Cylinder Pressure: Change in Length Calculation

This solution explains how to find the change in length ($\Delta L$) of a thin cylinder under internal pressure ($P$). We use the formulas for stress and strain in cylindrical pressure vessels.

Deriving Longitudinal Strain

For a thin cylinder, the stresses are:

• Hoop stress ($\sigma_h$): $\sigma_h = \frac{Pd}{2t}$
• Longitudinal stress ($\sigma_l$): $\sigma_l = \frac{Pd}{4t}$

The strain in the longitudinal direction ($\epsilon_l$) is given by Hooke's law, considering the effect of both stresses:

\(\epsilon_l = \frac{\sigma_l}{E} - \mu \frac{\sigma_h}{E}\)

Calculating Change in Length

1. Substitute the stress formulas into the strain equation:

   \(\epsilon_l = \frac{Pd/4t}{E} - \mu \frac{Pd/2t}{E}\)

2. Simplify the expression:

   \(\epsilon_l = \frac{Pd}{4tE} - \frac{\mu Pd}{2tE}\)

   \(\epsilon_l = \frac{Pd}{2tE} \left( \frac{1}{2} - \mu \right)\)

3. Relate longitudinal strain ($\epsilon_l$) to the change in length ($\Delta L$) and original length ($L$):

   \(\epsilon_l = \frac{\Delta L}{L}\)

4. Solve for the change in length ($\Delta L$):

   \(\Delta L = L \times \epsilon_l\)

   \(\Delta L = L \times \frac{Pd}{2tE} \left( \frac{1}{2} - \mu \right)\)

   \(\Delta L = \frac{PdL}{2tE} \left( \frac{1}{2} - \mu \right)\)

This result matches Option 1.