A NONLINEAR ANALYSIS METHOD OF STEEL FRAMES USING ELEMENT WITH INTERNAL PLASTIC HINGE

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1 Advnced Steel Constructon Vol. 4, o. 4, (2008) 341 A OLIEAR AALYSIS MEHOD OF SEEL FRAMES USIG ELEME WIH IERAL PLASIC HIGE Yu-Shu Lu 1,* nd Guo-Qng L 2 1 Lecturer, School of Cvl Engneerng, ongj Unversty, Shngh, Chn *(Corresondng uthor: E-ml: yslu@ tongj.edu.cn) 2 Professor, School of Cvl Engneerng, ongj Unversty, Shngh, Chn Receved: 18 October 2007; Revsed: 11 December 2007; Acceted: 12 December 2007 ABSRAC: A nonlner nlyss method of steel frmes usng element wth nternl lstc hnge s roosed. hs method cn nlyze the frme member led wth lterlly-dstrbuted lods only usng one element even tht lstc hnge ers wthn the member. By dvdng the member nto two segments t the locton of the mxmum moment, the ncrementl stffness mtrx of the two segments from tme t to t+ dt re derved, then the bem element stffness equton wth nternl lstc hnge fter the sttc condenston cn be obtned. Wht s more, ths method lso consders the nfluences of some geometrcl nd mterl nonlner fctors ncludng second-order effect of xl forces, sher deformton, cross-sectonl lstfcton, resdul stress nd ntl merfecton. hs method not only overcomes the tme-consumng dsdvntges of lstc zone method of frme members becuse of the fne mesh dscretzton but lso mkes u for the roblem of the trdtonl lstc hnge element tht lstc hnges must form t the elementl ends. Anlyss results show tht the roosed method s stsfctory. Keywords: Steel frmes; nonlner nlyss; nternl lstc hnge; lstc zone method; cross-secton lstfcton 1. IRODUCIO he members of steel frme my be subjected to lterlly-dstrbuted lods, so lstc hnges wll be formed wthn the members. he common method [1,2,3,4,5] to tret ths cse for nlyss of the frme s to rrnge node t the locton of the lstc hnge wthn the member to dvde the orgnl one element nto two or more elements reresentng the member. hs wll ncrese the number of nodes nd degrees of freedom for nlyss of the frme. Moreover, the trdtonl element wth lstc hnge formed t the end(s) must fx the loctons of nodes n dvnce, whch cn not sut the cse tht the loctons of the lstc hnge wthn the member wth lterlly-dstrbuted lods my vry durng the lodng rocess. In ths er, n roch for nonlner nlyss of steel frmes usng element wth nternl lstc hnge s roosed. hs roch cn use one element to smulte one member n frme even lstc hnge my form wthn the member. 2. OLIEAR AALYSIS MEHOD OF SEEL FRAMES 2.1 Plstc Zone Method nd Plstc Hnge Method Over the st 20 yers, scholrs hs develoed nd vldted vrous methods of erformng second-order nelstc nlyses on steel frmes [1,2,3,4,5]. Most of these studes my be ctegorzed nto one of two tyes: (1) lstc zone method; or (2) lstc hnge method bsed on the degree of refnement used to reresent yeldng. he lstc zone method uses the hghest refnement whle the elsto lstc hnge method llows for sgnfcnt smlfcton. he lod deformton chrcterstcs of the lstc nlyss methods re llustrted n Fgure 1. In the lstc zone method[1,2,3,4,5], frme member s dscretzed nto fnte elements, nd the cross-secton of ech fnte element s subdvded nto mny fbers. he deflecton t ech dvson long the member s obtned by numercl ntegrton. he ncrementl lod deflecton resonse t

