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Bending analysis of functionally graded sectorial plates using Levinson plate theory
Composite Structures 88 (2009) 548–557
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Bending analysis of functionally graded sectorial 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 7 July 2008 Keywords: FGMs Bending analysis Sector plate Levinson plate theory Boundary layer function
a b s t r a c t Based on the Levinson plate theory (LPT) and the first-order shear deformation plate theory (FST), the bending analysis of functionally graded (FG) thick circular sector plates is presented. The LPT solutions of FG sectorial plates are first expressed in terms of the solutions of the classical plate theory (CPT) for homogeneous sectorial plates and then presented using a direct method. It is assumed that the nonhomogeneous mechanical properties of plate, graded through the thickness, are described by a power function of the thickness coordinate. The results are given in closed-form solutions and verified with the known data in the literature. Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction Based on considering the effect of transverse 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 transverse shear deformation and (2) shear deformation plate theories are those in which the effect of the transverse 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 components through the thickness of plate is assumed to be known a priori, into two types: stress-based and displacement-based plate theories. Reissner [1] would appear to have been the first to consider shear deformations in a static plate theory (stress-based theories); while Mindlin [2] was the pioneer in developing dynamic plate theories which include the effects of transverse shear deformations and rotary inertia (displacement-based 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] presented an accurate simple theory for the static’s and dynamics of rectangular plates. He used a vector approach to derive his equations of equilibrium 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 * Address: No. 36, Ali Asghare Piri Alley, Chehel Metrie Sirous Avenue, Kermanshah, Iran. Tel.: +98 9188597151. E-mail address: [email protected] 0263-8223/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2008.05.014
and Kitipornchai [6] presented an exact frequency relationship between Levinson plate theory and Kirchhoff plate theory for homogeneous 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 based on the load equivalence and mathematical similarity of the governing equations of the both mentioned theories. Wang et al. [8] furnished a great book about shear deformation theories. This book studies the relationships between the solutions of classical theories of beams and plates with those of the first- and third-order shear deformation theories. A comprehensive 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 boundary layer phenomenon in bending analysis of laminated circular sector plates based on the FST. FGMs have attracted much attention as advanced structural materials in recent years because of their heat-shielding properties. FGMs were first introduced in 1984 by a group of material scientists 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 these materials are microscopically inhomogeneous. FGMs are made by combining two or more materials using powder metallurgy method, typically these materials are made from a mixture of ceramic and metal in which the ceramic component provides high-temperature resistance due to its low thermal conductivity; on the other hand, the ductile metal component prevents fracture due to thermal stress. FGMs were introduced as to take advantage of the desired material properties of each
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S. Sahraee / Composite Structures 88 (2009) 548–557
constituent material which leads to smooth distribution of stresses, without an abrupt change in the effective properties which may result in interface problems of traditional composite materials. FGMs are now considered as a potential structural material for high-speed spacecraft. Using the first-order theory of Mindlin, Reddy et al. [17] studied axisymmetric bending and stretching of functionally graded solid circular and annular plates. They presented the solutions for deflections, force and moment resultants based on the first-order plate theory in terms of those 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 based on the TST and those of homogeneous circular plates on the basis of CPT. In the present work, LPT is employed to analyze the pure bending of functionally graded sectorial plates. First, using an analytical method, the relationships between the solutions of the bending of functionally graded circular sector Levinson plates was derived in terms of the responses of homogeneous sectorial 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 deformation on the bending of the functionally graded sectorial plates have been all considered.
Using Eqs. (1) and (2), the material properties P of plate such as, Young’s modulus E can be written as follow:
n h 2z E ¼ ðEm Ec Þ þ Ec : 2h
Generally Poisson’s ratio m varies in a small range, for simplicity it is assumed to be a constant. It is apparent from Eq. (3) that the upper surface of plate (z = h/2) is purely ceramic and the lower surface (z = h/2) is purely metallic. 3. Equilibrium equations of Levinson plate theory Let (Ur, Uh, Uz) be the displacement components of an arbitrary point within the plate domain along the (r, h, z) coordinate directions in cylindrical coordinates, the equilibrium equations of LPT are derived based on the following displacement field as
U r ðr; h; zÞ ¼ zur az3 ður þ w;r Þ; 1 U h ðr; h; zÞ ¼ zuh az3 uh þ w;h ; r
ð4aÞ ð4bÞ
U z ðr; h; zÞ ¼ w;
ð4cÞ
where ur and uh are respectively, rotations of the middle surface (i.e., z = 0) of plate in h- and r-directions; w is the transverse deflection 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]:
2. Material properties Consider a solid FGM circular sector thick flat plate of radius b and thickness h with a subtended angle h0 subjected to transversely pressure q as shown in Fig. 1. The cylindrical coordinates (r, h, z) also shown in the figure, is used in the analysis. The r-coordinate 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 r h plane. The sectorial plate is simply supported at h = 0 and h = h0 while at r = b it may be simply 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 expressed for two different constituents as [19]
P ¼ Pc V c þ Pm V m ;
ð3Þ
1 Mrr;r þ ðM rh;h þ M rr M hh Þ Q r ¼ 0; r 1 Mrh;r þ ðM hh;h þ 2M rh Þ Q h ¼ 0; r 1 Q r;r þ Q r þ Q h;h þ q ¼ 0; r
ð5aÞ ð5bÞ ð5cÞ
where q denotes the transverse load; Mrr, Mhh are the bending moments; Mrh is the twisting moment; and Qi (i = r, h) are shear forces which are all expressed in terms of the stress components through the thickness of plate by the following expressions:
ð1Þ
where c and m refer to the ceramic and metal constituents, respectively; Pi (i = c, m) denotes the material property of the constituent material i; and Vi is the volume fraction of the constituent material i which can be expressed according to power law distribution as
Mi ¼ Qj ¼
Z
Z
h=2
ri z dz ði ¼ rr; hh; rhÞ;
ð6aÞ
h=2 h=2
rj dz ðj ¼ r; hÞ:
ð6bÞ
h=2
n h 2z V m ðzÞ ¼ ; 2h
ð2aÞ
At any edge of the plate with a normal ~ n ¼ nr e þnh e , the boundary r h conditions require the specifications of [7]
V c ðzÞ ¼ 1 V m ðzÞ;
ð2bÞ
w or Q r ;
ð7aÞ
ur or Mrr ; uh or Mrh :
ð7bÞ
*
where n denotes the power law index which takes values greater than or equal to zero.
ð7cÞ
b Simply supported radial edge
O
θ r
z
h /2
q
θ =0
θ = θ0
*
Circular edge may be simply supported, clamped or free
Fig. 1. A sectorial plate with simple supports along its radial edges.
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S. Sahraee / Composite Structures 88 (2009) 548–557
The constitutive equations of Levinson plate theory are given by
h
i
m
M rr ¼ D11 ur;r þ ður þ uh;h Þ r m 1 aF 11 w;rr þ w;r þ w;hh ; r r M hh
1 ¼ D11 mur;r þ ður þ uh;h Þ r 1 1 aF 11 mw;rr þ w;r þ w;hh ; r r
ð8aÞ
i a D44 h 1 M rh ¼ ur;h uh þ ruh;r 2F 44 w;rh w;h ; r r r
ð8bÞ
Mrh ¼ Dm1
1 w;h r
ð13cÞ
; ;r
Q Kr ¼ Dðr2 wK Þ;r ;
ð14aÞ
1 Q Kh ¼ Dðr2 wK Þ;h ; r
ð14bÞ
where superscript K refers to quantities in the Kirchhoff plate theory and r2 is the two-dimensional Laplace operator in polar coordinate defined as
1 r
r2 ðÞ ¼ ðÞ;rr þ ðÞ;r þ
1 ðÞ : r 2 ;hh
ð8cÞ
also D is the usual plate bending rigidity as follow:
ð9aÞ
D¼
3
Q r ¼ A44 ½ur þ w;r ; 1 Q h ¼ A44 uh þ w;h ; r
Eh : 12ð1 m2 Þ
ð9bÞ 5. Bending analysis
where A11, B11, etc. are the plate stiffness coefficients defined as
ðD11 ; F 11 Þ ¼
Z
h=2
h=2
ðA44 ; D44 ; E44 Þ ¼
Z
E ðz2 ; z4 Þdz; 1 m2 h=2
h=2
E ð1; z2 ; z4 Þdz; 2ð1 þ mÞ
ð10aÞ ð10bÞ
The governing equations of equilibrium of CPT may be expressed in terms of the moment sum MK and similarly equilibrium equations of Levinson plate theory may be written in terms of moment sum M which are both defined by the following expressions:
MKrr þ M Khh ; 1þm M rr þ M hh : M¼ 1þm
MK ¼
also
D11 ¼ D11 aF 11 ; D44 ¼ m1 D11 =2;
ð15aÞ ð15bÞ
The moment sums (15) are expressed in terms of the displacement functions as
A44 ¼ A44 bD44 ; with
MK ¼ Dr2 wK ; 11 — M¼D k aF 11 r2 w;
2
m1 ¼ 1 m; b ¼ 4=h :
ð16aÞ ð16bÞ
