Bending analysis of functionally graded ctorial plates using Levinson plate theory
S.Sahraee *
Department of Mechanical Engineering,Shahid Bahonar University of Kerman,Kerman,Iran
a r t i c l e i n f o Article history:
Available online 7July 2008Keywords:FGMs
Bending analysis Sector plate
Levinson plate theory Boundary layer function
a b s t r a c t
Bad on the Levinson plate theory (LPT)and the first-order shear deformation plate theory (FST),the bending analysis of functionally graded (FG)thick circular ctor plates is prented.The LPT solutions of FG ctorial plates are first expresd in terms of the solutions of the classical plate theory (CPT)for homogeneous ctorial plates and then prented using a direct method.It is assum
ed that the non-homogeneous mechanical properties of plate,graded through the thickness,are described by a power function of the thickness coordinate.The results are given in clod-form solutions and verified with the known data in the literature.
Ó2008Elvier Ltd.All rights rerved.
1.Introduction
Bad on considering the effect of transver shear strains through the thickness of plate two-dimensional plate theories can be categorized into two groups:(1)classical plate theory which is the simplest plate theory that neglects the effect of the transver shear deformation and (2)shear deformation plate the-ories are tho in which the effect of the transver shear strains is included.There are a number of shear deformation plate theories in literature.The simplest one is the first-order shear deformation plate theory (FST)which can be classified,depending on whether or not the expansion of displacement components or stress compo-nents through the thickness of plate is assumed to be known a pri-ori,into two types:stress-bad and displacement-bad plate theories.Reissner [1]would appear to have been the first to con-sider shear deformations in a static plate theory (stress-bad the-ories);while Mindlin [2]was the pioneer in developing dynamic plate th
eories which include the effects of transver shear defor-mations and rotary inertia (displacement-bad theories).Also,Reissner [3,4]was the first to determine that his three coupled equations can be uncoupled into two equations which called them edge-zone and interior equations.Levinson [5]prented an accu-rate simple theory for the static’s and dynamics of rectangular plates.He ud a vector approach to derive his equations of equi-librium of homogeneous plates and showed that his theory at least for one static problem provides a better approximation to the known elasticity solution than did Reissner plate theory.Wang
and Kitipornchai [6]prented an exact frequency relationship be-tween Levinson plate theory and Kirchhoff plate theory for homo-geneous plates of general polygonal shape and simply supported edges.Reddy et al.[7]derived the exact relationships between the bending solutions of the Levinson and Kirchhoff beam and plate theories bad on the load equivalence and mathematical similarity of the governing equations of the both mentioned theo-ries.Wang et al.[8]furnished a great book about shear deforma-tion theories.This book studies the relationships between the solutions of classical theories of beams and plates with tho of the first-and third-order shear deformation theories.A compre-hensive work on edge-zone equation of linear and non-linear shear deformation theories of symmetric laminated plates was done by Nosier and Reddy [9–12].Also,Nosier et al.[13]studied a bo
und-ary layer phenomenon in bending analysis of laminated circular ctor plates bad on the FST.
FGMs have attracted much attention as advanced structural materials in recent years becau of their heat-shielding proper-ties.FGMs were first introduced in 1984by a group of material sci-entists in Japan for developing thermal barrier materials [14–16].FGMs are spatial composites within which material properties vary continuously.The composition of constituent materials changes gradually usually in the thickness direction from point to point in which the materials are microscopically inhomogeneous.FGMs are made by combining two or more materials using powder metallurgy method,typically the materials are made from a mixture of ceramic and metal in which the ceramic component provides high-temperature resistance due to its low thermal con-ductivity;on the other hand,the ductile metal component pre-vents fracture due to thermal stress.FGMs were introduced as to take advantage of the desired material properties of each
0263-8223/$-e front matter Ó2008Elvier Ltd.All rights rerved.doi:10.pstruct.2008.05.014
*Address:No.36,Ali Asghare Piri Alley,Chehel Metrie Sirous Avenue,Kerman-shah,Iran.Tel.:+989188597151.
E-mail address:Shahab_ Composite Structures 88(2009)
548–557
Contents lists available at ScienceDirect
Composite Structures
j o u r n a l h o m e p a g e :w w w.e l s e v i e r.c o m /l o c a t e /c o m p s t r u c
t
constituent material which leads to smooth distribution of stres-s,without an abrupt change in the effective properties which may result in interface problems of traditional composite materi-als.FGMs are now considered as a potential structural material for high-speed spacecraft.Using thefirst-order theory of Mindlin, Reddy et al.[17]studied axisymmetric bending and stretching of functionally graded solid circular and annular plates.They pre-nted the solutions for deflections,force and moment resultants bad on thefirst-order plate theory in terms of tho obtained using the classical plate theory.As an extensive work,Ma and Wang[18]employed third-order shear deformation plate theory to solve the bending and buckling problems of functionally graded materials.They derived the relationships between the solutions of axisymmetric bending and buckling of FGMs bad on the TST and tho of homogeneous circular plates on the basis of CPT.
In the prent work,LPT is employed to analyze the pure bend-ing of functionally graded ctorial plates.First,using an analytical method,the relationships between the solutions of the bending of functionally graded circular ctor Levinson plates was derived in terms of the respons of homogeneous ctorial Kirchhoff plates. In follows,the unknown functions of LPT are again obtained using a direct method.The effects of the material distribution through the thickness and shear deform
ation on the bending of the func-tionally graded ctorial plates have been all considered.
2.Material properties
Consider a solid FGM circular ctor thickflat plate of radius b and thickness h with a subtended angle h0subjected to trans-verly pressure q as shown in Fig.1.The cylindrical coordinates (r,h,z)also shown in thefigure,is ud in the analysis.The r-coor-dinate is measured radially from the center and mid-plane of plate; h-coordinate is taken along the circumference;and the thickness coordinate z is perpendicular to the rÀh plane.The ctorial plate is simply supported at h=0and h=h0while at r=b it may be sim-ply supported,clamped or free.The material properties P of FGMs are a function of the material properties and volume fractions of the all constituent materials which can be expresd for two differ-ent constituents as[19]
P¼P c V cþP m V m;ð1Þwhere c and m refer to the ceramic and metal constituents,respec-tively;P i(i=c,m)denotes the material property of the constituent material i;and V i is the volume fraction of the constituent material i which can be expresd according to power law distribution as
V mðzÞ¼
hÀ2z
2h
n
;ð2aÞ
V cðzÞ¼1ÀV mðzÞ;ð2bÞwhere n denotes the power law index which takes values greater than or equal to zero.
Using Eqs.(1)and(2),the material properties P of plate such as, Young’s modulus E can be written as follow:
E¼ðE mÀE cÞ
hÀ2z
n
þE c:ð3Þ
Generally Poisson’s ratio m varies in a small range,for simplicity it is assumed to be a constant.It is a
pparent from Eq.(3)that the upper surface of plate(z=h/2)is purely ceramic and the lower surface (z=Àh/2)is purely metallic.
3.Equilibrium equations of Levinson plate theory
Let(U r,U h,U z)be the displacement components of an arbitrary point within the plate domain along the(r,h,z)coordinate direc-tions in cylindrical coordinates,the equilibrium equations of LPT are derived bad on the following displacementfield as
U rðr;h;zÞ¼z u rÀa z3ðu rþw;rÞ;ð4aÞU hðr;h;zÞ¼z u hÀa z3u hþ
1
w;h
;ð4bÞ
U zðr;h;zÞ¼w;ð4cÞwhere u r and u h are respectively,rotations of the middle surface (i.e.,z=0)of plate in h-and r-directions;w is the transver deflec-tion of mid-plane in z-direction;and a=4/3h2.In relations(4),a comma followed by a variable indicates partial differentiation with respect to that variable.
