1. Reference problem#

1.1. 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.

_images/10000201000006900000041A612D497526FF4849.png

Figure

Figure 1: Structure geometry

1.2. Material properties#

The material is isotropic elastic whose properties are:

      • \(>\)

      • \(>\)

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

  • Bottom and top coating \({c}_{\mathit{inf}}={c}_{\text{sup}}=4\mathit{cm}\)

  • Characteristic compressive strength of concrete \({f}_{\mathit{ck}}=35\mathit{MPa}\)

  • Work hardening limit characteristic of \({f}_{\mathit{yk}}=500\mathit{MPa}\) steel

  • Young’s modulus of \({E}_{\mathit{ys}}=210000\mathit{MPa}\) steel

  • Chart type (sant-e): “B2”

  • Coefficient of safety of steel at ELU Fundamental \({\mathrm{\gamma }}_{s}=\mathrm{1,15}\)

  • Coefficient of safety of concrete at the ELU Fundamental \({\mathrm{\gamma }}_{c}=\mathrm{1,5}\)

  • Coefficient of safety of steel at ELU Accidental \({\mathrm{\gamma }}_{s}=1\)

  • Coefficient of safety of concrete at ELU Accidental \({\mathrm{\gamma }}_{c}=\mathrm{1,2}\)

  • Boundary stress of concrete at ELS Characteristic \({\mathrm{\sigma }}_{\text{c,lim}}=\mathrm{0,6}\times 35=21\mathit{MPa}\)

  • Boundary stress of steel at ELS Characteristic \({\mathrm{\sigma }}_{\text{s,lim}}=\mathrm{0,8}\times 500=400\mathit{MPa}\)

  • Boundary stress of concrete at the ELS Quasi-Permanent \({\mathrm{\sigma }}_{\text{c,lim,qp}}=\mathrm{0,45}\times 35=\mathrm{15,75}\mathit{MPa}\)

  • Steel-concrete equivalence coefficient to ELS \({\mathrm{\alpha }}_{E}=\mathrm{15,0}\)

  • Steel class: “B”/\({\mathrm{\alpha }}_{\mathit{cc}}=\mathrm{1,0}\)

  • \(\mathit{FERR}\text{\_}\mathit{SYME}=\text{'}\mathit{NON}\text{'}\)/\(\mathit{FERR}\text{\_}\mathit{COMP}=\text{'}\mathit{OUI}\text{'}\)/\(\mathit{EPURE}\text{\_}\mathit{CISA}=\text{'}\mathit{NON}\text{'}\)/\(\mathit{FERR}\text{\_}\mathit{MIN}=\text{'}\mathit{NON}\text{'}\)/\(\mathit{UTIL}\text{\_}\mathit{COMPR}=\text{'}\mathit{NON}\text{'}\)

  • Density of \({\mathrm{\rho }}_{\mathit{acier}}=7800\mathit{kg}/{m}^{3}\) steel

  • Maximum crack opening allowed on the underside of ELS QP \({w}_{\text{max,inf}}=\mathrm{0,3}\mathit{mm}\)

  • Maximum crack opening allowed on the upper side of ELS QP \({w}_{\text{max,sup}}=\mathrm{0,3}\mathit{mm}\)

  • Load time coefficient for calculation at ELS QP \({K}_{T}=\mathrm{0,4}\)

  • Diameter of the bars following “X” for calculation in ELS QP \({\mathrm{\varphi }}_{X}=\mathrm{25,0}\mathit{mm}\)

  • Diameter of the bars following “Y” for calculation in ELS QP \({\mathrm{\varphi }}_{Y}=\mathrm{25,0}\mathit{mm}\)

For the posts:

  • Coatings \({c}_{y,\mathit{inf}}={c}_{y,\text{sup}}={c}_{z,\mathit{inf}}={c}_{z,\text{sup}}=4\mathit{cm}\)

  • Characteristic compressive strength of concrete \({f}_{\mathit{ck}}=35\mathit{MPa}\)

  • Work hardening limit characteristic of \({f}_{\mathit{yk}}=500\mathit{MPa}\) steel

  • Young’s modulus of \({E}_{\mathit{ys}}=210000\mathit{MPa}\) steel

  • Chart type (sant-e): “B2”

