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AN EXPERIMENTAL INVESTIGATION OFSTRESS CONCENTRATION FACTOR
ABSTRACT
In this project, investigation of stress concentration factor is carried out. Some
specimens with edge notches, multiple edge notches, and holes are fabricated.While two specimens are hand sawed to produce edge cracks. The stressconcentration factor (K) value is compared between the theoretical valuesfound in standard K value chart with our experimental results. Effect of multiple notches is considered and the distribution of stress in hand sawedcracks and notch is compared to deduce the significance of stressconcentration factor for the cracks. In the end, some recommendations aremade for future investigation.
Submitted byAhmad Loqman
Andiyanto Sutandar Julian Chan Hou Kan
Chen Guang ZeChou Shou Kang
SCHOOL OF MECHANICAL & PRODUCTION ENGINEERING NANYANG TECHNOLOGICAL UNIVERSITY
2000
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CONTENTS
Page No.TITTLE PAGE iABSTRACT iiACKNOWLEDGEMENTS iiiLIST OF ILLUSTRATIONS
List of Figure ivList of Table v
1. INTRODUCTION 12. OBJECTIVE 33. THEORY 34. EQUIPMENT AND MATERIALS 6
4.1 Equipment 64.2 Materials 6
5. PROCEDURES 86. RESULTS 97. DISCUSSION 138. CONCLUSION 199. RECOMMENDATION 2010. REFERENCES 21
APPENDICESAppendix A A1
Appendix B B1Appendix C C1Appendix D D1Appendix E E1
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Acknowledgements
Our Groups wishes to express our sincere appreciation to the following:
1. A/P Anand Krishna Asundi, the supervisor of this project, for his
invaluable advice and guidance throughout the project;
2. Liu Tong and Anil Kishen for helping us setting up the equipmentand advice they have given throughout the project;
3. The technical staff of CNC Lab I and II for the fabrication of our specimens 10 to 15
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LIST OF FIGURES
Page
Fig. 1 Location of zero fringe order 2
Fig. 2 Stress distribution in the vicinity of a circular hole 4Fig. 3 Normal stress in the tip of edge notches 4
Fig. 4 Cutting plan for specimen with a circular hole 7
Fig. 5 Cutting plan for specimen with edge notches 7
Fig. 6 Fringes appearance for specimen 3 and 4 under 40 kg of applied load 15
Fig. 7 Fringes appearance for specimen 3 and 4 under 40 kg of applied load 16
Fig. 8 Fringes appearances for specimen 10 and 11 without load 17
Fig. 9 Comparison between flow of stress in one U-shaped notch and
multiple U-shaped notches 18
Fig. 10 Fringe appearance of specimen 8 and 9 with 40 kg loading 19
LIST OF TABLES
Page
Table 1 Dimension of holes for specimen with a circular hole 7
Table 2 Dimension of curves radius for specimen with edge notches 7
Table 3 Calculation of K value for specimen with edge notches 8
Table 4 Calculation of K value for specimen with multiple edge notches 10
and cracks
Table 5 Calculation of K value for specimen with a circular hole 11
Table 6 Comparison of theoretical data with experimental data for
specimen 1 to 6 14
Table 7 Comparison of theoretical data with experimental data for 14
specimen 10 to 15
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1. INTRODUCTION
All crystal other than those of cubic crystal possesses one property called double
refraction or birefringence. Birefringence is an event when a ray of light, which is
incident on certain crystals, is split into two components. The two components are then
transmitted through the crystal in different directions. If the two rays that have passedthrough the crystal are observed through an analyzer, it is found that they are plane
polarized in mutually perpendicular planes.
A number of transparent amorphous materials which are optically isotropic
become optically anisotropic when stressed and exhibits similar characteristic to crystals
such as double refraction characteristic. The effect of double refraction will disappear
upon unloading.
The effect of birefringence on those materials was first observed by Sir David
Brewster in 1816. When such materials are loaded and observed in a polarized light field,
the temporary double refraction produces interference bands known as isochromatics, or
stress fringes. Each stress fringe denotes a locus of point of the same maximum shearing
stress in the plane of the specimen which is normal to the incident light beam.
If we observe the isochromatics effect by monochromatic filter, we will get a
changing of colour from dark to bright to dark, representing one optical cycle. The initial
dark fringe will be fringe order zero, the second fringe order one, the third fringe order
two, then three, four, etc. The bands of colour will form countour like pattern across the
specimen according to the irregularity in the shape of the material. Closely spaced bands
denotes high stress gradients region. Whereas broad spaced bands denotes low stress
gradients region.
