A dual system with sacrificial members that absorb the seismic energy prior to the beams and columns, e.g., a damage-tolerant structure (Wada et al., 2000), is one solution for creating structures with superior seismic performance. Unlike traditional earthquake-resistant structures, beams and columns in such dual systems are damage free (or have limited damage) after large earthquakes because most of the seismic energy input is absorbed by the sacrifice members. Therefore, buildings with such dual systems are more resilient than those without sacrificial members.

Steel damper columns (SDCs; Katayama et al., 2000) are energy-dissipating sacrificial members that are suitable for reinforced concrete (RC) buildings and are often used for multistory housing. The purpose of SDCs is to mitigate damage to beams and columns during strong seismic events. The author’s research group has been studying the seismic rehabilitation of existing RC buildings using SDCs (Fujii and Miyagawa, 2018; Fujii et al., 2019a) and the seismic design of new RC moment-resisting frames (MRFs) with SDCs (Mukoyama et al., 2021).

The peak deformation and cumulative strain energy are essential parameters in assessing the seismic performance of structural members. Specifically, the peak deformation is an essential parameter for RC members dominated by flexural behavior, as long as the story drift does not exceed 2.0% (Elwood et al., 2021). Meanwhile, both the peak deformation and the cumulative strain energy are important for the steel damper panels within SDCs. The residual displacement (Farrow and Kurama, 2003) is also important parameter, especially when the repair of the structures after earthquake is concerned. This is also important when the seismic sequence is considered (Ruiz-García and Negrete-Manriquez, 2011; Ruiz-García, 2012a, 2012b; Tesfamariam and Goda, 2015; Hoveidae and Radpour, 2021; Fujii, 2022). Specifically, Hoveidae and Radpour (2021) had pointed out that the large residual displacement after mainshock can significantly increase the peak response under aftershock.

In a previous paper, an energy-based prediction procedure for the peak and cumulative responses of an RC MRF building with SDCs was proposed (Fujii and Shioda, 2023). In this procedure, the building model is converted to an equivalent single-degree-of-freedom (SDOF) model that represents the first modal response. Then, two energy-related seismic intensity parameters are considered, namely, the maximum momentary input energy (Hori and Inoue, 2002) and the total input energy (Akiyama, 1985). The peak displacement is predicted by considering the energy balance during a half cycle of the structural response using the maximum momentary input energy. Meanwhile, the energy dissipation demand of the dampers is predicted considering the energy balance during an entire response cycle using the total input energy.

This procedure has been verified by comparing nonlinear time-history analysis (NTHA) results using non-pulse-like artificial ground motions (Fujii and Shioda, 2023) and 30 recorded pulse-like ground motions (Fujii, 2023b). However, the following issues remain.

I. In the presented procedure (Fujii and Shioda, 2023), the accuracy of the equivalent velocity of the maximum momentary input energy of the first modal response ( V Δ E 1 * )–equivalent displacement of the first modal response ( D 1 * ) relationship is essential for high quality predictions of the peak displacement. Accordingly, a monotonic pushover analysis was performed to evaluate the V Δ E 1 * – D 1 * relationship. However, the strain hardening effect observed in low-yield steel shear panels subjected to cyclic loading (Nakashima, 1995) cannot be considered in a monotonic pushover analysis.

The relationship between the energy and the peak deformation has been studied by several researchers. There are two main approaches: the first approach is to define a parameter that relates the cumulative input energy (or cumulative strain energy) and the peak deformation, and the second approach is to define an energy-based seismic intensity parameter that is directly related to the peak deformation. Akiyama (1988) stated that the cumulative inelastic deformation ratio should be assumed to be 4 times the peak inelastic deformation ratio for the seismic design of structures with elastic–perfectly plastic behavior, such as ductile steel MRFs. Then, the equivalent number of cycles can be formulated as the ratio of the cumulative inelastic deformation to the peak inelastic deformation in the simplified energy-based seismic design method (Akiyama, 1999). Manfredi and Cosenza (2003) investigated the relationship between the equivalent number of plastic cycles and the seismological parameters in the near field based on 128 near-fault and 122 far-fault records. They concluded that “the relative importance of the cyclic damage for structures grows at the higher distance from the fault, whereas in the near-source conditions structural response is governed by the peak demand, confirming the damage observations after destructive earthquakes.” Mota-Páez et al. (2021) noted that, for the seismic retrofit design of an RC soft-story building with a hysteresis damper under near-fault earthquakes, the equivalent number of cycles should be reduced. This is because, in the case of a near-fault earthquake, a large amount of seismic energy input occurs within a few cycles. Within the first approach, Fajfar (1992) proposed another dimensionless parameter γ normalizing the cumulative hysteretic dissipated (strain) energy by the peak deformation. This parameter γ has been applied to the pushover-based damage analysis method of RC MRFs (Gaspersic et al., 1992; Fajfar and Gaspersic, 1996) and the seismic design procedure of new RC MRFs (Teran-Gilmore, 1998). Decanini et al. (2000) studied the relationship between the cumulative input energy and the peak displacement of RC MRFs subjected to near-source earthquakes; they concluded that a reliable relationship between the cumulative energy and the peak displacement can be constructed, using either the cumulative hysteretic dissipated energy or the cumulative input energy. Mollaioli et al. (2011) analyzed the correlations between the energy and the peak displacement for linear and nonlinear SDOF and multi-degree-of-freedom (MDOF) models. Following these studies, Angelucci et al. (2023b) studied the relationship between the cumulative input energy and the peak displacement of RC MRFs with infills. Meanwhile, Benavent-Climent (Benavent-Climent et al., 2004; Benavent-Climent, 2011) proposed an energy-based assessment method for existing buildings; they focused on the strain energy under the monotonic loading of stories until the ultimate state, instead of the ultimate story drift.

