## 1 Introduction  
For the A572 Gr. 50 steel material, a probabilistic S-N curve does not exist. The main goal of the S-N curves analysis is related to the strength material behavior and its correlation with the defined stress ratio [1; 2; 3; 4]. Structural elements are subjected to a range of stress values, so to determine the probabilistic S-N curve, any material steel is necessary [1; 2; 3; 5; 6]. Despite this, currently, in structural design, two different approaches have been used, and they are: (1) the application of a failure theory to determine if the designed element is safe or not; and (2) the fatigue approach [1; 7]. Unfortunately, these approaches are not very efficient for predicting the reliability of a structural element. This mainly because, while the failure theory does not consider the time in the analysis, as it is the case of the Goodman, Elliptical and Soderberg failure theories given by [PERSON] and [PERSON] [8], in the fatigue strength analysis, the S-N curve only represents the strength average [9; 10]. Therefore, neither the failure theory nor the fatigue analysis is efficient to perform a probabilistic analysis [2; 11; 12]. Thus, in this paper based on the A572 Gr. 50 steel true stress-true strain analysis, the method to determine the probabilistic percentiles of the S-N curve is formulated, based on the two parameter Weibull distribution. Therefore, the efficiency of the proposed method to determine the probabilistic percentiles is based on the fact that the Weibull shape \(\beta\) and scale \(\eta\) parameters are both directly determined from the maximum and minimum material strength \((\sigma_{1},\sigma_{2})\) values, which are given from the true stress-true strain analysis [13]. Moreover, the analysis is based on the true stress-true strain approach, because the failure of a structure does not depend only on the applied axial stress, but it also dependson the instantaneous area (\(A_{i}\)) and on the instantaneous elongation (\(L_{i}\)) of the element [14]. Thus, to consider \(A_{i}\) and \(L_{i}\) in the analysis, it was performed based on the true stress-true strain analysis. Accordingly, due to the proposed probabilistic method, the Weibull \(\beta\) and \(\eta\) parameters are both determined directly from the maximum \(\sigma_{1}\) and minimum \(\sigma_{2}\) strength values, then based on the relationships between (\(\beta\) and \(\eta\)) with (\(\sigma_{1}\) and \(\sigma_{2}\)), and the corresponding log-mean (\(\mu_{x}\)) and the log-standard deviation (\(\sigma_{x}\)) values were both also estimated. Then, these \(\mu_{x}\) and \(\sigma_{x}\) values were used to determine the probabilistic S-N percentiles of the A572 Gr. 50 steel material. Fortunately, since the Weibull \(\beta\) and \(\eta\) parameters are directly fitted from the maximum and minimum strength (\(\sigma_{1}\) and \(\sigma_{2}\)) values, then \(\beta\) and \(\eta\) always represent them, implying the derived \(\mu_{x}\) and \(\sigma_{x}\) values are both unique [13]. Therefore, by applying the proposed methodology, \(\mu_{x}\) and \(\sigma_{x}\) are both unique, so they can be used to determine the corresponding S-N percentile for any other material where \(\sigma_{1}\) and \(\sigma_{2}\) are known. However, it is important to note that, if the proposed method is going to be used to determine the probabilistic percentiles for another material, we must first be sure the used \(\sigma_{1}\) and \(\sigma_{2}\) stresses values are those values that generate the failure in the material, or equivalent if the used data is a failure testing data [4; 6]. On the other hand, it is important to highlight that, according to the Canadian standard CSA G40.21, the A572 grade 50 steel is somewhat equivalent to the CSA G40.21 Gr. 350W steel. Thus, based on the excellent experimentation performed by [PERSON], [PERSON] and [PERSON] [15], in this paper the probabilistic A572 Gr. 50 steel S-N curve is determined by using the true stress-true strain experimental data determined from the CSA G40.21 Gr.