research report LRFD and LSD Resistance Factors for Cold- Formed Steel Compression Members RESEARCH REPORT RP10-5 JULY 2010

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1 research report RF ad S Resistace Factors for Cold- Fored Steel Copressio Mebers RESEARCH REPORT RP- 5 JUY Coittee o Specificatios for the esig of Cold-Fored Steel Structural Mebers Aerica Iro ad Steel Istitute

2 The aterial cotaied herei has bee developed by researchers based o their research fidigs. The aterial has also bee reviewed by the Aerica Iro ad Steel Istitute Coittee o Specificatios for the esig of Cold-Fored Steel Structural Mebers. The Coittee ackowledges ad is grateful for the cotributios of such researchers. The aterial herei is for geeral iforatio oly. The iforatio i it should ot be used without first securig copetet advice with respect to its suitability for ay give applicatio. The publicatio of the iforatio is ot iteded as a represetatio or warraty o the part of the Aerica Iro ad Steel Istitute, or of ay other perso aed herei, that the iforatio is suitable for ay geeral or particular use or of freedo fro ifrigeet of ay patet or patets. Ayoe akig use of the iforatio assues all liability arisig fro such use. Copyright Aerica Iro ad Steel Istitute

3 College of Egieerig The Charles Edward Via, Jr. epartet of Civil ad Eviroetal Egieerig Patto Hall (5) Blacksburg, Virgiia 46 54/ Fax: 54/ July, To: AISI Coittee Mebers Subject: Virgiia Tech Research Report No. /4 Resistace Factor for Cold-Fored Steel Copressio Mebers Please fid eclosed the fial research report that suarizes our efforts to calculate the resistace factor for cold-fored steel copressio ebers. A total of 675 colu tests were cosidered i this study. Mai Specificatio ad irect Stregth Method calculatios were perfored o the collected data ad the oial axial stregth each colu was predicted. The resistace factor was calculated per cross-sectio type, ultiate liit state, ad cosiderig partially ad fully effective colus accordig to both the RF ad S ethods. The observed treds deostrate that SM is a ore accurate stregth predictio approach tha the Mai Specificatio, ad that defiig SM resistace factors by liit state (i.e. local, distortioal, ad global bucklig) provide a viable optio for icreasig φ c above.85. Recoeded code revisios are suarized i Sectio 5. of this report. They are also provided here as a coveiece to the reader:. efie RF resistace factors by liit state for SM: where φ ce =.8, φ c =.9, ad φ cd =.95., efiig resistace factors by liit state allows us to take advatage of SM s stregth predictio accuracy for colus with partially effective cross-sectios. Higher resistace factor are supported by the calculatios ad data i this report - local-global bucklig iteractio goes to.9 ad distortioal bucklig goes to.95. The icreased resistace factors for locally sleder colus coe with a sall price - a resistace factor of.8 for colus with fully effective cross-sectios. This is a slightly usavory result (supported by the data) that could be a acceptable coproise cosiderig that ost cold-fored steel colus desiged with AISI-S-7 are partially effective. For copariso, the recoeded Mai Specificatio resistace factors (derived with the sae data used to evaluate SM) are also preseted: where φ ce =.8, φ c =.8, ad φ cd =.95. Ivet the Future VIRGINIA POYTECHNIC INSTITUTE AN STATE UNIVERSITY A equal opportuity, affirative actio istitutio

4 The lowered Mai Specificatio resistace factor for the local bucklig liit state reflects the observatio ade i this report that the Mai Specificatio is less accurate tha SM.. Move the AISI irect Stregth Method ito the Mai Specificatio. SM was deostrated throughout this research to be a ore accurate predictor of colu capacity tha the Mai Specificatio. Iproved predictio accuracy facilitates higher resistace factors, see Recoedatio #. 3. ower the S resistace factor to φ c =.7 to provide a uifor probability of failure cosistet with β=3.. Alteratively, the curret S resistace factor of φ c =.8 ca be aitaied by lowerig β fro 3. to.5 ad decreasig the / ratio fro /3 to /5 (see Sectio 4..9 of this report). SM resistace factors by S liit state could also be cosidered. 4. The S load effects coefficiet of variatio, V Q, could be decreased fro. to.9 i AISI-S-7, Chapter F, Sectio F.. As deostrated i Eq. (35) of this report, the coefficiet of variatio of the load effects is a fuctio of dead to live load ratio. For /=/5, V Q =.. For /=/3, V Q =.9. This reductio i COV is beeficial ad will icrease the calculated S resistace factor (ad available colu capacity) by approxiately 5%. We hope that our AISI COS frieds ad colleagues will carefully review our report ad seriously cosider its recoedatios. Thak you to our AISI task group for their advice ad guidace durig this research progra. Sicerely, Cris Moe coe@vt.edu Karthik Gaesa gkarthik@vt.edu VIRGINIA POYTECHNIC INSTITUTE AN STATE UNIVERSITY A equal opportuity, affirative actio istitutio

5 VIRGINIA POYTECHNIC INSTITUTE AN STATE UNIVERSITY The Charles E. Via, Jr. epartet of Civil ad Eviroetal Egieerig Blacksburg, VA 46 Structural Egieerig ad Materials RF ad S Resistace Factors for Cold-Fored Steel Copressio Mebers FINA REPORT CE/VPI-ST-/4 by Karthik Gaaesa Graduate Research Assistat r. Cristopher. Moe Assistat Professor for the Aerica Iro ad Steel Istitute 4 Coecticut Aveue, NW, Suite 75 Washigto,.C. 36 JUY

6 Resistace Factor for Cold-Fored Steel Copressio Mebers (ABSTRACT) This research ivestigates if the RF stregth reductio factor for cold-fored steel copressio ebers ca be icreased above its curret value of =.85, which was established by the RF Cold-Fored Steel esig Maual (99) o the basis of 64 colu tests. The resistace factor i the Caadia code for cold-fored steel copressio ebers is also evaluated.a total of 675 cocetrically loaded plai ad lipped C-sectio colus, plai ad lipped Z-sectio colus, hat ad agle colus, icludig ebers with holes, are cosidered i the study. The predicted stregths are calculated with the AISI-S-7 Mai Specificatio ad the AISI irect Stregth Method. The test-to-predicted stregth statistics are eployed with the first order secod oet reliability approach i AISI-S-7 Chapter F as well as a higher order ethod to calculate the resistace factor per cross-sectio type, ultiate liit state, ad cosiderig partially ad fully effective colus. SM is observed to predict the colu capacity ore accurately tha the Mai Specificatio, supportig a higher SM resistace factor for colus failig i local bucklig or distortioal bucklig liit states. The test-to-predicted ratios for plai ad lipped agle colus exhibit a high coefficiet of variatio ad becoe ore ad ore coservative as global slederess icreases. It is cocluded that fudaetal research o the echaics of agle copressio ebers is eeded to iprove existig desig ethods.

