This document describes an experiment to measure the deflection of curved bars (full circular, semi-circular, and quarter-circular) when loads are applied. The experiment uses a curved bar apparatus with load hangers, dial gauges, and weights. Deflection is measured experimentally and compared to theoretical calculations using Castigliano's theorem or the unit load method. The procedure for each type of curved bar is outlined, with observations and deflection measurements to be recorded in tables to calculate percentage errors between experimental and theoretical values. Practical applications of deflection in curved bars are noted.

1. 10.1 Objective:
Experiment # 10
To find the deflection in curved (full, semi & quarter circular) bar by using curved
bar apparatus.
10.2 Apparatus:
1. Curved bar apparatus
2. Load hangers and hook
3. Dial gauges
4. Wrench
5. Weight set
10.3 Theory:
Though curved bars are not commonly found as structures by themselves, they are usually part of
a mechanical member which has a combination of straight and curved elements. The study of how
curved bars deflect is thus, important so as to estimate the total mechanical displacement of
structures that incorporate curved sections. One of the more effective methods used to estimate
deflections in curved bars come from Castigliano’s theorem or from a unit-load method. This
experiment puts the unit load approach to the test. The test apparatus will demonstrate the actual
behavior of curved bars set up in different configurations and comparisons will be made to the unit
load method’s results. Sometimes, due to the complex nature of the integral involved, an
approximate method such as Simpson’s Rule can be used instead of an analytical integration.

2. Castigliano’s Theorem
– For displacements in a linearly elastic structure.
Castigliano's method for calculating displacements is an application of his second theorem,
which states:
If the strain energy of a linearly elastic structure can be expressed as a function of generalized
force Qi; then the partial derivative of the strain energy with respect to generalized force gives
the generalized displacement qi in the direction of Qi.
As above, this can also be expressed as:
qi=∂U/∂Q
10.4 Procedure:
For Ring
1. For the ring configuration, hang the hook from the bottom of the ring and attach the load
hanger to it. The vertical and horizontal dials are set in their positions.
2. Zero all three dials and read the initial zero-datum reading off them.
3. Next, load the ring by increments. At each increment record the deflection at each dial and
tabulate these in a table below.
4. Subtract your zero-datum reading from all your readings to get your actual
Fig.10.1: Deflection in circular ring

3. For Semi-circle
1. Attach the hanger onto the semi-circle and apply a weight on it. This will stabilize the system
and does not count as part of the weight.
2. Secure the removable dials onto the semi-circle experiment setup to measure vertical
displacement.
3. Record the zero-datum reading.
4. Record your results on a table
For Quadrant-circle
1. First, attach the hanger onto the quadrant and apply a weight on it. This is to
stabilize the system and will not be counted as part of the weight that we will apply
onto it.
2. Secure the removable dials onto the quadrant experiment setup to measure vertical
deflection of the curved (quadrant) member.
3. Zero the dial and record the zero datum reading.
4. Load the curved member with weight increments, at each
increment record your results on a table.
Fig.10.2: Deflection in semi-circle
Fig 10.3: Deflection in quadrant

4. 10.5 Observations & Calculations:
For Circular Ring
Sr.
#
Load
(N)
Experimental
Vertical
Deflection
(mm)
Experimental
Horizontal
Deflection
(mm)
Theoretical
Vertical
Deflection
(mm)
Theoretical
Horizontal
Deflection
(mm)
% Error
Vertical
Deflection
% Error
Horizontal
Deflection
1
2
3
4
5
Vertical: δv = 0.148 PR3 / EI E=180 GPa I= bd3/12
Horizontal: δh = 0.114 PR3 / EI
For Semi-circle
Sr
. #
Load
(N)
Experimental
Vertical
Deflection
(mm)
Experimental
Horizontal
Deflection
(mm)
Theoretical
Vertical
Deflection
(mm)
Theoretical
Horizontal
Deflection
(mm)
% Error
Vertical
Deflection
% Error
Horizontal
Deflection
1
2
3
4
5
Vertical δv =  PR3 / 2EI E=180 GPa I= bd3/12
Horizontal δh = 2PR3/EI

5. For Quadrant
Sr
. #
Load
(N)
Experimental
Vertical
Deflection
(mm)
Experimental
Horizontal
Deflection
(mm)
Theoretical
Vertical
Deflection
(mm)
Theoretical
Horizontal
Deflection
(mm)
% Error
Vertical
Deflection
% Error
Horizontal
Deflection
1
2
3
4
5
Vertical δv =  PR3 / 4EI E=180 GPa I= bd3/12
Horizontal δh = PR3 / 2EI

6. 10.6 Practical Applications:

7. 10.7 Results & Calculations: