40 J.M. Rotter tests are supported by a good understanding of the phenomena and by calculation of the effects of fire in reducing the member's strength, which extend the scope and confidence of the assessment far beyond the conditions actually tested. However, the structural environment of a member in such a fire test is not well related to the situation in the complete structure in a real compartment fire. It has long been recognised that the thermal scenario is unrealistic, but the greater shortcomings of the structural idealisation have not been properly identified. In a determinate structure, the pattem of internal forces and stresses can be determined using only equilibrium considerations, provided the displacements are small. Most fire tests on isolated members match this condition. By contrast, in a redundant structure, the pattern of internal forces and stresses depends on the relative stiffnesses of parts of the structure. In the training of structural engineers, the significance of lack of fit and imposed displacements in redundant structures is not strongly emphasised, and building structures are often portrayed as dominated by bending actions, accompanied by axial forces in the columns which are rather easily determined. There are good reasons for these choices, based on the theorems of plasticity. Whilst these ideas are effective in ambient temperature design, they do not carry over very well into the fire scenario. Figure 1: Runaway failure in determinate structure under fire At collapse, determinate and redundant structures are more sharply differentiated than the above simple definitions suggest. The determinate structure collapses when the most highly stressed region reaches the local strength, and this strength may be reduced by elevated temperatures. The concept of "runaway" failure in a structure under fire derives from this situation (Fig. 1) where the rapid deterioration of the properties of the material causes deflections to increase very rapidly when the temperature reaches the appropriate value (which naturally depends on the load level). However, in the redundant structure with adequate ductility and without instability, different stress paths may support additional load when the local strength is reached at a single location. This effect is classically defined as "plastic redistribution", but it is open to wider interpretation if different load carrying mechanisms can come into operation. Where a structure is very redundant and there are many alternative load paths, large deformations can develop without a loss of its capacity to carry the imposed loads, and it may be difficult to decide how to define "failure". The question of how to define failure is faced in structural engineering fields apart from fire; researchers in pressure vessels and rectangular storage structures are faced with the need for new failure definitions which can incorporate survival under large displacements. It should be noted that the theorems of plasticity on which structural engineering design depends so much depend not only on ductility and lack of instability, but they are strictly only valid for small displacements. Behaviour of Highly Redundant Multi-Storey Buildings 41 Structural engineering practice for the design of frames at ambient temperature is chiefly based on the concept that the forces in individual members can be found from a global elastic analysis, but the members are subsequently proportioned according to an ultimate strength assessment for each member alone. Thus, the inelastic and large deformation behaviour which may affect the member when alone is deemed to have little effect on the response of the complete structure. This design procedure cannot capture the phenomena which occur in highly restrained structural elements under fire. The studies described in this paper arose from attempts to understand the complex behaviours seen in calculations (Sanad et al., 1999) to model the Cardington full scale fire tests on a composite building (Kirby, 1997; Moore, 1997). Many conclusions concerning behaviour could be drawn directly from the tests (Martin, 1995; Newman, 1997), but those presented here are more difficult to extract from the experimental record. The paper is particularly concerned with the development of large displacements, since these permit the new load-carrying mechanism of tensile membrane action to come into play. EFFECTS OF IN-PLANE RESTRAINT IN COMPARTMENT FIRES When a compartment fire occurs in a large building, the effects are felt on the floor system above the fire and the columns of the fire floor. The columns are critical to the building's survival, and need fire protection; they are not discussed further here. For the floor system, the compartment boundaries effectively isolate the surrounding structure from really high temperatures, and the floor' s continuity in its own plane means that differential thermal expansions play a dominant role (Fig. 2). Figure 2: Plan view of floor with heated compartment under tire Under fire conditions, temperatures of the order of 800 or 1000~ are achieved, and the thermal strains are extremely large. In such highly redundant structures, the consequent lack of fit means that the cold structure imposes huge forces on the heated region, but these are relieved by two mechanisms, which are the subject of this paper: plastic straining (with decreasing material strength) and post-buckling 42 J.M. Rotter large displacements. The hot zone covers a limited area, determined by the compartment size, and the compressive stresses which develop within it are