Stress Analysis Methodology Comparison

1 ABSTRACT

Three stress analysis methodologies were conducted in order to obtain stress values in different

locations of a mild steel specimen containing an eccentric hole under in-plane uniaxial tension from

a 10kN applied load. These were; a theoretical analysis, using assumptions and analytical equations,

experimental analysis, using strain gauges and a tensile testing machine and finally a computational

analysis, using Finite Element Software called Abaqus. The stresses for the gross area, net area and

hole area were computed. Each methodology agreed on the stress distribution, with the maximum

present in the hole area, the second highest in the net area and the least in the gross area. Large

percentage differences existed in the gross area between theoretical and experimental, and

computational and experimental, 23.04% and 21.99% respectively. Retesting of the mild steel

specimen, rotating the specimen 180° between loadings, revealed the inducement of a bending

moment from the tensile testing machine. Averaging all the experimental results obtained and

comparing the results from each methodology, all results are seen to correlate well. Theory and

experimental differ within an average of 10%, theory and computational differing within an average

of 5.1% and computational and experimental with 4.8%.

2 NOMENCLATURE

A Cross Sectional Area m

E Young’s modulus Pa

E0 Bridge voltage V

F Applied force N/m2

H Width of the element m

M Bending moment N/m

Ks Gauge factor %

Ktg Gross section stress concentration factor -

Ktn Net area stress concentration -

R1-4 Resistance components of the Wheatstone bridge Ω

a Radius m

c Distance from hole centre to the closest edge of the specimen m

d Diameter m

e0 Wheatstone Bridge output voltage V

h Element thickness m

l Length m

n Misalignment error µε

β Angle of mounting angular error °

ε Strain με

εp Maximum principal strains με

εq Minimum principal strains με

ε(φ±β) Strain along gage axis with angular mounting error με

ν Poisson's Ratio -

σ Stress N/m2

σmax Maximum stress N/m2

σn Net stress N/m2

σnet A-B Average stress line A-B N/m2

Δl Change in length m

ΔR Change in resistance Ω

φ Angle between intended measurement axis and the principal axis °

3 INTRODUCTION

The analysis of stress and strain in a given structural material is often partaken as a method to

validate a design and to ensure structural integrity. Strain can be defined as the amount of

deformation an object undergoes in response to an applied force. This deformation induces stresses

in the material body, analytically expressed as the ratio of the applied force to the cross sectional

area of the material body. Stresses may be induced in a structural material from various factors,

such as manufacturing or fabrication processes, in-service repair or modification, and installation or

assembly procedures. Significant stress(es) and/or cyclic stress(es) may induce damage,

detrimentally leading to failure in a material. As a result the analysis of this parameter is of essential

importance in engineering.

Many methods of stress analysis exist, some of which include;

Theoretical analysis- uses assumptions and analytical equations to arrive at stress values,

Computational analysis- uses computer-aided design to model the component of interest

and appropriate mathematical models, solved using various methods i.e. finite element

method, to obtain stress values,

Digital image correlation- uses a full-field image analysis method, based on grey value digital

images that can determine the contour and the displacement of an object under load and in

turn stress (DIC, 2015).

Strain Gauges- bonded to the material surface, computing strain values through the variance

of electrical resistance proportional to displacement of its gage length as a result of strain in

the area of application,

The choice of stress analysis methods is dependent on various factors, such as geometry complexity,

ease of access to the points of interest, accuracy required, time, and so forth. Each method provides

its own advantages. DIC provides a fast and contactless full field stress analysis, which can prove

beneficial in pinpointing problem areas. Computational analysis methods, although timely, can

compute accurate stress values and many other parameters throughout the entire structure under

analysis. This method is, however, subjective to the skillset of the engineer conducting the analysis.

Theoretical calculations prove a quick and efficient tool in obtaining approximate stress values at

certain points in a structures material. Strain gauges, when applied accurately, prove an accurate

and reliable tool in computing the stress in a material. Identifying the areas of significant stresses

and applying strain gauges, the stress can be obtained with minimal error. The coupling of methods

is often conducted, for example the use of DIC or computational analysis to identify areas of

significant stress and employing strain gauges in these locations to obtain accurate stress values.

