A Study of a New Foundation Model By

Arno ld D. Ker r , N e w Y o r k , N . Y .

With 8 Figures

(Received October 1, 1964)

Summary. The characteristics of a new foundation model, consisting of two spring layers interconnected by a shear layer, are studied. The study is conducted on the classical problem of a foundation subiected to a rigid stamp. In order to reduce the number of foundation constants to an absolutely necessary minimum, special attention is given to a possible dependence of the constants of the upper and lower spring layer, particularly to the spring constant ratio three which is suggested by REISSNE~'S foundation model. A comparison of the obtained pressure distributions with relevant experimental data seem to support the adoption of this value, thus reducing the number of foundation constants to two. Advantages of the presented model over other foundation models are pointed out.

Zusammenfassung. Die Eigenschaften eines neuen Fundamentmodells, das aus zwei Federschichten besteht, die durch eine Schubschichte verbunden sind, werden studiert. Die Untersuchung folgt dem klassisehen Problem des starren Stempels auf elastischer Unterlage. Um die Zahl der Fundamentkonstanten auf das absolut Notwendige zu reduzieren, wh'd einer mSglichen Abh~ngigkei~ zwischen den Konstanten der oberen und der unteren Federschichte besondere Aufmerksamkeit gewidmet, wobei vor allem das Konstantenverh~ltnis 3 betrachtet wird, das dem REISSNERschen Modell entspricht. Ein Vergleich der errechneten Druckverteitung mit Versuchsergebnissen scheint diesen Wert zu bestiitigen, womit sich die Zahl der Konstanten auf 2 erniedrigt. Die Vorteile des vorgeschlagenen ~r werden hervorgehoben.

I. Introduction I n a r ecen t p a p e r t he a u t h o r d iscussed a n u m b e r of f o u n d a t i o n mode l s

p r e s e n t e d in t he l i t e ra tu re in t he p a s t severM decades [1]. One conc lus ion was t h a t t he PASTER~AK f o u n d a t i o n [2], cons is t ing of a WINKLER f o u n d a t i o n w i t h shear in te rac t ions , is m e c h a n i c a l l y t he m o s t logical ex t ens ion of t h e WINKL~R m o d e l a n d a n a l y t i c a l l y t he n e x t h igher a p p r o x i m a t i o n . T h e response of t he sur face of t he P A S T E ~ A K f o u n d a t i o n s u b j e c t e d t o a d i s t r i bu t ed load p(x, y) is desc r ibed b y

p = k e w - - G e V 2 w , (l)

where w is t he d i s p l a c e m e n t in t he z d i rec t ion , ]~p a n d Gp are t he f o u n d a t i o n

cons tan t s , a n d V ~ = ( ~2 ~x2 + - g ~ y 2 "~2)

136 A . D . K E n ~ :

In the same paper the I)ASTEI%NAK foundation was formulated in a manner suitable for further generalizations. Using these results a foundation model shown in Fig. l was introduced and formulated. The response of the foundation surface subjected to a continuously distributed load p(x, y) was found to be governed by the differential equation

1~- Z- p - - ~ - V ~ p = - k w - - G V 2 w , (2)

where c is the spring constant of upper spring layer, k is the spring constant of lower spring layer, and G is the constant of the shear layer.

I I IIIII Illll IIII1~iII IAH Hjj j

Fig. 1. Suggested foundation model

I t was subsequently pointed out tha t for c-~ 3k equation (2) coincides with the corresponding equation obtained by E. I~ElSSXER [3] if we set

k 4 E~. G 4 3 H ' ~ H G j ,

where E~ is You~G's Modulus and G~ the shear modulus of the foundation material, and H the thickness of the foundation. REISS~E~'S foundation model consists of an elastic layer resting on a rigid base, subjected to the assumptions tha t throughout the layer the in-plane stresses ax ~ av ~ ~ - - ~ 0 and additionally the horizontal displacements on the upper and lower surface are zero.

