Determination of the Statistical Power of Fatigue Characteristics in Relation to the Number of Samples

Abstract

The paper presents guidelines included in the ISO 12107:2003 standard regarding the number of samples for determining the fatigue characteristics in the high cycle range. The proposed normative values were compared with the classical statistical approach. Fatigue test results for S355J2+C steel for rotary bending were used for verification. In addition, the error and power of a statistical test were determined for characteristics with different sample sizes. It determined the number of specimens to estimate the fatigue curve, which required significant error and power of the statistical test.

1. Introduction

In order to determine the fatigue life or fatigue strength of a structural component, we use, depending on the assumed loading condition, an appropriate computational model. Using any computational model, we need to know the fatigue characteristics of the given element. This applies, for example, to the design of machine components such as cutting drum shafts [1], bicycle frames [2], railway axles [3], or bridge crane shafts [4]. Recommendations such as ISO [5] or ASTM [6] standards can be used to determine such characteristics. In the high-cycle range, we most often use a linear regression describing the Basquin equation [7], which describes the relationship of the stress amplitude to the number of cycles (Stress—Number of cycles; S–N) and is written as follows:

log(N) = m log(S_a) + c,

(1)

where S_a is the stress amplitude, N is the number of cycles, m is the slope coefficient and c is the intercept.

A scheme of the procedure for determining fatigue characteristics in the high-cycle range is shown in Figure 1. Since fatigue tests are costly and expensive, efforts are being made to keep them to a minimum. For example, the testing time for one sample (not including sample preparation, mounting, setting test parameters in the testing machine, etc.) for a frequency of 30 Hz and a fatigue life of 10⁵ cycles is less than an hour. However, for 10⁶ cycles, it is more than 9 h. On the other hand, the cost of conducting the test per sample was calculated by Shen [8] at USD 500 to USD 1000. Many researchers have made an effort to determine the minimum number of samples. Gope [9] determined that at least 10 samples are required for a 10% probability of failure and a 90% confidence level. In contrast, Lewis and Sadhasivini in their paper [10] propose seven samples for a 5% probability of failure and a 95% confidence level for each load level. A different approach was presented by Soh et al. in their paper [11], where they identified seven samples with a 50% probability of failure.

Various statistical methods have been developed to improve the accuracy of fatigue characteristic estimation. One of them is the backward statistical inference method presented by Xie et al. [12]. This method assumes that the probability distribution for each load level is the same. An improvement of the backward statistical inference method was presented by Li et al. in a paper [13]. Both methods require at least 15 samples. Another approach was presented by Liu and Sun in a paper [14], where non-invasive polynomial chaos was used to determine fatigue characteristics. This approach requires 16 samples. Meanwhile, in paper [15], Zu et al. proposed an α-S–N method based on uncertainty theory. This method obtained better results than the least-squares method. Unfortunately, it still requires 15 samples. It should be emphasized that the statistical methods presented above are intended to improve the accuracy of estimation from experimental data, but indicate that they cannot replace standard testing. This is related to many factors affecting fatigue life, such as material properties, geometry of the test object, surface condition, environmental conditions, etc.

This paper presents the guidelines contained in ISO 12107:2003 [16] for the number of samples for determining fatigue characteristics in the high-cycle range. The size of the experimental data set depends on the required accuracy of the estimated characteristics. The proposed normative values were compared with the classical statistical approach. In addition, the error was determined for characteristics with different data set counts.

The purpose of this study is to determine the statistical power of the estimated fatigue characteristics in the high-cycle range with respect to the number of samples. The representation of the required number of samples for the determination of fatigue characteristics was referred to as normative recommendations.

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 distribution equation was used to describe the damage probability for the linear regression represented by Equation (1) and has the following form:

$$f\left(N_i\right)=\frac{1}{0.2166\sqrt{2\mathsf{\pi }}}·\exp \left\{\frac{-{\left[\log \left(N_i\right)-\left(-16.8·\log (S_{ai})+48.02\right)\right]}^2}{2·{\left(0.2166\right)}^2}\right\}$$

(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 σ (for 0.72%C steel, it equals 0.2166). The variable log(N) has the mean value μ 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:

$$n=\frac{\ln \left(\alpha \right)}{\ln (1-P)},$$

(3)

where P is the probability of failure, n is the number of specimens and α 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]:

$$n=\frac{u_{\left(1-\mathsf{\alpha }\right)/2}^2}{4·P^2},$$

(4)

where u_(1−α)/2 is the value above which we find an area of α/2 under the standard normal curve.

