On the contact pressure of circular plates on elastic subgrade

O n the contact pressure of

circular plates on elastic subgrade S. S. Issa Department of Czvzl Engmeertng, Kuwait Umverstty, PO Box 5969 A l-Safat. Kuwatt (R ecezved March 1982, revised March 1984)

A mathematical model is presented based on combined methods for determining the contact pressure of circular plates on elastic subgrades. The proposed numerical approach considers two parameters of the subgrade simultaneously, the modulus of elastmity Esand the modulus of subgrade reaction K. This approach can be utilized in the design of circular foundations such as oil and water towers, silos, chimneys and other structures of similar nature. Key words: mathematical model, mrcular plates

Several factors such as soft density, compomt:on and stress lustory, and discontinuity of various layers of soft media affect the mechamcal properties of the subgrade. Due to the many uncertainties revolved, determination of the characteristics and the behavmur of the half space is extremely difficult and unpredictable. Accordingly, estabhshment of the contact pressure of plates on the elastic subgrade is usually based on idealized soil conditions. The methods rated m the hterature can be classified into linear elastic ~-a and nonhnear methods 4 In both cases the solutmns are based on one of the mechanical properties of the subgrade, Le the elastic parameters E s associated with Polsson's ratm v or the modulus of the subgrade reactmn K. Nevertheless, the implementation of either of these propert:es has its disadvantages. A detaded dlscussmn can be found m the work by Schultze, s m which it was coneluded that both methods do not yield the real contact pressure, which is believed to be somewhere m between Solutions based on E s produce high contact pressure along the edges which increases with mcreasmg ngi&ty of plates, those based on K yMd remarkably low contact pressures. The removal of these deficiencies m the classical approaches motwated the proposed combined model In the present study, a hnear elastic mathematical model for deterrrunmg the contact pressure of circular plates on elastic subgrade ~s developed. The proposed model combines the two sod parameters, Es and K, in an attempt to eliminate the disadvantages experienced when considering them separately The deformatmn pattern resulting from a uniform load on a mrcular area acting on a half space is described by an elliptical shape. 6 Thas can be determined with the help of the completed elhptleal integrals K(k, 7r/2) and E(k: 7r/2) 7

Since the contact pressure is not umform, ~t can be represented by concentric cylinders whose heights are proportional to the magnitude of the contact pressure The accuracy obtained for the contact pressure is directly proportional to the number of cylinders selected. In the present work, 16 concentric cyhnders are considered. It xs first necessary to deternune the deformation patterns of the subgrade due to cylinders of unit height. The coor&nates of these deformation patterns, referred to as the influence coefficients e(I, J), can be estabhshed as the difference of the deformations of two uniform unit loads on the circular area acting on the half space. The first covers the outer circle of the cylinder, the second covers the inner circle. The influence coefficients e(l, J) are gwen as a square array of the 16th order In estabhshmg the deformation patterns of the circular plate, it is considered to be smaply supported at its outermost c~rcle and subjected to the same cyhnders of umt umform load. With the heights of the pressure cyhnders taken as unknowns, the deformation pattern of the circular plate can be calculated and equated to those of the subgrade. Ttus yields 15 con&tlons, along with the static condition Z V= 0, hence, the necessary equatmns for determining the 16 unknowns and thus the contact pressure distribution are estabhshed Several researchers ~'3'~-~1have investigated some specml cases of ctrcular plates on an elastic subgrade by considering only one subgrade to be constant at a time In the prevmus work, 3 a combined method was proposed, however, the fluctuation in the contact pressure actueved and its alternatlng behavmur could not be explained physically The cases of loading thought to be of pracUcal use have been studied and the results illustrated m tins paper

0141-0296/85/01064-05[$03.00 64

Eng. Struet., 1985, Vol 7, January

© 1985 Butterworth & Co (Pubhshers) Ltd

Contact pressure of circular plates on elastic subgrade: S S. Issa

Theoretical considerations The area of the circular plate m contact with the subgrade was divided into 16 loaded rings (see Figure 1). The primary results obtained showed a high gradient of contact pressure around the centre and the outermost circle of the circular plate. This explains the variation in the width of the loading cylinders chosen. The settlement due to a uniform load on a circular area, as shown m Figure 2, acting on an elastic and lsotroplc half space of constant elastic modulus, is gwen by a

S(rt)=p g - X ~

(1)

-

Ftgure 3 Dmgrammatlc presentation of settlement-deflect=on relation

Considering Xi as a function of the complete elhptlcal integrals K(k, 7r/2) and E(k; ~r/2), the settlement is obtamed as:

a 4

[& 7r\

r~

=, g

a

= P " -E-~.~. X,

(2)

ri < a

r, > a

(4)

A]

(5)

It follows that

It follows that

x,

a4

/r, 2)

(3)

rz

~

~/

v,;211

The elements of the square array of influence coefficients e(16, 16) are determined by

and

rt 4 [ E ( a

n

a2\

/a

7r\']

& 1 2

q_ Xz =,_,.,_ a nt ~r 2]

3 4

5

6

7

8

9

10

11

12

1~'~-3x0

=,=

066£:~4x0

I

:, : 06

~-

0 55

='

I"--0 2333

Ftgure I Cross-sectmn along radius of mrcular plate showing cyl,nders selected o

I

-

]/

ei, 16 =

for c = 16

(7)

Circular load

i

P

(1111111111~1

~kt, 16

a c=16

s, = ~

05-~_

0 97,5 1

(6)

13 14 15 16 I

~Oo95 I=

for c~< 15

Thus, the settlement due to uniform cylinder loads becomes.

V/~,.\"~./A',.\"cV//.~',.\\",V///'~\\\\'~////~,,.\\",'V/./A,,.\\"~/~\~/~\~ xO 0 5 ~ ' ~ 3 x O O 6 6 6 - " ~ 2 x O

e,,c=Xl, c-Xl,(c+O

E e.e,c

(8)

e=l

This mathematical relation yields the settlement patterns of the subgrade, provided that the plate is in continuous contact with the subgrade. Obviously, the up-hft, if any, is dependent on the load case, load magnitude, and plate configuration. Furthermore, the parameters of the subgrade have an important role to play. In the present work a computer program for calculating the contact pressure was developed and can be utilized in evaluating the tensile contact pressure which cannot be taken by the subgrade. In an iteratlve method the diameter of the plate can be adjusted until zero contact pressure has been achieved. The deflection patterns of the circular plate are estabhshed for the loading concerned by use of the classical elastic approach cited in the hterature, n The contact pressure, Le. the heights of the 16 pressure cylinders, can be obtained through estabhshing 16 simultaneous hnear equations. Equating the deflection and the settlement patterns yields the relation (see Figure 3)

S, - $ 1 + wc v = WEt.

(9)

where a

Si= g

c=16

E Pceeic C=I

a

s,=~

c=16

£ ec,,,c C=I

(10)

c=16

F/gure 2 Settlement mould due to umform load on circular area acting on a half space

Wcp =

E wicPc ('=1

Eng. Struct, 1985, Vol. 7, January 65

pressure of cwcular plates on elastm subgrade S. S. Issa

Contact

Ep/Es = 305 92

~ - ~ D

Cb/Es=O8m~

p,

q-

'

Relations gwen in equation (10) are considered to rewrite equation (9) as

•

'

'

_L

h r

I

a

c=16

a

c :16

c=16

E PceEw----~ E Pceelc+ E wzcPc

Es c=1

Es c=l

c=l

r(m) 1

2

3

4

n=m

5

X w,.e,,

(13)

n=l

Ep/E s = 305 92 Cb/Es = 0 8 m

\

r

~_

7 =°g

J_ h

2

T r(m)

