Postbuckling analysis of rectangular orthotropic plates

Int. J . mech. Sc~. Pergamon Press. 1978. Vol. 15, pp. 81-97. Printed in Great Britain

POSTBUCKLING ANALYSIS OF RECTANGULAR ORTHOTROPIC PLATES RAlVIESH CHANDRA a n d B. BASAVA RAJU Structure Sciences Division, National Aeronautical Laboratory, Bangalore 17, India (Received 3 January 1972, and in revised form 1 June 1972) Summary--The postbuckling analysis of orthotropic simply supported rectangular plates of symmetric cross-sectlon is studied and curves of load-shortening and effective widths are presented for various cases of orthotropic plates having different elastic properties. The results obtained are compared with results already published. NOTATION plate length in x direction b plate width in y direction h plate thickness U, V displacement of point on middle surface in x and y directions respectively W deflexion of point on middle surface of plate in direction normal to undeformed plate Young's moduli in x and y directions respectively Zx~ ~ y G shear modulus /~,/z~x Poisson's ratios; first subscript denotes lateral direction, second denotes load direction 9Tt, number of half-waves in x direction n number of half-waves in y direction D1 J~EhS/12(E-tz~), flexural stiffness in x direction D3 J~D1, flexural stiffness in y direction Ds /~D1 ÷ (2Gh8/12) P total compressive load in x direction P0 critical load A0 end-shortening corresponding to critical load N~, N , and 1V~, stress-resultants in x, y co-ordinates s~ e~ and Yxv strains in middle surface in x, y co-ordinates Eb Young's modulus of the binder EI Young's modulus of the fibres Yb~ Vl Poisson's ratios of the binder and fibres respectively r~ binder content by volume a

O~

G/E

2 ~b/a A U(O ' ~) --

U( a, ~/)

1. I N T R O D U C T I O N NUMEROUS studies h a v e been m a d e of the postbuc~Hng b e h a v i o u r of flat r e c t a n g u l a r plates a n d some o f the m o s t i m p o r t a n t o f these investigations are described in refs. 1-31. The basic differential equations for a plate element undergoing large deflexions were presented b y y o n K a r m a n . 1 V o n K a r m a n et al. 2 i n t r o d u c e d t h e concept of effective width. A p p r o x i m a t e solutions for 81 6

82

RAMESH C ~ D R A

and B. BASAVA I:~AJU

postbuckling behaviour were presented by Cox, 3 Timoshenko, 4 Marguerre and Trefftz 5 and Marguerre, ~ where analysis was carried out by energy methods. Kromn and Marguerre ~ extended the results of Marguerre and Trefftz for simply supported infinity long plates in compression. Koiter s further extended this work to make it applicable far beyond buckling. Levy 9 obtained an exact solution to the large-deflexion equations of von K a r m a n for square plates. Stein 1° presented an elastic postbuekling analysis of rectangular plates, and compared it with an experimental load-shortening curve for an aluminium alloy flat plate. Stein n discussed the phenomenon of buckle wavelength change in elastic structures. Aalami and Chapman 12presented a finite difference analysis of the large deflexion behaviour of rectangular orthotropic (including isotropie) plates under symmetrical or antisymmetrical systems of transverse and in-plane loading. Initial lack of flatness and rotational, extensional and tangential boundary restraints were considered. Rushton 13,14 presented a dynamic relaxation method for analysing the postbuckling of variable thickness as well as constant thickness plates with various boundary conditions; the phenomenon of the change in the number of waves with a change in the load was investigated by using three different methods of loading the plate. The analytical determination of postbuckled plate behaviour in the plastic range has been investigated by Mayers and Budiansky for square plates ~5 and for rectangular plates by Mayers et al. ~6 The latter study served to explain the absence of a plate maximum strength in the case of a square plate in the analysis in ref. 15, and correlated well with the experimental data presented by Stein. 1° Mayers and Nelson 17 presented a postbuckling and maximum strength analysis of a uniformly shortened simply supported rectangular plate with straight unloaded edges employing Reissener's variational principle in conjunction with a deformation theory of plasticity. The increased interest in composite materials by the aerospace industries demands a better understanding of the strengths of these composite materials. Studies of the rudiments of anisotropic elasticity have been conducted by Hearmon 1S and Lekhnitskii. 19 Seydel, 2° Schulesko, ~1 Smith, 2~ Wittrick, 23 Chang e~ and Holston 26 have contributed to specially orthotropic plate buckling. Plates of general orthotropy were studied by March, ~6 Thielemann ~7 and Fraser. 2s Although March's experimental results are quite valuable, his mathematical analysis neglected some important terms in the strain energy. Thielemarm restricted himself to infinite plates and found exact closed-form solutions. Mandel129 presented experimental results of the buckling of anisotropic plates. Chan a° presented an elastic postbuekling analysis of laminated anisotropie clamped or simply supported rectangular plates; the solution was obtained utilizing the Raleigh-Ritz method with a two-mode approximation for the transverse displacement. In this paper the postbuekling analysis of orthotropic rectangular plates of symmetric cross-section has been carried out. Von K a r m a n large-displacements equations as applied to orthotropic plates are reduced to an infinite set of linear differential equations by expanding the displacements in a power series of a perturbation parameter. The first three equations of the infinite set correspond

