Cantilever Beam Experiment Lab Report
Name: Muhammed Masoud
Email: mm335@hw.ac.uk
HW ID: H00310657
Mechanical Engineering Science 3
Date Conducted: 21/10/2018
Group Number: 9
Date Submitted: 09/11/2018
Index
Sl. No.
Topic
Page No.
1
Abstract
3
2
Introduction
4
3
Theory
6
4
Apparatus
15
5
Procedure
17
6
Results and discussions
19
7
Conclusions
28
8
Recommendations
29
9
References And Citation
30
Abstract
In this experiment, an aluminium beam of E=70 GPa is clamped to the workshop bench to view it as a cantilever beam. We then add weights to find the values of strain and deflection. Experimental results are recorded.
This experiment is carried out to expound the strain experienced by a Cantilever beam under flexural stress using Hooke’s Law. Further, equations of bending moment, slope, deflection, stress and strain are derived. Then, the experimental results are compared with the theoretical values and the percentage error is found.
Introduction
• A cantilever beam is a brand of beam anchored at only one end. Like other beams, they do maintain their shape in spite of the large tensile and compressive forces, as well as shear, and therefore are relatively massive.
• Cantilever arms are very rigid, because of their depth.
• They have various useful applications in real life engineering. Some of them are described below:
Cantilever cranes
Cranes are a well-known application of cantilever beams. They are used to lift and relocate heavy objects such as concrete, steel bars, etc. The unique property of cantilever beam of withstanding humungous amount of load on the free end makes it a convenient device.
Parking Canopies
Parking canopies are designed to provide shade for the vehicles. It is a truss-system.
Balconies
Most of the balconies of buildings are supported by cantilever beams. The cantilever is extended to beams inside the floor-ceiling assemblies inside the building.
Objectives:
♣ To analyze the relationship between load, stress, strain, bending moment, slope, and deflection in a cantilever beam.
♣ To compare the theoretical and experimental values of strain and deflection and hence to find the percent error.
Theory
As mentioned above, this experiment is conducted to find the relationship between load, stress, strain, bending moment, slope, and deflection when a load W is applied on the far end of a cantilever beam, at a distance ‘x’ from the far end of the beam.
Figure 4
W- Load
X – distance from the far end where the dial gauge is positioned.
L- Total length of the beam
Free Body Diagram(FBD) of a cantilever beam:
Figure 5
Moment equilibrium,
Vertical equilibrium,
As we all know, Shear force Qx is the differential of Bending Moment. Thus, we differentiate BM to obtain the shear force.
Bending Moment(BM), = -Wx
Shear Force(SF),
Figure 6
Figure 2 shows us the bending moment and shear force diagrams of the cantilever beam.
Bending moment: Bending moment(BM) is an internally developed moment to counter act the externally applied loads and hence to attain equilibrium which is developed inside the body. Note that it is not an applied load moment on the body, it is only developed inside when the body is subjected to some external stimuli.
Shear force: Shear force is the force acting along a surface i.e., at zero degree inclination to surface.
This is the equation relating Bending Moment and shear stress which is known as the flexural formula.
Where,
M = Moment, Nm
I =Second moment of area, m4
= Stress, N/m2
E = Young’s modulus, N/m2
R = Radius of curvature, m
Slope: Slope is defined as the tangent of the angle made by the bending line with the positive x-axis.
Deflection: The deflection at any point on the axis of the beam is the length between its position before and after loading.
To find the deflection of the beam,
We consider a beam which is being bent
Figure 7
Equation of the circle,
In this equation x is negligibly small when compared to the radius of curvature, r. So we can safely disregard x.
Hence by differentiating w.r.t x,
by integrating,
Applying the boundary conditions,
At And
Substitute these values in the equations 3 and 4 to find the values of constants C1 and C2.
By plugging in the values of C1 and C2 in the equations 3 and 4,
Slope,
Deflection,
Strain,
Where,
M = Moment
W=Load
I =Second moment of area
= Stress
E = Young’s modulus
R = Radius of curvature
= Strain
Y=Half the height of the beam used
L=Total length of the beam
=Deflection
Experimental Setup
Figure 9
Apparatus
Figure 10
The equipments used are the workshop bench, an aluminium beam(E=70GPA), strain bridge amplifier, G clamps, weight hanger and weights, a dial gauge and a Vernier caliper.
Procedure
• Start the experiment by clamping any of the ends of the beam to the workshop bench using G clamps such that the cantilever is 500 mm away from the edge of the table.
• Attach the beam with strain gauges which is connected to the strain gauge bridge. Strain gauge bridge is further connected to a strain gauge amplifier.
• Designate the dial gauge at the far end of the beam to measure the deflection.
• Now, measure the dimensions of the beam using the Vernier caliper.
• Set the strain gauge amplifier to 0 using the offset knob and the dial gauge to 0 by turning the outer face of it.
• Now, position the weight hanger over the free end of the cantilever beam. Remember, the weight hanger contains the load of 100 grams.
• When the gauges are offset to zero and the weight hanger is connected, take down the readings of the strain from the strain gauge amplifier, and the deflection from the dial gauge.
• Now, keep on adding the weights until you reach 1000 grams. While adding each 100 grams load, note down the strain, and the deflection due to the applied load in a table. Remember that the dial gauge has 100 divisions; one whole rotation in the Dial gauge is 1 mm. The strain gauge amplifier will show negative values because the beam is deflecting in the downward direction. Always read the values in the dial gauge in the direction of rotation i.e., if the rotation is clockwise start measuring the divisions travelled by the pointer in the clockwise direction neglecting the sign.
• Now, Adjust the beam such that the cantilever is 400 mm away from the edge of the table and record data following the steps mentioned above.
