G. Monti 1 and S. Alessandri 2. Professor, Dept. of Structural Engineering and Geotechnics, Sapienza, University of Rome, Rome, Italy 2

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1 The 4 th World Coferee o Earthquake Egieerig Otober -7, 008, Beijig, Chia DESIGN EQUATIONS FOR EVALUATION OF STRENGTH AND DEFORMATION CAPACITY FOR UNSTRENGTHENED AND FRP STRENGTHENED RC RECTANGULAR COLUMNS UNDER COMBINED BIAXIAL BENDING AND AXIAL LOAD G. Moti ad S. Aleadri Profeor, Dept. of Strutural Egieerig ad Geotehi, Sapieza, Uiverity of Rome, Rome, Italy Dept. of Strutural Egieerig ad Geotehi, Sapieza, Uiverity of Rome, Rome, Italy giorgio.moti@uiroma.it, ilvia.aleadri@uiroma.it ABSTRACT: Whe aeig the eimi performae of exitig reifored orete buildig deiged aordig to obolete ode, oe a idetify potetially dagerou ituatio that ould reult i atatrophi failure. A typial iadequay lie i the o-alled trog beam-weak olum ituatio: olum are o o tregth to fore plati hige formatio i beam ad, if thi i exteded to all olum at a give floor, a lead to the developmet of a oft-torey mehaim. Suh weakee hould be elimiated by upgradig all weak olum i the zoe of potetial formatio of plati hige o to ireae their flexural apaity ad to fore plati hige formatio i the beam. With thi aim oe or more layer of FRP ould be wrapped logitudially alog the olum at the ed zoe. Neverthele, the flexural tregtheig a redue the elemet deformatio apaity. Thi paper, tartig from previou author work propoe a implified proedure for the aemet of flexural tregth ad deformatio apaity of r utregtheed ad FRP tregtheed reifored orete retagular olum. Approximate loed form equatio are developed by whih the flexural tregth ad etio ultimate urvature are evaluated a a futio of the ormalized atig axial load ad the geometrial ad mehaial etio parameter. The propoed approah i ompared with a fibre approah. KEYWORDS: RC olum, FRP tregtheig, iteratio domai, biaxial bedig, loed-form equatio, eimi upgrade. EXACT APPROACH.. Evaluatio of etio tregth apaity Claial method for hek of RC member uder ombied biaxial bedig ad axial load are baed o the otrutio of the D failure domai ( N, M, M ). The boudary of the failure domai (the o alled x y iteratio failure urfae ) defie the limit ter ( N Rd, M xrd, M yrd ) that aue ultimate limit tate ahievemet. It otrutio i performed poit by poit by itegratio of tree aoiated to the trai ditributio, orrepodig to a flexural failure mode for the etio. Reifored orete etio aalyi at the ultimate limit tate i baed o the followig uual hypothee: plae etio remai plae (liear trai), perfet bod betwee teel ad orete, o teile tregth i orete, o liear tre trai law for teel ad orete. The trai tate over the etio i therefore uiquely defied by the orete ompreio trai ad by the teel teile trai. Flexural failure our whe oe of the followig oditio i met: the orete ultimate trai,, or the teel teile ultimate trai, : u u

