Time-depending behavior of PC beams externally plated with prestressed FRP laminates: a mechanical model

Transcription

1 Tim-dpnding bhavior o PC bam xrnally plad wih prrd FRP lamina: a mchanical modl Luigi cion, Valnino Paolo Brardi, nna D pon Dparmn o Civil Enginring, Univriy o Salrno, aly l.acion@unia.i, brardi@unia.i, adapon@unia.i Kyword: PC rucur, prrd FRP ym, crp modl. SUMMRY. n hi papr h auhor prn a mchanical modl capabl o prdicing h long rm bhavior o PC bam xrnally plad wih prrd FRP lamina, aring rom h rhological propri o compoi. i bad on a mchanical approach ha aum hr i a prc adhion bwn concr cor and xrnal rinorcmn. 1 NTRODUCTON Th rnghning o xiing rucur hrough h u o ibr-rinorcd polymr (FRP) marial rprn on o h mo inring and promiing chniqu wihin h conx o Civil Enginring. Mor rcnly, prrd FRP ym ar bing adopd in h rhabiliaion o indurial rucur or bridg, whr PC bam nd rinorcmn inrvnion. Wihin hi conx, a paricularly imporan apc i rprnd by h accura modlling o h drrd bhavior o rinorcd lmn [1-6]. n uch ca, in ac, h dirn rhological propri o h componn o h rnghnd rucur (.g. concr cor and compoi rinorcmn) can lad o a migraion o FRP r oward PC bam. Th conqun incra o h r a in bam can compromi h icacy o h rnghning chniqu. Thorical and xprimnal udi, currnly availabl in liraur, rr o h ailur bhavior o h rinorcd lmn, whil hr ar ill w on hir long rm bhavior. Svral horical and xprimnal udi on h rhological bhavior o FRP hav bn prormd by h auhor, in ordr o characriz hir vicou propri [7-13]. Th guid-lin on h digning o FRP rnghning inrvnion drawn up wihin an inrnaional conx, including h rcnly publihd Guidlin CNR-DT 200/2004, ak ino conidraion h aormniond rhological phnomna hrough h inroducion o uiabl rduciv acor o h FRP dign r. n hi papr h auhor prn a mchanical modl capabl o prdicing h long rm bhavior o a PC bam rnghnd wih FRP prrd rinorcmn, onc h vicou coniuiv law o compoi i known. 2 MECHNCL MODEL Th mchanical modl chmaiz h vicou bhavior o a prniond concr bam, ubjc o a bnding momn and rnghnd wih a FRP prrd lamina (Fig. 1). Th baic hypoh ar: - plan cro cion; - prc adhion bwn FRP and prrd concr bam; - xrnal bnding momn, M x, unvarid ovr im; - linar vicolaic bhavior o h compoi;

2 - prc adhion bwn concr and l ndon; - laiciy moduli o h marial coniuing h cro cion, unvarid ovr im. - ngligibl concr crp bhavior. Th la aumpion i uaind by h ac ha h rnghning inrvnion i gnrally ralizd on a bam ha ha bn in rvic or vral yar. b2 c q Μx H x G L y Νp d0 b1=b Figur 1: Prrd concr bam rnghnd wih a FRP lamina. n rrnc o h cro cion in Figur 2, dining h cnroid o h ranormd cion, G, a h origin o h ax, x and y, h axial rain ovr im, ε, in h cro cion, i wrin: (, ) ( ) ( ) ε y = λ + µ y (1) whr λ and µ rprn h axial rain in G and h curvaur, rpcivly. n addiion, auming ha h concr ha an laic bhavior and ha h rnghning i characrid by an iniial Young modulu, E, quaion (1) can b rwrin or h hr marial coniuing h cro cion a ollow: (, y ) () () y () () ε = λ + µ = ε + ε ε ε (, y) = λ () + µ () y = ε() ε0 εc (, y) = λ () + µ () y = εc() v 0 (2) whr: - ε c and ε ar h concr and l rain, rpcivly; - ε and ε v ar h laic and vicou conribuion o FRP rain, rpcivly; - ε 0 and ε 0 ar h rain in ndon and in FRP corrponding o iniial civ prnion, rpcivly.