2 342 A onlner Anlyss Method of Steel Frmes Usng Element wth Internl Plstc Hnge ech lod ste, wth udted geometry, ctures the second-order effects. he resdul stress n ech fber s ssumed constnt snce the fbers re suffcently smll. he stress stte of ech fber cn be exlctly determned, nd the grdul sred of yeldng trced. A lstc zone nlyss elmntes the need for serte member ccty checks snce second-order effects, the sred of lstcty, nd resdul stresses re ccounted for drectly. As result, lstc zone soluton s consdered exct. Although the lstc zone soluton my be consdered exct, t s not lcble to dly use n engneerng desgn, becuse t s too comuttonlly ntensve nd too costly. Its lctons re lmted to: (1) the study of detled structurl behvor; (2) verfyng the ccurcy of smlfed methods; (3) rovdng comrsons for exermentl results; (4) dervng desgn methods or genertng chrts for rctcl use; nd (5) lcton to secl desgn roblems. A more smle nd effcent wy to reresent nelstcty n frmes s the elsto lstc hnge method [1,2,3,4,5]. Here the element remns elstc excet t ts ends where zero-length lstc hnges form. hs method ccounts for nelstcty but not the sred of yeldng through the secton or between the hnges. he effect of resdul stresses between hnges s not ccounted for ether. he elsto lstc hnge methods my be frst- or second-order. In frst-order lstc nlyss, nonlner geometrcl effects re consdered neglgble, nd not ncluded n the formulton of the equlbrum equtons. As result, ths method redcts the sme ultmte lod s conventonl rgd lstc nlyss would. In second-order lstc nlyss, the effect of the deformed she s consdered. he smlest wy to model the geometrcl nonlnertes s to use stblty functons. hese use only one bem-column element to defne the second-order effect of n ndvdul member. Stblty functons re n effcent nd economcl method of erformng frme nlyss. It hs dstnct dvntge over the lstc zone method for slender members (whose domnnt mode of flure s elstc nstblty) s t comres well wth lstc zone solutons. However, for stocky members (whch sustn sgnfcnt yeldng), the smle elsto lstc hnge method over-redcts the ccty of members s t neglects to consder the grdul reducton of stffness s yeldng rogresses through nd long the member. Consequently, modfctons must be mde before ths method cn be roosed for wde rnge of frmed structures. Frst-Order Elsto-Plstc Second-Order Elsto-Plstc Fgure1. Lod Deformton Chrcterstcs of Plstc Anlyss Methods

3 Yu-Shu Lu nd Guo-Qng L Refned Plstc Hnge Method A refned lstc hnge nlyss ncorortes consderton of geometrcl nd mterl nonlner fctors ncludng second-order effect of xl forces, sher deformton, cross-sectonl lstfcton, resdul stress nd ntl merfecton to the nlyss of steel frmes. he concet s outlned n the followng secton Column element L nd Shen [6] resented n mroved lstc hnge model, whch consdered the cross-secton lstfcton. Usng ths model the elsto-lstc ncrementl stffness equton of the frme column element s gven by [ k ]{ δ} { f} Δ = Δ (1) Where, { Δδ } nd { Δ f } refer to the ncrementl nodl dslcements nd forces, resectvely, [ k ] refers to the elsto-lstc stffness mtrx, nd tkes the followng form [ k ] [ k ] [ k ][ G][ E][ L][ E] [ G] [ k ] where [ L] = (2) 1 e e = [ E] [ G] ([ k ] + [ k e n e ])[ G][ E] [ k ] = dg[ α k, α k, α k, α k, α k, α k ] [ E] n 1 e11 1 e22 1 e33 2 e44 2 e55 2 e = x x x x G = dg, 0,,, 0, 1 M1 2 M2 [ ] [ e ] effect nd sherng deformton[6]. In mtrx [ ] k reresents the elstc stffness mtrx of the bem element ccountng for the second order G, x ( = 1, 2 ) denotes the ultmte yeld surfce functon of the secton. In ths er, the ultmte yeld surfce functon for I tye secton gven by reference [3] nd [7] s used here nd cn be wrtten s 1.3 M x = + = 1 (3) y M In mtrx [ k n ], α ( = 1, 2 ) denotes the elsto-lstc hnge rmeter of the two end sectons, reresents the lstfcton extent of the two end sectons nd cn be exressed s r α = (4) 1 r

4 344 A onlner Anlyss Method of Steel Frmes Usng Element wth Internl Plstc Hnge Where, ( = 1,2) s the restorng force rmeter of the two end sectons nd tkes the form s r r 1 M M = 1 M M β s s ( 1 β ) M M M s M M M M s M, M s nd M reresent the moment of the secton, the ntl yeld moment the ultmte yeld moment under the xl force, resectvely. β reresents the mterl strn hrdenng coeffcent, for norml low crbon steel nd low lloyed steel, β cn tke 0.01~0.02, nd M cn be gven by equton (3). he ntl yeld surfce equton [3,7] wthout ccountng for the nfluences of resdul stress s exressed s γm + y M = 1.0 (5) nd M s = ( 1.0 ) M / γ (6) y he ntl yeld surfce equton [3,7] ccountng for the nfluences of resdul stress s exressed s 0.8 γm y M = 1.0 (7) nd M = 0.9(1.0 ) M / γ (8) 0.8 y where, γ s the lstfcton coeffcent of the secton Bem element wth nternl lstc hnge Fgure 2 shows the bem element wth nternl lstc hnge. Referrng to Fgure 2, n nternl node C between elementl ends s nserted so tht the element s dvded nto two segments, the lengths of whch re L nd L b resectvely. Assume the mxmum bendng moment, M 1, t tme t s t oston C nd the mxmum bendng moment, M 3, t tme t+ dt s t oston C (see Fgure 3). For dervton of ncrementl stffness mtrx of the element durng t t+ dt, vrtul stte of moment, M 2, s conceved, whch s the bendng moment t the sme oston of M 3 t the tme t. he ncrementl stffness reltonsh of ech segment of the element cn be exressed s the stndrd form s

