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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l~ario. F. F. and Ha. K. Il. (1974). S:rndwich plate structure analysis by Iinitc element. A.SCI:‘I. Sfrucr. Diu. 100, 1’43 1’6’

Fol;. G. h:‘( 1970). Ucndinb of clamped orthotropic sandwich plates. XXE J. f%gng blc*clr. /Xv. 96. 243-265. I linton, E.. Rarzaquc, A.. Zicnkicwicz. 0. C. and D;lvics. J. D. (IY75). A simpls Iinite clcmcnt solution for plutcs

of homogcncous, sandwich and cellular construction. &or. Insrn C’ir. Emqrs, Purr 2 59.43 65. Jones, R. M. ( 1975). .W&~nic.r ~$‘Cc~~puvirr ~\Iu~;*ri&. McGraw-1 lill, New York. Kant. 7. (1 Wt), Numerical analysis of thick plates. Cwn~. &f&r. A& &I&r. &gng 31. I --IX. Kant. T. and Fandya. B. N. (19X7). Finite clement cv&ation of int~rl~min~r straws bascc on first and highcr-

order thcorics. I’rcrr. tr’llrk.~lrtrp-cu,N-S~‘~?~jf~ffr on ~~/ff~~ii~~~~rio~I in Cttr,ipr~&rr. Indian Institute of Science, B;tng:tlorc. India. pp. X5 103.

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Kant. T. and Fandya. B. N. (lO%Xh). A simple finite elcmcnt formulation of a hither-order theory for unsym- nl~tri~~Ily laminated composite plates. CW*I/W.*. Sfrwt. 9. 2 15 -246.

Kant. T. and Sahani. N. F. f 19X5). Fibrc reinforced plotus-some studies with 9.noded L;lsrangisn/Het~rosis elcmcnt. Trans. 8th Int. Conf. Struct. Mcch. Rcxtor Tech. (SMiRT-8). Brussels. Belgium. Paper BR/7. pp. 315-310.

Kant. T.. Owen. D. R. J. and Zicnkicwict, 0. C. (1982). A refined higher-order C plate bcndingelemcnt. Co~pur. .Qrrc<.r. 15. I77 - 1x3.

Kao. J. S. (1965). Bending of circular sandwich plates. ASCEJ. &gng ,tlcc/~. 91, 165-176. Kha&aa. T. F. and Chcune. Y. K. (1972). Trian&er element for multil~iy~r sandwich plates. ASCE J. Engng

,\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.

Lo. K. H.. Christenson. R. M. and Wu. E. M. (1977). A high-order theory of plate deformation. Part I : homogeneous plates : Part 2 : laminated plates. AXWE J. .4ppl. Mrch. 44.663476.

Martin. H. C. (1967). Stiffness matrix for a triangular element in bending. Jet Propulsion Lab. Report No. 32- 1158.

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