Buckling of Spherical Shells

31
Buckling of Spherical Shells
31.1 INTRODUCTION
By ‘‘spherical shell,’’ we mean complete spherical configurations, hemispherical heads (such as
pressure vessel heads), and shallow spherical caps. In analyses, a spherical cap may be used to
model the behavior of a complete spherical vessel with thickness discontinuities, reinforcements,
and penetrations.
Although the response of a spherical shell to external pressure has received considerable
attention from analysts, the calculation of collapse pressure still presents substantial difficulties in
the presence of geometrical discontinuities and manufacturing imperfections. The bulk of the
theoretical work carried out so far has had a rather limited effect on the method of engineering
design, and therefore much experimental support is still needed. At the same time, the application of
spherical geometry to the optimum vessel design has continued to be attractive in many branches
of industry dealing with submersibles, satellite probes, storage tanks, pressure domes, diaphragms,
and similar systems. This chapter deals with the mechanical response and working formulas
for spherical shell design in the elastic and plastic ranges of collapse, which could be used for
underground and aboveground applications. The material presented is based on state-of-the-art
knowledge in pressure vessel design and analysis.
31.2 ZOELLY–VAN DER NEUT FORMULA
R. Zoelly and A. Van der Neut conducted significant original theoretical work on the buckling of
spherical shells [1]. They used the classical theory of small deflections and the solution of linear
differential equations. Based upon this work, the elastic buckling pressure P CR for complete, thin
spherical shell was found to be
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
P CR ¼ 2E=m 2 3(1 n 2 )
(31:1)
where
E is the elastic modulus
n is Poisson’s ratio
m is the radius=thickness ratio (R=T)
For a typical Poisson’s ratio n of 0.3, Equation 31.1 becomes simply
P CR ¼ 1:21E=m 2 (31:2)
31.3 CORRECTED FORMULA FOR SPHERICAL SHELLS
At the time of the development of the classical theory, which led to Equation 31.1, no systematic
experimental work was done. Several years later, however, some tests reported at the California
Institute of Technology [2] showed that the experimental buckling pressure could be as low as 25%
of the theoretical value given by Equation 31.1. The value derived by means of Equation 31.1 was
then considered as the upper limit of the classical elastic buckling, while several investigators
embarked on special studies with the aim of explaining these rather drastic differences between the
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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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h
Local and=or overall out-of-roundness
Thickness variation
Residual stresses
Penetration and edge boundaries
These effects are likely to be more significant when spherical shells are formed by spinning
or pressing rather than by careful machining.
31.5 EFFECT OF INITIAL IMPERFECTIONS
In a subsequent series of collapse tests, Krenzke and Charles [7] aimed at evaluating the potential
applications of manufactured spherical glass shells for deep submersibles. Because of the anticipated
elastic behavior of glass vessels, the emphasis was placed on verifying the linear theory that
resulted in Equation 31.2. Prior to this series of tests, very limited experimental data existed, which
could be used to support a rational, elastic design with special regard to the influence of initial
imperfections.
The formula for the collapse pressure of an imperfect spherical shell can be expressed in terms
of a buckling coefficient K and a modified ratio m i as
P CR ¼ KE
m 2 i
(K 0:84) (31:5)
where, based upon the work of Krenzke and Charles [7], the modified radius=thickness ratio m i may
be approximated as
m i ¼ R i =h (31:6)
where Figure 31.1 illustrates the modified radius R i and thickness h.
According to the results obtained by Krenzke and Charles on glass spheres, the buckling
coefficient K in Equation 31.5 was about 0.84. Their study showed that the elastic buckling
strength of initially imperfect spherical shells must depend on the local curvature and the thickness
of a segment of a critical arc length, L c . For a Poisson’s ratio of 0.3, this critical length can be
estimated as
L c ¼ 2:42h(m i ) 1=2 (31:7)
L c
R i
T
R
FIGURE 31.1
Notation for defining a local change in wall thickness.
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In a related study conducted at the David Taylor Model Basin Laboratory, for the Department of
Navy, the effect of clamped edges on the response of a hemispherical shell was evaluated. The
relevant collapse pressure was found to be about 20% lower than that for a complete spherical shell
having the same value of the parameter m and the elastic modulus E. Although these tests on
accurately made glass spheres tended to support the validity of the small-deflection theory of
buckling, there appeared to be little hope that metallic shells would yield a similar degree of
correlation even under controlled conditions.
