Normalized buckling load and buckling mode shapes as functions of dimensionless length, L/r, of a cylindrical tube with various radius/thickness, r/t

From the same paper as the previous image.

Short shells buckle in shear; very long shells buckle locally in axial compression; shells of intermediate length show mostly shear buckling modes but the buckling load factors are significantly influenced by axial compression on the compressive side of the cantilevered shell loaded at the top by the shear force, V.

Adam J. Sadowski writes:

“VLBA, VGNA and VGNIA are … critical transverse shear forces obtained from a linear eigenmode analysis (LBA), a path-tracing ('Riks') analysis of the perfect shell (GNA) and imperfect shell (GNIA) respectively. Their algebraic *approximations* are Vk and Vcr, given by Eq. 11, 14 and 17 respectively,” [in the paper cited in the previous slide.]

N-sub-z,cl is the critical axial resultant from classical theory:

N-sub-z,cl = 0.605Et(t/r) for Poisson ratio = 0.3.

N-sub-max = maximum compressive axial resultant in the tube cross section.

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