When a plate with circular hole is subjected to axial loading, the region in the vicinity of hole is under high stress. To reduce this stress, normally one auxiliary hole is introduced on either side of the main hole. The present work studies how a semi-elliptical slot on each side of the hole effects the maximum stress induced in a isotropic plate with circular hole. Using finite element analysis method, stress calculation is carried out in ANSYS.

1. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),
ISSN 0976 – 6359(Online), Volume 5, Issue 9, September (2014), pp. 374-378 © IAEME
374
STRESS REDUCTION USING SEMI ELLIPTICAL SLOTS IN AXIALLY
LOADED PLATE HAVING CIRCULAR HOLE
HARDIK ACHARYA
Department of Mechanical Engineering, LDRP Institute of Technology & Research,
Gandhinagar, India.
ABSTRACT
When a plate with circular hole is subjected to axial loading, the region in the vicinity of hole
is under high stress. To reduce this stress, normally one auxiliary hole is introduced on either side of
the main hole. The present work studies how a semi-elliptical slot on each side of the hole effects the
maximum stress induced in a isotropic plate with circular hole. Using finite element analysis method,
stress calculation is carried out in ANSYS. The effect of variation in the geometric parameters of
ellipse on ‘maximum stress’ is presented in graphical form. Stress reduction of 42% is achieved for a
particular combination of geometric parameters. A relation for these optimum geometric parameters
is obtained with respect to the hole diameter.
Keywords: Slot, Stress Reduction, Semi-Elliptical, Optimization.
1. INTRODUCTION
As a general practice, in order to reduce the stress concentration around a circular hole,
auxiliary holes of smaller dimensions are drilled on either side. But this method has certain
limitations. Different approaches for stress reduction are examined and analysed in the literature.
H Acharya [1] obtained a relation for the optimum diameter and centre distance of auxiliary
hole for 21% stress reduction and showed that thickness and width of the plate does not affect the
optimum hole dimensions. N. K. Jain [2] carried out finite element study for reducing SCF by four
coaxial holes on either side of the main hole. Using this he revealed that holes helps to smooth flow
of the tensile stresses through the main hole. He also studied the effect of length and diameter of two
auxiliary holes on each side of the main hole in isotropic and orthotropic plates. V Manur [3] studied
the optimum location and diameter of auxiliary holes with respect to different diameter to plate width
ratios. Plotting the graph of stress concentration factor (SCF) v/s Central hole diameter-to-plate
width ratio, he found that SCF can be relieved by a factor of 19 to 21 percentage. Z Yang [4]
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2. International Journal of Mechanical Engineering and Technology (IJME
ISSN 0976 – 6359(Online), Volume 5, Issue
observed the sensitivity of the stress and strain concentration factor with respect to plate thickne
well as the Poisson’s ratio. While the structural dimensions like hole diameter and plate thickness
were varied to examine their effect directional stress concentration factor and net deflection of the
plate by S. Nagpal [5]. Studying the effect of f
concentration, B Endigeri [6] found that fibre oriented at 45
concentration. While N.K. Jain [7] analysed the distributions of stresses and deflection in simply
supported rectangular isotropic and orthotropic composite plates with central circular hole subjected
to transverse static loading using 3-
minimise the stress concentration factor, Mansfield [8] determi
edge reinforcement around a circular hole in a flat sheet subjected to uniaxial tension.
[9] presented a method in which transverse normal pressure created by bolt clamping at open holes
can be used to mitigate the hole stress concentration and increase the fatigue life of the plate.
D Gunwant [10] studied stress in a plate with elliptical hole for different aspect ratios and compared
the results of ANSYS with the analytical method.
As per these literatures, introducing circular holes, maximum stress reduction of only 21%
can be achieved. Also in order to achieve this reduction, auxiliary holes are to be drilled very near
the main hole making it weak.
