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COMPOSED BENDING (Eccentric tension/compression)

COMPOSED BENDING (Eccentric tension/compression). z. y. M y. Plane bending. Neutral axis for bending. z. x. z. y. + . Tension. Neutral axis for tension. Neutral axis for bending. z. x. - . z. y. M y. Combined bending. Plane bending. z. Neutral axis. z 0. =. +. +.

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COMPOSED BENDING (Eccentric tension/compression)

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  1. COMPOSED BENDING(Eccentric tension/compression)

  2. z y My Plane bending Neutral axis for bending z x

  3. z y +  Tension Neutral axis for tension Neutral axis for bending z x - 

  4. z y My Combined bending Plane bending z Neutral axis z0 = + + Neutral axis equation: „Eccentric” Squared inertia radius

  5. N / A Bi-plane combined loading Normal stress Eccentrics Normal stress in terms of normal force and eccentrics y0 ,z0 Non-dimensional normal stress Normal stress at neutral axis Neutral axis equation

  6. NEUTRAL AXIS MOVEMENT in (y,z) plane

  7. Side view Stress distribution Cross-section view

  8. + N/A N N Neutral axis in an „infinity” A Side view Stress distribution Cross-section view

  9. + N/A N Neutral axis already being „seen” A Side view Stress distribution Cross-section view

  10. + N/A N Neutral axis outside of cross-section A Side view Stress distribution Cross-section view

  11. + N/A N Neutral axis touching cross-section contour N N A Side view Stress distribution Cross-section view

  12. + - N/A N Neutral axis within cross-section A Side view Stress distribution Cross-section view

  13. Dual interpretation of neutral axis equation Eccentric co-ordinates Neutral axis co-ordinates

  14. Dual interpretation of neutral axis equation 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):

  15. Dual interpretation of neutral axis equation 2. If a number of neutral axis touches cross-section corner of given co-ordinates: 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:

  16. 2 cm 3 cm 3 cm 2 cm 8 cm 14 cm 4 cm 8 cm Example of cross-section kernel finding

  17. a d C B D A b c Example of cross-section kernel finding z Mode 1: Finding eccentrties y 7,22 cm

  18. e Example of cross-section kernel finding Mode 2: Finding normal force acting lines z E y 7,22 cm

  19. f Example of cross-section kernel finding z Mode 2: Finding normal force acting lines F y 7,22 cm

  20. g Example of cross-section kernel finding z Mode 2: Finding normal force acting lines y 7,22 cm G

  21. Example of cross-section kernel finding z y 7,22 cm

  22. Example of cross-section kernel finding z y 7,22 cm

  23. Cross-section kernel 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.

  24. stop

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