• the design resistance, defined as ratio of the resistance (of the specified model) and the appropriate partial safety factor, where the resistance is determined with (specified) material strength parameters
• the resistance of a corresponding model but with material strength parameters replaced by the corresponding design material strength parameters, obtained by dividing the material strength parameters by the partial safety factors of the resistance.
Usually the difference is very small, and it is recommended that the resistance in the second route may be used as design resistance; this recommendation has already been used in the examples. Performing a real GPD-check only, e.g. no limit actions are determined, the admissibility of specified actions acting on a specified structure is shown, if the maximum absolute value of the principal strains does not exceed 5%, under usage of a linear-elastic ideal-plastic material law in the FE-calculations.
If the FE-model consists of shell elements, usually the mid-surface of the structure is modelled. Therefore, the practical relevance of the results in points (nodes) on the intersection curve of two shells depends on the kind of geometry of the structure.
For example, for (cylindrical) main shell – (cylindrical) nozzle intersections the results in nodes of the intersection curve should not be used, since they do not correspond to points of the real structure with the real geometry under consideration of the reinforcement due to the weld - see Figure 3.1. Therefore, if usage of solid elements or of submodelling is not possible, the results in the so called “evaluation” cross-sections should be used for the determination of the 5% principal strain limit.
Figure 3.1: Shell intersection
As stated in the principle in prEN 13445-3 Annex B.9.2.1, the design resistance (limit action) should be obtained from calculations with proportional increase of all design actions. The limit action is independent of the action history, but with strain limitation, i.e. if the strain limitation governs, the limit will depend on the action’s history. In the case of constant moment load and varying internal pressure load (examples 3.1 and 3.2), where the strain limitation does govern, the deviation from the standard’s procedure – proportional loading – is the only sound one, the moment being constant during all action cycles.
3.3.2 Check against Progressive Plastic Deformation (PD)
3.3.2.1 General
Again, two different ways of performing calculations to check PD are possible: First, if the action cycles for a given structure are specified, a real check can be performed, showing that the actions are admissible (or not) under application of prEN 13445-3 Annex B. This procedure was applied within the PD-check of the examples 2, 5 and 6. The second possibility is to calculate the limit
actions (in the sense of PD) for a given structure, and afterwards using these limit actions to determine the maximum admissible actions according to prEN 13445-3 Annex B. The latter procedure will be useful, if the structure is to be used with extreme actions. This procedure was applied within the PD-check of the examples 1.1, 1.2, 1.3, 1.4, 3.1, 3.2 and 4.
Principally, the application rule in prEN 13445-3 Annex B.9.3.2 corresponds to the well-known criterion for the sum of the primary and secondary stresses in stress categorisation – the (often) so- called “3f–criterion” (where f stands for the allowable stress). This criterion is an upper bound criterion for shakedown, and, therefore, the requirement given in this application is only a necessary and not a sufficient condition for the fulfilment of the principle – prEN 13445-3 Annex B.9.3.1 [1],
[2]
. Usage of this application rule could be the easiest way of applying the check against PD if only one action is considered, but if more than one action and/or additional thermal stresses have to be considered, its usage could be difficult and uncertain. Therefore, and for guideline purposes, usually another possibility of fulfilling the principle – Melan’s shakedown theorem – is employed.
Melan’s shakedown theorem states [1], [2]:
The structure will shake down for a given cyclic action, if a time-invariant self-equilibrating stress field can be found such that the sum of this stress field and the cyclically varying stress field determined with the (unbounded) linear-elastic constitutive law for the given cyclic action is compatible with the yield condition – the equivalent stress nowhere and at no time exceeds the (yield) material parameter.
Using this theorem the principle is fulfilled, since Progressive Plastic Deformation (PD) and Alternating Plasticity (AP) are the two possible inadaption modes if a structure does not shake down under a cyclic load set.
One advantage of using Melan’s theorem, in comparison with the application rule, is, that the admissibility is shown for all points of the structure, if the determined self-equilibrating stress field is superposed onto the linear-elastic stress fields of the different states using the postprocessor of the FE-software.
A problem arises if the maximum allowable action given by the shakedown limit is lower than the one resulting from the check against GPD. In this case, detailed examination of the structure’s behaviour under a cyclic loading with the maximum allowable action according to the check against GPD, i.e. determination of the structures inadaption mode – progressive plastic deformation and/or alternating plasticity - would be necessary. If it can be shown that the inadaption mode is given by alternating plasticity only, the action would be admissible. Unfortunately, the possibility of such examinations is restricted, on one hand because of the hardware and time limits for performing cyclic calculations and on the other hand because of a lack of generally applicable theorems. Determining generally applicable theorems in this field is a present topic of research, and, therefore, they should be available in the future.
3.3.2.2 Problems in performing the shakedown check using shell elements in the FE-model [3]
Usage of stress resultants of technical theories of structures, i.e. generalised stresses, to verify that progressive plastic deformation (PD) does not occur is often not appropriate, because
• the validity of the corresponding theorem[4] seems to be restricted to passive (unloading) processes [2], [5], [6];