Long-period and pulse-like earthquake ground motions with large amplitudes, called near-fault pulse-like ground motions, were recorded near earthquake source faults. The characteristics of these types of ground motions and their effect on damage to building structures have been studied in the previous works (for example, Bertero et al., 1978; Sasani and Bertero, 2000; Sakai et al., 2000; Alavi and Krawinkler, 2001; Alavi and Krawinkler, 2004; Mavroeidis and Papageorgiou, 2003; Mavroeidis et al., 2004; Makris and Black, 2004; Kalkan and Kunnath, 2006; Xu et al., 2007; Takewaki and Tsujimoto, 2011; Minami and Hayashi, 2013; Khaloo et al., 2015; Okazawa et al., 2018). Such near-fault pulse-like ground motions are broadly classified into two types depending on their input directions (Sasani and Bertero, 2000). One is the fling-step input with a fault-parallel component of the near-fault pulse-like ground motion, and the second is the forward-directivity input with a fault-normal component. The main part of these inputs can be represented by a few series of wavelets, and their mathematical models have been proposed to evaluate seismic responses of building structures to these inputs in many papers (Sasani and Bertero, 2000; Sakai et al., 2000; Alavi and Krawinkler, 2001; Alavi and Krawinkler, 2004; Mavroeidis and Papageorgiou, 2003; Makris and Black, 2004; Xu et al., 2007; Takewaki and Tsujimoto, 2011; Minami and Hayashi, 2013; Okazawa et al., 2018). The one-cycle sinusoidal wave and three sinusoidal wavelets were employed as the mathematical models of the near-fault ground motions (Sasani and Bertero, 2000; Sakai et al., 2000; Makris and Black, 2004; Kalkan and Kunnath, 2006; Khaloo et al., 2015). Sakai et al. (2000) proposed a theory for predicting the acceleration amplitude and the period of one-cycle sinusoidal wave, which is regarded to be equivalent to actual recorded ground motions. Alavi and Krawinkler (2001) and Alavi and Krawinkler (2004) used the combination of square waves as a ground acceleration input to evaluate the buildings’ responses under the directivity pulse. Mavroeidis and Papageorgious (2003) compared various wavelets, for example, the Gabor wavelet, Berlage wavelet, and Ricker wavelet, in the modeling of the near-fault ground motions. Mavroeidis and Papageorgious (2003) also proposed a simple model of the near-fault ground motions, and Mavroeidis et al. (2004) evaluated the effects of its parameters on elastic and inelastic responses of single-degree-of-freedom (SDOF) systems. Xu et al. (2007) evaluated the performance of passive energy dissipation systems under near-fault ground motions using the Berlage wavelet. Takewaki and Tsujimoto (2011) employed Xu’s model as the near-fault ground motion and proposed its scaling method based on deformation and input energy. The Ricker wavelet has also been used as a simple model of the forward-directivity input (Minami and Hayashi, 2013; Okazawa et al., 2018). Okazawa et al. (2018) investigated the damage to super high-rise buildings under the Ricker wavelet, representing pulse-like ground motions.

Kojima and Takewaki (2015a) and Kojima and Takewaki (2015b) introduced the double and triple impulse inputs to model ground accelerations of the fling-step and forward-directivity inputs. These simple models enable the easy derivation of the critical elastic–plastic response under the near-fault ground motions. Actually, Kojima and Takewaki (2015a) and Kojima and Takewaki (2015b) derived the critical double-impulse and triple-impulse responses of undamped elastic–perfectly plastic SDOF systems in a closed form using the energy balance approach in terms of free vibration after each impulse. This innovative approach is based on the input model transformation rather than the structural model transformation, like an equivalent linearization (Takewaki and Kojima, 2021). The double impulse has one time interval, and the zero-restoring force timing characterized the critical timing of the second impulse for the SDOF system, which is equal to half of the resonant pulse period (Kojima and Takewaki, 2015a). It was shown that, by extending this theory, the critical response to the double impulse can be formulated for the SDOF system with bilinear hysteresis (Kojima and Takewaki, 2016). The approximate closed-form solutions of the critical double-impulse responses of the elastic–plastic SDOF system with viscous damping were also derived (Kojima et al., 2018; Akehashi et al., 2018a). In the derivation, both the approximation of the damping force–displacement relationship by a quadratic function and the assumption that the critical timing of the second impulse is the zero-restoring force timing even in the damping SDOF system played crucial roles in addition to an energy balance law. Furthermore, Akehashi and Takewaki (2019), Akehashi and Takewaki (2021), Akehashi and Takewaki (2022a), and Akehashi and Takewaki (2022b) extended these theories to elastic–plastic multi-degree-of-freedom (MDOF) systems under the critical double impulse. They proved that the critical timing of the second impulse is the timing when the sum of inertial forces at all floors becomes zero (Akehashi and Takewaki, 2019). They also proposed a pseudo-double impulse input, which can be expressed by external impulsive forces with a first-mode distribution, to simulate rather accurately the elastic–plastic response under the one-cycle sinusoidal wave (Akehashi and Takewaki, 2021) and derived approximate expressions in the closed form for the elastic–plastic responses of MDOF systems to the critical double impulse (Akehashi and Takewaki, 2022a; Akehashi and Takewaki, 2022b). Fujii (2024a) and Fujii (2024b) applied the theory of the pseudo-double impulse and pseudo-multiple impulse to analyze the critical response of reinforced concrete structures.

