developments on the design of cold-formed steel angles
TRANSCRIPT
Developments on the Design of Cold-Formed Steel AnglesNuno Silvestre1; Pedro B. Dinis2; and Dinar Camotim, M.ASCE3
Abstract: This paper presents procedures for the design of fixed- and pin-ended equal-leg angle columns with short-to-intermediate lengths.First, some remarks concerning the buckling and postbuckling behavior of the angle columns are presented that (1) illustrate the main differ-ences between the fixed- and pin-ended column responses and (2) demonstrate the need for specific design procedures. Then, the paper reportsan in-depth investigation aimed at gathering a large column ultimate strength data bank that includes (1) experimental values collected from theliterature and (2) numerical values obtained fromshellfinite-element analyses carried out in the codeABAQUS. The set of experimental results iscomprised of 41 fixed-ended columns and 37 pin-ended columns and the numerical results obtained include 89 fixed-ended columns and 28pin-ended columns; various cross-section dimensions, lengths, and yield stresses are considered. Finally, the paper closes with the proposal ofnew design procedures for fixed- and pin-ended angle columns based on the direct strength method (DSM). The two procedures adoptmodified global and local strength curves, and it is shown that the proposed DSM approach leads to accurate ultimate strength estimates forshort-to-intermediate columns covering a wide slenderness range.DOI: 10.1061/(ASCE)ST.1943-541X.0000670.© 2013 American Societyof Civil Engineers.
CEDatabase subject headings:Cold-formed steel; Columns; Buckling; Postbuckling; Ultimate strength; Finite element method; Design.
Author keywords: Cold-formed steel angle columns; Equal-leg angle columns; Fixed-ended columns; Pin-ended columns; Bucklingbehavior; Postbuckling behavior; Ultimate strength; Shell finite element analysis; Design procedures; Direct strength method (DSM).
Introduction
Thin-walled columns with cross sections that have all wall midlinesintersecting at a point (e.g., angle, T-section, and cruciform col-umns) are known to exhibit no primary warping but only secondarywarping. Thus, their torsional resistance is extremely low, whichrenders them highly susceptible to torsional or flexural-torsionalbuckling. Moreover, it is often hard to separate the torsional andlocal deformations, and thus to distinguish between local and tor-sional buckling. Because these two instability phenomena arecommonly associated with markedly different postcritical strengthreserves, it is fair to say that this distinction may have far-reachingimplications on the development of a rational model capable ofproviding accurate ultimate strength estimates for such columns.
The postbuckling behavior and strength of equal-leg angle col-umns has attracted the attention of several researchers in the past(Stowell 1951; Wilhoite et al. 1984; Gaylord and Wilhoite 1985;Kitipornchai andChan 1987; Popovic et al. 1999).Moreover,Young(2004), Ellobody and Young (2005), and Rasmussen (2005, 2006)performed experimental tests and shell finite-element analyses(SFEAs) on fixed-ended columns, aimed at obtaining the ultimate
loads and comparing them with predictions from the currentlyavailable design rules. Rasmussen (2005, 2006) and Young (2004)also put forward two approaches for the design of angle columns,which are based on the direct strengthmethod (DSM) (Schafer 2008)global curve to estimate the flexural and local strengths. Chodrauiet al. (2006) proposed a slightly different approach, which differedfrom the Rasmussen (2005, 2006) approach in that the columnglobal strength is the lower of the flexural and torsional values, bothobtained with the DSM global design curve. Following the work byChodraui et al. (2006), both Maia (2008) and Maia et al. (2008)reported experimental tests on fixed- and pin-ended angle columns,and Mesacasa (2011) performed a few further tests on fixed-endedangle columns. Mesacasa (2011) also carried out a fairly completecompilation of the experimental results available in the literatureconcerning both fixed- and pin-ended angle columns, explaining insome detail the main differences between the various test setups andprocedures. Also, very recently Shifferaw and Schafer (2011) in-vestigated the significant global postbuckling strength reserve ob-served in cold-formed steel angle column tests and provided somedesign guidance for locally slender cold-formed steel lipped andplain angles with fixed-end supports. Finally, fresh numericalinvestigations, carried out by means of generalized beam theory(GBT) (Camotim et al. 2010) and ABAQUS SFEAs (Dinis et al.2010a, b, 2011, 2012a, b; Dinis and Camotim 2011), shed new lighton how to distinguish between local and global buckling in equal-legangle columns. These studies showed that both fixed- and pin-endedcolumnswith short-to-intermediate lengths and buckling in flexural-torsional modes with very similar critical stresses exhibit differentpostbuckling behaviors and ultimate strengths (the amount of cornerflexural displacements was found to play a key role in separating thevarious behaviors), and that the critical slenderness is inadequate tomeasure the ultimate strength of equal-leg angle columns.
This work reports the numerical and experimental resultsobtained with the aim of developing a rational DSM-based designapproach for short-to-intermediate equal-leg angle columns—both
1Assistant Professor, Dept. of Civil Engineering and Architecture,Instituto Superior Técnico (ICIST), Technical Univ. of Lisbon, 1049-001Lisbon, Portugal (corresponding author). E-mail: [email protected]
2Assistant Professor, Dept. of Civil Engineering and Architecture,Instituto Superior Técnico (ICIST), Technical Univ. of Lisbon, 1049-001Lisbon, Portugal. E-mail: [email protected]
3Assistant Professor, Dept. of Civil Engineering and Architecture,Instituto Superior Técnico (ICIST), Technical Univ. of Lisbon, 1049-001Lisbon, Portugal. E-mail: [email protected]
Note. This manuscript was submitted on September 2, 2011; approvedon June 5, 2012; published online on August 10, 2012. Discussion periodopen until October 1, 2013; separate discussions must be submitted forindividual papers. This paper is part of the Journal of Structural Engi-neering, Vol. 139, No. 5, May 1, 2013. ©ASCE, ISSN 0733-9445/2013/5-680–694/$25.00.
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fixed- and pin-ended columns (but with end section secondarywarping prevented) are considered. After a brief overview on thecolumn buckling and postbuckling behavior (Dinis et al. 2010a,2011, 2012a, b; Dinis and Camotim 2011), the paper addresses theassembly of column ultimate strength data, intended to provide thebasis for the development, calibration, and validation of specificdesign procedures for equal-leg angle columns. This work consistedof (1) collecting and organizing experimental values available in theliterature and (2) determining numerical (SFEA) values by per-forming a fairly extensive parametric study. The output results of thiseffort are 76 experimental and 117 numerical ultimate load valuesconcerning columns with either pinned or fixed-end supports andvarious cross-section dimensions, lengths, and yield stresses.Finally, the column ultimate load data bank gathered is used topropose and assess the efficiency (accuracy and economy) of a DSMapproach to predict the ultimate strength of the angle columns underconsideration, which involves distinct procedures for fixed- andpin-ended columns.
Brief Overview of the Buckling, Postbuckling,and Strength Behaviors
This section provides a brief overview of the buckling, postbuck-ling, and ultimate strength behaviors of thin-walled steel anglecolumns. For further details, the interested reader is referred to Diniset al. (2010a, 2011, 2012a, b) and Dinis and Camotim (2011). Out ofthe results reported in these publications, the following deserve to bespecially mentioned:1. Both the fixed- and pin-ended columns (secondary warping
and major-axis rotation prevented at supports) display similarbuckling features: critical load Pcr decreases monotonicallywith length L and the critical buckling mode always exhibits asingle half-wave. Columns with short-to-intermediate lengthsare associated with well-defined critical stress plateaus in thesignature curvePcrðLÞ, corresponding to critical bucklingmodesthat combine torsion with major-axis flexure—the amount ofmajor-axis flexure grows as L increases. Long columns bucklein pure minor-axis flexural modes.
2. Within the length ranges corresponding to the plateaus, boththe fixed- and pin-ended columns exhibit postbucklingbehaviors characterized by the simultaneous occurrence ofcross-section torsional rotations and corner displacements—translations as a result of major- and minor-axis bending. Inparticular, the minor-axis flexural displacements have a strongimpact on the column postbuckling response; namely, on thepostcritical strength reserve and longitudinal normal stressdistributions. This impact is considerably more pronounced inthe pin-ended columns than in the fixed-ended columns.
3. Because of the relevance of the corner flexural displacements(mostly the minor-axis ones), the behavior of equal-leg fixed-ended angle columns cannot be viewed as the sum of twofixed-ended (longitudinally) pinned-free (transversally) longplates, unlike it would be tempted to anticipate. In particular,the cross-section longitudinal normal stress distributions be-come far from parabolic as postbuckling progresses. If thecorner displacements are prevented, the pinned-free long platepostbuckling behavior is recovered.
