Seismic performance of buckling‐restrained brace outrigger system in various configurations

Abstract

The outrigger system is deemed an effective solution for mitigating the seismic responses of tall core‐tube‐type buildings. By incorporating a buckling‐restrained brace (BRB) in the outrigger system (BRB‐outrigger), the BRB‐outrigger reduces seismic response not only through the outrigger mechanism but also through the seismic energy dissipation from the BRB’s hysteretic response. This study investigates the seismic behavior of structures with a single layer BRB‐outrigger and proposes three types of BRB‐outrigger configurations for practical design purposes that fit different architectural requirements. An analytical model, with heights of 64, 128, 256, and 384 m and different outrigger spans was used to investigate the optimal outrigger elevation and required outrigger stiffness for achieving minimum seismic response using spectral analysis and nonlinear response history analysis. The design indexes and design charts based on the analysis results are proposed for preliminary design. Design examples of structures with different BRB‐outrigger configurations utilizing the proposed design charts are demonstrated.

Introduction

The outrigger system has been an effective solution in tall core-tube-type buildings for mitigating seismic responses, and it has been widely adopted in tall buildings worldwide. When lateral loads are applied, the relatively stiff outrigger, which connects the core structure and the perimeter column, applies a resisting moment to the core structure to reduce roof drift, inter-story drift, and bending moment of the core structure. The design of the conventional outrigger that allows for all members to be elastic could be uneconomical. Therefore, the concept of a damped-outrigger was proposed. The most common damped-outrigger configuration has the dampers inserted at the outrigger truss end and the perimeter column. The dampers dissipate energy through relative movements between the outrigger truss end and the perimeter column. This damped-outrigger configuration incorporating viscous dampers has been used in studies to investigate optimal outrigger elevation to maximize the system damping ratio. The study indicates that the elevation of optimal damped-outrigger when viscous dampers are incorporated ranges from 50% to 80% of the building height. The damped-outrigger system incorporating viscous dampers has also been utilized in real construction projects to reduce wind load effect. The damped-outrigger incorporating a buckling-restrained brace (BRB) as the energy dissipation device has also been studied. The damped-outrigger incorporating a BRB (BRB-outrigger) functions as a conventional outrigger system during small earthquakes, and the BRB deforms elastically. During large earthquakes, the BRB dissipates seismic energy through its hysteretic response. Furthermore, the BRB can limit the maximum force demands on the perimeter column and the outrigger truss members. The arrangement of a BRB to function as an outrigger has also been adopted in real construction project. Most of the studies that investigate optimal outrigger elevations were determined using preselected possible elevations. The continuous seismic response distributions with respect to changes in outrigger elevation have not been demonstrated. Furthermore, the details of BRB-outrigger configurations to fit different architectural requirements have not been proposed.

Figure shows a laterally deformed structure with a BRB-outrigger. The core structure provides most of the lateral stiffness and lateral force resistance capacity. The BRB is arranged vertically between the outrigger truss ends and the perimeter columns to optimally impose axial deformation demand on the BRB. As shown in Figure , when the structure deforms laterally to the right, the BRB and perimeter column on the right-hand side are in compression, and in tension on the left-hand side. The outrigger truss, BRB, and perimeter columns act in series to provide a resisting moment to the core structure. When the BRB yields, the BRB starts dissipating energy. The maximum force demands for the outrigger truss members and perimeter columns are limited by the maximum axial force developed in the BRBs. However, when the outrigger span is long, the relative movement between the outrigger truss end and the perimeter becomes very large. Therefore, the BRB length should be very long (8-10 m) to meet the large deformation demand. Furthermore, the design of the outrigger truss is sometimes difficult and uneconomical because very large sections of the outrigger truss members are required to make the outrigger sufficiently stiff. These shortcomings could make a BRB-outrigger system solution an uneconomical and impractical seismic design solution.

Deformed structure with BRB-outrigger system

With this background, the aim of this research was to introduce alternative BRB-outrigger configurations for achieving economical and effective design results. The indexes of the stiffness provided by the BRB-outrigger to the core structure’s rotational stiffness, and to the perimeter column axial stiffness are introduced for comparison for seismic performance between structures with different BRB-outrigger configurations. These indexes and the outrigger elevation for minimizing seismic response were investigated. This study uses a simplified structure and increases the number of analytical models with different heights (64, 128, 256, and 384 m) and outrigger spans (12.8, 13.8, 14.5, and 16 m) for enhanced verification of optimal design results. The continuous seismic response distributions with respect to various outrigger elevations and different outrigger stiffness are demonstrated by performing spectral analysis (SA) and nonlinear response history analysis (NLRHA). The maximum roof drift ratio and the base overturning moment at the core structure base are adopted as seismic performance indicators. The optimal outrigger elevations in order to maximize outrigger effect, maximize BRB energy dissipation efficiency, and minimize roof drift ratio and base overturning moment responses are investigated. Design charts based on newly introduced design indexes are proposed to assist designers with selecting an appropriate outrigger elevation and determining the required outrigger stiffness at the preliminary design stage. Furthermore, three different BRB-outrigger configurations are proposed in this research. Methods for using the simplified structure to model structures with different BRB-outrigger configurations are proposed. Based on the analysis results, all three BRB-outrigger configurations are capable of achieving the desired seismic response. The suitability of each BRB-outrigger configuration for different architectural requirements is discussed.

Numerical Model

The structure with a single ordinary BRB-outrigger system (OB outrigger) can be simplified, as shown in Figure A. The core structure, which provides most of the lateral stiffness, is modeled using a cantilever column with a lateral flexural stiffness of EI. The outrigger truss has a flexural stiffness of k_t. The perimeter column with a length of h has an axial stiffness of k_c. The bases of the perimeter columns are free to rotate about the out-of-plane direction. The BRB has a bilinear force-deformation relationship with an axial initial stiffness of k_d and a post-yield stiffness ratio of 0.01. Each of the BRB connects to the outrigger truss end and perimeter column with pinned-connection detail. For simplicity, in this simplified model, it is assumed that only the BRBs can yield while the other elements deform elastically. The mass is assumed to be concentrated at the core structure and uniformly distributed along the core structure height. As shown in Figure A, if θ₁ is the core structure rotation at an outrigger elevation of α, the relationship between θ₁ and the flexural deformation of the outrigger truss (u_t), axial deformations of the BRB (u_d), and perimeter column below the outrigger elevation (u_c) is expressed as follows:[Image Omitted. See PDF]

Simplified structure with (A) OB outrigger configuration (B) DM model (c) MBM model

