Experimental investigation of cold-formed steel material at elevated temperatures

ARTICLE IN PRESS

Thin-Walled Structures 45 (2007) 96–110 www.elsevier.com/locate/tws

Experimental investigation of cold-formed steel material at elevated temperatures Ju Chen, Ben Young Department of Civil Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong Received 2 July 2006; received in revised form 26 October 2006; accepted 16 November 2006 Available online 19 January 2007

Abstract This paper presents the mechanical properties data for cold-formed steel at elevated temperatures. The deterioration of the mechanical properties of yield strength (0.2% proof stress) and elastic modulus are the primary properties in the design and analysis of cold-formed steel structures under fire. However, values of these properties at different temperatures are not well reported. Therefore, both steady and transient tensile coupon tests were conducted at different temperatures ranged approximately from 20 to 1000 1C for obtaining the mechanical properties of cold-formed steel structural material. This study included cold-formed steel grades G550 and G450 with plate thickness of 1.0 and 1.9 mm, respectively. Curves of elastic modulus, yield strength obtained at different strain levels, ultimate strength, ultimate strain and thermal elongation versus different temperatures are plotted and compared with the results obtained from the Australian, British, European standards and the test results predicted by other researchers. A unified equation for yield strength, elastic modulus, ultimate strength and ultimate strain of cold-formed steel at elevated temperatures is proposed in this paper. A full strain range expression up to the ultimate tensile strain for the stress–strain curves of cold-formed carbon steel at elevated temperatures is also proposed in this paper. It is shown that the proposed equation accurately predicted the test results. r 2006 Elsevier Ltd. All rights reserved. Keywords: Cold-formed steel; Elevated temperatures; Experimental investigation; Mechanical properties; Steady state test; Stress–strain curves; Transient state test

1. Introduction Material properties play an important role in the performance of steel structural members, hence, it is important to find out the mechanical properties of steel structural member for the purpose of design. In addition, mechanical properties are greatly affected by temperature, special attention must be given by the designer for extreme conditions below 30 1F (34 1C) and above 200 1F (93 1C) [1]. However, previous research on the material behaviour has been mainly focused on hot-rolled steel, and hence limited data is available for the mechanical properties of cold-formed light gauge steels at elevated temperatures [2]. The reduction factors for mechanical properties at elevated temperatures recommended by the Australian Standard AS

Corresponding author. Tel.: +852 2859 2674; fax: +852 2559 5337.

E-mail address: [email protected] (B. Young). 0263-8231/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.tws.2006.11.003

4100 [3], British Standard BS 5950 [4] Part 8 and European Code 3 [5] Part 1.2 are based on investigation of hot-rolled steel. In addition, the stress–strain curve models proposed in previous documents by EC3-1.2 [5], Olawale and Plank [6], Outinen [7], and Lee et al. [2] are mostly calibrated with hot-rolled steel that may be different from cold-formed steel. Therefore, it is important to investigate the behaviour of cold-formed steel material at elevated temperatures. Both steady and transient state test methods are commonly used in the small-scale tensile test of steel at elevated temperatures. 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 rises until the test specimen fails. Temperature would rise in a real fire, therefore, the transient state test method is more realistic in predicting the behaviour of a material under fire than the steady state test method.

ARTICLE IN PRESS J. Chen, B. Young / Thin-Walled Structures 45 (2007) 96–110

Nomenclature a, b, c coefficients for proposed equations and stress– strain equations specified in the EC3-1.2 E elastic modulus Enormal elastic modulus at normal room temperature ET elastic modulus at temperature, T 1C Ea,y elastic modulus at a given temperature specified in the EC3-1.2 Ey,T elastic modulus at yield strength at temperature, T 1C fp proportional limit fp,normal proportional limit at normal room temperature fp,T proportional limit at temperature, T 1C fp,y proportional limit at a given temperature specified in the EC3-1.2 fT stress at temperature, T 1C ft,u,T ultimate strength at temperature, T 1C obtained from transient state test fu,T ultimate strength fu,normal ultimate strength at normal room temperature fu,T ultimate strength at temperature, T 1C fy,normal yield strength at normal room temperature fy,T yield strength at temperature, T 1C fy,y yield strength at a given temperature specified in the EC3-1.2 f0.2 0.2% yield strength f0.2,normal 0.2% yield strength at normal room temperature f0.2,T 0.2% yield strength at temperature, T 1C f0.5 0.5% yield strength f0.5,normal 0.5% yield strength at normal room temperature f0.5,T 0.5% yield strength at temperature, T 1C f1.5 1.5% yield strength

This paper presents the details of an experimental study of cold-formed steel at elevated temperatures approximately up to 1000 1C. The purpose of this study is to investigate the behaviour of cold-formed steel material at different temperatures using both steady and transient test methods. Steady state tests were carried out on cold-formed steel grade G550 and G450 with plate thickness of 1.0 and 1.9 mm coupon specimens, respectively. Transient state tests were carried out on cold-formed steel grade G450 with plate thickness of 1.9 mm coupon specimens. The mechanical properties were determined from the test results of both transient and steady state test methods. A unified equation for yield strength, elastic modulus, ultimate strength and ultimate strain of cold-formed steel at elevated temperatures is proposed. Further more, a stress–strain curve model that accurately describes the cold-formed steel material at elevated temperatures is proposed in this paper. The proposed model is based on Mirambell and Real [8] and Rasmussen [9] stress–strain curve model for stainless steel at

97

f1.5,normal 1.5% yield strength at normal room temperature f1.5,T 1.5% yield strength at temperature, T 1C f2.0 2.0% yield strength f2.0,normal 2.0% yield strength at normal room temperature f2.0,T 2.0% yield strength at temperature, T 1C n coefficient for proposed equation and coefficient for proposed unified equation and Ramberg–Osgood equation nT coefficient for proposed stress–strain equation mT coefficient for proposed stress–strain equation T value of temperature b coefficient for Ramberg–Osgood equation s stress at a given temperature specified in the EC3-1.2 e strain at a given temperature specified in the EC3-1.2 ep,y proportional limit at a given temperature specified in the EC3-1.2 eT strain at temperature, T 1C et,y limiting strain for yield strength at a given temperature specified in the EC3-1.2 eu,normal strain corresponding to ultimate strength at normal room temperature eu,T strain corresponding to ultimate strength at temperature, T 1C eu,y ultimate strain at a given temperature specified in the EC3-1.2 ey,T strain corresponding to yield strength at temperature, T 1C ey,y yield strain at a given temperature specified in the EC3-1.2 etotal elongation (tensile strain) at the point of fracture based on gauge length of 25 mm

normal room temperature. The proposed equations are compared with cold-formed carbon steel test results. 2. Experimental investigation 2.1. Testing device The tensile testing machine used in this study was an MTS 810 Universal testing machine of 100 kN capacity. Testing machine was calibrated before testing. The installation of the coupon specimen and the testing device used are shown in Fig. 1. The heating device of MTS Model 653 high temperature furnace with a maximum temperature of 1400 1C was used. The furnace composed of upper and lower heating elements. 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

