LARGE DISPLACEMENT ANALYSIS OF SLENDER ARCHES

Transcription

1 Arch Brdg ARCH P. Roca and E. Oñat Ed CIMNE, Barclona, LARGE DISPLACEMEN ANALSIS OF SLENDER ARCHES M. Arc * and M. F. Granata * * Unvrtà dgl Std d Palrmo Dpartmnto d Inggnra Strttral Gotcnca Val dll Scnz - Palrmo -mal: arc@dg.npa.t, granata@dg.npa.t Ky word: ranfr matrc, arch, nonlnar bhavor. Abtract. A procdr for th analy of th lnar and nonlnar bhavor of arch trctr ng th tranfr matrx mthod MM hr prntd. Introdcd for tatc and dynamc lnar analy, th mthod can b xtndd to gomtrc nonlnar formlaton, wth larg dplacmnt and mall dformaton. Hnc t pobl to prform nonlnar analy of lndr arch trctr that follow th lnar tr-tran law bt prnt larg dplacmnt. h procdr can b fl whn t ncary to valat th trctral bhavor f th magntd of dplacmnt wth rpct to th trctral lmnt dmnon too hgh to nglct th nflnc of cond ordr ffct. In th papr th tranfr matrx obtand by th ntgraton of a Hamltonan ytm n whch th canoncal form of qlbrm and compatblty qaton drvd from th prncpl of tatonary total potntal nrgy.

2 M. Arc and M. F. Granata. INRODUCION Modrn arch trctr, pcally on n tl or almnm havng cton wth hgh lndrn, or cabl n tnl trctr, ar too dformabl to b tratd wth th frt-ordr latc thory, o that thr analy ha to b dvlopd n th larg dplacmnt fld. h poblty of practcal tatc and dynamc non-lnar analy of trctr drng th lat fw yar ha gratly progrd, d to th ntnv of dgtal comptr opratng on fnt lmnt modl. h dfnton of non-lnar gomtrc and matral thr-dmnonal lmnt can b vry fl for allowng gnral non-lnar analy. From th pont of vw th ntrodcton of 3D bam lmnt and 3D crvd bar lmnt can b vry mportant for applcaton to non-lnar analy of arch brdg or to crvd mmbr of brdg trctr. h tranfr matrx mthod, ntrodcd n th fld of dynamc analy and d n tatc lnar latc analy, ha bn xtndd to th fld of larg dplacmnt analy 3,. Hr a vratl procdr prntd for larg dplacmnt analy drvd from a prvo on dvlopd ntally a a mxd mthod by ng tranfr matrc n th fld of lnar analy wth mall dplacmnt. h procdr lad to a non-lnar ytm of qaton of ordr x whch do not dpnd on th nmbr of lmnt. h olton of th ytm obtand by man of an tratv modfd Nwton mthod. In th papr th procdr appld to D trctr plan arch loadd n th plan.. GOVERNING EUAIONS OF ELASOSAIC PROBLEM Wth rfrnc to a on-dmnonal pac crvd bar total lngth L, havng a mall ntal crvatr and contng of homogno otropc matral that dplay lnarly latc bhavor, lt b th crvlnar coordnat jonng th cntrod of th cro-cton, wth rpct to a fxd orgn =. Condr a local coordnat ytm formd by th nt r r r vctor n, b, t 3, normal, bnormal and tangnt to th bar ax. W am that th cntrod concd wth th har cntr of th cro-cton, that th prncpal ax of nrta ar r and r, and that th warpng of th cro-cton d to toron nglctd. h confgraton of th dformd bar flly dcrbd by th gnralzd dplacmnt vctor havng componnt, th frt thr collctd n th bvctor dplacmnt v = v v v,, 3 and th lat thr collctd n th bvctor ϕ =ϕ,ϕ,ϕ 3. h bar bjctd to appld dtrbtd load p = p, p, p and momnt 3 m = m, m, m that can b collctd n th gnralzd xtrnal forc vctor f. 3 hn: = v, ϕ, f = p, m. a,b

