deflection curve beam

TITLE : DEFLECTION OF CURVED BEAM

OBJECTIVE : Determine the vertical and horizontal

deflection of a curved beam

APPARATUS : refer to figure 3.0

Label	Part name
1	Semi-circle beam
2	Quarter circle beam
3	Fixed support
4	Dial Gauge
5	Load holder
6	Load

figure 3.0

THEORY :

Based on Castigliano`s theorem of horizontal and vertical deflections at free end of the curved beams as shown in Figure E4 are as follows:

Beam 1 δ_v= πWR³ ; δ_h= 2WR³

2EI EI

Beam 2 δ_v = πWR³ + WR²L δ_h = WR³ + WRL[R + L]

4EI EI 2EI EI 2

PROCEDURE :

1. The apparatus is set up as shown in Figure 3
2. The dimension of beam cross-sectional 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.

RESULTS :

Beam 1: Semi Circle

Radius = 150 mm I =134.75 mm⁴

E = 210000 N/mm²

Beam dimension: 22.3 mm x 4.17 mm x 500 mm

W (N)	Yv (mm) Exp	Yv(mm) Theory	Yb (mm) Exp	Yb(mm) Theory
2.5	0.20	0.46	0.14	0.59 1.1926
5	0.68	0.938	0.63	1.19
7.5	1.40	1.40	1.28	1.79
10	1.98	1.8783	1.68	2.39
12.5	3.3	2.342	2.2	2.98

Sample of calculation

At W=2.5N,

=0.46mm =0.59mm

GRAPH:

Beam 2: Quarter Circle

Radius of quarter circle beam, R=150mm.

L = 72mm I = 172.07 mm⁴

E = 210000 N/mm²

Beam dimension: 22.36 mm x 4.52 mm x 3 mm

W (N)	Yv (mm) Exp	Yv(mm) Theory	Yb (mm) Exp	Yb(mm) Theory
2.5	0.12	0.295	0.19	0.199
5	0.26	0.59	0.52	0.399
7.5	0.70	0.886	0.99	0.599
10	0.95	1.182	1.28	0.799
12.5	1.31	1.477	1.66	0.999

Sample of calculation

=0.295

=0.199

GRAPH :

DISCUSSION :

1) Compare the theoretical and experimental results.

From the experiments that have been conducted, the deflection for horizontal and vertical are different from the theoretical values. The factors that may contribute to these different values between the theory and experiment are:

• 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

2) Calculate the percentage of error.

Beam 1

Percentage of error (100%)
Vertical deflection	Horizontal deflection
-26.0	-50.8
-16.84	-29.41
0.07	-18.44
14.46	-12.55

Beam 2

Percentage of error (100%)
Vertical deflection	Horizontal deflection
-20.63	-63.63
-22.22	-12.5
-5.26	4.54
3.17	11.68

CONCLUSION:

The semi circular beam displays much deflection if compared to the quarter circular beam. This is due to the geometrical shape of the beam that provides much support for the quarter circular beam to withstand the load exerted onto it. However, when a semi circular bends on the upper part of the beam it continues to deflect at the lower part of the beam.

REFFERENCE:

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

Wiley Sixth Edition, 2002.

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

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

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