M. I. ANSARI et al.: DYNAMIC ANALYSIS OF FGM RHOMBIC PLATES WITH A VARIATION IN THE MASS
731–736
DYNAMIC ANALYSIS OF FGM RHOMBIC PLATES WITH A
VARIATION IN THE MASS
DINAMI^NA ANALIZA FGM ROMBSKIH PLO[^ RAZLI^NIH
MAS
Md Irfan Ansari
1
, Ajay Kumar
1
, Danuta Barnat-Hunek
2
, Przemyslaw Brzyski
2*
,
Wojciech Andrzejuk
3
1
National Institute of Technology Patna, Department of Civil Engineering, Patna, Bihar, India
2
Lublin University of Technlogy, Faculty of Civil Engineering and Architecture, Department of Construction, Nadbystrzycka 40, 20-618
Lublin, Poland
3
Pope John Paul II State School of Higher Education in Bia³a Podlaska, Faculty of Economics and Engineering, 95/97 Sidorska, 21-500 Bia³a
Podlaska, Poland
p.brzyski@pollub.pl
Prejem rokopisa – received: 2018-04-06; sprejem za objavo – accepted for publication: 2018-06-14
doi:10.17222/mit.2018.071
A dynamic analysis of shear-deformable rhombic plates from functionally graded material (FGM) with a variation in the mass is
presented. The present mathematical model incorporates a realistic cubic variation of the thickness coordinate in displacement
fields. Due to the parabolic variation of the transverse shear strains in the thickness, the shear-correction factor is eliminated.
The finite-element formulation of the present mathematical model is done using a two-dimensional C
0
element with seven nodal
unknowns. A computer code is written for the present finite-element formulation. The material properties vary in the thickness
direction of the FGM rhombic plate at any point according to the Mori-Tanaka scheme. The accuracy of the formulation is
demonstrated by comparing it with suitable examples from the literature. This is the first attempt at a dynamic study of FGM
rhombic plates with a variation in the mass and with various volume-fraction indices, thickness ratios and boundary constraints.
Keywords: functionally graded material, finite-element method, cutouts, additional mass
V ~lanku je predstavljena dinami~na analiza stri`no deformiranih rombskih plo{~ iz materiala s funkcionalno porazdeljenimi
lastnostmi (FGM, angl.: functionally graded material) razli~nih mas. Predstavljeni matemati~ni model vklju~uje realisti~ne
kubi~ne variacije koordinat debeline v premi~nem polju. Zaradi paraboli~nih variacij pre~nih stri`nih deformacij v debelini je
eliminiran korekcijski faktor striga. Formulacijo z metodo kon~nih elementov predstavljenega matemati~nega modela so avtorji
izvr{ili z uporabo dvodimenzionalnega C
0
elementa s sedmimi vozli{~nimi neznankami. Napisali so ra~unalni{ko kodo za
predstavljeno formulacijo kon~nih elementov. Materialne lastnosti se spreminjajo v smeri debeline rombske FGM plo{~e v vsaki
to~ki skladno s shemo Tanake. Natan~nost formulacije je v ~lanku predstavljena s primerjavo primernih primerov iz literature.
To je prvi pristop k dinami~nemu {tudiju rombskih FGM plo{~ s spremenljivo maso in s spreminjajo~imi kazalci volumskim
dele`ev, razmerji debeline in mejnimi omejitvami.
Klju~ne besede: material s funkcionalno porazdeljenimi lastnostmi, metoda kon~nih elementov, izrezi, dodatna masa
1 INTRODUCTION
In recent years, plates made of FGMs have gained
considerable attention in civil, aeronautical, mechanical
and marine engineering. A variation in the mass is
achieved using a cutout and additional mass in the
present model. Plates with cutouts are used to modify the
weight of a structural member, provide ventilation, attain
the appropriate connection between structural compo-
nents or alter the resonant frequency of a structure. The
additional mass is generally used to reduce the funda-
mental frequency to a desired value. Reddy
1
evaluated
the static behaviour of FGM plates based on a third-order
shear-deformation theory (TSDT). Abrate
2
studied the
complications of a free vibration analysis of FGM plates
using a classical laminated-plate model. The three-
dimensional solution for the vibration problem of a plate
from functionally graded material was presented by
Uymaz & Aydogdu
3
under various boundary conditions.
