A unified formulation for circle and polygon concrete-filled steel tube columns under axial compression

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1 uifid formulatio for irl ad polygo ort-filld stl tub olums udr axial omprssio Mi Yu a, Xiaoxiog Zha a,*, Jiaqiao Y b,*, Yutig Li a a. Shzh Graduat Shool, Harbi Istitut of Thology, Shzh 51855, Chia; b. Dpartmt of Egirig, Laastr Uivrsity, Laastr, L1 4YR. UK; bstrat: Currt dsig prati of ort-filld stl tub (CFST) olums uss diffrt formulas for diffrt stio profils to prdit th axial load barig apaity. It has always b a hallg ad pratially importat issu for rsarhrs ad dsig girs who wat to fid a uifid formula that a b usd i th dsig of th olums with various stios, iludig solid, hollow, irular ad polygoal stios. This has b driv by modr dsig rquirmts for otiuous optimizatio of struturs i trms of ot oly th us of matrials, but also th topology of strutural ompots. This papr xtds th authors prvious wor [1] o a uifid formulatio of th axial load barig apaity for irular hollow ad solid CFST olums to, ow, iludig hollow ad solid CFST olums with rgular polygoal stios. This is do by taig a irular stio as a spial as of a polygoal o. Fially, a uifid formula is proposd for alulatig th axial load barig apaity of solid ad hollow CFST olums with ithr irular or polygoal stios. I additio, laboratory tsts o hollow irular ad squar CFST log olums ar rportd. Ths rsults ar usful additio to th vry limitd op litratur o tstig ths olums, ad ar also as a part of th validatio pross of th proposd aalytial formulas. Kywords:Cort-filld stl tub (CFST), hollow ad solid stio, irular ad polygoal stio, load barig apaity Notatios f s ϕ ombid strgth of CFST s stability fator of CFST N strgth barig apaity of CFST, N = fs s N u load barig apaity of CFST, N = ϕ N = ϕ f η had ofiig offiit η had ofiig offiit for irular stio η s, had ofiig offiit for irular solid stio s,, ara of stl, ort ad hollow, rsptivly s ara of CFST stio, s = s + Ω solid ratio, Ω= ( + ) ψ hollow ratio, ( ) 1 β ratio of stl ara, β = ( + ) α stl ratio, α = s s ψ = + = Ω s s α solid stl ratio, α = ( + ) s s s s s s I s, I momt of irtia of stl, ort, rsptivly I s omposit momt of irtia, Is = Is + I f, f y haratristi strgth of ort ad stl, rsptivly *Corrspod author -mail addrsss: Zhaxx@hit.du., j.y2@laastr.a.u

2 ξ s solid ofiig offiit, ξs = αs fy f ξ ofiig offiit, ξ = α fy f E, E lasti modulus of ort ad stl, rsptivly s E omposit bdig modulus, = ( + ) s h K K L E EI EI I = s s s s ofimt fftivss offiit, h hollow ofimt fftivss offiit polygo ofimt fftivss offiit dg umbr, for irular ross stio, ta ifiity iitial imprftio offiit iitial imprftio offiit for irular CFST fftiv lgth of olum λ λ = L Is s sldrss ratio, λ o-dimsioal sldrss ratio, λ = λπ f E = L π N ( E I ) s s s s s s 1. Itrodutio ort-filld stl tub (CFST) olum is formd by fillig a stl tub with ort. ordig to th form of th ross-stio, CFST olums a b dividd ito diffrt groups, suh as irular, squar ad otago CFST olums, t. Th ross stios of ths olums a b ithr solid or hollow. solid ort-filld stl tub (S-CFST) olum is formd by pourig wt ort ito th tir spa losd by th stl tub, ad a hollow ort-filld stl tub (H-CFST) o is formd by pourig ort ito a stl tub usig th trifugal mthod. Figur 1-1 shows som of th ommoly usd ross stios of th ort-filld stl tub olums. a) Hollow squar b) Hollow otagoal ) Hollow irular d) Solid squar ) Solid otagoal f) Solid irular Figur 1-1 Commo stio typs of CFSTs xial load barig apaity of a CFST olum is a importat ad fudamtal dsig paramtr i ostrutio girig. Extsiv rsarh o solid CFST olums has b odutd ithr xprimtally or aalytially. Comprhsiv rsarh moographs hav b publishd by Zhog [2], Ha [3], Zhao [4], Zha [5] ad Chiai [6]. Thr ar also may publishd rsarh paprs o xprimtal studis of solid irular [7-9], lliptial [1], otagoal [11-13] ad squar CFST [14-18] olums. Numrial simulatios also playd a importat rol i studyig th bhavior of solid CFST olums udr axial omprssio [19-22] ad tri loadig [23-24]. Pratial dsig formulas wr proposd ad adoptd i, for xampl, CECS 254:29 [25] for hollow, Eurood 4 [26] for solid ad Ha [3] for irular ad squar solid CFST olums. From th abov, it a b oludd that most of th rsarh i th last fw dads fousd oly o solid CFST olums. Diffrt dsig formulas ad produrs wr rommdd for olums with diffrt stio profils. This is ot idal for modr strutural dsig whr strutural, matrial, arhittural, asthti ad viromtal paramtrs ar all dsigd i a otiuous mar to ahiv th bst possibl dsig. Th modr produr rquirs a otiuous hag of all dsig paramtrs, iludig stio *Corrspod author -mail addrsss: Zhaxx@hit.du., j.y2@laastr.a.u