5 Yu-Shu Lu nd Guo-Qng L 345 M 1 q(x) M 2 Q 1 C Δδ θ1 2 θ1c θ2c θ Q 2 L L b L Fgure2. Bem Element wth Internl Plstc Hnge C M 2 C M 1 tme t C tme t+ dt M 3 Fgure3. Poston of Mxmum Moment of Dfferent Lod Ste for the segment of L dq1 dδ dδ1 dm 1 dθ dθ 1 = K dq = 1c dδ1c dδ1c dm dθ dθ 1c 1c 44 1c (9) nd for the segment of L b dq2 c dδ2c b11 b12 b13 b14 dδ 2c dm 2c dθ 2c b22 b23 b 24 dθ 2c = K b dq = 2 dδ2 b33 b 34 dδ2 dm 2 dθ b 2 44 dθ 2 (9b) where K nd K b [6] re the elsto-lstc stffness mtrces for the segments of L nd L b of the element resectvely, j nd b j (, j = 1,2,3, 4 ) re the corresondng elements n such mtrces. It cn be seen from Fgure 2 tht the two segments of the elements shre the sme deformton comonents t ther juncton, nmely dδ1 c = dδ2c = dδc nd dθ1 c = dθ2c = dθc. Combnng Eq. 9 nd Eq. 9b, one hs

6 346 A onlner Anlyss Method of Steel Frmes Usng Element wth Internl Plstc Hnge dq dδ1 dm d 23 θ 24 1 dq 2 b33 b34 b13 b 23 dδ 2 = dm 2 b44 b14 b24 dθ2 dq1 c + dq 2c 33 + b b 12 dδ c dm + dm + b dθ 1c 2c c (10) For the urose of sttc condenston to elmnte the degree of freedom of the dslcements of nternl node, bove stffness mtrx s rttoned nto nternl nd externl degrees of freedom s dfe kee ke dδe = df ke k dδ (11) where { df e } nd { df } re the elementl end nd nternl force vectors resectvely, { dδ e } nd { dδ } re elementl end nd nternl deformton vectors resectvely. her exressons re s follows { dfe} = [ dq1, dm1, dq2, dm2],{ dδe} [ dδ1 dθ1 dδ2 dθ2] { df } = [ dq + dq, dm + dm ], { dδ } = [ dδ, dθ ], k 1c 2c 1c 2c ee = 0 0 b33 b , k b b =,,,, c c e =, k b13 b b b22 b b b + b = (12) Snce no externl forces re led t nternl node C, nmely { df } = { 0}, { d } be exressed wth { dδ e } 1 ( kee kek ke ){ dδe} { dfe} δ n Eq. (11) cn. he stffness equton condensed off nternl dslcement vector s s = (13) In bove dervton, t s ssumed tht the nternl lstc hnge occurs t oston of C t tme t, nd the moment ncreses from M 2 t t to M 3 t t+ dt. But ctully n the durton t t+ dt, the moment chnge should hve been from M 1 t oston of C to M 3 t oston of C. A stffness 1 1 mtrx modfcton ( kee kek k e - C, ' t kee kek k e ) my be suermosed to C, t roxmtely tke the effect from oston chnge of nternl lstc hnge nto ccount. he subscrts n the stffness mtrx modfcton ndcte the oston nd the tme of mxmum bendng moment. Assume the nternl lstc hnge occurs t the oston of mxmum bendng moment between two ends. he oston of the mxmum bendng moment between the two ends of the element, oston C, vres n lodng rocess. Hence, the rtonl wy to trce the nternl lstc hnge s to clculte the oston of the mxmum bendng moment t ech lodng ste fter elementl yeldng. wo common nternl lodng tterns for bem elements re concentrted lod nd unformly dstrbuted lod, s shown n Fgure 4.

