Strain Measurements
• Module goals…
– Introduce students to operating principles behind strain gauges
– Discuss practical issues regarding strain gauge installation and
usage.
– Understand how bridge circuits are used to determine changes
in gauge resistance --- and hence, strain.
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Experimental Stress Analysis
• Reasons for Experimental Stress Analysis
–
–
–
–
Material characterization
Failure analysis
Residual or assembly stress measurement
Acceptance testing of parts prior to delivery or use
• Some Techniques
– Photoelasticity
– Non-contact holographic interferometry
– Electrical Resistance Strain Gauges
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Stress vs. Strain
• Strain (e) is a measure of displacement usually in terms of microstrain such as micro-inches of elongation for each inch of specimen
length.
• Stress (s) is a measure of loading in terms of load per unit crosssectional area
• Stress and strain are related by a material property known as the
Young’s modulus (or modulus of elasticity) E.
s  eE
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Strain Defined
• Strain is defined as relative elongation in a particular
direction
T
ea= dL/L (axial strain)
et= dD/D (transverse strain)
m = et / ea (Poisson’s ratio)
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L
D
T
Strain gauges
• The electrical resistance of a conductor changes when it
is subjected to a mechanical deformation
T
T
Rbefore < Rafter
T
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T
Resistance = f(A…)
• Electrical Resistance (R) is a function of…
r
L
A
the resistivity of the material (Ohms*m)
the length of the conductor (m)
the cross-sectional area of the conductor (m2)
• R= r* L/A
• Note R increases with
–
–
–
–
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Increased material resistivity
Increased length of conductor (wire)
Decreased cross-sectional area (or diameter)
Increased temperatures (can bias results if not accounted
for)
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Deriving the Gauge Factor (GF)
• Since L and A both change as a wire is stretched it is reasonable to
think that we can rewrite the equation
R= r* L/A
to relate strain to changes in resistance.
• Start with the differential:
dR = d r* (L/A ) + r*d(L/A)
expanding with the chain rule again one gets:
dR = d r* (L/A ) + r/A*d(L)+ r*L*(-1/A2)*d(A)
• Divide left side by R and right side by equivalent (r* L/A ) to get:
dR dr dL dA



R
r
L
A
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…substituting into the equation
2
D
A     , so
2
also,
dA
dD
D
dA   (2) dD , or
2
 2e t
2
A
D
 
dL
 e a , so
L
dR dr

 e a  2e t
R
r
Noting the definition of Poisson’s ratio…
dR
dr
 e a (1  2 m ) 
,
R
r
or GF 
dR
ea
R  1  2 m  1 dr
ea r
Hence, we define the Gauge Factor GF as:
GF  1  2 m
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~ 0 for many
materials!
Using Gauge Factors with Strain Gauges
GF  1  2 m
1 R
e a strain
 is given by …
So, the axial
GF R
In most applications R and e are very small and so we use
sensitive circuitry (amplified and filtered bridge circuit)
contained within a strain-indicator box to read out directly in
units of micro-strain. Obviously this strain-indicator will require
both R (gauge nominal resistance) and GF (gauge factor)
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Typical Strain Gauge
Strain-relief wires
Solder terminals
for lead wires
Figure 1. A typical strain gage.
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Steps for Installing Stain Gauges
• Clean specimen – degreaser
• Chemically prepare gauge area – Wet abrading with MPrep Conditioner and Neutralizer
• Mount gauge and strain relief terminals on tape, align on
specimen and apply adhesive
• Solder wire connections
• Test
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Beam Loading Example
x
P
a
Displacement, v(x)
strain gage at
Beam length, L
x=b
Figure 2. Cantilever beam under load.
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Measuring Strain with a Bridge Circuit
• A quarter-bridge circuit is one in
which a simple Wheatstone bridge is
used and one of the resistors is
replaced with a strain guage.
• Vo may still be small such that
amplification (Amp>1.0) is usually
desirable
4 Vo 1
e
Amp Vex GF
• Note: Vo and Vex are also sometimes
labeled as Eo and Ei (or Eex)
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Non-linear term
Current (i) Limitations
• In general gauges cannot handle large currents
• The current through the gage will be driven by the
voltage potential across it.
• Note: Text denotes the excitation voltage as Vi. It is
also often labeled Ve or Vex.
VG
VEx
iG 

RG RG  R3
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Measuring Strain with a Strain-Indicator
• First install a strain gauge
• Connect the wires from the strain gauge to the strain
indicator.
• Apply loading conditions
• Read strain from strain indicator
– Note that the indicator always displays 4 digits and reads in
microstrain!
– Thus, 0017 means 17 micro-inches / inch of strain.
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Strain gauge bridge enhancements
• 3-wire configuration addresses lead
wire resistance issues
• Half-bridge configuration – with a
dummy gauge mounted transversely
addresses gauge sensitivity to
surface temperature
• Half bridge – amplification through
use of dual gauges
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Theoretical Determination of Strain in a
Loaded Cantilever Beam
•
•
•
•
•
You must either know the load P or the displacement (v)
Determine displacement (v) at x=a
Knowing beam dimensions and material (and hence EI) estimate the load P
Calculate stress at location of gauge
Calculate e from s=eE
s
 Px 2 (3a  x)

, so
6 EI
P
a
Displacement, v(x)
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x=b
a3
My P * b * h / 2

, where h  beam thickness
I
I
x
strain gage at
P3
EI
Beam length, L
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Strain Gauge Vibration Experiment Notes:
Cantilever Beam Damping
When the cantilever beam is “plucked” it will respond as a damped 2nd order
system. The amplitude of vibration has the general form:
Y (t ) Ce  nt sin d t 
Where the damped frequency (what you measure) is related to the natural frequency
(n) by:
d  n 1 
2
The damping ratio (zeta) can be determined by plotting the natural log of the
amplitude/magnitude (M) vs time:
So, the slope of the plot of ln(M) vs. t is (–  n)
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0.06
Amplitude (raw voltage)
M (t ) Ce nt so, ln( M )  C2  ( n )  t
0.04
0.02
0
-0.02
0
2
4
6
-0.04
-0.06
-0.08
-0.1
Time (s)
8
10
12
Additional Considerations for natural
frequency of “plucked” beams
• Note: Unless otherwise indicated, natural frequencies are expressed
in terms of radians/sec.
• The natural frequency of a uniform beam is given by:
n  (1.875)
2
EI
m' L4
• E is the modulus of elasticity, I is the moment of intertia about the
centroid of the beam cross-section (bh3/12), m’ is the mass per unit
length of the beam (ie kg/m), and L is the cantilevered beam length
• If the beam is not uniform…
– A mass at the end can be represented as an effective change in beam
mass per unit length
– A hole in the end can be accounted for in a similar fashion…
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