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Compression tests of high strength cold-formed steel channels with buckling interaction
Journal of Constructional Steel Research 65 (2009) 278–289
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Compression tests of high strength cold-formed steel channels with buckling interaction Young Bong Kwon a,∗ , Bong Sun Kim a , Gregory J. Hancock b a
Department of Civil and Environmental Engineering, Yeungnam University, Gyongsan, Republic of Korea
b
Department of Civil Engineering, University of Sydney, NSW 2006, Australia
article
info
Article history: Received 9 October 2007 Accepted 2 July 2008 Keywords: Compression tests Cold-formed steel sections Local buckling Distortional buckling Buckling interaction Direct Strength Method (DSM)
a b s t r a c t This paper describes a series of compression tests conducted on cold-formed simple lipped channels and lipped channels with intermediate stiffeners in the flanges and web fabricated from high strength steel plate of thickness 0.6 and 0.8 mm with the nominal yield stress 560 MPa. A range of lengths of lipped channel sections were tested to failure with both ends of the column fixed with a special capping to prevent local failure of column ends and influence from the shift of centroid during testing. The high strength cold-formed steel channel sections of intermediate lengths generally displayed a significant interaction between local and distortional buckling. A noticeable interaction between local and overall buckling was also observed for the long columns. A significant post-buckling strength reserve was shown for those sections that showed interaction between local and distortional or overall buckling. Simple design strength formulas in the Direct Strength Method for the thin-walled cold-formed steel sections failing in the mixed mode of local and distortional buckling have been studied. The strengths predicted by the strength formulas proposed are compared with the test results for verification. © 2008 Elsevier Ltd. All rights reserved.
1. Introduction Thin-walled cold-formed steel sections can be used efficiently as structural members of light-weight structures when hot-rolled sections or others are not efficient. However, since the thin-walled sections may undergo local, distortional, overall or mixed modes of buckling, the accurate prediction on the member strength of thin-walled cold-formed steel sections becomes more complex. Until recently, the conventional Effective Width Method (EWM) has been the only way to estimate the member strength for over 60 years. This method can take account of the interaction between local and lateral buckling and the post-buckling strength reserve in the local buckling mode. However, as structural shapes became more complex with additional lips and intermediate stiffeners, the accurate computation of the effective widths of individual elements of the complex shapes becomes more difficult and inaccurate. In order to overcome this problem, the Direct Strength Method (DSM) was developed by Shafer and Pekoz [1] in 1998 and was studied further by Hancock et al. [2]. North American Specification Supplement 1 (AISI, 2004) [3] and Australian/New Zealand Cold-Formed Steel Structures Standard (AS/NZS 4600) [4] recently adopted the Direct Strength Method
∗
Corresponding author. E-mail address: [email protected] (Y.B. Kwon).
0143-974X/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.jcsr.2008.07.005
as an alternative to the conventional Effective Width Method to estimate the compression and the flexural member strength, which can consider the interaction of local or distortional and overall buckling modes. The method uses the elastic buckling solutions for the whole section rather than for individual elements and the design strength curves developed on the basis of various test results. Research into the distortional buckling mode of thin-walled cold-formed open sections has widely been carried out in recent years since the first discussion by Hancock [5]. Lau and Hancock [6,7] tested a range of different channel and rack sections and proposed a preliminary set of design curves. Charnvarnichborikarn and Polyzois [8] modified the formula to predict the distortional buckling stress of Z-section columns based on the test results. Kwon and Hancock [9] conducted compression tests for the high strength cold-formed channel sections, which showed a substantial post-distortional buckling strength, and proposed a distortional buckling strength equation for the columns. Distortional buckling may govern the intermediate length column strength of open cold-formed steel sections and often interacts with local buckling depending on the section geometry. The interaction produces a significantly negative effect on the column strength despite the post-buckling strength reserve in both the local and distortional buckling modes. The interaction between local and distortional buckling of the cold-formed steel sections of intermediate lengths was reported by Kwon and Hancock [9], Yang and Hancock [10], Kwon et al. [11] and Silvestre et al. [12].
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279
Notations A b d1 d2 Fcr Fcrl Fcrd Fmax Fne Fu Fy h l ls Pne Pnld Pkh Pcrl S1 S2 t
λ λ1 λl
Cross-sectional area Width of web element Web stiffener depth Flange stiffener depth Critical buckling stress Elastic local buckling stress Elastic distortional buckling stress Ultimate strength determined in test Compression member design stress Ultimate tensile stress Yield stress Flange depth Length of specimen Lip stiffener depth Compression member design strength (= Fne × A) Design compressive strength Distortional buckling strength computed by Eqs. (2a) and (2b) Elastic local buckling load (= Fcrl × A) Web stiffener width Flange stiffener width Thickness of section p Distortional buckling slenderness factor (= Fy /Fcrd ) √ Slenderness factor (= Pkh /Pcrl ) √ Local buckling slenderness factor (= Pne /Pcrl )
Yang and Hancock [10] proposed a strength formula for the Direct Strength Method for cold-formed steel channel columns undergoing significant interaction between local and distortional buckling. The formula was based on the compression test results of G550 high strength cold-formed steel sections. However, extensive research into the interaction between local and distortional buckling has not been conducted yet. The main purpose of this paper is to describe the results of compression tests conducted on high strength cold-formed steel sections of thickness 0.60 and 0.80 mm, which had buckling mode interaction between local and distortional buckling, and to calibrate the Direct Strength Method formula proposed by Yang and Hancock [10]. Three types of high strength cold-formed lipped channel sections with intermediate stiffeners in the flange or web were tested under concentric compressive load to failure. The test sections were made from high strength (nominal yield strength Fy = 560 MPa) cold-formed steel plates by the brake-pressing process. The lengths of stub and intermediate columns tested ranged from 300 to 1200 mm. The difference between local and distortional buckling stresses ranged from negative to positive. The critical buckling stresses of the test sections were generally lower than half the yield stress of the material, so a significant postbuckling strength and buckling interaction were displayed after the initiation of local or distortional buckling. The Direct Strength Method formula proposed by Yang and Hancock [10] considering the interaction between local buckling and distortional buckling were calibrated with the test results, and a slight modification of the strength formula is proposed. The negative effect due to the interaction of the local, distortional and overall buckling modes for long columns is also investigated. 2. Section geometries and material properties 2.1. Material properties The structural steel grade of the test sections was SGC570 (KSD 3506) [13]. The minimum specified yield and ultimate stresses of
Fig. 1. Typical stress versus strain relation for 0.6 mm thick coupons.
the test sections of thickness 0.60 and 0.80 mm were 560 and 570 MPa, respectively. Tensile coupon tests were previously conducted for flat and corner coupons cut from the fabricated sections. All coupons were tested in a 250 kN capacity UTM (Schmazu AUTOGRAPH AG 250kNG) at a displacement rate 0.1 mm/min. Tensile coupon test results are given in Table 1 and the typical stress–strain relations are shown in Fig. 1. The experimental yield (0.2% offset) and ultimate stresses were higher than the nominal yield and ultimate stresses, respectively. The elongation ranged from 8.5% to 13.0% with the average being 10.4%, which is significantly lower than that of mild steel. The yield and ultimate stresses of the corner coupon were higher than those of the flat coupons by 6.0%–20.0%, due to the cold-working of the brake-pressed corners. The thinner coupons showed a larger difference between the yield and ultimate stresses than the thicker ones because of the amount of plastic straining. 2.2. Section design Simple lipped channels and lipped channels with intermediate stiffeners in the flanges or web and brake-pressed from galvanized SGC570 steel sheets of 0.6 and 0.8 mm thickness were chosen for the compression tests. The local buckling, distortional buckling and their interaction should be considered to predict the strength capacity of intermediate length columns in compression. The interaction of local and distortional buckling modes may significantly reduce the column strength, which is estimated on the assumption that the column fails in the distortional buckling mode without buckling mode interaction. The test sections were selected to focus on the interaction between local and distortional buckling. The schematics of the section geometries of three types of lipped channel sections are given in Fig. 2. The cross-sectional dimensions were suitably chosen for buckling interaction between the local and distortional modes. The critical local and distortional buckling stresses were lower than half the yield stress, and the difference between the local and distortional buckling stresses fall in the three categories of positive, negative and about zero. The detailed crosssectional dimensions are summarized in Table 2. The internal radii of the corners and intermediate stiffeners were measured to be less than 1.0 mm and so, were omitted in the table. The flange widths of the test sections were 40.0 or 50.0 mm, and the web depth ranged from 40.0 to 90.0 mm. The sizes of the edge stiffeners were 5.0, 8.0 and 10.0 mm, and the intermediate stiffeners in the flange and web were 5.0, 6.0, 7.0 mm in height and 10.0, 12.0, 14.0 mm in width, respectively. The test specimens were labelled so that the type of section, the thickness of the specimens, cross-sectional dimension types and the actual length of the specimen could be included. The first character of the specimen label indicated the type of
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Table 1 Mechanical properties Specimen t = 0.6 mm
Corner Flat Corner Flat
t = 0.8 mm
Yield stress Fy (MPa)
Ultimate tensile stress Fu (MPa)
Fu Fy
Elongation (%)
705.2 627.7 671.1 632.8
726.4 633.9 692.8 646.4
1.03 1.01 1.03 1.02
10.5 12.8 8.5 9.4
Table 2 Detailed dimensions of test channel sections t (mm)
b (mm)
h (mm)
A-6-1
0.6
40.0
40.0
10.0
83.0
A-6-2
0.6
50.0
50.0
8.0
98.0
B-6-1
0.6
40.0
40.0
12.0
6.0
10.0
86.0
B-6-2
0.6
80.0
40.0
14.0
7.0
10.0
110.0
C-6-1
0.6
90.0
40.0
10.0
118.0
A-8-1
0.8
50.0
40.0
10.0
117.0
A-8-2
0.8
40.0
40.0
10.0
109.0
A-8-3 A-8-4
0.8 0.8
80.0 80.0
40.0 40.0
10.0 5.0
141.0 133.0
B-8-1
0.8
50.0
50.0
10.0
137.0
(a) Type A.
s1 (mm)
d1 (mm)
s2 (mm)
10.0
12.0
d2 (mm)
5.0
6.0
(b) Type B.
ls (mm)
A (mm2 )
Specimens
L (mm) 400.0 800.0 1000.0 1200.0 1000.0 400.0 800.0 1000.0 1200.0 400.0 800.0 1000.0 1200.0 1000.0 400.0 800.0 1000.0 1200.0 400.0 800.0 1000.0 1200.0 1000.0 1000.0 400.0 800.0 1000.0 1200.0
(c) Type C.
