Minimum flexural reinforcement in rectangular and T-section concrete beams

Transcription

1 Tehnial Paper lberto Carpinteri Eria Cadamuro Mauro Corrado* DOI: /suo Minimum lexural reinorement in retangular and T-setion onrete beams The presriptions provided by odes o pratie or assessing the minimum reinorement amount or strength purposes in reinored onrete beams usually disregard the non-linear ontribution o onrete in tension and size-sale eets. In the present paper, these phenomena are taken into aount orretly in the desription o the lexural ailure in lightly reinored onrete beams by means o a numerial algorithm based on non-linear rature mehanis. In this ontext, the appliation o dimensional analysis permits a redution in the number o governing parameters. In partiular, it is demonstrated analytially that only two dimensionless parameters, reerred to as reinorement brittleness number and stress brittleness number, are responsible or the brittle-to-dutile transition in the mehanial response. ording to this approah, new ormulae suitable or evaluating the minimum reinorement in pratial appliations is proposed or both retangular and T-setions. omparison with experimental results demonstrates the eetiveness o the proposed model or dierent reinorement perentages and beam depths. Keywords: reinored onrete, ode provisions, minimum reinorement, dimensional analysis, size eets, ohesive rak 1 Introdution The minimum reinorement amount in onrete elements is usually determined by two dierent requirements: limiting rak width in the servieability limit state onditions and avoiding the hyper-strength phenomenon. s regards rak ontrol, this is required or dierent reasons, suh as appearane, imperviousness and durability [1]. From the point o view o durability, or instane, raks in the onrete over may lead to the ingress o various agents (e.g. oxygen, water and hlorides), whih may in turn lead to the onset and propagation o orrosion o the steel reinorement. These requirements are usually satisied by limiting the rak width to levels speiied by standards (see, or instane, setions to in ib Model Code or Conrete Strutures 2010 [2]). Sine rak opening is a untion o the tensile stress in the rebars, among other parameters, the upper limits to rak width * Corresponding author: mauro.orrado@polito.it Submitted or review: 19 July 2013 Revised: 2 January 2014 epted or publiation: 29 January 2014 beome lower limits or the area o steel reinorement. On the other hand, the hyper-strength phenomenon is avoided by imposing that the maximum raking load is lower than the ultimate load, given by the reation o the yielded reinorement multiplied by the moment arm. For this purpose, several ormulae are provided by dierent standards, although most o them are too simplisti. The present paper ouses on the strength riterion, with partiular emphasis on the issue o size eets, whih are urrently disregarded by the standards. 1.1 Code provisions Limit analysis o reinored onrete (RC) beams usually disregards the non-linear ontribution o onrete in tension in the evaluation o the load-arrying apaity. This assumption does not always lead to a sae design ondition, e.g. in the ase o lexural members that or arhitetural or other reasons have a larger ross-setion than that required or strength. With a very small amount o tensile reinorement, the tensile onrete strength in at makes a substantial ontribution to the deinition o the peak raking load, whih may turn out to be higher than the ultimate load. In this ase a sudden drop in the loadarrying apaity takes plae, rom the value o the maximum raking load to the ultimate load (hyper-strength phenomenon). This behaviour is haraterized by unstable rak propagation, with a derease in the rature moment while the rak extends. For this reason, all national and international odes o pratie impose limitations on the minimum reinorement amount in order to prevent sudden ailure. Most o them onsider only two parameters: onrete grade and steel yield strength, whereas the eets o other important parameters, e.g. beam depth, are ompletely negleted. ording to the approah o the CI 318 Building Code [3, 4], the ondition o minimum reinorement is deined by the equality φm u = M r, where φ is a resistane ator, assumed to be 0.9, M u is the nominal lexural resistane (or ultimate load) given by the reation o the yielded reinorement times the moment arm, and M r is the raking moment o the plain onrete setion evaluated on the basis o the lexural tensile strength t,l. By means o simple substitutions and some analytial manipulations (see Seguirant et al. [5] or more details), the value or the minimum reinorement area an be expressed thus: 2014 Ernst & Sohn Verlag ür rhitektur und tehnishe Wissenshaten GmbH & Co. KG, Berlin 361

