Notes on the AISC Provisions for Slender Compression Elements in Compression Members

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1 Nos on h AISC Provisions for Slndr Comprssion lmns in Comprssion Mmrs LOUIS. GSCHWINDNR and MATTHW TROMNR ABSTRACT Comprssion mmr srngh is onrolld h limi sas of flxural ukling, orsional ukling, and flxural-orsional ukling, as applial. Ths omprssion mmrs ma ukl gloall or loall, dpnding on h ovrall olumn slndrnss and h loal pla lmn slndrnss for h plas ha mak up h shap. If an of h pla lmns will ukl a a srss lowr han ha whih would aus h olumn o ukl gloall, h loal ukling of h pla will onrol h ovrall olumn srngh. Whn his ours, h olumn is said o omposd of slndr lmns. This papr will rifl disuss pas spifiaion provisions for slndr lmn omprssion mmrs and inrodu h nw provisions in h 16 AISC Spifiaion. I will prsn a simplifiaion ha rdus h numr of onsans ha mus usd and will prsn h spifiaion rquirmns in an alrna forma. Baus h 16 rquirmns rsul in diffrn srnghs han h 1 rquirmns, figurs ar providd o illusra h ovrall impa of hs hangs on olumn srngh. K Words: omprssion mmrs, pla ukling, slndr lmns, AISC Spifiaion. INTRODUCTION Comprssion mmr srngh is onrolld h limi sas of flxural ukling, orsional ukling and flxural-orsional ukling, as applial (AISC, 1). Ths omprssion mmrs ma ukl gloall or loall, dpnding on h ovrall olumn slndrnss and h loal pla lmn slndrnss for h plas ha mak up h shap. If an of h pla lmns will ukl a a srss lowr han ha whih would aus h olumn o ukl gloall, h loal ukling of h pla will onrol h ovrall olumn srngh. Whn his ours, h olumn is said o omposd of slndr lmns. This papr rifl disusss pas spifiaion provisions for slndr lmn omprssion mmrs and inrodus h nw provisions in h 16 AISC Spifiaion. I will prsn a simplifiaion ha rdus h numr of onsans ha mus usd and will prsn h spifiaion rquirmns in an alrna forma. Baus h 16 rquirmns rsul in diffrn srnghs han h 1 rquirmns, figurs ar providd o illusra h ovrall impa of hs hangs on olumn srngh. Louis. Gshwindnr, Ph.D, P.., Snior nginr, Providn nginring Corporaion, and Profssor mrius, Arhiural nginring, Th Pnnslvania Sa Univrsi, Sa Collg, PA (orrsponding). mail: lfg@psu.du Mahw Tromnr, nginring and Rsarh Inrn, Amrian Insiu of Sl Consruion, and Sudn, Arhiural and Sruural nginring, Illinois Insiu of Thnolog, Chiago, IL. mail: romnr@ais.org Papr No HISTORICAL PRSPCTIV Th AISC Spifiaion approah for drmining h lmn slndrnss a whih loal ukling gins o onrol olumn srngh has volvd ovr h ars. Prior o h 1961 AISC Spifiaion, a simpl, maximum, widh-ohiknss raio was spifid. or insan, in h 1949 Spifiaion, h projing lmns of singl-angl srus had a limiing widh-o-hiknss raio of 1. In h 1961 Spifiaion, h provisions wr rvisd o inlud rogniion ha nw marials wih diffrn ild srnghs wr ing usd and ha ild srngh of h marial hn plad a rol in drmining a wha srss lvl loal ukling should onsidrd. Th limi was hangd o,4, whr was akn in pounds pr squar inh. In 1969, h limi was ssniall unhangd u was prsnd as 76. wih now akn in kips pr squar inh. In ordr o onvr h 1993 LRD Spifiaion o mri unis, h 1994 Mri LRD Spifiaion s h limi as a unilss quaion rsoring h varial in h limi. Thus, his sam limi am.45. Ovr ha sam priod of im, svral nw lmns wr dfind. or h 1 Spifiaion, hr wr nin ass dfind in Tal B4.1a for h limiing widh-o-hiknss raios for omprssion lmns in mmrs suj o axial omprssion. Howvr, h aual limis wr ssniall h sam as h had n sin During his sam priod, h approah o aoun for h influn of lmns ha xdd hs limiaions also volvd. Prior o h 1969 Spifiaion, h prai was o rmov h widh of h pla ha xdd h limiaion. This approah rquird h sion propris o ralulad asd on his nw gomr, a umrsom and NGINRING JOURNAL / THIRD QUARTR / 16 / 137

