PHOTOELASTICTY AND
BEAMS UNDER THREE POINT
BENDING
Tuesday February 18th, 2014
Matthew Stevens – First Author, Results, Error Analysis, Abstract &
Introduction
Kanchan Bhattacharyya – Discussion, Data Analysis, and Conclusion
Ting Zhang – Specimens and Instrumentation, Experimental Procedure
Xie Zheng - Experimental Theory
Abstract
Birefringent materials are those which are normally optically isotropic and, when subjected to a
light beam and mechanical stresses, become optically anisotropic and undergo a phenomena known as
double refraction in which the incident beam produces one reflected beam and two refracted beams.
Light traveling through the material at varying velocities results in lights traveling at varying
wavelengths, resulting in light of varying color and patterns which with the proper instrumentation can
be seen through the transparent material. 1 In a realm of mechanics that is many times determined
theoretically, Photoelastic Analysis Methods in Engineering utilize these material properties to
qualitatively determine the stress aspects of the states of the stress of points within these materials and
provide a means of optically processing elusive concept of stress. Color patterns formed within the
birefringent material are analyzed in terms of their fringe order, which is a measure of the phase
difference between light rays as they pass through the material. Specific points within each color pattern
or fringe order are considered to possess the same relative stress magnitude, and these relative
magnitudes are found by graphical interpolation and geometric considerations. Experimental Analysis
and comparison of other stress-related phenomena yields in some case consistent results, consistent
enough to inspire advances in optical equipment and numerical methods to process optical results.2
While the concept of a visual yet analytical approach and the theory that gives its mathematical
validity have given insight into stresses and their distributions throughout a material, that validity is
strongly subjective to the accuracy of the mathematical models developed from experimental results
and the manner in which measurements are taken. 2 Appropriate scaling of the model and the
prototype, as well as interpretation of the isochromatic fringe boundaries form the framework for the
mathematical models developed during experimentation and their validity depends on concise
interpretation and analysis techniques.
1
Introduction
While the Photoelastic effect in birefringent materials was first noticed in the early 19th century
by Scottish scientist Sir David Brewster, its application to mechanical stress analysis was not
implemented with much vigor until the 20th century.1 While Photoelastic methods cannot completely
define a state of material stress, it does however impart useful information about them. When plane,
optically isotropic materials are made double refracting anisotropic (i.e. material properties change from
varying smoothly in all directions to being predominant in certain directions) from inducing stress and
subjected to plane polarizing light, points of zero transmission may be associated with points at which
one of the principal stresses are parallel to the axis of polarization of the polarizing device.2 Series of
these points formed due to alignment of principal stresses, therefore giving an indication as to the
position and direction of these principal stresses within a birefringent material. Interpretation of the
patterns that result from stresses can be used to develop relationships between fringe order for
different positions within the material as the applied forces and subsequent stresses are varied, and if
performed accurately can describe stress distributions and concentrations encountered in the physical
world. While the accuracy of the method is largely subjective to both physical and mathematical models
developed in analysis, it is still an effective method used with modern application in determination of
residual stresses during traditional manufacturing processes as well as emerging technologies such as 3D
printing and rapid prototyping via steolithography.3
The validity in the relationship that exists between fringe order and it’s variance with position
within a material and stress resulting from different physical scenarios can be verified experimentally.
For example, principal stresses associated with beams in pure three and four point bending have been
theoretically calculated based on bending moments and maximum distances from a neutral axis along
which one of the principal stresses is zero, with the maximum tensile or compressive stress occurring at
the distances farthest from that neutral axis about the moment center of material. Similarly, stress
2
concentrations are considered on a theoretical basis as ratios of varying stresses considering only the
measurable quantities of force and cross sectional area. These theories form the basis against which the
accuracy of the Photoelastic method can be established.
Specimens and Instrumentation
(a)
(b)
FIG. 1(a) LG Flatron M237WD 23” LCD Monitor (b) Sony XDR-160 Digital Camcorder
(a)
(b)
FIG. 2(a)Photoelastic beam used in three and four-point bending stress analysis (b) Photoelastic plate
with circular hole used in stress concentration analysis
3
FIG. 3(a) Measurements Group 061 Polariscope (b) Measurements Group P3500 Strain indicator
Theory
Part one: Determination of material stress fringe value ‘fσ’ for the given material.
In this photoelasticity experiment, the basic equation if the stress-optical law:
σ1 – σ2 =
𝑁𝑓𝜎
𝐷
(1)
Here, N is the fringe order that can be determined from the fringe pattern of the model. D is the
thickness of the material we used in this experiment and fσ is the material stress fringe value. The value
of fσ is always different for different photoelasticity materials. Due the differences of fσ between
materials, the determination of fσ, the material stress fringe value, is the key to find out the principle
stress of the model in photoelasticity experiment. In this experiment, we are using the four point
bending method to find out the material stress fringe value.
4
A. The theoretical prediction for pure bending beam.
Base on the assumption, a beam is under pure bending, there is no shearing force which means
the shearing force is zero at any cross section. As shown in Fig. 4, after doing the force balancing, we
find out that the reaction forces in this case are half of the load, P/2. By considering the equilibrium of
the part of the beam to the left mm which is taken out for consideration, the bending moment should
be statically equivalent to a couple equal and opposite to the bending moment Pa/2.
