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7/28/2019 Effect of Fiber Orientation on Stress Concentration Factor in a Laminate With Central Circular Hole Under Transverse Static Loading
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Indian Journal of Engineering & Materials SciencesVol. 15, December 2008, pp. 452-458
Effect of fibre orientation on stress concentration factor in a laminate with central
circular hole under transverse static loading
N D Mittal* & N K Jain
Department of Applied Mechanics, Maulana Azad National Institute of Technology, Bhopal 462 007, India
Received 12 April 2007; revised received 17 June 2008
The effect of fibre orientation (θ) on stress concentration factor (SCF) in a rectangular composite laminate with centralcircular hole under transverse static loading has been studied by using finite element method. The percent variations indeflection with fibre orientation are also compared with deflection in laminate without hole. Studies are carried out for three
D/A ratios (where D is hole diameter and A is plate width). The results are obtained for four different boundary conditions.
Three different types of materials are used for whole analysis to find the sensitivity of stress concentration with elastic
constants. A finite element study is made for whole analysis of laminate with a central hole under transverse static loading.Keywords: Finite element method, Stress concentration factor, Composite, Laminate, material properties, Fibre orientation,
Transverse loading
A laminated composite plate with central circular hole
have found widespread applications in various fields
of engineering such as aerospace, marine, automobile
and mechanical. Stress concentration arises from any
abrupt change in geometry of plate under loading. As
a result, stress distribution is not uniform throughout
the cross-section. Failures such as fatigue cracking
and plastic deformation frequently occur at points of
stress concentration. Hence, for the design of alaminated composite plate with central circular hole,
stress concentration factor plays an important role and
accurate knowledge of stresses and stress
concentration factor at the edges of hole under in
plane or transverse loading are required. Analytical
solutions are available in the literature for prediction
of SCF in different types of abrupt changes in shape.
Shastry and Raj1 have analysed the effect of fibre
orientation for a unidirectional composite laminate
with finite element method by assuming a plane stress
problem under in plane static loading. Paul and Rao2,3
presented a theory for evaluation of stressconcentration factor of thick and FRP laminated plate
with the help of Lo-Christensen-Wu higher order
bending theory under transverse loading. Xiwu
et al.4,5
evaluated stress concentration of finite
composite laminates with elliptical hole and multiple
elliptical holes based on classical laminated plate
theory. Iwaki6 worked on stress concentrations in a
plate with two unequal circular holes. Ukadgaonker
and Rao7 proposed a general solution for stresses
around holes in symmetric laminates by introducing a
general form of mapping function and an arbitrary
biaxial loading condition into the boundary
conditions. Ting et al.8
presented a theory for stress
analysis by using rhombic array of alternating method
for multiple circular holes. Chaudhuri9 worked on
stress concentration around a part through hole
weakening a laminated plate by finite elementmethod. Mahiou and Bekaou10
studied for local stress
concentration and for the prediction of tensile failure
in unidirectional composites. Toubal et al.11 studied
experimentally for stress concentration in a circular
hole in composite plate. Younis12 investigated by
reflected photoelasticity method that the assembly
stress are the result of contact and bearing stresses
between the bolts and member, contributes to
reducing stresses around the circular holes in a plate
under uniaxial tension. Peterson18 has developed good
theory and charts on the basis of mathematical
analysis and presented excellent mythology ingraphical form for evaluation of stress concentration
factors in isotropic plates with different types of
abrupt change, but no results are presented for
orthotropic and laminated plate.
In this paper, a study of rectangular laminated
composite plate with central circular hole for theeffect of fibre orientation on stress concentration
factor under transverse static loading is made. The
analytical treatment for such type of problem is very
difficult and hence the finite element method is__________*For correspondence (E-mail: [email protected])
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MITTAL & JAIN: STRESS CONCENTRATION FACTOR IN A LAMINATE 453
adopted for whole analysis. The purpose of thisresearch work is to investigate the effect of fibre
orientations on SCF for normal stress in X , Y
directions (σx, σy), shear stress in XY plane (τxy) andvon mises (equivalent) stress (σeqv) in a single layer
laminate plate with central circular hole. Three typesof different composite materials of different material
properties are used for analysis to find out the
sensitivity of SCF with respect to elastic constants.
