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Compression tests of welded section columns undergoing buckling interaction
Journal of Constructional Steel Research 63 (2007) 1590–1602 www.elsevier.com/locate/jcsr
Compression tests of welded section columns undergoing buckling interaction Young Bong Kwon a,∗ , Nak Gu Kim b , G.J. Hancock c a Department of Civil Engineering, Yeungnam University, Gyongsan, 712-749, Republic of Korea b Yeungnam University, Gyongsan, 712-749, Republic of Korea c Faculty of Engineering, University of Sydney, NSW 2006, Australia
Received 1 September 2006; accepted 31 January 2007
Abstract This paper describes a series of compression tests performed on welded H-section and channel section columns fabricated from a mild steel plate of thickness 6.0 mm with nominal yield stress of 240 MPa. The ultimate strength and performance of the compression members undergoing nonlinear interaction between local and overall buckling were investigated experimentally and theoretically. The compression tests indicated that the interaction between local and overall buckling had a significant negative effect on the ultimate strength of the thin-walled welded steel section columns. The Direct Strength Method (DSM), which was newly developed and adopted as an alternative to the effective width method for the design of cold-formed steel sections recently by NAS (AISI, 2004), was calibrated by using the test results for application to welded steel sections. This paper confirms that the Direct Strength Method can properly predict the ultimate strength of welded section columns when local buckling and flexural buckling occur simultaneously or nearly simultaneously. c 2007 Elsevier Ltd. All rights reserved.
Keywords: Direct strength method; Effective width method; Interaction between local and overall buckling; Ultimate strength; Performance; Welded sections
1. Introduction The compression and flexural members of hot-rolled shapes and welded sections fabricated from hot-rolled plate will normally buckle in local and flexural/flexural–torsional buckling mode [1], while the cold-formed steel sections will buckle in the distortional mode in addition to those modes [2]. However, whenever the local or distortional buckling stress is lower than the overall buckling stress, the interaction between local or distortional and overall buckling may occur and have a significant effect on the performance of the sections [3–5]. Since the interaction between local and overall buckling generally deteriorates the overall column strength, it is necessary to account for the negative effect of buckling interaction in the conservative prediction of the ultimate strength of columns.
∗ Corresponding address: Department of Civil and Environmental Engineering, 214-1 Daedong, 712-749 Gyongsan-si, Gyongbuk-do, Republic of Korea. Tel.: +82 53 810 2418, +82 11 802 2418; fax: +82 53 810 4622. E-mail address: [email protected] (Y.B. Kwon).
c 2007 Elsevier Ltd. All rights reserved. 0143-974X/$ - see front matter doi:10.1016/j.jcsr.2007.01.011
The ultimate strength of compression members, which are composed of thin plate elements, is dependent on both the width–thickness ratio of the plate elements and the slenderness ratio of the columns. When the local buckling stress is lower than the overall buckling stress or both types of buckling occur nearly simultaneously, local buckling may negatively affect the column strength. Because the local buckling mode has a post-buckling strength reserve, it has generally been considered in the design strength through the effective width concept [6–8]. However, since the computation of the effective width can be tedious and complicated, some countries such as Japan and Korea, do not account for the post-local-buckling strength reserve in the design strength of compression members to maintain simplicity. Instead of using the effective width concept, the overall column strength is simply reduced by the ratio of the design strength based on the local buckling stress to the yield stress in the Korean Highway Bridge Design Specifications [9]. The Direct Strength Method (DSM) has been developed by Schafer and Pekoz [10] and studied further by many researchers [11–13]. The method has been developed to
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Notation Abbreviation ASD DSM EWM LRFD
Allowable stress design Direct strength method Effective width method Load and resistance factor design
Symbols A b bf d Fcrl Fcr o Fne f nl Fl Fmax Fy l Pcrl Pl Pnl Pne Qa Qs r tf tw λ
Cross sectional area Clear width of plate element Flange width Web depth Elastic local buckling stress Elastic flexural/flexural–torsional buckling stress Overall column strength based on the overall buckling stress Design limiting stress accounting for interaction of local and overall buckling Nominal strength based on the local buckling stress Ultimate strength determined in test Yield stress Length of specimen Elastic local buckling load (=Fcrl × A) Resulting limiting load (=Fl × A) Design limiting strength (=Fnl × A) Compression member design strength (=Fne × A) Form factor for stiffened element Form factor for unstiffened element Radius of gyration Flange thickness Web thickness √ √ Slenderness ratio factor (= Fne /Fcrl , Pne /Pcrl )
overcome the weak points of the effective width method (EWM), which has been used in thin-walled steel section design for over 60 years. These weak points are the complication in accurate computation of effective width and the difficulty in consideration of elements interacting in isolation. Also, as coldformed steel sections become more complex with additional edge and intermediate stiffeners, calculations of effective width can become very complex. The interaction between local or distortional and overall buckling for cold-formed steel sections has been studied, and simple design formulas have already been presented. In 2004, the Direct Strength Method was adopted as an alternative to the effective width method, which has been used for thin section design by NAS Supplement 1 (AISI, 2004) [14] and the Australian/New Zealand Cold-Formed Steel Structures Standard AS/NZS 4600 [15]. However, there has been less research concerning the application of this method to hot-rolled and welded sections.
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This paper aims to develop the direct strength formulas of design for welded section compression members. The application of the Direct Strength Method to mild steel welded sections was studied experimentally and theoretically. A series of compression tests was performed on welded Hsections and C-sections fabricated from mild steel plates of thickness 6.0 mm with nominal yield stress 240 MPa to develop the direct strength formulas for the welded steel section compressive members undergoing interaction between local and flexural/flexural–torsional buckling. Nonlinear analyses of the sections tested have also been conducted to compare their results with the test results. The direct strength formulas of design were compared with the effective width method (EWM) and the allowable stress design method (ASD), which is currently used in Korea. The direct strength formulas proposed have been proven accurate and efficient to predict the ultimate strength of columns when local buckling and overall buckling occur simultaneously or nearly simultaneously. The results of David and Hancock [16] and Rasmussen and Hancock [17] are also included in the paper for comparison. 2. Test sections 2.1. General A series of compression tests were performed on mild steel H-sections and C-sections of thickness 6.0 mm with nominal yield and ultimate stresses 240 MPa and 400 MPa, respectively, which were fabricated by the continuous fillet welding of effective width on both sides of the flange–web joint. The size of fillet welds was determined as 6.0 mm according to the AISC specifications [7]. The geometry of the welded Hsection and the C-section tested are shown in Fig. 1. The dimensions of the test specimens were optimized in order to ensure that the local buckling stress was slightly lower than the flexural/flexural–torsional buckling stress and consequently, the interaction between local and overall buckling occurred before the ultimate load was reached. The cross sections tested were optimized by using a finite strip analysis program THINWALL [18], repeatedly. The width–thickness ratios of the webs and the flanges of the test sections were selected, such that the elastic local buckling stress of the section was lower than half the yield stress (Fy /2) and significant post-local-buckling strength reserve was displayed before the ultimate load, which is mainly dependent on the overall buckling strength, was, reached. The limiting width–thickness ratio p for Class 3 cross sections is expressed as d/tw ≤ 42 235/Fy for the web and p b/ t f ≤ 14 235/Fy for the flange of the sections according to Eurocode3 [8]. Since the width–thickness ratios of the webs or the flanges of sections tested are larger than the slenderness limit values, the sections selected can be classified as Class 4, where the effective width must be used to account for the local buckling effect. The dimensions of the H-sections and the Csections tested are summarized in Table 1. The thickness of all the specimens tested was 6.0 mm, and the dimensions of the webs and flanges of the sections were adjusted to ensure the
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(a) H-section.
(b) C-section. Fig. 1. Test sections.
Table 1 Dimensions of test sections Specimens
b f (mm)
d (mm)
t f (mm)
tw (mm)
d/tw
b/t f
l (mm)
k ∗ l/r
A (mm2 )
H-1 H-2 H-3 C-1 C-2 C-3
100.0 160.0 140.0 153.0 103.0 303.0
400.0 550.0 500.0 550.0 500.0 150.0
6.0 6.0 6.0 6.0 6.0 6.0
6.0 6.0 6.0 6.0 6.0 6.0
66.7 91.7 83.3 91.7 83.3 25.0
7.8 12.8 11.2 24.5 16.0 49.5
1400.0 2200.0 2400.0 2200.0 2200.0 2200.0
58.59 54.95 63.49 55.37 53.97 22.89
3600.0 5220.0 4680.0 5136.0 4236.0 4536.0
k ∗ = 0.7.
interaction between local buckling and overall buckling of the test sections. 2.2. Determination of overall length of specimens For the columns tested, the effect of the interaction between local and overall buckling was the focus of the tests. In order to determine the optimum lengths of the test sections, the local and overall buckling stresses of the specimens needed to be computed accurately. The elastic buckling stresses of the H2 and C-2 specimens, subjected to uniform compression, are illustrated in Fig. 2(a) and 2(b) as the buckling stress versus buckle half-wavelength, respectively. The lengths of the test specimens were determined by using the elastic buckling stress versus buckle half-wavelength curves obtained by the program THIN-WALL [18]. To ensure that the overall buckling stress is higher than the local buckling stress and that there exists a significant post-local-buckling strength reserve before the final collapse of the specimens, the test column lengths were determined. The lengths of test specimens should be in the hatched range shown in Fig. 2(a) and (b), which are longer than the half-wavelengths where the mixed mode of local and overall buckling occurs (point B) and shorter than the half-wavelengths (point D) where the overall buckling stresses are equal to the local buckling stresses. To avoid the failure due to only the local buckling without buckling interaction, the overall lengths of the test specimens were determined to be in range from 1400 to 2400 mm. The local buckling stress minima (point A) occurred in the curves at the half-wavelengths of 480 mm and 500 mm and the local buckling stresses of 118.7 MPa and 106.5 MPa for the
C-2 specimen and the H-2 specimen, respectively. The mixed mode buckling between local and overall buckling occurred at the half-wavelength of 1000 mm and 1200 mm for the C2 specimen and the H-2 specimen, respectively. The overall buckling modes of the H-2 specimen of 1540 mm in length (point C) was the flexural buckling mode in slight interaction with the local buckling mode, as shown in Fig. 2(b). Since the flexural buckling load about the unsymmetrical axis (xaxis) of the C-2 section was larger than the flexural–torsional buckling load about the symmetrical axis (y-axis), the overall buckling mode at the length of 1540 mm was the flexural mode about x-axis in slight interaction with the local buckling rather than the flexural–torsional mode about y-axis, as shown in Fig. 2(a). A noticeable interaction between overall buckling and local buckling was generally observed for the H-section and C-section columns at lengths between points B and D in Fig. 2(a) and (b). Since the unloaded and the loaded end of the test columns were a fixed and a hinged boundary condition, respectively, the effective length of the columns was taken as 0.7L. The slenderness ratios (k L/r ) of the test specimens ranged from 22.9 to 63.5. 2.3. Numerical analysis results The material and geometrical nonlinear analysis of the specimens selected for the compression test was conducted using the program LUSAS [19], to investigate the ultimate strength and the structural behavior. To study the effect of buckling interaction between local and overall modes on the ultimate strength of the columns, an elastic buckling analysis was conducted first to find out the local buckling mode.
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(a) C-2 section.
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(b) H-2 section. Fig. 2. Typical buckling stress versus half-wavelength curves.
