Engineering Structures 28 (2006) 229–239
www.elsevier.com/locate/engstruct
Stress–strain curves for stainless steel at elevated temperatures
Ju Chen, Ben Young∗
Department of Civil Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong
Received 17 February 2005; received in revised form 15 July 2005; accepted 27 July 2005
Available online 19 September 2005
Abstract
This paper presents the mechanical properties data of stainless steel at elevated temperatures. Accurate prediction of the material properties
of stainless steel at elevated temperatures is necessary for determining the load-carrying capacity of structures under fire conditions. However,
full utilization of the special feature of stainless steel has not been possible due to lack of technique data on the fire resistance of stainless
steel structural material. Therefore, both steady and transient tensile coupon tests were conducted at different temperatures ranging from
approximately 20 to 1000 ◦ C to obtained the material properties of stainless steel types EN 1.4462 (Duplex) and EN 1.4301 (AISI 304)
with plate thickness of 2.0 mm. The elastic modulus, yield strength obtained at different strain levels, ultimate strength, ultimate strain and
thermal elongation versus different temperatures are also plotted and compared with the prediction from the Australian, British and European
standards. The test results obtained from this study are also compared with the test results predicted by other researchers. A unified equation
for yield strength, elastic modulus, ultimate strength and ultimate strain of stainless steel at elevated temperatures is proposed in this paper. It
is shown that the proposed equation accurately predicted the yield strength, elastic modulus, ultimate strength and ultimate strain compared
with the test results. Furthermore, stress–strain curves at different temperatures are plotted and a stress–strain model is also proposed.
© 2005 Elsevier Ltd. All rights reserved.
Keywords: Elevated temperatures; Experimental investigation; Fire resistance design; Mechanical properties; Stainless steel; Steady state tests; Transient state
tests
1. Introduction
Stainless steel has high corrosion resistance and aesthetic
appearance as well as ease of maintenance and ease of
construction. Therefore, stainless steel members have been
increasingly used in the construction industry, especially
in architectural and structural applications. Specifications
for the design of stainless steel structural members are
available, such as the American Society of Civil Engineers
Specification [1], Australian/New Zealand Standard [2] and
European Code 3 Part 1.4 [3]. However, these design
specifications are mainly for structures at normal room
temperature. Design specifications for steel structures at
elevated temperatures are available, such as the Australian
Standard AS 4100 [4], British Standard BS 5950 Part
8 [5] and European Code 3 Part 1.2 [6]. The reduction
∗ Corresponding author. Tel.: +852 2859 2674; fax: +852 2559 5337.
E-mail address: young@hku.hk (B. Young).
0141-0296/$ - see front matter © 2005 Elsevier Ltd. All rights reserved.
doi:10.1016/j.engstruct.2005.07.005
factors for mechanical properties at elevated temperatures
are recommended in these design specifications, but they are
mainly based on investigation of hot-rolled carbon steel.
The material properties of stainless steel are significantly
different from those of carbon steel, and the standard
for carbon steel cannot be used for stainless steel
without modification. For carbon and low-alloy steels, the
proportional limit is assumed to be at least 70% of the
yield point, but for stainless steel the proportional limit
ranges from approximately 36% to 60% of the yield strength
[7]. Therefore, the material properties of stainless steel at
elevated temperatures should be known for the fire resistant
design of stainless steel structures.
The material properties of stainless steel, such as the yield
strength, elastic modulus, ultimate strength and ultimate
strain, can be obtained from the stress–strain curve. These
material properties are one of the important factors in the
design and numerical modelling of structures. Furthermore,
the development of an accurate non-linear numerical
230
J. Chen, B. Young / Engineering Structures 28 (2006) 229–239
Nomenclature
The following symbols are used in this paper:
a, b, c coefficients for proposed equation;
E
elastic modulus;
E normal elastic modulus at normal room temperature;
ET
elastic modulus at temperature T ◦ C;
elastic modulus at yield strength at temperaE y,T
ture T ◦ C;
f 0.2
0.2% yield strength;
f 0.2,normal 0.2% yield strength at normal room temperature;
f 0.2,T
0.2% yield strength at temperature T ◦ C;
0.5% yield strength;
f 0.5
f 0.5,normal yield strength corresponding to 0.5% strain
level at normal room temperature;
yield strength corresponding to 0.5% strain
f 0.5,T
level at temperature T ◦ C;
1.5% yield strength;
f 1.5
f 1.5,normal yield strength corresponding to 1.5% strain
level at normal room temperature;
yield strength corresponding to 1.5% strain
f 1.5,T
level at temperature T ◦ C;
2.0% yield strength;
f 2.0
f 2.0,normal yield strength corresponding to 2.0% strain
level at normal room temperature;
yield strength corresponding to 2.0% strain
f 2.0,T
level at temperature T ◦ C;
fT
stress at temperature T ◦ C;
ultimate strength at temperature T ◦ C
f t,u,T
obtained from transient state test;
f u,normal ultimate strength at normal room temperature
obtained from steady state test;
f u,T
ultimate strength at temperature T ◦ C
obtained from steady state test;
yield strength at temperature T ◦ C;
f y,T
n
coefficients for proposed equation;
coefficient for proposed stress–strain equanT
tion;
coefficient for proposed stress–strain equamT
tion;
T
value of temperature;
ε
elongation (tensile strain) at the point of
fracture based on gauge length of 25 mm;
strain at temperature T ◦ C;
εT
εu,normal strain corresponding to ultimate strength at
normal room temperature;
strain corresponding to ultimate strength at
εu,T
temperature T ◦ C;
ε y,T
strain corresponding to yield strength at
temperature T ◦ C.