In our case, the example of the zero fringe order used in the calculation can be
seen in figure 1. The zero fringe order will describe a zero bending moment region.
Zero fringe order
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This isochromatics (birefringence) effect has been applied widely to analyze
stress on materials. The method was termed photoelasticity. While the materials used as
photoelastic model materials are epoxy resins, polymethacrylate, polycarbonate, cellulose
nitrate, etc. The choice of materials will then depends on the objective of the experiment
to be performed.
Our experiment applied photoelasticity to investigate the stress concentration
factor for specimens with notches and holes. This experiment would only cover a small
part of the stress concentration factor as we only had 1 specimen with semi circular edge
notches, 4 specimens with curve type of edge notches, 1 specimen with U-shaped edge
notches, 1 specimen of multiple U-shaped edge notches, and 6 specimens of a circular
hole. Stress concentration obtained experimentally would be compared with the
theoretical value read from the standard graph provided in stress concentration handbook.
The effect of multiple notches would also be deduced. Besides that, we also examinewhether it is stress concentration factor or stress intensity factor or stress concentration
factor that we ought to apply on the specimens with cracks.
However, for specimen 7, 8, 9, we cannot find any standard graph to fit them in.
So, we are going to consider them qualitatively only.
The report begins with introduction. Then objective and next theory will be
presented. After that equipment and materials used are specified before procedures in
performing the experiments are stated. The results obtained will then be presented,
followed by the discussions of the results. Conclusion and recommendation for future
investigation will be presented before list of references. At the end of this report, some
stress concentration graphs will be attached as appendices.
2. OBJECTIVE
The experiment performed was aimed to investigate the stress concentration factor
for specimens with different radius and type of edge notches, to study the stress
concentration factor for specimens with different radius of holes cut at the middle part of
the specimens, and to classify and study the specimens with cracks that is created by
hand-saw.
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3. THEORY
A structural member under vertical uniaxial tension (figure 1) experiences what is
called normal stress. Stress is defined as intensity of forces per unit area.
t w P A P *== (1)
= stress (Pa) w = width of the member (m)
P = applied load (N) t = thickness of the member (m)
A = cross sectional area (m 2)
However, when the member contains discontinuity, such as hole, notch or a
sudden change in cross section, high localized stresses may also occur near the
discontinuity. As shown in figure 2, we can see that in member with holes, high stresses
occurs in a section passing through the center of the hole. While for notches, high stress
distribution will be at the tip of the notch (figure 3) where the cross sectional of the
member will be at the least.
Using photoelastic method, we will get the fringe order (N). The fringe order has
been experimentally related to the maximum in plane shearing stress for two dimensional
problems through stress optic law.
t
f N
2
*max
= (2)
while2
minmaxmax
= (3)
N = fringe order
max ave
Fig. 2 Stress distribution in the vicinity of a circular hole
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f = material fringe value (kPa/fringe/mm)
max = maximum shearing stress (kPa)
max = maximum inline principle stress (kPa)
min = minimum inline principle stress (kPa)
In our experiment, we will apply a uniaxial testing as shown below,
Then (3) will be left with
maxmax
max 20
=
= (4)
While (2) will become
t f N
*
max = (5)
Equation 5 is used only to determine the highest stress occurring in the specimen,that is at the tip of the notch or the middle of the hole. While for the part of specimen
which is far away from the discontinuity will experience average stress ( ave) (fig.2)
defined in (1).
Stress concentration factor is then defined as
ave
K
max= (6)
The stress concentration factor (K) is found to be independent of the size of the
material used. The value only depends upon the geometric parameters involved, i.e. upon
the ratio of r/d in the case of the circular hole.
Fig. 3 Normal stress in the tip of edge notches
1=max
2=0
1=max
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The K used above is for material with notches. However, we need to define
another parameter called stress intensity factor. This K value is used to define how
intensively material near the crack tip is being loaded. The stress intensity factor depends
on Y (a dimensionless parameter that depends on both crack and specimen sizes and
geometries, as well as the manner of load application), applied, and the size of the
crack.
aY K = (7)
We see from the equation that K value will increase as value increase and if
reaches f (fracture stress) value, the K will reach K C (critical value called fracture
toughness).
