The selection of the bolt material's membership function and the determination of its yield limit can be based on empirical methods or by analogy. According to GB309811) 82, the ultimate strength of the material is determined once the performance level of the fastener is selected. In this context, the lower limit Rsmin of the yield limit of the bolt material is defined as the strength limit of the material.
Knowing the average maximum working load Q6 and the number of bolts n in the bolt group, the average working load per bolt is calculated as Qp = Q6 / n. The mean stress caused by the working load on each bolt is given by Rp = 4Qp / (πd²), where d is the thread diameter in millimeters. Considering the coefficient of variation of the working load, VQ = σQ / Qp, the standard deviation of the stress becomes DRp = VQ × Rp. The pre-tightening stress Ri, due to the pre-tightening force, is also considered. Based on literature <2>, the coefficient of variation for pre-tightening stress VR is generally taken as 0.149, so the standard deviation DRi = 0.149 × Ri.
According to literature <1>, the total tensile force Q acting on an axially loaded tight bolt is given by Q = C1/(C1 + C2) × Qp + Qi, where Qp is the working load, Qi is the pre-tightening force, and C1, C2 are the stiffnesses of the connected and coupling members, respectively. The ratio C = C2/C1 is used to determine the total force, and the working stress mean is R = 1/(1 + C) × Rp + Ri. Finally, the membership function of the bolted joint is established. Based on the fuzzy reliability design criterion, the normal working state and failure state of a pre-tightened bolt under variable loads are not distinct but represent a gradual transition. Using the characteristics of the fuzzy variable of material strength, the semi-ladder distribution function is chosen to express the membership function of the yield limit: uA(x) = 1 - x/VR.
Given the vibration angular frequency X = 10216 sâ»Â¹, vibration amplitude A = 8 mm, and grinding tube mass (including medium) m = 4500 kg, 12 M24 high-strength bolts (grade 10.9) are used to connect the frame and the pressure code. The task is to calculate the fuzzy reliability of the bolt group connection R.
For the 10.9 high-strength bolt, the nominal tensile strength Rb ranges from 1000 to 1200 MPa. Considering the load characteristics of the vibrating mill’s pressure-coded bolt and the reliability of the connection, the critical plastic zone pre-tightening method is adopted, which is also applicable to high-strength bolts. At this point, the average pre-tightening stress is Ri = 0.7 × Rb = 630 MPa, with a standard deviation DDi = 0.149 × Ri = 93.187 MPa. Due to limited test data, the mean value of the joint stiffness coupling coefficient C = C2/C1 is treated as a fixed value, here set at C = 3. The standard deviation of C is Dc = Vc × C = 0.10 × 3 = 0.3 MPa. The final total working stress mean and standard deviation are then calculated. The fuzzy reliability of the bolt connection is obtained as RR = P(A) = [1 - b - a(bR)] / [5b - RDR] - [5a - RDR] - DR²e^[-(aR)² / 2DR²] - e^[-(bR)² / 2DR²] = 0.99775.
Conclusion: Through the fuzzy reliability design of the vibrating mill’s coded bolt connection, a basis for determining the preload is established. It is evident that when the pre-tightening 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 reliable operation.
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