Engineering Structures

Engineering Structures 33 (2011) 3432 3441 Contents lists ville t SciVerse ScienceDirect Engineering Structures journl homepge: www.elsevier.com/locte/engstruct Upper nd lower ounds for structurl design of RC memers with ductile response A. Crpinteri, M. Corrdo Politecnico di Torino, Deprtment of Structurl Engineering nd Geotechnics, Corso Duc degli Aruzzi 24, 10129, Torino, Itly r t i c l e i n f o s t r c t Article history: Received 27 Jnury 2011 Received in revised form 1 June 2011 Accepted 1 July 2011 Aville online 4 August 2011 Keywords: Reinforced concrete Rottionl cpcity Minimum reinforcement Size effects Frcture mechnics Dimensionl nlysis Code provisions In the present pper, some of the complex phenomen chrcterizing the flexurl ehviour of reinforced concrete ems, such s hyper-strength nd snp-ck nd snp-through instilities, re interpreted under unified pproch sed on nonliner frcture mechnics concepts. In prticulr, they re nlysed y mens of numericl lgorithm dopting the cohesive crck model for concrete in tension nd the overlpping crck model for concrete in compression. According to the ltter constitutive lw, fictitious interpenetrtion is ssumed to descrie the concrete dmge, nlogously to the fictitious crck opening used for tension. Such n integrted cohesive-overlpping crck model is pplied to ssess the minimum reinforcement mount necessry to prevent unstle tensile crck propgtion nd to evlute the rottionl cpcity of plstic hinges. The min novelty is given y the cpility to predict the size-scle effects evidenced y severl experimentl progrmmes ville in the literture. According to the numericl results otined, new prcticl design formule nd digrms re proposed, s well s, upper nd lower ounds to the reinforcement mount, the mteril properties nd the structurl dimensions re defined in order to void rittle filures. 2011 Elsevier Ltd. All rights reserved. 1. Introduction The glol response of reinforced concrete (RC) ems in ending during the loding process is chrcterized y complex phenomen due to mechnicl nonlinerities, s evidenced in the qulittive moment vs. rottion digrm shown in Fig. 1. In more detils, we refer to concrete frcturing, which produces hyper-strength in the incresing rnch, steel yielding, which is t the origin of ductile ehviour, nd concrete crushing, which determines decrese in the lod crrying cpcity nd, consequently, limit to the ultimte rottion. Although two filure modes re usully oserved in the flexurl ehviour, i.e., yielding of the steel reinforcement nd crushing of the compressed concrete, more complex phenomen cn ffect the lod vs. displcement reltionships: snp-through instility, defined s loss of stility in the controlled lod condition, nd snpck instility, representing loss of stility in the controlled displcement condition. Such phenomen re very generl, nd usully encountered in structurl prolems chrcterized y either geometricl or mechnicl nonlinerities. As n exmple, they my pper in the uckling response of elstic structures, s evidenced y von Kármán nd Tsien [1] for thin cylindricl shells under xil Corresponding uthor. Tel.: +39 011 5644873; fx: +39 011 5644899. E-mil ddress: muro.corrdo@polito.it (M. Corrdo). compression, nd y Crlson et l. [2] nd Kpln [3] for complete sphericl shells nd sphericl cps sujected to externl pressure. On the other hnd, snp-ck instilities cn e esily encountered when mterils exhiiting strin softening ehviours re considered. This is, for instnce, the cse of plin concrete sls in tension nd in ending, whose overll responses re highly influenced y the softening ehviour of the process zone hed of the rel crck tip. The detiled nlyticl nd numericl investigtions crried out y Crpinteri [4,5] y mens of the cohesive crck model put into evidence trnsition from softening to snp-ck instility either y incresing the specimen dimensions nd/or the mteril strength, or y decresing the mteril frcture energy. The virtul post-pek ctstrophic rnch, chrcterized y positive slope in the lod vs. displcement plne, cn e cptured only if the loding process is controlled y the crck mouth opening displcement. In this context, the ppliction of the cohesive crck model to three-point-ending tests on plin concrete ems hs permitted to descrie the size effects on the nominl flexurl tensile strength, σ n, function of the mximum lod in the hypotheses of liner strin distriution long the cross-section nd liner-elstic ehviour for concrete in tension nd compression. The numericl results, s function of the em height, re compred to the empiricl prescriptions provided y the Model Code 90 [6] nd the Eurocode 2 [7] in Fig. 2. The rtio σ n /σ u, where σ u is the limit stress tht the mteril cn loclly sustin nd, therefore, is mteril property, tends to unity only for very lrge heights, h, 0141-0296/$ see front mtter 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.engstruct.2011.07.007

A. Crpinteri, M. Corrdo / Engineering Structures 33 (2011) 3432 3441 3433 Nottions A s Steel reinforcement re; Thickness of the em; d Effective depth of the em; D M Coefficient of influence for the pplied moment; {D w } T Vector of the coefficients of influence for the nodl displcements; E c Elstic modulus of concrete; {F} Vector of nodl forces; G C Crushing energy of concrete; G F Frcture energy of concrete; {K M } Vector of the coefficients of influence for the pplied moment; K w Mtrix of the coefficients of influence for the nodl displcements; h Overll em height; L Spn of the em; l Length of the considered em element; M Applied ending moment; N C Stress rittleness numer in compression; N L P Lower limit for the reinforcement rittleness numer; N U P Upper limit for the reinforcement rittleness numer; P Applied lod; s Stress rittleness numer in tension; {w} Vector of nodl displcements; w c cr Criticl overlpping displcement; w t cr Criticl crck opening displcement; δ Mid-spn deflection; ε c,u Ultimte elstic compressive strin of concrete; ε t,u Ultimte elstic tensile strin of concrete; ϑ Loclized rottion of the considered em element; ϑ PL Plstic component of the loclized rottion; ρ (A s /h) 100, steel reinforcement percentge; σ c Averge compressive strength of concrete; σ n Nominl flexurl tensile strength of concrete; σ u Averge tensile strength of concrete; Tensile yield strength of steel. σ y wheres it is equl to three for h tending to zero. Such n increse in the pprent tensile strength is due to the comined effect of the post-pek softening ehviour nd the high strin grdient chrcterizing the ending condition. Trnsitions from softening to snp-ck instility lso chrcterize the overll response of qusi-rittle mterils in compression y vrying the specimen size nd/or slenderness (see the experimentl tests y vn Mier [8] nd Jnsen nd Shh [9]). In this context, recent nlyticl nd numericl studies hve een crried out y Crpinteri et l. [10,11] in the cse of concrete specimens sujected to unixil nd eccentric compression. Finlly, snp-through locl instilities re often evidenced in the cse of composite mterils, such s reinforced nd firereinforced concrete, due to the fct tht fires ct s crck-rrest mechnisms, producing glol ductile response. Experimentl evidence hs een found in cement mtrix specimens reinforced y glss-fire undles [12], reinforced concrete ems [13], nd lumin mtrix smples with SiC whiskers [14]. Very interesting interprettions of such phenomenon hve een proposed y mens of the ridged crck model [15 17]. All the forementioned spects hold fundmentl role in the study of RC memers, where the ppliction of simplified constitutive reltions for concrete do not permit to model ll Fig. 1. Nonliner contriutions involved in the flexurl ehviour of RC elements. Fig. 2. Size-effect on the flexurl tensile strength: ppliction of the cohesive crck model nd prescriptions of the design codes. the experimentlly oserved effects. In the stress strin lws proposed y Stndrds, in fct, the concrete tensile contriution s well s the softening rnch nd the locliztion of strin in the post-pek regime oth for tension nd compression, re usully neglected. In the cse of low reinforcement percentges, for instnce, the tensile concrete contriution, my determine hyper-strength with respect to the ultimte loding condition, nd consequent possile instility in the overll mechnicl response, the post-pek crcking moment eing monotonic decresing function of the crck length. For this reson, ll ntionl nd interntionl codes of prctice provide empiricl formule for the determintion of the minimum reinforcement mount which enle RC memers to prevent unstle crck propgtion. To this regrd, it is worth noting tht most of these prescriptions do not consider the size effects. Anlogously, the most common pproches used to nlyse the ehviour of over-reinforced concrete ems in ending, usully sed on stress vs. strin constitutive lws for concrete in compression, do not permit to descrie the decrese in the lod crrying cpcity due to concrete crushing nd the experimentlly oserved size effects on the plstic rottion cpcity [18 20]. It hs to e noticed tht, lso in this cse, the current prescriptions provided y codes of prctice completely disregrd the effects of the memer size. In the present pper, the min fetures of numericl method recently developed y Crpinteri et l. [21,22] le to descrie the nonliner ehviour of RC memers during oth frcturing nd crushing is riefly outlined. In prticulr, it will e shown tht the crushing process cn e efficiently nlysed ccording to the overlpping crck model [10], which considers fictitious mteril interpenetrtion in the post-pek regime. With the proposed lgorithm it is possile to completely cpture the moment vs. rottion response of severl experimentl tests ville in the literture. As