4. Althoughminor-axis flexure does not participate in the criticalflexural-torsional buckling modes of columns with short-to-intermediate lengths (flexural means major-axis flexure), itemerges in the postbuckling stage as a result of the longitudinalvariation of the torsional rotations. Moreover, it causes non-linear cross-section midline longitudinal stress distributions
that are associated with effective centroid shifts (toward thecross-section corner).
5. There are relevant differences between the characteristics ofthe minor-axis flexural displacement profiles of the fixed- andpin-ended columns in the postbuckling range;while the formerexhibits three inner half-waves, the latter apparently exhibitsjust one half-wave. Additionally, the minor-axis displace-ments significantly prevail over their major-axis counterpartsin the pin-ended columns. Conversely, in fixed-ended col-umns the minor- and major-axis displacements exhibit similarmagnitudes.
6. The differences outlined in the previous item stem from theabsence ofminor-axis endmoments in the pin-ended columns,making it impossible to counteract the minor-axis bendingcaused by the effective centroid shifts as a result of the cross-sectionnormal stress redistribution (e.g.,Young andRasmussen1999). Indeed, although the mechanical reasoning behindthe development of the three half-wave minor-axis displace-ment profile remains valid for both the fixed- and pin-endedcolumns, the predominance of the well-curved, single half-wave component largely overshadows it. Such predominanceis even clearer in the longer columns [near the end of thePcrðLÞplateau] because of the much more intense interaction withminor-axis flexural buckling (closer flexural-torsional andflexural buckling loads).
The markedly different postbuckling behaviors displayed by thefixed- and pin-ended short-to-intermediate equal-leg angle columnsprovides the explanation for the significant discrepancy betweentheir ultimate strengths Pu associated with a given yield stress.Because all these columns have virtually identical critical stresses—thus sharing a common critical slenderness l5 ðPy=PcrÞ0:5—theirPu=Py valuesmay exhibit a high vertical dispersionwith respect tol.Τhis behavioral feature must be adequately accounted for by anefficient design procedure for equal-leg angle columns.
Ultimate Strength Data: Test Results andNumerical Predictions
Following the findings reported by Dinis et al. (2010b, 2011,2012a, b) and Dinis and Camotim (2011), which were summarizedpreviously, it was decided to assess the performance of the existingdesign rules for cold-formed steel equal-leg angle columns. The firststep toward achieving this goal consisted of putting together a fairlylarge column ultimate strength data bank comprised of (1) experi-mental test results performed by other researchers and collected fromthe available literature, and (2) numerical results obtained by meansof the shell finite element model developed earlier.
The experimental results gathered concern 41 fixed-ended col-umns, tested by Popovic et al. (1999), Young (2004), and Mesacasa(2011); and 37 pin-ended columns (with cylindrical supports), testedby Wilhoite et al. (1984), Popovic et al. (1999), Chodraui et al.(2006), andMaia et al. (2008). The fixed-ended columns had nominalcross-section dimensions of 503 2:5, 503 4:0, and 503 5:0mm(Popovic et al. 1999); 703 1:2, 703 1:5, and 703 1:9mm (Young2004); and 603 2:0mm (Mesacasa 2011). The nominal cross-section dimensions of the pin-ended columns were 703 3:0mm(Wilhoite et al. 1984); 503 2:5, 503 4:0, and 503 5:0mm(Popovic et al. 1999); and 603 2:4mm (Chodraui et al. 2006; Maia2008). Further details concerning the measured specimen dimen-sions and steel properties can be found in the publications men-tioned previously, which are included in the reference section ofthis paper. It is worth mentioning that the four fixed-ended columnstested by Maia et al. (2008) were excluded from this investigation
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because the ultimate strengths reported did not seem plausible.Indeed, they are much lower than the numerical results obtained bythe same authors, adopting fair-to-high torsional imperfections (0.64and 1.55t). In fact, they are even lower than those also reported byMaia et al. (2008) for pin-ended columns with practically identicalgeometrical and material characteristics. In the authors’ opinion, theonly logical explanation for this apparent paradox is that these fixed-ended columns contain abnormally large initial imperfections and/orload eccentricities (maybe caused by the procedure adopted toensure the fixity of the column end sections).
For each cross-section geometry, specimen length L and mea-sured yield stress fy, (Table 1, fixed-ended columns, and Table 2,pin-ended columns) provide the test results; namely, the columnultimate stress fu. Concerning the information given in Table 2 (forpin-ended columns), it should be mentioned that the 9 pin-endedcolumns tested by Wilhoite et al. (1984) had only three differentlengths (823, 1,227, and 1,636 mm) because three specimens with(approximately) the same length were tested. The repeated testresults were included herein because they show the following pe-culiar feature: while the three shorter columns (L5 823mm)
Table 1. Fixed-Ended Column Experimental Ultimate Stresses and Their Estimates according to (1) the Design Method Developed by Young (2004) and(2) the Proposed DSM-Based Approach
Section L (mm)
Buckling analysis
fy (N/mm2)Test
fu (N/mm2)Young
(2004) fu=fp
DSM-Ffnle (N/mm2)
Test-to-predictedratio fu=fnlefcrl (N/mm2) fcre (N/mm2)
503 2:5a 150 376 38,074 396 308 1.30 329 0.94503 2:5a 550 198 2,832 396 225 1.01 250 0.90503 2:5a 970 186 910 396 173 0.90 214 0.81503 2:5a 1,379 180 450 396 154 1.00 172 0.89503 2:5a 1,747 174 281 396 130 1.11 133 0.98503 2:5a 2,199 167 177 396 110 1.44 89 1.24503 2:5a 2,598 159 127 396 93 1.70 63 1.46503 4:0a 150 961 38,826 388 424 1.26 385 1.10503 4:0a 970 460 928 388 314 1.19 286 1.10503 4:0a 1,381 430 458 388 250 1.33 216 1.16503 4:0a 1,743 398 288 388 178 1.41 152 1.17503 5:0a 150 1,431 37,819 388 414 1.15 385 1.07503 5:0a 970 667 904 388 307 1.20 288 1.06503 5:0a 1,378 602 448 388 216 1.21 213 1.02503 5:0a 1,749 531 278 388 180 1.56 148 1.22703 1:2b 250 37.6 28,143 550 143 1.16 177 0.81703 1:2b 1,000 22.3 1,759 550 113 1.03 128 0.88703 1:2b 1,500 21.7 782 550 92 0.99 107 0.86703 1:2b 2,000 21.4 440 550 76 1.04 84 0.90703 1:2b 2,500 21.3 281 550 70 1.28 62 1.13703 1:2b 3,000 21.2 195 550 48 1.10 49 0.99703 1:2b 3,500 21.1 144 550 35 0.96 40 0.87703 1:5b 250 61.8 27,852 530 189 1.26 208 0.90703 1:5b 1,000 36.6 1,741 530 148 1.12 151 0.98703 1:5b 1,500 35.6 774 530 120 1.07 127 0.94703 1:5b 2,000 35.2 435 530 83 0.93 100 0.83703 1:5b 2,500 34.9 279 530 75 1.13 74 1.01703 1:5b 3,000 35.6 193 530 62 1.17 58 1.07703 1:5b 3,500 34.4 142 530 55 1.24 47 1.16703 1:9b 250 96.7 28,379 500 212 1.22 237 0.90703 1:9b 1,000 57.0 1,774 500 180 1.15 174 1.03703 1:9b 1,500 55.4 788 500 134 1.00 147 0.91703 1:9b 2,000 54.6 443 500 102 0.94 118 0.86703 1:9b 2,500 54.0 284 500 84 1.03 89 0.95703 1:9b 3,000 53.4 197 500 56 0.87 68 0.82703 1:9b 3,500 52.8 145 500 54 1.03 55 0.98603 2:0c 400 103 7,411 350 177 — 191 0.93603 2:0c 600 93 3,294 350 166 — 179 0.93603 2:0c 900 89 1,464 350 137 — 166 0.83603 2:0c 1,200 87 823 350 128 — 152 0.85603 2:0c 1,800 84 366 350 88 — 118 0.74Mean 1.15 0.98SD 0.18 0.14aTested by Popovic et al. (1999).bTested by Young (2004).cTested by Mesacasa (2011).
