Journal of Emerging Trends in Engineering and Applied Sciences (JETEAS) 2 (5): 835-839
© Scholarlink Research Institute Journals, 2011 (ISSN: 2141-7016)
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Journal of Emerging Trends in Engineering and Applied Sciences (JETEAS) 2 (5): 835-839 (ISSN: 2141-7016)
Buckling of Fiber-reinforced Annular Disks
Subjected to a Concentrated Tangential Edge Load
V. Soamidas
D.E.I. Faculty of Engineering,
Dayalbagh, Agra 282110, India
___________________________________________________________________________
Abstract
Circular and annular disks are one of the most commonly used shapes of components in machinery and hence
the behavior of these is of utmost interest. Consequently, we find that circular disks have been analyzed
extensively. Yet some aspects of their buckling behavior have not been addressed. The aim of the present work
is to analyze polar and rectangular orthotropic annular disks subjected to a concentrated tangential load acting
on the outer edge. The effect of variation in dimensions, elastic properties, point of load application and
number of layers with different fiber orientations is considered. Non-dimensionalized results are presented in
the form of graphs and an attempt is made to analyze the observed behavior
_________________________________________________________________________________________
Keywords: annular disks, buckling, fiber reinforced, tangential load
__________________________________________________________________________________________
INTRODUCTION
critical speeds of moderately thick circular spinning
Fiber reinforced composites have been witnessing
disks. Bauer and Eidel (2007) determine the natural
steadily increasing acceptability and use in various
frequencies for a spinning disk for various boundaries
applications over the past few decades.
conditions. Hochlenert (2009) describes the nonCorrespondingly,
experimental
research
and
linear stability analysis of a realistic disk brake
theoretical analysis of composites have also seen
model.
tremendous growth. One of the most widely used
industrial components is the annular or circular disk.
A frequently occurring situation in annular disks is
Annular disks have been analyzed extensively both
that of a concentrated inplane load applied on the
while conventional isotropic materials were in
outer edge of the disk, e.g. wheels, gears, cams, etc.,
primary usage and also after composites have been
but the analysis of disks loaded in this manner has not
developed. Srinivasan and Ramamurti (1979, 1980a,
received adequate attention.
Srinivasan and
1980b) did an extensive analysis of isotropic annular
Ramamurti (1979, 1980a, 1980b) have considered the
disks. Later Gorman (1983), Satyamoorthy (1984),
concentrated inplane edge load for isotropic disks.
Lakshminarayana (1986), Gupta, et al (1986),
Similar work with respect to orthotropic disks has
Soamidas and Ganesan (1988, 1989a, 1989b, 1995),
been reported by Soamidas and Ganesan (1988,
Thyagarajan (1989), Malhotra (1989), Mermertas and
1989a, 1989b, 1995), but they too have not analyzed
Belek (1994), Chen and Jhu (1996), Lee, et al (1998),
the buckling of annular disks under concentrated
Arnold, et al (2002), Deshpande and Mote (2003),
edge loads.
DasGupta and Hagedorn (2005), Koo (2006), Eid and
Adams (2006), Bauer and Eidel (2007), Hechlenert
In this paper, a study of the buckling behavior of
(2009), Bashmal, et al (2009), and many others have
annular disks having polar or rectangular orthotropy
published various aspects of analysis of composite
is presented when subjected to a concentrated inplane
annular disks, including buckling.
tangential load at a point P on the outer edge. The
analysis is done by considering variation in material
In the work quoted here, buckling of annular and
properties (E2 /E1 and G12 /E1), in the aspect ratio
circular disks have also been covered. Gupta, et al
( = b/a), and in the thickness ratio (hx = ho /hi ). In
(1986) consider buckling under inplane hydrostatic
the case of disks with rectangular orthotropy,
forces. Mermertas and Halek (1994) study annular
multiple layer configurations (keeping the total
plates under periodic radial forces acting on the inner
thickness constant) been considered with variation in
as well as outer boundaries. Chen and Jhu (1996)
angle,, between the direction of E1 and the radial
analyze the stability of a spinning disk. DasGupta
line through P. Results are non-dimensionalized and
and Hagedorn (2005) investigate the effect of an
presented in graphical form. The inner edge the disk
external ring on the critical speeds of a spinning
is assumed to be clamped while the outer edge is
annular disk. Koo (2006) proposes an application of
assumed free.
composite materials to increase the critical speed of
rotating disks. Eid and Adams (2006) determine the
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Journal of Emerging Trends in Engineering and Applied Sciences (JETEAS) 2 (5): 835-839 (ISSN: 2141-7016)
METHODOLOGY
The finite element method was used for the analysis.
