This file contains a report for the 6th experiment in the Lab "Instrumentation and Dynamic Systems", taken by mechanical engineering students.

1. Jordan University of Science and Technology
Faculty of Engineering
Department of Mechanical Engineering
Instrumentation and Dynamic Systems Lab
Experiment #6: Strain Measurements 1

2. Abstract:
This experiment is dedicated to study the main objective of resistance strain gages and how they
are used to record the behavior of different materials, and to measure the stresses acting on a
certain part.
Using the proper instrumentation system, two cantilever beams of unknown materials (one
having a hole in the middle) were tested-by applying a predefined load on both of them- in order
to measure the modulus of elasticity, Poisson’s ratio, and stress concentration factor. The stress-
strain diagrams for both systems were plotted and modulus of elasticity for both materials was
measured to be (E =̃ 61 𝐺𝑝𝑎 ). Poisson’s ratio for the first material was 0.42, and the stress
concentration factor of the second is 1.54 compared to 2.2 when calculated theoretically. The
two specimens are most likely to be made from aluminum.
The results and sources of error have been discussed, and conclusions are stated briefly.
Introduction:
Strain is the percentage of dimensional change in the work-piece, with respect to the
original size. It's widely defined for a cylindrical bar when is subjected to an axial load, hence
the deformation per unit length for this rod.
Measuring strain is very useful in design concern, and for establishing the stress strain
diagram, which is one of the most important curves in mechanics science. Many methods are
introduced in measuring this dimensionless quantity, but the most used and accurate is the
electrical strain gages.
The electrical-resistance strain gage is the most widely used device for strain measurement,
its operation is based on the principle that electrical resistance of a conductor changes when it's
subjected to mechanical deformation. Typically, an inductor is bonded to the specimen with no-
load condition; a load is then applied, which produces deformation in both the specimen and the
resistance element, this deformation is indicated through a measurement of the change in
resistance of the element.
When a strain gage is mounted on a specimen, two notes of caution should be considered:
 The surface must be absolutely clean; this is usually done with cloth wetted with acetone.
 Sufficient time must be allowed for the cement - which clamps or bonds the gage to the
specimen – to dry completely.
Problems are also introduced with strain gage usage, these are:
 Temperature effect: these problems arise because of the different thermal expansion
between the specimen's material, and the gage's material. Also the temperature change
may change the Resistivity of the gage's material.
 Moisture effect: absorption of moisture may change the electrical resistance of the gage.
 Wiring problem: poorly soldered connections or inflexible wiring may pull gage loose
from test specimen.
Theory:

3. Electrical resistance of a piece of wire is directly proportional to the length and inversely to the
area of the cross section. Resistance strain gage is based on that phenomenon (see Sec.11.3
Resistance Strain Gauges, Text p.488-494 or similar reference). If a resistance strain gage is
properly attached onto the surface of a structure which strain is to be measured, the strain
gage wire/film will also elongate or contract with the structure, and as mentioned above, due
to change in length and/or cross section, the resistance of the strain gage changes accordingly.
This change of resistance is measured using a strain indicator (with the Wheatstone bridge
circuitry), and the strain is displayed by properly converting the change in resistance to strain.
Every strain gage, by design, has a sensitivity factor called the gage factor which correlates
strain and resistance as follows:
Gage factor (F) = (ΔR/R)/e
Where: R = Resistance of un-deformed strain gage
ΔR = Change in resistance of strain gage due to strain
e = Strain
As specified by the manufacturer of strain indicator, we set the initial gage factor (as 2.005 for
example) and take the measurements.
Equipments and Instruments:
 Strain gages.
 Digital gage indicator: digital device which measure the strain directly from the strain
gage on the specimen as a digit.
 Switcher: device used to make easier work; it enables the user to switch between different
strains gages on the specimen with the aid of witch, and appropriate connections.
 Specimen: rectangular, cross section specimen with a hole discontinuity.
Figure 1: Experiment Set up

4. Procedure:
1. Assemble the strain gages on the specimen.
2. Connect the digital gage with the switcher (considering the color guides.), with the strain
gages on the specimen.
3. Calibrate the digital gage; this is achieved with no-load condition on the specimen.
4. The digital will read some arbitrary reading for each strain gage, so set the reading to
zero for each one using the finer on the switcher, in order to eliminate the constant error caused
from constant deviation.
5. Load the specimen with mass of "100 g", and measure the strain on each strain gage.
6. Repeat the previous step with increasing the load up to "1000 g", with a step of "100 g"
gradually upward and downward, in order to study Hysterics.
Results:
Part 1: Rectangular Specimen
Figure 2: Part 1 Specimen dimensions
Table 1: Strain gages readings (Part 1)
Load
(Grams)
Stress
σ
(Mpa)
Upward Reading(micro)
Download
Reading(micro)
Modulus
of
Elasticity
(E)
Poisson’s
Ratio
(υ)Є1(Axial) Є2 (Lateral) Є1 Є2
100 1.52 22 10 3.880597 31 6.90E+10 0.454751
200 3.04 52 17 32.33831 38 5.87E+10 0.328558
300 4.55 81 25 60.79602 46 5.59E+10 0.306777
400 6.07 92 52 91.8408 54 6.61E+10 0.566197
500 7.59 118 61 120.2985 61 6.45E+10 0.518216
600 9.11 150 68 150.0498 69 6.07E+10 0.453183
700 10.63 179 75 179.801 76 5.95E+10 0.420151
800 12.14 210 83 206.9652 83 5.80E+10 0.396083
900 13.66 238 90 231.5423 90 5.74E+10 0.378135
1000 15.18 266 97 266.4677 97 5.70E+10 0.364022