FIG. 4 Free Body Diagram depicting the calculation of the bending moment
Since we need to find the distribution of the internal force over the cross section, we have to
consider the deformation of the beam. As for the beam which has rectangular cross section and two
adjacent vertical lines mm and pp are drawn on its sides. In the experiment, shown in Fig. 5, it shows
that these lines are still straight in the procedure of bending and rotate. In the theory below, the
assumption is based on the unchanged of the entire transverse section of the beam, not only the lines as
5
mm and pp. This experiment shows that the theory provides very accurate results for the deformation
of the beams and the strain of longitudinal fibers.
Fig. 5) Diagram depicting a member in pure positive bending as a result of the bending moments
applied on the free ends of the member.
In Fig. 5, nn1 is the trace of the surface which is called neutral surface and its intersection with any cross
section is called the neutral axis. The elongation s’s1 pf any fiber at distance y from the neutral surface is
obtained by drawing the line n1s1 parallel to mm. The unit elongation of the fiber ss’ is
(2)
According to (2), we can see that the distance y form the neutral surface is proportional to the strain of
the longitudinal and inversely proportional to the radius of curvature.
Then, according to Hooke’s law:
(3)
6
As shown in Fig. 6, we can see the distribution of the stresses and they are proportional to its distance
from the neutral axis nn. There are two unknowns, radius of curvature r and the position of the neutral
axis, now they could be determined according to the relationship between the distributed forces and a
resisting couple M(Fig. 5). Make dA as an element are of cross section at distance y from the neutral
axis, and then the force would be the product of the stress and the area.
FIG. 6. Distribution of tensile and compressive stresses about the neutral axis for a member in pure
bending.
Since there is a relationship between the force and couple, and the resultant of all the forces in the x
direction must be equal to zero, we can get
(4)
Also, we can get the moment of the force: (Ey/r) x dA x y. Get the sum of all the moments over the cross
section and they are equal to moment M of the external forces, we can get the following equations:
(5)
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where
(6)
Iz is the moment of inertial of the cross section with respect to the neutral axis z. When we ignore r in
equation (3) and ( 5), we can get an equation for stress:
(7)
Here, both M and y are positive values.
As we discussed before, in the case of rectangular cross section, we have the moment of inertial:
(8)
As for the circular cross section with d (diameter):
(9)
B. Determination of the material stress fringe value fσ by pure bending beam.
As shown in Fig. 7(a), it is a four point beam. We are putting two loads of P/2 on the beam symmetrically
and the distance is l1. Let l2 be the distance between supports. Therefore, the stress components are: σyy
= 0, τxy = 0, and σxx = My/I, Then, the moment M could be expressed as:
(10)
which is a constant. And y is the y coordinate of any point under the consideration.
8
FIG. 7. (a) Free Body Diagram of a simply supported beam subjected to four point bending (b) Free Body
Diagram of a simply supported beam subjected to three point bending
Because the σxx is the only fiber stress:
(11)
(12)
Part two: Determination of the fiber stresses along the top and bottom edges of a beam subjected to
three-point bending.
In this experiment, we are applying a central load P to a beam which is supported symmetrically by the
two supports with distance of l1 apart. Then, we can get the magnitude of bending moment:
(13)
Again, the normal stress at the outermost fiber is the only stress. So:
(14)
By using the fσ we get from previous experiment, we can have the fiber stress σxx at any cross section.
Part three: Determination of Stress concentration in a perforated sheet undergoing tensile load.
In this experiment, we are using a photoelastic material with a hole on it and let it subject to a tensile
stress. The stresses at both sides of the edges of the hole along x-axis will be much higher than the same
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strip without a hole under same load. In this case, the situation is called stress concentration. And there
is a factor in this condition which is called Stress Concentration Factor.
K c = σc / σ
(15)
Kc: stress concentration factor.
σc: stress at the edge of the hole along x-axis
σ : stress in the same strip without a hole under same loading condition.
Experimental Procedures
PART ONE -- Four-point Bending
1.Observed and checked the setting up of equipments and found that all the equipments had already
been exactly set up as the lab manual required including strain indicator was on, amplifier button was
turned on, output reading of the amplifier was adjusted to zero and strain gage factor was set to 3.94 and
optical arrangement was done as required.
2. Put photoelastic beam onto fixtures for four point bending the loading frame and adjusted the position
of the beam to let it have a clear image on the TV screen.
3. Initiated some load on the beam and saw colored isochromatic fringes. then, adjusted apparatus to get
a symmetric isochromatic fringe pattern
4. Observed the color sequence of colored isochromatic fringe from the center of the beam to the top
and bottom of the beam is yellow-red-green-yellow-red-green.
5. Used a Sony HDR-XR160 digital camcorder to take a picture of isochromatic fringes
6. Put a filter in front of the camcorder to change isochromatic fringe to mono color fringes which made
the fringe order easily countable
7. Increased the load slowly until the 3rd order isochromatic fringe just showed up at the top edge of the
beam.
