STRESS-STRAIN-TEMPERATURE RELATIONSHIP
STRUCTURAL STEEL
FOR
By K. W. Poh1
ABSTRACT: This paper presents a new mathematical relationship for representing the stress-strain behavior of
structural steel at elevated temperatures. The relationship is constructed by fitting two versatile, continuous
equations to experimental data. The first equation, a general stress-strain equation previously proposed by Poh,
is used for characterizing the stress-strain data. The second equation is proposed in this paper. It is used for
representing the variation of the stress-strain behavior with temperature. A simple two-stage procedure is used
for fitting the equations to the experimental data. This gives a set of curve fitting coefficients that can be simply
expressed in a matrix format. It is shown that the resulting relationship accurately represents the experimental
data. Furthermore, since the relationship comprises only two equations and a coefficient matrix, it can be easily
incorporated into computer analysis programs. It can also be readily used for establishing the properties of other
steels.
INTRODUCTION
In the analysis of structures, the stress-strain relationship of
the material must be represented mathematically. In the case
of steel, the stress-strain relationship is often assumed that to
be elastic-perfectly plastic and expressed accordingly as
再
⫺␴y;
␴=
when ε < ⫺␴y /E
Eε;
when ⫺␴y /E ⱕ ε ⱕ ␴y /E
␴y;
when ε > ␴y /E
(1)
where ␴ = stress; ε = strain; E = modulus of elasticity; and ␴y
= yield stress.
The shape of the curve described by (1) is controlled by
two parameters, E and ␴y. There are no established theories
available that can be used to derive the values of these parameters. They must be obtained experimentally. This is achieved
by fitting mathematical equations to the appropriate parts of
the experimental ␴-ε data and obtaining their values as the
curve fitting coefficients.
Provided that the maximum strain does not encroach substantially upon the strain-hardening portion of the ␴-ε curve,
(1) is a good approximation of the ␴-ε curve for steel
(Fig. 1).
However, as the temperature increases, the ␴-ε behavior
changes. Apart from a gradual decrease in both stiffness and
strength (i.e., E and ␴y), the shape of the ␴-ε curve also varies
significantly (Fig. 2). Consequently, it can be very complicated
to fully and accurately represent the behavior. To do so, a
three-dimensional stress-strain-temperature (␴-ε-T ) relationship must be used.
(Jeanes 1985; Contro et al. 1988), trilinear (Corraddi et al.
1990), or quadrolinear (Ianizzi and Schleich 1991).
Multilinear approximations tend to be somewhat coarse in
representing the complex shape of the steel’s ␴-ε curves.
Therefore, in order to more closely represent the steel behavior
others use a combination of linear and smooth curves to represent the ␴-ε curves. Examples of these are ones proposed
by Furamura et al. (1985), Anderberg (1988), Lie (1992), and
Eurocode 3 (CEC 1995).
For the purpose of general analysis and design, the Lie and
Eurocode 3 relationships are probably the most commonly
used. The Lie relationship is more widely accepted in North
America, whereas, amongst the European countries, the Eurocode 3 relationship tends to be recommended. The features
of these two relationships are outlined below:
1. The Lie relationship (see Appendix I) essentially uses a
bilinear curve, with a small transition between the linear
portions, to represent the ␴-ε behavior. The curve comprises two separate equations. The first describes the linear elastic portion, and the second describes the rest of
the ␴-ε curve. The variation of the ␴-ε behavior with
temperature is represented using another four separate
equations. Therefore, altogether the relationship uses six
separate equations and contains 13 independent coefficients.
2. The Eurocode 3 relationship (see Appendix II) is more
complicated than Lie’s in that it attempts to fit the vari-
EXISTING STRESS-STRAIN-TEMPERATURE
RELATIONSHIPS
For the purpose of analyzing the behavior of structures, various ␴-ε-T relationships are being used throughout the world.
The simplest are probably ones based on multilinear approximations of the steel’s ␴-ε behavior. This may be bilinear
1
Sr. Res. Fellow, Victoria Univ. of Technol., Ctr. for Envir. Safety and
Risk Engrg., BHP Fire and Constr. Res. Unit, Werribee Campus (W075),
P.O. Box 14428, Melbourne City MC, Victoria, Australia 8001.
Note. Associate Editor: Gary Fry. Discussion open until March 1,
2002. To extend the closing date one month, a written request must be
filed with the ASCE Manager of Journals. The manuscript for this paper
was submitted for review and possible publication on September 27,
1999; revised August 30, 2000. This paper is part of the Journal of
Materials in Civil Engineering, Vol. 13, No. 5, September/October, 2001.
