1 Introduction

The strong mainshock always triggers lots of aftershocks clustered both in time and space, forming mainshock–aftershock sequence-type ground motions. In the post-mainshock environment, it is impossible to repair the damaged structures before the occurrence of subsequent aftershocks due to the short intervals of time. The post-earthquake field reconnaissance (Elnashai et al. 2009; Augenti and Parisi 2010; Ceci et al. 2010; Jing et al. 2011) has confirmed that the strong aftershocks have the potential to increase the damage levels of the structures for the additional damage. Nonetheless, to the authors’ best knowledge, all the seismic design codes in the world are based on the single ‘design earthquake’ without taking into account the danger of aftershock.

Recently, many investigations have been accomplished to examine the influence of aftershock on the damage of structures. Some researchers (Sunasaka et al. 2002; Amadio et al. 2003; Das et al. 2007; Hatzigeorgiou and Beskos 2009; Hatzigeorgiou 2010a, b, c; Goda and Taylor 2012; Goda 2012; Di Sarno 2013; Zhai et al. 2013a, 2014) focused on the inelastic response spectra, such as inelastic displacement ratio, ductility demand, behavior factor and damage spectra for mainshock–aftershock sequence-type ground motions. In addition, several investigations (Fragiacomo et al. 2004; Lee and Foutch 2004; Li and Ellingwood 2007; Hatzigeorgiou and Liolios 2010; Moustafa and Takewaki 2010; Ruiz-García and Negrete-Manriquez 2011; Ruiz-García 2012; Ruiz-García et al. 2012, 2014; Faisal et al. 2013; Zhang et al. 2013; Efraimiadou et al. 2013; Han et al. 2014) have been conducted to study the effects of aftershock on multiple-degree-of-freedom structures.

In the modern seismic codes (CEN 2003; International Building Code (IBC) 2006), the structures are allowed to experience the damage under the moderate to severe earthquakes. The strength reduction factor R is widely used to determine the static seismic force from the elastic design acceleration spectra. The ductility-based strength reduction factor R μ is an important part in the definition of R by the seismic codes. Lots of researchers (Veletsos and Newmark 1960; Newmark and Hall 1969; Riddell and Newmark 1979a; Miranda 1993; Miranda and Bertero 1994; Miranda and Ruiz-García 2002b; Riddell et al. 2002; Chakraborti and Gupta 2005; Karakostas et al. 2007; Gillie et al. 2010; Qu et al. 2011; Palermo et al. 2013) have investigated the characteristics of R μ and proposed the corresponding simplified expressions as the design tool. In the all above investigations, just the investigations (Amadio et al. 2003; Hatzigeorgiou 2010a, c) focused on the influence of aftershock or multiple earthquakes on the R μ . Amadio et al. (2003) repeated the same ground motion two or three times to simulate the sequence-type ground motions. However, as pointed out in other researches (Ruiz-García and Negrete-Manriquez 2011; Zhai et al. 2013a), the sequence-type ground motions generated by this way is unrealistic because the identical ground motion is unlikely to occur two times on the same site. Hatzigeorgiou (2010a, c) studied the R μ for multiple far-fault and near-fault ground motions with bilinear system. No structural degrading behavior, such as stiffness degrading, strength degrading or pinching behavior, are considered. In addition, the results in investigations (Ruiz-García et al. 2012; Zhai et al. 2014) indicated that the relative intensity of the aftershock ground motion has the significant effect on the seismic performance of structure. However, the current investigations do not specially focus on the influence of relative intensity of the aftershock ground motion on the R μ .

Based on the above discussions, this manuscript studies the strength reduction factor R μ with four hysteretic models and lots of mainshock–aftershock sequence-type ground motions. The aftershock ground motions are scaled to have the different intensity levels relative to the mainshock ground motion. The effects of period, ductility factor, site condition, aftershock, hysteretic behavior and damping ratio on the R μ are studied statistically. Finally, a predictive model is proposed to facilitate the application of R μ in the seismic design.

2 Strength reduction factors

In this manuscript, the strength reduction factor is defined as the ratio of strength demand of elastic structure for the mainshock ground motion to the yield strength of corresponding inelastic structure for the certain ductility level and sequence-type ground motion. Mathematically it can be expressed as:

$$ R_{\mu } = \frac{{F_{\text{e,ms}} }}{{F_{\text{y}} }} $$
(1)

where F e,ms = strength demand on an infinitely elastic SDOF system during the prescribed mainshock ground motion, F y = yield strength of the corresponding inelastic SDOF system, making the system satisfy the criterion of ductility level under the mainshock–aftershock sequence-type ground motions.