Inoue and his research group (Hori et al., 2000; Inoue et al., 2000; Hori and Inoue, 2002) proposed the maximum momentary input energy as an energy-based seismic intensity parameter that is directly related to the peak displacement of RC structures. Note that a similar energy-based seismic intensity parameter was proposed by Kalkan and Kunnath (2007). The present authors formulated the time-varying function of the momentary energy input of an elastic SDOF model using Fourier series (Fujii et al., 2019b). Then, the concept of the momentary input energy was extended to bidirectional horizontal excitation (Fujii, 2021; Fujii and Murakami, 2021). In addition, Fajfar’s γ parameter was re-formulated using the maximum momentary input energy and the total input energy for RC structures (Fujii, 2021). Similarly, for base-isolated structures with hysteresis dampers, Akiyama’s equivalent number of cycles was reformulated using the maximum momentary input energy and the total input energy (Fujii, 2023a). Angelucci et al. (2023a) studied the relationship between the energy-related seismic intensity parameters proposed by Kalkan and Kunnath (2007) and the peak displacement of bare RC MRFs. The author thinks, there are two advantages in the concept of the maximum momentary input energy in comparison with the concept of equivalent number of cycles. The first is, the maximum momentary input energy can be evaluated from the complex Fourier coefficients of the input ground motion by using the time-varying function of the momentary energy input of an elastic SDOF model, as well as the cumulative input energy over the course of seismic input (Fujii et al., 2019a). The second is, theoretically, the concept of the maximum momentary input energy is easy to apply the structures with any type of hysteresis behavior, if the energy dissipation during a half cycle of structural response can be properly modeled: in the previous study by the author’s group (Fujii et al., 2021), the peak displacement of the RC structures with brittle members was predicted using the concept of the maximum momentary energy input. On the contrary, the equivalent number of cycles, investigated by Akiyama (1988), Akiyama (1999), Manfredi and Cosenza (2003), depends on so many parameters, e.g., the hysteresis models, ratio of the stiffness after yielding with respect to the initial stiffness, and type of ground motion (near-field or far-field). Therefore, the proposed equations of the equivalent number of cycles are empirical, which strongly relies on the numerical analysis results by using a certain ground motion set. In addition, from the author’s point of view, the prediction of the peak response based on the concept of maximum momentary energy input is straightforward, while its prediction based on the total energy input and the equivalent number of cycles is roundabout. Therefore, the author thinks that use of the two energy-based seismic intensity parameters, the maximum momentary input energy and the total input energy, is the best for evaluating the peak and cumulative response of building structures.

The above-discussed studies are based on NTHA results using recorded ground motions. Conversely, Takewaki and his research group (Kojima and Takewaki, 2015a; Kojima and Takewaki, 2015b; Kojima and Takewaki, 2015c; Kojima et al., 2015; Akehashi and Takewaki, 2021; Akehashi and Takewaki, 2022) studied simplifying the seismic input as a series of impulsive forces. First, Kojima et al. (2015) introduced the concept of the “critical double impulse input,” which represents the upper bound of the earthquake energy input for a given pulse velocity ( V p ). Next, Kojima and Takewaki introduced the double impulse input as a substitute for the fling-step near-fault ground motion Kojima and Takewaki (2015a). Following this study, they introduced the triple impulse input as a substitute for the forward-directivity near-fault ground motion Kojima and Takewaki (2015b) and the multiple impulse input as a substitute for long-duration earthquake ground motion Kojima and Takewaki (2015c). Then, Akehashi and Takewaki introduced pseudo-double impulse (PDI) Akehashi and Takewaki (2021) and pseudo-multi impulse (PMI) Akehashi and Takewaki (2022) analyses. In PDI and PMI analyses, the MDOF model oscillates predominantly in a single mode, considering the impulsive lateral force corresponding to a certain mode vector; when the impulsive lateral force corresponding to the first mode vector is considered, the MDOF model oscillates predominantly in the first mode.

