Strength reduction factor of self-centering structures under near-fault pulse-like ground motions

Abstract

Many studies on the strength reduction factor mainly focused on structures with the conventional hysteretic models. However, for the self-centering structure with the typical flag-shaped hysteretic behavior, the corresponding study is limited. The main purpose of this study is to investigate the strength reduction factor of the self-centering structure with flag-shaped hysteretic behavior subjected to near-fault pulse-like ground motions by the time history analysis. For this purpose, the smooth flag-shaped model based on Bouc-Wen model which can show flag-shaped hysteretic behavior is first described. The strength reduction factor spectra of the flag-shaped model are then calculated under 85 near-fault pulse-like ground motions. The influences of the ductility level, vibration period, site condition, hysteretic parameter, and hysteretic model are investigated statistically. For comparison, the strength reduction factors under ordinary ground motions are also analyzed. The results show that the strength reduction factor from near-fault pulse-like ground motions is smaller. Finally, a predictive model is proposed to estimate the strength reduction factor for the self-centering structure with the flag-shaped model under near-fault pulse-like ground motions.

Keywords

flag-shaped hysteretic model near-fault pulse-like ground motions predictive model self-centering structures strength reduction factor

Introduction

Conventional structures are normally designed to be able to suffer large plastic deformations without collapse based on the ductility-based design concept. However, these structures may experience severe residual displacement due to the accumulation of unrecoverable deformations. The existence of residual deformation after the earthquake may seriously reduce the structure’s resistance to aftershocks, increase the cost of post-earthquake structural retrofitting and maintenance, and even require the structure to be rebuilt.

Extensive research works have been carried out to reduce the large residual displacements of structures after severe earthquakes, and many methods have been proposed (Christopoulos et al., 2008; Lu et al., 2010, 2014; Xu et al., 2018b). These methods can be generally divided into two categories. In the first category, the post-tensioned rocking columns/joints are adopted (Han et al., 2019; Li et al., 2019; Nazarimofrad and Shokrgozar, 2019). Once the column/joint deviates from its original position during a strong earthquake, the pre-stressed tendons can provide large self-centering force to the column/joint and bring it back to its initial position, so that small residual displacement is achieved. In the second category, the self-centering energy dissipation braces (SCEBs) are adopted and installed to the structure to provide self-centering force (Dong et al., 2019a; Eatherton et al., 2014; Kitayama and Constantinou, 2016a; Xu et al., 2016). The SCEB system normally includes an energy dissipation group to provide stable energy dissipation capability and a self-centering group to help the structure to return to its original position. During a severe earthquake, the brace yields before the structure yielding so that most of the seismic energy is dissipated by the SCEB system and the key structure remains elastic or only experiences minor damage. Both of the two structures have the self-centering capability that can return the structure to its initial position after earthquakes, and they are hence called the self-centering structure. These self-centering structures normally exhibit typical flag-shaped (FS) hysteretic behavior with stable energy dissipation and excellent self-centering capacities (Araki et al., 2016; Dong et al., 2017; Xu et al., 2017, 2018a; Zhu and Zhang, 2008). Due to the excellent characteristics of self-centering structures, they are widely applied in engineering practices. Therefore, the seismic design method of such structures has attracted more and more attentions of the structural earthquake engineering community as well (Eatherton et al., 2014; Kitayama and Constantinou, 2016b; Liu et al., 2018; Qiu and Zhu, 2017a).

For the current seismic design codes, the conventional design spectrum of elastic single-degree-of-freedom (SDOF) structural systems is adopted to estimate the design forces or displacements of the system with specified period and damping in the structural design procedure (ASCE, 2005; Eurocode 8, 2003; GB 50011, 2010; Japan Road Association, 2002). Because structures may enter the plastic stage under strong earthquakes (Hatzigeorgiou and Beskos, 2009; Hatzigeorgiou and Liolios, 2010), the inelastic design spectrum is required for nonlinear seismically resistant structures in seismic design. The inelastic design spectrum is conventionally obtained by scaling down the elastic design spectrum through the strength reduction factor $(R_{μ})$ (Ordaz and Pérez-Rocha, 1998; Whittaker et al., 1999). The strength reduction factor is defined as the ratio of the elastic strength demand of elastic structure to the yield strength of the elastic-plastic structure corresponding to a specific displacement ductility factor $(μ)$ (Miranda and Bertero, 1994). The strength reduction factor is the modification factor for the seismic response, and it is affected by earthquake characteristics and structural properties. Newmark and Hall (1973) first presented that the strength reduction factor is mainly controlled by the displacement ductility level and the fundamental period of the SDOF system. Following studies (Ahmadi and Khoshnoudian, 2015; Arroyo-Espinoza and Teran-Gilmore, 2003; Kitayama and Constantinou, 2016c; Palermo et al., 2013; Thuat, 2014; Wu et al., 2019) investigated the influences of earthquake types, site conditions and structural models on the strength reduction factor for SDOF systems, and developed the corresponding predictive models. From these studies, it is found that the displacement ductility level, the structural period, and the hysteretic model have a significant effect on the strength reduction factor. In recent years, the study on the continuous accumulation of the strong earthquake data and the increased understanding of the earthquake mechanism indicated that the near-fault pulse-like ground motions may cause severe structural damage (Alavi and Krawinkler, 2004; Chopra and Chintanapakdee, 2001; Güneş and Ulucan, 2019; Hatzigeorgiou, 2010a). Some research works have been carried out to investigate the strength reduction factor for near-fault pulse-like ground motions (Ahmadi and Khoshnoudian, 2015; Gillie et al., 2010). It should be noted that the studies on the strength reduction factor mainly focus on structures with the conventional hysteretic models, namely, bilinear model, elasto-plastic model, Ramberg-Osgood model, origin-oriented trilinear model, degradation model, and Takeda model (Chakraborti and Gupta, 2005; Palermo et al., 2013; Zhai et al., 2015). However, the study on the strength reduction factor of the self-centering structure with FS hysteretic model is limited, especially under the near-fault pulse-like ground motions.