Furthermore, Kojima and Takewaki (2015c) proposed the multiple-impulse input to derive the critical steady-state responses of an elastic–plastic system. Previous studies provided the critical steady-state responses of undamped and damped bilinear hysteretic SDOF systems to the multiple-impulse input (Kojima and Takewaki, 2017; Akehashi et al., 2018b). Tamura et al. (2019) derived approximate expressions for the inelastic resonant responses of an elastic–perfectly plastic SDOF system with a nonlinear damper under the multiple-impulse input.

In contrast to the double impulse, it seems difficult to define the critical triple impulse for the elastic–plastic system because of the existence of two time intervals between the three consecutive impulses. Kojima and Takewaki (2015b) defined the critical triple impulse for an “undamped elastic–perfectly plastic system” as the triple impulse, in which the second impulse acts at the zero-restoring force timing, and the third impulse acts at the same time interval, following the second impulse. In this critical triple impulse, the two time intervals of the triple impulse have the same value. Given the simulation of directivity pulses and the capturing of the critical pulse period, this definition is reasonable phenomenologically. On the other hand, Kojima and Takewaki (2015b) proposed another triple impulse, with the second and third impulses acting at the zero-restoring force points. The derivation of the elastic–plastic responses to this triple impulse seems simple. Although this triple impulse has different time intervals among three impulses for the case in which the system yields after the first or second impulse, the elastic–plastic response of the undamped system can be derived by the energy balance law and obtained as a simple closed-form expression. Furthermore, this response is larger than that under the critical triple impulse (the former triple impulse) with the same time interval. Kojima and Hikita (2020) also used the energy balance approach to derive the approximate expressions of the critical responses of the SDOF “elastic–perfectly plastic system” with viscous damping to the triple impulse (the latter triple impulse), as well as a similar approximation of the damping force–displacement relationship in the previous works (Kojima et al., 2018; Akehashi et al., 2018a). Kojima and Takewaki (2024) named the triple impulse, in which the timings of the second and third impulses are the zero-restoring force timings (the latter triple impulse), Input Sequence 1 (IS1), and the triple impulse with the same time interval (the former triple impulse) Input Sequence 2 (IS2) for “an undamped bilinear hysteretic system.” The triple impulse (IS2) has the same time interval, and the critical triple impulse (IS2) was defined as the triple impulse with the critical time interval that maximizes the maximum displacement response for the varied time interval with the same value. Furthermore, they derived the elastic–plastic responses of the undamped SDOF systems with bilinear hysteresis under the triple impulse (IS1) and compared that with the critical elastic–plastic responses to the triple impulse (IS2).