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provided similar ultimate strengths, those concerning the in-termediate columns (L5 1,227mm) showed some scatter (higherand lower ultimate strength values, 11.5% apart) and the longercolumns (L5 1,636mm) showed even more scatter (22.5%difference).
In the tests by Wilhoite et al. (1984), a small clearance was builtinto the pin-ended bearings to avoid locking, and a load eccentricitymay have been induced by this clearance. Because the bearings weremanufactured to a tolerance that ensured the load eccentricity in-ducedwould not exceed 1=1:000 of the longer column length, it mayhave happened that nonnegligible load eccentricities prevailedin the tests and influenced the ultimate strength. Moreover, the503 2:5mm pin-ended columns tested by Popovic et al. (1999)
were not concentrically loaded, as they were reported to exhibitload eccentricities of roughly 6 L=1,000 along the major axis(1 L=1,000 and 2L=1,000 eccentricities increase compression atthe leg tips and corner, respectively). These eccentrically loadedcolumns are, in fact, beam-columns with similar lengths that exhibitdifferent ultimate strengths fu because of the eccentricity sign (thelower fu values correspond to 1 L=1,000 eccentricities). It is alsopossible to conclude that the percentage difference between the1 L=1,000 and 2L=1,000 eccentricity results also grows with L.Because the load eccentricity may be viewed as a geometricalimperfection (both similarly affect the column response), theseresults indicate that the pin-ended columns are also sensitive to theminor-axis initial imperfection sign.
Table 2. Pin-Ended Column Experimental Ultimate Stresses and Their Estimates according to the Design Method Developed by Rasmussen (2006) andthe Two Proposed DSM-Based Approaches
Section L (mm)
Buckling analysis
fy(N/mm2)
Testfu (N/mm2)
Rasmussen(2006) fu=fp
DSM-Ffnle (N/mm2)
Test to DSM-Fpredicted ratio fu=fnle
DSM-Pfnle (N/mm2)
Test to DSM-Fpredicted ratio fu=fnle
fcrl(N/mm2)
fcre(N/mm2)
703 3:0a 823 155 596 465 174 1.18 191 0.91 133 1.31703 3:0a 823 155 596 465 174 1.18 191 0.91 133 1.31703 3:0a 1,227 149 268 465 140 1.27 121 1.16 109 1.28703 3:0a 1,227 149 268 465 144 1.30 121 1.19 109 1.32703 3:0a 1,227 149 268 465 156 1.41 121 1.29 109 1.43703 3:0a 1,636 145 151 465 116 1.48 75 1.54 75 1.54703 3:0a 1,636 145 151 465 125 1.59 75 1.66 75 1.66703 3:0a 1,636 145 151 465 142 1.81 75 1.89 75 1.89503 2:5b 286 237 2,618 396 187 1.06 264 0.71 198 0.95503 2:5b 285 237 2,637 396 212 1.07 264 0.80 198 1.07503 2:5b 490 202 892 396 158 1.07 219 0.72 167 0.95503 2:5b 490 202 892 396 180 1.03 219 0.82 167 1.08503 2:5b 674 192 471 396 139 1.13 180 0.77 151 0.92503 2:5b 675 192 470 396 213 1.42 179 1.19 150 1.42503 2:5b 900 187 264 396 113 1.19 131 0.86 125 0.90503 2:5b 900 187 264 396 144 1.21 131 1.10 125 1.16503 2:5b 1,099 184 177 396 79 1.06 89 0.90 89 0.90503 2:5b 1,100 184 177 396 111 1.14 88 1.25 88 1.25503 4:0b 285 605 2,689 388 367 — 351 1.05 344 1.07503 4:0b 490 512 910 388 295 — 289 1.02 285 1.04503 4:0b 675 484 479 388 205 — 221 0.93 221 0.93503 5:0b 285 900 2,619 388 360 — 350 1.03 350 1.03503 5:0b 490 758 886 388 277 — 286 0.97 286 0.97503 5:0b 490 758 886 388 273 — 286 0.95 286 0.95503 5:0b 675 714 467 388 214 — 218 0.98 218 0.98503 5:0b 675 714 467 388 196 — 218 0.90 218 0.90603 2:4c 480 143 1,307 371 112 — 200 0.56 126 0.89603 2:4c 835 130 432 371 105 — 149 0.70 109 0.96603 2:4c 1,195 126 211 371 83 — 98 0.85 90 0.92603 2:4c 1,550 124 125 371 76 — 63 1.21 63 1.21603 2:4d 480 144 1,320 357 112 — 197 0.57 127 0.88603 2:4d 650 136 720 357 130 — 174 0.75 117 1.11603 2:4d 835 131 436 357 105 — 149 0.70 110 0.95603 2:4d 1,000 129 304 357 144 — 126 1.14 103 1.40603 2:4d 1,195 127 213 357 81 — 99 0.82 91 0.89603 2:4d 1,350 126 167 357 103 — 81 1.27 78 1.31603 2:4d 1,550 126 127 357 76 — 63 1.20 63 1.20Mean 1.26 1.01 1.13SD 0.21 0.29 0.25aTested by Wilhoite et al. (1984).bTested by Popovic et al. (1999).cTested by Chodraui et al. (2006).dTested by Maia et al. (2008).
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The numerical results were obtained through ABAQUS (DSSimulia 2008) SFEAs by adopting column discretizations into fine4-node isoparametric element meshes (with a length-to-width ratioclose to 1) and modeling the column supports by fully attaching themember end sections to rigid end plates, thus ensuring the fullsecondary warping and local displacement/rotation restraints. Bypreventing both the major- and minor-axis flexural rotations (fixedsupports) or only the major-axis flexural rotations (pinned supportsand cylindrical hinges), the torsional rotations were prevented inboth cases.
The SFEA results obtained concern 89 fixed-ended columns,displaying the cross-section dimensions of 703 1:2, 503 1:2, and503 2:6mm; and 28 pin-ended columns, all with the cross sectionsof 70 3 1.2 mm. The column lengths were selected to ensure thatcritical buckling occurs in the flexural-torsional modes (i.e., they fallwithin Pcr versus L curve horizontal plateaus). Their values were532, 980, 1,330, 1,820, 2,520, 3,640, 4,200, 5,320, 7,000, and 8,900mm (for the fixed-ended 703 1:2mm columns); 1,500, 2,000,2,500, 3,000, and 4,000 mm (for the fixed-ended 503 1:2mmcolumns); 1,000, 1,500, and 2,000 mm (for the fixed-ended503 2:6mm columns); and 532, 980, 1,330, 1,820, 2,520, 3,640,and 4,200 mm (for the pin-ended 703 1:2mm columns). In allanalyses the steel material behavior wasmodeled as elastic-perfectlyplastic (E5 210GPa, n5 0:3); critical-mode initial imperfectionswith amplitude equal to 10% of wall thickness t were considered;and both the residual stresses and rounded corner effects weredisregarded. The preliminary numerical studies showed that thecombined influence of strain hardening, residual stresses, androunded corner effects had little impact on the angle column ultimatestrength (all differences were below 3%), which is in line with thefindings reported by other authors; namely, Ellobody and Young(2005) and Shi et al. (2009). As mentioned previously, yield stressesfy were selected to cover a wide critical slenderness range, thusleading to the consideration of a few unrealistic (small) values;i.e., 30, 60, 120, 235, 400, and 500 N/mm2 (for the fixed-ended703 1:2mm columns); 120, 235, 400, and 500 N/mm2 (for thefixed-ended 503 1:2mm columns); 120, 235, and 400 N/mm2 (forthe fixed-ended 503 2:6mm columns); and 30, 60, 120, and 235N/mm2 (for the pin-ended 703 1:2mm columns).
Following the behavior observed in the experimentally testedcolumns described previously (namely, the length dependency ofthe imperfection sensitivity), a preliminary study was carried out toidentify the most detrimental imperfection shape; i.e., the criticalflexural-torsional shape and/or minor-axis flexural shape. For col-umn lengths corresponding to the left and intermediate parts of thePcrðLÞ curve horizontal plateaus, the flexural-torsional imperfec-tions were found to be the most detrimental ones (the columns werevirtually insensitive to the minor-axis flexural imperfections). Inthese (short-to-intermediate) columns, flexural-torsional imperfec-tions with amplitudes equal to 10% of wall thickness twere adopted.Conversely, for column lengths associated with the right part ofthe PcrðLÞ curve horizontal plateaus, the minor-axis flexural im-perfections were shown to be much more relevant that their criticalflexural-torsional counterparts. In these (intermediate-to-long) col-umns, the adopted initial imperfections combined a critical flexural-torsional component, with amplitudes equal to 10% of wall thicknesst; and a noncritical minor-axis flexural component, with amplitudesequal to L=750 (for the fixed-ended columns) or L=1,000 (for the pin-ended columns). These amplitudes are in line with the means of thevalues measured in the specimens tested by Popovic et al. (1999) andYoung (2004).