The present study aims to investigate the strength reduction factor for self-centering structures with the FS model under near-fault pulse-like ground motions based on the nonlinear dynamic analysis of a single degree of freedom (SDOF) system. Here, the near-fault pulse-like ground motions consider soil and rock sites, and the target ductility factors of the self-centering structure are from 2.0 to 10. To highlight the characteristics of the strength reduction factor for the FS model, the Bouc-Wen model is also analyzed. Moreover, the effects of the hysteresis parameters of the FS model including post-yielding stiffness ratio and energy dissipation ratio are investigated statistically. Comparisons of the strength reduction factor between the near-fault pulse-like ground motions and ordinary motions are also carried out. Finally, based on the statistical results, the predictive model of the strength reduction factor for the self-centering structure under near-fault pulse-like ground motions is presented and verified.

Analytical method

The proposed self-centering structure normally consists of two systems: the energy dissipation system and the self-centering system. The self-centering structure can be regarded as the self-centering system and the energy dissipation system assembled in parallel. Figure 1(a) shows the mass-spring-dashpot idealization of the self-centering structure. In this figure, $F_{S}$ and $F_{E}$ are the forces of the self-centering system and the energy dissipation system, respectively.

Figure 1.

The self-centering structure: (a) mass-spring-dashpot idealization; (b) simplified hysteretic model.

Many previous studies (Dong et al., 2017; Erochko et al., 2014) indicated that the self-centering structure shows the FS hysteretic behavior with the excellent self-centering capability. Figure 1(b) gives the simplified hysteretic model of the self-centering structure. The hysteretic behavior of the self-centering structure is the superposition of those of the energy dissipation system and the self-centering system. In Figure 1(b), $k_{SC 1}$ , $k_{E 1}$ , and $k_{S 1}$ are the initial stiffness of the whole self-centering structure, the energy dissipation system, and the self-centering system, $k_{SC 2}$ , $k_{E 2}$ , and $k_{S 2}$ are the post-yielding stiffness, $f_{SCy}$ , $f_{Ey}$ , and $f_{Sy}$ are the yield strength, $u_{SCy}$ , $u_{Ey}$ , and $u_{Sy}$ are these corresponding yield displacements, $f_{u}$ is the unloading force, and $f_{SC - S}$ is the self-centering force.

As shown in Figure 1(b), some parameters as mentioned above can be used to determine the FS model. Two key dimensionless parameters controlling the hysteretic behavior of the self-centering structure are defined in the present study.

1. Post-yielding stiffness ratio

α_{c} = \frac{k_{SC 2}}{k_{SC 1}}

(1)

2. Energy dissipation ratio

β = \frac{f_{u}}{f_{SCy}}

(2)

in which, the energy dissipation ratio represents the energy dissipation capability of the self-centering structure, and it also manages the self-centering ability of the structure. The energy dissipation capability gets better as the energy dissipation ratio increases, whereas the larger the energy dissipation ratio, the smaller the self-centering force $(f_{SC - S})$ .

