Issue 35

P. Bernardi et al, Frattura ed Integrità Strutturale, 35 (2016) 98-107; DOI: 10.3221/IGF-ESIS.35.12 98 Focussed on Crack Paths A non-linear procedure for the numerical analysis of crack development in beams failing in shear P. Bernardi, R. Cerioni, E. Michelini, A. Sirico Dept. of Civil, Environmental, Land Management Engineering and Architecture, University of Parma (Italy) patrizia.bernardi@unipr.it, roberto.cerioni@unipr.it, elena.michelini@unipr.it, alice.sirico@studenti.unipr.it A BSTRACT . In this work, a consistent formulation for the representation of concrete behavior before and after cracking has been implemented into a non-linear model for the analysis of reinforced concrete structures, named 2D-PARC. Several researches have indeed pointed out that the adoption of an effective modeling for concrete, combined with an accurate failure criterion, is crucial for the correct prediction of the structural behavior, not only in terms of failure load, but also with reference to a realistic representation of crack initiation and development. This last aspect is particularly relevant at serviceability conditions in order to verify the fulfillment of structural requirements provided by Design Codes, which limit the maximum crack width due to appearance and durability issues. In more details, a constitutive model originally proposed by Ottosen and based on non-linear elasticity has been here incorporated into 2D-PARC in order to improve the numerical efficiency of the adopted algorithm, providing at the same time an accurate prediction of the structural response. The effectiveness of this procedure has been verified against significant experimental results available in the technical literature and relative to reinforced concrete beams without stirrups failing in shear, which represent a problem of great theoretical and practical importance in the field of structural engineering. Numerical results have been compared to experimental evidences not only in terms of global structural response (i.e. applied load vs. midspan deflection), but also in terms of crack pattern evolution and maximum crack widths. K EYWORDS . Reinforced concrete; Constitutive modeling; Biaxial stress state; Cracking; FE analysis. I NTRODUCTION he analysis of the response up to failure of reinforced concrete (RC) structures through numerical techniques requires the adoption of effective material constitutive models, able to correctly represent the behavior of concrete and steel at the element level. This represents a quite complex task, since as loading increases, RC behavior is influenced by several non-linear mechanisms which often interact with each other, such as concrete cracking and crushing, aggregate interlock, bond-slip behavior, dowel action and yielding of reinforcement. Therefore, realistic simulations of RC structural behavior require a correct description of each of these phenomena into the adopted material model, which can be subsequently implemented within the Finite Element (FE) framework. In this context, among the several constitutive laws proposed in the past within smeared crack formulations ([1-6], just to mention some), 2D-PARC model [7] represents an effective tool to perform non-linear analyses up to failure of RC structures, since it is able to account for the most influencing aforementioned contributions. This model is structured in a modular framework, so that each mechanical phenomenon is indeed individually analyzed by using a proper constitutive law and then the corresponding contribution is inserted into a material stiffness matrix. In more detail, the present work focuses on the evaluation of concrete contribution before and after the development of cracking. As regards concrete modeling, the basic requirements concern the choice of an accurate failure criterion to be employed in conjunction with a proper non-linear stress-strain relationship. The Seventies have seen the development of several T