Scale-dependent maximum reinforcement percentage in reinforced concrete beams

The Cohesive/Overlapping Crack Model is able to describe the transition between cracking and crushing failures occurring in reinforced concrete beams by increasing beam depth and/or steel percentage. Within this Nonlinear Fracture Mechanics model, the tensile and compressive ultimate behaviors of the concrete matrix are modeled through two different process zones that advance independently one of another. Moreover, this model is able to investigate local mechanical instabilities occurring in the structural behavior of reinforced concrete structures: tensile snap-back and snap-through, which are due to concrete cracking or steel fracture, and the compressive snap-back occurring at the end of the plastic plateau, which is generated by the unstable growth of the crushing zone. In this context, the application of the Cohesive/Overlapping Crack Model highlights that the ductility, which is represented by the plastic rotation capacity of a reinforced concrete element subjected to bending, decreases as reinforcement percentage and/or beam depth increase. Thus, a scale-dependent maximum reinforcement percentage beyond which concrete crushing occurs prior to steel yielding is demonstrated to exist. In particular, the maximum steel percentage results to be inversely proportional to h 0.25 , h being the beam depth. In this way, a rational and quantitative definition of over-reinforcement is provided as a steel percentage depending on the beam depth.


| INTRODUCTION
The flexural behavior of a reinforced concrete (RC) beam is heavily influenced by nonlinear phenomena that are usually observed during laboratory tests (Figure 1), although not properly taken into account during the RC design process. More precisely, after the first cracking load, P cr, is reached, a loss of stability is revealed due to a Discussion on this paper must be submitted within two months of the print publication. The discussion will then be published in print, along with the authors' closure, if any, approximately nine months after the print publication. local reduction in the resistance capacity of the beam. This unstable branch represents a snap-back instability, [1][2][3][4][5][6] and its severity has been proved to be a function of the mechanical and geometrical characteristics of the structure, as demonstrated by Carpinteri 2,3,7 by means of the brittleness number s E . After concrete cracking, an ascending branch generated by the steel reinforcement activation is registered. Then, at the end of the plastic plateau ( Figure 1), a second snap-back can be revealed, due to the unstable propagation of the crushing zone in the concrete matrix.
In the study of RC fracturing process, the Bridged Crack Model 5,7-10 proved to be able to capture the loading drops due to cracking, particularly in the case of lightly reinforced or high-performance concrete elements. More recently, an extended version of the Bridged Crack Model 11,12 was applied to investigate the scaletransitional role of the shearing failure in RC beams.
Extensive experimental tests have demonstrated that the parameters influencing the ductility of RC elements may be summarized into three main groups: construction material; structural geometry; static system. 13 In this framework, a decrease in the plastic rotation capacity together with an increase in specimen depth and/or reinforcement percentage has been recognized. [14][15][16] Nevertheless, a large scatter in experimental data has been observed and a comprehensive theory that is able to predict thoroughly the plastic rotation capacity developed by RC members has not been still provided. 17,18 On the other hand, the Cohesive/Overlapping Crack Model is a Nonlinear Fracture Mechanics application that is able to describe the transition between cracking and crushing failure in RC beams, highlighting a strong correlation between structural behavior, scale, and reinforcement percentage. 6,[19][20][21] In the present paper, the Cohesive/Overlapping Crack Model together with a dimensional analysis approach are adopted in order to define an effective scale-dependent maximum reinforcement percentage for RC beams in flexure.

| THE COHESIVE/ OVERLAPPING CRACK MODEL
The Cohesive Crack Model has been widely applied to simulate the damage process zone ahead of the crack tip in concrete structures. [1][2][3][22][23][24] According to this model, the material behaves elastically during the first loading stage (Figure 2(a)), whereas in the zone where the principal stress reaches the tensile strength, σ t , the process zone starts developing. 25 Within this zone, a cohesive law (Figure 2(b)) in the form σ-w t , σ being the applied stress and w t the crack opening, is adopted. Stresses apply until the critical value of crack opening, w t cr, is reached: beyond this limit, the crack faces assume a stress-free condition.