Matsunaga
4
used a two-dimensional higher-order defor-
mation theory and Zhao et al.
5
implemented a first-order
shear-deformation theory (FSDT) while Fares et al.
6
used
a refined two-dimensional theory to estimate the vibra-
tions of FG plates under different boundary conditions.
The bending behaviour of an FGM plate using a higher-
order shear-deformation theory was studied by Taj et al.
7
The finite-element formulation based on a third-order
shear-deformation theory was used by Taj and Chakra-
barti
8
to analyse the static and dynamic behaviour of
skew plates from functionally graded material. Asemi et
al.
9
utilized the principle of minimum energy and Ray-
leigh-Ritz energy method for static and dynamic anal-
yses of FGM skew plates. Most of the earlier dynamic
analyses of the plates with cutouts were limited to iso-
tropic plates
10–11
and Reddy
12
reported on a laminated
composite plate with a cutout. Huang and Sakiyama
13
used a numerical method for the analysis of free vib-
rations of square plates with different types of cutouts.
The thermal effect on free vibrations of FGM non-uni-
Materiali in tehnologije / Materials and technology 52 (2018) 6, 731–736 731
UDK 620.1:620.172.21 ISSN 1580-2949
Original scientific article/Izvirni znanstveni ~lanek MTAEC9, 52(6)731(2018)
form straight-sided plates with different shapes of
cutouts was investigated by Janghorban and Zare.
14
Most of the finite-element (FE) software is based on
FSDT, which is not suitable for moderately thick plates
because FSDT requires the shear-correction factor. In the
present analysis, a parabolic transverse-shear-strain
deformation across the thickness is taken and conse-
quently the need for the shear-correction factor is elimi-
nated. From the literature review, it is clear that no result
of a dynamic analysis of FGM rhombic plates with a
variation in the mass is available. Hence, in the present
study, an attempt was made to carry out a dynamic
analysis of FGM rhombic plates with a cutout and
additional mass.
2 MATERIALS AND METHODS
2.1 Effective material properties
An FGM plate is a combination of two differently
dispersed constituents; its material is macroscopically
isotropic and material properties gradually change only
in the thickness direction. The effective property of the
FGM plate at any height x
3
can be expressed as
Px PV x PV x () () ()
333
=+
cc mm
(1)
Vx
x
h
n
c
()
3
3
1
2
=+
⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ,() 0≤≤ ∞ n andVx Vx
cm
() ()
33
1 += (2)
where P
m
and P
c
denote the material properties of the
metal and ceramic, respectively, V
c
and V
m
are called the
volume fractions of ceramic and metallic constituents,
respectively, and n is known as the volume fraction
index.
According to the Mori-Tanaka scheme,
15
the effective
bulk modulus (B), the effective shear modulus (G), the
modulus of elasticity (E) and Poisson’s ratio (v) can be
calculated at any point within the FGM plate.