Therefore, the moment applied by the OB outrigger (M_o,OB) is calculated as follows:[Image Omitted. See PDF]where l_t and k_rg,OB are the outrigger span and the rotational stiffness provided by the OB outrigger system, respectively. k_og,OB (1/(1/k_d + 1/k_t)), including the elastic stiffness of the BRB, is defined as the outrigger stiffness, which is the combined stiffness of k_d and k_t in the OB outrigger configuration. When the BRB yields, the corresponding core structure rotation at outrigger elevation θ_y can be calculated as follows:[Image Omitted. See PDF]where u_d,y is the BRB yield deformation. Figure B shows an analytical model developed using OpenSees to perform an analysis of a simplified structure with an outrigger. For simplicity, the BRB with a fixed length of 1 m is modeled using a truss element with bilinear material property. The post-yield stiffness ratio (p) is set as 0.01. The outrigger truss is modeled using an elastic beam column element, and its corresponding cross-sectional property is properly assigned so that its flexural stiffness is equal to k_t. The core structure is modeled using an elastic beam column element. The mass is assigned to the nodes that evenly distribute along the core structure height with a fixed 1 m spacing. The core structure base is fixed, while the perimeter column bases are free to rotate about the z-axis. This model is known as a discrete mass (DM) model. The effectiveness of using a DM model to represent the real building was verified using a member-by-member (MBM) model, as shown in Figure C. Therefore, the BRB element in the MBM model is modeled by using bilinear material property. The core structure of the MBM model is represented by a braced frame. The mass is concentrated on each floor. The details of the floor beams and outrigger truss are all included in Figure C.

Configurations of BRB-Outrigger

In the OB outrigger configuration, the outrigger truss should be stiff enough to generate adequate axial deformation demand for the BRB. However, it would be difficult to design the outrigger truss members if l_t is too long or the desired k_t is too large. Furthermore, if the required BRB yield deformation (approximately 1/1000 of the BRB length) is large, the BRB length can be longer than one story height (Figure ). To solve this problem, two alternative BRB-outrigger configurations are introduced.

Figure A shows the BRB-truss outrigger (BT outrigger) configuration. The BRBs are used as braces in the outrigger truss. The ends of the top and bottom chords that connect with the core structure and perimeter column can be designed with either shear or moment connection detail. As the BRBs and the outrigger truss members act in parallel, slight plastic deformation in the outrigger truss member may be allowed to prevent having too large an outrigger truss member size. When compared with the OB outrigger configuration, the outrigger truss member and BRB sizes can be reduced. Furthermore, the increased number of BRBs could provide a more stable energy dissipation mechanism. As shown in Figure A, the core structure rotation at outrigger elevation (θ₁) can be calculated as follows:[Image Omitted. See PDF]where u_c,BT and u_bt are the perimeter column axial deformation and the flexural deformation of the BT outrigger, respectively. Therefore, if k_bt is the flexural stiffness of the BT outrigger, the moment applied by the BT outrigger (M_o,BT) can be calculated as follows:[Image Omitted. See PDF]where k_rg,BT is the rotational stiffness provided by the BT outrigger system. When compared with Equation (2), the outrigger stiffness k_og,BT is equal to k_bt in the BT outrigger configuration. If u_bt,y is the BT outrigger flexural deformation when the first BRB yields, θ_y can be expressed as follows:[Image Omitted. See PDF]

Configurations of the (A) BT outrigger system and (B) GB outrigger system [Colour figure can be viewed at wileyonlinelibrary.com]

To use the DM model to analyze a structure with a BT outrigger, k_d, k_t, and u_d,y in Equations (2) and (3) have to be replaced with k_bt, an infinity value, and u_bt,y, respectively.

Figure B shows the giant BRB-outrigger (GB outrigger) system. The GB outrigger removes the outrigger truss but connects the core structure and the perimeter column using a giant BRB. When a BRB axial force of P_gb is developed, the BRB and the floor beam below the BRB act as a force couple on the core structure, as shown in Figure B. The force couple applies a resisting moment (M_o,gb) to the core structure. It should be noted that both the BRB and floor beam are acting as truss elements. Furthermore, the floor beam is usually strong enough to resist the maximum force developed by the BRB and is stiff enough that the shortening or elongation of the floor beam can be neglected. If u_c,GB and u_d,gb,v are the vertical deformations of the perimeter column below the outrigger elevation and the BRB in the GB outrigger configuration (BRB_GB), and k_d,gb is the axial stiffness of BRB_GB, the corresponding core structure rotation at outrigger elevation (θ₁) and M_o,gb can be calculated as follows:[Image Omitted. See PDF][Image Omitted. See PDF]where h_t and η (tan⁻¹(h_t/l_t)) are the vertical span and the inclined angle of BRB_GB, respectively. k_rg,GB is the rotational stiffness provided by the GB outrigger system. The outrigger stiffness k_og,GB is equal to k_d,gbsin²η in the GB outrigger configuration. If u_d,ygb is the axial yield deformation of BRB_GB, θ_y can be expressed as follows:[Image Omitted. See PDF]

When compared with Equations (2) and (3), the DM model can be used to model the structure with a GB outrigger by replacing k_d, k_t, and u_d,y with k_d,gbsin²η, an infinity value, and u_d,ygb/sin η, respectively. It is anticipated that the GB outrigger could conserve steel usage as the outrigger truss is not necessary. Furthermore, when the required BRB yield deformation is large, the long BRB in the GB outrigger configuration can be adopted easily. However, the BRB_GB may be required to span across more than one story, which may reduce usable floor areas. As indicated in Equation (8), the larger the value of η, the greater is M_o,GB. The seismic performance of structures with OB, BT, and GB outriggers can be estimated using the DM model (OB outrigger configuration) with modified parameters, as shown in Table , and its effectiveness is verified by the analysis results calculated using MBM models, which follow individual outrigger configuration details. The details of the MBM model for each outrigger configuration are introduced in the following sections. The DM model with an OB outrigger configuration is used to investigate the optimal outrigger elevation for minimizing maximum roof drift (θ_max) and core structure base overturning moment (M_c,max).

