SDOF model of mass *m,* stiffness *k,* and damping coefficient *c* as shown in Figure 5.1. It is assumed that the damping coefficient does not change, regardless of yielding. The parameters со, = *yjkhn*, T, = *2л1(о _{и}* and

*b = cl[ljkm*) denote the undamped natural circular frequency, the undamped natural period, and the damping ratio, respectively. On the other hand, the parameters coj = -y/l -frco, and T, = 2л7со) denote the damped natural circular frequency and the damped natural period, respectively. As for deformation and force parameters, let

*u, f*and

_{R}*f*denote the

_{D}*FigureS.I* Elastic-perfectly plastic SDOF model with viscous damping under double impulse.

displacement of the mass relative to the ground (deformation of the system), the restoring force in the spring, and the damping force in the dashpot, respectively. The parameters *d _{y}* and

*f*denote the yield deformation and the yield force, respectively. The deformation and force parameters will be treated as normalized parameters to capture the intrinsic relation between the input parameters and the elastic-plastic response.

_{y}## Elastic-plastic response of undamped system to critical double impulse

Kojima and Takewaki (2015a) derived a closed-form expression on the maximum elastic-plastic responses of an undamped EPP SDOF model under the critical double impulse (see Chapter 2). Those responses can be derived by an energy balance approach without solving the equation of motion directly. More specifically, the maximum deformation can be calculated by using the energy balance law, in which the kinetic energies given at the times of the first impulse and the second impulse are transformed into the sum of the elastic strain energy corresponding to the yield deformation and the energy dissipated during the plastic deformation (see Figures 1.5 and 1.7). The critical elastic-plastic response can be derived in closed form, and the critical time interval (corresponding to a half of the resonant period) can be derived automatically for the increasing input velocity level of the double impulse by using this method. Since it is expected that a similar theory can be developed for deriving the maximum elastic-plastic response of a damped SDOF system, the closed-form expression for the maximum deformation of the undamped EPP SDOF model that was derived in Kojima and Takewaki (2015a) and Chapter 2 is explained briefly in this section for later smooth connection.

The maximum deformations after the first and second impulses are denoted by w_{maxl} and *u _{maxl},* respectively, as shown in Figure 5.2, and the

*Figure 5.2* Maximum deformation of elastic-perfectly plastic model to critical double impulse: (a) CASE I: elastic range, (b) CASE 2: yielding after 2nd impulse, (c) CASE 3: yielding after I st impulse (•: I st impulse. A: 2nd impulse) (Kojima et al. 201 7, 2018).

maximum deformation under the critical double impulse is evaluated by *u _{max} =* max (и

_{тах}1,и

_{тах2}). Note that both parameters и

_{тах1}and и

_{тах2}are the absolute values. The plastic deformations after the first and second impulses are denoted by

*u*and

_{pl}*u*respectively. The maximum elastic-plastic response of the undamped EPP SDOF model under the critical double impulse can be classified into one of three cases, depending on the input velocity level (yielding stage). CASE 1 is the case of the elastic response even after the second impulse. CASE 2 is the case of the plastic deformation only after the second impulse. Finally, CASE 3 is the case of the plastic deformation after the first impulse. Figure 5.2 shows a schematic diagram of CASE 1, CASE 2, and CASE 3.

_{pl},Let *V _{y}(=w_{l}d_{)})* denote the input velocity level of the double impulse at which the maximum deformation of the undamped EPP SDOF model just attains the yield deformation after the first impulse. It is important to note that this parameter can be regarded as a strength parameter of the undamped EPP SDOF model. This parameter V

_{}}, is used for normalizing the input velocity level, and

*V/V*is simply called the input velocity level. The maximum deformations w

_{y}_{maxl}and м

_{тах2}with respect to

*V/V*in CASES 1-3 can be obtained as follows by using the energy balance law.

_{y}

Figure 5.3 shows the maximum deformation of the undamped EPP SDOF model under the critical double impulse normalized by the yield deformation with respect to input velocity level V/V... The critical timing of the second impulse (the critical time interval), which maximizes the maximum deformation и_{тах2} after the second impulse under a constant level of *V/V _{y}, *is characterized as the time when the restoring force attains zero in the unloading process after the first impulse (Kojima and Takewaki 2015a and Chapter 2). In CASES 1 and 2, since the response after the first impulse is in an elastic range, the critical time interval

*t*is a half of the initial natural period T, of the undamped EPP SDOF model. In CASE 3, since the undamped EPP SDOF model enters the yielding stage after the first impulse, it is necessary to derive the expression by solving the equation of motion. The critical time interval

_{0}^{c}*t*can be obtained by solving the equation of motion as follows.

_{0}^{c}*Figure 5.3* Maximum deformation *u _{max}ld_{y}* for input level

*VIV*(Kojima and Takewaki 201 5a).

_{y}*Figure 5.4* Critical impulse timing t_{0}^{c}/T, for input level *VIV _{y}* (Kojima and Takewaki 201 5a).

Figure 5.4 illustrates the critical time interval *t _{0}^{c},* normalized by

*T*with respect to input velocity level

_{b}*V/V*

_{y}.