# Rectangular mindlin plates on elastic foundations

## Rectangular mindlin plates on elastic foundations

Int, d, Mech. Sci. Vol. 31, No. 9, pp. 67% 692, 1989 0020-7403/89 S3.00+.00 © 1989 Pergamon Press plc Printed in Great Britain. RECTANGULAR MINDLI...
Int, d, Mech. Sci. Vol. 31, No. 9, pp. 67% 692, 1989

0020-7403/89 S3.00+.00 © 1989 Pergamon Press plc

Printed in Great Britain.

RECTANGULAR

MINDLIN PLATES ON ELASTIC FOUNDATIONS

HARUTOSHI KOBAYASHI a n d KEIICHIRO SONODA Department of Civil Engineering, Osaka City University, Osaka 558, Japan

(Received 27 May 1988; and in revised form 30 May 1989) Abstract--Rectangular plates on Winkler foundations are analysed on the basis of Mindlin's thick plate theory. The plates are simply supported on the two opposite edges and the other two edges may be arbitrarily restrained, e.g. simply supported, clamped or free. Solutions are presented in the Levytype single series forms, of which forms must be distinguished into three different forms depending upon the properties of plate materials and the modulus of foundation. The effects of shear deformation are first showed numerically for the deflections and stress resultants at major points of the plate. Furthermore, the twisting moment and shear force distributions along the edges and centre lines of the plate are illustrated graphically to demonstrate the principal difference between Mindlin's plate theory and classical thin plate theory.

NOTATION

Am,B,.,..., Fm constants of integration a, b, h

length, width and thickness of plate plate flexural rigidity E, G, v Young's modulus, shear modulus and Poisson's ratio K=(ka4/D) t/4 dimensionless foundation modulus k elastic foundation modulus

D=Eh3/[12(1-v2)]

I (D/k) TM Mx, Mr, Mxy moment resultants m integer Qx, Q~, shear resultants q(x,y) surface load q,., w.,, 4O,. Fourier coefficients R=2Mxr corner reaction of thin plate

s (kD)tlZ/DcGh W,.(y), ~,.(y), W,.(y) homogeneous solutions w transverse deflection wn(x, y), wp(x, y) homogeneous and particular solutions for w x,y rectangular coordinates ~z,., tim, - • •, r/,. parameters x=5/6 shear correction factor

lain mrt/a 4on(x, y), 4op(x, y) homogeneous and particular solutions for 4O

4o (oq,xlOx)+(a~,./Oy) (o¢~/ax)-(o~ j & ) ~ , ~br rotations of the normal V2 Laplace operator in rectangular coordinates

1. I N T R O D U C T I O N A m o n g t h e v a r i o u s t h i c k p l a t e t h e o r i e s a v a i l a b l e at p r e s e n t , t h e m o s t w i d e l y u s e d t h i c k p l a t e

theories are those of Reissner  and Mindlin . But both the theories are not so much different. If the effect of transverse normal stress is neglected in Reissner's plate theory and the shear correction factor is taken equal to 5/6 in Mindlin's plate theory, both the theories coincide with each other. Some analytical and numerical studies for the rectangular thick plates on an elastic foundation have been reported on the basis of Reissner's plate theory. Frederick  showed that both Navier- and Levy-type series solutions could be used for the analysis. Voyiadjis and Baluch  formulated a successive approximate technique represented in term of the deflection alone and solved a simply supported rectangular plate. 679

