It is known that concrete structures behave more fragile as their size increases. However, this phenomenon is ignored in many standards for the design of structures and the same design criteria are used for homothetic structures, a decision that is not on the side of safety. The research carried out on concrete, mainly in the last decade, has shown that this size effect can be predicted and calculated taking into account the concrete fracture process. When a designer knows how to model this phenomenon, it is possible to predict and evaluate the fragility of a structure based on its size. It can be thought that the increase in fragility with size is due to the fact that a larger element contains more defects and, therefore, is weaker. This effect -which can certainly contribute- does not justify the experimental results, as demonstrated by Z.P. Bazant (Planas, Bazant, & Jirasek, 2001). The true cause of fragility must be sought in the process of fracture, since this phenomenon - the increase of fragility with size - also occurs in ideally homogeneous materials; that is, with defects of a size that does not depend on the dimensions of the structure. The current work will analyze the numerical models of Zdenek P. Bazant, Carpinteri, and B. L. Karihaloo.
The Numerical model of Zdenek P. Bazant
The study of the effect size and scale effect on fracture of concrete and heterogeneous materials has been an important area of research throughout the ages. There are several parameters that must be considered in determining the size effect, such as the size of the aggregate that affects the apparent scale effect of the fragile materials, because it influences the microcracking process, leading to cracks (Bazant & Gettu, 1990). In the study of the scale effect, carried out by Planas, Bazant, and Jirasek (2001), the concrete resistance was related to the ratio between the dimensions of the specimen and the dimensions of the aggregate.
Law of Bazant (Size Effect Law - SEL)
Considering the equilibrium of the energy required for the propagation of a crack in a concrete structure, Planas, Bazant, and Jirasek (2001) proposed, in an analysis of reinforced concrete pieces of different dimensions and geometrically similar, the formula called SEL (size effect law) [1 and 2], as follows:
sn - nominal breaking voltage;
P - Maximum load;
b - Thickness;
D - Characteristic sample size;
f 't - Direct tensile strength;
da - Maximum size of the aggregate;
B e lo - Empirical constants.
The curve representing this equation (1) is given as shown in Figure 1.
In his studies, Bazant employed the theory proposed by Griffith which relies on the propagation of preexisting cracks (Bazant & Gettu, 1990). It was found that when the SEL is applied in non-cracked specimens the hypothesis becomes flawed. In the face of the numerous criticisms Z. Bazant introduced a term so that was designated as being the resistance of an infinitely large specimen:
Figure 1. Bazant scale effect law
In the technical literature it is observed that there are two theories about the scale effect: a proposal by Z. Bazant whose validity is limited and another by Carpinteri, which is based on the heterogeneity of the concrete and considers that for small concrete structures the size of the aggregate is large in relation to this, so the heterogeneity of the material is large and the scale effect will be accentuated (Bazant & Gettu, 1990). However, for large structures in relation to the size of the aggregate, the material becomes homogeneous and the scale effect disappears. The scale effect and size effect formulas for determining the mechanical strength properties of the materials cannot be used to evaluate these effects on large reinforced concrete structures, such as bridges, as they disappear into large specimens. When larger specimens are used, the exaggerated statement that Bazant transcribe to the present day, that is, as the dimensions of the pieces is increased their strength decreases.
The Numerical model of Carpinteri
According to Carpinteri (2006), the nominal tensile strength of concrete structures is usually constant for relatively large sizes, it however increases with the size for relatively large sizes and reduces with the size for small sized concrete structures. The experimental investigation is usually lower that the one order or magnitude in the scale range, resulting in a tangential slope in the bilogarithmic strength versus size diagram.
Law of Carpinteri (Multifractal Scaling Law - MFSL)
The formula representing this law is as follows :
Where: sn - nominal breaking voltage;
d - Characteristic dimension of the structure;
A and B - physical constants;
dmax - maximum size of the aggregate;
ao - constant.
The curve representing this equation (3) is shown in Figure 2.