In the present paper, a cohesive constitutive law in the form: is assumed. The area subtended by the σ-w t curve represents the fracture energy, G F (Figure 2(b)). Different experimental investigations carried out by Kotsovos,26 van Mier, 27 Vonk et al. 28 pointed out that concrete exhibits strain localization in compression in a similar way as in tension. In 1997, the RILEM Committee 148-SSC 29 developed an effective standard test for the assessment of the strain softening behavior of concrete under uniaxial compression. The test program, organized by this Committee, involved two different loading conditions and several specimen geometries with different concrete grades were investigated. It was demonstrated that the maximum compressive load in the σ-ε diagram can represent a bifurcation point for the equilibrium path depending on the specimen size and slenderness: an increment in the global ductility was observed with a decrement in the specimen slenderness. Thus, it was concluded that a single σ-ε relationship may not be assumed as an effective constitutive law for concrete in compression in the postpeak regime. On the other hand, it emerged that, if an overlapping relative displacement, w c , is considered rather than the compressive strain, all the curves converge within a narrow band. 30 The main conclusion of the RILEM round-robin was that two different constitutive laws must be adopted for concrete in compression. In the uncracked stage, the material behavior is described appropriately by a F I G U R E 1 Structural behavior of a reinforced concrete beam subjected to bending σ-ε constitutive law (Figure 3(a)) up to the compressive strength σ c . Beyond that limit, a σ-w c law must be adopted. The area subtended by the σ-w c curve (Figure 3(b)) may be interpreted, in close analogy to Figure 2(b), as the crushing energy, G c , i.e., a compressive fracture energy. 31 In such a context, Akiyama et al. 32 and Suzuki et al. 33 conducted several tests on plain and laterally-confined concrete specimens, proposing a formula for the determination of G c . In the case of plain concrete, they suggested the following expression: Based on these experimental evidence, Carpinteri et al. 34 introduced the Overlapping Crack Model to simulate the post-peak compressive behavior of RC beams in bending. Such model is analogous to the Cohesive Crack Model in tension, since in the elastic stage the concrete behavior is described by a σ-ε constitutive law ( Figure 3 (a)), whereas, beyond the peak load, a stress versus fictitious overlapping relationship, σ-w c , is adopted. This post-peak softening law can thoroughly describe concrete crushing and material expulsion by a fictitious interpenetration. 31 In the present paper, a linear overlapping law in the form: is adopted. Within the Cohesive/Overlapping Crack Model 21,35-37 the cracking process zone (Figure 4(a)) tends to propagate from the intrados to the extrados of the beam, whereas, vice-versa, the crushing process zone tends to propagate from the extrados to the intrados of the beam (Figure 4(b)). Thus, cracking and crushing failures are modeled as two separate mechanisms, independently evolving and mutually competing.
By means of this Nonlinear Fracture Mechanics Model, the RC beam cross-section is discretized into n different nodal points, being n = 161 in order to avoid numerical instabilities, as suggested by Carpinteri et al. 38 For these nodes, the following equation applies: {w} being the vector containing the crack opening/ overlapping displacements, [K F ] the matrix containing the coefficients of influence for the nodal displacements generated by the unit nodal forces, {K M } the vector containing the nodal displacements generated by a unit bending moment, and M the value of the applied bending moment. The number of the unknowns in Equation 4 is equal to (2n + 1): n crack opening/overlapping displacements, n nodal forces, and the applied bending moment, M ( Figure 5). In addition to Equation 4, the following conditions should be taken into account to describe the RC beam cross-section behavior: j being the real crack tip, m the fictitious crack tip, p the fictitious overlapping crack tip, q the real overlapping crack tip, and r the node where the reinforcement layer is located. Equation 5f correlates the crack opening at the level of reinforcement with the force exerted by the steel bar. This equation can be calibrated according to a bond-slip law proposed in Ruiz and Planas 24 , Ruiz 23 , Model Code 2010 39 . In Figure 6 it is possible to recognize the length 2L τ , along which concrete and steel mutually transfer the shearing stress τ(x). The value of L τ may be calculated by means of the