7 Table of Cotets ist of Figures... iv ist of Tables... vi Chapter : Itroductio.... oad ad Resistace Factor esig of Cold Fored Steel Copressio Mebers.... Code History Objective ad Scope of Research Overview of Thesis...7 Chapter : AISI Specificatio...9. AISI Mai Specificatio ocal-global Bucklig Iteractio iit State istortioal Bucklig iit State Capacity of Agle Colus irect Stregth Method... 5 Chapter 3: CFS Colu Test atabase Overview of atabase Experietal Progra etails... ii

8 Chapter 4: Resistace Factor Equatios ad Calculatios Resistace Factor erivatio Resistace Factor Results All Colus Colus with Holes Colus without Holes Partially ad Fully Effective Sectios Resistace Factors by iit State Copariso of Predictio Accuracy with Cross-sectio iesios Resistace Factors for Agle Colus Resistace Factor usig Modified Expressios for V Q ad V R Copariso of Resistace Factors for S... 6 Chapter 5: Coclusios Suary ad Coclusios Recoedatios for Code Revisios Future Work Refereces... 7 Appedix Appedix Appedix 3... iii

9 ist of Figures Figure. Probability distributios of the load effect, Q, ad the resistace, R.... Figure. History of AISI ad AISC colu curves...4 Figure 3. istortioal bucklig ode of a ipped C-sectio colu with holes....6 Figure 4. Out-to-out diesios of differet types of colus used i this study...7 Figure 5. Effective width ethod... Figure 6. Elastic bucklig curve geerated usig CUFSM... 6 Figure 7. Boudary coditios defiitio.... Figure 8. Web of the lipped C-sectio colu for Pu speciets (Pu et al. 999)... 5 Figure 9. The oral distributio curve... 9 Figure. Mai Specificatio test-to-predicted stregth as a fuctio of global slederess... 4 Figure. SM test-to-predicted stregth as a fuctio of global slederess... 4 Figure. Mai Specificatio test-to-predicted stregth as a fuctio of effective area-to-gross area... 4 Figure 3. SM test-to-predicted stregth as a fuctio of local-to-global bucklig Figure 4. Mai Specificatio test-to-predicted stregth as a fuctio of flage width-tothickess (B/t) Figure 5. SM test-to-predicted stregth as a fuctio of flage width-to-thickess (B/t) Figure 6. Mai Specificatio test-to-predicted stregth as a fuctio of lip width-to-thickess (/t) iv

10 Figure 7. SM test-to-predicted stregth as a fuctio of lip width-to-thickess (/t) Figure 8. Mai Specificatio test-to-predicted stregth as a fuctio of web height-to-thickess (H/t) Figure 9. SM test-to-predicted stregth as a fuctio of web height-to-thickess (H/t) Figure. Mai Specificatio test-to-predicted stregth as a fuctio of web height-to-flage width (H/B) Figure. SM test-to-predicted stregth as a fuctio of web height-to-flage width (H/B) Figure. Mai Specificatio test-to-predicted stregth as a fuctio of flage width-to-lip legth (B/)... 5 Figure 3. SM test-to-predicted stregth as a fuctio of flage width-to-lip legth (B/)... 5 Figure 4. Test-to-predicted stregth of plai agle colus with ad without P/ Figure 5. Mai Specificatio test-to-predicted stregth ratio as a fuctio of slederess v

11 ist of Tables Table. CFS colu test database... 9 Table. Resistace factors for colus with ad without holes (Mai Specificatio) Table 3. Resistace factors for colus with holes (Mai Specificatio) Table 4. Resistace factors for colus without holes (Mai Specificatio) Table 5. Resistace factors for colus without holes (SM) Table 6. Resistace factors for partially ad fully effective colus... 4 Table 7. Resistace factors by ultiate liit state Table 8. Mai Specificatio test-to-predicted stregth ratios for agle colus Table 9. Resistace factors for agle colus (Mai Specificatio) Table. Resistace factors for agle colus with (Mai Specificatio) Table. Resistace Factors for agle colus (SM) Table. Resistace factors for colus with ad without holes (Mai Specificatio) Table 3. Copariso of resistace factors for all colus for S (Mai Specificatio)... 6 Table 4. Modified Mai Specificatio liits Table 5. Modified SM prequalified liits vi

12 Chapter : Itroductio. oad ad Resistace Factor esig of Cold Fored Steel Copressio Mebers A liit state, as defied by Hsiao (99), is the coditio at which the structural usefuless of a load-carryig eleet or eber is ipaired to such a extet that it becoes usafe for the occupats of the structure. I sipler ters, the eber is uable to resist the applied load ad it fails. Cold-fored steel (CFS) copressio ebers ca fail due to yieldig or colu bucklig.elastic bucklig aalysis reveals at least three differet bucklig odes icludig local, distortioal ad Euler (flexural, torsioal or flexural-torsioal bucklig). Thus for the desig of CFS ebers, all the above liit states ust be cosidered.the stregth liit state of the load ad resistace factor desig (RF) ethod is expressed as R c Q, () i i where the oial resistace is R, is the resistace factor, γ i is the load factor ad Q i is the load effect. The oial resistace is the stregth of the eber for a give liit state. The resistace factor,, accouts for the ucertaities i the oial resistace, R. The load effect, Q i, isthe force (e.g., bedig oet, axial forces) actig o the eber. A liit state is violated whe the load effect, Q, is greater tha the oial resistace, R. The load effect, Q ad the resistace, R are rado paraeters whose distributios are ot typically kow ad oly their eas, Q ad R ad stadard deviatios σ Q ad σ R are kow. If the exact probabilitydistributios of Q ad R were kow, the probability of failure, i.e., the probability of (R - Q) < ca be deteried. Sice the probabilitydistributios of Q ad R are ot kow, the relative reliability of a desig is obtaied usig the reliability idex, β. The reliability idex is a relative easure of the safety of desig ad a higher value of β idicates a better desig. Figure

13 (a) presets the probabilitydistributios of Q ad R while Figure (b) presets the probability of failure. The area uder lr / Q represets the probability of failure. Figure. Probability distributios of the load effect, Q, ad the resistace, R.