The smooth FS model based on Bouc-Wen model has been developed to predict the unique FS hysteretic behavior of the self-centering structure (Dong et al., 2019b), which is expressed as

m \overset{\cdot\cdot}{u} + c \overset{\cdot}{u} + F (t) = - m {\overset{\cdot\cdot}{u}}_{g} (t)

(3)

$F (t)$ is the restoring force, which can be expressed as follows

F (t) = {\begin{matrix} ku, & for the elastic SDOF system \\ F_{E} + F_{S}, & or the inelastic SDOF system with the flag - shaped model \end{matrix}

(4)

in which

F_{E} = a_{E} k_{E 1} u + (1 - a_{E}) k_{E 1} z_{E}

(4-1)

F_{S} = {\begin{matrix} k_{S 1} u, \leq | u | < | u_{Sy} | \\ k_{S 1} u_{Sy} sgn (u) + k_{S 2} u, | u | \geq | u_{Sy} | \end{matrix}

(4-2)

where $a_{E}$ is the ratio of the post-yield stiffness $(k_{E 2})$ and pre-yield stiffness $(k_{E 1})$ of the energy dissipation element in the self-centering structure, ${\overset{\cdot}{z}}_{E}$ is an evolutionary variable to account for the hysteresis property considering the degradation, which can be expressed as

{\overset{\cdot}{z}}_{E} = \frac{1}{1 + δ_{η}} [(\overset{\cdot}{u} - (1 + δ_{v})) (ϕ \overset{\cdot}{u} {| z_{E} |}^{n - 1} z_{E} + γ {(\overset{\cdot}{u} | {\overset{\cdot}{z}}_{E} |)}^{n})]

(5)

where $ϕ$ is the parameter controlling the hysteresis shape, γ is the parameter controlling the loop size, n is the parameter controlling the loop smoothness, $δ_{η}$ and $δ_{v}$ , respectively, control the stiffness degradation and the strength degradation. Figure 2(a) and (b) present the hysteretic behavior of the Bouc-Wen model and FS model, respectively. As shown, the FS model shows excellent self-centering capability and stable energy dissipation ability. It should be noted that since the self-centering structure can be regarded as the self-centering system and the energy dissipation system assembled in parallel, the stiffness of the self-centering structure is the sum of that for the self-centering system and the energy dissipation system. Therefore, the period of the system with the FS model can be obtained by the following equation

T = 2 π \sqrt{\frac{m}{K_{SC 1}}} = 2 π \sqrt{\frac{m}{k_{E 1} + k_{S 1}}}

(6)

Figure 2.

(a) Bouc-Wen model; (b) smooth FS model.

The strength reduction factor is defined as the ratio of strength demand of elastic structure to the yield strength of the corresponding inelastic structure under ground motions as shown in Figure 3. The strength reduction factor can be expressed as

R_{μ} = F_{e} (μ = 1) / F_{y} (μ = μ_{i})

(7)

where $F_{e} (μ = 1)$ is the strength demand for the elastic SDOF system with $μ = 1$ , and $F_{y} (μ = μ_{i})$ is the yield strength of the corresponding inelastic SDOF system with the demand displacement ductility factor $(μ)$ under the earthquake. $μ$ is defined as the ratio of the maximum demand target displacement to the yield displacement. Figure 4 illustrates the flowchart used to calculate the strength reduction factor spectrum.

Figure 3.

Schematic representation of the strength reduction factor and ductility factor.

Figure 4.

Flowchart for the strength reduction factor spectrum.

Ground motions

A set of near-fault pulse-like earthquake records from 26 different earthquake events is used as inputs in the present study. The set of records is downloaded from the NGA West II and NGA East Database. These earthquake records cover a moment magnitude range from 5.0 to 7.5 and rupture distance range from 0.0 to 19.8 km. Table 1 lists information about these records. The total sample of earthquakes can be characterized as fairly broad since it ranges in terms of shear wave velocity between 163 and 2016 m/s. According to the ground soil conditions classified by United States Geological Survey (Surhone et al., 1998), it is rock soil conditions when the shear wave velocity is larger than 360 m/s; it is soil conditions for the shear wave velocity less than 360 m/s. In the present study, 42 ground motions on rock soil conditions and 43 ground motions on soil conditions are considered.

Table 1.

Characteristics of selected near-fault pulse-like ground motion records.

