Local and distortional buckling of thin-walled short columns

Thin-Walled Structures 34 (1999) 115–134 www.elsevier.com/locate/tws

Local and distortional buckling of thin-walled short columns Jyrki Kestia, J. Michael Daviesb,* a

Helsinki University of Technology, P.O. Box 2100, FIN-02015-HUT, Finland Manchester School of Engineering, University of Manchester, Manchester, M13 9PL, UK

b

Abstract This paper assesses the applicability of Eurocode 3 (EC3) to the prediction of the compression capacity of short fixed-ended columns with different cross-sections. This compression capacity is determined by combining the effective width of plane elements due to local buckling and the effective stiffener thickness due to distortional buckling. Numerical calculations have been carried out in order to compare alternative methods for determining the minimum elastic distortional buckling stress in compression. The method given in EC3 does not correlate as well as Lau and Hancock’s method with the results given by Generalized Beam Theory (GBT). The end boundary conditions have a significant influence on the distortional buckling strength, and thus also on the compression capacity of short columns. Selected experimental results from compression tests on C-, Hat- and rack upright-sections are compared with the predictions given by EC3. The procedure in EC3 was modified by determining the distortional buckling stress using GBT, taking into account the actual column length and the end boundary conditions. This lead to better agreement between the experimental results and the theoretical predictions.  1999 Elsevier Science Ltd. All rights reserved. Keywords: Local buckling; Distortional buckling; Column; Thin-walled structures; Cold-formed steel

1. Introduction The compression capacity of short open thin-walled columns of open cross-section is usually determined by local and distortional buckling. For example, the failure of * Corresponding author. Tel: ⫹ 44-(0)161-275-4434; fax: ⫹ 44-(0)161-275-4361; e-mail: jmdavies@ fst.eng.man.ac.uk 0263-8231/99/$ - see front matter  1999 Elsevier Science Ltd. All rights reserved. PII: S 0 2 6 3 - 8 2 3 1 ( 9 9 ) 0 0 0 0 3 - 8

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a short, concentrically compressed C-section is almost always the result of a combination of local bucklings of the thin plate elements and buckling of the edge stiffeners (distortional buckling). In the case of longer columns, the failure is often observed as a combination of global buckling and local or distortional buckling. In the codes and standards, such as in Eurocode 3, Part 1.3 [1] (EC3), the American AISI Specification [2] or the Standard of Australian and New Zealand AS/NZS 4600 [3], local buckling is taken into account by using effective widths for plane elements. In the AISI Specification and the AS/NZS 4600, the inability of an edge stiffener to prevent distortional buckling is taken into acccount by reducing the local buckling coefficient for the plate element supported by the stiffener to a value below the basic value of 4.0. This reduced buckling coefficient is then included in the “Winter” effective width formula that is used to compute the strength of the plate element. If the stiffness of the stiffener is adequate to prevent its deformation in a plane normal to the plane of the element, the adjacent element is assumed to be stiffened. This design method does not account for the restraint to distortional buckling provided by the web section. Furthermore, the design against a pure distortional mode of buckling is provided for in AS/NZS 4600, which includes a calculation method for the elastic distortional buckling stress and design curves. There is no similar provision in the AISI Specification. In EC3, the distortional mode of buckling is taken into account by assuming that both edge or intermediate stiffeners behave as a compressed strut on an elastic foundation. The elastic foundation is represented by a spring whose stiffness depends upon the bending stiffness of the adjacent parts of the plane elements of the crosssection and on the boundary conditions of the element. The preferred wave length for buckling is assumed to be free to develop. For the purposes of design, this buckling calculation leads to a reduced thickness of the stiffeners. EC3 includes several alternative “column curves” and the design strength is based on curve a0 with ␣ ⫽ 0.13. One purpose of this paper is to compare different methods for determining the minimum elastic distortional buckling stress in compression. The methods considered are those given in AS/NZS 4600 or EC3, and those found by using the Generalized Beam Theory (GBT). The influence of errors in the elastic buckling stress on the effective cross-section area according to EC3 is also studied. The effective area of a thin-walled column can be determined by a stub column test. However, the restricted end boundary conditions significantly restrict distortional buckling in stub column tests, and this leads to high distortional buckling stresses in comparison to those observed in tests on longer columns where multiple distortional buckling half-waves may occur. Thus, the stub column tests may give values for the effective area which are too optimistic. Selected experimental results for short fixed-ended columns with different crosssections were compared with the predictions given by EC3. Two different design methods were used for C-, Hat- and rack-sections. In the first method, the compression capacities were determined according to EC3. In the second method, the procedure in EC3 was modified by determining the distortional buckling stress using GBT taking into account the actual column length and the end boundary conditions.

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Another purpose of this paper is thus to demonstrate the importance of the support conditions on the strength of short columns.

