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Keywords:

  • buckling-restrained brace;
  • connections;
  • cyclic loading test;
  • buckling;
  • moment transfer capacity;
  • stability concepts

SUMMARY

  1. Top of page
  2. SUMMARY
  3. INTRODUCTION
  4. STABILITY CONDITION FOR BRBS INCLUDING CONNECTIONS
  5. CYCLIC BRB LOADING TESTS WITH OUT-OF-PLANE DRIFT
  6. CONCLUSIONS
  7. ACKNOWLEDGEMENTS
  8. REFERENCES

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 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 bending-moment transfer capacity at restrainer ends for various connection stiffness values with initial out-of-plane drifts, and a unified 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

  1. Top of page
  2. SUMMARY
  3. INTRODUCTION
  4. STABILITY CONDITION FOR BRBS INCLUDING CONNECTIONS
  5. CYCLIC BRB LOADING TESTS WITH OUT-OF-PLANE DRIFT
  6. CONCLUSIONS
  7. ACKNOWLEDGEMENTS
  8. REFERENCES

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 [13-17].

image

Figure 1. Global buckling of buckling-restrained brace including connections (MRL 1.0S2 in Table 4).

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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

  1. Top of page
  2. SUMMARY
  3. INTRODUCTION
  4. STABILITY CONDITION FOR BRBS INCLUDING CONNECTIONS
  5. CYCLIC BRB LOADING TESTS WITH OUT-OF-PLANE DRIFT
  6. CONCLUSIONS
  7. ACKNOWLEDGEMENTS
  8. 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 bending-moment 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 bending-moment transfer capacity is provided at the restrainer ends, the composite stability of the restrained zone and the connection zone are ensured [Figure 2(b)].

image

Figure 2. Buckling-restrained brace stability condition concepts (AIJ Recommendations for Stability Design of Steel Structures, 2009).

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For Concept (1), the following equations were proposed by Koetaka et al. [8].

The stability condition of the restrained zone:

  • display math(1)

The stability condition of the connection zone:

  • display math(2)

where inline image denotes yield bending strength of the restrainer, a denotes maximum imperfection along the restrainer, e denotes axial force eccentricity, sr denotes the core–restrainer clearance, Ncu 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, inline image denotes the Euler buckling strength of the restrainer, γJEIB denotes the bending stiffness of the connections, L0 denotes the total BRB length, and ξL0 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.

image

Figure 3. Connection with varying stiffness.

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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 Lin 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 ar = a + e + sr + (2sr/Lin)ξL0, as shown in Figure 5. Then, the relationship between the axial force N, and out-of-plane deformations yr at the end of the restrainer can be approximated by the following equation (Figure 6).

  • display math(3)

where inline image 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 inline image asymptotically for increasing values of yr; 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, Ncu, is lower than the stability limit, the brace is considered stable, and the BRB core exhibits regular hysteretic behavior. However, when Ncu 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.

image

Figure 4. Bending moment transfer at restrainer ends.

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image

Figure 5. Assumed initial imperfection.

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image

Figure 6. Buckling-restrained brace stability concepts and limits.

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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 KRg at both gusset plates. Then, an initial imperfection ar is modeled at restrainer ends. When the bending moment at the restrainer-end zone exceeds the restrainer moment transfer capacity inline image, the load on the BRB exceeds its ultimate strength, eventually resulting in buckling.

image

Figure 7. Collapse mechanisms with rotational springs at gusset plates.

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Firstly, the gusset plates are assumed to be rigid (KRg[RIGHTWARDS ARROW] ∞), 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).

  • display math(4)

Then, the strain energy stored in both connection zones is given as follows:

  • display math(5)

The rotation angle of the plastic hinges is expressed as follows:

  • display math(6)

Then, the plastic strain energy stored in the plastic hinges is as follows:

  • display math(7)

The axial deformation is as follows:

  • display math(8)

The work carried out is given as follows:

  • display math(9)

with the following balance of energy differential

  • display math(10)
  • display math(11)

Approximating 8/π2 ≈ 1, the following is obtained:

  • display math(12)

Similar calculations can be carried out for the asymmetrical buckling mode shown in Figure 7(b), as follows.