ID	Event	Year	Station
1	San Fernando	1971	Pacoima Dam
2	Coyote Lake	1979	Gilroy Array #6
3	Imperial Valley-06	1979	Aeropuerto Mexicali/Agrarias/Brawley Airport/EC County Center FF/EC Meloland Overpass FF/El Centro Array #10/El Centro Array #11/El Centro Array #3/El Centro Array #4/El Centro Array #5/El Centro Array #6/El Centro Array #7/El Centro Array #8/El Centro Differential Array/Holtville Post Office
4	Mammoth Lakes	1980	Long Valley Dam
5	Irpinia, Italy-01	1980	Sturno
6	Westmorland	1981	Parachute Test Site
7	Coalinga	1983	Oil City/Transmitter Hill/Coalinga-14th & Elm
8	Morgan Hill	1984	Coyote Lake Dam/Gilroy Array #6
9	N. Palm Springs	1986	North Palm Springs
10	San Salvador	1986	Geotech Investig Cent.
11	Whittier Narrows	1987	Santa Fe Springs-EJoslin
12	Superstition Hills	1987	Parachute Test Site
13	Loma Prieta	1989	Gilroy Array #2/Gilroy Array #3/Saratoga-W Valley Coll./ Saratoga-Aloha Ave
14	Erzican, Turkey	1992	Erzincan
15	Cape Mendocino	1992	Petrolia
16	landers	1992	Lucerne
17	Northridge	1994	Jensen Filter Plant/Jensen Filter Plant Generator/LA Dam/Newhall-W Pico Canyon Rd./Pacoima Dam (down str)/Pacoima Dam (upper left)/Rinaldi Receiving Sta/Sylmar-Converter Sta/Sylmar-Converter Sta East/Sylmar-Olive View Med FF
18	Kobe, Japan	1995	Takarazuka/Takatori
19	Kocaeli, Turkey	1999	Gebze
20	Chi-Chi, Taiwan	1999	CHY006/CHY035/CHY101/TCU036/TCU046/TCU049/TCU053/TCU054/TCU056/TCU060/TCU065/TCU068/TCU075/TCU076/TCU082/TCU087/TCU101/TCU102/TCU103/TCU104/TCU128/TCU136
21	Northwest China-03	1997	Jiashi
22	Chi-Chi, Taiwan-03	1999	CHY024/TCU076
23	Yountville	2000	Napa Fire Station #3
24	Parkfield-02, CA	2004	Slack Canyon/Parkfield—Cholame 2WA/Parkfield—Fault Zone 1/Parkfield—Fault Zone 12
25	Chuetsu-oki	2007	Joetsu Kakizakiku/Yoshikawaku Joetsu City/Kashiwazaki City Center/Kariwa
26	Christchurch, New Zealand	2011	PRPC/MQZ

Figure 5(a) shows the acceleration, velocity, and displacement time history curves of the near-fault pulse-like ground motion (i.e., Rinaldi, Northridge earthquake). As shown, this ground motion record displays an obvious large long period pulse in the acceleration history with a similar pulse in the velocity and displacement histories. However, such a significant characteristic does not exist in some other ground motion records, for example, the Imperial Valley record obtained from El Centro #9, 1940 as shown in Figure 5(b). To better understand the influence of the near-fault pulse-like ground motion on the strength reduction factor for the self-centering structure with the FS model, a set of ordinary ground motions without obvious large long period pulse listed in Table 2 is also used for comparison purposes.

Figure 5.

Ground motions: (a) Rinaldi, Northridge earthquake; (b) El Centro #9, 1940 Imperial Valley.

Table 2.

Characteristics of selected ordinary ground motion records.

ID	Earthquake	Year	Station
1	Imperial Valley	1940	El centro Array#9/ Holtville
2	Kern county	1952	Taft/ Barbara/ Santa Barbara Courthouse/ Delta/ El centro Array#11/ Compuertas/ Chihuahua
3	Coyote Lake	1979	Halls Valley
4	Coalinga	1983	Parkfield-Cholame 12W/ Parkfield-Stone Corral 1E/ Parkfield-Gold Hill 4W/ Parkfield -Gold Hill 1W/ Parkfield-Fault Zone 9/ Parkfield-Fault Zone 3/ Poe Road (temp)/ Calipatria Fire Station/ Salton Sea Wildlife Refuge
6	Loma Prieta	1989	Capitola/Gilroy Array #3/APEEL 2-Redwood City/SF-Presidio/Gilroy Array #1/Treasure Island/Foster City-Menhaden Court/Golden Gate Bridge/Hayward City Hall-North/Larkspur Ferry Terminal (FF)/Monterey City Hall
7	Manjil, Iran	1990	Abbar/ Qazvin/ Rudsar/ Tehran-Building&Housing/ Tehran-Sarif University/ Tonekabun
8	Cape Mendocino	1992	South Bay Union School/ Eureka-Myrtle&West/ Shelter Cove Airport/ Butler Valley Station/ College of the Redwoods
9	Landers	1992	Yermo Fire Station/ Coolwater/ Joshua Tree/ Amboy/ Anaheim-WBall Rd/ Arcadia-Campus Dr/ Baker Fire Station
10	Northridge	1994	Beverly Hills Mulhol/ Canyon W Lost Cany/ San Gabriel-EGr Ave/ LA-Wonderland Ave/ Canyon Country
11	Kobe, Japan	1995	Nishi-Akashi/Abeno/Chihaya/ OKA/TOT
12	Duzce, Turkey	1999	Bolu
13	Hector Mine	1999	Hector
14	Kocaeli, Turkey	1999	Duzce
15	Chi-Chi, Taiwan	1999	CHY101/ TCU045/ HWA056/ TAP103/ NST/ CHY041/ TAP003/ TCU040/
16	Lytle Creek	1970	Wrightwood
17	San Fernando	1971	LA-Hollywood Stor/ Castaic/ Castaic-Old Ridge Route
18	Northern Calif	1975	Cape Mendocino, Petrolia
19	Friuli, Italy	1976	Tolmezzo
20	Santa Barbara	1978	Santa Barbara Courthouse
21	Victoria, Mexico	1980	Cerro Prieto
22	Morgan Hill	1984	Gilroy Array #4
23	Westmorland	1981	Salton Sea Wildlife Ref.
24	Whittier Narrows	1987	Mt Wilson-CIT Station
25	N. Palm Springs	1989	San Jacinto-Soboba
26	Kobe	1995	Nishi-Akashi/ Kakogawa/ Westmorland Fire Sta/ Silent Valley