With c ~ 3 k equation (2) reduces to

4 G y p -- - ~ V ~ p =- k w --GV~w. (3)

Hence, although the PASTE~NAK foundation covered by a spring layer with constant c----3/c and REISS~n's foundation are mechanicMly two different models, the vertical displacements of their surfaces subjected to p(x, y), are governed essentially by the same differential equation.

In an at tempt to reduce the constants in a foundation model to an absolutely necessary minimum, one is tempted, due to this coincidence, to assume for the upper spring layer the value 3/c, thus reducing the

A Study of a New Foundation Model 137

numbe r of foundat ion constants to two: ]c and G. One advan tage of describing the response of the foundat ion surface b y only two constants is t h a t their exper imenta l de te rmina t ion can proceed along lines suggested by PaSTERNAK for his model. An advan tage of the suggested model over o ther models (such as the isotropic elastic cont inuum, REISS?CEa's or PAST~VAK'S foundat ion) is t ha t because of the upper spring layer no concent ra ted react ions or infinite react ion pressures can appear, not even along the edge of a rigid s tamp.

An addi t ional advan tage of the suggested model over the P A S T ~ A K model is t ha t an addi t ional bounda ry condit ion is available to specify the rest ra ints on the foundat ion (see [1 ]) which in some cases m a y not iceably affect the behavior of the s t ructure .

The present paper is pa r t of an a t t e m p t to establish a ra t ional representa t ion (mathemat ica l ly as well as mechanical ly) of the response of cont inuous foundat ions to s t ructures of various types. The s tudy herein is conducted on the classical problem of a foundat ion subjected to a rigid s tamp.

IL Foundation Loaded Centrally by a Rigid Stamp A rigid s tamp of width 2 L is rest ing on the foundat ion and is subjected

to a line l o a d / 5 as shown in Fig. 1. Since P acts centrally, the deflection under the s tamp, w0, is constant .

F o r the purpose of der ivat ion of the governing differential equat ions, we assume tha t the deflection a t a point on the foundat ion surface in Ix[ < L consists of two par ts

W : W e t • W s i , (4)

where wet and w~ ~ are the contract ions or extensions of the upper and lower spring layer, respectively. For the region I xl => L the index i is replaced by e.

Denot ing the contac t pressure between foundat ion and rigid body b y p(x, y), we m a y wri te

p = c wc i (5) and also

p = k w~ -- O V 2 w~. (6)

Per forming the opera t ion ( ~ G ) - - % - V 2 on equat ion (5), adding the

result ing equa t ion to (6), and not ing equa t ion (4) we obta in

(l d- ~) p - G V2p= (7)

This is the relat ion between the contac t pressure and the deflection of foundat ion surface in ]x] =< L. Not ing t ha t in Ix] G L we have w ~ w 0 ~- ~--const., and in the present case p and w are functions of x only, the differential equat ion for p(x) assumes, wi th c ~ n k, the form

d2P ( 1 + n) lc n k ~ ~ - p -- w0. (s) dx 2 G Since no bounda ry conditions can be prescribed on p(x) at x = 4 -L ,

the pressure dis t r ibut ion in the contac t area cannot be der ived di rect ly

138 A.D. KER~:

from (8). In the following this is achieved by formulat ing the problem in terms of w.~i and w~, obtaining the expression for w~ 4, and then inserting it into equat ion (6).

The differential equat ion for w~ is derived eliminating p from equations (5) and (6) and not ing tha t according to (4) w~i ~ w - - w~i. With c ----- n ]c the resulting equation assumes the form

]c /c [72w~i - - (1 § n ) - ~ w s i : - - n - f f w . (9)

we have w = wo const, and Wsi = W s i ( X ) , equat ion (9) Since in Ixl ~ L reduced to

k k d%~dx~ (1 § n) e -w~ = -- n - ~ w0. (10)

I ts general solution is

w~t ~- Cle+ ~, § C2e -~ , § -]~--d (11)

where )~ -~ ~ /~ § n) k/G. The deflections of the shear layer outside the contact area Ix[ >= L

are governed according to equat ion (6), by the differential equat ion

d2wse k dx ~ G w~e 11- O. (12)

