An integrated cohesive/overlapping crack model for...

Int J Fract (2010) 161:161–173DOI 10.1007/s10704-010-9450-4

ORIGINAL PAPER

An integrated cohesive/overlapping crack model for theanalysis of flexural cracking and crushing in RC beams

Alberto Carpinteri · Mauro Corrado ·Marco Paggi

Received: 30 July 2009 / Accepted: 14 January 2010 / Published online: 7 February 2010© Springer Science+Business Media B.V. 2010

Abstract In the present paper, a new fracture-mechanics based model is proposed for the analysis ofreinforced concrete beams in bending describing bothcracking and crushing growths taking place during theloading process by means of the concept of strain local-ization. In particular, the nonlinear behaviour of con-crete in compression is modelled by the OverlappingCrack Model, which considers a material interpenetra-tion when the elastic limit is overcome, in close anal-ogy with the Cohesive Crack Model, routinely adoptedfor modelling the tensile behaviour of concrete. On thebasis of different nonlinear contributions due to con-crete and steel, a numerical finite element algorithmis proposed. According to this approach, the flexuralbehaviour of reinforced concrete structural elements isanalyzed by varying the main geometrical and mechan-ical parameters. Particular regard is given to the role ofthe size-scale effects on the ductility of plastic hinges,which is available at the ultimate load conditions.

Keywords Reinforced concrete · Cohesive CrackModel · Nonlinear analysis · Finite element method ·Crushing · Ductility · Overlapping Crack Model

A. Carpinteri · M. Corrado (B) · M. PaggiDepartment of Structural Engineering and Geotechnics,Politecnico di Torino, Corso Duca degli Abruzzi 24,10129 Torino, Italye-mail: [email protected]

1 Introduction

Structural ductility is a fundamental property for thesafe design of reinforced concrete (RC) structures.From a practical point of view, this requirement issatisfied by complying with some empirical prescrip-tions provided by the existing design codes. In the caseof RC elements, in fact, an accurate evaluation of theavailable ductility is difficult to be achieved due to thesimultaneous presence of several nonlinear contribu-tions affecting the global mechanical behaviour, suchas concrete cracking in tension, steel yielding or slip-page, and concrete crushing in compression. It is worthnoting that, among the various forms of nonlinearity,concrete crushing, which highly depends on the size-scale of the specimen, is fundamental for the predictionof the ultimate rotation, since it delimits the plasticplateau of the moment versus rotation diagrams.

In structural design, the concrete contribution in ten-sion is usually totally neglected or just considered witha linear-elastic stress–strain law until the ultimate ten-sile strength is reached. A more accurate descriptionof the nonlinear behaviour of concrete has been pro-vided in the framework of the finite element method(see Owen and Hinton 1980 for a wide overview oncomputational aspects), where several models basedon plasticity (Pramono and Willam 1989; Owen et al.1983; De Borst 1986) or on a combination of dam-age and plasticity (Simo and Ju 1987; Grassl andJiràsek 2006) have been proposed. However, as pointedout by Morley (2008), the application of plasticity

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162 A. Carpinteri et al.

theories does not allow the study of size-scale effects,in spite of their apparently successful applications overa wide range of structures of ordinary size. More recentresearch trends show the integration of plasticity withfracture mechanics theories, such as the smeared crackmodel (Cervenka and Papanikolaou 2008), in order tocapture size-scale effects due to concrete cracking.

As far as the nonlinear constitutive behaviour ofconcrete in compression is concerned, it can be tra-ditionally modelled in the framework of elasto-plastic-ity (Barbosa and Ribeiro 1998). A hardening/softeningplasticity model based on the Menétrey and Willam(1995) three-parameter failure surface was used tosimulate concrete crushing. Papanikolaou and Kappos(2007) proposed a plasticity formulation for multiaxialcompression handling confinements effects. Alterna-tively, the local behaviour of the material can be gov-erned by the Drucker–Prager yield criterion. The lin-ear elastic behaviour governs the analysis until exceed-ing the compressive strength. Once the principal stressreaches the compressive strength, crushing takes placeand the crushed regions develop perpendicular to thedirection of principal stress. To model this nonlinearprocess, iterative algorithms with high performancecomputers have to be used. The main drawback of suchan approach is represented by the fact that the parame-ters defining the yield surface have to be updated dur-ing the loading process in order to capture the soften-ing response of the material. This was shown by Cela(2002), who proposed a material identification proce-dure for modelling the uniaxial compression tests ofconcrete. The extension of such an approach to bend-ing specimens is still an open point. With regard to thesteel behaviour, the most used constitutive laws are theelastic-perfectly plastic or the elastic-hardening stress–strain relationships.

