International Conference on Global Trends in Engineering, Technology and Management (ICGTETM-2016)
On The Accuracy of Finite Element Method for static and
Dynamic Problem
MayurArvind Jadhav#1, Pradeep D. Darade*2, Sanjay S. Deshpandeα3
#
PG Student, Mechanical Engineering, SITS, Pune, India
*
Department of mechanical engineering, SITS, Pune, India
α
Department of mechanical engineering, KSOE, Pune, India
Abstract:
Finite element Analysis (FEA) plays a very
important role in the overall mechanical design
process. It has been applied successfully to almost all
kinds of problems and the complexity of problem
ranges from static to dynamics and multyphysics
problems. Although we find lot of literature on static
analysis with respect to verification and validation of
results these is not always the case with respect to
dynamic problems in frequency and time domain due
to complexity of physics and numeric. The present
work bridges the gap and analyses a standard
configuration of a cantilever beam and we show that
the FEA results deviates much from exact solution for
natural frequency calculations. It is expected that the
users will benefit from the understanding we had from
this work.
Keywords —Static, Dynamic, FEA, Analysis
I. INTRODUCTION
Finite element analysis (FEA) is the modelling of
products and systems in a virtual environment, for the
purpose of finding and solving potential (or existing)
structural or performance issues. FEA is the practical
application of the finite element method (fem), which
is used by engineers and scientist to mathematically
model and numerically solve very complex structural,
fluid, and multy-physics problems. FEA software can
be utilized in a wide range of industries. It has
emerged the third dimension of engineering
supplementing the other two dimensions of pure
theory and experimentation. It is also become an
integral part of design process. Although much has
been talked
on
static
problems
and
several software’s/codesare available, however in
literature we
find
less
verification
and validation on standard problems with respect
to, dynamics.The present work analyses one standard
configuration of a cantilever beam and we put
forward
the
behaviour
of
several
finite
elements from one dimensions to 3- dimensions in
this paper . We discover that there are still lot of
unanswered questions when it comes to dynamic
simulations and these need to be taken care of by
the user of the software. Physics understanding is
much more important and its correlation to
ISSN: 2231-5381
the numerical with respect to element plays a very
important role. Further situation is more complex for
practical problems as several mesh quality parameters
such asjacobian, aspect-ratio, distortion of the
elements come into picture.
II. STANDARD CONFIGURATION AND E XACT
SOLUTION
The problem we have taken for analysis is a cantilever
beam made of steel (Young’s modulus= 2.1 x 105MPa,
ρ=7800kg/ m3). Cross-section of the beam in y-z
Plane is 1 x 3 mm.
1. Displacement
= 6.34 mm
Where, P = 1 N
l = 100 mm
E = 2.1 x 105
N/mm2
Izz= 0.25 mm4
2. Stress
= 200 MPa
Where, M = 100 Nm
y = 0.5 mm
Izz= 0.25 mm4
3. Natural Frequencies
1.
= 84.62 Hz
Where, E = 2.1 x 105
N/mm2
L = 100 mm
ρ = 7.8 x 10 mm
A = 3 mm2
Izz= 0.25 mm4
2.
= 253.88 Hz
Where, E = 2.1 x 105
N/mm2
L = 100 mm
ρ = 7.8 x 10 mm
A = 3 mm2
Iyy= 2.25 mm4
3.
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= 524.25 Hz
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International Conference on Global Trends in Engineering, Technology and Management (ICGTETM-2016)
Where, E = 2.1 x 105
N/mm2
L = 100 mm
ρ = 7.8 x 10 mm
A = 3 mm2
Izz= 0.25 mm4
4.
= 1572 Hz
Where, E = 2.1 x 105
N/mm2
L = 100 mm
ρ = 7.8 x 10 mm
A = 3 mm2
Iyy= 2.25 mm4
5.
= 1470.9 Hz
Where, E = 2.1 x 105
N/mm2
L = 100 mm
ρ = 7.8 x 10 mm
A = 3 mm2
Izz= 0.25 mm4
6.
= 4412.7 Hz
Where, E = 2.1 x 105
N/mm2
L = 100 mm
ρ = 7.8 x 10 mm
A = 3 mm2
Iyy= 2.25 mm4
III. FEA Model and Approximate Solution
For FEA analysis of given problems, we are
considering Hypermesh software with NASTRAN
solver. We are considering three types of meshing
system, viz., quadrilateral (quad), tetrahedron (tetra)
and triangular (tria).
1. QUAD
i. Static
a) Displacement = 6.26 mm
b) Stress = 177 MPa
Fig. 2 Stress, Contour plot for QUAD
ii. Dynamic
Mode: 1
Freq: 84.0 Hz
Maximum deformation is 0.130E+04 at grid
14.
Mode: 2
Freq: 251. Hz
Maximum deformation is 0.130E+04 at grid
15.
Mode: 3
Freq: 527. Hz
Maximum deformation is 0.126E+04 at grid
14.
Mode: 4
Freq: 0.150E+04 Hz
Maximum deformation is 0.118E+04 at grid
14.
Mode: 5
Freq: 0.157E+04 Hz
Maximum deformation is 0.126E+04 at grid
13.
Mode: 6
Freq: 0.307E+04 Hz
Maximum deformation is 0.103E+04 at grid
14.
2. TETRA
i. Static
a) Displacement = 6.41 mm
b) Stress = 131 MPa
Fig. 1 Displacement, Contour plot for QUAD
ISSN: 2231-5381
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International Conference on Global Trends in Engineering, Technology and Management (ICGTETM-2016)
Fig. 3 Displacement, Contour plot for TETRA
Fig. 5 Displacement, Contour plot for TRIA
Fig. 4 Stress, Contour plot for TETRA
ii. Dynamic
Frequencies
Fig. 6 Stress, Contour plot for TRIA
Mode: 1
Freq: 83.3 Hz
Maximum deformation is 0.131E+04 at grid
15.
Mode: 2
Freq: 249. Hz
Maximum deformation is 0.130E+04 at grid
42.
Mode: 3
Freq: 525. Hz
Maximum deformation is 0.131E+04 at grid
42.
Mode: 4
Freq: 0.148E+04 Hz
Maximum deformation is 0.132E+04 at grid
15.
Mode: 5
Freq: 0.157E+04 Hz
Maximum deformation is 0.131E+04 at grid
16.
Mode: 6
Freq: 0.296E+04 Hz
Maximum deformation is 0.138E+04 at grid
42.
3. TRIA
i. Static
a) Displacement = 6.22 mm
b) Stress = 178 MPa
ISSN: 2231-5381
ii. Dynamic
Frequencies
Mode: 1
Freq: 84.3 Hz
Maximum deformation is 0.130E+04 at grid
13.
Mode: 2
Freq: 526. Hz
Maximum deformation is 0.126E+04 at grid
13.
Mode: 3
Freq: 0.115E+04 Hz
Maximum deformation is 0.130E+04 at grid
15.
Mode: 4
Freq: 0.149E+04 Hz
Maximum deformation is 0.119E+04 at grid
13.
Mode: 5
Freq: 0.300E+04 Hz
Maximum deformation is 0.109E+04 at grid
13.
Mode: 6
Freq: 0.379E+04 Hz
Maximum deformation is 0.153E+04 at grid
15.
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International Conference on Global Trends in Engineering, Technology and Management (ICGTETM-2016)
IV.COMPARISON OF RESULTS
i. Static
PARAMETER
Displacement(mm)
2
Stress (N/mm )
EXACT
SOLUTION
QUAD
TETRA
TRIA
6.34
6.26
6.41
6.22
200
177
131
178
ii. Dynamic
PARAMETER
EXACT
FREQUENCY SOLUTION
MODE 1 (Hz)
MODE 2 (Hz)
MODE 3 (Hz)
MODE 4 (Hz)
MODE 5 (Hz)
MODE 6 (Hz)
FEA SOLUTION
84.62
253.88
524.25
1572
1470.9
4412.7
FEA SOLUTION
QUAD TETRA TRIA
84
251
527
1500
1570
3070
83.3
249
525
1480
1570
2960
84.3
526
1150
1490
3000
3790
I am also thankful to my family members and
friends for their continuous support.
Let me in the end express my sincere thanks to all
those persons from whom I received co-operation,
help and motivation directly or indirectly during the
preparation of this paper.
REFERENCES
[1]
[2]
[3]
[4]
Thomas J. R. Hughes-The Finite Element Method,
Linear Static and Dynamic Finite Element AnalysisPrentice Hall, 1987.
O. C.Zienkiewicz, R. L. Taylor- Finite Element Method,
Vol.
2 Solid Mechanics, 5th edition, Butterworth
Heinemann, 2000.
Robert D. Cook - Finite Element Modeling for Stress
Analysis, John Wiley & sons, 1995.
Tirupati R. Chandrupatla, Ashok D.Belegundu,
Introduction to Finite element in Engineering, 3rd edition,
Prentice hall of India, 2002.
V. CONCLUSION
It is observed that most of the elements perform
very well in static. It is well known fact that QUAD
and TETRA and TRIA are stiff elements and predict
the
displacements
to
a
low
value.
However the correct interpretation of how much stiff
comes from the dynamic results for natural frequency
as it is here that mass of the structure remains the
same and frequency value gives an indication of
the stiffness of the element.
If one expects that conventional elements would
predict
the
frequencies
especially
the
higher frequencies which are extremely important in
MEMS, then we observe that even though the low
frequencies are predicted to good accuracy, there is
much deviation against the exact solution values.
It is expected that the FEA user keeps in
mind the above deviations and this gives an idea of
the
deviations.
In
practical
problems the
meshing plays a very important role and this forms the
further part of our studies.
ACKNOWLEDGEMENT
I hereby take this opportunity to express profound
thanks and gratitude from the bottom of my hearts
towards my guide, Respected Prof. P. D. Darade
(Assistant Professor in Department of Mechanical
Engineering, SITS,Pune) for his valuable guidance
and untiring encouragement during the preparation of
this paper. Also, that he spared his valuable time from
the busy academic schedule for the expert suggestions.
I am also very much thankful to Prof. S. S.
Deshpande, (Assistant Professor in Department of
Mechanical Engineering, KSOE, Pune), for his
guidance, understanding, patience, and most
importantly, his friendship during my graduate studies.
His mentorship was paramount in providing a wellrounded experience consistent my long-term career
goals.
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http://www.ijettjournal.org
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