barden - 1962 - distribution of contact pressure under foundations

DISTRIBUTION OF CONTACT PRESSURE UNDER FOUNDATIONS by

LAING BARDEN, MSc., A.M.I.C.E.

SYNOPSIS A new approximate method of obtaining contact

pressure distributions is described, which can deal with almost all the common types of soil which occur in practice, including homogeneous anisotropy and isotropy, compressibility decreasing with depth, and certain cases of stratification such as laminations.

The solution is presented in the form of a single set of influence coefficients, governed by a dimensionless parameter 0, which is characteristic of the beam- soil system.

It is shown that the solution is also given with good accuracy, for all but rigid beams, by the simpler Winkler solution, provided the hypothetical modulus of subgrade reaction is calculated as indicated from the relevant elastic constants required to define the foundation material.

On decrit une nouvelle methode approximative pour arriver a des distributions de pression de contact, cette methode pouvant servir pour presque tous les types ordinaires de sols rencontres dans la pratique, y compris l’anisotropie et l’iso- tropie homogenes, la compressibilite decroissant avec la profondeur et avec certains cas de stratification, tels que les feuilletages.

La solution est presentee sous la forme d’un simple groupe de coefficients d’influence, gouverne par un parametre sans dimension $ qui est caracteristique du systeme de poutre-sol.

On montre que la solution est obtenue avec aussi une bonne precision pour toutes les poutres non rigides, par la solution plus simple Winkler, a condition que le module hypothetique de reaction du terrain de fondation soit calcule a partir des con- stantes Blastiques appropriees qui sont necessaires pour determiner le materiel de fondation.

INTRODUCTION

It is usually accepted that in contact pressure calculations a foundation model is best defined by its deformation under a unit point load.

Consider a beam resting on a semi-infinite medium or “half-space”, the beam being loaded by a number of vertical forces and moments. The beam will deflect under the action of the

known applied loads and moments and the unknown contact pressure distribution. The foundation medium will also deflect under the equal and opposite reaction to the contact pressure. Assuming that the beam does not lift, the deflected shapes of beam and soil must coincide ; hence the unknown contact pressure distribution may be solved.

The contact pressure consists of vertical normal stresses and horizontal shear stresses. The horizontal shear effect is not much understood, but is generally taken to be of secondary importance when compared with the vertical normal stresses. Since it would greatly com- plicate the analysis, the effect of the horizontal shear stresses has been neglected in that which follows.

To solve the problem in a simple manner the effect of the vertical contact pressure distribution has been assumed to be made up from a number of steps, or blocks, of uniform vertical pressure, as illustrated in Fig. 1 (a).

It is well known that the bending moments and particularly the deflexions of a beam, are not appreciably in error when calculated from such approximations. It appears reasonable to assume that the foundation displacements will similarly be of acceptable accuracy when calculated from this approximation, provided the number of steps or blocks is sufficiently high.

In order to equate the beam and foundation displacements, they must be measured from a common datum. The datum will be taken as the line connecting the ends of the beam.

CALCULATION OF BEAM DEFLEXIONS

Consider the loading diagram shown in Fig. 1 (a). The uniform intensities of pressure in the pressure blocks at 1,2,3, . . . 10 are given by $1, $2, $3 . . . $1,~. To illustrate the method the unit influence load W = 1 at point 4 should be considered.

+ The bending moment diagram resulting fn$ the loads applied by pi, P, . . . p,, and W

182 LAING BARDEN

must be of the form indicated in Fig. l(b). Treating this bending moment diagram as a loading diagram, the resulting bending moment diagram will be the required deflexion diagram, and is shown by the hatched area in Fig. (1~). The required deflexions 6 relative to the adopted datum XY are obviously as indicated in Fig. l(c).

Let L and b be the length and breadth of the beam XY. Since the number of blocks of

). lb1

X

id

Fig. 1. (a) Load diagram; (b) Bending moment diagram; (c) Deflexion diagram

pressure have been assumed to be 10 it is necessary to put 1 = L/10. Thus p,, p, , . . p,,, are the heights of pressure blocks of base area 1 x b, and the forces exerted by these blocks will therefore be blp,, blp,, etc.

The bending moments are as follows:

Ml=0 M, = Wzh) M, = bJ2(2P1 + Pz.) Mb = bWP1 + 2;62 + P,) M, = b12(4p, + 3P, + ZP, + $4 - 1) M, = bZ2(5p, + 4$, + 3p3 + 29, + p, - 2) M, = bP(6pl + $52 + 4P, + 3~54 + 2$, + $6 - 3) M, = bZ2(7@1 + 692 + 5~53 + 4P, + 3P, + ZP, + P, - 4) Ms = bJ2(3Pr + V2 + 6Ps + 5P4 + 4P, + 3p, + 2p, + p, - 5)

Ml, = 0

DISTRIBUTION OF CONTACT PRESSURE UNDER FOUNDATIONS

Treating the moments M, to Ml0 as loads the resulting deflexions are obtained:

