NATURAL VIBRATION ANALYSIS OF CONTINUOUS HORIZONTALLY CURVED GIRDER BRIDGES USING DQEM *

Transcription

1 Th 4 th World Confrnc on Earthquak Engnrng Octobr -7, 008, Bng, Chna TURL VIBRTIO LYSIS OF COTIUOUS HORIZOTLLY CURVED GIRDER BRIDGES USIG DQEM * LI Hongng, WG Tong and WEI Shuangk 3 Profssor, Collg of Cvl Engnrng, anng Unvrsty of Tchnology, anng, Chna M. S. Canddat, Collg of Cvl Engnrng, anng Unvrsty of Tchnology, anng, Chna 3 Mastr, Mdas Inforaton Tchnology Co., Ltd., Guangzhou, Chna Eal: harbnr@63.co, woton0006@63.co, wshuangk@dasusr.co BSTRCT : Th dffrntal quadratur lnt thod (DQEM) s appld to coputaton of th gnvalus of sall apltud natural vbraton for contnuous horzontally curvd grdr brdgs wth rgd prs n ths papr. Th DQEM uss th dffrntal quadratur tchnqu to dscrtz th govrnng dffrntal gnvalu quatons dfnd on ach br, th transton condtons dfnd on th nnr-lnt boundary of two adacnt brs and th nd condtons of th contnuous curvd grdr. atural frquncs ar calculatd for a two-qual-span, contnuous, curvd, unfor grdr brdg, and ar copard wth xstng xact solutons by anothr thod. Fnally, paratrc rsults for th ffcts of scton gyraton radus and flxur-torson stffnss rato on fv dffrnt nd condtons, to th out-of-plan natural frquncs of two-qual-span, contnuous, curvd grdr brdgs, ar prsntd n dnsonlss for. KEYWORDS: natural vbraton analyss, contnuous curvd grdr brdg, dffrntal quadratur lnt thod. ITRODUCTIO Wth th rapd dvlopnt of vhcl traffc and cty constructon, th nubr of curvd grdr brdgs s rsng rapdly. Thr s hug dffrnc btwn th curvd grdr brdgs and noral rctlnar grdr brdgs n dynac charactrstcs, bcaus of spac curvatur of th brdgs. Th torsonal otons wll b producd n th suprstructurs of th curvd grdr brdgs accopand wth th bndng rsponss undr xctatons of arthquak ground oton horzontally or vrtcally. Owng to thr portanc n structural dsgn, th dynac bhavor of curvd grdrs has bn th subct of a larg nubr of nvstgatons. Th total xtnt of th work n ths fld of natural vbraton of curvd grdrs s now too grat to rvw n dtal, but part of thos paprs that st th prsnt work n contxt ar dscrbd. Shor and Chaudhur (97) studd th fr vbraton of horzontally curvd bas usng closd-for solutons of th quatons of oton. Snydr and Wlson (99) calculatd th fr vbraton frquncs of contnuous horzontally curvd bas usng a nonxplct closd-for soluton of th partal dffrntal quatons of oton. Kang, Brt and Strz (996) analyzd th natural vbraton charactrstcs of sngl-span, horzontally curvd bas wth warpng usng th dffrntal quadratur thod (DQM). Howson and Jah (999) obtand th xact out-of-plan frquncs of curvd Toshnko bas usng dynac stffnss atrx, and dscussd th ffcts of shar dflcton and rotary nrta. Th dffrntal quadratur lnt thod (DQEM) s dvlopd on th bass of DQM (Chn 005, 008). Ths thod absorbs advantags of both th tradtonal DQM and th fnt lnt thod (FEM). For th nnr-lnt boundary condtons, ncountrd n contnuous curvd grdrs, whch hav bn dffcult probls for th tradtonal DQM, th DQEM can solv th ffctvly. Th DQEM adopts th dffrntal quadratur to dscrtz th govrnng dffrntal quatons dfnd on ach lnt, th transton condtons dfnd on th nnr-lnt boundary of two adacnt lnts and th nd condtons of th curvd contnuous grdr. Thn unt all th dscrt quatons and gv soluton to th. * Supportd by atonal Rsarch Proct n Ssc Profsson Undr Grant o ; atural Scnc Foundaton of Jangsu Provnc Undr Grant o. BK0055.

