Acta Mechanica 146, 183 197 (2001) ACTA MECHANICA Springer-Verlag 2001 Buclding and free vibration of elastic plates using simple and mixed shear deformation theories A. M. Zenkour, Kafr El-Sheikh, Egypt (Received January 18, 1999; revised October 15, 1999) Summary. Simple and mixed shear deformable theories accounting for the transverse shear, in the sense of Reissner-Mindlin's thick plate theory, are employed in the construction of variational statements for rectangular flat plates. Analytical solutions for natural frequencies and buckling loads of anisotropic plates under various boundary conditions are developed. A variety of simply-supported, clamped and free boundary conditions is considered, and comparisons with the existing literature are made. 1 Introduction The classical plate theory based on the Kirchhoff hypothesis for homogeneous and composite plates is inadequate, even if the plate is fairly thin [1]-[3]. It underestimates deflections and overestimates stresses, natural frequencies and buckling loads. Refined plate theories altowing one a more adequate description of structural response characteristics are needed. They should include transverse shear deformation and transverse normal stress and should account for the higher-order effects. They provide improved global response estimates for deflections, stresses, vibration frequencies and buckling loads of moderately thick plates when compared to the classical plate theory. In these theories the displacements or stresses (or both) are expanded as a linear combination of the thickness coordinate and undetermined functions of position in the reference surface to reduce the three-dimensional (3D) elasticity problem to a 2D one. The usual refined theory considered in the treatment of the dynamic response of flat orthotropic plates is the simple first-order transverse shear deformation plate theory (SFPT) [4]-[10]. The theory is given for statics by Reissner [4], [5] and extended to dynamics by Mindlin [6]. In this theory, the in-plane displacements are expanded up to the first term in the thickness coordinate, and the rotations of normals to the mid-surface are assumed to be inde- pendent of the transverse deflection. Related to SFPT, it is well known that: (i) it requires the introduction of transverse shear correction factor; (ii) disregards the effect of transverse normal stress; and (iii) involves a constant distribution of transverse shear stresses across the thickness which prevents from fulfilling the tangential conditions on the bounding planes. In the case of multilayered composite structures the same shortcoming appears in the fulfill- ment of continuity conditions at the layer interfaces (see [11]-[18]). For SFPT, any varia- tional principle may be used to derive a consistent set of differential equations governing the motion of the plate.
Buclding and free vibration of elastic plates using simple and mixed shear deformation theories
A. M. Zenkour, Kafr El-Sheikh, Egypt
(Received January 18, 1999; revised October 15, 1999)
Summary. Simple and mixed shear deformable theories accounting for the transverse shear, in the sense of Reissner-Mindlin's thick plate theory, are employed in the construction of variational statements for rectangular flat plates. Analytical solutions for natural frequencies and buckling loads of anisotropic plates under various boundary conditions are developed. A variety of simply-supported, clamped and free boundary conditions is considered, and comparisons with the existing literature are made.
1 Introduction
The classical plate theory based on the Kirchhoff hypothesis for homogeneous and composite
plates is inadequate, even if the plate is fairly thin [1]-[3]. It underestimates deflections and overestimates stresses, natural frequencies and buckling loads. Refined plate theories altowing
one a more adequate description of structural response characteristics are needed. They
should include transverse shear deformation and transverse normal stress and should account
for the higher-order effects. They provide improved global response estimates for deflections,
stresses, vibration frequencies and buckling loads of moderately thick plates when compared to the classical plate theory. In these theories the displacements or stresses (or both) are
expanded as a linear combination of the thickness coordinate and undetermined functions of position in the reference surface to reduce the three-dimensional (3D) elasticity problem to a 2D one.