where 4. Equilibrium equations of Kirchhoff plate theory The classical thin plate theory is based on following displacement field [20]
U r ðr; h; zÞ ¼
zwK;r ;
z U h ðr; h; zÞ ¼ wK;h ; r U z ðr; h; zÞ ¼ wK ;
ð11aÞ ð11bÞ ð11cÞ
where wK is the transverse deflection of a point on the mid-plane The above displacements is based on Kirchhoff hypothesis, implies that straight lines normal to r h plane before deformation remain straight and normal to the mid-plane after deformation which neglects both transverse shear and traverse normal effects. Using the principle of virtual displacements, equilibrium equations of CPT are derived as
1 K M Krr;r þ Mrh;h þ M Krr MKhh Q Kr ¼ 0; r 1 K K Mh;hh þ 2MKrh Q Kh ¼ 0; M rh;r þ r 1 K K Q r þ Q Kh;h þ q ¼ 0: Q r;r þ r
1 k ¼ ur;r þ ður þ uh;h Þ: — r From the equilibrium equations of the both theory one can easily obtain the following expressions:
r2 MK ¼ q;
ð17aÞ
2
r M ¼ q:
ð17bÞ
In view of the load equivalence (Eq. (17)), one can arrive at the following moment sums relationship as
M ¼ M K þ D11 r2 N;
ð18Þ
where N is a function such that it satisfies the biharmonic equation as follow:
r4 N ¼ 0:
ð19Þ
ð12aÞ
Substitution of Eqs. (9a) and (9b) into Eq. (5c), yields
ð12bÞ
k¼ —
ð12cÞ
Introduction of the above expression into Eq. (16b), gives
q A44
r2 w:
ð20Þ
D11
The constitutive equations for classical Kirchhoff plate theory are given by
M ¼ D11 r2 w
m K 1 K M Krr ¼ D wK;rr þ w;r þ w;hh ; r r 1 1 M Khh ¼ D mwK;rr þ wK;r þ wK;hh ; r r
From Eqs. (16a), (17a), (18), and (21) the following relationship between the deflection, w, of FG circular sector plates based on the Levinson plate theory and the deflection, wK, of homogeneous sectorial plates based on the classical Kirchhoff plate theory can be obtained as
ð13aÞ ð13bÞ
A44
q:
ð21Þ
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S. Sahraee / Composite Structures 88 (2009) 548–557
w¼
D K D11 w þ MK N þ W; D11 D11 A44
ð22Þ
where W is a function which satisfies the harmonic equation as
r2 W ¼ 0:
ð23Þ
From Eqs. (5a), (5b), (8a)–(8c) and (16b) one then obtains
Q r ¼ M ;r þ
m1 D11 2r
U;h ;
ð24aÞ
1 r
U ¼ ður;h uh þ r uh;r Þ:
fwðr; hÞ; wK ðr; hÞg ¼ fur ðr; hÞ; uh ðr; hÞg ¼
ð30aÞ
1 X
fwm ðrÞ; wKm ðrÞg sin am h;
m¼1 1 X
ð30bÞ
furm ðrÞ; uhm ðrÞg sin am h;
ð30cÞ
m¼1
Uðr; hÞ ¼
1 X
Um ðrÞ cos am h;
ð30dÞ
m¼1
ð25Þ where
Using Eqs. (9a), (9b), (18), (22) and (24) one can obtain the rotation functions of Levinson plate theory as
D K E2 w þ E1 M K;r þ C;r þ U;h ; D11 ;r r D 1 K 1 K 1 uh ¼ w þ E1 M ;h þ C;h E2 U;r ; D11 r ;h r r
ur ¼
aF 11 D11 A44 D11 2 A44
;
E2 ¼
m1 D11 2A44
ð26bÞ
2 2 d Um dUm am 2 þ þ c Um ¼ 0: dr 2 r dr r2
Um ¼ B1m Iam ðcrÞ þ B2m K am ðcrÞ;
M rr ¼ M Krr þ m1 D11 E2 RðUÞ þ m1 D11 C;rr þ D11 r2 N; 2
m1 D11 E2 RðUÞ D11 m1 C;rr þ D11 r N;
M rh ¼ M Krh þ m1 D11 RðCÞ þ
m1 2
Q r ¼ Q Kr þ D11 r2 ðr2 NÞ;r þ
D11 X þ D11 r2 N;
E2 A44 U;h ; r
1 Q h ¼ Q Kh þ D11 r2 ðr2 NÞ;h E2 A44 U;r ; r
ð27aÞ ð27bÞ ð27cÞ
ð28bÞ
C¼
D11 A44
ð29Þ
A1m r am sin am h;
ð34aÞ
m¼1
W¼
1 X
A2m r am sin am h;
ð34bÞ
w¼
1 X D K D11 r 2 A1m w þ MK þ A2m ram D11 4ðam þ 1Þ D11 A44 m¼1 sin am h:
ð35Þ
Rotation–slope relationships
1 X D K D a þ 2 am þ1 w;r þ E1 M K;r þ am 11 ram 1 þ m r A1m D11 4ðam þ 1Þ A44 m¼1 am am r am 1 A2m 2 Iam ðcrÞB1m sin am h; ð36aÞ c r
ur ¼
where
m1 D11
1 X
where A1m and A2m are constants which both with B1m to be determined 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 solutions of FG sectorial Levinson plates in terms of the solution of homogeneous Kirchhoff plates may be summarized below. Deflection relationship
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
sffiffiffiffiffiffiffiffiffiffiffiffi 2A44
For the sectorial plates considered here, the function r N and W are given by
m¼1
r2 N þ N W :
r2 U ¼ c2 U;
ð33Þ 2
ð28aÞ
also
1 U;hh U;rr ; r2
Um ¼ B1m Iam ðcrÞ:
r2 N ¼
1 1 RðÞ ¼ ðÞ;rh 2 ðÞ;h ; r r
1 r
ð32Þ
where B1m and B2m are constants; Iam and K am are the modified Bessel functions of the first and second kinds, respectively. To have finite displacement functions at r = 0 it is deduced that B2m = 0 and therefore
where R operator is defined as
X ¼ U;r þ
ð31Þ
The general solution of Eq. (31) is given by
;
Introduction of Eqs. (22) and (26) into Eqs. (8) and (9) yield
M hh ¼
2q0 ½1 ð1Þm : mp
Introduction of Eq. (30d) into Eq. (29) yields the following modified Bessel function as:
r N þ N W:
M Khh
m ¼ q
ð26aÞ
where
c¼
m sin am h; q
ð24bÞ
where U is a potential function referred to as the boundary layer function as follow
C¼
1 X
qðr; hÞ ¼
m¼1
1 m1 D11 U;r ; Q h ¼ M;h r 2
E1 ¼
load q0 will be studied. Assuming the simply supported radial edges and based on the Levy method of solution, the load distribution and displacement functions for both Kirchhoff and Levinson plate theories; and boundary layer function of LPT may be represented as
:
5.1. Levy solutions using relationships In this section, the bending problem of functionally graded circular sector plates subjected to transversely uniform distributed
1 X D E1 D r am þ1 A1m wK;h þ M K;h þ am 11 ram 1 þ rD11 r 4ðam þ 1Þ A44 m¼1 1 ð36bÞ am r am 1 A2m I0am ðcrÞB1m cos am h; c
uh ¼
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S. Sahraee / Composite Structures 88 (2009) 548–557
Moment relationships
M rr ¼
M Krr
1 X
("
am þ 2
D11
wK ¼
#
v
r am A1m þ þ 4 m1 r2 A44 ) D11 am Iam ðcrÞ 0 a 2 m mr a A2m þ Iam ðcrÞ B1m sin am h; cr D11 cr þ m1 D11
a m
m¼1
ð37aÞ M hh ¼ M Khh D11 m1
("
1 X
D11
am þ 2
1
#
r am A1m 4 m1 r2 A44 m¼1 ) D11 am Iam ðcrÞ 0 am 2 am r A2m þ Iam ðcrÞ B1m sin am h; cr D11 cr
a m
þ
ð37bÞ M Krh
M rh ¼
þ
þ D11 m1
D11 D11
1 X
("
a m
D11
r
am 2
þ
am
r
4 ) Iam ðcrÞ lm Iam ðcrÞ B1m cos am h; cr m¼1
A44
mr A1m a
am 2
A2m
where
cm ¼ ð16 a2m Þð4 a2m Þ: 5.1.2. Clamped circular sector plates (SSC) Suppose a solid circular sector plate which is clamped at r = b. The boundary conditions at r = b are
wK ¼ 0; w ¼ 0;
A1m
0
ð37cÞ
m¼1
Q h ¼ Q Kh þ
1 X m¼1
am D11 ram 1 A1m
A44 Ia ðcrÞB1m c2 r m
(
A am D11 ram 1 A1m 44 I0am ðcrÞB1m c
) sin am h; ) cos am h:
ð38bÞ
ð39aÞ
uh ¼ 0:
ð39bÞ
Satisfying the above boundary conditions at r = b give
A1m
1 a m I1 b i; ¼ a h 2 þ mm1 b m acbm m21 I1 a1m þ 2ð2aamm þ1 þ1Þ
A2m ¼ b
B1m
am
am 2
ð40aÞ
2
Hm þ
"
am
D11 A44
A1m ;
ð43bÞ 2
þ
b
#
! a
2ðam þ 1Þ
m ; b m A1m þ am Hm þ bE1 H
D11 M Km jr¼b ; D11 A44 cI0 ðcbÞ I 2 ¼ am : Iam ðcbÞ
Hm ¼
M Krr ¼ 0;
Hm
4ðam þ 1Þ
where
The CPT solution of the deflection of a SSC homogeneous plate under a uniform load q0 is given by [8]
b A1m ; 4ðam þ 1Þ
cam D11 D11 am ¼ 0 Hm þ b A1m ; bIam ðcbÞ D11 A44
ð40bÞ
ð40cÞ
1 r am 4 r am 2 m q 1X 2 þ ð2 am Þ ð4 am Þ D m¼1 2cm b b
wK ¼
5.1.1. Simply supported circular sector plates (SSS) Suppose a solid circular sector plate with a simply support at r = b. The boundary conditions at r = b are
Mrr ¼ 0;
2
ð38aÞ
This completes the derivation of the relationships between the solutions of FG sectorial plates based on the LPT in terms of the associated quantities of homogeneous circular sector plates on the basis of CPT. In what follows; a solid circular sector plate with various boundary conditions at its circular edge will be considered.
wK ¼ 0;
b
ð43aÞ
ð43cÞ
Shear–force relationships 1 X
Hm þ
c2 am Iam ðcbÞ
rc
Q r ¼ Q Kr þ
ð42bÞ
m abm2 þ Ib2 DD11 Hm þ aI2m DD11 E1 H 11 11 i; ¼ a h bD11 I2 am 2 b b m ðcbÞ 1 I 2 2 m a 2 a D ð1þ a Þ m m 11 m 1
B1m ¼
(
ð42aÞ
ur ¼ 0; uh ¼ 0:
am
a m ¼ am ðam 1Þ; 1 a 2 lm ¼ þ m :
w ¼ 0;
wK;r ¼ 0;
A2m ¼ b
where
2
ð41Þ
Satisfying the above boundary conditions at r = b give
# am
1 m q 1X ð2 am Þðam þ 5 þ mÞ r am 4 2þ D m¼1 2cm 2am þ 1 þ m b ð4 am Þðam þ 3 þ mÞ r am 2 4 r sin am h; 2am þ 1 þ m b
r 4 sin am h;
5.1.3. Free circular sector plates (SSF) Suppose a solid circular sector plate which is free at r = b. The boundary conditions at r = b are
1 Q Kr þ MKrh;h ¼ 0; r Q r ¼ 0;
ð45aÞ
M rh ¼ 0:
ð45bÞ
A1m
MKrh jr¼b þ 2II1 þ a2m II2 Q Krm jr¼b ; ¼ a D11 b m ð2 mÞ 2b ½am II1 þ II2
A2m ¼
Hm ¼
The CPT solution of the deflection of a SSS homogeneous plate under a uniform load q0 is given by [8,23]
Mrr ¼ 0;
M Krr ¼ 0;
Satisfying the above boundary conditions at r = b give
where
D11 M Km jr¼b ; D11 A44 Ia ðcbÞ : I1 ¼ m0 bcIam ðcbÞ
ð44Þ
B1m ¼
ð46aÞ
! 2 m D11 am b a 2 A þ II þ a m b2 A44 4 bm1 2 1m 2a b m 2 þ MKrh jr¼b þ II2 Q Krm jr¼b ; a m m1 D11 am b
am D44 Iam ðcbÞ
Q Krm jr¼b þ D11 am b
am 1
A1m ;
ð46bÞ
ð46cÞ
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S. Sahraee / Composite Structures 88 (2009) 548–557
where
Introduction of Eq. (30d) into Eq. (49a), yields the differential equation (31) which its solution given in Eq. (32) and substitution of Eqs. (30a) and (30b) into Eq. (49b), yields
0
Iam ðcbÞ 1 ; cIam ðcbÞ bc2 I0 ðcbÞ blm : II2 ¼ am cIam ðcbÞ II1 ¼
2
d 1 d a2m þ 2 r dr r 2 dr
The CPT solution of the deflection of a SSF homogeneous plate under a uniform load q0 is given by [8] ( 1 m ðam 4Þvm Am þ am ð3 þ mÞ 8ð3 þ mÞ 2va2m r am 4 1X q wK ¼ 2þ D m¼1 2cm b a2m ð1 am Þm1 ð3 þ mÞ ) ð4 am ÞAm r am 2 4 r sin am h; ð47Þ am ð1 þ am Þð3 þ mÞ b where
2
d wm 1 dwm a2m þ 2 wm r dr dr 2 r
! ¼
m q : D11
It must be noted the above aforementioned CPT solutions of the deflection of the SSS, SSC and SSF homogeneous plates, i.e. Eqs. (41), (44) and (47), are given for am – 2, 4. To obtain the FST or LPT solutions of the sectorial plates for am = 2, 4 the problem must be first solved based on the classical plate theory which may generates more computational efforts. Thus, using a direct method, the unknown functions of LPT are presented in the following section in order to have less computational efforts.