Bending equations of Levinson plate theory are obtained by integrating the stress equations of the three-dimensional elasticity as follow[5]:
M rr;rþ
1
r
M r h;hþM rrÀM hh
ðÞÀQ r¼0;ð5aÞM r h;rþ
1
M hh;hþ2M r h
ðÞÀQ h¼0;ð5bÞQ r;rþ
1
r
Q rþQ h;h
ÀÁ
þq¼0;ð5cÞwhere q denotes the transver load;M rr,M hh are the bending mo-ments;M r h is the twisting moment;and Q i(i=r,h)are shear forces which are all expresd in terms of the stress components through the thickness of plate by the following expressions:
M i¼
Z h=2
Àh=2
r
i
z d zði¼rr;hh;r hÞ;ð6aÞQ j¼
Z h=2
Àh=2
r
j
d zðj¼r;hÞ:ð6bÞ
At any edge of the plate with a normal~n¼n r e*
r
þn h e*
h
,the boundary conditions require the specifications of[7]
w or Q r;ð7aÞu
r
or M rr;ð7bÞu
h
or M r h:ð7cÞ
S.Sahraee/Composite Structures88(2009)548–557549
The constitutive equations of Levinson plate theory are given by
M rr¼D11u r;rþm
r
ðu rþu h;hÞ
h i
Àa F11w;rrþm
r
w;rþ
1
r
w;hh
!
;ð8aÞ
M hh¼D11m u r;rþ1
r
ðu rþu h;hÞ
!
Àa F11m w;rrþ1
r
w;rþ
1
r
w;hh
!
;ð8bÞ
M r h¼D44
r
u
r;h
Àu hþr u h;r
h i
À
a
r
2F44w;r hÀ
1
r
w;h
!
;ð8cÞ
Q r¼44½u rþw;r ;ð9aÞ
Q h¼A44u hþ1
r
w;h
!
;ð9bÞ
where A11,B11,etc.are the plate stiffness coefficients defined as
ðD11;F11Þ¼
Z h=2
Àh=2
E
1Àm2
ðz2;z4Þd z;ð10aÞ
ðA44;D44;E44Þ¼
Z h=2
Àh=2
E
2ð1þmÞ
ð1;z2;z4Þd z;ð10bÞ
also
11
¼D11Àa F11;
44
¼m111=2;
A44¼A44Àb D44;
with
m1¼1Àm;b¼4=h2:
4.Equilibrium equations of Kirchhoff plate theory
The classical thin plate theory is bad on following displace-mentfield[20]
U rðr;h;zÞ¼Àzw K
;r
;ð11aÞ
U hðr;h;zÞ¼Àz
r
w K
;h
;ð11bÞ
U zðr;h;zÞ¼w K;ð11cÞwhere w K is the transver deflection of a point on the mid-plane The above displacements is bad on Kirchhoff hypothesis,implies that straight lines normal to rÀh plane before deformation remain straight and normal to the mid-plane after deformation which ne-glects both transver shear and traver normal effects.
Using the principle of virtual displacements,equilibrium equa-tions of CPT are derived as
M K
rr;r þ
1
r
M K
r h;h
þM K
rr
ÀM K
hh
ÀQ K
r
¼0;ð12aÞ
M K
r h;r þ
1
r
M K
h;hh
þ2M K
r h
ÀQ K
h
¼0;ð12bÞ
Q K
r;r þ
1
r
Q K
r
þQ K
h;h
þq¼0:ð12cÞ
The constitutive equations for classical Kirchhoff plate theory are given by
M K
rr ¼ÀD w K
;rr
þ
m
w K
;r
þ
1
w K
;hh
!
;ð13aÞ
M K
hh ¼ÀD m w K;rrþ
1
w K
;r
þ
1
w K
;hh
!
;ð13bÞ
M r h¼ÀD m1
1
r
w;h
!