Postbuckling analysis of rectangular orthotropic plates

83

to small-deflexion theory, whose solution yields the buckling load. Solution of the succeeding equations gives the postbuckling solutions. The curves of loadshortening and effective widths are presented for various cases of plates having various elastic properties. The results obtained in this investigation are compared with the results of Stein 1° and Chan)° 2. G O V E R N I N G

EQUATIONS

Von K a r m a n large-deflexion equations for orthotropic rectangular plates are as

follows: N.,~+Nx¢,~ = O,

(la)

Nv,,~+N.u,. = O,

(lb)

L ( D ) w - N . w , ~ . - N u w , u . - 2 N . u w,xt, = O,

(lc)

where 3~ b* L(D) = D I . ~ x 4 + 2 D a ~ + D , ~ y

b* 4.

The s t r a i n - m e m b r a n e force relations are 1 [N~

Nv"~

1 {~__~N~, ~" = h \ E , -I~'xN--~]

(2b)

1 N~

Solving equations (2a), (2b) and (2c) for the membrane forces, we obtain EEh

N, = ~

(Beu +/ze,),

(3b)

N~I, = GhT~ ~,

(3e)

where E~ J~=~,

/z=,a~,j

and

E=E

z.

The strain-displacement relations are ex = u.~ + ~w~..

(4a)

e, = v,~ + ½w~,,

(4b)

y ~ = u,v + v,~ + w,~ w~.

3. M E T H O D

(40)

OF ANALYSIS

The displacement function u, v and w are expanded in a power series of an a r b i t r a r y perturbation p a r a m e t e r fl as below u = uo +f12 u2 +f14 u4 + ....

(Sa)

v = Vo+fl~ v2 +fit v4 + ....

(5b)

+fl' ~. + 3

(~e)

w =~wl

~ w , + ....

where

3~= ( p - Po)/Po.

84

I % ~ S H CHa_~D~ ~nd B. BASAVA RAJU

Substituting equations (5) into equations (4) and substitution of the expressions obtained for the strains in equations (3) yields the following expansions for the forces oo

where

co

~z=0,2,4

n=l,3

~=0,2,4

n=l,3 m~l,3

n=0,2,4

n=l,3 m~l,3

BEh N~n = ~~ - ~ ]~Eh

N=.m = Z - ~

EEh N u n -- ~ EEh

N~m = ~_~

m~l,3

(6d)

(un,~+t~vn,u) , 1

~ (win,=w.,= + ~w~,~ w.,~),

(6e)

(Evn,,~ + ~un,~),

(6f)

1

~ (Ew.,~ wm,v +/~w.,~ w~,~),

N~,,,~ = Gh(un,u + v~,~), N ....

(6g) (6h) (6i)

= Gh(wm, ~ w,,,,,).

