A REVIEW OF THE ACCURACY OF STRESS
CONCENTRATION FACTORS
By
MATTHEW L. MAUK
Bachelor of Science in Mechanical Engineering
Oklahoma State University
Stillwater, Oklahoma
1999
Submitted to the Faculty of the
Graduate College of the
Oklahoma State University
in partial fulfillment of
the requirements for
the Degree of
MASTER OF SCIENCE
December, 2010
ii
A REVIEW OF THE ACCURACY OF STRESS
CONCENTRATION FACTORS
Thesis Approved:
Dr. James K. Good
Thesis Adviser
Dr. Ronald D. Delahoussaye
Dr. Gary E. Young
Dr. Mark E. Payton
Dean of the Graduate College
iii
ACKNOWLEDGMENTS
I would like to thank Dr. James K. Good, for his support and sponsorship of this thesis, for which
it would not have been possible. I would also like to thank Dr. Ronald D. Delahoussaye and Dr.
Gary E. Young for their time and contributions. I would be remiss if I didn’t thank two very
important people in my life, my parents. They have, without question, sacrificed much in helping
me achieve my goals throughout life.
iv
TABLE OF CONTENTS
Chapter Page
I. INTRODUCTION ......................................................................................................1
II. LITERATURE SURVEY .........................................................................................2
2.1 Literature ............................................................................................................2
2.2 Research Objective ............................................................................................3
III. THE FINITE ELEMENT METHOD ATTACK .....................................................7
3.1 Introduction ........................................................................................................7
3.2 Software .............................................................................................................7
3.3 Setup ..................................................................................................................8
3.4 Mesh Convergence.............................................................................................9
3.5 Results ..............................................................................................................10
IV. THE PHOTOELASTIC METHOD ATTACK .....................................................14
4.1 Introduction ......................................................................................................14
4.2 Setup ................................................................................................................14
4.3 Samples and Geometry ....................................................................................15
4.4 Results ..............................................................................................................17
4.5 Errors................................................................................................................18
V. COMPARISON OF RESULTS .............................................................................20
5.1 Transverse Hole ...............................................................................................20
5.2 Semicircular Edge Notch .................................................................................21
5.3 Waist ................................................................................................................22
VI. CONCLUSIONS ...................................................................................................24
v
Chapter Page
REFERENCES ............................................................................................................26
APPENDICES .............................................................................................................28
A1 Mesh Convergence ...........................................................................................29
A2 Contour Plots....................................................................................................32
A3 Machining Process ...........................................................................................41
vi
LIST OF TABLES
Table Page
3.1 Transverse Hole Analysis Results and Gross Dimensions ..........................11
3.2 Semicircular Edge Notch Results and Gross Dimensions ...........................12
3.3 Waist Results and Gross Dimensions ..........................................................13
4.1 Parameters for Specific Samples Geometries .............................................17
4.2 Transverse Hole Photoelastic Results and Gross Dimensions ....................17
4.3 Semicircular Edge Notch Photoelastic Results and Gross Dimensions ......18
4.4 Waist Photoelastic Results and Gross Dimensions .....................................18
vii
LIST OF FIGURES
Figure Page
2.1 Semicircular Edge Notch Reference Comparisons .......................................4
2.2 Transverse Hole Reference Comparisons .....................................................5
2.3 Waist Reference Comparisons ......................................................................6
3.1 Mesh Convergence Graph .............................................................................9
3.2 Transverse Hole Symmetry and Loading ....................................................11
3.3 Transverse Hole Dimensions .......................................................................11
3.4 Semicircular Edge Notch Symmetry and Loading ......................................12
3.5 Semicircular Edge Notch Dimensions ........................................................12
3.6 Waist Symmetry and Loading .....................................................................13
3.7 Waist Dimensions ........................................................................................13
4.1 General Specimen Dimension used for (a) Transverse Hole (b) Semicircular
Edge Notch (c) Waist .........................................................................16
5.1 Transverse Hole Results Comparison .........................................................21
5.2 Semicircular Edge Notch Results Comparison ...........................................22
5.3 Waist Results Comparison ..........................................................................23
1
CHAPTER I
INTRODUCTION
Throughout the course of the design process engineers must be vigilant about their design to be
sure their designs don’t fail under normal conditions of operation. During the design of their
components they may encounter similar geometries on a regular basis. For instance, holes drilled
in plates in tension. This type of geometry recurs on a regular basis and must be analyzed for
potential failure. Because of the regularity of this geometry, stress concentration tables have been
established to help speed the analysis of the geometry. No need to “reinvent the wheel”.
