1. Reference problem#
1.1. Geometry#
It is a square slab, \(l\mathrm{=}1.8m\) long, \(h=0.12m\) thick, on simple bilateral supports. The flexure reinforcement is parallel to the edges; it is identical on each of both faces and in each of the two directions (\(\mathrm{dx}\), \(\mathrm{dy}\) being the spacings of the bars in the directions \(x\) and \(y\)). The coating of the longitudinal bars closest to the faces is \(22\mathrm{mm}\). The coating of the irons in relation to the lateral edges of the \(2\mathrm{cm}\) slab is neglected. The table below summarizes the reinforcement data. The geometric percentage of steel \(\mu\) is given for one face in one direction.
Reinforcement diameter |
Spacing |
Steel section/concrete section |
grid distance/mean slab area |
\(\Phi =\mathrm{0,01}\text{m}\) |
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We note \({a}_{x}=\frac{{A}_{x}}{{d}_{x}}\), \({a}_{y}\mathrm{=}\frac{{A}_{y}}{{d}_{y}}\) the reinforcement rates (here: \({a}_{x}\mathrm{=}{a}_{y}\mathrm{=}\mathrm{7,854}\mathrm{.}{10}^{\mathrm{-}4}m\)), with \({A}_{x}({A}_{y})\) being the section of an iron bar in the \(x(y)\) direction; \({e}_{z}\) the distance of the sheets from the mean surface.
1.2. Material properties#
The characteristics of steels are as follows:
Young’s Modulus \({E}_{a}\) |
Poisson’s Ratio |
Elastic limit at 0.2% \({\sigma }_{y}\) |
Fracture limit \({\sigma }_{r}\) |
Work hardening slope |
Elongation at break |
|
\(210000\mathrm{MPa}\) |
0.3 |
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Those of concrete are as follows:
Young’s Module \({E}_{b}\) |
Poisson’s Ratio |
Compressive Strength, \({\sigma }_{c}\) |
Tensile Strength, \({\sigma }_{t}\) |
\(35700\mathrm{MPa}\) |
0.22 |
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1.3. Boundary conditions and loads#
The boundary conditions boil down to simple supports: blocked vertical movement and free rotations on two opposite edges, the other two edges remaining free.
Uniform pressure: \(p=0.01\mathrm{MPa}\)
1.4. Initial conditions#
Not applicable.