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

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

Analytical solution [:ref:`bib1 <bib1>`]: movements in :math:`B` in the coordinate system :math:`(\mathit{Oxyz})` linked to the beam.

Single pull: :math:`{u}_{x}=\frac{{F}_{x}L}{ES}`

Simple bending: :math:`{u}_{y}=\frac{{F}_{y}{L}^{3}}{3E{I}_{z}}` :math:`{t}_{z}=\frac{{L}^{2}{F}_{y}}{2E{I}_{z}}`

Simple bending: :math:`{u}_{z}=\frac{{F}_{z}{L}^{3}}{3E{I}_{y}}` :math:`{\theta }_{y}=\frac{-{L}^{2}{F}_{z}}{2E{I}_{y}}`

Twist: :math:`{\theta }_{x}=\frac{{M}_{x}L}{G{J}_{x}}`

Pure flex: :math:`{u}_{z}=\frac{-{M}_{y}{L}^{2}}{2E{I}_{y}}` :math:`{\theta }_{y}=\frac{{M}_{y}L}{E{I}_{y}}`

Pure flex: :math:`{u}_{y}=\frac{{M}_{z}{L}^{2}}{2E{I}_{z}}` :math:`{\theta }_{z}=\frac{{M}_{z}L}{E{I}_{z}}`

Pressure: :math:`{u}_{r}=\frac{P{a}^{2}r}{E({b}^{2}-{a}^{2})}\left[(1-\nu )+(1+\nu )\frac{{b}^{2}}{{r}^{2}}\right]` calculated in :math:`r=\frac{a+b}{2}`

actually :math:`{u}_{r}\in \left[7.12E\text{-}06,7.78E\text{-}06\right]` for :math:`r\in \left[b,a\right]`

Here, the values are obtained with:

:math:`S=1.809557E\text{-}03{m}^{2}` :math:`{I}_{y}={I}_{z}=1.18707E\text{-}06{m}^{4}` :math:`{J}_{x}=2.37414E\text{-}06{m}^{4}` :math:`L=5m`

For generalized beam deformations, using the law of behavior, we obtain:

Single pull: :math:`{\epsilon }_{x}=\frac{{F}_{x}}{ES}`

Simple bending: :math:`{\gamma }_{\mathrm{xy}}=\frac{{F}_{y}}{GS}` :math:`{\kappa }_{z}=\frac{{F}_{y}(L-x)}{E{I}_{z}}`

Simple bending: :math:`{\gamma }_{\mathrm{xz}}=\frac{{F}_{z}}{GS}` :math:`{\kappa }_{y}=\frac{{F}_{z}(L-x)}{E{I}_{y}}`

Twist: :math:`{\kappa }_{x}=\frac{{M}_{x}}{G{J}_{x}}`

Pure flex: :math:`{\kappa }_{y}=\frac{{M}_{y}}{E{I}_{y}}`

Pure flex: :math:`{\kappa }_{z}=\frac{{M}_{z}}{E{I}_{z}}`

Gravity loading and line loading:

If :math:`p` refers to the distributed load, the moment at the origin is equal to: :math:`M(o)=\frac{p{L}^{2}}{2}` and the next displacement is set to :math:`z` at the end B is equal to: :math:`{u}_{z}(B)=\frac{p{L}^{4}}{8EI}`.

The thermal expansion load leads to an axial displacement (in the local direction :math:`x`):

:math:`{U}_{x}(B)=L(\alpha T)`

The free expansion deformations of the pipe surface are simply, as a local coordinate system:

:math:`{\epsilon }_{\mathrm{xx}}={\epsilon }_{\mathrm{yy}}=\alpha T`

Finally, to validate the calculation of the mass matrix, a modal analysis of the first 12 natural modes (with embedding in :math:`O`) must give, for the bending modes:

:math:`{f}_{i}={(\frac{{\lambda }_{i}}{L})}^{2}\sqrt{\frac{EI}{\rho S}}`


.. csv-table::

    "**Mode**", ":math:`{\lambda }_{i}` ", "**Frequency**"
    "1", "1.87510407", "2.9030234"
    "2", "4.69409113", "18.192937"
    "3", "7.85475744", "50.9407506"
    "4", "10.9955407", "99.8235399"
    "5", "14.1371684", "165.015464"
    "6", "17.2787596", "246.504532"
    "7", "20.4203522", "344.291453"
    "8", "23.5619449", "458.376195"
    "9", "26.7035376", "588.758758"
    "10", "29.8451302", "735.43914"
    "11", "32.9867229", "898.417343"
    "12", "36.1283155", "1077.69337"



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

• Movement to point :math:`B`, efforts, stresses and deformations in the vicinity of point :math:`O`.

• Generalized deformities.

• Natural frequencies


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

Analytical solution.


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


1. Validation manual, test SSLL102 Embedded beam subjected to unit efforts [:ref:`V3.01.102 <V3.01.102>`]