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I think that mu is the ratio of the hole radius to the wall thickness.
This and the next several slides show buckling of axially compressed cylindrical shells with holes of various sizes.
Some of the slides are from James H. Starnes Jr., "Effect of a circular hole on the buckling of cylindrical shells loaded by axial compression", AIAA Journal, Vol. 10, No. 11, November 1972
ABSTRACT: An experimental and analytical investigation of the effect of a circular hole on the buckling of thin cylindrical shells under axial compression was carried out. The experimental results were obtained from tests performed on seamless electroformed copper shells and Mylar shells with a lab joint seam. These results indicate that the character of the shell buckling is dependent on a parameter [called mu] which is proportional to the hole radius divided by the square root of the product of the shell radius and thickness. For small values of this parameter, there is no apparent effect of the hole on the buckling load. For slightly larger values of the parameter [mu], the shells still buckle into a general collapse configuration, but the buckling loads are sharply reduced as the parameter [mu] increases. For still larger values of the parameter [mu], the buckling loads are further reduced and the shells buckle into a stable local buckling configuration. The analytical solution is a simplified Rayleigh-Ritz type approximation. It provides an upper bound for the buckling stresses of the cylinders tested with hole radius less than 10 per cent of the shell radii, and verifies the dependence of the shell buckling characteristics on the parameter [mu] used to correlate the experimental results.
mu = 0.5x[12(1-nu^2)]^(1/4)x(hole radius)/[cylinder radius x cylinder wall thickness)]^(1/2)
P = axial buckling load
P(cl) = "classical" buckling load of the perfect cylindrical shell without any holes.
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