bending2(for adeq)

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TITLE : Bending In Beam

OBJECTIVE : To determine the elastic modulus (E) of beam specimen by method of

deflection.

APPARATUS : Dial gage, fixed support, load meter, beam specimen, Mild Steel beam,

Brass beam, Aluminum beam, Vernier caliper, ruler, and load.

DIAGRAM :

Figure 1 The instrument for calculating bending in beam.



Figure 2 The dial gage

THEORY : Pure Bending

2R =22 )2/()( LyR +−

= 4/2 222 LyyR ++−

Hence,

4/2 2LRy =

yLR 8/2=

I

M

R

E = Where, M = W(x)

I =12

3bh

Therefore; I

xW

L

yE )(8.

2=

)(

)8..(2xL

yI E W =∴



PROCEDURE :

1. The equipment is first set up as shown in Figure 1.

2. The dimension of the cross sectional beam is measured.

3. The dial gage is set to zero as the initial reading.

4. Next, apply the loads on the load holder.

5. Then, record all of the readings of the dial gage.

6. Repeat step 4 by increasing the load and record the readings at every increment

until 8 readings.

7. Step 6 is repeated by changing the beam to aluminum and brass beam.

EXPERIMENTAL RESULTS:

Load (W) Beam maximum deflection ()01.0 mm×

Mild Steel Aluminum Brass

2 23.4 22.0 14.5

4 47.6 48.2 31.2

6 73.6 71.5 47.6

8 98.2 93.4 64.5

10 123.2 119.6 82.4

12 148.8 147.8 99.4

14 174.4 170.2 117.0

16 198.8 191.6 133.8

Table 1.0 Readings for deflections in beams.

CALCULATIONS:



i. Mild Steel

Beam Dimension: mmmmmm 54.44.201000 ××

Modulus of Elasticity, E : 210 GPa I: 41010591.1 m−

×

L: 50cm x: 15cm

Load (W) Beam max. deflection (mm) E (GPa)

2 0.234 251.8

4 0.476 247.6

6 0.736 240.2

8 0.982 240.0

10 1.232 239.1

12 1.488 237.6

14 1.744 236.516 1.988 237.1

Table 1 Readings for beam maximum deflection for Mild Steel.



Deflection 0f mild steel beam

0

2

4

6

8

10

12

14

16

18

0 0.5 1 1.5 2

Load (N)

Dimension of beam for Mild steel

12

)1054.4)(0204.0(

12

333−

×

==bh

I

= 41010591.1 m−

×

For y = 23.4 x 10-4m , W = 2N

yI

xLW E .8.

)(. 2=

= 251.8 GPa

1000mm 20.4mm

4.54mm





Deflection of aluminum beam

0

2

4

6

8

10

12

14

16

18

0 0.5 1 1.5 2

Load (N)

Dimension of beam for Aluminum:

12

)100.6)(02.0(

12

333−

×

==bh

I

= 410106.3 m−

×

For y = 2.2 x 10-4m , W = 2N

yI

xLW E

.8.

)(.2

=

= 118.4 GPa

Average E = (118.4 + 108.1 + 109.3 + 111.5 + 108.9 + 105.7 + 107.1 + 108.7) GPa

1000mm 20.0mm

6.0mm



8

= 110 GPa

iii. Brass

Beam dimension : mmmmmm 6.68.211000 ××

Modulus of Elasticity, E :104.1GPa I : 1010222.5−

× m4

L : 50cm x : 15cm

Table 1.3 Readings for maximum deflection of Brass.

Load, W (N) Beam max. deflection (mm) E (GPa)

2 0.145 123.84 0.312 115.1

6 0.476 113.1

8 0.645 111.3

10 0.824 108.9

12 0.994 108.4

14 1.170 107.4

16 1.338 107.3



Deflection of brass beam

0

2

4

6

8

10

12

14

16

18

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Load (N)

Dimension of beam for Brass:

12

)106.6)(0218.0(

12

333−

×

==

bhI

= 41010222.5 m−

×

For y = 1.45 x 10-4m , W = 2N

yI

xLW E

.8.

)(. 2

=

= 123.8 GPa

1000mm 21.8mm

6.6mm



Average E = (123.8 + 115.1 + 113.1 + 111.3 + 108.9 + 108.4 + 107.4 + 107.3) GPa

8

= 112 GPa

DISCUSSION:

When the experiment have be done, the value of Modulus of Elasticity, E for each

beam is calculated. Since, there are distinction values of experiment by compared

to theoretical values. Comparing values is described below:

• Mild Steel

E ( Theory) = 210 GPa

E ( Experiment) = 241 GPa

Percentage of error is given by,

)(

)()(%

TheoryE

ExpE TheoryE Error

−=

= (241 – 210) x 100

210

= 14.8 %

• Aluminum

E ( Theory) = 70 GPa

E ( Experiment) = 110 GPa

Percentage of error is given by,





accuracy. For example, the dial gage is not consistent when the load was applied.

If the experiment repeat once again as same load, it showed a different readings.

This means that, the dial gages is not reliable in order to measure the readings.

The specimens also experience wear same as the dial gauge because its have been

used many times before this experiment is conducted. When the specimens is

measured by 1 meter ruler it is not straight as the 1 meter ruler.

One more precautions that could be useful to reduce the errors is when placing the

load. The experimenter must place it slowly and tend not to release it so hard as

the dial gages were so sensitive and may detect more loads that it supposed to be.

Besides, the errors may comes from the distance of x. There may have little

difference distance of x at right and left sides. It can cause the loaded was not

balance accurately.

CONCLUSION:

By using the method of deflection, we obtain that the values of elastic modulus is:

E (mild steel) = 241 GPa

E (aluminum) = 110 GPa

E (brass) = 112 GPa

REFERENCES:

1. Mechanics of materials, 3rd Edition, McGraw Hill by Ferdinand P. Beer, E. RussellJohnston Jr. and John T. DeWolf.

2. Mechanics of materials, 3rd Edition, McGraw Hill by R. C. Hibbeler.3. Statics and Strength of Materials, 2nd Edition, Mc-Graw Hill. Jensen and

Ehenoweth.

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bending2(for adeq)