Engineering students are taught about this early in their education and allowed to use it
throughout the course of their future careers. Reference books have been written that contain this
information to make it readily available for the engineer or student. As with any reference data, it
should be verified to be applicable to the application at hand, but due to deadlines, assumptions
may be made that the data in the reference books is correct if time and resources are not available
to verify the established data. Thus if the published data is incorrect it could lead to an
unacceptable design flaw that may fail at the most inopportune time.
2
CHAPTER II
LITERATURE SURVEY
2.1 Literature
There are several sources for stress concentration values. Two general references are “Peterson’s
Stress Concentration Factors” (Peterson’s) [12] and “Mechanical Engineering Design” (Shigley)
[16]. These sources are readily available and used widely throughout industry and collegiate
environments.
Mechanical Engineering Design is a textbook used by many Engineering schools for the
instruction of machine design and the analysis needed to assure that those design meet the
intended requirements. This text is used to introduce students to a few general forms of stress
concentrations. Students can then apply their skills to other forms of geometry in their field of
expertise.
As with any design there are certain features that repeat themselves throughout the design. For
example, fillets are used in corners to reduce the stresses in a component with two intersecting
structures. Grooves are cut into shafts to allow for O-rings to seal fluids from leaking past them
into the environment were they may cause harm to humans or the environment. With the
recurrence of these features, a graph can be selected from a reference [12, 16] to determine if a
3
localized stress in the area of a stress concentration will exceed the material capabilities in that
application.
Peterson’s Stress Concentration Factors is a reference book for Engineers and students concerned
with increasing stresses at the intersection of changing geometries. As opposed to Shigley it
delves deeper into the theory of stress concentrations. It also provides a much wider field of
geometries to reference for structural analysis.
2.2 Research Objective
In some cases the stress concentrations provided by these references deviate. This report looks at
the deviation in the plots for the semicircular edge notch. Two other cases are also compared, the
transverse hole and the waist geometry.
The semicircular edge notch data from Shigley and Peterson’s are compared in Figure 2.1. The
figure shows that the values for Shigley are different than those of Peterson’s. The Shigley data
taken from [13] was based on theoretical calculations of “Theory of Notch Stress” [11], which
were derived from the theory of elasticity. The Peterson’s data was based on calculations by [9]
and [10].
4
Figure 2.1 Semicircular Edge Notch Reference Comparisons
For the transverse hole configuration the comparison of the published sources shows good
correlation of the data as seen in Figure 2.2. These results were based on photoelastic models.
For Shigley’s data source [15], the photoelastic measurements data was taken from [6] and [17].
The Peterson’s data was based on theoretical calculations of a “successive approximation” for a
tension problem [8].
5
Figure 2.2 Transverse Hole Reference Comparisons
As seen in Figure 2.3 of the waist comparison, published sources are showing difference between
their plotted values. The Shigley values were based on [14] whose values were taken from the
photoelastic results of [6] and [7]. The Peterson’s values were also based on photoelastic results
[6] but included other refinements from [2] and [18], to improve the values for Kt as the original
data was showing lower values [6].
6
Figure 2.3 Waist Reference Comparisons
The three geometries are compared through different means of measurement. Photoelastic stress
analysis is used for one type of analysis. The finite element method is used for the second type of
analysis. These results will then be compared to the graphs in Shigley and Peterson’s.
