Experimental validation of a proposed numerical model for the FRP consolidation of masonry arches

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1 Historical Constructions, P.B. Lournço, P. Roca (Eds.), Guimarãs, Exprimntal validation of a proposd numrical modl for th FRP consolidation of masonry archs S. Briccoli Bati and L. Rovro Univrsity of Flornc, Dipartimnto di Costruzioni, Firnz, Italy ABSTRACT: In planning th rstoration and consolidation of historical buildings on must assss th ffcts on thir structural bhavior of applying advancd matrials and tchnologis to lmnts mad with traditional ons. Th prsnt papr dals with th consolidation of masonry archs through application of Carbon Fibr Rinforcd Polymrs (CFRP). A numrical modl for archs rinforcd with FRP shts is proposd, and its validity chckd through xprimntal trials. Th numrical modl rprsnts th arch as a rigid assmbly of blocks joind by lastic conncting rods with boundd tnsil strngth, which simulat th mortar joint, and tnsionrsistant conncting rods on th intrados and/or xtrados, which simulat th FPR rinforcmnt. Th numrical modl has bn chckd xprimntally through a sris of tsts on 1:2 scal modls of brick archs subjctd to th application of a load concntratd at th kyston. 1 INTRODUCTION Safguarding th world's historical buildings rquirs dfining appropriat mthodologis for thir rstoration and consolidation. In dfining such mthodologis, on must account for, on th on hand, structural safty, including sismic hazard, and on th othr, rspct for th original architctur from both th asthtic as wll as structural prspctivs. Th consolidation tchniqus rcommndd and widly usd in rcnt yars for masonry structurs includ th addition of rinforcd-concrt floors or cntring to th xtrados of vaultd structurs. Ths, howvr, rprsnt highly invasiv and irrvrsibl oprations. In fact, apart from altring th original stat and function of th structurs undr rstoration, such mthods oftn introduc xtranous lmnts that ar incompatibl with th pr-xisting convntional matrials and tchniqus, so much so that, at last in som cass, thir application winds up contributing dcisivly to ruining th structur thy wr supposd to prsrv. With rgard to th many issus associatd with historical building maintnanc and rstoration, th intrst of workrs in th sctor has incrasingly turnd to th dvlopmnt of advancd tchnologis and th adoption of innovativ matrials. Many innovativ matrials ar abl to satisfy mchanical rquirmnts not mt by traditional matrials and tchnologis. Of ths, composit matrials rprsnt a particularly valuabl group and hav alrady bn applid widly in aronautics, mchanics and sporting quipmnt. Th composit matrials most oftn usd in th filds of civil nginring and rstoration ar th FRP (Fibr Rinforcd Polymr).

2 1058 Historical Constructions FRP ar composit matrial mak by th union of fibrs and polymric matrix. Th fibrs ar th strong componnts and ar short and random arrangd or long and in on or mor dirction arrangd. Th fibrs ar ithr glasss fibrs (GFRP, Glasss Fibr Rinforcd Polymr), ithr aramidic fibrs (AFRP, Aramidic Fibr Rinforcd Polymr), or carbon fibrs (CFRP, Carbon Fibr Rinforcd Polymr). Th matrix protcts th fibrs, distributs th tnsions and givs shap to th composit. Th mor usd matrics ar th thrmostting polymric matrics. Th principal charactristics of FRP ar thir high strngth and stiffnss rlativ to thir wight, thir considrabl volumtric stability, good rsistanc to chmical action and low lctrical conductivity. As thy ar anisotropic, thy can b custom dsignd for spcific tasks by arranging th fibrs in such a way as to provid optimal load-baring capacity for th particular loading conditions. Th us of shts of composit matrial in th form of cladding, applid to ithr th intrados or th xtrados of