The usage of traditional approximating functions directly to the desired displacement vector of the internal point of a finite element to determine it through nodal unknowns in the form of displacement vectors and their derivatives is described. To analyze the stress state of a geometrically non-linearly deformable shell of rotation at the loading step, the developed algorithm for forming the stiffness matrix of a hexagonal finite element with nodal values in the form of displacement increments and their derivatives was used. To obtain the desired approximating expressions, the traditional interpolation theory is used, which, when calculated in a curved coordinate system, is applied to the displacement vector of the internal point of a finite element for its approximation of class C(1) through nodal displacement vectors and their derivatives. For the coordinate transformation, expressions of the bases of nodal points are obtained in terms of the basis vectors of the inner point of the finite element. After the coordinate transformations, approximating expressions of class C(1) are found for the components of the displacement vector of the internal point of the finite element, leading in a curved coordinate system to implicitly account for the displacement of the finite element as a rigid whole. Using calculation examples, the results of the developed method of approximation of the required values of the FEM with significant displacements of the structure as an absolute solid are obtained.

In the curvilinear coordinate system, an approximation of the finite element required quantities in the vector formulation is developed with the implementation of the stiffness matrix of the volumetric finite element of the shell of rotation taking into account the geometric nonlinearity.

Structures made of thin-walled shells are widely distributed elements in the environmental protection systems. In this connection, it is necessary to develop their effective and refined calculation. The point of the shell of the environmental protection structures is considered with a plane load in the initial position, deformed after j loading steps (displacement vector) and adjacent after (j + 1) loading step. The displacement increments and their derivatives are taken as nodal unknowns. The displacement vectors of the inner point of a finite element are represented in the initial basis, and their components are approximated through nodal unknowns using Hermite polynomials of the 3rd degree. To determine the deformed state of the shell, the algorithm of the method of discrete continuation with respect to the parameter in the vicinity of a singular point is used, in which the increment of loading at the step is the desired parameter. Based on the obtained dependencies, a step-by-step calculation procedure is organized, which allows you to get correct results if there is a special point.

The article discusses the objects of water management of the environment. An algorithm for forming a matrix of the stress-strain state of a prismatic finite element with a triangular cross-section in a mixed formulation is proposed to take into account the physical nonlinearity of water management objects in the form of rotation shells. The basic relations of the flow theory for arbitrary loading are used. The desired displacements and stresses of the inner point of the finite element were approximated by linear functions in terms of their nodal values. The stress-strain state matrix was formed using a functional expressing the equality of the actual work of external and internal work of external and internal forces, in which the actual work of internal forces was represented by the difference between the total and additional work of these forces.