Buckling of Spherical Shells

theory and experiment. There was no reason to doubt the classical theory of elasticity, which
worked well for flat plates, and it was soon suspected that the effect of curvature and spherical shape
imperfections could have been responsible for the discrepancies.
This thesis led to the realization that the classical theory must have failed to reveal the fact that
for a vessel configuration, not far away but somewhat different from the perfect geometry, lower
total potential energy was involved, and therefore a lower value of buckling load could be expected,
such as that indicated by tests. The theoretical challenge then became to formulate a solution
compatible with such a lower boundary of collapse pressure at which the spherical shell could
undergo the ‘‘oil canning’’ or ‘‘Durchschlag’’ process.
After making a number of necessary simplifying assumptions, von Kármán and Tsien [2]
developed a formula for the lower elastic buckling limit for collapse pressure, which for n ¼ 0.3
was found to be
P CR ¼ 0:37E=m 2 (31:3)
This level of collapse pressure may be said to correspond to the minimum theoretical load necessary
to keep the buckled shape of the shell with finite deformations in equilibrium. The lower limit
defined by Equation 31.3 appeared to compare favorably with experimental results, also given in
the literature [2]. On the other hand, the upper buckling pressure given by Equation 31.1 could be
approached only if extreme manufacturing and experimental precautions were taken. In practice, the
buckling pressure is found to be closer to the value obtained from Equation 31.3 and therefore this
formula is often recommended for design.
The exact calculation of the load–deflection curve for a spherical segment subjected to uniform
external pressure is known to involve nonlinear terms in the equations of equilibrium, which cause
substantial mathematical difficulties [3].
31.4 PLASTIC STRENGTH OF SPHERICAL SHELLS
Equations 31.2 and 31.3 may be regarded as design formulas based upon results using elasticity
theory. Bijlaard [4], Gerard [5], and Krenzke [6] conducted subsequent studies to determine
the effect of including plasticity upon the classical linear theory. To this end, Krenzke [6] conducted
a series of experiments on 26 hemispheres bounded by stiffened cylinders. The materials were
6061-T6 and 7075-T6 aluminum alloys, and all the test pieces were machined with great care at the
inside and outside contours. The junctions between the hemispherical shells and the cylindrical
portions of the model provided good natural boundaries for the problem. The relevant physical
properties for the study were obtained experimentally. The best correlation was arrived at with the
aid of the following expression:
P CR ¼ 0:84(E sE t ) 1=2
m 2 (31:4)
where E s and E t are the secant and tangent moduli, respectively, at the specific stress levels. These
values can be determined from the experimental stress–strain curves in standard tension tests. The
relevant test ratios of radius to thickness in Krenzke’s work varied between 10 and 100 with a
Poisson’s ratio of 0.3. The correlation based on Equation 36.4 gave the agreement between
experimental data and the predictions within þ2% and 12%.
The extension of the Krenzke results to other hemispherical vessels should be qualified.
Although his test models were prepared under controlled laboratory conditions, the following
detrimental effects should be considered in a real environment:
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