1. 1 /10 M.Chrzanowski: Strength of Materials SM2-04: Inclined bending INCLINED BENDING

2. 2 /10 M.Chrzanowski: Strength of Materials SM2-04: Inclined bending If bending moment does not coincide with any of principal central axis of cross-section inertia we have to deal with inclined bending. y z  Plane of load acting Direction of bending moment vector  y z y z +

3. 3 /10 M.Chrzanowski: Strength of Materials SM2-04: Inclined bending y z y z We use superposition principle For neutral axis: y z Neutral axis  J y >J z J y <J z y z y z

4. 4 /10 M.Chrzanowski: Strength of Materials SM2-04: Inclined bending Stress distribution: y z  Neutral axis is represented by the equation of the plane Tension zone Compression zone This plane cuts thorough cross-section plane at the neutral axis

5. 5 /10 M.Chrzanowski: Strength of Materials SM2-04: Inclined bending Animation of neutral axis movement dependent on bending moment direction

6. 6 /10 M.Chrzanowski: Strength of Materials SM2-04: Inclined bending y z 80 o 70 o 60 o 50 o 40 o 30 o 20 10 o 00 o h/b=1,41 J y /J z =2 Loading plane Neutral axis Bending moment vector Angle  

7. 7 /10 M.Chrzanowski: Strength of Materials SM2-04: Inclined bending y z h/b=1,41 J y /J z =2 Loading plane Neutral axis Bending moment vector  80 o 70 o 60 o 50 o 40 o 30 o 20 10 o 00 o

8. 8 /10 M.Chrzanowski: Strength of Materials SM2-04: Inclined bending y z b/h=1,41 J y /J z =1/2 Loading plane Neutral axis Bending moment vector  80 o 70 o 60 o 50 o 40 o 30 o 20 10 o 00 o

9. 9 /10 M.Chrzanowski: Strength of Materials SM2-04: Inclined bending y z Płaszczyzna obciążenia Oś obojętna Wektor momentu h=b=a J y /J z =1 y z Z max =a/2=0,5a Z max =a/2 ½ =0,7a Ideal shape: independent of plane of loading angle maximum stress is the same! y z

10. 10 /10 M.Chrzanowski: Strength of Materials SM2-04: Inclined bending  stop

11. 11 /10 M.Chrzanowski: Strength of Materials SM2-04: Inclined bending y z h/b=1,41 J y /J z =2 Loading plane Neutral axis Bending moment vector Angle   80 o 70 o 60 o 50 o 40 o 30 o 20 10 o 00 o

12. 12 /10 M.Chrzanowski: Strength of Materials SM2-04: Inclined bending y z h/b=1,41 J y /J z =2 Loading plane Neutral axis Bending moment vector Angle  80 o 70 o 60 o 50 o 40 o 30 o 20 10 o 00 o

13. 13 /10 M.Chrzanowski: Strength of Materials SM2-04: Inclined bending y z Płaszczyzna obciążenia Oś obojętna Wektor momentu h=b=a J y /J z =1 y z α =45° Z max =a/2=0,5a Z max =a/2 ½ =0,7a Ideal shape: independent of plane of loading angle maximum stress is the same!