Energy-based response prediction of reinforced concrete buildings with steel damper columns under pulse-like ground motions

Abstract

The response of structures under pulse-like ground motions is characterized by the large amount of energy input in a few cycles. Consequently, structures with insufficient strength may suffer severe damage owing to excessive deformation. In a previous paper, the energy-based prediction procedure for the peak and cumulative response of a reinforced concrete (RC) frame building with steel damper columns was proposed (Fujii and Shioda, Buildings, 2023, 13, 401). Although this procedure was verified by comparison to the nonlinear time-history analysis (NTHA) results, the performance of the proposed procedure with pulse-like ground motion records has not been verified yet. In this study, the accuracy of the energy-based prediction procedure for an RC frame building with steel damper columns was investigated by considering pulse-like ground motions. The numerical analysis results reveal that the accuracy of the predicted peak response is satisfactory, which agrees with the results of the author’s previous study. However, the accuracy of the predicted total input energy to the building model depends on the ratio of the pulse period of the ground motion to the effective fundamental period of the building model. The reasons for this underestimation of the total input energy are discussed in this paper.

1 Introduction

1.1 Background and motivation

Pulse-like ground motions have been observed in past earthquakes (1994 Northridge Earthquake, 1995 Kobe Earthquake, 1999 Kocaeli Earthquake, 1999 Chi-Chi Earthquake). The response of structures under such ground motions is characterized by a large amount of energy input in a few cycles. Consequently, structures with insufficient strength may suffer severe damage owing to excessive deformation. Therefore, the evaluation of the peak deformation of structures is essential in the case of pulse-like ground motions.

In a previous paper, the energy-based prediction procedure for the peak and cumulative response of a reinforced concrete (RC) frame building with steel damper columns (SDCs) was proposed (Fujii and Shioda, 2023). In this procedure, 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. Then, the energy dissipation demand of the dampers is predicted by considering the energy balance during an entire response cycle using the total input energy. Although this procedure has been verified by comparing the nonlinear time-history analysis (NTHA) results, the performance of this procedure in the case of pulse-like ground motion records has not been verified yet. Therefore, this study investigated the accuracy of the proposed procedure for buildings subjected to pulse-like ground motions.

1.2 Brief review of related studies

1.2.1 Studies on characteristics of near-fault ground motions

The characteristics of near-fault ground motions have been widely investigated (Somerville et al., 1997; Alavi and Krawinkler, 2000; Mavroeidis and Papageorgiou, 2003; Bray and Rodriguez-Marek, 2004; Baker, 2007; He and Agrawal, 2008; Ghahari et al., 2010; Shahi and Baker, 2014; Yang and Zhou, 2015; Quaranta and Mollaioli, 2019; Feng et al., 2021; Sugino et al., 2021; Ghanbari and Fathi, 2022). Somerville et al. (1997) pointed out that large velocity pulses can be observed in the normal-fault direction in near-fault records owing to the forward directivity effect. Many studies have modeled the velocity pulses (Alavi and Krawinkler, 2000; Mavroeidis and Papageorgiou, 2003; He and Agrawal, 2008). Alavi and Krawinkler (2000) modeled the velocity pulses using simple rectangular waves for structural analysis. Mavroeidis and Papageorgiou (2003) proposed a mathematical model for representing the velocity pulses as the product of two sine functions. He and Agrawal (2008) proposed a mathematical model based on the Belarge wavelet. Yang and Zhou (2015) and Sugino et al. (2021) modeled velocity pulses using the Gabor wavelet. The decomposition of the near-fault ground motion records into pulse components and (other) residual components has also been attempted by several studies. Ghahari et al. (2010) proposed a procedure for decomposing the near-fault ground motions into long-period pulses and relatively high-frequency background records using a moving average filtering technique; Quaranta and Mollaioli (2019); Feng et al. (2021) proposed a procedure for decomposing near-fault ground motions using the Variational Mode Decomposition (VMD) technique. Ghanbari and Fathi (2022) proposed a procedure for decomposing near-fault ground motions using empirical Fourier decomposition.