The first several local buckling loads and modes were nearly similar. Since the buckling interaction seemed to be quite sensitive to the imperfections, the sensitivity of the magnitude of imperfections was studied. The load versus shortening curves of the H-2 section obtained by the nonlinear analysis using the LUSAS with the different initial imperfections which were the first local buckling mode multiplied by the magnification factor 0.1, 0.01 and 0.001, respectively, are compared in Fig. 3(a) and 3(b). In the analysis, a triggering load was applied laterally at the column center to cause the interaction between local buckling and overall buckling. The ultimate loads of the H-2 section computed with different magnitudes of imperfections were nearly similar in the case of 0.1% triggering load applied as shown in Fig. 3(a). The ultimate loads obtained with 0.1% triggering load were slightly higher than those obtained with 0.5% triggering load. The ultimate loads and the structural behavior after the peak load became slightly different in the case of 0.5% triggering load applied as shown in Fig. 3(b). In consideration of the effect of the initial imperfections and the triggering load, the initial imperfections could be assumed as the first local buckling mode multiplied by the factor of 0.01 and the triggering load could be taken as 0.1% of the vertical reference load in the further analysis. The 4-node shell element (QTS4) was used for the numerical modeling of the section, and the loaded end was assumed as a hinged boundary condition with the vertical direction of the loaded end to be free to move to allow the vertical application of the load. The uniform displacement control technique was used at the loaded end to make a similar boundary condition to the compression test. The bottom end of the column was assumed as a fixed boundary condition as the test condition. The average yield stress obtained from the tensile tests of the coupons which were cut from the flange and web of the welded sections was approximately 260 MPa, which was slightly higher than the nominal yield stress of 240 MPa. The effect of the magnitude of the yield stress on the ultimate strengths of the C-2 and H-2 sections was not negligible as illustrated in Fig. 3(c). Therefore, the yield stress of the material was taken as 260 MPa in the further analysis of the specimens. Young’s modulus was assumed to be 2.0 × 105 MPa and Poisson’s ratio was taken as 0.3. The stress–strain relation of the material was assumed
elastic–perfectly plastic neglecting the strain-hardening and the von Mises yield criterion was applied for the plasticity theory of the material. The ultimate stresses of the sections obtained by the nonlinear analysis are summarized in Table 2. The elastic local and overall buckling stresses, and the local buckle halfwavelength obtained by the program THIN-WALL [18] are also given in the table for comparison. The elastic overall buckling modes obtained by THIN-WALL were in the interaction mode between local buckling and flexural/flexural–torsional buckling. Since the interaction between local buckling and flexural/flexural–torsional buckling negatively affected the column buckling stress, Euler buckling stresses FE were expected to be higher than the overall buckling stresses computed by THIN-WALL. The elastic local buckling stresses were lower than the maximum stresses and overall buckling stresses of the sections. Since the elastic overall buckling stresses of test columns were larger than the elastic local buckling stresses by the difference between 69.6 MPa for H1 and 112.9 MPa for C-2, it was supposed that most of test columns might fail in the mixed mode of local and overall buckling. 3. Compression tests 3.1. General The H-sections and the C-sections listed in Table 1 were chosen for the pseudo-static compression test, considering the maximum loading capacity and the dimension of the swivel head of the testing machine. The steel grade of the test sections was SM400 structural steel to KS D3515 [20] (equivalent to ASTM A36 Steel), of which the nominal yield stress and the ultimate tensile stress were 240 MPa and 400 MPa, respectively. The end plates of thickness 30 mm were welded to both ends of the specimens to minimize eccentric loading and to prevent the local failure of the specimen ends. The loaded end boundary condition was free about the x- and y-direction rotations and movable in the vertical direction, and the bottom end boundary condition of the specimens was fully fixed. The concentric compression test was conducted by using a 3000 kN MTS testing machine. Downward loading at
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Table 2 Maximum and buckling stress of sections Specimen
Maximum stress Fmax (MPa)
Elastic local buckling stress Fcrl (MPa)
Elastic overall buckling stress Fcr o (MPa)
Euler buckling stress FE (MPa)
Local buckle half-wavelength l (mm)
H-1 H-2 H-3 C-1 C-2 C-3
213.5 172.2 184.2 136.9 177.6 151.7
197.9 106.5 120.9 97.9 118.7 77.1
267.5 194.3 233.8 185.7 231.6 159.8
575.0 653.7 489.6 643.8 621.1 433.6
380 500 460 550 480 550
Fcr o : Elastic buckling stress computed by THIN-WALL.
(a) Effect of initial imperfections (0.1% triggering load) for H-2 section.
(a) Locations of gauges & LVDT.
(b) Test configuration.
Fig. 4. Test set-up (H-section).
(b) Effect of initial imperfections (0.5% triggering load) for H-2 section.
0.01 mm/s was controlled by the displacement control method. The vertical displacement was obtained from the machine directly and the horizontal displacements were measured by displacement transducers, which were attached at the center and the quarter points of the test specimens. Since the local buckling occurred in three half-waves, six strain gauges were attached at the centers of each local buckling half-wave expected, as shown Fig. 4(a). Typical compression test configurations for the Hsection are shown in Fig. 4(a) and (b). Since the C-section was assumed to buckle in the flexural mode about the minor axis, its test set-up was prepared to be quite similar to that of the H-section. 3.2. Test section behavior
(c) Effect of yield stress for H-2 and C-2 sections. Fig. 3. Effect of initial imperfections and yield stress.
Axial load versus displacement relations of H-sections and C-sections tested are shown in Fig. 5(a) and 5(b), respectively. The displacements in Fig. 5(a) and (b) were the axial shortening of the test columns measured by the LVDT, which was located at the top of the specimen. As shown in Fig. 5(a) and (b),
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(a) H-sections.
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(b) C-sections. Fig. 5. Axial load versus displacement curves.
(a) Side view.
(b) Front view. Fig. 6. Buckled shape of H-section.
the columns displayed a stable post-local-buckling behavior but some of the C-section did not show a stable behavior after the ultimate loads. As the load was increased, the column shortened elastically and gradually at the beginning of the test. At or near the elastic local buckling load, local buckling occurred as expected, first at the web of the section and then propagated to the flanges of the H-sections and C-sections, except the C-3 section which had a very large width–thickness ratio in the flange. For the H-2 sections, the test local buckling stress was much higher than that computed numerically. For the C1 section, the local buckling occurred at a load 71% lower than that expected. The premature local buckling was due to the concentration of the load applied. Other H-sections and Csections buckled at the load near the theoretical local buckling load within a range of approximately 10%. After the local buckling load was reached, a significant post-local-buckling strength reserve was observed before flexural buckling about the minor axis for the H-section and about the unsymmetrical axis for the C-section occurred, respectively. A significant buckling mode interaction between local buckling and flexural buckling was observed before the maximum load was reached. After the peak load, the horizontal
displacement continued to increase slowly as the load was decreased. Typically, the buckled shapes of the H-2-1 section and C-2-1 section are shown in Fig. 6(a), 6(b) and Fig. 7(a), 7(b), respectively. All the buckled shapes of the test sections agreed well with those of the numerical analysis. The numerical analysis results are also shown in Figs. 6(a), (b) and 7(a), (b) for comparison of the buckled shapes. The buckled shape of the H-section was a mixed mode of local buckling in three short half-waves and flexural buckling in a long half-wave about the minor axis. This mixed mode was similar to that obtained by the numerical analysis, as shown in Fig. 6(a) and (b). The overall buckling mode of the C-section was the flexural buckling about the unsymmetrical axis, and the flange and the web buckled in the local mode as shown in Fig. 7(a) and (b) additionally. The mode interaction between local buckling and flexural buckling observed during testing was quite similar to that obtained by a nonlinear analysis as shown in Fig. 7(a) and (b). 3.3. Test results The experimental local buckling stresses and the ultimate stresses of the test sections are summarized in Table 3 and
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(a) Side view.
(b) Front view. Fig. 7. Buckled shape of C-section.
Table 3 Ultimate stress and local buckling stress of specimens Specimen H-1-1 H-2-1 H-2-2 H-3-1 H-3-2 C-1-1 C-1-2 C-2-1 C-3-1
Ultimate stress (Fmax ) Test (MPa) Analysis (MPa)
Test/Analysis
Local buckling stress (Fcr ) Test (MPa) Analysis (MPa)
Test/Analysis
198.7 170.9 182.0 158.1 165.6 120.0 106.4 155.1 116.4
0.93 0.99 1.06 0.86 0.90 0.88 0.78 0.87 0.77
185.2 143.7 155.2 121.8 128.2 66.7 73.8 128.9 86.7
0.93 1.35 1.46 1.01 1.06 0.68 0.75 1.09 1.12
213.5 172.2 172.2 184.2 184.2 136.9 136.9 177.6 151.7
compared with the numerical analysis results obtained by the program THIN-WALL [18] and LUSAS [19], respectively. The test ultimate stresses were generally lower than the analysis results except for the H-2-2 specimen. As shown in Table 3, the test ultimate stresses of the C-1-2 and the C-3-1 were not in good agreement with those obtained by the numerical analysis. The comparison of the local buckling stresses of the test specimens showed a different trend from the ultimate stresses. The experimental elastic local buckling stresses of the C-1-1 and the C-1-2 specimens were lower than the theoretical elastic local buckling stresses by 25%–32% due to the local collapse of the web, which was caused by the concentration of load. The local buckling stress of the C-1 sections was very much lower than the yield stress and the translation of the effective centroid of the buckled section caused the premature failure of the web before the interaction between local buckling and overall buckling. Except for the H2-2 section, the experimental ultimate stresses of the columns tested were slightly lower than the numerical values because of the adverse effect of the interaction between local buckling and overall buckling of the specimens. For the C-3-1 section, the test ultimate stress was lower than the analysis result by 23.0%, while the test ultimate stress was higher than the analysis result by 6.0% for the H-2-2 section. However, the difference between test ultimate stress and local buckling stress
197.9 106.5 106.5 120.9 120.9 97.9 97.9 118.7 77.1
was 29.7 MPa and 26.8 MPa, respectively. All the specimens displayed more or less significant post-local-buckling strength reserve up to the ultimate load which was determined by the minor axis flexural buckling load. The differences between the experimental ultimate stresses and the local buckling stresses ranged from 13.5 MPa for the H-1-1 section to 53.3 MPa for the C-1-1 section. Since the local buckling stress of the column tested cannot be determined easily, the experimental buckling stress was taken as the stress at the location where the deterioration of stiffness on the load–displacement curves commenced. The local buckling stress was determined by two different methods. The first set of buckling stresses was determined by examining the change in the slope of the load–displacement curves. The second set of buckling stresses was determined by plotting the stress versus the square of the strain and subsequently fitting a line through the test results in the post-buckling region. The intersection of the fitted line and load or stress axis was assumed to be the experimental buckling stress [21]. Since the change in the overall flexural stiffness was subtle and the intersection point was difficult to choose, the average value of the two methods was taken as the experimental local buckling stress in Table 3. In cases where the slope change of the load–displacement curves was not clear and the local buckling stress was difficult to decide, the stress–strain curves, which
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Fig. 8. Stress versus strain curves (H-section).
