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


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

Analytical solution:

In short trips:


* in elastic regime (for :math:`F<{\sigma }_{Y}`) the theoretical critical value corresponds to the Euler load. In the context of beam kinematics, the critical load is equal to:


.. image:: images/Object_1.svg
    :width: 81
    :height: 44


.. _RefImage_Object_1.svg:

, so the critical pressure:

.. image:: images/Object_2.svg
    :width: 81
    :height: 44


.. _RefImage_Object_2.svg:

with

.. image:: images/Object_3.svg
    :width: 81
    :height: 44


.. _RefImage_Object_3.svg:

and

.. image:: images/Object_4.svg
    :width: 81
    :height: 44


.. _RefImage_Object_4.svg:

either

.. image:: images/Object_5.svg
    :width: 81
    :height: 44


.. _RefImage_Object_5.svg:


* under elastoplastic conditions, as uniform compression without elastic discharge is considered and due to the law of behavior, the critical buckling load is equal to:


.. image:: images/Object_6.svg
    :width: 81
    :height: 44


.. _RefImage_Object_6.svg:

or a critical pressure of:

.. image:: images/Object_7.svg
    :width: 81
    :height: 44


.. _RefImage_Object_7.svg:


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

Critical load values for both load cases.

In elastic mode, for :math:`F<4\mathrm{MPa}`, which is :math:`t<0.61538462`, we have to get: :math:`{P}_{\mathrm{cr}}=12.95\mathrm{MPa}`.

Under plastic conditions the critical buckling pressure value is: :math:`\mathrm{4,32}\mathrm{MPa}`.

The critical coefficients as a function of loading are:


.. csv-table::

    "**No time**", "**Surface force**
    **(en** :math:`\mathrm{MPa}` **)**", "**Critical Coefficient**", "**Critical Load**
    **(in** :math:`\mathrm{MPa}` **)**"
    "1", "0.65", "19.9290", "12.9539"
    "2", "1.3", "9.9645", "12.9539"
    "3", "1.95", "6.6430", "12.9539"
    "4", "2.6", "4.9823", "12.9539"
    "5", "3.25", "3.9858", "3.9858", "12.9539"
    "6", "3.9", "3.3215", "12.9539"
    "**7**", "4.55", "0.9490", "4.3180"
    "**8**", "5.2", "0.8304", "4.3180"
    "**9**", "5.85", "0.7381", "4.3180"
    "**10**", "**6.5**", "0.6643", "4.3180"