The investigations reviewed above may be of particular interest to designers dealing with
complete spherical vessels as well as domed-end configurations. From a practical point of view,
the most satisfactory method of predicting the collapse pressure would be to use a plot of
experimental data as a function of the following well-defined dimensional quantities:
Experimental collapse pressure, P e
Pressure to cause membrane yield stress, P m
Classical linear buckling pressure, P CR
31.6 EXPERIMENTS WITH HEMISPHERICAL VESSELS
Using experimental data for collapse of hemispherical vessels subjected to external pressure, Gill [8]
provides information for a nondimensional plot suitable for preliminary design purposes. Figure 31.2
shows this plot for the following dimensionless ratios:
0:83P e m 2
E
¼ P e
P CR
where
P e is the experimental collapse pressure
P CR is the classical linear buckling pressure
and
0:61E
mS y
¼ P CR
P m
(31:8)
0.6
0.5
0.83P e m 2 /E
0.4
0.3
0.2
0.1
1 2 3 4
0.61E/mS y
FIGURE 31.2
Lower-bound curve for hemispherical vessels under external pressure.
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m is the radius=thickness ratio (R=T)
E is the elastic modulus
S y is the yield stress
The accuracy with which the collapse pressure can be predicted on the basis of experimental data
must be influenced by the maximum scatter band involved. Since this scatter is sensitive to material
and geometry imperfections, their probable extent should be known before a more reliable, lowerbound
curve can be developed. The results given in Figure 31.2 include hemispherical vessels in the
stress-relieved and as-welded condition without, however, specifying the extent of geometrical
imperfections, which, in this particular case, were known to be less pronounced. It follows that
Figure 31.2 is applicable only to the design of hemispherical vessels, where good manufacturing
practice can be assured. Further research work is recommended to narrow the scatter band to assure
better correlation for the lower bound.
The dimensionless plot given in Figure 31.2 is sufficiently general for practical design purposes.
For example, consider a titanium alloy hemisphere with m ¼ 60, E ¼ 117,200 N=mm 2 , and the
compressive yield strength, S y ¼ 760 N=mm 2 . From Equation 31.8, we get 0.61E=mS y ¼ 1.57.
Hence, Figure 31.2 yields 0.83P e m 2 =E ¼ 0.36, from which P e ¼ 14.1 N=mm 2 .
It may now be instructive to look briefly at the empirical result in relation to the theoretical
limits defined by Equations 31.2 and 31.3 for the complete spherical vessels.
Making P e ¼ P CR ¼ 14.1 N=mm 2 and solving Equation 31.5 for the magnitude of the buckling
coefficient gives K ¼ 0.43. This value is close to the theoretical lower limit of 0.37 given by
Equation 31.3 for a complete spherical vessel, and it appears to suggest that certain portions of
such a vessel under uniform external pressure may behave in a manner similar to that of a complete
vessel. This observation may be of special importance in dealing with the spherical shells containing
local reinforcements and penetrations. It is also generally consistent with the elastic theory of shells,
according to which the influence of geometrical discontinuities is local and does not extend
significantly beyond the range determined by the value of the parameter T(m) 1=2 .
31.7 RESPONSE OF SHALLOW SPHERICAL CAPS
Consider a relatively thin and shallow spherical cap fully clamped at its edge and subjected to
uniform external pressure as represented in Figure 31.3 [9]. A key parameter characterizing a
spherical cap is l o ,defined as
l o ¼ 1,82a o
T(m) 1=2 or l o ¼ 2:57(H=T) 1=2 (31:9)
P cr
T
H
a o
a o
R
q
q
FIGURE 31.3
A spherical cap and notation.
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where
a o is the support radius
T is the shell thickness
m is the radius=thickness ratio (R=T)
R is the shell radius
H is the shell height above its support (see Figure 31.3)
The structural response of the cap for a typical Poisson ratio n of 0.3 may be described as
l o < 2:08
l o > 2:08
4l o > 6
continuous deformation with buckling
axisymmetric snap-through
local buckling
From Figure 31.3, the half-central angle u is related to a o , R, and H as
a o ¼ R sin u and H ¼ R(1 cos u) (31:10)
By squaring and adding these expressions we obtain, after simplification,
H 2 2HR þ a 2 o ¼ 0 (31:11)
Assuming that H is small, H 2 is considerably smaller than 2HR. Then by neglecting H 2 in Equation
31.11, the equation may be written as
H ¼ a2 o
2R
(31:12)
By substituting this expression for H into the second expression of Equation 31.9, we obtain the first
expression of Equation 31.9. Thus the two expressions of Equation 31.9 are equivalent for shallow
caps (that is, H considerably smaller than R).
As a guide, a spherical cap may be regarded as thin when m > 10. Shallow geometry is then
approximately defined as a o =H 8. Once the spherical cap parameter l o is calculated by either of
the equations in (Equation 31.9), we can estimate the critical buckling pressure by using the curve of
Figure 31.4. This curve is based upon numerical data quoted by Flügge [9].
Buckling load parameter, (0.91 p CRa 4 o)/(ET 4 )
300
250
200
150
100
50
2 4 6 8
Geometrical parameter, l 0
FIGURE 31.4
Design chart for a shallow spherical cap under external pressure.
