MECHANICS RESEARCH COMMUNICATIONS Voi.13(3),165-168, 1986. Printed in the USA 0093-6413 $3.00 + .00 Copyright (c) 1986 Pergamon Journals Ltd

STABILITY OF THIN RECTANGULAR ELASTIC PLATES UNDER A FOLLOWER FORCE

ASHWINI KUMAR and A.K. SRIVASTAVA Civil Engineering Department Indian Institute of Technology Kanpur, India

(Received 7 February 1986; accepted for print 21 April 1986)

Introduction

Studies of the stability of two-dimensional undamped structures under non-conservative forces have been rather scarce. Most of the work has dealt with one-dimensional problems, i.e., beams and columns. Leipholz EI,23 , Leipholz and Waddington [3] and more recently Leipholz and Pfendt i4,5] have studied the stability of rectangular plates, with different boundary conditions, subjected to non-conservative follower forces. Adali E63 determined the stability regions of a rectangular plate under a tangential follower force and a unidirectional axial force. In this investigation, a thin rectangular plate is considered with two opposite edges simply supported and the other two edges free; along the free edge acts the follower force which maintains a given angle of inclination with the horizontal. A study is made of flutter and divergence instabi- lities using an approach similar to that outlined in [6] .

Problem Statement

Consider an isotropic, rectangular plate of length a, with b,

thickness h, mass per unit area m, Poisson's ratio ¢, Young's

modulus E and flexural rigidity D. The edges x = 0, a are

simply supported while the edges y = ~ b/2 are free. The

plate is compressed in its middle plane by a uniformly

distributed force of intensity N along the free edges. The

line of action of this force maintains an angle of inclination

~.~w/Dy at y = ! b/2 with the horizontal: when~= 0 the force

165

166 ASHWINI KUMAR and A.K. SRIVASTAVA

remains horizontal and is conservative in nature, whereas

when ~ : 1 the force follows the direction of the tangent at

the free edge. In fact ~ is a measure of the nonconservative

nature of the force N.

' I I

I 8 '0 - , , i I

" ] I '

I I • I , I 6 '0 - I

I I '

\ •

f, ',/', 4.0 ~ / \ & k .

/ o \ / \

f 2.0

I \

I I~ Sym I \ \ - - - - - Ant isym I

/ I , I \

• I \ \ ' \

J \

• 1 \ •

', / ( oc - 0./_,)

f

J

\

"~. _____ ./'\

- - -~' " ~ ' " - - - " " ~ ' - ( 0 ' 2 )

" ~ " - - (00)

I I 1 I I I 0.5 1"5 2.5 3.5

b/o

FIG. 1

Variation of nondimensional load k vs aspect ratio b/a

ELASTIC PLATE UNDER FOLLOWER PLATE 167

Results

The variation of k(= Na2/D~ 2) with b/a for ~ = 0,0.2,0.4 is

shown in Fig.l when the loss of stability is due to divergence.

At the critical value of k, the mode of instability can be

anti-symmetric or symmetric in the y direction depending upon

b/a. In general for low values of b/a, the antisymmetric mode

8'0

6"0

4.0

2.0 0

\ Antisym static ~-~

(b/a :1.0)

/

/ /

/

/

/

f f

/

"~"---5ym, static (b/a:1.0)

/

./'i~'\// l.Antisym,static (b/a-3.5) ! " , / ' \ ~ 5 y m ,dynamic (b/a=l'O) I I" "~ . - -_ . ]/ [, ~ ~ l~ - __--..__=__._______. .......

~ , ~ \ Sym,static (b/a=3.5)

I 1"5

/ -i

"" ~ Antisym,static (b/a =1.0) j -

/ z " I I

0.5 1.0 oL.

2.0

FIG. 2

Variation of nondirnensional load k vs ~<for b/a = 1.0,3.5

168 ASHWINI KUMAR and A.K. SRIVASTAVA

is the governing one, when ~ = 0, the axial force remains

horizontal (and is therefore, conservative): the results

obtained are exactly the same as reported in Timoshenko and

Gere E73. An important feature of the results is that the

divergence load is minimum for a particular aspect ratio when

the force is purely conservative, i.e. for ~ : 0. This obser-

vation is in keeping with the findings of Culkowski and

Reismann E 8 ~ .

In order to obtain the stability boundary for a given

value of b/a, the variation of k with respect to ~ is to be

plotted using results of the static and the dynamic analysis.

A typical plot is shown in Fig.2. For b/a = 1.0, it is seen

that initially for low ~ the instability is due to divergence

in the antisymmetric mode. Then the flutter takes over but

after that the divergence instability condition again becomes

the governing one. However, the mode at this stage is first

symmetric and then antisymmetric. Interestingly, this is not

so for large b/a. For example, if b/a = 3.5, the flutter

case is missing (Fig.2) ; the instability is due to divergence

only. Initially, both the modes yield almost identical criti-

cal loads; then the symmetric mode dominates and the situation

is the same again beyond ~ = i.

References

1 °

2.

3. 4.

5.

6. 7.

8.

H. Leipholz, Trans. Can. Soc. Mech. Eng. 3, 25 (1975) H. Leipholz, Stability of Elastic Systems, Noordhoff,

Alphen aan Rijn (1980) H. Leipholz and D. Waddington, Mech. Res. Comm. 8, 223 1981) H. Leipholz and F. Pfendt, Comp. Meth. App. Mech. Engg.

30, 19 (1982) H. Leipholz and F. Pfendt, Comp. Mech. App. Mech. Engg.

37, 341 (1983) S. Adali, Int. J. Solids Struct. 18, 1043 (1982) S.P. Timoshenko and G.M. Gere, Theory of Elastic Stability,

McGraw-Hill, New York (1961) P.M. Culkowski and H. Reismann, ASME J. Appl. Mech. 44,

768 (1977)