governed by the lack of fit, the in-plane stiffness of the floor system around it, and the stress-relieving mechanisms of plasticity and post-buckling. Most importantly, the deflections which develop within the hot region are not controlled by material degradation, as was the case for a determinate structure (Fig. 1) but by restrained thermal expansion. No "runaway" collapse conditions occur, provided the building has adequate in-plane restraint. The development of large deflections limits the damage to the structure, and these large deflections permit different load carrying mechanisms to develop (other than small deflection bending). The differential thermal environment is not simply a contrast between the heated zone and the cold surroundings. Exposed steel members (low mass and high thermal conductivity) rapidly achieve high temperatures, but the concrete slab (high mass and low thermal conductivity) develops significant temperature gradients through its thickness, and with its high indeterminacy as a plate structure, acts as a major restraint against thermal expansion. As the slab is heated, its expansion must also be accommodated by the mechanisms described above, but its slenderness means that buckling, rather than plasticity, is the dominant phenomenon. Thermal gradients, both in the two dimensional horizontal plane, and vertically through the slab, strongly affect the deflections of the structure. Yielding under thermal expansion The floor system of a building is designed to carry load by bending and shear. The slab often spans between beams in something like one way action, and its behaviour is most easily understood by considering beam behaviours. As noted above, significant axial forces develop in a beam or slab if it is heated and fully or partially restrained against axial expansion (or contraction during cooling). Depending on the surroundings, these forces can be either beneficial or deleterious to the performance of the structure. When floor slabs expand, they can exert enormous forces on the surrounding structure. The first key aspect of the floor system behaviour under fire is therefore in the plane of the floor. If the floor system provides stiff restraint, the thermal expansion forces can become very large. A fully restrained steel element under thermal expansion reaches compressive yield at a temperature of only: % (1) ATe = Ec~ in which ATy is the temperature rise to cause yield, tx is the thermal expansion coefficient and E is the elastic modulus of steel. This relationship shows that a temperature change of 102~ for 250 grade steel and 142~ for 350 grade steel (ignoring any material degradation) is needed to achieve yield. Compared with the 800 or 1000~ which the fire may achieve, these temperatures are so low that there is plenty of scope for high stress development in real fires even when the restraint is only partial. Key understanding of responses to thermal expansion When heated, a structure displays a variety of responses. Structural engineers, trained at the outset to relate deflections to structural stiffnesses, stresses to deflections, and growing deflections to material degradation, are often surprised by the more complex responses arising from thermal expansion. Indeed, because the structural fire literature is mostly concerned with determinate structures in which these connections are valid, the importance of thermal expansion strains is often lost. To understand redundant structural behaviours under fire, attention should be focused on the strain state, since this is where thermal expansion, stress-strain relationships, and strain-displacement relationships can all be brought together. The key relationships needed for understanding are: Behaviour of Highly Redundant Multi-Storey Buildings 43 ~total -" ~thermal + ~mechanical (2) with: ~mechanical > 6 stresses (3) e,otat > 8 deflections (4) The total strains govern the deformed shape of the structure ~5, through kinematics or compatibility. The stress state in the structure cr (elastic or plastic) depends only on the mechanical strains. In a structure whose displacements are not restrained, thermal strains are free to develop in an unrestricted manner. If there are no external loads, axial expansion or thermal bowing results from: ~,o,al = ~,h~.n~ (5) [Stherma ! ~ 5 deflections (6) By contrast, if thermal strains are fully restrained without external loads, thermal stresses and plastification result from: 0 = Etherma I + [;mechanical (7) E:mechanical ~ Cr stresses (8) In real structures under fire, most situations have a complex mix of mechanical strains due to applied loading and mechanical strains due to restrained thermal expansion. These lead to combined mechanical strains (Eqn 8) which often far exceed the yield values, resulting in extensive plastification. The deflections of the structure, by contrast, depend only on the total strains, so these may be quite small if there is high restraint, but they are associated with extensive plastic straining. Alternatively, where less restraint exists, larger deflections may develop, but with a lesser demand for plastic straining and so less destruction of the stiffness properties of the materials. These relationships show that larger deflections may reduce material damage and may simultaneously correspond to higher structural stiffnesses. Alternatively, they show that high restraint may lead to smaller deflections with lower stiffnesses due to material damage. Thus they cause structural situations which appear to be quite counter-intuitive for most structural engineers. small transverse load P~I ++++++++++++~++++++++++++ ~P prebuckling state: expansion develops axial compression ~ecr L endsr n c, against axial ~ translation ~ postbuckled state: expansion produces deflections Figure 3: Beam with rigid axial end restraint subjected to increasing temperature Thermal buckling and post-buckling When an elastic beam with rigid axial restraint at its ends is uniformly heated (Figure 3), compressive stresses develop following Eqns 2 & 3. If the modulus E and thermal expansion coefficient ot are deemed independent of temperature, the beam reaches a bifm'cation point when the thermal thrust attains the classical Euler buckling load: 44 J.M. Rotter : ~2 I~! 