7
CHAPTER III
THE FINITE ELEMENT METHOD ATTACK
3.1 Introduction
The first method of attack on the stress concentration factors is Finite Element Analysis (FEA).
FEA has become a common tool for many engineers in the past decade. Software has been
written with them in mind to help further the designs they produce. It is a viable method of
determining stress concentrations in complex geometries as well as simple geometries as is the
case here. Abaqus was chosen as the software to perform these analyses. It is widely used in the
automotive industry as well as other fields.
3.2 Software
To begin with, the geometry for analysis was modeled directly in the Abaqus 6.9.1 software
itself. The graphical user interface (GUI), has a modest capability for modeling different
geometries. It also has the capability of importing geometry directly from computer aided design
(CAD) software as well. This wasn’t necessary in this case as the geometry was simple and easy
to create in the provided GUI. The geometry was modeled as a 2D surface. For the purposes of
this report the measurements of the photoelastic models was used to better correlate between
physical measurements and FEA results. The surface was then meshed using the CPS8R element
[1], which is an 8 node plane stress element with reduced integration. The thickness of the
geometry was controlled in the material section property.
8
3.3 Setup
For the given geometries, two of the three were modeled with quarter symmetry, transverse hole
and edge notch. In the third case the waist samples were modeled using half symmetry. Then the
material properties were applied to the model. For the PS-1 material the elastic modulus used for
all samples was E=360,000 psi and a poison ratio of υ=0.38 [4].
The applied load was chosen based on keeping the maximum stress at the stress concentration
below the yield point of the PS-1 material. The properties given [4] did not include yield
strength. Typically yield strength is determined from a 0.2% strain offset of the elastic modulus
and its intersect with the stress/strain plot. As an engineering judgment, the calculation of the
yield strength will be calculated as 0.2% of the Elastic modulus (3.1). This value will be
conservatively lower than the actual yield strength of the material.
. %
, . % (3.1)
The initial testing load was set at 10 lbf. Based on this load and the smallest cross sectional area
of sample 3 of the transverse hole, the maximum allowable stress concentration factor was
calculated (3.3).
,
!
."##” . %”
100 ()* (3.2)
+,,
-.
/012,234
56 789
789
7.2 (3.3)
From the FEA results, all the samples show that the maximum stress will be below yield as all the
calculated stress concentration values are below the maximum of 7.2. Finally, the load P as it is
applied to the FEA model is uniformly distributed across the edge of the model, Figure 3.2.
9
3.4 Mesh convergence
For mesh convergence, the geometries were analyzed with several different mesh sizes to
determine the best mesh size to meet convergence of the results at the stress concentration.
Looking at the transverse hole as an example, the mesh size was set to a global value of .050 in
for element size. The peak stress S11 was documented and then the mesh was refined in the area
of the geometry in question. The graph below shows the peak stress results at the edge of the
hole with respect to the different mesh densities. The percent difference between the global mesh
size of .01 in and the combined mesh size of .01 in global / .005 in local, resulted in a 0.27%
increase in recorded stress.
Figure 3.1 Mesh Convergence Graph.
10
The final mesh density selected for all analysis was a global value of .01 in and a local element
size of .005 in. This resulted in an overall change in peak stress between a global of .01 in and a
refined local of .005 in of 0.27%. Mesh and contour plots are located in Appendix A1. For all
models an element type of 8-node biquadratic plane stress quadrilateral, reduced integration
(CPS8R) was used [1]. The CPS8R element has two degrees of freedom in the X and Y
directions at each node.
3.5 Results
The first sample analyzed was the transverse hole configuration. Three models for the different
geometries given in Table 3.1 were developed. Symmetry was used in both the X and Y direction
to simplify the model. The boundary conditions were set to symmetry on X and Y with a load P
in the X direction, see Figure 3.2. Due to symmetry in the y direction the load was reduced in
half to 5 lbf. The resulting values from the analysis are shown in Table 3.1. All contour plots are
located in Appendix A2.
The stress concentration values Kt (3.4) are then calculated from the σmax at the hole and divided
by the net area stress σn (3.5).
