Reference problem
=====================


Geometry
---------


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**Figure 1.1-a: Geometry (tests ELSA carried out by CEA/EMSI) .**

Mesh group:

 :math:`\mathit{POUTRES}`: set of straight pipes

 :math:`\mathit{COUDES}`: set of bent pipes

 :math:`\mathrm{PATVAN},\mathrm{VANNE},\mathrm{ENCBIS1},\mathrm{ENC1},\mathrm{ENC2}`

 :math:`\mathrm{ENCBIS2},\mathrm{PATBIELA},\mathrm{PATBIELB}`

 :math:`\mathrm{PATBIELC},\mathrm{BIELA},\mathrm{BIELB},\mathrm{BIELC},\mathrm{CDGVAN}`

Node group:

 :math:`A,\mathrm{...},L`

Pipe geometry:

:math:`\mathit{SECTION}A`


* Mesh groups: :math:`\mathit{POUTRES}` :math:`\mathit{COUDES}` :math:`\mathit{PATVAN}`


    * :math:`R=8.485\times {10}^{\text{-2}}m` Outer radius


    * :math:`\mathrm{EP}=7.345\times {10}^{\text{-3}}m` Thickness


* Mesh groups: :math:`\mathit{BIELA}`, :math:`\mathrm{BIELB}`, :math:`\mathrm{BIELC}`


    * :math:`R=38.05\times {10}^{\text{-3}}m` Outer radius


    * :math:`\mathrm{EP}=4.5\times {10}^{\text{-3}}m` Thickness

:math:`\mathit{SECTION}B`


* Mesh groups: :math:`\mathrm{PATBIELC}`


    * :math:`R=4.55\times {10}^{\text{-2}}` Radius


* Mesh groups: :math:`\mathit{PATBIELA}`


    * :math:`\mathrm{R1}=8.6\times {10}^{\text{-2}}m` Radius at end 1


    * :math:`\mathrm{R2}=4.55\times {10}^{\text{-2}}m` Radius at end 2


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**Figure 1.1-b: Geometry modeling.**


Elastic properties of materials
-----------------------------------


* :math:`\mathrm{POUTRES}`:


*
    * Young's modulus: :math:`E=1.9\times {10}^{11}\mathrm{Pa}`


    * Poisson's ratio: :math:`\nu \mathrm{=}0.3`


    * Density: :math:`\rho =1.30273\times {10}^{4}\mathrm{kg}{m}^{\text{-3}}`


* :math:`\mathrm{COUDE1},\mathrm{COUDE2},\mathrm{COUDE3},\mathrm{COUDE4}`:


*
    * Young's modulus: :math:`E=1.9\times {10}^{11}\mathrm{Pa}`


    * Poisson's ratio: :math:`\nu =0.3`


    * Density: :math:`\rho =1.47373\times {10}^{4}\mathrm{kg}{m}^{\text{-3}}`


    * Slope of the traction curve: :math:`D\text{\_}\mathrm{SIGM}\text{\_}\mathrm{EPSI}=7.67\times {10}^{9}N{m}^{\text{-2}}`


    * Elastic limit: :math:`\mathit{SY}\mathrm{=}121.2\mathrm{\times }{10}^{6}N{m}^{\text{-2}}`


    * Constant of :math:`\mathrm{PRAGER}`: :math:`C=5.328434\times {10}^{9}`

The elastic limit has been reduced in order to plasticize sooner.


* :math:`\mathrm{PATVAN}`:


*
    * Young's modulus: :math:`E=1.9\times {10}^{11}\mathrm{Pa}`


    * Poisson's ratio: :math:`\nu \mathrm{=}0.3`


    * Density: :math:`\rho =0.0\mathrm{kg}{m}^{\text{-3}}`


* :math:`\mathrm{PATBIELA}`:


*
    * Young's modulus: :math:`E=1.8\times {10}^{11}\mathrm{Pa}`


    * Poisson's ratio: :math:`\nu \mathrm{=}0.3`


    * Density: :math:`\rho =4.43\times {10}^{3}\mathrm{kg}{m}^{\text{-3}}`


* :math:`\mathrm{PATBIELC}`:


*
    * Young's modulus: :math:`E=1.8\times {10}^{11}\mathrm{Pa}`


    * Poisson's ratio: :math:`\nu \mathrm{=}0.3`


    * Density: :math:`\rho =6.92\times {10}^{3}\mathrm{Kg}/{m}^{3}`


* :math:`\mathrm{BIELA},\mathrm{BIELB},\mathrm{BIELC}`:


*
    * Young's modulus: :math:`E=1.8\times {10}^{11}\mathrm{Pa}`


    * Poisson's ratio: :math:`\nu \mathrm{=}0.3`


    * Density: :math:`\rho =6.86\times {10}^{3}\mathrm{kg}{m}^{\text{-3}}`


* :math:`\mathrm{POUTRES}`:


*
    * Young's modulus: :math:`E=1.9\times {10}^{11}\mathrm{Pa}`


    * Poisson's ratio: :math:`\nu \mathrm{=}0.3`


    * Density: :math:`\rho =1.47373\times {10}^{4}\mathrm{kg}{m}^{\text{-3}}`


* :math:`\mathrm{PATBIELB}` mass added:

:math:`[M]=\left[\begin{array}{ccc}m& 0.& 0.\\ 0.& m& 0.\\ 0.& 0.& m\end{array}\right]` with :math:`m=2.46\mathrm{kg}`


* :math:`\mathrm{CDGVAN}` (center of gravity of the valve) added mass:

:math:`[M]=\left[\begin{array}{cccccc}m& 0.& 0.& 0.& -{\mathrm{m.e}}_{z}& -{\mathrm{m.e}}_{y}\\ 0.& m& 0.& {\mathrm{m.e}}_{z}& 0.& -{\mathrm{m.e}}_{x}\\ 0.& 0.& m& -{\mathrm{m.e}}_{y}& {\mathrm{m.e}}_{x}& 0.\\ 0.& 0.& 0.& {V}_{\mathrm{xx}}& {V}_{\mathrm{xy}}& {V}_{\mathrm{xz}}\\ 0.& 0.& 0.& 0.& {V}_{\mathrm{yy}}& {V}_{\mathrm{yz}}\\ 0.& 0.& 0.& 0.& 0.& {V}_{\mathrm{zz}}\end{array}\right]` with :math:`\begin{array}{c}{V}_{\mathrm{xx}}={I}_{\mathrm{xx}}+m({e}_{y}^{2}+{e}_{z}^{2})\\ {V}_{\mathrm{yy}}={I}_{\mathrm{yy}}+m({e}_{x}^{2}+{e}_{z}^{2})\\ {V}_{\mathrm{zz}}={I}_{\mathrm{zz}}+m({e}_{x}^{2}+{e}_{y}^{2})\\ {V}_{\mathrm{xy}}={I}_{\mathrm{xy}}-{\mathrm{m.e}}_{x}{e}_{y}\\ {V}_{\mathrm{yz}}={I}_{\mathrm{yz}}-{\mathrm{m.e}}_{y}{e}_{z}\\ {V}_{\mathrm{xz}}={I}_{\mathrm{xz}}-{\mathrm{m.e}}_{x}{e}_{z}\end{array}`


*
    * :math:`m=275\mathrm{kg}` mass


    * :math:`\left[\begin{array}{c}{I}_{\mathrm{xx}}=2.696123\\ {I}_{\mathrm{yy}}=3.81480\text{}\\ \text{}{I}_{\mathrm{zz}}=0.9166667\\ \text{}{I}_{\mathrm{xy}}={I}_{\mathrm{xy}}={I}_{\mathrm{yz}}=0.\end{array}\right]` mass inertia tensor values


    * :math:`{e}_{x}={e}_{y}={e}_{z}=0.` component of the mass eccentricity vector

The non-linearity is due to the material behavior in the bends.

The law of behavior used is: under COMORTEMENT: RELATION =' VMIS_ECMI_LINE ', which corresponds to elastoplasticity with mixed isotropic and linear kinematic work hardening.


Boundary conditions and loads
-------------------------------------

**Travel required:**


* :math:`\mathrm{ENC1},\mathrm{ENCBIS1}`:

:math:`\mathit{DX}\mathrm{=}\mathit{DY}\mathrm{=}\mathit{DZ}\mathrm{=}\mathit{DRX}\mathrm{=}\mathit{DRY}\mathrm{=}\mathit{DRZ}\mathrm{=}0`


* :math:`\mathrm{ENC2},\mathrm{ENCBIS2}`:

:math:`\mathit{DX}\mathrm{=}\mathit{DY}\mathrm{=}\mathit{DZ}\mathrm{=}\mathit{DRX}\mathrm{=}\mathit{DRY}\mathrm{=}\mathit{DRZ}\mathrm{=}0`


* :math:`\mathrm{PATBIEL4}`:

:math:`\mathrm{DX}=\mathrm{DY}=\mathrm{DZ}=\mathrm{DRZ}=0`


* :math:`\mathit{PATBIEL3}`:

:math:`\mathit{DRZ}\mathrm{=}0`

**Imposed links:**


* :math:`\mathrm{PATVAN},\mathrm{VANNE}`: LIAISON_SOLIDE


* :math:`\mathrm{PATBIEL3},\mathrm{PATBIEL2}`: LIAISON_UNIF (DX, DY, DZ)

**Pressure imposed:**


* :math:`\mathrm{COUDES},\mathrm{POUTRES}`:

:math:`\mathit{PRES}\mathrm{=}120.5\mathit{Pa}`


*

**Seismic loading:**


* The structure is subjected to seismic loading in direction X. The imposed accelerogram is characterized by a total duration of :math:`\mathrm{40,95}s` and a time step of :math:`\mathrm{0,01}s`.


Initial conditions and list of calculation times
--------------------------------------------------

The system is initially at rest.

To limit time CPU, we will only conduct the numerical simulation until a final moment worth :math:`\mathrm{0,17}s`. At this moment, the level of loading reached is such that plasticity has already occurred.

We build a list of optimized calculation times: the time step increasing from :math:`\mathrm{0,01}s` to 0.001 s in order to go linear quickly and to reduce the time step just before plasticization.