A thin-walled circular pressure vessel of diameter $D$ and wall thickness $t$ is subjected to internal pressure $p$. Under biaxial tension, the hoop stress is $\sigma_h = \frac{p D}{2t}$ and longitudinal stress is $\sigma_L = \frac{p D}{4t}$. If $\sigma_y$ is the uniaxial yield strength of the material, the ratio of the internal yield pressure predicted by the von Mises yield criterion to that predicted by the Tresca yield criterion $(p_{\text{Mises}} / p_{\text{Tresca}})$ is:

(A) $\frac{2}{\sqrt{3}} \approx 1.155$

(B) $\sqrt{3} \approx 1.732$

(C) 1.000

(D) $\frac{\sqrt{3}}{2} \approx 0.866$

Answer Key:

Verified in official key

### 1. Strategy & Theoretical Principles In the cylinder skin, principal stresses are $\sigma_1 = \sigma_h$, $\sigma_2 = \sigma_L = \frac{1}{2}\sigma_h$, and radial stress $\sigma_3 \approx 0$. 1. **Tresca Yield Criterion:** Yield occurs when the maximum shear stress reaches $\frac{\sigma_y}{2}$: $$\tau_{\max} = \frac{\sigma_1 - \sigma_3}{2} = \frac{\sigma_h - 0}{2} = \frac{\sigma_y}{2} \implies \sigma_h = \sigma_y$$ $$p_{\text{Tresca}} = \frac{2 t \sigma_y}{D}$$ 2. **von Mises Yield Criterion:** Yield occurs when equivalent stress reaches $\sigma_y$: $$\sigma_v = \sqrt{\sigma_1^2 - \sigma_1 \sigma_2 + \sigma_2^2} = \sigma_y$$ Substitute $\sigma_1 = \sigma_h$ and $\sigma_2 = \frac{1}{2} \sigma_h$: $$\sigma_v = \sqrt{\sigma_h^2 - \sigma_h\left(\frac{1}{2}\sigma_h\right) + \left(\frac{1}{2}\sigma_h\right)^2} = \sqrt{\sigma_h^2 \left(1 - \frac{1}{2} + \frac{1}{4}\right)} = \sqrt{\frac{3}{4}} \sigma_h = \frac{\sqrt{3}}{2} \sigma_h$$ Setting $\sigma_v = \sigma_y$: $$\frac{\sqrt{3}}{2} \sigma_h = \sigma_y \implies \sigma_h = \frac{2}{\sqrt{3}} \sigma_y$$ $$p_{\text{Mises}} = \frac{2 t}{D} \left(\frac{2}{\sqrt{3}} \sigma_y\right) = \frac{2}{\sqrt{3}} p_{\text{Tresca}}$$ ### 2. Analytical Ratio Calculation $$\frac{p_{\text{Mises}}}{p_{\text{Tresca}}} = \frac{2}{\sqrt{3}} \approx 1.1547 \approx 1.155$$ ### 3. Distractor Forensics - **Option A:** Correct factor $2/\sqrt{3} \approx 1.155$ representing the maximum divergence between Tresca and von Mises in biaxial tension. - **Option C:** Pure uniaxial stress state where both yield criteria coincide. - **Option D:** Inverted ratio ($p_{\text{Tresca}} / p_{\text{Mises}}$). ### 4. Exam Trap & Standard Reference - **Trap Alert:** von Mises allows up to $15.5\%$ higher allowable pressure than Tresca for cylindrical pressure vessels under internal pressure. - **Standard Reference:** Megson, T.H.G., *Aircraft Structures for Engineering Students*, Chapter 1.

← Prev Question Next Question →