Modified Energy-Based Design Method of the Precast Partially Steel-Reinforced Concrete Beam–CFST Column Eccentrically Braced Frame

Abstract

The eccentrically braced frame (EBF) is a typical structural system used in high-rise buildings. Current related design methods focus on the concrete and steel structures rather than on the complex composite structure. In addition, they tend to overlook the contribution of the energy-dissipation unit and its corresponding additional influence on the structure. In this study, a precast composite EBF structure is selected as a case study, including the partially steel-reinforced concrete (PSRC) beam and the concrete-filled steel tubular (CFST) column. A modified energy-based design method is proposed to leverage the excellent seismic performance of the precast composite EBF structure. The multi-stage energy-dissipation mechanism and the additional influence of the eccentric braces are systematically considered through the energy distribution coefficient and the layout of dampers. A case study of a 12-floor, three-bay precast composite EBF structure is conducted using a series of nonlinear time-history analyses. Critical seismic responses, including the maximum inter-story drift ratio, residual inter-story drift ratio, and peak acceleration, are systematically analyzed to evaluate the effectiveness of the proposed design theory. The distribution coefficient is recommended to range from 0.70 to 0.80 to balance the energy-dissipation contribution between the frame and the eccentric braces. In terms of the damper layout, the energy-dissipation contribution of the eccentric brace should differ among the lower, middle, and upper floors.

1. Introduction

Owing to the advantages of steel and concrete, the composite structure is gradually being accepted worldwide. As steel typically exhibits significant hysteretic behavior, it serves as an important energy-dissipation unit in the structure [1,2,3]. Considering that bond slip results in the pinching effect in traditional reinforced concrete (RC) components [4,5], researchers have proposed an embedded steel configuration at the ends of beams to achieve seismic performance comparable to that of steel structures [6,7]. The partially steel-reinforced concrete (PSRC) beam has been demonstrated as feasible in several forms of section steel in frame structures [8,9]. For frame structures, the concrete filled steel tubular (CFST) column is considered a suitable member to pair with the PSRC beam due to the ease of steel connection installation [10,11,12]. Experimental results of substructure tests have consistently supported the seismic performance of these structures. However, owing to the high bearing capacity and compact cross-section of the CFST column [13,14,15], its lateral stiffness is relatively limited, and hence, frame structures using CFST columns are typically applied as multi-story buildings.

To broaden the application of the PSRC beam–CFST column frame, an additional lateral force-resisting system is necessary. The braced frame structure is considered an evolution of frame structures in high-rise building design, including the concentrically braced frame (CBF) [16,17] and the eccentrically braced frame (EBF) [18,19]. The EBF structure can achieve the advantages of both the pure frame and the CBF in the event of the frequent earthquake (FE) and the design basis earthquake (DBE). In addition, it can absorb the excessive large earthquake energy through the damper and protect the frame from damage [20]. As the energy-dissipating unit can be easily replaced, the EBF structure possesses excellent post-earthquake repairability. The Y-shaped layout is one of the typical configurations in the EBF structure [21,22], which is recommended to work with a shear panel damper (SPD). Lin et al. [23] pointed out that the PSRC beam–CFST column EBF structure exhibits a two-stage failure mode, namely the failure of the SPD and the beam plastic hinge.

As for the traditional design method of the EBF system, the cross-sectional dimensions of the structural members are usually determined based on the assumption of elastic theory [24]. To overcome this shortcoming, an energy-based theory is developed to accurately quantify the input and absorbed earthquake energy [25,26]. Based on the transformation of structural energy dissipation, the base shear force and the lateral force distribution can be calculated. Chao et al. [27] proposed a distribution pattern for the lateral force to uniformly distribute the interstory drift ratio (IDR), which has been demonstrated to be valid in both the frame and CBF structures. Considering the ductile seismic mechanism during the maximum considered earthquake (MCE), most design methods only take into account the positive effect of plastic deformation of the energy-dissipating unit. The additional load from the eccentric brace to the frame is considered through designing the brace and frame member in sequence together with an amplification factor for the frame member. Consequently, the frame tends to be too stiff to achieve a necessary yielding IDR for the eccentric brace. On one hand, it is difficult for the brace to reach the design yielding force, and the corresponding energy-dissipation capacity cannot function as expected [28,29,30,31]. On the other hand, no matter which form of the connection configuration is used, the overestimated additional load will cause waste in construction materials due to excessive safety factors [32,33]. Therefore, it is important to design the energy-dissipation unit with respect to the expected additional load and post-damage degradation of the frame [23,34,35]. Xiong et al. [36] recommended an energy distribution coefficient for the eccentric brace in an EBF steel structure. It is worth noting that the RC structure tends to cause a serious pinching effect in the hysteretic behavior [37,38]. The concrete-based EBF structure may be characterized by a dramatic difference from the steel structure.

In this study, a modified energy-based design method is proposed for the application of the precast composite EBF structure in high-rise buildings. The multi-stage energy-dissipation mechanism and the additional influence of the eccentric brace are considered systematically through the distribution coefficient for the energy dissipation and the layout of the damper. A 12-floor, three-bay precast PSRC beam–CFST column EBF structure is designed. Through a series of nonlinear time-history analyses, the effectiveness of the modified method is evaluated through critical seismic responses, including the maximum interstory drift ratio (MIDR), the residual interstory drift ratio (RIDR), and the peak acceleration.