Statistical analyses

In this study, the inelastic SDOF system for the self-centering structure with the FS model is used as the target system. The FS model with $α_{c} = 0.05$ and $β = 0.7$ are adopted in the following analysis. The SDOF systems with a set of 50 periods between 0.1 s and 5.0 s with an interval of 0.1 s are considered. In this analysis, the short, medium, and long period regions that approximately corresponded to 0.1–0.5 s, 0.5–2.0 s and 2.0–5.0 s are defined for easily describing the characteristics of the results. Five ductility levels are selected to consider the effect of varying ductility targets. The low ductility levels including μ = 2, 4, and 6 commonly correspond to most new or existing structural systems, and the systems with μ = 8 and 10 are also considered for seismic isolated structures in this study. The viscous damping ratio is assumed constant with a value equal to 5%.

Strength reduction factor of the self-centering structure with FS model on different site conditions

The strength reduction factor of the self-centering structure with the FS model subjected to 42 near-fault pulse-like ground motions on the rock site and 43 near-fault pulse-like ground motions on the soil site as shown in Table 1 are calculated. The results are, respectively, plotted in Figure 6(a) and (b). The mean value of these strength reduction factors computed by averaging results for the two sites is also discussed as shown in Figure 6(c). The values of the strength reduction factors for difference ductility levels corresponding to the formulation by Newmark and Hall (1982) are included in these figures as a benchmark. It should be noted that the shape of the strength reduction factor provided in the research report by Jalali and Trifunac (2008) is different from that in this study. In the research report, they adopt a simplified approach and model the near-fault motions by smooth pulses. The results indicate that the strength reduction factor for the near-fault pulse-like ground motions is quite different from that for a simplified approach by smooth pulses.

Figure 6.

Strength reduction factors for the self-centering structure under near-fault pulse-like ground motions: (a) mean values on rock site; (b) mean values on soft site; (c) mean values on rock site and soft site; (d) COVs on rock site; (e) COVs on soft site; (f) COVs on rock site and soft site.

As shown in Figure 6, the strength reduction factor of the FS model is affected by the displacement ductility level, vibration period, and site condition. It is obvious that the displacement ductility factor is the most important factor determining the strength reduction factor for the FS model. During the period less than 5.0 s, the strength reduction factor for the self-centering structure increases with the increase of the ductility level. During the short-medium period region, the strength reduction factor is significantly dependent on the structural period, and it increases sharply with the increase of the period. For the long period, the effect of the period on the strength reduction factor becomes negligible. For systems with the period higher than 3.0 s, the strength reduction factor oscillates around the corresponding displacement ductility factor. Moreover, the strength reduction factor from this study shows a similar variation trend as the equations by Newmark and Hall with the increase of the vibration period or ductility level.

It is found that the strength reduction factors of the FS model on the two site conditions present a similar general trend. To further investigate the influence of the site condition on the strength reduction factor for the FS model, the ratios of the strength reduction factors for the FS model between different sites and all sites are calculated and shown in Figure 7. During the short period region, the ratio is larger than 1.0 for rock site, and the maximum value reaches 1.17. It means that the strength reduction factor of the FS model on all site conditions would underestimate that on rock site. For the medium-long period region, the ratios for rock site and soft site are both small and within the interval [0.93, 1.07]. The results mean that the strength demand of the FS model on rock site tends to be smaller than that on soft site for the short period. The result is reasonable. This is because the displacement response of the structure with the short period on rock site is generally smaller than that on soft site. Moreover, the ratio is not dependent on the ductility level. All these observations indicate that the effect of the site condition on the strength demand of the FS model is not significant.

Figure 7.

The ratio of strength reduction factors for the FS model between different sites and all sites: (a) rock site; (b) soft site.