(3.4)
(3.5)
n
Kt
σ
σ
= max
H d t
Force
net area
Force
n _ ( − ) *
σ = =
11
Figure 3.2 Transverse Hole Symmetry and Loading
Figure 3.3 Transverse Hole Dimensions
Table 3.1 Transverse Hole Analysis Results and Gross Dimensions
The next sample analyzed was the Semicircular Edge Notch configuration. The 3 different sized
models were input into the software. The gross dimensions are given in Table 3.2. Symmetry
was used in both the X and Y direction to simplify the model. The boundary conditions were set
to symmetry on X and Y with a load P in the X direction, see Figure 3.4. Due to symmetry in the
y direction the load was reduced in half to 5 lbf. The resulting values from the analysis are shown
in Table 3.2. All contour plots are located in Appendix A2.
12
The stress concentration values Kt for the edge notch samples are calculated similar to the
transverse hole with the exception of the σn which is calculated from equation (3.6).
(3.6)
Figure 3.4 Semicircular Edge Notch Symmetry and Loading
Figure 3.5 Semicircular Edge Notch Dimensions
Table 3.2 Semicircular Edge Notch Results and Gross Dimensions
Finally the last sample analyzed was the Waist configuration. The 3 different sized models were
input into the software. The gross dimensions of the models are given in Table 3.3. Symmetry
was used in the Y direction to simplify the model. The boundary conditions were set to
H r t
Force
net area
Force
n _ ( − 2 ) *
σ = =
13
symmetry on Y with a load P in the X direction, see Figure 3.6. Due to symmetry in the y
direction the load was reduced in half to 5 lbf. The resulting values from the analysis are shown
in Table 3.3. All contour plots are located in Appendix A2.
The stress concentration Kt (3.1) of the waist configuration is calculated straight from the
measured σmax and σn. Where σn is measured in the field of width d.
Figure 3.6 Waist Symmetry and Loading
Figure 3.7 Waist Dimensions
Table 3.3 Waist Results and Gross Dimensions
14
CHAPTER IV
THE PHOTOELASTIC METHOD ATTACK
4.1 Introduction
Photoelastic stress analysis is a unique form of structural analysis. It involves the use of light
passing through a plastic material that exhibits temporary double refraction, “optically isotropic
when free of stress but becomes optically anisotropic and display characteristics similar to
crystals when they are stressed” [3]. The light from the plastic is then passed through polarized
plates. The resulting vision is one of a fringe pattern that describes the stresses in the material it
is passing through. Decades ago these techniques were used in place of cumbersome FEA
software and even before computers were available to analyze 2 dimensional models. It is
considered more of an analog structural analysis versus today’s high end computer based digital
analysis.
4.2 Setup
For the photoelastic analysis, 9 different samples were produced to compare with other sources of
data. The sample were broken into 3 groups, transverse hole, semicircular edge notch and waist.
Within each group three different sized samples were produced to cover a broad range of
geometry ratios.
15
Each sample was placed in an apparatus to apply a predefined load of 10 lbf. A load transducer
was used to verify the correct force was applied. A polariscope was used to measure the stress
level in the reduced width field of the sample and the peak stress at the stress concentration.
From here the stress concentration factor Kt (4.1) can be calculated [12]. Kt is max normal stress
divided by normal stress based on net area.
(4.1)
Each sample was measured 5 times consecutively. Data was then averaged to obtain the results.
The peak stress level was measured on both sides of the part and then averaged together to get a
mean value. This was done to average out any bias from one side of the part to the other based on
any non-symmetry of the machining.
4.3 Samples and Geometry
The machining methods are given in detail in Appendix A3.
The material used for the machined samples was a high-modulus polymer PS-1 [4]. The material
was chosen for its ease of machining. The PS-1 properties are; Elastic Modulus E=360,000 psi,
Poisson’s ratio υ=0.38 and for these samples a general thickness of 0.120 inches with a tolerance
of +/-0.002 inches. The sheets were received with a silver backing and a protective paper
covering on the opposing side.
