MOIRE METHODS AND
STRAIN ANALYSIS
MARCH 24TH, 2014
GROUP #20
Matthew Stevens - First Author, Abstract, Introduction
Ting Zhang – Results and Error Analysis
Kanchan Bhattacharyya – Discussion and Results
Xie Zheng – Experimental Theory, Procedures and List of equipment
I.Abstract
Moiré Methods are an optical means of strain analysis that are relatively new in both academic
and industrial worlds. In these methods, transparent sheets possessing gratings of dark lines are applied
to a test material which are then used to perform strain analysis on a stressed material. By observing
fringes that arise from variations in these networks of line gratings, the strain induced in a material can
be determined. There are several moiré methods for strain analysis, two of which include the in-plane
moiré for in-plane deformation and shadow moiré for out-of-plane deformation. The in-plane moiré
method utilizes superposition of two gratings and the resulting interference patterns that result from
overlapping of their respective gratings to determine the normal and shear strains in a stressed material,
while the shadow- moiré utilizes a single grating and the grating’s shadow under illumination as a
second grating in deflection analysis. A powerful means of analysis, moiré techniques can be applied to
virtually any material regardless of its inherent material properties. [1].
II.Introduction
Unlike make engineering analysis techniques who are named after the person who discovered
them and found their practical application, the term moiré originated in early 17th century France to
describe silks with watery or wave-like patterns [2]. The moiré effect is a phenomenon occurring in
repetitive structures, such as gratings of closely spaced lines, which are superposed against each other.
When such repetitive structures are supposed against one another they produce repetitive dark and
bright areas, which are only visible when the two gratings are paired. These repetitive regions, referred
to as moiré fringes, are extremely sensitive to distortions in the grids being used and change with such
variances in the grids. It is this sensitivity that has made the family of moiré techniques useful in the
fields of mechanics, physics, mathematics, and color printing.
Moiré techniques are used in mechanics to analyze material behavior when subjected to stress.
There are several moiré stress analysis techniques, including the In-Plane and Shadow Moiré Method for
Out of Plane transformations which will be discussed here. As mentioned previously, the in-plane moiré
utilizes two gratings which are superimposed on one another. A reference grating is used with a grating
that is subjected to an in-plane loading, and the resulting moiré fringes and their geometries can give
insight into the deformation of the stressed grating as well as both the normal and shear strains. By
observing the distance between reference grating lines as well as the distances between adjacent
fringes running parallel and perpendicular to the direction of the reference grating, the normal and
shear strains can be compute. The shadow moiré uses a single grating as a reference and its shadow as
the specimen grating. When illuminated and subjected to out-of-plane deformation, the shadow grating
forms moiré fringes which radiate from the center of the specimen outward towards its edges. Through
observation of the reference grating pitch, the angles at which light is incident and reflects from the
sample, and the number of fringes that appear, the deflection resulting from the induced load can be
calculated.
The validity of the family of moiré methods has been proven in the comparability of their
rendered results with those predicted by the theories of solid mechanics such as stress-strain
relationships and bending, which will be explored here. Through geometric interpretation of the moire
fringes we can compute the same strains as those predicted by the aforementioned theories with the
advantage that results are physically observable, esthetically pleasing, and offer more insight than
traditionally numerically laden solutions.
III.List of Equipment
Part one: The In-plane Moiré Method
1. Plexiglas specimen with premounted
2. Loading machine, Tinius Olsen
grating for stress concentration test.
Universal digital testing, for pulling
plate. Model #: H5KS
3. Camera to take picture of the specimens
Model #: Digital.8
Part Two: Shadow Moiré Method.
1. Digital camcorder to take picture of the specimens
2. LVDT
Model #: SONY DCR-TRV840
3. He-Ne laser and optical components.
Serial # for laser: 0244BG
4. Digital Multimeter
Model #:22-813
5.Painted circular specimen for bending experiment
6. Micro meter to calibrate the relation
between deflection and voltage for LVDT
7. Loading frame to provide the bending to
painted circular specimen
8.
Light
Model #: TG-50TR
9. Supporter
Serial #: 657
Part Three: Stress distribution along a three-point bending beam using strain gage
measurements.
1. A aluminum beam with 8 pre-mount strain gages. We use this specimen to do the stress
distribution experiment.
2. P3500 Strain Indicator and P3 Digital Strain Indicator
3. PCs with P3 software
4. Loading machine. Model #: 162. Serial #: 192328
IV.Theory
Part One: The In-plane Moiré Method.
In-plane Moiré Method are using two moiré grating to overlap each other and generate an
interference pattern which is called moiré if there is either a mismatch in orientation or grating pitch.
For the In-plane moiré method, we are using a grating as a reference and the other is attach to a
specimen which would be subject to in-plane load, then the deformation could be find according to the
resulting moiré fringes. In addition, if there is no large rotation on the deformation field, the normal
strain ε and shear strain γ could be obtained from the equations:
(Eq. 1)
(Eq. 2)
where p is the pitch of the reference grating (distance between two grating lines)
δT is the distance between two adjacent fringes along the direction parallel to the grating lines.
Figures showing the pitches and also the generated fringes can be found be Appendices A1 and
A2.
Part Two: Shadow Moiré Method.
The shadow moiré method is used for measuring out-of-plane deformation. In the moiré
method, only one grating is used. The optical arrangement used can be found in Appendix A3.
When the plate is pressed at a certain point which results a transverse bending, the deflection at that
point could be calculated with the equation:
(Eq. 3)
Where N is the fringe order,
P is the grating pitch,
β and α are the illumination and receiving angles from the normal to the grating
plane.
In the case β=0, which means both the light source and recording camera are at infinity, the Eq.3 would
be reduced to
(Eq.4)
In this experiment, we are using this equation.
Part Three: Stress distribution along a three-point bending beam using strain gage measurements.
There are eight strain gages used in the experiment to examine the theoretical prediction of
stress distribution in a supported beam under concentrated load at its mid span. They are divided into to
two groups and attached to the top edge and bottom edge respectively. Appendix A4 shows the
principle of strain gages mounting and also the moment distribution
We are using Hooke’s law to calculate the stress:
σxx = Eεxx
(Eq.5)
We can obtain the εxx from strain gages and P should be relatively small to make sure the deflection does
not exceed the elastic range.
V.Procedure
Part One: The In-plane Moiré Method.
1. The lab is already set up and camcorder is in the proper position, check the connection
of the power of loading machine, Tinius Olsen Universal digital testing machine.
2. Zero the load on testing machine and then add initial load to specimen. And take that
reading on the loading machine as the initial reading for “zero” condition.
3. Start to add load to the specimen with increment of 50lb until it reaches around 320
(the initial load is about 20lb). Record the reading for every load and also take picture at
the same time.
4. Release the resting machine and repeat the steps 2 and 3 for another two trials.
5. After taking all the reading and pictures, release the specimen and measure the
diameter of the hole and its width for reference.
Part Two: Shadow Moiré Method.
1. All the optical components are placed at the proper position.
2. Use the micrometer to do the calibration for the LVDT first to find out the relation
between distance and voltage.
3. Place the LVDT at the proper position to make sure the deflection of the specimen is
within its measuring range.
4. Measure the angle α by measuring two right-angle sides and take the tangent ratio of
them.
5. Starting to load the plate with the micrometer and wait to see there is a black spot
appearing on the computer. Take that spot as first order of the fringes and take picture
of that. Record the reading on multimeter.
6. Continue to add load to the plate until the second, third and fourth black spot appears,
take that as the second, third and fourth order respectively, and take pictures of them.
Record the reading on multimeter for every load.
7. Release the load and repeat steps 5 and 6 for another two trials.
8. Save the pictures in memory stick to the computer and then burn it to the disk.
Part Three: Stress distribution along a three-point bending beam using strain gage
measurements.
1. The experiment is set up and check the connections. Measure the dimensions of the
beam and the position of the mounted gages.
2. For the P3 software set up:
Bal: Select Auto and check Zero.
3. Click Record, and select record manually, all the channels and save on this computer.
4. Zero the load and then click record and that would be the zero reading for the trial.
5. Adjust the load to 100lb and then record on P3 software.
6. Adjust the load to 200lb and then record on P3 software.
7. Release the load, repeat steps 4, 5 and 6 to get another two trials.
8. Release the load and save the data on a disk.
Results
Part I: In-Plane moiré
Fig. 1: In-Plane Moire Tensile Testing Specimens and Assembly
A reference grating superimposed onto a Plexiglas specimen with pre-mount grating to be subjected to
tensile loading. The superposition of the two grating produces the series of parallel bright bands above
and below the hole in the specimen.
Fig. 2: In-Plane Moire Tensile Testing Results for a load of 335 lbs.
A reference grating superimposed onto a Plexiglas specimen after being subjected to a tensile load of
approximately 335 lbs. In addition to the bright bands appearing from the superposition of the two test
gratings, a series of symmetrical fringes appear emanating from the hole as a result of the applied load.
Fig. 3: In-Plane Moiré Tensile Testing Analysis for a load of 335 lbs. – A magnified image giving the
calculated distance from the edge of the hole to the outermost fringe, the distance from the mid-plane
of the specimen to the outermost fringe, and the spacing between fringes.
Fig. 4: In-Plane Moire Tensile Testing Results for a load of 350 lbs.
A reference grating superimposed onto a Plexiglas specimen after being subjected to a tensile load of
approximately 350 lbs. In addition to the bright bands appearing from the superposition of the two test
gratings, a series of symmetrical fringes appear emanating from the hole as a result of the applied load.
Fig. 5: In-Plane Moiré Tensile Testing Analysis for a load of 350 lbs. – A magnified image giving the
calculated distance from the edge of the hole to the outermost fringe, the distance from the mid-plane
of the specimen to the outermost fringe, and the spacing between fringes.
Fig 6. : In-Plane Moire Tensile Testing Results for a load of 347 lbs.
A reference grating superimposed onto a Plexiglas specimen after being subjected to a tensile load of
approximately 347 lbs. In addition to the bright bands appearing from the superposition of the two test
gratings, a series of symmetrical fringes appear emanating from the hole as a result of the applied load.
Fig. 7: In-Plane Moiré Tensile Testing Analysis for a load of 347 lbs. – A magnified image giving the
calculated distance from the edge of the hole to the outermost fringe, the distance from the mid-plane
of the specimen to the outermost fringe, and the spacing between fringes.
D (in)
0.493
0.493
0.49
d (in)
0.94
0.931
0.927
Kc = d/D
1.907
1.888
1.892
Table 1: Calculation of Stress Concentration Factors the measured distances from the edge of the hole
and the mid-plane to the outermost fringe. The Stress concentration factor is calculated by the ratio of
the distance from the mid-plane to the outermost fringe to the distance from the edge of the hole to the
same outermost fringe.
Part II: Shadow-moiré
Trial 1 Output
Trial 2 Output
Trial 3 Output
Average
No. of 1/4 Turns Deflection (in)
Voltage (V)
Voltage (V)
Voltage (V)
Voltage (V)
0
0.025
6.77
6.78
6.79
6.78
1
0.05
5.54
5.54
5.56
5.55
2
0.075
4.28
4.29
4.30
4.29
3
0.1
3.03
3.05
3.04
3.04
4
0.125
1.78
1.78
1.78
1.78
5
0.15
0.52
0.52
0.52
0.52
6
0.175
-0.72
-0.72
-0.72
-0.72
7
0.2
-1.95
-1.95
-1.96
-1.95
8
0.225
-3.19
-3.20
-3.20
-3.20
9
0.25
-4.42
-4.43
-4.44
-4.43
10
0.275
-5.63
-5.66
-5.66
-5.65
11
0.3
-6.83
-6.83
-6.89
-6.85
Table 2: LVDT Calibration the recorded deflections and the corresponding LVDT output voltages used to
calibrate the true deflection induced with each loading during Shadow-Moiré testing. Three sets of
deflections and voltages were taken, from which an average was computed to determine an average
deflection per output voltage.
Deflection vs. Voltage
0.35
0.3
Deflection (in)
0.25
0.2
0.15
0.1
0.05
y = -0.0201x + 0.1611
0
-8.00
-6.00
-4.00
-2.00
0.00
2.00
4.00
6.00
8.00
Voltage (v)
Fig. 8: Graph plotting LVDT Calibration deflections vs. output voltages this graph plots the average
deflections against the corresponding LVDT output voltage. The slope of -0.0201 represents the
deflection per unit output voltage associated with each induced load.
Fig. 9: Shadow moiré Testing Results – Zero Order Fringe
an image of the shadow moiré testing results before a load was applied to the specimen. The fringe
order of zero is easily identified by the dark region in the center of the specimen (the fringe itself), with
the illuminated red regions emanating outwards.
Fig. 10: Shadow moiré Testing Results – First Order Fringe
An image of the shadow moiré testing after an initial load was applied, resulting in a shift of the zero
order fringe originally in the center of the specimen. The zero order fringe begins to spiral outwards
from the its original location in the center of the specimen towards the edges, with the newly formed
first order fringe forming in the center. The two fringes are separated by an illuminated region which,
making the occurrence of each order easy to identify.
Fig. 11: Shadow moiré Testing Results – Second Order Fringe
An image of the shadow moiré testing after a second load was applied, resulting in another outward
radial shift of both the zero and first order fringes from their previous positions. These shifts are
accompanied by the formation of the second order fringe at the center of the specimen. The resulting
fringes are ordered with decreasing order with increasing distance from the center, i.e. second order
fringes in the center with the zeroth order closest to the edge of the specimen, separated by the first
order fringe.
Fig. 12: Shadow moiré Testing Results – Third Order Fringe
An image of the shadow moiré testing after a third load was applied, resulting in another outward shift
of the zero, first, and second order fringes. In the center, we find the newly formed third order fringe as
a result of the final applied load.
Fringe Order
Trial 1 Voltage
Trial 2 Voltage
Trial 3 Voltage
Zero Order
2.771
2.76
2.69
First Order
3.85
3.802
3.79
Second Order
4.94
4.97
4.91
Third Order
6.2
6.19
6.19
Table 3: Recorded LVDT Output Voltage for Shadow-moiré testing the observed order corresponding to
each induced load, and the corresponding LVDT output voltage for Trials One, Two and Three. These
voltages were used in computation of the true deflection with each induced load.
Average
Average
Average
Trial 1
Trial 2
Trial 3
Absolute
Relative
Relative
Fringe Order
Deflection
Deflection Deflection Deflection (in) Deflection(in) Deflection(mm)
Zero Order
0.2167971
0.216576
0.215169
0.2161807
0.0000000
0.00000000
First Order
0.238485
0.2375202 0.237279
0.2377614
0.0215807
0.54814978
Second Order
0.260394
0.260997
0.259791
0.260394
0.0442133
1.12301782
Third Order
0.28572
0.285519
0.285519
0.285586
0.0694053
1.76289462
Table 4: Deflections computed during Shadow-moiré testing the recorded deflections using the
calibration factor determined by LVDT calibration corresponding to each of the recorded output
voltages listen in Table 3. An average deflection was computed for each fringe order using the individual
deflections computer from each set of results.
tanα
p(mm)
0.405247813
0.2
Table 5: Theoretical Parameters used to Compute Deflections the value computed for the tangent of
the angles existing between the sending and receiving light beams as light is incident on the testing
specimen, and the grating pitch (line spacing) for the reference grating used in Shadow-moiré testing.
These values were used to compute the theoretical deflections considering only moiré fringe pattern
and reference grating geometries.
Fringe Order
Theoretical Deflection (mm)
Zero Order
0
First Order
0.49352518
Second Order
0.98705036
Third Order
1.48057554
Table 6: Theoretical Deflections for Shadow-Moire Testing Results the computed theoretical
deflections in millimeters for each respective fringe order as per equation using the theoretical
parameters given in Table 5.
Part III: Stress Distribution along a three-point bending beam using strain gage measurements
Thickness (in)
height (in)
0.2565
1.545
Moment of Inertia I (in4)
0.0788
ymax (in)
0.7725
Table 7: Three-Point Bending Test Specimen Material Properties The physical properties of the testing
specimen, including the thickness and height used to calculate the Moment of Inertia 𝐼𝑧 =
𝑏ℎ 3
12
and the
distance ymax used in stress analysis.
𝐼𝑧 =
𝑏ℎ3 0.2565 𝑥 1.5453
=
= 0.0788 𝑖𝑛4
12
12
Fig. 13. Calculation of Moment of Inertia 𝐼𝑧 the moment of inertia about the z-axis calculated
considering the specimen height and thickness.
channel number
channel location(in)
stress under 100lbs
stress under 200lbs
1
4.4
2155.90487
4311.80975
2
3
4
3.8435
2.38
0.82
1883.2319 1166.14855 401.782272
3766.46381 2332.29709 803.564544
Table 8: Computed Theoretical Stresses under induced loading for Top-Side Strain Gages – Trial One
the calculated theoretical stresses in psi for the strain gages mounted to the top side of the testing
specimen for loads of 100 lbs. and 200 lbs., respectively.
channel number
channel location(in)
strain under 100lbs
strain under 200lbs
5
6
7
8
4.79
3.8
2.26
0.738
2346.99644 1861.91785 1107.35114 361.604045
4693.99288 3723.83569 2214.70228 723.208089
Table 9: Computed Stresses under induced loading for Bottom-Side Strain Gages the calculated
theoretical stresses in psi for the strain gages mounted to the bottom side of the testing specimen for
loads of 100 lbs. and 200 lbs., respectively.
Fig. 14: Example Calculation of Bending Moment and Theoretical Normal Stress for loads of 100 lbs.
and 200 lbs. Examples of how each bending moment and normal stress was calculated for each strain
gage position. The distance y and moment of Inertia Iz given in Table 7, are used in conjunction with the
bending moment and horizontal position of the strain gage to compute the normal stress. Examples are
given for loads of 100 lbs. and 200 lbs. for a strain gage located 4.4 in. from the fixed end, considering
readings from the top set of strain gages.
Compression Stress Distribution under 100lbs
(Theoretically)
2500
y = 489.98x + 2E-12
stress (psi)
2000
1500
1000
500
0
0
1
2
3
4
5
location (in)
Fig. 15: Graph plotting Theoretical Compressive Stress distribution under load of 100 lbs. – Trial One
The computed theoretical compressive stresses in psi vs. strain location. The plot takes a linear form
y = 489.98x + 2E-12. The individual values for stresses and locations can be found in Table 8.
stress (psi)
Compression Stress Distribution under 200lbs
(Theoretically)
5000
4500
4000
3500
3000
2500
2000
1500
1000
500
0
y = 979.96x + 4E-12
0
1
2
3
4
5
location (in)
Fig. 16: Graph plotting Theoretical Compressive Stress distribution under load of 200 lbs. – Trial One
The computed theoretical compressive stresses in psi vs. strain location. The plot takes a linear form
y = 979.96x + 4E-12. The individual values for stresses and locations can be found in Table 8.
Tensile Stress Distribution under 100lbs (Theoretically)
2500
y = 489.98x + 8E-13
stress (psi)
2000
1500
1000
500
0
0
1
2
3
4
5
6
location (in)
Fig. 17: Graph plotting Theoretical Tensile Stress distribution under load of 100 lbs. – Trial One
The computed theoretical compressive stresses in psi vs. strain location. The plot takes a linear form y =
489.98x + 8E-13. The individual values for stresses and locations can be found in Table 9.
Tensile Stress Distribution under 200lbs (Theoretically)
5000
y = 979.96x + 2E-12
4500
4000
stress (psi)
3500
3000
2500
2000
1500
1000
500
0
0
1
2
3
4
5
6
location (in)
Fig. 18: Graph plotting Theoretical Tensile Stress distribution under load of 200 lbs. – Trial One the
computed theoretical compressive stresses in psi vs. strain location. The plot takes a linear form y =
979.96x + 2E-12. The individual values for stresses and locations can be found in Table 9.
channel number
1
2
3
4
channel location(in)
4.4
3.8435
2.38
0.82
strain under 100lbs
217
184
117
37
strain under 200lbs
434
367
232
75
stress under 100lbs
2170
1840
1170
370
stress under 200lbs
4340
3607
2320
750
Table 10: Top Side-Strain Gage Compressive Micro-Strain Readings and Stress Calculations – Trial One
the strain readings as recorded by the strain indicator for loads of 100 and 200 lbs, and the
corresponding stresses computed via Hooke’s Law 𝜎 = 𝜀𝐸.
channel number
channel location(in)
strain under 100lbs
strain under 200lbs
strain under 100lbs
strain under 200lbs
5
4.79
235
446
2350
4460
6
3.8
194
384
1940
3840
7
2.26
115
227
1150
2270
8
0.738
43
84
430
840
Table 11: Bottom Side-Strain Gage Tensile Micro-Strain Readings and Stress Calculations – Trial One
the strain readings as recorded by the strain indicator for loads of 100 and 200 lbs, and the
corresponding stresses computed via Hooke’s Law 𝜎 = 𝜀𝐸.
Fig. 19: Example Calculation of Normal Stress using Strain Gage Readings – Trial One The normal stress
was calculated via Hooke’s Law, using the known modulus of elasticity for aluminum of 𝐸 =
10 𝑥 106 𝑝𝑠𝑖 and the various strain gage outputs. An example using the first 100 lb. strain reading from
the first top-side strain gage was used, which can be found in Table 10.