vaultd structurs, could obviat at last som of th inhrnt drawbacks of th consolidation tchniqus currntly in us. In ordr to assss th fficincy of such rinforcmnt tchniqus, w conductd an xprimntal analysis on 1:2 scal modls of brick archs consolidatd by application of CFRP shts and subjctd to th application of a load concntratd at th kyston. Th xprimntal rsults obtaind wr thn utilizd to chck a numrical modl that simulats th bhavior of masonry archs rinforcd with FRP cladding. Th arch is modld as a sris of rigid blocks joind at thir intrfacs by lastic conncting rods. In particular th voussoir is modld as rigid block; th mortar-joint is modld as a sris of lastic rods with boundd tnsil strngth; th FRP rinforcmnt, in corrspondnc to th intrados and xtrados, is simulatd by lastic tnsion-rsistant conncting rods ndowd with suitabl xtnsional stiffnss. 2. NUMERICAL MODEL 2.1 Dscription In Briccoli Bati t al. (1998) a numrical modl is introducd for calculations on masonry archs, undr th assumptions that th voussoirs ar rigid and th intrfacs lastic-cracking. Such modling furnishs rsults in prfct agrmnt with xprimntal rsults and abl to accuratly intrprt actual cass. In th prsnt work th numrical modl has bn modifid in ordr to nabl its application to calculations rlativ to archs rinforcd with FRP shts placd on th intrados and/or xtrados. A B C D RIGID BLOCK ELASTIC NO-TENSION ROD ELASTIC ROD ELASTIC ROD Figur 1: Numrical modl.

3 S. B. Bati and L. Rovro 1059 Th proposd modl concivs of th rinforcd arch as mad up of rigid blocks fixd to th intrfac by thr typs of appropriatly rigid lastic conncting rods. In Figur 1 th conncting rods indicatd by th lttrs A and B simulat th mortar joints: mor prcisly, typ A joints, tangnt to th arch axis, hav boundd tnsil C and absorb th normal componnt, whil typ B, orthogonal to th arch axis, rsists both tnsion and comprssion and absorbs th shar componnt. Conncting rods C, rsistant alon to tnsion, ar st on th intrados or xtrados and rprsnt th FRP rinforcmnt. Th rods A with bondd tnsil strngth wr liminatd, whn th intrnal forc is major or qual to ultimat tnsil strngth. Th limination of th rods fits th rduction of sction bcaus thr is a fractur du to th no-tnsion strngth of th masonry. Th stiffnss of th rod B corrsponding to liminatd rods A is rducd in proportion to th numbr of liminatd rods A in th sction. Th mchanical paramtrs of th modl ar th lastic modulus of th rods A and B, th ultimat tnsil strngth of th rods A and th lastic modulus of th rods C. Th lastic modulus and th tnsion limit of A rods dpnd on mchanical charactristics of joint mortar. Th lastic modulus of C rods dpnd on mchanical charactristics of th FRP rinforcmnt. Th lastic modulus of th rods B dpnds on th mchanical charactristics of joint mortar and on gomtric charactristics of th sction by mans of following formula: λtt GA sz = E NlB A rodb rodb with λ shar factor, T tangntial forc, t joint thicknss, G lastic tangntial modulus of th mortar, A sz surfac of th cross sction of th structur, N normal forc, l B lngth of rod B qual to hight of structur sction, A rodb surfac of th cross sction of th rod B. E rodb is dtrmind imposing that th longitudinal dformation of th rod B with normal forc is qual to th dformation producd by th tangntial forc T (qual to normal forc N ) in th sction of th structur. 