Fig. 2. Cross-sectional geometries.
cross section, the second indicated the thickness, and the third number indicated the different dimensions of the cross section, which varied slightly even for the same section type. The actual column length in mm for the test specimens was appended at the label end, such as the label B-6-1-400. The elastic buckling analysis for lipped channel sections was conducted to select the suitable section geometry and the test column length, where the local and the distortional buckling occurred simultaneously or nearly simultaneously. The elastic buckling analysis program Thin-Wall (Papangelis and Hancock [14]) and SFSM (Lau and Hancock [15]) were used to design a suitable cross section and to set the length for compression tests. The program Thin-Wall provides an elastic buckling stress versus buckling half-wavelength plot for a cross section assuming hinged end boundary conditions whereas the SFSM can incorporate fixed end boundary conditions. The buckling analysis results of the B6-2 and C-6-1 sections obtained using the program Thin-Wall and
SFSM are illustrated in Fig. 3(a) and (b). For the results obtained by SFSM, the abscissa indicates the actual column length rather than the buckle half-wavelength for the result obtained by Thin-Wall. In the buckling stress versus half-wavelength plot obtained by ThinWall, shown in Fig. 3(a) and (b), two distinct minima represent the local buckling and the distortional buckling. However, the buckling stress versus column length plot obtained by SFSM does not show such a minimum point. As shown in Fig. 3(a) and (b), the influence of the fixed ends was to enhance mainly the distortional buckling stress, which drops gradually with increasing column length. The local buckling stress of the sections, which buckles in at least three half-waves or more, is not affected significantly by the fixed ends. The local buckling stress of the B-6-2 section was higher than the distortional buckling stress, which was quite similar to the sections which were previously tested by Yang and Hancock [10]. The local buckling mode for the specimen B-6-2 has a minimum at around 40.0 mm in half-wavelength and the distortional mode
Y.B. Kwon et al. / Journal of Constructional Steel Research 65 (2009) 278–289
(a) B-6-2 section.
281
(b) C-6-1 section. Fig. 3. Buckling stress versus half-wavelength/column length curves.
has a minimum at 500.0 mm in half-wavelength. According to the SFSM analysis, up to the column length of 1000.0 mm, local buckling was the governing mode with a different number of halfwaves according to the column length. Beyond the length of 1000.0 mm, the section mainly buckled in the distortional mode with two to four half-waves. As the column length increased beyond 2000.0 mm, the flexural–torsional buckling mode became the critical buckling mode. From the analysis results illustrated in Fig. 3(a), the interaction between local buckling and distortional buckling was expected to occur in the compression tests of the intermediate length columns for the B-6-2 section. This section displays interaction between local buckling and distortional buckling since there is a certain amount of post-buckling strength reserve in the local and distortional modes. The buckling analysis results for the C-6-1 section showed that the distortional buckling stress was significantly higher than the local buckling stress as illustrated in Fig. 3(b). The local buckling mode for the specimen C-6-1 had a minimum at 70.0 mm in halfwavelength and the distortional mode had a minimum at 500.0 mm in half-wavelength. The SFSM analysis showed that since the local buckling stress was far lower than the distortional buckling stress, the critical buckling was local or overall buckling even at the intermediate length unlike the B-6-2 section. Distortional buckling was not a critical buckling mode of the sections, regardless of the column length. However, since the local buckling mode has a significant post-buckling strength, the interaction between local buckling and distortional buckling can also occur for the sections at intermediate lengths. If the ultimate stress of the section at an intermediate length is higher than the distortional buckling stress, as the load is increased, local buckling will occur first and will show a significant post-local-buckling strength reserve. However, when the overall buckling stress is lower than the distortional buckling stress, local buckling will interact with overall buckling. In some special cases where the distortional buckling stress is slightly lower than the overall buckling stress, since the distortional buckling mode has some post-distortionalbuckling strength, complex interaction among local, distortional and overall buckling may occur. From the plots, similar to those shown in Fig. 3(a) and (b), test column lengths were chosen with the intent that the local buckling and distortional buckling is prone to occur simultaneously or nearly simultaneously in the compression tests of the columns at intermediate lengths. Elastic local and distortional buckling stresses of simple lipped channel sections and channel sections with intermediate stiffeners in the flanges or web are summarized in Table 3. The local buckling stress of the sections is lower or higher than the distortional buckling stress, or nearly equal according to the section type. The difference between local and distortional buckling stresses ranged from −83.9 to 52.6 MPa for the sections of 0.6 mm in thickness and −99.3 to 43.1 MPa for the sections of 0.8 mm in thickness. The distortional buckle half-wavelengths ranged from 375.0 to 500.0 mm,
whereas the local buckle half-wavelengths ranged from 35.0 to 70.0 mm. Since the local and distortional buckling stresses of the test specimens were lower than the nominal half yield stress of the material except the B-8-2 section, the test sections were assumed to display elastic buckling behavior and severe interaction between local buckling and distortional buckling. 3. Compression tests A total of 28 stub and intermediate columns were tested under axial compressive load to the failure. The specimen lengths chosen were 400.0, 800.0, 1000.0 and 1200.0 mm. Pseudo-static concentric compression tests of lipped channel sections were performed using a UTM (Schmazu AUTOGRAPH AG 250kNG). Unlike the mild steel cold-formed channel sections (Kwon et al. [11]), high strength cold-formed channel sections generally have significant post-buckling strength reserve in the local and distortional mode (Kwon and Hancock [9]). However, a single symmetric section may have a shift in the line of axial force after local and distortional buckling, if it is loaded between pinned ends. The shift of centroid of sections will affect the ultimate strength of the columns significantly. To avoid this problem, fixed end boundary conditions were used in the tests using the specially designed capping system, which was made of unsaturated polyester resin. The procedure of capping on the column ends is illustrated in Fig. 4. The wooden frame was fabricated first, and a 4.0 mm thin round steel plate was installed at the bottom of the wooden frame. Then, the specimen was set upright at the center of the wooden frame and finally the unsaturated polyester resin mixed with hardening material was poured into the wooden frame up to 40.0 mm in thickness. The wooden frame was removed at least two weeks after the casting of the resin material to acquire the required strength. The compression test could be conducted immediately after removing the wooden frame. The unsaturated polyester resin used as the capping material has several advantages such as rapid hardening, high strength, high energy absorption capacity, and so on, in comparison with the pattern stone or cement mortar used by Kwon and Hancock [9]. It was not difficult to center the column at the loading plate of the UTM because the capping was semitransparent. The capping prevented local failure of the column ends and warping of the section. The typical test set-up for the B-6-1-400 and the B-6-3-800 section is shown in Fig. 5. Most of the test specimens were painted white and marked black in grid lines so that the complex deformed shapes due to the interaction of local and distortional buckling could be displayed clearly. The loading was applied downward very slowly by using a UTM up to the failure of the specimen. A pseudo-static compression test was conducted by the displacement control method with the velocity of 0.2 mm/min. The lateral displacement of the flange center point was measured
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Table 3 Buckling and ultimate stresses of columns Specimens
A-6-1 A-6-2 B-6-1
B-6-2 C-6-1 A-8-1
A-8-2 A-8-3 A-8-4 B-8-1
a b
Length (mm)
400 800 1000 1200 1000 400 800 1000 1200 400 800 1000 1200 1000 400 800 1000 1200 400 800 1000 1200 1000 1000 400 800 1000 1200
Tests
Thin-Wall
Fu (MPa)
Fcr (MPa)
Buckling modea
Fcrd b (MPa)
245.9 192.2 165.7 159.0 140.7 292.4 239.1 209.1 144.0 235.2 213.6 170.4 198.8 169.1 333.9 293.4 253.8 229.5 299.0 274.5 238.1 210.8 209.0 166.4 294.3 237.9 253.4 247.0
175.5 117.9 118.6 146.9 117.1 235.8 207.7 197.7 – 197.3 194.2 153.4 184.1 57.7 287.9 242.5 228.9 219.8 285.2 243.1 222.1 192.4 186.0 141.2 281.7 216.9 229.6 225.8
L(9) + D(1) L(16) + E L(18) + E L(20) + E L(14) + D(2) L(11) + D(1) L(16) + D + E L(18) + D(1) L(21) + D(1) L(10) + D(1) L(20) + D(2) L(11) + D(2) L(25) + D(2) L(13) + D(1) L(7) + D(1) L(16) + E L(12) + E L(20) + E L(3) + D(1) E L(6) + E L(4) + E L(13) + D(2) L(13) + D(3) L(7) + D(1) L(9) + D(1) L(8) + D(3) L(13) + D + E
440.1 – – – 448.2 316.8 223.2 212.0 – 345.1 159.6 145.2 151.1 145.3 446.5 – – – 499.1 – – – 254.9 115.4 404.3 198.7 195.5 219.8
SFSM Fcrl (MPa)
169.3 108.5 206.6
197.8 47.4 229.0
300.2 98.7 98.1 235.2
Fcr (MPa)
Buckling modea
171.1 172.3 170.8 170.7 110.4 208.7 209.3 209.1 210.3 196.8 197.6 196.2 197.1 48.2 231.5 229.8 229.9 230.0 301.1 301.2 301.7 259.2 98.6 97.9 237.9 236.0 236.2 233.9
L(8) L(18) L(24) L(30) L(18) L(11) L(24) L(30) L(35) L(9) L(21) L(25) L(30) L(13) L(9) L(18) L(22) L(28) L(9) L(19) L(24) E L(15) L(15) L(9) L(17) L(24) D(2)
L: local buckling, D: distortional buckling, E: overall (flexural/flexural–torsional) buckling, (): number of half-waves. Fcrd : the lesser of the two distortional buckling stresses at 0.5L or 0.25L in half-wavelength.
Fig. 4. Procedures for capping on column ends using resin.
using two LVDTs attached at the column center and at the quarter point of the column as shown in Fig. 5. The axial shortening of the specimen was attained in stroke mode of the testing machine. 4. Test results 4.1. Test section behavior As the load was increased, local buckling occurred first, regardless of the column length and section type. Then distortional
or overall buckling followed according to the column length. In some cases, distortional buckling followed local buckling for a short time, and overall buckling occurred and governed the deformation. The magnitude of the out-of-plane deformation due to local and distortional buckling increased, and the column ultimately failed in the mixed mode. The interaction of the local and the distortional buckling for the B-6-2-400 specimen is illustrated in Fig. 6. The deformation due to elastic local buckling (Fig. 6(a)) increased gradually, then the distortional buckling commenced (Fig. 6(b)), and the interaction between these two
Y.B. Kwon et al. / Journal of Constructional Steel Research 65 (2009) 278–289
(a) B-6-1-400 section.
