Reference problem ==== Geometry ---- It is a :math:`5m` long reinforced beam of which only a quarter can be modelled thanks to the symmetries. The dimensions are given in millimeters. +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ | | + .. image:: images/1000000000000441000001AB4ABB93A232EA7165.png + | :width: 5.65in | + :height: 2.2917in + | | + + | .. image:: images/10000000000002A90000033C04181522D8323B80.png | + :width: 2.1528in + | :height: 2.5472in | + + | | +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ .. _RefHeading__34154849: Initial condition and thermal loads ---- The initial temperature is uniform at :math:`20°C`. Concrete is subject to thermo-hydration. The temperature is set to :math:`20°C` on the face :math:`\mathrm{EFGH}` noted HAUT in the mesh. The temperature is constant at :math:`20°C` for up to 10 days then varies linearly from :math:`20\mathrm{°}\mathrm{C}` to :math:`40\mathrm{°}\mathrm{C}` between 10 days and 30 days and is then constant at :math:`40\mathrm{°}\mathrm{C}` on the :math:`\mathrm{ABCD}` side marked BAS in the mesh. .. _RefHeading__34165010: Initial condition and drying loads ---- The initial water concentration is uniform at :math:`120l\mathrm{/}{m}^{3}`. The water concentration is set to :math:`50l\mathrm{/}{m}^{3}` on the :math:`\mathrm{EFGH}` side marked HAUT. The water concentration is set to :math:`70l\mathrm{/}{m}^{3}` on the :math:`\mathrm{ABCD}` side. .. _RefHeading__34232174: Boundary conditions and mechanical loads ---- **Symmetry conditions** The plate is stuck next :math:`\mathrm{Ox}` on the side :math:`\mathrm{BCGF}` noted SYME_X and next :math:`\mathrm{Oz}` on the side :math:`\mathrm{ABFE}` noted SYME_Z. **Boundary condition** The plate is stuck next :math:`\mathrm{Oy}` on the side :math:`\mathrm{HE}` noted APPUI. **Loading** It is subjected to a force :math:`\frac{F}{4}=3840N` following :math:`\mathrm{Oy}` distributed on the side :math:`\mathrm{BC}`, denoted by FORCE, which is equivalent to a total force of :math:`F=15360N` on the entire beam. The force is applied at the final moment of each calculation (in modeling A to the only time step taken and in modeling B to :math:`t\mathrm{=}100\mathit{jours}`). Thermal properties of materials ---- The characteristics of concrete are: * Heat capacity :math:`\mathrm{\rho }{C}_{p}=2.4{e}^{6}J/{m}^{3}/°C`; * Conductivity :math:`\mathrm{\lambda }=1W/m/°C`; and the following characteristics relating to moisturizing behavior: * heat per degree of hydration: :math:`{Q}_{0}\mathrm{=}{1.14e}^{8}J\mathrm{/}{m}^{3}` * affinity as a function of hydration (polynomial evaluation of the known function by points) and of temperature (table): :math:`A(\mathrm{\xi },T)=(50.12{\mathrm{\xi }}^{6}-190.76{\mathrm{\xi }}^{5}+258.38{\mathrm{\xi }}^{4}-123.71{\mathrm{\xi }}^{3}-11.82{\mathrm{\xi }}^{2}+15.37\mathrm{\xi }+2.43)\mathrm{exp}(\frac{-{\mathit{QSR}}_{K}}{(273.15+T)})` * with Arrhenius constant: :math:`{\mathit{QSR}}_{K}=4000K`. **Note:** The Arrhenius constant is always expressed in degrees Kelvin. Temperatures are expressed in :math:`°C`. Properties of drying materials ---- We use the law of diffusion SECH_GRANGER: :math:`D(C,T)=A\cdot \mathrm{exp}(\mathit{BC})\frac{T}{{T}_{0}}\mathrm{exp}\left[-\frac{{Q}_{s}}{R}\left(\frac{1}{T}-\frac{1}{{T}_{0}}\right)\right]` * :math:`A=3.3e-13{m}^{2}/s`; * :math:`B=0.05`; * :math:`\mathit{QSR}=4000K` * :math:`{T}_{0}=293°K=20°C` Mechanical properties of materials ---- The behavior is elastic. The characteristics of steels are: * Young's module :math:`{E}_{a}=200000\mathrm{MPa}`; * Poisson's ratio :math:`\mathrm{\nu }=0.3`; The characteristics of concrete are: * Young's module :math:`{E}_{b}=32000\mathit{MPa}` for modeling A. In modeling B, it varies linearly as a function of temperature from :math:`{E}_{b}=30000\mathrm{MPa}` for :math:`T=0°C` to :math:`{E}_{b}\mathrm{=}40000\mathit{MPa}` for :math:`T=100°C`; * Poisson's ratio :math:`\mathrm{\nu }=0.2`; * Thermal expansion :math:`\mathrm{\alpha }=1.2e-6` at the reference temperature of :math:`{T}_{\mathrm{ref}}=20°C` * Desiccation shrinkage coefficient :math:`{K}_{\mathit{des}}\mathrm{=}8e-6` * Endogenous shrinkage coefficient :math:`{B}_{\mathit{endo}}\mathrm{=}9e-5`