The peak deformation, cumulative strain energy, and residual deformation are essential parameters in assessing the seismic performance of structural members. Two energy-based seismic intensity parameters—the maximum momentary input energy (Hori et al., 2000; Inoue et al., 2000; Hori and Inoue, 2002) and the total input energy (Akiyama, 1985)—are related to the peak and cumulative responses, respectively. According to a study by Hori and Inoue (2002), the peak displacement of a structure can be evaluated by considering the energy balance during a half cycle of the structural response using the maximum momentary input energy. Meanwhile, the cumulative strain energy of structural members can be evaluated by considering the energy balance during an entire seismic event using the total input energy.

The motivation for using energy dissipation devices (dampers) is to mitigate damage to beams and columns during strong seismic events. A dual system with dampers, e.g., a damage-tolerant structure (Wada et al., 2000), is one solution for creating structures with superior seismic performance. In such a dual system, dampers play important roles (a) to reduce the peak displacement of the system and (b) to reduce the cumulative damage to beams and columns by absorbing seismic energy before it reaches the beams and columns. Accordingly, a building with such a dual system is more resilient than one with a traditional system: in the case of traditional moment-resisting frames (MRFs), most of the seismic energy is absorbed by the plastic hinges at the beam ends. Conversely, in the case of a dual system, most of the seismic energy is absorbed by the dampers; therefore, the seismic energy absorbed by the beams and columns is much smaller than in the case of traditional MRFs. Steel damper columns (SDCs; Katayama et al., 2000) are dampers suitable for reinforced concrete (RC) multistory housing. A SDC consists of a damper panel made of low-yield-strength steel plate, which absorbs the hysteresis energy, and a roll-formed H-section column, which behaves elastically. Numerous studies have been conducted on the seismic rehabilitation of existing RC buildings using SDCs (Fujii and Miyagawa, 2018; Fujii et al., 2019) and the seismic design of new RC MRFs with SDCs (Fujii and Kato, 2021; Mukoyama et al., 2021).

The concept of energy balance is quite useful to understand how such dampers work to improve the seismic performance of buildings. Recent advances in energy-based earthquake engineering can be found in Benavent-Climent and Mollaioli (2021) and Varum et al. (2023). Following Akiyama’s theory (1985, 1999), Benavent-Climent and his research group proposed a simplified seismic retrofitting design method for RC frames using dampers (Benavent-Climent, 2011; Benavent-Climent and Mota-Páez, 2017; Mota-Páez, et al., 2021; Benavent-Climent et al., 2024).

Takewaki and his research group (Kojima et al., 2015; Kojima and Takewaki, 2015a, 2015b, 2015c; Akehashi and Takewaki, 2021, 2022) have introduced the concepts of critical double impulse (DI) and critical multi impulse (MI) as substitutes for near-fault and long-duration earthquake ground motions. First, the concept of the critical DI was introduced to derive the upper bound of the earthquake input energy to a building structure (Kojima et al., 2015). Following this study, the critical response of an undamped elastoplastic single-degree-of-freedom (SDOF) model subjected to near-fault and long-duration earthquake ground motions was examined (Kojima and Takewaki, 2015a, 2015b, 2015c). Then, Akehashi and Takewaki (2021, 2022) introduced pseudo-double impulse (PDI) and pseudo-multi-impulse (PMI) to form a multi-degree-of-freedom (MDOF) model. 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.

An energy-based prediction procedure for the peak and cumulative response of RC MRFs with SDCs has been proposed (Fujii and Shioda, 2023). In the presented procedure, the building model is converted to an equivalent SDOF model that represents the first modal response based on a monotonic pushover analysis result. Then, the peak displacement is predicted using the maximum momentary input energy (Hori and Inoue, 2002), while the cumulative energy dissipation demand is predicted using the total input energy (Akiyama, 1985). In this procedure, the accuracy of the equivalent velocity of the maximum momentary input energy of the first modal response ($V_{\mathsf{\Delta }E1}{}^{*}$)–peak equivalent displacement of the first modal response ($D_1{{}^{*}}_{\max }$) relationship is essential for high quality prediction of the peak displacement. This procedure has been verified by comparing nonlinear time-history analysis (NTHA) results using non-pulse-like ground motions (Fujii and Shioda, 2023) and 30 recorded pulse-like ground motions (Fujii, 2023). The accuracy of the $V_{\mathsf{\Delta }E1}{}^{*}$–$D_1{{}^{*}}_{\max }$ relationship (seismic capacity curve) has also been verified by comparing the critical PDI analysis results (Fujii, 2024). However, the following issues remain.

