2.6.4.7 Limitation de la contrainte de compression du béton

Limitation de la contrainte de compression du béton

In window 1.3 Surfaces, the concrete compressive stress is limited to σ_c,max = 0.45 ⋅ f_ck and the steel stress to σ_s,max = 0.80 ⋅ f_yk.

For concrete C30/37, the maximum (negative) concrete stress σ_c,max is thus determined:

• σ_c,max = 0.45 ∙ f_ck = 0.45 ∙ (-30.0) = -13.5 N/mm²

La compression du béton fournie est déterminée en supposant une distribution linéaire des contraintes car le nombre d'itérations nécessaires pour déterminer une direction viable de la bielle en béton comprimée imposerait un calcul trop long. Une distribution linéaire est suffisamment précise car à l'ELS

The maximum stress σ_c,max is to be compared with the provided stress of the concrete compression zone for both reinforcement directions.

The provided concrete compressive stress σ_c is determined as follows:

${\sigma }_c = \frac{m_{Ed}}{I_{i,II}} ·x$

avec

For the reinforcement direction φ₁, the following neutral axis depth x_-z,φ1 is thus obtained:

$x_{-z,{\varphi }_1} = \frac{6.061 · 11.31}{100} · \left(-1.0 + \sqrt{1.0 + \frac{2.0 · 100 · 17}{6.061 ·11.31}}\right) = 4.19\text{ cm}$

The same value and the related intermediate values can also be found in the details table.

For the reinforcement direction φ₂, the neutral axis depth x_-z,φ2 is obtained:

$x_{-z, {\varphi }_2} = \frac{6.061 · 11.31}{100} · \left(-1.0 + \sqrt{1.0 + \frac{2.0 · 100 · 15.8}{6.061 · 11.31}}\right) = 4.02 \text{cm }$

This value and the related intermediate values can also be found in the details.

For the two directions of reinforcement, the ideal moments of inertia I_i,II in state II (cracked section) are determined as follows:

$I_{i,II,-z,{\varphi }_1} = \frac{1}{3} · 100.0 · 4.{19}^3 + 6.061 · 11.31 · {\left(17 - 4.19\right)}^2 = 13701 {\text{cm}}^4$

$I_{i,II,-z,{\varphi }_2} = \frac{1}{3} · 100.0 · 4.{02}^3 + 6.061 · 11.31 · {\left(15.8 - 4.02\right)}^2 = 11 678 {\text{cm}}^4$

Thus, according to Equation 2.69, the following concrete compressive stresses σ_c are obtained in the concrete compression zone (i.e. at the top side of the surface) for the two reinforcement directions φ₁ and φ₂:

${\sigma }_{c,o,{\varphi }_1} = \frac{3 676 · 4.19}{13 701} = -11.24{\text{ N/mm}}^2$

${\sigma }_{c,o,{\varphi }_2} = \frac{2 773 · 4,02}{11 678} = -9.41 {\text{N/mm}}^2$

These values are also shown in Figure 2.92 (the program takes more decimal places into account).

The existing compressive stresses σ_c,+z,φ1 and σ_c,+z,φ2 are therefore smaller than the maximum concrete stress σ_c,max (see Figure 2.90). The governing quotient of existing and allowable concrete compressive stress is available in the reinforcement direction φ₁. The design is fulfilled.