Dissertation

Validation of Finite Element Analysis and
Peterson’s Stress Concentration Factors
through the use of the Photoelastic Method
By
Sam Joshua Cutlan
BEng (hons) Automotive Engineering
University of Wales Trinity Saint David Swansea
x

Sam Joshua Cutlan P134357
VALIDATION OF FINITE ELEMENT ANALYSIS AND PETERSON’S STRESS CONCENTRATION FACTORS
THROUGH THE USE OF THE PHOTOELASTIC METHOD
1
Table of Contents
1.0 Acknowledgments.......................................................................................................................3
2.0 Introduction ................................................................................................................................4
2.1 Photoelastic Method...............................................................................................................4
2.2 Finite Element Analysis...........................................................................................................4
2.3 Peterson’s Stress Concentration Factors................................................................................4
2.4 Project Aims............................................................................................................................4
3.0 Literature Review........................................................................................................................5
3.1 Materials .................................................................................................................................5
3.1.1 Perspex............................................................................................................................6
3.1.2 Testing Procedure...........................................................................................................7
3.2 The Photoelastic Method........................................................................................................9
3.2.1 The Nature of Light .........................................................................................................9
3.2.2 Polarised Light.................................................................................................................9
3.2.3 Refraction......................................................................................................................10
3.2.4 Double Refraction (Birefringence) ................................................................................10
3.2.5 Stress-optic Law ............................................................................................................11
3.2.6 Features of the Photoelastic Package (GFP 1400) ........................................................13
3.3 Derivation of Shearing Force in relation to Tensile Force.....................................................14
3.3.1 Finite Element Analysis.................................................................................................16
3.4 Peterson’s Stress Concentration Factors..............................................................................18
4.0 Methodology.............................................................................................................................20
4.1 Tensile Test Piece Manufacture for Denison Tensile Test ....................................................20
4.1.1 Denison Tensile Test .....................................................................................................22
4.2 Test Piece Manufacture for Photoelastic Stress Analysis .....................................................26
4.2.1 Photoelastic Stress Experimentation............................................................................28
4.3 FEA Set-up and Refinement..................................................................................................34
5.0 Results.......................................................................................................................................36
5.1 Experimentation Error ..........................................................................................................50
6.0 Analysis .....................................................................................................................................51
6.1 Photoelastic Stress Experimentation....................................................................................51
6.1.1 U-Shaped Semi-circular Notches ..................................................................................51
6.1.2 V-Shaped Triangular Notches .......................................................................................51
6.1.3 Two-dimensional Bolt Design .......................................................................................52
6.2 Limitations.............................................................................................................................52

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6.3 FEA ........................................................................................................................................52
7.0 Conclusion.................................................................................................................................53
8.0 References ................................................................................................................................54
9.0 Appendix ...................................................................................................................................56
9.1 U-Shaped Semi-circular Notches ..........................................................................................56
9.1.1 10N Load Case...............................................................................................................56
9.1.2 20N Load Case...............................................................................................................59
9.1.3 30N Load Case...............................................................................................................62
9.1.4 40N Load Case...............................................................................................................65
9.1.5 50N Load Case...............................................................................................................68
9.1.6 60N Load Case...............................................................................................................71

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1.0 Acknowledgments
I am grateful to my project supervisor, Malcolm McDonald, whose expertise, great understanding,
generous guidance and support made it possible for me to work on a topic that was of great interest
to me.
I would also like to thank Dr. Peter Charlton for his support and guidance in the photoelastic stages
of this project. The help with initial advice and guidance in setting up important test equipment was
crucial in obtaining accurate results for this project.
I would like to thank Daniel Butler for the fast production and advice when producing important test
pieces. And thank Gareth Owen for his help at the initial stages of this project.
Finally a special thank you to my family and friends for their moral support, who have put their faith
in me and have urged me to improve throughout this academic year.

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2.0 Introduction
2.1 Photoelastic Method
One of the main experimental methods for analysing stress and strain distributions in loaded
members is called the Photoelastic method. Photo meaning the use of optical methods and elasticity
meaning the interpretation of the theory of elasticity using experimental data acquired. From using
models made from transparent polymers, applications of this method have been extended to
inelastically deforming bodies (Khan & Wang, 2001). The method works on the principles of
birefringence caused when a body is strained, from this, an accurate value of stress within the body
can be determined. With the advancement of computers, the traditional methods of photoelasticity
have been replaced with more modern numerical approaches. However, stress or strain analysis are
on the increase in areas such as fracture mechanics, due to the advancements in fibre optics, lasers,
data acquisition and image analysis (Khan & Wang, 2001).
2.2 Finite Element Analysis
Finite Element Analysis (FEA) is a computerised method used to predict how a product will react in
real-world situations such as; forces, heat, vibrations, and other physical effects. FEA shows that the
design of a product is critical as this can drastically change how resistant the product is to breaking
or wearing out. From dividing a structure into small pieces called elements, each element can be
analysed using simple equations for stress and strain. As the number of elements increases, the
mesh density will increase, and the approximation solution, theoretically, becomes more accurate
(Mac Donald, 2011). However, increasing mesh density can increase convergence times, possibly
leading to inaccurate results, this will be discussed in detail later on.
2.3 Peterson’s Stress Concentration Factors
Stress concentration factors are obtained experimentally using the photoelastic method, from using
elasticity theory and computationally using FEA within Computer Aided Engineering Design software
such as SolidWorks. The U-shaped notch or circumferential groove is a geometrical shape with
significant interest within engineering. The shape itself occurs in machine elements such as blade
rows, turbine rotors and at seals. Other examples of this are found in a variety of shafts, for
example; a shoulder relief groove or a retainer for a spring washer (Pilkey & Pilkey, 2008). In areas of
fatigue, creep-rupture, and brittle fracture, the round bottomed V-shaped notch or circumferential
groove, and to a less significant extent the U-shaped notch are conventional contour shapes in stress
concentration test pieces. A multi-grooved member may also be considered (Pilkey & Pilkey, 2008).
2.4 Project Aims
This project aims to question the accuracy of the photoelastic method, FEA and Peterson’s Stress
Concentration Factors and whether or not the results obtained from these experimentations will
correlate. As well as this, the author wishes to determine the stress concentration factors in
alternate designs and how the design affects the magnitude of the stress concentration.

5
3.0 Literature Review
In modern times, stress calculations within a body are calculated using Computer Aided Engineering
Design tools such as FEA within SolidWorks. For most design engineers, it is their primary source for
designing and calculating displacements, stress, strain and more for a product. However, FEA is
extremely idealised such that the user of the software has to input all parameters such as loads,
constraints and material properties. These inputs highlight the following problems;
1. Loadings – Are they body, surface, or point loads, and in which direction are they being
applied?
2. Constraints – What degrees of freedom is the body allowed to rotate or translate?
3. Material Properties – Experimentally determining material properties such as the young’s
modulus of the material does not mean that any one piece of the same material will have
identical properties to the next.
Finally, most FEA solvers are known as linear solvers such that they obey Hooke’s Law. This being
that the force required to stretch a body is directly proportional to its displacement. This point is
also highlighted as not all deformation in the real-world is linear, and definitely not when the yield
point is reached.
It can be concluded that the biggest challenge within FEA is validation. Physical tests which are
closely monitored for accuracy and reliability can be created in order to determine whether the
physical reality and virtual reality line up. A general agreement among FEA analysists is that there
are no hidden disconnects between the model within the software and real-world testing because of
validation. Validation therefore ensures that, based on the principles of physics, the correct physical
properties are used and the properties are scrutinised correctly (Jiju, 2014).
Hence, the advantage of real-world practical testing such as photoelastic testing, is that it benefits
the problems still underlying within FEA. This is because, results obtained from practical testing will
be accurate if controllable input factors are evaluated, uncontrollable input factors are recognised to
understand the response of the experiment, the responses or output measures ensure the desired
effects are implemented and replication applied to determine random error (Jiju, 2014).
It is essential that the fundamentals and history of the photoelastic method are understood by the
author in order for this project to be a success. To do this, the following literature review concerns
the main points regarding the materials, software and the photoelastic method implemented.
3.1 Materials
An ideal photoelastic model material should meet the following requirements for the experiment to
be considered successful, these are summarised below;
1. The material should be relatively colourless and transparent to visible light. The material
should also be isotropic, meaning that the material is uniform in all directions. It should also
be homogeneous, meaning it has a uniform composition throughout.
2. The induction of stress or chipping should not be brought on when machining.
3. The material should be free from any residual stresses. To reduce the amount of residual
stresses, the material can be carefully annealed.

