Stress Analysis of Notches using Numerical Method

Transcription

International Journal of Innovative and Emerging Research in Engineering
Volume 2, Special Issue 1 MEPCON 2015
Available online at www.ijiere.com
International Journal of Innovative and Emerging
Research in Engineering
e-ISSN: 2394 - 3343
p-ISSN: 2394 - 5494
Stress Analysis of Notches using Numerical Method
Gaurav K. Agrawala , R. D. Palhadeb
a
PG Scholar, Shri Sant Gajanan Maharaj College of Engineering, Shegaon and India
b
Faculty, Shri Sant Gajanan Maharaj College of Engineering, Shegaon and India
ABSTRACT:
Notches are the most vulnerable case for fracture in parts. The factors accelerating fracture in parts are
notch geometry, size, flange angles, and depth etc. In this study the effect of various notch geometries on
maximum Von-Mises (VM) stress has been investigated. The numerical analysis has been done for various
notch geometries for same depth and area separately at axial loading condition.
Keywords: Axial loading, Notch, Finite Element Simulation, Von Mises Stress.
I. INTRODUCTION
Notches are inevitable part of many engineering structures and machines. It may be regarded as sharp, when the
radius of its tip is very small compared to the length of its side. In particular case cracks, holes can also be regarded
as notches. Screws, gears, bolts, keyways and nuts are some well-known machine element comprising of notches.
On the other hand, notches are the most susceptible case for fracture in parts. Thus a good understanding of stresses
in the vicinity of the notch tip is of prime importance for a reliable failure analysis of engineering component and
structures with notches.
M. H. El Haddad, T. H. Topper and K. N. Smith (1979), [1] found that the crack propagation from notches mainly
depends upon the geometry and size of the notches, in their study. In this work, N. A. Noda, M. Sera and Y. Takase
(1995), [2] have proposed formulas for stress concentration for round and flat test specimen with V-notch using
Neuber’s relation in order to evaluate the stress concentration in the notched component. T. Kondo, M. Kobayashi
and H. Sekine (2000), [3] proposed strain gage to determine stress intensities of sharp notched strips like components
in their research work. F.J. Gomez, M. Elices (2003), [4] performed various experiments to ascertain the influence of
V-Notch on the strength of different brittle materials. M. Zappalorto, P. Lazzarin, J.R. Yates (2008), [5] developed
closed-form solutions for the stress fields induced by circumferential hyperbolic and parabolic notches in
axisymmetric shafts under torsion and uniform antiplane shear loading. In this research work M.R. Ayatollahi and M.
Nejati (2011), [6] used V-Notched Brazilian Disk for the experimental evaluation of stress intensities around the sharp
notches for various notch angles under pure mode I, mode II, and mixed mode I/II loading conditions. M. Saravani
and M. Azizi (2011), [7] investigated various crack parameters on the V- Notch using photoelasticity techniques.
Recently, Dominic Tiedemann, Jürgen Bär and Hans-Joachim Gudladt (2014), [8] experimentally investigated the
influence of sharp notches and overloads on fatigue life of part. In more recent work, Alberto Sapora, Pietro Cornetti
and Alberto Carpinteri (2014), [9] have modifies the analytical expression for stress intensities related to crack
emanting from blunted V-Notch. N.O. Larrosa (2014), [10] calculated stress intensities in Mode I loading condition
for various notch geometries using Distributed Dislocations Techniques (DDT). A.R. Torabi and M.R. Ayatollahi
(2014), [11] proposed two closed-form expressions for the point stress (PS) and the mean stress (MS) failure criteria
to predict brittle fracture in engineering components weakened by a V notch with end hole (V-O Notch) under pure
compression in their study.
In all mentioned research work, investigation is mainly found in the stress parameters of V-notch geometry. Little
work is seen in the open literature on other notch geometries; therefore it seems necessary to study the stresses in case
of other notch geometries and their co relationship. In addition to this, despite of large number of analytical solutions
for calculating stress parameters currently available, the numbers of real-life problems that can be solved with the
close-form expressions are limited. For this reason, several numerical methods have been developed and available for
assessment of various required parameters.
Therefore, in this paper the numerical analysis of sharp V-notch, square notch and semi-circular notch are
performed for different depth and area by using commercially available finite element software ANSYSTM-14.5
software. The results are then correlated with the present cases.