3434 A. Crpinteri, M. Corrdo / Engineering Structures 33 (2011) 3432 3441 Fig. 3. Cohesive crck model: () liner-elstic σ ε lw; () post-pek softening σ w reltionship. Fig. 4. Overlpping crck model: () liner-elstic σ ε lw; () post-pek softening σ w reltionship. result of prmetric investigtion, new reltionship for the minimum reinforcement mount, which is ffected y size-scle effects, will e determined. It will e shown tht the prescriptions of some codes re not conservtive for smll structurl sizes, wheres other ones over-estimte the minimum mount, especilly in the cse of lrge structurl sizes. Then, plstic rottion vs. neutrl xis position curves will e determined, tht re found to e dependent on the structurl dimension nd the steel percentge. Also in this cse, it will e shown tht the Europen prescriptions do not consider the trnsition from ductile-to-rittle response y incresing the structurl size, leding to over-estimte the rottionl cpcity of lrge ems. 2. Numericl pproch In this section, the numericl lgorithm proposed y Crpinteri et l. [21,22] for the nlysis of the mechnicl ehviour of portion of n RC em sujected to constnt ending moment, M, is riefly descried. This element, hving spn to height rtio equl to unity, is representtive of the zone of em where plstic hinge formtion tkes plce. Such rtio hs een chosen ccording to the experimentl evidence tht most of the contriutions to the plstic rottion re generted within this portion (s n exmple, the Eurocode 2 [7] suggests to consider rtio equl to 1.2). Then, it is ssumed tht frcturing nd crushing processes re fully loclized long the mid-spn cross-section of the element, wheres the prt of the hinge outside the locliztion zone is ssumed elstic. This ssumption lso implies tht only one equivlent min tensile crck is considered. The loding process is chrcterized y crck propgtion in tension, steel yielding nd/or slippge s well s concrete crushing in compression. 2.1. Constitutive models In the proposed lgorithm, the ehviour of concrete in tension is descried y mens of the well estlished cohesive crck model [23], lrgely used, in the pst, to study the ductile-torittle trnsition in plin concrete ems in ending [5]. According to this model, the dopted constitutive lw is stress strin liner-elstic reltionship up to the chievement of the verge tensile strength, σ u, for the undmged zone (Fig. 3()), nd stress displcement reltionship descriing the process zone ehviour up to the criticl opening, w t cr, eyond which the trnsferred trction vnishes (Fig. 3()). The softening function, σ = f (w), is considered s mteril property, s well s the criticl vlue of the crck opening, w t cr, nd the frcture energy, G F. The shpe of f (w) my vry from liner to iliner or even more complicted reltionships depending on the chrcteristics of the mteril considered nd the prolem nlysed. The criticl vlue of the crck opening displcement is pproximtely equl to 0.1 mm, nd the frcture energy is ssumed to vry from 0.050 N/mm to 0.150 N/mm, depending on concrete strength nd mximum ggregte dimeter, ccording to the prescriptions given y the Model Code 90. As fr s modelling of concrete crushing filure is concerned, the overlpping crck model [10] is dopted. According to such n pproch, sed on the originl insights provided y Kotsovos [24] nd vn Mier [8], nd relted to the pioneering work y Hillerorg [25], the inelstic nd loclized deformtion in the post-pek regime is descried y fictitious interpenetrtion of the mteril, while the remining prt of the specimen undergoes n elstic unloding. As result, pir of constitutive lws for concrete in compression is introduced, in close nlogy with the cohesive crck model: stress strin reltionship until the compressive strength is chieved (Fig. 4()), nd stress displcement (overlpping) reltionship descriing the phenomenon of concrete crushing (Fig. 4()). The ltter lw, pproximted y liner softening reltionship for modelling purposes, descries how the stress in the dmged mteril decreses from its mximum vlue s the fictitious interpenetrtion increses. It is worth noting tht the crushing energy, G C, which is dissipted surfce energy, is defined s the re elow the post-pek softening curve in Fig. 4(). It cn e ssumed s true mteril property, since it is only slightly ffected y the structurl size, s shown in Ref. [11], where lrge vlidtion of the overlpping model for concrete-like mterils hs

A. Crpinteri, M. Corrdo / Engineering Structures 33 (2011) 3432 3441 3435 procedures of such n pproch hve een proposed y Ruiz et l. [28]. Typiclly, the reltionships otined re chrcterized y n scending rnch up to steel yielding, to which corresponds the criticl vlue of the crck opening for steel, w y (see Fig. 5). After tht, the steel rection is nerly constnt. As regrds the vlue of w y, it is ffected y the prmeters influencing the ond slip ehviour, such s the dimeter, the numer nd the roughness of the rers. In this study, n verge vlue equl to 0.3 mm hs een used for ny steel percentges. It cn e ssumed s the representtive for ried rs, commonly used in prcticl pplictions. The contriution of the steel yielding strin to w y is considered s negligile compred to the steel concrete slip. Due to lck of studies on the ehviour of reinforcing rs in compression, the sme σ w reltionship is dopted for oth tension nd compression. Fig. 5. Elsto-plstic σ w constitutive lw for steel rers. een proposed in the cse of specimens with different vlues of slenderness nd/or sizes. Certinly, it is worth noting tht the compressive dmge is more spred phenomenon, chrcterized y n energy dissiption within multiscle or frctl domin. On the other hnd, the vlue of the physicl