ð51Þ
The general solution of Eq. (51) may be represented as
wm ðrÞ ¼ wcm ðrÞ þ wpm ðrÞ;
ð52Þ
where wcm and wpm are the complementary and particular solutions of Eq. (51), respectively. For am – 2,4 these solutions are given as
wcm ðrÞ ¼ A1m ram þ A2m r 2þam þ A3m ram þ A4m r 2am ; m 4 q wpm ðrÞ ¼ r : cm D11
vm ¼ am m1 þ 2ð1 þ mÞ; Am ¼ 8 þ am ð5 þ mÞ þ ma2m :
ð53aÞ ð53bÞ
where cm is defined in Eq. (41). Since w must be finite at the center of sector plate, it is concluded that A3m ¼ A4m ¼ 0 and therefore
wm ðrÞ ¼ A1m r am þ A2m r2þam þ
m q
cm D11
r4 :
ð54Þ
For am = 2, 4 the complementary solution wcm remains valid but the particular solution wpm will be given as
wpm ðrÞ ¼ 5.2. Direct method
m 4 q r ln r 48D11
when am ¼ 2;
and h0 ¼
2p 6p ; ; 4 4 ð55aÞ
In what follows, using a direct method, the governing equation (5) will be solved to yield the deflection and boundary layer solutions of LPT and therefore the two other unknown displacement functions will be obtained in terms of them. The governing equation (5) can be expressed in terms of displacement functions (w, ur, uh) by substituting for the force and moment resultants from Eqs. (8) and (9). Thus, the governing equation (5) for FGM circular sector plates take the form
D11 — k;r þ
!
D44 U;h A44 ður þ w;r Þ aF 11 ðr2 wÞ;r ¼ 0 r
ð48aÞ
wpm ðrÞ ¼
ð48bÞ
A44 — k þ A44 r2 w þ q ¼ 0:
ð48cÞ
when am ¼ 4;
and h0 ¼
p 3p 5p 7p
; ; ; : 4 4 4 4 ð55bÞ
5.2.1. SSS circular sector plates Consider again a functionally graded circular sector plate with the boundary conditions defined in Eq. (39b). The boundary conditions give a set of three non-homogeneous algebraic equations of the form 3 X
D11 1 1 k;h D44 U;r A44 uh þ w;h aF 11 ðr2 wÞ;h ¼ 0; — r r r
m 4 q r ln r 96D11
; ij C j ¼ b a i
i ¼ 1; 2; 3
ð56Þ
j¼1
whose solutions yield the integration constants in which
C 1 ¼ B1m ;
C 2 ¼ A1m ;
C 3 ¼ A2m :
Following the procedure proposed by Nosier and Reddy [9], three coupled Eqs. (48) can be reduced to yield two uncoupled equations as:
11 ¼ 0; a
D44 r2 U ¼ A44 U;
ð49aÞ
21 a
ð49bÞ
22 ¼ D11 am ðam m1 Þbam 2 ; a
D11 r4 w ¼ q
D11 A44
r2 q:
1 r
D11 A44
uh ¼ w;h
ðr2 wÞ;r þ
Satisfying the boundary conditions (39b), the coefficients are
1 ¼ wpm j ; 12 ¼ bam ; a 13 ¼ bam þ2 ; b a r¼b a 1 m 0 b1 ¼D m1 cIam ðcbÞ; Iam ðcbÞ ; b b
b 1a 23 ¼ D11 ðam þ 2Þðam þ m1 Þbam þ 4 D m ðam þ 1Þm1 bam 2 ; a
Eqs. (49a) and (49b) referred to as the edge-zone (see Eq. (29)) and interior equation, respectively. It is obvious from Eq. (49) that the LPT totals order of differential equations is sixth which is the same as the FST and both lower than the Reddy’s higher-order theory [21] which is the eighth-order. Using Eqs. (25) and (48) one can obtain
ur ¼ w;r
ð57Þ
D44 1 D U;h 211 q;r ; A44 r A44
D11 1 2 D44 D 1 U;r 211 q;h ; ðr wÞ;h A44 r A44 A44 r
mD11 ¼ D w b 1 r2 wpm;rr j b wpm;r jr¼b D 2 11 pm;rr jr¼b r¼b b b 1 a2 2 mD m þ r wpm jr¼b r2 wpm;r jr¼b ; b b
ð50aÞ
ð50bÞ
D44
32 ¼ 0; cI0a ðcbÞ; a A44 m 3 ¼ D11 am r2 wpm j ; b r¼b A44 b
31 ¼ a
33 ¼ a
4D11 A44
ð58aÞ
ð58bÞ
am ðam þ 1Þbam 1 ; ð58cÞ
554
S. Sahraee / Composite Structures 88 (2009) 548–557
where
b 1 ¼ D11 D11 : D A44 5.2.2. SSC circular sector plates ij and In view of the boundary conditions (42b), the coefficients a j in Eq. (56) for SSC sectorial plates take the form b
11 ¼ 0; a 21 a
12 ¼ bam ; a
D44 am ¼ Iam ðcbÞ; A44 b
23 ¼ ðam þ 2Þbam þ1 þ a
13 ¼ bam þ2 ; a am 1
22 ¼ am b a 4D11 A44
1 ¼ wpm j ; b r¼b
ð59aÞ
;
am ðam þ 1Þbam 1 ;
D44
32 ¼ 0; cI0a ðcbÞ; a A44 m 3 ¼ D11 am r2 wpm j : b r¼b A44 b
33 ¼ a
ð59bÞ
4D11 A44
am ðam þ 1Þbam 1 ;
5.2.3. SSF circular sector plates ij and Based on the boundary conditions (45b), the coefficients a j in Eq. (56) for SSF sectorial plates are given as b
am D44 b D11
Iam ðcbÞ;
12 ¼ 0; a
where
13 ¼ 4am ðam þ 1Þbam 1 ; a
Note that FST analysis of FGM sectorial plate is not available in the open literature at the time of this writing. However, this is easily obtainable from the above aforementioned expressions by substituting a = b = 0 and imposing the shear correction factor 5/6 into the plate stiffness coefficient A44. 6. Results and discussion
ð59cÞ
11 ¼ a
ð60cÞ
b 2 ¼ D11 D44 : D A44
2 ¼ wpm;r j D11 r2 wpm;r j ; b r¼b r¼b A44 31 ¼ a
b 2a m bam 2 ; a m ðam þ 1Þbam 2 ; 32 ¼ 2D44 am ðam þ 1Þbam þ 8 D 31 ¼ 2D44 a a a 2 D2 c 33 ¼ 44 c2 I00am ðcbÞ þ m Iam ðcbÞ I0am ðcbÞ ; a b b A44 3 ¼ 2am D44 1 wpm j wpm;r j b r¼b r¼b b b b2 1 2 2am D þ r wpm jr¼b r2 wpm;r jr¼b ; b b
ð60aÞ
¼ r2 w j : b 1 pm r¼b b 1 am m1 cI0 ðcbÞ; 1 Ia ðcbÞ ; a 21 ¼ D 22 ¼ D11 a m m1 bam 2 ; a am b m b b 1a 23 ¼ D11 ðam þ 2Þðam þ m1 Þ ma2m bam þ 4 D m ðam þ 1Þm1 bam 2 ; a a 2 mD m ¼ D b mwpm D11 wpm;rr jr¼b 11 wpm;r jr¼b 2 11 b b r¼b 2 b b 1 r2 wpm;rr j þ m D 1 am r2 wpm j r2 wpm;r j D r¼b r¼b r¼b ; b b ð60bÞ
To validate the propose formulation, an Aluminum/Zirconia’s functionally graded material with m = 0.3 and Er = 0.4636 is considered here in which Er denotes the ratio of Young’s modulus of metal component to ceramic component which are taken from Praveen and Reddy [22]. Table 1 presents a comparison study of ¼ 103 wDm =ðq0 b4 Þ for SSS, the non-dimensional deflection in w SSC and SSF homogeneous sectorial plates from the present LPT results with the results from CPT given by Timshenko and Woinowsky-Krieger [23]; and with Abacus finite element program and FST by Wang et al., [8]. It is seen that the results of the present theory are in excellent agreement with the results of Abaqus finite element program and Mindlin theory. Fig. 2 shows the non-dimen of functionally graded SSS, SSC and SSF sional deflection, w, plates versus the power law index n for various values of thickness radius ratio h/b. The opening angle h0 is considered to be p/3. It can decreases with increasing the values of n and inbe seen that w creases with increasing the values of h/b due to the effect of trans for verse shear deformation. Also, it is concluded that w homogeneous plates is considerably more than the corresponding value for the FG sectorial plates. Variation of the non-dimensional deflection of functionally graded SSS, SSC and SSF plates versus the Er is demonstrated in Fig. 3 for various values of n with h/b = 0.1. The opening angle h0 and circumferential angle h are considered
Table 1 of SSS, SSC and SSF homogeneous sectorial plates Comparison of the non-dimensional deflection, w, h/b p=6Þ SSS plate, h0 ¼ p=3; wð0:75b;
p=6Þ SSC plate, h0 ¼ p=3; wð0:75b;
p=4Þ SSF plate, h0 ¼ p=2; wðb;
a b c
Thin platea
0.001
0.9248
Abaqusb 0.9248
0.1
–
1.0210
0.2
–
1.3097
0.001 0.1 0.2 0.001 0.1 0.2
0.4674 – – 63.2800 – –
0.4673 0.5872 0.9262 63.2800 64.6480 66.9260
Timshenko and Woinowsky-Krieger [23]. From Ref. [8]. From present article with n = a = b = 0 and imposing the shear correction factor 5/6 into the plate stiffness coefficient A44.