2. Elastic distortional buckling stress 2.1. Analytical and numerical methods Recently, a number of methods have been developed for determining the elastic distortional buckling stress of singly-symmetric cross-sections. Two analytical methods have been presented, namely the EC3 method, which is based on flexural buckling of the stiffener, and the model developed by Lau and Hancock [4], which is based on the flexural-torsional buckling of a simple flange including a stiffener. The latter method is used in AS/NZS 4600. The Generalized Beam Theory (GBT) provides a particularly good tool with which to analyze distortional buckling in isolation and in combination with other modes. The finite strip method has also proved to be a useful approach, because, like GBT, it also has a short solution time compared to the finite element method. The finite strip method assumes simply supported end boundary conditions and is applicable for longer sections where multiple half-waves occur in the section length. In this paper, the first three methods mentioned are briefly described and a numerical comparison between the different methods is carried out. 2.1.1. The method in Eurocode 3: part 1.3 (EC3) In EC3, the design of compression elements with either edge or intermediate stiffeners is based on the assumption that the stiffener behaves as a compression member with continuous partial restraint. This restraint has a spring stiffness that depends on the boundary conditions and the flexural stiffness of the adjacent plane elements of the cross-section. The spring stiffness of the stiffener may be determined by applying a unit load per unit length to the cross-section at the location of the stiffener, as illustrated in Fig. 1. In this figure, the rotational spring stiffness C␪ characterizes the bending stiffness of the web part of the section. The spring stiffness K per unit length may be determined from:

Fig. 1.

Determination of the spring stiffness K according to Eurocode 3.

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K ⫽ u/␦

(1)

where ␦ is the deflection of the stiffener due to the unit load u. The elastic critical buckling stress for a long strut on an elastic foundation, in which the preferred wave length is free to develop, is given by Timoshenko and Gere [5]:

␴cr ⫽

␲2EIs 1 ⫹ K␭ 2 As␭2 As␲2

(2)

where As and Is are the effective cross-sectional area and the second moment of area of the stiffener according to EC3, as illustrated in Fig. 2 for an edge stiffener. ␭ ⫽ L/m is the half-wavelength. m is the number of half-wavelengths. The preferred half-wavelength of buckling for a long strut can be derived from Eq. (2) by minimizing the critical stress:

␭cr ⫽

冪K

4

EIs

(3)

For an infinitely long strut, the critical buckling stress can be derived, after substitution, as:

␴cr ⫽

2√KEIs As

(4)

Eq. (4) is given in EC3, thus the EC3 method does not consider the length of the column but assumes that it is sufficiently long for integer half-waves to occur in the section length. 2.1.2. AS/NZS 4600 method The elastic distortional buckling stress is based on the flexural-torsional buckling of a simple flange, as shown in Fig. 3. The rotational spring k␾, represents the flexural restraint provided by the web which is in pure compression, and the translational spring kx, represents the resistance to translational movement of the section in the

Fig. 2.

Effective cross-sectional area of an edge stiffener.

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Fig. 3.

119

Lau and Hancock’s model for distortional buckling.

buckling mode. The model includes a reduction in the flexural restraint provided by the web as a result of the compressive stress in the web. In Lau and Hancock’s analysis [4], it is shown that the translational spring stiffness kx does not have much significance and it is assumed to be zero. The rotational spring stiffness can be expressed as: k␾ ⫽

冋

冉

b2w␭ Et3 1.1f ⬘od 1⫺ 5.46(bw ⫹ 0.06␭) Et3 b2w ⫹ ␭2

冊册 2

(5)

where f ⬘od is the compressive stress in the web at distortional buckling, computed assuming k␾ is zero. bw is the web depth, t is the thickness of the section, E is Young’s modulus and ␭ is the half-wavelength in buckling and is expressed, for a simple C-section, as:

冉 冊

␭ ⫽ 4.80

Eb2fbw t3

0.25

(6)

where bf is the flange width. The elastic distortional buckling stress then has the form: f od ⫽

冋

E (␣1 ⫹ ␣2) ⫺ √(␣1 ⫹ ␣2)2 ⫺ 4␣3 2A

册

(7)

where A is the cross-sectional area of the flange and stiffener and ␣1, ␣2 and ␣3 are characteristic values of some complexity which are given in Appendices D1 and D2 of AS/NZS 4600 and which are related to the k␾, ␭ and the geometry and dimensions of the flange and the lip. The computation process is iterative due to the incorporation of f⬘od in k␾, but only one iteration is required. This type of model proves to be sensitive to the value assumed for the rotational spring stiffness k␾. Davies and Jiang [6] proposed an improvement to the above method if the rotational spring stiffness k␾ is negative, i.e. the web buckles earlier than the flange. In this case the buckling stress can be obtained with k␾ as zero, whereas the buckling stress of the web plate is (Timoshenko and Gere [5]):

␴w ⫽

冉

␲2D b2w ⫹ ␭2 tb4w ␭

冊

2

(8)

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The final distortional buckling stress can be calculated approximately as the mean value of the buckling stresses of the web and flange:

␴cr ⫽

2fodAf ⫹ ␴wtbw A

(9)

where Af is area of the flange and stiffener and A is the area of the whole cross-section. 2.1.3. Generalized beam theory (GBT) The Generalized Beam Theory has been presented in more detail by, e.g. Davies and Leach [7,8], and only a short description of the solution is presented in this paper. A unique feature is that GBT can separate and combine individual buckling modes and their associated load components. In GBT, each mode has an equation and, in second-order format, ignoring the shear deformation terms, the equation for mode “k” is:

冘冘 n

EkCkV⬙⬙ ⫺ GkDkV⬙ ⫹ kBkV ⫹

n

␬(iWj V⬘)⬘ ⫽ kq for k ⫽ 1,2,...n

ijk

(10)

i⫽1 j⫽1

where the left superscript k denotes the mode k, and kC is the generalized warping constant, and kD is the generalized torsional constant and kB is the transverse bending stiffness. These are the generalized section properties which depend only on the cross-section geometry. In addition, ijk␬ are the second-order section properties which relate the cross-section deformations to the stress distributions, and E and G are the modulus of elasticity and shear modulus, respectively. kV and kW are the deformation resultant and stress resultant, kq is the uniformly distributed load and n is the number of modes in the analysis. The section properties and the ijk␬ values may be calculated manually, but in general, this task is best carried out by computer. If the right-hand side terms kq of Eq. (10) are zero, the solution gives the critical stress resultant iW. In general this requires the solution of an eigenvalue problem in which the analyst is free to choose which modes to include in his analysis. When a constant stress resultant is applied along the member, which is assumed to buckle in a half sine wave of wavelength ␭, GBT allows some particularly simple results to be obtained. Thus, the critical stress resultant for single-mode buckling is (Davies and Leach [8]): i,k

Wcr ⫽

冉

冊

␭2 1 ␲2 k E C ⫹ GkD ⫹ 2 kB 2 ␬ ␭ ␲

ikk

(11)

As the wavelength is varied, the minimum critical stress resultant is: i,k

Wcr ⫽

冉

1 2√EkCkB ⫹ GkD ␬

ikk

冊

(12)

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and the corresponding half-wavelength is k

冉 冊

␭⫽␲

EkC k B

0.25

(13)

This approach allows some particularly simple solutions to be obtained for distortional buckling problems. 2.2. Numerical comparison Numerical calculations have been carried out for a variety of C-section sizes under concentric compression in order to compare the minimum elastic distortional buckling values determined using the different methods discussed above. The dimensions of the C-sections are given in Table 1 for web height h, flange width b, stiffener width c and thickness t. A value of E ⫽ 210 000 N/mm2 was used for Young’s modulus. The GBT results were obtained using a computer program written by Davies and Jiang [9]. In these GBT analyses, the pin-ended conditions with a half sine wave shape function were used for distortional buckling. In all cases, the critical

Table 1 Comparison of elastic distortional stresses Section

t ⫽ 1.5 mm

h-b-c

AS

EC3

GBT

AS

EC3

GBT

AS/GBT EC3/GBT AS/GBT EC3/GBT

200-75-20 200-75-15 200-50-20 200-50-15 200-50-10 150-75-20 150-75-15 150-50-20 150-50-15 150-50-10 100-100-30 100-100-20 100-100-15 100-50-20 100-50-15 100-50-10 100-30-15 100-30-10

165 129 167* 135* 101* 217 176 295 243 173 234 182 146 420 372 287 503 401

179 129 331 251 163 203 144 373 283 184 209 151 103 438 332 216 725 501

168 136 179 149 113 225 183 290 247 189 258 193 152 441 383 296 493 417

230 183 236* 195* 153* 303 248 411 343 253 325 254 205 584 523 411 707 583

234 172 441 335 218 262 192 498 377 246 289 188 131 583 443 288 967 668

234 192 251 214 167 312 257 404 349 276 351 265 210 609 535 423 699 607 Mean St.dev

0.98 0.95 0.94 0.91 0.90 0.96 0.96 1.02 0.98 0.91 0.91 0.94 0.96 0.95 0.97 0.97 1.02 0.96 0.96 0.04

*

t ⫽ 2.0 mm

t ⫽ 1.5 mm

1.07 0.95 1.85 1.68 1.45 0.9 0.79 1.29 1.15 0.97 0.81 0.78 0.68 0.99 0.87 0.73 1.47 1.20 1.09 0.34

t ⫽ 2.0 mm

0.98 0.95 0.94 0.91 0.91 0.97 0.96 1.02 0.98 0.92 0.93 0.96 0.98 0.96 0.98 0.97 1.01 0.96 0.96 0.03

1.00 0.90 1.75 1.57 1.30 0.84 0.75 1.23 1.08 0.89 0.82 0.71 0.62 0.96 0.83 0.68 1.38 1.10 1.02 0.32

Values have been calculated according to proposed method by Davies and Jiang [9] when k␾ is negative.

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distortional buckling half-wave length was first determined, thus leading to the minimum distortional buckling stress. The AS/NZS-methods gives on average, 4% lower values of buckling stress than the GBT for both cases, t ⫽ 1.5 mm and t ⫽ 2.0 mm. All of the values are within 10% of the GBT-values. When compared with the GBT results, the EC3-methods give 9% higher values for t ⫽ 1.5 mm and 2% lower values for t ⫽ 2.0 mm. The variation in the EC3-method is, however, rather large. If the web buckles earlier than the flange (marked by * in Table 1), the EC3-method seems to give very high values of buckling stress compared to GBT. This is because the EC3-model does not include a reduction in the flexural restraint provided by the buckled web. In the case of wide flanges or short stiffeners, the EC3-methods gives rather low values. It should be noted that sections with b ⫽ 100 mm and t ⫽ 1.5 mm do not satisfy the b/t ⬍ 50 limit given in EC3 and the section with h ⫽ 100 mm, b ⫽ 100 mm and c ⫽ 15 mm does not satisfy the c/b > 0.2.

