# Why is the cracking moment Mcr smaller for biaxial bending than for uniaxial bending?

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 fctm 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 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:

$\begin{array}{l}M_{cr\;}=\sqrt{M_{cr,y}^2+M_{cr,z}^2}\\M_{cr,y\;}=k\times My\\k=\frac{f_{ctm}}{\sigma_M}\\\sigma_M=\frac{M_y}{W_y}+\frac{M_z}{W_z}=\\\end{array}$