In this report a mild steel bar with an eccentric transverse hole is under analysis. The specimen was

subject to in-plane uniaxial tension from the application of a 10kN load. The eccentric hole acts as a

stress concentrator, due to the discontinuity in the material i.e. reduction in cross-sectional area,

leading to a non-uniform stress pattern through the material. This non-uniformity will allow the

comparison of different stress analysis methods. The methods conducted were theoretical analysis,

computational analysis and experimental analysis i.e. strain gauges. The theoretical used

assumptions and known equations to compute the stress at various points in the specimen. Abaqus,

a finite element method software, was used to conduct the computational analysis. Finally a test

specimen, containing three strain gauges, was subjected to a uniaxial load within the elastic region.

The data was processed using a data acquisition logger and results were obtained. The details of

each analysis will be outlined further in the report. Each analysis generated results for comparison

and discussion.

3.1 OBJECTIVES The objectives of the work involved in this paper are as follows;

Obtain stress values through the use of assumptions and theoretical equations

Computationally model the specimen and conduct a stress analysis

Apply strain gauges to a mild steel specimen to compute stress values under experimental

uniaxial in-plane tension

Compare all results obtained

3.2 SPECIMEN SPECIFICATIONS The specimen under analysis was a mild steel bar containing an eccentrically positioned transverse

hole, illustrated in figure 1. Note the specimen has a thickness, t = 6mm, and material properties

displayed in table 1.

Figure 1: Mild steel specimen and dimensions

Young’s Modulus, E 200GPa

Poisons ratio, ν 0.33 Table 1: Properties of specimen

3.3 THEORY

3.3.1 Theoretical Analysis

Firstly, in order to conduct the theoretical analysis, certain assumptions were made about the

material. The material was assumed to be;

Linear

o Obeys Hooke’s law.

Homogeneous

o Material properties are the same at every point and are invariant upon translation.

Isotropic

o Material properties are invariant of rotation of axes at a point.

Elastic

o Deformations due to the applied load are completely and instantaneously reversible

upon load removal.

Under these assumptions, well known and well defined equations may be used to analysis different

properties of the stressed specimen.

Stress can be calculated at a given point of constant section i.e. in the absence of stress

concentrators, using the following equation:

𝜎 =

𝐹

𝐴

(Rees, 2000) [1]

Where:

𝜎 = Stress

F = Force Applied

A = Cross-Sectional Area

Strain can be calculated at a given point, in the absence of stress concentrators, in the geometry

using the following equation:

𝜀 =∆𝑙

𝑙 (Rees, 2000) [2]

Where,

ε = Strain

∆l = Change in Length

l = Original Length

The Young’s Modulus can be evaluated using the following equation:

𝐸 =𝜎

𝜀 (Rees, 2000) [3]

Where,

E = The Young’s Modulus

σ = Stress

ε = Strain

When the mild steel bar undergoes in-plane uniaxial tension, the eccentrically placed transverse

hole reduces the cross-sectional area in the middle of the specimen and results in localized high

stresses denoted stress concentration. This stress concentration is measured by the stress

concentration factor, which is defined as the ratio of the peak stress in the material body to a

reference stress. The stress concentration factor Ktg has a reference stress based on the gross cross

sectional area, and Ktn has the reference stress based on the net cross-sectional area. For a two

dimensional element with a single hole, the formulas for these stress concentration factors are:

𝐾𝑡𝑔 =𝜎𝑚𝑎𝑥

𝜎 (Walter & Pilkey, 2008) [4]

Where Ktg is the stress concentration factor based on the gross stress, σmax is the maximum stress at

the edge of the hole, σ is the stress on gross section far from the hole and,

𝐾𝑡𝑛 =𝜎𝑚𝑎𝑥

𝜎𝑛 (Walter & Pilkey, 2008) [5]

Where Ktn is the stress concentration factor based on net (nominal) stress, σn is the net stress σ/(1-

d/H), with d the hole diameter and H the width of element.

