ANALYTICAL BENDING ANALYSIS OF A CIRCULAR SANDWICH PLATE UNDER DISTRIBUTED LOAD

International Journal of Recent Advances in Mechanical Engineering (IJMECH) Vol.6, No.1, February 2017
DOI: 10.14810/ijmech.2017.6101 1
ANALYTICAL BENDING ANALYSIS OF A
CIRCULAR SANDWICH PLATE UNDER
DISTRIBUTED LOAD
Hamed Vahdati Zadeh and Masoud Tahani
Department of Mechanical Engineering,
Ferdowsi University of Mashhad, Mashhad, Iran
ABSTRACT
In this paper, bending analysis of a circular sandwich plate under distributed load with simply supported
and clamped boundary conditions is investigated. First, the governing equations of the circular sandwich
plate are obtained and they are solved using the Bessel functions. Then in order to validate the correctness
of analytical results, numerical finite element method is used and its results are presented in the forms of
contours and graphs. The results indicate that under distributed load, maximum deflection happens at 0.3
of outside radius, away from centre, and minimum deflection occurs at the outer edge of the circular
sandwich plate. The results from analytical and numerical methods are compared and it shows that
analytical method provides an acceptable accuracy.
KEYWORDS
Circular Sandwich Plate, Bending Analysis, Modeling, FEM, ABAQUS
1. INTRODUCTION
Circular and annular plates are popular in many engineering fields, including: mechanical
engineering and medical equipment’s engineering, and also they are classic subjects in the theory
of elasticity due to their relatively simple geometry. Unlike composite plates, sandwich plates are
usually composed of layers with quite different material properties and rigidities. Necessity of
using the mentioned diversity in the material properties usually stems from the simultaneous
different and in some circumstances, opposite design criteria that have to be met by the individual
layers. Gupta et al. [1] investigated the axisymmetric bending of circular sandwich panel under
traverse shear loading. Dynamic response of the circular sandwich plate was investigated under
dynamic loadings in reference [2]. In this study, the numerical finite element method was used to
verify the analytical relations. Ma and Wang [3,4] studied the post-buckling and nonlinear
bending behaviours of Functionally Graded Material circular plates based on classical plate
theory and third-order plate theory. Evaluation of stress and strain in the cells of the human spine
with the synthesized disk is a very complicated work and must be performed using numerical
simulations [5]. Sepahi [6] investigated the effect of large deflections of the FGM circular plate
under thermo-dynamic loading on a non-linear foundation using the square element. Saeedi et al.
Investigated the axi-symmetric shear and buckling of a thick FGM circular plate using the third
shear order theory [7]. Alipour M. M. and Shariyat M. studied the axisymmetric bending based
on the elasticity equilibrium equations of Zigzag theory [8]. The non-linear analysis of FGM
circular plate has been studied by Nosier A. And Fallah F. [9]. In this research the axis-symmetric

2
traverse loading was applied according to the first shear deformation theory based on non-linear
theory of Von-Karman.
In this paper, bending analysis of a circular sandwich plate under distributed load with simply
supported and clamped boundary conditions is investigated. First, the governing differential
equations are derived, and then are solved by means of Bessel function. ABAQUS software has
been used for the numerical analysis.
2. GOVERNING EQUATIONS
A circular sandwich plate with the thickness of h2 , and the core radius of a. The thickness of the
core and each of the shells is 12h and 2h , as it is seen in Figure (1).Therefore we have: 21 hhh  .
Figure 1. Cross-section of the circular sandwich panel
Regarding figure (1), the coordinates of the cylinder are: zr ,, , and in the mid-plane it is z= 0.
Two separators at the above and below surfaces are: hhz  ,1 .The two surfaces have same
materials, but the cores’ materials are different. All the quantities in the core, upper surface, and
the lower surface are shown by numbers 1 to 3, respectively. Taking into account that it is
assumed that bending of the circular plate happens in a non-symmetric manner, the displacement
along . Therefore, the displacement in the core and the shells in the cylindrical coordinates are
as in relation (1):
   
       
       
   
       
       
   






