Confined reinforced concrete beam. Soumis le 14/02/1999 Accepté le 03/06/2000

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1 Conine reinore onree beam Soumi le 4/0/999 epé le 03/06/000 Réumé Dan e arile, un moele pour le béon armé oniné e propoé. La relaion e Ken e Park e généraliée pour prenre en ompe l ee e armaure longiuinale, le moule élaiié u béon e elui e armaure ur la réiane e la uilié e élémen en béon armé. Une nouvelle approhe pour l analye e la branh eenane e la ourbe onraineéormaion u béon en ompreion e égalemen propoée. Mo lé: oninemen, uilié, réiane, béon armé, armaure longiuinale. bra In hi paper, a moel o onine reinore onree member i propoe. The Ken an Park relaion i generalie o ake ino aoun he longiuinal reinoremen ee on he rengh an uiliy o reinore onree member. new approah or analyi o ompreive re- rain urve o onree i alo propoe. Keywor: oninemen, uiliy, rengh, reinore onree, longiuinal reinoremen. I. ZRGU (). CHRIF () () Déparemen e Génie Civil Univerié Menouri Conanine (lgérie) () Iniu Hyraulique Univerié e Bana Bana (lgérie) T p pt G µ T ST S T O p D uiliy i an imporan aor in whole ruural elemen ha have a large eormaion wihou a grea lo in rengh. I i alo an imporan aor ine i i irely relae o he aey o he ruure. Many uie [,,3,4] on he uiliy o reinore onree beam ube o lexure have hown he ee o ome parameer on heir uiliy an rengh. There i no ye a unique relaion ha may be ue univerally. Rihar an al. [5] a well a Balmer an al. [6] oun ha he rengh o onree i aee by lui preure or irular piral oninemen. Oher uie have propoe moel or onree onine by ranvere reinoremen. Sargin an al. [7] have propoe a general equaion or re-rain urve relae o he onen, paing an yiel rengh o he ranvere eel an onree rengh. The Ken an Park [8] moel or onree onine by reangular hoop ombine many o he previouly propoe moel. Irawan an Maekawa [9] oun ha he reinoremen raio an paing an he lexural ine o laeral ie have a igniian beneiial ee on he oninemen aion. I ha been aume ha he rengh gain by laeral reinoremen i proporional o he prou o reinoremen raio an eel yiel rengh [0]. However, he longiuinal reinoremen ee on uiliy an rengh ha been neglee unil reenly. Some oe [] ake ino onieraion he ee o eel ype an eel raio on uiliy o beam. I ha been alo repore [] ha he longiuinal reinoremen may onine he onree elemen. Beaue he ranvere reinoremen provie he onining reaion o longiuinal reinoremen. Shehaa an Shehaa [3] oun ha he longiuinal eel raio igniianly ae he beam uiliy. oring o hee obervaion, a new moel or onine onree i propoe. MTRILS LWS Conree moel in ompreion The ompreion moel ue in hi uy i bae on he Ken an Park [8] moel. I i evelope o ake ino aoun he longiuinal reinoremen ee on he onine onree. The number o bar o