2 Th 4 th World Confrnc on Earthquak Engnrng Octobr -7, 008, Bng, Chna Th obctv of ths papr s to provd an approach of th fr vbraton bhavors for th contnuous horzontally curvd grdr brdgs wth rgd prs. atural frquncs ar calculatd for a two-qual-span, contnuous, curvd, unfor grdr brdg, and th calculatd frquncs of th brdg ar copard wth th xstng xact soluton. Paratrc rsults of th two-qual-span, contnuous, curvd grdr brdgs ar also prsntd n dnsonlss for.. DIFFERETIL QUDRTURE Th da of dffrntal quadratur, ntroducd by Bllan and Cast (97), s to approxat th valus of th drvatv at ach grd pont as wghtd lnar sus of th functon valus at all saplng grd ponts wthn th doan undr consdraton,.. d dx y ( ) ( x) = y( x ) x= x = for =,,, (.) Equaton (.) rlats th th-ordr drvatv of th functon y(x) at a saplng grd pont x=x to th functon valu y =y(x ) at x=x, wth and dnotng th corrspondng wghtng coffcnt and nubr of dscrt () ponts wthn th doan, rspctvly. Th wghtng coffcnts can b dtrnd such that Equaton (.) s satsfd xactly for lnarly ndpndnt tst functons. Ths functons can b polynoals, trgonotrc functons, or spln functons. If th Lagrang polynoals ar chosn, th wghtng coffcnts for th frst-ordr drvatv can b calculatd by () = k= k, ( x x ) ( x x ) for (.) k k= k k= k () = k () k for = (.3) whr th doan of ndpndnt varabl x s usually changd nto [0,] by rgularzaton transforaton. Thn th wghtng coffcnts for th th-ordr drvatv can b dtrnd by ] ( ) () = [ (.4) Th accuracy of th quadratur solutons s dctatd by th choc of th locatons of th saplng grd ponts. Equal spacng s a natural and convnnt choc. Howvr, non-unfor grd ponts hav bn donstratd to nhanc th accuracy of th rsults. n unqual for of coon us can b xprssd as x cos[( ) π /( )] = (.5) whr th doan of x s [0,]. 3. GOVERIG DIFFERETIL EQUTIOS Ths papr dals wth th contnuous horzontally curvd grdr brdgs wth rgd prs, constant radus and unfor sctons consdrng concdnc of torson cntr and gravty cntr. Basd on ths assuptons, th

3 Th 4 th World Confrnc on Earthquak Engnrng Octobr -7, 008, Bng, Chna brdg could b tratd ust as a horzontally curvd grdr wth dcouplng of th out-of-plan and n-plan oton. For convnnc, only th out-of-plan oton s consdrd hr. Fgur Coordnat of th curvd grdr Consdrng th coordnat syst of th curvd grdr shown n Fgur, n whch R, φ, and O ar radus, angl fro th bgnnng scton to a gnrc scton, and gravty cntr of th gnrc scton, rspctvly; V, M, and T ar shar forc paralll to axs y, bndng ont about th radal axs x, and torsonal ont about th cntrodal axs z, rspctvly; and u, α, and θ ar vrtcal dsplacnt of th gravty cntr, bndng slop, and twst angl, rspctvly. Consdrng th ffct of shar dflctons, th thr nnr forcs V, M, and T can b wrttn n trs of u, α, and θ as kg dv EI dα GJ dθ V = + Rα, M = + θ, T = α (3.) R dφ R dφ R dφ whr k s th shar corrcton factor; E and G ar odulus of lastcty and shar odulus, rspctvly;, I, and J ar th ara, scond ont of ara about axs x, and Sant-Vnant torson constant, rspctvly. Lt ρ and ω dnot th ass dnsty and natural frquncy, rspctvly. Basd on th assupton that th cross-sctonal shap s constant and doubly sytrc (Vlasov 97), and for rfrnc n th squl, th dffrntal gnvalu quatons of th curvd grdrs consdrng shar dforaton ar xprssd n dnsonlss for as s v + RΦs α + Φ λ v = 0 (3.) ( μ + s ) α + RΦ( + μ) θ + RΦ γ λ α = 0 s v + Rα RΦ (3.3) ( + ) α + μθ θ + ηφ γ λ θ = 0 μ (3.4) whr ach pr ( =d/dξ) dnots on dffrntaton wth rspct to th dnsonlss coordnatξ = φ / Φ ; Φ =opnng angl of th br; s =kgr /(EI); γ =I/(R ); μ=gj/(ei); η=i d /d; λ =ρr 4 ω /(EI); and I d = polar scond ont of ara of th cross scton. Th followng boundary condtons ar takn for sply supportd nds: no vrtcal dflcton, no bndng ont, and no torsonal rotaton; and for fr nds: no shar forc, no torsonal ont and no bndng ont. For clapd nds, v, α, and θ qual zro. Th boundary condtons for sply supportd, fr, and clapd nds ar, n dnsonlss for rspctvly v = α' = θ = 0 (3.5) v ' + RΦα = α' +Φθ = θ ' Φα = 0 (3.6)