Figure 4.1 shows the base configuration of the specimens which was specified to be 10 inches by
1.500 inches, with two .375 inch holes 8.5 inches on center. The thickness was a predefined
value based on the purchased material. Each sheet had its own specific thickness (t). The
samples were machined in accordance with the machining process in Appendix A3. The final
sample configuration is shown in Table 4.1, 9 samples in all.
n
Kt
σ
σ
= max
16
(a)
(b)
(c)
Figure 4.1 General Specimen Dimension used for (a) Transverse Hole (b) Semicircular Edge
Notch (c) Waist
17
Table 4.1 Parameters for Specific Sample Geometries
4.4 Results
The first sample analyzed was the transverse hole configuration. The samples were measured
with digital calipers in order to confirm their actual size listed in Table 4.2. Then measurements
were taken with the polariscope to obtain σmax,avg, which is read from the outer edge of the hole.
From here σn (4.2) was calculated to determining Kt.
(4.2)
Table 4.2 Transverse Hole Photoelastic Results and Gross Dimensions
H d t
Force
net area
Force
n _ ( − ) *
σ = =
18
For the semicircular edge notch samples, Kt is calculated very similar to that of the transverse
hole. σmax,avg is read from the inner edge of the notch and σn is calculated from the net area
(4.3). Results are shown in Table 4.3
(4.3)
Table 4.3 Semicircular Edge Notch Photoelastic Results and Gross Dimensions
Finally the results for the waist samples are given in Table 4.4. This case gives σn,avg as a
measured value as opposed to being calculated like the other samples. The reduced section of
width d is the nominal net area needed for the calculations of Kt.
Table 4.4 Waist Photoelastic Results and Gross Dimensions
4.5 Errors
As with any project that requires human interaction or machines designed and built by humans
there will be errors in the research results. Photoelastic analysis is no different and the following
errors were identified.
H r t
Force
net area
Force
n _ ( − 2 ) *
σ = =
19
The first errors were introduced during the machining process. The samples were specified to
have a predefined size as seen in Figure 4.1 with dimension from Table 4.1. Due to wear in the
machining equipment and error on the machinist’s part, the specimens’ final sizes were not what
was initial specified. More care should be taken by the machinist to minimize human error and
newer equipment with Computer Numerical Control is recommended.
For these samples the final sample dimension can be seen in Tables 4.2, 4.3, 4.4. These
dimensions were taken with a digital caliper with resolution to .001 inches. Some features
presented difficulties to measure, such as the edge notch and the fillet radius in the waist samples.
These features were assumed to be of nominal size. These features could be under or oversized.
To correct this, it is recommended to have the samples measured by a Coordinate Measurement
Machine. These would provide final dimensional data of the samples to minimize any calculated
error to be compared with the measured photoelastic data.
The next place error was introduced into the results was during the stress measurements. Here
there is the potential for equipment error as well as human error. The measurement equipment
was calibrated before the measurements were taken so any error there will be ignored. The
human error deals with the viewing of the fringes and the measurement point of the peak stress.
During the process the operator is adjusting the equipment to the point of termination of the black
fringes at the stress point. The first error here is the point at which the fringe disappears. This
point is open to interpretation by the operator, it was noticed that eye fatigue also played a part in
the difficulty of making the measurements. The next issue was the orientation of the sample.
The sample was being viewed on opposing sides and any rotation of the sample with respect to
the polariscope would show the machined edge directly or make it visible threw the plastic, thus
adding to the difficultly of read the extinction of the black fringe. The standard error was
quantified and listed in Tables 4.2, 4.3, 4.4.
20
CHAPTER V
COMPARISON OF RESULTS
The following graphs compare the values for Kt from the 4 different sources. The data plotted for
Peterson’s and Shigley was interpolated from the graphs located in their respective books. There
is a small amount of error present in these data sets represented here as human error factors into
the interpolations of the graphs. A machinist ruler was used to measure the distance between
major gridlines and interpolate to the closest values.
