This paper investigates the behavior of large span grid structure exposed to a localized fire. The localized fire may generate hot smoke and thus induce non-uniform temperature distribution in the grid structure. The thermal expansion of the heated members tend to be axially restrained by the adjacent cold members thus inducing additional forces on the critical members of the grid structure. The buckling resistance of axially restrained member at elevated temperature may be obtained based on second order analysis of member with initial lateral imperfection by considering force equilibrium at deformed geometry and cross section resistance being reached. The critical temperature of the member is reached when the axial force reaches its buckling resistance. It is found that the critical temperature of members with initial lateral imperfection was higher than that without such imperfection for chord members with large slenderness ratio and high axial restraint

Bo, Zeng

Liew, J.Y. Richard

Xiong, Mingxiang

Yong, Du

Zou, Jie

English

CTU Open Journal Systems (Czech Technical University, Prague / České vysoké učení technické v Praze)

Applications of Structural Fire Engineering, 15-16 October 2015, Dubrovnik, CroatiaBUCKLING RESISTANCE OF AXIALLY RESTRAINED CHORD MEMBEROF GRID STRUCTURE AT ELEVATED TEMPERATUREDu Yonga,b, J.Y. Richard Liewa,b,Mingxiang Xiongb, Jie Zoua, Zeng BocaNanjing Tech University, College of Civil Engineering, Jiangsu, ChinabNational University of Singapore, Department of Civil & Environmental Engineering, Singaporec Architectual Design and Research Institute of Nanjing Tech UniversityAbstractThis paper investigates the behavior of large span grid structure exposed to a localized fire. Thelocalized fire may generate hot smoke and thus induce non-uniform temperature distribution in thegrid structure. The thermal expansion of the heated members tend to be axially restrained by theadjacent cold members thus inducing additional forces on the critical members of the grid structure.The buckling resistance of axially restrained member at elevated temperature may be obtainedbased on second order analysis of member with initial lateral imperfection by considering forceequilibrium at deformed geometry and cross section resistance being reached. The criticaltemperature of the member is reached when the axial force reaches its buckling resistance. It isfound that the critical temperature of members with initial lateral imperfection was higher than thatwithout such imperfection for chord members with large slenderness ratio and high axial restraint.Keywords:axially restrained chord member, buckling temperature, initial lateral imperfection1 INTRODUCTIONSpatial grid structures are extensively employed by buildings with long span roof, which would besubjected to localized fires. Since the temperature distributions throughout the localized fire arenon-uniform, the longatinal thermal expansion of hotter chord members may be restrained by theless hotter chord members, and occur thermal stress in the hotter (e.g. Du Y. et al.,2014).In gerenal,the restraint would reduce the buckling resistance.Then, behaviours of restrained members of steelframes at elevated temperature are deeply analized(e.g. I. C. Neveset al.,1983, J. M. Franssen. 2000,Wang P. Jet al.,2009). Is there any difference between frame members and chord members? It isdistinct that chord members are always with larger slender ratio than frame members, especially fortension chord members which shall be inversed to the compression at elevated temperature (e.g. DuY. et al.,2014). It must be deeplydiscovered thatthe intial lateral imperfection acts on the bucklingcapacity and the axially restrained member with muchhigher slender ratioresponds under elevatedtemperature.Shown as Fig.1the buckling temperature at point A is dependent on the pre-buckling historystrongly. If the point A is pushed forwards by pre-buckling history, the post-buckling history will beFig. 