The pulse period (or pulse duration) is a key parameter for appropriately modeling velocity pulses. Several studies (Alavi and Krawinkler, 2000; Mavroeidis and Papageorgiou, 2003; Bray and Rodriguez-Marek, 2004; Baker, 2007; Shahi et al., 2014; Quaranta and Mollaioli, 2019) have pointed out that, although the definition of the pulse period may differ among researchers, the pulse period becomes longer when the moment magnitude () of earthquakes becomes larger.

1.2.2 Studies on response of buildings subjected to near-fault ground motions

The responses of structures under pulse-like ground motions were widely investigated after the 1994 Northridge and 1995 Kobe earthquakes (Hall et al., 1995; Alavi and Krawinkler, 2000; Huang, 2003; Alavi and Krawinkler, 2004; Mavroeidis et al., 2004; Akkar et al., 2005; Kalkan and Kunnath, 2006; Xu et al., 2007; Kojima and Takewaki, 2015a; Alonso-Rodríguez and Miranda, 2015; Kojima and Takewaki, 2015b; Alhan and Öncü-Davas, 2016; Güneş and Ulucan, 2019; Al Shawa et al., 2020; Yalcin and Dicleli, 2020; Mota-Páez et al., 2021). Hall et al. (1995) investigated the response of 20-story steel moment-resisting frame (MRF) building models and a three-story RC base-isolated building model using artificially generated pulse-like ground motions. They found that long-period pulse-like ground motions are critical to such flexible building structures. Alavi and Krawinkler (2000; 2004) investigated the response of generalized steel MRF models using a rectangular pulse wave model. They demonstrated that the response of MRF models strongly depends on the ratio of the pulse period () to the fundamental period () of the MRF model: if the ratio is larger than unity, the response of the MRF model is governed by the fundamental mode, while the contribution of the higher modal response to the entire response is obvious when is smaller than unity. Huang (2003) investigated the response of an elastic continuous shear-beam model, and reported that the influence of a higher modal response to the entire response is obvious when is larger than ; Mavroeidis et al. (2004) investigated the response of elastic and inelastic single-degree-of-freedom (SDOF) systems subjected to near-fault ground motions using the velocity pulse model proposed in their previous study (Mavroeidis and Papageorgiou, 2003). They pointed out that the pulse period () and amplitude () can be used to “effectively normalize the elastic and inelastic response spectra of SDOF systems subjected to actual near-fault records”. Akkar et al. (2005) investigated the applicability of a simplified procedure for estimating the local displacement demands in regular MRF responding in the elastic range. In their procedure, the local displacement demands are estimated based on the response of the fundamental mode. Their study demonstrated that this simplified procedure is sufficiently accurate when the ratio is less than 1.5; Kalkan and Kunnath (2006) investigated 4-, 7-, and 13-story steel MRF building models subjected to near-fault and far-fault ground motion records. They demonstrated that low-cycle fatigue is critical in the case of far-fault ground motion records, owing to the gradual increase of cumulative energy with longer duration, while excessive larger deformation is critical in the case of near-fault ground motion records, owing to the high-amplitude velocity pulses; Xu et al. (2007) considered the response of a SDOF model with dampers subjected to the velocity pulse model proposed in their study (He and Agrawal, 2008), and investigated the relationship between the energy response of the model and the ratio. Alonso-Rodríguez and Miranda (2015) investigated the elastic response of a continuous model formed by a flexural beam laterally coupled to a shear beam subjected to the velocity-pulse model proposed by Mavroeidis and Papageorgiou (2003). They reported that the pulse duration is the most critical parameter affecting both the acceleration and drift response. Kojima and Takewaki (2015a, 2015b) formulated the critical response of an undamped elasto-plastic SDOF model subjected to pulse-like ground motions. In their studies, the fling-step input and forward-directivity input were modeled as double- and triple-impulses, respectively. They demonstrated that the timing of critical pulses depends on the ductility of the elasto-plastic SDOF model. Alhan and Öncü-Davas (2016) investigated the response of a base-isolated building model subjected to the velocity-pulse model proposed by He and Agrawal (2008). In their study, the superstructure model was assumed to behave elastically, while the isolation layer model was assumed to have smoothed bilinear behavior. They demonstrated that “the ratio of the isolation period to the pulse period significantly affects the peak base displacement demands and peak floor acceleration demands”. Güneş and Ulucan (2019) investigated the nonlinear response of a 40-story RC building model subjected to near-fault and far-fault ground motion records. In their study, the ground motion records were divided into four groups: the near-fault ground motions were divided into three groups depending on the pulse duration () defined by Shahi and Baker (2014), while the far-fault ground motions formed a single group. They demonstrated that the response of a tall reinforced concrete (RC) building depends on the ratio of to the first mode period (); the responses of the upper stories become obvious when the ratio is less than unity, while the responses of the lower stories becomes obvious when the ratio is larger than unity. Al Shawa et al. (2020) investigated the nonlinear response of SDOF models with different hysteresis models subjected to pulse-like ground motions in terms of energy responses. They demonstrated that the cumulative input energy to the long-period structures becomes larger as the moment magnitude () of the earthquake increases, which is consistent with the pulse duration phenomenon of the pulse-like ground motions becoming longer as increases. Yalcin and Dicleli (2020) compared the nonlinear response spectrum of the long-period pulses obtained using the moving average filtering technique to that of the original records. They reported that, in the case of flexible structures subjected to ground motions with larger , the influence of relatively high-frequency background records may be negligible. Mota-Páez et al. (2021) investigated the applicability of the energy-based response prediction procedure to an RC building with hysteresis dampers installed in the soft-story under near-fault ground motions. Their procedure is based on the simplified procedure proposed by Akiyama (1985). They demonstrated that, to better predict the peak response, the equivalent number of cycles should be smaller than the non-pulse-like ground motions.

