## Shear resistance of fractured concrete

After the initiation of fracture process on a plane perpendicular to the direction of major principal tensile stress, it is not unlikely to expect the rotation of principal stress directions under varying deformation modes. Hence, shear deformation may take place on the partially formed rough fracture plane. The definition of shear stress-strain behaviour of concrete during the fracture process seems to be difficult. Bazant and Oh (1983) ignored the shear deformation on the fracture plane, and the material stiffness matrix was derived for normal strains only. This formulation is not compatible with the linear isotropic stiffness matrix of initial state. In another proposition, Bazant and Gambarova (1984) proposed a nonlinear mathematical model, named the crack band microplane model, to account for shear in the partially formed crack, propagating as a blunt front of micro-cracks. Gambarova and Va-lente (1990) retained the initial shear modulus unchanged until the complete fracture had taken place and then applied an aggregate interlock model. de Borst and Nauta (1985) and Gajer and Dux (1990) applied the concept of simple shear retention factor, fl, proposed by Suidan and Schnobrich (1973), to derive the tangent shear modulus of the fracture plane. The simplified approach of applying a shear retention factor ignores the shear dilation and the dependence of crack shear stiffness on the crack opening displacement (COD). Dahlblom and Ottosen (1990) assumed a linear relationship between shear resistance and COD. The models proposed by Bazant and Gambarova (1980), Chen and Schnobrich (1981) Reinhardt and Wairaven (1982), Riggs and Powell (1986), and Wairaven (1981) represent the shear resistance and dilation on crack open surfaces of concrete. The models are very often computationally inconvenient because of the nonsymmetric stiffness matrix. A comparative study on different rough crack models is available in Feenstra et al. (1991a, 1991b). A special numerical problem associated with the shear retention in smeared crack finite element model is the spurious tension stiffness in the direction across the fracture plane (El-Aidi and Hall 1989a). This happens due to the application of continuous shape functions in deriving the finite element stiffness matrix. Discontinuous shape functions proposed by Ortiz et al. (1986) for localized failure analysis may be a solution to this problem.

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### 5.4 Post-fracture behaviour of concrete

The fracture direction is generally fixed based on the principal stress direction that initiates the first crack. An additional crack plane is allowed to form only when the stress reaches the tensile strength on the plane orthogonal to the first fracture plane. This type of model is stated as fixed or stationary crack model . In the "rotating crack" model, proposed by Cope et al. (1980), the orthotropic material reference axis system is rotated when the principal stress direction deviates by a certain amount from the direction that initiated the fracture process. The rotation of physical crack direction does not seem to be acceptable from common perception. After early opposition to the rotating crack model, Bazant et al. (Bazant 1983; Bazant and Lin 1988) adopted the concept based on the argument that the cracks of one direction may close and lock in shear while cracks of another direction may form. A special numerical technique to represent non-orthogonal multiple crack formation has been perfected by de Borst et al. (de Borst 1987; de Borst and Nauta 1985).

A very special post-fracture problem, associated with dynamic analysis, is the modelling of contact-impact phenomenon occurring upon closing and reopening of the cracks. Special numerical techniques to simulate the impact behaviour in discrete crack models have been proposed in the literature (Ayari and Saouma 199 1; Pekau et al. 199 1). El-Aidi and Hall (1989a) presented a discussion on numerical difficulties arising from high velocity closing and reopening of cracks in smeared crack analysis No rigorous procedure has been reported in the literature to deal with shock wave generated by the sudden change in stiffness resulting from the closing and reopening of the smeared cracks. And the consequences of the shock-wave phenomenon in smeared crack analysis are not conclusively known yet.

5.5 Material parameters for fracture propagation analysis

The present development of mathematical models is far ahead of the current knowledge of material behaviour, specially under transient conditions. Material parameter data determined from reliable experimental studies is limited in the literature for dam concrete. Recent experimental investigations, such as the one by Briihwiler (1990) and Briihwiler and Wittmann (1990) are revealing significant differences in the mechanical properties of structural concrete and mass concrete. Ideally, the selection of material properties for safety analysis of concrete dams should be dealt with on a case-to-case basis, because the material properties are widely varying from dam to dam. However, a review of literature is presented here to establish a reasonable limit of parametric values.

Poisson's ratio, v, and elastic modulus, E, are applied to represent the elastic behaviour of concrete in all analyses irrespective of constitutive models selected for propagation of cracks. Jansen (1988) suggested the Poisson's ratio for 1-year old dam concrete between 0.17 and 0.28. A value of 0.20 has been applied almost universally in the past studies. Briihwiler (1990) observed the reduction of Poisson's ratio with increasing compressive strain rate applied to the concrete cylinder. However, the influences of rate sensitive v may be insignificant in comparison to the influences of other material parameters. The static modulus for 1 -year old dam concrete was suggested by Jansen (1988) in the range of 28000 - 48000 MPa. A significant feature of concrete constitutive behaviour is the dynamic magnification of the elastic modulus under rapidly varying loading condition. A review of this phenomenon along with the tensile strength property of concrete is presented in the following section. Separate reviews are presented for commonly used material parameters in three major crack propagation criteria strength of material, linear elastic fracture mechanics, and nonlinear fracture mechanics. The shear resistance parameter of fractured concrete is also discussed.

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