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

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~

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

~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

(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

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

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