In this paper, a triple impulse input is introduced as one of the mathematical models of forward-directivity inputs of near-fault earthquake ground motions (Kojima and Takewaki, 2015a; Kojima and Takewaki, 2015b; Kojima and Hikita, 2020; Kojima and Takewaki, 2024), and the critical responses are evaluated for the SDOF systems with bilinear hysteresis and viscous damping under the triple impulse. Two sequences of the triple impulse input are employed, as shown in the previous works (Kojima and Takewaki, 2024): Input Sequence 1 (IS1), in which the timings of the second and third impulses are the zero-restoring force timings, and Input Sequence 2 (IS2) with the same time interval. These two types of triple impulse and the SDOF system used in this paper are defined in Sections 2, 3. Section 4 shows the elastic–plastic responses under the triple impulse with IS1 and the critical elastic–plastic responses under the triple impulse with IS2 calculated by the time-history response analysis and compared for the SDOF system with bilinear hysteresis and viscous damping. It is then showed that the response of the damped SDOF bilinear system under the triple impulse (IS1) is the same or slightly larger than the critical response under the triple impulse with IS2. It should be remarked that, although Kojima and Takewaki (2015b) defined the critical triple impulse as the triple impulse with the same time interval, in which the second impulse acts at the zero-restoring force timing, the critical triple impulse (IS2) is defined in this paper as the triple impulse with the critical time interval that maximizes the maximum displacement response for the varied time interval with the same value. Section 5 shows the approximate expressions derived for the elastic–plastic responses of the SDOF system with bilinear hysteresis and viscous damping under the triple impulse in IS1. In the derivation, the quadratic function approximation of the damping force–displacement relationship and the energy balance approach are employed. Section 6 shows the validity of the triple impulse as the mathematical forward-directivity input model investigated by comparing the critical triple-impulse responses in IS1 and IS2 with the responses under the equivalent three wavelets of sinusoidal waves (TWSW) and the Ricker wavelet. Then, it is confirmed that the elastic–plastic responses under the equivalent TWSW and the Ricker wavelet can be estimated by the approximate expression of the triple-impulse responses in IS1 obtained in Section 5. The conclusions are summarized in Section 7.

The triple impulse was proposed recently to model the forward-directivity input of near-fault ground motions (Kojima and Takewaki, 2015a; Kojima and Takewaki, 2015b; Kojima and Hikita, 2020; Kojima and Takewaki, 2024). Figure 1 shows the ground acceleration, ground velocity, ground displacement, and the Fourier amplitude of ground acceleration of the triple impulse (IS2); TWSW; and the Ricker wavelet, which are commonly used as forward-directivity pulse models (Sasani and Bertero, 2000; Kalkan and Kunnath, 2006; Minami and Hayashi, 2013; Khaloo et al., 2015; Okazawa et al., 2018). It was reported that the most remarkable property of the forward-directivity input of near-fault ground motions is the existence of the impulsive wave consisting of a few sinusoidal components, and such input could cause a large deformation at certain stories of buildings, especially at lower stories depending on the relationship between their natural frequencies and the input frequency (Kalkan and Kunnath, 2006; Minami and Hayashi, 2013; Khaloo et al., 2015; Okazawa et al., 2018). Since it is well-known in the past works that the most influential part of the forward-directivity input of near-fault ground motions can be modeled by TWSW or the Ricker wavelet, the triple impulse that can represent these two models is used in this paper. The amplitudes of ground accelerations of the TWSW and the Ricker wavelet are scaled so that their maximum Fourier amplitudes are equal, as shown in Figure 1D. As shown in Figures 1A–C, u ¨ g TI t , u ˙ g TI t , u g TI t denote the ground acceleration, velocity, and displacement of the triple impulse (IS2), respectively; u ¨ g TWSW t , u ˙ g TWSW t , u g TWSW t indicate those of the TWSW; and u ¨ g RW t , u ˙ g RW t , u g RW t denote those of the Ricker wavelet. t denotes the time, and a dot on the variable means the time derivative. The starting time of the ground accelerations u ¨ g TSWS t and u ¨ g RW t shown in Figures 1A–C moves forward in order to match the timing of their peak accelerations with the second-impulse timing of the triple impulse (IS2). The ground acceleration u ¨ g TI t , u ¨ g TWSW t of the triple impulse and TWSW is explained in this section, and the ground acceleration u ¨ g RW t of the Ricker wavelet is explained in Section 6.

The ground acceleration of the triple impulse u ¨ g TI t and TWSW u ¨ g TWSW t , which are simple forward-directivity input models as shown in Figure 1A, can be expressed as follows: u ¨ g TI t = 0.5 V δ t − V δ t − t 0 + 0.5 V δ t − 2 t 0 , (1) u ¨ g TWSW t = 0.5 A p ⁡ sin ω p t 0 ≤ t ≤ 0.5 T p , T p ≤ t ≤ 1.5 T p A p ⁡ sin ω p t 0.5 T p ≤ t ≤ T p , , (2) where δ t is the Dirac delta function; 0.5V and V are the given initial velocities by the first and third impulses and the second impulse; t ₀ is the time interval among three impulses; A _p and ω _p (=2π/T _p ) are the acceleration amplitude and the circular frequency of TWSW, respectively; and T _p = 2t ₀. As shown in Figure 1B, 0.5 V is the peak ground velocity (PGV) of the triple impulse and V is called the input velocity amplitude of the triple impulse in this paper. The following equation expresses the relationship between the input velocity amplitude V of the triple impulse and the velocity amplitude V _p (= A _p /ω _p ) of the TWSW and was obtained with the aid of the equivalence of the peak Fourier amplitude of these two ground acceleration values (Kojima and Takewaki, 2015b; Kojima and Hikita, 2020). V p / V = A p / ω p / V = 0.62235722 ... (3)

Equation 3 is used in Section 6 to adjust the input levels of these two inputs.

