Seismic design of buckling-restrained knee-braced frames considering cumulative damage

Abstract

A buckling-restrained knee-braced frame (BRKBF) is an efficient structural system utilizing short buckling-restrained braces (BRBs) as knee bracing elements. Because of the shorter yielding core length compared to those of conventional BRBs, one of the key parameters concerning the performance of this system is the low cycle fatigue cumulative damage of the BRBs. This paper presents a seismic design method considering the cumulative damage, performance evaluation, and collapse assessment of 3-story, 6-story, and 9-story archetype BRKBFs. An energy-based plastic design method was utilized to design the example frames. The frames were then evaluated by nonlinear static analysis and multiple-stripe analysis (MSA). The results indicated that BRKBs were able to accommodate seismic motions with limited damage. The validity of the design methodology considering cumulative damage was also confirmed.

Keywords

buckling-restrained knee-braced frame low cycle fatigue cumulative damage energy-based plastic design method nonlinear static analysis multiple-stripe analysis

Introduction

Buckling-restrained braces (BRBs) are well-known energy dissipation devices installed in new or existing buildings. The BRBs are the main designated yielding elements to dissipate the seismic energy. BRBs were first developed to eliminate the buckling issue of conventional braces under compression loading to provide full hysteretic characteristics (Watanabe et al., 1988). BRBs were conventionally used in concentrically braced frames. Later, numerous experimental tests and numerical simulations on different BRB configurations were conducted in many countries to widen the applications. Carden et al. (2006) fabricated relatively small BRBs to fit in the end cross frame of a bridge and investigated the performance of the small BRBs by uniaxial cyclic tests. The results indicated that even relatively small BRBs could still perform satisfactorily with stable hysteretic behavior.

A recent application of BRBs for building structures is to utilize relatively short BRBs or buckling-restrained knee braces (BRKBs) to replace bracing elements (Leelataviwat et al., 2011; Shin et al., 2012; Wongpakdee et al., 2014; Junda et al., 2018). This system, called the buckling-restrained knee brace frame (BRKBF), is shown in Figure 1. The BRKBs were used in seismic design applications because of less obstruction in the bay and ease of repairability. The system is designed so that the BRKBs will yield while the beams and columns remain fully elastic. The inelastic activities are confined to the BRKBs only. The beams are connected to the columns using single plate shear connections (SPSCs) which can increase the ease of construction and reparability after an earthquake. Experimental and analytical study by Junda et al. (2018) showed that BRKBs could improve the hysteretic behavior of the frame under cyclic loading and enhance the ductility and robustness of the structure. In addition, a new kind of BRKB based on shape memory alloy (SMA) has also been investigated for improved seismic performance. SMA material was used as the yielding core of BRKBs for its self-recentering ability, high stiffness, and good energy dissipation (Fahiminia and Zahrai, 2020; Qui et al., 2022). Because of the high cost of SMA material, short-length BRKBs are well suited for optimum use of the material. The performance of BRKBFs with SMA cores was evaluated and the results showed that the residual deformation was greatly reduced compared with that of a typical BRKBF.

Figure 1.

Buckling-restrained knee brace frame system (Junda et al., 2018).

One critical issue affecting the performance of BRBs is low-cycle fatigue damage from cyclic loading. The cycling loading produces cumulative damage in every loading cycle, eventually leading to failure. Nakamura et al. (2000) conducted fatigue tests for BRBs subjected to various constant strain amplitude levels of cyclic loading. BRB specimens with different core steel grades, core sections and core lengths were tested. The resulting strain-life curve (S-N curve) showed that the BRB specimens with different core steel grades and core sections had similar characteristics, but the shorter BRBs showed lower fatigue life as expected. Past research on BRKBFs by Junda et al. (2018) included only the fracture limit state based on maximum brace strain. Cumulative fatigue damage has not been investigated. Therefore, it is still an open question whether the low-cycle fatigue damage would affect the performance of BRKBFs. As this is a new structural system, this aspect is crucial for the further development and adoption of this system. Furthermore, it is well known that the performance of a BRB can vary greatly depending on the design and fabrication details; therefore, having a design procedure that systematically and explicitly considers low-cycle fatigue as part of the design check is very important.

To include the fatigue cumulative damage in the design and assessment, an energy-based design method such as the performance-based plastic design (PBPD) (Goel et al., 2009; Wongpakdee et al., 2014; Junda et al., 2018) is most suitable. In the PBPD method, the design base shear strength for a selected hazard level and a chosen target drift is derived using the energy balance concept. The required frame strength is computed by equating the work needed to push the structure monotonically up to the target drift that required by an equivalent elastic-plastic single degree of freedom system to achieve the same state.

This paper presents a design methodology for BRKBFs considering cumulative damage due to low-cycle fatigue. The design method was based on extending the PBPD method to include cumulative damage due to energy dissipation in the yielding elements. The cumulative damage model for BRKBs and the design method for BRKBFs are first presented. Archetype frames including 3-, 6-, and 9-story BRKBFs were then designed, and analyzed to validate the design methodology. Nonlinear analysis procedures were utilized to evaluate the seismic performance and assess the damage state of the frames. The response of the structures under different seismic loading intensities was assessed by multiple-stripe analysis (MSA) (Jarayer and Cornell, 2009). The MSA is an alternative to the incremental dynamic analysis (IDA) (Lin and Baker, 2013). The MSA assessment is conducted with ground motion sets that are consistent with each intensity level. The MSA avoids issues with ground motion scaling and can reduce the amount of analysis in the conventional IDA method (Jarayer and Cornell, 2009). Based on the analysis results, the effectiveness of the design method considering the cumulative damage and the performance of the BRKBFs are discussed.

BRKB damage model

Low cycle fatigue damage models are normally based on either a strain-based fatigue model or an energy-based damage model (Chai and Romstad, 1997). The cumulative damage can be expressed in terms of damage index (DI). The DI is defined so that it is equal to 1 when the component reaches the failure point.

For the strain-based model, the failure in a component and the level of damage depend on the strain demand and the number of loading cycles (Lee et al., 2005). For BRKBs, the damage can be calculated by the fatigue life of the core component. The strain-based low cycle fatigue model assumes that fatigue damage is generated by plastic strain in each loading cycle. The damage can be calculated by the ratio of the number of loading cycles to fatigue life called Miner’s rule as illustrated in equation (1) (Chai and Romstad, 1997)

DI = {\sum_{i = 1}^{n} (\frac{N}{N_{f}})}_{i}

(1)

where N is the number of loading cycles at each strain amplitude i, and N_f is the number of loading cycles to failure of each corresponding strain amplitude called the fatigue life as predicted by the Manson and Coffin equation (Manson, 1953; Coffin, 1954). The Manson and Coffin equation provides a strain-fatigue life (S-N) curve of the component (Figure 2), and can be written as in equation (2)

{Δ ε = ε}_{f}^{'} {(N_{f})}^{- c}

(2)

where

Δ ε

is the strain amplitude,

ε_{f}^{'}

is the fatigue ductility coefficient, and c is the fatigue ductility exponent varying between −0.3 and −1.0 for metal (Lee et al., 2005).

Figure 2.

The strain-life (S-N) curve (Lee et al., 2005).

The number of loading cycles at certain strain amplitudes can be obtained by the rainflow counting technique (Lee et al., 2005) and the cumulative DI can be calculated cycle by cycle corresponding to the loading history. For BRBs, several low cycle fatigue tests with different BRBs have been performed to generate a suitable S-N curve, and many Manson and Coffin type equations have been proposed (Razavi et al., 2012). In this study, the equation by Nakamura et al. (2000) was selected. The equation was derived by the fatigue tests of the BRBs subjected to various constant strain amplitudes. The equation was given by

Δ ε = 0.2048 {(N_{f})}^{- 0.49}

(3)

For this study, it was assumed that the deformation of BRBs was mostly concentrated in the core region. Hence, the above equation indicates the relationship between the strain amplitude of the core ( $Δ ε$ ) calculated by the loading displacement divided by the core length of the BRKBs and the fatigue life.

For the energy-based damage model, the well-known linear damage model by Park and Ang (1985) defines the DI as a function of deformation and cumulative energy dissipation demands as follows:

DI = \frac{δ_{M}}{δ_{u}} + \frac{β_{DI}}{Q_{y} δ_{u}} E_{h}

(4)

where

δ_{M}

is the maximum deformation,

δ_{u}

is the deformation capacity under monotonic loading of the component,

E_{h}

is the hysteretic energy of the component,

Q_{y}

is the yield force of the component, and

β_{DI}

is the nonnegative parameter. In this study, it is assumed that all parameters are associated with the core region of BRKBs. The parameter

δ_{u}

corresponding to 35% core strain, and

β_{DI}

of 0.23 were assumed (Andrew et al., 2008) in this study.

Ideally, both the S-N curve and the parameters for the damage model should be specifically calibrated for the type of BRBs used in the design. However, the assumed parameters should be sufficient for this study, as the focus is on the overall frame behavior and the assessment of the proposed design procedure.

BRKBs differ significantly from conventional BRBs in that they have relatively short core lengths compared to those of BRBs. As such, only a limited number of short BRBs representing the characteristics of BRKBs have been tested. To investigate the variability of the DI provided by the above strain-based and energy-based damage models, selected cyclic tests with BRB specimen characteristics close to those of BRKBs were collected from the literature. Based on a previous study by Junda et al. (2018), the angle of the knee brace (φ) significantly influences the brace strain demand. It was found that the appropriate angle lies between 30° and 60°. Assuming the length of the beam in the knee region (L_kb) (the distance between the column and the brace-to-beam connection) lies between 0.2 and 0.3 of the span length (L_b), the practical range of BRKB length is estimated to be between 2 and 3 m long depending on the span length (Figure 3). Hence, in this study, the test specimen data focused only on the BRBs with lengths less than 3 m. A total of 52 test specimens tested using both constant and variable amplitude loading were selected from the literature. The characteristics of the selected specimens are shown in Table 1. It should be noted that for subassemblage test specimens, axial strain was approximately obtained by converting subassemblage test deformation to BRB deformation based on the kinematics of the test specimen.

Figure 3.

Deformation of the BRKBF.

Table 1.

The characteristics of BRKB specimens from previous literature.