2. Design Method

To effectively predict structural energy absorption, the seismic input energy is considered fixed, depending on the target IDR during the MCE. To ensure structural safety, the total energy dissipation of the whole structure should exceed the expected input energy, and the energy should be distributed rationally among structural components, especially under the maximum considered earthquake, so that each element can contribute effectively [25,26,39]. As shown in Figure 1, the design process for the precast composite EBF structure consists of a total of nine steps.

(1)

Determination of the cross-sectional dimensions and design parameters

The cross-sectional dimensions of the PSRC beam and the CFST column can be estimated based on the floor height and span. A series of seismic design parameters should be determined, including the fundamental period T, the yield IDR θ_y, the plastic IDR θ_p, the ultimate IDR θ_u, the seismic influence coefficient α₁, and the damping ratio ζ.

(2)

Determination of the energy dissipation

Considering that the structure is in an elastic–plastic stage during the MCE, the input earthquake energy is equal to the energy dissipated by all structural members. To quantify the energy absorption, the dissipated energy is calculated as the work of the earthquake-equivalent static loading on the target displacement. The formula is as follows:

$$E_{\mathrm{e}}+E_{\mathrm{p}}={\gamma }_{\mathrm{s}}{E}_{\mathrm{I}}$$

(1)

where E_e, E_p, and E_I denote the elastic energy absorption, the plastic energy absorption, and the input earthquake energy, respectively. The relationship among the three indices is plotted in Figure 2. The term γ_s is the modification factor for the input energy, accounting for the structural damping; K_s and α_s are determined as the initial lateral stiffness and post-yield lateral stiffness hardening ratio, respectively; and V_y, Δ_y, and Δ_u represent the design base shear force, the yield displacement, and the ultimate displacement during the MCE, respectively. Correspondingly, V_e and Δ_e represent the base shear force and elastic displacement for the equivalent elastic structure. Let the ductility reduction factor be R_μ = V_e/V_y, and the displacement ductility coefficient μ_s = Δ_u/Δ_y. Newmark and Hill [40] proposed Formula (2) to define the relationship among R_μ, μ_s, and T. Therefore, the energy dissipation can be calculated through Formulas (3)–(5).

$${\gamma }_{\mathrm{s}}={\displaystyle \frac{2{\mu }_{\mathrm{s}}-1+{\alpha }_{\mathrm{s}}{\left({\mu }_{\mathrm{s}}-1\right)}^2}{R_{\mathsf{\mu }}^2}}$$

(2)

$$E_{\mathrm{I}}={\displaystyle \frac{1}{2}}M{\left({\displaystyle \frac{T}{2\pi }}{\alpha }_{1}g\right)}^2$$

(3)

$$E_{\mathrm{e}}={\displaystyle \frac{1}{2}}M{\left({\displaystyle \frac{T}{2\pi }}·{\displaystyle \frac{V}{W}}·g\right)}^2$$

(4)

$$E_{\mathrm{p}}={\displaystyle \frac{W{T}^{2}g}{{8\pi }^2}}\left[{\gamma }_{\mathrm{s}}{\alpha }_1-{\left({\displaystyle \frac{V}{W}}\right)}^2\right]$$

(5)

where M and W denote the total mass and weight of the structure and V represents the structural design base shear.

(3)

Determination of the lateral force

To utilize the excellent seismic performance of the EBF structure, the distribution pattern for the lateral force proposed by Chao et al. [27] is adopted to maintain the IDR uniform during the structural plastic stage. The lateral force distribution coefficient β_i for the ith floor can be represented as follows:

$${\beta }_{\mathrm{i}}={\displaystyle \frac{V_{\mathrm{i}}}{V_{\mathrm{n}}}}={\left({\displaystyle \frac{\sum _{j=i}^n{W}_{\mathrm{j}}{h}_{\mathrm{j}}}{W_{\mathrm{n}}{h}_{\mathrm{n}}}}\right)}^{0.75T^{-0.2}}$$

(6)

where W_j and W_n denote the representative gravity load of the jth floor and the top floor, respectively; V_j and V_n represent the shear force of the jth floor and the top floor, respectively; and h_j and h_n denote the distance from the jth floor or the top floor to the bottom, respectively. The analytical model is shown in Figure 3. Then, the shear force of the ith floor can be obtained as follows:

$$V_{\mathrm{i}}=\left({\beta }_{\mathrm{i}}-{\beta }_{\mathrm{i}+1}\right)V_{\mathrm{n}}=\left({\beta }_{\mathrm{i}}-{\beta }_{\mathrm{i}+1}\right){\left({\displaystyle \frac{W_{\mathrm{n}}{h}_{\mathrm{n}}}{\sum _{j=i}^n{W}_{\mathrm{j}}{h}_{\mathrm{j}}}}\right)}^{0.75T^{-0.2}}$$

(7)

According to θ_p, the plastic energy dissipation corresponding to the IDR can be obtained as follows:

$$E_{\mathrm{p}}=\sum _{i=1}^n{V}_{\mathrm{i}}{h}_{\mathrm{i}}{\theta }_{\mathrm{p}}$$

(8)

Substituting the above equation into Equation (5), the structural design base shear can be calculated as follows:

$$V={\displaystyle \frac{-\lambda +\sqrt{{\lambda }^2+4{\gamma }_{\mathrm{s}}{\alpha }_1^2}}{2}}W$$

(9)

$$\lambda =\left({\displaystyle \frac{8{\pi }^2{\theta }_{\mathrm{p}}}{T^{2}g}}\right)\left(\sum _{i=1}^n\left({\beta }_{\mathrm{i}}-{\beta }_{\mathrm{i}+1}\right)h_{\mathrm{i}}\right){\left({\displaystyle \frac{W_{\mathrm{n}}{h}_{\mathrm{n}}}{\sum _{j=i}^n{W}_{\mathrm{j}}{h}_{\mathrm{j}}}}\right)}^{0.75T^{-0.2}}$$

(10)

(4)

Determination of the distribution coefficient ψ for the energy dissipation

The index can be considered the ratio of the energy dissipation of the damper to the total energy absorbed by the structural system. For the composite structure, on one hand, the excessive contribution of the damper will underestimate the energy-dissipation capacity of the frame itself. On the other hand, a damper with a large energy absorption can transfer a relatively large additional force to the frame. Additionally, the concrete damage and bond slip potentially result in a degradation of the seismic performance. Therefore, the value for ψ is recommended within 0.70–0.80, which will be discussed in Section 3.4.

