The primary stresses developed in thin-walled pressure vessels are hoop stress (also known as circumferential stress) and longitudinal stress. These stresses arise from the internal pressure acting on the vessel walls.
Hoop Stress (Circumferential Stress)
Hoop stress acts in a direction tangential to the circumference of the vessel. It's essentially the stress resisting the bursting force caused by the internal pressure trying to split the vessel along its length. The formula for hoop stress (σh) is:
σh = (P * r) / t
Where:
- P = Internal pressure
- r = Radius of the vessel
- t = Thickness of the vessel wall
Longitudinal Stress (Axial Stress)
Longitudinal stress acts along the longitudinal axis (length) of the vessel. It resists the forces trying to pull the vessel apart end-to-end due to the internal pressure acting on the end caps. The formula for longitudinal stress (σl) is:
σl = (P r) / (2 t)
Where:
- P = Internal pressure
- r = Radius of the vessel
- t = Thickness of the vessel wall
Key Observations:
- Hoop stress is generally twice the longitudinal stress. This means a thin-walled pressure vessel is more likely to fail due to bursting along its length than due to being pulled apart end-to-end.
- The thin-walled assumption is valid when the ratio of the vessel radius to its wall thickness (r/t) is greater than or equal to 10. For thicker vessels, more complex stress analyses are required.
- These formulas provide the average stress values. Stress concentrations can occur at points of discontinuity, such as nozzles or supports, which need to be considered in the design.
- The material of the vessel must be able to withstand both hoop and longitudinal stresses with an adequate safety factor.
In summary, thin-walled pressure vessels experience both hoop stress (circumferential) and longitudinal stress (axial) due to internal pressure, with the hoop stress typically being the larger of the two.
