The V.I. Slivker's semi-shear theory finite elements research for calculation of thin-walled closed profile rods

The object of research is the finite element (FE) proposed earlier by the author of the article as part of the theory of V.I.Sliver's semi-shear theory, which differs from other FE by the approximation method of unknown functions: 3-nodal finite element having 6 degrees of freedom square-law approximation of torsional angle functions and warping functions. The subject of research is the convergence for the considered FE of displacements functions of both types (torsional angle and warping) and internal forces (bimoment, sectorial torsion moment and pure torsion moment) that occur during bending torsion and are the important components for calculating stresses according to the standarts. Method of research is mathematical modeling of parameters (stiffness matrix, load column) and determination of the unknowns of equations of the FE-method. Results. The test problems of thin–walled rods bending torsion for a number of boundary conditions are solved on the example of a closed profile; the main advantage of V.I.Slivker's theory (universality for open and closed profiles) and, as a consequence, the advantage of the previously proposed finite elements are demonstrated on concrete examples. It is shown that the FE with quadratic approximation has the convergence acceptable for engineering calculations for rods of not only open, but also closed profiles. Also obtained in the V.I.Slivker's semi-shear theory the expression for the influence parameter of thin-walled rectangular  profiles shape is creamed and the spectrum of its values is investigated.