The stiffness and geometric stiffness matrices for a
sector element were assembled to obtain the global
stiffness and geometric stiffness matrices ([K] and
[KG] respectively) for the structure. The buckling
load P was obtained as the lowest eigen value of the
following eigen-value problem:
[K] {} = – P [KG] {}
The solution to the equation was obtained by the
simultaneous iteration technique, described by
Ramamurti (1988).
makes an angle  with the radial line through P, the
point of application of the concentrated load on the
outer edge of the disk. The two cases of rectangular
orthotropy considered,  = 0 and  = 90, are shown in
figure 1(d, e). Boundary conditions used in the
analysis have the inner edge clamped and the outer
edge free.
Variation in parameters E2 /E1, G12 /E1, , hx and
are considered over the ranges shown in the table
below:
Parameter E2 /E1
G12 /E1

:
Limits:
0.1 – 0.5 0.05 – 0.4 0.1 – 0.5
Fiber reinforced composites may have fibers in
different layers oriented in different directions. For
the purpose of this paper, it is assumed that there is a
single layer with fiber distribution giving rise to
elastic moduli E1 and E2 with E1 being larger than E2 .
Different values of ratios E2 /E1 and G12 /E1 can be
obtained by changing fiber density in the direction of
E2 .
hx

0.2 – 1.0
0° – 90°
The analysis has been done as a parametric study for
some assumed values of composite material
properties and annular disk dimensions. To study the
effect of varying the moduli ratio, the values of E2 /E1
are varied over the range shown above and tangential
buckling loads are determined. Results are presented
in graphical form. Similarly values of G12 /E1, , hx
and  are varied and buckling loads obtained are
presented.
RESULTS AND DISCUSSION
Two cases of polar orthotropy ( = 0 and  = 90)
and two cases of rectangular orthotropy ( = 0 and
 = 90) have been considered to determine how
variables E2 /E1 , G12 /E1,  and hx affect the buckling
load. In addition multi layered composite disks with
rectangular orthotropy have been analyzed with
varying values of .
Effect of Material Properties: The three elastic
constants E1 , E2 and G12 were considered. Variation
in these was incorporated by changing E2 and G12,
one at a time. Results are presented in graphical
form.
Variation in E2 /E1: E2 was changed while E1 and
G12 were maintained constant to obtain the buckling
loads presented in Figure 2.
The following
observations can be made from the results:
The geometry of the structure is described with the
help of Figure 1. In figure 1, E1 is in the direction of
the lines shown while E2 is in the direction
perpendicular to the lines. Figure 1(a) indicates that
 = 0 refers to an annular disk having its larger
modulus of elasticity E1 in the radial direction.
Similarly, figure 1(b) indicates that  = 90 refers to a
disk with E1 in the circumferential direction. Figure
1(c) shows a disk with rectangular orthotropy, i.e.
fibers are laid out straight, with E1 and E2 oriented
along two mutually perpendicular directions as in a
Cartesian coordinate system. The direction of E1
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Journal of Emerging Trends in Engineering and Applied Sciences (JETEAS) 2 (5): 835-839 (ISSN: 2141-7016)
1. In all cases it is seen that an increase in the value
of E2 causes the buckling load to become larger. This
is expected as the disk stiffness is directly affected by
E2.
2. In polar as well as rectangular orthotropy, it is
seen that buckling loads are larger if E1 is in the
radial direction at the point of load application
( = 0 and = 0). This behavior can be explained as
follows – a tangential load causes rotation and, with
the inner edge clamped, the structure will have a
tendency to fold along a line approximately parallel
to the load. The modulus of elasticity opposing this
bending will be the dominant factor in determination
of the buckling load. Thus the  = 90 and  = 90
cases have lower values of the buckling load as E2 is
the dominant modulus resisting the bending.
3. In the  = 90 case the magnitude of the buckling
load is smaller than in all the other cases. It is seen
(figure 1e) that at all locations of the disk the
direction of E2 is perpendicular to the load and is the
dominant modulus resisting bending. This is the
reason for the disk being weaker than the others.
Further, an increase in E2 increases the dominant
modulus that directly resists buckling, thus giving
rise to the higher rate of increase in the buckling load.
Variation in G12 /E1: Results of variation in G12 are
shown in Figure 3. The following observations can
be made:
1. As in the case of variation in E2 , it is seen
that buckling load increases as G12 is
increased
2. The rate at which buckling load increases is
nearly the same for both cases of polar
orthotropy, viz.  = 0 and  = 90
3. Among the cases of rectangular orthotropy it
is seen that the effect of raising G12 is
significantly lower for  = 90. A possible
explanation could be that G12 provides
resistance to shear deformation. When
 = 90 shear strains might be low and
changes in G12 would not affect them to the
same extent as in the other cases.
