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

Calculation method used for the reference solution
--------------------------------------------------------

A loading path is reproduced on an element in each of the 3 domains (elastic, plastic, limit) of a unidirectional behavior relationship (local direction :math:`y`). The parameters are described in the attached Figure 1.

The charging path comprises 12 steps defined as follows:


.. image:: images/10000DD200001F51000012EA189C4DD3F5FE5626.svg
    :width: 403
    :height: 244


.. _RefImage_10000DD200001F51000012EA189C4DD3F5FE5626.svg:


Benchmark results
----------------------

Direct calculations on the limit curve of the behavioral relationship:

:math:`{F}_{y}={k}_{\mathrm{el}}\cdot {U}_{y}` if :math:`{U}_{y}<{d}_{e}`

:math:`{F}_{y}={k}_{\mathrm{el}}\cdot {d}_{e}+{k}_{\mathrm{pl}}({U}_{y}-{d}_{e})` if :math:`{U}_{y}\in \left[{d}_{e},{d}_{l}\right]`

:math:`\begin{array}{}\mathrm{Vari}={U}_{y}-{d}_{e}\\ \mathrm{Varimax}={d}_{l}-{d}_{e}\end{array}`

:math:`{F}_{y}={k}_{\mathrm{el}}\cdot {d}_{e}+{k}_{\mathrm{pl}}({d}_{l}-{d}_{e})+{k}_{G}({U}_{y}-{d}_{l})` if :math:`\mathrm{Vari}=\mathrm{Varimax}`


Uncertainty about the solution
---------------------------

Exact solution: :math:`\mathrm{Fy}` imposed and :math:`\mathrm{Uy}` deduced directly from relationships in [:ref:`§2.2 <§2.2>`].


Bibliographical references
---------------------------

Note HM-77/94/368, G. DEVESA. "Dynamic study of conductor failure and frost discharge on an experimental medium voltage line".