To the author’s understanding, the ratio of the pulse period to the fundamental period of structures ( or ) is a key parameter for investigating the response of a building subject to pulse-like ground motions. Because the pulse period () depends on the moment magnitude (), as discussed above, the ratio is essential for investigating the accuracy of the procedure proposed by the authors (Fujii and Shioda, 2023) for buildings subjected to pulse-like ground motions.

1.3 Objectives

In this study, the accuracy of the energy-based prediction procedure for an RC building with SDCs was investigated by considering pulse-like ground motions. To answer the questions stated above, 8- and 16-story RC MRFs with SDCs were considered. Additionally, 30 pulse-like ground motion records were used. The pulse-like ground motions were divided into two groups in accordance with the pulse period defined in the NGA-West2 ground motion database (Shahi et al., 2014).

The rest of this paper is organized as follows. Section 2 outlines the procedure proposed by the authors (Fujii and Shioda, 2023). Section 3 presents two RC MRFs with SDCs, and introduces the ground acceleration data used in the NTHA. The 30 ground motions are grouped into two groups and scaled such that the predicted peak equivalent displacement of the first modal response reaches the predetermined value. Section 4 describes the validation of the seismic demand predictions. Section 5 discusses the accuracy of the predicted peak equivalent displacement of the first modal response, and the contribution of the first modal response to the cumulative energy input. Then, the accuracy of the cumulative input of the first modal response is investigated. The conclusions drawn from this study and the directions of future research are discussed in Section 6.

2 Outline of prediction procedure

Figure 1 outlines the prediction procedure proposed in previous work by the author (Fujii and Shioda, 2023). As can be seen, this procedure consists of three stages, as summarized below.

FIGURE 1

Fujii and Shioda, 2023).

In Stage 1, the pushover analysis of the building model is carried out to obtain the restoring force–displacement relationship. From this result, the equivalent displacement () and equivalent acceleration of the RC MRF and SDCs ( and , respectively) are calculated for each loading step. For simplicity, the – and – relationships are idealized as bilinear curves. Then, the equivalent velocity of the maximum momentary input energy corresponding to () is calculated. In this procedure, the – relationship is referred to as the capacity curve. The effective period corresponding to is calculated as follows:where is the complex damping ratio of the equivalent linear system considered in the calculation of the maximum momentary input energy spectrum ( spectrum) and total input energy spectrum ( spectrum). In this study, was set to 0.10 based on the results obtained in previous work by the author (Fujii and Shioda, 2023).