Substituting the forces and displacement b y their respective expansions in equations (1), we obtain an infinite set of equations as follows : N~o,x+Nxuo,u -- 0, t

(7a)

N~o,~ + N ~ o , ~ = O, ] L ( D ) w 1 - (N~o w i , ~ + N~o wi,v~ + 2Nz~ o wi,~v) = 0,

Nx2,z + Nxu2,u

-- _ (Nxll, x + N~ull,V),

N~.~,+N~,2.~

(hT~n,~ + N ~ n . = ) ,

f

(7b)

(7c)

L ( D ) w a - (Nxo w 3 , ~ + Nvo wa,vv + 2N~v 0 wa,~) = (N~2 +N~n) Wl,x~

+ ( N ~ + Nvn ) wl.u~ + 2(N~u 2 + N ~ l i ) wl,~v,

(7d)

N~4,= + N~v4,~ = - (2N~la, ~ + N~13,~ + N~ai,~), Nua,v + Nxua, z : -- (2Nv13, v + Nxyl3, y ~- Nxy31, ?1), i

(70)

L ( D ) w 5 - (N~o w s , ~ + Nvo ws, vv + 2 N ~ 0 ws,~ ) = ( N ~ + N~n ) wa,~

+ (N,~ + Nvn ) wa,vu + 2(N~** + N~vn) w~,=. + (N~, + 2N~i~) wi,~ + (Nua + 2Nvla) wi,v, + 2(N~v, + N.vt~ + N.uat ) wt,,v.

(7f)

4. B O U N D A R Y C O N D I T I O N S The following b o u n d a r y conditions are imposed on the edges of the plate (see Fig. 1) : Zero deflexion : w(O, y) = w(a, y) = w ( x , O) = w ( x , b) -- O. (8a) Zero bending moment: w :~(O, y) = w **(a, y) = w . , , ( x , O) = w, vv(x, b) = O.

(8b)

Constant in-plane displacement: u,~(0, y) = u u(a, y) = v,~(x, O) = v ~ ( x , b) = O.

(8c)

v,,(O, y) = v,~(a, y) = u . , ( x , O) = u , , ( x , b) = 0.

(Sd)

Zero shear stress:

Postbuckling analysis of rectangular orthotropic plates

Fro. 1. Orthotropic plate. Boundary conditions (Sa) and (Sb) imply that all the four edges of the plate are simply supported. The plate is subjected to uniform compressive load of total magnitude P in the z direction. The force boundary conditions on the edges of the plate are as follows: Loaded edge :

sb 0

Unloaded edge :

W,L,,,

a W,L,,, s 0

dy = -I’.

(84

ds~

WI

=

0.

Substituting N, and N, by their series expansions (6a) and (Bb) and P by PO+/3* POin the above equations, the following boundary condition corresponding to different approximations in fi are obtained. Loaded edge :

sb sb sb 0

0

and

0

Unloaded edges :

(N,),_,,,dy+P,

(94

= 0,

Wz,+N,n),=o,a dy+P, = 0

(9b)

W, + 2NmL0,. dy = 0.

(9c)

(104 (lob)

sa 0

W,d + 2N,,,L,,,

dx = 0.

(1Oc)

5. SOLUTION Solutions of equations (78) provide the pre-buckling stresses in the plate.Equations (7a) are now rewritten in terms of u,, and et, as follows :

AN,,

+ ~h~o,v,,+ %,w

= 0

VW

Bu,,,, = 0,

Wb)

and #Au,,,, + (%,,,+

86

RAMESH CHANDRA and B. BASAVA RAJU

where

BEh A = B_~;

R = p~4+Gh.

Solutions of equations ( l l a ) and ( l l b ) satisfying t h e b o u n d a r y conditions (8c), (8d) and (9a) are Po [ a\

v0=/Zh~

N,o=

y-

Po ---~-;

,

(12b)

N,o = N x , o = 0.

(13)

Using the expression (13), e q u a t i o n (7b) can be w r i t t e n as

L(D) w t + (Po/b ) wl,xr = 0.

(14)

The expression for w 1 in order to satisfy the b o u n d a r y conditions (8a) and (8b) is assumed as below : w 1 = A1 sin m~rx s i. n -n~y -. a b

(15)

S u b s t i t u t i n g expression (15) in e q u a t i o n (14), we obtain

----(m~r/a)~

Po

~--~-) -I-D, \ b ] J"

The lowest buckling l e n a is d e t e r m i n e d b y t h e choice of m and n for a p a r t i c u l a r l e n g t h w i d t h ratio a/b. The v a l u e of A 1 is d e t e r m i n e d l a t e r on. E q u a t i o n (7c) is now solved for u S a n d v s. R e w r i t i n g e q u a t i o n (7e) in u s a n d vs:

Au"'x+Ghu~"'+Bv""-'--(-~-)A---~A[-(m~i'+l~('~) ' \ La / A~ A

-

+(B+Gh)

-4-

sin

] sin----~---2m'x

cos-a

(17a) b

'

a

i,,b)

"

Solutions of the a b o v e equations satisfying b o u n d a r y conditions (8c), (Sd) and (9b) are u,

r po

A,~ A~ [(m~'/a)'~-t,e(nv/b) ' 16

vs = I.Ehb E

L'

~

2minx

~,

sin--a

.