6
4. At fairly low loads the material should be sensitive enough to produce comparatively more
fringe numbers. Hence, the material fringe value should be relatively low.
5. Under applied loads the Young’s Modulus should be as high as possible, so that the shape of
the model does not differ significantly to the structural element. If the Young’s Modulus was
relatively low this would induce possible error in experimental results.
6. To improve the accuracy when determining the stress, the value of the proportional limit
should also be as high as possible such that it produces a higher-order fringe pattern. This
means that the safety of the model cannot be endangered when introducing higher loads.
7. The material should have a minimum-time edge effect, which means that moisture absorbed
at the edges where it has been machined are kept at a minimum. The induced stress at the
edges produce fringes where there is no stress applied.
8. The material should be relatively free of creep between stress and optical response within
the working time limit of taking screen captures and introducing higher loads (Khan & Wang,
2001).
In reality although all points are desirable, no one ideal model material exists in order to meet the
needs of the requirements outlined above.
3.1.1 Perspex
Acrylics are a group of vinyl plastics and the most important one being polymethyl methacrylate
(PMMA) which is better known as ‘Perspex’ under its trade name here in the UK. It is a clear-like
plastic developed in the Second World War, and was used in aircrafts. Perspex can easily be
moulded and it is much tougher and lighter than glass and will also transmit more than 90% daylight.
It is produced by the polymerisation of methyl methacrylate (Higgins, 1997). As Perspex meets most
of the requirements listed above it was used throughout this project. The properties of Perspex at
room temperature are shown below in Table 1.
Table 1 – Approximate Properties of Perspex, a Photoelastic Model at Room Temperature
Material Stress
Fringe
Value
𝑀𝑓
(kN/m)
Young’s
Modulus
E (MPa)
Poisson’s
Ratio µ
Proportional
Limit 𝜎 𝑝𝑙
(MPa)
Sensitivity
Index S
(mm)
Figure
of
Merit
Q
(mm)
Time-
Edge
Effects
Creep Machinability
Perspex 105 2,760 0.38 Not available Not
available
26 Excellent Excellent Good
(Khan & Wang, 2001).
Perspex is a well-known polymer. Polymers are materials which have long-chain molecules
containing carbon-to-carbon bonds. All materials known as plastics are known as polymers. Perspex
is said to be an amorphous thermoplastic. Amorphous meaning that its structure is random and
disordered when it is in a solid state or the long chained molecules being all entangled and without
form. Thermoplastic meaning that the material becomes plastic every time its heated, as opposed to
thermosets, and will retain its shape only when cooled (Dowling, 2013). Below its glass transition
temperature 𝑇𝑔 (temperature region where polymer transitions from a hard, glassy material to a soft
rubbery material) Perspex tends to be glassy and brittle, this goes for other amorphous polymers
which all have a Young’s Modulus, E of around the order 3GPa (Dowling, 2013). Although labelled as
an amorphous polymer, because of the long chains, a small percentage of the chains arranged

7
themselves in an ordered pattern. This results in small areas of the polymer having ordered regions
called crystallites (see Figure 1)
Figure 1 – ‘Crystallites’ in a Solid Plastic Material
(Higgins, 1997).
Therefore in a solid state, polymers consist of both crystalline and amorphous regions. Highly
crystalline polymers can have up to 90% crystalline regions whilst others such as Perspex are almost
completely amorphous (Higgins, 1997).
3.1.2 Testing Procedure
Before any physical tensile testing was completed on the Perspex models, it was important that the
author understood the correct procedure when tensile testing plastic materials. If the testing of the
tensile test pieces was to have any meaning then a permanent testing temperature was needed, the
temperature being at 23 ± 2°C with an atmospheric humidity of 50 ± 5%. The tensile test pieces
would have to remain at these conditions for 88 hours prior to when the testing was performed
(Higgins, 1997).
In terms of mechanical testing, plastic materials are very time-sensitive because the total
deformation depends on the following;
1. Carbon-carbon bonds in the polymer chains bending, which is shown by the ordinary
elasticity and is an immediate deformation (OE).
2. The polymer chains unravelling, this results in high elasticity (HE).
3. The polymer chains slipping past each other, which is known as plastic flow and is
irreversible (VISC).
The total deformation (D) of the above points are known as viscoelasticity, and the mechanical
properties of the material is greatly affected by the rate of strain, therefore;
Equation 1
𝐷 = 𝐷 𝑂𝐸 + 𝐷 𝐻𝐸 + 𝐷 𝑉𝐼𝑆𝐶
𝐷 𝑂𝐸 – Being instantaneous and time-independent.
𝐷 𝐻𝐸 – Very time dependent.
𝐷 𝑉𝐼𝑆𝐶 – Both time dependent and irreversible.
(Mathur & Bhardwaj, 2003).

8
Also, in comparison to metals testing, the geometry of testing plastics greatly differs. This is because
sudden changes in shape of the tensile test piece causes stress concentrations and can cause failure,
therefore the tensile test piece generally come in the form shown below (Higgins, 2016) (see Figure
2).
Figure 2 – Principal form of a tensile test piece used for testing of plastic materials
Since many plastics do not obey Hooke’s Law, it is impossible to calculate the Young’s Modulus of
the material because the stress is not directly proportional to the strain as the tensile test piece
deforms. The Young’s Modulus only applies to materials with Hookean characteristics and therefore
as an alternate method, the Secant modulus is calculated. The Secant Modulus is defined as the ratio
of nominal stress at the corresponding strain at a particular point (Higgins, 1997) (see Figure 3).
Figure 3 – Derivation of the Secant Modulus for non-Hookean Plastic Polymers
As can be seen from Figure 3, the secant modulus related to a strain of 0.2% is the slope of the line
OS. To ‘take up the slack’ to straighten the tensile test piece, prior to performing the whole test, an
initial force ‘w’ is applied. The force ‘w’ is typically about 10% of the expected force in order to
produce 0.2% of the strain. With the initial force set, the extensometer is set to zero. A strain rate is
specified by the user and the force is increased until the necessary force, ‘W’, is reached to produce
0.2% of the strain in the gauge length.
Therefore the Secant Modulus can be derived (see Equation 2).
Equation 2
𝐸𝑙𝑎𝑠𝑡𝑖𝑐 (𝑆𝑒𝑐𝑎𝑛𝑡)𝑀𝑜𝑑𝑢𝑙𝑢𝑠 =
𝑆𝑡𝑟𝑒𝑠𝑠
𝑆𝑡𝑟𝑎𝑖𝑛
= (
𝑊 − 𝑤
𝐴
) ÷ 0.002
=
𝑊 − 𝑤
0.002𝐴

9
Where ‘A’ equals the initial cross-sectional area of the tensile test piece at the gauge length (Higgins,
1997).
3.2 The Photoelastic Method
3.2.1 The Nature of Light
Light is a disturbance which propagates through space. Light can be represented as waves acting
along a stretched piece of string (see Figure 4)
Figure 4 – The Disturbance of Light
Light waves belong to the class of transverse waves, where the displacement of each particle along
the string at any point is at right angles to the string. Or it may be longitudinal which means that the
displacement is in the direction of the length of the string. In the case of light, it is a disturbance and
is a vector known as a light-vector. In modern physics, light is looked at as an electromagnetic
disturbance propagated through space. In this theory there are two light-vectors both acting at right
angles to the direction of propagation but equally perpendicular (Coker & Filon, 1957).
3.2.2 Polarised Light
An unpolarised beam of light consists of many transverse waves which are randomly orientated,
whose vibrations are transverse to a straight line of propagation. Consequently, a polarised light
beam consists of many transverse waves which have a preferred orientation (refer to Figure 5). In
practice there are multiple ways of polarising light from a natural source. Polarised light can be
attained from using Polaroid sheets, birefringence (or double refraction) and by reflection or
refraction (Khan & Wang, 2001). If the light is monochromatic, meaning of pure colour, all of the rays
have the same wavelength. With white light, rays of all visible wavelengths exist, however they will
be vertically polarised (Juvinall, 1967).
Figure 5 – Polarisation of Light

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After the light source passes through the polarising filter it will then pass through the strained
model.
3.2.3 Refraction
When a light source passes through a medium, the light source will bend due to the change in speed
when entering the medium. The refraction of the light is bent towards the normal to the boundary
between the fast and slow medium (refer to Figure 6). The amount of bending is dependent on the
indices of refraction, ‘n’ of a material. ‘n’ is a dimensionless number that defines the propagation of
light through a medium. For example, the refractive index in a vacuum = 1, water = 1.33 at 20°C and
Perspex = 1.4893-1.4899 (Nave, 2016).
Figure 6 – Refraction of Light through a Glass Medium
3.2.4 Double Refraction (Birefringence)
Double refraction is exhibited by certain transparent materials and coatings. When a ray of light
passes through a transparent material it experiences two refractive indices, otherwise known as
birefringence. Many optical crystals exhibit the property of birefringence. However, photoelastic
materials only experience the property of birefringence when stress is applied and the magnitude of
the refractive indices is directly proportional to the stress at each point in the material (Li, 2010).
When polarised light exits the photoelastic material, it is resolved along two principal stress
directions. Each of these constituent stresses experience alternate magnitudes of refractive indices.
The phase difference, or phase retardation, of the two waves is related to the difference in the
refractive index (Li, 2010). The magnitude of the phase retardation is known as the stress optic law
shown in Figures 7 and 8, (refer to Equation 3).

