experimental verification of deflection of beam using ... deflection distance of a beam under a load...
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International Journal of Advance Research in Engineering, Science & Technology(IJAREST),
ISSN(O):2393-9877, ISSN(P): 2394-2444, Volume 2, Issue 3, March- 2015
All Rights Reserved, @IJAREST-2015
1
Experimental verification of deflection of beam using theoretical and numerical
approach
Biltu Mahato1, Anil
2, Harish H.V
3
1, 2 Pre-final Year Undergraduate Students Department of Aeronautical Engineering, NMIT Bangalore, [email protected]
3 Assistant Professor Department of Aeronautical Engineering, NMIT Bangalore, [email protected]
Abstract
This study investigates the maximum deflection of simply supported beam and cantilever beam under point loading. Experiments
on these beams have been carried out and maximum deflection has been noted. The experiment has been carried out for different
loads. The results obtained have been validated through theoretical and numerical approach. Numerical approach includes
mathematical and simulation approach. Euler–Bernoulli beam equation is considered for theoretical, finite element methods
(FEM) for mathematical and ANSYS 14.0 for simulation approach. The results obtained through theoretical, FEM and simulation
is very near to experimental results.
Keywords: Simply supported beam, Cantilever beam, Maximum deflection, FEM, ANSYS
I. INTRODUCTION
A beam is a member subjected to loads applied transverse to
the long dimension, causing the member to bend. A beam
which is fixed at one end and free at other end is known as
cantilever beam. Simply supported beam is a beam
supported or resting freely on the supports at its both ends.
The deflection distance of a beam under a load is directly
related to the slope of the deflected shape of the member
under the load. It can be calculated by integrating the
function that mathematically describes the slope of the
member under that load. Deflection can be calculated by
standard formula calculated using Euler–Bernoulli beam
equation, virtual work, direct integration, Castiglione’s
method and Macaulay's method or the direct stiffness
method.
Gargi Majumder et al [1] have conducted finite element
analysis of the beam considering various types of elements
under different loading conditions in ANSYS 11.0. The
various numerical results were generated at different nodal
points by taking the origin of the Cartesian coordinate
system at the fixed end of the beam. The nodal solutions
were analyzed and compared. On comparing the
computational and analytical solutions it was found that for
stresses the 8 node brick element gives the most consistent
results and the variation with the analytical results is
minimum.
Amer M. Ibrahim et al [2] have described a nonlinear finite
element analyses which have been carried out to investigate
the behavior up to failure of simply supported composite
steel-concrete beams with external pre-stressing, in which a
concrete slab is connected together with steel I-beam by
means of headed stud shear connectors, subjected to
symmetrically static loading. ANSYS computer program
(version 12.0) has been used to analyze the three
dimensional model. They studied load deflection behavior,
strain in concrete, strain in steel beam and failure modes.
The nonlinear material and geometrical analysis based on
Incremental-Iterative load method, is adopted. Three models
have been analyzed to verify its capability and efficiency.
The results obtained by finite element solutions have shown
good agreement with experimental results.
From literature survey, we identified that theoretical and
ANSYS simulations have only been carried out. Further in
this study we have taken FEM as a mathematical approach
to validate the results. In this approach we have considered
global stiffness matrix to find the maximum deflection [3].
Along with this Euler–Bernoulli beam equation is
considered for theoretical validation [4…7] and ANSYS
14.0 workbench for simulation [8, 9].
II. METHODOLOGY
The experiment and validation processes have been carried
out in two steps. For experiment, beam deflection test rig
setup was made as shown in figure 1 and figure 2. Steel
beam with rectangular cross section was used. Maximum
deflection was measured at different locations for different
loading varying from 0.2 Kg to 1.0 Kg. The corresponding
deflection was noted at loading and unloading condition
with the help of digital dial gauge for all setup of beam. All
experiment was repeated for various numbers of times at
different time and conditions. The setup and condition
giving minimum percentage error is considered here for
validation.
III. EXPERIMENTAL SETUP AND
RESULTS
The specifications of beam experimental setup are given in
table 1.
International Journal of Advance Research in Engineering, Science & Technology(IJAREST),
ISSN(O):2393-9877, ISSN(P): 2394-2444, Volume 2, Issue 3, March- 2015
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Table 1. Specifications of Experimental Setup
Parameters Dimensions
Material Steel
Young's Modulus 2.1*105 N/mm
2
Thickness 5 mm
Breadth 25 mm
Length 600 mm
Moment of Inertia 260.41667 mm4
Figure 1. Simply supported beam experimental setup
Figure 2. Cantilever beam experimental setup
The experimental results obtained for simply supported and
cantilever beam at different load are presented in the table 2,
3 and 4.
