Introduction to strain gauges and beam bending
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Transcript of Introduction to strain gauges and beam bending
Introduction to strain gauges and beam bending
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.
Beam bending
Galileo, 1638 (though he wasn’t right)
Normal stress (σ) and strain (ε)
P P
L
A
P
L
δ
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?
Hooke’s law
steelfor GPa 200E
modulus sYoung'or modulus elastic is
E
E
What is the strain just before steel yields?
Strain gauge
6.4x4.3 mm
Gauge factor
R is nominal resistanceGF is gauge factor. For ours, GF = 2.1
Need a circuit to measure a small change in resistance
Wheatstone bridge
R4
R 1
Vmeas
VCC
R2
R3
+ -
Our setup
120
1 2 0
Vmeas
5 V
120
120+dR (strain gauge)
Strain gauge
Proportional to strain !
In practice we need variable R.Why?
115
1 2 0
Vmeas
10
5 V
120
120+dR (strain gauge)
Strain gauge
Demo
Left side only provides 2.5 V We already have that available
2.5 (V ref)
10
120 + dR
115
Vmeas
5 V
115
1 2 0
Vmeas
10
5 V
120
120+dR (strain gauge)
=Strain gauge
Strain gauge
A quantitative test
2.5 (V ref)
10
120
115
Vmeas
5 V
100 K
Conclusion – part 1
• Bridge circuit + instrumentation amp. allow us to detect small changes in resistance.
• Mechanical strain proportional to ΔR/R.
Beams in bending
Beam in pure bending
MM
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.'
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
Beam in pure bending
Fig 5-7, page 304
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!
y
(1/radius) curvature theis
This relation is easy to prove by geometry
Neutral axis
Cantilever beam
L
P
Isolate the beam – Find the reaction forces
PR Sum forces
L
P
R
Also need a bending moment
PLM
PR
Sum forces
L
P
R
M
Sum moments
State of stress inside the beam
L
P
P
P L
State of stress inside the beam
L
P
P
P L
Imagine a cut in the beam
State of stress inside the beam
PP
P L
x
)( xLPPxPLM
M
Calculate shear and bending moment to hold at equilibrium
Shear and bending moment diagram
X
X
Shear
MP(L- x)
P
PP
P L
x
ML
Normal stress in bending
M
Take a slice through the beam
σ
Neutral axis is the centroid
y
Flexure formulaWill derive this in Mechanics of Solids and Structures
EI
MyI
My
b
h
12
3bhI
CrossSection
y
Strain in cantilever – at support
L
P
P
PL
3
121
2)(
bhE
hPL
EI
My
h
b