A new stiffness-strength-relationship-based design approach for global buckling prevention of buckling-restrained braces

Abstract

This paper proposes a new stiffness-strength-relationship-based design approach that can pinpoint the target design solution for steel buckling-restrained braces (BRB). First, a stiffness–strength requirement interaction curve (the design criterion) with a very simple and easy-to-use form is derived based on a second-order analysis. This interaction curve clearly illustrates the opposing stiffness and strength requirements of the restraining system. Second, based on the geometrical parameters and material properties, a stiffness–strength relationship curve of the BRB restraining system is established. This second relationship curve is expressed by a linear function for a uniform steel BRB. By using the two analytical curves, the point of intersection defines the target design point. A straightforward design procedure for steel BRBs is then developed. A design example of steel BRBs is considered to demonstrate this easy-to-use design procedure for obtaining economical BRB designs. The design is verified and discussed by a rigorous finite element analysis.

Keywords

buckling-restrained brace (BRB)design restraining system stability stiffness–strength relationship

Introduction

Buckling-restrained braces (BRBs) have been widely used as lateral load resistant and energy-dissipating members for building structures (Takeuchi and Wada, 2017). A BRB is composed of a steel core and an external restraining member. When a BRB is axially loaded, the axial force is fully resisted by the core and the restraining member is designed to restrain lateral deformation of the core and to prevent its global buckling. Therefore, if a restraining member is well designed with adequate flexural rigidity, the BRB can exhibit axial yielding under both tensile and compressive loads, thus showing stable hysteretic performance under earthquake effects (Architectural Institute of Japan [AIJ], 2009; Kim and Choi, 2004; Takeuchi, 2018; Wada and Nakashima, 2004).

In recent years, researchers proposed steel BRBs with all-steel configurations to reduce self-weight and to achieve more economical design over conventional BRBs, which are encased by concrete-filled steel tubes (CFSTs) (Della Corte et al., 2015; Hosseinzadeh and Mohebi, 2016; Hoveidae and Rafezy, 2013; Mirtaheri et al., 2011; Momenzadeh et al., 2017), as shown in Figure 1. Since the steel core and restraining member is directly assembled for steel BRBs and no concreting process is involved, a more convenient fabrication procedure is achieved and the labor cost during fabrication is greatly reduced. Therefore, the steel BRBs have been more widely used over the past few years.

Figure 1.

Sectional view of typical types of steel BRBs.

To obtain safe design of a BRB, its restraining member should be capable of providing adequate lateral support for the steel core, such that global buckling failure of the BRB is prevented. The design criterion for global buckling prevention of BRBs has drawn the attention of numerous researchers (Black et al., 2004; Chou and Chen, 2010; Guo et al., 2017b; Tong and Guo, 2017; Tong et al., 2019). One of the most popular design approaches for global buckling prevention of BRBs was proposed by Fujimoto et al. (1988, 1990), called “restraining ratio” approach. This approach is based on a key parameter referred to as the restraining ratio ζ which is defined as

ζ = \frac{P_{cr}}{P_{y}}

(1)

where P_cr is the elastic buckling load of the restraining system, and P_y the axial yield load of the core.

In the “restraining ratio” approach, the criterion of preventing BRBs from global buckling failure is described by ζ ≥ ζ_req, where ζ_req is the required restraining ratio. This required restraining ratio should be defined for different BRBs based on analytical derivations or numerical/experimental investigations. For steel BRBs, the required restraining ratio could be obtained according to a second-order analytical model, which was given as (AIJ, 2009; Iwata et al., 2000; Tong and Guo, 2018)

ζ_{req} = \frac{η M_{r}}{M_{r} - η P_{y} v_{0}}

(2)

where η = P_max/P_y is the enhancement factor of the core material, and $η = β ω$ where β is the tension strength adjustment factor of the core and ω the compression strength adjustment factor of the core. P_max is the maximum compressive load, P_y the initial yield load of the steel core, M_r the bending moment resistance (i.e. flexural yield strength) of the restraining system, and v₀ the amplitude of the geometrical initial imperfection of the restraining system.