Table 2. Experimental results for simply supported beam
SI No. Load in Kg Maximum deflection in mm
1. 0.2 0.17
2. 0.4 0.33
3. 0.6 0.51
4. 0.8 0.68
5. 1.0 0.86
Table 3. Experimental results for cantilever beam with
center load
SI No. Load in Kg Maximum deflection in mm
1. 0.2 0.77
2. 0.4 1.54
3. 0.6 2.35
4. 0.8 3.36
5. 1.0 4.23
Table 4. Experimental results for cantilever beam with end
load
SI No. Load in Kg Maximum deflection in mm
1. 0.2 2.63
2. 0.4 5.07
3. 0.6 8.12
International Journal of Advance Research in Engineering, Science & Technology(IJAREST),
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4. 0.8 10.83
5. 1.0 13.44
IV. VALIDATION
Validation of the above experimental results was done by
two approaches. They are:
4.1. Theoretical Approach
First uniform rectangular cross-sectional beams of linear
elastic isotropic homogeneous material have been
considered. The beam is taken mass-less and inextensible
hence have developed no strains. It is subjected to a vertical
point load at the tip of its free end and centre [3…6].
Using the Bernoulli-Euler’s elastic curve equation [3] the
following relationship is obtained:
EI (d2 y/dx
2) =M (1)
Where E is modulus of elasticity which is of constant value,
I is moment of inertia=bh3 /12, b=width of beam, h=height
or thickness of beam, y=deflection due to loading, M=
moment due to applied force.
On solving and applying boundary condition on equation (1)
for simply supported beam as shown in figure 3 we get
y =
(2)
Where W=Force applied on the beam, L=Length of the
beam.
Figure 3. Schematic representation of simply supported
beam
For cantilever beam with centered loading as shown in
figure 4 we get
y =
(3)
Figure 4. Schematic representation of cantilever beam at
center load
For cantilever beam with end loading as shown in figure 5
we get
y =
(4)
Figure 5. Schematic representation of cantilever beam at
end load
4.2. Numerical Approach
4.2.1. Mathematical validation
Mathematical validation means validation using FEM. In this
approach we use global stiffness equation to get the
deflection at the nodes [3].
{f} = [K] {q} (5)
Where {f} is force vector = {F1 M1 F2 M2 ... Fn Mn} T, [K]
is stiffness matrix, {q} is displacement vector = {q1 ɵ1 q2
ɵ2 ... qn ɵn} T, n = number of nodes, F1…Fn is force at node
1…n, M1…Mn is moment due to applied force at node 1…n,
q1… qn is linear deflection at nodes 1…n, ɵ1… ɵn is angular
deflection.
4.2.2. Simulation
Simulation has been performed using ANSYS 14.0
workbench tool [8, 9].
V. RESULTS AND DISSCUSSIONS
The results obtained from validation approach are shown in
table 6, 7 and 8 for simply supported and cantilever beam
for centre and end loading respectively. Figure 6 to figure 21
shows the results obtained using ANSYS simulation.
Similarly all the values obtained from experimental,
theoretical, mathematical and simulation are plotted in load
vs. deflection graph as shown in figure 21, 22 and 23.
