Mohrscher Spannungskreis Beispiel Essay

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Wenn es um den Mohr’schen Spannungskreis geht, werden in der Regel folgende Aufgabentypen behandelt:

(i) Ermittlung von Hauptspannungen

(ii) Ermittlung der Spannungen in gedrehten Koordinatensystemen

Gegeben sei der ebene Spannungszustand . Zu den typischen Aufgabentypen schauen wir uns folgende Lösungsschritte an (vgl. Rolf Mahnken, Lehrbuch der Technischen Mechanik – Elastostatik, Springer Verlag, 1. Auflage, 2015).

Lösungsschritte zu (i):

  1. Achsen zeichnen – positiv nach unten!
  2. Eintragen der Punkte: und 
  3. Schnittpunkt der Verbindungslinie mit -Achse liefert Kreismittelpunkt
  4. Kreis um mit Radius zeichnen
  5. Hauptspannungen aus Schnittpunkt mit -Achse abgreifen
  6. Doppelten Hauptspannungswinkel ablesen

Lösungsschritte zu (ii):

  1. Verbindungen von mit und legen -Achsen fest! Richtungssinn von beliebig, unter Beachtung eines Rechtssystems folgt der Richtungssinn von .
  2. Von -Achse ausgehend für gegebenen Winkel die -Achse (\xi = Xi) zeichnen
  3. Unter Beachtung des Richtungssinnes folgt die -Achse (= Eta)

Merke: Aus wird Xi und aus wird Eta!

  1. Schnittpunkte der -Achse mit Kreis legen Punkte und fest
  2. Abgreifen der Spannungen  und 

Rechnerische Bestimmung:

(i) Hauptnormalspannungen (kurz: Hauptspannungen)


 In Hauptspannungsrichtung verschwindet Schubspannung!

  • Winkel der maximalen/minimalen Hauptspannungsrichtung:


  • Kontrolle über Invarianten:


(ii) Hauptschubspannungen


mit den Winkeln der Hauptschubspannungsachsen:



Video typische Belastungsfälle beim Mohrschen Spannungskreis


Aufgabe zum Mohr’schen Spannungskreis

Gegeben ist der folgende ebene Spannungszustand .

Bestimme zeichnerisch/rechnerisch

  1. die Hauptspannungen,
  2. die maximale Schubspannung,
  3. den Hauptspannungswinkel,
  4. die Spannungen für ein um 45° gedrehtes Koordinatensystem.

Welche Vergleichsspannungshypothesen gibt es und in welchen Bereichen finden die jeweiligen Hypothesen Anwendung?


Video Mohrscher Spannungskreis ähnliches Beispiel

Schubspannung infolge Querkraft

Mohrscher Spannungskreis – typische Belastungsfälle – Technische Mechanik 2

Mohrscher Spannungskreis – Hauptspannungen – Technische Mechanik 2

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