The linear and non-linear equilibrium equations for thin elastic shells according to the Kirchhoff-Love hypotheses

Abstract

The currently accepted bending theory of thin elastic shells contains equilibrium equations derived on the quite general premise that the shell can withstand an unspecified transverse shear load. The transverse shear stress resultants are then eliminated between the force-equilibrium and moment-equilibrium equations. This procedure is, of course, acceptable. But it is shown to be inconsistent with the universal practice of relating these equilibrium equations to the displacement components and their derivatives by means of strain-displacement relations formulated according to the Kirchhoff-Love hypothesis that normals to the shell middle surface before deformation remain normal after deformation, which prescribes zero transverse shears, both strains and stresses. Furthermore, it is revealed that in actually relating the stress resultants and couples to the strains, the accepted practice includes inconsistencies in the stress couple terms which are of the same order of magnitude as (and in some cases identical to) the terms retained. A new formulation of the thin-shell equilibrium equations, under the Kirchhoff-Love hypotheses, is presented so that a complete self-consistent theory of thin elastic shells is produced. This new shell theory is used elsewhere to analyse the buckling of thin cylindrical shells under axial compression. The analysis is a linear infinitesimal deformation solution for the actual buckling load of an ideal shell, not the post-buckling minimum load capacity, and is in good agreement with experiments, both from the viewpoint of the buckling stress and the deformation pattern. The buckling stress is found to be one-half of the classical value. The inconsistencies in the currently accepted thin-shell theories are thus seen to be too important to overlook.