# Study of the Rotary Bending Fatigue Resistance of 30 MnB5, 41 CrS4 and 30 MnVS6 Steels
## 3 Results and Discussion
### Fatigue Limit  
For B1 and MV steels the run-out data were used (Specimens surviving the \(10^{6}\)-cycle program) to evaluate the limit of resistance to fatigue. This limit represents the load threshold for fatigue stress states, below which it can be assumed that the useful life of the material is infinite. The statistical calculation procedure used was the maximum likelihood method, which is based on the hypothesis of normality in the fatigue limit distribution [48], and that the probability of observing what has been obtained experimentally will be:  
\[\mathrm{V}=\prod_{i=1}^{n}[1-\mathrm{P}(\mathrm{S}_{i},\,\overline{x},\,\, \sigma)]^{\mathrm{I}_{i}}[1-\mathrm{P}(\mathrm{S}_{i},\overline{x},\sigma)]^{ \mathrm{I}_{i}} \tag{4}\] \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Steel** & **Key** & **S** & \(\overline{\mathrm{N}}\) & **D** & \(\mathbf{D_{\overline{N}}}\) \\ \hline \multirow{3}{*}{30 MnB5} & \multirow{3}{*}{B1} & 723 & 7570 & 3646 & 1823 \\  & & 596 & 2558 & 12,927 & 6463 \\  & & 511 & 41,272 & 27,451 & 19,411 \\ \hline \multirow{3}{*}{30 MnB5} & \multirow{3}{*}{B2} & 777 & 4018 & 461 & 266 \\  & & 717 & 5133 & 110 & 78 \\  & & 653 & 12,360 & 2359 & 1668 \\  & & 555 & 3292 & 9081 & 5243 \\  & & 368 & 197,278 & 33,871 & 16,935 \\ \hline \multirow{3}{*}{41 CrS4} & \multirow{3}{*}{EC} & 836 & 3671 & 671 & 387 \\  & & 685 & 14,446 & 911 & 526 \\  & & 585 & 36,579 & 8269 & 4135 \\  & & 493 & 58,825 & 13,922 & 6961 \\  & & 393 & 166,047 & 45,099 & 22,550 \\ \hline \multirow{3}{*}{30 MnVS6} & \multirow{3}{*}{MV} & 581 & 26,758 & 20,361 & 9106 \\  & & 452 & 104,264 & 97,701 & 43,693 \\ \cline{1-1}  & & 388 & 169,139 & 130,124 & 65,062 \\ \cline{1-1}  & & 324 & 198,301 & 221,258 & 110,629 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Statistical values obtained for the experimental fatigue data. Figure 5: Fatigue resistance for each of the four types of steels evaluated. where \(\rm r_{i}\) is the number of runouts at the i-th load level (\(\rm S_{i}\)), \(\rm f_{i}\) is the number of failures observed at that same stress level, \(\rm P\) is the cumulative probability of failure, \(\rm\overline{x}\) is the mean value and \(\rm\sigma\) the standard deviation. The value of \(\rm P\) is calculated according to Equation (5). \[\rm P(S,\rm\overline{x},\sigma)=\int_{-\infty}^{S}\frac{1}{\sigma\sqrt{2\pi}} \exp\left[-\frac{1}{2}\left(\frac{\rm\overline{x}-x}{\sigma}\right)^{2} \right]\rm dx \tag{5}\]  
To maximize the likelihood, the values of \(\rm\overline{x}\) and \(\rm\sigma\) that maximize the probability are calculated, for which Equation (4) is expressed as a natural logarithm. \[\rm{lnV}=\sum_{i=1}^{n}r_{i}\ln(1-P_{i})+f_{i}\rm{lnP_{i}} \tag{6}\]  
where \(\rm P_{i}\) es \(\rm P\) (\(\rm S_{i}\), \(\rm\overline{x}\), \(\rm\sigma\)). Using the Matlab program, the partial derivatives with respect to \(\rm\overline{x}\) and \(\rm\sigma\) are calculated and set equal to zero, thus determining the values of the fatigue limit, \(\rm\overline{S_{i}}\), and the corresponding standard deviation, \(\rm D\)[49]. The results for the 30 MnB5-B1 and 30 MnVS6-MV steels are shown in Table 8.  
According to these data, the fatigue limit that represents the maximum load threshold, which guarantees the infinite life of the material in fatigue, is 326 MPa for quenched and tempered steel (B1) and 233 MPa for microalloyed steel (MV). Comparing the results presented, the fatigue limit of microalloyed steels represents a decrease of 28% in the load capacity in fatigue with respect to quenched and tempered steels. In both cases these values represent approximately 38% of their respective yield strengths. On the other hand, the fatigue index, which represents the relationship between the limit of fatigue resistance and tensile strength, is 45% and 40% for B1 and MV steels, respectively. The resistance reduction is more significant, from this point of view, for B1 than for MV.