Benchmark solution ===================== Calculation method ------------------ The deformation is estimated from the relative elongation of the beam. The elongation of a beam of length :math:`L` due to a longitudinal force :math:`F` is written as: :math:`\Delta L=\frac{FL}{ES}` In our case, we apply a force per unit area :math:`{F}_{B}` to the free end of the beam, so the relative elongation of the beam is in the following form: :math:`\frac{\Delta L}{L}={\epsilon }_{\mathrm{xx}}=\frac{{F}_{B}}{E}` For this test case, the results from a static calculation, a harmonic calculation, a transitory calculation and a modal calculation are calculated. For the static case, we get: :math:`{\varepsilon }_{\mathrm{xx}}=\frac{{F}_{B}}{E}` and :math:`{\varepsilon }_{\mathrm{yy}}={\varepsilon }_{\mathrm{zz}}=-\nu {\varepsilon }_{\mathrm{xx}}` For dynamic cases, the system is governed by the following equation: :math:`M\frac{{\partial }^{2}u}{\partial {t}^{2}}+Ku={F}_{\mathrm{ext}}` For a girder in tension - compression, if we consider a model that contains only one element, the matrices of mass :math:`M` and stiffness :math:`K` take the following form: :math:`M=\frac{\rho SL}{6}\left[\begin{array}{cc}2& 1\\ 1& 2\end{array}\right]` :math:`K=\frac{\mathrm{ES}}{L}\left[\begin{array}{cc}1& -1\\ -1& 1\end{array}\right]` with: :math:`u=\left[\begin{array}{c}{u}_{1}\\ {u}_{2}\end{array}\right]` :math:`{u}_{1}` and :math:`{u}_{2}` are the movements of the element's nodes. In harmonic pulse response :math:`\omega`, the movements of the nodes of the element are governed by the following relationship: :math:`\frac{\mathrm{ES}}{L}\left[\begin{array}{cc}1& -1\\ -1& 1\end{array}\right]\left[\begin{array}{c}{u}_{1}\\ {u}_{2}\end{array}\right]-{\omega }^{2}\frac{\rho SL}{6}\left[\begin{array}{cc}2& 1\\ 1& 2\end{array}\right]\left[\begin{array}{c}{u}_{1}\\ {u}_{2}\end{array}\right]=\left[\begin{array}{c}{F}_{1}\\ {F}_{2}\end{array}\right]` By exploiting the second line of this relationship, and applying the boundary conditions (:math:`{u}_{1}=0` and :math:`{F}_{2}=F={F}_{B}S`), we obtain: :math:`{u}_{2}=\frac{{F}_{B}}{\frac{E}{L}-{\omega }^{2}\frac{\rho L}{3}}=\Delta L` The deformation at the free end of the beam is written as: :math:`{\varepsilon }_{\mathrm{xx}}=\frac{\Delta L}{L}=\frac{{F}_{B}}{E-{\omega }^{2}\frac{\rho {L}^{2}}{3}}` and :math:`{\varepsilon }_{\mathrm{yy}}={\varepsilon }_{\mathrm{zz}}=-\nu {\varepsilon }_{\mathrm{xx}}` The reference solution for the transitory solution can be obtained in the same way. If we apply a longitudinal force :math:`F(t)=Ft={F}_{B}St` with an initial condition of a beam in balance (zero initial displacement and zero initial speed), we obtain: :math:`{u}_{2}(t)=\frac{3{F}_{B}}{\rho L{\omega }_{0}^{2}}\left[t-\frac{\mathrm{sin}({\omega }_{0}t)}{{\omega }_{0}}\right]=\Delta L` with: :math:`{{\omega }_{0}}^{2}=\frac{3E}{\rho {L}^{2}}` And the deformation at the free end of the beam is written as: :math:`{\varepsilon }_{\mathrm{xx}}=\frac{\Delta L}{L}=\frac{3{F}_{B}}{\rho {L}^{2}{\omega }_{0}^{2}}\left[t-\frac{\mathrm{sin}({\omega }_{0}t)}{{\omega }_{0}}\right]` and :math:`{\varepsilon }_{\mathrm{yy}}={\varepsilon }_{\mathrm{zz}}=-\nu {\varepsilon }_{\mathrm{xx}}` In the case of modal calculation, a non-regression test is carried out on the deformations calculated at the middle point of the beam. We also simulate a 90-degree rotation in order to check the change of frame of reference in OBSERVATION. Reference quantities and results ----------------------------------- We test the value of the mean deformation on the surfaces: :math:`\mathrm{surf3}`, :math:`\mathrm{surf4}`, :math:`\mathrm{surf5}` and :math:`\mathrm{surf6}`. The results obtained are then projected onto the "measurement" model which includes only the nodes :math:`\mathrm{P3}`, :math:`\mathrm{P4}`, :math:`\mathrm{P5}` and :math:`\mathrm{P6}` associated with the surfaces :math:`\mathrm{surf3}`, :math:`\mathrm{surf4}`, :math:`\mathrm{surf5}` and :math:`\mathrm{surf6}`. For the validation of the static solution, we chose: :math:`{F}_{B}=1000.N/{m}^{2}` For the validation of the harmonic solution, we chose: :math:`{F}_{B}=1000.(1+\mathrm{2j})N/{m}^{2}` and: :math:`\omega =2\pi 200\mathrm{rd}{s}^{-1}` For the validation of the transitory solution, we chose: :math:`{F}_{B}=1000.tN/{m}^{2}` and the solution is tested at the moment :math:`t=1s` The values of the fields obtained by mixed observation are also tested. Uncertainties about the solution ---------------------------- Analytical solution for the static case, the harmonic case and the transitory case. A non-regression solution is proposed for the case of modal deformation. Bibliographical reference ------------------------- [:ref:`R3.08.01 `] "Exact" elements of beams (straight and curved).