2.2 Th no-tnsion problm s quations Th constitutiv modl for a stuctur consistd of rigid blocks and lastic roods with boundd tnsil strght is drivd from linar lastic modl through th corrction of th static solution and through th following dtrmination of th kinmatic solution consistnt with th nw static solution. Th static solution is dtrmind adding slfquilibratd strss to lastic strss to annul th not accptabl tnsil trss. Th kinmatic solution is dtrmind considring th slfquilibratd strss as intrnal forcs rsultd from applid strains that giv validity to kinmatic quations. Th applid strains ar as fracturs causd of th boundd tnsil strnght. In particular th quation of th problms ar: for a structur mad up of n rigid bodis connctd by m lastic conncting rods (fig.2), of which z with boundd tnsil strngth lim TEN, th quations for th static and kinmatic problm ar AX + F = O A x + KX + = O X T Z lim TEN whr: A R m 3n is th quilibrium oprator or topological matrix; X R m is th vctor of th normal forcs in th conncting rods; z ar th rods absorbing th normal componnt of th n mortar joints and thrfor with boundd tnsil strngth (X Z lim TEN); F R 3 is th vctor of th forcs, body forcs and xtrnal forcs, applid to th barycntr of th bricks; (1)

4 1060 Historical Constructions T 3n m A R is th congrunc oprator, th transpos matrix of th quilibrium oprator; x R 3n is th displacmnt vctor of th rigid body's barycntr; K R m m is th complianc matrix of th rods; R m is th vctor of th inlastic strain which must b introducd in ordr to gnrat th suitabl strss which will annul th tnsil strsss unaccptabl in th z rods simulating th mortar joints; vctor contains th strains ncssary to rstablish congrunc of th lastic strains stmming from th solution conforming to th sign constraint on th z rods. Th strains containd in vctor may b thought of as "cracks" du to th lack of tnsil rsistanc in th rods modling th mortar joints. Solution (X, x ) and corrsponding vctor ar dtrmind through an itrativ procdur in which: i) initially th lastic solution of th systm ( X, x ) is dtrmind by: AX A T x + F = O + KX = O Th lastic solution is thn succssivly modifid by acting on a singl conncting rod until th solution satisfying condition X Z lim TEN is rachd. Vctor X o, corrsponding to th first stp, is obtaind through: in which Xo = X + CXN (3) 1 T 1 T 1 C = I K A ( AK A ) A is th oprator of orthogonal projction of null spac N (A), and (2) m R on th m X N R is an unknown vctor dtrmind by imposing th condition that within th st of strsss on th z conncting rods, strss X oj in rod j undr maximum tnsion major than boundd tnsil strngth lim TEN must b qual to zro: X oj = X j + CXNj = 0 (4) Th quations for th problm thus bcom: whr AX + F = O o T A x + KX + = O o o o 0 = KX is th vctor to b introducd in ordr to mak th problm's kinmatic n quations congrunt with th nw solution X = X CX, ). ii) solution i ( + 0 N x0 X is succssivly modifid by mans of th corrction X i+ 1 = X i + CXN( i+ 1) (6) m in which C is th sam as in th forgoing stp and X N ( i+ 1) R is an unknown vctor dtrmind by imposing th condition that th maximum strss from amongst thos of th z rods with strss major than lim TEN and all thos that hav bn annulld in prvious stps b qual to zro: X X + CX 0 (7) i + 1 = i N ( i+ 1) = (5) At stp ( i +1) solution ( X i + 1, xi + 1) and th vctor i + 1 satisfy th quations AX + F = O i+ 1 T A x + KX + = O i+ 1 i+ 1 i+ 1 (8)

5 S. B. Bati and L. Rovro 1061 whr i+ 1 = KX N ( i+ 1) is th vctor to b introducd in ordr to mak th problm's kinmatic quations congrunt with th nw solution X X CX, x ). ( i + 1 = i + N ( i + 1) i + 1 Th procdur nds whn th strss is lss than or qual to lim TEN for vry z conncting rod. Th Colontti thorm guarants that th linarly indpndnt trms that can b introducd to modify th strss stat in any givn structur is qual, at most, to its dgr of hyprstaticity. Th algorithm has bn implmntd in Matlab (5.3R11). It is possibl to input dfinitions for th gomtry of th structur and th loads and stiffnss of th conncting rods. Th stiffnss charactrizs th various typs of conncting rods (A,B and C in Fig. 1) and, in th cas of typ C (rprsnting th rinforcmnt), th stiffnss srvs to modl th width and thicknss of th sht. As output th algorithm furnishs th valus of th forcs in th conncting rods and th displacmnts of th blocks' barycntrs, and displays th straind configuration and prssur curv corrsponding to th applid loads. 