Tailor Made Concrete Structures – Walraven & Stoelhorst (eds)© 2008 Taylor & Francis Group, London, ISBN 978-0-415-47535-8

Improvement in the plastic rotation evaluation by means offracture mechanics concepts

M. Corrado, A. Carpinteri, M. Paggi & G. ManciniPolitecnico di Torino, Department of Structural Engineering and Geotechnics, Torino, Italy

ABSTRACT: The well-known Cohesive Crack Model describes strain localization with a softening stress vari-ation in concrete members subjected to tension. Based on the assumption that strain localization also occurs incompression, the Overlapping Crack Model, analogous to the cohesive one, is proposed to simulate materialcompenetration due to crushing. By applying this model, it is possible to describe the size effects in com-pression in a rational way. The two aforementioned elementary models are then merged into a more complexalgorithm based on the finite element method, able to describe both cracking and crushing growths duringloading processes in RC members. With this algorithm in hand, it is possible to investigate on the influ-ence of the reinforcement percentage and/or the structural size of RC beams, with special attention to theirrotational capacity. The obtained results evidence that the prescriptions concerning the plastic rotations pro-vided by codes of practice, not taking into account the scale effects, are not conservative in the case of largestructural sizes.

1 INTRODUCTION

The development of considerable ductility in the ulti-mate limit state is a key parameter for the designof reinforced concrete (RC) beams in bending. Theinterest in ductility was formerly connected with thediffusion of plastic analysis in the design of rein-forced concrete structures (Macchi 1969). In thiscontext, in fact, the rotational capacity is required toallow the bending moment redistribution in staticallyindeterminate structures. The ductility contributes tosatisfy many other requirements, absolutely necessaryin order to guarantee the structural safety, as, e.g., toprovide robustness, to give warning of incipient col-lapse by the development of large deformation priorto collapse and to enable major distortions and energydissipation during earthquakes.

Due to its complexity, this phenomenon was firstlyanalyzed from an experimental point of view. As aresult of the “Indeterminate Structures Commission”of the Comité Européen du Béton (1961), published inBaker & Amakarone (1967), an empirical hyperbolicrelationship between plastic rotation and relative neu-tral axis depth, x/d, was proposed to solve the problemof plastic rotation evaluation for practical purposes.A second fundamental contribution comes from theexperimental and analytical research carried out inthe early 1980s at the University of Stuttgart, by thegroup coordinated by Professor Eligehausen (Langer

1987, Eligehausen & Langer 1987). With respect tothe foregoing empirical relationship, two aspects wereintroduced: firstly, an increasing branch for low val-ues of x/d, due to steel rupture, and, secondly, theexplicit consideration of two classes of ductility forthe reinforcement.

Further improvement in the analysis of thebehaviour of RC members in bending was markedby the pioneering paper by Hillerborg (1990) whointroduced the concept of strain localization in con-crete in compression. According to this approach,when the ultimate compressive strength is achieved,a strain localization takes place within a characteris-tic length proportional to the depth of the compressedzone. This model permits to address the issue of sizeeffects, although the definition of the length overwhich the strain localization occurs is a free parameterand its value is not defined on the basis of theo-retical arguments. Afterwards, several models havebeen developed on the basis of the Stuttgart and theHillerborg proposals, emphasizing some more specificaspects (Cosenza et al. 1991, Tue et al. 1996, Bigaj &Walraven 2002, Fantillli et al. 2007).

In order to assess the rotational capacity of RCbeams, the Eurocode 2 (2003) provides a design dia-gram relating the admissible plastic rotation to therelative neutral axis position. From the analysis ofthese curves, it appears that the size-scale effects onthe rotational capacity of RC beams are not considered,

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Figure 1. Compression crushing with overlapping (a),analogous to tensile fracture with the cohesive zone (b).

although the dependence on the structural dimensionwas recognized in several experimental tests (Mattock1965, Corley 1966, Siviero 1974, Bosco & Debernardi1992, Bigaj & Walraven 1993).

