oo_xx7&?3’.9n s3.00+ .a0 c 19811 Pcrgnmon Press pk
HIGHER-ORDER SHEAR DEFORMABLE THEORIES FOR FLEXURE OF SANDWICH PLATES-FINITE
ELEMENT EVALUATIONS
B. N. PANDYAt and T. KANT Department of Civil Engineering. Indian Institute of Technology. Powai.
Bombay 400 076. India
(Ruceiwd I5 Jww 1987)
Abstract-A simple isoparametric finite element formulation based on a higher-order displacement model for Rexure analysis of multilayer symmetric sandwich plates is presented. The assumed displacement model accounts for non-linear variation of inplane displacements and constant vari- ation of transverse displacement through the plate thickness. Further. the present formulation does not require the fictitious shear correction coefficient(s) generally associated with the first-order shear deformable theories. Two sandwich plate theories are developed: one. in which the free shear stress conditions on the top and bottom bounding planes are imposed and another, in which such conditions are not impoud. The validity of the prescnl development(s) is establishcxl through. numerical evaluations for dcflcctions/stresses/stress-resultants and their comparisons with the avail- able three-dimensional analyses/closed-form/other finite element solutions. Comparison of results
from thin plate. Mindlin and present analyses with the exact thrcu-dimensional analyses yields some important conclusions regarding the clT&& of the assumptions made in the CPT and Mindlin type thcorics. Thccomparativc study further cstahlishcs the ncccssity of a highrr-order shear dcformahle theory incorporating warping of the cross-.sc%tion particularly for sandwich plates.
I. INTKODUCTION
A multilaycr sandwich plate is a special form of advanced librc-rcinforccd composite
laminate. The litcraturc available in the field of laminated composite plates is enormous
and the rclcvant avaik~blc litcraturc concerning bending stress analysis has been published
rcccntly (Kant and Pandya. 1987). We examine hcrc the available litcruture spkfically
rclcvant to the bending problems of sandwich plates.
Rcissncr (1948) formulated the small deflection theory for the bending of isotropic
sandwich type structures. Since this initial publication, a number of papers have been
published on various aspects of sandwich bending theory. Kao (1965) developed the govern-
ing difkrcntial equations for the non-rotationally symmetrical bending of isotropic circular
sandwich plates by means of a variational thcorcm. The governing equations for an ortho-
tropic clumped sandwich plate are dcrivcd using the variational principle of minimum
potential energy by Folie (1970). The most important contributions were from Srinivas and
Rao (1970) and Pagan0 (1970). who presented exact three-dimensional elasticity solutions
for laminated composite/sandwich plates. Whitney (1972) presented a theory analogous to
Mindlin’s (1951) first-order shear deformation theory for stress analysis of laminated
composite/sandwich plutcs. Later. Lo c/ ~1. (1977). Murthy (1981). Reddy (1984) and
Murty (1985) presented analytical solutions for laminated plate problems using highcr-
order thcorics. Thcsc thcorics include warping of the transverse cross-sections. However,
they have not prcscntcd sandwich plate problems whcrc the effect of warping of the cross-
stxztion is predominant. These analytical solutions are limited to a few simple gcomctrics.
loading and boundary conditions. This limitation is ovcrcomc by adopting the finitcelcmcnt
method as a gcncralizcd numerical solution technique for practical laminated/sandwich
plate problems.
Monforton and Schmit (1969) prcscnted displacement based finite element solutions
for sandwich plates using I6 degrees of freedom. 4 noded rectangular elements. Martin
t On deputation from Sardar Pate1 College of Engineering. Bombay 400 0%. India.
I x7
1168 B. N. PANDYA and 1. KANT
(1967) adopted 9 degrees of freedom, 3 noded triangular elements with assumed dis- placement fields. Cook (1972) developed a I2 degrees of freedom. 4 noded general quadri- lateral element including transverse shear deformation. Finite element solutions for mufti- layer sandwich plates have also been presented by Khatua and Cheung (1972, 1973) using triangular and rectangular plate bending elements. Their formulation considered the ideal type of sandwich construction in which the core layers contribute only to the shear rigidity of the plate. Fazio and Ha (1974) presented finite element solutions by explicit derivation of stiffness matrices for bending and membrane actions ofa rectangular three layer sandwich plate element using the assumed stress distribution approach. Mawenya and Davies (1974) presented a general fo~ufation for an 8 noded quadratic. isoparametric, multilayer plate bending element which permits the layers to deform focally and incorporates the effects of transverse shear deformation in each layer. Hinton et al. (IY75), Reddy and Chao (1981) and Putcha and Reddy (1984) adopted assumed displacement. penalty function and mixed methods. respectively. to develop the finite element formulations. Kant and Sahani (1985) presented a displacement based finite element formulation using a 9 noded Lagrangian/ Heterosis element. These formulations were based on a first-order shear deformable theory (FOST) which is based on the assumption of the constant shear strain distribution through the laminate thickness and requires the use of shear correction coefficients. Recently, Phan and Reddy (1985). Putcha and Reddy (1986) and Rcn and Hinton (1986) presented various finite efemcnt formulations of a higher-order theory for laminated plates. Howcvcr. they have not applied it to sandwich plate problems.