The time intervals between the first and second impulses and between the second and third impulses of the triple impulse shown in Equation 1 are the same; the triple impulse with the same two time intervals was called Input Sequence 2 (IS2) (Kojima and Takewaki, 2024). Although the two time intervals need to be the same for capturing the critical frequency of the forward-directivity input, the triple impulse (IS1), in which the second and third impulses act at zero-restoring force timing, is employed to drive an approximate solution of the critical triple-impulse responses using similar procedures in the previous works (Kojima and Takewaki, 2015b; Kojima and Hikita, 2020; Kojima and Takewaki, 2024). Therefore, the triple impulse, which has different time intervals, is defined as Equation 4: u ¨ g IS 1 t = 0.5 V δ t − t 1 − V δ t − t 2 + 0.5 V δ t − t 3 , (4) where t ₁ (= 0), t ₂, and t ₃ are the acting timings of the first, second, and third impulses, respectively. The first and second time intervals ΔT ₁ (= t ₂ − t ₁) and ΔT ₂ (= t ₃ − t ₂) are defined as the time intervals between the first and second impulses and between the second and third impulses, respectively.

Since the characteristics of the forward-directivity input of near-fault ground motions are quite uncertain and the analysis of their effects on the structural response is also versatile and time-consuming, only the critical response producing the maximum effect is taken into account in this paper from the viewpoint of the reliability of the analysis.

A single-degree-of-freedom (SDOF) system with bilinear hysteresis and linear viscous damping of mass m, initial stiffness k, and damping coefficient c is considered, as shown in Figure 2. The relative displacement response, restoring force, and damping force are expressed by u, f _R , and f _D , respectively. d _y , f _y (= kd _y ), and α indicate the yield deformation (limit deformation of the elastic range), the yield force corresponding to the yield deformation, and the ratio of the post-yield stiffness (second stiffness) to the elastic stiffness of the bilinear hysteresis, respectively. Let ω 1 = k / m , T ₁ (= 2π/ω ₁), and h = c / 2 m k denote the natural circular frequency, natural period, and damping ratio, respectively. V _y (= ω ₁ d _y ) is the initial velocity of the undamped elastic SDOF system to the single impulse, satisfying equation 0.5mV _y ² = 0.5kd _y ², where 0.5mV _y ² represents the kinetic energy by the single impulse and 0.5kd _y ² represents the elastic strain energy at the yield deformation. The parameter V _y is employed to normalize the input velocity amplitude V.

In this section, the maximum elastic–plastic responses of the damped bilinear hysteretic SDOF system under the triple impulse in IS1 are compared with the critical triple impulse responses in IS2. The exact solutions of the elastic–plastic responses of the damped SDOF system to the triple impulse are difficult to derive. Therefore, the triple-impulse responses are calculated by the time-history response analysis (THRA) in this section. The critical triple impulse in IS2 is defined as the triple impulse with the critical time interval t ₀ ^c , which maximizes the maximum displacement response, and the critical time interval t ₀ ^c is captured by changing the time interval t ₀ from 0.1T ₁ to 1.0T ₁ in Equation 1 for the specific input velocity amplitude V/V _y .

