ENGR 244 Lab 5
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Lab 5:
Deflection of Beams
Lab Section CI – X
Winter 2022
Professor Ahmed Soliman
Concordia University
Montreal, QC, Canada
Tuesday March 8
th
, 2022
Table of Contents
Nomenclature pg 3
List of Tables pg 3
List of Figures pg 3
Objective pg 4
Introduction pg 4
Procedure pg 4-5
Results pg 5-9
Discussion and Conclusion pg 9-10
References pg 10
Original Data pg 11
Nomenclature
ρ = radius of curvature
b = width
E = modulus of elasticity
h = height
I = moment of inertia
L = length of span
P = load
x = distance along span
y = deflection
List of Tables
Table 1: Experimental and Theoretical Values of Deflection for Brass Beam at L/2
p. 5
Table 2: Experimental and Theoretical Values of Deflection for Brass Beam at L/4
p. 5
Table 3: Experimental and Theoretical Values of Deflection for Steel Beam at L/2
p. 6
Table 4: Experimental and Theoretical Values of Deflection for Steel Beam at L/4
p. 6
Table 5: Experimental and Theoretical Values of Deflection for Aluminum at L/2
p. 6
Table 6: Experimental and Theoretical Values of Deflection for Aluminum at L/4
p. 6
Table 7: Experimental and Theoretical Values of Deflection for Cantilever Brass at L/2
p. 7
Table 8: Experimental and Theoretical Values of Deflection for Cantilever Brass at L
p. 7
Table 9: Experimental and Theoretical Values of Deflection for Cantilever Steel at L/2
p. 8
Table 10: Experimental and Theoretical Values of Deflection for Cantilever Steel at L
p. 8
Table 11: Experimental and Theoretical Values of Deflection for Cantilever Aluminum at L/2
p. 8
Table 12: Experimental and Theoretical Values of Deflection for Cantilever Aluminum at L
p. 8
List of Figures
Figure 1: Theoretical vs Experimental Deflection of the Beams at L/2
p.7
Figure 2: Theoretical vs Experimental Deflection of the Cantilever Beams at L/2
p.9
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Objective:
The goal of this experiment is to determine the modulus of elasticity using the
relationship between simply supported beams and load deflection[ CITATION Con \l 4105 ].
Introduction:
Subjecting a beam to a transverse load will cause deflection in the beam.
Different types of loads can induce deflection [CITATION Des22 \l 4105 ], but in this experiment only
transverse loading will be tested. The length of the beam, the magnitude of the loads and the
positioning of the loads will make a difference in the magnitude of the deflection. Deflection is an
important aspect in engineering as many building codes require certain limits on
deflections[ CITATION Con \l 4105 ]. These limits can include things such as deflections from
earthquakes or winds [ CITATION Mov22 \l 4105 ]. The data collected in this experiment is important
as it allows us to calculate the curvature of the beam about the neutral axis through this relation
[ CITATION Con \l 4105 ]:
1/ρ = M(x)/EI
The equation above is only valid within the elastic range where ρ is the radius of curvature, M(x) is the
bending moment at distance x, E is the modulus of elasticity of the assumed material, and I is the
moment of inertia of the entire cross-section [ CITATION Con \l 4105 ]. Various beam cross-sections
give different advantages when it comes to beam deflection [ CITATION Mov22 \l 4105 ], but for this
experiment, a full rectangular cross-section beam is used, so the value of I can be expressed as:
I = (1/12)bh
3
Though the radius of curvature, ρ, is related to the deflection of the beam, it does not allow us to
calculate it directly. The curvature has the following equation that will allow us to solve for the
deflection at any point on the beam:
1/ρ = (d
2
y/dx
2
)/(1 + (dy/dx)
2
)
3/2
Since the beam is only subjected to deflection in the elastic region, it can be said that the value of the
slope, dy/dx, is negligible and therefore our new relation between the curvature and the deflection will
be [ CITATION Con \l 4105 ]:
1/ρ = (d
2
y/dx
2
)
Substituting 1/ρ with M(x)/EI, we get:
d
2
y/dx
2
= M(x)/EI
(d
2
y/dx
2
)EI = M(x)
This equation will allow us to calculate the deflection of the beam at any point once we integrate twice
[ CITATION Con \l 4105 ].