The mofivation for the present development comes from the work of Kant (1982) and Kant ct cd. (1982). which was fimitcd to thick isotropic plates. Pundya and Kant (1987. f9XHu c) and Kitnt and Patldyii (f9XSa.b) cxtcndcd those dcvcfopmcnls for orthotropic :111d
fiuninatctf composite/sandwich pl;~k~. This paper spccificitlly dcitfs with the dcvclopmcnt ilMf application of a C is0pi~rilmCtriC finite cfcmcnt for hcnditig it!WlySiS Of I~~llltifilyCr
symmetric Sitndwicfl pkttcs by ~;ssL~tllin~ ;i high~r-~~r~~r ~Iispl~t~~Il~~nt modcf hitherto not cons&rod. The theory fca<fs to ;I rcafistic (parabolic) variation of’ transvcrsc shcitr strcsscs through the plate thicknsss. It is appficabfc to an I/-faycrcd sandwich plate with [(I/+ I)/?] stifl’foycrsund [(N - 1)/Z] altcrnitting weak cores. The!, nodcd Lagrangian quadraticefcmcnt devcfopcd has 5 cfcgrccs ol’ f’rccdom per node.
The prcscnt higher-order shear deformation theory for symmetric sandwich/faminatcd pfatcs has been d~cfopd by assuming the dispfaccmcnt field in the following form :
in which IV@ rcprcscnts the transverse dispfacemcnt of thr midpfanc and #,r, .!I,,. are the rotations of normals to the midpfanc about the J- and .Y-axes, rcspcctivefy, as shown in Fig. 1. The parameters O:, Q,;are the higher-order terms accounting for the flexuraf mode of deformation in the Taylor s&s expansion and arc also d&cd at the midpfanc. The conditions that the transvcrsc shear strcsscs vanish on the top and bottom facts of the plate arc cquivalcnt to the rcquircmcnt that the corresponding strains be xcro on these surfaces. The transvcrsc shear strains arc given by
Equating ~,_(.Y, _s, f Ait) and *t , r._(~~, _v. +/z/Z) to zero. we obtain
Finite clement evaluations 1269
Fig. I.
( 1.2.3 I - Lomino r*frrwm 01”
(3)
Murthy (19X1) and more recently Rcddy (1984) used conditions (3) to eliminate U.:and Oj!
liom the displaccmcnt field. which contains additional inplanc degrees of freedom (u,, o,,).
In the prcscnt theory. WC proceed with the displacement tieid given by cqns (I) and
conditions (3) arc introduced later in the shear rigidity matrix.
By substitution of eqns (I) in the strain displaccmcnt equations of the classical theory
of elasticity. the following relationships are obtained :
in which
(4)
(5)
The material constitutive relations for the Lth layer can be written as
1270 B. N. PANDYA and T. KANT
(6)
where (a ,. 0:. r , :. t:?. T,~) are thestressand (E,, E:. yIz, y13, y,,) the linear straincomponents referred to the Inmina coordinate axes (1. 2, 3) as shown in Fig. 1 and C,,‘s the reduced material stiffnesses of the Lth lamina and the following relations hold between these and the engineering elastic constants :
The stress-strain relation for the Lth lamina in the laminate coordinate axes (x. y, z) are written as
(8)
in which
are the stress and linear strain vectors with reference to the laminate axes and Q,,‘s are the transformed reduced elastic coefficients in the plate (laminate) axes of the Lth lamina. The transformation of the stresses/strains between the lamina and the laminate coordinate systems follows the usual transformation rule given in Jones (1975).
The total potential energy n of the plate is given by
in which A is the mid-surface arcn of the plate. Y the plate volume, F the intensity of the force vector corresponding to the degrees of freedom S defined as
Thcexprcssions for the strain components given by relations (4) are substituted in expression (IO). The functional given by expression (IO) is then minimized while carrying out explicit integration through the plate thickness. This leads to the following ten stress-resultants for the !I-layered laminate :
Finite clement evaluations 1’71
After integration. these relations are written in a matrix form which defines the stress- resultant/strain relations of the laminate and is given by
M II --- M* Q
x
= [ -- 9 oh: ’ 3, -- 0 1 Q'
iI --- x* *
CD*
or
Q*= {Q:Q:;'; cD* = '(I)* cDt!' 1 1. 11
whcrc
Hi = f (k&h~._ I). i= 1,3.5,7
H= (H,-H,;y). H* = (H5-H,;).
(13)
(14)
(15)
The shear rigidity matrix 3, given by eqn (I 5) is evolved by incorporating an alternate form of conditions (3). namely
1272 B. N. PA~YA and T. KAST
(16)
in it and the resulting theory. higher-order shear deformation theory satisfying zero trans-
verse shear conditions on top and bottom bounding planes of the plate (HOSTI). becomes
consistent in the sense that it satisfies zero transverse shear stress conditions on the top
and bottom boundary planes of the plate. If the conditions, given by eqns (16). are not
incorporated. the resulting non-consistent theory, higher-order shear deformation theory
without satisfying above referred zero transverse shear conditions (HOSTZ). does not
satisfy the zero transverse shear stress conditions on the top and bottom boundary planes
of the plate. In this case the shear rigidity matrix 9; is defined as
(17)
The transvcrsc shear strcsscs rtZ and rt: arc not cvaIuatcd from eqn (8) as the continuity conditions at the intcrfaccs of the fact sheet and the core arc not satisfied. For this reason
the intcrlaminar shear (s$. 5:;) bctwccn layer (I,) and layer (L+ I) at : = h,, are obtained
by integrating the equilibrium equations ofclasticity for each layer over the lamina thickness
and summing over layers I. through N as fdlows :
Substitution of strcsscs in terms of midplanc strains using relations (8) and (4), the integrals
of eqns (18) lead to the following expressions for intcrlaminar shear stresses:
1273 Finite element evaluations
in which. Hz, H, and Qi, have already been defined.
