Reference solution
----------------------


Calculation method
~~~~~~~~~~~~~~~~~~~~~~~

The reference solution comes from example 4 in
the Griffiths article [:ref:`1 <ref-1-v603507>`].

The safety factor was calculated using both method SRM ("strength reduction method")
based on the finite element method and the traditional method
assuming the sliding circles [:ref:`2 <ref-2-v603507>`].

The results of both methods are consistent.


Benchmark result
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

By varying the parameter :math:`\lambda` between 0.25 and 4,
the variation in the safety factor is plotted as a function of the foundation-slope ratio,
as illustrated in :numref:`fig2-FS-varie`.


.. figure:: images/10000201000002050000021A87FA547025CCC787.png
 :name: fig2-fs-varies
    :width: 40%

    **Plot of the change in FS as a function of the foundation-slope ratio**

.. _RefImage_10000201000002050000021A87FA547025CCC787.png:


During the hardening of the foundation, two mechanisms were identified of
breakdowns shown by :numref:`fig3a-mecanisme` and :numref:`fig3b-mecanisme`.

In the case of a weak foundation, the sliding circle deepens in
the foundation so that the slope slides in conjunction with the foundation.
The safety factor increases proportionally.
The transition state corresponds to :math:`\lambda =1.5`,
at which point the safety factor no longer depends on the ownership of the foundation.


.. figure:: images/10000201000002C7000000CEA02EC2FC4359B17B.png
 :name: fig3a mechanism
    :width: 70%

    **The two mechanisms of rupture:** :math:`c_u2/c_u1 = 0.6`


.. figure:: images/10000201000002CA000000D15E55F50A4ADD856F.png
 :name: fig3b-mechanism
    :width: 70%

    **The two mechanisms of rupture:** :math:`c_u2/c_u1 = 2.0`