The combined flexural-torsional 1 minor-axis flexural initialimperfections were considered for the following lengths: 4,200,5,320, 7,000, and 8,900 mm (for the fixed-ended 703 1:2mm
columns); 3,000 and 4,000 mm (for the fixed-ended 503 1:2mmcolumns); 1,000, 1,500, and 2,000 mm (for the fixed-ended503 2:6mm columns); and 1,820, 2,520, 3,640, and 4,200 mm(for the pin-ended 703 1:2mm columns). All cross-section di-mensions, lengths, yield stresses fy, and numerical (SFEA) ultimatestresses fu are given in Tables 3 and 4 (for 89 fixed-ended columns)and in Table 5 (for 28 pin-ended columns).
Direct Strength Method Design Considerations
Regarding the existing design provisions for concentrically loadedequal-leg angle columns, the earlier American Iron and Steel In-stitute (AISI 1996, 2001) specifications prescribed ultimate strengthestimates of the form
Pn ¼ Ae � fn ð1Þ
where Ae 5 effective cross-section area and fn 5 column globalstrength, given by
fn ¼fy�0:658l
2c
�if lc# 1:5
fy
0:877l2c
!if lc. 1:5
with lc ¼ffiffiffiffiffiffifyfcre
s8>>><>>>:
ð2Þ
where fy 5 yield stress, fcre 5 critical global buckling stress, andlc 5 global slenderness. Because fn is based on the minimumbetween theflexural-torsional (major-axis) andflexural (minor-axis)buckling stresses andAe is based on the local (or torsional) bucklingstress, Popovic et al. (2001) showed that the aforementionedprocedure led to overly conservative Pn values as a result of thetorsional buckling stress coming into play twice (through fn andAe). To achieve more accurate (but still safe) ultimate strengthpredictions, Popovic et al. (2001) proposed the following modi-fication: base fn on the flexural (minor-axis) buckling stress aloneand base Ae on the local (torsional) buckling stress. Subsequently,Young (2004) tested fixed-ended angle columns and showedthat the modified AISI estimates were still conservative for stockycolumns and unsafe for slender columns. To obtain more accurateestimates, Young (2004) proposed the use of a modified globalstrength curve, given by
fne ¼fy�0:5l
2c
�if lc# 1:4
fy
0:5l2c
!if lc. 1:4
where lc ¼ffiffiffiffiffiffifyfcre
s8>>><>>>:
ð3Þ
where fcre 5minor-axis flexural buckling stress. The column ultimatestrength is still determined on the basis of Eq. (1); however, fnis replaced by fne. Rasmussen (2005) followed a different path todesign slender pin-ended angle columns, arguing that the angle singlesymmetry called for the consideration of an additional moment be-cause of the effective centroid shift. Quantifying this additionalmoment required calculating an angle cross-section effectivemodulusfor minor-axis bending and using an N-M interaction formula. Thisapproach was shown to yield more accurate ultimate strength esti-mates than its predecessors.
In the last decade, the DSM emerged as a simple and reliableapproach to design cold-formed steel members, and it has alreadybeen included in the most recent AISI (2007) and StandardsAustralia/StandardsNewZealand (SA/NZS2005) cold-formed steel
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specifications. The DSM approach is based on theWinter-type localstrength curve (Schafer 2008)
fnl ¼
fy if ll # 0:776
fy
fcrlfy
!0:4
12 0:15
fcrlfy
!0:424
35 if ll . 0:776
8>>>><>>>>:
with ll ¼ffiffiffiffiffiffifyfcrl
sð4Þ
where fcrl and fnl 5 local buckling stress and strength. However,because the column local and global failures often interact, thecurrent DSM combines Eq. (4) for local failure with Eq. (2) for
global failure, where fy is replaced by fne in Eq. (4). The currentDSM curve for local/global interactive collapse is then given by
fnle ¼fne if lle# 0:776
fne
�fcrlfne
�0:4"12 0:15
�fcrlfne
�0:4#
if lle. 0:776
8>><>>:
with lle ¼ffiffiffiffiffiffifnefcrl
rð5Þ
where fnle 5 local/global interactive strength; fne 5 global strength,obtained from Eq. (2); and fcrl 5 critical local buckling stress. Thecolumn ultimate load is given by
Table 3. Fixed-Ended Column Numerical Ultimate Stresses and Their Estimates according to the Proposed DSM-Based Approach
Section L (mm)
Buckling analysis
fy (N/mm2)Numericalfu (N/mm2)
DSM-Ffnle (N/mm2)
Numerical-to-predictedratio fu=fnlefcrl (N/mm2) fcre (N/mm2)
703 1:2 532 27.5 5,981 30 25.5 24.7 1.03703 1:2 980 24.8 1,762 30 24.3 23.7 1.02703 1:2 1,330 24.2 957 30 24.1 23.4 1.03703 1:2 1,820 23.9 511 30 24.0 23.0 1.04703 1:2 2,520 23.6 267 30 23.9 22.4 1.07703 1:2 3,640 23.3 128 30 23.7 21.0 1.13703 1:2 4,200 23.2 96 30 19.3 20.3 0.95703 1:2 5,320 22.8 60 30 18.1 18.4 0.98703 1:2 7,000 22.1 35 30 14.5 15.4 0.94703 1:2 8,900 21.1 21 30 11.3 11.3 1.00703 1:2 532 27.5 5,981 60 36.7 38.9 0.94703 1:2 980 24.8 1,762 60 34.7 37.1 0.93703 1:2 1,330 24.2 957 60 33.3 36.3 0.92703 1:2 1,820 23.9 511 60 31.7 35.3 0.90703 1:2 2,520 23.6 267 60 29.7 33.5 0.89703 1:2 3,640 23.3 128 60 28.6 29.9 0.96703 1:2 4,200 23.2 96 60 20.6 27.8 0.74703 1:2 5,320 22.8 60 60 19.3 23.2 0.83703 1:2 7,000 22.1 35 60 15.5 16.4 0.95703 1:2 8,900 21.1 21 60 12.8 10.7 1.20703 1:2 532 27.5 5,981 120 65.7 60.5 1.09703 1:2 980 24.8 1,762 120 63.8 57.0 1.12703 1:2 1,330 24.2 957 120 61.8 55.1 1.12703 1:2 1,820 23.9 511 120 57.7 52.3 1.10703 1:2 2,520 23.6 267 120 49.2 47.3 1.04703 1:2 3,640 23.3 128 120 35.5 37.9 0.94703 1:2 4,200 23.2 96 120 24.1 32.9 0.73703 1:2 5,320 22.8 60 120 21.4 23.2 0.92703 1:2 7,000 22.1 35 120 15.6 15.9 0.98703 1:2 8,900 21.1 21 120 12.8 10.7 1.20703 1:2 532 27.5 5,981 235 110 91.7 1.20703 1:2 980 24.8 1,762 235 106 84.7 1.25703 1:2 1,330 24.2 957 235 93.7 79.9 1.17703 1:2 1,820 23.9 511 235 71.8 72.5 0.99703 1:2 2,520 23.6 267 235 55.1 60.0 0.92703 1:2 3,640 23.3 128 235 37.9 39.1 0.97703 1:2 4,200 23.2 96 235 29.7 31.9 0.93703 1:2 5,320 22.8 60 235 23.5 23.2 1.01703 1:2 7,000 22.1 35 235 15.6 15.9 0.98703 1:2 8,900 21.1 21 235 12.8 10.7 1.20703 1:2 532 27.5 5,981 400 145 126.3 1.15703 1:2 980 24.8 1,762 400 122 113.4 1.08703 1:2 1,330 24.2 957 400 98 103.5 0.95
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Pn ¼ A � fnle ð6Þ
where A5 gross cross-section area. In Eq. (1), the local and globalbuckling effects are dealt with separately by means of effective areaAe and global buckling strength fn, respectively. Conversely, theyare handled simultaneously in Eq. (6) through local/global inter-active strength fnle.