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where f nl = design axial column strength accounting for the interaction of local buckling and overall buckling; f ne = overall column strength not considering the local buckling strength; Fl = nominal strength based on the local buckling stress; Fy = nominal yield stress of the material. The nominal strengths are obtained by multiplying the allowable stresses by the factor of safety of 1.7, which is basically used in the specifications. Eq. (1) incorporates the concept that the compressive strength of columns is based on the overall buckling stress and is reduced by the ratio of the nominal strength based on the local buckling stress to the yield stress (Fl /Fy ) in order to consider the local buckling of the composing elements. Whenever the nominal stress based on the local buckling stress is higher than the yield stress of the section, the strength formula can predict reasonable design strengths of the section. However, if the nominal local buckling stress is lower than the yield stress, Eq. (1) is liable to produce unreasonable strengths. In the case where the overall column strength and the design local strength are lower than the yield stress, whether the overall column strength is larger than the design local buckling strength or not, the design axial column strength computed by Eq. (1) is extremely low. Consequently, Eq. (1) cannot reasonably include the negative effect of buckling mode interaction between local and overall buckling. In the cases where the local buckling stress is larger than the flexural/flexural–torsional buckling stress, the strength of the column should be determined by the overall buckling stress. However, if the local buckling stress is smaller than the overall buckling stress, the post-local-buckling strength reserve should be considered in the design strength to some extent. To incorporate the negative effect of mode interaction for the compression members undergoing interaction of local buckling and overall buckling, the strength formula in Eq. (1) should be revised properly. The design column strength (Pnl ) can be obtained by multiplying the axial column strength ( f nl ) by the unreduced cross section area (A). The resulting alternative formulation in Eq. (1) in terms of force is
were obtained by the strain gauges attached, were additionally used to determine the local buckling stress. The local buckling stresses of the H-2 sections were higher than those obtained by the numerical analysis by approximately 40.0%. The C-1 sections displayed a local buckling load that was approximately 28.0% lower than that of the numerical analysis. However, the local buckling stresses of all the other sections agreed well with those obtained by the numerical analysis. The typical axial stress versus strain curves for H-sections are shown in Fig. 8. The stress–strain curves for H-sections tested are similar to one another. At the intersection point where the slope changed, local buckling commenced at the web of the columns. Before the ultimate load, a significant post-local-buckling strength reserve was observed on the load–displacement and stress–strain curves except for H-1-1 section, whose width–thickness ratio was comparatively lower than those of other specimens. In Fig. 8, the stress was the average stress obtained by dividing the applied load by the unreduced gross section area, and the strain was the average strain, which was calculated from the gauge values measured at the column center. As shown in Fig. 8, the local buckling started at the strain of approximate 0.001. In some cases of H-2-2 and H-3-1 sections, the local buckling stress could be decided from the stress–strain curves more easily than the load–displacement curves. A significant post-local-buckling strength reserve similar to that in the load–displacement curves was observed. The H-2-2 and H-3-2 sections displayed a continuous increase of strain after the ultimate stress. However, the H-2-1 and H-3-1 sections showed a slightly unstable stresssoftening behavior.
where pnl = design column strength accounting for interaction of local buckling and overall buckling AFne , Pl = resulting limiting load AFl , and Py = yield load AFy .
4. Column design methods
4.2. Effective width method
4.1. Korean highway bridge design specifications
The NAS (AISI 2001) [6] and Eurocode 3 [8] have used the same effective width formula to include the post-localbuckling strength reserve in the design strength of compression members, where the width–thickness ratio is larger than the limit values specified. Therefore, even if the overall column strength formulas are slightly different, the design strengths of the compression members computed according to the NAS and the EC3 are quite similar. The AISC LRFD [7] provides different buckling coefficient K values, and the
The design compressive strength defined in terms of allowable stress in KHBDS [9] (Korean Highway Bridge Design Specifications, 2005) and KS D 3515 [20] can be modified in terms of nominal strength as f nl = f ne ×
Fl Fy
(1)
Pnl = Pne ×
Pl Py
(2)
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calculation method of the effective width is also different from other specifications mentioned. Since the section is generally composed of stiffened and unstiffened elements, the design procedure according to the AISC LRFD is as follows: the reduction factor Q s for the unstiffened elements with adequate K values specified should be calculated first, and then the reduction factor Q a for the stiffened elements is computed with the assumed stress level in the iterative method. The NAS and the EC3 adopted the same effective width formula for both stiffened and unstiffened elements for simplicity. The formula was originally proposed for the stiffened element and produces more conservative values than that for the unstiffened element. Consequently, the AISC LRFD may produce more or less higher design strengths than the NAS and EC3 in general. The buckling coefficient K values 4.0 and 0.43 are used to calculate the effective widths of the stiffened and unstiffened elements in isolation, respectively. The unreduced gross area is used to compute the limiting load in the AISC LRFD rather than the effective area which is used in the NAS and EC3. 4.3. Direct strength method The effective width method (EWM), which can consider the post-buckling strength reserve into the design, may need iterative calculations to compute the effective width of the composing element separately. As the section shape becomes more complicated, the accurate computation of the effective width becomes more difficult and the separation of each element of complicated sections becomes unreasonable. To overcome these problems, the direct strength method (DSM) for cold-formed steel sections has been developed by Shafer and Pekoz [10] and further studied by Hancock et al. [2,12]. The method incorporates the empirical formulae and the elastic local or distortional buckling stress obtained by the rigorous buckling analysis or reliable strength formulas. The application of the direct strength method to a welded section was studied recently by Kwon et al. [13]. The direct strength formulas considering the interaction between local buckling and overall buckling for cold-formed steel sections, which have been proposed by Shafer and Pekoz [10], are given by for λ ≤ 0.776 f nl = Fne .
(3a)
For λ > 0.776 f nl = 1 − 0.15
Fcrl Fne
0.4 !
Fcrl Fne
0.4 Fne
(3b)
in the Eurocode 3 [8] or the equations in Table 3.3.2 of the KHBDS [9]. The elastic local buckling stress Fcrl can be computed by the rigorous Finite Element Method (FEM) or the Finite Strip Method (FSM). The exponent 0.4 was used instead of 0.5, which was used in Von Karman [22] and Winter formulae [23], for the effective width of elements to reflect a higher post-local-buckling strength reserve for the unreduced gross section than for an element in isolation. Eqs. (3a) and (3b) were recently adopted as an alternative design method to the EWM by the NAS [14] and Australian Standards [15]. The DSM is much simpler to apply than the EWM since it uses gross section properties rather than effective section properties. It has already been proven quite accurate in comparison with the EWM for cold-formed steel sections [2]. 4.4. Proposed direct strength equations The direct strength formula considering the interaction between local buckling and overall buckling can be reasonably used for welded sections and hot-rolled sections. However, the strength formula should be modified to account for the different characteristics between cold-formed and welded steel sections. First, the effect of interaction between local buckling and overall buckling on the strength of the welded compressive members might be less significant, and the post-local-buckling strength reserve is smaller than that for the cold-formed steel section since the width–thickness ratios of commonly used welded sections are comparatively smaller than those of coldformed steel sections. Secondly, unlipped welded channel sections are commonly used as compression members, while lipped channel sections are common for cold-formed steel sections. However, in the case of single symmetrical sections such as the unlipped C-section, of which the local buckling stress was very much lower than the yield stress, the transition of the effective centroid of the locally buckled section might cause a premature failure. To account for these phenomena, the modified Winter formula [23] was proposed rather than Eqs. (3a) and (3b). When the exponent 0.5, as used in Winter formulas, is used, the coefficient 0.15 in Eq. (3b) is adopted rather than 0.22 in Winter formulas, which reflects a higher post-local-buckling strength reserve in the inelastic buckling range of a material for the intermediate length column. The equations for the limiting stress f nl considering the interaction between local buckling and overall buckling for the welded section are given by for λ ≤ 0.816 f nl = Fne .
(4a)
For λ > 0.816
√
where λ = Fne /Fcrl ; f nl = limiting stress accounting for local buckling and overall buckling (MPa) ; Fcrl = elastic local buckling stress (MPa) ; Fne = overall column strength based on the overall failure mode (MPa), which is determined from the minimum of the elastic flexural, torsional, and flexural–torsional buckling stresses. The overall column strength Fne can be calculated from Eqs. (C4-2) and (C43) of the NAS (AISI) [6] or Eqs. (6.47), (6.48) and (6.49)
f nl = 1 − 0.15 where p λ = Fne /Fcrl .
Fcrl Fne
0.5 !
Fcrl Fne
0.5 Fne
(4b)
(4c)
The results predicted by the strength formulas proposed are compared with tests results of the welded H-sections and
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Fig. 9. Comparison between DSM curves and test results.
the C-sections, and Eqs. (3a) and (3b) in Fig. 9. As shown in Fig. 9, Eqs. (3a) and (3b) predict reasonable strengths for the columns undergoing the interaction between local buckling and overall buckling in comparison with the test results, as long as the slenderness ratio factor is small. However, when the slenderness ratio factor λ becomes larger than 1.5, Eq. (3b) produces very optimistic strengths in comparison with the test results of channel sections. The strength curve equation (4b) proposed can predict slightly higher design strengths than Eq. (3b) when the slenderness ratio factor λ remains between 0.776 and 1.0. However, when the slenderness ratio factor λ becomes larger than 1.0, Eq. (4b) produces slightly lower values than Eq. (3b). The difference in the strengths predicted by Eqs. (3b) and (4b) becomes more significant as the slenderness ratio factor becomes large. The ultimate strengths of C-sections tested are under the design strength curves of Eq. (4b) proposed. However, Eq. (3b) predicts the ultimate strengths of C-sections too optimistically. As shown in Fig. 9, the proposed design curves have a less satisfactory fit to the ultimate strength of H-sections than Eq. (3b), but predict a less optimistic ultimate strength for C-sections than Eq. (3b) does. Since the single symmetric C-section generally shows a premature failure due to the translation of the effective centroid of the locally buckled section, the design strength curves is forced to predict somewhat optimistically according to the slenderness of the plate elements. Consequently, it can be concluded that the proposed ultimate strength curves of Eqs. (4a) and (4b) can predict more reliable ultimate strengths for the welded section columns than Eqs. (3a) and (3b) when local buckling and overall buckling occur simultaneously or nearly simultaneously. 4.5. Comparison with test results To compare the DSM with the EWM directly, the strength equations should be expressed in terms of load. If the distortional buckling does not occur, the design strength of compression members can be taken as the nominal strength Pnl , which is obtained by multiplying the limiting stress ( f nl ) in Eqs. (4a) and (4b) by the full unreduced section area (A). Therefore, the alternative formulations of Eqs. (4a) and (4b) in terms of load are given by
Fig. 10. Comparison between DSM (based on NAS) and test results.
for λ ≤ 0.816 Pnl = Pne .
(5a)
For λ > 0.816 Pnl = 1 − 0.15
Pcrl Pne
0.5 !