ARTICLE IN PRESS J. Chen, B. Young / Thin-Walled Structures 45 (2007) 96–110

98

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 ranged from 3% to 28%. The temperature accuracy of the internal and the external thermal couples was 1.0 and 70.1 1C. The heating rate of the furnace is 100 1C/min. The fast heating rate resulted of the temperatures overshoot slightly, but the overshoot stabilizes within a minute. The maximum overshoot was approximately 40 1C at low temperatures and decreases at higher temperatures. When the temperature beyond 700 1C, the overshoot was less than 20 1C. An MTS Model 632.53F-11 of axial extensometer was used to measure the strain of the middle part of the coupon specimen. Gauge length of the extensometer was 25 mm with range limitation of 72.5 mm. The extensometer was also calibrated before testing. The extensometer was reset when it approaches the range limit during testing, hence a complete strain of coupon specimen can be obtained. 2.2. Test specimen The coupon test specimens were taken from the central of the web plate of finished cold-formed steel specimens brake-pressed from structural steel sheets. The test coupon dimensions conformed to the ASTM Standard E 21 [10]

and Australian Standard AS 2291 [11] using 6 mm wide coupons and a gauge length of 25 mm. The specimens consisted of two different steel grades and thicknesses. Steel grade of G550 with plate thickness of 1.0 mm (G550 1.0 mm) and steel grade of G450 with plate thickness of 1.9 mm (G450 1.9 mm) were used. The steel grades of G550 and G450 have the nominal yield strength (0.2% proof stress) of 550 and 450 MPa at normal room temperature, respectively. A total of 63 tests (46 steady state tests and 17 transient state tests) was conducted in this study. The chemical compositions of the test specimens are presented in Table 1. The total metal thickness and the base metal thickness without zinc coating were measured using a micrometer and the zinc coating was removed by 1:1 hydrochloric acid before testing. The base metal thickness was used to determine the cross sectional area of each coupon. 2.3. Testing procedure 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 tensile load during the heating process. After reaching the pre-selected temperature, it needs approximately 2 min for the temperature to stabilize and after another 7 min, then the tensile load 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 1C (73 1C) during the tests. 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 0.00570.002/min as specified by the ASTM Standard E21-92 [10]. 2.3.2. Transient state test In the transient state tests, the specimen was under constant tensile load while the temperature was raised. The stress levels selected in the test were 2, 30, 60, 90, 120, 150, 180, 210, 240, 270, 300, 330, 360, 390, 420, 450 and 480 MPa. The temperatures specified in the temperature controller ranged from 100 to 900 1C at the interval of 100 1C. The strain of the specimen at a given temperature was recorded using the extensometer 6 min after the

Fig. 1. Testing device.

Table 1 Chemical properties Grade

C (%)

P (%)

Mn (%)

Si (%)

S (%)

Ni (%)

Cr (%)

Mo (%)

Cu (%)

Al (%)

Ti (%)

Nb (%)

Sn (%)

N (%)

V (%)

G550 G450

0.055 0.055

0.007 0.008

0.20 0.21

o0.005 0.005

0.015 0.014

0.026 0.027

0.013 0.011

0.002 0.002

0.009 0.009

0.037 0.038

o0.003 o0.003

0.001 0.001

0.002 0.003

0.0038 0.0028

o0.003 o0.003

Note: Percentage of element by mass.

ARTICLE IN PRESS J. Chen, B. Young / Thin-Walled Structures 45 (2007) 96–110

99

temperature reached the specified value. The ultimate strength of the specimen is defined when the strain keeps 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.

can be determined from the heat flow-temperature curves. The heat flow is measured in milliwatt (mW). Chemical reaction takes place when the temperature beyond approximately 320 1C for both materials. When the temperature reached approximately 800 1C, a major chemical reaction took place and the materials were also changed, as shown in Fig. 2. Comparison of the heat flowtemperature curves of the two test materials indicated that there was not a significant difference between these materials.

3. Differential thermal analysis

4. Steady state test results

A differential thermal analysis (DTA) was conducted to investigate the chemical reaction of the tested steel materials during the heating process. A DTA 7 Perkin Elmer machine was used for the DTA tests. The DTA test results are shown in Fig. 2. Since the material gives out or absorb heat in chemical reaction process, chemical reaction

4.1. Determination of strength and elastic modulus

a

4

0 -2

1

-4

f0.2 Stress, σ

Heat Flow (mW)

2

In this study, 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 (f0.2) is the intersection point of the stress–strain curve and the proportional line offset by 0.2% strain. Meanwhile, the yield strengths of f0.5, f1.5 and f2.0 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. 3.

-6 -8

0

250

500 750 Temperature (°C)

1000

E fu

f2.0

f1.5

f0.5

1250

b

0.2

0.5

1.5

εu

2.0

Strain, ε (%)

4 2 fu

0

Static drop Stress, σ

Heat Flow (mW)

b

-2 -4 0

250

500 750 Temperature (°C)

1000

1250

Fig. 2. Differential thermal analysis results. (a) G550 1.0 mm; (b) G450 1.9 mm.

Strain, ε (%) Fig. 3. Definition of symbols.

ε

ARTICLE IN PRESS J. Chen, B. Young / Thin-Walled Structures 45 (2007) 96–110

100

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 (Young’s modulus) was determined from the stress–strain curve based on the tangent modulus of the initial elastic linear curve. 4.2. Static drop In the tensile test at the normal room temperature (22 1C), a static drop of the stress–strain curve is obtained by pausing the applied strain for 1 min, as shown in Fig. 3. This allowed the stress relaxation associated with plastic strain to take place; hence, the effect of loading rate can be eliminated. A series of tests has been conducted to investigate the static drop for different temperatures. The ratio of the static drop to the ultimate strength obtained from the steady state tests for G550 1.0 mm and G450 1.9 mm specimens are presented in Table 2. When the temperature reached 760 1C, the drop caused the static ultimate strength fall to zero stress. Therefore, those values are not reported. The ratio of static drop to the ultimate strength obtained from the steady state tests increase as the temperature increases. It should be mentioned that all test results did not include the static drop, except for the investigation of the static drop as described in this section. 4.3. Yield strength The mechanical properties of the cold-formed steel G550 1.0 mm and G450 1.9 mm at normal room temperature is Table 2 Ratio of static drop to ultimate strength obtained from steady state tests Temperature (1C)

Drop/fu,T (%) (G450 1.9 mm)

Drop/fu,T (%) (G550 1.0 mm)

22 80 140 220 320 400 450 500 550 660 760

2.0 2.9 3.2 5.2 4.8 11.2 14.7 13.2 21.0 22.3 —

2.8 1.7 1.3 3.0 10.7 11.2 14.1 21.5 23.0 — —

presented in Table 3. The reduction factors (f0.2,T/f0.2,normal, f0.5,T/f0.5,normal, f1.5,T/f1.5,normal, f2.0,T/f2.0,normal) determined from the ratio of different yield strengths to normal room temperature (22 1C) at different temperatures for the four strain levels of 0.2%, 0.5%, 1.5% and 2.0%, respectively, are presented in Table 4. The test results of G550 1.0 mm and G450 1.9 mm materials are plotted in Fig. 4. The vertical axis of the graph plotted the reduction factor f0.2,T/f0.2,normal and the horizontal axis plotted against different temperatures. It is shown that the test results of G550 1.0 mm and G450 1.9 mm materials are different. However, a unified equation is proposed for the reduction factor of 0.2% yield strength for G550 1.0 mm and G450 1.9 mm cold-formed steel at elevated temperatures, as shown in Eq. (1). The coefficients a, b, c and n of the equation are presented in Table 5, and T is the temperature in degree Celsius (1C). It is demonstrated that the values of reduction factor f0.2,T/f0.2,normal predicted by the equation compared well with the test results of G550 1.0 mm and G450 1.9 mm, as shown in Fig. 4. Proposed equation for yield strength: f 0:2;T f 0:2;normal

¼a

ðT  bÞn , c

(1)

where f0.2,normal is the yield strength at normal room temperature. The reduction factor of 0.2% yield strength obtained from the tests were compared with the Australian Standard AS 4100 [3] prediction and also compared with the test results conducted by Lee et al. [2], as shown in Fig. 4. The comparison shown that the AS 4100 provides conservative prediction for G450 1.9 mm from 220 to 550 1C and for G550 1.0 mm from 220 to 400 1C. The test results obtained from this study are far below than the AS 4100 prediction for G450 1.9 mm at 660 1C and G550 1.0 mm from 450 to 800 1C. The unconservatism will be discussed in the section of stress–strain curve of this paper. It is interesting to note that the test results obtained from this study are significant different from the test results conducted by Lee et al. [2] for temperatures ranged from 450 to 800 1C. The reduction factors for the strain levels of 0.5%, 1.5% and 2.0% are compared with the EC3-1.2 [5] and BS 59508 [4] for hot-finished steel and cold-formed steel prediction. The reduction factors are also compared with the test results conducted by Lee et al. [2], as shown in Figs. 5–7. The reduction factor of 0.5% yield strength predicted by the BS 5950-8 for hot-finished steel and cold-formed steel

Table 3 Mechanical properties of cold-formed steel G550 1.0 mm and G450 1.9 mm at normal room temperature

G550 1.0 mm G450 1.9 mm

Enormal (GPa)

f0.2,normal (MPa)

f0.5,normal (MPa)

f1.5,normal (MPa)

f2.0,normal (MPa)

fu,normal (MPa)

etotal (%)

200.3 203.0

598 524

599 525

600 527

601 535

608 551

9.8 11.3

ARTICLE IN PRESS 0.022 0.023 0.025 0.027 0.326

f0.2,T /f0.2,normal

0.033 0.037 0.037 0.037 0.517 0.065 0.070 0.080 0.082 0.551

0.040 0.042 0.046 0.045 0.182

970

0.107, 0.115, 0.142, 0.148, 0.642,

660 550

0.532, 0.553, 0.571, 0.557, 0.816, 0.727 0.754 0.786 0.774 0.925 0.851, 0.870, 0.943, 0.936, 0.887, 0.933 0.933 1.047 1.050 0.854

450 400

400 600 Temperature (oC)

800

1000

0.969, 0.977, 1.011, 1.015, 0.916, 0.971 0.975 1.003 0.994 1.079

4.4. Elastic modulus

 Second test.

0.987 0.990 1.000 0.998 1.042 1.000 1.000 1.000 1.000 1.000 f0.2,T/f0.2,normal f0.5,T/f0.5,normal f1.5,T/f1.5,normal f2.0,T/f2.0,normal ET/Enormal

200

are both conservative for G550 1.0 mm for temperatures lower than 450 1C and G450 1.9 mm for temperatures lower than 660 1C. However, the reduction factors of 1.5% and 2.0% yield strength predicted by the BS 5950-8 for hot-finished steel and cold-formed steel are generally unconservative, except for temperatures lower than 320 1C. EC3-1.2 only provides reduction factor of 2.0% yield strength and the prediction are similar to the BS 59508 for hot-finished steel. The reduction factors of 0.5%, 1.5% and 2.0% yield strength for G550 1.0 mm from 450 to 800 1C and G450 1.9 mm at 660 1C are far lower than the BS 5950-8 prediction and the test results conducted by Lee et al. [2].

0.987 0.990 1.032 1.054 0.907 320 180 80 22 Temperature (1C) G450 1.9 mm

0

Fig. 4. Comparison of reduction factor of 0.2% yield strength prediction from AS 4100 and proposed equation with test results.

0.823 0.905 0.844 0.877 0.901

500

0.607 0.637 0.653 0.647 0.759

0.176, 0.196, 0.251, 0.265, 0.762, 0.204 0.232 0.293 0.316 0.798 0.722 0.784 0.727 0.777 0.827 0.943 1.054 0.951 1.023 0.989 0.905, 0.908, 0.978, 0.987, 0.929, 0.968 1.008 0.972 0.970 0.969 0.962 1.008 0.963 0.972 0.961 1.003, 1.002, 1.023, 1.038, 0.951, 0.987 1.017 0.993 0.992 1.053 0.985 0.992 0.997 0.987 1.036 1.000 1.000 1.000 1.000 1.000

0.4

AS 4100 G550 1.0mm G450 1.9mm Lee et al.(2003) G550 0.42mm Lee et al.(2003) G550 0.6mm Lee et al.(2003) G550 0.95mm Lee et al.(2003) G500 1.2mm Proposed Equation for G550 1.0mm Proposed Equation for G450 1.9mm

0.111 0.116 0.144 0.148 0.675

0.122, 0.138, 0.162, 0.170, 0.721,

500

f0.2,T/f0.2,normal f0.5,T/f0.5,normal f1.5,T/f1.5,normal f2.0,T/f2.0,normal ET/Enormal