Parameters used in the DM model for each outrigger configuration

Configuration	Outrigger stiffness (kog)	kt	kd	ud , y
OB outrigger	$k_{og,OB}=\frac{k_d{k}_t}{k_d+k_t}$	kt	kd	ud , y
BT outrigger	kog,BT = kbt	∞	kbt	ubt , y
GB outrigger	kog,GB = kd,gb sin2η	∞	kd,gbsin2η	$\frac{u_{d,ygb}}{sin\eta }$

Analysis Methods Index definitions

In each BRB-outrigger configuration, the relationship between outrigger stiffness (including BRB axial stiffness) and perimeter column stiffness is critical. Therefore, two dimensionless indexes are newly defined for the parametric study. Instead of the index S_bc described in the previous study, the outrigger effect (S_cc) is defined as the ratio of rotational stiffness provided by the outrigger when k_t and k_d are infinity (k_cl_t²/α) to the core structure’s rotational stiffness (EI/h), and can be expressed as follows:[Image Omitted. See PDF]

The value of S_cc when α equals 0.7 (S_cc07, outrigger effect factor) is used to indicate the magnitude of the outrigger effect. The stiffer perimeter column (larger k_c value) and longer outrigger span (larger l_t value) can enhance the outrigger effect. For taller structures, the EI can significantly increase because of the larger seismic demand. Therefore, the S_cc07 would be smaller for taller structures. In design practice, the k_c should be determined primarily by the gravity load demands, and the l_t determined by the architectural plan. Therefore, the outrigger effect factor S_cc07 also reflects the suitability of adopting an outrigger in certain buildings. The structure with larger S_cc07 value suggests that the efficiency of the mitigating seismic response would be higher when an outrigger system is adopted. In addition to the indexes R_dc and R_bc described in the previous study, the outrigger stiffness ratio (R_oc) is also newly defined as the ratio of outrigger stiffness, k_og, to the perimeter column axial stiffness, k_c, and is expressed as follows:[Image Omitted. See PDF]

R_oc indicates the stiffness provided by the outrigger system. After the perimeter column size is determined, R_oc can provide engineers with a rough estimate of the required BRB sizes and outrigger truss stiffness according to the selected R_oc value.

Analytical models

A total of seven types of analytical models with different structural plans and building heights were used for the parametric study. Figure shows the floor framing plan of the floor with an outrigger for the analytical models. The two outrigger elevations are designed to resist horizontal loads in EW direction. The dead load, which is also the mass source, is 0.8 tonf/m². Table shows the details of the analytical models. The values of EI were selected such that the fundamental vibration periods of the models without an outrigger (Core model) were close to 0.03 h. The R_oc values were fixed at 0.09, 0.45, 0.91, 1.36, 1.82, 2.27, and 2.73. The S_cc07 values were set to be smaller for taller structure models. Table and Figure show the ranges and distributions of the S_cc07 and R_oc values for each analytical model. The S_cc07 values were properly selected to create a dense and uniform distribution, as shown in Figure . Based on the distributions shown in Figure , the analysis results in this study are valid when the S_cc07 and R_oc range from 0 to 4 and from 0 to 3, respectively. For each analytical model, the values of k_c and k_og can be calculated based on the selected S_cc07 and R_oc values from Equations (10) and (11), and the DM model can be constructed accordingly.

Floor framing plan of the analytical model

Parameters of the analytical models

Model	h (m)	EI (kN-m2)	lt (m)	ws (m)	Scc 07	Fundamental period of the core structure (sec)	Roc
16-story	64	4.1 × 109	16	13	3.17-7.36	1.74	0.09-2.73
16-storyB	64	4.1 × 109	14.5	14	2.60-6.02	1.74	0.09-2.73
16-storyC	64	4.1 × 109	12.8	15.3	2.02-4.69	1.74	0.09-2.73
32-story	128	1.6 × 1010	16	13	1.45-3.35	3.50	0.09-2.73
32-storyD	128	1.6 × 1010	13.8	14.5	1.07-2.48	3.50	0.09-2.73
64-story	256	6.5 × 1010	16	13	0.69-1.60	6.92	0.09-2.73
96-story	384	2.2 × 1011	16	13	0.31-0.73	9.76	0.09-2.73

Distribution of Scc07 value with respect to Roc for each analytical model

BRB yield deformation

The method of calculating the BRB yield deformation (u_d,y) has been introduced in the previous study. The force-deformation relationship of the BRB in the analytical model is bilinear with a post-yield stiffness ratio of 0.01. The axial deformation of the BRB in the OB outrigger when the building laterally deforms in the first mode shape until the roof drift reaches θ_r is defined as the u_d,y. Figure shows the distributions of u_d,y with respect to S_cc07 and α when R_oc is 0.5, 1, 2, and 3 for the cases when l_t = 12 m and θ_r = 1/750. The α that results with the largest u_d,y is approximately 0.5-0.7, and is higher when the value of R_oc is smaller. The value of u_d,y changes primarily with α. The stiffer outrigger (larger R_oc value) results in smaller u_d,y values, and the value of S_cc07 has less effect on u_d,y. Under the same θ_r, it appears that the outrigger elevation with a larger u_d,y value is more efficient for utilizing the axial deformation of the BRB in minimizing seismic response. For design practice, the u_d,y is approximately 0.001 of the BRB length. Therefore, the u_d,y shown in Figure can be used to estimate the required BRB length at the preliminary design stage.

Distribution of ud,y with respect to α and Scc07 (unit: mm)

Spectral analysis

The SA procedure is used to evaluate the seismic performance of the BRB-outrigger system. To apply the SA to different outrigger configurations, the SA proposed in the previous study has been modified. As the BRBs in a BT outrigger configuration may not yield simultaneously, this study uses the DM model to perform a modal pushover analysis (MPA) using OpenSees to obtain a more accurate base shear and roof displacement relationship. Figure A shows the MPA result of the i^th mode, where y_top,i is the roof displacement when the first BRB yields, K_i is the elastic modal stiffness, and K_eq,i is the equivalent stiffness when the roof displacement reaches its maximum of y_max,i. The lateral force pattern used in the MPA is assumed to keep the same elastic mode shape even when the BRB yields. The equivalent damping ratio (h_eq,i) of the i^th mode response with a ductility of µ_i is calculated as follows:[Image Omitted. See PDF]where E_d(y) and E_s(y) are the energy dissipated by the BRB-outrigger per loop, and the strain energy with a roof displacement of y (as shown in Figure B), respectively. h₀ (0.02) is the inherent damping ratio. The response spectrum is then adjusted, because of the increased damping ratio, using the reduction factor D_h,i expressed as follows:[Image Omitted. See PDF]

Relationship between (A) base shear and roof displacement obtained from MPA (B) Ed and Es

If S_d(T,h_d) is the spectral displacement at period T and damping ratio h_d, the maximum roof displacement (y′_max,i) of the response of the i^th mode can be estimated as follows:[Image Omitted. See PDF]where T_eq,i is the equivalent vibration period, Γ_i is the i^th modal participation factor, and ϕ_i(h) is the roof displacement in the i^th mode shape. The SA calculation is an iterative procedure and should be continued until the y_max,i used in computing h_eq,i is sufficiently close to the y′_max,i obtained from Equation (14). It is anticipated that the yielding of BRBs only results in a marginal decrease in the stiffness of the entire structure. Thus, it is assumed that the modal superposition principle based on elastic mode shapes remains applicable. The responses of the first four modes are calculated separately and then combined using the square root of the sum of the squares (SRSS) rule. If ψ(x) is the lateral deformation at an elevation of x in the SRSS combined deformed shape, the maximum roof drift (θ_max) and the maximum overturning moment at the core structure base (M_c,max) can be calculated as follows:[Image Omitted. See PDF]

Nonlinear response history analysis

The DM model constructed using OpenSees was also used to perform the NLRHA. The NLRHA was performed using eight ground motions (seven observed and one artificial), as shown in Figure . The spectral accelerations of the ground motions are scaled so that the mean of the spectral accelerations fits the design spectral acceleration within a range of 0.2T₁ to 1.5T₁, where T₁ is the first mode period. A Rayleigh damping ratio of 0.02 for the first and second modes was applied for all NLRHA. The means of the NLRHA results obtained from the eight ground motions are used to verify the SA results.

