A Simple Finite Element Formulation of a Higher-order Theory ...repository.ias.ac.in/15877/1/343.pdfComposite Structures 9 (1988) 215-246 A Simple Finite Element Formulation of a Higher-order

Composite Structures 9 (1988) 215-246

A Simple Finite Element Formulation of a Higher-order Theory for Unsymmetrically Laminated Composite Plates

T. Kant and B. N. Pandya

Department of Civil Engineering, Indian Institute of Technology, liT Powai, (3ombay 400 076, India

ABSTRACT

A higher-order theory which satisfies zero transverse shear stress conditions on the bounding planes of a generally laminated fibre-reinforced composite plate subjected to transverse loads is developed. The displacement model accounts for non-linear distribution of inplane displacement components through the plate thickness and the theory requires no shear correction coefficients. A C O continuous displacement finite element formulation is presented and the coupled membrane-flexure behaviour of laminated plates is investigated. The nodal unknowns are the three displacements, two rotations and two higher-order functions as the generalized degrees of freedom. The simple isoparametric formulation developed here is capable of evaluating transverse shears and transverse normal stress accurately by using the equilibrium equations. The accuracy of the nine-noded Lagrangian quadrilateral element is then established by comparing the present results with the closed-form, three-dimensional elasticity and other finite element available solutions.

1 I N T R O D U C T I O N

It has long been recognized that classical plate theory must be modified to include certain higher-order effects like warping of the cross-section. The first generalization of the classical theory was given by Reissner 1 and Mindlin. 2 Since then, there have been many further generalizations. Perhaps the first higher-order theory, based upon the principle of stationary potential energy, resulting in eleven second-order partial differential

215 Composite Structures 0263-8223/88/$03" 50 © 1988 Elsevier Applied Science Publishers Ltd, England. Printed in Great Britain

216 1. Kant, B. N. Pandya

equations to determine the eleven functions in the assumed displacement model, was given by Lo et al. 3"4 A sub-set of the displacement model used in Ref. 3, which neglects the strain energy due to transverse normal stress, has been adopted by Levinson, 5 Murthy 6 and Reddy. 7 Later, Reddy and co- workers presented the displacement 8 and the mixed 9 finite element models of the theory developed earlier. 7 In these, 7-9 Reddy has modified the displacement model of Lo et al., by neglecting the transverse normal stress effect and satisfyilag the zero transverse shear stress conditions on the bounding planes of the plate, thereby expressing the three displacements of a point in plate space in terms of only five physical midplane displacement quantities. With this, the displacement model adopted by Reddy gives rise to second-order derivatives of transverse displacement in the energy expression, and hence displacement based finite element formulation requires C ~ continuous shape functions which are computationally inefficient and are not amenable to the popular and widely used isoparametric formulation in present day finite element technology.

The aim of the present work is to develop a simple isoparametric finite element formulation. The formulation presented here differs from that of Reddy and co-workers ~'~ in three ways:

(i) Only a part of the conditions for vanishing of the transverse shear stresses on the top and bottom bounding planes of the plate as given by eqn (3) is introduced in the assumed displacement model given by eqn (1).

(ii) The remaining conditions given by eqn (4) are introduced later in the shear rigidity matrix as given by eqn (18).

(iii) The general isoparametric displacement finite element formulation is developed.

The validity of the formulation is established by comparing the present numerical results with other finite element, 9 closed-form 1° and three- dimensional elasticity" solutions.

2 THEORY

The development of the present higher-order shear deformation theory begins with the assumption of the displacement field in the following form (Fig. 1)

u(x ,y , z) = Uo(X,y) + zOx(x,y) + Z2Uo*(X,y) + z30*x (x,y)

v(x ,y , z) = Vo(X,y) + zOy(x,y) +zZv*o(X,y) + z30~(x,y)

w(x, y , z ) = Wo(X, y) (1)

A higher-order theory for unsymmetrically laminated composite plates 217

A y v °

/

,o, , r/?'," o;: ,y-- X~Uo

- 0 ° 4 I//

Fig. 1. Geometry of a four-layer symmetric laminate.

