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


Calculation method
------------------

The deformation field is obtained from the displacements and the stress field is obtained using Hooke's law. The elastic energy :math:`{E}_{\mathit{elas}}` is then calculated by the following formula:

:math:`{E}_{\mathit{elas}}\mathrm{=}{\mathrm{\int }}_{V}\frac{1}{2}\sigma \mathrm{:}\epsilon \mathit{dV}` equation 2.1-1


Modeling A
~~~~~~~~~~~~~~~

The homogeneous deformation and stress fields on the element are shown in the.


.. csv-table::

    "Component", ":math:`X` "," :math:`Y` "," "," :math:`Z` "," "," :math:`\mathit{XY}` "," :math:`\mathit{YZ}` "," :math:`\mathit{ZX}`"
    "Deformation", "0.001", "-0.002", "0.002", "0.003", "0.003", "0.005", "-0.0001", "-0.0003"
    "Constraint (:math:`\mathit{Pa}`)", "1.3888889E6", "-1.111111E6", "3.0555556E6", "4.1166667E5", "-8.3333333E5", "-8.3333333E4", "-2.5E5"


**Table** 2.1.1-1 **: Strain and stress field**


Modeling B: plane deformations
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The homogeneous deformation and stress fields on the element are shown in the.


.. csv-table::

    "Component", ":math:`X` "," :math:`Y` "," "," :math:`Z` "," "," :math:`\mathit{XY}` "," :math:`\mathit{YZ}` "," :math:`\mathit{ZX}`"
    "Deformation", "0.001", "-0.002", "0.002", "0.0", "0.0", "0.0", "0.0"
    "Constraint (:math:`\mathit{Pa}`)", "5,5555556 E 5", "-1, 9444444 E6", "-2,7777778 E 5", "4,1666667E5", "-8,3333333E5", "-8,3333333E4", "-2,5E5"


**Table** 2.1.2-1 **: Strain and stress field**


C (plane stresses) and D (shell and plate elements) modeling
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The homogeneous deformation and stress fields on the element are shown in the. For shell and plate elements, this is membrane deformation, with the curvature being zero.


.. csv-table::

    "Component", ":math:`X` "," :math:`Y` "," "," :math:`Z` "," "," :math:`\mathit{XY}` "," :math:`\mathit{YZ}` "," :math:`\mathit{ZX}`"
    "Deformation", "0.001", "-0.002", "-0.002", "-0.00025", "0.0005", "0.0", "0.0"
    "Constraint (:math:`\mathit{Pa}`)", "6.25 E 5", "-1, 875 E5", "-1, 875 E6", "0.0", "0.0", "0.0", "0.0"


**Table** 2.1.3-1 **: Strain and stress field**


E modeling: axisymmetric.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The homogeneous deformation and stress fields on the element are shown in the.


.. csv-table::

    "Component", ":math:`X` "," :math:`Y` "," "," :math:`Z` "," :math:`\mathit{XY}`"
    "Deformation", "0.0", "0.0001", "0.0", "0.0"
    "Constraint (:math:`\mathit{Pa}`)", "0.0", "1.0", "1.0E5", "0.0", "0.0"


Table 2.1.4-1: Deformation and Stress Field

The elastic energy is calculated according to the by expressing the elementary volume :math:`\mathit{dV}` in the cylindrical coordinate system for a slice of infinitesimal thickness :math:`d\theta`:

:math:`{E}_{\mathit{elas}}\mathrm{=}{\mathrm{\int }}_{V}\frac{1}{2}\sigma \mathrm{:}\epsilon \mathit{dV}\mathrm{=}\frac{1}{2}\sigma \mathrm{:}\epsilon {\mathrm{\int }}_{{R}_{1}}^{{R}_{2}}\mathit{rdr}{\mathrm{\int }}_{{H}_{1}}^{{H}_{2}}\mathit{dz}`

The terminals of the integral are: :math:`{R}_{1}\mathrm{=}\mathrm{1m}`, :math:`{R}_{2}\mathrm{=}\mathrm{2m}`,, :math:`{H}_{1}\mathrm{=}\mathrm{0m}`, and :math:`{H}_{2}\mathrm{=}\mathrm{1m}`.


Reference quantities and results
-----------------------------------

The elastic energy calculated analytically for each of the models is shown in the.


.. csv-table::

    "Modeling", ":math:`A` "," :math:`B` "," "," :math:`C` "," :math:`D` "," :math:`E`"
    "Elastic energy", "6680,555556 J", "2430.555556 J", "2395,833333 J", "2395,833333 J", "2395,833333 J", ":math:`\mathrm{7,5}{\mathit{J.rad}}^{1}`"


Table 2.2-1: Elastic energy


Uncertainties about the solution
----------------------------

None. It is an analytical solution.


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