9.2.6.4 Mean Curvatures

Mean Curvatures

The mean curvatures arising with the selected Tension-Stiffening approach are determined from the calculations for pure state I and pure state II.

The underlying Tension Stiffening model described in book 525 [6] considers the concrete's tension stiffening effect between the cracks by means of a reduction of the steel strain. The required parameters are determined as follows.

${\sigma }_{s1,II} = 242.27 \mathrm{N}/{\mathrm{mm}}^2 \ge 1.3 · {\sigma }_{sr1,II} = 215.96 \mathrm{N}/{\mathrm{mm}}^2$

Hence, we will have a closer look at the final crack state.

• ε_sm = ε_s2,II - β_t ∙ (ε_sr,II - ε_sr,I)

• ε_sm = 1.211 - 0.306 ∙ (0.8306 - 0.199) = 1.0177 ‰

where

• ε_s2,II = 1.211 ‰ : steel strain in state II
• ε_sr1,II = 0.8306 : steel strain for crack internal force in state II
• ε_sr1,I = 0.199 ‰ : steel strain for crack internal force in state I
• β_t = 0.306 : load duration factor of available action

${\left(\frac{1}{r}\right)}_{z,m} = \frac{\left({\epsilon }_{sm} - {\epsilon }_c\right)}{d} = \frac{1.0177 + 0.6378}{0.135} = 12.26 \frac{\mathrm{mm}}{\mathrm{m}} = 1.226 · {10}^{-2} {\mathrm{m}}^{-1 }$

From the mean curvature (1/r)_z,m and the relation

${\left(\frac{1}{r}\right)}_{z,m} = \frac{M}{I_{y,m} · E}$

the secant stiffness in the corresponding node results.

$I_{y,m} · E = \frac{M_y}{{\left(1/r\right)}_{z,m}} = \frac{0.01764}{1.226 · {10}^{-2}} = 1.43883 {\mathrm{MNm}}^2 = 1438.83 {\mathrm{kNm}}^2$

where

• M_y = 17.64 kNm : available moment
• (1/r)_z,m = 1.226 ⋅ 10^-2m^-1 : steel strain for crack internal force in state II