A, =0

A2=&(M1;Mz) 1.112 = & (M, + M,)

A,=~(M1;Mz)Z.;+~(MP;M3)Z.Z/2=&(3M1+4Mz+Ma)

183

etc. Summarizing :

MI Mz M3 M4 M, M, M7 M, M,

A, = 0 A2 = Zs/4EI( 1 + 1 1 A3 = Zz/4EI( 3 + 4 + 1 A4 = Zs/4EI( 5 + 8 + 4 + 1 ; A, = 12/4EI( 7 + 12 + 8 + 4 + 1 A, = Zs/4EI( 9 + 16 + 12 + 8 + 4 + 1 ; A7 = 12/4EI(ll + 20 + 16 + 12 + 8 + 4 + 1 A3 = Zs/4E1(13 + 24 + 20 + 16 + 12 + 8 + 4 + 1 ; A9 = Z2/4EI(15 + 23 + 24 + 20 + 16 + 12 + 8 + 4 + 1)

A,, = Zs/4EI(17 + 32 + 28 + 24 + 20 + 16 + 12 + 8 + 4)

Relative to datum XY

S1 = 0/9A1,, - Al = 0 82 = 1/9A,, - A2 S3 = 2/9A1,, - A3

Etc., etc.

S1, = 9/9Alo - Al,, = 0

Thus :

Si =0 M2 M, M4 M, MB M, M, M, 82 = Zs/4EI(2-55 + 3.11 + 2.67 + 2.22 + l-78 + 1.33 + 0.89 + O-44) 6, = Zs/4EI(3*11 + 5.22 + 5.33 + 4-44 + 3.55 + 2.67 + l-78 + O-89) S, = Z2/4EI(2-67 + 5.33 + 7.00 + 6.67 + 5.33 + 4.00 + 2.67 + l-33) 6, = Zs/4EI(2.22 + 4.44 + 6.67 + 7.89 + 7.12 + 5.34 + 3.56 + 1.78) S6 = Zz/4EI(l-78 + 3.56 + 5.34 + 7.12 + 7.89 + 6.67 + 4.44 + 2.22) S7 = Zs/4EI(1.33 + 2.67 + 4.00 + 5.33 + 6.67 + 7.00 + 5.33 + 2.67) 6, = Zz/4EI(O-89 + l-78 + 2.67 + 3-55 + 4.44 + 5.33 + 5.22 + 3.11) 6, = Zs/4EI(O-44 + O-89 + 1.33 + 1.78 + 2.22 + 2.67 + 3.11 + 2.55)

610 = 0

Putting M2, MS, M4 . . . M, in terms of p,, p,, 9, . . . pi, we obtain:

sr =o

where

and

and

and *+

62 = b~414EV2,P, + Q2P2 + Q93 + . . Qu,P,, - Q) 63 = bZ4/4EI(R1P1 + R2p2 + R3P3 + . . . R,,P,, - R) etc.

610 = 0

Qr = 2-55 x 1 + 3-11 x 2 + 2.67 x 3 + . . . + O-44 x 8 = 52-29

Qz = 3.11 x 1 + 2.67 x 2 + . . . + O-44 x 7 = 37.30 etc.,

R1=3~11x1+5~22x2+5~33x3+...+0~89x8=100~65

R, = 5.22 x 1 + 5.33 x 2 + . . . + O-89 x 7 = 73.66 etc.

184 LAING BARDEN

The values of Q, R, S, etc., depend on the position of the unit influence load W.

When W is at 1 it can be seen that Q = Q1, R = R, etc.

When W is at 2 it can be seen that Q = Qz, R = Rz etc.

This method of calculating beam deflexions will also apply if the loading, instead of being a unit influence point load W, is a unit influence bending moment M, applied at any point along the beam.

Thus most of the usual practical loadings which consist of combinations of vertical point loads, vertical uniformly distributed loads and bending moments, can be dealt with by superimposing the solutions for an influence load W and influence moment M.

Solutions for EI varying along the beam, although perfectly straight forward, will require much tedious calculation and are not likely to prove popular in practice.

CALCULATIONS OF FOUNDATION DEFORMATIONS

As stated previously the foundation is, in this context, best described by its deformation under a vertical point load. Consider a half-space loaded at the surface by an isolated vertical point load P. The deformations can be obtained at any point if the stress distribution and the variation of the modulus of elasticity E and Poisson’s ratio Y throughout the material are known.

Generally the vertical strain at any depth z is given by

SW v.uz v.uy ----_ 4 = z = Eyv EzH EzH . . . . . .

where w is the vertical deformation,

o,, oZ and uU are the axial stresses, Ez~ and E,H are the values of E at a depth z in the vertical and horizontal directions, respectively.

w is obtained by integrating equation (1) with respect to .z, and the vertical deformation at the surface w. is obtained by putting z = 0 in the general expression for w.

The value of w. will be a function of r, the radial distance from the point load P, the value of w. obviously decreasing with distance from P.

