Querkraftbemessung (Bsp.)

Querschnittsabmessungen und Betongüte (Balken)

${\begin{array}{l}b/h=30/80cm\\d=74cm\\C35/45\end{array}}$

Einwirkende Bemessungskräfte und vorhandene Längsbewehrung

${\begin{array}{l}{{V}_{Ed}}=500kN\\{{N}_{Ed}}=50kN\\{{A}_{s,vorh}}=15,71cm{}^{\text{2}}\end{array}}$

Querkrafttragfähigkeit ohne Querkraftbewehrung

${\begin{array}{l}{{C}_{Rdc}}={\frac {0,15}{{\gamma }_{c}}}={\frac {0,15}{1,5}}=0,1\\k=1+{\sqrt {\frac {200}{740}}}\leq 2\\\ \ =1+{\sqrt {\frac {200}{740}}}=1,52\\{{\rho }_{l}}={\frac {{A}_{sl}}{{{b}_{w}}\cdot d}}={\frac {15,71}{30\cdot 74}}=0,007\\{{\sigma }_{cp}}={\frac {{N}_{Ed}}{{A}_{c}}}={\frac {50000}{\left(300\cdot 800\right)-1571}}=0,21N/mm{}^{\text{2}}\end{array}}$

${\begin{array}{l}{{V}_{Rd,c}}=\left[{{C}_{Rdc}}\cdot k\cdot {{\left(100\cdot {{\rho }_{l}}\cdot {{f}_{ck}}\right)}^{\frac {1}{3}}}+0,12\cdot {{\sigma }_{cp}}\right]\cdot {{b}_{w}}\cdot d\\\quad \ \ \ =\left[0,1\cdot 1,52\cdot {{\left(100\cdot 0,007\cdot 35\right)}^{\frac {1}{3}}}+0,12\cdot 0,21\right]\cdot 300\cdot 740\\\quad \quad =103,6kN\quad \leq \quad 500kN\end{array}}$

Anordnung senkrechter Bügel

Winkel der Betondruckstrebe

${\begin{array}{l}{{V}_{Rd,cc}}=0,5\cdot 0,48\cdot f_{ck}^{\frac {1}{3}}\cdot \left(1-1,2\cdot {\frac {{\sigma }_{cd}}{{f}_{cd}}}\right)\cdot {{b}_{w}}\cdot z=0,5\cdot 0,48\cdot {{35}^{\frac {1}{3}}}\cdot \left(1-1,2\cdot {\frac {0,21}{0,85\cdot {\frac {35}{1,5}}}}\right)\cdot 300\cdot 0,9\cdot 740\\\quad \quad =154,861kN\\\cot \Theta ={\frac {1,2-1,4\cdot {\frac {{\sigma }_{cp}}{{f}_{cd}}}}{1-{\frac {{V}_{Rd,cc}}{{V}_{Ed}}}}}={\frac {1,2-1,4\cdot {\frac {0,21}{0,85\cdot {\frac {35}{1,5}}}}}{1-{\frac {154,861}{500}}}}=1,72\end{array}}$

Nachweis der Betondruckstrebe

${\begin{array}{l}{{\nu }_{1}}=0,75\cdot \left(1,1-{\frac {{f}_{ck}}{500}}\right)\leq 0,75\\\quad =0,75\cdot \left(1,1-{\frac {35}{200}}\right)=0,69\\{{V}_{Rd,\max }}={{\nu }_{1}}\cdot {{f}_{cd}}\cdot {{b}_{w}}\cdot z\cdot {\frac {\cot \Theta }{1+\cot {}^{\text{2}}\Theta }}=0,69\cdot \left(0,85\cdot {\frac {35}{1,5}}\right)\cdot 300\cdot \left(0,9\cdot 740\right)\cdot {\frac {1,72}{1+1,72{}^{\text{2}}}}\\\quad \quad \ \ \ =1188kN\geq 500kN\end{array}}$

Nachweis der Zugstrebe

${\begin{array}{l}{{V}_{Rd,s}}={{a}_{sw}}\cdot {{f}_{yd}}\cdot \sin \alpha \cdot z\cdot (\cot \Theta +\cot \alpha )\\{{a}_{sw}}\geq {\frac {{V}_{Ed}}{{{f}_{yd}}\cdot \sin \alpha \cdot z\cdot \left(\cot \Theta +\cot \alpha \right)}}\\\quad \,\,\geq {\frac {500}{{\frac {50}{1,15}}\cdot \sin 90\cdot 0,9\cdot 74\cdot \left(1,72\right)}}=0,10039cm{}^{\text{2}}/cm=10,04cm{}^{\text{2}}/m\end{array}}$

gewählt: zweischnittig ø10/15cm, ${{a}_{sw}}=12,56cm{}^{\text{2}}/m$

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