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Circular plates on elastic foundations modelled with annular

plates

Mehmet Utku *, Ergin C tpto!glu 1, Ilker Inceleme

Department of Civil Engineering, Middle East Technical University, Ankara 06531, Turkey

Received 23 October 1998; accepted 20 October 1999

Abstract

In this paper, a new formulation is presented for the analysis of circular plates supported on elastic foundations. The

formulation is based on the exibility and stiness methods of structural analysis. Classical thin plate theory for small

deformations is applied to obtain the exibility and stiness coecients. The circular plate is represented as a series of

simply supported annular plates resting on support springs along their common edges. The computer implementation

of the method is given, and solutions obtained for an illustrative case are discussed. 2000 Civil-Comp Ltd. and

Elsevier Science Ltd. All rights reserved.

Keywords: Circular plate; Annular plate; Elastic foundation; Mixed formulation

1. Introduction

The theory of plates supported on elastic foundations

has been the subject of numerous investigations. Ex-

amples of this type of plates appear in a wide variety of

practical engineering problems, such as concrete roads,

airport runways, and mat foundations for buildings. It is

known that the response of a plate on soil not only

depends on the exibility of the slab, but also on the soil

behaviour. Since the behaviour of soil media is very

complex, several idealized models have been introduced

to represent it.The simplest approach depends on the assumption

that the intensity of the foundation reaction at any

point is proportional to the deection of the plate at

that point. As an extreme case, the foundation is rep-

resented as elastic half-space. In most investigations, the

bonded contact between the foundation and the plate is

assumed and, hence, compressive as well as tensile re-

actions are considered to be admissible. However, re-

gions of no-contact develop under plates that are

supported on tensionless foundations. Mathematically,

the main problem that arises in the case of tensionless

foundations is to determine the extent of the contact

region.

The problem of a laterally loaded circular plate on

an elastic foundation subjected to a concentrated load

was rst discussed by Hertz (1895) and Foppl (1920)

(quoted by Celep [1]). This problem was described in a

general setting in Timoshenko and Woinowsky-Krieger

[2]. In these references, compressive as well as tensile

foundation reactions were considered to be admis-

sible.Weitsman [3] extended the solution by considering a

plate of innite extent supported on an elastic founda-

tion that reacts in compression only.

Gladwell and Iyer [4] considered the unbonded con-

tact between a thin circular plate of nite radius pressed

by means of rotationally symmetric distributed load and

its own weight against the surface of an elastic half-

space. The contact was assumed frictionless and unb-

onded.

Celep [1] presented a solution to the analysis of the

lift-o problem of a circular plate supported on an

elastic foundation of Winkler type that reacts in com-

pression only. The plate was assumed to be subjected to

Computers and Structures 78 (2000) 365374

www.elsevier.com/locate/compstruc

*

Corresponding author.1 Deceased.

0045-7949/00/$ - see front matter 2000 Civil-Comp Ltd. and Elsevier Science Ltd. All rights reserved.

PII: S0 0 4 5 -7 9 4 9 (0 0 )0 0 0 6 3 -8

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eccentric concentrated load and moment as well as

uniformly distributed load.

Other quite recent investigations of this problem

have examined instead the use of approximate numeri-

cal methods of the nite element type. Cheung and

Zienkiewicz [5] suggested a method of solving the in-teraction of plates on soil by combining the nite ele-

ment method with the Boussinesq solution for point

loads. Melerski [6] presented a variational formulation

of the nite-dierence procedure for elastic analysis of

thin circular plates under various axisymmetric loading

and support conditions. Recently, a hybrid method us-

ing a combined nite element method and analytical

method has been suggested by Chandrashekhara and

Antony [7] for solving annular slab-soil interaction

problems.

In the present study, a new formulation is proposed

for the analysis of axisymmetrically loaded circularplates supported on elastic foundations. The method is

based on the classical plate theory. The circular plate

is modelled as a series of concentric annular plates

in such a way that each annular plate is considered

as simply supported along the inner and outer edges.

