2514600209550
DEPARTMENT OF BIOMEDICAL ENGINEERING
FACULTY OF ENGINEERING
UNIVERSITY OF MALAYA
LABORATORY REPORT
BEAM DEFLECTION
MATRIX NUMBER: KIB180046
GROUP: GROUP 11
LAB ADVISOR: DR. CHAN CHOW KHUEN
Abstract
Objective
To measure deflections in a simply supported aluminium, brass and copper beam.

To use measured and theoretical deflections to evaluate Young’s Modulus of the materials.

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To compare the analytical and experimental values of deflection in the simply supported and cantilever beam.

Hypothesis

Introduction
Deflection can be defined as the movement of a beam or a node from its original position due to the forces that were applied to it. It can happen in beams, trusses and frames and other structures but in this experiment we focused on the beam deflection.
Beam is an element that able to withstand load by resisting the bending from the forces. The degree of structural deformation of the beam can be affected by the type of the beam, span of the beam, shape of the beam and the magnitude of load acting on the beam. There are a few type of beams such simple beam, cantilever beam, overhanging beam and continuous which all of the beams have different degree of bending.

The type of beam used in this experiment is a simple supported beam. This beam has one end with pinned support and a roller support at the other end. The types of beam that were used in the experiment were copper beam, brass beam and aluminium beam and concentrated load and uniform distributed were tested on them to measure the beam deflections. From the measured beam deflection, the maximum load that a beam can withstand can be calculated.

In this experiment, we will calculate the theoretical value of deflection by using Equation 1 and Equation 2 to compare with the experimental value of deflection.

y=Px12EI(3l24-x2)Equation 1: Deflection for concentrated load
y=?x24EI(l3-2lx2+x3)Equation 2: Deflection for distributed load
Where P and w is the weight of load, E is the Young Modulus, l is length of beam, x is the distance of dial gauge from the initial point, and I is the moment of inertia.

Apparatus and Materials
Beam deflection apparatus
Vernier caliper and meter ruler
Aluminium beam
Brass beam
Copper beam
Hanger together with 9 different loads
Rectangular uniform distributed load with 12N
Allen key
Procedure
Deflection of a beam by applying a concentrated load
The clamping length (L) of the beam was set to 700mm.

The width and height of each beam were measured using a caliper and the value was recorded.

The copper beam was placed on the bearers.

One end of the beam was set as fixed end by tightening the screw.

Two dial gauges were moved to the copper beam. The height of the gauge was adjusted so that the needle touched the beam.

The dial gauge 1 (G1) was moved to 175mm and the dial gauge 2 (G2) was moved to 525mm.

1N load was added to middle of the beam and the dial gauge reading was recorded. The value of dial gauge reading was taken three times to obtain the average value.

Step 7 was repeated with 2N, 3N, 4N, 5N, 6N, 7N, 8N, 9N, 10N load. All of the gauge reading was recorded.

All of the loads were removed.

Step 1 to 9 was repeated with brass and aluminium beam.

The experimental deflection was compared with theoretical value.

Deflection of a beam by applying a uniform distributed load
The clamping length (L) of the beam was set to 700mm.

The copper beam was placed on the bearers.

One end of the beam was set as fixed end by tightening the screw.

Two dial gauges were moved to the copper beam. The height of the gauge was adjusted so that the needle touched the beam.

The dial gauge 1 (G1) was moved to 175mm and the dial gauge 2 (G2) was moved to 525mm.

The 12N load (distributed load) was placed onto the beam. The total length of the load on the beam was measured and recorded. The dial gauge reading was recorded.

Step 6 was repeated 3 times to obtain the average value of the dial gauge reading.

Step 1 to 7 was repeated with brass and aluminium beam.

The experimental deflection was compared with theoretical value.

Results
Type of beam Length,l (m) Width,w (m) Height,h (m) Moment of Inertia,I (m4) Young Modulus (GPa)
Copper 1.002 0.026 6.51×10-3 5.978×10-10 117
Brass 1.001 0.025 6.37×10-3 5.385×10-10 102
Aluminium1.001 0.026 6.61×10-3 6.257×10-10 70
Table 1: Dimension of the beam
The moment of inertia,I value was obtained from Equation 3 and the theoretical value of Young Modulus was obtained from our reference.

I=wh312Equation 3: Moment of inertia formula
Part A
Weight of load (N) Deflection of beam (mm)
G1 G2
1 2 3 Average 1 2 3 Average
1.0 0.015 0.01 0.015 0.013 0.02 0.03 0.02 0.023
2.0 0.04 0.04 0.04 0.04 0.06 0.05 0.07 0.05
3.0 0.07 0.07 0.07 0.07 0.11 0.115 0.11 0.112
4.0 0.09 0.09 0.09 0.09 0.14 0.14 0.14 0.14
5.0 0.115 0.115 0.11 0.113 0.18 0.18 0.18 0.18
6.0 0.135 0.135 0.14 0.137 0.21 0.21 0.215 0.212
7.0 0.16 0.16 0.16 0.16 0.26 0.25 0.25 0.253
8.0 0.18 0.185 0.18 0.182 0.29 0.29 0.3 0.293
9.0 0.21 0.21 0.21 0.21 0.33 0.34 0.33 0.333
10.0 0.235 0.23 0.235 0.233 0.375 0.37 0.375 0.373
Table 2: Gauge reading for copper
Weight of load (N) Deflection of beam (mm)
G1 G2
1 2 3 Average 1 2 3 Average
1.0 0.01 0.01 0.015 0.012 0.015 0.02 0.02 0.018
2.0 0.03 0.03 0.025 0.028 0.045 0.045 0.04 0.043
3.0 0.04 0.04 0.04 0.04 0.07 0.065 0.065 0.067
4.0 0.06 0.06 0.06 0.06 0.1 0.095 0.09 0.095
5.0 0.075 0.07 0.075 0.073 0.12 0.115 0.115 0.117
6.0 0.09 0.085 0.09 0.088 0.145 0.14 0.14 0.142
7.0 0.1 0.1 0.1 0.1 0.17 0.165 0.165 0.167
8.0 0.115 0.12 0.12 0.118 0.19 0.19 0.19 0.19
9.0 0.135 0.135 0.14 0.137 0.21 0.21 0.215 0.212
10.0 0.15 0.15 0.15 0.15 0.23 0.23 0.23 0.23
Table 3: Gauge reading for brass
Weight of load (N) Deflection of beam (mm)
G1 G2
1 2 3 Average 1 2 3 Average
1.0 0.04 0.04 0.04 0.04 0.06 0.065 0.055 0.06
2.0 0.08 0.08 0.08 0.08 0.13 0.14 0.14 0.137
3.0 0.12 0.12 0.125 0.122 0.21 0.2 0.21 0.207
4.0 0.17 0.165 0.17 0.168 0.29 0.28 0.29 0.287
5.0 0.21 0.21 0.21 0.21 0.36 0.36 0.365 0.362
6.0 0.25 0.26 0.25 0.253 0.425 0.44 0.42 0.428
7.0 0.3 0.3 0.3 0.3 0.51 0.5 0.51 0.507
8.0 0.35 0.35 0.35 0.35 0.59 0.59 0.595 0.592
9.0 0.395 0.4 0.395 0.397 0.67 0.675 0.66 0.668
10.0 0.44 0.45 0.44 0.443 0.73 0.75 0.74 0.74
Table 4: Gauge reading for aluminium
Figure 1: Graph of deflection of a beam by applying concentrated load
Part B
Length of load from initial point (cm)
Deflection of beam (mm)
G1 G2
1 2 3 Average 1 2 3 Average
26.0 0.27 0.27 0.27 0.27 0.36 0.35 0.35 0.353
28.0 0.28 0.28 0.28 0.28 0.38 0.38 0.38 0.38
30.0 0.28 0.28 0.28 0.28 0.405 0.4 0.395 0.4
32.0 0.29 0.29 0.29 0.29 0.43 0.43 0.43 0.43
34.0 0.29 0.29 0.29 0.29 0.44 0.45 0.45 0.447
36.0 0.285 0.29 0.29 0.288 0.47 0.48 0.49 0.48
38.0 0.28 0.28 0.285 0.282 0.5 0.495 0.5 0.498
Table 5: Gauge reading for copper
Length of load from initial point (cm)
Deflection of beam (mm)
G1 G2
1 2 3 Average 1 2 3 Average
26.0 0.27 0.27 0.27 0.27 0.22 0.22 0.22 0.22
28.0 0.175 0.175 0.175 0.175 0.24 0.24 0.24 0.24
30.0 0.18 0.18 0.18 0.18 0.26 0.26 0.26 0.26
32.0 0.18 0.18 0.18 0.18 0.275 0.275 0.275 0.275
34.0 0.18 0.18 0.18 0.18 0.29 0.29 0.28 0.287
36.0 0.18 0.18 0.18 0.18 0.3 0.295 0.29 0.295
38.0 0.17 0.17 0.17 0.17 0.3 0.295 0.305 0.3
Table 6: Gauge reading for brass
Length of load from initial point (cm)
Deflection of beam (mm)
G1 G2
1 2 3 Average 1 2 3 Average
26.0 0.505 0.49 0.49 0.495 0.67 0.66 0.66 0.663
28.0 0.505 0.505 0.5 0.503 0.72 0.7 0.69 0.703
30.0 0.525 0.51 0.52 0.525 0.775 0.775 0.77 0.773
32.0 0.535 0.53 0.535 0.533 0.81 0.82 0.82 0.817
34.0 0.54 0.545 0.545 0.543 0.895 0.885 0.885 0.888
36.0 0.53 0.53 0.535 0.552 0.89 0.89 0.9 0.893
38.0 0.52 0.53 0.53 0.527 0.92 0.92 0.93 0.923
Table 7: Gauge reading for aluminium
Figure 2: Graph of deflection of beam by applying distributed load
From the all of the tabulated data, we can calculate the experimental modulus elasticity of each beam by using Equation 4.

