In order to obtain the strain along any direction, we have to see the state of stress at any point.
At any point in a thin cylindrical shell with an internal pressure p, we have obtained the expressions for stresses along longitudinal direction and circumferential direction.
In the three mutually perpendicular directions, the stresses are as follows :
The state of stress is shown in Figure
- These are the principal stresses acting at the point considered. However, when d/t is very large making the shell thin, the radial pressure p will be very small compared to the longitudinal and hoop stresses.
- Hence, this compressive stress can be neglected at any point for the purpose of working out the strain, which is going to be still smaller.
- This assumption leaves only the two tensile stresses at any point, mutually perpendicular to each other.
- If E is the Young’s modulus of the material of the shell and v, its Poisson’s ratio, then the expression for the strains in the two direction are obtained as follows :
Using these expressions, we may proceed to obtain the changes in length and diameter of the cylinder.
Change in length = Longitudinal strain × Original length =
However, it can be noted that since the circumference is a constant product of diameter, i.e. C = π d, the diametrical strain will be the same as the circumferential strain.
Thus, change in diameter = Hoop strain × Original diameter.