FSTb – 0.9247c – 1.0209 – 1.3094 0.4674 0.5872 0.9262 63.2800 64.6474 66.9254
Present results 0.9247 1.0208 1.3095 0.4671 0.6079 0.9926 63.3217 65.2234 68.0697
555
S. Sahraee / Composite Structures 88 (2009) 548–557
a
b 0.951
0.9
0.9
SSS
SSC
0.85
0.8
SSF
11
0.8 0.75
h/b=0.001, h/b=0.1, h/b=0.2.
0.65
⎯w
0.6
10
h/b=0.001, h/b=0.1, h/b=0.2.
0.7 0.6
h/b=0.001, h/b=0.1, h/b=0.2.
9
⎯w
0.7
⎯w
c 12
0.55
0.5
8
0.5
θ0=π/3, θ=π/6, r=0.75b.
0.4
0.45
θ0=π/3, θ=π/6, r=0.75b.
0.4
0.3
0.35
0.2
0.25
θ0=π/3, θ=π/6, r=b.
7
0.3
0
10
20
30
40
50
6 0
10
20
n
30
40
50
0
10
20
30
40
50
n
n
of FG sectorial Levinson plates versus the power law index n for various values of thickness radius ratio h/b with h0 = p/3. (a) SSS, Fig. 2. Non-dimensional deflection, w, p=6Þ, (b) SSC, wð0:75b; p=6Þ and (c) SSF, wðb; p=6Þ. wð0:75b;
a
b 1.2
2.5
1
n=1, n=2, n=5.
3
n=1, n=2, n=5.
0.9
n=1, n=2, n=5.
2
0.8
SSS
2
0.7
⎯w
⎯w
2.5
⎯w
c
1.1
3.5
1.5
SSC
0.6
SSF
0.5
1.5
1
0.4 1
r=0.75b, h=0.1b, θ0=π, θ=π/2.
r=b, h=0.1b, θ0=π/4, θ=π/8.
r=0.75b, h=0.1b, θ0=π, θ=π/2.
0.3
0.5
0.2 0.5 0.1
0.2
0.3
0.4
0.5
0.1
0.1
0.2
Er
0.3
0.4
0.5
0.1
0.2
0.3
Er
0.4
0.5
Er
of FG sectorial Levinson plates versus Er for various values of n with h/b = 0.1. (a) SSS, h0 ¼ p; wð0:75b; Fig. 3. w p=2Þ, (b) SSC, h0 ¼ p; wð0:75b; p=2Þ and (c) SSF, ðb; p=8Þ. h0 ¼ p=4; w
to be p and p/2 for SSS and SSC plates, respectively, and also p/4 increases as Er inand p/8 for SSF plates. It is observed that w creases. Fig. 4 depicts the non-dimensional deflection of functionally graded SSS, SSC and SSF plates versus the opening angle h0 for various values of n with h/b = 0.1. The circumferential angle h is increases as h0 inconsidered to be h0/2. It is concluded that w of SSF plates more highly increases creases. Also, it is seen that w as h0 tends to p. It is because of the SSF plates are not globally in equilibrium when h0 = p. Figs. 5 and 6 shows, respectively, the varrz ¼ srz =q and iation of the non-dimensional shear stresses, s shz ¼ shz =q, of functionally graded SSS, SSC and SSF plates obtained
a
by the constitutive law versus the non-dimensional thickness coordinate variable z ¼ z=h for various values of n and h/b with h0 = p/3. The circumferential angle h is considered to be p/6 for Fig. 5 and p/4 for Fig. 6. Note that the value of shz at h = h0/2 is zero due to symmetry of circular sector plates about the line h = h0/2. It is seen hz decreases with increasing the rz and s that both shear stresses s values of h/b. Also, it is observed that the maximum non-dimensional shear stresses of FG sectorial plates are greater than the corresponding values of the homogeneous plates and do not occur at z ¼ 0 due to non-homogenous mechanical properties of the FGMs.
b 1.8
c 300
1.6
5
SSS
1.2
250
1
150
0.8 2
0.6
1
θ0
2
0.2
3
0
θ=θ0/2, r=b, h=0.1b.
100
θ=θ0/2, r=0.75b, h=0.1b.
0.4
θ=θ0/2, r=0.75b, h=0.1b.
1
SSF
⎯w
3
0
n=0, n=2, n=5.
SSC 200
⎯w
⎯w
4
n=0, n=2, n=5.
1.4
n=0, n=2, n=5.
50
0 1
θ0
2
3
0.5
1
θ0
1.5
2
of FG sectorial Levinson plates versus h0 for various values of n with h/b = 0.1. (a) SSS, 0:1p 6 h0 6 p; wð0:75b; Fig. 4. w h0 =2Þ, (b) SSC, 0:1p 6 h0 6 p; wð0:75b; h0 =2Þ and (c) h0 =2Þ. SSF, 0:1p 6 h0 6 0:7p; wðb;
556
S. Sahraee / Composite Structures 88 (2009) 548–557
b
a0 n=0, h=0.1b, n=2, h=0.1b, n=5, h=0.1b; n=0, h=0.2b, n=2, h=0.2b, n=5, h=0.2b.
-0.1
-0.2
c0
0 -0.2 θo=π/3, θ=π/6, r=0.75b.
-0.4
n=0, h=0.1b, n=2, h=0.1b, n=5, h=0.1b; n=0, h=0.2b, n=2, h=0.2b, n=5, h=0.2b.
-0.05
-0.6
-0.1
-0.4
n=0, h=0.1b, n=2, h=0.1b, n=5, h=0.1b; n=0, h=0.2b, n=2, h=0.2b, n=5, h=0.2b.
-0.8 -1
θo=π/3, θ=π/6, r=0.75b. -0.5
-1.2
-0.6
-1.4
SSS
-0.7 -0.5
⎯τrz
⎯τrz
⎯τrz
-0.3 -0.15
θo=π/3, θ=π/6, r=0.75b.
-0.2
SSC
SSF
-0.25
-1.6
-0.25
0
0.25
-0.5
0.5
-0.25
0
0.25
0.5
-0.5
-0.25
⎯z
⎯z
0
0.25
0.5
⎯z
rz ¼ srz =q, of FG sectorial Levinson plates versus the non-dimensional thickness coordinate variable z ¼ z=h for various values of n and Fig. 5. Non-dimensional shear stress, s p=6Þ, (b) SSC, wð0:75b; p=6Þ and (c) SSF, wðb; p=6Þ. h/b with h0 = p/3. (a) SSS, wð0:75b;
b
0
n=0, h=0.1b, n=2, h=0.1b, n=5, h=0.1b; n=0, h=0.2b, n=2, h=0.2b, n=5, h=0.2b.
-0.1 -0.2 -0.3
-0.3 -0.4
-0.5
-0.5
-0.6
-0.6
-0.7
θo=π/3, θ=π/4, r=0.75b.
-0.8
0 θo=π/3, θ=π/4, r=b.
-1 -2 -3
-0.7
θo=π/3, θ=π/4, r=0.75b.
-0.8
-0.9
n=0, h=0.1b, n=2, h=0.1b, n=5, h=0.1b; n=0, h=0.2b, n=2, h=0.2b, n=5, h=0.2b.
-4 -5 -6
-0.9
-1
-1
-1.1
-1.1
SSS
-1.2 -0.5
n=0, h=0.1b, n=2, h=0.1b, n=5, h=0.1b; n=0, h=0.2b, n=2, h=0.2b, n=5, h=0.2b.
-0.2
⎯τθz
⎯τθz
-0.4
c
0 -0.1
⎯τθz
a
-7
SSS
SSF
-8
-1.2 -0.25
0
0.25
0.5
-0.5
-0.25
⎯z
0
⎯z
0.25
0.5
-9 -0.5
-0.25
0
0.25
0.5
⎯z
hz ¼ shz =q, of FG sectorial Levinson plates versus the non-dimensional thickness coordinate variable z ¼ z=h for various values of n and Fig. 6. Non-dimensional shear stress, s p=4Þ, (b) SSC, wð0:75b; p=4Þ and (c) SSF, wðb; p=4Þ. h/b with h0 = p/3. (a) SSS, wð0:75b;
7. Conclusions The pure bending problem of functionally graded circular sector plates is considered in the present work based on the Levinson plate theory (LPT). This theory captures the higher-order effect by assuming the same third-order polynomials in the expansion of the in-plane displacements through the thickness of plate as in the Reddy’s higher-order theory and therefore it no needs to have shear correction factor. In this article the unknown displacement functions of the LPT are represented in two types: (1) in terms of the response of homogeneous sectorial plates based on the classical plate theory and (2) by solving the two independent edge-zone and interior equations, directly. An excellent rate of convergence was showed and the results were in good agreement with the results of other theories. Also, it is seen that the LPT solution of the deflection for homogenous sectorial plates are significantly higher than those for functionally graded ones; especially for the thick plates. Thus, the gradients in material properties play an important role in determining the response of the FGM plates. References [1] Reissner E. The effect of transverse shear deformation on the bending of elastic plates. J Appl Mech 1945;12:69–77. [2] Mindlin RD. Influence of rotary inertia and shear on flexural motions of isotropic, elastic plates. J Appl Mech 1951;18:31–8. [3] Reissner E. On the theory of bending of elastic plates. J Math Phys 1944:184–91.
[4] Reissner E. On bending of elastic plates. Quart Appl Math 1947;5:55–68. [5] Levinson M. An accurate, simple theory of the static’s and dynamics of elastic plates. Mech Res Commun 1980;7(6):343–50. [6] Wang CM, Kitipornchai S. frequency relationship between Levinson plates and classical thin plates. Mech Res Commun 1999;26(6):687–92. [7] Reddy JN, Wang CM, Lim GT, Ng KH. Bending solutions of Levinson beams and plates in terms of the classical theories. Int J Solid Struct 2001;38:4701–20. [8] Wang CM, Reddy JN, Lee KH. Shear deformable beams and plats, relationships with classical solutions. UK: Elsevier; 2000. [9] Nosier A, Reddy JN. On boundary layer and interior equations for higher-order theories of plates. ZAMM 1992;72:657–66. [10] Nosier A, Reddy JN. A study of non-linear dynamics equation of higher order shear deformation plate theories. Int J Non-Linear Mech 1991;22:233–49. [11] Nosier A, Reddy JN. On vibration and buckling of symmetric laminated plates according to shear deformations theories. Part I. Acta Mech 1992;94:123–44. [12] Nosier A, Reddy JN. On vibration and buckling of symmetric laminated plates according to shear deformations theories. Part II. Acta Mech 1992;94: 145–69. [13] Nosier A, Yavari A, Sarkani S. On a boundary layer phenomenon in Mindlin– Reissner plate theory for laminated circular sector plates. Acta Mech 2001;151:149–61. [14] Yamanouchi M, Koizumi M, Hirai T, Shiota I. Functionally gradient materials. In: Proceedings of the first international symposium; 1990. [15] Fukui Y. Fundamental investigation of functionally gradient material manufacturing system using centrifugal force. Int J Jpn Soc Mech Eng Ser III 1991;34:144–8. [16] Koizumi M. The concept of FGM. Ceram Trans Function Gradient Mater 1993;34:3–10. [17] Reddy JN, Wang CM, Kitipornchai S. Axisymmetric bending of functionally graded circular and annular plates. Eur J Mech A/Solid 1999;18:185–99. [18] Ma LS, Wang TJ. Relationship between axisymmetric bending and buckling solutions of FGM circular plates based on third-order plate theory and classical plate theory. Int J Solid Struct 2004;41:85–101. [19] Loy CT, Lam KY, Reddy JN. Vibration of functionally graded cylindrical shells. Int J Mech Sci 1999;41:309–24.