11
Figure 7 - Diagram showing the effect of Birefringence
Where;
H = Horizontally Polarised
V = Vertically Polarised
𝛿 = Phase Retardation
Figure 8 – Loaded Test Piece Viewed in a Plane Polariscope Arranged in a ‘Cross-linked’ Setup
3.2.5 Stress-optic Law
The phase retardation is expressed as the following;
Equation 3
𝛿 =
2𝜋𝑡
𝜆
× 𝐶 × (𝜎 𝑝 − 𝜎 𝑞)
Notice that the direction in
which the rays of light exit the
birefringent material are in the
direction of the principal
stresses and are also
perpendicular to one another.
Also, the magnitude of the
principal stresses are
proportional to the relative
speed of the light rays exiting
the body (Li, 2010).

12
Where;
𝛿 = Induced Retardation
t = material thickness (mm)
C = Stress-optic Coefficient
𝜆 = Vacuum Wavelength
𝜎 𝑝 = First principal stress
𝜎 𝑞 = Second principal stress
Since two-dimensional photoelasticity allows the measurement of retardation to obtain further
values such as maximum shear stress, stress-separation techniques are required. Stress-separation
techniques such as Mohr’s circle can be used to determine shear0, shear45 and shear max.
This stress separation technique was used by Delta Vision, the software used for the photoelastic
testing performed throughout this project. The software determined shear0, shear45 and shear max
below respectively (see Figures 9-11).
Figure 9 – Mohr’s Circle determination of shear0
Figure 10 – Mohr’s Circle determination of shear45

13
Figure 11 – Mohr’s Circle determination of shear max
(Stress Photonics, 2016).
Shear max is then determined from the two principal stresses and represented by the following
equation.
Equation 4
2𝜏 𝑚𝑎𝑥 = |𝜎 𝑝 − 𝜎 𝑞| =
𝜆
2𝜋𝑡𝐶
𝛿
(Li, 2010)
Or the preferred equation;
Equation 5
𝜏 𝑚𝑎𝑥 =
1
2
(𝜎 𝑝 − 𝜎 𝑞)
(Hearn, 2013)
The values obtained from the photoelastic testing of the maximum shear stress can be directly
converted into a tensile stress to compare with the FEA models tested within this project.
3.2.6 Features of the Photoelastic Package (GFP 1400)
The following applicable features of the photoelastic package used throughout this project are;
 Automated full-field strain measurement
 Simple Static Loading tests
 No fringes to analyse
 Fully computerised digital system
(Stress Photonics, 2016)
The package is a strain measurement system based on photoelasticity and although the instrument
is very different to those used in the past, it is well established on the fundamentals of
photoelasticity that have been used for decades.

14
3.3 Derivation of Shearing Force in relation to Tensile Force
As the package shown in Figures 9-11 used for the photoelastic testing can only output the shear
stress occurring in the model, it is important that the shear stress is converted directly into a tensile
stress so that the photoelastic testing can be compared with the FEA testing performed within
SolidWorks. To do this, a first principal approach was undertaken by the author as shown in Figure
12.
Figure 12
Considering the following circular bar which is cut at angle 𝜃 to its axis;
When;
Resolving normal to the surface: 𝜎 × (𝐴/𝑠𝑖𝑛𝜃) = 𝑃𝑠𝑖𝑛𝜃 → 𝜎 = (𝑃/𝐴) × 𝑠𝑖𝑛2
𝜃
Resolving parallel to the surface: 𝜏 × (𝐴/𝑠𝑖𝑛𝜃) = 𝑃𝑐𝑜𝑠𝜃 → 𝜏 = (𝑃/𝐴) × 𝑠𝑖𝑛𝜃𝑐𝑜𝑠𝜃
Since;
The maximum shear stress occurs at 45°, When 𝜃 = 45°, 𝜏 = (𝑃/𝐴) × sin(45) × cos(45) = 0.5𝑃/𝐴
The maximum tensile stress occurs at 90°, when 𝜃 = 90°, 𝜎 = (𝑃/𝐴) × sin(90)2
= 𝑃/𝐴
(Williams & Todd, 2000)
Proof;
Therefore, 𝑃𝑠ℎ𝑒𝑎𝑟 = 𝑃𝑐𝑜𝑠45

15
𝑃𝑠ℎ𝑒𝑎𝑟 = 𝑃𝑐𝑜𝑠45
𝑃𝑠ℎ𝑒𝑎𝑟 = 𝑃 𝑥
2
√2
𝑃𝑠ℎ𝑒𝑎𝑟 =
𝑃
√2
𝐿
𝐿𝑠ℎ𝑒𝑎𝑟
= 𝐶𝑜𝑠45
𝐴𝑠ℎ𝑒𝑎𝑟 =
𝐴
𝑐𝑜𝑠45
∴ 𝜏 𝑚𝑎𝑥 =
𝑃𝑠ℎ𝑒𝑎𝑟
𝐴𝑠ℎ𝑒𝑎𝑟
=
𝑃𝑐𝑜𝑠45
𝐴
𝑐𝑜𝑠45⁄
=
𝑃
𝐴
× 𝑐𝑜𝑠2
(45)
∴ 𝜏 𝑚𝑎𝑥 = 𝜎 𝑚𝑎𝑥 × 𝑐𝑜𝑠2
(45)
=𝜎 𝑚𝑎𝑥 ×
1
√2
=
𝜎 𝑚𝑎𝑥
2
∴ 𝜏 𝑚𝑎𝑥 =
1
2
𝜎 𝑚𝑎𝑥
Or
2𝜏 𝑚𝑎𝑥 = 𝜎 𝑚𝑎𝑥

16
3.3.1 Finite Element Analysis
When creating a model within a static study simulation, there are a few important questions in
which the user has to consider before resolving the model. These questions are;
1. What makes a good quality mesh?
2. How fine should the Finite Element (FE) mesh be in order to obtain accurate results?
3.3.1.1 Mesh Quality
So, what makes a good quality mesh? Important areas to consider are areas in which the change in
stress from one element to the next is large. In cases such as these, the user can set-up the model to
have a manual elemental size or for the software to determine the elemental size from using the h-
adaptive method. The h-adaptive method will generate a finer mesh in an area where the tolerance
is outside of the Error Norm, or in other words where the change is stress from one element to the
next is very high. Before refining the mesh in areas where the change in stress is high, the quality of
the mesh has to be considered. Since there is not a conclusive answer to what makes the perfect
mesh, a good starting point is to look at the general mistakes when generating a mesh (Lake, 2015).
Consider the following in Figure 13.
Figure 13 – various meshes for a ¼ symmetry plane stress analysis of a plate with a hole
(Mac Donald, 2011)
Scenario A – Only three elements have been used around the curve of the hole. The highly curved
geometry means that more elements need to be used to correctly model it. The small geometry
means that there will be a stress concentration around the hole and three elements are not fine
enough to determine this correctly. The ratio between the largest and the smallest mesh should not
be greater than 10:1, generally it is bad practice to do this (Mac Donald, 2011).
Scenario B – This scenario is using the h-adaptive method, where the software will automatically
generate a mesh for the user. However, this method has generated a poor mesh as there are many
poorly shaped elements. Again, the size ratio is too great between the largest and the smallest
element. As the transition between large and small elements happens quickly, it is considered that
this will generate inaccurate results (Mac Donald, 2011).
Scenario C – There are many poorly and misshaped elements in this mesh due to the automatic
refinement. The refined region has localised a lot around the hole and is becoming a problem. The
transition between small and large elements happens quickly (Mac Donald, 2011).
Scenario D – The transition between small and large elements is smooth. There is a refined mesh
generated around the hole. The elements are near square shaped, are organised and smooth (Mac
Donald, 2011).