In this analysis, the coefficient of variation (COV) is used to evaluate the dispersion of the strength reduction factor for the FS model under near-fault pulse-like ground motions. The COV is defined as the ratio of the standard deviation to the mean. Figure 6(d) to (f) shows the COVs of the strength reduction factor for the FS model on two site conditions. In general, the COVs of the strength reduction factor for the two site conditions show a similar trend. However, they present different trends during the short period region. For the short period region, the COV on rock site is large and reaches to be 0.7, while the COV on soil site is small. For the medium and long period regions, the COVs are within the interval [0.0, 0.5]. Moreover, it is found that the COVs are relatively sensitive to the displacement ductility factor and increase with the increase of the displacement ductility level. For example, the COVs are about 10% for $μ = 2.0$ and increase to 40% approximately for $μ = 10$ , as shown in Figure 6. This is reasonable because the seismic response uncertainty becomes severe when the structures are driven into the serious nonlinear state.

Influence of hysteretic models

To highlight the influence of the FS model on the strength reduction factor, the ratio of the strength reduction factor between the FS model and the Bouc-Wen model, denoted as $R_{μ, FS} / R_{μ, B - W}$ , is calculated and shown in Figure 8. As shown, the value of $R_{μ, FS} / R_{μ, B - W}$ is affected by the period and the ductility level. For the structure with the period less than 1.3 s, the difference between $R_{μ, FS} and R_{μ, B - W}$ is sensitive to the ductility level, and it increases with the increase of the ductility level. When the period is larger than 1.3 s, the ductility level has minor effect on the value of $R_{μ, FS} / R_{μ, B - W}$ . For periods less than 2.0 s, the value of $R_{μ, FS} / R_{μ, B - W}$ is significantly dependent on the structural period, and it increases first and then decreases with the increase of the period. When the period is larger than 2.0 s, the value tends to be stable, implying the $R_{μ, FS} / R_{μ, B - W}$ is not sensitive to the structural period.

Figure 8.

Ratio of the strength reduction factor for the FS model and that for the Bouc-Wen model: (a) on rock site; (b) on soil site; (c) on rock sites and soft sites.

As shown in Figure 8, all the value of $R_{μ, FS} / R_{μ, B - W}$ are smaller than 1.0. In short-medium period region (the range 0.1–2.0 s, which corresponds to the periods for most of the new and existing structures in practical applications), the values of $R_{μ, FS} / R_{μ, B - W}$ are within the range of 0.7–1.0, and the minimum value reaches 0.7, meaning that the strength reduction factors for the Bouc-Wen model are larger than those for the FS model by at least 30%. During the long period region, the values of $R_{μ, FS} / R_{μ, B - W}$ are within the range of 0.85–1.0. The phenomenon means that the strength demand of the FS model tends to be larger than that of the Bouc-Wen model over the whole period region. The result indicates that it is unsafe for the seismic design for the self-centering structure when the strength reduction factor of the Bouc-Wen model is used, particularly for the short-medium period. For this reason, the effect of the FS model on the strength reduction factor should be well considered in the seismic design of the self-centering structure.

Influence of hysteretic parameters

Hysteretic parameters of the self-centering structure can appreciably influence the seismic performances of the self-centering structure (Araki et al., 2016; Qiu and Zhu, 2017b). Without loss of generality, the strength reduction factor for the self-centering structure also may be affected by the hysteretic parameter. To examine the effect of the hysteretic parameter on the strength reduction factor, the effects of the post-yielding stiffness ratio and the energy dissipation ratio are discussed in this section.

The strength reduction factor for the different ductility level self-centering structures with $α_{c} = 0.01$ , 0.05, 0.10, 0.20, and 0.30 are calculated and shown in Figure 9. As shown, the value of the strength reduction factor of the self-centering structure increases with the increase of $α_{c}$ , meaning that the strength reduction factor is larger for the self-centering structure with larger post-yielding stiffness. This phenomenon indicates that the required strength is smaller for the self-centering structure with larger post-yielding stiffness (Cheng et al., 2017).

Figure 9.

(a) Strength reduction factor spectra of the self-centering structures with various post-yielding stiffness ratios, (b) strength reduction factor spectra of the self-centering structures with various post-yielding stiffness ratios when the ductility factor is 6, (c) the effect of the energy dissipation on the ratio of strength reduction factor for the systems between α_c = i (i > 0.01) and α_c = 0.01, (d) the effect of the ductility factor on the ratio of strength reduction factor for the systems between α_c = i (i > 0.01) and α_c = 0.01.

