Schiefe Biegung Beispiel Essay

Gerade Biegung ohne Normalkraft

 Lastebene = Verformungsebene

Voraussetzungen:

  • : -Achse durch den Schwerpunkt verläuft
  • : Querschnitt hat mindestens eine Symmetrieachse

Formeln:

   

mit als maximalen Randfaserabstand und als Biegesteifigkeit.

Lösungsschritte (vgl. Rolf Mahnken, Lehrbuch der Technischen Mechanik – Elastostatik, Springer Verlag, 1. Auflage, 2015)

(i) für statisch bestimmte Systeme

a. Statisches System

b. Schnittgrößen mittels Gleichgewicht am geschnittenen System  TM 1, wobei häufig nur benötigt wird!

c. Querschnittswerte

1. Berechnung der Schwerachse, so dass gilt.

2. Überprüfung, ob mindestens eine Symmetrieachse vorliegt, so dass – Wenn keine Symmetrieachse vorliegt, hat man schiefe Biegung!

3. berechnen.

d. Normalspannungen: maximale Spannung in Randfläche mit größtem Abstand

e. Biegelinie

1. als Funktion der Koordinate für Einzelbalken

2. Rand- und Übergangsbedingungen für und

3. Integration der Differentialgleichung

f. weitere Aufgabenstellungen, z.B. Spannungs-/Verformungsnachweis, Bauteilsicherheit etc.

(iI) für statisch unbestimmte Systeme

a. Statisches System

b. Querschnittswerte

1. Berechnung der Schwerachse, so dass gilt.

2. Überprüfung, ob mindestens eine Symmetrieachse vorliegt, so dass – Wenn keine Symmetrieachse vorliegt, hat man schiefe Biegung!

3. berechnen.

c. Biegelinie, unter Beachtung von Rand- und Übergangsbedingungen für berechnet man für jeden Einzelbalken:

   

d. weitere Aufgabenstellungen, z.B. , Dimensionierung etc.

Kochrezept bei Aufgaben zum Biegebalken

Vorgehen bei statisch unbestimmten Systemen

Beispiel Biegung ohne Normalkraft

 

Gerade Biegung mit Normalkraft

  • gleichgültig ob statisch bestimmt oder unbestimmt!
  • Vorgehen gemäß jeweiligen Lösungsschritten von oben
  • Resultierende Verschiebungen und Spannungen über Superposition
  • Voraussetzung: ;

Formeln:

1. Normalspannung:

2. Verschiebung:

3. Gleichgewicht:

 

Schiefe Biegung

 Lastebene  Verformungsebene

  • Belastung schräg zur Symmetrieachse oder bei unsymmetrischen Querschnitten
  • Auch hier gibt es statisch bestimmte und unbestimmte Systeme!

Insgesamt sind 4 Fälle zu unterscheiden:

1. Koordinatensystem im Schwerpunkt

Normalspannung:

   

Verschiebungen:

   

Resultierende Verschiebung:

2. Transformation auf Hauptachsen

In Hauptachsen sind maximal und

(i) Berechnung der Hauptträgheitsmomente und der Hauptachsen mit den Winkeln

(ii) Transformation der Momente und der Koordinaten bestimmter Punkte auf Hauptachsensystem .

   

Normalspannung:

Verschiebungen:

   

Fazit: Berechnung der Spannungen und Verschiebungen wesentlich einfacher, wenn man vorher auf die Hauptachsen transformiert!

3. Koordinatenachsen sind gleichzeitig Hauptachsen

Normalspannung: 

Verschiebungen:

   

Charakteristische Winkel

Winkel des resultierenden Momentes. Formel:

   

Winkel der Spannungsnulllinie. Formel:

   

 

4. Gerade Biegung ohne Normalkraft

-Hauptachsen im Schwerpunkt;

Fall 1: (Vergleich Gerade Biegung)

   

Fall 2:

   

Vorgehen bei schiefer Biegung

Flächenträgheitsmomente

Die Technische Torsionstheorie

Kochrezept bei Aufgaben zum Biegebalken – Technische Mechanik 2

Biegung – Biegelinie – bei statisch unbestimmten Systemen – Technische Mechanik 2

Balken – Biegelinie – Technische Mechanik 2

Vorgehen bei schiefer Biegung – Technische Mechanik 2

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