On the reduction of the interfacial stresses in a repaired beam with an adhesively FRP plate

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1 On he reducion of he inerfcil sresses in reired em wih n dhesively FRP le Kernou ssim, Khlil Belkhdr D éremen de génie civil e hydrulique, Universié de Sid (Algérie) : nssimkernougeniecivil@yhoo.fr Déremen de génie civil e hydrulique, Universié de Sid (Algérie) : e.khlil@gmil.com ABSTRACT. Severe ging nd deeriorion of he rnsorion infrsrucure hs moived exensive reserch in develoing srucurl reir echniques using FRP or seel les. As resul, revious reserchers hve develoed severl nlyicl mehods o redic he inerfce erformnce of onded reirs. An imorn feure of he reinforced em is he significn sress concenrion in he dhesive he ends of FRP le. n his er, numericl soluion using finie difference mehod is used o clcule he inerfcil sress disriuion in ems srenghened wih FRP le hving ered ends. These ler, cn significnly reduce he sress concenrions. umericl resuls from he resen nlysis re resened o demonsre he dvnges of use he ers in design of srenghened ems. KY WORDS: Ple onding; FRP comosie; nerfcil sresses; Reired em; Design; Ter. nroducion Over he s severl decdes, exensive reserch nd develomen in he field of merils engineering nd science hve een crried ou wih fire reinforced lsic (FRP) comosies leding o wide rnge of rcicl licions [, ]. As nion s infrsrucure ges, one of he mjor chllenges he consrucion indusry fces is h he numer of deficien srucures coninues o grow. The licions of using exernlly onded FRP les o reinforced concree (RC) or seel srucures hve shown h he echnique is sound nd efficien nd offers rcicl soluion o his ressing rolem. Rerofiing using exernlly onded les is quick, esy wih resec o meril hndling, cuses miniml sie disruion nd roduces only lile chnges in secion size. n recen yers, mny sudies hve een crried ou on he ehviour nd srengh of such rerofiing mehod [3 5]. However, he disdvnge of his echnique is h he srenghened memers re susceile o sresses concenrion ner he le end, which my cuse ol deonding eween he memer nd he FRP le. Some reserchers noe h reducing he FRP or seel le hickness ner he le end is n effecive mehod o minimize he inerfcil sresses [0]. As furher develomen of he soluions y Srford [0], Smih [9] nd Tounsi [], his er resens simle numericl soluion for oining he sher nd he norml sresses in he dhesive lyer of rerofied em under exernlly lods. The mehod cn e used o design srenghened ems wih secion roeries h chnge long he em such s ered les. Finlly, rmeric sudy ws conduced o comre he resuls of models wih differen geomeries.. Theoreicl roch The derivion of he resen soluion is descried in erms of dherends nd, where dherend is he em nd dherend is he soffi le (Fig ). The ssumions doed in he resen soluion re summrized elow:. The em, dhesive, nd FRP merils ehve elsiclly, linerly, nd isoroiclly.. The sher sress in he inerfce is roorionl o he sher sli.

2 3. Since he hickness of he dhesive lyer is smll, oh he sher nd eeling sresses in he dhesive re ssumed consn cross is hickness. 4. The wo onded odies hve he sme ending curvure he sme secion. This ssumion is no mde in he eel sress soluion. Fig.. Simly suored em srenghened wih onded FRP le.. quilirium nd consiuive relionshis in he em nd le A differenil segmen of led em is shown in Fig., where he inerfcil sher nd norml sresses re denoed y (x) nd (x), resecively.

3 q (x) (x) + d (x) (x) Concree () (x) +d (x) V (x) V (x) + dv (x) τ(x) σ(x) (x) (x) τ(x) V (x) σ(x) Ple () V (x) + dv (x) (x) + d (x) (x) +d (x) Fig.. Forces in infiniesiml elemen of soffi led em Fig. lso shows he osiive sign convenion for he ending momen, sher force, xil force nd lied loding. Considerion of equilirium gives: nd z () Where z is he lever rm eween he cenroids of he le nd he em z () Where, nd re he deh of em, le nd he hickness of dhesive. The suscris, nd denoe em, le nd dhesive, resecively. The srins he se of he em nd he o of he le re given s: A (3) A (4) 3

