2. Benchmark solution#

2.1. Calculation method used for the reference solution#

Analytical solution [bib1] and [bib2].

2.1.1. Recessed case-free, unit loads at the end#

Single pull \({u}_{x}=\frac{{F}_{x}L}{ES}\)

Simple bending \({u}_{y}=\frac{{F}_{y}{L}^{3}(4+{\phi }_{y})}{12E{I}_{z}}\) \({\theta }_{z}=\frac{{L}^{2}{F}_{y}}{2E{I}_{z}}\) \({\phi }_{y}=\frac{12E{I}_{y}}{{L}^{2}G{A}_{y}^{\text{'}}}\)

Simple bending \({u}_{z}=\frac{{F}_{z}{L}^{3}(4+{\phi }_{z})}{12E{I}_{y}}\) \({\theta }_{y}=\frac{-{L}^{2}{F}_{z}}{2E{I}_{y}}\) \({\phi }_{z}=\frac{12E{I}_{z}}{{L}^{2}G{A}_{z}^{\text{'}}}\)

Twist \({\theta }_{x}=\frac{{M}_{x}L}{G{J}_{x}}\)

Pure flex \({u}_{z}=-\frac{{M}_{y}{L}^{2}}{2E{I}_{y}}\) \({\theta }_{y}=\frac{{M}_{y}L}{E{I}_{y}}\)

Pure flex \({u}_{y}=\frac{{M}_{z}{L}^{2}}{2E{I}_{z}}\) \({\theta }_{z}=\frac{{M}_{z}L}{E{I}_{z}}\)

Note 1:

For the angle section, as the center of shear is not confused with the center of gravity \(({e}_{y}\ne 0)\) , the torsional moment must be added: \({M}_{x}={F}_{z}{e}_{y}\) to the load \({F}_{z}=1\) .

This changes the movement:

\({u}_{z}=\frac{{F}_{z}{L}^{3}(4+{\phi }_{z})}{12E{I}_{y}}+{\theta }_{x}{e}_{y}\) \({\theta }_{x}=\frac{{M}_{x}L}{G{J}_{x}}\)

In the same way, loading \({M}_{x}=1\) results in a move \({u}_{z}={\theta }_{x}{e}_{y}\).

Linear distributed loading:

\(\begin{array}{cc}{u}_{y}(x)=\frac{px}{360LEI}(3{x}^{4}-10{L}^{2}{x}^{2}+7{L}^{4})& {u}_{y}^{\mathrm{max}}=\frac{0.00652p{L}^{4}}{EI}\\ & \mathrm{en}x=0.519L\end{array}\)

Note 2:

As far as modeling A is concerned, the beam is carried by the vector \({e}_{1}=\frac{1}{\sqrt{3}}(\begin{array}{}1\\ 1\\ 1\end{array})\) . The other vectors of the local coordinate system are: \({e}_{2}=\frac{1}{\sqrt{2}}(\begin{array}{}-1\\ 1\\ 0\end{array})\) and \({e}_{3}\mathrm{=}\frac{1}{\sqrt{6}}(\begin{array}{c}\mathrm{-}1\\ \mathrm{-}1\\ 2\end{array})\)

The components of the displacement vector in the global coordinate system are obtained by:

\({u}_{G}\mathrm{=}(\begin{array}{ccc}\frac{1}{\sqrt{3}}& \frac{\mathrm{-}1}{\sqrt{2}}& \frac{\mathrm{-}1}{\sqrt{6}}\\ \frac{1}{\sqrt{3}}& \frac{1}{\sqrt{2}}& \frac{\mathrm{-}1}{\sqrt{6}}\\ \frac{1}{\sqrt{3}}& 0& \frac{2}{\sqrt{6}}\end{array}){u}_{\text{local}}\)

Generalized efforts and constraints in \(O\):

\(N(O)={F}_{x}\) \({\sigma }_{\mathrm{xx}}=\frac{N}{S}\)

\({M}_{z}(O)={T}_{y}L\) \({T}_{y}={F}_{y}\) \({\sigma }_{\mathrm{xx}}(y)=\frac{{M}_{z}y}{{I}_{z}}\) \({\sigma }_{\mathrm{xy}}=\frac{{T}_{y}}{{k}_{y}S}\)

\({M}_{y}(O)=-{T}_{z}L\) \({T}_{z}(O)={F}_{z}\) \({\sigma }_{\mathrm{xx}}(y)=\frac{-{M}_{y}z}{{I}_{y}}\) \({\sigma }_{\mathrm{xz}}=\frac{{T}_{z}}{{k}_{z}S}\)

\({M}_{x}(0)={M}_{x}(B)\) \({\sigma }_{\mathrm{xy}}={\sigma }_{\mathrm{xz}}=\frac{{M}_{x}{R}_{T}}{{J}_{x}}\)

\({M}_{y}(0)={M}_{y}(B)\) \({\sigma }_{\mathrm{xx}}(z)=\frac{{M}_{y}z}{{I}_{y}}\)

\({M}_{z}(0)={M}_{z}(B)\) \({\sigma }_{\mathrm{xx}}(y)=\frac{{M}_{y}y}{{I}_{z}}\)

2.1.2. Case: linear distributed transverse loading#

\(y\) \({\mathrm{f}}_{y}\mathrm{.}L\)

\(x\)

\(O\) \(B\)

The equilibrium in rotation around \(O\) gives the value of the reaction, supported by \(B\):

\({R}_{\mathit{By}}=-{\int }_{0}^{L}{\mathrm{f}}_{y}\mathrm{.}{x}^{2}\mathit{dx}=-\frac{1}{3}{\mathrm{f}}_{y}\mathrm{.}{L}^{2}=-12000\); then: \({R}_{\mathit{Oy}}=-6000\).

Then we have:

\(\begin{array}{ccc}{M}_{z}(x)=\frac{-1000}{6}\left({L}^{2}x-{x}^{3}\right)& {V}_{y}(x)=\frac{1000{L}^{2}}{6}-\frac{1000{x}^{2}}{2}& {\sigma }_{\mathit{xx}}^{\mathit{max}}=\frac{{M}_{z}^{\mathit{max}}R}{{I}_{z}}\\ \phantom{\rule{2em}{0ex}}& \phantom{\rule{2em}{0ex}}& \mathit{en}x=\frac{L\sqrt{3}}{3}\end{array}\)

2.2. Benchmark results#

2.2.1. Recessed case-free, unit loads at the end#

Moving point \(B\),

Generalized efforts at point \(O\),

Point \(O\) constraints.

2.2.2. Case: linear distributed transverse loading#

Shearing forces and transversal reactions at point \(O\) and point \(B\). Maximum bending moment at abscissa point \(x=\frac{L\sqrt{3}}{3}\).

2.3. Uncertainty about the solution#

Analytical solution.

2.4. Bibliographical references#

  1. J.L. BATOZ, G. DHATT: « Modeling structures by finite elements » - Volume 2 Ed. HERMES.

  2. N.D. PIKLEY: « Formulas for Stress, Stain & Structural Matrixes » Ed. John Wiley & Sons.