Response spectra of ground motions adopted in NLRHA

Analysis Results Optimal outrigger elevations

As the BRB-outrigger applies a resisting moment to the core structure, the structure becomes stiffer when the outrigger effect is more significant. Therefore, the decrease in the first mode vibration period of the elastic system when compared with the Core model without an outrigger (D_T1) is used to estimate the effectiveness of the BRB-outrigger. The smaller D_T1 value suggests that the outrigger effect is more significant. Figure A shows the distribution of α when the value of D_T1 is smallest (α_opt,T1) with respect to R_oc and S_cc07. The distribution of D_T1 when the outrigger locates at α_opt,T1 is shown in Figure B. When the value of S_cc07 is larger, the ratio of the perimeter column axial stiffness to the core structure’s flexural rigidity becomes larger. When the value of R_oc is larger, the ratio of the stiffness of the outrigger with BRB to the perimeter column axial stiffness becomes larger. Then, the outrigger effect is more significant as the D_T1 is smaller. The α_opt,T1 ranges from 0.5 to 0.8, and is lower when the values of R_oc and S_cc07 are larger. Based on the analysis results, the relationship between α_opt,T1, R_oc, and S_cc07 can be approximately fitted using a polynomial with least square method as follows:[Image Omitted. See PDF]

(A) αopt,T1 distribution and (B) DT1 distribution, with respect to Scc07 and Roc calculated from SA

Within the valid ranges of S_cc07 and R_oc, the adjusted R-square value of Equation (16) is 0.94. The α_opt,T1 calculated from using Equation (16) is shown in Figure A.

After the BRB yields, the BRB dissipates energy through its hysteretic response. The value of the equivalent damping ratio (h_eq,i) calculated using Equation (12) can be used to identify the energy dissipation efficiency of the BRB-outrigger system. As the first mode dominates the seismic response, the equivalent damping ratio calculated from the first mode (h_eq,1) is used. The larger h_eq,1 value developed by the BRB-outrigger suggests the energy dissipation efficiency is higher. Figure A shows the distribution of the outrigger elevation when the value h_eq,1 is maximum (α_opt,heq1) with respect to S_cc07 and R_oc. The distribution of h_eq,1 when the outrigger locates at α_opt,heq1 is shown in Figure B. The larger R_oc and S_cc07 values impose greater outrigger effect and thus result in a greater h_eq,1 value. The distribution of α_opt,heq1 is similar to α_opt,T1. The α_opt,heq1 ranges from 0.6 to 0.9, and is lower when the values of R_oc and S_cc07 are larger. Based on the analysis results, the relationship between α_opt,heq1, R_oc, and S_cc07 can be fitted using a polynomial with least square method as follows:[Image Omitted. See PDF]

(A) αopt,heq1 distribution and (B) heq,1 distribution, with respect to Scc07 and Roc calculated from SA

Within the valid ranges of S_cc07 and R_oc values, the adjusted R-square value of Equation (17) is 0.96. The α_opt,heq1 calculated from using Equation (17) is shown in Figure A.

Figure shows the distributions of the outrigger elevation when θ_max is minimum (α_opt,θ) with respect to R_oc and S_cc07 calculated from SA and NLRHA. The distributions of θ_max when the outrigger locates at α_opt,θ calculated from SA and NLRHA are shown in Figure . Both the SA and NLRHA results indicate that the larger values of R_oc and S_cc07 result in a smaller θ_max response. The distribution of θ_max calculated from NLRHA and SA is similar. As the NLRHA results are sensitive to different ground motions, the α_opt,θ calculated from NLRHA results does not exhibit a similar distribution to the SA results. However, both the SA and NLRHA results suggest that α_opt,θ ranges approximately from 0.6 to 0.8. α_opt,θ is smaller when the values of R_oc and S_cc07 are larger, which is similar to α_opt,T1 and α_opt,heq1. Based on the SA results, the relationship between α_opt,θ, R_oc, and S_cc07 can be fitted using a polynomial with least square method as follows:[Image Omitted. See PDF]

Distribution of αopt,θ with respect to Scc07 and Roc calculated from (A) SA and (B) NLRHA

Distribution of θmax with respect to Scc07 and Roc calculated from (A) SA and (B) NLRHA

Within the valid ranges of S_cc07 and R_oc values, the adjusted R-square value of Equation (18) is 0.94. The α_opt,θ calculated from using Equation (18) is shown in Figure A.

Figure shows the distributions of the outrigger elevation when M_c,max is minimum (α_opt,Mc) with respect to R_oc and S_cc07 calculated from SA and NLRHA. The distributions of M_c,max when the outrigger locates at α_opt,Mc calculated from SA and NLRHA are shown in Figure . The M_c,max calculated from NLRHA is similar to SA. The M_c,max decreases with increasing S_cc07 and R_oc values. However, the M_c,max stops decreasing when S_cc07 is greater than around 1-2 and when R_oc is greater than 1. The SA results indicate that α_opt,Mc drops to around 0.2 when R_oc and S_cc07 are greater than approximately 0.5. However, the NLRHA results suggest that α_opt,Mc drops to around 0.3 when R_oc and S_cc07 are greater than around 2 and 3, respectively. The differences between the SA and NLRHA results could be due to the SA calculation of M_c,max being based on elastic mode shape and linearly elastic force-deformation relation, and the NLRHA results could be sensitive to different ground motions. However, both the SA and NLRHA show the trend that the α_opt,Mc would be lower when the values of S_cc07 and R_oc are larger. Based on the SA results, the relationship between the α_opt,Mc, R_oc, and S_cc07 can be fitted using a polynomial with least square method as follows:[Image Omitted. See PDF]

Distribution of αopt,Mc with respect to Scc07 and Roc calculated from (A) SA and (B) NLRHA