II. NOTCHED GEOMETRIES
For the present study, the thin rectangular plate specimen with sharp V, square and semi-circular notch geometries
are chosen. A schematic diagram of the thin rectangular plate specimen with sharp V, square and semi-circular notch
geometries are shown in Fig. 1, 2 and 3 respectively. The geometrical characteristics of the test specimen are length
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(l) 200 mm, width (w) 50 mm and thickness (t) 5 mm. Aluminium 6061 was considered for investigation. For Sharp
V-notch, the flange angle (α) is kept fixed at angle 60°. In this present stress analysis using finite element simulation
in order to find out stresses and its effects, varying depth and area of notch are considered for these three cases.
Figure 1. Sharp V-notch with flange angle (α) = 60°
Figure 2. Square Notch
Figure 3. Semi-circular Notch
The Mechanical properties and chemical compositions of the test specimen material Aluminium 6061 are shown in
Table 1 and 2, respectively.
Table 1. Mechanical Properties of Aluminium 6061
Young’ Modulus
Poisson’s Ratio
69 GPa
Table 2. Chemical Composition of Aluminium 6061
(By Weight)
Aluminium
95.85%–98.56%
Silicon
0.4% - 0.8%
Iron
0% - 0.7%
Copper
0.15% - 0.40%
Manganese
0% - 0.15%
0.33
Brinell Hardness
95
Ultimate Tensile
Strength
310 MPa
Tensile Yield Strength
276 MPa
Fatigue Strength
96.5 MPa
Magnesium
0.8% - 1.2%
Machinability
50 %
Chromium
0.04% - 0.35%
Shear Modulus
26 GPa
Zinc
0% - 0.25%
Shear Strength
207 MPa
Titanium
0% - 0.15%
III. NUMERICAL MODEL
In this paper commercially available FE software ANSYSTM-14.5 is used for the 2-D stress-strain calculations. The
test specimen model is generated with the help of key points later joined by straight lines and then forming 2D area
model. The test specimen model is meshed using element Plane-182. The element plane-182 is found the best suitable
element for this type of geometry as well as for present structural analysis in Ansys TM14.5 program. The Plane-182
element consists of the following characteristics. Plane-182 is used for 2-D modelling of solid structures. The element
can be used as either a plane element (plane stress, plane strain or generalized plane strain) or an axisymmetric element.
It is defined by four nodes having two degrees of freedom at each node: translations in the nodal x and y directions.
The element has plasticity, hyperelasticity, stress stiffening, large deflection, and large strain capabilities. The
geometry and node locations for this element are shown in Fig. 4.
The applied free mesh model consists of 702 nodes and 633 elements for the of case semi-circular notch specimen
geometry with 10 mm of depth. The element input data includes eight nodes, structural material properties are as given
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in Table 1. To investigate stress parameters in an elastic range, the plates are modelled as a linear elastic material. The
Fig. 5 shows the loading and boundary condition model of test specimen. As Kondo et al. [3] applied an incremental
axial force in range of 1000-3000 N. Therefore, an axial force of 2400 N was considered for the present work and it
was applied on right side edge-line all nodes of test specimen, while the left side edge-line all nodes are considered
fixed as a zero degree freedom. Finally, the structural static finite element analysis is performed.
Figure 4. Geometry and Node Locations for Plane-182
Figure 5. Test Specimen with Square Notch and its loading condition
IV. RESULTS AND DISCUSSIONS
This paper discusses the application of structural static FE analysis for the modelling of the three different notch
geometries on plate specimen by keeping depth and area same. The profile of the test specimen plate is obtained after
transferring the actual data of the test part to the computer. Subsequently model is meshed and the corresponding loads
and boundary conditions are applied. The displacement distributions throughout, stress-strain and Von Mises (VM)
stress developed in the test part are evaluated by using FE simulation. The results obtained for three different notch
geometries keeping same depth and area are separately discussed in the following sections.
A. Same Depth
As mentioned previously, the three different notch geometries (i.e. Sharp V, Square and Semi-circular) are analyzed
keeping depth of each notch at 0, 5, 10, 15 and 20 mm. This section discusses the variation of maximum VM stress
for different notch geometries with respect to the depths. All other factor (i.e. loads, boundary condition etc.) remains
the same. The maximum VM stress distributions for test specimen without notch is shown in Fig. 6 and with different
notch geometries bearing depths of magnitude 5 and 20 mm are shown in Fig. 7 (a) to (c) and 8 (a) to (c), respectively
and corresponding comparison of results are shown in Table 3. The figures for depth 10 and 15 are not provided here,
however, the only VM magnitude is recorded in the comparison in Table 3.
