Issue 35

P. Bernardi et al, Frattura ed Integrità Strutturale, 35 (2016) 98-107; DOI: 10.3221/IGF-ESIS.35.12 99 failure criteria derived from different failure data (e.g., [8-11]), which have pointed out that concrete under a biaxial (or triaxial) state of stress exhibits different stiffness, strength and ductility than under uniaxial loading. Consequently, the strength characteristics of concrete under a general multi-axial state of stress cannot be reproduced into a material model by directly using experimental uniaxial stress-strain curves. To represent the problem, several mathematical material models have been proposed, which can be substantially divided into orthotropic, non-linear elastic, plastic or endochronic ones [12]. With reference to orthotropic formulations, a fairly simple solution can be obtained by referring to the concept of equivalent uniaxial strain, as originally proposed by Darwin and Pecknold in [13] and implemented into 2D-PARC model [7]. An interesting model based on non-linear elasticity was instead proposed by Ottosen [14]. This formulation provides non-linear stress-strain relations for concrete by only properly changing the secant values of Young modulus and Poisson ratio. In this way, even if the model is able to realistically represent concrete behavior under a general stress state, its calibration is quite simple, only requiring experimental data obtained by standard uniaxial tests. For its flexibility and numerical feasibility, it has immediately appeared suitable for the implementation into FE programs, and applicable to the analysis up to failure of different types of RC structures [15, 16]. Basing on the approach proposed by Ottosen [14] and on the implementation performed by Barzegar [16], concrete modeling into 2D-PARC has been here revised. This modified formulation has been verified with reference to the analysis of RC beams without shear reinforcement [17, 18], where concrete modeling assumes particular relevance both before and after crack pattern development. Furthermore, this represents a structural problem of significant theoretical and practical importance, subjected in the past to great experimental efforts [19, 20], since the comprehension of the mechanisms of shear transfer across cracks and failure in beams without shear reinforcement can improve the knowledge of concrete contribution on shear strength also in beams with web reinforcement. NUMERICAL MODEL he behavior of reinforced concrete (RC) beams without shear reinforcement has been herein studied through a non-linear finite element (NLFE) procedure. In order to account for reinforced concrete mechanical non-linearity, a suitable constitutive model, named 2D-PARC [7], has been adopted. This constitutive model is based on a smeared-fixed crack approach and its theoretical basis, which has been deduced for a RC membrane element subjected to general in-plane stresses, can be found in details in [7] and in [21, 22] with reference to its extension to the case of steel- fiber reinforced concrete (SFRC) elements or to the 3D case. In the uncracked stage, concrete and steel are schematized like two material working in parallel, by assuming perfect bond between them. When the principal maximum stress violates the failure envelope in the cracking region, crack pattern is assumed to develop with a constant spacing a m1 . Afterwards, a strain decomposition procedure is adopted, by subdividing the total strain into two components, respectively related to RC between cracks and to all the resistant mechanisms that develop after crack formation (i.e. aggregate bridging and interlock, tension stiffening and dowel action). These resistant contributions are expressed as a function of two main variables, namely crack width w 1 and sliding v 1 , and included into the crack stiffness matrix [ D cr1 ]. The behavior of RC between cracks is instead described by adopting the same approach used in the uncracked stage, even if a slight modification is operated on both concrete and steel stiffness matrices, [ D c ] and [ D s ], so as to account for the degradation induced by cracking. The cracked RC stiffness matrix is then expressed in the following form: 1 1 1 1 1                                                            c cr c s D D D I D D (1) where [ I ] is the identity matrix. This formulation has been successfully applied to the analysis of different types of structures (such as panels, beams, slabs, etc…), providing a good prediction of experimental evidences both in terms of load-deformation response up to the ultimate capacity of the considered element, and in terms of crack pattern evolution and failure mode (e.g. [7, 23, 24]). However, the proposed algorithm is quite complex and requires long calculation times, which are mainly related to the approach followed in the evaluation of concrete stiffness matrix [ D c ]. Both in the uncracked and cracked stage, concrete is indeed modeled as an orthotropic, non-linear elastic material subjected to a biaxial state of stress, which is duplicated by means of two equivalent uniaxial curves, following Darwin and Pecknold approach [13]. These uniaxial curves for concrete in compression and in tension report the actual stress as a function of an equivalent uniaxial strain, which is in turn determined according to [25]. These curves are characterized by maximum strength values derived from an analytical T