Int. J. mech. SC;., Vol. 19. pp. 317-323. Pergamon Press 1977. Printed in Great Britain

THE VLASOV FOUNDATION MODEL

R. JONES

Department of Applied Mathematics, University of the Witwatersrand, Johannesburg, South Africa

and

J. XENOPHONTOS

Department of Civil Engineering, Swinburne College of Technology, Victoria, Australia

(Received 30 April 1976, in revised form 19 August 1976)

Summary-An alternative variational formulation of Vlasov’s two parameter foundation model is presented. This formulation provides a rigorous theoretical basis for the present form of the vertical deformation profiles, which have previously been assumed, and which form the basis of the Vlasov model. An experimental investigation of the Vlasov model is also presented.

NOTATION

stress components horizontal and vertical displacements strain energy potential energy flexural rigidity of the plate Poisson’s ratio of the plate Youngs modulus, and Poissons ratio of the foundation foundation thickness the vertical deformation profile unknown parameter in the vertical deformation profile surface deformation the modulus of subgrade reaction, and the shear parameter for the soil surface load half lengths of the tested beam

INTRODUCTION

A number of attempts have been made to describe the behaviour of an elastic

foundation under surface load. Generally, the foundation is described either as a continuum, for which linear stress-strain relations are assumed valid or as a system of

continuously distributed independent springs which offer resistance in the direction of their axes only. The later assumption was first introduced by Winkler” who assumed that the reactive forces of the foundation, carrying a loaded beam, were proportional at

every point to the deflection of the beam at that point. Winkler’s assumption, in spite of

its simplicity, leads to satisfactory results for the surface deformations of beams on an elastic foundation, see FGppl’ and H&tCnyl.4 However the model is less satisfactory when applied to loaded surface area.

More recent analyses assume that some interaction occurs between spring elements. Filonenko-Borodich’ assumed that the top ends of the spring elements were connected to a membrane which could sustain tensile stresses. Pasternak” provided for shear interactions between the spring elements by connecting the ends of the springs to a beam or plate consisting of incompressible vertical elements, which deformed only by transverse shear.

Vlasov and LeontCv9 also considered the shear interactions in a foundation and formulated their problems by using a variational method. Their approach has been widely used by a number of recent workers3.‘,” and is now extensively used in the design of structures upon soil foundations. It has all of the advantages of a continuum approach as well as the simplicity of the coupled spring model. However this model strongly depends upon the assumed form of the vertical deformation profile.

IS Vol. 19. No. hA 317

318 R . . ] O N E S a n d J. XENOPHONTOS

The Vlasov model has the added advantage, as shown by the authors, '~ that by the correct choice of the vertical deformat ion profile it reduces to a model identical to the Kerr and the Reissner foundat ion models.

A more detailed survey of the available literature is given by Kerr , ' where the Kerr and Reissner models are discussed in detail.

The present paper uses a similar approach to that of Vlasov, but the formulation is based upon a different variational principle. The advantage of this formulat ion is that it both yields the Vlasov model and provides a rigorous theoretical basis for the form of the vertical deformation profile. A limited experimental investigation is then undertaken in which the form of the vertical deformat ion profile is examined. The theoretically predicted profile is found to be in excellent agreement with the experimental ly determined vertical deformat ion profile.

T H E M O D E L

Let us consider an elastic foundation of thickness H, resting on a rigid base. A plate of flexural rigidity D lies upon the upper surface of this foundation, and is subject to vertical load q(x, y) (Fig. I). Take the oxy plane at the upper surface with the z-axis

0 a"

X I J

l ' E Qstl:f°undatl°n H, i '

FIG. 1.

directed positively downwards. Then the total strain energy of the foundat ion and the plate is

fff[ ( (0.,+0q 1 au+ av+ aw a v + a U ~ + r ~

3w

"[32wO2w (O2w]2]]ds, (1) - 2 ( 1 - vo)~-~x 2 ~ \3yay/ /J

where u, v, w are the two horizontal and the vertical displacements, respectively, ~r,, or,.. cr=, -r~, r~, %.. are the stresses in the foundation, and v, is the Poissons ration of the plate. Here the double integration is over that portion of the upper surface of the foundat ion on which the plate lies. The volume integration is over the entire volume of the foundation.