The ductility factor is defined as:

$$ \mu = \frac{{x_{\text{m}} }}{{x_{\text{y}} }} $$
(2)

where x m = maximum displacement demand of inelastic structure under the ground motion, x y = yield displacement of inelastic structure.

The following four hysteretic models, which have been used in the references (Zhai et al. 2013b, 2014; Wen et al. 2014), are used to investigate the effect of hysteretic behavior on the R μ : (1) Elastic–Perfectly-Plastic (EPP) model, representing the non-degrading systems; (2) Modified Clough (MC) model, simulating the flexural behavior that exhibit stiffness degradation at reloading; (3) Pinching model, simulating the reinforced concrete structures with crack opening and closure, or the steel structures with connection slip behavior; (4) Stiffness Strength Degradation (SSD) model based on the three parameter model, representing global behavior of systems exhibiting stiffness degradation and strength deterioration during reloading branches.

Riddell and Newmark (1979b) pointed out that several deficiencies exist in the original Clough model, resulting from the lack of constitutive rules for treating incomplete and small amplitude loops. These deficiencies can cause unrealistic seismic responses, such as releasing energy instead of dissipating energy under some unloading–reloading circumstances. Thus the modified Clough model was proposed by the past researches (Mahin and Bertero 1975, 1976; Riddell and Newmark 1979b) to avoid the above deficiencies, and this model is extensively used to study the structural response (Miranda and Ruiz-García 2002a, b; Iervolino et al. 2006; Tong and Zhao 2007). The pinching model used here was developed by Rahnama and Krawinkler (1993). The detailed hysteretic rules of this model were introduced and described in the literature (Rahnama and Krawinkler 1993). The stiffness strength degradation model is based on the three parameter model (Kunnath et al. 1990; Valles et al. 1996), which was calibrated with the experimental results by Valles et al. (1996) and was included in the software IDARC.

The inelastic SDOF systems with a set of 60 periods between 0.1 and 6.0 s with an interval of 0.1 s are considered in this manuscript. The viscous damping ratio is assumed to be 5 %. Five ductility factors μ = 2, 3, 4, 5 and 6 are selected to consider the different ductility performances. Figure 1 shows the flowchart for the computation of the ductility-based strength reduction factor for the mainshock–aftershock sequence-type ground motion. The strength reduction factor R μ is calculated by gradually reducing the applied strength of SDOF system from the corresponding elastic strength demand F e,ms until the specified μ is achieved within a tolerance (1 % is used in this manuscript).

Fig. 1
figure 1

The flowchart for the computation of the ductility-based strength reduction factor for the mainshock–aftershock sequence-type ground motion

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In this manuscript, the in-house codes are developed with Fortran language to simulate the different hysteretic models. Then these in-house codes are used to compute the response of SDOF system under the mainshock–aftershock sequence-type ground motions.

3 Ground motions

When constructing the mainshock–aftershock sequence-type ground motions, several researchers (Ruiz-García and Negrete-Manriquez 2011; Goda and Taylor 2012; Zhang et al. 2013) selected all the available aftershock ground motions (satisfying their own criterions) in the sequence-type ground motion. The problem of this method is that, in the final selected database, many sequence-type ground motions (possess a big part in the database) include only one aftershock ground motion, while others contain multiple aftershock ground motions. The statistical results of these sequence-type ground motions would mix the effect of one aftershock on the seismic response of structure with that of multiple aftershocks. In this case, many investigations (Lee and Foutch 2004; Li and Ellingwood 2007; Ruiz-García et al. 2012, 2014; Han et al. 2014) just select one aftershock ground motion in the sequence-type ground motions. The results in these investigations also indicated that these sequence-type ground motions can provide valuable information about the influence of aftershock on the seismic response of structure. Thus, in this manuscript, just one aftershock ground motion is selected in the sequence-type ground motion.

In this manuscript, the recorded mainshock–aftershock sequence-type ground motions are selected from the Pacific Earthquake Engineering Research Center (PEER) Next Generation Attenuation (NGA) relationships database (http://peer.berkeley.edu/nga/). For the seismic sequence recorded on a given station, if the mainshock ground motion and one of aftershock ground motions both satisfy the characteristics specified in this manuscript, the mainshock ground motion and this aftershock ground motion will be used to generate a sequence-type ground motion. The specified characteristics of the ground motions are: (1) recorded on accelerographic stations where enough information about the geological and geotechnical conditions at the site is available; (2) recorded on free field stations or on the ground level of the structures; (3) the peak ground acceleration (PGA) of mainshock ground motion and aftershock ground motions is >0.1 g, respectively. Finally, a total of 218 mainshock–aftershock sequence-type ground motions are obtained with this approach. Table 1 shows the information of the mainshock–aftershock sequences used in this paper and the number of ground motions in each seismic sequence.

Table 1 Information of the mainshock–aftershock sequences used in this paper and the number of ground motions in each seismic sequence
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It should be pointed out that all the above 218 sequence-type ground motions are recorded on the site classes B and C, according to the site classification method of United States Geological Survey (USGS). In order to supplement the sequence-type ground motions on site classes A and D, the artificial mainshock–aftershock sequence-type ground motions are used in this manuscript. The details of the generating method are introduced in the following paragraphs.

For the given site class (site class A or D), a group of recorded ground motions with PGA being larger than 0.1 g are selected as seed ground motions from the PEER–NGA database. Then the artificial sequence-type ground motions are generated by randomly combining two different seed ground motions. It is known that the interdependence exists between mainshock and aftershock ground motions, and the real sequence-type ground motions can reflect this interdependence perfectly. However, there is no enough valuable information (e.g. ground motion parameters relationship between mainshock and aftershock) to quantitatively and clearly describe this interdependence. This randomly combining method has been applied in several researches (Hatzigeorgiou 2010a, b; Faisal et al. 2013; Zhai et al. 2013a; Ruiz-García et al. 2014). Though the artificial sequence-type ground motions may not perfectly reflect this interdependence and errors may be induced by using these artificial sequence-type ground motions to compute the structural response, the results in these investigations still showed that the artificial sequence-type ground motions generated by this method have the acceptable potential to study the influence of multiple successive ground motions on the seismic response of structures. Thus this randomly combining method is also used in this manuscript.

Table 2 summarizes the information of the single ground motions used to generating the sequence-type ground motions on site classes A and D. Table 3 presents the number of recorded and artificial sequence-type ground motions on different site classes.

Table 2 The information of seed ground motions for simulating the mainshock–aftershock sequence-type ground motions on site class A and D
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Table 3 The number of recorded and artificial sequence-type ground motions on different site classes
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The relative peak ground acceleration of aftershock ground motion \( \nabla PGA \) is defined as:

$$ \nabla PGA{ = }\frac{{PGA_{\text{as}} }}{{PGA_{\text{ms}} }} $$
(3)

where PGA as is the peak ground acceleration of aftershock ground motion, PGA ms is the peak ground acceleration of mainshock ground motion.

The \( \nabla PGA \) is used to represent the relative intensity level of aftershock ground motion with respect to the mainshock ground motion. In order to study the influence of relative intensity of aftershock ground motion on the strength reduction factor R μ , the aftershock ground motion is scaled to have four levels of relative intensity (i.e. \( \nabla PGA\, = \,0.5, \) 0.8, 1.0 and 1.5) for each sequence-type ground motion. It is the fact that the intensity of aftershock ground motion is generally smaller than the mainshock ground motion, due to the lower magnitude of aftershock. But the aftershock ground motions with greater intensity with respect to that of mainshock ground motions do exist in the past seismic sequences (Zhai et al. 2014), though the number of these aftershock ground motions is pretty lower than that of other aftershock ground motions. In order to provide a design tool considering aftershock ground motions with various relative intensities which exist in the past seismic sequences, the value of \( \nabla PGA \) being 1.5 is used to simulate the extreme case of aftershock ground motion.

4 Statistical analyses

4.1 Mean strength reduction factors

A total of 137,400 strength reduction factors R μ of EPP systems are computed as part of this investigation, for 458 sequence-type ground motions, 60 periods and 5 ductility factors. Mean R μ are then computed by averaging results for each period, each ductility factor μ and each site class. It should be noted that the quantification of the different spectral regions can facilitate the analysis and discussion. However, in the realistic case, it is difficult to strictly define the boundary between the different spectral regions. In this manuscript, the short, medium and long period regions are approximately corresponded to (0.0–0.5 s), (0.5–1.5 s) and (1.5–6.0 s). But in some cases, the above definition is not exact and proper to describe the characteristics of the results, and thus the specifically period range is used to analyze the results. Due to the space limitation, just the results of limited cases are presented in the following sections, while other cases having the similar results are not presented.