Elliptical holes can be used instead of c
of main hole will increase the machining cost. Hence a semi
2. DESCRIPTION OF PROBLEM
Fig 1: Model of plate with semi
Nomenclature
D = Main hole diameter
2a = Length of major axis
2b = Length of minor axis
l = Centre distance between ellipse and circle
As shown in fig. 1 a plate of width 100 mm and thickness
an infinitely long plate, length 2000 mm is taken. While the main hole diameter is D = 40 mm and
slot width is kept 5mm. Keeping these dimensions constant, the values of ‘a’, ‘b’ and ‘l’ are varied.
A tensile force of 1000N is applied on one end of the plate while keeping the other end f
International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976
6359(Online), Volume 5, Issue 9, September (2014), pp. 374-378 © IAEME
375
observed the sensitivity of the stress and strain concentration factor with respect to plate thickne
well as the Poisson’s ratio. While the structural dimensions like hole diameter and plate thickness
were varied to examine their effect directional stress concentration factor and net deflection of the
plate by S. Nagpal [5]. Studying the effect of fiber orientation of composite materials on stress
concentration, B Endigeri [6] found that fibre oriented at 450
and 1350
results in minimum stress
concentration. While N.K. Jain [7] analysed the distributions of stresses and deflection in simply
rectangular isotropic and orthotropic composite plates with central circular hole subjected
-D finite element method. Using numerical analysis, in order to
minimise the stress concentration factor, Mansfield [8] determined optimum distributions of compact
edge reinforcement around a circular hole in a flat sheet subjected to uniaxial tension.
[9] presented a method in which transverse normal pressure created by bolt clamping at open holes
gate the hole stress concentration and increase the fatigue life of the plate.
D Gunwant [10] studied stress in a plate with elliptical hole for different aspect ratios and compared
the results of ANSYS with the analytical method.
ese literatures, introducing circular holes, maximum stress reduction of only 21%
can be achieved. Also in order to achieve this reduction, auxiliary holes are to be drilled very near
Elliptical holes can be used instead of circular ones but very large elliptical holes on both side
of main hole will increase the machining cost. Hence a semi- elliptical slot is studied in this paper.
DESCRIPTION OF PROBLEM
Model of plate with semi-elliptical slots
l = Centre distance between ellipse and circle
As shown in fig. 1 a plate of width 100 mm and thickness 10 mm is taken. In order to study
2000 mm is taken. While the main hole diameter is D = 40 mm and
slot width is kept 5mm. Keeping these dimensions constant, the values of ‘a’, ‘b’ and ‘l’ are varied.
A tensile force of 1000N is applied on one end of the plate while keeping the other end f
T), ISSN 0976 – 6340(Print),
© IAEME
observed the sensitivity of the stress and strain concentration factor with respect to plate thickness as
well as the Poisson’s ratio. While the structural dimensions like hole diameter and plate thickness
were varied to examine their effect directional stress concentration factor and net deflection of the
iber orientation of composite materials on stress
results in minimum stress
concentration. While N.K. Jain [7] analysed the distributions of stresses and deflection in simply
rectangular isotropic and orthotropic composite plates with central circular hole subjected
D finite element method. Using numerical analysis, in order to
ned optimum distributions of compact
edge reinforcement around a circular hole in a flat sheet subjected to uniaxial tension. P. K. Mallick
[9] presented a method in which transverse normal pressure created by bolt clamping at open holes
gate the hole stress concentration and increase the fatigue life of the plate.
D Gunwant [10] studied stress in a plate with elliptical hole for different aspect ratios and compared
ese literatures, introducing circular holes, maximum stress reduction of only 21%
can be achieved. Also in order to achieve this reduction, auxiliary holes are to be drilled very near
ircular ones but very large elliptical holes on both side
elliptical slot is studied in this paper.
10 mm is taken. In order to study
2000 mm is taken. While the main hole diameter is D = 40 mm and
slot width is kept 5mm. Keeping these dimensions constant, the values of ‘a’, ‘b’ and ‘l’ are varied.
A tensile force of 1000N is applied on one end of the plate while keeping the other end fixed.