䉷ASCE, ISSN 0899-1561/01/0005-0371–0379/$8.00 ⫹ $.50 per page.
Paper No. 21953.
FIG. 1. Stress-Strain Curve of Steel and Elastic-Perfectly Plastic
Approximation
JOURNAL OF MATERIALS IN CIVIL ENGINEERING / SEPTEMBER/OCTOBER 2001 / 371
and is presented in this paper. For the ease of curve fitting,
this second equation is also formulated as a continuous curve.
The aim of this paper is to present a new ␴-ε-T relationship
constructed based on the two versatile, continuous equations
described above.
PROPOSED STRESS-STRAIN-TEMPERATURE
RELATIONSHIP
Curve-Fitting Equations
For the purpose of constructing the ␴-ε-T relationship, the
Poh general stress-strain equation is used to characterize the
␴-ε data. It expresses ␴ explicitly in terms of ε in a continuous
curve, and is written below in a parametric format:
FIG. 2.
Variation of Stress-Strain Behavior of Steel with Temperature
ous portions of the ␴-ε curve. It uses seven linear and
parabolic equations to represent the ␴-ε curve, including
the strain-hardening portion. The variation of the ␴-ε behavior with temperature is represented using another 19
separate linear equations. In total, the relationship uses
26 separate equations and contains 42 independent coefficients.
It is important to note that all ␴-ε-T relationships are derived
based on specific sets of experimental data. Therefore, in situations where the steel properties differ from those of the original data, the relevant coefficients in the relationships must be
reevaluated. For example, in the cases of Lie and Eurocode 3
relationships, the values of the coefficients in Appendices I
and II, respectively, must be redetermined.
DATA REPRESENTATION
In choosing a relationship for representing the experimental
data, the desired qualities are: (1) its accuracy in representing
the data; and (2) its ease in fitting to the data. The ideal choice
is obviously one that is both accurate and easy to use. However, review of the existing relationships shows that these two
qualities appear to conflict with each other. One that is more
accurate tends to be more difficult to fit to data. The reverse
is also true.
The reason for the above conflict associated with the existing relationships lies in their reliance on greater number of
equations for achieving higher accuracy. To increase the accuracy, more equations are involved, and the curve-fitting operation becomes more complex. The curve-fitting operation is
particularly complicated when multiple equations are to be fitted to individual data sets. This can be a very tedious, iterative
process which involves successive data division and curve fitting.
In order to resolve the above conflict, a general stress-strain
equation has been previously proposed (Poh 1997). This equation is highly versatile, and it expresses ␴ explicitly in terms
of ε in a single continuous curve. It has been formulated to
include the specific features of steel’s ␴-ε curve, including the
linear elastic, first yield peak, plastic plateau, and the strainhardening portions. This allows the equation to accurately represent steel behavior. At the same time, its continuity allows
it to be easily fitted to experimental data in one single operation, without dividing the data into parts.
The proposed general stress-strain equation, however, can
be used to construct only part of the ␴-ε-T relationship. In
order to construct a complete ␴-ε-T relationship, the equation
must be further extended to include the third variable, T. To
this end, a second versatile equation of variable T is proposed
␴=
ε
2兩ε兩
再
a ⫺ 兩a兩 ⫺ ␤6
⫹
冋冉
⫻
冋
冋
(␤2 ⫺ ␤3)b
冏
(␤2 ⫺ ␤3)b
1⫹
␤5
1⫹
1⫹
冏冊
␤9
册
兩(␤1兩ε兩 ⫺ ␤4 ⫺ ␤1␤7)兩 ⫺ 兩a兩
␤6 ⫺ ␤1␤7
1/␤9
⫹ ␤3b
冎
册
册
兩b兩 ⫺ 兩(b ⫺ ␤10)兩
␤10
(2)
where
a = ␤1兩ε兩 ⫹ ␤4 ⫹ ␤6
b = 兩ε兩 ⫺ ␤8 ⫺
␤4
␤1
in which, referring to Fig. 3, ␤1, ␤2, ␤3, . . . ␤i, . . . ␤10 are the
parameters that control the shape of the ␴-ε curve.
Since the ␴-ε behavior of steel changes significantly with
T, each parameter ␤i would also vary accordingly. In order to
construct a ␴-ε-T relationship, these variations must also be
expressed mathematically.
