2. Benchmark solution#

2.1. Calculation method used for the reference solution#

Analytical solution, isostatic structure.

The elastic deflection, the elastic axial stresses and deformations, and the bending moment at any point on the \(x\) axis are given by:

\({M}_{y}(x)={F}_{y}\mathrm{.}L(1-x/L)\)	(= \(E\mathrm{.}{I}_{z}\mathrm{.}{u}_{y}\text{'}\text{'}(x)\) in elasticity)
\({u}_{y}(x)={F}_{y}L\mathrm{.}{x}^{2}\mathrm{.}(3-x/L)/(6\mathrm{.}{\mathrm{E.I}}_{z})\)	(in elasticity)
\({\varepsilon }_{\mathrm{xx}}(x,y)=–{F}_{y}\mathrm{.}L(1-x/L)\mathrm{.}y/({\mathrm{E.I}}_{z})\)	(in elasticity)
\({\sigma }_{\mathrm{xx}}(x,y)=-{F}_{y}\mathrm{.}L(1-x/L)\mathrm{.}y/{I}_{z}\)	(in elasticity)

Load 2: \({C}_{z}=1\) torque or \({\mathrm{dr}}_{z}={C}_{\mathrm{z.L}}/({\mathrm{E.I}}_{z})\) rotation

\({M}_{y}(x)={C}_{z}\)	(= \(E\mathrm{.}{I}_{z}\mathrm{.}{u}_{y}\text{'}\text{'}(x)\) in elasticity)
\({u}_{y}(x)={C}_{\mathrm{z.}}{x}^{2}/(2\mathrm{.}E\mathrm{.}{I}_{z})\)	(in elasticity)
\({\sigma }_{\mathrm{xx}}(x,y)=–{C}_{z}\mathrm{.}y/{I}_{z}\)	(in elasticity)
\({\varepsilon }_{\mathrm{xx}}(x,y)=-{C}_{z}\mathrm{.}y/(E\mathrm{.}{I}_{z})\)	(in elasticity)

with:

\(L=\mathrm{L1}+\mathrm{L2}+\mathrm{L3}=30\mathrm{mm}\)

\({I}_{z}=a\mathrm{.}{h}^{3}/12=0.25{\mathrm{mm}}^{4}\)

\({\mathrm{dr}}_{z}=0.0006\)

Deflections, axial stresses and deformations and bending moments at 4 points on the axis of the beam.

Analytical solution.