Out-of-plane stability of buckling-restrained braces including moment transfer capacity

SUMMARY Buckling-restrained braces (BRBs) are widely used as ductile seismic-resistant and energy-dissipating 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 ﬂ exural 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 bending-moment transfer capacity at restrainer ends for various connection stiffness values with initial out-of-plane drifts, and a uni ﬁ ed simple equation set for ensuring BRB stability is proposed. Moreover, a series of cyclic loading tests with initial out- of-plane 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.


INTRODUCTION
Buckling-restrained braces (BRBs) are expected to exhibit stable hysteresis when subjected to in-plane cyclic axial loading, as described in seismic provisions as AIJ 2009 [1] and AISC 341-10 [2]. As determined in these provisions, various conditions are required to ensure the stable BRB hysteresis. Takeuchi et al. investigated the relation between the cumulative deformation capacity and the applied loading history [3] of a BRB and discussed the influence of local buckling on BRB strength and ductility [4]. Furthermore, several past studies have highlighted the risk of global BRB buckling induced by connection failure such as plastic hinges introduced at connection zone prior to core yielding ( Figure 1). For instance, Tsai et al. [5] and Lin et al. [6] reported the phenomenon of global buckling related to connection failure in BRB frame tests, and Wigle et al. [7] discussed the effect of connections on BRB performance. Koetaka et al. [8] discussed the conditions leading to instability, under which bending-moment transfer capacity at the restrainer ends is lost and gusset plates are designed to be rotationally rigid. Zao et al. [9,10] investigated the stability conditions of BRBs with pin-ended connections. Hikino et al. [11] carried out shaking table tests on BRB frames, and Okazaki et al. [12] suggested stiffness requirements for gusset plates under one-sided buckling considering the loss of restrainer moment transfer capacity and confirmed the requirements by the test results. In addition, the stiffness and strength requirements at connections have also been investigated through several other BRB frame tests [13][14][15][16][17].
However, in previous studies, stability concepts including the effects of bending-moment transfer capacity at the restrainer ends for varying rotational stiffness values of the gusset plates have not yet been clarified. The effect of story drift in the out-of-plane direction for simulating the transverse component of ground motion was not considered either. In this paper, the stability requirements for BRBs including the aforementioned conditions are discussed, and a simple set of equations covering these conditions is proposed. In addition, cyclic axial loading tests of BRBs with initial out-of-plane drift are performed, and the accuracy of the proposed equation set is verified.

STABILITY CONDITION FOR BRBS INCLUDING CONNECTIONS
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   Table IV).
The stability condition of the restrained zone: The stability condition of the connection zone: where M B y 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, N R cr 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 restrainer-end 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 corerestrainer 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 out-of-plane deformations y r at the end of the restrainer can be approximated by the following equation ( Figure 6).
where N B cr 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 N B cr 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 restrainer-end zone exceeds the restrainer moment transfer capacity M r p , 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 out-of-plane 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).
y ¼ a r x ξL 0 þ y r 1 À cos πx 2ξL 0 ! (4) Figure 6. Buckling-restrained brace stability concepts and limits. Then, the strain energy stored in both connection zones is given as follows: The rotation angle of the plastic hinges is expressed as follows: Then, the plastic strain energy stored in the plastic hinges is as follows: The axial deformation is as follows: The work carried out is given as follows: with the following balance of energy differential Approximating 8/π 2 ≈ 1, the following is obtained: Similar calculations can be carried out for the asymmetrical buckling mode shown in Figure 7(b), as follows.
When ξ = 0.25, N can be approximated as follows: When M r p = 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: 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: The gusset plate spring rotation, Δθ s , plastic hinge rotation, Δθ r , and axial deformation, Δu g can be expressed as follows: Then, the energy stored in the springs, U s , energy stored in the hinges, U p , and the total work done, T, can be evaluated respectively as follows: With the balance of energy differential ∂(U ε + U s + U p À T)/∂y r = 0, Equation (25) may be expressed as Equation (26) when the rotational stiffness of the gusset plates is negligible ( ξ κ Rg ≈ 0).
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).
Accordingly, one-sided buckling mode strength, shown in Figure 7(c), can be determined using the following expression.
Similar to Equation (25), Equations (27) and (28) cover asymmetrical or one-sided 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: where N r cr 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 out-of-plane displacement y r . As shown in Figure 6, when the elastic axial force and the out-of-plane 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 y r ¼ a r N= N B cr À N À Á from Equation (3) into Equation (29), the required restrainer moment transfer capacity, M p r , can be approximated as follows: When the brace is subjected to an axial force along with an out-of-plane drift, a bending moment is generated as shown in Figure 8. The restrainer moment transfer capacity is considered to be reduced by this secondary moment M r 0 . From Figure 8, this value can be estimated as follows: Figure 8. Additional bending moment under out-of-plane drift.
where δ 0 denotes the assumed maximum story out-of-plane drift, and L in denotes the insert zone length. Including M r 0 in Equation (29), the following equation for defining the ultimate strength is obtained: When the aforementioned N is substituted for the axial force in Equation (30), the stability condition can be expressed as follows: where M r p À M r 0 should be taken as zero if the difference is negative. In Equation (33), N B cr can be estimated by eigenvalue analysis; however, when γ j ≈ 1, the lowest value decided by the elastic symmetrical mode can be approximated as N B cr using the following expression: where L κ Rg is the rotational stiffness of the gusset plate normalized by total length. For satisfying Equation (33), two approaches can be followed.
(1) When ξ κ Rg is sufficiently large, the left part of Equation (33) tends to be small or zero. In order to satisfy Equation (33) N r cr should be larger than N cu . This approach corresponds to Equation (2), which allows for hinges at the restrainer ends (Figure 2(a)).
(2) Decrease M 0 r and N r cr by decreasing ξ κ Rg and providing sufficient bending strength M r p to satisfy Equation (33). This approach corresponds to transferring the bending moment at the restrainerend zone (Figure 2(b)).
With the use of these two approaches, Equation (33) covers the two design concepts discussed in Figure 2.
The proposed stability condition in Equation (33) is based on the condition that the gusset plates remain elastic. When plastic hinges are produced at the gusset plates, a different global buckling mode as shown in Figure 9 comes into play. The stability condition for this collapse mode can be expressed as follows.
where M g p is the plastic bending strength of the gusset plate including the axial force effect, and 1 À 2ξ ð ÞM g p À M r 0 or M r p À M r 0 should be taken as zero if the difference is negative.