4 Where is he elsic modulus, A he cross secionl re, nd is he second momen of re. The curvure in he em nd he le cn e exressed s:, z (5).. quilirium nd consiuive relionshis cross he dhesive join The men sher sress (x) nd eel sress (x) cing cross he dhesive join re ssumed o c ou he mid lne of he dhesive lyer. By considering equilirium of shor lengh of he le (Fig. ): d (6) Where is he widh of he dhesive lyer. V d d (7) The consiuive resonse of he dhesive is descried y dv (8).3. Sher comiiliy of dhesive lyer d, G G (x) The sher sress, (x), cross he dhesive inerfce is deermined y exmining sher comiiliy cross he dhesive join. The sher srin in he dhesive lyer cn e wrien s (9) u v y x (0) Where u nd v re he horizonl nd vericl dislcemens resecively ny oin in he dhesive lyer. Assuming h he dislcemen field vries linerly hrough he dhesive lyer, he verge sher srin in he dhesive lyer is given y: u u d v d x d v d x () Where u nd u re he longiudinl dislcemens he o of he le nd he se of he em, resecively. Differeniing he ove exression wih resec o x gives: d () 4

5 5 Where v / d is he curvure in he em nd he le. Susiuing for he srins using he consiuive relionshis (qs. (), (3), (4), (5) nd (9)) gives he comiiliy equion for sher srin cross he dhesive inerfce z A A d G The unknown vriles, nd re couled. To simlify he soluion, he difference eween he em nd le curvures is ssumed negligile for he uroses of deermining : z Hence z cn now e elimined from q. (3), giving he governing equion for 0 f d f Where G f, z A A f.4. orml dhesive sress The norml sress ) (x is found y exmining he hrough hickness srin, in he dhesive, he difference in vericl dislcemen cross he dhesive join: v v y v Rewriing in erms of sress nd differeniing wice wih resec o x d Susiuing for he curvures using q. (5) gives he comiiliy equion for norml sress: z d Using qs. (7) nd (8), he equion (0) cn e rewrien s (3) (4) (5) (6) (7) (8) (9) (0)

6 4 * d * 4 3 () Where,, z / 3 () And * is he rnsformed momen ou he le dhesive inerfce, * (3) 3. nerfcil sresses wih er The governing differenil equions for sher sress (q. (6)) nd norml sress (q. ()) re vlid for led ems wih geomeric nd meril roeries h vry long is lengh, however, he coefficiens ( f, nd 3 ) vry in x, nd closed form soluion is no ossile. A finie difference mehod cn e used o find he dhesive sresses for cses wih vrying secion roeries. A finie difference soluion using consn node scing,, is oulined elow. The nodes re numered i = n (from x = 0 o x = L/, he cenre of he em). Virul nodes (-, 0, n +, n + ) re used o llow derivives o e defined nodes nd n. Suerscris re used o define node numers in he following equions. 3.. Sher sress wih er The governing equion for he le force (q. (6)) cn e wrien ech node (i = n) long he em: Considering he oundry condiions: f i i i i i f 0 (4). Due o symmery, he sher sress mid sn is zero ( d / 0 ), i.e. n n. The xil force in he le osiion x = 0 is zero. 0 (5) x0 0 (6) These n + simulneous equions re solved exlicily o find he le force ech of he n + nodes. An imlici soluion mehod (for exmle, y vrying he sher sress node so s o sisfy oundry condiion (5) node n) will no e successful, due o he sensiiviy of condiions node n o smll chnges node. 6

7 The sher sress disriuion follows from q. (6): i i i i (7) 3.. orml sress wih er A fourh order finie difference soluion is required o find he rnsformed le momen. The governing equion for norml sress (q. ()) is wrien ech node (i = n) long he em: *( i) *( i) * i *( i) *( i) i * i i 4 3 i (8) A he end of le, he ending momen nd sher force re known. Hence, he oundry condiions cn e wrien s:. Boundry condiion (): he lied ending momen nd he xil force x = 0 re zero. * * x0 0 (9) * /. Boundry condiion (): V d V x = 0 x0 *( ) 8 *( ) 8 *() *() V x0 (30) * 3. Boundry condiion (3): V d / 0 he cenre of symmeric em: *( n) 8 *( n) 8 *( n) *( n) 0 (3) 3 * 4. Boundry condiion (4): d / 0 he cenre of symmeric em: 3 *( n) *( n) *( n) *( n) 0 (3) These n + 4 simulneous equions re solved exlicily for he le force ech of he n + 4 nodes. The norml sress disriuion follows qs. (7), (8) nd (3): i i *( i) 6 *( i) 30 * i 6 *( i) *( i) (33) 7

