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.
Understanding Engineering: Pressure Vessel Stress
The thin cylinder formulas come from balancing forces on an imagined cut through the wall. Picture a pipe sliced lengthwise into two halves. Internal pressure pushes the halves apart across the projected rectangular area of the cut.
The metal on each cut edge pulls back in tension. A larger radius gives pressure more leverage because the projected area grows. A thicker wall provides more metal to carry the pulling load.
This force balance explains why the circumferential, or hoop, direction is usually the critical direction in a cylindrical shell. Cracks that run lengthwise can be especially dangerous because hoop stress tends to pull them open.
The axial direction needs separate thought because it depends on the ends. Pressure on a closed end cap produces a strong outward force. That force must pass from the end cap into the cylindrical wall, creating axial tension.
An open pipe does not automatically carry this same end load in its wall. Its supports, flanges, anchors, or connected equipment may carry it instead. Engineers therefore examine the full system, not only a short piece of pipe.
Expansion joints, bends, valves, and rigid supports can add loads caused by temperature change or constrained movement. These loads combine with pressure stresses.
The simple thin-wall model assumes stress is nearly uniform across the wall thickness. This becomes less accurate when the wall is relatively thick, when pressure is very high, or near geometric changes. In a thick vessel, the stress near the inner surface can be much greater than the stress near the outer surface.
Holes for gauges and nozzles interrupt the flow of force through the metal. Welds, threads, sharp corners, and changes in diameter can concentrate stress too.
Vessel heads need care because their curved shape changes how forces travel. A smooth rounded head spreads load better than a flat plate, which bends strongly under pressure.
Real designs use material data that reflects more than a single pull test. Strength can fall at high temperature. Corrosion can slowly reduce wall thickness.
Repeated filling and emptying can cause fatigue cracks even when each individual pressure cycle seems safe. Weld quality matters because tiny flaws can grow under cycling. A hydrostatic pressure test often uses water since water stores far less compressible energy than gas.
Students should carefully track units, use the correct internal radius, and distinguish gauge pressure from absolute pressure. They should treat calculated stress as a starting point for a safety check, then consider wall loss, joints, temperature, cycles, inspection access, and the consequences of failure.
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.