Researcher	No. of BRKBs Specimen	Overall length, L_o (mm)	Core region			Loading/Testing
Researcher	No. of BRKBs Specimen	Overall length, L_o (mm)	Length, L_c (mm)	Yield strength, F_y (MPa)	Section shape	Loading/Testing
Nakamura et al. (2000)	1	2,000	470	239	Cruciform	Constant/Uniaxial
	1	2,000	470	222	Rectangle
	1	2,000	960	95	Rectangle
	2	2,000	960	259	Rectangle
	1	2,000	1180	95	Cruciform
Carden et al. (2006)	3	968	438	225	Rectangle	Constant/Uniaxial
Carden et al. (2006)	1	968	438	225	Rectangle	Variable/Uniaxial
Vargas and Bruneau (2009)	3	1,524	994	235	Rectangle	Variable/Uniaxial
Takeuchi et al. (2010)	2	2,000	1,000	261	Rectangle	Variable/Uniaxial
Dunai (2011)	1	2,960	2,000	235	Rectangle	Variable
Usami et al. (2011)	4	2,015	1,375	291	Rectangle	Constant/Uniaxial<
	4			251		Constant/Uniaxial<
	1			291		Variable/Uniaxial
	2			251		Variable/Uniaxial
Shin et al. (2012)	4	1638.7	800	278.3	Rectangle	Variable/Subassemblage
Tsai et al. (2014)	3	1,560	1,000	380	Rectangle	Variable/Uniaxial
Khampanit et al. (2014)	1	1,000	760	417.47	Rectangle	Variable/Subassemblage
Lin et al. (2016)	1	2,850	1,520	388	Rectangle	Variable/Uniaxial
	1	2,850	1,520	270
	1	2,810	1,677	359
	1	2,810	1,677	379
	1	2,600	1,860	393
	1	2,600	1,860	398
	2	2,850	2,000	302
	1	2,850	2,000	270		Constant/Uniaxial
Maurya et al. (2016)	3	680	610	450	Rectangle	Variable/Uniaxial
Maurya et al. (2016)	3	984	914	437	Rectangle	Variable/Uniaxial
Junda et al. (2018)	1	1,300	800	368	Cruciform	Variable/Subassemblage
Junda et al. (2018)	1	1,300	800	344	Cruciform	Variable/Subassemblage

Figure 4 shows the selected S-N curve (equation (3)) and the selected cyclic testing data. It should be noted that only those testing with constant amplitude loading were used in Figure 4 to be consistent with the definition of the S-N curve. The fatigue life of the BRKBs with core lengths not exceeding 3 m can be predicted well by the selected S-N curve even though they have different core characteristics.

Figure 4.

The S-N curve with various constant amplitude cyclic loading test data.

The DIs at the end of all the tests were calculated using Equations (1) and (4). The histograms of DIs for both the strain-based and energy-based models are shown in Figures 5(a) and 5(b). The DIs calculated by the strain-based model range from 0.1 to 1.8 with a mean of 0.76. For the energy-based model, the DIs range from 0.3 to 5 with a mean of 1.30.

Figure 5.

Damage index histograms: (a) strain-based low cycle fatigue model, (b) energy-based linear damage model, and (c) maximum damage indices calculated by the strain-based and energy-based models.

As expected, there is a large variation in the DIs in both damage models, as it is well known that the various factors affect the cumulative damage. In addition, damage parameters should be specifically calibrated for the type of BRKBs being used to minimize the variation in the calculated DIs (Park and Ang, 1985).

To account for the variation, the DI used for the performance evaluation could be selected based on the maximum damage indices calculated from the strain-based (Figure 5(a)) and energy-based (Figure 5(b)) models. The histograms of the maximum DIs calculated from the two models are shown in Figure 5(c). The DIs from the energy-based model tend to be larger, resulting in a conservative estimate of the damage level. Therefore, for design purposes, the energy-based model can be used as the primary DI to gauge the performance of structures. This will be discussed further in the next section.

To further account for the variation in the data, the performance of BRKBs used in this research is determined based on DI levels proposed by Oviedo-Amezquita et al. (2021), as shown in Table 2. Using these assessment criteria, the DI provides a safety factor of approximately 1.86 with respect to the mean DI values at failure (1.30/0.7).

Table 2.

Damage index levels (Oviedo-Amezquita et al., 2021).

Damage level	Range	Description
Severe	DI ≥0.7	Failure
Moderate	0.3 < DI <0.7	Moderate damage, and should be further investigated for the safety
Slight	DI ≤0.3	Minimum or no damage.

Seismic design of BRKBFs considering cumulative damage

The design of BRKBF hinges on two important aspects: controlling the deformation demand of BRKBs and designing beams and columns to remain elastic under forces produced by the BRKBs (Junda et al., 2018). Only the BRKBs are designated as yielding elements and are allowed to deform in the inelastic range. Due to the geometry of the BRBKF, the BRKB may experience a greater axial deformation demand for a given frame drift compared to that of a conventional BRB braced frame (Leelataviwat et al., 2017). Therefore, the deformation demand can be controlled by limiting the frame drift to within a target value. In addition, since the damage is governed by the deformation demand and cumulative energy, an energy-based design procedure is best suited for the seismic design of BRKBFs.

The BRKBF system in Figure 3 shows the configuration of the BRKBs and the frame kinematics. The core strain demand (ε_b) of the BRKBF can be estimated from the frame deformation and the brace angle (φ) based on the deformed shape of the frame where the frame is assumed to be pushed up to the target plastic drift (θ_p) (Junda et al., 2018)

ε_{b} = \frac{θ_{p} \sin (2 φ)}{2 (L_{c} {/ L}_{o})}

(5)

where L_c is the core length of the BRKB, and L_o is the overall length of the BRKB.

