On the Buckling of Axially Restrained Steel Columns in Fire. University of Bath, Bath BA2 7AY, UK. University of Sheffield, Sheffield S1 3JD, UK

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1 Title On the Bukling of Axially Restrained Steel Columns in Fire Authors Affiliations P.G. Shepherd a * and I.W. Burgess a Department of Arhiteture & Civil Engineering University of Bath, Bath BA2 7AY, U Department of Civil & Strutural Engineering University of Sheffield, Sheffield S 3JD, U *Corresponding Author P. G. Shepherd Corresponding Address Dept. of Arhiteture & Civil Eng, University of Bath, BA2 7AY, U Corresponding p.shepherd@ath.a.uk Corresponding Tel Corresponding Fax

2 ABSTRACT This paper desries the ehaviour of restrained steel olumns in fire. It follows the introdution of extra load into the olumn through the axial restraint of the surrounding ooler struture and the onsequential ukling. ey to this understanding is the post-failure ehaviour and re-stailisation of the olumn, whih is disussed with referene to a finite element model and an analytial model. Through i-diretional ontrol of the temperature, the finite element model allows the snap-ak ehaviour to e modelled in detail and the effets of varying slenderness and load ratio are investigated. The analytial model employs strutural mehanis to desrie the ehaviour of a heated strut, and is apale of explaining oth elasti and fully-plasti post-ukling ehaviour. Through this detailed explanation of what happens when a heated olumn ukles, the onsequenes for steel framed uilding design are disussed. In partiular, the need to provide roustness is highlighted, in order to ensure alternative load-paths are availale one a olumn has ukled and re-stailised. Without this roustness, the dynami shedding of load onto surrounding strutures may well spread failure from a fire s origin and lead to progressive ollapse. eywords: Steel, Columns, Bukling, Fire resistane, Axial Fores, Nonlinear analysis

3 NOTATION E A l I P F Elasti (Young s) modulus Memer ross-setional area Memer length Memer seond moment of area External fore at olumn top Internal fore in olumn 2 2 F E = π EI / l Column Euler ukling fore at amient temperature EA/ l ET yt M p Restraint stiffness at olumn top Beam Flexural ending stiffness = Column axial stiffness ρ = / Restraint ratio F E Elasti modulus redution with temperature fator Yield strength redution with temperature fator Plasti moment apaity of olumn (assumed invariant with axial fore) ρ = l / Restraint ratio (normalised with respet to the Euler ukling fore) δ Axial defletion of the top of the strut δ = δ / l Normalised axial defletion of the top of the strut Lateral defletion of the mid-height of the strut = / l Normalised lateral defletion of the mid-height of the strut λ F E Column slenderness ratio µ = P / External load ratio (normalised with respet to the Euler ukling fore) = P / EA µ External load ratio (normalised with respet to axial stiffness) φ = F / Internal load ratio (normalised with respet to the Euler ukling fore) F E 2 γ = /( π EI / l) Plasti moment of olumn (normalised with respet to Euler x length) M p 3

4 INTRODUCTION. Context As the full-sale fire tests at Cardington have shown [], whilst the majority of eams in a steel-framed uilding an e designed to funtion without the need for fire protetion, olumns are so ritial to the load arrying apaity of a uilding that they must remain proteted. A failure of a olumn on one floor an have a great effet on the floors aove, meaning fire ompartments an e reahed and there is a danger of disproportionate ollapse. It is also the ase that all olumns in uilding frames are sujet to axial restraint, sine their purpose is to support struture aove and this struture will have a vertial stiffness. It is therefore hugely important that the ehaviour of olumns in fire is understood, and in partiular, the role that axial restraint plays in this ehaviour. A numer of researhers have looked at this role, from oth an experimental and analytial point of view. For example Ali et al. [2], Rodrigues et al. [3] and Tan et al. [4] have all performed physial experiments on axially restrained olumns at elevated temperatures. Shepherd et al. [5], Franssen [6] and Huang & Tan [7] have simulated suh tests using finite element analysis tehniques and extended these simulations to further investigate the role of axial restraint. In this paper, the theoretial explanation ehind the ehaviour of steel olumns in fire is presented, alongside the results of finite element analyses in order to illustrate partiular harateristis of this explanation. Speifially, the differenes in ehaviour of olumns pre- and post-ukling are ontrasted and the details of snap-ak ukling examined..2 Modelling The ehaviour of olumns outlined in this paper is demonstrated through finite element analysis using a simplified model of an axially restrained olumn (see Figure ). All test ases model a 203x203x52UC pin-ended olumn of Grade 43 steel and length 5.6m to give a minor-axis slenderness of 00. The olumn was divided into eight finite elements along its length and given an initial geometri imperfetion and load ratio of 0.6 aording to EC3 design rules. The finite element analysis was performed using the Vulan program ( aessed Fe 200) to inorporate oth material- and geometrial non-linearities. Axial restraint was provided y an axial spring element at the same end of the olumn as the applied load. This elasti restraint was quantified y the use of the Relative Restraint Ratio, whih is the ratio of the axial stiffness of the restraint to the axial stiffness of the heated olumn as defined y Wang and Moore [8]. All olumns were uniformly heated, oth in ross-setion and along their length. All fores reported are the axial fores in the olumn element at the top of the olumn and any displaements refer to the axial displaement of the top of the olumn, measured from the initial unloaded datum position. 4