  • display math(13)
  • display math(14)
  • display math(15)

When ξ = 0.25, N can be approximated as follows:

  • display math(16)

When inline image = 0 and ar ≪ yr, 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, ξκRg can be defined as follows:

  • display math(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 yrs. Given that the deformation because of the connection zone bending, yre, becomes equivalent to yrs when ξκRg = 3, the strain energy stored in a specific spring can be approximated as follows:

  • display math(18)

The gusset plate spring rotation, Δθs, plastic hinge rotation, Δθr, and axial deformation, Δug can be expressed as follows:

  • display math(19)
  • display math(20)
  • display math(21)

Then, the energy stored in the springs, Us, energy stored in the hinges, Up, and the total work done, T, can be evaluated respectively as follows:

  • display math(22)
  • display math(23)
  • display math(24)

With the balance of energy differential ∂(Uε + Us + Up − T)/∂yr = 0,

  • display math(25)

Equation (25) may be expressed as Equation (26) when the rotational stiffness of the gusset plates is negligible (ξκRg ≈ 0).

  • display math(26)

Conversely, if ξκRg[RIGHTWARDS ARROW] ∞, 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).

  • display math(27)

Accordingly, one-sided buckling mode strength, shown in Figure 7(c), can be determined using the following expression.

  • display math(28)

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 ar and yr values. Equation (27) yields lower values than do Equation (25) or Equation (28), thus implying that the asymmetrical mode governs stability.

Here, if ar ≪ yr, Equation (27) can be approximated as follows:

  • display math(29)

where inline image 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 yr. 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 inline image from Equation (3) into Equation (29), the required restrainer moment transfer capacity, Mpr, can be approximated as follows:

  • display math(30)

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 inline image. From Figure 8, this value can be estimated as follows:

  • display math(31)

where δ0 denotes the assumed maximum story out-of-plane drift, and Lin denotes the insert zone length. Including inline image in Equation (29), the following equation for defining the ultimate strength is obtained:

  • display math(32)
image

Figure 8. Additional bending moment under out-of-plane drift.

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When the aforementioned N is substituted for the axial force in Equation (30), the stability condition can be expressed as follows:

  • display math(33)

where inline image should be taken as zero if the difference is negative.

In Equation (33), inline image can be estimated by eigenvalue analysis; however, when γj ≈ 1, the lowest value decided by the elastic symmetrical mode can be approximated as inline image using the following expression:

  • display math(34)

where LκRg is the rotational stiffness of the gusset plate normalized by total length. For satisfying Equation (33), two approaches can be followed.

  1. When ξκRg is sufficiently large, the left part of Equation (33) tends to be small or zero. In order to satisfy Equation (33) inline image should be larger than Ncu. This approach corresponds to Equation (2), which allows for hinges at the restrainer ends (Figure 2(a)).
  2. Decrease M0r and inline image by decreasing ξκRg and providing sufficient bending strength inline image to satisfy Equation (33). This approach corresponds to transferring the bending moment at the restrainer-end 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.

  • display math(35)

where inline image is the plastic bending strength of the gusset plate including the axial force effect, and inline image or inline image should be taken as zero if the difference is negative.

image

Figure 9. Collapse mechanism with plastic hinges at gusset plate ends.

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CYCLIC BRB LOADING TESTS WITH OUT-OF-PLANE DRIFT

  1. Top of page
  2. SUMMARY
  3. INTRODUCTION
  4. STABILITY CONDITION FOR BRBS INCLUDING CONNECTIONS
  5. CYCLIC BRB LOADING TESTS WITH OUT-OF-PLANE DRIFT
  6. CONCLUSIONS
  7. ACKNOWLEDGEMENTS
  8. REFERENCES

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 in-plane 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 1. The core plate material was JIS-SN400B (average yield strength: 270 MPa), and the core cross section size, Ac = 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 κRg = 0.04) and the stiffened type (ξ κRg = 0.3). The insert length of the stiffened part of the core plate into the restrainer, Lin, 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).

image

Figure 10. The cyclic loading test setup.

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image

Figure 11. The loading protocol.

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Table 1. Test matrix.
SpecimenAc (mm2)σcy (N/mm2)EIB (Nmm)σry (N/mm2)KRg (Nmm)γJEIB (Nmm)L0 (mm)Lin (mm)Lp (mm)sr (mm)ξL0 (mm)ξ'L0 (mm)
MRL1.0S1H1080266.05.81 × 1011305.06.90 × 1081.20 × 101223929013801416506
MRL2.0S11080266.85.81 × 1011385.89.73 × 1071.20 × 1012239218012001416596
MRL2.0S21080266.85.81 × 1011391.59.73 × 1071.20 × 1012239218012002416596
MCL2.0S21080269.77.14 × 1011365.79.73 × 1071.20 × 1012239218012002416596
MRL1.0S11080266.85.81 × 1011391.59.73 × 1071.20 × 101223929013801416506
MRL1.0S21080266.85.81 × 1011391.59.73 × 1071.20 × 101223929013802416506

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/Lp) 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 Lp. The expected values of the initial imperfection angles for each specimen are summarized in Table 2.

image

Figure 12. The loading protocol.