3. Effective cross-section area according to EC3

According to EC3, local buckling is taken into account by using effective widths for plane elements, whereas distortional buckling is considered by using a reduced thickness of the stiffener (see Fig. 2). This thickness reduction is based upon the design curve a0 with ␣ ⫽ 0.13. The effective area of a column with an edge or intermediate stiffener is always determined by checking both the local buckling and the distortional buckling. If the modified slenderness, ␭, of a plane element in local buckling is less than 0.673, the plane element width is fully effective. In distortional buckling, no lower limit has been given for the modified slenderness. The above comparison showed that the EC3-method sometimes gives elastic distortional buckling stresses which differ considerably from the values determined using GBT. The effect of the distortional buckling stress on the effective crosssection area is studied in Table 2. Columns 2 and 3 of this table give the ratios between elastic distortional buckling stresses determined according to EC3 and using GBT. The effective areas, determined according to EC3, are given in columns 4 and 6. Columns 5 and 7 give the values determined according to EC3, but using elastic distortional buckling stress given by GBT. The ratios between the values achieved using these two different methods are shown in the last two columns. As can be seen from the results, the significant errors in the elastic distortional buckling stress cause rather small errors in the effective area of the sections studied. For example, for section 200-50-20, an 85% error in the distortional buckling stress causes only a 14% error in the effective area. Of course, the influence is dependent on the distortional buckling stress level.

200-75-20 200-75-15 200-50-20 200-50-15 200-50-10 150-75-20 150-75-15 150-50-20 150-50-15 150-50-10 100-100-30 100-100-20 100-100-15 100-50-20 100-50-15 100-50-10 100-30-15 100-30-10 Mean St.dev

1.07 0.95 1.85 1.68 1.45 0.90 0.79 1.29 1.15 0.97 0.81 0.78 0.68 0.99 0.87 0.73 1.47 1.20 1.09 0.34

1.00 0.90 1.75 1.57 1.30 0.84 0.75 1.23 1.08 0.89 0.82 0.71 0.62 0.96 0.83 0.68 1.38 1.10 1.02 0.32

EC3/GBT

EC3/GBT 255 233 263 240 213 260 235 267 243 215 269 249 227 268 246 216 215 196

EC3[mm2]

t ⫽ 1.5mm

t ⫽ 2.0 mm

t ⫽ 1.5 mm

h-b-c

252 235 230 215 200 265 247 254 236 216 286 262 244 270 252 230 208 193

431 393 411 381 341 435 396 411 381 342 476 422 384 404 377 338 319 297

EC3-m[mm2] EC3[mm2]

t ⫽ 2.0 mm

Effective area (fy ⫽ 350N/mm2)

Difference in elastic dist. buckling stress

Section

Table 2 Comparison of effective cross-section areas

431 402 373 350 326 453 422 400 377 349 497 453 422 405 384 359 315 295 Mean St.dev

EC3-m[mm2] 1.01 0.99 1.14 1.12 1.06 0.98 0.95 1.05 1.03 0.99 0.94 0.95 0.93 0.99 0.98 0.94 1.03 1.02 1.01 0.06

EC3/EC3-m

t ⫽ 1.5 mm

1.00 0.98 1.10 1.09 1.05 0.96 0.94 1.03 1.01 0.98 0.96 0.93 0.91 1.00 0.98 0.94 1.01 1.00 0.99 0.05

EC3/EC3-m

t ⫽ 1.5 mm

Difference in effective area J. Kesti, J.M. Davies / Thin-Walled Structures 34 (1999) 115–134 123

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4. Short columns and effective area 4.1. Conditions for stub column tests The effective area of a section can be determined by means of a stub column test. According to the AISI specification, the stub column tests should be conducted between rigid end plattens and the length of the specimen should be at least three times the widest plate element but not more than twenty times the minor radius of gyration. According to EC3, the tests should also be conducted between rigid plattens, but the testing machine should be equipped with ball bearings at each end. The length of the specimen cannot be less than the expected buckling length of the stiffeners. If the overall length of the specimen exceeds 20 times the least radius of gyration, imin, intermediate lateral restraint should be supplied at spacing of not more than 20 imin. With both methods, warping is prevented in the ends of the specimen and the end boundary conditions can be assumed to be rigid with respect to distortional buckling. 4.2. Selected experimental research Selected experimental results on columns with different cross-sections have been compared with the predictions given by EC3 and modified EC3. In all of the column tests, warping was prevented by the welded or pattern stone-filled end-plates or flat plattens. The selected results are mainly for short columns, with L/imin ⬍ 20. Some longer column results were also selected, but for cases where the influence of global buckling is not very significant. The test results were collected from the published literature. Results given by the following researchers were used: —Lau and Hancock: C-, Hat- and rack-sections [10] CH-, HA-, RA- and RL-17,20,24 sections —Young and Rasmussen: C-sections [11] L36 and L48 sections —Mulligan and Pekoz: C-sections [12] SLC sections —Weng and Pekoz: C-sections [13] RFC, PBC and P-sections —Zaras and Rhodes: C- and Hat-sections [14] LC and HAT-sections —Lim: Hat-sections [14] A-, B- and C-sections —Kwon and Hancock: C- and web-stiffened C-sections [15] CH1 and CH2 sections The dimensions, measured tensile yield stress, Young’s modulus and ultimate load, are given in Appendix A. All of the dimensions have been converted to mid-line

J. Kesti, J.M. Davies / Thin-Walled Structures 34 (1999) 115–134

Fig. 4.