S. Sahraee / Composite Structures 88 (2009) 548–557 [20] Reddy JN. Theory and analysis of elastic plates. Philadelphia, PA: Taylor & Francis; 1999. [21] Reddy JN. A simple higher order-theory for laminated composite plates. J Appl Mech 1984;51(4):745–52.
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[22] Praveen GN, Reddy JN. Nonlinear transient thermo-elastic analysis of functionally graded ceramic–metal plates. Int J Solid Struct 1998;35:4457–76. [23] Timshenko S, Woinowsky-Krieger S. Theory of plates and shells. 2nd ed. McGraw-Hill Book Company; 1959.
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Composite Structures journal homepage: www.elsevier.com/locate/compstruct
Bending analysis of functionally graded sectorial 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 7 July 2008 Keywords: FGMs Bending analysis Sector plate Levinson plate theory Boundary layer function
a b s t r a c t Based on the Levinson plate theory (LPT) and the first-order shear deformation plate theory (FST), the bending analysis of functionally graded (FG) thick circular sector plates is presented. The LPT solutions of FG sectorial plates are first expressed in terms of the solutions of the classical plate theory (CPT) for homogeneous sectorial plates and then presented using a direct method. It is assumed that the nonhomogeneous mechanical properties of plate, graded through the thickness, are described by a power function of the thickness coordinate. The results are given in closed-form solutions and verified with the known data in the literature. Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction Based on considering the effect of transverse 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 transverse shear deformation and (2) shear deformation plate theories are those in which the effect of the transverse 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 components through the thickness of plate is assumed to be known a priori, into two types: stress-based and displacement-based plate theories. Reissner [1] would appear to have been the first to consider shear deformations in a static plate theory (stress-based theories); while Mindlin [2] was the pioneer in developing dynamic plate theories which include the effects of transverse shear deformations and rotary inertia (displacement-based 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] presented an accurate simple theory for the static’s and dynamics of rectangular plates. He used a vector approach to derive his equations of equilibrium 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 * Address: No. 36, Ali Asghare Piri Alley, Chehel Metrie Sirous Avenue, Kermanshah, Iran. Tel.: +98 9188597151. E-mail address: [email protected] 0263-8223/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2008.05.014
and Kitipornchai [6] presented an exact frequency relationship between Levinson plate theory and Kirchhoff plate theory for homogeneous 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 based on the load equivalence and mathematical similarity of the governing equations of the both mentioned theories. Wang et al. [8] furnished a great book about shear deformation theories. This book studies the relationships between the solutions of classical theories of beams and plates with those of the first- and third-order shear deformation theories. A comprehensive 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 boundary layer phenomenon in bending analysis of laminated circular sector plates based on the FST. FGMs have attracted much attention as advanced structural materials in recent years because of their heat-shielding properties. FGMs were first introduced in 1984 by a group of material scientists 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 these materials are microscopically inhomogeneous. FGMs are made by combining two or more materials using powder metallurgy method, typically these materials are made from a mixture of ceramic and metal in which the ceramic component provides high-temperature resistance due to its low thermal conductivity; on the other hand, the ductile metal component prevents fracture due to thermal stress. FGMs were introduced as to take advantage of the desired material properties of each
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S. Sahraee / Composite Structures 88 (2009) 548–557
constituent material which leads to smooth distribution of stresses, without an abrupt change in the effective properties which may result in interface problems of traditional composite materials. FGMs are now considered as a potential structural material for high-speed spacecraft. Using the first-order theory of Mindlin, Reddy et al. [17] studied axisymmetric bending and stretching of functionally graded solid circular and annular plates. They presented the solutions for deflections, force and moment resultants based on the first-order plate theory in terms of those 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 based on the TST and those of homogeneous circular plates on the basis of CPT. In the present work, LPT is employed to analyze the pure bending of functionally graded sectorial plates. First, using an analytical method, the relationships between the solutions of the bending of functionally graded circular sector Levinson plates was derived in terms of the responses of homogeneous sectorial 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 deformation on the bending of the functionally graded sectorial plates have been all considered.
Using Eqs. (1) and (2), the material properties P of plate such as, Young’s modulus E can be written as follow:
n h 2z E ¼ ðEm Ec Þ þ Ec : 2h
Generally Poisson’s ratio m varies in a small range, for simplicity it is assumed to be a constant. It is apparent from Eq. (3) that the upper surface of plate (z = h/2) is purely ceramic and the lower surface (z = h/2) is purely metallic. 3. Equilibrium equations of Levinson plate theory Let (Ur, Uh, Uz) be the displacement components of an arbitrary point within the plate domain along the (r, h, z) coordinate directions in cylindrical coordinates, the equilibrium equations of LPT are derived based on the following displacement field as
U r ðr; h; zÞ ¼ zur az3 ður þ w;r Þ; 1 U h ðr; h; zÞ ¼ zuh az3 uh þ w;h ; r
ð4aÞ ð4bÞ
U z ðr; h; zÞ ¼ w;
ð4cÞ
where ur and uh are respectively, rotations of the middle surface (i.e., z = 0) of plate in h- and r-directions; w is the transverse deflection 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]:
2. Material properties Consider a solid FGM circular sector thick flat plate of radius b and thickness h with a subtended angle h0 subjected to transversely pressure q as shown in Fig. 1. The cylindrical coordinates (r, h, z) also shown in the figure, is used in the analysis. The r-coordinate 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 r h plane. The sectorial plate is simply supported at h = 0 and h = h0 while at r = b it may be simply 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 expressed for two different constituents as [19]
P ¼ Pc V c þ Pm V m ;
ð3Þ
1 Mrr;r þ ðM rh;h þ M rr M hh Þ Q r ¼ 0; r 1 Mrh;r þ ðM hh;h þ 2M rh Þ Q h ¼ 0; r 1 Q r;r þ Q r þ Q h;h þ q ¼ 0; r
ð5aÞ ð5bÞ ð5cÞ
where q denotes the transverse load; Mrr, Mhh are the bending moments; Mrh is the twisting moment; and Qi (i = r, h) are shear forces which are all expressed in terms of the stress components through the thickness of plate by the following expressions:
ð1Þ
where c and m refer to the ceramic and metal constituents, respectively; Pi (i = c, m) denotes the material property of the constituent material i; and Vi is the volume fraction of the constituent material i which can be expressed according to power law distribution as
Mi ¼ Qj ¼
Z
Z
h=2
ri z dz ði ¼ rr; hh; rhÞ;
ð6aÞ
h=2 h=2
rj dz ðj ¼ r; hÞ:
ð6bÞ
h=2
n h 2z V m ðzÞ ¼ ; 2h
ð2aÞ
At any edge of the plate with a normal ~ n ¼ nr e þnh e , the boundary r h conditions require the specifications of [7]
V c ðzÞ ¼ 1 V m ðzÞ;
ð2bÞ
w or Q r ;
ð7aÞ
ur or Mrr ; uh or Mrh :
ð7bÞ
*
where n denotes the power law index which takes values greater than or equal to zero.
ð7cÞ
b Simply supported radial edge
O
θ r
z
h /2
q
θ =0
θ = θ0
*
Circular edge may be simply supported, clamped or free
Fig. 1. A sectorial plate with simple supports along its radial edges.
