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--- en:berechnungen:festigkeitsberechnung [2024/10/17 20:17] – [Strength calculation] neelest
+++ en:berechnungen:festigkeitsberechnung [2025/09/03 12:27] (aktuell) – [Shear Stress] neelest
@@ Zeile 1: / Zeile 1: @@
-======Strength calculation======
+======Stress Analysis======
-===== Strength calculation =====
+-> [[..:grafische_darstellung_der_ergebnisse:spannungsanalyse|For graphical representation of the strength calculation]]
-The strength calculation determines the maximum load of the screw with the shear
+===== Equivalent Stress=====
-stress hypothesis according to TRESCA:
+The strength calculation determines the maximum load on the screw using the shear stress hypothesis according to TRESCA:
 $$σ_v = \sqrt {(σ_x - σ_y)^2 + 4τ_{xy}^2}$$
-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:
+$$\tau_{nominal} = \frac{M_t}{W_t} = \frac{M_t \cdot a_{max}}{I_p}$$
+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:
+$$\alpha_{\tau} = 1+ \frac{1}{\sqrt{3,4 \cdot \frac{r}{t} + 38 \cdot \frac {r}{d} (1+ 2 \cdot \frac{r}{d})^2 + (\frac{r}{d})^2 \cdot \frac{d}{D}}}$$
-$$τ_{max} = \frac{M_t}{W_t} = \frac{M_t \cdot a_{max}}{l_p}$$
+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 $D$.
-Another influencing factor for the calculation of the screw strength is the radius, which
+From the shape factor, the notch effect factor $\beta_{\tau}$ (DIN 743-2) can be calculated as:
-indicates the transition from the bottom of the screw to the flight; a shape number is
-defined for this geometric influence:
-$$K_{t,f} = 1+ \frac{1}{\sqrt{3,4 \cdot \frac{r}{t} + 38 \cdot \frac {r}{d} (1+ 2 \cdot \frac{r}{d})^2 + (\frac{r}{d})^2 \cdot \frac{d}{D}}}$$
+$$\beta_{\tau} = \frac{\alpha_\tau}{n}$$
+$$\text{with}$$
+$$n=1+\sqrt{G' \cdot mm} \cdot 10^{-0,7}$$
+$$\text{and}$$
+$$G'=\frac{1,15}{r}$$
-In combination with the notch sensitivity number $q$:
+The total influence factor $K_\tau$ (DIN 743-1) is calculated as follows:
-$$q = \frac{1}{1+ \frac{8mm}{r} \cdot (1- \frac{R_{p0,2}}{R_m})^3}$$
+$$K_\tau = \left( \frac{\beta_\tau}{K_2(d)}+\frac{1}{K_{F,\tau}}-1 \right)\cdot \frac{1}{K_V}$$
+$$\text{with}$$
+$$K_2(d)=1-0,2 \frac{log(d/7,5\,mm)}{log(20)} \text{     with     } K_2(d>150\,mm)=0,8$$
-it is possible in according to THUM to calculate the notch coefficient $K_f$. The following
+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.
-equation describes the influence of the notch on the shear stress:
+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.
-$$K_f = 1+ (K_t-1) \cdot q$$
+The resulting shear stress, taking into account these influencing factors, is given by:
-Finally, the shape number $K_t$ is multiplied by the calculated shear stress to obtain the
+$$\tau_{xy} = \tau_{max}=\tau_{nominal} \cdot K_\tau$$
-maximum stress or rather the characteristic value oft he screw strength.
+===Further topics===
+  * [[en:berechnungen:einfache_berechnung|]]
+  * [[en:berechnungen:prozess_iterieren]]
+  * [[en:berechnungen:durchsatz|]]
+  * [[en:berechnungen:druckverlauf|]]
+  * [[en:berechnungen:aufschmelzverlauf|]]
+  * [[en:berechnungen:temperaturverlauf|]]
+  * [[en:berechnungen:leistung_und_schubspannungen|]]
+  * [[en:berechnungen:schergeschwindigkeit]]
+  * [[en:berechnungen:verweilzeit|]]
+  * [[en:berechnungen:verweilzeitverteilung|]]
+  * [[en:berechnungen:materialabbau|]]
+  * [[en:berechnungen:faserlaengenabbau|]]
+  * [[en:berechnungen:entgasungskennzahlen|]]
+  * [[en:berechnungen:festigkeitsberechnung|]]
+  * [[en:berechnungen:schlepp-druckstroemung|]]
+  * [[en:berechnungen:feststofffoerderung|]]
+  * [[en:berechnungen:verarbeitung_von_mischungen|]]
+  * [[en:berechnungen:wandgleitende_materialien|]]
+  * [[en:berechnungen:nutbuchsenberechnung|]]
+  * [[en:berechnungen:kompressionsverhaeltnisse|]]