For this study, the core length ratio of the BRKBs (L_c/L_o) was selected to be 0.7. The core strain demand could be calculated for different target plastic drifts and brace angles by equation (5). The results are shown in Figure 6. The drift limit for the design (target drift) can be selected from Figure 6 depending on the strain capacity of the braces.

Figure 6.

Core strain demand of the BRKB corresponding to different brace angles (Junda et al., 2018).

To ensure that the deformation demand of the BRKBs and cumulative energy dissipation stay within acceptable targets, the PBPD is utilized to design the BRKBF and control the key design aspects. The PBPD method is an energy-based design method that can be applied for various structural systems including the BRKBF system (Goel et al., 2009; Wongpakdee et al., 2014; Junda et al., 2018). The design frame strength is calculated based on a selected target drift. Once the target drift is selected, the BRKBF is designed to ensure that the frame drift does not exceed the target drift. In this paper, the PBPD method is also extended to include the cumulative energy consideration.

The PBPD method is based on the modified energy balance concept (Goel et al., 2009) as shown in Figure 7. The concept is similar to the energy balance concept introduced by Housner (1956) where the input energy (assumed to be equal to $\frac{1}{2} {MS}_{v}^{2}$ where M is the mass and S_v is the spectral velocity) that contributes to damage is taken equal to the sum of the elastic energy and the plastic work done by a structure. The energy balance concept assumes that the maximum earthquake input energy can be estimated from the design spectral velocity (S_v). In the PBPD, the energy balance is applied between the fraction of the energy (E) and the work needed to push the structure monotonically up to the target drift.

γ E = γ \frac{1}{2} {MS}_{v}^{2} {= E}_{e} {+ E}_{p}

(6)

where E_e and E_p are the elastic and plastic energy needed to push the structure up to the target drift, M is the total mass of the system, S_v is the design spectral velocity, and γ is the energy factor given as

γ = \frac{2 μ - 1}{R_{μ}^{2}}

(7)

where μ is the ductility ratio, and R_μ is the yield force reduction factor. The energy factor is a function of the ductility, the yield force reduction factor, and the period. The energy factor can be obtained from the R_μ-μ-T relationships proposed by Newmark and Hall (1982). The energy balance equation establishes the relationship between the strength of the structure and the deformation. For this concept, the required structural strength is calculated for the given input energy, while the deformation is limited within the target drift. The approximate structural strength of a multi-degree-of-freedom structural system can be obtained by this equation (Leelataviwat et al., 2009). In addition, the energy factor can also be modified for different hysteretic systems (Leelataviwat et al., 2020).

Figure 7.

The modified energy balance concept (Goel et al., 2009).

Based on the modified energy balance concept, the base shear corresponding to the target drift in each intensity level can be calculated by equation (8) (Goel et al., 2009)

\frac{V}{W} = \frac{- α + \sqrt{α^{2} {- 4 γ C}_{e}^{2}}}{2}

(8)

where W is the seismic weight of the structure, C_e is the design spectral acceleration from the seismic design code, γ is the energy modification factor, and α is the dimensionless parameter calculated by the following equation:

α = (\sum_{i = 1}^{n} λ_{i} h_{i}) \frac{θ_{p} {8 π}^{2}}{T^{2} g}

(9)

where h_i is the height from floor i to the ground, θ_p is the target plastic drift calculated by the difference between the maximum target drift and the yield drift (θ_t-θ_y), and T is the fundamental period. The base shear is distributed as lateral forces along the height of the frame. In this study, the lateral forces proposed by Chao et al. (2007) were used as follows:

F_{i} {= λ}_{i} V

(10)

and

λ_{i} = (β_{i} {- β}_{i + 1}) {(\frac{w_{n} h_{n}}{\sum_{j = 1}^{n} w_{j} h_{j}})}^{{0.75 T}^{- 0.2}}

(11)

where λ_i is the lateral force distribution factor, w_n and h_n are the seismic weight at floor n (top floor) and the height from the ground to the top floor, respectively, and β_i is a proportioning factor related to the floor strength at level i calculated by equation (12).

β_{i} = {(\frac{\sum_{j = 1}^{n} w_{j} h_{j}}{w_{n} h_{n}})}^{{0.75 T}^{- 0.2}}

(12)

The value β_i is defined to have a value of one at the roof level and increases for the lower stories. The strength of the BRKB at each floor level compared to the strength of the BRKB at the roof level can also be established according to β_i.

Once the design base shear and the lateral forces at the roof level have been obtained, the required axial strength of BRKBs corresponding to the lateral forces and the yield mechanism shown in Figure 3 can be determined by the work equation illustrated in equation (13)

\sum_{i = 1}^{n} (F_{i} h_{i} θ_{p}) = \sum_{i = 1}^{n} (N_{bay} {\times 2 β}_{i} Q_{BRB} δ_{p}) + (N_{bay} {\times 2 M}_{pc} θ_{p})

(13)

where δ_p is the plastic axial deformation of the BRKB calculated from the core strain demand (equation (5)), M_pc is the required plastic moment capacity of the columns at the bases calculated by preventing the soft-story mechanism (Goel et al., 2009), n is the number or floor levels, N_bay is the number of bays, and Q_BRB is the yield strength of the BRKBs at the roof level. In equation (13), it is assumed that the strength of the interior column is taken as twice that of the exterior column (M_pcex = M_pc and M_pcin = 2M_pc). The variation in the BRKB strength along the height of the frame is defined following the distribution given by β_i.