To obtain the shape of the deformed surface under a uniformly loaded rectangular area, or “block” of pressure, it is necessary to obtain the double integral

JJ wodx.dy . . . . . . . . . (2) between the correct limits.

The principle of St Venant indicates that the surface deformation at a distance from the block of pressure will be accurately given by an equivalent point load P, which means that the simpler expression for w. can be used. It remains to be seen how close the block of pressure can be approached before the more complex equation (2) has to be employed.

Once the equation for surface deformations caused by a single rectangular area of pressure has been obtained, this equation can be employed as an “influence shape”, and the total settlements A, of the foundation obtained along the length of the beam by applying the principle of elastic superposition.

As indicated in Fig. 2 the settlements relative to the datum XY are given as follows :

13~ = d, - [0/9.d,, + g/9.4,] = 0 6, = A, - [1/9.Alo + 8/9.A,] 83 = A3 - [2/9.A,, + 7/9.A,]

Etc.

6, = Ll, - [S/9.4,, + l/9.4,] 610 = 0

DISTRIBUTION OF CONTACT PRESSURE UNDER FOUNDATIONS 185

Fig. 2. Soil deflexions

Thus

61 =o S2 = Arti, + &A? + . . * + ~,,P,, 83 = BlPl + B2P2 + . . - + %,P,,

Let it be assumed that AI, As. B1, BB, etc., are such that the following can be written :

61 = 0

82 = owl + %&2 + @3P3 + * * * + @lcJPlo) 83 = ZhP, + b&t + bP3 + . . . + b,oP,,)

where Z is some parameter with dimensions and aI, as, a3, bl, b2 and b3, etc., are simple dimensionless numerical factors.

METHOD OF EQUATING BEAM AND SOIL DEFLEXIONS

On equating the expressions for the beam and soil deflexions (relative to the same XY datum) obtained at points 1 to 10 the following is obtained

Sr@eam) = S,(soil) = 0

Ss(beam) = & (52.29pr + 37.3OP2 + . . . + @OPlO + Q2)

= s2W) = Z(alpl + ati2 + . . . + ~~~~~~~

S3@eam) = & (100.65p, + 73.66p2 + . . . + @OPlO + Q3)

= s&oil) = Z(bdl + b2p2 + . . . + bloplo) etc.

Sro(beam) = S,,(soil) = 0

These equations can be rearranged and written as :

(52.294 - al)pl + (37.304 - a2)p2 + . . . (0.04 - aIo)plo = 9Q2 (100654 - b,)p, + (73.664 - b,)p, + . . . (0.04 - b,o)plo = 4Q3

etc.

b14 1 . where cj = 4m.2 IS a parameter characteristic of the beam-soil system.

This gives only eight simultaneous equations to solve for the ten unknown pressure ordinates $1 to $10, and so two further equations must be provided. This is done by considering the vertical equilibrium of the beam and by taking moments about one end of the beam.

Thusxp,,=W=l

and ~$,,.I = 0. **+

186 LAING BARDEN

If the important parameter $ can be obtained in a dimensionless form the ten simultaneous equations need be solved (on an electronic digital computer) only once, to yield influence coefficients of the contact pressure distribution. By using the methods outlined in the following sections, almost all of the important foundation models are made to yield their respective parameter 4 in a dimensionless form, and also the values of ai, a2, bl and b2 etc., as dimensionless numerical factors, and hence lead to a solution in the form of influence coefficients.

Should a particular foundation model fail to meet this condition, the ten simultaneous equations would have to be solved, after substitution of the relevant numerical values. Such extremely tedious “one-off” solutions are of little practical importance.

For symmetrical loading the number of equations is reduced to five.

HOMOGENEOUS ANISOTROPIC ELASTIC HALF-SPACE MODEL

Consideration of the manner in which a soil is usually deposited-by sedimentation-leads to the conclusion that its properties will be different in the horizontal and vertical directions, with generally no preferred direction in the horizontal plane. It has been shown that such axially symmetrical anisotropy is defined by five elastic constants-Ev and Ea the moduli of elasticity in the vertical and horizontal directions respectively, and three values of Poisson’s ratio, ZJ~ horizontal on horizontal, ‘us horizontal on vertical, and va vertical on horizontal.

The stresses and displacements in such an anisotropic medium have been obtained in terms of the five elastic constants by the Author (1962) 1 who adapted and modified the rigorous mathematical solutions of Michell (1900). In the present contribution the most important equation is that for the surface deformation under a vertical concentrated load P. This is given (Mitchell, 1900) by:

P

J

A [(d/AC + L)2 - (F + _Lyq*

w=2rrr’ TJ AC - Fz . . . . (3)

In reference (2) the following values are obtained :

A=+v2v3) c = 5 (1 - vr2)

F = E v,(l + vi) 45

cj = (1 + Vi)(l - 01 - 2v2v,) and E, = E and EH = nE.

On putting numerical values for n, vi and v2 and va equation (3) can be written as:

PJ w=E.r . . . . . . . .

where J is simply a dimensionless numerical factor. Now the important case of isotropy is simply a special case of anisotropy where n

vr = 7Je = va = 0. Thus for the isotropic state the following is obtained:

A = C = (1 $;1”“2v) F = (1 + v)T?- 2v) L = 2(1”+ v)

(4)

1 and

P(1 - v2) Substituting in equation (3) w = --

nEr ’ which is identical with the well-known Boussinesq

expression, is obtained. Referring to equation (4) it is shown that for the isotropic case

1 - vs J=, -a dimensionless factor.

1 The references are given on p. 198


d.r

-b

Fig. 3. Settlement due to uniformly loaded rectangular area

So far have been obtained the surface deformation ze, at a radial distance Y from a con-

PJ centrated load P as ze, = -. Er

To obtain the surface deformation due to a uniformly loaded

rectangular area it is necessary to proceed as follows : In Fig. 3 the settlement at 0 due to the uniformly distributed load 4 over the rectangular

area is :

II -b

which can be shown to give :

A :-~{Clog [; + J($‘-t- l] + blog [; + &)a + I]

_ blog [; + &)” + I] - alog [; + J(i)’ + l]} @)

As expected from the principle of St Venant the accurate vaIue of d given by equation (5)

PJ is given quite closely by w = -; and this greatly simplified expression will be used to provide Er

the “influence shape” necessary to obtain (by superposition) the deformed shape of the surface of the soil under the blocks of pressure making up the contact pressure distribution.

At Y = 0 equation (4) breaks down, due to the artificial concept of a point load, and the finite value of A0 to be adopted depends on the LIB ratio of the rectangular areas (and hence of the beam itself) and can be obtained from equation (5).

For a beam with L/B = IO the ten rectangular areas of uniform pressure will in fact be

squares and from equation (5) a mean value for A, will be d, =

The soil deflexions are given by:

A =y~P1+f.P2+&.p3+;p4+... 1 1 8

+ gP10 ]

A 1 2 = y [;pl + ; p, + ;.p, + ;.p4 + . . .

s + gj.PlO

-J

etc.

LAING BARDEN 188

Hence 61 = 0

83 = y [-1*6789pr + 2.0972$, + 05397$a + . . . - 0~307opr~] 8

83 = ; [-1+358O~r + 0*1944ps -I- 2*5794p, + . . . - 0.6101 p,,] II

Equating soil and beam deflexions as outlined earlier the following is obtained :

Now 101 = L, thus

b14 1 b14 E, C=,,,*,=myJ

E,L4 1 -.- 1044 = 4EI J . . . . . . . . .

and C$ is therefore a dimensionless parameter, characteristic of the beam soil system, and the ten simultaneous equations can therefore be solved once and for all to give the influence coefficients of the contact pressure distribution for values of (b covering the practical range. These coefficients are given in Tables I-10, which are Tables of influence coefficients for homogeneous anisotropic elastic half-space. The corresponding coefficients for beams with LIB ratios of 5 and 20 are fairly close to the given L/B = 10 values, and so Tables l-10 can be applied to any beam.

Table 1 (1049 = 0)

U.D.L. O.OL O.lL 0.2L _______ ______

$1 1.61 1.61 2.36 3.11 P2 0.83 0.83 1.09 1.34 Pa 0.89 0.89 1.11 1.32 P4 0.84 0.84 0.95 1.07 P5 0.83 0.83 0.87 0.91 P6 0.83 0.83 0.79 0.75 P, 0.84 0.84 0.72 0.60 Pa 0.89 0.89 0.67 0.46 P9 0.83 0.83 0.57 0.32 PlO 1.61 1.61 0.86 0.12

Eccentricity of load

Table 2

(1044 = 10)

0.3L

3.85 1.60 1.54 1.19 0.95 0.71 0.49 0.24

+0.06 -0.63

- -- I

0.4L

4.60 1.86 1.75 1.31 0.99 0.67 0.37

$0.02 -0.20 -1.38

U.D.L. p___

Pl 1.60

$: 0.83 0.89 P4 0.84 P5 0.84 P, 0.84 P7 0.84 P8 0.89 PS 0.83 PI0 1.60 I

Eccentricity of load

O.OL

1.53 0.82 0.90 0.87 0.88 0.88 0.87 0.90 0.82 1.53

O.lL 0.2L ______

2.28 3.05 I.08 1.35 1.12 1.34 0.99 1.10 0.91 0.94 0.83 0.77 0.74 0.61 0.68 0.46 0.57 0.31 0.80 0.08


-

-_

-

U.D.L.* O.OL

1.55 1.24 0.83 0.80 0.90 0.94 0.86 0.98 0.86 1.03 0.86 1.03 0.86 0.98 0.90 0.94 0.83 0.80 1.55 1.24

I- ~-

U.D.L. -~

1.50 0.83 0.91 0.88 0.89 0.89 0.88 0.91 0.83 1.50

U.D.L.

