Bending Deformation of a Straight Member

1. Lecture No. 17 Bending Deformation of a Straight Member Objective: To study the behavior of the member under bending; ● To find out the type of stress that the bending is causing ● To find Stress / Moment relationship

2. y x + Strain-Curvature Relationship • Strain varies linearly with y • +ve moment causes compression in the • upper half and tension in the lower half.

3. The Flexure Formula This wood specimen failed in bending due to its fibers being crushed at its top and torn apart at its bottom.

4. -y dA The Flexure Formula... Upon compression (5) & (6)

5. y z Summary • Neutral axis which passes through the centroid of the cross section • divides the section to two areas, one is subjected to compression • above the NA and the other below which is tension. • Need always centroid and moment of inertia • about bending axis if not given need to find them. • Apply the flexure formula to find stresses • at any point. (fraction of y not z). • Maximum stress happened at maximum • moment at the top and bottom • of the cross-section y z

6. Example: A beam has a rectangular cross section and is subjected to the stress distribution. Determine the internal moment M at the section caused by the stress distribution (a) using the flexure formula, (b) by finding the resultant of the stress distribution using basic principles.

7. Solution:

8. Solution (cont):

9. Example: The simple supported beam has the cross-sectional area as shown in the following figures. Determine the absolute maximum bending stress in the beam and draw the stress distribution over the cross section at this location.

10. Solution: Section property.

11. Solution (cont): Bending Stress.

12. Example: The beam has a cross-sectional area in the shape of a channel. Determine the maximum bending stress that occurs in the beam at section a-a.

13. Solution: Internal Moment.

14. Solution (cont): Section Property. Maximum Bending Stress.

15. The enD