283
(b) B-6-3-800 section. Fig. 5. Test set-up.
types of buckling occurred as shown in Fig. 6(c). The specimens displayed a significant post-buckling strength reserve in the local and the distortional modes. Even after interaction, significant postbuckling strength was displayed. However, the magnitude of the displacement due to the distortional buckling was much larger than that due to the local buckling. Near the maximum load, a kink formed in the flange and lip stiffener as shown in Fig. 6(d), and then, the load began to decrease. As the load decreased gradually after the maximum point, the other flange and the web failed, and ultimately, the section of intermediate length failed mainly in the distortional mode with interaction between local and distortional buckling. The final failure shapes of the intermediate length columns tested were mainly in the distortional mode which interacted with the local buckling mode as shown in Fig. 7(a)–(f). For some columns such as B-6-2-1200 and B-8-1-400 shown in Fig. 7(a) and (b), the flange lip moved inwards and the final local buckling failure occurred in the web. However, for the other specimens, the flange lip moved outward and the final local failure occurred in the flange lip, as shown in Fig. 7(c)–(f). In both cases, the interaction between local and distortional buckling was significant. Even if local buckling occurred first for the intermediate length columns, distortional buckling followed immediately and the displacement due to the distortional buckling dominated the final and total deformation. For some columns where the distortional and overall buckling stresses were nearly the same, the interaction between local, distortional and overall buckling occurred. However, since the distortional buckling was overshadowed by overall buckling, the column seemed to fail in the overall buckling mode with interaction with the local buckling mode. Local buckling interacted nonlinearly with distortional and overall buckling, but the interaction between the distortional and the overall buckling seemed to be linear at the similar buckle half-wavelength. In the case of the long columns whose flexural/flexural–torsional buckling stress was far lower than the distortional buckling stress, distortional buckling did not occur, but overall buckling did occur and interacted with the local buckling. The failure modes of the A-8-1-800 and the A-6-1-800, whose flexural buckling mode interacted with the local buckling mode, are shown in Fig. 8(a) and (b), respectively. However, the interaction between local, distortional and overall buckling also occurred for the B-8-1-1200 section, as shown in Fig. 8(c). In this case, distortional buckling occurred first as the load increased, then local buckling followed, and finally, overall buckling occurred and interacted with the local
and the distortional modes. However, since the distortional and the overall buckling interacted linearly, the distortional buckling mode is not shown clearly in Fig. 8(c). The ultimate strengths of the columns that showed interaction between local, distortional and overall buckling might more significantly be lowered than those of the columns that showed interaction between only local buckling and overall buckling. 4.2. Test results Load versus axial shortening curves for the B-6-2 sections are drawn in Fig. 9. The load versus axial shortening graphs showed a linear relationship after the initial take-up until the load approached the critical buckling load as shown in Fig. 9. For the B-6-2-800, 1000 and 1200 sections, a significant loss of the stiffness occurred after the local buckling load was reached. The B-6-2 sections displayed significant interaction between local and distortional buckling in the post-buckling stage. However, no special indication for buckling interaction can be observed in the load-shortening graphs for the test sections. The structural behavior of the test sections after the ultimate loads was stable and abrupt failure did not occur. It can be observed from the test results of the B-6-2-1000 and the B-6-2-1200 that the ultimate strength of the intermediate length column failing in distortional mode did not decrease according to the increase of the column length. This may be due to imperfection sensitivity for sections failing in interaction buckling. The test buckling and test ultimate stresses of the sections are summarized in Table 3. The distortional buckling stress of the sections obtained by Thin-Wall and the critical buckling stress obtained by SFSM are also given in Table 3 for comparison. The differences between the test buckling stresses and the ultimate stresses varied from 11.5 to 74.3 MPa according to actual column length and cross-section geometry. Generally, stub columns displayed larger discrepancy than intermediate or long columns. The ratio of ultimate strength to buckling stress ranged from 1.29 to 1.52 for the lipped channel sections. The short columns of 400 mm length, which buckled in the distortional mode and interacted with local buckling, had higher ultimate strengths and larger postbuckling strength reserve than the intermediate length columns. The ultimate stress of the columns which failed in the interacted mode of local and overall buckling such as the A-6-1, the A-8-1 and the A-8-2 sections decreased with the increase of column lengths. However, the test ultimate stress of the columns which
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(a) Elastic local buckling.
(b) Elastic local and distortional buckling.
(c) Increased elastic deformation.
(d) Final inelastic failure mode (front view & rear view). Fig. 6. Buckling behavior of B-6-2-400 specimen.
buckled and failed mainly in a distortional mode interacted with local buckling such as the B-6-2 and the B-8-1 sections, did not decrease consistently accordingly as the column length increased from 800 to 1200 mm. The test buckling stress of the sections was determined by investigation of the slope change on the load versus displacement curves. Even if the post-buckling strength reserve was significant, since the change of stiffness was too subtle, the accurate buckling point could not easily be found on the load versus axial shortening curves for some specimens. In these specimens, the load versus lateral displacement curve was used to decide the buckling stress. The intermediate stiffener in the web increased the ultimate strength of the sections considerably. This increase of the strength capacity can clearly be seen in comparison of the buckling and ultimate stress between A-6-1 and B-6-1, and A-8-1 and B-8-1, respectively. The experimental buckling stresses of the sections undergoing local and distortional buckling interaction agreed well with the numerical buckling stresses, which were local or distortional.
5. Direct strength method 5.1. Direct strength method for local and overall buckling interaction The direct strength method was developed by Schafer and Pekoz [1]. It uses the elastic buckling stress of the whole section and the strength curves developed based on the tests of various sections. The direct strength formulas were adopted in the NAS Supplement 1 by AISI (2004) [3] and in the Australian/New Zealand Cold-Formed Steel Structures Standard (AS/NZS 4600) by SA (2005) [4]. The direct strength equations accounting for the interaction between local and overall buckling (L + E) and the interaction between distortional buckling and yielding (D + Yielding) were only adopted by the AISI in 2004 and SA in 2005, respectively. The equations for Pnl considering the interaction between local and overall buckling (L + E ) are expressed in Eqs. (1a) and (1b). Pnl = Pne
for λl ≤ 0.776
(1a)
Y.B. Kwon et al. / Journal of Constructional Steel Research 65 (2009) 278–289
(a) B-6-2-1200.
(b) B-8-1-400.
(c) B-6-1-400.
(d) B-6-2-800.
(e) C-6-1-1000.
(f) A-8-3-1000.
285
Fig. 7. Distortional failure modes interacted with local buckling.
" Pnl = Pne 1 − 0.15
Pcrl Pne
0.4 #
Pcrl
0.4
Pne
for λl > 0.776
(1b)
where
p λl = Pne /Pcrl Pcrl = elastic local buckling load Pne = column design strength(= AFne ).
(1c)
The column strength Pne is based on the overall failure mode that is determined from the minimum of the elastic flexural, torsional, and flexural–torsional buckling stress. The overall column strength Pne can be calculated from Eqs. (C4-2) and (C43) of the NAS (AISI 2001) [16] or Eqs. (6.1), (6.2a) and (6.2b) in the Eurocode 3 (2001) [17].
The strengths predicted by Eqs. (1a) and (1b) are compared in Fig. 10 with the test results of columns which underwent the interaction between local and overall buckling. The direct strength equations predict conservatively the ultimate strengths for most sections in comparison with test results. The B-6-1800 and B-8-1-1200 sections which are marked as a circle in Fig. 10 showed complex interaction between local, distortional and overall buckling. However, considerable strength reductions were not shown in comparison with the direct strength equations. Therefore, in addition to the strength reduction due to L + E interaction, an extra reduction of the ultimate strength due to distortional buckling interaction did not occur. Consequently, the strength equations for Pnl considering the L + E interaction can be used for the columns undergoing complex interaction between local, distortional and overall (L + D + E ) buckling and failing in the overall mode.
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(a) A-8-1-800.
(b) A-6-1-800.
(c) B-8-1-1200.
Fig. 8. Overall failure modes interacting with local (and distortional) buckling.
buckling strength formulas are given by for λ ≤ 0.561
Pkh = A · Fy
" Pkh = A · Fy 1 − 0.25
Fcrd Fy
(2a)
0.6 #
Fcrd
0.6
Fy
for λ > 0.561 (2b)
where
p λ = Fy /Fcrd , Fcrd : elastic distortional buckling stress, Fy : nominal yield stress, A : gross section area. Fig. 9. Load versus end shortening curves for B-6-2 sections.
Fig. 10. Comparison between Design Curves (L + E) and test results.
5.2. Direct strength method for local and distortional buckling interaction 5.2.1. Distortional buckling strength equation The modified Winter formula [18] was proposed by Kwon and Hancock [9] to predict the distortional buckling strength for the cold-formed steel columns of intermediate length, which failed in the distortional mode. The elastic distortional buckling stress in the formulas can easily be calculated by the rigorous FE analysis or Semi-Analytical Finite Strip (SAFS) buckling analysis program such as Thin-Wall [13]. It can predict the ultimate strength of columns undergoing distortional buckling. The distortional
(2c)
The elastic distortional buckling stress, Fcrd , in Eqs. (2b) and (2c) can be calculated by the program SFSM [13] (Kwon and Hancock [9]). A general FE program can also be used instead of SFSM, which can account for the boundary conditions other than the simple boundary conditions of the columns. However, for sections whose local buckling stress is lower than the distortional buckling stress, since the distortional buckling stress cannot be obtained by the SFSM easily as illustrated in Fig. 3(b), Thin-Wall can be used simply instead of SFSM. In this case, since the lowest buckling stresses computed by SFSM are the local buckling stress rather than the distortional buckling stress, the calculation of the distortional buckling stress accounting for the fixed end boundary conditions is almost impossible with the general FE analysis program or SFSM. It could be necessary to search amongst higher modes which can be difficult. The elastic distortional buckling stress for the B-8-1-1200 section, where the critical buckling mode was distortional buckling, was computed by the SFSM considering the fixed end boundary conditions. All the B-8-1 sections failed in the distortional buckling mode except the B-6-1-1200, which finally failed in the overall buckling mode interacting with local mode. Since the distortional buckling stress did not decrease with the increase of the column length, the distortional buckling stress of the B-8-1-1200 section could be taken as the distortional buckling stresses in Eqs. (2b) and (2c) for the B-8-1 sections. In the case of B-6-2 sections, which failed in the distortional buckling mode with interaction with the local mode, the distortional buckling stress could not be obtained by SFSM, because the
Y.B. Kwon et al. / Journal of Constructional Steel Research 65 (2009) 278–289
fixed end boundary conditions increased the distortional buckling stress significantly and the critical buckling stress was the local buckling stress as shown in Table 3. For the other sections whose distortional buckling stresses were higher than the local buckling stresses or nearly equal under the assumption of hinged boundary conditions, most of these sections except the short columns of 400 mm in length failed in the overall buckling mode which interacted with the local buckling mode. For these sections, since the lower buckling stresses of the intermediate length columns obtained by the SFSM were local buckling stresses, the distortional buckling stresses could not be obtained by the SFSM. Therefore, the elastic distortional buckling stresses obtained by the program Thin-Wall with the assumption of hinged end boundary conditions were used in Eqs. (2b) and (2c). However, the lower of the buckling stresses corresponding to 0.5L and 0.25L in half-wavelengths for single and two buckle half-waves was taken as the elastic distortional buckling stress to account for the fixed end boundary conditions. Since the distortional buckling stress computed by SFSM is always higher than that computed by Thin-Wall, if the distortional buckling stress computed by Thin-Wall is used, the distortional strength of the sections will be conservatively predicted by Eqs. (2a) and (2b) than that predicted by using the elastic distortional buckling stress obtained by the SFSM. For example, the elastic distortional buckling stress of the B-8-1-1200 section obtained by the Thin-Wall and SFSM was 219.8 MPa and 233.9 MPa, respectively as shown in Table 3. The corresponding distortional buckling slenderness λ was 1.596 and 1.547, the ultimate strength Pkh was 273.9 and 282.5 MPa and the deviation from the test maximum stress was 26.9 and 35.5 MPa, respectively. The difference is approximately 3.0% of the test maximum stress. The test results of the simple lipped channel sections and the channel sections with intermediate stiffeners in flanges or web, which showed interaction between local buckling and distortional buckling and failed in the distortional mode, are compared with the ultimate strengths predicted by Eqs. (2a) and (2b) in Fig. 11. The test results of medium strength cold-formed steel channel and hat sections with the nominal yield stress of 300 MPa, which were executed previously at the Yeungnam University (Kwon et al. [11]), are also included for comparison. As shown in Fig. 11, all the sections which failed in the mixed mode of local buckling and distortional buckling had more or less lower maximum stresses than those which were predicted by the distortional buckling strength formulas. It is clearly shown that the local buckling significantly reduced the ultimate strength of the sections which failed in the distortional mode. Since the local buckling, which occurred earlier or later than the distortional buckling, reduced the stiffness of the sections significantly, the distortional buckling stress and the post-distortional-buckling strength reserve were reduced significantly. Therefore, the ultimate strength of the sections undergoing interaction between local buckling and distortional buckling predicted by the distortional buckling strength formulas without considering the buckling interaction became unconservative. It was clearly seen that the interaction between local buckling and distortional buckling had an adverse effect on the ultimate strength of the columns of intermediate lengths, which failed in the distortional buckling mode. It could also be concluded that the distortional buckling strength formulas given by Eqs. (2a) and (2b) generally produce optimistic predictions for the columns which undergo interaction between local buckling and distortional buckling. 5.2.2. Strength formulas for direct strength method For an intermediate length compression member, distortional buckling will dominate the buckling behavior according to the section geometry as known in the test results conducted. Therefore, local and distortional (L + D) interaction rather than
287
Fig. 11. Comparison between distortional strength curve and test results.