The author believes that PDI is suitable to discuss the above two issues (I and II) for the following reasons: 1) the momentary input energy can easily be calculated as the increment of the energy input as a result of the pseudo impulsive lateral force. And 2) the interval of a half cycle of the structural response ( Δ t ) can easily be evaluated and is expected to be stable because the MDOF model oscillates predominantly in a single mode.

Given the above-outlined background, this study addresses the following questions.

(i) What is the V Δ E 1 * – D 1 max * relationship when considering the response of an RC MRF with SDCs subjected to critical PDI input? Does it agree with the predicted V Δ E 1 * – D 1 * relationship from the simplified equation proposed in the author’s previous study (Fujii and Shioda, 2023)?

In this study, critical PDI analyses of RC MRF models are performed. These critical PDI analyses are based on studies by Akehashi and Takewaki (2021) with one modification: in this study, the change in the first mode shape in the nonlinear range is considered to maintain consistency with the assumptions applied in the procedure (Fujii and Shioda, 2023). Six 8- and 16-story RC MRFs with and without SDCs are analyzed considering various intensities of the pulse velocity V p . Then, the predicted V Δ E 1 * – D 1 * and T 1 e f f – D 1 * relationships calculated according to the procedure (Fujii and Shioda, 2023) are compared with those obtained from the critical PDI analysis results.

Several researchers (McCormick et al., 2007; Tremblay et al., 2008) have investigated the response of structures with self-centering energy-dissipative devices to minimize the residual displacement after earthquake. In their studies, the behavior of devices is characterized by the flag-shaped hysteresis responses. The behavior of structures with such devices is out of scope of this study because the behavior of RC MRFs with SDCs is the main target of the following discussions. However, the author thinks the proposed procedure in the author’s previous study (Fujii and Shioda, 2023) can be easily extended to the structures with such self-centering energy-dissipative devices: only the modeling of the hysteretic dissipated energy during a half cycle of structural response is needed.

The rest of this paper is organized as follows. Section 2 outlines the critical PDI analysis based on Akehashi and Takewaki (2021). Section 3 presents the six RC MRFs with and without SDCs and the analysis methods. Section 4 describes the responses of the six RC MRFs obtained from the critical PDI analysis results. Section 5 discusses the comparisons with the predicted results based on the author’s previous study (Fujii and Shioda, 2023) and the critical PDI analysis results, focusing particularly on (i) the V Δ E 1 * – D 1 * relationship and (ii) the T 1 e f f – D 1 * relationship. The conclusions drawn from this study and the directions of future research are discussed in Section 6.

Figure 1 outlines the critical PDI analysis. This analysis is based on the studies by Akehashi and Takewaki (2021), Akehashi and Takewaki (2022), and one modification is made to maintain consistency with the assumptions applied in the procedure (Fujii and Shioda, 2023): in this study, the change in the first mode vector ( Γ 1 φ 1 ) in the nonlinear range is considered for the calculation of the first modal response at time t and the second pseudo impulsive lateral force.

Consider a planer frame building model (number of stories, N ) subjected to a pseudo impulsive lateral force proportional to the first mode vector (pulse velocity: V p ). Here, M is the mass matrix of the building model; d t , v t , and a t are the relative displacement, velocity, and acceleration vectors, respectively; and f R t and f D t are the restoring force and damping force vectors, respectively. The equivalent displacement ( D 1 * t ), equivalent velocity ( V 1 * t ), and equivalent relative acceleration ( A r 1 * t ) of the first modal response are defined as Eqs 1–3, respectively: D 1 * t = Γ 1 φ 1 T Md t M 1 * , (1) V 1 * t = D ˙ 1 * t = Γ 1 φ 1 T Mv t M 1 * , (2) A r 1 * t = D ¨ 1 * t = Γ 1 φ 1 T Ma t M 1 * , (3) M 1 * = Γ 1 2 φ 1 T M φ 1 , (4) where M 1 * defined as Eq. 4 is the effective first modal mass. Note that Γ 1 φ 1 and M 1 * depend on the local maximum equivalent displacement within the range 0 , t . In this study, the first mode vector at time t is updated assuming that Γ 1 φ 1 is proportional to the displacement vector at the time when the maximum equivalent displacement occurs ( t max ). The first mode vector at time t is updated according to Eq. 5: Γ 1 φ 1 ← 1 D 1 * t max d t max . (5)

