31
By
‘‘ spherical shell,
’’ we mean complete spherical con fi gurations, 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 dif fi culties 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.
R. Zoelly and A. Van der Neut conducted signi fi cant original theoretical work on the buckling of spherical shells [1]. They used the classical theory of small de fl ections 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
P
CR
¼
2 E
= m
2 p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
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
:
21 E
= m
2
( 31
:
2 )
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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2008 by Taylor & Francis Group, LLC.

theory and experiment. There was no reason to doubt the classical theory of elasticity, which worked well for fl at 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 con fi guration, 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
:
37 E
= 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 fi nite deformations in equilibrium. The lower limit de fi ned 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
– de fl ection 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 dif fi culties [3].
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 s
E t
)
1
=
2 m 2
( 31
:
4 ) where E s and E t are the secant and tangent moduli, respectively, at the speci fi c 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
þ
2% and 12%.
The extension of the Krenzke results to other hemispherical vessels should be quali fi ed.
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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2008 by Taylor & Francis Group, LLC.

Local and
= or overall out-of-roundness
Thickness variation
Residual stresses
Penetration and edge boundaries
These effects are likely to be more signi fi cant when spherical shells are formed by spinning or pressing rather than by careful machining.
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 in fl uence of initial imperfections.
The formula for the collapse pressure of an imperfect spherical shell can be expressed in terms of a buckling coef fi cient K and a modi fi ed ratio m i as
P
CR
¼
KE m i
2
( K 0
:
84 ) ( 31
:
5 ) where, based upon the work of Krenzke and Charles [7], the modi fi ed radius
= thickness ratio m i be approximated as may m i
¼
R i
= h ( 31
:
6 ) where Figure 31.1 illustrates the modi fi ed radius R i and thickness h .
According to the results obtained by Krenzke and Charles on glass spheres, the buckling coef fi cient 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
:
42 h ( m i
)
1
=
2
( 31
:
7 )
L c
R h
R
T
FIGURE 31.1
Notation for de fi ning a local change in wall thickness.
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2008 by Taylor & Francis Group, LLC.

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-de fl ection 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 con fi gurations. 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-de fi ned dimensional quantities:
Experimental collapse pressure, P e
Pressure to cause membrane yield stress, P m
Classical linear buckling pressure, P
CR
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
:
83 P e m 2
E
¼
P e
P
CR and
0
:
61 E mS y
¼
P
CR
P m
( 31
:
8 ) where
P e is the experimental collapse pressure
P
CR is the classical linear buckling pressure
0.4
0.3
0.2
0.1
0.6
0.5
1 2 3
0.61
E
/ mS y
4
FIGURE 31.2
Lower-bound curve for hemispherical vessels under external pressure.
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2008 by Taylor & Francis Group, LLC.

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 in fl uenced 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 suf fi ciently general for practical design purposes.
For example, consider a titanium alloy hemisphere with m
¼
60, E
¼
117,200 N
= mm compressive yield strength, S
Hence, Figure 31.2 yields 0.83
y
P
¼
760 N
= mm
2
. From Equation 31.8, we get 0.61
E
= mS y e m
2 =
E
¼
0.36, from which P e
¼
14.1 N
= mm
2
.
2
, and the
¼
1.57.
It may now be instructive to look brie fl y at the empirical result in relation to the theoretical limits de fi ned 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 coef fi cient 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 in fl uence of geometrical discontinuities is local and does not extend signi fi cantly beyond the range determined by the value of the parameter T ( m )
1 = 2
.
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
, de fi ned as l o
¼
1,82 a o
T ( m )
1 = 2 or l o
¼
2
:
57 ( H
=
T )
1
=
2
( 31
:
9 )
P cr
T a o q q a o
R
FIGURE 31.3
A spherical cap and notation.
H
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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 l o
4 l o
<
2
:
08 continuous deformation with buckling
>
2
:
08 axisymmetric snap-through
>
6 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
)
By squaring and adding these expressions we obtain, after simpli fi cation,
( 31
:
10 )
H
2
2 HR
þ a
2 o
¼
0 ( 31
:
11 )
Assuming that H is small, H
2 is considerably smaller than 2 HR . Then by neglecting H
2 in Equation
31.11, the equation may be written as
H
¼ a 2 o
2 R
( 31
:
12 )
By substituting this expression for H into the second expression of Equation 31.9, we obtain the fi rst 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 de fi ned 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].
4 o
300
250
200
150
100
50
2 4 6
Geometrical parameter, l
8
0
FIGURE 31.4
Design chart for a shallow spherical cap under external pressure.
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2008 by Taylor & Francis Group, LLC.

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 fi tting 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
0
4 l 4
:
15
0 e
0
:
095 l
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
E
¼
117,200 N
= mm
2
. From this data, we obtain
¼
31.8 mm, T
¼
2.1 mm, and m
¼
R
=
T
¼
60
:
5 and n
0
¼ a o
=
T
¼
15
:
1 ( 31
:
14 )
Then from the fi rst 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
:
0026 u
)( 1 0
:
000175 m ) 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 20
8 and 50
8
.
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.
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
2
3 P
0
R 3 o
R 3 o
R i
3 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
3 P
0
R i
R
3 o
( 1 n
)
2 E R 3 o
R i
3
( 31
:
17 )
( 31
:
18 )
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where
E is the elastic modulus n is Poisson
’ s ratio
The corresponding displacement of the outer surface is u o
¼
P
2 E R 3 o
0
R o
R i
3
( 1 n
) 2 R
3 o
R i
3
2 n
R
3 o
R i
3
( 31
:
19 )
For a solid sphere subjected to external pressure, the amount of radial compression in the elastic range becomes u o
¼
P
0
R o
( 1 2 n
)
E
( 31
:
20 )
h
K
L c m m i
P
CR
P e
P m
P o
R a o
E
E s
E t
H
R i
R o
S
S y
T u i u o l o n
Support radius
Elastic modulus
Secant modulus of elasticity
Tangent modulus of elasticity
Depth of spherical cap
Reduced thickness of shell (
see Figure 31.1
)
Buckling coef fi cient
Critical arc length (see Figure 31.1)
Radius
= thickness ( R
=
T ) ratio
Mean radius
= local thickness ratio
Elastic buckling pressure
Experimental collapse pressure
Membrane yield stress
External pressure
Shell radius
Inner radius
Outer radius
Stress
Yield strength
Shell thickness
Inner surface displacement
Outer surface displacement
Shallow cap parameter
Poisson
’ s ratio
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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2008 by Taylor & Francis Group, LLC.