Table 5. Validation Results for Simply Supported Beam
SI No. Load in Kg Maximum deflection in mm
Theoretical Mathematical Simulation
1. 0.2 0.1614 0.1614 0.1615
2. 0.4 0.3229 0.3229 0.3231
3. 0.6 0.4843 0.4843 0.4829
4. 0.8 0.6458 0.6458 0.6461
International Journal of Advance Research in Engineering, Science & Technology(IJAREST),
ISSN(O):2393-9877, ISSN(P): 2394-2444, Volume 2, Issue 3, March- 2015
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5. 1.0 0.8072 0.8072 0.8076
Table 6. Validation Results for Cantilever Beam with
Center Loading
SI No. Load in Kg Maximum deflection in mm
Theoretical Mathematical Simulation
1. 0.2 0.8072 0.8072 0.7996
2. 0.4 1.6144 1.6144 1.5991
3. 0.6 2.4217 2.4217 2.3987
4. 0.8 3.2289 3.2289 3.1983
5. 1.0 4.0361 4.0361 3.9978
Table 7. Validation Results for Cantilever Beam with End
Loading
SI No. Load in Kg Maximum deflection in mm
Theoretical Mathematical Simulation
1. 0.2 2.5831 2.5831 2.6972
2. 0.4 5.1662 5.1662 5.1393
3. 0.6 7.7493 7.7493 7.7031
4. 0.8 10.3325 10.3325 10.2710
5. 1.0 12.9156 12.9156 12.8380
Figure 6. Deflection of simply supported beam at 0.2 Kg
load
Figure 7. Deflection of simply supported beam at 0.4 Kg
load
Figure 8. Deflection of simply supported beam at 0.6 Kg
load
International Journal of Advance Research in Engineering, Science & Technology(IJAREST),
ISSN(O):2393-9877, ISSN(P): 2394-2444, Volume 2, Issue 3, March- 2015
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Figure 9. Deflection of simply supported beam at 0.8 Kg
load
Figure 10. Deflection of simply supported beam at 1.0 Kg
load
Figure 11. Deflection of cantilever beam with 0.2 Kg load at
center
Figure 12. Deflection of cantilever beam with 0.4 Kg load at
center
International Journal of Advance Research in Engineering, Science & Technology(IJAREST),
ISSN(O):2393-9877, ISSN(P): 2394-2444, Volume 2, Issue 3, March- 2015
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Figure 13. Deflection of cantilever beam with 0.6 Kg load at
center
Figure 14. Deflection of cantilever beam with 0.8 Kg load at
center
Figure 15. Deflection of cantilever beam with 1.0 Kg load at
center
Figure 16. Deflection of cantilever beam with 0.2 Kg load at
end
International Journal of Advance Research in Engineering, Science & Technology(IJAREST),
ISSN(O):2393-9877, ISSN(P): 2394-2444, Volume 2, Issue 3, March- 2015
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Figure 17. Deflection of cantilever beam with 0.4 Kg load at
end
Figure 18. Deflection of cantilever beam with 0.6 Kg load at
end
Figure 19. Deflection of cantilever beam with 0.8 Kg load at
end
Figure 20. Deflection of cantilever beam with 1.0 Kg load at
end
International Journal of Advance Research in Engineering, Science & Technology(IJAREST),
ISSN(O):2393-9877, ISSN(P): 2394-2444, Volume 2, Issue 3, March- 2015
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Figure 21. Load vs. deflection graph for simply supported
beam
Figure 21. Load vs. deflection graph for cantilever beam
with center load
Figure 22. Load vs. deflection graph for cantilever beam
with end load
VI. CONCLUSION
From above validation results, experimental results have
been validated where maximum deflection profiles are
clearly matching. There is a good agreement between the
experimental, theoretical and numerical approach results for
maximum deflection. Although there are some small
discrepancies due to some experimental imperfection,
effects of temperature, creep and shrinkage. The final result
shows an error of around 7% for simply supported beam and
around 5% error for cantilever beam. Though FEM is an
approximation method its results are exactly matching with
the theoretical results whereas structural analysis using
ANSYS 14.0 gives result with an error of less than 1%.
Further from load vs. deflection graph it was clearly
observed that deflection was more in experimental results
when compared to that of the theoretical and numerical
approach results. As the error is within acceptable range we
conclude that FEM and ANSYS simulation tool that can be
used in the future for structural analysis.
ACKNOWLEDGEMENT
The authors want to acknowledge the department of
aeronautical engineering, Nitte Meenakshi Institute of
technology, Bangalore for providing the technical support
regarding the experimental setup and the faculties like Dr.
Vivek Sanghi, Srikant H.V., Mahendra M.A., and Nishant
Deshai for their proper guidance.
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International Journal of Advance Research in Engineering, Science & Technology(IJAREST),
ISSN(O):2393-9877, ISSN(P): 2394-2444, Volume 2, Issue 3, March- 2015
All Rights Reserved, @IJAREST-2015
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REFERANCES
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“Finite element modeling of composite steel-concrete
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[3] S. B. Halesh, Finite Element Methods, 1st edition, Sapna
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[4] Dr. R. K. Bansal, A textbook of Strength of Material
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[5] Timoshenko, S.P. and D.H. Young. Elements of Strength
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[6] E.A. Witmer (1991-1992). "Elementary Bernoulli-Euler
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[7] Unified Engineering Course Notes.Ballarini, Roberto,
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Mechanical Engineering Magazine Online. Retrieved
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[8] ANSYS 14.0 documentation.
[9] Victor Debnath, Bikramjit Debnath, “Deflection And
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