As mentioned earlier, the damage of the structure depends on both the maximum deformation and the energy dissipation. Similar to the input energy, the hysteretic energy component has also been studied by several investigators. The cumulative hysteretic energy is generally believed to be correlated to the total input energy (equal to the sum of kinetic energy, strain energy, and damping energy). Fajfar et al. (2005) showed that the ratio of the cumulative hysteretic energy to the input energy is approximately constant and varies between 0.5 and 0.8 depending on the ductility. The spectrum for hysteretic energy can also be used to estimate the cumulative hysteric energy based on the PGA and soil type (Dindar et al., 2015).

Assuming that the deformation is relatively uniform following the selected mechanism, the hysteretic energy is distributed among the BRKBs in proportion to the work done in the structure as given in equation (13). The ratio of the energy dissipation of each story (E_i) to the overall hysteretic energy (E_H) can be calculated based on equation (14).

\frac{E_{i}}{E_{H}} = \frac{β_{i} Q_{BRB} δ_{p} N_{BRB}}{\sum_{i = 1}^{n} (F_{i} h_{i} θ_{p})}

(14)

where N_BRB is the number of BRKBs in each story. Using this equation and the energy-based damage model in equation (4), the DI of each BRKB can be calculated as

DI = \frac{δ_{t}}{δ_{u}} + \frac{(\frac{E_{i}}{N_{BRB}})}{(β_{i} Q_{BRB}) δ_{u}} β_{DI} = \frac{δ_{t}}{δ_{u}} + \frac{δ_{p} E_{H}}{(\sum_{i = 1}^{n} (F_{i} h_{i} θ_{p})) δ_{u}} β_{DI}

(15)

where δ_t is the target deformation of the BRKBs corresponding to the yield mechanism (δ_y + δ_p where δ_y is the yield displacement, and δ_p is the plastic deformation of the BRKBs), and E_H is the hysteretic energy of the whole structure. The hysteretic energy is correlated to the total input energy of the structure. The hysteretic energy is approximately 0.5–0.8 of the input energy depending on the structural ductility (Fajfar et al., 2005). In this study, the total input energy was determined by using the equivalent velocity given by Alici and Sucuoğlu (2016) as follows:

V_{eq} = ({a \times e}^{- bT} + c) S_{v}

(16)

where parameters a, b, and c are based on the damping and period of the structure (Alici and Sucuoğlu, 2016), and S_v is the pseudospectral velocity. The hysteretic energy (E_H) can be approximated by

E_{H} {= α}_{e} \frac{1}{2} {MV}_{eq}^{2}

(17)

where M is the seismic mass of the structure, V_eq is the equivalent velocity corresponding to the spectral velocity value, and α_e is the numerical factor ranging between 0.5 and 0.8. Once E_H is determined, the DI corresponding to the maximum drift, and seismic hazard can be obtained using equation (15). If the DI value exceeds the desired limit, the frame base shear strength or the target drift can be reduced leading to a lower DI value.

Finally, after BRKB sizes have been determined, the columns can be designed. One of the concerns regarding the use of BRKBFs is the soft-story mechanism under seismic excitation caused by the large column moment produced by the knee brace. However, the columns can be accurately designed using modern methods of structural analysis techniques and the capacity design concept (AISC 360-16, 2016). They can be designed by using the pushover analysis. The frame can be pushed up to the target drift assuming elastic behavior except at the bases. The member forces at the target drift from the analysis are used to determine the sizes of the beam and column members. The analysis can be iterated until the member sizes satisfy the specified yield mechanism i.e., the inelastic activities are confined only to the BRKBs and the column bases.

Performance evaluation of example structures

To evaluate the performance of BRKBFs and the design procedure, BRKBFs with varying heights are designed and evaluated. Performance assessment was carried out by using nonlinear static and dynamic analyses. The low cycle fatigue damage model was incorporated in the design to limit the damage of the structure as described by the previous procedure.

Archetype structures

Three BRKBFs were chosen as archetype models including 3-, 6-, and 9-story frames. The models were assumed to be office buildings with elevation plan views as shown in Figures 8 and 9. The BRKBFs were assumed to be in the N-S direction of the frame. These frames were designed based on Design Category D with design acceleration parameters S_DS = 0.81 g and S_D1 = 0.44 g and soil site V_s30 = 270 m/sec. (ASCE 7-16, 2017). The seismic mass per floor was 5992 kN for the typical floors and 5284 kN for the roof. The target drifts (θ_t) were chosen to correspond to ductility levels of 2 and 3 for the design basis earthquake (DBE) and maximum considered earthquake (MCE) levels, respectively. With the yield drift of approximately 0.7% (Wongpakdee et al., 2014; Junda et al., 2018), this corresponded to target drifts of approximately 1.4% and 2.1% for the DBE and MCE levels, respectively. Based on the above criteria, the DBE level with the target drift of 1.4% governs the design. The design base shear coefficients (V/W) were found to be 0.22, 0.16, and 0.082 for the 3-, 6-, and 9-story frames respectively. The design base shear, the target plastic drift, and the expected (target) maximum drift corresponding to different ground motion intensity levels are summarized in Table 3. The drifts corresponding to different ground motion intensity levels in Table 3 were calculated by using equation (8) with the selected design base shear coefficient. The governing design base shear values shown in Table 3 were used to design the members. The BRKB capacity of each floor and section size for the beams and columns are illustrated in Table 4.