Several cross-section geometries (e.g., lipped channels, Z-sections, rack sections, or hat sections) are currently prequalified
for the application of the DSM. Despite their extreme geometricalsimplicity, angle sections have not yet achieved such status; i.e., theyare not prequalified for the application of the current DSM designcurves. Nevertheless, Rasmussen (2006) and Chodraui et al. (2006)proposed distinct DSM-based approaches for the design of con-centrically loaded angle columns. While the former explicitlyincorporates the eccentricity caused by the effective centroidshift, which amounts to treating the columns as beam-columns, thelatter ignores the previous eccentricity, exploring instead different
Table 4. Fixed-Ended Column Numerical Ultimate Stresses and Their Estimates according to the Proposed DSM-Based Approach (Continued from Table 3)
Section L (mm)
Buckling analysis
fy (N/mm2)Numericalfu (N/mm2)
DSM-Ffnle (N/mm2)
Numerical-to-predictedratio fu=fnlefcrl (N/mm2) fcre (N/mm2)
703 1:2 1,820 23.9 511 400 72 87.9 0.82703 1:2 2,520 23.6 267 400 55 64.0 0.86703 1:2 3,640 23.3 128 400 36 38.4 0.94703 1:2 4,200 23.2 96 400 30.3 31.9 0.95703 1:2 5,320 22.8 60 400 24.1 23.2 1.04703 1:2 7,000 22.1 35 400 15.6 15.9 0.98703 1:2 8,900 21.1 21 400 12.8 10.7 1.20703 1:2 532 27.5 5,981 500 163 144.1 1.13703 1:2 980 24.8 1,762 500 135 127.1 1.06703 1:2 1,330 24.2 957 500 105 113.6 0.92703 1:2 1,820 23.9 511 500 72 92.9 0.78703 1:2 2,520 23.6 267 500 55 62.5 0.88703 1:2 3,640 23.3 128 500 36 38.4 0.94703 1:2 4,200 23.2 96 500 30.3 31.9 0.95703 1:2 5,320 22.8 60 500 24.1 23.2 1.04703 1:2 7,000 22.1 35 500 15.6 15.9 0.98703 1:2 8,900 21.1 21 500 12.8 10.7 1.20503 1:2 1,500 46.3 383.8 120 58 63.9 0.91503 1:2 2,000 45.6 215.9 120 56 57.0 0.98503 1:2 2,500 45 138.2 120 54 49.2 1.10503 1:2 3,000 44.3 96.0 120 36.6 41.1 0.89503 1:2 4,000 42.6 54.0 120 29.1 26.6 1.10503 1:2 1,500 46.3 383.8 235 88 86.3 1.02503 1:2 2,000 45.6 215.9 235 67 69.4 0.97503 1:2 2,500 45 138.2 235 55 52.4 1.05503 1:2 3,000 44.3 96.0 235 38 39.7 0.96503 1:2 4,000 42.6 54.0 235 29.1 26.6 1.10503 1:2 1,500 46.3 383.8 400 89 100.2 0.89503 1:2 2,000 45.6 215.9 400 67 69.5 0.96503 1:2 2,500 45 138.2 400 55 50.8 1.08503 1:2 3,000 44.3 96.0 400 38.9 39.7 0.98503 1:2 4,000 42.6 54.0 400 29.1 26.6 1.10503 1:2 1,500 46.3 383.8 500 89 103.0 0.86503 1:2 2,000 45.6 215.9 500 67 68.3 0.98503 1:2 2,500 45 138.2 500 55 50.8 1.08503 1:2 3,000 44.3 96.0 500 39 39.7 0.98503 1:2 4,000 42.6 54.0 500 29.1 26.6 1.10503 2:6 1,000 214.2 863.6 120 111 109.0 1.02503 2:6 1,500 205 383.8 120 100 96.6 1.03503 2:6 2,000 194.6 215.9 120 87.8 81.6 1.08503 2:6 1,000 214.2 863.6 235 202 170.7 1.18503 2:6 1,500 205 383.8 235 176 143.5 1.23503 2:6 2,000 194.6 215.9 235 128 110.5 1.16503 2:6 1,000 214.2 863.6 400 230 222.8 1.03503 2:6 1,500 205 383.8 400 192 168.1 1.14503 2:6 2,000 194.6 215.9 400 132 110.7 1.19Mean 1.01SD 0.11
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relationships between the local (flexural-torsional) and global(minor-axis flexural) buckling stresses. At this stage, it is worthmentioning that Dinis et al. (2011) have shown that the straight-forward use of the current DSM design curves [a combination ofEqs. (2) and (5)] leads to a significant number of poor ultimatestrength predictions, which is just an obvious and natural cause/consequence of the fact that angle columns are not prequalified forthe DSM application.
While in Eqs. (2) and (3) it is mandatory to calculate minor-axisflexural buckling stress fcre, Eqs. (4) and (5) require the knowledgeof local buckling stress fcrl, replaced herein (in the equal-leg anglecolumns) by the flexural-torsional buckling stress; i.e., localbuckling stress fcrl must be equated to the flexural-torsional bucklingstress. As a result of the flexural component presence, the localbuckling of equal-leg angles cannot be viewed as the usual localbuckling exhibited by other cross sections. Because the cornerflexural displacements have been found to play a key role in thecolumn postcritical strength, the local bucklingmechanics cannot beequated solely to torsional buckling and the angle column behaviorcannot be viewed as the sum of two pinned-free long plates. Thus,the buckling mode flexural component should not be omitted, evenif its contribution does not significantly alter the design strengthpredictions.
In this work, these two buckling stresses were determined bymeans of the code GBTUL (Bebiano et al. 2008), taking into ac-count the column actual end support conditions (fixed or pin-ned end sections) and the experimentally measured cross-section
dimensions and steel properties—the fcrl and fcre values are givenin Tables 1 and 2 (for the experimental ultimate strengths) andTables 3–5 (for the numerical ultimate strengths). It is worthmentioning that the signature fcrlðLÞ and fcreðLÞ curves shown inFigs. 1 and 2were obtained bymeans of two separate GBT analyses:(1) those that included the major-axis bending mode, torsion mode,and antisymmetric local mode, to obtain the fcrlðLÞ curve, and (2)those that included only the minor-axis bending mode, to obtain thefcreðLÞ curve. The fcrlðLÞ curve corresponds to clamped supportconditions for participatingmodes (major-axis bending, torsion, andantisymmetric local), both in the pin- and fixed-ended columns. ThefcreðLÞ curve was obtained considering clamped or pinned supportconditions for the minor-axis bending mode, respectively, for thefixed- and pin-ended columns.
Figs. 1(a and b) and 2(a and b) show the signature fcrðLÞ curves ofthe fixed-ended columns tested by Popovic et al. (1999) [Fig. 1(a)]and Young (2004) [Fig. 1(b)], and the pin-ended columns tested byPopovic et al. (1999) [Fig. 2(a)], Chodraui et al. (2006), and Maiaet al. (2008) [Fig. 2(b)]; thefixed-ended columns tested byMesacasa(2011) are not included in Figs. 1(a and b) and 2(a and b). The dashedand thinner solid curves correspond to flexural-torsional bucklingstress fcrl and the thicker solid curve corresponds to minor-axisflexural buckling stress fcre; the column critical stresses correspondto the lower fcrl and fcre. Figs. 1(a and b) and 2(a and b) each depictone flexural-torsional curve per cross-section geometry and aminor-axis flexural curve common to all cross sections sharing the samemidline dimensions, which amounts to neglecting the contributions
Table 5. Pin-Ended Column Numerical Ultimate Stresses and Their Estimates according to the Two Proposed DSM-Based Approaches
Section L (mm)
Buckling analysis
fy (N/mm2)Numericalfu (N/mm2)
DSM-Ffnle (N/mm2)
Numerical toDSM-F predicted
ratio fu=fnle
DSM-Pfnle (N/mm2)
Numerical toDSM-F predicted
ratio fu=fnlefcrl (N/mm2) fcre (N/mm2)
703 1:2 532 27.4 1,496 30 24.6 24.5 1.00 21.1 1.17703 1:2 980 24.8 441 30 23.6 23.2 1.02 19.4 1.22703 1:2 1,330 24.2 234 30 23.3 22.4 1.04 18.9 1.23703 1:2 1,820 23.9 128 30 20.7 21.2 0.98 18.3 1.13703 1:2 2,520 23.7 67 30 18.7 19.1 0.98 17.3 1.08703 1:2 3,640 23.4 32 30 15.1 15.1 1.00 14.6 1.03703 1:2 4,200 23.2 24 30 13.2 12.6 1.05 12.6 1.05703 1:2 532 27.4 1,496 60 27.1 38.4 0.71 24.2 1.12703 1:2 980 24.8 441 60 24.6 35.4 0.69 21.9 1.12703 1:2 1,330 24.2 234 60 23.9 33.3 0.72 21.3 1.12703 1:2 1,820 23.9 128 60 21 30.1 0.70 20.6 1.02703 1:2 2,520 23.7 67 60 18.8 24.7 0.76 19.3 0.97703 1:2 3,640 23.4 32 60 15.3 15.6 0.98 15.0 1.02703 1:2 4,200 23.2 24 60 13.4 12.0 1.12 12.0 1.12703 1:2 532 27.4 1,496 120 29.6 58.9 0.50 25.8 1.15703 1:2 980 24.8 441 120 26.9 52.1 0.52 23.2 1.16703 1:2 1,330 24.2 234 120 25.5 46.5 0.55 22.5 1.13703 1:2 1,820 23.9 128 120 21 38.2 0.55 21.6 0.97703 1:2 2,520 23.7 67 120 18.8 25.8 0.73 19.6 0.96703 1:2 3,640 23.4 32 120 15.3 15.3 1.00 14.8 1.03703 1:2 4,200 23.2 24 120 13.4 12.0 1.12 12.0 1.12703 1:2 532 27.4 1,496 235 39.2 87.1 0.45 26.6 1.48703 1:2 980 24.8 441 235 30.3 71.1 0.43 23.8 1.27703 1:2 1,330 24.2 234 235 25.7 57.4 0.45 23.0 1.12703 1:2 1,820 23.9 128 235 21 39.5 0.53 21.7 0.97703 1:2 2,520 23.7 67 235 18.8 25.3 0.74 19.5 0.97703 1:2 3,640 23.4 32 235 15.3 15.3 1.00 14.8 1.03703 1:2 4,200 23.2 24 235 13.4 12.0 1.12 12.0 1.12Mean 0.80 1.10SD 0.24 0.11
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of the wall’s own inertia to the cross-section minor moment ofinertia. The open and closed circles located on the fcrlðLÞ and fcreðLÞcurves, respectively, identify the lengths of the columns tested by thevarious researchers.