125

Notations for cross-sections.

and the radius is the inside corner radius. These dimensions are in mm and are explained in Fig. 4. fy and E are in N/mm2 and the failure load Fu is in kN. 4.3. Influence of end boundary conditions and specimen length on the effective area The influence of the end boundary conditions on the elastic buckling stress for section CH17 is shown in Fig. 5. Four curves were calculated using GBT: “Fixsingle” is the pure distortional buckling mode for the column with fixed ends. “Fixcoupled” considers the interaction of all the buckling modes of the fixed-ended column. Pin-curves are correspondingly for the pin-ended columns. Local buckling modes were ignored in this analysis. The AUS-line and EC3-lines give the distortional buckling stress according to AS/NZS and EC3. It can be seen that the influence of the end boundary conditions is considerable for short columns. If the stub column length is chosen, according to EC3, to be as long as the critical buckling length of the stiffener ( 苲 500 mm), the elastic distortional buckling stress of the fixed-ended

Fig. 5.

Elastic buckling stresses for the section CH17.

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column is still about 2.3 times that of the pin-ended column. For shorter columns the influence is even greater. The distortional buckling stress of the fixed-ended column reaches that of the pin-ended column only with multiple distortional buckling half-waves. Fig. 6 illustrates the influence of the specimen length on the effective cross-section area. The effective area is determined according to EC3, but using the distortional buckling stress given by the GBT- results using fixed-ended conditions. The curves in Fig. 6 show the ratio between these values and the values determined according to the original EC3 approach. The difference between the effective area of short column and long columns is in the range 5–20% for the sections studied. It can also be seen that, for longer columns when the effect of the end boundary conditions is insignificant, the original EC3 methods usually gives lower effective area values than the modified method. This was expected, because most of the sections studied had quite a wide flange which caused an underestimation of the distortional buckling stress, as can be seen in Table 1. 4.4. Comparison of calculated and test results The compression resistance for all the selected C- and Hat-sections was determined using both of the above-mentioned methods, i.e., according to the original EC3 method and modified EC3-method using distortional buckling stresses determined by GBT taking into account the actual column length and the fixed-ended boundary conditions. In both cases, the local buckling was taken into account using effective widths for plane elements using buckling factors given in EC3. The partial factor value ␥M1 ⫽ 1.0 was used in all of the comparisons. Most of the columns were so short that global buckling did not occur, but in all cases global buckling was also

Fig. 6.

The effect of column length on effective area for fixed-ended columns.

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Fig. 7.

127

Comparison of test results and theoretical values for C-sections.

checked according to EC3. The results are compared in Fig. 7 for the C-sections and in Fig. 8 for the Hat-sections. The abscissa indicates the slenderness of the section in distortional buckling, fy indicates the measured yield stress and ␴cr is the elastic distortional buckling stress determined using GBT. As shown in Figs. 7 and 8, the EC3 method gives conservative values for the compression resistance of fixed-ended C- and Hat-sections. This was expected, because the EC3 model for distortional buckling assumes free buckling of the stiffener. The modified method predicts the ultimate resistance rather well. The mean

Fig. 8.

Comparison of test results and theoretical values for Hat-sections.

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Fig. 9.

Comparison of test results and theoretical values (EC3) for RA, RL and CH2-sections.

resistance ratio Ntest/Ncalc for C-sections is 1.03 and for Hat-sections 1.01. In both cases the standard deviation is 0.06. The C-sections with the most unconservative values have a high slenderness value. In this case, the stiffener widths are very small compared to flange width (c/b ⫽ 0.05–0.07). These CH1-sections belong to the Kwon and Hancock [15] test series. The results for CH2-, RA- and RL-sections are shown in Fig. 9 where the theoretical values are determined according to EC3, and in Fig. 10 where the modified method has been used. In the case of the web-stiffened sections CH2, the web stiff-

Fig. 10.

Comparison of test results and theoretical values (modified EC3) for RA, RL and CH2-sections.

J. Kesti, J.M. Davies / Thin-Walled Structures 34 (1999) 115–134

Fig. 11.

129

Comparison of test results and theoretical values for all sections.

ener thickness was reduced according to the original EC3 in both cases. It should be noted that the EC3 method is not valid for RL-sections with complicated stiffeners, but the same procedure was used in this research. The predicted resistances according to the modified method provides quite a good correlation with the test results. The comparison of test results and predicted values for all of the test specimens using the modified EC3-method is shown in Fig. 11. The abscissa indicates the reduction factor, ␹, for stiffener thickness. The mean resistance ratio Ntest/NEC3-mod for all of the studied sections is m ⫽ 1.03 and standard deviation s ⫽ 0.06. Fig. 12

Fig. 12.

Comparison of test results and theoretical values for all sections.

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J. Kesti, J.M. Davies / Thin-Walled Structures 34 (1999) 115–134

presents the same relation between the test results and predicted values that are now related to the plastic resistance of the cross-section. m ⫺ 2s and m ⫹ 2s straight lines are also given in Fig. 12 in order to describe the scatter of the results.

