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Strength calculation
The strength calculation determines the maximum load of the screw with the shear stress hypothesis according to TRESCA:
$$σ_v = \sqrt {(σ_x - σ_y)^2 + 4τ_{xy}^2}$$
The screw fails above this stress. The hypothesis is simplified, because the rotation of the screw generates only torsion stress, neither bend nor normal forces occur, therefore just shear stress takes into account. The occurring shear stress is calculated according to the following formula:
$$τ_{max} = \frac{M_t}{W_t} = \frac{M_t \cdot a_{max}}{l_p}$$
with the torque $M_t$, the section modulus $W_t$, the polar area moment of inertia $I_p$ and the maximum vertical distance $a_{max}$ between the edge fibre and the neutral (stress-free) fibre.
Another influencing factor for the calculation of the screw strength is the radius, which indicates the transition from the bottom of the screw to the flight; a shape number is defined for this geometric influence:
$$K_{t,f} = 1+ \frac{1}{\sqrt{3,4 \cdot \frac{r}{t} + 38 \cdot \frac {r}{d} (1+ 2 \cdot \frac{r}{d})^2 + (\frac{r}{d})^2 \cdot \frac{d}{D}}}$$
with the radius at the web $r$, the radius difference $t = \frac{D}{2} - \frac{d}{2} = h$, the screw core diameter $d$ and the screw outer diameter $D$.
Together with the notch coefficient $\beta_{\sigma,\tau}$ (DIN 743-2):
$$\beta_{\tau} = \frac{\alpha_\tau}{n}$$ $$\text{with}$$ $$n=1+\sqrt{G' \cdot mm} \cdot 10^{-0.7}$$ $$\text{and}$$ $$G'=\frac{1,15}{r}$$
the total influence factor $K_\tau$ can be calculated (DIN 743-1):
$$K_\tau = \left( \frac{\beta_\tau}{K_2(d)}+\frac{1}{K_{F,\tau}}-1 \right)\cdot \frac{1}{K_V}$$ $$\text{with}$$ $$K_2(d)=1-0,2 \frac{log(d/7,5\,mm)}{log(20)} \text{ with } K_2(d>150\,mm)=0.8$$
and with the influence of the surface roughness $K_{F,\tau}=1$ and the influence of the surface hardening $K_V = 1.15$ for nitrided surfaces.
The stress occurring under consideration of the influencing factors results in:
$$\tau_{max,korr.}=\frac{\tau_{max}}{K_\tau}$$