3 profils of th CFST olums. It is obvious that a uifid formulatio for th alulatio of axial load barig apaity of CFST olums with various stio profils will bfit both aalytially ad omputatioally th ovrall dsig pross. Fortuatly, obtaiig a uifid dsig solutio for all th stios show i Fig. 1 is possibl si (a) matrially, th diffr btw a solid ad a hollow stio is th hollow ratio ad a solid stio a b viwd as a spial hollow stio with a hollow ratio of zro; ad (b) gomtrially, th diffr btw a irular ad a rgular polygoal stio is th umbr of sids ad a irular stio a b viwd as a spial polygoal stio with a ifiit umbr of sids. O th basis of th aformtiod spial ass, this papr attmpts to xtd th axial load barig apaity formula of a irular CFST olum to th olums with polygoal stios, ad a uifid formula is fially obtaid for both hollow ad solid irular ad polygoal stios. 2. Uifid formulatio of th strgth for irl ad polygo CFST olums 2.1 Simplifiatio of th uifid formula for irular stios uifid formula for both solid ad hollow stios was proposd i rfr [1] to prdit th strgth of a CFST olum through dompositio of th lasti dformatio of a irular ort filld stl tub ito a uiaxial omprssio ad a pla strai problm. Displamt ompatibility ad th solutio of thi-walld ylidr wr th itrodud to driv th strgth formula that is appliabl for both solid ad hollow irular stios. Th formula was validatd by xprimtal rsults, ad is show blow [1] : ( ) fs = 1 + η (1 β) f + β f y ξs (2-1a) Ωξs i whih η = (2-1b) f 2.Ω+.5ξs ξsω ( Ω+ ξ s ) f y whr, η is th had ofiig offiit for irular stio; β is th ratio of stl ara; Ω is th solid ratio; ad ξs is th solid ofiig offiit. Th dfiitios of all th symbols ar list i th Notatios. From th dfiitios of th ofiig offiit ξ, solid ratio Ω ξ ad solid ofiig offiit s, o has = Ω ξ (2-2) Isrtig Eq.(2-2) ito Eq. (2-1b) yilds: ξ η =.5 (2-3a) 1 + ξ 2 whr = (2-3b) f y αs +.5αs 1 f Ω I girig prati, th most ommoly usd stl varis from Q235 to Q42, ad th ort grad from C3 to C8. Th valus of fy f, thrfor, is somwhr btw 4.7 ad 2.9. α For solid CFST olums, th stl ratio s is btw.4 ad.2, ad Ω= 1. pproximatly, th had ofiig offiit for a solid irular stio is th: η s, 2 ξ ξ = α 1+ ξ 1+ ξ s Th rlatioship btw th offiit ad th solid ratio Ω i Eq.(2-3b) is show by Figur 2-1. (2-4) *Corrspod author -mail addrsss: Zhaxx@hit.du., j.y2@laastr.a.u

4 fy/f=2.9 =Ω αs= Solid ratio,ω a) f f = 2.9 b) y αs=.2 =Ω fy/f= Solid ratio,ω α =.2 Figur 2-1 Th rlatioship btw offiit ad solid ratio Ω Furthr umrial tsts show that, for ay ombiatio of th paramtrs, is always smallr tha Ω. Thus, by assumig, =Ω= 1 ψ (2-5) a simplifid Eq.(2-3) a b obtaid, whih lads always to a osrvativ dsig. It will b s latr that this approximatio dos ot afft sigifiatly th auray of th prditios. Eq. (2-3a) ow boms: s ξ η =.5 Ω =Ω η 1 + ξ Th strgth ad th axial load barig apaity of a irular CFST olum ar, rsptivly: f s ( ) Ω ξ = f 1+ α s, (2-6) (2-7a) ξ ξ ad N = fs s = + Ω ( fy s + f ) (2-7b) whr, α is th stl ratio. 2.2 Extsio of th strgth formula to polygo stios For a CFST olum with a polygoal ross stio, th ommo prati i dsig is to fid th solutio of a olum with a quivalt irular stio. Th solutio is th modifid by a orrtio fator. Th orrtio fator is usually osidrd i th ofiig offiit [25]. Rsarh also showd that th ofimt of a squar stl tub o th ort a b dividd ito a fftiv had zo ad o-had zo [27], sparatd by boudaris of paraboli shap. Similarly, assumig that th sam priipl applis to a rgular polygoal stio ad, with th iras of th umbr of sids, th fftiv had zo approahs that of a irular stio. Basd o th fftiv ara mthod, Madr [28] proposd a ostitutiv modl for ort riford with stirrups by usig a quivalt modl of uiform ostrait. Followig this approah, th o-uiform ofimt prssur o th ort from a polygoal stl tub a b quivalt to a uiform ofimt prssur from a irular stl tub, i..: P = P (2-8) whr, P is th fftiv ofiig prssur; P is th ofiig prssur from a irular stl tub, ad is uiformly distributd; is th ofimt fftivss offiit. Eq.(2-8) is valid for olums with solid stios. For a olum with a hollow stio, a similar approah *Corrspod author -mail addrsss: Zhaxx@hit.du., j.y2@laastr.a.u

