A simple model for the fire resistance of axially loaded members—comparison with experimental results

A simple model for the fire resistance of axially loaded members—comparison with experimental results

J. Construct. Steel Res. Vol. 37, No. 3, pp. 175-204, 1996 S0143-974X(96)00008-9 Copyright © 1996 Elsevier Science Ltd Printed in Great Britain. All...

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J. Construct. Steel Res. Vol. 37, No. 3, pp. 175-204, 1996

S0143-974X(96)00008-9

Copyright © 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0143-974X/96 $15.00 + 0.00

ELSEVIER

A Simple Model for the Fire Resistance of Axially Loaded MembersmComparison with Experimental Results Jean-Marc Franssen, a Jean-Baptiste Schleich, b Louis-Guy Cajot b & Wenceslao Azpiazu c alnstitut du G6nie Civil, Universit6 de Liege, Quai Banning 6, B-4000 Liege, Belgium bprofilARBED Recherches, Route de Luxembourg, Esch/Alzette, Grand Duch6 de Luxembourg CLABEIN, Exp. Mech. Dpt., Cuesta de olabeaga, 16, 48.013 Bilbao, Spain (Received 27 January 1995; revised version received 26 September 1995; accepted 19 January 1996)

ABSTRACT A general model, i.e. a non linear computer code, has been extensively used to determine the buckling load of axially loaded members, considering that the material model behaves at elevated temperatures according to the hypotheses of Eurocode 3 Part 1.2, and the main results of this numerical investigation are summarised in this paper. The main parameters and the results of 59 experimental tests found in the literature are reported, as well as the results of 21 original tests made within this research project. Those test results are used to evaluate the severity factor of the analytical formula deduced from the numerical simulations. This factor is chosen in order to obtain an analytically calculated ultimate load which is, in the average, the same as the experimental load. The ultimate load or ultimate temperature can be determined by the proposed analytical formula or directly by interpolation in tables which give the ratio between the ultimate load and the plastic load at room temperature. Copyright © 1996 Elsevier Science Ltd.

NOTATION a

B e

E

fp

Thickness of the web (mm) Width of a profile (mm) Thickness of the flanges (mm) Young's modulus (MPa) Limit of proportionality (MPa) 175

176

fy fyfl fyw H KE(T)

Km x(T) L M N

T Tu

#, /3 g E

J.-M. Franssen et al.

Yield strength (MPa) Yield strength in the flange (MPa) Yield strength in the web (MPa) Depth of a profile (mm) Relative Young's modulus at T Relative yield strength at T Relative limit of proportionality at T Length of a column (mm) Bending moment (Nmm) Axial load (N) Ultimate buckling load (N) Plastic load (N) Temperature (°C) Ultimate temperature (°C) Cross-sectional area (mm 2) Overall buckling coefficient Imperfection factor Severity factor Buckling coefficient ~35/fy Relative slenderness

1 INTRODUCTION In addition to experimental tests, different solutions are available for the estimation of the fire resistance of compression members submitted to elevated temperatures. The first method relies on the use of numerical calculation tools based on acknowledged principles and assumptions of the theory of structural mechanics, taking into account the effect of temperature. The application of this method requires the utilisation of non-linear computer programs. It is very powerful in the sense that various geometrical configurations and loading situations can be analysed. It has the obvious shortcoming that every individual case needs to be solved by a non-linear numerical calculation, which are not yet very common, especially for elevated temperatures. Among various methods of analysis allowing the calculation of steel members at elevated temperatures, some have been more particularly dedicated to the analysis of instability in columns: • Ossenbruggen et al. 1 proposed an analytical method for the calculation of steel columns at elevated temperatures. The material model was based

Fire resistance of axially loaded members









• •



177

on stress addition, with the sharp yielding behaviour and the horizontal yield plateau of the stress-strain curve at room temperature assumed to be valid at elevated temperatures. The method was based on the determination of moment--curvature curves. Residual stresses--assumed to remain constant--and linear temperature gradients through the depth of the section could be considered. Aribert & Abdel Aziz 2 presented a method for the simulation of columns at elevated temperatures. Eccentricities of the load, different end conditions, initial geometrical imperfections and thermal gradients--longitudinal as well as transversal--could be considered. The material model was hybrid, based on strain addition for thermal, stress-related and creep strain, with an addition of the residual stress, supposed to vary in the same way as the yield strength at elevated temperature. The stress-strain relationship was of the elastic-plateau type. Olawale & Plank 3 have used the finite strip method to find the bifurcation load of elements in compression. In their study, both local and overall buckling modes, as well as interaction between them, are admitted. Setti 4 used a numerical method where the column is made of rigi d parts connected by elementary cells in which both axial and flexural flexibility are concentrated. The material was assumed to have linearly elastic perfectly plastic behaviour. Abbas 5 used the method of transfer matrix described in Ref. 2 to analyse steel columns buckling at elevated temperatures in the plane of the major axis, with the hypothesis of an elastic perfectly plastic stress-strain relationship. Skowronski6 has formulated and solved the differential equations of buckling steel columns during fire. Burgess & Najjar 7 have made numerical calculations based on the 'PerryRobertson' principles to investigate the behaviour of steel columns in fire conditions, including the effects of thermal gradients in the section. Poh & Bennets s,9 present a method based on moment-thrust-curvature relationships for the member cross-section, which they verify against 20 tests made by Aasen. 1°

The other solution is to rely on analytical formulas, derived from the results of experimental tests or from the application of calculation models. They can be applied with the help of pocket calculators. Their main shortcoming is that these formulas have been derived by extrapolations of the formula available for ambient temperature, with limited scientific justifications, this absence being covered by the introduction of some correction factors. • Culver et al. 1, utilised the analytical method described in Ref. 1 to calculate the ultimate load of steel columns at elevated temperatures. In the

178









J.-M. Franssen et al.