These stress concentration factors have been solved analytically by many investigators and are

available schematically for given criterion. The chart used in order to obtain the stress concentration

factors is presented in figure 2. The average stress present from the line A-B, the minimum net

section, schematically present on the RHS of figure 2, can be calculated from the nominal stress as;

𝜎𝑛𝑒𝑡 𝐴−𝐵 =𝜎𝑐ℎ√1−(

𝑎

𝑐)2

ℎ(𝑐−𝑎)[1−(𝑒

𝑐)(√1−(

𝑎

𝑐)

2)]

(Walter & Pilkey, 2008) [6]

Where 𝜎 is the gross area stress, c is the distance from the centre of the eccentric hole to the closest

edge of the specimen, h is the thickness of the specimen, a is the radius of the hole and e is the

distance of the centre of the hole to the furthest edge of the specimen.

Figure 2: Stress concentration factors for the tension of a finite-width element having an eccentrically loaded circular hole (based on mathematical analysis of Sjostrom 1950 (Walter & Pilkey, 2008))

3.3.2 Experimental analysis

3.3.2.1 Strain gauge theory

It is not possible to measure stress directly from experimentation; however it is possible to measure

strain through changes in length resulting from an applied load, i.e. equation 2. However the

displacement of a simple metal specimen, loaded within the elastic limit, is often small and hard to

measure. In order to accurately measure this displacement the application of strain gauges is

necessary. Various types of strain gauges exist for a given application, which predominantly function

as transducers, varying electrical resistance in proportion to the amount of strain in a given

component. The most widely used strain gauge is the bonded metallic strain gauge. This type of

gauge consists of a fine wire, fabricated from metallic foil, arranged in a grid pattern. The grid

arrangement maximises the amount of foil subject to strain, thus increasing accuracy. The grid is

bonded to a thin backing, referred to as a carrier, which is directly attached to the specimen. The

strain induced in the specimen lengthens the grid length, increasing electrical resistance linearly.

Due to the fact that strain is significantly small in value, a Wheatstone is required to accurately

convert the resistance change of the displaced gauge length into a voltage change. The output

voltage e0 of a Wheatstone bridge may be determined from the following equation:

𝑒0 =𝑅1𝑅3−𝑅2𝑅4

(𝑅1+𝑅2)(𝑅3+𝑅4). 𝐸0 (KYOWA, 2015) [7]

Where R1-4 are resistance components of the Wheatstone bridge in figure 3, and E0 is the Bridge

voltage.

Letting R1=ΔR, the change in strain gauge resistance, and taking the initial condition that

R1=R2=R3=R4, the above equation becomes;

𝑒0 =𝑅2+𝑅∆𝑅−𝑅2

(2𝑅+∆𝑅)2𝑅. 𝐸 (KYOWA, 2015) [8]

As R may be considered extremely larger than ΔR, the equation reduces to;

𝑒0 =1

4.

∆𝑅

𝑅. 𝐸 =

1

4. 𝐾𝑠. 𝜀. 𝐸 (KYOWA, 2015) [9]

Where Ks is the gauge factor, i.e. the coefficient expressing strain gauge sensitivity. Thus what is

obtained is an output voltage that is proportional to the change in resistance.