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S. Sahraee / Composite Structures 88 (2009) 548–557
The constitutive equations of Levinson plate theory are given by
h
i
m
M rr ¼ D11 ur;r þ ður þ uh;h Þ r m 1 aF 11 w;rr þ w;r þ w;hh ; r r M hh
1 ¼ D11 mur;r þ ður þ uh;h Þ r 1 1 aF 11 mw;rr þ w;r þ w;hh ; r r
ð8aÞ
i a D44 h 1 M rh ¼ ur;h uh þ ruh;r 2F 44 w;rh w;h ; r r r
ð8bÞ
Mrh ¼ Dm1
1 w;h r
ð13cÞ
; ;r
Q Kr ¼ Dðr2 wK Þ;r ;
ð14aÞ
1 Q Kh ¼ Dðr2 wK Þ;h ; r
ð14bÞ
where superscript K refers to quantities in the Kirchhoff plate theory and r2 is the two-dimensional Laplace operator in polar coordinate defined as
1 r
r2 ðÞ ¼ ðÞ;rr þ ðÞ;r þ
1 ðÞ : r 2 ;hh
ð8cÞ
also D is the usual plate bending rigidity as follow:
ð9aÞ
D¼
3
Q r ¼ A44 ½ur þ w;r ; 1 Q h ¼ A44 uh þ w;h ; r
Eh : 12ð1 m2 Þ
ð9bÞ 5. Bending analysis
where A11, B11, etc. are the plate stiffness coefficients defined as
ðD11 ; F 11 Þ ¼
Z
h=2
h=2
ðA44 ; D44 ; E44 Þ ¼
Z
E ðz2 ; z4 Þdz; 1 m2 h=2
h=2
E ð1; z2 ; z4 Þdz; 2ð1 þ mÞ
ð10aÞ ð10bÞ
The governing equations of equilibrium of CPT may be expressed in terms of the moment sum MK and similarly equilibrium equations of Levinson plate theory may be written in terms of moment sum M which are both defined by the following expressions:
MKrr þ M Khh ; 1þm M rr þ M hh : M¼ 1þm
MK ¼
also
D11 ¼ D11 aF 11 ; D44 ¼ m1 D11 =2;
ð15aÞ ð15bÞ
The moment sums (15) are expressed in terms of the displacement functions as
A44 ¼ A44 bD44 ; with
MK ¼ Dr2 wK ; 11 — M¼D k aF 11 r2 w;
2
m1 ¼ 1 m; b ¼ 4=h :
ð16aÞ ð16bÞ
where 4. Equilibrium equations of Kirchhoff plate theory The classical thin plate theory is based on following displacement field [20]
U r ðr; h; zÞ ¼
zwK;r ;
z U h ðr; h; zÞ ¼ wK;h ; r U z ðr; h; zÞ ¼ wK ;
ð11aÞ ð11bÞ ð11cÞ
where wK is the transverse deflection of a point on the mid-plane The above displacements is based on Kirchhoff hypothesis, implies that straight lines normal to r h plane before deformation remain straight and normal to the mid-plane after deformation which neglects both transverse shear and traverse normal effects. Using the principle of virtual displacements, equilibrium equations of CPT are derived as
1 K M Krr;r þ Mrh;h þ M Krr MKhh Q Kr ¼ 0; r 1 K K Mh;hh þ 2MKrh Q Kh ¼ 0; M rh;r þ r 1 K K Q r þ Q Kh;h þ q ¼ 0: Q r;r þ r
1 k ¼ ur;r þ ður þ uh;h Þ: — r From the equilibrium equations of the both theory one can easily obtain the following expressions:
r2 MK ¼ q;
ð17aÞ
2
r M ¼ q:
ð17bÞ
In view of the load equivalence (Eq. (17)), one can arrive at the following moment sums relationship as
M ¼ M K þ D11 r2 N;
ð18Þ
where N is a function such that it satisfies the biharmonic equation as follow:
r4 N ¼ 0:
ð19Þ
ð12aÞ
Substitution of Eqs. (9a) and (9b) into Eq. (5c), yields
ð12bÞ
k¼ —
ð12cÞ
Introduction of the above expression into Eq. (16b), gives
q A44
r2 w:
ð20Þ
D11
The constitutive equations for classical Kirchhoff plate theory are given by
M ¼ D11 r2 w
m K 1 K M Krr ¼ D wK;rr þ w;r þ w;hh ; r r 1 1 M Khh ¼ D mwK;rr þ wK;r þ wK;hh ; r r
From Eqs. (16a), (17a), (18), and (21) the following relationship between the deflection, w, of FG circular sector plates based on the Levinson plate theory and the deflection, wK, of homogeneous sectorial plates based on the classical Kirchhoff plate theory can be obtained as
ð13aÞ ð13bÞ
A44
q:
ð21Þ
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S. Sahraee / Composite Structures 88 (2009) 548–557
w¼
D K D11 w þ MK N þ W; D11 D11 A44
ð22Þ
where W is a function which satisfies the harmonic equation as
r2 W ¼ 0:
ð23Þ
From Eqs. (5a), (5b), (8a)–(8c) and (16b) one then obtains
Q r ¼ M ;r þ
m1 D11 2r
U;h ;
ð24aÞ
1 r
U ¼ ður;h uh þ r uh;r Þ:
fwðr; hÞ; wK ðr; hÞg ¼ fur ðr; hÞ; uh ðr; hÞg ¼
ð30aÞ
1 X
fwm ðrÞ; wKm ðrÞg sin am h;
m¼1 1 X
ð30bÞ
furm ðrÞ; uhm ðrÞg sin am h;
ð30cÞ
m¼1
Uðr; hÞ ¼
1 X
Um ðrÞ cos am h;
ð30dÞ
m¼1
ð25Þ where
Using Eqs. (9a), (9b), (18), (22) and (24) one can obtain the rotation functions of Levinson plate theory as
D K E2 w þ E1 M K;r þ C;r þ U;h ; D11 ;r r D 1 K 1 K 1 uh ¼ w þ E1 M ;h þ C;h E2 U;r ; D11 r ;h r r
ur ¼
aF 11 D11 A44 D11 2 A44
;
E2 ¼
m1 D11 2A44
ð26bÞ
2 2 d Um dUm am 2 þ þ c Um ¼ 0: dr 2 r dr r2
Um ¼ B1m Iam ðcrÞ þ B2m K am ðcrÞ;
M rr ¼ M Krr þ m1 D11 E2 RðUÞ þ m1 D11 C;rr þ D11 r2 N; 2
m1 D11 E2 RðUÞ D11 m1 C;rr þ D11 r N;
M rh ¼ M Krh þ m1 D11 RðCÞ þ
m1 2
Q r ¼ Q Kr þ D11 r2 ðr2 NÞ;r þ
D11 X þ D11 r2 N;
E2 A44 U;h ; r
1 Q h ¼ Q Kh þ D11 r2 ðr2 NÞ;h E2 A44 U;r ; r
ð27aÞ ð27bÞ ð27cÞ
ð28bÞ
C¼
D11 A44
ð29Þ
A1m r am sin am h;
ð34aÞ
m¼1
W¼
1 X
A2m r am sin am h;
ð34bÞ
w¼
1 X D K D11 r 2 A1m w þ MK þ A2m ram D11 4ðam þ 1Þ D11 A44 m¼1 sin am h:
ð35Þ
Rotation–slope relationships
1 X D K D a þ 2 am þ1 w;r þ E1 M K;r þ am 11 ram 1 þ m r A1m D11 4ðam þ 1Þ A44 m¼1 am am r am 1 A2m 2 Iam ðcrÞB1m sin am h; ð36aÞ c r
ur ¼
where
m1 D11
1 X
where A1m and A2m are constants which both with B1m to be determined 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 solutions of FG sectorial Levinson plates in terms of the solution of homogeneous Kirchhoff plates may be summarized below. Deflection relationship
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
sffiffiffiffiffiffiffiffiffiffiffiffi 2A44
For the sectorial plates considered here, the function r N and W are given by
m¼1
r2 N þ N W :
r2 U ¼ c2 U;
ð33Þ 2
ð28aÞ
also
1 U;hh U;rr ; r2
Um ¼ B1m Iam ðcrÞ:
r2 N ¼
1 1 RðÞ ¼ ðÞ;rh 2 ðÞ;h ; r r
1 r
ð32Þ
where B1m and B2m are constants; Iam and K am are the modified Bessel functions of the first and second kinds, respectively. To have finite displacement functions at r = 0 it is deduced that B2m = 0 and therefore
where R operator is defined as
X ¼ U;r þ
ð31Þ
The general solution of Eq. (31) is given by
;
Introduction of Eqs. (22) and (26) into Eqs. (8) and (9) yield
M hh ¼
2q0 ½1 ð1Þm : mp
Introduction of Eq. (30d) into Eq. (29) yields the following modified Bessel function as:
r N þ N W:
M Khh
m ¼ q
ð26aÞ
where
c¼
m sin am h; q
ð24bÞ
where U is a potential function referred to as the boundary layer function as follow
C¼
1 X
qðr; hÞ ¼
m¼1
1 m1 D11 U;r ; Q h ¼ M;h r 2
E1 ¼
load q0 will be studied. Assuming the simply supported radial edges and based on the Levy method of solution, the load distribution and displacement functions for both Kirchhoff and Levinson plate theories; and boundary layer function of LPT may be represented as
:
5.1. Levy solutions using relationships In this section, the bending problem of functionally graded circular sector plates subjected to transversely uniform distributed
1 X D E1 D r am þ1 A1m wK;h þ M K;h þ am 11 ram 1 þ rD11 r 4ðam þ 1Þ A44 m¼1 1 ð36bÞ am r am 1 A2m I0am ðcrÞB1m cos am h; c
uh ¼
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S. Sahraee / Composite Structures 88 (2009) 548–557
Moment relationships
M rr ¼
M Krr
1 X
("
am þ 2
D11
wK ¼
#
v
r am A1m þ þ 4 m1 r2 A44 ) D11 am Iam ðcrÞ 0 a 2 m mr a A2m þ Iam ðcrÞ B1m sin am h; cr D11 cr þ m1 D11
a m
m¼1
ð37aÞ M hh ¼ M Khh D11 m1
("
1 X
D11
am þ 2
1
#
r am A1m 4 m1 r2 A44 m¼1 ) D11 am Iam ðcrÞ 0 am 2 am r A2m þ Iam ðcrÞ B1m sin am h; cr D11 cr
a m
þ
ð37bÞ M Krh
M rh ¼
þ
þ D11 m1
D11 D11
1 X
("
a m
D11
r
am 2
þ
am
r
4 ) Iam ðcrÞ lm Iam ðcrÞ B1m cos am h; cr m¼1
A44
mr A1m a
am 2
A2m
where
cm ¼ ð16 a2m Þð4 a2m Þ: 5.1.2. Clamped circular sector plates (SSC) Suppose a solid circular sector plate which is clamped at r = b. The boundary conditions at r = b are
wK ¼ 0; w ¼ 0;
A1m
0
ð37cÞ
m¼1
Q h ¼ Q Kh þ
1 X m¼1
am D11 ram 1 A1m
A44 Ia ðcrÞB1m c2 r m
(
A am D11 ram 1 A1m 44 I0am ðcrÞB1m c
) sin am h; ) cos am h:
ð38bÞ
ð39aÞ
uh ¼ 0:
ð39bÞ
Satisfying the above boundary conditions at r = b give
A1m
1 a m I1 b i; ¼ a h 2 þ mm1 b m acbm m21 I1 a1m þ 2ð2aamm þ1 þ1Þ
A2m ¼ b
B1m
am
am 2
ð40aÞ
2
Hm þ
"
am
D11 A44
A1m ;
ð43bÞ 2
þ
b
#
! a
2ðam þ 1Þ
m ; b m A1m þ am Hm þ bE1 H
D11 M Km jr¼b ; D11 A44 cI0 ðcbÞ I 2 ¼ am : Iam ðcbÞ
Hm ¼
M Krr ¼ 0;
Hm
4ðam þ 1Þ
where
The CPT solution of the deflection of a SSC homogeneous plate under a uniform load q0 is given by [8]
b A1m ; 4ðam þ 1Þ
cam D11 D11 am ¼ 0 Hm þ b A1m ; bIam ðcbÞ D11 A44
ð40bÞ
ð40cÞ
1 r am 4 r am 2 m q 1X 2 þ ð2 am Þ ð4 am Þ D m¼1 2cm b b
wK ¼
5.1.1. Simply supported circular sector plates (SSS) Suppose a solid circular sector plate with a simply support at r = b. The boundary conditions at r = b are
Mrr ¼ 0;
2
ð38aÞ
This completes the derivation of the relationships between the solutions of FG sectorial plates based on the LPT in terms of the associated quantities of homogeneous circular sector plates on the basis of CPT. In what follows; a solid circular sector plate with various boundary conditions at its circular edge will be considered.