2 E1 EA ot AT = n 2 7 EA (9) where g is the effective length of the beam and depends on the end flexural restraint conditions. The critical buckling temperature rise ATer for unchanging elastic modulus E is thus (Figure 3) ~2 (~/2 (AT)r = g (10) For structural elements of the slenderness commonly found in slabs, this critical temperature can easily be as low as 100 or 200~ The phenomenon is thus also likely to occur in most fires. Any material degradation (deterioration with temperature rise or yielding or cracking) reduces the temperature. If the elastic modulus and/or expansion coefficient are accepted as temperature dependent, the relationship is not so simply defined, since the thrust is a nonlinear integral of the thermal expansion and elastic modulus, whilst the stability is governed by a tangent modulus condition: ~cr = a(T) E,r(T,g) dT = n 2 E.r(Tcr,g,~r) (11) To in which E.r(T,~) is the tangent modulus which varies with the temperature T and stress state cr and or(T) is the thermal expansion coefficient which changes with the phase of the material. Whilst buckling may occur at quite a low temperature, the phenomenon is unlike that in a classical column; the force in the beam is controlled by constrained thermal expansion (the beam is too long), not by an imposed force. Thus, the large displacements which rapidly follow bifurcation phenomena under ambient temperature static loading do not occur here. The post-buckling axial shortening 8x and transverse deflection ~y of an axially loaded pinned beam may be approximated (Euler, 1744) by: 8x = L ~-~+ 2 -1 (12) ~)y 2"Xf-2L~P = rc P-'~E- 1 (13) Figure 4: Deflection of heated axially restrained elastic beam Behaviour of Highly Redundant Multi-Storey Buildings 45 Applying Eqn 10 and the thermal expansion Lo~AT in the post-buckled state (Figure 3), the post- buckling transverse deflection 8y becomes: 2@L ]o~AT - (nr I g)2 2@L ~ AT/ATcr- 1 8y = n "~ 2 +- (-~rl-g)2 = ~ __ {2/(otATcr)} +1 (14) The prediction of Eqn 14 for the post-buckling deflection of the beam is shown in Figure 4, together with the prediction from a large displacement finite element calculation using ABAQUS (1997) for the response in the presence of a small transverse load. The load smoothes the bifurcation phenomenon slightly, but the critical temperature can be clearly identified (matching Eqn 10), and the post-buckling response involving rapid growth of deflections into a large deformation state matches Eqn 14 The key feature of this behaviour is that the increasing deflections in the post-buckling state permit the thermal expansion to be accommodated in member curvature, thus reducing the stresses present but inducing large deflections. Here, post-buckling is not, in any sense, an unstable condition. The magnitudes of the deflections are very substantial compared with the length of the beam. Figure 5: Axial force development in heated axially restrained elastic beam The axial force developing in the beam under increasing temperature is shown in Figure 5. As assumed above, this force almost constant in the post-buckling region and additional thermal expansion is all absorbed in additional deflection, instead of causing increased stresses (Eqns 2-4). For local fires in real structures, this is a helpful effect as it limits the additional forces generated by the restrained thermal expansion and thus reduces damage to adjacent parts of the structure. Thus, buckling is good for this structure! This is perhaps a rather unexpected conclusion. Figure 6: Response of heated axially restrained elastic-plastic beam 46 J.M. Rotter In real structures, the elastic modulus and yield stress are affected by temperature rise. Thus, steel yielding and concrete cracking may be expected to damage the simple responses seen above. It is then no longer easy to perform algebraic analyses, but the corresponding ABAQUS calculation is shown in Figure 6. The axial force developed in the beam declines, controlled by development of plastic hinges at midspan and the ends. Increasing deflections mean that the axial force must fall even if the moment were to remain constant, but the yield surface permits an increasing moment with falling axial force. However, the magnitude of the deflection and its rapid growth are little changed. Initially, this is another surprising result for structural engineers, but it is easily understood in terms of thermal expansion; the expanded beam length is very precisely proscribed, and is accommodated by deflection. This effect indicates that plasticity in the expanding structure may be a good phenomenon, since it reduces the forces to which other parts of the structure are subjected and thus reduces mechanical damage. This point is raised again later. Finite axial restraint against thermal expansion Rigid axial restraint is generally impossible to achieve, so the above represents only a limit; real structures offer only finite axial restraint. Assuming