5. w=25.4
mm
l=201mm
R=3.18mm
Figure 4: Poisson’s Ratio
Sample of Calcultaion:
𝜎 = 6 𝑃 𝐿
𝑤 𝑡2⁄ = 6*0.5*9.81*0.26/(0.254*0.0063^2) = 7.59 Mpa
𝐸 = 𝜎
𝜖⁄ = 7.59 *10^6 / (118*10^-6) = 64.5 Gpa
υ = 𝜖2/𝜖1 = 61 / 118 = 0.52
Average values:
E = 60.4 Gpa, υ = 0.45,
Aluminum is best suiting this material properties
Aluminum properties: E = 71.7 Gpa, υ = 0.33 (Richard G.Budynas, 2011)
Part 2: Rectangular Beam with a Hole:
0.00
2.00
4.00
6.00
8.00
10.00
12.00
14.00
16.00
0 50 100 150 200 250 300
Stress(Mpa)
Strain (micro)
Figure 3: Stress-Strain diagram
Upward
Downward
0
50
100
0 50 100 150 200 250 300

6. Figure 5: Part 2 Specimen dimensions
Table 2: Strain gages readings (Part 2)
Load
(Grams)
Stress
σ(Mpa)
Reading(micro) Modulus
of
Elasticity
(E)
Stress
Concentration
Factor
(kt)
Є1 Є2 Є3 Є4
100 1.34 17 20 25 34 6.71E+10 2.00
200 2.68 39 44 47 65 6.10E+10 1.67
300 4.02 61 66 72 95 6.10E+10 1.56
400 5.37 83 88 95 124 6.10E+10 1.49
500 6.71 105 110 119 154 6.10E+10 1.47
600 8.05 127 132 142 184 6.10E+10 1.45
S.G 4 is the closest strain gage to the discontinuity, and gives the maximum stress.
S.G 1 is the furthest strain gage from the discontinuity and gives the nominal stress.
Modulus of Elasticity (E) = 6*500*0.201/(0.0222*0.0063^2) = 61 Gpa
Stress Concentration Factor (Kt) =
𝝈 𝑴𝒂𝒙
𝝈 𝒏𝒐𝒎𝒊𝒏𝒂𝒍
=
𝑬∗ 𝝐 𝟒
𝑬∗ 𝝐 𝟏
= 124 / 83 = 1.49
0.00
1.00
2.00
3.00
4.00
5.00
6.00
7.00
8.00
9.00
0 50 100 150 200
Stress(Mpa)
Strain (µm/m)
Figure 6: Stree-Strain Diagram (Part 2)
S.G 1
S.G 2
S.G 3
S.G 4

7. Average Values: E = 620 Gpa, Kt = 1.53
Theoretical stress concentration factor = 2.2 ( d / t = 0.5, d / w = 0.125 )
(Richard G.Budynas, 2011)
DiscussionofResults:
In part 1, a load P is applied on the cantilever beam and the two strain gages measures the
developed strain in the specimen axially and laterally. The data are recorded Table 1. At the first
3 readings the errors are very high; this can be referred to the zero offset of the gage indicator
device that alters the measurement considerably at low stresses. Afterward, the readings become
more logical. The zero-offset error resulted from the improper balancing of all the channels at
one time, because balancing a channel will shift the balance pint of another one, and when taking
the readings from all channels simultaneously, error generates.
Figure 3 and 4 illustrate the stress-strain diagram and Poisson’s ratio for the specimen
respectively. The average value of: modulus of elasticity is (60.4 Gpa), and Poisson’s ratio is
(0.42). When comparing these values with many known materials properties we can conclude
that the material of the specimen is aluminum.
While the experiment where about to exam the hysteresis behavior of the system, the two loading
schemes didn’t show an obvious difference in between.
Table 2 show the recorded values for a rectangular cantilever beam with a discontinuity in the
middle. The readings of the four mounted gages are listed in the table, with gage 4 being the
closest to the continuity and 1 the furthest. The reduction in stress through the different gages is
clearly shown in Figure 6. The measured modulus of elasticity of the material is 620 Gpa which
is close to the value measured in part 1. The material is anticipated to be aluminum alloy, and the
deviation in the values of Poisson’s ratio and modulus of elasticity is mainly referred to the
impurities existing in the specimen structure, and secondarily to the errors generated when
balancing the bridges channels and in the strain gages themselves.
The stress concentration factor is measured to be 1.54, while the theoretical predicted value
referring to (Richard G.Budynas, 2011) is 2.2. This difference may be due to the errors
generating in the strain gages and in the digital gage indicator. It could be also a result for poor
finishing conditions.
Conclusions:
1- The maximum stress in a cantilever beam could be at the point of discontinuity
(hole, notch, …etc), or at the root of the beam furthest from the point of application
of force.
2- Poisson’s ratio is a measure of how the material deforms laterally, due to an applied
stress parallel to its axis.
3- The modulus of elasticity defines the stiffness of the material, which is how stiff is
the material against an axial stress acting on it, the more is the resistance to change
in length, the more stiff is the material.

8. 4- Discontinuities in the part cross section are places of stress concentrations. Stress
concentration factors are defined at these points where the stress is larger than the
nominal measured stress.
5- Stress concentration factors can be anticipated theoretically-depending on the
discontinuity geometry- by the aid of figures found in Mechanic’s of materials and
Mechanical engineering design books.
References:
Richard G.Budynas, J. K. (2011). Shigley's Mechanical Engineering Design, Ninth Edition in SI
Units. In J. K. Richard G.Budynas, Shigley's Mechanical Engineering Design, Ninth Edition in
SI Units (pp. 1005-1058). Singapore: Mc Graw Hill.