17
Therefore, it can be concluded that the appropriate mesh for this problem lies with scenario D. The
mesh density increases nearer to the hole, and is not too high in the upper left hand section where
the stress can be assumed to be pretty constant for this problem.
3.3.1.2 Mesh Convergence
Now, considering question 2 – How fine should the mesh be in order to obtain accurate results?
Again, there is not a definitive answer when considering this question. ‘Mesh convergence’ is the
term denoted to the answer when determining an appropriate mesh size. The number of elements
in a model is inversely proportional to the size of the elements in a model. The consequence of this
is that the accuracy of the results as the mesh density increases should also increase. This is the
principal of mesh convergence (Awang, et al., 2015).
There are a number of rudimentary practices that can be performed in order to check how accurate
the results are. The first is to generate an initial mesh with a uniform medium density and then
compare the results with a mesh that is twice as small as the initial mesh in critical regions with the
highest change in stress. Then if the results do not differ greatly, the initial mesh can thus be
considered acceptable for the problem. The second is to possibly compare the FE results with
available and accurate experimental results and then alter the mesh density in regions where the FE
model does not conform (Mac Donald, 2011).
It is important that the mesh density is acceptable for the model. If the mesh is too coarse, then this
may lead to some inaccurate results as the mesh may not be refined enough in places where there is
a large stress concentration, leading to poor mesh resolution. On the other hand, if the user was to
generate an overly-fine mesh, then the model will take a lot of time and computation resources to
solve, this solution may be inaccurate as the model may be overly-converged (Awang, et al., 2015).
An example of mesh convergence can be seen below in Figure 14.
Figure 14 – Number of elements in comparison with stress
The convergence curve shows that the accuracy of the results increases when the number of
elements increase as well. Not only does mesh convergence apply to obtaining true values for

18
displacements, but it also applies for obtaining the exact solution for stress and strain. However, for
different sets of results the mesh may converge at different mesh sizes. This will not be a problem in
this project as tensile stress is only concerned throughout (Mac Donald, 2011).
When creating an FE model, there are a number of assumptions made such as; geometry, material
properties, loadings and constraints. With the geometry it is assumed that the model is exact to the
engineering drawings, when in reality due to manufacturing errors it is not. It is assumed that the
material of the model is linear elastic when in the real world no material will behave in this way.
Then it is assumed that the loadings and constraints are constant, which is also not the case. Finally
when a mesh is introduced, there is a further uncertainty about the quality of the mesh itself.
Therefore, when generating an FE model, the total uncertainty is;
Equation 6
𝑈𝑡𝑜𝑡𝑎𝑙 = 𝑈𝑔𝑒𝑜𝑚𝑒𝑡𝑟𝑦 + 𝑈 𝑚𝑎𝑡𝑒𝑟𝑖𝑎𝑙 𝑚𝑜𝑑𝑒𝑙 + 𝑈𝑙𝑜𝑎𝑑𝑠+𝑏𝑜𝑢𝑛𝑑𝑎𝑟𝑦 𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛𝑠 + 𝑈 𝑚𝑒𝑠ℎ
(Mac Donald, 2011)
If the choices, when creating the FE model are as educated as possible, it will eliminate a lot of
uncertainty when generating the results. Thus, if this is true then the total uncertainty is equal to the
above equation.
3.4 Peterson’s Stress Concentration Factors
The objective that Rudolph Earl Peterson wanted to achieve by the application of introducing stress
concentration factors to engineering designs was to attain more balanced designs of structures and
machines. Stress concentration factors are acquired analytically from the elastic theory,
computationally from FEA, and experimentally using methods such as strain gages or the
photoelastic method. Existing information on stress concentration factors is to be recognised as an
approximation. This is due to formulas including assumptions such as a material being isotropic and
homogeneous. When in reality, this is never the case (Pilkey & Pilkey, 2008).
If a body, similar to the design in Figure 24 in section 4.0 is loaded in tension or in compression, a
stress concentration is created at the notch in the centre of the minimum cross section. If the
change in geometry is more sudden, the localised stress concentration increases accordingly. Hence,
the more sudden the change in geometry, the greater the change in stress concentration factor.
Equation 7 can be applied for the design in Figure 24 and the tensile test pieces used throughout the
experiments;
Equation 7
𝐾𝑡 ≈ 𝐵 × (
𝑟
𝑑
)
𝑎
(Pilkey & Pilkey, 2008)

19
Where;
B = Constant (refer to the table below)
r = radius (m)
d = minimum cross sectional distance (m)
a = Constant (refer to the table below)
(Pilkey & Pilkey, 2008).
Table 2 represents the respective ‘B’ and ‘a’ values from the above equation in relation to ‘D/d’
Table 2
Peterson’s Stress Concentration Factors for a notch in tension
D/d B a
2 1.1 -0.321
1.5 1.077 -0.296
1.15 1.014 -0.239
1.05 0.998 -0.138
1.01 0.977 -0.107
(Pilkey & Pilkey, 2008).
As can be seen from Table 2, the values of ‘B’ and ‘a’ are dependent on the ratio of the maximum
and minimum cross sections, or ‘D/d’ of the test piece. These results can then be interpolated and
the respective ‘B’ and ‘a’ values obtained for a particular test piece design. From then using the
above formula, the stress concentration of the test piece can be determined for a specific radius.
The following example defines the process undertaken by the author for determining the stress
concentration factor for a specific radius using mathematical interpolation;
Radius = 1.8mm
𝐷
𝑑
=
10
10 − (1.8 × 2)
= 1.5625
𝐵 = 1.077 + (
𝐷
𝑑
− 1.5
2 − 1.5
) × (1.1 − 1.077) = 1.079875
𝑎 = −0.296 + (
𝐷
𝑑
− 1.5
2 − 1.5
) × (−0.321 + 0.296) = −0.299125
𝑟
𝑑
=
1.8
6.4
= 0.28125
∴ 𝐾𝑡 = 1.079875 × (0.28125)−0.299125
= 1.578205976

20
4.0 Methodology
In this section, the methods used to perform all of the practical experiments are outlined and
described in detail as well as the manufacture of important test pieces. The experiments include;
 Denison Tensile Test
 Photoelastic Testing
 Finite Element Analysis
4.1 Tensile Test Piece Manufacture for Denison Tensile Test
The material chosen for these experiments was Perspex as stated beforehand. Perspex is known for
its temporary birefringent properties. In order to determine the Secant Modulus in the Perspex, ten
individual tensile test pieces were created in SolidWorks. The designs of the tensile test pieces
consisted of a normal ‘dog bone’ shape similar in design to Figure 2 in section 3.1.2. The geometry of
the tensile test pieces can be seen in Figure 15 below.
Figure 15 – Geometry of Tensile Test Pieces within SolidWorks
The tensile test pieces were designed in this way so that when they were put under tension in the
Denison tensile testing machine they would fail in the centre where the smallest cross section
occurs. Following the design of the tensile test piece within SolidWorks, the designs were generated
as a 2D Dwg (*dwg) file so they could be cut from the water jet cutting machine (see Figure 16). This
machine cuts to a high accuracy of up to 0.3mm and works on a two-axis system. The machine uses a
high pressure system so that it can cut various amounts of materials, including metals of up to 4mm.
The high pressure and accuracy ensures that the formation of chipping and induction of stress will
not appear. This is essential as Perspex and other plastics alike are known to be brittle and the
formation of cracks can easily appear.

21
Figure 16 – Water Jet Cutting of Tensile Test Pieces
The author manufactured all test pieces from the same cut of Perspex throughout this project so
that the Secant modulus would remain pretty constant and therefore as accurate as possible.
Figure 17 shows the manufactured tensile test pieces;
Figure 17 – Tensile Test Pieces
As can be seen in Figure 17, the tensile test pieces still had some flashing occurring at the edges, this
was later removed by the author using wet & dry P2000 grit paper.

22
4.1.1 Denison Tensile Test
Prior to testing using the Denison, it was important that the tensile test pieces remained at room
temperature prior to performing the test for at least 88 hours following initial research undertaken.
Following this, the test was performed on all 10 test pieces using the same strain rate setting of
10mm/min, ensuring that the distance between the grips (𝑙𝑜) remained a constant 100mm, the
cross-sectional area of 21mm² was inputted and the force was zeroed before the test was carried
out. Figure 18 shows the tensile test piece clamped prior to performing the test, note that the
picture was taken prior to setting up the specified distance of 100mm between the grips.
Figure 18 – Denison Tensile Test Set-up
Figure 19 – Data from Failed Tensile Test
Figure 19 shows part of the initial test which displays slipping of the Denison grips up to a force of
250N. As can be seen, there is clear steps of increased displacement when the Denison is losing grip
on the tensile test piece. This was rectified by increasing the clamping force on the Denison
following the second test.
-50
0
50
100
150
200
250
300
-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Force(N)
Displacement (mm)
Graph showing Force vs Displacement in Tensile Test Piece
Dension
It is important to note that the first
two initial tests failed due to the
clamping force on the grips being
too low. This caused a fair amount
of slipping which altered the
distance between the grips and
therefore the outcome of the
results. Data from one of the failed
test results can be seen below in
Figure 19.