wK ¼ 0;
b
ð43aÞ
ð43cÞ
Shear–force relationships 1 X
Hm þ
c2 am Iam ðcbÞ
rc
Q r ¼ Q Kr þ
ð42bÞ
m abm2 þ Ib2 DD11 Hm þ aI2m DD11 E1 H 11 11 i; ¼ a h bD11 I2 am 2 b b m ðcbÞ 1 I 2 2 m a 2 a D ð1þ a Þ m m 11 m 1
B1m ¼
(
ð42aÞ
ur ¼ 0; uh ¼ 0:
am
a m ¼ am ðam 1Þ; 1 a 2 lm ¼ þ m :
w ¼ 0;
wK;r ¼ 0;
A2m ¼ b
where
2
ð41Þ
Satisfying the above boundary conditions at r = b give
# am
1 m q 1X ð2 am Þðam þ 5 þ mÞ r am 4 2þ D m¼1 2cm 2am þ 1 þ m b ð4 am Þðam þ 3 þ mÞ r am 2 4 r sin am h; 2am þ 1 þ m b
r 4 sin am h;
5.1.3. Free circular sector plates (SSF) Suppose a solid circular sector plate which is free at r = b. The boundary conditions at r = b are
1 Q Kr þ MKrh;h ¼ 0; r Q r ¼ 0;
ð45aÞ
M rh ¼ 0:
ð45bÞ
A1m
MKrh jr¼b þ 2II1 þ a2m II2 Q Krm jr¼b ; ¼ a D11 b m ð2 mÞ 2b ½am II1 þ II2
A2m ¼
Hm ¼
The CPT solution of the deflection of a SSS homogeneous plate under a uniform load q0 is given by [8,23]
Mrr ¼ 0;
M Krr ¼ 0;
Satisfying the above boundary conditions at r = b give
where
D11 M Km jr¼b ; D11 A44 Ia ðcbÞ : I1 ¼ m0 bcIam ðcbÞ
ð44Þ
B1m ¼
ð46aÞ
! 2 m D11 am b a 2 A þ II þ a m b2 A44 4 bm1 2 1m 2a b m 2 þ MKrh jr¼b þ II2 Q Krm jr¼b ; a m m1 D11 am b
am D44 Iam ðcbÞ
Q Krm jr¼b þ D11 am b
am 1
A1m ;
ð46bÞ
ð46cÞ
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S. Sahraee / Composite Structures 88 (2009) 548–557
where
Introduction of Eq. (30d) into Eq. (49a), yields the differential equation (31) which its solution given in Eq. (32) and substitution of Eqs. (30a) and (30b) into Eq. (49b), yields
0
Iam ðcbÞ 1 ; cIam ðcbÞ bc2 I0 ðcbÞ blm : II2 ¼ am cIam ðcbÞ II1 ¼
2
d 1 d a2m þ 2 r dr r 2 dr
The CPT solution of the deflection of a SSF homogeneous plate under a uniform load q0 is given by [8] ( 1 m ðam 4Þvm Am þ am ð3 þ mÞ 8ð3 þ mÞ 2va2m r am 4 1X q wK ¼ 2þ D m¼1 2cm b a2m ð1 am Þm1 ð3 þ mÞ ) ð4 am ÞAm r am 2 4 r sin am h; ð47Þ am ð1 þ am Þð3 þ mÞ b where
2
d wm 1 dwm a2m þ 2 wm r dr dr 2 r
! ¼
m q : D11
It must be noted the above aforementioned CPT solutions of the deflection of the SSS, SSC and SSF homogeneous plates, i.e. Eqs. (41), (44) and (47), are given for am – 2, 4. To obtain the FST or LPT solutions of the sectorial plates for am = 2, 4 the problem must be first solved based on the classical plate theory which may generates more computational efforts. Thus, using a direct method, the unknown functions of LPT are presented in the following section in order to have less computational efforts.
ð51Þ
The general solution of Eq. (51) may be represented as
wm ðrÞ ¼ wcm ðrÞ þ wpm ðrÞ;
ð52Þ
where wcm and wpm are the complementary and particular solutions of Eq. (51), respectively. For am – 2,4 these solutions are given as
wcm ðrÞ ¼ A1m ram þ A2m r 2þam þ A3m ram þ A4m r 2am ; m 4 q wpm ðrÞ ¼ r : cm D11
vm ¼ am m1 þ 2ð1 þ mÞ; Am ¼ 8 þ am ð5 þ mÞ þ ma2m :
ð53aÞ ð53bÞ
where cm is defined in Eq. (41). Since w must be finite at the center of sector plate, it is concluded that A3m ¼ A4m ¼ 0 and therefore
wm ðrÞ ¼ A1m r am þ A2m r2þam þ
m q
cm D11
r4 :
ð54Þ
For am = 2, 4 the complementary solution wcm remains valid but the particular solution wpm will be given as
wpm ðrÞ ¼ 5.2. Direct method
m 4 q r ln r 48D11
when am ¼ 2;
and h0 ¼
2p 6p ; ; 4 4 ð55aÞ
In what follows, using a direct method, the governing equation (5) will be solved to yield the deflection and boundary layer solutions of LPT and therefore the two other unknown displacement functions will be obtained in terms of them. The governing equation (5) can be expressed in terms of displacement functions (w, ur, uh) by substituting for the force and moment resultants from Eqs. (8) and (9). Thus, the governing equation (5) for FGM circular sector plates take the form
D11 — k;r þ
!
D44 U;h A44 ður þ w;r Þ aF 11 ðr2 wÞ;r ¼ 0 r
ð48aÞ
wpm ðrÞ ¼
ð48bÞ
A44 — k þ A44 r2 w þ q ¼ 0:
ð48cÞ
when am ¼ 4;
and h0 ¼
p 3p 5p 7p
; ; ; : 4 4 4 4 ð55bÞ
5.2.1. SSS circular sector plates Consider again a functionally graded circular sector plate with the boundary conditions defined in Eq. (39b). The boundary conditions give a set of three non-homogeneous algebraic equations of the form 3 X
D11 1 1 k;h D44 U;r A44 uh þ w;h aF 11 ðr2 wÞ;h ¼ 0; — r r r
m 4 q r ln r 96D11
; ij C j ¼ b a i
i ¼ 1; 2; 3
ð56Þ
j¼1
whose solutions yield the integration constants in which
C 1 ¼ B1m ;
C 2 ¼ A1m ;
C 3 ¼ A2m :
Following the procedure proposed by Nosier and Reddy [9], three coupled Eqs. (48) can be reduced to yield two uncoupled equations as:
11 ¼ 0; a
D44 r2 U ¼ A44 U;
ð49aÞ
21 a
ð49bÞ
22 ¼ D11 am ðam m1 Þbam 2 ; a
D11 r4 w ¼ q
D11 A44
r2 q:
1 r
D11 A44
uh ¼ w;h
ðr2 wÞ;r þ
Satisfying the boundary conditions (39b), the coefficients are
1 ¼ wpm j ; 12 ¼ bam ; a 13 ¼ bam þ2 ; b a r¼b a 1 m 0 b1 ¼D m1 cIam ðcbÞ; Iam ðcbÞ ; b b
b 1a 23 ¼ D11 ðam þ 2Þðam þ m1 Þbam þ 4 D m ðam þ 1Þm1 bam 2 ; a
Eqs. (49a) and (49b) referred to as the edge-zone (see Eq. (29)) and interior equation, respectively. It is obvious from Eq. (49) that the LPT totals order of differential equations is sixth which is the same as the FST and both lower than the Reddy’s higher-order theory [21] which is the eighth-order. Using Eqs. (25) and (48) one can obtain
ur ¼ w;r
ð57Þ
D44 1 D U;h 211 q;r ; A44 r A44
D11 1 2 D44 D 1 U;r 211 q;h ; ðr wÞ;h A44 r A44 A44 r
mD11 ¼ D w b 1 r2 wpm;rr j b wpm;r jr¼b D 2 11 pm;rr jr¼b r¼b b b 1 a2 2 mD m þ r wpm jr¼b r2 wpm;r jr¼b ; b b
ð50aÞ
ð50bÞ
D44
32 ¼ 0; cI0a ðcbÞ; a A44 m 3 ¼ D11 am r2 wpm j ; b r¼b A44 b
31 ¼ a
33 ¼ a
4D11 A44
ð58aÞ
ð58bÞ
am ðam þ 1Þbam 1 ; ð58cÞ
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S. Sahraee / Composite Structures 88 (2009) 548–557
where
b 1 ¼ D11 D11 : D A44 5.2.2. SSC circular sector plates ij and In view of the boundary conditions (42b), the coefficients a j in Eq. (56) for SSC sectorial plates take the form b
11 ¼ 0; a 21 a
12 ¼ bam ; a
D44 am ¼ Iam ðcbÞ; A44 b
23 ¼ ðam þ 2Þbam þ1 þ a
13 ¼ bam þ2 ; a am 1
22 ¼ am b a 4D11 A44
1 ¼ wpm j ; b r¼b
ð59aÞ
;
am ðam þ 1Þbam 1 ;
D44
32 ¼ 0; cI0a ðcbÞ; a A44 m 3 ¼ D11 am r2 wpm j : b r¼b A44 b
33 ¼ a
ð59bÞ
4D11 A44
am ðam þ 1Þbam 1 ;
5.2.3. SSF circular sector plates ij and Based on the boundary conditions (45b), the coefficients a j in Eq. (56) for SSF sectorial plates are given as b
am D44 b D11
Iam ðcbÞ;
12 ¼ 0; a
where
13 ¼ 4am ðam þ 1Þbam 1 ; a
Note that FST analysis of FGM sectorial plate is not available in the open literature at the time of this writing. However, this is easily obtainable from the above aforementioned expressions by substituting a = b = 0 and imposing the shear correction factor 5/6 into the plate stiffness coefficient A44. 6. Results and discussion
ð59cÞ
11 ¼ a
ð60cÞ
b 2 ¼ D11 D44 : D A44
2 ¼ wpm;r j D11 r2 wpm;r j ; b r¼b r¼b A44 31 ¼ a
b 2a m bam 2 ; a m ðam þ 1Þbam 2 ; 32 ¼ 2D44 am ðam þ 1Þbam þ 8 D 31 ¼ 2D44 a a a 2 D2 c 33 ¼ 44 c2 I00am ðcbÞ þ m Iam ðcbÞ I0am ðcbÞ ; a b b A44 3 ¼ 2am D44 1 wpm j wpm;r j b r¼b r¼b b b b2 1 2 2am D þ r wpm jr¼b r2 wpm;r jr¼b ; b b
ð60aÞ
¼ r2 w j : b 1 pm r¼b b 1 am m1 cI0 ðcbÞ; 1 Ia ðcbÞ ; a 21 ¼ D 22 ¼ D11 a m m1 bam 2 ; a am b m b b 1a 23 ¼ D11 ðam þ 2Þðam þ m1 Þ ma2m bam þ 4 D m ðam þ 1Þm1 bam 2 ; a a 2 mD m ¼ D b mwpm D11 wpm;rr jr¼b 11 wpm;r jr¼b 2 11 b b r¼b 2 b b 1 r2 wpm;rr j þ m D 1 am r2 wpm j r2 wpm;r j D r¼b r¼b r¼b ; b b ð60bÞ
To validate the propose formulation, an Aluminum/Zirconia’s functionally graded material with m = 0.3 and Er = 0.4636 is considered here in which Er denotes the ratio of Young’s modulus of metal component to ceramic component which are taken from Praveen and Reddy [22]. Table 1 presents a comparison study of ¼ 103 wDm =ðq0 b4 Þ for SSS, the non-dimensional deflection in w SSC and SSF homogeneous sectorial plates from the present LPT results with the results from CPT given by Timshenko and Woinowsky-Krieger [23]; and with Abacus finite element program and FST by Wang et al., [8]. It is seen that the results of the present theory are in excellent agreement with the results of Abaqus finite element program and Mindlin theory. Fig. 2 shows the non-dimen of functionally graded SSS, SSC and SSF sional deflection, w, plates versus the power law index n for various values of thickness radius ratio h/b. The opening angle h0 is considered to be p/3. It can decreases with increasing the values of n and inbe seen that w creases with increasing the values of h/b due to the effect of trans for verse shear deformation. Also, it is concluded that w homogeneous plates is considerably more than the corresponding value for the FG sectorial plates. Variation of the non-dimensional deflection of functionally graded SSS, SSC and SSF plates versus the Er is demonstrated in Fig. 3 for various values of n with h/b = 0.1. The opening angle h0 and circumferential angle h are considered
Table 1 of SSS, SSC and SSF homogeneous sectorial plates Comparison of the non-dimensional deflection, w, h/b p=6Þ SSS plate, h0 ¼ p=3; wð0:75b;
p=6Þ SSC plate, h0 ¼ p=3; wð0:75b;
p=4Þ SSF plate, h0 ¼ p=2; wðb;
a b c
Thin platea
0.001
0.9248
Abaqusb 0.9248
0.1
–
1.0210
0.2
–
1.3097
0.001 0.1 0.2 0.001 0.1 0.2
0.4674 – – 63.2800 – –
0.4673 0.5872 0.9262 63.2800 64.6480 66.9260
Timshenko and Woinowsky-Krieger [23]. From Ref. [8]. From present article with n = a = b = 0 and imposing the shear correction factor 5/6 into the plate stiffness coefficient A44.