For ductile material, stress intensity factor is expected to be relatively large while
for brittle material, the value will be low since there is not possible for the material to
experience appreciable plastic deformation in front of advancing crack tip. Thus, brittle
material is vulnerable to brittle fracture.
4. EQUIPMENT AND MATERIALS
4.1. Equipment
1) One 060 Series Modular Transmission Polariscope System in the Laboratory
2) One Personal Computer
3) One Strain gauge
4) One Digital camera
5) One Monochromatic filter
6) EDC-1000HR imaging software for windows, installed in the personalcomputer
4.2. Materials
We used one photoelastic sheet, made by Measurement Group, Inc., Raleigh,
N. C., USA, with specification:
a. Type PSM-1 10x20
b. Item Code 17022
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c. Thickness .250 in. nominal
d. Lot No. 1829
e. C value 40/psi/fr/in.
The photoelastic sheet was made into 9 specimens as stated below:
a. 7 specimens with holes drilled.
b. 2 specimen with cracks, which is manually cut
The rest of the specimens, 6 specimens with notches and 1 specimen with multiple
notches, were supplied by our supervisor.
The holes specimens were made based on cutting plan shown in figure 4
while the specification of the is tabulated in table 1.
Specimen No. 10 11 12 13 14 15
Diameter of hole , (mm) 2 3 4 5 7 10
Table 1 Dimension of holes for specimen with a circular hole
Fig. 4 Cutting plan for specimen with a circular hole
250
25
9 9
125
12.5
10.5
10.5
12.5 12.5
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The specimens with notches have the cutting plan shown in fig.5 and the
notchs radius is shown in table 2.
Specimen No. 1 2 3 4 5 6
Radius, r (mm) 6.25 200 100 50 25 1.5
5. PROCEDURES
a) The polariscope, personal computer, strain gauge, and the digital camera are turned
on b) Image capturing software is started
c) The specimen labeled 1 is then mounted onto the polariscope system and the
loading is set to zero, as seen in the strain gauge.
d) The exposure time, gain, and bias are the set to get the best image meaning that
there is distinct contrast between the background and the specimen
e) A picture, to be used as a reference is captured and contrast in the image is
maximized before being saved on the computer.
f) A gradual tension of 20 N was then added onto specimen 1.
g) A picture of specimen 1 under a tension was captured and saved on the computer
h) Then the tension is gradually increased by another 20 N.
Table 2 Dimension of curves radius for specimen with edge notches
250
25
9 9 10.5
10.5
12.5 12.5r
r
12.5
Fig. 5 Cutting plan for specimen with edge notches
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h) A picture of specimen 1 under a tension of 20 N is captured and contrast in the
image is also maximized. After that the image is saved into the harddisk drive inside
the computer.
i) Further increase by the interval of 20 N is made until we reach 100 N. After each
increment of 20 N, the picture is taken and image is contrasted before we save the
image into the harddisk drive installed on the computer.
1. After the picture of specimen 1 under 100 N loading has been taken, the specimen is
unloaded and replaced by the next specimen
2. Step c) to j) is repeated for specimen 2 to 15
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6. RESULTS
6.1 Specimens with notches
Table 3 CALCULATION OF K VALUE FOR SPECIMENWITH EDGE NOTCHES
