To determine the absolute maximum shear stress on the inner surface of the thin-walled cylinder, we follow these steps:

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Step 1: Assumptions

The cylinder has thin walls. For a thin-walled pressure vessel, the radial stress is negligible compared to the hoop (circumferential) and axial stresses.

Step 2: Dimensions and Parameters

• Internal diameter, $d_{\text{in}} = 1\, \text{m}$
• Outer diameter, $d_{\text{out}} = 1.04\, \text{m}$
• Wall thickness, $t = \frac{d_{\text{out}} - d_{\text{in}}}{2} = \frac{1.04 - 1}{2} = 0.02\, \text{m}$
• Internal pressure, $p = 4\, \text{MPa}$

Step 3: Thin-Walled Cylinder Stress Analysis

(a) Hoop Stress ($\sigma_h$):

$\sigma_h = \frac{p \cdot r_{\text{in}}}{t}$ Where $r_{\text{in}}$ is the inner radius of the cylinder: $r_{\text{in}} = \frac{d_{\text{in}}}{2} = \frac{1}{2} = 0.5\, \text{m}$ Substitute values: $\sigma_h = \frac{4 \times 10^6 \cdot 0.5}{0.02} = 100 \times 10^6 = 100\, \text{MPa}$

(b) Axial Stress ($\sigma_a$):

$\sigma_a = \frac{p \cdot r_{\text{in}}}{2t}$ Substitute values: $\sigma_a = \frac{4 \times 10^6 \cdot 0.5}{2 \cdot 0.02} = 50 \times 10^6 = 50\, \text{MPa}$

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Step 4: Maximum Shear Stress on Inner Surface

The maximum shear stress is calculated using the formula: $\tau_{\text{max}} = \frac{\sigma_h - \sigma_a}{2}$ Substitute the hoop and axial stresses: $\tau_{\text{max}} = \frac{100 - 50}{2} = 25\, \text{MPa}$

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Final Answer

The absolute maximum shear stress on the inner surface is: $\boxed{25\, \text{MPa}}$

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Would you like a more detailed explanation or further assistance? 😊

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Further Questions

1. What is the formula for maximum shear stress in a thick-walled cylinder?
2. How does the stress distribution differ in thin vs. thick-walled pressure vessels?
3. Can we calculate radial stress for this thin-walled cylinder?
4. What assumptions make thin-wall theory applicable here?
5. What are common materials used for cylinders under high internal pressure?

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Tip

Always validate the thin-wall assumption by ensuring $t \ll r_{\text{in}}$. For this case, $t / r_{\text{in}} = 0.02 / 0.5 = 0.04$, which is valid.