In structural stability designs, the strength requirement may play an equally important role as the stiffness requirement. For instance, in stability bracing designs, the consideration of both stiffness and strength requirements proposed by Winter (2006) has been widely accepted, for example, (Pan et al., 2017, 2018). Equation (2) for calculating the restraining ratio requirement of a BRB has taken both the strength and stiffness requirements of the restraining member into consideration, however, the equally important roles of these requirements were not adequately addressed by previous researchers. Hence, “restraining ratio” design approach using constant restraining ratio requirements was more widely used in practical designs of BRBs. For instance, Wada and Nakashima (2004) and Takeuchi (2018) recommended the design criterion expressed by equation (3), which is applicable for BRB with an initial imperfection v₀/L ≤ 1/500 and a relatively slender restraining member with L/D_r ≥ 20.

ζ = \frac{P_{cr}}{P_{y}} \geq 1.5 η

(3)

The required restraining ratio is a variable depending on the geometrical dimensions, cross-sectional properties, and material properties of the BRB. For different BRBs, their required restraining ratios could be considerably different. This design approach that adopts a constant required restraining ratio is very convenient for engineers to use, but it lacks a theoretical basis. Therefore, with a focus on the stiffness requirement of the restraining system, the “restraining ratio” approach considering constant restraining ratio requirement may lead to uneconomical designs of BRBs.

This paper proposes a new form of the global buckling design criterion to provide clarity on the equal importance of the stiffness and strength requirements of the restraining member. On the basis of this new criterion, a stiffness-strength-relationship based design approach is developed to pinpoint an economical target design solution for steel BRBs. Then, a design procedure is developed and a design example is given to illustrate this procedure. The design result is compared with that obtained from conventional design approach to reveal the economical efficiency of the proposed approach. Finally, rigorous finite element nonlinear analysis, which considers both material and geometrical nonlinearities, is conducted to examine and further discuss the design results.

Stiffness-strength-relationship-based design approach

Review of design criterion of BRBs based on second-order analysis

The second-order analysis and the yield criterion of the outermost steel fiber in the restraining member are the basis for establishing a design criterion of steel BRBs (Takeuchi and Wada, 2017). In a BRB with an inner steel core restrained by an external restraining system, the load transferring mechanism is as follows. The axial compressive load applied to the inner core leads to a lateral deformation of the core. This deformation produces the lateral extruding forces between the core and the restraining system. If the restraining system can provide adequate lateral supports to the core, the lateral deformations associated with global buckling mode of the core could be prevented. Thus, the core could achieve a full cross-sectional yielding under a compressive load.

To establish the design requirement, a second-order analysis is conducted, as shown in Figure 2. The BRB is assumed to have an initial imperfection v₀ [with an assumed trigonometric imperfection v₀sin(πx/L)] and is subjected to an axial compressive load P, which is given by

P = β ω P_{y} = η P_{y}

(4)

where ω is the tension strength adjustment factor of the core, β the compression strength adjustment factor of the core (American Institute of Steel Construction [AISC], 2010), and η the enhancement factor of the core.

Figure 2.

BRB model for second-order analysis.

As shown in Figure 2, a commonly-used BRB with a uniform cross-section along its length is analyzed. The BRB exhibits a second-order effect under the axial compressive load P. The load effect of the bending moment at the mid-span cross-section is given by an amplification 1/(1 − P/P_cr) from the initial load effect Pv₀ as (Timoshenko and Gere, 1961)

M_{mid} = Pv = P v_{0} \cdot \frac{1}{1 - P / P_{cr}}

(5)

In view of equation (5), the buckling load of the BRB is conservatively taken as the elastic buckling load P_cr of the restraining system (Guo et al., 2017b). This is because the steel core is designed to exhibit full cross-sectional yielding, and hence it may have a significant stiffness degradation. Therefore, the contribution of the steel core to the elastic buckling load of the BRB is found to be negligible (Takeuchi and Wada, 2017). It is worth noting that equation (5) is suitable for calculating the mid-span bending moment of a BRB with either fixed-ended or pinned-ended cores. The boundary conditions for the inner core do not influence the analytical result in equation (5) since most of the bending resistance is handled by the restraining member. For the restraining member, the ends are always pinned (Takeuchi and Wada, 2017).

By adopting the strength criterion, the design requirement of the restraining system for prevention of the global buckling of BRBs may be expressed as (Takeuchi and Wada, 2017)

P v_{0} \cdot \frac{1}{1 - P / P_{cr}} \leq M_{r}

(6)

where M_r is the bending moment resistance of the BRB at its mid-span cross-section. M_r is conservatively considered as the bending moment resistance of the external restraining system only.