14 The reliability idex, β, is the distace of the failure surface fro the ea i stadard deviatios ad this ca be observed i Figure (b). The reliability idex, β, is used as a easure of safety i structural reliability. The cocept of the resistace factor, ad the reliability idex, β is discussed i greater detail i Chapter 4.. Code History This sectio discusses the evolutio of the Aerica Iro ad Steel Istitute (AISI) North Aerica Specificatio for the esig of Cold-Fored SteelStructural Mebers(AISI-S 7) sice its iceptio. A review of the Mai Specificatio seffective width ethod for predictig colu capacity as well as the irect Stregth Method (SM) is also preseted. I 99, AISI ipleeted the RF approach for CFS ebers for the first tie(aisi 99). Hsiao (99) developed the RF criteria for CFS siilar to the way the Aerica Istitute of Steel Costructio (AISC) developed RF criteria for hot-rolled steel(aisc 986). Usig AISC s RF criteria for hot-rolled steelas a basis, AISI adopted a value of.85 for the resistace factor, reliability idex,.for a give represetative dead-to-live load ratio of /5, a value of.5 for the was adopted. The 99 Specificatio deteried the oial axial stregth (P ) usig P A F, () e wherea e is the effective cross-sectioal area at the oial colu bucklig stress, F, calculated with the followig expressio: For F F e y, F F 4F, y e F e F, F F. (3) y e 3

15 The critical elastic global buckligstress, is the iiu of the critical elastic flexural, torsioal, or flexural-torsioal bucklig stress ad F y is thesteel yield stress. The ext editio of the AISI Specificatio waspublished i 996(AISI 996). This editioodified the equatios used to calculate the oial global bucklig capacity to atch the 993 editio of the AISC RF Specificatio (AISC 993): For.5, c c F c.658 Fy,.877F,.5, F (4) y c.5 where slederess,, is give by F F. Peköz ad Süer(99) studied 99 c y colu ad bea-colu tests ad showed that the revised colu desig equatios were ore accurate tha Eq. (3). These equatios also accout for the iitial crookedess ad thus provide a better fit to test results. Figure copares the curves produced usig Eq. (3) ad Eq. (4). e.9 AISC 986 & AISI 99 Specificatios AISC 993, AISI 996, & 7 Specificatios F / F y c = (F y /F e ).5 Figure.History of AISI ad AISC colu curves 4

16 It is to be oted that although the equatios used to calculate the oial global bucklig capacity were odified, the resistace factor ad the safety idex were left utouched ad they cotiue to be used eve ow. I, all AISI stadards were uified uder the baer of the North Aerica Specificatio (AISI ). While the RF ethod was used i the Uited States ad Mexico, Caada adopted the iit State esig (S) ethod. It is to be oted that while the desig philosophy used for RF ad S is the sae, the two ethods differ i the load factors, load cobiatios, assued dead-to-live load ratios ad the reliability idices. Asuppleet to the Specificatio, published i 4(AISI 4), itroduced a ew stregth predictio ethod called the irect Stregth Method (SM)(Schafer ). SM predicts colu stregth usig the elastic bucklig behavior of the whole cross-sectio. Ulike the Mai Specificatio, SM does ot quatify the cross-sectio istabilities eleet-byeleet. Istead, the log colu stregth (P e ) is reduced based o the elastic local bucklig load of the cross-sectio (P ). More iforatio about the two desig ethodsis preseted i Chapter. The ost recet editio of the North Aerica Specificatio was published i 7 (AISI- S 7). The distortioal bucklig ode of failure, which was itroduced i the 4 Suppleet (AISI 4) cotaiig SM, was added to the Mai Specificatio i the 7 editio. This bucklig ode occurs i ope cross-sectios adis characterized by the istability of a copressed flage (Figure 3). istortioal bucklig occurs at half-wavelegths betwee the local ad the flexural or the flexural-torsioal bucklig odes. 5

17 A et /A g Figure 3. istortioal bucklig ode of a ipped C-sectio colu with holes..3 Objective ad Scope of Research The 99 Specificatio(AISI 99), which itroduced the RF ethod for CFS for the first tie, established the resistace factor o the basis of a total of 64 colu tests. However, uerous colu tests have bee coducted sice 99 ad with the availability of ew data, a fresh look at the resistace factor is warrated. This research attepts to gather all available data ad ivestigate the suitability of usig =.85 as a appropriate value for the resistace factor ad cosider a possible icrease to =.9 so as to ake CFS ore copetitive with hot-rolled steel. While this fors the priary otivatio, this study also evaluates the viability of providig resistace factors o the basis of liit states or o the basis of the cross-sectio slederess. It also addresses the calibratio of the Caadia resistace factor for cold-fored steel copressio ebers, which uses a reliability idex of β = 3.. I this study, C-sectios, Z- sectios, hat sectios as well as agle sectios, both lipped ad ulipped, iclusive of sectios with holes are cosidered. This study draws its coclusios based o a total of 675 colu tests icludig plai ad lipped C-sectios, plai ad lipped Z-sectios, hat ad agle sectios. Figure 6

18 4 pictorially presets the otatios used for the out-to-out diesios of each type of colu used i this study. Figure 4. Out-to-out diesios of differet types of colus used i this study..4 Overview of Thesis This thesis begis with the presetatio of the 7 AISI Specificatio (AISI-S 7)for calculatig the capacity of CFS copressio ebers i Chapter, icludig both the Mai Specificatio as well as the irect Stregth Method. The capacity predictio of CFS agle 7