Figure 2 shows the mean R μ of EPP system for the four site classes and the sequence-type ground motions with \( \nabla PGA\, = \,0.5. \) In general, the mean R μ shows the same general trend regardless of site conditions. In the short period region, mean R μ is strongly dependent on the period and increases sharply with the increase of period. In the medium-long period region, the influence of period on the mean R μ is not significant. The mean R μ in medium-long period region is approximately period independent in comparison with that in short period region. In the long period region, the mean R μ varies approximately around the corresponding μ when the period increases.

Fig. 2
figure 2

The mean R μ of EPP system for the four site classes and the sequence-type ground motions with \( \nabla PGA = 0. 5 \): a site class A, b site class B, c site class C, d site class D

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In the whole period region, mean R μ increases with the increase of μ. In the medium-long period, the ductility factor μ has the significant effect on the R μ . In the short period region, the influence of μ on R μ reduces gradually as the period decreases. Take the mean R μ on site class B as the example, the difference between the mean R μ of μ = 2 and 4 is about 80 % for the structure with period being 0.5 s, while this difference decreases to 30 % for the structure with period being 0.1 s.

The limiting period dividing the period region whether the mean R μ is period independent or not depends on the ductility factor μ and site conditions. In general, this limiting period increases as the μ increases. For the R μ on site class A, the limiting period increases from 0.3 to 0.8 s as the μ increase from 2 to 6. For the R μ of μ = 4, the limiting period increases from 0.5 to 1.2 s as the site class changes from B to D.

It can be noted that the curves in Fig. 2 are smoother for site classes B and C in comparison to the ones that correspond to A and D site classes. This phenomenon may due to the errors induced by using the artificial sequence-type ground motions to compute the structural response, as pointed out in Sect. 3.

It is common to give the general information of strength reduction factors regardless of the site conditions. Figure 3 presents the mean R μ of EPP system for all sequence-type ground motions with \( \nabla PGA\; = \,0.5. \) It is obvious that the mean R μ in Figs. 2 and 3 shows the similar trend with the period and μ. The limiting period increases from 0.3 to 1.0 s as μ increases from 2 to 6.

Fig. 3
figure 3

The mean R μ of EPP system for all sequence-type ground motions with \( \nabla PGA = 0. 5 \)

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4.2 Dispersion of strength reduction factors

In this section, the coefficient of variation (COV) is used to evaluate the dispersion of the R μ . The COV is defined as the ratio of the standard deviation to the mean. Figure 4 illustrates the COVs of R μ on four site classes for EPP system and sequence-type ground motions with \( \nabla PGA\, = \,0.5. \)

Fig. 4
figure 4

The COVs of R μ on four site classes for EPP system and sequence-type ground motions with \( \nabla PGA = 0. 5 \): a site class A, b site class B, c site class C, d site class D

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In the whole period region, COVs do not significantly change with the variation of the period, and thus are approximately period independent. Except a few COVs on site class D, all the COVs in Fig. 4 are smaller than 0.5. In general, COVs of R μ on different site classes show the similar trend and the differences of COVs on different site classes are small. However, it can be found that COVs are relatively sensitive to the μ and increase with the increase of μ. For example, the COVs are about 20 % for μ = 2 and increase to 40 % approximately for μ = 6, as shown in Fig. 4.

4.3 Effect of site conditions

In order to investigate the influence of site conditions on the R μ , the mean R μ on different site classes are normalized by the mean R μ on all site classes, respectively. In this way, the error produced by ignoring the influence of site conditions when evaluating the R μ on different site classes can be investigated quantitatively. Figure 5 presents the normalized mean R μ of EPP system for different site classes and sequence-type ground motions with \( \nabla PGA\, = \,0.5. \)

Fig. 5
figure 5

The normalized mean R μ of EPP system for different site classes and sequence-type ground motions with \( \nabla PGA = 0. 5 \): a site class A, b site class B, c site class C; d site class D

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It can be seen that the normalized mean R μ in Fig. 5a can reach 1.2, indicating that the mean R μ on all site classes would underestimate the mean R μ on site class A at a maximum level of 20 %. All the values of normalized mean R μ in Fig. 5b, c are within the interval [0.9, 1.1]. This phenomenon means that the errors are within 10 % when evaluating the R μ on site classes B or C with the mean R μ on all site classes. The normalized mean R μ in Fig. 5b, c is not clearly dependent on the μ.