8. Read and recorded the load from the strain indicator display.
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9. Decreased the load to 0, and repeated step 7 to 8 two more times
10. Used the Sony HDR-XR160 digital camcorder to take a picture of mono color fringes when 3rd
isochromatic fringe just showed up
11. Took the specimen from polariscope
12. Measured the length, width and thickness of the specimen
Measured the distance between two loading points
Measured the distance between two supports of the beam
Part 2--Three-point Bending
1. Changed the upper fixture for three-point bending
2. Placed the specimen into a new set of fixtures.
3.Added some load to the beam and adjust the position of the specimen to get a symmetric isochromatic
fringe pattern.
4.Started from corners using color sequence of colored isochromatic fringe to determine the fringe order
on the top and bottom edges of the beam.
5. Used Sony HDR-XR160 digital camcorder to take the color photo of fringe pattern
6. Put the filter in front of camcorder and apply load to obtain a clear fringe pattern.
7. Read and recorded the load from the strain indicator
8. Found out the fringe order of the isochromatic fringes and the fringe locations along the top and
bottom edges of the beam
9.Used Sony HDR-XR160 digital camcorder again to take photos of filtered fringe patterns
10. Unloaded the specimen and gave the specimen a same load as the previous reading.
11. Repeated the experiment two more times
11
Part 3--Stress Concentration Undergoing Tensile Load
1. Measured the width and the thickness of specimen with a central hole
2. Changed fixtures for stress concentration experiment
3. Placed the specimen onto the fixtures
4. adjusted the location of the moving frame in order to make the image of the central hole in the
specimen can be easily observed
5.Set strain indicator to zero before adding load
6. Slowly increase the load (maximum less than 150lbs) and observe the isochromatic fringe patterns as
they appear on the TV.
7. Determined the fringe by observing fringe color orders through the projection on TV (The fringe order
is increasing if the the color consequence is yellow-red-green-yellow-red-green. )
8. Took a color photo of isochromatic fringes with Sony HDR-XR160 digital camcorder
9.Put the filter in the front of camcorder to see a clear image of fringes
10. Increased the load until Nth order of mono color fringes just appear at the edge of the hole along x axis.
11. Read and recorded the load from the strain indicator.
12.Took a photo of mono color fringes.
13. Repeated the experiment three times
12
Results
FIG 8. A monochromatic light filtered image of a photoelastic beam under four-point bending, displaying
the isochromatic fringe patterns in a greenish tint so as to emphasize isoclinic lines across the beam
under loading. Beginning at the centermost black region an emanating in either direction we count
three isoclinic regions corresponding to at most third order fringes as a result of the stress induced.
FIG 9. The third order isochromatic fringe pattern resulting from stresses induced from four-point
bending. While the isochromatic fringes make identifying isoclinic zones a more tedious task, they allow
for easier identification of the beginning and ending points of respective fringe orders. With each
13
successive passing of a reddish-pink region into a blue region another fringe order has been completed
indicating the beginning of a new order. As anticipated, we also notice erratic behavior near the force
application points along the upper surface of the beam.
FIG 10. Resolution of the isochromatic fringe patterns above the neutral axis of the beam with
identification of the respective fringe orders identified by the obtained patterns. Using digital measuring
and scaling techniques, approximations were made for the relative starting position of each fringe order.
Specifically, regions where the blue regions transcended into yellow regions constituted the beginning of
each order.
(a)
N
1
2
3
y (in)
0.088
0.331
0.476
(c)
(b)
a
b
fσ
ua
ub
-0.0906
0.1944
1.079
0.06061485
0.028059223
b (in)
h (in)
D (in)
Iz (in4)
l1 (in)
l2 (in)
M(x) (lb∙in)
5.000
1.000
0.250
0.417
3.000
4.000
9.250
Table 1(a). The experimentally determined values for the fringe order corresponding to various
distances from the neutral axis at y = 0 increasing towards y = +h/2, (b) the experimentally determined
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values for the linear approximation variables, their associated uncertainties, and the value calculated for
the material stress fringe value fσ, and (c) The measurements taken for the width b, height h, thickness
D, Moment of Inertia Iz, the spans between the applied forces and supports, and bending moment of the
cross section between the span.
Distance from Neutral Axis vs. Fringe Order
0.600
Distance from Neutral Axis (in.)
0.500
y = 0.1944x - 0.0906
0.400
0.300
0.200
0.100
0.000
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
3
Fringe Order (N)
Distance from Neutral Axis vs. Fringe Order
FIG. 11. – Graph plotting the vertical distance from the neutral axis at y = 0 to y = +h/2 against the
determined fringe order corresponding to each respective distance. We notice a linear increase in the
fringe order with increasing vertical displacement from the neutral axis.
Determination of the fiber stresses along the top and bottom edges of a beam subjected to three-point
bending
15
FIG 12. Third order Isochromatic Fringe patterns obtained from subjecting a photoelastic beam to threepoint bending. The stress concentration near the center of the upper portion of the beam indicates the
position of the applied force, with fringe order being determined by counting the repetitions in yellowred-blue-green sequence from the neutral axis of the beam (y = 0) towards either free end (y = +- h/2)
FIG 13. A monochromatic light filtered image of the photoelastic beam under three-point bending,
displaying the isochromatic fringe patterns in a greenish tint so as to emphasize isoclinic lines across the
16
beam under loading. Similar to the isochromatic images we find the stress concentration near the force
application point as well as clearly distinguishable regions for the zeroth, first, second, and third order
fringe patterns.
