Bibliography
=============

[:ref:`1 <1>`] J.L. LILIEN: Electromechanical constraints and consequences related to the passage of current intensity in cable structures. Thesis. University of Liège (1983).

[:ref:`2 <2>`] J.C. SIMO, L. VU- QUOC: A three-dimensional finite-strain rod model. Part II: computational aspects, Comput. Meth. call. Mech. Engng, Vol. 58, p.79-116 (1986).

[:ref:`3 <3>`] A. CARDONA, M. GERADIN: A beam finite element nonlinear theory with finite rotations, Int. Mr Numer. Meth. Engng. Vol. 26, p.2403-2438 (1988).

[:ref:`4 <4>`] R.L. WEBSTER: On the static analysis of structures with strong geometric non-linearity. Computers & Structures 11, 137-145 (1980).

[:ref:`5 <5>`] N. GREFFET, F. VOLDOIRE, M. AUFAURE: Dynamic nonlinear algorithm. Document R5.05.05.

[:ref:`6 <6>`] K.J. BATHE: Finite element procedures in engineering analysis. Prentice-Hall (1982).

[:ref:`7 <7>`] H.M. HILBER, T.J.R. HUGHES, R.L., R.L. TAYLOR: Improved numerical dissipation for time integration algorithms in structural dynamics. Earthq. Engng Struct. Dyn. 5, 283-292 (1977).

[:ref:`8 <8>`] F. DURAND: Three-phase line range (102m) for THT stations. Test report. EDF (1990).

**Document version history**

.. csv-table::

    "**Doc index**", "**Version**
    **Aster**", "**Author (s) or**
    **contributor (s),**
    **organization**", "**Description of changes**"
    "A", "3", "M. AUFAURE 1
    G. DEVESA1
    1 EDF /R & D", "Initial text"
    "B", "9.4", "J.L. FLEJOU
    EDF /R &D/ AMA ", "The temperature is no longer affected by the AFFE_CHAR_MECA command but by affe_materiau/affe_varc"


.. csv-table::

    "C", "12.3", "F. VOLDOIRE
    EDF /R &D/ AMA ", "Some corrections of form (§ 1, 3, 4...) and references in nonlinear dynamics § 10."




: Calculation of Laplace forces between conductors
Any conductor that a current flows through creates a magnetic field in its vicinity. This magnetic field, acting on the current carried by another conductor, induces on this one a so-called Laplace force.


.. image:: images/1000000000000182000000E4E83A96A032DE10B1.png
    :width: 4.0126in
    :height: 2.3681in


.. _RefImage_1000000000000182000000E4E83A96A032DE10B1.png:

Figure 5: Arrangement of two neighboring conductors

Let's take a conductor :math:`(1)` through which the current :math:`{i}_{1}(t)` [:ref:`Figure A1-a <Figure A1-a>`] flows, located in the vicinity of the conductor :math:`(2)` through which the current :math:`{i}_{2}(t)` flows. At the point P of conductor :math:`(1)`, where the unit tangent oriented in the direction of the current is :math:`{e}_{1}`, the linear Laplace force induced by conductor :math:`(2)` is:

:math:`f(P)={\text{10}}^{-7}{i}_{1}(t){i}_{2}(t){e}_{1}\times {\int }_{{\Gamma }_{2}}^{}{e}_{2}\times \frac{r}{{r}^{3}}{\mathrm{ds}}_{2}`

We are only interested in the forces due to very intense short-circuit currents, the Laplace forces under normal conditions being negligible.

:math:`f(P)` can obviously be put in the form of the product of a function of time by a function of space.

**A1.1** **Function of the time of the Laplace forces**

This function :math:`g(t)`, except for one factor, is the product of the intensities in conductors :math:`(1)` and :math:`(2)`:

 :math:`g(t)=2.{\text{10}}^{-7}{i}_{1}(t){i}_{2}(t)` [:ref:`12 <12>`]

where:

 :math:`{i}_{j}(t)=\sqrt{2}{I}_{\text{ej}}\left[\text{cos}(\omega t+{\varphi }_{j})-{e}^{-t/\tau }\text{cos}{\varphi }_{j}\right]` [:ref:`13 <13>`]

with:

.. csv-table::

    ":math:`{I}_{\text{ej}}`:", "effective current strength j;"
    ":math:`\omega`:", "current pulsation (:math:`\omega =100\pi` for a current of 50Hz);"
    ":math:`{\varphi }_{j}`:", "phase depending on when the short circuit occurs;"
    ":math:`\tau`:", "time constant of the short circuit line depending on its electrical characteristics (self, capacitance and resistance)."