2 unonomial approah. Wih h 1969 Spifiaion (AISC, 1969), a nw approah was inrodud ha followd h approah usd in h 1969 AISI Spifiaion for h Dsign of Cold-ormd Sl Sruural Mmrs (AISI, 1969). A rduion faor, Q, was dfind as h raio of h loal ukling srss o h ild srss for mmrs wih slndr lmns. In h olumn srngh quaions, was rplad Q. Two spara approahs wr usd for drmining Q. On was for unsiffnd lmns, whih wr assumd o rah hir limi sa whn h lmn rahd is loal ukling srss. Th ohr was for siffnd lmns, whih mad us of hir pos-ukling srngh. or unsiffnd lmns, Q was dirl drmind hrough spifiaion quaions asd on marial and gomri propris of h lmns. or siffnd lmns, an ffiv widh was drmind, and h raio of h ffiv ara o h gross ara was usd o salish Q. This approah was asd on h aual srss in h mmr undr h ukling load rahr han h ild srss as was usd for unsiffnd lmns. Th provisions in h 16 AISC Spifiaion us h ffiv widh approah for oh siffnd and unsiffnd lmns following h prai usd AISI for oldformd mmrs sin 1 (AISI, 1). 16 SLNDRNSS PROVISIONS To drmin if on mus vn onsidr lmn slndrnss in drmining olumn srngh, hr nds o som valu agains whih h lmn widh-o-hiknss raio an ompard. As has n h as sin h 1961 Spifiaion, whn was inrodud as par of h limiing raio, h assumpion usd o drmin ha limi is ha h mmr an uniforml srssd o h ild srss vn hough omprssion mmrs ar rarl srssd o his lvl. This limi, whn xdd, is usd o dir h dsignr o Sion 7, Mmrs wih Slndr lmns, of h Spifiaion (AISC, 1). This assumpion ausd som dsignrs diffiul whn h susqunl drmind, afr following all h rquirmns of Sion 7, ha h sion srngh was no rdud du o lmn slndrnss. This an undrsood rognizing ha h mmr is no srssd o h ild srss, as originall assumd o dir h dsignr o hs provisions, so h lmn is lss likl o ukl. Alhough h limis shown in Sion 7 for 16 now inlud h iial srss for h olumn drmind wihou onsidraion of slndr lmns, i is sill h widh-o-hiknss limi asd on from Spifiaion Tal B4.1a ha lls h dsignr o onsidr h slndr lmn provisions. Th 16 provisions ar wrin in a unifid form for oh siffnd and unsiffnd lmns using h ffiv widh formulaion for all u round HSS. This hang is no so muh h rsul of nw rsarh as i is a rinrpraion of h foundaional work of von Kármán al. (193), Winr (1947), and Pköz (1987), as summarizd in Zimian (1). Th ffiv widhs ar usd o drmin h ffiv ara, and ha ara is muliplid h iial srss, drmind wihou onsidraion of slndr lmns, o oain h nominal omprssiv srngh. Th 16 provisions, xp for round HSS, ar givn as: (a) Whn λ λ () Whn λ > λ r r = 1 = 1 l (16 Sp. q. 7-) l (16 Sp. q. 7-3) whr is h lmn widh, is h lmn ffiv widh, and is h iial srss drmind in aordan wih Sion 3 or 4 wihou onsidraion of slndr lmns. Th limiing slndrnss, λ r, is akn from Tal B4.1a and, in all ass, is a funion of. Th widh-ohiknss raio, λ, is, aording o Tal B4.1a, /, d/ or h/, dpnding on h lmn ing onsidrd. Thus, h widhs in quaions 7- and 7-3 will also akn as, d or h, dpnding on h lmn ing onsidrd. Th lasi loal ukling srss, l, from lassi pla ukling hor (Zimian, 1) is l = k π 1 1 ν ( ) whih is wrin in h 16 Spifiaion as λr l = λ (1) (16 Sp. q. 7-4) Th onsan 1 is h mpirial orrion faor assoiad wih imprfion snsiivi and is a onsan drmind 1 alon and usd onl for onvnin. Th onsans 1 and, givn in 16 Spifiaion Tal 7.1, ar 138 / NGINRING JOURNAL / THIRD QUARTR / 16

3 Tal 7.1 ffiv Widh Imprfion Adjusmn aor, 1 and aor Cas Slndr lmn 1 (a) Siffnd lmns xp walls of squar and rangular HSS () Walls of squar and rangular HSS () All ohr lmns Tal B4.1a Cas Tal 1. Consans for ffiv Widh quaion Tal 7.1 Appndix A Cas k quaion Numr 1 () A-3 () k A-5 3 () A-9 4 () A-7 5 (a) A-11 6 () A-15 7 (a) A-13 8 (a) A SLNDRNSS PROVISIONS SIMPLIID Baus hs provisions rquir h us of h aulad limiing slndrnss raio from Tal B4.1a and h onsans 1 and from Tal 7.1 ah im a pariular p lmn is onsidrd, i ma hlpful for h usr o omin hm all on im and hn us his nw quaion. To aomplish his simplifiaion, h limis from Tal B4.1a k ar akn as λ r = 3, so ha h rsuling quaion an usd for all ass ovrd in ha al xp for round HSS. Th varial k is akn as 1. for all ass in Tal B4.1a, xp Cas (flangs of uil-up I-shapd sions and plas or angls projing from uil-up I-shapd sions), whr i an var from.35 o.76 (no hang from arlir Spifiaions). Thus, h limi on appliaion of quaion 7-3 oms k k = λ > λ r = 3 = 3 () Thn drmin l in rms of 3. Thus, λr l = λ = 3 k = 3 k (3) Susiuing l from quaion 3 ino quaion 7-3 ilds = 1 1 = 1 1 l whih simplifis o 3 l k k = ( ) 3 Comining h onsans in quaion 5 ilds k 5 = 4 1 ( ) k (4) k (5) k (6) whr 4 = 3 and 5 = 1 3. or all ass in Tal B4.1a, xp for round HSS for whih h spifiaion provisions ar diffrn and rmain ssniall unhangd from 1, h onsans ar as aulad in Tal 1. NGINRING JOURNAL / THIRD QUARTR / 16 / 139

4 Tal. Dsipion of Shaps Usd o Dvlop igurs igur Shap, ksi 1 W3 9 5 lmn Slndrnss h = 57.5 w Limiing Slndrnss for Loal Bukling 1.49 = 35.9 HSS16 4 x 46 3 WT L h = 89 d = 31.5 w d = 1.4 = = = Buil-up I-shap (slndr flang) lang: 4 in. ½ in. W: 4 in. ¾ in. 5 r f = 4 h = 3 w.56 = = Buil-up I-shap (slndr flang and w) lang: 4 in. ½ in. W: 4 in. ½ in. 5 r f = 4 h = 48 w.56 = = 35.9 Thus, for ws of doul smmri rolld I-shapd sions Cas 5 in Tal B4.1a and Cas (a) in Tal 7.1 h following onsans ar drmind: 1 =.18 = = = =.351 and quaion 7-3 oms, from quaion 6, = ( ) (7) This ffiv widh quaion is vr los o quaion 7-17 from h 1 Spifiaion, wih h onsans onl slighl diffrn. In addiion, hr is h sam as ƒ in quaion Th 16 provisions ar rwrin using quaion 6 and prsnd in full in his papr s Appndix. Th sam omparison o h 1 Spifiaion anno mad for unsiffnd lmns aus h ffiv widh approah in h 16 Spifiaion is a nw approah for hos lmns. IMPACT O 16 PROVISIONS I is h inn of hs nw 16 provisions o rdu h omplx naur of h prvious slndr lmn provisions and o prsn a unifid approah for oh siffnd and unsiffnd lmns. In som insans, h hangs implmnd for 16 will hav lil o no impa on h srngh of slndr lmn omprssion mmrs, whil in ohr insans, h ma ild a signifian inas in prdid srngh. Whr signifian srngh inas is sn wih h 16 provisions, h ovrl onsrvaiv naur of h prvious provisions has n rdud. igurs 1 hrough 6 illusra h nominal srngh for svral slndr lmn omprssion mmrs, showing h rsuls of h 1 provisions and hos of h 16 provisions. As an aid o undrsanding h ovrall signifian of slndr lmns on rduing olumn srngh, h nominal srngh, wih h rduion for slndr lmns ignord, is also shown. Th shaps usd for igurs 1 hrough 6 and hir lmn slndrnss valus ar aulad in Tal. In ah of h ass prsnd, h rolld shap was sld aus i is h on wih h mos slndr lmn for ha shap. Th uil-up shap was sld as an xrm as o illusra h signifian of h nw provisions for 14 / NGINRING JOURNAL / THIRD QUARTR / 16

