Benchmark solution
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Analytical expressions
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Starting from the embedded end, the expression for the bending moment is: :math:`M(x)=F(x-L)`

The arrow at the loaded end of the beam is :math:`f=\frac{{\mathit{FL}}^{3}}{{\mathit{EI}}_{{G}_{0}}}` where :math:`{I}_{{G}_{0}}` is the quadratic moment calculated at the barycenter of the section :math:`{G}_{0}`: :math:`{I}_{{G}_{0}}={\int }_{S}{(y-{y}_{{G}_{0}})}^{2}\mathit{dS}`.

The curvature at a point located at a distance :math:`x` from the embedment is :math:`{\chi }_{s}(x)=-\frac{M(x)}{{\mathit{EI}}_{{G}_{0}}}`. Because of the eccentricity of the reference axis, the elongation of the beam (at the level of this axis) is equal to:

:math:`{ϵ}_{s}(x)=-\frac{{A}_{G}}{S}{\chi }_{s}(x)` where :math:`S` is the area of the section and :math:`{A}_{G}` the static moment of the section with respect to an axis passing through :math:`G`: :math:`{A}_{G}={\int }_{S}\mathit{zdS}`.

The deformation of a point with coordinates :math:`(x,y,z)` is: :math:`ϵ={ϵ}_{s}(x)-{\chi }_{s}(x)z`, and the stress at the same point is: :math:`\sigma =Eϵ`


Calculation of the characteristics of the straight section
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In order to eliminate the uncertainty of the approximate numerical calculation of the geometric characteristics of a straight section (low number of fibers), the values used in the reference solution are calculated as in the numerical calculation:

:math:`S={\sum }_{\mathit{fibres}}{S}_{i}` :math:`{A}_{G}={\sum }_{\mathit{fibres}}{z}_{i}{S}_{i}` :math:`{I}_{G}={\sum }_{\mathit{fibres}}{z}_{i}^{2}{S}_{i}` :math:`{I}_{{G}_{0}}={\sum }_{\mathit{fibres}}{({z}_{i}-{z}_{{G}_{0}})}^{2}{S}_{i}`

where :math:`{z}_{i}` is the ordinate of the center of the :math:`i` fiber and :math:`{S}_{i}` is the area of this fiber.