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Introduction to strain gauges and beam bending

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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.

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

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

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Stress-strain Yield stress in “ordinary” steel, 400 Mpa - How much can 1 x 1cm bar hold in tension? Yield stress in “ordinary” Aluminum 100 Mpa – How much can ½ x 1/16 inch bar hold in tension?

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

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Strain gauge 6.4x4.3 mm

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Gauge factor R is nominal resistance GF is gauge factor. For ours, GF = 2.1 Need a circuit to measure a small change in resistance

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Wheatstone bridge +-

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Our setup Strain gauge Proportional to strain !

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In practice we need variable R. Why? Strain gauge

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Demo

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Left side only provides 2.5 V We already have that available = Strain gauge

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A quantitative test

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Conclusion – part 1 Bridge circuit + instrumentation amp. allow us to detect small changes in resistance. Mechanical strain proportional to ΔR/R.

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Beams in bending

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Beam in pure bending MM

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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.'

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Beam in pure bending MM “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 y=0

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Beam in pure bending Fig 5-7, page 304

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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! This relation is easy to prove by geometry Neutral axis

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Cantilever beam L P

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Isolate the beam – Find the reaction forces Sum forces L P R

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Also need a bending moment Sum forces L P R M Sum moments

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State of stress inside the beam L P P P L

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State of stress inside the beam L P P P L Imagine a cut in the beam

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State of stress inside the beam P P P L x M Calculate shear and bending moment to hold at equilibrium

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Shear and bending moment diagram X X Shear M P(L- x) P P P P L x M L

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

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Flexure formula Will derive this in Mechanics of Solids and Structures b h Cross Section y

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Strain in cantilever – at support L P P PL h b

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