I ts general solution is w~ ~- C~ e + "~ § C4 e - ' z , (13)

where # = ~/k-/G. Because of s ymmet ry with respect to x = 0, it is sufficient to t rea t the

problem only in x ~ 0. The constants C1, Ce, Cs, and C4 are determined from the following four conditions:

w~ is finite at x = c~; w~i(L) = w~(L ) |

Clwsidx x=L ~- ~d-x--dws~ x=Z; dWsidx ~ = 0 : 0. / (14)

Subst i tut ion of the resulting w~ into equat ion (6) yields the pressure distr ibution in 0 ___ x _~ L

{ n cosh (~ x) / n~ w0 I t - - - �9 p~(x) = l + ~ oo~h (~ L) + 1 / i~ ~ sigh (~L) ~ (15)

The relation between P and w 0 is obtained from the equilibrium equat ion of the s tamp in the z-direction

L P = 21 pp(x) dx. (16)

o

Performing the integrat ion i t follows

El iminat ion of w0 from equat ion (15) by means of equat ion (17) results in

2 L p p ( x ) = ~ L [cosh ()~ L) + 1/1 + n sinh (~ L) + n cosh (Z x)] (18) P Z L [cosh ()~ L) + ~/1 + n sinh (Z L)] + n sinh (Z L) "

A Study of a New Foundation Model 139

The corresponding react ions in case of a PASTERNAK foundat ion consist of a uni form pressure in the contac t area

2 ~ L V ~ G ~ (19) F p~ = L Y~laP +

and concent ra ted react ions P 1

= -2- 1 + L ]/kp---~p ' (20 )

along x = • L (see [4], p. 101). At this point it should be no ted t h a t when c--~ co, i . e . n--~ c% the suggested model reduces to t h a t of a PASTERNAK foundat ion.

The dis t r ibut ion of react ions for the case of an elastic (linear) founda t ion is

2L 2 (21) p p p ( x ) = z ] /1- - (x/L)~

and does no t depend upon constants of the founda t ion mater ia l [5]. The results of the numerical eva lua t ion of equat ion (18) for var ious

values of the parameters n and L ~ k/G and of equat ion (21) are presented in Fig. 2 and Fig. 3.

-1,o -0,5 u 4,~ i,o ;~

I f

J

lsol,,'opib Pou#det/'on @u. (z/)

ZL

I

Fig. 2. Pressure distribution under centrally loaded stamp

140 A .D . KER~:

In Fig. 2 pressure distr ibutions are shown for L ~ ]c/G = 5 and var ious values of n. I t can be seen t h a t for large values of n (let us say n 1000) the pressure dis t r ibut ion is constant under the s tamp except in the vicini ty of the edges where it increases ve ry rapidly, and where wi th fu r the r increasing n it approaches the representa t ion of concent ra ted line reactions. I t can be easily verified t ha t the pressure dis t r ibut ion for n = 1000 agrees a l ready closely with the dis tr ibut ion obta ined f rom (19) assuming k~/a~ = k/a.

-I,o -M o d5 I [

g,5

z,5

z,o

r 7g

'0 L

Fig. 3. Pressure distribution under centrally loaded stamp

In this connect ion it should be poin ted out t h a t V. Z. VLASOV and N. N. LI~ONTIEV [6], [4] obta ined an expression similar to (18), considering a two layered foundat ion and assuming t ha t the upper layer is ve ry th in and hence its shear interact ions negligible. Discussing thei r results t h ey concluded t ha t when c ~ oo the pressure dis t r ibut ion approaches t h a t of a semi-infinite elastic solid which, in view of the above findings, is incorrect .

In Fig. 3 pressure distr ibutions are shown for n = 3 and different values of L 2k/G. I t can be seen tha t when k/G-~o% i.e. G--,O, the pressure dis tr ibut ion approaches the cons tant dis tr ibut ion corresponding to a WI~KLE~ model with an equivalent spring constant k c/(k 4-c).

Comparing the pressure distr ibutions presented in Fig. 2 and Fig. 3 as well as the corresponding ratios p(L)/p(O) with some pressure dis t r ibut ions

A Study of a :New Foundation Model 141

ob ta ined f rom exper iments , i t seems t ha t for a number of foundat ion materials , like clay ( immediate ly af ter load application) [7] and well compac ted granular media [8], c ~ 3 k is indeed a reasonable assumption.