5.1 Transverse Hole
Figure 5.1 shows the data plots for Peterson’s, Shigley, FEA and photoelastic samples. There is
good correlation between Peterson’s, Shigley and FEA results. At the lower d/H value (sample 1)
there is a small amount of deviation from Shigley, which could be the result of interpolation error.
The data for the photoelastic specimens parallel the results but show a higher Kt values and thus
only provide comparison based on trend. The error bars for the FEA results have not been shown
as the error is much less than the photoelastic models.
21
Figure 5.1 Transverse Hole Results Comparison
5.2 Semicircular Edge Notch
The semicircular edge notch results show good correlation between FEA results and data
interpolated from Peterson’s. Shigley is significantly lower than those two data sets, refer to
Figure 5.2. There is a slight departure between the two at the midpoint, which is a potential error
in the interpretation of the graph from Peterson’s. The photoelastic results do not follow the data
of the other two results and thus cannot be used for comparison.
22
Figure 5.2 Semicircular Edge Notch Results Comparison
5.3 Waist
Finally, the waist data presented in Figure 5.3 shows correlation among Peterson’s, Shigley and
FEA. There is a small amount of shift in the Kt values at each of the data points; this is most
likely a result of how the data was obtained for each source. The photoelastic results again show
some discrepancies compared with the other data sources. At lower values of Kt the photoelastic
models show a general correlation but for the higher r/d value results in a flyer compared to the
rest of the data. Both Peterson’s and Shigley’s data are derived from photoelastic tests results;
the Peterson’s date has been refined as the original values were showing lower results. As well,
some of the difference between these two sources, seen in Figure 5.3, may be the result of
23
interpolation error from the plots. It should be noted that the plot in Shigley is significantly
smaller compared to Peterson’s, thus presenting the possibility for more error in interpolation.
Figure 5.3 Waist Results Comparison
24
CHAPTER VI
CONCLUSIONS
The results for the transverse hole geometry shows good correlation between Peterson’s, Shigley
and FEA results. There is a small deviation in Shigley at the lower d/H, but that may be
attributed to interpolation error of the graph. The photoelastic model has a similar trend, but the
Kt values are higher.
The semicircular edge notch case presents the most significant error between the published
sources. In Figure 5.2 it was shown that Shigley’s Kt values were significantly lower than
Peterson’s and FEA results. This shows a level of error in the theoretical calculations used to
create the plots. Results for the photoelastic samples again show a variation across the board and
cannot be used as a comparison. FEA results correlated well to the data from Peterson’s.
The last configuration looked at was the waist geometry. There are comparable trends among the
plots for Peterson’s, Shigley and FEA results. Though there is a shift in values between all three
sources as seen in Figure 5.3. Both Peterson’s and Shigley values are based from the same
photoelastic results, though Peterson’s has refined the values based on newer sources, as the
original Kt values had been shown to be low. Photoelastic results are markedly better in this
geometry at the lower r/d levels but sample 3 is a flyer, as its Kt result is significantly higher.
25
Through analysis it has been shown that the stress concentration results published in Shigley, for
the semicircular edge notch, are in error compared with Peterson’s and FEA analysis. The stress
concentration values for the transverse hole geometry compare well, Figure 5.1. Finally, the
waist geometry shows good trends but there is a small shift in the values between both published
data and the FEA results. Photoelastic results can provide acceptable results when great care is
taken to produce and measure the samples. This researcher’s photoelastic results show
significantly different values from published data and FEA results in most cases. These errors are
attributed to the accuracy of the machining and the human error in data acquisition.
Photoelastic results for this report have shown that it can be significantly impacted by human
error and that it requires a high level of accuracy and control in machining and reading the fringe
results. Photoelastic measurement can be used to determine stress concentration values but great
care is needed to assure good data. FEA is an excellent alternative that can be used in the future
to calculate Kt values. In the past FEA was sophisticated software that required significant
knowledge to obtain satisfactory results. As the software has matured the necessity of being an
expert in it has subsided. It can provide an acceptable level of results to quantify and correlate
results for stress concentration factors with basic knowledge of the software.
