An Efficient Method of Estimating Spectral Fatigue Damage for Low RMS Stress Ranges and Arbitrary Fatigue Curves
Authors: Francis Benjamin , Mair David 
Published: November 2022

Abstract
In recent years API 579 has provided the analyst with a detailed outline of cycle counting techniques for uni-axial loading (the Rainflow Cycle Counting (RCC) method: ASTM Standard No. E1049 three-point method) and multi-axial loading (the Wang-Brown algorithm (WBCC)). However, for vibration-based fatigue, in the absence of any time history at all; it is common in industry to assess fatigue using frequency domain techniques.
The most accurate frequency domain techniques, such as the ever-popular Dirlik’s method, are optimized for a very restricted class of fatigue curve. In closed form Dirlik’s method is only applicable to the class of fatigue curves that exhibit a constant fatigue stress exponent over the number of cycles. In more general settings the validity of the Dirlik probability density is most accurate when the curve power (i.e. ‘m’ where m ≡ h−1 and ‘h’ is found in API 579 Table 14B.3 or ASME VIII Div. 2 Table 3-F.2) is ∼3.0, and is arguably only applicable between 2 to 5.
API 579 Method A provides the ‘smooth bar’ fatigue curves, which are described by a polynomial relationship in which m will often approach 20 at very large numbers of cycles. The alternative technique of API 579 Method C for assessing welds, does comply with the fatigue curve restrictions (i.e. m = 3.13 for ferritic and stainless steel and m = 3.61 for Aluminum). However, this method could arguably be augmented with an increased stress exponent at large numbers of cycles and beyond that an infinite life (e.g. BS EN 13445-3 where N = 5 × 106 is infinite life for monotonic loading and a transition to m = 5 for variable amplitude loading followed by infinite life at N = 108). While it is not the claim of this paper, this would be conceptually consistent with the minimum propagating crack size of fracture mechanics, which is the theoretical basis for the Method C approach.
This paper follows on from previous work (PVP2020-21392 [1]) and presents a detailed algorithm for constructing fatigue curve specific cycle count correlations in the spirit of the Dirlik cycle counting. As such these correlations are primarily sensitive to the spectral moments. These correlations are based on specific functions of the spectral moments, functions that have been found to produce reliably low scatter with respect to RCC. In addition to the traditional 5 spectral moments, we show that, at very large fatigue curve stress exponents, the spectral entropy can be used to enhance the accuracy of the estimated cycle count.
These parameters (5 spectral moments and spectral entropy) are very cheap to calculate in the spectral domain, making this method very computationally efficient. The algorithm also makes it possible for the user to choose the confidence interval on the scatter data. In this way, with some care, the user can naturally account for the inherent hyper-sensitivity of the high cycle part of the fatigue curve to atypically large stress events. Both of these characteristics make this technique suitable for rapid virtual prototyping and subsequent design optimization in real world quick turn-around fitness for service remediation applications.