5. Conclusions Distortional buckling is often the critical buckling mode for cold-formed steel columns. The compression capacity of short or laterally restrained open-section thinwalled columns according to EC3 is determined by using the effective width due to local buckling and the effective stiffener thickness due to distortional buckling. The applicability of the EC3 method to different cross-sections is evaluated in this paper. A comparison is made between several alternative methods for determining the minimum elastic distortional buckling stress. The Generalized Beam Theory (GBT) provides a particularly appropriate tool with which to analyze distortional buckling in isolation and in combination with other modes. Lau and Hancock’s relatively simply analytical expressions are found to give a good prediction of the distortional buckling stress. However, the method given in EC3 does not correlate as well as Lau and Hancock’s method with the results given by GBT. The error in the distortional buckling stress leads to a consequential error in the effective cross-sectional area depending on the distortional buckling stress level. The influence of the specimen length and end boundary conditions on elastic distortional buckling and the effective cross-sectional area is considerable. Usually, the buckling length for distortional buckling is much longer than that for local buckling. This leads to the important conclusion that conventional stub column tests can give values for the effective area of the cross-section which are too high. Selected experimental results on fixed-ended short columns with different crosssections were compared with the predictions given by EC3. Because the EC3 method is designed for longer columns, where several distortional buckling half-waves may occur, a modified design method was also used. In this modified method, the distortional buckling stress was determined using GBT, taking into account the actual column length and the end boundary conditions. The original EC3 methods gave conservative compression resistance values for the studied sections as expected, because the EC3 model for distortional buckling assumes free buckling of the stiffener. The modified method predicts the ultimate resistance with adequate accuracy for all of the sections studied. The study showed that the effective width approach for local buckling and the effective thickness approach for distortional buckling gives reasonable results for the compression resistance of the short columns if the distortional buckling stress is determined, taking into account the actual column length and the end boundary conditions. For longer columns, the influence of the end boundary conditions is smaller and the applicability of the original EC3 procedure to predict the effective area is dependent on its ability to predict the minimum distortional buckling stress, which may vary depending on the section dimensions.

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131

Acknowledgements This paper was prepared while the first author was on study leave at the Manchester University supported by the Academy of Finland. The facilities made available by the Division of Civil Engineering are gratefully acknowledged.

Appendix A: Test specimen dimensions, material properties and failure loads

Table 3 C-sections h SLC1 6⫻3 SLC1 9⫻3 SLC1 12⫻3 SLC1 6⫻6 SLC2 6⫻6 SLC1 12⫻6 SLC2 12⫻6 SLC1 18⫻6 SLC2 18⫻6 SLC1 24⫻6 SLC2 24⫻6 SLC3 24⫻6 SLC1 6⫻9 SLC2 6⫻9 SLC1 9⫻9 SLC 2 9⫻9 SLC1 18⫻9 SLC2 18⫻9 SLC4 27⫻9 SLC5 18⫻9 SLC1 27⫻9 SLC2 27⫻9 SLC1 36⫻9 L36-1 L48-1 CH17-1 CH17-2 CH20-1 CH20-2 CH24-1 CH24-1 RFC13-1 RFC13-2 RFC14-1

81.4 115.5 153.0 78.7 78.2 150.2 149.6 227.0 226.6 298.8 299.4 298.8 79.6 79.0 113.7 113.4 279.5 279.9 222.4 221.9 330.3 329.5 439.6 95.82 95.63 90.27 90.27 89.7 89.7 87.99 87.99 75.46 75.46 74.25

b 40.4 40.3 40.8 80.5 80.4 80.8 81.1 80.4 80.7 80.7 80.8 80.6 113.0 113.1 113.0 112.9 143.8 144.6 113.8 113.6 113.6 114.1 112.4 35.52 47.53 69.07 69.07 69.1 69.1 65.39 65.39 38.94 38.94 42.7

c

t

r

L

A

fy

E

9.2 9.3 9.0 16.6 16.7 16.7 16.3 16.9 17.0 16.7 17.0 16.7 18.6 18.6 18.7 18.7 33.4 32.6 18.3 18.9 18.4 18.4 18.6 11.76 11.47 13.84 13.84 13.6 13.6 13.8 13.8 16.94 16.94 16.68

1.219 1.219 1.219 1.194 1.219 1.194 1.194 1.194 1.194 1.194 1.194 1.194 1.143 1.143 1.143 1.143 1.549 1.549 1.270 1.295 1.270 1.270 1.245 1.48 1.47 1.67 1.67 1.996 1.996 2.394 2.394 2.438 2.438 1.905

2.59 2.64 3.10 1.88 1.75 1.98 1.93 1.88 2.13 1.98 2.41 1.98 2.51 2.67 2.64 2.57 2.21 2.41 2.95 3.07 2.95 2.95 2.77 0.85 0.85 2.8 2.8 2.5 2.5 2.1 2.1 3.962 3.962 5.563

303.5 304.3 278.4 457.7 457.7 456.7 456.7 559.3 685.0 558.8 913.6 558.0 647.7 647.2 646.9 647.2 891.3 891.5 647.7 648.0 762.0 971.3 761.7 280 300 300 700 300 700 300 800 254 254 254

214.19 254.84 301.29 322.58 325.16 409.03 406.45 500.64 496.77 582.58 585.16 584.51 384.52 383.87 420.64 423.87 978.06 974.19 612.90 620.00 748.39 750.32 865.80 280 314 417.4 417.4 497 497 576.5 576.5 456.5 456.5 367.6

