Conical Shells Assignment Help

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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.

2379_Conical Shells.png

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  )

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