Strictly speaking, for best representation, each ␤i (i = 1 to
10) requires a specific equation to portray its variation with T.
However, for the sake of brevity, it was considered desirable
to use only one general ␤i-T equation. This means that the
adopted equation must be sufficiently flexible to accommodate
the temperature variation of each ␤i. To this end, a versatile,
continuous ␤i-T equation is formulated. It is essentially a tri-
FIG. 3.
372 / JOURNAL OF MATERIALS IN CIVIL ENGINEERING / SEPTEMBER/OCTOBER 2001
Poh General Stress-Strain Equation
1. Stage I: Fit (2) to the individual experimental ␴-ε data
set to determine the values of ␤i (i = 1 to 10) for various T.
2. Stage II: Plot ␤i against T and fit (3) to each set of ␤i-T
results to determine the coefficients of ␥ij ( j = 1 to 6).
In order to further simplify the procedure, the following
assumptions are also made:
1. Stage I
• Plastic plateau of the ␴-ε curve vanishes at T = 500⬚C.
• ␤2 = ␤1 when T > 500⬚C. This ensures no sudden
change in slope at the end of elastic portion of the ␴ε curve when the plastic plateau vanishes.
• ␤4 = ␤5 when T > 500⬚C. This prevents ␤4 from dropping suddenly to zero after the plastic plateau vanishes.
• ␤6 = 0 when T > 500⬚C. This suppresses the upper
yield peak.
• ␤10 = 0.001. This gives a sharp transition from the
elastic plateau to the strain hardening portion of the ␴ε curve.
2. Stage II
• ␤1 = E20 and ␤4 = ␴y20 when T = 20⬚C, where ␴y20 and
E20 are the yield stress and the Young’s modulus, respectively, of the steel at T = 20⬚C. This forces the
resulting relationship to give the correct room temperature values.
• ␤1 = 0 and ␤4 = 0 when T = 1,200⬚C. This forces the
␴-ε curve to vanish at T = 1,200⬚C.
• ␥i6 = 5. This gives a gradual transition in the ␤i-T relationships.
Proposed ␤i-T Equation
FIG. 4.
linear curve that smoothly transitions at the intersections of
the linear portions. It expresses ␤i explicitly in terms of T in
a continuous curve and is written as
(␥i2 ⫺ ␥i1)(2T ⫺ ␥i3 ⫺ ␥i4)
␤i =
2
冉 冏
冏冊
2T ⫺ ␥i3 ⫺ ␥i4
1⫹
␥i4 ⫺ ␥i3
␥i6
1/␥i6
⫹
(␥i1 ⫺ ␥i2)(␥i3 ⫺ ␥i4)
2
⫹ ␥i1T ⫹ ␥i5
(3)
where ␥i1, ␥i2, ␥i 3, . . . ␥ij , . . . ␥i6 are curve fitting coefficients
that control the shape of the curve (Fig. 4).
Curve-Fitting Coefficients
Construction of ␴-␧-T Relationship
Once the curve-fitting equations are defined, the construction of the ␴-ε-T relationship is simply a systematic process
to determine the curve-fitting coefficients. The final product is
a set of coefficients, i.e., ( j = 1 to 6) in (3), for each ␤i (i =
1 to 10) in (2). These coefficients can also be expressed as a
10 ⫻ 6 matrix as shown below:
[␥] =
冋
册
␥1,2
␥1,2
⭈⭈⭈
␥1,10
␥2,1
␥2,2
⭈⭈⭈
␥2,10
⭈⭈
⭈
␥6,1
⭈⭈
⭈
␥6,2
⭈⭈
⭈
⭈⭈
⭈
⭈⭈⭈
The data from an experimental program conducted by Poh
and Skarajew (1995a,b, 1996) are used for constructing the ␴ε-T relationship. These data were obtained by means of elevated temperature tensile tests conducted on specimens taken
from various grades of structural steel. The chemical compositions of the steels are summarized in Table 1. The tests were
carried at room temperature (20⬚) and at every 100⬚C between
100⬚ and 1,000⬚C, inclusive. A strain-rate of 8 ⫻ 10⫺4 was
adopted for these tests. This corresponds to the target strain
rate recommended by AS 1391 (SA 1991) for tensile testing
of metals at room temperature. A typical set of results is shown
in Fig. 2.