Based on the yield criterion of the outermost steel fiber, the bending moment resistance M_r is formulated as

M_{r} = f_{yr} \frac{I}{z}

(7)

where f_yr is the yield stress of the restraining member, I the cross-sectional moment of inertia of the restraining member, and z the distance from the neutral axis to the outermost fiber of the restraining member.

Establishment of stiffness–strength requirement interaction curve of restraining system

In departure from conventional approach, this section establishes and discusses a very simple and easy-to-use stiffness-strength-relationship-based expression for the design requirement of the BRB restraining system. The design requirement of the restraining system derived in equation (6) may be rearranged in a symmetric form as

(\frac{P_{cr}}{P} - 1) (\frac{M_{r}}{v_{0} P} - 1) \geq 1

(8)

With the buckling load P_cr as the numerator, the term P_cr/P (a similar form as the restraining ratio ζ = P_cr/P_y) can be viewed as a stiffness requirement of the restraining system. With the bending moment resistance M_r as the numerator, the term M_r/(v₀P) can be viewed as a strength requirement. Therefore, equation (8) gives the interaction between stiffness and strength requirements of the restraining system.

The derivation of equation (8) is based on the classical second-order analysis of columns which has been well accepted. It now forms the basis for existing design criteria for axial-flexural members (e.g. columns, beam–columns, BRBs). In addition, based on the examination of yielding in the outermost steel fiber, the design equation (7) can be directly recommended for the global buckling prevention of steel BRBs, see for example in references (Takeuchi, 2018; Takeuchi and Wada, 2017; Wada and Nakashima, 2004).

Based on equation (8), the interaction between stiffness and strength requirements of the restraining system is plotted in Figure 3. The dark shaded area shows the safe region that satisfies the design requirements. Figure 3 shows that the strength requirement M_r/(v₀P) decreases with respect to increasing stiffness requirement P_cr/P. Therefore, the stiffness and strength requirements of the restraining system have an interaction of negative correlation.

Figure 3.

Stiffness–strength requirement interaction curve and safe region for BRB design.

The boundary curve of the safe region is defined by

(\frac{P_{cr}}{P} - 1) (\frac{M_{r}}{v_{0} P} - 1) = 1

(9)

Equation (9) specifies the minimum stiffness and strength requirements of the restraining system. This boundary curve is defined as the stiffness–strength requirement interaction curve of BRBs. It is shown that the boundary curve is in a form of inverse proportional function with two asymptotic lines P_cr/P = 1 and M_r/(v₀P) = 1.

It is noted that both multipliers in equation (9) should be greater than zero. This is because P_cr/P − 1 > 0 specifies that the elastic buckling load of the restraining system should be greater than the applied axial compressive load. This ensures that the BRB would not exhibit elastic buckling failure when subjected to axial compressive load. In addition, M_r/v₀P − 1 > 0 specifies that the bending moment resistance of the restraining system should be greater than the initial load effect. This ensures that the restraining system would not exhibit yielding when subjected to the load effect produced by the axial compressive load P on the initial imperfection v₀.

To provide a better understanding of equation (9) that defines the stiffness–strength requirement interaction curve, its left-hand side term is defined as an interaction factor κ:

κ = (\frac{P_{cr}}{P} - 1) (\frac{M_{r}}{v_{0} P} - 1)

(10)

Figure 4 shows the interaction curves of the stiffness and strength requirements of the restraining system with different κ values. The right-top area of the lines P_cr/P = 1 and M_r/(v₀P) = 1 is associated with positive κ values. For κ > 1, the interaction curve lies in the safe region, satisfying both stiffness and strength requirements of the restraining system. An interaction factor κ < 1 corresponds to BRB designs with an inadequate restraining capability against the global buckling. For 0 < κ < 1, the interaction curve lies in the area between the stiffness–strength requirement interaction curve and the lines P_cr/P = 1 and M_r/(v₀P) = 1. In addition, a value of κ < 0 leads to the condition of either P_cr/P < 1 or M_r/(v₀P) < 1, which corresponds to elastic buckling failure or yielding of the restraining system when the core is subjected to axial compressive load P.

Figure 4.

Interaction curves of stiffness and strength requirements with different κ values.

Stiffness–strength relationship curve of restraining system

In the quest to establish a more comprehensive design procedure, this section establishes the relationship between the stiffness and strength terms (P_cr/P and M_r/(v₀P)) of the restraining system. A stiffness–strength relationship curve can be used together with the stiffness–strength requirement interaction curve to find a target design point for the BRB design.

