Buckling of Spherical Shells

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.
ß 2008 by Taylor & Francis Group, LLC.