Figures 3–5 show the comparison of the maximum displacement responses u _max/d _y normalized by the yield deformation under the triple impulse in IS1 and that under the critical triple impulse in IS2 for α = 0.01, 0.1, and 0.5 and h = 0, 0.01, 0.02, 0.05, 0.1, and 0.2 with respect to the normalized input velocity amplitude V/V _y . As can be seen from Figures 3, 5, when α = 0.01 and 0.5 and h = 0, 0.01, 0.02, and 0.05, the maximum triple-impulse responses in IS1 are larger than the critical responses in IS2 in the larger input level. On the other hand, as shown in Figures 3–5, in cases except those mentioned above, the maximum triple-impulse responses in IS1 can provide the critical response in IS2. The reasons for this are summarized below. For α = 0.01, in both the triple-impulse responses in IS1 and the critical responses under the triple impulse with IS2, the maximum response just after the second impulse, corresponding to u _max2 shown in Section 5, determines the maximum displacement. In IS2 for h = 0, the third impulse acts in the loading process after the second impulse and makes the maximum response just after the second impulse smaller, as stated in Kojima and Takewaki (2015b) and Kojima and Takewaki (2024). A similar situation occurs for h = 0.01, 0.02, and 0.05 and the larger input level. In this situation, the critical timing of the second impulse in IS2 is longer than the zero-restoring force timing, and the maximum displacement in IS1 becomes slightly larger than that in the critical IS2. When α = 0.01 and h = 0.1 and 0.2, the third impulse acts in the unloading process after the maximum displacement just after the second impulse, and the critical timing of the second impulse is determined as the zero-restoring force timing in both IS1 and IS2. In this case, the maximum responses in IS1 and the critical IS2 are the same. When α = 0.1, both the maximum responses in IS1 and the critical IS2 are determined by the maximum displacement just after the second impulse, and the third impulse acts in the unloading process after the maximum displacement just after the second impulse, regardless of the damping ratio. Therefore, the critical timing of the second impulse is determined as the zero-restoring force timing in both IS1 and IS2, and the response in IS1 is equal to that in the critical IS2. When α = 0.5, the maximum response just after the third impulse, corresponding to u _max3 shown in Section 5, becomes the maximum displacement to the triple impulse in all input levels for h = 0, as shown in Kojima and Takewaki (2024), and in the larger input level for h = 0.01, 0.02, 0.05. The critical timing in IS2 is determined to maximize the maximum displacement after the third impulse and is longer than the zero-restoring force timing. Therefore, the maximum displacement in IS1 becomes somewhat larger than that in IS2. On the other hand, when α = 0.5 and h = 0.1 and 0.2, the maximum displacement in both IS1 and the critical IS2 is determined by the maximum displacement just after the second impulse due to the large damping ratio. The critical timing of the second impulse in both input sequences is the zero-restoring force timing, and the maximum displacements are the same.

Figure 6 shows the maximum displacement response u _max normalized by the yield deformation d _y under the triple impulse in IS2 with respect to the time interval t ₀ shown in Equation 1 for a specific input velocity amplitude V/V _y for the SDOF system with α = 0.1 and 0.5 and h = 0.02 and 0.1. It can be seen that, as the input velocity amplitude becomes larger, the time interval attaining the maximum displacement response moves to a longer range depending on the post-yield stiffness ratio and the damping quantity.

To investigate the validity of modeling the forward-directivity input by the triple impulse, the elastic–plastic responses under the triple impulse in IS1 and the critical triple impulse in IS2 are compared with the elastic–plastic response to TWSW and the Ricker wavelet, which have been employed to simulate the ground acceleration of the forward-directivity pulse (Sasani and Bertero, 2000; Kalkan and Kunnath, 2006; Minami and Hayashi, 2013; Khaloo et al., 2015; Okazawa et al., 2018).

To compare the elastic–plastic responses under these inputs, the TWSW and the Ricker wavelet equivalent to the triple impulse in IS2 in Equation 1 are set according to the following procedure. As mentioned in Section 2, the pulse period T _p of the TWSW in Equation 2 is twice of the time interval t ₀ of the triple impulse in IS2 in Equation 1, and the ground acceleration amplitude A _p in Equation 2 can be determined from the velocity amplitude V of the triple impulse and ω p = 2 π / T p by using Equation 3 based on the equivalence of the maximum Fourier amplitudes of the TWSW and the triple impulse in IS2 (Kojima and Takewaki, 2015b; Kojima and Takewaki, 2024; Kojima and Hikita, 2020). Equation 3 has been obtained from the equivalence of the maximum Fourier amplitudes of the TWSW and the triple impulse.