23
A total of eight correctly performed tests were recorded. Figure 20 shows all of the test pieces
following the tensile test.
Figure 20 – Tensile Test Pieces following the Denison Tensile Test
As can be seen, five out of a total of eight tensile test pieces failed near to the start of the radius of
curve close to where the grips are held in place. This is due to there being a larger stress
concentration factor near to the curve and is proven from a quick static analysis within FEA (see
Figure 21). Since all test pieces failed at the minimum cross-section and also had relatively the same
extension before failing, they can be considered a success.
Figure 21 – Stress concentration Near Curve

24
Figure 22 shows the eight successful tensile tests completed using the Denison.
Figure 22 – Data from Successful Tensile Tests
From Figure 22, it can be seen that there is great correlation between the tensile test pieces when
concerning the secant modulus. There is also great correlation at the point of yield (at the top of the
curve where plastic deformation occurs), all test pieces yield between 55 – 58MPa. As can be seen
following the yield point, all tests pieces display necking. Necking is the term denoted to when there
is a local decrease in the cross-sectional area at a point along the length of the test piece. As the
stress vs strain curve shows (for most test samples), the nominal stress decreases after the yield
point and settles at a relatively constant value as the neck extends along the test piece. The neck
extending is a process known as cold drawing, this is when the polymer chains unravel and align
themselves parallel to the direction of the applied stress. The cold drawing phase can be seen to be
large for Test 3, but relatively small for Test 4 and 9, all other tests are quite comparable. The only
anomaly that can be described from the above graph is that none of the test pieces display any
strain hardening. Strain hardening occurs after the whole test piece has necked and there is a
sudden increase in stress until the test piece has fractured. Strain hardening occurs due to the
parallel orientation of the polymer chains (Univeristy of Cambridge , 2015).
As it can be concluded from the Stress vs Strain curve that Perspex does not obey Hooke’s law, the
formula for the Secant Modulus was used as a replacement for calculating the Young’s Modulus of
the material.
Hence;
𝑆𝑒𝑐𝑎𝑛𝑡 𝑀𝑜𝑑𝑢𝑙𝑢𝑠 =
𝑊 − 𝑤
0.002𝐴
= 2.47𝐺𝑃𝑎
-10
0
10
20
30
40
50
60
70
-0.05 0 0.05 0.1 0.15 0.2 0.25 0.3
NominalStress(MPa)
Nominal Strain
Deformation of Perpsex Test Pieces
Test 3
Test 4
Test 5
Test 6
Test 7
Test 8
Test 9
Test 10

25
As can be seen from Figure 22, the young’s modulus of Perspex does not change up to 105N of
applied force (see Equation 8).
Equation 8
𝜎 =
𝐹
𝐴
5 × 106
𝑃𝑎 =
𝐹
21 × 10−6
∴ 𝐹 = 105𝑁
Therefore when performing the photoelastic experiments it will be ensured that 105N is not
exceeded in order to guarantee that the Young’s Modulus does not change. This is highlighted as a
definitive value of the Young’s Modulus has to be inputted into Delta Vision in order to calculate the
shear stress of the model as it is consequently strained as the load is applied (see Table 3).
Table 3 – Summary of Tensile Testing
Material Secant
Modulus
(GPa)
Average
UTS
(MPa)
Abide
Hooke’s law
Maximum force
prescribed for
photoelastic testing
(N)
Material
Thickness (mm)
Perspex
(PMMA)
2.47 56.5 No 105 2.1

26
4.2 Test Piece Manufacture for Photoelastic Stress Analysis
Prior to the creation and manufacture of the photoelastic test pieces, to ensure that the photoelastic
experiment could be undertaken, two steel plugs were manufactured on a lathe (see Figure 23).
Figure 23 – Steel Plugs
These steel plugs were created so that an even distribution of force could be applied to the test
piece. Notice that each plug has a step in it, this is to ensure that the tensile test piece did not slip
and remained still throughout the experiment. Each plug was drilled and tapped so that an M8 bolt
could be screwed into position. The author then tied metal reinforced string to the ends of each
bolt. The masses would then be hung from the larger lower plug whilst the smaller plug was
attached to a frame. This would set-up would simulate a static tensile test which could be
reproduced within FEA and compared.
Following the manufacture of the two steel plugs, the photoelastic test pieces were created within
SolidWorks and outputted using the same Dwg (*dwg) file as before so that they could be cut on the
water jet. The geometry of the tensile test pieces are as follows (see Figure 24).
Figure 24 – Geometry of tensile test pieces within SolidWorks

27
As can be seen, the author has left gaps for the plugs so that they can be inserted. The diameter of
the holes are exactly the same as the plugs to ensure a tight fit and limited movement of the tensile
test piece. As well as this, there is a notch of a 2mm radius in the centre of shaft of the test piece.
The notch radius would be decreased in 0.2mm increments for each test piece (see Table 4);
Table 4 – Test Piece Notch Radius
Test Piece Number Notch Radius (mm)
1 0.4
2 0.6
3 0.8
4 1.0
5 1.2
6 1.4
7 1.6
8 1.8
9 Base model (no notch) - Control
As can be seen a base model was also created which contained no notch, this acted as the control
and would be useful in the setup of the photoelastic testing to ensure the reliability of the test each
time. Notice that the minimum notch radius is 0.4mm, this is due to the water jet being unable to
cut a 0.2mm notch as the diameter of the water jet is too large.
Following these test pieces, three other similar designs were manufactured as well as a male and
female bolt design (see Figure 25).
Figure 25 – Further Designs
The top design consists of a notch which is cut 2mm into the maximum cross section and has a 45°
angle, the other two angles were increased to 90° and 135° to determine if the stress concentration
would decrease as the geometry changed less abruptly. The second lower design consisted of a 2D
male and female bolt. The radius of the threads were 2mm.

28
4.2.1 Photoelastic Stress Experimentation
This section describes in detail how the photoelastic equipment works, although working on the
same fundamentals as linear polarisation explained in section 3.2.4, the equipment uses a circular
setup (see Figure 26). This section also describes the setup of equipment and software to ensure
that accurate results are produced when performing the photoelastic experiment.
4.2.1.1 Circular Polarisation
When two right-angled electric field components of circularly polarised light pass through a
birefringent test piece, they are realigned with the planes of the principal stresses within the
material. The magnitude of the principal stresses change the amplitude of the two components.
Each component is transmitted at an alternate speed and therefore exit the test piece out of phase.
The consequential electromagnetic field vector as a result of the birefringent material is elliptically
polarised and the magnitude of each perpendicular wave changes dependent on the degree of
birefringence (Hamblyn, 2011). Each wave plane can be calculated using a polarising analyser, which
will measure the magnitude of the principal stresses and therefore from the use of Mohr’s circle, the
maximum shear stress can be determined (refer back to Figures 9-11).
Figure 26 – Circular Polariscope

29
4.2.1.2 Experimental Setup of Equipment
To ensure that the equipment will output accurate results, the set-up is a very important step and
must be performed correctly. The photoelastic equipment depends highly on ambient lighting
conditions being kept to a minimum in the room. This would ensure that the camera captures
reliable shots so that the stress in the model can be examined correctly. The camera can be seen in
Figure 27.
Figure 27 – Camera Set-up
When set-up correctly the camera will take 10 still images of the test piece and produce a single
photo displayed through the Delta Vision software, the user can then analyse the stresses occurring
at different sections in the model. It is important that the camera be placed in-line and facing the
model to capture accurate results. The light source (normal white light) is transmitted from the
white box and fed through an optical fibre cable in which it passes through a projector fitted with a
polariser (see Figure 28).
Figure 28 – Light Source

30
A diffuser was used so that the light source was distributed evenly over the specimen (see Figure
29). The diffuser would also ensure that the light source would not over saturate the camera when
capturing results leading to irregular and incorrect results.
Figure 29 – Diffuser
Finally, the test piece is suspended and attached to the frame so that masses can be hung from the
lower end, straining the model and therefore putting model under stress ready for a capture to be
taken by the camera (see Figure 30).
Figure 30 – Suspended Test Piece

31
Figure 31 shows the entire set-up of the equipment prior to calibrating the Delta Vision software;
Figure 31 – Set-up of all Equipment
So that the testing is repeatable, it is important that the equipment is arranged in exactly the same
position each time a test is carried out. Hence, the author marked the table ensuring that the
equipment was placed in the same position each time. This was a very important step as the
equipment would sometimes be moved due to human error. The distance between the light source
and the test piece ensures that no oversaturation occurs whilst calibrating the software. Due to the
camera taking a number of captures in order to produce one still image, it is important that the test
piece stay still for the entirety of the test. Otherwise, the results would be incorrect and the capture
distorted.

32
4.1.2.3 Experimental Set-up of Delta Vision
Once the equipment is correctly set-up, it is now important that the software is correctly calibrated
(see Figure 32). When opening the software the first important step is to select which material the
user is testing. Then, the user will input the following values;
 E = Young’s Modulus (2.47GPa)
 K = Photo Elastic Constant (0.152)
 v = Poison’s Ratio (0.38)
 h = Material Thickness (2.1mm)
 Output Data = Shear Stress (MPa)
Figure 32 – Calibration
To ensure that the camera capture is not oversaturated, and the output of shear stress is correct;
the red, green and blue (RGB) light gains need to be calibrated so that they are the same. If for
example, the green light is much higher than the red and the blue light gains, the camera will be
oversaturated with green light. Therefore, when taking a capture, the maximum shear stress
readout will be incorrect. To ensure that the output data is correct, the user altered the RGB gains
via the following calibration setting seen in Figure 33 below.
Figure 33 – RGB Gains Calibration
As can be seen, the green light is oversaturated in comparison to the red
and the blue. Hence the green gain was brought down to the red and
blue light level. From highlighting the ‘light button’ once a test shot was
taken, the RGB gains were seen. If they were the same then the software
and equipment was fully set-up and calibrated correctly.