Figure 9(c) shows the ratios of the strength reduction factors for the self-centering structures with $α_{c} = 0.05$ , 0.10, 0.20, and 0.30 to that with $α_{c} = 0.01 (R_{μ, α_{c} = i} / R_{μ, α_{c} = 0.01})$ . As shown, this ratio changes with the variation of $α_{c}$ during the whole period region. The larger the $α_{c}$ , the larger the $R_{μ, α_{c} = i} / R_{μ, α_{c} = 0.01}$ . It should be noted that these ratios are generally within the interval [1.0, 1.5]. In the short period region, the ratios are large, and the maximum value is up to 1.5 when the structural period is 0.1 s for $R_{μ, α_{c} = 0.3} / R_{μ, α_{c} = 0.01}$ , meaning that the influence of $α_{c}$ on the strength reduction factor is remarkable, whereas for the self-centering structure with periods greater than 1.3 s, the ratios become small and tend stable. The phenomenon means that $α_{c}$ has little effect on the strength reduction factor during the long period region. It can be seen from Figure 9(d), the influence of $α_{c}$ is more significant for the structure with larger ductility level. This result indicates that the strength reduction factor for the self-centering structure should consider the effect of $α_{c}$ , particularly for the structural period less than 1.3 s.

Figure 10 shows the strength reduction factor of the self-centering structures with $β = 0.6$ , 0.8, 1.0, 1.2, and 1.4. As shown, the strength reduction factor for the self-centering structure increases as the value of the $β$ increases, meaning the strength reduction factor is larger when the self-centering structure possesses better energy dissipation capability. This phenomenon indicates that the required strength is small for the self-centering structure with large $β$ . This result is reasonable, since the seismic responses of the structure can be reduced by enhancing energy dissipation capacity (Bazaez and Dusicka, 2016).

Figure 10.

(a) Strength reduction factor spectra of the self-centering structures with various energy dissipation ratios, (b) strength reduction factor spectra of the self-centering structures with various dissipation energy ratios when the ductility factor is 6, (c) the effect of the energy dissipation on the ratio of strength reduction factor for the systems between β= i (i > 0.6) and β = 0.6, (d) the effect of the ductility factor on the ratio of strength reduction factor for the systems between β = i (i > 0.6) and β = 0.6.

Figure 10(c) illustrates the ratios of the strength reduction factor for the self-centering structures with $β = 0.8$ , 1.0, 1.2, 1.4 to that for $β = 0.6 (R_{μ, β = i} / R_{μ, β = 0.6})$ . As shown, when the structural period is less than 2.0 s (i.e. the short-medium period region), these ratios are large and within the interval [1.0, 1.16], indicating that the $β$ has a significant effect on the strength reduction factor of the self-centering structure, whereas for the structure with the long period, these ratios are small and within the interval [1.0, 1.06], indicating the effect of $β$ on the strength reduction factor is not significant. It should be noted that the ratio is relatively sensitive to $β$ , and it increases with the increase of $β$ . Moreover, the ratio is less dependent on the ductility level as shown in Figure 10(d). These observations indicate that the influence of $β$ on the strength reduction factor for the self-centering structures is not significant and the corresponding errors are within 16% during the whole period region.

Comparisons of strength reduction factor spectra between near-fault pulse-like and ordinary ground motions

To investigate the influence of near-fault pulse-like ground motions on the strength reduction factor of the FS model, the comparisons of the strength reduction factor between near-fault pulse-like motions and ordinary motions are investigated in this section. Near-fault pulse-like ground motion records and ordinary ground motion records as listed in Tables 1 and 2, respectively, are utilized to calculate.

Figure 11 shows comparisons of the strength reduction factor for the FS model between near-fault pulse-like ground motions and ordinary ground motions. As shown in Figure 11(a), with the increase of the period, the general variation tendencies of both strength reduction factor for the two sets of ground motion records are similar. It should be noted that the strength reduction factor of the FS model under near-fault pulse-like ground motions are smaller than that under ordinary motions. This phenomenon for the FS model is consistent with that for the Bilinear model (Gillie et al., 2010).

Figure 11.

(a) Strength reduction factor spectra for near-fault pulse-like motions and ordinary motions, and (b) ratio of the strength reduction factor between the pulse-like motions and the ordinary motions.

The ratio of the strength reduction factor for near-fault pulse-like motions to that for ordinary motions $(R_{μ, pulse - like} / R_{μ, ordinary})$ is explored to quantify the effect of the near-fault pulse-like ground motions on the strength reduction factor of the FS model, as shown in Figure 11(b). As shown, the effect of near-fault pulse-like motions on the strength reduction factor is related to the period. When the periods are less than 0.7 s, the ratios decrease with the increase of the period, and the minimum value decreases to reach 0.54 for the structure with the ductility factor of 10. The difference decreases with the increase of the structural period for the period larger than 0.7 s. In general, the values of $R_{μ, pulse - like} / R_{μ, ordinary}$ are within the range of 0.54–1.0. This result indicates that the required strength of the self-centering structure under near-fault pulse-like motions is larger compared with that under ordinary motions, namely, it is unsafe to design the self-centering structure for near-fault pulse-like ground motions if the strength reduction factor obtained from the ordinary motions is used. This is consistent with the fact that near-fault pulse-like ground motions may lead to more severe damages of the structure compared with ordinary ground motions as mentioned by some researches (e.g. Alavi and Krawinkler, 2004).