case of rolled sections, they considered the temperatures of 204, 371 and 538°C, and three idealised cross-sections for A36 steel. An analytical formula was proposed for the critical stress at elevated temperature as a function of the critical stress at 20°C, of the temperature and of the slenderness at 20°C. Culver recognised that the method provided for a variable factor of safety as a function of slenderness. Aribert & Randriantsara 12 made some analytical calculations based on a material model which incorporates implicit creep to derive the buckling curve of the HEAl00 section. Abbas 5 used his calculation method to analyse eight different profiles. As the shape of the material law does not change whatever the temperature, the buckling curves depend very marginally on the temperature. A very simple analytical formula is proposed as a conclusion to the numerical calculations. An analytical method 13 had been proposed by the ECCS, based on the concept of buckling curve. The buckling curve c (see Eurocode 3 Part 1.114) was used irrespective of the cross-section type and buckling axis, with the relative slenderness evaluated at ambient temperature. The buckling curve was divided by a coefficient of 0.85 allowing, among other things, for the discrepancy between nominal and yield stress and reducing the discrepancies with the critical temperatures measured in experimental tests. The same analytical formula has been introduced in Eurocode 3 Part 1015 and Eurocode 3 Part 1.216 although the law for the variation of the effective yield strength with temperature is not the same as in the ECCS recommendations. The buckling curve is this time divided by a correction factor of 1.20. The relative slenderness was initially evaluated at room temperature but it has recently been decided to calculate this slenderness at the ultimate temperature.

In a previous paper, 17 the authors applied their own numerical model on virtually all the combinations of yield strength, buckling axis, buckling length, ultimate temperature and cross-section for the basic case of axially loaded, simply supported steel columns. The effects of initial deflections, residual stresses and large displacements have been considered and the non-linear stress-strain relationships were taken from EC3 Part 1.2. Different conclusions emerged from those simulations. A new proposal was made for a buckling curve valid at elevated temperatures. The relative slenderness is evaluated at the ultimate temperature. The buckling coefficient depends on the yield strength at 20°C and on a severity factor which has to be chosen to ensure the appropriate safety level. It has been shown that a severity factor of 1.20 leads to a result that is on the safe side compared to the results provided by the numerical calculations.

Fire resistance of axially loaded members

179

It has to be underlined, however, that those numerical simulations have been made with characteristic values for the residual stresses and for the geometrical imperfection, which are unlikely to be simultaneously present in a test or in a real fire. Also, one certain thing that can be said about the stressstrain relationships of EC3 is that they do not correspond to reality (provided that such a deterministic being as the real stress-strain relationship is supposed to exist). Indeed, important efforts have been made to ensure that the stressstrain curve at a defined temperature is continuous. The elliptic part is tangent to the linear elastic and plastic parts. Yet, the variation with the temperature of the three parameters used in the analytical expression of the stress-strain law (i.e. the Young modulus, the proportionality limit and the yield strength) is supposed to be made of multilinear segments, with the limit between adjacent segments being located exactly at 100, 200, 300, 400...°C (see Fig. 1). It would be amazing if nature had provided us with a material which has such a gentle variation of its parameters. Moreover, the variation of the parameters have been derived by recalculation of experimental tests made on bent elements, 18 where the stiffness in the elastic domain was not critical, whereas the Young modulus is known to be determinant in buckling. It was therefore decided to compare the proposed formula with available experimental test results from the literature. This should allow the severity factor to be used to be determined in order to obtain the desired safety level• Only tests when the actual yield strength had been measured were con-

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3.

180

J.-M. Franssen et al.

sidered. Other tests where the nominal yield strength is reported are available. 19,20 The tests made by Aasen m could not be used here because of the high non-uniformity of the longitudinal temperature distribution. Those tests are nevertheless highly valuable for the validation of computer codes, 9 provided that those codes are able to handle a non-uniform temperature distribution. When not measured, the nominal values have been used for the geometrical dimensions of the cross-section. A recent sensitive analysis by Talamona2~ has shown that the yield strength is largely predominant before other characteristics such as geometrical imperfections, cross-sectional area, residual stresses, etc. As only a limited number of tests have been made on simply supported axially loaded columns, the results of the tests made with other end conditions were supposed to represent a test made on an equivalent simply supported column with its length equal to the buckling length of the actual element. The analyses that will be detailed in this paper have been repeated with the results sorted in three groups according to the supports conditions, pinned-pinned, pinned-fixed and fixed-fixed, and no significant difference emerged between the three groups, which seems to be an indication that it was valid to consider the tests that are not simply supported. Tests when the load has been applied with a defined eccentricity have been considered provided that this eccentricity is sufficiently small. This was judged by the fact that, when the column is calculated by an interaction formula such as the one from Eurocode 3,14 the stress produced by the axial load is at least equal to 75% of the total stress. Tests where the stress produced by the bending moment is larger than 25% of the total stress were considered to be too sensitive to the interaction and were dismissed. It has also been verified at the end of this project that the 33 tests with a small eccentricity lead to the same conclusion as the 40 tests made on an axially loaded column. Only full-scale tests were considered. As the majority of available furnaces have a height in the range 3-4 m, a large number of different cross-section types have been tested, but very often with the same length of column. In order to circumvent that fact, it was decided to perform an original set of experimental tests made on the same cross-section, but for different length and different load factors. Those results will be reported here.

2 RESULTS OF THE NUMERICAL SIMULATIONS The results of the numerical simulations have been largely reported in a previous publication, Ref. 17. The main results are summarised here for clarity reasons.

Fire resistance of axially loaded members

181

More than 200,000 numerical tests have been performed, taking into account: 2 yield strengths: 235 and 355 MPa, 339 hot rolled H sections, classified as class 1, 2 or 3 under axial loading according to Ref. 14, 2 buckling axes, considered separately, 10 different slenderness, 12 different applied loads. 2 thermal situations: uniform temperature, and unprotected submitted to the ISO curve. For each test, the buckling coefficient has been calculated according to eqn 1:

Nu(T) Nu(T) X -

-

Ofy(T)

-

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The results were sorted and the following conclusions could be drawn. • The horizontal plateau of the buckling curve considered at room temperature up to a relative slenderness of 0.20 tends to disappear for ultimate temperatures of 400°C and beyond. The plastic load cannot be obtained even for small slenderness. This fact had already been reported by Burgess et al. 22 • The scatter in the buckling coefficient is smaller when the results are presented as a function of the relative slenderness calculated at the ultimate temperature than when the relative slenderness at room temperature is put on the horizontal axis. • The buckling coefficient is higher for $355 than for $235 steel grade. • The classification that is made in Eurocode 3, Part 1.1 ~4 between different curves labelled a, b, c or d tends to vanish at elevated temperatures, as reported by Vandamme & Janss. 2° • It is not possible to separate the results in terms of one buckling curve for each ultimate temperature and it seems reasonable to cover the whole temperature range from 400°C to 900°C by one single equation. Based on the above-mentioned observations, the following proposal could be made for the buckling coefficient of steel hot rolled H profiles submitted to fire. For T--< 100°C X is calculated according to Eurocode 3 Part 1.1,14 as for T = 20°C. For T --> 400°C

J.-M. Franssen et al.

