Discussion:
From this experiment, we’ve studied the Young’s modulus of 3 different kinds of material with varying
F
dimensions. The Young’s modulus of one material can be calculated by having the longitudinal stress ( )
A
∆L
acting on a material divided by the strain ( ) that it’s experiencing:
L
Stress
E=
Strain
We tested 5 scenarios in total with three different materials: steel, brass, and aluminum, which are
grouped into 3 parts:
Part 1: We performed the experiment with beams made of each of the three materials. Different amounts
of force were applied to the center of the beams, with one end fixed and the other simply supported. The
vertical deflection of the beam was recorded for each force applied.
Part 2: The experiment was similar to Part 1, but both ends of the beams were simply supported instead
of being fixed.
Part 3: This part was divided into two sub-parts:
In the first sub-part, the clamping length of the beams was reduced from 800mm to 500mm,
while the width and thickness remained the same.
In the second sub-part, in addition to reducing the length to 500mm, the width of the beams was
also increased and the thickness was reduced.
Part 3 illustrates how changes in the dimensions of the beam affect the magnitude of deflection.
Part 3a Part 3b
20 20
15 15
Load (N)
Load (N)
10 10
5 5
0.1 0.3 0.5 0.7 0.1 0.5 0.9 1.3
Deflection (mm) Deflection (mm)
Figure 4.0: Load versus Deflection graph of Part 3.
When we compare the results from Part 3b to the results from Part 3a; where the former have beams with
the same length, but increased width and reduced thickness, it becomes clear that thinner beams are more
susceptible to bending. This is because resistance to bending depends on how the material of the cross-
section is distributed relative to the bending axis. The plank on the left has more material located further
from the bending axis, which makes it much stiffer (The Efficient Engineer, 2024). This resistance to
bending can be quantified by calculating the area moment of inertia of the cross-section, І.
Figure 4.1: Area moment of inertia values in (mm4) for 3 shapes.
Cross-section that has majority of material away from the bending axis have higher area moment of
inertia, therefore more resistant to bending. The moment of inertia for a rectangular shaped material
relative to the horizontal bending axis can be calculated by using the following equation:
3
(Width )(Thickness)
I x=
12
Which again, proved that thicker beam is more resistant to bending due to higher moment of inertia.
Material
Part 1 Part 2
20 20
15 15
Load (N)
Load (N)
10 10
5 5
0 1 2 3 0 2 4
Deflection (mm) Deflection (mm)
Part 3a Part 3b
20 20
15 15
Load (N)
Load (N)
10 10
5 5
0.1 0.3 0.5 0.7 0.1 0.5 0.9 1.3
Deflection (mm) Deflection (mm)
Figure 4.3: Load versus Deflection graph of all Parts
From the calculations, we’ve identified that the relationship between Force over Deflection (F/mm) is
directly proportional to the Young's modulus of the material. The overall graph of our experiment shows
that steel consistently exhibits a higher Force over Deflection trend, corresponding to its higher Young's
modulus, while brass consistently has a Young's modulus value between that of steel and aluminum, and
aluminum consistently has the smallest Young's modulus value among the three materials tested in our
experiment.
Clamping method
Part 1 Part 2
20 20
15 15
Load (N)
Load (N)
10 10
5 5
0 1 2 3 0 2 4
Deflection (mm) Deflection (mm)
Figure 4.2: Load versus Deflection graph of Part 1 (one fixed end) and Part 2 (both ends simply
supported)
We can also see that beams with one simply supported end and one fixed end tend to deflect less
compared to beams with two simply supported ends. This is because a simply supported end have no
moment resistance (Machine Design, 2016), allowing the beam to deflect more, whereas the fixed end in
Part 1 can resist a moment.
Theoretical
Steel 200 GPa
Brass 100GPa
Al 70 GPa
Experimental
Steel 200.094 GPa 0.05%
Brass 78.813 GPa 21.19%
Al 57.064 GPa 18.48%
The theoretical Young's modulus values for steel, brass, and aluminum beams are 200 GPa, 102 GPa, and
69 GPa, respectively. In the experiment, the measured values for steel, brass, and aluminum beams in Part
1 are 175.493 GPa, 65.889 GPa, and 57.064 GPa, respectively. The percentage errors when comparing
these experimental values to the theoretical ones are 12%, 34%, and 18%, respectively. In Part 2, the
experimental values for steel, brass, and aluminum beams are 200.094 GPa, 78.813 GPa, and 80.49 GPa,
resulting in percentage errors of 0%, 21.19%, and 16.65%, respectively. For Part 3(a), the measured
values are GPa for steel, GPa for brass, and GPa for aluminum, with percentage errors of %, %, and %,
respectively. In Part 3(b), the experimental Young's modulus values are GPa for steel, GPa for brass, and
GPa for aluminum. The differences between these experimental values and the theoretical ones are 44.89
GPa, GPa, and GPa, respectively.
To avoid or reduce the percentage error and human error, there are some precautions that can be taken to
obtain a more accurate result. First, ensure the apparatus is set up on a stable, level surface. All
components should be securely fastened to avoid any movement during the experiment. Secondly, make
sure that all measuring instruments, like vernier calipers, and weights, are calibrated correctly. Third, take
care to measure the dimensions of the specimen accurately (length, diameter, etc.) using appropriate
instruments. Also, before applying the load, zero the measuring instruments such as the micrometer or
dial gauge to account for any initial offsets. Other than that, when reading scales or gauges, ensure your
line of sight is perpendicular to the scale to avoid parallax error. Lastly, apply loads incrementally and
wait for a short period after each increment to allow the material to reach equilibrium before taking
measurements. This helps in avoiding dynamic effects and ensures that the stress-strain relationship is
accurately captured.