0.6

0.0

0.211 0.367 0.233 0.295 0.750

550

0.8

0.2

450 400 320 220 180 140 80 22 Temperature (1C) G550 1.0 mm

Table 4 Reduction factors of yield strength and elastic modulus of cold-formed steel G 550 1.0 mm and G450 1.9 mm

101

1.0

0.135 0.238 0.152 0.176 0.674

800 660

970

J. Chen, B. Young / Thin-Walled Structures 45 (2007) 96–110

The elastic modulus is an important factor determining the buckling stress for thin-walled structures. The reduction factor (ET/Enormal) of elastic modulus at elevated temperatures determined from the steady state tests for G450 1.9 mm and G550 1.0 mm materials are presented in Table 4. The reduction factor was determined from the ratio of elastic modulus to normal room temperature (22 1C) at different temperatures. The test results obtained from this study for the reduction factor of the elastic modulus are compared with the AS 4100 and EC3-1.2 prediction and compared with the test results conducted by Lee et al. [2] in Fig. 8. The reduction factor predicted by the AS 4100 and EC3-1.2 are conservative. The test results obtained from this study for the reduction factor of elastic modulus are comparatively higher than the test results conducted by Lee et al. [2].

ARTICLE IN PRESS J. Chen, B. Young / Thin-Walled Structures 45 (2007) 96–110

102

Table 5 Coefficients of proposed equation for yield strength Eq. (1) G550 1.0 mm

22pTo300

300pTo450

450pTo1000

A B C n

1.0 22 2.78  103 1

0.9 300 4.8  106 3

0.02 1000 9  108 3

Temperature (1C)

22pTo300

300pTo650

650pTo1000

a b c n

1.0 22 5.56  103 1

0.95 300 1.45  105 2

0.105 650 5  103 1

1.0

1.0

0.8

0.8

f0.5,T /f0.5,normal

f0.5,T /f0.5,normal

G450 1.9 mm

Temperature (1C)

0.6 0.4

0.4 0.2

0.2 0.0

0.6

0.0 0

200

400 600 Temperature (oC)

800

1000

0

200

G550 1.0mm G450 1.9mm BS 5950-8 Hot-finished steel BS 5950-8 Cold-formed steel Lee et al.(2003) G550 0.42mm Lee et al.(2003) G550 0.60mm Lee et al.(2003) G550 0.95mm Lee et al.(2003) G500 1.20mm

400 600 Temperature (oC)

800

1000

G550 1.0mm G450 1.9mm BS 5950-8 Hot-finished steel BS 5950-8 Cold-formed steel Lee et al.(2003) G550 0.42mm Lee et al.(2003) G550 0.60mm Lee et al.(2003) G550 0.95mm Lee et al.(2003) G500 1.20mm

Fig. 5. Comparison of reduction factor of 0.5% strength predicted by BS5950-8 with test results.

Fig. 6. Comparison of reduction factor of 1.5% strength predicted by BS5950-8 with test results.

4.5. Ultimate strength

generally conservative compared with the test results obtained from this study for cold-formed steel G550 1.0 mm and G450 1.9 mm, as shown in Fig. 9. Proposed equation for ultimate strength is

The reduction factor of ultimate strength at different temperatures to normal room temperature (fu,T/fu,normal) obtained from the tests are plotted in Fig. 9. The vertical axis of the graph plotted the reduction factor fu,T/fu,normal and the horizontal axis plotted against different temperatures. The values of the reduction factor of ultimate strength of cold-formed steel G450 1.9 mm are generally higher than those of the cold-formed steel G550 1.0 mm for temperatures greater than or equal to 320 1C. A unified equation for the prediction of ultimate strength is proposed, as shown in Eq. (2). The coefficients a, b, c and n of the equation are calibrated with the cold-formed carbon steel test results, and the values of coefficients are presented in Table 6. It is shown that the prediction of reduction factor of ultimate strength using Eq. (2) is

f u;T f u;normal

¼a

ðT  bÞn , c

(2)

where fu,normal is the ultimate strength at normal room temperature. 4.6. Ultimate strain The ultimate strain is defined as the strain corresponding to the ultimate strength. The reduction factor of ultimate strain at different temperatures to normal room temperature (eu,T/eu,normal) obtained from the tests are plotted in

ARTICLE IN PRESS J. Chen, B. Young / Thin-Walled Structures 45 (2007) 96–110

1.0

0.8 fu,T / fu,normal

f0.5,T /f0.5,normal

1.0

0.6 0.4 0.2 0.0

103

0.8 0.6 0.4 0.2

0

200

400 600 Temperature (oC)

800

1000 0.0 0

G550 1.0mm G450 1.9mm BS 5950-8 Hot-finished steel BS 5950-8 cold-formed steel EC3-1.2 Lee et al. G550 0.42mm Lee et al. G550 0.60mm Lee et al. G550 0.95mm Lee et al. G500 1.20mm

200

400 600 Temperature (oC)

800

1000

Test results (G550 1.0mm) Test results (G450 1.9mm) Proposed Equation for G550 1.0mm Proposed Equation for G450 1.9mm

Fig. 9. Comparison of ultimate strength obtained from test results with prediction using the proposed equation.

Fig. 7. Comparison of reduction factor of 2.0% strength predicted by BS5950-8 and EC3-1.2 with test results.

Table 6 Coefficients of proposed equation for ultimate strength Eq. (2)

ET /E normal

1.0

G550 1.0 mm

0.8 0.6 0.4 0.2 0.0

G450 1.9 mm

0

200

400

600

800

1000

Temperature (°C) G550 1.0mm G450 1.9mm EC3-1.2 AS 4100 Lee et al.(2003) G550 0.42mm Lee et al.(2003) G550 0.60mm Lee et al.(2003) G550 0.95mm Lee et al.(2003) G500 1.20mm

Fig. 8. Comparison of elastic modulus predicted by AS 4100 and EC3-1.2 with test results.