5 2 LOADING 2. Explanation When a steel olumn is heated it thermally expands. If this olumn is part of a uilding frame, it will exert an upwards fore on the ends of any eams framing in to its top and any olumns diretly aove. Firstly let us onsider the single-storey (or top-storey) ase where there are no olumns aove. If the olumns at the other ends of the eams whih frame-in (i.e. one ay away) remain ool (see Figure 2a), the eams will undergo differential vertial movement of their ends, induing flexure and they will therefore exert a restraining fore on the heated olumn in question, resisting its vertial expansion. If however the surrounding olumns are also heated at a similar rate (i.e. they are within the same fire-ompartment) (see Figure 2) then oth ends of the eams will undergo the same vertial movement and no restraining fore will e introdued. There will of ourse generally e a restraining fore on the surrounding olumns. Only when an entire floor is heated at exatly the same rate (see Figure 2) will all the olumns esape the addition of a restraint fore, sine they will all thermally expand together and no relative vertial movement will e present to ause flexure in the onneting eams. If the presene of an upper-floor is also taken into aount, the situation is similar. But fore due to the thermal expansion of the olumn (or group of olumns) is also passed up through the olumns of the floor aove (see Figure 3) into the eams aove. Sine the olumns aove have an axial stiffness, only a proportion of the movement reahes the eams aove, ut one it does, this movement is resisted through flexure in exatly the same way as the eams elow. Simple mathematial models exist [9, 0] whih an e used to alulate the level of axial restraint likely to e experiened y a olumn, and an take into aount effets of eams, olumns, omposite floors, upper storeys, rotational restraint, et. This allows a omplex frame arrangement to e analysed using a simplified finite element model, with a single spring element representing the axial restraint applied to the olumn. In the ase of a single storey, the restraint stiffness experiened y a heated olumn, r, is simply the sum of the vertial stiffnesses of the eams whih frame in to the top. For example, the twodimensional ase shown in Figure 2a has a restraining stiffness given y Equation. restrainin numerof g eams r = i = i= 2 () For example, the stiffness of the heated 254x254x67UC olumn shown in Figure 4,, an e found from Equation 2 EA 20,000 x 2,300 = =,242, 500 N L 3600 = (2) mm 5

6 The stiffness experiened y the heated olumn from eah lower floor 305x65x54UB restraining eam,, an e alulated as Equation 3 2EI 2 x x = = 365 N 3 3 L 6000 = (3) And the stiffness, 2, from eah upper floor eam when the largest, 94x49x388UB setion is present an e alulated as Equation 4 2EI 2 x x = = = N mm (4) 3 3 L 6000 Whih gives the relative restraint ratio for the heated olumn in Figure 4, sujet to two of eah eamtype, as Equation 5 2 x 2 x 2 2 x365 2 x83957 = ρ = = (5) For a two-storey struture, as shown in Figure 3, the restraint experiened y a heated ground-floor olumn inludes the stiffness of the restraining eams on eah floor aove as well as the axial stiffness of the seond floor olumn. In this ase the stiffness of the upper olumn and the seondfloor eams are omined in series, as shown in Equation 6. mm r = floor = ( 2 ) (6) olumn2 floor 2 2 ( ) 3 4 As shown with dotted lines in Figure 3, this system an e extended to any numer of floors aove the heated olumn y generating more deeply nested ontinued frations. In the ommon ase where a frame onsists of a single setion size for all the olumns and another for all the eams, an upper ound an e estalished for the restraint stiffness y alulating the restraint that would e theoretially applied if there were infinitely many upper storeys. If represents the vertial restraining stiffness of all the eams on a single floor that frame in to a olumntop, and represents the axial stiffness of eah olumn, then the restraint provided y infinite floors, would e as shown in Equation 7 elow. = (7) 6

