Stability of Steel Columns with Non-Uniform Cross-Sections

Transcription

1 The Open Cnstructn and Buldng Technlgy Jurnal, 1, 6, Stablty f Steel Clumns wth n-unfrm Crss-Sectns Open Access Thedre G. Knstantakpuls, Ianns G. Raftyanns* and Gerge T. Mchaltss Department f Cvl Engneerng, atnal Techncal Unversty f Athens, Athens 1578, Greece Abstract: In ths wrk, nn-unfrm steel members wth r wthut ntal gemetrcal r ladng mperfectns, that are laded by axal frces appled cncentrcally r eccentrcally and by cncentrated mments appled at the ends r at ntermedate pnts, are studed. Mre specfcally, steel members wth varyng crss-sectns, tapered r stepped r members cnsstng by tw dfferent tapered parts are cnsdered. The frmulatn presented n ths wrk s based n slvng the gvernng equatn f the prblem thrugh a numercal methd where the egenshapes f the member are emplyed. A falure plastcty crtern s ntrduced fr members especally the shrt nes that wll never reach the elastc crtcal bucklng lad. Althugh nly the smply supprted beam-clumn case s studed heren, t s bvus that the methd can be extended t mult-span beams and frames, by emplyng the crrespndng egenshapes. Useful dagrams are presented fr bth the crtcal bucklng lads and the equlbrum paths shwng the nfluence f the man characterstcs f the beam-clumn. Keywrds: Steel clumns, stablty, nn-unfrm members, bucklng lads, mperfectns, yeld falure. ITRODUCTIO The use f steel members wth nn-unfrm crsssectns ether as clumns r as dstressed parts f a structure wth r wthut bendng mments s very cmmn n steel cnstructns. There s a wde varety f structures such as buldngs frames, brdge members, masts r cranes, etc, whch are desgned wth members f nn-unfrm crsssectns n rder t mnmze the requred materal. The sgnfcance f usng such members n structures and the necessty t study ther structural behavr has been realzed snce the begnnng f the 19 th century by A. Föppl, as t was referenced by Tmshenk [1]. A frst apprach and study f the abve-mentned prblems f clumns wth varable crss-sectns was made by Dnkk n 1914 and n The man results f these studes were translated n Englsh by Malets [, 3]. The same prblem was studed by Ostwald [4], by On [5], by Stedman [6] and by Mrley [7, 8]. On the hstry f early studes n these tpcs, ne can refer t Tmshenk [9]. Blech [1] studed cmpressn members the crsssectnal mment f nerta f whch was varyng by a halfsne curve. On the ther hand, the sgnfcance f the ntal mperfectns was nted very early and studed manly expermentally by Marstn [11], by Jensen [1] and by Llly [13], the studes f whch were gathered by Salmn [14]. A sgnfcant study f cmpressn stepped clumns cnsstng by tw parts thrugh the use f the Galerkn methd has been presented by Dmtrf [15]. There s a relatvely large number f theretcal r expermental publcatns n tapered r stepped clumns wth r wthut mperfectns [16-31]. In the present paper, *Address crrespndence t ths authr at the Cvl Engng Dept., atnal Techncal Unversty f Athens, 1578 Greece; Tel: ; Fax: ; E-mal: raft@central.ntua.gr nn-unfrm steel members wth r wthut mperfectns (f any frm), laded by axal frces (cncentrcally r eccentrcally appled) and by cncentrated mments appled at ts ends r ntermedate pnts are studed. The steel