Bending of plates based on improved theory

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Voi.I0(4),205-211,1983o Printed in the USA. Copyright (c) 1983 Pergamon Press Ltd

BENDING OF PLATES BASED ON IMPROVED THEORY

K. Shirakawa Department of Mechanical Engineering, University of Osaka Prefecture, Sakai, Japan

(Received 17 January 1983; accepted for print 24 May 1983)

Introduction

The plate theory which is not based on the Kirchhoff assumptions has been developed by Reissner [1,2] and Ambartsumyan [3] by taking into account the effects of shear deformation and normal stress. This so-called improved theory is the one to complement the classical plate theory and to extend its application range. In this paper the Ambartsumyan theory is used, but the resulted equation is quite the same as the Reissner theory. The statical equation including both effects is presented in terms of a single deflection, which seems useful to the analysis of practically important problems. In the improved theory, even for a simply supported edge there are two ways of supporting, that is, in addition to the classical boundary condition (a) rotation angle=O or (b) twisting moment=0. It is in general not easy to obtain solutions in the case of (b). In this work the analytical method of the basic problem for a rectangular plate with any boundary condition is presented and the effects of shear deformation and normal stress on the deflection are examined compared with the classical values.

Basic Equations

The governing equations of an isotropic plate including the effects of shear deformation and normal stress are given after some modification as [3,4] ¢'x

+~'y

- p'

(i-i)

DV2W'x- (D/-D)(~'xx+ a~'yy) + ~- B(D/-D)~'xy= -A°(h2/lO)P 'x'

(1-2)

DV2W,y - 6( D/D)~,xy - (D/D)(c~ ,xx+ ~ ,yy) + ~ = -A°(h2 /lO )p 'y'

(1-3)

where ( ),x = 3(

)/3x, ( ),y

3( )/3y, w:

deflection, @ and ~: the unknowns

which are equivalent to the transverse shear resultants 205

Qx

and

Qy, h:

thick-

206

KAORU

P:Eh3/12(l-v2): flexural

~:~: of plate, Poisson's

ratio, T): surface

ficient relatin~ and D :
SH I R A K A W A

r~dity,

E: Youn~'~ modulus,

load, V 2 : ~2/~X2 + ~ 2 / ~ y 2

to the effect of normal

in which K= 5/6 ~s the shear coefficient

CO(!f-

A0 = - ~ / ( l - ~ ) :

~
~:

8 = (i + ~)/2

and C [~; the shear

modulus. As in the case of the classical

[)late theory

tion in terms of the deflection

w. Elimination

the following

V4[l-

~(D/D)V2]w : [ 1 -

tion and normal

a(D/D)V2][1-

of plate which

stress.

V4w= [ i - ( D / D ) V

(D/D)V 2 -

as a part

(3)

and A0 = -~/(I- ~).

The relations

equation, but in this work it is

that the nezlect

Ill I

between w, ¢, ~ a n d

Mx = - D(W ~xx + VW,yy) +

from the following relations

the stress couples M ~

(D/F)(#,x+

V~,y) +

(4)

My, Mm~ are + ~,y) '

+~(D/F)(¢,y+~,x).

(5--1)

(s-2) (5-3)

Conditions

In the improved plate theory the relaxed boundary conditions applied.

ziven by

Ao(h2/lO)(¢,x+ ~,y)

My = - D(W,yy + vw,z x) + (D/D)(~,y + re,x) + Jo(h2/lO)(¢,x

Boundary

is

of both effects yield

(#D/D+ Aoh2llO)p].

m/muJ [DV2w+

= -

Mxy=- (1 - V)DW'xy

solution

equation.

Once w is found, then ~ and ~ are obtained

=]

[5]

In general the term

~n w such as in (2) when the complementary

classical

a(D/F)V

(2)

(3) is the same one as was given by Reissner

It is easily understood

-

Ao(h2110)V2]p/D.

2-Ao(h2/lO)v2]p/D.

D/D= h2/5(i-~)

the well-known

I1

to

includes the effects of shear deforma-

[i- a(D/D)V 2] is used as a second fundamental

needed.

of ¢ and ~ in (i) leads

The main part of (2) is

It can be found that equation

considered

to have the egua-

equation

Th~s ~s the equation

if we put

it is useful

That is, along the edge of x = const,

are expressed

the boundary

[3] are

conditions

as follows:

(a) simple support

IS:

w= 0,

[

0,

S': W

Mx= 0, M

=0, X

-W,y+f(zo)~/-D M

(S-l) (e-2)

=0 Xy

= O,

"

B E N D I N G OF PLATES B A S E D ON I M P R O V E D T H E O R Y

(b) clamp

C :

w = 0,

-

w, x+f(zo)~/D = O,

(c) built-in

C':

w : 0,

-

w, x+ f(zo)~/D

(d) free

F :

Qx

= 0,

In the above expression cal models

to describe

M

x

: 0,

M

the complicated

f(zo)

as one of the mathemati-

h/2,

in

of the plate, where z~ is taken of a line element originally

is given by

(zo/h) 2.