dimension of such domins, closer to 2.0 thn 3.0 (out 2.3 for concrete, ccording to the nlysis performed y Crpinteri nd Corrdo [26]), mkes cler the effectiveness of the simplified overlpping crck model, which, in fct, ssumes dissiption over surfce. By vrying the concrete verge compressive strength from 20 to 90 MP, the crushing energy rnges from 30 to 58 N/mm. The criticl vlue for the crushing interpenetrtion, w c cr, is experimentlly found to e pproximtely equl to 1 mm, nd it is decresing function of the compressive strength. When lterl confinement is exerted y stirrups, the empiricl eqution recently proposed y Suzuki et l. [27] cn e pplied to clculte the crushing energy. Both G C nd w c cr re incresing functions of concrete confinement. As fr s the ehviour of the steel reinforcement is concerned, it is impossile to dopt the clssicl σ ε lws, since the kinemtics of the mid-spn cross-section of the RC memer is descried y mens of displcements, insted of strins. Therefore, we ssume tht the reinforcement cts through concentrted forces, functions of the reltive opening displcement, equilirting the generlized stress torque in the cross-section. To this im, constitutive reltionships etween the reinforcement rection nd the crck opening displcement re otined y mens of preliminry studies crried out on the interction etween the reinforcing r nd the surrounding concrete. The integrtion of the differentil slips over the trnsfer length, l tr, is equl to hlf the crck opening t the reinforcement level, wheres the integrtion of the ond stresses gives the reinforcement rection. Simplified 2.2. Numericl lgorithm The RC memer is considered s constituted y two symmetricl elements chrcterized y n elstic ehviour, nd connected y mens of (n) pirs of nodes (Fig. 6()). In this pproch, ll the mechnicl nonlinerities re loclized in the mid-spn crosssection, where cohesive nd overlpping stresses re replced y equivlent nodl forces, F i, y integrting the corresponding stresses over the nodl spcing. Such nodl forces depend on the nodl opening or closing displcements ccording to the cohesive or overlpping softening lws previously introduced. With reference to Fig. 6(), the horizontl forces, F i, cting t the ith node long the mid-spn cross-section cn e computed s follows: {F} = [K w ]{w} + {K M }M, (1) where: {F} is the vector of nodl forces, [K w ] is the mtrix of the coefficients of influence for the nodl displcements, {w} is the vector of nodl displcements, {K M } is the vector of the coefficients of influence for the pplied moment M. Eq. (1) constitutes liner lgeric system of (n) equtions nd (2n + 1) unknowns, {F}, {w} nd M. With reference to the generic sitution reported in Fig. 6(), (n) dditionl equtions cn e introduced y considering the constitutive lws for concrete in tension nd compression nd for the reinforcement in the node r (see [22] for more detils). The lst dditionl eqution derives from the strength criterion dopted to govern the propgtion processes. At ech step of the loding process, in fct, we cn set either the force in the fictitious crck tip, m, equl to the ultimte tensile force, F u, or the force in the fictitious crushing tip, p, equl to the ultimte compressive force, F c. It is importnt to note tht the condition for crck propgtion (corresponding to the chievement of the tensile strength t the fictitious crck tip, m) does not imply tht the compressive strength is reched t the Fig. 6. Finite element nodes (); nd force distriution with cohesive crck in tension nd crushing in compression () long the mid-spn cross-section.

3436 A. Crpinteri, M. Corrdo / Engineering Structures 33 (2011) 3432 3441 () ρ = 0.085%. () ρ = 0.256%. (c) ρ = 0.653%. Fig. 7. Comprison etween numericl nd experimentl [13] lod vs. mid-spn deflection curves for em height h = 0.1 m. () ρ = 0.064%. () ρ = 0.190%. (c) ρ = 0.490%. Fig. 8. Comprison etween numericl nd experimentl [13] lod vs. mid-spn deflection curves for em height h = 0.2 m. () ρ = 0.043%. () ρ = 0.128%. (c) ρ = 0.327%. Fig. 9. Comprison etween numericl nd experimentl [13] lod vs. mid-spn deflection curves for em height h = 0.4 m. corresponding overlpping crck tip, p, nd vice vers. Hence, the driving prmeter of the process is the tip tht in the considered step hs reched the limit resistnce. Only this tip is moved when pssing to the next step. Finlly, t ech step of the lgorithm, it is possile to clculte the loclized em rottion, ϑ, s follows: ϑ = {D w } T {w} + D M M, (2) where {D w } is the vector of the coefficients of influence for the nodl displcements nd D M is the coefficient of influence for the pplied moment. It hs to e oserved tht the coefficients entering Eqs. (1) nd (2) re computed priori using liner finite element nlysis. They re connected y simple reltions of proportionlity to the structurl dimension, nd, therefore, it is not necessry to repet the finite element nlysis for ny different considered em size. 3. Comprison of numericl predictions nd experimentl results In this section, comprison etween the numericl predictions using the cohesive/overlpping crck model nd the results of two experimentl