The calculated values according to Formulas (1) and (2) are shown in

Table 1. The number of specimens required so that the minimum value of test data can be expected to fall below the true value for the population at a given level of probability of failure at various confidence levels.

Probability P [%]	Confidence Level, 1 – α [%]
	50	90	95
	Number of Specimens [−]
50	1 (1)	3 (3)	4 (4)
10	12 (7)	68 (22)	97 (28)
5	46 (13)	271 (45)	385 (58)
1	1138 * (69)	6764 * (229)	9604 * (298)
()—values acc. ISO 12107:2003 Equation (3)

* Such a large number of samples is not economical and is hard to realize; these values are presented only to illustrate the requirements of the classical statistics when a preliminary test is not made.

. 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 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 Gope 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:

$$\delta =\frac{t_{\alpha }s\sqrt{{\frac{1}{n}+u}_P^2\left(k^2-1\right)}}{\stackrel{\mathrm{-}}{x}+u_{p}k\sigma },$$

(5)

where u_P is the quantile of order P from the standardized normal distribution, P is the maximum estimation error, t_α is the quantile of order α from distribution t(n − 1), ͞x is the mean value, x is log(N) and k is the coefficient according to the following formula:

$$k=\sqrt{\frac{n-1}{2}}\frac{\mathsf{\Gamma }\left(\frac{n-1}{2}\right)}{\mathsf{\Gamma }\left(\frac{n}{2}\right)},$$

(6)

where Γ is the gamma function.

The considerations outlined above involved taking into account type I error, that is, rejecting the null hypothesis H₀^, 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. The occurrence of each error is shown in

Table 2. Possible situations for testing a statistical hypothesis.

	H0 Is True	H0 Is False
Do not reject H0	Correct decision	Type II error
Reject H0	Type I error	Correct decision

. It is desirable that both errors be as small as possible. For the S–N curve, the power of determining the fatigue characteristics depending on the number of samples can be determined by the following formula [18,19]:

$$1-\beta =P_{H_0}\left(\frac{\overline{x}-\mu }{\sigma }\sqrt{n} \frac{k_1-\mu }{\sigma }\sqrt{n}\right)+P_{H_0}\left(\frac{\overline{x}+\mu }{\sigma }\sqrt{n} \frac{k_1+\mu }{\sigma }\sqrt{n}\right),$$

(7)

where k₁ is the coefficient calculated according to the following formula:

$$k_1=\mu +\frac{\sigma t_{\alpha }}{\sqrt{n}} .$$

(8)

The power of the statistical test M of the test is 1 − β. It should be emphasized that if the probability of making a type I error is decreased, then the probability of a type II error is increased. To minimize these errors, the number of samples must be increased. To the author’s knowledge, there is no study on the influence of the type II error on the number of samples in fatigue tests.

3. Materials and Methods

Fatigue tests were conducted for S355J2+C low-carbon construction steel with a ferritic–pearlitic structure, which is commonly used for construction and machine-building purposes [22]. The geometry and dimensions of the specimens are shown in Figure 3. The material was bought as a drawn rod with a 10 mm diameter with tolerance h10 (min. 10 mm, max. 10.058 mm) made according to standard [23]. The specimens were turned into the shape presented in Figure 3 and ground at the measuring area using 80, 320, 600, 1200, and 2400 grit sanding sheets. The tests were conducted on a testing machine used for rotary bending, which was verified in paper [24], and the diagram is presented in Figure 4. The test was interrupted when the specimen was broken. Similar tests can be found in the paper [25]. The maximum measurement error of the bending moment was 1.15%, which is less than the permissible value according to ISO-1143 [26] of 1.3%. This error is the sum of an error applied to the mass m and the length of the arm l. The influence of the error of applied stress in rotary bending was analyzed in the paper [27], where the scatter of applied stress was less than half of the standard deviation. The load was applied at a frequency of 28.5 Hz. The tests were planned according to standard ISO 12107:2003 [16]. The tests were performed on 21 samples. A comparative study with a small number of samples, amounting to nine, was carried out by Zawadzki in his dissertation [28] on the same test stand. Both tests were performed with the same material (specimens were made earlier in advance), in the same room (the same temperature equals ~21 °C, humidity ~35%, etc.), but performed by different researchers. The two datasets were then combined, resulting in the amount of data recommended for reliability tests.

The tensile test was carried out on fifteen specimens on an Instron 8874 testing machine according to the recommendations of ISO 6892-1 [29]. The extension rate was set up to 0.0198 mm/s, which is commonly used in such tests (see [30]). Figure 5 presents the geometry of the specimens used for tensile tests.

