SUMMARY
 Top of page
 SUMMARY
 INTRODUCTION
 STABILITY CONDITION FOR BRBS INCLUDING CONNECTIONS
 CYCLIC BRB LOADING TESTS WITH OUTOFPLANE DRIFT
 CONCLUSIONS
 ACKNOWLEDGEMENTS
 REFERENCES
Bucklingrestrained braces (BRBs) are widely used as ductile seismicresistant and energydissipating structural members in seismic regions. Although BRBs are expected to exhibit stable hysteresis under cyclic axial loading, one of the key limit states is global flexural buckling, which can produce an undesirable response. Many prior studies have indicated the possibility of global buckling of a BRB before its core yields owing to connection failure. In this paper, BRB stability concepts are presented, including their bendingmoment transfer capacity at restrainer ends for various connection stiffness values with initial outofplane drifts, and a unified simple equation set for ensuring BRB stability is proposed. Moreover, a series of cyclic loading tests with initial outofplane drifts are conducted, and the results are compared with those of the proposed equations. © 2013 The Authors. Earthquake Engineering & Structural Dynamics published by John Wiley & Sons Ltd.
STABILITY CONDITION FOR BRBS INCLUDING CONNECTIONS
 Top of page
 SUMMARY
 INTRODUCTION
 STABILITY CONDITION FOR BRBS INCLUDING CONNECTIONS
 CYCLIC BRB LOADING TESTS WITH OUTOFPLANE DRIFT
 CONCLUSIONS
 ACKNOWLEDGEMENTS
 REFERENCES
In AIJ Recommendations for Stability Design of Steel Structures [1], the implementation of the following two concepts is presented for preventing global instability as shown in Figure 2.
 Concept (1)
Although plastic hinges without bendingmoment transfer capacity is allowed at the restrainer ends, the stability conditions are satisfied individually for the restrained zone and the connection zone [Figure 2(a)].
 Concept (2)
When bendingmoment transfer capacity is provided at the restrainer ends, the composite stability of the restrained zone and the connection zone are ensured [Figure 2(b)].
For Concept (1), the following equations were proposed by Koetaka et al. [8].
The stability condition of the restrained zone:
 (1)
The stability condition of the connection zone:
 (2)
where denotes yield bending strength of the restrainer, a denotes maximum imperfection along the restrainer, e denotes axial force eccentricity, s_{r} denotes the core–restrainer clearance, N_{cu} denotes the maximum axial strength of the core plates, normally estimated as 1.2–1.5 times the yield strength of the core plates, including hardening, denotes the Euler buckling strength of the restrainer, γ_{J}EI_{B} denotes the bending stiffness of the connections, L_{0} denotes the total BRB length, and ξL_{0} denotes the connection zone length.
Equation (2) is based on the assumption that the ends of the connection zones are rigidly fixed against rotation; in practice, however, substantially stiff gusset plates are required to satisfy this condition (e.g., stiffened gusset plates similar to the one shown in Figure 3(c)). Moreover, rotation of the beam to which BRBs are connected should be prevented using stiff secondary beams.
For Concept (2), Matsui et al. [17] indicated that the restrainerend zone can transfer bending moment as large as the smaller of the bending strength of the restrainer or the neck, if the extension of the insert zone L_{in} into the restrainer is more than the width of the core plate (Figure 4). In Figure 4, the neck represents the portion of the cruciform sectioned core that extends beyond the restrainer case. When the rotational stiffness of the gusset plate is relatively low as in Figure 3(a), the core–restrainer clearance generates an initial imperfection that can be assumed as a_{r} = a + e + s_{r} + (2s_{r}/L_{in})ξL_{0}, as shown in Figure 5. Then, the relationship between the axial force N, and outofplane deformations y_{r} at the end of the restrainer can be approximated by the following equation (Figure 6).
 (3)
where denotes the global elastic buckling strength of BRB, including the effects of the connection zone's bending stiffness, and the gusset plates' rotational stiffness. In Figure 6, the axial force approaches asymptotically for increasing values of y_{r}; however, the elastic buckling process is interrupted when the brace reaches its ultimate strength as computed by collapse mechanisms resulting from the plasticity at the restrainer ends. The intersection point of the elastic buckling path and the ultimate strength path is defined as the stability limit. In cases where the maximum axial strength of the core, N_{cu}, is lower than the stability limit, the brace is considered stable, and the BRB core exhibits regular hysteretic behavior. However, when N_{cu} exceeds the stability limit, the plastic hinges are produced at the insert zone, and the hysteretic behavior is considered to be interrupted by the global buckling collapse mechanisms.