model requires the stress–strain curve of the material. An
investigation on the effect of stress–strain relationships on
the fire performance of steel beams indicated that beam
behaviour is very sensitive to the stress condition relative to
the temperature-reduced proportional limit and yield stress
[8]. Hence, the stress–strain relationship of a material is
an important piece of data in the non-linear numerical
model of structures for both the normal room and elevated
temperatures. It is important to investigate the stress–strain
relationship of stainless steel at elevated temperatures for the
fire resistant design of stainless steel structures.
The purpose of this study is to investigate the material
properties of stainless steel at different temperatures using
both steady and transient tests. In steady state tests, the
test specimen is heated up to a specified temperature then
a tensile test is carried out, whereas in transient state tests
the load remains constant and the temperature is raised until
the test specimen fails. Temperature would rise in a real
fire; therefore, the transient state test is more realistic in
predicting the behaviour of a material under fire than the
steady state test. Tensile coupon tests were conducted for
stainless steel types EN1.4462 (Duplex) and EN 1.4301
(AISI 304) for temperatures ranging from approximately
20 to 1000 ◦ C. The material properties were obtained
from the test results of steady and transient state tests.
A unified equation for yield strength, elastic modulus,
ultimate strength and ultimate strain for stainless steel at
elevated temperatures is proposed in this paper. Furthermore,
a stress–strain curve model that accurately describes the
stainless steel material at elevated temperatures is also
proposed in this paper.
2. Experimental investigation
2.1. Testing device
An MTS 810 Universal testing machine of 100 kN
capacity was used for tensile coupon tests. The testing
machine was calibrated before testing. The installation of the
coupon specimen and the testing device used in this study
are shown in Fig. 1. The heating device of an MTS Model
653 high temperature furnace with a maximum temperature
of 1400 ◦ C was used. The furnace composed of upper
and lower heating elements, as shown in Fig. 2. Heat was
generated by the heating elements for each heating zone. An
MTS model 409.83 temperature controller was used. Two
internal thermal couples are located inside the furnace to
measure the air temperature. Since there is a distance from
the internal thermal couples to the specimen, the temperature
detected by the internal thermal couples is higher than
the surface temperature of the specimen. Therefore, an
external thermal couple was used to measure the surface
temperature of the specimen, and the measured temperature
was considered as the real temperature of the specimen
in this paper. The differences between the temperatures
detected by the internal and external thermal couples were
in the range from 3% to 28%. The temperature accuracy
of the internal and the external thermal couples was 1.0 ◦ C
J. Chen, B. Young / Engineering Structures 28 (2006) 229–239
231
(a) Stainless steel section.
Fig. 1. Testing device.
(b) Coupon test specimen dimension.
Fig. 3. Test specimen.
A total of 45 tests (29 steady state tests and 16 transient state
tests) was conducted in this study. The actual dimensions
of the cross sectional areas of the coupon specimens within
the gauge length were measured using a micrometer. The
measured dimensions were used to determine the cross
sectional area of each coupon.
2.3. Testing procedure
Fig. 2. High temperature furnace (MTS Model 653).
and ±0.1 ◦ C. The heating rate of the furnace is 100 ◦ C/min.
The fast heating rate resulted in the temperatures
overshooting slightly, but the overshoot stabilizes within a
minute. The maximum overshoot was approximately 40 ◦ C
at low temperatures and it decreases at higher temperatures.
When the temperature was beyond 700 ◦ C, the overshoot
was less than 20 ◦ C.
An MTS Model 632.53F-11 of axial extensometer was
used to measure the strain of the middle part of the coupon
specimen. The gauge length of the extensometer was 25 mm
with the range limitation of ±2.5 mm. The extensometer was
also calibrated before testing. The extensometer was reset
when it approached the range limit during testing; hence a
complete strain of coupon specimen could be obtained.
