Hierarchical derivation of orthogonal cross-section modes for thin-walled beams with arbitrary sections

A new systematic approach is presented to derive the cross-section deformation modes of thin-walled beams with arbitrary sections within the framework of a higher-order beam theory (HoBT). New sets of higher-order modes, in this case warping and distortion, are derived hierarchically from the lowest mode set by considering the consistency between the strain field and the stress field generated by the modes in lower sets. Warping modes are derived by the shear stress of in-plane modes while distortion modes are induced by out-of-plane deformations via Poisson's effect. Higher-order modes are shown to be built as a linear combination of the integrated functions of lower-order modes, where the combination coefficients are determined by the orthogonality condition among the higher-order modes. Because the proposed method does not require any approximation when determining sectional mode shapes, no cross-section discretization, commonly used in existing studies, is needed. The effectiveness of the proposed mode derivation process is verified by comparing the static and modal analysis results of thin-walled beams with open, closed, and flanged cross-sections obtained by the proposed method, other HoBTs, and shell finite elements.