• Plot graphs of load vs strain and load vs. deflection on graph paper.
• Calculate the values of strain and deflection theoretically and compare it with the values obtained through the experiment. Thus, calculate the percentage error.
Results and Discussions
⎫ The results of strain and deflection for different weights ranging from 100g to 1000g.
Length=500mm
Load
Deflection (in mm)
Strain
0g
0
0
100g
0.10 mm
8
200g
0.28 mm
20
300g
0.47 mm
32
400g
0.66mm
45
500g
0.84mm
58
600g
1.07 mm
75
700g
1.21 mm
85
800g
1.50 mm
100
900g
1.68 mm
115
1000g
1.82 mm
124
Table 1
⎫ The results of strain and deflection for different weights ranging from 100g to 1000g.
Length=400mm
Load
Deflection (in mm)
Strain
0g
0
0
100g
0.08 mm
12
200g
0.19 mm
23
300g
0.30 mm
39
400g
0.38 mm
49
500g
0.51 mm
64
600g
0.58 mm
74
700g
0.69 mm
88
800g
0.79 mm
104
900g
0.85 mm
113
1000g
0.97 mm
126
Table 2
Graphing strain and deflection vs load for both the lengths –
• Strain of 500mm beam vs strain of 400mm beam
Graph 1
• Deflection of 500mm beam vs Deflection of 400mm beam
Graph 2
To find the theoretical results, we use the equation mentioned in the theory section above,
Theoretical value of deflection,
where the values of,
• E = 70 X 109 Pa
• I = = = 4.57 X 10-9 m4
• W = Weight on the far end (100g – 1000g)
• L = Total Length of the beam(0.4m/0.5m)
Theoretical value of strain,
where the values of,
• E = 70 X 109 Pa
• I = = = 4.57 X 10-9 m4
• W = Weight on the far end (100g – 1000g)
• L = Total Length of the beam(0.4m/0.5m)
• Y=Half the height of the beam used (h/2=0.013/2=0.0065m)
After calculating the theoretical values of strain and deflection, percentage error is calculated using the formula:
Comparison of the experimental results with the theoretical values:
Length=500mm
Weight
Experimental Deflection
Theoretical deflection
Percentage error
0
0 mm
0 mm
0%
100
0.10 mm
0.12mm
16.67%
200
0.28 mm
0.25mm
12%
300
0.47 mm
0.38mm
23.6%
400
0.66mm
0.51mm
29.4%
500
0.84mm
0.63mm
31.8%
600
1.07 mm
0.76mm
40.7%
700
1.21 mm
0.89mm
35.9%
800
1.50 mm
1.02mm
47%
900
1.68 mm
1.14mm
47.3%
1000
1.82 mm
1.27mm
43.3%
Table 3
Average percentage error: 32.77 %
Length=400mm
Weight
Experimental Deflection
Theoretical deflection
Percentage error
0
0
0 mm
0%
100
0.08 mm
0.06mm
33.33%
200
0.19 mm
0.13mm
46.15%
300
0.30 mm
0.19mm
57.8%
400
0.38 mm
0.26mm
46.15%
500
0.51 mm
0.32mm
59.37%
600
0.58 mm
0.39mm
48.7%
700
0.69 mm
0.45mm
53.3%
800
0.79 mm
0.52mm
51.9%
900
0.85 mm
0.59mm
44%
1000
0.97 mm
0.66mm
45.7%
Table 4
Average percentage error: 48.64 %
Length=500mm
Weight
Experimental Strain
Theoretical Strain
Percentage error
0
0
0
0%
100
8
9.95
19.5%
200
20
20.12
0.59%
300
32
29.86
7.16%
400
45
39.82
13%
500
58
49.78
16.5%
600
75
59.73
25.5%
700
85
69.69
21.9%
800
100
79.64
25.5%
900
115
89.60
28.3%
1000
124
99.56
24.5%
Table 5
Average percentage error: 18.24%
Length=400mm
Weight
Experimental Strain
Theoretical Strain
Percentage error
0
0
0
0%
100
12
7.96
50.7%
200
23
16.2
41.9%
300
39
24.9
56.6%
400
49
32.5
50.7%
500
64
40.2
59.2%
600
74
48.8
51.6%
700
88
58.5
50.4%
800
104
64.7
60.7%
900
113
66.1
70.9%
1000
126
79.6
58.2%
Table 6
Average percentage error: 55.09%
Graph 3
Graph 4
Basically, deflection is linearly related to the load applied. Experimental results are slightly different from the theoretical values. This is due to different errors occurring in the workshop.
The sources of error are:
¬ Personal error: Errors in reading due to parallax.
¬ Accuracy of values: Young’s modulus value is approximated and hence values after calculation get approximated.
¬ Device error: Precision of the device is less as compared to the calculation. Tolerance level is low.
When a load is applied on a body, some small changes occurs in dimensions of the body, then the ratio of change in dimensions of body and original dimensions of body known as strain. A strain is hence developed on the beam. The error in the experimental values are due to:
o Instrumental error: Vibration of the parts in the strain gauge amplifier and strain gauge bridge.
o Elasticity of the beam
Conclusions
References and Citation
1. https://www.britannica.com/technology/cantilever
2. https://en.wikipedia.org/wiki/Deformation_(mechanics)
3. https://reference.wolfram.com/applications/structural/BendingofCantileverBeams.html
a. Hibbeler, Russell Charles. Statics and Mechanics of Materials. Prentice Hall, 2011.
b. Bansal, R. K. Engineering Mechanics and Strength of Materials. Laxmi Publications, 1998.
c. Craig, Roy R. Mechanics of Materials. Wiley, 2011.