3 EXPERIMENTAL TESTS Bfor prforming tsts on th 1:2 scal modl masonry archs consolidatd with FRP shts, w conductd mchanical trials of th matrials mployd in constructing th modls thmslvs, as wll as tsts of th adhrnc btwn th brick and FRP shts. 3.1 Mchanical charactristics of th matrials Th mchanical charactristics of th bricks and mortar usd to mak th arch modls wr dtrmind by monoaxial comprssion, dirct tnsil and bnding strss tsts. Th comprssion tsts wr prformd on prismatic brick spcimns, 2.2 x 2.2 x 4.5 cm, mad from brick blocks furnishd by th firm Latrizi S. Marco, Vnzia, Italia, as wll as on 4 x 4 x 4 cm cubic spcimns of cmnt-lim mortar whos mixtur was constitutd by on part hydratd "Fior" typ mortar, on part Portland composit cmnt (typ II/a-L UNI ENV R 32,5), ight parts Ticino sand and two parts watr. Uniaxial comprssiv tsts ar prformd also on 25 x 12 x 20 cm masonry sampls. Th dirct tnsil tsts wr conductd on 4 x 1 x 8 cm prismatic brick spcimns with symmtrically arrangd notchs, approximatly 1 cm in lngth. Th bnding strss tsts wr carrid out on 11 x 25 x 5 cm prismatic brick sampls and 4 x 4 x 16 cm prismatic lim-cmnt mortar sampls. Th mchanical paramtr valus dtrmind by ths tsts ar prsntd in Tabl I Th "Mbrac Fibrs C1-30" high-strngth carbon fibr shts, "Mbrac" primr and twocomponnt poxy-bas adhsiv that mak up th MBrac FRP systm wr furnishd by th firm MAC S.p.A. Trviso, Italia, who also providd th tchnical spcifications shown in Tabl2. Tabl 1: Mchanical charactristics of th bricks, mortar and masonry mployd for making th modl archs. Spcific wight Elastic modulus Comprssiv strngth Dirct tnsil strngth Bnding tnsil strngth [kg/m 3 ] (Mpa) (Mpa) (Mpa) (Mpa) Brick Mortar Masonry

6 1062 Historical Constructions Tabl2: Tchnical and mchanical charactristics of th CFRP componnts (Mbrac systm of MAC spa, Trviso, Italy). Spcific Lngthning at wight failur Tnsil lastic modulus Bnding lastic modulus Dirct tnsil strngth Bnding tnsil strngth [kg/l] [Mpa] [Mpa] [Mpa] [Mpa] [%] Fibr C > Primr 1067 >700 >580 >12 >24 3 Adhsiv 1020 >3000 >3500 >50 > Adhrnc tsts In ordr to chck that th adhsiv systm of FRP Mbrac was also suitabl for cladding th brick masonry support, w prformd som dirct tnsil tsts on spcimns mad up of two bricks connctd nd-to-nd on to th othr through a pair of fibr shts of varying dimnsions. Fiftn spcimns wr prpard, ach mad up of two paralllpipd 11 x 11 x 4 cm brick lmnts placd nd to nd with small facs abutting, and hld togthr on both larg facs by CFRP shts of varying lngth and width. Obviously, with incrasing surfac ara of th applid shts, thr is also a proportional incras in th sampl's braking load. Sinc dtachmnt of th fibr shts from th brick support nvr occurs, it can b statd that th adhrnc strss, calculatd roughly as th ratio of braking load to anchoring surfac, rsults in all cass gratr than 1 Mpa Tsts on modl archs Thr masonry dprssd 1:2 scal archs wr prpard with doubl-layrd brick and limcmnt mortar, with a span of 150 cm, ris of cm and both hight and thicknss of 10 cm. Ths wr subjctd to th sam conditions of incrasing load up to collaps: arch n.1 without rinforcmnt and archs n.2 and n.3 with CFRP rinforcmnt shts applid according to two diffrnt rinforcmnt arrangmnts. Arch n.2 wr rinforcd with 1.25 cm-wid CFRP sht applid along th ntir lngth of th intrados and arch n.3 wr rinforcd with 1.25 cm-wid CFRP sht applid along th ntir lngth of th xtrados (Figur 3). Figur 3: Arch n2 with CFRP sht on intrados and arch n3 with CFRP sht on xtrados.

