Time-dependent behaviour of inhomogeneous Restructures: application to long term analysis of R C arch and arch-frame bridges

Transcription

1 Timedependen behaviur f inhmgeneus Resrucures: applicain lng erm analysis f R C arch and archframe bridges E Mla* Pliecnic di Milan aly F Pigni Pliecnic di Milan aly 26h Cnference n OUR WORLD N CONCRETE & STRUCTURES: Augus 2001 Singapre Aricle Online d: The nline versin f his aricle can be fund a: hp://cipremier.cm/ This aricle is brugh yu wih he suppr f Singapre Cncree nsiue All Righs reserved fr C Premier PTE LTD Yu are n Allwed re disribue r re sale he aricle in any frma wihu wrien apprval f C Premier PTE LTD Visi Our Websie fr mre infrmain

2 26h Cnference n Our Wrld in Cncree & Srucures: Augus 2001 Singapre Timedependen behaviur f inhmgeneus Resrucures: applicain lng erm analysis f R C arch and archframe bridges E Mla* Pliecnic di Milan aly F Pigni Pliecnic di Milan aly Absrac The paper appraches he prblem f he evaluain f he sae f sress and defrmain in srucures exhibiing differen rhelgical inhmgeneiies r delayed resrains. The mahemaical prblem assciaed he sluin f Vlerra inegral equains is slved by means f an algrihm based n he cncep f marix f slving kernels: i cvers all cases f rhelgical inhmgeneiy and generaes as special cases her mahemaical appraches in paricular he Reduced Relaxain Funcin Mehd. The analyical apprach is hen applied he sudy f arch bridges and archframe bridges deriving basic sluins fr permanens lads impsed displacemens and delayed resrains. A general herem regarding he behaviur f R.e. arch bridges incrpraing seel russes is finally saed. Keywrds: creep rhelgical inhmgeneiy addiinal resrains arch bridges archframe bridges 1. nrducin Mdern srucural arrangemens fr P.C. and R.C. bridges invlve members exhibiing marked rhelgical inhmgeneiies due varius facrs e.g. differen ages ninal hickness presence f elasic resrains amun f reinfrcing and presressing seel. Mrever he cnsrucinal echniques fen inrduce changes in saical scheme by impsing addiinal resrains. The evaluain f he sae f sress and defrmain in such srucure represens a quie cmplex ask as he srucural behaviur is gverned by sysems f Vlerra inegral equains [1]. A general mehd slve he prblem can be based n he cncep f marix f slving kernels which applies bh srucures exhibiing rhelgical inhmgeneiies disribued alng he srucural axis and in he ransverse secin. The main prerequisies f he mehd are discussed in deail in he nex pins. poins. 2. Analysis f srucures inhmgeneus alng he lngiudinal axis Referring he saically deerminae scheme f fig.1 indicaing by () 2S() he clumn vecr f he reacins f he redundan resrains we can wrie (s)= (S)+T(S)() (s)= (s)+!t(s)2s() (1) wih (s ) vecr f inernal acins (s ) vecr f he inernal acins due exernal lads Fig.1. Saical scheme in he saically deerminae srucure and! T (s) influence marix f he inernal acins generaed by he uni vecr 2S() () =.. Accrding he McHenry Principle f Superpsiin [2] fr he vecr f secinal defrmain we derive (s)= = fe()1()a "') a " 1') J('s)d'+e(s) 1' d1'+e (s ) (2) by applying he Principle f Virual Wrks we hen bain 457

