TITLE : CONTINUOUS BEAM

OBJECTIVE : To determine the supporting force.

APPARATUS : Refer to figure 1.0

Label

Part name

1 & 2

Roller support

3

Fixed support

4

Dial Gauge

5

Beam

6

Dynamometer

7

Load holder

Figure 1.0

THEORY :

The reaction of forces at points 1, 2, and 3 can be determined by using the moment equation as follows




∑ F = 0 N

Resultant of total forces




∑ MA = 0 Nm

Resultant of total moment

Basic Sign Convention System



















= (+ve) = (-ve) = (+ve) = (-ve)

W2b2 – R2L2 + W1 (b1+L2) –R1 (L1+L2) + mg (L1+ L2) = 0

2

PROCEDURE :

1) The apparatus is set up as shown in the figure above.

2) Distance L1, L2, a1, a2, b1 and b2 is measured.

3) The scale on the dial gauges and dynamometers is set to zero.

4) The beam is being loaded at (7).

5) The readings on the dial gauges and dynamometers are recorded in the table given.

6) The load is increased by 2.5N until the load reaches 22.5N and the reading of dial gauges and dynamometer are recorded in the table given.

7) Step 6 is repeated to get at least four readings.

RESULTS :

a1 = 0.235 m b2 = 0.250 m

a2 = 0.180 m L1 = 0.500 m

b1 = 0.265 m L2 = 0.430 m

Beam Dimension = 868.00 mm x 20.30 mm x 6.20 mm

Young’s Modulus = 210.00 GPa

Data:

W1 (N)

W2 (N)

R1 (N)

(Theory)

R1 (N)

(Exp)

R2 (N)

(Theory)

R2 (N)

(Exp)

2.50

2.50

0.42

0.50

4.58

4.50

7.50

7.50

1.28

1.30

13.72

13.68

12.5

12.5

2.12

2.16

22.88

22.81

17.5

17.5

2.98

3.00

32.02

32.10

22.5

22.5

3.82

3.82

41.18

41.16

Example of calculations:

When W1 and W2 = 2.5 N

2.5N 2.5N

2 3

1

C

0.408 m 0.46 m

R1 R2

∑ Fy = 0

R1 + R2 – W1 –W2 = 0

R1 + R2 – 2.5 – 2.5 = 0

R1 = 5.0 – R2 ---eqn (1)

+ ∑ M3 = 0

-R1(0.93) + W1(0.695) - R2(0.43) + W2(0.25) = 0

-R1(0.93) + 2.5(0.695) - R2(0.43) + 2.5(0.25) = 0

0.93R1 + 0.43R2 = 2.36 ---eqn (2)

(1) (2):

R1=0.42N

R2=4.58N

DISCUSSION :

1. The comparison between the theoretical and actual results slightly differs due to certain factors :

  • Observation error @ Parallax errors due to reading taken
  • The dial gauge may not calibrated
  • Small vibration and movement interferences which effects the reading on dynamometers
  • Slightlty inclined workbench which may cause vectored load into 2 axis components

Percentages of error:




R1 = R1(exp) R1(theory) x 100%

R1(theory)

For W1, W2 = 2.5 N;

R1 = R1(exp) R1(theory) x 100% R2 = R2(exp) R2(theory) x 100%

R1(theory) R2(theory)

= (0.50 – 0.42) x 100% = 19.05% = (4.62 – 4.58) x 100% = 0.87%

0.42 4.58

For W1, W2 = 7.5 N;

R1 = R1(exp) R1(theory) x 100% R2 = R2(exp) R2(theory) x 100%

R1(theory) R2(theory)

= (1.30 – 1.28) x 100% = 1.56 % = (13.76 – 13.72) x 100% = 0.29%

1.28 13.72

For W1, W2 = 12.5 N;

R1 = R1(exp) R1(theory) x 100% R2= R2(exp) R2(theory) x100%

R1(theory) R2(theory)

= (2.16 – 2.12) x100% = 1.87 % = (22.92 – 22.88) x 100% = 0.17%

2.12 22.88

For W1, W2 = 17.5 N;

R1 = R1(exp) R1(theory) x 100% R2 = R2(exp) R2(theory) x 100%

R1(theory) R2(theory)

= (3.02 – 2.98) x 100% = 1.34 % = (32.08 – 32.02) x 100% = 0.19 %

2.98 32.02

For W1, W2 = 22.5 N;

R1 = R1(exp) R1(theory) x 100% R2= R2(exp) R2(theory) x 100%

R1(theory) R2(theory)

= (3.86 – 3.82) x 100% = 1.05 % = (41.22 – 41.18) x 100% = 0.10%

3.82 41.18

Average R1 = 4.97% Average R2 = 0.32 %

CONCLUSION :

It can be conclusively said that the reaction away from the cantilever displays much reaction force compared to the one that is closer to it. Based on the observation the experiment has shown that there will be more deflection at the other end of the beam as the beam gets longer away from cantilever point. Even though there some errors or indifferences in the results compared to theoretical, however the principal idea shows that both theoretical and experimental shows the same concept of cantilever deflection which causes higher reaction force as it moves away from the cantilever point.

REFFERENCE :

1. Mechanics of materials, Ferdinand P. beer, E. Russell Johnton, Jr, John T. deWolf, Mc Graw Hill, Third edition in SI units.

2. Advance Mechanics of Materials, Arthur P Boresi, Richard J. Schmidt

Wiley Sixth Edition, 2002.