On stiffness matrix construction for naturally twisted rod

Abstract

Here we considered the Saint Venant problem for a naturally twisted rod with a rectangular cross-section. Research was conducted on the basis of homogeneous solutions in conjunction with the finite element method. The general solution is constructed as a linear combination of elementary solutions corresponding to three four-roots eigenvalues of spectral problem on the cross section. Elementary solutions determining the stress-strain state of Saint Venant type contain unknown eigenvectors and associated vectors. To determine of unknown solutions the authors formulated boundary value problems and their variational formulations previously. They are correspond to problems of stretching-torsion, pure bending and bending of the lateral force. Variational problems were solved using FEM. The stress-strain state of the rod was studied numerically and non-zero elements of the stiffness matrix were found in the case of square and rectangular cross-sections of the rod for the different values of twist. Graphically shows the numerical results for a wide range of change of the twist parameter τ. Calculations showed that the identified patterns are consistent with the corresponding behavior untwisted rods (for small twist parameter), and with the growth of the twist, new effects, which confirm the hypothesis proposed earlier. Refs 17. Figs 3.