THREE-PARAMETER ELASTIC FOUNDATION MODEL OF FRP STRENGTHENED CONCRETE BEAMS

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1 As-Pcfc onference on RP n Structures (APS 7) S.T. Smt (ed) 7 nterntonl nsttute for RP n onstructon THREE-PARAMETER EAST OUNATON MOE O RP STRENGTHENE ONRETE BEAMS J.. Wng eprtment of vl onstructon nd Envronmentl Engneerng Te Unversty of Alm Tuscloos Alm USA. Eml:Jwng@eng.u.edu ABSTRAT Eternl ondng of RP pltes or seets s ecome populr metod for strengtenng renforced concrete structures. eondng long te PR-concrete nterfce cn led to premture flure of te structures. or ts reson stresses long te RP-concrete nterfce re of gret reserc nterest. Estng solutons mnly dopt te clsscl solutons of desvely onded jont n wc te desve lyer s essentlly modeled s twoprmeter elstc foundton. losed-form solutons of nterfce ser nd norml stresses cn e otned n ts model. However ts model cn t stsfy te zero ser stress oundry condton ecuse degree-offreedom of te desve lyer s gnored n te two-prmeter elstc foundton model. n ts pper we propose n nnovtve tree-prmeter elstc foundton model to smulte te desve lyer. Ts new model regns te mssng degree-of-freedom n te two-prmeter foundton model of te desve lyer y ntroducng te trnsverse dsplcement of te desve lyer s new prmeter. Eplct closed-form epressons of nterfce stresses nd em forces re otned. Te new model not only stsfes ll oundry condtons ut lso predcts correctly tensle norml stress long te desve-concrete nterfce nd compressve norml stress long te RP plte-desve nterfce ner te end of te plte. Ts new model s verfed y fnte element nlyss. Te new tree-prmeter elstc foundton model provdes n effectve nd effcent tool for nlyss nd desgn of generl desve jonts. KEYWORS er renforced plstcs; Strengtenng; oncrete; nterfce deondng; nterfce stress; Adesvely onded jont; Tree-prmeter elstc foundton NTROUTON eondng long te desvely onded nterfce s common flure mode for RP strengtened renforced concrete (R) structures. Accurte crcterzton of te ond strengt s necessry for desgn of structures eternlly onded wt RP compostes. Etensve reserces ve een conducted nd vrous models ve een proposed to study te yrd mterl nterfce. A compreensve revew on tese models ws gven y Smt nd Teng (). Most of tese teoretcl models re te decedents of te clsscl soluton of desvely onded jont proposed y Golnd nd Ressner (9) wc models te desve lyer essentlly s twoprmeter elstc foundton. Smple eplct closed-form epressons of nterfce stresses nd em forces cn e otned y ts model s demonstrted y mny resercers (elle et l. 9 Smt nd Teng Wng 3). Te nterfce stresses predcted y te two-prmeter elstc foundton model rec good greements wt tose otned troug contnuum nlyss suc s fnte element nlyss (EA) (Teng et l. ) ecept smll zone t te vcnty of te edge of te desve lyer. n ts smll zone contnuum nlyss (Rnovc nd rostg Teng et l. ) revels tt te norml stresses long te desve-concrete (A) nterfce nd long te RP plte-desve (PA) nterfce re sgnfcntly dfferent. Te norml stress long te A nterfce s tensle wle te one long te PA nterfce s compressve. Ts s n mportnt feture ecuse t predcts deondng sould occur. n ts study we present new soluton of nterfce stresses of RP strengtened em y drectly etendng te two-prmeter elstc foundton model to n nnovtve treeprmeter elstc model. Ts s mde possle y ntroducng new term relted to te deflecton of te desve lyer wc s usully gnored n te two-prmeter elstc foundton model to te epresson of te ser strn n te desve lyer. All te egt oundry condtons re stsfed n ts model. Snce te formulton of ts study s n te smlr fson of two-prmeter elstc foundton (elle et l. 9 Wng 3

