## 1 Introduction  
Fatigue is one of the principal causes of failure in mechanical and structural elements. Determining an equivalent slope of the stress-cycles S-N curve is essential when an element presents distinct fatigue slopes. Then, the equivalent slope is used in the Basquin model to characterize the bilinear fatigue behavior as in the ASTM 739 standard [1] and the Eurocode 3 norm [2]. Unfortunately, because the Basquin model is efficient only for linear behavior, its use to represent bilinear behavior is inefficient [3]. In S-N curves, bilinear behavior represents a transition between two competing failure modes or a step of variant stress. In the fatigue frame, this transition is called a knee-point [4]. Although its consideration improves the analysis, it does not incorporate the probabilistic behavior [5].  
The S-N curve is used as an input to evaluate the reliability and quality of mechanical elements subjected to stress. Since the S-N curve represents median stress values and mechanical elements fail with a certain degree of random dispersion, it is practical to consider failure percentiles around the median stress value in the analysis. These percentiles are known as the P-S-N field, and they provide us with a probabilistic representation that aids in decision-making for fatigue analysis. Therefore, given the inherent randomness of fatigue, it is crucial to develop a probabilistic P-S-N curve method to obtain more reliable failure predictions [6].  
During the last few decades, several fatigue probabilistic models have been proposed. [PERSON] and [PERSON] [7] developed the Weibull fatigue model. In [8], a model for fatigue crack growth was introduced. In [9], a fatigue model was given based on the statistical characteristics of the cycles to failure. Similar research is found in [10]. Unfortunately, none of them focused on bilinear behavior. Among the research focused on bilinear behavior, we found [11]. In this research, the authors mention that to improve fatigue life prediction, a bilinear model with a probabilistic focus must be used to model the uncertainties associated with the deterioration process produced by the fatigue phenomenon. In [12], an application of the bilinear model to fourteen types of alloys was performed. They show that this model provides a better representation of fatigue behavior than the linear model. However, despite their utility, both linear and bilinear models lack probabilistic information.  
The novelty of this paper lies in that the proposed method lets us incorporate probabilistic behavior into bilinear fatigue analysis and generate the corresponding P-S-N field. This approach is based on the statistical treatment of fatigue life data, considering the median stress value and its reliability index. The methodology involves determining the Weibull parameters and using them to estimate the P-S-N percentiles. Furthermore, the relationship between the confidence CL level and the P-S-N percentiles is formulated. Finally, by using 42 CrMo4 data, a zero-test plan is designed and numerically evaluated to validate that the element presents the minimum required reliability.  
The paper is organized as follows. Section 2 outlines the general background on bilinear fatigue and Weibull/IPL analysis. In Section 3, the proposed method is detailed step by step. In Section 4, the numerical application is performed. Finally, Section 5 presents the conclusions derived from the study case.