Concerning the fixed-ended columns [Figs. 1(a and b)], it isclear that all the columns tested by Young (2004) are located faraway (to the left) from the intersection between the fcrlðLÞ andfcreðLÞ curves, which means that fcre is always much higher thanfcrl—the lowest fcre=fcrl ratio is equal to 2.7, corresponding to thelongest column (L5 3,500mm) with the stockiest cross section(703 1:9mm). Conversely, most of the fixed-ended columnstested by Popovic et al. (1999) have lengths placing them on theright side of the fcrlðLÞ curve horizontal plateaus; i.e., near theirintersection with fcreðLÞ, which means fcre=fcrl values close to 1.0.The few exceptions concern some 503 2:5mm column lengthslocated on the left side of the fcrlðLÞ curve horizontal plateau.Moreover, in some of these columns the minor-axis flexuralbuckling mode becomes critical (i.e., fcre=fcrl , 1:0). Concerningthe pin-ended columns [Figs. 2(a and b)], most of their lengthsplace them on the right side of the fcrlðLÞ curve horizontal pla-teaus. The exceptions are the shortest 503 2:5mm columns testedby Popovic et al. (1999) and the columns tested by Chodraui et al.(2006), which are located on the left side of the fcrlðLÞ curvehorizontal plateaus.
It will be shown subsequently that in both the fixed- and pin-ended columns the ultimate strength is strongly affected by the lo-cation of the column length; i.e., the closeness between the fcrl and
fcre values. As mentioned previously, the shorter columns locatedon the left side of the plateaus have clearly stable postcritical be-haviors because they exhibit very small corner displacements.Conversely, the longer columns located on the right side of theplateaus possess a minute/negligible postbuckling strength becausethey exhibit significant corner displacements stemming predomi-nantly fromminor-axisflexure (evenwith fcre=fcrl . 1:0); i.e.,minor-axis flexural buckling is noncritical.
After having determined the fcrl and fcre values of all the columns(given in Tables 1–5), it becomes possible to calculate their minor-axis flexural and flexural-torsional slenderness values, given bylc 5
ffiffiffiffiffiffiffiffiffiffiffiffify=fcre
pand ll 5
ffiffiffiffiffiffiffiffiffiffiffify=fcrl
p. Figs. 3(a and b) (fixed-ended
columns) and Figs. 4(a and b) (pin-ended columns) show the varia-tionof theultimate-to-yield stress ratiovalues (fu=fy) obtained from theexperimental (open circles) and numerical (closed circles) results,withbothlc [Figs. 3(a) and 4(a)] andll [Figs. 3(b) and 4(b)]. The thin solidline in Fig. 4(b) identifies a strength curve proposed by Rasmussen(2005), which will be addressed subsequently. The solid lines inFigs. 3(a andb) and 4(a andb) correspond to the global [fne=fy; Eq. (2)]and local [fnl=fy; Eq. (4)] DSM design curves. The observations fromFigs. 3(a and b) and 4(a and b) prompts the following comments:1. In Figs. 3(a and b) and 4(a and b) the clouds of open and closed
circles share nearly the same location, showing that the ex-perimental and numerical ultimate strengths exhibit similaroverall tendencies.
2. Regardless of the global or local slenderness range, the vastmajority of fu=fy values fall well below the DSM curves, thus
Fig. 1. Variation of fcr with L for the fixed-ended columns tested by (a) Popovic et al. (1999) and (b) Young (2004)
Fig. 2. Variation of fcr with L for the pin-ended columns tested by (a) Popovic et al. (1999) and (b) Chodraui et al. (2006) and Maia (2008)
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showing/confirming that the DSM pure global and localstrength curves consistently overestimate the angle columnultimate strengths by fairly large margins.
3. The fu=fy values are widely spread for low-to-moderate globalslenderness values lc; however, they become less scattered forhigher lc values. Figs. 3(a) and 4(a) also show the globalstrength curve [the dashed line; Eq. (3)] proposed by Young(2004). It is clear that this curve follows the tendency of thefu=fy values much more closely than the current DSM globalstrength curve [the solid line; Eq. (2)], particularly formoderate-to-high global slenderness values lc.
4. Because of the small variation (drop) of fcrl with L within thefcrlðLÞ curve horizontal plateau, the fu=fy values concerning thecolumns with the same yield stress are clearly grouped to-gether in Figs. 3(b) and 4(b). As fy increases, the correspondinggroup is associated with a higher local slenderness ll and
a lower strength (i.e., moves down and to the right)—withineach group, the slenderness increases with length L.
5. Within each group, the variation of fu=fy with ll is markedlydifferent for the fixed- and pin-ended columns. While thefixed-ended columns [Fig. 3(b)] exhibit a high vertical dis-persion, thus implying a very significant variation of fu=fywith L (even if fcrl remains practically unaltered), those con-cerning the pin-ended columns [Fig. 4(b)] are rather packedtogether and located considerably below the DSM localcurve [Eq. (4)]. This behavioral difference is mainly causedby the influence of the effective centroid shift, which wasshown to be much more relevant in pin-ended columns than intheir fixed-ended counterparts. This effect was considered inthe rational design methodology developed by Rasmussen(2005, 2006) and based on the beam-column concept; theadditional moment caused by the compressive force action
Fig. 3. Fixed-ended columns: variation of fu=fy with (a) lc and (b) ll
Fig. 4. Pin-ended columns: variation of fu=fy with (a) lc and (b) ll
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on the eccentricity as a result of the effective centroid shiftwas taken into account by using a beam-column interactionequation.
6. Despite the quite pronounced qualitative and quantitativedifferences detected in the elastic postbuckling behaviors ofthe pin-ended columns (see the results presented previously),the differences between their ultimate strengths are onlymoderate. The fu=fy values corresponding to the pin-endedcolumns [displayed in Fig. 4(b)] are only slightly groupedtogether and show a tendency to vary with the local slender-ness ll. Thus, it seems possible (and may be advantageous) tofit a new strength curve for the design against local (flexural-torsional, in reality) failure of pin-ended angle columns, whichis given by
fnl ¼fy if ll # 0:71
fy
fcrlfy
!"12 0:25
fcrlfy
!#if ll . 0:71
8>>><>>>:
with ll ¼ffiffiffiffiffiffifyfcrl
sð7Þ
Fig. 4(b) depicts this curve and it is clear that it follows fairlywell the trend of the fu=fy values. Moreover, because theinfluence of the effective centroid shift is directly taken intoaccount, it should lead to an efficient design procedure forpin-ended angle columns without the need to resort to thebeam-column concept. It should also be emphasized that thiscurve falls well below the current DSM local strength curve(applicable to columns with various prequalified cross-section shapes). This fact just shows that the interaction be-tween flexural-torsional buckling and minor-axis flexuralbuckling in pin-ended angle columns ismuchmore severe thanthe interaction between local and global (flexural or flexural-torsional) buckling in columns with other cross-sectionshapes. This stems from the extremely high sensitivity to theeffective centroid shift, which was clearly demonstrated pre-viously in the paper and can be physically explained by the factthat both angle cross-section walls (legs) are outstands—in allthe columns with cross sections prequalified to the applicationof the DSM, local buckling is virtually always triggered byinternal walls, which entails a considerably less severe in-teraction with global buckling.