The residual displacement (Farrow and Kurama, 2003) is another essential parameter that is important to discuss in the repair of structures after earthquakes. The residual displacement is also important when the seismic sequence is considered (Ruiz-García and Negrete-Manriquez, 2011; Ruiz-García, 2012a, 2012b; Tesfamariam and Goda, 2015; Hoveidae, N., Radpour, 2021; Fujii, 2022). Specifically, Ruiz-García (2012b) pointed out that the residual displacement of a stiffness-degrading SDOF model is smaller than that of an elastoplastic SDOF model, even though the peak displacement of a stiffness-degrading SDOF model is larger than that of an elastoplastic SDOF model. In addition, Hoveidae and Radpour (2021) found that the large residual displacement after a mainshock can significantly increase the peak response under an aftershock. In Fujii (2024), the residual displacement obtained from the critical PDI analysis of RC MRFs with SDCs is larger than that of RC MRFs without SDCs: the residual equivalent displacement reaches close to 30% of the peak equivalent displacement in the case of RC MRFs with SDCs. This is larger than that obtained in the NTHA considering the ground motion records (Fujii, 2022). Therefore, the residual displacement obtained from the critical PDI analysis may be the upper bound. Accordingly, the influence of the number of impulsive inputs ($N_p$) on the residual displacement should be investigated.

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

In this article, the seismic capacities of RC MRFs with and without SDCs are evaluated using incremental critical pseudo-multi impulse analysis (ICPMIA). How the seismic capacity of a structure corresponds to $D_1{{}^{*}}_{\max }$ is defined in terms of $V_{\mathsf{\Delta }E1}{}^{*}$. In the ICPMIA, the structure is subjected to various intensities of pulsive input. From the ICPMIA results, (a) the influence of the number of impulsive inputs ($N_p$) and the pinching behavior of the RC members on $D_1{{}^{*}}_{\max }$ and the residual equivalent displacement and (b) the influence of $N_p$ on the cumulative strain energies of the RC MRFs and SDCs are investigated. Then, the $V_{\mathsf{\Delta }E1}{}^{*}$–$D_1{{}^{*}}_{\max }$ relationships (seismic capacity curves) obtained from the ICPMIA results are compared with the predicted results based on the simplified equations.

The rest of this paper is organized as follows. Section 2 outlines the critical PMI analysis and ICPMIA. Section 3 presents four RC MRFs with and without SDCs and the analysis methods. Section 4 describes the responses of the RC MRFs obtained from the critical PDI and PMI analysis results, focusing in particular on (i) the pulse velocity ($V_p$)–peak equivalent displacement ($D_1{{}^{*}}_{\max }$) relationship, (ii) the hysteresis loop and residual displacement of the first modal response, and (iii) the cumulative strain energies of the RC MRFs and SDCs. Section 5 focuses on comparisons with the predicted results based on the study of Fujii and Shioda (2023) and the ICPMIA results. First, the simplified equations for calculating the energy dissipation capacity during a half cycle of the structural response are formulated. Next, the seismic capacity curve is predicted using the pushover analysis results. Then, the predicited seismic capacity curve is compared with the $V_{\mathsf{\Delta }E1}{}^{*}$−$D_1{{}^{*}}_{\max }$ plot obtained from the ICPMIA results. The conclusions drawn from this study and the directions of future research are discussed in Section 6.

First, an outline of the critical PMI analysis is described as follows. Note that this analysis is based on the critical PDI analysis presented in Fujii (2024). Figure 1 outlines the critical PMI analysis.