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Table 2. Initial imperfection angle.
SpecimenLin (mm)sr (mm)θ0 = 2sr/Lin (rad)
MRL1.0S1H9010.02
MRL2.0S118010.01
MRL2.0S218020.02
MCL2.0S218020.02
MRL1.0S19010.02
MRL1.0S29020.04

The hysteresis loops obtained by cyclic loading tests for each specimen are shown in Figure 13. The normalized cumulative plastic deformation ΣΔεp = ΣΔδp/Lp, and normalized cumulative absorbed energy χw = Ed/σyAc 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-of-plane 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.

image

Figure 13. Axial force-deformation relationship.

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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 inline image, of each specimen was estimated using the following equation proposed by Matsui et al. [17]:

  • display math(36)

where inline image represents the restrainer moment transfer capacity determined by the cruciform core plate at the neck as follows:

  • display math(37)

where inline image denotes the yield axial force of the cruciform core plate at the web zone, inline image denotes the ultimate strength of the cruciform core plate at the neck, Zcp denotes the plastic section modulus at the neck, and σcy denotes the yield stress of the core plate. In Equation (36), inline image represents the restrainer moment transfer capacity determined by the restrainer section at rib end as follows:

  • display math(38)

where Zrp denotes the plastic section modulus of the restrainer tube, σry denotes restrainer yield stress, KRr1 denotes the restrainer elastic rotational stiffness about the rib end, θy1' denotes the pseudo initial yield angle of the rectangular restraint tube, KRr2 denotes the post-yielding rotational stiffness of the restrainer about the rib end, θy2 denotes the angle at which the plastic hinge occurs, θy denotes the yield angle of the circular restraint tube, and Bc denotes the core plate width.

In most cases, inline image is determined by the bending strength of the neck section, inline image, given by Equation (37) when the insert zone length is more than 2.0 times the core plate width. The obtained values of inline image for each specimen are summarized in Table 3. The expected failure axial force assuming elastic springs for gusset plates Nlim1 is obtained by solving Equation (39), which is derived from Equation (33) and summarized in Table. 4.

  • display math(39)
Table 3. Bending capacities at restrainer ends.
SpecimenYield bending strength of cruciform zone (kNm)Yield bending strength of restrainer (kNm)Mpr (kNm)
MRL1.0S1H2.462.972.46
MRL2.0S15.508.385.50
MRL2.0S26.568.566.56
MCL2.0S26.5035.686.50
MRL1.0S16.784.284.28
MRL1.0S26.784.284.28
Table 4. Stability evaluations using proposed equation.
SpecimenNcrB(kN)ar (mm)Ncrr (kN)Ncu (kN)M0r (kNm)Failure cycle at experimentStability limit (kN)Failure axial force in experiment (kN)
       Nlim1 (Equation (39))Nlim2 (Equation (41)) 
  • *

    Ncu is estimated as 1.5Acσcy (approximately equivalent to axial forces of the first 3% normalized axial deformation).

  • ( ): Exceptionally, MRL1.0S1H did not fail, so only the maximum applied axial force is shown above.

MRL1.0S1H188011.46954310.00None8181390(452)
MRL2.0S111586.80824320.093.0%-12cycle520520535
MRL2.0S2115812.4824320.003.0%-2cycle419410507
MCL2.0S2138912.4824370.001.0%-2cycle440432375
MRL1.0S1115811.41114320.000.5%-1cycle367345362
MRL1.0S2115821.71114320.000.5%-1cycle264217300

Here, inline image 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.

  • display math(40)

where ic is the radius of gyration at the connection zone.

Similarly, the expected failure axial force that assumes plastic hinges at the gusset plates, Nlim2, is determined by solving Equation (41), which is derived from Equation (35).

  • display math(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 Nlim1 and Nlim2 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 Ncu (assumed as Ncu = 1.5 × Ac × σcy). The other specimens were designed such that the stability limit failed to exceed the Ncu, 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 Nlim1 or Nlim2 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.

image

Figure 14. Axial force versus out-of-plane displacement relationship.

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A comparison of the estimated failure axial forces Nlim, which is the smaller of Nlim1 and Nlim2, 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.

image

Figure 15. Distribution of proposed/experimental accuracy by specimen type.