Procedure:
Point load on a simply supported beam:
Make sure to measure and record the cross section of each
beam and its span. The span should be 455mm. Place the brass, steel or aluminum sample on the
supports of the test machine and ensure that load is at the center of the sample. Place the two
deformation gauges at the center and quarter portions of the beam and ensure it measure the vertical
deformation. Add loads in increments of 200N, up to 1000N and record the deflections at the middle
and quarter portions for each load. Repeat these steps for the other two metal beams.
Point load on cantilever beams:
Measure the cross section of your steel, brass or aluminum sample
and record it. Ensure that the span of the beam is at 250mm. Add increments of 100g loads to the beam,
up until 500g, and record the deformation at half the span and the full span. Repeat steps for the other
metal beams.
Results:
Sample Calculations
Finding the expression for the deflection at any point:
(d
2
y/dx
2
)EI = M(x)
(d
2
y/dx
2
)EI = Px/2 since load P is applied at x/2. Now… (dy/dx)EI = Px
2
/4 + C
solve for C using condition y
1
(L/2) = 0
(0)EI = P(L/2)
2
/4 + C
C = -PL
2
/16
(dy/dx)EI = Px
2
/4 -
PL
2
/16. Integrate again to get… yEI = Px
3
/12 – (PL
2
/16)x +C
1
and use
y(0) = 0 to solve for C
1
0 =
0 – 0 + C
1
so C
1
= 0
yEI = Px
3
/12 – (PL
2
/16)x
Final equation: y(x) = (Px(4x
2
-3L
2
))/48EI
Formulas used:
I = (1/12)bh
3
For beam:
y(x) = (Px(4x
2
-3L
2
))/48EI
and
E = (Px(4x
2
-3L
2
))/48I(y(x))
For cantilever beam:
y(x) = (Px
2
(x-3L))/6EI
and
E = (Px
2
(x-3L))/6I(y(x))
Calculations for Brass Beam Load 1:
Beam: I
brass
= (1/12)(0.01907m)(0.01295m)
3
= 3.451x10
-9
m
4
y(L/2) = ((200N)(0.2275m)(4(0.2275m)
2
- 3(0.455m)
2
))/48(105x10
9
GPa)(3.451x10
-9
m
4
) =
-1.08mm (Negative sign since deflection is downwards, mark as positive displacement in data)
E = ((200N)(0.2275m)(4(0.2275m)
2
- 3(0.455m)
2
))/48(3.451x10
-9
m
4
)(0.00187m) = 60.819GPa
Table for brass x=L/2:
Load (N)
200
400
603
799
999
Theoretical(mm)
1.083
2.167
3.266
4.327
5.410
Experimental(mm
)
1.87
3.54
5.25
6.88
8.58
Table 1: Experimental and Theoretical Values of Deflection for Brass Beam at L/2
Table for brass x=L/4:
Load (N)
200
400
603
799
999
Theoretical(mm)
0.745
1.489
2.245
2.975
3.720
Experimental(mm
)
1.14
2.30
3.48
4.62
5.80
Table 2: Experimental and Theoretical Values of Deflection for Brass Beam at L/4
E
ave brass
=66.191GPa
vs
105GPa
I
steel
= 3.472x10
-9
m
4
Tables for steel x=L/2:
Load(N)
205
399
597
800
1008
Theoretical(mm)
0.579
1.128
1.687
2.261
2.849
Experimental(mm
)
0.66
1.29
1.93
2.60
3.29
Table 3: Experimental and Theoretical Values of Deflection for Steel Beam at L/2
Table for steel x=L/4:
Load(N)
205
399
597
800
1008
Theoretical(mm)
0.398
0.775
1.160
1.554
1.958
Experimental(mm
)
0.50
0.96
1.42
1.89
2.37
Table 4: Experimental and Theoretical Values of Deflection for Steel Beam at L/4
E
ave steel
= 168.625GPa
vs
200GPa
I
aluminum
= 3.438x10
-9
m
4
Table for Aluminum x=L/2:
Load(N)
201
400
599
800
1000
Theoretical(mm)
1.639
3.262
4.884
6.523
8.154
Experimental(mm
)
1.67
3.47
5.29
7.11
8.92
Table 5: Experimental and Theoretical Values of Deflection for Aluminum at L/2
Table for Aluminum x=L/4:
Load(N)
201
400
599
800
1000
Theoretical(mm)
1.127
2.242
3.358
4.485
5.606
Experimental(mm
)
1.43
2.69
3.96
5.23
6.49
Table 6: Experimental and Theoretical Values of Deflection for Aluminum at L/4
E
ave aluminum
= 55.546GPa
vs
70GPa
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0
1
2
3
4
5
6
7
8
9
10
0
200
400
600
800
1000
1200
Theoretical vs Experimental Deflection of Beams at L/2
Theoretical Brass
Linear (Theoretical Brass)
Experimental Brass
Linear (Experimental Brass)
Theoretical Steel
Linear (Theoretical Steel)
Experimental Steel
Linear (Experimental Steel)
Theoretical Aluminum
Linear (Theoretical Aluminum)
Experimental Aluminum
Linear (Experimental Aluminum)
Deflection (mm)
Load (N)
Figure 1: Theoretical vs Experimental Deflection of the Beams at L/2
Calculations for Cantilever Brass Beam Load 1:
I
brass
= (1/12)(0.0194m)(0.00323m)
3
= 54.479x10
-12
m
4
y(L) = ((119.81g x 9.81m/s
2
)(0.250m)
2
(0.250m-3(0.250m)))/6(105x10
9
Pa)(54.479x10
-12
m
4
) =
-1.070mm
(Negative sign since deflection is downwards, mark as positive displacement in data)
E = ((0.11981kg x 9.81m/s
2
)(0.250m)
2
(0.250m-3(0.250m)))/6(54.479x10
-12
m
4
)(0.00127m) =
88.477GPa
Brass x=L/2:
Load(g)
119.81
218.76
319.58
422.89
521.96
Theoretical(mm)
0.334
0.611
0.892
1.180
1.457
Experimental(mm
)
0.44
0.77
1.17
1.59
1.95
Table 7: Experimental and Theoretical Values of Deflection for Cantilever Brass at L/2
Brass x=L:
Load(g)
119.81
218.76
319.58
422.89
521.96
Theoretical(mm)
1.070
1.954
2.854
3.777
4.662
Experimental(mm
)
1.27
2.33
3.53
4.77
5.91
Table 8: Experimental and Theoretical Values of Deflection for Cantilever Brass at L
E
ave brass
= 82.694GPa
vs
105GPa
I
steel
= 81.065x10
-12
m
4
Steel x=L/2:
Load(g)
119.21
219.21
320.51
421.63
523.41
Theoretical(mm)
0.117
0.216
0.316
0.415
0.515
Experimental(mm
)
0.21
0.38
0.57
0.76
0.97
Table 9: Experimental and Theoretical Values of Deflection for Cantilever Steel at L/2
Steel x=L:
Load(g)
119.21
219.21
320.51
421.63
523.41
Theoretical(mm)
0.376
0.691
1.010
1.329
1.649
Experimental(mm
)
0.65
1.16
1.72
2.32
2.89
Table 10: Experimental and Theoretical Values of Deflection for Cantilever Steel at L
E
ave steel
= 113.258GPa
vs
200GPa
I
aluminum
= 68.346x10
-12
m
4
Aluminum x=L/2:
Load(g)
119.61
218.51
319.21
422.41
521.31
Theoretical(mm)
0.399
0.729
1.065
1.410
1.740
Experimental(mm
)
0.48
0.91
1.35
1.79
2.23
Table 11: Experimental and Theoretical Values of Deflection for Cantilever Aluminum at L/2
Aluminum x=L:
Load(g)
119.61
218.51
319.21
422.41
521.31
Theoretical(mm)
1.277
2.334
3.409
4.511
5.567
Experimental(mm
)
1.47
2.79
4.13
5.50
6.83
Table 12: Experimental and Theoretical Values of Deflection for Cantilever Aluminum at L
E
ave aluminum
= 57.098GPa
vs
70GPa
0
1
2
3
0
100
200
300
400
500
600
Theoretical vs Experimental Deflection of Cantilever Beams at L/2
Cantilever Brass Theoretical
Linear (Cantilever Brass Theoretical)
Cantilever Brass Experimental
Linear (Cantilever Brass Experimental)
Cantilever Steel Theoretical
Linear (Cantilever Steel Theoretical)
Cantilever Steel Experimental
Linear (Cantilever Steel Experimental)
Cantilever Aluminum Theoretical
Linear (Cantilever Aluminum Theoretical)
Cantilever Aluminum Experimental
Linear (Cantilever Aluminum Experimen-
tal)
Deflection (mm)
Load (g)
Figure 2: Theoretical vs Experimental Deflection of the Cantilever Beams at L/2
Discussion and Conclusion:
If we compare the theoretical and experimental values of
deflection, we can see that though they are within the same orders of magnitude, they are not
comparable. This can be seen when we look at the error percentage. At its lowest, the error percentage
is 15.11% for the cantilever aluminum specimen and at its highest, and error value of 75.26% is
calculated for the cantilever steel beam. Considering that error percentage is expected to be between
5%-10%, it can be assumed that the specimen used were either compromised or alloys with different
moduli of elasticity. Since the specimen used for the experiment had already been for multiple other
experiments, it was observed that several of the beams were already subject to some plastic
deformation. It should also be noted that uniform, rectangular beams were used as specimen since
different configurations are more resistant to deflection. An example of this would be “I” beams which
are more resistant to bending and therefore buckling[ CITATION Mov22 \l 4105 ]. These
configurations are more resistant due to the flanges resisting the bending moment along with the rest of
the beam [ CITATION Mov22 \l 4105 ].
The experimental values for aluminum were the closest to its theoretical results, but this can be
due to several different reasons. It is possible that the aluminum specimen were the purest metals out of
the three, but it is also possible that the aluminum specimen had the least plastic deformation. This is
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not an unreasonable assumption as aluminum has the lowest value for its modulus of elasticity,
meaning that it can be subject to more elastic deformation before reaching plastic deformation.
For steel and brass, it was observed that the specimen had already undergone some plastic
deformation before the experiment. This is most likely the cause for the large error percentage, but
possibly not the only cause. Both brass and steel are alloys that can have variable percentages of
different metals in them. This causes fluctuations in their respective moduli of elasticity. The
information regarding these specimens’ compositions were unavailable to us, but it is likely that the
low moduli of elasticity’s are due to these alloys having a larger part of softer metals.
Human error while using the machines and taking the measurements is always a possibility, but
for this experiment it can be considered as negligible as the magnitudes of our calculated values are
much higher than those of our measured values.
In conclusion, even if the error percentage is so high, we know the experiment was carried out
properly as our graphing shows very clear linear relations between the theoretical and experimental
values. This experiment successfully showed that we could accurately calculate the modulus of
elasticity using its relation to deflection.
Bibliography
[1] C. U. E. a. C. Science, ENGR 244 Mechanics of Materials Lab Manual, Montreal: Concordia University.
[2] Designing Buildings, "Deflection," Designing Buildings, 11 01 2022. [Online]. Available:
https://www.designingbuildings.co.uk/wiki/Deflection. [Accessed 13 03 2022].
[3] Movie Cultists, "Why beam deflection is important," MovieCultists, [Online]. Available:
https://moviecultists.com/why-beam-deflection-is-important. [Accessed 13 03 2022].
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