2.2 Mathematical formulation
The geometry of the FGM plate with a cutout used in
present study is shown in Figure 1. The length of the
plate is taken as a, the width is b and the total thickness
is h. The middle section of the plate from functionally
graded material is taken as the reference. The displace-
ment field for the FGM rhombic plate is considered to
derive the mathematical model based on Reddy
1
:
ux x x u x
x
h
w
ux x x
xx x
(,,) ( )
(,,)
, 123 03
3
3
2 0
123
11 1
4
3
=+ − +   
=+ − +
=
vx
x
h
w
wx x x w
xx x 03
3
3
2 0
123 0
22 2
4
3
   ()
(,,)
,
(3)
where u, v and w are displacements of any generic point
in the plate geometry, u
0
, v
0
and w
0
are displacements at
the mid-plane and   x1
,   x2
are the bending rotations
defined at the mid-plane about the x
2
and x
1
axes, res-
pectively. For the condition of field the variables are
continuous within the element, while for the C
0
conti-
nuity problem the out-of-plane derivatives are substi-
tuted by incorporating the following relations in
Equation (3):
  
xx x
w
11 1
0
=+ ()
,
,  
xx x
w
22 2
0
=+ ()
,
(4)
The strain-displacement relationships can be ex-
pressed as
      x
x
xx
x
x
xx
u
v
uv
1
2
12
1
2
21
0
0
00
⎧ ⎨ ⎪ ⎩ ⎪ ⎫ ⎬ ⎪ ⎭ ⎪ =
+
⎧ ⎨ ⎪ ⎩ ⎪ ⎫ ⎬ ,
,
,,
⎪ ⎭ ⎪ +
+
⎧ ⎨ ⎪ ⎩ ⎪ ⎫ ⎬ ⎪ ⎭ ⎪ +
+
x
x
h
xx
xx
xx xx
3
3
3
2
11
22
12 21
4
3
      
,
,
,,
      
    xx
xx
xx xx
xx
xx
11
22
12 21
23
31
,
,
,,
,
,
+
⎧ ⎨ ⎪ ⎩ ⎪ ⎫ ⎬ ⎪ ⎭ ⎪ ⎧ ⎨ ⎩ ⎫ ⎬ ⎭ =
+
+
⎧ ⎨ ⎩ ⎫ ⎬ ⎭ −
⎧ ⎨ ⎩ ⎫ ⎬ ⎭ w
w
x
h
x x
x x
x
x
0
0
3
3
2
2 2
1 1
2
1
4
3
,
,
        (5)
Further, the expression of the strain vector   can be
correlated with the displacement vector   using the
following relationship:
{} [ ] {}
   = B (6)
where B is known as the strain-displacement matrix,
involving the derivatives of shape-function terms.
2.3 Finite-element formulation
2.3.1 Element description
A nine-nodded C
0
isoparametric Lagrangian element
is utilized in the present investigation. It has a total of
sixty-three degrees of freedom and each node has seven
degrees of freedom.
M. I. ANSARI et al.: DYNAMIC ANALYSIS OF FGM RHOMBIC PLATES WITH A VARIATION IN THE MASS
732 Materiali in tehnologije / Materials and technology 52 (2018) 6, 731–736
Figure 1: Geometry of an FGM plate with a cutout having a zero
skew angle with the x
2
axis
2.3.2 Skew boundary transformation
For the rhombic plate shown in Figure 2, the edges
of the boundary elements are not parallel to the global
axes of the rhombic plate. Hence, the transformation
matrix T is required to transform the element matrices
from the global to the local axes.
Transformation matrix
[]
T
cs
sc
cs
sc
cs
=
−
−
−
00000
00000
0010000
000 00
000 00
00000
00000sc
⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ (7)
where c = cos  , s = sin  and  is the skew angle of the
plate.
2.3.3 Governing equation for the free-vibration
analysis
The acceleration at any point within the element may
be expressed in terms of the mid-surface displacement
parameters (u
0
, v
0
, w
0
)as
{} {}
[]
{} f
t
f
u
v
w
Ff == −
⎧ ⎨ ⎪ ⎩ ⎪ ⎫ ⎬ ⎪ ⎭ ⎪ =−
∂
∂
2
2
22
   (8)
where the vector f represents the nodal unknowns and is
ofthe7×1order, containing the terms of Equation (3).
The vector f is decoupled into the matrix C that con-
tains the shape functions (N
i
) and the global displace-
ment vector  .
{}
[] {}
fC =   (9)
By utilizing Equations (8) and (9), the mass matrix of
an element may be expressed as
[] [][] []
mCL C A
A
T
  ∫ ∫
d (10)
where the expression of the matrix L can be written as
[] [][]
LFF z
T
A
  
∫
d (11)
where   is the density of the FGM estimated from
Equation (1). Hence, the governing equation for the
free-vibration analysis becomes
[] [] () {}{}
KM X −=   2
0 (12)
where M and K are the mass matrix and linear-stiffness
matrix, respectively. The solution of the above equation
(eigenvalue problem) provides the vibration characte-
ristics, that is, the frequency parameters for the free vib-
ration of the functionally graded rhombic plate.