cmpigns is presented. First, the three-pointending tests crried out y Bosco et l. [13] on reinforced highstrength concrete ems to investigte the size-scle effects on the minimum reinforcement mount re considered. Three different size-scles were nlysed, chrcterized y overll height, h, equl to 0.1, 0.2 nd 0.4 m, nd constnt width,, equl to 0.15 m. The spn to depth rtio ws equl to 6. Five different steel percentges, ρ, were considered for ech em size. The model prmeters hve een set equl to the ctul concrete nd steel properties, tht cn e deduced from Ref. [13]. In the numericl simultions, the RC element of Fig. 6() is ssumed to e representtive of the mid-spn portion of the em sujected to three-point-ending test. As result, the mid-spn deflection is otined s the sum of the loclized rottion given y Eq. (2), nd the elstic contriution, ccording to the following expression: δ = δ loc + δ el = ϑl PL 3 4 + 1 48 E c I (3) where L is the em spn, P is the pplied lod, E c is the concrete elstic modulus, nd I is reduced moment of inerti of the crosssection, introduced to tke into ccount the stiffness reduction due to smered crcking long the em spn. The hypothesis of locliztion of the nonlinerities long the symmetry crosssection, in fct, does not permit to consider this effect. More in detils, the vlue of I is empiriclly selected in order to otin the stiffness of the incresing rnch of the numericl lod vs. deflection curves equl to the experimentl ones (see Figs. 7 9). In the present study, the prolem hs not een deepened more

A. Crpinteri, M. Corrdo / Engineering Structures 33 (2011) 3432 3441 3437 () h = 0.2 m. () h = 0.4 m. (c) h = 0.6 m. Fig. 10. Comprison etween numericl nd experimentl results [20] for different em heights: h = 0.2 m (); h = 0.4 m (); h = 0.6 m (c). since it does not influence the minimum nd mximum steel percentges, defined on the sis of lod crrying cpcities. Some of the numericl simultions compred to the corresponding experimentl results, in terms of pplied lod vs. mid-spn deflection curves, re shown in Figs. 7 9. Such curves evidence trnsition from n overll softening response to hrdening response y incresing the steel percentge, with the ppernce of locl snp-through instilities. Furthermore, it is worth noting tht the condition for which the pek-crcking lod is equl to the ultimte lod occurs for vlues of reinforcement percentge decresing with the em size (curves in Figs. 7 9). Such phenomenon evidence size-scle effect. Aiming t further vlidtion of the proposed model in the cse of high steel percentges, the experimentl nlysis crried out y Bosco nd Deernrdi [20] on RC ems to investigte the sizescle effects on the rottionl cpcity is lso considered. In order to otin consistent comprison, the numericl simultions hve een crried out y modelling the em portion positioned t the mid-spn of the em. This element is chrcterized y spn to height rtio equl to one. The rottions of such portion, where the lrgest mount of ductility is developed, were experimentlly determined s functions of the pplied ending moment. Also in this cse, the mechnicl nd geometricl prmeters re set equl to the experimentl vlues. Numericl nd experimentl moment rottion curves re compred in Fig. 10() (c) for different em heights nd different steel percentges. Such digrms put into evidence tht the mximum rottion is decresing function of the tensile reinforcement rtio nd of the em height. In the cse of low steel percentges, the mechnicl ehviour is chrcterized y the reinforcement yielding nd the mechnicl response is lmost plstic. By incresing the reinforcement mount, the contriution of concrete crushing ecomes more nd more evident with the ppernce of softening rnch t the end of the plstic plteu. This is n importnt feture of the proposed model, which lso permits to follow snp-ck rnches y controlling the loding process through the length of the tensile crck nd the extension of the fictitious crushing zone, rther thn y the externl lod or the centrl deflection. Good greement ws otined etween numericl nd experimentl results for ll the tested ems. 4. Numericl results nd prcticl prescriptions for design 4.1. Minimum reinforcement In this section, new reltionship etween the minimum reinforcement nd the mechnicl nd geometricl prmeters is proposed on the sis of wide prmetric nlysis. To this im, different vlues of the em height, h, rnging from 0.1 nd 3.2 m, nd different vlues of the concrete verge compressive strength, σ c, rnging from 16 to 76 MP, hve een considered. All the other mechnicl properties of concrete, s, for instnce, the tensile strength nd the frcture energy, hve een evluted ccording to the reltionships provided y the Model Code 90 [6] nd reported in Tle 1. As regrds the steel reinforcement, yield strength σ y = 600 MP, nd n elstic modulus E s = 200 GP hve een ssumed. The rtio etween effective nd overll depth, d/h, hs een fixed equl to 0.9. For ech of the considered ems, severl simultions hve een crried out y vrying the steel percentge, in order to find the minimum reinforcement mount. In prticulr, such vlue is determined when the pek crcking lod, P cr, is