When analyzing the stress-strain state of thin-walled structural elements that have the shape of an ellipsoid, it becomes necessary to calculate the geometric characteristics of the ellipsoidal surface. When using the canonical ellipsoid equation, regions of uncertainty appear in the Cartesian coordinate system at the intersection points of the ellipsoid surface with the horizontal coordinate plane. To exclude these areas of uncertainty, we propose an expression of the radius vector of an ellipsoidal surface whose projections are functions of two parametric representations in mutually perpendicular planes. One of the planes is the vertical plane XOZ, and the other plane is the plane perpendicular to the axis O at the point with the x coordinate. The parameter T of the ellipse obtained from the intersection of the ellipsoid with the XOZ plane was chosen as the argument of the first parametric function. The argument of the second parametric function t is the parameter of an ellipse formed as a result of the intersection of an ellipsoidal surface with a plane perpendicular to the abscissa axis at a distance of x from the origin. The proposed representation of the ellipsoidal surface allowed us to exclude uncertainties at the intersection points of the ellipsoid with the HOWE coordinate plane. By differentiating the proposed radius-vector expression at an arbitrary point on an ellipsoidal surface, we obtain relations for the basis vectors of an arbitrary point and their derivatives represented by components in the same local basis. These relations are necessary for the development of algorithms for numerical analysis of deformation processes of engineering structures that have ellipsoidal surfaces.

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Comparative analysis of the use of the defining equations of plasticity theories obtained at the loading step in three ways is performed. In the first method, the relations between strains increments and stresses increments are obtained by differentiating the governing equations of the small elastic-plastic deformations theory between full stresses and strains. In the second method, the authors based on the proportionality hypothesis between the component deviators of strains increments and the component deviators of stresses increments without separating the incremental strain into elastic and plastic parts obtain the determining equations at the loading step. In the third method, the relations between the incremental strain and the stresses increment of the plastic flow theory are used on the basis of the hypothesis about the proportionality of the plastic deformations increments to the components of the stress deviator. Based on the analysis of algorithms for obtaining the constitutive relations and the analysis of the numerical results of the calculation example, preference is given to the second method of obtaining expressions between stress increments and strain increments without separating the latter into elastic and plastic parts.

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Using the shells of triangular finite elements in calculations with a choice of nodal unknowns in the displacements form and their first derivatives, the continuity between the elements is provided only by displacements. Continuity in the derivative values is performed only at the nodal points and is absent at the boundaries between the elements. This circumstance often leads to a very slow convergence of the computational process. In this article, when using a prismatic finite element with a triangular base, improvement of the computational process by ensuring the equality condition of the derivatives normal displacements in the sides middle of adjacent triangular bases using uncertain Lagrange multipliers. Correctness and efficiency of the developed algorithm are shown in terms of the calculation.

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The article presents a comparative analysis of the effectiveness of the use of finite elements of various dimensions in the study of the stress-strain state (SSS) of objects of the agro-industrial complex (AIC). To determine the strength parameters of the AIC objects, which can be attributed to the class of thinwalled, it is proposed to use a two-dimensional finite element in the form of a fragment of the middle surface of a triangular shape with nodes at its vertices. To improve the compatibility of a two-dimensional finite element at the boundaries of adjacent elements, it is proposed to use the Lagrange multipliers introduced in additional nodes located in the middle of the sides of the triangular fragment as additional unknowns. It is proposed to use a three-dimensional finite element in the form of a prism with triangular bases to study the SSS of agricultural objects of medium thickness and thick-walled. To improve the compatibility of the prismatic element, Lagrange multipliers in the middle of the sides of the upper and lower bases are also used. On the example of calculating a fragment of a cylindrical pipeline rigidly clamped at the ends loaded with internal pressure, the effectiveness of the developed two-dimensional and three-dimensional finite elements with Lagrange multipliers was proved. The validity of the use of a twodimensional element for researching the SSS of agricultural objects belonging to the class of thin-walled was proved.