Every air compressor tank, propane cylinder, water heater, and hydraulic accumulator you've ever seen is a pressure vessel — a closed container designed to hold fluids at a pressure substantially different from ambient. Getting the wall thickness wrong means a catastrophic rupture, so engineers use straightforward formulas to size them correctly.

Thin-wall assumption: When the wall thickness t is small relative to the internal radius r (rule of thumb: r/t > 10), stress distributes nearly uniformly through the wall. This simplification covers the vast majority of tanks you'll encounter in practice — propane tanks, air receivers, pipelines, and fire extinguishers.

Two critical stresses arise in a cylindrical vessel under internal pressure P:

• Hoop (circumferential) stress: σ_h = Pr / t — This acts around the circumference, trying to split the cylinder lengthwise. It's the dominant stress and the one that governs design.
• Longitudinal (axial) stress: σ_l = Pr / 2t — This acts along the cylinder's axis, trying to pop the end caps off. Notice it's exactly half the hoop stress.

This 2:1 ratio explains why pressurized cylinders always fail along a longitudinal seam first — and why a hot dog splits lengthwise when you overcook it.

Worked example: You're sizing a compressed-air receiver tank. Internal pressure is 150 psi (1.03 MPa), inside diameter is 12 inches (radius = 6 in / 152 mm), and you're using SA-516 Gr.70 steel with an allowable stress of 17,500 psi. What minimum wall thickness do you need?

Using the hoop stress formula rearranged: t = Pr / σ_allowable = (150 × 6) / 17,500 = 0.051 inches. In practice, you'd add a corrosion allowance (typically 1/16" or 1/8"), apply a weld joint efficiency factor (0.85 for spot-examined welds per ASME code), and round up to the next available plate thickness. With a joint efficiency of 0.85: t = 900 / (17,500 × 0.85) = 0.060", plus 0.0625" corrosion allowance = 0.123", so you'd likely specify 1/8" (0.125") plate.

Spherical vessels have uniform stress in all directions: σ = Pr / 2t — the same as the longitudinal stress in a cylinder. This makes spheres the most material-efficient shape for pressure containment, which is why large LNG storage tanks are spherical (Horton spheres), despite being harder to fabricate.

Design codes matter: Real pressure vessel design follows ASME Boiler and Pressure Vessel Code (Section VIII) in the US, or PED (Pressure Equipment Directive) in Europe. These codes specify allowable stresses, inspection requirements, safety factors, and mandatory testing. Never field-engineer a pressure vessel — the stored energy in compressed gas makes failures explosive, literally.