2 . Carpinteri/E. Cadamuro/M. Corrado Minimum lexural reinorement in retangular and T-setion onrete beams where: C k b w h d K 12 h K 7.5 d 2 (MPa) bwd(mm 2) k C 5.1 (1) (2) multiplier to adjust setion modulus or dierent beam shapes harateristi ompressive strength o onrete yield strength o steel width o beam overall depth o beam eetive depth o beam For T-beams with the lange in ompression, using a value C = 1.5 (it is 1.0 or retangular members), and h/d = 1.2, parameter K assumes a value slightly larger than 3. Thereore, the expression adopted in CI [3], and still inluded in CI [4], is 0.25 (MPa) (mm 2 b w ) 1.4 wd b d (mm 2) k The expression provided by the 2004 Euroode 2 [6] and ib Model Code 2010 [2] has a similar derivation. The relation proposed or both retangular and T-beams is based on the average uniaxial tensile strength o onrete tm instead o the lexural tensile strength t,l : 0.26 tm bwd bwd It is worth noting that the oeiient 0.26 in Eq. (4) is derived by assuming a ratio h/d = 1.2 and a resistane ator φ=0.9. Only the Norwegian Standard NS 3473 E [7] aounts or the eet o member size. The reinorement should have a total ross-setional area equal to = 0.35k w tk,0.05 / (5) where tk,0.05 is the lower harateristi tensile strength o onrete and is the gross ross-setional area. The size eets are taken into aount by means o ator k w, equal to 1.5 h/h 1 1, where h is the beam depth in m and h 1 = 1.0 m. In this ase only retangular members are onsidered. 1.2 Models or omputing minimum reinorement From the modelling point o view, signiiant ontributions to the evaluation o the minimum lexural reinorement were derived rom the appliation o the bridged rak model [8, 9] an approah based on linear elasti rature mehanis (LEFM) to the study o rak propagation in the presene o reinorement. More preisely, the model onsiders an RC beam element haraterized (3) (4) by a retangular ross-setion with a layer o steel reinorement and an edge rak. The stress intensity ator at the rak tip K I is a untion o the externally applied bending moment and the reinorement reation by means o suitable shape untions. The raking moment is obtained when K I reahes its ritial value K IC, and the orresponding reinorement reation is determined by setting a kinematial ompatibility ondition on the rak opening (rak opening displaement equal to zero beore steel yielding). Both o these are untions o the relative rak length. ording to the analytial ormulation o the bridged rak model, the overall behaviour is only a untion o the reinorement brittleness number N P, deined on the basis o the mehanial and geometrial parameters [8]: N P b h s w h 0.5 K IC where: steel yield strength b w width o beam h overall depth o beam K IC onrete rature toughness s area o tension reinorement lthough originally deined as a brittleness number, N P represents a dutility parameter: the larger its value, the more dutile is the mehanial behaviour. In partiular, the yielding and the ultimate bending moments are inreasing untions o N P. Thereore, a brittle response, with a sotening branh ater the peak raking load, is expeted or low values o N P, whereas by inreasing its value (it typially ranges rom 0.1 to 10), a dutile response is predited, with a hardening behaviour ater the raking load and a large inelasti displaement due to steel yielding. However, it is worth noting that by inreasing N P beyond an upper limit value, whih is a untion o the strutural sale, the overall response beomes brittle again due to the appearane o a rushing ollapse beore steel yielding. s ar as the minimum reinorement amount is onerned, it is possible to deine N PC as the ritial value separating the brittle response rom the dutile one. The ollowing empirial equation has been proposed by Boso and Carpinteri [10, 11] to express the dependene o N PC on the average onrete ompressive strength m : N PC = m (7) ording to Eq. (7), higher values o N PC are obtained by inreasing the onrete ompressive strength. The ormula or the minimum reinorement area an be obtained by mathing the expression o N P (Eq. (6)) to that o N PC (Eq. (7)) and solving or s : K IC m bwh 0.5 Eq. (8) reveals how depends on the mehanial and geometrial properties. In partiular, it is proportional to the square root o the beam depth h. (6) (8) 362

3 . Carpinteri/E. Cadamuro/M. Corrado Minimum lexural reinorement in retangular and T-setion onrete beams Several other models have been proposed or evaluating the minimum reinorement amount. Most o them reveal a derease in the minimum amount by inreasing the size sale, although there is a large variation in the results. This is mainly due to the dierent deinitions assumed or the minimum reinorement as well as the dierent approahes used to takle the problem. Hawkins and Hjorteset [12], or instane, studied the problem with the ohesive rak model, and proposed an expression or the minimum reinorement on the basis o a size-dependent nominal lexural tensile strength o onrete. Gerstle et al. [13] proposed an analytial ormula obtained by onsidering the equilibrium o tensile and ompressive ores and the deormation o onrete. They deined the minimum reinorement amount as the quantity below whih rak propagation beomes unstable, i.e. the moment versus rak length urve assumes a dereasing trend. Ruiz et al. [14] evaluated taking into aount the bond-slip behaviour between onrete and steel along the transer length and by adopting the ohesive rak model to desribe the rature propagation. Finally, ppa Rao et al. [15] proposed an approah based on LEFM onepts, using the two-parameter rature model proposed by Jenq and Shah [16] or onrete. ll the ormulae obtained by the aorementioned models are given in Table 1, with details o the parameters involved. In the present paper, the problem o the minimum reinorement amount is analysed by means o a numerial approah based on non-linear rature mehanis [17]. First, its eetiveness is proved by a omparison between numerial simulations and the results o an experimental ampaign involving three-point bending tests on RC beams. parametri analysis is then arried out in order to evaluate the minimum reinorement amount by varying onrete grade and beam size. Finally, dimensional analysis is applied in order to ombine the dierent variables haraterizing the phenomenon into a redued number o governing parameters (N P and s). s a onsequene, the results o the parametri analysis align peretly along a hyperboli urve in the diagram N PC vs. s, making the deinition o a new design ormula easier, based on the results o the numerial simulations. With respet to the ormulae available in the literature, the one proposed is very syntheti and suitable or pratial purposes without gross simpliiations o the set o variables governing the problem. 