The stresses in the foundat ion are related to the deformat ions by the formulae, given by Vlasov ~ (p. 30)

E }0.,+ /a. I - - /2 - - q - - -

The Vlasov foundation model 319

z~, - 2(1 + u) ~x-x +

E I Ow+ Ou

E I Oy+ O~ ~"~ 2(1 + u) Ov (2)

where, as in Vlasov (p. 31), E and v are related to the Young 's modulus of the foundation, E . and the Po isson ' s ratio of the foundation, v~, by

Es l)s v - . (3) E - l _ v ~ , l - v ~

Let us now adopt the assumptions, put forward by Vlasov," that since there is no horizontal loading then the horizontal displacements are negligible in compar ison with the vertical displacement, i.e. u = v = O. The vertical deformat ion w(x, y, z) is also taken in the usual (Viasov) form, viz:

w(x, y, z ) = w,(x, y),Nz) (4)

where w,(x, y) is the vertical deformat ion of the foundations upper surface and &(z) is a function of the vertical distribution of displacements, chosen in accordance with nature of the problem.

Based upon experimental evidence, details of which are not readily available, the function &(z) was taken by Vlasov ~ in the form

&(z) = sinh [T(H - z)l/sinh (TH) (5)

where 7 is an unknown constant determining the variation, with depth, of the vertical displacements.

Substituting for w and setting u = v = 0 in equation (1) gives

1 . Ow, O w , ]

dx dy (6)

while the stresses ~r~-xz, and ~-,~ reduce to

E d4~ o~z = 1 - ] ~ w, d-- ~

aW~ "rx~ = G4~ Ox

Ow ~',,~ = G & -~y .

Consequent ly , the strain energy V is finally expressible as

v=½f f f (~@~ w,2(~)" + ~21vw,12)dx d. dz+--~2

× f~ f ( ,~2w,)2- 2('- ~°,t ~ oxo, .~--~. j, dx d,

(7)

(8)

320 R. JONES and .1. XENOPH()NIOS

The requirement that the total potential energy I is minimized, see Timoshenko et al. ~ (p. 251), now yields

which leads to the following partial differential equations in dJ and w,, viz

) 2dzw' f " E " ( d ~ E ch~dzV2w,+DV%v,=q(x,v) (10) (1 Z~ 2) \~-z 2(1~ p) ,

and

d-~d~ E 2 " E dz'-( l-v )~ f w'~dxdy- '52(l+v)~ I lW'd:dxdy =0 ( I I )

where the double integration is over the entire upper surface of the foundation. It is important to note that equation (10) is precisely the same equation as derived by Vlasov ~ for the response of a plate lying on an elastic foundation.

It is usual to denote as k. and 2t

and

k - I--1,2 k~zz/ "" (12)

2t - E f ," 2(1+ v) ch'-dz, (13)

so that equation (10) may be written as

DV4w, + kw, - 2tV:w, = q(x, y). (14)

From equation (10) it is apparent that the surface deformation, i.e. w,(x, y) will be dependent upon the values taken for the parameter k, and 2t, which are inturn dependent upon the form of the vertical deformation profile, i.e., q~(z). Consequently, if Vlasov's guessed form for q~(z), as given by equation (5) is in error then this will adversely effect the computed values of w,.

In equation (14) the coefficient k characterizes the compressive strain in the foundation, and is equivalent to a Winkler spring constant (or modulus of subgrade reaction). The coefficient t characterizes the shearing strain in the foundation.

The advantage of the present formulation is that, unlike Vlasov's analysis, it yields a differential equation from which the vertical deformation profile may be determined. Indeed noting that since the foundation rests on a rigid base we must have d~ = 0 on z = H, and that on z = 0 we have

61: , , = l

then equation (11) has as its solution

where

4~(z) = sinh 7(H - z)/sinh 7H (16)

(1- p) I f lVw,l: dx dy (17) 2

~ w," dx dv

The Vlasov foundation model 321

This form for ~b agrees exactly with the previously guessed form of the vertical deformat ion profile. Consequent ly we have established a rigorous theoretical basis for the vertical deformat ion profile 4~(z), which in turn gives added strength to the Vlasov two parameter soil model. Fur thermore , on the basis of equation (17) we are able to deduce several important factors which will influence the value of % Let us consider that the upper surface of the soil is of infinite extent so that away from the plate the deflections decay to zero. Then using Green 's theorem equation (17) reduces to

(v-l)ffw,V2w, dxdy 2

y = (18) 2ffw,2dxdy which making use of equation (14) gives

(l-,,)[f f (qwl-DWw,)/2t dx dy- f f kw, dx dy/2t] y2 = (19)

2 f f w , dxdy

Here the subscript s under the integral sign means that the integration is over that portion of the surface on which the plate lies, while the unsubscripted double integral is over the entire upper surface of the foundation. Hence equation (19) shows that y will be dependent upon both the applied load q, and the shape and flexural rigidity of the plate. In the special case of surface loading only (i.e. D = 0) then y will only depend on the load and the shape of the loaded region. This is an important result since in all previous publications it has been assumed that y was independent of such consideration.