For a steel BRB with a uniform cross-section, the stiffness term P_cr/P of the restraining system is given by

\frac{P_{cr}}{P} = \frac{π^{2} EI}{L^{2} P}

(11)

where P_cr is given by the Euler buckling load of the external restraining member of the BRB, L is the length of the restraining member, which is the same as the length of the core in typical BRB analytical models (Takeuchi, 2018; Takeuchi and Wada, 2017; Wada and Nakashima, 2004).

On the other hand, the strength term M_r/(v₀P) of the restraining system is formulated as

\frac{M_{r}}{v_{0} P} = \frac{f_{yr} I}{z v_{0} P}

(12)

where M_r is the yielding moment resistance formulated in equation (7).

By selecting an intermediate parameter, equations (11) and (12) combine to give a parametric function for the relationship between the stiffness and strength terms of the restraining system. For a steel BRB, the intermediate parameter is selected as the moment of inertia I of the restraining system. The stiffness and strength terms have a linear relationship, and thus the stiffness–strength relationship curve is given by a straight line as shown in Figure 5. This linear relationship can be described by the slope R, which is given by

R = \frac{M_{r} / v_{0} P}{P_{cr} / P} = \frac{1}{π^{2}} \frac{f_{y} L^{2}}{E v_{0} z}

(13)

Figure 5.

Determination of target design point based on two curves,

where z = h/2 (or D_r/2) for a symmetric steel cross-section, and h (or D_r) is the cross-sectional height of the restraining member.

Target design point based on the two curves

By using the stiffness–strength requirement interaction curve (equation (9)) and the stiffness–strength relationship curve (equation (13)), the point of intersection defines the target design point, as shown in Figure 5.

By considering the formulas of the two curves (equations (9) and (13)), the stiffness and strength requirements specified by the target design point is derived in an explicit form as

\begin{matrix} \frac{P_{cr}}{P} = 1 + \frac{1}{R} \\ \frac{M_{r}}{v_{0} P} = 1 + R \end{matrix}

(14)

where P_cr/P and M_r/(v₀P) specify the x- and y-coordinates of the target design point, respectively.

Design procedure

Based on the target design point which is derived by using equations (13) and (14), this section develops a straightforward design procedure for steel BRBs. For designing the BRB, it is assumed that the following parameters are given: (i) overall target BRB characteristics, including L, P_y, ω, and β; (ii) material properties, E, f_yr, and f_yc; and (iii) initial geometrical imperfection, v₀/L.

A step-by-step procedure for the stiffness-strength-relationship-based BRB design is summarized in the flowchart as shown in Figure 6 and presented below.

Figure 6.

Design procedure for steel BRBs.

Step (1) Sizing of restraining member after initial design of core

An initial design of the core is conducted by using its calculated cross-sectional area

A_{c} = \frac{P_{y}}{f_{yc}}

(15)

Before the slope R of the stiffness–strength relationship curve can be calculated by using equation (13), the distance z should first be determined. For this, the cross-sectional height h of the restraining member is selected based on an amplification of the core cross-section, and the distance z is obtained as h/2 for a symmetric cross-section of the restraining member.

Step (2) Establishing stiffness-strength relationship and determining target design point

The stiffness-strength relationship curve is characterized by the slope R defined by equation (13). The target design point with its coordinates (P_cr/P, M_r/(v₀P)) is determined by equation (14).

Step (3) Design of restraining member based on target design point

By using the target design point, the intermediate parameter I (cross-sectional moment of inertia of the restraining member) is determined by

I_{target} = \frac{P_{cr}}{P} \frac{P L^{2}}{E π^{2}}

(16)

The restraining member is designed on the basis of this intermediate parameter I and the selected cross-sectional height h in Step (1).

After Steps (1)–(3), some other requirements (e.g. the economical and local buckling requirements) should be checked for the restraining member. For steel BRBs, the plate width-to-thickness ratio of the restraining member should be checked for an economical design and for local buckling prevention.

Step (4) Final design and detailing

To ensure a gap between the restraining member and the core, the cross-section of the core may be redesigned using the required cross-sectional area obtained from equation (15). The outer diameter of the core for a circular steel BRB is specified by

D_{c} = D_{r} - 2 t_{r} - 2 g

(17)

where g is the gap size along the interface of the restraining member and the core. If the calculated outer diameter of the core is in agreement with the initial selected outer diameter, the gap size could be redesigned instead.

