The cracking moment of a concrete cross-section is calculated from the mean tensile strength of the concrete and the ideal section modulus. The cracking moment describes the internal force that occurs when the tension stress f
ctm is reached in the outermost fiber of the cross-section and crack formation occurs.
For uniaxial bending, it is possible to calculate the cracking moment analytically. For biaxial bending, the introduction of a weight factor k is helpful to determine Mcr from the components Mcr,y and Mcr,z.
Calculation for the attached example:
Bending moment My = 20 kNm
Bending moment Mz = 20 kNm
Ideal section modulus Wy = 3,081 cm3
Ideal section modulus Wz = 3,081 cm3
Mean tensile strength of the concrete fctm = 0.290 kN/cm²
Member 1: Uniaxial bending My:
$\begin{array}{l}M_{cr\;}=f_{ctm}\times W_y\\M_{cr\;}=0.29\;\frac{kN}{cm^2}\times3,081\;cm^3\\M_{cr\;}=893\;kNcm\;=\;8.9\;kNm\end{array}$
Member 2: Uniaxial bending Mz:
$\begin{array}{l}M_{cr\;}=f_{ctm}\times W_z\\M_{cr\;}=0.29\;\frac{kN}{cm^2}\times3,081\;cm^3\\M_{cr\;}=893\;kNcm\;=\;8.9\;kNm\end{array}$
Member 3: Biaxial bending My and Mz:
$\begin{array}{l}M_{cr\;}=\sqrt{M_{cr,y}^2+M_{cr,z}^2}\\M_{cr,y\;}=k\times My\\k=\frac{f_{ctm}}{\sigma_M}\\\sigma_M=\frac{M_y}{W_y}+\frac{M_z}{W_z}=\\\end{array}$
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