1.38 0.82 0.93 o-93 0.94 0.94 0.93 0.93 0.82 1.38

-- 1 -

Table 3

(1044 = 50)

O.lL 0.2L

2.00 2.85 1.08 1.36 1.17 1.39 1.12 1.19 1.06 1.02 0.95 0.83 0.82 0.65 0.70 0.46 0.54 $0.29 0.57 -0.04

* Uniformly distributed load.

Table 4

(104f# = 100)

O.OL O.lL

0.97 1.72 0.78 1.07 0.97 1.23 l-10 1.24 1.19 1.20 1.19 1.07 1.10 0.89 0.97 0.71 0.78 0.51 0.97 0.35

-- I -

Table 5

(104fj = 300)

O.OL

0.32 0.71 1.03 1.35 1.59 1.59 I .35 1.03 0.71 0.32

I

-

O.lL 0.2L

1.03 2.10 1.05 1.43 1.38 1.66 1.58 1.57 1.57 1.32 1.35 1 .oo 1.03 0.69 0.70 0.41

+0.42 $0.20 -0.12 -0.37

-

-

0.2L

2.65 1.37 1.46 1.29 1.11 0.89 0.67 0.45

$0.27 -0.15

-- I -

0.3L 0.4L

3.80 4.83 1.63 1.87 1.57 1.72 1.21 1.21 0.95 0.87 0.70 0.56 0.47 +0.29 0.23 -0.01

$0.05 -0.18 -0.61 -1.15

0.3L 0.4L ~~

3.75 5.01 1.66 1.89 1.61 1.69 1.24 1.13 I 0.95 0.78 0.69 0.47 0.45 +0.22 0.21 -0.03

1-0.04 -0.17 -0.59 -0.99

190 LAING BARDEN

Table 6

(lO%$ = 500)

PI Pa Pa P4 P6 PI P7 Pa P9 PlO

Pl Pa

p”: P, PO P7 P8 PB PlO

Pl P2 Pa P4 PO PB P7 PO PO PI0

-

U.D.L. O.OL O.lL

1.31 0.01 0.66 1.75 0.82 0,66 1.04 1.48 0.94 1.04 1.47 1.81 095 1.48 1.79 1.76 0.97 1.81 1.79 1.43 0.97 1.81 1.48 1.03 095 1.48 1.06 0.66 0.94 1.04 0.66 0.36 0.82 0.66 to.36 +0.15 1.31 0.01 -0.31 -0.43

U.D.L. O.OL O.lL

1.28 -0.16 0.42 0.82 $0.61 1.03 0.95 1.03 1.53 097 1.56 1.95 0.99 1.96 1.94 0.99 1.96 1.56 0.97 1.56 1.06 0~95 1.03 0.62 0.82 +0.61 10.31 1.28 -0.16 -0.40

U.D.L.

1 a24 0.83 0.96 0.98 1 .oo 1 .oo 0.98 0.96 0.83 1.24

-

--

-

Table 7

(lO%# = 700)

0.2L 0.3L ______

1.51 3.28 1.53 l-95 1.94 1.95 1.91 1.43 1.50 092 1.03 0.54 0.62 0.27 0.31 +0.08

+0.10 -0.02 -0.44 -0.38

Table 8

(lO%j = 1,000)

O.OL

-

-_ O.lL

-0.31 0.18 +0.56 1.01

1.01 1.60 1.62 2.12 2.12 2.10 2.12 1.62 1.62 1.03 1.01 0.55

$0.56 $0.25 -0.31 -0.45

--

-

0.2L

1.23 3.10 5.88 1.59 2.06 2.00 2.10 2.09 1.53 2.08 1.49 0.70 1.57 090 0.28 1 .oo 0.48 +0.03 0.55 0.20 -0.05 0.24 $0.04 -0.10

to.06 -0.04 -0.09 -0.42 -0.31 -0.19

0.3L

0.4L

5.77 1 a7 1.56 0.78 0.36

t-o.09 -0.03 -0.10 -0.11 -0.29

0.4L

DISTRIBUTION OF CONTACT PRESSURE UNDER FOUNDATIONS

Table 9

(1044 = 5,000)

191

Pl PZ P.3 P4 P, Pa P7 P8 P9 PlO

Pl 1.09 -0.24 -0.41 -0.24 1.40

P2 0.88 +0.02 +0.48 $1.80 3.52

P3 I.01 0.32 1.56 3.40 3.39

P4 I.01 1.57 3.39 3.37 1.60

P5 1.01 3.34 3.28 1.56 $0.30

P.5 1.01 3.34 1.60 $0.31 -0.06

P7 1.01 1.57 j-o.31 -0.05 -0.06

P8 1.01 0.32 -0.05 -0.08 -0.04

P9 0.88 +0.02 -0.07 -0.05 -0.05

PlO 1.09 -0.24 -0.09 -0.03 0.00

F -

-

U.D.L.