7 7 Sine this is an infinite ontinued fration, the fration an e sustituted in to itself, as shown in the first part of Equation 8. = = = (8) And this an e solved for as follows in Equations. 9-: = (9) = 2 (0) 0 2 = () Equation is quadrati in and an therefore e solved in the usual way as shown in Equation ± = (2) Sine the stiffness values and are oth positive, the quantity inside the square-root is > 2, and therefore the only solution whih makes physial sense (i.e. gives a positive value for ) is when the positive square-root is used, giving Equation = (3) This value is the restraint experiened y a olumn with infinitely many floors aove it, it is an upperound on the axial restraint provided to the ground floor olumn, and for that matter any olumn within the frame. 2.2 Modelling The test olumn was analysed with a range of restraint spring stiffness to model the effets of Restraint Fators ranging from the very low (0.004) to the very high (0.38). These values do not neessarily aim to model realisti values of restraint seen in uilding frames, whih are usually in the range of [8], ut more to show the differene in ehaviour from olumns with very low and very high restraint levels. They are ased on previous work y Bailey and Newman [] whih takes a simple 2D two-storey frame as a ase model and hanges the setion size of the top eam to vary the restraint applied to the olumn as shown in Figure 4.

8 The axial fore present in the olumns is shown in Figure 5 for the pre-ukling heating phase. The initial part of the urves (at amient 20 C) shows that the higher the axial restraint, then lower the fore in the olumn. This is to e expeted, sine the axial springs are introdued efore loading has een applied, and the higher level of axial restraint is provided y a stiffer axial spring, whih will attrat a greater proportion of the applied load. A less-stiff restraint spring will shed more of the applied load onto the olumn itself. The main feature of the graph is learly that the olumns with higher restraint inrease their fore quiker than the less-restrained olumns. This is ovious from the fat that the inrease in temperature results in a thermal strain, whih is diretly onverted into a stress (and therefore fore) through the stiffness of the restraint. Another, less ovious feature is that all the urves pass through a single point around 50 C when the thermal expansion has overome the initial shortening due to applied load. Even though the highly restrained olumns egan with less fore, this fore inreases quiker with temperature until they reah the exat same value of fore as the less restrained olumns. The explanation of this eomes lear when it is noted that at this point the thermal strain has exatly overome the shortening due to loading, therefore the restraint spring is exatly ak to its original unstressed length. The spring has therefore not een moilised, and may as well not even e present. Its stiffness must therefore e irrelevant and eah test ase with a different spring must reah this point at the same temperature. 3 BUCLING 3. Explanation As heating progresses, the fore in the olumn ontinues to inrease until the ukling load of the olumn is reahed. Although the olumn setions are all the same, and therefore their amient ukling loads would e idential, this load is ahieved at different temperatures, depending on how quikly the restraint fore is inreasing. Sine the highly restrained olumns gain restraint fore quikly, they reah the ukling load sooner (i.e. at a lower temperature) and sine at this lower temperature the material properties (stiffness and yield stress) have not degraded muh, the ukling load is relatively high and a high level of olumn fore an e ahieved. The lower restrained olumns don t reah their ukling load until they are muh hotter, ut at this stage the material properties have degraded suffiiently to result in a muh lower value of ukling load. As the temperature rises, at the same time as the restraint fore is inreasing due to axial restraint, the ukling load is dereasing due to material properties hanging, and at some point these to ome together to initiate a ukling failure in the olumn. When ukling ours, the analysis shows the olumns undergo a sudden shortening and eome shorted than their original length. This results in a shedding of load ak onto the restraint spring and the fore in the olumn is seen to suddenly redue. Sine the analysis steps up in small temperature inrements, this exhiits itself y what seems like snap-through ehaviour. Figure 6 desries this ehaviour in the Displaement-Temperature domain where urve a shows the inrease in displaement due to thermal expansion against the axial restraint and urve shown this sudden 8