Before curve fitting, the experimental ␴-ε data were first
converted to a dimensionless format, by dividing the experimental ␴ values by ␴y20 and the ε values by (␴y20 /E20). This
was necessary because the data were obtained from steels of
different grades, i.e., different ␴y20.
A least-square method was then used obtain the best fit in
each stage. Fig. 5 shows examples where (2) is fitted to the
experimental ␴-ε data in Stage I; Fig. 6 shows examples where
(3) is fitted to the ␤i-T results in Stage II.
The resulting coefficients are contained in the matrix below:
(4)
␥6,10
It will be seen later that, by making some simplifying assumptions, the size of the resulting matrix can be significantly
reduced.
Curve-Fitting Procedures
Since the relationship is made up of only two equations, (2)
and (3), a simple two-stage approach can be used to determine
the values of all the coefficients ␥ij . The procedure is outlined
below:
TABLE 1.
Steel
specimen
1A
2A
2B
3A
3B
Steel
standard
AS
AS
AS
AS
AS
3679.1
3679.2
3679.2
3679.2
3679.2
Steel Specimens
Percentage of Element by Massa
Form
Steel
grade
C
P
Mn
Si
S
Ni
Cr
Cu
Sn
Mo
V
Ti
Nb
Al
N
Section
Plate
Plate
Plate
Plate
300
300
300
400
400
0.16
0.145
0.145
0.165
0.160
0.018
0.018
0.015
0.013
0.018
1.46
1.080
1.070
1.19
1.31
0.15
0.180
0.170
0.22
0.20
0.006
0.010
0.009
0.012
0.012
0.003
0.021
0.020
0.022
0.021
0.009
0.013
0.019
0.012
0.019
0.013
0.010
0.013
0.008
0.007
0.002
—
—
—
—
0.003
0.001
0.02
0.002
0.002
0.004
0.006
0.004
0.007
0.007
0.001
0.003
0.003
0.015
0.013
0.001
—
—
0.029
0.026
0.024
0.031
0.037
0.040
0.028
0.004
—
—
—
—
a
Chemical compositions taken from ladle analysis of steel heat.
JOURNAL OF MATERIALS IN CIVIL ENGINEERING / SEPTEMBER/OCTOBER 2001 / 373
i=1
i=2
1.6 ⫻ 10
0
⫺1.4 ⫻ 10
[␥] =
⫺3
0.0135
i=3
⫺4
4 ⫻ 10
i=4
⫺6
⫺6 ⫻ 10
⫺5
⫺2 ⫻ 10
i=5
⫺4
⫺1.45 ⫻ 10
⫺3
i=6
⫺2 ⫻ 10
5 ⫻ 10
⫺4
⫺3
0
⫺1.2 ⫻ 10
⫺3
i=7
i=8
0
0
⫺0.05 ⫺3.3 ⫻ 10
i=9
i = 10
1 ⫻ 10⫺3
0
⫺3
⫺2 ⫻ 10
⫺3
j=1
1 ⫻ 10⫺3
j=2
350
450
520
90
50
420
300
450
300
1 ⫻ 10⫺3
j=3
1080
700
610
720
200
320
100
0
450
1 ⫻ 10⫺3
j=4
1.004
0.03
5
5
2.5 ⫻ 10
5
⫺3
1.028
0.509
0
0
0.1
0.5
5
5
5
5
5
5
where ␥2j = ␥1j and ␥5j = ␥4j when T > 500⬚C.
These coefficients are dimensionless because the data from
which they were obtained are also dimensionless.
Resulting Relationship
Fig. 7 shows the resulting ␴-ε curves calculated using (2),
(3), and the coefficients in (5) against the experimental data.
It can be seen that the computed ␴-ε curves represent the experimental data well.
The resulting ␴-ε-T relationship, unlike the original experimental data, which are discrete 2-dimensional curves, is, mathematically, a continuous surface over the ␴-ε-T domain. Fig.
8 shows a perspective view of the ␴-ε-T surface over the positive ε domain. The surface is symmetrical over the negative
ε domain, i.e., (2) gives negative ␴ for ε.
DISCUSSIONS
Effects of Curve Fitting
Fig. 6 shows some scatter in the values of ␤i obtained. This
reflects the fact that the data were obtained from various steel
FIG. 5. Examples of Experimental and Fitted ␴-ε Curves—Stage I Fitting: (a) 200⬚C; (b) 500⬚C
1 ⫻ 10
⫺3
1 ⫻ 10⫺3
j=5
j=6
(5)
grades, for which some differences are to be expected. Nevertheless, fitting (3) through each set of ␤i-T data results in a
set of coefficients that somewhat averages the behavior of the
steels (Fig. 7). Hence, the curve-fitting procedure has provided
an effective means of averaging the behavior of a group of
data.