E=Px12?I(3l24-x2)Equation 4: Young Modulus formula
Where P is the weight of load, ? is the beam deflection, l is length of beam, x is the distance of the dial gauge from the initial point, and I is the moment of inertia.

For copper,
E=100.52512(0.37×10-3)5.978×10-103(0.7)24-(0.5252)=181.73GPaFor brass,
E=100.52512(0.23×10-3)5.385×10-103(0.7)24-(0.5252)=324.54GPaFor aluminium,
E=100.52512(0.74×10-3)6.257×10-103(0.7)24-(0.5252)=86.81GPa Discussion
Weight of
load (N) Experimental Deflection (mm) Theoretical deflection (mm)
G1 G2 G1 G2
A B C A B C A B C A B C
1 0.01 0.01 0.04 0.02 0.02 0.06 0.07 0.09 0.11 0.06 0.07 0.09
2 0.04 0.03 0.08 0.05 0.04 0.14 0.14 0.18 0.22 0.11 0.15 0.15
3 0.07 0.04 0.12 0.11 0.07 0.21 0.21 0.27 0.34 0.17 0.22 0.28
4 0.09 0.06 0.17 0.14 0.10 0.29 0.28 0.36 0.45 0.23 0.29 0.37
5 0.11 0.08 0.21 0.18 0.12 0.36 0.35 0.45 0.56 0.29 0.37 0.46
6 0.14 0.09 0.25 0.21 0.14 0.43 0.42 0.54 0.67 0.34 0.44 0.55
7 0.16 0.1 0.3 0.25 0.17 0.51 0.49 0.63 0.79 0.40 0.51 0.64
8 0.18 0.12 0.35 0.29 0.20 0.59 0.56 0.72 0.90 0.46 0.59 0.73
9 0.21 0.14 0.40 0.33 0.21 0.67 0.63 0.80 1.01 0.52 0.66 0.83
10 0.23 0.15 0.44 0.37 0.23 0.74 0.70 0.89 1.12 0.57 0.73 0.92
Table 8: Comparison of experimental and theoretical deflection
Where A is copper, B is brass and C is aluminium.

The theoretical deflection per unit load force for each beam was calculated using Equation 1. From the theoretical value and experimental value, the percentage error can be calculated using Equation 5.

Percentage error%=theoretical value-experimental valuetheoretical value×100%Equation 5: Percentage error formula
(N)

Percentage
Error (%)
Copper Brass AluminiumG1 G2 G1 G2 G1 G2
1 85.71 66.67 88.89 71.43 63.63 33.33
2 71.43 54.55 83.33 73.33 63.64 6.67
3 66.67 35.29 85.19 68.18 64.71 25.0
4 67.86 26.09 83.33 65.52 62.22 21.62
5 68.57 37.93 82.22 67.57 62.5 21.73
6 66.67 38.24 83.33 68.18 62.69 21.82
7 67.35 37.5 84.13 66.67 62.03 20.32
8 67.86 36.96 83.33 66.10 61.11 19.18
9 66.67 36.54 82.5 68.18 60.4 19.28
10 67.14 35.09 83.15 68.49 60.71 19.57
Table 9: Percentage error of the beam deflection
We can see that most of the value of the percentage error above was more that 30%. In addition the percentage error for the Young Modulus of copper, brass and aluminium were 54.7%, 21.82% and 23.1% respectively. So, it can be justified there were errors when the experiment was conducted as there was significant difference for the measured value compared to theoretical value.

One of the sources of error in this experiment was the system error which caused by the inaccurate dial gauge. The dial gauge value was not acceptable as there are errors when we were using it. The spindle of the dial gauge was in a poor condition as sometimes it cannot move to read the deflection. Next, the beams were already deformed as the beams were used many times in previous experiment. The errors also can be caused by parallax error when we record the reading of the dial gauge.
However, in Figure 1, the graph of deflection against weight of load is plotted. It is shown that aluminium was the hardest beam followed by copper and brass. So the theory of the beam deflection is still met although there are errors in the experiment when all the beams has different value of deflection according its type and this also follows the Young Modulus. The deflection of the beam also increased when the weight of load increased.
In Figure 2, the graph of deflection against length of load from initial point is plotted. It is shown that the deflection also depends on the point of application of force. The deflection increase when the length of load from initial point increase.
Beam deflection is important in designing biomedical engineering equipment because the engineer should know how much the beam can deflect and which type of beam is stronger. This will make the equipment is more durable and can work efficiently.
The beam deflection can be predicted as shown in Equation 1 and Equation 2. It is useful in designing a beam as we can know how much a beam can deflect at certain point of a beam. So, the engineer can predict where the force should be applied on the beam that will cause less deflection.
The main importance of measuring beam deflection is to ensure the beam will not exceed the maximum deflection or have a specification of allowable deflection when the force is applied. For example, we need to know beam deflection when designing biomedical engineering equipment such as artificial limbs.

Conclusion