.. _v3.05.104-solution_reference:


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

Internal movements and efforts
--------------------------------

At node :math:`A` all movements are blocked.
At nodes :math:`B` and :math:`C` the load is an effort :math:`F` along the axis of the element.
The temperature is assigned to the nodes: :math:`Temper`.

The displacement due to thermal expansion in the axis of the element is:

.. math::
 U_ {th} =\ alpha_ {th}\ left (T - T_ {ref}\ right) L

The displacement due to the force in the axis of the element is:

* For beams: :math:`U_p =\frac{F L}{E S}`
* For the discreet: :math:`U_d =\frac{F}{K_x}`

The stiffness of the discrete is chosen so that the movements of the beams and the discretes are identical: :math:`K_x = \frac{E S}{L}`.

When the load is both the temperature and the effort:

* The total displacement is: :math:`U_{total} = U_{th} + U_{d} = U_{th} + U_{p}`.
* The internal effort is: :math:`F_{normal} = F`.
* The supporting reactions are:

  - For the following elements :math:`AB`: :math:`F_x = F`
  - For the following elements :math:`AC`: :math:`\left[ F_x, F_y, F_z \right] = -F \left[ \cos(\beta) cos(\alpha), \cos(\beta) sin(\alpha), \sin(\beta) \right]`



Uncertainties about solutions
------------------------------

None