1. Reference problem#

1.1. Geometry#

The study concerns a pipe comprising two straight pipes and an elbow [Figure 1.1-a].

The geometric data for the problem is as follows:

  • the length \({L}_{G}\) of the two straight pipes is \(3m\),

  • the radius \(\mathit{Rc}\) of the elbow is \(0.6m\),

  • the angle \(\theta\) of the elbow is \(90°\),

  • the \(\text{e}\) thickness of the straight pipes and the elbow is \(0.02m\),

  • and the outside radius \(\text{Re}\) of the straight pipes and the elbow is \(0.2m\).

_images/Object_1.svg

Figure 1.1-a

Note:

The geometry of the problem is symmetric with respect to the plane \((A,X,Y)\) .

1.2. Material properties#

For all models A, B, C and D:

Isotropic linear elastic material. Material properties are those of \(\mathrm{A42}\) to \(20°C\) steel:

  • Young’s module \(E=204000.\times {10}^{+6}N/{m}^{2}\),

  • Poisson’s ratio \(\mathrm{\nu }=0.3\).

For thermo-elastic calculation (modeling A):

  • the thermal expansion coefficient \(\mathrm{\alpha }=1.096\times {10}^{–5}/°C\),

  • heat conduction \(\mathrm{\lambda }=54.6W/m°C\),

  • volume heat \(\mathrm{\rho }\mathit{Cp}=3.71\times {10}^{6}J/{m}^{3}°C\),

For dynamic calculation (B, D models):

  • density \(\rho =7800\mathrm{kg}/{m}^{3}\),

  • the depreciation of clean modes will be taken to \(\text{5\%}\) for modes.

1.3. Boundary conditions and loads#

The boundary conditions for all models are as follows:

  • embedding at the level of section \(A\),

  • during static loads, embedding at the level of section \(A\) and section \(B\).

As far as static calculations are concerned, the loads applied are of three types:

  • internal pressure (on the inner side) \(P={15.10}^{+6}N/{m}^{2}\) (shell or 3D modeling),

  • thermo-mechanical loading with a temperature transient imposed on the inner face of the pipe (rise from \(20°C\) to \(70°C\) in \(\mathrm{10s}\)) and a zero exchange condition on the outer face of the pipe (thermal insulation) (modeling A only).

As far as dynamic calculation is concerned, the load applied is a transitory force (in Newtons):

\(\mathit{FY}(t)=10000000.\mathrm{sin}(2\mathrm{\pi }\mathit{Freq}1\mathrm{.}t)\) directed along the \(Y\) axis and applied to section \(B\) with \(\mathit{Freq}1=20\mathit{Hz}\).