182

X is calculated according to eqn 2

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~ = tIE is the imperfection factor, with/3 to be chosen in order to ensure the appropriate safety level.

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E = ~f235/fy fy in MPa is the nominal yield strength, i.e. not reduced according to EN 1002523depending on the flange thickness. For 100°C < T < 400°C X is linearly interpolated between the values calculated for 100 and 400°C. The buckling curves are represented on Fig. 2, in the case of $355 steel for a section buckling according to the curve c at ambient temperature. Although there is one unique analytical expression for temperatures beyond 400°C, see eqn 2, this expression is given as a function of the slenderness at the ultimate temperature. When plotted as functions of the slenderness at room temperature as on Fig. 2, the curves are different for each temperature. The value of 1.20 has been chosen for the severity factor/3. This value guarantees that the ultimate load provided by the analytical formula is lower than the ultimate load that could be calculated by the numerical simulation with characteristic struc1.0"



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Fire resistance of axially loaded members

183

tural imperfections. This must therefore be considered as a very severe condition. The value to be given to the severity factor in case of practical design will be discussed below in this paper.

3 RESULTS OF THE EXPERIMENTAL TESTS

3.1 Test from Borehamwood One test result was found in the Compendium 24 published by BSC and BRE in the U.K. Columns with blocked-in web which are also reported were not considered here because of the high thermal gradient existing in the section. The test which could be used is labelled as 41 in Ref. 24. It is labelled as 117 in Table 2 of this paper. The actual dimensions of the section have been measured, as well as the yield strength in the flange. The ends of the column were encased in concrete caps and, as the element was restrained in rotation by the loading system, end fixity was assumed and an effective length of half of the exposed length has been adopted in this paper in order to be consistent with the interpretation of the tests made in Gent, although 0.70 of the exposed length was chosen in the interpretation of Ref. 24. Figure 3 shows the experimental buckling coefficients plotted versus the relative slenderness at the ultimate temperature. Two test results are not visible on the figure because the relative slenderness was higher than 2. The result of Borehamwood is highlighted on this figure and it appears that it is one of the more favourable results. Had the test been interpreted as having a buckling length of 0.70 L as in Ref. 24, this point would have been shifted to the right

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184

J.-M. Franssen et al.

and would have appeared as even more favourable. The analytical curve has been drawn with the yield strength of the Borehamwood test and with the severity factor ensuring that the ratio between the analytical results and the experimental results has a mean value of 1.00 (see Section 4). The general trend of all the experimental tests is very similar to the analytical curve.

3.2 Tests from Gent Sixteen results from Gent are considered. The results have been taken from the original test reports from the laboratory of Professor Minne at the University of Gent and some values may differ slightly from what has been reported elsewhere. The test labelled as No. 2.4 in Ref. 25 has not been considered here because of the uncertainty mentioned in the original test report on the position of the load resulting from problems in the welding of the end plates on the specimen. The test labelled as No. 2.18 in Ref. 26 has not been considered because of the very severe longitudinal gradients existing in the temperature distribution. A difference of almost 300°C is found in the test report between two different cross-sections. The geometrical dimensions have been measured in each element and the yield strength was measured for each section type. In the evaluation of each fire test, the mean value of the available tension test result was used. The columns were clamped in special end fixtures intended to provide a perfect rotational restraint at both ends. This hypothesis of rotational fixity was experimentally verified by four tests made at room temperature, 2° and was also confirmed by the shape of the elements observed after the fire tests. The tests were therefore interpreted as tests made on simply supported elements with a length equal to 0.50 of the exposed length of the actual elements, and buckling around the minor axis. Figure 4 shows the buckling coefficients derived from the tests of Gent as a function of the relative slenderness at the ultimate temperature.

3.3 Tests from Germany Twenty-five test results from the Technische Universit~it Braunschweig and three test results from Stuttgart were obtained. The results of those tests have been collected and transmitted to the authors by Dorn, 26 with the exception of test No. 1 (see Table 2) taken from Ref. 27. Most of the results from Braunschweig have been published in the reports 28 of Sonderforschungsbereich 148. Figure 5 shows how the tests made in Braunschweig are located on the graph whereas Fig. 6 refers to the tests made in Stuttgart.

Fire resistance of axially loaded members

185

1.2 1.0

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3.4 Tests from Rennes Fourteen axially loaded simply supported columns were tested at the INSA in Rennes and reported by Aribert & Randriantsara. 12 All the tests were made on H E A l 0 0 sections, half of them under increasing temperature and constant load, half of them at constant temperature and increasing load. The length of all the elements was 1.994 m, which gives for one slenderness the influence of the load level on the ultimate temperature. One measured value is reported for the yield strength as well as the second moment of area and the sectional area.

J.-M. Franssen et al.

186 1.0 0.9

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The fact that all the tests have been made on the same cross-section type and for one buckling length is clearly reflected on Fig. 7. The points are not exactly at the same horizontal distance because the relative slenderness is evaluated at the ultimate temperature and this one was not the same in the different tests. The fact that the higher values from Rennes are shifted to the left and the lower values to the right when the slenderness is evaluated at the ultimate temperature indeed reduce the scattering from the analytical curve.