In the short period region, the normalized mean R μ in Fig. 5d is smaller than 1.0 and the minimum value can reach 0.7, meaning that the mean R μ on all site classes would overestimate the mean R μ on site class D at a maximum level of 30 %. The normalized mean R μ in this period region decreases as the μ increases, indicating that the effect of site condition on R μ of site class D is more obvious for structures with larger inelastic deformation. Thus, for μ smaller than 3 the errors produced by neglecting the influence of site condition are commonly smaller than 20 %, while the corresponding errors can reach 30 % for μ = 6, as can be seen in Fig. 5d.

In the medium-long period region, the normalized mean R μ in Fig. 5d varies within the interval [0.9, 1.1]. This phenomenon indicates that the difference between the mean R μ on site class D and the mean R μ on all site classes is within 10 % for structures with medium-long periods. Similar to the results in Fig. 5a–c, the normalized mean R μ in this region is also not clearly dependent on the μ.

For the structure with short period, the seismic response on site class A (rock site) is generally smaller than that on other site classes, while the seismic response on site class D (soft soil site) is generally larger than that on other site classes (Zhai et al. 2013a). Thus in the short period region, the mean R μ on site class A is greater than that on all site classes, while the mean R μ on site class D is smaller than that on all site classes.

4.4 Effect of aftershock

The effect of aftershock on the R μ is studied in this section. The ratio between the R μ of sequence-type ground motion and that of corresponding mainshock ground motion, denoted as R μ,seq/R μ,ms, is computed for each sequence-type ground motion. Then the mean R μ,seq/R μ,ms of all sequence-type ground motions with constant \( \nabla PGA \) is computed for each period and each μ.

Figure 6 shows the mean R μ,seq/R μ,ms for the sequence-type ground motions with \( \nabla PGA\, = 0.5 \) and different hysteretic models. It can be seen that all the mean R μ,seq/R μ,ms values in Fig. 6 are within the interval [0.97, 1.0], indicating that the aftershock ground motion with \( \nabla PGA = 0.5 \) would decrease the strength reduction factor at a level of <3 %. Thus the effect of aftershock ground motion with\( \nabla PGA\, = \,0.5 \) on the R μ can be ignored in the engineering practice.

Fig. 6
figure 6

The mean R μ,seq/R μ,ms for the sequence-type ground motions with \( \nabla PGA = 0. 5 \) and different hysteretic models: a EPP model, b MC model, c PH model, d SSD model

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Figure 7 presents the mean R μ,seq/R μ,ms for the sequence-type ground motions with \( \nabla PGA\, = \,1.0 \) and different hysteretic models. For systems with periods >0.5 s, the mean R μ,seq/R μ,ms varies slightly with the change of period and is approximately period independent. The values of mean R μ,seq/R μ,ms in the whole period region are generally within the interval [0.9, 0.95], meaning that the aftershock ground motion with \( \nabla PGA = 1.0 \) would decrease the strength reduction factor R μ of structure at a level of <10 %.

Fig. 7
figure 7

The mean R μ,seq/R μ,ms for the sequence-type ground motions with \( \nabla PGA = 1.0 \) and different hysteretic models: a EPP model, b MC model, c PH model, d SSD model

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From the Figs. 6 and 7, it can be seen that the values of mean R μ,seq/R μ,ms are generally comparable for different hysteretic models. This phenomenon indicates that the aftershock ground motion has similar effects on the R μ for different hysteretic models.

In order to compare the influences of aftershock ground motions with different relative intensities \( \nabla PGA \) on the R μ more clearly, Fig. 8 illustrates mean R μ,seq/R μ,ms for system with μ = 4 and sequence-type ground motions with different \( \nabla PGA \). It is clear that the mean R μ,seq/R μ,ms in Fig. 8 decreases with the increase of \( \nabla PGA \). For the mean R μ,seq/R μ,ms of the sequence-type ground motion with \( \nabla PGA\, = 1.5 \), the corresponding values in the short period region can be smaller than 0.8, while the corresponding values in the medium-long period region are always >0.85. These phenomena indicate that the aftershock ground motion with \( \nabla PGA = 1.5 \) can decrease the R μ of structure with short period at a level of more than 20 %, while this decrease magnitude is always <15 % for structure with medium-long period. The above results reveal that the effects of aftershock on the mean R μ are related to the period, ductility factor and intensity of aftershock ground motion.