In the case of concrete specimens subjected to uni-axial compression, several experiments (Van Vliet &Van Mier 1996, Jansen & Shah 1997) reveal that thesize-scale effects are due to two interconnected phe-nomena: the strain localization after the peak load andthe consequent energy dissipation over a surface, thevalue of which referred to a unitary surface can beconsidered as a material parameter. Based on theseevidences, Carpinteri et al. (2007) have recently pro-posed to model the process of concrete crushing usingan approach analogous to the Cohesive Crack Model,which is adopted for the tensile behaviour of concrete.The former approach is referred to as the OverlappingCrack Model (Carpinteri et al. 2007), which assumesa stress-displacement law for the post-peak behaviour.In tension, the localized displacement is representedby a tensile crack opening, while in compression it isrepresented by a compenetration, as clearly shown inFigure 1.

In this paper, we propose a new numerical methodable to describe the nonlinear behaviour of RC mem-bers during both fracturing and crushing. Firstly, theCohesive Crack Model for concrete in tension andthe Overlapping Crack Model for concrete in com-pression are introduced. Then, a numerical algorithmbased on the finite element method is presented forthe analysis of intermediate situations ranging frompure concrete members to over-reinforced beams. Itis assumed that the fracturing and crushing processesare fully localized along the mid-span cross-sectionof the representative structural element in bending.This assumption, fully consistent with the physics ofthe crushing phenomenon, also implies that only oneequivalent main tensile crack is considered. In orderto validate the proposed model, a comparison betweenthe numerical predictions and the experimental resultsfor the beams tested by Bosco and Debernardi (1992) iscarried out. Finally, as a result of a parametric inves-tigation on the influence of the structural dimensionand of the steel percentage, a practical diagram show-ing the size-scale effects on ductility is reported, interms of plastic rotation as a function of the relativeneutral axis position.

Figure 2. Nonlinear behaviour of concrete in tension andcompression.

2 NUMERICAL INVESTIGATION

In this section, a new model based on FractureMechanics concepts is proposed for the evaluation ofthe rotational capacity of RC beams in bending. Letus consider a portion of a RC beam subjected to abending moment M . This element, having a span todepth ratio equal to one, is representative of the centralzone of the beam where a plastic hinge formation takesplace. Note, incidentally, that the analysis of this struc-tural element is also consistent with the prescriptionsreported in the Eurocode 2 (2003). We also assume thatthe mid-span cross-section of this element is fully rep-resentative of its mechanical behaviour. The stress dis-tribution in this cross-section is linear-elastic until thetensile stress at the intrados reaches the concrete tensilestrength. When this threshold is reached, a cohesivecrack propagates from the beam intrados towardsits extrados. Correspondingly, the applied momentincreases. Outside the crack, the material is assumed tobehave linear-elastically. According to the well-knownCohesive Crack Model (Carpinteri 1985), the stressesin the cohesive zone are assumed to be decreasingfunctions of the crack opening displacement until acritical value of crack opening is reached.

On the other hand, concrete crushing takes placewhen the maximum stress in compression reaches theconcrete compressive strength.After that, damage willbe described by a compenetration of the two half-beams, representing in this way the localization of thedissipated energy (Fig. 2b). Larger is the compenetra-tion, also referred to as overlapping in the sequel, lowerare the transferred forces along the damaged zone.

2.1 Cohesive Crack Model for the description ofconcrete fracturing

A pioneering model for the analysis of nonlinear crackpropagation in concrete was proposed by Hillerborget al. (1976) with the name of Fictitious Crack Model.Subsequently, an updated algorithm was implementedby Carpinteri (1985, 1989), with the terminology ofCohesive Crack Model, in order to study the ductile tobrittle transition in plane concrete beams in bending.According to this model, the constitutive law used for

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the non-damaged zone is a σ − ε linear-elastic rela-tionship up to the tensile strength, σt,u. In the processzone, the damaged material is still able to transfer atensile stress across the crack surfaces. The cohesivestresses are considered to be decreasing functions ofthe crack opening, wt, as follows:

where wt is the crack opening, wtc is the critical value

of the crack opening corresponding to the conditionσt = 0, and σt,u is the ultimate tensile strength ofconcrete.

The area defined by the stress vs. displacementcurve represents the fracture energy, t

F .

2.2 Overlapping Crack Model for the description ofconcrete crushing

The most frequently adopted constitutive laws for con-crete in compression describe the material behaviourin terms of stress and strain (elastic-perfectly plas-tic, parabolic-perfectly plastic, Sargin’s parabola, etc).Such approaches imply that the energy is dissipatedin a volume, whereas experimental results reveal thatthe energy is substantially dissipated over a surfaceas a result of strain localization, (Van Vliet & VanMier 1996, Jansen & Shah 1997). Hillerborg (1990)firstly proposed to model the crushing phenomenonas a strain localization over a length proportional tothe depth of the compressed zone. This condition,however, does not permit to formulate a material con-stitutive law fully describing the mechanical responseof concrete in compression.