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Fire resistance of axially loaded members

187

3.5 Tests from Bilbao Twenty-one original tests were conducted in the LABEIN laboratory of Professor Ramirez and reported by Azpiazu & Unanue. 29 The sections were HEAl00 profiles. Residual stresses, geometrical dimensions of the section, geometrical imperfections and yield strength were measured for each tested element. The load was applied before the test and kept constant during the heating. It was applied with a well defined eccentricity of 5 m m through a very sharp knife support inducing bending around the minor axis. The end rotation around this axis was free, and the rotation around the major axis was supposed to be restrained by the action of the knife. Five different buckling lengths from 510 to 3510 m m were considered, each of them with different load levels. The specimens were electrically heated by means of ceramic mat elements at a rate of 5°C/min. Automatic control on different heating devices was present in order to ensure a uniform temperature distribution along the length of the elements. The temperature field has been measured with thermocouples welded on the specimens. The number of measurement points has been from 17 for 510 mm length specimens up to 23 for the length of 3510 mm. When somewhat lower temperatures were recorded near the supports, the failure temperature of the element was estimated as the mean temperature of the thermocouples located in the central part of the column. The load as well as the axial expansion and horizontal displacements at mid level were monitored during the test. Table 1 is a summary of the tests made in Bilbao. In this table, No. is the number of the test in Ref. 9, i2 is the imperfection at L/4, i3 is the imperfection at 2L/4, a n d / 4 is the imperfection at 3L/4. Note: • The values given for the ultimate temperature, the yield strengths and the dimensions of the section result from the average between several measures on the specimen. • The elements were placed vertically. They were turned in such a way that the effect of the imperfection was added to the effect of the load eccentricity. • The measured residual stresses were in the order of magnitude of 0.10 × 235 MPa. Test Nos ALl, AL3, SL43 and AL6 were performed at ambient temperature and will not be considered in the evaluation of the analytical formula for buckling at elevated temperatures. They were conducted to verify the loading equipment and to provide reference points at 20°C.

J.-M. Franssen et al.

188

TABLE 1 Results of the Experimental Tests Made by LABEIN No.

L (mm)

Tu (°C)

N fy w fy fl (kN) (MPa)(MPa)

B (ram)

H (mm)

a e i2 i3 i4 (mm) (rnm) (ram) (mm) (mm)

ALl BL1 CL1 DL1 AL3 BL3 CL3 DL3 SL40 SIAl SLA2 SIA3 SIA4 AL5 BL5 CL5 DL5 AL6 BL6 CL6 DL6

513 513 513 513 1270 1272 1271 1269 2020 2026 2020 2021 2023 2770 2772 2771 2772 3510 3510 3510 3510

20 532 694 863 20 390 474 749 525 509 485 20 495 457 587 587 886 20 446 493 727

537 362 110 40 490 292 251 24 170 174 171 366 173 127 73 34 7.7 176 105 90 11.5

101.90 101.85 101.78 102.28 101.95 101.93 101.90 102.15

99.20 98.85 99.07 99.12 99.08 98.90 99.25 99.15

6.10 5.92 6.43 6.13 5.97 5.97 6.13 6.02

7.80 7.61 7.80 7.68 7.67 7.64 7.82 7.73

101.84 101.82 101.84 101.68 101.94 101.76 102.03 102.15 101.99 101.88 102.05 101.68

98.97 99.04 98.89 99.17 99.06 98.95 99.25 99.16 99.08 98.93 99.12 99.17

5.73 5.76 5.80 5.73 5.78 5.76 5.98 5.96 5.79 5.93 5.94 5.73

7.58 0.60 0.70 0.70 7 . 6 1 0.90 1 . 7 0 0.90 7.57 -0.40 0.00 0.00 7.60 0.50 1 . 1 0 0.60 7.68 - 0 . 0 7 - 0 . 0 4 0.60 7.62 0.30 1.00 0.80 7.76 0.70 0.80 0.80 7.72 0.80 1.60 0.80 7.66 - 0 . 7 0 - 0 . 4 0 0.60 7.63 0.80 1.00 0.30 7 . 7 1 0.70 0.80 0.80 7.60 0.80 1.60 0.80

300 300 316 309 300 300 316 309 286 286 286 286 286 300 300 316 309 300 300 316 309

280 286.5 292.5 282.5 280 286.5 292.5 282.5 280 280 280 280 280 280 286.5 292.5 282.5 280 286.5 292.5 282.5

0.00 0.00 0.00 0.00 0.00 0.20 0.40 0.30

Test No. SL40 was a preliminary test performed to verify the heating equipment. It was performed on an element which had not been precisely measured. The specimen No. DL3 was accidentally heated when the elongation was restrained in the loading system. This was recognized early and the temperature was decreased before the test was restarted with free axial displacement. Due to this preheating history, the specimen had an initial sinusoidal imperfection of 7 mm before the actual test. If an imperfection of HI1000 = 2 mm could be regarded as a usual structural imperfection, it can be considered that the load had been applied with a first order eccentricity of 5 + (7 - 2) = 10 mm. The test can therefore be used for the validation of computer programs or for M - N interaction formula but in this case, it leads to a bending stress that is too important before the axial stress and this test can no more be regarded as representative of axial buckling (see Section 1). In test No. CL5, the temperature was significantly higher in the regions near the supports than in the middle of the column. This was confirmed by the fact that the buckling sections were 360 nun approximately from the ends. Because of the uncertainty on the real buckling length, this test will not be used in the comparison with the simple model.

Fire resistance of axially loaded members

189

TABLE 2 # 1 7 8 9 11 12 13 14 15 16 17 18 19 20 21 22 23 31 32 33 34 35 35 54 56 56 57 56 60 61 62 78 79 82 68 87 68 89 90 94 95 96 97 98 103 104 105 106 107 108 109 110 111 112 113 114

Section

Ax~

HD 210x198 W HEB300 W IPE160 W IPE150 W IPE200 W IPE200 W HEB120 W HEB120 W HEB180 W HEB180 W HEA200 W HEA3(X) W HEA220 W HEB200 W HEB140 W HEB140 W IPE220 W HEB120 S HEM220 S HEB220 S HEA220 S HEM220 S HEB120 S HEB180 S HEB180 S HEB220 S HEB220 S HEB220 S HEB160 S HEB160 S HEB160 S HEB240 W HEB240 W HEMI00 W IPE360 W IPE360 W IPE300 W IPE300 W IPE180 W HEB180 S HEB180 S HEB180 S HEB180 S HEB150 S HEAIO0 W HEAl00 W HEAl00 W HEAl00 W HEAl00 W HEAl00 W HEAl00 W HEAl00 W HEAIO0 W HEAl00 W HEAIO0 W HEAIO0 W'

fy Mpa 364 271 271 271 277 271 260 260 275 275 279 269 252 216 247 247 273 257 269 261 309 269 257 267 250 269 250 263 262 259 249 229 221 332 287 273 248 248 433 250 244 283 278 277 300 300 300 300 300 300 300 300 300 300 300 300

