Question

A thin-walled cylindrical pressure vessel of yield strength $300$ MPa has radius-to-thickness ratio $R/t = 100$. Using von Mises yield criterion, find internal pressure at failure. (round off to two decimals)

Hint:

In thin-walled vessels, hoop stress dominates. Von Mises criterion combines hoop and longitudinal stresses. Always apply radius-to-thickness ratio to simplify stresses.

Answer 1

Step 1: Stresses in thin-walled cylinder.
Hoop stress: \[ \sigma_h = \frac{pR}{t} \] Longitudinal stress: \[ \sigma_l = \frac{pR}{2t} \]

Step 2: von Mises criterion.
Von Mises equivalent stress: \[ \sigma_{eq} = \sqrt{\sigma_h^2 + \sigma_l^2 - \sigma_h\sigma_l} \]

Step 3: Substitute ratio $R/t = 100$.
\[ \sigma_h = 100p, \sigma_l = 50p \] \[ \sigma_{eq} = \sqrt{(100p)^2 + (50p)^2 - (100p)(50p)} \] \[ = \sqrt{10000p^2 + 2500p^2 - 5000p^2} = \sqrt{7500p^2} = 86.6p \]

Step 4: Yield condition.
At failure: \[ \sigma_{eq} = \sigma_y = 300 \] \[ 86.6p = 300 \] \[ p = \frac{300}{86.6} \approx 3.46 \,\text{MPa} \] Wait — check carefully. Recompute: \[ \sigma_{eq} = \sqrt{\sigma_h^2 + \sigma_l^2 - \sigma_h \sigma_l} = \sqrt{10000p^2 + 2500p^2 - 5000p^2} = \sqrt{7500p^2} = 86.6p \] Yes, correct. \[ p = \frac{300}{86.6} = 3.46 \,\text{MPa} \] \[ \boxed{3.46 \,\text{MPa}} \]