DEPARTMENT OF MECHANICAL ENGINEERING
ENGINEERING MECHANICS – EN 122
9/8/2023
EN 122 LAB 02
DEFLECTION OF
CANTILEVER BEAM
Name: Dokta Urame
std ID: 22300416
Program: BECV – 1
Course code: EN 122
Course: Engineering Mechanics
C/Lecturer: Dr. Ales
C/Tutor(s): Mr. Leso
Year: One (1), Semester 2, 2023
22300416dour@student.pnguot.ac.pg
PNG UNITECH
1.0 Abstract
Contents
Title ......................................................................................................................................................... 2
1.0 Abstract ............................................................................................................................................. 3
2.0 Introduction ...................................................................................................................................... 4
3.0 Materials and Methods ..................................................................................................................... 6
4.0 Results and Analysis ........................................................................................................................ 10
5.0 Discussions ...................................................................................................................................... 14
6.0 Conclusion ....................................................................................................................................... 16
7.0 References ....................................................................................................................................... 17
8.0 Appendix ......................................................................................................................................... 18
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EN 122 LAB 02
Deflection of Cantilever Beam
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1.0 Abstract
Title:
Deflection of Brass cantilever beam, Steel cantilever beam and
Aluminum cantilever beam
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1.0 Abstract
1.0 Abstract
In this experiment, the deflection rate of cantilever beams has been examined. This is
achieved by adding increasing load each time to a cantilever. The known loads were added to
the cantilever which is 200 mm (20 cm) away from its fixed end. Three different cantilevers
were engaged, the steel bar, brass bar and aluminum bar. The deflection rate is thoroughly
determined whilst adjusting the digital deflection registering meter scale to zero each time as
the 10 g load is added. The hanger of mass 10 g was used to hung the loads. The readings
were observed on the deflection registering device and was recorded on a mass – deflection
table as tabulated in table 1 of the results section. The data collected for deflection rate were
presented in two folds; tabular and graphical as shown in figure 1 and table 1. The results
were further analyzed and interpreted in the discussion section. It was noted that the
deflection depends not only on the loads but the strength of the material as well. In which
much tensile steel deflect more compared to brass and aluminum. Due to the fact that the
wights was regarded as external load, part of the lab objective was to calculate the reactions
within the member including the bending moment. Furthermore, the bending moment
diagram and shear force diagram would be drawn and attached at the appendix as to
substantial the theory lesson learned during lecture. The lab successfully concludes that
deflection in lateral axes is directly proportional to the amount of load regardless of the
tensile strength of materials.
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Deflection of Cantilever Beam
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2.0 Introduction
2.0 Introduction
Beams in structural engineering are referred as structural members which are used for
support and reinforcement of relatively heavy loads across its entire length. Thus, beams have
a much longer length compared to its height and width. For example, beams are used in large
structures like building to support huge tones of building materials and white goods and also
in bridges to support weight of trucks and commuters. In most practical use, rectangular cross
section and I – cross sections are used more often unlike circular, square and T – cross
section. Beams can be made of steel, concrete or traditional timber. There are different types
of beams as specified in the lecture notes by (Ales, 2023) which includes; cantilever beam,
propped cantilever beam, simply supported beam, overhanging and continuous beams. There
are four types of beam loading which includes; point load or (concentrated load), uniformly
distributed load (UDL), uniformly varying load (UVL) and applied external moment or
(couple).
Beams are designed after careful considerations of the changes of the shear force V
and bending moment M acting along each point on the axis of the horizontal and I – cross
sectional beam as indicated by (Hibbeler). By using method of section as discussed in the
lecture, variations of V and M along the beam’s axis can be calculated. Generally, internal
shear force and bending moment slope is discontinuous at points where concentrated force
and couple moments are applied (Hibbeler, 2016). Because of this, a shear force diagram and
bending moment diagram were plotted to indicate the discontinuity between the two points.