3. International Journal of Mechanical Engineering and Technology (IJME
ISSN 0976 – 6359(Online), Volume 5, Issue
3. FINITE ELEMENT ANALYSIS
A quadrilateral element is selected with maximum face length of 2mm. The element has eight
nodes with each node having two degree of freedom, giving a total of sixteen degree of freedom per
element. Proximity and curvature function is used in meshing to give fine meshing near the holes.
The study is done for isotropic material. So the material taken is structural steel with Young's
modulus 2E+11 Pa and Poisson's ratio 0.3.
4. RESULTS AND DISCUSSION
To observe the effect of change in the geometric parameters, a total of 185 design points with
different combination of major axis, minor axis and centre distance are calculated for maximum
stress in the plate. Using the optimization module of ANSYS, it is found that minimum value
‘Equivalent maximum stress’ is 2.1516 MPa at a= 100mm, b= 20mm, l= 20mm. When the same
plate without semi-elliptical slots is used under the same force the ‘Equivalent maximum stress’ is
3.7698 Mpa. Hence by introducing the slots of optimized geometric
42.92% can be achieved.
As there are three input and one output parameters, it is not possible to show the relation of
all the four parameters in one graph. Hence a graph of variation in maximum stress with respect to
‘a’ is shown in fig 2. In graph (i), the curved lines represent corresponding values of ‘b’ while for
l= 20mm in graph (ii) the curved lines represent corresponding ‘l’ values for b= 20mm. From both
the graphs, it can be seen that stress goes on de
or ‘l’, increasing ‘a’ beyond 100mm does not yield appreciable decrease in stress value hence
optimum value of a= 100mm (2.5 D) is taken
(i) With respect to ‘b’
The variation in maximum stress with respect to ‘b’ is shown in fig 3. In graph (i), the curved
lines represent corresponding values of ‘a’ for l= 20mm while graph (ii) the curved lines represent
corresponding ‘l’ values for a= 100mm. From both the graphs, it can be seen that stress is minimum
for b= 20 – 21mm thus a relation b = 0.5 D is obtained.
International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976
6359(Online), Volume 5, Issue 9, September (2014), pp. 374-378 © IAEME
376
FINITE ELEMENT ANALYSIS
A quadrilateral element is selected with maximum face length of 2mm. The element has eight
nodes with each node having two degree of freedom, giving a total of sixteen degree of freedom per
function is used in meshing to give fine meshing near the holes.
The study is done for isotropic material. So the material taken is structural steel with Young's
modulus 2E+11 Pa and Poisson's ratio 0.3.
RESULTS AND DISCUSSION
hange in the geometric parameters, a total of 185 design points with
different combination of major axis, minor axis and centre distance are calculated for maximum
stress in the plate. Using the optimization module of ANSYS, it is found that minimum value
‘Equivalent maximum stress’ is 2.1516 MPa at a= 100mm, b= 20mm, l= 20mm. When the same
elliptical slots is used under the same force the ‘Equivalent maximum stress’ is
3.7698 Mpa. Hence by introducing the slots of optimized geometric parameters, a stress reduction of
As there are three input and one output parameters, it is not possible to show the relation of
all the four parameters in one graph. Hence a graph of variation in maximum stress with respect to
is shown in fig 2. In graph (i), the curved lines represent corresponding values of ‘b’ while for
l= 20mm in graph (ii) the curved lines represent corresponding ‘l’ values for b= 20mm. From both
the graphs, it can be seen that stress goes on decreasing with increasing ‘a’. But for any value of ‘b’
, increasing ‘a’ beyond 100mm does not yield appreciable decrease in stress value hence
optimum value of a= 100mm (2.5 D) is taken
(ii) With respect to ‘l’
Fig 2: Stress Vs ‘a’
The variation in maximum stress with respect to ‘b’ is shown in fig 3. In graph (i), the curved
lines represent corresponding values of ‘a’ for l= 20mm while graph (ii) the curved lines represent
responding ‘l’ values for a= 100mm. From both the graphs, it can be seen that stress is minimum
21mm thus a relation b = 0.5 D is obtained.