L + E interaction may occur more often than not according to the width-to-thickness ratio and cross-section geometry. The interaction between local and distortional buckling cannot be accounted for by the effective width method at the moment. In 2004, strength formulas for DSM have been proposed to account for the interaction between local buckling and distortional buckling for thin-walled cold-formed steel channels by Yang and Hancock [10]. The design strength equations were based on the direct strength formulas for the overall mode with local instability. The distortional buckling strength of Eqs. (2a) and (2b) was treated as overall column design strength in the DSM which was developed by Schafer and Pekoz [1]. The direct strength formulas to determine the nominal axial strength of the lipped channel sections at intermediate lengths undergoing interaction between local and distortional buckling proposed by Yang and Hancock [10] are given by for λ1 ≤ 0.776
Pnld = Pkh
" Pnld = Pkh 1 − 0.15
Pcrl Pkh
0.4 #
(3a) Pcrl Pkh
0.4
for λ1 > 0.776
(3b)
where
p λ1 = Pkh /Pcrl , Pnld : design compressive strength, Pcrl : elastic local buckling load (= Fcrl · A), Pkh : distortional buckling strength computed by Eqs. (2a) and (2b).
(3c)
The elastic local buckling stress used in Eqs. (3b) and (3c) can be computed by the Thin-Wall. Yang and Hancock [10] proposed that the distortional strength Pkh should be computed using the elastic distortional buckling stress Fcrd computed by the SFSM, which can account for the fixed end boundary conditions. However, in most cases of test sections where the elastic distortional buckling stress could not be obtained by the SFSM, Thin-Wall was used to obtain the elastic distortional buckling stress as explained in the previous sections of this paper. For the B-8-1-1200 section, since the distortional buckling stress at the intermediate length was the critical buckling stress, the distortional buckling stress Fcrd could be easily obtained by the SFSM, which accounted for the fixed end boundary conditions. However, in the cases of the other sections where the local buckling stress was the critical buckling stress, since the elastic distortional buckling stress could not be obtained easily by SFSM, the elastic distortional buckling stress obtained by Thin-Wall, which cannot account for the fixed end boundary conditions directly and assumes the hinged boundary conditions, was taken as Fcrd in Eq. (3c). Therefore, to account for the fixed end boundary conditions indirectly, the effective column length was chosen as 0.5L for the single half-wave and 0.25L for the two halfwaves. This method seems to be a simple and only way to obtain the elastic distortional buckling stress as discussed earlier.
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Fig. 12. Comparison between Design Curves (L + D) and test results.
The strengths of the test sections predicted by the direct strength formula are compared with the test results in Fig. 12. The test results of high strength cold-formed steel channels with the nominal yield stress of 550 MPa, which were executed previously at the University of Sydney (Yang and Hancock [10]), are also included for comparison. The test results of medium strength coldformed steel channel and hat sections with the nominal yield stress of 300 MPa, which were conducted previously at the Yeungnam University (Kwon et al. [11]), are also included. The distortional strengths of test sections predicted by Eqs. (3a) and (3b) were conservative for most test results of high strength cold-formed steel sections except for four sections. However, the direct strength formula predicts the ultimate strengths of the medium strength cold-formed steel sections optimistically as shown in Fig. 12. It seems that the adverse effect for the medium strength cold-formed steel section was more severe than that for the high strength cold-formed steel sections. It may be due to the difference in the magnitude of the post-buckling strength. The high strength cold-formed steel sections showed larger post-buckling strength reserve than the medium strength cold-formed steel sections. The adverse effect of the interaction between local and distortional buckling on the axial compression member strength seemed to be more significant than that of the interaction between local buckling and overall buckling. It seemed to result from the similarity of the local and distortional buckling modes. In order to give a lower bound of the test results for high strength cold-formed steel sections undergoing interaction between local and distortional buckling than Eq. (3b), and to predict conservative strengths for the medium strength coldformed steel sections, the coefficient of 0.15 in Eq. (3b) was substituted by 0.2, while the exponent 0.4 used was unchanged. The modified strength formulas are given in Eqs. (4a) and (4b). for λ1 ≤ 0.667
Pnld = Pkh
" Pnld = Pkh 1 − 0.2
Pcrl Pkh
0.4 #
(4a) Pcrl Pkh
0.4
for λ1 > 0.667.
(4b)
The modified design strength formulas proposed are compared with the strength curves developed by Yang and Hancock [10] and test results in Fig. 12. The modified strength formula predicts slightly more conservative strengths than those obtained by Eq. (3b) for the range of slenderness factor λ1 from 0.667 to 1.5. The generalized test ultimate strengths of stub columns obtained with the distortional buckling strength Pkh are too conservative. This is due to the fact that the elastic distortional buckling stresses of the stub columns obtained by Thin-Wall were under-estimated. This resulted from the fact that the accurate elastic distortional buckling stress of some sections could not be computed by the buckling stress analysis programs which are available currently. Since the accurate distortional buckling stress, if it can be obtained, moves the generalized test results to the downward direction in Fig. 12, the regression seem to be more agreeable. However, it can
be concluded that the modified strength formulas can account for the interaction between local buckling and distortional buckling reasonably and predict the ultimate strength of the compression members undergoing interaction between local and distortional buckling conservatively. The design member strength of intermediate length columns that may show simultaneous or nearly simultaneous interaction between local buckling and distortional buckling should be taken as the lowest strength among the strengths which account for the adverse interaction between local and overall buckling, distortional and overall buckling, and local and distortional buckling, respectively. To account for the interaction between local buckling and distortional buckling when it is likely to occur, the modified design method can be used reasonably for intermediate length columns. However, further investigation on this problem will be required for practical use. The rigorous computational procedures for the elastic distortional buckling stress should also be developed. In particular, ranges for the use of these equations should be provided. 6. Conclusions A series of compression tests on the cold-formed steel lipped channel sections and channel sections with intermediate stiffeners in the flanges and web to study the ultimate strength of intermediate length columns were conducted. Three types of lipped channel sections made of SGC570 (nominal yield stress, Fy = 560 MPa) plates were tested to failure. In the case of sections where local buckling was the critical buckling, local buckling and distortional buckling occurred simultaneously only for the stub columns. For the intermediate and long columns where local buckling was the critical buckling, the interaction between local buckling and overall buckling was the final failure mode. However, for the sections where distortional buckling was the critical buckling, the interaction modes of local and distortional buckling were the critical failure mode for the stub and intermediate length columns. In some cases, the interaction between local, distortional and overall buckling occurred. The design strength formula developed by Yang and Hancock [10]was modified slightly to account for the significant interaction between local and distortional buckling of intermediate length columns. From the experimental study, the following conclusions are drawn: 1. The adverse interaction between local and distortional buckling reduced significantly the ultimate strength of the intermediate length cold-formed steel columns, which failed in the distortional mode. 2. Lipped channel sections undergoing interaction between local buckling and distortional buckling displayed a significant postbuckling strength reserve, regardless of the critical buckling mode, local or distortional. 3. The modified strength formula for the Direct Strength Method can reasonably predict the ultimate strength of intermediate length columns undergoing the interaction between local buckling and distortional buckling and failing in the distortional mode. 4. The strength equations for the DSM for L + E interaction predicted conservatively the ultimate strength of the lipped channel sections tested which demonstrated interaction between local and overall buckling. 5. The ultimate strength for the compression members should be taken as the minimum strengths among the ultimate strengths accounting for the interaction between local and overall, distortional and overall, and local and distortional buckling, respectively.
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Acknowledgement This research was conducted at the Structural Testing Laboratory at Department of Civil Engineering, Yeungnam University and supported by the 2007 research fund of Yeungnam University.
[9] [10]
[11]
References [1] Schafer BW, Pekoz T. Direct strength prediction of cold-formed steel members using numerical elastic buckling solutions. In: Shanmugan NE, Liew JYR, Thevendran V, editors. Thin-walled structures, research and development. Elsevier; 1998. p. 137–44. [2] Hancock GJ, Murray TM, Ellifrit DS. Cold-formed steel structures to the AISI specifications. Marcel Dekker, Inc; 2001. [3] American Iron and Steel Institute, North American specifications for the design of cold-formed steel structural members, Supplement no. 1, Washington (DC, USA); 2004. [4] Standards Australia, Cold-formed steel structures AS/NZS 4600: 2005, Sydney, NSW, Australia, 2005. [5] Hancock GJ. Distortional buckling of steel storage rack columns. J SE, ASCE 1985;111(12):2770–83. [6] Lau SCW, Hancock GJ. Distortional buckling tests of cold-formed channel sections. In: Proceedings of 9th international specialty conference on coldformed steel structures. 1988 p. 45–73. [7] Lau SCW, Hancock GJ. Inelastic buckling of channel columns in the distortional mode. Thin-Walled Structures 1990;10(2):59–84. [8] Charnvarnichborikarn P, Polyzois D. Distortional buckling of cold-formed steel
[12]
[13] [14] [15]
[16] [17]
[18]
289
z-section columns. In: Proceedings of 11th international specialty conference on cold-formed steel structures. 1992. p. 353–60. Kwon YB, Hancock GJ. Strength design curves for thin-walled sections undergoing local and distortional buckling. J SE, ASCE 1992;118(7):1786–803. Yang D, Hancock GJ. Compression tests of high strength steel channel columns with interaction between local and distortional buckling. J Struct Eng, ASCE 2004;130(12):1954–63. Kwon YB, Kim NK, Kim BS. A study on the direct strength method for compression members undergoing mixed mode buckling. In: Proceedings of steel structures ISSS’05. Seoul (Korea): Korean Society of Steel Construction. vol. 1. 2005. p. 108–19. Silvestre BW, Dinis PB, Camotim D. Direct strength method for lipped channel columns and beams affected by local-plate/distortional interaction. In: Proceedings of 18th international specialty conference on cold-formed steel structures. 2006. p. 1–18. Korean Standard Association KSD3506, hot-dip zinc coated steel sheets and coils. 2006 [in Korean]. Lau SCW, Hancock GJ. Buckling of thin flat-walled structures by a spline finite strip method. Thin-Walled Structures 1986;4(4):269–94. Papangelis JP, Hancock GJ. THIN-WALL (ver. 2.0). Sydney (Australia): Center for Advanced Structural Engineering, Dept. of Civil Engineering, Univ. of Sydney; 1998. American Iron and Steel Institute, North American specifications for the design of cold-formed steel structural members. Washington (DC, USA); 2001. European Committee for Standardization (ECS), Eurocode 3 part 1-3: General rules for cold-formed thin gauge members and sheeting, Brussels (Belgium); 2001. Winter D. Strength of thin steel compression flanges. Transactions ASCE 1947; 112:527–76. paper No. 2305.