680

HARUTOSHIKOBAYASHIand KEIICHIROSONODA

Ariman  derived the governing equations for an orthotropic plate and analysed an infinite strip with two opposite sides simply supported. Numerical study using FEM based on thick triangular plate elements was developed by Svec 1,6]. On the other hand, the authors I-7, 8] presented the double series solutions by means of eigenfunction expansions for Mindlin plates having two opposite edges simply supported. A numerical solution method which is applicable to Mindlin plates with any boundary condition and of variable thickness has been proposed by Matsuda and Sakiyama . In all the investigations described above, the attention has been focused mainly upon the effect of shear deformation on the deflection and stress resultants, and also the range of applicability of thin plate theory. Since thick plate theories, however, require the satisfaction of three boundary conditions along each plate edge rather than the two required for thin plate theory, it is of interest to show that a principal difference between the results predicted with thin and thick plate theories arise in the twisting moment and shear force distributions along and near the edges. In the present study, the bending response of rectangular Mindlin plates on Winkler foundation is analysed. Main attention is paid to the twisting moment and shear force distributions along the edges and centre lines of the plate. The plates are simply supported on the two opposite edges and the other two edges may be arbitrarily restrained, e.g. simply supported, clamped or free. Although the double series solutions for the current plate bending problems have been presented by the authors as mentioned earlier, the Levy-type single series solutions are derived for the convenience of numerical computations and also for the sake of further researches. To this end, the original coupled governing equations are first separated into two uncoupled partial differential equations with respect_to the deflection and stress function, respectively, one of which is of fourth order and the other of second order. Then, the Levy-type single series solutions may be obtained directly from these sixth-order uncoupled differential equations under consideration of the prescribed boundary conditions. 2. BASIC E Q U A T I O N S

The equilibrium equations of Mindlin plate on Winkler foundation are given in terms of displacement components as follows I-7, 8]:

&b Ow 2 [ ( 1 - v, V2 ~k,+ (1 +v)-~y]+XGh(--~y-~kj,)=O

(lb)

xGh(V2w- gp)+ q- kw = 0,

(lc)

where ~=

-~ a y '

(2)

and w(x, y) is the transverse deflection; ~x(x, y) and qy(x, y) are the rotations of the normal about x- and y-axes, respectively; D is the plate flexural rigidity; E, G and v are Young's modulus, shear modulus and Poisson's ratio, respectively; h is the plate thickness; x(= 5/6) is the shear correction factor; k is the elastic foundation modulus; q(x, y) is the surface load intensity; and V 2 is the Laplace operator in rectangular coordinates. Moment and shear resultants are expressed as

Mxy=-- 2 D

+

(3c)

Rectangular Mindlin plates on elastic foundations

681

Qx= xGh (~xx - ¢:,)

(4a)

Qy=xGh(~yy -qly).

(4b)

and

Using Marguerre and Woernle's formulation  for the plates without the presence of a foundation, coupled equations (1) can be transformed into the following three uncoupled equations:

kD 2w

DV2V2W -- - ~ - ~ V

D

"q- k w = q-- x - ~ V

1

dp= V2w + ~G~(q - kw) 2 xGh V2~

2q

(5) (6) (7)

1 - v D ~k=0,

where

0~

O¢x

= Ox

(8)

Oy "

At the same time, two rotations ~ and ~byare expressed in the following functional forms:

Ow . D ["O~b l - v Od/I ~b~=~x-x+x---~~xx

_

Ow

2

~y /

D fOdp 1-vO~O) 2

(9a) (9b)

Mindlin's plate equations are sixth-order differential equations and, therefore, three boundary conditions must be specified along each plate edge. For a simply supported edge in addition to the classical boundary conditions one of two further conditions must be enforced, i.e. (1) the angular rotation = 0 or (2) the twisting moment = 0. Thus, for example on an edge, x = constant, the boundary conditions are described as follows: (S):

w=M~=~ky=0,

(10)

(S'):

w=M:=Mxr=O.

(11)

(2) Clamped edge

(c):

w=q,x=¢,=0.

(12)

(3) Free edge

(F):

(1) Simply supported edge

Qx=M~,=Mxy=O.

(13)

Similar conditions may also be specified for an edge of y = constant. The designations, S, S', C and F, above will be used later for the sake of brevity. 3. A N A L Y S I S

The plate considered herein has S-simple support conditions (10) along the two edges x = 0 and x = a and the other two edges y = + b/2 are subjected to a variety of boundary conditions given in equations (10) to (13) as shown in Fig. 1. The applied surface load is assumed to be independent of y and may be expressed in Fourier series form:

q(x,y)=q(x)= ~ qmsin/.tmx,

(14)

m=l

where/a m=

mn/a and Fourier coefficient qm is given by 2

qm-- a I ° q(x)sin/~,.x dx. 30

(15)

682

HARUTOSHI KOBAYASHI a n d KEIICHIRO SONODA

or,trory !1T l_ b/2

FIG. 1. Coordinate system.

