Out-of-plane stability of buckling-restrained braces including their connections

Transcription

1 Out-of-lane stability of buckling-estained baces including thei connections T. Takeuchi, R. atsui & T. Tada Tokyo Institute of Technology, Jaan K. Nishimoto Nion Steel Engineeing Co., Ltd., Jaan SUARY: uckling-estained baces (Rs) ae widely used in seismic counties as ductile seismic-esistant elements and enegy dissiating elements. One of the key limits of Rs is oveall flexual buckling, and they ae equied to exhibit stable hysteesis unde cyclic axial loading with out-of-lane difts, thei stability unde such conditions being essential. Howeve, many eseaches ae indicating that thee ae isks of oveall buckling including connections befoe the Rs yield. In this ae, the stability conditions of Rs including thei connections ae discussed based on cyclic loading tests with out-of-lane difts, and a unified simle equation fo evaluating thei stability is oosed. Keywods: uckling-estained bace, Connections, Cyclic loading, uckling, Stability condition 1. INTRODUCTION uckling-estained baces (Rs) ae exected to exhibit stable hysteesis unde cyclic axial loading with out-of-lane difts, and thei stability unde such conditions is essential. Howeve, many eseaches have indicated that thee ae isks of oveall buckling including connections befoe the baces yields, when lastic hinges ae intoduced at the ends of the estaines (Fig.1.1). In this ae, the stability conditions of Rs including thei connections ae discussed and equations evaluating such conditions ae oosed. Cyclic axial loading tests of Rs with initial out-of-lane dift ae caied out, and the validity of the oosed equation is confimed. Connection Plastic Hinge Plastic Zone Restaine JEI L JEI > EI EI l L ending oment Tansfe EI K Effective uckling Length kl Plastic Zone JEI L JEI > EI Elastic Zone K Figue 1.1. Oveall uckling of R including connections (a) Plastic Hinge Concet (b) ending oment Tansfe Figue.1. R Stability Condition

2 . STAILITY CONDITION FOR RS INCLUDING CONNECTIONS In AIJ Recommendation fo Stability Design of Steel Stuctues (9), following two concets fo R design to sustain stability including connections ae indicated, as shown in Fig..1. 1) Plastic hinges ae allowed at the estaine-ends, and the stability conditions ae given fo the estained at and connections individually [Fig..1(a)]. ) ending moment tansfe is exected at the estaine-ends, and comosite stability of the estained at and connections is confimed [Fig..1(b)]. Pin-ends tyes ae included in this categoy. Fo the concet 1), the following equations ae oosed by Kinoshita et.al (7). The stability condition of the estained at; an cu y (.1) 1 N N cu c The stability condition of connections; J 1 JEI N cu (.) ( L ) whee, y : bending stength of the estaine; a : exected imefections; sum of a (estaine imefection), e (axial foce eccenticity), and s (cleaance between coe and estaine); N cu :maximum axial foce of coe lates, nomally estimated times of yield foce of the coe late including hadening; N c:eule buckling stength of the estaine; γ J EI :ending stiffness of the connections; and ξl :connection length. This concet is based on the condition that the ends of the connections ae igidly fixed against otation; howeve, vey stiff gusset lates ae equied to satisfy this condition. Fo examle, a stiffened gusset late as in Fig.. (c) is essential. oeove, eventing the otation of the beam whee Rs ae connected, lage stiffening beam in out-of-lane diections ae equied as shown in Fig... The othe design concet of Rs is the tansfe of bending moment at the estaine-ends as in the concet ) in AIJ ecommendation, which confim the stability of the estained at and the connections comositely as shown in Fig..1(b). Tekeuchi et.al (9) indicated that the estaine-ends can tansfe the bending moment u to the bending stength of the estaine o stiffened coe section, if the stiffened ends of the coe lates ae inseted into the estaine by moe than two times the coe late width (L in > c in Fig..4(a)). Whee, they oduce an initial imefection a =a+e+s +(s / L in )ξl (Fig..4(b)) and the ocess of oveall buckling can be descibed by a simle model as in Fig..5. As in the figues, the R ae modelled as a bending element with otational sings K (K = fo in-ends) at both ends and initial imefection a. When the bending moment eaches the bending stength of the estaine-ends, the bace collases. Fistly, the ends of the connections ae assumed to be igid (K = ) and out-of-lane defomations of the connections in the mechanism hase ae assumed to be cosine cuves as in Fig..5(a); a x y y 1cos L L Then the stain enegy stoed in both connection is; L 4 J EI d y JEIy U dx dx ( L ) (.4) Connected eam (.) Stiffene Rib (a)low Stiffness (b)edium Stiffness (c)high Stiffness eam Stiffening beam R R Figue.. Connection with vaious stiffness Figue.. Rotational stiffene at R connections