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Journal of Constructional Steel Research journal homepage: www.elsevier.com/locate/jcsr
Compression tests of high strength cold-formed steel channels with buckling interaction Young Bong Kwon a,∗ , Bong Sun Kim a , Gregory J. Hancock b a
Department of Civil and Environmental Engineering, Yeungnam University, Gyongsan, Republic of Korea
b
Department of Civil Engineering, University of Sydney, NSW 2006, Australia
article
info
Article history: Received 9 October 2007 Accepted 2 July 2008 Keywords: Compression tests Cold-formed steel sections Local buckling Distortional buckling Buckling interaction Direct Strength Method (DSM)
a b s t r a c t This paper describes a series of compression tests conducted on cold-formed simple lipped channels and lipped channels with intermediate stiffeners in the flanges and web fabricated from high strength steel plate of thickness 0.6 and 0.8 mm with the nominal yield stress 560 MPa. A range of lengths of lipped channel sections were tested to failure with both ends of the column fixed with a special capping to prevent local failure of column ends and influence from the shift of centroid during testing. The high strength cold-formed steel channel sections of intermediate lengths generally displayed a significant interaction between local and distortional buckling. A noticeable interaction between local and overall buckling was also observed for the long columns. A significant post-buckling strength reserve was shown for those sections that showed interaction between local and distortional or overall buckling. Simple design strength formulas in the Direct Strength Method for the thin-walled cold-formed steel sections failing in the mixed mode of local and distortional buckling have been studied. The strengths predicted by the strength formulas proposed are compared with the test results for verification. © 2008 Elsevier Ltd. All rights reserved.
1. Introduction Thin-walled cold-formed steel sections can be used efficiently as structural members of light-weight structures when hot-rolled sections or others are not efficient. However, since the thin-walled sections may undergo local, distortional, overall or mixed modes of buckling, the accurate prediction on the member strength of thin-walled cold-formed steel sections becomes more complex. Until recently, the conventional Effective Width Method (EWM) has been the only way to estimate the member strength for over 60 years. This method can take account of the interaction between local and lateral buckling and the post-buckling strength reserve in the local buckling mode. However, as structural shapes became more complex with additional lips and intermediate stiffeners, the accurate computation of the effective widths of individual elements of the complex shapes becomes more difficult and inaccurate. In order to overcome this problem, the Direct Strength Method (DSM) was developed by Shafer and Pekoz [1] in 1998 and was studied further by Hancock et al. [2]. North American Specification Supplement 1 (AISI, 2004) [3] and Australian/New Zealand Cold-Formed Steel Structures Standard (AS/NZS 4600) [4] recently adopted the Direct Strength Method
∗
Corresponding author. E-mail address: [email protected] (Y.B. Kwon).
0143-974X/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.jcsr.2008.07.005
as an alternative to the conventional Effective Width Method to estimate the compression and the flexural member strength, which can consider the interaction of local or distortional and overall buckling modes. The method uses the elastic buckling solutions for the whole section rather than for individual elements and the design strength curves developed on the basis of various test results. Research into the distortional buckling mode of thin-walled cold-formed open sections has widely been carried out in recent years since the first discussion by Hancock [5]. Lau and Hancock [6,7] tested a range of different channel and rack sections and proposed a preliminary set of design curves. Charnvarnichborikarn and Polyzois [8] modified the formula to predict the distortional buckling stress of Z-section columns based on the test results. Kwon and Hancock [9] conducted compression tests for the high strength cold-formed channel sections, which showed a substantial post-distortional buckling strength, and proposed a distortional buckling strength equation for the columns. Distortional buckling may govern the intermediate length column strength of open cold-formed steel sections and often interacts with local buckling depending on the section geometry. The interaction produces a significantly negative effect on the column strength despite the post-buckling strength reserve in both the local and distortional buckling modes. The interaction between local and distortional buckling of the cold-formed steel sections of intermediate lengths was reported by Kwon and Hancock [9], Yang and Hancock [10], Kwon et al. [11] and Silvestre et al. [12].
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279
Notations A b d1 d2 Fcr Fcrl Fcrd Fmax Fne Fu Fy h l ls Pne Pnld Pkh Pcrl S1 S2 t
λ λ1 λl
Cross-sectional area Width of web element Web stiffener depth Flange stiffener depth Critical buckling stress Elastic local buckling stress Elastic distortional buckling stress Ultimate strength determined in test Compression member design stress Ultimate tensile stress Yield stress Flange depth Length of specimen Lip stiffener depth Compression member design strength (= Fne × A) Design compressive strength Distortional buckling strength computed by Eqs. (2a) and (2b) Elastic local buckling load (= Fcrl × A) Web stiffener width Flange stiffener width Thickness of section p Distortional buckling slenderness factor (= Fy /Fcrd ) √ Slenderness factor (= Pkh /Pcrl ) √ Local buckling slenderness factor (= Pne /Pcrl )
Yang and Hancock [10] proposed a strength formula for the Direct Strength Method for cold-formed steel channel columns undergoing significant interaction between local and distortional buckling. The formula was based on the compression test results of G550 high strength cold-formed steel sections. However, extensive research into the interaction between local and distortional buckling has not been conducted yet. The main purpose of this paper is to describe the results of compression tests conducted on high strength cold-formed steel sections of thickness 0.60 and 0.80 mm, which had buckling mode interaction between local and distortional buckling, and to calibrate the Direct Strength Method formula proposed by Yang and Hancock [10]. Three types of high strength cold-formed lipped channel sections with intermediate stiffeners in the flange or web were tested under concentric compressive load to failure. The test sections were made from high strength (nominal yield strength Fy = 560 MPa) cold-formed steel plates by the brake-pressing process. The lengths of stub and intermediate columns tested ranged from 300 to 1200 mm. The difference between local and distortional buckling stresses ranged from negative to positive. The critical buckling stresses of the test sections were generally lower than half the yield stress of the material, so a significant postbuckling strength and buckling interaction were displayed after the initiation of local or distortional buckling. The Direct Strength Method formula proposed by Yang and Hancock [10] considering the interaction between local buckling and distortional buckling were calibrated with the test results, and a slight modification of the strength formula is proposed. The negative effect due to the interaction of the local, distortional and overall buckling modes for long columns is also investigated. 2. Section geometries and material properties 2.1. Material properties The structural steel grade of the test sections was SGC570 (KSD 3506) [13]. The minimum specified yield and ultimate stresses of
Fig. 1. Typical stress versus strain relation for 0.6 mm thick coupons.
the test sections of thickness 0.60 and 0.80 mm were 560 and 570 MPa, respectively. Tensile coupon tests were previously conducted for flat and corner coupons cut from the fabricated sections. All coupons were tested in a 250 kN capacity UTM (Schmazu AUTOGRAPH AG 250kNG) at a displacement rate 0.1 mm/min. Tensile coupon test results are given in Table 1 and the typical stress–strain relations are shown in Fig. 1. The experimental yield (0.2% offset) and ultimate stresses were higher than the nominal yield and ultimate stresses, respectively. The elongation ranged from 8.5% to 13.0% with the average being 10.4%, which is significantly lower than that of mild steel. The yield and ultimate stresses of the corner coupon were higher than those of the flat coupons by 6.0%–20.0%, due to the cold-working of the brake-pressed corners. The thinner coupons showed a larger difference between the yield and ultimate stresses than the thicker ones because of the amount of plastic straining. 2.2. Section design Simple lipped channels and lipped channels with intermediate stiffeners in the flanges or web and brake-pressed from galvanized SGC570 steel sheets of 0.6 and 0.8 mm thickness were chosen for the compression tests. The local buckling, distortional buckling and their interaction should be considered to predict the strength capacity of intermediate length columns in compression. The interaction of local and distortional buckling modes may significantly reduce the column strength, which is estimated on the assumption that the column fails in the distortional buckling mode without buckling mode interaction. The test sections were selected to focus on the interaction between local and distortional buckling. The schematics of the section geometries of three types of lipped channel sections are given in Fig. 2. The cross-sectional dimensions were suitably chosen for buckling interaction between the local and distortional modes. The critical local and distortional buckling stresses were lower than half the yield stress, and the difference between the local and distortional buckling stresses fall in the three categories of positive, negative and about zero. The detailed crosssectional dimensions are summarized in Table 2. The internal radii of the corners and intermediate stiffeners were measured to be less than 1.0 mm and so, were omitted in the table. The flange widths of the test sections were 40.0 or 50.0 mm, and the web depth ranged from 40.0 to 90.0 mm. The sizes of the edge stiffeners were 5.0, 8.0 and 10.0 mm, and the intermediate stiffeners in the flange and web were 5.0, 6.0, 7.0 mm in height and 10.0, 12.0, 14.0 mm in width, respectively. The test specimens were labelled so that the type of section, the thickness of the specimens, cross-sectional dimension types and the actual length of the specimen could be included. The first character of the specimen label indicated the type of
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Table 1 Mechanical properties Specimen t = 0.6 mm
Corner Flat Corner Flat
t = 0.8 mm
Yield stress Fy (MPa)
Ultimate tensile stress Fu (MPa)
Fu Fy
Elongation (%)
705.2 627.7 671.1 632.8
726.4 633.9 692.8 646.4
1.03 1.01 1.03 1.02
10.5 12.8 8.5 9.4
Table 2 Detailed dimensions of test channel sections t (mm)
b (mm)
h (mm)
A-6-1
0.6
40.0
40.0
10.0
83.0
A-6-2
0.6
50.0
50.0
8.0
98.0
B-6-1
0.6
40.0
40.0
12.0
6.0
10.0
86.0
B-6-2
0.6
80.0
40.0
14.0
7.0
10.0
110.0
C-6-1
0.6
90.0
40.0
10.0
118.0
A-8-1
0.8
50.0
40.0
10.0
117.0
A-8-2
0.8
40.0
40.0
10.0
109.0
A-8-3 A-8-4
0.8 0.8
80.0 80.0
40.0 40.0
10.0 5.0
141.0 133.0
B-8-1
0.8
50.0
50.0
10.0
137.0
(a) Type A.
s1 (mm)
d1 (mm)
s2 (mm)
10.0
12.0
d2 (mm)
5.0
6.0
(b) Type B.
ls (mm)
A (mm2 )
Specimens
L (mm) 400.0 800.0 1000.0 1200.0 1000.0 400.0 800.0 1000.0 1200.0 400.0 800.0 1000.0 1200.0 1000.0 400.0 800.0 1000.0 1200.0 400.0 800.0 1000.0 1200.0 1000.0 1000.0 400.0 800.0 1000.0 1200.0
(c) Type C.