N Tu kN "C 1100.0 656 1999.8 588 104.2 564 151.3 475 265.9 394 323.5 250 362.9 519 267.0 585 602.7 616 892.9 560 676.8 556 1507.3 561 972.0! 502 681.01 549 541.7 516 372.0' 576 319.0 522 317.8 560 1267.5 600 767.1 590 783.8 560 681.8 650 105.0 685 891.0 475 876.0 446 1489.0 335 1528.0 230 1136.0 476 755.0 232 590.0 412 650.0 290 1230.0 425 1195.0 547 147.0 685 768.0 290 610.0 355 320.0 255 280.0 316 50.0 541 847.0 515 857.0 466 928.0 513 928.0 540 958.0 523 337.0 365 318.0 400 250.0 510 143.0 550 110.0 600 61.0 680 57.0 750 360.0 235 320.0 440 250.0 450 200.0 480 150.0 552

L mm 5700 1890 1890 1890 1890 1915 1890 1890 1890 18~0 1890 1890 1915 lg15 1915 1915 1915 3800 3800 3800 3800 4800 4800 3860 3860 3700 3700 3700 4700 4700 4700 3700 1850 2850 3700 3700 4700 4700 4700 2702 2702 1930 1930 1930 Ig94 1994 I994 1984 I994 1994 1994 1984 1994 1994 1994 1994

TOP

BOTTOM

FREE FIXED FIXED FIXED FIXED FIXED FIXED FIXED FIXED FIXED FIXED FIXED FIXED FIXED FIXED FIXED FIXED FREE FREE FREE FREE FREE FREE FREE FREE FREE FREE FREE FREE FREE FREE FREE FIXED FIXED FREE FREE FREE FREE FREE FIXED FIXED FIXED FIXED FIXED FREE FREE FREE FREE FREE FREE FREE FREE FREE FREE FREE FREE

FREE FIXED FIXED I~IXED FIXED FIXED FIXED FIXED FIXED FIXED FIXED FIXED FIXED FIXED FIXED FIXED FIXED FREE FREE FREE FREE FREE FREE FREE FREE FREE FREE FREE FREE FREE FREE FREE FIXED FIXED FREE FREE FREE FREE FREE FREE FREE FIXED FIXED FIXED FREE FREE FREE FREE FREE FREE FREE FREE FREE FREE FREE FREE

• mm 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 12 12 12 14 12 0 0 12 12 12 12 12 12 0 0 0 9 g 11 11 25 10 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

lab. Braur~ Gent Gent Gent Gent Gent Gent Gent Gent Gent Gent Gent Gent Gent Gent Gent Gent Brau~ig Braunschweig Brau~ Braunschweig Braunschweig Braunschweig Braunschweig Stuttgart Braunschweig Braunscl'r~ Braunschwetg Brau~ Braunschweig Braunschwelg Bmunscl'N,¢~ Braummhwelg Braunschw~g Bmunschweig Braunschweig Braunschwei9 Braunschweig Bmunschweig Braunschweig Stuttgalt Braunschweig Bmunschweig Stut~att Rennes Rennes Rennes Rennes Rennes Rennes Rennes Rennes Rennes Renrms Rennes Rennes

190

J.-M. Franssen et al.

TABLE 2 (cont.) 115 116 117 118 119 120 121 122 124 125 126 127 128 130 131 132 133

HEAl00 HEAl00 UC 203 HEAl00 HEAl00 HEAl00 HEAl00 HEAl00 HEAl00 HEAl00 HEAl00 HEAl00 HEAI00 HEA100 HEAl00 HEAl00 HEAl00

W W W W W W W W W W W W W W W W W

300 300 349 290 298 289 290 296 281 281 281 285 290 289 290 296 289

100.0 48.0 550.0 361.5 110.0 40.1 292.5 251.0 174.2 170.8 172.9 127.1 72.7 6.42 105.3 g0.4 11.5

618 701 688 532 694 863 390 474 509 485 495 457 587 858 446 493 727

1994 1994 1500 513 513 513 1272 1271 2026 2020 2023 2770 2772 2772 3510 3510 3510

FREE FREE FIXE FREE FREE FREE FREE FREE FREE FREE FREE FREE FREE FREE FREE FREE FREE

FREE FREE FIXE FREE FREE FREE FREE FREE FREE FREE FREE FREE FREE FREE FREE FREE FREE

0 0 0 5 5 5 5 5 5 5 5 5 5 5 5 5 5

Rennes Rennes Bomhamwood Bilbao Bilbao Bilbao Bilbao Bilbao Bilbao Bilbao Bilbao Bilbao Bilbao Bilbao Bilbao Bilbao Bilbao

Due to this severe examination of the 17 hot tests, 14 tests performed at elevated temperature remain for consideration in the data base. They are plotted in Fig. 8. 3.6 S u m m a r y of the available tests

Table 2 gives a summary of the parameters of the 73 experimental test results that will be considered for the evaluation of the analytical formula and the determination of the severity factor. Note: • In addition to those 73 tests, 19 experimental tests made in the 1970s in 1.2 • 1.0 0.8

l.~bein

I

. , ~

- -

°o D []

0.6 []

0.4

[] 0.2 Relative slenderness at Tu 0.0

0.0

[

I

0.5

1.0

Fig. 8. Tests of Bilbao.

1.5

2.0

Fire resistance of axially loaded members

191

Berlin 3°,31 have recently been introduced in the data base. They confirm the conclusions of this paper, provided that the buckling length of the elements is estimated as 0.50 L, as already pointed out by Witteveen and Twilt, 32 instead of 0.70 L as in the original interpretation of the authors of the tests.

4 DETERMINATION OF THE SEVERITY FACTOR If a value of the severity factor is chosen, say/31, it is possible to calculate for each test the ratio between the value of the ultimate load calculated with the use the analytical formula (see eqn 1-5) and the experimental load. The analytical load is of course evaluated at the ultimate temperature of the test. This ratio is also the ratio between the analytical and the experimental buckling coefficient.

Xi-

N~u/alyt /~aalyt. .~rex'P',ut - - ~/xp.

i = 1,...,73.

