Reference solution
=====================

Torsional moment
------------------

An analytical solution for the torsional moment is easily found by calculating the Strength of Materials.

Let the :math:`\mathit{AB}` beam of length :math:`L`, embedded in :math:`A`, if we apply a torsional moment :math:`\mathit{mt}` at a point :math:`C` of :math:`\mathrm{[}\mathit{AB}\mathrm{]}` then the moment resulting in :math:`A` is :math:`\mathit{mt}`. So the reaction at the moment is :math:`\mathrm{-}\mathit{mt}`.

By applying a linear torsional moment, distributed over the beam, equal to :math:`{\mathit{mt}}_{A}` in :math:`A` and to :math:`{\mathit{mt}}_{B}` in :math:`B`, we obtain the moment reaction :math:`{M}_{A}` in :math:`A`:

:math:`{M}_{A}\mathrm{=}\mathrm{-}{\mathrm{\int }}_{0}^{L}{\mathit{mt}}_{A}+\frac{({\mathit{mt}}_{B}\mathrm{-}{\mathit{mt}}_{A})}{L}x\mathit{dx}`

:math:`{M}_{A}\mathrm{=}\mathrm{-}L\frac{({\mathit{mt}}_{A}+{\mathit{mt}}_{B})}{2}`


Bending moment
------------------

The Material Strength Forms provide reference results for a moment according to :math:`Z` applied to point :math:`C` of a :math:`\mathit{AB}` beam of length :math:`L` embedded in :math:`A` and supported according to :math:`Y` in :math:`B`.


.. image:: images/1000000000000248000000CF1D036B7855322E72.png
    :width: 4.5984in
    :height: 1.6299in


.. _RefImage_1000000000000248000000CF1D036B7855322E72.png:

:math:`{R}_{A}\mathrm{=}\mathrm{-}{R}_{B}\mathrm{=}\frac{3\mathit{mfz}({L}^{2}\mathrm{-}{b}^{2})}{{\mathrm{2L}}^{3}}`

:math:`{M}_{A}\mathrm{=}\frac{\mathit{mfz}({L}^{2}\mathrm{-}{\mathrm{3b}}^{2})}{{\mathrm{2L}}^{2}}`

where :math:`{R}_{A}` is the supporting reaction and :math:`{M}_{A}` is the moment, in :math:`A`.

By applying a linear bending moment, distributed over the beam, equal to :math:`{\mathit{mf}}_{A}` in :math:`A` and to :math:`{\mathit{mf}}_{B}` in :math:`B`, we obtain:

:math:`{R}_{A}=-{R}_{B}=\frac{3}{{\mathrm{2L}}^{3}}\underset{0}{\overset{L}{\int }}({\mathit{mf}}_{A}+\frac{({\mathit{mf}}_{B}-{\mathit{mf}}_{A})}{L}x)({L}^{2}-{(L-x)}^{2})\mathit{dx}`

:math:`{M}_{A}=\frac{1}{{\mathrm{2L}}^{2}}\underset{0}{\overset{L}{\int }}({\mathit{mf}}_{A}+\frac{({\mathit{mf}}_{B}-{\mathit{mf}}_{A})}{L}x)({L}^{2}-3{(L-x)}^{2})\mathit{dx}`

What results after integration:

:math:`{R}_{A}\mathrm{=}\mathrm{-}{R}_{B}\mathrm{=}\frac{3{\mathit{mf}}_{A}+5{\mathit{mf}}_{B}}{8}`

:math:`{M}_{A}\mathrm{=}L\frac{{\mathit{mf}}_{B}\mathrm{-}{\mathit{mf}}_{A}}{8}`

Note: If you go into plan :math:`\mathit{XOZ}` with the application of a moment according to :math:`Y`, you have to multiply the reactions by :math:`\mathrm{-}1`.


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

None.