19 colus usig RF is also discussed at depth. Chapter 3explores the developet of acfs colu test database that will be used later i the calculatio of the resistace factor,. It discusses the source of each data set, the boudary coditios used i each experietal progra ad how each progra is differet fro the other. Chapter 4itroduces the cocept of a resistace factor ad presets a derivatio of the resistace factor for CFS copressio ebers.resistace factors are calculated for all the test data o the basis of liit states ad also o the basis of the cross-sectio slederess (partially ad fully effective sectios) for both the Mai Specificatio ad the irect Stregth Method. The resistace factors are calculated for both the RF ad the S. This chapter also presets the test-to-predicted stregth ratios ad resistace factor results obtaied for sigle aglecolus. A odified ethod of calculatig the resistace factor wherei the coefficiet of variatio is chaged to iclude a additioal ter i Taylor Series expasio is also preseted i this chapter. Chapter 5 suarizes the results obtaied i the resistace factor studies ad also draws coclusios. Areas that require further study as well as recoedatios for code chages are also addressed i this chapter. Appedix suarizes the etire database with details about each copressio eber such as cross-sectio diesios ad yield stresses alog with the test-to-predicted stregth ratios for both the Mai Specificatio ad the irect Stregth Method. Appedix presets a custo MATAB (Mathworks 9) code that this research uses to calculate the capacity of the colus accordig to the Mai Specificatio ad SM. Appedix 3 presets the atheatical derivatio of the resistace factor i a geeral for without the use of uerical values. 8

20 Chapter : AISI Specificatio This chapter presets the procedure forcalculatig the capacity of CFS copressio ebers usig both the Mai Specificatio as well as the irect Stregth Method i accordace with the 7 AISI Specificatio (AISI-S 7).. AISI Mai Specificatio The Mai Specificatio cosiders two liit states, local-global bucklig iteractio (icludig flexural, torsioal or flexural-torsioal bucklig) ad distortioal bucklig. The oial colu capacity, P, is cosidered to be the iiu of the two liit states... ocal-global Bucklig Iteractio iit State The Mai Specificatio (AISI-S 7) calculates the oial axial capacity (P ) of a colu usigeq. (), wherei, the oial colu bucklig stress, F, is deteried usig Eq. (4). I order to use Eq. (4), slederess,, as expressed by the equatio, F F, be deteried. The critical elastic global buckligstress, F e, is the iiu of the critical elastic flexural, torsioal, or flexural-torsioal bucklig stress ad F y is thesteel yield stress. c y e.5 ust The elastic flexural bucklig stress, F e, for doubly-syetric sectios, closed cross-sectios, ad other cross-sectios that are ot subjected to either torsioal or flexural-torsioal bucklig, ca be calculated as follows: F e E, (5) K r 9

21 where E is the odulus of elasticity of steel, K is the effective legth factor, is the lateral ubraced legth of the eber ad r is the radius of gyratio of the full ureduced cross-sectio about the axis of bucklig. If the sectio is also subject to torsioal bucklig, which is the failure of a colu due to a twist without ay bedig, the the critical elastic global buckligstress, F e, is take as the iiu of F e, as calculated by Eq. (5) ad the torsioal bucklig stress, σ t, calculated as follows: EC w t, GJ (6) Aro Ktt where A is the full cross-sectioal area, r o is the polar radius of gyratio of the cross-sectio about the shear ceter, G is the shear odulus, J is the Sait Veat torsio costat of the crosssectio, E is the odulus of elasticity, C w is the torsioal warpig costat of the cross-sectio, adk t t is the effective legth for twistig. This ode of failure is possible for poit syetric shapes such as doubly syetric I-shapes whose shear ceter ad cetroids coicide. However, if the sectio is subject to flexural-torsioal bucklig, where the colu fails due to siultaeous bedig ad twistig, the the critical elastic global buckligstress, F e, is take as the iiu of F e, as calculated by Eq. (5) ad the flexural-torsioal bucklig stress, which is calculated as follows: Fe ex t ex t 4 o ex t (7)

22 where σ ex is the flexural Euler bucklig stress about the x-axis, σ t is the torsioal bucklig stress, ad x r o, where x o is the distace betwee the cetroid ad the shear ceter ad r o is the polar radius of gyratio of the cross-sectio. Flexural-torsioal bucklig is a possible ode of failure of sigly-syetric sectios i which the shear ceter ad cetroid do ot coicide. Now that the critical elastic global buckligstress, F e, has bee foud, the slederess, λ c, ad cosequetly the oial colu bucklig stress, F, ca be foud. However, i order to calculate the oial axial capacityof a colu,p, usigeq. (), the effective area, A e of the colu at F, ust be deteried. I order to calculatea e, the Mai Specificatio uses the effective width ethod.i this ethod, the o-uifor distributio of stress over the etire width, w, of a sleder buckled eleet is assued to be uiforly distributed over a fictitious effective width, b, of the eleet as show i Figure 5. It was Vo Kara(93), who first suggested that the stress distributio at the cetral sectio of a stiffeed plate be replaced by two widths of (b/) o each side of the plate, each subjected to a uifor stress, f ax, as show i Figure 5. Figure 5. Effective width ethod

23 The effective width of a uiforly copressed stiffeed eleet accordig to the 7 AISI Specificatio (AISI-S 7) is expressed as b w b w whe.673, whe.673, (8) wherew is the flat width of the eleet ad ρ is the local reductio factor give by Here,λ is the slederess factor give by.. (9) f, () ad f is the stress i the copressio eleet. The colu bucklig stress,f cr is give by F cr F cr E k t w, () where the plate bucklig coefficiet, k = 4 for a log siply supported plate,e is the odulus of elasticity of steel, t is the thickess of uiforly copressed stiffeed eleet ad µ is Poisso s ratio.for the case of a uiforly copressed ifiitely log ustiffeed eleet, the plate bucklig coefficiet is take as k =.43. Oce, the effective width, b, is deteried, the effective area, A e, ca be foud by suig theeffective width, b over all the eleets... istortioal Bucklig iit State istortioal bucklig is characterized by istability of a copressed flage ad ivolves rotatio at the juctio of the web ad the flage i ope cross-sectios, e.g., C-sectios ad Z- sectios. istortioal bucklig occurs at half-wavelegths betwee the local ad the flexural or the flexural-torsioal bucklig odes. The distortioal bucklig stregth is calculated as follows:

24 For λ d.56; P P, y For λ d.56; P d P.5 P crd y.6 P P crd y.6 P, y () where d Py Pcrd, Py Ag Fy ad Pcrd Ag Fd, where A g is the gross area of the crosssectio ad F d is the elastic distortioal bucklig stress. The Mai Specificatio (AISI-S 7) allows the use of a ratioal bucklig aalysis to calculate the elastic distortioal bucklig stress with freely available fiite strip progras, for exaple, CUFSM (Schafer ad Ádày 6), which is used i this research. The Specificatio (AISI-S 7) also provides siplified equatios to calculate the elastic distortioal bucklig stress...3capacity of Agle Colus AISI provides additioal desig cosideratios for predictig the stregth of agle colus which wereadded to the 986 Specificatio (AISI 986) o the basis of recoedatios ade by Peköz (987). urig his research, Peköz foud the possibility of a reductio i the colu stregth due to the iitial out-of-straightess (sweep) of agle sectios ad he recoeded the use of a iitial out-of-straightess of /. However, Popovic (999) foud that the iclusio of the additioal oet due to the iitial out-of-straightess ade the predictios too coservative ad recoeded that the additioal oet be applied oly for agle sectios whose effective area (A e ) at stress F y is less tha A g, or i other words, a sleder agle sectio. The agle colu capacity, P, is calculated icludig the copressive axial force ad oet with a iteractio equatio: 3

25 cp P c o cp.. M b y (3) Here, P o is the oial axial capacity ad about the y-axis (see Figure 4). M y is the flexural stregth of the gross cross-sectio The equal leg agle is a sigly-syetric sectio ad hece oial flexural stregth, M, correspodig to global bucklig is calculated accordig to equatio C3...- of the Mai Specificatio (AISI-S 7): M S F (4) c c where, S c is the elastic sectio odulus of the effective sectio calculated relative to the extree copressio fiber at F c. The stress, F c, is the critical global bucklig stress ad is deteried as follows: For.78F y F.56F ; e e For F.56F ; y y Fy Fc Fy F 9 36 e F F c e (5) where F e is the elastic critical global bucklig stress calculated accordig to equatio C3...- of the Mai Specificatio (AISI-S 7): F e Cs A C S ex j Cs j ro TF f t ex (6) E (7) ex K r x x x EC w t GJ (8) Aro Ktt C TF.6.4M M. (9) 4

26 The effective legth factors,k x adk y are for bedig aboutthe cetroidalx-axis ad y-axis respectively,k t is the effective legth factor for twistig, r x ad r y are the radii of gyratio of the cross-sectio about the cetroidal pricipal axes,r o is the polar radius of gyratio of the crosssectio about the shear ceter, C s = + for oet causig copressio o the shear ceter side of cetroid ad C s = - for oet causig tesio o the shear ceter side of cetroid, M ad M are the saller ad larger bedig oets at the eds of the ubraced legth i the plae of bedigrespectively, ad x, y ad t are the ubraced legths for bedig about x ady axes ad twistig respectively.. irect Stregth Method The AISIirect Stregth Method (SM)uses cross-sectio elastic bucklig behavior to predict colu stregth. Three elastic bucklig odes are cosidered for CFS copressio ebers local, distortioal ad global, wherei the global ode icludes flexural, torsioal, or flexural-torsioal bucklig. The local bucklig ode is deteried for a eber as a whole ad ot o a eleet-by-eleet basis as i the effective width ethod described i Eqs. (8) to (). This research akes use of the fiite strip aalysis to perfor the cross-sectio stability aalysis, which is a specialized variat of the fiite eleet ethod(schafer ad Ádày 6). Figure 6 presets a elastic bucklig curve for a lipped C-sectio i pure copressio obtaied usig CUFSM, a freely available progra that eploys the fiite strip ethod to perfor elastic bucklig aalysis of a CFS eber. 5

27 P cr (kips) load factor 8 6 ocal bucklig istortioal bucklig P crd crd Global bucklig 4 P cr Pcr P cre Pcre 3 half-wavelegth (i.) Figure 6. Elastic bucklig curve geerated usig CUFSM It ca be observed fro Figure 6 that for thecosidered lipped C-sectio i pure copressio, the elastic bucklig load for local bucklig, P cr, is lower tha the elastic bucklig load for distortioal bucklig, P crd. ocal bucklig occurs at short half-wavelegths ad bucklig odes occurrig at loger half-wavelegths are either distortioal or global i ature. Accordig to Appedix of the 7 AISI Specificatio (AISI-S 7), the oial axial stregth (P ) of a CFS colu usig SM is: P, i P, Pd Pe () where P, P d ad P e are the oial axial stregths for local, distortioal ad flexural, torsioal or flexural-torsioal bucklig failures respectively. The oial global capacity, P e is calculated with the sae colu curve described i Eq.(4), where P e = F A g. The oial axial stregth for local bucklig, P, is calculated as follows: For.776, P Pe, () 6

28 .4.4 P.776,.5 cr P For P cr Pe, Pe P () e where P e P cr where, P cr is the elastic local bucklig load deteried usig elastic bucklig aalysis. The distortioal bucklig capacity, P d, is calculated usig Eq. () described earlier i Sectio... Chapter 3 presets the CFS colu test database which cosists of 675 colus testsfro differet experietal progras. The stregth of each colu preset i the database is predicted usig the AISI Mai Specificatio ad SM equatiospreseted i this chapter. 7

29 Chapter 3: CFS Colu Test atabase 3. Overview of atabase This chapter presets the CFS colu test database which will be used to calculate the resistace factor, c. Whe the resistace factor, c, was first calculated for CFS ebers(aisi 99), test results fro 64 colu tests were used.give that sice the, uerous colu tests have bee coducted, this study aied at collectig as uch data as possible. Thus the otivatio behid this study was to expad the existig colu data set with colutests of every kid short, log ad iterediate legth colus that are either sigly syetric (Csectios, agle sectios ad hat sectios)orati-syetric (Z-sectios). A effort has also bee ade to collect data fro experietal progras cosiderig colus with holes. These holes are of differet shapes circular(ortiz-colberg 98; Sivakuara 987), square(pu et al. 999; Sivakuara 987), oval (Sivakuara 987), rectagular (Miller ad Peköz 994) ad slotted (Moe ad Schafer 8). The CFS colu test database cotais a total of 675 colus testsfro differet experietal progras. Plai ad lipped C-sectios, Z-sectios, plai ad lipped agle sectios ad hat sectios, iclusive of ebers with holes, have bee cosidered i this study. oubly syetric colus such as built-up I-sectios (Weg ad Pekoz 99), (ewolf et al. 974) ad box sectios (ewolf et al. 974)have ot bee cosidered i this study. Eccetrically loaded colus(oh ad Peköz 985) are also ot cosidered i this study. Table provides a suary of the experietal progras icluded i the database. It also presets the axiu ad iiu ratios of cross-sectioal diesios. 8

