The selection of the bolt material and its yield limit can be determined through empirical methods or by analogy. According to GB309811) 82, the ultimate strength of the material is defined based on the selected performance level. In this context, the lower limit of the yield strength (Rsmin) is considered as the strength limit of the bolt material.
Knowing the average maximum working load Q6 and the number of bolts n in the group, the average load per bolt is calculated as Qp = Q6 / n. The corresponding mean stress caused by this load is given by Rp = 4Qp / (πd²), where d is the thread diameter in millimeters. The coefficient of variation for the working load is defined as VQ = σQ / Qp, leading to the standard deviation DRp = VQ × Rp. The pre-tightening stress Ri due to the pre-tension force is considered, and according to literature, the coefficient of variation for the pre-tightening stress VR is generally taken as 0.149. Hence, the standard deviation DRi = 0.149 × Ri.
According to literature, the total tensile force Q acting on an axially loaded bolted joint is expressed as Q = C1/(C1 + C2) × Qp + Qi, where Qp is the working load, Qi is the pre-tightening force, and C1, C2 are the stiffness of the connected and coupling members, respectively. The ratio C = C2/C1 is used to calculate the total load as Q = [1/(1 + C)] × Qp + Qi. The mean working stress R is then given by R = [1/(1 + C)] × Rp + Ri. Finally, the membership function for the bolted joint is established using a semi-step distribution function to represent the yield limit of the material, considering the fuzzy statistical characteristics of the material strength.
Given the vibration angular frequency X = 10216 sâ»Â¹, vibration amplitude A = 8 mm, and the grinding tube mass (including media) m = 4500 kg, 12 M24 high-strength bolts (grade 10.9) are used to connect the frame and pressure code. The goal is to determine the fuzzy reliability of the bolt group connection R.
For the 10.9 grade high-strength bolt, the nominal tensile strength Rb ranges from 1000 to 1200 MPa. Considering the load characteristics and the reliability of the vibrating mill's pressure-coded bolt connection, the critical plastic zone pre-tightening method is recommended, which is also acceptable for high-strength bolts. At this point, the average pre-tensioning stress Ri is set at 0.7 × Rb = 630 MPa, with a standard deviation DDi = 0.149 × Ri = 93.187 MPa. Since there are no test results available for reference, the mean value of the joint stiffness coefficient C = C2/C1 is treated as a fixed value of 3, resulting in a standard deviation Dc = 0.10 × C = 0.3 MPa.
Ultimately, the total mean and standard deviation of the working stress are calculated. The fuzzy reliability of the bolt connection is obtained as RR = P(A) = [1 / (b - a)] × [exp(- (aR)^2 / (2DR²)) - exp(- (bR)^2 / (2DR²))] = 0.99775, indicating that the probability of failure is less than 0.225%.
Conclusion: Through the fuzzy reliability design of the vibrating mill’s pressure-coded bolt connection, a reliable basis for determining the preload is established. It is evident that when the pre-tensioning stress is set at 0.7 × Rb, the tightness rigidity between the grinding pipe, frame, and pressure code is satisfied, and the fuzzy reliability of the bolt group connection remains above 0.9977, ensuring safe and stable operation under varying loads.
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