2.2. Test specimen
The coupon test specimens were taken from stainless
steel sections cold-rolled from structural steel sheets, as
shown in Fig. 3(a). The tensile coupon specimens were taken
from the centre of the face at 90◦ angle from the weld in
the longitudinal direction of the stainless steel sections. The
test specimens were prepared in accordance with the ASTM
Standard E21-92 [9] and Australian Standard AS 2291 [10],
as shown in Fig. 3(b). Stainless steel of types EN 1.4462
and EN 1.4301 with plate thickness of 2.0 mm was used.
2.3.1. Steady state test
In the steady state tests, the specimen was heated up
to a specified temperature then loaded until it failed while
maintaining the same temperature. In this study, thermal
expansion of specimen was allowed by maintaining zero
tension load during the heating process. After reaching the
pre-selected temperature, it needed approximately 2 min for
the temperature to be stabilized, and after another 7 min
the tensile load was applied to the specimen. This would
allow the heat to transfer into the specimen. The external
thermal couple indicated that the variation of the specimen
temperature within the gauge length was less than 6 ◦ C
(±3 ◦ C) during the tests. In the steady state tests, the strain
control was used in the tensile testing machine. A constant
tensile loading rate of 0.2 mm/min was used and the strain
rate obtained from the extensometer was approximately
0.006/min, which is within the range 0.005 ± 0.002/min
as specified by the ASTM Standard E21-92 [9].
2.3.2. Transient state test
In the transient state tests, the specimen was under a
constant tensile stress level while the temperature was raised.
The stress levels selected in the tests were 2, 50, 100, 150,
200, 250, 300, 350, 400, 450, 500, 550, 600, 650, 700
and 750 MPa. In the transient state tests, the load control
was used in the tensile testing machine. The temperatures
specified in the temperature controller ranged from 100 to
232
J. Chen, B. Young / Engineering Structures 28 (2006) 229–239
Fig. 4. Definition of symbols.
Fig. 5. Stress–strain curves of stainless steel type EN 1.4462 at different
temperatures obtained from transient state test results.
900 ◦ C at intervals of 100 ◦ C. The strain of the specimen
at a given temperature was recorded using the extensometer
6 min after the temperature reached the specified value. The
ultimate strength of the specimen was defined as the point at
which strain kept increasing at a given value of temperature.
In the tests, there are two reasons for the temperature to rise
step by step. Firstly, there is a rapid loss of strength for the
loaded specimen and the loading machine could not follow
the sudden load drop under load control. Secondly, the strain
data for different specified temperatures should be obtained,
because the results of the transient state tests need to be
converted to stress–strain curves.
3. Test results
Fig. 6. Comparison of thermal elongation predicted by BS 5950-8 and EC31.2 with test results of stainless steel type EN 1.4462.
3.1. Determination of strength and elastic modulus
The yield strengths at strain levels of 0.2%, 0.5%, 1.5%
and 2.0% were obtained for the purpose of comparison
since these strain levels are widely accepted. The 0.2% yield
strength ( f 0.2 ) is the intersection point of the stress–strain
curve and the proportional line off-set by 0.2% strain.
Meanwhile, the yield strengths of f 0.5 , f 1.5 and f 2.0 at
the strain levels of 0.5%, 1.5% and 2.0%, respectively, are
those values corresponding to the intersection points of the
stress–strain curve and the non-proportional vertical line
specified at a given strain level, as shown in Fig. 4. Serration
of the stress–strain curve was observed at high temperatures
and the intersection point was the mean value determined
from the serration. The elastic modulus was determined from
the stress–strain curve based on the tangent modulus of the
initial elastic linear curve.
For transient state tests, the results are firstly converted
into stress–strain curves, as shown in Fig. 5. The specimen
was loaded to a given stress level, and the elastic modulus at
different temperatures of each specimen can be determined
from the stress–strain curves obtained from the transient
state tests. The data of the specimen at each temperature
are normalized with respect to the initial elastic modulus
at normal room temperature of each specimen, so that the
influence of elastic modulus variation can be eliminated.
Some repeat tests were conducted and the deviations
between these tests results were quite small, with a
maximum difference of 3%.
3.2. Thermal elongation in transient state test
Thermal elongation of the specimens was determined
at a tensile stress level of 2 MPa, which is close to
free thermal expansion, and compared with the thermal
elongation calculated according to British Standard 5950-8
[5] and European Code 3 Part 1.2 [6] in Fig. 6. The thermal
elongation of the strain in percentage (%) is plotted in the
vertical axis of the graph and the horizontal axis is plotted
against different temperatures. The comparison indicates
that the test values of thermal elongation of stainless steel
type EN 1.4462 are less than the values predicted by the
BS 5950-8 [5] and EC3-1.2 [6] at temperatures lower than
or equal to 660 ◦ C, and the test values are higher than
the predicted values for temperatures higher than 660 ◦ C.