The ultimate strength values of the possible collapse mechanisms are estimated using the models shown in Figure 7. As can be seen in this figure, the BRB is modeled with rotational springs of stiffness K_{Rg} at both gusset plates. Then, an initial imperfection a_{r} is modeled at restrainer ends. When the bending moment at the restrainerend zone exceeds the restrainer moment transfer capacity , the load on the BRB exceeds its ultimate strength, eventually resulting in buckling.
Firstly, the gusset plates are assumed to be rigid (K_{Rg} ∞), and outofplane deformations of the connection zone during the mechanism phase are assumed to be of sinusoidal shape, as shown in Figure 7(a) and given in Equation (4).
 (4)
Then, the strain energy stored in both connection zones is given as follows:
 (5)
The rotation angle of the plastic hinges is expressed as follows:
 (6)
Then, the plastic strain energy stored in the plastic hinges is as follows:
 (7)
The axial deformation is as follows:
 (8)
The work carried out is given as follows:
 (9)
with the following balance of energy differential
 (10)
 (11)
Approximating 8/π^{2} ≈ 1, the following is obtained:
 (12)
When ξ = 0.25, N can be approximated as follows:
 (16)
When = 0 and a_{r} ≪ y_{r}, Equation (16) turns into Equation (2). As indicated by Equations (12) and (16), the global buckling strength is generally determined by the asymmetrical mode when the ends of the connections are fixed rigidly.
Secondly, considering the normalized rotational stiffness of the gusset plates, _{ξ}κ_{Rg} can be defined as follows:
 (17)
Additional displacement due to the rotation of the gusset plate (represented by the end spring shown in the detail of Figure 7(a)) is defined as y_{rs}. Given that the deformation because of the connection zone bending, y_{re}, becomes equivalent to y_{rs} when _{ξ}κ_{Rg} = 3, the strain energy stored in a specific spring can be approximated as follows:
 (18)
With the balance of energy differential ∂(U_{ε} + U_{s} + U_{p} − T)/∂y_{r} = 0,
 (25)
Equation (25) may be expressed as Equation (26) when the rotational stiffness of the gusset plates is negligible (_{ξ}κ_{Rg} ≈ 0).
 (26)
Conversely, if _{ξ}κ_{Rg} ∞, Equation (25) is restored to match Equation (12). Hence, Equation (25) covers the symmetrical buckling strength for rotational stiffness values ranging from those of pinned ends to those of rigid ends.
The asymmetrical collapse mechanism strength can be derived following a similar process (Figure 7(b)) to obtain Equation (27).
 (27)
Accordingly, onesided buckling mode strength, shown in Figure 7(c), can be determined using the following expression.
 (28)
Similar to Equation (25), Equations (27) and (28) cover asymmetrical or onesided buckling strength for varying gusset plate rotational stiffness values. Equation (28) yields results similar to those of Okazaki et al. [12] for relatively small a_{r} and y_{r} values. Equation (27) yields lower values than do Equation (25) or Equation (28), thus implying that the asymmetrical mode governs stability.
Here, if a_{r} ≪ y_{r}, Equation (27) can be approximated as follows:
 (29)
where is the global elastic buckling strength of the connection zone with pinned conditions at the restrainer ends.
Equation (29), which describes the ultimate strength path shown in Figure 6, indicates that the axial force decreases for increasing values of the outofplane displacement y_{r}. As shown in Figure 6, when the elastic axial force and the outofplane displacement relationship considering initial imperfections expressed by Equation (3) exceed the limits of the aforementioned possible mechanisms, it is considered that the BRB is expected to start undergoing global buckling. Substituting from Equation (3) into Equation (29), the required restrainer moment transfer capacity, M_{p}^{r}, can be approximated as follows:
 (30)
When the aforementioned N is substituted for the axial force in Equation (30), the stability condition can be expressed as follows:
 (33)
where should be taken as zero if the difference is negative.
With the use of these two approaches, Equation (33) covers the two design concepts discussed in Figure 2.
CYCLIC BRB LOADING TESTS WITH OUTOFPLANE DRIFT
 Top of page
 SUMMARY
 INTRODUCTION
 STABILITY CONDITION FOR BRBS INCLUDING CONNECTIONS
 CYCLIC BRB LOADING TESTS WITH OUTOFPLANE DRIFT
 CONCLUSIONS
 ACKNOWLEDGEMENTS
 REFERENCES
For confirming the proposed stability conditions, cyclic loading tests were performed on BRBs with outofplane drifts. This test program simulated the worstcase scenario in which the maximum inplane story drift occurs at the same as the 1% outofplane story drift. The test configuration with the specimen is shown in Figures 10 and 11, and the test matrix is summarized in Table 1. The core plate material was JISSN400B (average yield strength: 270 MPa), and the core cross section size, A_{c} = 12 mm × 90 mm. The restrainer is either a mortar filled square box section with a side length of 125 mm and thickness of 2.3 mm, or a circular tube with an external diameter of 139.8 mm and tube wall thickness of 3.2 mm. Two types of gusset plates are used in the tests, namely, the regular type (_{ξ} κ_{Rg} = 0.04) and the stiffened type (_{ξ} κ_{Rg} = 0.3). The insert length of the stiffened part of the core plate into the restrainer, L_{in}, is chosen to be 90 mm and 180 mm, which are equal to 1.0 and 2.0 times the core plate width, respectively. In addition, the core plate–restrainer clearance varies from 1.0 mm to 2.0 mm. The specimens are labeled as M (R: rectangular, C: circular), L (insert zone length to core plate width ratio), S (clearance), and H (stiffened type gusset plate).