Fig. 10. The vertical axis of the graph plotted the reduction factor eu,T/eu,normal and the horizontal axis plotted against different temperatures. Some of the test results in the temperatures ranged from 450 to 660 1C are relatively higher than expected. This could be due to the different type of stress–strain curves as explained below. The proposed unified Eq. (3) for ultimate strain is identical to the unified equation for yield strength, elastic modulus and ultimate strength, except for the different values of coefficients. The values of coefficients a, b, c and n of the

Temperature (1C)

22pTo320

320pTp1000

A B C N

1.0 22 1.5  104 1

0.026 1000 2.24  1011 4

Temperature (1C)

22pTo450

450pTp1000

a b c n

1.0 22 5.6  108 3

0.043 1000 1.12  1011 4

equation are calibrated with the cold-formed carbon steel test results in this study, and the coefficients are presented in Table 7. It is shown that the prediction of the reduction factor of ultimate strain using Eq. (2) is generally conservative compared with the test results obtained from this study for cold-formed steel G550 1.0 mm and G450 1.9 mm, as shown in Fig. 10. Proposed equation for ultimate strain is u;T u;normal

¼a

ð T  bÞ n , c

(3)

where eu,normal is the ultimate strain at normal room temperature. 4.7. Stress–strain curve 4.7.1. General The stress–strain curves obtained from the tests can be categorized into three types, as shown in Fig. 11. The

ARTICLE IN PRESS J. Chen, B. Young / Thin-Walled Structures 45 (2007) 96–110

104

2.0

Curve a (22oC)

1.2 0.8

Stress, σ

εu,T /εu,normal

1.6

0.4

Curve b (450oC)

0.0 0

200

400 600 Temperature (oC)

800

1000

Curve c (660oC)

Test results (G550 1.0mm) Test results (G450 1.9mm) Proposed Equation for G550 1.0mm Proposed Equation for G450 1.9mm

Fig. 10. Comparison of ultimate strain obtained from test results with prediction using the proposed equation.

Curve c (970oC) Strain, ε

b Curve a (22oC)

Table 7 Coefficients of proposed equation for ultimate strain

Curve b (450oC)

G550 1.0 mm

G450 1.9 mm

Stress, σ

Eq. (3) Temperature (1C)

22pTp1000

a b c n

0.2 1000 1.1  1018 6

Temperature (1C)

22pTo550

550pTp1000

a b c n

0.25 550 3.7  105 2

0.25 550 3600 1

Curve c (970oC)

stress–strain curve changes from ‘‘a’’ type to ‘‘b’’ type and finally to ‘‘c’’ type when the temperature increases. For both G450 1.9 mm and G550 1.0 mm specimens, curve of ‘‘a’’ type represents the stress–strain curve model for low temperatures (22 1CpT 1Cp320 1C). However, G450 1.9 mm and G550 1.0 mm specimens have different temperature range for ‘‘b’’ type stress–strain curve. Typically, the range for ‘‘b’’ type curve are 320 1CoT 1Cp550 1C for G450 1.9 mm specimens and 320 1CoT 1Cp400 1C for G550 1.0 mm specimens. For even higher temperatures of ‘‘c’’ type stress–strain curve, the range are 550 1CoT 1Cp970 1C for G450 1.9 mm specimens and 400 1CoT 1Cp970 1C for G550 1.0 mm specimens. A sudden drop of yield strength for the tests of G550 1.0 mm and G450 1.9 mm is shown in Fig. 4. This sudden drop may be due to the change of stress–strain curve from ‘‘b’’ type to ‘‘c’’ type. An obvious feature of ‘‘c’’ type curve is the serrations in the stress–strain curve. The serrations in the stress–strain curve may occur under

Curve c (660oC)

Strain, ε Fig. 11. Stress–strain curves at different temperatures of steady state tests for G450 1.9 mm: (a) complete stress–strain curves; (b) initial part of stress–strain curves.

certain combination of strain rate and temperature that the interstitial atoms can be dragged along with dislocations or dislocations can alternately break away and be re-pinned [12]. 4.7.2. Comparison of stress–strain curve model proposed by EC3-1.2 with test results The stress–strain curve model proposed by the EC3-1.2 [5] is based on hot-rolled steel so that it may not accurately represent the behaviour of cold-formed steel material at elevated temperatures. The mechanical properties of coldformed steel obtained from the tests were substituted into the equations describing the stress–strain relationship proposed by EC3-1.2 and compared with the stress–strain curves of G550 1.0 mm and G450 1.9 mm coupon specimens obtained from the tests. The complete stress– strain curves at different temperatures are plotted in Figs. 12(a) and 13(a), while the initial parts of the stress–strain curves at different temperatures are compared

ARTICLE IN PRESS J. Chen, B. Young / Thin-Walled Structures 45 (2007) 96–110

22°C

Test results EC3-1.2

80°C

400°C 600

22°C Stress (MPa)

Stress (MPa)

600 450 80°C 400°C

400°C

300

550°C 550°C 0

4

450 400°C

16

20

22°C 600 80°C

80°C

450

Test results EC3-1.2

300 150

550°C

0 0.0

Stress (MPa)

Stress (MPa)

600

400°C

550°C

300

550°C 1.0 Strain (%)

1.5

2.0

0

550°C

4

8 12 Strain (%)

16

20

80°C

450

150

0.5

80°C

300

0

8 12 Strain (%)

Test results EC3-1.2

80°C

150

150 0

105

0 0.0

400°C Test results EC3-1.2

550°C 0.5

1.0 Strain (%)

1.5

2.0

Fig. 12. Comparison of stress–strain curves predicted by EC3-1.2 with test results (G550 1.0 mm): (a) complete stress–strain curves; (b) initial part of stress–strain curves.

Fig. 13. Comparison of stress–strain curves predicted by EC3-1.2 with test results (G450 1.9 mm): (a) complete stress–strain curves; (b) initial part of stress–strain curves.

in Figs. 12(b) and 13(b). The comparison indicates the stress–strain curves predicted by the EC3-1.2 are quite different from the curves obtained from the tests, as shown in Figs. 12 and 13. The stress–strain curve model proposed by the EC3-1.2 is shown in Eq. (4). The mechanical properties of steel at elevated temperatures are given in EC3-1.2. Stress–strain curve model proposed by the EC3-1.2 [5] is: 8 E a;y for pp;y ; > > > h i0:5 >     > 2 > > f p;y  c þ b=a a2  y;y   for p;y ooy;y ; > > > > >   t;y > >  > f y;y 1   for t;y oou;y ; > >  > u;y  t;y > > > :0 for  ¼ u;y

where s is the stress at a given temperature, Ea,y, the slope of the linear elastic range at a given temperature, fp,y, the proportional limit at a given temperature; fy,y, the effective yield strength at a given temperature; e, the yield strain at a given temperature; ep,y, the strain at the proportional limit at a given temperature; et,y, the limiting strain for yield strength at a given temperature; eu,y, the ultimate strain at a given temperature; and ey,y, the yield strain at a given temperature. According to Buchanan et al. [13] ‘‘Beam behaviour is very sensitive to the stress condition relative to the temperature-reduced proportional limit and yield stress.’’ both the proportional limit and yield stress play an important role in the behaviour of steel beams at elevated temperatures. The proportional limit presented in Table 3.1 of the EC3-1.2 Code [5] was compared with the test results of flat coupons of G550 1.0 mm and G450 1.9 mm, as shown in Fig. 14. The vertical axis of Fig. 14 shows the proportional limit (fp,y) at different temperatures relative to yield strength (fy) at normal room temperature, and the horizontal axis plotted against different temperatures. It can be seen that the reduction factors of proportional limit predicted by EC3-1.2 for different temperatures are generally unconservative, especially for temperatures less than 200 1C. The difference is excepted since the EC3-1.2 was based on the hot-rolled steel. The unified equation for