3. FINITE ELEMENT FORMULATION
In the stilndilrd finite element technique. the total solution domain is discrctized into
NE subdomains (elements) such thllt
.V h n(S) = c KC(S) (20)
c- I
where II and IL’ ;lre the total potcntinl of the system and the elcmcnt. respectively. The
element potential can bc cxprcsscd in terms of intcrnnl strain energy U’ and the cxtcrnal
work done ZV’ for an clement “1,” ;Is
n’(S) = u’- W’ (31)
in which S is the vector of unknown disploccment v:lriiibles in the problem and it is defined
by eqn (I I). If the same interpolation function is used to dclinc all the components of the
generalized displacement vector 6, we can write
.v,v S= c N,S, (27)
I- I
in which N, is the interpolating (shape) function associiited with node i. S, the villue of S
corresponding to node i and NN the numbcr of nodes in an element.
The bending curvatures (x. x’) ilnd the transverse shear strains (Q, CD*) are written in
terms of the dcgrccs of freedom S by making use of eqns (5) as follows :
(23)
Subscripts b and s refer to bending and shear. respectively. and matrices .Yiob and 2, are
defined as follows :
1271 B. N. PANDYA and T. KAST
0 spx 0 0 0
0 0 Z’?v ,_ 0 0
0 S$_r c’/s.r 0 0
yb=oo 0 dfQ?.r 0
00 0 0 c?;l?y
-0 0 0 s/zy d/Sx
y,= :
s/G?s 1 0 0 0 spy 0 1 0 0
0 0 0 3
0 0 0 0 3 1 0.
With the generalized displacement vector 6 known at all points within the element, the
generalized strain vectors at any point are determined with the aid ofeqns (24) and (22) as
follows :
NV ,V.V
= Y,S = y’, c NJ, = c 8,,8, = :39,d i-l I- I
(251)
whcrc
and
d’=!# 6’ I I* 2, . . . , ‘k,v). b (25b)
For the elastostatic analysis, the internal strain energy of an clement due to bending and
shear can be determined by integrating the products of moment stress-resultants and
bending curvatures, and shear stress-rcsuitants and shear strains over the area of an element
(26)
Impiementin~ the stress resultants given by cqn (13) in the strain energy expression (26).
we obtain
(27j
Substitution of eqn (25a) for bending and shear strains into eqn (27) leads to the strain
energy expression in terms of the nodal displacements which is given as follows :
Finite ckmcnt evaluations 1275
This can be written in a concise form as
W’ = l[d’ ;y’ d] (29)
in which X’ is the stiffness matrix for an element “e” which includes bending and the
transverse shear effects and is given by
The computation of the element stiffness matrix from eqn (30) is economized by explicit
multiplication of the S,, 4% and .9?, matrices instead of carrying out the full matrix mul-
tiplication of the triple product. In addition, due to symmetry of the stiffiress matrix. only
the blocks X, lying on one side of the main diagonal are formed. The integral is evaluated
using the Gauss quadrature
(31)
in which W,, and W,, arc weighting cocfficicnts, _CJ the number of numerical quadrature
points in each of the two directions (X and yf and j,fl the determin~~nt of the standard
Jacobian matrix. Subscripts i and,j vary from I to a numtir of nodes Fr cfcmcnt (NN).
Matrix 9 is dclincd by cqn (13) and ma&r& Pi and 2, arc given by
J tii = [ 9; 1 and
99 9, = [ ‘* 1
99 .
II
(32)
For the problem of bending of sandwich plates, the applied external forces may consist of
concentrated nodal loads F,, each corresponding to nodal degrees of freedom, a distributed
load y acting over the clement in the z-direction and a sinusoidal distributed load P_ acting
over the element in the z-dirtytion. The total external work done by these forces may be
expressed as follows :
W =d’F,+d’ f
{N~,O,O,0,O,N~,O,0,0,O,N ,,..., hl;yN,O,O,O,Of’(~+P,,)dA. (33) .I
The integral in eqn (33) is evaluated numcric~ffy using Gauss quadrature as follows :
P= i i Ct/,W,I/I{1~,,0,0,0,0,N~.0,0,0,0 t..., N,vN,O,O,O,Ofr Y- I b- I
in which a and h arc the plate dimensions; .r and y are the Gauss point coordinates and rrt
and n are the usual harmonic numbers.
4. NUMERICAL EXAMPLES AND DISCUSSION
Validity of the finite element formulations of the higher-order theories is established
by comparing results for laminated and sandwich plate problems with those available in
1276 8. N.
the form of exact, closed form and other finite element solutions. The element properties
in the isoparametric finite element formulation presented here are evaluated through Gauss quadrature. The selective integration scheme. namely 3 x 3 for flexure and 2 x 2 for shear contributions, has been employed. The geometrical and material properties for two different composite plate problems are as follows.
Material I
c , , = 0.999781 ; Cj, = 0.26293 I
c ,r = Cr, = 0.231192; Cad = 0.266810
Czt = 0.524886 ; cs5 = 0.159914
h, = 0.01. hz = 0.08, h, = 0.01, a = O”.q = I. (35)
Material 11 Face sheets
El G12 - = 25; E = 0.5; G2.l Ez z
E = 0.2 z
Ez=lO”. G,,=G,:, ~,~=0.25
h, = h3 = 0. I h. a = 0”. pm. = I.