equilibrium condition of the steel bar. Thus, in the case of overreinforced concrete beams and neglecting the tension stiffening effect, we have: 40,41 The variation of τ along L τ depends on concrete strength, steel surface condition, and slippage between these two materials. Thus, in order to simplify Equation 6, it is possible to consider the mean value of the shearing stresses, τ m , leading to: where ℓ is the beam span. Then, the value of the crack opening, w t , may be obtained as: Neglecting the concrete contribution and referring to a linear variation in the steel stress along L τ (Figure 6), it is possible to find: In the present paper, Equation 9 is used to calculate the crack opening that generates steel yielding, w t y , considering the following simplified elastic-perfectly plastic constitutive law for the steel reinforcement: σ y being the yield strength of steel. Furthermore, it is possible to evaluate the crack opening that generates steel fracture as: Equation 10 and Equation 11 describe the steel constitutive law as depicted in Figure 7. It is important to note that Equation 5f coincides with Equation 10 only if the node r lies within the real crack length and if w r < w t y Within the Cohesive/Overlapping Crack Model numerical procedure, at each calculation step, the value of M is set as the minimum value of the bending moment that is able to generate the ultimate tensile force, F t , in the fictitious tensile crack tip, m, or the ultimate compressive force, F c , in the fictitious overlapping crack tip, p. Only the tip that firstly reaches the ultimate condition is moved to the following nodal point and calculation step, allowing the propagation of the cracking or crushing zone.
Then, the RC beam rotation is computed as: {K M } being the vector containing the nodal displacements generated by a unit bending moment (Betti's Theorem) and D M the coefficient of influence for M = 1.

| NUMERICAL VERSUS EXPERIMENTAL COMPARISON
In Figure 8, numerical curves obtained through the application of the Cohesive/Overlapping Crack Model to RC beams in bending are presented. The curves describe the structural behavior of beams with thickness, b, equal to 0.2 m, and depth, h, varying between 0.1 and 2.0 m. The concrete matrix is assumed to have a compressive strength σ c = 40 MPa, and a tensile strength σ t = 4 MPa, whereas the fracture energy, G F , and the crushing energy, G c , are assumed equal to 0.08 and 30 N/mm, respectively. The steel yield strength σ y = 400 MPa and the crack opening that generates steel yielding w t y = 0.4 mm have been assumed. On the other hand, the critical value generating steel fracture w t u = 18.5 mm has been calculated for a class B reinforcement (Eurocode 2). 42 In the numerical study of Figure 8, the reinforcement ratio, ρ, varies between 0.5% and 3.0%, and, for each reinforcement percentage, the ratio of the effective beam depth to the beam depth, d/h, is set equal to 0.9. Figure 8(a) shows the curves obtained for ρ = 0.5%. It is possible to observe that the rotational capacity, represented by the extension of the plastic plateau, decreases as the beam depth increases. For large beam depths, at the end of the plastic plateau it is possible to observe a catastrophic drop in the bearing capacity, generated by unstable propagation of the concrete crushing zone (snap-back), as it is also described in Figure 1. In Figure 8(b), the curves obtained for ρ = 1.0% are reported. This figure reveals a severe reduction in the extension of the plastic plateau for all the considered geometries, as well as a transition towards a more brittle behavior. The beams having h = 1.5 m and 2.0 m present a completely unstable behavior, since, once the maximum load is reached, a catastrophic loss in the bearing capacity occurs without steel yielding.
The reduction in the rotational capacity is also evident in Figure 8(c), where beams with ρ = 2.0% are analyzed: only the curves A-D present a plastic plateau. In Figure 8(d), all the curves present an unstable crushing behavior, a short plastic plateau being present only for h = 0.1 m.
In order to validate the abovementioned results, a numerical versus experimental comparison is performed by means of the Cohesive/Overlapping Crack Model. The experimental tests carried out by Carpinteri et al. 12 in order to investigate the ductile-to-brittle transition in RC beams are considered. These tests were performed on specimens having a cross-section of 100 Â 200 mm 2 and a span of 1400 mm. The beams were casted using a concrete matrix characterized by σ c = 36.75 MPa, E = 33,058 MPa, and G F = 0.112 N/mm. Four different reinforcement percentages were investigated in the range ρ = 0.25-3.14%.
