Closed-form moment-curvature relations for reinforced concrete cross sections under bending moment and axial force

The paper develops a computer strategy for the analysis of cracked reinforced concrete cross sections within the scope of Eurocode 2, using a symbolic algebraic manipulator. The nonlinear constitutive behaviours for either concrete or steel are considered. For concrete, a nonlinear constitutive equation, function of the concrete class, is adopted, as indicated in Eurocode 2 for structural analysis. This equation is more general than the parabola equation used in many models for both ultimate and service design. For the steel, the present model uses a continuous and smooth function that approximates both the elastic and plastic ranges with hardening, instead of the traditional multi-linear forms. The moment-curvature relations are obtained by imposing equilibria for the bending moments and longitudinal forces, assuming that cross sections remain plane after deformation and that there is a perfect adherence between concrete and steel. In addition, closed-form solutions are also proposed and the cross section's collapse and rotation capacity is determined by the reaching of the ultimate strains for concrete or steel. The paper ends with the presentation of illustrative examples, for rectangular and T- cross sections.