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.

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

load onto the steel beam

·

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

fully unloaded.

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

Load (N)

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

Load (N)

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.