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cantilever beam

By atap_aje

TITLE: CANTILEVER BEAM

OBJECTIVE:

To determine the deflection of cantilever beams

APPARATUS:

Num

Part Name

Num

Part Name

1

Fixed Support

4

Dial Gauge

2

Cantilever beam

5

Weight (W)

3

Clamping lever

6

Base Plate

L

4

1 2

5

6

Figure E2

THEORY:

The deflection at free end of a cantilever beam is given by this relation


y max =

where F=


PROCEDURE:

  1. The apparatus is set up as shown in Figure E2.
  2. The distance of L is measured.
  3. The scale on the dial gauge is set to zero
  4. The beam is loaded with 5 N.
  5. The dial gauge is read and recorded.
  6. The load is increased and the reading is recorded.
  7. Step 6 is repeated to get at least four reading.
  8. Step 1 to 7 is repeated by using the beam with the different cross-sectional area.

RESULT:

Beam 1: I- Cross sectional

L = 550 mm I = 12736 mm4

E = 68000 N/mm2 Beam dimension: 25 mm x 25 mm x 3 mm

W (N)

Yb (mm) Exp

Yb (mm) Theory

5

0.300

0.352

10

0.670

0.704

15

1.032

1.057

20

1.450

1.409

25

1.690

1.761

Beam 2: L - Cross sectional

L = 550 mm I = 13030 mm4

E = 68000 N/mm2 Beam dimension: 25 mm x 25 mm x 3 mm

W (N)

Yb (mm) Exp

Yb (mm) Theory

5

0.290

0.344

10

0.620

0.689

15

0.970

1.033

20

1.320

1.377

25

1.650

1.721

Beam 3: U- Cross sectional

L = 550 mm I= 19970 mm4

E = 68000 N/mm2 Beam dimension: 25 mm x 25 mm x 3 mm

W (N)

Yb (mm) Exp

Yb (mm) Theory

5

0.260

0.225

10

0.470

0.449

15

0.700

0.674

20

0.830

0.898

25

1.140

1.123

The graph of deflection against weight.

A) Beam 1: I- Cross sectional

Graph Deflection (y) versus Weight (F) – Experiment




Graph Deflection (y) versus Weight (F) – Theory




B) Beam 2: L - Cross sectional

Graph Deflection (y) versus Weight (F) – Experiment




Graph Deflection (y) versus Weight (F) – Theory




C) Beam 3: U- Cross sectional

Graph Deflection (y) versus Weight (F) – Experiment




Graph Deflection (y) versus Weight (F) – Theory




DISCUSSION:

  1. Sample of calculation (theory)
    • for I-cross sectional


ymax = FL3

3EI

Where, F = 1.1(W)

ymax = 1.1(W) (550)3

3(68000) (12736)

Where, W = 5N

ymax = 1.1(5) (550)3

3(68000) (12736)

ymax = 0.352 mm

Percentage of error, %δ

when W = 5N

%δ = yv (theory) – yv (exp) x 100%

yv (theory)

= 0.352 – 0.320 x 100%

0.352

= 9.1%


  1. Compare the theoretical and experimental results.

From the experiments results, we can see they are more different than the theoretical values we get from the calculation. Some factors of errors were determined in this experiment:

  • The measuring instrument that is not perfect errors when reading the scale on the measuring instrument such as meter ruler. In addition, the zero errors where the tools are damage and ‘zero’ point cannot be seen. So, take the errors and correct it. A few instruments which is too old should be replace with the new one or recondition to remove the corrosive surfaces and make the scale more clear as they will cause the imperfect experiment.
  • Parallax error due to reading taken from the experiment.
  • Inadequate pre-load made between the dial gauge and the load
  • The workbench might not in a flat position which contributes to unbalanced position of specimen and as a result the readings obtained were not accurate and precise as expected.

  1. The percentage of error.

% of error =

when W = 5N

= 0.300 – 0.352 x 100%

0.352

= -14.77%

CONCLUSION:

It can be concluded that the objective of the experiment have been achieved.

However, the values from the experiment not accurate because the percentage error is more than 5%, but we can minimize these errors by avoid relax parallax and make several adjustment on the apparatus and also replaced old apparatus.

REFFERENCE:

1. Advanced Mechanics of Materials, Robert D. Cook, Warren C. Young, Prentice Hall, Second Edition, 1999.

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

 

continuous beam

By atap_aje

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.