2 3) nd te solutons re n eplct closed-forms te present model cn e followed nd mplemented convenently y oter resercers. THREE-PARAMETER EAST OUNATON MOE OR AHESVEY BONE B- AYERE BEAM onsder concrete em (derend ) renforced y n RP plte (derend ) troug tn desve lyer (gure ). Bot te derends nd desve re lner elstc nd ortotropc mterls to ccount for te most generl stuton. Te derends re modeled s two ems wt tckness nd respectvely nd re connected y n nterfce of tn desve lyer wt tckness of. A smply support em s consdered (gure ) n ts pper for smplcty. Oter oundry nd lodng condtons cn lso e solved y ts model wt very lttle modfcton. Adesvely Bonded B-yered Bem System onsderng typcl nfntesml solted ody of te -lyered em system (g. ). Te deformton of te frp plte nd concrete em cn e wrtten s: U z u z φ W z w () Were u () w () nd ( ) ( ) ( ) ( ) ( ) φ ( ) re te l trnsverse dsplcements nd rotton of te neutrl s of em respectvely; u (z ) nd w (z ) ( ) re te l nd trnsverse dsplcements of em respectvely; suscrpt represent te em (concrete em) nd (frp plte) n fgure respectvely; nd z re te locl coordntes of em wt -s long te neutrl es of te em. By mkng use of te consttutve equtons of ndvdul lyers we cn relte em forces nd dsplcements of ems s: du du N N () P z z z z dw Q ( ) dw Q φ ( ) φ () B B E G dφ dφ M M (c) N () nd N () Q () nd Q () nd M () nd M () re te nternl l forces trnsverse ser forces nd endng moments n em nd em respectvely; B nd ( ) re te l ser nd endng stffness respectvely nd tey re epressed s 5 E B G E 6 E nd G ( ) re te longtudnl Young s modulus nd ser modulus of em respectvely; s te wdt of em. Assumng tt te ser stress s constnt troug te tckness of te desve lyer we cn estls te followng equlrum equtons y usng free ody dgrm sown n fgure : dn dn τ τ (3) dq dq σ σ (3) dm dm Q τ Q τ (3c) σ () σ () re te norml stresses long te A nterfce nd te norml stress long te PA nterfce respectvely; τ() s te ser stresses n te desve; Note tt te overll equlrum condton requres (gure ) 3 N NT Q Q Q QT M M N M T N E G E G gure. An desvely onded -lyered em system under concentrted lods Q Δ Q oncrete N Δ N Δ N M Q τ M ΔM N M τ Q τ Adesve RP Q Δ Q N Δ N M Δ M ( -c) σ σ τdτ Q T M T gure. ree ody dgrm of lyered em system N T APS 7 3

3 N T Q T nd M T re te correspondng resultng forces wt respect to te neutrl s of te RP plte; Q () s te ser force of te desve lyer wc s gven y τ(). Tree-Prmeter Elstc oundton Model of te Adesve yer Assume tt te desve lyer cn e tree-prmeter elstc foundton model (Kerr 965). As sown n g.3 te desve lyer cn e vewed s two lner norml sprng lyers wt stffness of K ntercollected y ser lyer wt constnts of G. By usng ts model te strn-stress reltons of te desve lyer re ten gven y: E σ ( w w ) E W σ ( w w ) (5) K E / G dw G W τ u φ u φ G (5) K E / W E nd G re young s modulus nd ser modulus of te desve respectvely. gure 3 Tree-prmeter elstc foundton model of desve lyer model gnore te l nd endng moment of te desve lyer equlrum condton of te desve lyer requres (g. ): dτ σ σ (6) Eq. (6) descres te ntercton etween te norml nd ser stresses wtn te desve lyer. Notng tt τ() cnges drstclly t te vcnty of te RP plte end Eq. (6) suggests tt σ () nd σ () re sgnfcntly dfferent t te vcnty of te end of te RP plte. n two-prmeter elstc foundton model σ () s ssumed equl to σ () even toug te left-nd sde of Eq. (6) s not zero. Terefore te force equlrum condton of te desve lyer s not stsfed n two-prmeter elstc foundton model. Governng fferentl Equton Susttutng te frst equton n Eq. (3) nto Eq. (5) nd dfferenttng ot sdes twce gves: G N M N M d w d N G (7) fferenttng te frst equton of Eq. (5) twce we ve d w d w d σ E () d w dq dφ d M d N M B B (9) d σ Susttutng Eqs. () to () nto Eq. (7) yelds: : d M 3 ( ) ( ) d Q d M d N 3 d N d N d M A M A NT A MT A A A A N () η A A ξ ( ) E G B G A () E E A η B E ξ 5 E E A A6 A 7 () Bsed on Eqs. (5) nd () we ve: ( ) ( ) d σ ( ) ( ) d σ E ( ) ( ) ( ) ( ) d w d w E dq d dq d φ φ () B B APS 7 33