9 snap-through and if heating is ontinued, the olumn would follow urve. Most existing experimental studies [6, 2] have exhiited this type of sudden ukling ehaviour eause, during a real fire test, it is impossile to predit exatly when a olumn might fail and then adjust the heating regime suffiiently to ontrol the ehaviour of the olumn during ukling. 3.2 Modelling For a given level of restraint, an initial analysis was performed to assess the ehaviour of the olumns up-to and eyond ukling. The results in the fore domain are shown on Figure 7 and similar ehaviour to the defletion urves a, and of Figure 6 an e seen. After ukling, the olumns re-stailise in a state where their shortening y ukling is more than their lengthening through thermal expansion and they are overall shorter than their original length. In this state they are essentially hanging from their restraint springs and arry little axial fore. As the temperature ontinues to inrease their fore dereases in line with their redution in stiffness as they shed more and more load onto the restraint spring. The little extra thermal expansion they undergo, whih would tend to inrease their axial fore as they pik up load from the restraint is negligile ompared to the more dominant redution in stiffness. The kink in the urves of Figure 7 at 400 C diretly reflets the hange in material properties that ours at this same temperature in the Euroodes and also therefore Vulan. By 200 C hardly any strength remains in the olumns themselves and the fore tails off towards zero. The analyses of the four olumns with least axial restraint are not ale to find numerial post-ukling solutions due to huge numerial instaility assoiated with this jump from one stale solution to another. Sine the restraint stiffness is muh lower, the overall stiffness present in the system is also lower, the gloal stiffness matrix therefore eomes ill-formed and the solution proedure fails to onverge. 4 SNAP-BAC 4. Explanation Most existing experimental studies [6, 2] have exhiited the type of sudden snap-through ukling ehaviour outlined aove sine it is impossile to ontrol the ehaviour of the olumn during ukling. With an analytial model however, full ontrol over the temperature parameters is availale at every stage and the analysis is exatly repeatale, allowing this ukling to e studied in more detail. In partiular the effet of reduing the temperature of the olumns after ukling an e investigated. As the temperature is redued, the fore in the olumn inreases, sine the material properties of the olumn are restored. This leads to the olumn stiffness inreasing, therey taking a greater proportion of the load from the restraint spring. This is represented as urve d in Figure 6 and ontinues until there no longer exists this post-ukled solution. If the temperature is lowered even further, the solution would snap-up along urve e to the initial loading solution, wherey it would ontinue to unload along urve f ak to amient temperature. This initial loading solution is stale and 9

10 therefore if the model were to e heated one again at this stage, it would ontinue up the loading path along urve g one again. It an e seen that in the temperature range from urve e to in Figure 6 there exist two stale solutions, one pre-ukled and the other post-ukled, in whih the olumn an exist. By putting the urrent state of the finite element analysis into one of the regions where only one stale state exists (either lower temperatures than urve e or higher than urve ) the analysis an e fored into one or other of the two states, and this state an e followed up or down in temperature. It an e inferred that the true solution path of a heated olumn would e the path a - h - d - in Figure 6 if only it were possile to oserve it diretly. 4.2 Modelling For those analyses where stale post-ukled solutions had een found, eah was repeated up the ukling temperature of eah olumn (the ottom point of urve in Figure 6) and then the temperature was redued to follow urve d ak through equilirium points not measurale during physial tests. Eventually the analyses snapped-up and egan to go ak down the original loading path. The results of these ooling phases have then een inorporated into those of the heating analyses y reversing the results from the urve d parts of the analyses and inserting them etween the top and ottom of the urve parts of the analyses. The resulting trae of equilirium positions in the fore domain is shown in Figure 8. The orresponding displaement domain results are shown in Figure 9 for ompleteness. It an e seen from Figure 9 that at 200 C, when there is very little strength left in the olumn itself, the vertial displaement of the top of the olumn is wholly dependent on the stiffness of the restraint spring that arries the applied load. It an also e seen that there exists a seond point around 80 C where all the urves pass through a single point. This again represents the point at whih the olumn eomes equal to its original length and therefore the spring is not ative and its stiffness is irrelevant. This is an exat parallel of what happened during loading, ut is slightly less well defined on the graph sine results either side of ukling are interpolated with straight lines through this point. 5 SIMPLE RATIONALISATION The snap-through ehaviour seen in the finite element results is not neessarily intuitively understandale, and it is therefore worthwhile to use normal strutural mehanis to attempt to model the ehaviour of a heated olumn, axially loaded in ompression at its top and under the influene of elasti axial restraint to thermal expansion. Consider a perfetly straight simple elasto-plasti steel strut of length l and ross-setional area A, shown in Figure 0(a), pinned to allow free rotation at its top and ottom and restrained against axial movement at its top y an elasti spring of stiffness whih is a proportion ρ of the axial stiffness of the strut at 20 C, so = ρ. The steel of the strut has a onstant oeffiient of thermal expansion α, so as its temperature is raised from 20 C to T C its natural length inreases to l( α ( T 20)). It is assumed that the elasti 0

