Engineering task and calculation objective
The ENAF module determines the allowable external pressure for cylindrical shells with out-of-roundness exceeding the tolerance, to DIN EN 13445-3 Annex F. The regular external pressure rules of clause 8 assume that the departure from the ideal circular shape, measured from the true centre, does not exceed 0.5% of the radius. If this tolerance is exceeded, for example on existing vessels, after repairs, or due to manufacturing-related shape deviations, Annex F allows the allowable external overpressure to be calculated on the basis of the actually measured vessel shape, instead of rejecting the vessel outright or re-rounding it at great expense.
The method evaluates a roundness survey with measuring points distributed evenly around the circumference: the measured radii are decomposed into harmonic components of the departure from the mean circle by Fourier analysis. For each harmonic number, the pressure is checked at which the combination of elastic buckling behaviour and the additional bending stress caused by the initial imperfection reaches the elastic limit of the shell. The smallest of these values, compared with the lower bound estimate of the collapse pressure and the allowable pressure of the ideally round shell, determines the allowable external overpressure.
Typical applications are vacuum vessels, jacketed vessels with heating or cooling jackets, and the reassessment of older vessels after a survey during periodic inspections. The measurement data can be imported from the UNRD module.
Standard and calculation basis: DIN EN 13445-3: 2021-12 Anhang F
Calculation workflow
- Record the roundness survey: The basis is a measurement of the radii r(Φ) at a sufficient number of evenly distributed measuring points at the governing cross-section; the angular interval and the number of measuring points enter the evaluation. The measurement data can be imported from a chapter of the UNRD module.
- Define geometry and material properties: Required are the mean radius and the minimum calculation thickness of the cylindrical shell (after deducting corrosion and manufacturing allowances), the unsupported shell length between effective supports taking into account the depths of adjacent dished ends, as well as the modulus of elasticity, Poisson's ratio and the 0.2% proof strength at design temperature; adjusted elastic limits apply to austenitic steels.
- Fourier analysis of the shape deviation: From the measured radii, the Fourier coefficients of the shape deviation are determined via the sums of the sine and cosine components. This yields the maximum deviation from the mean circle, with the initial imperfection broken down by harmonic number.
- Calculate the pressure for each harmonic number: For each harmonic number, the pressure is determined at which the circumferential membrane stress, plus the bending stress caused by the harmonic imperfection as the pressure approaches the buckling pressure, reaches the allowable elastic limit of the shell. The smallest value over all considered positions and harmonics governs.
- Establish the allowable external pressure: The allowable external overpressure results from the minimum of the harmonic checks, limited by the lower bound estimate of the collapse pressure of the cylindrical shell and by the allowable pressure of the same shell with out-of-roundness below 0.5%. A safety factor according to equation (8.5.2-8) can be taken into account.
Input quantities
| Quantity | Symbol | Unit |
|---|---|---|
| Fourier series coefficients | ancyl | - |
| Fourier series coefficients | bncyl | - |
| Allowable external pressure | Pra | MPa |
| Allowable pressure for an otherwise similar cylinder within 0.5% tolerance | Pa | MPa |
| Lower bound estimate of the collapse pressure of cylinder | Pq | MPa |
| Harmonic value | ncyl | - |
| Allowable pressure | P | MPa |
| Mnimum calculation thickness | ea | mm |
| Modulus of elasticity | E | N/mm² |
| External heights of dished ends | h' | mm |
| External heights of dished ends | h'' | mm |
| Unsupported length of the shell | L | mm |
| Cylinder length between tangent lines | Lcyl | mm |
| Number of measuring points | N | - |
| 0.2% proof strength of the shell | Rp0,2/T | N/mm² |
| Mean radius of cylindrical shell, spherical shell or dome | Rm | mm |
| Maximum deviation from the mean circle | wmax | mm |
| Sum ncyl | Σncyl | - |
| Sum Rsin | ΣRsin | - |
| Sum Rcos | ΣRcos | - |
| Poisson's ratio | ν | - |
| Angular interval of the measurements | Φ | ° |
| Parameter in calculation | Z | - |
| Material | WNr | - |
Calculated results
| Quantity | Symbol | Unit |
|---|---|---|
| Allowable pressure at position r for ncyl | Pm(ncyl) | MPa |
| Mean elastic circumferential strain at collapse | ε | – |
| Nominal elastic limits for the shell | σbr | N/mm² |
| Measurement of radius at position rΦ | RrΦ | mm |
| Number of the measurement | r | - |
Frequently asked questions
At what point do I have to calculate to Annex F instead of clause 8?
The external pressure rules of clause 8 apply to shells whose departure from the mean circle, measured from the true centre, does not exceed 0.5% of the radius. If the measured out-of-roundness lies above this, the allowable pressure must be determined to Annex F on the basis of the real geometry. The result is always less than or equal to the value for the shell within tolerance.
Why is the shape deviation decomposed into Fourier series?
The buckling behaviour of a cylindrical shell under external pressure depends on the wave number: each harmonic imperfection grows preferentially in its own buckling mode under pressure, and does so more strongly the closer the pressure is to the elastic buckling pressure of that wave number. Only the decomposition into harmonics makes it possible to pair each wave number with its associated buckling pressure and to find the truly critical combination. A single figure such as the maximum ovality would not correctly capture the behaviour of higher wave numbers.
What requirements does the method place on the measurement?
The measuring points must be distributed evenly around the full circumference, and their number must be sufficient to resolve the relevant harmonics; by the sampling theorem, at least twice as many measuring points as the highest wave number to be evaluated are needed, and the standard requires a minimum number. Measurement is taken at the most unfavourable cross-section, typically where the visible shape deviation is greatest. Systematic measurement errors, for example from tilting of the measuring equipment, distort the low wave numbers in particular.
Can I use Annex F to qualify an out-of-round vessel for a higher pressure than clause 8 allows?
No. The allowable pressure determined to Annex F is explicitly limited by the value obtained for the same shell with out-of-roundness within the 0.5% tolerance. The method serves to assess shells with shape deviations realistically and, where appropriate, to keep them in service at a reduced allowable pressure, not to outdo the regular design rules.