7 Yu-Shu Lu nd Guo-Qng L 347 P q M 1 M 2 Q 1 Q 2 b L ( ) concentrted lod M 1 M 2 Q 1 Q 2 x L (b) unformly dstrbuted lod Fgure 4. Lod Ptterns wthn Bem Sn If one concentrted lod s led wthn the bem sn (see Fgure 4), the oston of the mxmum moment wthn sn s certnly the lodng oston. But, f unformly dstrbuted lod s led(see Fgure 4b), the oston of the mxmum moment wthn sn s chngeble. he condton of the mxmum moment wthn the bem sn s tht d M( x) = 0 or Q ( x) = 0 (14) dx he sher force t end 1 cn be exressed s Q M 1 M 2 1 = ql (15) L And lettng the sher force be equl to zero yelds the oston of the mxmum moment desred M 1 M 2 1 x = + L (16) ql 2 As for the bem element wth both concentrted lod nd unformly dstrbuted lod wthn sn, one cn dvde ths element nto two segments t the oston where the concentrted lod led. he mxmum moment oston of ech segment cn be determned ccordng the method for the unformly dstrbuted lod cse s bove-mentoned. Wth comrson of the mxmum moments of two segments of the element nduced by the unformly dstrbuted lod nd the bendng moment where the concentrted lod led, the rel mxmum moment of ths bem element cn be obtned wth the mxmum of the bove three moments Resdul stress When the rto of xl force to sqush lod s lrge for member n comresson, resdul stresses cn nfluence the lstcty dstrbuton long element length. A trnsent elstc modulus concet, nmely the concet of tngent modulus, s roosed to tke ths effect nto ccount. he CRC column strength equtons [4,5] cn be emloyed n dervng the tngent modulus. he rto of the tngent modulus to the elstc modulus E (see Fgure 5) s roosed to be E t E t =1.0 P 0. 5P E (17) E t 4P P = 1 when P > 0. 5P E P y y P y (17b)

8 348 A onlner Anlyss Method of Steel Frmes Usng Element wth Internl Plstc Hnge Intl merfectons Fgure5. CRC ngent Modulus here re three methods to ccount for the nfluences of ntl merfectons n the nelstc nlyss of steel structures, they re drect modelng, equvlent nomnl lod method nd reduced CRC tngent modulus method [2]. Among ll the methods, the thrd method s the most convenent nd drect method whch consders the stffness reducton cused by ntl merfectons of the members by multlyng reducton coeffcent. Some reserch llustrted tht the resonble results cn be obtned by tkng the reducton coeffcent s In ths er, the reduced CRC tngent modulus method s used. 3. UMERICAL EXAMPLES he structure exmned s four-story frme wth md-sn concentrted lods s shown n Fgure 6. ble 1 gves the frme member sze. he mterl elstc modulus E of steel s 206k/mm 2, the sectons of ll the bems re W16 40, the sectons of the frst storey columns re W12 79 nd the sectons of the other columns re W he horzontl dslcement versus lod fctor curves both obtned by nlyss wth the elements wth nternl hnge roosed n ths er nd wth the norml elements through dvdng the frme bem nto two elements [8] re shown n Fgure 7. he ultmte lod fctor λ obtned wth the roosed elements s 1.03 whle tht wth norml elements [8] s he sequence of lstc hnges formed n the frme s llustrted n Fgure 8. /2 /2 /2 A Δ H=14.64m = = k 9.15m Fgure 6. Four-story Frme wth Concentrted Lods t Bems

9 Yu-Shu Lu nd Guo-Qng L 349 ble 1. Dmensons of All the Comonents of Four-storey Steel Frme Secton H(mm) B(mm) t w (mm) t f (mm) A(mm 2 ) I( 10 6 mm 4 ) W W W Lod fctor 荷载因子 文献 Results 3.66of reference [8] 系列 Results 3 of roosed method A 点水平位移 (mm) Horzontl dslcement of node A(mm) Fgure 7. Lod-dslcement Curve of Four-storey Steel Frme Fgure 8. Aerng Sequence of Plstc Hnges of Four-storey Steel Frme