The equivalent acceleration A 1 * t is defined as Eq. 6: A 1 * t = Γ 1 φ 1 T f R t M 1 * . (6)

Consider the energy response of the equivalent SDOF model representing the first modal response subjected to the ground acceleration ( a g t ). The cumulative input energy of the first modal response per unit mass over the course of the entire seismic event ( E I 1 * / M 1 * ) is calculated from the time derivative of the equivalent displacement ( D ˙ 1 * t = V 1 * t ) and the ground acceleration ( a g t ) as Eq. 24: E I 1 * M 1 * = − ∫ 0 t e n d a g t ⋅ D ˙ 1 * t d t = − ∫ 0 t e n d a g t ⋅ V 1 * t d t . (24)

According to Hori and Inoue (2002), the momentary input energy of the first modal response per unit mass ( Δ E 1 * / M 1 * ) is calculated as Eq. 25: Δ E 1 * M 1 * = − ∫ t t + Δ t a g t ⋅ V 1 * t d t . (25)

In Eq. 25, t and t + Δ t are the beginning and ending times of a half cycle of the structural response, respectively. The maximum momentary input energy per unit mass ( Δ E 1 max * / M 1 * ) is defined as the maximum value of Δ E 1 * / M 1 * over the course of the entire seismic event.

Following the study by Kojima and Takewaki (2015a), the ground acceleration ( a g t ) in the case of a critical DI analysis can be written as Eq. 26: a g t = V p δ t − t p 1 − δ t − t p 2 . (26)

In Eq. 26, δ · is the Dirac delta function, which satisfies Eq. 27: δ t = lim ε → + 0 0 t > ε 1 2 ε t ≤ ε ∫ − ∞ ∞ δ t d t = 1 ∫ − ∞ ∞ δ t f t d t = f 0 . (27)

Next, the momentary input energy of the first modal response per unit mass at the first and second half cycles ( Δ E 1 * / M 1 * 1 and Δ E 1 * / M 1 * 2 , respectively) are calculated from Eqs 25, 26. Assuming that the intervals of the first and second half cycles of the structural response are 0 , t p e a k 1 and t p e a k 1 , t p e a k 2 , respectively, as shown in Figure 1, Δ E 1 * / M 1 * 1 and Δ E 1 * / M 1 * 2 can be calculated as Eqs 28, 29: Δ E 1 * M 1 * 1 = − ∫ 0 t p e a k 1 a g t ⋅ V 1 * t d t , (28) Δ E 1 * M 1 * 2 = − ∫ t p e a k 1 t p e a k 2 a g t ⋅ V 1 * t d t . (29)

Note that, in Eq. 28, the interval of integration is changed from t p 1 , t p e a k 1 to 0 , t p e a k 1 to calculate the integrals that contain the Dirac delta function. To calculate Eqs 28, 29, the equivalent velocities ( V 1 * t ) at times t = t p 1 and t = t p 2 are rewritten as Eqs 30, 31, respectively: V 1 * t p 1 = 1 2 V 1 * t p 1 − 0 + V 1 * t p 1 + 0 = − 1 2 V p , (30) V 1 * t p 2 = 1 2 V 1 * t p 2 − 0 + V 1 * t p 2 + 0 = V 1 * t p 2 − 0 + 1 2 V p . (31)

Therefore, Δ E 1 * / M 1 * 1 and Δ E 1 * / M 1 * 2 can be calculated as Eqs 32, 33, respectively: Δ E 1 * M 1 * 1 = − V p ⋅ V 1 * t p 1 = 1 2 V p 2 , (32) Δ E 1 * M 1 * 2 = − − V p ⋅ V 1 * t p 2 = 1 2 V p 2 1 + 2 V 1 * t p 2 − 0 V p . (33)

The calculated Δ E 1 * / M 1 * 1 and Δ E 1 * / M 1 * 2 shown in Eqs 32, 33 are consistent with the above-shown increments of the energy input (Eqs 9, 20, respectively). This implies that, in the case of a critical PDI analysis, the momentary input energy is calculated as the increment of the energy input as a result of the pseudo impulsive lateral force.