where Uo, Vo and Wo denote the displacements of a point (x,y) on the midplane and 0~ and Oy are the rotations of normals to midplane about the y and x axes respectively. The parameters Uo*, Vo*, 0x* and 0~' are the corresponding higher-order terms in the Taylor's series expansion and are also defined at the midplane. 12,13 The condition that the transverse shear stresses vanish on the plate's top and bottom faces is equivalent to the requirement that the corresponding strains be zero on these surfaces. The transverse shear strains are given by

Ov Ow Ow o = q'- = Oy + 2zv* 2 * - - + 3 z Oy 4 - - -

Yyz Oz Oy Oy

Ou Ow Ow o Yxz = - - + = O~ + 2ZU*o + 3z 20*x + - -

Oz Ox Ox (2)

Equating •yz(X, y , +_h/2) and yx~(X, y , +_h/2) to zero, we obtain

* * o ( 3 ) Y o = U o =

and 4(+0wo) 4(0wo) 0~ - 3h 2 0 y Oy ; 0 " - 3h 2 0 x + ~ (4)

In t roduct ion of condition (3) in eqn (1) yields a compact displacement form:

u = Uo+ZOx+z30 *

v = V o + Z O y + z 3 0 t

w = Wo ( 5 )

218 I. Kant, B. N. Pandya

Murthy 6 and more recently Reddy and co-workers 7-~ have used conditions (4) to eliminate 0x* and 0* from eqn (5) and have obtained a modified displacement model

z

w = Wo (6)

In the present formulation, we proceed with the displacement field given by eqn (5) and conditions (4) are introduced later in the shear rigidity matrix.

The strains associated with the displacements in eqn (5) are:

Ex = Exo + zkx+ zak *

Ey = ~'yo -~- z k y + Z 3 k*

E z ~ 0

. = + Z k x y "Yxv Exy ° + Z k x y 3 ,

"Yyz ~- ,by ~- Z 2 ,b3, *~

vx~ = 6x + z: ,b *~ (7)

w h e r e

Ex ° - - OU o c)V o Ott o OV o

OX ' ~-yo Oy ' e~yo Oy + Ox

OOx OOy OOx OOy k x - , k y - , k x ~ , - - - 4

Ox Oy Oy c?x

Ox ' " Oy ' k ' y - O---y40x

~y = Oy + OWo ay ' `b~ = Ox

,by = 3Or*, ,b~* = 3¢*

OW o + - -

Ox

(8)


The constitutive equations for the L 'h layer can be written as

/I)L Icll el2 0 0"2 C12 C22 0

7"12 0 0 C33

EIIL E2

"Y12

T23 0 Y23

"?13 C55 ")/13 (9)

where (O"1, 0"2, T12, '/'23, 7"13) are the stress and (el, e2, y12, y23, y13) are the linear strain components referred to as the lamina co-ordinates (1, 2, 3), as shown in Fig. 1, and Cijs are the plane stress reduced elastic constants of the L 'h lamina. The following relations hold between these and the engineering elastic constants:

El 1)12 E 2 E 2 C I 1 - ~. C12 - , C22 -

1 - - /212 1-'21 1 - - V12/"21 1 - - /)12/"21

C33 = G12, C44 = G23, (755 = GI3 (lo) The stress-strain relations for the L 'h lamina in the laminate co-ordinates (x, y, z) are written as

/0-x / L Qll 0-y = 012

7xy 013 { y}L I0. 7"xz Q45

Q12 QI3 L, (.} Q22 Q23 / Ey

Q23 Q33 ] yxr

Q45] L (YYZ} L Q55 J yxz

(11)

in which

o-= (0-x,%,rxy,'ry~,rxz)'

22(1 T. Kant, B. N. Pandya

and

IE ~- ( e x , if.y, ")/ xy , ")/ y z ' "Y xz ) t (12)

are the stress and linear strain vectors with respect to the laminate axes and Qijs are the plane stress reduced elastic constants in the plate (laminate) axes of the L th lamina. The transformation of the stresses/strains between the lamina and laminate co-ordinate systems follows the usual transformation rule. 14

The total potential energy rr of the plate is given by

L 77" = ~ E t ( r d V - d t p d A (13)

in which A is the mid-surface area of the plate, V is the plate volume, P is the equivalent load vector corresponding to the seven degrees of freedom and d is defined as

d = (Uo, Vo,wo,Ox,O.O*,O*~) ' (14)

The expressions for the strain components given by eqn (7) are substituted in eqn (13). The functional given by eqn (13) is then minimized while carrying out explicit integration through the plate thickness. This leads to the following thirteen stress resultants for the n-layered laminate.

