thickwalll cylinder has an inner diameter of 100mm and outer diameter of 200mm. the yield strength of material is 200mpa. the internal pressure is 120 mpa. calculate the outer pressure so that the materials will not fail due to yield based on maximum shear stress theory.

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

A thick-walled cylinder has an inner diameter of 100mm and outer diameter of 200mm. The yield strength of the material is 200MPa. The internal pressure is 120MPa. Calculate the outer pressure so that the material will not fail due to yield based on the maximum shear stress theory.

Calculate the outer pressure required to prevent failure of a thick-walled cylinder with a 100mm inner diameter, 200mm outer diameter, 200MPa yield strength, and 120MPa internal pressure. This solution applies the maximum shear stress theory (Tresca Criterion) and uses Lame’s equations for stress analysis.

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