Figure 3: Wheatstone bridge configuration (KYOWA, 2015)

3.3.2.2 Strain gauge misalignment theory

In bonding strain gauges, angular errors with respect to the intended axis of strain can occur. This

misalignment will incur inaccurate results and must be factored for. The error induced as a result of

angular misalignment may be calculated from equation;

𝑛 =𝜀𝑝−𝜀𝑞

2[𝑐𝑜𝑠2(𝜙 ± 𝛽) − 𝑐𝑜𝑠2𝜙] (VPG, 2015) [10]

Where n is the error (με), εp and εq are the maximum and minimum principal strains respectively, 𝜙

is the angle between intended measurement axis and the principal axis, and 𝛽 is the angle of

mounting angular error ie. angle of strain gauge to intended axis. The actual strain may then be

found from the following equation;

𝑛 = 𝜀(𝜙±𝛽) − 𝜀𝜙 (VPG, 2015) [11]

Where 𝜀(𝜙±𝛽) is the strain along gage axis with angular mounting error of ±𝛽, 𝜀𝜙 is the strain along

the axis of intended measurement at the angle 𝜙 from the principal axis.

Figure 4: Polar strain distribution corresponding to uniaxial stress, illustrating the error in indicated strain when a gauge is misaligned by ±𝛽 from the intended angle 𝜙 (VPG, 2015)

4 METHODOLOGY

4.1 EXPERIMENTAL ANALYSIS

4.1.1 Strain gauge application

Three 120Ω Kyowa gauges were used in the experimental analysis. Two gauges were 5mm strain

gauges with a gauge factor of 2.1±1.0% and the other was a 2mm with a gauge factor of 2.13±1.0%.

Gauge 6 was placed in the gross section i.e. area of maximum section, gauge 7 was positioned in the

net area i.e. area of minimum section and gauge 10 was positioned on the inside of the hole where

the highest stress is to be expected.

Figure 5: Mild steel test specimen with applied strain gauges 6,7 & 10

Gauge 6

Gauge 7

Gauge 10

The strain gauges were applied in the following manner;

The area of application on the specimen was prepared by initially sanding the area, to

minimise imperfections, and secondly wiped with acetone, to clean the area.

Conditioner A and Neutraliser where applied with a medical gaze to further clean the

surface.

Guidelines were implemented on the test specimen to ensure application accuracy. Note

these lines were not applied to the application area, however in close proximity.

A strain gauge was positioned on a glass slide with a tweezers and attached to Mylar tape to

minimise handling.

The Mylar tape containing the Strain gauge was

o Positioned to the clean surface, ensuring alignment with the guidelines.

o Pulled at a shallow angle to allow the application of glue under the strain gauge.

o Depressed with a finger and allowing sufficient time for the glue to dry.

o The tape was then removed

A terminal was positioned on the specimen and soldered to the lead wires of the strain

gauge.

An appropriate plug, for connection to the data acquisition system, was then soldered to the

terminal.

To ensure no electrical shorting, as a result of the soldering process, an ohmmeter was used

to read the resistance through the unstrained gauge.

4.1.2 Experimental Apparatus & Procedure

The experimental analysis was conducted using a Tinius Olsen H25KS Tensile Testing Machine,

illustrated in figure 6.

Figure 6: Tensile Testing Machine

With the strain gauges connected to the data acquisition system, the test specimen was;

o Secured in the tensile testing machine via the two 8mm diameter grip holes.

o The specimen was then loaded from 0 to 10kN at a displacement rate of 1mm/min.

The data acquisition system converts the change in resistance into strain, which it logs and saves for

extraction. The Tenius Olsen computer control system logs and saves the load-displacement

behaviour for extraction.

4.2 COMPUTATIONAL ANALYSIS

4.2.1 Abaqus, Finite Element Modelling

Abaqus/CAE 6.14-3 (Finite Element Analysis Software) was used to computationally compute the

strains within the specimen. The specimen was modelled in the following manner:

1. The specimen was modelled using the computer-aided design feature in Abaqus. Availing of

symmetry, a quarter of the specimen was modelled. The part was created using the material

properties in table 1.

2. The part was partitioned appropriately in order to generate an accurate mesh.

Figure 7: modelled and partitioned specimen

3. Boundary conditions were created, constraining the bottom surface in y and z-direction and

the right hand surface in the x and z-direction. A reaction point was created offset from the

part.

Figure 8: Boundary conditions and reaction points

4. The reaction point was then constrained to the inside of the quartered modelled grip hole

i.e. displaces proportionally to the reaction point. This will allow for the load to be applied.

