# Critical earthquake response of an elastic–perfectly plastic SDOF model under double impulse as a representative of near-fault ground motions

## Introduction

Critical excitation problems were posed independently by Dr. Drenick and Dr. Shinozuka in 1970 (Drenick 1970, Shinozuka 1970) and acknowledged as one of the important fields in applied mechanics. More detailed description on the essential features of the critical excitation problems can be found in Takewaki (2007), which is the first and unique monograph on critical excitation and includes its historical sketch. In contrast to the linear elastic response in the early stage, some critical excitation methods for elastic-plastic responses were treated comprehensively using the equivalent linearization methods in the reference (Takewaki 2007). Furthermore, deterministic critical excitation methods for elastic-plastic responses were dealt with by using the mathematical programming (SQP) in the same reference. In both approaches, repetitive computations were needed. Compared to these approaches, a more straightforward critical excitation method enabling the derivation of closed-form expressions on the critical elastic-plastic responses is explained in this chapter. This approach is based on an innovative concept in nonlinear structural dynamics, i.e., the substitution of earthquake ground motions by the impulse input and the capture of the critical timing of the impulse. This style is consistent throughout this book.

As pointed out in Chapter 1, there are two types of earthquake ground motions in general. One is the near-fault ground motion and the other is the long-period, long-duration ground motion. In this chapter, the former one is discussed, and the double impulse input is regarded as its simplification.

## Double impulse input

Consider a ground acceleration *u _{g}(t*) in terms of double impulse expressed

by

where V is the given initial velocity (velocity amplitude), 5(t) is the Dirac delta function, and *t _{0}* is the time interval between two impulses.

## SDOF system

Consider an undamped elastic-perfectly plastic (EPP) single-degree-of-free- dom (SDOF) system of mass *m* and stiffness *k.* The yield deformation and yield force are denoted by *d* and *f _{y},* respectively (see Figure 2.1). Let со, =

*yjklm*,

*u,*and

*f*denote the undamped natural circular frequency, the displacement of the mass relative to the ground, and the restoring force of the system, respectively. The time derivative is denoted by an over-dot. In Section 2.4, these parameters will be dealt with in a dimensionless or normalized form to derive the relation of permanent interest between the input and the elastic-plastic response. However, numerical parameters will be used partially in Section 2.5 to demonstrate an example setting of actual parameters.

## Maximum elastic-plastic deformation of SDOF system to double impulse

The elastic-plastic response of the EPP SDOF system to the double impulse can be described by the continuation of free-vibrations with sudden velocity change in the mass at the action of impulse. Different from the linear elastic system, the superposition principle for both impulses cannot be used for the elastic-plastic system. The maximum deformation after the first impulse is denoted by *u _{maxl}* and that after the second impulse is expressed by

*u*(w

_{maxl }_{maxl}and

^{w}max2 are the absolute value) as shown in Figure 2.1. It is noted that, if и

_{тах1}is large, и

_{тах2}may become smaller for the second impulse. This may be related to the drift of the origin in the restoring-force characteristic. The input of each impulse results in the instantaneous change of velocity of the structural mass. Such responses can be derived by an energy balance approach without resorting to solving the equation of motion directly. The kinetic energy given at the initial stage (at the time of action of the first impulse) and at the time of action of the second impulse is transformed into the sum of the hysteretic dissipation energy in terms of the plastic deformation and the strain energy in terms of the yield deformation. The corresponding schematic diagram was shown in Section 1.3 in Chapter 1. By using this rule, the maximum deformation can be obtained in a simple manner.

It should be emphasized that, while the resonant equivalent frequency can be computed for a specified input level by changing the excitation frequency parametrically in the conventional methods dealing with the sinusoidal input (Caughey 1960a, b, Roberts and Spanos 1990, Liu 2000, Moustafa

*Figure 2.1* Prediction of maximum elastic-plastic deformation under double impulse based on energy approach: (a), (b) CASE I: Elastic response; (c), (d) CASE 2: Plastic response after the second impulse; (e), (f) CASE 3: Plastic response after the first impulse (• : first impulse, A : second impulse) (Kojima and Takewaki 2015).

er al. 2010), no iteration is required in the method for the double impulse explained in this chapter. This is because the resonant equivalent frequency (resonance can be proved by using energy investigation: see Appendix 1) can be obtained directly without the repetitive procedure. As a result, the critical timing of the second impulse can be characterized as the time corresponding to zero restoring force.

While resonance curves were drawn in the conventional methods, only critical response (upper bound) 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 double impulse. One of the original points of the approach explained 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). Once the frequency and amplitude of the critical double impulse are computed, the corresponding one-cycle sinusoidal motion as a representative of the fling- step motion can be identified (see Section 1.2 in Chapter 1).

Let us explain the evaluation method of и_{тах1} and w_{max2} in the following. The plastic deformation after the first impulse is expressed by *u _{pi}* and that after the second impulse is denoted by

*u*as shown in Figure 2.1. There are three cases to be considered depending on the yielding stage as shown in Figure 2.1. Let us introduce a new parameter

_{p2}*V*for expressing the input level of the velocity of the double impulse at which the EPP SDOF system just attains the yield deformation after the first impulse. This parameter enables the nondimensional expression of the input velocity.

_{y}(=co-_{l}d_{y})Let us refer to Figure 1.5(a) for the energy balance law. Figures 2.1(a) and 2.1(b) show the maximum deformations after the first and critical second impulses, respectively, for the elastic case (CASE 1) during the whole stage. The maximum deformation и_{тах1} after the first impulse can be obtained from the following energy balance law (also conservation law in this case).

On the other hand, the maximum deformation и_{тах2} after the critical second impulse can be computed from another energy balance law (also conservation law in this case).

As explained earlier in Chapter 1, the critical timing of the second impulse is the time when the zero restoring force is attained in the unloading process. The velocity V induced by the second impulse is added to the velocity V at the zero restoring-force timing induced by the first impulse (full recovery of the velocity at the zero restoring-force timing due to zero damping). Since the strain energy does not exist at the zero restoring-force point, the idea in Figure 1.5(a) can be applied to this case.

Consider secondly the case (CASE 2) where the system goes into the yielding stage after the second impulse. Figures 2.1(c) and 2.1(d) show the schematic diagram of the response in this case. As in CASE 1, the maximum deformation и_{тах}, after the first impulse can be obtained from the energy balance law (see Figure 1.5(a)).

On the other hand, the maximum deformation *u _{max2}* after the second impulse can be computed from another energy balance law (mechanical energy is not conserved in this case).

As in the above case, the velocity V induced by the second impulse is added to the velocity V at the zero restoring-force timing induced by the first impulse. Please see Figure 1.7 except for the experience of plastic deformation after the first impulse.

Consider finally the case (CASE 3) where the system goes into the yielding stage even after the first impulse. Let us refer to Figures 1.5(b) and 1.7. Figures 2.1(e) and 2.1(f) present the schematic diagram of the response in this case. The maximum deformation w_{maxl} after the first impulse can be obtained from the energy balance law.

On the other hand, the maximum deformation и_{тах2} after the second impulse can be computed from another energy balance law.

where *v _{c}* is the velocity of mass at the zero restoring-force point induced by the first impulse and characterized by

*mv?l2*=

*f*and

_{y}dJ2*v*=

_{c}*V*A part of the input kinetic energy provided by the first impulse is dissipated by the plastic deformation and the strain energy corresponding to the yield deformation, which is stored at the maximum deformation point is transformed into the kinetic energy expressed by the velocity

_{y}.*v*at the zero restoring force point in the unloading process. The plastic deformation

_{c}*u*after the second impulse is characterized graphically by и

_{pl}_{тах2}+ (и

_{тах1}-

*d*+

_{y})= d_{y}*u*(г<

_{p2}_{тах1}is the absolute value). In other words, w

_{max2}can be obtained from

As in the above case, the velocity V by the second impulse is added to the velocity *v _{c}* (the maximum velocity in the unloading stage) induced by the first impulse.

