EN 13480 Head (End Closure) Thickness Calculator

Calculate Minimum Wall Thickness for Ellipsoidal (torispherical) Heads per EN 13480-3

A head (end closure) seals the end of a vessel or pipe and is subject to the same internal pressure as the rest of the system, but its curved geometry introduces three distinct failure modes that must all be checked: membrane stress, knuckle yield at the junction between the crown and the knuckle, and knuckle buckling of the thin curved region.

This online calculator determines the minimum required wall thickness of a Torispherical or ellipsoidal head in accordance with EN 13480-3 (7.1.4) — Design and calculation of industrial piping systems. It is built for piping and pressure vessel engineers who need to verify that a head is thick enough to carry the design pressure without overstressing or buckling the knuckle.

Head Geometry - the Inner Factor

The shape of the head is defined by the inner factor f, the ratio of the inner head height h to the inner diameter D_i (f = h / D_i). A larger inner factor corresponds to a deeper head. The default value is 0.25.

From the inner factor the calculator first computes the geometric factor K:

(7.1.4)

and then derives the two defining radii of the head:

(7.1.4-1)

(7.1.4-2)

• R — the large (crown) radius of the head
• r — the knuckle radius of the curved transition at the junction

A deeper head (larger inner factor) produces a larger knuckle radius relative to the diameter and a smaller crown radius. These two radii are then used in the three thickness checks below.

The Three Thickness Checks

The required head thickness is the largest of three values, each covering a different failure mode.

1. Membrane Thickness

The membrane check covers the uniform hoop stress in the crown of the head:

(7.1.3-1)

where σ_d is the design stress and z is the joint efficiency.

2. Knuckle Yield Thickness

The knuckle is the region of highest bending stress at the junction between the crown and the cylindrical portion. Its yield strength is checked using a β factor that depends on the head geometry:

(7.1.3-2)

The β factor is determined from the geometry (the Y, Z, X, and N factors) by interpolation over the knuckle radius to diameter ratio. The knuckle yield thickness and the β factor are solved iteratively, because β depends on the thickness. The full determination of the β factor is covered in The Beta Factor.

3. Knuckle Buckling Thickness

Thin knuckles can fail by elastic buckling rather than by yield. This is checked using the design stress for buckling, a separate material property:

(7.1.3-3)

where σ_db is the design stress for buckling.

Governing Thickness

The minimum required head thickness is the largest of the three checks:

(7.1.3)

The Beta Factor

The β factor scales the membrane pressure term to represent the additional stress concentration in the knuckle. It is a pure function of the head geometry, captured by four intermediate factors — Y, Z, X, and N:

(7.1.5-1)

(7.1.5-2)

(7.1.5-3)

(7.1.5-4)

• Y — the thickness to crown radius ratio, capped at 0.04
• Z — the base-10 logarithm of the inverse of Y
• X — the knuckle radius to inner diameter ratio (r / D_i); the same ratio referenced in the applicability limits
• N — a normalising factor that approaches 1.006 as Y grows

Three reference β values are then evaluated at the standard knuckle radius ratios of 0.06, 0.1 and 0.2:

(7.1.5-5)

(7.1.5-7)

(7.1.5-9)

The final β is read from these reference values by linear interpolation on the actual X ratio:

• X = 0.06 — β = β_0.06
• 0.06 < X < 0.1 — interpolate between β_0.06 and β_0.1:

(7.1.5-6)

• X = 0.1 — β = β_0.1
• 0.1 < X < 0.2 — interpolate between β_0.1 and β_0.2:

(7.1.5-8)

• X = 0.2 — β = β_0.2
• X > 0.2 — the geometry is outside the applicability range and the calculation stops

Because the β factor depends on the thickness e (through Y), and the knuckle yield thickness depends on β (through the e_ky formula above), the two are solved iteratively: the calculator starts from an initial thickness estimate, computes β, recomputes the knuckle yield thickness, and repeats until the thickness converges.

Design Stress

The membrane and knuckle yield checks use the standard design stress from the material and design temperature, exactly as for a straight pipe. For how that value is determined per material, see the design stress page.

The knuckle buckling check uses a separate design stress for buckling, which is a head-specific material property:

where Rp0.2 is the 0.2% proof stress at the design temperature. This buckling stress is equal to or higher than the ordinary design stress, and it is the value that controls the knuckle buckling check above.

Applicability Limits

The calculation is only valid for head geometries within the range supported by the standard. The calculator enforces this by limiting the knuckle radius to diameter ratio (X factor, r / D_i) — if the ratio exceeds 0.2 the calculation cannot define the beta factor and a clear error is raised.

[ FILL IN CODE CHAPTER ]

Analysis Thickness - Accounting for Tolerance and Corrosion

The head uses an analysis thickness that accounts for manufacturing tolerance, corrosion allowance, and thread allowance (if applicable):

1. Manufacturing Tolerance — The negative deviation permitted by the manufacturing standard
2. Corrosion Allowance — Additional thickness reserved for expected material loss over the design life
3. Thread Allowance — Depth of threading for threaded connections (if applicable)

Code Compliance and Utilisation

The calculator evaluates the head by comparing the calculated minimum thickness to the analysis thickness:

• Code Compliant (PASS): e_{calc} \leq e_{analysis} — the head wall is sufficient for the design conditions
• Non-Compliant (FAIL): the head wall is insufficient; a thicker head, a larger knuckle radius, or a higher-grade material is required

The utilisation percentage quantifies how close the head is to its limit:

A utilisation of 85% means the head uses 85% of its available wall capacity, leaving a 15% margin. Values above 100% indicate the head does not meet code requirements.

Step-by-Step Calculation Process

The calculator performs the following steps for each head and pressure/temperature case:

1. Determine head geometry — Derive the spherical radius R and knuckle radius r from the inner factor
2. Check applicability limits — Confirm the head geometry is within the supported range
3. Determine Analysis Thickness — Subtract manufacturing tolerance, corrosion allowance, and thread allowance from the nominal wall thickness
4. Determine Design Stresses — Look up the design stress and the design stress for buckling at the design temperature
5. Determine the beta factor — Compute the Y, Z, X, and N factors from the head geometry, then interpolate to find β (see The Beta Factor)
6. Calculate the three thicknesses — Evaluate membrane, knuckle yield (iterative, using β), and knuckle buckling
7. Select the governing thickness — Take the largest of the three
8. Evaluate compliance — Compare the governing thickness against the analysis thickness and report utilisation

Each step is documented in the calculation report with formulas, intermediate values, and units for full traceability.

Key Inputs

Frequently Asked Questions

Why does a head need three separate thickness checks?

A head has multiple failure modes because of different types of stresses. Membrane tension, membrane compression and bending. This means every failure mode has to be checked. For thinner Heads stability and buckling is often an issue, while thicker heads under higher pressure can have issues with pure membrane stresses.

What is the difference between a torispherical and a hemispherical head?

A hemispherical head has a single radius like a Hemisphwere, or half of a globe. while a torispherical head is built up from a knuckle and a crown whith different radii that flow into each other.

Is this calculator suitable for certification and documentation?

The calculator produces a detailed calculation report showing all input parameters, intermediate values, formulas applied, and code compliance status. The output is structured to support engineering documentation and design review. For formal certification, always verify results against the latest published version of the applicable standard.