;r
;ð13cÞ
Q K
r
¼ÀDðr2w KÞ;r;ð14aÞ
Q K
h
¼À
1
r
Dðr2w KÞ
;h
;ð14bÞ
where superscript K refers to quantities in the Kirchhoff plate the-
ory and r2is the two-dimensional Laplace operator in polar coor-
dinate defined as
r2ðÁÞ¼ðÁÞ
;rr
þ
1
r
ðÁÞ
;r
þ
1
r2
ðÁÞ
;hh
:
also D is the usual plate bending rigidity as follow:
D¼
Eh3
m:
5.Bending analysis
The governing equations of equilibrium of CPT may be ex-
presd in terms of the moment sum M K and similarly equilibrium
equations of Levinson plate theory may be written in terms of mo-
ment sum M which are both defined by the following expressions:
M K¼
M K
rr
þM K
hh
1þm
;ð15aÞ
M¼
M rrþM hh
1þm
:ð15bÞ
The moment sums(15)are expresd in terms of the displacement
functions as
M K¼ÀD r2w K;ð16aÞ
M¼ D11—kÀa F11r2w;ð16bÞ
where
—k¼u r;rþ
1
ðu rþu h;hÞ:
From the equilibrium equations of the both theory one can easily
obtain the following expressions:
r2M K¼Àq;ð17aÞ
r2M¼Àq:ð17bÞ
In view of the load equivalence(Eq.(17)),one can arrive at the fol-
lowing moment sums relationship as
M¼M KþD11r2N;ð18Þ
where N is a function such that it satisfies the biharmonic equation
as follow:
r4N¼0:ð19Þ
Substitution of Eqs.(9a)and(9b)into Eq.(5c),yields
—
k¼
44
r2w:ð20Þ
Introduction of the above expression into Eq.(16b),gives
M¼ÀD11r2wÀ11
A44
q:ð21Þ
From Eqs.(16a),(17a),(18),and(21)the following relationship be-
tween the deflection,w,of FG circular ctor plates bad on the
Levinson plate theory and the deflection,w K,of homogeneous c-
torial plates bad on the classical Kirchhoff plate theory can be ob-
tained as
550S.Sahraee/Composite Structures88(2009)548–557
w¼
D
11w Kþ
D11
1144
M KÀNþW;ð22Þ
where W is a function which satisfies the harmonic equation as
r2W¼0:ð23ÞFrom Eqs.(5a),(5b),(8a)–(8c)and(16b)one then obtains
Q r¼M;rþm1D11
2r
U;h;ð24aÞ
Q h¼1
r
M;hÀ
m111
2
U;r;ð24bÞ
where U is a potential function referred to as the boundary layer function as follow
U¼1
r
ðu r;hÀu hþr u h;rÞ:ð25Þ
Using Eqs.(9a),(9b),(18),(22)and(24)one can obtain the rotation functions of Levinson plate theory as
u
r ¼À
D
11
w K
;r
þE1M K
;r
þC;rþ
E2
U;h;ð26aÞ
u
h ¼À
D
D11
1
r
w K
;h
þE1
1
r
M K
;h
þ
1
r
C;hÀE2U;r;ð26bÞ
where
E1¼
a F11
D1144
;E2¼
m1D11
244
;
C¼D11
A44
r2NþNÀW:
Introduction of Eqs.(22)and(26)into Eqs.(8)and(9)yield
M rr¼M K
rr
þm111E2RðUÞþm111C;rrþD11r2N;ð27aÞ
M hh¼M K
hh
Àm1D11E2RðUÞÀD11m1C;rrþD11r2N;ð27bÞ
M r h¼M K
r h þm1D11RðCÞþ
m1
2
D11XþD11r2N;ð27cÞ
Q r¼Q K
r þD11r2ðr2NÞ;rþ
E2
A44U;h;ð28aÞ
Q h¼Q K
h þ
1
r
D11r2ðr2NÞ;hÀE2A44U;r;ð28bÞ
where R operator is defined as
RðÁÞ¼1
r
ðÁÞ
;
r h
À
1
r2
ðÁÞ
;h
;
also
X¼1
r
U;rþ
1
r2
U;hhÀU;rr;C¼
D11
A44
r2NþNÀW:
Eliminating the moment sum M between Eq.(24)and introducing the Eqs.(9a)and(9b)into the resulted equation,one then obtains the following expression referred to as the edge-zone equation[9] as
r2U¼c2U;ð29Þwhere
c¼
ffiffiffiffiffiffiffiffiffiffiffiffi
2A44 m111 s
:
5.1.Levy solutions using relationships
In this ction,the bending problem of functionally graded cir-cular ctor plates subjected to transverly uniform distributed load q0will be studied.Assuming the simply supported radial edges and bad on the Levy method of solution,the load distribu-tion and displacement functions for both Kirchhoff and Levinson plate theories;and boundary layer function of LPT may be repre-nted as
qðr;hÞ¼
X1
m¼1
q
m
sin a m h;ð30aÞf wðr;hÞ;w Kðr;hÞg¼
X1
m¼1
f w mðrÞ;w K
m
ðrÞg sin a m h;ð30bÞf u rðr;hÞ;u hðr;hÞg¼
X1
m¼1
f u rmðrÞ;u h mðrÞ
g sin a m h;ð30cÞUðr;hÞ¼
X1
m¼1
U mðrÞcos a m h;ð30dÞwhere
q
m
¼
2q
m p
½1ÀðÀ1Þm :
Introduction of Eq.(30d)into Eq.(29)yields the following modified
Besl function as:
d2U m
þ
d U m
À
a2
mþc2
U m¼0:ð31ÞThe general solution of Eq.(31)is given by
U m¼B1m I a
m
ðcrÞþB2m K a
m
ðcrÞ;ð32Þ
where B1m and B2m are constants;I a
m
and K a
m
are the modified Bes-l functions of thefirst and cond kinds,respectively.To havefi-nite displacement functions at r=0it is deduced that B2m=0and therefore
U m¼B1m I a
m
ðcrÞ:ð33ÞFor the ctorial plates considered here,the function r2N and W are given by
r2N¼
X1
m¼1
A1m r a m sin a m h;ð34aÞW¼
X1
m¼1
A2m r a m sin a m h;ð34bÞ
where A1m and A2m are constants which both with B1m to be deter-mined using the boundary conditions at the edge r=b.
In view of the above derived relationships for the displacement functions and stress resultants between the two theories,the solu-tions of FG ctorial Levinson plates in terms of the solution of homogeneous Kirchhoff plates may be summarized below.
Deflection relationship
w¼
D
D11
w Kþ11
D11A44
M KÀ
X1
m¼1
r2A1m
4ða mþ1Þ
þA2m
&'
r a m
Âsin a m h:ð35ÞRotation–slope relationships
u
r
¼À
D
D11
w K
;r
þE1M K
;r
þ
X1
m¼1
a m D11
44
r a mÀ1þ
a mþ2
4ða mþ1Þ
r a mþ1
!
A1m
&
Àa m r a mÀ1A2mÀ
a m
c2r
I a
m
ðcrÞB1m
'
sin a m h;ð36aÞ
u
h
¼À
D
11
w K
;h
þ
E1
M K
;h
þ
X1
m¼1
a m D11
44
r a mÀ1þ
r a mþ1
a m
!
A1m
&
Àa m r a mÀ1A2mÀ
1
c
I0a
m
ðcrÞB1m
'
cos a m h;ð36bÞ
S.Sahraee/Composite Structures88(2009)548–557551
Moment relationships
M rr¼M K
rr þm1D11
X1
m¼1
a m D11
r244
þ
a mþ2
þ
v
m1
"#
r a m A1m
(
À a m r a mÀ2A2mþD11
11
a m
I
a mðcrÞÀI0
a m
ðcrÞ
!
B1m
)
sin a m h;
ð37aÞ
M hh¼M K
hh ÀD11m1
X1
m¼1
a m11
r2A44
þ
a mþ2
4
À
1
m1
"#
r a m A1m
(
À a m r a mÀ2A2mþ11
D11
a m
cr
I
a mðcrÞ
cr
ÀI0a
m
ðcrÞ
!
B1m
)
sin a m h;
ð37bÞ
M r h¼M K
r h þD11m1
X1
m¼1
a m D11
A44
r a mÀ2þ
a m
4
r a m
"#
A1mÀ a m r a mÀ2A2m (
þD11
D11
I0a
m
ðcrÞ
cr
Àl m I a mðcrÞ
!
B1m
)
cos a m h;ð37cÞ
where
a m¼a mða mÀ1Þ;
l m ¼
1
þ
a m
2!