3 f02$ T (S )(S )ds + i) 2$ T (Sr ).6. r () = 02$ T '{() + 02$ T T '{A () r=1 nsering eq. (2) in eq. (3) we can finally wrie f fe( s(s)1(s)d(s' ')J( 'ss + f fe(s)(s)1(s). T (s)d2$(')j( 's )ds dsr).6.r()+ f(s)e(s)ds='{()+t'{o() r=1 where '{() is he vecr f he impsed displacemens in he direcin f he redundan reacins;.6.() is he vecr f he relaive displacemens impsed in he crss secin f abscissa Sr; '{A () is he vecr f he impsed displacemens in he remaining resrains; ( s) is he siffness marix f he ransverse secin; J( ' s) is he creep funcin variable alng s. When he rhelgical inhmgeneiy varies by seps eq. (4) can be wrien in he shr frm [3] nc nc fei(oi ) e/i) d2$(')jl ') = E() fei(oiqei(i' ')Ji( ') Eq. (5) represening a sysem f Vlerra inegral equains can be slved by means f he marix f slving kernels. T d his we inrduce he cupling marix fr he ih par and he vecrs f he elasic sluins fr saical and gemeric lads i ( Oi ) = 1 () ei ( Oi ) 2$eg () = 1 () E() d2$eqi (') = _1 () dqei (Oi') (6) nrducing he equaliies nc ( ' )= i()ei(oi)ji( ') i=1 2$()= nc fd2$eqi(') Ei(O)Ji( ') i=10 eq. (5) becmes f ( ' ) d2$( ') = 2$eg ()+ 2$ () (8) The sluin f eq. (8) accrding he Principle f Superpsiin can be wrien 2$() = f( ' ) d[2$eg (')+ 2$ (')] (9) where marix!( ' ) is he marix f slving kernels saisfying he se f Vlerra inegral equains fc( " )alj(" ' ) d" = = ' 0 a". (10) Eq. (10) is a sysem wih n 2 unknwns and can be slved applying sandard numerical algrihms [4] [5] [6]. n fig.2 w cmpnens f marix!( ' ) are represened in seremeric frm. Every elemen f his marix represens a surface in he dmain (') ( ') s ha he descripin f he lngerm behaviur f a nredundan nnhmgeneus srucure requires evaluae n 2 surfaces each ne represening he slving kernel f a creep marix bained as linear cmbinain f he creep marices f every nnhmgeneus srucural par. 2.1 Paricular case f srucural inhmgeneiy f he srucure invlves nly w pars wih differen rhelgical behaviur he marix f slving kernels can be diagnalized by means f a linear ransfrmain in he space f he redundan reacins. n his case he mehd f sluin is called he Reduced Relaxain Funcins Mehd. Accrding eq. (6) we can wrie Q2(O)=!Q1(O) (11) s ha eq. (7) becmes (3) (4) (5) (7) 458

4 ( ' ) = E1(01 ) / )J1( ')+ E2(02) Q 1 ( )2( ') (12) 2S () = f 2S:q1 (')E1 ( 01 )J1 ( ')+ d2s:q2 (')E 2 (02 )J2 ( ')} Hll "H12 (13) '" Fig.2. Cmpnens f he marix slving kernels We inrduce he linear ransfrmain 2S() = ( )y() where marix s( ) is he mdal marix f marix 1 (). Subsiuing eq.(14) in eq.(8) we derive f * ( ' y(') = Yeg ()+ Y () 9() = s 1 ( )/ )s() where * ( ' )= E1 (01 )8(O)J1 ( ')+ E2 (02) 8( )2 ( ') Marix () is he specral marix (eigenvalues diagnal marix) f /). n [7] i was prved ha he elemens wj() f marix () lie in he inerval 0wj(};1 (i= n) and are derived by slving he fllwing equain delw() Q 1 ( )j= 0 (17) The sluin f eq.(15) can s be wrien in he fllwing frm y() = ff ( ' [Yeg (')+ Y (')] (18) wih JE1(1)J(")aH':"'')d"=1 i=12... n Hij("')=O ij= wih i*j (19) J( ' )= E1(01i(O)J1(')+E2(02X1Wi(O)]J2(') (20) ' 0 E1 (01 ) Finally he sluin expressed in vecr 2S() becmes 2S() = fs()j* ( ' 1( [2Seg(')+ 2S(')] (21) f ne par f he srucure is viscelasic and he her is elasic we hen bain C*(' )=E( )n( )J(')+n()] J(' )=E1(01i(O)J1(l')+[1Wi(O)] (22) = = 0 1 e = 0 0 E1 ( 01 ) Funcins J( ' ) are varied creep funcins while he cmpnens H (14) (15) (16) f marix!j* ( ' ) are he s called Reduced Relaxain Funcins. is s necessary evaluae nly n surfaces sudy he lngerm behaviur f viscelasic hmgeneus srucures ineracing wih elasic resrains. Mrever as he variain f parameer 0) in he inerval 00)1 cvers all he pssible kinds f 459