In Stage 2, the and spectra are calculated from the time-varying function (TVF) proposed in a previous study by the author (Fujii et al., 2019). First, the maximum momentary input energy per unit mass () and the total input energy per unit mass () of the equivalent linear system (natural period , complex damping ratio ) are calculated using the TVF. The equivalent velocities of the maximum momentary input energy () and total input energy () are calculated as follows:

The equivalent displacement of the equivalent linear system () is calculated as follows:

In this procedure, the – relationship is referred to as the demand curve. The peak response point of the equivalent SDOF model (; ) is located at the intersection point of the capacity and demand curves. Then, the equivalent velocity of the cumulative input energy of the first modal response () is obtained from the spectrum, as follows:where is the effective period corresponding to the peak response point, and is calculated as follows:

The total input energy of the entire building model is evaluated as follows:where is the effective first modal mass corresponding to the peak response point, and is the total mass. The cumulative strain energy of the RC MRF and SDCs, and cumulative damping dissipated energy are calculated such that the total cumulative energy is equal to .

In Stage 3, the local seismic demand of the building model is predicted using the peak and cumulative response of the equivalent SDOF model and the pushover analysis results.

More details on the procedure can be found in a previous paper by the author (Fujii and Shioda, 2023).

3 Building and ground motion data

3.1 Building data

The two planar building models analyzed in this study are 8- and 16-story RC MRFs with SDCs, which are the same as those used in the author’s previous study (Fujii and Shioda, 2023). Figure 2 shows the simplified structural plan and elevation of the RC MRF building models with SDCs. Details on the two structural models can be found in the author’s previous study (Fujii and Shioda, 2023). In this study, the viscous damping ratio of the first modal response of the RC MRFs in the elastic range () was set to 0.03.

FIGURE 2

Fujii and Shioda, 2023).

Figure 3 shows the capacity curve of the two models calculated based on the pushover analysis results. In this study, the input ground motions were scaled such that the predicted reaches the predetermined value: the target was set to 0.252 m for the 8-story model and 0.479 m for the 16-story model. The and corresponding to target are 1.220 m/s and 1.318 s, respectively, for the 8-story model, and 1.187 m/s and 2.576 s, respectively, for the 16-story model.

FIGURE 3

3.2 Ground motion data

As has been shown by the previous studies discussed in Section 1.2, the response of a building structure subjected to pulse-like ground motions is obviously affected by the ratio of the pulse period of the ground motion () to the fundamental period of the structure (). Therefore, two ground motion groups are considered in this study: in group 1, the pulse period () ranges between 1.0 s and 2.0 s; in group 2, ranges between 2.0 s and 4.0 s. A total of 30 horizontal ground motion sets (15 horizontal ground motion sets in each group) from the NGA-West2 ground motion database of the Pacific Earthquake Engineering Research Center were used. The ground motion sets were selected based on the following criteria: (i) the moment magnitude () is larger than 6.0; (ii) the closest distance from the rupture plane () is smaller than 20 km. These values were obtained from the NGA-West2 ground motion database. Table 1 presents the ground motion records. In group 1, ranges from 6.0 to 7.1, ranges from 0.3 km to 10.2 km, ranges from 1.02 s to 1.81 s, and (the time-averaged shear-wave velocity in the top 30 m at the recording sites) ranges from 139 m/s to 2016 m/s. In group 2, ranges from 6.2 to 7.9, ranges from 1.0 km to 12.8 km, ranges from 2.02 s to 3.77 s, and ranges from 198 m/s to 553 m/s.

It is important to show the range of the ratio of each ground motion group for both models. The range of the ratio for group 1 is 0.775–1.370 for the 8-story model, and 0.397–0.701 for the 16-story model. The range of the ratio for group 2 is 1.535–2.863 for the 8-story model, and 0.785–1.465 for the 16-story model.