26
REFERENCES
1) Abaqus Analysis User’s Manual, Version 6.9, Vol. 4, 2009.
2) Appl, F. J., and Koerner, D. R., “Stress concentration factors for U-shaped, hyperbolic
and rounded V-shaped notches”, ASME Pap. 69-DE-2, 1969, American Society of
Mechanical Engineers, New York.
3) Dally, J. W., and Riley, W. F., Experimental Stress Analysis, 3rd ed., McGraw Hill 1991.
4) Document No. 11222, “PhotoStress Coating Materials and Adhesives”, Micro-
Measurements, 2010 March 29. <http://www.micro-measurements.com>
5) Document No. 11223, “Instructions for Bonding Flat and Contoured PhotoStress Sheets”,
Micro-Measurements. 2007 December 03. <http://www.micro-measurements.com>
6) Frocht, M. M., “Factors of stress concentration photoelastically determined”, Trans.
ASME Appl. Mech. Sect., 1969, Vol. 57, p. A-67.
7) Frocht, M. M., and Landsberg, D., “Factors of stress concentration in bars with deep
sharp grooves and fillets in torsion”, Trans. ASME Appl. Mech. Sect., 1951, Vol. 73, p.
107.
8) Howland, R. C. J., “On the stresses in the neighborhood of a circular hole in a strip under
tension”, Philos. Trans.R. Soc.(London) Ser. A, 1929-1930, Vol. 229, p. 67.
9) Isida, M., “On the tension of the strip with semi-circular notches”, Trans. Jpn. Soc. Mech.
Eng., 1953, Vol. 19, p. 5.
27
10) Ling, C.-B., “On stress concentration factor in a notched strip”, Trans. ASME Appl.
Mech. Sect., 1968, Vol. 90, p. 833.
11) Neuber, H., 1937, Kerbspannungslehre, Springer, Berlin; translation, 1945, Theory of
Notch Stress, J.W. Edwards Co., Ann Arbor, MI 1946.
12) Pilkey, W. D., and Pilkey, D. F. Peterson’s Stress Concentration Factors, 3rd ed., John
Wiley & Sons, Inc. 2007.
13) Peterson, R. E., Design Factors for Stress Concentration, Machine Design, 1951, Vol. 23,
no. 3, p. 161.
14) Peterson, R. E., Design Factors for Stress Concentration, Machine Design, 1951, Vol. 23,
no. 6, p. 173.
15) Peterson, R. E., Design Factors for Stress Concentration, Machine Design, 1951, Vol. 23,
no. 7, p. 155.
16) Shigley, J. E., and Mischke, C. R., Mechanical Engineering Design, 5th ed., McGraw
Hill, 1989.
17) Wahl, A. W., and Beeuwkes, R., “Stress concentration produced by holes and notches”,
Trans. ASME Appl. Mech. Sect., 1934, Vol. 56, p. 617.
18) Wilson, I. H., and White, D. J., “Stress concentration factors for shoulder fillets and
grooves in plates”, J. of Strain Anal., 1973, Vol. 8, p. 43.
28
APPPENDICES
29
A1 – FEA Mesh convergence
Figure 1 - Mesh size global .05
30
Figure 2 - Mesh size global .01
31
Figure 3 - Mesh size global .01 / local .005
32
A2 – FEA Results
Hole 1
33
Hole 2
34
Hole 3
35
Edge Notch 1
36
Edge Notch 2
37
Edge Notch 3
38
Waist 1
39
Waist 2
40
Waist 3
41
A3 – Machining Process
For the results of this report, coupons had to be generated for physical testing. These specimens
were machined from sheets of photoelastic sheets. The sheets are made of plastic PS-1A [4] with
measured values for “K” Factor and “f” values that are used in the calculations and measured
fringe plots. Before beginning with the manufacturing of the samples, a few items of concern
must be addressed. The first is the material is sensitive to heat and thus should be kept at room
temperature at all times. Second, due to the plasticity of the material the sheets should always be
stored in a way that minimizes warping. Finally, stress and heat can be introduced during the
machining process and as such the machining techniques should take this into account.
