The stress-strain behaviour of carbon steel
at high temperatures is essentially different from that at ambient
temperature, without a clear yield plateau but strain hardening
occurring all the way in the plastic range. Figure 6 shows the
stress-strain curves of grade 43A (i.e. S275) steel at elevated
temperature, plotted from British Steel data (Kirby & Preston
1988). British Steel Corporation (now named as Corus) carried
out an extensive small-scale tensile test programme in 1980s
on BS4360: Grades 43A and 50B steels
to provide elevated temperature data for structural fire engineering
design applications. To represent the behaviour of beams and
columns in large scale tests, the heating rates were set at the
range 5 to 20°C/min. The stress-strain curves shown in Figure
1 were derived from the transient-state tests on grade 43A
steel with a heating rate of 10°C/min.

The test results show that carbon steel begins
to lose strength at temperatures above 300°C and reduces
in strength at a steady rate up to 800°C. The well defined
yield plateau at 20°C is replaced by a gradual increase of
strength with increasing strain (or strain-hardening) at high
temperatures. Such characteristics make it very difficult to
define the strength of steel at high temperatures which is an
important parameter in fire structural design. Instead of fixing
a single strain limit for yield strength at elevated temperatures
(such as the 0.2% proof strength at 20°C), BS5950-8 (2003)
suggests three strain limits of 0.5%, 1.5% and 2% according to
the member types (see
Table 1).

For a given strain, the reduced strength of
steel at a particular temperature can be determined from the
experimental stress-strain curve at that temperature. For instance,
the reduced steel strengths of approximate 171, 208 and 213 N/mm2
corresponding to the strain limits of 0.5%, 1.5% and 2% respectively
have been obtained from the stress-strain curve of steel at 500°C
in Figure 1. The strength reduction factors (i.e. the ratio of
the steel strength at a temperature relative to its yield strength
at 20°C) can then be calculated. Figure
2 presents the strength reduction factors of Grade 43A steel
with increasing temperature at the three strain limits, as recommended
by BS5950-8.

The use of the British Steel data in BS5950-8
was justified by large scale beam and column tests. The loaded
fire tests on bare steel beams showed that high strains in excess
of 3% were developed. Excluding the thermally-induced curvature,
the stress-related strains were of the order of 2 to 3%. Thus
a conservative strain limit of 1.5% has been selected for the
fire design of steel beams (Lawson & Newman 1990). Consequently,
the strain limit was increased to 2% in composite beams design
because higher strains were normally developed in composite beams
than in steel beams at a given deflection, due to the composite
action of the slabs supported by the beams. For columns and structural
members protected with fire protection materials of poor stickability,
a strain limit of 0.5% was considered to be appropriate.

Based on the British Steel data, EN1993-1.2
derives the reduction factors for effective yield strength, proportional
limit and slope of linear elastic range as given in Table
2. The effective yield strength is related to 2% strain limit. Figure
3 illustrates the variation of the reduction factors with
temperature. The strength reduction factor at 2% strain of BS5950-8
is also plotted for comparison.

The definitions of effective yield strength,
proportional limit and slope of linear elastic range are established
on the basic characteristic of the stress-strain model for steel
at high temperatures proposed by EN1993-1.2. Figure
4 shows that the first part of the curve is a linear line
progressing up to the proportional limit `f`_{p,θ} and
the elastic modulus `E`_{a,θ} is equal
to the slope of this straight-line segment. The second part of
the curve depicts the transition from the elastic to the plastic
range. This region is formulated by an elliptical progression
up to the effective yield strength `f`_{y,θ}.
The third part of the curve is a flat yield plateau up to a limiting
strain for yield strength. The last part of the curve is characterised
by a linear line decreasing to zero stress at the ultimate strain.

Basically, the slope of linear elastic range
governs the steel stiffness, whereas the effective yield strength
governs the strength. Comparing their reduction factors at elevated
temperatures, it can be seen that the stiffness of steel reduces
earlier and more rapid than the strength. This indicates that
the failure mode of steel members may change at elevated temperatures.
For instance, a steel beam made of slender I-section, which is
designed to plastic-hinge failure under ultimate load at ambient
temperature, may experience the premature failure of web buckling
at elevated temperatures.

EN1993-1-2 provides detailed mathematical
formulae of stress-strain relationships of steel at elevated
temperatures as shown in Figure
4. The effect of creep is implicitly considered and the material
models are applicable for heating between 2 and 50 K/min.

Based on the mathematical formulae provided
by EN1993-1-2, a series of stress-strain curves at elevated
temperatures have been constructed for S273 steel as shown in Figure
5. The original British Steel data are also included for
comparison. Although the formulae cannot provide perfect fitting
with the test data at all temperatures, the correlation at temperatures
above 400°C is in good agreement. Generally, the lack of
accuracy at low temperatures below 400°C will not hinder
the accurate prediction of fire resistance of steel structures
in practice. This is because the actual loads applied to most
buildings are commonly below 60% of the ultimate loads they are
designed for at ambient temperature. That means the structures
will generally have a minimum inherent fire resistance of 500°C.

EN1993-1-2 further extends the stress-strain
relationship to include strain-hardening for steel temperatures
below 400°C, providing local or overall buckling does not
lead to premature collapse. In this case, the mathematical formulae
in Figure
4 need to be modified according to Figure
6. Figure
7 shows the stress-strain relationships for S275 steel
at elevated temperatures, allowing for strain hardening.