Very often, we replace the complete function :math:`g(t)` [eq] and [eq] by its mean - which is called the continuous part - by neglecting the terms :math:`\mathrm{cos}(\omega t+\mathrm{...})` and :math:`\mathrm{cos}(2\omega t+\mathrm{...})`. Taking these terms into account would require a very small step of time and the corresponding forces, at :math:`50\mathrm{Hz}` and :math:`\mathrm{100 }\mathrm{Hz}`, have almost no effect on cables whose oscillation frequency is of the order of one Hertz.

So:

:math:`{g}_{\text{continue}}(t)=2{I}_{\mathrm{e1}}{I}_{\mathrm{e2}}\left[\frac{1}{2}\text{cos}({\varphi }_{1}-{\varphi }_{2})+{e}^{-(\frac{t}{{\tau }_{1}}+\frac{t}{{\tau }_{2}})}\text{cos}{\varphi }_{1}\text{cos}{\varphi }_{2}\right]`

**A1.2** **Space function**

This function is:

:math:`h(P)=\frac{1}{2}{e}_{1}\times \int {e}_{2}\times \frac{r}{{r}^{3}}{\mathrm{ds}}_{2}`

The integral is calculated analytically when conductor :math:`{\Gamma }_{2}` is divided into rectilinear elements. Along with such an element :math:`{M}_{1}{M}_{2}` [:ref:`Figure A1.2-a <Figure A1.2-a>`], we have an effect:

:math:`\begin{array}{}{r}^{3}={({y}^{2}+{r}_{m}^{2})}^{}\\ {e}_{2}\times r={e}_{2}\times {r}_{m}\\ {\mathrm{ds}}_{2}=\text{dy}\end{array}`


.. image:: images/10000000000001A10000012DA56BE206C07472A8.png
    :width: 4.5252in
    :height: 3.2689in


.. _RefImage_10000000000001A10000012DA56BE206C07472A8.png:

Figure 6: Laplace force induced by a rectilinear conductor element

Like:

:math:`{\int }_{{y}_{1}}^{{y}_{2}}\frac{\text{dy}}{{({y}^{2}+{r}_{m}^{2})}^{}}=\frac{1}{{r}_{m}^{2}}{\left[\frac{y}{{({y}^{2}+{r}_{m}^{2})}^{}}\right]}_{{y}_{1}}^{{y}_{2}}`

we have:

:math:`\frac{1}{2}{e}_{1}\times {\int }_{{M}_{1}}^{{M}_{2}}{e}_{2}\times \frac{r}{{r}^{3}}\mathrm{dy}=\frac{1}{2{r}_{m}^{2}}{e}_{1}\times {e}_{2}\times {r}_{m}{\left[\frac{y}{{({y}^{2}+{r}_{m}^{2})}^{}}\right]}_{{y}_{1}}^{{y}_{2}}`

The hook on the second member is also equal to:

:math:`\text{sin}{\alpha }_{2}-\text{sin}{\alpha }_{1}`.

**A1.3** **Realization in***Code_Aster*

The space function :math:`h(P)` defined above is calculated by an elementary routine that evaluates, for each of the elements of the conductor :math:`(1)`, the contribution of all the elements of the conductor :math:`(2)` that act on it.

This contribution is evaluated at the Gauss points (only 1 for 2-node elements) of the conductor element, :math:`(1)` by the INTE_ELEC keyword of the AFFE_CHAR_MECA command.

The basic routine has 2 input parameters:


* the loading map of the conductor element :math:`(1)` including the list of the cells of the conductor :math:`(2)` acting on it;


* the name of the geometry, which varies over time, which allows quantities :math:`{r}_{m},\text{sin}{\alpha }_{1},\text{sin}{\alpha }_{2}` to be evaluated at any moment.

The time function :math:`g(t)` is calculated by the operator DEFI_FONC_ELEC which produces a function-like concept.

**A1.4** **Use in***Code_Aster*


* Definition of the function of time :math:`g(t)`


.. csv-table::

    "**Command**", "**Factor Key**", "**Keyword**", "**Argument**"
    "", "", "", "", "'COMPLET'"
    "DEFI_FONC_ELEC ", "", "SIGNAL ", "or"
    "", "", "", "", "'CONTINU'"
    "", "COUR "," INTE_CC_1 "," :math:`{I}_{{e}_{1}}`"
    "", "", "TAU_CC_1 "," :math:`{\tau }_{1}`"
    "", "", "PHI_CC_1 "," :math:`{\varphi }_{1}`"
    "", "", "INTE_CC_2 "," :math:`{I}_{{e}_{2}}`"
    "", "", "TAU_CC_2 "," :math:`{\tau }_{2}`"
    "", "", "PHI_CC_2 "," :math:`{\varphi }_{2}`"
    "", "", "INST_CC_INIT ", "Short-circuit start time."
    "", "", "INST_CC_FIN ", "Short-circuit end instant."