T), ISSN 0976 – 6340(Print),
© IAEME
A quadrilateral element is selected with maximum face length of 2mm. The element has eight
nodes with each node having two degree of freedom, giving a total of sixteen degree of freedom per
function is used in meshing to give fine meshing near the holes.
The study is done for isotropic material. So the material taken is structural steel with Young's
hange in the geometric parameters, a total of 185 design points with
different combination of major axis, minor axis and centre distance are calculated for maximum
stress in the plate. Using the optimization module of ANSYS, it is found that minimum value of
‘Equivalent maximum stress’ is 2.1516 MPa at a= 100mm, b= 20mm, l= 20mm. When the same
elliptical slots is used under the same force the ‘Equivalent maximum stress’ is
parameters, a stress reduction of
As there are three input and one output parameters, it is not possible to show the relation of
all the four parameters in one graph. Hence a graph of variation in maximum stress with respect to
is shown in fig 2. In graph (i), the curved lines represent corresponding values of ‘b’ while for
l= 20mm in graph (ii) the curved lines represent corresponding ‘l’ values for b= 20mm. From both
creasing with increasing ‘a’. But for any value of ‘b’
, increasing ‘a’ beyond 100mm does not yield appreciable decrease in stress value hence
(ii) With respect to ‘l’
The variation in maximum stress with respect to ‘b’ is shown in fig 3. In graph (i), the curved
lines represent corresponding values of ‘a’ for l= 20mm while graph (ii) the curved lines represent
responding ‘l’ values for a= 100mm. From both the graphs, it can be seen that stress is minimum

4. International Journal of Mechanical Engineering and Technology (IJME
ISSN 0976 – 6359(Online), Volume 5, Issue
(i) With respect to ‘a’
The variation in maximum stress with respect to ‘l’ is shown in fig 4. In graph (i), the curved
lines represent corresponding values of ‘a’ for b= 20mm while graph (ii) the curved lines represent
corresponding ‘b’ values for a= 100mm. From both the
for l= 20 – 22mm thus a relation l = 0.5 D is obtained.
(i) With respect to ‘a’
5. CONCLUSION
In this study, by using semi-
of 42% can be achieved which is double of what can be achieved using auxiliary
(21%). The relation of optimum geometric parameters
hole diameter is: a= 2.5D, b= 0.5D and l= 0.5D.
International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976
6359(Online), Volume 5, Issue 9, September (2014), pp. 374-378 © IAEME
377
(ii) With respect to ‘l’
Fig 3: Stress Vs ‘b’
The variation in maximum stress with respect to ‘l’ is shown in fig 4. In graph (i), the curved
lines represent corresponding values of ‘a’ for b= 20mm while graph (ii) the curved lines represent
corresponding ‘b’ values for a= 100mm. From both the graphs, it can be seen that stress is minimum
22mm thus a relation l = 0.5 D is obtained.
(i) With respect to ‘a’ (ii) With respect to ‘b’
Fig 4: Stress Vs ‘l’
-elliptical slots it is seen that maximum possible stress reduction
of 42% can be achieved which is double of what can be achieved using auxiliary
. The relation of optimum geometric parameters for stress minimization with respect to main
hole diameter is: a= 2.5D, b= 0.5D and l= 0.5D.
T), ISSN 0976 – 6340(Print),
© IAEME
(ii) With respect to ‘l’
The variation in maximum stress with respect to ‘l’ is shown in fig 4. In graph (i), the curved
lines represent corresponding values of ‘a’ for b= 20mm while graph (ii) the curved lines represent
graphs, it can be seen that stress is minimum
elliptical slots it is seen that maximum possible stress reduction
of 42% can be achieved which is double of what can be achieved using auxiliary circular holes
imization with respect to main

5. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),
ISSN 0976 – 6359(Online), Volume 5, Issue 9, September (2014), pp. 374-378 © IAEME
378
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