The work also illustrates the variation of SCF versus
D/A ratio in a lamina at different fibre orientations.The deflections in transverse direction (U z) for
different cases are also calculated.
Description of Problem
To study the influence of fibre orientation upon
deflection and SCF for different stresses, a laminated
composite plate of dimension 200 mm × 100 mm × 1
mm with a central circular hole of diameter D
subjected to a total transverse static load of P Newton
(which is uniformly distributed on whole plate) for all
cases is analysed by finite element method. The
analysis is carried out for three different D/A ratios.
Figure 1a shows the basic model of the problem.
Finite Element Analysis
An eight nodded linear layered structural 3-D shell
element with six degrees of freedom at each node
(specified as Shell99 in ANSYS package) was
selected based on convergence test and used through
out the study. Each node has six degrees of freedom,
making a total 48 degrees of freedom per element. In
order to construct the graphical image of thegeometries of the three different models for different
D/A ratios, a laminated plate examined using the
ANSYS (Advanced Engineering Simulation). It was
necessary to input the basic geometric elements such
as points, lines and arcs. Mapped meshing are usedfor all models so that more elements are employed
near the hole boundary. Due to the un-symmetricnature of different models investigated, it was
necessary to discretize the full laminated plate for
finite element analysis. Main task in finite elementanalysis is selection of suitable element type.
Numbers of checks and convergence test are made forselection of suitable element type from different
available elements and to decide the element length.
Results were then displayed by using post processor
of ANSYS programme. For some simple problems of
plates, the finite elements results are also assessedwith available theoretical and experimental results in
literature and it in concluded that the finite elements
results are acceptable. Figure 1b provides the example
of the discretized models for D/A =0.2, used in study.
Results and Discussion
Numerical results are presented for three different
D/A ratio as 0.1, 0.2 and 0.5. Three different
orthotropic composite materials are used for analysis.
The material properties are given in Table 1.
Where; E , G and µ represent modulus of elasticity,
modulus of rigidity and poisson’s ratio respectively.
Four types of plates (a)-(d) are analysed. In plate (a)
all edges are simply supported, in plate (b) one edge is
Fig. 1a — Details of model analysed in study (A laminated plate
with central hole under uniformly distributed static loading of P
Newton in transverse direction)
Table 1—The material properties
Materials
Properties
Boron/
aluminium
Silicon carbide/
ceramic
Woven glass/
epoxy
E x
E y
E z
Gxy
Gyz
Gzx
µxy
µyz
µzx
235 GPa
137 GPa
137 GPa
47 GPa
47 GPa
47 GPa
0.3
0.3
0.3
121 GPa
112 GPa
112 GPa
44 GPa
44 GPa
44 GPa
0.2
0.2
0.2
29.7 GPa
29.7 GPa
29.7 GPa
5.3 GPa
5.3 GPa
5.3 GPa
0.17
0.17
0.17
Fig. 1b — Typical example of finite element mesh for D/A=0.2
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INDIAN J. ENG. MATER. SCI., DECEMBER 2008454
fixed, in plate (c) two edges are simply supported andtwo edges are fixed, in plate (d) all edges are fixed.
Figure 2 provides the boundary conditions at all edges
of plates (a), (b), (c) and (d).The variation of SCF for different stresses and
percent variation in U z with different fibreorientations are presented in Figs 3-11. It has been
noted that these are the maximum values in the plates.
In case of plates (a) and (c), the maximum stress
concentration for all stresses is always occurred on
boundary of hole, i.e., values of SCF for differentstresses are plotted for boundary of hole, where, in
case of plates (b) and (d), the maximum stress
concentration is occurred on supports, i.e., values of
SCF for different stresses are plotted for supports.