10 350 A onlner Anlyss Method of Steel Frmes Usng Element wth Internl Plstc Hnge Vogel sx-story frme [9] usully ers n benchmrk study of lnr steel frmes. he frme sze nd lod nformton re llustrted n Fgure 9 nd the frme member szes re lsted n ble 2. he mterl elstc modulus E of steel s 206k/mm 2 nd the yeld strength f y s 235/mm 2. he horzontl dslcement of rght-uer corner (ode A) versus lod fctor curve by the elsto-lstc hnge model resented n ths er s comred wth the results n reference [9] wth lstc zone method n Fgure 10. he ultmte lod fctor λ obtned by the method roosed s 1.15 whle tht by reference [9] s he xl force nd moment dgrms n the ultmte stte re shown n Fgure 11, where the fnl lstc hnge dstrbuton s dotted n the moment dgrm. H 2 =10.23k H 1 =20.44k H 1 H 1 H 1 H 1 q 2 =31.7k/m IPE240 q 1 =49.1k/m IPE300 HEB200 HEB200 q 2 q 1 q 1 q 1 HEB240 IPE300 q 1 q 1 IPE330 HEB240 q 1 q 1 HEB260 IPE360 q 1 q 1 IPE400 HEB260 HEB160 HEB220 HEB160 HEB220 HEB220 HEB220 A m 2 6.0m Fgure 9. Vogel Sx-storey Steel Frme ble2. Dmensons of ll the Comonents of Vogel Sx-storey Steel Frme Secton H(mm) B(mm) t w (mm) t f (mm) A(mm 2 ) I( 10 6 mm 4 ) S( 10 3 mm 3 ) HEA , HEB , HEB , HEB , HEB , HEB , HEB , IPE , IPE , IPE , IPE , IPE ,

11 Yu-Shu Lu nd Guo-Qng L 351 Lod fctor 荷载因子 文献 Results 3.57 of 计算结果 reference [9] clfrme6 Results of roosed method Horzontl A 点水平位移 dslcement (mm) of node A(mm) Fgure10. Lod-dslcement Curve of Vogel Sx-storey Steel Frme () xl forces (k) (b) moments (k m) Fgure 11. Internl Forces of Vogel sx-storey Steel Frme 4. COCLUSIO An roch for nonlner nlyss of steel frmes usng element wth nternl lstc hnge s roosed n ths er. hs roch cn use just one element to smulte one member n frme even lstc hnge my form wthn the member when subjected to lterlly-dstrbuted lods. Wht s more, ths roch lso consders the nfluences of some geometrcl nd mterl nonlner fctors ncludng second-order effect of xl forces, sher deformton, cross-secton lstfcton, resdul stress, ntl geometrcl merfecton. he numercl results show tht the roosed roch s effcent nd stsfctorly ccurte, nd s sutble for the nonlner nlyss of steel frmes.

12 352 A onlner Anlyss Method of Steel Frmes Usng Element wth Internl Plstc Hnge REFERECES [1] Km, Seung-Eock, Km, Moon-Kyum nd Chen, W-Fh, Imroved Refned Plstc Hnge Anlyss Accountng for Strn Reversl, Engneerng Structures, 2000, Vol. 22, [2] Km, S.E. nd Chen, W.F., Prctcl Advnced Anlyss for Unbrced Steel Frmes Desgn, Journl of Structurl Engneerng, ASCE, 1996, Vol. 122, o. 11, [3] Kng, W.S., Whte, D.W. nd Chen, W.F., A Modfed Plstc Hnge Method for Second-order Inelstc Anlyss of Rgd Frmes. Structurl Engneerng Revew, 1992, Vol. 4, o. 1, [4] Lew, J.Y.R., Second-order Refned Plstc Hnge Anlyss of Frme Desgn Prt 1, Journl of Structurl Engneerng, 1993, Vol. 119, [5] Lew, J.Y.R., Second-order Refned Plstc Hnge Anlyss of Frme Desgn Prt 2, Journl of Structurl Engneerng, 1993, Vol. 119, [6] L, Guo-qng, Shen, Zu-yn. Elstc nd Elsto-lstc Anlyss nd Comuttonl heory for Steel Frme Systems, Shngh, Shngh Scence nd echnology Press, (n Chnese) [7] Dun, L. nd Chen, W.F., Desgn Intercton Equton for Steel Bem-columns. Journl of Structurl Engneerng, 1989, Vol. 115, o. 5, [8] Chen, W.F. Anlyss nd Desgn of Bem-column, Volume (I), Plnr Problem Chrcterstc nd Desgn. Bejng, Chn Communctons Press, (n Chnese) [9] Chn, S.L. nd Chu, P.P.., on-lner Sttc nd Cyclc Anlyss of Steel Frmes wth Sem-rgd Connectons, Oxford, Elesver, 2000.