If d tends to infinity, the nominal resistance tends to a constant value other than zero (threshold resistance). On the contrary, when d tends to zero, the nominal resistance tends to infinity. This means that the scale effect for some concrete structures is accentuated only for some size ranges, which may be large or small, depending on the type of situation (Carpinteri & Puzzi, 2006). In equation (3) the constant B depends on the characteristics of the dimensions of the structure. For structures with d> B the scale effect tends to disappear, for example, the structure fails at the beginning of cracking - fragile rupture. However, for d <B, the scale effect is accentuated and the rupture is ductile.
In the study of scale effect and size effect, the transition zone at the pulp - aggregate interface should be considered, which results in a reduction of the concrete resistance to larger aggregates, since there is an increase in the amount of water in this region increasing the porosity and therefore compromising the compressive strength of the concrete.
The numerical model of B. L. Karihaloo
The evaluation of a load of failure by diagonal traction (failure by shear) in reinforced concrete elements is a problem not solved in a completely satisfactory way within the scientific-technological field. Karihaloo states that the shear strength in concrete elements decreases with increasing deformation in the longitudinal reinforcement (Karihaloo & Xiao, 2002). If the deformation of these bars is considered a function of size, the scale effect observed in the tests is reproduced implicitly by the proposed formulations. Gamino, Borges, and Bittencourt (2006) other methodologies that provide expressions from theoretical models and that include new parameters such as the position of the failure section, the orientation of the same, the tensile strength of the concrete and the stress state inside the concrete element. Karihaloo and Xiao (2002) add that the design formulas to determine shear failure load should be based on theories based on Fracture Mechanics, given the relationship between the progress of the fissures, the failure of the element and the scale effect that is revealed in this type of failure. It is considered that the diagonal traction failure occurs in a fragile way due to the unstable growth of a fissure that originates in an area between the point of application of the load and the support. It is appreciated that after reaching the maximum load there is an abrupt failure of the reinforced concrete element.
This model is able to explain and predict most of the experimental results obtained with concrete specimens. In addition, this model can be simplified - within a reasonable range of sizes - by approximating it by means of an equivalent elastic crack and its associated curve D. The model, in its simplest version, assumes that the material, outside the crack, behaves in an elastic and linear way. (It can be generalized for other behaviors). Secondly, that the fissure starts at the point where the maximum principal tension reaches the value d. The fissure is formed in the plane perpendicular to the direction of this tension. Thirdly, that the fissure propagates when the two faces of this plane are separated. It is assumed that a voltage - cohesive tension - is transmitted between them, which ao is a function of the history of the relative displacements between the two faces w. The size effect on the fracture of concrete decreases significantly with decreasing time to failure. When the loading rate increases, the effective process zone length increases, resulting in an increase in the brittleness. There is an increase in the fracture toughness with decreasing time to failure, and this is why the strength decreases.
Bazant, Z. P., & Gettu, R. (1990). Size effect in concrete structures and influence of loading rate. In Serv Durability Constr Mater Proc First Mater Eng Congr (pp. 1113-1123).
Carpinteri, A., & Puzzi, S. (2006). Fractals, statistics and size-scale effects on concrete strength. Polytechnic of Torino, Department of Structural Engineering and Geotechnics, Torino, Italy.
Gamino, A., Borges, J., & Bittencourt, T. (2006.). Size effect of concrete: Uniaxial and Flexural compression. Measuring, Monitoring, and Modeling Concrete Properties, 107-113. doi:10.1007/978-1-4020-5104-3_13
Karihaloo, B. L., & Xiao, Q. Z. (2002). Size effect in the strength of concrete structures. Sadhana, 27(4), 449-459. doi: 10.1007/bf02706993
Planas, J., Bazant, Z. P., & Jirasek, M. (2001). Reinterpretation of Karihaloo's size effect analysis for notched quasi-brittle structures. International Journal of Finance 111 (1), 17-28.
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