33
It is important to note why the equipment outputs RGB light; it is known that they are the three
main primary colours, and when added together with various degrees of intensity many colours can
be produced. However, when they are mixed together with the same intensity, white light is
obtained (see Equation 9).
Equation 9
∴ 𝑅 + 𝐺 + 𝐵 = 𝑊
From correctly calibrating the software to produce similar intensities of RGB light, a very close
representation to the true maximum shear stress was obtained from determining the average of the
three colours.
Finally, to perform the experiment, various masses were hung from the lower end of the test piece
(see Table 5). It was important that the test piece remained still throughout the experiment, the
light levels in the room remained constant and the temperature remained constant. The
temperature changing in the room could possibly alter the material properties of the tensile test
piece and therefore the windows were kept shut throughout testing to keep the temperature
change to a minimum.
Table 5 – Masses used for Experiment
Experiment 1 2 3 4 5 6
Mass
(grams)
1019.716 2039.432 3059.148 4078.864 5098.581 6118.297
Mass (N) 10 20 30 40 50 60

34
4.3 FEA Set-up and Refinement
So that the results produced from the static analysis within FEA were as accurate as possible, the
author used the finest mesh possible whilst also trying to keep the run times to a minimum. The
author followed a process within the static analysis similar to that of section 3.3.1 previously
described. However, the author used the h-adaptive method which automatically refined the mesh
where there were areas of a higher stress concentration and coarsened the mesh where the change
in stress was not large. This ensured that the results were reliable and therefore comparable to the
results obtained from the photoelastic experiments. Hence, a mesh convergence analysis was
undertaken for each model to ensure a high accuracy of the results (see Figures 34 and 35).
Figure 34 – 1.8mm radii model of tensile test piece
Figure 35 – Mesh Convergence Plot 1.8mm Radii Model
Here it can be seen that for loop number 2, the mesh has been refined enough so that the value of
the von Mises stress has plateaued by the time loop numbers 3 and 4 have been calculated.
Therefore, loop number 2 ensures that a value closest to the true stress occurring in the model is
calculated. Hence, a total of 2 loops for the h-adaptive method were used for the remaining tensile
test pieces to ensure that the accuracy of the results were high, as well as keeping the run times
down to a minimum. The stress calculated is the maximum tensile stress occurring in the model, and
can be seen in Figure 36, it occurs at the notch created in the centre of the minimum cross-section.

35
Figure 36 – Stress Occurring at Notch
A reason for using the h-adaptive method was because the method worked to a high accuracy when
generating the results. In this case, the method was inputted to be 99% accurate by the user. It is
also recommended that for single parts, the user specifies the default mesh size using the h-adaptive
method (Higgins, 2016).

36
5.0 Results
In the following section, the results from the photoelastic testing are displayed, analysed and
compared with the results from FEA and the experimental results from Peterson’s Stress
Concentration Factors. As there is a large number of results for the photoelastic testing, only Figures
37-39 are analysed by the author. Specific test pieces were chosen to be analysed, based on their
interesting points. Following this, all of the results obtained for each tensile test piece at every load
case is summarised in Figures 40-49.
Figure 37 – Base Model Test Piece at 10N
The above figure shows a screen shot of the stressed base model using the photoelastic software -
Delta Vision, as well as an exported Excel graph which represents the maximum shear stress
occurring across the centre of the model. This test piece was very important during the setup of the
testing as it would ensure the accuracy of the entirety of the test for the remaining test pieces. At
the start of each test, the author would ensure that similar results were obtained from the base
model each time. Hence, the error for each test could be taken into account, this will be discussed
later on. The graph is obtained by clicking and dragging the arrow across the test piece. It is
important to notice that the red, green and blue light sources start very closely to a value of zero.
This ensures that the test is accurate as this means that no one light source is over saturating the
image. As can be seen, a shear max value of 0.2MPa occurs across the test piece.

37
Figure 38 – 1.0mm Radii Test Piece at 40N
The above figure shows the 1.0mm radii test piece with a load of 40N being hung from the lower
end. The red, green and blue light sources all converge closely to zero at the start of where the
arrow is displaced across the model. This is similar to the base model test as this ensures that the
test is accurate again. The graph shows that the nominal shear stress occurring across the model is
1.2MPa and has a maximum shear stress of 2MPa. What may come to a surprise is that the notch on
the left hand side of the test piece displays no increase in stress concentration unlike the notch on
the right hand side of the model.
Figure 39 – 1.8mm Radii Test Piece at 60N
The above figure shows the 1.8mm radii test piece with a load of 60N being applied. Here shows a
good example of the accuracy of the software as it shows an increase in stress concentration at both
edges of the notch. As can be seen the stress has increased a lot when compared to the base model
having a load of 10N, compared to 60N for this test piece. A maximum shear stress of 3.8MPa is
shown at the notch on the left hand side, and a nominal shear stress of 2.2MPa is shown across the
centre of the test piece.

38
Figures 40-49 show the maximum tensile stress values obtained from the photoelastic testing of the
tensile test pieces of different radii. These values are also compared with the static analysis within
FEA for each tensile test piece.
Figure 40 – Graph Showing 0.4mm Test Piece Max Tensile Stress Values
The data obtained from FEA and the photoelastic experiments are summarised in the above graph.
The graph shows the maximum tensile stress for the 0.4mm radius test piece. From converting the
maximum shear stress into the maximum tensile stress using the methods shown in section 3.3, the
two methods can be compared. Due to FEA being a linear solver, the line is shown to be increasing
by the same amount as the force is increases. On the other hand, the results obtained from the
photoelastic experiments show to increase at different amounts each time the force increases.
However, there is a direct correlation between the two methods; the greatest difference being
0.4MPa at 40N.
0
1
2
3
4
5
6
7
0 10 20 30 40 50 60 70
MaxTensileStress(Mpa)
Force (N)
Max Tensile Stress for 0.4mm Radius Model
FEA Photoelastic

39
Figure 41 - Graph Showing 0.6mm Test Piece Max Tensile Stress Values
Figure 41 shows great correlation between the FEA and photoelastic test for the 0.6mm radius test
piece. Figure 42 shows the visual correlation between the FEA and photoelastic tests at 20N.
Figure 42 – Visual Correlation between FEA and Photoelastic tests
Figure 42 shows some visual correlation at the notch on the left hand side of the photoelastic model
in comparison with the FE model. However, although the value of the maximum shear stress in the
photoelastic test is showing to be close to half of the tensile stress for the FE model; there is no
symmetry in the photoelastic model whereas the FE model shows symmetry at both notches.
0
1
2
3
4
5
6
7
0 10 20 30 40 50 60 70
MaxTensileStress(MPa)
Force (N)
FEA Photoelastic

40
Figure 43 shows great correlation between the magnitudes of stress between the two methods
despite the lack of visual symmetry in the photoelastic testing.
Figure 44 shows that due to the increasing notch radius and the cross section becoming smaller, the
maximum tensile stress is increasing within the test piece for both methods. The graph shows good
correlation for most of the points apart from at 40N. At the 40N load case, there is a difference of
0.73MPa. On the other hand, both data sets show that they are increasing at the same rate and both
have a maximum tensile stress value of 7.4MPa for the photoelastic test and 7.1MPa for the FE
model.
0
1
2
3
4
5
6
7
0 10 20 30 40 50 60 70
Force (N)
FEA Photoelastic
0
1
2
3
4
5
6
7
8
0 10 20 30 40 50 60 70
Force (N)
FEA Photoelastic

41
Figure 45 shows the data obtained from both methods. Both data sets show an increase in stress at
very similar rates. However, from 40-50N the photoelastic method shows an increase in stress of
2.08MPa as appose to 1.2MPa shown from the FE model.
Figure 46 shows the data obtained for the 1.4mm radius test piece for both methods. Again both
methods show a very similar rate of rise in the maximum tensile stress. This graph differs from the
others so far as the results of the photoelastic test is constantly greater than the results of the FE
model. On the other hand, some interesting and accurate comparisons from both methods can be
compared (see Figure 47).
0
1
2
3
4
5
6
7
8
0 10 20 30 40 50 60 70
Force (N)
FEA Photoelastic
0
1
2
3
4
5
6
7
8
9
0 10 20 30 40 50 60 70
Force (N)
FEA Photoelastic

42
Figure 47 – Visual Correlation between FEA and Photoelastic tests
Figure 48 - Graph Showing 1.6mm Test Piece Max Tensile Stress Values
Here, a visual representation of how the shear stress changes
for the photoelastic test piece and how the tensile stress
changes for the FE model can be compared. The photoelastic
test piece shows great symmetry for this load case, which
highlights how accurate the software can be given the correct
conditions. The symmetry in the FE model shows that both
methods correlate since;
2𝜏 𝑚𝑎𝑥 = 𝜎 𝑚𝑎𝑥
Hence; 2 × 3.4𝑀𝑃𝑎 ≈ 6.3𝑀𝑃𝑎
0
1
2
3
4
5
6
7
8
0 10 20 30 40 50 60 70
Force (N)
FEA Photoelastic

