A Method Based on the Principle of Critical Energy for Calculating Flange Joints

An analysis has been made of the behavior of flange joint components under initial stress loading (preclamping) and in service. There have been highlighted the drawbacks in calculating flange joints to be found in currently acknowledged methods that do not take into account: the influence of the bending moment produced by the total hydrostatic end force on bolt strength and flange rotation; the presence of cracks in the welding or in its thermal influence zone; the residual stresses; – the relaxation of the sealing gasket over time. The calculation method proposed in the paper takes into account all these particularities of loading. Furthermore, alongside the strength calculation of bolts and flanges, a proposal has been put forth for the calculation of the flange joint leak-tightness.

Pressure vessels and industrial pipelines are provided with flange joints. Figure 1 shows a flange joint of the kind usually found in some pressure vessels.
The sealing of this removable assembly is achieved by bolting (6) together two flanges, 1 and 2, with a gasket (3) between them to provide a seal. The sealing gasket is compressed as much as necessary in order to ensure the leaktightness of the flange joint while in service when the vessel is undergoing an internal pressure, p.
The internal pressure causes the total hydrostatic end force FH, which causes the flanges to rotate and deforms the shells to which they are joined. The flanges rotation has two effects ( Fig. 2): -applies further stress upon bolts in bending; -determines the variation of the sealing gasket thickness and reduces the effective sealing gasket width.
The bolt undergoes a tensile stress caused by the clamping force applied by fitting in bolting in nut. On top of this, while in service, one should add the bending moment produced by the total hydrostatic end force, FH (Fig. 2).
The flanges undergo a bending stress caused by bolt force which tends to rotate them around the circumference of the circle diameter D3 (characteristic of the gasket) and, separately, by the total hydrostatic end force, FH, which rotates them around the circumference passing through the axes of the bolts (D2). The total rotation is 2θ (Fig. 2).
The following issues arise: -ensuring the mechanical strength of the bolts under traction load in service with force Fb,s; -ensuring flange rigidity and end stiffness of each shell, such that flange rotation, 2θ, and flange deformation between bolts might be sufficiently small so that the thickness variation of the gasket in service should be insignificant and should not affect the tightness; -the deformation of the sealing gasket produced by the clamping force should ensure leak-tightness throughout the -prescribed period, also taking into account its relaxation over time.
Current calculation methods [1][2][3] do not take into account: -the presence of cracks in the welding or in its thermal influence zone; -residual stresses in flange joint components; -the relaxation of the sealing gasket over time, which affects its sealing capability; -the influence of the mean stress on flange joint life in the case of variable stress loading (fatigue). Further down there is proposed a method of calculating flange joints in order to eliminate the reported drawbacks. With this end in view, the principle of critical energy has been used [4][5][6][7].

Stress loading of flange joint components
a. The force of the bolts in initial clamping at ambient pressure (pre-clamping) is calculated by the relationship (Figure 1), (1) where b is the effective or computation width of the sealing gasket, which depends on its reference width, b0 ; q0 is the gasket pressure on the initial clamping recommended by the gasket manufacturer, or taken from the calculation norm..
b. The force of the bolts in service, under the vessel pressure, p, if the temperature difference between the flanges (Tf) and the bolts (Tb) is null, where, is the total hydrostatic end force; is the compression load on gasket the ensure tight joint;  The total hydrostatic end force, FH, consists of the unclamping force determined by the action of the inner pressure on the cross-sectional area of the vessel, FD, and on the flange, between diameters D3 and D1, written as FT, where ( Fig. 1,b), The bending moment applied to the bolts, determined by the partial unclamping forces FD and FT has the expression (Fig 1, b), where, for the case represented in Figure 1, (9) where F(ΔT) is calculated with the relation given in works [8][9][10]. The total force in the bolts in service, becomes, . (10) d. Flange stress loading. First (Fb,0), the force in the bolts is applied during pre-clamping -the internal pressure is introduced, the stress loading caused by it emerges, which generates the bending moment (Mb,H).
When clamping the bolts, force causes the flanges to rotate around the circumference of diameter D3, characteristic of the sealing gasket location. A bending moment is created, (11) where for the case in Figure 1, (12) When in service (under pressure), each flange is subjected to a total bending moment, when Optimally, when one acquires the total bending moment applied to the flanges, (13), one replaces FG with * G F and one gets the total bending moment,

e. Stresses in bolts and flanges
 Bolts undergo tensile stresses featuring force Stresses in bolts have the following expressions: The maximum bending stress in the flange is, where Mb,t is given by the relation (13) or (14), and the modulus of strength of the flange section is, The maximum flange bending stress can be calculated with some relationships to be found in works [10][11][12][13]. For example, if: -the width of the flange plate is relatively small compared to the median diameter of the flange, the corresponding stress can be calculated with the relationship, -the width of the flange plate is not small as compared to the median diameter of the flange plate,

