Issue 35

P. Bernardi et al, Frattura ed Integrità Strutturale, 35 (2016) 98-107; DOI: 10.3221/IGF-ESIS.35.12 102 Similarly, 1 3 is the value of   2 c f J f for uniaxial compressive loading. As regards the evaluation of the secant value of Poisson coefficient to be inserted into Eq. 2, it has been experimentally observed that in presence of compressive stresses concrete tends first to compact, while after the appearance of micro- cracks it tends to dilate. As a consequence, its value should be properly adjusted during the analysis so as to correctly represent this behavior. Following the procedure proposed in [14, 16], the Poisson coefficient is kept fixed until  reaches a “limit value”  a equal to 0.8, and afterwards it is updated by applying the following relation:   2 1 1                   a f f i a , (9) where  i is the initial value of the Poisson coefficient (assumed equal to 0.2) and  f represents its secant value at peak (approximately equal to 0.36). The so obtained secant values of concrete elastic modulus and Poisson coefficient have been inserted into the concrete stiffness matrix [ D c ] (Eq. 2); in this way, only the constitutive relation adopted for concrete modeling has been modified, by keeping unchanged the global structure of 2D-PARC algorithm. It should be observed that all the above described procedure is applied until either cracking or crushing occurs. Cracking takes place when the current stress state reaches or violates the failure envelope in the cracking region, i.e.  1c ≥  1max in tension-compression or simply  1c ≥ f ct in case of tension-tension, being f ct the tensile strength of concrete. In this case, the fixed-smeared crack approach previously described is applied (see [7] for further details), even if the modeling of concrete between cracks has been properly modified as described in the following Subsection. On the contrary, when the stress state reaches or violates the failure envelope in the crushing region, i.e.  ≥ 1 in case of biaxial compression or  2c ≤  2max in case of compression-tension, a very simplified procedure is applied herein. For the considered case studies, only small portions of the modeled structures can enter indeed in the post-crushing regime during the analysis (e.g. near concentrated loads or at supports) without affecting the global behavior, which is instead much more influenced by the appearance of tensile cracks. However, concrete crushing should be included in the model formulation so as to avoid numerical difficulties or wrong predictions of the ultimate failure load. To this aim, in 2D-PARC model a reduced Young modulus E c is simply considered, equal to 25% that of undeformed concrete, as suggested also in [27]. Modeling of concrete behavior in the cracked stage The above described formulation has been applied also in the cracked stage for the evaluation of the stiffness matrix of concrete between cracks, [ D c ]. However, in order to include damaging due to the presence of cracks, a proper reduction in the concrete compressive strength and stiffness has been operated. To this aim, the biaxial concrete failure envelope shown in Fig. 1 has been properly reduced, following the relation suggested in [26]: * * max 0 1 0.2 , 0.8                     c c c c c c f f f f f , (10) where  max is the current maximum principal tensile strain in cracked concrete and * c f is the modified value of the uniaxial compressive strength at peak. The corresponding modified strain *  co can be in turn calculated as a function of the initial value of the strain  c0 corresponding to peak stress in uniaxial compression, through the expression: * max 1 0.1                    co co co . (11) COMPARISONS WITH EXPERIMENTAL OBSERVATIONS he capability of the proposed model to describe RC global behavior and crack pattern evolution has been verified against the results of two well-documented experimental programs concerning beams without shear reinforcement [17, 18]. Among several research projects, the one carried out by Vecchio and Shim [17] has been selected owing T