Following a study by Kojima and Takewaki (2015c), the ground acceleration ($a_g\left(t\right)$) in the case of the critical PDI and PMI analysis can be written as

In Equation (1), $N_p$ (≥ 2) is the number of pseudo impulsive lateral forces, ${}_k\mathsf{\Delta }{V}_g$ is the ground motion velocity increment of the $k$-th pulse, ${}_{k}t{}_p$ is the time when the pseudo impulsive lateral force acts, and $\delta \left(·\right)$ is the Dirac delta function that satisfies Equation (2). In the case of the PDI analysis ($N_p$= 2), ${}_k\mathsf{\Delta }{V}_g$ is defined as in Equation (3): where $V_p$ is the pulse velocity. Similarly in the case of the PMI analysis ($N_p$≥ 3), ${}_k\mathsf{\Delta }{V}_g$ is defined as in Equation (4):

Next, consider a planer frame building model (number of stories, $N$) subjected to a pseudo impulsive lateral force proportional to the first mode vector. Here, $M$ is the mass matrix of the building model; $d\left(t\right)$, $v\left(t\right)$, and $a\left(t\right)$ are the relative displacement, velocity, and acceleration vector, respectively; and $f_R\left(t\right)$ and $f_D\left(t\right)$ are the restoring and damping force vectors, respectively. The equivalent displacement ($D_1{}^{*}\left(t\right)$), equivalent velocity ($V_1{}^{*}\left(t\right)$), and equivalent relative acceleration ($A_{r1}{}^{*}\left(t\right)$) of the first modal response are defined in Eqs. (5)–(8): where $M_1{}^{*}$ is the effective first modal mass. Note that ${\mathsf{\Gamma }}_1{\phi }_1$ and $M_1{}^{*}$ depend on the local maximum equivalent displacement within the range $\left(0,t\right)$. In this study, the first mode vector at time $t$ is updated assuming that ${\mathsf{\Gamma }}_1{\phi }_1$ is proportional to the displacemt vector at the time when the maximum equivalent displacement occurs ($t_{\max }$). The first mode vector at time $t$ is updated via Equation (9): The equivalent acceleration $A_1{}^{*}\left(t\right)$ is defined as in Equation (10):

Figure 2 shows the flow of the critical PMI analysis. This flow is based on the flow of the critical PDI analysis presented in Fujii (2024). The analysis procedure is implemented by the computer code used in Fujii and Miyagawa (2018).

Incremental critical pseudo-multi impulse analysis (ICPMIA) is a parametric analysis method used to evaluate the nonlinear response of a structure by performing a critical PMI analysis considering various pulse velocities ($V_p$). In the ICPMIA analysis, $V_p$ varies from small to large levels until the structural response reaches a predetermined damage level (e.g., Life Safety of Collapse Prevention). This analysis is similar to the incremental dynamic analysis (IDA) (Vamvatsikos and Cornell, 2002) and the incremental N2 method (Dolšek and Fajfar, 2004). The difference between the three analysis methods is the nonlinear structural analysis method. In IDA, an NTHA of the model is performed using the time-history of the ground accelerations. The IDA results are the most rigorous of the three methods. However, the IDA result is very complex to understand with respect to the nonlinear structural characteristics because the IDA result is intricately intertwined with the nonlinear structural characteristics and the ground motion characteristics. Meanwhile, in IN2, a nonlinear static (pushover) analysis of the model is performed to obtain the nonlinear structural characteristics. Then, the seismic response (the seismic intensity corresponds to a certain peak displacement) is evaluated based on the inelastic spectra. The IN2 result is the most simple and easiest of the three to understand. However, its accuracy strongly depends on several assumptions, that the structure oscillates predominantly in a fundamental mode and that the spectra are inelastic. In ICPMIA, a critical PMI analysis is performed to obtain the nonlinear structural characteristics. Because ICPMIA can directly include the influence of cyclic loading, the influence of the duration of ground motions can be considered in ICPMIA by adjusting the number of pulsive inputs ($N_p$). In addition, because the ground motion in ICPMIA is simplified as the critical pulses determined automatically from the structural response, the ICPMIA result is still simple to understand with respect to the nonlinear structural characteristics.

The $V_{\mathsf{\Delta }E1}{}^{*}$−$D_1{{}^{*}}_{\max }$ plot is obtained from the ICPMIA result. In this study, the $V_{\mathsf{\Delta }E1}{}^{*}$−$D_1{{}^{*}}_{\max }$ curve is referred to as the “seismic capacity curve.” Note that the peak equivalent displacement ($D_1{{}^{*}}_{\max }$) may not occur at the end of the half cycle of the structural response corresponding to the maximum momentary input energy per unit mass ($\mathsf{\Delta }{E}_1{{}^{*}}_{\max }/M_1{}^{*}$). However, because $D_1{{}^{*}}_{\max }$ occurs at the end of a half cycle of the structural response, corresponding to $\mathsf{\Delta }{E}_1{{}^{*}}_{\max }/M_1{}^{*}$ in most cases analyzed herein, the relationship between the as-obtained $V_{\mathsf{\Delta }E1}{}^{*}$ and the as-obtained $D_1{{}^{*}}_{\max }$ is simply plotted in this study.

The four planar building models analyzed in this study are 8- and 16-story RC MRFs with and without SDCs. Figure 3 shows the simplified plan and elevation of the RC MRF building models. The two models labeled Type Dp (8Story-Dp and 16Story-Dp) are the same as those used in Fujii and Shioda (2023). The two models made from Type Dp by removing all SDCs are referred to as Type O (8Story-O and 16Story-O). All RC MRFs analyzed herein were designed according to the strong-column/weak-beam concept, except at the foundation level beam and in the case of steel damper columns 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 beam–column joints is not considered because it is assumed that sufficient reinforcement is provided. The natural periods of the first modal response in the elastic range ($T_{1e}$) of the 8-story models are 0.740 s and 0.561 s for Types O and Dp, respectively. Similarly, the $T_{1e}$ values of the 16-story models are 1.41 s and 1.12 s for Types O and Dp, respectively.

The nonlinear behavior of the RC members and SDCs is modeled as in previous studies (Mukoyama et al., 2021; Fujii, 2022; Fujii and Shioda, 2023), except the hysteresis rule used for the RC members. Figure 4 shows the nonlinear force–deformation relationship. In this study, the pinching behavior of the RC members is considered. The pinching model is assumed to be a linear combination of perfectly non-pinching and perfectly pinching models. The perfectly non-pinching model is identical to the stiffness degradation model used for RC members in previous studies (Mukoyama et al., 2021; Fujii, 2022; Fujii and Shioda, 2023). Meanwhile, the perfectly pinching model is a model that has no energy hysteresis energy dissipation in symmetric loading. A parameter $c$ (0 ≤ $c$ ≤ 1) is introduced to control the pinching behavior. When $c$ is 0, its behavior is that of a perfectly pinching model; when $c$ is 1, its behavior is that of a perfectly non-pinching model. In this study, four different pinching behaviors are considered: the parameter $c$ was set to 0.25, 0.50, 0.75, and 1.00, as shown in the bottom of Figure 4. For the damper panel in the SDCs, the same hysteresis model (trilinear model) is used. Other details concerning the four 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 ($h_{1f}$) was set to 0.03.

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 ($H_1{}^{*}$). The target $D_1{{}^{*}}_{\max }$ was set to 0.252 m for the 8-story models, while for the 16-story models the target $D_1{{}^{*}}_{\max }$ was set to 0.479 m. The total number of pseudo impulsive lateral forces ($N_p$) was set to 4, 6, and 8. A critical PDI analysis ($N_p$ = 2) of each model was performed for the comparisons. In each analysis, the ending time of the analysis ($t_{end}$) was determined as the ending of the 32nd half cycle of free vibration after the action of the second pseudo impulsive lateral force.