(5)

Design of the frame beam

Although the total energy should be dissipated through the plastic hinge deformation in the beam member to prevent the global structural failure, it is difficult to avoid all the column members from participating in the energy dissipation. From a construction cost perspective, the limited contribution of the bottom column is considered feasible under the MCE. As shown in Formula (11), the energy dissipation of the frame is considered equal to the work done by both the beam plastic hinge and the bottom column.

$$\left(1-\psi \right)\sum _{i=1}^n{V}_{\mathrm{i}}{h}_{\mathrm{i}}{\theta }_{\mathrm{p}}=\sum _{i=1}^{n}2\eta M_{\mathrm{p}\mathrm{b}\mathrm{i}}{\theta }_{\mathrm{p}}+2M_{\mathrm{p}\mathrm{c}}{\theta }_{\mathrm{p}}$$

(11)

where M_pbi and M_pc denote the yield moment of the beam end at the ith floor and the bottom column, respectively. Note that the scenario where all beam ends yield under the MCE condition does not align with the actual seismic damage patterns of frame structures. Reference [36] introduces an energy-dissipation reduction coefficient η to account for the contribution of the partial beam plastic hinges. The recommended value is set between 0.5 and 1.0. According to the structural yield mechanism, M_pc can be calculated as follows:

$$2M_{\mathrm{p}\mathrm{c}}\theta ={\displaystyle \frac{1}{2}}\phi V{h}_1\theta$$

(12)

where θ is the rotation of the bottom column. φ represents the over-strength of the bottom column. As for the PSRC beam–CFST column, φ can be taken as 1.20 according to the experimental result of six corresponding joint specimens [8]. Additionally, according to Reference [11], the PSRC beam can be initially designed as the steel-reinforced concrete beam. Then, the cross-sectional dimensions of the embedded section steel can be determined according to the constructional requirements, which depend on the following two considerations:

(1) There is weakness around the connection area between the embedded section and the CFST column. According to AISC 358-16 [41] with FEMA-350 [42], the connection should be set with a distance of a + 0.5b to the column side, where a = (0.5~0.75)b_f and b = (0.65~0.85)h. The terms b_f and h represent the width and height of the embedded section, respectively;

(2) The section steel should be inserted into the reinforced beam with adequate embedded length to avoid local failure. Guo et al. [43] suggest that the embedded length should be no less than 200 mm. Therefore, the distance is set as (a + b + 200) from the end of the embedded section to the column edge.

(6)

Design of the frame column

The moment and axial force of the side and middle columns are shown in Figure 4 as an isolated unit. Therefore, the shear forces can be expressed as follows:

$$V_{\mathrm{cn},\mathrm{S}}={\displaystyle \frac{\sum _{i=1}^n{{\xi }_{\mathrm{i}}M}_{\mathrm{p}\mathrm{b}\mathrm{i}}+\frac{h_{\mathrm{c}}}{2}\sum _{i=1}^n{V}_{\mathrm{p}\mathrm{b}\mathrm{i}}+M_{\mathrm{p}\mathrm{c}}}{\sum _{i=1}^n\left({\beta }_{\mathrm{i}}-{\beta }_{\mathrm{i}+1}\right)h_{\mathrm{i}}}}$$

(13)

$$V_{\mathrm{cn},\mathrm{M}}={\displaystyle \frac{\sum _{i=1}^n{\xi }_{\mathrm{i}}\left(M_{\mathrm{p}\mathrm{b}\mathrm{i},\mathrm{L}}+M_{\mathrm{p}\mathrm{b}\mathrm{i},\mathrm{R}}\right)+\frac{h_{\mathrm{c}}}{2}\sum _{i=1}^n\left(V_{\mathrm{p}\mathrm{b}\mathrm{i},\mathrm{L}}+V_{\mathrm{p}\mathrm{b}\mathrm{i},\mathrm{R}}\right)+M_{\mathrm{p}\mathrm{c}}}{\sum _{i=1}^n\left({\beta }_{\mathrm{i}}-{\beta }_{\mathrm{i}+1}\right)h_{\mathrm{i}}}}$$

(14)

where h_c represents the height of the column cross-section and ξ_i denotes the amplification factor for the moment of the beam end; the amplification factor can be set between 1.0 and 1.1, according to Reference [44].