A completely different approach can be pursuedby applying fracture mechanics concepts to model-ling crack growth and crushing processes. The BridgedCrack Model, for instance, has been largely usedto study the mechanical stability of under-reinforcedconcrete beams and fibre reinforced structural ele-ments (Carpinteri 1984; Carpinteri and Massabò 1997;Carpinteri et al. 2004). Recent applications permit topredict the mutual transitions between the differentcollapse mechanisms—flexural, shear or crushing fail-ures—by varying the governing parameters (Carpinteriet al. 2007a). However, such a model is not able to fullydescribe the crushing process, which strongly influ-

ences the ultimate condition, making impossible theevaluation of the structural ductility. In this paper, inorder to overcome this drawback, a different approach,still based on nonlinear fracture mechanics models, isproposed. In particular, the tensile cracking phenom-enon is described by means of the well establishedCohesive Crack Model, whereas the crushing process ismodelled according to the Overlapping Crack Model,which considers a material interpenetration in the post-peak regime (Carpinteri et al. 2007b). As far as RCbeams in bending are concerned, in fact, the crush-ing process may lead either to material expulsion tothe upper side or to fragmentation. This is therefore theresult of a severe strain localization which is analogousto what happens in uniaxial compression. Hence, theuse of a fictitious interpenetration instead of a strain isfully justified by the fact that energy dissipation in com-pression takes place over a surface, rather than withina volume. Although this interpenetration in compres-sion can be considered as an artefact introduced formodelling purposes, it permits to describe in a simpli-fied way the actual failure mode. In fact, the results byJansen and Shah (1997) show that, regardless of thefailure mode of concrete specimen in uniaxial com-pression tests, the concept of overlapping crack can beprofitably applied to predict the mechanical responseeven when inclined shear bands develop. In the presentpaper, this concept is generalized to the analysis of thebehaviour of RC beams in bending. In this field, mostof the available models require a detailed description ofthe triangular-shaped wedge which is expelled duringcrushing (e.g., Fantilli et al. 2007). With respect to suchapproaches, the proposed model permits to avoid theelaborate description of the kinematics of the failuremode.

Consequently, the analysis of the mechanical behav-iour of RC structural elements will be carried out bymeans of a complex algorithm obtained by mergingthe two aforementioned elementary fracture mechan-ics models, along with a suitable constitutive law forsteel. Diagonal shear cracks are not considered in thepresent work. With this algorithm, based on the finiteelement method, we will show that it is possible to com-pletely capture the moment versus rotation responseunder monotonic loadings, taking into account all themain nonlinear contributions. A parametric analysiswill provide a detailed quantification of the effect ofthe most important parameters entering the proposedmodel on the ductility of RC beams in bending, with

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An integrated cohesive/overlapping crack model 163

special regard on size-scale effects. A wide experi-mental comparison will show the effectiveness of theproposed approach.

2 Mathematical formulation

Let us consider a portion of a RC beam subjected toa bending moment M. This element, having a span todepth ratio equal to unity, is representative of the zonewhere a plastic hinge formation takes place. We alsoassume that fracturing and crushing processes are fullylocalized along the mid-span cross-section of this ele-ment. This assumption, fully consistent with the crush-ing phenomenon, also implies that only one equivalentmain tensile crack is considered.

2.1 Cohesive Crack Model

In the proposed algorithm, the behaviour of concrete intension is described by means of the well establishedCohesive Crack Model (Hillerborg et al. 1976; Carpin-teri 1985, 1989), largely used, in the past, to study theductile-to-brittle transition in the mechanical responseof plain concrete beams in bending. According to thismodel, the constitutive law for the undamaged zone is aσ − ε linear-elastic relationship up to the achievementof the tensile strength, σu (Fig. 1a). In the process zone,the damaged material is still able to transfer a tensilestress across the crack surfaces. The cohesive stressesare considered to be linear decreasing functions of thecrack opening, wt (see Fig. 1b):

σ = σu

(1 − wt

wtcr

), (1)

where wtcr is the critical value of the crack opening

corresponding to σ = 0, and σu is the ultimate tensilestrength of concrete. The area under the stress versusdisplacement curve in Fig. 1b represents the fractureenergy, GF, which ranges from 0.050 to 0.150 N/ mmfor concrete.