Although the 2 MPa tensile stress was almost negligible
at normal room temperature for determining the thermal
elongation, it slightly affected the elongation when the
temperature increases. Since the thermal elongation was
J. Chen, B. Young / Engineering Structures 28 (2006) 229–239
233
Table 1
Material properties of stainless steel types EN 1.4462 and EN 1.4301 at normal room temperature
Stainless steel
f 0.2,normal
(MPa)
f 0.5,normal
(MPa)
f 1.5,normal
(MPa)
f 2.0,normal
(MPa)
f u,normal
(MPa)
εu,normal
(%)
E normal
(GPa)
EN 1.4462 (Duplex)
EN 1.4301 (AISI 304)
731
398
729
409
817
443
824
452
870
709
15.8
60.6
227
187
Table 2
Reduction factors of yield strength and elastic modulus of stainless steel types EN 1.4462 and EN 1.4301
EN 1.4462 (Duplex)
Temperature (◦ C)
f 0.2,T / f 0.2,normal
f 0.5,T / f 0.5,normal
f 1.5,T / f 1.5,normal
f 2.0,T / f 2.0,normal
E T /E normal
22
1.000
1.000
1.000
1.000
1.000
80
0.943, 0.923a
0.936, 0.920a
0.949, 0.919a
0.947, 0.922a
0.947, 0.949a
180
0.834
0.834
0.834
0.836
0.863
320
0.758
0.742
0.802
0.814
0.714
450
0.728
0.717
0.752
0.765
0.696
550
0.683
0.678
0.718
0.729
0.690
660
0.528
0.539
0.542
0.541
0.643
760
0.354
0.351
0.324
0.325
0.409
870
0.170
0.177
0.163
0.161
0.300
960
0.031, 0.038a
0.033, 0.041a
0.031, 0.037a
0.032, 0.036a
0.060, 0.063a
EN 1.4301 (AISI 304)
Temperature (◦ C)
f 0.2,T / f 0.2,normal
f 0.5,T / f 0.5,normal
f 1.5,T / f 1.5,normal
f 2.0,T / f 2.0,normal
E T /E normal
22
1.000
1.000
1.000
1.000
1.000
80
0.839
0.848
0.860
0.863
1.045
180
0.764
0.773
0.786
0.788
0.939
320
0.698
0.704
0.720
0.730
1.036
450
0.678
0.687
0.693
0.704
0.951
550
0.595
0.599
0.619
0.635
0.900
660
0.523
0.543
0.562
0.577
0.862
760
0.357
0.359
0.357
0.347
0.574
870
0.181
0.176
0.163
0.159
0.452
960
0.116, 0.101a
0.115, 0.098a
0.106, 0.088a
0.104, 0.089a
0.340, 0.351a
a Second test.
determined for specimens loaded at a stress level of 2 MPa,
the elastic modulus obtained from the transient state tests
were slightly underestimated.
3.3. Yield strength
The material properties obtained from the tests for
stainless steel types EN 1.4462 and EN 1.4301 at
normal room temperature are presented in Table 1.
The reduction factors ( f 0.2,T / f 0.2,normal , f 0.5,T / f 0.5,normal,
f 1.5,T / f 1.5,normal , f 2.0,T / f 2.0,normal) determined from the
ratio of different yield strengths to normal room temperature
(22 ◦ C) at different temperatures for the four strain levels of
0.2%, 0.5%, 1.5% and 2.0%, respectively, are presented in
Table 2. The test results of 0.2% yield strength of stainless
steel types EN 1.4462 and EN 1.4301 are plotted in Fig. 7.
The vertical axis of the graph plotted the reduction factor
f 0.2,T / f 0.2,normal and the horizontal axis plotted against
different temperatures. It is shown that the test results of
stainless steel types EN 1.4462 and EN 1.4301 obtained
from the steady state and transient state tests are similar.
The reduction factor of 0.2% yield strengths obtained
from the tests were compared with the Australian Standard
AS 4100 [4] prediction and also compared with the test
results conducted by Ala-Outinen [11], Sakumoto et al.
[12] and Ala-Outinen et al. [13], as shown in Fig. 7. The
comparison shows that AS 4100 provides a conservative
prediction for the test results of stainless steel types EN
1.4462, EN 1.4301 and EN 1.4571 from 400 to 900 ◦ C,
but an unconservative prediction for temperatures less than
400 ◦ C. The AS 4100 conservatively predicted the stainless
steel type SUS 304 for temperatures in the range from 600 to
Fig. 7. Comparison of reduction factor of 0.2% yield strength predicted by
AS 4100 and proposed equation with test results.
900 ◦ C, but unconservatively predicted for temperature less
than 600 ◦ C. It can be seen that the test results obtained from
this study are generally close to the test results conducted
by Ala-Outinen [11], Sakumoto et al. [12] and Ala-Outinen
et al. [13].
Since the reduction factors of 0.2% yield strength
predicted by AS 4100 are unconservative for temperatures
in the range from 80 to 500 ◦ C, a new equation is needed for
the prediction of reduction factor of 0.2% yield strength. The
unified equation for the prediction of yield strength proposed
by Chen and Young [14] for cold-formed carbon steel at
elevated temperatures is used in this study for stainless steel.
The coefficients a, b, c and n of the equation are calibrated
with all the stainless steel test results, and the coefficients are
presented in Table 3, where T is the temperature in degrees
Celsius (◦ C).
234
J. Chen, B. Young / Engineering Structures 28 (2006) 229–239
Table 3
Coefficients of the proposed equation for yield strength and elastic modulus of stainless steel types EN 1.4462 and EN 1.4301
Yield strength Eq. (1)
Temperature (◦ C)
a
b
c
n
Elastic modulus Eq. (2)
Temperature (◦ C)
a
b
c
n
22 ≤ T < 300
1.0
22
45
0.5
300 ≤ T < 850
0.63
300
5.7 × 105
2
850 ≤ T < 1000
0.1
850
600
0.8
22 ≤ T < 922
1.0
22
900
1
Fig. 8. Comparison of reduction factor of 0.5% strength predicted by
BS5950-8 with test results.
Fig. 9. Comparison of reduction factor of 1.5% strength predicted by
BS5950-8 with test results.
Proposed equation for yield strength:
f 0.2,T
f 0.2,normal
=a−
(T − b)n
c
(1)
It is demonstrated that the values of reduction factor
f 0.2,T / f 0.2,normal predicted by the proposed Eq. (1)
compared well with all of the stainless steel test results,
as shown in Fig. 7. The reduction factors of yield strength
for the strain levels of 0.5%, 1.5% and 2.0% are compared
with the EC3-1.2 [6] and BS 5950-8 [5] prediction for hotfinished steel and cold-formed steel, as shown in Figs. 8–10.
The reduction factors of 0.5%, 1.5% and 2.0% yield strength
predicted by the BS 5950-8 for hot-finished steel are
conservative for both stainless steel types EN 1.4462 and EN
1.4301 for temperatures in the range from 550 to 960 ◦ C,
but unconservatively predicted for temperatures less than
550 ◦ C. The reduction factors of 0.5%, 1.5% and 2.0% yield
strength predicted by the BS 5950-8 for cold-formed steel
are conservative for both stainless steel types EN 1.4462
and EN 1.4301 for temperatures in the range from 450 to
960 ◦ C, but unconservatively predicted for temperatures less
than 450 ◦ C. The reduction factor of 2.0% yield strength
predicted by EC3-1.2 is conservative for temperatures in the
range from 550 to 960 ◦ C, but unconservatively predicted
for temperatures less than 550 ◦ C.
Fig. 10. Comparison of reduction factor of 2.0% strength predicted by
BS5950-8 and EC3-1.2 with test results.
3.4. Elastic modulus
The reduction factor of the elastic modulus of stainless
steel types EN 1.4462 and EN 1.4301 was compared with the
AS 4100 [4], EC3-1.2 [6] and EN 10088-1 [15] predictions,
as shown in Fig. 11. It can be seen that the reduction factor
of the elastic modulus of stainless steel type EN 1.4462
obtained from the steady state tests generally agree well
with the results obtained from the transient state tests, except
J. Chen, B. Young / Engineering Structures 28 (2006) 229–239
Fig. 11. Comparison of elastic modulus predicted by AS 4100, EC3-1.2 and
EN 10088-1 with test results.
for temperatures 300 and 660 ◦ C. The reduction factor
of the elastic modulus of stainless steel type EN 1.4301
is higher than those of type EN 1.4462. It is shown that
the reduction factors predicted by EC3-1.2 are generally
conservative, except for stainless steel type EN 1.4462 for
temperatures in the range from 80 to 320 ◦ C. The AS
4100 [4] predictions of the elastic modulus are generally
conservative, except for stainless steel type EN 1.4462 for
temperatures in the range from 80 to 450 ◦ C. The EN 100881 [15] predictions are conservative for stainless steel type EN
1.4301, but unconservative for stainless steel type EN 1.4462
for temperatures in the range from 80 to 450 ◦ C, as shown
in Fig. 11.
The reduction factors of the elastic modulus obtained
from the tests are also compared with the test results
conducted by Ala-Outinen [11], Sakumoto et al. [12]
and Ala-Outinen et al. [13], as shown in Fig. 12. It
can be seen that the results are significantly different
from each other. Chen and Young [14] also proposed a
unified equation for the prediction of the elastic modulus
for cold-formed carbon steel at elevated temperatures
that is used in this study for stainless steel, as shown
in Eq. (2). The coefficients a, b, c and n of the
equation are calibrated with all of the stainless steel
test results, and the coefficients are presented in Table 3.
