Beams and Strain Gages
Cantilever Beam
One End Fixed , One End Free
Everything is a Spring
Beams Bend When Loaded
Tip Load Creates Internal
Moments and Forces
x-section
Isolate a section of the beam
(NOT physically separated)
F
F
M(x)
x
L-x
V
Distribution of moment, M, and (shear) force, V, along beam
x
V(x)
At any point along the beam, there
is a reaction force and moment
(from contact with material to the
left of x-section) to maintain static
equilibrium.
How Does the Force Deform the Beam?
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.'
Galileo-1638
Incorrectly claimed that the root of beam was in
tension.
(give him a break-Newton was born a year after he died)
Consider a Free-Free Beam
Loaded by a Moment
For long, thin beams loaded by a transverse force, deformation is predominately
due to internal moment as compared to internal force. (Euler-Bernoulli Beam Assumption)
M
M
neutral surface
The upper portion is in compression.
The lower portion is in tension.
‘neutral surface’ does not change in length
Transverse Force Causes
Axial Deformation
How do we relate Load to Deformation?
Load
Stress
Strain
Deformation
Stress is a Quantitative Measure of
Load Acting on a Material
In general stress is a tensor quantity
-> represented by a matrix of scalars
stress matrix
For Simple Cases,
Stress Can Be Represented by a Single
Scalar Component*
Force applied normal to a cross-section (axially)
F
F
Ao
s=
F
Ao
average
normal stress
over the
cross-section
*In the stress matrix, only one term will have significant nonzero magnitude.
The other eight terms ~ 0. [s] s11 = F/A0
9
Strain = Normalized Deformation
x
Lo
F
F
Ao
e=
x
Lo
Average normal
engineering strain
over the
cross-section
10
How are Normal Stress
and Strain Related?
Uniaxial Tension Test
test specimen
extensometer
(measures strain)
load cell
(measures force)
11
s
Linear Elastic Behavior
sy
When loading is released, material
returns to its original length.
yield stress
slope = E (Young’s Modulus)
property of a material
Aluminum (6061-T6): E = 40 ksi = 275 MPa
linear elastic
deformation
e
12
s
Typical Ductile Material
su
ultimate tensile strength
“necking”
fracture
stress
sy
yield stress
“strain hardening”
linear elastic
deformation
uniform plastic
deformation
nonuniform plastic
deformation
e
13
Bending (Normal) Stress in a Beam
Will derive this in Mechanics of Solids and Structures
Mzy
s =
I zz
Mz = Moment about the z-axis (out of page) acting on
the x-section. Constant over the x-section.
y = distance from neutral surface (in y-direction).
Izz = area moment of inertia of the x-section about z-axis.
y
σ (-compression)
Mz
O
Neutral surface passes through
the centroid of cross-section
x
Mz
σ (+tension)
Specified cross-section
Stress distribution is
linear over x-section
Area Moment of Inertia
Concept-measure of how hard a cross-section is to rotate.
y
z
y
O = centroid
h
z
O = centroid
(always lies on lines
of symmetry)
b
I zz =  y dA
2
A
3
bh
I zz =
12
*effect from changing
h >> b
Strain Gages Measure Axial Strain
The resistance of gage changes
as it stretches or compresses.
Strain gages must be well attached
to the surface of a material so that it
deforms as the material deforms.
6.4 x 4.3 mm
Strain is Proportional to the Change of
Resistance of the Gage
R is nominal resistance
GF is gage factor-a calibration constant. Ours have GF = 2.1
Change in resistance is very small-need a circuit to measure.
Wheatstone Bridge
VCC
R3
R1
+
R2
-
V
meas
R4
Invented by Samuel Hunter Christie ~1833
Improved by Sir Charles Wheatstone ~1843
A Possible Setup
5V
120
Strain gage
R4
120+dR (strain gauge)
V
meas
120
120
Proportional to strain !
In Practice,
Why Do We Want Variable R?
5V
Strain gage
120
120+dR (strain gauge)
V
meas
120
115
10
Now That You Know About Beams and
Strain Gages
You Can Make a Calibrated Scale
F
Mzy
s =
I zz
s = Ee
Relate the force applied to the strain*
Compare analytical predictions to measurements
* depends on both geometry and material