Two adjacent annular plates are supported by a spring

system, as shown in Fig. 1, which corresponds to the

foundation represented by the Winkler model of linear

springs. The vertical deection wi of the plate and the

internal bending moment Mi are considered as un-

knowns at each of the interior support points. Since the

slope discontinuity is at variance with the compati-

bility requirements for the original circular plate, itmust be corrected in order to obtain the correct solu-

tion.

This correction is accomplished by introducing ap-

propriate compatibility and equilibrium equations,

which results in two coupled equations for each interior

support in terms of (wi, Mi) at the interior support i, and

(wi1YMi1) and (wi1YMi1) at the left and right interiorsupports, respectively. Hence, the formulation becomes

a ``mixed formulation'', since the coupled equations in-

volve both exibility and stiness terms, which are ob-

tained from the classical plate theory equations for

annular plates.

2. Formulation of the problem

2.1. Circular plates on an elastic foundation

It will be assumed that the foundation is the Winkler

type and that consequently the foundation reaction isassumed to be proportional to the deection w of the

plate:

q xYy kswX 1

The constant ks, expressed in kilonewtons per cubic

meter, is called the modulus of the foundation (or the

modulus of subgrade reaction). The constant ks has

a wide range of values, and can best be determined

through bearing tests of the actual foundation material

[8]. For the circular plate, the basic dierential equation

becomes

d2

dr2

1

r

d

dr

!d2w

dr2

1

r

dw

dr

!w

1

Dq xYy kswY 2

or

Dr4w ksw q xYy Y 3

where q is the axisymmetrically distributed lateral load

per unit area and D is the exural rigidity of the plate

given by D Et3a121 m2.

2.2. Analysis of circular plates on elastic foundations byannular plates

Discussions of the theory of bending of circular

plates on elastic foundations can be found in Timo-

shenko [2]. Derivation of the governing dierential

equation (2) is straightforward, but its solution is rather

complex and far from simple. Approximate analytical

techniques are tedious and lack generality [9]. Accord-

ingly, one has to resort to approximate numerical

methods of the nite dierence or nite element type.

In the present study, a new formulation is presented

for an easy and eective analysis of circular plates sup-ported on elastic foundations. The formulation is based

on the classical plate theory and does not involve any

approximation for the solution variables. The circular

plate with a circular hole at the centre shown in Fig. 2 is

supported on an elastic foundation and carries a uni-

formly distributed load. The circular plate is modelled as

a series of concentric annular plates in such a way that

each annular plate is considered as simply supported

along the inner and outer edges. In addition to that, two

adjacent annular plates are supported by the spring

system, as shown in Fig. 1, which corresponds to the

foundation represented by the Winkler model of linear

springs. Therefore, adjacent annular plates have theFig. 1. Circular plate on an elastic foundation modelled as a

series of concentric annular plates.

366 M. Utku et al. / Computers and Structures 78 (2000) 365374

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same transverse deection along their outer and inneredges, respectively.

If the plate is represented by n annular plates, the

model shown in Fig. 1 is then equivalent to selecting the

internal moments at the n 1 interior spring supportedges as the redundant. For this selection, the model can

be interpreted as the primary structure used in the ex-

ibility (or force) method. Hence, the primary structure is

a series of n simply supported annular plates resting on

spring supports, and in this case, compatibility for the

given structure is relaxed at each of the interior support

edges where the continuity of radial slope is violated.

Motivated by the formulation referred to as thethree-moment equation (see, for instance, West [10]),

attention is now focused on the two adjacent annular

plates that reach over the supports i 1, i, and i 1, asshown in Fig. 3a. When the internal moments are re-

moved and the given plate is transformed into a series of

simply supported annular plates, there is a slope dis-

continuity over each of the interior supports. For the

actual loading on the primary structure, this disconti-

nuity is hio, as indicated in Fig. 3b. Since this disconti-

nuity is at variance with the compatibility requirements

for the original structure, it must be corrected in order to

obtain the correct solution. This correction is accom-

plished by introducing, through three separate loading

conditions, unit values of Mi1, Mi, and Mi1. The dis-

continuities in the slope at point icorresponding to eachof these unit moment cases are shown as fmiYi1, f

miYi , f

miYi1

respectively, which are the exibility coecients for the

primary structure.