2 nalytial and numerial investigation 2.1 Integrated ohesive/overlapping rak model Let us onsider a reinored onrete beam element with a retangular ross-setion o width b w, depth h, and eetive depth d. The beam segment has a length l equal to its depth and is subjeted to the external bending moment M. The middle ross-setion is representative o the mehanial behaviour o the whole element, sine all the non-lin- Table 1. Minimum reinorement aording to dierent models [11 15] uthors Model ormula Boso and Carpinteri [11] (1992) Hawkins and Hjorteset [12] (1992) Gerstle et al. [13] (1992) K IC m bwh h tm tm h w d b h E tmh/ Ew bwh E s Ruiz et al. [14] (1999) ppa Rao et al. [15] (2007) ( ) 1 1 b 1 y * wh 1 where: β 1 = h/(α l h ); l h = E G F / 2 tm; α=( d max /d 0 )/170; d 0 = 8 mm; d max = maximum aggregate size; γ = /h; * y = / tm ; η 1 = 15, ϕ =(β γ 1 ) s,min 0.01 d k w b d minimum steel area K IC onrete rature toughness d eetive beam depth h overall beam depth E onrete elasti modulus tm average onrete uniaxial tensile strength steel yield strength m average onrete ompressive strength E s steel elasti modulus ρ min minimum steel perentage w ritial rak width reinorement over G F onrete rature energy b w beam width onrete lexural tensile strength (size-dependent) 363

4 . Carpinteri/E. Cadamuro/M. Corrado Minimum lexural reinorement in retangular and T-setion onrete beams earities, i.e. onrete raking in tension, steel yielding and rushing o onrete in ompression, are loalized in this setion, whereas the outer parts exhibit an elasti response. The numerial model proposed here to desribe the rature behaviour o lightly reinored RC beams is derived by the more general algorithm introdued by Carpinteri et al. [17] or modelling the mehanial response o all the possible situations ranging rom plain to over-reinored onrete beams. It is based on a numerial proedure similar to those proposed by Carpinteri [18] and Planas and Elies [19] or implementing a ohesive rak. 2.2 Constitutive models The mehanial response o onrete in tension is desribed by the ohesive rak model [18, 20], whih onsiders a damaged and miroraked proess zone ahead o the real rak tip, partially stithed by inlusions, aggregates or ibres, where non-linear and dissipative phenomena take plae. linear-elasti stress-strain relationship is assumed or the undamaged phase (see Fig. 1a), whereas a sotening stress-rak width relationship desribes the proess zone up to the ritial opening w t r being reahed (see Fig. 1b). The sotening untion σ =(w) is a material property, likewise the uniaxial tensile strength tm, the ritial value o the rak opening w t r and the rature energy G F. The shape o (w) may vary rom linear to bilinear or even more ompliated relationships depending on the harateristis o the material onsidered and the problem being analysed. For instane, when plain onrete subjeted to a high strain gradient is being studied, a simple linear sotening law an be suiient to obtain aurate results. On the other hand, bilinear relationships with long tails are neessary to desribe ibre-reinored onrete elements, taking into aount the losing trations exerted by ibres or large rak opening values. s ar as modelling onrete rushing ailure is onerned, the overlapping rak model introdued by Carpinteri et al. [17, 21] is adopted. ording to suh an approah, the inelasti deormation in the post-peak regime is desribed by a ititious interpenetration o the material, while the remaining part o the speimen undergoes an elasti unloading. Thereore, a pair o onstitutive laws is onsidered, in lose analogy with the ohesive rak model: a stress-strain relationship until the ompressive strength is ahieved, and a stress-displaement (i.e. overlapping) relationship desribing the phenomenon o Fig. 1. Cohesive rak model: a) linear-elasti σ ε law, and b) post-peak sotening σ w relationship onrete rushing. The latter law, usually assumed as a linear dereasing untion, desribes how the stress in the damaged material dereases rom its maximum value down to zero as the ititious interpenetration inreases rom zero to the ritial value w r. The rushing energy G C, whih is a dissipated surae energy deined as the area beneath the post-peak sotening urve, an be assumed as a true material property sine it is almost independent o the strutural size. The interation between steel and onrete along the rebar is not modelled speiially. The reinorement reation is inluded in the numerial algorithm proposed in the next setion as an external applied ore, untion o the rak opening displaement aording to a suitable onstitutive law. On the basis o the bond-slip relationship provided by ib Model Code 2010 [5], and by imposing equilibrium and ompatibility onditions, it is possible to orrelate the reinorement reation to the relative slip at the rak edge, whih orresponds to hal the rak opening displaement. Typially, the relationship obtained is haraterized by an asending branh up to steel yielding, orresponding to whih there is a ritial value o the rak opening w y. ter that, the steel reation is nearly onstant. It has been shown that the value o w y varies in the range mm depending on steel