7 S. B. Bati and L. Rovro 1063 Th xprimntal trials (Figur 4) wr prformd at th Official Laboratory for Tsting of Matrials and Structurs of th Construction Dpartmnt of th Univrsity of Flornc, Italy. Th tsting apparatus consistd of a rigid stl tst fram that allowd imparting vrtical displacmnt to th modl's right springing, horizontal displacmnt to lft springing and concntratd vrtical loads at any point on th xtrados. Th load was applid gradually by mans of a flywhl-opratd scrw jack abl to impart incrasing displacmnts, whil th corrsponding load incrass wr masurd by mans of a load cll with a capacity of 5KN and stp of 50N, for th unrinforcd arch, and a capacity of 100 kn and stp of 100N for th consolidatd archs. Figur 4: Th xprimntal trials. Arc n.1 was brought to collaps in th absnc of any rinforcmnt. Th first stag of th loading procss transformd th archs into isostatic structurs through th formation of thr hings: on at th kyston on th xtrados appard at a load of 300N; and two othrs formd on th intrados nar th haunchs at a load of 450N. In th scond stag of th loading procss collaps nsud du to th mrgnc of two furthr hings on th xtrados in corrspondnc to th springings (Figur 5). Figur 5: Arch n.1 at load of 300N. Archs n.2 was damagd bforhand through a loading procss that rndrd thm isostatic through th formation of thr hings, which occurrd in corrspondnc to a load of 450N. Aftr unloading, th archs wr thn consolidatd through application of an CFRP sht on th ntir lngth of th intrados and wr onc again subjctd to th narly static application of loads at th ky. Two hings appard on th intrados nar th haunchs in corrspondnc to a load of 1000N. Failur occurrd du to dtachmnt of th FRP sht in corrspondnc to th

8 1064 Historical Constructions kyston with a pak load of 3560N. Aftr th almost complt dtachmnt of th FRP sht, th arch rsumd th classical collaps mchanism by th mrgnc of thr altrnating hings at th kyston and haunchs, th sam as that which occurrd in th archs with no FRP-sht rinforcmnt (Figur 6). Figur 6 : Arch n.2 at load of 3560N. Archs n.3 was damagd bforhand through a loading procss that rndrd thm isostatic through th formation of thr hings, which occurrd in corrspondnc to a load of 450N. Aftr unloading, th archs wr thn consolidatd through application of an FRP sht on th ntir lngth of th xtrados and wr onc again subjctd to th narly static application of loads at th ky. A hing appard on th xtrados at th kyston in corrspondnc to a load of 750N. Subsquntly two hings appard on th xtrados nar th springings in corrspondnc to a load of 2100N. Collaps nsud du to th mrgnc of two furthr hings on th intrados nar th haunchs bcaus of th local dtachmnt of th FRP sht. Th collaps load was 3400N. Figur 7: Arch n.3 at load of 750N.