I •

2

3

4

I

I

I

I

L

h=010

~ = 0 1 5 -~3---#

1

Ep = 3Ox lO6 kNIm 2

Es =gB l x103kNIm2

Cb = 7 8 5 * l O B k N / m 3

~= 0 19

P~

= 0 25

Figure 4 Contact pressure due to uniform load acting over whole mrcular plate

\ 2¸

Ep/E s = 3 0 5 9 2 Cb/E s = 0 8 m 1 r=

cL

•

C

3

I 2 t

3 i

4 I

/)=010

D h ~ = 0 1 5

r(m) 1 I

I

L •

5

.~o---

I

J

Ep=3Ox106kNIm 2

Es = 9 8 1 . 103 kN/m 2

Cb = 7 8 5 x l O 6 k N / m 3

p- = 0 19

h =0 25

Figure 6 Contact pressure due to r,ng load

-0-1

Ep/E$= 3 0 5 9 2 Cb/Es = 0 8 m

\

~

r

r=o05

~,~r

~

3_ h

•

I •(m) 1 3

I

I

•

h = 010 D h =0 1 5 - - - - o - - ~ ~---o --h

L

Ep = 30~106 k N / m 2

2

3

4

5

I

Es = 9 8 1 × 10 ? k N / m z

Cb=785x106kNIm 3 ~ = 0 1 9

=025

Ftgure 5 Contact pressure due to umform load acting concentr,cally over part of mrcular plate

\

rl=m

WEL =, E

wmPn

n=l

(11)

Pc = Pce + K "Sz

Substituting S~ from equation (10) into equation (11) yields g a c=16 Pc = Pce +

66

, Es

~_, pcee,c c= 1

Eng. Struct., 1 9 8 5 , Vol. 7, January

23 •

I ~h = 0 1 0

..........,,~-~ ~ = 0 15 ~ - ~

(12)

k

I

1

Ep-3OxlO6kNIm 2 Es=981*103kNIm 2

Cb=785×106kNIm~ # = 0 1 9

h =0 25

Ftgure 7 Contact pressure due to concentrated load actmg on the centre of c,rcular plate

Contact pressure o f ctrcular plates on elastic subgrade: S. S. Issa

Inserting the expression for Pc from equation (12) mto (13), a relationstup with Pce as the umque unknown can be established C=16

C=16

E

E

c=l

c=1 C

6

+K

[ c = 16

g 's

C=16

a

c=1

-I

ww

E w, P. E s n =m

=--

c=1

Middle East. The sod parameters used in the accompanying

apphcatlons are therefore based on rme sand. The material of the circular plate is considered to be reinforced concrete of average quality. Soft parameters as well as mechanical properties of the plate used are given m Figures 4-8. From the figures it can be seen that the results obtained are close to the contact pressure distribution associated with Boussinesq as the rigidity (hiD) of the circular plate

winP n (14)

a

increases.

=1

The effect of different plate stiffness expressed by hiD has been studied. It was found that an increase in hiD above 0.20 had only a trivml effect on the contact pressure. Generally, the contact pressure Is umformly distributed with a small dewatlon at the outermost rmg-like strip for ratios of hiD > 0.2 It appears that, given the parameters of the particular soil, any excess of plate tluckness h wtuch increases hiD beyond 0.2 is wasteful.

The conditions reqmred for the simultaneous linear equations have now been estabhshed. Since the condition of statics, namely the sum of all vertical forces being equal to zero, is given by.

ZV=O it follows that t=16

~, P,C, = Z F t=l

Conclusions

Practical a p p l i c a t i o n s

The numencal solution outlined here enables the designers of foundations for storage tanks, resting directly on the elastic subgrade, to determine the contact pressure with sufficient accuracy. The accuracy could be further Improved by varying E s as a function of the depth and varying K as a function of the pressure level, this second consideration being time consuming. Further, it Is questionable whether the increase in accuracy would, for practacal purposes, justify the extra effort needed. The proposed method can be utiltzed successfully m determining the proportions of the required circular foundation

(hi or water tanks on sandy soft represent a typical sltuat]on of importance to Kuwait as well as the whole of the