30 Table.CFS colu test database Referece Sectio type Holes B/t H/t /t /B h hole /H λ c i ax i ax i ax i ax i ax i ax Thoasso 978 ipped C oughla 979 ipped C at 98 ipped C esod et al. 98 ipped C esod et al. 98 Hat Ortiz-Colberg 98 ipped C Ortiz-Colberg 98 ipped C Mulliga 983 ipped C Wilhoite et al. 984 Plai Agles Sivakuara 987 ipped C Sivakuara 987 ipped C Polyzois,. et al. 993 Plai Z Polyzois,. et al. 993 ipped Z Miller ad Peköz 994 ipped C Miller ad Peköz 994 ipped C Moldova 994 Plai C Moldova 994 ipped C Abdel-Raha ad Sivakuara 998 ipped C Youg ad Rasusse 998a ipped C Youg ad Rasusse 998b Plai C Popovic et al. 999 Plai Agles Pu et al. 999 ipped C Pu et al. 999 ipped C Shauga ad haalakshi Plai Agles Youg ad Hacock 3 ipped C Youg 4 Plai Agles Chodraui et al. 6 Plai Agles Youg ad Che 8 ipped Agles Moe ad Schafer 8 ipped C Moe ad Schafer 8 ipped C I all, there are a total uber of 455 lipped C-sectios, 7 lipped Z-sectio colus, 49 plai C-sectio colus, 3 plai Z-sectio colus, 5 plai agle colus, 5 lipped agle colus ad hat colus. Of the 455 lipped C-sectio colus, 6 cotai holes. The Z- sectio, hat ad aglecolus do ot cotaiholes. etails about each experietal progra icludig the boudary coditios, rage of diesios, ad experietal set up are provided i Sectio 3.. 9

31 3. Experietal Progra etails I this study, the predicted stregths were calculated for all the colu test results suarized itable with the Mai Specificatio ad SM. This sectio provides details about each experietal progra. Full details of each study, icludig the diesios ad ed restraits are provided i Appedix. Figure 7 presets a pictorial represetatio of the boudary coditios. Figure 7. Boudary coditiosdefiitio. The first set of colu tests i the database were perfored by Thoasso(978)ad cosisted of a total of 3 lipped C-sectio colus. The web heights, lip ad flage legths of the colus were suarized by Peköz (987) ad Schafer (). The legth of all the

32 ebers was kept costat at 5.9 iches.while there are iute differeces i the web height, flage legth ad lip legth of the ebers, there is a cosiderable differece i specie thickess (.5 i. to.55 i.) ad this affects thelocal bucklig behavior.colu tests were coducted assuig that the weak axis was pied (warpig free) ad the other axes were fixed (warpig fixed), i.e., effective legth factors were K x =.5, K y =., ad K t =.5. The ext set of data was produced by oughla (979) as suarized by Schafer () cosists of 33 lipped C-sectio colus differig i web height, lip ad flage legths. The thickess of all the exaied sectios raged betwee.3 i. ad.64 i. All the exaied ebers were of iterediate legths betwee 5 i. ad 75 i. Stregth predictios were ade by assuig that the weak axis was pied (warpig free) ad the other axes were fixed (warpig fixed), i.e., effective legth factors were K x =.5, K y =., ad K t =.5. Research byat (98)cosisted of 43lipped C-sectio colus without holes. The colus were ade by roll-forig ad by press-brakig ad were called rolled-fored chaels (RFC) ad press-baked chaels (PBC) respectively. The thickess of all the exaied sectios was betwee.73 i. ad.9 i. The legths used i this study vary betwee i. ad i. I the actual tests, the load was applied usig the static ethod, wherei the load was slowly icreased ad stabilized at every load icreet.at eployed ed fixtures that acted as kife edges allowig rotatio oly i the y directio. Thus for this experietal progra, stregth predictios were ade by assuig that the y-axis was pied, while the other axes are fixed, i.e. K x =.5, K y =., ad K t =.5. ata fro esod et al.(98)cosisted of 7 lipped C-sectio colus ad hat sectio colus without holes. Aog the hat sectio colus, 5 had lips that were iclied at a agle of 45 degrees. The sectios were all stub colus with legths less tha i. ad were

33 fixed eded (warpig fixed) with effective legth factorsk x =.5, K y =.5, ad K t =.5.This experietal progra studied the effect of edge stiffeers o the local bucklig behavior of the flage. I the sae year, Ortiz-Colberg (98) preseted his Master s thesis ocolu tests of 3 lipped C-sectio colus with holes ad lipped C-sectio colus without holes. The thickess of all the exaied sectios was betwee.49 i. ad.76 i. The holes were all circular i shape ad the data cosisted of stub colus as well as iterediate ad log colus. The stub colus were fixed-eded (warpig fixed) with effective legth factorsk x =.5, K y =.5, ad K t =.5 while the iterediate ad log colus were weak axis pied, i.e. K x =.5, K y =., ad K t =.5. The ext experietal progra coducted by Mulliga (983) preseted37 lipped C-sectio colus without holes. Therewere4 stub colus ad 3 log colus. The thickess of all the exaied sectios was approxiately.45 i. The stub colus were fixed-eded (warpig fixed) with effective legth factorsk x =.5, K y =.5, ad K t =.5 while the log colus were weak axis pied, i.e. K x =.5, K y =., ad K t =.5. Soo after Mulliga, Wilhoite et al. (984), aalyzed 7 agle sectio colus. The colus were all equal legged ad ade of high stregth press-braked steel. The thickess of all the exaied sectios was approxiately.7 i. The log colus were weak axis pied with effective legth factors K x =.5, K y =., ad K t =.5. Sivakuara (987)preseteda total of 48 lipped C-sectio colus, of which 4 sectios had holes ad 6 sectios did ot cotai holes. The holes were circular, square or oval i shape.the holes sizes raged fro % to 6% of the web flat width. The colus legths