Figure 6. Maximum VM Stress for test specimen without notch
Table 3 shows the maximum VM stress for the aluminium 6061 test specimen plates with different notch geometries
at five different depths. However, for all of the cases consistently, the stresses increase as the depth of the notch
increased (excluding the initial condition for sharp V, square notch). The maximum stress of value 63.4976 MPa is
seen at sharp V notch with depth 20 mm.
Fig. 9 shows the maximum VM stress with different notch geometries for five different depths. The graph shows a
steep and consistent rise of the maximum stress for the semi- circular notch, whereas the maximum stress for square
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and sharp V notch are an initial fall of 0.19 and 0.18 MPa respectively. This is due to the reduction of stress
concentration by introduction of specific notch [12].
After the initial condition, the maximum stress magnitude rises for all cases consistently till depth 15 mm is
reached then the sharp V notch surpasses the square and semi-circular notch in between 15 to 20 mm of depth. This is
because the maximum stress is concentrated at the tip of the sharp V notch, whereas in the cases of others notch
geometries blunt or flat surface are present. In addition to this the depth (15-20 mm) of the notch has reached 30 to
40% of its width (50 mm).
(a) Sharp V notch
(a) Sharp V notch
(b) Square notch
(b) Square notch
(c) Semi-circular notch
Figure 7. Maximum VM stress for test specimen
notch depth 5 mm
Figure 8. Maximum VM stress for test specimen
notch depth 20 mm
Table 3. Comparison of Maximum Von-Mises Stress
Max. Von Mises Stress (MPa)
Depth (mm)
Sharp V
Square
Semi-circular
0
21.1153
21.1153
21.1153
5
20.9447
20.9271
29.841
10
28.2817
32.0101
36.4729
15
41.3387
43.3172
46.9095
20
63.4976
58.6574
61.9937
Figure 9. Maximum VM Stress with respect to
depth for different geometries
64
Fig. 10 (a) to (c) show the maximum VM stress on the specimen with respect to its length are obtained through
ANSYSTM-14.5. All three graphs are plotted for three different notch geometries at the same depth of 10 mm. It is
observed in all three graph the maximum stress at the middle of specimen, as the path was defined tangentially to the
tip or surface of the notch while plotting the graph.
(a) Sharp V notch
(b) Square notch
Figure 10. Maximum VM stress with respect to length of test specimen for 10 mm depth
B. Same Area
In order to analyzed stresses with respect to area, again three different notch geometries (i.e. sharp V, square and
semi-circular) are considered keeping areas of each notch geometry at 0, 50, 100, 150 and 200 mm². This section
discusses the variation of maximum VM stress in different geometries with respect to their areas. All other factor (i.e.
loads, boundary condition etc.) are taken the same.
The maximum VM stress distribution for different notch area geometry of 50 mm2 and 200 mm2 are shown in Fig.11
(a) to (c) and 12 (a) to (c), respectively. However, the comparison of results is given in Table 4. The value for maximum
VM Stress for all geometries for area 0 mm² (i.e. without notch) is taken directly from Figure 6.
Figure 11. Maximum VM stress distribution for
different notch area geometry of 50 mm2
(a) Sharp V notch
(a) Sharp V notch
(b) Square notch
(b) Square notch
65
Figure 12. Maximum VM stress distribution for
different notch area geometry of 200 mm2
As shown in Fig. 13 sharp V notch and semicircular notch have a steep rise in maximum stress, whereas the
maximum stress of square notch falls a bit till 50 mm². After that there is a sudden rise in maximum stress of square
and sharp V notch till 100 mm², on the other hand maximum stress of semi-circular notch rises consistently till 200
mm². Then maximum stress of sharp V and square notch both rises gradually till 150 mm². Then the maximum VM
stress of sharp V notch shows a steep rise till 200mm² whereas the maximum stress of square and semicircular notch
rises gradually. The steep rise in maximum stress of sharp V notch is because, when the area of sharp V, square and
semi-circular notch are kept constant, the relative rise in the depth of sharp V notch is much more than square and
semi-circular notch.
Figures 14 (a) to (c) show the maximum VM stress on the specimen with respect to its length are obtained through
AnsysTM14.5. All the three graphs are plotted for three different notch geometries for the same area of 100 mm². These
three graphs show the maximum stress at the middle of specimen, as the path was defined tangentially to the tip or
surface of the notch.
