Issue 35

P. Bernardi et al, Frattura ed Integrità Strutturale, 35 (2016) 98-107; DOI: 10.3221/IGF-ESIS.35.12 100 biaxial strength envelope based on the one suggested by Kupfer et al. [8] (see also [7] for further details), and by material secant moduli in the two orthotropic directions, to be inserted into the matrix [ D c ]. Moreover, in the cracked stage the terms of matrix [ D c ] are adequately softened through an empirical damage coefficient  related to crack width w 1 , whose expression can be still found in [7]. The need of adopting a so refined model for the description of concrete behavior, instead of a simple linear-elastic matrix, is mainly aimed to a correct simulation of those elements made of plain concrete or reinforced in a single direction, such as, i.e., beams without shear reinforcement. In these cases, the structural behavior is indeed mainly governed by concrete performances, and consequently its correct constitutive modelling is mandatory for a realistic prediction of both element stiffness and strength. This work illustrates an alternative procedure that still allows a quite sophisticated representation of concrete behavior but requires a lower computational effort. Its implementation into the 2D-PARC model is also briefly discussed. Modeling of concrete behavior in the uncracked stage The constitutive relation herein adopted for concrete modeling represents a specialized 2D form, according to Barzegar [16, 26], of the 3D non-linear elastic model originally proposed by Ottosen [14, 15]. The main advantages of this model lay on its simple definition, since the stress-strain relation is expressed as a function of only two parameters, i.e. the secant values of the Young modulus E c and of the Poisson coefficient  , which are properly modified to account for material non-linearity. As a consequence, concrete stiffness matrix [ D c ] can be written as: 2 1 0 1 0 1 1 0 0 2                           c c E D (2) in the global x - y co-ordinate system. Hence, with respect to the previous formulation implemented into 2D-PARC relation, this model depends on a lower number of parameters and allows to bypass the evaluation of the two equivalent uniaxial strains, so reducing the required computational effort. Moreover, this model is very flexible, since it can be used in conjunction with any failure criterion, by simply modifying a single parameter, the so-called “nonlinearity index”, as discussed in the following. Similarly to the original approach of 2D-PARC, the failure envelope proposed by Kupfer et al. [8, 9] has been still considered, even if its analytical expression has been slightly modified according to [16] in the region corresponding to tension-compression, so as to avoid possible discontinuities in those points where the maximum principal tensile stress is close to zero. Fig. 1 shows the adopted failure envelope and the analytical expressions describing each considered region (tension-tension, tension-compression and compression-compression); the bold line indicates the part of the curve effectively implemented into the 2D-PARC model, according to its conventions (that is  1c ≥  2c ). Concrete secant elastic modulus E c is computed using a parameter called “nonlinearity index”,  , which depends on how far the current stress point is from failure and is then related to the amount of non-linearity in the stress-strain curves [14, 16]. In case of biaxial compression this parameter can be evaluated through the expression: 2 2     c fin (3) where  2c is the maximum principal compressive stress and  2fin represents its corresponding value on the failure envelope, determined by keeping fixed the other principal compressive stress  1c (being  1c ≥  2c , and assuming compressive stresses as negative). Thus,  < 1,  = 1 and  > 1 respectively correspond to stress states located inside, on, and outside the considered failure curve. When tensile stresses are present, the nonlinearity index is instead computed in terms of effective stresses. To this aim, the actual state of stress (  1c ,  2c ), where at least  1c is a tensile stress, is properly turned into an “equivalent compressive case”, by superposing an hydrostatic pressure -  1c to the existing stress field. In this way, a new state of stress (  ' 1c ,  ' 2c ) = (0,  2c -  1c ) is obtained and the nonlinearity index is evaluated as: