Thin-wall cylinder: internal pressure

From the free body to the equation

A pressure is a force per projected area. Cut a unit-length cylinder along its axis and keep one half. Pressure pushes on a projected rectangle of width D. Two cut wall strips resist with 2tσh, so 2tσh = pD. This gives the hoop membrane stress σh = pD/(2t). The length cancels.

For a closed end whose thrust is carried by the cylindrical shell, pressure acts over πD²/4. In this inner-diameter thin-wall approximation the resisting wall area is πDt. Equating the two gives σz = pD/(4t), half the hoop stress. The exact annular area differs; this calculator deliberately keeps the stated first-order approximation.

An open or independently restrained end does not automatically guarantee zero axial stress. The zero-axial mode requires that no end pressure thrust, support restraint, thermal restraint or other axial load enters the wall. If a separate frame carries a sealed end’s pressure thrust, that frame must carry pπD²/4; the tool reports this equivalent diverted thrust. For a truly uncapped pipe, no attached cap exists; the displayed pA is a reference capped-end thrust, not a calculated anchor load.

The model uses the inner radius ri = D/2 throughout. It requires t/ri < 0.1 as a teaching-model gate; thinning the wall increases membrane stress and does not prove adequacy. Inner, mean and outer diameter conventions must not be interchanged in formulas, especially near the edge of a thin-wall approximation.

Tensile membrane stress is positive. Radial stress is compressive at the inner wall and follows the pressure boundary condition; with an exterior gauge reference its surface values are −p and 0. Plane stress neglects this smaller component. The displayed t/ri quantifies |σr,inner|/σh when p > 0; it does not assert that radial stress is literally absent.

The optional-equivalent interpretation is purely algebraic: for principal stresses (σh, σz, 0), σeq = √[((σh−σz)² + σz² + σh²)/2]. There is no in-plane shear here. This scalar is not a stress direction, material strength, allowable pressure or verdict. No material database, yield strength or factor of safety is invented.

Worked maritime-context example

Synthetic training case: a straight section of a shipboard compressed-air receiver is idealised far from its heads, with inner D = 800 mm, t = 8 mm and internal differential p = 0.6 MPa. Closed ends transfer pressure thrust into the shell. This is not an actual receiver specification or an operating-pressure recommendation.

Calculated response

Closed ends · shell carries the end thrust

Displayed results are rounded. Validation and CSV retain the full converted floating-point precision.

σeq assumes principal plane stress (σh, σz, 0) and zero shear. Radial stress is neglected. No strength or safety comparison is made.

Active analytic equations

• ri=D/2; t/ri<0.1
• 2 sigma_h t ell=p D ell => sigma_h=pD/(2t)
• closed: sigma_z (pi D t)=p(pi D^2/4) => sigma_z=pD/(4t)
• open / externally carried end thrust: sigma_z=0
• sigma_eq=sqrt(((sigma_h-sigma_z)^2+sigma_z^2+sigma_h^2)/2)
• sigma_r(inner)=-p; sigma_r(outer)=0 relative to exterior gauge reference; plane-stress model takes sigma_r approximately 0

Substituted calculation

1. D = 800 × 0.001 = 0.8 m; t = 8 × 0.001 = 0.008 m
2. p = 0.6 × 1,000,000 = 600,000 Pa
3. ri = D/2 ≈ 0.4 m; t/ri ≈ 0.008/0.4 ≈ 0.02
4. The geometric gate uses unrounded converted inputs. t < D/20: 0.008 m < 0.04 m.
5. σh = 600,000 × 0.8 / (2 × 0.008) = 30 MPa
6. σz = 600,000 × 0.8 / (4 × 0.008) = 15 MPa
7. σeq = √[30² − (30 × 15) + 15²] = 25.9807621 MPa
8. Fend = 600,000 × π × 0.8² / 4 = 301.592895 kN

Residuals compare the pressure resultants with the membrane-force resultants. They verify the chosen approximation’s equilibrium, not vessel adequacy.

Assumptions and boundaries

• Uniform circular membrane region, away from ends, nozzles, changes of thickness, weld details and supports. End-cap geometry and local stresses are not solved.
• Only internal p ≥ 0, and ri/t > 10. The strict geometric gate is a simplified educational applicability convention, not an acceptance or safety boundary.
• The gauge differential model neglects a common ambient hydrostatic compression. High external confining pressure, subsea pressure hulls, external-pressure collapse and vacuum service require another model.
• Hoop and axial values are first-order membrane estimates. Through-thickness variation and radial stress are neglected in the equivalent-stress calculation. It is not a thick-cylinder or full three-dimensional stress result.
• No yield comparison, allowable stress, corrosion or weld efficiency, fatigue, creep, temperature dependence, stability or pressure-vessel code certification.
• Numerical implementation bounds: SI D 10⁻⁶–10⁴ m, t 10⁻⁹–10² m, p zero or 10⁻¹²–10¹⁰ Pa, hoop stress ≤10¹⁶ Pa. These broad bounds are not physical operating limits.

Teaching calculation only. No capacity, safety or regulatory approval is implied.

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Sources and curriculum connection

Curriculum connection: the cylinder is a bounded application of the combined-stress and stress-analysis topics in İTÜ MUK204. The course page does not claim this specific calculator; the cylinder equations are supported separately by the primary mechanics references.

• MIT OCW · David Roylance · Pressure Vessels (2001), pp. 2–4, equations 2–3
• KFUPM · CE203 · Thin-Walled Pressure Vessels, section 8.6
• MIT · Paul A. Lagace · Unified Engineering (2007), Unit M5-3, p. 11
• MIT OCW · Pressure Vessel Stress Analysis, section 7 and Appendix C1: radius conventions
• İTÜ · MUK204 · Strength of Materials II

Primary source content checked 9 October 2026. Historical mechanics publications support the equations; they are not current design codes.