Then, the load-bearing capacity of the column can be calculated as follows:

$$M_{\mathrm{ci},\mathrm{S}}=\sum _{j=i}^n{{\xi }_{\mathrm{j}}M}_{\mathrm{p}\mathrm{b}\mathrm{j}}+{\displaystyle \frac{h_{\mathrm{c}}}{2}}\sum _{j=i}^n{V}_{\mathrm{p}\mathrm{b}\mathrm{j}}+\sum _{j=i}^n\left(h_j-h_{j-1}\right)V_{\mathrm{cj},\mathrm{S}}$$

(15)

$$M_{\mathrm{ci},\mathrm{M}}=\sum _{j=i}^n{\xi }_j\left(M_{\mathrm{p}\mathrm{b}\mathrm{j},\mathrm{L}}+M_{\mathrm{p}\mathrm{b}\mathrm{j},\mathrm{R}}\right)+{\displaystyle \frac{h_{\mathrm{c}}}{2}}\sum _{j=i}^n\left(V_{\mathrm{p}\mathrm{b}\mathrm{j},\mathrm{L}}+V_{\mathrm{p}\mathrm{b}\mathrm{j},\mathrm{R}}\right)+\sum _{j=i}^n\left(h_j-h_{j-1}\right)V_{\mathrm{cj},\mathrm{M}}$$

(16)

$$N_{\mathrm{ci},\mathrm{S}}=\sum _{j=i}^n{V}_{\mathrm{p}\mathrm{b}\mathrm{j}}+\sum _{j=i}^n{N}_{\mathrm{W}\mathrm{j},\mathrm{S}}$$

(17)

$$N_{\mathrm{ci},\mathrm{M}}=\sum _{j=i}^n\left(V_{\mathrm{p}\mathrm{b}\mathrm{j},\mathrm{L}}-V_{\mathrm{p}\mathrm{b}\mathrm{j},\mathrm{R}}\right)+\sum _{j=i}^n{N}_{\mathrm{W}\mathrm{j},\mathrm{M}}$$

(18)

where M_ci,S and M_ci,M denote the moment of the end of the side column and middle column, respectively; N_ci,S and N_ci,M represent the axial force of the end of the side column and middle column, respectively; and N_wj,S and N_wj,M denote the vertical load transferred from the jth floor to the side column and middle column, respectively.

(7)

Design of the eccentric brace system

Different forms of SPDs can be used as the energy-dissipation unit in the eccentric brace system. Reference [45] recommends a modified strip model to predict the hysteretic behavior and critical mechanical properties of SPD. Then, the cross-sectional dimensions of the SPD, namely the height, width, and thickness of the shear panel, can be determined according to the total energy-dissipation demands calculated from the distribution coefficient, ψ. The formula is listed as follows:

$$\psi \sum _{i=1}^n{V}_{\mathrm{i}}{h}_{\mathrm{i}}{\theta }_{\mathrm{p}}=\sum _{i=1}^n{W}_{\mathrm{i},\mathrm{d}\mathrm{a}\mathrm{m}\mathrm{p}\mathrm{e}\mathrm{r}}$$

(19)

where W_i,damper represents the energy absorbed by the damper in the ith floor with respect to the target displacement. As the eccentric brace remains elastic during the whole process, its dimensions can be calculated according to the ultimate shear force of the damper.

(8)

Determination of the additional influence of the eccentric brace system

The additional influence mainly manifests as the moment and axial force on the beam and the column. With respect to the layout of the eccentric brace, namely symmetrical and asymmetrical, the degree of the additional influence tends to be different between the side and middle columns. Therefore, the load-bearing capacity of the frame member should be compared with the sum of the internal force calculated in steps (5) and (6) and the additional influence based on the ultimate force of the damper.

(9)

Check the deformation requirement

The time-history analysis should be conducted on the precast composite EBF structure. The critical seismic response should meet the coded deformation requirement, including the MIDR, the RIDR, and the peak acceleration.

3. Case Study

3.1. Benchmark Model

To verify the aforementioned design method, a 12-floor, three-bay precast PSRC beam–CFST column EBF office structure is designed for a building on a site with an intensity 8 earthquake (see Figure 5). Heights of the first floor and other floors are set as 4000 mm and 3300 mm, respectively. The seismic design group and site class are Group III and Class II according to CB50011-2010 [24]. Dead loads are taken as 5.00 kN/m² for both the standard floor and the roof. Live loads of the standard floor and the roof are set as 2.00 kN/m² and 0.45 kN/m², respectively. The factors of the representative gravity load are taken as 1.0 and 0.5 for the dead load and the live load [46]. The shear panel damper is made of Q235B, while other steel members are all made of Q355B. Strength grades of the concrete and the reinforcement are taken as C40 and HRB400, respectively. According to ASCE 7-16 [46], the fundamental period of the structure is set as 1.20 s. Other design parameters are summarized in

Table 1. Design parameter.

Design Parameter	Value	Design Parameter	Value
T	1.20 s	α1	0.325
θy	0.005	Rμ	2.5
θp	0.015	μs	2.5
θu	0.020	γs	0.688
αs	0.135	ζ	0.05

.

3.2. Numerical Model

Reference [23] experimentally investigates the seismic performance of the PSRC beam–concrete encased CFST column frame with a duplex assembled I-shaped SPD. Therefore, the composite EBF structure is selected as the actual seismic system in this section. Nowadays, many advanced techniques have been used to comprehensively simulate the seismic response of a frame structure using a 3D solid finite element method [47,48]. However, considering the analysis efficiency and hardware dependency, the numerical model (see Figure 6) is established as a fiber beam element model through OpenSees [49] to characterize the hysteretic behavior of the EBF system. Considering that the force-based element could achieve ideal accuracy with few elements [50], the beam and the column models are represented by the forceBeamColumn elements with five integration points set along the longitudinal direction. The cross-section of each element consists of unconfined concrete, confined concrete, and steel through the built-in Concrete01 and Steel02 materials. As the plastic damage and the confinement effect significantly influence the simulation accuracy [51,52,53], compressive strengths of the confined tube concrete and core beam concrete are regarded as 1.57 times and 1.19 times higher than the compressive strength of the unconfined concrete, respectively. Material properties are listed in

Table 2. Material properties.