FIG 14. Resolution of the monochromatic image depicting stress distribution within the upper portion
(region from x ≥ 0, 0 ≤ y ≤ +h/2) beam during three-point bending, and identification of the points used
during fringe order determination and stress analysis.
FIG 15. Resolution of the monochromatic image depicting stress distribution within the upper portion
(region from x ≥ -h/2≤ y ≤ 0) beam during three-point bending, and identification of the points used
during fringe order determination and stress analysis.
17
(a)
N
1
1
2
2
3
3
4
4
x (in)
1.956
1.650
1.425
1.300
0.650
0.488
0.125
0.084
M(x) (lb∙in)
0.506
4.025
6.613
8.050
15.525
17.393
21.563
22.034
σ = My/I (psi)
0.607
4.830
7.935
9.660
18.630
20.871
25.875
26.441
(b)
a
b
ua
ub
σ =NFσ/D (psi)
4.904
4.904
9.808
9.808
14.713
14.713
19.617
19.617
19.995
-8.059
0.703
0.600
Table 2(a). The various horizontal displacements from x = 0 taken from the top part of the beam at y =
+h/2, the fringe orders associated with these positions, the moment induced from the applied load at
each respective position, and the theoretical and experimentally determined values for the normal
stresses at each respective point. We notice that fringe order decreases linearly with displacement from
x = 0 in, with corresponding linear increases in the induced bending moment and normal bending
stresses (b) The calculated parameters quantifying the linear relationship that exists between the
normal stresses and the horizontal displacement from x = 0 and their respective uncertainties.
Normal Stress vs. Horizontal Distance (y = + h/2)
Normal Stress, σ (psi)
30.000
25.000
20.000
15.000
10.000
y = -8.059x + 19.995
5.000
0.000
0.000
0.500
1.000
1.500
2.000
Horizontal Distance (inches)
Theoretical Stress Values
Experimental Stress Values
Linear (Experimental Stress Values)
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2.500
FIG. 16. Graph plotting the theoretical and experimentally determined normal stresses versus the
horizontal displacement from x = 0 at y = h/2. The experimental normal stress distribution exhibits near
linear behavior, as shown by the linear trend-line. The theoretical stresses predicted prior to x ≈ 1.375
are large than those determined from experimental results, with experimental stresses becoming larger
than theoretical stresses for x ≥ 1.375.
(a)
N
1
1
2
2
3
3
4
4
x (in)
1.799
1.674
1.324
1.174
0.774
0.665
0.199
0.158
M(x) (lb∙in)
2.316
3.754
7.779
9.504
14.104
15.350
20.716
21.189
σ = My/I (psi)
2.779
4.504
9.334
11.404
16.924
18.420
24.859
25.427
σ =NFσ/D (psi)
4.904
4.904
9.808
9.808
14.713
14.713
19.617
19.617
(b)
a
b
ua
ub
21.3164376
-9.3308945
0.434
0.38265017
Table 3(a). A table giving the various horizontal displacements from x = 0 taken from the top part of the
beam at y = -h/2, the fringe orders associated with these positions, the moment induced from the
applied load at each respective position, and the theoretical and experimentally determined values for
the normal stresses at each respective point. We notice that fringe order decreases linearly with
displacement from x = 0 in, with corresponding linear increases in the induced bending moment and
normal bending stresses (b) gives the calculated parameters quantifying the linear relationship that
exists between the normal stresses and the horizontal displacement from x = 0 and their respective
uncertainties.
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Normal Stress vs. Horizontal Distance (y = - h/2)
Normal Stress, σ (psi)
30.000
25.000
20.000
15.000
y = -9.3309x + 21.316
10.000
5.000
0.000
0.000
0.200
0.400
0.600
0.800
1.000
1.200
1.400
1.600
1.800
2.000
Horizontal Distance, x (in.)
Theoretical Stress Values
Experimental Stress Values
Linear (Experimental Stress Values)
FIG. 17. Graph plotting the theoretical and experimentally determined normal stresses versus the
horizontal displacement from x = 0 at y = -h/2. The experimental normal stress distribution exhibits near
linear behavior, as shown by the linear trend-line. As in the previous measurements at y = +h/2, the
theoretical stresses predicted prior to x ≈ 1.375 are large than those determined from experimental
results, with experimental stresses becoming larger than theoretical stresses for x ≥ 1.375. We also
notice that in comparison to considering the upper portion of the beam the theoretical and
experimental results are slightly more accurate, most of which can be attributed to the erratic behavior
near the application point at y = +h/2
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Determination of Stress Concentration in a perforated sheet undergoing tensile load
FIG. 18. Isochromatic Fringe patterns around a hole in beam in bending. We notice increasing fringe
order near the vicinity of the hole, with lower order fringe patterns more prevalent with increasing
distance away from the edge of the hole.
FIG. 18. Isochromatic Fringe patterns around a hole in beam in bending. We notice increasing fringe
order near the vicinity of the hole, with lower order fringe patterns more prevalent with increasing
distance away from the edge of the hole.
