Design of heads and closures: flat, torispherical, ellipsoidal

Why it matters

Every cylindrical vessel needs two closures, and the head type decides how thick they are, how much they cost to form, how much headroom they take and whether they can be opened. Choosing a flat cover where a dished head belongs can make the cover several times thicker than the shell; choosing a hemispherical head for a low-pressure tank wastes money on pressing. Head design is also where most students first meet discontinuity stresses.

Key ideas

Why curvature matters. A curved head carries pressure mainly by membrane tension, like the shell. A flat plate has no curvature, so it must carry the pressure by bending, which is far less efficient: its thickness grows with the square root of pressure and in proportion to diameter, and quickly becomes very large.

The common head types, from strongest to weakest shape:

• Hemispherical: half a sphere. Membrane stress P·D/(4t) is half the cylinder's hoop stress, so it needs roughly half the shell thickness. Deepest and most expensive to form (pressed in segments); used for high-pressure vessels and large spheres.
• 2:1 semi-ellipsoidal: depth (inside) = D/4, major-to-minor axis ratio 2. Its required thickness is almost exactly the shell thickness, so shell and head can be the same plate. The usual choice above roughly 1–1.5 MPa.
• Torispherical (dished and flanged): a spherical crown of radius Rc joined to the cylinder by a toroidal knuckle of radius rk. Standard proportions are Rc ≈ D (crown radius not more than the outside diameter) and rk ≥ 6 % of the crown radius. Shallow and cheap to press, but the tight knuckle raises the stress, so it needs to be thicker than the shell — typically about 1.7–1.8 times for standard proportions. The most common head for vessels up to about 1–1.5 MPa.
• Conical: used for hoppers, bottoms that must drain solids, and transitions between column sections. Stress increases with the half-apex angle α; above about 30° a knuckle or reinforcement at the junction is needed.
• Flat: flat covers, blind flanges, manholes and small or low-pressure vessels; also where a removable cover is needed. Designed by plate-bending formulas with a code factor C that depends on how the plate is attached.

Discontinuity stresses. A cylinder and a head expand by different amounts under pressure; at the junction they must stay together, so local bending and shear appear. A hemispherical head and a cylinder of the same thickness expand differently; making the head thinner reduces the mismatch. In torispherical heads the knuckle develops a high compressive hoop stress, which is why the code formula has a stress-intensification factor and why very thin large torispherical heads can buckle in the knuckle.

Selecting a head. Up to moderate pressure: torispherical. Above it: 2:1 ellipsoidal. High pressure or very thick walls: hemispherical. Small openings and covers: flat. Headroom, column skirt height and the volume below the tangent line also influence the choice. Forming reduces thickness (especially in the knuckle), so the plate ordered must allow for forming thinning as well as the corrosion allowance.

Formulas

(ASME VIII Div. 1 forms, internal pressure; IS 2825 gives equivalent expressions. t excludes corrosion allowance and forming allowance.)

t = P·Di / (4·S·E − 0.4·P) (hemispherical head) t = P·Di / (2·S·E − 0.2·P) (2:1 semi-ellipsoidal head) t = P·L·M / (2·S·E − 0.2·P), M = ¼·(3 + √(L / r)) (torispherical head)

• For standard proportions (L = Do, r = 0.06·L): M ≈ 1.77, giving t ≈ 0.885·P·L / (S·E − 0.1·P).

t = P·Di / (2·cos α·(S·E − 0.6·P)) (conical head or section, α ≤ 30°) t = d·√(C·P / (S·E)) (flat unstayed head)

• P = design pressure (MPa); Di = inside diameter (mm); S = allowable stress (MPa, from code book); E = joint efficiency (–); L = inside crown radius (mm); r = inside knuckle radius (mm); M = stress-intensification factor (–); α = half-apex angle; d = effective diameter of the flat head (mm); C = attachment factor (–), from the code's sketches (roughly 0.17–0.33, take from your code book).

h = Di / 4 (inside depth of a 2:1 ellipsoidal head) V = π·Di³ / 24 (2:1 ellipsoidal head), V = π·Di³ / 12 (hemispherical head) h = Rc − √((Rc − rk)² − (Di/2 − rk)²) (inside depth of a torispherical head)

Worked examples

Example 1 (standard): comparing heads for one shell Given: P = 1.5 MPa, Di = 2000 mm, S = 130 MPa, E = 1.0 (seamless heads). Torispherical: L = 2000 mm, r = 120 mm (6 %).

1. Shell (for reference): t = P·R/(S·E − 0.6·P) = 1.5 × 1000 / (130 − 0.9) = 11.6 mm.
2. Hemispherical: t = 1.5 × 2000 / (520 − 0.6) = 3000 / 519.4 = 5.78 mm.
3. 2:1 ellipsoidal: t = 3000 / (260 − 0.3) = 3000 / 259.7 = 11.55 mm (same as the shell).
4. Torispherical: M = ¼ × (3 + √(2000/120)) = ¼ × (3 + 4.08) = 1.77; t = 1.5 × 2000 × 1.77 / 259.7 = 20.5 mm.
5. Hemispherical 5.8 mm, ellipsoidal 11.6 mm, torispherical 20.5 mm (before corrosion and forming allowances). At 1.5 MPa the ellipsoidal head is the sensible choice.

Example 2 (GATE level): why flat covers are kept small Given: a welded flat cover of effective diameter d = 600 mm, P = 1.0 MPa, S = 120 MPa, E = 1, attachment factor C = 0.33 (given; take from your code book in practice).

1. Flat: t = d·√(C·P/(S·E)) = 600 × √(0.33 × 1.0 / 120) = 600 × 0.05244 = 31.5 mm.
2. A 2:1 ellipsoidal head of the same diameter: t = 1.0 × 600 / (240 − 0.2) = 2.50 mm.
3. The flat cover is about 12.6 times thicker. Doubling d to 1200 mm would double the flat cover to 63 mm, which is why flat heads are limited to small diameters, manways and low pressures.
4. Depth and volume of a 2:1 ellipsoidal head for Di = 2000 mm: h = 2000/4 = 500 mm; V = π × 2³/24 = 1.047 m³.

Common mistakes

• Using the cylinder formula for a hemispherical head (it needs about half) or for a torispherical head (it needs more, because of M).
• Assuming torispherical heads are as strong as ellipsoidal ones because they look similar.
• Ignoring forming thinning in the knuckle when ordering plate.
• Using crown radius larger than the outside diameter or knuckle radius below 6 % — both outside code limits.
• Treating flat heads with membrane formulas; they are governed by bending.
• Confusing semi-axes with full axes when computing head volumes.

For GATE CH

Expect ranking questions (which head needs the least thickness for a given pressure), standard proportions (2:1 head depth D/4, torispherical crown and knuckle limits), quick thickness calculations with a given formula, and head volume or depth calculations used in tank-capacity problems. Practise the ellipsoidal and torispherical formulas and the volume expressions.

Quick check

1. Rank hemispherical, 2:1 ellipsoidal and torispherical heads from thinnest to thickest for the same P and D.
2. What is the inside depth of a 2:1 ellipsoidal head on a 1.6 m vessel?
3. Why is a flat head so much thicker than a dished head?
4. What are the usual crown and knuckle radii of a standard torispherical head?

Answers: 1. Hemispherical, 2:1 ellipsoidal, torispherical. 2. 0.4 m. 3. It carries pressure by bending instead of membrane tension. 4. Crown radius about equal to the outside diameter; knuckle radius at least 6 % of the crown radius.