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The paper discusses the process of non-linear deformation of shell structures made of reinforced concrete. A mathematical model of deformation in the form of the functional of full potential deformation energy is provided. The model is based on the Kirchhoff–Love hypotheses, and allows accounting for structure reinforcement with stiffeners. An orthogonal network of stiffeners, located from the concave side, is considered as the structure support. Type of load — external, uniformly distributed. The Ritz method is applied to the functional to reduce the variational problem of the functional minimum to a system of nonlinear algebraic equations. Then, for each load value, the problem is solved using iterative methods. Analysis of strength and stability of shallow shells of double curvature and rectangular planform is performed. Values of critical loads, deflection and stress fields are obtained. Curves of deflection depending on load are provided. All results are given in dimensionless parameters. The Mohr–Coulomb criterion was used to analyze concrete strength, and the Lyapunov criterion was used for stability analysis. Influence of the number of stiffeners reinforcing the shell on the resulting stress values is shown. It has been revealed that with account for physical non-linearity of concrete, when the dependence of stresses and deformations is curvilinear, deformations (and deflections as well) of shells increase in comparison with the linear-elastic solution. It has been also found that when nonlinearity is taken into account, redistribution of stresses over the shell field occurs (the maximum stresses shift towards the shell contour).

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