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FIG. 19. Resolution of the respective fringe order patterns for determination of linear variation of fringe
order with horizontal distance away from the edge of the hole. The visibility of the fourth order fringe
pattern near the edge of the hole with decreasing order with increasing distance due to stress
concentration near the hole edge region.
(b)
(a)
X (in)
0.486
0.393
0.348
0.321
0.295
0.277
0.268
0.259
N
1
1
2
2
3
3
4
4
σc =NFσ/D (psi)
4.904181818
4.904181818
9.808363636
9.808363636
14.71254545
14.71254545
19.61672727
19.61672727
Kc = σc /σ∞
0.72208463
0.72208463
1.444169259
1.444169259
2.166253889
2.166253889
2.888338518
2.888338518
a
b
ua
ub
7.072
-13.819
0.959
2.832
(c)
l (in)
W (in)
d1 (in)
10.125
2.9375
0.5
A0 (i2
29.742
Table 4 (a) The respective positions from the x axis (x=0) at y = 0, their corresponding fringe order and
calculated value for the stress in the beam at that position, as well as the stress concentration factor
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calculated considering both the experimentally determined stress as well as the same induced in a beam
with uniform solid cross section (b) gives the calculated parameters quantifying the linear relationship
that exists between the stress concentration factor and the horizontal displacement along
y = 0. (c) Gives values measured for the length width of the beam in consideration, the diameter of the
hole that was drilled in the beam, as well as the area of the beam without the hole used in calculating
the stress concentration factor
Fringe Order vs. Horizontal Distance (y = 0)
5
Fringe Order N
4
3
2
y = -13.819x + 7.0724
1
0
0.25
0.30
0.35
0.40
0.45
0.50
Horizontal Distance, x (in.)
Fringe Order vs. Distance
Linear (Fringe Order vs. Distance)
FIG. 20. Graph plotting the Fringe order N against the Horizontal Distance x from the center of the beam.
Values were measured outward from approximately 0.260 inches, with the void space in the hole
experiencing zero stress. There is a general linear relationship between the fringe order and the
displacement away from the edge of the hole, with a first order fringe pattern developing as soon as
0.39 inches. This implies that higher order fringes develop very close to the edge of the hole and
propagate quickly before reaching the edge itself.
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Stress Concentration Factor, Kc
Stress Concentration Factor vs. Horizontal Distance
4.500
4.000
3.500
3.000
2.500
2.000
1.500
1.000
0.500
0.000
-0.5000.250
-1.000
y = -9.9783x + 5.1069
0.300
0.350
0.400
0.450
0.500
Horizontal Distance, x (in)
Stress Concentration Factor vs. Horizontal Distance
Linear (Stress Concentration Factor vs. Horizontal Distance)
FIG. 21. Graph plotting the Stress Concentration Factor Kc against the horizontal displacement away
from the center of the beam. As expected, we find that the stress concentration factor is maximum near
the edge of the hole approaching a value of 3.0. The concentration factor decreases linearly with
horizontal distance away from the x-axis.
DISCUSSION
Part I – Determination of Material Stress Fringe Value Fsigma
Each photoelastic material is expected to have a characteristic material stress fringe value fσ.
fσ is calculated using (12). The thickness D and the moment of inertia I are properties of the beam’s
geometry and are constant. Within the two loading points of the four point bending test – a distance
defined as L1, the moment M is also constant. The only remaining quantity is “y/N”, which defines the
vertical distance between any two fringes. It’s important for this to also remain constant on average
otherwise the Fsigma cannot be reliably applied to different loading situations. Qualitatively, looking at
Fig. (8), each fringe appears evenly spaced and distanced from the neutral axis. This is confirmed by
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more meticulous scaling efforts in Fig. 10 and by the linear plot in Fig. 11. An fσ = 1.079 is obtained and
is subsequently applied in all future calculations. It yields comparable stress values to the theoretical
values as seen in Table 2(a).
Part II – Determination of the fiber stresses along the top & bottom edges of beam subject to 3-pt
bending
After Part I calibration leads to fσ = 1.079, the setup is changed to a 3 point bending test with a
single loading point at the center of the top edge, which is defined as the “zero” of the x-axis. The
bending moment is now described by (13). The objective is to compare stresses predicted by simple
beam bending theory with those obtained from photoelastic methods.
Qualitatively, looking Fig. 13, the first failure of simple bending theory is made clear. The
asymmetry between the top and bottom edges of the beam is not predicted – significant warping of
fringes are observed on the top edge (y = +h/2) near the center loading point which act as a site for
stress concentration. By comparison, the bottom edge (y = -h/2) which is not directly impacted by the
loading point shows fairly regular semicircular fringes.
Before an in-depth analysis of the Fig. 16 and Fig. 17 to confirm those observations, it should be
noted that the yellow curves representing the experimental stress values from the photoelastic method
appear to have “plateaus” where the stress appears to be constant over a small range of x values. This
range which is actually the width of the fringe. Recall that stress from the photoelastic method is
calculated by (1). For x-values falling within the width of the fringe, the stress calculated is constant
because the same fringe order N is associated with that range. These fringes are also discrete or
“quantized” with some distance between them – stresses for x-values between fringes must be obtained
by interpolation. These experimental flaws do limit our analysis but can be minimized by more
sophisticated experimental setups involving fractional fringe orders to account for stress behavior in
between whole number fringes N = 1,2,3,…etc.