5 is followd to alulat th fftiv ofiig prssur. This approah has two stps as show i Figur 2-2. Stp 1 Stp 2 P P = P h P = P = P h (a) Hollow polygoal stio (b) Solid polygoal stio () Solid irular stio Figur2-2 Equivalt hollow ad polygoal stios Th first stp fids a quivalt solid stio (b) of th hollow o (a). Both hav th sam form of th xtral boudary ad th sam ross stioal ara. Thus, a orrtio fator, h is itrodud btw (a) ad (b). Th sod stp fids a solid irular stio () that has th sam ross stioal ara as th polygoal stio (b), ad h itrodus aothr orrtio fator. I th followig stios, th two orrtio fators ar also dfid as thir rsptiv fftivss offiits. By followig ths stps, th o-uiform ofiig prssur of a hollow polygoal stio a b quivalt to a uiform ofiig prssur o a quivalt irular solid stio, i..: whr, P fftiv ofiig prssur; P ofiig prssur from th stl tub; ofimt fftivss offiit; h hollow ofimt fftivss offiit; polygo ofimt fftivss offiit. P = P = P h (2-8) Eq.(2-8) proposs a quivalt systm that rquirs itrodutio of a fftivss offiit that aouts for th fft of th hollow ad polygoal stio profil o th ofiig prssur. Similar to th solutio proposd i [1] for irular stios, a formula for a polygoal stio a b obtaid i th followig form: (, ) fs = 1 + η s (1 β) f + β f y (2-9) Th ovrall ofiig offiit is ow th ofiig offiit of th solid stio, η s,, multiplid by th fftivss offiit. Thrfor, th had ofiig offiit for both a irular ad a polygoal stio a b grally writt as: For a irular stio, = 1. H: ξ h = hs, = h hs, = h ξ ξ h = h ξ From Eq.(2-6) ad Eq.(2-11), th ofimt fftivss offiit, *Corrspod author -mail addrsss: Zhaxx@hit.du., j.y2@laastr.a.u (2-1) (2-11) h,of a hollow stio a b,

6 obtaid by dividig th solid ara with th sum of th solid ad hollow aras h =Ω= + + : (2-12) For a polygoal solid stio, is obtaid by dividig th ara of th fftiv had zo with th total ara of ort [28] of th solid stio, i.., = (2-13) It a b s from th abov that it is sstial to dfi th boudary btw th fftiv had ad th o-had zos as show i Figur 2-3. Madr [28] assumd that th boudary btw th two zos is a parabola, with a agl of 45 dgrs at th itrstio of th tagt of th boudary ad th sid of th polygo, as show i Figur 2-3. B O Efftiv had zo OO1 y B O = a O θ 1 No-had zo R x Paramtr dsriptio: 1. B is a sid of th polygo, ad O is th 1 middl poit of th sid; 2. BO = BO = 45 ; 3. R is th radius of th quivalt irl of th polygo havig th sam ara. r is th radius of th hollow. a is th radius of iirl of th polygo; 4. OO1 = θ. Figur 2-3 Efftiv had zo I Figur 2-3, th oordiats of poit is ( a taa, a ), whr α = π ad is th umbr of sids. Th o-had ara rlatd to sid B, whih is th ara losd by th sid ad th parabola, is 2 2 o 2 3a ta θ =. Th ara of ort is fftivss offiit is = a 2 taa. For th tir stio, thrfor, th 2 o 2 ta θ = = 1 (2-14) 3 ta a It is lar from Eq.(2-14) that th fftivss offiit dpds solly o th positios of ad B. i., how th quivalt irular stios ar dfid. Two ass ar osidrd i this papr: Cas 1: Th ara of th quivalt irular stio is th sam as that of th polygoal stio. I this as th irular boudary has two itrstios with ah sid of th polygo. Th fftivss offiit of this as is alulatd blow. Si th qual ross stioal aras: Thus, a R 2 2 = π 1 a taa = taa, a θ =, R From Figur 2-3, os Thus, a ta a taa = πr, 2 2 a ta = 1. a os 2 θ a =, ad th 2 ta θ *Corrspod author -mail addrsss: Zhaxx@hit.du., j.y2@laastr.a.u