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CONCLUSIONS

  1. Top of page
  2. SUMMARY
  3. INTRODUCTION
  4. STABILITY CONDITION FOR BRBS INCLUDING CONNECTIONS
  5. CYCLIC BRB LOADING TESTS WITH OUT-OF-PLANE DRIFT
  6. CONCLUSIONS
  7. ACKNOWLEDGEMENTS
  8. REFERENCES

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.

ACKNOWLEDGEMENTS

  1. Top of page
  2. SUMMARY
  3. INTRODUCTION
  4. STABILITY CONDITION FOR BRBS INCLUDING CONNECTIONS
  5. CYCLIC BRB LOADING TESTS WITH OUT-OF-PLANE DRIFT
  6. CONCLUSIONS
  7. ACKNOWLEDGEMENTS
  8. REFERENCES

The authors would like to acknowledge the support of Nippon Steel and Sumikin Engineering Co. Ltd for this study.

NOTATIONS
a:

maximum imperfection along the restrainer

ar:

total initial imperfection

e:

axial force eccentricity

ic:

radius of gyration at connection zone

lB:

restrained zone length

sr:

clearance between core and restrainer

yr:

out-of-plane deformation at restrainer end

yre:

additional out-of-plane deformation due to connection zone bending

yrs:

additional deformation due to end spring rotation

Ac:

core plate cross section

Bc:

core plate width

EIB:

bending stiffness of restrainer

Ed:

absorbed hysteretic energy until instability or fracture

KRg:

rotational spring stiffness at gusset plate

KRr1:

elastic rotational stiffness of restrainer about rib end

KRr2:

post-yielding rotational stiffness of restrainer about rib end

Lin:

insert zone length

Lp:

plastic zone length of core plate

inline image:

additional bending moment derived from story out-of-plane drift

inline image:

bending strength of restrainer

inline image:

plastic bending strength of gusset plate including axial force effect

inline image:

restrainer moment transfer capacity

inline image:

restrainer moment transfer capacity determined by cruciform core plate at neck

inline image:

restrainer moment transfer capacity determined by restrainer section at rib end

N:

axial force

Ncu:

maximum axial strength of core plate

inline image:

global elastic buckling strength of BRB including effect of gusset plate rotational stiffness

inline image:

global elastic buckling strength with pin conditions at restrainer ends

inline image:

Euler buckling strength of restrainer

inline image:

expected failure force assuming elastic gusset plates

inline image:

expected failure force assuming plastic hinges at gusset plates

inline image:

ultimate axial strength of cruciform core plate at neck

inline image:

yield axial force of cruciform core plate at web zone

Ny:

yield axial force of core plate

T:

external work

Up:

plastic strain energy stored in plastic hinges

Us:

energy stored in springs

Uε:

strain energy stored in both connection zones

Zcp:

plastic section modulus at neck

Zrp:

plastic section modulus of restrainer tube

γJEIB:

bending stiffness of connection zone

δ0:

story out-of-plane drift

δ:

axial deformation of BRB

δp:

axial plastic deformation of BRB

ξL0:

connection zone length

ξ'L0:

length between gusset plate end and rib end ( = ξL0 + Lin )

Δθr:

rotational angle of plastic hinge at restrainer end

Δθs:

rotation at gusset plate

Δug:

axial deformation caused by global buckling

χw:

normalized cumulative absorbed energy (=ΣΔδp/Lp)

φ0:

rotational angle at gusset plate due to out-of-plane drift

ξκRg:

normalized rotational stiffness for gusset plate

LκRg:

normalized rotational stiffness for gusset plate by total length

λr:

equivalent slenderness ratio for global elastic buckling strength with pin conditions at restrainer ends

θ0:

initial imperfection angle of connection zone

θy:

yield angle for circular restrainer tube

θ ' y1:

pseudo initial yield angle for rectangular restrainer tube

θy2:

angle at which plastic hinge occurs at restrainer end

ΣΔεp:

normalized cumulative plastic deformation (=Ed/σyAc)

σcy:

yield stress of core plate material

σry:

yield stress of restrainer tube material

REFERENCES

  1. Top of page
  2. SUMMARY
  3. INTRODUCTION
  4. STABILITY CONDITION FOR BRBS INCLUDING CONNECTIONS
  5. CYCLIC BRB LOADING TESTS WITH OUT-OF-PLANE DRIFT
  6. CONCLUSIONS
  7. ACKNOWLEDGEMENTS
  8. REFERENCES
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