The maximum momentary input energy of the first modal response per unit mass ( Δ E 1 max * / M 1 * ) is obtained as Eq. 34: Δ E 1 max * M 1 * = ⁡ max Δ E 1 * M 1 * 1 , Δ E 1 * M 1 * 2 = Δ E 1 * M 1 * 2 = 1 2 V p 2 1 + 2 V 1 * t p 2 − 0 V p . (34)

The cumulative input energy of the first modal response per unit mass ( E I 1 * / M 1 * ) is calculated as Eq. 35: E I 1 * M 1 * = Δ E 1 * M 1 * 1 + Δ E 1 * M 1 * 2 = V p 2 1 + V 1 * t p 2 − 0 V p . (35)

The equivalent velocity of the maximum momentary input energy of the first modal response ( V Δ E 1 * ) is calculated as Eq. 36: V Δ E 1 * = 2 Δ E 1 max * M 1 * = V p 1 + 2 V 1 * t p 2 − 0 V p . (36)

Similarly, the equivalent velocity of the cumulative input energy of the first modal response ( V I 1 * ) is calculated as Eq. 37: V I 1 * = 2 E I 1 * M 1 * = 2 ⋅ V p 1 + V 1 * t p 2 − 0 V p . (37)

In this study, the response period of the first modal response ( T 1 r e s ) is defined as twice the interval between the two local peaks ( D 1 p e a k * 1 and D 1 p e a k * 2 ) as Eq. 38: T 1 r e s = 2 Δ t = 2 t p e a k 2 − t p e a k 1 . (38)

Figure 2 shows the flow of the critical PDI analysis. In this flow, the damping force increment resulting from the velocity vector changing at analysis step n ( v n ) is treated as the unbalanced force to be corrected in the next step. In addition, the timing of the second pseudo impulsive lateral force is determined by checking the sign of the equivalent relative acceleration ( A r 1 * n ). The timing of the second pseudo impulsive lateral force is determined according the following condition: A r 1 * n A r 1 * n + 1 ≤ 0 . (39)

When Eq. 39 is satisfied, the second pseudo impulsive lateral force acts.

The analysis procedure was implemented in the computer code used in the previous analysis (Fujii and Miyagawa, 2018).

The six planar building models analyzed in this study are 8- and 16-story RC MRFs with and without SDCs. Figure 3 shows the simplified plans and elevations of the RC MRF building models. The two models labeled Type B are the same as those used in the previous study (Fujii and Shioda, 2023). Meanwhile, the two models made from the Type B models by removing all SDCs are referred to as Type O. The models referred to as Type A were made from the Type B models by reducing the number of SDCs. All RC MRFs analyzed herein were designed according to the strong-column/weak-beam concept, except for the foundation level beam and in the case of SDCs installed in an RC frame. In the latter case, at the joints between an RC beam and a steel damper column, the RC beam was designed to be sufficiently stronger than the yield strength of the steel damper column considering strain hardening. Sufficient shear reinforcement of all RC members was provided to prevent premature shear failure. The failure of the beam–column joints was not considered because it was assumed that sufficient reinforcement was provided. The natural periods of the first modal response in the elastic range ( T 1 e ) of the 8-story models are 0.740 s, 0.627 s, and 0.561 s for the Type O, A, and B models, respectively. Similarly, the T 1 e values of the 16-story models are 1.41 s, 1.21 s, and 1.12 s for the Type O, A, and B models, respectively.

The nonlinear behavior of the RC members and SDCs was modeled as in previous studies (Mukoyama et al., 2021; Fujii, 2022; Fujii and Shioda, 2023), except for the hysteresis rule used for the SDCs. Figure 4 shows the hysteresis rule. The same hysteresis model (stiffness degradation model) was used for the flexural springs in the RC members. Meanwhile, for the damper panel in the SDCs, two hysteresis models were considered to investigate the influence of strain hardening on the energy response. The first model was the normal bilinear model (LB); its yield strength is set to the initial yielding strength of the damper panel ( Q y D L ), and the strain hardening effect is neglected. The second model was the trilinear model used in previous studies (Mukoyama et al., 2021; Fujii, 2022; Fujii and Shioda, 2023) with the strain-hardening effect (SH); its upper bound strength ( Q y D U ) was set such that the ratio Q y D U / Q y D L equaled 300/205 = 1.46. Other details concerning the six structural models can be found in previous studies (Fujii, 2022; Fujii and Shioda, 2023). In this study, the viscous damping ratio of the first modal response of the RC MRFs in the elastic range ( h 1 f ) was set to 0.03.

