Engineering task and calculation objective
Jacketed vessels — such as agitated vessels with a heating or cooling jacket — need a clean transition from the outer jacket to the inner vessel at the top and bottom of the jacket. These transitions are executed as external shell knuckles: curved plate zones, either drawn in or dished, which close off the annular space and transfer the jacket forces into the vessel wall. Anyone who wants to calculate an external shell knuckle must evaluate the bending and membrane stresses occurring there, because the knuckle region is loaded considerably more highly than the undisturbed shell due to the redirection of forces.
Module AKR determines the stresses in the external shell knuckle and at the transition to the pressure vessel on the basis of the German AD 2000 pressure vessel code and compares them with the allowable stresses σzul, which follow from the material strength value, the safety factor and the weld joint efficiency (e.g. v = 0.7 for a knuckle welded on one side only). Drawn-in and dished knuckles are distinguished, as are the upper and lower end knuckles.
The verification is needed in apparatus engineering for every double-walled vessel with jacket heating or cooling, particularly when the annular space is pressurized: the jacket pressure acts on the inner vessel as external pressure, and the knuckle must safely transmit both load cases.
Standard and calculation basis: AD 2000
Calculation workflow
- Record the geometry of knuckle and transition: First the geometric conditions are defined: diameters of the inner vessel and outer jacket, wall thicknesses, knuckle radius and the type of knuckle (drawn-in or dished), separately for the upper and lower end knuckle. The graphical representation in the module assigns each input quantity to the geometry.
- Define the loading and the type of attachment: Governing are the pressure in the annular space of the jacket and the type of welded connection between knuckle and vessel wall. For a knuckle welded on one side only, the weakening coefficient is reduced to v = 0.7, which lowers the allowable loading accordingly.
- Determine the allowable stresses: From the material strength value at design temperature and the safety factor, the allowable stresses σ<sub>zul</sub> are formed for the individual verification locations — in the knuckle arc, in the adjoining jacket shell and at the transition to the vessel wall.
- Calculate and compare the stresses: The module determines the stresses occurring in the knuckle region and at the vessel transition and compares them with the allowable values. Exceedances indicate that the wall thickness or the knuckle radius must be adjusted.
- Assess both end knuckles: Since the upper and lower knuckles may have different geometries and attachment conditions, the verification is carried out and documented separately for both ends.
Input quantities
| Quantity | Symbol | Unit |
|---|---|---|
| D_a1 | Da1 | mm |
| D_i3 | Di3 | mm |
| e_2 | e2 | mm |
| D_a2 | Da2 | mm |
| D_m1 | Dm1 | mm |
| s_1e | s1e | mm |
| D_i1 | Di1 | mm |
| D_m2 | Dm2 | mm |
| s_2e | s2e | mm |
| D_i2 | Di2 | mm |
| e_1 | e1 | mm |
| s_ke | ske | mm |
| Delta_1 | δ1 | °C |
| K_2 | K2 | N/mm² |
| Delta_2 | δ2 | °C |
| p_1 | p1 | bar |
| K_1 | K1 | N/mm² |
| p_2 | p2 | bar |
| v =0.7 if welded on one side: | v | – |
| Material 1 | 1 | – |
| Material 2 | 2 | – |
Calculated results
| Quantity | Symbol | Unit |
|---|---|---|
| Sig_b1 | σb1 σz1 σv1 | N/mm² |
| Sig_z1 | σb1 σz1 σv1 | N/mm² |
| Sig_v1 | σb1 σz1 σv1 | N/mm² |
| σall1 | S1 | – |
| Sig_b2 | σb2 σz2 σv2 | N/mm² |
| Sig_z2 | σb2 σz2 σv2 | N/mm² |
| Sig_v2 | σb2 σz2 σv2 | N/mm² |
| σall1 | S1 | – |
| σall1 | S1 | N/mm² |
| σall1 | S1 | N/mm² |
Frequently asked questions
What is the difference between a drawn-in and a dished knuckle?
With a drawn-in knuckle, the end of the outer jacket is drawn inwards towards the vessel and welded there; the curve is produced by drawing in the jacket plate. A dished knuckle, on the other hand, is cold or hot formed like the knuckle region of a dished end and has a defined knuckle radius. The two designs differ in their stress distribution and are treated differently in the verification.
Why is the knuckle region loaded more highly than the cylindrical shell?
In the knuckle, the meridional force of the outer jacket is redirected over a small radius and introduced into the vessel wall. In addition to the membrane stresses, this generates considerable bending stresses, similar to the knuckle region of dished ends. The undisturbed cylinder, by contrast, carries almost pure membrane stresses — which is why the knuckle verification must be carried out separately.
What role does the pressure in the annular space play for the inner vessel?
The pressure in the heating or cooling jacket acts on the inner vessel as external overpressure. In addition to the knuckle verification, a stability check against elastic buckling (AD 2000 B6) is therefore regularly required for the inner vessel — particularly in the most unfavourable load case of "jacket pressurized, vessel unpressurized or under vacuum".
When must v = 0.7 be applied?
When the knuckle is welded to the vessel wall on one side only and the weld is not fully penetrated or not executed so that it can be inspected from both sides. The reduced coefficient accounts for the lower load-bearing capacity and inspectability of such attachments; for welds made from both sides with full penetration, a higher value may be applied.