Figure 9: Grip hole constrained to reaction point

5. A quadrilateral mesh was applied with a seed size of 0.1, containing 9463 nodes and 9179

elements. 9164 of the elements were linear quadrilateral (CPS4R) and 15 were linear

triangular (CPS3).

Figure 10: Meshed Model

6. A load of -5000N was applied to the reaction point in the x-direction. The load was halved to

factor in the symmetrical modelling of the specimen.

5 RESULTS

5.1 THEORETICAL ANALYSIS Three theoretical calculations were made: (1) stress in gross area, (2) stress at the area of minimum

section and (3) stress at the edge of the eccentric hole.

1. Gross area

a. Calculating the CSA as width x height yields 2.28x10-4m2

b. Knowing the applied load, 10kN, and through use of equation 1, the stress is

calculated to be 43.86MN/m

2. Net area

a. Through use of the gross section stress, and geometric dimensions of the test

specimen, the averaged stress from the edge of the hole to the side of the specimen

(presented schematically as line A-B in figure 1) was calculated using equation 6. The

minimum area stress was calculated to be 63.4MN/m

3. Hole area

a. Using the chart in figure one the stress concentration, Ktg, was calculated to be 3.6.

b. With knowledge of the gross area stress, and rearranging equation 4 the maximum

stress was calculated to be 157.9MN/m

From these stresses, rearranging equation 3, the corresponding strains may be calculated, and are

presented in table 3, section 5.4.

5.2 EXPERIMENTAL ANALYSIS

Figure 11: Load versus displacement of mild steel bar under uniaxial tension

Figure 11 displays the extension of the mild steel load cell under uniaxial in-plane tension. The

specimen experiences a maximum displacement of 1.01mm @ 10kN. The spcimen displaced

proportionally with the induced load. This linear fashion, and the absence of yielding or plastic

deformation, verifies that the specimen was loaded within the elastic region, confirming that the

specimen is elastic and obeys Hooke’s laws, and further confirming the appropriate use of

theoretical equations.

Figure 12: Stress versus strain of load cell under uniaxial in-plane tension

Figure 12 presents the experimental results obtained from the strain gauges, adjusted for angular

misalignment. As expected, strain gauge 10 experienced the highest stress, 129.8 MN/m, as localised

stresses are most dominant at the perimeter of the hole nearest the edge of the load cell. Strain

gauge 7 experienced the second highest stress, 62.83 MN/m, as it was positioned in the area of

0

0.2

0.4

0.6

0.8

1

1.2

0 2000 4000 6000 8000 10000 12000

Dis

pla

cem

ent

(mm

)

Load (N)

0.E+00

2.E+07

4.E+07

6.E+07

8.E+07

1.E+08

1.E+08

1.E+08

0.E+00 1.E-04 2.E-04 3.E-04 4.E-04 5.E-04 6.E-04 7.E-04

Stre

ss (

N/m

)

Strain

Strain gauge 6

Strain gauge 7

Straing gauge 10

minimum section. Strain gauge 6 experienced the lowest stress, 34.8 MN/m, as it is positioned in the

gross area.

5.3 COMPUTATIONAL ANALYSIS Computation

Figure 13: Computational Strain Scalar Scene In the x direction

Figure 13 displays the strain results from the computational analysis. The intensity of the colour on

the specimen indicates the severity of the local strain and can be estimated from the legend on the

left hand side. Note that this strain is in the x-direction, normal to the applied load. The colour

contour verifies the stress trend obtained from the theoretical and experimental analysis.

Figure 14: Strain gauge placement approximations

Approximate probes were implemented to obtain strain values comparable to the theoretical and

experimental results. These values are located in table 2, section 5.4.

