Sizing the wall thickness of a pressure vessel component under internal pressure is one of the most frequent calculations in pressure vessel engineering — and one governed by a different formula for almost every shape. This guide walks through the ASME Section VIII Division 1 formula for each common component type — cylindrical shell, hemispherical head, ellipsoidal head, torispherical head, and conical shell — with a fully worked numeric example for each, using the same formulas behind this site's free ASME VIII Div.1 Thickness Calculator.
Design Basis and Common Variables
Every formula below uses the same set of design variables, and every formula adds corrosion allowance (CA) to the pressure-governed thickness at the end — CA is a fixed metal allowance for expected future corrosion/erosion over the vessel's design life, not part of the stress calculation itself.
- P — Internal design pressure (MPa)
- R — Inside radius, corroded (mm) — for inside-diameter input, \(R = D_i/2 + CA\); for outside-diameter input, \(R = D_o/2\)
- S — Maximum allowable stress at design temperature, from the applicable material chart in ASME Section II Part D (MPa)
- E — Joint efficiency, per UW-12 (dimensionless, ≤ 1.0)
- CA — Corrosion allowance (mm)
All the worked examples below use the same base conditions so the results are directly comparable to each other: P = 1 MPa, inside diameter Di = 3000 mm, S = 138 MPa (SA-516 Gr. 70 at 200°C), E = 0.85 (spot radiography), CA = 1.6 mm.
Cylindrical Shell (UG-27)
A cylindrical shell under internal pressure is checked against two independent stress directions — circumferential (hoop) stress and longitudinal stress — because the two directions see different stress magnitudes for the same pressure. The larger of the two thicknesses governs.
Worked example (R = 3000/2 + 1.6 = 1501.6 mm):
| Direction | Formula | Result |
|---|---|---|
| Longitudinal | \( \dfrac{1 \times 1501.6}{2(138)(0.85)+0.4(1)}+1.6 \) | 7.99 mm |
| Circumferential | \( \dfrac{1 \times 1501.6}{(138)(0.85)-0.6(1)}+1.6 \) | 14.47 mm ← governs |
Circumferential stress governs here, as it almost always does for realistic pressure/stress combinations — the denominator is smaller (subtracting rather than adding the pressure term), which makes the circumferential thickness the larger, controlling value.
Hemispherical Head (UG-32f)
A hemispherical head has equal stress in every direction (a sphere has no "weak axis" the way a cylinder does), which is exactly why it needs roughly half the thickness of a cylindrical shell of the same diameter and pressure — but the doubly-curved shape is the most expensive to form.
Worked example (same R = 1501.6 mm):
| Formula | Result |
|---|---|
| \( \dfrac{1 \times 1501.6}{2(138)(0.85)-0.2(1)}+1.6 \) | 8.01 mm |
Compare 8.01 mm here to 14.47 mm for the cylindrical shell above — almost exactly half, confirming the textbook rule of thumb.
Ellipsoidal Head (UG-32d)
The standard 2:1 semi-elliptical head is the most common head type on process vessels — a good balance of forming cost and pressure efficiency. Head height is fixed at one-quarter of the diameter, and the formula introduces a shape factor K that accounts for the ellipse's changing curvature.
Worked example (D = corroded diameter = 3003.2 mm):
| Step | Formula | Result |
|---|---|---|
| Head height | \( h = 3003.2/4 \) | 750.8 mm |
| K factor | \( K = 0.167[2+(3003.2/(2 \times 750.8))^2] \) | 1.002 |
| Thickness | \( \dfrac{1 \times 3000 \times 1.002}{2(138)(0.85)-0.2(1)}+1.6 \) | 14.42 mm |
For a standard 2:1 ellipsoidal head, K is almost always very close to 1.0 — which is why the ellipsoidal head thickness here (14.42 mm) lands so close to the cylindrical shell's governing circumferential thickness (14.47 mm). This is a useful sanity check: an ellipsoidal head should size out very close to its matching cylindrical shell, not dramatically thicker or thinner.