9 S. B. Bati and L. Rovro 1065 load (N) FRP on intrados FRP on xtrados displacmnt (mm) Figur 8: Load-displacmnt diagrams rcordd during th tsts for archs n.2 n.3. Figur 8 shows th load-displacmnt diagrams rcordd during th tsts for th consolidation modls. Th load-displacmnt diagram rvals th followings charactristic points: th start of th linar sgmnt (li), th nd of th linar sgmnt (l), th point of intrsction btwn th xtnsion of th linar sgmnt and th point on th ordinat corrsponding to th maximum load (l'), th maximum load (m) and th convntional ultimat load (u), which corrsponds to 2/3 of th maximum. Using th valus of th charactristic points rcordd during th tsts, th valus of th following mchanical paramtrs wr calculatd: tangnt stiffnss Kt=(yl-yli)/(xl-xli), kinmatic ductility µc=xm/xl and availabl kinmatic ductility µcd=xm/xu. Ths hav bn summarizd in Tabl 3, togthr with th valus of th pak loads rcordd. Tabl 3: mchanical paramtrs of th archs calculatd using th valus of th charactristic points rcordd during th tsts. ARCHES Pak load (N) Tangnt stiffnss (N/µm) Kinmatic ductility Availabl kinmatic ductility FRP on intrados FRP on xtrados COMPARISON OF EXPERIMENTAL AND NUMERICAL RESULTS Figurs 9 and 10 show th numrical and xprimntal load-displacmnt diagrams for th archs n.2 and n.3. rspctivly: th paramtrs of numrical modl com from th mchanical paramtrs of th matrials mployd to build th masonry arch modls. Th comparison btwn th xprimntal and numrical quilibrium pattrn brings to th for a vry good fit, with th only xclusion of th valus nar th pak load. For ths valus of th load, th numrical modl is stiffr than xprimntal. In corrspondnc of th pak load, th stiffnss rduction of th xprimntal modl is du to th damag of th masonry.

10 1066 Historical Constructions load (N) xprimntal numrical displacmnt (mm) Figur 9: numrical and xprimntal load-displacmnt diagrams for th archs n.2. load (N) xprimntal numrical displacmnt (mm) Figur 10: numrical and xprimntal load-displacmnt diagrams for th archs n.3. REFERENCES Briccoli Bati S., Paradiso M., Tmpsta G Analysis of Masonry structurs modlld by a st of rigid blocks and lastic unilatral contact constraints. In Pand G.N t al. (d.), Computr Mthods in Structural Masonry-4; Proc. Intrn. Conf., Firnz. Briccoli Bati S., Rovro L Consolidation of masonry archs with carbon-fibr rinforcd plastics. In 12 TH Intrnational BRICK/BLOCK Masonry Confrnc, Madrid. Briccoli Bati S., Rovro L., Consolidation of masonry archs througt shts of long carbon fibrs composits, In Di Tommaso A. (d.), Mchanics of masonry structurs strngthnd with FRP-matrials; Proc. Nat. Conf., Vnzia. Padova: Edizioni Librria Cortina. Christnsn J.B., Gilstrap J., Dolan C Composits matrial Rinforcmnt of xisting masonry walls. Journal of Architctural Engng vol 2 n2, p Ehsani M.R., Saadatmansh H., Vlazquz-Dimas Bhavior o Rtrofittd URM walls undr simulatd arthquak loding. Jour of Composits for Construction vol 3, No 3, p Kolsch H., Carbon fibr cmnt ovrlay systm for masonry strngthning. J. Composits for Construction, ASCE 2(5), p Kolsch H Carbon Fibr Cmnt Matrix (CFCM) Ovrlay Sistm for Masonry Strngthning. Jour of Composits for Construction vol 2, No 2, p Luciano R., Sacco E Strngthning of masonry panls with FRP composits. Int. J. Solids Structurs 35(15), p Nanni A Fibr_Rinforcd_Plastic Rinforcmnt for Concrt Structurs:Proprtis and Applications. Amstrdam: Elsvir. Saadatmansh, H., Fibr composits for nw and xisting structurs, Structural J.,ACI, 91(3), pp , Triantafillou T. C., Fardis M.N Strngthning of hystoric masonry structurs with composit matrials. Matrials and Structurs RILEM, vol 30, n 202, p Triantafillou T. C Strngthning of masonry structurs using poxy-bondd FRP laminats. J. Composits for Construction, 2(2) ASCE, p