In addition to the discontinuities in the slope at point

i due to the released interior moments, there is an ad-

ditional slope discontinuity at support i due to support

displacements associated with spring supports. If the

supports displace as shown in Fig. 4, where upward

displacements are taken as positive, then the disconti-

nuities in the slope at point i corresponding to unit

values of wi1, wi, and wi1 are shown as fs

iYi1, fs

iYi and

Fig. 3. (a) Statically indeterminate continuous plate with non-

yielding support, (b) statically determinate primary structure,

and (c) unit values of redundant moments.

Fig. 4. Slope discontinuity from support displacements.

Fig. 2. Circular plate with a circular hole at the centre sup-

ported on an elastic foundation.

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fsiYi1, respectively. These discontinuities must also be

included in the compatibility equation at point i.

Applying superposition, each of these discontinuities

is multiplied by the actual value of the respective re-

dundant moment or displacement, and then they are

combined with the discontinuity hio to obtain the totaldiscontinuity, hi X Since hi is zero in the given plate, the

nal compatibility equation at point i becomes

fsiYi1wi1 fs

iYiwi fs

iYi1wi1 fm

iYi1Mi1 fm

iYi Mi

fmiYi1Mi1 hio hi 0X 4

Each of the n 1 compatibility equations involvesthree moments and three displacements; the moment

and the displacement at the point where compatibility is

being considered, and the moments and the displace-

ments at the far ends of the spans to the left and to the

right. Therefore, the compatibility equation involves twounknowns, the bending moment Mi and the displace-

ment wi, at each interior spring support. The necessary

additional equation is obtained from the equilibrium of

forces at point i, which includes the edge loads and the

spring force along the common boundary between two

adjacent annular plates.

The forces at point i corresponding to unit values of

the moments Mi1, Mi and Mi1 are shown as kmiYi1, k

miYi

and kmiYi1, respectively; and those corresponding to unit

values of the displacements wiYi1, wi and wiYi1 are

shown as ksiYi1, ksiYi and k

siYi1, respectively, which are the

stiness coecients for the series of annular plates. Thenal equilibrium equation at point i becomes

ksiYi1wi1 ksiYi kiwi k

siYi1wi1 k

miYi1Mi1

kmiYiMi kmiYi1Mi1 Rio 0Y 5

where ki is the spring stiness and Rio is the reaction at i

for the actual loading on the primary structure. There-

fore, two coupled Eqs. (4) and (5) must be applied at

each interior support i with i 2Y 3Y F F F Y n 1.

2.3. Flexibility coecients and compatibility equations

In order to obtain the exibility coecients fsiYi1, fs

iYi

and fsiYi1 for any point i, rotations need to be calculated

at this point due to the support settlements at points

i 1, i and i 1, respectively.An extensive catalog of solutions for the deection,

slope, radial bending moment and shear force of a cir-

cular plate with a circular hole at the centre is given in

Young [9]. These formulas will be needed for the deri-

vations of exibility and stiness coecients that will be

introduced now.

The rotation at the inner edge due to a unit upward

deection at the inner edge is obtained as

hifb

1

a

L9

C1L9 C7L3Y 6

where C and L are the plate constants and loading

functions, respectively, given in Ref. [9].

The rotation at the outer edge due to a unit upwarddeection at the inner edge is

hifa

1

a

C4L9 C7L6C1L9 C7L3

X 7

Similarly, the rotation at the inner edge due to a unit

upward deection at the outer edge is

hofb

1

a

C9

C1C9 C7C3X 8

Finally, the rotation at the outer edge due to a unitupward deection at the outer edge is given as

hofa

1

a

C4C9 C7C6C1C9 C7C3

X 9

The discontinuities in the slope at point i corre-

sponding to each of these unit displacement cases are

shown as fsiYi1, fs

iYi and fs

iYi1, respectively, which are the

exibility coecients for the primary structure shown in

Fig. 5. Hence, the expressions for the exibility coe-

cients are

Fig. 5. Flexibility coecients due to unit displacements.


