Elastoplastic analysis of knee bracing frame

784Huang et al. /J Zhejiang Univ SCI 2005 64(8):784-789Jourmal of Zhejiang University SCIENCEISSN 1009-3095htp://ww.zju.edu.cn/jzusJzuSE-mail: jzus@zju.edu.cnElastoplastic analysis of knee bracing frameHUANG Zhen (黄真)"， LI Qing-song (李庆松), CHEN Long-zhu (陈龙珠)*(Department of Civil Engineering, Shanghai Jiao Tong University, Shanghai 200030, China)'E-mail: huang. zhen_ sh@263.net; lzchen@milsjtu.edu.cnReceived Apr. 18, 2004; revision acepted Aug. 15, 2004Abstract: The knee bracing steel frame (KBF) is a new kind of energy dissipating frame, which combines excellent ductility andlateral siffness. As the structural fuse of the frame, the knee element will yield first during a severe earthquake so that no damageoccurs to the major structural members and the rehabilitation is easy and economical. To help fully understand the relations be-tween its seismic performance and the structural parameters, systematic eclastoplastic analysis of the KBF structure with finiteelement method was conducted in this work. Finally, general design recommendations were made according to the results of theanalysis.Key words: Steel frame, Knee bracing frame, Elastoplatic analysisdoi: 10.163 1/jzus.2005.A0784Document code: ACLC number: TU31INTRODUCTIONyielding of the shear link in a severe earthquake, theTrame provides reliable protection from buckling.Steel framed structures are widely used in in-However, as the major part of a frame, the beamdustrial and commercial buildings. According to the should not be severely damaged in view of the diff-different lateral load resisting system, the steel frames culties and costs required for rehabilitation of thecan be mainly divided into four kinds(Fig.1a): the beam. A new braced frame, called knee bracing framemoment-resisting frame (MRF)， concntrically (KBF), having all the favorable features of the abovebraced frame (CBF)， eccentrically braced frameframes but without having the deficiencies, was pre-(EBF), and knee bracing frame (KBF).sented t, Aristizabal-Ochoa (1986) and further in-Fig.lb shows the difference in the lateral per- vestigated by Sam et al.( 1995), Mofid and Khosraviformances of the above frames that have similar(2000)，Balendra et al.(2001) and William etstructural parameters. Although the MRF is an ex- al.(2002). The KBF uses a secondary structuralcellent energy dissipating system, its members have to member (the knee member) instead of the shear linkbe designed with uneconomically large sections toas the“structural fuse”" to ensure enough ductility, butmeet the drift requirement. The CBF is much stiffer achieves excellent lateral stiff ness through the settingthan the MRF, but it cannot meet the ductility re- of the diagonal brace. By limiting the plastic hingesquirement due to the buckling of the brace. To over-formed in the knee only, the major parts of the .come the deficiencies of the MRF and the CBF，structure are safe and the rehabilitation may then beReoder and Popov (1978) proposed a new structural easy.system, named EBF. It combines sufficient stiffnessThe knee bracing steel frame (KBF) is a newand excellent ductility by setting the brace eccentri- kind of energy dissipating frame which combinescally to the beam to form a shear link. Due to the excellent ductility with lateral stiffiness. As thestructural fuse of the frame, the knee element will"Project (No. 2002CB412790) supported by the National Basicyield first duringrthruake cn_ _that noResearch Program (973) of Chinadamage occurs to中国煤化工:rs andYHCNM HGHuang et al. /J Zhejiang Univ SCI 2005 6.4(8):784-789785the rehabilitation is easy and economical. To help shear module, G= 7.69x10' N/m2; and the yieldingfully understand the relations between its seismie strength, f=2.1x108 Nm2. The profiles of the struc-performance and the structural parameters, systematic tural elements are wide flange H. The horizontal loadelastoplastic analysis of the KBF structure with finite is on point A (Fig.2a).element method was conducted in this work. Finally,As shown in Fig.2b, the yielding procedure ofgeneral design recommendations were suggested by KBF could be divided into two stages. First, yieldingthe analysis results.will occur in the knee member under the action of thelateral force F. At the moment, plastic hinges in theknee-column and the knee-beam connections, and theBASIC PARAMETERS OF KBFmidpoint of the knee will develop simultaneously.From then on, the structure turns into the energy dis-Fig.2a is a typical knee bracing frame with basic sipating stage of the knee, which means that the braceresearch parameters. Fig.2b is the corresponding system has reached its ultimate bearing capacity, andforce-displacement curve. All materials used in