3 RESULTS AND DISCUSSION
The vibration behaviour of the FGM rhombic plates
with additional mass and cutouts was analysed con-
sidering different combinations of ceramic and metal
constituents, the boundary condition, skew angle, cutout
size and additional mass with several geometric para-
meters. An FE code was implemented based on the
present formulation. Seven unknowns per node for the
present model were utilized for the nine-nodded isopara-
metric elements to discretize the FGM rhombic plate.
The non-dimensional quantities used are:
The non-dimensional frequency parameter
      =
a
h
E
2
c
c
The non-dimensional additional mass
M
M
ha
=
  c
2
The boundary conditions used in the present study
are as follows:
Clamped and simply supported (CCSS):
At x
1
=0, a uvw
xxxx
= ======     
1212
0
At x
2
=0, b uw
xx
== = =   
11
0
Clamped and free (CCFF):
At x
1
= 0,a
At x
2
= 0,b uvw
xxxx
= =====≠     
1212
0
3.1 Convergence and validation
Since there is no study of a free-vibration analysis of
an FGM plate with additional mass available, a compari-
son with an FGM plate without additional mass was
made. The material properties of the FGM components
specified at the normal temperature and utilized for the
calculation used in present study are provided below.
M. I. ANSARI et al.: DYNAMIC ANALYSIS OF FGM RHOMBIC PLATES WITH A VARIATION IN THE MASS
Materiali in tehnologije / Materials and technology 52 (2018) 6, 731–736 733
Figure 2: Plan of a plate with a skew angle (  ) with the x
2
axis
FGM (Al/Al
2
O
3
): E
c
= 380 GPa, E
m
=70GPa,v
c
= 0.3,
v
m
= 0.3,  c
= 3800 kg/m
3
,   m
= 2707 kg/m
3
Table 1 shows a convergence-and-validation study of
an FGM plate without additional mass where the present
finite-element formulation is validated with a three-
dimensional solution by Uymaz and Aydogdu
3
.Itisno -
ticed that a 20 × 20 mesh is satisfactory for the free-vib-
ration analysis of the FGM plate. For various thickness
ratios, the present results compare well with the previous
results. The dimensionless frequency parameter of an
FGM rhombic plate made of SUS304 and Si
3
N
4
is shown
in Table 2. The side-to-thickness ratio a/h = 10 and skew
angles   = 15° and 30° were considered. For various
volume-fraction indices, the non-dimensional frequency
parameter applying up to 4 modes was compared with
the result obtained by Zhao et al.
5
and reasonable agree-
ment between the results was found. The numerical
results for the non-dimensional frequency parameter of a
simply supported isotropic plate for v = 0.3 and a/h=50
are presented in Table 3. The presented results were
checked against those obtained by Ali and Atwal
11
;we
found the results were close, confirming a high accuracy
of the present model.
3.2 Results and discussion
Table 4 shows the effect of the non-dimensional
additional mass and volume-fraction index (n) on the
non-dimensional frequency parameter for the simply
supported (SSSS) FGM rhombic plate. The results are
computed for a/h = 10, a/b = 1 and M = 0.5, 1, 2. It is
observed that the rise in the volume fraction (V
c
=0to1)
results in a decrease in the dimensionless frequency. The
reason for this is the fact that an FGM plate with a larger
volume fraction (near to 1) implies that the plate has a
smaller ceramic component and thus the stiffness is
reduced. Apart from this, the dimensionless-frequency
parameter increases with an increase in the skew angle.
Due to the fact that the increase in the skew angle
reduces the length of the shorter diagonal, which leads to
an enhancement in the stiffness of the rhombic plate, the
frequency increases.
Table 4: Variation in the frequency parameter for a simply supported
FGM rhombic plate with additional mass (a/b=1,a/h = 10)
nM
Skew angle
15° 30° 45° 60°
Ceramic
0.5 3.2506 3.6066 4.3476 5.8281
1 2.4517 2.6919 3.1919 4.1979
2 1.7940 1.9580 2.3008 2.9959
0.2
0.5 2.6773 2.9714 3.5869 4.8309
1 2.0159 2.2149 2.6314 3.4789
2 1.4738 1.6099 1.8961 2.4825
0.5
0.5 2.3343 2.5889 3.1220 4.2024
1 1.7531 1.9255 2.2869 3.0240
2 1.2798 1.3979 1.6465 2.1572
1
0.5 2.0933 2.3180 2.7873 3.7336
1 1.5671 1.7192 2.0374 2.6837
2 1.1419 1.2462 1.4652 1.9133
10
0.5 1.6393 1.8046 2.1445 2.8105
1 1.2156 1.3272 1.5574 2.0129
2 0.8810 0.9576 1.1162 1.4325
Metal
0.5 1.4470 1.5962 1.9065 2.5281
1 1.0730 1.1742 1.3852 1.8113
2 0.7778 0.8473 0.9930 1.2893
Figure 3 illustrates the effect of additional mass on
the frequency parameter of the FGM rhombic plate
under various boundary conditions and Figure 4 shows
the effects of side-to-thickness ratios on the frequency
parameter. It can be noticed that the value of the dimen-
sionless frequency parameter decreases with an increase
in the additional mass. It is interesting to notice that a
very low effect of additional mass is observed for the
CFCF (clamped and free) type of boundary condition. It
is also noticed that the frequency parameter increases
when constraints on the boundaries increase. The CCCC
type exhibits the highest frequency parameter while the
CFCF type exhibits the lowest frequency parameter
M. I. ANSARI et al.: DYNAMIC ANALYSIS OF FGM RHOMBIC PLATES WITH A VARIATION IN THE MASS
734 Materiali in tehnologije / Materials and technology 52 (2018) 6, 731–736
Table 1: Convergence of linear frequency with the volume-fraction
index for an FGM (Al/ZrO
2
) square plate under the clamped-boundary
condition
a/h
Mesh
size
n
0 0.5 1 2 5 10
10
4 × 4 3.3290 2.9372 2.8051 2.7061 2.6063 2.5284
8 × 8 3.3026 2.9131 2.7824 2.6850 2.5867 2.5094
12 × 12 3.3010 2.9116 2.7837 2.6836 2.5854 2.5082
16 × 16 3.3006 2.9113 2.7806 2.6834 2.5852 2.5080
18 × 18 3.3005 2.9112 2.7805 2.6833 2.5851 2.5079
20 × 20 3.3005 2.9112 2.7805 2.6833 2.5851 2.5079
Uymaz
and
Aydogdu
3
3.3496 3.0249 2.8809 2.7658 2.6645 2.5923
Table 2: Comparison of frequency parameters for an FGM
(SuS
3
O
4
/Si
3
N
4
) clamped rhombic plate (a/h = 10, a/b=1)
n Mode
Skew angle (  )
15° 30°
Present Zhao et al.
5
Present Zhao et al.
5
1
1 6.2833 6.2043 7.4459 7.3546
2 11.2783 11.1789 12.3863 12.2774
3 12.6163 12.5160 15.4895 15.3673
4 16.1050 15.9364 17.2137 17.0489
Table 3: Comparison of frequency parameters for a simply supported
square plate with a square cutout at the centre
Cutout size
Ali and
Atwal
11
FEM
Ali and
Atwal
11
Rayleigh’s
method
Present
(TSDT)
No cutout 19.816 19.739 19.7133
0.1a × 0.1a 18.491 19.427 19.4282
0.2a × 0.2a 18.446 19.274 19.1095
0.3a × 0.3a 19.126 19.549 19.4235
0.4a × 0.4a 20.650 20.705 20.7270
among all combinations of the end supports and all the
skew angles. The dimensionless frequency parameter for
the other end supports (CCSS, CSCS, CCFF and CFCF)
is between CCCC and CFCF. The frequency parameter
increases with an increase in the a/h ratio up to a/h = 50;
beyond that no significant change in the frequency
parameter is noticed. The variation in the non-dimen-
sional frequency for the simply supported FGM rhombic
plate with a central cutout is presented in Table 5. The
results are computed for a/h =1 0a n da/b =1 .T h e
dimensionless frequency parameter first decreases, then
it increases with the increase in the cutout size at the
centre. The increase in the cutout size results in an
increase in the frequency parameter of the FGM rhombic
plate due to the reduction in mass. However, this is not
always the case due to the fact that the position and size
of the cutout change the mass as well as the flexure
rigidity of the FGM plate.