The usual refined theory considered in the treatment of the dynamic response of flat orthotropic plates is the simple first-order transverse shear deformation plate theory (SFPT)
[4]-[10]. The theory is given for statics by Reissner [4], [5] and extended to dynamics by Mindlin [6]. In this theory, the in-plane displacements are expanded up to the first term in the thickness coordinate, and the rotations of normals to the mid-surface are assumed to be inde- pendent of the transverse deflection. Related to SFPT, it is well known that:
(i) it requires the introduction of transverse shear correction factor; (ii) disregards the effect of transverse normal stress; and
(iii) involves a constant distribution of transverse shear stresses across the thickness which prevents from fulfilling the tangential conditions on the bounding planes.
In the case of multilayered composite structures the same shortcoming appears in the fulfill- ment of continuity conditions at the layer interfaces (see [11]-[18]). For SFPT, any varia- tional principle may be used to derive a consistent set of differential equations governing the motion of the plate.
184 A.M. Zenkour
In the mixed first-order transverse shear deformation plate theory (MFPT), both the dis- placements and stresses must be considered arbitrary. For this reason a mixed variational for-
mula should be used [19]-[24]. The utilization of the mixed variational principles allows one to treat the plate problems by introducing kinematics assumptions with any power of the thickness coordinate. Also the transverse shear stresses are consistent with the surface condi- tions. So, the rationale for the shear correction factor required to the SFPT is obviated. In addition, the effect of transverse normal stress is taken into account.
In the present study, a relationship between the simple and mixed first-order transverse shear deformation theories is presented. The free vibration and buckling problems of ortho- tropic and transversely-isotropic (transtropic) flat plates with various boundary conditions are developed. The edges y = 0, b will be considered simply-supported while the remaining ones, x = 0, a, having arbitrary combinations of edge conditions. Numerical results of natural frequencies and critical buckling loads of such plates are presented. Comparisons with some of the available results (obtained for simply-supported edge conditions) are undertaken and appropriate conclusions concerning the various effects are formulated.
2 Derivation of governing equations
Consider a rectangular plate of length a, width b and uniform thickness h. The mid-plane of the plate will be taken as the xy plane, and the x- and y-axes are directed along the edges. The z-axis is taken perpendicular to the mid-plane and positive in a downward direction. Let the plate be subjected to a distributed transverse load f as well as the in-plane edge forces $1 and $2 (applied in the directions x and y, respectively, and considered positive in tension).
We start in the usual way by proposing the first-order displacement field:
Ul(x ,y , z , t ) : u (x ,y , t ) 4- z r
where (u, v, w) denote the displacements of a point (x, y) on the mid-plane, and ~p and ~ are the rotations of normals to mid-plane about the y- and x-axes, respectively.
The strain field for the assumed displacement field follows immediately as
Ou O~ Ov O~ s :~4-Z = - - , Oz ' c22 ~ y + Z Oy
Ow Ow e23 ~ + Oy ' e13 = ~ 4- 0~ ,
s = 0
0v 0r "
(2)
For SFPT, the stress-strain relations of an orthotropic body are given by
i +oo 0"2S2 C12 C22 0 0
0"$3 = 0 C44 0 0
CrlS 3 0 0 c55 0
O'$2 0 0 0 C66
o[+11} 0 ~22
g'23 :
s
s
(3)
Simple and mixed shear deformation theories 185
in which c~j are the t ransformed material constants. Note that the transverse normal stress eras a
is dropped. The stress resultants in this case are given by
+h/2 {N;Sq, M~;Sq}= f @q{1 , z }dz ,
+h/2 @s 3 = f Kp0.Sadz, (p,q = 1,2) ,
-h/2
(4)
where Kp (K1 = K2 = k) are shear factors to correct for the errors stemming from (2) that 0.s 3
and 0.s a are constants over the thickness of the plate (i.e., are not functions of z). For M F P T , the stress and displacement fields are taken to be arbitrary. Then, the non-.
vanishing stress field is assumed to be of the form [19], [22]:
% ~ = G ( ~ y) 1 - z p3 \ ~
4
0"33 ~ 33 k ~ r = l .
(p,q = 1,2).
