Determining its permanent fixtures are dead loads, while consideration

Determining Young Modulus for
Various Materials Using a Computer Simulation

Introduction

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When given the option to select to
do a laboratory report on whatever I wished, I was drawn to the investigation
of Young Modulus of materials. As an individual particularly interested in
engineering fields, I am very interested in the mechanical properties of solid
materials and how this shapes design in all facets of engineering. After
participating in a job shadowing opportunity in civil engineering, I was drawn
in particular to the mechanical properties of concrete and industrial steel.
Concrete has the ability to resist a tremendous amount of compressive force but
has very low tensile strength whereas steel has a large elastic range under
stress. Therefore, concrete that has been reinforced with rebar is one of the
best composite construction materials in the world. Having discussed Young
Modulus to great depth in my IB Design Technology course, I was curious as to
how the differing slopes of each material at various points on the stress
strain curve could result in the determination of single values for the
ultimate tensile strength and yield strength of each material. This curiosity
prompted me to select the focus of my Internal Assessment to be determining
Young Modulus for various materials using a computer simulation so that I could
compare these found values with the generally accepted values for the selected
materials as seen in textbooks, in order to give insight on the use of
composite materials in engineering and construction.

 

 

Background

Youngs
Modulus is a measure of stiffness of an elastic material and is used to
describe the elastic properties of objects when stretched or compressed. When a
load is applied to a solid material, the material is deformed because of the
weight of its load. Materials often undergo deformation due to dead loads and
live loads, both of which must be considered in construction and often in
manufacturing as well. Dead loads are static, permanent forces that materials
undergo. When considering reinforced concrete, as an application of this
investigation, the weight of a concrete building and its permanent fixtures are
dead loads, while consideration of live loads—the forces exerted on a material
with varying magnitude and intensity—take into account the weight of people in
a building and the tensile strength required of the reinforced concrete to bear
wind, seismic shifts, and weather conditions such as snow and ice that
contribute to greater loads on the structure. Youngs Modulus measures the
material’s ability to return to its original shape when these loads are
removed, resulting in the stress-strain curve, and represents the constant, k, in
Hooke’s law, which relates the stress and the strain, though law is only applicable
under the assumption of an elastic and linear response. For most materials, the
stress-strain curve only remains linear under a limited amount of deformation. Hooke’s
law states that the force F needed to extend or compress a spring by some
distance x has a linear relationship with that distance. F=kx, where k is a
constant: the stiffness of the spring, and the value of x is minor compared to
the total possible deformation of the spring. Real materials
eventually fail when placed under a very large force, so this model doesn’t
hold true for large distances.

Hooke’s
law can be expressed in terms of stress and strain where stress is the force on
a unit area of a material as a result of the externally applied force, while
strain is the relative deformation this causes.

Extension Produced ? Load Attached

Stress ? Strain ?  k

 

Research Question

The
primary aim of this investigation is to model Young Modulus for a variety of
materials to compare to established values of said materials using a computer
simulation of Searle’s apparatus.

I
chose to use a computer simulation because this experiment involves specialized
equipment that is not readily available. Furthermore, using a computer
simulation provides better accuracy and precision because it provides a closed
system that lacks confounding variables such as pressure and temperature.

Design and Methodology

The set-up of Searle’s Method is
illustrated in figures 2 and 3 utilizing spherometers.

The
apparatus utilized in Searle’s Method consists of two wires of the same
material, length and diameter, each of which have one end of the wire tightened
in torsion screws above to a fixed support, and the other attached below: the
first to a weighted control mass and the latter to the masses applied to vary
the stress of the load on the wire of material. Thus, the first wire (a) becomes
the auxiliary wire while the second (b) becomes the experimental wire. Each
wire is then loaded equally with 1 kg. The pitch and the least count of the
spherometer are then determined according to the following:

Pitch, P =

Least Count, LC =  

The
central screw is then adjusted so that the bubble in the spirit level on the
apparatus is exactly at the center, indicating that the weights are aligned
horizontally to one another. The scale reading of the spherometer is then
calibrated to display zero weight being hung from the experimental wire.
Weights are then added to the experimental wire in half-kg increments, causing
the air bubble in the level to shift away from the center. Between the addition
of each weight, the spherometer screw is adjusted so that the bubble returns to
the initial position and corresponding spherometer readings for each weight
observed. The same procedure of noting the spherometer reading and adjusting
the level is then repeated for unloading the weights in the same half-kg
increments. From these spherometer readings, the extension of the wire material
can be determined and Young’s modulus found.

Simulation Selection

In selecting a simulation for this
investigation, I was searching for a program that would resemble the Searle’s
Method of determination of conducted in the real-world. I required the ability
to hold variables constant, to measure the extension of the wire with
precision, and to exchange the material making up the wire to calculate Youngs
Modulus for multiple metals. This lead me to select a simulation from Amrita
University which had fairly precise measuring instruments and the ability to
keep multiple components of the simulation constant, greatly reducing error
that would cause inconsistencies in my results.

Deriving Equations

If
a wire with length L and radius r has a load of mass M under the acceleration
due to gravity g and the extension produced is denoted by l, then,

Normal Stress =  ,
where Force F= mg and Cross-Sectional Area A= , then

Normal Stress =  , and

Longitudinal Strain = , then

Longitudinal Strain =  .
Thus, as

Young Modulus  .
Therefore,

Error = , and

% Error =

The
program I used for this investigation provided me with the ability to keep
variables L, r, and g constant and in the simulation, I set these variables as
follows for all trials:

·        
Wire
length, L= 5.0m

·        
Wire
radius, r= .0001m

·        
Acceleration
due to gravity, g= 9.8m/

Data

Observation
 

Load on Hanger, M (kg)

Spherometer Screw
Reading (mm)

Extension, l for load (mm)

Youngs Modulus, Y (GPa)

Loading, x (mm) ± .5

Unloading, y (mm) ± .5

Mean (mm)

1

0.0

0.42

0.42

0.42

0.00

 

2

0.5

4.83

4.83

4.83

4.41

176.9285

3

1.0

8.24

8.24

8.24

7.82

199.5537

4

1.5

12.65

12.65

12.65

12.23

191.3953

5

2.0

17.06

17.06

17.06

16.64

187.5612

6

2.5

20.47

20.47

20.47

20.05

194.5773

7

3.0

24.88

24.88

24.88

24.46

191.3953

8

3.5

29.29

29.29

29.29

28.87

189.1854

9

4.0

32.71

 

32.71

32.29

193.3118

 

 

 

18.35

 

 

190.4886

Given
Youngs Modulus for Material A: 1.90 x  Pa = 190 GPa

 = 190.4886
GPa

Slope
=  = 8.1443

 = 191.6075727 GPa

By
calculation, the Youngs Modulus of the material of the given wire is 190.4886 GPa.

From
the graph, Youngs Modulus of the material of the given wire is 191.6075727 GPa.

Observation
 

Load on Hanger, M (kg)

Spherometer Screw
Reading (mm)

Extension, l for load (mm)

Youngs Modulus, Y
(GPa)

Loading, x (mm) ± .5

Unloading, y (mm) ± .5

Mean (mm)

1

0.0

0.42

0.42

0.42

0.00

 

2

0.5

11.55

11.55

11.55

11.13

70.10375

3

1.0

22.68

22.68

22.68

22.26

70.10375

4

1.5

34.81

34.81

34.81

34.39

68.06526

5

2.0

45.94

45.94

45.94

45.52

68.56369

6

2.5

57.07

57.07

57.07

56.65

68.86626

7

3.0

68.20

68.20

68.20

67.78

69.06947

8

3.5

79.34

79.34

79.34

78.92

69.20658

9

4.0

90.78

 

90.78

90.36

69.07966

 

 

 

50.88

 

 

69.1323

Given
Youngs Modulus for Material B: 6.90 x  Pa = 69.0 GPa

 = 69.1323
GPa

Slope
=  = 22.609

 = = 69.02161 GPa

By
calculation, the Youngs Modulus of the material of the given wire is 69.1323 GPa.

From
the graph, Youngs Modulus of the material of the given wire is 69.02161 GPa.

Observation
 

Load on Hanger, M (kg)

Spherometer Screw
Reading (mm)

Extension, l for load (mm)

Youngs Modulus, Y
(GPa)

Loading, x (mm) ± .5

Unloading, y (mm) ± .5

Mean (mm)

1

0.0

0.42

0.42

0.42

0.00

 

2

0.5

7.09

7.09

7.09

6.67

116.9797

3

1.0

13.75

13.75

13.75

13.33

117.0675

4

1.5

20.42

20.42

20.42

20.00

117.0382

5

2.0

27.09

27.09

27.09

26.67

117.0236

6

2.5

33.75

33.75

33.75

33.33

117.0499

7

3.0

40.42

40.42

40.42

40.00

117.0382

8

3.5

47.09

47.09

47.09

46.67

117.0299

9

4.0

53.76

 

53.76

53.34

117.0236

 

 

 

30.00

 

 

117.0313

Given
Youngs Modulus for Material C: 1.17 x  Pa = 117 GPa

 =
117.0313 GPa

Slope
=  = 13.334

 = = 117.03236 GPa

By
calculation, the Youngs Modulus of the material of the given wire is 117.0313 GPa.