Figure 5 shows the equivalent acceleration–equivalent displacement relationships of the six models. In this study, a displacement-based mode-adaptive pushover (DB-MAP) analysis (Fujii, 2014) was applied to obtain the relationship between the equivalent acceleration (whole building: A 1 * , RC MRF: A 1 f * , and SDCs: A 1 d * ) and the equivalent displacement ( D 1 * ). The pushover analyses were performed until the equivalent displacement ( D 1 * ) reached 1/75 of the assumed equivalent height ( H 1 * ): the assumed H 1 * values were 18.9 m and 35.9 m for the 8- and 16-story models, respectively. The target equivalent displacement for the pushover analysis was 0.252 m for the 8-story model, while that for the 16-story model was 0.479 m. Note that, in the pushover analyses of the Type A and B models, the influence of the strain hardening of the SDCs was neglected. The idealized A 1 * – D 1 * curves for the Type O models and the idealized A 1 f * – D 1 * and A 1 d * – D 1 * curves for the Type A and B models are shown in Figure 5. The values of D 1 * and A 1 * at the “yield point” ( D 1 y f * , A 1 y f * for RCF and D 1 y d * , A 1 y d * for SDC) are also shown. The bilinear idealization of each curve was performed following the methods shown in Fujii (2022).

The following observations can be drawn from Figure 5.

• The “yield displacement” of RCF ( D 1 y f * ) in Type A and B models is smaller than that in Type O models. The equivalent drift ratio at the “yielding of RCF” ( D 1 y f * / H 1 * ) of the Type O 8-story model is 1/170, while those of the Type A and B models are 1/196 and 1/205, respectively. Similarly, the ratio D 1 y f * / H 1 * of the Type O 16-story model is 1/187, while those of the Type A and B models are 1/215 and 1/218, respectively. This is because of the shortening of the beam span resulting from the presence of the SDCs.

Note that the D 1 y d * / D 1 y f * ratio influences the effectiveness of the SDCs with respect to seismic energy absorption. This is discussed in the analysis results.

In this study, the pulse velocity ( V p ) was set from 0.10 m/s, with an interval of 0.05 m/s, until D 1 max * was close to 1/75 of the assumed equivalent height ( H 1 * ). In each analysis, the ending time of the analysis ( t e n d ) was determined as the ending of the 32nd half cycle of free vibration following the action of the second pseudo impulsive lateral force.

Figure 6 compares the relationships between the seismic intensity parameters ( V p , V I 1 * , and V Δ E 1 * ) and the peak equivalent displacement ( D 1 max * ).

The following conclusions can be drawn from Figure 6.

• The seismic intensity parameters ( V p , V I 1 * , and V Δ E 1 * ) increase as D 1 max * increases. The V I 1 * – D 1 max * and V Δ E 1 * – D 1 max * curves are very similar to the V p – D 1 max * curve of the same model: the V I 1 * / V p ratio ranges from 1.8 to 2.0, while the V Δ E 1 * / V p ratio ranges from 1.5 to 1.7. The differences in the V I 1 * / V p and V Δ E 1 * / V p ratios between models are very small.

Figure 7 compares the ratios of the cumulative energy at the end of the simulation ( t = t e n d ), i.e., the ratios of the kinetic energy ( E K / E I ), damping dissipated energy ( E D / E I ), cumulative strain energy of the RC MRF ( E S f / E I ), and cumulative strain energy of the SDCs ( E S d / E I ). In this figure, the vertical dotted lines indicate the “yield displacement” of the SDCs ( D 1 y d * ) and the RC MRF ( D 1 y f * ).

The following conclusions can be drawn from Figure 7.

• In all cases, the ratio of the kinetic energy ( E K / E I ) is close to zero.

The differences in the E S d / E I ratios for the 8- and 16-story models can be explained by the differences in the D 1 y d * / D 1 y f * ratios shown in Figure 5. As shown in Figure 5, the D 1 y d * / D 1 y f * ratios in the 8-story models are 0.563 (Type A) and 0.598 (Type B), while those in the 16-story models are 0.731 (Type A) and 0.780 (Type B). Therefore, the effectiveness of the SDCs in 8-story models is better than that in 16-story models because the D 1 y d * / D 1 y f * ratio is smaller for the 8-story models than for the 16-story models.

Figure 8 shows the hysteresis loops of the first modal response (the A 1 * t – D 1 * t relationship) for each model. In this figure, the points at which the first and second pseudo impulsive lateral forces act ( Δ 1 E 1 * and Δ 2 E 1 * , respectively), and the points of the local peak responses ( D 1 p e a k * 1 and D 1 p e a k * 2 , respectively) are shown. The A 1 * – D 1 * curves obtained from the pushover analysis results (as in Figure 5) are also shown Figure 8.