Fig. 8
figure 8

The mean R μ,seq/R μ,ms for different hysteretic models with μ = 4 and sequence-type ground motions with different \( \nabla PGA \): a EPP model, b MC model, c PH model, d SSD model

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For the structure which has been damaged by the mainshock, the subsequent aftershocks may induce the additional damage due to the damage accumulation. In order to satisfy the given ductility factor, the structure under the seismic sequence needs greater yield strength in comparison to the structure under the mainshock alone. Moreover, the stronger aftershocks are included in the seismic sequence, the greater yield strength is needed. Thus the aftershock has the potential to reduce the R μ with respect to the mainshock alone, and the reduction depends on the intensity of the aftershock ground motion, just as revealed by the above results.

Similarly, the effects of aftershock on the dispersion of the R μ are investigated through the ratios of the COVs of R μ,seq to those of the R μ,ms. Figure 9 shows the COVseq/COVms for the sequence-type ground motions with \( \nabla PGA = 1.0 \) and different hysteretic models. Except the periods smaller than 0.3 s, the ratios in Fig. 9 are generally >1.0, and the maximum ratio exceeds 1.20. This phenomenon indicates that the strong aftershock with the same intensity with that of mainshock may increase the dispersion of the R μ in a maximum level of 20 %. The ratios of COV in Fig. 9 depend on the ductility factor μ, and this dependence would gradually vanish with the increase of the μ. The COV is the ratio of the mean R μ to the standard deviation of the R μ . In comparison with the results for the mainshock alone, the mean response of structure and the corresponding standard deviation are both enhanced by the strong aftershocks. When the enhancement of the mean response exceeds that of the standard deviation, the aftershock increases the COV of the structural responses. The influences of aftershock on the COV of the R μ depend on the period, ductility factor and intensity of aftershock ground motion (explained in the next paragraph).

Fig. 9
figure 9

The mean COV seq/COV ms for the sequence-type ground motions with \( \nabla PGA = 1.0 \) and different hysteretic models: a EPP model, b MC model, c PH model, d SSD model

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Figure 10 illustrates COVseq/COVms for system with μ = 4 and sequence-type ground motions with different \( \nabla PGA \). The ratios correspond to \( \nabla PGA \) being 0.5 and 0.8 are within the interval [0.9, 1.1], indicating that the influences of aftershock ground motion with \( \nabla PGA \) smaller than 1.0 are approximately within 10 %. The ratios in Fig. 10 generally increase as the intensity of aftershock ground motion increases. The results in Figs. 8 and 10 show the consistence that the stronger aftershock has more obvious influences.

Fig. 10
figure 10

The mean COV seq/COV ms for different hysteretic models with μ = 4 and sequence-type ground motions with different \( \nabla PGA \): a EPP model, b MC model, c PH model, d SSD model

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4.5 Effect of hysteretic behavior

In this section, the influence of degrading behavior on the R μ is investigated. In this manuscript, the degrading systems are simulated by the MC model, PH model and SSD model. The ratio between the R μ of the degrading system and that of EPP system, denoted as R μ,degrading/R μ,EPP, is computed for each sequence-type ground motion. Then the mean R μ,degrading/R μ,EPP of all sequence-type ground motions with constant \( \nabla PGA \) are calculated for each period and each μ.

Figure 11 shows the mean R μ,degrading/R μ,EPP for sequence-type ground motions with \( \nabla PGA = 0.5 \) and different hysteretic models. For the periods being smaller than 1.0 s, the mean R μ,degrading/R μ,EPP has strong dependence on the period and increases as the period increases. When the period is >1.0 s, the mean R μ,degrading/R μ,EPP region has weaker dependence on the period. When the period is very small (e.g. T = 0.2 s), the degrading behavior decreases the R μ . When the period is beyond the limitation period, the degrading behavior increases the R μ . This limitation period varies for different degrading models (i.e. 0.3 s for MC model, 0.8 s for PH model and 0.5 s for SSD model).

Fig. 11
figure 11

The mean R μ,degrading/R μ,EPP for sequence-type ground motions with \( \nabla PGA = 0. 5 \) and different hysteretic models: a MC model, b PH model, c SSD model

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All the values of mean R μ,degrading/R μ,EPP in Fig. 11 are generally within the interval [0.8, 1.2]. This phenomenon means that degrading behavior would decrease the R μ of structure with short period at a magnitude of <20 %, while increase the R μ of structure with medium-long period as a maximum level of 20 %. For the degrading systems with long periods, the smaller hysteretic loops are more than those of non-degrading systems. For the case of constant μ, the seismic response of degrading system with long periods tends to be smaller than that of non-degrading system. Thus the R μ factors of degrading system are greater than those of non-degrading system in long period region. These above results and phenomena are consistent with the conclusions stated by past researches (Riddell and Newmark 1979a; Riddell et al. 2002).

