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650:349 MECHANICAL MEASUREMENTS
GROUP 4B
Strain Gage Measurements of
Constant Stress and Cantilever
Beams
MAY 11, 2017
has one clamped end and one free end, on
which the load is applied.
Your Name Here*, Norwood Fisher, Lisa Grant, Walter
Kibby, Kendall Jones, Chris Dowd, Angelo Moore
Department of Mechanical
and Aerospace Engineering
Rutgers University, Piscataway, New Jersey 08854
Axial and transverse strain is measured in both
constant stress standard cantilever beams using
single strain gages and strain gages rosettes for
several different metals. The Young’s moduli of
steel, brass, and aluminum are measured to be 299
GPa, 44 GPa, and 22 GPa respectively. The
Poisson’s ratio of steel, brass, and aluminum are
measured to be 0.54, 0.31, and 0.22 respectively.
Strain measurements using a standard Wheatstone
Bridge/Digital Volt Meter arrangement are
compared to those made using a Strain Gage
Indicator. Finally, multi-axis strain measurements
are used to determine the principal stresses in a
cantilever beam.
Figure 1. Constant Stress Beam.
Two different types of strain gage sensors
as used in this laboratory. Two single axis
strain gages are employed to measure both
axial and transverse strain in the constant stress
beam. For better accuracy when measuring
strain in multiple directions, a strain rosette is
used. An example of a rectangular strain gage
rosette sensor is shown in Figure 2.
INTRODUCTION
Strain measurements are commonly used to
determine the local deformation of materials to
applied loads. These are frequently used in
mechanical engineering and materials science
applications. The strain gage is a small device
which changes its resistance when stretched in
its primary axis. By incorporating the strain
gage element into a Wheatstone bridge (a
circuit which can be made sensitive to small
changes in resistance), a varying voltage can be
recorded which is proportional to the strain.
This laboratory uses two different kinds of
beams. The first is a constant stress beam, with
a width which varies linearly along its length.
Due to this variation, the axial stress is
constant along the length of the beam, which is
not the case in a traditional cantilever beam.
An example constant stress beam is shown in
Figure 1. The second kind of beam in use is a
typical cantilever beam. This is a beam which
*
Corresponding Author
YOUR NAME HERE
Figure 2. Schematic of a rectangular
strain gage rosette.
Both strain gage elements measure strain
by the change in electrical resistance, Equation
1, where R is the resistance and F is the gage
factor.
1 R
[1]

F R
When strain gage output is measured using a
Wheatstone bridge, the voltage reading is
proportional to the change in resistance
(Equation 2), which is directly related to the
strain from Equation 1.
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650:349 MECHANICAL MEASUREMENTS
R
V
R

V
4  2R
GROUP 4B
[2]
R
Using this rosette, one can determine the
principal strains using geometry and the strain
transformation equations. Thus strains
recorded by the three gages can give the
principle strains of the material, using Equation
1.
 
1
[3]
 2   3 2   3   1 2
 A, B  2 1 
2
2
Once the principal strains are known, using the
material properties, one can also calculate the
stresses using Equation 2 a and b, where E is
the Young’s modulus and  is the Poisson’s
ratio for the material. Then, using equations
from the theory of cantilever beams, we can
compare the measured stresses to those
calculated using standard formulas.
A 

E
1    
2
A
  B 
[4a,b]
E
 B   A 
B 
1 2

MAY 11, 2017

Figure 3. Axial and transverse strain for a
constant stress aluminum beam.
Comparison of the axial and transverse
strains from the constant stress beams gives the
material’s Poisson ratio. These strains are
plotted in Figure 4 and the negative slope of
the line is equal to the Poisson ratio. For mild
steel, the Poisson ratio is 0.28. For brass and
aluminum, the Poisson ratios are 0.27 and 0.33
respectively. These values agree fairly well
with the theoretical ratios of 0.27-0.3 for steel,
0.33 for aluminum, and 0.34 for brass [1]
RESULTS AND DISCUSSION
Axial and transverse strain measurements
were recorded from constant stress beams
made of steel, brass, and aluminum. Strain is
recorded using a zeroed Wheatstone
bridge/digital volt meter as well as a strain
gage indicator (SGI). Example data are shown
in Figure 3 for the aluminum constant stress
beam.
There is good agreement between the two
measurement methods for the transverse strain,
but there is significant difference between the
two methods for the axial strain. This could be
due to drift in the Wheatstone bridge or in the
strain gage. However, the strain recorded
increases linearly with applied load as expected
from materials theory. No significant
hysteresis is present between loading and
unloading the beam.
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Figure 4. Comparison of axial strain to
transverse strain to determine Poisson’s ratio of
several materials.
Using cantilever beam theory, we can
calculate the stress in the constant stress beam
and compare that to the axial strain. The slope
of the line from this plot gives the Young’s
modulus of elasticity (E), Figure 5. For mild
steel, E is found to be 155 GPa. For brass and
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650:349 MECHANICAL MEASUREMENTS
GROUP 4B
aluminum, it is found to be 62.5 and 45.6 GPa,
respectively. The values are about 30% lower
than the accepted values (200 GPa for steel,
100 GPa for brass, and 70 GPa for brass). The
source of this error could be the position of the
clamp holding the beam to the table during the
experiment. The clamp should have been as
close to the edge of the table as well as to the
point of the beam where the rectangular base
meets the triangular shaped section of the
beam. This idealized setup what not achieved
in the lab and probably led to the discrepancies
between the calculated and theoretical values.
MAY 11, 2017
Table 1. Measured and calculated stresses and
strains of a cantilever beam.
A

mm mm/mm
0
2.5
5
7.5
10
A
MPa
B
mm/mm
Cantilever
MPa
calc
0.2
11.9
23.7
35.6
47.4
meas
-4.14E-07
-4.22E-05
-8.68E-05
-1.31E-04
-1.76E-04
calc
0
11.8
23.6
35.4
47.2
meas.
2.41E-06
1.61E-04
3.23E-04
4.85E-04
6.47E-04
CONCLUSIONS
Strain gages were used to measure axial and
transverse strains in constant stress and standard
cantilever beams. Measured values for Young’s
modulus and Poisson’s ratio were close to accepted
values; however, errors in the way the beam was
clamped may have caused these differences. Strain
measurements were made with a Wheatstone
bridge/Digital Voltmeter system as well as a strain
gage indicator. Generally, the SGI was more
accurate. For the cantilever beam, measured
principal stains were in excellent agreement with
theoretical values.
REFERENCES
Figure 5. Calculated axial stress versus
measured axial strain to determine Young’s
Modulus for several constant stress beams.
[1] Gere, J.M., Mechanics of Materials, 5th ed,
Chapman and Hall, London, 2000.
The second part of this study looked at the
stress in a simple cantilever beam. A strain
rosette (as pictured in Figure 2) is used to
measure three strain values at 45 degrees from
each other for a beam of aluminum with
dimension (9 x 1 x 0.1 inches) that is deflected
a known amount () using a micrometer.
Principal strains and stresses are then
computed using Equation 3 and 4a,b and are
shown in Table 1. Very good agreement is seen
between the calculated principal stress and the
calculated stress in the cantilever beam.
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