7 Substitutig th abov ito Eq.(2-14) yilds = 1 3 a ta a, π α = (2-15) Cas 2: Th quivalt irular stio is ta as th irumirl of th polygo. I this as th irl passs through all th vrtis of th polygo, i.., ad B oiid, rsptivly, with ad B. Thus, θ = α, ad Eq.(2-14) is rdud to: 2 π = 1 taa, α = (2-16) 3 From Eqs. (2-15) ad (2-16), it a b s that th orrtio offiit is a futio of th umbr of sids ad th hollow ratio. Wh th umbr of sids approahs ifiity, approahs uity for a irular stio. Th of th two ass for diffrt umbr of sids ar show i Figur 2-4, whr th urvs of ass 1 ad 2 wr obtaid from Eqs.(2-15) ad (2-16), rsptivly. Th urv of th simplifid formula was plottd by usig th followig approximat quatio: 2 4 = (2-17) Cas 1 Cas 2 simplifid formula Numbr of dgs, Figur 2-4 vs. urvs for th two ass Eq. (2-17) is formd from a rgrssio aalysis o th rsults of th two ass by osidrig th fats that: (a) wh th umbr of sids is gratr tha 16, Figur 2-4 shows that is los to 1, ad (b) wh th umbr is smallr tha 8, Cas 2 provids a safr dsig. It is vid that Eq.(2-17) is simpl ad lis dirtly to th umbr of sids of a polygo. Th quatio a b usd i dsig i a straightforward mar. Thus, th uifid ombid strgth formula ad th axial load barig apaity of a gral polygoal CFST olum ar, rsptivly: 1+ ( 1+.5 ) ξ fs = f 1+ α ξ N = fs s = 1+.5 fy s + f 1 + ξ whr: ad ( ) ξ ofiig offiit, ξ = f s y f ; f y, f haratristi strgth valus for stl ad ort rsptivly; s,, ara of stl, ort ad hollow, rsptivly; ofimt fftivss offiit, h h = ; hollow ofimt fftivss offiit, =Ω= 1 ψ ; *Corrspod author -mail addrsss: Zhaxx@hit.du., j.y2@laastr.a.u h (2-18a) (2-18b)

8 2 2 polygo ofimt fftivss offiit, ( 4) ( 2) = +, is th sid umbr. Ths uifid formulas apply to all forms of solid ad hollow stios of ort filld stl tub olums. For a solid irular CFST olum, = = 1, ad for othr stio profils, th olum strgths ar obtaid from thir rsptiv quivalt solid irular CFST os. h 2.3 Vrifiatio of th uifid strgth formula for CFST olums udr uiaxial omprssio To vrify th uifid strgth formula, th irular CFST olums ivstigatd i Tabls 1 ad 2 of [1] wr studid hr first usig Eq.(2-18). Th prditd axial load barig apaity was ompard with th xprimtal rsults. Th omparisos, whih ar ot prstd hr, showd that th avrag ratios of N N wr.963 ad 1.55, with a varia of.16 ad.7, rsptivly. This dmostrats that th tst uifid formula a provid good prditios to th strgth of irular CFST olums. ftr this sussful validatio for irular CFST olums, th uifid strgth formula was usd to prdit th axial load barig apaity of otagoal ad squar CFST short olums udr uiaxial omprssio. Nw xprimtal tsts o short olums wr also arrid out for validatios. Th omparisos ar prstd i Tabl 2-1 ad Tabl 2-2. It is vidt that th prditd load barig apaity agrs vry wll with th tst rsults for all th ass. Th avrag ratios of N N, ar.987 ad.957, ad th varia tst ar.12 ad.6, rsptivly, for th otagoal hollow ad solid olums. For th squar olums, th rsptiv avrag ratios N N of th hollow ad solid stios ar 1.66 ad.94, ad th varia ar tst.14 ad.7. From ths rsults, it appars that th umbr of sids of th stios dos ot hav sigifiat iflu o th auray of th prditios. It is otid that th prditd load barig apaity a b ithr gratr or smallr tha th tstd valus, whih should b awar of wh th formula is usd i dsig. Tabl 2-1 Compariso of formula basd ad xprimtal rsults for otagoal hollow ad solid CFST Typ otagoal hollow CFST otagoal solid CFST Rf. [2] [5] [11] Explaatio of umbrig NO. Numbrig Lgth of sid Gomtri paramtrs Matrial Tsts Cal. Ratio Stl thiss Radius of hollow B/mm t/mm ro/mm fy /Mpa f' /Mpa f /Mpa Ntst /KN N /N N/Ntst 1 1C C C C C C C C C CFST CFST CFST P G P *Corrspod author -mail addrsss: Zhaxx@hit.du., j.y2@laastr.a.u

9 [12] P P G HN HN HN MN MN MN LN LN LN Tabl 2-2 Compariso of formula basd ad xprimtal rsults for squar hollow ad solid CFST Srial Gomtri paramtrs Matrial Tsts Cal. Ratio Typ Rf. NO. Numbrig Lgth of sid Thiss Radius of hollow B/mm t/mm ro/mm squar Hollow CFST squar Solid CFST [2] [14] [15] [16] [17] fy /Mpa f' /Mpa f /Mpa Ntst /KN N /N N/Ntst 1 1D D D D D D D D D R R R R R R R R S S S S S *Corrspod author -mail addrsss: Zhaxx@hit.du., j.y2@laastr.a.u

