Exact flexural-torsional buckling analysis of non-funicular thin-walled beam-columns using matrix stiffness method

This study presents an extended matrix stiffness method for the exact flexural–torsional buckling analysis of non-funicular thin-walled beam-columns under general loading conditions. The formulation is built upon the well-established second order stiffness matrix proposed by Yang and McGuire, which is widely recognised as a robust and accurate benchmark for the geometric nonlinear analysis of thin-walled members. However, the original formulation does not explicitly account for the effects of uniformly distributed transverse loads at arbitrary heights on the cross section. To overcome this limitation, in this study, a superposition principle is introduced to incorporate the eccentricity of the distributed load into the total potential energy expression. The validity of this superposition approach is further confirmed by an independent variational derivation in the Appendix. The refined total potential energy is then used to derive an improved element second order stiffness matrix. The proposed method is implemented within a standard matrix stiffness framework and validated using three representative problems: beam columns under combined axial and bending loads, beams subjected to moment gradients, and beams carrying uniformly distributed loads applied at varying heights. The results demonstrate excellent agreement with classical analytical solutions and high-fidelity shell finite element models. Thus, the proposed matrix stiffness method provides a consistent and computationally efficient tool for the exact stability analysis of non-funicular thin-walled structural systems.