equl to the ultimte lod, P u, s shown in Fig. 11(). In order to etter clrify the effects of ech of the vriles involved in the physicl phenomenon, s well s their interction to govern the glol response, reduction of the primry vriles is otined y comining them into dimensionless groups y mens of dimensionl nlysis. According to the numericl model proposed in the previous section, the generl functionl reltionship mong the quntities tht chrcterize the phenomenon is the following: M = φ σ u, G F, σ c, G C, E c, σ y, ρ, h; h, lh, ϑ. (4) When the flexurl ehviour of lightly-reinforced concrete ems is studied, the prmeters descriing the ehviour of concrete in compression, σ c nd G C, cn e omitted, since the crushing filure is not involved in the filure mechnism. On the other hnd, only the em height, h, cn e considered if the geometricl rtios of the smples, /h nd l/h, re ssumed to e constnt. This ssumption permits to investigte the size effects ll the em dimensions vrying with the em height wheres the effects of the width nd the slenderness re not tken into ccount. More precisely, the resistnt moment is ssumed to e independent of the slenderness, nd linerly dependent on the em width. Under these hypotheses, the ppliction of Buckinghm s Π- Theorem [29] for physicl similrity nd scle modelling yields to the following reltionship: M σu h 0.5 h 2.5 = Φ 1, ρ σ yh 0.5, ϑ E ch 0.5, (5) G F E c GF E c GF E c GF E c if h nd G F E c, which corresponds to the mteril toughness K IC, re ssumed s the dimensionlly independent vriles. As consequence, the dimensionless functionl reltion for the proposed model ecomes: M = Φ 2 (s, N P, ϑ n ), (6) where: s = nd K IC σ u h 0.5 (7) N P = ρ σ yh 0.5 K IC (8)

3438 A. Crpinteri, M. Corrdo / Engineering Structures 33 (2011) 3432 3441 Fig. 11. Definition of minimum reinforcement (); nd est-fit reltionship of numericl results (not filled-in symols) etween N L P nd s. Filled-in symols refer to the experimentl results from [13]. Tle 1 Mechnicl nd geometricl prmeters of the ems considered in the numericl simultions. σ c (MP) σ u (MP) G F (N/mm) E c (MP) G C (N/mm) h (mm) ρ min (%) s N L P ρ mx (%) N C N U P 16 1.2 0.042 25 147 30.0 30 2.4 0.065 31 008 30.0 40 3.0 0.079 34 129 30.0 65 4.5 0.111 40 124 43.8 76 5.0 0.124 42 271 49.1 100 0.070 2.698 0.130 1.430 0.184 0.099 200 0.063 1.908 0.165 1.320 0.261 0.129 400 0.056 1.349 0.206 1.160 0.368 0.160 800 0.050 0.954 0.262 0.980 0.521 0.191 1600 0.046 0.675 0.344 0.820 0.737 0.227 3200 0.042 0.477 0.440 0.654 1.042 0.256 100 0.123 1.867 0.165 2.240 0.311 0.139 200 0.110 1.320 0.207 1.960 0.440 0.172 400 0.099 0.933 0.265 1.650 0.622 0.205 800 0.090 0.660 0.341 1.350 0.880 0.238 1600 0.083 0.467 0.447 1.095 1.244 0.272 3200 0.075 0.330 0.571 0.937 1.760 0.330 100 0.150 1.733 0.173 2.700 0.395 0.160 200 0.135 1.225 0.220 2.270 0.559 0.190 400 0.122 0.866 0.282 1.880 0.791 0.223 800 0.111 0.613 0.362 1.550 1.118 0.260 1600 0.102 0.433 0.471 1.300 1.581 0.308 3200 0.100 0.306 0.653 1.106 2.236 0.0371 100 0.215 1.484 0.193 3.900 0.490 0.177 200 0.194 1.050 0.246 3.260 0.693 0.209 400 0.176 0.742 0.316 2.700 0.981 0.244 800 0.162 0.525 0.412 2.290 1.387 0.293 1600 0.146 0.371 0.526 1.940 1.961 0.351 3200 0.132 0.262 0.672 1.600 2.774 0.410 100 0.237 1.448 0.196 4.400 0.528 0.183 200 0.215 1.024 0.252 3.680 0.747 0.217 400 0.195 0.724 0.323 3.000 1.056 0.250 800 0.179 0.512 0.419 2.530 1.493 0.298 1600 0.161 0.362 0.535 2.200 2.111 0.367 3200 0.160 0.256 0.750 1.810 2.986 0.427 re the governing nondimensionl numers, M is the nondimensionl ending moment, nd ϑ n is the normlized locl rottion. Eqs. (7) nd (8) define the stress nd the reinforcement rittleness numers, introduced y Crpinteri [30,31]. As result, ech numericl simultion is completely descried y different couple of vlues s nd N P. In prticulr, the vlue of N P reltive to the condition of minimum reinforcement is referred to s N L P, where superscript L stnds for lower, since it will define the lower limit to the rnge of ductile response. The vlues of s nd N L P for the numericl simultions crried out in this study re shown in Fig. 11() nd reported in Tle 1. In the rnge of interest for common structurl pplictions, h rnging from 0.1 m up to 3.2 m, the trend otined cn e descried with very good pproximtion (goodness of fit r 2 = 0.99) y the following hyperolic curve: N L P = 0.26 s 0.71. (9) By sustituting Eqs. (7) nd (8) into (9), the following reltionship etween the minimum reinforcement percentge nd the mechnicl nd geometricl properties of the em is otined: ρ min = 0.26 σu h 0.5 K IC 0.71 K IC u K = 0.26σ0.71 0.29 IC 0.5 σ y h σ y h 0.15. (10) The new proposed formul is compred to the prescriptions of the design codes in Fig. 12, where the vlues of minimum reinforcement re reported s functions of the em height, h, for concrete verge compressive strength σ c = 40 MP, nd steel yielding strength σ y = 450 MP. In prticulr, ρ min is defined s A s,min /h, where A s,min is the minimum reinforcement mount, in the hypothesis of rtio etween effective nd overll depth, d/h, equl to 0.9. It cn e seen tht only the Norwegin Stndrds NS 3473 E (1989) ccount for the effect of the memer size. On the contrry, ll the other considered curves completely disregrd the size-scle effects. With respect to the proposed pproch, more complete study of the prolem, for definitive Stndrds improvement, should lso consider the vriility of the reinforcement cover,