Fig. 2. Cross-sectional geometries.
cross section, the second indicated the thickness, and the third number indicated the different dimensions of the cross section, which varied slightly even for the same section type. The actual column length in mm for the test specimens was appended at the label end, such as the label B-6-1-400. The elastic buckling analysis for lipped channel sections was conducted to select the suitable section geometry and the test column length, where the local and the distortional buckling occurred simultaneously or nearly simultaneously. The elastic buckling analysis program Thin-Wall (Papangelis and Hancock [14]) and SFSM (Lau and Hancock [15]) were used to design a suitable cross section and to set the length for compression tests. The program Thin-Wall provides an elastic buckling stress versus buckling half-wavelength plot for a cross section assuming hinged end boundary conditions whereas the SFSM can incorporate fixed end boundary conditions. The buckling analysis results of the B6-2 and C-6-1 sections obtained using the program Thin-Wall and
SFSM are illustrated in Fig. 3(a) and (b). For the results obtained by SFSM, the abscissa indicates the actual column length rather than the buckle half-wavelength for the result obtained by Thin-Wall. In the buckling stress versus half-wavelength plot obtained by ThinWall, shown in Fig. 3(a) and (b), two distinct minima represent the local buckling and the distortional buckling. However, the buckling stress versus column length plot obtained by SFSM does not show such a minimum point. As shown in Fig. 3(a) and (b), the influence of the fixed ends was to enhance mainly the distortional buckling stress, which drops gradually with increasing column length. The local buckling stress of the sections, which buckles in at least three half-waves or more, is not affected significantly by the fixed ends. The local buckling stress of the B-6-2 section was higher than the distortional buckling stress, which was quite similar to the sections which were previously tested by Yang and Hancock [10]. The local buckling mode for the specimen B-6-2 has a minimum at around 40.0 mm in half-wavelength and the distortional mode
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(a) B-6-2 section.
281
(b) C-6-1 section. Fig. 3. Buckling stress versus half-wavelength/column length curves.
has a minimum at 500.0 mm in half-wavelength. According to the SFSM analysis, up to the column length of 1000.0 mm, local buckling was the governing mode with a different number of halfwaves according to the column length. Beyond the length of 1000.0 mm, the section mainly buckled in the distortional mode with two to four half-waves. As the column length increased beyond 2000.0 mm, the flexural–torsional buckling mode became the critical buckling mode. From the analysis results illustrated in Fig. 3(a), the interaction between local buckling and distortional buckling was expected to occur in the compression tests of the intermediate length columns for the B-6-2 section. This section displays interaction between local buckling and distortional buckling since there is a certain amount of post-buckling strength reserve in the local and distortional modes. The buckling analysis results for the C-6-1 section showed that the distortional buckling stress was significantly higher than the local buckling stress as illustrated in Fig. 3(b). The local buckling mode for the specimen C-6-1 had a minimum at 70.0 mm in halfwavelength and the distortional mode had a minimum at 500.0 mm in half-wavelength. The SFSM analysis showed that since the local buckling stress was far lower than the distortional buckling stress, the critical buckling was local or overall buckling even at the intermediate length unlike the B-6-2 section. Distortional buckling was not a critical buckling mode of the sections, regardless of the column length. However, since the local buckling mode has a significant post-buckling strength, the interaction between local buckling and distortional buckling can also occur for the sections at intermediate lengths. If the ultimate stress of the section at an intermediate length is higher than the distortional buckling stress, as the load is increased, local buckling will occur first and will show a significant post-local-buckling strength reserve. However, when the overall buckling stress is lower than the distortional buckling stress, local buckling will interact with overall buckling. In some special cases where the distortional buckling stress is slightly lower than the overall buckling stress, since the distortional buckling mode has some post-distortionalbuckling strength, complex interaction among local, distortional and overall buckling may occur. From the plots, similar to those shown in Fig. 3(a) and (b), test column lengths were chosen with the intent that the local buckling and distortional buckling is prone to occur simultaneously or nearly simultaneously in the compression tests of the columns at intermediate lengths. Elastic local and distortional buckling stresses of simple lipped channel sections and channel sections with intermediate stiffeners in the flanges or web are summarized in Table 3. The local buckling stress of the sections is lower or higher than the distortional buckling stress, or nearly equal according to the section type. The difference between local and distortional buckling stresses ranged from −83.9 to 52.6 MPa for the sections of 0.6 mm in thickness and −99.3 to 43.1 MPa for the sections of 0.8 mm in thickness. The distortional buckle half-wavelengths ranged from 375.0 to 500.0 mm,
whereas the local buckle half-wavelengths ranged from 35.0 to 70.0 mm. Since the local and distortional buckling stresses of the test specimens were lower than the nominal half yield stress of the material except the B-8-2 section, the test sections were assumed to display elastic buckling behavior and severe interaction between local buckling and distortional buckling. 3. Compression tests A total of 28 stub and intermediate columns were tested under axial compressive load to the failure. The specimen lengths chosen were 400.0, 800.0, 1000.0 and 1200.0 mm. Pseudo-static concentric compression tests of lipped channel sections were performed using a UTM (Schmazu AUTOGRAPH AG 250kNG). Unlike the mild steel cold-formed channel sections (Kwon et al. [11]), high strength cold-formed channel sections generally have significant post-buckling strength reserve in the local and distortional mode (Kwon and Hancock [9]). However, a single symmetric section may have a shift in the line of axial force after local and distortional buckling, if it is loaded between pinned ends. The shift of centroid of sections will affect the ultimate strength of the columns significantly. To avoid this problem, fixed end boundary conditions were used in the tests using the specially designed capping system, which was made of unsaturated polyester resin. The procedure of capping on the column ends is illustrated in Fig. 4. The wooden frame was fabricated first, and a 4.0 mm thin round steel plate was installed at the bottom of the wooden frame. Then, the specimen was set upright at the center of the wooden frame and finally the unsaturated polyester resin mixed with hardening material was poured into the wooden frame up to 40.0 mm in thickness. The wooden frame was removed at least two weeks after the casting of the resin material to acquire the required strength. The compression test could be conducted immediately after removing the wooden frame. The unsaturated polyester resin used as the capping material has several advantages such as rapid hardening, high strength, high energy absorption capacity, and so on, in comparison with the pattern stone or cement mortar used by Kwon and Hancock [9]. It was not difficult to center the column at the loading plate of the UTM because the capping was semitransparent. The capping prevented local failure of the column ends and warping of the section. The typical test set-up for the B-6-1-400 and the B-6-3-800 section is shown in Fig. 5. Most of the test specimens were painted white and marked black in grid lines so that the complex deformed shapes due to the interaction of local and distortional buckling could be displayed clearly. The loading was applied downward very slowly by using a UTM up to the failure of the specimen. A pseudo-static compression test was conducted by the displacement control method with the velocity of 0.2 mm/min. The lateral displacement of the flange center point was measured
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Table 3 Buckling and ultimate stresses of columns Specimens
A-6-1 A-6-2 B-6-1
B-6-2 C-6-1 A-8-1
A-8-2 A-8-3 A-8-4 B-8-1
a b
Length (mm)
400 800 1000 1200 1000 400 800 1000 1200 400 800 1000 1200 1000 400 800 1000 1200 400 800 1000 1200 1000 1000 400 800 1000 1200
Tests
Thin-Wall
Fu (MPa)
Fcr (MPa)
Buckling modea
Fcrd b (MPa)
245.9 192.2 165.7 159.0 140.7 292.4 239.1 209.1 144.0 235.2 213.6 170.4 198.8 169.1 333.9 293.4 253.8 229.5 299.0 274.5 238.1 210.8 209.0 166.4 294.3 237.9 253.4 247.0
175.5 117.9 118.6 146.9 117.1 235.8 207.7 197.7 – 197.3 194.2 153.4 184.1 57.7 287.9 242.5 228.9 219.8 285.2 243.1 222.1 192.4 186.0 141.2 281.7 216.9 229.6 225.8
L(9) + D(1) L(16) + E L(18) + E L(20) + E L(14) + D(2) L(11) + D(1) L(16) + D + E L(18) + D(1) L(21) + D(1) L(10) + D(1) L(20) + D(2) L(11) + D(2) L(25) + D(2) L(13) + D(1) L(7) + D(1) L(16) + E L(12) + E L(20) + E L(3) + D(1) E L(6) + E L(4) + E L(13) + D(2) L(13) + D(3) L(7) + D(1) L(9) + D(1) L(8) + D(3) L(13) + D + E
440.1 – – – 448.2 316.8 223.2 212.0 – 345.1 159.6 145.2 151.1 145.3 446.5 – – – 499.1 – – – 254.9 115.4 404.3 198.7 195.5 219.8
SFSM Fcrl (MPa)
169.3 108.5 206.6
197.8 47.4 229.0
300.2 98.7 98.1 235.2
Fcr (MPa)
Buckling modea
171.1 172.3 170.8 170.7 110.4 208.7 209.3 209.1 210.3 196.8 197.6 196.2 197.1 48.2 231.5 229.8 229.9 230.0 301.1 301.2 301.7 259.2 98.6 97.9 237.9 236.0 236.2 233.9
L(8) L(18) L(24) L(30) L(18) L(11) L(24) L(30) L(35) L(9) L(21) L(25) L(30) L(13) L(9) L(18) L(22) L(28) L(9) L(19) L(24) E L(15) L(15) L(9) L(17) L(24) D(2)
L: local buckling, D: distortional buckling, E: overall (flexural/flexural–torsional) buckling, (): number of half-waves. Fcrd : the lesser of the two distortional buckling stresses at 0.5L or 0.25L in half-wavelength.
Fig. 4. Procedures for capping on column ends using resin.
using two LVDTs attached at the column center and at the quarter point of the column as shown in Fig. 5. The axial shortening of the specimen was attained in stroke mode of the testing machine. 4. Test results 4.1. Test section behavior As the load was increased, local buckling occurred first, regardless of the column length and section type. Then distortional
or overall buckling followed according to the column length. In some cases, distortional buckling followed local buckling for a short time, and overall buckling occurred and governed the deformation. The magnitude of the out-of-plane deformation due to local and distortional buckling increased, and the column ultimately failed in the mixed mode. The interaction of the local and the distortional buckling for the B-6-2-400 specimen is illustrated in Fig. 6. The deformation due to elastic local buckling (Fig. 6(a)) increased gradually, then the distortional buckling commenced (Fig. 6(b)), and the interaction between these two
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(a) B-6-1-400 section.