Figure 8.

Geometry of the archetype BRKBFs.

Figure 9.

Plan view of the archetype BRKBFs.

Table 3.

Key parameters in the PBPD method for the seismic design of the example BRKBFs.

Level	Sa (g)	Yield drift (θ_y) (%)	V/W (Eq. (8))	Target plastic drift (θ_p) (%)	Expected (target) maximum drift (θ_u)
3 (%)-story
DBE	0.6	0.7	0.22	0.70	1.40
1.5DBE	0.9			1.32	2.02
2.5DBE	1.5			2.57	3.27
3.0DBE	1.8			3.19	3.90
4.0DBE	2.4			4.46	5.16
6-Story
DBE	0.34	0.7	0.116	0.70	1.40
1.5DBE	0.51			1.30	2.00
2.5DBE	0.85			2.53	3.23
3.0DBE	1.02			3.17	3.87
4.0DBE	1.36			4.41	5.11
9-Story
DBE	0.25	0.7	0.082	0.70	1.40
1.5DBE	0.375			1.30	1.99
2.5DBE	0.625			2.51	3.21
3.0DBE	0.75			3.12	3.82
4.0DBE	1			4.33	5.03

Table 4.

BRKB capacity of each floor and beam and column sections.

3-Story frame
Story	BRKBs capacity (kN)	Beam section	Column section
1	910	B1: W24 × 76 B2: W27 × 94 B3: W27 × 102	C1: W14 × 176 C2: W14 × 211
2	780
3	498
6-Story frame
Story	BRKBs capacity (kN)	Beam section	Column section
1	1,110	B1: W24 × 62 B2: W27 × 84 B3: W27 × 102 B4: W30 × 108 B5: W30 × 116	C1: W14 × 283 C2: W14 × 176 C3: W14 × 257 C4: W14 × 193
2	1,071
3	990
4	865
5	685
6	425
9-Story frame
Story	BRKBs capacity (kN)	Beam section	Column section
1	1,237	B1: W24 × 62 B2: W27 × 84 B3: W27 × 94 B4: W30 × 108 B5: W30 × 116 B6: W33 × 118	C1: W14 × 398 C2: W14 × 233 C3: W14 × 109 C4: W14 × 283 C5: W14 × 176
2	1,219
3	1,183
4	1,125
5	1,044
6	936
7	801
8	627
9	395

Given the selected design base shear strength in Table 3, the expected target drifts and DIs for different intensity levels can be recalculated by Equations (8) and (15) for any intensity level. Figure 10 shows examples of the story drifts and DIs for intensity levels corresponding to 1, 1.5, 2.5, 3, and 4 times the DBE levels. The DIs were calculated by equation (15) for various values of α_e. The relationship between the maximum interstory drift and DIs is expected to increase when the structure is subjected to a higher ground motion intensity. Hence, the DIs can be reduced by designing for higher frame strength and the story drift is also reduced.

Figure 10.

The predicted damage indices at each ground motion intensity level corresponding to various α_e.

Model calibration and performance evaluation

To assess the effectiveness of the above design procedure, the 3-, 6-, and 9-story frames were analyzed using PERFORM-3D (2006). Prior to carrying out the full building analysis, the BRKB and damage model were calibrated using existing test results. The force-deformation characteristic of BRKB was calibrated by the subassemblage test by Junda et al. (2018) and uniaxial test by Maurya et al. (2016). The BRKB was modeled as an inelastic bar with end hinges. A tri-linear load-deformation curve was used in the BRB modeling. The post-yield stiffness was obtained from calibration as 7.5% of the initial stiffness for the tension side and 12.5% for the compression side. The tri-linear load-deformation curve reached the maximum load obtained by using the compression strength adjustment factor (β) of 1.15 and the strain hardening adjustment factor (ω) of 1.3 at the deformation of 5 times the yield deformation. It should be noted that the post-yield stiffness of a BRB varies depending on the type, the configuration, and the restraining mechanism. The post-yield stiffness ratio and the tri-linear load deformation BRB model used in this study were meant to represent a wide range of BRBs. Ideally, the load-deformation model must be specifically calibrated with the type of BRKBs being used in a given project. In the case of the subassemblage cyclic test (Figure 11(a)), the beam and column were modeled as lumped plasticity elements (FEMA 356, 2000). The load-deformation curve as well as the cumulative DI were computed from the analysis. The calibration results of the uniaxial and subassemblage cyclic tests are shown in Figures 11 and 12, respectively. The damage indices of the BRKB were computed cycle-by-cycle until the same displacement when the test specimens failed. The obtained load-deformation response curve compared well with the test results. The maximum damage indices calculated by the damage model for both BRKBs were in the range of 0.7–0.85 and were in line with the DI levels used in this study, where any DI exceeding 0.7 would be considered failure as proposed by Oviedo-Amezquita et al. (2021). Both the force-deformation characteristic and the damage model of BRKB were used in the subsequent full building analysis.