Core
EX=~y=0.4x10~; G~:=G,V:=0.6~10s
G my = 0.16 x IO’ ; v,, = 0.25, hz = 0.811
452 v2, = fi v,2; directions I and x arccoincidcnt.
In both the cxamplcs that follow, the plate is square and simply supported along all four edges. Except for the convcrgcncc study the plate is discretized with four, 9 noded quadrilateral elements in a quarter plate. The finite element evaluations of stresses are at the nearest Gauss points. The dcflcction and stresses presented here are nondimensionalizcd using the following multipliers :
100/t’Ez 11’ h I C, , (core) MI ZT’ _*
p,.a ’ rn2 =
P,,d ’ n, --.
’ -pmna’ nl --’
4-q’ ms =
hy * (37)
Superscripts “e” and “c” used in Tables l-8 represent stress predictions from equilibrium and constitutive relations, respectively. The two examples considcrcd arc described below.
4. I. Example I : symmetric luminated plate under un$orm trunsrerse pressure This example is selected from Srinivas and Rao (1970). The set of material and
gcomctrical propcrtics given by relations (35) arc used. The full (6 x 6) material stitfncss matrix given in Srinivas and Rao (1970) is reduced (5 x 5) to suit the present theories, by assuming b, = 0 and eliminating E: from the stress-strain constitutive relations. The final material stiffness cocflicicnts adopted arc given by relations (35). All the stiffness matrix coetlicicnts for top and bottom laminae are some constant multiplier (modular ratio, R) times the corresponding stiffness matrix coefficients for the middle lamina. The numerical results showing convergence of deflection and stresses with mesh rctinemcnt are given in Table I. The convergence of transverse shear stress value with mesh refinement is shown
in Fig. 2. The transverse deflection and stresses at different locations in the thickness direction and for various modular ratios (R = 5. IO, 15. 25. 50. 100) are given in Tables 2-5. The effect of varying modular ratio (R) on transverse deflection is shown in Fig. 3. The effect of modular ratio on inplane normal stresses in the .v- and y-directions at c = 0.05
Finite clement evaluations 1277
Table I. Convergence of maximum stresses and displacement in a simply supported square laminated plate (material I. a/h = 10. R = 5)
Mesh size in quarter b.1 Xm, b,I Xm, T.“I Xm, rC,..: X m, r:,, X m, w0xm5
Source plate (~2. ~3, h/Z) (a?. o/2. hi?) (0.0. h/Z) (O.a/Z. 0) (a/2.0.0) (u/2, 412.0)
2x2 62.38 38.93 -33.22 - HOSTI ix,: 60.31 38.43 34.08
60.54 38.57 -33.98 5X5 60.35 38.26 -34.41
2X2 61.03 38.78 -33.81 HOST2 :;1: 60.65 38.58 - 34.35
60.55 38.53 -34.57 5x5 60.52 38.52 - 34.69
Srinivas and Rao (1970) -
60.353 38.491 - 4.36451 - 258.97
3.089 2.541 256.13 3.652 2.814 256.47 3.832 3.069 256.38 3.954 3.179 256.43
3.259 2.539 257.78 3.634 2.879 257.44 3.833 3.068 257.38 3.953 3.188 257.37
CLT - 61.141 36.622 - 4.5899 - 216.94
is shown in Figs 4 and 5, respectively. The following general observations are made from
the results presented in Tables I - 5 and Figs 2-5.
(I) Deflection and inplnne stresses can be accurately predicted without refining the
mesh, as the 2 x 2 mesh in a quarter plate gives sufficiently accurate results. The refined
mesh (5 x 5 in a quarter plate or more) is ntTessary for accurate prediction of transverse
shear strcsscs. (2) Errors in stress and dctlcction predictions increase with increasing value of modular
ratio (R). The difkrcnccs in the first (FOST) and higher-order shear deformation theories
(HOSTI. fiOST2) arc very high for a large value of modular ratio. say R = 100.
(3) CI’T and FOST undcrprcdict dcflcctions considerably. Deflections obtained using
higher-order thcorics agree well with exact solutions.
(4) Out of the two highcr-order shear deformation theories prcscnted. the one which
dots not satisfy free transvcrsc shear stress conditions on top and bottom boundary planes
of the plate (HOST2) is prcfcrrcd as its agreement with exact solutions is superior than the
other one (HOST I).
/
7. Error. Awroa. - Exacl I loo
Lzoct
I 1 1 I J 5 10 IS 20 26
- NO. OF ELEMENTS IN A QUARTER PLATE
Fig. 2. Convergence of transverse shear stress with the mesh refinement for a simply supported square laminated plate under uniform transverse load (u/h = IO).
Tab
le 2
. Max
imum
st
ress
es a
nd d
ispl
acem
enl
in a
sim
ply
supp
orte
d sq
uare
la
min
ated
pl
ate
(mat
eria
l I,
u/h
=
IO, R
=
5)
Sour
ce
(I,,
xm,
0‘2
x m
, Gz
3XM
r 0,
I
x fl*
, b,z
xm,
$tX”
r sr
., x
111,
G2xm
C
Z,X
tti+
w
,x??
t, (u
/Z.
uil,
k/2
) (d
’, U
!2,4
JI/lO
) (u
j2,1
r.K
!, 4J
l:lo)
(u
$!, u
:2.
h/t)
(~
2,
a 2.
