Engineering task and calculation objective
Dished ends close pressure vessels at their extremities. Three shapes dominate in practice: the Klöpper head (torispherical, crown radius equal to the outside diameter), the deeper Korbbogen head (approximately semi-ellipsoidal) and the hemispherical head. Engineers who want to calculate a torispherical (Klöpper) or Korbbogen head work to AD 2000-Merkblatt B3 of the German AD 2000 pressure vessel code — it governs the wall thickness determination for internal and external pressure.
Module B3 determines the required wall thickness separately for the crown region and for the more highly loaded knuckle region, as well as the required cylindrical skirt height of the head. The options select the head type, the pressure direction, the manufacturing method and the allowance rule. For the existing wall thickness, the module additionally states the largest unreinforced opening for which no separate verification is required — important for central nozzles and manways in the head.
The calculation is needed in practically every vessel project: for new designs, when ordering heads (the manufacturing method — cold or hot formed, in one piece or welded from segments — affects the verification), and for re-rating with measured wall thicknesses, since pressed heads thin out in the knuckle region as a result of forming.
Standard and calculation basis: AD 2000 B3: 2021-06
Calculation scope
- Korbbogen type ends under internal and external pressure
- Kloepper type ends under internal and external pressure
- Hemispherical ends under internal and external pressure
Calculation workflow
- Select the head type and load case: First the head type (Klöpper, Korbbogen or hemispherical head), the pressure direction (internal or external pressure), the manufacturing method and the allowance rule are defined. The head type fixes the crown radius and knuckle radius as set ratios to the outside diameter.
- Calculate the wall thickness in the crown region: The crown is verified as a spherical shell with its crown radius; from the pressure, the strength value K, the safety factor S and the joint efficiency v follows the required wall thickness of the spherical cap.
- Determine the wall thickness in the knuckle region with the coefficient β: The knuckle is loaded in bending by the redirection of the wall forces. Its required wall thickness is determined with the design coefficient β, which depends on the head type, on the ratio of wall thickness to diameter and on any opening in the knuckle — the determination is iterative, via the charts of the code sheet or their stored functions.
- Determine the skirt height and opening limits: The module states the required cylindrical skirt height so that the weld to the shell lies outside the knuckle discontinuity zone, and gives, for the selected wall thickness, the largest unreinforced opening permissible without separate verification.
- Add the allowances and fix the execution: After adding the allowances for undertolerance and corrosion, the nominal wall thickness is selected. For pressed heads it must be noted that the ordered wall thickness has to cover the forming-induced thinning in the knuckle; for external pressure, the buckling resistance of the crown must additionally be verified.
Input quantities
| Quantity | Symbol | Unit |
|---|---|---|
| Required knuckle wall thickness operation | Krempe | mm |
| Required crown wall thickness operation | Kalotte | mm |
| Outside diameter of head | Da | mm |
| Maximum nozzle diameter outside 0.6·Da *) | di | mm |
| Design pressure | p | bar |
| Nominal design strength operation | K | N/mm² |
| Safety factor operation | S | – |
| Joint efficiency | v | – |
| Wall thickness manufacturing tolerance | c1 | mm |
| Corrosion / erosion allowance | c2 | mm |
| Run-out length | x | mm |
| Shape factor | β | – |
| Final wall thickness | se | mm |
| Final crown wall thickness | sK | mm |
| Skirt height | h | mm |
| Weight (empty) | Mantelgewicht | kN |
| Weight (operating) | Betriebsgewicht | kN |
| Weight (water-filled) | m.Wasserfüllung | kN |
| Volume (contents) | Teilvolumen | m³ |
| External design pressure | pa | bar |
| Safety factor testing | S' | – |
| Nominal design strength testing | K' | N/mm² |
| Test pressure | p' | bar |
| Modulus of elasticity at design temperature | E | N/mm² |
Calculated results
| Quantity | Symbol | Unit |
|---|---|---|
| Maximum allowable buckling pressure (according to 8.2.2) | Überdruck | bar |
| Required knuckle wall thickness (according to 8.2) | Überdruck | mm |
| Required crown wall thickness (according to 8.1.1) | Überdruck | mm |
| Required crown wall thickness (according to 8.1.1) | Probedru | mm |
| Required knuckle wall thickness (according to 8.2) | Probedruck | mm |
| Maximum allowable buckling pressure (according to 8.2.2) | Probeüberdruck | bar |
| Safety factor crown for testing (=2.4) | S'min | – |
Calculation options
Type of head
Torispherical head (Kloepper type) · Korbbogenboden-type torispherical head · Hemispherical head
Manufacturing
End with knuckle and crown of equal wall thickness · End with knuckle and crown of unequal wall thickness
Pressure
Internal pressure · External pressure
Allowances acc. to
Standard AD · DIN 2801x · Minimum wall thickness · Free input
Frequently asked questions
Klöpper head or Korbbogen head — what is the difference?
For the Klöpper head to DIN 28011, the crown radius is R = Da and the knuckle radius r = 0.1 · Da; for the Korbbogen head to DIN 28013, R = 0.8 · Da and r = 0.154 · Da. The Korbbogen head is deeper and its knuckle redirects the forces more gently — it therefore manages with a smaller knuckle wall thickness and is preferred at higher pressures. The Klöpper head is shallower and cheaper to manufacture. Internationally, both are usually filed under torispherical heads.
Why is the knuckle the critical location of the head?
In the knuckle region, the wall curvature changes over a small radius from the crown to the cylindrical skirt. The redirection of the meridional forces generates bending stresses there and, under internal pressure, circumferential compressive stresses that can cause the head to buckle in the knuckle. The coefficient β captures this intensification; it grows when the wall thickness is small relative to the diameter or when an opening lies near the knuckle.
What is the purpose of the cylindrical skirt height?
The skirt creates distance between the bending-loaded knuckle and the circumferential weld to the shell. If the weld lay in the discontinuity zone, weld imperfections and knuckle bending would superimpose unfavourably. The minimum skirt height depends on the wall thickness and is laid down in the head standards and in the code sheet.
Why do I have to consider the thinning of a pressed head?
During cold forming (pressing, flanging), the plate is stretched in the knuckle and can become several percent thinner there than the starting blank. The verification applies to the minimum wall thickness after forming — which is why heads are ordered with a guaranteed minimum wall thickness, or the blank thickness is chosen correspondingly thicker. The manufacturing option in the module accounts for this relationship.