5 prblems we can cnclude ha he surface envelpe bained by slving eq.(19) assuming c as parameer represens he ms pwerful design aid fr he lng erm analysis f hmgeneus srucures ineracing wih elasic resrains. n fig.3 he inersecin f he surface envelpe wih he plane =7 days is repred fck= 50 N/mm2 1 :;:: 2.8 RH = 80% (0 =0 i (0 =0 0.8 === 2Ac/u = 400 mm (0 =01 (0 =01 (0 = i =7 9 (0 =03 00 N 0.6 (0 =02 (0 =04 w H! N w "0 2 :::; 1.6 8=1 i 1/ f : l " ".' (0 =05 "0 0.5 (0 =06 (0 = ' f i fck=50 N/mm2 (0 =08! 0.3 (0 =09 RH =80%.::::: 2Aclu=400mm (0 =1 0.2 ff =7 9 T a) Fig.3. Varied Creep Funcins a) and Reduced Relaxain Funcins b) l = (0 =03 (0 =04 (0 =05 (0 =0'6 8:6 (0 =0'9 (0 =1' 3. Cmpsie seelcncree srucures These srucures exhibi secins invlving a viscelasic maerial cncree and an elasic ne srucural seel. Cmpaibiliy and equilibrium equains geher wih he cnsiuive law fr he w maerials lead he fllwing expressin fr he secinal defrmain fc () + s Ec ( )J( ')F(S' ') = f[d (s ') + T (s )d2$(')c ( )J( ') (23) 0 nrducing he curling marix fr he secin Q(O)=c(O)+sr1c() (24) and changing he crdinae sysem by means f he linear ransfrmain (s ) = ( (s ) we bain [R*( ')] ()= fe(s()1 S()Ec() (d (S')+(S)d2$(')) (25) By applying he Principle f Virual Wrks we can wrie he cmpaibiliy equain in he fllwing cmpac frm f*( ')d2$(')= Q{) (26) The sluin f eq. (26) can be bained by direc inversin (27) 2$()= f*(')dq(') where * ( ') is he marix f slving kernels. 3.1 Paricular case f dubly symmerical secin n [8] i was prved ha if he secin des n vary wih crdinae s exhibis duble symmery and he rai beween he gyrain radius f seel and he ne f cncree has uni value eq. (25) becmes b) R*(')] ( R*(')] ljl( () ' ((0( A :f! d (s (' H;( s ))(((')) dljl (s (' A 5(() (28) wih e () 2S e () unknwn vecrs calculaed in he elasic dmain. The parameer A = 1 +nps depends bh n he seel rai (Ps = As/ Ac) and n he mdular rai (n = Es/ Ec). As he vecr \f varies in an 460

6 affine way wih respec is elasic value 4J fr he redundan acins he simple fllwing relainship e mus hld 2$() = 2$e () (29) Eqs. (28) (29) represen a generalizain f he firs herem f linear viscelasiciy which saes he affiniy beween he secinal creep defrmain and he elasic ne geher wih he invariance f he sae f sress wih respec he elasic sage under saical acins. n his case he variain f sresses in he seel par f he secin and in he cncree ne is 1_Ff() 1_R*() s(ys) =1+ Ec() c(ys) =1nps Ec() se(ys ) )1 ce(ys) )1 Under he same hypheses fr gemeric acins eq. (25) becmes A R('f)] (5 ) = J ( { ;1(;) T (30) (5)d(')) J. ( T (5)d(') =. (5 ) (31 ) while fr he redundan acins we can wrie is( ) = ± [(HiS()+ f dis (1'):: i: 1 (32) Eqs. (31) (32) represen a generalizain f he secnd herem f linear viscelasiciy which saes he affiniy beween he sae f sress and he elasic ne geher wih he invariance f he defrmain wih respec he elasic sage under gemeric acins. n his case he variain f sresses in he w maerials is ( ) ( {R{) 1 c{ys) _R{) flc ys =ce ys O Ec{ )1 (33) Oce{ys) Ec{) s ha here is n variain f sress in he seel par. 4. Effecs f delayed resrains in hmgeneus srucures The reacins arising in delayed resrains reach heir highes values fr hmgeneus srucures. n rder evaluae he upper bunds f he srucural behaviur under delayed resrains we shall cnsider hmgeneus srucures. n is iniial cnfigurain he srucure can be saically deerminae r can exhibi n redundan resrains. Afer sme ime during which exernal lads and he riginal redundan reacins sar acing n he srucure sme new redundan resrains are added. As saed in [9] fr cnsan acins and delayed resrains all added a he same ime he reacins are X() = X ( )S8CP(' ) R{ ') d' = X ( ( * ) (34) e. a' E( ) _e f 0 a Eq. (34)derived in [9] was generalized in [8] evaluae he effecs f delayed resrains in nnhmgeneus srucures saisfying he hypheses f 3.1. n his case fr = i resuls X() = 2$e () A 1 S8CP(' ) R( '\..!' = 2$e () S:( + +) )1 A a' f'j A" 0' a The funcin ( ) gverning he ime develpmen f he reacins f he delayed resrains calculaed accrding CEB MC '90 Mdel [10] is represened in fig.4. f he addiinal redundan resrains are added a differen imes 1 s 2 S... S i S... S n as usual in he cnsrucin pracice he prblem can sill be slved hrughly fllwing he evluin f he redundan acins in ime; during he generic ime inerval Oj s S j+1 he sluin has he frm "s :: 0.4 UJ' fck=30 N/mm2 RH = 50%./ V :: 2Ac/u = 200 mm V V / // r/ // v / 1/ / v V / / Fig.4. Funcin ( ) (35) f f r

7 j Xk()=Xke(O)(kO)+ Xhe(O)f3hk(O)(hO) 1::;k::;j. Xj()=Xje((j) (36) h=k+1 where he elasic influence cefficiens hk(o) mus be calculaed referring he srucural scheme in he hh ime inerval. Eq. (36) shws ha here is muual ineracin beween he addiinal redundan acins bu ha nly he newer nes influence he lder nes wihu being influenced by he laer. 5. Case Sudies 5.1 Arch bridges Parameric analyses were made in [8] fr a cncree arch wih parablic axis spanning 100 m and exhibiing a rise f 15 m; he crss secin is recangular (160*180 cm); hree differen ses f rhelgical disribuins were cnsidered: hmgeneus srucure symmeric inhmgeneus srucure dissymmeric inhmgeneus srucure. Bh saical and gemeric acins were cnsidered. Sme f he ms ineresing resuls are nw discussed. Referring a whinge arch wih an elasic half and he her hmgeneus viscelasic subjeced a cnsan hriznal displacemen impsed a he lef abumen fig.5 shws ha rhelgical inhmgeneiy des n affec significanly he relaxain f he bending mmen in ime nr he dyssimmery f he rhelgical inhmgeneiy reflecs in a dissymmeriy f he bending mmens. Fr he displacemens n he cnrary here is a marked dyssimmerizain in ime. The effec f delayed resrains is illusraed in fig.6. n his case he arch is assumed rhelgically hmgeneus and i is iniially saically deerminae (hree hinges). The hree hinges are hen clsed a differen imes Oi s ha in is final cnfigurain he arch becmes clamped. n fig 6a) he iniial bending mmen he ne fr preexising resrains and he ne fr delayed resrains are repred while fig. 6b) shws he muual ineracin beween he redundan acins. Lengh [m] i:"'''''' L7 : \\ =Oda/s.' / 5 80 \ = days E 100 C : m160 Lengh [m] r'rrr'' E " ';: 0.2 c G E 0.4 ;;;8 " CS 1:: C. 0.6 G.!!l > 'C 0.8 =O days " = days Fig.5. Bending mmen and verical displacemen A marked endency f he sysem assume he sae f sress crrespnden he presence f he addiinal resrains befre he applicain f exernal lads can be bserved E 1000 z 800 1nn 1nn = c 600 G E 400 Z' = 7 days 1. 