1. 1 /23 M.Chrzanowski: Strength of Materials SM2-05: Composed bending COMPOSED BENDING (Eccentric tension/compression)

2. 2 /23 M.Chrzanowski: Strength of Materials SM2-05: Composed bending x y z z MyMy Neutral axis for bending Plane bending

3. 3 /23 M.Chrzanowski: Strength of Materials SM2-05: Composed bending x y z z Neutral axis for bending Neutral axis for tension +  -  Tension

4. 4 /23 M.Chrzanowski: Strength of Materials SM2-05: Composed bending y z z MyMy Neutral axis + = + Neutral axis equation: „Eccentric” Squared inertia radius Plane bending Combined bending z0z0

5. 5 /23 M.Chrzanowski: Strength of Materials SM2-05: Composed bending Normal stress Normal stress at neutral axis A N / Eccentrics Normal stress in terms of normal force and eccentrics y 0,z 0 Non-dimensional normal stress Neutral axis equation Bi-plane combined loading

6. 6 /23 M.Chrzanowski: Strength of Materials SM2-05: Composed bending NEUTRAL AXIS MOVEMENT in (y,z) plane

7. 7 /23 M.Chrzanowski: Strength of Materials SM2-05: Composed bending Side view Stress distribution Cross-section view

8. 8 /23 M.Chrzanowski: Strength of Materials SM2-05: Composed bending A + N/A Neutral axis in an „infinity” N N Side view Stress distribution Cross-section view

9. 9 /23 M.Chrzanowski: Strength of Materials SM2-05: Composed bending N A + N/A Side view Stress distribution Cross-section view Neutral axis already being „seen”

10. 10 /23 M.Chrzanowski: Strength of Materials SM2-05: Composed bending N A N/A + Side view Stress distribution Cross-section view Neutral axis outside of cross-section

11. 11 /23 M.Chrzanowski: Strength of Materials SM2-05: Composed bending N A + N/A N N Side view Stress distribution Cross-section view Neutral axis touching cross-section contour

12. 12 /23 M.Chrzanowski: Strength of Materials SM2-05: Composed bending N A + - N/A Side view Stress distribution Cross-section view Neutral axis within cross-section

13. 13 /23 M.Chrzanowski: Strength of Materials SM2-05: Composed bending Dual interpretation of neutral axis equation Neutral axis co-ordinates Eccentric co-ordinates

14. 14 /23 M.Chrzanowski: Strength of Materials SM2-05: Composed bending 1. If neutral axis coincides with cross-section contour edge given by the equation: then from the transformed equation of neutral axis: one can find co-ordinates of normal force position (eccentricity): Dual interpretation of neutral axis equation

15. 15 /23 M.Chrzanowski: Strength of Materials SM2-05: Composed bending then by inserting these co-ordinates into neutral axis equation one can obtain eqaution of a straigth line defining the position of a normal force: 2. If a number of neutral axis touches cross-section corner of given co-ordinates: Dual interpretation of neutral axis equation

16. 16 /23 M.Chrzanowski: Strength of Materials SM2-05: Composed bending 4 cm 14 cm 2 cm 8 cm 2 cm 3 cm 8 cm 3 cm Example of cross-section kernel finding

17. 17 /23 M.Chrzanowski: Strength of Materials SM2-05: Composed bending 7,22 cm y z a b c d D B A C Example of cross-section kernel finding Mode 1: Finding eccentrties

18. 18 /23 M.Chrzanowski: Strength of Materials SM2-05: Composed bending 7,22 cm y z e E Example of cross-section kernel finding Mode 2: Finding normal force acting lines

19. 19 /23 M.Chrzanowski: Strength of Materials SM2-05: Composed bending 7,22 cm y z f F Example of cross-section kernel finding Mode 2: Finding normal force acting lines

20. 20 /23 M.Chrzanowski: Strength of Materials SM2-05: Composed bending 7,22 cm y z G g Example of cross-section kernel finding Mode 2: Finding normal force acting lines

21. 21 /23 M.Chrzanowski: Strength of Materials SM2-05: Composed bending 7,22 cm y z Example of cross-section kernel finding

22. 22 /23 M.Chrzanowski: Strength of Materials SM2-05: Composed bending 7,22 cm y z Example of cross-section kernel finding

23. 23 /23 M.Chrzanowski: Strength of Materials SM2-05: Composed bending Cross-section kernel defines normal force positions (eccentrities) such that resulting normal stress in the whole cross-section area has the same sign (plus for N>0 and minus dla N<0). Cross-section kernel has always a convex form. Cross-section kernel

24. 24 /23 M.Chrzanowski: Strength of Materials SM2-05: Composed bending  stop