2 The 4 th World Coferee o Earthquake Egieerig Otober -7, 008, Beijig, Chia 0 u, u ultimate (.) 0 u, u The limit ter N Sd, M xrd, M yrd, orrepodig to boudary poit o the etio failure urfae are alulated otiuouly modifyig the eutral depth for every value of the eutral axi agle ad olvig at eah tep the equilibrium equatio: N da da (.) Sd A A M yda yda (.) xrd A A yrd A A M xda xda (.4) where i the orete tre, i the tre i teel reiforemet, ad A the teel reiforemet area... Evaluatio of etio ultimate urvature at otat axial load A i the orete ompreed area Setio urvature aoiated with a axial load ad bedig momet a be evaluated o the bai of the ame hypothei ued for determiatio of reitig momet ad from the requiremet of trai ompatibility ad equilibrium of fore. The ultimate urvature, u, a be evaluated a: m u (.5) u h where u i the o dimeioal eutral axi depth of the ompreed zoe at failure, h i the etio height, meaured orthogoally to the eutral axi, ad m i the trai at the extreme ompreed fiber: u orete failure m u (.6) u teel failure u To evaluate the ultimate urvature the eutral axi depth, u, mut be alulated by iteratively olvig the tralatioal equilibrium equatio (.).. APPROXIMATE APPROACH.. Evaluatio of etio tregth apaity... Utregtheed etio Cotrutio of the D failure domai ivolve ome omputatioal diffiultie maily due to the itegratio of tre over the ompreive portio of orete ad to the aalytial ad graphial repreetatio of the urfae ad the ompario with the atig exteral fore. A implified method, whih allow to avoid the umerial itegratio required for olutio of the equilibrium equatio, i fully deribed i Moti et al., 006; it i baed o the aalytial approximatio of the failure urfae by etio at otat axial load propoed by Breler, 960, ad expreed a: where m m ux m m uy 0x 0y m ux, m uy = ormalized uiaxial reitig momet uder the ormalized applied axial load Sd ; m 0x, (.)

3 The 4 th World Coferee o Earthquake Egieerig Otober -7, 008, Beijig, Chia m 0 y = ormalized reitig momet about the etio mai axi x, y, give by: NSd M M x y, mx, m f b h y f b h f b h d d where: b etio width, h etio height, fd deig ompreive tregth of orete (0.85 aout for log term load), expoet depedig o the ro etio geometry, the teel reiforemet peretage ad the axial load. The expoet i evaluated a a futio of the ro etio mehaial ad Sd geometrial parameter ad the ormalized applied axial load Sd : b x y x y (.) h where Sx, Sy = mehaial ratio of teel reiforemet laid parallel to x ad y etio axi give, repetively, by: Ax fyd Ay fyd x, y, (.4) f b h f b h d with A Sx, A Sy = area of reiforemet laid parallel to x ad y axi; f yd = deig yield tregth of teel. The value of parameter i the equatio (.) have bee obtaied through the leat quare method; they are how i Table.. Table.. Parameter for alulatio of expoet for utregtheed etio Sd x y > = < Fig. how the ompario betwee the orrepodig failure domai for everal value of the bai parameter ( bh, x, y, Sd ). It a be ee that the implified equatio orretly repreet the iteratio diagram of the etio.... FRP-tregtheed etio FRP tregtheed RC etio aalyi at the ultimate limit tate i baed o the ame uual hypothee adopted for the utregtheed etio, with the additio of the followig: perfet bod betwee FRP ad orete, o ompreive tregth i FRP, liear tre trai law for FRP. The trai tate over the etio i uiquely defied by the orete ompreio trai ad by the FRP teile trai. Flexural failure our whe oe of the followig oditio i met: the orete ultimate trai, f, or the FRP ultimate trai, u f : 0 u, fd f ultimate (.5) 0 u, f fd The expliit relatiohip developed for the expoet i equatio (.) i here modified to ilude the FRP reiforemet mehaial parameter: fd b x y fx fy x y fx fy h (.6) where fx ad fy are the mehaial ratio of FRP-tregtheig laid parallel to x ad y etio axi d d (.)