(5)
The functions G (~ , G(~ ) and G~ ~ may be got easily from the point that the stresses apMq and 0.~;~a ~ satisfy the following stress resultants:
+h/2
-h/2
QplV~ = f 0.~,~dz, ( p , q = 1,2). -h/2
(6)
M satisfies the Also the functions G ~ ) arise f rom the point that the transverse normal stress 0-33 following conditions:
+h/2 +h/2 0.3~ lz=_h/~ = -- f 0.~ , y dz:O, f z0. dz:O. (7)
-5/2 -h/2
Therefore, the final expressions for the stress components can be written in terms of their resultants and the thickness coordinate z:
Pq ~- z ~ - 3 Q p 3 1-- M % h ~ ' - 2--h- '
0.33=~- 1 - 2 - 5 1 - z , (p,q = 1,2) .
(s)
It is to be noted that the transverse shear stresses 0.~a~ and 0.~ are functions of z and vanish on the bounding planes (z = •
186 A.M. Zenkour
2.1 Equations o f motion
The simple and mixed variational formulations based upon Hamilton's principle are given, respectively, by [19], [22]:
] 0 = f fff (~i%6ui + 6U s) dv + 6V s dr, hi . V
(9)
t2F ] 0 = f fff (~gi6ui + 5(a~sij) - 6R ~) dv + 6V M dr,
t l k v (lO)
where (h, t2) is a time interval; ~ is the density of the undeformed body, U s is the strain energy density, and R M is the complementary energy density. Note that quantities with superscript "S" refer to the SFPT while quantities with "M" refer to the MFPT. The potential energy V ~ (a = S, M) of the applied loads can be defined as a function of displacement field ui and the applied loads as follows:
v s : -fffB{u{ dv - ffFiui ds, (11) v s~
V M = - f f f B i u i dv - f fF~u i ds - f f n j c~g(u i - u~) ds , v so s~
(12)
where nj are the components of the unit vector along the outward normal to the total surface S~ + S~; B~ are the body forces measured per unit volume of the undeformed body; F~ are the prescribed components of the stress vector, per unit area of the surface S~, and u* are the prescribed components of the displacements of the remaining surface S,. In the absence of the body forces and the prescribed displacements, we have for the first varia- tion of V ~
j r ( awa owa ) gV ~ = S, -~z-~z +$2 O~ O y - f 6was . s~
(13)
The next step in deriving the governing equations consists of the substitution of Eqs. (1), (2), (3), (8), and (13) into the two variational formulations (9) and (10). The extremum conditions give the following dynamic equations:
oNgl 0N~2 ~h~ = ~ + Oy
oN~: ONe2
o ~h /~=-~ -x + ~ - y + f + ~ x x SI~X- x +~yy $2 ,
h 3 .. 0M~1 OM~s ~ h 3 OM~2 0M~2 o ~ ~ - o x ~--~-y - Q13 , Q -i~ ~ = Ox +--O~-y - Q23 "
(14)
Simple and mixed shear deformation theories 18~
2.2 Boundary conditions
The form of the geometric and force boundary conditions is also given below:
G e o m e t r i c (Es sen t ia l ) F o r c e ( N a t u r a l )
v Ng2n~ + N~2%
(o Q~% + s~ ~ ~ : + Q~ + s~ ~ ~ OyJ y
M~2nx + M~2n~
(15)
where (n~, nv) denote the direction cosines of a unit normal to the boundary of the mid-plane.