members wth crss-sectn that may vary alng the length, can be tapered r stepped r they can be members cnsstng by tw unequal tapered parts. The mperfectns cnsdered may have any frm. The frmulatn presented n ths paper s based n slvng the gvernng equatn f the prblem thrugh the Galerkn methd usng the egenshapes f the member. A plastcty falure crtern s ntrduced fr stub r shrt members that wll never reach the elastc crtcal bucklng lad. Althugh n ths paper nly the smply supprted sngle-span beam-clumn s studed, t s bvus that the frmulatn presented may be extended t any type f frame members r frames usng the crrespndng egenshapes. The results are presented n the frm f dagrams ether fr the crtcal bucklng lads r fr the equlbrum paths, shwng the nfluence f the man member s characterstcs, as fr example the crss-sectnal varatn law, the ntermedate lads and bendng mments r the exstng mperfectns n the abve mentned bucklng lads and equlbrum paths. Althugh these dagrams are derved fr a smply supprted beam-clumn they can be readly emplyed fr the desgn f steel frames wth such members thrugh the use f the equvalent bucklng length factr cncept. GOVERIG EQUATIOS Let us cnsder the beam-clumn shwn n Fg. (1), where the crss-sectn vares alng the length (x-axs) accrdng t a knwn shape (parablc, tapered, stepped etc). The cnsdered member s laded by frces and mments as shwn n Fg. (1). Mrever, the member may have an ntal mperfectn w (x) /1 1 Bentham Open

2 The Open Cnstructn and Buldng Technlgy Jurnal, 1, Vlume 6 Knstantakpuls et al. where we have set: P = /μ (4b) Fg. (1). Frces and mments n a beam-clumn. In addtn, we assume that the member s laterally supprted and hence, t s prtected aganst the pssblty f bucklng abut the weak axs. The dfferental equatn fr bucklng f the beam-clumn s gven as fllws: [E I(x) w] + (w + w ) + + P (w + w )[1 H(x )] (1) m (x ) = where EI s the bendng rgdty f the member, w s the deflected shape, w s the ntal mperfectn, s the externally appled axal frce, P and m are cncentrated frces and mments appled at ntermedate pstns ver the length, H(x) s the Heavsde unt step functn and (x) the Drac delta functn. The abve equatn may be wrtten as fllws: EIw + EIw + E Iw + w + + P w[1 H(x )] = w () P w [1 H(x )]+ m (x ) n-exstence f mments and ntal mperfectn leads t a hmgeneus dfferental equatn (wthut secnd member), whch gves the bucklng crtcal lads, whle exstence f mments and ntal mperfectns leads t a cmplete dfferental equatn, whch allws us t study the equlbrum paths. We are searchng fr a slutn n the fllwng frm: w(x) = c 1 X 1 (x)+ c X (x)++ c n X n (x) (3) where X n (x) = sn(n x / ) are the egenshapes f the smply supprted beam and c n are cnstants t be determned. Intrducng eq(3) nt eq() we btan: EIc 4 sn x E Ic 3 cs x E I c sn x c sn x c μ sn x [1 H(x )] = w w [1 H(x )]+ m (x ) μ (4a) where μ are the rats f the lads P ver. BUCKLIG LOADS By settng w = and m =, eq(4a) becmes: EIc μ 4 sn x E Ic 3 cs x E I c sn x c sn x c sn x [1 H(x )] = Applyng the Galerkn prcedure n eq(5) and takng nt accunt the rthgnalty cndtn we btan: c 1 (A k1 k1 )+ c (A k k )++ (6a) + c n (A kn kn ) = wth k = 1tn where: A k = E E E 4 I(x) I(x) sn x 3 I(x) cs x sn x sn k x dx sn k x dx sn k x dx k = + μ (sn x ) [1 H(x )]dx ( = k) k = + μ sn x sn k x [1 H(x )]dx ( k) (5) (6b) In rder that the abve lnear hmgeneus system eq(6a) has nn-trval slutns the determnant f the ceffcents f the unknwns c must be equal t zer. Ths cndtn leads t the fllwng egenvalue prblem expressed by the fllwng equatn: A k k = (7) Equatn (7) gves the spectrum f the crtcal bucklng lads cr. EQUILIBRIUM PATHS Equatn (4) allws us t determne equlbrum paths. Applyng nce agan the Galerkn prcedure n eq(4) and takng nt accunt the rthgnalty cndtn we btan: c 1 (A k1 k1 )+ c (A k k )++ (8a) +c n (A kn kn ) = B k wth k = 1tn where A k and k are gven n eq(6b), whle B k s gven by the fllwng equatn: B k = w sn k x dx μ w sn k x dx k m cs k (8b)

3 Stablty f Steel Clumns wth n-unfrm Crss-Sectns The Open Cnstructn and Buldng Technlgy Jurnal, 1, Vlume 6 3 Frm the system f equatns (8a), we btan the unknwn cnstants c (), ( = 1tn) and frm eq(3) the equlbrum paths as fllws w(x) = c 1 ()sn x ++ c n x ()sn (9) n FAILURE CRITERIO It s bvus that there are cases (especally fr shrt r stub clumns), where the member wll never reach the elastc crtcal bucklng lad cr. Thus, t s necessary t ntrduce a plastcty crtern, whch can be expressed as fllws: A(x) + M(x) W(x) r f + A(x) I(x) z(x)m(x) f A(x) r f A(x) z(x) (x) M(x) = z(x) M(x) A(x) r, f [(x) / ] fnally f A(x) z(x) (x) M(x) (1) where z(x) s the dstance f the extreme fber f the member s crss-sectn at pstn x, and (x) s the slenderness f the member. IITIAL IMPERFECTIOS Intal mperfectns may appear n a member due t a bad stackng durng the transprtatn f steel members and, rather rarely, due t cnstructnal causes. In any case, ntal mperfectns have usually a frm resemblng t a parablc type that can be expressed as fllws: x w (x) = 4f + 4f x where f s the maxmum defrmatn at x = /. (11) CROSS-SECTIOAL TYPES The beam-clumn crss sectn may have ne f the frms shwn n Fg. (). In Fg. (a), ne can see a clumn the crss-sectn f whch changes parablcally. In Fg. (b) a tapered member s shwn, whle n Fg. (c) ne can see a member cmpsed by tw tapered parts wth dfferent length. Fnally, n Fg. (d), ne can see a stepped beamclumn. In the cases f Fg. (a,b and c), the crss-sectn at any pnt x has usually cnstant flanges (b t f ) but a web wth varable depth z. In all the abve cases, the fllwng relatn s vald: I(x)= 4 3 t w z3 (x)+ bt f z(x)+ t f (1) The Parablc Beam If the web depth z r the mment f nerta I(x) change parablcally, t wll be (see Fg. a and e): Fg. (). Steel members wth dfferent frms. z(x)= 4( f) x 4( f) x + (13a) I(x) = 4(I I f ) x 4(I I f ) x + I (13b) The Tapered Beam Fr the case f Fg. (b), and assumng that 1 =n, we wll have: z(x)= (n 1) x + (14) The Beam wth Tw Tapered Parts Fr the case f Fg. (c), and assumng that and = n 1 we wll have: 1 = n1 z(x) = (n 1 1) x 1 + fr x 1 (15a) z(x) = n 1 (n 1) x 1 + n 1 fr 1 x (15b) 1 Fr members cnsstng f tw tapered parts we nte that the dscntnuty at x= 1, affects the results btaned by usng sme cmmercal mathematcal manpulatrs. Fr the case f a beam cmpsed by tw parts wth lengths 1 and, and flanges AB and BC (Fg. c) that have the same nclnatn, the fllwng apprachng frmula that remves the afrementned dscntnuty s suggested: z(x)= wth n = 1 n 1 1 (x 1 ) n The Stepped Beam-Clumn In ths case (Fg. d) we have: 1 (15c) I (x) = I = cnst. fr x (16) =1 =1