0 M./ E r.' >< "01= 7+ days./. C.: r "02= 30 days 'C N "03= 90 days /.. c 200 >< 400 G m 400 :::: p.(':. /' 600 >< : T ) b) Lengh [m] a Fig.6. Bending mmen and redundan acins in ime 5.2 Arch frame bridges An arch frame bridge wih free span f 150 m rise f 70 m bx crss secin bh fr he deck and fr he pylns was sudied in deail in [8]. Tw differen ses f rhelgical characerisics were cnsidered: hmgeneus srucure and symmeric inhmgeneiy (viscelasic cener beam and 462

8 alms elasic laeral pars f he srucure). The permanen graviy lad was applied he whle srucure r nly pars f i in rder clearly pin u he effecs f he rhelgical inhmgeneiy n he imedependen behaviur f he srucure. Fig.? shws ha he srucure des n exhibi a significan variain f he bending mmen because f he cmbined presence f w disribuins f bending mmen evlving in ppsie ways in ime. M" rlu ' Fig.? Tal lad: bending mmen and verical displacemen The firs repred in fig.s is he mmen disribuin peraining an elasic srucure wih a viscelasic resrain. n his case he bending mmen has a relaxing behaviur in ime. The secnd illusraed in fig.9 is he ne relaed a viscelasic srucure wih an elasic resrain. n his case he bending mmen increases in ime s he final resul is a cmpensain which leads a behaviur alms independen n he rhelgical inhmgeneiies in erms f sresses while in erms f dtile7mens here are SignifiCjllaria"TTiOT1n"TSrT' rf'i<:'.llll.l.llllll..llf;!rrnrtttttt' r Fig.S. Exernal lad: bending mmen and verical displacemen 4 ) \ """'LM+.rL..L"':::'" Fig.9. nernal lad: bending mmen and verical displacemen. ". =" ''<"<. /1_=Omys \ \ " '... =:DXXldays' "_'_ =Odays. 5.3 Arch bridges wih seel russ reinfrcemen The arch is he same as in (5.1) while he crss secin saisfies he hypheses f (3.1). Three differen percenages f seel were cnsidered (Ps=2%; 4%; 6%) while he exernal saical acin cnsidered was he graviy lad. Fig.10 a) b) shws he ime variain f he verical displacemen f he cenral pin and f he sresses in seel and cncree prduced by permanen lads. We bserve ha an increase f A assciaed an increase f Ps significanly reduces he displacemens wih respec he hmgeneus case represened by A= /. :::::: /. /. /' a) 1..5=1 14=11418 '"=12836 '"=14254 V r' Fig.10. Verical displacemen and sresses fr saical lads r b) 1..4= = =14254 A 1 =a;a.=o 1..2= = =11418 The increase f he vecr.w. in an affine way prduces a crrespnden increase f he seel sress as repred in fig.1 b) and des n mdify he hrus s ha he vecr 9 remains cnsan in ime and he sress in cncree mus decrease. n fig.11 a) b) c) (where he same values f A as in fig.10 are adped) he resuls relaed he impsiin f a hriznal cnsan displacemen f he lef abumen are illusraed. Fig.11a) shws he reducin in ime f he hrus due he cncree relaxain. The presence f seel reduces he lss f hrus which can be abu ne half f he ne peraining a rhelgically hmgeneus arch. As he vecr.w. is cnsan in ime he sress in he 463

