The end of the last article on probabilistic analysis described one aspect of a typical technique for assessing fabricated structures under fatigue loading, accounting for failure probability of the welded joints.
Most current approaches for assessing such structures do not employ the sort of probabilistic analysis the last article described, as there is only one in one empirically derived probability distribution parameter into which all the variables that could affect failure are ‘lumped’ together. The parameter is the principal stress at, or near, the weld location. Each type of weld, or other connection where failure could arise, is identified as a specified class, defined in standards such as BS 7608, 8118, or 5500, from which the corresponding S-N curve for various levels of failure probability can be identified. Basic calculations or conventional FEA can be deployed to predict the weld stresses, which are then compared with the allowable value for a given fatigue life and probability of failure from the standards.
A stress corresponding to a low failure probability of, say 1%, seems a sensible allowable value by which to assess the welds in the structure under fatigue loads. In theory, this means that a structure could be qualified or deemed acceptable if every weld in it had a stress corresponding to just below a 1% failure probability. The point touched on briefly in the July 2005 article was that, if there are several such structures, each with a hundred welds, there will be a mean of one failure per structure. The only reason this may not happen in practice is because the stresses in most welds in the structure do not approach the 1% failure stress.
This point nonetheless highlights a limitation of attempting to qualify a structure in this way. It could be argued that the number of welds on which to base failure probability could be the number, not on a single structure, but on all structures built. However, reducing the allowable stress to a corresponding failure probability for all structures would result in impossible designs in many cases.
The NAFEMS publication ‘Procedural Benchmarks for Common Fabrication Details in Plate/Shell Structures ’ discusses problems associated with weld modelling and is recommended to anyone wishing to establish a sensible method for representing and qualifying fabricated structures with FEA. One issue this publication highlights is the disparate responses which several industry experts gave to a theoretical fabrication FE problem, posed by the authors for research purposes. This should serve as an effective reminder that FEA results in real applications are not automatically correct, and it can be detrimental to regard them as such. The fact that answers from experts may disagree is not too worrying providing the analyses satisfy some important conditions – to be valid, a modelling strategy should be conservative, consistent and correlated (with structural tests and/or failures in the field).
The allowable failure probability stress approach described above is likely to be quite conservative. Coupling this with the generally accepted conservatism in many published weld strength standards can give premature failure predictions.
A refinement to the above approach involves identifying which welds in the structure are most critical; the implication being that many joints in the structure could develop cracks without affecting structural performance. In other words, redundancy in the structure would allow some welds to fail without leading to unstable collapse. A method for assessing this would be to quantify the critical regions in a structure by sequentially removing individual finite elements in which the stress has exceeded the fatigue limit for the particular class of joint represented. Implementation can take the form of manually removing elements or connections around the failed joint and re-running the analysis. This approach is useful for a single problem region, but for a full understanding of redundancy in a structure, automated removal of failed elements from the model during solution is required.
Although for high cycle fatigue, a linear material model would be sufficient in the analysis, consideration should be given to buckling and large displacements that may require a non-linear analysis in some form.
Such a technique can quantify the relative significance of predicting localised failure in different parts of the structure, using the failure probability fatigue stress. Some areas are likely to be critical only under one loadcase, so it is important to consider all loading scenarios. This approach shares some similarities with plastic limit-state analysis, used for ultimate loads, which will be described in a future article.
Having established the levels of redundancy (‘criticality’ of a joint could be seen as the opposite of redundancy) this can be used to relax the allowable failure probabilities for some locations within the structure. Allowable stresses can be raised for regions where local weld failure has been shown to be self-contained. The increase in allowable stress should not be seen as inferring this region of the structure has a lower allowable fatigue life of curse, but that an individual failure in this region of the structure is tolerable because of local redundancy.