1. Reference problem#
1.1. Geometry#
The structure studied is similar to the curved zone of a steam generator tube. It is supposed to be embedded at its two ends \(A\) and \(C\); the latter are supposed to correspond to the passage of the tube in the upper spacer plate of a steam generator. At point \(B\) at the top of the hanger (also called apex), the structure is guided into a game support.
Figure 1.1-a: Diagram of the curved zone of a GV tube, supposed to be embedded in the last spacer plate, guided into a game support
The structure is similar to a beam with a hollow circular cross section with a total length of \(\mathrm{1,74329}m\) and a straight \(\mathrm{250,43 }\mathrm{mm}\) at each end of the tube.
Outer tube diameter: \(\mathrm{22,22 }\mathrm{mm}\)
Inner tube diameter: \(\mathrm{19,68 }\mathrm{mm}\)
1.2. Material properties#
The values of the characteristics of the various elements of the structure are as follows:
Inconel 600 tube
\(E={\mathrm{2,02 10}}^{11}N/{m}^{2}\) |
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Internal fluid: |
the modelled internal fluid is water under pressure at high temperature; its density \({\rho }_{i}\) is assumed to vary linearly along the curved zone, between \(A\) and \(C\), between and, from \(\mathrm{738,58 }\mathrm{kg}/{m}^{3}\) to \(\mathrm{731,16 }\mathrm{kg}/{m}^{3}\). |
External fluid: |
the tube is supposed to be immersed in a two-phase mixture over its entire length; the flow of the mixture is transverse to the hanger at all points. The equivalent density of the mixture is obtained using the formula: |
\({\rho }_{e}\mathrm{=}\alpha {\rho }_{\mathit{gaz}}+(1\mathrm{-}\alpha ){\rho }_{\mathit{liquide}}\), where \(\alpha\) refers to the volume of gas.
This density appears to be between \(84\mathrm{kg}/{m}^{3}\) and \(150\mathrm{kg}/{m}^{3}\). An equivalent density is assigned to the dynamic system when calculating its modal fluid base at rest; this equivalent density includes the density of the internal fluid, that of the structure and that of the external fluid; the inertial effect of the latter is evaluated by means of an added mass coefficient.
1.3. Boundary conditions and loads#
The structure is embedded at points \(A\) and \(C\). A game stand is positioned at point \(B\) (\(\mathrm{jeu}=\mathrm{1,20 }\mathrm{mm}\)). A distributed random load, transverse to the tube, is imposed on \((A-C)\). This loading is defined, on the one hand using a speed profile along the excited zone, and on the other hand using a dimensionless excitation spectrum.
1.4. Initial conditions#
The tube is initially at rest, which results in zero displacements and velocities at the initial moment.