Pcrl Pne
0.5 Pne
(5b)
√ where λ = Pne /Pcrl , Pcrl = elastic local buckling load AFcrl , pne = nominal column design strength AFne . The Direct Strength Method (DSM) curves based on the overall column strength formula provided by the NAS (AISI) and Eurocode 3 are compared with the test results in Fig. 10 and Fig. 11, respectively. Since the nominal column strengths computed by the NAS and EC3 column strength formula are slightly different, the test results were generalized by the two different design strengths and compared. The test results of the welded I-section and channel section columns with the nominal yield stress of 350 MPa, which were executed previously at the University of Sydney [16,17], are also included in Figs. 10 and 11. For most of the columns, the DSM curve predicts the ultimate strengths fairly conservatively. However, in the case of very slender unlipped channel sections, the ultimate strengths are predicted optimistically by the DSM curve. The DSM curve predicts the ultimate strengths too optimistically in comparison with the test results of very slender channel sections, which were executed by Rasmussen and Hancock [17]. However, it can be concluded that the ultimate strengths of welded Hsections and C-sections, which are not too slender, can be predicted fairly reasonably by the DSM. The column design strengths, which are calculated according to the Effective Width Method (EWM), the NAS [6] and the EC3 [8], and the DSM proposed have been summarized and compared with test results in Table 4. Since the web of the C-1-1 and C-1-2 sections failed prematurely in the local mode before interaction between local buckling and overall buckling, the test results of the C-1-1 and C-1-2 sections were not compatible with the design strengths predicted by the specifications. For the H-sections tested, the ratio of the maximum test strength to NAS ranged from 1.33 to 1.62, the ratio of the maximum test strength to EC3 from 1.43 to 1.83 and the ratio of the maximum test strength to DSM ranged from
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Table 4 Comparison of tests and design strengths Specimen H-1-1 H-2-1 H-2-2 H-3-1 H-3-2 C-1-1 C-1-2 C-2-1 C-3-1
Design strengths (kN) NAS EC3
AISC
DSM
513.0 585.7 585.7 584.8 584.8 497.6 497.6 504.6 323.1
575.0 766.9 766.9 682.7 682.7 596.9 596.9 632.6 251.9
550.2 566.2 566.2 564.3 564.3 529.7 529.7 505.5 400.8
444.0 519.6 519.6 519.0 519.0 447.9 447.9 442.8 334.3
Test ultimate strength (kN) 715.3 892.1 950.0 739.9 775.0 502.9 444.1 697.8 519.9
Test/Design NAS EC3
AISC
DSM
1.39 1.52 1.62 1.27 1.33 1.01 0.89 1.38 1.61
1.24 1.16 1.24 1.08 1.14 0.84 0.74 1.10 2.06
1.30 1.58 1.68 1.31 1.37 0.95 0.84 1.38 1.30
1.61 1.72 1.83 1.43 1.49 1.12 0.99 1.58 1.56
Table 5 Comparison between design methods
Fig. 11. Comparison between DSM (based on EC3) and test results.
1.30 to 1.68. The ratio of the test ultimate strength to the AISC ranged from 1.08 to 1.24. For the C-sections tested, the ratios of the ultimate test strength to NAS were 1.38 and 1.61, the ratios of the ultimate test strength to EC3 1.56 and 1.58 and the ratios of the maximum test strength to DSM were 1.30–1.38. The ratios of the test ultimate strength to the AISC were 1.10 and 2.06. For H-sections and C-sections, the difference between the test ultimate strength and design strength was fairly constant. However, for the C-section, the difference for AISC was highly variable in magnitude. The DSM and EWM predicted the ultimate strength for the H-sections fairly conservatively. In the case of the C-sections, the predictions of the ultimate strength by the specifications and the DSM were fairly conservative. However, it can also be concluded that the ultimate strength of the channel sections such as the C-3 specimen which had exceptionally slender flange was predicted too conservatively by the EWM but fairly by the DSM. Generally speaking, the test results for H-sections and C-sections validate the DSM clearly. The comparison of predictions for the ultimate strength of the sections among the design specifications and the DSM proposed is summarized in Table 5. As shown in Table 5, since the provisions to take account of the local buckling stress on the column strength are not adequate and the post-local-buckling strength is not considered at all, the KHBDS [9] cannot closely predict the ultimate strength of the slender welded section columns undergoing interaction between local buckling and flexural/flexural–torsion buckling. The average ratio of KHBDS to DSM was 0.19. Even if the width–thickness of the test sections is beyond the limit prescribed in the specification, the
Specimen
NAS/DSM
EC3/DSM
AISC/DSM
KHBDS/DSM
H-1 H-2 H-3 C-1 C-2 C-3 Average Standard deviation
0.93 1.03 1.04 0.94 1.00 0.81 0.97 0.07
0.81 0.92 0.92 0.85 0.88 0.83 0.88 0.04
1.05 1.35 1.21 1.13 1.25 0.63 1.15 0.15
0.25 0.19 0.19 0.20 0.21 0.10 0.19 0.03
predictions of the ultimate strength by the KHBDS are absurdly lower than the test ultimate strengths for the sections which may have the buckling interaction between local buckling and overall buckling. The slight difference between NAS, EC3 or AISC and DSM might result from the magnitude of post-local-buckling strength considered in the specifications and the effect of interaction between the elements composing the whole cross section. Unlike the EWM, which estimates the effective width of each element in isolation, the DSM handles the whole section at a one time. Thus, the difficulty in computing the effective width of the each element in isolation for the EWM has been exempted in the DSM. The design strengths predicted by the DSM proposed agree quite closely with those estimated by the NAS. The ratio of the design strengths predicted by the NAS to those obtained according to the DSM ranged from 0.81 to 1.04 with the average 0.97 and the standard deviation 0.07, as shown in Table 5. The C-3 section has an exceptional width–thickness ratio of the flange, which is four times the buckling limit of 14.0. In comparison with EC3, the DSM estimates the design strength higher than the EC3 by an average of 12%. The prediction by the EC3 is slightly more conservative than that by the NAS except for the C-3 specimen. The ultimate strengths predicted by the AISC are higher than those predicted by the DSM except for the C-3 by an average of 15%. The fact that the AISC uses a different effective width formula between stiffened and unstiffened elements made the difference in the predictions. The difference in the ultimate strengths predicted by the specifications is mainly the result of the single, effective width formula for both stiffened and unstiffened elements adopted by the NAS and the EC3 with the advantage of simplicity. Since the formula for the unstiffened element
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has more post-local-buckling strength reserve and produces a higher value than that for the stiffened element, the NAS and EC3 predict the ultimate strength less optimistically than the AISC. From the comparison of the ultimate strengths predicted by the design specifications based on the effective width concept and the DSM, it can be concluded that the design ultimate strength of welded sections can reasonably be predicted by the DSM. The DSM proposed can be used as an alternative method to the conventional EWM, which has been used for the welded section columns. However, in the case of the welded sections such as a lipped channel section, which is liable to undergo distortional buckling, even if the sections are not of common shape, the distortional limiting strength should also be considered to predict the overall column strength. Therefore, the DSM should determine the ultimate strength for the interaction between local buckling and overall buckling, and distortional buckling and overall buckling and then take the lesser of the two as the column strength. The DSM should also be carefully applied to very slender single symmetric sections such as a channel section. In that case, if the shift of the centroid of the effective area relative to the center of gravity of the gross section is not prevented by constructional arrangements, the possible shift of the centroid and the resulting additional moment should be determined, and the beam–column strength formula for the DSM should be used. The beam–column strength formula for the DSM is under preparation at the moment. 5. Conclusions The experimental study for the application of the direct strength method (DSM) to the thin-walled welded section columns undergoing interaction between local buckling and flexural/flexural–torsional buckling was conducted. The local buckling, which occurs prior to the overall column buckling and has a significant post-local-buckling strength reserve, deteriorates the overall column strength to some extent. This phenomenon should be considered appropriately to predict conservatively and accurately the ultimate strength of the welded sections as well as the cold-formed sections. The direct strength formulas, which were adopted for the design of the cold-formed steel sections, have been modified to take into account the effect of the interaction between local buckling and overall buckling and post-local-buckling strength reserve for the welded sections. The member strengths from the direct strength formulas proposed were compared with the test results of welded H-sections and C-sections fabricated from the hot-rolled steel plates with nominal yield stresses of 240 and 350 MPa. The ultimate strengths predicted by the direct strength method were also compared with those estimated by the effective width method adopted in the NAS, AISC and EC3. The comparison of the ultimate strengths predicted by the design specifications and the test results has proven the reliability of the direct strength method. The direct strength formulas proposed have been proven easier to apply and accurate to predict the ultimate strength of the columns
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that undergo the interaction between local buckling and overall buckling and fail in the mixed mode between local buckling and flexural–torsional buckling. However, detailed provisions as to the width–thickness ratio limit of single symmetric sections, such as a channel section, should be prepared immediately. The proposed direct strength formulas should be further verified and calibrated against the test results of various kinds of sections for practical use. Acknowledgements This research was supported by the Korea Bridge Design and Engineering Center under the sponsorship of 2004 Development of Core Construction Technology Project of Korea Ministry of Construction and Transportation. References [1] Kemp AR. Inelastic lateral and local buckling in design codes. J Struct Eng, ASCE 1996;122(4):374–82. [2] Hancock GJ, Murray TM, Ellifritt DS. Cold-formed steel structures to the AISI specification. Marcel Dekker, Inc.; 2004. [3] Hancock GJ, Kwon YB, Bernard ES. Strength design curves for thinwalled members. J Constr Steel Res 1994;31(3):169–86. [4] Mulligan GP. The influence of local buckling on the structural behavior of singly-symmetric cold-formed steel columns. Ph.D. thesis. Cornell University; 1983. [5] Yang D, Hancock GJ. Compression tests of high strength steel channel columns with interaction between local and distortional modes. J Struct Eng, ASCE 2004;130(12):1954–63. [6] American Iron and Steel Institute (AISI). North American specifications for the design of cold-formed steel structural members. Washington (DC, USA); 2001. [7] American Iron and Steel Construction (AISC). Load and resistance factor design specification for steel structural buildings. Chicago (Il, USA); 1999. [8] European Committee for Standardisation (ECS). Eurocode 3: Design of steel structures, Part 1-1: General rules and rules for buildings. Brussels (Belgium); 2003. [9] Korean Association of Highway and Transportation Officials. Korean highway bridge design specifications (KHBDS). Korea; 2005. [10] 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. Elsevier; 1998. [11] Schafer BW. Advances in direct strength design of thin-walled members. In: Hancock GJ, Bradford M, Wilkinson T, Uy B, Rasmussen K, editors. Proceedings advances in structures conference, ASSCCA03. 2003. [12] Hancock GJ. Developments in the direct strength design of cold-formed steel structural members. In: Proceedings, 3rd international symposium on steel structures, KSSC. 2005. p. 120–31. [13] Kwon YB, Kim NG, Kim BS. A study on the direct strength method for compression members undergoing mixed mode buckling. In: Proceedings, 3rd international symposium on steel structures, KSSC. 2005. p. 120–31. [14] American Iron and Steel Institute (AISI). Supplement 2004 to the North American specifications for the design of cold-formed steel structural members. Washington (DC, USA); 2004. [15] Standards Australia, Cold-formed steel structures AS/NZS 4600: 2005. Sydney (NSW, Australia); 2005. [16] Davids AJ, Hancock GJ. Compression tests of long welded I-section columns. J Struct Eng, ASCE 1986;112(10):2281–97. [17] Rasmussen KJR, Hancock GJ. Compression tests of welded channel section columns. J Struct Eng, ASCE 1989;115(4):789–808. [18] Papangelis JP, Hancock GJ. THIN-WALL (ver. 2.0). Sydney (Australia): Center for Advanced Structural Engineering, Dept. of Civil Engineering, Univ. of Sydney; 1998.
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[19] FEA Co., Ltd. Lusas element reference manual & user’s manual (version 13.4). 2002. [20] Korean Standards. Welded structural steel, KSD3515. Seoul (Korea). [21] Venkataramaiah KR, Roorda J. Analysis of local plate buckling experimental data. In: Proc. 6th int. specialty conf. on cold-formed steel
structures. St. Louis (Mo, USA): Univ. of Missouri-Rolla; 1982. p. 45–74. [22] Von Karman T, Sechler EE, Donnell LH. The strength of thin plates in compression. Transactions, ASME, vol. 54. 1932. MP 54–5. [23] Winter G. Strength of thin steel compression flanges. Transactions, ASCE, vol. 112. 1947. p. 527–76. Paper no. 2305.
Compression tests of welded section columns undergoing buckling interaction Young Bong Kwon a,∗ , Nak Gu Kim b , G.J. Hancock c a Department of Civil Engineering, Yeungnam University, Gyongsan, 712-749, Republic of Korea b Yeungnam University, Gyongsan, 712-749, Republic of Korea c Faculty of Engineering, University of Sydney, NSW 2006, Australia
Received 1 September 2006; accepted 31 January 2007
Abstract This paper describes a series of compression tests performed on welded H-section and channel section columns fabricated from a mild steel plate of thickness 6.0 mm with nominal yield stress of 240 MPa. The ultimate strength and performance of the compression members undergoing nonlinear interaction between local and overall buckling were investigated experimentally and theoretically. The compression tests indicated that the interaction between local and overall buckling had a significant negative effect on the ultimate strength of the thin-walled welded steel section columns. The Direct Strength Method (DSM), which was newly developed and adopted as an alternative to the effective width method for the design of cold-formed steel sections recently by NAS (AISI, 2004), was calibrated by using the test results for application to welded steel sections. This paper confirms that the Direct Strength Method can properly predict the ultimate strength of welded section columns when local buckling and flexural buckling occur simultaneously or nearly simultaneously. c 2007 Elsevier Ltd. All rights reserved.
Keywords: Direct strength method; Effective width method; Interaction between local and overall buckling; Ultimate strength; Performance; Welded sections
1. Introduction The compression and flexural members of hot-rolled shapes and welded sections fabricated from hot-rolled plate will normally buckle in local and flexural/flexural–torsional buckling mode [1], while the cold-formed steel sections will buckle in the distortional mode in addition to those modes [2]. However, whenever the local or distortional buckling stress is lower than the overall buckling stress, the interaction between local or distortional and overall buckling may occur and have a significant effect on the performance of the sections [3–5]. Since the interaction between local and overall buckling generally deteriorates the overall column strength, it is necessary to account for the negative effect of buckling interaction in the conservative prediction of the ultimate strength of columns.
∗ Corresponding address: Department of Civil and Environmental Engineering, 214-1 Daedong, 712-749 Gyongsan-si, Gyongbuk-do, Republic of Korea. Tel.: +82 53 810 2418, +82 11 802 2418; fax: +82 53 810 4622. E-mail address: [email protected] (Y.B. Kwon).
c 2007 Elsevier Ltd. All rights reserved. 0143-974X/$ - see front matter doi:10.1016/j.jcsr.2007.01.011
The ultimate strength of compression members, which are composed of thin plate elements, is dependent on both the width–thickness ratio of the plate elements and the slenderness ratio of the columns. When the local buckling stress is lower than the overall buckling stress or both types of buckling occur nearly simultaneously, local buckling may negatively affect the column strength. Because the local buckling mode has a post-buckling strength reserve, it has generally been considered in the design strength through the effective width concept [6–8]. However, since the computation of the effective width can be tedious and complicated, some countries such as Japan and Korea, do not account for the post-local-buckling strength reserve in the design strength of compression members to maintain simplicity. Instead of using the effective width concept, the overall column strength is simply reduced by the ratio of the design strength based on the local buckling stress to the yield stress in the Korean Highway Bridge Design Specifications [9]. The Direct Strength Method (DSM) has been developed by Schafer and Pekoz [10] and studied further by many researchers [11–13]. The method has been developed to
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Notation Abbreviation ASD DSM EWM LRFD
Allowable stress design Direct strength method Effective width method Load and resistance factor design
Symbols A b bf d Fcrl Fcr o Fne f nl Fl Fmax Fy l Pcrl Pl Pnl Pne Qa Qs r tf tw λ
Cross sectional area Clear width of plate element Flange width Web depth Elastic local buckling stress Elastic flexural/flexural–torsional buckling stress Overall column strength based on the overall buckling stress Design limiting stress accounting for interaction of local and overall buckling Nominal strength based on the local buckling stress Ultimate strength determined in test Yield stress Length of specimen Elastic local buckling load (=Fcrl × A) Resulting limiting load (=Fl × A) Design limiting strength (=Fnl × A) Compression member design strength (=Fne × A) Form factor for stiffened element Form factor for unstiffened element Radius of gyration Flange thickness Web thickness √ √ Slenderness ratio factor (= Fne /Fcrl , Pne /Pcrl )
overcome the weak points of the effective width method (EWM), which has been used in thin-walled steel section design for over 60 years. These weak points are the complication in accurate computation of effective width and the difficulty in consideration of elements interacting in isolation. Also, as coldformed steel sections become more complex with additional edge and intermediate stiffeners, calculations of effective width can become very complex. The interaction between local or distortional and overall buckling for cold-formed steel sections has been studied, and simple design formulas have already been presented. In 2004, the Direct Strength Method was adopted as an alternative to the effective width method, which has been used for thin section design by NAS Supplement 1 (AISI, 2004) [14] and the Australian/New Zealand Cold-Formed Steel Structures Standard AS/NZS 4600 [15]. However, there has been less research concerning the application of this method to hot-rolled and welded sections.
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This paper aims to develop the direct strength formulas of design for welded section compression members. The application of the Direct Strength Method to mild steel welded sections was studied experimentally and theoretically. A series of compression tests was performed on welded Hsections and C-sections fabricated from mild steel plates of thickness 6.0 mm with nominal yield stress 240 MPa to develop the direct strength formulas for the welded steel section compressive members undergoing interaction between local and flexural/flexural–torsional buckling. Nonlinear analyses of the sections tested have also been conducted to compare their results with the test results. The direct strength formulas of design were compared with the effective width method (EWM) and the allowable stress design method (ASD), which is currently used in Korea. The direct strength formulas proposed have been proven accurate and efficient to predict the ultimate strength of columns when local buckling and overall buckling occur simultaneously or nearly simultaneously. The results of David and Hancock [16] and Rasmussen and Hancock [17] are also included in the paper for comparison. 2. Test sections 2.1. General A series of compression tests were performed on mild steel H-sections and C-sections of thickness 6.0 mm with nominal yield and ultimate stresses 240 MPa and 400 MPa, respectively, which were fabricated by the continuous fillet welding of effective width on both sides of the flange–web joint. The size of fillet welds was determined as 6.0 mm according to the AISC specifications [7]. The geometry of the welded Hsection and the C-section tested are shown in Fig. 1. The dimensions of the test specimens were optimized in order to ensure that the local buckling stress was slightly lower than the flexural/flexural–torsional buckling stress and consequently, the interaction between local and overall buckling occurred before the ultimate load was reached. The cross sections tested were optimized by using a finite strip analysis program THINWALL [18], repeatedly. The width–thickness ratios of the webs and the flanges of the test sections were selected, such that the elastic local buckling stress of the section was lower than half the yield stress (Fy /2) and significant post-local-buckling strength reserve was displayed before the ultimate load, which is mainly dependent on the overall buckling strength, was, reached. The limiting width–thickness ratio p for Class 3 cross sections is expressed as d/tw ≤ 42 235/Fy for the web and p b/ t f ≤ 14 235/Fy for the flange of the sections according to Eurocode3 [8]. Since the width–thickness ratios of the webs or the flanges of sections tested are larger than the slenderness limit values, the sections selected can be classified as Class 4, where the effective width must be used to account for the local buckling effect. The dimensions of the H-sections and the Csections tested are summarized in Table 1. The thickness of all the specimens tested was 6.0 mm, and the dimensions of the webs and flanges of the sections were adjusted to ensure the
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(a) H-section.
(b) C-section. Fig. 1. Test sections.
Table 1 Dimensions of test sections Specimens
b f (mm)
d (mm)
t f (mm)
tw (mm)
d/tw
b/t f
l (mm)
k ∗ l/r
A (mm2 )
H-1 H-2 H-3 C-1 C-2 C-3
100.0 160.0 140.0 153.0 103.0 303.0
400.0 550.0 500.0 550.0 500.0 150.0
6.0 6.0 6.0 6.0 6.0 6.0
6.0 6.0 6.0 6.0 6.0 6.0
66.7 91.7 83.3 91.7 83.3 25.0
7.8 12.8 11.2 24.5 16.0 49.5
1400.0 2200.0 2400.0 2200.0 2200.0 2200.0
58.59 54.95 63.49 55.37 53.97 22.89
3600.0 5220.0 4680.0 5136.0 4236.0 4536.0
k ∗ = 0.7.
interaction between local buckling and overall buckling of the test sections. 2.2. Determination of overall length of specimens For the columns tested, the effect of the interaction between local and overall buckling was the focus of the tests. In order to determine the optimum lengths of the test sections, the local and overall buckling stresses of the specimens needed to be computed accurately. The elastic buckling stresses of the H2 and C-2 specimens, subjected to uniform compression, are illustrated in Fig. 2(a) and 2(b) as the buckling stress versus buckle half-wavelength, respectively. The lengths of the test specimens were determined by using the elastic buckling stress versus buckle half-wavelength curves obtained by the program THIN-WALL [18]. To ensure that the overall buckling stress is higher than the local buckling stress and that there exists a significant post-local-buckling strength reserve before the final collapse of the specimens, the test column lengths were determined. The lengths of test specimens should be in the hatched range shown in Fig. 2(a) and (b), which are longer than the half-wavelengths where the mixed mode of local and overall buckling occurs (point B) and shorter than the half-wavelengths (point D) where the overall buckling stresses are equal to the local buckling stresses. To avoid the failure due to only the local buckling without buckling interaction, the overall lengths of the test specimens were determined to be in range from 1400 to 2400 mm. The local buckling stress minima (point A) occurred in the curves at the half-wavelengths of 480 mm and 500 mm and the local buckling stresses of 118.7 MPa and 106.5 MPa for the
C-2 specimen and the H-2 specimen, respectively. The mixed mode buckling between local and overall buckling occurred at the half-wavelength of 1000 mm and 1200 mm for the C2 specimen and the H-2 specimen, respectively. The overall buckling modes of the H-2 specimen of 1540 mm in length (point C) was the flexural buckling mode in slight interaction with the local buckling mode, as shown in Fig. 2(b). Since the flexural buckling load about the unsymmetrical axis (xaxis) of the C-2 section was larger than the flexural–torsional buckling load about the symmetrical axis (y-axis), the overall buckling mode at the length of 1540 mm was the flexural mode about x-axis in slight interaction with the local buckling rather than the flexural–torsional mode about y-axis, as shown in Fig. 2(a). A noticeable interaction between overall buckling and local buckling was generally observed for the H-section and C-section columns at lengths between points B and D in Fig. 2(a) and (b). Since the unloaded and the loaded end of the test columns were a fixed and a hinged boundary condition, respectively, the effective length of the columns was taken as 0.7L. The slenderness ratios (k L/r ) of the test specimens ranged from 22.9 to 63.5. 2.3. Numerical analysis results The material and geometrical nonlinear analysis of the specimens selected for the compression test was conducted using the program LUSAS [19], to investigate the ultimate strength and the structural behavior. To study the effect of buckling interaction between local and overall modes on the ultimate strength of the columns, an elastic buckling analysis was conducted first to find out the local buckling mode.