{Nx} N , =

Nxy

My

Mxy

Oy

L÷ I O-y d z

Txy

! M~*i'~

' My~-=

Mx*y:)

hL

o',, [ z i? ]dz L = 1 L+I Txy

Q*xt: Q~*J

f Txz t [1 " z2]dz L = I 1.+1

Tyz ) 05)

A higher-order theory for unsymmetrically laminated composite plates 22 1

After integration, these relations are written in a matrix form which defines the stress resultant/strain relations of the laminate and is given by

N A ', B ', 0 k -

M B t Da 0

I I

0 0 ~ Ds Q* ' (16)

in w h i c h

N :{Nx ,Ny ,Nxy} t

M = ( m x , my, mxy} t

M* - { M ; , * , t M;, Mxy}

O = {Qx, Qy}t

Q* ={Ox,ay} t

Eo = { •xo, Eyo, Exyo } t

k = {k~,ky,k~y}t

k* = {k*,ky,kxy} t

,t, = {+x,6y} ' ~* * * t = {,t,x ,6y }

The superscript t denotes the transpose of a vector/matrix and

(17)

A =

QIIH1 QIEH1 QI3HI-

Q22 HI Qz3 Ht L=I

L_ Symmetric Q33 H1

t th Layer

B =

-Qn H2

Q12 Hz L=I

-Qa3H2

QI2Hz Q13Hz QllH4 QI2H4

Q22Hz Q23H2 QIEH4 QE2H4

Q23/-/2 Q33 H2 QI3 H4 Q23 H4

Q13 H4 7 Ltn Layer

Q23 H4

Q33 H4

L=I

0 . / 4 3

Symmetric

QlzH3 Q13H3 QllH5 QI2H5 Qt3H5

Q22H3 Qz3H3 Q12H5 QzzH5 Q23Hs

Q33/-/3 Q t3/-/5 Q23 H5 Q33 H5

QIIH7 Q,2H7 QI3H7

QzzH7 Qz3H7

O~3H7

',L th Layer

222 T. Kant, B. N. Pandya

U S z

Q55 H Q45 H

Q44H

L = 1

Symmetric

0 0 -

0 0

Q55 H* Q45 H*

Q44H*

L th Layer

(18)

where

Hj = (hL-- hL+l), H2 = ~(hL-hL+l )

1 3 . ~ = 5 ( h c - h[+ ~),

1 5 H~ = g(hL - h i + , ) ,

1 4 H4 = ~(hL -- h4L+ 1)

I h7 H7 = ~ ( L - - h ~ . . , )

(19)

( ) ( h2) 4 H* = H5 -- H3 × 5 - H = H 1 - H 3 × ~-~ ,

The shear rigidity matrix Ds in eqn (18) is evolved by incorporating an alternate form of the conditions (4), viz.

h 2

4'v +5-4,.,,* = o

h 2 +x+G--4 4'~* = o

(20)

and the resulting theory becomes consistent in the sense that it satisfies zero transverse shear stress conditions on the bounding planes of the plate. If the conditions given by eqn (20) are not incorporated, then the resulting non-consistent theory does not satisfy the zero transverse shear stress conditions on the bounding planes of the plate. In this case, the shear rigidity matrix D* is defined as S

L = 1

Q55H1 QasHI Q55H3 Q45H3 ~ L th layer

Q44 H1 Q45 H3 Q44 H3

Q55 H5 Q45 H5

Symmetric Q44 H5 (2 1)


The transverse shear stresses ~ and ~yz cannot accurately be given by eqn (11) as the continuity condition at the interfaces of any two layers is not satisfied for laminated plates. For this reason, the interlaminar shear and normal stresses L L (~'xz, "1"yz, o-L) between layer (L) and (L + 1) at z = hL+l are ob ta ined by integrating the three equilibrium equations of elasticity for each layer over the lamina thickness and summing over layers L through n as follows:

,rxLz = - - z=hL+ 1 L=I L+I \ OX Oy

ILI ~ fh hL (Oo'L~-OTLy)H z Z = -- z = h L + 1 L = I L + I \ OY Ox

Or L = - -

z = h L + l L = 1 L + ~ 3x Oy

3 FINITE E L E M E N T F O R M U L A T I O N

We follow the standard finite element technique in which the total solution domain is discretized into N E sub-domains (elements) such that

NE ~-(d) = ~ 7re(d) (23)

e = l

where 7r and rr e are the total potential of the system and the element respectively. The element potential can be expressed in terms of internal strain energy U e and the external work done W e for an element e as,

~-e(d) = U ~ - W ~ (24)

in which d is the vector of unknown displacement variables in the problem and is defined by eqn (14). If the same interpolation function is used to define all the components of the generalized displacement vector d, we can write

NN d = F N,d, (25)

i=1

where Ni is the interpolating (shape) function associated with node i, di is the value of d corresponding to node i and N N is the number of nodes in the element .

224 T. Kant, B. N. Pandva

The extensional strains ~ , the bending curvatures (k ,k*) and the transverse shear strains (~b,~b*) can be written in terms of the nodal displacements d by referring to eqn (8). The result can be written in matrix form as follows:

~o = LEd

(26)

The subscripts E, B and S refer to extension, bending and shear respectively and the vector of element nodal displacements is given by eqn (14). The matrices LE, LB and Ls attain the following form

t E

a 0 ax

0 0 - -

ay

o a

ay ax

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

L B =

0 0 - - 0 0 0 8x

a 0 0 0 - - 0 0

ay

a a 0 0 0 0

ay 8x

0 0 0 0 - - 0 8x

0 0 0 t) 0 a ay

a a 0 0 0 0

ay ax -

A higher-order theory for unsymmetrically laminated composite plates 2 2 5

t S =

0 0 0 ~ 1 0 0 0

Ox

(9 0 0 - - 0 1 0 0

Oy

0 0 0 0 0 3 0

0 0 0 0 0 0 3 (27)

With the generalized displacement vector d known at all points within the e lement , the generalized strain vectors at any point are determined with the aid of eqns (27) and (25), as follows:

NN NN

Eo = LE d = LE E Nidi = ~ B,~ di = BE a i = l i = l

= L n d = LB N i d i : B iad i = B a a k * i = l i = l

= Ls d = Ls Ni di = B,s di = Bs a * i = 1 i = 1

(28)

where

NN

BiE = L E N i , BE = ~ B,~ i = 1

BiB = LBNi NN

, BB = Y~ B,B i = 1

NN

B~s = L s N i , Bs = ~ B,s i = 1

and

t t a = {dl,d2 . . . . . d~vN} t (29)

For the elastostatic analysis, the internal strain energy of an e lement due to extension, bending and shear can be determined by integrating the


products of inplane stress resultants and extensional strains, moment stress resultants and bending curvatures and shear stress resultants and shear strains over the area of an element.

1 I lk/, i.lt U~ = 2 fa Ct°N+ k* {M,M*}+ 6* [Q,Q*} dA (30)

Implement ing the stress resultants given by eqn (16) in the strain energy expression (30) gives

t U e ~A¢o+ k*

, tkl Bt¢o + etoB k*

k +

k*

t

DB k*j

+ • Ds * dA (31 )

Substituting eqn (28) for extension, bending and shear strains into eqn (31) leads to the internal strain energy expression in terms of the nodal displacements as follows:

U ~ = -~ { a tB~ABEa+atB tBBtB~a+

a t B~ BBa a + a t B~ DB Br~ a +

atBtsDsBsa } dA (32)

Expression (32) can be written in concise form as

ue 1 t = ~[a Kea] (33)

where K ° is the stiffness matrix for an element e and includes extension, bending and the transverse shear effects and is given by

g e = J'A { B~ABE + B~BtBE + B~BBs + B~DB BB + B~Dsns} dA (34)

The computat ion of the element stiffness matrix from eqn (34) is economized by explicit multiplication of the Bi, D and Bj matrices instead of


carrying out the full matrix multiplication of the triple product. In addition, due to the symmetry of the stiffness matrix, only the blocks K o lying on one side of the main diagonal are formed. The integral is evaluated using the Gauss quadrature

