Static problem
=================

This problem is that of seeking the balance of a cable structure in any position and subject to a system of given forces.




Iterative algorithm
-------------------

The equilibrium equation, the discretized form of [eq] and [eq] without the inertia term, which must be satisfied at each node, is:

 :math:`{F}_{\text{int}}={F}_{\text{ext}}` [:ref:`10 <10>`]

Suppose we've just calculated the cable displacement field, :math:`{u}^{n}({s}_{o})`, in iteration :math:`n`:


* if this field makes it possible to satisfy, with one tolerance, [:ref:`éq 9.1-1 <éq 9.1-1>`], the line is considered to be:


*

:math:`x({s}_{o})+{u}^{n}({s}_{o})`

is the balance figure of the cables;


* otherwise, displacement corrections :math:`\Delta {u}^{n+1}` are calculated by the linearized system:


*

:math:`\left[{K}_{M}^{n}+{K}_{G}^{n}\right]\Delta {u}^{n+1}={F}_{\text{ext}}-{F}_{\text{int}}^{n}`

For this purpose, the nonlinear, quasistatic algorithm described in [:external:ref:`R5.03.01 <R5.03.01>`] is used, and which corresponds to the STAT_NON_LINE command. The move in iteration :math:`(n+1)` is:

:math:`{u}^{n+1}={u}^{n}+\Delta {u}^{n+1}`

We see if [eq] is satisfied by field :math:`{u}^{n+1}` and so on.




example
-------

We want to calculate the equilibrium figure of a heavy cable [:ref:`fig 9.2-a <fig 9.2-a>`] whose one end :math:`A` is fixed and whose other end, :math:`B` at level with :math:`A`, is subject to a given horizontal force.

This problem is addressed in [bib], where it is considered to be highly nonlinear.


.. image:: images/100000000000025A00000103994B3D839D9D39E1.png
    :width: 6.2598in
    :height: 2.6874in


.. _RefImage_100000000000025A00000103994B3D839D9D39E1.png:

Extensional stiffness (E.A): 4.45 x 105 N, Line weight: 1.46 N/m

Figure 2: Balance of a heavy cable subjected to horizontal tension

Initially, the cable, modelled by 10 elements of the 1st degree, is assumed to be weightless and has a rectilinear horizontal position :math:`{\mathrm{AB}}_{o}`. It is simultaneously subjected to the action of gravity and to the horizontal force :math:`F` applied in :math:`{B}_{o}`. Static equilibrium position :math:`\mathrm{ACB}` is reached in only 8 iterations.