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(a) C-2 section.
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(b) H-2 section. Fig. 2. Typical buckling stress versus half-wavelength curves.
The first several local buckling loads and modes were nearly similar. Since the buckling interaction seemed to be quite sensitive to the imperfections, the sensitivity of the magnitude of imperfections was studied. The load versus shortening curves of the H-2 section obtained by the nonlinear analysis using the LUSAS with the different initial imperfections which were the first local buckling mode multiplied by the magnification factor 0.1, 0.01 and 0.001, respectively, are compared in Fig. 3(a) and 3(b). In the analysis, a triggering load was applied laterally at the column center to cause the interaction between local buckling and overall buckling. The ultimate loads of the H-2 section computed with different magnitudes of imperfections were nearly similar in the case of 0.1% triggering load applied as shown in Fig. 3(a). The ultimate loads obtained with 0.1% triggering load were slightly higher than those obtained with 0.5% triggering load. The ultimate loads and the structural behavior after the peak load became slightly different in the case of 0.5% triggering load applied as shown in Fig. 3(b). In consideration of the effect of the initial imperfections and the triggering load, the initial imperfections could be assumed as the first local buckling mode multiplied by the factor of 0.01 and the triggering load could be taken as 0.1% of the vertical reference load in the further analysis. The 4-node shell element (QTS4) was used for the numerical modeling of the section, and the loaded end was assumed as a hinged boundary condition with the vertical direction of the loaded end to be free to move to allow the vertical application of the load. The uniform displacement control technique was used at the loaded end to make a similar boundary condition to the compression test. The bottom end of the column was assumed as a fixed boundary condition as the test condition. The average yield stress obtained from the tensile tests of the coupons which were cut from the flange and web of the welded sections was approximately 260 MPa, which was slightly higher than the nominal yield stress of 240 MPa. The effect of the magnitude of the yield stress on the ultimate strengths of the C-2 and H-2 sections was not negligible as illustrated in Fig. 3(c). Therefore, the yield stress of the material was taken as 260 MPa in the further analysis of the specimens. Young’s modulus was assumed to be 2.0 × 105 MPa and Poisson’s ratio was taken as 0.3. The stress–strain relation of the material was assumed
elastic–perfectly plastic neglecting the strain-hardening and the von Mises yield criterion was applied for the plasticity theory of the material. The ultimate stresses of the sections obtained by the nonlinear analysis are summarized in Table 2. The elastic local and overall buckling stresses, and the local buckle halfwavelength obtained by the program THIN-WALL [18] are also given in the table for comparison. The elastic overall buckling modes obtained by THIN-WALL were in the interaction mode between local buckling and flexural/flexural–torsional buckling. Since the interaction between local buckling and flexural/flexural–torsional buckling negatively affected the column buckling stress, Euler buckling stresses FE were expected to be higher than the overall buckling stresses computed by THIN-WALL. The elastic local buckling stresses were lower than the maximum stresses and overall buckling stresses of the sections. Since the elastic overall buckling stresses of test columns were larger than the elastic local buckling stresses by the difference between 69.6 MPa for H1 and 112.9 MPa for C-2, it was supposed that most of test columns might fail in the mixed mode of local and overall buckling. 3. Compression tests 3.1. General The H-sections and the C-sections listed in Table 1 were chosen for the pseudo-static compression test, considering the maximum loading capacity and the dimension of the swivel head of the testing machine. The steel grade of the test sections was SM400 structural steel to KS D3515 [20] (equivalent to ASTM A36 Steel), of which the nominal yield stress and the ultimate tensile stress were 240 MPa and 400 MPa, respectively. The end plates of thickness 30 mm were welded to both ends of the specimens to minimize eccentric loading and to prevent the local failure of the specimen ends. The loaded end boundary condition was free about the x- and y-direction rotations and movable in the vertical direction, and the bottom end boundary condition of the specimens was fully fixed. The concentric compression test was conducted by using a 3000 kN MTS testing machine. Downward loading at
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Table 2 Maximum and buckling stress of sections Specimen
Maximum stress Fmax (MPa)
Elastic local buckling stress Fcrl (MPa)
Elastic overall buckling stress Fcr o (MPa)
Euler buckling stress FE (MPa)
Local buckle half-wavelength l (mm)
H-1 H-2 H-3 C-1 C-2 C-3
213.5 172.2 184.2 136.9 177.6 151.7
197.9 106.5 120.9 97.9 118.7 77.1
267.5 194.3 233.8 185.7 231.6 159.8
575.0 653.7 489.6 643.8 621.1 433.6
380 500 460 550 480 550
Fcr o : Elastic buckling stress computed by THIN-WALL.
(a) Effect of initial imperfections (0.1% triggering load) for H-2 section.
(a) Locations of gauges & LVDT.
(b) Test configuration.
Fig. 4. Test set-up (H-section).
(b) Effect of initial imperfections (0.5% triggering load) for H-2 section.
0.01 mm/s was controlled by the displacement control method. The vertical displacement was obtained from the machine directly and the horizontal displacements were measured by displacement transducers, which were attached at the center and the quarter points of the test specimens. Since the local buckling occurred in three half-waves, six strain gauges were attached at the centers of each local buckling half-wave expected, as shown Fig. 4(a). Typical compression test configurations for the Hsection are shown in Fig. 4(a) and (b). Since the C-section was assumed to buckle in the flexural mode about the minor axis, its test set-up was prepared to be quite similar to that of the H-section. 3.2. Test section behavior
(c) Effect of yield stress for H-2 and C-2 sections. Fig. 3. Effect of initial imperfections and yield stress.
Axial load versus displacement relations of H-sections and C-sections tested are shown in Fig. 5(a) and 5(b), respectively. The displacements in Fig. 5(a) and (b) were the axial shortening of the test columns measured by the LVDT, which was located at the top of the specimen. As shown in Fig. 5(a) and (b),
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(a) H-sections.
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(b) C-sections. Fig. 5. Axial load versus displacement curves.
(a) Side view.
(b) Front view. Fig. 6. Buckled shape of H-section.
the columns displayed a stable post-local-buckling behavior but some of the C-section did not show a stable behavior after the ultimate loads. As the load was increased, the column shortened elastically and gradually at the beginning of the test. At or near the elastic local buckling load, local buckling occurred as expected, first at the web of the section and then propagated to the flanges of the H-sections and C-sections, except the C-3 section which had a very large width–thickness ratio in the flange. For the H-2 sections, the test local buckling stress was much higher than that computed numerically. For the C1 section, the local buckling occurred at a load 71% lower than that expected. The premature local buckling was due to the concentration of the load applied. Other H-sections and Csections buckled at the load near the theoretical local buckling load within a range of approximately 10%. After the local buckling load was reached, a significant post-local-buckling strength reserve was observed before flexural buckling about the minor axis for the H-section and about the unsymmetrical axis for the C-section occurred, respectively. A significant buckling mode interaction between local buckling and flexural buckling was observed before the maximum load was reached. After the peak load, the horizontal
displacement continued to increase slowly as the load was decreased. Typically, the buckled shapes of the H-2-1 section and C-2-1 section are shown in Fig. 6(a), 6(b) and Fig. 7(a), 7(b), respectively. All the buckled shapes of the test sections agreed well with those of the numerical analysis. The numerical analysis results are also shown in Figs. 6(a), (b) and 7(a), (b) for comparison of the buckled shapes. The buckled shape of the H-section was a mixed mode of local buckling in three short half-waves and flexural buckling in a long half-wave about the minor axis. This mixed mode was similar to that obtained by the numerical analysis, as shown in Fig. 6(a) and (b). The overall buckling mode of the C-section was the flexural buckling about the unsymmetrical axis, and the flange and the web buckled in the local mode as shown in Fig. 7(a) and (b) additionally. The mode interaction between local buckling and flexural buckling observed during testing was quite similar to that obtained by a nonlinear analysis as shown in Fig. 7(a) and (b). 3.3. Test results The experimental local buckling stresses and the ultimate stresses of the test sections are summarized in Table 3 and
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(a) Side view.
(b) Front view. Fig. 7. Buckled shape of C-section.
Table 3 Ultimate stress and local buckling stress of specimens Specimen H-1-1 H-2-1 H-2-2 H-3-1 H-3-2 C-1-1 C-1-2 C-2-1 C-3-1
Ultimate stress (Fmax ) Test (MPa) Analysis (MPa)
Test/Analysis
Local buckling stress (Fcr ) Test (MPa) Analysis (MPa)
Test/Analysis
198.7 170.9 182.0 158.1 165.6 120.0 106.4 155.1 116.4
0.93 0.99 1.06 0.86 0.90 0.88 0.78 0.87 0.77
185.2 143.7 155.2 121.8 128.2 66.7 73.8 128.9 86.7
0.93 1.35 1.46 1.01 1.06 0.68 0.75 1.09 1.12
213.5 172.2 172.2 184.2 184.2 136.9 136.9 177.6 151.7
compared with the numerical analysis results obtained by the program THIN-WALL [18] and LUSAS [19], respectively. The test ultimate stresses were generally lower than the analysis results except for the H-2-2 specimen. As shown in Table 3, the test ultimate stresses of the C-1-2 and the C-3-1 were not in good agreement with those obtained by the numerical analysis. The comparison of the local buckling stresses of the test specimens showed a different trend from the ultimate stresses. The experimental elastic local buckling stresses of the C-1-1 and the C-1-2 specimens were lower than the theoretical elastic local buckling stresses by 25%–32% due to the local collapse of the web, which was caused by the concentration of load. The local buckling stress of the C-1 sections was very much lower than the yield stress and the translation of the effective centroid of the buckled section caused the premature failure of the web before the interaction between local buckling and overall buckling. Except for the H2-2 section, the experimental ultimate stresses of the columns tested were slightly lower than the numerical values because of the adverse effect of the interaction between local buckling and overall buckling of the specimens. For the C-3-1 section, the test ultimate stress was lower than the analysis result by 23.0%, while the test ultimate stress was higher than the analysis result by 6.0% for the H-2-2 section. However, the difference between test ultimate stress and local buckling stress
197.9 106.5 106.5 120.9 120.9 97.9 97.9 118.7 77.1
was 29.7 MPa and 26.8 MPa, respectively. All the specimens displayed more or less significant post-local-buckling strength reserve up to the ultimate load which was determined by the minor axis flexural buckling load. The differences between the experimental ultimate stresses and the local buckling stresses ranged from 13.5 MPa for the H-1-1 section to 53.3 MPa for the C-1-1 section. Since the local buckling stress of the column tested cannot be determined easily, the experimental buckling stress was taken as the stress at the location where the deterioration of stiffness on the load–displacement curves commenced. The local buckling stress was determined by two different methods. The first set of buckling stresses was determined by examining the change in the slope of the load–displacement curves. The second set of buckling stresses was determined by plotting the stress versus the square of the strain and subsequently fitting a line through the test results in the post-buckling region. The intersection of the fitted line and load or stress axis was assumed to be the experimental buckling stress [21]. Since the change in the overall flexural stiffness was subtle and the intersection point was difficult to choose, the average value of the two methods was taken as the experimental local buckling stress in Table 3. In cases where the slope change of the load–displacement curves was not clear and the local buckling stress was difficult to decide, the stress–strain curves, which
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Fig. 8. Stress versus strain curves (H-section).