5.4 RESULTS COMPARISON Theoretical Strain

(µε) Experimental Strain

(µε) Computational Strain

(µε)

Gross area 219.3 174 217

Hole area 789.5 650 710

Net area 317 314 305 Table 2: Strain as obtained from the methods employed

Theory & Experimental

Theory & Computational

Computational & Experimental

Gross area 23.04% 1.05% 21.99%

Hole area 19.38% 10.6% 8.82%

Net area 0.95% 3.85% 2.91% Table 3: Percentage difference between results obtained

6 DISCUSSION

Figure 12 illustrates the stress and strain experienced by the strain gauges, as the 10kN was applied.

It is evident that strain gauge 10 experienced the highest strain, with strain gauge 7 second and

strain gauge 6 the lowest. This is expected due to the stress concentration induced by the eccentric

hole. The decrease in section induces localised stress concentrations as a resulting from the fact that

stress is inversely proportional to CSA.

The computed strain scalar scene, displayed in figure 13, allows for the visual interpretation of the

induced stresses within the specimen material. Evidently, a stress gradient formulates from the

discontinuity in geometry and the maximum stress is seen to be located at the perimeter of the hole,

nearest the specimen edge. Along the line, represented by line A-B in figure X, as the distance from

the centre of the hole increases, the stress distribution recovers to a uniform distribution. In a line

parallel to the specimen edge, running through the centre of the hole, the stress present is lower in

magnitude than the gross stress. As the distance from the centre of the hole progresses, the stress

eventually reaches uniform distribution. The computational analysis proves a useful tool in

computing a full view of the stress distribution.

Comparing all the data for strain in the gross, net and hole area, it is clear that they all agree upon

the stress distribution. However percentage differences are present. The computational and

experimental analyses correlate well. However, a 21.99% difference is present in the gross area.

Additionally, a 23.04% difference also exists between the theoretical and experimental results in the

gross area, while the theory and computational analysis differ by 1.05%. Note that in the three

cases, the strain gauge results are lower in magnitude. The experimental difference might loan itself

to the positioning of strain gauge 6 (see figure 5). Strain gauge 6 was positioned in the gross area,

although its proximity was significantly close to the hole and in turn lowered localised stress area.

This location may have affected the strain gauge reading; however comparison of the computational

scalar scene depicts that the strain gauge was sufficiently distant from the low stress gradient.

Further investigation into the reasoning revealed that the tensile testing machine was inducing some

form of bending moment on the specimen that in turn affected the strain gauge readings. Retesting

of the specimen was conducted by Daniel Powers, Meng Mechanical Engineer University of Limerick

Postgraduate, revealing a 40% difference in strain readings in the gross area when the specimen

initially tested and then rotated 180 degrees in the tensile testing machine, these results are shown

in appendix A, table 3. Analysing the results generated, strain gauge 6 readings indicate that the

specimen experienced a compressive force and tensile force on either surface, analogous to figure

15, indicating the presence of a bending moment. This can be concluded as the strain was 40%

higher when the specimen was orientated at 180 degrees, as opposed to 0 degrees. Therefore, in

order to obtain strain values comparable to the other methods, the averaged values from all the

tests conducted will be used. The averaged values can be found at the bottom of table 3, and the

revised percentage difference table for all the methods is present in Appendix B, table 4.

Figure 15: Induced bending moment on specimen during tensile test

Studying table 4, it is evident that the averaged strain values better agree with the computed

theoretical and computational results. Computational and experimental results complement one

another, with a maximum percentage difference of 8.15% in the gross area, which may be a factor of

the induced bending moment. Likewise, theoretical and experimental agree, with a maximum

percentage difference of 11.44% in the hole area. Note, the strain gauge results in the hole area are

lower in magnitude and this factor may loan itself to the fact that the strain gauge averages the

strain over the 2mm gauge length. The stress is a maximum at a specific point, therefore averaging

the strain over the 2mm length, and not specifically the point of maximum stress, would decrease its

value. Finally, computational and theoretical results correlate, with a maximum difference of 10.60%

in the hole area. The difference could be a function of the computational modelling i.e.

partition/seed/mesh choice or the idealisation of the theoretical calculations i.e. assumptions.