43
Figure 48 represents the two data sets obtained from testing the 1.6mm radius test piece. As can be
seen from the above graph, the correlation between the two sets of data is at its best here with each
point only differing slightly in magnitude. At load cases 10, 30 and 50N the maximum difference in
the magnitudes is 0.175MPa, the closest being within 0.01MPa.
Finally, Figure 49 shows the data sets obtained from the two methods for the 1.8mm radius test
piece. The graph shows correlation as the stress increases, however the magnitudes of the stresses
are not the same following the 20N load case for this test; as can be seen from the results, the
greatest difference in stress is 1.65MPa for the 60N load case. The FE model shows a maximum
tensile stress of 7.65MPa whereas the photoelastic testing states a value of 6MPa.
Table 6 Summarises the Stress concentration for each tensile test piece. The stress concentration
obtained from FEA, Peterson’s Stress Concentration Factors and the photoelastic testing are
summarised in Figure 50.
Table 6 – Stress Concentration Factors
0
1
2
3
4
5
6
7
8
9
0 10 20 30 40 50 60 70
Force (N)
FEA Photoelastic
0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
Peterson's 1.739572405 1.85264464 1.819462096 1.754792096 1.705567905 1.666180677 1.633310702 1.578205976
FEA 1.713893654 1.71403913 1.714795918 1.714152265 1.713895216 1.714190115 1.714201741 1.714213368
Photoelastic 1.804746483 1.87383178 1.760874593 1.735874536 1.668972708 1.682142188 1.659087614 1.641220438

44
Figure 50 – Stress Concentration Factors
As can be seen, FEA is a linear solver and therefore the stress concentration factor stays constant.
The stress concentration factor for the photoelastic testing was determined from Equation 10.
Equation 10
𝐾𝑡 =
𝜎 𝑚𝑎𝑥
𝜎 𝑛𝑜𝑚
(Pilkey & Pilkey, 2008)
As can be seen, the stress concentration factors obtained from the photoelastic testing follow a
similar trend to the stress concentration factors obtained from Petersons. They both show to
increase at first and then decrease as the notch radius increases.
Following these set of results, the three designs in Figure 25 which contained the triangular shaped
notches at angles 45°, 90° and 135° were tested. The aim of these designs was to determine if the
stress concentration factor increased with how abruptly the geometry changed. For this reason, the
test was performed by firstly using the base model as a control and using a load of 10-60N in 10N
increments.
Due to these designs not following a typical trend within Peterson’s Stress Concentration Factors,
and with the stress concentrations of the previous photoelastic models following a similar trend to
Petersons; the author set out to determine the stress concentration factors of the triangular shaped
notches (see Figure 51).
1.55
1.6
1.65
1.7
1.75
1.8
1.85
1.9
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
StressConcentrationFactor(Kt)
Notch Radius (mm)
Stress Concentration Factor (Kt)
Peterson's FEA Photoelastic

45
Figure 51 – Stress Concentration Factors of the Triangular Shaped Notches
The above graph shows that the stress concentration factor decreases very linearly for these type of
designs. The maximum value being 2.7 for the 45° notch, which is much higher than the maximum
value in the original semi-circular designs. The minimum value being 1.85 for the 135° notch. The
values obtained in the graph are a result of calculating the average stress concentration factor across
all of the load cases.
For the authors experimental work, the 2 Dimensional bolt design was tested through the use of
photoelasticity (see Figures 52-55). As the method has showed very similar results to FEA for the
semi-circular notch designs, the author wished to use the photoelastic software to determine the
true shear stress occurring throughout the bolt. The bolt was tested from 10N up to 40N in 10N
increments. The test was not completed up to the 60N load case as the female bolt would slip due to
the increase in load. This was not due to an improper fitment as the top end of the female test piece
splayed outward.
1.5
1.7
1.9
2.1
2.3
2.5
2.7
2.9
45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135
StressConcentrationFactor(Kt)
Notch Angle (degrees)
Stress Concentration Factor (Kt)
Photoelastic Triangular Notches

46
Figure 52 – Bolt Test at 10N
Due to the male and female bolt mating improperly at the top section, a stress concentration did not
occur there. Hence the entire load was held at the lower end resulting in increased stress in this
section. Prior to testing, the female bolt cracked at the lower end due to a tight fitment between the
two test pieces. Therefore, a rise in stress was located at the crack.
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0 50 100 150 200 250 300 350
ShearStress(MPa)
Arrow Displacement (mm)
Bolt Test at 10N
Red Green Blue
Location of Crack.

47
Very similar results were obtained for the 20N load case when concerning the locations of stress.
However, it can be seen that the stress has increased in these locations. Likewise to the first load
case, there is an increase in stress where the threads come into contact.
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
0 50 100 150 200 250 300 350
ShearStress(MPa)
Bolt Test at 20N
Red Green Blue

48
Due to the female bolt slipping, the author tied the top end with string to hold the bolt in place. The
location of the string can be seen at the top end of the threads. Increased stress can be seen to
occur at the location of the crack. The location of the increased stress can be seen in the above
graph on the right hand side. Again, similarity is seen with the locations of increased stress between
the load cases.
0.00
0.20
0.40
0.60
0.80
1.00
1.20
1.40
0 50 100 150 200 250 300 350
ShearStress(MPa)
Bolt Test at 30N
Red Green Blue

49
The screen shot undertaken from applying the 40N load shows the location of the crack in the
female model the best. 40N was the maximum load case that could be attached to the model whilst
testing. This was due to the female slipping as the load was increased. The slipping can be seen in
the above graph and is represented by the spike of rapidly decreasing stress. Again an increase in
stress can be seen where the threads come into contact with the female bolt design.
0.00
0.20
0.40
0.60
0.80
1.00
1.20
1.40
0 50 100 150 200 250 300 350
ShearStress(MPa)
Bolt Test at 40N
Red Green Blue

50
5.1 Experimentation Error
As previously mentioned, the control to the photoelastic testing was used to calibrate the
photoelastic equipment and software at the start of each test. The shear stress determined from
each control was noted so that an average could be calculated. From this average, a percentage of
error could be calculated for each test piece. The error for each test piece is summarised in Tables 7-
9.
Table 7 – Experimental Error – (U Shaped Semi-circular Notches)
Test Piece Number Radius (mm) Control Shear Stress
(MPa)
Experimental Error
(%)
1 0.4 0.21 3.99
2 0.6 0.195 3.29
3 0.8 0.208 3.06
4 1.0 0.22 8.35
5 1.2 0.186 7.75
6 1.4 0.214 5.78
7 1.6 0.202 0.19
8 1.8 0.178 11.72
Average = 0.201625
Table 8 – Experimental Error – (V Shaped Triangular Notches)
Test Piece Number Notch (degrees) Control Shear Stress
(MPa)
Experimental Error
(%)
1 45 0.21 2.38
2 90 0.196 4.39
3 135 0.209 1.91
Average = 0.205
Table 9 – Experimental Error – (2 Dimensional Bolt)
Test Piece Number Control Shear Stress
(MPa)
Experimental Error (%)
1 0.23 9.13
1 0.185 11.48
1 0.226 7.52
1 0.195 6.70
Average = 0.209

51
6.0 Analysis
In the following section, the results from the photoelastic software are discussed. This will include
the accuracy and set up of the experiment as well as the stress distribution throughout the tensile
test pieces.
6.1 Photoelastic Stress Experimentation
Calibrating the photoelastic equipment and software has proved difficult as extensive time was
required in the lab to correctly generate accurate results. Although the magnitude of the stress
determined within the photoelastic testing correlated with FEA, rarely did the photoelastic results
show signs of symmetry. On the other hand, FEA always showed the distribution of stress
throughout the model was symmetrical.
6.1.1 U-Shaped Semi-circular Notches
When correctly calibrated such that no one light source over-saturated the camera, the photoelastic
testing proved very accurate when compared with FEA for these designs. The largest possible error
calculated throughout testing was for the 1.8mm radius test piece having an experimental error of
11.72% (see Table 7). The magnitude of the stress between the photoelastic testing and FEA
generally correlated perfectly well for most test pieces. Both FEA and photoelastic testing showed
similar rates of increasing stress as the load was increased. However, the stress did not always
correlate; seen in the 1.8mm radius test piece seen in Figure 49. The load case of 60N showed a
difference in stress of 1.65MPa and was out by 21.57% when compared to FEA (see Figure 49). This
stress difference is a result of some uncontrollable input factors such as the light and the heat in the
room changing.
6.1.2 V-Shaped Triangular Notches
The aim of producing these test pieces was to determine if the stress concentration increased with a
more abruptly changing geometry design. The testing performed can be granted as successful as
experimental error was kept low. However, due to there not being comparable results within
Peterson’s Stress Concentration Factors, these results rely solely on the photoelastic software being
accurate. The reason being that these results are not comparable is that the V-Shaped notches
within Peterson’s show a slightly more curvaceous or inclined notch design when compared to the
manufactured test pieces (Pilkey & Pilkey, 2008). From the previous experiment comparing the U-
Shaped notches in photoelastic testing and FEA (see Figures 40-49) it can be seen the tensile stress
results roughly have a 10% difference. Due to this relatively small difference the test can be
considered quite accurate. As well as this, the U-Shaped notches showed a very similar trend to
Peterson’s Stress Concentration Factors, giving confidence in the V-Shaped notches results. Although
this is the case, without other experimental data showing correlation, the only factor determining
how correct the results are is the accuracy of the photoelastic software (not taking into account the
limitations).