3 M. Arc and M. F. Granata Lt dnot th tran and crvatr bvctor a = ε, ε, ε3 = k, k k. h can b collctd n th gnrald tran vctor q k, 3 ε and. In th am way th tr rltant vctor ha componnt, th frt thr collctd n th bvctor,, 3 = M, M M. hn: M, 3 = and th lat thr collctd n th bvctor = k, =, M q ε,. a,b Whn thr ar xtrnal mpod tran q, th vctor a can b dcompod nto two part: q = q + q 3, whr q ar th latc tran. Lt E b th dagonal tffn matrx of th latc bar: E = dagga/χ, GA/χ, EA, EJ, EJ, GJ whr for =, χ ar th har factor, G and E th har and th ong modl, A th cro-cton ara, J =, th prncpal nrta momnt and J th toronal contant. h q contttv qaton can b wrttn: = E. 5 h prncpl of tatonary total potntal nrgy can b wrttn a: l δ Π p = δ Wd =, whr: W = q Eq f, 7 th total potntal nrgy dnty, a fncton of arbtrary bt tabl dplacmnt and tran rpctng th followng fld compatblty qaton : ' = B + q + q. h ymbol ' rprnt th drvatv wth rpct to th crvlnar coordnat and th matrx B a qar matrx rprntng th gradnt matrx of th dplacmnt, namly : B B B =, 9 B whr τ τ k B =, k B =, a,b 3

4 M. Arc and M. F. Granata and τ and k ar th ntal toron and th ntal crvatr of th bar ax 5. Introdcng nto 7, w obtan th total potntal nrgy dnty, a a fncton of th vctor, and of th vctor of th xtrnal acton f and q : W, ', = ' + B q E ' + B q f. 7* In th total potntal nrgy prncpl for tatc problm th fncton W, wrttn n th form 7*, play th am rol that Lagrangan fncton play n th Hamlton prncpl for dynamc problm. h varatonal problm can b tranformd nto a dobl t of frtordr dffrntal qaton, by applyng th Lgndr tranformaton to th xpron 7*. Amng a tranformaton actv varabl th componnt of th vctor, and nw corrpondng canoncal varabl momnt, that condrng 5and concd wth th componnt of th vctor, w obtan th dal Hamltonan fncton of W. It follow that: W = H,, = 'W a,b ' h varatonal problm can now b wrttn n th nw canoncal form : l δ ' H d = h corrpondng ytm of th Elro-Lagrang qaton now: d W W W H d W W d H = = ' + =, = + =. 3a,b d ' d ' d h ytm a canoncal Hamltonan ytm of dffrntal qaton govrnng th tatc bhavor. h Hamltonan fncton ng b,7*,5 can b wrttn n xplct form: H, = E whch can b rdcd to th compact form: B + f + q, a H z = z Az + d z, b dfnng th qar ymmtrc matrx A of ordr, and th vctor z and d : B A =, B E z =,, d = f, q. 5a,b,c It worth nothng that btttng th Hamltonan fncton xprd n xplct form a nto, th fnctonal that hold b tatonary am th mxd form of th Hllngr- Rnr prncpl. h Elro-Lagrang ytm 3 can b rwrttn n th form:

5 M. Arc and M. F. Granata H ' = = B + E + q, H ' = = B f, a,b whr th frt qaton how th fld compatblty qaton, n whch appar th contttv qaton, and th cond on th fld qlbrm qaton. In a mor compact form th ytm can b wrttn: H z' = J, c z ng th kw ymmtrc mplctc matrx: I J =. 7 I 3. INEGRAION OF HE EUAION SSEM Wth rfrnc to a pac plan bar, havng toron τ =, w mt fnd th olton of th ytm c. Frt, lt condr th rdcd ytm a that can b olvd knowng th tranfr matrx of th ytm 7. h olton tak on th form: = o + q d o. h frt trm on th rght-hand d rprnt th homogno olton of th ytm a, whl th cond trm th partclar olton wth zro ntal condton. h homogno olton, whch obtand by ttng th tran vctor q =, rprnt th t of fncton that dcrb th rgd body dplacmnt of th lmnt; hnc th tranfr matrx can b obtand drctly by wrtng th rgd dplacmnt componnt of th bar n th cton havng crvlnar coordnat a a fncton of th dplacmnt componnt at th ntal cton. hn th tranfr matrx tak on th form: =, = I. 9a,b h olton of th ytm b can b obtand drctly bca t homogno ytm th adjont ytm of th homogno ytm corrpondng to a; hnc t tranfr matrx. hn: = f d. If th gradnt matrx do not dpnd on th crvlnar coordnat,.. f th bar ha contant ntal crvatr, th tranfr matrx concd wth th matrx -B, and po th followng proprt: =, =. a,b In th ca th olton and can b wrttn mor mply: 5

