Conjugate Approximation of Thin-Walled Rods Internal Forces Functions in Bending Torsion
The goal of the work is to increase the convergence rate of bending torsion internal forces (bimoment, moment of warping torsion, moment of pure torsion) in finite elements in the calculation of thin-walled rods using V.I. Slivker's semi-shear theory. The object of research is the finite elements (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 5 degrees of freedom with square-law approximation of torsional angle function and linear approximation of warping function and 3-nodal finite element having 6 degrees of freedom with square-law approximation of torsional angle and warping functions. The subject of research is the convergence of internal forces in thin-walled rods, determined using the conjugate approximation method. Method of research is mathematical modeling of parameters (stiffness matrix, load column) and determination of the unknowns of two systems equations: of the FE-method and of the conjugate approximation method. Results. The formulas of the conjugate approximations method of are proposed in 2 variants: linear and quadratic conjugate approximations. On particular cases of two-sided fixed and cantilever beams, it is shown that when calculating open-type profiles, an acceptable 5% engineering error is provided by a linear conjugate approximation of the bimoment. For closed profiles, due to the special pattern of the bimoment distribution near the fixed supports, the linear conjugate approximation cannot provide engineering accuracy: it is necessary to use the deviding of the rod into 32 finite elements or more and use the quadratic conjugate approximation to refine the bimoment values on fixed supports.