Engineering task and calculation objective
The UNRD module assesses the out-of-roundness of cylindrical and conical shells to DIN EN 13445-3, Annex E. Real vessel courses are never perfectly circular: rolling and welding processes leave deviations from the design contour, such as flats or peaking at the longitudinal weld seam, or oval cross-sections. Under internal pressure, these shape deviations generate additional circumferential bending stresses that go beyond the pure membrane stress and can become fatigue-relevant under cyclic loading.
In practice, this calculation is needed when the measured out-of-roundness exceeds the tolerances of EN 13445-4 and it must be judged whether the component can still be used — typically at fabrication acceptance, in reassessments of existing vessels, or when evaluating dents and flats after a damage event. From the radii measured around the circumference, the coefficients of a Fourier series expansion are determined, and from these the mean radius, the out-of-roundness and the resulting additional local stress.
The module thus provides a code-compliant basis for evaluating shape deviations quantitatively instead of rejecting or accepting them wholesale.
Standard and calculation basis: DIN EN 13445-3: 2021-12, Anhang E
Calculation workflow
- Survey of the cross-section: On the affected course, the radii or diameters are recorded at a sufficient number of measuring points distributed evenly around the circumference. The measuring plane is located where the largest shape deviation was observed.
- Fourier analysis of the contour: From the measured values, the coefficients of the harmonic series expansion of the contour deviation are determined. The zeroth term yields the mean radius; the higher terms describe ovality and local flats or peaks (e.g. peaking at the longitudinal weld seam).
- Determine out-of-roundness: From the coefficients, the out-of-roundness is calculated as the characteristic measure of the shape deviation and compared with the fabrication tolerances of EN 13445-4.
- Calculate additional stresses under internal pressure: For each harmonic, the associated circumferential bending stress under the design pressure is determined; superposition with the membrane stress yields the total local stress at the most unfavourable location.
- Assessment: The total stress is assessed against the allowable values (static, and where applicable fatigue to Clauses 17/18 of EN 13445-3). The result is a statement of whether the measured out-of-roundness is acceptable for the intended service.
Input quantities
| Quantity | Symbol | Unit |
|---|---|---|
| 1 | 1 | - |
| 2 | 2 | - |
| 3 | 3 | - |
| 4 | 4 | - |
| 5 | 5 | - |
| 6 | 6 | - |
| 7 | 7 | - |
| 8 | 8 | - |
| 9 | 9 | - |
| 10 | 10 | - |
| 11 | 11 | - |
| 12 | 12 | - |
| 13 | 13 | - |
| 14 | 14 | - |
| 15 | 15 | - |
| 16 | 16 | - |
| 17 | 17 | - |
| 18 | 18 | - |
| 19 | 19 | - |
| 20 | 20 | - |
| 21 | 21 | - |
| 22 | 22 | - |
| 23 | 23 | - |
| Grad | 0 | - |
Frequently asked questions
Why is out-of-roundness a problem under internal pressure at all — doesn't the pressure round out the shell?
The internal pressure does indeed act as a restoring force and reduces the deviation with increasing pressure, but it is precisely this rounding-out that generates bending stresses in the shell wall. In thin-walled shells with a large diameter-to-wall-thickness ratio, these bending stresses can reach several times the membrane stress. Statically this is often still tolerable because of their secondary stress character, but under pulsating pressure loading it quickly becomes fatigue-critical — particularly at a longitudinal weld seam with peaking.
How does ovality differ from local peaking in the assessment?
Ovality (2nd harmonic) is a global, gentle deviation with comparatively moderate bending stresses. Peaking at the longitudinal weld seam is a local deviation with a small radius of curvature; it corresponds to high orders of the Fourier series and, at the same amplitude, produces significantly higher peak stresses directly at the weld — the most fatigue-critical location anyway. Annex E covers both cases via the coefficients of the series expansion.
When may I exceed the tolerances of EN 13445-4?
The fabrication tolerances are acceptance criteria, not absolute limits. If they are exceeded, the code permits an individual calculated assessment: if the calculation to Annex E demonstrates that the additional stresses remain acceptable — including in the fatigue assessment over the intended number of load cycles — the component may be used. The decision must be documented and, where applicable, agreed with the notified body.
Does the calculation also apply to external pressure?
No — caution is required here: under external pressure, the out-of-roundness is not self-correcting but promotes buckling — the shape deviation grows with increasing pressure. This case is covered by the stability checks of EN 13445-3 Clause 8, into which the out-of-roundness enters as an imperfection. Annex E assesses the additional stresses under internal pressure.