In general, the solution of equation (5) involves the combination of a particular solution wp and a homogeneous solution wh which are obtained from 2 2 kD DV V w p - - ~ V

2

D 2 wp+kwp=q---~V q

2 2 kD 2 DV V w h - - - ~ V wh+kWh=O.

(16) (17)

Therefore, equation (6) yields two solutions, ~p and ~h, which correspond to wp and wh, respectively. On the other hand, equation (7) for ~, has only homogeneous solution. For the assumed surface load distributions, the particular solutions, wp and t~p, which satisfy the S-simple support condition on the two edges x = 0 and x = a, may be taken in the form Wp(X)= ~ w~ sin #mx

(18)

m=l

(19)

~p(x) = ~ q~msin #rex. m=l

The coefficient w~ is determined from equation (16) to be 2 D

Substituting equations (18) and (19) into equation (6) together with the expression (20) leads to qmP~

~. =

.

(21)

D#4m+ k (1 + #~-~--~) The S-type simply supported condition along the two edges x = 0 and x = a may be satisfied by the following Wh(X, y)= ~

Wm(y)sin#mx

(22)

~h(x, Y)= ~ ~m(y)sin#mX

(23)

~bh(x,y)= ~ ~Fm(y)cos#mx.

(24)

m=l

m=l

m=l

If equation (22) is substituted into equation (17), the following differential equation is

Rectangular Mindlin plates on elastic foundations

683

obtained for Win(y): _d'Wm __(2g2 dy 4

+

k "~d2Wm [ 4

k

2

k'~

w.:o.

(25)

Assuming

Wm(y)=exp(ry)

(26)

the characteristic equation of the solution of equation (25) is obtained as ,

4 /

2

k \2

{4

k

2

k\

(27)

The roots of this equation are 1 2 rl,2,3,4 = +_Trrl,.+s-I-(s2-1)1/2"]1/2,

(28)

/D\I/2

(29)

where

qm=la,.l,

l=t-k )

(kD)I/2

,

s= 2xGh "

Therefore, the solution of differential equation (25) obviously depends on the conditions of the cases: s < 1, s = 1 and s > 1. For each case of s, four roots expressed by equation (28) are reduced to two pairs of conjugate complex roots, two double real roots and four real roots, respectively. Hence the solution can be written as follows: Case 1, s < l :

Win(y) = Amcosh 0troyCOSflmY+ B~ cosh ~tmysin fl,.y + Cmsinh amyCOS~my+ Dmsinh amy sin flmY,

(30)

where ctm = ~ 1 [(1 + 2Sq2 + r/g) 112_+(s + r/2)] 1/2. (~-~m)

(31)

Win(y) = Am cosh y,. y + Bm),mycosh )'mY+ Cm sinh ),my+ Dm),mysinh),my,

(32)

Case 2, s = 1:

where I

2

),m=7(1 +r/.)

112

.

(33)

Case 3, s > 1:

Win(y) = At. cosh 5my +Bm sinh dimy+ Cmcosh e~y + Dr. sinh e,.y,

(34)

( ~ ) = ~ I-s + r/2 + (s2-- 1)t/2l 1/2

(35)

where

In the foregoing expressions for Win(y), A,. to Dm are the constants of integration to be determined from the boundary conditions along the two edges y = + b/2. Next, substituting equations (22) and (23) into equation (6) together with q = 0, we have ~,.(y)=

d2Wra(y) dY2

( 2 k ) #,,+-~--~ W,.(y).

(36)

In view of the form of W,.(y) given in equations (30), (32) and (34), the explicit form of ~,.(y) is readily obtained for each case of s from the above relation.

684

HARUTOSHI KOBAYASHI and KEIICHIRO SONODA

On the other hand, substituting equation (24) into equation (7) and solving the resulting ordinary differential equation of second order, the expression for q~=(y) is determined as follows:

qJ,,,(y)= Emcosh (,,,y + Fmsinh (mY,

(37)

where E,, and F= are the constants of integration and

( 2 xGh ) 1/2. C"= /z2-~1--v

(38)

4. N U M E R I C A L R E S U L T S

We shall now consider only the practically important case of 0 ~
TABLE 1. DEFLECTIONS, MOMENTS AND SHEARS OF UNIFORMLY LOADED SQUARE SSPLATES ON ELASTIC FOUNDATIONS (v = 0.3)

w•

h/a

x=a/2, x=a/2, x=a/2,

Mxb

Mx~,b x=0,

y=0

y=0

Myb y=0

y=a/2

Q:, x=0, y=0

x=a/2, x=O, y = a / 2 y=a/2

QyC

Rb

(1) K = 1 0 0.01 0.02 0,05 0.10 0.15 0.20

4.052 4.054 4.060 4.104 4.261 4.522 4.888

4.775 4.775 4.775 4.775 4.774 4.773 4.772

4.775 4.775 4.775 4.775 4.774 4.773 4.772

3.241 3.241 3.241 3.241 3.240 3.240 3.239

0.337 0.337 0.337 0.337 0.337 0.337 0.337

-0.337 -0.337 -0.337 -0.337 --0.337 --0.337 -0.337

6.482

(2) K = 3 0 0.01 0.02 0.05 0.10 0.15 0.20

3.347 3.349 3.353 3.381 3.483 3.648 3.873

3.875 3.875 3.874 3.865 3.834 3.784 3.716

3.875 3.875 3.874 3.865 3.834 3.784 3.716

2.751 2.751 2.751 2.746 2.728 2.699 2.660

0.293 0.293 0.293 0.292 0.291 0.288 0.284

--0.293 -0.293 - 0.293 --0.292 -0.291 -0.288 -0.284

5.502

(3) K = 5 0 0.01 0.02 0.05 0.10 0.15 0.20

1.506 1.506 1.507 1.509 1.519 1.534 1.55I

1.541 1.540 1.538 1.526 1.482 1.414 1.328

1.541 1.540 1.538 1.526 1.482 1.414 1.328

1.463 1.462 1.461 1.452 1.421 1.373 1.311

0.176 0.176 0.176 0.175 0.172 0.168 0.162

--0.176 -0.176 --0.176 -0.175 -0.172 -0.168 --0.162

2.926

• lO-aqa4/D; b lO-2qa2; ~ qa.

Rectangular Mindlin plates on elastic foundations

685

TABLE 2. DEFLECTIONS, MOMENTS AND SHEARSOF UNIFORMLY LOADED SQUARE S'S'-PLATESON ELASTICFOUNDATIONS(v = 0.3) w"