It can also be seen from Fig. 8 that, as a result of the curve
fitting, ␴ can be determined for any given ε and T values,
including those for which experimental data are not directly
available. Therefore, apart from averaging the behavior, it has
also provided an efficient means of data interpolation.
Upper Yield Peak
The general stress-strain equation, (2), has been formulated
incorporating an upper yield peak (Fig. 3). This upper yield
peak is usually ignored in analysis. However, it is important
to include it in the curve-fitting equation since the equation is
to be directly fitted to experimental data in a single operation.
FIG. 6.
Fitting
Examples of Experimental and Fitted ␤i-T Curves—Stage II
374 / JOURNAL OF MATERIALS IN CIVIL ENGINEERING / SEPTEMBER/OCTOBER 2001
FIG. 7.
Comparison of Experimental and Computed Curves: (a) 20⬚C; (b) 200⬚C; (c) 400⬚C; (d) 600⬚C; (e) 800⬚C; (f ) 1,000⬚C
It helps to eliminate bias in the curve-fitting results. Without
it, a higher plastic plateau will result in the fitted curve.
However, once all the curve-fitting coefficients have been
obtained, the upper yield peak can be ignored in the resulting
relationship. It is achieved simply by setting ␤6 = 0 in (2).
Comparison
For the purpose of comparison, the resulting relationship is
reexpressed in Appendix III, in a similar format to the Lie and
Eurocode relationships, as contained in Appendices I and II,
respectively. In the expressions in Appendix III, ␤6 has been
set to zero to remove the upper yield peak. The values of the
coefficients in (5) are then substituted into (3), before (3) is
expanded into separate equations, (15)–(21).
It can be seen in Appendix III that, in the expanded format,
the resulting relationship contains a total of eight individual
equations and 40 coefficients, which is more than those in the
Lie relationship (six equations and 13 coefficients), but less
than those in the Eurocode 3 relationship (26 equations and
42 coefficients).
A comparison of the resulting ␴-ε curves with those calculated using the Lie and Eurocode 3 relationships is shown
in Fig. 9. It can be seen from the figures that at low temperatures (20⬚C, 200⬚C) the Lie relationship, due to its bilinear
nature, is intrinsically limited in its capability to closely represent the ␴-ε behavior of steel. Beyond the linear elastic portion, it attempts to represent both the plastic plateau and the
strain-hardening portion using a single straight line. This results in a much higher stress for any given strain, and is hence
unconservative. The Eurocode 3 relationship, on the other
hand, is capable of representing the various portions of the ␴ε behavior. However, it does not allow for the gradual drop in
the plastic plateau over this temperature range. As a result, it
also gives higher stress for a given strain over the plastic plateau region, so is again unconservative.
At intermediate temperatures (400⬚C, 600⬚C), the ␴-ε be-
JOURNAL OF MATERIALS IN CIVIL ENGINEERING / SEPTEMBER/OCTOBER 2001 / 375
FIG. 8. Perspective View of Resulting Stress-Strain-Temperature Relationship
havior of steel tends to smoothly transition from the linear
elastic portion to the strain hardening portion. The Lie relationship transits more abruptly and hence is unable to closely
represent the behavior over that region. This tends to result in
unconservative estimates for lower strain levels and conservative estimates for high strain levels. The Eurocode 3 relationship, on the other hand, transits more gradually, but to a
perfectly flat plateau. Strain-hardening is ignored at and above
400⬚C. Again, this tends to give unconservative estimates for
lower strain levels and conservative estimates for high strain
levels.
At higher temperatures (800⬚C, 1,000⬚C), both the Lie and
Eurocode 3 relationships appear to be conservative for all
strain levels.
Advantages of Proposed Relationship
One major advantage of the proposed relationship is its capability to closely represent the ␴-ε behavior of steel. This can
be very particularly important in situations where a particular
set of experimental data are to be used to accurately analyze
the behavior of a structure. The proposed relationship provides
a means of accurately transforming the data to a readily useable format for analysis.
The previous section has alluded to the differences in the
number of equations and coefficients. As far as analyses are
concerned, these differences are somewhat irrelevant, because
analyses are invariably conducted using computers. It is
inconceivable that any analysis would be conducted by
hand, using the proposed or any other existing ␴-ε-T relationships.
