Toward Mathematical Rigor for the Conventional Flexural-Torsional Buckling Theory of Thin-Walled Members

Based on the variational principle for buckling problems in solid mechanics， the author proposed a new flexural-torsional buckling theory for thin-walled members in 2002. The present paper returns to the conventional flexural-torsional buckling theory of thin-walled members， which has evolved over the past 170 years， and seeks to provide it with mathematical and mechanical rigor. This paper makes the following contributions： the conventional flexural-torsional buckling theory is reformulated under the assumption that the total shear strain， including its nonlinear part， vanishes on the middle surface of the thin-walled member， and based on this assumption， an accurate warping displacement valid for large-displacement analysis is derived； it is shown that， under this assumption together with the rigid cross-section assumption， the shear stresses and transverse normal stresses on the middle surface make no contribution to the total potential energy； based on the derived warping displacement， the load potential of the external bending moments， which has not previously been successfully incorporated into the total potential energy， is established； and the complete total potential energy is thereby obtained. It has been verified that， under various external loads and boundary conditions， the total potential based on the variational principle for buckling in solid mechanics is transformed into the total potential of the present paper in any load and any boundary conditions. This confirms the complete equivalence between the two formulations.