7. Conversely, the differences between the fixed-ended columnultimate strengths are rather sharp. Indeed, most of them arelocated in almost vertical line segments; thus, columns sharingthe same yield and critical stresses (but with different lengths)exhibit quite distinct fu=fy values. This somewhat paradoxicalbehavior appears to indicate that local slenderness ll does notprovide an adequate measure of the column ultimate strength.Recalling that most of these columns buckle in flexural-torsional modes almost akin to a local mode (the word almoststems from the presence of corner flexural displacements), itseems fair to say that, within the fcrlðLÞ curve horizontalplateau, the fixed-ended column ultimate strength naturetravels from local to global as the length increases—anefficient design procedure for these columns must take thisfact into account.
8. The strength curve proposed by Rasmussen (2005) for pin-ended columns [the solid thin line in Fig. 4(b)], which isgiven by
fnl ¼ r ×b × fy ð8Þ
r ¼ Ae
A¼
8><>:
1 if ll # 0:673
ll2 0:22l2l
if ll . 0:673and
b ¼
8><>:
1 if ll # 1:22
0:68ðll2 1Þ0:25
if ll . 1:22
and takes into account the bending as a result of the effectivecentroid shift, through parameter b, and the local (torsional)buckling, through effective area reduction factor r. Althoughthis curve also provides fairly accurate ultimate strengthpredictions, it is clear that it leads to slightly higher and lessaccurate ultimate strength predictions than the curve proposedin Eq. (7).
The DSM distortional buckling curve is not considered in thiswork because this bucklingmode does not occur in plain angles. Theinterested reader is referred to the work of Silvestre and Camotim(2010) for a mechanical definition of distortional buckling.
Proposal of a Direct Strength Method–BasedDesign Approach
To enable the application of the DSM philosophy to the design offixed- and pin-ended equal angle columns, the following DSM-basedapproach is proposed:1. To adopt different procedures/design curves to estimate the
ultimate strength of fixed- and pin-ended columns, they are de-signated as DSM fixed (DSM-F) and DSM pinned (DSM-P),respectively.
2. The DSM-F procedure combines Eq. (3), which is the globalstrength curve proposed by Young (2004), with Eq. (5), whichis the current DSM design curve for local/global interactivefailure. Although this procedure was developed specificallyfor fixed-ended columns, its application to pin-ended columnsis also assessed.
3. The DSM-P procedure combines Eq. (3), as before, with aproposed/new DSM curve for local/global interactive failure,defined by [see Fig. 4(b)]
fnle ¼fne if lle# 0:71
fne
�fcrlfne
��12 0:25
�fcrlfne
��if lle. 0:71
8><>:
with lle ¼ffiffiffiffiffiffifnefcrl
rð9Þ
which only differs from Eq. (5) in the fact that the yieldstress fy is replaced by the global strength fne. It is worthemphasizing again that this procedure is applied solely to pin-ended columns, which constitute the specific target of itsdevelopment.
Attention is now turned to assessing the performance of the pro-posedDSM-based approach; i.e., theDSM-F andDSM-P procedures.The corresponding column ultimate strength predictions (the fnlevalues) are included in Tables 1–5. The ultimate-to-predictedstrength ratios (fu=fnle) are also given, where ultimate strengths fucorrespond to the test values (Tables 1 and 2) and numerical values(Tables 3–5). In addition, Table 1 also gives the test-to-predictedratio fu=fn values provided by the application of the methodology
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developed by Young (2004), which combines Eqs. (1) and (3) to theset of fixed-ended columns tested by Popovic et al. (1999) andYoung (2004). On the other hand, Table 2 also presents the test-to-predicted ratios fu=fn obtained by Rasmussen (2006), with hisbeam-column methodology, for the pin-ended columns tested byWilhoite et al. (1984) and Popovic et al. (1999).
The close observation of the ultimate strength estimates pre-sented in Tables 1–5 leads to the following remarks:1. The DSM-F procedure leads to fairly accurate estimates of the
fixed-ended column experimental ultimate strength values(see Table 1); i.e., fu=fnle average and SD of 0.98 and 0.14,respectively. The methodology developed by Young (2004)leads to considerably more conservative results; i.e., fu=fnaverage and SD of 1.15 and 0.18, respectively.
2. The DSM-F procedure also provides fairly accurate predic-tions of the fixed-ended column numerical ultimate strengths(see Tables 3 and 4); i.e., fu=fnle average and SD of 1.01 and0.11, respectively.
3. Not surprisingly, the DSM-F procedure leads to accurate (onaverage) but widely scattered estimates of the pin-ended col-umn experimental ultimate strengths (see Table 2); i.e., fu=fnleaverage and SD of 1.01 and 0.29, respectively. However, 16(out of 37) predictions have errors higher than 20% (both safeand unsafe). This is a result of the lack of proper accounting forthe effective centroid shift effect—the DSM-F procedureadopts the current DSM strength curve for local/global in-teractive failure [Eq. (5)]. Conversely, the DSM-P procedureleads to reasonably accurate and safe (but fairly scattered)estimates of the pin-ended column experimental ultimatestrengths; i.e., fu=fnle average and SD of 1.13 and 0.25, re-spectively. The design approach developed by Rasmussen(2006) yields more conservative and slightly less-scatteredultimate strength predictions; i.e., fu=fn average and SDof 1.26and 0.21, respectively (the lower scatter is also related to thesmaller number of results involved).
4. The DSM-F procedure also predicts rather poorly the pin-ended column numerical ultimate strengths (see Table 5),which are mostly largely overestimated; i.e., fu=fnle average
and SD of 0.80 and 0.24, respectively. As for the estimatesprovided by the DSM-P procedure, they are slightly conser-vative and exhibit a fairly low scatter; i.e., fu=fnle average andSD of 1.10 and 0.11, respectively.
5. The ultimate-to-predicted (where ultimate means test or nu-merical) stress ratios fu=fnle can be viewed and compared inFig. 5(a) for fixed-ended columns and Fig. 5(b) for pin-endedcolumns. The open and closed circles again stand for theexperimental (test) and numerical results, respectively. Withthe exception of four less-accurate pin-ended column ultimatestrength estimates [open circles in Fig. 5(b); the four under-estimations concerning eccentrically loaded columns tested byWilhoite et al. (1984)], all the pin- and fixed-ended columnfu=fnle values exhibit an acceptable scatter and vary randomlyaround 1.01 (fixed-ended columns) and 1.12 (pin-endedcolumns).
6. While the fixed-ended column fu=fnle values are spread alonga fairly wide local/global slenderness range, their pin-endedcolumn counterparts aremostly accumulated in a limited local/global slenderness range (0:5, lle , 1:5); only numericalfu=fnle values fall outside this range. To properly assess theaccuracy of the DSM-P procedure in the full lle range, it isnecessary to perform experimental tests on pin-ended columnswith high local/global slenderness values (lle . 1:5). In thisrespect, it is worthmentioning thatmost of the (few) numericalultimate strengths associated with high lle values are under-estimated by the DSM-P procedure; this underestimation islarger for the two columns exhibiting lle values higher than 2.5.