The sheets come in many sizes and may need to be reduced into a size more suitable for testing.
The first step in creating the specimen is to cut the initial coupon from the photoelastic sheets.
For this analysis the coupon size of choice was 10 in x 1.5 in. To obtain this initial size, the
specimen was machined through several steps. The first step was to use a band saw [5] to cut the
samples into strips that are 10 in long and 1.625 in wide. The added width is to allow the
machining of the edges to remove any heat affected edge conditions from the band saw. At this
point the specimens were then place in the machining fixture to reduce their width to the required
1.500 inches. It may be necessary to machine both long edges to remove the heat affected areas
from both sides. The short ends were not machined as they are outside the area of interest.
The next phase of machining is the incorporation of specific features. All the samples have 2
holes 0.375 inches in diameter that are spaced 8.500 inches apart and on centerline long ways. A
.375 in diameter center cut end mill was used to cut these holes. The end mill is plunged into the
part to machine out the hole. An alternate method to creating these holes would be to drill to a
slightly smaller size and then ream the hole to the correct size. For simplifications of machining
the center cut end mill technique was used.
42
Finally, the geometry of interest is machined into the samples. The first set of samples has a hole
placed at the center of the part. These holes are machined using the same techniques as described
for the fixture holes but of their unique size. The second set of samples has an edge hole on
either side of the sample on center and machined into the part exactly one half the diameter of the
hole. The plunge cut method was also used on these samples.
The final set of samples has a necked region machined into it. These samples utilize a constant
1.000 inch internal width with differing corner radiuses. These samples required the most care
need to machine of any of the samples. After several test runs it was found, the best way to
machine the waist samples was to make several passes with a decreasing amount of material
being removed. The final pass should remove no more than .001/.002 of an inch to obtain the
desired shape. This becomes more important as the fillet radius r decreases as the resulting stress
from machining begins to play a larger role in the analysis with respect to edge effects.
VITA
Matthew Lee Mauk
Candidate for the Degree of
Master of Science
Thesis: A REVIEW OF THE ACCURACY OF STRESS CONCENTRATION
FACTORS
Major Field: Mechanical Engineering
Biographical:
Education:
Completed the requirements for the Master of Science in Mechanical
Engineering at Oklahoma State University, Stillwater, Oklahoma in December,
2010.
Completed the requirements for the Bachelor of Science in Mechanical
Engineering at Oklahoma State University, Stillwater, Oklahoma in 1999.
ADVISER’S APPROVAL: Dr. James K. Good
Name: Matthew Lee Mauk Date of Degree: December, 2010
Institution: Oklahoma State University Location: Stillwater, Oklahoma
Title of Study: A REVIEW OF THE ACCURACY OF STRESS CONCENTRATION
FACTORS
Pages in Study: 42 Candidate for the Degree of Master of Science
Major Field: Mechanical Engineering
Scope and Method of Study:
Engineers rely on published results and data to be correct. If it wasn’t, then they would
be “reinvent the wheel” every time. This research will review the published results of
stress concentration factors for semicircular edge notch geometry. It will be shown that
two published sources show significantly different values for the stress concentrations of
the semicircular edge notch geometry. Transverse holes and waist geometry will also be
reviewed. The references will be compared with photoelastic models as well as FEA
results.
Findings and Conclusions:
Results show that there exists a discrepancy between the published sources. There is a
significant difference in the resulting stress concentration values for the same geometries.
Photoelastic results were inconclusive based on error in the manufacturing and
photoelastic analysis performed for this report and could have been responsible for errors
in earlier sources. It is also noted that Finite Element Analysis is a viable alternative for
characterizing the stress concentration values of geometry, as it has become a common
tool for engineers.