3.3 Analysis method

For the NTHA of building structures subjected to near-fault ground motions, the selection of the axis of the horizontal component is important. According to research on near-fault ground motions, the horizontal component of the fault normal/fault-parallel (FN/FP) directions is critical to structures (Somerville et al., 1997). However, Kalkan and Kwong demonstrated that rotating the ground motions to the FN/FP directions does not always provide the maximum responses at all angles (Kalkan and Kwong, 2013). Güneş and Ulucan (2019) analyzed a 40-story reinforced concrete building model subjected to near-fault pulse-like ground motions. In their study, the direction of the maximum pseudo-velocity spectrum was used instead of the FN direction, because large velocity pulses were observed in the FP direction in the Yarimca records of the 1999 Kocaeli earthquake. Therefore, it is likely that the FN/FP directions cannot be used as the critical axis of the horizontal ground motion.

In this study, the horizontal component axis was calculated based on the author’s previous study (Fujii, 2022). The procedure is described below.

FIGURE 4

4 Analysis results

4.1 Peak response

In the following discussion, the peak response obtained from the pushover analysis results corresponding to the target is referred to as “the predicted peak response”.

Figure 5 compares the predicted peak responses of the 8-story model and the NTHA results; the following local response quantities are compared: (i) the peak relative displacement, (ii) the peak story drift, (iii) the peak plastic rotation at the beam end at the right of column X2 (), and (iv) the peak shear strain of the damper panel (). In addition to the NTHA results for each ground motion, the mean, maximum, and minimum value of the NTHA results for the 15 ground motions are compared with the predicted results.

FIGURE 5

FIGURE 6

4.2 Cumulative response

FIGURE 7

The reasons for having unconservative predicted total input energy for the 16-story model in group 1 would be (i) the contribution of a higher modal response is large, and (ii) the cumulative energy input of the first modal response at the end of the seismic event cannot satisfactorily predicted using the equivalent linear system. These reasons will be discussed below.

4.3 Summary of analysis results

Importantly, differences in the accuracy of each analysis case may occur owing to the pulse period of the ground motions. As noted in Section 3.2, the ratio of the ground motion sets in group 1 for the 16-story model, which is the most inaccurate estimation of the total input energy, is less than unity. The difference in the energy response of the first modal response of each case is discussed below.

5 Discussion

This section focuses on (i) the accuracy of the predicted peak equivalent displacement of the first modal response, (ii) the contribution of the first modal response to the cumulative energy input, and (iii) the accuracy of the predicted cumulative input energy of the first modal response. The equivalent velocities of the maximum momentary input energy and cumulative energy ( and ) and the peak equivalent displacement are calculated from the NTHA results according to the procedure described in a previous paper by the author (Fujii, 2022).

5.1 Accuracy of predicted peak equivalent displacement of first modal response

FIGURE 8

Therefore, as far as is concerned, the prediction accuracy is satisfactory for both the 8- and 16-story models: the influence of the ratio on the accuracy of the predicted is limited.

5.2 Contribution of first modal response to cumulative energy input

In the prediction procedure, the contribution of the higher modal response to the total input energy is approximated by assuming the following relationship.

Eq. 14 suggests that the total input energy can be evaluated from the equivalent velocity of the cumulative input energy of the first modal response () and total mass (). Therefore, the accuracy of the predicted depends on (i) the validity of the assumed relationship (Eq. 14), and (ii) the accuracy of the predicted from the spectrum. Therefore, (i) the validity of Eq. 14 is evaluated first.

FIGURE 9

The trends shown in Figure 9 are consistent with the results shown in Figure 7. Thus, one of the reasons for the predicted being less accurate in the case of the 16-story model subjected to the ground motion group 1 is that, in this case, the contribution of a higher modal response to the total input energy is large. This observation is consistent with the conclusions drawn by previous studies (Huang, 2003; Alonso-Rodríguez and Miranda, 2015).

5.3 Accuracy of predicted cumulative input energy of first modal response

FIGURE 10

FIGURE 11

FIGURE 12

Therefore, the reason for the predicted from the spectrum underestimating the NTHA results for the 16-story model subjected to ground motion group 1 is the difference of the time-history of the TVF and NTHA. In this case, the cumulative input energy of the first modal response at the end of the seismic event cannot be satisfactorily predicted using the equivalent linear system (effective period , complex damping = 0.10). To better predict , the maximum value of over the course of a seismic event calculated from the TVF should be used instead of the value at the end.

5.4 Summary of discussion

6 Conclusion

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