From
the graph, Youngs Modulus of the material of the given wire is 117.03236 GPa.

 

To
better demonstrate the accuracy of attributing a linear model—such as that
Youngs Modulus provides—to the relationship between mass of load and extension
of wire material, I calculated and graphed the residuals of the
experimentally-found l and the l? predicted by the least squared
regression line (LSRL) for each of the three materials selected in the
simulation.

 

Mass (kg)

Extension (mm)

0.5

4.41

4.072

0.338

11.13

11.305

-0.175

6.67

6.667

0.003

1.0

7.82

8.144

-0.324

22.26

22.609

-0.349

13.33

13.334

-0.004

1.5

12.23

12.216

0.014

34.39

33.913

0.477

20.00

20.001

-0.001

2.0

16.64

16.289

0.351

45.52

45.218

0.302

26.67

26.668

0.002

2.5

20.05

20.361

-0.311

56.65

56.523

0.127

33.33

33.335

-0.005

3.0

24.46

24.433

0.027

67.78

67.827

-0.047

40.00

40.002

-0.002

3.5

28.87

28.505

0.365

78.92

79.132

-0.212

46.67

46.669

0.001

4.0

32.29

32.577

-0.287

90.36

90.436

-0.076

53.34

53.336

0.004

 

 

 

 

 

 

 

 

 

 

0.9991

0.9999

1.0000

 

Evaluation

The
above demonstrates that LSRLs calculated using Youngs Modulus are good models
for the data because the range of the residuals is minimal in comparison to the
data, the residual plot is scattered, and the  value is very
close to 1.00 for all materials within the investigation, meaning that nearly
100% of the variance in wire extension can be attributed to the LSRL of wire
extension and mass of load for each material studied.

According
to my results, the Youngs Modulus for each of the three materials studied was
found to be within ±00.9% of the given Youngs Modulus value for the material,
indicating a confirmation of this model’s validity.

While
within the range of mass loads studied, there is an extremely strong positive
correlation between mass of load and extension of wire with a slope equating to
the Youngs Modulus of the material studied, it is important to note that to
assume this same correlation for all loads of mass M would be to extrapolate
upon the results of this investigation. As noted in the “Background” section of
this investigation, materials fail when placed under large forces, so this
model doesn’t hold true when extension l
is very large due to large Mg.

Limitations

Although
this investigation was conducted with the use of a computer simulation, the
accuracy and precision of the results found were in fact limited chiefly by the
precision of the spherometer utilized to calculate extension, which was accurate
to the nearest millimeter. Errors may have arisen from the limited ability to
measure extension because of this, resulting in the presence of residuals when
comparing experimental and predicted values for the extension of the wire.
However, as a simulation, many of the errors that could have resulted in the
experimental setting were not present. Had this investigation taken the form of
an experiment rather than simulation, instrumental errors could have resulted
from the linear scale zero having not been calibrated to exactly coincide with
the circular scale zero on the apparatus. Additionally, wire outside of a
simulated setting may not be uniform throughout in composition or diameter,
resulting in error when radius of the wire is measured and used to calculate
cross-sectional area. Therefore, error within this investigation was minimized
to a great degree through the utilization of computer simulation.

Conclusion

As
this investigation was performed by a computer simulation, its purpose proved
to primarily act as an exercise in understanding one component of material
indices that allows for the calculation of material performance. This allows
for engineers, manufacturers, those in the construction industry, etc. to
enhance the selection of materials for applications in analysis of component
functions. In calculating Young Modulus using Searle’s Method, I was able to
determine the value of the materials selected quite accurately.

When
considering ways in which I could improve or extend this investigation, I would
like to be able to calculate Youngs Modulus in multiple ways. There are various
methods of finding Young Modulus, including Searle’s Method, the Tabletop
Method, and the Beam Deflection Method. The multitude of options prompted me to
wonder which of these methods of determining Young Modulus proves most accurate
when compared to the generally accepted value, such as that found in a IB
Physics textbook. Additionally, I would like to physically perform this
experiment. The computer simulation utilized for this investigation was
programmed to achieve specific results, and so the accuracy of the values found
for Youngs Modulus is incredibly high. A physical experiment, however, while
perhaps less accurate, would better reflect the conditions of the real world,
outside of a closed system, where other factors must be taken into
consideration in the realm of design and engineering such as temperature and
pressure, which could lead to further investigation of the benefits of cold
rolling, annealing, and other metal-working techniques.