2. Benchmark solution

2.1. Analytical expressions

Starting from the embedded end, the expression for the bending moment is: \(M(x)=F(x-L)\)

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

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

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

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

2.2. Calculation of the characteristics of the straight section

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:

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

where \({z}_{i}\) is the ordinate of the center of the \(i\) fiber and \({S}_{i}\) is the area of this fiber.