283
(b) B-6-3-800 section. Fig. 5. Test set-up.
types of buckling occurred as shown in Fig. 6(c). The specimens displayed a significant post-buckling strength reserve in the local and the distortional modes. Even after interaction, significant postbuckling strength was displayed. However, the magnitude of the displacement due to the distortional buckling was much larger than that due to the local buckling. Near the maximum load, a kink formed in the flange and lip stiffener as shown in Fig. 6(d), and then, the load began to decrease. As the load decreased gradually after the maximum point, the other flange and the web failed, and ultimately, the section of intermediate length failed mainly in the distortional mode with interaction between local and distortional buckling. The final failure shapes of the intermediate length columns tested were mainly in the distortional mode which interacted with the local buckling mode as shown in Fig. 7(a)–(f). For some columns such as B-6-2-1200 and B-8-1-400 shown in Fig. 7(a) and (b), the flange lip moved inwards and the final local buckling failure occurred in the web. However, for the other specimens, the flange lip moved outward and the final local failure occurred in the flange lip, as shown in Fig. 7(c)–(f). In both cases, the interaction between local and distortional buckling was significant. Even if local buckling occurred first for the intermediate length columns, distortional buckling followed immediately and the displacement due to the distortional buckling dominated the final and total deformation. For some columns where the distortional and overall buckling stresses were nearly the same, the interaction between local, distortional and overall buckling occurred. However, since the distortional buckling was overshadowed by overall buckling, the column seemed to fail in the overall buckling mode with interaction with the local buckling mode. Local buckling interacted nonlinearly with distortional and overall buckling, but the interaction between the distortional and the overall buckling seemed to be linear at the similar buckle half-wavelength. In the case of the long columns whose flexural/flexural–torsional buckling stress was far lower than the distortional buckling stress, distortional buckling did not occur, but overall buckling did occur and interacted with the local buckling. The failure modes of the A-8-1-800 and the A-6-1-800, whose flexural buckling mode interacted with the local buckling mode, are shown in Fig. 8(a) and (b), respectively. However, the interaction between local, distortional and overall buckling also occurred for the B-8-1-1200 section, as shown in Fig. 8(c). In this case, distortional buckling occurred first as the load increased, then local buckling followed, and finally, overall buckling occurred and interacted with the local
and the distortional modes. However, since the distortional and the overall buckling interacted linearly, the distortional buckling mode is not shown clearly in Fig. 8(c). The ultimate strengths of the columns that showed interaction between local, distortional and overall buckling might more significantly be lowered than those of the columns that showed interaction between only local buckling and overall buckling. 4.2. Test results Load versus axial shortening curves for the B-6-2 sections are drawn in Fig. 9. The load versus axial shortening graphs showed a linear relationship after the initial take-up until the load approached the critical buckling load as shown in Fig. 9. For the B-6-2-800, 1000 and 1200 sections, a significant loss of the stiffness occurred after the local buckling load was reached. The B-6-2 sections displayed significant interaction between local and distortional buckling in the post-buckling stage. However, no special indication for buckling interaction can be observed in the load-shortening graphs for the test sections. The structural behavior of the test sections after the ultimate loads was stable and abrupt failure did not occur. It can be observed from the test results of the B-6-2-1000 and the B-6-2-1200 that the ultimate strength of the intermediate length column failing in distortional mode did not decrease according to the increase of the column length. This may be due to imperfection sensitivity for sections failing in interaction buckling. The test buckling and test ultimate stresses of the sections are summarized in Table 3. The distortional buckling stress of the sections obtained by Thin-Wall and the critical buckling stress obtained by SFSM are also given in Table 3 for comparison. The differences between the test buckling stresses and the ultimate stresses varied from 11.5 to 74.3 MPa according to actual column length and cross-section geometry. Generally, stub columns displayed larger discrepancy than intermediate or long columns. The ratio of ultimate strength to buckling stress ranged from 1.29 to 1.52 for the lipped channel sections. The short columns of 400 mm length, which buckled in the distortional mode and interacted with local buckling, had higher ultimate strengths and larger postbuckling strength reserve than the intermediate length columns. The ultimate stress of the columns which failed in the interacted mode of local and overall buckling such as the A-6-1, the A-8-1 and the A-8-2 sections decreased with the increase of column lengths. However, the test ultimate stress of the columns which
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(a) Elastic local buckling.
(b) Elastic local and distortional buckling.
(c) Increased elastic deformation.
(d) Final inelastic failure mode (front view & rear view). Fig. 6. Buckling behavior of B-6-2-400 specimen.
buckled and failed mainly in a distortional mode interacted with local buckling such as the B-6-2 and the B-8-1 sections, did not decrease consistently accordingly as the column length increased from 800 to 1200 mm. The test buckling stress of the sections was determined by investigation of the slope change on the load versus displacement curves. Even if the post-buckling strength reserve was significant, since the change of stiffness was too subtle, the accurate buckling point could not easily be found on the load versus axial shortening curves for some specimens. In these specimens, the load versus lateral displacement curve was used to decide the buckling stress. The intermediate stiffener in the web increased the ultimate strength of the sections considerably. This increase of the strength capacity can clearly be seen in comparison of the buckling and ultimate stress between A-6-1 and B-6-1, and A-8-1 and B-8-1, respectively. The experimental buckling stresses of the sections undergoing local and distortional buckling interaction agreed well with the numerical buckling stresses, which were local or distortional.
5. Direct strength method 5.1. Direct strength method for local and overall buckling interaction The direct strength method was developed by Schafer and Pekoz [1]. It uses the elastic buckling stress of the whole section and the strength curves developed based on the tests of various sections. The direct strength formulas were adopted in the NAS Supplement 1 by AISI (2004) [3] and in the Australian/New Zealand Cold-Formed Steel Structures Standard (AS/NZS 4600) by SA (2005) [4]. The direct strength equations accounting for the interaction between local and overall buckling (L + E) and the interaction between distortional buckling and yielding (D + Yielding) were only adopted by the AISI in 2004 and SA in 2005, respectively. The equations for Pnl considering the interaction between local and overall buckling (L + E ) are expressed in Eqs. (1a) and (1b). Pnl = Pne
for λl ≤ 0.776
(1a)
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(a) B-6-2-1200.
(b) B-8-1-400.
(c) B-6-1-400.
(d) B-6-2-800.
(e) C-6-1-1000.
(f) A-8-3-1000.
285
Fig. 7. Distortional failure modes interacted with local buckling.
" Pnl = Pne 1 − 0.15
Pcrl Pne
0.4 #
Pcrl
0.4
Pne
for λl > 0.776
(1b)
where
p λl = Pne /Pcrl Pcrl = elastic local buckling load Pne = column design strength(= AFne ).
(1c)
The column strength Pne is based on the overall failure mode that is determined from the minimum of the elastic flexural, torsional, and flexural–torsional buckling stress. The overall column strength Pne can be calculated from Eqs. (C4-2) and (C43) of the NAS (AISI 2001) [16] or Eqs. (6.1), (6.2a) and (6.2b) in the Eurocode 3 (2001) [17].
The strengths predicted by Eqs. (1a) and (1b) are compared in Fig. 10 with the test results of columns which underwent the interaction between local and overall buckling. The direct strength equations predict conservatively the ultimate strengths for most sections in comparison with test results. The B-6-1800 and B-8-1-1200 sections which are marked as a circle in Fig. 10 showed complex interaction between local, distortional and overall buckling. However, considerable strength reductions were not shown in comparison with the direct strength equations. Therefore, in addition to the strength reduction due to L + E interaction, an extra reduction of the ultimate strength due to distortional buckling interaction did not occur. Consequently, the strength equations for Pnl considering the L + E interaction can be used for the columns undergoing complex interaction between local, distortional and overall (L + D + E ) buckling and failing in the overall mode.
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(a) A-8-1-800.
(b) A-6-1-800.
(c) B-8-1-1200.
Fig. 8. Overall failure modes interacting with local (and distortional) buckling.
buckling strength formulas are given by for λ ≤ 0.561
Pkh = A · Fy
" Pkh = A · Fy 1 − 0.25
Fcrd Fy
(2a)
0.6 #
Fcrd
0.6
Fy
for λ > 0.561 (2b)
where
p λ = Fy /Fcrd , Fcrd : elastic distortional buckling stress, Fy : nominal yield stress, A : gross section area. Fig. 9. Load versus end shortening curves for B-6-2 sections.
Fig. 10. Comparison between Design Curves (L + E) and test results.
5.2. Direct strength method for local and distortional buckling interaction 5.2.1. Distortional buckling strength equation The modified Winter formula [18] was proposed by Kwon and Hancock [9] to predict the distortional buckling strength for the cold-formed steel columns of intermediate length, which failed in the distortional mode. The elastic distortional buckling stress in the formulas can easily be calculated by the rigorous FE analysis or Semi-Analytical Finite Strip (SAFS) buckling analysis program such as Thin-Wall [13]. It can predict the ultimate strength of columns undergoing distortional buckling. The distortional
(2c)
The elastic distortional buckling stress, Fcrd , in Eqs. (2b) and (2c) can be calculated by the program SFSM [13] (Kwon and Hancock [9]). A general FE program can also be used instead of SFSM, which can account for the boundary conditions other than the simple boundary conditions of the columns. However, for sections whose local buckling stress is lower than the distortional buckling stress, since the distortional buckling stress cannot be obtained by the SFSM easily as illustrated in Fig. 3(b), Thin-Wall can be used simply instead of SFSM. In this case, since the lowest buckling stresses computed by SFSM are the local buckling stress rather than the distortional buckling stress, the calculation of the distortional buckling stress accounting for the fixed end boundary conditions is almost impossible with the general FE analysis program or SFSM. It could be necessary to search amongst higher modes which can be difficult. The elastic distortional buckling stress for the B-8-1-1200 section, where the critical buckling mode was distortional buckling, was computed by the SFSM considering the fixed end boundary conditions. All the B-8-1 sections failed in the distortional buckling mode except the B-6-1-1200, which finally failed in the overall buckling mode interacting with local mode. Since the distortional buckling stress did not decrease with the increase of the column length, the distortional buckling stress of the B-8-1-1200 section could be taken as the distortional buckling stresses in Eqs. (2b) and (2c) for the B-8-1 sections. In the case of B-6-2 sections, which failed in the distortional buckling mode with interaction with the local mode, the distortional buckling stress could not be obtained by SFSM, because the
Y.B. Kwon et al. / Journal of Constructional Steel Research 65 (2009) 278–289
fixed end boundary conditions increased the distortional buckling stress significantly and the critical buckling stress was the local buckling stress as shown in Table 3. For the other sections whose distortional buckling stresses were higher than the local buckling stresses or nearly equal under the assumption of hinged boundary conditions, most of these sections except the short columns of 400 mm in length failed in the overall buckling mode which interacted with the local buckling mode. For these sections, since the lower buckling stresses of the intermediate length columns obtained by the SFSM were local buckling stresses, the distortional buckling stresses could not be obtained by the SFSM. Therefore, the elastic distortional buckling stresses obtained by the program Thin-Wall with the assumption of hinged end boundary conditions were used in Eqs. (2b) and (2c). However, the lower of the buckling stresses corresponding to 0.5L and 0.25L in half-wavelengths for single and two buckle half-waves was taken as the elastic distortional buckling stress to account for the fixed end boundary conditions. Since the distortional buckling stress computed by SFSM is always higher than that computed by Thin-Wall, if the distortional buckling stress computed by Thin-Wall is used, the distortional strength of the sections will be conservatively predicted by Eqs. (2a) and (2b) than that predicted by using the elastic distortional buckling stress obtained by the SFSM. For example, the elastic distortional buckling stress of the B-8-1-1200 section obtained by the Thin-Wall and SFSM was 219.8 MPa and 233.9 MPa, respectively as shown in Table 3. The corresponding distortional buckling slenderness λ was 1.596 and 1.547, the ultimate strength Pkh was 273.9 and 282.5 MPa and the deviation from the test maximum stress was 26.9 and 35.5 MPa, respectively. The difference is approximately 3.0% of the test maximum stress. The test results of the simple lipped channel sections and the channel sections with intermediate stiffeners in flanges or web, which showed interaction between local buckling and distortional buckling and failed in the distortional mode, are compared with the ultimate strengths predicted by Eqs. (2a) and (2b) in Fig. 11. The test results of medium strength cold-formed steel channel and hat sections with the nominal yield stress of 300 MPa, which were executed previously at the Yeungnam University (Kwon et al. [11]), are also included for comparison. As shown in Fig. 11, all the sections which failed in the mixed mode of local buckling and distortional buckling had more or less lower maximum stresses than those which were predicted by the distortional buckling strength formulas. It is clearly shown that the local buckling significantly reduced the ultimate strength of the sections which failed in the distortional mode. Since the local buckling, which occurred earlier or later than the distortional buckling, reduced the stiffness of the sections significantly, the distortional buckling stress and the post-distortional-buckling strength reserve were reduced significantly. Therefore, the ultimate strength of the sections undergoing interaction between local buckling and distortional buckling predicted by the distortional buckling strength formulas without considering the buckling interaction became unconservative. It was clearly seen that the interaction between local buckling and distortional buckling had an adverse effect on the ultimate strength of the columns of intermediate lengths, which failed in the distortional buckling mode. It could also be concluded that the distortional buckling strength formulas given by Eqs. (2a) and (2b) generally produce optimistic predictions for the columns which undergo interaction between local buckling and distortional buckling. 5.2.2. Strength formulas for direct strength method For an intermediate length compression member, distortional buckling will dominate the buckling behavior according to the section geometry as known in the test results conducted. Therefore, local and distortional (L + D) interaction rather than
287
Fig. 11. Comparison between distortional strength curve and test results.