6 M. Arc and M. F. Granata + = d q, = d f. a,b h complt ytm c fnally ha th followng olton obtand by ng qaton, 3 and 5: + = = N N Z z, 3 whr: = d E Z, = d f N, [ ] µ µ µ + µ µ = d N E q N. 5a,b In th ca of concntratd xtrnal acton, δ-drac fncton for f and q can b ntrodcd nto 5. Smlar xpron ar dvlopd by Fj and Gong. Eqaton a,b clarly how that th dplacmnt n th gnrc cton th m of on part d to a rgd moton pl a cond latc part d to th latc and mpod dformaton, whl th vctor obtand from th qlbrm condton only. In th ca of bar loadd n th plan arch th actv dgr of frdom and th ntrnal forc ar only th odd x componnt of th tat vctor z, whl n th ca of loadng prpndclar to t plan crvd bam thy ar th vn x componnt. h lmnt tranfr matrc and th tat vctor for both th D problm can b obtand by dltng th odd or th vn row and colmn from th gnral on.. ELASIC ANALSIS OF COMPLEX SRUCURE Whn a crvd bar mad p of vral lmnt havng dffrnt bt contant gomtrc and latc proprt, lngth L, th trctr can b dvdd nto <n> lmnt. For ach lmnt t pobl wth qaton 3 to xpr th tat vctor z n th gnrc cton coordnat rfrrd to th vctor z n th ntal cton of th lmnt <>. Whn om nod I prnt mpod dcontnt n th gomtrc ax or appld concntratd forc P =, th can b takn nto accont by condrng th compatblty and qlbrm of th acton on th lft and rght of th nod. Fgr Crvd bar wth vral lmnt

7 M. Arc and M. F. Granata D S W hold hav: z = z + z, z =, 7a,b Intrnal rla or addtonal pport n th nod cold alo b condrd, bt thy wll not b condrd n th papr. It convnnt to add on componnt on th tat vctor z and on row and on colmn on th tranfr matrc, takng thr dmnon to +. It th pobl, by man of qaton 3, to dfn th nw tat vctor S = F S by ntrodcng th nw tranfr matrx F for th lmnt <> and th nodal pont matrx P : S z =, Z N N I z F =, P = 9a,b,c Nod amd to b at th lft nd of th trctr. Condrng th frt lmnt and th ntal bondary condton, t pobl to dfn th tat vctor S of cton = ; th vctor wll hav + known componnt and nknown. In th lmnt <> th tat vctor S can b xprd by th rcrv formla: S F P F L... P F L S =, 3 Procdng p to th fnal rght nod N, and mpong th bondary condton at th rght nd, th lnar ytm of qaton on th nknown qantt fnally obtand: S n L F L P F L... P F L S n = n n n n n, 3 Onc th ytm olvd, th vctor S compltly known and by man of qaton 3 t pobl to calclat th tat vctor n ach lmnt and th vctor and for th ntr trctr. Any pac crvd bar can b bdvdd nto a bg nmbr of mall crclar lmnt, R rad, amng th rad of ntal cvatr and tabl matrc P. 5. FINIE DISPLACEMENS Amng th followng hypoth: a th contttv law ar lnarly latc; b th ffct of th dformaton of th cton ar nglgbl; c th dplacmnt and rotaton of th cton can b larg; d th dformaton mt b vry mall vn f th dplacmnt ar larg. It pobl to xtnd th procdr xpondd n th prvo cton for mall dplacmnt to nonlnar gomtrc problm wth fnt dplacmnt. Rfrrng to qaton a n th ca of larg dplacmnt th frt trm on th rght hand, rlatd to rgd dplacmnt, ha to b modfd throgh th ntrodcton of th f matrx rathr than. h f matrx can b fond drctly throgh th rlatonhp btwn th larg rgd dplacmnt of th gnrc cton and th ntal dplacmnt vctor for th 7

