Fatigue phenomena when designing industrial steel structures – Part 2

Predicting the rate of crack growth during load cycling helps replace damaged elements before the crack reaches a critical length. The crack growth rate can be correlated with the cyclic variation in the stress intensity factors:

da/dN= A.DK^{m [3]}

where da/dN is the fatigue crack growth rate per cycle, DK = Kmax – Kmin is the stress intensity factor range during the cycle, and A and m are parameters that depend on the material, environment, frequency, temperature and stress ratio. The equation is also known as Paris’ law (Fig. 1).

S-N curves (Wöhler curves) are charts with a constant sloping linear line, plotted using a log-log relationship (Fig. 2). S-N curves are used to help evaluate the fatigue life of a part during the design stage. S-N curves are based on the part’s stress range, combined with the cycles to failure and material properties of the part. In high cycle fatigue mode, it is common to apply the S-N curve to determine element behaviour and the impact of cyclic loads on the element. Over time, these models have been developed to describe the relationships between the cyclic load and the number of cycles to failure.

To estimate an S-N chart plot, data points from fatigue tests regarding cycles to failure must be inserted. The relation between cycles to failure and the applied force is viewed as linear since the plotting method is usually listed as log vs. log, provided from the regression model.

Most of today’s design code restricts the fatigue upper-stress limit to a specific number and defines different log-log stress-loading number curves for different loading conditions. The S-N curve should more properly be a stress-cycle-probability (S-N-P) curve to capture the probability of failure after a given number of certain stress cycles.

Damage in fatigue is a process of cycle-by-cycle accumulation in a material undergoing fluctuating stresses and strains. The fatigue life may be calculated based on the S-N fatigue approach under the assumption of linear cumulative damage (Palmgren-Miner rule), especially when the applied fatigue load is not constant. When the long-term stress range distribution is expressed by a stress histogram, consisting of a convenient number of constant stress range blocks (Δσ_i), each with a number of stress repetitions (n_i), the fatigue criterion reads:

Where:

D = accumulated fatigue damage

a = intercept of the design S-N curve with the log N axis

m = negative inverse slope of the S-N curve

k = number of stress blocks

n_i = number of stress cycles in stress block i

N_i = number of cycles to failure at constant stress range Δσ

i = usage factor

When applying a histogram to express the stress distribution, the number of stress blocks (k) should be large enough to ensure reasonable numerical accuracy.

If fatigue-related problems are encountered, a repair or retrofit strategy must be employed to rectify the issue and help prevent additional problems. While the specifics of the repair strategy are dependent upon the nature of the detail and problem, there are several general repair and retrofit strategies that are common for virtually all fatigue details. These general strategies include understanding the source of cracking, designing the repair detail and validating the repair detail.

Although the specifics of these three steps are dependent upon the unique fatigue characteristics of the affected element, each step is essential for any successful repair or retrofit strategy. In addition, validating the repair solution to prevent creating a new problem, or intensifying the crack propagation in other parts of the element, is very important to approve the repair detail (Fig. 3 and 4).

Detailed finite element analysis of structures may make it difficult to evaluate the “nominal stress” to use together with the S-N curves, as some of the local stress due to detail is accounted for in the S-N curve. In many cases, it may be more convenient to use an alternative approach to calculate fatigue damage when local stresses are obtained from finite element analysis. In developing the finite element model, the mesh size needs to be fine enough to properly predict the hot spots in the element subject to fatigue. It is also important to have a continuous, and not too steep, change in the density of the element mesh in the areas where the hot spot stresses are to be calculated. In addition, the size of the model should be so large that the calculated results are not significantly affected by assumptions made for boundary conditions and the application of loads.

After calculating the hot spots, stress points within a distance of 0.5 t and 1.5 t away from the hot spot need to be evaluated, where “t” is the plate thickness (plate thickness at the weld toe where the hot spot at the connection point is evaluated). These locations are also denoted as stress read-out points. These hot spot stress values need to be compared with the specified fatigue upper limit in a linear analysis (Fig. 5).

In another approach, and in case the need arises to model stationary discontinuities, such as a crack, the conventional finite element method requires the mesh to conform to the geometric discontinuities. Therefore, considerable mesh refinement is needed in the crack tip area to capture the singular asymptotic fields adequately.

Modelling a growing crack is even more cumbersome because the mesh must be updated continuously to match the geometry of the discontinuity as the crack progresses. The extended finite element method (XFEM) alleviates the shortcomings associated with meshing crack surfaces. It is an extension of the conventional finite element method, which allows local enrichment functions to be easily incorporated into a finite element approximation. The presence of discontinuities is ensured by the special enriched functions in conjunction with additional degrees of freedom (Fig. 6). However, the finite element framework and its properties, such as sparsity and symmetry, are retained.

We have now explained many influences on steel structure fatigue life and provided strategies as well as ways to calculate fatigue life. Of course, there is so much more involved in designing steel structures. We invite you to contact our experts for more information or for support in solving any issues you may be encountering with your steel structures. We are here and ready to help.

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