,,,
,,,,
,,,,
,,,
,,,,
,,,,
,,,
21113
21112
11
21113
21112
11
rwzrw
rhzrhzrv
rhzrhzrv
rzzrv
rhzrhzru
rhzrhzru
rzzru
i 






(1)
In the above equation we have i=1,2,3.
The total strain energy for an isotropic sandwich plate in the polar coordinate system is written as
follows:
  drrdzdV
i
a y
x
riirziziiiriri 

   

3
1 0
2
0
1
1
(2)

3
In the equation (2), the limits of the integration for xi to yi (i=1) is –h1 to h1, for (i=2) –h to –h1,
and for (i=3) h1 to h.
The strains results obtained from following equation:
   dzkkzzzQQMMNN
i
i
y
x
zisrzisiririiriiriiriiri    ,,,,,,,,,,, (3)
variations of the total potential energy of the system, using the surface load of  ,rp is:
  drwrdrpU
a


 
0
2
0
, (4)
For a system in equilibrium, variations of the total potential energy ( VUT  ) would be
insignificant with regards to the dependent parameters of 2121 ,,,, w . In other words, we
have:
0 VUT  (5)
Equilibrium equations with the use of displacement term are achieved as follows:
    
 
   
    
    02
1
1
1
131
3
21
1
3
3
21
1
1
13
1
3
2
3
3
2
,1,223
2
2
,12232111,223
2
2
,123211,2223
2
2
,12232
11112
2
132
2
21
2
132211
1





R
R
R
w
R
GGH
R
GGHGHH
R
GH
R
GHHH
R
GGH
R
GGH
GHHGGHGGHGH
H









(6)
   
      02
1
1
3
21
1
1
1
3
21
1
11
3
2
3
2
,2,222
2
2,12221
,222
2
2,12221
,1221,22
2
22
2
12
2
21
2
1221



r
RR
w
R
GH
R
GHH
R
GH
R
GHH
R
HH
R
HGHGHH






   
    
      
  0
1
2
1
1
1
131
3
21
1
1
131
3
21
1
3
3
2
3
3
2
,1,2223
2
2
,12232111,2223
2
2
,122321111,2232
2
2
,123221112
2
13
2
21
2
13211


















w
RR
GGH
R
GGHGHH
R
GGH
R
GGHGHH
R
GGH
R
GGHGHHGHGHHH
R
R

4
   
   
0
1
2
1
1
3
21
1
1
1
3
21
1
1
3
21
3
2
,2
,222
2
2,12221
,222
2
2,12221
,222
2
2122212
2
1
2
21121


















w
R
R
GH
R
GHH
R
GH
R
GHH
R
GH
R
GHHHHH
RR
 
  0
2
,11
1111
,22,232
,11,11,2,,321




















P
Rp
RR
GH
RR
Hw
R
w
R
wGHH
R
RRRR





In the above equations:
PP
aPwwPP
GGG
ahaharR
RdR
d
RdR
d
/,/
/,/,/
/,/,/
/H,/H,/
11
212111
1212111
123222111
2211
22
2
2
1








(7)
Also in the above P is the average pressure on the surface. Regarding above equations, the
following parameters could be defined:












































0
0
,2,22
0
,1,11
0
2,22
0
1,11
cos
sin
sin
cos
cos
m
m
m
RmRm
m
RmRm
m
mRm
m
mRm
mWw
mTS
R
m
mTS
R
m
mT
R
m
S
mT
R
m
S





(8)
In the equations (8), mS1 , mS2 , mT1 , mT2 , and mW are a function of R. The load can also be
definedas:
(9)
Using the above equations, the following equations achieved:

5
(10)
 
 
 
    pRpRpHaHHaGHa
GHHGHaaHaGGHaGHa
GHHaGGHaGGHHGHa
R
m
dR
d
RdR
d
TaTa
mm
mm
/,3/2,,
3/32,,,3/2
,,;3/32
1
02
2
2102193
2
28
3211
2
1751632252
2
24
221332
2
2232211
2
11
2
2
2
2
2
2
2
101
4
9





The solution of equations (10) has two parts: Complementary Solution, and Particular Solution.
In the following equation mF replaces the three variables of mS1 , mS2 , and mW [1]:
0)( 4
2
2
1  mFbb (11)
Therefore, the complementary solution is in a way that imA , imB , and imC are the coefficient of
integration.
)()(
)()(
)()(
33
2
21
33
2
212
33
2
211
RnJCRRCCW
RnJBRRBBS
RnJARRAAS
mm
m
mmm
mm
m
mmm
mm
m
mmm



(12)
And the particular solution for the equations will be reached at as:









n
mnm
n
mm
n
mnm
n
mm
n
mnm
n
mm
RKJCW
RKJBS
RKJAS
)(
)(
)(
12
11
(13)
By eliminating one of the variables, equation answer in term of Bessel functions is as follows:
)()(
)()(
25142
25141
RnJBRnJBT
RnJARnJAT
mmmmm
mmmmm

 (14)
Therefore, the final equations achieved as bellow:
)(
)2(
)(
)2(
2
28
52
275
2
18
42
174
na
A
naB
na
A
naB
m
m
m
m


(15)
The load distribution over the sector in figure (2) is uniform, then in order to Calculate n
mD for
 2 , equation (16) is resulted.
Figure 2. Choosing one point for solution of equations

6
 
 
dRRKJRpR
KJ
D
KJDdRRKRJDdRRKJRpR
mnmm
mnm
n
m
mnm
n
mmnm
n
mmnmm
)()(
)(
2
)(
2
1
)()()(
1
0
2
2
1
0
2
1
0





(16)
n
mD obtained as follows:
 
dRRKRJ
KJbRm
m
D
R
R
mnm
mnm
n
m )(
)(
)2/sin(4 2
1
2 


 (17)
If the loading on the sector section is linear, then the coefficient of n
mD is calculated herein:
    


2
1
)()1(
)()(26
)2/sin(24
2
21
2
2
2
1
R
R
mnm
mnm
n
m dRRKRJR
KJRRRRRbm
m
D 

 (18)
3. MODELING
3.1. Analytical model
The schematic of the sandwich disk is presented in figure (3). The average load P and the
dimensional values have been chosen as bellow.
Figure 3. The schematic of the sandwich disk
 240,100,50,1,5 21 KNPmmammhmmh
Table 1. Mechanical properties of different parts of the analytical model
Analytical modelE(MPa)
M450 Shells)polyester resin/glass fibres(55000.3
Core)polypropylene(8960.41
3.2. Numerical model
ABAQUS software is used for numerical analysis. Assuming that symmetrical loading, only half
of the geometry of disk is modeled and symmetric boundary conditions is used for the shear
plane. In following models 6063 S4R 4-linear node (with an approximate size of 1.5mm for
100mm disc diameter) is used.

7
Figure 4. ABAQUS model and cross section of circular sandwich
4. RESULTS
4.1. 100KN Pressure loading Analysis With Clamped boundary
The results of analysis of static stress distribution for finite element model is presented in
ABAQUS software. As can be seen with clamped conditions, for shell model, the maximum Von-
Mises stress is 164Mpa and maximum deflection is 0.87mm.
Figure 5. Stress and deflection distribution-pressure analysis under 100KN load with clamped boundry
4.2. 100KN pressure loading analysis with simply supported boundary
Regard to the results, with clamped boundary condition for shell model, maximum Von-Mises
stress is 115MPa and maximum deflection is 0.89mm. As can be seen in figure (6), compared to
the exact model with clamped boundary conditions, only the stress amount changed, but
deflection is very similar.
Figure 6. Stress and deflection distribution-pressure analysis under 100KN load with simply supported
boundry
4.3. Analytical method results
Figure (7) shows the deflection distribution. Maximum deflection with value 1.3mm occurs at
14mm away from the center of the disk; whereas in finite element method of the shell model

8
maximum deflection that happened at 10mm away from the center was 0.89mm. Figure (7) that
obtained by analytical approach, is showing decrease-increase trend in the deflection behavior.
Figure 7. Results of analytical method in MATLAB software
4.4. Comparison between the analytical and the numerical results
Figure (8) displays deflection distribution at ( 0 50 , 0, 0)r to mm z    for two numerical
analysis and analytical analysis with considering 100KN distributed load on the disk. As its
shown, results from analytical method (MATLAB) has smaller error value than shell model and
maximum deflection happens at 65mm away from center, which means 0.3 of outer radius. Also
the minimum deflection occurs at outer edge of the disk.
Figure 8. Comparison between analytical method result and results from numerical analysis from
ABAQUS at ( 0 50 , 0, 0)r to mm z   