2,~,

a Sm--a

~".'Yl'

cos 0 j

(lSa)

8 A~ rJ~(n~r/b)~-I~(m'/a)2 sm " -2nrry 16 [ E(nar/b) b

- -

( 7 1 cos 2m~rx sin ~ ] a

.

( 18b )

U s i n g expressions (18) and (15), e q u a t i o n (7d) becomes

Po Po ( m ~ 2 _ E h A ~ L(D) w8 + T w3,,, = A~ { T \ a ] 16 +-iT

~a)

s~

a

m,rr i

sin--i- +-i~-~Y)

n~" *

. mlrx . n:rry

sm-i-s~

a

•

(19)

Postbuckling analysis of rectangular orthotropic plates

87

The particular solution of the above equation which satisfies the boundary conditions does not exist unless the coefficient of sin (nvrrx]a) sin (n~y/b) is zero. Thus

(20)

16 ro [nv~r~ [ ( m ~ ' ~ ' + 8 (n~r~'l-1 A ~ =-E'h"b ~ a ] [\ a ]

~b / J

"

The complete solution of equation (19) is now written as wa=Aasin

m~'x . n.z'y

a

.

. m~rx . 3n~rnj

sm--~-+Aalsm

a

sm--~

.

.

+A33sm

3mTrx . n.rry

a

sm--~--,

(21)

where

~h'A 13

-- - w

( - ~ ) 3 /3nTl'\ 2 - ( 3"D~fl'~4 Po (~.'J'r~21-1

(V) 4

['.

-

IT)

,

(22)

A~2 = --i-6-Equation (70) with the associated boundary conditions (8c), (Sd) and (9c) is solved for ui and va. The hitherto u n k n o w n constant A a is determined from equation (Tf) in the same way in which A x is determined. The complete solution to the problem in u, v and w up to the terms containing fla is now obtained b y adding zeroth-, first- and second-order approximation solutions. As the final expressions for u, v and w are quite lengthy only a few of the important results derived from them are presented. The expression for total end shortening is given by A 3 ( 8 - F 2 ) b~

Pb8

r~a{1-~'A~

8~A3

I

1~--I

3\

B--1

_

~ . 1

where

83=

( 8 - ~ ~) h 2

= 8 (Pb/D1 ~r2) )~8_ IrA" + 2{[~8 + 2~(8-F~)]/8} A 3 n ~ + En4]] h 4+ 8 n ~ A4 8

z~a1 = ~

[(~+

2a(8__~2)) ~3 n + 582 n4] -1,

8n a

-~a3 = 8(9A 4 _ 8 n , ) , Xa = _3 X81{A 4 + [SB2(8 - 1 ) / ( 8 - ~3)] n4} + 8X3~ n4 2 A~+ En4 + [ 2 8 ( 8 - 1)/(8-F3)J n ~ ( E n 3 - ~,~2)

The effective width b, is defined as p

A

hE~ b~

a

°

(25)

Using equation (24) in the above expression, we obtain b,

Pb

E-1

[ PbJE

_

~3A2

~a I l i a / 1 - F

3-- l ' - X

~ "~

]E-1

\

(26)

6. D I S C U S S I O N The theory discussed in this paper is applied to a few specific examples of plates A - N whose elastic constants are given in Table 1. The elastic constants for plates A - L are taken fromref. 31 and for plates M and N refer to plates 6 a n d 9 ofref. 30. Plates A - F represent stiffness ratio, 8 less t h a n unity, whereas plates G-L represent stiffness ratio, J~ higher than