34 raged betwee 8. i. ad.3 i. The tests were coducted uder flat pied ed coditios i.e. K x =.5, K y =., ad K t =.5. The first ad the oly set of tests o Z-sectios cosidered i this study was perfored by Polyzois et al. (993). This progra cosisted of 85 Z-sectio colus, of which 3 were plai Z-sectio colus ad 7 were lipped Z-sectio colus. The colus were 8, 4 or 48 i. log ad had a average thickess of approxiately.58 i. The colus i this progra were tested with fixed-fixed ed coditios i.e. K x =.5, K y =.5, ad K t =.5. Miller et al.(994)perfored a experietal progra that cosisted of a total of 8 lipped C-sectio colus of which 37 had holes. Aog the 8 lipped C-sectios, 44 were stub colus. Holes were preset i of the 44 stub colus. The stub colus raged betwee i. ad i. log while the log colus were betwee 47 i.ad i. log. The holes were rectagular i shape ad varied i uber, with soe sectios cotaiig as ay as 4 holes. The legth, depth ad the spacig of the holes are preseted i Appedix. The stub colus were assued to be fixed i.e. K x =.5, K y =.5, ad K t =.5 ad for the log colus, rotatio at the eds was free about oe axis ad fixed about the other for each test i.e. K x =.5, K y =., ad K t =.5 ad also K x =., K y =.5, ad K t =.5. Moldova(994)tested 64 C-sectio colus,of which 35 were plai C-sectio colus ad 9 were lipped C-sectio colus. Of the 64 colus, 7 were stub colus with legths ragig betwee 9 i. ad 5 i. The log colu legths raged betwee 4 i. ad 78 i. The thickess of the lipped C-sectio colus varied betwee.7 i. ad. i. while for plai C-sectio colus it varied betwee.7 i. ad.6 i. The stub colus were assued to be fixed i.e. K x =.5, K y =.5, ad K t =.5, while the log colus were assued to have their weak axis pied i.e. K x =.5, K y =., ad K t =.5. 3

35 Abdel-Raha et al. (998)coducted tests o8 lipped C-sectio coluswith holes. All 8 colus were stub colus with legths ragig betwee 9 i. ad 8 i. The holes were circular, square, rectagular or oval i shape. The thickess varied betwee.5 i. ad.7 i. The colus are all fixed-eded with effective legth factors, K x =.5, K y =.5, ad K t =.5. I 998, Youg ad Rasusse published two papers, oe cotaiig lipped C-sectio colus(998a) ad the other cotaiig 4 plai C-sectio colus(998b), both without ay holes i the species. The colus raged fro stub colus to log colus i.e. betwee. i.ad 8. i. log. The thickess of all the colus was approxiately.58 i. ad all colus were fixed-eded coluswith effective legth factors, K x =.5, K y =.5, ad K t =.5. Popovic (999)coducted tests o plai agle sectio colus. The legth of the agles varied betwee i. ad i. while the thickess raged fro.9 i. to.8 i. The agles were equal-legged ad their legs (B ad B) were.95 i.log. All agles i this progra were fixed-eded with effective legth factors, K x =.5, K y =.5, ad K t =.5. The database also cotais 36 lipped C-sectio colu tests coducted by Pu et al.(999).of the 36 lipped C-sectio colus i this experietal progra, 3 cotaied holes i the. Although 63 lipped C-sectio colus are preseted, oly 36 are cosidered i the preset study. Colus with edge holes have bee eglected while colus with a hole i the ceter of the web have bee cosidered. Colus with holes that are a quarter of the web width away fro the edge o the sae side were cosidered as just oe hole that exteds fro oe side to the other as show i Figure 8 whe calculatig the capacity (see Appedix ). The colus were all stub colus approxiately 4 iches i legth ad the holes were all square i shape. All the 36 lipped C-sectios were assued to be fixed-eded with effective legth factors, K x =.5, K y =.5, ad K t =.5. 4

36 Shauga () tested 3 plai agle sectio stub colus with equal legs with legths ragig fro 5 i to i. All the colus were assued to be fixed eded with effective legth factors, K x =.5, K y =.5, ad K t =.5. Web as tested Web assued i calculatios Figure 8. Web of the lipped C-sectio colu forpu speciets(pu et al. 999) Youg ad Hacock (3)coducted a experietal prograo 4 lipped C-sectio colus. All the colus were of approxiately the sae legth of 58 i. The colus had a oial thickess that raged betwee.6 i. ad.9 i. The colus cotaied edge stiffeers that were iclied at differet agles betwee 3 degrees ad 5 degrees. Both iward ad outward edge stiffeers were cosidered ad the colus were copressed betwee fixed eds i.e.k x =.5, K y =.5, ad K t =.5. Youg(4) also perfored experiets o 4 plai agle colus. The legth of the colus raged betwee 9 i. ad 36 i. The agles were all equal-legged ad their thickess raged fro.45 i. to.73 i. The agle colus had fixed-fixed ed coditios i.e.k x =.5, K y =.5, ad K t =.5. 5

37 Chodraui (6)tested 4 plai agle colus. The colu legths varied betwee 4 i. ad 66 i. Agai, all the 4 agles were equal-legged ad had a thickess of.9 iches. Stregth predictios of all colus i this experietal prograwere ade by assuig that the weak axis was pied, i.e. K x =.5, K y =., ad K t =.5. Youg ad Che(8)perfored tests o 5 lipped agle colus of uequal flage width. The legths of the agle colus raged betwee 9 i.ad 7 i. The agle colus had a oial thickess that raged betwee.38 i. ad.73 i. All the agle colus i this progra were fixed-eded with effective legth factors, K x =.5, K y =.5, ad K t =.5. Moe ad Schafer(8)tested a total of 4 lipped C-sectio colus, of which had slotted holes i the. The legth of the colus raged betwee 4 i. ad 48 i. The thickess of the colus was about.4 iches ad they were all fixed-eded with effective legth factors, K x =.5, K y =.5, ad K t =.5. The stregth of each colu discussed i this chapter is predicted usig the AISI Mai Specificatio ad SM. I Chapter 4, the cocept of a resistace factor is itroduced ad evetually it will be observed that resistace factor value depeds o the predicted capacities. Further, Chapter 4 presets the calculated resistace factors for all the colus o the basis of cross-sectio type, liit state ad partially ad fully effective cross-sectios. 6