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where f nl = design axial column strength accounting for the interaction of local buckling and overall buckling; f ne = overall column strength not considering the local buckling strength; Fl = nominal strength based on the local buckling stress; Fy = nominal yield stress of the material. The nominal strengths are obtained by multiplying the allowable stresses by the factor of safety of 1.7, which is basically used in the specifications. Eq. (1) incorporates the concept that the compressive strength of columns is based on the overall buckling stress and is reduced by the ratio of the nominal strength based on the local buckling stress to the yield stress (Fl /Fy ) in order to consider the local buckling of the composing elements. Whenever the nominal stress based on the local buckling stress is higher than the yield stress of the section, the strength formula can predict reasonable design strengths of the section. However, if the nominal local buckling stress is lower than the yield stress, Eq. (1) is liable to produce unreasonable strengths. In the case where the overall column strength and the design local strength are lower than the yield stress, whether the overall column strength is larger than the design local buckling strength or not, the design axial column strength computed by Eq. (1) is extremely low. Consequently, Eq. (1) cannot reasonably include the negative effect of buckling mode interaction between local and overall buckling. In the cases where the local buckling stress is larger than the flexural/flexural–torsional buckling stress, the strength of the column should be determined by the overall buckling stress. However, if the local buckling stress is smaller than the overall buckling stress, the post-local-buckling strength reserve should be considered in the design strength to some extent. To incorporate the negative effect of mode interaction for the compression members undergoing interaction of local buckling and overall buckling, the strength formula in Eq. (1) should be revised properly. The design column strength (Pnl ) can be obtained by multiplying the axial column strength ( f nl ) by the unreduced cross section area (A). The resulting alternative formulation in Eq. (1) in terms of force is
were obtained by the strain gauges attached, were additionally used to determine the local buckling stress. The local buckling stresses of the H-2 sections were higher than those obtained by the numerical analysis by approximately 40.0%. The C-1 sections displayed a local buckling load that was approximately 28.0% lower than that of the numerical analysis. However, the local buckling stresses of all the other sections agreed well with those obtained by the numerical analysis. The typical axial stress versus strain curves for H-sections are shown in Fig. 8. The stress–strain curves for H-sections tested are similar to one another. At the intersection point where the slope changed, local buckling commenced at the web of the columns. Before the ultimate load, a significant post-local-buckling strength reserve was observed on the load–displacement and stress–strain curves except for H-1-1 section, whose width–thickness ratio was comparatively lower than those of other specimens. In Fig. 8, the stress was the average stress obtained by dividing the applied load by the unreduced gross section area, and the strain was the average strain, which was calculated from the gauge values measured at the column center. As shown in Fig. 8, the local buckling started at the strain of approximate 0.001. In some cases of H-2-2 and H-3-1 sections, the local buckling stress could be decided from the stress–strain curves more easily than the load–displacement curves. A significant post-local-buckling strength reserve similar to that in the load–displacement curves was observed. The H-2-2 and H-3-2 sections displayed a continuous increase of strain after the ultimate stress. However, the H-2-1 and H-3-1 sections showed a slightly unstable stresssoftening behavior.
where pnl = design column strength accounting for interaction of local buckling and overall buckling AFne , Pl = resulting limiting load AFl , and Py = yield load AFy .
4. Column design methods
4.2. Effective width method
4.1. Korean highway bridge design specifications
The NAS (AISI 2001) [6] and Eurocode 3 [8] have used the same effective width formula to include the post-localbuckling strength reserve in the design strength of compression members, where the width–thickness ratio is larger than the limit values specified. Therefore, even if the overall column strength formulas are slightly different, the design strengths of the compression members computed according to the NAS and the EC3 are quite similar. The AISC LRFD [7] provides different buckling coefficient K values, and the
The design compressive strength defined in terms of allowable stress in KHBDS [9] (Korean Highway Bridge Design Specifications, 2005) and KS D 3515 [20] can be modified in terms of nominal strength as f nl = f ne ×
Fl Fy
(1)
Pnl = Pne ×
Pl Py
(2)
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calculation method of the effective width is also different from other specifications mentioned. Since the section is generally composed of stiffened and unstiffened elements, the design procedure according to the AISC LRFD is as follows: the reduction factor Q s for the unstiffened elements with adequate K values specified should be calculated first, and then the reduction factor Q a for the stiffened elements is computed with the assumed stress level in the iterative method. The NAS and the EC3 adopted the same effective width formula for both stiffened and unstiffened elements for simplicity. The formula was originally proposed for the stiffened element and produces more conservative values than that for the unstiffened element. Consequently, the AISC LRFD may produce more or less higher design strengths than the NAS and EC3 in general. The buckling coefficient K values 4.0 and 0.43 are used to calculate the effective widths of the stiffened and unstiffened elements in isolation, respectively. The unreduced gross area is used to compute the limiting load in the AISC LRFD rather than the effective area which is used in the NAS and EC3. 4.3. Direct strength method The effective width method (EWM), which can consider the post-buckling strength reserve into the design, may need iterative calculations to compute the effective width of the composing element separately. As the section shape becomes more complicated, the accurate computation of the effective width becomes more difficult and the separation of each element of complicated sections becomes unreasonable. To overcome these problems, the direct strength method (DSM) for cold-formed steel sections has been developed by Shafer and Pekoz [10] and further studied by Hancock et al. [2,12]. The method incorporates the empirical formulae and the elastic local or distortional buckling stress obtained by the rigorous buckling analysis or reliable strength formulas. The application of the direct strength method to a welded section was studied recently by Kwon et al. [13]. The direct strength formulas considering the interaction between local buckling and overall buckling for cold-formed steel sections, which have been proposed by Shafer and Pekoz [10], are given by for λ ≤ 0.776 f nl = Fne .
(3a)
For λ > 0.776 f nl = 1 − 0.15
Fcrl Fne
0.4 !
Fcrl Fne
0.4 Fne
(3b)
in the Eurocode 3 [8] or the equations in Table 3.3.2 of the KHBDS [9]. The elastic local buckling stress Fcrl can be computed by the rigorous Finite Element Method (FEM) or the Finite Strip Method (FSM). The exponent 0.4 was used instead of 0.5, which was used in Von Karman [22] and Winter formulae [23], for the effective width of elements to reflect a higher post-local-buckling strength reserve for the unreduced gross section than for an element in isolation. Eqs. (3a) and (3b) were recently adopted as an alternative design method to the EWM by the NAS [14] and Australian Standards [15]. The DSM is much simpler to apply than the EWM since it uses gross section properties rather than effective section properties. It has already been proven quite accurate in comparison with the EWM for cold-formed steel sections [2]. 4.4. Proposed direct strength equations The direct strength formula considering the interaction between local buckling and overall buckling can be reasonably used for welded sections and hot-rolled sections. However, the strength formula should be modified to account for the different characteristics between cold-formed and welded steel sections. First, the effect of interaction between local buckling and overall buckling on the strength of the welded compressive members might be less significant, and the post-local-buckling strength reserve is smaller than that for the cold-formed steel section since the width–thickness ratios of commonly used welded sections are comparatively smaller than those of coldformed steel sections. Secondly, unlipped welded channel sections are commonly used as compression members, while lipped channel sections are common for cold-formed steel sections. However, in the case of single symmetrical sections such as the unlipped C-section, of which the local buckling stress was very much lower than the yield stress, the transition of the effective centroid of the locally buckled section might cause a premature failure. To account for these phenomena, the modified Winter formula [23] was proposed rather than Eqs. (3a) and (3b). When the exponent 0.5, as used in Winter formulas, is used, the coefficient 0.15 in Eq. (3b) is adopted rather than 0.22 in Winter formulas, which reflects a higher post-local-buckling strength reserve in the inelastic buckling range of a material for the intermediate length column. The equations for the limiting stress f nl considering the interaction between local buckling and overall buckling for the welded section are given by for λ ≤ 0.816 f nl = Fne .
(4a)
For λ > 0.816
√
where λ = Fne /Fcrl ; f nl = limiting stress accounting for local buckling and overall buckling (MPa) ; Fcrl = elastic local buckling stress (MPa) ; Fne = overall column strength based on the overall failure mode (MPa), which is determined from the minimum of the elastic flexural, torsional, and flexural–torsional buckling stresses. The overall column strength Fne can be calculated from Eqs. (C4-2) and (C43) of the NAS (AISI) [6] or Eqs. (6.47), (6.48) and (6.49)
f nl = 1 − 0.15 where p λ = Fne /Fcrl .
Fcrl Fne
0.5 !
Fcrl Fne
0.5 Fne
(4b)
(4c)
The results predicted by the strength formulas proposed are compared with tests results of the welded H-sections and
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Fig. 9. Comparison between DSM curves and test results.
the C-sections, and Eqs. (3a) and (3b) in Fig. 9. As shown in Fig. 9, Eqs. (3a) and (3b) predict reasonable strengths for the columns undergoing the interaction between local buckling and overall buckling in comparison with the test results, as long as the slenderness ratio factor is small. However, when the slenderness ratio factor λ becomes larger than 1.5, Eq. (3b) produces very optimistic strengths in comparison with the test results of channel sections. The strength curve equation (4b) proposed can predict slightly higher design strengths than Eq. (3b) when the slenderness ratio factor λ remains between 0.776 and 1.0. However, when the slenderness ratio factor λ becomes larger than 1.0, Eq. (4b) produces slightly lower values than Eq. (3b). The difference in the strengths predicted by Eqs. (3b) and (4b) becomes more significant as the slenderness ratio factor becomes large. The ultimate strengths of C-sections tested are under the design strength curves of Eq. (4b) proposed. However, Eq. (3b) predicts the ultimate strengths of C-sections too optimistically. As shown in Fig. 9, the proposed design curves have a less satisfactory fit to the ultimate strength of H-sections than Eq. (3b), but predict a less optimistic ultimate strength for C-sections than Eq. (3b) does. Since the single symmetric C-section generally shows a premature failure due to the translation of the effective centroid of the locally buckled section, the design strength curves is forced to predict somewhat optimistically according to the slenderness of the plate elements. Consequently, it can be concluded that the proposed ultimate strength curves of Eqs. (4a) and (4b) can predict more reliable ultimate strengths for the welded section columns than Eqs. (3a) and (3b) when local buckling and overall buckling occur simultaneously or nearly simultaneously. 4.5. Comparison with test results To compare the DSM with the EWM directly, the strength equations should be expressed in terms of load. If the distortional buckling does not occur, the design strength of compression members can be taken as the nominal strength Pnl , which is obtained by multiplying the limiting stress ( f nl ) in Eqs. (4a) and (4b) by the full unreduced section area (A). Therefore, the alternative formulations of Eqs. (4a) and (4b) in terms of load are given by
Fig. 10. Comparison between DSM (based on NAS) and test results.
for λ ≤ 0.816 Pnl = Pne .
(5a)
For λ > 0.816 Pnl = 1 − 0.15
Pcrl Pne
0.5 !