Compression Stress Distribution under 100lbs
2500
y = 495.44x - 29.893
stress (psi)
2000
1500
1000
500
0
0
1
2
3
location (in)
4
5
Fig. 20: Graph plotting Compressive Stress distribution under load of 100 lbs. – Trial One
The calculated stresses under a 100 lb. load as per the procedure described in Fig. 19. using the top-side
strain gage strain measurements given in Table 10.
Compression Stress Distribution under 200lbs
5000
4500
y = 980.21x - 49.997
4000
stress (psi)
3500
3000
2500
2000
1500
1000
500
0
0
1
2
3
4
5
location (in)
Fig. 21: Graph plotting Compressive Stress distribution under load of 200 lbs. – Trial One The
calculated stresses under a 200 lb. load as per the procedure described in Fig. 19. using the top-side
strain gag strain measurements given in Table 10.
Tensile Stress Distribution under 100lbs
2500
y = 479.56x + 78.221
stress (psi)
2000
1500
1000
500
0
0
1
2
3
4
5
6
location (in)
Fig. 22: Graph plotting Tensile Stress distribution under load of 100 lbs. – Trial One The calculated
stresses under a 100 lb. load as per the procedure described in Fig. 19. using the bottom-side strain gage
strain measurements given in Table 11.
Tensile Stress Distribution under 200lbs
5000
y = 913.88x + 204.98
4500
4000
stress (psi)
3500
3000
2500
2000
1500
1000
500
0
0
1
2
3
location (in)
4
5
6
Fig. 23: Graph plotting Tensile Stress distribution under load of 200 lbs. – Trial One
The calculated stresses under a 200 lb. load as per the procedure described in Fig. 19. using the bottomside strain gage strain measurements given in Table 11.
Compression Stress Distribution under 100lbs
2500
y = 489.98x + 2E-12
Stress (psi)
2000
y = 495.44x - 29.893
1500
1000
500
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
location (in)
Theoretical data
Experiment data
Fig. 24: Graph Plotting the Theoretical and Experimentally Determined Normal Compressive Stresses
against Strain Gage location under a load of 100 lbs. – Trial One
The normal compressive stresses induced by the application of a 100 lb. force, considering readings
from the strain gages mounted to the top of the test specimen. The theoretically determined stresses
given in Table 9 are plotted in blue, with the orange points representing the experimentally determined
stresses computed in Table 11.
5
Compression Stress Distribution under 200lbs
5000
4500
y = 979.96x + 4E-12
4000
Stress (psi)
3500
3000
y = 980.21x - 49.997
2500
2000
1500
1000
500
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
location (in)
Theoretical data
Experiment data
Fig. 25: Graph Plotting the Theoretical and Experimentally Determined Normal Compressive Stresses
against Strain Gage location under a load of 200 lbs. – Trial One
The normal compressive stresses induced by the application of a 200 lb. force, considering readings
from the strain gages mounted to the top of the test specimen. The theoretically determined stresses
given in Table 9 are plotted in blue, with the orange points representing the experimentally determined
stresses computed in Table 11.
5
Tensile Stress Distribution under 100lbs
2500
pt
Stress (psi)
2000
y = 479.56x + 78.221
1500
1000
500
0
0
1
2
3
4
5
6
location (in)
Theoretical data
Experiment data
Fig. 26: Graph Plotting the Theoretical and Experimentally Determined Normal Tensile Stresses against
Strain Gage location under a load of 100 lbs. – Trial One
The normal compressive stresses induced by the application of a 200 lb. force, considering readings
from the strain gages mounted to the top of the test specimen. The theoretically determined stresses
given in Table 8 are plotted in blue, with the orange points representing the experimentally determined
stresses computed in Table 10.
Tensile Stress Distribution under 200lbs
5000
y = 979.96x + 2E-12
Stress (psi)
4000
y = 913.88x + 204.98
3000
2000
1000
0
0
1
2
3
4
5
6
location (in)
Theoretical data
Experiment data
Fig. 27: Graph Plotting the Theoretical and Experimentally Determined Normal Tensile Stresses against
Strain Gage location under a load of 200 lbs. – Trial One The normal compressive stresses induced by
the application of a 200 lb. force, considering readings from the strain gages mounted to the top of the
test specimen. The theoretically determined stresses given in Table 8 are plotted in blue, with the
orange points representing the experimentally determined stresses computed in Table 10.
channel number
1
2
3
4
channel location(in)
4.4
3.8435
2.38
0.82
stress under 100lbs
2155.904873
1883.2319
1166.14855 401.782272
stress under 200lbs
4311.809747 3766.46381 2332.29709 803.564544
Table 12: Computed Theoretical Stresses under induced loading for Top-Side Strain Gages – Trial Two
the calculated theoretical stresses in psi for the strain gages mounted to the top side of the testing
specimen for loads of 100 lbs. and 200 lbs., respectively. The same procedure for calculating the
respective stresses follows from Fig. 14.
channel number
5
6
7
8
channel location(in)
4.79
3.8
2.26
0.738
stress under 100lbs
2346.99644 1861.91785 1107.35114 361.604045
stress under 200lbs
4693.99288 3723.83569 2214.70228 723.208089
Table 13: Computed Theoretical Stresses under induced loading for Top-Side Strain Gages – Trial Two
The calculated theoretical stresses in psi for the strain gages mounted to the top side of the testing
specimen for loads of 100 lbs. and 200 lbs., respectively. The same procedure for calculating the
respective stresses follows from Fig. 14.
Compression Stress Distribution under 100lbs (Theoretically)
2500
y = 489.98x + 2E-12
stress (psi)
2000
1500
1000
500
0
0
1
2
3
4
5
location (in)
Fig. 28: Graph plotting Theoretical Compressive Stress distribution under load of 100 lbs. – Trial Two
The computed theoretical compressive stresses in psi vs. strain location. The plot takes a linear form
y = 489.98x + 2E-12. The individual values for stresses and locations can be found in Table 12.
Compression Stress Distribution under 200lbs (Theoretically)
5000
4500
y = 979.96x + 4E-12
4000
stress (psi)
3500
3000
2500
2000
1500
1000
500
0
0
1
2
3
4
5
location (in)
Fig. 29: Graph plotting Theoretical Compressive Stress distribution under load of 200 lbs. – Trial Two
The computed theoretical compressive stresses in psi vs. strain location. The plot takes a linear form
y = 979.96x + 4E-12. The individual values for stresses and locations can be found in Table 12.
Tensile Stress Distribution under 100lbs (Theoretically)
2500
y = 489.98x + 8E-13
stress (psi)
2000
1500
1000
500
0
0
1
2
3
4
5
6
location (in)
Fig. 30: Graph plotting Theoretical Tensile Stress distribution under load of 100 lbs. – Trial Two
The computed theoretical compressive stresses in psi vs. strain location. The plot takes a linear form y =
489.98x + 8E-13. The individual values for stresses and locations can be found in Table 13.