2.2 Overlapping Crack Model

In order to model the concrete crushing failure, theOverlapping Crack Model is adopted (Carpinteri et al.2007b). Based on the original insights provided by

Fig. 1 Cohesive Crack Model for concrete in tension: linear-elastic σ − ε law (a); post-peak softening σ − ε relationship (b)

Fig. 2 Evaluation of the localized interpenetration, wc, from thetotal shortening of the specimen, δ

Kotsovos (1983), van Mier (1984) and Jansen and Shah(1997), the interpenetration wc is computed by sub-tracting the elastic elongation, caused by the reductionof the applied stress in the post-peak regime, δel, andthe pre-peak plastic deformation, δpl, from the σ − δ

diagram (see Fig. 2).Typical stress-interpenetration curves obtained from

specimens with different size and slenderness areshown in Fig. 3 (see also Carpinteri et al. (2010) for acomprehensive analysis of these curves for quasi-brittlematerials). The collapse of the different curves onto avery narrow band proves the size-scale and slendernessindependences of the overlapping crack law. Althoughthis is nonlinear, the use of a simplified linear softeningexpression is considered for modelling purposes (seethe dashed line in Fig. 3 for a visual comparison). Thisis a reasonable approximation because only the range0.0 < wc/wc

cr < 0.5 is usually utilized in the numeri-cal simulations of RC beams in bending.

According to this assumption, in fact, the Overlap-ping Crack Model permits to describe in a simplifiedway the actual failure mode that can be very com-plex and may range from shear crack propagation toconcrete fragmentation (RILEM TC 148-SSC 1997;Carpinteri et al. 2001). Hence, this concept permits

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Fig. 3 Experimental and linear-softening approximation(dashed curve) for uniaxially compressed concrete specimenswith different sizes and slendernesses

to avoid the elaborate description of the kinematics ofthese failure modes (Carpinteri et al. 2009). As a result,a pair of constitutive laws for concrete in compressionis introduced, in close analogy with the Cohesive CrackModel: a stress–strain relationship until the compres-sive strength is achieved, and a stress–displacement(overlapping) relationship describing the phenomenonof concrete crushing (see the dashed line in Fig. 3).The latter law describes how the stress in the damagedmaterial decreases from its maximum value to zero asthe interpenetration increases from zero to the criticalvalue, wc

cr:

σ = σc

(1 − wc

wccr

), (2)

where wcis the interpenetration, wccr is the critical value

of the interpenetration corresponding to σ = 0 in thelinear softening approximation, and σc is the compres-sive strength. It is worth noting that the crushing energy,GC, defined as the area below the post-peak softeningcurve of the dashed line in Fig. 3, can be consideredas a material property, since it is not affected by thestructural size. This is in full agreement with the recentfractal analyses by Ferro and Carpinteri (2008) andCarpinteri and Corrado (2009). The following empir-ical equation for calculating the crushing energy hasrecently been proposed by Suzuki et al. (2006), tak-ing into account the confined concrete compressivestrength by means of the stirrups yield strength andthe stirrups volumetric content:

GC = GC,0 + 10,000k2

a pe

σc, (3)

where σc is the average concrete compressive strength,GC,0 is the crushing energy for unconfined concrete:

GC,0 = 80 − 50kb, (4)

ka is a parameter depending on the stirrups strength andvolumetric content:

ka = 1 + ke

(fsy − fs,c

)fsy

, (5)

and pe is the effective lateral pressure:

pe = keρw fs,c. (6)

The parameter kb depends on the concrete compressivestrength:

kb = 40

σc≤ 1.0, (7)

fs,c is the stress in transverse reinforcement at the peakstrength:

fs,c = Es

[0.45εc0 + 6.8

(keρw

σc

)9/10]

≤ fsy, (8)

ke is the effective confinement coefficient:

ke =(

1−∑ (

w′i

)2

6bcdc

)(1− s′

2bc

)(1− s′

2dc

)/ (1−ρcc) ,

(9)

ρw is the geometric ratio of transverse reinforcement,Es is the modulus of elasticity of transverse steel, εc0 =(0.0028 − 0.0008kb), fsy is the steel yield strength, w′

iis the spacing between longitudinal steel bars, s′ is thespacing between stirrups, bc and dc are the widths ofthe concrete compressed zone, and ρcc is the longitu-dinal reinforcement ratio in the compression side. Theunits of measure in Eqs. (3)–(9) are N and mm.