8 M. Arc and M. F. Granata cton =. h dplacmnt magntd mpl a chang only n th bmatrx n 9a. h matrx, who lmnt lnk th dplacmnt componnt n th cton wth rgd rotaton n =, mt b modfd n th matrx f n whch ach trm bcom a fncton of th rotaton φ =,, 3 n th ntal cton. For th hypoth of mall dformaton, th cond trm on th rght hand of qaton a only ha to b modfd n ch a way a to tak nto accont th rgd ax rotaton, now no longr nglgbl. Eqaton b qlbrm condton mt b modfd to tak nto accont th rgd rotaton of th gomtrc ax on th componnt of th xtrnal and ntrnal forc vctor too. Nw xpron 3, and 5 of th tat vctor, tranfr matrx and load vctor of ach lmnt for fnt dplacmnt can b fond. o mantan th mall tran hypoth, t ncary to bdvd th ntr trctr nto mch mallr lmnt. Wth th chang n th matrx xpron, t pobl to apply th procdr n n th prvo cton, obtanng th x-ordr qaton ytm 3, whch bcom non-lnar whn th dplacmnt ar larg.. APPLICAION O CIRCULAR PLANE ARCH ELEMEN For a crclar plan arch lmnt, wth ntal crvatr rad R, th tat vctor rdcd to only x componnt: z =, v = v ϕ 3, = N. 3a,b,c M h tranfr matrx 9a now of ordr 3 and condrng = Rϑ, wth ϑ anglar coordnat, t can b obtand drctly condrng rgd moton for fnt dplacmnt: f ϑ co ϑ = n ϑ n ϑ co ϑ R R n ϕ ϑ co ϕ n + co ϑ ϕ ϕ nϕ ϕ ϑ co co n ϑ ϕ ϕ h rato n ϕ / ϕ and ϕ / ϕ rotaton mall dplacmnt, all th trm of f ar ndpndnt from ϕ bng ϕ wll mply b n ϕ / ϕ and ϕ / ϕ. 33 co xpr th nflnc of th larg ϕ on th trm of th matrx wthot any lmt abot t ampltd. In th ca of, bca of th rato = co =. o condr rctlnar bam lmnt of lngth L, both for mall and larg dplacmnt, w mt hav L = R γ, whr γ th anglar coordnat of th xtrmal cton; o n qaton 33 w wll condr: = Rϑ, R ϑ. For nmrcal analy wth larg dplacmnt t convnnt to xpr th tat vctor n ach cton of th gnrcal

9 M. Arc and M. F. Granata lmnt th dplacmnt vctor and th ntrnal forc vctor wth rfrnc to a fxd ytm of rctanglar Cartan coordnat rathr than a a fncton of th crvlnar coordnat. h can b don by ng mpl traformaton matrc. 7. NUMERICAL APPLICAION A a nmrcal applcaton th procdr appld to a plan crclar arch wth two hng and a cntrd vrtcal forc P dcd by Hddlton. o compar th rlt, har dformaton ar nglctd and th followng dmnonl paramtr ar takn nto accont: h/l and C = I/AL th nvr of th qar of lndrn. A a fncton of th dmnonl load paramtr PL /EI, thr ar hown th dagram of th rato v /L wth v th vrtcal dplacmnt of th mdpan cton, of ntal rotaton ϕ and of th dmnonl horzontal racton HL /EI. Fgr. Plan crclar arch Fgr 3. Fnclar rctlnar arch PL /EI PL /EI C=, h/l=,5 C=, h/l=,5 -rod C=, h/l= -rod C=, h/l= C=, h/l=,5 C=, h/l=,5 -rod C=, h/l= -rod C=, h/l= rav app. h/l=,5,,5,,5,3,35,,5,5 v/l P- v dagram for crclar arch,5,,5,,5,3,35,,5,5 v/l P- v dagram for rctlnar arch PL /EI PL /EI C=, h/l=,5 C=, h/l=,5 -rod C=, h/l= -rod C=, h/l= C=, h/l=,5 C=, h/l=,5 -rod C=, h/l= -rod C=, h/l= rav app. h/l=,,,,,, ϕ,,,,,, ϕ 9

