# Bending and shear and axial force (6.2.10)

The von Mises stress is often used in determining whether an isotropic and ductile metal will yield when subjected to a complex loading condition. This is accomplished by calculating the von Mises stress and comparing it to the material's yield stress, which constitutes the von Mises Yield Criterion. Von Mises Stress in 2-D applications $(\sigma_{zz} = \tau_{xz} = \tau_{yz} = 0)$

$$\sigma_\text{VM} = \sqrt{\sigma^2_{xx} + \sigma^2_{yy} - \sigma_{xx} \sigma_{yy} + 3 \, \tau^2_{xy} }$$

The above formula is a general form from Mechanics of elastic solids. For the elastic verification, the Eurocode gives us the following yield criterion for a critical point:

$$\begin{equation} \left(\frac{\sigma_{x,Ed}}{f_y/\gamma_{M0}}\right)^2 + \left(\frac{\sigma_{z,Ed}}{f_y/\gamma_{M0}}\right)^2 - \left(\frac{\sigma_{x,Ed}}{f_y/\gamma_{M0}} \cdot \frac{\sigma_{z,Ed}}{f_y/\gamma_{M0}}\right) + 3\left(\frac{\tau_{Ed}}{f_y/\gamma_{M0}}\right)^2 \le 1 \end{equation}$$
• If $V_{Ed} \le 50\ \%\ V_{pl,Rd}$ and no buckling is threatened then shear load is small and might be ignored.
• Otherwise, for great shear forces (that means if $V_{Ed} \gt 50\ \%\ V_{pl,Rd}$) the calculations of allowance use reduced yield strength
$$\begin{equation} f_{y, reduced} = (1-\rho)f_y,\hskip2em \text{where} \hskip2em \rho = (2V_{Ed}/V_{pl,Rd} - 1)^2 \end{equation}$$

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