By varying the concrete compressive strength from20 to 90 MPa, Eq. (4) gives a crushing energy rang-ing from 30 to 58 N/ mm. It is worth noting that GC

is between two and three orders of magnitude higherthan GF. The crushing energy, in fact, may be inter-preted as the sum of the fracture energy dissipated bythe tensile microcracks taking place during the crushingphenomenon and the frictional energy. Typical criticalvalues for crushing interpenetration and crack openingfor normal strength concrete are approximately equalto wc

cr ≈ 1 mm (see also Jansen and Shah 1997) andwt

cr ≈ 0.1 mm, respectively. Finally, we remark that, in

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the case of concrete confinement, the crushing energy,computed using Eq. (3), and the corresponding crit-ical value for crushing interpenetration, considerablyincrease.

2.3 Steel–concrete interaction

In the proposed model, it is not possible to adopt theclassical σ −ε laws for steel, since the kinematics of themid-span cross-section of the RC member is describedby means of displacements, instead of strains. Conse-quently, a constitutive relationship between the rein-forcement reaction and the crack opening displace-ment has to be introduced. In the past, this problem hasbeen overcome by assuming a rigid-plastic behaviourfor steel (Bosco and Carpinteri 1991). This assumptioninvolves that the crack opening is zero up to steel yield-ing. Then, the reinforcement reaction is constant. In theproposed model, an approach closer to the actual behav-iour is adopted, following the procedure proposed byRuiz et al. (1999). The typical slip between reinforc-ing bar and surrounding concrete, in fact, allows fora crack opening before steel yielding. This phenom-enon is taken into account by considering the rela-tionship between the tangential stress along the steel–concrete interface and the relative tangential displace-ment, as proposed in Model Code 90 (Comité Euro-International du Béton 1993). By imposing equilibriumand compatibility conditions, it is possible to correlatethe reinforcement reaction, given by the integration ofthe bond stresses acting along the bar, to the relative slipat the crack edge, which corresponds to half the crackopening displacement. Typically, the obtained relation-ship is characterized by an ascending branch up to steelyielding, after which the reaction is nearly constant. Inthis way, the actual bond-slip shear stress distribution isreplaced by a pair of concentrated forces acting along

the crack faces, functions of the opening displacementat the reinforcement level.

3 Numerical algorithm

A discrete form of the elastic equations governing themechanical response of the two half-beams is hereinintroduced. The mid-span cross-section of the beam canbe subdivided into finite elements by n nodes (Fig. 4a).In this scheme, cohesive and overlapping stresses arereplaced by equivalent nodal forces, Fi, by integratingthe corresponding tensile stresses over each elementsize. Such nodal forces depend on the nodal openingor closing displacements according to the cohesive oroverlapping softening laws previously introduced.

The horizontal forces, Fi, acting at the i-th nodealong the mid-span cross-section can be computed asfollows:

{F} = [Kw] {w} + {KM } M (10)

where {F} is the vector of nodal forces, [Kw] is thematrix of the coefficients of influence for the nodal dis-placements, {w} is the vector of nodal displacements,{KM } is the vector of the coefficients of influence forthe applied moment M.

In the generic situation shown in Fig. 4b, the follow-ing equations can be considered:

Fi = 0; for i = 1, 2, . . . , ( j − 1) ; i �= r (11a)

Fi = Fu

(1− wt

i

wtcr

); for i = j, . . . , (m − 1) ; i �= r

(11b)

wi = 0; for i = m, . . . , p (11c)

Fi = Fc

(1 − wc

i

wccr

); for i = (p + 1) , . . . , n (11d)

Fr = f (wr) ; for i = r. (11e)

Fig. 4 Finite element nodes(a), and force distributionwith cohesive crack intension and crushing incompression (b) along themid-span cross-section

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where: j represents the real crack tip, m represents thefictitious crack tip, p is the fictitious overlapping tipand r is the node corresponding to the steel reinforce-ment. A bi-linear softening cohesive crack law can beeasily introduced by modifying Eq. (11b). The rein-forcement contribution is included in the nodal forcecorresponding to the r-th node (see Eq. (11e)). It isworth noting that, in this framework, it is possible toinsert a reinforcement in compression by introducinganother stress–displacement constitutive law in the cor-responding node. Due to lack of studies on the behav-iour of reinforcing bars in compression, the σ −w rela-tionship adopted is the same as that used for steel intension.