(6)

The analytical proposal is safe if it leads to values of xi, lower than 1, and unsafe for values higher than 1. For each chosen severity factor, the average value (Mx) and the standard deviation (dx) of the 73 x,. are calculated:

73 M x _ i=l

73

/(x~ -_ Mx) ~ dx=

~/

72

"

(7)

Figure 9 shows how the mean value M x varies depending on the choice that has been made for the severity factor/3. This figure confirms what has already been mentioned, that the value of 1.20 is very safe because it leads to an analytical load that is equal, on average, to 82.5% of the experimental load. In order to calibrate the analytical proposal to obtain, on average, the same ultimate loads as in experimental tests, /3 has to be given the value of 0.65. On the base of the mean value and the standard deviation, and assuming a normal distribution, it is possible to calculate the proportion of tests which are safely calculated by the analytical formula. Figure 10 shows the evolution of this percentage with the severity factor. It comes immediately from Fig. 9

J.-M. Franssen et al.

192

1.6 1.4 . 1.2-

~

1.00.8

0.6 0.4 0.20.0

0.0

I

I

I

I

I

I

I

I

I

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

20

Fig. 9. Evolution of the mean value with/3. 100 p

70

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . . . . . .

1o1-t OI 0.0

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

+

I

t

t

I

r

t

r

I

t

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

1 |

2.0

Fig. 10. Percentage of safe results. that a severity factor of 0.65 leads to 50% of safe results. It can be noticed that the value of 1.20, which ensures that the formula is safe with respect to the numerical simulations performed with characteristic imperfections, also leads to a safe result with respect to the tests in 88% of the cases. In order to reach a safety level of 95%, /3 has to be given the value of 1.46, which leads to the very conservative value of 0.77 for the average result. It is the opinion of the authors that for practical applications a value of 0.65 should be chosen, in order to represent the test results as close as possible and that no additional safety margin should be added when the fire resistance of an element is assessed by means of an analytical formula. The argument that the requirements are often unrealistically severe (ISO curve, irrespective

Fire resistance of axially loaded members

193

of the fire load or of any active protection measure) cannot be put forward. First of all, this is not the case in every country and some realistic fire safety concepts are already in use, such as the SIA 81 in Switzerland. 33 It has also to be mentioned that things are slowly changing and that the trend is toward more realistic fire safety concepts, including the so-called natural fire curves. It is hoped that the excessive requirements should gradually disappear from the regulations. In fact, our job as structural engineers is to provide a method for the evaluation of the fire resistance in which we believe, whatever the requirements. Introducing an unsafe structural method in order to counterbalance the requirements that we think are too severe would not be a very sound practice. The first reason for choosing the mean value when establishing the analytical formula is that the same objective is in fact used when the assessment of the fire resistance is made by experimental testing. When a new structural system or a separating element has to be tested against fire, it is seldom the case that a statistically significant series of tests is performed and the element is required to succeed in 95% of the tests. Usually, one single positive result is enough for the authorities. The other reason is that the formula has been calibrated using the actual yield strength as measured on the tested specimens. In a real life use of the formula, the nominal yield strength of the material would be used, which for commonly delivered elements introduces an additional safety margin. Recognising that a discrepancy existed when the fire resistance of a steel column was determined on one hand by a standard fire resistance test and on the other hand by the ECCS analytical approach, 34'35 based on characteristic values, Petterson and Witteveen 36 had already developed a correction procedure in order to obtain improved consistency between analytically and experimentally determined fire resistance values. When the analytical load was calculated according to EC3 part 1.2 as proposed in Ref. 16, i.e. with the buckling curve c divided by 1.20, then the average value of the ratio between the analytical and the experimental load was equal to 1.06. Figure 11 illustrates that this proposal tended to be unsafe and that the safety level was not uniform with the slenderness, decreasing with higher buckling length (one test, not seen on Fig. 11, has a ratio of 2.1 for a slenderness of 1.7). When the relative slenderness is evaluated at the ultimate temperature as it has been recently decided in an EC3 meeting in Dublin (the document has not been published and cannot be referenced), then the average value of the ratio between the analytical and the experimental load is equal to 0.93. Figure 12 illustrates that this proposal tends to be excessively safe. Due to the fact that the relative slenderness is evaluated at the ultimate temperature, the safety level is more uniform, slightly increasing for increasing slenderness.

194

J.-M. Franssen et al.

1.75

.....t o

d. 1.50



K

~ 1.25

..............

| Unear r~msion r ........ T o

<>

E C 3 P a r t 1.2. <;-+-

...... +_+.

.............................................................

......................

~ .....

~ ..................................

~1.~ 0.75 S l e n d e r n e s s at

0.50 0.0

I

I

I

0.5

1.0

1.5

20* 2.0

Fig. 11. EC3 proposal.

1.75

[XJblin

J

1.5o ...... *,,,, U , ~ - , ~ o n ~""

~ =

+



r ....................................................................

~ ............

1.25 ...............................



O

~

80

1.00 .............. - ~ - - + - g --V'--x-n--.=+~O~<> .........}- .......
o.75 . . . . . . . . . . . . .

o.5o 0.0

+*-0-..~

.............

- . - ~ - . . . . . . . . . . . . . . . . . . . . + ........ - ~ - - . + . . . . . . . . . . . . . . . . . . .

.

,.

o.5

1.o



-I

sp.~m~azr~ 1.s

zo

Fig. 12. Dublin proposal. The values 1.06 and 0.93 indicate that those analytical methods had also been calibrated in such a way that the calculated loads are, more or less, equal to the experimental loads. The value of/3 = 0.65 has been used to draw the analytical line on Figs 3 8. It has to be mentioned that the analytical curve depends on the yield strength of steel (see eqn 5). In each of those figures, a choice had to be made and the mean value of the yield strength for the tests highlighted on the figure was considered. Figure 13 summarises all the test results as well as the analytical proposal, drawn with the characteristic values of the severity factor,/3 = 1.20, as well as with the value for practical applications, /3 = 0.65. The curves have been drawn with a yield strength of 281 MPa, average value of all the experimental tests.

Fire resistance of axially loaded members

195

1.2 o* .

1.0 0.8

Test= I beta = 0.651

. .

"~

**

° * ".¢,

...... bea=l.2O I

.

8 0.6

---,.,~..~.

.

0.4 0.2 Relative slenderness at T u

0.0

I

I

I

0.5

1

1.5

2

Fig. 13. All the test results.