2.3 Constitutive equations
For the SFPT, the stress resultants are related to the strains by Eq. (3) while for the M F P T
they will be derived f rom the extremum condition of the mixed variational formulat ion (10). In general, the constitutive equations are given by:
01 N~2 = ~ A2~ 0 ~yy ,
NI~ 0 A~6 Ov Ou
M~2 = 2 z~ 0 y '
M~2 0 D~6 0p 0r
where
Q 2~ = A~ ( ~ + O~y ) ,
QI~ = A5a5 ( r + 0~xW) ,
(16.1)
(16.2)
V ] [ 1-1 A1N~ AIM : h all al2 M _ 5h M h L A~2 A~{ L. a12 a22 .J ' A,.,. 6a~r ' A66 a66
Here, Ei, G# and u~j stand for Young's moduli, shear moduli and Poisson's ratios, respec-
tively. In the case of a transtropic body, the specialized counterparts of (17) become:
E Ev E 5 1 1 = C22 1 - - //' 2 ' C 1 2 1 - - 1,,2 ' C4 4 = 5 5 5 = G ' , e66 = G = 2(1 + u) (18)
1 u 1 1
511 =522 :-- : 512-- ~ a44 =555 = ~ ~ a66 =-- �9 E E G
3 Solution procedure
The simple and mixed variational formulations will be extended here in order to analyze the
free vibration and the buckling problems of rectangular plates. The S F P T and M F P T will be
established in this section by suppressing the in-plane displacement degrees of freedom.
The following boundary conditions are considered:
(i) at edges y = 0, b:
Simply-supported (S): w = #? = M~2 = 0,
(ii) at edges x = 0, a:
Simply-supported (S): w = ~ = M~I = 0,
Clamped (C): w = ~ = ~ = 0, Ow
Free (F): M~I = M~2 = Qla3 q- S1-0~ z = 0.
3.1 The free vibration problem
The following representation for the displacement quantities (deleting stretching effects) is
appropriate in the analysis of the free vibration problem ($1 = 5'2 = f = 0):
r = gJm~ X'(AmX) sin#ny ~ e i~t ,
qe.~n X(A,~x) cos PnY )
(19)
where w(=cz,~n) denotes the eigenfrequency associated with the (m,n) th eigenmode, #n = nrc/b, and W,~, kP,~n and ~ , ~ being arbitrary parameters. The function X(A,~x) can be
constructed for any combination of simply-supported (S), clamped (C) and free (F) bound- ary conditions on the edges x = 0, a. The different forms of X (Amx) and the corresponding
values of A,~ are defined in the Appendix [25] (see also [21]). Substitution of Eqs. (2), (3), (8), and (16) considered in conjunction with (19) in:th6 varia-
tional formulations (9) and (10) yields a system of algebraic equations expressed in a compact
form as
([L] - w2[R]) {A} = {0}, (20)
Simple and mixed shear deformation theories
where {A} denotes the column
{zi} ..... .....
The elements of the symmetric matrices [L] and [R] are expressed as:
~2 Oo~ c 2 ~ a Lla = #nA44{1, Lu2 = m 1193 -- #nD66g2 + Asa~2,
L23 = Ara#n(D66{2 - D12~'4 ) ,
ha t t l l ohm1, R22 = 0 ~ ~2 ,
where
: 2
0
~:3 = ) [ J ( H( /~mZ) ]2 dx, 0
2 a 2 a c~' L33 = /~mD66~2 -}- #nD22~1 + A44~1 ;
h 3 R33 = ~o ~-~ ~1 , R12 = Rla = R2a = 0,
~2 = )" [X'(Amx)] 2 dx , 0
0
The condition expressed by det ([L] - ~2 [R]) = 0 yields the eigenfrequencies w.
189
(21)
(22)
(2a)
3.2 The static buckling problem
The representation in (19) specialized for a~ -+ 0 is appropriate in dealing with the (static) compressive buckling problem. The simplest case, to derive some results which concern the
buckling of rectangular plates, is obtained when the forces 5'1 and $2 are constant throughout
the plate, and the dynamic as well as the transversal load terms are dropped. Assuming that
there is a given ratio between these forces so that & = - /3 and $2 = -~/3; ,7 = 52/$1, we get
([L 1 - ,8[5]) {zi} = {0}. (24)
The matrices {A} and [L] are defined in Eqs. (21) and (22), respectively, while the elements
Solving (24) for the boundary conditions, we shall find that the assumed buckling of the plate
is possible only for definite values of/3. The smallest of these values determines the desired cri- tical value.
