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Featured Collections. Many equations have been proposed to describe the S-N curve. In the absence of tested data, the fatigue strength coefficient, , and the fatigue strength exponent, b , can be estimated using the monotonic tensile properties or Brinell hardness [ 4 , 9 — 12 ]. This method will be hereafter referred to as indirect method. Based on the indirect method, the fatigue strength coefficient and the fatigue strength exponent can be predicted by six existing estimation methods [ 4 , 9 — 12 ] firstly, and then the fatigue limit is calculated using equation 6.
The six existing estimation methods considered in this paper are listed in Table 4. As analyzed above, the direct methods proposed by Roessle and Fatemi [ 4 ] give better predictions than the methods shown in References [ 1 , 3 ]. Therefore, the direct methods proposed by Roessle and Fatemi [ 4 ] are evaluated by using the tested data of steels.
For comparison purpose, the same tested data are also used to evaluate the aforementioned six indirect methods Figure 4. As for the HB-based direct method, it may be said to provide the best estimation results among all the eight estimation methods.
Most of the work done in comparing both the direct and indirect methods use the scatter band criterion where the percentage of predicted fatigue limit versus experimental data falling within a scatter band is compared Table 5. It can be seen that the HB-based direct method has the most data with all deviation factors. For example, In contrast, the best results predicted by MM-based method among the considered indirect methods from the literature are It should be noted here that one of the shortcomings of the scatter band criterion is that the trend of the correlation between prediction and experiment cannot be captured.
To overcome this drawback, the standard deviation criterion is introduced to compare different estimation methods: where n p is the number of experimental data.
It can be seen that the bigger the S D is, the more scattered the predictions are. The comparisons using the standard deviation criterion are also tabulated in Table 5. It is shown that the HB-based direct method provides the best predictions.
In order to have a better quantitative comparison, the comparisons for different estimation methods are also given in the format of fitted probability density functions PDF for the predicted errors. For the statistical study, each data set is sorted in ascending order, and then each data point is associated with its mean rank.
After that, each data set is fitted using the selected probability distribution. Here, the log-logistic distribution is selected based on the recommendation of Meggiolaro and Castro [ 12 ]. The cumulative distribution function of the log-logistic distribution function is given by the following equation [ 13 ] where p is a shape parameter and a is a scale parameter.
The corresponding probability density function PDF of the log-logistic distribution is given by [ 13 ]. The cumulative probability P f of estimated data is evaluated by the mean rank method as where i is the rank number and I is the total number of considered steels. Optimum values of the parameters a and p are obtained by applying the least squares method [ 14 ].
The curvilinear lines indicate the results of fitting Figure 6. The optimum values of a and p are also presented in Figure 6. Then, the best-fitted distributions of different estimation methods are shown in Figure 7. The HB-based direct method provides reasonable predictions while somewhat underestimations are shown in Figure 7. In contrast to the considered estimation methods, the HB-based direct method on the basis of Brinell hardness provides the best predictions for the fatigue limits of most of steels.
Although the HB-based direct method presents the best predictions among the considered estimation methods, it is known that, for advanced steels, the predictability of the fatigue limit based on monotonic properties often fails.
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