K~ = B~DBjIJIde d , l 1

g~ = ~ ~ WaWblJIBIDBj (35) a=l b=l

where Wa and Wb are weighting coefficients, g is the number of numerical quadrature points in each of the two directions (x and y) and I JI is the determinant of the standard Jacobian matrix. The subscripts i and j vary from one to a number of nodes per element. The matrices Bi and D are defined as

BiE ]

B i = BiB , D =

L-i ij I A B 0

B t DB 0

0 0 Ds

(36)

and B/is obtained by replacing i by j. For the problem of bending of laminated anisotropic plates, the applied

external ~forces F consist of concentrated nodal loads F¢, each corresponding to a nodal degree of freedom, a distributed load q acting over the element in the z direction and a sinusoidally distributed load Pmn acting over the element in the z direction. The total external work done by these forces may be expressed as follows:

We = -21 atF¢ + -21 at fA {Ntq+Ntp.,n}dA (37)

The integral of eqn (37) is evaluated numerically using the Gauss quadrature. The result is

P ~ ~ WaWblJlNi{O010000} t {q+Pm, sin mrcx " nrty } = •sm T (38) a=l b=l a

where a and b are the plate dimensions, x and y are the Gauss point co-ordinates and m and n are the usual harmonic numbers.

2 2 8 T. Kant, B. N. Pandya

4 N U M E R I C A L EXAMPLES AND DISCUSSION

Performance of the present finite element formulation is demonstrated by comparing results for various laminate geometries with those obtained using elasticity, closed-form and other finite element formulations. The selective integration scheme, namely 3 x 3 for membrane and flexure and 2 x 2 for shear contributions, has been employed. In all the examples considered, the individual laminae are taken to be of equal thickness and the following set of material properties is used for each lamina:

El GI2 G 2 3 - 25, = 0"5, = 0"2, ul: = 0.25

E2 E: E2 (39)

It is assumed that Gl3 = G , 2 and v21 = (E2 /EOvl2 . The simply supported square plate is discretized with four nine-noded quadrilateral elements in a quarter plate except for the convergence study. The values of stress resultants and stresses are at the nearest Gauss points. The deflection, stresses and stress resultants are presented here in the non-dimensional form using the following multipliers.

1 0 0 h 3 E 2 1 1 m l - m 2 - - _ m 3 - - qa 4 ' qa 2 ' qa

h 2 h lOOh 2 m 4 = - - m 5 ---- - - , m 6 - ( 4 0 ) q a 2 ' qa qa 2

The superscripts e and c used in the various tables that follow, represent values of stresses obtained from equilibrium and constitutive relations, respectively.

The following three unsymmetrically laminated simply supported square plate problems are considered.

4.1 Example 1

This example is considered to bring out the effect of mesh refinement on the deflection and stress predictions with the present element. The numerical results are compared with the exact solutions. The following three cases with different loading conditions and lamination schemes are considered for this purpose.

(i) A two layer unsymmetric cross-ply (00/90 °) square plate subjected to sinusoidal transverse load.


(ii) A two layer unsymmetric cross-ply (00/90 °) square plate subjected to uniform transverse load.

(iii) An eight layer unsymmetric angle-ply ( 4 5 ° / - 4 5 ° / . . . 8 layers) square plate subjected to sinusoidal transverse load.

The numerical results for the above three cases showing convergence of deflection and stresses are given in Tables 1-3 respectively. The convergence of maximum transverse shear stress with the number of elements is shown graphically for the first case in Fig. 2. The following points are noted from this study:

mMaximum transverse deflection and inplane stresses predicted with 2 × 2 mesh are reasonably accurate and little further improvement is observed with mesh refinement.

- - A n accurate estimation of transverse shear stresses through equilibrium equations needs a more refined mesh, as is evident from Fig. 2.

4.2 Example 2

This example is selected to establish the accuracy of stress predictions through the thickness in the present development. The numerical results are compared with three-dimensional elasticity solutions. A two-layer unsymmetric cross-ply (00/90 °) square plate under sinusoidal transverse load is considered for this purpose. The numerical results for a square plate with side-to-thickness ratios of 4, 10, 50 and 100 are given respectively in Tables 4-7. The maximum stress resultant and deflection values are given in Table 8. The variation of maximum transverse deflection with different side-to-thickness ratios is shown in Fig. 3. The normal stress (o'x) and the transverse shear stress variation (rx~) through the plate thickness is shown graphically in Figs 4 and 5 respectively.

The following important observations are made from the numerical results presented in Tables 4--8 and Figs 3-5.

m F o r a thick plate (a/h = 4), errors in the inplane normal stress (o'x) computations are higher in comparison with the inplane shear (rxy) and the transverse shear stress 0-x~).

mAl l stress components evaluated are reasonably accurate for moderately thick-to-thin plates (a/h >- 10).

- -The results for stress resultants presented in Table 8 should serve as a bench-mark for future comparative studies. They will also be useful to designers of composite laminates.

- - T h e curve in Fig. 3 shows the inadequacy of the classical lamination theory to predict the displacements accurately.

231) 1". Kant, B. N. Pandya

r -

<

X ~ ml

2

~ × ~"

' r =

0 X +1

m

e -

0

~= ~ ~ ~

I I

t I

o4

= o $ $ o $ $ ~ $ o o $

I I I l i t /

t k I I I

I / I I I I

x x x × I I

¢=


e~

c~

. I

X ~ I'¢'q

X ~ I 'eq

X

II

~ x

¢-

N ~ x

N

._o

E ~g~ -~ ~ ~ ~"

N

~ k

e- o

,,-, I ~- , ~ ~ 0", I ~-

I ~.

I I

I I I I

• ~ ~ ~ I I I

6 6 6 o 6 ~ 6 6 I t I I

X X X X X

232 7". Kant, B. N. Pandya

=

i

-g- t¢'3

e -

<

e -

2 ~

.2 ~d

E

.~_

r -

C

X ~ 1 ~

X ~ I ~

x

X

t"q r'q C'q r"q 5 ~ 5

~ ¢q 30

,r~ ¢'q ¢'q ¢q

5 5 6 5

I I I I

g gggggg

I I I I

X X X X

E

P__,

C,

2


0.33

0'32

io E

N (~ 0.3!

t 0'30

0.2!

Pagano 11

P r e s e n t

/ lJ .... Ji I i ~ 1'o ,5 ,o ;~

No. of e lements in a quar te r plr, te

Fig. 2. Convergence of transverse shear stress (~'xz) with the mesh refinement.

2.0

T wXml

1.5'

IC

,PRESENT

CLT

o ~b 4'o 6'o ~o ,;o alh -

Fig. 3. Effect of plate side-to-thickness ratio (a/h) on the centre deflection (w × ml) of a simply supported (0°/90 °) cross-ply square plate under sinusoidal load.


e-

o

p

e-

E

= [ . -

m N

E =

e-,

e-,

¢-

a ,...,

g

0

X

b.

X ~ I~J

X

X

× ~2

I~1 X

I ~ 1 1 1

• ~ . ' " 7 . @ ~ . ~ . ~ . ~ . . . "7.

I I I I I I I I I

I I I I I

i i i r l l l l l

I i l 1 1 1


e-.

e'~

.=_

, 1 .-.,

E ~ E ~

e-.

e-,

i -

e -

¢- [...,

e . -

._q

~ L e ' q )<

I,.-

×

b..

×

× ~ 1 TM

)<

)<

I.-.

X ~ l e ' q

. ~ , . . , _ I _ . : ~ ~

I I I I I

v-. • ,..y. ~ . ~ . ~ . . . . . . .

I I I I 1 I I I I

• ...-, e q ,--, " 7 . . . . • • f'q.. • • •

I I I I I

I I I I I I I I I

I I I I I I

2 3 b I'. Kant, B. N. Patzdva

.<

o

.=_ B

t--

¢3

..+..

+.-

+,..,

+,.

e -

.2

>

X +

X

~ . , ~ I ~1

X ~ ++"I

,~ i tr+, I

,-+.

X

I-,.

~ ,

I +"-,,i X

~ I ~ , I b

m i,+-,i X

+ + + + ~ m I + + + + ~ . . . . . . ~ ~ .

l i l ] l

P l !

i i l 1 1 1 1 1 t

t t~ ' . ~ .,,+r. ~ _ _ ~ ~ - - t'--'l re'+ ~ ',f',

i I I i i I


e.

t ~

.=.

ua~

¢-

[. ~D

¢-

[-

¢-

i

X

I-.

×

X ~ l ex l

>(

b.