88

R/L~ESH CHANDRA and B. BASAVA RAJU

unity. I n fact t h e values of elastic m o d u l u s E~ a n d E , for plates G--L are t h e same as E~ a n d E~ for plates A - F . The stiffness ratios, ~ , a a n d Poisson's ratio F~v are shown for plates A - N in Table 1. I n Figs. 2(a)-(d), t h e critical stress is p l o t t e d for plates C, F, I and L a n d in Fig. 2(e) t h e critical stress for an isotropic plate is plotted. The portions of t h e curves defining the lowest critical values of the load are shown b y full lines. F o r plate L h a v i n g a stiffness ratio, E ~ 13, two half-waves are formed for a square plate, Fig. 2(d). The transition from m to ( m + 1) half-waves occurs w h e n

a/b = ~/[m(m + 1)]/(E) °'~s. T A B L E 1. E L A S T I C CONSTANTS FOR T H E VARIOUS CASES OF COMPOSITE PLATES A - N

( A - L refer to t h e elastic properties given b y ROSEN et al. ;31 M and N refer to plates 6 and 9 of CHAN3°) Design a t i o n of different c~ses

E/*

E~*

E~*

G*

E

a

t~

A B C D E F G H I J K L M 1~

2 5 10 20 50 100 2 5 10 20 50 100

1.887 3.761 7.068 14.130 36.390 69.660 1.5772 1.648 3.578 4.282 4.788 5.071 2'6 7"5

1-5772 2.648 3.578 4.282 4.788 5'071 1,887 3.761 7.068 14.13 36.39 69.66 7.5 2.6

0.6123 1.061 1-409 1-714 1.959 2.062 0.6123 1.061 1.409 1.714 1-959 2.062 2-1 1.1

0.8358 0.7041 0"5064 0.3031 0.1315 0.0728 1-1964 1.4202 1.9747 3.2292 7.6045 13.7362 2.89 0.347

0.3245 0"2823 0.1994 0.1213 0.0538 0"0296 0.3882 0.4009 0"3937 0.4002 0.4091 0"4065 0"424 0.147

0.3 0.3 0.3 0.3 0.3 0-3 0.25074 0.21123 0.15192 0.09093 0.03945 0.02184 0.08675 0.25

* 106 lb/in 2. PLATE C (E" : O. 5064) m=l

m:2

m:3

m:4

m:5

4-

b2

(N,)cr~,3-

2-

a/b

FIG. 2(a).

Critical stress vs. a/b ratio for plate C (N = 0.51).

F~

0.25074 0-21123 0.15192 0.09093 0.03945 0.02184 0-3 0.3 0.3 0"3 0.3 0.3 0.25 0.08673

Postbuckling analysis of rectangular orthotroplc plates

6"~

PLATE

89

F (i~ -" 0 . 0 7 2 8 )

5m=l

m:2

m=3 m=4 m=5

4-

(N=)Cr ~ ,

3-

0

'

~

2.72

2

4.72

r 6

4

6.61

8.62 8

o/b

FIQ. 2(b).

Critical stress vs. a/b ratio for plate F ( ~ = 0.072).

7-

/

o~

~

P L A T E I 1~" : 1.97471

,,,9

I

;,206

2.,3~,

~.Ta

;,

Qlb

FIG. 2(c). Critical stress vs. a/b ratio for plate I (• = 1-97).

i

I0

90

RAMESH CHANDRA a n d B. BASAVA RAJU

18-

,

,,

',

16-

I

:

¢

s

II i

,'

t

m:4/

/

'

I

,"

I"

,s

,

(N=)c,~--~, ~0

',

'~

,,

m=l

,/

/

~

~

•

m=5

!

/

-

12-

/

/

t

/

~

,

re=S/

s

',

,

/

m=21

i' J

',

/

,

m=ll

,

14-

,

/,

/

/

7 I

*

/ //

~ PLATE

L (1~ = 13. 73621

6-

42

0.735

0

1.27

a/b

FIG. 2(d). Critical stress vs.

a/b ratio

for p l a t e L ( E = 13.73).

7m:l

m:2

m:S \

6-

/ /i

~

\

5-

4~ (N=)cr 3-

ISOTROPIC PLATE

(E" = 1.01

alb FIG.

2(e). Critical stress vs. a/b ratio for isotropic p l a t e ( E - 1-00).