However, the number of equations does matter as far as
curve fitting is concerned. Curve fitting to experimental data
can be a very labor-intensive exercise, even with the use of
computers. To a great extent, the complexity of the curvefitting process depends on the number of equations used. Essentially, the number of curve-fitting operations required
equals the number of equations used. In addition, when multiple equations are to be fitted to individual data sets, an iterative procedure may be necessary, particularly when the points
of division between the separate equations are also unknowns
to be determined. In such instances, each individual data set
must first be divided into parts by trial and error. The separate
equations must then be individually fitted to the relevant parts.
The collective results will thereby reveal whether the data di-
vision has resulted in an overall best fit. If not, the data must
be redivided and the whole process repeated until an overall
best fit is obtained.
Therefore, when using multiple-equation curves to construct
a ␴-ε-T relationship, the aforementioned iterative procedure
may be required to fit the individual experimental ␴-ε data
sets, and a similar iterative procedure used for representing the
variation of the ␴-ε behavior with temperature. This can easily
compound to a very large number of curve-fitting operations.
It therefore clearly highlights another major advantage of the
proposed relationship, i.e., its curve-fitting capability. This
arises from the versatility and particularly the continuity of the
equations used. No data division is required because each
equation can be easily and directly fitted to a data set in one
operation.
Another advantage of the proposed relationship is that once
the coefficients are obtained, the resulting relationship can be
readily and easily incorporated into analysis programs. This is
because it comprises only two equations, (2) and (3), and a
matrix, (5). Furthermore, other sets of coefficients can also be
easily incorporated in the program to represent steel or other
materials with different properties.
CONCLUSIONS
This paper has presented a new mathematical relationship
for representing the stress-strain behavior of structural steel at
elevated temperatures. It has shown that the resulting relationship can accurately represent the stress-strain behavior of steel.
It has also shown that the proposed relationship is more versatile than the existing relationships. It can be easily used for
establishing the properties of other steels and can also be readily incorporated into computer programs for analyzing the behavior of steel structures at an elevated temperature.
APPENDIX I. LIE STRESS-STRAIN-TEMPERATURE
RELATIONSHIP
For temperature range of 0–1,000⬚C:
␴=
再
when ε ⱕ εp
Eε
(c1ε ⫹ c2)␴y ⫺
c3(␴y)2
E
when ε > εp
(6)
where
␴y
=
␴y0
再
冋再
εp =
冋
冉 冊册
T
1.0 ⫹
c4 ln
c6 ⫺ c7T
T ⫺ c8
T
c5
冉 冊册
T
c9 ln
c11 ⫺ c12T
T ⫺ c13
when 0⬚C < T ⱕ 600⬚C
(7)
when 600⬚C < T ⱕ 1,000⬚C
1.0 ⫹
E
=
E0
c2␴y ⫺ c3(␴y)2
E ⫺ c1␴y
T
c10
when 0⬚C < T ⱕ 600⬚C
(8)
when 600⬚C < T ⱕ 1,000⬚C
where the coefficients c1 = 12.5; c2 = 0.975; c3 = 12.5; c4 =
900; c5 = 1,750; c6 = 340; c7 = 0.34; c8 = 240; c9 = 2,000; c10
= 1,100; c11 = 690; c12 = 0.69; and c13 = 53.5.
376 / JOURNAL OF MATERIALS IN CIVIL ENGINEERING / SEPTEMBER/OCTOBER 2001
FIG. 9.