I f the forces & and 52 are not constant, the problem becomes more involved, since Eq. (24) has in this case variable coefficients, but the general conclusion remains the same. Let, for example [26]
5 1 = - ~ ( 1 - C 0 b ) , 5 2 = 0 , (26)
where co is a numerical factor. Equation (24) is still the same while the elements of [S] become
as well as for the transtropic material defined by:
E = 20.83 Msi, U = 10.00 Msi, G ~ = 3.71 Msi, u = u ~ = 0.44.
The numerical applications are done, unless otherwise stated, using M F P T for homo-
geneous flat plates with various boundary conditions. The designations SS, CC, CS, and CF refer to the edge conditions at x = 0, a, only. The edges y = 0, b are invariably assumed to be
simply-supported. For the cases of the orthotropic and transtropic plates and for the input
data presented above, Tabs. 1 - 4 display the eigenfrequencies and critical buckling loads
obtained in the framework of various theories. The exact three-dimensional elasticity solu-
tions of Srinivas and Rao [27] for simply-supported square plates are used to asses the
improvement in the prediction of frequencies. The results obtained in [28], as per the higher-
order shear deformation theory (HSDT) developed by Reddy, and of the theory referred to as
DT, specialized for 6A = 0 developed in [29] by Librescu et al. (see also [30]), are also used in
the comparisons.
The present results obtained using M F P T are found to be in excellent agreement with
their counterparts available in the field literature, obtained for SS edge conditions (see
Table 1. Eigenfrequencies, aJ = aJh,~/ev, of a simply-supported orthotropic square plate (a/h = 10)
SFPT s
3 ~_2 5 m n Exact s HSDT b DT ~ MFPT d k = 1 k = ~- k = ~ /c = ~-
Results obtained in [27] using the 3D elasticity theory, b Results reported in [28] using HSDT. c Results reported in [29] using DT. a Results obtained in this paper.
Simple and mixed shear deformation theories 19 [
Table 2. Nondimensional fundamental frequencies c~ = a,'(a2/h) ~ / E 2 of a simply-supported ortho- tropic rectangular plate
Figures l, 3 and 4 display the variation of dimensionless fundamental frequency (�9 and
respectively of biaxial and uniaxial buckling loads (/~) vs the plate thickness ratio (= a/h). They reveal that the variations of c? and/~ are sensitive to the variation of the plate thickness
ratio and this irrespective of the considered boundary conditions. However the CF instance
constitutes an exception in the sense that the considered variation of a/h has little effect on
the variation ofc~ and/~. This effect may vanish for a/h >_ 5. Figures 2 and 5 display respectively the variation of cO and o f / J vs the plate aspect ratio
( - a/b). They reveal also the sensitivity of cO and/~ to the variation of a/b parameter. Figure 6
contains plots of the critical buckling loads (/3) as functions of transverse-to-axial buckling
ratio q/(= $2/$1). It reveals that 7 has a great effect on/~ irrespective of the boundary condi-
tions. Table 5 emphasizes the effects of the aspect ratio (a/b) and mode number (m) on the
natural frequencies of orthotropic plates. The frequencies increase with the increase of a/b and m. The CC case shows the highest sensitivity in the context of the considered edge condi-
tions. Finally, Table 6 contains the dimensionless critical buckling loads/3 of transtropic rectan-
gular plates under combined bending and compression, $ 1 = - - / 3 ( 1 - - Coy~b), $ 2 = 0. It reveals
that the variation of the critical buckling loads is very sensitive to the variation of the plate
aspect ratio. In addition, these critical buckling loads are increased with increasing the numer-
ical factor co, the thickness ratio a/h, and the aspect ratio a/b. In the case of a/b = 2 only,
and for other values of a/h and co, the CF instance shows the highest sensitivity.