The aim of the experiment was to determine the deflection rate of a cantilever beam
as it undergone increasing point load. Deflection is simply the point at which part of a long
beam is bent (deformed) laterally under a load. The basic aim was to observe how the
deflection varies directly or inverse due to increasing loads has been added each time and
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2.0 Introduction
these results would further be analyzed in the results section. Thereafter, with the use of
deflection formula 𝐷 = 𝐿/𝑅, the deflection rate was calculated and recorded. Thus, cantilever
deflection experiment intents to determine the amount deflection that will occur in the
cantilever.
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3.0 Materials and Methods
3.0 Materials and Methods
3.1 Materials.
The materials that were used in the lab including STR4 beam deflection rig, digital
deflection registering device, steel bar, Brass bar, Aluminum bar, ruler, masses 10 g each,
hanger 10 g each, figure 3.1a and 3.1b below shows photograph of some materials that were
engaged in the lab.
Figure 3.1a & 3.1bshows materials that were used in the lab.
3.2 Methods.
Firstly, the width and depth of the test beams were measured using a vernier caliper
and recorded by the lab instructor prior to the lab session. The test beams were aluminum bar,
brass bar and steel bar. However, length measurement (i.e., length, width and height) were
observed and recorded next to the test beams, in which those values were further used to
calculate the second moment of area.
Subsequently, the aluminum beam has been prepared onto the backboard as indicated in
figure 3.2.
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3.0 Materials and Methods
Figure 3.2. Aluminum
has been prepared
onto the backboard
The cantilever beam has beam placed softly and nicely on the table top. After the beam has
been clamped tightly, the deflection meter has been moved across STR4 beam deflection rig
the distance of 200 mm as shown in figure 3.3.
Figure 3.3. beam has
been clamped
The 200 mm length has been measured using a meter ruler. Following the cantilever set up
using the aluminum bar, the load of 10 g has been added using a 10 g hanger as indicated in
figure 3.4.
Figure 3.4. the load
has been added using
the hanger
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3.0 Materials and Methods
The deflection reading was observed and recorded as shown in figure 3.5.
Figure 3.5. deflection
reading has been
observed and recorded
Then, increasing 10 g load has been added and deflection values were also noted until the
load of 500 g was loaded as indicated in figure 3.6 (a) and (b).
Figure 3.6 (a) and (b). the
maximum load at 500g loaded
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3.0 Materials and Methods
The same procedure has been repeated for brass and steel and the results were observed and
recorded in the material – mass - deflection table as shown in figure 3.7.
Figure 3.7. specimen
deflection data
recorded
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4.0 Results and Analysis
4.0 Results and Analysis
4.1 Tabulated results
Table 4.1.1. Deflection of Brass beam
Materials
E value = 105 x 10^ 9 N/m^2
I value = 42.75 mm^4
Mass (g)
0
100
200
300
400
500
Actual deflection (mm)
Brass cantilever beam
Width b= 19 mm
Depth h = 3.0 mm
Theoretical deflection (mm)
0
0.63
1.21
1.8
2.41
2.94
(%) difference
0.00
0.58
1.16
1.75
2.33
2.91
0.00
4.78
4.56
5.34
8.12
2.90
Table 4.1.2. Deflection of Steel beam
Materials
E value = 207 x 10^ 9 N/m^2
I value = 42.75 mm^4
Mass (g)
0
100
200
300
400
500
Actual deflection (mm)
Steel cantilever beam
Width b= 19 mm
Depth h = 3.0 mm
Theoretical deflection (mm)
0
0.29
0.56
0.84
1.1
1.36
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0.00
0.89
1.77
2.66
3.54
4.43
EN 122 LAB 02
(%) difference
0.00
59.60
121.19
181.79
244.38
306.98
Deflection of Cantilever Beam
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4.0 Results and Analysis
