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

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

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

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

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

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

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

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

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