Download - Introduction to beam bending
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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).
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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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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.
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Beam bending
Galileo, 1638 (though he wasn’t right)
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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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Normal stress (σ) and strain (ε)
P P
L
A
P
L
δ
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Stress-strain
Yield stress in “ordinary” steel, 200 MpaHow much can 3 x 0.25 bar hold in tension?
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Hooke’s law
steelfor GPa 200E
modulus sYoung'or modulus elastic is
E
E
What is the strain just before steel yields?
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Shear stress
• P28 fig 1-24, 1-28
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Shear stress in tension/compression
P P
P P
NV
-80 -60 -40 -20 0 20 40 60 80-0.5
0
0.5
1
Nor
mal
/She
ar
N
V
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Beams in bending
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Simply supported beam
LbLa
P
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How do we find reaction forces?
LbLa
P
Ra Rb
)( baba
ba
LLRPL
PRR
Sum forces
Sum moments
L
LPR
L
LPR
ab
ba
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State of stress inside the beam
LbLa
P
Ra Rb
L
LPR
L
LPR
ab
ba
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State of stress inside the beam
LbLa
P
Ra RbImagine a cut in the beam
L
LPR
L
LPR
ab
ba
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State of stress inside the beam
VM
Calculate shear and bending moment to hold at equilibrium
x
P Lb/L
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State of stress inside the beam
VMx
P Lb/L
X
X
V
M
PLb/L x
P Lb/L
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State of stress inside the beam
V
Mx
P Lb/L
X
X
V
M
P P Lb/L
P La/L
P La Lb/L
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Shear and bending moment diagramLb
La
X
P Lb/L
P La/L
P La Lb/L
V
M
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Shear and bending diagram examples
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Beam in pure bending
MM
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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!
y
(1/radius) curvature theis
This relation is easy to prove by geometry
Neutral axis
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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
y=0
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Normal stress in bending
yE
M
Take a slice through the beam
σ
0)( ydAEydAEdAy
Neutral axis is the centroid
y
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Normal stress in bending
M
σ
interia ofmoment is
)( 22
I
IEM
MdAyEdAyEydAy
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Flexure formula
EI
MyI
My
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Moment of inertia, I
b
h
12
3bhI
CrossSection
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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