HI. Foundation Loaded Eccentrically by a Rigid Stamp In this case the load acting on the rigid s tamp consists of a centra l ly

act ing load P and a momen t M. Because of the l inear i ty of the formulat ion, a displacement w m a y be considered consisting of a displacement due to a t rans la t ion and a rota t ion, each par t t r ea ted separately, and the effects

i I

I

J -

l P

l . . . . . . : i

M=Pe

8 i i

__ 2

T . . . . . . . . .

Fig. 4. Stamp subjected to an eccentric load P

added as shown symbolical ly in Fig. 4. Since the centra l ly loaded case was t r ea t ed in the previous section, the rigid s tamp subjected to a m o m e n t is invest igated in the following.

The procedure is as before. The governing differential equat ions are (10) and (12), not ing t ha t in the present case in Ix[ ~ L, assuming re la t ively small angles of ro ta t ion a,

w--~ w 0 ~ ~ x. (22)

The corresponding general solutions are

n --~ ~ x (23) w~i B l e §

and w~ ~- B a e + ~z ~_ Ba e - ' ~ . (24)

Since the problem is an t i - symmetr ic with respect to x ~ 0, it is sufficient to t r ea t i t only for x 2 0 . The constants B1, B2, Bs, and B 4 are de te rmined f rom the following four condit ions:

w~ is finite at x ~ c~; w~i(L) ~ w ~ ( L ) | ~= ~=L; / (25)

dWsi - - dWse W~i(O ) ~ O. dx L -- ~ - x

142 A.D. KEmp:

Subst i tut ing the resulting w~ into equat ion (6) we obtain the contact pressure in 0 < x g L

{ 1 _} (26} n ~ k (o~ L) (1 -j- te L) sinh (2 x) _~_ n . pM(X) -- ]--+--n (te L) [sinh (2 L) + Vl+-~cosh (AL)]

The relation between M and ~ is derived from the moment equilibrium equat ion of the s tamp

L . ~ = 2 1 pM x d x (27)

0 which yields

2 n k (~ L) 2 (o~ L) {1 ~- 3 n (te L -}- 1) [(,% L) cosh (2 L) - - sinh (A L)] / ~ - - 32'(1 q-n) i-# L ) i ~ ~ i ~ (2/~ + ]/1 -4- n cosh ( ) ~ ] J"

(2s)

Eliminat ion of ~ from equat ion (26) by means of equat ion (28) and not ing tha t M = P e, results in

2L = p M ( x ) = P

3~- (2 L)2(n (1 + te L) sinh (2 2) + [sinh )~ L) ~- ~/1- -4- n cosh (~. L)] (re x)}

(teL) (~L)~[sinh (),L) + ~ l - ~ c o s h (2L)] + 3n(1 + #L)[(~L)cosh (~L)--sinh(ZL)] "

(29)

The corresponding reactions in case of a PASTE~AK foundat ion (see [4], p. 102) consists of a l inearly varying pressure in the contact area

3 e (te L) (te x) 2 L pM(x) = (30) p L [(te Z)' + 3 (te L) + 33

and concentrated reactions

3 ~ e ( 1 + teL) (31) = 2L[ (gL) ' + 3(teL) + 3]

Mong x = :k L. The distr ibution of reactions for the case of an elastic (linear) solid is [9]

2 L p i ( X ) = 4 e (x/L) (32} T~ L , l l/V~ (x/L)'

and does not depend upon constants of the foundat ion material. In Fig. 5 pressure distributions are shown for L 2 k/G = 5 and various

values of n. I t can be seen tha t for large values of n the pressure distr ibution varies l inearly under the s tamp, except in the vicini ty of the edges where i t increases very rapidly, and where wi th fur ther increasing n i t approaches the representat ion of concentrated line reactions. I t can be verified t h a t for example the pressure distr ibution for n = 1000 agrees a l ready closely wi th the distr ibution obtained from (30) assuming kp/Ge = k/G.