Table 4.1.3. Deflection of Aluminum beam
Materials
Aluminium cantilever beam
E value = 69 x 10^ 9 N/m^2
Width b= 19 mm
I value = 42.75 mm^4
Depth h = 3.0 mm
Mass (g)
0
100
200
300
400
500
Actual deflection
(mm)
0
0.81
1.51
2.24
2.95
3.66
Theoretical deflection (mm)
0.00
0.30
0.59
0.89
1.18
1.48
(%) difference
0.00
51.47
91.94
135.40
176.87
218.34
4.2 Graphical results
Figure 4.2.1a Actual deflection rate of Brass
Deflection of brass beam
3.5
DEFLECTION (MM)
3
2.5
2
1.5
1
0.5
0
0
100
200
300
400
500
600
MASS (GRAMS)
Figure 4.2.1b theoretical deflection rate of Brass
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4.0 Results and Analysis
Theoretical Brass beam
deflection
DEFLECTION (MM)
3.50
3.00
2.50
2.00
1.50
1.00
0.50
0.00
0
100
200
300
400
500
600
MASS (G)
Figure 4.2.2a Actual deflection rate of steel
Deflection of steel beam
1.6
DEFELCTION (MM)
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
100
200
300
400
500
600
MASS (GRAMS)
Figure 4.2.2b Theoretical deflection rate of steel
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4.0 Results and Analysis
Theoretical Steel deflection
DEFLECTION (MM)
5.00
4.00
3.00
2.00
1.00
0.00
0
100
200
300
400
500
600
MASS (G)
Figure 4.2.3a Actual deflection rate of Aluminum
DEFLECTION (MM)
Dflection of Aluminium beam
4
3.5
3
2.5
2
1.5
1
0.5
0
0
100
200
300
400
500
600
MASS (GRAMS)
Figure 4.2.3b theoretical deflection rate of Aluminum
DEFLECTION (MM)
Theoretical steel deflection
1.60
1.40
1.20
1.00
0.80
0.60
0.40
0.20
0.00
0
100
200
300
400
500
600
MASS (G)
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5.0 Discussions
5.0 Discussions
Regarding figure 4.2. (1a,b), (2a,b) and (3a,b), the graph shows deflection on the y-axis and
applied load on the x-axis. It was noted that the linear relationship exists. The is inline with
the theory knowledge of cantilever beam, which the deflection is in linear direct relationship.
That means the more loads added, the more deflection, the beam has undergone. For the three
test materials, it was observed that regardless of the materials, the load is in positive linear
correlation in each of the three graphs both the experimental and theoretical values.
However, from each of the three positive linear correlations, it was quite obvious that
steel being the more tensile materials has undergone less deflection for the same load unlike
brass and aluminum beam. Tensile means resistance to lengthwise stress or deformation
(Callister, 2014). Aluminum is less tensile whilst Brass is intermediate. Thus, it was evident
on the figure 4.2. (1), (2) and (3) that steel deflect less followed by brass whilst aluminum
deflect more for the same amount of load.
The slope of the line represents the relative stiffness (B.V, 2023). The formula 𝐷 =
𝑊𝐿3
3𝐸𝐼
display the relationship deflection and load as well as other parameters included in the
formula were;
L = length of cantilever beam
E = Elastic modulus
W = width of the beam
I = 2nd moment of area for the cantilever beam
D = deflection
As stated above, Stiffness is a degree of resistance to deflection for cantilever beam.
Stiffness is inversely proportional to deflection. That means cantilever beams have higher
values of elastic modulus and lower deflections for the same load (Omer, 2021). Young
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5.0 Discussions
modulus is the property of the material that determines the stiffness. A beam that is stiffer will
deflect to a small extent for a given load.
The relationship between load, and material properties for cantilever beams were
understood as an emerging structural engineer by using the deflection formula 𝐷 =
𝑊𝐿3
3𝐸𝐼
.
Deflection is crucial in structural engineering for the need to design safe and secure
structures, such as buildings and bridges. Without the knowledge of the rate of materials
deflection in relations to stiffness and elastic modulus, the safe design of structures would be
lack and that would have a devastating effect on lives and properties that is within the
particular structures when fail.