FSTb – 0.9247c – 1.0209 – 1.3094 0.4674 0.5872 0.9262 63.2800 64.6474 66.9254
Present results 0.9247 1.0208 1.3095 0.4671 0.6079 0.9926 63.3217 65.2234 68.0697
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S. Sahraee / Composite Structures 88 (2009) 548–557
a
b 0.951
0.9
0.9
SSS
SSC
0.85
0.8
SSF
11
0.8 0.75
h/b=0.001, h/b=0.1, h/b=0.2.
0.65
⎯w
0.6
10
h/b=0.001, h/b=0.1, h/b=0.2.
0.7 0.6
h/b=0.001, h/b=0.1, h/b=0.2.
9
⎯w
0.7
⎯w
c 12
0.55
0.5
8
0.5
θ0=π/3, θ=π/6, r=0.75b.
0.4
0.45
θ0=π/3, θ=π/6, r=0.75b.
0.4
0.3
0.35
0.2
0.25
θ0=π/3, θ=π/6, r=b.
7
0.3
0
10
20
30
40
50
6 0
10
20
n
30
40
50
0
10
20
30
40
50
n
n
of FG sectorial Levinson plates versus the power law index n for various values of thickness radius ratio h/b with h0 = p/3. (a) SSS, Fig. 2. Non-dimensional deflection, w, p=6Þ, (b) SSC, wð0:75b; p=6Þ and (c) SSF, wðb; p=6Þ. wð0:75b;
a
b 1.2
2.5
1
n=1, n=2, n=5.
3
n=1, n=2, n=5.
0.9
n=1, n=2, n=5.
2
0.8
SSS
2
0.7
⎯w
⎯w
2.5
⎯w
c
1.1
3.5
1.5
SSC
0.6
SSF
0.5
1.5
1
0.4 1
r=0.75b, h=0.1b, θ0=π, θ=π/2.
r=b, h=0.1b, θ0=π/4, θ=π/8.
r=0.75b, h=0.1b, θ0=π, θ=π/2.
0.3
0.5
0.2 0.5 0.1
0.2
0.3
0.4
0.5
0.1
0.1
0.2
Er
0.3
0.4
0.5
0.1
0.2
0.3
Er
0.4
0.5
Er
of FG sectorial Levinson plates versus Er for various values of n with h/b = 0.1. (a) SSS, h0 ¼ p; wð0:75b; Fig. 3. w p=2Þ, (b) SSC, h0 ¼ p; wð0:75b; p=2Þ and (c) SSF, ðb; p=8Þ. h0 ¼ p=4; w
to be p and p/2 for SSS and SSC plates, respectively, and also p/4 increases as Er inand p/8 for SSF plates. It is observed that w creases. Fig. 4 depicts the non-dimensional deflection of functionally graded SSS, SSC and SSF plates versus the opening angle h0 for various values of n with h/b = 0.1. The circumferential angle h is increases as h0 inconsidered to be h0/2. It is concluded that w of SSF plates more highly increases creases. Also, it is seen that w as h0 tends to p. It is because of the SSF plates are not globally in equilibrium when h0 = p. Figs. 5 and 6 shows, respectively, the varrz ¼ srz =q and iation of the non-dimensional shear stresses, s shz ¼ shz =q, of functionally graded SSS, SSC and SSF plates obtained
a
by the constitutive law versus the non-dimensional thickness coordinate variable z ¼ z=h for various values of n and h/b with h0 = p/3. The circumferential angle h is considered to be p/6 for Fig. 5 and p/4 for Fig. 6. Note that the value of shz at h = h0/2 is zero due to symmetry of circular sector plates about the line h = h0/2. It is seen hz decreases with increasing the rz and s that both shear stresses s values of h/b. Also, it is observed that the maximum non-dimensional shear stresses of FG sectorial plates are greater than the corresponding values of the homogeneous plates and do not occur at z ¼ 0 due to non-homogenous mechanical properties of the FGMs.
b 1.8
c 300
1.6
5
SSS
1.2
250
1
150
0.8 2
0.6
1
θ0
2
0.2
3
0
θ=θ0/2, r=b, h=0.1b.
100
θ=θ0/2, r=0.75b, h=0.1b.
0.4
θ=θ0/2, r=0.75b, h=0.1b.
1
SSF
⎯w
3
0
n=0, n=2, n=5.
SSC 200
⎯w
⎯w
4
n=0, n=2, n=5.
1.4
n=0, n=2, n=5.
50
0 1
θ0
2
3
0.5
1
θ0
1.5
2
of FG sectorial Levinson plates versus h0 for various values of n with h/b = 0.1. (a) SSS, 0:1p 6 h0 6 p; wð0:75b; Fig. 4. w h0 =2Þ, (b) SSC, 0:1p 6 h0 6 p; wð0:75b; h0 =2Þ and (c) h0 =2Þ. SSF, 0:1p 6 h0 6 0:7p; wðb;
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S. Sahraee / Composite Structures 88 (2009) 548–557
b
a0 n=0, h=0.1b, n=2, h=0.1b, n=5, h=0.1b; n=0, h=0.2b, n=2, h=0.2b, n=5, h=0.2b.
-0.1
-0.2
c0
0 -0.2 θo=π/3, θ=π/6, r=0.75b.
-0.4
n=0, h=0.1b, n=2, h=0.1b, n=5, h=0.1b; n=0, h=0.2b, n=2, h=0.2b, n=5, h=0.2b.
-0.05
-0.6
-0.1
-0.4
n=0, h=0.1b, n=2, h=0.1b, n=5, h=0.1b; n=0, h=0.2b, n=2, h=0.2b, n=5, h=0.2b.
-0.8 -1
θo=π/3, θ=π/6, r=0.75b. -0.5
-1.2
-0.6
-1.4
SSS
-0.7 -0.5
⎯τrz
⎯τrz
⎯τrz
-0.3 -0.15
θo=π/3, θ=π/6, r=0.75b.
-0.2
SSC
SSF
-0.25
-1.6
-0.25
0
0.25
-0.5
0.5
-0.25
0
0.25
0.5
-0.5
-0.25
⎯z
⎯z
0
0.25
0.5
⎯z
rz ¼ srz =q, of FG sectorial Levinson plates versus the non-dimensional thickness coordinate variable z ¼ z=h for various values of n and Fig. 5. Non-dimensional shear stress, s p=6Þ, (b) SSC, wð0:75b; p=6Þ and (c) SSF, wðb; p=6Þ. h/b with h0 = p/3. (a) SSS, wð0:75b;
b
0
n=0, h=0.1b, n=2, h=0.1b, n=5, h=0.1b; n=0, h=0.2b, n=2, h=0.2b, n=5, h=0.2b.
-0.1 -0.2 -0.3
-0.3 -0.4
-0.5
-0.5
-0.6
-0.6
-0.7
θo=π/3, θ=π/4, r=0.75b.
-0.8
0 θo=π/3, θ=π/4, r=b.
-1 -2 -3
-0.7
θo=π/3, θ=π/4, r=0.75b.
-0.8
-0.9
n=0, h=0.1b, n=2, h=0.1b, n=5, h=0.1b; n=0, h=0.2b, n=2, h=0.2b, n=5, h=0.2b.
-4 -5 -6
-0.9
-1
-1
-1.1
-1.1
SSS
-1.2 -0.5
n=0, h=0.1b, n=2, h=0.1b, n=5, h=0.1b; n=0, h=0.2b, n=2, h=0.2b, n=5, h=0.2b.
-0.2
⎯τθz
⎯τθz
-0.4
c
0 -0.1
⎯τθz
a
-7
SSS
SSF
-8
-1.2 -0.25
0
0.25
0.5
-0.5
-0.25
⎯z
0
⎯z
0.25
0.5
-9 -0.5
-0.25
0
0.25
0.5
⎯z
hz ¼ shz =q, of FG sectorial Levinson plates versus the non-dimensional thickness coordinate variable z ¼ z=h for various values of n and Fig. 6. Non-dimensional shear stress, s p=4Þ, (b) SSC, wð0:75b; p=4Þ and (c) SSF, wðb; p=4Þ. h/b with h0 = p/3. (a) SSS, wð0:75b;
7. Conclusions The pure bending problem of functionally graded circular sector plates is considered in the present work based on the Levinson plate theory (LPT). This theory captures the higher-order effect by assuming the same third-order polynomials in the expansion of the in-plane displacements through the thickness of plate as in the Reddy’s higher-order theory and therefore it no needs to have shear correction factor. In this article the unknown displacement functions of the LPT are represented in two types: (1) in terms of the response of homogeneous sectorial plates based on the classical plate theory and (2) by solving the two independent edge-zone and interior equations, directly. An excellent rate of convergence was showed and the results were in good agreement with the results of other theories. Also, it is seen that the LPT solution of the deflection for homogenous sectorial plates are significantly higher than those for functionally graded ones; especially for the thick plates. Thus, the gradients in material properties play an important role in determining the response of the FGM plates. References [1] Reissner E. The effect of transverse shear deformation on the bending of elastic plates. J Appl Mech 1945;12:69–77. [2] Mindlin RD. Influence of rotary inertia and shear on flexural motions of isotropic, elastic plates. J Appl Mech 1951;18:31–8. [3] Reissner E. On the theory of bending of elastic plates. J Math Phys 1944:184–91.
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