Global Journal of Researches in Engineering, I: Numerical Methods, Volume 23 Issue 1

Table 2: Gaussian distribution parameters (large sample) (Where s is sample standard deviation) Table 3: Weibull distribution parameters (large sample) Fig. 3: Histogram of original data (large sample) and fitting diagram of Gaussian and Weibull distribution Fig. 4: Histogram after logarithm of the original data (large sample) and fitting diagram of Gaussian and Weibull distribution "back" to the original state, only the median can "recover" (see Table 2, line 3), and the mean is left- skewed, the relative coefficient and the coefficient of determination is improved. Nevertheless, it is still not possible to obtain a 100% safe lifetime. In contrast, the fit with the Weibull distribution, as seen in Table 3, is a fairly good fit. Even after taking the logarithm, the fit is almost the same as that of the Gaussian distribution. From the data in row 2 of the Weibull distribution parameters in Table 3 and (6) and (7), we can calculate that μ ^=5.7137; σ ^=0.1005 And this result is almost the same as the data in row 2 of Table 2. In this sense the Weibull distribution is indeed more general than the Gaussian distribution, which can be seen as a first-order approximation to the Weibull distribution. It can be seen that using the Weibull distribution to fit this set of fatigue life data does not require any logarithm of the data at all and the physical meaning of each parameter is very clear. Example 4: Looking again at the case of a small sample, 20 data for the life of a structure using Table 8-4 of P136 in [2]. Again, this can be obtained by Python code as follows Fatigue life (raw data) N= [3.5, 3.8, 4.0, 4.3, 4.5, 4.7, 4.8, 5.0, 5.2, 5.4, 5.5, 5.7, 6.0, 6.1, 6.3, 6.5, 6.7, 7.3, 7.7, 8.4] (10^5cycle) Also the following parameter table and histogram can be obtained. Table 4: Gaussian distribution parameters (small sample) (Where s is sample standard deviation) Table 5: Weibull distribution parameters (small sample) Fig. 5: Histogram of original data (small sample) and fitting diagram of Gaussian and Weibull distribution Global Journal of Researches in Engineering Volume XxXIII Issue I Version I 4 Year 2023 © 2023 Global Journals ( ) I From Gaussian Distribution to Weibull Distribution As seen in Fig. 3, the histogram of the original data is asymmetric and left-skewed, and fitting it with a Gaussian distribution would be less appropriate, as in fact demonstrated with the chi-square test [4] . At this point it would be more appropriate to use the Weibull distribution. Looking at the logarithm of the data, we can see from Fig. 4 that the data do appear to be symmetric, and the Gaussian distribution is indeed a good fit. The problem is that the fatigue life PDF left-skewed features are lost, and the physical meaning of safe life is lost. Even if the results obtained in the logarithmic case