The following conclusions can be drawn from Figure 8.

• In all models, larger equivalent displacements occur in the positive direction: D 1 p e a k * 2 is larger than D 1 p e a k * 1 . This means that the peak equivalent displacement of the first modal response over the course of the entire seismic event ( D 1 max * ) occurs at the end of the second half cycle of the response, where the momentary energy input Δ 2 E 1 * occurs.

Figure 9 shows the input energy ratio ( η E ) and the local peak equivalent displacement ratio ( η D ). Here, the η E ratio is defined as Eq. 40: η E = Δ 1 E 1 * / Δ 2 E 1 * . (40)

The η E ratio is 1/3 if the structure exhibits undamped linear elastic behavior. Meanwhile, the η E ratio is 1 if the structure exhibits rigid–perfectly plastic behavior, that is, no strain energy is released after the first local peak ( D 1 p e a k * 1 ) occurs.

The η D ratio is defined as Eq. 41: η D = D 1 p e a k * 1 / D 1 p e a k * 2 . (41)

The η D ratio is 0.5 if the structure exhibitis undamped linear elastic behavior. Meanwhile, the η D ratio is 1 if the structural response is symmetric in the positive and negative directions.

The following conclusions can be drawn from Figure 9.

• In all models, the η E ratio is nearly constant: most of the plots are distributed within a narrow range between 1/3 and 0.4.

Figure 10 shows the response periods of the first modal response ( T 1 r e s ) and the ratios of the effective period of the first modal response ( T 1 e f f ) and the response period ( T 1 r e s ). Here, T 1 e f f is defined as in Fujii and Shioda (2023) as Eq. 42: T 1 e f f = 2 π 4 + 7 π β 6 D 1 max * V Δ E 1 * . (42)

In Eq. 42, β is the complex damping ratio of the equivalent linear system. Here, β is set to 0.10.

The following conclusions can be drawn from Figure 10.

• In all models, T 1 r e s increases as D 1 max * increases.

Figure 11 shows the residual equivalent displacement ratio ( r r e s D ). Here, the r r e s D ratio is defined as Eq. 43: r r e s D = D 1 * t e n d / D 1 max * . (43)

The following conclusions can be drawn from Figure 11.

• In all models, the r r e s D ratio increases as D 1 max * increases. In all 8-story models, r r e s D is close to zero when D 1 max * is smaller than 0.08 m. Similarly in all 16-story models, r r e s D is close to zero when D 1 max * is smaller than 0.16 m.

Figure 12 compares the peak responses of all model types for V p = 0.55 m/s. The following local response quantities are compared: 1) the peak relative displacement; 2) the peak story drift; 3) the peak plastic rotation at the beam end ( θ p ⁡ max ); and 4) the peak shear strain of the damper panel ( γ D ⁡ max ). Note that, because the span length is different due to the presence of SDCs, θ p ⁡ max for the beam end at the right of column X2 is shown for the Type O and B models; meanwhile, θ p ⁡ max for the beam end at the left of column X2 is shown for the Type A models. The plot of γ D ⁡ max for the Type O models is not shown because no SDCs were installed in this model type.

The following conclusions can be drawn from Figure 12.

• For both the 8- and 16-story models, the responses of the Type O models are the largest, while those of the Type B models are the smallest.

Figure 13 compares the normalized cumulative strain energies of all the model types for V p = 0.55 m/s. The following local response quantities are compared: 1) the normalized cumulative strain energy at the beam end ( N E S b ) and 2) the normalized cumulative strain energy of the damper panel ( N E S d ). Here, N E S b is defined as Eq. 44: N E S b , k = E S b , k M y b , k ⋅ θ y b , k . (44)

In Eq. 44, M y b , k and θ y b , k are the yield moment and the chord rotation, respectively, at the yielding of the k th beam end and E S b , k is the cumulative strain energy of the k th beam end.

N E S d is defined as Eq. 45: N E S d , k = E S d , k Q y D L , k ⋅ γ y D L , k ⋅ h d 0 , k . (45)

In Eq. 45, Q y D L , k and γ y D L , k are the initial yield shear force and the initial yielding shear strain, respectively, of the k th damper panel; h d 0 , k is the height of the k th damper panel; and E S d , k is the cumulative strain energy of the k th damper panel.

The following conclusions can be drawn from Figure 13.