10 Uifid formulatio of stability barig apaity for irl ad polygoal log CFST olums 3.1 Modifid formula of stability fator of irl stio Th abov approah for dvlopig th uifid strgth formula also applis to th formulatio of stability fator of log CFST olums. By osidrig a ort-filld stl tub olum as a olum mad of a omposit matrial, th stability fator has b obtaid from Prry-Robrtso formula i rfr [1]. Th fudamtal assumptio is that th quivalt iitial imprftio offiit is dirtly proportioal to th stl ratio β, ad th stability fator of a irular stio [1] is, 1 ϕ = λ + K λ + 1 λ + λ + 1 4λ ( K ) s 2 s s s s s 2λ s (3-1) whr, λs is th o-dimsioal sldrss ratio, CFST olum. I rfr [1], it was assumd that K =.25β. K is th iitial imprftio offiit of th irular 1. Stability fator φ Eurood 3, Bulig urv "a" Eurood 3, Bulig urv "b" DBJ/T (solid irl CFST).2 GB (stl tub typ "b") K=.16. K= No-dimsioal sldrss λs Figur 3-1 th stability fator vs. th o-dimsioal sldrss Figur 3-1 prsts th stability fator urvs agaist th o-dimsioal sldrss from various dsig ods ad formula Eq.(3-1). It a b s from th figur that wh K =.16, th stability fator urv from Eq.(3-1) agrs wll with th bulig urv a of Eurood 3 ad th bulig urv for solid *Corrspod author -mail addrsss: Zhaxx@hit.du., j.y2@laastr.a.u

11 CFST olums from th thial spifiatio DBJ/T It is foud also that wh K =.25, th prditd stability fator urv agrs wll with th bulig urv b of Eurood 3 ad th bulig urv of GB for stl tubs. Thrfor, th valu of th quivalt iitial imprftio offiit ( K ) is alulatd by liar itrpolatio of th hollow ratio ( ψ ) ragd from a irular hollow CFST ( K =.16 ) to a stl tub ( K =.25 ). Thus, K = ψ =.25.9Ω=.25.9 (3-2) 3.2 Extsio of th stability fator to polygo stios By followig th sam produr dsribd i Stio 3.1 for irular stios, Th valu of th imprftio offiit for a polygoal stio a b alulatd by furthr assumig that th umbr of sids affts also th iitial imprftio offiit K. Thus, o th basis of Eq.(3-2), th followig formula is proposd: K=.25.9 =.25.9 (3-3) h whr, is th ofimt fftivss offiit. Wh approahs ifiity (irular CFST), th formula is rdud to th stability fator of a irular CFST olum. Wh Ω= (stl tub oly), K =.25, whih is th valu rommdd by GB From th abov aalysis, th uifid formulatio for prditig axial load barig apaity of log irular ad polygoal CFST olums ar as follow: h N u = ϕ N (3-4a) s whr: 1 ϕ = λ + Kλ + 1 λ + λ + 1 4λ ( K ) s 2 s s s s s 2λ s N strgth barig apaity of CFST, usig Eq.(2-18b); ϕ s stability fator of CFST; s ara of CFST stio; λ o-dimsioal sldrss ratio, λ L π N ( E I ) s K iitial imprftio offiit, usig Eq.(3-3). = ; s s s (3-4b) 3.3 Vrifiatio of th stability load barig apaity formula for uiaxial omprssio For irular CFST olums, this was do by ralulatig th load barig apaity of th log olums tstd i Tabl 3 of rfr [1] usig Eq.(3-4) ad omparig th prditd rsults with th tst os. It was foud that th avrag ratio of N N was.912, ad th varia was.8. Th ompariso shows that tst th simplifid formula is still suffiitly aurat. I ordr to validat th appliatio of th uifid formula for polygoal CFST olums, availabl xprimtal rsults from Zhog [2], Cao, t al [13] ad Guo, t al [18] wr ompard with th prditios from Eq.(3-4) i Figur 3-1. Si availabl tsts rsults o log olums ar vry limitd, th fiit lmt (FE) rsults of otagoal CFST olums du to Sh [29] wr also usd i th validatio show i Fig.3-1. Th omparisos show that th avrag ratio of th prditd load barig apaity from th simpl formula (3-4) ad th rsults from othr rsarhrs is.956 ad th varia is.8 for otagoal CFSTs. Th avrag ratio is 1.24 ad th varia is.8 for squar solid CFST. From th *Corrspod author -mail addrsss: Zhaxx@hit.du., j.y2@laastr.a.u