amount, rebar diameter and number o rebars (see also [22]). In the present study, a value o 0.2 mm has been assumed, more onsistent with the small reinorement perentages onsidered. However, it is worth noting that this parameter has a limited eet on the evaluation o the minimum reinorement amount sine it barely aets the maximum raking load M r. In the present algorithm, this stress-displaement law is introdued in the input together with the ohesive and overlapping onstitutive laws. 2.3 Numerial algorithm The mid-span ross-setion o the element shown in Fig. 2a is subdivided into n nodes, where ohesive and overlapping stresses are replaed by equivalent nodal ores F i whih depend on the orresponding relative nodal displaements aording to the ohesive or overlapping post-peak laws (Fig. 2a). These horizontal ores an be omputed as ollows: {F} = [K w ]{w} + {K M } M (9) where: {F} vetor o nodal ores [K w ] matrix o oeiients o inluene or nodal displaements {w} vetor o nodal displaements {K M } vetor o oeiients o inluene or applied moment M Eq. (9) permits the rature and rushing phenomena to be studied by taking into aount the elasti behaviour o the RC member. To this end, all the elasti oeiients are omputed a priori using a inite element analysis. For symmetry, only a hal-element is disretized through quadrilateral plane stress elements with uniorm nodal spaing. Horizontal onstraints are then applied at the nodes along 364

5 a) b). Carpinteri/E. Cadamuro/M. Corrado Minimum lexural reinorement in retangular and T-setion onrete beams Fig. 2. Sheme o the mid-span ross-setion: a) inite element nodes and b) ore distribution with ohesive rak in tension, rushing in ompression and reinorement losing ores the vertial symmetry edge. Eah oeiient o inluene Kw, i,j whih relates the nodal ore F i to the nodal displaement w j, is omputed by imposing a unitary displaement on the orresponding onstrained node. On the other hand, the oeiients KM i are omputed by imposing a unitary external bending moment. In the generi situation shown in Fig. 2b, the ollowing equations an be onsidered, taking into aount, respetively, the stress-ree rak (Eq. (10a)), the ohesive sotening law (Eq. (10b)), the steel onstitutive law (Eq. (10)), the undamaged zone (Eq. (10d)) and the overlapping sotening law (Eq. (10e)): F i = 0; or i = 1, 2,, (j 1); i r F i = (w i ); or i = j,, (m 1); i r F i = h(w i ); or i = r w i = 0; or i = m,, p F i = g(w i ); or i = (p + 1),, n where: j real rak tip m ititious rak tip p ititious overlapping zone tip r reinorement layer level (10a) (10b) (10) (10d) (10e) Eqs. (9) and (10) onstitute a linear algebrai system o (2n) equations in (2n+1) unknowns, namely {F}, {w} and M. s regards the ompatibility requirement between steel and onrete, the displaement o the rebars and that o the surrounding onrete are assumed to be equal. On the other hand, the neessary additional equation derives rom the strength riterion adopted to govern the propagation proesses. We an set either the ore in the ititious rak tip m, equal to the ultimate tensile ore F u, or the ore in the ititious rushing zone tip p, equal to the ultimate ompressive ore F. It is important to note that raking and rushing phenomena are physially independent o eah other. s a result, the situation that is loser to one o these two possible onditions is hosen to establish the prevailing phenomenon. The driving parameter in the proess is the position o the ititious tip that has reahed the limit resistane in the step onsidered. Only this tip is moved to the next node when passing to the next step. The loalized rotation ϑ is omputed as ollows: ϑ={d w } T {w} + D M M (11) where {D w } is the vetor o the oeiient o inluene or the nodal displaements and D M is the oeiient o inluene or the applied bending moment M. s the present study relates to the analysis o beams o dierent sizes, it should be noted that the elasti oeiients in Eqs. (9) and (11) are onneted to the strutural dimension by simple relations o proportionality. Thereore, it is not neessary to repeat the inite element analysis or any dierent beam sizes onsidered. 2.4 ppliation o dimensional analysis to lightly reinored RC members When the lexural behaviour o RC beams is studied, then aording to the numerial model proposed in the previous setion, the untional relationship among the quantities haraterizing the phenomenon is M g ( tm, F, k, C, E,, s, h ; b w l G G h,, ) h where: M resistant bending moment tm average uniaxial tensile strength G F rature energy k harateristi ompressive strength G C rushing energy E elasti modulus o onrete yield strength o tension reinorement s area o tension reinorement h beam depth ϑ loal rotation o the element (12) and b w /h and l/h deine the geometry o the sample. Sine we are interested in the mehanial response o lightly reinored RC beams, the set o variables an be redued as ollows: M = g ( tm, G F, E,, s, h; ϑ) (13) 365

6 . Carpinteri/E. Cadamuro/M. Corrado Minimum lexural reinorement in retangular and T-setion onrete beams where the parameters desribing the behaviour o onrete in ompression, k and G C, are not expliitly onsidered sine the rushing ailure is not involved in the ailure mehanism. On the other hand, the geometrial ratios o the samples, b w /h and l/h, are anelled out sine they are assumed to be onstant. This assumption permits an investigation o the size-sale eets (all the beam dimensions vary with beam depth), whereas the eets o width and slenderness are not taken into aount. The appliation o Bukingham s Π-Theorem [23] or physial similitude and sale modelling permits a minimization o the dimension spae o the primary variables by ombining them into dimensionless groups as ollows: h 2.5 (14) i the beam depth h and the onrete rature toughness G F E = K IC are assumed to be the dimensionally independent variables. s a onsequene, the dimensionless untional relation or the proposed model beomes M = g 2 (s, N P, ϑ n ) (15) where: s M tmh 0.5 1, s h 0.5 Eh g, 0.5 G E G E bwh G E G E K F IC tmh 0.5 F (16) and N P, deined in Eq. (6), are the governing dimensionless numbers, M is the dimensionless bending moment and ϑ n is the normalized loal rotation. It is worth noting that the presene o ohesive losing stresses along the ititious proess zone, in addition to a steel reinorement layer, means that the strutural response is governed by two dimensionless numbers instead o only N P, as in the ase o the bridged rak model. In partiular, the matrix strength and toughness are onsidered by the stress brittleness number s introdued by Carpinteri [24]. In partiular, the raking load is a dereasing untion o the dimensionless number s. Thereore, or a ixed value o N P, by inreasing the value o s (it typially ranges rom 0.2 to 3.0), the response beomes more and more stable. In general, a transition rom brittle to dutile response is obtained by inreasing N P and/or s (a detailed analysis o the transitions ruled by N P and s is reported in [25]). Physial similarity in the dimensionless moment vs. normalized rotation diagrams is obtained when both N P and s are kept onstant. 3 Comparison o preditions and experimental results The experimental investigation onsidered here was arried out by Boso et al. [10, 12] in 1990 with the purpose o veriying the existene o size eets in the lexural behaviour o RC beams. To this end, three-point bending tests were perormed on initially unnothed and unraked reinored and plain onrete retangular beams varying in depth rom 100 to 800 mm. The eetive depth d was set to 0.9h, and the span-to-depth ratio was set to 6. The speimens were made with two dierent onrete F F Table 2. Material properties or the beams tested by Boso et al. [10, 12] Conrete grade 2 4 k (MPa) E (GPa) G F (N/mm) tm (MPa) Steel used in onrete Grade 2 Grade 4 Φ (mm) s (MPa) y (MPa) Table 3. Craking loads P r and ultimate loads P u or onrete grade 2 beams [12] Case h b Steel s /b w h P r P u (mm) (mm) (%) (kn) (kn) φ φ φ φ φ φ φ φ φ φ φ φ grades and dierent steel grades as shown in Table 2. The loading proess was strain-ontrolled beore raking and rak mouth opening displaement-ontrolled ater raking. The details o the steel reinorement in Tables 3 and 4 show that the perentage was not assumed to be onstant by varying the speimen size; on the ontrary, it was varied by keeping the brittleness number onstant N P, deined in Eq. (6). Test results in terms o raking loads P r and ultimate loads P u are shown in Tables 3 and 4. The eetiveness o the proposed model is demonstrated by a omparison between numerial preditions and experimental results or beams ontaining onrete grade 4 [10] in terms o applied load vs. mid-span deletion urves (Figs. 3 5). No ase 4 in Table 4 is onsidered sine eah reers to a reinorement amount onsiderably larger than the minimum quantity. In the numerial simulations the RC element o Fig. 2a is assumed to be representative o the mid-span portion o the beam subjeted to a three-point bending test. s a result, the mid-span deletion is obtained as the sum o the loalized rotation given by Eq. (11) and the elasti ontribution aording to the ollowing expression: 366

7 . Carpinteri/E. Cadamuro/M. Corrado Minimum lexural reinorement in retangular and T-setion onrete beams Table 4. Craking loads P r and ultimate loads P u or onrete grade 4 beams [10] Case h b Steel s /b w h P r P u (mm) (mm) (%) (kn) (kn) φ φ φ φ φ φ φ φ φ φ φ φ L 1 PL lo el 4 48 EI 3 (17) where L is the beam span and I is the moment o inertia o the ross-setion. s reported in Table 2, the value o the rature energy determined on pre-nothed beams, aording to the RILEM TC 50-FCM reommendation [26], is G F = 0.09 N/mm. However, the value o the rature energy adopted in the simulations in order to ahieve a best it or the experimental results reers to the beams without rein- orement (Figs. 3a, 4a, 5a) and is G F = 0.18 N/mm. Suh a dierene is due to the at that, unlike the nothed beams onsidered by the RILEM reommendation, initially unnothed beams suh as those tested by Boso et al. [10, 12] are haraterized by the development o a ew miroraks around the marorak responsible or inal ollapse. This phenomenon leads to an inrease in the deormation and energy dissipation, without signiiant eets on the raking load, given by the propagation o the marorak. Thereore, the inrease in the value o G F permits a better desription o the experimental results in terms o deormation and energy dissipation even i a single equivalent ohesive rak is onsidered, with negligible eets on the raking load (the tensile strength has not been inreased). The urves in Figs. 3 5 show generally good agreement between numerial and experimental results. However, it should be noted that the hypothesis o setional loalization haraterizing the proposed model beomes less onsistent with the real mehanial behaviour o RC beams in the ase o high steel perentages. In at, in these ases the rature phenomenon is spread more along the beam length L, whih in the present model is onsidered only in the elasti ontribution related to the load vs. displaement response. 4 Parametri analysis and disussion The results o an extensive numerial study arried out in order to determine a relationship between the minimum reinorement and the mehanial and geometrial parameters o a retangular ross-setion beam are presented in this setion. Eight dierent values o beam depth h, rom 25 to 3200 mm, and ive dierent values o onrete ompressive strength k, rom 20 to 80 MPa, have been onsidered. ll the other mehanial properties o on- Fig. 3. Comparison between numerial and experimental [10] applied load vs. mid-span deletion urves or beam depth h = 100 mm and dierent amounts o tension reinorement 367