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QUESTION#1: Comparison of Stress vs. X Relationship for y = +h/2 & y = -h/2
These columns show key aspects of the yellow curves from Fig. 16 and Fig. 17
Top Edge: (y = +h/2)
Bottom Edge: (y= -h/2)
N = 4; stress = 19.617 psi; 0.084-0.125”
N = 4; stress = 19.617 psi; 0.158-0.199”
N = 3; stress = 14.713 psi; 0.4876-0.6500”
N = 3; stress = 14.713 psi; 0.665-0.774”
N = 2; stress = 9.808 psi; 1.3-1.425”
N = 2; stress = 9.808 psi; 1.174-1.324”
N = 1; stress = 4.904 psi; 1.65-1.956”
N = 1; stress = 4.904 psi; 1.674-1.799”
The fringe order decreases from N = 4 to N = 1 from the center zero to the ends of the beam in
the abs x direction. Note that both the top and bottom edges have the same number of fringes with the
same associated stresses as a consequence of using (12). However, notice that those fringes and their
associated stresses appear at different ranges in the top edge and in the bottom edge.
For example, in the top edge N=4; stress = 19.617 psi appears at 0.084” and in the bottom edge
it will appear at 0.158”, nearly twice the distance from the center. This is also true for N=3. Clearly,
stress and therefore, the fringes are more concentrated near the loading point on the top edge. Fringes
N=2 and N=1 appear far enough from the loading point in the top edge that the effects of stress
concentration is not as great and for these fringes the ranges roughly coincide with those of the bottom
edge. In short, the photoelastic method is able to provide a more accurate model of how stress varies in x
from a loading point by actually showing the stress concentration effect on the fringes. The simple
bending theory can only predict how stress drops off linearly as a function of distance with no
consideration for very high stress concentrations at loading points, which can be sites of failure. Also,
both graphs show that the experimental stress values are consistently higher than the values predicted
by simple bending theory.
26
QUESTION 2: Description of How to Determine Stress Along Free Boundary of Photoelastic Model
The free boundary is a region where both principal stresses are zero and thus N = 0 results in a
large dark fringe covering these boundaries until the first fringe going towards the bounded region.
However, this is not the case for the portion of the free boundary close to the supports. In these areas
as shown in Fig. 12, similar to the loading point on the top edge, stress concentrations are visible near
the supports. Therefore, just like how the scaling and calculations were done for the fringes near the top
loading point in the beginning of Part II, starting from the middle region of N = 0 or from the edge of the
beam which is also N=0, fringes can be counted concentrating near the support point.
After the fringes are counted, the axis can be set on the edge of the free boundary and used to
determine the position of each fringe. With these two, the stress along the free boundary close to
supports can be mapped even though the theoretical stress would be equal to zero because the
moment is zero outside the support points. This is because (1) relies only on the fringe order N, the
material stress fringe value fσ, and the thickness D.
PART III – Determination of Stress Concentration in a Perforated Sheet Under Tensile Load
Mechanical defects or imperfections such as notches, cracks, and holes can be sources of
significant stress concentrations. In our experiment, a perforated sheet with a circular hole is chosen to
undergo tensile load and its stresses are analyzed using the photoelastic method. The top and bottom
edges of the hole experience tensile forces while the sides collapse inwards as the hole becomes an oval
and thus the sides experience compressive stresses. Compared to the normal cross sectional stress in
the sheet without a hole, the theoretical case of a holed plate with a finite width and infinite length
predicts a stress concentration of 3x.
Qualitatively, looking at Fig. 18 (the fringes visibly broaden significantly with increasing distance
in x from the edge of the hole, resulting in spectacular “butterfly” fringes near the ends. This broadening
27
of the fringes and the drop in Kc (stress concentration factor) will be analyzed with the help of Fig. 20
and Fig. 21.
Essential Data from Both Graphs:
FRINGE WIDTH
N = 4; 0.259-0.268” = 0.009”; Kc = 2.888x
N = 3; 0.277-0.295” = 0.018”; Kc = 2.166x
N = 2; 0.321-0.348” = 0.027”; Kc = 1.444x
N = 1; 0.393-0.486” = 0.093”; Kc = 0.722x
DROPS BETWEEN FRINGES
N = 4 to N = 3 (2.888 to 2.166x); 0.268-0.277” = 0.009”
N = 3 to N = 2 (2.166x to 1.444x) 0.295-0.321” = 0.026”
N = 2 to N = 1 (1.444x to 0.7224x) 0.348-0.393” = 0.045”
Shows width of fringe greatly increasing as we go farther out in x direction, reflecting broader
bands of stress. This is also combined with shallower drops between fringes in terms of stress as well.
Near the hole fringes tend to be thinner and there is a large stress gradient drop between them while
farther out broader fringes are present with shallower drops in stress. The data accounts for the broad
“butterfly” bands seen in our qualitative analysis as well as the tiny localized fringes near the surface.
Now to compare with theory, where the case of a hole in a plate of finite width and infinite
length was considered, the maximum stress concentration was predicted to be 3x. In our experimental
results, we obtained a factor of 2.888x near the edge of the hole at x = 0.259”. This matches just
underneath the 3x stress concentration predicted in the theoretical case of finite width and infinite
length for a perforated sheet.
