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Each pair of buckling modes, which have the same eigenvalue, has the same mode shape, with the second mode of the pair rotated about the axis of revolution by one-half of a circumferential half-wave. (In this respect Mode 5 in the figure is incorrect, as its mode shape is simply the negative of that of Mode 4. Mode 5 should be oriented eiither vertically or horizontally, not at 45 deg.)
FROM:
Adina-Ana Muresan, Mihai Nedelcu and Rodrigo Goncalves,
“GBT-based FE formulation to analyse the buckling behaviour of isotropic conical shells with circular cross-section”, Thin-Walled Structures, Vol. 134, pp 84-101, January 2019, https://doi.org/10.1016/j.tws.2018.07.032
ABSTRACT: The following paper presents a Finite Element formulation based on the Generalized Beam Theory (GBT), to analyse the buckling behaviour of isotropic conical shells under various loading and boundary conditions. The formulation offers the solution of the 1st order analysis from which the pre-buckling stresses are computed, including stress concentrations, and the linear (bifurcation) buckling solution. Due to the variable cross-section of conical shells, the mechanical and geometrical properties are no longer constant along the bar’s axis as they are in the case of cylindrical structures and thin-walled prismatic bars. Special focus is given to the effect of the pre-buckling stress concentrations and non-conventional cross-section deformation modes. The proposed formulation is validated by comparing results obtained from GBT and Shell Finite Element Analyses.
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