8 . Carpinteri/E. Cadamuro/M. Corrado Minimum lexural reinorement in retangular and T-setion onrete beams Fig. 4. Comparison between numerial and experimental [10] applied load vs. mid-span deletion urves or beam depth h = 200 mm and dierent amount o tension reinorement Fig. 5. Comparison between numerial and experimental [10] applied load vs. mid-span deletion urves or beam depth h = 400 mm and dierent amounts o tension reinorement rete have been evaluated aording to the relationships provided by ib Model Code 2010 [2] (see Table 5). s regards the steel reinorement, a yield strength = 450 MPa and an elasti modulus E s = 200 GPa have been assumed. The ratio o eetive depth to overall depth was set to 0.9. In order to ind the value orresponding to the transitional mehanial response between brittle and dutile behaviour, i.e., the minimum amount rom the design point o view, several simulations were arried out or eah beam by varying the steel perentage. More preisely, suh a transitional oniguration is determined when the peak raking load P r equals the ultimate load P u (see 368

9 Table 5. Mehanial parameters or the retangular beams onsidered in the numerial simulations. Carpinteri/E. Cadamuro/M. Corrado Minimum lexural reinorement in retangular and T-setion onrete beams k tm G F E h s /b w h N PC s (MPa) (MPa) (N/mm) (GPa) (mm) (%) Fig. 6). The /b w h ratios obtained or the beams onsidered, and the orresponding values o s and N P (in this ase N PC sine it ontains the ritial value o the reinorement area), are reported in Table 5. The results o N PC vs. s are also shown in Fig. 6, where a derease in the ritial value o the reinorement brittleness number is seen as the stress brittleness number inreases. In the range o interest or ommon strutural appliations, the trend obtained an be desribed with a very good approximation by the ollowing hyperboli urve: Fig. 6. Best-it relationship o numerial results ( symbols) between N PC and s; symbols are experimental results [10, 12] N PC = 0.27s 0.70 (18) By substituting Eqs. (6) and (16) in (18), the relation or the minimum reinorement area reads as ollows: The minimum reinorement perentages ρ min = / b w d obtained rom Eqs. (19) and (20) are ompared with the presriptions o the design odes [2, 4, 6, 7] in Fig. 7 or k = 35 MPa, and = 450 MPa. The reinorement ratio given by the Norwegian Standard is slightly larger than that given by Eq. (19) or any beam depth. The provisions o Euroode 2 and ib Model Code 2010 are unsae or beam depths < 540 mm, whereas they overestimate the reinorement ratio or d > 540 mm. It is worth noting that, or the same value o d/h and or a resistane ator equal to unity, the urve proposed or retangular beams (Eq. (19)) should tend to the provisions o Euroode 2 and ib Model Code 2010 or large beam depths, i.e. when the ohesive ores an be negleted. s regards the CI provision, it is onsiderably higher than the urve pro K 0.27 ( ) ( ) tm 0.70 IC 0.30 b h 0.85 (19) ording to Eq. (19), is an inreasing untion o the tensile strength and toughness o onrete, and o the width and depth o the beam, whereas it is a dereasing untion o the steel yield strength. The minimum reinorement amount derived by the experimental tests o Boso et al. [10, 12] or onrete grades 2 and 4 are indiated in Fig. 6 by the symbols. The values in some ases obtained by interpolating the data reported in Tables 3 and 4 onirm the trend o the numerial results. The same alulations have also been perormed or a T-beam with the lange in ompression having a width 8b w and depth 0.20h. Suh geometrial ratios determine a setion modulus 1.5 times larger than that o a retangular setion having the same depth and a width b w (the same ase onsidered by CI 318 [4]). The expression obtained or the minimum reinorement area is K 0.34 ( ) ( ) tm 0.84 IC 0.16 (20) w b h w

10 . Carpinteri/E. Cadamuro/M. Corrado Minimum lexural reinorement in retangular and T-setion onrete beams Fig. 7. Comparison o minimum reinorement ratios alulated by dierent odes [2, 4, 6, 7] or = 450 MPa and k = 35 MPa Fig. 8. Minimum reinorement vs. eetive beam depth aording to various models [11 15] or = 450 MPa and k = 35 MPa posed or T-beams (Eq. (20)). In this ase too, the proposed urve does not tend to the CI provision or large beam depths due to a dierent value or the d/h ratio and the appliation o the resistane ator. Furthermore, in the CI approah the lexural tensile strength is adopted or onrete instead o the diret tensile strength used in the ohesive model. ording to the merian standard, the lexural strength is greater than the diret tensile strength and is independent o size, whereas it is atually a dereasing untion o the beam depth. This assumption results in a greater overestimation o the minimum reinorement or large beam depths. Finally, the urve in Fig. 7 labelled MC10-modiied is obtained by substituting the uniaxial tensile strength o onrete tm in Eq. (4) by the lexural tensile strength t,l whih, aording to ib Model Code 2010, is given