Pressure Vessel End Closures
Flat ends, domed ends, hemispherical ends, ellipsoidal forms, pressure end load transfer, local bending at the shell-to-head junction and load path considerations.
Technical provenance
Applicable standards / specifications
- ASME BPVC Section VIII Division 1 (2025) — Rules for Construction of Pressure Vessels
- ASME BPVC Section VIII Division 2 (2025) — Alternative Rules for Construction of Pressure Vessels
- EN 13445 — Unfired pressure vessels — Relevant European pressure-vessel code family where specified by the project.
References
- ASME Boiler and Pressure Vessel Code — 2025 edition — Primary code family reference for pressure-vessel design where ASME BPVC is the governing basis.
- Moss, D. R. & Basic, M. — Pressure Vessel Design Manual — Background engineering reference for pressure-vessel load paths, stresses and design checks.
Function of end closures
An end closure (head) seals the end of a pressure vessel and transfers the pressure end load into the cylindrical shell. The head must contain the pressure without leakage and transfer the axial force into the shell wall as longitudinal membrane stress. The head geometry determines how efficiently this transfer occurs and what local stresses are generated at the shell-to-head junction.
Flat ends
A flat end plate carries the pressure load primarily in bending. The plate behaves like a circular plate under uniform pressure, with bending moments that are highest at the centre and at the edge depending on the boundary condition. Flat ends are structurally inefficient because bending is far less efficient than membrane action for carrying pressure. Flat ends are used where space constraints or access requirements make a dished head impractical, but they require significantly greater thickness than a domed head of equivalent pressure capacity.
Domed ends
Domed ends carry the pressure load primarily in membrane action, which is structurally efficient. The membrane stress in a hemispherical head equals the longitudinal stress in the cylinder (p * r / (2 * t)), making the hemisphere the most efficient head geometry. Torispherical and ellipsoidal heads are less efficient than hemispherical but more practical to manufacture. The head geometry affects the local stresses at the shell-to-head junction.
Hemispherical ends
A hemispherical head has the same membrane stress as a spherical vessel: p * r / (2 * t). This is half the hoop stress in the cylinder, meaning the hemisphere can be thinner than the cylinder for the same pressure. However, at the junction between the hemisphere and the cylinder, there is a mismatch in radial displacement under pressure (the hemisphere expands differently from the cylinder), which generates local bending stresses at the junction.
Ellipsoidal-type forms
Ellipsoidal heads (typically 2:1 semi-elliptical) are a compromise between the structural efficiency of a hemisphere and the manufacturing simplicity of a shallower dome. The membrane stress in an ellipsoidal head varies with position — it is highest at the crown and varies towards the knuckle region. The knuckle region (where the head curves to meet the cylinder) is a discontinuity that generates local bending and can be a critical stress location, particularly for heads with a tight knuckle radius.
Pressure end load transfer
The pressure end load (F = p * pi * r^2) is transferred from the head into the shell as longitudinal membrane stress. In a domed head, the transfer occurs through membrane action in the head and shell, with a local bending disturbance at the junction. In a flat head, the transfer occurs through bending in the plate and membrane in the shell, with a more severe discontinuity at the junction. The load path must be continuous — any weakness in the head, the junction or the shell can become the governing failure location.
Local bending at the junction
At the shell-to-head junction, the difference in radial displacement under pressure creates a local bending moment and shear force. These discontinuity forces decay exponentially with distance from the junction over a characteristic length of approximately sqrt(r * t). The local stresses can be several times the nominal membrane stress. The junction design — the head geometry, the transition radius, any reinforcement and the weld — must manage these local stresses.
The shell-to-head junction is one of the most common locations for failure in pressure vessels. The local bending and stress concentration at this discontinuity must be assessed in addition to the nominal membrane stresses in the head and shell.
Choosing a closure geometry
Closure selection is a structural, manufacturing and inspection decision rather than a question of pressure capacity alone. Hemispherical heads minimise membrane stress but are deep and expensive to form. Ellipsoidal and torispherical forms package more efficiently but introduce stronger curvature changes and therefore larger local bending near the knuckle and shell junction. Flat closures may be attractive for access covers or compact equipment, yet the load path becomes bending dominated and thickness rises rapidly with span. The assessment should therefore compare pressure level, diameter, available depth, fabrication route, weld arrangement, inspection access, cyclic duty and any attached hardware before selecting a geometry.
Load-path equilibrium at the end of a vessel
A useful independent check is to cut the vessel immediately behind the closure and balance the pressure resultant against the shell force. Internal pressure acting over the projected internal area produces an axial thrust that must be transferred through the closure and its attachment into longitudinal shell stress. This equilibrium check remains valid regardless of head shape and is extremely effective at finding missing pressure faces, omitted end-cap loads or inappropriate symmetry constraints in an FE model. The same principle also applies to blind flanges, plugs and removable covers: the pressure does not disappear at a boundary; its resultant must close through a physical structural path.
Local versus global assessment
The closure requires two distinct checks. The first is global pressure containment: membrane or bending stress, gross plasticity and overall deformation. The second is local integrity at the junction, weld, knuckle, bolt circle or gasket interface. A head can have comfortable nominal membrane stress while a transition detail controls fatigue or ratcheting. Conversely, a local elastic peak at a theoretical sharp corner may not control static collapse. Separating global and local mechanisms prevents a single contour maximum from being treated as the entire assessment.
FE modelling and verification
For routine heads, shell elements are usually sufficient if thickness is small relative to radius and the junction is represented correctly. Solid elements become useful where through-thickness gradients, thick sections, complex weld preparations or contact interfaces matter. Mesh refinement should be driven by curvature and stress gradients rather than by a uniform element size. Verify the FE result against pressure-resultant equilibrium and simple membrane solutions away from discontinuities, then inspect whether the local junction response converges to a physical structural stress rather than a mesh-dependent mathematical peak.