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The curve of Figure 31.4 is smoothed out somewhat in the midregion of the parameter l o , which
involves a transition between the theoretical and experimental data in simplifying the curve fitting
process. By using the curve of Figure 31.4, the following expression for the critical buckling
pressure can be developed:
P CR ¼ 0:075 En 4
0 l4:15 0 e 0:095l 0
(31:13)
where n 0 is the dimensionless ratio a o =T.
As an example application of Equation 31.13 let R ¼ 127 mm, a o ¼ 31.8 mm, T ¼ 2.1 mm, and
E ¼ 117,200 N=mm 2 . From this data, we obtain
m ¼ R=T ¼ 60:5 and n 0 ¼ a o =T ¼ 15:1 (31:14)
Then from the first equation of Equation 31.9 we obtain l o as
l o ¼ 3:53 (31:15)
Finally, by substituting the data and results into Equation 31.13, we obtain
P CR ¼ 22:7 N=mm 2 (31:16)
In a special situation where a spherical cap is very thin, with a range of m values between 400 and
2000, the following empirical formula has been suggested for the relevant buckling pressure [10]:
P CR ¼
(0:25 0:0026u)(1 0:000175m)E
m 2 (31:17)
where u is the half central angle of Figure 31.3 in degrees. In Equation 31.17, u is intended to have
values between 208 and 508.
Although Equation 31.17 is useful within the indicated brackets of m, it may not be quite
suitable for bridging the boundaries between the shallow caps and hemispherical shells without a
careful study. Ideally, the formula for the collapse pressure of a spherical shell should be reduced to
the form of Equation 31.5 with the K value representing a continuous function of the shell geometry
and manufacturing imperfections. For inelastic behavior, the parameter (E s E t ) 1=2 appears to have the
best chance of success for a meaningful correlation of theory and experiment. In the interim,
however, the formulas given in this chapter are recommended for the preliminary design and
experimentation.
31.8 STRENGTH OF THICK SPHERES
When a thick-walled spherical vessel is subjected to an external pressure P 0 , the maximum stress S
occurs at the inner surface as
S ¼
3P 0R 3 o
2 R 3 o R 3 (31:17)
i
where R i and R o are the inner and outer sphere radii.
The displacement of the inner surface toward the center of the vessel is
u i ¼ 3P 0R i R 3 o (1 n)
(31:18)
2E R 3 o
R 3 i
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where
E is the elastic modulus
n is Poisson’s ratio
The corresponding displacement of the outer surface is
P 0 R o
u o ¼
2E R 3 o
R 3 i
(1 n) 2R 3 o R 3
i
2n R 3 o
R 3 i
(31:19)
For a solid sphere subjected to external pressure, the amount of radial compression in the elastic
range becomes
u o ¼ P 0R o (1 2n)
E
(31:20)
SYMBOLS
a o Support radius
E Elastic modulus
E s Secant modulus of elasticity
E t Tangent modulus of elasticity
H Depth of spherical cap
h Reduced thickness of shell (see Figure 31.1)
K Buckling coefficient
L c Critical arc length (see Figure 31.1)
m Radius=thickness (R=T) ratio
m i Mean radius=local thickness ratio
P CR Elastic buckling pressure
P e Experimental collapse pressure
P m Membrane yield stress
P o External pressure
R Shell radius
R i Inner radius
R o Outer radius
S Stress
S y Yield strength
T Shell thickness
u i Inner surface displacement
u o Outer surface displacement
l o Shallow cap parameter
n Poisson’s ratio
REFERENCES
1. S. P. Timoshenko and J. M Gere, Theory of Elastic Stability, 2nd ed., McGraw Hill, New York, 1961,
pp. 512–519.
2. T. von Kármán and H. S. Tsien, The buckling of thin cylindrical shells under axial compression, Journal of
Aeronautical Sciences, 8, 1941, pp. 303–312.
3. C. B. Biezeno, Über die Bestimmung der Durchschlagkraft einer schmach gekrümmten kreisförmigen
Platte, AAMM, Vol. 19, 1938.
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4. P. P. Bijlaard, Theory and tests on the plastic stability of plates and shells, Journal of the Aeronautical
Sciences, 16(9), 1949, pp. 529–541.
5. G. Gerard, Plastic stability of thin shells, Journal of the Aeronautical Sciences, 24(4), 1957, pp. 269–274.
6. M. A. Krenzke, Tests of Machined Deep Spherical Shells Under External Hydrostatic Pressure, Report
1601, David Taylor Model Basin, Department of the Navy, 1962.
7. M. A. Krenzke and R. M. Charles, The Elastic Buckling Strength of Spherical Glass Shells, Report 1759,
David Taylor Model Basin, Department of the Navy, 1963.
8. S. S. Gill, The Stress Analysis of Pressure Vessels and Pressure Vessel Components, Permagon Press,
Oxford, 1970.
9. W. Flügge, Handbook of Engineering Mechanics, McGraw Hill, New York, 1962.
10. K. Kloppel and O. Jungbluth, Beitrag zum Durchschlagproblem dünnwandiger Kugelschalen, Stahlbau,
1953.
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