  • Coefficient of safety of steel at ELU Fundamental \({\mathrm{\gamma }}_{s}=\mathrm{1,15}\)

  • Coefficient of safety of concrete at the ELU Fundamental \({\mathrm{\gamma }}_{c}=\mathrm{1,5}\)

  • Coefficient of safety of steel at ELU Accidental \({\mathrm{\gamma }}_{s}=1\)

  • Coefficient of safety of concrete at ELU Accidental \({\mathrm{\gamma }}_{c}=\mathrm{1,2}\)

  • Boundary stress of concrete at ELS Characteristic \({\mathrm{\sigma }}_{\text{c,lim}}=\mathrm{0,6}\times 35=21\mathit{MPa}\)

  • Boundary stress of steel at ELS Characteristic \({\mathrm{\sigma }}_{\text{s,lim}}=\mathrm{0,8}\times 500=400\mathit{MPa}\)

  • Boundary stress of concrete at the ELS Quasi-Permanent \({\mathrm{\sigma }}_{\text{c,lim,qp}}=\mathrm{0,45}\times 35=\mathrm{15,75}\mathit{MPa}\)

  • Steel-concrete equivalence coefficient to ELS \({\mathrm{\alpha }}_{E}=\mathrm{15,0}\)

  • Steel class: “B”/\({\mathrm{\alpha }}_{\mathit{cc}}=\mathrm{1,0}\)

  • \(\mathit{FERR}\text{\_}\mathit{SYME}=\text{'}\mathit{NON}\text{'}\)/\(\mathit{FERR}\text{\_}\mathit{COMP}=\text{'}\mathit{OUI}\text{'}\)/\(\mathit{EPURE}\text{\_}\mathit{CISA}=\text{'}\mathit{NON}\text{'}\)/\(\mathit{FERR}\text{\_}\mathit{MIN}=\text{'}\mathit{NON}\text{'}\)/\(\mathit{UTIL}\text{\_}\mathit{COMPR}=\text{'}\mathit{NON}\text{'}\)

  • Density of \({\mathrm{\rho }}_{\mathit{acier}}=7800\mathit{kg}/{m}^{3}\) steel

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

  • Maximum crack opening allowed on the upper side (Y and Z) at ELS QP \({w}_{\text{max,sup}}=\mathrm{0,3}\mathit{mm}\)

  • Load time coefficient for calculation at ELS QP \({K}_{T}=\mathrm{0,4}\)

  • Diameter of the bars following “Y” for calculation in ELS QP \({\mathrm{\varphi }}_{Y}=\mathrm{25,0}\mathit{mm}\)

  • Diameter of the bars following “Z” for calculation in ELS QP \({\mathrm{\varphi }}_{Z}=\mathrm{25,0}\mathit{mm}\)

1.3. Boundary conditions and loads#

The 4 posts are embedded in their bases.

In addition, 3 loading cases will be considered:

  • Own weight \(G\) of the structure

  • Surface pressure \({P}_{r}=15000N\) applied to the slab

  • A horizontal translation force following “X” \({F}_{t}=800000N\) 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 — \({P}_{r}\) dominant: \(\mathrm{1,35}G+\mathrm{1,5}{P}_{r}+\mathrm{1,5}\times \mathrm{0,6}{F}_{t}\)

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

  • Comb 3 - ELU ACCIDENTEL: \(G+{P}_{r}+{F}_{t}\)

  • Comb 4 - ELSCARACTERISTIQUE — \({P}_{r}\) dominant: \(G+{P}_{r}+\mathrm{0,6}{F}_{t}\)

  • Comb 5 - ELSCARACTERISTIQUE — \({F}_{t}\) dominant: \(G+\mathrm{0,7}{P}_{r}+{F}_{t}\)

  • Comb 6 - ELSQUASI PERMANENT — \({P}_{r}\) dominant: \(G+{P}_{r}+\mathrm{0,15}{F}_{t}\)

  • Comb 7 - ELS QUASI PERMANENT — \({F}_{t}\) dominant: \(G+\mathrm{0,3}{P}_{r}+{F}_{t}\)

1.4. Initial conditions#

The structure is initially at rest.