Equations (10) and (11) constitute a linear algebraicsystem of (2n) equations in (2n+1) unknowns, namely{F}, {w} and M. A possible additional equation canbe chosen: we can set either the force in the fictitiouscrack tip, m, equal to the ultimate tensile force, or theforce in the fictitious crushing tip, p, equal to the ulti-mate compression force. In the numerical scheme, wechoose the situation which is closer to one of these twopossible critical conditions. This criterion will ensurethe uniqueness of the solution on the basis of physi-cal arguments. The driving parameter of the process isthe tip that in the considered step has reached the limitresistance. Only this tip is moved when passing to thenext step.

It is important to remark that the present problem isnonlinear even if linear softening laws in tension andcompression are considered. At each loading step, infact, the size of the fictitious process zones in tensionand in compression are unknown and have to be founditeratively. Therefore, the proposed numerical schemeis general and does not depend on the shape of thecohesive and overlapping crack laws being considered.

The two fictitious tips advance until they con-verge to the same node. So forth, in order to describethe descending branch of the moment-rotation diagram,the two tips can move together towards the intrados ofthe beam. As a consequence, the crack in tension closesand the overlapping zone is allowed to extend towardsthe intrados. This situation is quite commonly observedin over-reinforced beams, where steel yielding does nottake place.

Finally, at each step of the algorithm, it is possibleto calculate the localised rotation, ϑ, as follows:

ϑ = {Dw}T {w} + DM M (12)

where {Dw} is the vector of the coefficients of influencefor the nodal displacements and DM is the coefficientof influence for the applied moment.

A feature of the proposed model is that it is unneces-sary to introduce any assumptions on the deformationfield as, for instance, the Bernoulli hypothesis. On thecontrary, the longitudinal nodal displacements alongthe mid-span cross-section obtained through the step-by-step numerical solution are nonlinear. The nodal dis-placement profiles, in the case of a beam depth equalto 0.4 m and for an amount of tensile reinforcementequal to 2%, are shown in Fig. 5a, for different val-ues of the applied bending moment. For low values ofthe bending moment, up to approximately 60% of theyielding value, the mechanical behaviour is character-ized by a progressive advancing and opening of the ten-sile crack (see the crack tip position corresponding tow = 0). Concrete crushing starts developing just beforethe yielding moment and it becomes more and moreevident by approaching the ultimate moment, when thenodal positions of the fictitious crack and overlappingtips coincide. The corresponding stress distributionsalong the mid-span cross-section are plotted in Fig. 5b.

3.1 Computation of the elastic coefficients

Equation (10) permits to analyse the fracturing andcrushing processes of the mid-span cross-section takinginto account the elastic behaviour of the RC member.To this aim, all the coefficients are computed a prioriusing a finite element analysis. In the present work, theFE code FEAP, developed by Prof. R. Taylor at the Uni-versity of California, Berkeley, has been used. Due tothe symmetry of the problem, a homogeneous concreterectangular region, corresponding to half the testedspecimen shown in Fig. 4a, is discretized by means ofquadrilateral plane stress elements with uniform nodalspacing. Horizontal constraints are then applied to thenodes along the vertical symmetry edge. The coeffi-cients K i,j

w entering Eq. (10), which relate the nodalforce Fj to the nodal displacement wi, have the physicaldimensions of a stiffness and are computed by imposinga unitary horizontal displacement to each of the con-strained nodes. On the other hand, by applying a unitaryexternal bending moment, it is possible to compute thecoefficients K i

M . They have the physical dimensions of[L]−1. At the same time, each coefficient of influencefor the nodal displacement on the global rotation, Di

w,

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Fig. 5 Nodal displacement profiles (a), and concrete stress distributions (b) at different applied moments for h = 0.4 m and ρt = 2%

entering Eq. (12), is given by the rotation of the freeedge corresponding to a unitary displacement of the i-th constrained node and it has the physical dimensionsof [L]−1. Finally, DM is the rotation of the free edgecorresponding to a unitary external bending momentand it has the physical dimensions of [F]−1[L]−1.