4h,
10)
(u,‘Z
, UC
, Jh
’lO)
(0,
u/2,
SJr
jlO)
(0, U
P, 0
) (0
, uJ2
, -4
JtllO
) (u
/2,
ul2.
0)
in f
ace
shee
t in
cor
e in
Ike
sh
eet
in c
ore
HO
ST
I 62
.38
46.9
1 9.
382
(3.3
6)
(0.6
2)
(0.4
5)
HO
ST
2 61
.03
41.3
2 9.
463
(1.1
2)
(1.4
9)
(1.3
2)
FO
ST
61
.87
49.5
0 9.
899
(2.5
1)
(6.1
7)
(5.9
9)
Srin
ivas
and
R
ao (
1970
) 60
.353
46
.623
9.
340
CL
T
6).lJ
I 48
.913
9.
783
(1.3
1)
(4.9
1)
(4.7
4)
..~_
38.9
3 (1
.11)
38.7
8 (0
.75)
36.6
5 (-
4.78
)
38.4
91
36.6
22
(-4.
86)
30.3
3 (0
.77)
30.4
2 (1
.07)
29.3
2 (-
2.58
)
30.0
97
29.2
97
(-2.
66)
6.06
5 2.
566
(-
1.56
) (-
31.0
)
6.08
3 2.
422
(-
1.27
) ( -
34.
9)
5.86
4 2.
444
(-4.
82)
( - 3
4.29
)
6.16
1 3.
7194
5.86
0 (-
4.89
) 3.
3860
(-
8.96
)
3.08
9 ( -
29.
2)
3.25
9 (-
25.3
)
3.31
3 (-
24.1
)
4.36
41
4.58
99
(5.1
7)
2.56
6 (-
21.5
)
2.42
2 ( -
25.
9)
2.44
4 ( -
25.
2)
3.26
75
3.38
60
(3.6
3)
256.
13
(-1.
1)
257.
78
( - 0
.46)
236.
IO
( - 8
.83)
258.
97
216.
94
( -
16.2
3)
Tab
le 3
. Max
imum
st
ress
es a
nd d
ispl
acem
ent
in a
sim
ply
supp
orte
d sq
uare
la
min
ated
pl
nte
(mat
eria
l 7,
ufl
r =
IO
. R
= IO
)
Q.1
xnt
4 fl,
x x
ng,
6.1
xm,
0,‘
x m
a o,
,I!
xm,
u,,x
nf,
Zr.
,, xm
, (a
,!‘,
trQ
, h!
2)
(q?,
U
/3,4
h~lO
) (u
,‘2, u,
‘?, -
lJIi
lO)
(u2.
U
A2.
h;2)
(0
;2,
u:’
4Ji:
lO)
(a.”
cl
.‘,
4h;lO
) (0
, u/2
,4Jr
/lO
) -9
.-*
So
urce
in
fac
e sh
wt
in c
ore
in f
xe
shee
t in
cor
e --
~---
----
~_.-
-_--
_.-.
--_.
.___
_ _
_.._
_._~
____
__
__I_
____
HO
ST
I
64.6
5 51
.31
5.13
1 42
.83
33.9
7 3.
397
2.58
7 (-
I.&
r)
(5.0
2)
(4.6
5)
(-1.
69)
(1.6
7)
(-
2.94
) (-
34.1
)
I lO
ST
2 66
.23
50.0
0 5.
000
43.7
8 33
.81
3.38
1 2.
629
(1.3
7)
(2.3
4)
(l.U
kq
(0.4
9)
(1.1
9)
(-3.
4)
(-33
.1)
FO
ST
67
.10
54.2
4 5.
424
4O.I
O
32.0
8 3.
208
2.67
6 (3
.78)
(1
1.02
) (1
0.63
) (-
7.
96)
(-3.
99)
(-8.
34)
(-31
.9)
Srin
ivas
and
R
ao (
1970
) 65
.332
48
.857
4.
903
43.5
66
33.4
13
3.50
0 3.
9285
CL
T
66.9
47
53.5
57
5.35
6 40
.099
32
.079
3.
208
3.70
75
(2.4
7)
(9.6
2)
(9.2
4)
(-
7.96
) ( -
3.
99)
(-8.
34)
(-5.
63)
3.14
7 (
- 23
.2)
3.07
3 ( -
25
.0
3.15
2 (-
23.0
)
4.05
9
4.36
66
(6.6
1)
r:,,x
nh
wax
nil
(0, u
J2.
-4/l/
10)
(u/2
, rr
J2.0
)
__._
--
..__
____
_ __
2.58
7 15
2.33
( -
26.
4)
(-4.
42)
2.62
9 15
6.18
(-
25.2
) (-
2.01
)
2.67
6 l3
l.W5
( -
23.9
) (-
17.7
5)
3.51
54
159.
31
3.70
75
118.
77
(5.4
6)
( - 2
5.48
)
Tab
le 4
. h
iaa
imu
m
s~w
sse
s an
d d
isp
lac
em
en
t in
a s
imp
ly s
up
po
rte
d s
qu
are
lam
ina
ted
pla
te (m
ate
rial
I, u
/h =
IO
, R =
IS
) l_
_-P
--..p
____
-
---~
-~
- ..-
-
u,,
xm,
0,:
x ffl
, U
,JX
ffl,
(u/2
, ui2
, hi
?)
