Is this an ASME or EN 13445 pressure-vessel design calculation?

No. It is a general mechanics worksheet for idealized membrane stress. It does not implement the complete construction, material, fabrication, examination, testing or design rules of a pressure-vessel code.

Which pressure belongs in the equations?

Use the positive internal-minus-external pressure difference across the wall. Do not enter absolute internal pressure unless external absolute pressure is zero.

Why is the axial result limited to closed ends?

The axial expression assumes pressure end thrust is carried by the cylindrical wall being analyzed. Another load path or additional axial load changes that result.

Does radius divided by thickness of ten prove that a vessel is safe?

No. It is only the applicability gate used by the cited mechanics source for its thin-wall idealization, not a code acceptance or safety criterion.

Is the displayed equivalent stress a complete wall stress?

No. It is the von Mises combination of the two uniform in-plane membrane components. Radial stress, local stresses, discontinuities and other loads are outside the worksheet.

General mechanics · not code design

Closed-End Thin-Cylinder Membrane-Stress Worksheet

Estimate uniform hoop and pressure-induced axial membrane stress in the straight cylindrical wall, far from discontinuities. The model requires positive internal-over-external pressure and a closed-end thrust path through the analyzed wall.

Δp = p_inside − p_outsider_i/t ≥ 10 model gateNo pass/fail verdict

Model outputs

—

Hoop membrane σθ

—

Closed-end axial membrane σz

—

In-plane membrane equivalent σeq

—

Internal radius rᵢ

—

Geometry ratio rᵢ/t

—

Relative wall thickness t/rᵢ

—

Not an acceptance result: these are idealized membrane components, not allowable stresses, required wall thickness, maximum allowable working pressure or proof of compliance. Do not compare them to yield strength alone and declare a vessel safe.

Uniform thin-cylinder membrane components

r_i = D_i/2
σ_θ = Δp r_i/t = Δp D_i/(2t)
σ_z,p = Δp r_i/(2t) = Δp D_i/(4t)
σ_eq,mem = √(σ_θ² − σ_θσ_z,p + σ_z,p²)

Here Δp is internal pressure minus external pressure; D_i is internal diameter; r_i is internal radius; and t is wall thickness. Pressure and stress share a unit. When MPa and mm are used, stress is MPa. The US route uses exactly 1 in = 25.4 mm and 1 psi = 0.006894757293168361 MPa; displayed stress is converted back to psi.

The hoop expression follows transverse force equilibrium of a longitudinally cut segment. The axial expression follows equilibrium of pressure force on a closed end against the membrane force in the cylindrical wall. It therefore does not apply when another structure carries end thrust or when an additional axial load must be included.

Radial stress omitted: actual wall stress also includes a radial component that changes through the thickness, plus possible local and bending stresses. The equivalent value above combines only the two stated in-plane membrane components; it is not a complete stress analysis or code utilization.

Geometry gate

The cited Purdue mechanics source uses radius at least ten times wall thickness as an assumption for its educational thin-wall analysis. This worksheet therefore calculates only when r_i/t ≥ 10. Equality is accepted; a smaller ratio is rejected.

Not a safety boundary: r_i/t ≥ 10 does not prove thin-shell accuracy for every purpose and does not satisfy any pressure-vessel code. It only prevents use outside the stated classroom model. A required accuracy, diameter convention, stress classification or governing code may demand another method.

Outside this worksheet

External-over-internal pressure (negative Δp), vacuum and buckling/stability.
Open ends, restrained ends, force transferred through another component, or any added axial, bending, thermal, dead-weight, wind, seismic, nozzle or support load.
Heads, cones, spheres, junctions, openings, nozzles, flanges, supports, weld details, defects and other local discontinuities.
Plasticity, large deformation, anisotropy, laminates, residual stress, fatigue, creep, fracture, corrosion/erosion allowance or cyclic service.
Allowable stress, joint efficiency, tolerances, forming/fabrication effects, examination, testing, relief protection and jurisdictional requirements.

Purdue ME 323 Homework Set 10, Problem 10.3

The published solution uses r = 3000 mm, t = 20 mm and p = 2 MPa. It reports r/t = 150, axial stress 150 MPa and hoop stress 300 MPa. Enter D_i = 6000 mm to represent that radius. The worksheet independently reproduces those membrane components and gives 259.807621 MPa for the stated in-plane equivalent combination.

Input / output	कीमत	Role
Δp, D_i, t	2 MPa, 6000 mm, 20 mm	Published example, with diameter entered as twice the published radius
r_i/t	150	Published solution
σ_z,p	150 MPa	Published solution
σ_θ	300 MPa	Published solution
σ_eq,mem	259.807621 MPa	Independent plane-stress combination, not a reported Purdue answer

Open mechanics source

Purdue University ME 323 Lecture 30: Thin-walled pressure vessels states the r ≥ 10t, insignificant through-thickness strain variation, plane-stress, linear-elastic and small-deformation assumptions and derives the axial and hoop forms for a cylindrical pressure vessel. Purdue ME 323 Homework Set 10 (Fall 2025), Problem 10.3 supplies the published numerical example used above.

NIST SP 811 Appendix B.8 gives the international inch and pound-force SI factors used to derive the exact psi route; the pressure factor is independently cross-checked against Appendix B.9.

Pressure-vessel codes are not implemented

ASME BPVC Section VIII, Division 1 (2025) covers design, fabrication, inspection, testing and certification within its scope; the full rules are not reproduced by this three-input equilibrium worksheet.

NBN EN 13445-3:2026, an official national adoption of EN 13445-3:2026, is active and replaces NBN EN 13445-3:2021+A1:2026. It covers design of unfired pressure vessels within the EN 13445 series. Its licensed detailed rules are not claimed or implemented here. Applicable editions, amendments, jurisdiction and contractual requirements must be established for the real project. Sources/status accessed 13 July 2026.

वर्गीकरण: the displayed equations are general thin-wall mechanics formulas under the declared assumptions. They are not labelled as an ASME, EN or ISO design formula, and the result is not a normative conformity calculation.

No. It is an idealized mechanics worksheet. Code design requires the applicable licensed rules and many inputs and checks absent here.

Enter internal absolute pressure minus external absolute pressure. A gauge reading may equal that difference only when its reference actually represents the external pressure acting on the wall.

The axial pressure-membrane expression balances pressure thrust on a closed end against membrane force in this wall. A different thrust path or extra axial load changes the axial component.

No. It is only this worksheet’s model-applicability gate from the cited mechanics source. It is not an allowable-stress, stability or code-compliance check.

External-over-internal pressure introduces stability and buckling concerns outside this internal-pressure membrane worksheet. Zero Δp is accepted and produces zero pressure-induced components.