4 ormalized bedig momet my ormalized bedig momet my The 4 th World Coferee o Earthquake Egieerig Otober -7, 008, Beijig, Chia give, repetively, by: fx Afx ffd, f b h d fy Afy ffd, (.7) f b h with A fx, A fy =area of FRP reiforemet laid parallel to x ad y axi; f fd = deig tregth of FRP. The value of the parameter i the equatio (.6) have bee obtaied through the leat quare method; they are how i Table.. Table.. Parameter for alulatio of expoet for FRP-tregtheed etio x y fx fy > Fig. how the ompario betwee the orrepodig failure domai for everal value of the bai parameter ( bh, x, y, fx, fy, Sd ). It a be ee that the implified equatio orretly repreet the iteratio diagram of the etio. d Utregtheed etio FRP-tregtheed etio Fig.. Compario betwee exat (fiber method) ad approximate approahe for a RC etio uder ombied biaxial bedig ad axial load... Evaluatio of etio ultimate urvature The approximate approah for reitig momet evaluatio a be exteded to evaluatio of etio ultimate urvature. Setio ultimate urvature aoiated to the atig axial load Sd ad reitig bedig momet m Rd a be evaluated by: where: bh bh ormalized bedig momet mx exat approah approximate approah 0. bh u u yu xu x0u Sd y0u Sd bh Sd bh f bh.0 f f f 0. f 0.05 f 0. Sd 0. 0 Sd 0. bh.0 bh ormalized bedig momet mx xu, yu etio urvature about the etio mai axi related to the ompoet m xrd, m yrd of the ormalized reitig bedig momet m Rd uder the atig axial load Sd x0u y0u etio urvature about the etio mai axi at ultimate. The u expoet a be evaluated a a futio of the ro-etio mehaial ad geometrial parameter ad the ormalized applied axial load Sd : exat approah approximate approah (.8) ;,

5 The 4 th World Coferee o Earthquake Egieerig Otober -7, 008, Beijig, Chia u b x u yu u u x y utregtheed etio h u (.9) u b x u y u fx u fyu u u x y fx fy FRP tregtheed etio h The value of the parameter i the equatio (.9) have bee obtaied through the leat quare method for differet kid of retagular ro etio (obtaied by varyig the bai parameter); thee value are how i Table.. ad Table.4. Table..Parameter for alulatio of u expoet for utregtheed etio Sd x y > = < Table.4. Parameter for alulatio of u expoet for FRP-tregtheed etio x y fx fy > Cloed-form equatio for uiaxial bedig apaitie of etio with double ymmetri teel reiforemet For appliatio of equatio (.) the uiaxial reitig momet mut be evaluated beforehad. To thi aim the tralatioal equilibrium equatio mut be iteratively olved to fid the eutral axi depth uder the atig axial load Sd. I order to avoid the iterative olutio, the eutral axi depth a be expreed a a futio of the atig axial load by implified loed form equatio. Aordig to the otatio of Fig., a implified model with a equivalet area of reiforig teel uiformly ditributed aroud the etio ide i ued; by thi way the equilibrium equatio, for a etio with two-way teel reiforemet ad FRP heet aroud eah ide, a be writte i a o dimeioal form a follow: f f f f f (.) m 0. k. k. k (.) 0. 5 f f f 0. 5 k f f 0. 5 f f I the previou equatio the o dimeioal parameter defie the depth of the equivalet tre blok ormalized with repet to the etio height h ; the ubript idetifie the material ( for orete, for teel ad f for FRP), the uperript idetifie the ompreio or the teio, while the ymbol ad defie the diretio with repet to the eutral axi. The ymbol idetify the equivalet tre blok, while k are for the relevat reultat depth. The ymbol idiate the over ratio evaluated orthogoally to the eutral axi. The parameter ad, f ad orthogoal teel ad FRP reiforemet, repetively, where f repreet the mehaial ratio of parallel ad A ad A, Af ad Af are their area.

6 The 4 th World Coferee o Earthquake Egieerig Otober -7, 008, Beijig, Chia h y x y x d'h d h h = A A f y A f A x A A f y + - y f d + f + yd f - - yd A f f fdd A f yd yd b orete teel FRP model whit with uiformly etio trai tree ditributed teel reiforemet ' f d f yd f yd - - f f f fdd Fig.. No-dimeioal quatitie depitig the geometry ad the trai ad tre tate (i orete, teel ad FRP) of a RC etio with two-way teel reiforemet. The oeffiiet i the equilibrium equatio (.) ad (.) have bee alulated for the differet mode by whih etioal failure a our (their value/expreio are give i Table.5, Table.6 ad Table.7) defiig, for eah mode, a implified expreio to evaluate the o dimeioal eutral axi depth. Three differet etioal failure mode a be defied: u ad t u (mode ); ad u t yd (mode); u u ad 0 (mode). yd where t i the trai at the extreme teile fiber ad u i the ultimate teile trai ( u u, for utregtheed etio ad u fd for FRP- tregtheed etio). Mode a be ubdivided ito two ub mode: mode a ad mode b, whih differ i the ompreio teel tate, either elati or yielded; mode i oly for utregtheed etio, beaue of FRP-tregtheig i effetivee oly for yielded etio. Table.5. Value ad expreio of the oeffiiet i equatio (.) ad (.) for differet failure mode Failure Mode b 0.8 u yd u yd u 0.5 yd 0.5 yd u 0.5 yd u 0.5 yd 0.5 u 0.5 u yd Table.6. Value ad expreio of the oeffiiet i equatio (.) ad (.) for differet failure mode Failure Mode k k ( ) k ( )