3 The otation angle of the lastic hinges is; dy a y (.5) dx L L x L Then the lastic stain enegy stoed in the hinges ae; y U L (.6) The axial deflection is; L 1 dy a y ay ug dx dx L 8 L L (.7) The wok done is; ( y 16 ay / ) T Nug N 8 L (.8) with the incile of stationay total otential enegy; 4 ( U U T) JEIy y ( y 16 ay / ) y 16( L ) L 8L (.9) 4 JEIy y (.1) ( L ) y 8 a / ( y 8 a / ) Aoximating 8/π as 1, we obtain the following. J EI 4 (.11) ( L ) y a Simila calculation can be caied out in an asymmetical mode as shown in Fig..5(b), as follows. Reinfocement Zone Cucifom Zone Lin c (a) Detail (b) odel Figue.4. ending oment Tansfe at Restaine End Plastic hinge Coe Plate Restaine s Cleaance s Lin ξl θ s Restaine s ξ L N Ime -fection Plastic hinge Imefection (a) Symmetic mode (b) Asymmetic mode Figue.6. Assumed Pocess of Oveall uckling K Plastic hinge Imefection Collase L a +y γ k (a) Symmetical (b) Asymmetical (c) One-side Figue,5. Oveall uckling with Rotational Sings L L in L s K (1 ) L Lin Figue.7. Additional ending oment by Out-of-lane Dift

4 U y 4 L 1 (.1) N T ( y 16 ay / ) (.1) 8 (1 ) L (1 ) JEIy y 4(1 4 / ) (.14) ( L ) y 8 a / ( y 8 a / ) When ξ =.5, it can be aoximated as follows: (1 ) J EI y (.15) ( L ) y a y a When = and a <<y, Eq.(.15) aoaches Eq.(.). As indicated by Eqs.(.11) and (.15), the oveall buckling stength is detemined by the asymmetical mode when the ends of the connections ae igidly fixed. Next, conside otational stiffness K. Define nomalized otational stiffness κ as follows: K L (.16) J EI As in Fig..5(a), additional defomation by the otation of the end sing is defined as y s. As defomation by connection bending y e become equivalent to y s when κ =, the stain enegy stoed in the sings is estimated as; 4 JEIy U (.17) ( L ) The sing otation Δθ s, lastic hinge otation Δθ, and axial defomation can be exessed as follows: y s (.18) L y 6 y y (.19) L L L y 4 / ay 1 y ay ug (.) L 16 L 8 Then the enegy stoed in the sings and hinges and the woks done can be evaluated, esectively; JEIy U s (.1) ( L ) y 6 U (.) L ( y ) 4 / ay T N (.) 8L Fom the condition ( U U T)/ y, 6/ JEIy y 4 (.4) ( L ) 4 / y a y a 4 / In the above equation (.4) become Eq.(.5) when the connection ends ae inned (κ = ). (.5) y a On contay, Eq.(.11) can be estoed when κ =. Hence Eq.(.4) coves symmetical buckling stength fo vaious otational stiffness fom in-ends to igid-ends. Asymmetical stength can be deived by simila ocess as; N c (1 ) ( L ) 4 / JEIy y y a y a (.6)

5 Similaly, stength fo one-side buckling mode as shown in Fig..5(c) can be deived as follows. (1 ) J EI y (.7) ( L ) (1 )( 4 ) y a y a Eqs.(.6), (.7) aoaches Eq.(.5) when κ = and Eq.(.15) when κ =, also coveing asymmetical buckling stength fo vaious otational stiffness. Eq.(.6) gives slightly lowe values than Eq.(.7). Eqs.(.4), (.6) and (.7) all indicate that the axial foce deceases as the out-of-lane dislacement y inceases. When the elastic axial foce and out-of-lane dislacement elationshi exessed by Eq. (.8) eaches this condition, the bace is consideed to be collased [Fig..6(a)]. ym y N (.8) amym a y whee, N c is oveall elastic buckling stength which can be deived as follows EI L L L L 14L 64 (.9) L L K EI Substituting /( y an N) into Eq. (.6), the equied bending stength can be deived as follows: a (1 ) JEIy y a N NN c 1 N/ ( L ) 4/ y a 1 N N (.) c When the stuctue defoms out-of-diection not only axial defomation, additional bending moment is distibuted as in Fig..7. The initial bending moment at the estaine-end can be estimated as; s (1 ) (1 ) K (.1) L Lin whee, δ is exected stoy dift in an out-of-lane diection. ending moment stength at estaine-ends ae consideed to be educed by this initial moment. Consequently, the stability condition can be exessed as follows, with the condition that the coss oint of Eqs.(.8) and (.) (see Fig..8) exceeds the exected maximum axial foce N cu. a Ncu whee, (.) 1 Ncu whee, :Ultimate bending stength of estaine-ends, : Initial bending moment at estaine- ends [Eq.(.1)], a : Initial imefections =a + e + s + (s / L in )ξl, N cu : Exected maximum axial foce of R = Yield foce multilied by hadening facto, N c: Elastic oveall buckling stength [Eq.(.9)], N c: Elasto-lastic buckling stength caused by connections using the following equivalent slendeness atio by Eq.(.). In this equation (.), ξ' in Fig..4(b) instead of ξ should be used estimating lastic hinges can be oduced at the neck of the einfoced zone of the coe late. ' L 4 / (.) i (1 ') whee N c must satisfy the following limit to eventing the yield at oute ends of the gusset lates: 1 g g N c (.4) a 1 ' whee, g c : ending stength of the oute ends of the gusset lates including the effect of axial foce. To satisfy Eq.(.), two aoaches can be used fo R design. 1)Decease and N c, by deceasing κ, and ovide enough bending stength at the estaineend. This concet coesonds to tansfeing bending moment at the estaine-ends. [Fig..1(b)]. )When κ is lage, the left at of Eq.(.) becomes small o zeo, so satisfy Eq.(.) by designing N c lage than N cu. This concet coesonds to Eq. which allows hinges at the estaine-ends. [Fig..1(a)]. As above, Eq.(.) coves both design concets discussed in Fig..1.

6 . CYCLIC LOADING TEST OF R WITH OUT-OF-PLANE DISPLACEENT To confim the stability including the connections, cyclic loading tests of the R with out-of-lane dislacement ae caied out. The test configuation with secimens is shown in Figs..1 and., the loading ogam ae shown in Fig.., and the test matix is shown in Table.1. The coe lates ae JIS-SN4 (aveage yield stength: 7Pa) 1mm thick and 9mm wide, the estaines ae mota in-filled box section of 15mm squae and.mm thick o cicula tube of 19.8 dia. and.mm thick. The inset length of the stiffened at of the coe lates into the estaines L in can be 18mm, 9mm, 45mm, which is. times, 1. times and.5 times of the coe width, esectively. In addition, the cleaance between the coe late and the estaines ae vaied fom 1.mm to.mm, and 6 diffeent secimens ae tested. The secimens ae labelled as -(R:Requtangula, C:Cicula)-L(Inset length atio) -S-(Cleaance). The same gusset lates ae used in all secimen which have a small otational stiffness (κ.4). Initial imefection angles in each secimen ae summaized in Table.. N c N Yield Stength (.) a y ' Stability limit N cu Elasto-lastic Path Failue Path a Elastic Path y N a y (.8) Figue.8. Load-Deflection Relationshi a y Initial Dift Angle 1/1 15mm b =1/18 Figue.1. Cyclic Loading Test Setu Lin Lin 16 a b CoePlate:PL-1(SN4) C L c 1 ota Edge Cleaance s a b 8-18 c 4 8 A-A View Stiffene:PL-1(SS4) 4-1 Long Nut (L=5) Long Nut (L=5) 75 4 GPL-1(SS4) A Restaine: -15.(STKR4) Lin Cushion Lin 16 a CL b c CoePlate:PL-1(SN4) Edge Cleaance s a ota - View b 4-18 c Stiffene:PL-1(SS4) A Ue Gusset Plate GPL-1(SS4) PL-1 1 PL-1 a-a section ota 15 9 b-b section -15. ota 18 1 c-c section PL-1-1 Long Nut (L=5) 7-1 Long Nut (L=5) Lowe Gusset Plate Figue.. Test Secimen