The numerical versus experimental comparison is presented in Figure 9. In Figure 9(a) and (b), the casestudies with ρ = 0.25% and ρ = 0.50% are analyzed, suggesting that the Cohesive/Overlapping Crack Model has a high capability in predicting concrete cracking, local snap-back instabilities, as well as steel yielding. Moreover, for large reinforcement percentages (Figure 9 (c) and (d)), a clear ductile-to-brittle transition is predicted by this Fracture Mechanics model. 43

| SCALE-DEPENDENCY OF THE MAXIMUM REINFORCEMENT PERCENTAGE
The numerical and experimental studies described in the previous section make evident the existence of an upper bound to the steel percentage, which is intended to avoid crushing and over-reinforcement. Such a steel percentage results to be scale-dependent (Carpinteri and Corrado 21 ; Carpinteri et al. 35 ). Currently, within the international standard requirements, a lack of knowledge exists about scale effects on RC maximum reinforcement conditions, since the maximum reinforcement percentage (also called "balanced reinforcement percentage") is trivially identified through the simplified Stress-block Model ( Figure 10).
Hence, defining ε c as the maximum deformation in concrete, and ε y the deformation in steel at yielding, it is possible to find the neutral axis position in the section ( Figure 10) as: Indicating the depth of the stress block with βx, and imposing the equilibrium, it is possible to identify the maximum, or balanced, reinforcement percentage (Stress-block Model): On the other hand, for a more thorough vision of the problem, it is possible to perform a dimensional analysis F I G U R E 1 0 RC beam cross-section with balanced reinforcement percentage (Euler-Bernoulli theory) through Buckingham's Theorem (Carpinteri and Accornero). 44 The resistance moment M developed by RC beams may be written as a function of several variables: E being the concrete elastic modulus and ℓ the beam span.
If we intend to avoid over-reinforced concrete beams, it is possible to neglect the variables G F and σ t , since the mechanical behavior of the abovementioned elements is mainly governed by the compressive strength and by the crushing energy of concrete. Thus, Equation 15 may be simplified as follows: Assuming (G c E) 0.5 and h as the independent variables, it is possible to rewrite Equation 16 as: where it is possible to recognize the Matrix Brittleness Number, s c = (G c E) 0.5 /σ c h 0.5 , and the Reinforcement Brittleness Number, N U P = ρσ y h 0.5 /(G c E) 0.5 . Hence, through the application of the Cohesive/Overlapping Crack Model, it is possible to set up a parametric analysis in order to find the maximum reinforcement T A B L E 1 Mechanical and geometrical parameters of the beams considered in the parametric study condition in the diagram s c -N U P . The mechanical and geometrical parameters of the RC beams investigated in the present paper are reported in Table 1. Five different concrete grades were considered, and several beam depths were analyzed, spanning from h = 0.1 m to h = 3.2 m. For each RC beam, the ratio d/h is fixed at the value 0.9, whereas the steel yield strength is σ y = 450 MPa. The fracture energy, G F , was estimated through Model Code 2010 39 , whereas the crushing energy, G c , was calculated through Equation 2.
The calculated values of s c and N U P are reported in Figure 11, where a best-fitting relationship provides: Considering the definitions of s c and N U P , it is possible to find: where an evident scale effect on the maximum reinforcement percentage is defined through the h À0.25 power-law.
In Table 2, the β parameters and the ultimate strain of concrete, ε c , provided by different international codes of practice are reported. It is possible to observe that all the considered provisions are rather similar, except for IS456:2000 45 , which provides a smaller equivalent stressblock at Ultimate Limit State. Accordingly, in Figure 12 a comparison between the maximum reinforcement percentages provided by the above-mentioned standards and by Equation 19 are reported. This diagram clearly shows that all the current international codes do not take into account the scale effects on RC beams, providing a maximum reinforcement percentage far from that predicted by the Cohesive/Overlapping Crack Model and leading to a wrong assessment of the RC ductility.

| CONCLUSIONS
The application of the Cohesive/Overlapping Crack Model to the study of the maximum reinforcement percentage in RC beams provides a power law proportional to h À0. 25 , being h the beam depth. The proposed formula for the assessment of an effective scale-dependent maximum reinforcement condition is compared to the current provisions given by several international standards,