Figure 5 compares the relationship between the pulse velocity ($V_p$) and the peak equivalent displacement ($D_1{{}^{*}}_{\max }$). The following conclusions can be drawn.

Figure 6 compares the peak story drift. Here, the cases $c$ = 0.25 (significant pinching) and $c$ = 1.00 (perfectly non-pinching) are selected.

The following conclusions can be drawn from Figure 6.

Figure 7 shows the hysteresis loops of the first modal response ($A_1{}^{*}\left(t\right)$−$D_1{}^{*}\left(t\right)$ relationship) for each model. The hysteresis loops obtained from the critical PDI analysis ($N_p$ = 2) and critical PMI analysis ($N_p$ = 8) are shown in this figure; the hysteresis loops for $c$ = 0.25 (significant pinching) and $c$ = 1.00 (perfectly non-pinching) are compared. In Figure 7, the beginning and ending points of the half cycle of the structural response when the maximum momentary input energy per unit mass ($\mathsf{\Delta }{E}_1{{}^{*}}_{\max }/M_1{}^{*}$) occurs is shown by the red curve. The numbers in the figure indicate the number of the local peak. The points at which the pseudo impulsive lateral force acts ($\mathsf{\Delta }{E}_1{{}^{*}}_{\max }$) and the point at the end of the simulation ($t_{end}$) are also shown.

The following conclusions can be drawn from Figure 7.

Figure 8 shows the residual equivalent displacement ratio ($r_{resD}$). Here, the $r_{resD}$ ratio is defined as shown in Equation (33):

The following conclusions can be drawn from Figure 8.

Figure 9 compares the ratios of the cumulative strain energy of the entire frame model ($E_S/E_I$) at the end of the simulation.

For Type O, the following conclusions can be drawn from Figure 9.

In addition, the following conclusions can be drawn from Figure 9 for Type Dp.

Next, the discussion focuses on the cumulative strain energy ratio of the RC MRF ($E_{Sf}/E_I$) and the cumulative strain energy ratio of the SDCs ($E_{Sd}/E_I$) for Type Dp models. Figure 10 compares the ratios $E_{Sf}/E_I$ and $E_{Sd}/E_I$.

For 8story-Dp, the following conclusions can be drawn from Figure 10.

In addition, the following conclusions can be drawn from Figure 10 for 16story-Dp.

This section summarizes the responses of the RC MRF models with and without SDCs as obtained from the critical PMI analysis results.

It is assumed that the peak equivalent displacement of the first modal response ($D_1{{}^{*}}_{\max }$) occurs when the maximum momentary energy input ($\mathsf{\Delta }{E}_1{{}^{*}}_{\max }/M_1{}^{*}$) occurs. Following Fujii and Shioda (2023), the energy dissipation capacity during a half cycle of the structural response is expressed as

In Equation (34), $\mathsf{\Delta }{E}_{\mu 1f}{}^{*}/M_1{}^{*}$ and $\mathsf{\Delta }{E}_{\mu 1d}{}^{*}/M_1{}^{*}$ are the contributions from the hysteretic dissipated energies of the RC MRFs and SDCs, respectively, while $\mathsf{\Delta }{E}_{D1}{}^{*}/M_1{}^{*}$ is the contribution from viscous damping.

Figure 11 shows the simplified model for calculating $\mathsf{\Delta }{E}_{\mu 1f}{}^{*}/M_1{}^{*}$ and $\mathsf{\Delta }{E}_{\mu 1d}{}^{*}/M_1{}^{*}$. In the figure, $A_{1f}{}^{*}$ and $A_{1d}{}^{*}$ are the contributions of the RC MRFs and SDCs, respectively, to the equivalent acceleration of the first modal response ($A_1{}^{*}$). Here, the $A_{1f}{}^{*}$−$D_1{}^{*}$ and $A_{1d}{}^{*}$−$D_1{}^{*}$ relationships are idealized by bilinear curves, where the “yield” point of the idealized $A_{1f}{}^{*}$−$D_1{}^{*}$ relationship is Y_F($D_{1yf}{}^{*}$, $A_{1yf}{}^{*}$) and that of the idealized $A_{1d}{}^{*}$−$D_1{}^{*}$ relationship is Y_D($D_{1yd}{}^{*}$, $A_{1yd}{}^{*}$). In addition, ${\mu }_f$ and ${\mu }_d$ denote the global ductilities of the RC MRFs and SDCs, respectively, which are defined in Equation (35):