Figure 5 describes the variation in the frequency
parameter with the cutout size for the FGM rhombic
plate under various boundary conditions. It can be seen
that the increasing boundary constraints increase the
non-dimensional frequency parameter as expected. It is
also noticed that the dimensionless-frequency parameter
increases with an increase in the cutout size for SSSS,
CCCC, CCSS, CSCS, but not for the CFCF boundary
condition. Figure 6 shows the deviation in the non-
dimensional frequency parameter of the FGM rhombic
clamped plate with the side-to-thickness ratio for diffe-
rent cutout sizes. The results are computed for various
cutout sizes, keeping a/b = 1 and n=1.
M. I. ANSARI et al.: DYNAMIC ANALYSIS OF FGM RHOMBIC PLATES WITH A VARIATION IN THE MASS
Materiali in tehnologije / Materials and technology 52 (2018) 6, 731–736 735
Figure 5: Variation in the frequency parameter with the cutout size for
the FGM rhombic plate for various boundary conditions
Figure 3: Variation in the frequency parameter with non-dimensional
additional mass for the FGM rhombic plate for a/h=10
Figure 6: Variation in the frequency parameter with the side-to-thick-
ness ratios for the FGM rhombic plate with a cutout
Figure 4: Variation in the frequency parameter with non-dimensional
additional mass for the simply supported FGM rhombic plate
Table 5: Variation in the frequency parameter with the cutout size for
the simply supported FGM rhombic plate
n c/a = d/a
Skew angle
15° 30° 45° 60°
Ceramic
0.1 5.9877 7.0664 9.6664 16.5884
0.2 5.8644 6.8498 9.1699 15.1024
0.3 5.9319 6.8652 9.0272 14.4389
0.4 6.2847 7.2101 9.3307 11.4766
0.2
0.1 5.0072 5.9121 8.0967 13.9318
0.2 4.9085 5.7363 7.6878 12.6871
0.3 4.9690 5.7536 7.5728 12.1298
0.4 5.2677 6.0461 7.8302 9.9944
0.5
0.1 4.4485 5.2523 7.1928 12.3751
0.2 4.3604 5.0956 6.8284 11.2642
0.3 4.4136 5.1102 6.7245 10.7632
0.4 4.6781 5.3687 6.9505 9.0363
1
0.1 4.0793 4.8147 6.5878 11.3109
0.2 3.9976 4.6698 6.2521 10.2930
0.3 4.0452 4.6816 6.1548 9.8303
0.4 4.2860 4.9164 6.3583 8.2377
10
0.1 3.4273 4.0397 5.5100 9.3908
0.2 3.3606 3.9208 5.2353 8.5733
0.3 3.4024 3.9333 5.1590 8.2046
0.4 3.6061 4.1322 5.3329 6.2411
Metal
0.1 3.0466 3.5955 4.9183 8.4400
0.2 2.9839 3.4853 4.6657 7.6840
0.3 3.0183 3.4931 4.5931 7.3465
0.4 3.1977 3.6686 4.7475 5.8376
4 CONCLUSIONS
The following general conclusions are made from the
present study considering various side-to-thickness ra-
tios, volume-fraction indices, additional-mass amounts,
cutout sizes and boundary conditions.
The frequency parameter decreases with the increase
in the volume-fraction index irrespective of the boundary
condition, side-to-thickness ratio, skew angle, cutout size
and additional mass.
The effect of additional mass on the vibration of the
FGM plate under the CFCF boundary condition is
negligible.
The frequency parameter increases with the skew
angle.
Under the CFCF boundary condition, the frequency
parameter decreases with the increase in the cutout size.
5 REFERENCES
1
J. N. Reddy, Analysis of functionally graded plates, Int. J. Numer.
Meth. Eng., 47 (2000), 663–684, doi:10.1002/(SICI)10970-207
(20000110/30)47
2
S. Abrate, Free vibration buckling and static deflections of function-
ally graded plates, Compos. Sci. Technol., 66 (2006), 2383–2394,
doi:10.1016/j.compscitech.-2006.02.032
3
B. Uymaz, M. Aydogdu, Three-dimensional vibration analysis of
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736 Materiali in tehnologije / Materials and technology 52 (2018) 6, 731–736