the the succeeding load should be carried by the mainanalysis are supposed to have ideal elastic-plastic frame until further plastic hinges occur in the columnsproperties: the elastic module, E=2x10" Nm2; the or the beam, after which a secondary energy dissipat-00( CBF▲Buckling point of diagonal braceYield point of column/beamYield point of kneeTkx7. = 20203040506070(b)H 16100x10x10Fig.4 Lateral performance of frames with different kneeelements(a) Force-displacement curves of frames with different x val-ues; (b) Force-displacement curves of frames with differentknee sectionsFig.3 Basic research parametersIn Fig.4a, x (x=B2/B=H2/H) is 0.15~0.5. ThisPosition of knee bracingfigure shows that the position of the knee has influ-The research ofAristizabal-Ochoa (1986) ence on the behavior of the KBF structure. Decreaseshowed that the cross section of knee bracing should of x greatly increases the ultimate structural bearingmeet the requirement that the yield moment of knee capacity and ductility. Research by William etbracing is smaller than 50% of the frame column yieldal.(2002) revealed that in order to have good energymoment. But he did not give the details of, and dis- dissipating capacity, the knee element should be incuss and explain the limitation of the knee cross sec- bending failure mode rather than shearing failuretion. The research of Mofid and Khosravi (2000)mode, that is, Eq.(1) should be satisfied.yielded only an option parameter x without suggestedvalue. The x value and the lateral performance ofkx24Mpvp中国煤化工(1)MHCNMHGHuang et al. /J Zhejiang Univ SCI 2005 6.4(8):784-789787where k is the length of knee,Mp is the bending also increases the difficulty of construction. Com-moment, Vp is the shear force.pared to them, the inclined brace member of KBF hasIfx is not less than 0.15, the above equation can many advantages. First, the KBF can be protectedbe satisfied. The ultimate load reduces as x increases, with only small cross section knee elements. Second,and tends to a certain value when.x is larger than 0.30. from Fig.5a, increasing the cross section of the in-At the same time, the ductility tends to decrease. With clined brace members cannot improve the lateralfurther increasing of x, thelateral stiffness of the stiffness of the structure. So for economy and con-structure in the elastic stage appears somewhat small venience of construction, the cross sectional area ofand the safety of the major structural members is inclined brace members of KBF should be smalldifficult to control. Therefore it is better to choosex rather than unduly large in order to satisfy the re-(the position of knee) of 0.15~0.30.quirement of stability.Fig.4b shows the force-displacement curves ofKBF frames with different knee sections. In Fig.4bthe sectional dimensions are H 40x40x4x4~H 100x0E100x7x7, that is IK/Ic=1.5%~ 44%, where Ik is mo-60ment of inertia of knee element and Ic is moment ofinertia of column. The ultimate bearing capacity isreduced as the area of the knee element is reduced,-日- H 185x185x15x15一-- H 100x100x10x10especially in elastic stage. According to the researchoF1results of Aristizabal-Ochoa (1986)， the yielding0日that of the column. Same result was obtained in thiso日research.' When the sectional area of the knee element1020304050607Cis near that of the column, the yielding point of theShift at point A (mm)knee is near the yielding point of the beam or column.(aTherefore it is difficult to ensure that the knee elementyields first in frame. If Ik/Ic <40%, the failure momentW/2 .of the frame should be more than 1.2 times theyielding moment of the knee element, and so effc-Psinβtively guarantees the safety of the main frame. But ifthe knee element is too small, the behavior of the totalP。 PcosBKBF frame should be more like the behavior of an1)MRF with low stiffness and high ductility. In suchcase, the knee element easily fails under conditions ofPsinpl2normal wind loads, frequent earthquake, or severeP.sinpB/2earthquake. The analysis indicates that, whenI0.13,N+- M__1(4)suitable inclined brace cross section: .NN。1.15M。Pr2 >yPk .(2)where Mp, Np are the section yielding moment andwhere γ is the safety factor, Pr is the yielding load ofyielding axial force respectively.inclined brace member which yield along the weakLateral performance of frames with different col-axis; Pk is the yielding load when the knee memberumns and beamsyields. According to the simplified model shown inThe section stiffness of beams and columns inFig.5b, the axial force in knee member is Pksin.