A beam is a structural element utilized in the supporting of heavy loads that make various structures. The huge loads supported by the beam always tend to bend the beam while the beam reacts by resisting the bending caused by the load. The bending characteristics of a beam are dependent on the type of beam, the material of beam and the shape of the beam.
The elastic modulus is a mechanical property of a material that describes the ratio of tensile stress and strain. A greater value of the elastic modulus means the material is stiffer hence the material will deflect less for greater forces. The elastic modulus can be calculated as
Where
is the elastic modulus
is the stress
is the strain
A body under stress tend to change in its shape and dimension a phenomenon called deflection. The change in the dimensions of the body is called strain. The deflection of a body can be described using the following equation
Where
is the force
is the length of the beam
is the constant based on the position
is the elastic modulus
is the second moment of area
Equipment
Figure 1: The set up for the deflection of the beams and the cantilevers
Procedure
The experiment entailed the application of a series of loads to a beam and determination the deflections. The loading will be point and uniform distributed the load.
A beam was selected, i.e. either red, yellow or blue which were made from various materials that were unknown. The support system was also selected, and the unsupported length was also determined. The beam was then inserted in the support and secured.
The dimensions of the steel section were measured. These dimensions will be vital in the calculation of the moment of inertia.
The magnitude of the point load was determined and then applied to the beam using the knife-edge load hanger.
The deflection along the length of the beam L was measured and located.
The magnitude of the uniform load was determined and then applied to the beam using equally placed point loads along the span.
The deflections of the beam length were measured. The beams were then unloaded.
Result
F
8 N
LF
400 mm
Figure 2. Details of point load experiment
W
1200 g
Single Load
200 g
Number of single Loads
6
Figure 3. Details of uniformly distributed load experiment
Location (mm)
60
120
180
240
300
360
420
480
540
550
Deflection (mm)
Point Load
1.87
3.56
4.94
5.87
6.18
5.88
4.59
2.81
0.01
0
Uniform Load
2.68
4.96
6.70
7.66
7.79
7.12
5.52
3.29
0.85
0
Table 1. Experimental deflections
The equation for the calculation of the theoretical deflection
Parameters
Location (mm)
60
120
180
240
300
360
420
480
540
550
Deflection (mm)
Point Load
4.66
6.88
7.86
8.87
9.14
8.43
6.85
5.25
2.0
0
Uniform Load
5.55
7.68
8.76
8.98
9.35
8.78
7.81
6.56
3.21
0
Table 2a: Theoretical deflections for aluminum
Parameters
Location (mm)
60
120
180
240
300
360
420
480
540
550
Deflection (mm)
Point Load
3.06
4.67
6.52
7.90
8.20
7.68
5.47
4.43
3.20
0
Uniform Load
4.06
5.46
6.80
7.94
8.45
7.77
6.62
5.56
3.54
0
Table 2b: Theoretical deflections for Brass
Parameters
Location (mm)
60
120
180
240
300
360
420
480
540
550
Deflection (mm)
Point Load
1.55
2.45
3.54
4.35
6.64
4.65
3.33
2.67
1.23
0
Uniform Load
2.34
3.76
4.29
5.13
7.56
5.60
4.23
3.49
2.03
0
Table 2c: Theoretical deflections for steel
Value of the elastic modulus
Error
Discussion
The experimental results for the deflections are close to the theoretical values of brass. The resultant elastic modulus of the material selected was 171921 MPa which was close to the theoretical value of brass, i.e. 207000 MPa. The second moment of area for the beam was maintained constant as its variation would significantly affect the resistance of the materials to deflection. The experiment was aimed at testing the elastic modulus of the beam hence the second moment of inertia would cause a smaller deflection than expected. The experimental elastic modulus was lower than the theoretical value by 16.95%. The main source errors are instrumental and human errors. The instrumental errors might be the inaccuracy in the measurement gauges and the imbalance in the horizontal surface of the beam. The beam might also be deformed. The human error may be substantially due to the parallax in reading the values.