8 4. Resuls nd discussion 4.. FRP Ple wih consn hickness (wihou er) Finie elemen (F) nlysis hs een emloyed o vlide he resuls of he ove rocedure. A qudric rick elemen is emloyed in he resen clculion. The hlf of em is nlyzed ecuse of he symmery. The refined mesh is rrnged ner he end of he FRP le. The geomeric nd meril roeries of he em re shown in Fig. 3 nd le. A consn CFRP-le hickness is nlyzed firsly using finie difference mehod nd F mehod. The em ws sujeced o wo cses of loding. Uniformly disriued lod (UDL) q=500k/m². id oin lod P=500k The oined resuls re comred wih n nlyicl soluion crried ou y Smih nd Teng [9]. Figs. 4 nd 5 show he inerfcil sresses vs. he disnce from he FRP le end. is shown h oh finie difference soluion (FD) nd F clculion re greele well wih he nlyicl curve. There re sress concenrions of norml nd sher sresses he end of FRP le, which cuse he inerfcil delminion or filure..3mm 0.5m 5m.7mm 544.5mm.9mm Fig. 3: Geomeric roeries of he comosie em Comonen Tle Dimensions nd meril roeries of he comosie em Widh [mm] Deh [mm] Young s modulus [GP] Poisson s rio Sher modulus [G] Bem Adhesive lyer FRP le

9 0 5 [9] F soluion FD soluion Sress (P) 0 5 Sher sress 0 orml sress Disnce from he le end(mm) Fig. 4: Comrison of inerfcil sher nd norml sress for em wih onded FRP soffi le sujeced o UDL 0 5 [9] F soluion FD soluion Sress (P) 0 5 Sher sress 0 orml sress Disnce from he le end(mm) Fig. 5: Comrison of inerfcil sher nd norml sress for em wih onded FRP soffi le sujeced o mid oin lod. 4.. FRP Ple wih er Herein he effec of FRP le wih generlly vrile hickness (Fig. 6) is resened. The hickness of he FRP le (x) is descried y rirry funcion of he longiudinl coordine x; hence he cross secionl re nd he second momen of re of he FRP le, A (x) nd (x), re lso funcions of he coordine x. Four riculr cses re exmined. () Tye A: consn hickness 9

10 () Tye B: liner vriion for L 0 x (34) e x e for 0 x (35) for x L (35) (c) Tye C: Tngen rolic vriion e x e for 0 x (36) for x L (36) (d) Tye D: Prolic vriion x e e x e for 0 x (37) for x L (37) Tye A: Consn hickness Tye B: Liner vriion (i) e i Tye C: Tngen rolic vriion Tye D: Prolic vriion x Fig. 6: Thickness erns of FRP le. The FRP le wih consn hickness (ye A) will e regrded s he reference cse. A numericl sudy is hen conduced o suly informion on he conriuion of he she of he hickness rofile on he reducion of edge inerfcil sresses. n his sudy, he lengh of he er is = 00 mm nd he hickness he end of he e is e = mm. From he resuls resened in le, we cn oserve h he decrese of he hickness of he 0

11 FRP le in he edge region leds o reducion in he edge sresses. We cn lso conclude h FRP le of ye C gives n imorn reducion in edges inerfcil sresses. Figures 7 nd 8 descrie he yicl sress field o clrify he comrison of he resuls wih hose of he reference cse (FRP le wih consn hickness). Hence, reducing he hickness of he FRP le in he edge region my e considered s n effecive wy for reducing he mgniude of he edge sresses involved. Tle nerfcil edge sresses for differen she of er x sher sress τ [P] x norml sress σ [P] ( ) TyeA TyeA ( ) TyeA TyeA [%] [%] Tye A Tye B % 55.48% Tye C % 64.85% Tye D % 46.98% Fig. 7: nerfcil sher sress for vrious hickness rofiles of FRP le.

12 Fig. 8: nerfcil norml sress for vrious hickness rofiles of FRP le Prmeric sudy Vrious rmeers influence he mximum vlues of he sher nd norml sresses in he onding region. n his sudy we used he er of ye C. For rerofied ems, he mos imorn ones re he hickness nd sher modulus of he dhesive, nd he hickness nd he elsic modulus of he FRP le. These rmeers were sudied y severl uhors [7, o 4]. n he resen sudy, we inend o show how he mximum dhesive sresses re influenced y he dimension of he er. The imorn rmeers of he er re: he lengh of he er () nd he hickness he end of he er ( e ). Figures 9 nd 0 lo he inerfcil sresses he ered end of he le versus he lengh of he er nd he hickness end, resecively. The rmeric sudy indices he eneficil effec of hving hin ered end nd long er. For he ler, he enefi ers o hve sured when he lengh of he er is eyond 500 mm.