As regards the analysis of size-scale effects, the coef-ficients entering Eqs. (10) and (12) are connected bya simple relation of proportionality to the structuraldimension. This means that it is not necessary to repeatthe finite element analysis for any different consid-ered beam size. More precisely, if all the three spec-imen sizes (depth h, span L, thickness b) are multipliedby a factor k, then the coefficients are transformed asfollows:

K i,jw (kd) = kK i,j

w (d), (13a)

K iM (kd) = 1

kK i

M (d), (13b)

Diw(kd) = 1

kDi

w(d), (13c)

DM (kd) = 1

k3 DM (d). (13d)

On the other hand, if only depth and span are multipliedby the factor k, the thickness being kept constant, thenEqs. (13a,13d) are modified as follows:

K i,jw (kd) = K i,j

w (d), (14a)

DM (kd) = 1

k2 DM (d), (14b)

while Eqs. (13b, c) are unchanged.

4 Parametric analysis and experimentalcomparison

In this section, a parametric analysis is carried outto investigate on the influence of the main parame-ters affecting the global mechanical behaviour of RCbeams in bending, namely the beam size, h, the ten-sile and compression steel percentages, ρt and ρc, theconcrete compressive strength, σc, and the stirrups per-centage, ρw. The percentages ρt, ρc and ρw are defined

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as the ratios between the steel area and the beam cross-section area contained in the plane perpendicular tosteel.

The slenderness, λ, and the thickness, b, are keptconstant in the following examples and equal, respec-tively, to 1 and 0.2 m. The tensile yield strength ofsteel, σy , is set equal to 400 MPa. Moreover, detailednumerical–experimental comparisons are proposed onthe basis of the results found in the Literature. Thelabels used to identify the different experimentalcurves coincide with those reported in the originalpapers.

4.1 Size-scale effects

The available ductility, measured by means of the rota-tion at failure, progressively decreases by increasing thebeam size. In the case of high tensile steel percentage,as, e.g., ρt = 2%, the typical trend obtained by varyingthe beam depth is shown in Fig. 6. Similar trends havebeen noticed for the other values of ρt . A plateau isobserved in the nondimensional bending moment ver-sus rotation diagram when steel yielding occurs. Thesecurves put into evidence that the beam rotation at fail-ure is progressively diminished by increasing the beamdepth, with the appearance of sharper and sharper snap-back branches.

Such a trend is confirmed by the experimentalevidences obtained by Corley (1966), Bosco andDebernardi (1993) and Bigaj and Walraven (1993). Inparticular, a close numerical–experimental comparison

Fig. 6 Moment versus rotation diagrams for different beamdepths, h

Fig. 7 Effect of the size of the tested beams: numerical(solid lines) versus experimental (dashed lines) moment-rotationcurves. The experimental data are from Bosco and Debernardi(1993)

is shown in Fig. 7. The experimental data are fromBosco and Debernardi (1993).

4.2 Effect of the tensile steel reinforcement

In order to analyse the effect of the tensile steel per-centage on the ductility, numerical simulations havebeen carried out by varying such a parameter, all theother geometrical and mechanical parameters beingkept constant. The moment versus rotation curves forh=0.4 m, σc = 40 MPa, GC = 30 N/mm, σu = 4 MPaand GF = 0.08 N/mm are shown in Fig. 8a. Theultimate rotation increases by increasing the tensilereinforcement from 0.1 up to 0.5%, and it decreasessubsequently. This result is in good agreement withthe prescriptions of design codes (Comité Euro-International du Béton 1993), where a similar trend isproposed in terms of plastic rotation versus relative neu-tral axis depth (which is directly related to the amountof tensile reinforcement).