The least squares method was used to determine the parameters of Equation (1). The equations for this method can be found in normative documents, e.g., ISO 12107:2012 [5]. All calculations were performed using R software version 4.3.1 [31].

4. Results

The mean tensile strength of the tested material was 815 MPa, while the mean yield strength at 0.2% plastic elongation was 745 MPa. The standard deviations of tensile strength and yield strength were 14 MPa and 50 MPa, respectively. Figure 6 presents the obtained engineering stress-strain curve for all specimens. It is worth mentioning that the scatter of stress increases after reaching maximum force. A similar effect (increased dispersion of test results after reaching maximum force) can be found in the paper [32] for DC04 low-carbon steel. Additionally, statistical analysis of the tensile test of S355 steel can be found in [33], where the standard deviation for 1089 specimens was 25.1 MPa and 25.4 MPa for tensile strength and yield strength, respectively. However, min. and max. values were 461–665 MPa for the tensile strength and 325–483 MPa for the yield strength.

S355J2+C steel does not have a clear yield strength, which was expected for the material after plastic deformation such as rod drawing (more details about it can be found in [34,35,36]). The obtained values meet the requirements of PN-EN 10,277 [37], which are as follows: tensile strength 630 ÷ 950 MPa and yield strength min. 520 MPa.

The fatigue test results are shown in Figure 7, Figure 8 and Figure 9. The solid lines show the characteristics of a 50% probability of failure. The dashed lines indicate characteristics for a confidence level of α = 5%. The power value of determining the fatigue characteristics 1 − β according to Formula (7) and the error according to Formula (5) are shown in

Table 3. Value of the estimation error and the power of the statistical test of S–N characteristics.

Number of Specimens n	Slop Coeff. m [−]	Intercept c [−]	Standard Dev. σ [−]	Confidence Level α [%]	Probability of Error Type II Error β [%]	Power of Test M [−]	Error δ [%]
9	−10.16	32.2	0.108	5	62.7	0.372	6.3
9	−10.16	32.2	0.108	10	49.2	0.508	6.3
21	−9.91	31.3	0.175	5	12.2	0.737	5.0
21	−9.91	31.3	0.175	10	16.7	0.833	5.0
30	−9.68	30.8	0.183	5	12.2	0.878	3.8
30	−9.68	30.8	0.183	10	6.8	0.932	3.8

. A comparison of characteristics is shown in Figure 10. The power of the test was calculated to detect differences in mean values of 15%, while the standard deviation σ was taken for characteristics n = 30 of 0.183.

The first test was made on 21 specimens. At the beginning of the test, one specimen was tested on the stress amplitude level Sa. Then, the preliminary slope coefficient m was determined. In the next step, stress level intervals were defined for the full test to determine the S–N curve equal to 45 MPa. The different slope coefficient m leads to different stress level intervals, which was discussed in the paper [38]. This is why, in Figure 8, there are few single specimens between stress amplitude levels. However, the test for nine specimens was made later, so stress levels were known. For this test, there are no single specimens between stress levels.

Additionally, correlation coefficients were calculated for all S–N curves by the following equation [5]:

$$R^2=\frac{{\left(\sum _{i=1}^n{X}_i{Y}_i\frac{\sum _{i=1}^n{X}_i\sum _{i=1}^n{Y}_i}{n}\right)}^2}{\left[\left(\sum _{i=1}^n{X_i}^2-\frac{{\left(\sum _{i=1}^n{X}_i\right)}^2}{n}\right)\left(\sum _{i=1}^n{Y_i}^2-\frac{{\left(\sum _{i=1}^n{Y}_i\right)}^2}{n}\right)\right]}$$

(9)

where X_i is a vector of log(N_i) and Y_i is a vector of log(S_ai).

The obtained values are equal to 0.956, 0.9323 and 0.9131 for a number of specimens n = 9, 21 and 30, respectively. Obtained values fulfill the recommendations of ISO 12107:2012 [5], which is defined as ‘a value of 0.9 or better is indicative of a good fit’.

5. Discussion

Figure 11 shows the probability distributions obtained from the fatigue characteristics for S355J2+C material for a stress amplitude S_a equal to 450 MPa and the number of samples n equal to 9, 21 and 30. It can be seen that the scatter of test results is lower for the smallest number of samples, n = 9. This is due to the lower value of the standard deviation σ than the other characteristics, which are shown in Table 3. It can be observed that this value increases as the number of samples increases. However, the S–N curve with a small number of specimens is more susceptible to being shifted by adding a new tested point. Additionally, the mean value is more incredible, which can be seen in Figure 11, where the mean value of fatigue life is underestimated for the fatigue curve with 9 specimens.