In the present formulation, we adopt the stress-displacement relationship proposed by Carpinteri et al.(2007) and then considered also in Corrado (2007)between the compressive stress and the compenetra-tion, in close analogy with the cohesive model. Themain hypotheses are:

1. The constitutive law used for the undamaged mate-rial is a linear-elastic stress-strain relationship, seeFigure 3a.

2. The crushing zone develops when the maximumcompressive stress reaches the concrete compres-sive strength.

3. The process zone is perpendicular to the maincompressive stress.

4. The damaged material in the process zone isassumed to be able to transfer compressivestresses between the overlapping surfaces. Theyare assumed to be decreasing functions of thecompenetration, wc (see Fig. 3b):

Figure 3. Concrete constitutive law in compression: σ−ε

linear-elastic law (a); σ−w post-peak softening law (b).

where wc is the compenetration, wcc is the criti-

cal value of the compenetration corresponding tothe condition of σc = 0, and σc,u is the ultimatecompressive strength.

It is worth noting that the crushing energy, cF ,

defined as the area below the post-peak softeningcurve, is now a true material parameter, since it is notaffected by the structural dimension.

An empirical formulation for calculating the crush-ing energy has recently been proposed by Suzuki et al.(2006), based on the results of compression tests car-ried out on plane and transversal reinforced concretespecimens. In this study, the crushing energy is com-puted according to the following empirical equation,which considers the confined concrete compressivestrength by means of the stirrups yield strength andthe stirrups volumetric content:

where σc,0 is the average concrete compressivestrength, ka is a parameter depending on the stirrupsstrength and on the volumetric percentage, and pe isthe effective lateral pressure.

The crushing energy for unconfined concrete,cF ,0, can be calculated using the following

expression:

where the parameter kb depends on the concretecompressive strength (see Suzuki et al. 2006).

It is worth noting that cF is between 2 and 3 orders

of magnitude higher than tF . Finally, we remark that

the critical values for crushing compenetration andcrack opening are approximately equal to wc

c ≈ 1 mm(see also the experimental results by Jansen and Shah1997) and wt

c ≈ 0.1 mm, respectively.

2.3 Numerical algorithm

A discrete form of the elastic equations governingthe mechanical response of the two half-beams isherein introduced in order to develop a suitable algo-rithm for the analysis of intermediate situations where

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both fracturing and crushing phenomena take place.According to the finite element method, the mid-spancross-section of the beam is subdivided into finite ele-ments with n nodes (Fig. 4). In this scheme, cohesiveand overlapping stresses are replaced by equivalentnodal forces by integrating the corresponding trac-tions over each finite element size. Such nodal forcesdepend on the nodal opening or closing displacementsaccording to the cohesive or overlapping softeninglaws previously introduced.

With reference to Figure 4, the horizontal forces Facting along the mid-span cross-section are given by:

where {F} is the vector of nodal forces, [Kw] is thematrix of the coefficients of influence for the nodaldisplacements, {w} is the vector of nodal displace-ments, {Km} is the vector containing the coefficientsof influence for the applied moment, and M is theapplied moment.

The coefficients of influence, Kwi, present the phys-ical dimension of a stiffness and are computed a prioriwith a finite element analysis by imposing a unitdisplacement to each of the nodes shown in Figure 4.

In the generic situation shown in Figure 5, thefollowing equations can be considered:

Equation (5e) represents the relationship between theclosing force exerted by the reinforcing steel and thecrack opening at the reinforcement level. Such a lawis determined on the basis of the bond-slip behaviourof concrete and steel.

Equations (4) and (5) constitute a linear algebraicsystem of (2n) equations and (2n + 1) unknowns,namely {F}, {w} and M . A possible additional equa-tion can be introduced: we can set either the force inthe fictitious crack tip, m, equal to the ultimate ten-sile force, or the force in the fictitious crushing tip, p,equal to the ultimate compressive force. In the numer-ical scheme, we choose the situation which is closerto one of the two possible critical conditions. This cri-terion will ensure the uniqueness of the solution onthe basis of physical arguments. The driving parame-ter of the process is the tip that in the considered stephas reached the limit resistance. Only this tip is movedwhen passing to the next step.

