Engineering task and calculation objective
Once the diameter ratio of a cylinder exceeds the range of validity of AD 2000-Merkblatt B1, the stress distribution across the wall is no longer approximately constant: at the inner fibre, significantly higher stresses occur than at the outside. Such thick-walled cylindrical shells — typical of high-pressure apparatus, reactors in the chemical industry, hydraulic components and ultra-high-pressure piping — are calculated to AD 2000-Merkblatt B10 of the German AD 2000 pressure vessel code.
Module B10 determines the required wall thickness of thick-walled cylinders under internal overpressure on the basis of the stress theory for thick-walled tubes (Lamé equations with an equivalent stress hypothesis). It also calculates the equivalent stresses at the inner and outer fibre resulting from the internal pressure, as well as the thermal stresses that arise when there is a radial temperature gradient across the wall — for thick walls and rapid temperature changes, frequently a governing part of the loading.
In practice, this verification is needed whenever high pressure meets a comparatively small diameter — for example in synthesis apparatus, autoclaves or injection systems — or when start-up and shutdown operations with steep temperature ramps have to be evaluated.
Standard and calculation basis: AD 2000 B10: 2000-10
Calculation workflow
- Define the configuration and geometry: The case under consideration is selected via the type option; then the inside or outside diameter and the existing or required wall thickness are entered. The diameter ratio Da/Di decides whether B10 is applicable at all.
- Determine material values and allowable stress: From the strength value K at design temperature and the safety factor S follows the allowable stress. For the thermal stress calculation, the modulus of elasticity, the coefficient of thermal expansion and Poisson's ratio of the material are added.
- Determine the required wall thickness from the internal pressure: Using the B10 relation for thick-walled cylinders, the wall thickness is determined at which the equivalent stress at the most highly loaded inner fibre does not exceed the allowable stress. Allowances for tolerance and corrosion are added.
- Calculate the equivalent stresses at the inner and outer fibre: For the executed wall thickness, the pressure-induced stress components (tangential, radial and axial stress) are evaluated at the inner and outer fibre and combined into the equivalent stress.
- Superimpose the thermal stresses: For a radial temperature gradient — for example due to heating, cooling or temperature changes during start-up and shutdown — the module calculates the associated thermal stresses at both fibres and superimposes them on the pressure stresses to obtain the total loading.
Input quantities
| Quantity | Symbol | Unit |
|---|---|---|
| Outside diameter | Da | mm |
| Inside diameter | Di | mm |
| Design pressure | p | bar |
| Nominal design strength | K | N/mm² |
| Safety factor | S | – |
| Wall thickness manufacturing tolerance | c1 | mm |
| Corrosion / erosion allowance | c2 | mm |
| Actual wall thickness | se | mm |
| Modulus of elasticity | E | N/mm² |
| Design temperature | T | °C |
| Material | Wk | – |
| Coefficient of linear thermal expansion | α | 1/°C |
| Inside temperature | ϑi | °C |
| Outside temperature | ϑa | °C |
| Poisson's ratio | ν | – |
| Innen | Innen | – |
| außen | außen | – |
| Bauform | Bauform | – |
Calculated results
| Quantity | Symbol | Unit |
|---|---|---|
| Required wall thickness | s | mm |
| Maximum stress at internal surface | σi | N/mm² |
| Maximum stress at external surface | σa | N/mm² |
| Equivalent stress of internal surface | σvi | N/mm² |
| Equivalent stress of external surface | σva | N/mm² |
| Thermal stress of internal surface | σwi | N/mm² |
| Thermal stress of external surface | σwa | N/mm² |
| Auxiliary value | A | – |
| Auxiliary value | B | – |
| Ratio | Da / Di η | – |
| Allowable stress for required thickness | K/S | N/mm² |
| Allowable pressure | pall | bar |
Calculation options
Bauform
Walls without significant temperature difference · Walls with significant temperature difference
Frequently asked questions
From when on is a cylinder considered thick-walled?
The usual limit is a diameter ratio Da/Di of about 1.2: below it, the membrane formula to AD 2000 B1 gives sufficiently accurate results; above it, the non-uniform stress distribution across the wall must be taken into account to B10. At very high pressures, the B10 verification is worthwhile even below this limit, to know the actual inner-fibre loading.
Why is the inner fibre the critical location?
According to the Lamé equations, the tangential stress and the radial stress difference reach their maximum at the inner wall; the equivalent stress decreases towards the outer wall. In a thick-walled cylinder, the inner-fibre stress can be several times the mean membrane stress — designing on mean values alone would therefore be unsafe.
When do thermal stresses become relevant for design?
Thermal stresses grow with the temperature difference across the wall, and this difference grows with the wall thickness: thick-walled components are precisely the ones affected. Critical are rapid start-up and shutdown operations and thermal shocks (e.g. cold water injection into hot components). Since thermal stresses are constraint stresses, they are classified differently in the assessment from primary pressure stresses — for frequent temperature cycles, a fatigue analysis (AD 2000 S2) is additionally indicated.
Cooling or heating — where do tensile and compressive stresses arise?
If the inner wall is hotter than the outer wall (heating from the inside), the inner fibre wants to expand more and is thereby loaded in compression, the outer fibre in tension. When cooling from the inside, the signs reverse: tensile stresses then arise at the inner fibre, which superimpose unfavourably on the pressure-induced tensile stresses — this load case is frequently design-governing.