β/2 andthe frame is determined by their length and crossthe maximum moment in knee member is PklkcosB/8.sections. Their influences on the lateral performancePk can be deduced with the fllowing formula of the of the frame can be seen in Fig.6 and Fig.7.total section yielding theory (Shen et al, 2000). .Fig.6a shows the force-displacement curves offrames with different beam lengths. Fig.6b showsN≤0.133) force-displacement curves of frames with differentVp~beam sections. These figures show that the changing0fi 60---B-4.0 m- H120x 120x8x8。-B-3.2m-母H 110x110x8x8H 100x 100x7x7B-2.4 mH 90x90x7x7ofH 80>80*6x60203040506070Shift at point A O (mm)Shift at point A△(mm)a)(bFig.6 Lateral performance of frames with different beams. Force-displacement curves of frames withdifferent beam lengths (a) and with different beam sections (b)互80f80[os- H=2.4m- H 135x 135x 10x10_ H8m--0- H 125x 125x8x8-H=3.6 moHH 110x110x8x8.H=4.0 m. H 100x 100x8*80580Fig.7 Lateral performance of frames with different column length (a) an中国煤化工,YHCNMHGHuang et al. 1J Zhejiang Univ SCI 2005 6.4(8):784-789789of beam stiffness has only lttle influence on the lat- tematicanalysiseral bearing capacity and the ductilty of KBF.structure conducted in this work yielded results ap-Fig.7a shows the force-displacement curves ofplicable for general design and suggesting that:frames with different column length. Fig.7b shows the1. The position and stiffiness of the knee is theforce-displacement curves of frames with different most important factor affecting the lateral resistingcolum cross section. Evidently, the changing ofability of KBF and has great influence on its energycolumn length and cross section area has much more dissipating behavior. According to the investigation,influence than that of beam length and cross section the value of IK/Ic should be 20%~ 40% and the x valuearea. The most obvious influence can be seen with theshould be 0.15~0.3. In that way the structure haschanging of column length. Fig.7a shows that with the enough lateral stiffiness and excellent ductility, and atincreasing of column length, the lateral stiffness in the same time the failure of knee element under fre-elasticity and plasticity stages reduced greatly; the quent earthquake can be avoided.ultimate bearing capacity decreased linearly. But the2. Inclined bracing member of KBF can be de-ductility was much improved while its stiffness was signed with Per> pyPk. Too large cross section area ofdeclining. Fig.7b shows that increasing of cross sec-inclined brace can make construction diffcult anction area had almost no influence on the lateral stiff- waste material, and does not improve the structuralness at the elasticity stage, but the plastic behavior lateral resisting ability.changed a lot. Therefore, the lateral ability and energy3. As the main members of frame, the beam anddissipating ability of the steel frame can be feasibly columns have influence on the lateral behavior ofadjusted by changing column stiffness.KBF frame. Changing the column section is muchIn a building, because the length of beams and more effective than changing the beam cross sectioncolumns cannot be changed easily, the lateral behav- area.ior of the frame can be improved through adjustingthe knee elements and the cross sectional dimensionsReferencesof beam and columns. As the main frame element, Aristizabal-Ochoa, J.D., 1986. Disposable knee bracing: im-changing the cross section area of column is muchprovement in seismic design of steel frames. Journal ofStructural Engineering, 112(7):1544-1552.more effective than changing the beam. As the mainlateral force resisting member, the knee element plays ,Balendra, T, Yu, C.Y, Xiao, Y, 2001. An economical struc-tural system for wind and earthquake loads. Engineeringan important role. With suitable cross section area andSructures, 23:491-501.position of knee element, the KBF structure hasenough lateral stiffness and good ductility even inposable knee bracing. Computers & Structures, 75: 65-72.severe earthquake.Roeder, C.W, Popov, E.P., 1978. Ecentrically braced steelframes for earthquakes. Journal of Structural Div ASCE,104(3):391-412.am， M.T， Balendra, T，Liaw， C.Y， 1995. Earth-CONCLUSIONquake-resistant steel frames with energy dissipating kneeelements. Engineering Structure, 17(5):334-343.As an energy disspating system, the knee brac- Shen, z.Y, Chen, Y.Y,, Chen, Y.Y, 2000. The Principle ofing frame combines excellent ductility and lateralSteel Structure. Chinese Industrial Building Press, Bejing,p.84-89 (in Chinese).stiffness and is easy for application to rehabiltation ifearthquake damaged buildings. With the protection ofWilliam, M.S., Blakeborough, A., Clement, D., Bourahla, N.,2002. Seismic behavior of knee braced frames. Proceed-the knee elements, no damage occurs to the majorings of the Institution of Civil Engineers: Structures andstructural members during a severe earthquake. Sys-Buildings, 152(2):147-155.中国煤化工MHCNM HG