Torispherical Head (UG-32e)
Also called a "flanged and dished" (F&D) head, the torispherical head uses two radii — a large crown radius L (typically equal to the shell diameter) and a smaller knuckle radius r (typically 6% of L) — making it cheaper to form than a true ellipse, at the cost of needing more thickness for the same pressure.
Worked example (L = 3003.2 mm, r = 0.06 × 3003.2 = 180.19 mm):
| Step | Formula | Result |
|---|---|---|
| M factor | \( M = 0.25(3+\sqrt{3003.2/180.19}) \) | 1.771 |
| Thickness | \( \dfrac{1 \times 3003.2 \times 1.771}{2(138)(0.85)-0.2(1)}+1.6 \) | 24.29 mm |
Notice this is significantly thicker than the ellipsoidal head (24.29 mm vs. 14.42 mm) at the exact same pressure and diameter — the M factor (1.771) is much larger than the ellipsoidal head's K factor (1.002), directly reflecting the torispherical head's less efficient stress distribution at the knuckle. This is the standard trade-off: cheaper to form, but a heavier, thicker head is required.
Conical Shell (Appendix 1-4e)
Conical shells are used for diameter transitions — a reducer between two cylindrical sections, or the bottom of a hopper/silo vessel. Thickness depends on the half apex angle α: a steeper cone (larger α) requires more thickness for the same pressure, since the formula divides by \(\cos\alpha\), which shrinks as α grows.
Worked example (D = 3003.2 mm, α = 30°, so \(\cos 30° = 0.866\)):
| Direction | Formula | Result |
|---|---|---|
| Longitudinal | \( \dfrac{1 \times 3003.2}{4(0.866)(138 \times 0.85+0.4)}+1.6 \) | 8.97 mm |
| Circumferential | \( \dfrac{1 \times 3003.2}{2(0.866)(138 \times 0.85-0.6)}+1.6 \) | 16.46 mm ← governs |
As with the cylindrical shell, circumferential stress governs. Notice the cone's governing thickness (16.46 mm) is somewhat larger than the cylindrical shell's (14.47 mm) at the same base diameter and pressure — the \(1/\cos\alpha\) term always makes a cone at least as thick as an equivalent cylinder, growing quickly as the half apex angle increases beyond about 30°.
Comparing Head Types at the Same Conditions
Putting all four worked results side by side, at identical pressure, diameter, material, and joint efficiency:
| Component | Governing Thickness |
|---|---|
| Hemispherical head | 8.01 mm (thinnest, most efficient — most expensive to form) |
| Ellipsoidal head (2:1) | 14.42 mm |
| Cylindrical shell | 14.47 mm |
| Torispherical head | 24.29 mm (thickest — cheapest to form) |
This ordering — hemisphere thinnest, torispherical thickest, with ellipsoidal landing almost exactly at the cylinder's thickness — holds broadly true across most practical pressure vessel design conditions, and is a useful mental model when choosing a head type early in a design: the "cheap to form" options cost you wall thickness (and therefore weight and material cost), while the "efficient shape" options cost you fabrication complexity.
✓ Formula Source
All formulas above are taken directly from ASME Boiler & Pressure Vessel Code, Section VIII, Division 1: UG-27 (cylindrical shells), UG-32 (formed heads), and Mandatory Appendix 1-4 (conical shells and conical shell-to-shell junctions without a knuckle).
Try It Yourself
Reproduce any of the worked examples above and then vary one input at a time to build intuition for how sensitive each component type is to pressure, diameter, and joint efficiency:
- Open the ASME VIII Div.1 Thickness Calculator.
- Select a component type and set Diameter Type to "Inside Diameter."
- Enter P = 1 MPa, D = 3000 mm, S = 138 MPa, Joint Type = Spot Radiography (E = 0.85), CA = 1.6 mm.
- Click Calculate and compare your result to the tables above.
- Switch component types with the same inputs to reproduce the full comparison table.
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