3 εc(o) εc() ε> ε> c x G Μx µ(o) λ(o) µ() λ() y Νp y Ν εci(o) ε0 εci() ε0 ε0 ε0 Figur 2: rain and curvaur variaion ovr im in h cro cion. From rlaion (2), h ollowing xprion o h r in h marial ar obaind: ( ) () () () () () σ = E ε = E λ + µ y ε v + ε 0 σ () = E ε() = E λ () + µ () y+ ε0 σc () = Ec εc() = Ec λ () + µ () y (3) Saring rom h quilibrium quaion o h cro cion: σ () σ () σ () d + c dc + d = 0 c σ () y d + σ () yd + σ () yd = M c c c and uing h xprion (3), h ollowing i obaind: (ranlaion) (roaion abou h x axi) (4) ( λ () µ () ) ε () ε 0 λ () µ () E + y v + d + Ec + y dc + c + E λ () + µ () y d = 0 E ( λ () µ () y ) ε v () ε yd + Ec λ () + µ () y+ ε0 ydc + c + E λ () + µ () y yd = M x

4 Such quaion can b wrin a ollow: ( E + Ecc + E) λ () + ( ES + EcSc + ES) µ () E ε v () d + + Eε0 + Eε 0 = 0 ( ES + EcSc + ES) λ () + ( E + Ecc + E) µ () E ε v () yd + + Eε0S + Eε 0S = Mx (5) whr: - Α, Α c and Α ar h FRP, concr and l ara, rpcivly; - S, S c and S ar h FRP, concr and l ir momn o ara, rpcivly; -, c and ar h FRP, concr and l momn o inria abou h x axi, rpcivly. nroducing h ollowing ymbol: E E - n =, n = ; c c E E - = + n + n (ara o h ranormd cion); ( c c ( c c ) ( c c ) - S = S + n S + n S = 0 (ranormd cion ir momn o ara abou h x axi); - = + n + n (ranormd cion momn o inria abou h x axi); - Np = Eε 0 (prrd ndon conribuion o axial orc); - N = Eε 0 (prrd FRP lamina conribuion o axial orc); - M p = Eε 0S (bnding momn du o Np h x axi); - M = E ε S (bnding momn du o N h x axi). 0 h quaion (5) bcom: () () () E λ + ES µ = E ε v d ( Np + N ) ES λ () + E µ () = Mx + E ε v () yd ( M p + M ) (6) Th vicou dormaion o h rnghning, ε ( ) 0 ( τ) v, prn h ollowing xprion: σ ε v () = ( τ, ) dτ, (7) E

5 in which: Φ( τ, ) - ( τ, ) = E ; τ 1 - Φ ( τ, ) = ( 1 + ϕ( τ, ) ) (crp uncion); E - ϕτ, (crp coicin). ( ) Taking ino accoun h quaion (3), h rlaion (7) bcom: ( ) ( ) ( ) ( ) ( ) ( ) ε = λτ + µτ y ε τ + ε 0 τ, dτ. v v 0 (8) Subiuing (8) in h quaion (6), h ollowing i obaind by impl algbra: S λ () + 1 λ( τ) ( τ, ) dτ µ ( τ) ( τ, ) dτ = λ + λ ( τ, ) dτ 0 0 S µ () λ ( τ) ( τ, ) dτ + 1 µ ( τ) ( τ, ) dτ = µ + µ ( τ, ) dτ (9) whr: ( N + N ) M ( M + M ) - λ =, µ = ; p x p E E N M M - λ =, µ =. E p x p E Equaion (9) rprn a coupld ym o wo ingral Volrra quaion in h unknown λ() and µ(). Such a ym can b olvd uing h Laplac ranormaion chniqu:

6 S L λ () = L ( ) (, ) d 1 L ( ) (, ) d µ τ τ τ λ τ τ τ λ + + λ L ( τ, ) dτ 0 S L µ () = L λ ( τ) ( τ, ) dτ 1 L µ ( τ) ( τ, ) dτ µ + + µ L ( τ, ) dτ 0 (10) uming 0 =0 and bing vidnly λ( 0 )=0, µ( 0 )=0 or < 0, (-τ)=0 or <τ, h convoluion horm allow: S ( ) ( ) 1 ( ) ( ) L d L d L ( ) d µ τ τ τ λ τ τ τ λ τ τ = S F( ) = µ ( ) F( ) 1 λ ( ) F( ) + λ (11) S L λ ( τ) ( τ) dτ 1 L µ ( τ) ( τ) dτ + µ L ( τ) dτ = S F( ) = λ ( ) F( ) + 1 µ ( ) F( ) + µ whr F(), λ() and µ() rprn h Laplac ranorm o h uncion (-τ), λ() and µ(), rpcivly. Subiuing (11) in h quaion (10), h unknown uncion λ() and µ() can b obaind by h ollowing quaion ym: S F( ) λ λ ( ) = µ ( ) F( ) 1 λ ( ) F( ) + λ + S F( ) µ µ ( ) = λ ( ) F( ) 1 µ ( ) F( ) + µ + (12) Th invr Laplac ranorm o λ() and µ() upply h oluion o h vicou problm in h im domain.

7 3. NUMERCL SMULTONS numrical analyi on h long rm bhavior o prniond concr bam rnghnd wih prrd FRP lamina ha bn dvlopd. n paricular, i ha bn aumd ha h rinorcmn inrvnion i rquird o rplac h ir boom row o ndon, du o hir high corroion. Wihin hi udy, dirn plaing yp, characrizd by h am iniial prrd axial orc (N = N), hav bn analyzd; h gomrical and mchanical propri, loading condiion and rnghning gomry ar hown in Figur 3. q = N/m ndon 1/2'' L = m G 270 M x=mmax = N m Concr (C 40/50): = 40 N/mm E = N/mm ck c 2 2 Tndon: pk = 1900 N/mm E = N/mm FRP rinorcmn 10 ndon 1/2'' 12 ndon 1/2'' 12 ndon 1/2'' Figur 3: Dail o h rnghnd PC bam (dimnion in mm). Th long rm bhavior o adopd lamina hav bn characrizd by uing a micromchanical modl rcnly propod by h auhor [7-13], in h ild o linar vicolaiciy, capabl o prdicing h vicou propri o a FRP lamina aring rom ho o h ingl pha (marix and ibr). Th vicolaic paramr o h maric and aramid ibr, rpord in Tabl 1 and 2, hav bn valuad by uing daa o Makimov [14]. nad, a linar laic coniuiv law ha bn aumd in h ca o carbon ibr (Tabl 2). Typ o Fibr Tabl 1: Rhological propri o h marix. E 1,m [MPa] E 2,m [MPa] η 1,m [MPa h] η 2,m [MPa h] Tabl 2: Rhological propri o h ibr. E 1,m E 2,m η 1,m η 2,m [MPa h] [MPa] [MPa] [MPa h] ramid E E+08 Carbon E E E+49

8 Th numrical imulaion, or a im priod o 50 yar, ha bn carrid ou on 4 yp o lamina (Tabl 3), characrizd by dirn volumric racion o ibr, mad o am maric (Tabl 1) and dirn ibr (Tabl 2). Th iniial r valu ha bn aumd variabl bwn 10 and 40% o h ailur rngh o h ibr, aumd boh or aramid and carbon ibr qual o: k = 3500 MPa. Tabl 3: Examind ca. Typ o Typ o V V m σ (0)/ k σ (0) lamina ibr [%] [%] [%] [MPa] Carbon Carbon ramid V ramid For ach ca, h iniial and drrd r in h compoi and in h boom ibr o concr, σ ci, hav bn valuad (Tabl 4). Tabl 4: nananou r in h concr cor and h FRP rinorcmn. Typ o σ ci (0) σ ci (50 yar) σ (0) σ (50 yar) σ /σ (0) lamina [MPa] [MPa] [MPa] [MPa] [%] V