1.12 0.86 1.00 1.01 1.02 1.02 1.01 1.00 0.86 1.12

U.D.L. O.OL O.lL 0.2L 0.3L

-- I -

O.OL

-0.39 +0.19

0.59 1.68 2.93 2.93 1.68 0.59

+0.19 -0.39

T

-

O.lL 0.2L

-0.43 0.06 +0.74 1.85

1.69 3.00 2.98 2.98 2.90 1.67 1.69 0.58 0.58 +0.09

$0.10 -0.05 -0.03 -0.06 -0.23 -0.11

Table 10

(1044 = 10,000)

T

-

-

-

0.3L 0.4L

1.96 6.06 2.98 2.31 2.96 1.48 1.67 0.37 0.56 +0.01

+0.08 -0.11 -0.05 -0.03 -0.07 -0.07 -0.06 -0.02 -0.04 0.00

T

-

0.4L

5.97 2.59 1.45

$0.20 -0.05 -0.13 +0.04 -0.06 -0.01

0.00

A number of typical pressure distributions are plotted in Fig. 4 to illustrate the general form of the solutions.

1 - 11s As mentioned earlier the special case of isotropy is obtained by putting J = -

7r

7rE,L4 which gives 1044 = 4El(l _ v2).

The large proportion of clay soils can be considered as being homogeneous over the shallow depth affected by the pressure bulb of a foundation beam. If it is assumed that there is a factor of safety of three against failure it can also be considered that Hook’s Law applies over the small stress range involved. Thus the majority of clay soils should give good agreement when analysed on the basis of the homogeneous anisotropic elastic half-space model, with isotropy considered as a special case.

1. Stratijcation

NON-HOMOGENEOUS SOILS

In general stratification presents a problem that is very difficult to treat mathematically. The problem of stresses and dispIacements in a layered elastic system has been treated, notably by Burmister (1943). He has considered two- and three-layer systems, but so far only the stresses and displacements along the centre-line of the load have been obtained.

Attempts to calculate the stresses away from the centre-line have been made by Fox (1948) using a relaxation technique, but the method is very tedious and it has proved impossible to

192 LAING BARDEN

Fig. 4. Typical examples of contact pressure distributions

obtain expressions for the influence shape of the surface under a point load. Even if a solution were obtained it is likely to prove, judging from Burmister’s centre-line values, too complex to be of practical use in the present method of analysis.

In the case of beams, as opposed to slabs, the pressure bulb will usually be predominantly in the top stratum ; so the lack of a solution should not prove too much of a handicap when considering contact pressures.

Consider the special case of stratification consisting of alternating layers of equal thickness and of moduli E, and E2. It can be shown that :

Ev = s and EH = E1 1;L”” 1 2

The foundation can then be treated as homogeneous and anisotropic by the method just given. Another special case of stratification occurs in the foundation model treated by Wester-

gaard (1938). Westergaard considered a laminated or varved clay to be a homogeneous material reinforced by thin horizontal sheets, infinitely flexible in the vertical direction, but completely undeformable horizontally. He derived the following expression for the vertical stress at any point :

P G.2 (sz = -

271 (~2 + 6222)3/z

where

G = J 1 - 2v

2(1 - V)

From equation (1) since Ez~ = infinity


aw P GZ

aZ = $! = BrrE’p + Gw9312

After integration and putting z = 0 at the surface :

P 1

w = mG.7

Put into the form of equation (4) the following is obtained for Westergaard’s model :

J=&

and this can then be put into the expression

E,L4 1 -.- lo44 = 4EL J

and use made of the computed influence values of the contact pressure distribution.

NON-HOMOGENEOUS SOILS

2. Decrease of compressibility with depth

By integrating equation (1) it is possible theoretically to obtain the surface deformation under a concentrated load for any distribution of stress and for any regular variation in modulus.

The only soil in which modulus has been found to alter appreciably with depth is sand. It can be shown that under conditions of perfect lateral confinement E is proportional to applied stress or E = Q . CT. Even under imperfect lateral confinement E = Qu is found to apply with reasonable accuracy to sands. This leads to the conclusion that sand is anisotropic with

the coefficient of earth pressure at rest ; and that

E=Qu=Q.~.z=C.Z . . . . . . . (7)

or modulus is proportional to depth. Since no theory has yet been developed to give the stresses and displacements in an aniso-

tropic material in which modulus increases with depth, we shall have to proceed in the following semi-empirical manner.

Frohlich (1934) has modified in a semi-empirical manner Boussinesq’s expressions for the stresses in an incompressible (V = 0.5) soil to obtain :

mP c?z = -. COP+2 e

2nz2 .......

mP u,=-~cosmOsinsO

2rr.zs .......

cs8 = 0 . . . . . . . . . (84 where m is called the “concentration index”, and it can be theoretically demonstrated that

m=2+J-. Ko

Nearly all the measured values of a, in sand have been found to be given by equation (Sa) with 4 < m < 6 and it is generally accepted that a close approximation to the actual stress distribution in a sand is obtained by using Frohlich’s equations (8~) to (8~) with m = 6. Thus to obtain the surface deformation under a concentrated load in the case of a sand equations (8~) to (SC) will be used, and it can be assumed from equation (7) that E = C.Z.