Design example to illustrate the design procedure

Design using the established design procedure

In this section, a design example of a steel BRB is presented to illustrate the design steps. Two design cases are considered: Case 1—BRB comprising core and restraining member with circular cross-sections as shown in Figure 7(a), and Case 2—BRB comprising core with an I-shaped cross-section and restraining member with a square cross-section as shown in Figure 7(b).

Figure 7.

Cross-sections of two design cases in the design example: (a) case 1, and (b) case 2.

For this BRB design example, the following parameters are assumed: (i) overall target BRB characteristics, L = 3000 mm, Py = 1620 kN, ω = 1.598, and β = 1.017 (Guo et al., 2017a); (ii) material properties, E = 200 GPa, f_yr = 400 MPa, f_yc = 235 MPa; and (iii) initial geometrical imperfection, v₀/L = 0.002, for example (Guo et al., 2017a; Takeuchi, 2018; Takeuchi and Wada, 2017).

For the design Case 1, the design procedure in Figure 6 is also followed.

Step (1) Sizing of restraining member after initial design of core

The cross-sectional area of the core is calculated as

A_{c} = \frac{P_{y}}{f_{yc}} = \frac{1620 kN}{235 MPa} = 6894 m m^{2}

(18)

The core is designed as a steel tube with initial selected dimensions D_c × t_c = 130 mm × 20 mm. The outer diameter (cross-sectional height) of the restraining member may be selected as D_r = 158 mm considering an amplification of the core cross-section.

Step (2) Establishing stiffness–strength relationship and determining target design point

To establish the stiffness–strength relationship curve, the slope R is calculated as

R = \frac{1}{π^{2}} \frac{f_{y} L^{2}}{E v_{0} z} = \frac{1}{π^{2}} \frac{400 \cdot 3000}{2 \times 10^{5} \cdot 0.002 \cdot 158 / 2} = 3.85

(19)

The x-coordinate of the target design point is determined as

\frac{P_{cr}}{P} = 1 + \frac{1}{R} = 1 + \frac{1}{3.85} = 1.26

(20)

Step (3) Design of restraining member based on target design point

By using the target design point, the target cross-sectional moment of inertia I of the restraining member is calculated as

\begin{matrix} I_{target} = \frac{P_{cr}}{P} \frac{P L^{2}}{E π^{2}} = 1.26 \times \frac{2633 \times 10^{3} \times 3000^{2}}{2 \times 10^{5} \times π^{2}} \\ = 1.513 \times 10^{7} m m^{4} \end{matrix}

(21)

where P is determined by

P = β ω P_{y} = 1.017 \times 1.598 \times 1620 = 2633 kN

(22)

The cross-section of the restraining member is designed based on equation (21) and the selected size h (or D_r) in step (1), as shown in Tables 1 and 2. The plate width-to-thickness ratio of the restraining member is checked by

\frac{D_{r}}{t_{r}} = \frac{158}{12.5} \approx 12.6 < 50 \frac{235}{f_{yr}} = 29.4

(23)

Table 1.

Design results of circular-shaped BRB: Case 1.

Component	Outer diameter (mm)	Thickness (mm)	Area (mm²)	Moment of inertia (mm⁴)
Core	130	20	6894	1.080 × 10⁷
Restraining member	158	12.5	5714	1.523 × 10⁷

Table 2.

Design results of square-shaped BRB: Case 2.

Component	Height (mm)	Width (mm)	Flange thickness (mm)	Web thickness (mm)	Area (mm²)	Momentof inertia (mm⁴)
Core	116	116	25	17	6922	6.531 × 10⁶
Restraining member	140	140	10.1	10.1	5248	1.485 × 10⁷

The designed cross-sections of the restraining member correspond to the Class 1 classification specified by the Eurocode 3 for steel structures (British Standards Institution [BSI], 2005), which shows that the adopted thickness t_r of the restraining member satisfies the local buckling criterion.

Step (4) Final design and detailing

By first specifying the gap size along the interface of the core and the restraining member as 2 mm, the outer diameter of the core is calculated as

D_{c} = D_{r} - 2 t_{r} - 2 g = 158 - 2 \times 12.5 - 2 \times 2 = 129 mm

(24)

This outer diameter D_c of the core is close to the initial selected outer diameter of 130mm. So there is no need to redesign the dimensions of the core. The final design solution for Case 1 is shown in Table 1, and the gap size between the core and the restraining member is considered as 1.5 mm.