Material	Type	Nominal Diameter/Thickness (mm)	Yield Strength (Mpa)	Ultimate Strength (Mpa)	Elastic Modulus (Mpa)
Steel	Stirrup	6	614.1	709.8	208.1
	Longitudinal bar	10	542.5	635.2	205.3
		14	441.4	615.0	201.9
	Shear plate in SPD	8	299.2	443.5	202.0
	Tube	7	430.0	562.3	204.8
	Prestressed strand	10.8	1633.1	1751.3	203.1
Unconfined concrete	Encasing concrete	-	-	30.3	29.9
	Precast concrete	-	-	38.0	32.1
	In situ concrete	-	-	39.6	32.5

. The pretension method is characterized by one truss element and two elasticBeamColumn elements in each beam. The second-order effect of gravity is considered by introducing the P-delta coordinate transformation. The scissor model, which consists of one zeroLength element and four elasticBeamColumn elements, is adopted to characterize the shear behavior of the beam-column joint zone. The rotational constitutive relationship of the core zeroLength element is determined according to the superposed model [54]. Rayleigh damping is typically assumed for the first two modes of the building.

For the eccentric brace, as the brace maintains elasticity throughout the process, the elasticBeamColumn element is used to represent the load-transfer mechanism. With respect to the duplex-configured I-shaped SPD, the modified strip model is used to characterize the hysteretic behavior according to Reference [45] and is represented by twoNodeLink elements consisting of the parallel material combining Steel01 and Hysteretic materials. With respect to the additional moment from the damper to the beam, an elasticBeamColumn element is adopted with the same height as the damper.

The numerical seismic performance is plotted in Figure 7 from aspects of both the hysteresis curve and the energy-dissipation curve. As there is little difference between the numerical model and the tested specimen, the model establishment can be demonstrated to be valid.

3.3. Selection of the Ground Motion

As the environment of the case is similar to Site Class D in the United States, the ground motion suggested by FEMA-P695 [55] is used in the nonlinear time-history analysis. Twenty-two groups of ground motions are obtained from the PEER database [56] and listed in

Table 3. Ground motion.

No.	Name	Monitoring Station	Magnitude	Seismic Component	PGA (g)
GM1	Northridge	Beverly Hills-Mulhol	6.7	MUL009	0.416
GM 2	Northridge	Canyon Country-WLC	6.7	LOS000	0.410
GM 3	Duzce, Turkey	Bolu	7.1	BOL090	0.822
GM 4	Hector Mine	Hector	7.1	HEC090	0.338
GM 5	Imperial Valley	Delta	6.5	H-DLT262	0.238
GM 6	Imperial Valley	El Centro Array#11	6.5	H-E11140	0.364
GM 7	Kobe, Japan	Nishi-akashi	6.9	NIS090	0.503
GM 8	Kobe, Japan	Shin-Osaka	6.9	SHI000	0.243
GM 9	Kocaeli, Turkey	Duzce	7.5	DZC180	0.312
GM 10	Kocaeli, Turkey	Arcelik	7.5	ARC000	0.219
GM 11	Landers	Yermo Fire Station	7.3	YER270	0.244
GM 12	Landers	Coolwater	7.3	CLW-TR	0.417
GM 13	Loma Prieta	Capitola	6.9	CAP090	0.443
GM 14	Loma Prieta	Gilroy Array#3	6.9	G03000	0.555
GM 15	Manjil, Iran	Abbar	7.4	ABBAR-L	0.515
GM 16	Superstition Hills	El Centro Imp. Co.	6.5	B-ICC090	0.238
GM 17	Superstition Hills	Poe Road	6.5	B-POE360	0.300
GM 18	Cape Mendocino	Rio Dell Overpass	7.0	RIO270	0.385
GM 19	Chi-chi, Taiwan	CHY101	7.6	CHY101-E	0.340
GM 20	Chi-chi, Taiwan	TCU045	7.6	TCU045-N	0.507
GM 21	San Fernando	LA-Hollywood Stor	6.6	PEL090	0.210
GM 22	Friuli, Italy	Tolmezzo	6.5	A-TMZ270	0.315

. To reflect different intensities of the FE, DBE, and MCE, the corresponding peak ground accelerations (PGAs) are modulated to 70 gal, 200 gal, and 400 gal [24], respectively.

3.4. Determination of the Distribution Coefficient for the Energy Dissipation

To investigate the effective value of the distribution coefficient, ψ, the range recommended in Reference [36] for the steel structure is expanded to 0.60–0.90. Maintaining the total energy dissipation identical across all models, a total of seven tested models (ψ = 0.60, 0.65, 0.70, 0.75, 0.80, 0.85, 0.90) are designed according to the method proposed in Section 2. The details of the cross-sectional dimensions are listed in Appendix A. Each model is analyzed under 22 ground motion records listed in Table 3 with the modulated PGAs for the FE, DBE, and MCE.

3.4.1. Maximum Interstory Drift Ratio

The distribution of the MIDR for each model under various seismic actions is plotted in Figure 8. To accurately analyze the overall structural response under 22 ground motion records, both the mean value (μ) and the mean value plus standard deviation (μ + σ) are displayed as the solid line and the dashed line, respectively. In general, the mean values of the MIDR are less than the IDR limit required according to Reference [24], namely 0.4% for the FE, 1.0% for the DBE, and 2.0% for the MCE. With the increase in the floor level, the MIDR initially increases and then decreases. The peak MIDR is found to appear from the second floor to the sixth floor.