11 modulus of the strut s steel redues with temperature as shown in Figure (a), following the redution fator k ET, in a smoother fashion than if it were in aordane with the assumption of EN For the purposes of this illustrative example the degradation of elasti modulus with temperature is expressed as k =.0 (T 00 C) ET k ET = /( T ) (T > 00 C) The yield strength is assumed to deline, again in a smoother fashion than EN demands, as shown in Figure (): k =.0 (T 400 C) yt k yt = /( T 00.0) (T > 400 C) 5. Pre-ukling The redution of the free thermal expansion due to the fore it auses in the restraint spring is δ T =. The olumn is loaded (Figure 0()) at its top with a onstant axial α( T 20) l /( k / ρ) ET fore P, whih an e expressed as a proportion µ of the elasti ukling load δ = P /( EAk / l 2 π EI / l 2. This auses an extra redution of the free thermal expansion given y ). This gives a total axial defletion (where δ is positive downwards) of Equation 4. P ET δ = ( α ( T 20) EAk = ( α ( T 20) k ET ET / ) /( EAk ET l Pl / EA) /( l / EA k / l) P /( EAk ET ) ET / l) (4) In dimensionless terms this an e expressed as δ = µ / k α( T 20)) /( ρ / k ). As the ( ET ET steel temperature rises from 20 C to T C the fore F arried y the olumn is given y Equation 5. F = P δ (5) 2 2 If φ is the internal fore ratio F /( π EI / l ) then φ µ ρδ when φ = ket. 5. Elastially post-ukled = and the olumn ukles elastially After elasti ukling ommenes, post-ukling defletion hardly auses any hange in the olumn fore from the elasti ukling load, although this redues with the elasti modulus redution fator k ET. The asi equilirium of Equation (5) always applies, so in dimensionless terms the normalised axial defletion is given y δ = µ k ) / ρ. ( ET It is worth examining the amplitude of the lateral defletion whih orresponds to this axial shortening of the olumn. Assuming that the elasti ukle shape is a single half-sine wave, the first-order

12 relationship etween the normalised axial and lateral defletions shown in Figure 0() is δ ( ρ / k ) = π / 2 α( T 20) µ /. 2 2 ET k ET 5.2 Plasti post-ukling Although the transition from elastiity to a plasti hinge mehanism requires too muh detailed information aout the ross-setional properties of the olumn to e worth studying in detail, it is possile to look simply at the ehaviour of the fully-plasti mehanism. It is assumed that the plasti moment apaity of the ross-setion hanges only with the yield strength redution fator k yt, shown in Figure (), whih is a reasonale assumption for slender olumns ut over-estimates moment apaity somewhat for stoky olumns. The fully-plasti mehanism inludes a single plasti hinge at the olumn s mid-height, as shown in Figure 0(d). Equilirium now gives P δ = M p k yt /. In dimensionless terms this is equivalent to 2 µ ρδ = / where γ = /( π EI / l). γk yt M p The relationship etween the axial olumn-top defletion and the mid-span defletion is approximately 2 δ = ( µ / k α( T 20) 2 ) /( ρ / ). In terms of the normalised lateral displaements this ET k ET produes a ui equation shown in Equation 6. 3 µ γk yt ( α( T 20)) ( ρ / k 2 ρ 2ρ ET ) = 0 (6) This equation produes either one or three real roots for at any temperature, depending on the values of the oeffiients in Equation (6). The single root is invalid, sine it is negative, and therefore the plasti moments are in the wrong diretion. The other pair of roots, when they exist, are valid and represent, respetively: Very small lateral defletion, so the olumn fore is inreased y its restrained negative defletion aused y thermal expansion. This is just equilirated y the plasti moment, ut is unstale if any positive inrement of displaement ours. Large lateral defletion, so that the olumn fore is dereased y restrained positive defletion to the extent that the plasti moment an reate equilirium. This is a stale equilirium state. It an e seen from Figure 2, whih plots the lateral displaement (or ukling amplitude) of the olumn, that the plasti mehanism phase produes a urve in whih a destailization ours shortly after elasti ukling ours, whih is re-stailized at higher amplitude. When translated to the equivalent axial displaement of the olumn top, shown in Figure 3, the effet is seen to eome muh more uspid, and very similar to Figure 9, with the unstale plasti urve eoming almost tangential to the pre-ukled urve at the ukling temperature. 2