The maximum deformations for CASEs 1-3 can be summarized as follows:

(CASE 1)

(CASE 2)

(CASE 3)

Figure 2.2 presents the schematic diagram of the employed energy balance law. Figure 2.2(a) shows the correspondence of the response of the SDOF model to the critical double impulse with the trajectory in the force- deformation relation. On the other hand, Figure 2.2(b) indicates the energy balance law after the first and second impulses. It can be found that the kinetic energy given by the first impulse is transformed into the strain energy of the SDOF model attaining the maximum displacement. It should be pointed out that, since the critical second impulse giving the maximum displacement after the second impulse is characterized by the point of the zero restoring force, the computation of the strain energy after the second impulse is simple.

Figure 2.3 shows the comparison between the critical timing of the second impulse and the noncritical timing of the second impulse. It can be observed that the critical input of the second impulse leads to the maximum strain energy (maximum deformation after the second impulse). It may be said by analogy that when a person is pushed by the same power in the state of the fastest running speed, the person can attain the furthest point. The proof of the critical timing of the second impulse is shown in Appendix 1.

Figure 2.4 illustrates the maximum deformation *u _{m?x}/d_{y}=* ma

*x(u„*

_{?xl}ld_{y},^{M}max2

*fd*with respect to the input level. Three regions exist corresponding to

_{y})CASEs 1-3. In CASEs 1 and 2, the maximum deformation *u _{max2}/d_{y}* after the second impulse is larger than the maximum deformation

*u*after the

_{nm}Jd_{y}*Figure 2.2*** Schematic diagram for employed energy balance law, (a) Response of building to critical double impulse, (b) Energy balance law after first and second impulses (kinetic energy is transformed into strain energy).**

first impulse. On the other hand, in CASE 3, two regions exist for the boundary case of M_{maxI} = *u _{imx2}.* While

*u*is larger than и

_{max2}ld_{y}_{пмх1}

*ld*in the smaller input level,

_{y}*u*is larger than

_{maxl}/d_{y}*u*in the larger input level. The boundary input level can be calculated by

_{maxl}id_{y}*u*=

_{maxX}*u*and is obtained as

_{maxl}*= 1 + л/З.*

**V/V**_{y}Figure 2.5 presents the normalized critical timing f_{0}/T, (T, = *2nlw _{x})* of the second impulse with respect to the input level. In CASEs 1, 2, the SDOF system exhibits an elastic response before the second impulse. Therefore, the critical timing

*tJT*= 0.5. On the other hand, in CASE 3, the SDOF system shows the elastic-plastic response before the second impulse. The time from the initial state to the input of the second impulse can be expressed as the sum of the time of the initial elastic region, that of the plastic region and that

_{x}*Figure 2.3* Comparison between critical timing of second impulse and noncritical timing of second impulse.

*Figure 2.4* Maximum normalized elastic-plastic deformation under double impulse with respect to input level (Kojima andTakewaki 201 5).

*Figure 2.5* Critical interval time of double impulse between first and second impulses with respect to input level (Kojima andTakewaki 2015).

of the unloading region. As stated before, the critical timing of the second impulse coincides with the time corresponding to the zero restoring force in the first unloading process (see Figure 2.1). The passing times of the above three regions can be obtained by solving the differential equation under the initial and terminal conditions. After some manipulation (see Appendix 2), it can be obtained as

Figure 2.5 shows that the timing is delayed due to plastic deformation as the input level increases. It is important to state again that only critical response giving the maximum value of *u _{nml}ld_{y}* is sought by the method explained in this chapter and the critical resonant frequency is obtained automatically for the increasing input level of the double impulse. One of the original points in this chapter is the tracking of only the critical elastic- plastic response.