L + E interaction may occur more often than not according to the width-to-thickness ratio and cross-section geometry. The interaction between local and distortional buckling cannot be accounted for by the effective width method at the moment. In 2004, strength formulas for DSM have been proposed to account for the interaction between local buckling and distortional buckling for thin-walled cold-formed steel channels by Yang and Hancock [10]. The design strength equations were based on the direct strength formulas for the overall mode with local instability. The distortional buckling strength of Eqs. (2a) and (2b) was treated as overall column design strength in the DSM which was developed by Schafer and Pekoz [1]. The direct strength formulas to determine the nominal axial strength of the lipped channel sections at intermediate lengths undergoing interaction between local and distortional buckling proposed by Yang and Hancock [10] are given by for λ1 ≤ 0.776
Pnld = Pkh
" Pnld = Pkh 1 − 0.15
Pcrl Pkh
0.4 #
(3a) Pcrl Pkh
0.4
for λ1 > 0.776
(3b)
where
p λ1 = Pkh /Pcrl , Pnld : design compressive strength, Pcrl : elastic local buckling load (= Fcrl · A), Pkh : distortional buckling strength computed by Eqs. (2a) and (2b).
(3c)
The elastic local buckling stress used in Eqs. (3b) and (3c) can be computed by the Thin-Wall. Yang and Hancock [10] proposed that the distortional strength Pkh should be computed using the elastic distortional buckling stress Fcrd computed by the SFSM, which can account for the fixed end boundary conditions. However, in most cases of test sections where the elastic distortional buckling stress could not be obtained by the SFSM, Thin-Wall was used to obtain the elastic distortional buckling stress as explained in the previous sections of this paper. For the B-8-1-1200 section, since the distortional buckling stress at the intermediate length was the critical buckling stress, the distortional buckling stress Fcrd could be easily obtained by the SFSM, which accounted for the fixed end boundary conditions. However, in the cases of the other sections where the local buckling stress was the critical buckling stress, since the elastic distortional buckling stress could not be obtained easily by SFSM, the elastic distortional buckling stress obtained by Thin-Wall, which cannot account for the fixed end boundary conditions directly and assumes the hinged boundary conditions, was taken as Fcrd in Eq. (3c). Therefore, to account for the fixed end boundary conditions indirectly, the effective column length was chosen as 0.5L for the single half-wave and 0.25L for the two halfwaves. This method seems to be a simple and only way to obtain the elastic distortional buckling stress as discussed earlier.
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Fig. 12. Comparison between Design Curves (L + D) and test results.
The strengths of the test sections predicted by the direct strength formula are compared with the test results in Fig. 12. The test results of high strength cold-formed steel channels with the nominal yield stress of 550 MPa, which were executed previously at the University of Sydney (Yang and Hancock [10]), are also included for comparison. The test results of medium strength coldformed steel channel and hat sections with the nominal yield stress of 300 MPa, which were conducted previously at the Yeungnam University (Kwon et al. [11]), are also included. The distortional strengths of test sections predicted by Eqs. (3a) and (3b) were conservative for most test results of high strength cold-formed steel sections except for four sections. However, the direct strength formula predicts the ultimate strengths of the medium strength cold-formed steel sections optimistically as shown in Fig. 12. It seems that the adverse effect for the medium strength cold-formed steel section was more severe than that for the high strength cold-formed steel sections. It may be due to the difference in the magnitude of the post-buckling strength. The high strength cold-formed steel sections showed larger post-buckling strength reserve than the medium strength cold-formed steel sections. The adverse effect of the interaction between local and distortional buckling on the axial compression member strength seemed to be more significant than that of the interaction between local buckling and overall buckling. It seemed to result from the similarity of the local and distortional buckling modes. In order to give a lower bound of the test results for high strength cold-formed steel sections undergoing interaction between local and distortional buckling than Eq. (3b), and to predict conservative strengths for the medium strength coldformed steel sections, the coefficient of 0.15 in Eq. (3b) was substituted by 0.2, while the exponent 0.4 used was unchanged. The modified strength formulas are given in Eqs. (4a) and (4b). for λ1 ≤ 0.667
Pnld = Pkh
" Pnld = Pkh 1 − 0.2
Pcrl Pkh
0.4 #
(4a) Pcrl Pkh
0.4
for λ1 > 0.667.
(4b)
The modified design strength formulas proposed are compared with the strength curves developed by Yang and Hancock [10] and test results in Fig. 12. The modified strength formula predicts slightly more conservative strengths than those obtained by Eq. (3b) for the range of slenderness factor λ1 from 0.667 to 1.5. The generalized test ultimate strengths of stub columns obtained with the distortional buckling strength Pkh are too conservative. This is due to the fact that the elastic distortional buckling stresses of the stub columns obtained by Thin-Wall were under-estimated. This resulted from the fact that the accurate elastic distortional buckling stress of some sections could not be computed by the buckling stress analysis programs which are available currently. Since the accurate distortional buckling stress, if it can be obtained, moves the generalized test results to the downward direction in Fig. 12, the regression seem to be more agreeable. However, it can
be concluded that the modified strength formulas can account for the interaction between local buckling and distortional buckling reasonably and predict the ultimate strength of the compression members undergoing interaction between local and distortional buckling conservatively. The design member strength of intermediate length columns that may show simultaneous or nearly simultaneous interaction between local buckling and distortional buckling should be taken as the lowest strength among the strengths which account for the adverse interaction between local and overall buckling, distortional and overall buckling, and local and distortional buckling, respectively. To account for the interaction between local buckling and distortional buckling when it is likely to occur, the modified design method can be used reasonably for intermediate length columns. However, further investigation on this problem will be required for practical use. The rigorous computational procedures for the elastic distortional buckling stress should also be developed. In particular, ranges for the use of these equations should be provided. 6. Conclusions A series of compression tests on the cold-formed steel lipped channel sections and channel sections with intermediate stiffeners in the flanges and web to study the ultimate strength of intermediate length columns were conducted. Three types of lipped channel sections made of SGC570 (nominal yield stress, Fy = 560 MPa) plates were tested to failure. In the case of sections where local buckling was the critical buckling, local buckling and distortional buckling occurred simultaneously only for the stub columns. For the intermediate and long columns where local buckling was the critical buckling, the interaction between local buckling and overall buckling was the final failure mode. However, for the sections where distortional buckling was the critical buckling, the interaction modes of local and distortional buckling were the critical failure mode for the stub and intermediate length columns. In some cases, the interaction between local, distortional and overall buckling occurred. The design strength formula developed by Yang and Hancock [10]was modified slightly to account for the significant interaction between local and distortional buckling of intermediate length columns. From the experimental study, the following conclusions are drawn: 1. The adverse interaction between local and distortional buckling reduced significantly the ultimate strength of the intermediate length cold-formed steel columns, which failed in the distortional mode. 2. Lipped channel sections undergoing interaction between local buckling and distortional buckling displayed a significant postbuckling strength reserve, regardless of the critical buckling mode, local or distortional. 3. The modified strength formula for the Direct Strength Method can reasonably predict the ultimate strength of intermediate length columns undergoing the interaction between local buckling and distortional buckling and failing in the distortional mode. 4. The strength equations for the DSM for L + E interaction predicted conservatively the ultimate strength of the lipped channel sections tested which demonstrated interaction between local and overall buckling. 5. The ultimate strength for the compression members should be taken as the minimum strengths among the ultimate strengths accounting for the interaction between local and overall, distortional and overall, and local and distortional buckling, respectively.
Y.B. Kwon et al. / Journal of Constructional Steel Research 65 (2009) 278–289
Acknowledgement This research was conducted at the Structural Testing Laboratory at Department of Civil Engineering, Yeungnam University and supported by the 2007 research fund of Yeungnam University.
[9] [10]
[11]
References [1] Schafer BW, Pekoz T. Direct strength prediction of cold-formed steel members using numerical elastic buckling solutions. In: Shanmugan NE, Liew JYR, Thevendran V, editors. Thin-walled structures, research and development. Elsevier; 1998. p. 137–44. [2] Hancock GJ, Murray TM, Ellifrit DS. Cold-formed steel structures to the AISI specifications. Marcel Dekker, Inc; 2001. [3] American Iron and Steel Institute, North American specifications for the design of cold-formed steel structural members, Supplement no. 1, Washington (DC, USA); 2004. [4] Standards Australia, Cold-formed steel structures AS/NZS 4600: 2005, Sydney, NSW, Australia, 2005. [5] Hancock GJ. Distortional buckling of steel storage rack columns. J SE, ASCE 1985;111(12):2770–83. [6] Lau SCW, Hancock GJ. Distortional buckling tests of cold-formed channel sections. In: Proceedings of 9th international specialty conference on coldformed steel structures. 1988 p. 45–73. [7] Lau SCW, Hancock GJ. Inelastic buckling of channel columns in the distortional mode. Thin-Walled Structures 1990;10(2):59–84. [8] Charnvarnichborikarn P, Polyzois D. Distortional buckling of cold-formed steel
[12]
[13] [14] [15]
[16] [17]
[18]
289
z-section columns. In: Proceedings of 11th international specialty conference on cold-formed steel structures. 1992. p. 353–60. Kwon YB, Hancock GJ. Strength design curves for thin-walled sections undergoing local and distortional buckling. J SE, ASCE 1992;118(7):1786–803. Yang D, Hancock GJ. Compression tests of high strength steel channel columns with interaction between local and distortional buckling. J Struct Eng, ASCE 2004;130(12):1954–63. Kwon YB, Kim NK, Kim BS. A study on the direct strength method for compression members undergoing mixed mode buckling. In: Proceedings of steel structures ISSS’05. Seoul (Korea): Korean Society of Steel Construction. vol. 1. 2005. p. 108–19. Silvestre BW, Dinis PB, Camotim D. Direct strength method for lipped channel columns and beams affected by local-plate/distortional interaction. In: Proceedings of 18th international specialty conference on cold-formed steel structures. 2006. p. 1–18. Korean Standard Association KSD3506, hot-dip zinc coated steel sheets and coils. 2006 [in Korean]. Lau SCW, Hancock GJ. Buckling of thin flat-walled structures by a spline finite strip method. Thin-Walled Structures 1986;4(4):269–94. Papangelis JP, Hancock GJ. THIN-WALL (ver. 2.0). Sydney (Australia): Center for Advanced Structural Engineering, Dept. of Civil Engineering, Univ. of Sydney; 1998. American Iron and Steel Institute, North American specifications for the design of cold-formed steel structural members. Washington (DC, USA); 2001. European Committee for Standardization (ECS), Eurocode 3 part 1-3: General rules for cold-formed thin gauge members and sheeting, Brussels (Belgium); 2001. Winter D. Strength of thin steel compression flanges. Transactions ASCE 1947; 112:527–76. paper No. 2305.