7 The 4 th World Coferee o Earthquake Egieerig Otober -7, 008, Beijig, Chia Table.7. Value ad expreio of the oeffiiet for FRP-tregtheig i equatio (.) ad (.) for differet failure mode Failure Mode a b f u fd f u 0.5 fd The equatio of the eutral axi depth are give for differet failure mode i Table.8; i failure mode a, ad they are get by applyig the eat method. The parameter A i give by the followig expreio: ( ) u ( ) yd f ( ) f fd fd A (.) ( b ) f 0.5 Table.8. Value/expreio of o-dimeioal eutral axi depth for differet failure mode. Failure mode a b k f d f f f ab d f f f d f f A f A u Sd yd yd u yd u yd To defie the relevat failure mode of the etio the ormalized atig axial load Sd mut be ompared with the value, i, at the failure mode boudarie, that a be evaluated by uig the expreio give i Table.0, with the value of the eutral axi depth give i Table.0. Oe the relevat failure mode of the etio ha bee defied ad the eutral axi depth m a be evaluated uig equatio (.) with oeffiiet give i Table.5. ha bee evaluated, the ormalized reitig momet 0 Sd Table.9. Value ad expreio of o dimeioal eutral axi depth for differet failure mode boudarie. Failure mode 0 a a b 4 boudarie yd d u u d u 0 yd u u u u yd

8 The 4 th World Coferee o Earthquake Egieerig Otober -7, 008, Beijig, Chia Table.0. Expreio of o dimeioal axial load for differet failure mode boudarie. Failure mode boudarie i 0 a a b 0. 8 ab a b The ultimate urvature, give by: b 0. 8 b b yd u yd u u, a be evaluated a a futio of the o-dimeioal atig axial load, m u h u (.4) where u i the o dimeioal eutral axi depth of the ompreed zoe at failure ad m i the trai at the extreme ompreed fiber.. CONCLUSIONS A method ha bee propoed that arrive at defiig loed-form equatio for performig the aemet of exitig RC olum with two-way teel reiforemet, uder ombied biaxial bedig ad axial load, ad the deig of the FRP flexural tregtheig. Startig from the load otour method origially propoed by Breler (960) ad from a previou author work (Moti et. al. 006), a effiiet proedure for etimatig the tregth/deformatio etio apaity ha bee developed. I additio, imple loed-form equatio for omputig etio uiaxial reitig momet ad ultimate urvature ha bee defied. The reult obtaied tetig the approximate approahe o retagular r etio with differet geometrial ad mehaial harateriti are ompared with that obtaied from a exat oe, whih make ue of the diretizatio fibre method. The urve obtaied with the firt how very little deviatio from the exat oe. The propoed method led itelf to a traightforward aemet of retagular orete olum: tartig from the aiged axial load, the failure mode i diretly foud ad the orrepodig momet/urvature apaity omputed. Sd. It i ACKNOWLEDGEMENTS Thi work ha bee arried out uder the program Dipartimeto di Protezioe Civile Coorzio RELUIS, iged o (. 540), Reearh Lie, whoe fiaial upport wa greatly appreiated. REFERENCES Moti, G., S. Aleadri. (006). Aemet of RC Colum Uder Combied Biaxial Bedig ad Axial Load. The Seod fib Cogre. 5-8 Jue 006, Naple, Italy. Breler, B. (960). Deig riteria for reifored olum uder axial load ad biaxial bedig. Joural of the. Ameria Corete Ititute, 57(5), Farmigto Hill, Mi., November 960,