Tensile Stress Distribution under 200lbs (Theoretically)
5000
4500
y = 979.96x + 2E-12
4000
stress (psi)
3500
3000
2500
2000
1500
1000
500
0
0
1
2
3
4
5
6
location (in)
Fig. 31: Graph plotting Theoretical Tensile Stress distribution under load of 200 lbs. – Trial Two
The computed theoretical compressive stresses in psi vs. strain location. The plot takes a linear form y =
979.96x + 2E-12. The individual values for stresses and locations can be found in Table 13.
channel number
1
2
3
4
channel location(in)
4.4
3.8435
2.38
0.82
strain under 100lbs
201
183
116
39
strain under 200lbs
410
369
233
76
stress under 100lbs
2010
1830
1160
390
stress under 200lbs
4100
3690
2330
760
Table 14: Top Side-Strain Gage Compressive Micro-Strain Readings and Stress Calculations – Trial Two
The strain readings as recorded by the strain indicator for loads of 100 and 200 lbs, and the
corresponding stresses computed via Hooke’s Law, 𝜎 = 𝜀𝐸. The procedure for the calculating the stress
follows Fig. 19.
channel number
5
6
7
8
channel location(in)
4.79
3.8
2.26
0.738
strain under 100lbs
234
193
117
47
strain under 200lbs
468
386
230
87
stress under 100lbs
2340 1930 1170
470
stress under 200lbs
4680 3860 2300
870
Table 15: Bottom Side-Strain Gage Tensile Micro-Strain Readings and Stress Calculations – Trial One
the strain readings as recorded by the strain indicator for loads of 100 and 200 lbs, and the
corresponding stresses computed via Hooke’s Law 𝜎 = 𝜀𝐸. The procedure for the calculating the stress
follows Fig. 19.
Compression Stress Distribution under 100lbs
2500
y = 457.66x + 38.2
stress (psi)
2000
1500
1000
500
0
0
1
2
3
4
5
location (in)
Fig. 20: Graph plotting Compressive Stress distribution under load of 100 lbs. – Trial Two
The calculated stresses under a 100 lb. load as per the procedure described in Fig. 19. using the top-side
strain gage strain measurements given in Table 14.
Compression Stress Distribution under 200lbs
4500
y = 939.71x + 31.6
4000
stress (psi)
3500
3000
2500
2000
1500
1000
500
0
0
1
2
3
4
5
location (in)
Fig. 33: Graph plotting Compressive Stress distribution under load of 200 lbs. – Trial Two
The calculated stresses under a 200 lb. load as per the procedure described in Fig. 19. using the top-side
strain gage strain measurements given in Table 14.
Tensile Stress Distribution under 100lbs
2500
y = 466.13x + 127.11
stress (psi)
2000
1500
1000
500
0
0
1
2
3
4
5
location (in)
Fig. 34: Graph plotting Tensile Stress distribution under load of 100 lbs. – Trial Two
6
The calculated stresses under a 100 lb. load as per the procedure described in Fig. 19. using the bottomside strain gage strain measurements given in Table 15.
Tensile Stress Distribution under 200lbs
5000
y = 950.93x + 172.66
4500
4000
stress (psi)
3500
3000
2500
2000
1500
1000
500
0
0
1
2
3
4
5
6
location (in)
Fig. 35: Graph plotting Tensile Stress distribution under load of 100 lbs. – Trial Two
The calculated stresses under a 200 lb. load as per the procedure described in Fig. 19. using the bottomside strain gage strain measurements given in Table 15.
Compression Stress Distribution under 100lbs
2500
y = 489.98x + 2E-12
Stress (psi)
2000
y = 457.66x + 38.2
1500
Theoretical data
1000
Experiment data
500
0
0
1
2
3
4
5
location (in)
Fig. 36: Graph Plotting the Theoretical and Experimentally Determined Normal Compressive Stresses
against Strain Gage location under a load of 100 lbs. – Trial Two
The normal compressive stresses induced by the application of a 100 lb. force, considering readings
from the strain gages mounted to the top of the test specimen. The theoretically determined stresses
given in Table 12 are plotted in blue, with the orange points representing the experimentally
determined stresses computed in Table 14.
Compression Stress Distribution under 200lbs
5000
4500
y = 979.96x + 4E-12
4000
Stress (psi)
3500
y = 939.71x + 31.6
3000
2500
2000
1500
1000
500
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
location (in)
Theoretical data
Experiment data
Fig. 36: Graph Plotting the Theoretical and Experimentally Determined Normal Compressive Stresses
against Strain Gage location under a load of 200 lbs. – Trial Two
The normal compressive stresses induced by the application of a 200 lb. force, considering readings
from the strain gages mounted to the top of the test specimen. The theoretically determined stresses
given in Table 12 are plotted in blue, with the orange points representing the experimentally
determined stresses computed in Table 14.
Tensile Stress Distribution under 100lbs
2500
y = 489.98x + 8E-13
Stress (psi)
2000
y = 466.13x + 127.11
1500
1000
500
0
0
1
2
3
4
5
6
location (in)
Theoretical data
Experiment data
Fig. 37: Graph Plotting the Theoretical and Experimentally Determined Normal Tensile Stresses against
Strain Gage location under a load of 100 lbs. – Trial One
The normal compressive stresses induced by the application of a 100 lb. force, considering readings
from the strain gages mounted to the top of the test specimen. The theoretically determined stresses
given in Table 13 are plotted in blue, with the orange points representing the experimentally
determined stresses computed in Table 15.
Stress (psi)
Tensile Stress Distribution under 200lbs
5000
4500
4000
3500
3000
2500
2000
1500
1000
500
0
y = 979.96x + 2E-12
y = 950.93x + 172.66
0
1
2
3
4
5
6
location (in)
Theoretical data
Experiment data
Fig. 38: Graph Plotting the Theoretical and Experimentally Determined Normal Tensile Stresses against
Strain Gage location under a load of 200 lbs. – Trial Two
The normal compressive stresses induced by the application of a 200 lb. force, considering readings
from the strain gages mounted to the top of the test specimen. The theoretically determined stresses
given in Table 13 are plotted in blue, with the orange points representing the experimentally
determined stresses computed in Table 15.
channel number
1
2
3
4
channel location(in)
4.4
3.8435
2.38
0.82
stress under 100lbs
2155.90487
1883.2319
1166.14855 401.782272
stress under 200lbs
4311.80975 3766.46381 2332.29709 803.564544
Table 16: Computed Theoretical Stresses under induced loading for Top-Side Strain Gages – Trial
Three the calculated theoretical stresses in psi for the strain gages mounted to the top side of the
testing specimen for loads of 100 lbs. and 200 lbs., respectively. The same procedure for calculating the
respective stresses follows from Fig. 14.
channel number
5
6
7
8
channel location(in)
4.79
3.8
2.26
0.738
stress under 100lbs 2346.99644 1861.91785 1107.35114 361.604045
stress under 200lbs 4693.99288 3723.83569 2214.70228 723.208089
Table 17: Computed Theoretical Stresses under induced loading for Bottom-Side Strain Gages – Trial
Three the calculated theoretical stresses in psi for the strain gages mounted to the bottom side of the
testing specimen for loads of 100 lbs. and 200 lbs., respectively. The same procedure for calculating the
respective stresses follows from Fig. 14.
