STRUCTURES LAB - II
FORCES IN TRUSS MEMBERS
[Experiment Number: 08]
{October 4th 2018}
{Gopalabhatla V S K S S Avinash}
{16AE30003}
{Aerospace Engineering}
{Structures Lab-II }
{Indian Institute of Technology Kharagpur}
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STRUCTURES LAB - II
Forces in a Truss
Aim
➢ To measure the axial strain and hence force in each member of a pin-jointed
truss for comparison with calculated theoretical values
➢ To compare experimental results with the member forces calculated by
resolution of the forces at joints
➢ Measure strains exerted on each truss member
➢ To view compressive and tensile forces/strains.
Apparatus
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Trusses setup
Load Cell
4 Strain gauges
Vernier Callipers
Experimental Setup
Figure 1 showing different components of the setup
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STRUCTURES LAB - II
Theory
A t r u s s t h a t i s a s s u m e d t o c o m p r i s e m e m b e r s t h a t a r e connected
by means of pin joints, and which is supported at both ends by means of hinged
joints or rollers, is described as being statically determinate. Newton's Laws
apply to the structure as a whole, as well as to each node or joint. In order for any
node that may be subject to an external load or force to remain static in
space all the vertical forces, horizontal forces all a s w e l l a s a l l m o m e n t s
acting about the node should be equal zero.
Analysis of these conditions at each node yields the magnitude of the forces in
each member of the truss. These may be compression or tension forces. Trusses that
are supported at more than two positions are said to be statistically indeterminate and
applying Newton's Laws alone is not sufficient to determine the member
forces. In order for a truss with pin -connected members to be
s t a b l e , i t m u s t b e e n t i r e l y c o n s i s t i n g o f t r i a n g l e s . I n mathematical
terms, we have the following necessary condition for stability:
M +R ≥ 2
Member 1: F* sqrt(2)
Member 2: -F
Member 3: -F
Member 4: F
Member 5: F* sqrt(2)
Member 6: -2*F
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STRUCTURES LAB - II
Procedure
• Setup is initiated and the load cell is initially set to a particular force which
acts on the truss system.
• For every member the Longitudinal strain is obtained and the readings of all
these strains and applied forces are noted down in a sheet.
• The dimensions and material properties of the trusses are also taken
Analysis of Results
Error Calculation:
The error comes out to be
Member 1
24.8%
Member 2
12.6%
Member 3
23.3%
Member 4
8.02%
Member 5
12%
Member 6
20.3%
Discussions
• The strain readings are too fluctuating to be noted. So a small change in strain
even contributes in significant change in determined force
• In order to maintain planar symmetry alternate members were split into 2 on
either side of the pins.
•
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STRUCTURES LAB - II
Vibration in a Beam
Aim
➢ To measure the axial strain and hence force in each member of a pin-jointed
truss for comparison with calculated theoretical values
➢ To compare experimental results with the member forces calculated by
resolution of the forces at joints
➢ Measure strains exerted on each truss member
➢ To view compressive and tensile forces/strains.
Apparatus
➢
➢
➢
➢
Trusses setup
Load Cell
4 Strain gauges
Vernier Callipers
Experimental Setup
Figure 2 showing different components of the setup
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STRUCTURES LAB - II
Theory
A continuous solution can be obtained for the cantilever beam or an analytical
model can be developed using the finite element technique in a software package such
as MATLAB. Using some basic strength of materials approximations along with the
continuous beam vibration equation, an equivalent model can be developed for
analysis purposes. The mode shapes for a continuous cantilever beam are given as
Figure 3 Natural Frequencies of first 3 modes of vibrations
The governing equation and boundary conditions are given below
Figure 4 Schematic of Experimental setup
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STRUCTURES LAB - II
Procedure
➢ First the electrodynamic shaker along with the power amplifier and the
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oscillator give the vibrational force to the beam.
Oscillation signal is picked up from the location of beam and signal is amplified
and sent to the millivoltmeter.
This reading is taken on 1 axis of oscilloscope and the other axis is given signal
from oscillator.
The first 3 modes of frequencies are to be noted from the oscilloscope by
continuously increasing the frequency
Whenever the moving ellipse becomes circular the frequency gives mode
frequency
Observations
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Length - 30 cm
Width - 24.82 mm
Height - 1.20 mm
Accelerometer - 26.5 cm
First Mode Frequency – 10.64Hz
Second Mode Frequency – 66.24Hz
Third Mode Frequency – 174.6Hz
Moment of Inertia, I Young’s Modulus – 2500 N/mm2
Calculations
To find the natural frequencies of the beam the governing equation is
supposed to be solved with the boundary conditions.
The modes of shapes for a continuous cantilever beam are given by
After using this in the governing equations and finding the solution we get the
natural frequencies for first three modes as
𝜔1 = 1.8752 √(
𝑔𝐸𝐼
𝑔𝐸𝐼
𝑔𝐸𝐼
) ; 𝜔2 = 4.6942 √( 4 ) ; 𝜔3 = 7.8552 √( 4 )
4
𝑊𝐿
𝑊𝐿
𝑊𝐿
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STRUCTURES LAB - II
Results
Theoretical Natural Frequencies:
f1Theo= 10.9053 Hz
f2Theo= 68.3474 Hz
f3Theo= 191.397 Hz
Experimental Natural Frequencies:
F1exp= 10.64 Hz
F1exp= 66.24 Hz
F1exp= 174.6 Hz
Error for each frequency is
First Mode
: 2.43 %
Second Mode
: 3.08 %
Third Mode
: 8.77 %
Discussions
• The boundary conditions play a very important role in the determination of
the natural frequencies
• The equations can be solved in 4th degree polynomials or 2nd degree based on
the number of boundary conditions given.
• The nodal shapes were seen in stroposcope when the beam vibrates with the
natural frequency then there is a phase lag of 900 between the excitation and
the output signal.
• Error can be due to approximation of ellipse to be perfect circle during the
determination of natural frequency
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