by t,l tm 0.7 ( h ) 0.06h0.7 (21) The shape o the urve obtained is very similar to the proposed one, even i it overestimates the minimum reinorement. This is due to the at that the oeiient 0.26 in Eq. (4) inludes a resistane ator, as speiied in setion 1.1. In ase a unitary resistane ator is onsidered, the proposed urve or retangular ross-setions and the ib Model Code 2010 modiied presription would be almost oinident. The minimum reinorement perentage vs. beam depth urves aording to the dierent models available in the literature (see Table 1) are ompared with Eq. (19) in Fig. 8 or k = 35 MPa and = 450 MPa. The eetive beam depth has been kept at 0.9h at any sale. The model by Gerstle et al. [13] presents an inrease in the minimum steel perentage as the beam depth inreases. ll the other urves learly highlight a dereasing trend. The proposed urve is very lose to those related to the studies by Hawkins and Hjorsetet [12] and Ruiz et al. [14], also based on the ohesive rak model. It should be noted, however, that the appliation o dimensional analysis permits better lariiation o the eets o eah o the variables involved in the physial phenomenon, as well as their interation in the global response, or both retangular and T-beams. Contrary to the bridged rak model, the proposed approah an be applied to unreinored onrete beams too. When N P = 0 due to s = 0, the overall mehanial behaviour is in at a untion o the dimensionless number s. Small values or beam depths mean high values or s and thereore a dutile response. For the sake o ompleteness, it should be remarked that Eqs. (19) and (20) an only be applied within the range o beam depths onsidered in the present study. Their asymptotes or a beam depth tending to zero and to ininity are in at inorret. For h tending to zero, the minimum reinorement ratio is approximately equal to ( tm h 2 )/(2 d 2 ) sine very small beams ail in a dutile manner, and the losing trations along the ohesive rak an be assumed onstant, even or large rotational hinges. On the other hand, or h tending to ininity, ρ min is equal to ( tm h 2 )/(6 d 2 ) or retangular setions and ( tm h 2 )/(4 d 2 ) or the T-setion onsidered due to the at that the ohesive ontribution an be negleted. 5 Summary and onlusions New ormulae or evaluating the minimum reinorement have been derived by applying dimensional analysis to the results o a numerial algorithm proposed to study the lexural behaviour o lightly reinored RC beams with retangular and T-shaped setions. Suh an approah permits the main mehanial and geometrial parameters aeting the phenomenon being studied namely the onrete rature toughness and tensile strength, the steel yield strength and the strutural size to be onsidered by means o two dimensionless numbers, N P and s. Other eets, suh as the steel-onrete interation, the size and spaing o the rebars and the onrete over, have not been analysed in the present study sine they aet the dependene o the minimum reinorement amount on the beam dimensions only marginally. s ar as size-sale eets are onerned, it should be noted that the presene o ohesive losing stresses determines a variation in ρ min with the beam size whih is less pronouned than that predited by the bridged rak 370

11 . Carpinteri/E. Cadamuro/M. Corrado Minimum lexural reinorement in retangular and T-setion onrete beams model. It turns out to be a untion o h 0.15 or retangular beams and h 0.08 or T-beams, instead o the h 0.50 obtained by LEFM. Suh a dierene is learly shown in Fig. 8, where the urve o Boso and Carpinteri [11] is ompared with the present proposal. The proposed ormulae, Eqs. (19) and (20), an be urther rearranged or pratial purposes by expressing tm and K IC as untions o k aording to relationships available in the literature and/or in design odes (see, or example, ib Model Code 2010 [2]). lternatively, very similar results an be obtained by applying the presription provided by Euroode 2 and ib Model Code 2010 (Eq. (4)), in whih the uniaxial tensile strength is replaed with the lexural tensile strength given by Eq. (21). The ormula obtained is ertainly suitable or pratial purposes. Finally, it should be remarked that, in order to takle the problem o minimum reinorement ully, the servieability onditions should also be studied aurately. ording to the approah o ib Model Code 2010, or instane, more restritive presriptions or the minimum amount o reinorement may be derived rom limitations to the rak mouth opening displaement neessary to prevent steel orrosion and improve durability. Size eets are also expeted to inluene the rak opening in this ase, and thereore the minimum reinorement, as revealed in experiments by Yasir lam et al. [27]. knowledgements The inanial support provided by the Ministry o University and Sientii Researh (MIUR) or the projet dvaned appliations o Frature Mehanis or the study o integrity and durability o materials and strutures is grateully aknowledged. Notation gross ross-setional area s steel reinorement area b w width o tension side o beam d eetive depth o beam D M oeiient o inluene or applied moment {D w } T vetor o oeiients o inluene or nodal displaements E elasti modulus o onrete E s elasti modulus o steel k harateristi ompressive strength o onrete m average ompressive strength o onrete tm average uniaxial tensile strength o onrete t,l lexural tensile strength o onrete harateristi tensile yield strength o steel {F} vetor o nodal ores G C rushing energy o onrete G F rature energy o onrete h overall depth o beam K IC onrete rature toughness {K M } vetor o oeiients o inluene