For the design Case 2, the design procedure in Figure 6 is also followed.

Step (1) Sizing of restraining member after initial design of core

The core cross-section is an I-shaped section with h_c × b_c × t_w × t_f = 116 mm × 116 mm × 17 mm ×25 mm. The cross-sectional height of the restraining member may be selected as h = 140 mm.

Step (2) Establishing stiffness–strength relationship and determining target design point

To establish the stiffness–strength relationship curve, the slope R is calculated as

R = \frac{1}{π^{2}} \frac{f_{y} L^{2}}{E v_{0} z} = \frac{1}{π^{2}} \frac{400 \cdot 3000}{2 \times 10^{5} \cdot 0.002 \cdot 140 / 2} = 4.34

(25)

The x-coordinate of the target design point is determined as

\frac{P_{cr}}{P} = 1 + \frac{1}{R} = 1 + \frac{1}{4.34} = 1.23

(26)

Step (3) Design of restraining member based on target design point

By using the target design point, the target cross-sectional moment of inertia I of the restraining member is calculated as

\begin{matrix} I_{target} = \frac{P_{cr}}{P} \frac{P L^{2}}{E π^{2}} = 1.23 \times \frac{2633 \times 10^{3} \times 3000^{2}}{2 \times 10^{5} \times π^{2}} \\ = 1.477 \times 10^{7} m m^{4} \end{matrix}

(27)

\frac{h}{t_{r}} = \frac{140}{10.1} \approx 13.9 < 33 \sqrt{\frac{235}{f_{yr}}} = 25.3

(28)

The designed cross-sections of the restraining member correspond to the Class 1 classification specified by the Eurocode 3 for steel structures (BSI, 2005).

Step (4) Final design and detailing

The height (or width) of the I-shaped cross-section of the core is calculated as

\begin{matrix} h_{c} (= b_{c}) = h - 2 t - 2 g = 140 - 2 \times 10.1 - 2 \times 2 \\ = 115.8 mm \end{matrix}

(29)

This cross-sectional height (or width) of the core is in agreement with the initial design dimension of 116mm. So there is no need to redesign the dimensions of the core. The final design solution for Case 2 is shown in Table 2, and the gap size between the core and the restraining member is considered as 1.9 mm.

Designs based on conventional restraining ratio approach

The design example in Section 4.1 can be carried out by using the simplified design formula of equation (3) proposed by Wada and Nakashima (2004). The cross-sectional moment of inertia I of the restraining member is calculated as

\begin{matrix} I = 1.5 β ω P_{y} \frac{L^{2}}{E π^{2}} = 1.5 \times 1.017 \times 1.598 \times 1620 \\ \times 10^{3} \times \frac{3000^{2}}{2 \times 10^{5} \times π^{2}} = 1.801 \times 10^{7} m m^{4} \end{matrix}

(30)

On the basis of equation (30), the restraining member and then the core of the circular-shaped BRB can be designed as shown in Table 3.

Table 3.

Design results of circular-shaped BRB based on a recommended restraining ratio in equation (3).

Component	Outer diameter (mm)	Thickness (mm)	Area (mm²)	Moment of inertia (mm⁴)
Core	130	20	6894	1.080 × 10⁷
Restraining member	162	14.5	6719	1.845 × 10⁷

By comparing equations (20) and (21) with equations (3) and (30), and comparing Table 1 with Table 3, it is shown that a specified constant restraining ratio requirement may be very conservative and hence lead to uneconomical designs. In contrast, by following the established design procedure, Section 4.1 illustrates that it is easy for engineers to pinpoint an economical target design solution. This is because the established design procedure seeks a design result in the interaction curve between stiffness and strength requirements of the restraining system and hence it is more ecomonical than a specified constant restraining ratio requirement. The cross-sectional moment of inertia of the restraining member in Table 3 is 21.1% more than its cross-sectional moment of inertia in Table 1 (which is determined based on the established design procedure in Figure 6).

Examination of design result by rigorous finite element second-order analysis

Examination of design result

In this section, the design result obtained in Section 4.1 is assessed by results obtained using a rigorous finite element (FE) analysis that considers the second-order effect. The design result for Case 1 is analyzed here as model M-1.

