## Abstract by simple bending. Young’s modulus is

Abstract

The purpose of the
experiment was to determine a value for Young’s modulus E for a steel beam by
simple bending. Young’s modulus is used to describe and measure deflection of a
material. A simple supported steel beam of different thicknesses was provided.
The base and height of the steel beam was measured at three different points,
then an average was determined using these.

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The main test
was conducted to determine the amount of deflection in the steel beam. Couples of
2N were placed on both sides of the beam simultaneously and the deflection was
measured. This was repeated till the load reached 8N on both sides of the beam
(16N in total). The deflection was also measured as the couples were unloaded. These
results were then put into an equation used to determine a value for Young’s
Modulus E.

1.) Introduction and Background

The theory
states that when couples (a pair of forces, in this case weights) are applied
to the end of a beam, the beam will deflect into a circular arc, this results
in the beam being in a state of simple bending.

The aim of the
experiment (as mentioned in the abstract) was to determine Young’s modulus for
a steel beam by simple bending. Conducting this would describe the elasticity
of the material (steel), and how much force it could withhold before deflecting.

This is
important as in manufacturing, engineers may require specific properties (e.g. more
brittle, high ductility etc.) and conducting these experiments will provide the
specific properties of the material.

Deflection

Deflection is the movement
of a body (material) from its original position when a force, load or weight is
applied directly to it.

If for example an
engineer is constructing a bridge, the engineer would want the material used in
the structure to hold the bridge up, conducting these experiments on the
material would help to realise the amount of load the material can handle
before deflecting.

2.) Experiential procedure

Equipment

The experiment was
conducted to measure deflection on a steel beam by bending. The equipment used
to carry out the experiment includes:

·
A steel beam of different thickness – used to
determine Young’s modulus E

·
Rule – used to measure the steel beam

·
A set of couples (16N in total) – used to apply

·
Dial gauge – used to measure deflection of the
steel beam

Method

Initially the steel beam
was placed onto a support structure which held it in place. A dial gauge was
placed with the steel beam to measure deflection.

Firstly, the base and
height of the steel beam was measured at three different points, using these an
average was determined. Two hooks were placed on both sides of the steel beam, 200mm
from the end.

Making sure the dial gauge
was set to zero, couples (2N) were placed simultaneously on both sides of the
hook. As the load was applied, the measurements displayed on the dial gauge were
noted. This was repeated 8 times till the load reached 16N, at this point the
load was unloaded 2N at a time and the measurements were recorded till it was

Figure 1

3.) Results and Calculations

Initially the steel beam
was measured for its base and height at three different points. The results
gathered are as follows:

Table 1

All measurements are in
mm

Point 1

Point 2

Point 3

Average

Base

24.98

25.23

25.01

25.07

Height

4.00

4.07

4.17

4.08

Working out for the
average:

Base- 24.98 + 25.23 + 25.01
= 75.22 ¸ 3 = 25.07

Height- 4.00 + 4.07 + 4.17
= 12.24 ¸ 3 = 4.08

Once the steel bar was
measured, deflection was measured using a dial gauge, couples of 2N were placed
at a time till the load reached 16N, at this point the load was unloaded 2N at
a time and the measurements were recorded till it was fully unloaded. The
results gathered are as follows:

Table 2

Deflection (mm)

D (actual deflection)
(mm)

0

8.27

0

2

8.92

8.92 – 8.27 = 0.65

4

9.58

9.58 – 8.27 = 1.31

6

10.23

10.23 – 8.27 = 1.96

8

10.89

10.89 – 8.27 = 2.62

10

11.55

11.55 – 8.27 = 3.28

12

12.20

12.20 – 8.27 = 3.93

14

12.89

12.89 – 8.27 = 4.62

16

13.53

13.53 – 8.27 = 5.26

Table 3

Deflection (mm)

D (actual deflection) (mm)

14

12.98

12.98
– 8.27 = 4.71

12

12.31

12.31
– 8.27 = 4.04

10

11.65

11.65
– 8.27 = 3.38

8

10.98

10.98
– 8.27 = 2.71

6

10.29

10.29
– 8.27 = 2.02

4

9.63

9.63
– 8.27 = 1.36

2

8.96

8.96
– 8.27 = 0.69

0

8.28

8.28
– 8.27 = 0.01

The theory of bending gives
the relationship: =  =

s is the longitudinal stress
in the beam at a distance y from its axis; R is the radius of the curve, M is
the bending moment, E is the Young’s Modulus for the steel beam.  is the second moment of area of the cross
section of the beam about its neutral axis and: =

Using this formula will
provide the second moment of area. As mentioned previously the average base
measures at 25.07mm and the average height measures at 4.08mm. Adding these to
the formula will give.

=

=  =

With this equation  , d (deflection
of the beam) and W (Weight), are two constants, these are instead represented
as k, this is also the point the gradient is discovered on the graph. The new
equation with k as the subject becomes,

This
equation can be rearranged to make E (Young’s modulus) the subject, this
will give a value for Young’s modulus E for the steel beam. The new
equation with E as the subject becomes,

The gradient of the graph equals
= 0.33Nmm, this can also be written as.

The value for L as shown in Figure 1 measures at
200mm or 0.2m. Similarly, the value for W measures at 600mm or 0.6m.

Putting these values into the equation will give a
value for Young’s Modulus.

Figure
2

d

4.) Discussion

The aim of the experiment
was to measure Young’s modulus E for a
steel beam by simple bending. Conducting this experiment required bending the
steel beam using couples (up to 16N). The
results of this experiment demonstrate that there is a positive correlation
between the load of the couples and the deflection of the steel beam. As the
load applied to the steel beam increases, the amount of deflection on the steel
beam also increases. When the initial load of 2N was applied, the beam
deflected by 0.65mm, and when the load of 4N was applied, the beam deflected by
a 0.66mm. When the load reached 6N however, the beam deflected by another
0.65mm. These results show that the deflection of the beam is constant when the
same amount of load is applied. One source of error which could have
potentially affected the results would be the dial gauge, to get an exact
reading, the dial gauge has to be set to 0 exactly, any small objects near the
set up (i.e. on the table) could potentially affect the results. Another source
of error could be due to the way couples were applied to the beam, as the load
has to be applied simultaneously, any small difference in time between the two
weights could potentially affect the results. One potential solution to get
more of an accurate reading would be to conduct the experiment multiple times.

5.) Conclusion

In conclusion, the experiment proves that there
is a positive correlation between the deflection of a steel beam and loads
applied. This shows that, the higher the number of loads applied onto the beam,
the deflection of the beam is greater. The graphs as shown in the results page
as Figure
2 show that the trend lines are drawn in ascending order which shows a
positive relationship between the loads applied and the deflection of the beam.
The gradient of the graph, k, is determined to be . Therefore, using the formula,  elastic modulus of the
beam is determined as which
was simplified to. An online database which
mainly holds measurements of Young’s modulus E for various materials shows that
this steel beam has a higher value for Young’s modulus E than aluminium, iron, copper
and titanium alloys. This means that this steel beam is brittle yet strong.