Mxb

Myb

Qx¢

QxO

QyC

x = a/2,

x = a/2,

x = a/2,

x = O,

x = a/2,

y=0

y=0

y=0

y=0

x = O, y=a/2

h/a

y=a/2

(1) K = 1 0.01 0.02 0.05 0.10 0.15 0.20

4.071 4.095 4.193 4.442 4.797 5.252

4.794 4.813 4.871 4.967 5.056 5.131

4.787 4.799 4.837 4.904 4.977 5.057

0.338 0.338 0.341 0.345 0.345 0.352

- 10.3 -5.21 -2.12 - 1.07 -0.711 -0.526

-0.419 -0.418 -0.417 -0.413 -0.408 -0.403

(2) K = 3 0.01 0.02 0.05 O.lO 0.15 0.20

3.360 3.376 3.443 3.603 3.825 4.098

3.887 3.898 3.927 3.957 3.961 3.935

3.882 3.887 3.899 3.906 3.899 3.878

0.293 0.294 0.295 0.296 0.295 0.294

-8.77 -4.41 - 1.78 -0.891 -0.584 --0.424

-0.360 -0.359 -0.356 -0.350 -0.342 -0.333

(3) K = 5 0.01 0.02 0.05 0.10 0.15 0.20

1.508 1.51I 1.521 1.541 1.563 1.586

1.542 1.541 1.533 1.497 1.437 1.357

1.539 1.537 1.522 1.477 1.413 1.337

0.176 0.176 0.176 0.173 0.169 0.163

-4.66 --2.34 -0.934 --0.452 --0.284 -0.197

-0.204 --0.203 -0.201 -0.195 -0.187 --0.177

,

l O - 3 q a 4 / D ; b lO-2qa2;C qa.

TABLE 3. DEFLECTIONS.MOMENTSAND ~IEARS OF UNIFORMLYLOADEDSQUARECCPLATESON ELASTICFOUNDATIONS(v = 0.3) w~ h/a

Mxb

M~.b

x=a/2,

x=a/2,

x=a/2,

Myb x = a/2,

Mxyb x=O,

Q:C x=O,

x=a/2,

Q~

3, = 0

y=0

y =0

y = a/2

y = a/2

y=0

y = a/2

--0.516

(1) K = 1

0 0.01 0.02 0.05 O.lO 0.15 0.20

1.915 1.918 1~927 1.989 2.206 2.551 3.015

2.435 2.437 2.441 2.472 2.575 2.728 2.915

3.320 3.320 3.320 3.321 3.321 3.315 3.298

-6.976 --6.974 --6.969 -6.931 -6.789 -6.553 -6.257

0 0.101 0.198 0.466 0.850 1.169 1.433

0.244 0.244 0.244 0.245 0.245 0.251 0.256

-0.513 -0.509 -0.500 -0.488 -0.474

(2) K = 3 0 0.01 0.02 0.05 0.10 0.15 0.20

1.742 1.744 1.752 1.802 1.976 2.245 2.590

2.184 2.184 2.187 2.207 2.272 2.362 2.463

2.990 2.989 2.988 2.978 2.941 2.878 2.790

- 6.425 -6.422 -6.414 -6.359 --6.156 -5.832 -5.426

0 0.095 0.187 0.437 0.787 1.064 1.278

0.229 0.229 0.230 0.230 0.231 0.231 0.231

- 0.482 -0.480 -0.479 -0.474 -0.462 -0.445 -0.425

(3) K = 5 0 0.01 0.02 0.05 0.10 0.15 0.20

1.068 1.069 1.072 1.088 1.141 1.212 1.289

1.212 1.212 1,211 1.204 1.181 1.145 1.098

1.711 1.709 1.705 1.673 1.569 1.421 t.252

--4.270 -4.266 -4.253 --4.163 -3.860 -3.420 --2.921

0 0.073 0.143 0.326 0.556 0.705 0.788

0.172 0.172 0.172 0.171 0.167 0.161 0.154

-0.348 -0.347 -0.346 -0.340 -0.322 -0.297 -0.269

• lO-3qa4/D; b 10-2qa2;

c qa.

-0.515

HARUTOSHI KOBAYASHIand I(EIICHIRO SONODA

686

TABLE 4. DEFLECTIONS, MOMENTS AND SHEARS OF UNIFORMLY LOADED SQUARE FF-PLATES ON ELASTIC FOUNDATIONS (V= 0.3)

w'

w"