1 Axial load-temperaturecurveforan axially restrained membersrevised. Therefore, pre-buckling behaviors of the axially restrained chord members with initiallateral imperfections and large slender ratio under elevated temperature must be distinct asfollowing.2 AXIAL BEHAVIOR OFEND-RESTRAINED CHORD MEMBERS WITH INITIALLATERAL IMPERFECTThe proceeding of deformation for a axially restrained chord member with lateral initialimperfection is shown as Fig. 2graphically.Figure 2(a) shows a simply supported chord member with an initial bow imperfectionwithmaximum magnitude v0at the mid-height. The vertical displacement u0, caused by axial loadP0, issuperimposed byum,0caused by thecurvature effect due to bending at ambient temperature. When thechord member is further exposed to fire, the restraint effect due to adjacent structure can beMeanwhile, the axial load causes a lateral deformation, vm,0 at the mid-height as shown in Fig.2(b).When the chord member is further exposed to fire, the restraint effect due to adjacent structure canbe represented by a restraining spring attached at one end, as shown in Fig. 2(c).As shown in Figure2(d), an upward displacement of us due to thermal expansion is observed, and a downwarddisplacement of uc due to initial force and the additional restrained force, P+ΔP upon the freedomthermal expansion chord member, as shown in Fig. 2(e).3 MECHANICAL RESPONSE OF AXIALLY RESTRAINED CHORD MEMBERSWITH INITIAL LATERAL IMPERFECTIONAt ambient temperature, the total vertical deformation, , as following= + , = , + , (1)where , the axial stiffness of a chord member at ambient temperature.At elevated temperature, the longitudinal thermal expansion, us, of a chord member is under thespring with restrained stiffness, ks and initial forceP0. Then, the loading, ∆P, caused by spring canbe given ass sP k u  (2)andthe total loading, PT, is given asEq. (3) considering of Eq. (2)T 0 0 s sP P P P k u     (3)longitudinal thermal expansion, ∆l,  in Fig.3(e) is given asT th( )=l l T l    (4)where T thermal expansion coefficient,[1.4×10-5/°C],∆ step of temperatureincrement.Fig.2 Deformation for a chord member with initial imperfectionat ambient temperature and elevated temperaturesIfloading PT on the freedom chord member, see Fig. 3(e), the longitudinal deformation is representby uc, which combines with axially loading action and lateral bending action. The deformationcaused by axially loading action can be given asc m 0 c( ) /u u P P k    (5)where, mu the longitudinal deformation caused by lateral bending at elevated temperature,ck axial stiffness of a chord member at elevated temperature.substitutingEq. (2) into Eq. (5), thenc m 0 c( ) /s su u P k u k   (6)let Fig. 3(d) compare with Fig. 3(b) and Fig. 3(e), there is a relationship of deformation as below0th m,0c,0s cPu u l uk    (7)substitutingEq.(6)into Eq.(7), thenc 0 0th ,0 ms c ,0( )s mc ck p pu l u uk k k k     (8)let 0 0mec,0c cp puk k  , thencth mec m,0 ms c( )sku l u u uk k    (9)substitutingEq. (9) into Eq. (3), thens cT 0 th mec m,0 ms c( )k kP P l u u uk k     (10)in order to consider of the effect of initial lateral imperfect, v0,for longitudinal deformation, therelationship between initial lateral imperfect and longitudinal deformation should be established asbelow. Shown as Fig.3, based on the mechanical modelof second order analysis for the chordmember with lateral initial imperfection, the function of displacement for chord membersisillustrated as following.The function of lateraldisplacement for initial lateral imperfect is given by Eq.(11)0 0πsin xy vl (11)where l initiallength of a chord member,0v initiallateral imperfection, 0 /1000v l .The force differential equation of the lateral displacement is given by Eq.(12).0EIy P y EIy   (12)where EI flexural rigidity of  a chord member at ambient temperature.let 2P kEI and substituting Eq.(11) into Eq. (12), as followingFig. 3 Analysis model of deflectioncurve for steel membersPEIPMxyy22 02π πsin( )v xy k yl l    (13)the general solution of partial differential Eq. (13)as below202 2 2π πsin( ) cos( ) sin( )πv xy A kx B kxk l l    (14)whereA, B undetermined coefficients.inducing boundary conditions to Eq. (14), i.e.ifx=0 or x=l, then y=0, the function of lateraldeflection equation with initial lateral imperfect can be obtained202 2 2π πsin( )πv xyk l l   (15)Then, the longitudinal displacements of a axially restrained chord membercaused by bending actionat ambient temperature and elevated temperature as following2m,0 001 ( )d2lu y P x  (16)2m T01 ( )d2lu y P x  (17)substitutingEq. (16)and Eq. (17)into Eq. (10), then2 2s c0 th mec 0 T T0 0s c1 1[ ( )d ( )d ] 02 2l lk kP l u y P x y P x Pk k          (18)integralEq. (18), then2 6 22 2s c 0 0 T0 th mec T2 2 2 2s c 0 0 T Tπ{ [( ) ( ) ]} 04 π πk k I v E EP l u Pk k l Pl E I P l E I        (19)the transient axial force of chord members can be gotten by solving Eq. (19) at any temperaturepoint with MATLAB calculation software.4 INITIAL LATERAL IMPERFECTIONACTION ON BUCKLING RESISTANCEReference of Eq. (10), the axial force of achord member without initial lateral imperfection can begiven as belows cT 0 th mecs c( )k kP P l uk k    (20)letEq. (20)minusEq. (10), then the initial lateral imperfect action, Pb, is given as belows cb T T m m,0s c= ( )k kP P P u uk k    (21)givenrestraint ratio l s ck