Table 1. Test matrix.Specimen  A_{c} (mm^{2})  σ_{cy} (N/mm^{2})  EI_{B} (Nmm)  σ_{ry} (N/mm^{2})  K_{Rg} (Nmm)  γ_{J}EI_{B} (Nmm)  L_{0} (mm)  L_{in} (mm)  L_{p} (mm)  s_{r} (mm)  ξL_{0} (mm)  ξ'L_{0} (mm) 

MRL1.0S1H  1080  266.0  5.81 × 10^{11}  305.0  6.90 × 10^{8}  1.20 × 10^{12}  2392  90  1380  1  416  506 
MRL2.0S1  1080  266.8  5.81 × 10^{11}  385.8  9.73 × 10^{7}  1.20 × 10^{12}  2392  180  1200  1  416  596 
MRL2.0S2  1080  266.8  5.81 × 10^{11}  391.5  9.73 × 10^{7}  1.20 × 10^{12}  2392  180  1200  2  416  596 
MCL2.0S2  1080  269.7  7.14 × 10^{11}  365.7  9.73 × 10^{7}  1.20 × 10^{12}  2392  180  1200  2  416  596 
MRL1.0S1  1080  266.8  5.81 × 10^{11}  391.5  9.73 × 10^{7}  1.20 × 10^{12}  2392  90  1380  1  416  506 
MRL1.0S2  1080  266.8  5.81 × 10^{11}  391.5  9.73 × 10^{7}  1.20 × 10^{12}  2392  90  1380  2  416  506 
Prior to each test, an outofplane displacement equivalent to 1% radian story drift was applied to each specimen. For cyclic loading, up to 3% normalized axial deformation (δ/L_{p}) was applied, according to the loading protocol shown in Figure 12. Here, the normalized axial deformation, which is approximately equivalent to the story drift angle, is the ratio of the axial deformation to the plastic length of the core plate L_{p}. The expected values of the initial imperfection angles for each specimen are summarized in Table 2.
Table 2. Initial imperfection angle.Specimen  L_{in} (mm)  s_{r} (mm)  θ_{0} = 2s_{r}/L_{in} (rad) 

MRL1.0S1H  90  1  0.02 
MRL2.0S1  180  1  0.01 
MRL2.0S2  180  2  0.02 
MCL2.0S2  180  2  0.02 
MRL1.0S1  90  1  0.02 
MRL1.0S2  90  2  0.04 
The hysteresis loops obtained by cyclic loading tests for each specimen are shown in Figure 13. The normalized cumulative plastic deformation ΣΔε_{p} = ΣΔδ_{p}/L_{p}, and normalized cumulative absorbed energy χ_{w} = E_{d}/σ_{y}A_{c} until instability are also noted in each figure. Specimen MRL1.0S1H (Figure 12(a)) with stiffened gusset plates showed stable hysteretic behavior until core plate fracture after 6 cycles of over 3.5% (48.3 mm) normalized axial deformation. Similarly, MRL2.0S1 (Figure 12(b)) with regular gusset plates showed stable hysteresis until 12 cycles of 3% (36 mm) normalized axial deformation. This performance is considered satisfactory for energydissipating braces. MRL2.0S2 (Figure 12 (c)), which has slightly larger initial imperfection compared with that ofMRL2.0S1, showed stable hysteresis until the second cycle at 3% (36 mm) normalized axial deformation, after which outofplane instability occurred. MCL2.0S2 (Figure 12(d)), using a mortar filled circular steel tube, showed stable hysteresis until the second cycle at 2% (27.6 mm) normalized axial deformation, at which time outofplane instability occurred. MRL1.0S1 (Figure 12(e)) reached the yield strength of the core plate and showed stable hysteresis up to the second cycle of 1.0% (13.8 mm) normalized axial deformation, after which it experienced global buckling associated with hinging at the neck. MRL1.0S2 (Figure 12(f)) showed stable a hysteresis loop for only one cycle of 0.5% (6.9 mm) normalized axial deformation, then experienced global buckling associated with hinging at the neck.