7 Table.1. Test atix Secimen A c (mm ) σ cy (N/mm ) EI (Nmm) σ y (N/mm ) K (Nmm) γ JEI (Nmm) L ξl ξ ξ'l ξ' RL.S RL.S CL.S RL1.S RL1.S CL1.S Axial Stain (%) Cycle Figue.. Loading Potocol Table.. Initial Imefection Angle Secimen L in s θ = L in / s (ad) RL.S1 1.1 RL.S 18 CL.S. RL1.S1 1. RL1.S 9 CL1.S.4 efoe the test, out-of-lane defomation equivalent to the stoy dift of 1% adian is alied to each secimen, and then axial cyclic defomation equivalent u to 1-% of the lastic length of the coe late is alied. This nomalized axial stain is oughly equivalent to in-lane stoy dift angle. The hysteesis loos obtained fo each secimen ae shown in Fig..4 to Fig..1. RL.S1 (Fig..4) showed stable hysteesis u to 1 cycles of % nomalized stain, until out-of-lane instability aeaed. This efomance is consideed to be satisfactoy fo enegy-dissiation baces. RL.S (Fig..5) which has slightly lage initial imefection than evious one, showed stable hysteesis until cycles u to % nomalized stain, then out-of-lane instability aeaed. CL.S (Fig..6) is constituted by a cicula mota in-filled steel tube, showed stable hysteesis until cycle u to % nomalized stain, until aeaance of out-of-lane instability. RL1.S1 (Fig..7) eached the yield stength of the coe late and showed stable hysteesis u to the nd cycle of 1.% nomalized stain, then exeienced oveall buckling hinged at the estaine-ends. RL1.S (Fig..8 and Fig..1) showed a hysteesis loo fo only one cycle of.5% nomalized stain, then exeienced oveall buckling hinged at the estaine-ends. CL1.S (Fig..9) exhibited a hysteesis loo fo only one cycle of.5 nomalized stain, then undegoes oveall buckling hinged at the estaine-ends. 4. COPARISON WITH THE PROPOSED EQUATION These test esults indicate that the stabilities of Rs ae stongly affected by the inset length atio and cleaance, which is exected fom the oosed Eq.(.). In the following, each secimen is evaluated using Eq.(.). Fo the evaluation, bending stength of each secimen at the estaine-ends needs to be estimated. Takeuchi et.al (9) oosed the following equations fo the tested tyes of R: coe est min, (4.1) est eesents the bending stength of the estaine end as follows: min, 1 1' 1' Rectangula Tube est Zy KR y KR y y (4.) min Z, K Cicula Tube y R1 y

8 Axial Foce (kn) Axial Foce (kn) Axial Foce (kn) 6 Out-of-lane buckling (Ue side,.% 1-cycle) Out-of-lane buckling (Ue side, 1.% -cycle) Dislacement Rotational Angle (ad) Dislacement Rotational Angle (ad) (a) Foce-Defomation (b) Foce-Angle (a) Foce-Defomation (b) Foce-Angle Figue.4. RL.S1 Figue.7. RL1.S1 6 Out-of-lane buckling (Ue side,.% -cycle) Out-of-lane buckling -4 (Ue side,.5% 1-cycle) Dislacement Rotational Angle (ad) Dislacement Rotational Angle (ad) (a) Foce-Defomation (b) Foce-Angle (a) Foce-Defomation (b) Foce-Angle Figue.5. RL.S Figue.8. RL1.S Out-of-lane buckling Out-of-lane buckling (Cucifom Pat Yield,.% 1-cycle) (Ue side,.5% 1-cycle) Dislacement Rotational Angle (ad) Dislacement Rotational Angle (ad) Axial Foce (kn) Axial Foce (kn) Axial Foce (kn) (a) Foce-Defomation (b) Foce-Angle Figue.6. CL.S Table 4.1. ending Caacities at the Restaine Ends Axial Foce (kn) Axial Foce (kn) Axial Foce (kn) Axial Foce (kn) Axial Foce (kn) Axial Foce (kn) (a) Foce-Defomation (b) Foce-Angle Figue.9. CL1.S Yield Stength of Yield Stength of Secimen Cucifom Zone Restaine (Nmm) (Nmm) (Nmm) RL.S RL.S CL.S RL1.S1 RL1.S CL1.S (a) uckling (b) uckling Zone Figue.1. RL1.S Collase ode whee, Z is the lastic section modulus of the estaine, σ y is the yield stess of the estaine, K R1 is the elastic otational stiffness at the estaine-ends, θ y1 is the seudo initial yield angle fo the ectangula estaint tube, K R is the otational stiffness at the estaine-ends afte yielding, θ y is the angle that lastic hinge occus, and θ y is the yield angle fo the cicula estaint tube. coe eesents the bending stength of the cucifom coe late as follows: c Ncu N wy 1 c c Nu N wy Z coe c cy (4.)

