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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 ⫺ 17)兩 ⫺ 兩a兩 6 ⫺ 17 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 c2y ⫺ c3(y)2 E ⫺ c1y 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 ⫹ c3y ⫺ c1u 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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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