Eq. (2) for elastic modulus prediction is identical to Eq. (1)
for yield strength prediction, except for the different values
of the coefficients.
Proposed equation for elastic modulus:
ET
E normal
=a−
(T − b)n
c
(2)
It is shown that the prediction of the reduction factor of
the elastic modulus using Eq. (2) is generally conservative
compared with the test results obtained from this study for
stainless steel types EN 1.4462 and EN 1.4301, and the test
results obtained by other researchers, as shown in Fig. 12.
235
Fig. 12. Comparison of elastic modulus obtained using the proposed
equation with test results.
Fig. 13. Comparison of ultimate strength obtained using the proposed
equation with test results.
3.5. Ultimate strength
The reduction factor of the ultimate strength to normal
room temperature ( fu,T / f u,normal ) obtained from the steady
state tests at different temperatures are plotted in Fig. 13.
The vertical axis of the graph plotted the reduction factor
f u,T / f u,normal and the horizontal axis plotted against different temperatures. The values of the reduction factor of the
ultimate strength of stainless steel type EN 1.4462 are generally higher than those of stainless steel type EN 1.4301
for all temperatures. A unified equation for the prediction of
ultimate strength is proposed, as shown in Eq. (3). The coefficients a, b, c and n of the equation are calibrated with the
stainless steel test results, and the coefficients are presented
in Table 4. Eq. (3) for ultimate strength prediction is identical to Eqs. (1) and (2) for yield strength and elastic modulus
prediction, respectively, except for the different values of the
coefficients. It is shown that the predictions of the reduction
factor of the ultimate strength using Eq. (3) agree well with
the test results obtained from this study for stainless steel
types EN 1.4462 and EN 1.4301, as shown in Fig. 13.
Proposed equation for ultimate strength:
(T − b)n
fu,T
(3)
=a−
f u,normal
c
236
J. Chen, B. Young / Engineering Structures 28 (2006) 229–239
Table 4
Coefficients of the proposed equation for ultimate strength
EN 1.4462 (Duplex)
Temperature (◦ C)
a
b
c
n
22 ≤ T < 450
0.85
450
9.6 × 1013
5
450 ≤ T < 660
0.85
450
1.3 × 105
2
660 ≤ T ≤ 960
0.51
660
200
0.8
EN 1.4301 (AISI 304)
Temperature (◦ C)
a
b
c
n
22 ≤ T < 450
0.7
450
4.8 × 1013
5
450 ≤ T < 660
0.7
450
1.92 × 105
2
660 ≤ T ≤ 960
0.06
960
−2.2 × 105
2
Ultimate strength Eq. (3)
Table 5
Comparison of ultimate strength of stainless steel type EN 1.4462 obtained from transient state and steady state tests
Temperature (◦ C)
22
80
140
220
320
450
550
660
760
870
960
f t,u,T (MPa) (Transient)
>750
>750
700–750
700–750
700–750
650–700
600–650
400–450
50–100
2–50
2–50
f u,T (MPa) (Steady)
870
847
804a
–
–
765
754
692
453
277
136
26
30a
a Second test.
The ultimate strength of the specimen in transient state
tests is defined at a specified load when the temperature
reaches a certain value and the specimen undergoes a
continuous elongation at an appreciable rate. This specified
load was considered as the ultimate strength of the specimen
at that particular temperature in the transient state tests. In
Table 5, the ultimate strength obtained from the transient
state tests ( f t,u,T ) is compared with the ultimate strength
obtained from the steady state tests ( fu,T ). The results of
ultimate strength obtained from the steady state tests are
generally higher than the results obtained from the transient
state tests, as shown in Table 5.
3.6. Ultimate strain
The ultimate strain is defined as the strain corresponding
to the ultimate strength. The reduction factors of the ultimate
strain to normal room temperature (εu,T /εu,normal ) obtained
from the steady state tests at different temperatures are
plotted in Fig. 14. The vertical axis of the graph plotted
the reduction factor εu,T /εu,normal and the horizontal axis
plotted against different temperatures. The values of the
reduction factor of ultimate strain of the stainless steel
type EN 1.4462 are higher than those of the stainless steel
type EN 1.4301 for all temperatures. The proposed unified
Eq. (4) for ultimate strain is identical to the unified equation
for yield strength, elastic modulus and ultimate strength,
except for the different values of the coefficients. The
coefficients a, b, c and n of the equation are calibrated
with the stainless steel test results in this study, and the
coefficients are presented in Table 6. It is shown that the
predictions of the reduction factor of ultimate strain using
Eq. (4) generally agree with the test results obtained from
Fig. 14. Comparison of ultimate strain obtained using the proposed equation
with test results.
this study for stainless steel types EN 1.4462 and EN 1.4301,
as shown in Fig. 14.
