The maximum shear stress theory (also known as the Tresca criterion) states that failure occurs when the maximum shear stress in a material reaches half of its yield strength. We can use this criterion to determine the required outer pressure so that the material does not fail.

Here's the detailed solution:

Given Data:

• Inner diameter, $d_i = 100 \, \text{mm}$
• Outer diameter, $d_o = 200 \, \text{mm}$
• Yield strength of the material, $\sigma_y = 200 \, \text{MPa}$
• Internal pressure, $p_i = 120 \, \text{MPa}$
• Outer pressure, $p_o = ?$

Step 1: Convert the Diameters to Radii

• Inner radius $r_i = \frac{d_i}{2} = \frac{100}{2} = 50 \, \text{mm}$
• Outer radius $r_o = \frac{d_o}{2} = \frac{200}{2} = 100 \, \text{mm}$

Step 2: Maximum Shear Stress Theory (Tresca Criterion)

The maximum shear stress $\tau_{\text{max}}$ in a thick-walled cylinder under internal and external pressure is given by: $\tau_{\text{max}} = \frac{\sigma_{\text{max}} - \sigma_{\text{min}}}{2}$ where $\sigma_{\text{max}}$ and $\sigma_{\text{min}}$ are the maximum and minimum principal stresses in the cylinder wall.

For a thick-walled cylinder under pressure, the radial and tangential (hoop) stresses at any radius $r$ are given by Lame’s Equations:

1. Radial Stress: $\sigma_r = \frac{p_i r_i^2 - p_o r_o^2}{r_o^2 - r_i^2} + \frac{(p_o - p_i) r_i^2 r_o^2}{(r_o^2 - r_i^2) r^2}$
2. Hoop (Circumferential) Stress: $\sigma_{\theta} = \frac{p_i r_i^2 - p_o r_o^2}{r_o^2 - r_i^2} - \frac{(p_o - p_i) r_i^2 r_o^2}{(r_o^2 - r_i^2) r^2}$

At the inner radius $r = r_i$, the stresses are:

• Radial stress $\sigma_r(r_i) = -p_i$ (since it's subjected to the internal pressure)
• Hoop stress $\sigma_{\theta}(r_i) = \frac{p_i r_o^2 + p_o r_i^2}{r_o^2 - r_i^2}$

Step 3: Applying the Tresca Criterion for Yield

According to the Tresca criterion, failure will occur if: $\tau_{\text{max}} = \frac{\sigma_{\theta}(r_i) - \sigma_r(r_i)}{2} = \frac{\sigma_y}{2}$ Substitute $\sigma_r(r_i) = -p_i$ and $\sigma_{\theta}(r_i) = \frac{p_i r_o^2 + p_o r_i^2}{r_o^2 - r_i^2}$ into this equation: $\frac{\frac{p_i r_o^2 + p_o r_i^2}{r_o^2 - r_i^2} + p_i}{2} = \frac{\sigma_y}{2}$ Solving for $p_o$: $\frac{p_i r_o^2 + p_o r_i^2 + p_i (r_o^2 - r_i^2)}{r_o^2 - r_i^2} = \sigma_y$

Substitute values to calculate