Distribution of Mc,max with respect to Scc07 and Roc calculated from (A) SA and (B) NLRHA

Within the valid ranges of S_cc07 and R_oc values, the adjusted R-square value of Equation (19) is 0.76. The α_opt,Mc calculated from using Equation (19) is shown in Figure A. It should be noted that the polynomials shown in Equations (16)-(19) are only used to demonstrate the relationships between optimal outrigger elevations in order to achieve different optimal responses, which is introduced in the next section. Based on the analysis results, the ranges of α_opt,T1, α_opt,heq1, and α_opt,θ are similar (0.6-0.8), but the range of α_opt,Mc (0.2-0.8) is much different from the others. This is because that the BRB-outrigger applies a resisting moment to the core structure to reduce seismic response. The core structure can result in greater rotational demand on BRB-outrigger at higher elevation, however, the axial stiffness provided from the perimeter column is smaller. Therefore, the BRB-outrigger elevation of 0.6-0.8 should be the best elevation to result in largest outrigger effect and to reduce θ_max response. On the other hand, as the maximum bending moment of core structure develops at the foundation, if the resisting moment applied by BRB-outrigger is closer to the core structure base, it is more efficient in reducing M_c,max response.

Effects of outrigger effect factor and outrigger stiffness ratio

Figure shows the reductions in θ_max (D_θ) when compared with the Core model when R_oc is 0.5, 1, 2, and 3 calculated from SA and NLRHA. The SA and NLRHA results are similar and indicate that the α that best reduces θ_max is approximately between 0.6 and 0.8, and is lower when the value of R_oc is larger (stronger outrigger effect). Under a fixed R_oc value, the larger S_cc07 value leads to greater reductions in θ_max when α is higher than 0.5. This suggests that when α is lower than 0.5, θ_max cannot be effectively reduced by increasing the S_cc07 value. The dashed lines in Figure show α_opt,T1, α_opt,heq1, α_opt,θ, and α_opt,Mc calculated from Equations (16)-(19). The similar distributions of α_opt,T1, α_opt,heq1, and α_opt,θ indicate that the optimal α to maximize outrigger effect, equivalent damping ratio, and to minimize θ_max is approximately 0.7-0.9. Furthermore, the α_opt,T1, α_opt,heq1, and α_opt,θ distributions calculated from SA well match the D_θ distributions calculated from both SA and NLRHA. Figure shows the reductions in M_c,max (D_Mc) when compared with the Core model when R_oc is 0.5, 1, 2, and 3. Both the SA and NLRHA suggest that the larger S_cc07 value results in greater reduction in M_c,max. The α that results in the greatest reduction in M_c,max calculated from SA is lower than the one calculated from NLRHA. However, the α_opt,Mc can still provide satisfactory outrigger elevation to reduce the M_c,max response, according to the NLRHA results. Furthermore, as shown in Figure B, the influence on D_Mc because of changes in α is insignificant when α is greater than 0.4. Therefore, the α selected from α_opt,θ, α_opt,heq1, and α_opt,T1 distributions can also provide satisfactory D_Mc response.

Dθ distribution chart calculated from (A) SA and (B) NLRHA

DMc distribution chart calculated from (A) SA and (B) NLRHA

Based on the analysis results, both the D_θ and D_Mc responses indicate that the larger R_oc value offers a stiffer outrigger (stronger outrigger effect) and results in a smaller seismic response. However, the reductions in seismic responses are not proportional to the increasing R_oc. For example, when comparing the cases where R_oc increases from 1 to 2, the required outrigger stiffness k_og is doubled, but, the D_θ and D_Mc are increased by approximately 5% only. Therefore, to efficiently reduce seismic response utilizing a BRB-outrigger, selecting α approximately between 0.6 and 0.8, and an S_cc07 value greater than 1 would be more efficient than increasing R_oc. In summary, based on the analysis results, to efficiently mitigate the seismic response, the optimal α is around 0.6-0.8, and the recommended values of S_cc07 and R_oc are greater than 1 and around 0.5 and 1, respectively. Those recommended design parameters in order to minimize seismic responses are similar to the previous study, in which the optimal design parameters were verified by performing a series of NLRHA. Figures and can be used as design charts to assist the designer in selecting the outrigger elevation and outrigger stiffness ratio for achieving the desired seismic response at the preliminary design stage. The examples with different outrigger configurations designed by utilizing the design charts are introduced in the following sections.

Design recommendation

The optimal values of α and R_oc in order to minimize seismic response are studied. Figure shows the recommended design flow chart. For design practice, the building lateral stiffness (core structure flexural rigidity, EI) should be mainly determined based on the code specifications. The perimeter column sizes (k_c) should be determined according to the floor framing plan and gravity load demands. The recommended design procedure is as follows,

1. If α is not restricted for architectural reasons, select α between 0.6 and 0.8, and calculate the S_cc07 value.
2. Target R_oc value as large as possible within the range of 0.5 and 1.
3. Based on the selected R_oc, calculate the k_og and design the BRB and outrigger members. If the k_og is too large to design the BRB or outrigger members, select suitable BRB and outrigger member sizes and update the corresponding k_og and R_oc values.
4. Determine the BRB yield deformation (for example, 0.001 of the BRB element length). Then, perform the first mode MPA.
5. Confirm if the roof drift when BRB yields (θ_r) is within suitable range. If θ_r is too large (θ_r > 1/300), decrease k_og. If θ_r is too small (θ_r < 1/800), increase k_og.
6. After all the parameters are determined, perform the analysis and proceed to member design. As the outrigger effect results in additional force demands on the perimeter column, the perimeter column axial force demand should include the maximum BRB axial force capacity.

Flow chart of design recommendation

Effects of BRB-Outrigger Configurations Introduction of example models

A 40-story model (h = 160 m) is used to demonstrate design examples for the structures with OB, BT, and GB outriggers. The structural plan is shown in Figure with l_t and w_s equal to 12 m and 5 m, respectively. The core structure span is 5 m. The dead (mass source) and live loads are 0.8 tonf/m² and 0.3 tonf/m², respectively. The core structure flexural rigidity (EI) is 4 × 10⁹ kN-m². Based on the dead and live load demands in the first story, the perimeter column with size of Box 900 × 900 × 75 mm, made from SN490 grade steel (material yield stress = 325MPa), was designed. The compressive demand-to-capacity ratio (DCR) of the perimeter column in the first story is 0.35. Therefore, the value of k_c is 309 375 kN/m, and the outrigger effect factor S_cc07 of 2.55 can be calculated from Equation (10). The design details and the seismic performance of each configuration are introduced in the following sections. Based on the analysis results in previous sections, the value of R_oc is set to be approximately 0.45, and the α is set at 0.7. Therefore, the required outrigger stiffness (k_og) is approximately 139 219 kN/m, and the required u_d,y is approximately 16-18 mm. As shown in Figures A and A, the case of R_oc = 0.45 and S_cc07 = 2.55 when α = 0.7 suggests that the reductions in θ_max and M_c,max are approximately 39% and 23%, respectively. With this condition, the seismic performance of structures with different BRB-outrigger configurations of OB, BT, and GB is compared.