9 an xampl, or yp V lamina, ubjc o an iniial prrd axial r o σ (0)= MPa, h diagram o h inananou r in h compoi (Figur 4) and in h boom ibr o concr (Figur 5) ar rpord σ [MPa] [h] Figur 4: nananou r in h FRP compoi. (V = 0.4, V m = 0.6) σ ci [MPa] [h] Figur 5: nananou r in h boom ibr o concr. (V = 0.4, V m = 0.6) 5 CONCLUSON Thi papr ha prnd a micromchanical modl, ormulad by h auhor, ha allow, wihin h conx o linar vicolaiciy hory, h long rm bhavior o prrd bam rnghnd wih FRP lamina o b analyd. n paricular, i i capabl o obaining h drrd global bhavior o a rnghnd lmn, aring rom h rhological characriaion o h FRP lamina. Th rul obaind wihin numrical imulaion hav highlighd a ngligibl r migraion in h ca o CFRP lamina.

10 On h conrary, a markd r variaion ha bn obrvd wih rrnc o h FRP rinorcmn, alhough h iniial r in h compoi do no xcd h r limi, uggd by inrnaional guid lin. ugg o alo analyz h im-dpnding bhavior o uch rnghnd lmn, whn rinorcmn ar ralizd by uing FRP or GFRP lamina, du o hir ponial markd vicou bhavior. Th propod modl could rprn a valid ool o prorm uch a vriicaion, giving h xac oluion o h problm inad o mor approximad on obaind hrough numrical approach availabl in liraur. Rrnc [1] M. Savoia, D. Frri, C. Mazzoi, Crp bhavior o RC nil lmn rroid by FRP pla, CC '02 Conrnc Procding, San Francico (2002). [2] M. Savoia, B. Frracui, C. Mazzoi, Crp dormaion o FRP-plad R/C nil lmn uing olidiicaion hory, Comp. Modlling o Concr Srucur EURO-C, uria (2003). [3] Brardi, V.P., Di Nardo, E., Giordano,., On h crp bhavior o rinorcd concr bam rnghnd wih FRP: a numrical invigaion, Compoi in Conrucion nrnaional Conrnc CCC2003, Rnd (CS), aly (2003). [4] cion, F., Brardi, V.P., Fo, L., Giordano,., ndagin primnal ul comporamno vicoo di laminai pulrui in ibra di carbonio, XV MET Congr, CD-ROM Procding (2005). [5] cion, F., Brardi, V.P., Fo, L., Giordano,., On h crp bhavior o CFRP pulrudd lamina: an xprimnal udy, Th Scond ib Congr, Napl (2006). [6] cion, F., Mancui, G., Long-rm bhavior o CFRP lamina: an xprimnal udy, CCE Conrnc (2006). [7] cion, L., Brardi, V.P., D pon,., Un modllo mccanico pr lo udio dl comporamno dirio di laminai compoii ibrorinorzai, Colloquium Lagrangianum, Scilla (2006). [8] cion, L., Brardi, V.P., D pon,., Vicou coniuiv law o FRP Marial, PMM, Salrno (2007). [9] cion, L., Brardi, V.P., D pon,., Th durabiliy o ibr rinorcd polymr marial in rucural rroiing and upgrading, Thrmo-mchanial modling o olid, Ecol Polycniqu, Parigi (2007). [10] cion, L., Brardi, V.P., D pon,., Vicou bhavior o FRP marial: a mchanical modl, MET Congr, Brcia (2007). [11] cion, L., Brardi, V.P., D pon,., l comporamno a lungo rmin di mariali compoii ibrorinorzai: un modllo micromccanico, CP Conrnc, Salrno (2007). [12] cion, L., Brardi, V.P., D pon,., Problmi di durabilià di mariali compoii ibrorinorzai nl conolidamno ruural, Colloquium Lagrangianum, Ecol Polychniqu, Parigi (2007). [13] cion, L., Brardi, V.P., D pon,., Vicou coniuiv law or FRP marial,.c.m.m.s. Congr, Bangalor (2008). [14] R.D. Makimov, E. Plum, Long-Trm crp o hybrid aramid/gla-ibr-rinorcd plaic, Mchanic o Compoi Marial, Vol. 37, No.4 (2001).