Postbuckling analysis of rectangular orthotropic plates

91

The different a/b ratios at which the transition from m to ( m + 1) half-waves takes place are shown in Figs. 2(a)-(e). Thus for a high stiffness ratio (E ~, 13), the transition from single to double half-wave takes place for a n a/b ratio of 0.735. The corresponding figures are 1.19, 1.68, 2.72 and ~/2 for values of J~ equal to 1.97, 0.51, 0.07 and 1.0, respectively. The last case represents the isotropic plate. A n isotropic plate of a/b ratio 2 will buckle in two half-waves whereas plates L, I, F and C of the same side ratio will buckle in four, two, two half-waves and one half-wave respectively. Therefore the values of the parmneter, in these cases are 2, 1, 1 and 0-5. I n Fig. 3, the variation of the buckling load p a r a m e t e r

4 9

3-

~ - - L

Pob

..------ K

TROPIC

I-

0

o

,:0

,~z

~,~

~le

118

21o

X FIe. 3. Variation of buckling load p a r a m e t e r with ~ for various cases.

Po/4zr~D~ is plotted for various values of A. The buckling load parameter monotonically increases with ~ for the isotropic plate a n d plates A - F , whereas for plates L, K, J, I, H a n d G, the p a r a m e t e r decreases first and then again increases as A is increased. Figs. 4-7 represent load-shortening curves from which the following conclusions can be drawn. The postbuckling strength in the case of orthotropic plates depends on the ratios, J~, ~, /zx~ and the p a r a m e t e r ~. Plates G - L ( ~ > 1) have a higher postbuckling strength compared to plates A - F (E < 1). F o r A -- 1, plates G - L have distinct postbuckling strengths, whereas the postbuckling strength of plates A - D is the same as for isotropic case (see Fig. 4). As the value of the p a r a m e t e r A increases the postbuckling strengths of plates A - F are quite different from t h a t of isotropie case (see Fig. 6). F o r the infinitely long plate, plates A - D exhibit a very close postbuckling strength whereas plates G - L have distinct postbuckling strengths (see Fig. 7). Fig. 8 is a plot of non-dimensional effective width vs. non-dimensional loads.

92

RAMESH ~HANDRA a n d B. BASAVA RAJU

T h e r ~ u l t s o f t h i s i n v e s t i g a t i o n are c o m p a r e d w i t h t h e available r e s u l t s o f o t h e r i n v e s t i g a t o r s . F o r t h e isotropie ease, ~ a s s u m e s a u n i t v a l u e a n d t h e r e s u l t s (for J~ = 1) s h o w n in Figs. 3, 5, 6 a n d 7 c o m p a r e e x a c t l y w i t h t h e r e s u l t s o f Stein. l° T h e r e s u l t s o f S t e i n for t h e p a r t i c u l a r case o f t h e p l a t e o f a/b r a t i o 1.25 h a v i n g a n e n f o r c e d d i s p l a c e m e n t of 8-5 a n d b u c k l e d as a double w a v e are c o m p a r e d w i t h t h e results o b t a i n e d for t h e s a m e A,B,C,D F

2-

P Po

I-

0

! 2

0

!

,4

6

A

&o FIG. 4. L o a d - s h o r t e n i n g c u r v e s for t h e ease ~ = 1.0. L

/ /

2-

/ /

ISOT~DOPIC

E~

J

F

I-

6

8

Ao

FIG. 5. L o a d - s h o r t e n i n g curves for t h e case A = 1.33.

I0

12

Postbucklmg analysis of rectangular orthotropic plates

3"

L

K ISOTROPIC C

98

O

E

2-

i'o

,'s

~

2'5

_A

Ao

FIG. 6. Load-shortening curves for the case ~

3"

C,D B J~ISOTROPIC

J

=

2"0.

K

L

2-

I-

z~ A

Fzo. 7. Load-shortenhag curves for the infinitely long plate.