Mxb

M~~

My b

Q C

x = a/2,

x = a/2,

x = a/2,

x = a/2,

x = a/2,

x = 0,

x = 0,

Rb x = 0,

y = 0

y = a/2

y = 0

y = a/2

y = 0

y = 0

y = a/2

y = a/2

h/a

QxC

(1) K = 1 0 0.01 0.02 0.05 0.10 0.15 0.20

12.95 12.96 12.97 13.05 13.31 13.75 14.37

14.85 14.88 14.91 15.05 15.43 15.98 16.69

12.12 12.12 12.12 12.11 12.11 12.12 12.14

12.96 12.93 12.89 12.79 12.62 12.44 12.27

2.679 2.667 2.654 2.613 2.536 2.447 2.345

0.465 0.464 0.464 0.463 0.461 0.459 0.457

0.401 8.03 4.27 2.01 1.25 0.994 0.865

-4.759

(2) K = 3 0 0.01 0.02 0.05 0.10 0.15 0.20

6.988 6.988 6.990 7.007 7.075 7.188 7.343

7.955 7.966 7.978 8.023 8.131 8.273 8.446

6.399 6.395 6.390 6.367 6.300 6.201 6.074

6.776 6.757 6.735 6.661 6.505 6.314 6.092

1.456 1.449 1.442 1.418 1.366 1.301 1.224

0.296 0.296 0.296 0.294 0.291 0.287 0.282

0.251 4.44 2.38 1.13 0.712 0.566 0.488

-2.612

(3) K = 5 0 0.01 0.02 0.05 0.10 0.15 0.20

1.680 1.680 1.680 1.678 1.676 1.674 1.671

1.851 1.852 1.852 1.854 1.853 1.849 1.840

1.321 1.320 1.319 1.311 1.286 1.249 1.200

1.322 1.320 1.317 1.303 1.270 1.224 1.169

0.341 0.340 0.339 0.335 0.322 0.305 0.283

0.143 0.143 0.142 0.142 0.140 0.137 0.134

0.116 1.29 0.716 0.369 0.249 0.206 0.105

--0.728

• lO-aqa4/D;

b 10-2qa2; c qa.

more than three significant figures. The deflections converge rapidly at the first several terms of the series, while for the moment resultants terms below 10 (say m = 19) are satisfactory. The shear force Qy are not so rapidly convergent, so that more than 20 (m = 39) terms are needed to obtain reasonable accuracy. In the case of the shear force Qx at the centre of the edge x = 0, the convergence is very slow, therefore the convergent values may be obtained by taking some hundred terms. The numerical results presented in Tables 1 to 4 indicate that the effect of shear deformation exhibits a tendency toward enlargement of the deflection and reduction of the stress resultants, however, it is interesting to note that for certain boundary conditions (S'S', CC and FF) the shear deformation has the effect of decreasing the deflection and increasing the stress resultants at certain values of thickness-to-span ratio and elastic foundation modulus. Next, to demonstrate a principal difference between Mindlin theory and thin plate theory, the shear force Qx and twisting moment Mxy distributions along the edge x = 0 and the shear force Qy distributions along the centre line x=a/2 are computed for the case of dimensionless foundation modulus K ---3. The results are illustrated in Figs 2-10 for the plates with S'S'-, CC- and FF-boundary conditions. In these figures, the results obtained from thin plate theory are represented by dashed lines. The following observations may be made from these figures.

1. SS-plate The S-type simply supported condition implies M~y~0, and this condition is closely approximated to thin plate theory. Although the distributions of Q~, Qy and M~y are not presented herein, two theories agree very closely in the evaluations of these quantities.

2. S'S'-plate The shear force Q~ predictions of both Mindlin and thin plate theories agree reasonably well as shown in Fig. 2, though significant differences arise near the S'-type edge. Figure 3

Rectangular Mindlin plates on elastic foundations

687

-0,40

-0,36 s'

0,1

s

s " x

-0,30

y

SI

-0,24 Mlndl In

Klrchhoff

¢~/

/~~~0.i 0,02~,

h/o =

-0,18

jS

-0,12

0

0.i

0,2

0,3

0,5

0,4

ylo

FIG. 2. Variations of Q), along the centre line x =

a/2 of S'S'-plate.

