Explored orthotropic shallow shells of double curvature, as well as cylindrical panels that are reinforced from the concave side by an orthogonal grid of stiffeners. The external transverse load acting on the structure is uniformly distributed and has a linear dependency on time. A geometrically nonlinear variant of the model which also takes into account orthotropy of the material and transverse shears are considered. The model is presented as a functional of total deformation energy of the shell. The algorithm for studying the mathematical model is based on the L.V. Kantorovich method and the Rosenbrock method. The proposed algorithm was implemented in the analytical computing environment Maple 2016. The calculations showed a significant increase in critical load values for the loss of stability when the shell is reinforced with stiffeners.