>(

t~

i

i l i l i

l i i

i l i l L t l l i

i l l l i l

2 3 8 T. Kant, B. N. Pandya

r ~

t -

,-1

o9,..

p r~

r..

f i

,,.-!

r-,,

r~

0

*d

..1

X ~ l t " l

X

X

X ~

x ~ (,.~

• . o . o .

I I I I


I

<

o

~ g

2

e..,

.~_

.o

8

0

~s

II

F

X

×

X

X ~ 1 ~

I I I I I I I I

I 1 [ t 1 1 1 1 1

• . . - . . ~ . ~ . ~ . ~ . ~ . ~ . - . . . ~ .

0 ,,D ¢q r--- 0 ~ ',~

• t"-,I ¢,,I t"-,I ¢'q • o 6 6 o 6 6 6 6 o

6 6 o 6 6 6 6 ~ 5 6 6 o o o 6 6 ~ 5 I I I I I I

I I I I I I I I I I I

I I I I

241) T. Kant, B. N. Pandya

o,s-~ Zlh 0011~0.4+

J ' Present (From Constitutive Law) / o.1+ • P a g a n o 1 1 0 1

j . . . . . o', x m+

-0.6 -0.4 -0.2 I0 0.2 04. 0 . ~ r I --~- ~ ÷ i J6 == 0"8 1 'JO

- - 0 . 1

- - 0 . 2

- - 0 . 3

• - 0 . 4

- -0 .5

Fig. 4. Variation of the normal stress (Crx × m4) through the laminate thickness for a simply supported cross-ply (00/90 °) square plate under sinusoidal transverse load (a/h = 10).

- - I t is clear from Fig. 5 that the transverse shear stress variation through the thickness of plate is correctly given by equilibrium equations. As anticipated, the maximum value occurs at the level of neutral surface and the stress is continuous at the interface. But this is not true when transverse shear stress is computed using stress-strain constitutive relations.

4.3 Example 3

This example is an extension of Example 2 for an 8 layer unsymmetric angle-ply ( 4 5 ° / - 4 5 ° / . . . 8 layers) square plate under sinusoidal transverse load. The numerical results for stresses are presented in Tables 9 and 10 and


0.4 " " L ~ .

0"3

0"2

0.1

0

-0.1

-0,2

-0.3

- 0 ' 4

-0.5

/

, ,d > i

i /

/

. • . - , " • / " \ i i ~xzXm5

I t ~ I • 0"2 0'3 0"4

Present (From Equilibrium Equations)

. . . . Present (From Constitutive Law) . . . . . Pa9ano11

Fig. 5. Variation of the transverse shear stress (rxz × ms) through the laminate thickness for a simplysupportedcross-ply(O°/90°)squareplateundersinusoidaltransverseload(a/h = 10).

maximum deflection and stress resultant values are given in Table 11. The variation of maximum transverse deflection with different side-to-thickness ratios is shown in Fig. 6. The normal stress (o-x) and the transverse shear stress variation (rxz) through the plate thickness is shown graphically in Figs 7 and 8 respectively. These results should serve as a bench-mark for future investigations.

5 C O N C L U S I O N S

A simple C o isoparametric formulation of a higher-order theory which satisfies the zero transverse shear stress conditions on the top and bo t tom

242 T. Kant, B. N. Pandva

[

r - <

e,,

¢-

.2

. , -

>

X '~ I~1

X ~ lexJ

I ,"~ X

X

b-

l e ' l X

q~

::2", tt'-~ : ~ r ~ c ' l ['-- "d" ""5" ~1 i'-- 9,0 re), :::7", t/'h

, ,~ ~ ~ , ~ ~ o , ~ r,,'-, "t"

"7'. ~ . ~ . ~ . . '.~. . . . . . . . . . . ~ "

",D ~ ~ ¢'q 0¢, ¢"q " ~ ",-"g

I I I I I I I I

I I I I I I I I I

['~ t t ) t"q ~ tth

I I I I

E


¢O

I

q%

¢- <

.=_.

E

a ~

e.

re

~D

r~

E

Z~

~ l t . q X

Z~

X ~ l ~ q

~a

t 'q 0¢ ~ q%

t I I i

I I I t

eD ~J

e- e- e-

1.5

1"0

1 wXm 1

0.5

PRESENT

CLT

o ~ 4'o 6~ ~o ~o a / h ~

Fig. 6. Effect of plate side-to-thickness ratio (a/h) on the centre deflection (w x rnl) of a simply supported ( 4 5 ° / - 4 5 ° / . . . 8 layers) angle-ply square plate under sinusoidal load.