Figure 4. Finite element nodes along the middlecross-section.

Figure 5. Force distribution with cohesive crack in tensionand crushing in compression.

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

where {Dw} is the vector of the coefficients of influ-ence for the nodal displacements, and Dm is thecoefficient of influence for the applied moment.

3 EXPERIMENTAL COMPARISON ANDNEW PROPOSAL FOR DESIGN CODES

In this section, a comparison between the numericalpredictions using the cohesive/overlapping model andthe experimental results of the tests carried out byBosco and Debernardi (1992) is proposed. In orderto obtain a consistent comparison, the numerical sim-ulations have been carried out by modelling the beamportion positioned at the mid-span of the beam. Suchan element is characterized by a span to depth ratioequal to one. The rotations of such portion wereexperimentally determined as functions of the appliedbending moment.

Numerical and experimental moment-rotationcurves are then compared in the Figures 6a,b,c fordifferent beam depths and different steel percentages.These diagrams put into evidence that the maximumrotation is a decreasing function of the tensile rein-forcement ratio and of the beam depth. In the case

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Figure 6. Comparison between numerical and experi-mental results for different beam depth: h = 200 mm (a);h = 400 mm (b); h = 600 mm (c).

of low steel percentages, the mechanical behaviour ischaracterized by the reinforcement yielding and themechanical response is almost plastic. By increasingthe amount of reinforcement, the contribution of con-crete crushing becomes more and more evident withthe appearance of a softening branch at the end of theplastic plateau. This is an important feature of the pro-posed model, which also permits to follow unstablesoftening branches with positive slope, (snap-back).This is possible by controlling the loading processthrough the length of the tensile crack and the exten-sion of the fictitious crushing zone, rather than by theexternal load. A good agreement is obtained betweennumerical and experimental results for all the testedbeams.

Figure 7. Numerical plastic rotation as a function of therelative neutral axis position for different beam depths.

With reference to the moment-rotation diagrams,the plastic component of the total rotation can beobtained as the difference between the ultimate rota-tion and the rotation corresponding to the reinforce-ment yielding. According to the definition proposedby Hillerborg (1990) and Pecce (1997), the ultimaterotation is the rotation beyond which the moment startsdescending rapidly.

The results of the parametric analysis can be sum-marized in a plastic rotation vs. relative neutral axisposition, x/d, diagram (see Fig. 7). This is also consis-tent with the practical prescriptions of the Eurocode2. As can be readily seen, the rotational capacity is anonlinear function of the relative neutral axis position.In a first stage, for x/d up to approximately 0.05, theplastic rotation is an increasing function of x/d, sincebeam failure is mainly governed by steel yielding.Then, when crushing comes into play by increasingthe amount of reinforcement, the plastic rotation turnsout to be a decreasing function of x/d. Moreover, notethat size-scale effects are particularly relevant, sincethe higher is the beam depth, the lower is the rotationalcapacity.

4 CONCLUSIONS

In the present paper, a numerical algorithm has beenproposed for the analysis of the behaviour of RCelements in bending. To this aim, all the principalnonlinear contributions are taken into account for anaccurate evaluation of the moment-rotation diagram.With this tool in hand, the simulation of the mechani-cal behaviour of all the intermediate situations rangingfrom pure concrete to over-reinforced concrete beamscan be made. The following main conclusions canbe drawn from the comparison between numericalpredictions and experimental results:

1. The introduced constitutive law for concrete incompression through the Overlapping Crack Model

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permits to describe the nonlinear behaviour ofconcrete considering the effect of the structuraldimension and to describe the descending branch ofthe moment-rotation diagram, as shown in Figure 6.For large structural sizes and/or high reinforce-ment percentages, this softening branch can havea positive slope, leading to a snap-back instability.

2. Referring to Figure 6, it is possible to state thatthe proposed algorithm catches the experimentalresults (Bosco & Debernardi 1992) by varying boththe structural dimension and the steel percentage.

3. Independently of the reinforcement ratio, thebehaviour becomes more and more brittle byincreasing the beam depth, with a progressivelyreduction of the ultimate rotation.

4. According to the Eurocode 2 (2003), the plasticrotation of RC beams is considered to be a func-tion of the neutral axis position only. In orderto improve the code provisions, the effect of thestructural dimension should be explicitly taken intoaccount by considering different design curves, as,for instance, those proposed in Figure 7.

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