cantilever beam
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:
 The apparatus is set up as shown in Figure E2.
 The distance of L is measured.
 The scale on the dial gauge is set to zero
 The beam is loaded with 5 N.
 The dial gauge is read and recorded.
 The load is increased and the reading is recorded.
 Step 6 is repeated to get at least four reading.
 Step 1 to 7 is repeated by using the beam with the different crosssectional area.
RESULT:
Beam 1: I Cross sectional
L = 550 mm I = 12736 mm^{4}
E = 68000 N/mm^{2} 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 mm^{4}
E = 68000 N/mm^{2} 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 mm^{4}
E = 68000 N/mm^{2} 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:
 Sample of calculation (theory)
 for Icross sectional
y_{max }= FL^{3}
3EI
Where, F = 1.1(W)
y_{max }= 1.1(W) (550)^{3}
3(68000) (12736)
Where, W = 5N
y_{max }= 1.1(5) (550)^{3}
3(68000) (12736)
y_{max }= 0.352 mm
Percentage of error, %Î´
when W = 5N
%Î´ = y_{v} (theory) – y_{v} (exp) x 100%
y_{v} (theory)
= 0.352 – 0.320 x 100%
0.352
= 9.1%
 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 preload 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.
 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
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

∑ M_{A }= 0 Nm
Resultant of total moment
Basic Sign Convention System
= (+ve) = (ve) = (+ve) = (ve)
W_{2}b_{2} – R_{2}L_{2} + W_{1} (b_{1}+L_{2}) –R_{1} (L_{1}+L_{2}) + mg (L_{1}+ L_{2}) = 0
2
PROCEDURE :
1) The apparatus is set up as shown in the figure above.
2) Distance L_{1}, L_{2}, a_{1}, a_{2}, b_{1} and b_{2} 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 :
a_{1} = 0.235 m b_{2} = 0.250 m
a_{2} = 0.180 m L_{1} = 0.500 m
b_{1} = 0.265 m L_{2} = 0.430 m
Beam Dimension = 868.00 mm x 20.30 mm x 6.20 mm
Young’s Modulus = 210.00 GPa
Data:
W_{1 }(N)  W_{2 }(N)  R_{1 }(N) (Theory)  R_{1 }(N) (Exp)  R_{2} (N) (Theory)  R_{2} (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 W_{1 }and W_{2} = 2.5 N
2.5N_{ }2.5N
2 3
1
C
0.408 m 0.46 m
R_{1 } R_{2}
_{ }
∑ Fy = 0
R_{1 }+ R_{2} – W_{1 }–W_{2 }= 0
R_{1 }+ R_{2 }– 2.5 – 2.5 = 0
R_{1 }= 5.0 – R_{2} eqn (1)
+ ∑ M_{3 }= 0
R_{1}(0.93) + W_{1}(0.695)  R_{2}(0.43) + W_{2}(0.25) = 0
R_{1}(0.93) + 2.5(0.695)  R_{2}(0.43) + 2.5(0.25) = 0
0.93R_{1 }+ 0.43R_{2} = 2.36 eqn (2)
(1) (2):
R_{1}=0.42N
R_{2}=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:
R_{1 }= R_{1(exp)} – R_{1(theory) }x 100%
R_{1(theory) }
For W_{1,} W_{2 }= 2.5 N;
R_{1 }= R_{1(exp)} – R_{1(theory)}_{ }x 100% R_{2 }= R_{2(exp)} – R_{2(theory)}_{ }x 100%
R_{1(theory) } R_{2(theory) }
_{ }
= (0.50 – 0.42) x 100% = 19.05% = (4.62 – 4.58) x 100% = 0.87%
0.42 4.58
For W_{1,} W_{2 }= 7.5 N;
R_{1 }= R_{1(exp)} – R_{1(theory)}_{ }x 100% R_{2 }= R_{2(exp)} – R_{2(theory)}_{ }x 100%
R_{1(theory) } R_{2(theory) }
= (1.30 – 1.28) x 100% = 1.56 % = (13.76 – 13.72) x 100% = 0.29%
1.28 13.72
For W_{1}, W_{2 }= 12.5 N;
R_{1 }= R_{1(exp)} – R_{1(theory)}_{ }x 100% R_{2}= R_{2(exp)} – R_{2(theory)}_{ }x100%
R_{1(theory) }R_{2(theory) }
= (2.16 – 2.12) x100% = 1.87 % = (22.92 – 22.88) x 100% = 0.17%
2.12 22.88
For W_{1,} W_{2 }= 17.5 N;
R_{1 }= R_{1(exp)} – R_{1(theory)}_{ }x 100% R_{2 }= R_{2(exp)} – R_{2(theory)}_{ }x 100%
R_{1(theory) }R_{2(theory) }
= (3.02 – 2.98) x 100% = 1.34 % = (32.08 – 32.02) x 100% = 0.19 %
2.98 32.02
For W_{1,} W_{2 }= 22.5 N;
R_{1 }= R_{1(exp)} – R_{1(theory)}_{ }x 100% R_{2}= R_{2(exp)} – R_{2(theory)}_{ }x 100%
R_{1(theory) }R_{2(theory) }
= (3.86 – 3.82) x 100% = 1.05 % = (41.22 – 41.18) x 100% = 0.10%
3.82 41.18
Average R_{1 }= 4.97% Average R_{2 }= 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.