(4) and p;y ¼ f p;y =E a;y ;

y;y ¼ 0:02;

t;y ¼ 0:15;

   a2 ¼ y;y  p;y y;y  p;y þ c=E a;y ,   b2 ¼ c y;y  p;y E a;y þ c2 ,  2 f y;y  f p;y    , c¼ y;y  p;y E a;y  2 f y;y  f p;y

u;y ¼ 0:20;

ARTICLE IN PRESS J. Chen, B. Young / Thin-Walled Structures 45 (2007) 96–110

106

1.5

1.0

Test results (G550 1.0mm) Test results (G450 1.9mm) EC3-1.2

fp,T/fp,normal

fp,θ /fy

0.8 0.6 0.4 0.2 0.0

0

200

400

600 800 Temperature (°C)

1000

0.6

0.0

1200

0

Table 8 Coefficients of proposed equation for proportional limit Eq. (5) G550 1.0 mm

G450 1.9 mm

Temperature (1C)

22 p To450

450pTp1000

a b c n

1.0 22 9.26  106 2.6

0.25 450 20 0.25

Temperature (1C)

22 p To450

450pTp1000

a b c n

1.0 22 4.58  105 2

0.05 1000 -5  1016 6

the prediction of yield strength is used for the prediction of proportional limit, as shown in Eq. (5). The coefficients a, b, c and n of the equation are calibrated with the coldformed carbon steel test results of proportional limit, and the values of coefficients are presented in Table 8, where T is the temperature in degree Celsius (1C). The comparison between the values obtained from the proposed equation and the test results is shown in Fig. 15. The vertical axis is plotted against the reduction factor fp,T/fp,normal, while the horizontal axis is plotted against temperature. It can be seen that the proposed equation is conservative. Proposed equation for proportional limit is f p;normal

0.9

0.3

Fig. 14. Comparison of reduction factors for proportional limit predicted by EC3-1.2 with test results.

f p;T

Test results (G550 1.0mm) Test results (G450 1.9mm) Proposed Equation for G550 1.0mm Proposed Equation for G450 1.9mm

1.2

¼a

ð T  bÞ n , c

(5)

where fp,normal is the proportional limit at normal room temperature. 4.7.3. Comparison of stress–strain curve model proposed by other researchers with test results Ramberg and Osgood [14] proposed a three parameters model describing the stress–strain curve at normal room

200

400 600 Temperature (°C)

800

1000

Fig. 15. Comparison of the proportional limit obtained from the proposed equation with test results.

temperature. Stress–strain curve models at elevated temperatures were proposed by Olawale and Plank [6] for hotrolled steel, Outinen [7] for S355 hot-rolled steel and Lee et al. [2] for light gauge steel that are based on the Ramberg–Osgood stress–strain curve model. The basic form of the Ramberg–Osgood stress–strain curve model for elevated temperatures is shown in Eq. (6). A parameter b in Eq. (3) was taken as a constant value of 3/7 by Olawale and Plank [6] and 6/7 by Outinen [7], while Lee et al. [2] determine the value in accordance with temperature variation. Another parameter n was determined with temperature variation by Olawale and Plank [6] and Outinen [7], while Lee et al. [2] took it as a constant value. Basic form of the Ramberg–Osgood [14] equation for elevated temperatures is !n   f y;T fT fT T ¼ þb , (6) ET ET f y;T where eT, is the strain at temperature T1C fT, the stress at temperature T1C fy,T, the yield strength at temperature T1C ET, the elastic modulus at temperature T1C and b the coefficient; and n, the coefficient. The stress–strain curves predicted using the stress–strain curve models proposed by Olawale and Plank [6], Outinen [7] and Lee et al. [2] were compared with the test results obtained in this study, as shown in Figs. 16–18, respectively. Generally, the stress–strain curve models proposed by Olawale and Plank [6], and Outinen [7] are unconservatively predicted when compared with the cold-formed carbon steel test results. This is because their models were calibrated with hot-rolled steel test results. The stress– strain curve model proposed by Lee et al. [2] provides a closer predication with the cold-formed steel test results in comparison with the models proposed by Olawale and Plank [6], and Outinen [7]. However, the model proposed by Lee et al. [2] is generally conservative. 4.7.4. Proposed stress–strain curve model The stress–strain curve models of steel at elevated temperatures in the EC3-1.2 [5] and the models proposed

ARTICLE IN PRESS J. Chen, B. Young / Thin-Walled Structures 45 (2007) 96–110

80°C

700 600

500

500

400

400°C

300

Test results Proposed by Olawale & Plank (1988)

500°C

200

550°C

100 0

600

Stress (MPa)

600

0

660°C 4 6 Strain (%)

2

Stress (MPa)

400

200

Test results Proposed by Olawale & Plank (1988)

0

1

2 Strain (%)

3

660°C 4

0

2

320°C

660°C 4 Strain (%)

6

8

450°C

500 400 500°C

300

660°C

200

Test results Proposed by Outinen (1999)

0 5

Fig. 16. Comparison of stress–strain curves predicted by Olawale and Plank [6] with test results: (a) G550 1.0 mm; (b) G450 1.9 mm.

by Olawale and Plank [6], Outinen [7], and Lee et al. [2] are not accurate for cold-formed carbon steel for temperature ranged from 22 to 660 1C. Therefore, a new model describing the stress–strain curve up to the ultimate strength is proposed in this paper. This model is based on the stress–strain curve model for stainless steel at normal room temperature proposed by Mirambell and Real [8] and Rasmussen [9], which is originally based on the Ramberg–Osgood [14] equation. Proposed stress–strain curve model is:  nT 8 fT fT > þ 0:002 for f T pf y;T ; < ET f y;T   mT T ¼ f f f f > : TE y;Ty;T þ u;T f T fy;T þ y;T for f T 4f y;T u;T

y;T

(7) and E y;T ¼

550°C

100

100 0

200

600

500°C

550°C

400°C

700

500

300

Test results Proposed by Outinen (1999)

300

10

450°C

320°C

80°C

400

0 8

80°C

100

Stress (MPa)

