A study on the extreme value distribution of the minimum tensile strength of bolt

. Taking M12.5 × 1.25 × 65-10.9 bolt as an example, this paper studies the extreme distribution of the minimum value of bolt tensile strength in order to evaluate the reliability and stability of bolt product quality and to verify the production process. Through data collection, parameter estimation, distribution test and extreme prediction, it is concluded that: 1) the distribution of the minimum value of bolt tensile strength conforms to Gumbel extreme distribution; 2) when the return period is 10000, the predicted minimum tensile strength is 1071.7 MPa ± 3.2 MPa, k = 2. It is higher than the minimum value of 1040 MPa required by ISO 898-1 standard.


Introduction
Fasteners are the most widely used standard parts in the mechanical industry. According to ISO 898-1:2013 standard [1], the tensile strength of fasteners cannot be lower than the specified value. When the chemical composition of the raw material is determined, the tensile strength of the fastener depends on the standardized production process, and the process evaluation is required before the standardized production process is determined.
The general requirements for nondestructive testing and metallographic structure defects of metal materials shall be less than or equal to the specified value, i.e. the maximum value shall not be exceeded, which belongs to the application research of maximum value distribution. The mechanical properties of metal materials, such as strength, plasticity, hardness, impact absorbed energy, fatigue and so on, are generally required to be greater than or equal to the specified value, i.e. not less than the minimum value, which belongs to the application research of minimum value distribution. The application research of minimum value has important guiding significance in process design, process evaluation, process control, product quality evaluation and so on.
In this paper, samples were randomly collected from M12.5 × 1.25 × 65-10.9 bolts made of 35CrMo by a specific process. The distribution of the minimum tensile strength of the bolts was studied. Predict the minimum tensile strength that may occur for the production department to evaluate the reliability and stability of bolt product quality and to verify the specific production process.

Data collection
Twenty four groups of testing data of M12.5 × 1.25 × 65-10.9 bolts were collected from the testing history samples. Five tensile strength bolts were tested in each group. The testing was conducted according to Clause 9.2 of ISO 898-1:2013 standard [1]. See Table 1 for the data of tensile strength samples.
The minimum values of 24 groups of tensile strength in Table 1 are arranged in non descending order, which are x(1)~x(24), and x(1)≤x(2)≤…≤x(24), see the "Data collection" column of Table 2.
The sample average value (Avg) of the minimum value is 1130.3 MPa, the sample standard deviation(S) is 8.42 MPa, the sample skewness coefficient (bs) is -0.4 ≤ 0, and the sample kurtosis coefficient (bk) is 2.0 ≤ 3. See the last row of Table 2. From the skewness coefficient and kurtosis coefficient of the sample, we can see that: 1) the distribution of the minimum tensile strength at the low value range tends to deviate from the center more than that at the high value range; 2) it may have a few small extreme values; 3) the minimum value is not concentrated near the average value; 4) it has a long left tail [11][12].