12 omparisos, it a b s that th uifid formula ad th xprimtal rsults agr rasoably wll. Nu from Eq.(3-4) (N) Cao B Z, Zhag Y C,t al(25) Sh C Y(26) Nu from othr mthods (N) Nu from Eq.(3-4) (N) Guo L H, Zhag S M,t al(25) Zhog S T(26) Nu from othr mthods (N) a) Otagoal hollow ad CFST b) squar solid CFST Figur 3-1 Comparisos of th aalytial, th tst ad th FE rsults for log ad short CFSTs 4. Exprimtal study ad vrifiatio of th barig apaity of hollow CFSTs 4.1 Tst spim ad produr of log CFST olums udr axial omprssio Extsiv xprimtal study o th load barig apaity of both short ad log solid CFSTs has b arrid out ad wll rportd i th litratur. Howvr, rportd xprimtal studis o hollow CFST log olums, spially squar hollow CFSTs ar vry limitd. Th purpos of this stio is to st up th tst produr for log, hollow CFST olums with irular ad squar stios. Ths tst rsults ar w ad a b usd i th futur to validat w umrial modls. I this papr, th tst rsults ar also usd to furthr validat th appliability of th formulas that wr drivd idpdtly i th prvious stios. I total, 6 irular ad 6 squar hollow-cfst log olums udr axial omprssio wr tstd. Th irular stl tub has a diamtr (D) of 219mm ad a thiss (t) of 3.8mm. Th lgth of th sid of th squar tub (B) was 2mm ad th thiss (t) was 3.9mm. Th hollow ratio of th olums ragd from.25 to.65. Th ovrall lgth of th olums was 381mm, iludig th thiss of th two d plats, 2mm ah, attahd to th ds of th olums. th top d Spim LVDT th bottom d Strai Gaugs *Corrspod author -mail addrsss: Zhaxx@hit.du., j.y2@laastr.a.u

13 Figur 3-2 rragmt of tsts Th tst st-up is show i Figur 3-2. Thr LVDTs wr usd to masur th trasvrs dfltio of th olums, of whih o was for masurig th displamt at th mid spa whil th rmaiig two wr moutd at a dista of L/3 from th two ds, rsptivly. dditioally, ight strai gaug rostts, four o o sid ad th othr four o th opposit sid, wr plad symmtrially at a vrtial dista of L/2 from th top to masur th strais i th stl at ths loatios. Th top ad bottom ds of th olum wr otd to th supports allowig oly th rotatioal displamts of th ds. Th axial load was applid through a loadig ll that has a maximum load apaity of 5to. 4.2 Exprimtal rsults ad disussio Bfor th olums wr tstd, six ort ubs of (1mmx1mmx1mm) wr tstd to masur thir omprssiv strgth. It was foud that avrag 1mm ub omprssiv strgth was 56.14Mpa, whih was ovrtd to a stadard 15mm ub omprssiv strgth (fu) of 53.3MPa. Th stadard omprssiv strgth (f) is 34.4Mpa, thrfor, aordig to th Chis Stadard GB To dtrmi th stl matrial proprtis, thr tsio oupos wr ut from th squar ad irl stl tubs ad tstd. From ths tsts, th avrag yild strgth (fy) of th irular tubs was MPa, ad th avrag yild strgth (fy) of th squar tubs was Mpa. Th load -mid spa dfltio urvs rordd from th tsts ar show i Figur 3-3, ad th dformatio ad failur mods of th hollow CFST olums ar show i Figur 3-4. xial load(n) S1-S S1-S-2 S2-S S2-S-2 C1-S-1 12 S3-S-1 9 C1-S-2 S3-S-2 C2-S C2-S C3-S-1 C3-S Dfltio at midhight (mm) Dfltio at midhight (mm) a) Cirular hollow CFST b) Squar hollow CFST xial load(n) Figur 3-3 xial load Midspa displamt urvs of tsts. Dtaild iformatio ad failur mods of th tstd olums ad th omparisos btw th prditd ad tstd axial load apaity ar prstd i Tabl 3.1. Two major typs of failur mods wr obsrvd i th tsts: o is global bulig as show i Figur 3-4a-b, ad th othr is d rushig show i Figur 3-4-d. For th irular hollow CFST olums, C1-S-1, C1-S-2 ad C2-S-1, th domiatig failur mod was global bulig ad for C2-S-1, C3-S-1 ad C3-S-2, th failur was ausd by d rushig. For th squar hollow CFST olums, S1-S-1, S1-S-2, S2-S-1 ad S2-S-2 faild du to global bulig, whil for S3-S-1 ad S3-S-2, rushig ourrd at th ds of th olums. Durig th tsts, it was obsrvd that th radius of hollow had sigifiat fft o th failur mods. I gral, wh th thiss of ort is small, th olum d tds to rush suddly. For th olums faild from global bulig, it a b s from Figur 3-3 that th irular hollow CFST olums show mor dutility i ompariso with th squar os. From Tabl 3.1, it is otid that as th hollow ratio irass, th failur mods td to hag from global buig to d rashig for both irular ad squar olums. Thus, spial osidratio should b ta i dsigig th hollow stios, si ay brittl failur, suh as th d rash of th olums, must b avoidd. dditioal riformt or strgthig at th d may b osidrd i dsig. *Corrspod author -mail addrsss: Zhaxx@hit.du., j.y2@laastr.a.u