It comes as no surprise on Fig. 14 that the average value of the ratio between the analytical and the experimental load is equal to 1.00 when the analytical load is calculated according to the present proposal and with/3 = 0.65. More significant is the fact that the analytical expression deduced from the extensive numerical simulations '7 leads to a safety level that is totally independent of the slenderness.

1.75

o

I °~'

~ 1 . ~ ......l ~ U n ~ r r ~ r ~ = l

1.25

I..........................................................................

...................f-..~-..;.;;~. ............. tI~.

~1.00

.0

--= 0,~

..........

6

~w

O~

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

,,~,

=.

i:'_~rt 8 _ ~ [ _ _ ; !

0.0

~,

.......~ ......... _~ ....... *_............... o

0.50

~

i

i

0.5

1.0

o

Fig. 14. The present proposal.

S I ~ i 1.5

at 20"C 2.0

196

J.-M. Franssen et al.

5 OVERALL BUCKLING COEFFICIENT As already stated in Ref. 17, the ultimate load o f a steel c o l u m n at e l e v a t e d t e m p e r a t u r e can b e directly related to its plastic load at r o o m t e m p e r a t u r e b y the overall b u c k l i n g coefficient, taking into a c c o u n t the d e c r e a s e o f the material strength as well as the instability. E q u a t i o n 1 can be rewritten in the form:

Nu(

=

(8)

w h e r e ~ ( T ) = x(T)Km~x(T) is the overall b u c k l i n g coefficient, with x(T) d e t e r m i n e d b y eqns 2 - 5 , Kmax(T)=fy(T)/fy t a k e n f r o m Fig. 1. T a b l e 3 gives the overall b u c k l i n g coefficient for $235 steel and T a b l e 4 can b e used for $355 steel.

TABLE 3

Overall Buckling Coefficient for $235 Steel Relative slenderness at 20°C

400°C

500°C

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90 2.00

1.000 0.927 0.860 0.794 0.728 0.663 0.598 0.535 0.476 0.422 0.374 0.332 0.296 0.264 0.237 0.213 0.192 0.175 0.159 0.145 0.133

0.780 0.726 0.675 0.626 0.578 0.529 0.480 0.433 0.388 0.346 0.309 0.275 0.246 0.220 0.198 0.179 0.162 0.147 0.1344 0.123 0.113

Ultimate temperature 600°C 700°C

0.470 0.435 0.402 0.370 0.339 0.307 0.275 0.245 0.217 0.192 0.170 0.150 0.133 0.119 0.106 0.096 0.086 0.078 0.071 0.065 0.060

0.230 0.211 0.194 0.178 0.161 0.144 0.128 0.112 0.098 0.086 0.075 0.066 0.058 0.052 0.046 0.041 0.037 0.034 0.031 0.028 0.025

800°C

900°C

0.110 0.103 0.096 0.089 0.082 0.076 0.069 0.062 0.056 0.050 0.045 0.040 0.036 0.032 0.029 0.026 0.024 0.022 0.020 0.018 0.017

0.060 0.057 0.053 0.050 0.047 0.044 0.041 0.038 0.035 0.032 0.029 0.026 0.024 0.022 0.020 0.018 0.017 0.015 0.014 0.013 0.012

Fire resistance of axially loaded members

197

6 DESIGN EXAMPLES The proposed examples apply to the following element: • Section HE220 A • Radius of gyration iz = 5.51 cm • Sectional area g] = 64.34 cm 2 • Buckling axis minor • Length L = 3.30 m • Yield strength f y = 355 M P a Preliminary calculations are;

e. =

A(20oc) -

//235 = 0.814, [see eqn (5)] A AE(20°C)

L/i z

3300/55.1

- 93.91e - 93.91 x 0.814

= 0.784.

Example 1 Nd = 150 kN, determine the ultimate temperature.

Method 1: overall buckling coefficient (Table 4) Nu(T) Na 150,000 to = Np, - g2fy - 6434 x 355 = 0.066 [see eqn (8)]. The ultimate temperature has to be found by interpolations in Table 4. At 700°C:

At 800°C:

for for for for for for

A(20 °) A(20 °) A(20 °) h(20 °) A(20 °) A(20 °)

= 0.70, to= 0.121 = 0.80, to = 0.106 -- 0.784, qJ = 0.108 = 0.70, to = 0.067 = 0.80, tO = 0.060 = 0.784, qJ = 0.061

for A(20 °) = 0.784 and tO = 0.066,

T = 790°C.

198

J.-M. Franssen et al. TABLE 4 Overall Buckling Coefficient for $35 Steel

Relative slenderness at 20°C

400°C

500°C

Ultimate temperature

600°C

700°C

800°C

900°C

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90 2.00

1.000 0.940 0.882 0.825 0.765 0.703 0.639 0.575 0.512 0.454 0.402 0.356 0.316 0.281 0.251 0.225 0.203 0.184 0.167 0.152 0.139

0.780 0.735 0.692 0.650 0.606 0.560 0.512 0.464 0.417 0.373 0.332 0.295 0.263 0.235 0.211 0.189 0.171 0.155 0.141 0.129 0.118

0.470 0.441 0.413 0.385 0.356 0.326 0.295 0.264 0.234 0.207 O.182 0.161 0.142 0.126 0.113 0.101 0.091 0.082 0.075 0.068 0.062

0.230 0.215 0.200 0.185 0.170 0.153 O.137 0.121 0.106 0.092 0.081 0.071 0.062 0.055 0.049 0.044 0.039 0.035 0.032 0.029 0.027

O.110 0.104 0.098 0.092 0.086 0.080 0.073 0.067 0.060 0.054 0.048 0.043 0.039 0.035 0.031 0.028 0.025 0.023 0.021 0.019 0.018

0.060 0.057 0.054 0.052 0.049 0.046 0.043 0.040 0.037 0.034 0.031 0.028 0.026 0.023 0.021 0.019 0.018 0.016 0.015 0.014 0.012

Method 2: analytical expression Nu(T) = xOfy(T) = xl2Kma~(T)fy [see eqn (1)]

is written in the form N~(T) Kmax(T) - X~,-]fy.

First calculation with the initial assumption; A(T) -- 1.20 A(20°C) -- 1.20 × 0.784 = 0.941. The calculation is then made as shown hereafter.

