Introduction to beam bending

1. Introduction to beam bending There are no circuits in what follows. I will not use the words voltage, current, or op-amp today (well, maybe just once or twice if you count this slide).

2. Labs and work • You need to let us know (while working, not after the fact) if labs are taking too long. • You should be working, but it should be reasonable (3 credit course, ~9hrs/week total). • Waiting to the night before, means you are up all night.

3. Lab etiquette • Clean up, clean up, everybody let’s clean up. • Keep your op-amp. Do not take them out of the breadboard and leave on the desk. • Throw away old wires, resistors, and scraps. • Cups, plates, and food? Really? • Solderless breadboard, www.digikey.com part 438-1045-ND. $8.98. 438-1046-ND is $13.82 and comes with a little box of wires.

4. Beam bending Galileo, 1638 (though he wasn’t right)

5. DaVinci-1493 "Of bending of the springs: If a straight spring is bent, it is necessary that its convex part become thinner and its concave part, thicker. This modification is pyramidal, and consequently, there will never be a change in the middle of the spring. You shall discover, if you consider all of the aforementioned modifications, that by taking part 'ab' in the middle of its length and then bending the spring in a way that the two parallel lines, 'a' and 'b' touch a the bottom, the distance between the parallel lines has grown as much at the top as it has diminished at the bottom. Therefore, the center of its height has become much like a balance for the sides. And the ends of those lines draw as close at the bottom as much as they draw away at the top. From this you will understand why the center of the height of the parallels never increases in 'ab' nor diminishes in the bent spring at 'co.'

6. Normal stress (σ) and strain (ε) L δ P P

7. Stress-strain Yield stress in “ordinary” steel, 200 Mpa How much can 3 x 0.25 bar hold in tension?

8. Hooke’s law What is the strain just before steel yields?

9. Shear stress • P28 fig 1-24, 1-28

10. Shear stress in tension/compression P P N V P P N V

11. Beams in bending

12. Simply supported beam P La Lb

13. How do we find reaction forces? P La Lb Ra Rb Sum forces Sum moments

14. State of stress inside the beam P La Lb Ra Rb

15. State of stress inside the beam P La Lb Ra Rb Imagine a cut in the beam

16. State of stress inside the beam x M V P Lb/L Calculate shear and bending moment to hold at equilibrium

17. State of stress inside the beam P Lb/L V x M V X P Lb/L PLb/L x M X

18. State of stress inside the beam P P Lb/L V x M X V P La/L P Lb/L P La Lb/L M X

19. Shear and bending moment diagram La Lb P Lb/L V X P La/L P La Lb/L M X

20. Shear and bending diagram examples

21. Beam in pure bending M M

22. Beam in pure bending Fig 5-7, page 304

23. Beam in pure bending Lines, mn and pq remain straight – due to symmetry. Top is compressed, bottom expanded, somewhere in between the length is unchanged! Neutral axis This relation is easy to prove by geometry

24. Beam in pure bending M M y=0 “If a straight spring is bent, it is necessary that its convex part become thinner and its concave part, thicker. This modification is pyramidal, and consequently, there will never be a change in the middle of the spring.” DaVinci 1493

25. Normal stress in bending M σ y Take a slice through the beam Neutral axis is the centroid

26. Normal stress in bending M σ

27. Flexure formula

28. Moment of inertia, I Cross Section h b

29. Lab • Calculate shear/moment diagram for your beam, now know M(x). • Calculate I, look up E. • Calculate strain, ε(x). • Stand on beam, measure strain. Strain gages