Determine the radial and circumferential stresses:
The internal & external diameters of a thick hollow cylinder are 80 mm & 120 mm respectively. This is subjected to an external pressure of 40 N/mm^{2} and an internal pressure of 120 N/mm^{2}. Determine the circumferential stress at the external & internal surfaces and determine the radial & circumferential stresses at the mean radius.
Solution
We know that
σ_{r } = b /r^{2} - a r
Thus, at r = 40, σ_{r} = 120 N/mm^{2}, and at r = 60, σ_{r} = 40 N/mm^{2}. Putting these values, we obtain
120 = (b/40^{2}) - a
And 40 = (b /60^{2}) - a
On solving out , we get a = 25 and b = 230400. Circumferential stress is provided by,
σ_{h } = b/r^{ 2} + a
Therefore, at r = 40, σ _{h} = (230400/40^{2}) + 24 = 168 N/mm^{2}
at r = 60, σ = (230400 /60^{2 })+ 24 = 88 N/mm^{2}
At the mean radius, i.e. (40 + 60)/2 = 50 mm .
Radial stress = (230400/50^{2} ) - 24 = 68.16 N/mm^{2} , and
Circumferential stress = (230400/50^{2}) + 24 = 116.16 N/mm^{2} .