Determine the maximum stress - thick cylinder:
A thick cylinder of steel with an internal diameter of 100 mm & external diameter of 200 mm is subjected to an internal pressure of 80 N/mm^{2}. Determine the maximum stress induced in the material and the alteration in the external diameter.
Take Young's Modulus = 2 × 10^{5} N/mm^{2} and Poisson's ratio = 0.3.
Solution
By using Lame's expressions for the inside and outside pressure, we get
80 = b/50^{2} - a
and
0 = (b/100^{2}) - a
On solving out, we obtain, b = 266666.67 and a = 26.667.
Therefore, the maximum stress (Hoop stress at the inner surface of the cylinder)
= (b/50^{2}) + a = (266666.67/50^{2} ) + 26.667
= 133.33 N/mm^{2}
To discover the strain at the outer surface.
Hoop stress at surface = (266666.67 /100^{2) +} 26.667 = 53.33 N/mm^{2}
As, pressure outside is zero.
Hoop strain = 53.33 / E = 53.33 / (2 × 10^{5})
= 2.666 × 10^{- 4}
Therefore, increase in external diameter = 2.666 × 10^{- 4} × 200 = 0.053 mm .