Since the value of ψ denotes the design contribution of the displacement-based damper, a limited IDR inevitably has a negative influence on the energy dissipation of the EBF system, especially for the upper floor (e.g., the floor higher than the eighth floor). When ψ ranges from 0.70 to 0.80, the average MIDR under the MCE varies from 0.75% to 1.25%. The drift distribution remains uniform, with no localized weak stories. For ψ = 0.60, the main structure absorbs a higher proportion of energy, leading to slightly larger deformations (e.g., an average drift of 0.91% under the MCE). The maximum IDR tends to occur on the sixth and seventh floors. With ψ exceeds 0.85, the distribution of the IDR becomes gradually uneven with the earthquake intensity increasing. Therefore, the energy distribution coefficient ψ is recommended to be within 0.70–0.80 to achieve a uniform MIDR.

3.4.2. Residual Interstory Drift Ratio

The RIDR is an important seismic performance indicator to evaluate the post-earthquake structural reparability. Reference [57] suggests the RIDR of a braced frame structure should be less than 0.50%. The distribution of RIDR of each model under various seismic actions is plotted in Figure 9. It can be found that RIDRs of most curves are less than the required limit, except for the dashed curve of ψ at 0.85 and 0.90 in Figure 9c. The phenomenon results in the excessive contribution of the damper with a large ψ. Since its energy dissipation results from the plastic deformation of the shear panel, the corresponding excessive restoring force prevents the frame from returning to the initial state. Therefore, the energy distribution coefficient ψ is recommended to be less than 0.85 to achieve acceptable post-earthquake repairability.

3.4.3. Peak Acceleration

The peak acceleration of each model is displayed in Figure 10. Regardless of the intensity of the input earthquake, there are limited differences in curves with respect to floors below the seventh floor. Note that the peak acceleration in a common frame usually rises with the increase in the floor level. For the FE curve, peak accelerations of the upper floors are significantly decreased, except for the top floor owing to the whipping effect. Therefore, it can be concluded that the eccentric brace system is conducive to controlling the acceleration of the floors close to the top floor. The influence of the distribution coefficient can be observed in curves of the DBE and the MCE with the floor above the eighth floor. The phenomenon indicates that the EBF system gradually functions through plastic deformation. For the DBE curve, there is a dramatic difference between ψ = 0.60 and other values. For the MCE curve, the tendency of curves can be sorted into four groups, namely 0.60, 0.65 to 0.70, 0.75 to 0.80, and 0.85 to 0.90. An increase in ψ can dramatically enhance the contribution of the damper, and hence, control the seismic response of the upper floor. Considering that curves of ψ ≥ 0.65 characterize a similar peak acceleration in the top floor, the range can be regarded as suitable with respect to the seismic performance and the cost of damper.

3.5. Layout of Dampers

According to Figure 8, it can be found that the distribution of the MIDR varies with the change in the distribution coefficient ψ and the floor level. A small ψ results in a uniform MIDR in all floors, which is consistent with the assumed expectation proposed by Chao et al. [27]. As for a large ψ, the seismic response tends to appear in a local uniform style. For example, under different intensities, the MIDR is mainly concentrated in the middle and lower parts, e.g., the second, third, and fifth floors, while the peak acceleration is concentrated in the fourth and top floors. The MIDR and RIDR achieve relatively small values from the eighth floor to the top floor. The phenomenon mainly results from a totally consistent layout of dampers along each floor. To stress the uniformity, the layout of the damper should be adapted according to the floor levels. Therefore, the building is divided into three categories for the convenience of engineering design, namely the lower floors (first to third), the middle floors (fourth to seventh), and the upper floors (eighth and twelfth). Considering the value of ψ recommended in Section 3.4, only tested models with a ψ ranging from 0.70 to 0.80 are updated in this section. The damper configuration is redesigned as shown in

Table A4. Cross-sectional dimensions for the damper, considering the layout.

Floor	Distribution Coefficient ψ
	0.70	0.75	0.80
12	328-245-7	328-260-7	374-260-8
11	328-245-7	328-260-7	374-260-8
10	328-245-7	328-260-7	374-260-8
9	328-245-7	328-260-7	374-260-8
8	328-245-7	328-260-7	374-260-8
7	374-280-8	374-280-8	374-300-8
6	374-280-8	374-280-8	374-300-8
5	374-280-8	374-280-8	374-300-8
4	374-280-8	374-280-8	374-300-8
3	374-280-8	374-340-8	422-340-9
2	374-280-8	374-340-8	422-340-9
1	374-280-8	374-340-8	422-340-9

. The energy-dissipation capacity of dampers in lower, middle, and upper floors accounts for 30%, 35%, and 35% of the total energy dissipated in the eccentric brace system, respectively. The beam and the column are identical to those listed in

Table A1. Cross-sectional dimensions for the column.

