i

AN EXPERIMENTAL INVESTIGATION OFSTRESS CONCENTRATION FACTOR

ABSTRACT

In this project, investigation of stress concentration factor is carried out. Somespecimens 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 ofmultiple 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 ENGINEERINGNANYANG TECHNOLOGICAL UNIVERSITY

2000

i

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 A1Appendix B B1Appendix C C1Appendix D D1Appendix E E1

i

Acknowledgements

Our Groups wishes to express our sincere appreciation to the following:

1. A/P Anand Krishna Asundi, the supervisor of this project, for hisinvaluable 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 ourspecimens 10 to 15

i

LIST OF FIGURES

Page

Fig. 1 Location of zero fringe order 2

Fig. 2 Stress distribution in the vicinity of a circular hole 4

Fig. 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 curve’s 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

i

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 passed

through 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

i

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 examine

whether 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.

i

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.

tw

P

A

P

*==σ (1)

σ = stress (Pa) w = width of the member (m)

P = applied load (N) t = thickness of the member (m)

A = cross sectional area (m2)

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

fN

2

*max

στ = (2)

while2

minmaxmax

σστ

−= (3)

N = fringe order

σmax σave

Fig. 2 Stress distribution in the vicinity of a circular hole

i

fσ = material fringe value (kPa/fringe/mm)

τmax = maximum shearing stress (kPa)

σmax = maximum inline principle stress (kPa)

σmin = 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 σστ =−= (4)

While (2) will become

t

fN σσ*

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 (σave) (fig.2)

defined in (1).

Stress concentration factor is then defined as

ave

Kσσ 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

σ1=σmax

σ2=0

σ1=σmax

i

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), σ applied, and the size of the

crack.

aYK πσ= (7)

We see from the equation that K value will increase as σ value increase and if σ

reaches σf (fracture stress) value, the K will reach KC (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 10”x20”

b. Item Code 17022

i

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 φ is tabulated in table 1.

Specimen No. 10 11 12 13 14 15

Diameter of hole , φ (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

φ

9

125

12.5

10.5

10.5

12.5 12.5

i

The specimens with notches have the cutting plan shown in fig.5 and the

notch’s 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 curve’s radius for specimen with edge notches

250

25

9 910.5

10.5

12.5 12.5

r

r

12.5

Fig. 5 Cutting plan for specimen with edge notches

i

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

i

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 * fσ ) / t

Max. Stress Constants: f = 7.008 N / fringe. mmK =

Aver. Stress

Specimen No: 1

Load (p)N

Width (w)mm

Thickness (t)mm

FringeNumber (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 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

FringeNumber (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

FringeNumber (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

i

0.98Specimen No: 4

Load (p)N

Width (w)mm

Thickness (t)mm

FringeNumber (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

FringeNumber (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

FringeNumber (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

i

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

FringeNumber (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

i

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

i

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

i

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 2

K (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

i

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

Ø 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 specimens’

fringe 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 Comparison of theoretical data with experimental data for specimen 10 to 15

Specimen3

Specimen4

Fig. 6 Fringes appearance for specimen 3 and 4 under 40 kg of applied load

i

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 we’ve 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.

i

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

i

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 a

very 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 multipleU-shaped notches

i

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)

i

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 to

give 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.

i

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. (2nd 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. (4th

Ed.). USA: John Wiley & Sons, Inc.