Question

In a thick-walled cylinder pressurized from inside, the Hoop stress is maximum at

• A. the inner radius
• B. the centre of the wall thickness
• C. the outer radius
• D. both the inner and outer radii

Hint:

Hoop stress decreases from inner to outer radius in thick cylinders.

Answer 1

The Correct Option is A

In a thick-walled cylinder subjected to internal pressure, the distribution of stress varies across the wall thickness. The stress type known as "Hoop stress" or "circumferential stress" is a key aspect to consider, particularly in how it changes from the inner to the outer radius of the cylinder.

For such a cylinder, Hoop stress (\(\sigma_h\)) is influenced predominantly by the radial position within the wall. The general formula for Hoop stress in a thick-walled cylinder is derived from Lame's equations and is expressed as:

σ_h = (p_ir_i² - p_or_o²)r² + (r_o²r_i²(p_o - p_i))/(r_o² - r_i²)

Where:

• p_i is the internal pressure.
• p_o is the external pressure (often set to zero in typical problems).
• r is the radial position where stress is being calculated.
• r_i is the inner radius of the cylinder.
• r_o is the outer radius of the cylinder.

The maximum Hoop stress typically occurs where the internal pressure affects the cylinder most directly. Mathematically, this is evidenced by evaluating the stress gradient with respect to the radius. It confirms that the Hoop stress is maximum at the inner radius \(r = r_i\).

Therefore, the maximum Hoop stress in a thick-walled cylinder pressurized from the inside is found at the inner radius. This insight is crucial for ensuring the integrity and safety of pressure vessels in aerospace engineering applications.

Correct Answer: the inner radius