O,035 Ss

SL x

0.030

s,

h/a =

o.o5,~

.~

0.025

/'\

Klrchhoff

%

0,020

0.015 0.010

/

/

O.050 3

0,i

0.3

0,2

0,4

0,5

y/o

FIG. 3. Variations of Mxy along the edge x = 0 of S'S'-plate. MS 31:9-D

688

HARIYrOSHIKOBAYASHIand KEilCHIROSONODA 1,0 0,02

-1.0

-2.0

8

SS

S

S ~x

-3,0

-tl,O -

-

ly

S'

--

Mlndlln

-

KI rchhoff -5,0

-6,0

0

0,1

0,2

0,3

O,q

0,5

y/o

FIG. 4. Variations of Qx along the edge x = 0 of S'S'-plate.

displays the twisting moment Mxy distributions and significant differences between two theories are evident near to the plate corner. There is a shape fall from the peak values at the corner. As the h/a ratio increases, the locations of the peak twisting moment shift towards the centre of the edge and reduce in value. The peak values of Mxy approach the thin plate theory peak value as the h/a ratio decreases. Figure 4 shows the shear force Qx distributions. It follows from this figure that Mindlin theory agrees quite well with the thin plate theory except in the vicinity of the plate corner. In the proximity of the corner, the Qx distributions predicted by Mindlin theory exhibits an abrupt change from a positive value to a negative value. For a very small h/a value, the nature of Q~ increases very rapidly at the corner. Mindlin theory does not produce the concentrated reaction R at the corner of the plate which acts there according to the thin plate theory.

3. CC-plate Differences between the Qy distributions predicted by two theories are not apparent as shown in Fig. 5. Figure 6 indicates that the distributions of Mxy along the S-type simply supported edge. When the thickness of the plate is very small Mindlin and thin plate theories give different values only in a very narrow zone close to the plate corner, but these differences are not significant. As the h/a ratio increases, however, the differences between the predictions of the two theories are clearly evident over most of the edge. Since the corner value of M~y gradually increases as the h/a ratio increases, the difference in the predicted values near the corner become significant. Moreover, it is clear that when the h/a value exceeds some value the maximum twisting moment occurs at the corner. Figure 7 shows a very different distribution for Q~ near the corner. At the corner the Qx values predicted by Mindlin and thin plate theories are zero and a negative constant value, respectively.

4. FF-plate It may be seen from Fig. 8 that the Qr curves exhibit abrupt changes at distances from the free edge approximately equal to the plate thickness and Qy reaches its maximum value near the free edge, where it is zero. When the thickness of the plate increases, the maximum value

689

Rectangular Mindlin plates on elastic foundations -0,;

-0,f

C

h/o :

S

S • x

-0,5

Y

-O,q

-

0,02\

2

C Mlndlln Klrchhoff

-0,3

-0.

S

f

0

0.1

0.2

0,3

0,4

0,5

y/o FIG. 5. Variations of Qr along the centre line

x=a/2 of CC-plate.

0,014

j

0,012

....._y

O,010

%

/

0,008

-

:E

s

x 1~,./ O'

0,006

y

O,004

Mlndl In KIrchhoff 0,002 0.021

0,1

0,2

0,3

0,4

0,5

y/o FIG. 6. Variations of Mxy along the edge x = 0 of CC-plate.

690

HARUTOSHI KOBAYASHIand KEIICHIRO SONODA

0.3

0.2

0,1

hlo =

~..

s c j-x

-0,1

\~os

-0,2 Mlndl In

0 •02Y~"

Klrchhoff -0.1

0

0,1

0,2

0.3

0.4

0.5

Y/O

FIG. 7. Variations of Q~ along the edge x=0 of CC-plate.

-0,035 F

hi(I=

S

S

O'02~x~/

-0.030

,y

F

~

-0.025

/

Kirchhoff

-0,020

-0.015

/ 0

/ 0.1

0.2

0.3

OJ1

0.5

YlO

F1o. 8. Variations of Qy along the centre line x=a/2 of FF-plate.

Rectangular Mindlin plates on elastic foundations

691

-0.14

-0.12

h/o=,~

F

-0. i0

S

,S x ry

%

0.05".

F

-0.08 In

,/

--

Mlndl

-.

Klrchhoff

/'

-0.06

-0.04

-0.02

0 0

o.i

0.2

o.3

0,4

o.s

y/o

FIG. 9. Variations of Mxy along the edge x = 0 of FF-plate.

