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Sealing joints with elastic edge

It was cylindrical shell elements that were mainly considered, as technologically simpler ones. The angle of the cone α is usually accepted to be 10-15°, in order to avoid self-braking (jamming) of the loaded sealing joint. The long service life of the sealing joint implies the absence of plastic deformations of the shell element (excluding, of course, the deformation area of the sealed joint, i.e. the sealing line), therefore, usually hardened steel, bronze, forming a wear-resistant friction pair with the spool valve, are used as the material of the shell element.


Introduction
Nowadays, in the valve industry in metal-to-metal seals, there are mainly two types of sealing joints used: flat and conical. Flat surfaces are technologically easier to process, achieving accurate shape and minimal roughness, using grinding-in of sealing surfaces [1][2][3][4][5][6]. In conical sealing joints, that are more demanding on manufacturing accuracy, technological flaws can be compensated by increased contact pressures due to the wedge effect. Despite the presence of recommendations and regulatory documents, the question of the width of the seal band for both flat and conical sealing joints has not been unambiguously resolved. Thus in the works [7,11] the question of minimizing the sealing pressure is raised by choosing the width of the sealing area and the angle of taper. In any case, even with regard to wear, the authors are talking about the contact surface, which is very difficult to maintain in the required condition during operation due to thermal cycling, installation loads, impact effects, and technological features of operation.
As a result, the ground sealing surfaces are deformed, deviations of form and waviness appear, that requires the corresponding increase in sealing force, the use of special overlay welding, etc.
A fundamentally different sealing scheme is used in sealing joints with shell elements or simply with shells. Here the sealing surface is not formed in advance; seal is not flat but is carried out along a line. Due to the fact that, during the operation of the valve, the parameters of the working medium (pressure) change, the sealing force changes (the parameters of the drive systems are also not constant), the shell is de-formed, and thus it is possible to speak about the non-constancy of the sealing line. Each time the valve is triggered or the flange connection is loaded, different parts of the sealing elements are in contact.
Therefore, grinding-in of sealing joints using deformable shell elements is meaningless. Shell elements can be used as a spool valve or seat, as shown in Figure 1. To implement self-sealing, the cone may come in contact with the shell element along the inner or outer surface, as shown in Figure 2. On the review of sealing joints with the use of thin-walled elements, more than 70 home-produced and foreign seals are mentioned, which indicates the interest of valve manufacturers to this type of sealing joints.

Results and Discussion
The first publication where sealing joints with a shell element are considered is the paper [11], the static and dynamic calculations of the shell element are given in more detail in the work [7]. It was cylindrical shell elements that were mainly considered, as technologically simpler ones. The angle of the cone α is usually accepted to be 10-15°, in order to avoid self-braking (jamming) of the loaded sealing joint. The long service life of the sealing joint implies the absence of plastic deformations of the shell element (excluding, of course, the deformation area of the sealed joint, i.e. the sealing line), therefore, usually hardened steel, bronze, forming a wear-resistant friction pair with the spool valve, are used as the material of the shell element.
In contrast to a flat or conical sealing joint warped as a result of, for example, thermal cycling, the shell element easily assumes the form of a spool valve, fits tightly to it, therefore the sealing efforts for that kind of a connection are closer to the metal-polymer seal than to the flat or conical metal-to-metal seals. For a proper fit of the shell to the mating conical surface, the shell should be made as thin-walled as possible (while reducing the required sealing force), but ensuring the work of the shell in an elastic zone. If for flat and tapered sealing joints there are recommendations for choosing the geometrical dimensions of the deformation area of the sealing joint, then for the shell element in each specific case it is necessary to perform a strength calculation. The basic computational scheme of the shell seat is shown in Figure  3. The determination of the intensity of the loads T and Q for various design variants of the sealing joint is considered in [7].  The solution of the equation (1) is sought in the form: The structure of the Krylov functions as a combination of elementary functions, including the and the stresses on the shell surface are determined by the expressions: Further, using the accepted hypothesis of strength, it is concluded that the choice of geometrical dimensions is correct.
If the shell element works under shock loading (valve), then the adopted computational scheme of the shell is the same, however, T and Q are set taking into account the loading dynamics [8][9][10][12][13][14][15][16]. In general, the maximum dynamic load that occurs when the valve is triggered can be determined from the expression: 2 12 2( )( tg tg( )), where F is the static sealing force; In the derivation of (6), 0 of E  was assumed for simplicity, , which is used in the safety margin. The definition of оf E with the well-known circular plot of the end of a shell element is described in [7], оf E can be used as a criterion for the rejection of warped shells.
One of the ways to reduce the maximum dynamic load that occurs during shock loading of the shell seat in the valves is to reduce the unit stiffness of the sealing joint. This can be obtained by using the shell-plate seat shown in Figure 4. If the stiffness of the plate element is denoted as 3 c , then the maximum dynamic load during the shock landing of the spool valve will be determined by the expression: In fact, there is a sequential attachment of the stiffness  (7)) under the condition that the permissible stress in the shell and plate elements is not exceeded.
When dynamically calculating the valve's impact response, the loading of the shell element by the pressure of the working medium is not taken into account, because in a fraction of a second, while the transient process of shock loading lasts, the pressure of the working medium does not load the shellplate element.
However, after a while the loading scheme will change. The forces T and Q will take their stationary (static) value, but there will be a pressure load of the working medium. Therefore, for selected optimal dimensions of the shell-plate seat, a verification calculation should be performed to check the strength under the new operating conditions of the sealing joint.
If the shell is loaded not dynamically, but statically (a flange connection), then its strength can be determined not by T and Q, but by pressure p. In this case it is more profitable to make the shell shorter than In the works [11,[17][18] an attempt was made to link the length of the shell element with the pressure of the working medium, but the recommendations given were obtained on the basis of solving particular problems.
To experimentally verify the correctness of the developed analytical model for determining the stiffness parameters of a shell-plate seat, an experimental installation was performed, shown in Fig. 5. In addition to experimental studies, finite element modeling was performed in APM WinMachine and MSC vN4W ( Figure 6), which showed a satisfactory agreement between the analytical, numerical and experimental results.