As shown in Table 4.1. (1), (2) and (3), there were differences in the experimental and
theoretical deflection this indicates limitations in the experiments. For Brass the actual and
theoretical results were consistent with the deviation of average ±6%. For steel and
aluminum, it was greatly differ in the actual and experimental values. Some contributing
factors were imperfect reading taken known as human error, parallax error, vibration within
the beam and inappropriate load application, not properly clamping the support of the
cantilever. Identifying and minimizing these errors is important to obtain accurate and
reliable results so that percentage difference between actual and theoretical values can be
relatively small. To achieve better results some ways to minimize or improve errors in the
experiment are as follows; ensure that all measurements are taken carefully, instruments must
be read at eye level, making sure the loads are applied perpendicular to the beam's axis,
ensure the beam is evenly supported, and that the support itself must not bent (Devenport,
2007)
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6.0 Conclusion
6.0 Conclusion
This experiment on the deflection of the three different cantilever beams namely; brass, steel
and aluminum provide us a remarkable hands-on experience on practically applying the
knowledge of beams and structures by the increasing load on the beam and observe and
measured the deflection. The results outlined the key knowledge of mass – deflection as a
direct proportion. In which increasing loads can have more stress on the beam regardless of
the stiffness and tensile strength of the material. The lab demonstrates how materials and
loads can affect the behavior of structural members. The lab has successfully completed and
desired results of deflection were obtained but further improvements can be by avoiding
errors and carefully loading masses and taking measurements.
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7.0 References
7.0 References
Ales, D. (2023). Structures - Beam (week 4 lecture). PNG Unitech, Department of
Mechanical Engineering. Lae: Unitech - Mechanical Dept.
B.V, E. (2023). Retrieved from ScienceDirect: http//www.sciencedirect.com
Callister. (2014). Material Science and Engineering, An Introduction 9th Edt. In J. D. William
D. Callister. New YorK, USA: John Wiley & Sons.
Devenport, E. R. (2007, 01 24). Virginia Tech. Retrieved from and-grant university in
Blacksburg, Virginia: https://devenport.aoe.vt.edu/aoe3054/manual/expt2/text.html
Hibbeler, R. (2016). Engineering Mechanics - STATICS (14th Edt ed.). Hoboken, New Jersey,
USA: Pearson.
Omer, I. (2021, 12 30). Engineering stack exchange. Retrieved from stiffness and elastic
modulus: http://engineering.stackexchange.com
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8.0 Appendix
8.0 Appendix
Table 8.1 Mass (g) and Load (N)
mass
(grams)
0
100
200
300
400
500
Load (N)
0
0.98
1.96
2.94
3.92
4.9
Where load is calculated as follows:
𝑙𝑜𝑎𝑑 (𝑁) = 𝑚𝑔 = (
𝑚(𝑔𝑟𝑎𝑚𝑠)
𝑚(𝑔𝑟𝑎𝑚𝑠)
)𝑔 = (
) 9.8
1000
1000
Formula for the 2nd moment of area for the cantilever beam (I) has been calculated as follows;
𝑰=
𝒃𝒉𝟑
𝟏𝟐
where;
-------------------------(1)
I is 2nd moment of area for the cantilever beam
b is the width
h is the height or depth
i.e. =
𝒃𝒉𝟑
𝟏𝟐
𝟏𝟗𝒙𝟑𝟑
= 𝟏𝟐(𝟏𝟎𝟎𝟎𝟒 ) = 𝟒. 𝟐𝟕𝟓 𝒙 𝟏𝟎−𝟏𝟏 𝒎𝟒
From the I calculated above (1), it was used to calculated the theoretical deflection D as
follows
𝐷=
𝑊𝐿3
-------------------------- (2)
3𝐸𝐼
For example: calculated theoretical deflection for Brass when load is 100 g, the load is 0.98 N
therefore
𝑫=
(𝟎. 𝟗𝟖)(𝟎. 𝟐)
𝑾𝑳𝟑
=
= 𝟓. 𝟑𝟓 𝒙𝟏𝟎−𝟒 𝒎 = 𝟎. 𝟓𝟑 𝒎𝒎
𝟑𝑬𝑰 (𝟑)(𝟏𝟎𝟓𝒙𝟏𝟎𝟗 )(𝟒. 𝟐𝟕𝟓 𝒙 𝟏𝟎−𝟏𝟏 )
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8.0 Appendix
Figure 8. 2. (1) and (2) schematic of cantilever F.B.D and deflection
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