Proposed equation for ultimate strain:
εu,T
εu,normal
=a−
(T − b)n
c
(4)
4. Stress–strain curve
The stress–strain relationship of stainless steel material
at elevated temperatures is necessary in the analysis and
simulation of stainless steel structures under fire. However,
most stress–strain curve models are based on hot-rolled
steel, and very few stress–strain curve models based on
stainless steel have been reported. Therefore, a stress–strain
curve model for stainless steel at elevated temperatures is
proposed in this paper. Chen and Young [16] proposed an
accurate stress–strain curve model for cold-formed carbon
J. Chen, B. Young / Engineering Structures 28 (2006) 229–239
237
Table 6
Coefficients of the proposed equation for ultimate strain
EN 1.4462 (Duplex)
Temperature (◦ C)
a
b
c
n
22 ≤ T < 180
1.0
22
480
1
180 ≤ T < 660
0.67
180
5.42 × 1021
8
660 ≤ T ≤ 960
0.15
660
5000
1
EN 1.4301 (AISI 304)
Temperature (◦ C)
a
b
c
n
22 ≤ T < 180
1.0
22
247
1
180 ≤ T < 660
0.36
180
6.1 × 1016
6
660 ≤ T ≤ 960
0.16
660
2000
1
Ultimate strain Eq. (4)
steel material at elevated temperatures. This model is used
and calibrated with the steady state test results of stainless
steel types EN 1.4462 and EN 1.4301. The model is based
on the stress–strain curve model for stainless steel at normal
room temperature proposed by Mirambell and Real [17]
and Rasmussen [18], which is originally based on the
Ramberg–Osgood [19] equation.
The proposed stress–strain curve model is as follows:

fT
fT nT


+ 0.002

ET
f y,T
εT =
f T − f y,T m T
f T − f y,T



+ εu,T
+ ε y,T
E y,T
f u,T − f y,T
for f T ≤ f y,T
(5)
for f T > f y,T
(a) Initial part of stress–strain curves.
and
ET
(6)
1 + 0.002 n T E T / f y,T
√
n T = 6 + 0.2 T
(7)

T
5.6 −

for stainless steel type EN 1.4462

200
(8)
mT =

T

2.3 −
for stainless steel type EN 1.4301
1000
E y,T =
where f T is the stress, f u,T is the ultimate strength, f y,T is
the yield strength, E T is the elastic modulus at temperature
T in degrees Celsius (◦ C), E y,T is the elastic modulus at
yield strength at temperature T ◦ C, εT is the strain, εu,T is
the strain corresponding to ultimate strength and ε y,T is the
strain corresponding to yield strength at temperature T ◦ C.
The comparisons of the stress–strain curves obtained
using Eq. (5) with the test results for stainless steel types EN
1.4462 and EN 1.4301 at different temperatures are shown
in Figs. 15 and 16, respectively. The proposed stress–strain
curves plotted using Eq. (5) were based on the material
properties obtained from the test results for the full strain
range up to the ultimate tensile strain. Generally, the
proposed stress–strain curve model accurately predicted the
stainless steel type EN 1.4462 and conservatively predicted
the stainless steel type EN 1.4301 for the temperature in the
range from 22 to 960 ◦ C. In order to plot a stress–strain curve
for stainless steel at a given temperature, the yield strength
( f y,T ), elastic modulus (E T ), ultimate strength ( f u,T ), and
ultimate strain (εu,T ) are needed. The equations for f y,T
(b) Complete stress–strain curves.
(c) Complete stress–strain curves.
Fig. 15. Comparison of stress–strain curves predicted using the proposed
stress–strain model with test results for stainless steel type EN 1.4462.
and E T , f u,T , and εu,T have been proposed in this study
as shown in Eqs. (1), (2), (3) and (4), respectively.
238
J. Chen, B. Young / Engineering Structures 28 (2006) 229–239
(a) Initial part of stress–strain curves.
modulus predicted by the Australian standard and European
standards are generally unconservative for temperatures in
the range from 80 to 450 ◦ C, but conservatively predicted
for higher temperatures up to approximately 1000 ◦ C.
Hence, the standards provide unconservative predictions for
relatively low temperatures. A unified equation to determine
the yield strength, elastic modulus, ultimate strength and
ultimate strain of stainless steel types EN 1.4462, EN 1.4301
and EN 1.4571 for temperatures ranging from approximately
20 to 1000 ◦ C has been proposed in this paper. It is shown
that the proposed equation conservatively predicted the yield
strength, elastic modulus, ultimate strength and ultimate
strain of stainless steel material at elevated temperatures.
