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en:berechnungen:festigkeitsberechnung [2025/02/18 09:28] – cschallen:berechnungen:festigkeitsberechnung [2025/09/03 12:27] (aktuell) – [Shear Stress] neelest
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 ======Stress Analysis====== ======Stress Analysis======
  
-The strength calculation determines the maximum load of the screw with the shear +-> [[..:grafische_darstellung_der_ergebnisse:spannungsanalyse|For graphical representation of the strength calculation]] 
-stress hypothesis according to TRESCA: + 
 +===== Equivalent Stress===== 
 + 
 +The strength calculation determines the maximum load on the screw using the shear stress hypothesis according to TRESCA:
  
 $$σ_v = \sqrt {(σ_x - σ_y)^2 + 4τ_{xy}^2}$$ $$σ_v = \sqrt {(σ_x - σ_y)^2 + 4τ_{xy}^2}$$
  
-The screw fails above this stress. The hypothesis is simplified, because the rotation +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. 
-of the screw generates only torsion stress, neither bend nor normal forces occur, +As bending of the screw can also be neglected, the normal stress in the y-direction  
-therefore just shear stress takes into account. The occurring shear stress is +$\sigma_y$​ may likewise be disregarded. 
-calculated according to the following formula: +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:
  
-$$τ_{max} = \frac{M_t}{W_t} = \frac{M_t \cdot a_{max}}{l_p}$$+$$\tau_{nominal} = \frac{M_t}{W_t} = \frac{M_t \cdot a_{max}}{I_p}$$
  
-with the torque $M_t$, the section modulus $W_t$, the polar area moment of inertia $I_p$ and the maximum vertical distance $a_{max}$ between the edge fibre and the neutral (stress-free) fibre.+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 the calculation of the screw strength is the radius, which +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:
-indicates the transition from the bottom of the screw to the flight; a shape number is +
-defined for this geometric influence: +
  
-$$K_{t,f} = 1+ \frac{1}{\sqrt{3,4 \cdot \frac{r}{t} + 38 \cdot \frac {r}{d} (1+ 2 \cdot \frac{r}{d})^2 + (\frac{r}{d})^2 \cdot \frac{d}{D}}}$$+$$\alpha_{\tau} = 1+ \frac{1}{\sqrt{3,4 \cdot \frac{r}{t} + 38 \cdot \frac {r}{d} (1+ 2 \cdot \frac{r}{d})^2 + (\frac{r}{d})^2 \cdot \frac{d}{D}}}$$
  
-with the radius at the web $r$, the radius difference $t = \frac{D}{2} - \frac{d}{2} = h$, the screw core diameter $d$ and the screw outer diameter $D$.+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 $D$.
  
-Together with the notch coefficient $\beta_{\tau}$ (DIN 743-2):+From the shape factor, the notch effect factor $\beta_{\tau}$ (DIN 743-2) can be calculated as:
  
 $$\beta_{\tau} = \frac{\alpha_\tau}{n}$$ $$\beta_{\tau} = \frac{\alpha_\tau}{n}$$
 $$\text{with}$$ $$\text{with}$$
-$$n=1+\sqrt{G' \cdot mm} \cdot 10^{-0.7}$$+$$n=1+\sqrt{G' \cdot mm} \cdot 10^{-0,7}$$
 $$\text{and}$$ $$\text{and}$$
 $$G'=\frac{1,15}{r}$$ $$G'=\frac{1,15}{r}$$
  
-the total influence factor $K_\tau$ can be calculated (DIN 743-1):+The total influence factor $K_\tau$ (DIN 743-1) is calculated as follows:
  
 $$K_\tau = \left( \frac{\beta_\tau}{K_2(d)}+\frac{1}{K_{F,\tau}}-1 \right)\cdot \frac{1}{K_V}$$ $$K_\tau = \left( \frac{\beta_\tau}{K_2(d)}+\frac{1}{K_{F,\tau}}-1 \right)\cdot \frac{1}{K_V}$$
 $$\text{with}$$ $$\text{with}$$
-$$K_2(d)=1-0,2 \frac{log(d/7,5\,mm)}{log(20)} \text{ with } K_2(d>150\,mm)=0.8$$+$$K_2(d)=1-0,2 \frac{log(d/7,5\,mm)}{log(20)} \text{     with     } K_2(d>150\,mm)=0,8$$
  
-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.+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 stress occurring under consideration of the influencing factors results in:+The resulting shear stress, taking into account these influencing factors, is given by:
  
-$$\tau_{max,korr.}=\frac{\tau_{max}}{K_\tau}$$+$$\tau_{xy} = \tau_{max}=\tau_{nominal} \cdot K_\tau$$
  
 ===Further topics=== ===Further topics===
   * [[en:berechnungen:einfache_berechnung|]]   * [[en:berechnungen:einfache_berechnung|]]
 +  * [[en:berechnungen:prozess_iterieren]]
   * [[en:berechnungen:durchsatz|]]   * [[en:berechnungen:durchsatz|]]
   * [[en:berechnungen:druckverlauf|]]   * [[en:berechnungen:druckverlauf|]]
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   * [[en:berechnungen:temperaturverlauf|]]   * [[en:berechnungen:temperaturverlauf|]]
   * [[en:berechnungen:leistung_und_schubspannungen|]]   * [[en:berechnungen:leistung_und_schubspannungen|]]
 +  * [[en:berechnungen:schergeschwindigkeit]]
   * [[en:berechnungen:verweilzeit|]]   * [[en:berechnungen:verweilzeit|]]
   * [[en:berechnungen:verweilzeitverteilung|]]   * [[en:berechnungen:verweilzeitverteilung|]]