= 5/4-

If we put z0 =

(9)

so as to satisfy the condition

of the rotation

normal to the plate's midsurface,

(8)

supporting method for a real plate,

over the thickness

a measure

0, (7)

Mxy : 0,

the quantity z0 is introduced

the region of -z0 ~ z { z 0

f(zo),

- W,y+f(zo)~/D=

= 0.

xy

which can be seen in [3], and is selected

O ~ z o ~h/2.

= 0,

20"}

(I0)

we have

f(zo ) = i. In this case -

which

W,x+

~/D= 0

- W,y+ ~/~= O,

and

show that the rotations

all the thickness

of the plate.

tions are equivalent given by Reissner

[5].

The value of

by putting

boundary

D~

along x = const,

of the rotation

f(zo)

in equation

conditions

for the classical is a u t o m a t i c a l l y

and for the clamped and the built-in of

over

that these rela-

for the clamped edge (i0) is the modified

result.

; -w,y+f(zo)~/D= 0

the known condition

are restricted

It can also be r e c o g n i z e d

to the conditions

one of the A m b a r t s u m y a n ' s The geometrical

of the section x = const,

W,x= O.

Mxy

in (6-1) and

theory are obtained satisfied

since

conditions

(8) is excluded

W,y

we reach in the

classical

theory.

For the free edge the well known condition proposed by

Kirchhoff

is used

[5].

The boundary

conditions

along y = const,

0

are given in the simmilar manner.

Analysis

The general analytical m e t h o d load is presented.

The side length of the plate

usual m a n n e r the deflection ticular

for a r e c t a n g u l a r

solution w

w is assumed

and the c o m p l e m e n t a r y P

plate subjected

to a surface

is denoted by a and b. In a

to be given by the sum of the parone w . c

208

KAORU

SH I P A K A W A

If p is expressed in the form of Fourier series, the particular solution is obtained from (3) in the form OO

W

=

where W

OO

[ [ W

(i+ @)sin(m~x/a)sin(n~y/b),

(ii)

corresponds to the deflection for the classical theory and 6

mno

represents the correction term due to shear deformation and normal stress, which is given by

= IT~[D/Db ~ + A o ( h / b ) 2 / l O ] ( p ~ + n2),

p:mb/a.

(12)

It is found that w is the solution of a plate with all edges simply supported P for the case of (6-I)~ if we assure that M

: 0 and ~

: 0.

For other type of the boundary conditions we should consider the complementary solution Wc. Ambartsumyan developed the solutions for plates with simply supported edges at the opposite sides, but here in order to obtain the solutlon: for more general problems we consider ~

expressed in the following form

[A~icoshl.x + ,4~21.X sinhl.x + A~ 3coshq~x

Wc : n

+A~4sinhX~x

+ A. sl~mcoshh.x

+ A~6sinhQ.x]sin(nzy/b)

+ ~ [B~lcOShSmy + Bmzs~ysinhs~y + B ~ 3 e o s h ~ y m

+Bm~sinhsmy

+Bmss.ycoshs~y

+B~6sinh~my]sin(mITx/a),

(13

where In, q~, s. and ~m are the roots of equation (2), which are given by X.=n~/b,

q. = / (nw/b) 2 + (D/aD),

(14

s.=mz/a,

~.: / (m~/a) 2 + (~/aD).

(15

In view of the form of we, the corresponding form of ~c with coefficients Cni, Dm~ and ~c with coefficients En~, F ~

is determined.

Applying the

equations (4) and employing the relations (14), (15) and (i], then we can obtain the following relations among the unknown constants. A~3

: A ~ 6 : B ~ 3 : B ~ 6 : C ~ 2 : C ~ 5 : Ore2 : Dins : E ~ 2 : E ~ 5 : F ~ 2 : Fro5 : O,

C~I : E~I = -21~3DAn2,

C~4 : E~4 : -21~3DA.5,

D~I = F~I = -2s~3DB~2 ,

Dm~ : F ~

E~3: (n./X.)C~3,

E.6 = ( q ~ / h . ) C . 6 ,

: -2S~DB~s, F~3 = ( s . / ~ ) D ~ 3 ,

F~6 = ( s . / ( . ) D ~ 6 .

Therefore all the quantities together with the stress couples can be

BENDING OF PLATES BASED ON IMPROVED THEORY

represented with the unknowns Anl, A~2, C.3, A ~ ,

209

A~s, C~6 and B~I. B~2.