14 ) C3-S-1 a) C2-S-1 b) S2-S-2 d) S3-S-2 Figur 3-4 Failur mods of olums It has to b mtiod that th uifid formulas proposd i this papr osidrd oly th strgth of short olums ad global bulig of log olums. Prditios to th failur du to th obsrvd d rush wr ot iludd. This is dmostratd by th good agrmt btw th tstd ad th prditd axial load barig apaity of th olums faild du to global bulig. Howvr, th omparisos show also good agrmt for most of th olums faild du to d rash. This may b baus th rashd ds ratd additios joits ar th two ds, whih prhaps oly wad slightly th pid d supports ad this did ot afft th ovrall axial load barig apaity sigifiatly. Tabl 3-1 Tst rsults ad alulatio rsult of 12 log olums Gomtri paramtrs Matrial Tsts Cal. Shap Cirular Hollow CFST Spim Dimsio or Stl Radius Numbr lgth Ntst N lgth of sid Thiss of hollow fy/mpa f/mpa /KN /N D or B /mm t/mm ro/mm L/mm C1-S C1-S C2-S C2-S C3-S C3-S Fail mod Global bulig Ed rush *Corrspod author -mail addrsss: Zhaxx@hit.du., j.y2@laastr.a.u

15 S1-S Squar Hollow CFST Global bulig S1-S S2-S S2-S S3-S Ed S3-S I Tabl 3-1, th avrag ratio of N N is 1.5 ad th varia is.18 for th irular hollow tst olums, ad ths ar rsptivly.995 ad.3 for th squar hollow os. 5. Summary of th uifid formatio for irl ad polygo CFST olum udr axial load Si a rathr larg umbr of quatios or formulas hav b prstd i th prvious stios, ad ot all of thm ar rquird i a typial dsig alulatio, this stio summarizs th most importat formulas ad th y stps that should b followd i th alulatio of axial load barig apaity of solid, hollow, irl ad polygoal ort-filld stl tub olums udr axial omprssio: rush Stp 1: Calulat strgth apaity of a CFST stio by: ( η )( ) N = + f + f (5-1) 1 y s ξ whr, η had ofiig offiit, η = ξ ; ξ ofiig offiit, ξ = f s y f ; 2 2 ofimt fftivss offiit, ( 1 ψ )( 4) ( 2) = +. Stp 2: Calulat th stability fator by 1 ϕ = λ + Kλ + 1 λ + λ + 1 4λ ( K ) s 2 s s s s s 2λ s (5-2) whr, λ s ormalizd sldrss ratio λ L N s = ; π EsIs K iitial imprftio offiit, K =.25.9 ; E I th omposit bdig rigidity, EsIs = EI + EI s s. s s Stp 3: Calulat th stability load barig apaity by *Corrspod author -mail addrsss: Zhaxx@hit.du., j.y2@laastr.a.u

16 N u = ϕ N (5-3) s 6. Coludig rmars uifid formulatio for prditig load barig apaity of both irular ad polygoal CFST olums has b proposd i th papr. This was basd o th simplifid form of th strgth formula for irular CFST olums proposd i th authors prvious study [1] ad th xtsio of th formulas to olums with polygoal stios. modifid formula of th stability fator for irular CFST olum was also proposd. Th liar itrpolatio thiqu was usd to stimat th stability offiit for olums with diffrt hollow ratios. Th fator was th itrodud ito th uifid formula to alulat th stability load barig apaity of both irular ad polygoal CFST olums. Th proposd uifid formulas wr validatd through omparisos with availabl tst rsults ad th w tst rsults of irular ad squar hollow CFST log olums udr axial load rportd also i this papr. Th omparisos showd satisfatory agrmt ad suggstd that th simplifid uifid formulas had pottial to b usd i pratial dsig. Futur wor is dd to xtd th load barig apaity formulas of CFST olums to ilud th fft of tmpratur lvatio. Rfrs [1] Yu M, Zha XX, Y JQ. t al. Uifid Formulatio for Hollow ad Solid Cort-Filld Stl Tub Colums udr xial Comprssio [J]. Egirig struturs. 21, 32(4):146~153. [2] Zhog ST. Rsarh ad ppliatio hivmt of Cort-Filld Stl Tubular (CFST) Struturs [M]. Bijig: Tsighua Uivrsity Prss, Chia; 26. (i Chis) [3] Ha LH. Cort filld stl tubular struturs from thory to prati [M]. Bijig: Si Prss, 27. (i Chis) [4] Zhao XL, Ha LH, Lu H. Cort-Filld Tubular Mmbrs ad Cotios. Taylor & Frais, 21. [5] Zha XX. Hollow ad solid ort-filld stl tub struturs[m]. Bijig: Si Prss, 21. (i Chis) [6] Matsui Chiai. Prforma ad dsig of ort-filld stl tubular strutur. Ohmsha, 29. [7] Ha LH, H SH ad Liao FY. Prforma ad alulatios of ort filld stl tubs (CFST) udr axial tsio [J]. Joural of Costrutioal Stl Rsarh. 211, 67(11): [8] Wag YC. Tsts o Sldr Composit Colums[J]. Joural of Costrutioal Stl Rsarh. 1999, 64(11): [9] Yu Z W, Dig F X, Cai C S. Exprimtal Bhavior of Cirular Cort-Filld Stl Tub Stub Colums [J]. Joural of Costrutioal Stl Rsarh, 27,63(2): [1] Yag H, Lam D ad Gardr L. Tstig ad aalysis of ort-filld lliptial hollow stios [J]. Egirig Struturs. 28, 3(2): [11] Zhag YC, Wag QP, Mao XY ad Cao BZ. Rsarh o Mhais Bhavior of Stub -olum of Cort-filld Thi -walld Stl Tub udr xial Load[J]. Buildig Strutur, 25,35(1): (i Chis) [12] Tomii M, Yoshimura K, Morishita. Y. Exprimtal Studis O Cort-Filld Stl Tubular Stub Colums udr Cotri Loadig[C]//Itratioal Colloquium o Stability of Struturs udr Stati ad Dyami Loads, Washigto DC, May 17-19, 1977: [13] Cao BZ, Zhag YC, Zhao YM. Exprimtal Rsarh O Cort Filld Thi-Walld Stl Tub Log Colums[J]. Ky Egirig Matrials, 29,4-42: *Corrspod author -mail addrsss: Zhaxx@hit.du., j.y2@laastr.a.u