Maximum U z is always occurred at boundary of hole,hence, the percent variation in U z is plotted for
boundary of hole in all the cases.
Variations of SCF for σx, σy, τxy for different D/A ratios with respect to fibre orientations in plates (a),
(b), (c), and (d) made of different composite materialsare shown in Figs 3-5. Following observation can be
Fig. 2 — Boundary conditions at all edges of plates (a), (b), (c)
and (d)
Fig. 3 — Variation of SCF (for σx, σy, τxy) versus fibreorientations in plates (a), (b), (c) and (d) of boron/aluminum
material
Fig. 4 — Variation of SCF (for σx, σy, τxy) versus fibreorientations in plates (a), (b), (c) and (d) of silicon
carbide/ceramic material
Fig. 5 — Variation of SCF (for σx, σy, τxy) versus fibreorientations in plates (a), (b), (c) and (d) of woven glass/epoxy
material
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MITTAL & JAIN: STRESS CONCENTRATION FACTOR IN A LAMINATE 455
made from Figs 3-5. In case of plate (a); for D/A=0.1and 0.2, maximum SCF is obtained for σx for almost
all the values of θ and attaining maximum at θ=90°,
but for D/A=0.5 maximum SCF is obtained for τxy foralmost all the values of θ and attaining maximum at
θ=90° for all materials. Figures illustrate that at anyfibre orientation, SCF for σx, σy, τxy decrease with
increase of D/A ratio for all materials. It is also
clear from figures that SCF for σx, σy, τxy obtained
maximum when θ=90° for all D/A ratios andmaterials. For all D/A ratios and materials, it has been
seen that SCF for σy is always lesser then SCF for σx
at almost all the values of θ. Maximum value of SCFis coming as 3.5 in case of woven glass/epoxy
composite material at θ=90° for D/A=0.1 for σx. Incase of plate (b); maximum SCF is obtained for τxy for
almost all the values of θ and attaining maximum
value at θ=90° for all D/A ratios and materials. For all
Fig. 6 — Variation of SCF (for σeqv) versus fibre orientations inplates (a), (b), (c) and (d) of boron/aluminum material
Fig. 7 — Variation of SCF (for σeqv) versus fibre orientations in
plates (a), (b), (c) and (d) of silicon carbide/ceramic material
Fig. 8 — Variation of SCF (for σeqv) versus fibre orientations inplates (a), (b), (c) and (d) of woven glass/epoxy material
Fig. 9 — Percent variation in U z versus fibre orientations in plates
(a), (b), (c) and (d) of boron/aluminum material
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MITTAL & JAIN: STRESS CONCENTRATION FACTOR IN A LAMINATE 457
made from figures. In case of boron/aluminiummaterial, SCF increases continuously when θ changes
from 0° to 15°, decreases when θ changes from 15° to
30°, again increases when θ changes from 30° to 75°,attaining a maximum value when orientation is at 75°
and then decreases when θ changes from 75° to 90°for all D/A ratios. In case of silicon carbide/ceramic
material, SCF decreases continuously when θ changes
from 0° to 45°, attaining a minimum value when
orientation is at 45° and then again increases when θ
changes from 45° to 90°, attaining a maximum valuewhen orientation is at 90° for all D/A ratios. In case of
woven glass/epoxy, SCF increases continuously when
θ changes from 0° to 15°, decreases when θ changes
from 15° to 45°, again increases when θ changes from
45° to 75°, attaining a maximum value when
orientation is at 75° and then decreases when θ
changes from 75° to 90° for all D/A ratios. Maximum
SCF are coming as 2.2 at θ=75°, 1.9 at θ=90° and 2.1
at θ=75° for D/A=0.1 in boron/aluminium, silicon
carbide/ceramic and woven glass/epoxy composite
materials respectively. It is observed that the SCF for
σeqv also follows a symmetric trend with respect to
90° in all cases. For woven glass/epoxy laminate, SCF
follows a symmetric trend with respect to 45° when
orientation changes from 0°
to 90° and to135°
when
orientation changes from 90° to 180°. It is clear from
figures that, for all materials and D/A ratios,
maximum stress concentration occurred in case of
plate (a) for all values of θ and for plate (a) SCF
varied from 1.3 to 2.3 for different cases. It is also
observed that, in case of plate (c), some significant
stress concentration occurred. But in case of plates (b)
and (c), the effect of stress concentration is much
small, and in case of plate (d), it is almost negligible
for all cases. For plate (d), the variation of SCF with
respect to θ is also negligible for all D/A ratios and
materials; SCF is fluctuated near about 1 for all cases.