***+

194

If E =C.z and

LAING BARDEN

WLP p+2

a,== m+2 (r2 + 22)-T-

mP zm.r2 u’r = gp In+2

(r2 + .22)-T

then from equation (1) is obtained:

and gB = 0

L3W mp p-1 - y2.zm-3 -=-. ~ az 2~c

[ 7n+2

(r2+z2)2 1 After integration and putting z = 0 the surface deformation will be found to be:

ze,=_. l--v_- mP

[

V

27rCr2 m mf2 1 Putting v = O-5 and m = 6 is obtained:

P 1 w = GC’F . . . . . . . . .

The surface deformation due to a uniformly loaded rectangle is, referring to Fig. 3, given by :

+b A+. ‘. ._ s s dx dy

?r x2 + y2 a -b

which is given to a close approximation by :

. . . . . . .

As expected from St Venant’s principle, equation (10) is followed quite closely by equation (9) and at r = 0 where (9) breaks down, from (10) is obtained:

Comparing equations (4) and (9) it will be seen that instead of i it is now f, and so a new

set of equations for the displacements of the soil are necessary. They are produced just as in the case of the homogeneous anisotropic model. On equating beam and soil displacements ten simultaneous equations are again obtained, and the dimensionless parameter 4 is in this case given by :

(11)

The equations can therefore be solved once and for all, for values of 4 covering the practical range, yielding influence coefficients for the contact pressure distribution.

WINKLER MODEL

To illustrate the versatility of this method of solution it will be applied to the discontinuous Winkler model ; and the accuracy of this “approximate ” method can be assessed by comparing the results with those of a rigorous solution to the Winkler model. The Winkler model is defined by9 = ky where k is the modulus of subgrade reaction. The foundation displacements are then given by:


thus

4 = ;* [ l.Pl + o.p, + o.p, + . . . + O.PlO 1

4 = ;* b.Pl + 1.p, + o.p3 + . . . + O-P10 1

etc.

-s,*P1 + 1.9, + o.p3 + o.p, + . . . + O-P9 - &ho 1

-;-PI + o.p, + 1.p3 + o.p, + . . . 2

+ O-p9 - fj*p,, 1 etc.

On equating beam and soil displacements is obtained :

bkL4 Thus with L = 101 is obtained 1044 = 4EI a dimensionless parameter, which will be

recognized as the fourth power of Hetenyi’s (1946) parameter AL = 0 = 4 J-

$i.

The ten simultaneous equations were again solved for various values of 4 covering the practical range. Making use of the relationship 1044 = 04, these results were compared with the rigorously computed solutions for the corresponding value of 0. The rigorous solutions were obtained on the DEUCE electronic computer by programming Hetenyi’s exact solution

of the Winkler equation EI $$ + ky = w(x).

It was found that the “approximate” solution was generally indistinguishable from the “rigorous” solution over the entire range of 4.

RESEMBLANCE BETWEEN WINKLER AND ANISOTROPIC HALF-SPACE SOLUTIONS

Inspection of the general form of the ten simultaneous equations, obtained on equating beam and soil displacements, indicates that for large values of 4 the beam displacement term completely outweighs the soil displacement term. This suggests that no matter what the foundation model the solutions will be similar for large values of 4 - that is for flexible beams.

Hetenyi (1946), Biot (1937), and Vesic (1961) demonstrated a special case of this more general conclusion when they showed that for infinitely long beams the isotropic elastic half- space solution was given approximately by the Winkler solution.

It therefore appears that the solutions for complex foundation models will be given for finite flexible beams by the simple Winkler model, provided a suitable value of modulus of subgrade reaction is used.

By comparing the rigorously computed Winkler solutions with the Author’s approximate solution for an anisotropic elastic half-space over the practical range, it was hoped to discover :

(a) Is there indeed a close resemblance between the solutions? (b) What portion of the practical range does it apply to? (c) What is the form of the relationship between the hypothetical modulus of subgrade

reaction and the five elastic constants necessary to define anisotropy? In the Winkler solution

196 LAING BARDEN

and in the anisotropic half-space solution

E,L4 1 -.- lo44 = @I J

1044 Let it be supposed that 104+ = N%4 or N = ga It was discovered that for 9 2 2.75 the

two solutions were almost indistinguishable for N = 4.8. For stiffer beams (% < 2.75) the development of edge stresses in the anisotropic solution prevented the two solutions being similar.

Thus :

and

EL4 1 S._ = 4.8% 4EI J

The above was deduced for beams with an L/b ratio of 10.

discovered that the modulus of subgrade reaction was given Vogt (1925) expression :

_ 0.45 E, .