94

RAMESH CHANDRA and B. BASAVA RAJU 1.0-

0.8-

b._.e.e 06-~ b "

D

0.4-

0.2

~

EISOTROPICA

~F

I

,4

I

,8

P P0

2'z

2'6

~o

FIO. 8. Effective width vs. load curves for the case ~ - 1.0.

case by R u s h t o n J 4 Rushton uses a dynamic relaxation technique, and compares the resultant applied load, m a x i m u m deflexion and edge displacement. He reports good agreement between these results. I n Fig. 9, the postbuekling strengths are compared for the cases M and N with plates 6 and 9 of Chan. a° Chan considers two distinct cases of boundary conditions on the unloaded edges, namely, v displacements were restrained or unrestrained whereas in our ease the unloaded edges were partially restrained in such a way that the v displacement was uniform all along the edge. Also the curves of Chan were based on a two-mode approximation for the transverse displacement, w, whereas in this investigation a three mode approximation was used for the transverse displacement. The curves for plates M and N lie in between the curves of Chan. In the postbuckling of plates, one of the most important phenomenon is the change in buckling pattern due to increase of load or increase of end shortening. Fig. 10 shows the load-shortening curves (for J~ = 1) where the load and endshortening are not nondimensionalized with respect to the critical load and critical end shortening. These are similar to those obtained by Stein. 1° Stein has reported t h a t the intersection of the curves indicates possible changes in buckling pattern. However, experiments by Stein have shown that the change in buckling p a t te r n does not necessarily occur at the intersection points of these lines for different numbers of waves. Rushton la demonstrates t h a t a solution to the postbuckling problem is possible by using the dynamic relaxation method. However, it is essential to distinguish between the different postbuckling configurations and to a t t e m p t to find how the postbuckled form depends on the loading path. Rushton uses the dynamic relaxation method to investigate the effect of the loading path, and describes three experiments, namely (a) gradual increase in enforced displacement, (b) gradual increase in applied stress, and (c) change in waveform caused by lateral loads. I n the first method, solutions by dynamic relaxation have been obtained in which the edge displacements are gradually increased (or decreased) through the value at which the applied stress-displacement curves for single and double waves intersect. For a plate of side ratio 1 : 1½ it has been demonstrated that with increasing displacements the profile starts

Postbuokling analysis of rectangular orthotropic plates 3"

PLATE No.6 (WITH UNLOAI~D EDGES- RESTRINED)-- CHAN

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PLATE No.9 (WITH UNLOADEDEDG£S REGTRAINED)-CHAN PLATE'No.9 (WITH UNLOADED EDGES U N R E S T R A I N E D ) - CHAN

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96

RAMESH CHANDRAand B. BASAVARAJU

as a single half-wave and remains as a single half-wave, whereas under decreasing displacements the profile changes from a double wave to a single wave. I n the second method of studying the effect of changing waveform, the average applied boundary stress is gradually increased and the change in the deflexion profile from a single to double wave was studied b y a dynamic relaxation method in plates of side ratios 1 : 2 and 1 : 1½. The third method of investigating the change from one waveform to another is to apply additional lateral loads while the plate remains under a constant in-plane load. The change in waveform will only take place if the lateral loads provide sufficient energy. Thus the dynamic relaxation method offers a new technique for the postbuekling analysis of orthotropic and composite plates. 7. C O N C L U S I O N S In this paper the postbuekling analysis of orthotropic rectangular plates of symmetric cross-section has been performed. The curves of load-shortening a n d e f f e c t i v e w i d t h s a r e p r e s e n t e d f o r v a r i o u s cases o f o r t h o t r o p i c p l a t e s h a v i n g various elastic properties . The results obtained in this investigation are comp a r e d w i t h t h e r e s u l t s o f S t e i n l° a n d C h a n . a° T h e a n a l y s i s p r e s e n t e d i n t h i s p a p e r is t o b e c o n s i d e r e d a s t h e first p h a s e o f t h e s t u d y o f p o s t b u e k l i n g o f c o m p o s i t e p l a t e s . F u r t h e r w o r k is d e s i r a b l e o n t h e p o s t b u c k l i n g o f o r t h o t r o p i c p l a t e s o f t h e e c c e n t r i c a l l y s t i f f e n e d t y p e a n d also o t h e r b o u n d a r y c o n d i t i o n s . Acknowledgements--The authors t h a n k the Director of the National Aeronautical Laboratory for permitting this work to be undertaken under project 534. The authors t h a n k Mr. S. P r a d h a n for his kind help with the computational work. Thanks are also due to Mr. B. R. Sheshadri for his help in the preparation of the manuscript. REFERENCES