28
Error Analysis
Determination of the Material Fringe Stress Value
N
1
2
3
y (in)
0.088
0.331
0.476
xy
0.088
0.661
1.429
xx
1
4
9
Sx
6
Sy
0.8946
Sxy
2.178
Sxx
14
a+bxi
0.1038
0.2982
0.4926
(y-[a+bxi])2
0.00026244
0.00104976
0.00026244
a
b
fσ
ua
ub
-0.091
0.194
1.079
0.061
0.028
theta
0.00157464
Table 5 Experimental uncertainty calculations for determining the relationship between vertical
displacement from the neutral axis y and the fringe order N. The results yield a linear relationship very
limited uncertainties.
N
1
1
2
2
3
3
4
4
X (in)
1.956
1.650
1.425
1.300
0.650
0.488
0.125
0.084
Sx
7.6776
σ = My/I (psi)
0.607
4.830
7.935
9.660
18.630
20.871
25.875
26.441
σ =NFσ/D (psi)
4.904
4.904
9.808
9.808
14.713
14.713
19.617
19.617
Sy
98.08363636
xy
9.593
8.092
13.977
12.751
9.563
7.174
2.452
1.648
Sxy
65.24915825
xx
3.826
2.723
2.031
1.690
0.423
0.238
0.016
0.007
a+bxi
4.231
6.697
8.511
9.518
14.756
16.065
18.987
19.318
(y-[a+bxi])2 a
0.453
b
3.215
ua
1.684
ub
0.084
0.002
1.829
0.396
0.089
19.995
-8.059
0.703
0.600
Sxx
10.95199576
Table 6 Experimental uncertainty calculations for determining the relationship between the normal
bending stresses in the beam during three point bending along the upper edge of the beam. Again, we
find a linear relationship existing between the fringe order N and the position away from the midpoint
of the beam with uncertainties higher than those calculated analyzing the four point bending model.
29
N
1
1
2
2
3
3
4
4
x
1.799
1.674
1.324
1.174
0.774
0.665
0.199
0.158
M(x)
2.316
3.754
7.779
9.504
14.104
15.350
20.716
21.189
σ = My/I
2.779
4.504
9.334
11.404
16.924
18.420
24.859
25.427
Sx
Sy
7.764 98.084
Sxy
69.676
Sxx
10.270
σ =NFσ/D
4.904
4.904
9.808
9.808
14.713
14.713
19.617
19.617
xy
8.821
8.208
12.982
11.511
11.382
9.787
3.896
3.090
xx
3.235
2.801
1.752
1.377
0.598
0.442
0.039
0.025
a+bxi
4.534
5.700
8.966
10.366
14.098
15.110
19.463
19.847
(y-[a+bxi])2
0.137
0.634
0.709
0.311
0.378
0.158
0.024
0.053
a
b
ua
ub
21.316
-9.330
0.434
0.383
theta
2.403
Table 7 Experimental uncertainty calculations for determining the relationship between the normal
bending stress in the beam during three point bending along the lower edge of the beam. Again, we find
a linear relationship existing between the fringe order N and the position away from the midpoint of the
beam with uncertainties higher than those calculated analyzing the four point bending model. We also
note that with the absence of the erratic behavior near the concentrated stress region near the
application point the uncertainties are reduced by a factor of two.
30
a)
x
0.486
0.393
0.348
0.321
0.295
0.277
0.268
0.259
N
1
1
2
2
3
3
4
4
σc =NFσ/D
4.904
4.904
9.808
9.808
14.713
14.713
19.617
19.617
Kc = σc /σ∞
0.722
0.722
1.444
1.444
2.166
2.166
2.888
2.888
xy
0.486
0.393
0.696
0.643
0.884
0.830
1.071
1.035
xx
0.237
0.154
0.121
0.103
0.087
0.077
0.072
0.067
(y-[a+bxi])2
0.422
0.414
0.068
0.398
0.000
0.061
0.396
0.255
c)
b)
Sx
2.65
a+bxi
0.351
1.644
2.261
2.631
3.001
3.248
3.371
3.495
Sy
20
Sxy
6.04
Sxx
0.92
a
b
ua
ub
theta
2.01
7.0723823
-13.81869
0.959
2.83243906
Table 8 a) Experimentally determined values, their respective products for linear regression analysis,
evaluation of the linear approximation function b) Summations of variables and their products used in
linear regression analysis c) the parameters characterizing the linear relationship between the fringe
order N and the horizontal displacement from the edge of the hole and decreasing stress concentration.
x
0.486
0.393
0.348
0.321
0.295
0.277
0.268
0.259
Kc = σc /σ∞
0.722
0.722
1.444
1.444
2.166
2.166
2.888
2.888
xy
0.351
0.284
0.503
0.464
0.638
0.600
0.774
0.748
xx
0.237
0.154
0.121
0.103
0.087
0.077
0.072
0.067
Sx
2.647
Sy
14.442
Sxy
4.361
Sxx
0.918
a+bxi
0.253
1.187
1.633
1.900
2.167
2.345
2.434
2.524
(y-[a+bxi])2
0.220
0.216
0.035
0.208
0.000
0.032
0.206
0.133
theta
0.371
31
a
b
fσ
ua
ub
5.107
-9.978
0.000
0.412
1.216
Table 9 Uncertainty analysis of the linear approximation made describing the relationship between the
stress concentrations as a function of horizontal displacement from the center of the hole. We notice an
unusually high error associated with this approximation, which can be attributed to the sheer difficulty
of obtaining an exact numerical solution from an experimental linear regression approximation with
limited terms.