9
5. CONCLUSIONS
In this study analytical method used to obtain deflection in circular sandwich plate with 3 layers
and the results are compared with outcomes of the finite element method which is done by
ABAQUS. According to the complexity of the equations in analytical method, only the deflection
distribution along radius of the circular sandwich plate was obtained. Using of the Bessel
functions, derivative and roots of them in expanding set of equations is the main source of error at
analytical method which minimum error obtains in this method by increasing the number of the
sentences in the Bessel function and convergence of the answer. Error in numerical method
depends on the kind and number of elements and properties of the intersection. Stress distribution
results for shell and solid models, from analysis of the bending composite disk, don’t show a
noticeable difference in a problem with different constraint type.
Also results for numerical method of shell model show that applying simply supported boundary
and clamped boundary affect stress and deflection results for the presented problem in a way that
replacing simply supported boundary with clamped boundary leads to smaller deflection and
greater stress.
REFERENCES
[1] Anand P.Gupta, Kameshwar P.Sharma. (1982). Asymmetric bending of circular sandwich plate
including transverse shear in facings, Indian J.Pure appl.Math., 13(12):1537-1550.
[2] V. S. Deshpande ,N. A. Fleck. (2004). Dynamic Response of a Clamped Circular Sandwich Plate
Subject to Shock Loading, J. Appl. Mech 71(5), 637-645.
[3] L.S. Ma, T.J. Wang. (2003). Nonlinear bending and post-buckling of a functionally graded circular
plate under mechanical and thermal loadings, Int. J. Solids Struct. 40, 3311–3330.
[4] L.S. Ma, T.J. Wang. (2004). Relationships between axisymmetric bending and buckling solutions of
FGM circular plates based on third-order plate theory and classical plate theory, Int. J. Solids Struct.
111 , 85–101.
[5] Bono.Ch., Garvin S.R. ( 2004). History and evolution of disc replacement, Spine J., 4, 145–150.
[6] Sepahi O., Forouzan M. R., Malekzadeh P. (2010). Large deflection analysis of thermo-mechanical
loaded annular FGM plates on nonlinear elastic foundation via DQM. Composite Structures, 92:
2369-2378.
[7] Saidi A.R, Rasouli A. ,Sahraee S. (2009). Axisymmetric bending and buckling analysis of FG circular
plates using third order shear deformation theory. Composite Structures, 89:110-119
[8] Alipour MM, Shariyat M. (2012). An elasticity-equilibriumbased zigzag theory for axisymmetric
bending and stress analysis of the functionally graded circular sandwich plates, using a Maclaurin-
type series solution. European Journal of Mechanics - A/Solids, 34: 78-101.
[9] Nosier A., Fallah F. (2009). Nonlinear analysis of functionally graded circular plates under
asymmetric transverse loading. International Journal of Non-Linear Mechanics, 44: 928-942.

10
AUTHORS
Hamed Vahdati Zadeh received his B.S. degree in Mechanical Engineering from
Ferdowsi University of Mashhad, Iran, in 2011. He is currently studying his M.S. degree
in Solid Mechanics. His main research interests include composite materials such as
FGM and sandwich plates.
Masoud Tahani is currently a Professor and Head of the Department of Mechanical
Engineering at Ferdowsi University of Mashhad, Iran. He received his B.S. in
Mechanical Engineering from Ferdowsi University of Mashhad, Iran, in 1995. He then
received his M.S.and Ph.D. degrees from Sharif University of Technology, Iran, in 1997
and 2003, respectively. Dr. Tahani’s research interests include design of structures using
advanced composites, mechanics of anisotropic materials, smart materials and structures,
nanomechanics, and biomechanics.

ANALYTICAL BENDING ANALYSIS OF A CIRCULAR SANDWICH PLATE UNDER DISTRIBUTED LOAD

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