that the restraint to axial expansion can be represented by a linear translational spring of stiffness kt (Figure 7), the compressive axial stress developed by thermal expansion in an elastic beam with unchanging modulus becomes: E ct AT ~=(I+EA-k~ ) (15) The critical buckling temperature increment (AT)c r is modified from the Eqn 10 value to (Figure 8): (A~. = ~ 1 + (16) From this relationship it can be seen that buckling and post-buckling phenomena should be observable at moderate fire temperatures (say 300~ in structures with translational restraint stiffnesses kt which are quite comparable with the axial stiffness of the member (EA/L). This axial stiffness itself is reduced by heating through the reduction in ET, so these post-buckling phenomena should be observed in slabs and beams in typical fires. p length L, effective length gefr properties E, A, I k p t prebuckling state: expansion develops axial compression ~" __ ~ with stiffness k postbuckled state: expansion produces deflections against axial translation Figure 7: Elastic axial restraint to beam expansion under increasing temperature Not only are the buckling temperatures reduced by elastic-plastic material degradation, but as shown in Figure 6a, the forces imposed by the post-buckled beam decline rapidly. Because these forces become smaller, even relatively modest elastic restraint stiffnesses become effective and act in a manner similar to rigid axial restraints. For this reason, large post-buckling deflections can be expected in large buildings under compartment fires, even when the compartment is in a edge or comer position. Behaviour of Highly Redundant Multi-Storey Buildings 47 The moments developing in the beam have not been shown for space reasons, but it should be noted that the thermal expansion effects rapidly swamp the load-carrying primary bending effects. Figure 8: Bifurcation temperature for partially axially restrained beams THERMAL GRADIENTS THROUGH THE THICKNESS The above discussion assumed a uniform temperature distribution through the slab or beam thickness. The concrete slab is heated from below and a high temperature gradient develops through its thickness. Temperature gradients also produce some surprising consequences. For clarity, it is helpful first to study the gradient alone, before recombining the effects of the gradient and a uniform rise into a realistic distribution. The temperature differential leads to thermally induced bending or to thermal bowing (Eqns 2-4 apply again). The differential induces either bending moments or additional deflections or both in the slab. Where the bending moment causes cracking, the stiffness again declines and encourages post-buckling large displacement effects. 8 ~-~ cold: contra~ ~hermal] ~ains ] hot: expansion xxxxx cold: contraction 8 = 0 ,~ hot: expansion "q ai a' Figure 9: Thermal gradient, and the effect of rotational boundary conditions The natural starting point is a simply supported beam (axially and rotationally flee), subject to a linear dT dT through-thickness thermal gradient ~ (Figure 9). A uniform curvature d~ = tx~ ~y is caused by thermal expansion (and contraction). No stresses develop and the hot lower surface leads to downward bowing. If instead, the beam is rigidly restrained against end rotations (but axially free to translate), no deflections develop at all in the beam! It remains perfectly straight. Instead, a constant bending hogging moment is induced throughout the beam (Eqns 2-4 again), given by M = E1 cti dy" The hot 48 J.M. Rotter lower surface is thus in compression, and first cracking in concrete occurs on the top unheated surface (a counter-intuitive result for most structural engineers). More importantly, where the beam is composite, the steel joist at the bottom can become fully yielded throughout its length in compression under extreme fire conditions, causing engineers trained in conventional design to ask how the composite beam structure can possibly still carry its loads. The thermal curvature qb due to a uniform gradient (with no net temperature rise), causes a deflection gy in an axially free beam of: I~y = ~" 1 - cos 7 (17) and, in a large displacement evaluation, this causes the distance between the supports to reduce by: ~Sx =L-2~ (~-~) (sin txL dTyy~ )-~- (18) If the beam ends are now axially restrained, the loss of length in arc shortening 6x must be replaced by a stress-related extension, which requires a uniform axial tension closely modelled by (EA/L) 6x. Thus, for axially restrained but rotationally free beams (close to real conditions), a thermal gradient produces axial tension. By contrast, a uniform temperature rise produces axial compression. Thus, the observed deformed shape of the structure is a poor indicator of whether part of the structure is in axial tension or compression, and a real temperature distribution with both thermal gradient and centroidal temperature rise can cause either axial tension or axial compression, with quite similar deformations. Some of these forces participate in load-carrying mechanisms (under large displacement regimes), whilst others are purely self-stressing in character. The effects of reduced flexural restraint at the ends of the beam is discussed by Rotter et al. (1999). In a composite building, an expanding heated steel joist beneath a slab is restrained by the colder slab (a vertical thermal gradient) throughout the fire period and can become severely plastified in compression (Fig. 9) if large deflections do not occur. The slab is a major cause of thrust developing in the steel joist. The thermal