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fsiYi1 hifa

LY

fsiYi hofa

L

hifb

RY

fsiYi1 hofb

RY

10

where subscripts L and R refer to an evaluation of thequantity (slope in this case) for the annular plate, which

is either on the left or on the right of point i, respectively.

The exibility coecients fmiYi1, fm

iYi and fm

iYi1 are

shown in Fig. 6, which express the rotations at point i

that are caused by unit values of Mi1, Mi and Mi1,

respectively. These coecients are obtained from the

equations of an annular plate given in Ref. [9], which is

simply supported along the inner and outer edges, with a

uniform line moment M0 at a radius r0.

Calculation of these exibility coecients is based on

the radial location r0 of the unit moments. Hence,

fmiYi1 ha LY 11

where ha L haM0 1r0 bL

fmiYi haL hbRY 12

where ha L haM0 1r0 aLX

hb R hbM0 1r0 bR

Y

fmiYi1 hb RY 13

where hb R hbM0 1r0 aR

.The nal exibility coecient, fio, required for the

compatibility equation is related to the actual loading on

the primary structure that is given by the expression:

fio hio ha L hb RX 14

It reects the slope discontinuity over each of the inte-

rior supports due to the rotations formed at the outer

edge of the left span and at the inner edge of the right

span. Fig. 7 shows the generation of fio and the direc-

tions of the rotations.

Finally, setting the total discontinuity hi equal to

zero, the compatibility equation at point i becomes

fsiYi1wi1 fs

iYiwi fs

iYi1wi1 fm

iYi1Mi1 fm

iYi Mi

fmiYi1Mi1 fio 0X 15

2.4. Stiness coecients and equilibrium equations

The stiness coecients ksiYi1, ksiYi and k

siYi1 corre-

sponding to unit values of the displacements wi1, wi and

wi1, respectively, will be considered rst. The equations

given in Ref. [9] for an annular plate with a uniform

annular line load V will be used in the following deri-vations.

Consider rst the annular plate with outer edge

simply supported and inner edge free, as shown in Fig. 8.

The force Rifb , required to produce a unit upward de-

ection, wb 1, at the inner edge is

Rifb V D

a3C7

C1C9 C7C3X 16

Equilibrium gives Rifa as

Rifa b

aRifb

Fig. 6. Flexibility coecients due to unit moments.

Fig. 7. Slope discontinuities at point i for the actual loading.

Fig. 8. Outer edge simply supported inner edge free.


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or

Rifa Db

a4C7

C1C9 C7C3Y 17

where Rifa can be viewed as a reaction to the applied load

Rifb .For the second case, the outer edge of the annular

plate is considered free and the inner edge simply sup-

ported as indicated in Fig. 9. The force Rofa that causes a

unit upward deection at the outer edge is obtained by

the same procedure:

Rofa V Db

a4C7

C1C9 C7C3X 18

From equilibrium,

Rofb a

bRofa

D

a3C7

C1

C9

C7

C3

Y 19

where Rofb is the reaction at the inner edge to the applied

load Rofa . Thus, the expressions for stiness coecients

related to unit displacements are given as follows, and

their development is shown in Fig. 10.

ksiYi1 Rifa

LY

ksiYi Rofa

L

Rifb

RY

ksiYi1 Rofb

RX

20

For the calculation of stiness coecients kmiYi1, kmiYi

and kmiYi1 due to unit values of the moments Mi1, Mi and

Mi1, respectively, formulas for annular plate of both

edges simply supported with a uniform line moment M0at a radius r0 will be used. Substituting M0 1 and usingthe appropriate value of the radial location r0 for the

unit moments yield the related stiness coecients:

kmiYi1 Qa LY 21

where Qa L QaM0 1r0 bL

kmiYi Qa L Qb RX 22

where Qa L QaM0 1r0 aL

Qb R QbM0 1r0 bR

Y

kmiYi1 Qb RY 23

where Qb R QbM0 1r0 aR

.