228.9 228.9 228.9 233.1 233.1 233.1 233.1 233.1 233.1 233.1 241.7 233.1 226.2 226.2 227.5 227.5 417.4 417.4 201.2 201.2 201.2 205.8 203.8 515 550 393.1 393.1 220.2 220.2 488.6 488.6 357.5 357.5 379.8

203000 203000 203000 203000 203000 203000 203000 203000 203000 203000 203000 203000 203000 203000 203000 203000 203000 203000 203000 203000 203000 203000 203000 210000 200000 225000 225000 215000 215000 225000 225000 203400 203400 203400

Fu 46.28 44.72 45.17 58.74 60.52 57.85 60.52 56.96 56.96 56.96 53.40 56.07 51.18 52.51 52.96 53.40 138.62 139.73 61.41 64.97 60.52 62.30 55.63 100.2 111.9 128.7 123.7 106.8 101.9 256.2 231.1 155.7 161.9 130.8

132

J. Kesti, J.M. Davies / Thin-Walled Structures 34 (1999) 115–134

Table 3 Continued h RFC14-2 RFC14-3 PBC13-1 PBC13-2 PBC14-1 PBC14-2 PBC14-3 P11(b) P16(b) LC1 LC2 CH-1-5-800 CH-1-6-800 CH-1-7-400 CH-1-7-600 CH-1-7-800

b

74.25 74.25 74.62 74.62 74.43 74.43 74.43 124.9 65.55 148.7 148.7 118.5 118.7 119.7 119.3 119.4

42.7 42.7 38.94 38.94 39.65 39.65 39.65 60.2 33.35 48.85 73.71 88.57 88.67 88.61 88.46 88.4

c

t

r

L

A

fy

E

Fu

16.68 16.68 14.39 14.39 14.47 14.47 14.47 20.76 15.01 19.39 19.78 4.258 5.458 6.453 6.453 6.45

1.905 1.905 2.21 2.21 1.803 1.803 1.803 3.073 1.626 0.965 0.97 1.085 1.085 1.095 1.095 1.1

5.563 5.563 3.962 3.962 3.962 3.962 3.962 3.175 2.388 2.44 2.74 1.3 1.3 1.3 1.3 1.3

254 254 254 254 254 254 254 432 254 449 450 800 800 400 600 800

367.6 367.6 384.5 384.5 318.1 318.1 318.1 856.8 254.8 271.2 321.1 326.45 329.54 335.76 334.99 336.52

379.8 379.8 264.8 264.8 250.3 250.3 250.3 231.7 221.1 288.7 288.7 590 590 590 590 590

203400 203400 203400 203400 203400 203400 203400 203400 203400 202500 202500 210000 210000 210000 210000 210000

133 131.7 104.5 104.5 82.3 78.7 79.2 201.5 58.7 43.99 41.88 45.64 45.84 50.87 49.38 47.62

Table 4 Hat-sections

HA-17-1 HA17-2 HA20-1 HA20-2 HA24-1 HA24-2 A2 A3 A4 A5 A6 B2 B3 B4 B5 B6 C2 C3 C4 C5 C6 HAT3 HAT4 *

h

b

c

t

r

L

A

fy

E

Fu

91.35 90.47 92.58 88.57 85.27 85.06 77.16 76.77 76.99 77.13 78.00 77.75 78.18 78.66 78.73 78.24 77.49 78.8 78.42 78.42 78.65 148.6 148.6

78.45 79.57 78.98 78.87 87.37 89.66 77.81 77.77 77.55 78.21 76.83 78.38 78.6 78.59 78.81 78.84 77.92 78.07 77.52 77.52 78.07 98.9 123.8

13.83 13.74 13.29 13.69 13.69 13.78 7.27 12.74 20.17 25.84 32.59 7.472 13.37 20.08 26.15 32.46 6.775 13.02 19.44 25.86 31.5 19.5 19.5

1.652 1.67 1.98 1.97 2.373 2.363 0.621 0.626 0.625 0.622 0.625 1.236 1.238 1.233 1.235 1.243 0.890 0.894 0.893 0.889 0.892 0.966 0.959

2.8 2.8 2.5 2.5 2.1 2.1

300 802 300 800 300 802 609 606 611 610 607 610 610 610 607 610 458 458 459 458 458 449 449

445.6 452.4 536.9 526.6 668.6 677.1 153.6* 161.4* 170.3* 177.4* 185.5* 308.3* 324.5* 340.3* 356.5* 373.9* 219.7* 233.3* 243.2* 253.5* 265.6* 368 413.3

393.1 393.1 220.2 220.2 488.6 488.6 270.0 270.0 270.0 270.0 270.0 278.8 278.8 278.8 278.8 278.8 297.64 297.64 297.64 297.64 297.64 288.7 288.7

225000 225000 215000 215000 225000 225000 200000 200000 200000 200000 200000 200000 200000 200000 200000 200000 200000 200000 200000 200000 200000 202500 202500

119.91 118.39 114.95 105.32 271.45 230.89 11.86 17.64 18.69 19.57 18.55 48.49 63.18 66.73 68.96 66.07 23.84 33.08 37.82 37.37 40.95 42.92 44.08