In case of plate (b); it has been seen that the effect of
D/A ratio on SCF is negligible for all values of θ and
materials. In case of all plates, the trend of variation
of SCF with respect to θ is different for different
material, i.e., variation of SCF depends up on elastic
constants. In case of plate (a); SCF obtained always
greater then 1.0 for all values of θ, D/A ratios and
materials but in case of plates (b), (c), and (d), SCF
obtained less then 1.0 in some cases.
The variation of percent variation in U z for
different D/A ratios with respect to θ in plates (a), (b),
(c), and (d) made of different composite materials are
shown in Figs 9-11. The percent variation in U Z hasbeen calculated with respect to laminate without hole
for same case. Following observation can be made
from Figs 9-11. In case of plates (a), (b) and (c), U zincreases with increase in D/A ratio, but in case of
plate (d) U z increases when D/A ratio increase from0.1 to 0.2 and then decreases when D/A ratio increase
from 0.2 to 0.5 for all values of θ and materials. For
boron/aluminium and silicon carbide/ceramic plates
(a), (c) and (d), the maximum and minimum
deflection occurred at θ=90° and θ=0° respectively,but in case of plate (b) maximum deflection occurred
when θ=0° and minimum occurred when θ=90°. In
case of woven glass/epoxy material; percent variation
in U z is almost constant with respect to θ for all D/A
ratios and plates (maximum variation is obtained up
to 5%). It has been observed that maximum percent
variation occurred for plate (a) and minimum
occurred for plate (d). It has been also seen that, per
cent variation in U z is obtained less then 0% at some
values of θ for all D/A ratios, plates and materials.
Conclusions
In general; for plates (a) and (c), the maximum
stress concentration is always occurred on hole
boundary and in case of plates (b) and (d), the
maximum stress concentration is occurred on
supports. The SCF for σx, σy, σeqv play an importantrole in plate (a), a significant role in plate (c) and
negligible role in plates (b) and (d). The SCF for τ xy
plays, an important role in plates (b), (c), (d) and a
significant role in plate (a). It has been observed that
SCF for all stresses decrease with increase in D/A
ratio, where deflection increases with increase in D/A
ratio for almost all values of θ, materials and plates.
For plates (a), (c) and (d), maximum U z always
occurred at θ=90° and for plate (b), maximum U z
always occurred at θ=0° for all D/A ratios. Maximum
SCF for τ xy always occurred at θ=90° for all cases. It
is also observed that SCF for all stresses anddeflection follow a symmetric trend with respect to
90° fibre orientation. In case of composite materials
those have same modulus of elasticity in X and Y
directions SCF for all stresses and deflection follow a
symmetric trend with respect to 45° when orientation
changes from 0 or 90° and to135° when orientation
changes from 90° or 180°. In case of all plates, the
trend of variation of SCF with respect to θ is different
for different material, i.e., variation of SCF depends
up on elastic constants. It has been also seen that the
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INDIAN J. ENG. MATER. SCI., DECEMBER 2008458
SCF is most sensitive to material properties anddirectly depend on the ratio of E x / E y and E x / Gxy. The
results obtained, show that for higher values of these
ratios, SCF for all stresses may also be higher.
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