## 2 Standard Methodology for Estimating the Number of Specimens  
An example fatigue characteristic graph for 0.72%C steel (data taken from [17]) is shown in Figure 2. The fatigue tests were performed under axial loading with asymmetry cycles \(R=-1\) on a multi-type fatigue testing machine at Meiji University at 80 Hz. A tensile test of 0.72%C steel was performed on twenty-six specimens. The tensile strength is equal to 930 MPa with a standard deviation of 80 MPa. The diameter in the measuring place of the used hourglass specimens was 5 mm. Eighty-nine points were used to estimate the parameters of the S-N curve. The tests were performed on five stress levels. The normal  
Figure 1: Diagram for determining S-N characteristics. distribution equation was used to describe the damage probability for the linear regression represented by Equation (1) and has the following form:  
\[f(N_{i})=\frac{1}{0.2166\sqrt{2\pi}}\cdot\exp\left\{\frac{-\left[\log(N_{i})-(-16. 8\cdot\log(S_{ai})+48.02)\right]^{2}}{2\cdot\left(0.2166\right)^{2}}\right\} \tag{2}\]  
Note that the description of fatigue characteristics by the normal distribution shown in Figure 2 and recommended in standards [5; 6] assumes a constant standard deviation \(\sigma\) (for 0.72%C steel, it equals 0.2166). The variable \(\log(N)\) has the mean value \(\mu\) described by Equation (1) and depends on the value of the stress amplitude \(S_{a}\). Further calculation of the required number of samples will assume a normal distribution of fatigue characteristics. To determine the number of samples required, ISO 12107:2003 [16] can be used, where the following formula is recommended:  
\[n=\frac{\ln(\alpha)}{\ln(1-P)}, \tag{3}\]  
where \(P\) is the probability of failure, \(n\) is the number of specimens and \(\alpha\) is the confidence level. In the absence of data regarding the expected value and variance, the classical statistical approach expressed by the following formula can be used [18; 19]:  
\[n=\frac{u_{(1-\alpha)/2}^{2}}{4\cdot P^{2}}, \tag{4}\]  
where \(u_{(1-\alpha)/2}\) is the value above which we find an area of \(\alpha/2\) under the standard normal curve. The calculated values according to Formulas (1) and (2) are shown in Table 1. It must be emphasised that values for 1% probability (rarely required in engineering) are a very large number of samples (especially for Equation (4)), which are not economical and hard to  
Figure 2: S–N curve for 0.72%C steel; points—experimental data, surface—PDF acc. Equation (2). realize. These values are presented only to illustrate the requirements of classical statistics when a preliminary test is not made. For example, the probability of failure shall be at max 5% probability of failure for railway bogie frames acc. standard [20]. So, the number of samples to estimate the S-N curve of the material used to build of bogie frame should be at least 385 acc. Equation (4) or 58 acc. Equation (3). A different approach was presented by [PERSON] in paper [21], which calculates the error of a prediction of fatigue life. The following formula for the error [21] can be used when performing preliminary tests:  
\[\delta=\frac{t_{a}s\sqrt{\frac{1}{n}+u_{p}^{2}(k^{2}-1)}}{\bar{x}+u_{p}\delta \sigma}, \tag{5}\]  
where \(u_{p}\) is the quantile of order \(P\) from the standardized normal distribution, \(P\) is the maximum estimation error, \(t_{a}\) is the quantile of order \(\alpha\) from distribution \(t(n-1)\), \(\overline{x}\) is the mean value, \(x\) is log(\(N\)) and \(k\) is the coefficient according to the following formula:  
\[k=\sqrt{\frac{n-1}{2}}\frac{\Gamma\Big{(}\frac{n-1}{2}\Big{)}}{\Gamma\Big{(} \frac{n}{2}\Big{)}}, \tag{6}\]  
where \(\Gamma\) is the gamma function. The considerations outlined above involved taking into account type I error, that is, rejecting the null hypothesis \(H_{0^{\prime}}\) even though it is true. On the other hand, there is a type II error, namely the mistake of not rejecting the null hypothesis even though it is false. In the case of a type II error, the power of the test is calculated.