Thin-Walled Pressure Vessel Stress Calculator

Hoop and longitudinal stress in a thin-walled cylinder or sphere under internal pressure (a study calculation, not a pressure vessel design).

Calculated from standard textbook formulas, with every step shown. The formulas are fixed, so the working does not need checking: what matters is the numbers you type in. Use these calculators only if you are competent to choose the inputs (loads, material strength, dimensions, supports) and check your units and assumptions. For learning and checking your own working: design and sign-off are for a qualified engineer. Drafting Central does not store the numbers you type.

How this calculator works

In a thin-walled cylinder under internal pressure p, the hoop (circumferential) stress is σ = p r / t and the longitudinal stress is half of that. In a sphere the stress is p r / (2 t) in every direction. The theory holds when the radius is about ten times the wall thickness or more.

Real pressure vessels, pipes and boilers are designed to the pressure vessel and piping standards by a qualified engineer, with design stresses well below yield, testing and inspection. Use this calculator to understand the stresses, not to design a vessel.

Formulas used

  • \sigma_h = \frac{p r}{t}
  • \sigma_l = \frac{p r}{2 t}
  • \sigma_{sphere} = \frac{p r}{2 t}

Worked example

With the default numbers in the calculator above:

  1. Hoop (circumferential) stress: 41.7 MPa
  2. Longitudinal (axial) stress: 20.8 MPa
  3. Check the thin-wall assumption: 83.3 (thin-wall theory is reasonable)

Change the numbers in the calculator to see the full working, every step, for your own case.

Common questions

Why is the hoop stress twice the longitudinal stress?

Because the wall resists the pressure on the end caps along its length with the full circumference, but resists the splitting force along a strip with only two wall thicknesses.

Is this valid for thick walls?

No. If r / t is below about 10, use thick-wall (Lame) theory.

Can I use this to design a pressure vessel?

No. Pressure vessels need the design standard, weld and nozzle checks and a qualified engineer.

Related calculators

Further reading: Hibbeler, Mechanics of Materials; Gere and Goodno, Mechanics of Materials; Shigley, Mechanical Engineering Design.

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