The general finite element program ANSYS (version 15.0) (2015) is used for the second-order analysis of the BRB model. Figure 8 shows the FE model. The core and the restraining member are both modeled by beam element BEAM188 in ANSYS. The BEAM188 element is a classical beam element in ANSYS, and it was recommended by the ANSYS user’s manual for conducting second-order analyses of beam–columns (ANSYS, 2015). This element has also been validated to accurately capture the cyclic test results of BRBs, see for example (Guo et al., 2016, 2017c; Tong and Guo, 2017). Since this paper focuses on the design against the global buckling of BRBs, it is justifiable to use beam elements to model the BRB. The design against other failure modes of BRBs, for example local buckling failure (Takeuchi, 2018; Takeuchi and Wada, 2017), failure associated with higher buckling modes (Dehghani and Tremblay, 2017), and failure at the connection between BRB and frame member) should be investigated by using other more refined structural models.

Figure 8.

Description of FE model: (a) FE model, (b) First-order buckling mode, and (c) Boundary conditions and interactions between elements.

The initial geometrical imperfection is assumed to be proportional to the first-order buckling shape of the BRB (Figure 8(b)). This geometrical imperfection is corresponding to the assumed trigonometric imperfection v₀sin(πx/L) in Section 2.1, and the amplitude v₀ is specified as L/500.

Figure 8(c) shows the boundary conditions and the connections between the core and the restraining member. To simulate the restraining effect of the restraining member, the translational displacements of corresponding nodes on the core and the restraining member are coupled (whereas their axial displacements are not coupled). It is noted that the beam elements for the core and the restraining member are established on the same positions, and in Figure 8(c) the offset between them is used to illustrate their connections. The axial compressive load P is applied to the core following a cyclic loading protocol. The displacement levels correspond to axial displacements that are equal to ±0.25%, ±0.5%, ±0.75%, ±1.0%, ±1.25%, and ±1.5% of the BRB length L (AISC, 2010).

For the material properties, a Young’s modulus E = 200 GPa is used. The yield stress of the core and the restraining member are 235 MPa and 400 MPa, respectively. For the restraining member, a bilinear elastoplastic hardening model is adopted, and the tangent modulus after yielding is set to 0.5%E.

The material properties for the core was validated by sub-assemblage tests of BRB members (Guo et al., 2017a). A combination of the CHAB nonlinear kinematic hardening rule and the BISO isotropic hardening rule in ANSYS is adopted (ANSYS, 2015). The back stress α₁ for the kinematic hardening rule is expressed as

α_{1} = \sum_{k = 1}^{n} \frac{C_{k}}{γ_{k}} (1 - e^{- γ_{k} ε_{pl}})

(31)

where ε_pl is the equivalent plastic strain; n the superimposed number of kinematic hardening models; and C_k and γ_k the kinematic parameters obtained from cyclic coupon tests. The CHAB kinematic parameters are set as C₁ = 16 GPa, γ₁ = 300; C₂ = 6 GPa, γ₂ = 150; and C₃ = 0.5 GPa, γ₃ = 20 to be corresponding to the design parameters in Section 4.1 (Guo et al., 2017a).

To define the BISO isotropic hardening rule, a maximum change in the yield stress of Q_∞ = 90 MPa and a rate factor of 4.2 are used. A strain hardening effect corresponding to the design parameters ω = 1.598 and β = 1.017 in Section 4.1 could be achieved (Guo et al., 2017a), if the specimen exhibits a stable hysteretic response (i.e. without any degradation in the resistance during the loading process).

The simulation results of the BRB model are presented in Figure 9. The hysteretic curves, and the deflection and von Mises stress distributions of the restraining member at the final loading step are presented. This model could exhibit stable hysteretic responses. With an interaction factor κ = 1.044 (which is slightly larger than 1.0), this BRB model has an adequate restraining system, and the BRB will not suffer any large lateral deformations associated with the global buckling failure mode. The strain hardening of the core with P_max/P_y = 1.625 could be achieved. Because this BRB model is designed to be slightly safer than the target design point (with I/I_target = 1.01), its restraining member exhibits yielding in a small region at the mid-span.

Figure 9.

FE results of BRB model M-1.