Before considering any experimental results, we entered the lab with the understanding that
the Photoelastic method in itself is only as effective and accurate as the model that Is developed to
approximate the behavior of the prototype.3 This technique requires either hand or computer measured
determined coordinates to approximate a point where we arbitrarily decide to interpret as the
boundary of two neighboring fringes, in addition to use of appropriate scaling techniques to transpose
the magnified images down to the scale of the material in testing.
In determining the material stress fringe value fσ by four-point our results rendered a linear
relationship between the fringe order and the vertical displacement from the neutral axis, with
uncertainties in the slope and intercept of the equation of 0.028 and 0.061, respectively. While we could
still manage to decrease this uncertainty with closer approximations, it certainly isn’t too concerning
considering the error we had anticipated. When considering the three-point bending scenario in which
we related fringe order and horizontal displacement along the upper free boundary we found increasing
errors in our linear approximation terms of 0.600 in the slope and 0.703 in the intercept, which instinct
would attribute to the stress concentration in the vicinity of the load application point. Further
inspection of the linear approximation modeled from the measurements taken along the bottom free
surface yielded more accurate results to validate this assumption, with resulting errors of 0.382 in the
slope and 0.434 in the intercept. Considering the stress concentration factor analysis, our linear
approximations for the stress distribution along the x-axis yielded relatively accurate results, with
32
uncertainties of 2.832 in the slope and 0.959 in the intercept. Similarly, when considering the
distribution of the stress concentration factor along the x-axis our approximations yielded uncertainties
of 1.216 in the slope and 0.412 in the intercept, respectively.
CONCLUSION
As illustrated in the 3 point bending test, the photoelastic method can account for stress
concentration near loading points or supports to describe the “stress-distance” relationship more
accurately compared to the simple linear model from first-order beam bending theory. This can be
observed qualitatively via the monochrome images as well as through scaling and graphical analysis
where the location and density of “fringe plateaus” gives an indication that a stress concentration site is
nearby, distorting the normal spacing and shape of the fringes. This allowed us to justify the asymmetry
between the top and bottom edges of the beam in that experiment as well, which the first-order theory
would have considered symmetric. Furthermore, it can provide relatively simplified analyses of cases
such as the perforated sheet which does not have a simple theory to describe the stress field.
The apparent disadvantages of the photoelastic method were touched upon earlier and can be
summed up as: (1) Model can handle max load < 150 lbs and this creates a very limited number of
discrete fringes – in this experiment only 4 were achieved at max load. (2) Stresses are calculated as
constant over the x-values of the fringe and are guesstimated for x-values in between fringes using
interpolation. These two limitations had an impact on accuracy by having an Fsigma dependent on only
3-4 data points and in modeling the behavior of “stress vs. x” in the 3 point bending tests which may not
be a perfectly linear relationship.
These experimental flaws can be minimized using more sophisticated experimental methods
involving fractional fringe orders to account for stress behavior in between whole number fringes N =
1,2,3,…etc. Some of these methods can achieve precision of 1/50th of a fringe order which would provide
33
far greater accuracy of the Fsigma value and in mapping the “stress-distance” relationship with higher
resolution and less need for interpolation and prevalence of “stress plateaus”. The continuity of the
simple beam bending theory over all x-values is one advantage over the photoelastic method used in
this experiment. By adopting such methods, this can become a strength of the photoelastic method as
well. Two methods that can be used are the Tardy method and the Senarmont method.4
In particular, the Senarmont method utilizes quarter-wave plates inserted between the model
and the analyzer, then rotated till a dark field appears. Then the analyzer is moved while everything else
is kept steady, which moves the fringes around. It’s then possible to align the lowest order fringe at any
given point and by knowing the angle needed to rotate the analyzer to move the fringe there, fractional
fringe orders at that point can be obtained from “n + (𝜃 ÷ 180°)”. Using this technique, multiple
fractional fringe orders can be identified at x-values between whole number fringes, lending a great deal
of accuracy to the whole enterprise.
REFERENCES
1. R.C. Dove and P.H. Adams, Experimental Stress Analysis and Motion Measurement, Charles E.
Merrill Books, Inc., Columbus Ohio, 1964, p300-308
2. J.W. Dally and W.F. Riley, Experimental Stress Analysis, Second Edition, McGraw-Hill, 1978,
p.406-446
3.
A.Asundi, Recent Advances In Photoelastic Applications, 1996,
http://www.ntu.edu.sg/home/masundi/optical-methods/photoelasticity/recent_advances.html
(February 16th, 2014)
4. Hearn, E.J. 2001. Mechanics of Materials 2: The Mechanics of Elastic and Plastic Deformation of
Solids and Structural Materials, Volume 2, 3rd Ed. Massachusetts: Butterworth-Heinemann.
34