A Study of ~ New Foundation Model 143

In Fig. 6 pressure distributions are shown for n ---- 3 and different values of L 2 k /G. I t can be seen that when k / G ~ co, i. e. G --~ O, the pressure

L

lea. (JB)

-o,,r o

~o

3,0

tl.~,] 7 - -

L 2 ~ - = 5

Fig. 5. Pressure distribution under stamp subjected to moment M

distribution approaches the linearly varying distribution corresponding to a WINKLER model with an equivalent spring constant /c c/(Ic ~ c).

The pressure distribution for the case of an eccentrically loaded rigid stamp is obtained superposing the corresponding pressures from equation (18) and equation (29). In this connection it should be noted

144 A.D. KEtCR:

t ha t for large eccentricities tension will occur in the contac t area. For si tuations when no tension can take place (for example when a soil foundat ion is loaded by a s tamp) for large e-values the above analysis will have to be modified.

- ~ 0 -

I i 1,o e5 o ~ o,5 § 1,o

1,o 1/ . . ~

ogZ

J,O z

Fig. 6. Pressure distribution under stamp subjected to moment ,~//

IV. S o m e A d d i t i o n a l Cha rac t e r i s t i c s of the Sugges t ed F o u n d a t i o n Mode l

In a recent paper M . I . GORBV~ov-PosAI)OV [10] reviewed some short-comings of the weightless isotropie elastic con t inuum when used as a soil foundat ion model, par t icular ly when subjected to s t ructures wi th large bearing areas. He demons t ra ted his major point - - the effect of the

A Study of a New Foundation Model 1 4 5

bearing area upon *he vertical displacements -- on an example of a semi- infinite foundation subjected to a rigid stamp of square cross-section.

Based on experimental data of H. P~ESS, F. KbGLER, I). E. 1)OLSHIN, and others, he presents for soil foundations a typical stamp size versus vertical displacement curve for a constant average bearing pressure p~, ~ P/a 2 which is reproduced in :Fig. 7. For very small bearing areas this curve is monotonically decreasing which is due to the diminishing predominance of "plastic" deformations in the base along the stamp edge, with increasing bearing area the curve rises linearly which agrees with the so- lution of an isotropic elastic solid, but then with further increase of the bearing area the curve levels off which contradicts the solution of the isotropic elastic semi- infinite solid, but agrees with the results for the WI~KLER foundation or the iso- tropic foundation of finite depth.

After discussing various approaches to bring in agreement the analytical results of the weightless semi-infinite elastic base with the experimental

Fig. 7. Typical stamp size ver- sus displacement graph for soils

(a n is contact area of stamp)

data, GORBUNov-PosADOV finally suggests as solution the treatment of the semi-infinite elastic continuum subjected simultaneously to the structure and gravity forces; - -genera l ly a rather complex problem.

In this connection it is of interest to investigate the corresponding behavior of the foundation model discussed in the present paper. For this purpose equation (17) is rewritten as follows:

w o ]c __ 1 + n I (33) Pay n 1 -~ n '

(2 L) [ctgh (2 L) + V1 + n]

where Pa, = / ~ 2 L is the average bearing pressure. I t can be seen that as L - ~ 0, equation (33) reduces to

as L-~ co it reduces to

and that

wok t 1

( ~ o k t _ 1 + . ( 3 5 ) Pay l L=cx:) n

w0]c ] __( Wok ] : 1. (36) Pay / L ~ c ~ ~ Pay ]L=O

The numerical evaluation of equation (33) for n ~ 3 and k/G ~-- 5 is shown in Fig. 8. Comparing Fig. 7 and Fig. 8, it can be concluded that except for very small bearing areas, where plastic effects predominate, the response of the presented foundation model is similar to that observed in experiments

Acta Mech. I/2 10

146 A . D . KEtCR:

with soil foundations. This result is not unexpected since the suggested model, like the WI~;LER and PASTE~AK foundations, is essentiMly model of finite depth.

In this connection it should be noted that ]c and G, like the foundation modulus of the WI~KLER foundation, depend upon the depth of the foundation layer H. I t seems, however, that with increasing H/L, the effect of H on /c and G will rapidly diminish; a hypothesis which should be verified by experiments.