Compression Stress Distribution under 100lbs
(Theoretically)
2500
y = 489.98x + 2E-12
stress (psi)
2000
1500
1000
500
0
0
1
2
3
4
5
location (in)
Fig. 39: Graph plotting Compressive Stress distribution under load of 100 lbs. – Trial Three
The calculated stresses under a 100 lb. load as per the procedure described in Fig. 19. using the top-side
strain gage strain measurements given in Table 16.
Compression Stress Distribution under 200lbs (Theoretically)
5000
4500
y = 979.96x + 4E-12
4000
stress (psi)
3500
3000
2500
2000
1500
1000
500
0
0
1
2
3
4
5
location (in)
Fig. 40: Graph plotting Compressive Stress distribution under load of 200 lbs. – Trial Three
The calculated stresses under a 100 lb. load as per the procedure described in Fig. 19. using the top-side
strain gage strain measurements given in Table 16.
Tensile Stress Distribution under 100lbs (Theoretically)
2500
y = 489.98x + 8E-13
stress (psi)
2000
1500
1000
500
0
0
1
2
3
4
5
6
location (in)
Fig. 41: Graph plotting Tensile Stress distribution under load of 100 lbs. – Trial Three
The calculated stresses under a 100 lb. load as per the procedure described in Fig. 19. using the bottomside strain gage strain measurements given in Table 17.
Tensile Stress Distribution under 200lbs (Theoretically)
5000
y = 979.96x + 2E-12
4500
4000
stress (psi)
3500
3000
2500
2000
1500
1000
500
0
0
1
2
3
4
5
6
location (in)
Fig. 42: Graph plotting Tensile Stress distribution under load of 100 lbs. – Trial Three
The calculated stresses under a 100 lb. load as per the procedure described in Fig. 19. using the bottomside strain gage strain measurements given in Table 17.
channel number
1
2
3
4
channel location(in)
4.4
3.8435
2.38
0.82
strain under 100lbs
211
184
117
38
strain under 200lbs
427
369
234
76
stress under 100lbs
2110
1840
1170
380
stress under 200lbs
4270
3690
2340
760
Table 18: Top Side-Strain Gage Compressive Micro-Strain Readings and Stress Calculations – Trial
Three The strain readings as recorded by the strain indicator for loads of 100 and 200 lbs, and the
corresponding stresses computed via Hooke’s Law, 𝜎 = 𝜀𝐸. The procedure for the calculating the stress
follows Fig. 19.
channel number
5
6
7
8
channel location(in)
4.79
3.8
2.26
0.738
strain under 100lbs
235
193
116
44
strain under 200lbs
469
386
230
87
stress under 100lbs
2350
1930
1160
440
stress under 200lbs
4690
3860
2300
870
Table 19: Bottom Side-Strain Gage Compressive Micro-Strain Readings and Stress Calculations – Trial
Three The strain readings as recorded by the strain indicator for loads of 100 and 200 lbs, and the
corresponding stresses computed via Hooke’s Law, 𝜎 = 𝜀𝐸. The procedure for the calculating the stress
follows Fig. 19.
Compression Stress Distribution under 100lbs
2500
y = 480.86x - 0.6662
stress (psi)
2000
1500
1000
500
0
0
1
2
3
4
5
location (in)
Fig. 43: Graph plotting Compressive Stress distribution under load of 100 lbs. – Trial Three
The calculated stresses under a 100 lb. load as per the procedure described in Fig. 19. using the top-side
strain gage strain measurements given in Table 18.
Compression Stress Distribution under 200lbs
4500
y = 972.94x - 18.447
4000
stress (psi)
3500
3000
2500
2000
1500
1000
500
0
0
1
2
3
4
5
location (in)
Fig. 44: Graph plotting Compressive Stress distribution under load of 200 lbs. – Trial Three
The calculated stresses under a 200 lb. load as per the procedure described in Fig. 19. using the top-side
strain gage strain measurements given in Table 18.
Tensile Stress Distribution under 100lbs
2500
y = 475.65x + 92.042
stress (psi)
2000
1500
1000
500
0
0
1
2
3
4
5
location (in)
Fig. 45: Graph plotting Tensile Stress distribution under load of 100 lbs. – Trial Three
6
The calculated stresses under a 100 lb. load as per the procedure described in Fig. 19. using the bottomside strain gage strain measurements given in Table 19.
Tensile Stress Distribution under 200lbs
5000
y = 952.93x + 169.37
4500
4000
stress (psi)
3500
3000
2500
2000
1500
1000
500
0
0
1
2
3
4
5
6
location (in)
Fig. 46: Graph plotting Tensile Stress distribution under load of 200 lbs. – Trial Three
The calculated stresses under a 200 lb. load as per the procedure described in Fig. 19. using the bottomside strain gage strain measurements given in Table 19.
Compression Stress Distribution under 100lbs
2500
y = 489.98x + 2E-12
Stress (psi)
2000
y = 480.86x - 0.6662
1500
1000
500
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
location (in)
Theoretical data
Experiment data
Fig. 47: Graph Plotting the Theoretical and Experimentally Determined Normal Compressive Stresses
against Strain Gage location under a load of 100 lbs. – Trial Three
The normal compressive stresses induced by the application of a 100 lb. force, considering readings
from the strain gages mounted to the top of the test specimen. The theoretically determined stresses
given in Table 16 are plotted in blue, with the orange points representing the experimentally
determined stresses computed in Table 18.
Compression Stress Distribution under 200lbs
5000
4500
y = 979.96x + 4E-12
4000
Stress (psi)
3500
y = 972.94x - 18.447
3000
2500
2000
1500
1000
500
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
location (in)
Theoretical data
Experiment data
Fig. 48: Graph Plotting the Theoretical and Experimentally Determined Normal Compressive Stresses
against Strain Gage location under a load of 200 lbs. – Trial Three
The normal compressive stresses induced by the application of a 100 lb. force, considering readings
from the strain gages mounted to the top of the test specimen. The theoretically determined stresses
given in Table 16 are plotted in blue, with the orange points representing the experimentally
determined stresses computed in Table 18.
Tensile Stress Distribution under 100lbs
2500
y = 489.98x + 8E-13
2000
Stress (psi)
y = 475.65x + 92.042
1500
Theoretical data
1000
Experiment data
500
0
0
1
2
3
4
5
6
location (in)
Fig. 49: Graph Plotting the Theoretical and Experimentally Determined Normal Tensile Stresses against
Strain Gage location under a load of 100 lbs. – Trial Three
The normal tensile stresses induced by the application of a 100 lb. force, considering readings from the
strain gages mounted to the bottom of the test specimen. The theoretically determined stresses given in
Table 17 are plotted in blue, with the orange points representing the experimentally determined
stresses computed in Table 19.
Tensile Stress Distribution under 200lbs
5000
y = 979.96x + 2E-12
4500
4000
y = 952.93x + 169.37
Stress (psi)
3500
3000
2500
2000
1500
1000
500
0
0
1
2
3
4
5
6
location (in)
Theoretical data
Experiment data
Fig. 50: Graph Plotting the Theoretical and Experimentally Determined Normal Tensile Stresses against
Strain Gage location under a load of 200 lbs. – Trial Three
The normal tensile stresses induced by the application of a 200 lb. force, considering readings from the
strain gages mounted to the bottom of the test specimen. The theoretically determined stresses given in
Table 17 are plotted in blue, with the orange points representing the experimentally determined
stresses computed in Table 19.