Analysis of models with different cross-sections of restraining member

For comparisons, three additional BRB models M-2∼M-4 with modified cross-sections of the restraining member are also analyzed. The properties of the models are listed in Table 4 and their satisfaction of the design requirement is shown in Figure 10. For the inner core, the four FE models use the same properties in order to be consistent. Models M-2 and M-3 have the same outer diameter D_r as M-1, and hence they have the same slope R for the stiffness–strength relationship curve, as shown in Figure 10. The ratios between the selected moment of inertia I for models M-2 and M-3 and the target moment of inertia I_target (equation (21)) are 1.25 and 0.99, which correspond to interaction factors κ = 2.995 and 0.932, respectively. In addition, model M-4 with an interaction factor κ < 1 is also analyzed. To discuss the disadvantage of the “restraining ratio” criterion, model M-4 is designed to have a relatively large P_cr/P ratio (and hence a larger restraining ratio than M-1) but have an unsatisfactory interaction factor κ < 1.

Table 4.

Properties and FE results of analyzed BRB models.

Model	Properties of the restraining member						FE results
Model	I/I_target	Outer diameter D_r (mm)	Thickness t (mm)	P_cr/P	M_r/v₀P	κ	Status	P_max/P_y
M-1	1.01	158	12.50	1.269	4.882	1.044	Stable	1.625
M-2	1.25	158	17.08	1.587	6.105	2.995	Stable	1.626
M-3	0.99	158	12.20	1.246	4.793	0.932	Unstable	1.605
M-4	–	283	2.00	1.452	3.119	0.957	Unstable	1.556

Figure 10.

Satisfaction of design requirements for the analyzed models.

Figure 11 shows the simulation results of models M-2∼M-4. Model M-2 could exhibit stable hysteretic responses, and the strain hardening of the core with P_max/P_y = 1.626 is achieved (see Table 4). The restraining member of M-2 with κ = 2.995 could maintain elastic during the whole loading process with a maximum von Mises stress of 38.6 MPa. Comparing with the von Mises stress distribution for M-1, the elastic von Mises stress distribution for M-2 indicates that a larger interaction factor κ corresponds to a stronger restraining capability of the restraining member.

Figure 11.

FE results of BRB models M-2~M-4: (a) FE results of M-2 (κ = 2.995; P_cr/P = 1.587), (b) FE results of M-3 (κ = 0.932; P_cr/P = 1.246), and (c) FE results of M-4 = 0.957; P_cr/P = 1.452).

Models M-3 and M-4 (with κ values that are slightly less than 1.0) exhibit global buckling failure during the loading process. Model M-3 has a degradation in the resistance during the compressive load step of 1.5%, and model M-4 has a degradation in the resistance during the load step of 1.25%. Their failure modes are characterized by the yielding in a large proportion of the restraining member and the significant mid-span deformations associated with the global buckling failure mode.

By comparing the simulation results of Models M-1 and M-3, it is found that BRB models having an interaction factor κ that is less than 1.0 will start to exhibit a global buckling failure before full axial compressive yielding of the inner core. This result is consistent with the design criterion in equation (8). In addition, the simulation results of Model M-4 give an illustration that global buckling failure may occur in the case of κ < 1.0 even when the BRB model has a large P_cr/P ratio.

Conclusions

This paper proposes a new stiffness-strength-relationship-based design approach that can easily pinpoint the target design solution for BRBs. The following conclusions may be drawn:

Based on a second-order analysis of the BRB, the new stiffness-strength-relationship-based design criterion is established. By denoting the stiffness and strength terms as P_cr/P and M_r/(v₀P), respectively, the stiffness–strength requirement interaction curve takes a very simple and easy-to-use form. The curve is expressed by an inverse proportional function with two asymptotic lines P_cr/P = 1 and M_r/(v₀P) = 1 (see Figure 3).

The stiffness–strength relationship curve of the restraining system is expressed by a parametric function. By selecting the intermediate parameter as the cross-sectional moment of inertia I, the stiffness and strength terms have a linear relationship. The relationship curve is plotted as a straight line starting from the origin of the coordinates system.

The proposed design procedure for BRBs is based on the pinpointed target design solution. This solution is defined by the intersection point of the stiffness–strength requirement interaction curve and the stiffness–strength relationship curve (see Figure 5).

An interaction factor κ is defined and it can be used to evaluate the restraining capability of BRB designs. A safe design corresponds to an interaction factor of κ ≥ 1.0, and a larger interaction factor κ corresponds a stronger restraining capability of the restraining system.

Footnotes

Appendix

Declaration of conflicting interests

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

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The writers gratefully acknowledge the financial support provided by the National Key R&D Program of China (No. 2016YFC0701201 and 2016YFC0701204).

ORCID iD

Jing-Zhong Tong

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