:
Shear–force relationships
Q r¼Q K
r þ
X1
m¼1
a m D11r a mÀ1A1mÀA44
c2r
I a
m
ðcrÞB1m
()
sin a m h;ð38aÞ
Q h¼Q K
h þ
X1
m¼1
a m D11r a mÀ1A1mÀA44I0a
m
ðcrÞB1m
()
cos a m h:ð38bÞ
This completes the derivation of the relationships between the solu-tions of FG ctorial plates bad on the LPT in terms of the associ-ated quantities of homogeneous circular ctor plates on the basis of CPT.In what follows;a solid circular ctor plate with various boundary conditions at its circular edge will be considered.
5.1.1.Simply supported circular ctor plates(SSS)
Suppo a solid circular ctor plate with a simply support at r=b.The boundary conditions at r=b are
w K¼0;M K
rr
¼0;ð39aÞw¼0;M rr¼0;u h¼0:ð39bÞSatisfying the above boundary conditions at r=b give
A1m¼
H m a m
b2
1Àa m I1
ÂÃ
b a m a m
cb
ÀÁ22
m1
I1À1a
m
þ2a mþ1
2ða mþ1Þ
þm m
1
h i;ð40aÞ
A2m¼ÀbÀa m H mþ
b2
a m A1m;ð40bÞ
B1m¼
c a m
bI0a
m
ðcbÞ
D11
11
H mþ
D11
44
b a m A1m
;ð40cÞ
where
H m¼
D11
D11A44
M K
m
j
r¼b
;
I1¼
I a
m
ðcbÞbcI0a
m
ðcbÞ
:
The CPT solution of the deflection of a SSS homogeneous plate un-der a uniform load q0is given by[8,23]w K¼
1X1
m¼1
q
m
c
m
2þ
ð2Àa mÞða mþ5þmÞ
a m m
r a mÀ4
&
À
ð4Àa mÞða mþ3þmÞ
2a mþ1þm
r
b
a mÀ2'
r4sin a m h;ð41Þwhere
c
m
¼ð16Àa2mÞð4Àa2mÞ:
5.1.2.Clamped circular ctor plates(SSC)
Suppo a solid circular ctor plate which is clamped at r=b. The boundary conditions at r=b are
w K¼0;w K
;r
¼0;ð42aÞw¼0;u r¼0;u h¼0:ð42bÞSatisfying the above boundary conditions at r=b give
A1m¼
Àa m
b2
þI2D11
11
H mþI2
a m
D11
11
E1 H m
b a m a m
ðcbÞ2
2
m1
1Àb a
m
2
ÀbD11I2
a m11a m
h i;ð43aÞA2m¼ÀbÀa m H mþ
b2
4ða mþ1Þ
A1m;ð43bÞB1m¼
c2
a m a
m
a m D11
44
þ
b2
a m
"#
b a m A1mþa m H mþbE1 H m
!
;
ð43cÞwhere
H m
D
D11A44
K
m
j
r¼b
;
I2¼
cI0a
m
ðcbÞ
I a
m
ðcbÞ
:
The CPT solution of the deflection of a SSC homogeneous plate un-der a uniform load q0is given by[8]
w K¼
1
D
X1
m¼1
q
m
2c m
2þð2Àa mÞ
r
b
a mÀ4
Àð4Àa mÞ
r
b
a mÀ2
&'Âr4sin a m h;ð44Þ
5.1.3.Free circular ctor plates(SSF)
Suppo a solid circular ctor plate which is free at r=b.The boundary conditions at r=b are
Q K
r
þ
1
r
M K
r h;h
¼0;M K
rr
¼0;ð45aÞQ r¼0;M rr¼0;M r h¼0:ð45bÞSatisfying the above boundary conditions at r=b give
A1m¼
M K
r h
j
r¼b
þ2II1þ2a
m
II2
Q K
rm
j
r¼b
D11b a mð2ÀmÞÀ2b½a m II1þII2
ÀÁ;ð46aÞA2m¼
b2
a m
a m
b2
11
A44
þ
a m
4
þ
2
b m1
II2
!
A1m
þ
b2Àa m
a m m111M
K
r h
j
r¼b
þ
2
a m II2Q
K
rm
j
r¼b
;ð46bÞ
B1m
a m D44I a
m
ðcbÞ
Q K
rm
j
r¼b
þD11a m b a mÀ1A1m
;ð46cÞ
552S.Sahraee/Composite Structures88(2009)548–557
