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Stress Analysis
Equivalent Stress
The strength calculation determines the maximum load on the screw using the shear stress hypothesis according to TRESCA:
$$σ_v = \sqrt {(σ_x - σ_y)^2 + 4τ_{xy}^2}$$
Since the rotation of the screw is purely a torsion process, the hypothesis can be simplified by considering only the shear stress resulting from torsion. Additionally, bending of the screw can be neglected, meaning the normal stress in the y-direction can also be disregarded. The normal stress in the x-direction is determined solely by the pressures at the beginning and end of the screw.
$$σ_v = \sqrt {σ_x^2 + 4τ_{xy}^2}$$
The screw's strength is ensured when the resulting equivalent stress $\sigma_v$ is less than the permissible stress. The permissible stress typically corresponds to the yield strength $R_{p0,2}$.
Normal Stress
The normal stress in the x-direction within the screw results from the pressures at the beginning and end of the screw. In this context, the pressure at the hopper generally plays a minor role and is usually only relevant for melt extruders.
$$\sigma_x = \sigma_{x,Hopper} + \sigma_{x,ScrewTip}$$ $$\text{with}$$ $$\sigma_{x,Hopper} = \frac{p_{Hopper} \cdot A_{projected}}{A_{screw core}} = p_{Hopper}\cdot \frac{D^2-d^2}{d^2}$$ $$\text{and}$$ $$\sigma_{x,ScrewTip} = \frac{p_{Backpressure} \cdot A_{projected}}{A_{screw core}} = p_{Backpressure}\cdot \frac{D^2}{d^2}$$
With the outer diameter $D$ and the screw core diameter $d$.
The acting force results from the pressure and the projected area on which the pressure acts. It is assumed that the compressive stress is transmitted solely through the screw core.
Shear Stress
The shear stress resulting from the applied torque is calculated using the following formula:
$$\tau_{nominal} = \frac{M_t}{W_t} = \frac{M_t \cdot a_{max}}{I_p}$$
with the torque $M_t$, the section modulus $W_t$, the polar moment of inertia $I_p$ and the maximum perpendicular distance from the outer fiber to the neutral (stress-free) fiber $a_{max}$.
Another influencing factor for calculating the screw strength is the radius that describes the transition from the screw core to the flight. For this geometric influence, a shape factor (stepped round bar under torsion, DIN 743-2) is defined as:
$$\alpha_{\tau} = 1+ \frac{1}{\sqrt{3,4 \cdot \frac{r}{t} + 38 \cdot \frac {r}{d} (1+ 2 \cdot \frac{r}{d})^2 + (\frac{r}{d})^2 \cdot \frac{d}{D}}}$$
with the radius at the flight $r$, the radius difference $t = \frac{D}{2} - \frac{d}{2} = h$, the screw core diameter $d$ and the outer diameter $D$.
From the shape factor, the notch effect factor $\beta_{\tau}$ (DIN 743-2) can be calculated as:
$$\beta_{\tau} = \frac{\alpha_\tau}{n}$$ $$\text{mit}$$ $$n=1+\sqrt{G' \cdot mm} \cdot 10^{-0,7}$$ $$\text{und}$$ $$G'=\frac{1,15}{r}$$
The total influence factor $K_\tau$ (DIN 743-1) is calculated as follows:
$$K_\tau = \left( \frac{\beta_\tau}{K_2(d)}+\frac{1}{K_{F,\tau}}-1 \right)\cdot \frac{1}{K_V}$$ $$\text{with}$$ $$K_2(d)=1-0,2 \frac{log(d/7,5\,mm)}{log(20)} \text{ width } K_2(d>150\,mm)=0,8$$
with the influence of the surface roughness $K_{F,\tau}=1$ and the influence of surface hardening $K_V = 1,15$ for nitrided surfaces.
Finally, the shear stress, taking all influence factors into account, is given by:
$$\tau_{xy} = \tau_{max}=\tau_{nominal} \cdot K_\tau$$