As a general remark, it is possible to state that theeffect of the shape of the cohesive crack law is not of pri-mary importance when the stress field is characterizedby a strong gradient, as analytically and numericallyshown in (Williams and Hadavinia 2002; Carpinteriet al. 2008) for double cantilever beam tests. More-over, the examples proposed in this section focus onthe evaluation of the post-peak branch of the moment-rotation curve, where the nonlinearity due to crushingand steel yielding or slippage are highly prevailing.The shape of the cohesive crack law influences the rel-ative maximum of the ascending branch of the moment-rotation diagram before steel yielding or slippage (see

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Fig. 8 Moment versusrotation diagrams fordifferent tensilereinforcement percentages,ρt (a), and theirmagnification forM ≤ 60 kNm (b)

Fig. 9 Moment versusrotation diagrams (a), andnodal displacements (b),for different amounts ofreinforcement incompression, ρc

also Carpinteri et al. 1989). However, this part of thediagram has a negligible effect on the overall behav-iour of beams with reinforcement ratios higher than0.25% (see Fig. 8b, where a magnification of Fig. 8afor M ≤ 60 kNm is provided).

4.3 Effect of the steel reinforcement in compression

With regard to the introduction of reinforcement incompression, it determines a reduction in the com-pressed zone, particularly in the case of high ten-sile steel percentage. This reduction determines anincrement in the crack opening displacement at theultimate condition with a subsequent increment ofthe rotational capacity, as experimentally evidencedby Burns and Siess (1966). Numerical simulationshave been carried out on RC beams with h = 0.4 m,ρt = 2%, σc = 40 MPa, GC = 30 N/mm, σu = 4 MPaand GF = 0.08 N/mm. As shown in Fig. 9a, the ulti-mate rotation turns out to be an increasing function ofthe compression reinforcement percentage, ρc, whichhas been varied from 0 up to 1%. This trend canbe clearly deduced by the displacement profiles ofthe points along the vertical cross-section shown inFig. 9b. It is worth noting that the neutral axis depthconsiderably decreases by increasing the compression

Fig. 10 Effect of the reinforcement in compression: numerical(solid lines) versus experimental (dashed lines) load-deflectioncurves. Experimental data are from Burns and Siess (1966)

steel percentage, whereas the maximum interpenetra-tion remains almost constant.

A numerical–experimental comparison to assessthe reliability of the model predictions is proposedin Fig. 10. The experimental data are from Burnsand Siess (1966) and refer to RC beams with h =0.25 m, ρt = 2.04%, ρw = 0.47%, and differentvalues of ρc. The moment-rotation diagrams obtainedfrom the numerical method are converted a posteri-ori into load-displacement curves for a direct com-parison with experiments. A good agreement in termsof ultimate rotation is noticed, whereas the numeri-

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Fig. 11 Moment versusrotation diagrams (a), andnodal displacements (b) fordifferent concretecompressive strengths, σc

Table 1 Overlapping law parameters for different concrete com-pressive strengths, σc

σc ( MPa) GC (N/ mm) σu ( MPa) GF (N/ mm)

20 30.0 1.6 0.049

40 30.0 3.0 0.080

60 40.8 4.1 0.105

80 50.6 4.6 0.130

100 56.5 5.0 0.148

cal results underestimate the maximum load carryingcapacity. This has to be ascribed to the actual hardeningbehaviour of steel which has not been modelled in thepresent study.

4.4 Effect of the concrete compressive strength

The rotational capacity is also an increasing function ofthe concrete compressive strength (see the experimen-tal results from Pecce and Fabbrocino 1999; Shin et al.1989; Markeset 1993). Analogously to the influence ofthe reinforcement in compression, in fact, the incre-ment of the compressive strength determines a reduc-tion in the crushing zone and a correspondent incrementin the crack opening displacement. In order to investi-gate on the influence of such a parameter, the numeri-cal simulations whose results are shown in Fig. 11 havebeen carried out by varying the compressive strengthfrom 20 to 100 MPa, and computing GC by means ofEq. (4) and σu and GF according to the prescriptions inModel Code 90 (Comité Euro-International du Béton1993) (see Table 1).

A numerical–experimental comparison is proposedin Fig. 12. The experimental data are from Pecceand Fabbrocino (1999) and refer to RC beams withh = 0.18 m, ρt = 2.23%, ρw = 0.25%, ρc = 0.08%

Fig. 12 Effect of the concrete compressive strength: numeri-cal (solid lines) versus experimental (dashed lines) load-rotationdiagrams. The experimental data are from Pecce and Fabbrocino(1999)

and two different values of σc. The moment-rotationdiagrams obtained from the numerical method are con-verted a posteriori into load-rotation curves for a directcomparison with experiments. A good agreement hasto be noticed both for the ultimate rotation and themaximum load carrying capacity.