Floor	Distribution Coefficient ψ
	0.60		0.65		0.70		0.75		0.80		0.85		0.90
	Side	Middle	Side	Middle	Side	Middle	Side	Middle	Side	Middle	Side	Middle	Side	Middle
12	430 × 8	529 × 8	430 × 8	430 × 8	430 × 8	430 × 8	430 × 8	430 × 8	430 × 8	430 × 8	430 × 8	430 × 8	430 × 8	430 × 8
11	430 × 8	529 × 8	430 × 8	430 × 8	430 × 8	430 × 8	430 × 8	430 × 8	430 × 8	430 × 8	430 × 8	430 × 8	430 × 8	430 × 8
10	430 × 8	529 × 8	430 × 8	430 × 8	430 × 8	430 × 8	430 × 8	430 × 8	430 × 8	430 × 8	430 × 8	430 × 8	430 × 8	430 × 8
9	430 × 8	529 × 8	430 × 8	430 × 8	430 × 8	430 × 8	430 × 8	430 × 8	430 × 8	430 × 8	430 × 8	430 × 8	430 × 8	430 × 8
8	430 × 8	630 × 8	430 × 8	529 × 8	430 × 8	529 × 8	430 × 8	430 × 8	430 × 8	430 × 8	430 × 8	430 × 8	430 × 8	430 × 8
7	430 × 8	630 × 8	430 × 8	529 × 8	430 × 8	529 × 8	430 × 8	529 × 8	430 × 8	430 × 8	430 × 8	430 × 8	430 × 8	430 × 8
6	430 × 8	630 × 8	430 × 8	529 × 8	430 × 8	529 × 8	430 × 8	529 × 8	430 × 8	529 × 8	430 × 8	430 × 8	430 × 8	430 × 8
5	529 × 8	630 × 8	430 × 8	630 × 8	430 × 8	529 × 8	430 × 8	529 × 8	430 × 8	529 × 8	430 × 8	430 × 8	430 × 8	430 × 8
4	529 × 8	720 × 10	430 × 8	630 × 8	430 × 8	630 × 8	430 × 8	529 × 8	430 × 8	529 × 8	430 × 8	430 × 8	430 × 8	430 × 8
3	529 × 8	720 × 10	430 × 8	630 × 8	430 × 8	630 × 8	430 × 8	529 × 8	430 × 8	529 × 8	430 × 8	529 × 8	430 × 8	430 × 8
2	529 × 8	720 × 10	529 × 8	630 × 8	529 × 8	630 × 8	430 × 8	529 × 8	430 × 8	630 × 8	430 × 8	529 × 8	430 × 8	529 × 8
1	529 × 8	720 × 10	529 × 8	630 × 8	529 × 8	630 × 8	430 × 8	630 × 8	430 × 8	630 × 8	430 × 8	529 × 8	430 × 8	529 × 8

and

Table A2. Cross-sectional dimensions for the beam.

Floor	Distribution Coefficient ψ
	0.60	0.65	0.70	0.75	0.80	0.85	0.90
12	450 × 300 × 11 × 18	400 × 300 × 10 × 16	400 × 300 × 10 × 16	400 × 300 × 10 × 16	400 × 300 × 10 × 16	400 × 300 × 10 × 16	350 × 300 × 9 × 14
11	450 × 300 × 11 × 18	400 × 300 × 10 × 16	400 × 300 × 10 × 16	400 × 300 × 10 × 16	400 × 300 × 10 × 16	400 × 300 × 10 × 16	350 × 300 × 9 × 14
10	450 × 300 × 11 × 18	450 × 300 × 11 × 18	450 × 300 × 11 × 18	400 × 300 × 10 × 16	400 × 300 × 10 × 16	400 × 300 × 10 × 16	400 × 300 × 10 × 16
9	450 × 300 × 11 × 18	450 × 300 × 11 × 18	450 × 300 × 11 × 18	400 × 300 × 10 × 16	400 × 300 × 10 × 16	400 × 300 × 10 × 16	400 × 300 × 10 × 16
8	500 × 300 × 11 × 18	450 × 300 × 11 × 18	450 × 300 × 11 × 18	450 × 300 × 11 × 18	400 × 300 × 10 × 16	400 × 300 × 10 × 16	400 × 300 × 10 × 16
7	500 × 300 × 11 × 18	450 × 300 × 11 × 18	450 × 300 × 11 × 18	450 × 300 × 11 × 18	400 × 300 × 10 × 16	400 × 300 × 10 × 16	400 × 300 × 10 × 16
6	500 × 300 × 11 × 18	500 × 300 × 11 × 18	450 × 300 × 11 × 18	450 × 300 × 11 × 18	400 × 300 × 10 × 16	400 × 300 × 10 × 16	400 × 300 × 10 × 16
5	500 × 300 × 11 × 18	500 × 300 × 11 × 18	450 × 300 × 11 × 18	450 × 300 × 11 × 18	400 × 300 × 10 × 16	400 × 300 × 10 × 16	400 × 300 × 10 × 16
4	600 × 300 × 12 × 20	500 × 300 × 11 × 18	500 × 300 × 11 × 18	450 × 300 × 11 × 18	450 × 300 × 11 × 18	400 × 300 × 10 × 16	400 × 300 × 10 × 16
3	600 × 300 × 12 × 20	500 × 300 × 11 × 18	500 × 300 × 11 × 18	450 × 300 × 11 × 18	450 × 300 × 11 × 18	400 × 300 × 10 × 16	400 × 300 × 10 × 16
2	600 × 300 × 12 × 20	500 × 300 × 11 × 18	500 × 300 × 11 × 18	450 × 300 × 11 × 18	450 × 300 × 11 × 18	400 × 300 × 10 × 16	400 × 300 × 10 × 16
1	600 × 300 × 12 × 20	500 × 300 × 11 × 18	500 × 300 × 11 × 18	450 × 300 × 11 × 18	450 × 300 × 11 × 18	400 × 300 × 10 × 16	400 × 300 × 10 × 16

. The different analysis results are listed in

Table 4. Comparison of seismic responses.

ψ	Seismic Intensity	Value Type	MIDR		RIDR		Peak Acceleration
			Original	Updated	Original	Updated	Original	Updated
0.70	FE	Average	0.10	0.10	0.011	0.012	0.98	0.98
		CoV	15.37%	12.65%	33.89%	26.14%	12.85%	12.77%
	DBE	Average	0.31	0.31	0.02	0.021	2.43	2.43
		CoV	30.14%	26.67%	35.91%	30.63%	11.19%	11.26%
	MCE	Average	0.69	0.69	0.039	0.039	4.65	4.65
		CoV	25.96%	23.07%	29.55%	28.27%	9.99%	10.15%
0.75	FE	Average	0.11	0.11	0.015	0.016	0.98	1.01
		CoV	14.18%	20.16%	21.81%	24.36%	12.73%	14.95%
	DBE	Average	0.31	0.31	0.03	0.028	2.42	2.45
		CoV	32.93%	26.11%	22.02%	20.51%	10.38%	10.93%
	MCE	Average	0.68	0.66	0.074	0.064	4.58	4.61
		CoV	35.65%	27.46%	53.94%	52.23%	9.51%	9.26%
0.80	FE	Average	0.11	0.11	0.015	0.017	0.98	1.02
		CoV	9.45%	20.50%	17.83%	28.75%	13.18%	14.82%
	DBE	Average	0.3	0.31	0.029	0.03	2.42	2.45
		CoV	31.78%	26.48%	31.9%	24.79%	11.05%	11.44%
	MCE	Average	0.67	0.67	0.063	0.06	4.58	4.6
		CoV	32.73%	26.50%	37.6%	35.69%	9.89%	9.75%

with respect to the average value and the coefficient of variation (CoV). It can be found that there is little difference in the average value between the original and updated models. However, note that the CoV significantly decreases, especially for the cases under the DBE and MCE.

Figure 11, Figure 12 and Figure 13 compare the seismic performance indicators before and after the model update, including the MIDR, RIDR, and peak acceleration. Under the FE, it should be mentioned that the updated layout of the damper leads to a significant increase in the upper floors, especially when ψ is equal to 0.75 or 0.80. However, as the corresponding values are relatively small and far from the limit, the amplification effect can be regarded as acceptable.

As for the DBE, MIDR curves (see Figure 12a) of the updated models exhibit a trend of generally vertical development, except for the top three floors. In the same range, a larger RIDR can be observed in curves of updated models than those of original models (see Figure 12b). The phenomenon results from an increased energy absorption contribution of the plastic deformation of the damper, which potentially prevents the unchanged frame from returning to its initial state. According to Figure 12c, as curves are close to each other, it can be concluded that the layout has little influence on the peak acceleration.

For the MIDR curve under the MCE (see Figure 13a), an obvious decrease in the lower floors can be observed together with a slight increase in the middle and upper floors. Taking curves with ψ = 0.75 as an example, although the peak MIDR appears on the second floor in both original and updated models, the value decreases from 1.13% to 0.95%. There is no doubt that the drop of 15.93% can dramatically relieve the negative influence of the P-delta effect, especially for the frame member in the lower floors. The result demonstrates the effectiveness of increasing the energy-dissipation capacity around the bottom of the structure. Given the minimal difference observed between the original and updated models (see Figure 13b,c), the adjustment in the damper layout can be identified as the dominant contributor to mitigating the MIDR.

Figure 14 plots the comparison between the updated and original models of the case with ψ = 0.75, where dashed lines denote the average value of the corresponding sample. Generally, the MIDR does not change sharply between each pair of adjacent floors under the DBE and MCE, which meets the assumption proposed in Step (4). It can be found that the MIDRs of the middle floors are close to the average value for the updated model in the FE, DBE, and MCE, while a similar phenomenon is only observed in the original sample under the MCE. This phenomenon results from the excessive stiffness of the SPD designed for the middle and high floors of the original model. Therefore, the regional energy distribution strategy benefits the participation of all eccentric braces.

4. Conclusions

This study presents a modified energy-based design method for the precast PSRC beam–CFST column EBF structure. The multi-stage energy-dissipation mechanism is systematically considered from the perspective of structural plastic energy absorption. The additional influence of the damper in the eccentric brace is accounted for, addressing the limitations in previous analysis. Based on the case study of a 12-floor, three-bay precast composite EBF structure, the following conclusions can be drawn:

(1) The traditional energy-based design method has been modified through a parallel design process that integrates both the eccentric brace and the frame members. Owing to both the energy absorption and additional load of the eccentric brace, the MIDR, the RIDR, and the peak acceleration consistently meet the code requirements, demonstrating the feasibility of the proposed method.

(2) To represent the multi-stage energy-dissipation mechanism, the distribution coefficient ψ is introduced to balance the energy absorption between the frame and the eccentric brace. In terms of limiting the seismic response and construction cost, a range of 0.70 to 0.80 is recommended for ψ.

(3) The uniformity of the cross-sectional dimensions of the damper negatively impacts the overall seismic response. The excessive stiffness of the SPD on the upper floors leads to a reduction in the contribution of the energy absorption.

(4) A simplified layout, which divides the structure into the lower, middle, and upper floor range, is proposed to optimize the energy-dissipation contribution of the damper. This layout helps reduce the MIDR on the lower floors under the MCE while maintaining a uniform MIDR distribution across the middle floors.

Although the modified energy-based design method is primarily proposed for the precast PSRC beam–CFST column EBF structure in this study, it is expected to be applicable to other composite EBF structure with a steel beam or a steel-reinforced concrete beam. The damper layout method is a simplified approach designed for convenience in engineering practice. To further optimize the seismic performance of the precast EBF structure from a research perspective, the energy-dissipation contribution of each eccentric brace can be individually taken into account with respect to the floor level through a fragility analysis.