Issue 37

P. Bernardi et alii, Frattura ed Integrità Strutturale, 37 (2016) 15-21; DOI: 10.3221/IGF-ESIS.37.03 16 provided by embedded bars produces indeed a not uniform distribution of induced tensile stresses in concrete, with a consequent initial warping of the member. Comparisons between numerical and experimental results prove that the proposed approach is able to correctly catch the effects of concrete shrinkage strains on member deflection, as well as on cracking strength and development. N UMERICAL M ODEL n this work, the original formulation of 2D-PARC constitutive relation is properly revised so as to include concrete shrinkage effects in short-term analyses. As already mentioned, this model, which is based on a smeared-fixed crack approach, was developed for a concrete membrane element containing n reinforcing layers with different orientations, subjected to plane stress conditions. The main features of 2D-PARC, as well as its governing equations can be found in [10-12], to which reference is made. In the following Sections the attention will be only focused on the main changes made to the original structure of the model in order to include concrete shrinkage. Uncracked stage In the uncracked stage, perfect bond is assumed between steel bars and the surrounding concrete, so the total strain vector { ε } is coincident with the strain in concrete { ε c } and steel { ε s }:       s c      , (1) The total stress vector { σ } can be then simply evaluated as the sum of the stresses acting in concrete, { σ c }, and in steel reinforcement, { σ s }. To take into account shrinkage effects, concrete stresses { σ c } are here computed as a function of concrete net strains ({ ε c } - { ε sh }), being { ε sh } the free shrinkage strain vector. According to 2D-PARC conventions, free shrinkage strains are assumed as negative, since they cause concrete shortening. The terms of vector { ε sh } can be either set equal to the shrinkage strain values measured during experimental tests, if available, or properly calculated according to classical formulations obtained from technical literature (e.g. [15-17]). In any case, the component of { ε sh } associated to shear strains is usually assumed equal to zero. The equilibrium equation for the uncracked RC element can be then written as:                   s s sh c c s c D D            , (2) where [ D c ] and [ D s ] respectively represent concrete and steel stiffness matrix, whose construction has been discussed elsewhere ([10] and [12], in a more recently revised form). Cracked stage Crack formation takes place when the current state of stress violates the concrete failure envelope in the cracking region (see [12]). In presence of shrinkage, the transition from uncracked to cracked stage occurs in correspondence of a lower load level, since the restraint provided by the embedded reinforcement causes the appearance of tensile stresses in concrete even before the application of any external load. Crack pattern is hypothesized to develop with a constant spacing a m 1 and a strain decomposition procedure is adopted, so leading to the following compatibility condition:       1cr c      , (3) where the total strain vector { ε } is obtained as the sum of the strains { ε c } in RC between two adjacent cracks (still intact, even if damaged), and those in the fracture zone, { ε cr 1 }, related to all the kinematics that develop after crack formation. As known, crack opening and sliding activate indeed several resistant mechanisms, such as aggregate bridging and interlock, tension stiffening and dowel action, which provide strength and stiffness. According to the procedure described in [10], the two strain vectors, { ε c } and { ε cr 1 }, are obtained by inverting the equilibrium conditions in the uncracked RC between cracks and at crack location, respectively. Moreover, the stresses in RC between adjacent cracks, { σ c }, and those in the crack, { σ cr1 }, are assumed to be coincident with each other and consequently to the total stress vector { σ }. In more detail, the equilibrium condition in RC between cracks is formally identical to Eq. 2, even if concrete and steel stiffness matrices, [ D c ] and [ D s ], are slightly modified with respect to the uncracked stage, so as to consider the degradation I