Pressure vessels are tanks, pipes, boilers, and cylinders designed to hold fluids or gases at a pressure different from the outside environment. The pressure creates tensile stresses in the vessel wall that must be predicted so the vessel does not leak, bulge, crack, or burst. For thin-walled cylindrical vessels, engineers use simple formulas that connect internal pressure, radius, wall thickness, and stress.
These formulas matter in mechanical, chemical, aerospace, and civil engineering because pressure equipment often stores large amounts of energy.
Key Facts
- Thin-wall assumption: t <= r/10, where t is wall thickness and r is inside radius.
- Hoop stress in a thin cylindrical vessel: sigma_h = pr/t.
- Longitudinal stress in a closed-end thin cylindrical vessel: sigma_L = pr/(2t).
- Hoop stress is twice the longitudinal stress for a closed thin cylinder: sigma_h = 2 sigma_L.
- For a thin spherical pressure vessel, membrane stress is sigma = pr/(2t).
- A basic design check is sigma_working <= sigma_allowable, where sigma_allowable often includes a safety factor.
Vocabulary
- Pressure vessel
- A pressure vessel is a container designed to hold a gas or liquid at an internal or external pressure different from its surroundings.
- Hoop stress
- Hoop stress is the circumferential tensile stress that acts around the cylinder and tends to split it lengthwise.
- Longitudinal stress
- Longitudinal stress is the axial tensile stress that acts along the length of a closed cylinder and is caused by pressure pushing on the end caps.
- Thin-walled vessel
- A thin-walled vessel is one whose wall thickness is small compared with its radius, commonly t <= r/10.
- Allowable stress
- Allowable stress is the maximum stress permitted in design after accounting for material strength, safety factor, temperature, corrosion, and codes.
Common Mistakes to Avoid
- Using diameter instead of radius in sigma_h = pr/t is wrong unless the formula has been rewritten for diameter. If D is used, the hoop stress formula becomes sigma_h = pD/(2t).
- Forgetting that hoop stress is larger than longitudinal stress is wrong for a closed thin cylinder. The hoop stress is twice the longitudinal stress, so it often controls wall thickness.
- Applying thin-wall formulas to thick vessels is wrong when t is not small compared with r. Thick-walled vessels need stress distributions that vary through the wall thickness.
- Ignoring end conditions is wrong because longitudinal stress depends on whether the vessel has closed ends. An open pipe section under pressure has hoop stress, but it does not carry the same pressure-induced axial stress as a closed vessel.
Practice Questions
- 1 A thin cylindrical pressure vessel has internal pressure p = 2.0 MPa, inside radius r = 0.50 m, and wall thickness t = 10 mm. Calculate the hoop stress and longitudinal stress.
- 2 A steel cylinder must operate at p = 1.5 MPa with inside radius r = 0.40 m. If the allowable tensile stress is 120 MPa, find the minimum wall thickness based on hoop stress.
- 3 A long crack forms parallel to the axis of a pressurized cylindrical vessel. Explain which stress most directly opens this crack and why that stress is usually the critical one.