9 seel russ cann change while in cncree he sress varies accrding he relaxain funcin R f he hmgeneus maerial. Finally when a delayed resrain is inrduced he increase in ime f he added resrain is smaller as he presence f he elasic par prevens he free develpmen f he delayed displacemens which he grwh in ime f he reacin is cnneced >< ! i! w "'' i "rw 'ru Hi il " :' O.4 A2 b A3 0.2 A4 A5 0 a) 1 ' 'n '" ii ii ' '! i! ' ' J b) f f :+ r Y ii! c) Fig.11. Redundan acin and sress in cncree fr gemeric lad and redundan acin fr delayed resrain 6. Cnclusins The new erecin echniques available fr he building f arch and archframe bridges have made his kind f srucures cmpeiive and widespread bu have inrduced new prblems relaed he rhelgical inhmgeneiy f hese srucures due he differen imes f erecin f heir differen pars and als relaed he frequen changes in heir saical scheme during heir differen erecin phases. These prblems can be sudied and slved in he general hery which is based n he cncep f marix f slving kernels and is varius specializains. n his paper his echnique was applied he parameric analyses f arch and archframe bridges. The parameric analyses have shwn ha rhelgical inhmgeneiy influences he sae f defrmain bh in arch and archframe bridges while fr cnsan saical lads he sae f sress is n grealy influenced. Fr gemeric acins he degree f relaxain is s high whaever he rhelgical inhmgeneiy ha i's n useful ry and beer he sae f sress by impsing displacemens he resrains bh fr arch and archframe bridges. Delayed resrains significanly influence he sae f sress fr bh kind f bridges: here is a marked endency f he srucure assume he sae f sress crrespnden he siuain in which he resrains are applied befre he exernal acins. As his saical scheme can be very differen frm he iniial ne grea aenin mus be paid in designing and erecing srucures which exhibi changes in heir saical scheme. Fr bridges wih cmpsie secin he presence f seel changes in ime he secinal sae f sress and he redundan reacins. The ime evluin f he srucural behaviur can be sudied effecively by applying he generalizains f he fundamenal herems f linear viscelasiciy bained in (3.1) under nn resricive hypheses. 7. References [1] Vlerra V. Lecns sur les fncins de ligne GauhierVillars Paris (1913) [2] McHenry D. N A new aspec f creep in cncree and is applicain design Prc. AST M 43 (1943) [3] Mla F. Viscelasic analysis f srucures inhmgeneus alng heir axis Sudi e Ricerche Vl. 9 alcemeni Bergam (1987) (in alian) [4] Bazan l.p Thery f creep and shrinkage in cncree srucures: a Precis f recen develpemens Mechanics Tday Pergamn Press Lndn (1975) [5] Bazan l.p. Mahemaical Mdels fr creep and shrinkage f cncree Creep and Shrinkage in cncree srucures. Ed. Bazan l.p. Wimann F. H. Jhn Wiley & Sns Chicheser (1982) [6] Pisani M. A Numerical Analysis f Creep prblems Cmpuers & Srucures Vl. 51 N. 1 (1994) [7] Mla F. General viscelasic analysis f srucures and secins wih nnhmgeneus rhelgical behaviur Sudi e Ricerche Vl. 8 alcemeni Bergam (1986) (in alian) [8] Mla E. Pigni F. Time dependen behaviur f inhmgeneus R. C. srucures: applicain lng erm analysis f R. C. arch and arch frame bridges Graduae Thesis in Srucural Engineering Pliecnic di Milan (2001) (in alian) [9] CEB Revisin f he design aids f he CEB design Manual "Srucural Effecs f Timedependen Behaviur f Cncree" CEB Bullein n. 215 Lausanne (1993) [10] FPCEB Mdel Cde 1990 Design Cde Thmas Telfrd Lndn (1993) 464