13 6 CONCLUSIONS The ehaviour of axially-restrained steel olumns sujet to heating has een explained in detail and this explanation has een validated against a non-linear finite element analysis using the Vulan software. By reduing the temperature of ukled olumns, the snap-ak ehaviour of olumns an e oserved. This shows the ranges of the two stale solutions. Whilst an isolated olumn has een modelled with a spring element providing axial restraint, this work is equally appliale to the ase of a olumn as part of a frame, wherey the axial restraint is provided y eams framing in to the olumn. In these analytial models olumns are shown to hang from the restraining spring after ukling. In a real uilding fire, a olumn would similarly exhiit unstale ukling ehaviour efore reahing a limit to its axial defletion y transferring load onto adjaent struture and hanging from its eams, providing that there is an alternative load path, along the eams and down the surrounding olumns. However, in a fire senario, these surrounding olumns may also e hot, and may e just aout to ukle. There is therefore the real possiility of a progressive ollapse senario, wherey a ukling olumn sheds its load through its eams onto the surrounding olumns, whih in turn then egin to ukle, shedding their load and that of the first olumn onto even more surrounding struture. The load-shedding is partiularly likely to initiate ukling in the surrounding olumns sine it is dynami. An understanding of this hange in load path, and the role that axial restraint plays in supporting the olumn after ukling, are extremely important in the design of steel frames. Often designs are optimised suh that no one part of the struture is partiularly weaker than another, and so when something fails it is likely that the surrounding struture is also lose to failure. It has een shown that olumn ukling represents an extreme failure, where olumns dynamially pass through an unstale region efore restailising through support from their axial restraint where availale. Sine it has een shown that olumn ukling an lead to a hard failure whih is sudden and an initiate progressive ollapse, it is therefore suggested that steel frames are designed for roustness, with alternative load-paths provided whih an aept the dynami loads aused y olumn ukling. In partiular, the effets of the spread of failure in fire from an origin should e onsidered. 3

14 REFERENCES [] Newman G.M, Roinson J.T & Bailey C G. (2006). "Fire Safe Design: A new Approah to Multi-storey Steel-framed Buildings (2nd Edition) SCI Puliation P288", The Steel Constrution Institute (SCI). [2] Ali, F. A., Simms I., O Connor D. (997). Effet of axial restraint on steel olumns ehaviour during fire, Pro. 5 th Int. Fire Safety Conf., Melourne, Australia. [3] Rodrigues, J. P. C., Neves, I. C., Valente, J. C. (2000). Experimental researh on the ritial temperature of ompressed steel elements with restrained thermal elongation, Fire Safety J., 35 (2), pp [4] Tan, -H., Toh, W-S., Huang, Z-F. (2007). Strutural responses of restrained steel olumns at elevated temperatures. Part : Experiments, Engineering Strutures, 29 (8), pp [5] Shepherd, P.G., Burgess, I.W., Plank, R.J. and O Connor, D.J. (997). The Performane in Fire of Restrained Steel Columns in Multi-Storey Constrution. Pro. 4 th erensky Int. Conf., Hong ong, pp [6] Franssen, J-M. (2000). Failure temperature of a system omprising a restrained olumn sumitted to fire, Fire Safety J., 34 (2), pp [7] Huang, Z-F., Tan, -H. (2007). Strutural responses of restrained steel olumns at elevated temperatures. Part 2: FE simulation with fous on experimental seondary effets, Engineering Strutures, 29 (9), pp [8] Wang, Y. C. and Moore, D. B. (994). Effet of thermal restraint on olumn ehaviour in a frame, Pro. 4 th Int. Symp. On Fire Safety Siene, T. ashiwagi, ed., Ottawa, pp [9] Shepherd, P. G. (999). The Performane in Fire of Restrained Columns in Steel- Framed Constrution, PhD Thesis, University of Sheffield, U. [0] Wong, M.B. (2005). Modelling of axial restraints for limiting temperature alulation of steel memers in fire, J. Constr. Steel Res. 6 (5), pp [] Bailey, C. G., Newman, G. M. (996). The ehaviour of steel olumns in fire, Steel Constrution Institute Doument RT524. [2] Wang, Y. C. (2004). Postukling Behavior of Axially Restrained and Axially Loaded Steel Columns under Fire Conditions, J. Strut. Eng. 30(3), pp