4 4 The Open Cnstructn and Buldng Technlgy Jurnal, 1, Vlume 6 Knstantakpuls et al. Table 1. IPE Steel Prfles Prfle IPE IPE 4 IPE 6 z (m) b (m) t w (m) t f (m) A (m ) I 1-4 (m 4 ) Table. HEB Steel Prfles Prfle HEB HEB 4 HEB 6 z (m) b (m) t w (m) t f (m) A (m ) I 1-4 (m 4 ) Fg. (4). Crtcal lads fr parablc beams made frm (a) IPE and (b) HEB prfles wth lengths 15,, 5 and 3m. Fg. (3). Cnvergence study fr IPE, (a) L=3m and (b) L=1m. UMERICAL RESULTS In rder t study the bucklng behavr f steel members wth nn-unfrm crss sectn, we wll use members havng at x= the characterstc prpertes gven by the fllwng Tables 1 and whch refer t IPE and HEB standard prfles, respectvely, accrdng t Eurpean rms fr characterzatn f standard steel sectns. COVERGECE STUDY OF THE METHOD Studyng frst the cnvergence f the methd, we see frm the plts f Fg. (3) that the results arsng frm a slutn usng the frst three egenshapes and anther slutn usng the frst sx egenshapes cncde fully. We ntce nly Fg. (5). Crtcal lads fr tapered beams made frm (a) IPE4 and (b) HEB4 prfles wth lengths 15,, 5 and 3m. a slght msmatch less than.1% fr values f n 1. 7 and fr beam length L=1m. Ths very small dfference s prbably due t the numercal apprach f the prgram used. We see als that fr n=1 we recver the Euler crtcal bucklng lads P cr = EI/L crrespndng t smply supprted axally cmpressed clumns. DETERMIATIO OF THE CRITICAL LOAD Applyng the prpsed frmulae, we determne the crtcal lads cr versus n fr clumns wth dfferent lengths and crss-sectnal types. In the fllwng fgures, the plts f crtcal lads are shwn fr beams wth parablc frm made frm IPE and HEB standard prfles (Fg. 4), wth tapered frm made frm IPE4 and HEB4 prfles (Fg. 5), fr a beam

5 Stablty f Steel Clumns wth n-unfrm Crss-Sectns The Open Cnstructn and Buldng Technlgy Jurnal, 1, Vlume 6 5 Fg. (6). Crtcal lads fr beams wth tw tapered peces made frm (a) IPE4 and (b) HEB4 prfles wth lengths 15,, 5 and 3m. Fg. (8). Equlbrum paths fr a parablc beam made frm IPE wth (a) n=1.5 and (b) n=.. Fg. (7). Crtcal lads fr a stepped beam made frm IPE wth ttal length and 3m, wth (a) μ 1 =1, μ =1 and (b) μ 1 =5, μ =3. cmpsed by tw tapered parts (Fg. 6), and fr a stepped beam (Fg. 7) cmpsed by three equal length parts wth equal n =n, where the part wth the smaller crss-sectn s IPE and s laded by a frce at, by P 1 =μ 1 at 1 and by P =μ at. In Fg. (5a) t s μ 1 =, μ =, whle n Fg. (5b) t s μ 1 =5, μ =3. Frm Fgs (4 t 7), we can easly see that the crtcal lads cr are sgnfcantly hgher than the nes crrespndng t prsmatc members (unfrm crss-sectn) dependng n the taper rat n. As the taper rat n ncreases the crtcal lad als ncreases fr all types f crss-sectnal varatn. Fg. (9). Equlbrum paths f a tapered beam wth (a) n=1.5 and (b) n=. and ntal mperfectns f. Equlbrum Paths In the plts f Fg. (8), the equlbrum paths versus the axal lad are shwn fr a beam wth parablc frm, length m and n=1.5 (Fg. 8a) and n= (Fg. 8b) whch s laded by mments f 1, and 3 km. The Effect f Intal Imperfectns In the plts f Fg. (9), the equlbrum paths versus f a tapered beam wth length m and f=l/3, L/5, L/1 wth n=1.5 (Fg. 9a) r n= (Fg. 9b) are shwn.