expansion strains are absorbed as large compressive plastic strains, causing significant shortening of the joist. On cooling, this length reduction is not easily recovered, especially because the cooling steel gains stiffness and strength faster than the tensile stresses develop. Thus, very high tensile stresses develop during cooling, which can cause rupture damage to the connections unless these are designed to be ductile under joist tension, even though they occur in positions where the ambient temperature designer believes that hogging bending is occurring. In the design of highly redundant buildings, fire design should not ask "How is the load being carried?", but "Can large deflections develop well?", and "Must greater ductility be provided for the cooling phase?". LARGE DEFLECTIONS AND MEMBRANE ACTION Two separate structural stress pattems in slabs are termed "membrane action". Both involve axial forces in the plane of the slab (membrane forces). Both require the boundaries of the slab to be restrained in the plane of the slab (this was termed axial restraint above). At small displacements, compressive membrane action occurs (Figure 10). When cracking occurs in concrete, the neutral axis or zero strain axis is displaced in the direction of the compression face. The middle plane of the slab is thus effectively subjected to an expansion. Such an effect can occur at both midspan in sagging bending and at supports in hogging bending, giving additive expansive displacements. Where these expansions are resisted by a stiff boundary, additional compressive forces develop, and where the slab is thick, the eccentricity of the compressive force transmission produces an arching action which can Behaviour of Highly Redundant Multi-Storey Buildings 49 carry a greatly increased load. This mechanism is present in steel-concrete composite beams in highly redundant structures even under ambient conditions, due to the very large disparity between their hogging and sagging neutral axes. However, this action is more powerfully demonstrated in the thermally expanding slab of the composite structure, because the thermal expansions are very large and can cause major changes in the load-carrying mechanism. The load-carrying mechanism is promoted by any effect which assists the development of large deflections. At large displacements, tensile membrane action begins (Figure 11). In tensile membrane action, the large deformations lead to a new load-carrying mechanism by change of geometry; effectively a small component of the tension carries transverse load directly. Under ambient conditions, such large displacements mean that large mechanical strains have developed, and there is a danger of rupture due to loss of ductility. Under fire conditions, thermal expansion provides much of the required deformation (Eqns 2-4), reducing the need for mechanical strains and ductility. The post-buckling deformations described above promote large displacements, and the 2D slab with a continuous displacement field, permits tensile membrane action to develop even adjacent to zones in a post- buckling compressive state. Because buckling restricts the compression forces and promotes increased deformations, tensile membrane action is more readily achieved. These membrane mechanisms make the floor slabs the strongest elements in the building since, under extreme conditions, they possess considerably greater strength than is required to carry the design loads in bending. ], sagging neutral axis high ~ i end ined ~ ~ ~ }1 ~ against axial b ~ i translation ] hogging neutral axis low [ axially restrained: compression due to changing NA location Figure 10: Compressive membrane action ~ Shear, V Axial tension, T Figure 11: Tensile membrane action at large deflections The worst scenario for a fire in a composite frame building structure is compartment breach. Structural fire design should define compartment breach as an "ultimate limit state" and ensure that it is prevented. The only structural member in a composite frame that acts as a compartment boundary is the composite floor slab. A compartment breach of the slab is unlikely because it is mostly in membrane compression throughout the fire. Appropriate reinforcement should be provided to ensure that through thickness cracks cannot develop in the slab. CONCLUSIONS Composite multi-storey building structures are highly redundant, and their floor systems exhibit high in-plane stiffness. When a compartment fire occurs beneath the floor, the behaviour of the floor system is dominated by restraint to thermal expansion, with middle surface heating and through thickness gradients causing quite different effects. The restraint to thermal expansion can easily lead to buckling and large post-buckling displacements, which are both stable and beneficial. Runaway failures are not seen in these redundant structures because the large displacements permit compressive and tensile membrane action to carry the loads in place of bending. Almost all the phenomena . expansion Lo~AT in the post-buckled state (Figure 3), the post- buckling transverse deflection 8y becomes: 2@L ]o~AT - (nr I g)2 2@L ~ AT/ATcr- 1 8y = n "~ 2 +- (-~ rl-g)2 = ~ __ {2/(otATcr)}. deflections in the post-buckling state permit the thermal expansion to be accommodated in member curvature, thus reducing the stresses present but inducing large deflections. Here, post-buckling is. thermally induced bending or to thermal bowing (Eqns 2-4 apply again). The differential induces either bending moments or additional deflections or both in the slab. Where the bending moment