17 [14] Liu DL. Tsts O High-Strgth Rtagular Cort-Filld Stl Hollow Stio Stub Colums[J]. Joural of Costrutioal Stl Rsarh, 25, 61(7): [15] Liu DL, Gho WM. xial Load Bhaviour of High-Strgth Rtagular Cort-Filld stl Tubular Stub Colums[J]. Thi-Walld Struturs, 25, 43(8): [16] Shidr SP. xially Loadd Cort-Filld Stl Tubs[J]. Joural of Strutural Egirig, 1998, 124(1): [17] Zhag SM, Guo LH, Y ZL, t al. Bhavior of Stl Tub ad Cofid High Strgth Cort for Cort-Filld RHS Tubs[J]. dvas i Strutural Egirig, 25, 8(2): [18] Guo LH, Zhag SM, Wag YY, t al. Exprimtal d alytial Rsarh O xially Loadd Sldr High Strgth Cort-Filld RHS Tubs[J]. Idustrial Costrutio, 25,35(3):75-79.(i Chis) [19] Ellobody E ad Youg B. Noliar aalysis of ort-filld stl SHS ad RHS olums [J]. Thi-Walld Struturs. 26, 44(8): [2] Hu HT, Huag CS, Wu MH. t al. Noliar alysis of xially Loadd Cort-Filld Tub Colums with Cofimt Efft [J]. Joural of Strutural Egirig. 23, 129(1):1322~1329. [21] Tao Z, Uy B, Liao FY ad Ha LH. Noliar aalysis of ort-filld squar stailss stl stub olums udr axial omprssio [J]. Joural of Costrutioal Stl Rsarh. 211, 67(11): [22] Dai X ad Lam D. Numrial Modllig of th xial Comprssiv Bhaviour of Short Cort-Filld Elliptial Stl Colums [J]. Joural of Costrutioal Stl Rsarh. 21, 66(4):542~555. [23] Portolés, JM, Romro ML, Filippou FC ad Bot JL. Simulatio ad dsig rommdatios of trially loadd sldr ort-filld tubular olums[j]. Egirig Struturs, 211, 33(5): [24] Liag QQ ad Fragomi S. Noliar aalysis of irular ort-filld stl tubular short olums udr tri loadig [J]. Joural of Costrutioal Stl Rsarh. 21, 66(2): [25] CECS 254:29. Thial spifiatio of hollow ort-filld stl tubular struturs [S]. Chia assoiatio for girig ostrutio stadardizatio, 29. (i Chis) [26] Eurood 4. BS EN :24. Dsig of omposit stl ad ort struturs. Part 1-1: Gral ruls ad ruls for buildigs[s]. British Stadards Istitutio, 24. [27] Varma H, Saus R, Rils JM. t al. Dvlopmt ad Validatio of Fibr Modl for High-Strgth Squar Cort-Filld Stl Tub Bam-Colums [J]. CI strutural joural. 25, 12(1):73~84. [28] Madr JB, Pristly M. ad Par R. Thortial Strss-Strai Modl for Cofid Cort [J]. Joural of Strutural Egirig. 1988, 114(8):184~1826. [29] Sh CY. Rsarh o th Strgth ad Stability of Hollow Polygoal Cort-Filld Stl Tubs [D]. Harbi istitut of Thology, 28. (i Chis) *Corrspod author -mail addrsss: Zhaxx@hit.du., j.y2@laastr.a.u