OBJECTIVE
The objective of this experiment is to evaluate the deformation of a structure using four
test structures; quarter circle, semicircle, curved davit, and angled davit in horizontal and
vertical deflections, to investigate the relationship between load and curve deflection and to
make a comparison between the experimental and theoretical results.
INTRODUCTION
Curved bars are not commonly found as structures in and of themselves; instead, they
are generally part of a mechanical member that includes both straight and curved elements. It
is crucial to understand how curved bars deflect in order to quantify the total mechanical
displacement of structures with curved parts. Castigliano's first theorem or a unit-load method
are two of the most effective techniques for estimating deflections in curved bars. The unitload method is put to the test in this experiment. The test equipment would show the real action
of curved bars set up in various configurations, and the findings of the unit-load process will
be compared.
THEORY
Castigliano's theorem can be used to measure deflections in curved bars, semicircles,
quarter circles, curved davits, and angle davits. It is also a method for calculating a structure's
deflections using strain energy. When yielding has not yet occurred in the structure's support
and the temperature of the structure is defined, Castigliano's theorem is correct. As deformation
is induced by external forces, the integral energy of a linearly elastic system equals the strain
energy. A beam such as shown in Figure 1 that subjected to transverse loads, the strain energy
associated with the normal stress is given by
U =
L
0
M2
dx =
2EI
(1)
where, M = bending moment, E = Young’s Modulus for aluminium alloy, I = second moment
of area for the curved beam and L = length.
Load,
Figure 1 A Simply Supported Beam
According to Castigliano's theorem states that the deflection x j , of the point of
application of a load Pj determined along the line of action of Pj is proportional to the partial
derivative of the strain energy U of the structure with respect to the load Pj . It can be shown
that
xj =
U
Pj
(2)
The theorem can be used to measure deflections and slopes at different points on a given
structure. The use of “dummy or imaginary” loads enabled us to include points where no actual
load was applied. If the separation of U with respect to the load Pj is performed before the
integration, the estimation of a deflection x j can be simplified. In the case of a beam, the
deflection can be written as using equations (1) and (2):
xj =
L M M
U
=
dx
0 EI P
Pj
j
(3)
Shape
 H (Horizontal Defl.)
 V (Vertical Defl.)
R = 150 mm
2 PR 3
EI
PR 3
2 EI
R = 150 mm
PR 3
2 EI
PR 3
4 EI
Dimensions
P
a) Semi circle
P
b) Quarter circle
P
R = 75 mm
L = 150 mm
PRL
(2 R + L )
2 EI
PR
2
4 EI
(4 L + R)
c) Curved davit
P
L1 = 150 mm
L2 = 105 mm
PL1 L2 (0.707L1 + L2 ) PL2
+
2 EI
6 EI
3
2
PL2
(3L1 + L2 )
6 EI
d) Angled davit
Table 1 Shows Formula of Horizontal and Vertical Deflections For Few Types of Curved
Beam Made of Aluminium Alloy
Note that, Young’s Modulus for aluminium alloy, 𝐸𝑎𝑙 = 69 GN/m2, P = load, R =
radius, I = second moment of area for the curved beam and L = length.
APPARATUS
Figure 3 Test Frame
Figure 5 Vernier Calliper
Figure 2 Test Structures; Semicircle, Quarter Circle,
Curved Davit & Angle David
Figure 6 Hanger & Masses
Figure 7 Pair of Dial Indicators
Figure 4 Hanger Load Holder
PROCEDURES
1. The apparatus for the first test specimen, a semi-circle structure was arranged. The
semi-circle frame is tightly placed and clamped on both sides.
2. The section’s width and thickness were measured and recorded, using an average of
different spots on the test structure. Then, the second moment of area (I) of the cross
section was calculated.
3. The load hanger to the free end of the test structure has been clipped.
4. The dial indicator has been placed in such a way that it touches horizontally and
vertically and most moves in both directions. The reading on the indicator has been set
to zero.
5. A weight of 100 g was applied to the load hanger. To reduce the effects of friction, the
test frame was tapped. The readings of both dial indicators were recorded.
6. The weight was increased by a total of 100 g to 700 g, the test frame was tapped each
time the weight was added. Both dial indicators have recorded their readings.
7. The above procedure was repeated with other structures (quarter circles, curved lines
and right angles).
DATA AND RESULT
Width, b = 19.1 mm = 19.1 × 10-3 m
Thickness, h = 3.2 mm = 3.2 × 10-3 m
𝑏ℎ3
𝑆𝑒𝑐𝑜𝑛𝑑 𝑀𝑜𝑚𝑒𝑛𝑡 𝑜𝑓 𝐴𝑟𝑒𝑎, 𝐼 =
12
(19.1𝑥10−3 )(3.2𝑥10−3 )3
𝐼=
12
𝐼 = 5.2156 × 10−11 𝑚4
Semicircle (mm)
Weight (g)
Load (N)
100
200
300
400
500
600
0.981
1.962
2.943
3.924
4.905
5.886
Δ𝐻
Theo.
1.84
3.68
5.52
7.36
9.20
11.04
Quarter Circle (mm)
Δ𝑉
Exp.
