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


Geometry
---------

The structure used is described in the following:

- a slab with dimensions dx=10m and dy=6m in the plane (x, y), and with a thickness ez = 30cm.

- 4 posts on which the slab rests, 5m high, and with a rectangular section of dimensions: hy = 50cm and hz = 20cm. The local 'y' axis of the columns coincides with the global X axis of the geometry, while the local 'z' axis coincides with the global Y axis of the geometry.


.. image:: images/10000201000006900000041A612D497526FF4849.png
    :width: 6.3291in
    :height: 3.2618in


.. _RefImage_10000201000006900000041A612D497526FF4849.png:

Figure

Figure 1: Structure geometry


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

The material is isotropic elastic whose properties are:


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    *
        * :math:`>`


        * :math:`>`

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


* Bottom and top coating :math:`{c}_{\mathit{inf}}={c}_{\text{sup}}=4\mathit{cm}`


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


* Work hardening limit characteristic of :math:`{f}_{\mathit{yk}}=500\mathit{MPa}` steel


* Young's modulus of :math:`{E}_{\mathit{ys}}=210000\mathit{MPa}` steel


* Chart type (sant-e): 'B2'


* Coefficient of safety of steel at ELU Fundamental :math:`{\mathrm{\gamma }}_{s}=\mathrm{1,15}`


* Coefficient of safety of concrete at the ELU Fundamental :math:`{\mathrm{\gamma }}_{c}=\mathrm{1,5}`


* Coefficient of safety of steel at ELU Accidental :math:`{\mathrm{\gamma }}_{s}=1`


* Coefficient of safety of concrete at ELU Accidental :math:`{\mathrm{\gamma }}_{c}=\mathrm{1,2}`


* Boundary stress of concrete at ELS Characteristic :math:`{\mathrm{\sigma }}_{\text{c,lim}}=\mathrm{0,6}\times 35=21\mathit{MPa}`


* Boundary stress of steel at ELS Characteristic :math:`{\mathrm{\sigma }}_{\text{s,lim}}=\mathrm{0,8}\times 500=400\mathit{MPa}`


* Boundary stress of concrete at the ELS Quasi-Permanent :math:`{\mathrm{\sigma }}_{\text{c,lim,qp}}=\mathrm{0,45}\times 35=\mathrm{15,75}\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{'}`/:math:`\mathit{UTIL}\text{\_}\mathit{COMPR}=\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{0,3}\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{25,0}\mathit{mm}`


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

For the posts:


* Coatings :math:`{c}_{y,\mathit{inf}}={c}_{y,\text{sup}}={c}_{z,\mathit{inf}}={c}_{z,\text{sup}}=4\mathit{cm}`


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


* Work hardening limit characteristic of :math:`{f}_{\mathit{yk}}=500\mathit{MPa}` steel


* Young's modulus of :math:`{E}_{\mathit{ys}}=210000\mathit{MPa}` steel


* Chart type (sant-e): 'B2'


* Coefficient of safety of steel at ELU Fundamental :math:`{\mathrm{\gamma }}_{s}=\mathrm{1,15}`


* Coefficient of safety of concrete at the ELU Fundamental :math:`{\mathrm{\gamma }}_{c}=\mathrm{1,5}`


* Coefficient of safety of steel at ELU Accidental :math:`{\mathrm{\gamma }}_{s}=1`


* Coefficient of safety of concrete at ELU Accidental :math:`{\mathrm{\gamma }}_{c}=\mathrm{1,2}`


* Boundary stress of concrete at ELS Characteristic :math:`{\mathrm{\sigma }}_{\text{c,lim}}=\mathrm{0,6}\times 35=21\mathit{MPa}`


* Boundary stress of steel at ELS Characteristic :math:`{\mathrm{\sigma }}_{\text{s,lim}}=\mathrm{0,8}\times 500=400\mathit{MPa}`


* Boundary stress of concrete at the ELS Quasi-Permanent :math:`{\mathrm{\sigma }}_{\text{c,lim,qp}}=\mathrm{0,45}\times 35=\mathrm{15,75}\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{'}`/:math:`\mathit{UTIL}\text{\_}\mathit{COMPR}=\text{'}\mathit{NON}\text{'}`


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


* Maximum crack opening allowed on the underside (Y and Z) at ELS QP :math:`{w}_{\text{max,inf}}=\mathrm{0,3}\mathit{mm}`


* Maximum crack opening allowed on the upper side (Y and Z) at 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 'Y' for calculation in ELS QP :math:`{\mathrm{\varphi }}_{Y}=\mathrm{25,0}\mathit{mm}`


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


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

The 4 posts are embedded in their bases.

In addition, 3 loading cases will be considered:


* Own weight :math:`G` of the structure


* Surface pressure :math:`{P}_{r}=15000N` applied to the slab


* A horizontal translation force following 'X' :math:`{F}_{t}=800000N` applied to the 4 posts

The reinforcement calculation will then be carried out in relation to 7 combinations of these 3 forces, corresponding to different calculation limit states:


* Comb 1 - ELU FONDAMENTAL — :math:`{P}_{r}` dominant: :math:`\mathrm{1,35}G+\mathrm{1,5}{P}_{r}+\mathrm{1,5}\times \mathrm{0,6}{F}_{t}`


* Comb 2 - ELU FONDAMENTAL — :math:`{F}_{t}` dominant: :math:`\mathrm{1,35}G+\mathrm{1,5}\times \mathrm{0,7}{P}_{r}+\mathrm{1,5}{F}_{t}`


* Comb 3 - ELU ACCIDENTEL: :math:`G+{P}_{r}+{F}_{t}`


* Comb 4 - ELSCARACTERISTIQUE — :math:`{P}_{r}` dominant: :math:`G+{P}_{r}+\mathrm{0,6}{F}_{t}`


* Comb 5 - ELSCARACTERISTIQUE — :math:`{F}_{t}` dominant: :math:`G+\mathrm{0,7}{P}_{r}+{F}_{t}`


* Comb 6 - ELSQUASI PERMANENT — :math:`{P}_{r}` dominant: :math:`G+{P}_{r}+\mathrm{0,15}{F}_{t}`


* Comb 7 - ELS QUASI PERMANENT — :math:`{F}_{t}` dominant: :math:`G+\mathrm{0,3}{P}_{r}+{F}_{t}`


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

The structure is initially at rest.