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/mm² and an internal pressure of 120 N/mm². 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 /r² - a r

Thus, at r = 40, σ_r = 120 N/mm², and at r = 60, σ_r = 40 N/mm². Putting these values, we obtain

120 = (b/40²) - a

And 40 =   (b /60²) - 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  = (230400/40²) + 24 = 168 N/mm²

at r = 60,  σ  = (230400 /60² )+ 24 = 88 N/mm²

At the mean radius, i.e. (40 + 60)/2  = 50 mm .

Radial stress = (230400/50² ) - 24 = 68.16 N/mm² , and

Circumferential stress = (230400/50²) + 24 = 116.16 N/mm² .