Parameter estimation
The distribution function and probability density function of Gumbel minimum distribution are formula (1) and formula (2), respectively. The empirical guarantee function of Gumbel is formula (3), and the natural logarithm maximum likelihood function of Gumbel distribution samples is formula (4). (3) and (4), where ∞＜x＜∞, ∞＜λ ＜∞, σis the scale parameter, λ is the location parameter, X is the random variable, x is the value of X, n is the number of samples used for the evaluation of extreme distribution parameters, pi is the empirical cumulative density [2,[11][12][13].
Set: (2) and (5) can be used to deduce: Formula (6) and (7) can further deduce: (11), xp is called the p-quantile of distribution function F (x). p is the probability associated with the p-quantile of F (x) [12].
If T is the return period, the minimum strength of the bolt predicted at the return period T is: In formula (10)(11)(12), there are two unknown parameters, scale parameter σ and location parameter λ. Two unknown parameters were evaluated by the maximum likelihood method [2][3][4][5][6][7][8][9][10][11]. The specific methods are as follows: 1) The minimum value samples of bolts in Table 1 are arranged in non descending order.
2) The empirical cumulative density pi of each sample is calculated by formula (3).
3) yi is calculated from formulas (3) and (9). 4) Set the average value as the initial value of λ, and the standard deviation as the initial value of σ, that is, λ = 1130.3 MPa, σ = 8.42 MPa. From formula (4), the component L(xi) of the maximum likelihood function of the natural logarithm of the sample is calculated in turn.
5) The maximum likelihood function LL(x, λ,σ) is calculated from formula (4). 6) Using the Excel's "planning solving" function, the σ and λ that make LL get the maximum value are calculated by Newton iterative method. From this, It can be concluded that: LL(x, λML,σML) =-83.863, σML=6.79, λML=1134.2. σML and λML are scale and location parameters calculated by maximum likelihood method.
The above keyabbreviations, symbols and datas are in the "parameter estimation" column and the last row in Table 2.
Through the parameter evaluation, the extremum distribution function and probability density function of the minimum value of bolt tensile strength are formula (13) and formula (14)  Whether the theoretical distribution function and the theoretical density function of the minimum tensile strength of bolts are consistent with the actual needs distribution test. In this paper, K-S test method [2][3][4][5][6][7][8][9] is used. See Table 2 for the process data of K-S test. The specific methods are as follows: 1) Calculate the empirical distribution function Fn(xi)=(i-1)/n;

Distribution test
2) Calculate the theoretical distribution function Fo(xi) according to formula (13).
3 It can be seen that Dn=0.1166<D24,0.05=0.2776, so it can be determined that the theoretical distribution function and theoretical density function formula (13)(14) of the minimum value of the bolt tensile strength are consistent with the actual extreme value distribution.
The above key abbreviations, symbols and datas are in the " K-S test " column and the last row in Table 2.
The distribution curve of the minimum value of the bolt tensile strength obtained according to formula (13)(14) is shown in Figure 1

Mimimum value prediction
According to the theoretical extreme value distribution function formula (13)(14) and formula (12) of minimum value of bolt tensile strength, it can be concluded that the prediction function of minimum value of bolt strength with return period of T is formula (14): According to formula (14), the theoretical value corresponding to 24 samples can be calculated, as shown in xi(T) in Table 2. The fitting standard deviation between the theoretical value and the actual value of the sample is calculated according to formula (15) [14], then the expanded uncertainty (U) of the theoretical value is ±k×SE. After calculation, SE = 1.6Mpa, 95% confidence interval: U= ±3.2MPa，k=2. Figure 2 is the quantile plot of the actual value and theoretical value of the sample, i.e. Q-Q Plot. The dotted line in the figure is the upper and lower limit of the expanded uncertainty.

Fig.2. Q-Q Plot
Through the calculation and analysis of the extreme value of 24 groups of samples, it can be predicted that when the return period T = 10000, the minimum value of bolt tensile strength (Rm) is 1071.7 MPa ± 3.2 MPa, k = 2. That is to say, for M12.5 × 1.25 × 65-10.9 bolt made of 35CrMo, the technical requirement [1] is Rm ≥ 1040 MPa. When the return period is 10000, the minimum Rm is higher than the technical requirement. It can be determined that the quality of the bolt is stable and reliable and the specific production process is reliable. The prediction results can be explained as follows: For every 10000 samples, the probability of Rm ≤ 1071.7 MPa ± 3.2 MPa is 1 / 10000, i.e. 0.01%. Or, Rm ≤ 1071.7 MPa ± 3.2 MPa will appear in every 10000 samples; Or, in every 10000 samples, Rm of each sample will be less than 1071.7 MPa ± 3.2 MPa with a probability of 0.01%.

Conclusion
Through the research, analysis and calculation of the minimum value distribution of 24 groups of bolts, it can be concluded that: 1) the distribution of the minimum value of bolt tensile strength conforms to Gumbel extreme distribution; 2) when the return period is 10000, the predicted minimum tensile strength is 1071.7 MPa ± 3.2 MPa, k = 2.