(Y!?
, u$2
,4h;
lO)
(Ui2
.0,~
2, Ill,
10)
SO
UF
X
in f
ac
e sh
ee
t in
co
re
(~_“
__~
. --
.-.-
-
I IO
ST
1 51
.97
3.44
5 (7
.6w
(7
.01)
WO
ST
2 67
.88
49.9
4 3.
329
(I.6
41
(3.4
0)
(2.8
1)
FO
ST
70
.04
56.0
3 3.
735
(4.8
7j
~16.
00)
(15.
35)
Srin
iva
s a
nd
R
ae
( 1
970)
66
.787
48
.299
3.
238
(I,,
x ffi
l,
(u.2
, u 2
, h, 2
)
---
44.9
2 (-
3.
24)
46.4
5 (0
.W
4t.3
9 (-
10
.81)
46.4
24
u,2x
nr,
@,J
xnf4
(u
2.0
.2,4
/i 10
) ((
1,2,
u,‘2
,4h,
~10)
in
fa
ce
she
et
in c
ore
Cl,
x m
W
”Xff
l,
(0,4
2,
-4h/
lO)
(4%
u/2
,0)
35.5
1 2.
361
2.6Y
I (1
.30)
(-
5.33
) (-
31.9
8)
35.3
6 2.
357
2.69
3 (1
.16)
(-
5.
4Y)
(-31
.92)
33.1
1 2.
2(x?
2.
764
I-
5.28
) (-
ll.4
7f
(-30
.13)
34.9
55
2.49
4 3.
9559
3.03
5 (-
23.4
3)
2.98
9 (-
24.5
9)
3.O
Yt
( -
22.0
2)
3.96
38
2.69
1 (-
24.7
7)
2.69
3 (-
24.7
1)
2.76
4 ( -
22
.72)
3.57
68
110.
43
( -
9.21
)
117.
14
( -
3.76
)
Ya.
85
(-25
.36)
121.
72
CL
T
69.1
35
55.3
08
3.68
7 41
.410
33
.128
2.
209
3.82
87
4.28
25
3.82
117
MI.
768
(3.5
2)
(14.
51)
( 13.
87)
(-
10.8
0)
(-
5.23
) f-
11.4
3)
(-3.
22)
(11.
04)
(7.0
4)
( -
32.8
2)
Tab
le 5
. M
axi
mu
m
stre
sse
s a
nd
dis
pla
ce
me
nt i
n a
sim
ply
su
pp
ort
ed
sq
ua
re la
min
ate
d p
late
(ma
teria
l I,
a/h
=
IO)
0.2
x 01
, 0,
) X
Ul4
U
.bJ x
n14
C,t
XnJ
, W
”XI(
I,
R 6,
(
x M
l‘ @
‘2,
a/2.
1/1:
IO
) (a
&!,
u,‘2
,4h
/lO)
(u.2
. a,
!2, h
/2)
(0,
ul&
01
w,
ul2,
O)
(ug?
. up
, h/
2)
in M
e
she
et
in c
ore
in
fa
ce
she
et
in c
ore
HO
sTl
I IO
ST
2 I:o
sT
HO
ST
1 H
OS
T2
FO
ST
HO
ST
f H
OS
T2
FU
ST
46h6
53
.03
2.12
1 46
.64
37.0
6 i .
482
2.74
4 2,
Y73
72
.741
75
6W
.rnY
51
1.27
I.
YJi
JY
.88
37.0
3 I.
481
2.72
6 2.
W97
ll
?.U
6 71
.94
57.5
5 2.
302
42.4
9 33
.99
1.36
2.
831
3.04
0 56
.33
I
67.3
7 52
.75
1.05
5 48
.54
38.3
9 0.
7678
2.
791
2.89
8 39
.137
3 50
69
.14
43,5
7 0.
8714
55
.04
38.8
9 0.
7779
2.
708
2.78
2 s3
.301
73
.44
58.7
5 1.
175
43.3
s 34
.48
0.69
35
2.89
7 3.
000
21,9
04
67.3
0 52
.57
0.52
57
49.5
4 39
.33
0.39
33
2.80
8 2.
861
21.1
66
tOQ
69
.18
37.L
S
0.37
15
60.6
3 40
.15
0.#0
15
2.65
0 2.
677
34.5
2f
74.2
2 59
.37
0.59
37
43.7
9 35
.03
0.35
03
2.92
7 2.