Figure A shows the design details of the OB outrigger. The top and bottom chords of the outrigger truss locate at the 28th and 27th floors, respectively. The two ends of the brace and column members in the outrigger truss are designed with moment connection detail. The connections between the top and bottom chords to the core structure are rigid connections. The top chord end near the perimeter column connects with the BRB, which is arranged vertically, with a length of 8 m. The bottom end of the BRB connects to the perimeter column at the 26th floor. Both two ends of the BRB are pinned connections. The 26th-floor beam is spliced adjacent to the lower BRB end. The value of k_t is 187 987 kN/m, which is calculated using OpenSees. Table shows details of the BRB design in the OB outrigger configuration (BRB_OB). Figure B shows MBM model of OB outrigger. The members in the outrigger truss are modeled using beam column elements. The BRB_OB, which is modeled using a truss element, connects to the outrigger truss end at Node A and to the perimeter column at Node C. The perimeter column at the 28th floor level is separated at Node B, which shares the same coordinates with Node A but moves independently of Node A. The bilinear material model with a post-yield stiffness ratio of 0.01 is used for all the outrigger truss and BRB members in order to confirm if the structural members deform inelastically.

Design details of the BRB in each outrigger configuration

[Table omitted. See PDF]

(A) Design detail and (B) MBM model of the OB outrigger

Figure A shows the details of the BT outrigger. The top and bottom chords with both ends of shear connection detail locate at the 28th and the 27th floors, respectively. The member size of the top and bottom chords are the same as those of the OB outrigger (BH 700 × 500 × 50 × 70 mm), but the column size in the outrigger truss is designed to be smaller (RH 700 × 300 × 13 × 24 mm). The two ends of the outrigger truss columns are designed with moment connection detail. Four identical BRBs (BRB_BT), are arranged along the BT outrigger with an equal span of 3 m, as shown in Figure A. The design details of the BRB_BT are presented in Table . Figure B shows the BT outrigger in the MBM model. The top and bottom chords are modeled using beam column elements. The ends at Node A, B, C, and D are free to rotate about the out-of-plane direction using the equalDOF command in OpenSees. The ideal inelastic behavior of the BT outrigger is to let the BRB yield first and then dissipate majority of the input seismic energy. Slight inelastic deformations in the top and bottom chords and the columns in the BT outrigger would be permitted. The bilinear material model with a post-yield stiffness ratio of 0.01 is used for all the elements in the OB outrigger in order to confirm if the structural members deform inelastically.

(A) Design detail and (B) MBM model of the BT outrigger

Figure A shows the details of the GB outrigger. The top and bottom ends of the BRB (BRB_GB) connect to the core structure at the 28th floor and to the perimeter column at the 26th floor, respectively. Both BRB_GB ends are pinned connections. The floor beams at the 26th and 28th floors are spliced adjacent to the BRB connections. The details of the BRB_GB design are presented in Table . As the floor beam at the 26th floor is required to sustain the maximum axial force developed in the BRB_GB and has to be stiff enough to prevent excessive axial deformation, the axial force demand for the 26th floor beam is estimated as 15 861 kN (N_y × 1.1 × 1.3 × 1.15 × cosη, where N_y is the axial yield force of the BRB_GB, and the perimeters 1.1, 1.3, and 1.15 are the factors accounting for material overstrength, strain hardening, and compression strength adjustment, respectively). It is assumed that sufficient lateral support is provided on the 26th-floor beam. Therefore, size BH 820 × 400 × 22 × 50 mm, made from SN490 grade steel, can be designed with a compression DCR of 0.97. The axial stiffness of the 26th-floor beam is 2.6 times the axial stiffness of the BRB_GB in a horizontal direction. Thus, it should be stiff enough to prevent excessive axial deformation. Figure B shows the GB outrigger in the MBM model; the 27th-floor beam is not included. Both the BRB and floor beams are modeled using truss elements. The bilinear material model with a post-yield stiffness ratio of 0.01 was used for the BRB member.

(A) Design detail and (B) MBM model of the GB outrigger

Seismic response of the example models

Figure A shows the relationship between the vertical force applied at the outrigger end (P_v) and the corresponding vertical deformation (u_v) in each outrigger configuration calculated from the vertical pushover analysis using OpenSees. Figure B illustrates the vertical pushover analysis of each configuration. Based on the analysis results, the elastic outrigger stiffness (k_og) of the OB, BT, and GB outriggers is 139 344, 132 384, and 159 077 kN/m, respectively. The vertical deformation when BRB yields are 46, 51, and 40 mm for the OB, BT, and GB outriggers, respectively. Figures A and C show the sequence of BRB yielding and the flexural plastic hinges forming in the outrigger truss columns. BRB1 and BRB4 (Figure C) yield first when vertical deformation (u_v) reaches 51 mm (0.4% rad. deflection). BRB2 and BRB3 yield when u_v reaches 90 mm (0.8% rad. deflection). The flexural plastic hinges form at the two ends of outrigger truss columns when u_v reaches 120 and 159 mm (1% and 1.3% rad. deflection), respectively. The post-yield stiffness of the OB outrigger is slightly larger than that of the others. However, as the elastic stiffness and the yield deformation of the OB, BT, and GB outriggers are similar, it is anticipated that the structure with the different outrigger configurations would exhibit similar seismic response. Table shows the parameters used in the DM model for the structure with different outrigger configurations. The value of 9 × 10⁹ is used for infinity k_t in the DM model for BT and GB outrigger configurations.

