Keywords: reinforced concrete membrane and shell structures, verification, design, multidirectional reinforcement, non-linear analysis.

Typically reinforcement layout in two-dimensional concrete structures (plates and shells) is designed as two steel bar families placed in orthogonal directions. However, in some cases, particularly in zones where high principal stresses do not coincide with the reinforcement directions it is usual to design an extra level of reinforcement oblique to the general reinforcement. Also, in cases of biased plates may be more convenient, from the construction point of view, to design the direction of the bars reinforcement parallel to the sides of the plate than an orthogonal bar layout. However, these situations, that often occur in actual design of reinforced concrete structures, are not contemplated in the majority of concrete Codes and Recommendations. Usually in current Codes reinforcement design and verification of structures different to beams, like two dimensional structures, is scarcely treated.

In the methodology used in this work the standard elasticity equations, namely, equilibrium, compatibility and constitutive equations, among the known stress resultants, stresses and strains of the two materials, concrete and steel, are applied. In the constitutive equations for the concrete its tension strength is not considered and in order to be specific a parabolic-rectangle diagram with a descendent branch is used. For the steel an elastic-plastic behavior is assumed, that is represented by a bilinear stress-strain diagram and in this way hardening effects can be simulated.

In order to carry out the proposed procedure to verify the reinforcement and the concrete the use of a desk-top computer generally is required. By application of a simple computer program it is possible to obtain in a few seconds the curves representing the paths of the strains and stresses of each reinforcement family and the principal stresses in the concrete as function of the amplification factor. In this way the safety level at the point under study of the reinforced concrete structure can be obtained.

In this work, an unified approach to verify the reinforcement in two dimensional reinforced concrete structures is presented. First, the application of this approach to structures type membrane plate and membrane shell, i. e. structures subjected to in-plane stresses, longitudinal and tangential stresses, is shown. As is standard, the stress resultants, that are obtained by means a linear elastic analysis, are amplified by a set of safety coefficients in order to reach the so called verification stress resultants. In this paper the reinforcement verification at a point is assumed to be designed with the maximum generality, i.e. it can have either an arbitrary number of curtains of reinforcement, one in the middle plane or two or more symmetrically placed respect to the middle plane of the structure and for each curtain of reinforcement two or more bar directions. The above described methodology is applied to the general case of plates and shells subjected to bending and axial stress resultants and the potentialities of the method are emphasized by means of some illustrative examples, including some design charts. Finally using these results a tentative concrete failure surface for the reinforced concrete is proposed.

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