38 Chapter 4: Resistace FactorEquatios ad Calculatios This chapter presets a detailed derivatio of the resistace factor for both the RF ad S ethods i Sectio 4.. The calculated resistace factors for colus i the CFS colu test database accordig to both the Mai Specificatio ad the irect Stregth Method are preseted i Sectio Resistace Factor erivatio The resistace factor accouts for ucertaities i diesios, aterial properties ad stregth predictio accuracy. Both the load effect,q, ad the resistace,r, are rado paraeters ad their probability distributios are geerally ukow. Oly the eas, Q ad R ad the stadard deviatios ad are kow. Usig these kow values, the relative easure of safety of a desig ca be obtaied usig the reliability idex,. The followig steps illustrate how the resistace factor, defied i Chapter F of the Specificatio (AISI-S 7), is derived startig with the defiitio of the reliability idex. Give that the exact probability distributios of the load effect,q, ad the resistace,r, areukow, it is assued that they follow a logoral probability distributio ad are idepedet. Thus the followig trasforatios are defied: X l R Y l Q. (3) Eq. (3) is of the for Y = g(x, X,..., X), where X, X ad so o are rado variables. The fuctio, g(x, X,..., X) is expaded usig a Taylor Series expasio about the ea values: 7

39 Y g g g X, X,... X Xi Xi X i Xi X j Xj. (4) x x x i i i j i j The Taylor series expasio i Eq. (4) is trucated at liear ters to obtai a first order approxiatio for the ea ad the variace. Thus the eas based o a first order approxiatio are obtaied as X Y l R l Q. (5) Agai, the variace of Xbased o first order approxiatio is obtaied by differetiatig the ea, X, with respect to the ea of the resistace, R : X X d dr R l R R R V R, (6) where, V R is the coefficiet of variatio of resistace. Siilarly, it ca be show thatvariace of Y is also equal to the square of the coefficiet of variatio of the load effect, i.e =. Failure,Z, i ters of the ea of the resistace ad the load effect isdefied as Z lr lq lr Q. (7) Sice, the load effect, Q, ad the resistace, R, are logoral distributios, l Q ad l R will becoe orally distributed. Thus failure, Z, is also orally distributed. The probability of failure, p f, is expressed as p f P Z Z, Z (8) 8

40 where, the stadard deviatio of z, Z V R V Q. The ter, Φ(z), represets the area uder the oral curve util the value of z. If the value of z is egative, the usig the syetric property of the oral curve, Φ(-z) ca be deoted by -Φ(z), as show i Figure 9. Substitutig Eq. (7) i Eq. (8), Figure 9. The oral distributio curve lr Q p f (9) VR VQ Thus, the reliability idex,, as established by Ravidra ad Galabos (978), accordig to first order approxiatio, is expressed as R The resistace of the copoets of a structure deteries its load carryig capacity. The resistace is iflueced by aterial properties such as the aterial stregth, odulus of elasticity as well as diesios of the copoets. This depedece of resistace o the aterial properties ad the diesios of copoets iduce a eleet of ucertaity. Three differet 9 l R Q (3) V V Q

41 paraeters, M, F ad P are used to odel the resistace icludig this ucertaity. Specifically, M accouts for the variatio i the stregth of the aterial, Faccouts for the fabricatio errors that result i variatios i diesios ad P is used to accout for the ucertaity arisig fro the chose approxiate ethod of stregth predictio. I this research, the chose ethods of stregth predictio are the Mai Specificatio ad SM, which are used to predict the stregths of all the colus i the test database discussed i Chapter 3 ad the test-to-predicted stregth ratio provides a easure of the Professioal factor, P. The product of the three paraeters (M, F ad P) ad the oial resistace (R ), which is the stregth of a eleet coputed for oial sectio properties ad the specified aterial properties for a particular liit state, represets the actual resistace R. Thus the ea resistace, R ca be expressed as R R M F P. (3) The ea value of the load effect ca be expressed as a fuctio of the ea values of dead ad live load itesities, ad as follows: Q C, (3) where, C is the deteriistic ifluece coefficiet.thus the coefficiet of variatio of resistace, V R ca be expressed i ters of the coefficiets of variatio of aterial properties, geoetric properties ad the test to predicted stregth ratio, V M, V F ad V P respectively ad V R is expressed as V R M F P V V V. (33) The coefficiet of variatio of the load effect, V Q ca be expressed i ters of the coefficiets of variatio of the dead ad the live loads (V ad V ), ad the ea values of the dead ad live load itesities, ad as 3

42 VQ V V. (34) Hsiao(99)developedthe RF criteria for CFS ebers ad suggested use of the values M., V., F., V F. 5,. 5, V.,, V. 5.It ca be see that the coefficiet of variatio of the live load(v ) is higher tha the coefficiet of variatio of the dead load (V ) idicatig that the ucertaity with respect to the live load is higher. Usig the values recoeded by Hsiao, the expressios for the ea resistace, R, i Eq. (3), the coefficiet of variatio of resistace, V R, i Eq. (33)ad the coefficiet of variatio of the load effect, V Q, i Eq. (34), ca be expressed as R R,.P V R.5VP,.5..5 V Q. (35).5. The resistace factor, c, the oial resistace, R, the oial values of the dead ad live loads, ad, ad the load factors, α adα are related by the followig equatio: c R C, (36) 3

43 3 where the deteriistic ifluece coefficiet, C, trasfors the dead ad live load itesities ito load effects. Usig the expressio for the oial resistace, R, i Eq. (35), Eq. (36) ca be expressed as c C P R. c P R C. (37) Usig the values recoeded by Hsiao, the ea value of the load effect, i Eq. (3), ca be expressed as.5 C Q..5 Q C (38) Equatig Eqs. (37)ad (38), the followig expressio is obtaied: c P R Q..5 (39) Fro Eq. (39), the ratio of the eas of the resistace (R ) to the load effect (Q ) ca be expressed as

44 33.5. c P Q R (4) The ratio of the ea resistace to the ea load effect (R /Q ) fro Eq. (4), the coefficiet of variatio of resistace, V R, ad the coefficiet of variatio of the load effect, V Q, fro Eq. (35)are substitutedi Eq. (3) ad the followig expressio is obtaied: l P c V P (4) Thus, rearragig the ters i Eq. (4), the expressio for the resistace factor, of a coldfored steel colu is as follows: exp.5. P c V P (4) Equatio(4) is aapproxiate equatio for the resistace factor,, ad is a fuctio of the oial dead-to-live load ratio ( / ), the load factors, α ad α, the reliability idex,, the