Keywords: damage mechanics, reinforced concrete, beams, grids, building floor frames, shear force effects, finite element reinforced concrete modelling.

Structural analysis of reinforced concrete building floor frames can be significantly improved when appropriate material models are assumed to represent the actual global behaviour and consequently to improve the ultimate and serviceability limit state verifications. Often, the stiffness reduction and the evaluation of the ultimate strain values are carried out by means of simple models based on bending moment-curvature diagrams and therefore the shear effects are neglected. For instance, the models proposed by Galli-Favre [1] and Debernardi [2], have already demonstrated to be capable of representing accurately the stiffness reduction of reinforced concrete elements. These models can represent very well the beam behaviour treating the element as a continuous and uniform element therefore without requiring the definition of cracks and their position. The actual distribution of strains, as well as the resulting curvatures, is averaged along the element, considering the intact cross-sections (State I) and cracked cross-sections (State II-Naked). Better elaborated models to predict the stiffness reduction could lead to more reliable results in engineering practice. The continuum damage mechanics concepts is perhaps the more efficient way to lead to appropriate reinforced concrete beam models. However, for the majority of the reinforced concrete beam models proposed so far takes into account only the bending behaviour, either to verify the ultimate load capacity or to obtain the actual rigidity in service. The effects of shear stresses are often neglected. Thus, Further reduction of beam and consequently the structural system is neglected. The behaviour of the concrete material in shear is very particular due to the presence of the micro-crack distribution.

In this work, an improved model to compute either the stiffness reductions or the ultimate loads has been proposed to take into account the non-linear effects due to the shear components. The finite element method for Timoshenko's beam with three degree polynomial approximation has been chosen to drive the beam model. A damage model is adopted to govern the stress and strain fields across the cross-section of the reinforced concrete member. The damage parameter used to reduce the beam stiffness as well the ultimate stresses is evaluated by considering the complete stress tensor, i.e., by taking into account the shear components. Then, a simple mechanism to transfer the shear stresses not sustained by the concrete material to the shear reinforcements. As the shear forces carrying capacity of the elements is severely reduced, a simple mechanism based on the Mörsch's truss concept to transfer the shear stresses to the shear reinforcements is developed. The shear reinforcement shear force resultant is computed according to the stain component measured in the tensile principal direction. The model was also extended to deal with slabs. The model proposed by Mazars [3], particularly develop to simulate concrete behaviour, has been implemented to govern the concrete behaviour both based on the continuum damage mechanics, while an elasto-plastic model was assumed for the bending and shear reinforcements. Two practical examples to illustrate the model efficiency are shown and whenever possible the results are compared with experimental values.

1

Ghali, A., Favre, R., "Concrete Structures: Stresses and Deformations", Chapman & Hall, 1th edition. 1986.

2

Debernardi, P.G., "Behaviour of Concrete Structures in Service", Journal of Structural Mechanics, 115, 32-50, 1989. doi:10.1061/(ASCE)0733-9445(1989)115:1(32)

3

Mazars, J., "Application de la Mécanique de l'Endommagement au Comportement Nonlineaire et a la Rupture du Béton de Structure", Thése de dotoract d'etat, Université Paris 6, Paris, 1984.

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