4 Susttutng Eqs. (6) (3) nd (c) nto Eq. () we ve B d M d N d N d M B M B MT B B B BN 5 7 () E B B E E B5 B7. Elmntng M () rom Eqs. () nd () gves E B B B 6 d N d N d N E B d N 6 5N 6 T 7 T N M (5) : E E E E E E E 5 E B A B A B A B5 A5 E 5 A6 6 B B 7 A5B AB5 5 A A 7 A6B AB AB 6 A7B B7 A 5 5 AB AB 6 6 AB AB Eq. (5) s te governng equton of te desvely onded -em system sown n gure sed on treeprmeter elstc foundton model. ompred wt tt sed on two-prmeter elstc foundton model (Wng 3) Eq. (5) s two orders ger nd llows for mplementng ll te oundry condtons. nterfce stresses Te governng equtons Eq. (5) cn e solved troug te crcterstc equton nd tey re epressed for te forces n em : N N 5 S nd T re gven y: ΔN N M ΔM M Q ΔQ Q ΔN (6) R R R c e ΔM c S e ΔQ c T e E 6 NT M 6 E NT 7 5 E6 MT Q 5 E 5 (7) dn dm () 6 S ER ER ER E T SR R (9) R ( ) re egt roots of te crcterstc equton of Eq. (5). c ( ) re egt coeffcents to e determned y oundry condtons. N () M () nd Q () re prtculr solutons of Eq. (5) wc re te nternl forces of em f te wole desvely onded em s treted s composte em (Wng nd Qo ). Eq. (6) sows tt te nternl forces of te em consst of two prts: te eponentl terms (ΔN () ΔM () nd ΔQ ()) wc s te locl dsturnce of te reltve soft desve lyer nd stedy terms (N () M () nd Q ()) wc re te nternl forces of te em sed on composte em teory. By usng Eq. (3) te nterfce stresses re otned s σ Δσ σ Δσ ( ) ( ) ( ) ( ) ( ) τ Δτ τ σ σ () APS 7 3

5 Δ c T R R R σ e Δ R σ e Δ e c T R c R c R τ () dq dn τ () dq ( ) σ σ ( ) ( ) Smlr to te nternl forces te nterfce stresses re lso composed of two prts te eponentl terms representng te locl stresses concentrton ner te plte ends nd te stedy terms representng te composte em solutons. Boundry ondtons To determne te coeffcents c n te ove solutons necessry oundry condtons re needed. onsder te tree-pont endng test sown n g. n wc concentrted force P s ppled to te em t dstnce P from te left end of RP plte. To smplfy te clculton we cn only tke te onded zone s sown n g. () for nlyss. Te eternl forces ppled to te concrete em (N M Q N R M R nd Q R ) cn e esly determned y equlrum nlyss. n conventonl wy (Rnovtc nd rostg Smt nd Teng ) ts em s dvded nto two segments nd s sown n g. 3(). ndvdul solutons re wrtten for ec segment nd totl egt oundry condtons nd egt contnuty condton t te lodng pont re used to determne te coeffcents n te solutons. lerly te ove pproc s very tedous especlly wen more tn one dscontnutes of lodng estng n te onded zone. To elmnte ts tedous procedure used n te estng studes we propose new pproc to mplement oundry condtons y usng te prncple of superposton. As sown n Eqs. (6) nd () te em forces nd nterfce stresses (g. ()) cn e otned y superposng te composte em solutons (g. ()) nd te locl dsturnce (g. (c)). n g. () te oundry forces wt suscrpt t te left nd rgt ends of te em re gven y Eq. (). Besdes te forces ppled t two ends tere s no eternl force ppled to te em wtn te onded regon n g. (c). Terefore tere s no need to dvde te onded zone nto two segments nd tus te comple clculton nduced y mplementng contnuty condtons s voded. n ts cse we only to determned egt coeffcent c usng te followng oundry condtons: Δ N ( ) N N ( ) Δ M ( ) M M ( ) Δ Q ( ) Q Q ( ) τ ( ) Δ N ( ) N N ( ) Δ M ( ) M M ( ) Δ Q ( ) Q Q ( ) ( ) R R R τ (3) Te ove procedure s pplcle to generl lodng condtons nd re prtculrly effcent wen more tn one eternl lodng dscontnutes estng wtn te onded zone. VERATON N