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 represents a purely torsional load, the hypothesis can be simplified and only the shear stress resulting from torsion needs to be considered. As bending of the screw can also be neglected, the normal stress in the y-direction $\sigma_y$ may likewise be disregarded. The normal stress in the x-direction $\sigma_x$ therefore results solely from the pressures at the inlet and outlet 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:
$$\beta_{\tau} = \frac{\alpha_\tau}{n}$$ $$\text{with}$$ $$n=1+\sqrt{G' \cdot mm} \cdot 10^{-0,7}$$ $$\text{and}$$ $$G'=\frac{1,15}{r}$$
The total influence factor $K_\tau$ (DIN 743-1) is calculated as follows:
$$K_\tau = \left( \frac{\beta_\tau}{K_2(d)}+\frac{1}{K_{F,\tau}}-1 \right)\cdot \frac{1}{K_V}$$ $$\text{with}$$ $$K_2(d)=1-0,2 \frac{log(d/7,5\,mm)}{log(20)} \text{ with } K_2(d>150\,mm)=0,8$$
as well as with the influence of surface roughness $K_{F,\tau}=1$ for polished surfaces and the influence of surface hardening $K_V$. $K_V$ ranges from 1.15 to 1.25 for nitrided surfaces, 1.2 to 2.1 for case-hardened surfaces, and 1.2 to 1.6 for induction-hardened surfaces. The values apply up to a diameter of 25 mm and decrease linearly beyond this value down to 1.0 at a diameter of 40 mm. For diameters above 40 mm, the value remains constant at 1.0.
The resulting shear stress, taking into account these influencing factors, is given by:
$$\tau_{xy} = \tau_{max}=\tau_{nominal} \cdot K_\tau$$