Device lift bolt presets according to simulation criteria

The design criterion for the fuzzy reliability of pre-tightened bolt connections can be expressed as: the material strength limit of the bolt must exceed the fuzzy probability of its working stress, and it should be greater than or equal to the required design fuzzy reliability R₀, i.e., σ > σ₀ ≥ R₀ (2). The mean value of the stress caused by the load is given by σ = 4Q / (πd²) (3), where d is the thread diameter in mm. The coefficient of variation of the working load is known as V_PQ = 3, and the dispersion is calculated as δ_p = V_PQ × σ_P (4). The pre-tightening stress induced by the preload is σ_i. According to reference [2], the coefficient of variation for the pre-tightening stress, σ_V, is generally taken as 0.149, so δ_i = 0.149 × σ_i (5). Based on reference [1], the total tensile force Q of the bolt connection under axial load is Q = C₁Q_p + C₂Q_i, where Q_p is the working load, Q_i is the pre-tightening force, and C₁, C₂ are the stiffnesses of the connected and coupling members, respectively. The ratio is C = C₁/C₂, so Q = C₁(Q_p + Q_i). The mean total working stress is then σ = C₁σ_p + σ_i (6), and the standard deviation is σ_cc = 2Cδ. In the formula, δ_c is the standard deviation of the mean value of the stiffness ratio coefficient C, which can be determined by the following expression from reference [2]: δ_c = 0.1 × V_C (8). As a fuzzy variable, the ultimate strength is equivalent to the shear strength. The membership function for the reduced half-normal distribution is μ(x) = A(b - x) for x ≤ b (9), where a and b are distribution coefficients that can be determined using the amplification coefficient method.

The fuzzy reliability design for bolts subjected to shear is crucial for working condition B, especially when pre-tightening forces are neglected. In conventional design, the shear stress of the bolt is τ = 4KQ / (πd²) ≤ (12), where τ is the shear stress in N/mm², Q is the theoretical shear load per bolt in N, K is the uneven load distribution coefficient, and d is the diameter of the shear-resistant part in mm.

During fuzzy reliability design, Q, K, and d are considered independent random variables, often assumed to follow a normal distribution, as supported by the central limit theorem. When none of these variables dominates, the shear stress is also expected to follow a normal distribution closely. The mean and standard deviation are calculated based on first-order reliability theory. The shear stress is given by τ = 4KQ / πd² (13), and the standard deviation is σ_τ = 1/2 (14). The probability density function of the normal distribution is f(τ) = (1 / (σ_τ√(2π))) * exp(-(τ - μ_τ)^2 / (2σ_τ²)) (15). As a fuzzy variable, the shear strength requires an intermediate transition from fully allowed to partially allowed states, typically described by a membership function. Common forms include half-normal, half-trapezoidal, half-ridge, and half-rectangle distributions. For shear strength, a half-trapezoidal distribution is commonly used due to its simplicity and practicality. The membership function for the half-trapezoid is defined as follows: μ(τ) = 1 for a ≤ τ < b, μ(τ) = (b - τ)/(b - a) for τ ≥ b, and μ(τ) = 0 otherwise. In the formula, (a, b) are distribution parameters.

In the membership function, a₁ is selected as the allowable value from conventional design, while a₂ is determined using the amplification coefficient method, with values ranging from 1.05 to 1.3. The stress is treated as a random variable with probability density f(τ), while the strength is a fuzzy variable with membership function μ(τ). Fuzzy reliability is defined as the probability of the fuzzy event: PR = ∫_{τ ≥ μ} f(τ) dτ (17). Using formulas (11) and (17), the fuzzy reliability for both shear and tension can be calculated, and the lower of the two is taken as the failure-free condition for the bolt. Based on this, the relevant dimensions of the bolt, such as diameter d and safety factor n, can be determined.

Conclusion (1) For critical equipment like spreaders, introducing fuzzy reliability design helps designers better understand the safety level of lifting operations, offering a more intuitive and clear measure of reliability compared to average safety factors. (2) Although many researchers have applied fuzzy reliability concepts, certain parameters—such as a₁, a₂, a, b, and K in the formulas—require further refinement. Strengthening experimental research in this area will make fuzzy reliability design more practical and widely applicable.

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