10 M. Arc and M. F. Granata P-ϕ dagram for crclar arch P-ϕ dagram for rctlnar arch PL /EI PL /EI C=, h/l=,5 C=, h/l=,5 -rod C=, h/l= -rod C=, h/l= C=, h/l=,5 C=, h/l=,5 -rod C=, h/l= -rod C=, h/l= HL /EI P-Η dagram for crclar arch HL /EI P-Η dagram for rctlnar arch No aymmtrcal dformd hap and nap-throgh phnomna hav bn condrd. h nmrcal olton wa prformd olvng th nonlnar ytm of ordr thr throgh an tratv bt not ncrmntal mthod, amng om ntal attmpt val for nknown ϕ, V, H and mpong th followng nonlnar condton bondary condton at th rght nd cton: or L/= f ϕ,v,h =, ϕl/= f ϕ,v,h =, VL/ P/ = f 3 ϕ,v,h =. h olton wa obtand throgh a modfd Nwton mthod 9. h calclaton wr dvlopd condrng th half-arch dvdd nto 5 mallr crclar lmnt fndng th am rlt gvn n rfrnc. o ovrcom th problm of addl crv of P, v dagram t wa bttr to mpo agnd val for v at th mdpan cton and to calclat th val VL/ rathr than to mpo val for th load P bca th tratv procdr convrg too lowly or do not convrg. Anothr analy wa alo prformd on a fnclar arch wth two rctlnar lmnt. Wth th val h/l = th ca of an xtnbl t-rod wa alo condrd, th am rlt bng fond a for th crclar arch havng h/l =. h rlt concd wth tho obtand by Hddlton and Dowd for th lat ca. It ntrtng to not that, for crclar arch wth dffrnt val of th paramtr C, th crv PL /EI, v /L ntrct ach othr acro th am pont. h am occr for th crv PL /EI, ϕ. h occrrnc can b obrvd for rctlnar arch too.. CONCLUSIONS A calcl procdr wa dvlopd bad on th tranfr matrx mthod, makng t pobl to dal wth on-dmnonal latc trctr wth crvatr contant n th varo lmnt, n th larg dplacmnt fld, both n th plan and n pac. h on-dmnonal lmnt can hav axal, flxonal, har and D Sant Vnant-typ toronal flxblty too. h procdr vratl and for pac crvd trctr nd only th olton of a nonlnar qaton ytm of ordr x.

11 M. Arc and M. F. Granata 9. REFERENCES. Ptl E. C., Lck F. A., Matrx mthod n latomchanc, Mc Graw-Hll, Nw ork, 93.. Arc M., Mragla N., R V., Elmnt tffn matrx of pac crvd bar, I Intrnatonal Confrnc on Arch Brdg, Bolton U.K., ohda H., Imoto., Inlatc latral bcklng of rtrand bam, J. Engrg. Mch. Dv., ASCE, 99, 33-3, Fj F., Gong S. X., Fld tranfr matrx for nonlnar crvd bam, J. Strct. Dv. ASCE,, 75-9, Arc M., Rcprocal conjgat mthod for pac crvd bar, J. Strct. Dv. ASCE, 5 S, 5-575, 99.. Lanczo C., h varatonal prcpl of mchanc, Math. Expoton n.. Unv. Of oronto Pr, oronto, Pp L. A., Hovanan S.A., Matrx-comptr mthod n ngnrng, J. Wly & Son, Nw ork, 99.. Hddlton J.V., Fnt dflcton and nap-trogh of hgh crclar arch, J. App. Mch.Dv. ASCE, 35,73-79, Ortga J. M., Rhnboldt W.C., Itratv olton of nonlnar qaton n vral varabl, Acadmc Pr N.., 97. Hddlton J.V., Dowd J.P., A nonlnar analy of xtnbl -Rod, J. Strct. Dv. ASCE, 5, 5-, 979.