Stress (MPa)

700

107

ET , 1 þ 0:002nT E T =f y;T

(8)

pffiffiffiffi nT ¼ 20  0:6 T ,

(9)

mT ¼ 1 þ T=350,

(10)

where eT, is the strain at temperature T1C eu,T, the strain corresponding to ultimate strength at temperature T1C ey,T,

0

1

2 3 Strain (%)

4

5

Fig. 17. Comparison of stress–strain curves predicted by Outinen [7] with test results: (a) G550 1.0 mm; (b) G450 1.9 mm.

the strain corresponding to yield strength at temperature T1C fT, the stress at temperature T1C fu,T, the ultimate strength at temperature T1C fy,T, the yield strength at temperature T1C ET, the elastic modulus at temperature T1C Ey,T, the elastic modulus at yield strength at temperature T1C and T, the value of temperature in degree Celsius (1C). The definition of symbols used in the proposed stress– strain curve model Ey,T, mT, and nT are given by Eqs. (8)–(10), respectively. Fig. 19 shows the comparison of the stress–strain curves obtained using Eq. (7) with the experimental results for G550 1.0 mm and G450 1.9 mm at different temperatures. The proposed stress–strain curves plotted using the Eq. (7) were based on the mechanical properties obtained from the test results. Generally, the proposed stress–strain curve model accurately predicted the cold-formed carbon steel material for the temperature ranged from 22 to 660 1C, as shown in Fig. 19. Hence, the proposed model is applicable for normal room temperature of 22–660 1C. The proposed expression for the full-range stress–strain curve in Eq. (7) involves three parameters (ET, fy,T, nT) in the first part of the curve for fTpfy,T, and three additional parameters (eu,T, fu,T, mT) in the second part of the curve for fT4fy,T. The values of nT and mT are function of temperature that can be determined from Eqs. (9) and (10),

ARTICLE IN PRESS J. Chen, B. Young / Thin-Walled Structures 45 (2007) 96–110

108

80°C

600

180°C

500 400

Test Results Proposed by Lee et al. (2003)

300

500°C

400°C

200

Stress (MPa)

Stress (MPa)

600

100 0.2 0.4

0.6

0.8 1.0 1.2 Strain (%)

1.4

1.6

400

500°C

400°C

300 200

550°C

0

4 6 Strain (%)

8

300

Stress (MPa)

400

500°C

200

Test Results Proposed by Lee et al. (2003)

100 0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

10

140°C

500

400°C

0.2

2

600

180°C

0 0.0

660°C

0

1.8 2.0

22°C

500 Stress (MPa)

Test results Proposed

500

100

0 0.0

600

80°C

700

700

320°C

400

450°C

300

500°C

660°C

200

Test results Proposed

100 0

2.0

0

1

2

3 Strain (%)

Strain (%) Fig. 18. Comparison of stress–strain curves predicted by Lee et al. [2] with test results: (a) G550 1.0 mm; (b) G450 1.9 mm.

4

5

80°C (G550 1.0mm) 600

5. Transient state test results

Test results Proposed

500 Stress (MPa)

respectively. In the second part of the curve of Eq. (7) for fT4fy,T, the elastic modulus at yield strength (Ey,T) can be determine from Eq. (8), and the strain corresponding to yield strength (ey,T) is the last point of the first part of the curve. Hence, in order to plot a stress–strain curve for a particular grade of cold-formed steel at a given temperature, the yield strength (fy,T), elastic modulus (ET), ultimate strength (fu,T) and ultimate strain (eu,T) are needed. Equations of yield strength, elastic modulus, ultimate strength and ultimate strain at elevated temperatures are proposed in this paper.

6

22°C (G450 1.9mm) 550°C (G450 1.9mm)

400 300 200

450°C (G550 1.0mm)

100 0

0

1

2

3

4 5 Strain (%)

6

7

8

9

Fig. 19. Comparison of stress–strain curves obtained from test results with prediction using the proposed stress–strain curve model: (a) G550 1.0 mm; (b) G450 1.9 mm; (c) G550 1.0 mm and G450 1.9 mm.

5.1. General Thermal elongation of the specimens was determined at a tensile stress level of 2 MPa that is close to free thermal expansion and compared with the thermal elongation calculated according to BS 5950-8 and EC3-1.2 in Fig. 20. 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 thermal elongation of G450 1.9 mm steel is generally less than the values predicted by the BS 5950-8 and EC3-1.2. Although the 2 MPa tensile stress was

almost negligible at normal room temperature for determining the thermal elongation, but it slightly affected the elongation when the temperature increases. When the temperature reached 870 1C, the specimen would undergo continuous deformation at a tensile stress of 2 MPa. Since the thermal elongation was determined for specimens loaded at a stress level of 2 MPa, the elastic modulus obtained from the transient state tests were slightly underestimated. In the transient state test, the specimen was loaded to a given stress level, and the elastic modulus of each specimen can be determined from the stress–strain curve of the

ARTICLE IN PRESS J. Chen, B. Young / Thin-Walled Structures 45 (2007) 96–110

109 AS 4100 EC3-1.2 Outinen et al.(2001) G450 1.9mm Steady State Test G450 1.9mm Transient State Test Proposed equation

1.4 EC3-1.2 BS 5950-8 G450 1.9mm

1.0 0.8

1.2

0.6

1.0

0.4 0.2 0.0 0

200

400 600 Temperature (°C)

800

1000

Fig. 20. Comparison of thermal elongation predicted by BS 5950-8 and EC3-1.2 with test results of G450 1.9 mm.

ET / Enormal

Strain (%)

1.2

0.8 0.6 0.4 0.2 0.0 0

loading process. The elastic 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%.

200

400 600 Temperature (°C)

800

1000

Fig. 21. Comparison of elastic modulus predicted by AS 4100, EC3-1.2 and proposed equation with test results for transient and steady state tests.