5 W3 9 wih slndr w, = 5 ksi w/o Slndr lmns ffiv Lngh, L (KL), f ig. 1. Comparison of 1 and 16 slndr lmn olumn srngh, W3 9, = 5 ksi HSS16 4 x wih slndr walls, = 46 ksi w/o Slndr lmns ffiv Lngh, L (KL), f ig.. Comparison of 1 and 16 slndr lmn olumn srngh, HSS16 4 x, = 46 ksi. NGINRING JOURNAL / THIRD QUARTR / 16 / 141

6 WT15 45 wih slndr sm, = 5 ksi w/o Slndr lmns ffiv Lngh, L, (KL) ig. 3. Comparison of 1 and 16 slndr lmn olumn srngh, WT15 45, = 5 ksi L5 3 4 wih slndr lg, = 36 ksi ffiv Lngh, L, (KL) 1 16 w/o Slndr lmns ig. 4. Comparison of 1 and 16 slndr lmn olumn srngh, L5 3 ¼, = 36 ksi. 14 / NGINRING JOURNAL / THIRD QUARTR / 16

7 4 4 uil-up I-shap wih slndr flang, = 5 ksi w/o slndr lmns ffiv Lngh, L, (KL) ig. 5. Comparison of 1 and 16 slndr lmn olumn srngh, 4 4 uil-up I-shap, = 5 ksi. 4 4 uil-up I-shap wih slndr flang and slndr w, = 5 ksi w/o slndr lmns ffiv Lngh, L, (KL) ig. 6. Comparison of 1 and 16 slndr lmn olumn srngh, 4 4 uil-up I-shap, slndr w and slndr flang, = 5 ksi. NGINRING JOURNAL / THIRD QUARTR / 16 / 143