9 whee, N cu is the maximum axial foce of the coe late, N wy c is the yield axial foce of the cucifom coe late at the web zone, N cu is the ultimate stength of the coe late, Z c is the lastic section modulus of the coe late, and σ cy is the yield stess of the coe late. The study indicates that is decided the cucifom section whose stength given by Eq.(4.) when the inset length atio exceeds aound.. The obtained values of in each secimen ae summaized in Table 4.1. The conditions fo each secimen ae evaluated using the safety Index of Eq.(4.4), and the esults ae summaized in Table 4.. Out of six secimens, only RL.S1 satisfies the condition, fo which safety indices given by; a ( ) Ncu (4.4) 1 Ncu is The safety index of RL.S and CL.S is.6 and.68 esectively, which is slightly Table 4.. Stability Evaluations using the Poosed Equation N c a N cu N c Safety Exeimantal Secimen (kn) (kn) (kn) (knm) Index Result RL.S %-1cycle RL.S 8.6.%-cycle 1.4 CL.S %-cycle RL1.S %-1cycle RL1.S %-1cycle 1.7 CL1.S %-1cycle Axial Foce (kn) Estimated Stability Limit Eq. (9) Restaine Stength Eq. (6) Cucifom Stength Eq. (7) Test Result Axial Foce (kn) Eq. (9) Cucifom Stength Eq. (7) Restaine Stength Eq. (6) Axial Foce (kn) Axial Foce (kn) Out-of-lane Dislacement y +a (a) RL.S1 1 Restaine Stength Eq. (6) 8 Eq. (9) 6 Cucifom Stength Eq. (7) 4 Test Result Out-of-lane Dislacement y +a (b) RL.S Eq. (9) Restaine Stength Eq. (6) Cucifom Stength Eq. (7) Test Result Out-of-lane Dislacement y +a Axial Foce (kn) Axial Foce (kn) Out-of-lane Dislacement y +a Restaine Stength Eq. (6) Eq. (9) Cucifom Stength Eq. (7) Test Result Out-of-lane Dislacement y +a 1 Restaine Stength Eq. (6) (d) RL1.S1 (e) RL1.S Eq. (9) Cucifom Stength Eq. (7) Test Result Out-of-lane Dislacement y +a (c) CL.S (f) CL1.S Figue 4.1. Axial Foce vs. Out-of-lane Dislacement

10 unsatisfactoy fom Eq.(.). All othe secimens have much lowe values, which indicate that thei oveall stabilities ae not guaanteed. In total, the given safety values satisfactoy estimate the efomance of each secimen obtained in the cyclic loading tests and theefoe ae consideed to be valid. Fig.4.1 shows the measued axial foce-dislacement elationshis comaed with the equations discussed in Sec.. The test esults ae well estimated by the oosed equations so the oosed equations ae consideed to be valid. 5. CONCLUSIONS The oveall stabilities of Rs ae discussed and confimed by cyclic loading test with out-of-lane dislacement. The conclusions eached ae summaized as follows. 1) The stability conditions fo Rs can be exessed by a single equation using a simle hinge model with end sings. This equation coves both design concets of Rs discussed in AIJ ecommendation 9. ) In the cyclic loading tests, secimens with lesse inset length at the estaine-ends exeience oveall buckling befoe achieving stable hysteesis, which is not satisfactoy as the standad efomance of a R. In contast, secimens with lage inset length showed stable hysteesis u to % ) The oosed equation exlains well the efomance of each secimen in the test, and is consideed to be valid. AKCNOWLEDGEENT The authos would like to acknowledge that the eseach is suoted by Nion Steel Engineeing Co. Ltd., and JFE Engineeing Co. Ltd. REFERENCES Achitectual Institute of Jaan (9) Recommendation fo Stability Design of Steel Stuctues, Sec..5 uckling Restained aces. Takeuchi, T., Yamada, S., Kitagawa,., Suzuki, K., and Wada, A. (4) Stability of uckling Restained aces Affected by the Out-of-lane Stiffness of the Joint Element, Jounal of Stuctue and Constuctional Engineeing, 575, (in Jaanese) Kinoshita, T., Koetaka, Y., Inoue, K. (7): Citeia of uckling Restained aces to Pevent Out-of-lane uckling, Jounal of Stuctue and Constuctional Engineeing, 1, (in Jaanese) Chou, C. C., Chen, P. J. (9) Comessive behavio of cental gusset late connections fo a buckling-estained baced fame, Jounal of Constuctional Steel Reseach, 65, atsui, R., Takeuchi, T., Nishimoto, K., Takahashi, S., and Ohyama, T. (1) Effective uckling Length of uckling Restained aces Consideing Rotational Stiffness at Restaine Ends, 7th Intenational Confeence on Uban Eathquake Engineeing & 5th Intenational Confeence on Eathquake Engineeing Poceedings, Hikino, T., Okazaki, T., Kajiwaa, K., Nakashima,. (11) Out-of-Plane stability of buckling-estained baces, Poceeding of Stuctual Congess, ASCE, 11