2 eel, he paing o reangular hoop an Young moulu o onree an eel are oniere in hi new moel. The relaion are given a ollow: Wih: Wih: : p p u p p : z z uner onene orm, where: an b p b In whih:, : uniaxial re an rain o onree;, :Young elai moulu o eel an onree repeively; " : peak ompreive re o onine onree; p : rain a peak ompreive re u : maximum rain; ; : raio o volume o ranvere reinoremen o volume o onree ore meaure o ouie o hoop; : ummaion o iameer o longiuinal reinoremen; : paing o hoop; z : Parameer eining he lope o he linear eening branh o he ompreive re- rain urve o onree; b : wih o onree ore meaure o ouie o hoop; b, : imenion o ro eion o beam elemen; Tenion iening I i aume ha he onree will arry ome enile ree when i reahe i ulimae enile rengh. Thi approah ha been ue by many auhor [4,5]. bilinear re-rain urve i aope, wih a linear aening branh an a linear oening branh or onree aer raking (ig. ). Figure : Bilinear re-rain relaion or onree in enion. Thi behaviour may be haraerie a ollow: p : : p : 0 In whih : : ire enile rengh; : angen rain-oening moulu; : rain a peak enile re; p : inal rain when he enile re i reue o zero. Seel reinoremen The eel i aume a elai perely plai, haraerie by Young elai moulu an uniaxial yiel re y. FINIT LMNT NLYSIS reinore onree beam i ivie ino many imenional beam elemen. ah elemen i aume o have hree egree o reeom. I reangular eion i ivie ino a iree number o layer in he ireion perpeniular o he axi o ymmery. The geomery o eah layer i eine by i lengh, wih an he iane beween i mi-heigh an he reerene axi y i (ig. ). In eah ro eion he ine marix i obaine by ummaion o imilar marie o eah layer. Figure : Layere moel an rain iribuion or onree eion. The uual Bernoulli-Navier hypohei ha a plane ro eion o he beam remain plane an orhogonal i aope. Furher, i i aume ha he average rain in eel equal he average rain in onree a he ame level. The lineariy o rain iribuion require ha 0. ky : rain a h layer, 0 rain a axi reerene, k : urvaure, y : iane beween reerene axi an he mi-heigh o layer. The ine marix elemen are given a ollow: M D M I p

3 : ean axial ine, I : ean lexure ine, M :ean oupling ine, I n n M Y n n n Y n Y Y n, n : Number o onree layer an eel layer in a ro eion repeively; : Young moulu o elaiiy o layer ; : rea o onree layer. : rea o eel layer NON-LINR SOLUTION PROCDUR Newon-Raphon non-linear meho i aope in whih he ean ine marie are upae a he beginning o eah ieraion. The onvergene rierion ue in hi uy i he one ue by hme an al. [6]. The ieraive yle are repeae unil he onvergene. The onvergene i reahe when he ierene beween he axial rain a he reerene axi ( 0 ) an he urvaure () eermine in wo ueive yle are maller han a erain olerane, an hen he loa i inremene. The onvergene rierion i eine by he ollowing relaion: 0 I I u x i L I i, : L I : elemen lengh. are he roaion in noe i an repeively, Deerminaion o he maximum loa an analyi o he eening branh o re-rain urve During he ieraive proeure, i he maximum loa i exeee, he onvergene will no ake plae an he maximum number o ieraion i reahe. ah ime hi number i reahe, he program goe bak o he previou loaing an hange he loa inremen one again o hal he inremen ep. When he value o he ep ( p i ) beome maller han a erain olerane, he program give he maximum loa (ig. 3). The eermine maximum loa orrepon o he peak o re rain urve. The oorinae o hi poin will be ueul or he analyi o he eening branh o hi urve. One he peak i eermine, he program hange he oorinae yem in orer o be able o analye he eening branh o he re rain urve. The hange o oorinae yem i illurae in igure 3. The eening branh o he urve, in he oorinae yem ( - ), beome an aening one, in he oorinae yem ( - ). Figure 3: Repreenaion o he hange o axi a he peak in re-rain urve. Comparion wih e aa Two numerial example eribing boh ingly an oubly reinore onree imply uppore beam ubee o a onenrae loa, ee by Pera [7] an lami an Ferguon [8] are preene in able, where we alulae he loa eleion iagram. (MPa) y (MPa) (MPa) (MPa) L(mm) b(mm) h(mm) Table : Mehanial an geomerial properie o beam ee by Pera [7] an lami an Ferguon [8]. Comparion o he preiion o our moel wih hee aa are hown in igure 4, an igure 5. Maerial parameer orreponing o he hown urve are ummarie in able. The value o he parameer neee in hi moel were hoe repore by hee reearher. Figure 4 an igure 5 how he omparion o he e reul wih he alulae reul regaring loa-eleion relaionhip. In igure 4, experimenal aa an alulae reul obaine by lami an Ferguon [8] an hoe obaine by he preen moel are repore. i an be een rom hi igure, here i no ierene beween he preen moel an he one obaine rom he experimenal aa. The ee o longiuinal reinoremen ha no appeare. The longiuinal eel raio in hi ae i maller han ha ue in he ae o he beam ee by Pera [7]. The preen

