Conical Shells:
Stresses
In the shape of conical containers Shells shall normally be supported around their upper rim. Let a conical shell of total height h and supported at the top as illustrated in Figure.
Figure
Let any level XX at a height y from the apex O.
At this level, r2 =∞
By using the expression for stresses in a doubly curved shell,
f1 /r1 + f2 /∞ = p/t
f1= pr1 /t
where r1 is radius of the circumferential curvature. From the geometry curvature of the shell, we get
p = w (h - y)
r1= y ( tan α / cos α)
∴ f1 = (w (h - y) )/ t × ( y tan α /cos α)
It is the hoop stress at the level XX. For this stress to be maximum,
df1 / dy = (w tan α / t cos α )(h - 2 y) = 0
∴ y=h/2
Substituting y = h /2 , we get
Maximum hoop stress = w(h-(h/2))(h/2)tan α / t cos α
= w h2 tan α / (4t cos α)
Stress in the meridional direction f2 may be calculated by letting the total weight of water contained in the shaded portion of the vessel as illustrated in Figure .
This weight is resisted by the vertical component of the induced material tension on the circumference XX.
Therefore,
f2 2π y tan α t cos α= w (h - y) π y2 tan 2 α+ w π y2 tan2 α (y/3)
∴ f2 = (w tan α/2t cos α) (hy - (2/3)y2)
For this stress to be maximum,
df2 /dy= (w tan α/2t cos α) (h - (4y/3))
∴ y = 3h
Maximum meridional stress,
= ( w tan α/2t cos α) ( 3h2 /4 - 3h2/8)
= (3/16) (wh2 tan α/ t cos α )