52
6.1.3 Two-dimensional Bolt Design
This design brought on some complications to the testing and cannot be considered successful for
the following reasons;
 The male and female test pieces were a tight fit which in turn resulted in the generation of
the crack in the female test piece.
 Improper fitment between the two test pieces resulted in increased stress in other regions.
 Improper fitment resulted in only 4 out of 6 load cases being able to be tested due to
slippage.
From the captures taken (see Figures 52-55), the bolt shows that there is no change in stress at the
top half. This is due to the female test piece splaying outwards at the top end. This then caused
greater stress occurring at the lower half. The entire load acted on the last 6 threads which resulted
in increased stress. As there was an improper fitment, the generation of the rapidly decreasing stress
spike for the 40N load case was a result of the threads no longer coming into contact with one
another (see Figure 55).
Although the above regions determine that the stress generated was incorrect, some good points of
the test can be determined and are as follows;
 Increased stress where the threads come into contact.
 Increased stress at crack.
As expected, the stress within the bolt increased where the threads of the male come into contact
with the threads of the female. The crack formed due to the stress concentration increasing at the
corner of the female test piece. As well as this, the stress can be seen to rapidly rise in this region as
a result of the formation of the crack (see Figures 52-55).
6.2 Limitations
The ambient light levels in the room were kept as constant and as low as possible as the software is
very dependent on this factor for producing accurate results. Although the aim was to keep the light
in the room constant, this was not the case. As the lab contained large windows, and no black out
blinds, the light levels would constantly change. Due to this uncontrollable factor having an effect on
the results, the generation of the true stress occurring in the test piece can never be exact. As well
as this, the properties of the tensile test pieces could change due to the temperature in the room
changing. Although this was recognised and as previously mentioned, controlled to an extent, the
temperature was never truly constant. This therefore would have an effect on the results obtained.
6.3 FEA
The set up and refinement of the static analysis within FEA proved no difficulty and showed very
similar results to the photoelastic tests. The h-adaptive method proved very helpful as the mesh was
refined in regions where the stress changed dramatically whilst also limiting convergence times.
However, FEA is known as a linear solver which is unrealistic as not one real-world material deforms
linearly. This was especially proven when comparing the stress concentration within FEA showing no
change when compared to the photoelastic method and Peterson’s. Although this is the case, FEA
proved extremely accurate as itself and the photoelastic method generated very close results.

53
7.0 Conclusion
Comparisons between the photoelastic method, FEA and Peterson’s Stress Concentration Factors
were made throughout this report. These methods were compared in order to determine whether
the photoelastic method undertaken was accurate. Although the locations of stress rarely showed
symmetry within the photoelastic test pieces, the magnitude of the maximum shear stress agreed
with the magnitude of the maximum tensile stress within FEA. The comparisons were made
following the conversion of maximum shear force in relation to a maximum tensile force in section
3.3. FEA would always show symmetry along the midsection of the test piece, showing how idealised
the simulation setup actually is.
Comparing the stress concentration factors with Peterson’s showed very similar results and thus
proved that the experiment was indeed very accurate as it now agreed with the two alternate
methods questioned within this project. Although Rudolph Earl Peterson has been Mr. Stress
Concentration for the past half century, the formulas used throughout Peterson’s Stress
Concentration Factors are also idealised. The formulas assume that the material is isotropic and
homogeneous, which is never the case (Pilkey & Pilkey, 2008).
The author believes this project to be a success as the photoelastic results proved accurate when
compared with FEA and Peterson’s.

54
8.0 References
Awang, M., Ibrahim, E. M. & Muhammad, D., 2015. Finite Element Modeling of Nanotube Structures:
Linear and Non-linear Models. 1st ed. Chicago: Springer.
Coker, E. G. & Filon, L. N. G., 1957. A Treatise on Photo-Elasticity. 2nd ed. Cambridge: Universit Press,
Cambridge .
Dowling, N. E., 2013. Mechanical Behaviour of Materials. 4th ed. Essex: Pearson Education Limited .
Hamblyn, A., 2011. Photo-elastic Stress Analysis , Swansea: s.n.
Hearn, E. J., 2013. Mechanics of Materials: An Introduction to the Mechanics of Elastic and Plastic
Deformation of Solids and Structural Components. 2 ed. Exeter: Elsevier.
Higgins, J., 2016. Using Adaptivity for automatic mesh convergence. [Online]
Available at: http://www.javelin-tech.com/newsletter/tech_old/2008/july/article_adaptivity.htm
[Accessed 8 April 2016].
Higgins, R. A., 1997. Materials for the Engineering Technician. 3rd ed. New York: John Wiley & Sons,
Inc.
Jiju, A., 2014. Design of Experiments for Engineers and Scientists. 2nd ed. s.l.:Elsevier.
Juvinall, R. C., 1967. Engineering Considerations of Stress, Strain and Strength. 1st ed. New York:
McGraw-Hill, Inc.
Khan, A. S. & Wang, X., 2001. Strain Measurements and Stress Analysis. 1st ed. London: Prentice-Hall
Inc .
Lake, K., 2015. Advanced Computational Methods. [Online]
Available at: http://moodle.swanseamet.uwtsd.ac.uk/course/view.php?id=778
Li, F., 2010. Study of Stress Measurement Using Polariscope, Georgia: Georgie Institute of
Technology.
Mac Donald, B. J., 2011. Practical Stress Analysis with Fintite Elements. 2nd ed. Dublin: Glasnevin
Publishing .
Mathur, A. B. & Bhardwaj, I. S., 2003. Testing and Evaluation of Plastics. 1st ed. Mumbai: Allied
Publishers.
Nave, R., 2016. Refraction of Light. [Online]
Available at: http://hyperphysics.phy-astr.gsu.edu/hbase/geoopt/refr.html
Pilkey, W. D. & Pilkey, D. F., 2008. Peterson's Stress Concentration Factors. 3rd ed. United States of
America: John Wiley & Sons, Inc .
Stress Photonics, 2016. GFP - 1000 PSA. [Online]
Available at: http://www.stressphotonics.com/PSA/PSA_Intro.html
[Accessed 5 March 2016].

55
Stress Photonics, 2016. GFP 1000 - PSA. [Online]
Available at: http://www.stressphotonics.com/PSA/PSA_Intro.html
Univeristy of Cambridge , 2015. Polymer Stress-Strain Curve. [Online]
Available at: http://www.doitpoms.ac.uk/tlplib/polymers/stress-strain.php
Williams, M. S. & Todd, J. D., 2000. Structures Theory and Analysis. 1st ed. London: MACMILLAN
PRESS LTD.

56
9.0 Appendix
9.1 U-Shaped Semi-circular Notches
9.1.1 10N Load Case
0.4mm Radius
0.6mm Radius
0.8mm Radius

57
1.0mm Radius
1.2mm Radius
1.4mm Radius

58
1.6mm Radius
1.8mm Radius

59
9.1.2 20N Load Case
0.4mm Radius
0.6mm Radius
0.8mm Radius

60
1.0mm Radius
1.2mm Radius
1.4mm Radius

61
1.6mm Radius
1.8mm Radius

62
9.1.3 30N Load Case
0.4mm Radius
0.6mm Radius
0.8mm Radius

63
1.0mm Radius
1.2mm Radius
1.4mm Radius

64
1.6mm Radius
1.8mm Radius

65
9.1.4 40N Load Case
0.4mm Radius
0.6mm Radius
0.8mm Radius

66
1.0mm Radius
1.2mm Radius
1.4mm Radius

67
1.6mm Radius
1.8mm Radius

68
9.1.5 50N Load Case
0.4mm Radius
0.6mm Radius
0.8mm Radius

69
1.0mm Radius
1.2mm Radius
1.4mm Radius

70
1.6mm Radius
1.8mm Radius

71
9.1.6 60N Load Case
0.4mm Radius
0.6mm Radius
0.8mm Radius

72
1.0mm Radius
1.2mmm Radius
1.4mm Radius

73
1.6mm Radius
1.8mm Radius

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