A nonlinear axial-flexural (P-M₁-M₂) interaction formula for cold-formed steel members: Development via trigonometric polynomial interpolation

Cold-formed steel members under combined compression ( P ) and bending ( M 1 , M 2 ) exhibit complex buckling behaviors, and the existing linear interaction approach fails to adequately account for the coupling effects between axial compression and bending, resulting in conservative estimates of load-bearing capacity. In various national design codes, the development of nonlinear formulations has been initiated. This study proposes a nonlinear P - M 1 - M 2 interaction formula for cold-formed steel members using trigonometric interpolation polynomials. First, typical members subjected to combined compression and bending loads are selected, and extensive parametric finite element analyses are conducted. Subsequently, based on the linear interaction approach, a trigonometric polynomial P - M 1 - M 2 interaction formula is constructed by introducing radial functions. Then, the undetermined coefficients of the trigonometric polynomial are derived through regression analysis, establishing a nonlinear P - M 1 - M 2 interaction formula and plotting the corresponding failure surface. Finally, the proposed nonlinear P - M 1 - M 2 interaction formula is verified by comparison with experimental results and predictions from existing design codes. The developed nonlinear P - M 1 - M 2 interaction formula accurately reflects the combined compression and bending behavior of cold-formed steel members, providing robust support for the direct analysis of cold-formed steel structures.