Comparison of Existing and Proposed Relationships: (a) 20⬚C; (b) 200⬚C; (c) 400⬚C; (d) 600⬚C; (e) 800⬚C; (f ) 1,000⬚C
APPENDIX II. EUROCODE 3 STRESS-STRAINTEMPERATURE RELATIONSHIP
where
For a temperature range of 0–1,200⬚C and with the strainhardening option:
a2 = (c1 ⫺ ␴p /E )(c1 ⫺ ␴p /E ⫹ c/E )
b2 = c(c1 ⫺ ␴p /E )E ⫹ c2
Eε; when ε ⱕ (␴p /E )
b
␴p ⫺ c ⫹ 兹a2 ⫺ (c1 ⫺ ε)2 when (␴p /E ) < ε < c1
a
(␴u ⫺ ␴y)ε ⫹ c3␴y ⫺ c1␴u
when c1 < ε < c3 and T < 400⬚C
(c3 ⫺ c1)
␴=
␴u when c3 < ε ⱕ c4 and
T < 400⬚C
␴u(c5 ⫺ ε)
when c4 < ε < c5 and
(c5 ⫺ c4)
␴y when c2 < ε ⱕ c4 and
冋
␴y 1 ⫺
0
册
(ε ⫺ c2)
(c3 ⫺ c2)
when ε ⱖ c5
T < 400⬚C
T ⱖ 400⬚C
when c4 < ε < c5 and
T ⱖ 400⬚C
(9)
c=
(␴y ⫺ ␴p)2
(c1 ⫺ ␴p /E )E ⫺ 2(␴y ⫺ ␴p)
1 when 0⬚C ⱕ T < 100⬚C
c6T ⫹ c7 when 100⬚C ⱕ T < 500⬚C
c8T ⫹ c9 when 500⬚C ⱕ T < 600⬚C
E
= c10 T ⫹ c11 when 600⬚C ⱕ T < 700⬚C
E0
c12 T ⫹ c13 when 700⬚C ⱕ T < 800⬚C
c14 T ⫹ c15 when 800⬚C ⱕ T < 1,200⬚C
0 when T ⱖ 1,200⬚C
(10)
JOURNAL OF MATERIALS IN CIVIL ENGINEERING / SEPTEMBER/OCTOBER 2001 / 377
1 when 0⬚C ⱕ T < 100⬚C
16
17
c T⫹c
when 100⬚C ⱕ T < 400⬚C
18
19
c T⫹c
when 400⬚C ⱕ T < 500⬚C
20
21
p
c T⫹c
when 500⬚C ⱕ T < 600⬚C
␴
y0 =
22
23
␴
c T⫹c
when 600⬚C ⱕ T < 700⬚C
24
25
c T⫹c
when 700⬚C ⱕ T < 800⬚C
26
27
c T⫹c
when 800⬚C ⱕ T < 1,200⬚C
0 when T ⱖ 1,200⬚C
1 when 0⬚C ⱕ T < 400⬚C
28
29
c T⫹c
when 400⬚C ⱕ T < 500⬚C
30
31
c T⫹c
when 500⬚C ⱕ T < 600⬚C
32
33
y
c T⫹c
when 600⬚C ⱕ T < 700⬚C
␴
y0 =
34
35
␴
c T⫹c
when 700⬚C ⱕ T < 800⬚C
36
37
c T⫹c
when 800⬚C ⱕ T < 900⬚C
38
39
c T⫹c
when 900⬚C ⱕ T < 1,200⬚C
0 when T ⱖ 1,200⬚C
␴
y0 =
␴
u
再
40
c
␤5
=
␴y20
(12)
when T < 300⬚C
c T⫹c
41
1
(11)
when 300⬚C ⱕ T < 400⬚C
42
(13)
when T ⱖ 400⬚C
where the coefficients c1 = 0.02; c2 = 0.02; c3 = 0.04; c4 =
0.15; c5 = 0.20; c6 = ⫺1 ⫻ 10⫺3; c7 = 1.1; c8 = ⫺2.9 ⫻ 10⫺3;
c9 = 2.05; c10 = ⫺1.8 ⫻ 10⫺3; c11 = 1.39; c12 = ⫺4 ⫻ 10⫺4;
c13 = 0.41; c14 = ⫺2.25 ⫻ 10⫺4; c15 = 0.27; c16 = ⫺1.933 ⫻
10⫺3; c17 = 1.193; c18 = ⫺0.6 ⫻ 10⫺3; c19 = 0.66; c20 = ⫺1.8
⫻ 10⫺3; c21 = 1.26; c22 = ⫺1.05 ⫻ 10⫺3; c23 = 0.81; c24 = ⫺2.5
⫻ 10⫺4; c25 = 0.25; c26 = ⫺1.25 ⫻ 10⫺4; c27 = 0.15; c28 = ⫺2.2
⫻ 10⫺3; c29 = 1.88; c30 = ⫺3.1 ⫻ 10⫺3; c31 = 2.33; c32 = ⫺2.4
⫻ 10⫺3; c33 = 1.91; c34 = ⫺1.2 ⫻ 10⫺3; c35 = 1.07; c36 = ⫺5
⫻ 10⫺4; c37 = 0.51; c38 = ⫺2 ⫻ 10⫺4; c39 = 0.24; c40 = 1.25;
c41 = ⫺2.5 ⫻ 10⫺3; and c42 = 2.