Ave. Stress = p / ( w * t ) Max. Stress = ( N * f ) / t
Max. Stress Constants: f = 7.008 N / fringe. mm K =Aver. Stress
Specimen No: 1
Load (p) N
Width (w)mm
Thickness (t)mm
Fringe Number (N)
Ave. StressMpa
Max. StressMpa
K
0.00 12.50 6.35 0.00 0.000 0.000 N.A
196.20 12.50 6.35 1.50 2.472 1.655 0.67392.40 12.50 6.35 5.55 4.944 6.125 1.24588.60 12.50 6.35 7.70 7.415 8.498 1.15784.80 12.50 6.35 8.30 9.887 9.160 0.93981.00 12.50 6.35 10.75 12.359 11.864 0.96
0.99
Specimen No: 2
Load (p) N
Width (w)mm
Thickness (t)mm
Fringe Number (N)
Ave. StressMpa
Max. StressMpa
K
0.00 12.50 6.35 0.00 0.000 0.000 N.A196.20 12.50 6.35 2.55 2.472 2.814 1.14392.40 12.50 6.35 5.10 4.944 5.628 1.14588.60 12.50 6.35 7.60 7.415 8.388 1.13784.80 12.50 6.35 10.40 9.887 11.478 1.16981.00 12.50 6.35 12.73 12.359 14.049 1.14
1.14
Specimen No: 3
Load (p) N Width (w)mm Thickness (t)mm Fringe Number (N) Ave. StressMpa Max. StressMpa K
0.00 12.50 6.35 0.00 0.000 0.000 N.A196.20 12.50 6.35 2.00 2.472 2.207 0.89392.40 12.50 6.35 4.00 4.944 4.414 0.89588.60 12.50 6.35 7.00 7.415 7.725 1.04784.80 12.50 6.35 9.00 9.887 9.933 1.00981.00 12.50 6.35 12.00 12.359 13.243 1.07
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0.98Specimen No: 4
Load (p) N
Width (w)mm
Thickness (t)mm
Fringe Number (N)
Ave. StressMpa
Max. StressMpa
K
0.00 12.50 6.35 0.00 0.000 0.000 N.A196.20 12.50 6.35 2.00 2.472 2.207 0.89392.40 12.50 6.35 4.00 4.944 4.414 0.89588.60 12.50 6.35 6.00 7.415 6.622 0.89784.80 12.50 6.35 8.00 9.887 8.829 0.89981.00 12.50 6.35 10.00 12.359 11.036 0.89
0.89
Specimen No: 5
Load (p) N
Width (w)mm
Thickness (t)mm
Fringe Number (N)
Ave. StressMpa
Max. StressMpa
K
0.00 12.50 6.35 0.00 0.000 0.000 N.A196.20 12.50 6.35 2.00 2.472 2.207 0.89392.40 12.50 6.35 4.00 4.944 4.414 0.89588.60 12.50 6.35 6.00 7.415 6.622 0.89784.80 12.50 6.35 8.00 9.887 8.829 0.89981.00 12.50 6.35 10.00 12.359 11.036 0.89
0.89
Specimen No: 6
Load (p) N
Width (w)mm
Thickness (t)mm
Fringe Number (N)
Ave. StressMpa
Max. StressMpa
K
0.00 12.50 6.35 0.00 0.000 0.000 N.A196.20 12.50 6.35 2.00 2.472 2.207 0.89392.40 12.50 6.35 5.00 4.944 5.518 1.12588.60 12.50 6.35 6.00 7.415 6.622 0.89784.80 12.50 6.35 9.00 9.887 9.933 1.00981.00 12.50 6.35 11.00 12.359 12.140 0.98
0.98
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Table 4 CALCULATION OF K VALUE FOR SPECIMENWITH MULTIPLE EDGE NOTCHES AND CRACKS
Specimen No: 7 (multiple notches)
Load (p) N
Width (w)mm
Thickness (t)mm
Fringe Number (N)
Ave. StressMpa
Max. StressMpa
K
0.00 12.50 6.35 0.00 0.000 0.000 N.A196.20 12.50 6.35 3.00 2.472 3.311 1.34392.40 12.50 6.35 6.00 4.944 6.622 1.34588.60 12.50 6.35 9.00 7.415 9.933 1.34784.80 12.50 6.35 12.00 9.887 13.243 1.34981.00 12.50 6.35 15.00 12.359 16.554 1.34
1.34
Specimen No: 8 ( 5mm cracks )
Load (p) N
Width (w)mm
Thickness(t) mm
Fringe Number (N)
Ave. StressMpa
Max. StressMpa
K
0.00 15.00 6.35 0.00 0.000 0.000 N.A196.20 15.00 6.35 3.00 2.060 3.311 1.61392.40 15.00 6.35 5.00 4.120 5.518 1.34588.60 15.00 6.35 8.00 6.180 8.829 1.43784.80 15.00 6.35 11.00 8.239 12.140 1.47981.00 15.00 6.35 14.00 10.299 15.451 1.50
1.47
Specimen Number: 9 ( 8mm cracks )
Load (p) N
Width (w)mm
Thickness(t) mm
Fringe Number (N)
Ave. StressMpa
Max. StressMpa
K
0.00 9.00 6.35 0.00 0.000 0.000 N.A196.20 9.00 6.35 4.00 3.433 4.414 1.29392.40 9.00 6.35 8.00 6.866 8.829 1.29588.60 9.00 6.35 12.00 10.299 13.243 1.29784.80 9.00 6.35 15.00 13.732 16.554 1.21981.00 9.00 6.35 20.00 17.165 22.072 1.29
1.27
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Table 5 CALCULATION OF K VALUE FOR SPECIMENWITH A CIRCULAR HOLE
Specimen No: 10
Load (p) N
Width (w)mm
Thickness(t) mm
Fringe Number (N)
Ave. StressMpa
Max. StressMpa
K
0.00 23.00 6.35 0.00 0.000 0.000 N.A196.20 23.00 6.35 1.00 1.343 1.104 0.82392.40 23.00 6.35 2.00 2.687 2.207 0.82588.60 23.00 6.35 3.00 4.030 3.311 0.82784.80 23.00 6.35 3.00 5.374 3.311 0.62981.00 23.00 6.35 4.00 6.717 4.414 0.66
0.75
Specimen No: 11
Load (p) N
Width (w)mm
Thickness(t) mm
Fringe Number (N)
Ave. StressMpa
Max. StressMpa
K
0.00 22.00 6.35 0.00 0.000 0.000 N.A196.20 22.00 6.35 1.00 1.404 1.104 0.79392.40 22.00 6.35 3.30 2.809 3.642 1.30588.60 22.00 6.35 5.00 4.213 5.518 1.31784.80 22.00 6.35 6.20 5.618 6.842 1.22981.00 22.00 6.35 7.60 7.022 8.388 1.19
1.16
Specimen No: 12
Load (p) N
Width (w)mm
Thickness(t) mm
Fringe Number (N)
Ave. StressMpa
Max. StressMpa
K
0.00 21.00 6.35 0.00 0.000 0.000 N.A196.20 21.00 6.35 2.00 1.471 2.207 1.50392.40 21.00 6.35 4.40 2.943 4.856 1.65588.60 21.00 6.35 6.00 4.414 6.622 1.50784.80 21.00 6.35 7.80 5.885 8.608 1.46981.00 21.00 6.35 8.80 7.357 9.712 1.32
1.49
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Specimen No: 13
Load (p) N
Width (w)mm
Thickness(t) mm
Fringe Number (N)
Ave. StressMpa
Max. StressMpa
K
0.00 20.00 6.35 0.00 0.000 0.000 N.A196.20 20.00 6.35 2.00 1.545 2.207 1.43392.40 20.00 6.35 4.00 3.090 4.414 1.43588.60 20.00 6.35 6.00 4.635 6.622 1.43784.80 20.00 6.35 8.00 6.180 8.829 1.43981.00 20.00 6.35 10.00 7.724 11.036 1.43
1.43
Specimen No: 14
Load(p) / N
Width(w) / mm
Thickness(t) / mm
Fringe Number (N)
Aver. Stress/ Mpa
Max. Stress /Mpa
K
0.00 18.00 6.35 0.00 0.000 0.000 N.A196.20 18.00 6.35 2.40 1.717 2.649 1.54392.40 18.00 6.35 6.00 3.433 6.622 1.93588.60 18.00 6.35 0 5.150 0.000 0.00784.80 18.00 6.35 9.00 6.866 9.933 1.45981.00 18.00 6.35 12.00 8.583 13.243 1.54
1.62
Specimen No: 15
Load (p) N Width (w)mm Thickness(t) mm Fringe Number (N) Ave. StressMpa Max. StressMpa K
0.00 15.00 6.35 0.00 0.000 0.000 N.A196.20 15.00 6.35 3.00 2.060 3.311 1.61392.40 15.00 6.35 5.55 4.120 6.125 1.49588.60 15.00 6.35 7.70 6.180 8.498 1.38784.80 15.00 6.35 8.30 8.239 9.160 1.11981.00 15.00 6.35 10.75 10.299 11.864 1.15
1.35
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7. DISCUSSION
Specimens with notches and holesUpon obtaining the experimental values of the each of the various stress
concentration (K) for individual specimens, it was thus necessary for us to compare the
values with their corresponding theoretical ones. For this matter, the book Stress
Concentration Factors gave quite a number of examples of specimens of different
geometrical shape and situations, with their corresponding K values.
For specimen 1, we use standard stress concentration chart for opposite U notches
(appendix A). While for specimen 2, standard stress concentration chart for semi circular
edge notches are applied (appendix B). The result for curve type edge notches found in
specimen 3 to 6 will be compared with standard K value chart found in appendix C.
Lastly, for holes standard chart found in appendix D is used.However, in our case, for specimens Nos. 7, 8, and 9 (the multiple notches and the
hand cut cracks), their respective K values were not available in the stress concentration
handbook. Thus, for those three cases, we will try to investigate the significance of the K
value.