or applied moment [K w ] matrix o oeiients o inluene or nodal displaements L span o beam l length o beam portion onsidered M M u M r N P N PC P P r P u s {w} wr wr t δ ϑ ρ φ Reerenes applied bending moment nominal lexural resistane (reation o yielded reinorement x moment arm) raking moment o plain onrete setion reinorement brittleness number ritial value or reinorement brittleness number, orresponding to minimum reinorement amount applied load maximum raking load ultimate load with respet to yielded steel and ompletely raked ross-setion stress brittleness number vetor o nodal displaements ritial overlapping displaement ritial rak opening displaement mid-span deletion loalized rotation o beam portion onsidered ( s /b w d) 100, steel reinorement perentage resistane ator 1. Balázs, G. L.: Design or SLS aording to ib Model Code Strutural Conrete, 2013, 14, No. 2, pp Federation International du Beton (ed.): Model Code 2010 First omplete drat, vol. 1, Thomas Telord Ltd, Lausanne, ib Bulletin No. 55, merian Conrete Institute Committee 318 (ed.): Building Code Requirements or Strutural Conrete (CI ) and Commentary (CI 318R-95), Detroit, MI, merian Conrete Institute Committee 318 (ed.): Building Code Requirements or Strutural Conrete (CI ) and Commentary (CI 318R-08), Farmington Hills, MI, Seguirant, S. J., Brie, R., Khaleghi, B.: Making sense o minimum lexural reinorement requirements or reinored onrete members. PCI Journal, 2010, 55, No. 3, pp European Committee or Standardization (ed.): Euroode 2: Design o Conrete Strutures, Part 1-1: General Rules and Rules or Buildings, Brussels, Norwegian Standard, NS 3473 E (English translation): Conrete Strutures, Design Rules, Norwegian Counil or Building Standardization, Olso, Norway, Carpinteri,.: Stability o raturing proess in RC beams. Journal o Strutural Engineering, 1984, 110, No. 3, pp Boso, C., Carpinteri,.: Sotening and snap-through behavior o reinored elements. Journal o Engineering Mehanis, 1992, 118, No. 8, pp Boso, C., Carpinteri,., Debernardi, P. G.: Minimum reinorement in high-strength onrete. Journal o Strutural Engineering, 1990, 116, No. 2, pp Boso, C., Carpinteri,.: Frature mehanis evaluation o minimum reinorement in onrete strutures. In: ppliations o Frature Mehanis to Reinored Conrete, Carpinteri,. (ed.), Elsevier pplied Siene, London, 1992, pp Hawkins, N. M., Hjorsetet, K.: Minimum reinorement requirements or onrete lexural members. In: ppliations o Frature Mehanis to Reinored Conrete, Carpinteri,. (ed.), Elsevier pplied Siene, London, 1992, pp Gerstle, W. H., Dey, P. P., Prasad, N. N. V., Rahulkumar, P., Xie, M.: Crak growth in lexural members a rature me- 371

12 . Carpinteri/E. Cadamuro/M. Corrado Minimum lexural reinorement in retangular and T-setion onrete beams hanis approah. CI Strutural Journal, 1992, 89, No. 6, pp Ruiz, G., Elies, M., Planas J.: Size eets and bond-slip dependene o lightly reinored onrete beams. In: Minimum Reinorement in Conrete Members, Carpinteri,. (ed.), Elsevier Siene Ltd., Oxord, UK, 1999, pp ppa Rao, G., ravind, J., Eligehausen, R.: Evaluation o minimum lexural reinorement in r beams using ititious rak approah. Journal o Strutural Engineering (Madras), 2007, 34, No. 4, pp Jenq, Y. S., Shah, S. P.: Shear resistane o reinored onrete beams a rature mehanis approah. In: Frature Mehanis: ppliation to Conrete (speial report CI SP- 118), Li, V. C., Bazant, Z. P. (eds.), merian Conrete Institute, Detroit, 1989, pp Carpinteri,., Corrado, M., Manini, G., Paggi, M.: Size-sale eets on plasti rotational apaity o r beams. CI Strutural Journal, 2009, 106, No. 6, pp Carpinteri,.: Size eets on strength, toughness and dutility. Journal o Engineering Mehanis, 1989, 115, pp Planas, J., Elies, M.: symptoti analysis o a ohesive rak: 1. theoretial bakground. International Journal o Frature, 1992, 55, No. 2, pp Hillerborg,., Modeer, M., Petersson, P. E.: nalysis o rak ormation and rak growth in onrete by means o rature mehanis and inite elements. Cement and Conrete Researh, 1976, 6, pp Carpinteri,., Corrado, M., Manini, G., Paggi, M.: The overlapping rak model or uniaxial and eentri onrete ompression tests. Magazine o Conrete Researh, 2009, 61, No. 9, pp Tammo, K., Thelandersson, S.: Crak widths near reinorement bars or beams in bending. Strutural Conrete, 2009, 10, No. 1, pp Bukingham, E.: Model experiments and the orm o empirial equations. SME Transation, 1915, 37, pp Carpinteri,.: Noth sensitivity in rature testing o aggregative materials. Engineering Frature Mehanis, 1982, 16, No. 4, pp Corrado, M., Cadamuro, E., Carpinteri,.: Dimensional analysis approah to study snap bak-to-sotening-to-dutile transitions in lightly reinored quasi brittle materials. International Journal o Frature, 2011, 172, pp RILEM TC 50-FCM: Determination o the rature energy o mortar and onrete by means o three-point bend tests on nothed beams. Materials and Strutures, 1985, 18, pp Yasir lam, S., Lenormand, T., Loukili,., Regoin, J. P.: Measuring rak width and spaing in reinored onrete members. Pro. o FraMCoS-7, Oh, B. H. (ed.), Korea Conrete Institute, Seoul, 2010, pp Pro. lberto Carpinteri Department o Strutural Geotehnial and Building Engineering Politenio di Torino Corso Dua degli bruzzi, Torino, Italy Dr. Eria Cadamuro Department o Strutural Geotehnial and Building Engineering Politenio di Torino Corso Dua degli bruzzi, Torino, Italy Dr. Mauro Corrado Corresponding author Department o Strutural Geotehnial and Building Engineering Politenio di Torino Corso Dua degli bruzzi, Torino, Italy Tel Fax mauro.orrado@polito.it 372