Benchmark solution
=====================


Equations of balance
---------------------

Effort at point :math:`\mathit{A1}`

:math:`\mathit{Fa}\mathrm{=}`

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Effort at point :math:`\mathit{B1}`

:math:`\mathit{Fb}\mathrm{=}`

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    :width: 601
    :height: 77


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Effort due to the wind


* Wind speed at one point :math:`M\mathrm{\in }\mathit{barre}`

:math:`{V}_{r}\mathrm{=}`

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with :math:`\mathit{Vvx}`, :math:`\mathit{Vvy}`: wind speed along the :math:`x` axis and the :math:`y` axis.


* Relative speed perpendicular to the bar at point :math:`M`:

:math:`{V}_{p}\mathrm{=}`

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* Force due to wind at one point :math:`M`

:math:`{\mathit{Fvent}}_{(M)}\mathrm{=}{\mathit{Fcx}}_{(M)}`

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In our case we choose :math:`{\mathit{Fcx}}_{(M)}\mathrm{=}\mathrm{\parallel }{V}_{p}\mathrm{\parallel }`

So we get :math:`{\mathit{Fvent}}_{(M)}\mathrm{=}{V}_{p}`


* Resultant of the force due to the wind on the bar

Wind =

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Equation of balance: :math:`\mathit{Fa}+\mathit{Fb}+\mathit{Fvent}\mathrm{=}0`


Reference quantities and results
-----------------------------------

Displacement of points :math:`\mathit{A1}` and :math:`\mathit{B1}` at the times: :math:`1.s`, :math:`1.05s` and :math:`2.s`. These times correspond to wind speeds of :math:`10`, :math:`15` and :math:`\mathrm{20m}\mathrm{/}s` respectively.

The resolution of the 3 equilibrium equations, projection of :math:`\mathit{Fa}+\mathit{Fb}+\mathit{Fvent}\mathrm{=}0`, is done by iterations. The 3 unknowns of the problem are the position of the center of gravity of the bar :math:`G`: :math:`(x,y)` and the variation of the angle: :math:`\theta`.

In *Code_Aster*, the wind effect is taken into account by a force distributed along the line element. The expression for the modulus of this distributed force is as follows:

:math:`{\mathit{Fcx}}_{(v)}\mathrm{=}\frac{1}{2}\mathrm{.}\rho \mathrm{.}{V}^{2}\mathrm{.}\mathit{Cx}(v)\mathrm{.}{D}_{h}`

.. csv-table::

    "where", ":math:`{\mathit{Fcx}}_{(v)}`: is the modulus of the force distributed along the cable in :math:`N\mathrm{/}m`, depending on the speed."
    "", ":math:`\rho`: is the density of the air in :math:`\mathit{kg}\mathrm{/}{m}^{3}`.
    :math:`V`: is the relative speed of the cable in :math:`m\mathrm{/}s`.
    :math:`\mathit{Cx}(v)`: is the drag coefficient of the cable, depending on the relative speed.
    :math:`{D}_{h}`: is the hydraulic diameter of the cable in :math:`m`."


To obtain a simple analytical reference solution, the function :math:`{\mathit{Fcx}}_{(\mathit{Vp})}` is taken to be equal to :math:`\mathrm{\parallel }{V}_{p}\mathrm{\parallel }`. In the *Code_Aster* command file the :math:`\mathit{Fcxv}` function is therefore defined as follows:


.. code-block:: text

    FCXV = DEFI_FONCTION (

    NOM_PARA =' VITE ',

    VALE =( 0.0, 0.0,

    10.0, 10.0),

    PROL_GAUCHE =' LINEAIRE ',

    PROL_DROITE =' LINEAIRE ',

    )


Uncertainties about the solution
----------------------------

None. The balance equation is solved by iterations with an error less than :math:`1.0E\mathrm{-}09`.


Bibliographical reference
-------------------------


*HM77 /01/046/A. "Project M7-01-70. Evolution of the*Code_Aster* to better take into account dynamic wind loads on line elements".