k  , thenb c m m,0l1= 1 ( )1P k u u    (22)at ambient temperature, let 20 0P kEI and substituting Eq. (15)into Eq. (16), then222 20m,0 2 2 2 003202 2 2032 62 2 20π1 π π( ) cos ( )d2 ππ1 ( )4 ππ( ) 104 πlv xu xk l l lvl k llk l       (23)at elevated temperature, let 2T TTP kE I and substituting Eq. (15)into Eq. (17), similar to Eq. (23)asbelow32 6m 2 2 2Tπ( ) 104 πluk l  (24)becausek0, v0 and l should be given for a chord member at ambient temperature, the value of axialdeformation, m,0u , caused by bending can be gotten fromEq. (23).Then, According to Eq. (22), theinitial lateral imperfect action, Pb, is mainly dependent on mu and cl111k   .According to Eq. (24),the factor, 2Tk , should be noted firstly. Then,calculating PT withEq. (19),the history of 2Tk atelevated temperature is shown as Fig. 4, which is for a chord member with agiven slender ratio, 200l I A  .LetT blk yTP P f A  (24)where blkP buckling resistance force at fire limit state,yTf yield strength at elevated temperature,A size of section for chord member.Fig. 4illustrates graphically that 2Tk is always below 2 2l . Then, the2Tk increases with elevatedtemperature pre-buckling.See Eq. (24),the term of 2 2 2Tπ k l reduces with kT increasing, whileumincreases. According to Eq. (22), for given m,0u , the Pb increases at elevated temperature under anyrestrained ratio.In the meantime, with the l increasing, the 2Tk increases more rapidly,while umincreases morerapidly also, shown as Fig. 4. Finally, the initial lateral imperfect influences on the axial loading ofchord members more strongly with the higher restrained ratio.Shown as Fig. 5, there is a chord member with slender ration 200 and loading ratio 0.5. in case 1,theaxially restrained chord member without initial lateral imperfect and axial loading can be gotten byEq. (22), in case 2,the axially restrained chord member with initial lateral imperfect and axialloading can be gotten by Eq. (19). Under higher axially restrained stiffness ratio there is aremarkable distinction of buckling temperature between case 1 and case 2.Fig. 4 Thehistory of  T TP E I at elevated temperaturevaried with restrained stiffness ratio0 100 200 300 400 500 600 700 800 9002.53.03.54.04.55.05.5l=0.01l=0.02l=0.03l=0.04l=0.05l=0.08l=0.1Temperature [℃]blkTPE I2 2π / lP T/(ETI)[×10-8m]It can be concluded that the initial lateral imperfect provides larger lateral displacement at elevatedtemperature, in a way, released thermal loading for chord members. However, it postpones thebuckling temperature for the chord member with higher restrained stiffness ratio. Therefore, theinitial lateral imperfect should be noted enough when evaluating the bulking resistance at elevatedtemperature for chord members with higher restrained stiffness ratio.5 CONCLUSIONSThe main conclusions of the work presented as following- Based on the second order analysis of members with initial lateral imperfection, thecalculation method of transient axial loading for axially restrained chord members atelevated temperature has been derived.- The initial lateral imperfection improves the buckling resistance of chord member withhigher restrained stiffness ratio.ACKNOWLEDGMENTSThis study is apart of the project, whichis supported by National Natural Science Foundation ofChina.REFERENCEDu Y., Li X., Gu J., 2014. Fire-resistance of Regular Pyramidia Gird Structures Exposed to Localized Firewith New Temperature-time Curve, in 8th international conference on structure in fire, 11-13 June 2014,Shanghai, China, p. 935-942I. C. Neves. 1983. The Column-structure Interaction on the Behavior of Steel Columns under LocalFire.PhD thesis, The Instituto Superior T´ecnico.J. M. Franssen. 2000. Failure Temperature of a System Comprising a Restrained ColumnSubmitted to Fire.Fire Safety Journal, 34(2), p.191–207Wang P. J., Li G. Q., Wang Y.C.2009. Parametric Studies on Restrained SteelColumnin Fire :Under AxialLoad Only. Journal of Tongji University: Natural Science , 37(12),p.1571-1575.Fig. 5Initial axial imperfect action on the buckling resistancefor a chord memberat elevated temperaturevaried with restrained stiffness ratio0 100 200 300 400 500 600 700 800 9000100200300 Case 1l=0.01l=0.02l=0.05l=0.1 Case 2l=0.01l=0.02l=0.05l=0.1Temperature [°C]Axial load[kN]Buckling resistance resulted fromChina code, CECS 200:2006Slenderness ratio 200LoadratioN=0.5

Buckling Resistance of Axially Restrained Chord Members of Grid Structure at Elevated Temperatures

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Buckling Resistance of Axially Restrained Chord Members of Grid Structure at Elevated Temperatures

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