The contribution of the hysteretic dissipated energy of the RC MRFs is calculated using Eq (36):

The function $\widetilde{f_F}\left({\mu }_f\right)$ is defined by Equation (37): where ${\eta }_D$ is the ratio of the displacements in the positive and negative directions. Given the left panel of Figure 11, $f_F\left({\mu }_f,{\eta }_D\right)$ is calculated as

Note that, in Equation (38), the influence of the pinching behavior of the RC members on the hysteretic energy dissipation in a half cycle of the structural response is considered by the parameter $c$. This is an updated version of this equation, as compared with previous studies (Fujii, 2022; Fujii and Shioda, 2023). By substituting Equation (38) into Equation (37), the function $\widetilde{f_F}\left({\mu }_f\right)$ is calculated as

Note that Equation (39) is consistent with the equation in Fujii and Shioda (2023): by substituting $c$ = 1 into Equation (39), the same equation for the perfectly non-pinching RC MRFs (Fujii and Shioda, 2023) is obtained. Similarly, the contribution of the hysteretic dissipated energy of the SDCs is calculated using Eq (40):

The function $\widetilde{f_D}\left({\mu }_d\right)$ is defined as in Equation (41):

It is assumed that the energy dissipation of SDCs is independent of the pinching behavior of the surrounding RC beams, as shown in the right panel of Figure 11. Therefore, $f_D\left({\mu }_d,{\eta }_D\right)$ is calculated such that

By substituting Equation (42) into Equation (41), the function $\widetilde{f_D}\left({\mu }_d\right)$ is calculated as shown in Equation (43):

The contribution of the viscous damping is calculated using Equation (44), as in Fujii (2022): where $h_{1f\max }$ is the damping ratio corresponding to the peak equivalent displacement ($D_1{{}^{*}}_{\max }$) and $A_{1f}{{}^{*}}_{\max }$ is the contribution of the RC MRF to the equivalent acceleration at $D_1{{}^{*}}_{\max }$.

Next, the seismic capacity curve (the $V_{\mathsf{\Delta }E1}{}^{*}$−$D_1{{}^{*}}_{\max }$ curve) is predicted based on the formulated equations. First, a pushover analysis of the $N$-story MRF model is performed. Then, the equivalent displacement at loading step $n$ (${}_{n}D{{}_1}^{*}$) and the equivalent acceleration at step $n$ (${}_{n}A{{}_1}^{*}$) are calculated as in Eqs. (45) and (46), given Equation (47), assuming that the displacement vector at the loading step $n$ (${}_{n}d$) is proportional to the first mode vector at step $n$ (${}_n\mathsf{\Gamma }{}_1{}_n\phi {}_1$):

In Equation (46), ${}_{n}f{}_R$ denotes the restoring force of the entire MRF model. The contributions of the equivalent accelerations of the RC MRFs and SDCs (${}_{n}A{{}_{1f}}^{*}$ and ${}_{n}A{{}_{1d}}^{*}$, respectively) are calculated using Eqs. (48) and (49):

Here, ${}_{n}f{}_{Rf}$ and ${}_{n}f{}_{Rd}$ denote the restoring forces of the RC MRFs and SDCs, respectively. The restoring force vector ${}_{n}f{}_R$ is equal to the sum of ${}_{n}f{}_{Rf}$ and ${}_{n}f{}_{Rd}$, which are calculated from the shear forces of the RC columns and SDCs, respectively. Then, the ${}_{n}A{{}_{1f}}^{*}$−$D_1{}^{*}$ and ${}_{n}A{{}_{1d}}^{*}$−$D_1{}^{*}$ relationships are idealized by bilinear curves.