As could be expected, a lower value of the standard deviation was obtained, followed by a higher value of the correlation coefficient. However, the error calculated by Equation (5) and the error type II β calculated by Equation (7) are lower for a larger number of test points. So, the correlation coefficient cannot be a criterion for judging the quality of the estimated S–N curve.

Wormsen et al., in their paper [39], conducted tests for low alloy forged AISI 8630 M steel in a quenched and tempered condition of the same grade supplied by different suppliers. The tests were performed for alternating axial loading on specimens. They obtained a standard deviation value σ for the S–N characteristics of 0.159 ÷ 0.214. The σ for 0.72% steel, presented in Chapter 1, was 0.2166, so it is higher than was obtained for AISI 8630 M and S355J2+C steels. On the other hand, in the recommendations of the International Institute of Welding [40], one can find recommendations for n < 10 to adopt a σ value of 0.178 for geometrically simple structures between 10⁴ ÷ 10⁵ cycles and 0.25 for complex structures up to 10⁶ cycles. Based on the presented σ values, it can be concluded that the obtained results are in line with the literature data. In addition, it can be concluded that for a dataset smaller than 10 samples, the resulting scatter of test results (estimated σ value) is unreliable. Figure 11 shows the probability distribution for the S–N characteristics of S355J2+C steel for n = 9, with a standard deviation σ of 0.25 indicated by the International Institute of Welding [40]. Based on this distribution, with a probability of 5%, a value of 65,770 cycles (the logarithm value is 4.818) is obtained. However, for the probability distribution of S–N characteristics for S_a = 450 MPa and the number of samples n = 30, the fatigue life is 59,538 cycles (the logarithm value is 4.774). This gives a difference of 10% estimation error (the logarithm value is 0.9%). In contrast, the difference in the mean values of these two distributions yields a difference of 42% (169,528 for n = 9 and 119,035 for n = 30), and it is higher than the error according to Equation (5), which equals 6.3%. However, the difference in mean logarithm values is 3.03%.

It is worth noticing that components with sharp notches have a smaller dispersion of fatigue life, which was shown in a paper [41]. So, using a large value of the standard deviation can reduce the overestimation/underestimation of mean values for notched elements.

In addition, the graph in Figure 11 marks the area of rejection of the null hypothesis H₀ stating that the mean value of the experiment is equal to the population value. Two-sided testing was adopted for α equal to 10%.

Figure 12 shows the dependence of test power on the number of samples. According to the statistical recommendations presented in the paper [18], the power of the test should be 0.8, that is, the β error equals 20%. For material S355J2+C and α equal to 10%, the required number of samples is 20. For α equal to 5%, the value of n is 25. Additionally, the graph presented in Figure 12 shows the calculations for the standard deviation σ of 0.1 and 0.25. Based on these calculations, it can be concluded that the scatter of test results (expressed by the standard deviation σ) has a significant impact on the power of the statistical test performed.

6. Conclusions

Based on the presented research and calculations, the following conclusions can be drawn:

• The error difference according to Formula (5) of determination of characteristics when using different numbers of samples (9, 21 and 30) is 2.5% and 1.3%.
• The characteristic obtained from 9 samples falls within the scatter of the error of estimation of characteristic for 30 samples and can be adopted for estimating fatigue life when taking the standard deviation according to the recommendations of the International Institute of Welding [40], i.e., σ = 0.25. However, fatigue life at 5% probability for 9 specimens without correction of standard deviation is larger than fatigue life at 5% probability for 30 specimens (discussed in Chapter 5 and presented in Figure 11) and could not be used for engineering purposes.
• The power of the test M increased more than two times between the number of samples (9 and 30).
• The number of samples should be determined based on consideration of the required accuracy of fatigue life/strength estimation and economic justification.
• The determination of the accuracy of the estimation should be preceded by an analysis of the damage caused after damage to a given structural element, i.e., loss of health, property damage, risk of environmental contamination, etc.
• The value of the standard deviation has a significant effect on the power of the statistical test M and the error δ of fatigue life estimation, according to Formula (5).
• Recommendations according to ISO 12107:2003 [16] should be considered reasonable, but the final number of samples should be determined after conducting preliminary tests (min. eight specimens [16]) and verifying the estimating error δ according to Equation (5), or the power of the statistical test M according to Equation (7). If the obtained error δ or power of the statistical test M is lower than required, the number of specimens must be increased by one sample on the stress level. Then the error δ is calculated according to Equation (5) or the power of the statistical test M is calculated according to Equation (7). The obtained value is checked. When the error δ or the power of the statistical test M meets the requirements, the test is stopped.