Beyond Vlasov: Towards a Generalized Modal Theory of Warping in Beam Structures

Classical thin-walled beam theory, following Vlasov, decomposes the longitudinal normal stress field into axial, bending and a single sectorial warping contribution governed by the bimoment B and the warping constant I_w. This work explores a natural generalization of that framework: a modal decomposition of warping in which the classical Vlasov sectorial mode is treated as the first member of a broader, orthogonal family of warping modes. Each additional mode φᵢ(y, z) is defined through orthogonality conditions with respect to area, bending and the sectorial coordinate, carries its own generalized bimoment qᵢ(x) and generalized warping inertia Iᵢ, and induces, through longitudinal equilibrium, a self-equilibrated shear flow governed by a Neumann-type Poisson problem on the cross-section. The resulting strain energy decomposes additively over modes, preserving full compatibility with energy methods. The physical origin of the higher-order modes is discussed, with eigenfunctions of a Laplacian cross-sectional operator, finite element cross-sectional eigenanalysis, and variational energy minimization proposed as candidate generating mechanisms. The framework is formulated independently of any thin-wall or mid-line kinematic hypothesis, making it directly applicable to thick-walled and solid cross-sections as well as to classical thin-walled members. Under this view, Vlasov warping theory emerges as the fundamental mode of a richer modal warping basis, analogous to the role of the fundamental mode in structural vibration theory.