For the other specimens with corresponding standard stress concentration value
(specimen 1-6, 10-15), we list the comparison in the following tables list the theoretical
values of K:-
Comparison of K value for specimens 1 to 6 (with notches)
Specimen 1 2 3 4 5 6
Width D 25 25 25 25 25 25
r 1.5 6.25 25 50 100 200
r/d 0.12 - 2 4 8 16
2r/D - 0.5 - - - -
D/d 2 - 2 2 2 2K (theoretical) 2.82 1.62 1.16 1.08 1.025 1.02
K (experimental) 0.99 1.14 0.99 0.89 0.89 0.98
Error -64.9% -29.6% -14.6% -17.6% -0.13% -0.04%
Table 6 Comparison of theoretical data with experimental data for specimen 1 to 6
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Note: We used the line for D/d= to determine the K value for specimen 3-6, line for D/d=2 should lie
between the line for D/d= and D/d =1.10
Comparison of K value for specimens 10 to 15 (with a circular hole)
Specimen 10 11 12 13 14 15
Width w 25 25 25 25 25 25
of hole 2 3 4 5 7 10
A/w 0.08 0.12 0.16 0.2 0.28 0.4
K (theoretical) 2.76 2.68 2.60 2.52 2.38 2.24
K (experimental) 0.75 1.16 1.49 1.43 1.62 1.35
Error -72.8% -56.7% -42.7% -43.2% -31.9% -39.7%
Worth noticing before we proceed with the discussion is that not all specimensfringe numbers are extrapolated using methods shown in appendix E. The specimens
using the extrapolation are specimen 1 to 2 and 10 to 15. As for specimen 3 to 6, we
observe the propagation of the fringe number and directly counted the fringe number as
we apply the load to the specimens. The very reasons for not applying extrapolation
method for specimen 3 to 6 is the lack of fringe (only 1 to 2 fringes) to get reasonable
accuracy result from the picture as shown in fig. 6.
Table 7 C omparison of theoretical data with experimental data for specimen 10 to 15
Specimen Specimen
Fig. 6 Fringes appearance for specimen 3 and 4 under 40 kg of applied load
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Specimen 1(40 kg load)
Specimen 2(40 kg load)
Fig. 7 Fringes appearance for specimen 1 and 2 under 40 kg of applied load
We notice that there is large errors for specimen 1 and 2. The discrepancies are
due to the error in extrapolation performed. For specimen 1, it could be seen from fig. 7 in
the next page that fringes near the tip of the notches are very closely propagated. Thus it
is very difficult to assign a fringe number to a particular fringe.
Nevertheless, it could be observed that though there are discrepancies when comparing
the experimental values of the stress concentration obtained for specimen 3 to 6 and the
theoretical ones found in the handbook (table 3), the errors are acceptable taking note that the
theoretical K value is taken at D/d= , while the actual D/d=2. Thus, we are bound to get
lower value of K as the value of K decrease as D/d decrease (Appendix C).
Another large discrepancies are very evidential in the case of the specimens with the
holes with them. Stress concentration factors that weve have obtained are by experiment
using photo elasticity, and by mathematical calculation. It should emphasized that when
the experiment work is conducted, sufficient precision are needed in order to have an
excellent agreement with the well-established mathematical stress concentration factors.
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There could be various reasons as to the different values that was obtained; the most
significant one being the number of fringe order (N) of the specimens under the tensile
load.
The number of fringe order for some of the specimens with a circular holes were
not clearly distinguished and a difficulty arise when we try to allocate the fringe number
to the fringes, thus error in the computation of the respective experimental values of K.
Especially in the using of extrapolation method to estimate the N; this is a major reason as
to why there are such a different in the values of the experimental and theoretical K.
Another reason is the presence of residual stress resulted from machining process.
The residual stress is seen as a white region around the hole in figure 8. The residual
stress acts a hindrance for the elastic deformation occurring during the tensile loading.
We lose one image of specimen 14 with 60 kg loading. Nonetheless, the values of
K for four other loadings nearly correspond to one another. Thus, the value of K obtained
experimentally will not be affected at all since the average value is taken between the four
loadings.