4 Th 4 th World Confrnc on Earthquak Engnrng Octobr -7, 008, Bng, Chna v = α = θ = 0 (3.7) 4. DQEM FORMULTIO OF COTIUOUS CURVED GIRDERS For splcty of statnt, ths scton taks an n-span contnuous curvd grdr, wth all supports sply supportd, to llustrat th analyss procss of th out-of-plan natural vbraton of contnuous curvd grdr brdgs wth rgd prs usng th DQEM. Takng ach span of th grdr as an ndpndnt lnt, th nubr of all th lnts s n. Lt Φ dnot cntral angl th th lnt. Lt φ b th coordnat varabl of th local coordnat syst wth th orgn locatd at nod of th lnt, and φ [0, Φ ]. Lt ξ qualφ / Φ, so th rang of ξ s [0,]. Thn w hav k k d y d y = (4.) k k k d( φ ) ( Φ ) dξ Introducng Equatons (.) and (4.) to Equatons (3.)-(3.4), on ay hav s = () v + R = s = () () v + RΦ s = () α + ( Φ ) λ v = 0 (4.) () ( μ + s ) α + RΦ ( + μ) θ + R( Φ ) γ λ α = 0 α R( Φ ) (4.3) () () ( + ) α + μ θ ( Φ ) θ + η( Φ ) γ λ θ = 0 = μ (4.4) for =,..., -, whr dnots th nubr of th ponts on th th lnt. = Usng Equaton (.), th condtons of th two nds of th contnuous curvd grdr can b xprssd as whr = whl r=, and =n whl r= n. = () v = 0, α = 0, θ = 0 (4.5) r = r Usng Equaton (.), th transton condtons of th nnr-lnt boundary of two adacnt brs and + can b wrttn as r v = 0, v 0 =, θ = 0, θ 0 =, α = α + () + (), α = α, +, Φ = Φ = (4.6) ftr untng Equatons (4.)-(4.6) and gvng soluton to th, on can obtan th valus of natural frquncy paratr λ. 5. EXMPLE D COMPRISO n xapl of a two-qual-span contnuous curvd grdr s prsntd to llustrat th applcaton of th DQEM. s shown n Fgur, all of th thr supports ar spl. Th gotrcal and physcal condtons of th two spans,.. B and BC, ar dntcal, and th opnng angl of ach span s π/, th rotary nrta γ=/3.39, shar corrcton factor k=0.83, and Posson s rato =0.3. Thn th natural frquncy paratr λ s calculatd usng th

