The formula for axial stress in a thin-walled cylinder with closed ends subjected to internal pressure is: σa = Pr / 2t
Here's a breakdown of the formula and its components:
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σa: Axial stress (also known as longitudinal stress). This is the stress acting along the length of the cylinder. It is often measured in Pascals (Pa) or pounds per square inch (psi).
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P: Internal pressure. This is the pressure exerted by the fluid or gas inside the cylinder. It is measured in Pascals (Pa) or pounds per square inch (psi).
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r: Internal radius of the cylinder. This is the radius of the cylinder's inner surface, measured in meters (m) or inches (in).
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t: Wall thickness of the cylinder. This is the thickness of the cylinder wall, measured in meters (m) or inches (in). It is important that the cylinder meets the "thin-walled" criteria for this formula to be accurate. This generally means that r/t > 10.
Derivation and Explanation:
The formula is derived by considering the equilibrium of forces acting on a transverse section of the cylinder. The internal pressure acts on the circular area enclosed by the cylinder, creating a force that tends to pull the cylinder apart along its longitudinal axis. This force is resisted by the axial stress in the cylinder wall acting over the cross-sectional area of the wall.
Specifically, consider a transverse section of the cylinder. The force due to the internal pressure acting on the end cap is:
F = P * πr2
This force is resisted by the axial stress acting on the cross-sectional area of the cylinder wall:
Resisting Force = σa * 2πrt
For equilibrium, these forces must be equal:
P πr2 = σa 2πrt
Solving for σa, we get:
σa = (P * πr2) / (2πrt) = Pr / 2t
Important Considerations:
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Thin-Walled Assumption: This formula is valid only for thin-walled cylinders, where the wall thickness is significantly smaller than the radius (typically, r/t > 10). For thick-walled cylinders, more complex stress analysis methods are required.
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Closed Ends: This formula applies to cylinders with closed ends, where the internal pressure acts on the end caps, contributing to the axial stress.
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Circumferential Stress (Hoop Stress): It's important to note that the axial stress is half the magnitude of the circumferential stress (also known as hoop stress) in a thin-walled cylinder, which is given by σc = Pr / t. This means the cylinder is more likely to fail due to circumferential stress than axial stress.
Example:
A thin-walled cylinder has an internal pressure of 5 MPa, an internal radius of 0.5 m, and a wall thickness of 0.01 m. The axial stress can be calculated as follows:
σa = (5 MPa 0.5 m) / (2 0.01 m) = 125 MPa
Therefore, the axial stress in the cylinder wall is 125 MPa.