* Definition of the function of space :math:`h(P)`


.. csv-table::

    "**Command**", "**Factor Key**", "**Keyword**", "**Argument**"
    "AFFE_CHAR_MECA ", "", "MODELE ", "Model name."
    "", "INTE_ELEC "," GROUP_MA ", "Name of the group of elements for conductor :math:`(1)`."
    "", "", "GROUP_MA_2 ", "Name of the group of elements of the :math:`(2)` conductor."



* Taking into account Laplace's strengths


.. csv-table::

    "**Command**", "**Factor Key**", "**Keyword**", "**Argument**"
    "DYNA_NON_LINE "," EXCIT "," CHARGE ", "name of :math:`h(P)`"
    "", "", "FONC_MULT ", "name of :math:`g(t)`"




: Calculation of the force exerted by the wind
**A3.1** **Formulation**

It is assumed that a wind of speed :math:`V` exerts in the vicinity of the point :math:`P` of a cable [Figure] an aerodynamic linear force :math:`f` having the following characteristics:


* :math:`f` has the direction and direction of the :math:`{V}_{n}` wind speed component in the normal plane of the cable.


* :math:`f` has a modulus that is proportional to the square of that of :math:`{V}_{n}`.


.. image:: images/1000000000000146000000D92E924E1B9BD49B1B.png
    :width: 3.3382in
    :height: 1.9252in


.. _RefImage_1000000000000146000000D92E924E1B9BD49B1B.png:

Figure 7: Wind speed in the vicinity of a cable

The rules for calculating lines define the strength of a wind by the pressure :math:`p` it exerts on a flat surface normal to its direction. For a cable, normally placed in the direction of the wind, these regulations prescribe to take the linear force as follows: :math:`{f}_{\frac{\pi }{2}}\mathrm{=}p\mathrm{\varnothing }`,

:math:`\varnothing` being the diameter of the cable. This is equivalent to considering that the cable offers the wind a flat surface equal to its master torque. An increase in force is thus obtained because the cable, which is cylindrical, has less air resistance than a flat surface.

If wind speed :math:`V` makes an angle :math:`\theta` with the cable, its component in the plane perpendicular to the cable has the module: :math:`\parallel {V}_{n}\parallel =\parallel V\parallel \mathrm{.}\mid \text{sin}\theta \mid`.

So the linear force is: :math:`{f}_{\theta }\mathrm{=}p\mathrm{\varnothing }{\text{sin}}^{2}\theta`.

Of course, the linear force exerted by the wind depends on the position of the cable: it is "following".

**A3.2** **Use in***Code_Aster*

Here's how to introduce wind force in*Code_Aster*. The unit vector with the direction and direction of wind speed has components :math:`{v}_{x},{v}_{y},{v}_{z}`.


.. csv-table::

    "**Command**", "**Factor Key**", "**Keyword**", "**Argument**"
    "AFFE_CHAR_MECA "," FORCE_POUTRE "," TYPE_CHARGE ", "'VENT'"
    "", "", "FX", ":math:`p\varnothing {v}_{x}`"
    "", "", "FY", ":math:`p\varnothing {v}_{y}`"
    "", "", "FZ", ":math:`p\varnothing {v}_{z}`"




: Modeling cable installation
A cable being laid in a canton (several spans between posts) [Figure] is fixed to one of the stop supports. It rests on pulleys placed at the bottom of the alignment insulators and is retained by a force at the level of the second stop support.


.. image:: images/10000000000001B200000162A1FA24FBAD8C93B8.png
    :width: 3.889in
    :height: 2.8339in


.. _RefImage_10000000000001B200000162A1FA24FBAD8C93B8.png:

Figure 8: Laying a cable in a two-span township

By playing on this force - or by moving its point of application - we adjust the arrow of one of the ranges, the one that is most subject to environmental constraints. Then the pulleys are removed and the cable is fixed to the insulators: the length of the cable in the various ranges is then fixed. It is on this configuration that additional components are possibly mounted: spacers, equipment descents, point masses,... to give the canton its final shape.

This modeling is described in the reference documentation for elements CABLE_POULIE [:external:ref:`R3.08.05 <R3.08.05>`].