Pcrl Pne
0.5 Pne
(5b)
√ where λ = Pne /Pcrl , Pcrl = elastic local buckling load AFcrl , pne = nominal column design strength AFne . The Direct Strength Method (DSM) curves based on the overall column strength formula provided by the NAS (AISI) and Eurocode 3 are compared with the test results in Fig. 10 and Fig. 11, respectively. Since the nominal column strengths computed by the NAS and EC3 column strength formula are slightly different, the test results were generalized by the two different design strengths and compared. The test results of the welded I-section and channel section columns with the nominal yield stress of 350 MPa, which were executed previously at the University of Sydney [16,17], are also included in Figs. 10 and 11. For most of the columns, the DSM curve predicts the ultimate strengths fairly conservatively. However, in the case of very slender unlipped channel sections, the ultimate strengths are predicted optimistically by the DSM curve. The DSM curve predicts the ultimate strengths too optimistically in comparison with the test results of very slender channel sections, which were executed by Rasmussen and Hancock [17]. However, it can be concluded that the ultimate strengths of welded Hsections and C-sections, which are not too slender, can be predicted fairly reasonably by the DSM. The column design strengths, which are calculated according to the Effective Width Method (EWM), the NAS [6] and the EC3 [8], and the DSM proposed have been summarized and compared with test results in Table 4. Since the web of the C-1-1 and C-1-2 sections failed prematurely in the local mode before interaction between local buckling and overall buckling, the test results of the C-1-1 and C-1-2 sections were not compatible with the design strengths predicted by the specifications. For the H-sections tested, the ratio of the maximum test strength to NAS ranged from 1.33 to 1.62, the ratio of the maximum test strength to EC3 from 1.43 to 1.83 and the ratio of the maximum test strength to DSM ranged from
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Table 4 Comparison of tests and design strengths Specimen H-1-1 H-2-1 H-2-2 H-3-1 H-3-2 C-1-1 C-1-2 C-2-1 C-3-1
Design strengths (kN) NAS EC3
AISC
DSM
513.0 585.7 585.7 584.8 584.8 497.6 497.6 504.6 323.1
575.0 766.9 766.9 682.7 682.7 596.9 596.9 632.6 251.9
550.2 566.2 566.2 564.3 564.3 529.7 529.7 505.5 400.8
444.0 519.6 519.6 519.0 519.0 447.9 447.9 442.8 334.3
Test ultimate strength (kN) 715.3 892.1 950.0 739.9 775.0 502.9 444.1 697.8 519.9
Test/Design NAS EC3
AISC
DSM
1.39 1.52 1.62 1.27 1.33 1.01 0.89 1.38 1.61
1.24 1.16 1.24 1.08 1.14 0.84 0.74 1.10 2.06
1.30 1.58 1.68 1.31 1.37 0.95 0.84 1.38 1.30
1.61 1.72 1.83 1.43 1.49 1.12 0.99 1.58 1.56
Table 5 Comparison between design methods
Fig. 11. Comparison between DSM (based on EC3) and test results.
1.30 to 1.68. The ratio of the test ultimate strength to the AISC ranged from 1.08 to 1.24. For the C-sections tested, the ratios of the ultimate test strength to NAS were 1.38 and 1.61, the ratios of the ultimate test strength to EC3 1.56 and 1.58 and the ratios of the maximum test strength to DSM were 1.30–1.38. The ratios of the test ultimate strength to the AISC were 1.10 and 2.06. For H-sections and C-sections, the difference between the test ultimate strength and design strength was fairly constant. However, for the C-section, the difference for AISC was highly variable in magnitude. The DSM and EWM predicted the ultimate strength for the H-sections fairly conservatively. In the case of the C-sections, the predictions of the ultimate strength by the specifications and the DSM were fairly conservative. However, it can also be concluded that the ultimate strength of the channel sections such as the C-3 specimen which had exceptionally slender flange was predicted too conservatively by the EWM but fairly by the DSM. Generally speaking, the test results for H-sections and C-sections validate the DSM clearly. The comparison of predictions for the ultimate strength of the sections among the design specifications and the DSM proposed is summarized in Table 5. As shown in Table 5, since the provisions to take account of the local buckling stress on the column strength are not adequate and the post-local-buckling strength is not considered at all, the KHBDS [9] cannot closely predict the ultimate strength of the slender welded section columns undergoing interaction between local buckling and flexural/flexural–torsion buckling. The average ratio of KHBDS to DSM was 0.19. Even if the width–thickness of the test sections is beyond the limit prescribed in the specification, the
Specimen
NAS/DSM
EC3/DSM
AISC/DSM
KHBDS/DSM
H-1 H-2 H-3 C-1 C-2 C-3 Average Standard deviation
0.93 1.03 1.04 0.94 1.00 0.81 0.97 0.07
0.81 0.92 0.92 0.85 0.88 0.83 0.88 0.04
1.05 1.35 1.21 1.13 1.25 0.63 1.15 0.15
0.25 0.19 0.19 0.20 0.21 0.10 0.19 0.03
predictions of the ultimate strength by the KHBDS are absurdly lower than the test ultimate strengths for the sections which may have the buckling interaction between local buckling and overall buckling. The slight difference between NAS, EC3 or AISC and DSM might result from the magnitude of post-local-buckling strength considered in the specifications and the effect of interaction between the elements composing the whole cross section. Unlike the EWM, which estimates the effective width of each element in isolation, the DSM handles the whole section at a one time. Thus, the difficulty in computing the effective width of the each element in isolation for the EWM has been exempted in the DSM. The design strengths predicted by the DSM proposed agree quite closely with those estimated by the NAS. The ratio of the design strengths predicted by the NAS to those obtained according to the DSM ranged from 0.81 to 1.04 with the average 0.97 and the standard deviation 0.07, as shown in Table 5. The C-3 section has an exceptional width–thickness ratio of the flange, which is four times the buckling limit of 14.0. In comparison with EC3, the DSM estimates the design strength higher than the EC3 by an average of 12%. The prediction by the EC3 is slightly more conservative than that by the NAS except for the C-3 specimen. The ultimate strengths predicted by the AISC are higher than those predicted by the DSM except for the C-3 by an average of 15%. The fact that the AISC uses a different effective width formula between stiffened and unstiffened elements made the difference in the predictions. The difference in the ultimate strengths predicted by the specifications is mainly the result of the single, effective width formula for both stiffened and unstiffened elements adopted by the NAS and the EC3 with the advantage of simplicity. Since the formula for the unstiffened element
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has more post-local-buckling strength reserve and produces a higher value than that for the stiffened element, the NAS and EC3 predict the ultimate strength less optimistically than the AISC. From the comparison of the ultimate strengths predicted by the design specifications based on the effective width concept and the DSM, it can be concluded that the design ultimate strength of welded sections can reasonably be predicted by the DSM. The DSM proposed can be used as an alternative method to the conventional EWM, which has been used for the welded section columns. However, in the case of the welded sections such as a lipped channel section, which is liable to undergo distortional buckling, even if the sections are not of common shape, the distortional limiting strength should also be considered to predict the overall column strength. Therefore, the DSM should determine the ultimate strength for the interaction between local buckling and overall buckling, and distortional buckling and overall buckling and then take the lesser of the two as the column strength. The DSM should also be carefully applied to very slender single symmetric sections such as a channel section. In that case, if the shift of the centroid of the effective area relative to the center of gravity of the gross section is not prevented by constructional arrangements, the possible shift of the centroid and the resulting additional moment should be determined, and the beam–column strength formula for the DSM should be used. The beam–column strength formula for the DSM is under preparation at the moment. 5. Conclusions The experimental study for the application of the direct strength method (DSM) to the thin-walled welded section columns undergoing interaction between local buckling and flexural/flexural–torsional buckling was conducted. The local buckling, which occurs prior to the overall column buckling and has a significant post-local-buckling strength reserve, deteriorates the overall column strength to some extent. This phenomenon should be considered appropriately to predict conservatively and accurately the ultimate strength of the welded sections as well as the cold-formed sections. The direct strength formulas, which were adopted for the design of the cold-formed steel sections, have been modified to take into account the effect of the interaction between local buckling and overall buckling and post-local-buckling strength reserve for the welded sections. The member strengths from the direct strength formulas proposed were compared with the test results of welded H-sections and C-sections fabricated from the hot-rolled steel plates with nominal yield stresses of 240 and 350 MPa. The ultimate strengths predicted by the direct strength method were also compared with those estimated by the effective width method adopted in the NAS, AISC and EC3. The comparison of the ultimate strengths predicted by the design specifications and the test results has proven the reliability of the direct strength method. The direct strength formulas proposed have been proven easier to apply and accurate to predict the ultimate strength of the columns
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that undergo the interaction between local buckling and overall buckling and fail in the mixed mode between local buckling and flexural–torsional buckling. However, detailed provisions as to the width–thickness ratio limit of single symmetric sections, such as a channel section, should be prepared immediately. The proposed direct strength formulas should be further verified and calibrated against the test results of various kinds of sections for practical use. Acknowledgements This research was supported by the Korea Bridge Design and Engineering Center under the sponsorship of 2004 Development of Core Construction Technology Project of Korea Ministry of Construction and Transportation. References [1] Kemp AR. Inelastic lateral and local buckling in design codes. J Struct Eng, ASCE 1996;122(4):374–82. [2] Hancock GJ, Murray TM, Ellifritt DS. Cold-formed steel structures to the AISI specification. Marcel Dekker, Inc.; 2004. [3] Hancock GJ, Kwon YB, Bernard ES. Strength design curves for thinwalled members. J Constr Steel Res 1994;31(3):169–86. [4] Mulligan GP. The influence of local buckling on the structural behavior of singly-symmetric cold-formed steel columns. Ph.D. thesis. Cornell University; 1983. [5] Yang D, Hancock GJ. Compression tests of high strength steel channel columns with interaction between local and distortional modes. J Struct Eng, ASCE 2004;130(12):1954–63. [6] American Iron and Steel Institute (AISI). North American specifications for the design of cold-formed steel structural members. Washington (DC, USA); 2001. [7] American Iron and Steel Construction (AISC). Load and resistance factor design specification for steel structural buildings. Chicago (Il, USA); 1999. [8] European Committee for Standardisation (ECS). Eurocode 3: Design of steel structures, Part 1-1: General rules and rules for buildings. Brussels (Belgium); 2003. [9] Korean Association of Highway and Transportation Officials. Korean highway bridge design specifications (KHBDS). Korea; 2005. [10] 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. Elsevier; 1998. [11] Schafer BW. Advances in direct strength design of thin-walled members. In: Hancock GJ, Bradford M, Wilkinson T, Uy B, Rasmussen K, editors. Proceedings advances in structures conference, ASSCCA03. 2003. [12] Hancock GJ. Developments in the direct strength design of cold-formed steel structural members. In: Proceedings, 3rd international symposium on steel structures, KSSC. 2005. p. 120–31. [13] Kwon YB, Kim NG, Kim BS. A study on the direct strength method for compression members undergoing mixed mode buckling. In: Proceedings, 3rd international symposium on steel structures, KSSC. 2005. p. 120–31. [14] American Iron and Steel Institute (AISI). Supplement 2004 to the North American specifications for the design of cold-formed steel structural members. Washington (DC, USA); 2004. [15] Standards Australia, Cold-formed steel structures AS/NZS 4600: 2005. Sydney (NSW, Australia); 2005. [16] Davids AJ, Hancock GJ. Compression tests of long welded I-section columns. J Struct Eng, ASCE 1986;112(10):2281–97. [17] Rasmussen KJR, Hancock GJ. Compression tests of welded channel section columns. J Struct Eng, ASCE 1989;115(4):789–808. [18] Papangelis JP, Hancock GJ. THIN-WALL (ver. 2.0). Sydney (Australia): Center for Advanced Structural Engineering, Dept. of Civil Engineering, Univ. of Sydney; 1998.
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