/,J

4~

o t i i I I I~ 0 7 Z d # d z'L

Fig. 8. S t a m p size versus d i sp lacement g raph of sugges ted mode l for n ~ 3 and I~/G = 5

In the present paper the ana- lytical results were compared with data of only a few experiments found in the literature on Soil Mechanics. In order to establish the range of validity of the sug- gested model for a number of foun- dation materials, a comparative study with systematic carefully conducted tests is in order. Preli- minary tests are presently being conducted which should contri- bute to the clarification of some problems mentioned above.

In conclusion it should be mentioned that the suggested model may be easily extended into the viscoelastic range [11].

Because of the above features and the fact that the response of the suggested foundation model is described by a relatively simple mathematical expression, the model discussed in the present paper seems to be very suitable for the analytical t reatment also of continuously supported flexible beams, plates, and shells of various shapes.

Acknowledgement This paper is part of the consulting activities of the writer for the

Cold Regions Research and Engineering Laboratory (CRREL) of the U. S. Army. The writer wishes to thank Dr. A. Assu~, Scientific Advisor, CRREL, and D. E. Nevel, l%esearch Civil Engineer, CRREL, for reviewing the manuscript before publication.

References

[1] K x ~ , A. D. : Elas t ic and Viscoelastic F o u n d a t i o n Models. Journa l of Appl ied Mechanics 31, 491--498 (1964).

[2] PAST]~RNA~:, :P. L. : On a New Method of Analysis of an Elas t ic F o u n d a t i o n by Means of Two F o u n d a t i o n Cons tan ts (in ~uss ian) . Gosud. Izd . Li t . po St ro i te ls tvu i Arkh i t ek tu re , Moscow, USSI~ 1954.

[3] I~ElSSbrER, E. : A Note on Deflect ion of P la tes on a Viscoelast ic Founda t ion . Jou rna l of Appl ied Mechanics 25, 144--145 (1958).

[4] VLASOV, V. Z., and N. N. LEO~TIEV: Beams, Pla tes , and Shells on Elas t ic F o u n d a t i o n (in Russian) . F I Z M A T G I Z : Moscow, U S S R 1960.

[5] SADOWSKI, M. : Zweidimensionale P rob leme der Elas t iz i t s Zeitschr. f. Angew. Math . Mech. 8, 107 (1928).

A Study of a New Foundat ion Model 147

[6] VLASOV, V. Z., and N. N. LEom~r~v: Technical Theory of Analysis of Foundations on an Elastic Base (in Russian). MISI Sbornik Trudov Nr. 14, Moscow, USSR 1956.

[7] FAJ3EX, O. : Pressure Distribution under Bases and Stability of Foundations. Structural Engineer. 1933.

[8] SIE~O~SEN, F. : Die Lastaufnahmekr&fte im Baugrund und ihre Auswlrkung auf die Spannungen in einem Fundament . Abhandlungen fiber Bodenmechanik und Grundbau. Berlin, Bielefeld, Detmold: Erich Schmidt Verlag 1948, 120. See also Die Bauteehnik 159 (1941); 329 (1942).

[9] FLORIN, V. A. : Analysis of Structures on Weak Foundations (in Russian). Sbornik Nr. 1 Gidroenergoprojecta, Vyp. I (1936). [l%ef. in ,,Osnovy Mekhaniki Gruntov" by V. A. FLORIN (1959) GOSSTROIIZDAT, Leningrad, Moscow, Vol. I, p. 271.]

[10] GOI%BI:YI~OV-I~ M. I. : On Ways of Development of the Theory of Structures on Elastic Foundat ion (in Russian). Osnovania, Fundamenty i Mekhanika Grunter , No. 1, p. 1 (1963).

[11] KE!aR, A. D.: Viscoelastic Winkler Foundation with Shear Interactions. Proceeding ASCE 87, EM 3, 13--30 (1961).

Arnold D. Kerr, Associate Fro]essor, Department o/ Aeronautics and Astronautics,

iVew York University, I e w York, 2r Y .

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