All in all, theory and experimental differ within an average of 10%, theory and computational

differing within an average of 5.1% and computational and experimental 4.8%. It is clear that each

method is affective in computing the distributed stress, and the choice of method, outside of

scholarly applications, comes down to factors discussed in the introduction.

M

Tension Compression

Surface of strain gauge

application M

7 CONCLUSION

From the report it may be concluded that:

The mild steel specimen behaved as expected during the experimentation, with the linearity

of the load-displacement plot confirming that the specimen was loaded within the elastic

limit, further validating the use of the theoretical formulae.

High percentage discrepancies between the experimental and theory, and the experimental

and computational stresses in the gross area revealed that the tensile testing machine was

not applying in-plane tension.

Averaging the strain gauges results and comparing all the stress obtaining methodologies, it

was seen that they all agreed on the stress trend, with the highest stress in the hole area,

the next highest at the net area and the least in the gross area.

Computational analysis displayed a full view of the stress gradient present, revealing the

presence of a stress region lower in magnitude than the stress in the gross area of the

specimen. This area is at an angle of π/2 radians from the maximum stress.

Averaging the percentage differences between the methodologies, theory and experimental

differ within an average of 10%, theory and computational differing within an average of

5.1% and computational and experimental with 4.8%.

From the conduction of the report, theoretical calculations proved the quickest method in

computing stress values in the specific points of interest. Strain gauges were also non time

consuming and proved useful tool, as they revealed problems with the tensile testing machine

through discrepancies in readings when the specimen was rotated 180 degrees between testing.

Computation analysis, although more time dominant, gave a full material view of the distributed

stress gradient resulting from the hole. To finally conclude, each methodology in obtaining stress

prove effective, all correlating well, and the choice of methodology is a function of various

parameters discussed briefly in the introduction, such as geometry complexity, ease of access, time,

and so forth.

8 REFERENCES

1. Digital Image Correlation (DIC) Measurement Principles. 2015. Digital Image Correlation

(DIC) Measurement Principles. [ONLINE] Available at:

http://www.dantecdynamics.com/measurement-principles-of-dic. [Accessed 16 October

2015].

2. David W A Rees, 2000. Mechanics of Solids and Structures. 1st Edition. World Scientific

Publishing Company.

3. Walter D. Pilkey, 2008. Peterson's Stress Concentration Factors. 3 Edition. Wiley.

4. Principle of Strain Measuement | KYOWA. 2015. Principle of Strain Measuement | KYOWA.

[ONLINE] Available at: http://www.kyowa-

ei.com/eng/technical/strain_gages/measurement.html. [Accessed 1 November 2015].

5. VPG - Micro-Measurements - Strain Gage Knowledge Base - Technical Notes. 2015. VPG -

Micro-Measurements - Strain Gage Knowledge Base - Technical Notes. [ONLINE] Available

at: http://www.vishaypg.com/micro-measurements/stress-analysis-strain-gages/technotes-

list/. [Accessed 22 October 2015].

9 APPENDIX

9.1 APPENDIX A

Retesting of Mild Steel Specimen

Strain gauge 6 (µε)

[Gross area]

Strain gauge 7 (µε)

[Net area]

Strain gauge 10 (µε)

[Hole area] 235 298 631

206 271 634

Rotated test

151 249 607

137 228 605

Percentage difference between initial and rotated test

43.52% 17.91% 3.87%

40.23% 17.23% 4.68%

Averaged Strain Values

200 289 704 Figure 16: Retest of specimen, two tests computing strain reading and two tests of the specimen rotated 180°. Retesting

conducted by Daniel Powers, University of Limerick Postgraduate. Note the averaged strains are an average of all experimental data obtained for the report.

9.2 APPENDIX B

Theory & Experimental

Theory & Computational

Computational & Experimental

Gross area 9.20% 1.05% 8.15%

Hole area 11.44% 10.60% 0.85%

Net area 9.24% 3.85% 5.38%

Table 4: Revised percentage differences using averaged strain gauge data