Understanding Stresses in Thin Cylindrical Shells

Thin cylindrical shells subjected to internal pressure experience two primary types of stresses acting on the wall:

1. Hoop stress (or circumferential stress): This stress acts tangentially around the circumference of the cylinder. It is caused by the internal pressure tending to burst the cylinder along its longitudinal seam.
2. Longitudinal stress: This stress acts parallel to the axis of the cylinder. It is caused by the internal pressure tending to pull the end caps off the cylinder.

Formulas for Hoop and Longitudinal Stress

For a thin cylindrical shell with diameter 'd', subjected to an internal pressure 'p', and having a wall thickness 't' (where 't' is significantly smaller than 'd', typically d/t > 20), the formulas for these stresses are derived based on equilibrium principles.

The formula for Hoop Stress ($\sigma_h$) is:

$\sigma_h = \frac{pd}{2t}$

The formula for Longitudinal Stress ($\sigma_l$) is:

$\sigma_l = \frac{pd}{4t}$

Calculating the Ratio of Longitudinal Stress to Hoop Stress

The question asks for the ratio of longitudinal stress to the hoop stress ($\sigma_l / \sigma_h$). We can find this ratio by dividing the formula for longitudinal stress by the formula for hoop stress.

Ratio = $\frac{\text{Longitudinal Stress}}{\text{Hoop Stress}} = \frac{\sigma_l}{\sigma_h}$

Substitute the formulas for $\sigma_l$ and $\sigma_h$:

Ratio = $\frac{\frac{pd}{4t}}{\frac{pd}{2t}}$

To simplify the expression, we can multiply the numerator by the reciprocal of the denominator:

Ratio = $\frac{pd}{4t} \times \frac{2t}{pd}$

We can cancel out the common terms 'p', 'd', and 't' from the numerator and the denominator:

Ratio = $\frac{\cancel{pd}}{\cancel{4t}} \times \frac{\cancel{2t}}{\cancel{pd}}$

Ratio = $\frac{2}{4}$

Simplify the fraction:

Ratio = $\frac{1}{2}$

Therefore, the ratio of longitudinal stress to hoop stress in a thin cylindrical shell under internal pressure is 1/2.

Revision Table: Thin Cylinder Stress Formulas

Additional Information: Thin Cylinder Stress Analysis

The analysis of thin cylindrical shells under internal pressure makes certain assumptions:

• The wall thickness 't' is small compared to the diameter 'd' (typically d/t > 20).
• The stress distribution across the thickness is uniform.
• The material is isotropic and homogeneous.
• Stress concentration effects at joints or openings are neglected.

The hoop stress is always twice the longitudinal stress in a thin cylindrical shell under internal pressure. This is why longitudinal joints (like welds running along the length) are typically designed to be stronger or more efficient than circumferential joints (like welds running around the diameter), as they have to withstand the larger hoop stress.