As a consequence of this dependence the parameters

fo Ey(sinh 7H cosh y H + y H ) _

k (I --~~ \dz/ 2(i - v 2) sinh 2 7H ' (20)

and

fo/q E q52 E(sinh 7H cosh yH - y H ) 2 t - 2 ( 1 + v ) d z = 4(l+v) sinh2yH , (21)

will also depend on the load, the shape, and interestingly, the flexural rigidity of the plate.

Indeed such a dependence is encountered when one at tempts to experimental ly determine the value of the modulus of subgrade reaction, i .e .k .

EXPERIMENTAL INVESTIGATION

A limited experimental investigation was undertaken to investigate this behaviour. A sheet of gum rubber was used to simulate an elastic foundation material. This material was chosen mainly because of the limited finance available, but also because of its low Young 's modulus (approximately 1000 K N / m 2) and because the propert ies of the rubber were easily obtained. Poisson ' s ratio, u, was found to be 0.48.

For simplicity a two dimensional model was tested. The load was transmitted to the foundat ion via rigid beams of the same width as the foundation. Five beams of lengths 12.5, 25, 37.5 and 75 mm were tested, the depth H of the foundat ion being 150 mm while the thickness d of the foundat ion is 6.25 mm. Part iculars of the testing apparatus are given in Fig. 2. The deformat ions were measured at various points beneath the point of application of the load, and were measured using a travelling microscope to within an accuracy of 0.00! ram.

322 R , J O N E S a n d J . X E N O P H O N ' r ( } s

P

mm

i ~ 450mm

225mm

Rigid support

J /

Gum rubber

P

I

[

i

6 25mm

Rigid b e a m

Fro. 2, Geometry of the test apparatus.

The results of these measurements are shown in Figs. 3(a) and (b) where c is the half length of the beam. In each case the deformation profiles were found to be a hyperbolic sine function as predicted in the previous section. Indeed the agreement between the theory and the experimental results is very good. The values of 3' obtained from these deformation profiles is given in Table 1 for each of the five beams tested, As predicted the value of 3' may be seen to depend upon the length of the beam, In the case of the 75mm beam (c/d = 6 ) the value of 3' was small (=0 .0087mm ') and the vertical deformation profile was very nearly linear. Table 1 also presents the values of the modulus of subgrade reaction k, as computed from equation (20) using the experimentally determined values of 3'.

¢ 0 0 2 0 4 0 6 0 8 I 0

] ] I / / i x " 7 ~

I 8 ,x t

,x / !¢.; X /~ <"

I g × /

!C ×

/ 18 X /

2C --x

@

0 0 2 0 4 0 6 0 8 IO

.,,..Z.".? 2x

,, xl ]: y : 3

L / '

/ i Theory

6i i / ×Experimentoiresults

7 i

FIG. 3. Vertical deformation profiles.

TABLE 1.

e (ram) 6.25 12.5 18.75 25.00 37.5 c /d 1 2 3 4 6

3, (mm ') 0.0175 0.0157 0.0138 0.0122 0.0087 k (MN/m ~) 11.78 10.97 10.26 9.69 8.90

R E F E R E N C E S

I. M. M. FILENKO-BORODICH, Vohenyie Zapiski Moskovskyo Gosudar-stuenno Universieta Mechanica, No. 46, p. 3 (1940).

2. A. FOPPL, Vorlesungen uber Technische Mechanik, 9th edn, Vol. 3, p. 258, Leipzig (1922). 3. M. E. HARR, A , S . C . E J. Soil M e c h . F o u n d a t i o n s Dic. 95, 933 (1969). 4. M. HETENYE, B e a m s on E las t i c F o u n d a t i o n s . University of Michigan Press, Ann Arbor (1946)

The Vlasov foundation model 323

5. A. D. KERR, Trans. ASME, series E, 31,491 (1964). 6. P. L. PATERNAK, Gosuedarstvennoe Izadateslstve Literaluri po stroitelstvu Arkhitekture. Moscow (1954). 7. S.-F. CHEN, Proc. Jap. Soc. Cir. Engrs p. 42 (1972). 8. S. P. TIMOSHENKO and J. N. GOODIER, Theory o[ Elasticity. 3rd edit., McGraw-Hill, New York (1970). 9. V. Z, VLASOV and N. N. LEONT'EN, Beams Plates and Shells on Elastic Foundations. Israel Program for

Scientific Translations, Tel Aviv (1966). 10. E. WINKLER, Die Lehre yon der Elastizitat und Festigkeit. (1867). 11. T. Y. YANG, Compters & Struct. 2, 593 (1972). 12. R. JONES and J. XENOPHONTOS, Acta Mechanica Vol. 25 (1976).