The ground acceleration of the forward-directivity pulse modeled by the Ricker wavelet is expressed by (Minami and Hayashi, 2013; Okazawa et al., 2018) u ¨ g RW t = A p RW 0.5 ω p 2 t − T p 2 − 1 exp − 0.25 ω p 2 t − T p 2 , (52) where T _p = 2t ₀ and A _p ^RW is the maximum ground acceleration of the Ricker wavelet. The Fourier amplitude of the Ricker wavelet in Equation 52 attains the maximum at ω = ω _p , and its maximum Fourier amplitude is 4 π e − 1 A p / ω p . Similar to the scaling of the acceleration amplitude of the TWSW, the amplitude A _p ^RW is scaled so that the maximum Fourier amplitude of the Ricker wavelet becomes equal to that of the triple impulse in IS2. Then, A _p ^RW can be calculated by using V and t ₀ of the triple impulse in Equation 1 as Equation 53 (Kojima and Hikita, 2020): A p = π e 2 t 0 V . (53)

Figure 1D shows the Fourier amplitudes of the triple impulse, the TWSW, and the Ricker wavelet for V = 1.0 m/s and t ₀ = 0.5 s, whose ground acceleration amplitudes A _p and A _p ^RW are adjusted so that their maximum Fourier amplitudes are the same.

Figure 14 shows the comparison of the ductility (the maximum displacement response) to the triple impulse in IS1, the critical triple impulse in IS2, the TWSW, and the Ricker wavelet. The time interval t ₀ ^c , which maximizes u _max under the triple impulse in IS2, is adopted as t ₀ (= T _p /2) for the TWSW and the Ricker wavelet. The time-history response analysis is used to calculate the responses under the triple impulse (IS1 and IS2), the TWSW, and the Ricker wavelet in this section. The response to the triple impulse (IS1) is also obtained by the approximate expressions (AE) derived in Section 5.2. When α = 0.01 and h = 0.01, the triple-impulse response in IS1 by AE and THRA can evaluate the response under the Ricker wavelet with reasonable accuracy, and the response to the critical triple impulse in IS2 corresponds well to the response to the TWSW. When α = 0.1 and 0.5 and h = 0.01, the responses under the triple impulse in IS1 (THRA and AE), the critical triple impulse in IS2, TWSW, and the Ricker wavelet are almost the same. On the other hand, when h = 0.1, the response under the Ricker wavelet is slightly larger than both the triple-impulse responses (IS1 and critical IS2). This is because, while the response under the Ricker wavelet is a forced vibration, the triple-impulse response is a series of free vibrations. Although the response under the Ricker wavelet is slightly larger than that under the triple impulse (IS1) and the critical triple impulse (IS2) for the system with h = 0.1, the responses of the system with bilinear hysteresis and viscous damping to the TWSW and the Ricker wavelet can be estimated by the approximate expressions of the triple-impulse response (IS1).

Two types of sequences of a triple impulse, as one of the mathematical models of forward-directivity inputs, were proposed. One is the triple impulse, in which the timings of the second and third impulses are the zero-restoring force timings. This is called Input Sequence 1 (IS1). The other is Input Sequence 2 (IS2) with the same time interval. Then, the elastic–plastic responses of an SDOF system with bilinear hysteresis and linear viscous damping under the triple impulse (IS1) and the critical triple impulse (IS2) were calculated and compared. The critical triple impulse (IS2) was defined as the triple impulse with the critical time interval that maximizes the maximum displacement response when the time interval with the same value was changed. Furthermore, approximate expressions were derived for the elastic–plastic responses of the SDOF system with bilinear hysteresis and viscous damping under the triple impulse (IS1). To validate the appropriateness of the triple impulse as the mathematical model of the directivity pulse, the responses under the triple impulse (IS1) and the critical triple impulse (IS2) were compared with those under the equivalent three wavelets of sinusoidal waves (TWSW) and the Ricker wavelet. The conclusions are summarized as follows:

(1) The elastic–plastic responses of the damped bilinear hysteretic SDOF system under the triple impulse (IS1) were calculated for various normalized input velocity levels by the time-history response analysis and compared with those under the critical triple impulse (IS2) calculated by the time-history response analysis for the post-yield stiffness ratio α = 0.01, 0.1, and 0.5 and the damping ratio h = 0, 0.01, 0.02, 0.05, 0.1, and 0.2. The critical time interval of the triple impulse (IS2) was calculated by changing the same time interval and captured as the time interval that maximizes the maximum displacement response to the triple impulse (IS2). When the second stiffness ratio α is 0.01 and 0.5 and the damping ratio h is 0, 0.01, 0.02, and 0.05, the elastic–plastic responses under the triple impulse (IS1) were larger than those under the critical triple impulse (IS2) in the larger input velocity level. On the other hand, for the systems except those mentioned above, the elastic–plastic responses under the triple impulse (IS1) and the critical triple impulse (IS2) were the same even in the larger input level.