6 6 The Open Cnstructn and Buldng Technlgy Jurnal, 1, Vlume 6 Knstantakpuls et al. Fg. (1). Equlbrum paths f a parablc beam laded eccentrcally wth (a) e=.5m and (b) e=1.m. Eccentrcally Appled Lads Fnally, the plts f Fg. (1) shw the equlbrum paths fr the specal case f a parablc beam wth length 1,, r 3m and n=, whch s laded by an axal frce actng eccentrcally at e=.5m (Fg. 1a) and e=1.m (Fg. 1b). Sme f the abve results have been verfed va the fnte element methd. Mre specfcally, a tapered beam wth lengths m and 3m made frm IPE prfle and taper rat n=1.5 has been mdeled and analyzed fr lnear bucklng usng the SAP- v11 cmmercal FE cde. The devatn between analytcal and FE results s.7% fr L=m and 1.3% fr L=3m (Fg. 5a). Mrever, a stepped beam wth lengths m and 3m made frm IPE prfle and taper rat n=1.5 wth μ 1 =μ =1 has als been mdeled and analyzed wth the same cde. In ths case, the devatn between analytcal and FE results s 1.64% fr L=m and 1.15% fr L=3m (Fg. 7a). SUMMARY AD COCLUSIOS In ths paper, a smple and effcent methd fr the study f nn-unfrm steel members wth r wthut mperfectns, laded by axal frces cncentrcally r eccentrcally appled and by mments at ts ends r at ntermedate pnts s presented. The gvernng equatn f the prblem ncludng all the abve parameters s slved by the Galerkn methd usng the egenshapes f the member. The varatn f the members crss-sectn may be whchever as well as the type f ntal mperfectns. The accuracy f the methd s prven excellent usng nly the three frst egenshapes. A plastcty crtern s appled n rder t predct materal falure due t bucklng defrmatn. The exstence f bendng mments r ntermedate frces reduces nt nly the crtcal bucklng lad, but als the lad arsng frm the falure crtern. The numercal results and the dagrams presented n ths study refer t a number f crss-sectns made frm IPE and HEB standard prfles that are cmmnly used n steel structures. The presented frmulae can be easly frmulated n a persnal cmputer usng ne f the avalable cmmercal prgrams (such as Maple, Mathematca, Matlab, etc). Althugh n ths wrk nly the smply supprted sngle-span beam-clumn s presented, t s bvus that the frmulatn can be extended t any type f frame members r frames usng the crrespndng egenshapes. The results are presented n the frm f dagrams ether fr the crtcal bucklng lads r fr the equlbrum paths, shwng the nfluence f the man member s characterstcs, as fr example the crss-sectnal varatn law, the ntermedate lads and bendng mments r the exstng mperfectns n the abve mentned bucklng lads and equlbrum paths. Althugh these dagrams are derved fr a smply supprted beam-clumn they can be readly emplyed fr the desgn f steel frames wth such members thrugh the use f the equvalent bucklng length factr cncept. COFLICT OF ITEREST ne declared. ACKOWLEDGEMET ne declared. REFERECES [1] S.P. Tmshenk, and J.M. Gere, Thery f Elastc Stablty, ew Yrk: Mc-Graw Hll, [] J. Malets, Trans. ASME, vl. 51, 195. [3] J. Malets, Trans. ASME, vl. 54, 193. [4] S. Ostwald, Klassker der exakten Wssenschaften, 175, Lepzg, 191. [5] W. On, Mem. Cll. Eng. Kyushu Imp. Unv. Fakuka, vl. 1, Japan, [6] B.E.W. Stedman, Engneerng, vl. 98, Lndn, [7] F. Mrley, Engneerng, vl. 97, Lndn, [8] F. Mrley, Engneerng, vl. 14, Lndn, [9] S.P. Tmshenk, Hstry f Strength f Materals, ew Yrk: Mc-Graw Hll, [1] F. Blech, Thery and Berechnung der esernen Brücken, Berln, 194. [11] A. Marstn, Trans. ASCE, vl. 39, [1] C. Jensen, Engneerng, vl. 85, Lndn, 198. [13] F. Llly, Engneerng, vl. 9, Lndn, [14] E.H. Salmn, Clumns, Lndn, 195. [15]. Dmtrv, Ermttlung knstanter Ersatzträghets mmente für Druckstäbe mt feränderlchen Querschntt, Baungeneur, vl. 1, p. 8, [16] J.W. Harrey, Bucklng lads fr stepped clumns, J. Struct. Dv. ASCE, vl. 9, n. ST, pp. 1-1, [17] C.K.Wang, Stablty f rgd frames wth nn-unfrm members, J. Struct. Dv. ASCE, vl. 93, n. ST1, pp , [18] D.J. Fraser, Desgn f tapered member prtal frames, J. Cnstr. Steel Res., vl. 3, n. 3, pp. -6, [19] T. Galambs, Ed., Gude t Stablty Crtera fr Metal Structures, 4 th ed. Readng: Wley, ew Yrk, [] S.Y. Lee, and Y.H. Ku, Elastc stablty f nn-unfrm clumns, J. Sund Vb., vl. 148,. 1, pp. 11-4, [1] H.R. Rnagh, and M.A. Bradfrd, Elastc dstrtnal bucklng f tapered I beams, Eng. Struct., vl. 16, n., pp , 1994.

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