0.50
1.70
2.73
3.91
4.53
5.58
Theo.
1.45
2.89
4.34
5.78
7.23
8.67
Δ𝐻
Exp.
0.56
1.49
3.00
4.71
5.24
6.83
Theo.
0.46
0.92
1.38
1.84
2.30
2.76
Curved Davit (mm)
Weight (g)
Load (N)
100
200
300
400
500
600
0.981
1.962
2.943
3.924
4.905
5.886
Δ𝐻
Theo.
0.46
0.92
1.38
1.84
2.30
2.76
Theo.
0.32
0.64
0.96
1.28
1.60
1.92
Exp.
1.05
1.79
2.43
2.90
3.62
4.87
Theo.
0.72
1.45
2.17
2.89
3.61
4.34
Exp.
0.77
1.27
1.66
2.02
2.43
3.23
Angle Davit (mm)
Δ𝑉
Exp.
0.24
0.98
1.52
1.96
2.52
3.01
Δ𝑉
Δ𝐻
Exp.
0.15
0.56
0.87
1.12
1.66
1.74
Theo.
0.51
1.01
1.52
2.02
2.53
3.04
Δ𝑉
Exp.
0.40
0.51
0.91
1.30
1.91
2.77
Theo.
0.28
0.56
0.84
1.11
1.39
1.67
Table 2 Experimental and Theoretical Results of Semicircle, Quarter Circle, Curved
Davit & Angle Davit
Exp.
0.30
0.34
0.66
0.95
1.39
1.97
ANALYSIS AND DISCUSSION
GRAPH OF SEMICIRCLE DEFLECTIONS
SE MI CI RCLE T HEO RE T I CA L & E XPE RI ME NTA L G RA PH
Δ𝐻 Theo.
Δ𝑉 Theo.
Δ𝐻 Exp.
1.962
2.943
3.924
Δ𝑉 Exp.
12
DEFLECTION (MM)
10
8
6
4
2
0
0.981
4.905
5.886
LOAD (N)
GRAPH OF QUARTER CIRCLE DEFLECTIONS
SE MI CI RCLE T HEO RE T I CA L & E XPE RI ME NTA L G RA PH
Δ𝐻 Theo.
Δ𝑉 Theo.
Δ𝐻 Exp.
1.962
2.943
3.924
Δ𝑉 Exp.
6
DEFLECTION (MM)
5
4
3
2
1
0
0.981
LOAD (N)
4.905
5.886
GRAPH OF CURVED DAVIT DEFLECTIONS
CURVE D DAVI T T HEO RE T I CA L & E XPE RI ME NTA L G RA PH
Δ𝐻 Theo.
Δ𝑉 Theo.
Δ𝐻 Exp.
Δ𝑉 Exp.
1.962
2.943
3.924
4.905
3.5
DISPLACEMENT (MM)
3
2.5
2
1.5
1
0.5
0
0.981
5.886
LOAD (N)
GRAPH OF ANGLE DAVIT DEFLECTIONS
A N G LE DAVI T T HEO RE T I C A L & E XPE RI ME N TA L G RA PH
Δ𝐻 Theo.
Δ𝑉 Theo.
Δ𝐻 Exp.
Δ𝑉 Exp.
1.962
2.943
3.924
4.905
3.5
DISPLACEMENT (MM)
3
2.5
2
1.5
1
0.5
0
0.981
LOAD (N)
5.886
As the load was added by attaching more loads to the hanger during the experiment,
there was a deflection in the horizontal and vertical axes. The dial indicator was used to
measure the deflection, and the deflection for each structure was recorded. The deflection value
varies depending on the structure used i.e., semicircle, quarter circle, curved davit, and angular
davit. The graph shows a linear effect, which means that as the number of loads increases, so
the deflection of the structure also increases. This also occurs to the other structures. The
semicircle has the greater value of deflection, while the curved davits have the lowest value, as
shown by the theoretical value obtained using the given formula. When comparing the results,
there is a small disparity between the theoretical and experimental values. This is attributed to
a mistake made during the experiment
The mistake is caused by the incorrect location of the dial indicator. If the dial indicator
is not in contact with the structure until positioning the load, the effect would be inaccurate. To
avoid the mistake, make sure the dial indicator is correctly set up and that the indicator is
touching the circular form of the structure used. Besides that, an unnecessary force is exerted
as the load is placed on the hanger. This is how various people place the load, causing the
reading to differ. To solve this issue, make sure that the load is placed by the same person each
time for a more reliable reading.
CONCLUSION
The experiment determined the relationship between load, horizontal deflection, and
vertical deflection for a semicircle, a quarter circle, curved davit, and angle davit. The
horizontal and vertical deflections differ depending on the arrangement used. According to the
graph, the greater the applied load, the greater the deflection of the structure.
REFERENCES
Bao-lian, F. (1984). On the modified castigliano"s theorem. Applied Mathematics and
Mechanics, 5(2), 1263-1272. doi:10.1007/bf01895122
Castigliano’s principle of minimum strain-energy. (1936). Proceedings of the Royal Society
of London. Series A - Mathematical and Physical Sciences, 154(881), 4-21.
doi:10.1098/rspa.1936.0033