13 ximum sress (P) ximum sress (P).000 Sher sress orml sress Lengh of er, (mm) Fig. 9: nerfcil edge sresses for differen lengh of he er ( e = mm) Sher sress orml sress Ple end hickness e (mm) Fig. 0: nerfcil edge sresses for differen ered end hicknesses ( = 00 mm). 5. Conclusion 3

14 The resen sudy hs develoed relively simle rocedure for conrolling he high sher nd norml sress concenrions h occur he edges of he FRP le in exernlly srenghened ems. F nlysis hs een emloyed o vlide he resuls from he nlyicl nd he numericl soluions, nd he greemen eween he resuls oined from he differen soluions is good, which demonsres h resen rocedure is simle ye ccure. High sress concenrions occur he free ends of dhesively onded les. The er, however, reduce he mximum sher sresses y ou 40% nd he mximum norml sresses y ou 60%. n ddiion, i hs een shown h he she of he er hs n imorn effec on he reducion of such sresses. Hence, he er is very eneficil for voiding deonding of he FRP les from he ems. The dimensions of he er lso hve n influence on he inerfcil edge sresses. The mximum sher nd norml sresses decrese s he hickness of he end of he er decreses nd he lengh of he er increses. However, here is no furher chnge in sress if he lengh of he er is incresed eyond 500 mm. References. Seile F, Priesley J, Hegemier GA, nnmoro D. Seismic rerofi of RC columns wih coninuous cron fire jckes. J. Com Consr 997; (): o YL, Tsi SP, Lee S. Seismic erformnce ehviour of em column connecions in re sressed concree ridges. J er Sruc 998; 3: Vilny, O. The nlysis of reinforced concree ems srenghened y eoxy onded seel les The n. J Cemen Comos. nd Lighweigh Concree, 0(), 73 78, Roers, T.. Aroxime nlysis of sher nd norml sress concenrions in dhesive lyer of led RC ems The Sruc. ngr., London, 67(), 9 33, Roers, T.., nd Hji Kzemi, H. A heoreicl sudy of he ehviour of reinforced concree ems srenghened y exernlly onded seel les Proc., nsn. of Civ. ngrg., 87, Pr, 39 55, Tljsen, B. Srenghening of ems y le onding J.. in Civ. ngrg., ASC, 9(4), 06, Lu, K. T., Du, P. K., Zhou, L.., nd Hui D. echnics of onds in FRP onded concree em Comosies, Pr B, 3(6), 49 50, lek, A..; Sdmnesh, H.; hsni,. R., Predicion of Filure lod of R/C Bems Srenghened wih FRP Ple Due o Sress Concenrion he Ple nd AC Srucurl Journl, Vol 95(), 998, Smih, S.T., nd Teng, J.G. nerfcil sresses in led RC ems ngrg Sruc., 3(7), , Srford, T.; Cdei, J., lsic Anlysis of Adhesion Sresses for he Design of Srenghened Ple Bonded o Bem Consrucion nd Building erils, Vol 0, 006, Tounsi, A., mroved Theoreicl Soluion for nerfcil Sresses in Concree Bems Srenghened wih FRP Ple n. J. Solids nd Srucures, Vol 43, 006, Tounsi, A., nd Benyoucef, S. nerfcil Sresses in xernlly FRP Pled Concree Bems n. J. Adhesion & Adhesives, Vol 7, 007,

15 3. Benyoucef S, Tounsi A, efh SA, Add Bedi A. Aroxime nlysis of he inerfcil sress concenrions in FRP RC hyrid ems. Comosie nerfces 006; 3(7): S. Benyoucef, A. Tounsi, SA. efh A. Add Bedi Cree nd Shrinkge ffec on Adhesive Sresses in RC Bems Srenghened wih Comosie Lmines Comosies Sciences nd Technology 007; 67: S. Benyoucef, A. Tounsi, K. H. Benrhou,. A. Add edi. Time Deenden Behviour of RC Bems Srenghened wih xernlly Bonded FRP Ples: nerfcil Sresses ffecs. echnics of Time Deenden erils Journl 008; :