Not given in reference

2.09 1.97

J. Kesti, J.M. Davies / Thin-Walled Structures 34 (1999) 115–134

133

Table 5 Web-stiffened C-sections CH2

CH-2-7-800 CH-2-7-1000 CH-2-8-1000 CH-2-10-1000 CH-2-12-1000 CH-2-14-1000

h⬘

b

c

bs

r

t

L

A

fy

E

Fu

49.70 49.95 49.43 47.45 49.95 47.70

89.05 88.70 88.86 88.85 88.85 88.75

6.45 6.65 7.45 9.45 11.45 13.45

13.39 13.41 13.49 13.59 13.20 13.59

1.3 1.3 1.3 1.3 1.3 1.3

1.100 1.100 1.095 1.105 1.100 1.105

800 1000 1000 1000 1000 1000

348.31 348.62 348.1 351.58 359.03 360.86

586.00 586.00 586.00 586.00 586.00 586.00

210000 210000 210000 210000 210000 210000

66.07 64.46 63.98 68.14 70.08 75.28

h

b1

D1

b2

r

t

L

A

fy

E

Fu

83.35 82.13 85.69 81.49 87.89 87.50 85.89

39.45 39.03 38.39 40.69 34.59 35.40 37.69

20.75 20.83 20.99 20.89 20.19 19.00 19.09

28.32 29.72 29.00 29.49 30.30 33.20 31.99

2.8 2.8 2.5 2.5 2.1 2.1 2.1

1.645 1.634 1.992 1.987 2.39 2.4 2.385

300 800 300 800 300 800 1100

413.6 415.8 505.2 505.8 596.6 609.8 608.3

393.1 393.1 220.2 220.2 488.6 488.6 488.6

225000 225000 215000 215000 225000 225000 225000

144.76 133.35 116.30 105.21 272.05 251.54 232.37

L

A

fy

E

Fu

456.1 469.2 548.0 563.7 645.9 630.9 637.0

393.1 393.1 220.2 220.2 488.6 488.6 488.6

225000 225000 215000 215000 225000 225000 225000

153.71 143.81 124.89 122.55 290.66 258.80 250.91

Table 6 Rack-sections RA

RA17-300 RA17-800 RA20-300 RA20-800 RA24-300 RA24-800 RA24-1100

Table 7 Rack-sections RL

RL17-300 RL17-800 RL20-300 RL20-800 RL24-300 RL24-800 RL24-1100

h

b1

d1

b2

d2

r

t

92.65 92.57 88.80 87.92 88.80 86.68 87.58

44.05 43.57 44.70 45.52 33.30 35.68 35.58

20.05 18.37 21.00 20.02 21.10 18.78 19.58

27.45 30.77 26.90 29.02 34.20 32.58 32.88

6.43 7.64 6.20 7.21 7.40 7.59 7.69

2.8 2.8 2.5 2.5 2.1 2.1 2.1

1.650 300 1.670 800 1.997 300 2.020 800 2.396 300 2.380 800 2.376 1100

134

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References [1] Eurocode 3. CEN ENV 1993-1-3, Design of steel structures—supplementary rules for cold formed thin gauge members and sheeting. Brussels, 1996. [2] AISI. The specification for the design of cold-formed steel structural members. Washington DC: American Iron and Steel Institute, 1996. [3] AS/NZS 4600. Australian/New Zealand standard for cold-formed steel structures. Sydney: Standards Australia, 1996. [4] Lau SCW, Hancock GJ. Distortional buckling formulas for channel columns. J Struct Eng, ASCE 1987;113(5):1063–78. [5] Timoshenko SP, Gere JM. Theory of elastic stability, 2nd ed. New York: McGraw-Hill, 1961. [6] Davies JM, Jiang C. Design of thin-walled columns for distortional buckling. Proc. of the Second Int. Conference on Coupled Instability in Metal Structures CIMS’96, Liege (Belgium), 1996:165–72. [7] Davies JM, Leach P. First-order generalized beam theory. J. Const. Steel Res. 1994;31:187–220. [8] Davies JM, Leach P. Second-order generalized beam theory. J. Const. Steel Res. 1994;31:221–41. [9] Davies JM, Jiang C. GBT-Computer program, public domain. University of Manchester, 1995. [10] Lau SCW, Hancock GJ. Strength tests and design methods for cold-formed channel columns undergoing distortional buckling. Research Report No. R579, The University of Sydney, School of Civil and Mining Engineering, 1988. [11] Young B, Rasmussen KJR. Design of lipped channel columns. J. Struct. Eng., ASCE 1998;124(2):140–8. ¨ [12] Mulligan GB, Pekoz T. Local buckling interaction in cold-formed columns. J. Struct. Eng., ASCE 1987;113(3):604–20. ¨ [13] Weng CC, Pekoz T. Compression tests of cold-formed steel columns. Proceedings of the Ninth International Specialty Conference on Cold-Formed Steel Structures, 1988:1–20. [14] Chou SM, Seah LK, Rhodes J. The accuracy of some codes of practice in predicting the load capacity of cold-formed columns. J. Const. Steel Res. 1996;37(2):137–72. [15] Kwon YB, Hancock GJ. Tests of cold-formed channels with local and distortional buckling. J. Struct. Eng., ASCE 1992;117(7):1786–803.