DISCUSSION
The in-plane moiré method was used to test for the stress concentration near a hole. The setup
involved a rectangular Plexiglas specimen with a center hole cut out, where the effects of tensile loading
produces a visible full field fringe pattern as the result of the specimen’s grating stretching and being
viewed against the superimposed reference grating. The stress concentration can be calculated by
taking the ratio of the fringe spacing far from the hole and of the fringe spacing near the hole. This is an
inversion of the normal ratio of stress near the hole and of stress far from the hole. This is because
stress is not being directly calculated in the in-plane moiré method but fringe spacing is and fringe
spacing tends to be larger farther from the hole and smaller due to bunching of fringes near the hole,
which is opposite of the stress values.
The Kc values obtained Table 1 are 1.907, 1.888, and 1.892 across three trials for a loading of
approx. 347 lbs each time. This matches the theoretical prediction of stress concentration near a hole
which can be no larger than 3x for any combination of hole diameter and width. In general, these values
indicate that the stress near the hole is roughly 2x the stress at a distance 2.5D away from the hole,
which is appropriate for this specimen. The variation in precision over the range of 0.004 to 0.015 is
acceptable given the subjectivity of relative scaling and image quality as well as the differences in
images given that trial loads ranged from 347-350 lbs. Rather it is remarkable that a full-view method
can produce such precise results in comparison to the other full-view method studied – photo-elasticity.
The out-of-plane shadow moiré method was used to measure the deflection of a plate out of
the plane, meaning if the plate is in the xy plane, it is now being stressed in the z-direction. The stress is
applied using a LVDT calibrated at D = -0.0201V + 0.1611, (Fig 8) (where D = deflection in inches and V =
voltage in volts) to produce concentric fringes of N = 1, 2, and 3. Using Eq. 3. Which neglects the angle β,
the deflection can be predicted from the fringe spacing in the shadow moiré images obtained and then
compared to the actual deflection measured by the calibrated LVDT. From Table 4, the average absolute
deflections from the calibrated LVDT were: 0th order = 0.2161807”, 1st order = 0.2377614”, 2nd order =
0.260394”, 3rd order = 0.285586”. From Table 6, the deflections predicted by shadow moiré testing are:
0th order = 0, 1st order = 0.49352518”, 0.98705036”, 1.48057554”.
At first glance it is apparent that the actual deflection predicted by the LVDT ranged from 0.20.3” over N = 0 to N = 3, which is a small range of small deflections. However, the shadow moiré
predicted very large deflections of a range of 0-1.481”. Furthermore, because Eq. 3 has a linear
dependence on N the number of fringes, each new fringe results in a huge 0.493”, or roughly 0.5”
increase in deflection. Since N is only a multiplier, the 0.493” step size comes from the other terms
which is the pitch grating p and the tan 𝛼 below. 𝑡𝑎𝑛𝛼 is only 0.405 and this causes the step size to be so
large. For the equation of shadow moiré to more accurately predict the actual deflections observed by
the calibrated LVDT, it appears it’s necessary to include the 𝑡𝑎𝑛𝛽 term. Looking at Appendix A3, it is
clear that as the specimen surface deflects towards the camera and light source, 𝛼 and 𝛽 will grow
larger. This is a significant effect given that the assumption that the camera and light source are
infinitely away and 𝛽 = 0 does not hold given the small size of the setup. If the 𝛽 term is included, the
denominator of Eq. 3 will be larger and the step size will become more reasonable. This is especially
important at higher fringe values as the shadow moiré method predicts linear increase in deflection
whereas in reality the deflection is much shallower between fringes and the difference might become
smaller at very high absolute deflections. However, as is, the predicted deflections from the shadow
moiré method grossly deviate from the true values from the calibrated LVDT, because of the fringe
multiplier N in Eq. 3 and the omission of the 𝑡𝑎𝑛𝛽 term in the denominator causing the step size being
too large.
The last part of the experiment involved doing a 3 point bend test with strain gages on a beam,
4 below and 4 above, using the readouts from the P3500 Strain Indicator and the P3 Digital Strain
Indicator, to compare the theoretical stresses obtained from beam bending theory with the actual
stresses calculated using the stress-strain version of Hooke’s law, plugging in the experimentally
obtained strains. Table 8 and Table 9 list the stresses calculated from the strains measured at each strain
gauge location along the top side and the bottom side of the beam respectively. Comparing the “stress
under 100 lbs” in Table 8 and 9, it appears that at x = 4.4”, top side stress is 2155.90487 and the bottom
side stress is 2346.99644, at x = 3.835”, the top side stress = 1883.2319 and the bottom side stress is
1861.91785, at x = 2.38”, top side stress is 1166.14855 and the bottom side stress is 1107.35114, and at
x = 0.82 top side stress is 401.782272 while bottom side stress is 361.604045.
In general, aside from x = 4.4 which is nearest the loading point, for all locations farther from
the loading point, the top side stress is higher than the bottom side stress. This makes sense considering
that the top surface is subject to localized stress from the loading point and will sustain higher stresses.
The bottom surface is not in direct contact with the loading point and sustains lower stresses.
Furthermore, differences between stresses on the top and bottom grow larger when points farther from
the loading point are considered. This phenomenon was also seen in the photoelastic method where
fringes bunched together near the loading point due to the stress concentration.
To compare experimental and theoretical values using Fig 15-18 and 20-21 for the 100 lb load,
the theoretical compressive load (top surface) has an equation y = 489.98x + 2 x 10^-2 while the
experimental compressive load has an equation y = 495.44x - 29.893. The experimental compressive
load on the top surface is higher than the theoretical given the stress concentration near the loading
point. For the theoretical tensile load (bottom surface), it has an equation y = 489.98x + 8E-12 while the
experimental has an equation of y = 479.56x + 78.221. Here the experimental tensile load on the bottom
surface is lower than the theoretical, which again matches our prediction that the stress concentration
from the loading point will result in higher stresses on the top face and lower stresses on the bottom
face, although theoretically they should be the same in magnitude.
ERROR ANALYSIS
Error factor
Instrument error, Measurement error.
It is the main error occurrence factor. When we measured distance, there was no
measurement error because we measured only one time. In measurement with calibrator, we
have a resolution error of 0.005 in in ruler. When we did this lab, we repeated it 3 times. This in
turn made a large value of standard deviation in measurement error. We also have
manufacturer error of all devices.
Properties of the specimen
The properties of specimen can be changed by temperature. The metal bar is more
likely to expand when temperature is high. If the inside of specimen was perfect, the bar would
have same expansion in every direction of inside. But we do not have perfect bar. We supposed
that it is perfect bar and it has constant young’s modules.
Uncertainty Analysis:
The final expression of a measurement should be where X' = 𝑋̅ ± ∆X (P%)
where X' is the true value, 𝑋̅ is the most probable estimate, ∆X is the uncertainty in this
estimate, and P% is the confidence interval.
The uncertainty analysis is the determination of 𝑋̅ and ∆X. The total uncertainty includes
instrument uncertainty and measurement uncertainty. In this report, we mainly use statistics
method to obtain uncertainty.