8 oh slndr flangs and slndr ws. Th shaps ha show h mos signifian hang ar h uil-up I-shap, WT and angl. Ths ar all mmrs wih unsiffnd slndr lmns. Th W-shap and h HSS show lss hang, illusraing h rlaivl minor impa on olumns wih slndr siffnd lmns. CONCLUSIONS Th 16 Spifiaion provisions for slndr omprssion lmns in omprssion mmrs ras siffnd and unsiffnd lmns in a similar fashion hrough h sam govrning quaion. I also aouns for h fa ha olumns ar no dsignd o srssd o h ild srss, so limiing widh-o-hiknss raios nd no asd on a limi salishd using h ild srss. A omparison wn h 1 Spifiaion and 16 Spifiaion for six slndr lmn mmrs shows ha h hang in srngh an signifian for mmrs wih slndr unsiffnd lmns. Two alrna approahs hav n prsnd ha produ h sam rsuls as h nw 16 Spifiaion. quaion 6, wih h onsans givn in Tal 1, ma usd for all slndr lmn mmrs xp round HSS, or h xpandd prsnaion givn in h Appndix ma usd. RRNCS AISC (1969), Spifiaion for h Dsign, ariaion, and rion of Sruural Sl for Buildings, Amrian Insiu of Sl Consruion, Nw York, NY. AISC (1), Spifiaion for Sruural Sl Buildings, ANSI/AISC 36-1, Amrian Insiu of Sl Consruion, Chiago, IL. AISI (1969), Spifiaion for h Dsign of Cold-ormd Sl Sruural Mmrs, Amrian Iron and Sl Insiu, Washingon, DC. AISI (1), Norh Amrian Spifiaion for h Dsign of Cold-ormd Sl Sruural Mmrs, Amrian Iron and Sl Insiu, Washingon, DC. Pköz, T. (1987), Dvlopmn of a Unifid Approah o Dsign of Cold-ormd Sl Mmrs, Amrian Iron and Sl Insiu, Washingon, DC. von Kármán, T., Shlr,.. and Donnll, L.H. (193), Th Srngh of Thin Plas in Comprssion, Transaions of h ASM, Vol. 54. Winr, G. (1947), Srngh of Thin Sl Comprssion langs, Transaions of ASC, Vol. 11, p Zimian, R.D., d. (1), Guid o Saili Dsign Criria for Mal Sruurs, 6h d., John Wil & Sons In., Hookn, NJ. APPNDIX This prsnaion rorganizs Sion 7 of h 16 AISC Spifiaion wih spifi quaions givn for ah as, similar o h 1 Spifiaion. Th onsans from Tal 7.1 and Tal 1 hav n inludd in h quaions. Wih h 16 Spifiaion, ah im a pariular shap is onsidrd, h sam onsans will nd o usd and h sam quaion will vnuall rsul. Thus, wriing ou h quaions on for ah as, as don hr, ma a simplifiaion usful o h dsignr. 7. MMBRS WITH SLNDR LMNTS This sion applis o slndr-lmn omprssion mmrs, as dfind in Sion B4.1 for lmns in uniform omprssion. Th nominal omprssiv srngh, P n, shall h lows valu asd on h applial limi sas of flxural ukling, orsional ukling, and flxural-orsional ukling. P n = A (A-1) whr A = summaion of h ffiv aras of h osssion asd on h rdud ffiv widh, or d, in. (mm ), or as givn quaions A-16 or A-17 = iial srss drmind in aordan wih Sion 3 or Sion 4, ksi (MPa) 1. Slndr Unsiffnd lmns Th ffiv widh, or d, for slndr unsiffnd lmns is drmind as follows: (a) or flangs, angls and plas projing from rolld olumns or ohr omprssion mmrs: (i) Whn (ii) Whn.56 >.56 = = ( ) (A-) (A-3) () or flangs, angls and plas projing from uil-up I-shapd olumns or ohr omprssion mmrs: 144 / NGINRING JOURNAL / THIRD QUARTR / 16

9 (i) Whn.64 (ii) Whn >.64 k = (A-4) k k = ( ) k (A-5) whr 4 k = and shall no akn lss han.35 nor hw grar han.76 for alulaion purposs () or sms of s: (i) Whn d.75 (ii) Whn d d = d (A-6) >.75 d = dph of, as dfind in Sion B4.1, in. (mm) = hiknss of lmn, as dfind in Sion B4.1, in. (mm). Slndr Siffnd lmns Th ffiv widh,, for slndr siffnd lmns is drmind as follows: (a) or all shaps xp ovr plas, diaphragm plas, walls of squar and rangular HSS and round HSS: (i) Whn (ii) Whn 1.49 > 1.49 = = ( ) () or ovr plas and diaphragm plas: (i) Whn 1.4 (A-1) (A-11) d = ( d ) (A-7) (d) or singl angls, doul angls wih sparaors, and all ohr unsiffnd lmns: (i) Whn.45 (ii) Whn > 1.4 = = ( ) (A-1) (A-13) (ii) Whn >.45 = (A-8) () or walls of squar and rangular HSS: (i) Whn 1.4 = ( ) (A-9) (ii) Whn > 1.4 = (A-14) whr = widh of unsiffnd omprssion lmn, as dfind in Sion B4.1, in. (mm) = ( ) (A-15) NGINRING JOURNAL / THIRD QUARTR / 16 / 145

10 (d) or round HSS, h ffiv ara is drmind as follows: (i) Whn D.11 A = A g (ii) Whn.11 D < <.45 (A-16) A.38 = + A ( D ) 3 whr D = ousid diamr of round HSS, in. (mm) = hiknss of wall, in. (mm) g (A-17) 146 / NGINRING JOURNAL / THIRD QUARTR / 16