2. Benchmark solution

2.1. Calculation method

The reference solution uses the results below:

\(\delta c\) and \(\delta b\) represent the movement at points \(C\) and \(B\).

\(\mathrm{EI.}\delta c=\frac{\mathrm{Fc}\mathrm{.}L}{3}+\frac{5\mathrm{.}\mathrm{Fb}\mathrm{.}{L}^{3}}{48}\) [1]

\(\mathrm{EI.}\delta b=\frac{5\mathrm{.}\mathrm{Fc}\mathrm{.}{L}^{3}}{47}+\frac{\mathrm{Fb.}{L}^{3}}{24}\) [2]

The supportive reaction \(\mathrm{Fa}=-\mathrm{Fc}-\mathrm{Fb}\) [3]

The boundary conditions that are used during the validation are: a move to \(C\): \(\delta c\) and a possible effort to \(B\): \(\mathrm{Fb}\). Solving the equations [], [], [] gives the following expressions:

\(\begin{array}{}\delta b=\frac{7.\mathrm{Fb.}{L}^{3}}{768.\mathrm{EI}}+\frac{5.\delta c}{16}\\ \mathrm{Fc}=\frac{3.\delta \mathrm{c.}\mathrm{EI}}{{L}^{3}}-\frac{5.\mathrm{Fb}}{16}\\ \mathrm{Fa}=-\frac{3.\delta \mathrm{c.}\mathrm{EI}}{{L}^{3}}-\frac{\mathrm{11.Fb}}{16}\end{array}\)

2.2. Reference quantities and results

The calculations performed on the 4 beams differ either by boundary conditions imposed on their nodes \(B\) and \(C\), or by a different definition of the local coordinate system. The boundary conditions imposed during movement or effort make it possible to determine the theoretical solutions of the various cases considered.

Characteristics of the beam:

\(3.\mathit{EIy}\mathrm{=}10E+07{\mathit{N.m}}^{2}\) \(\mathrm{3.EIz}\mathrm{=}4.0E+07{\mathit{N.m}}^{2}\)

Beam name: \(\mathit{Pout01}\)

Data:

Vect_y: \((-0.01.00.0)\)

\(\delta y\) in \(C\) (in the local coordinate system): \(2.0E\mathrm{-}03m\)

\(\delta z\) in \(C\) (in the local coordinate system): \(1.0E-03m\)

Theoretical results:

\(\mathrm{FY}\) in \(A\) (in the global frame of reference): \(\frac{-3\mathrm{.}\delta y\mathrm{.}\mathrm{EIz}}{{L}^{3}}\)

\(\mathrm{FZ}\) in \(A\) (in the global frame of reference): \(\frac{-3\mathrm{.}\delta z\mathrm{.}\mathrm{EIy}}{{L}^{3}}\)

Beam name: \(\mathit{Pout02}\)

Data:

Vect_y: \((-0.00.01.0)\)

\(\delta y\) in \(C\) (in the local coordinate system): \(2.0E-03m\)

\(\delta z\) in \(C\) (in the local coordinate system): \(1.0E-03m\)

Theoretical results:

\(\mathrm{FY}\) in \(A\) (in the global frame of reference): \(\frac{3.\delta \mathrm{z.}\mathrm{EIz}}{{L}^{3}}\)

\(\mathrm{FZ}\) in \(A\) (in the global frame of reference): \(\frac{-3.\delta \mathrm{y.}\mathrm{EIy}}{{L}^{3}}\)

Beam name: \(\mathit{Pout03}\)

Data:

Vect_y: \((-1.01.00.0)\)

\(\delta y\) in \(C\) (in the local coordinate system): \(2.0E\mathrm{-}03m\)

\(\delta z\) in \(C\) (in the local coordinate system): \(1.0E\mathrm{-}03m\)

Theoretical results:

\(\mathit{FX}\) in \(A\) (in the global frame of reference): \(\frac{-3\sqrt{2}\mathrm{.}\delta \mathrm{y.}\mathrm{EIz}}{{\mathrm{2.L}}^{3}}\)

\(\mathit{FY}\) in \(A\) (in the global frame of reference): \(\frac{3\sqrt{2}\mathrm{.}\delta \mathrm{y.}\mathrm{EIz}}{2.{L}^{3}}\)

\(\mathit{FZ}\) in \(A\) (in the global frame of reference): \(\frac{-3\mathrm{.}\delta \mathrm{z.}\mathrm{EIy}}{{L}^{3}}\)

Beam name: \(\mathit{Pout04}\)

Data:

Vect_y: \((-1.01.00.0)\)

\(\delta y\) in \(C\) (in the local coordinate system): \(2.0E\mathrm{-}03m\)

\(\delta z\) in \(C\) (in the local coordinate system): \(0\)

\(\mathit{FZ}\) in \(B\) (in the global frame of reference): \(1000.0\)

Theoretical results:

\(\mathit{FX}\) in \(A\) (in the global frame of reference): \(\frac{-3\sqrt{2}\mathrm{.}\delta y\mathrm{.}\mathrm{EIz}}{{\mathrm{2.L}}^{3}}\)

\(\mathit{FY}\) in \(A\) (in the global frame of reference): \(\frac{-3\sqrt{2}\mathrm{.}\delta \mathrm{y.}\mathrm{EIz}}{{\mathrm{2.L}}^{3}}\)

\(\mathit{FZ}\) in \(A\) (in the global frame of reference): \(\frac{-11\mathrm{.}\mathrm{Fb}}{16}\)

\(\mathit{DZ}\) in \(B\) (in the global frame of reference): \(\frac{7\mathrm{.}\mathrm{Fb}\mathrm{.}{L}^{3}}{768.\mathrm{EI}}\)