1. The In-plane moiré method
Trial 1
C=D/d
1.907
Trial 2
C=D/d
1.888
Table 20: Stress Concentration Factor
∆𝑋𝑖𝑛𝑠𝑡𝑟𝑢𝑚𝑒𝑛𝑡 = 0.05
𝑋̅ =
1.907 + 1.888 + 1.892
= 1.896
3
Trial 3
C=D/d
1.892
∑3𝑖=1(𝑥𝑖 − 𝑥̅ )2
(1.896 − 𝑥̅ )2 + (1.896 − 𝑥̅ )2 + (1.896 − 𝑥̅ )2
√
√
𝑆𝑥 =
=
= 0.01
𝑛−1
2
Corresponding tv.p= t2,0.95 =4.303
∆Xmeasurement =
𝑡2,0.95 × 𝑆𝑥
√𝑛
= 0.0248, and ∆Xinstrument = 0.005
1/2
Therefore, ∆𝑋𝑐𝑜𝑛𝑐𝑒𝑛𝑡𝑟𝑎𝑡𝑖𝑜𝑛 𝑓𝑎𝑐𝑡𝑜𝑟 = [∆𝑋𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑚𝑒𝑛𝑡 2 + ∆𝑋𝑖𝑛𝑠𝑡𝑟𝑢𝑚𝑒𝑛𝑡 2 ]
= 0.056
Part2
average voltage(v)
6.78
5.546666667
4.29
3.040333333
1.779333333
0.522333333
-0.717666667
-1.952666667
-3.197
-4.43
-5.65
-6.85
deflection (in)
0.025
0.05
0.075
0.1
0.125
0.15
0.175
0.2
0.225
0.25
0.275
0.3
xixi
45.9684
30.76551
18.4041
9.243627
3.166027
0.272832
0.515045
3.812907
10.22081
19.6249
31.9225
46.9225
xiyi
0.1695
0.277333
0.32175
0.304033
0.222417
0.07835
-0.12559
-0.39053
-0.71933
-1.1075
-1.55375
-2.055
N
12
Sx
-0.83867
Sy
1.95
Sxx
220.8392
Sxy
-4.57832
a
0.161094
b
-0.02012
θ
1.12E-06
ua
9.68E-05
ub
2.26E-05
Table 21: Calibrate LVDT via Linear Regression Methods
a+bxi
0.02468
0.049495
0.074779
0.099922
0.125294
0.150585
0.175533
0.200382
0.225418
0.250226
0.274772
0.298916
[y-(a+bx)]2
1.02144E-07
2.54958E-07
4.87526E-08
6.00728E-09
8.63263E-08
3.4182E-07
2.84572E-07
1.45659E-07
1.74423E-07
5.08954E-08
5.1984E-08
1.17506E-06
In order to calibrate LVDT, we use linear regression method to get the relation between deflection and
voltage, we get y = -0.0201x + 0.1611. This linear expression takes the form y = a+bxi with ucertainties in
the constants a and b of ua = 12 and ub = 37.6x103 in (1), ua =6.83 and ub = 20.3x103 in (2), and ua=9.68E05 and ub = 2.26E-05. The uncertainties associated with these values are very low, which indicates we
have a pretty accuracy.
Part 3
channel number
Theoretical σ (psi)
Experimental σ (psi)
error (%)
Trial 1 top (100lb)
1
2
2155.905 1883.232
2170
1840
0.653792
-2.29562
3
1166.149
1170
0.330271
4
401.7823
370
-7.91032
channel number
Theoretical σ (psi)
Experimental σ (psi)
error (%)
Trial 1 top (200lb)
1
2
4311.81
3766.464
4340
3607
0.653792
-4.23378
3
2332.297
2320
-0.52725
4
803.5645
750
-6.66587
channel number
Theoretical σ (psi)
Experimental σ (psi)
error (%)
Trial 1 bottom (100lb)
1
2
2346.996 1861.918
2350
1940
0.127975 4.193641
3
1107.351
1150
3.851431
4
361.604
430
18.9146
Trial 1 bottom (200lb)
channel number
1
2
3
Theoretical σ (psi)
4693.993 3723.836 2214.702
Experimental σ (psi)
4460
3840
2270
error (%)
-4.98494
3.11948
2.496847
Table 22: Theoretical stress, experimental stress and error
4
723.2081
840
16.14914
CONCLUSION
This experiment involved three parts, first using the in-plane moire method to test for the stress
concentration near a hole in a tensile stressed plexiglass specimen which gave accurate results of 1.81.9x, likely working well due to the fact that stress concentration is a relative comparison of the stress
near the hole and far from the hole, which is reflected in the fringe spacing.
The second part was to use the out-of-plane moire method to test the deflections predicted by
the moire method’s Eq. 3 against the calibrated LVDT which gave poor results because the predicted
values scaled linearly with N, the number of fringes and the step size between fringes that comes from
the pitch grating p divided by 𝑡𝑎𝑛𝛼 was nearly 0.5”.
To reduce this, a recommendation is to measure and include 𝑡𝑎𝑛𝛽 which would increase the
magnitude of the denominator and result in a smaller step size that would cause lower order fringes to
correspond well with the LVDT readings. However, given that the method involves a linear relationship
with N, this is not enough. The use of fractional fringes would be preferred to study the relationship and
come up with a more accurate equation, as it appears that the deflections predicted grow too quickly
which does not reflect how it tends to asymptote at large deflections in reality.
The final part was not using the moire method but using 4 LVDT strain gauges on the top and
bottom surface of a beam under 3 point bending and testing the difference in stresses between the top
and bottom surfaces at different distances in x. The results were that the effect of the stress
concentration near the loading point on the top surface resulted in a stress curve with a higher slope
compared to the theoretical and the linear stress curves for the bottom surface were consistently lower
in stress as they were not in direct contact with the loading point. The difference between the two
surfaces increased as cross sections at farther x-distances were examined. This experiment could be
improved with the use of more strain gauges in order to fully test and see if the stress curve is linear in
reality as the increase in the linear fit could have been due to a slight exponential growth towards the
location of the loading point, which was not examined and could be better defined with more data
points.
APPENDIX
A1: Determination of normal strain by moire fringes A diagram illustrating how to determine the
normal strain induced via grid geometry using the moiré method.
A2: Determination of shear strain by moiré fringes A diagram illustrating how to determine the shear
strain induced via grid geometry using the moiré method.
A3: Optical Arrangement used in Shadow-Moire Method
A diagram illustrating the geometric interpretation of the Shadow-Moire technique including light
sending and receiving angles.
A4: Principle Strain-Gage Mounting and Bending Moment Distribution in Three-Point Bending Beam
A diagram illustrating the positions of the strain gages used in Part III, as well as the corresponding
bending moments associated with each strain gage position.
REFERENCES
1. F.P. Chiang and T.Y. Hsu, Experiments in Solid Mechanics for MEC317 Mechanical Engineering Lab
II, Spring 2014, p.62-68
2. www.dictionary.com
3. Amidror, Isaac, The Theory of the Moire Phenomenon: Volume I: Periodic Layers, Springer; 2nd ed.
2009 edition (July 23, 2009) p.1-5.