Specimen 10(no load)
Specimen 11(no load)
Fig. 8 Fringes appearances for specimen 10 and 11 without load
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Specimen with multiple notches and cracks
Specimen 7 and specimen 1 have the same notch radius. And it is clearly seen
from figure 9 that the stress distribution is pretty much different with zero stress found in
the part of specimen that separate two notches in one row. In multiple notches case, the
flow is smoother and thus the concentration factor is supposed to decrease. However,
due to large error in the calculation for specimen 1 as explained some paragraph before,
we get a higher K value for specimen 7 (multiple notches) which is 1.34 compared to the
experimental value of K in specimen 1 which is only 0.99. However, if we compare the
value of K for specimen 7 obtained in this experiment with the theoretical value of K for
specimen 1, it is obvious that the K value for multiple notches is smaller (1.34 compared
to 2.82).
We see from the crack cases as shown in figure 10 found in the next page that
there is avery different type of stress distribution compared to the notches type. We also clearly
cannot apply stress concentration factor to this two cases due to different stress
distribution in the vicinity of the crack compared with the stress distribution in the tip of
general notches.
Specimen 1(40 kg load)
Specimen 7(40 kg load)
Fig. 9 Comparison between flow of stress in one U-shaped notch and multiple
U-shaped notches
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The two high stress regions shown in figure 10 indicate that there are two very
small notches or even sharp tip in the corner of the hand-sawed crack. The extrapolation
and manual counting of fringe number cannot be performed at all in this case since there
is a very intense and closely spaced fringe number at the tip of the crack.
8. C
ON
CL
USI
ON
O
ur grou
p
has
foun
d
that
diff
eren
t type of specimens cut from photoelastic specimen will give us different fringe numbers
under different load. Each specimen with surface discontinuities will produce a different
fringe patterns corresponding to its radius and width (be it a notch or holes).
We can observe that as the load is increased, the fringe number is almost
impossible to distinguish by the naked eye. As a result, some of the experimental results
and the theoretical results differ by a large value. Though for specimen 3 to 6 which is U-
shaped edge notches case, we get an acceptable errors. And thus, we can conclude that
the K value found in standard stress concentration factor graph is verified. The presence
of residual stress induced in the machining process has accounted largely for our errors in
specimen with a circular hole.
Specimen 8(40 kg load)
Fig. 10 Fringe appearance of specimen 8 and 9 with 40 kg loading
Specimen 9(40 kg load)
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From our result and discussion, we can also conclude that there is a reduction in
the stress concentration in the case of multiple notches compared to a single notch of the
same type due to smoother flow of stress.
For the crack which is produced from hand sawing, we need to apply stress
intensity factor, not stress concentration factor due to obvious difference in stress
distribution found between general notch and our cracks.
With the above mentioned, we would like to conclude that the stress
concentration values of each specimen is dependent on the fringe numbers. We can also
say that the cracks can be said to be with radius very large. Extrapolation method could
only be performed if there are enough fringes available in the vicinity of the crack (3or 4
at least). And if there are too many fringes, extrapolation could not be performed, too.
9. RECOMMENDATIONS
Our group suggested some changes to be made in future investigation of this
type of experiments.
Firstly, magnification of picture for analysis should be performed carefully. It
would be better to get the counting of fringe number done during the application of
loading. Perhaps, some sort of magnifying devices should be installed in the polariscope
to help the counting process.
Secondly, decrease in interval load especially for specimens that are expected togive us closely packed fringes. Increment of load by 10 or 5 kg is recommended instead
of 20 kg as what we did.
Thirdly, decrease in the range of load should also be carried out. Since large
loading also results in a closely arranged fringes which in turn cause difficulty in fringe
counting.
Lastly, careful machining must be carried out. If large residual stress is found,
annealing is recommended. While for the case of crack investigation, sharp and thinner
cutter should be use to produce sharper and more valid type of crack.
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10. REFERENCES
Kuske, Albrecht & Robertson, George (1977). Photoelastic Stress Analysis. Bristol: John
Wiley & Sons.
Sih, G.C. (Ed.). (1981). Experimental Evaluation of Stress Concentration and Intensity
Factors. The Hague: Martinus Nijhoff Publishers.Beer, F. P. & Johnston, Jr., E. R. (1992). Mechanics of Materials. (2 nd ed.). Singapore:
McGraw-Hill.
Peterson, R. E. (1974). Stress Concentration Factors. Wiley-Interscience Publication
Callister, Jr., Wiiliam D. (1997). Materials Science and Engineering An Introduction. (4 th
Ed.). USA: John Wiley & Sons, Inc.