Set-theoretic Approach to Determining the Strength Characteristics of a Discrete Model of a Cylindrical Shell Based on Vector Approximation

Abstract

To determine the stress-strain state of the shell structure, a discrete model of an elliptical cylinder was used. The entire study area was divided into a finite number of disjoint sets (finite elements), interacting only at nodal points. The derivation of basic geometric relationships involves two calculation options: counting the angle of rotation of the normal from its initial state and from its deformed position. The components of the displacement vector, their first derivatives, and the components of the normal rotation angle vector are taken as nodal unknowns. Approximating expressions between the components of the displacement vector, their derivatives, the vector of the normal rotation angle of the internal point of the finite element and the components of the displacement vectors, their derivatives, the vectors of the normal rotation angles of its node points were obtained on the basis of a vector interpolation procedure, which makes it possible to automatically take into account the displacements of the cylindrical shell as absolutely solid body. This paper presents an algorithm for the strength calculation of an elliptical cylinder when measuring the angle of inclination of the normal from its deformed state, taking into account the vector interpolation procedure. A specific example shows the high efficiency of the proposed algorithm, which solves the problem of taking into account the displacement of a finite element as a rigid whole.