(A) Vertical force and vertical deformation relationship. (B) Illustrations for performing vertical pushover analysis for the OB, BT, and GB outriggers. (C) Illustration of plastic hinge locations of the BT outrigger

Parameters used in DM model for the design example with different outrigger configurations

Configuration	kog (kN/m)	kt (kN/m)	kd (kN/m)	ud,y (mm)	P
OB outrigger	139 344	187 987	538 518	11.8	0.01
BT outrigger	132 384	∞	132 384	51.0	0.01
GB outrigger	159 077	∞	159 077	40.4	0.01

Table shows the first four mode vibration periods calculated from using DM and MBM models. The vibration periods calculated using the DM model are slightly larger than those from using the MBM model. Figure shows the roof drift histories of the structures with OB, BT, and GB outriggers under BCJ-L2 ground motion (scale factor = 1). The roof drift history results obtained using DM and MBM models are close to each other. Furthermore, the roof drift responses between the three structures with different outrigger configurations are similar. The differences between the analysis results calculated using DM and MBM models could be due to the fact that the span of the core structure is not included in the DM model, the cantilever column in the DM model could not perfectly resemble the braced core structure in the MBM model, and the different distribution of mass along the building height in the DM and MBM model. The modal analysis and roof drift history results suggest that the DM model with modified parameters shown in Table can be used to model the structure with BT and GB outrigger configurations. Furthermore, the structures with different BRB-outrigger configurations but sharing the same S_cc07 and R_oc values exhibit very close seismic response. This indicates that the proposed indexes (S_cc07 and R_oc) can effectively reflect seismic performance for a structure when any one of the OB, BT, or GB outrigger configurations is adopted.

Vibration periods calculated using DM and MBM models for the design example with different outrigger configurations

Configuration	Model	Vibration period (sec)
		1st mode	2nd mode	3rd mode	4th mode
OB outrigger	DM	4.383	0.840	0.316	0.160
	MBM	4.252	0.880	0.344	0.178
BT outrigger	DM	4.410	0.841	0.316	0.160
	MBM	4.390	0.890	0.344	0.178
GB outrigger	DM	4.312	0.835	0.316	0.160
	MBM	4.184	0.880	0.344	0.178

Roof drift history of models with OB, BT, and GB outrigger calculated using DM and MBM models

Figures and show the maximum roof drift (θ_max) and the maximum overturning moment at the core structure base (M_c,max) calculated from NLRHA with the original observed ground motions using DM and MBM models, respectively. Figure shows the cumulative plastic deformation ratio (R_CPD) for the BRB in each outrigger configuration calculated from NLRHA using the MBM model. The locations of the four BRBs in the BT outrigger are shown in Figure A. The zero values of R_CPD indicate that the BRB deforms elastically. The analysis results show that θ_max and M_c,max responses between the structures with OB, BT, and GB outrigger configurations are only slightly different. The model without an outrigger (Core model) generally exhibits greater θ_max and M_c,max than the models with a BRB-outrigger. However, under Tohoku, El Centro, Taft, and Kumamoto ground motions, the BRB-outrigger only slightly improves the seismic response when compared with the Core model. This is because the increased stiffness resulting from the outrigger effect might increase seismic demand, and the BRB deforms elastically (R_CPD = 0) or exhibits only slight inelastic deformation (low R_CPD values) and thus results in a low energy dissipation efficiency. Figure shows the percentages of energy dissipated by the BRB (E_BRB) to the total input energy. The reductions in the θ_max and M_c,max are greater when the value of E_BRB is larger. The reductions in θ_max (the average of OB, BT, and GB outriggers), when compared to the Core model are approximately 27% and 50% under ChiChi and BCJ-L2 ground motions, respectively. Furthermore, the reductions in M_c,max (the average of OB, BT, and GB outriggers) when compared with the Core model are approximately 40% and 30% under ChiChi and BCJ-L2 ground motions, respectively. According to Figures and , the SA results indicate that the reductions in θ_max and M_c,max are around 39% and 23%, respectively. This suggests that the SA could provide appropriate estimations of the seismic response if the BRBs develop sufficient hysteretic responses. As shown in Figure , the R_CPD values also indicate the ductility demand for the BRB. The BRB_OB exhibits the largest R_CPD value as the vertical BRB arrangement imposes a large amount of axial deformation demand on the BRB_OB. The R_CPD values of the BRB_GB are smaller than those of the BRB_OB. However, the seismic response and the E_BRB of the models with OB and GB outrigger configurations are similar. This suggests that the GB outrigger configuration could be a better alternative configuration for preventing excessive ductility demand on the BRB, as a BRB with too large an R_CPD value could easily fracture before the end of an earthquake. For the BT outrigger, because of the outrigger arrangement, the ductility demands for the BRBs near the two outrigger truss ends (BRB_BT(1) and BRB_BT(4)) are greater than that for the BRBs in the mid-span of the outrigger truss (BRB_BT(2) and BRB_BT(3)). In design practice, the sizes of the BRB in the BT outrigger configuration could be properly adjusted to reduce steel usage. For instance, BRB_BT(2) and BRB_BT(3) with low R_CPD values in the design example could be replaced with ordinary elastic steel braces.

θmax responses calculated from NLRHA with the original observed ground motions

Mc,max responses calculated from NLRHA with the original observed ground motions

The RCPD calculated from NLRHA with the original observed ground motions

The EBRB calculated from NLRHA with the original observed ground motions

Comparison between OB, BT, and GB outrigger configurations

Table shows the steel usage of the OB, BT, and GB outriggers (single span within the core structure and perimeter column) for the design example models. The weight of the 26th-floor beam in the GB outrigger, which is designed to sustain the maximum force developed by the BRB, is included. The OB outrigger consumes the most steel. Although the force demands on the OB outrigger truss members can be effectively limited by the maximum force capacity of the BRB_OB, the OB outrigger truss member sizes can be determined to create sufficient k_t value, instead of being determined by force demands. Furthermore, when the outrigger truss span (l_t) becomes longer, the required outrigger truss member size must be sharply increased. Therefore, the OB outrigger configuration would be suitable only when the outrigger truss span is short. However, the OB outrigger truss can be designed to occupy only one story; and only one BRB is required. The BT outrigger configuration requires more than one BRBs. Slight plastic deformations are allowed in the BT outrigger truss members to prevent too large a member size. Therefore, the design of the BT outrigger configuration is more flexible but more complicated than the OB outrigger configuration. Furthermore, as the BRB and outrigger truss act in parallel, the post-yield stiffness could be properly adjusted by selecting different sizes of the outrigger truss member. The larger post-yield stiffness ratio would be beneficial, as it avoids sudden stiffness drops, as seen in the OB and GB outrigger configurations when the BRB yields. The GB outrigger consumes the least amount of steel. However, the GB outrigger requires a very long BRB when the l_t is long, and the BRB_GB may be required to span more than two story heights to generate sufficient outrigger stiffness (k_og), which could reduce usable floor area. Based on the analysis results, all three outrigger configurations can achieve satisfactory seismic response and could be selected by designers to fit individual architectural requirements using the proposed indexes and design charts.