I --0"2

0"500

0"375

O. 2¢rJO ~

O' 125"

O" 125

" 0 " 2 5 0

' ' 0 " 3 7 5

S "0"500

d'x Xm 4 I i 01 01.2

Fig. 7. Variation of the normal stress (o-x x m4) through the laminate thickness for a simply supported angle-ply ( 4 5 ° / - 4 5 ° / . . . 8 layers) square plate under sinusoidal transverse load

(a/h = 10).


I z /h

O. 500 ~ " ~

0.375

From Equilibrium Equations

- - - From Const i tu t ive Relation

0"250

0.125

I

- 0.125

\

0.1 0-2 I ] 0.3 / /

-0.25C

- 0.375

-0.5¢

f /

Fig. 8. Variation of the transverse shear stress (zxz x ms) through the laminate thickness for a simply supported angle-ply (45°/-45°/ . . . 8 layers) square plate under sinusoidal

transverse load (a/h = 10).

surfaces of a plate is presented. Such an approach is commercially attractive due to ease in software development and implementation in an existing general purpose programme which is usually based on isoparametric formulation. The nine-noded Lagrangian element of the isoparametric quadrilateral family developed here is a fairly simple element. The results obtained here have proved the simplicity and reliability of this element in a general stress analysis. The displacement model used here is the simplest in the family of higher-order models and the theory does not require the usual shear correction coefficient(s) generally associated with first-order shear deformable theories of Reissner and Mindlin.

246 I. Kant, B. N. Pandya

A C K N O W L E D G E M E N T S

Partial support of this research by the Aeronautics Research and Deve lopmen t Board, Ministry of Defence, Government of India through its Grant No. AERO/RD-134/100/84-85/362 is gratefully acknowledged.

R E F E R E N C E S

1. Reissner, E., The effect of transverse shear deformation on the bending of elastic plates, ASME Journal of Applied Mechanics, 12 (1945) A69-A77.

2. Mindlin, R. D., Influence of rotatory inertia and shear deformation on flexural motions of isotropic elastic plates, ASME Journal of Applied Mechanics, 18 (1951)31-8.

3. Lo, K. H., Christensen, R. M. and Wu, E. M., A higher-order theory of plate deformation: Part 1--homogeneous plates, ASME Journal of Applied Mechanics, 44 (1977) 663-8.

4. Lo, K. H., Christensen, R. M. and Wu, E. M., A higher-order theory of plate deformation: Part 2--laminated plates, ASME Journal of Applied Mechanics, 44 (1977) 669-76.

5. Levinson, M., An accurate, simple theory of the statics and dynamics of elastic plates, Mechanics Research Communications, 7 (1980) 343-50.

6. Murthy, M. V. V., An improved transverse shear deformation theory .for laminated anisotropicplates, NASA Technical Paper 1903, 1981.

7. Reddy, J. N., A simple higher-order theory for laminated composite plates, ASME Journal of Applied Mechanics, 51 (1984) 745-52.

8. Phan, N. D. and Reddy, J. N., Analysis of laminated composite plates using a higher-order shear deformation theory, International Journal/or Numerical Methods in Engineering, 21 (1985) 2201-19.

9. Putcha, N. S. and Reddy, J. N., A refined mixed shear flexible finite element for the nonlinear analysis of laminated plates, Computers and Structures, 22 (1986) 529-38.

10. Ren, J. G., A new theory of laminated plate, Composites Science and Technology, 26 (1986) 225-39.

11. Pagano, N. J., Exact solutions for rectangular bidirectional composites and sandwich plates, Journal of Composite Materials, 4 (1970) 20-34.

12. Kant, T., Numerical analysis of thick plates, Computer Methods in Applied Mechanics and Engineering, 31 (1982) 1-18.

13. Kant, T., Owen, D. R. J. and Zienkiewicz, O. C., A Refined higher-order C O plate bending element, Computers and Structures, 15 (1982) 177-83.

14. Jones, R. M., Mechanics of Composite Materials, McGraw-Hill, New York, 1975.

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