Reference problem
=====================

Geometry
---------

We consider a rectangular plate with the following dimensions:


* :math:`B=30\mathit{cm}` thickness


* length :math:`H=3m`


* height :math:`L=12m`


.. image:: images/100002010000032000000116E3EC5DBFAE1C86E2.png
    :width: 6.0043in
    :height: 1.8217in


.. _RefImage_100002010000032000000116E3EC5DBFAE1C86E2.png:

Figure 1. Geometry of the rectangular plate.

Material properties
----------------------

**Concrete** is isotropic elastic with the following material properties:


* :math:`E=30000N/{\mathit{mm}}^{2}`


* :math:`\nu =0.3`

For the calculation of reinforcement (in Eurocode 2), the following set of properties will be considered:


* Bottom and top coating :math:`{c}_{\mathit{inf}}={c}_{\text{sup}}=20\mathit{mm}`


* Characteristic compressive strength of concrete :math:`{f}_{\mathit{ck}}=35\mathit{MPa}`


* Work hardening limit characteristic of :math:`{f}_{\mathit{yk}}=450\mathit{MPa}` steel


* Chart type (sant-e): 'B1'


* Steel safety factor at ELU (Fundamental and Accidental) :math:`{\mathrm{\gamma }}_{s}=\mathrm{1,15}`


* Concrete Safety Coefficient at ELU (Fundamental and Accidental) :math:`{\mathrm{\gamma }}_{c}=\mathrm{1,5}`


* Boundary stress of concrete at ELS Characteristic :math:`{\mathrm{\sigma }}_{\text{c,lim}}=\mathrm{1,0}\times 35=35\mathit{MPa}`


* Boundary stress of steel at ELS Characteristic :math:`{\mathrm{\sigma }}_{\text{s,lim}}=\mathrm{1,0}\times 450=450\mathit{MPa}`


* Boundary stress of concrete at the ELS Quasi-Permanent :math:`{\mathrm{\sigma }}_{\text{c,lim,qp}}=\mathrm{1,0}\times 35=35\mathit{MPa}`


* Steel-concrete equivalence coefficient to ELS :math:`{\mathrm{\alpha }}_{E}=\mathrm{15,0}`


* Steel class: 'B'/:math:`{\mathrm{\alpha }}_{\mathit{cc}}=\mathrm{1,0}`


* :math:`\mathit{FERR}\text{\_}\mathit{SYME}=\text{'}\mathit{NON}\text{'}`/:math:`\mathit{FERR}\text{\_}\mathit{COMP}=\text{'}\mathit{OUI}\text{'}`/:math:`\mathit{EPURE}\text{\_}\mathit{CISA}=\text{'}\mathit{NON}\text{'}`/:math:`\mathit{FERR}\text{\_}\mathit{MIN}=\text{'}\mathit{NON}\text{'}`


* Density of :math:`{\mathrm{\rho }}_{\mathit{acier}}=7800\mathit{kg}/{m}^{3}` steel


* Maximum crack opening allowed on the underside of ELS QP :math:`{w}_{\text{max,inf}}=\mathrm{0,25}\mathit{mm}`


* Maximum crack opening allowed on the upper side of ELS QP :math:`{w}_{\text{max,sup}}=\mathrm{0,3}\mathit{mm}`


* Load time coefficient for calculation at ELS QP :math:`{K}_{T}=\mathrm{0,4}`


* Diameter of the bars following 'X' for calculation in ELS QP :math:`{\mathrm{\varphi }}_{X}=\mathrm{20,0}\mathit{mm}`


* Diameter of the bars following 'Y' for calculation in ELS QP :math:`{\mathrm{\varphi }}_{Y}=\mathrm{25,0}\mathit{mm}`


Boundary conditions and loads
-------------------------------------

The plate is loaded in its plane by a shear force :math:`{F}_{Z}=-\mathrm{1,5}\cdot {10}^{6}N` in the first load case (nodal load "**Shearld1**" in the command file). This load case will be tested at ELS QUASI PERMANENT.

A second load case (nodal load "**Shearld2**" in the command file) is obtained by applying twice this force, i.e. :math:`{F}_{Z}=-3\cdot {10}^{6}N`. This load case will be tested at ELS CARACTERISTIQUE.

A third load case (nodal load "**Shearld3**" in the command file) is obtained by applying triple this force, i.e. :math:`{F}_{Z}=-\mathrm{4,5}\cdot {10}^{6}N`. This load case will be tested at ELU.

The plate is then loaded into its plane by a :math:`{F}_{Y}=1\cdot {10}^{6}N` traction force ("**tractLD1**" node loading in the command file). This load case will be tested at ELS QUASI PERMANENT.

A second load case (nodal load "**tractLD2**" in the command file) is obtained by applying twice this force, i.e. :math:`{F}_{Y}=2\cdot {10}^{6}N`. This load case will be tested at ELS CARACTERISTIQUE.

A third load case (nodal load "**tractLD3**" in the command file) is obtained by applying triple this force, i.e. :math:`{F}_{Y}=3\cdot {10}^{6}N`. This load case will be tested at ELU.

The plate is finally loaded in its plane by a compression force :math:`{F}_{Y}=-\mathrm{1,5}\cdot {10}^{6}N` in a seventh load case ("**ComprLD1**" node loading in the command file). This load case will be tested at ELS CARACTERISTIQUE.

A last load case (nodal load "**ComprLD2**" in the command file) is obtained by applying twice this force, i.e. :math:`{F}_{Y}=-3\cdot {10}^{6}N`. This load case will be tested at ELU.

The plate is embedded on the left side, at the opposite end of the point of application of the force (); thus, all degrees of freedom are set to zero.


.. image:: images/1000020100000227000000DC547B1B1F538A532A.png
    :width: 4.9492in
    :height: 1.5008in


.. _RefImage_1000020100000227000000DC547B1B1F538A532A.png:

Figure 2. Blocking and loading for 'ShearlD' cases

Initial conditions
--------------------

The structure is initially at rest.