Predictive model

The predictive model of the strength reduction factor is an appealing tool used to determine the strength demand for the seismic design of the structure. In this study, a two-step nonlinear regression analysis is carried out (Hatzigeorgiou, 2010b): first, the relation of the strength reduction factor spectra and the structural period and ductility factor is regressed from the statistical analysis as discussed above; second, the effect of the structural and ground motion characteristic is considered. Based on the general tendencies of the strength reduction factor spectra of FS model for near-fault pulse-like ground motions as above mentioned, it is found that the effects of the site conditions and energy dissipation ratio are not significant and the corresponding errors are not less than 20%, and the post-yield stiffness ratio has a significant effect. Therefore, the simplified equation of the strength reduction factor spectra for the self-centering structure with the FS model which only considers the effect of the post-yielding stiffness ratio is developed as

R_{μ} = \frac{1}{A + \frac{B}{μ} + C {(lnT)}^{2} + DlnT}

(8)

A = A_{1} α_{c}

(8-1)

B = B_{1} α_{c}

(8-2)

C = C_{1} α_{c}

(8-3)

D = D_{1} α_{c}

(8-4)

where A, B, C, and D are coefficients which take into account the effect of the post-yield stiffness ratio, A₁, B₁, C₁, and D₁ is the corresponding parameters.

The boundary conditions of the strength reduction factor confirmed by many researchers which are applied to equation (6) can be expressed as

R_{μ} (T \to 0, μ) = 1

(9)

R_{μ} (T \to \infty, μ) = μ

(10)

The function expression of the strength reduction factor for a self-centering structure ( $a_{c} = 0.05$ , $β = 0.7$ ) under near-fault pulse-like ground motions, which is obtained from nonlinear fitting, is given as

R_{μ} = \frac{1}{0.05 + \frac{0.098}{μ} + 0.047 {(lnT)}^{2} - 0.11 lnT}

(11)

The comparison results of the strength reduction factor for the FS model between the predicted model and the statistical results are shown in Figure 12(a). As shown, the developed predicted model can provide a good estimation of the strength reduction factor for self-centering structure under near-fault pulse-like ground motions.

Figure 12.

(a) Comparisons of the predicted strength reduction factor with the statistical results of the self-centering structure, (b) sample mean error $E_{T, μ}$ .

To evaluate the accuracy of the developed predicted model (i.e. equation (6)), the sample mean error $(E_{T, μ})$ is selected, which can be expressed as follows

E_{T, μ} = \frac{1}{n} \sum_{i = 1}^{n} [\frac{{\tilde{R}}_{T, μ}}{{(R_{T, μ})}_{i}}]

(12)

in which, n is the number of the ground motions, $(R_{T, μ})_{i}$ is the strength reduction factor for ith ground motion, ${\tilde{R}}_{T, μ}$ is the strength reduction factor from the predicted model proposed in the present study. The sample mean error of the predicted model of the strength reduction factor for self-centering structure under near-fault pulse-like motions is shown in Figure 12(b). As shown, during the vibration period less than 5.0 s, the errors are generally with the interval [0.83, 1.16]. This result indicates the predictive model provides the rather accurate estimation of the strength reduction factor in this region.

Conclusions

The main objective of the article was the statistical and analytical characterization of the strength reduction factor of the self-centering structure. Some conclusions drawn from this study are as follows:

The strength reduction factor for the FS model under near-fault pulse-like motions is relatively sensitive to the ductility factor and vibration period. The effect of the site condition on the strength reduction factor of the FS model for the near-fault pulse-like ground motions is not significant.

During the period range 0.1–5.0 s, the strength reduction factor of the FS model is smaller than that of the Bouc-Wen model, and the minimum value of $R_{μ, FS} / R_{μ, B - W}$ can reach 0.7. This result indicates the effect of the FS model on the strength reduction factor should be considered in the seismic design of self-centering structures.

The post-yielding stiffness ratio has a significant effect on the strength reduction factor for the FS model, in particular for the short period region. The effect of the energy dissipation ratio on the strength reduction factor is within 20% and not significant during the whole period region.

The strength reduction factor of the FS model for near-fault pulse-like motions is obviously smaller than that for ordinary motions. The phenomenon indicates that it is unsafe for the structural design of the self-centering structure under near-fault pulse-like motions if the strength reduction factor from the ordinary motions is used, particularly for the short-medium period region.

The proposed predicted model can provide a good estimation of the strength reduction factor for the self-centering structure with FS model under near-fault pulse-like ground motions.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors acknowledge the partial support from the National Science Foundation of China (No. 51908325 and No. 51778023), Postdoctoral Science Foundation of China (2018M641363), and Beijing Municipal Education Commission (No. IDHT20190504 and KZ202010005001).

ORCID iDs

Qiang Han

Xiuli Du

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