f. Stress in the shell to which the flange is welded
In order to check the shell thickness (4 or 5 in Fig. 1) to which the flange is welded, the bending moment theory [12][13][14] is used. For example, for the flange in Figure 1, one calculates the stress in the shell wall whose calculation thickness is , (15) where M0 is the bending moment acting in the junction section between the flange and the shell (between 4 and 1 in Figure 1) and which can be calculated with the Eq. [12], is the damping coefficient and  is Poisson's coefficient for the flange material. For steel, with

Calculation for testing flange joints
In design, a flange joint is first selected based on the service parameters. Then its components are verified. At present [1][2][3] -total deterioration to the assembly components, Such an approach to the calculation of strength is possible due to the use of the principle of critical energy [4][5][6][7]. In addition to the calculation of strength, our work has separately put forth a leak-tightness calculation as well. a. Bolt strength testing. Bolt and flange materials are believed to behave linearly-elastic, according to Hooke's law, (16) where σ is the normal stress,  -strain, E -the modulus of longitudinal elasticity (Young).
Stress loading must not exceed the admissible state, which means that the maximum stresses are lower than the yield limit and, consequently, the behavior of the material is linear -elastic according to the law (16).
In service, the bolts undergo a tensile force and a bending moment. This is a case of stress superposition to be solved on the basis of Energonics [4;7], using the principle of critical energy [4][5][6][7] and the law of equivalence of processes and phenomena [15] from Energonics.
 The calculation is based on the use of the concept of participation of the specific energy introduced by stress loading, in relation to the admissible state, and the admissible participation concept, both of which are nondimensional variables dependent on the behavior of the material under stress.
For bolts the following relationships are obtained: -the total contribution of the specific energies corresponding to the stress loadings, with respect to the allowable status, (17) where b  is the tensile stress in the bolt produced by force Fb and   (20) From relations (17) and (20), according to the law of equivalence of processes and phenomena [4; 7; 15], there results the equivalent bolt bending stress, The condition of stress loading admissibility is, in this case, where, however, for generality   where   where   where   c

. Leak-tightness condition
The leak tightness condition results from the observance of the first relationship (4). It should be remembered that the gaskets made of elastoviscous materials relax over time. Gasket relaxation is equivalent to the reduction in time of factor   t m m  in the relationship of sealing pressure   sp p . It should be also considered that the sealing material often behaves non-linearly [18; 19].
As a result, the sealing force, FG or * G F , decreases over time. This causes the bolt force to decrease over time, which explains the fact that, in some cases, sometimes "inexplicably", the flange joint loses its tightness. At present, design rules [1][2][3] prescribe unique time independent values for the m factor of each type of sealing gasket. This problem, regarding the dependence of m on service time must be investigated and remedied.
After an operating time t, the effective pressure on the gasket is, The leak -tightness condition is verified if: e. Fatigue life prediction. In the case of cyclic loading may be calculate the fatigue life using the results obtained in the paper [20], concerning the simultaneous cyclic loading with blocks of normal and shear stresses.
Analysis of the proposed calculation method Practically, the proposed calculation of flange joints is a calculation meant to test the strength of the bolts, flanges and leak-tightness.
The proposed calculations in this paper are more general than those in official regulations, but they are easier to apply because they rely directly only on primary variables that are easy to understand and process by the designer.
The proposed calculation method comprises the following sequence: -set the regime parameters (p; Tf; Ts) and calculate pressure ( 24)); -test the strength of the shell to which the flange is welded (rel. (28)); -calculate the pressure on the gasket required by pre-clamping (rel. (4)); -calculate the effective pressure on the gasket (qef(t)) after operating for a time, t, according to relationship (rel. (26)); -test the leak-tightness condition (rel. (27)).

Conclusions
New relationships have been proposed for calculating the strength of flange joints corresponding to actual conditions.
When calculating the stress in the bolts: -one also took into consideration the bending moment caused by the unclamping force; -the superposition of stress loads caused by the bolt pressure and bolt force based on the principle of critical energy.
The admissible reference condition for bolts also includes the influences of residual stresses and deterioration. When calculating flange strength, one took into consideration, in writing the expression for allowable stress, the possible influence of residual stresses as well as the flange deterioration.
One also introduced, distinctly, a calculation for testing the flange joint leak-tightness as well as the strength of the shell to which the flange was welded.
The sequel of calculations presented in the paper allows the implementation of the proposed relations and represents a new complete method for calculating flange joints.