The stiness coecients kmiYi1, kmiYi and k

miYi1 can be

viewed as the reactions at point i corresponding to unit

values of Mi

1, Mi and Mi

1, respectively. This is illus-

trated in Fig. 11.Fig. 9. Outer edge free inner edge simply supported.

Fig. 10. Stiness coecients due to unit displacements.

Fig. 11. Stiness coecients due to unit moments.


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The support reactions Rb and Ra (force per unit of

circumferential length) at the inner and outer edges of an

annular plate with a uniform load q are obtained as

Rb

qaC1L17 C7L11

C1C9 C3C7 Y 24

Ra Rbb

a

q

2aa2

b2X 25

Therefore, the reaction at point i for the actual

loading on the primary structure (Fig. 12) is

Rio Ra L Rb RX 26

The basis of the Winkler concept for soil deformation

was discussed in Section 2.1. A set of support springs

can be used to model a Winkler support. Fig. 13 shows

an annular plate on such a spring support equivalent.The springs are given a stiness ki (stiness per unit of

circumferential length)

ki ksA

2priY 27

where ks is the modulus of subgrade reaction, and A is

the tributary area for two adjacent annular plates with a

common edge at a radius ri, which is given by

A pri ri1

2

2

ri1 ri

2

2

!X 28Having dened the stiness coecients, Eq. (20), due

to unit displacements and the stiness coecients, Eqs.

(21)(23), due to unit moments, the forces caused by

these eects can be calculated at point i and they are

shown in Fig. 14.

In Fig. 15, the connection at point i between the two

adjacent simply supported annular plates and the sup-

port spring is illustrated. The free-body diagram of this

connection is shown in Fig. 16. On this free-body dia-

gram, the forces exerted by the adjacent annular plates

are those given in terms of displacements and moments,

and Rio, due to distributed load on the plates, and the

reactive force produced by the support spring is given as

kiwi.

Applying the equilibrium condition to the force sys-

tem acting on the connection gives the equilibrium

equation at point i:Fig. 12. Support reactions of an annular plate with outer and

inner edges simply supported.

Fig. 13. Annular plate on spring support model.

Fig. 14. Forces at i due to (a) deection wi1, (b) deection wi,

(c) deection wi1, (d) moment Mi1, (e) moment Mi, and (f)

moment Mi1.


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ksiYi1wi1 ksiYi kiwi k

siYi1wi1 k

miYi1Mi1

kmiYiMi kmiYi1Mi1 Rio 0X 29

3. Numerical studies

In order to verify the new formulation and its im-

plementation, the problem of circular plate with a cir-cular hole at the centre is solved and discussed in this

section. The circular plate shown in Fig. 17 is supported

on an elastic foundation and carries a uniformly dis-

tributed load q.

The geometrical and material properties of the plate

and the modulus of subgrade reaction ks are as follows:

a 5X0 m, b 2X5 m, t 0X25 m, m 0X2, E 2X7 107

kN/m2, q 20 kN/m2, ks 10000 kN/m3.

As no analytical results are available, the results

obtained using the present approach are compared with

the results obtained from general-purpose nite element

analysis program SAP90 [11].The plate shown in Fig. 17 is represented as a series

of annular plates and discretized into 2, 4, 6, 8 and 10

annular plates of equal span. Elastic foundation is

modelled by springs along each common edge at r ribetween the two adjacent annular plates i and i 1.Along this common edge, which corresponds to outer

radius of annular plate i and inner radius of annular

plate i 1, the spring stiness ki per unit of circumfer-ential length is obtained from Eq. (27).

Partial results obtained using the present approach

are plotted in Figs. 18 and 19 for an increasing number

of annular plates. The rst of these gures shows the

results of a convergence study for midspan deection.

Fig. 17. Test problem.

Fig. 18. Deection w at the midspan of a circular plate.

Fig. 19. Bending moment Mat the midspan of a circular plate.

Fig. 15. Connection at point i.

Fig. 16. Free-body diagram at point i.


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Points on the graph correspond to negative (or down-ward) deections in accordance with the chosen coor-

dinate directions. Results of the convergence study for

midspan radial bending moment are shown in Fig. 19.