CYCLIC BRB LOADING TESTS WITH OUT-OF-PLANE DRIFT
For confirming the proposed stability conditions, cyclic loading tests were performed on BRBs with out-of-plane drifts. This test program simulated the worst-case scenario in which the maximum inplane story drift occurs at the same as the 1% out-of-plane story drift. The test configuration with the specimen is shown in Figures 10 and 11, and the test matrix is summarized in Table I. The core plate material was JIS-SN400B (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). Prior to each test, an out-of-plane 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 II. 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 energy-dissipating 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 out-ofplane 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 out-of-plane 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 M r p , of each specimen was estimated using the following equation proposed by Matsui et al. [17]: where M rÀneck p represents the restrainer moment transfer capacity determined by the cruciform core plate at the neck as follows: where N c wy denotes the yield axial force of the cruciform core plate at the web zone, N c u 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), M rÀrest p represents the restrainer moment transfer capacity determined by the restrainer section at rib end as follows: 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 post-yielding 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. In most cases, M r p is determined by the bending strength of the neck section, M rÀneck p , given by Equation (37) when the insert zone length is more than 2.0 times the core plate width. The obtained values of M r p for each specimen are summarized in Table III. The expected failure axial force assuming elastic springs for gusset plates N lim1 is obtained by solving Equation (39), which is derived from Equation (33) and summarized in Table. IV.  Here, N r cr is estimated as the elasto-plastic 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.
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).
The least of the two failure forces obtained from Equations (39) and (41) is considered the failure axial force of the specimen. In Table IV, 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.

CONCLUSIONS
When BRBs exhibit global buckling behavior, a certain amount of bending moment is transferred between the core section and restrainer tube end, which is defined as restrainer moment transfer capacity in this study. The restrainer moment transfer capacity depends on the internal structure of the BRB, as explained throughout the paper, and significantly influences BRB stability. This study proposes a compact set of equations for evaluating global BRB stability considering restrainer moment transfer capacity. The proposed equations cover a wide range of BRB connections from rigid end to pinned end.
Cyclic loading tests were performed on full-scale BRBs with initial out-of-plane drifts and varying connection stiffness values to confirm the proposed stability conditions and validating the proposed equations. The conclusions of this study are as follows: (1) The BRB stability conditions were expressed by a set of simple equations as functions of restrainer moment transfer capacity, rotational stiffness at gusset plate ends, initial imperfections governed by core-plate clearances, and expected out-of-plane story drifts. This equation set covers the stability conditions of all BRB types regardless of the gusset plate rigidity or restrainer moment transfer capacity.
(2) In the cyclic loading tests with initial out-of-plane drifts, specimens with shorter insert zones and larger clearances experienced global buckling before achieving stable hysteresis, whereas specimens with longer insert zones and smaller clearances showed more stable hysteresis, which may be attributed to the fact that the restrainer moment transfer capacity significantly influences BRB stability.
In addition, the specimens with stiffened gusset plates exhibited greater stable hysteresis and higher axial capacity even with shorter insert zones. These results agree with the stability conditions anticipated using the proposed stability equations.
(3) The proposed equation set can be applied for directly evaluating the stability limit strength of BRBs. The results have shown that the evaluated stability limit strengths agree well with the experimental results of buckled specimens, thus validating the proposed equations.
Stability conditions for different types of BRBs such as pin-connected BRBs or ones with alternative restrainer shapes will be discussed in future papers, the studies corresponding to which are in progress.