Table 4. Comparison of Maximum Von-Mises Stress
Max. Von Mises Stress (MPa)
Area
(mm²)
Sharp V
Square
Semicircular
0
21.1153
21.1153
21.1153
50
32.3315
21.0147
31.8642
100
40.4435
32.0101
34.2567
150
40.9368
35.3311
36.3651
200
54.1484
40.2598
38.5236
(a) Sharp V notch
Figure 13. Maximum VM Stress with respect to
varying area for different geometries
(b) Square notch
Figure 14. Maximum VM stresses distribution for notch geometry with respect to the length of specimen at
100 mm² of area.
IV.
CONCLUSION
The study presents maximum Von-Mises stress of notched aluminium 6061 plates with various geometries. The
observations made through the study are:
When the notch length is 30% of the specimen’s width semi-circular notch is more prone to crack than the other
notch as the Von-Mises stress is more. Afterwards sharp V and semi-circular notch both possess approximately equal
threat to fracture.
Comparatively sharp V notch is the most vulnerable notch when the notch area is same.
66
REFERENCES
[1] M. H. El Haddad, T. H. Topper and K. N. Smith, “Prediction Of Non Propagating Cracks,” Engineering Fracture
Mechanics, vol.11, 1979, pp. 573–584.
[2] N. A. Noda, M. Sera and Y. Takase, “Stress concentration factors for round and flat test specimen with notches,”
Int. J. Fatigue, vol.17, 1995, pp. 163–178.
[3] T. Kondo, M. Kobayashi and H. Sekine, “Strain Gage Method for Determining Stress Intensity of Sharp-Notched
Strips,” Experimental Mechanics, vol.41, no.1, 2000, pp. 1-7.
[4] F.J. Gomez and M. Elices, “Fracture of components with V-shaped notches,” Engineering Fracture Mechanics
70, 2003, pp. 1913-1927.
[5] M. Zappalorto, P. Lazzarin and J.R. Yates, “Elastic stress distributions for hyperbolic and parabolic notches in
round
shafts under torsion and uniform antiplane shear loadings,” International Journal of Solids and Structures 45,
2008, pp. 4879–4901
[6] M. R. Ayatollahi and M. Nejati, “Experimental evaluation of stress field around the sharp notches using
Photoelasticity,” Materials and Design 32, 2011, pp. 561–569.
[7] M. Saravani and M. Azizi, “The Investigation of Crack’s Parameters on the V-Notch using Photoelasticity
Method” World Academy of Science, Engineering and Technology, vol.5, 2011, pp. 666-671.
[8] Dominic Tiedemann, Jürgen Bär and Hans-Joachim Gudladt, “The crack propagation rate according to notches
and overload levels,” Procedia Materials Science 3, 2014, pp. 1359 – 1364.
[9] Alberto Sapora, Pietro Cornetti and Alberto Carpinteri, “Analytical Stress Intensity Factors for cracks at blunted
V-notches,” Procedia Materials Science 3, 2014, 738 – 743.
[10] N. O. Larrosa, “A dislocation based method using generative algorithms to model notch geometries:
Determination of Mode I SIFs,” Engineering Fracture Mechanics 127, 2014, pp. 327–335.
[11] A. R. Torabi and M. R. Ayatollahi, “Compressive brittle fracture in V-notches with end holes,” European Journal
of Mechanics A/Solids 45, 2014, 32-40.
[12] V. B. Bhandari, “Design of Machine Elements,” third edition, Tata McGraw Hill Education Private Limited, 2011,
pp. 145-149.
AUTHORS PROFILE
Gaurav K. Agrawal has received B. E. in Mechanical Engineering from Rashtrasant Tukadoji Maharaj Nagpur
University, Nagpur, Maharashtra, India in 2012. He is currently a student of M. E. in Advance Manufacturing and
Mechanical System Design from Sant Gadge Baba Amravati University, Amravati, Maharashtra, India.
R.D. Palhade is an Associate Professor in Mechanical Engineering Department at the Shri Sant Gajanan Maharaj
College of Engineering, Shegaon, Dist- Buldana, (MS), India. He received his BE in Production Engineering, DBM
and PG-DMM from the Sant Gadge Baba Amravati University, Amravati, India, and M Tech in Mechanical
Engineering with specialisation in manufacturing processes engineering, from the Indian Institute of Technology, (IIT)
Kharagpur, India. He has presented more than 20 papers in international/national journals, an international conference
and national conference. He is a member of the Institute of Engineers (India), (IE), Kolkata and Indian Society of
Technical Education, New Delhi (ISTE).
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Stress Analysis of Notches using Numerical Method

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