A. P. Nikolaev

Comparison of scalar and vector FEM forms in the case of an elliptic cylinder

An invariant vector approximation of unknown quantities is proposed and implemented to construct the stiffness matrix of a quadrilateral curved finite element in the form of a fragment of the mid-surface of an elliptic cylinder with 18 degrees of freedom per node. Numerical examples show that the vector approximation has significant advantages over the scalar one as applied to arbitrary shells with considerable mid-surface curvature gradients.

The finite element approximation of vector fields in curvilinear coordinates

We demonstrate that it is possible to express each component of the displacement vector for the interior point of the finite element (FE) through all components of nodal unknowns in curvilinear coordinates. The effectiveness of the valid technique of vector approximation for displacement fields has been verified on an example.

The finite element analysis of shells of revolution with a branching meridian

We present an algorithm for calculating the shells of revolution with the branching meridian using the triangular finite elements, the rigidity matrices of which are formed based on the vector method of displacement interpolation [1]. The correct kinematic and static conditions of shell conjugation on the line of their coupling have been developed. The shell structure consisting of a cylinder and adjacent shells is calculated under various conditions of support.

Comparison of variants of the triangular discrete element shape functions for the vector method of shell displacement interpolation

Different variants of finding shape functions in a triangular finite element are considered. The efficiency of the vector method proposed by the authors for obtaining approximating functions is proved by specific examples.

Comparative analysis of the results of finite element calculations based on an ellipsoidal shell

In the curvilinear coordinate system, we describe an algorithm for generating the stiffness matrix of the quadrilateral element of the middle surface of a thin shell based on Kirchhoff’s hypothesis with nodal unknowns in the form of displacements and their derivatives using a vector approximation of the displacement fields. The developed finite element is verified by calculating the shell with the middle surface in the form of a triaxial ellipsoid in the two-dimensional formulation. The results are compared with the calculation results obtained using ANSYS software. The efficiency of the vector approximation of the displacement fields for the calculation of arbitrary thin shells in the moment stress state is shown.

A comparative evaluation of the scalar and vector approximations of sought quantities in the finite-element method of arbitrary shells

New versions of formulas that specify the median surfaces of an ellipsoid, elliptical cylinder, and cone, which allow one to perform calculations without features of the definitional domains of the considered shells, are proposed. The algorithm for forming stiffness matrices of quadrangular curvilinear finite elements with eighteen degrees of freedom at a node is given. The designed algorithm is verified based on the example of the calculation of a circular cylinder using scalar and vector approximations of displacements. The efficiency of the vector approximation of displacement fields in calculations of the elliptical cylinder with different fastening conditions and its displacement as a rigid body is shown.

Analysis of a shell of revolution subjected to axisymmetric loading taking into account geometric nonlinearity on the basis of the mixed finite element method

An algorithm is presented to obtain the deformation matrix for the annular finite element with a quadrilateral cross-section. The displacement and strain increments are taken as nodal unknowns. For numerical realization of the algorithm, we take the functional obtained using the principle of equal works for the internal and external forces at a step of loading. A newly developed version of approximating the unknown quantities allowed us to solve the problem of taking into account the finite element displacement as a rigid body.

Analysis of an arbitrarily loaded shell of revolution based on the finite element method in a mixed formulation

The volume finite element in the form of hexahedron with nodal unknowns as components of the displacement vector and stress tensor has been developed to analyze the shells of revolution. The displacement vector components for the inner point of the finite element and the components of its stress tensor are expressed through the nodal unknowns using the method of vector and tensor fields interpolation by the trilinear shape functions; that provides taking into account the finite element displacement as a whole solid. The variational principle in a mixed formulation is applied to form the matrix of hexahedron deformation. The efficiency of the proposed method for approximating the values being sought as vector and tensor fields in comparison with the traditional method for approximating the values being sought as scalar fields is confirmed by a numerical example.

To the question on continuous parameterization of spatial figures having an ellipse in a section

When solving a problem of mechanics of shells by numerical methods, inevitably arises a problem of continuous parameterization of shell construction in question. This problem requires a calculation of necessary geometrical descriptions in an arbitrary point of examined shell. At that, position of the point must be completely defined, and geometrical parameters used must have clear geometrical interpretation. We propose new variations of formulas which allow to obtain continuous parameterizations of spatial figures having an ellipse in a section, moreover, these formulas possess clear geometric interpretation.

Finite-element calculations of thin-walled structures of concatenated shells with different material stress–strain properties

The algorithm for calculating the stressed-deformed state (SDS) of thin-walled structures of concatenated shells with different material stress–strain properties on the finite-element method’s base with scalar and vector interpolation of displacement fields has been outlined. The quadrangular curvilinear finite-element with nine degrees of freedom in the node has been used as a discretization element. The SDS analysis of a thin-walled shell structure formed by a cylinder and two jointed shells of dissimilar materials has been performed.