4.6 Effect of damping

In order to investigate the influence of damping on the R μ , the R μ of EPP system with damping ratio ζ = 0.02 and 0.1 is calculated. Then R μ of EPP system with ζ = 0.02 and 0.1 is normalized by the R μ of EPP system with ζ = 0.05 for each sequence-type ground motion. In final, the mean normalized R μ of all sequence-type ground motions with constant \( \nabla PGA \) is computed for each period and each μ.

Figure 12 presents the mean normalized R μ of all sequence-type ground motions with \( \nabla PGA = 0.5. \) It can be seen that the mean normalized R μ in Fig. 12a is >1.0, while that in Fig. 12b is smaller than 1.0. This phenomenon means that the strength reduction factor R μ decreases as the damping ratio ζ increases. It is well known that the response of elastic and inelastic structure will be reduced when the structural damping increases. All the input energy of elastic system is dissipated by the structural damping, while just part of input energy of inelastic system is dissipated by the structural damping (the other part of input energy is dissipated by the inelastic deformation). The response reduction of elastic structure is larger than that of inelastic structure when the structural damping increases. Thus, in order to achieve the same target ductility, the strength reduction factor R μ should decrease as the structural damping increases (Riddell and Newmark 1979a).

Fig. 12
figure 12

The mean normalized R μ of all sequence-type ground motions with \( \nabla PGA = 0. 5 \): a ζ = 0.02, b ζ = 0.1

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The most values of mean normalized R μ in Fig. 12a are smaller than 1.2, and the most values of mean normalized R μ in Fig. 12b are >0.9. Thus take the R DI of ζ = 0.05 as the benchmark, the influence of damping is commonly within 20 and 10 % for ζ = 0.02 and 0.1 respectively.

5 Predictive model

The predictive model of the R μ is an appealing tool to determine the R μ in the seismic design. Based on the statistical results in Sect. 4 and boundary conditions of R μ confirmed by many researchers which will be discussed in the following paragraphs, the predictive model is proposed as:

$$ R_{\mu } = \frac{{[a \cdot (\mu - 1) + 1] \cdot T^{2} + b \cdot T^{0.5} }}{{T^{2} + b \cdot T^{0.5} }} $$
(4)

where, T is the period, μ is the ductility factor, a and b are regression parameters depending on the site classes, hysteretic models and relative peak ground acceleration of aftershock ground motion \( \nabla PGA. \) The data of the statistical mean R μ is used for the regression analysis. Parameters a and b are calculated by a nonlinear least-square regression analysis with the Levenberg–Marquardt method (Levenberg 1944; Marquardt 1963) for each site class, each hysteretic model and each \( \nabla PGA \). Table 4 summarizes resulting values of these regression parameters.

Table 4 The regression parameters of Eq. (4) for relative intensity being 0.0 (i.e. mainshock), 0.5, 0.8, 1.0 and 1.5
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It should be noted that the derivation of the predictive model was heavily guided by the characteristics of R μ and inspection of the data. The basic mathematical form of the predictive model is firstly developed based on the boundary conditions of R μ , as explained in the next paragraph. Then the tentative exponents of T in the first and second terms of numerator and denominator are included in the initial form and the nonlinear least-square regression analysis is performed. These tentative exponents are adjusted based on the residuals, until the predictive model can give the good estimation of mean R μ for different cases. With this procedure, the predictive model can be obtained with the good regression fit and fewer regression parameters.

The Eq. (4) rigorously satisfies the following boundary conditions:

$$ R_{\mu } (T \to 0,\mu ) = 1 $$
(5)
$$ R_{\mu } (T,\mu = 1) = 1 $$
(6)

Many researches (Veletsos and Newmark 1960; Newmark and Hall 1969; Riddell and Newmark 1979a; Miranda 1993; Miranda and Bertero 1994) suggested that the R μ of the extreme flexible structure tends to be the value of target ductility factor, because the maximum relative displacement tends towards the maximum ground displacement. This condition is not rigorously satisfied by the Eq. (4). However, for the long periods investigated in this manuscript, the predicted R μ of Eq. (4) tends to be close to the target ductility factor and the differences are generally within 10 %.

Figure 13 illustrates the comparison of predicted mean R μ using Eq. (4) with the statistical results for EPP system and sequence-type ground motion with \( \nabla PGA = 0.5 \). It can be concluded that the Eq. (4) provide the good estimation of the mean R μ .

Fig. 13
figure 13

The comparison of predicted mean R μ using Eq. (4) with the statistical results for EPP system and sequence-type ground motion with \( \nabla PGA = 0. 5 \): a site class A, b site class B, c site class C, d site class D

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It is important to compare the prediction results of Eq. (4) with the actual R μ dataset of selected sequence-type ground motions. Based on the definition of FEMA 440 (2005), the following two error measures (i.e. sample mean error \( E_{T,\mu } \) and the standard deviation of the error \( \sigma_{T,\mu } \)) are used to evaluate the accuracy of the Eq. (4).

$$ E_{T,\mu } = \frac{1}{n}\sum\limits_{i = 1}^{n} {\left[ {\frac{{\tilde{R}_{T,\mu } }}{{(R_{T,\mu } )_{i} }}} \right]} $$
(7)
$$ \sigma_{T,\mu } = \sqrt {\frac{1}{n - 1}\sum\limits_{i = 1}^{n} {\left[ {\frac{{\tilde{R}_{T,\mu } }}{{(R_{T,\mu } )_{i} }} - E_{T,\mu } } \right]^{2} } } $$
(8)

where, n is the number of the ground motions, \( (R_{T,\mu } )_{i} \) is the result of ith ground motion computed by the nonlinear response history analysis, \( \tilde{R}_{T,\mu } \) is the result predicted by the Eq. (4). The \( E_{T,\mu } \) can provide the average bias of Eq. (4) and the\( \sigma_{T,\mu } \) can provide the measure of the dispersion of the errors, when the Eq. (4) is used to estimate the R μ . Figure 14 presents the error measures of Eq. (4) for the EPP system under sequence-type ground motions with \( \nabla PGA = 0.5 \) on all site classes. It can be seen that the Eq. (4) tend to underestimate the actual R μ in the short period region at a level of <20 %, thus the Eq. (4) provides the conservative predicting results in short period region. In the medium-long period, the average error of Eq. (4) is generally within 10 %. In general, \( \sigma_{T,\mu } \) has strong dependence on the μ and increases with the increase of μ, meaning that the greater dispersion error would be induced when Eq. (4) is used to estimate the R μ of greater μ.

Fig. 14
figure 14

The error measures of Eq. (4) for the EPP system under sequence-type ground motions with \( \nabla PGA = 0. 5 \) on all site classes

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In the seismic design practice, the relative intensity of aftershock ground motion in the given seismic zone should be determined through the seismic hazard analysis (i.e. to determine the hazard level of aftershock in the given seismic zone).

6 Conclusions

The purpose of this investigation is to propose the ductility-based strength reduction factor R μ for the mainshock–aftershock sequence-type ground motions. The R μ is computed with 458 sequence-type ground motions and four hysteretic models. The aftershock ground motions are scaled to have different relative intensity levels. The effect of aftershock on the R μ is specially studied. The following conclusions are drawn from this investigation:

  1. 1.

    The R μ in short period region is strongly dependent on the period, while that in medium-long period is approximately period independent. In the whole period region, mean R μ increases with the increase of ductility factor μ. The influence of μ on the R μ in short period region reduces gradually as the period decreases. The COV is approximately period independent.

  2. 2.

    The site condition has more obvious effect on the R μ in short period region in comparison with the medium-long period region. In short period region, the mean R μ on all site classes would overestimate the mean R μ on site class D at a maximum level of 30 %, while mean R μ on all site classes would underestimate the mean R μ on site class A at a maximum level of 30 %.

  3. 3.

    The effect of aftershock ground motion with\( \nabla PGA = 0.5 \) on the R μ of EPP system is negligible. The aftershock ground motion has similar effects on the R μ for different hysteretic models. The aftershock ground motion with \( \nabla PGA = 1.5 \) can decrease the R μ of structure with short period at a level of more than 20 %. The effects of aftershock on the COV of the R μ can exceed 20 %. The influences of aftershock on the R μ (mean and dispersion) depend on the period of structure, ductility factor and intensity of the aftershock ground motion.

  4. 4.

    The degrading behavior would decrease the R μ of structure with short period at a magnitude of <20 %, while it would increase the R μ of structure with medium-long period at a maximum level of 20 %. The R μ decrease as the structural damping increases and the effect level of damping on R μ decreases as the structural damping increases.

  5. 5.

    The predictive model of the R μ is proposed as a function of period, ductility factor. The regression parameters are dependent on the site classes, hysteretic models and relative intensity of aftershock ground motion \( \nabla PGA. \) The predictive model can provide the good estimation of R μ .