Fully analytical solution framework for general thin-walled composite beams with mixed variational approach

Abstract

AbstractA variationally consistent analytical beam model that describes the theory in a Timoshenko-Vlasov level is developed based on Reissner’s mixed variational theorem. Starting from a shell theory, all the field-governing equations (equilibrium and continuity) and the boundary conditions of the shell wall are derived in closed form, and the mixed method enables finding the explicit forms of the reactive stresses and sectional warpings which are evaluated progressively depending on the level of beam model to be analyzed. The stress recovery part is incorporated in the post-stage of the analysis to compute the layer-wise distribution of stresses over the beam cross-section. The present analysis is validated against numerous benchmark examples available in the literature, including beams with multi-layered strip section, thin-walled anisotropic box sections with elastic couplings, and two-cell airfoil section. The comparison study demonstrates excellent correlations with the results from detailed three-dimensional finite element analysis and other up-to-date beam approaches. Also presented are symbolically expressed stiffness coefficients and the sectional warping modes of coupled composite beams to demonstrate the strength of the proposed beam model.Keywords: Beamsection analysiswarpingstress recovery; stiffness matrix Nomenclature a=Local shell radius of curvatureFx=Axial force along x axisFy, Fz=Shear forces along y and z axesMx=Torsional moment about x axisMy, Mz=Bending moments about y and z axesMω=Torsional bi-momentMxx, Mss, Mxs=Bending and twisting couples of the shell wallNxx, Nss, Nxs=In-plane stress resultants of the shell wallNxn, Nsn=Transverse shear stress resultants of the shell wallU, V, W=Translational displacements of beam sectional reference origin along x, y, z axesu, v, w=Translational displacements of an arbitrary material point of beam section along x, y, z axesux, us, un=Translational displacements of the shell wall along x, s, n axesβy, βz=Sectional rotation angles about y and z axesγxn,γsn=Transverse shear strains of the shell wallγxn,γsn=Transverse shear strains of the beam in x-y, x-z planesγxs=In-plane shear strain of the shell wall\isinxx,\isinss=In-plane normal strains of the shell wallκxx,κss,κxs=Curvatures of the shell wallϕ=Sectional rotation angle about x axisψx, ψs=Rotation angles of the shell wall about s, x axesωx=Contour warping function along x axisSubscripts=(),x, (),s=∂()/∂x, ∂()/∂sSuperscripts=()T=Transpose of an array()−1=Inversion of an arrayDisclosure statementThe authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.Additional informationFundingThis work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2022R14A1018884). This work was supported by Korea Research Institute for defense Technology planning and advancement (KRIT) grant funded by the Korea government (DAPA (Defense Acquisition Program Administration) (21-107-E00-007, 2023)). This paper was written as part of Konkuk University’s research support program for its faculty on sabbatical leave in 2022.