979
14,6
47
(O’te-
1 -
- 5620’0
09SI’O
260’0
-
8SSl’O
SW0
-
- -
OLZSO
’O
(O’lf-1
LSSI’O
V
OE
ZO
’O
9f9f0’0
(C’S1 -1
LRO
Z.0 zw
so’o 29W
O.O
(CL1
-1 ftoz’o
f 6SSo’O
99fuJO
(O’S)
- tZ
f’0
0111’0 -
Zlll’O
-
- C
QO
C‘O
&L-I
2111’0 6L
LZ
.0
(S’OI -1
WJK
’O
5892‘0
(O’O
I -1 1 P82’0
OO
LZ’O
(8’8f-) (8‘0s -)
Ifto’O-
Etw
O
EE
SO’O
- 9LLo’O
SESO
’O-
PLLO’O
LOLW
O-
Ml 1’0
(8’lZ-) (O
’LZ-) ZESSO
’O-
LSO80’0
(I’Z-)
(L’P-I 2690’0
- Z
SOI’O
(8X-I W
L-) 9990’0-
LIO
I’O
(8’62) R
L8’0
-
829’0
w7i)
S6D8’0
(‘~76) 8L89.0
(9.81) SW
L‘O
ha-1 L6O
.I
Llo’l
SW1
fSl‘1
.I.13
sa3-41861~ oeq.2 P”E
XPPW
Kklc-tl86l)
*=I3
PUF A
PPatl
(OL
61) *“eaad
(b’L--I Z
9o’I J.sozi
(I*11 991’1
(LX-) 01 I’I
ZJSO
H
1.Ls011
:: (0 ‘0 ‘Z
/N
22 r 141 K “1 3
‘p,Ip:;$’ aJJn
oS
8
I yyf’
5 li:Y
?~,‘pz!D
) (O
ll~&~~~W
) (Z
/V ‘Z
/n ‘ZiN
‘I&
i x ‘a -----
c; W
(01 =
y/n ‘11 py.w118) wtld
q+%pulrs
a~wb
s pam
ddnr kldur!s t! %
I! wxuaselds!p pue
s3ssa~~s tunmy?C
y ‘L
a[qcL
9
5 e --
d
-_ (S’ZL-I
- (9’Sf)
(6.69-j -
(1’6L-)
(S’6Z -
)
i S6Z
o’O
t.Zf.0
EE
WO
- Z
KO
’O
8L9’0 L6O
’I 113
cd LPLP’O
W
LI’O
- E
M0
- LL80’0-
OZS 1’0
- O
L9R‘O
s;13-4
I861 1 *v3
P”E iPP3’a)l
i9Lp’O
W
LI’O
- M
&O
-
8L80’0- L
ISI’O
- os9iR
’o l(r’3+
-l1861) “=
q3 PU
G A
PPW
- -
ZLOI’O
-
06f Z’O
L
ftl’o- S6SZ
’O
OfC
Z’O
- 9SS.l
(OL
61) *ur@l
(p‘ftf-1 t8.b)
G’9f -)
(;‘6f-1 18’It-1
SSLP’O
9ftwO
f0990’0
5660’0 SO
SZ’O
Z
l60’0- U
LSI’O
t-t-C
L‘0 9506’0
.Lsozl
(O’LI
-1 (6’L-)
(I.1 -1 (O
’L-) (IX
-1 09lL
’O
Lfli’O
8688W
O
OSLZ’O
O
OZZ’O
61blm
0- fItT
O
OZ
IO’O
- O
EZ
S’I Z
JSOH
ft.61 -) 11’9-1
(S’9-1 (6’6-1
l6’61-) L
P69.0
Zfl I’0
fS980’0
Z8fZ
.O
SVZ
Z.0
fFZI’O
- 8ffZ
‘O
9lK’O
O
LVZ’ I 1.L
soti
Tab
le 8
. M
axim
um
SWL
SXS a
nd d
ispl
acem
ent
in a
sim
ply
supp
orte
d sq
uare
san
dwic
h pl
ate
(mat
eria
l II
, u/
h =
IOU
) -
a,
Xrn
? 0,
x n
r, u,
x #
I*
Sour
ce
r,,
x m
? r’
,,XB
Fl,
?:lX
m,
T;lX
t?t,
$,
XW
iJ
(42,
u/
l, h,
‘2)
(u,‘_
7,
u/Z
, 4h
!tO)
(~‘2
. a
,‘l,
h/2
) w
,xm
, (0
.0,
h:‘2
) (0
, u/
z,
0)
u-4
u/z
. 0)
la
40
) (u
/2,4
0)
(u
/2, u
/2. w
tiO
STI
I.10
8 0.
8852
0.
055-
t -o
.@uO
0.
2880
0.
3001
0.
0270
3 0.
0336
2 O
.UtlY
I (0
.Y)
(1.2
) (0
.7)
(0.7
) (-
11.1
) (-
9.0)
t IO
ST
2 I.
109
0.88
47
0.05
54
-o.o
uo
0.28
80
0.36
27
0.02
704
0.03
322
0.08
9 I
(1.0
) (1
.1)
(0.7
) (0
.7)
(-11
.1)
(-9.
0)
FO
ST
I.
104
0.88
36
0.05
46
-0.0
435
0.28
75
O.li
52
0.02
695
0.01
767
0.08
83
(0.5
) (1
.0)
(-0.
7)
(-0.
5)
(-11
.3)
(-9.
3)
Paga
no (
197
0)
I.09
8 0.
875
0.05
50
-0.0
437
0.32
.W
- 0.
0297
0 -
-
Red
dy a
nd C
huo
(IY
IIt)
-FE
hl
1.06
3 -
0.05
30
- 0.
032
I -
0.1
I58
- 0.
072
0.01
182
Red
dy a
nd C
hno
(tY
8 I )
--C
FS
I .06
7 -
0.05
3 I
- 0.
0420
-
0.11
49
- 0.
069
0.08
85
CL
T
1.09
7 0.
878
0.05
43
- 0.
0433
0.
3240
0.
0295
0 (-
0.1)
10
.3)
(-1.
3)
(-0.