A. Crpinteri, M. Corrdo / Engineering Structures 33 (2011) 3432 3441 3439 Fig. 12. Minimum reinforcement percentge, defined s A s,min /h, vs. em depth ccording to vrious design codes. of the steel concrete interfce, s well s of the size nd the numer of the rers. In this pper, it is ssumed tht they only mrginlly ffect the rittle-to-ductile trnsition, tht is the min investigted topic, together with the size effects. Furthermore, it is worth noting tht more restrictive prescriptions for the minimum mount of reinforcement my rise from the limittions to the crck mouth opening displcement in the serviceility conditions, necessry to prevent steel corrosion nd improve durility. Also in this cse, size-effects re expected, s experimentlly oserved y Ysir Alm et l. [32]. In this context, the proposed model will permit to evlute the crck mouth opening displcement under the serviceility lods y tking into ccount the closing contriutions exerted y the cohesive forces long the fictitious crck. However, the interction etween crcks long the em spn nd the steel concrete interction will ply n importnt role in determining the correct vlue of the crck opening. The former effect cn e considered y nlysing em portion hving length equl to the crck spcing evluted, for instnce, ccording to the procedure proposed t the Pr. 7.3.4. of the Eurocode 2 Prt 1-1 [7]. Such length, function of the r dimeter nd the effective steel percentge, will influence the trnsfer length for the sher stresses etween rers nd surrounding concrete nd, therefore, the steel concrete interction. At this regrds, more relile results will e otined y improving the lgorithm with step-y-step nlysis of the ond slip ehviour together with the crck propgtion process. 4.2. Plstic rottion cpcity A second detiled numericl study is proposed to nlyse the effect of ech prmeter to the plstic rottion cpcity. With reference to the typicl moment versus rottion curve otined y the ppliction of the proposed lgorithm nd shown in Fig. 13(), the plstic component of the totl rottion cn e otined s the difference etween the rottion eyond which the moment strts descending rpidly nd the rottion corresponding to the reinforcement yielding. It is worth noting tht the softening or even snp-ck rnches t the end of the plstic plteu re usully due to concrete crushing, nd cn e cptured only if the crushing zone extension is ssumed to govern the loding process. The results of severl numericl simultions, crried out y considering different em heights nd reinforcement percentges, re summrized in the plstic rottion, ϑ PL, vs. reltive neutrl xis position, x/d, digrm shown in Fig. 13(). Such digrm is consistent with the prcticl prescriptions of the Eurocode 2 [7], of whom, tht reltive to high ductility steel nd concrete compressive strength less thn or equl to 50 MP is lso provided (dshed curve in Fig. 13()). Bems with height equl to 0.2 m hve rottionl cpcity greter thn tht suggested y the code. On the other hnd, y incresing the em height up to 0.8 m, the rottions provided y the code pper to e not conservtive. It is worth noting tht the numericl results for h = 0.4 m re in good greement with the curve provided y the code, which represents the 5%-frctile of the plstic rottions of ems or sls with height of out 0.3 m (see [33] for more detils). The new proposed curves cn e esily used in structurl design. In the cse of plstic structurl nlysis, in fct, the designer hs to verify tht the rottion required for the moment redistriution is lower thn the dmissile one. To this im, for given vlue of x/d otined from the ppliction of the ultimte stte nlysis, he cn enter Fig. 13() nd determine the dmissile plstic rottion s function of the em size. It is evident from the digrms in Fig. 13() tht the plstic rottion cpcity tends to zero s the neutrl xis reltive position coordinte increses, i.e. the tensile reinforcement percentge increses. In prticulr, it is possile to define n upper limit to the reinforcement mount eyond which the steel does not yield, nd the em collpses in compression, without the development of significnt ductility. Such limit, function of ll the vriles involved in the phenomenon, cn e otined y mens of dimensionl nlysis, s previously done for the minimum reinforcement. Since now we re interested in over-reinforced concrete ems, the Buckinghm s Π-Theorem is pplied to the generl functionl reltionship expressed in Eq. (4) y omitting the prmeters descriing the ehviour of concrete in tension, σ u nd G F. When h nd G C E c re ssumed s the dimensionlly independent vriles, the dimensionless functionl reltionship ecomes: M = Φ 2 (N C, N P, ϑ n ), (11) where: Fig. 13. Definition of plstic rottion (); nd predicted plstic rottion for different em heights (solid lines) compred with the Eurocode 2 prescription (dshed line) ().