1. TH. VO~ KA~M_~, Festigkeitsprobleme in Maschinenbau 4, 311; Encykl der Math. Wiss. 4, 311 (1910). 2. TH. voN KARMAN, E. E. SECHLER and L. H. D O ~ E L , A . S . M . E . Trans. APM-54-5 54, 53 (1932). 3. H. L. Cox, The Buckling of Thin Plates in Compression. R. & M. No. 1554, British A.R.C. (1933). 4. S. TIMOSHENKO, Theory of Elastic Stability, pp. 390-395. McGraw-Hill, New York (1936). 5. K. MARGUEREE and E. TREF~Z, Z . f . a . M . M . 17, 85 (1937). 6. K. MARGUERRE, The Apparent Width of the Plate in Compression. NACA TM 833 (1937). 7. A. KROMM and K. MARGUEttRE, Behaviour of a Plate Strip Under Shear and Comprezsive Stresses Beyond the Buckling Limit. NACA TM 870 (1938). 8. W. T. KOITER, De meedragende breedte bij groote overschrijding der knikspanning voor verschillende inklemming der p l ~ t r a n d e n (The effective width of infinitely long, flat rectangular plates under various conditions of edge restraint). Rep. S. 287, National Luehtva~rtlaboratorium, Amsterdam (Dec. 1943). 9. S. LEVY, Bending of Rectangular Plate~ with Large Deflections. NACA Rep. 737, (1942). 10. M. S ~ I ~ , Loads and Deformations of Buckled Rectangular Plates. NASA Tech. Rep. R-40 (1959). 11. M. S ~ I N , The Phenomenon of Change in Buckle Pattern in Elastic Structures. NASA Teeh. Rep. R-39 (1959). 12. B. ~ I and J. C. CHAPMAN, Proc. Inst. civil Engrs 42, 347 (1969). 13. K. R. RUSHTON, lnt. J. mech. Sci. 11, 461 (1969). 14. K. R. RUSHTO~, Aeronaut. Q. 21, 163 (1970).

Postbuckling analysis of rectangular or~ho~ropic plates

97

15. J. MAYERS and B. BUDIANSKY, Analysis of Behaviour of Simply Supported Flat Plates Compressed Beyond Buckling into the Plastic Range. NACA Tech. Note 3368 (1955). 16. J. MAYERS, E. NELSON and L. B. SMrrH, Maximum Strength Analysis of Postbuckled Rectangular Plates. S U D A A R No. 215 (1964). 17. J. MAYE~S and E. NELSON, Elastic and Maximum Strength Analysis of Postbuckled Rectangular Plates Based upon Modified Versions of Reissener's Variational Principle. SUDAAR No. 262 (1966). 18. R. F. S. HEAaMON, An Introduction to Applied Anisotropic Elasticity. Oxford University Press (1961). 19. S. LEKHNITSKII,Theory of Elasticity of an Anisotropic Elastic Body. Holden-Day, San Francisco (1963). 20. E. SEYDEL, The Critical Shear Load of Rectangular Plate. N A C A T M 705 (1933); translation of A usbeuli~chblast rectackiger Platten, Z. f .M. 78 (1933). 21. P. SHULESKO, Aeronaut. Q. 3, 145 (1957). 22. R. C. T. SMITH, The Buckling of Flat Plywood Plates in Compression. Australian Council for Aeronautics Report 12 (1944). 23. W. H. WITrRICK, Aeronaut. Q. 4, 83 (1952). 24. F. CHANG, Scientia Sinica VII, 716 (1958). 25. A. HOLSTON, JR., A I A A J. 8, 1332 (1970). 26. H. MARCH, Buckling of Flat Plywood Plates in Compression, Shear, or Combined Compression and Shear. Forest Products Lab. Rep. 1316 (1942). 27. W. T H I E L V . ~ , Contributions to the Problem of Buckling of Orthotropic Plates, with Special Reference to Plywood. NACA TM 1263 (1950). 28. H. FRASER, JR., Bifurcation t y p e buckling of generally orthotropic plates. Ph.D. thesis, University of Illinois (1968). 29. J. M. MANDELL,An experimental s t u d y of the buckling of anisotropic plate. M.S. thesis, Case Western Reserve University (1968). 30. D. P. CHAN, An analytical study of the postbuckling of laminated, anisotropie plates. Ph.D. thesis, Case Western Reserve University (1970). 31. B. W. ROSEN, N. F. D o w and Z. HASHIN, Mechanical Properties of Fibrous Composites. NASA CR-31 (April 1964).