5 Th 4 th World Confrnc on Earthquak Engnrng Octobr -7, 008, Bng, Chna DQEM forulaton prsntd n Scton 4. Th rsults ar copard wth th analytcal solutons as shown n Tabl. For splcty of analyss, th nubr of saplng grd ponts of ach br s chosn th sa. Fgur two-qual-span contnuous curvd grdr Tabl Coparson btwn th DQEM and xact solutons atural Frquncy paratr λ frquncy Exact nubr soluton =5 =9 =3 =7 =0 = Fro Tabl, t can b sn that thr s a good agrnt btwn th nurcal and analytcal rsults, and th accuracy of th nurcal soluton and th nubr of frquncs rsolvd ncrass wth ncrasng. For gnral ngnrng probls, 3 ponts on ach lnt wll b nough to satsfy th rqurd prcson, so th DQEM s of hghr ffcncy copard wth th FEM for slar probls. 6. PRMETRIC LYSIS Th odl for th prsnt paratrc analyss s alost th sa as th on n Scton 5 xcpt that th condtons of th two nds can chang, that s, th ddl support of th ba s sply supportd for vr. Th ffcts of scton gyraton radus and flxur-torson stffnss rato on fv dffrnt nds condtons ar prsntd n Fgur 3-7, n dnsonlss for. In ths analyss, th saplng ponts ar unqually spacd by Equaton (.5), and th nubr of saplng ponts on ach br s 0. Fro Fgur 3-7, t can b sn that th fr vbraton frquncs of th grdr bco sall wth rducng th rstrants of two nds; th ffct of th flxur-torson stffnss rato can b nglctd whl μ>0; th ffct of scton gyraton radus s sall whl γ<0. f th rstrants of two nds ar lss, such as th cas of havng on sply supportd and on fr nd, but th rang rducs wth ncrasng rstrants of two nds.

6 Th 4 th World Confrnc on Earthquak Engnrng Octobr -7, 008, Bng, Chna Fg. 3 Paratrc rsults wth two clapd nds Fg. 4 Paratrc rsults wth on clapd and on sply supportd nd Fg. 5 Paratrc rsults wth two sply supportd nds Fg. 6 Paratrc rsults wth on clad and on fr nd Fg. 7 Paratrc rsults wth on sply supportd and on fr nd

7 Th 4 th World Confrnc on Earthquak Engnrng Octobr -7, 008, Bng, Chna 7. COCLUSIOS Basd on th assupton of rgd prs, th DQEM s usd to calculat th out-of-plan natural vbraton frquncs of contnuous horzontally curvd grdr brdgs consdrng th ffct of shar dforaton. atural frquncs of a two-qual-span contnuous curvd grdr ar prsntd and copard wth xstng xact rsults. Coparson donstrats hgh ffcncy of th DQEM. Fnally, th ffcts of scton gyraton radus and flxur-torson stffnss rato on dffrnt nd condtons, to th out-of-plan natural vbraton of two-qual-span contnuous curvd grdrs ar analyzd usng th DQEM. Rsults ndcats that th fr vbraton frquncs of th grdr bcos sall wth rducng th rstrants of two nds; th ffct of flxur-torson stffnss rato can b nglctd whl μ>0; th ffct of scton gyraton radus s sall whl γ<0. f th rstrants of two nds was lss, but th rang rducs wth ncrasng th rstrants of two nds. REFERECES Bllan, R. E. and Cast, J. (97). Dffrntal quadratur and long-tr ntgraton. Journal of Mathatcal nalyss and pplcatons 34, Chn, C.. (005). DQEM analyss of n-plan vbraton of curvd ba structurs. dvancs n Engnrng Softwar 36:6, Chn, C.. (008). DQEM analyss of out-of-plan vbraton of nonprsatc curvd ba structurs consdrng th ffct of shar dforaton. dvancs n Engnrng Softwar 39:6, Howson, W. P. and Jah,. K. (999). Exact out-of-plan natural frquncs of curvd Toshnko bas. Journal of Engnrng Mchancs 5:, 9-4. Kang, K., Brt, C. W. and Strz,. G. (996). Vbraton analyss of horzontally curvd bas wth warpng usng DQM. Journal of Structural Engnrng :6, Shor, S. and Chaudhur, S. (97). Fr vbraton of horzontally curvd bas. Journal of th Structural Dvson 98:3, Snydr, J. M. and Wlson, J. F. (99). Fr vbraton of contnuous horzontally curvd bas. Journal of Sound and Vbraton 57:, Vlasov, V. Z. (96). Thn-Walld Elastc Bas, atonal Scnc Foundaton, Washngton, D.C., US Wang, X. W. (995). pplcaton of dffrntal quadratur thod n structural chancs. dvancs n Mchancs 5:, 3-40.