Dm3, Bm~, Bms, ~)m6. When we consider the boundary

conditions

along x = const.

the conditions

for

w, Mm, - w,y+f(zo)~/D

the conditions

for

xMy" Qx" -W,x+f(zo)¢/D

Take M

xy

~

as an example,

(x= 0 or x = a),

can readily be described,

it may be expressed

are considerably

but

complicated.

as

k eos~.~ n

P

+ ~ [alAn1

+a2A.2

+a3C~3+a4A.~

+asA~s+aeC~e]cos~y

n i

+ ~ [b~sinhsmyB ~ + (b2sinhs~y + b29 coshs~y)B~2 m

Din3+ b4coshs~yBm4

+ b3 s l n h ~ y

t

+ (bscoshs~y+bsy sinhs~y)B~s+becosh~yD~6]= 0, where k

p

b (i = 1 ---,6),

, ai,

i

from t h e g i v e n c o n d i t i o n . (16),

b2

'

which s a t i s f i e s

It

the c o n d i t i o n

slnhs~y,

y coshs~y,

are expanded

in Fourier series the condition

are t h e known q u a n t i t i e s

i s needed t o f i n d t h e e q u a t i o n ,

this purpose

we can describe

z

b5

'

calculated instead of

a l o n g the edge r e g a r d l e s s o f y.

sinh~my,

in cosl,y.

coshs~y,

For

y sinhs~y and c o s h ~ y

By using the results obtained thus

along x = const.

In this way we can arrive at the simultaneous the unknowns

(16)

for any combination

equations

of the boundary

to determine

condition

of

all

a thick

plate.

Results The deflection distributed and normal

at the center of a rectangular

load is examined

in considering

the effects

stress for the boundary conditions

and S'),clamped The deflection the plates with

(C) and built-in

(C').

ratios to the classical

h/a: 0.i,

that as the thickness from the classical

0.2 and 0.3.

of the plate

value becomes

of shear deformation

such as simply supported

(S

The effect of z0 is also examined. values are shown in Figure i for It is observed

increases

large,

plate subjected to uniformly

from the figures

the difference

especially

of the deflection

for a square plate, and

210

KAORU SHIRAKAWA

1.6

,

,

I

I

2

3 b/a 4

C

I

I

s [----]s

zo/h-0.5

s

1.4~\

? . 2 ~

1.0 2.5

s

0.1

2

r 3

I

I

zo/h =0

C

S[

IS

_

i ~ 2

3 b/a 4

I

I

~

1

c"

.

.

.

.

3 b/a &

I

I

C'

2

IS C'

_

•

1.0

.

2

2.0

1.5 ~_X~xX~x~h/a ~0.3

.

zo/h-0.5 S[

I c,l I_1

c

=0.3

15

25

I

_

b/a 4

2.0

10

S'

-"

__ 0.2

~'~--r

'X

3 b/a 4-

I

FTG. ] (con* inued)

,

2

,

-~-

:

,-~

3 b/o

/+

BENDING

3.0

_~~_~_zo/'h=

2.5

OF PLATES

0

I

h/a- 0.3

BASED

ON

IMPROVED

THEORY

211

z0;h0 cl ic

c

c

2.0

~ _

0=2_.

_

. . . .

'~__

1.5

02

0.1

01

I 2

1.0

I 3 b/a

I

4

_ I

2

3 b/a 4

FIG. I Deflection ratio to classical value. - including shear deformation and normal stress; ..... including shear deformation only.

the deflection

is largely

influenced

normal stress reduces the deflection of the edge conditions

by the shear deformation, slightly.

the deflection

while the

As may be expected,

in view

in the case of S' is large compared

with that of S. The effects of shear deformation

and normal stress appear

more remarkably

plates rather than for the

for the clamped and the built-in

simply supported plate,

and it is seen that for a plate of h / ~

all edges clamped the deflection with the classical the deflection so slight,

0.2 with

becomes more than 1.5 times larger compared

value for all the aspect ratios b/a. The difference

due to the clamped and the built-in

although a small difference

conditions

is recognized

of

can be said

for a thick plate of

h/G= 0.3. Figures also show an expected tendency that the increase

of s0

reduces the deflection.

R~,fereces i. 2. 3. 4. 5.

E. Reissner, J. Appl. Mech. 12, A69 (1945) E. Reissner, Int. J. Solids and Structures ii, 569 (1975) S. A. Ambartsumyan, Theory of Anisotropic Plates, Technomic, New York (1970) K. Shirakawa, Ing.-Arch. 50, 165 (1981) S. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells, McGraw-Hill Book Company Inc., New York (1959)