2.8

2,4 F s

2,0

s -x F

Y 1.6

Mlndl In

°°

x

KI rchhoff

1.2

0.8

= ~ ~ 0.2\

h/o

0.4

0 - 0.i 0.2

toO.05~ ~" 0,02 ~ R

0 0

0.1

0.2

0.3

O.q

0.5

y/o

FIG. 10. Variations of Q~ along the edge x = 0 of FF-plate.

692

HARUTOSHIKOBAYASHIand KEIICHIROSONODA

of Qy predicted by Mindlin theory decreases. As the h/a ratio decreases, the maximum value of Qy tends toward that predicted by thin plate theory. A similar tendency for Mxy along the S-type simply supported edge x = 0 may be observed in Fig. 9. Substantial differences also occur near to the corner, where Mindlin theory allows the required boundary conditions to be satisfied whereas thin plate theory does not. In the case of the shear force Qx shown in Fig. 10, the two theories agree except in the region near the corner. However, the Qx distributions change rapidly over the narrow zone close to the corner and high Q~ value arise at the corner as the h/a ratio decreases. This trend is very similar to the case of Q~,for the S'S'plate (see Fig. 4). 5. C O N C L U S I O N S

A method of analysis is established for the bending problems of rectangular plates on Winkler foundations on the basis of Mindlin's thick plate theory. The plates are simply supported on the two opposite edges and the other two edges may be arbitrarily restrained, e.g. simply supported, clamped or free. The sixth-order uncoupled system of governing equations are solved and the Levy-type single series solutions are presented. Their solution forms are distinguished into three different forms depending upon the properties of plate materials and the foundation modulus. Numerical results for uniformly loaded square plates subjected to various boundary conditions show the effect of shear deformation on the deflection and stress resultants. Furthermore, the illustrated twisting moment and shear force distributions along the edges and centre lines of the plate have revealed the principal differences between the results predicted by Mindlin and thin plate theories. Acknowledgements--The authors are indebted to Mr T. Okuda, Master Course Student, Department of Civil Engineering, Osaka City University, for preparing the figures. The authors also wish to thank the reviewers for their constructive criticism.

REFERENCES 1. E. REISSNER,The effect of transverse shear deformation on the bending of elastic plates. J. appl. Mech. 12, 69-77 (1945). 2. R. D. MINDLXN,Influence of rotatory inertia and shear on flexural motion of isotropic, elastic plates..L appl. Mech. 18, 31-38 (1951). 3. D. FREDERICK,Thick rectangular plates on an elastic foundation. Trans. ASCE 122, 1069-1085 (1957). 4. G. Z. VOY1ADJI\$and M. H. BALUCH,Thick plates on elastic foundations: one variable formulation, d. Enflng Mech. Div. ASCE 105, 1041-1045 (1979). 5. T. ARIMAN,On orthotropic thick elastic plates on an elastic foundation. Die Bautechnik 45, 230-234 (1968). 6. O. J. SvEc, Thick plates on elastic foundation by finite elements. J. Engng Mech. Div. ASCE 102, 461-477 (1976). 7. H. KOBAYASHIand K. SONODA,Rectangular thick plates on linear viscoelastic foundations. Proc. Jap. Soc. Cir. Engrs No. 341, 33-39 (1984). 8. K. SONODAand H. KOBAYASm,Timoshenko beams and Mindlin plates on linear viscoelastic foundations. In Civil Engineering Practice, Vol. 2, Chap. 23. Technomic Publishing, Lancaster, PA (1988). 9. n. MATSUDAand T. SAKIYAMA,Bending analysis of rectangular plate on non-uniform elastic foundations. Structural Eng.~Earthquake Eng. 4 (Proc. Jap. Soc. Cir. Engrs No. 380/I-7), 51s-595 (1987). 10. K. MARGOERREand H.-T. WOERNLE, Elastic Plates. Blaisdell Publishing, Waltham, MA (1969).