Next, the determination of LRFD resistance factor f for theproposed DSM-based approaches is briefly addressed. According tothe most recent North American cold-formed steel specification(AISI 2007), resistance factorf can be calculated using the followingformula (given in Section F.1.1 of Chapter F of the specification):
f ¼ CfðMmFmPmÞ e2b0
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiV2M1V2
F1CPV2P1V2
Q
pwith CP ¼
�1 þ 1
n
�m
m2 2ð10Þ
(a) (b)
Fig. 5. Variation of fnle=fu with lle for the (a) DSM-F (fixed-ended columns) (b) and DSM-P (pin-ended columns) procedures
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whereCf5 calibration coefficient (Cf 5 1:52 for LRFD);Mm 5 1:0and Fm 5 1:005mean values of the material and fabrication factor,respectively; b0 5 target reliability index (b0 5 2:5 for structuralmembers in LRFD); VM 5 0:10, VF 5 0:05, and VQ 5 0:215coefficients of variation of thematerial factor, fabrication factor, andload effect, respectively; andCP 5 correction factor that depends onthe number of tests (n) and degrees of freedom (m5 n2 1). Toevaluate resistance factorf for each DSM-based procedure (DSM-Fand DSM-P), it is necessary to calculate Pm and VP, which are themean and SD values of the exact-to-predicted stress ratios fu=fnle.Here, exact means either the test fu values, numerical fu values, orboth the test and numerical fu values.
Table 6 gives the n, CP, Pm, VP, and f values obtained for thecolumn ultimate strength estimates provided by the DSM-F (fixed-ended columns) and DSM-P (pin-ended columns) procedures usingthe test, numerical, and overall (test plus numerical) data. The re-sistance factor values associated with each of the two proposedDSM-based procedures are as follows: (1) for the experimental data,f5 0:81 (for the fixed-ended columns) and f5 0:78 (for the pin-ended columns); and (2) for the numerical data, f� 0:89 (for thefixed-ended columns) and f� 0:95 (for the pin-ended columns).When all experimental and numerical datawere considered together,the overall application of the DSM-F and DSM-P procedures led tof5 0:87 and 0.85, which are values practically coincident with thef5 0:85 value recommended by AISI (2007). Therefore, it may bereadily concluded that the f5 0:85 value, which is used whenapplying the current DSM, can also be safely adopted with theproposed DSM-F and DSM-P procedures. Additionally, the DSM-Fapproach was also tested for the pin-ended columns, which usesEqs. (3) and (5) and differs from the current DSM in that the globalstrength curve is replaced by the one proposed byYoung (2004). Theresistance factor values obtained were (see Table 6) f5 0:65(experimental), f5 0:56 (numerical), and f5 0:60 (experimentaland numerical), which are well below the f5 0:85 value recom-mended by AISI (2007). Therefore, the DSM-F approach can beadopted for pin-ended plain angle columns iff5 0:60 is adopted (aneven lower value would be required to enable the application of thecurrent DSM),which significantly lowers the strength of any columnbecause of the high scatter of the DSM-F predictions. The authorsrather prefer the use of a new local buckling curve (i.e., the DSM-Papproach), leading to less scattered predictions, together with therecommended resistance factor off5 0:85. This it is amore rationalapproach, reflecting the different mechanics of the pin- and fixed-ended plain angle behaviors.
Finally,Ganesan andMoen (2010, 2012) recently conducted 264column tests to study the LRFD strength reduction factor for DSMdesign of cold-formed steel compression members in order to in-vestigate whether this factor can be increased above its current valueof f5 0:85, given that this specific value was established about20 years ago (AISI 1991). They considered a much larger set ofconcentrically loaded column tests; namely, 675 tests involving
columns with plain channels, lipped channels, angles, Z-sections,and hat sections. In the particular case of concentrically loaded(i.e., without the eccentricity L=1,000) angle columns, Ganesan andMoen (2010) found that the current DSM resistance factor, based on75 test results (50 and 25 for plain and lipped angles, respectively) isf5 0:71. They argued that this very low value is a result of the highcoefficient of variation of test-to-predicted ratios for angle columnsand concluded that “. . .fundamental research on the mechanics ofangle compression members is needed to improve existing designmethods. . . The low values for the resistance factor for angle col-umns indicate that the fundamental behavior of angle section col-umns is yet to be completely understood and there is a need for moreresearch in the future.” The work reported in this paper shows thatthe proposed DSM-based approach is accurate and, above all, is alsomechanically sound because (1) it separates the provisions for fixed-and pin-ended columns because of the qualitative and quantitativedifferences involving the effective centroid shift responsible forthe interaction between flexural-torsional (local) and (minor-axis)flexural buckling, and (2) it incorporates a recently developed (forfixed-ended columns only) global strength curve. This curve, whichcan be applied to both pin- and fixed-ended columns, is able tocapture the strong influence of minor-axis flexural buckling (eithercritical or noncritical) on the column postbuckling behavior andstrength. The fact that these relevant features are not contemplated inthe current DSM is at the root of the inadequacy (Dinis et al. 2012a)and the very low load resistance factor (Ganesan and Moen 2010,2012) associated with its application to equal-leg angle columns; inparticular, the local strength curve does not adequately capture thepeculiar behavioral features associated with the flexural-torsionalbuckling of angle columns.
It is still worth mentioning that the angle column problemaddressed in this work is the simplest one; i.e., concentricallyloaded plain equal-leg angle columns. Yet, the ultimate strengthestimates provided by the current design methods (the main spec-ification and DSM) are not as good as those obtained for morecomplex sections, such as plain or lipped channels. The considerationofother relevant aspects—such as load eccentricities, leg asymmetries,or stiffened legs—only hides the pure, but rather singular, behavior ofconcentrically loaded plain equal-leg angle columns, which deservesto be investigated on its own. It is expected that the aforementionedeffects can be successfully handled by a DSM approach similar to theone proposed in this work within specified limits, provided that thebuckling loads involved are evaluated rigorously and account forall the relevant effects; the DSM prequalification procedure will bevery helpful in the specification of those limits.
Conclusion
After summarizing the recent findings concerning the buckling,postbuckling, and ultimate strength behaviors of fixed- and
Table 6. LRFD Resistance Factors f Calculated according to AISI (2007): DSM-F and DSM-P Procedures
LRFDCoefficients
Fixed-ended column (DSM-F) Pin-ended column (DSM-F) Pin-ended column (DSM-P)
Test NumericalTest andnumerical Test Numerical
Test andnumerical Test Numerical
Test andnumerical
n 41 89 130 37 28 65 37 28 65CP 1.078 1.035 1.024 1.087 1.119 1.048 1.087 1.119 1.048Pm 0.980 1.023 1.010 1.007 0.800 0.918 1.133 1.103 1.120VP 0.145 0.105 0.120 0.288 0.237 0.285 0.245 0.111 0.198f 0.81 0.89 0.87 0.65 0.56 0.60 0.79 0.95 0.86
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pin-ended short-to-intermediate equal-leg angle columns, the paperaddressed their design bymeans of aDSM-based approach. Becauseshort-to-intermediate angle columns buckle in flexural-torsionalmodes almost akin to the local buckling modes exhibited by col-umns with other cross-section shapes, the first step consistedof adopting the DSM concept of local/global interactive failure.The curve proposed by Young (2004), within the context of fixed-ended columns, was adopted to estimate the column globalstrength because it was found to provide more accurate fixed- andpin-ended column ultimate strength estimates than the currentDSM curve.
The recent disclosure of distinct mechanical features in thepostbuckling and ultimate strength behaviors of fixed- and pinned-ended columns, showing that the latter are much more prone andsensitive to the occurrence of interaction between flexural-torsional(local) and global buckling modes, led to the adoption of a differentlocal strength curve for each end support condition. While the cur-rent DSM local strength curve is retained for fixed-ended columns,a new strength curve is proposed for pin-ended columns. This lastcurve makes it possible to capture the effective centroid shift effectsquite accurately, which are much more relevant in pin-ended col-umns than in fixed-ended columns.
The DSM-based approach proposed for the design of fixed- andpin-ended equal-leg angle columns (1) adopts the local/global in-teractive failure concept and (2) uses distinct procedures (localstrength curves) for fixed-ended (DSM-F) and pin-ended (DSM-P)columns, thus reflecting more closely the actual angle columnbehavior. This design approach was shown (1) to provide fairlyaccurate ultimate strength predictions for a wide column slendernessrange, while retaining the simplicity of the current DSM application;and (2) to exhibit an overall performance that compares favorablywith those displayed by the other methods available in the liter-ature to design angle columns. Moreover, it was also shown thatthe LRFD resistance factor of f5 0:85, used in the current DSM,can also be safely adopted when applying the proposed DSMapproach.
Acknowledgments
The authors gratefully acknowledge Mr. Enio Mesacasa Jr., for thework carried out during his stay in Lisbon and also for sharing someof his recent research results; and the financial support of the Portu-guese Foundation for Science and Technology (FCT), through theresearch project Generalised Beam Theory (GBT) - Development,Application and Dissemination (PTDC/ECM/108146/2008).
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