Figure 11.

Calibration of the BRKB characteristics (subassemblage cyclic test by Junda et al. (2018)).

Figure 12.

Calibration of the BRKB characteristic (uniaxial cyclic test by Maurya et al. (2016)).

For performance assessment purposes, the limit state of the BRKBF is governed mainly by the BRKBs. The seismic performance can be expressed by the damage of the BRKBs. In the computer model, the strain capacity of the BRKB elements was assigned to be sufficiently large. In the frame modeling for the performance assessment, beams and columns were also modeled as lumped plasticity elements (FEMA 356, 2000) except column bases where the plastic hinges were considered inelastic behavior. The beam-to-column connection stiffness had a small effect in the damage investigation. Therefore, pinned connections were assumed (Junda et al., 2018). The foundation was modeled as a fixed support. Each frame was also modeled with a P-Δ column. The P-Δ column was designed to carry the gravity forces of the interior columns and was required to deform with the main frame. The strength and stiffness of the P-Δ column was determined by the properties of all the gravity columns divided by the number of frames resisting the seismic loading (Sabelli, 2001).

Pushover analysis

Nonlinear static or pushover analysis was utilized to evaluate the seismic performance of the structure including the strength and yield mechanism. The results of the pushover analysis are shown in Figure 13. The pushover curves also show the key response points including BRBK yielding and column yielding at the bases. The first BRKBs started yielding at approximately 0.55% drift for all structures, with the overall frame yield drift at approximately 0.75% for all the frames as assumed in the design. This indicated that the yield drift was a fairly constant parameter and could be reasonably assumed for low-to mid-rise BRKBFs. After the peak strength, the base shear gradually decreased due to P-Δ effects. It should be noted that the BRKB fracture was not explicitly modeled in the pushover analysis.

Figure 13.

Relationship between the base shear coefficient and roof drift for three BRKBFs.

Nonlinear dynamic analysis

For nonlinear dynamic analysis, the MSA method was utilized to assess the performance and collapse behavior of the structure. In this procedure, engineering demand parameters (EDPs) are estimated from sets of ground motions associated with selected intensity measure (IM) levels. The ground motions are selected corresponding to each IM level and reflecting the hazard properties of each level (Baker, 2015). In this research, the building location was hypothetically taken to be near San Diego, USA which provided S_DS and S_D1 parameters close to those used in the design of the archetype frames. The hazard curves in Figure 14 were obtained from the USGS (2021) database. Five IM levels were selected with their return periods and corresponding S_a values at the fundamental periods as illustrated in Table 5.

Figure 14.

Hazard curve for three BRKBFs.

Table 5.

Spectral acceleration in each IM level for three BRKBFs

IM level	Return period (years)	Spectral acceleration, S_a(T₁) (g)
IM level	Return period (years)	3-Story (T₁ = 0.86 s)	6-Story (T₁ = 1.57 s)	9-Story (T₁ = 2.25 s)
1	500	0.514	0.312	0.202
2	1,000	0.745	0.452	0.297
3	2,500	1.189	0.733	0.482
4	5,000	1.59	1.012	0.665
5	10,000	2.058	1.332	0.904

Twenty ground motions were utilized for each IM level and were selected from the PEER NGA-West2 database (Ancheta et al., 2013). They were based on the selected damping of 5%, magnitude 6–8 with 0–20 km., and V_s30 between 180 and 540 m/sec. No more than five ground motions were selected from the same seismic event (Mazzoni et al., 2020). For IM levels larger than the 2500-year return period where ground motions could not be directly obtained, the same ground motion set from the 2500-year return period was scaled to increase their spectral acceleration (Jarayer and Cornell, 2009). To reduce any bias from the selection, the ground motion set was scaled with scaling factors less than four for each IM level (Kohrangi et al., 2020). The response spectra of the scaled ground motions for the 2500-year return period level are shown in Figure 15.

Figure 15.

Ground motion set at the 2500-year return period level for three BRKBFs

In the analysis, the results including the energy dissipation, DIs of the BRKBs, maximum core strain, and maximum interstory drifts were extracted. The failure limit states were post-calculated from the nonlinear time history results. The DIs were obtained by the maximum DIs calculated by the strain-based and energy-based models. The frames were assumed to have reached the collapse state when the following conditions occurred: 1) DI of one BRKB exceeded 0.7; 2) BRKB core strain exceeded 4%; 3) plastic rotation of the column reached 0.07; and 4) maximum interstory drift exceeded 6%. The 6% interstory drift limit was based on the connection failure as observed in the experiments by Junda et al. (2018). The SPSC was designed primarily to carry the shear force and was unable to accommodate the rotation of the beam beyond a certain limit. In the experiments by Junda et al. (2018), the SPSC failed at the rotation corresponding to the story drift between 5% and 6%. The 6% limit imposed in this study was based on this observation. The MSA results are shown in Figure 16. Ground motions associated with collapse are indicated in the plots.

Figure 16.

Results of the MSA (a) maximum damage index of BRKBs, (b) maximum interstory drift, and (c) maximum core strain of BRKBs.

