Engineering task and calculation objective
The EN12 module calculates bolted domed ends (spherically dished covers) to DIN EN 13445-3 clause 12. A bolted domed end consists of a spherical crown that merges into a flange ring and is connected to the mating flange of the vessel by a bolted joint. The dome can be convex to the pressure (pressure on the concave side) or concave to the pressure; the gasket can be a continuous full-face gasket or a narrow-face gasket located inside the bolt circle.
Designers typically calculate such bolted domed ends as removable covers of reactors, filters, heat exchanger channels and autoclaves: they combine the favourable membrane behaviour of a dished end with the detachability of a flange joint and, at larger diameters, are considerably lighter than flat blind flanges. The calculation rules of EN 13445-3 go back to the well-established Taylor-Forge methods and treat crown and flange ring together, because the ring cross-section is loaded by the combination of the inclined pull from the dome and the bolt moment.
Input quantities include the design pressure and design temperature for each load case, the inside or mean radius of curvature and the nominal wall thickness of the dome, and the actual flange thickness at the thinnest cross-section. The module checks all governing load cases including the bolting-up condition.
Standard and calculation basis: DIN EN 13445-3/12: 2021-12
Calculation workflow
- Define geometry and configuration: First, the configuration (dome convex or concave to the pressure), the gasket type (full face or narrow face) and the main dimensions are recorded: inside and mean radius of curvature of the dome, nominal thickness of the dome, and the flange ring dimensions including the actual flange thickness at the thinnest cross-section.
- Define load cases and design values: For each load case, design pressure and design temperature are specified; from these the nominal design stresses of the end, flange and bolt materials follow. In addition to the operating condition, the bolting-up condition with the required gasket seating stress must always be verified.
- Verify the dome thickness: The required wall thickness of the spherical dome is determined from pressure, radius of curvature and allowable stress. If the dome is concave to the pressure (pressure on the convex side), an additional buckling check against collapse is required.
- Calculate bolt loads and gasket: From the pressure area, gasket geometry and gasket factors, the required bolt loads for the operating and bolting-up conditions are determined, and from these the required bolt cross-sectional area.
- Verify the flange ring: The flange ring is verified for the combined loading from the bolt moment and the radial-horizontal component of the dome pull. The thinnest ring cross-section governs; the standard limits the stresses there for all load cases.
Input quantities
| Quantity | Symbol | Unit |
|---|---|---|
| Design pressure | P | N/mm² |
| Design temperature | T | °C |
| Load case | Lastfall | – |
| Inside radius of curvature of dome | R | mm |
| Nominal wall thickness of dome | e1n | mm |
| Material | Werk | – |
| Mill tolerance | c1 | mm |
| Corrosion allowance | c2 | mm |
| Strength | KOp | N/mm² |
| KTest | KTest | N/mm² |
| Safety factor | SOp | - |
| STest | STest | - |
| Nominal design stress | fD | N/mm² |
| fDTest | fDTest | N/mm² |
| Modulus of elasticity | EOp | N/mm² |
| ETest | ETest | N/mm² |
| Density | ρ | kg/m³ |
| Actual flange thickness at thinnest cross-section | e | mm |
| Mean radius of curvature of dome | Rm | mm |
Calculated results
| Quantity | Symbol | Unit |
|---|---|---|
| Required thickness of spherical dome section | eD | mm |
| Geometrical condition | BedGeo | – |
| Strength condition | BedFest | – |
| Strength condition for external pressure acc. to EN 13445-3 section 8 | 8 | – |
| Analysis wall thickness of spherical dome section | e1a | mm |
| Maximum permissible pressure of spherical dome section | Pmax | MPa(p) |
Worked example
For a bolted domed cover of a vessel, the required wall thickness of the spherical dome is to be determined to DIN EN 13445-3 clause 12 — a worked example of a spherically dished cover calculation. The dome is convex to the pressure (internal pressure on the concave side). The flange and bolt verification is not included here.
Given values
| Design pressure P | 1.0 MPa (10 bar) |
| Inside radius of curvature of the dome R | 800 mm |
| Nominal design stress f | 150 N/mm² |
| Corrosion allowance c | 1.0 mm |
Solution
Required dome thickness from the membrane formula
For the spherical dome of a bolted domed end, EN 13445-3 clause 12 gives:
e = 5 · P · R / (6 · f)
e = 5 · 1.0 N/mm² · 800 mm / (6 · 150 N/mm²) = 4000 / 900 = 4.44 mm
Allowances and selected thickness
With the corrosion allowance c = 1.0 mm, the minimum required thickness becomes:
e + c = 4.44 mm + 1.0 mm = 5.44 mm
Taking the plate under-tolerance into account, a nominal thickness of 6 mm is selected. For the complete verification of the bolted domed end, the flange ring, bolts and gasket must additionally be verified for the bolting-up and operating conditions to clause 12.
Result
| Required dome thickness e | 4.44 mm |
| Required thickness incl. corrosion allowance | 5.44 mm |
| Selected nominal thickness | 6 mm |
All values are illustrative. The applicable standard and project-specific boundary conditions remain authoritative.
Frequently asked questions
How does a bolted domed end differ from a dished end to clause 7?
A dished end to EN 13445-3 clause 7 (Klöpper or Korbbogen type) is permanently welded to the shell and is verified through the shell geometry alone. The bolted domed end of clause 12, by contrast, is a detachable bolted component: the dome calculation is supplemented by a complete flange and bolt calculation, and the flange ring experiences, in addition to the bolt bending, a ring loading from the horizontal component of the dome force.
What role does the gasket location play?
With a narrow-face gasket inside the bolt circle, a large lever arm develops between bolt load and gasket reaction, producing a high bending moment in the flange ring; in return, the gasket seating stress is well controlled. With a continuous full-face gasket, the ring bears on the entire gasket face, the ring moment drops considerably, but higher bolt loads are needed to seat the large gasket area. The standard provides separate calculation routes for both cases.
Why must the bolting-up condition be verified separately?
In the bolting-up condition, the full bolt pre-load acts on ring and gasket without any relieving internal pressure. Particularly with soft gaskets requiring high seating stress, this condition can govern the design of the flange ring. EN 13445-3 therefore requires separate verification of the bolting-up and operating conditions, each with its associated allowable stresses.
What must be considered when the dome is concave to the pressure (external pressure on the dome)?
If the convex side of the dome is under pressure, the shell is loaded in compression and can buckle elastically or plastically. In addition to the strength check, a stability check is then required, which works with considerably higher safety factors. The allowable load is therefore noticeably below that of an equally thick dome convex to the pressure.