4 moel how a ligh inreae in ine a ompare o he reul obaine by Merabe [9] (ig. 5). Thi inreae o ine i he ee o he longiuinal reinoremen oninemen. CONCLUSION moel o onine reinore onree member i propoe. I ake ino aoun he longiuinal reinoremen ee on he rengh an uiliy o member. The omparion beween he propoe moel an he experimenal reul how a goo agreemen beween hem an valiae he preen moel. Thi moel may rae he behaviour o reinore onree elemen up o he maximum loa. The hange o he ompreive re-rain urve axi proue a he peak re an be eaily ue o analye he eening branh o he re- rain urve. Figure 4: Comparion o preen moel wih eleion beam ee by lami an Ferguon [8]. Figure 5: Comparion o preen moel wih eleion beam ee by Pera [7]. In igure 6, he plain line repreen he reul rom he preen moel whih ake ino aoun he longiuinal reinoremen ee on he uiliy an rengh o he onree member. On he ame igure, he reul preie by Ken an Park moel [8] or ompreive re-rain urve are alo hown. The eening branh o he urve preen a ligh inreae o uiliy in our moel. The inreae o rengh an uiliy an be obviouly aribue o he longiuinal reinoremen ee. Figure 6: Comparion beween he monooni ompreive rerain urve preie by Ken an Park [8] an he propoe moel. RFRNCS []- Lelie K.., Raagopalan S., verar N.J., CI Journal Proeeing, V.73, n 9 (976), pp []- Tognon G., Urella P., Coppei G., CI Journal Proeeing, V. 77, n 3 (980), pp [3]- Waon S., Journal o he Sruural Diviion, SC, 0, n 6 (994), pp [4]- Légeron F. Paulre P., 4 h inernaional ympoium on Uilizaion o High-rengh /High-perormane onree, Pari (996). [5]- Rihar F.., Branzaeg., Brown R.L., Univeriy o Illinoi ngineering xperimenal Saion, Bullein n 90, (99), 74 p. [6]- Balmer G.G., Sruural Reearh Laboraory, Repor n SP- 3,U. S. Bureau o relamaion, (949),3 p. [7]- Sargin M., Ghoh S.K., Hana V.K., Magazine o onree reearh, V. 3, n 75-76, June-epember (97), pp [8]- Ken D.C. Park R., Journal o he Sruural Diviion, SC, V. 97, ST 7, (97), pp [9]- Pallewaa J.M., Irawan P. an Maekawa K., J. Maerial, Con. Sru., Pavemen, V. 8, n 50 (995), pp [0]- So B.D., Park R., Prieley M.J.N., CI Journal Proeeing, V. 79 (98), pp []- Teor T., "Comié uro-inernaional u béon, CB FIP, Moel Coe 990", (993), 457 p. []- Park R. Pauley T., "Reinore onree ruure", John Wiley an Son, (975), 769 p. [3]- Shehaa I...M., Shehaa L.C.D., 4 h Inernaional Sympoium on Uilizaion o High-rengh /Highperormane onree, Pari, (996). [4]- Sanlon., Murray D.W., SC Journal o he ruural iviion, 00 (ST9), (974), pp [5]- Cope R.J., Rao P.V., Clark L.., CSC-SC-CI-CB Inernaional Sympoium, Univeriy o Waerloo, On., (979), pp [6]- hme M.H., Ibrahim M.M.I., Moaa, K.Z., min S.., Thir rab Sruural ngineering Conerene, U.. mirae Univeriy, (989), pp [7]- Pera J., Thee e Doeur Ingénieur, INS Lyon, (973), 86 p. [8]- lami Z.Y., Ferguon P.M., CI Journal Proeeing, V.60, n (963), pp [9]- Merabe O., Thèe e Doora, INS Lyon, (990), 67 p.

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