4.5 Effect of the stirrups confinement

Finally, the effect of the transversal reinforcement, usu-ally introduced in RC beams in order to prevent shearfailure, is investigated. The presence of stirrups, infact, also influences the ductility, since it increases theconcrete confinement, with a consequent increment inthe compressive strength and the ultimate strain (seethe experimental results from Burns and Siess 1966;Somes 1966; Shin et al. 1989). In the proposed model,this effect is taken into account by varying the crush-ing energy, and consequently the critical value of theoverlapping displacement, according to Eq. (3). Thevalues used in the numerical simulations are reported

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Fig. 13 Moment versusrotation diagrams (a), andnodal displacements (b)for different stirrupscontents, ρw

Table 2 Overlapping law parameters for different stirrups con-tents, ρw

ρw (%) σc ( MPa) GC (N/ mm) wccr ( mm)

0.00 40.0 30.00 1.5

0.25 40.0 51.96 2.6

0.50 40.0 79.73 4.0

0.75 40.0 112.49 5.6

1.00 40.0 149.74 7.5

Fig. 14 Effect of the stirrups content: numerical (solid lines) ver-sus experimental (dashed lines) load-deflection diagrams. Theexperimental data are from Burns and Siess (1966)

in Table 2. The numerical results are shown in Fig. 13aand confirm that the available ductility considerablyincreases by varying the stirrups percentage from 0to 1%. The nodal displacement diagrams shown inFig. 13b evidence the corresponding increment of thecritical value of the crushing interpenetration.

A numerical–experimental comparison is proposedin Fig. 14. The experimental data are from Burns andSiess (1966) and refer to RC beams with h = 0.35 m,ρt = 1.46%, ρc = 0.00% and two different valuesof ρw. The moment-rotation diagrams obtained fromthe numerical method are converted a posteriori intoload-displacement curves for a direct comparison with

experiments. Again, a good agreement has to be noticedfor the predicted ultimate rotations.

5 Discussion and conclusions

Differently from the limit analysis, which yields onlythe ultimate load, the proposed model is able to inves-tigate on size-scale effects, instability phenomena andductile-to-brittle transitions in the mechanical responseof RC beams. In particular, the softening behaviourin the increasing branch of the moment versus rota-tion diagram due to the cohesive contribution of con-crete in tension can be captured, as well the softening(or snap-back) behaviour due to concrete crushing, asclearly evidenced in Figs. 6, 7, 8, 9, 10, 11, 12, 13 and14. In all of these cases, the virtual snap-back branchesare captured by governing the simulations with thecrack and the crushing lengths as the driving param-eters, rather than the applied load or the beam rotation.

A systematic application of the numerical algorithmby varying the main model parameters permits to cor-rectly describe most of the experimental tests carriedout in the last decades to investigate on the rotationalcapacity of RC beams (Burns and Siess 1966; Corley1966; Somes 1966; Shin et al. 1989; Markeset 1993;Bosco and Debernardi 1993; Bigaj and Walraven 1993;Pecce and Fabbrocino 1999). In particular, we can con-clude that the available ductility is an increasing func-tion of the compression steel percentages, the concretecompressive strength, the stirrups contents, whereasit decreases as the tensile steel percentage and/or thestructural dimension increase.

It is worth noting that the assumption of linear con-stitutive relationships, as well as the use of linear-elasticfinite elements, permits to correctly describe very com-plex phenomena by means of simple calculations. Fur-thermore, the proposed model can be easily extended

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to characterise the mechanical behaviour of fibre-rein-forced structural elements (Cotterell et al. 1992; Liand Liang 1986; Wecharatana and Shah 1983; Visal-vanich and Naaman 1983). The increment of the matrixtoughness due to the presence of fibres in concrete,in fact, may be globally considered by modifying thecohesive law. On the other hand, the analysis of deepRC beams characterized by reinforcements distributedover the whole web may be carried out by consideringmultiple reinforcement levels in tension. The introduc-tion of an axial force in addition to the applied bend-ing moment will also permit to study the mechanicalbehaviour of eccentric axial loading elements, as, forexample, columns and pre-stressed beams.

Acknowledgments The Authors would like to acknowledgethe financial support of the European Union to the Leonardoda Vinci Project “Innovative Learning and Training on Fracture(ILTOF)”, where Prof. Carpinteri and Dr. Paggi are involved,respectively, as the Project Coordinator and the ScientificSecretary.

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