1. 1 /14 M.Chrzanowski: Strength of Materials SM2-03: Bending B E N D I N G

2. 2 /14 M.Chrzanowski: Strength of Materials SM2-03: Bending N=0, Q y =0, Q z =0 M y = M y z x M M M x =0, M y ≠ 0, M z =0 M y (x)=M=const MyMy Q z (x)=0 N=0, Q y =0, Q z =0 M x =0, M y =0, M z ≠ 0 „Pure” bending Formal definition: the case when set of internal forces reduces solely to the moment vector which is perpendicular to the bar axis Example: a straight bar loaded by concentrated moments applied at its ends. or

3. 3 /14 M.Chrzanowski: Strength of Materials SM2-03: Bending  Remarks on terminology

4. 4 /14 M.Chrzanowski: Strength of Materials SM2-03: Bending NORMAL (proste)INCLINED (ukośne) y z x y z x M x =0, M y ≠ 0, M z ≠0M x =0, M y ≠ 0, M z =0 90 o <90 o 90 o <90 o

5. 5 /14 M.Chrzanowski: Strength of Materials SM2-03: Bending PURE (czyste) y z x M x =0, M y ≠ 0, M z =0 N=0, Q y =0, Q z =0 IMPURE („nie-czyste”) y z x M x =0, M y ≠ 0, M z =0 N=0, Q y =0, Q z ≠ 0 NON-UNIFORM (poprzeczne)

6. 6 /14 M.Chrzanowski: Strength of Materials SM2-03: Bending  End of remarks

7. 7 /14 M.Chrzanowski: Strength of Materials SM2-03: Bending E.Mariotte (1620-1684) Galileo (1564-1642) EXPERIMENTAL approach

8. 8 /14 M.Chrzanowski: Strength of Materials SM2-03: Bending EXPERIMENTAL approach Galileo (1564-1642) Jacob Bernoulli (1654-1705) x z D D’ P uDuD wDwD l h M=M(x) Q=Q(x) M x =0, M y ≠ 0, M z =0 N=0, Q y =0, Q z ≠ 0 For h<<l shear forces can be neglected N=0, Q y =0, Q z = 0 u is linear function of z !  is linear function of z and does not depend on x if M=const| x

9. 9 /14 M.Chrzanowski: Strength of Materials SM2-03: Bending tension compression Hooke law: Continuum Mechanics application y z x

10. 10 /14 M.Chrzanowski: Strength of Materials SM2-03: Bending x y z z MyMy ? Continuum Mechanics application

11. 11 /14 M.Chrzanowski: Strength of Materials SM2-03: Bending x y z MyMy ? y-axis is the central inertia axis of cross- section area y-z axes are central principal inertia axes of cross-section area z Equilibrium conditions

12. 12 /14 M.Chrzanowski: Strength of Materials SM2-03: Bending Axes x – which coincides with bar axis y,z – which are central principal inertia axes of the bar cross-section area are principal axes of strain and stress matrices Continuum Mechanics application

13. 13 /14 M.Chrzanowski: Strength of Materials SM2-03: Bending x y z z MyMy where W y is called section modulus For z =0 (i.e. along y- axis ) there is and section of y-axis within bar cross-section is called neutral axis (for normal stress and strain) Neutral axis Neutral axis coincides with only non-zero bending moment component M y Pure plane bending

14. 14 /14 M.Chrzanowski: Strength of Materials SM2-03: Bending Important remarks 1. All above formulas are valid only for principal central axes of cross-section inertia 2. If moment vector coincides with any of two principal axes we have to deal with plane bending. If this is not the case – we have to deal with inclined bending and derived formulas cannot be used. 3. Bar axis ( x -axis) is one of the principal axis of strain and stress matrices. As two remaining principal stress and strains are equal therefore any two perpendicular axes lying in the plane of bar cross-section are also principal axes. 4.The neutral axis for normal stress and strain coincides with bending moment vector.

15. 15 /14 M.Chrzanowski: Strength of Materials SM2-03: Bending  stop