Based on the results, there was no severe damage in the BRKBs as indicated by the maximum DIs only between 0.5 and 0.6 for the 3-story, 6-story, and 9-story frames (Figure 16(a)). The collapse of the frames was detected only for ground motions beyond the 2500-year return period level. The most severe limit state that triggered collapse detection was found to be the story drift and joint rotation limit. In general, BRKBs are the main designated energy dissipation elements for seismic events. Hence, the performance of the BRKBs is the key to the robustness of the system. Based on the DI damage levels in Table 2, all the BRKBs at the 2500-year return period level can be classified as having only slight to moderate damage in only certain cases. From the maximum interstory drift, no collapse was detected for intensity levels below the 5000-year return period except for the 9-story frame (Figure 16(b)) where only one case was detected. The core strain of the BRKBs was also less than 3% for the 3- and 6-story frames and 4.1% for the 9-story frame at the 2500-year return period level as shown in Figure 16(c). The median of the maximum interstory drift shown in Figure 17 is lower than the expected drift. All results implied that there was only a low probability of collapse for the 2500-year return period level. The overall analysis results are similar to those reported by Wongpakdee et al. (2014) and Junda et al. (2018) where only the core strain limit was applied in the analysis. The results of this study indicate that low-cycle fatigue does not significantly affect the performance of BRKBFs, and the performance is most likely governed by the deformation limit of the connections caused by story drift and joint rotation.

Figure 17.

Maximum interstory drift at the 500-year and 2500-year return period levels.

The plots between the frame maximum story drifts and DIs from all the ground motions are shown in Figure 18. The damage indices predicted by the PBPD method with different α_e values are also shown in the plots. The relationship between the maximum interstory drift and DIs is similar to the relationship predicted by the PBPD method with the value of the parameter α_e in the vicinity of 0.8 as shown in Figure 18. There is a large scattering of the data because of variation in the scaled ground motions especially in the short period for the 3-story frame and at high intensity levels such as 5,000- and 10,000-year return periods. Nevertheless, the results confirmed that cumulative damage could be included in the PBPD method, and it is a valid seismic design approach for the BRKBFs in this study.

Figure 18.

Comparison between the resulting damage indices and predicted damage indices.

Finally, Figure 19 shows the energy distribution along the height of the frames compared to the energy dissipation assumed in the PBPD method. In the plot, the normalized energy dissipation was obtained by the sum of the energy dissipated by the BRKBs in each story divided by the total energy dissipated by all BRKBs. The normalized energy dissipation of the PBPD method was defined as E_i/E_H in equation (14). It was found that the energy dissipation tended to concentrate in the lower stories where the large story shear was larger. A previous study indicated that the frame deformation depends on the column and beam strength distribution along the height of the structures (Wongpakdee and Leelataviwat, 2017). To obtain uniform damage, the strength distribution can be adjusted or an optimization method (Nabid et al., 2020) could be applied to yield a more uniform DI along the height.

Figure 19.

Energy dissipation distribution at the 500-year and 2500-year return period levels.

Conclusions

This paper presents a seismic design considering cumulative damage due to low-cycle fatigue. The low-cycle fatigue of BRKBs was investigated. The PBPD method was extended to include the damage. 3-story, 6-story, and 9-story archetype BRKBFs were model to validate the design method and were then evaluated by nonlinear analysis. The seismic performance assessment of the frames were investigated, and the results can be concluded as follows:

- The low cycle fatigue damage of the BRKBs represented by DI was investigated by strain-based low cycle fatigue and energy-based linear damage models. The fatigue damage of the BRKBs is governed mainly by the energy-based model. Hence, the energy-based model can be included in the design to express the damage in the BRKBs

- Based on the MSA results, there was no severe damage in the BRKBs with maximum DIs between 0.5 and 0.6. The performance of the BRKBFs is governed mainly by the connection deformation limit. The low cycle fatigue damage from the BRKBs does not significantly affect the performance of the structures. Therefore, deformation limits can be used as the main key to assess the collapse of the structure.

- The cumulative damage can be included in the PBPD framework. The modified energy balance concept shows that the deformation, strength, and cumulative damage can all be considered in a unified approach. The validity of this design methodology was confirmed by the analysis results.

Although the presented design framework was verified by the type of frames, the BRKBs, and the ground motions assumed in this study, for practical design application, the DI calculation should be specifically calibrated with the type of BRKBs being used in the project. In addition, design iterations with rigorous analyses based on nonlinear time history should always be carried out for a final design check.

Footnotes

Acknowledgements

The authors gratefully acknowledge the financial support from the Petchara Prajom Klao PhD research scholarship from King Mongkut’s University of Technology Thonburi. Supplementary funding was provided by TRF Senior Research Scholar under Grant RTA 6280012 and Thailand Research and Innovation Fundamental Fund 2023 (Advanced Construction Towards Thailand 4.0 Project). The authors also acknowledge the assistance from Ms Penpichcha Khongpermgoson.

Author contributions

Natakan Naiyana: Conceptualization, Methodology, Software, Investigation, Writing-original draft, Final draft approval. Sutat Leelataviwat: Conceptualization, Methodology, Supervision, Writing-review & editing, Final draft approval, Funding acquisition. Suchart Limkatanyu: Conceptualization, Supervision, Writing-review & editing, Final draft approval, Funding acquisition.

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) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Sutat Leelataviwat

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