Effect of Dimensions: Dimensions considered for
variation are the inner radius b, and the thickness of
the disk. To consider different values of b the ratio
b/a is varied. Thickness variation is considered by
maintaining a constant equivalent thickness (heq)
while the disk thickness at the inner and outer edges
(hi , ho) are changed so as to obtain different values of
a thickness ratio variable (hx = ho /hi ). The values of
hx that are used give either a disk with a uniform
thickness (hx = 1) or one having a thin outer edge
and a thick inner edge (hx < 1).
Variation in inner radius b: Keeping the outer radius
a constant, b is varied to obtain different values of
the radii ratio. Results are plotted and shown in
figure 4. In all situations considered there is a nearly
uniform increase in the buckling load. As an
explanation, consider a column. The buckling load is
inversely proportional to the square of the length. In
the present case, the width of the disk between the
free outer edge and the clamped inner edge reduces as
b is increased, thus contributing to the observed steep
increase in the buckling load.
Variation in thickness ratio hx: Keeping the
equivalent thickness heq constant, thickness at both
outer and inner edges are changed to obtain a desired
value of their ratio, hx. The tangential buckling loads
obtained are presented in figure 5. It is observed that
as hx increases, the buckling load reaches a peak and
further increase in hx causes the buckling load to
reduce. Physically, an increase in hx corresponds to
reducing the thickness at the clamped inner edge and
increasing that at the free outer edge. These opposing
trends have opposite effects on the buckling load and
which one is dominant at any particular point
determines whether there will be an increase or
decrease in the buckling load. A further observation
is that the rate of increase of buckling load is greater
than the rate at which it falls off as hx is increased.
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Journal of Emerging Trends in Engineering and Applied Sciences (JETEAS) 2 (5): 835-839 (ISSN: 2141-7016)
Using a larger number of layers shows some
improvement in the buckling load at  = 0. The
minimum value of the buckling load is drastically
improved, as can be observed in figure 6. Further,
there are intermediate local minima and maxima at
values of  between 45 and 60. This is more
prominent in the 8-layered case where layers with
fibers oriented at 45° are encountered.
CONCLUSION
Tangential buckling loads on annular composite disks
for various values of material properties, dimensions
and fiber orientations have been determined. It is
suggested that the observed trends can be used
effectively in designing the material and dimensions
of parts and component that can be modeled as
annular disks and that are subjected to tangential
loads or loads that have a major tangential
component.
Effect of location of the point of application of load:
Application of the load P at different locations on the
outer free boundary does not have any effect on the
buckling load when the disk has polar orthotropy.
This variation is considered only for the case of disks
with rectangular orthotropy, where the position of the
point of application of P changes the value of 
(figure 1c). The variation of buckling load as  is
varied is shown in figure 6. For a single layered disk,
it is seen that buckling load is high when  = 0 but
this falls drastically when  is increased and reaches a
very small value by 45° and remains approximately at
this low value right upto 90°. Considering a larger
number of layers while maintaining the same total
thickness, it is seen that the value of low buckling
load at high values of  is improved substantially. In
the present study, a symmetric (0,90)S 4-layered
composite and a symmetric (0,45,135,90)S 8-layered
composite were considered.
NOMENCLATURE
a, b
E1
E2
G12
heq
hi, ho
outer and inner radii, respectively, of the annular disk
Modulus of elasticity in the direction of lines in Fig. 1
Modulus of elasticity in the direction normal to
the lines in Fig. 1
Modulus of rigidity of the composite material
equivalent thickness of the annular disk, thickness of
uniformly thick annular disk having same inner and
outer radii and containing the same volume of material
thickness of the annular disk at inner and outer edges
respectively
th ic k n e s s a t o u te r e d g e
th ic k n e s s a t in n e r e d g e
hx
ho/hi = thickness ratio =
n
Number of layers
P
Load, also point of application of load on outer boundary
PO,
RO
Polar orthotropy, Rectangular orthotropy

Parameter to describe the major fiber orientation in
the annular disk
 = 0 for radial fiber orientation,  = 90 for
circumferential fiber orientation


is angle between direction of E1 and the
radial line through P
(In each case, a small quantity of fibers may be
present in the direction normal to the major fiber
direction, so as to control the value of E2 )
radii ratio, b/a
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