In addition, a stress–strain curve model for stainless steel
at elevated temperatures has been proposed in this paper.
The proposed model covers the full strain range up to the
ultimate tensile strain for temperature in the range from 22 to
960 ◦ C. Generally, it is shown that the proposed stress–strain
curve model accurately predicted the stainless steel test
results.
Acknowledgment
(b) Complete stress–strain curves.
The authors are grateful to STALA Tube Finland for
supplying the test specimens.
References
(c) Complete stress–strain curves.
Fig. 16. Comparison of stress–strain curves predicted using the proposed
stress–strain model with test results for stainless steel type EN 1.4301.
5. Conclusions
A test program on the material properties of stainless
steel at elevated temperatures has been presented. The test
program included two types of stainless steel, EN 1.4462
(Duplex) and EN 1.4301 (AISI 304), with plate thickness
of 2.0 mm. Steady and transient state tests were conducted
at different temperatures. The yield strengths, elastic
modulus and thermal elongation obtained from the tests
were compared with the Australian, British and European
predictions. Generally, it is shown that the yield strengths
predicted by the Australian, Britain and European standards
are unconservative for temperatures in the range from 80 to
450 ◦ C, but conservatively predicted for higher temperatures
up to approximately 1000 ◦ C. It is also shown that the elastic
[1] ASCE. Specification for the design of cold-formed stainless steel
structural members. SEI/ASCE-8-02. Reston (Virginia): American
Society of Civil Engineers; 2002.
[2] AS/NZS 4673:2001. Cold-formed stainless steel structures. Sydney
(Australia): Australian/New Zealand Standard; 2001.
[3] EC3. Eurocode 3: Design of steel structures – Part 1.4: General
rules – Supplementary rules for stainless steels. Brussels: European
Committee for Standardization, ENV 1993-1-4, CEN; 1996.
[4] AS 4100:1998. Steel structures. Sydney (Australia): Standards
Australia; 1998.
[5] BS 5950-8. Structural use of steelwork in building - Part 8: Code of
practice for fire resistant design. British Standard BS 5950-8:1990.
British Standards Institution (BSI); 1998.
[6] EC3. Eurocode 3: Design of steel structures – Part 1.2: General
rules –Structural fire design. Brussels: European Committee for
Standardization. DD ENV 1993-1-2:2001, CEN; 2001.
[7] Yu WW. Cold-formed steel design. 3rd ed. New York: John Wiley and
Sons, Inc.; 2000.
[8] Buchanen A, Moss P, Seputro J, Welsh R. The effect of stress–strain
relationships on the fire performance of steel beams. Engineering
Structures 2004;26(11):1505–15.
[9] ASTM E21-92. Standard test methods for elevated temperature
tension tests of metallic materials. In: Annual book of ASTM
standards. Metals-mechanical testing; Elevated and low-temperature
tests; Metallography, vol. 03.01. West Conshohochken (PA):
American Society for Testing and Materials; 1997.
[10] AS 2291: 1979. Methods for the tensile testing of metals at elevated
temperatures. Sydney (Australia): Standards Australia; 1979.
[11] Ala-Outinen T. Fire resistance of austenitic stainless steels Polarit
725 (EN 1.4301) and Polarit 761 (EN 1.4571). VTT research notes.
Finland: Technical Research Center of Finland ESPOO; 1996.
J. Chen, B. Young / Engineering Structures 28 (2006) 229–239
[12] Sakumoto Y, Nakazato T, Matsuzaki A. High-temperature properties
of stainless steel for building structures. Journal of Structural
Engineering, ASCE 1996;122(4):399–406.
[13] Ala-Outinen T, Oksanen T. Stainless steel compression members
exposed to fire. VTT research notes. Finland: Technical Research
Center of Finland ESPOO; 1997.
[14] Chen J, Young B. Mechanical properties of cold-formed steel at
elevated temperatures. In: Proceedings of 17th international specialty
conference on cold-formed steel structures. 2004, p. 437-65.
[15] EN 10088-1. Stainless steels-Part 1: List of stainless steels. Brussels:
European Committee for Standardization, EN 10088-1:1995E,
239
CEN; 1995.
[16] Chen J, Young B. Effects of elevated temperatures on the mechanical
properties of cold-formed steel. In: International symposium on coldformed steel metal structures. 2004, p. 139–58.
[17] Mirambell E, Real E. On the calculation of deflections in structural
stainless steel beams: an experimental and numerical investigation.
Journal of Constructional Steel Research 2000;54(1):109–33.
[18] Rasmussen KJR. Full-range stress–strain curves for stainless steel
alloys. Journal of Constructional Steel Research 2003;59(1):47–61.
[19] Ramberg W, Osgood WR. Description of stress–strain curves by three
parameters. NACA technical note 902. 1943.