M M () N () N () M () Q () Q () M -M () Q -Q () N -N () -N () -M () -Q () Q R -Q () M R -M () N R -N () -N () -Q () -M (R) As verfctons R em strengtened y tn RP plte under tree-pont endng (gs. nd ) studed y Smt nd Teng () s emned. Te smply supported R em wt spn of 3 mm s sujected to md-spn lod of P 5 kn. Te dstnce from te support to te end of RP plte s 3 mm. Mterl propertes re gven s: Adereds E 3 MP (concrete) E MP (RP); Adesve E MP ν.35. Te geometres of te em re gven y: 3 mm mm mm Q () () (c) Q R Q () Q () M R N R M () N () N () M () gure. mplement oundry condtons usng te prncple of superposton APS 7 35

6 mm. Numercl solutons y EA re crred out s selne for comprson wt te commercl fnte element pckge ANSYS. soprmetrc egt-node qudrlterl elements re used to generte te mes. Ser nd norml stresses otned y te present metod nd EA re presented n g. 5. As comprson stresses predcted y two-prmetrc elstc foundton model (Smt nd Teng ; Wng 3) re lso presented n te fgure. t cn e oserved tt ecellent greement wt EA predcton s een ceved y te present model ecept very smll regon t te end of te RP plte cused y te stress sngulrtes estng t te edge corners. g. 5 verfes tt te present tree-prmeter elstc foundton model overcomes te drwcks of te well-known two-prmeter model successfully nd cptures two mportnt fetures of te nterfce stress of te RP strengtened concrete em. ser stress (MP) () stnce from te RPP end (mm) present model Wng 3 Smt nd Teng EA centerlne EA A nterfce EA PA nterfce norml stress (MP) () present model A nterfce present modle PA nterfce Wng 3 Smt nd Teng EA A nterfce EA PA nterfce stnce from te RP end (mm) gure 5. omprson of nterfce stress otned y dfferent metods: () ser stress; () norml stress ONUSON n ts study n nnovtve tree-prmeter elstc foundton model s proposed to predct te nterfce stresses of te RP strengtened concrete em. Ts model s drect etenson of te two-prmeter model used wdely n desvely onded jont nlyss. Te new model consders te deflecton of te desve lyer wc s mssng n two-prmeter elstc foundton model. Te ntercton etween te norml nd ser nterfce stress s lso counted for troug te equlrum condton of te desve lyer. n ts wy n egt-order governng dfferentl equton s reced wc mkes t possle to mplement ll te egt vlle oundry condtons Te slent fetures of te re s follows: ) t predcts correctly tensle norml stress dstruton long te A nterfce nd compressve norml stress long te PA nterfce t te vcnty of te RP plte end; ) t stsfes ll oundry condtons ncludng te zero ser stress t te edge of te desve lyer; 3) ts solutons re n eplct closed-form nd terefore cn e esly mplemented for nlyss nd desgn of RP strengtened ems. Te ccurcy of te present model s een verfed y ts good greements wt EA solutons. t sould e ponted out tt te present model cn lso e drectly used to nlyze generl desvely onded jonts. REERENES elle. Erdogn. nd Aydnoglu M.N. (9). Stress n desvely onded jonts: closed-for soluton Journl of omposte Mterls Golnd M. nd Ressner E. (9). Te Stresses n emented Jonts Journl of Appled Mecncs 66 A7- A7. Kerr A.. (965). A study of new foundton model Act Mecnc / 5 7. Rnovtc O. nd rostg Y.. losed-form g-order nlyss of R ems strengtened wt RP strps Journl of ompostes for onstructons ASE Smt S.T. nd Teng J.G. (). nterfcl stresses n plted ems Engneerng Structures Teng J.G. Zng J.W. nd Smt S.T. (). nterfcl stresses n R ems onded wt sofft plte: fnte element study onstructon nd Buldng Mterls 6() -. Wng J. (3) Mecncs nd frcture of yrd mterl nterfce ound P.. Tess eprtment of vl Engneerng te Unversty of Akron OH. Wng J. Qo P. (). nterfce frcture etween two ser deformle elstc lyers Journl of te Mecncs nd Pyscs of Solds 5(): 9-95 APS 7 36