Steel usage for the OB, BT, and GB outriggers with single outrigger span

Configuration	Outrigger truss (tonf)	BRB (tonf)	Total (tonf)
OB outrigger	43.6	3.8	47.4
BT outrigger	20.6	11.2 (2.8 tonf for each BRB_BT)	31.8
GB outrigger	5.3 (including the 12 m-long floor beam)	9.2	14.5

Conclusions

This study investigated the seismic performance of structures incorporating the BRB-outrigger as a lateral force resisting system using SA and NLRHA procedures. Three different BRB-outrigger configurations were proposed with common design indexes and charts. Based on the analysis results using the proposed simplified models, the conclusions of this study are as follows:

1. The outrigger elevation strongly affects seismic response. The ranges of outrigger elevation (α) in order to enhance outrigger effect, to increase equivalent damping ratio, and to reduce θ_max response are 0.5-0.8, 0.6-0.9, and 0.6-0.8, respectively. The optimal outrigger elevation (α) should be approximately 0.6-0.8. Based on the NLRHA results, the maximum overturning moment at the core structure base can be efficiently reduced when the outrigger locates at its optimal elevation.
2. The outrigger effect factor S_cc07 can be used to indicate the efficiency of utilizing the BRB-outrigger as a seismic resistance system to improve seismic performance. A large S_cc07 value suggests the efficiency of mitigating the seismic response of the BRB-outrigger is higher. Based on the analysis results, the recommended value of S_cc07 should be larger than 1.0.
3. The outrigger stiffness ratio R_oc indicates the required outrigger stiffness. A large R_oc value leads to better reductions in the seismic response but also increases the cost. According to the analysis results, when the value of S_cc07 is greater than 1.0, selecting an outrigger elevation α of 0.6-0.8 should be the first priority. The R_oc is then selected to fit the desired seismic response.
4. When the outrigger locates at approximately α = 0.6 to 0.8, and when the values of S_cc07 and R_oc are greater than 1.0 and 0.5, respectively, both maximum roof drift ratio and maximum overturning moment at the core structure base can be reduced by approximately 20%-30%, if the BRBs develop full hysteretic responses.
5. Three different BRB-outrigger configurations were designed and calibrated using the proposed indexes and charts, and their seismic performance was investigated in this study. According to the analysis results, the proposed indexes (S_cc07 and R_oc) can effectively reflect the seismic performance of a structure with a BRB-outrigger when any one of the OB, BT, and GB outrigger configurations is adopted.
6. From the viewpoint of BRB design, the OB and GB outrigger configurations are suitable when the outrigger span is short. The OB and BT outrigger configurations utilize the building interior space best. The GB outrigger could be the most economical solution as outrigger truss members are not necessary. All three BRB-outrigger configurations are capable of achieving the desired seismic response. Designers can select suitable BRB-outrigger configurations to fulfill both architectural requirements and economical design using the proposed design charts and indexes.

Disclosure

The authors have no conflicts of interest to declare.

AppendixList of symbols

D_h,i: reduction factor for response spectrum

D_Mc: drop of the M_c,max when compared with the Core model without outrigger

D_T1: drop of the first mode vibration period of the elastic system when compared with the Core model without outrigger

D_θ: drop of the θ_max when compared with the Core model without outrigger

E_d: energy dissipated by the BRB-outrigger per loop

EI: core structure flexural rigidity

E_s: strain energy of the system

h: building height

h₀: inherent damping ratio

h_eq,i: equivalent damping of the i^th mode response

h_eq,i: equivalent damping ratio of the i^th mode response

h_t: vertical span of the BRB in the GB outrigger configuration

k_bt: flexural stiffness of the BT outrigger

k_c: axial stiffness of perimeter column with length of h.

k_d,gb: axial stiffness of the BRB in the GB outrigger configuration

k_d: axial stiffness of the BRB in the OB outrigger configuration

K_eq,i: equivalent stiffness when the roof displacement reaches its maximum of y_max,i

K_i: elastic modal stiffness of the i^th mode

k_og,BT: flexural stiffness combined with outrigger truss and BRB in the BT outrigger configuration

k_og,GB: flexural stiffness provided by the GB outrigger

k_og,OB: flexural stiffness combined with outrigger truss and BRB in the OB outrigger configuration

k_rg,BT: rotational stiffness provided by the BT outrigger system

k_rg,GB: rotational stiffness provided by the GB outrigger system

k_rg,OB: rotational stiffness provided by the OB outrigger system

k_t: flexural stiffness of outrigger truss in the OB outrigger configuration

l_t: outrigger span

M_c,max: maximum overturning moment at core structure base

M_o,BT: moment applied by the BT outrigger until the core structure rotation at outrigger elevation reaches θ₁

M_o,GB: moment applied by the GB outrigger until the core structure rotation at outrigger elevation reaches θ₁

M_o,OB: moment applied by the OB outrigger until the core structure rotation at outrigger elevation reaches θ₁

p: post-yield stiffness ratio of BRB in OB outrigger configuration

P_gb: axial force of the BRB in the GB outrigger configuration

S_cc: outrigger effect when α varies from 0 to 1

S_cc07: outrigger effect factor (when α = 0.7)

S_d: spectral displacement

T_eq,i: equivalent vibration period of the i^th mode

u_bt,y: flexural deformation of the BT outrigger when BRB yields

u_bt: flexural deformation of the BT outrigger

u_c,BT: axial deformation of perimeter column below outrigger elevation in the BT outrigger configuration

u_c,GB: axial deformation of perimeter column below outrigger elevation in the GB outrigger configuration

u_c: axial deformation of perimeter column below outrigger elevation in the OB outrigger configuration

u_d,gb,v: vertical component of the axial deformation of the BRB in the GB outrigger configuration

u_d,y: axial yield deformation of BRB in the OB outrigger configuration

u_d,ygb: axial yield deformation of BRB in the GB outrigger system

u_d: axial deformation of the BRB in the OB outrigger configuration

u_t: flexural deformation of the outrigger truss in the OB outrigger configuration

y_max,i: maximum roof lateral displacement

y_top,i: roof lateral displacement when BRB yields in the i^th mode shape

Γ_i: modal participation factor of the i^th mode

T₁: the first mode vibration period

α: ratio of outrigger elevation to building height

α_opt,heq1: optimal outrigger elevation for maximizing the first mode equivalent damping ratio

α_opt,Mc: optimal outrigger elevation for minimizing M_c,max

α_opt,T1: optimal outrigger elevation for maximizing outrigger effect

α_opt,θ: optimal outrigger elevation for minimizing θ_max

ϕ_i: mode shape of the i^th mode

η: inclined angle of the BRB in the GB outrigger configuration

µ_i: ductility ratio of the i^th mode response

θ₁: core structure rotation at the height of outrigger elevation

θ_max: maximum roof drift ratio

θ_r: roof drift ratio of core structure when BRB yields

θ_y: core structure rotation at the height of outrigger elevation when BRB yields

ψ: SRSS combined deformed shape