APPENDIX III. PROPOSED STRESS-STRAINTEMPERATURE RELATIONSHIP
(EXPANDED FORMAT)
␴=
ε
2兩ε兩
冋
再 冋冉
a ⫺ 兩a兩 ⫹
⭈ 1⫹
(␤2 ⫺ ␤3)b
冏
(␤2 ⫺ ␤3)b
1⫹
␤5
冎
冏冊
␤9
1/␤9
⫹ ␤3b
册
册
兩b兩 ⫺ 兩(b ⫺ 0.001)兩
0.001
(14)
where
a = ␤1兩ε兩 ⫹ ␤4
b = 兩ε兩 ⫺ ␤8 ⫺
␤4
␤1
and
␤2
=
E20
再
␤1
c1T ⫹ c2
=
⫹ c5
E20 (1 ⫹ 兩c3 T ⫹ c4兩5)0.2
c6T ⫹ c7
⫹ c10 T ⫹ c11
(1 ⫹ 兩c8 T ⫹ c9兩5)0.2
␤1
E20
(15)
when T ⱕ 500⬚C
when T > 500⬚C
(16)
再
␤3
c12 T ⫹ c13
=
⫹ c16 T ⫹ c17
E20 (1 ⫹ 兩c14T ⫹ c15兩5)0.2
(17)
␤4
c18 T ⫹ c19
=
⫹ c22 T ⫹ c23
␴y20 (1 ⫹ 兩c20 T ⫹ c21兩5)0.2
(18)
c24T ⫹ c25
⫹ c28 T ⫹ c29
(1 ⫹ 兩c26 T ⫹ c27兩5)0.2
␤4
␴y20
when T ⱕ 500⬚C
when T > 500⬚C
(19)
␤8
c30 T ⫹ c31
=
⫹ c34
(␴y20 /E20) (1 ⫹ 兩c32 T ⫹ c33兩5)0.2
(20)
c35 T ⫹ c36
⫹ c39 T ⫹ c40
(1 ⫹ 兩c37 T ⫹ c38兩5)0.2
(21)
␤9 =
where the coefficients c1 = ⫺1.4 ⫻ 10⫺3; c2 = 1.001; c3 = 2.74
⫻ 10⫺3; c4 = ⫺1.959; c5 = 0.493; c6 = 0.01334; c7 = 7.6705;
c8 = 8 ⫻ 10⫺3; c9 = ⫺4.6; c10 = 1.6 ⫻ 10⫺4; c11 = ⫺1.6975;
c12 = ⫺6.4 ⫻ 10⫺5; c13 = 0.03616; c14 = 0.0222; c15 = ⫺12.556;
c16 = 4 ⫻ 10⫺6; c17 = ⫺3.8 ⫻ 10⫺4; c18 = ⫺1.25 ⫻ 10⫺3; c19
= 0.50625; c20 = 3.175 ⫻ 10⫺3; c21 = ⫺1.2857; c22 = ⫺2 ⫻
10⫺4; c23 = 0.63425; c24 = 7.4 ⫻ 10⫺3; c25 = ⫺0.925; c26 =
0.0133; c27 = ⫺1.667; c28 = ⫺2.4 ⫻ 10⫺3; c29 = 0.046; c30 =
⫺0.05; c31 = 10; c32 = 0.01; c33 = ⫺2; c34 = 5; c35 = ⫺3 ⫻
10⫺3; c36 = 1.125; c37 = 0.0133; c38 = ⫺5; c39 = 1 ⫻ 10⫺3; and
c40 = 0.275.
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378 / JOURNAL OF MATERIALS IN CIVIL ENGINEERING / SEPTEMBER/OCTOBER 2001
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NOTATION
The following symbols are used in this paper:
ci = ith curve-fitting coefficient;
E0 = modulus of elasticity of steel at T = 0⬚C;
E20
T
␤i
␥ij
ε
␴
␴y0
␴y20
=
=
=
=
=
=
=
=
modulus of elasticity of steel at T = 20⬚C;
temperature;
ith parameter controlling shape of stress-strain curve;
jth parameter controlling shape of ␤i-T curve;
strain;
stress;
yield stress at T = 0⬚C; and
yield stress at T = 20⬚C.
JOURNAL OF MATERIALS IN CIVIL ENGINEERING / SEPTEMBER/OCTOBER 2001 / 379