Bearing joint rigidity in designing thin-walled beams with two axes of symmetry when bending with torsion

The article is devoted to the mechanics of thin-walled beams. A solution to the Vlasov equation for bending with torsion of a thin-walled beam with two axes of symmetry is provided, the effect of bearing joint rigidity being taken into account.

Introduction. The current state of the issue of designing thin-walled beams subjected to transverse bending with torsion is described.

Materials and methods. The solution of the system of differential equations of stability of the plane bending form of V.Z. Vlasov for thin-walled beams under transverse bending with torsion is presented, the effect of bearing joint rigidity being taken into account. The original equations of V.Z. Vlasov for bending and torsion of a thin-walled beam with two axes of symmetry are transformed into a right-hand coordinate system. Next, from two differential equations of V.Z. Vlasov, a system of 12 equations is obtained for all calculated forces and deformations in a thin­walled beam. Boundary conditions were also obtained that take into account the relationship between forces and deformations in the support section. The results of solving the specified system of equations using the Euler method are presented.

Results. The solution of Vlasov system of equations for the stability of thin-walled beams under transverse bending is obtained, taking into account the rigidity (malleability) of the support nodes by the Euler method and the general form of the function of the angle of rotation of the cross section. The solution is obtained for beams with any support nodes, from a pure hinge to absolutely rigid nodes. The paper presents the results of numerical verification and draws conclusions on the accuracy of the obtained solution. During verification, a special case of beams of various cross-sections with absolutely rigid support units was considered. The difference between the numerical solution in the LIRA-CAD PC and the solution proposed in the article is within 12 per cent.

Discussion and conclusions. Conclusions have been drawn on the accuracy of the developed mathematical model. The difference is caused by the accuracy in determining the moments of inertia of the section for pure torsion and the stiffness of the support units.

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