• For both 8- and 16-story models, the N E S b values of the Type O models are the largest, while those of the Type B models are the smallest. The largest N E S b values are observed at the forth floor level in the 8-story models and at the seventh floor level in the 16-story models.

This section summarizes the responses of the RC frame building models with and without SDCs subjected to a pseudo impulsive lateral force proportional to the first mode vector. The analysis results can be summarized as follows.

A) In the critical PDI analysis results shown herein, the peak equivalent displacement of the first modal response over the course of the entire seismic event ( D 1 max * ) occurs at the end of the second half cycle of the response, when the second pseudo impulsive lateral force acts. The momentary input energy corresponding to the second pseudo impulsive lateral force ( Δ 2 E 1 * ) is larger than that corresponding to the first pseudo impulsive lateral force ( Δ 1 E 1 * ).

Point (A) is important for discussing the relationship between the maximum momentary input energy ( Δ E 1 max * ) and the peak displacement ( D 1 max * ). In the prediction procedure for the peak displacement of Hori and Inoue (2002), as well as that of Fujii and Shioda (2023), the peak displacement is calculated considering the energy balance during the half cycle of the structural response: it is assumed that the peak displacement ( D 1 max * ) occurs at the end of the response, when the maximum momentary energy input occurs. Therefore, point (A) is consistent with the assumption of the prediction procedure.

Point (B) indicates that the A 1 * – D 1 * curve obtained from the pushover analysis results agrees well with the critical PDI analysis results, as far as the model without the strain hardening effect is concerned. This implies that, for the bare RC MRF studied herein, the A 1 * – D 1 * curve constructed from the critical PDI analysis results of various V p by plotting the peak response point will be the same as the A 1 * – D 1 * curve obtained from the pushover analysis.

Point (C) indicates that T 1 e f f calculated from D 1 max * and the equivalent velocity of the maximum momentary input energy of the first modal response ( V Δ E 1 * ) via Eq. 42 is clearly related to the response period ( T 1 r e s ). This may indicate that Eq. 42 is valid for calculating T 1 e f f when evaluating V Δ E 1 * and V I 1 * from the V Δ E and V I spectra as discussed in previous studies (Fujii, 2023a; Fujii and Shioda, 2023).

Point (D) indicates that the residual displacement after earthquake may be noticeable in case of the large number of SDCs are installed in RC MRFs. One reason why r r e s D becomes larger as the number of SDCs increases can be explained as follows: the unloading stiffness after the second peak equivalent displacement ( D 1 p e a k * 2 ) of Type B is larger than that of Types O and A as shown in Figure 8. This is consistent with the observations pointed out by Ruiz-García (2012b). It should be pointed out that the value of r r e s D obtained from the critical PDI analysis results is larger than that obtained in the NTHA considering the ground motion records (Fujii, 2022). This is because no seismic input occurs after D 1 p e a k * 2 occurs and the building model oscillates without external forces (free vibration) until t = t e n d in case of the critical PDI analysis, while in case of the NTHA considering the ground motion records the seismic input continues after the peak response occurs. Therefore, the residual displacement obtained from the critical PDI analysis may be the upper bound.

Figure 14 shows comparisons between the predicted results and the critical PDI analysis results for the Type O models. The upper two panels show comparisons of the predicted V Δ E 1 * – D 1 * curve and the V Δ E 1 * – D 1 max * plots obtained from the critical PDI results, while the lower two panels show comparisons of the predicted T 1 e f f – D 1 * curve and the T 1 r e s – D 1 max * plots obtained from the critical PDI results.

The following conclusions can be drawn from Figure 14.

• The predicted V Δ E 1 * – D 1 * curves are slightly below the V Δ E 1 * – D 1 max * plots obtained from the critical PDI results. More specifically, the predicted V Δ E 1 * – D 1 * curve of the 8-story model underestimates the critical PDI results when D 1 max * is smaller than 0.111 m (= D 1 y f * ), while the predicted V Δ E 1 * – D 1 * curve becomes closer to the critical PDI results as D 1 * increases. Similar observations can be made for the 16-story model.

Figure 15 shows comparisons between the predicted results and the critical PDI analysis results for the Type A and B models. Similar to Figure 14, the upper two panels show comparisons of the predicted V Δ E 1 * – D 1 * curves with the V Δ E 1 * – D 1 max * plots obtained from the critical PDI results, while the lower two panels show comparisons of the predicted T 1 e f f – D 1 * curves with the T 1 r e s – D 1 max * plots obtained from the critical PDI results. In this figure, the critical PDI results obtained by the models without the strain hardening effect (LB) and with the strain hardening effect (SH) are shown for comparison.