Table 9 Coefficients of proposed equation for elastic modulus Eq. (11)

5.2. Elastic modulus The transient state test results were firstly converted into stress–stain curves, and the reduction factor of elastic modulus (ET/Enormal) for different temperatures was determined. The reduction factor of elastic modulus determined from the transient state tests was compared with the steady state test results for G450 1.9 mm specimens, as shown in Fig. 21. The transient state test results are also compared with the AS 4100 and EC3-1.2 prediction as well as compared with the transient state tests conducted by Outinen et al. [15], as shown in Fig. 21. It can been seen that the reduction factor of elastic modulus obtained from the transient state tests in this study agree well with the EC3-1.2 prediction and the test results obtained by Outinen et al. [15] for temperatures ranged from 320 to 450 1C. For the temperatures ranged from 80 to 320 1C and from 550 to 660 1C, the values of the reduction factor of elastic modulus obtained from the transient state tests are smaller than those predicted by the EC3-1.2 and the tests conducted by Outinen et al. [15]. The AS 4100 predictions of elastic modulus are unconservative compared with the transient state test results. It should be noted that the reduction factor of elastic modulus obtained from the transient state tests is quite different from the steady state tests. A unified equation is proposed for G450 1.9 mm cold-formed steel at elevated temperatures, as shown in Eq. (11). The coefficients a, b, c and n of the equation are presented in Table 9, and T is the temperature in degree Celsius (1C). Proposed equation for elastic modulus: ET E normal

¼a

ð T  bÞ n , c

(11)

Temperature (1C)

22pTo450

450pTo650

a b c n

1.0 22 1.25  103 1

0.11 860 2.2  105 2

where Enormal is the elastic modulus at normal room temperature. It is shown that the test results obtained from the transient state tests compared well with the values of elastic modulus predicted by the proposed equation for G450 1.9 mm, as shown in Fig. 21. 5.3. Ultimate strength The ultimate strength of the specimen is defined at a specified load when the temperature reached a certain value and the specimen undergoes a continuous elongation at an appreciate rate. This specified load was considered as the ultimate strength of the specimen at that particular temperature in the transient state tests. In Table 10, the ultimate strength obtained from the transient state tests (ft,u,T) is compared with the ultimate strength obtained from the steady state tests (fu,T) with and without consideration of the static drop. The ultimate strength obtained from steady state tests with consideration of the static drop are closer to the results obtained from the transient state tests than the results obtained from the steady state tests without consideration of the static drop, as shown in Table 10.

ARTICLE IN PRESS J. Chen, B. Young / Thin-Walled Structures 45 (2007) 96–110

110

Table 10 Comparison of ultimate strength obtained from transient and steady state tests Temperature (1C)

ft,u,T (MPa)

fu,T (MPa)

fu,TDrop (MPa)

22 80 140 220 320 450 550 660 760 870

4480 4480 4480 4480 4480 420–450 180–210 30–60 2–30 o2

551 545 538 540 566 509 347 92 — —

540 529 521 512 539 440 274 71.5 — —

temperatures. Furthermore, equation for ultimate strength and ultimate strain of the cold-formed steel at elevated temperatures is also proposed, and the predicted values obtained from the proposed equation compared well with the test results. A full strain range expression up to the ultimate tensile strain for the stress–strain curves of coldformed carbon steel at elevated temperatures is also proposed. Acknowledgements The authors are grateful to BHP Steel Lysaght Singapore (now BlueScope Lysaght) for supplying the test specimens.

6. Conclusions References An experimental investigation on the mechanical properties of cold-formed steel at elevated temperatures has been presented. The test programme included two steel grades of G550 and G450 with nominal yield strengths of 550 and 450 MPa, respectively. The plate thicknesses of the coupon test specimens were 1.0 and 1.9 mm. Both steady and transient state tests were conducted at different temperatures. A differential thermal analysis was also performed in order to determine the chemical reaction of the tested materials during the heating process. The yield strengths, elastic modulus and thermal elongation obtained from the tests were compared with the Australian, British and European predictions. The test results obtained from this study were also compared with the test results obtained by other researchers. Generally, it is shown that the yield strengths predicted by the Australian, British and European standards are conservative, except for G550 1.0 mm steel from 450 to 970 1C and G450 1.9 mm steel at 660 1C. Hence, the standards provide unconservative predictions for high temperatures. It is also shown that the elastic modulus predicted by the Australian and European standards are conservative for the steady state tests, but generally unconservative for the transient state tests. In this paper, a unified equation has been proposed to determine the yield strength of G550 1.0 mm and G450 1.9 mm cold-formed steel for temperatures ranged from 22 to 1000 1C. In addition, the unified equation to determine the elastic modulus of G450 1.9 mm cold-formed steel for temperatures ranged from 22 to 650 1C based on the transient state tests has been also proposed in this paper. The yield strength and elastic modulus calculated from the proposed equation were compared with the test results. It is shown that the proposed equation accurately predicted the yield strength and elastic modulus of the cold-formed steel at elevated

[1] Yu WW. Cold-formed steel design. 3rd ed. New York: Wiley; 2000. [2] Lee JH, Mahendran M, Ma¨kela¨inen P. Prediction of mechanical properties of light gauge steels at elevated temperatures. J Construct Steel Res 2003;59(12):1517–32. [3] AS 4100:1998. Steel structures. Sydney, Australia: Standards Australia; 1998. [4] BS 5950-8: 1990. Structural use of steelwork in building—part 8: code of practice for fire resistant design. British Standards Institution (BSI), British Standard BS, 1998. [5] EC3. Eurocode 3: design of steel structures—part 1.2: general rules— structural fire design. European Committee for Standardization, DD ENV 1993-1-2:2001, CEN, Brussels, 2001. [6] Olawale AO, Plank RJ. The collapse analysis of steel columns in fire using a finite strip method. Int J Numer Methods Eng 1988;26(17): 2755–64. [7] Outinen J. Mechanical properties of structural steels at elevated temperatures. Licentiate thesis, Helsinki University of Technology, Finland, 1999. [8] Mirambell E, Real E. On the calculation of deflections in structural stainless steel beams: an experimental and numerical investigation. J Construct Steel Res 2000;54(1):109–33. [9] Rasmussen KJR. Full-range stress–strain curves for stainless steel alloys. J Construct Steel Res 2003;59(1):47–61. [10] ASTM E21-92. Standard test methods for elevated temperature tension tests of metallic materials. Annual book of ASTM standards, vol. 03.01: metals-mechanical testing; elevated and low-temperature tests; metallography. West Conshohochken, PA: American Society for Testing and Materials; 1997. [11] AS 2291:1979. Methods for the tensile testing of metals at elevated temperatures. Sydney, Australia: Standards Australia; 1979. [12] ASM. Heat-resistant materials. In: Davis JR, editor. ASM specialty handbook. ASM International, Handbook Committee; 1997. [13] Buchanan A, Moss P, Seputro J, Welsh R. The effect of stress–strain relationships on the fire performance of steel beams. Eng Struct 2004;26(11):1505–15. [14] Ramberg W, Osgood WR. Description of stress–strain curves by three parameters. NACA technical note 902, 1943. [15] Outinen J, Kaitila O, Ma¨kela¨inen P. High-temperature testing of structural steel and modelling of structures at fire temperatures. Research report TKK-TER-23, Helsinki University of Technology Laboratory of Steel Structures, Helsinki, Finland, 2001.