9)
- (f
-W
- (-
0.7)
-
B. N. hwxa and T. KAST
* HOST 1
- UOSTZ ----FOST
0 EXACT Sriniuos L Rae (1970)
2 (BOTTOM Of Top PCVI 3 ( TOP OF MiDWE FLY 1
_4-“- c-- ---------
tic- K HOST%
#cJ I..+*c - HOST 2
/ ---FOST /
/’
0 EXACT Srinivas 0 Roo
,
2( BOTTOM OF TOP PLY 1 3( TOP OF MIOOLE PLY)
“2 sO.06 .*-- ._,_.
1970 )
I 50 R--. DO
Fig. 4, Et&t o~moduhr ratio (top or bottom/middle) on mlrximum inplanc normai sIfCss (at kvd
i in .r-direction) fOF a simply SuppQFted. SymmClFiCUl~y bIminaWd squrtrc p&W under UnifOFRt
trlilS%‘CFSZ kXId (U/h = to).
Fide dmmt cvahatioru 12113
Fig. 5. Effect of modular ratio (top or bottomlmiddlc) on maximum inplane normal stress (st levet I in ydrcclion) for a simply supported, symmetrically IaminaM square plate under uniform
tmnsvcr3c load (U//J = IO).
4.2. E*uimpk $ _ : smt&+z phi* rut&T .~~~t~.~~~~~~~~~ ~~~.rr~~btif~b~~ kd This cxamptc is sclcetcd from Puguno (1970). The propertics given by relations (36)
MC: used for the analysis. The clitstic propcrtics given by Papano (1970) are modified
xcordingly by introducing thcrcin thy assumptiun o1.0~ = 0. The results for dcflcction and
stresses with pcrccntap errors spccifia,i within pxcnthcscs for q% = 4, tO and 100 arc
prc~~~ntcd in Tzbfcs 6-8, rcspcxztivc~y. The &l&t ofpfatc side-to-thiekncss ratio on transverse
dcfltxtion Es shown in Fig, 6. The variation of inpinno displacement along the x-direction
(II) through the plate thickness is shown in Fig. 7. The effect of plate sideto-thickness ratio
on transvarse shear stresses (T,.,) and inplanr normal stresses (a,) are shown in Figs 8 and
0.7
0’6
05
g 0.4
*
P
I
04
v au -aih
100
Fig. 6. Effect of plate side-to-thickness ratios on Ihe transverse deflections for a simply supported square sandwich piate under sinusoidal (ransvcrse load.
I284 8. N. PANDYA and T. K.Asr
I HOST3 - HOST2
Fig. 7. Variation of inplane displacement along x-axis for a simply supported square sandwich plate (U//I = 4) under sinusoidal transverse load.
o.t4
0-U t
I HOST 1
- HOST2
--- FOST
* 30-EUSTfCITY Pogano(1970f
002 -
0 ,
50 - 0th
1 100
Fig. 8. Effect of plate side-to-thickness ratios on the transverse shear strcsscs for a simply supported square sandwich plate under sinusoidal transverse load.
9, respectively. The following observations are made from the results presented in Tables 6-8 and Figs 6-9.
(I) For thick (a/h = 4) and moderately thick (cl,‘h = IO) plates, the deflection itnd stresses predicted by CPT and FOST are grossly in error.
(2) Ah the theories agree well with each other for thin plates (~//f = 100). (3) The transverse cross-section warping phenomenon which will be predominant for
a thick sandwich plate is evident in the present higher-order theories (Fig. 7). (4) The first and the last observations made in Example 1 are true for this example
too.
5. CONCLUSIONS
The results from the higher-order two-dimensional plate theories developed here com- pare well with three-dimensional elasticity solutions. The theories lead to realistic parabolic
Finite element evaluations 1285
x HOST 1
- ItOST 2
--- FOST
. 30-ELsSTlct7Y
Popam ( lsf0 1
l'b -
;;:
2 l
w
E” I
b”
I 1.0 - /”
4
50 1 100 - a/h
Fig. 9. Et!&% of plate side-to-thickness ratios on the inplunc normal stresses for ;L simply supported yuarc sandwich plalc under sinusoidal tcm*versc load.
varkttion of transvcrso shear strcsscs through the plate thickness. thus they do not require
the use of shcitr correction cocllicicnts. The simplifying rtssm~~ptions m:dc in CYT and
C:OST ikx r&xztcd by high pcrccntagc error in the results of thick sandwich or ktmini~td
p&s with highly stitT fittings. It is Micvd that the improved shrsr ~~l~rrn~lti~~n theory
presented hcrc is csscntid for rcliahlc analyses of sandwich typt’ laminated composite plates.
Finally. the general isoparamctric finite clcmsnt formulation of thcss thcorios presented c;ln
hc ilpplkl to illlillySC ilfly practical plllk structures.
,.fr.~nrrtl./~,~!,r~i~.~tr --Partial support of this research by the Aeromtutics Rc.scnroh snd Dcvctopment Board, Ministry of Dcfcncc. Gov~rnn~~nt of India, through its Grant No. A~rc~/Rl~-l34~13013~84--85~362 is grzWidly acknowlcd&. The authors pr~tcfully acknowledge the constructive: .uup&ons made by the referees.
REFERENCES
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,\Il‘&. 98 I ?%-I 23x. . --
1286 B. N. PANDYA and T. KAST
Khatua. T. P. and Cheung. Y. K. (1973). Bending and vibration of multilayer sandwich beams and plates. Im. 1. ,Vumur. Merlh. En.qng 6. I I-24.
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