Research and Analysis on the Determination of Bolt Cyclic Stress

There are various types of gasket constitutive equations, and their performance can vary significantly. Through extensive mechanical property tests conducted at both ambient and high temperatures, it has been observed that the loading and unloading curve of a gasket—also known as the compression-rebound curve—is nonlinear and non-conservative. At a constant temperature, the displacement of the gasket during preloading and operation is given by the following expressions: Dk = (Sk / Ac)1Nc(1) Dg = (Sg / As)1Ns + Dp(2) Here, Sk and Sg represent the pre-tightening stress and the residual stress during operation, respectively. Ac, As, Nc, and Ns are regression coefficients, while Dp denotes the plastic deformation after full unloading. The value of Dp depends on the type and material of the gasket, the initial pre-tightening pressure, and the operating temperature. When considering the bolt lengths during preloading and operation, we have: lb1 = l0 + qbW1(4) lb2 = l0 + qtbW2 + 2 + bcl0 + tb(T2 - T1)l0(5) In these equations, l0 is the initial length of the bolt; qb and qtb are the elastic coefficients of the bolt under pre-tightening temperature T1 and working temperature T2, respectively; tb is the linear expansion coefficient of the bolt material at the working temperature; bc represents the creep strain of the bolt at the working temperature; W1 and W2 are the bolt loads during preloading and operation, respectively, defined as: W1 = AgSk W2 = AgSg + Dm²p/4 where Ag is the full area of the gasket in mm², Dm is the average diameter of the gasket in mm, and p is the operating pressure in MPa. Based on the deformation compatibility condition of the flange connection system, the following equation holds: Dk - (Dg + Dgc) = (lb2 - lb1) + 2(Df2 - Df1) - 2tf(9) Substituting equations (1) through (8) into equation (9), we obtain the deformation coordination equation for the high-temperature flange connection system: (Sk / Ac)1Nc <1 - (a + bT2)1nt> - (Sg / As)1Ns - Dp + 2tftf(T2 - T1) - - 2(qtfM2 + qpp - qfM1) = 0(10) Assuming that the temperature difference between the bolt and the flange is zero during pressure and temperature fluctuations, and only considering the effect of the difference in linear expansion coefficients between the two materials, the deformation compatibility equation for an unstable condition can be written as: (Sk / Ac)1Nc <1 - (a + bT*)1nt> - (Sg* / As)1Ns - Dp + 2tftf(T* - T1) - - 2(qtfM* + qpp* - qfM1) = 0(11) By subtracting equation (11) from (10), we get: (Sg* / As)1Ns - (Sg / As)1Ns + qtb(W* - W2) + Tbl0(T* - T2) + 2qtf(M* - M2) + 2qp(p* - p) - 2tftf(T* - T2) = 0(12) Which simplifies to: Sg* / As1Ns - Sg / As1Ns qtbW + tbl0T + 2qtfM + 2qpp - 2tftft = 0(13) where W = W* - W2 = Dm²p/4 + Ag(Sg* - Sg)(14) The change in flange bending moment caused by temperature and pressure fluctuations is: M = M* - M2 = Di²pl1 / 4 + (Dm² - Di²)pl2 / 4 + Ag(Sg* - Sg)l3 Due to the nonlinearity of the gasket's constitutive relationship, equations (13) and (14) alone are insufficient to solve for Sg, Sg*, and W. To resolve this, the residual pressure Sg of the gasket under stable conditions must first be determined using equation (10) based on the initial conditions, such as the known bolt preload. Once Sg is obtained, equations (13) and (14) can then be used to calculate Sg* and W.

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