15 Figure Captions Figure. Figure 2. Figure 3. Figure 4. Figure 5. Figure 6. Figure 7. Figure 8. Figure 9. Analytial model Single floor heating senarios Multiple floor heating senario, with equivalent- and omined- spring representation Heating senario of hosen restraint values Fore vs Temperature during loading phase Shemati of olumn ehaviour Fore vs Temperature urves eyond snap-through phase Full Fore vs Temperature urves exhiiting snap-ak Axial displaement of top of olumn Figure 0. Simplified restrained olumn representations: (a) efore loading or heating, () preukling, () elasti ukling, (d) plasti ukling. Figure. Assumed degradation of elasti modulus and yield strength with temperature. Figure 2. Lateral displaements of olumn mid-span. Figure 3. Axial displaements of olumn top. 5

16 Load Length Imperfetion Figure Analytial model 6

17 Restraining Beams a) Single Heated Column Restraining Beams ) Three Heated Columns ) Whole Floor Heated Figure 2 Single floor heating 7

18 Beam Stiffness Beam Stiffness Beam Stiffness Column Stiffness Beam Stiffness Column Stiffness Figure 3 Multiple floor heating senario, with equivalent- and omined-spring representation 8

19 3.6m UB Setion Varies Aording To Tale 305x65x54UB 305x65x54UB 254x254x67UC Heated 6m Top Beam Setion Restraint Ratio 305x27x48 UB x52x60UB x20x22 UB x305x79 UB x267x73 UB x267x97 UB x292x94 UB x305x20 UB x305x224 UB x305x253 UB x305x289 UB x49x388 UB 0.38 Figure 4 Heating senario of hosen restraint values 9

20 Fore (kn) High Restraint Low Restraint Restraint Ratio ρ Temperature ( C) Figure 5 Fore vs Temperature during loading phase 20

21 Displaement a f e g h Temperature d Figure 6 Shemati of olumn ehaviour 2

22 Fore (kn) High Restraint Low Restraint Restraint Ratio ρ Temperature ( C) Figure 7 Fore vs Temperature urves eyond snap-through phase 22

23 Fore (kn) High Restraint Low Restraint Restraint Ratio ρ Temperature ( C) Figure 8 Full Fore vs Temperature urves exhiiting snap-ak 23

24 Temperature ( C) Displaement (mm) Figure 9 Vertial displaement of top of olumn 24

25 P P P δ F F F l 20 T T T M pt M pt F F F (a) () () (d) Figure 0 Simplified restrained olumn representations: (a) efore loading or heating, () preukling, () elasti ukling, (d) plasti ukling 25

26 .200 EC3 modulus.200 EC3 yield.000 Assumed modulus.000 Assumed yield Redution fator ket Redution fator kyt Temperature ( C) Temperature ( C) Figure Assumed degradation of elasti modulus and yield strength with temperature 26

27 000 µ = 0.2 ρ = 0.02 α =.20Ε 05 λ =00 γ = Bukled elasti 600 Unstale plasti Temperature ( C) Imperfet transition Stale plasti Normalised lateral displaement Figure 2 Lateral displaements of olumn mid-span 27

28 Normalised axial displaement Unstale plasti Bukled elasti Imperfet Temperature ( C) transition Stale plasti µ = ρ = 0.02 α =.20Ε λ = γ = 0.0 δ Figure 3 Axial displaements of olumn top 28