Figs. 20 and 21 give the variation of the deection w and

the radial bending moment M in the radial direction,

respectively. Results are obtained by using 10 annular

plates.

4. Conclusions

In this paper, a new formulation for the analysis ofcircular plates supported on elastic foundations has been

presented. The circular plate is represented as a series of

simply supported annular plates resting on support

springs along their common edges. The method is based

on the classical plate theory and does not involve any

approximation for the solution variables. The deection

and radial bending moment are chosen as the solution

variables at each interior support. Application of ap-

propriate compatibility and equilibrium conditions at

each interior support results in a mixed formulation. The

implementation of the formulation is veried by solving

a circular plate with a circular hole at the centre sup-

ported on an elastic foundation.

The following conclusions can be drawn from the

obtained results based on this study.

As no analytical results are available, the results

obtained from the solutions by using the present for-

mulation were compared with the results obtained from

general-purpose structural analysis nite element pro-gram SAP90. When the plate is represented with 10

annular plates by using the proposed model, 22 vari-

ables are introduced in terms of deections and bending

moments. On the other hand, the nite element mesh for

quarter of the plate corresponding to the same number

of annular plates consists of 110 nodes and 90 plate

elements with three degrees of freedom at each node. It

has been observed that the de ections and bending

moments obtained using the present approach agree

well with the results of the nite element solution, de-

spite the small number of solution variables used to

characterize the plate. A simpler ``axisymmetric'' niteelement model consisting of individual plane stress ele-

ments may also be employed for the SAP90 solution;

however, only deection values can be compared for

this case.

The formulation does not involve any approximation

for the solution variables and for the representation of

the plate domain. Classical plate theory is used to obtain

the exibility and stiness coecients of the coupled

equations. As a result, the proposed formulation pro-

vides a valuable and practical alternative to the nite

element method for this class of problems. The approach

can be used for the solution of circular plates with no

hole at the centre by assuming the central hole to be ofnegligible radius.

Although the analysis is carried out for uniformly

distributed loading, any type of symmetric loading can

be handled very easily by simply redening the exibility

coecient fio associated with the actual loading on the

primary structure.

When the plate is supported on tensionless founda-

tions, there exists the possibility that, under certain

loading conditions, regions of no-contact develop under

the plate. The present approach can be used successfully

to compute the location and extent of no-contact regions

by examining the sign of plate deections obtained froma series of numerical experiments in which the number of

annular plates is systematically increased. Hence, these

regions can be represented using annular plates with no

elastic supports. However, this technique needs further

verication and research.

References

[1] Celep Z. Circular plate on tensionless Winkler foundation.

J Engng Mech ASCE 1988;114:172339.

[2] Timoshenko SP, Woinowsky-Krieger S. Theory of plates

and shells, 2nd ed. New York: McGraw-Hill, 1959.

Fig. 20. Variation of de ection in the radial direction (for

n 10).

Fig. 21. Variation of bending moment in the radial direction

(for n 10).


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[3] Weitsman Y. On foundations that react in compression

only. J Appl Mech 1970;37:101930.

[4] Gladwell GML, Iyer KRP. Unbonded contact between

a circular plate and an elastic half-space. J Elasticity

1974;4:11530.

[5] Cheung YK, Zienkiewicz OC. Plates and tanks on elastic

foundationsan application of nite element method. Int

J Solids Struct 1965;1:45161.

[6] Melerski ES. Circular plate analysis by nite dier-

ences: energy approach. J Engng Mech ASCE 1989;115:

120524.

[7] Chandrashekhara K, Antony J. Elastic analysis of an

annular slab-soil interaction problem using a hybrid

method. Comput and Geotech 1997;20:16176.

[8] Bowles JE. Foundation analysis and design, 4th ed. New

York: McGraw-Hill, 1988.

[9] Young WC. Roarks formulas for stress and strain, 6th ed.

New York: McGraw-Hill, 1989.

[10] West HH. Fundamentals of structural analysis. New York:

Wiley, 1993.

[11] Wilson EL, Habibullah A. SAP90-Structural analysis

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