(9)

Fire resistance of axially loaded members

'(

t0(T) = ~

199

)

1 + aX(T) + X2(T) [see eqn (4)1

1 to(T) = ~ (1 + 0.65 x 0.814 x 0.941 + 0.9412) = 1.191

X(T) =

1

[see eqn (2)]

¢(T) + x/ 2(T) x(T)=

1

= 0.520

1.191 + ~/1.1912- 0.9412 Kmax(T) = Nu(T) X---~ [see eqn (9)] 150,000 Kmax(T) = 0.520 X 6434 X 355 = 0.126. From the values of Fig. 1, Km~x= 0.126 gives T = 786°C. For this temperature, Fig. 1 yields KE = 0.095. The relative slenderness at 786°C is then calculated, according to eqn (3), as;

A(796°C) = [ " " " " 0.794 = 0.902. ~/0.095 As the first calculation has been made with the assumption of )t(T)= 0.941, an iteration will be made. Iteration 1:

to(T) = ~1 (1 + 0.65 x 0.814 x 0.902 + 0.9022) = 1.145 x(T) =

1

= 0.540

1.145 + x/1.1452 - 0.9022 150 000 /(max(/') = 0.540 × 6434 × 355 = 0.122.

From the values of Fig. 1, Kmax= 0.122 gives T = 790°C. For this temperature, Fig. 1 yields KE = 0.094.

J.-M. Franssen et al.

200

The relative slenderness at 790°C is then calculated, according to eqn (3), as:

~(790°C) = ~ 0.784 = 0.892. ¥ 0.094

Iteration 2:

1

~o(T) = ~ (1 + 0.65 x 0.814 × 0.892 + 0.8922) = 1.134 x(T) =

Kmax(T)

-

1

, = 0.545 1.134 + ~/1.1342 - 0.8922 150 000 = 0.120 0.545 x 6434 x 355

From the values of Fig. 1, Km~x= 0.120 gives T = 791°C. For this temperature, Fig. 1 yields KE = 0.093. The relative slenderness at 791°C is then calculated, according to eqn (3), as:

~(791°C) = ~

0.784 = 0.890.

This value of 791°C is the converged solution of the analytical equations. The utilisation of the overall buckling coefficient as well as the analytical calculation limited to one iteration provide answers with a precision of 1°C.

Example 2 Tu = 653°C, determine the ultimate load. N.B.: this temperature exists in the unprotected HE 220 A section after an exposure of 15 minutes to the ISO fire.

Method 1: overall buckling coefficient (Table 4) The ultimate temperature has to be found by interpolations in Table 4. At 600°C: for A(20 °) = 0.70, qJ = 0.264 for A(20 °) = 0.80, q~= 0.234 for A(20 °) = 0.784, ~0= 0.239 At 700°C: for A(20 °) = 0.70, qJ= 0.121

Fire resistanceof axially loadedmembers

201

for )~(20 °) = 0.80, ~b= 0.106 for )t(20 °) = 0.784, ~ = 0.108 =*

for )t(20 °) = 0.784 and T = 653°C,

• = 0.170.

Nu(T) = ~T)Np~ = 0.170 x 6434 x 355 = 387 579 N [see eqn (8)]

Method 2: analytical expression From the values of Fig. 1, T = 653°C gives Kmax = 0.343 and KE = 0.215. ~(20oc ) [see eqn (3)] ~(653°C) = /Kmax(T) ~/

0 " ~ 4 3 0.784 = 1.264 x 0.784 = 0.991

--V0 i

q~(T) = ~(1 + a ~ ( T ) + ~2(T))[see eqn (4)] q~(T) = ~(1 + 0.65 x 0.814 x 0.991 + 0.9912) = 1.253 X(i0 = x(T) =

1 q (T) +

- X2(T)

[see eqn (2)]

1

1.253 + ql.2532 - 0.9912

Nu(T) = xOfy(T)

=

= 0.495

Xg-]gmax(T)fy [see

eqn (1)]

Nu(653°C) = 0.495 x 6434 x 0.343 x 355 = 387736 N. No iteration is required in this case where the ultimate temperature belongs to the data. The utilisation of the overall buckling coefficient gives virtually the same ultimate load as the use of the analytical formula.

7 CONCLUSIONS The buckling of steel columns at elevated temperatures has been the subject of quite a lot of investigations and publications in the last decades. The most distinctive aspects of the work that has been reported here and in Ref. 17 are: • The numerical study has considered virtually every possible combination

202

J.-M. Franssen et al.

of all the parameters such as the profile section, the yield strength, the buckling axis, the thermal distribution, the slenderness and the load level. • The numerical study was based on non-linear stress-strain relationships, whereas most of the previous work on the subject stuck to bi-linear behaviour. The influence of the residual stresses has been considered on the base of principles of structural mechanics. 37 • The calibration of the proposed analytical solution was founded on a significant number of carefully scrutinized experimental results from six different laboratories. Of those tests, 80% come from sources that are independent from the authors. A simple analytical proposal has been made, similar in its form to what exists for room temperature in the Eurocode 3 Part 1.1.14 This ultimate load of columns at elevated temperatures can be determined with the help of pocket calculators, or directly related to the plastic load at 20°C by tables giving the overall buckling coefficient. The present study is limited to the basic case of simply supported, axially loaded, symetrically heated H columns. It was the opinion of the authors that this basic case had to be thoroughly investigated before more complicated effects could be envisaged. The study on the interaction between axial load and bending moments or thermal gradients is presently under way within the same ECCS research program, involving the TNO and the CTICM.

ACKNOWLEDGEMENTS This work was sponsored by the European Convention for Coal and Steel. 38 This research project involved Mr L. Twilt from TNO in the Netherlands and Dr J. Kruppa and D. Talamona from CTICM in France who are investigating the interaction between axial loads and bending. The authors acknowledge their contribution to the discussions on the subject reported in this paper. The Spanish activity of the project was also partially financed by project MAT931139-CE of the Interministerial Commission of Science and Technology CICYT of Spain.

REFERENCES 1. Ossenbruggen, P. J., Aggarwal, V. & Culver, C. G., Steel column failure under thermal gradients. ASCE, J. Struct. Division, 99(ST4) (1973) 727-739. 2. Aribert, J. -M. & Abdel Aziz, M., Simulation du comportement h rincendie de poteaux comprimrs et flrchis en prrsence de gradients quelconques de temprrature. CTICM, Construction M~tallique, 2 (1987).

Fire resistance of axially loaded members

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