3440 A. Crpinteri, M. Corrdo / Engineering Structures 33 (2011) 3432 3441 By sustituting Eqs. (14) nd (15) into (16), the mximum reinforcement cn e expressed s function of the mechnicl nd geometricl properties, s follows: ρ mx = σ 0.49 0.51 c GC E c 0.25. (17) σ y h 0.25 5. Discussion nd conclusions Fig. 14. Best-fit reltionship of numericl results (not filled-in symols) etween N U P nd N C. M M = h 2.5, (12) G C E c ϑ n = ϑ E ch 0.5 GC E c, (13) N P = ρ σ yh 0.5 GC E c (14) nd N C = σ ch 0.5 GC E c. (15) N P nd N C re two nondimensionl numers defined for overreinforced concrete ems. The ems considered in the previous section for the evlution of the minimum reinforcement re now nlysed in cse of high reinforcement mount. In prticulr, for ech of the ems, severl numericl simultions hve een crried out in order to find the limit vlue of the reinforcement percentge eyond which the steel does not yield. The vlues of N P corresponding to this sitution, referred to s N U P, where superscript U stnds for upper limit, re reported together with the vlues of N C in Tle 1. In the considered rnge for the em height, the reltionship etween N U P nd N C cn e descried with very good pproximtion (goodness of fit r 2 = 0.99) y the following power lw (see Fig. 14): N U P = 0.25N 0.49 C. (16) In the present pper, numericl method le to descrie the nonliner ehviour of RC memers during oth tensile frcturing nd compression crushing hs een presented. With the proposed lgorithm, sed on nonliner frcture mechnics concepts, it is possile to completely cpture the moment vs. rottion response of ll the intermedite situtions rnging from plin concrete to over-reinforced concrete ems under monotonic lodings. In prticulr, the prolem of minimum reinforcement nd the ehviour of plstic hinges hve een nlysed in order to highlight the limits of the code prescriptions nd provide new esy-to-use design formule nd digrms. According to the new proposl expressed y Eq. (10), the minimum reinforcement percentge, ρ min, is n incresing function of the concrete tensile strength nd toughness, wheres it decreses s the steel yielding strength nd the em height increse. As regrds the size-scle effects, the presence of cohesive closing stresses determines vrition in ρ min with the em size descried y the power h 0.15. With reference to the existing code provisions, the proposed formul permits to sve steel reinforcement in the cse of lrge structures. As fr s the plstic rottion cpcity is concerned, the numericl results otined, summrized in Fig. 13(), show tht ϑ PL is not only dependent on the neutrl xis position. This ssumption, in fct, leds to over-estimte the rottionl cpcity of deep ems. In order to improve the code provisions, the effect of the structurl dimension should e explicitly tken into ccount y considering different design curves s, for instnce, those proposed in Fig. 13(). As result of the comprehensive investigtion proposed in this pper, the conditions for the structurl design of RC elements exhiiting ductile response cn e highlighted. From qulittive point of view, the decrese in one prmeter mong h, ρ, nd σ y, or the increse in σ u or G F, ll the other prmeters eing kept constnt, determines trnsition from ductile response to unstle tensile crck propgtion, s represented in Fig. 15(). On the other hnd, the increse in h, or ρ, or σ y, or the decrese in σ c or G C, ll the other prmeters eing kept constnt, produces trnsition towrds crushing filure without steel yielding (see Fig. 15. Conditions for structurl design of RC memers with ductile response (); nd corresponding limit vlues for the tensile steel reinforcement ().

A. Crpinteri, M. Corrdo / Engineering Structures 33 (2011) 3432 3441 3441 Fig. 15()). The exct vlues of the upper nd the lower ounds to the reinforcement percentge given y Eqs. (10) nd (17) re shown in Fig. 15() s function of the em height, for n verge concrete compressive strength σ c = 40 MP, nd steel yield strength σ y = 450 MP. Finlly, lthough the proposed pproch hs een proved to e very effective in the situtions chrcterized y high grdient in the ending moment digrm, such s in the three-pointending prolem or in the portion over n intermedite support of continuous em, it cn e profitly pply lso to more generl frme clcultion. The moment vs. rottion reltionships representtive of the ehviour of the plstic hinge regions, in fct, cn e dopted in step-y-step plstic structurl nlysis, replcing the usully used elsto-perfectly plstic lw without limittion to the mximum rottion. In this cse, the ending moment redistriution, nd, therefore, the ultimte lod crrying cpcity, could e determined y the chievement of the sizedependent rottionl cpcity in one of the plstic hinges, efore the formtion of perfectly-plstic collpse mechnism. Acknowledgements The finncil supports provided y the Ministry of University nd Scientific Reserch (MIUR) to the project Advnced pplictions of Frcture Mechnics for the study of integrity nd durility of mterils nd structures, nd y the Regione Piemonte to the RE-FRESCOS project Preservtion, sfegurd nd vloriztion of msonry decortions in the rchitecturl historicl heritge of Piedmont, re grtefully cknowledged. References [1] von Kármán T, Tsien HS. The uckling of thin cylindricl shells under xil compression. J Aerosol Sci 1941;8:303 12. [2] Crlson RL, Sendleeck RL, Hoff NJ. Experimentl studies of the uckling of complete sphericl shells. Exp Mech 1967;7:281 8. [3] Kpln A. Buckling of sphericl shells. In: Fung YC, Sechler EE, editors. Thin shell structures, theory, experiment, nd design. Englewood Cliffs (NJ): Prentice-Hll Inc.; 1974. p. 248 88. [4] Crpinteri A. Interprettion of the Griffith instility s ifurction of the glol equilirium. In: Shh SP, editor. Proc. NATO dvnced reserch workshop on ppliction of frcture mechnics to cementitious composites. Dordrecht (The Netherlnds): Mrtinus Nijhoff Pulishers; 1985. p. 287 316. [5] Crpinteri A. Size effects on strength, toughness, nd ductility. J Eng Mech, ASCE 1989;115:1375 92. [6] Comité Euro-Interntionl du Béton. CEB-FIP model code 1990. Lusnne: Thoms Telford Ltd.; 1993. [7] CEN TC/250. Eurocode 2: design of concrete structures, prt 1 1: generl rules nd rules for uildings. Brussels. 2004. [8] vn Mier JGM. Strin-softening of concrete under multixil loding conditions. Ph.D. thesis. Eindhoven University of Technology. 1984. [9] Jnsen DC, Shh SP. Effect of length on compressive strin softening of concrete. J Eng Mech 1997;123:25 35. 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