5 Conclusions
The simple and mixed variational formulations are used to develop both the analytical and numerical solutions for anisotropic elastic plates based on Reissner-Mindlin's thick plate
theory. The plate is considered to be subjected to a distributed transverse load as well as in-plane edge forces. Numerical results are presented for natural frequencies and critical buckling loads of rectangular plates subjected to various edge conditions. Comparisons are
Simple and mixed shear deformation theories
12
O9
lO
8
6
4
2
a / b = 1 / C C
CF / f I i I i I i t i I
4 8 12 16 20
a / h
24
20
16
(0 12
8
4
q
0
0.25
a / h = L0 / .
CC CS SS
CF , I r I , I h I i I r i i I , I i
0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5
a / b
5
4
2
1
0
y = 1 ~ _ - ~ - C C
a / b = l
CS
SS
CF
, I i r , r i F ~ r
0 4 8 12 16 20
a / h
t923
Fig. 1. Effect of plate thickness ratio (a/h) on the nondimensional funda- mental frequency ~ = ,~(a2/h) ~/-~/E2
Fig. 2. Effect of plate aspect ratio (a/b) on the nondimensional funda- mental frequency ca = co(a2/h) ~ / E 2
Fig. 3. Effect of a/h on the biaxial critical buckling load ~ = 13a2/(haF_,2)
made to show that the results obtained using the present theories are in close agreement with
the available solutions in the literature. The results of M F P T and SFPT themselves are com-
pared to show the effect of the shear correction factors. Both M F P T and SEPT (with a proper
shear correction factor) allow one to treat the response in buckling and free vibration of
homogeneous plates for a variety of boundary conditions. In general, SFPT predicts frequen-
194 A.M. Zenkour
lO
9
8
7
5
4
3
2
1
7 = 0 a / b = l
i I i I h I i I
4 8 12 16
I C C
CS
SS
CF
i I
2O
a / h
6
5
4
2
CC
CS
SS
1 CF a / h = 10
0 T i ~ I , l i i T I T f t r I I
0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5
a / b
8 CC
7
6 CS
_ 5 SS
4 CF
3
2
1
0 i ] l l l l l l l ] i l l ] l
-0.5 0 0.5 l 1.5 2 2.5 3
a / h = l O
a / b = l
i I i r J
Fig. 4. Effect of a/h on the uniaxial critical buckling load/3 =/3a2/(haE2)
Fig. 5. Effect of plate aspect ratio (a/b) on the biaxial critical buckling load fl = fla2/(haE2)
Fig. 6. Effect of transverse-to-axial 3.5 4 4.5 5
buckling ratio (3') on the critical y buckling load fi = fla2/(h3E~)
cies and buckling loads significantly different from those of M F P T even if the transverse
shear correct ion factor (for SFPT) is assumed to be 5/6. This puts into evidence the great role
played by M F P T in the model ing of homogeneous plate theories, which in contrast to S F P T
does not require the incorpora t ion of a shear correct ion factor, since it is considered as a
consistent t reatment of transverse shear deformat ions in the anisotropic plates.
Simple and mixed shear deformation theories
l 'able 5. Eigenfrequencies a: ~h X/~/E2 of an orthotropic rectangular plate (a/h = 10, n = 1)
SS sin A~,~x CC sin A.,x - sh A.~x - ~m(cos Ar~x - chAmx) (shAm - sin A.O/(ch Am -- COS Am) CS sin A~x - sh A~.:c - ~jm(cos A~z - chA.~z) (sh Xm § sin $,n)/(ch X m -}- COS/~m)
CF sin A~,x - sh A.~x - ~7.,~(cos Ar~x - chA~.x) (shA~ + sin A.~)/(ch Am + cos )~m)
196 A.M. Zenkour
The values of ~ (= A~a) corresponding to the function X(A~z) for various boundary conditions are given by:
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Author's address: Dr. A. M. Zenkour, Department of Mathematics, Faculty of Education, Tanta Univer- sity, Kafr El-Sheikh, Egypt (E-mail: [email protected])
