Pressure vessels on the PE Mechanical exam

Pressure-vessel stress is a high-frequency Machine Design & Materials topic and a quick win once the thin-wall relations are automatic. The whole thing turns on a few formulas and one judgment call: thin wall or thick wall.

What you actually need to own

• Thin-wall cylinder (valid when wall t < ~r/10):
  • Hoop (circumferential) stress: σ_h = p·r / t = p·D / (2t).
  • Longitudinal (axial) stress: σ_l = p·r / (2t) = p·D / (4t).
  • So hoop is twice longitudinal — the cylinder wants to split along its length first.
• Thin-wall sphere: σ = p·r / (2t) in every direction (which is why spheres are efficient pressure shapes).
• Thick-wall cylinder (t not small vs r): use the Lamé equations — radial and tangential stresses vary across the wall, peaking at the inner surface. Know to switch to these when the thin-wall assumption fails.
• Combined effects: internal pressure plus axial load or thermal/interference stress — superpose, then apply a failure theory if asked.

Where people lose points

• Using radius vs diameter inconsistently — σ = p·r/t and p·D/(2t) are the same; mixing them isn't.
• Swapping hoop and longitudinal, or forgetting hoop is the larger (governing) stress.
• Applying thin-wall to a thick wall (or vice versa) — check t/r before choosing the relation.
• Gauge vs absolute pressure in the stress.

How to study it

Memorize the thin-wall hoop and longitudinal relations cold and the t < r/10 check, then practice a few problems that combine pressure with axial or thermal stress so the superposition is automatic. Work one thick-wall/Lamé problem so you recognize when the chart-relations don't apply.

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