Nomenclature

Comments The characteristics of the contact pressure distribution obtained from the study case (Figures 4-8), are in good agreement with those m the literature, 13 these being calculated from models of a single parameter, i.e. the models are based on e]ther the elastic modulus E s or the modulus of subgrade reaction K. The slight differences in magmtude of the contact pressure verifies the statement made by Schultze. s

i~

D

--~

,

Cb/Es = 0 8 m -~

a

_1

Ep/Es =305 92

c,

C, i

rnr=lONm

Ep, E~ r(m) 1 i

2 i

3

4

5

E's K m

P

Pc

-02

P, -01

em ZF r

~.o

r, S1 St

01

S(ri) V I • ~

h=o

I 10

=0 15

I

Ep= 3 0 x l O 6 k N / m 2

wcv

I E s = 9 8 lx103 kN/m 2

Cb = 78 5 x l O 6 k N / m 3 ~= 0 19

Figure 8 C o n t a c t p r e s s u r e d u e t o b e n d i n g m o m e n t acting along outermost mrcle

~EL

m r = 1 0 Nm CI, C

radius of circular umformly distributed load geometncal dependent coefficients subscripts refer to cylindrical loading concerned and location of settlement, respectwely elastic moduli of plates and subgrade elastic modulus of subgrade Es/( 1 --:v 2) modulus of subgrade reaction number of load cases circular uniform load total combined contact pressure elastic portion of combined pressure contact pressure as deFmed by heights of 16 cylinders average contact pressure sum of applied loads radius of circular plate location of settlement settlement of outermost circle of circular plate actual settlement beneath point i due to external load settlement at r z vertical forces deformation of simply supported circular plate due to contact pressure deformation of simply supported circular plate due to external load deflection at i due to cylinder contact pressure c defleclaon at t due to external load n influence coefficients

Eng. Struct., 1985, Vol. 7, January

67

Contact pressure of circular plates on elastic subgrade S. S. Issa ~.~ v

influence factor o f s e t t l e m e n t due to circular u n i f o r m l y distributed load at r J a f r o m plate centre Polsson's ratio

References 1

Grasshoff, H 'Die Sohldruckverteflung unter zentralsymmetnsche belasteten, elastlschen Krelsplattenfundamenten', Dze Bautechmk 1953, 12, 352-358 2 Schultze, E 'Die Kombmatlon yon Bettungszahl- und Stelfezahlverfahren', Mitt. VGB 48, Technlsche Hochschule, Aachen, W. Germany, 1970, pp 1-43 3 Shertf, G and Goepfert, N 'Krelsplatten nach dem StelfezahlBettungszahlverfahren', Mitt. VGB 51, Techmsche Hochschule, Aachen, W. Germany, 1970, pp 363-380 4 Suklje, L 'Stresses and stratus m nonqmear viscous sods', Int J for Numer and Analyt Meth tn Geomech 1958, 2, 129 5 Schultze, E. 'Bettungszahl oder Stetfezahl', Konstrucktlver Ingemeurbau, Werner Verlag, 1967, pp 269-285

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Eng. Struct, 1985, Vol. 7, January

6

Tmaoshenko, S P and Goodler, J N 'Theory of elasticity', (3rd ed ), McGraw-Hall, New York, 1970, pp. 403-409 7 Jahnke, E , Emde, F. and Losch, F 'Tables of higher functions', McGraw-Hall, New York, 1960 8 Datta, S 'Large deflection of a circular plate on elastic foundation under a concentrated load at the center', J Appl Mech 1975, 7, 503 9 Chu, K H and Afandl, O 'Analysis of ctrcular and annular slabs for chmaney foundations', J Amer Concrete Inst 1966, 12, 1425 10 Chattopadhyay, R and Ghosh, A 'Analysis of cLtcular plates on semi-infinite elastic subgrade', lndzan J Technol 1969, 12, 312 11 Chakravorty, A K. and Ghosh, A 'Finite difference solution for circular plates on elastic foundations', Int J Numer Meth m Eng 1975, 9, 73 12 Tlmoshenko, S P and Womowsky-Kneger, S 'Theory of plates and shells', (2nd ed ), McGraw-Hall, Kogakusha Co, Tokyo, Japan, 1959, pp 51-77 13 Grasshof, H. 'Das Stelfe Bauwerk aufnachgleblgem Untergrund', Verlag yon Wdhelm Ernst and Sohn, Berlin, W. Germany, 1966