These test results indicate that BRB stability is significantly affected by the length of the insert zone and the clearance, as is expected from the proposed Equations (33) and (35). In order to confirm the validity of the proposed equations, each specimen is evaluated using Equations (33) and (35). To this end, restrainer moment transfer capacity , of each specimen was estimated using the following equation proposed by Matsui et al. [17]:
 (36)
where represents the restrainer moment transfer capacity determined by the cruciform core plate at the neck as follows:
 (37)
where denotes the yield axial force of the cruciform core plate at the web zone, denotes the ultimate strength of the cruciform core plate at the neck, Z_{cp} denotes the plastic section modulus at the neck, and σ_{cy} denotes the yield stress of the core plate. In Equation (36), represents the restrainer moment transfer capacity determined by the restrainer section at rib end as follows:
 (38)
where Z_{rp} denotes the plastic section modulus of the restrainer tube, σ_{ry} denotes restrainer yield stress, K_{Rr1} denotes the restrainer elastic rotational stiffness about the rib end, θ_{y1}' denotes the pseudo initial yield angle of the rectangular restraint tube, K_{Rr2} denotes the postyielding rotational stiffness of the restrainer about the rib end, θ_{y2} denotes the angle at which the plastic hinge occurs, θ_{y} denotes the yield angle of the circular restraint tube, and B_{c} denotes the core plate width.
Table 3. Bending capacities at restrainer ends.Specimen  Yield bending strength of cruciform zone (kNm)  Yield bending strength of restrainer (kNm)  M_{p}^{r} (kNm) 

MRL1.0S1H  2.46  2.97  2.46 
MRL2.0S1  5.50  8.38  5.50 
MRL2.0S2  6.56  8.56  6.56 
MCL2.0S2  6.50  35.68  6.50 
MRL1.0S1  6.78  4.28  4.28 
MRL1.0S2  6.78  4.28  4.28 
Table 4. Stability evaluations using proposed equation.Specimen  N_{cr}^{B}(kN)  a_{r} (mm)  N_{cr}^{r} (kN)  N_{cu} (kN)  M_{0}^{r} (kNm)  Failure cycle at experiment  Stability limit (kN)  Failure axial force in experiment (kN) 

       N_{lim1} (Equation (39))  N_{lim2} (Equation (41))  


MRL1.0S1H  1880  11.4  695  431  0.00  None  818  1390  (452) 
MRL2.0S1  1158  6.80  82  432  0.09  3.0%12cycle  520  520  535 
MRL2.0S2  1158  12.4  82  432  0.00  3.0%2cycle  419  410  507 
MCL2.0S2  1389  12.4  82  437  0.00  1.0%2cycle  440  432  375 
MRL1.0S1  1158  11.4  111  432  0.00  0.5%1cycle  367  345  362 
MRL1.0S2  1158  21.7  111  432  0.00  0.5%1cycle  264  217  300 
Here, is estimated as the elastoplastic buckling strength obtained by substituting the equivalent slenderness ratio of Equation (40) into the column curves. In Equation (40), ξ' from Figure 4 instead of ξ should be used for assuming whether plastic hinges can be produced at the rib ends.
 (40)
where i_{c} is the radius of gyration at the connection zone.
Similarly, the expected failure axial force that assumes plastic hinges at the gusset plates, N_{lim2}, is determined by solving Equation (41), which is derived from Equation (35).
 (41)
The least of the two failure forces obtained from Equations (39) and (41) is considered the failure axial force of the specimen. In Table 4, the estimated N_{lim1} and N_{lim2} values are compared with the maximum axial loads in the experiments. It is shown that the stability limit of specimens MRL1.0S1H and MRL2.0S1 exceed the expected N_{cu} (assumed as N_{cu} = 1.5 × A_{c} × σ_{cy}). The other specimens were designed such that the stability limit failed to exceed the N_{cu}, thus indicating that their global stability is not guaranteed. Figure 14 shows a comparison of the measured axial force–displacement relationships with those obtained using Equations (3) and (32). In the figure, the ultimate strength paths for the collapse mechanisms are given by R1, C1, R2, and C2. Here R stands for rib end, C stands for the neck cruciform section; 1 and 2 stand for N_{lim1} or N_{lim2} failure axial forces, respectively. Although the test results that exceeded the stability limit have better force–displacement relationships compared with those obtained using the proposed equations, their paths tend to be parallel to the estimated collapse path.
A comparison of the estimated failure axial forces N_{lim}, which is the smaller of N_{lim1} and N_{lim2}, with the peak axial force from experimental results, is shown in Figure 15. It can be seen that the results of the proposed equations are consistent with the experimental results with some variation. In general, the given safety condition successfully estimates the performance of each test specimen and is therefore considered valid.