Engineering task and calculation objective
The S2 module performs the detailed fatigue analysis (analysis of fatigue behaviour under cyclic loading) to AD 2000-Merkblatt S2 of the German AD 2000 pressure vessel code. Unlike the simplified method of S1, S2 works with the actual stresses at the governing component region: from the extreme values of the cyclic stresses or of the sums of principal stresses, the equivalent stress range and the mean equivalent stress are formed and assessed against the S-N (Woehler) curves of the code for welded and non-welded component regions.
The module calculates in both directions: for a known number of load cycles it delivers the allowable stress range, and for a known stress range the allowable number of load cycles. Overelastically stressed regions are covered as well – there, the governing equivalent stress range is determined from the elastically calculated range via a plasticity correction, so that notch locations with local yielding remain assessable.
The verification to AD 2000 S2 is needed when the simplified verification to S1 is insufficient or too conservative: at high cycle counts, under thermal cycling, with superimposed external loads, or when real stress ranges are available from a detailed calculation (e.g. FEA) and are to be exploited economically.
Standard and calculation basis: AD 2000 S2: 2012-07
Calculation workflow
- Determine the governing stress cycles: For the component region under consideration, the maximum and minimum cyclic stress, or the maximum and minimum of the sum of principal stresses, are determined from the load states of the cycle – from hand formulas of the codes or from an FE analysis.
- Form stress range and mean stress: From the extreme values, the equivalent stress range and the mean equivalent stress of the cycle follow. For non-welded regions, a compressive mean stress improves the endurable stress range, whereas for welded regions it is largely disregarded because of welding residual stresses.
- Correct for overelastic loading: If the elastically calculated stress range exceeds the yield limit of the cycle, the governing equivalent stress range is increased via the plasticity correction of the Merkblatt in order to represent the actual strain range in the overelastically stressed region.
- Classify the component region: The region is classified as non-welded or welded (with the associated weld class); influence factors for wall thickness, temperature and surface condition reduce the endurable stress range.
- Carry out the verification: Depending on the task, either the allowable stress range for the required number of load cycles or the allowable number of load cycles for the existing stress range is determined from the S-N curve. With several cycle types, the total damage is assembled via linear damage accumulation.
Input quantities
| Quantity | Symbol | Unit |
|---|---|---|
| Constant acc. table 4 | B1 | - |
| DeltaS12max | Δσ12max Δσ12min | N/mm² |
| DeltaS12min | Δσ12max Δσ12min | N/mm² |
| DeltaS23max | Δσ23max Δσ23min | N/mm² |
| DeltaS23min | Δσ23max Δσ23min | N/mm² |
| DeltaS31max | Δσ31max Δσ31min | N/mm² |
| DeltaS31min | Δσ31max Δσ31min | N/mm² |
| Mean stress factor | fM (19 & 21) | - |
| otherwise: here: | fM* = 1 , fM* | - |
| Correction factor for surface influence | fo (15 & 16) | - |
| Temperature influence factor | fT* (24 & 25) | - |
| Amplification factor | ke (8 & 9) | - |
| Wall thickness factor | fd (17 & 18) | - |
| Amplification factor for thermal stresses beyond yield | kv (11) | - |
| Number of load cycles | Nall | - |
| Weld class (from 0 to 3) | K | - |
| Tensile strength at room temperature | Rm | N/mm² |
| 0.2% - yield strength at T* | Rp02/T* | N/mm² |
| Surface roughness | Rz | µm |
| Maximum of the sum of principal stresses | σijmax | N/mm² |
| Minimum of the sum of principal stresses | σijmin | N/mm² |
| Maximum cyclic stress | σmax | N/mm² |
| Minimum cyclic stress | σmin | N/mm² |
| Mean equivalent stress | σvquer | N/mm² |
Frequently asked questions
What is the practical advantage of S2 over S1?
S1 estimates the stress range from the pressure cycle in a blanket manner and is deliberately conservative. S2 uses the actual equivalent stress range of the governing detail and accounts for the mean stress influence and the plasticity correction in a differentiated way. The same design therefore often obtains a considerably higher allowable number of load cycles under S2 – at the price that the stresses at the detail must be known.
Why is the mean stress treated differently at welds than in the parent material?
Welds contain tensile residual stresses up to the level of the yield strength. As a result, the actual cycle oscillates in the high tensile range regardless of the calculated mean stress, so a favourable (low or compressive) mean stress may not be credited. For non-welded regions with low residual stresses, on the other hand, the mean stress influence may be accounted for via the mean equivalent stress.
What does the plasticity correction mean for overelastic loading?
At sharp notches, the elastically calculated stress range can exceed twice the yield limit; the material then yields locally, and the actual strain range is larger than the elastic calculation indicates. The correction increases the governing stress range accordingly, so that the assessment against the strain-based S-N curve stays on the safe side. Without this correction, the fatigue life in the low-cycle regime would be overestimated.
How are several different load cycles combined?
For each cycle type (e.g. start-up/shutdown, pressure fluctuations in operation, temperature transients), the allowable number of load cycles is determined separately. The partial damages n/N are summed linearly (Miner's rule); the sum must remain less than or equal to 1. Correct cycle counting – for instance recognizing that a large cycle can enclose several small ones – is a frequent source of error here.