“=Jq@&

. . . . . . . (12)

For other 4 ratios it was

fairly closely by the modified

. . . . . . . (13)

For L/b = 10 equation (13) gives equation (12). 1 - 212

For the special case of isotropy J = y

and equation (12) gives :

bk = 0.65 & . . . . . . . Wa)

For infinitely long beams on an isotropic half-space Vesic (1961) gives

Since

bk = O-65.12 Al-

%.A

12 E,b4 J- EI

cannot depart much from unity there is close agreement with Vesic’s findings. Vesic’s limiting value of % also agrees closely with the Author’s value of % > 2.75.

EXPERIMENTAL VERIFICATION OF THE APPROXIMATE SOLUTION Since no exact solutions exist for any of the elastic half-space models, the accuracy of the

method could be checked only experimentally. In one series of experiments (Barden, 1962), the contact pressure distributions were

measured on a sand foundation by means of vibrating-wire earth pressure cells set flush in the beam-sand interface. It was found impossible to obtain a reliable value for the compression modulus E, of the sand foundations; so the corresponding value of the modulus of subgrade reaction could not be predicted by equation (13), but had to be obtained directly by dividing the measured pressures by the measured displacements.

The experimental contact pressure distributions were shown to be given very closely by


the corresponding Winkler solution for all but the stiffest beams, thus providing general confirmation to the theoretical deductions.

In a second series of experiments the contact pressure distributions were measured on a homogeneous isotropic elastic foundation employing the photo-elastic techniques. Because the theoretical foundation is considered to be a half-space the usual two-dimensional photo- elastic method was inadequate. Of the two main three-dimensional photo-elastic techniques the “sandwich” method was selected in preference to the more tedious “frozen-stress” method. The homogeneous isotropic elastic half-space was provided by a sheet of Araldite D cemented between two blocks of Perspex, and this foundation was used to support small centrally loaded mild-steel beams, machined to dimensions giving 104$ = 10,50,100,200,300, 500, 1,000, 5,000, 10,000. The analysis of the resulting fringe and isoclinic patterns is rather difficult at the beam-araldite interface, but a reliable approximate analysis has been applied to the photo-elastic results to yield the experimental contact pressure distributions.

In Fig. 5 the photo-elastic experimental contact pressure distributions have been plotted

Fig. 5. Comparison of theoretical and photo-elastic experimental results Dhoto-elastic

________ ieoretical

as the full lines and the corresponding theoretical distributions plotted as the broken lines. Good agreement was obtained over the entire practical range of 1044 = 10 to 10,000. The occasional discrepancies between the measured and theoretical curves are as likely to be caused by inaccuracies in the analysis of the photo-elastic results, as in the tests themselves or in the new approximate solution.

DISCUSSION The new approximate solution yields for the first time pressure distributions for finite

beams resting on most of the soil foundations, that are commonly encountered in practice. The vast majority of foundations are solved using a single set of influence coefficients

(Tables l-10) governed by a dimensionless parameter 4, characteristic of the beam-soil system. Isotropy is treated as a special case of the more general anisotropic state.

Alternatively, the solution for all but the stiffest beams is given by the influence coefficients derived for the much simpler Winkler model, using the equation (13) to calculate the hypothetical modulus of subgrade reaction from the elastic constants required to define the foundation material.

198 LAING BARDEN

Theoretical support of the method is given by the excellent agreement with the results of Vesic (1961), which were derived by a rigorous method applicable to infinitely long beams.

Experimental support is provided by both the sand and photo-elastic experiments. The foregoing leads to the conclusion that the method is accurate as well as versatile. The

presentation of the results as a single set of influence coefficients should make the method particularly suitable for use by design engineers.

The simple principles underlying the method can similarly be applied to the more complex cases of networks of foundation beams and to slabs. The solution for flexible slabs is more difficult and is being investigated more fully by the Author.

The solution for a rigid slab is, however, quite simple as the deflected shape of the slab and soil is planar : it can be shown that the great majority of isolated pad footings can be treated as rigid.

Raft foundations are usually a complex mixture of slabs and networks of beams and would be very difficult to analyse accurately by the new approximate method. In such cases it is suggested that the modulus of subgrade reaction be calculated as indicated from the relevant elastic constants, and the Winkler model applied as explained by Baker (1945) using his Soil Line method.

ACKNOWLEDGEMENT

The work was carried out in the Civil Engineering Department, Liverpool University.

REFERENCES

BAKER, A. L. L., 1945. ” Raft foundations-the soil line method.” Concr. Publ. Ltd. BARDEN, L., 1962. “Stress distributions and displacements in an anisotropic soil.” In preparation. BARDEN, L., 1962. “Winkler model and its application to soil.” To be published in Struct. Eng. BIOT. M. A.. 1937. “Bending of an infinite beam on an elastic foundation.” Trans. Amer. Sot. ciu. Engrs,

59 : AI-7. BURMISTER, D. H., 1943. “The theory of stresses and displacements in layered systems.” Proc. Highw.

Res. Brd, 23 : 126-144. Fox, L., 1948. “Computation of traffic stresses in simple road structures.” Road Res. Tech. Pap. No. 9

Land. 1948. FROHLICH, 0. K.. 1934. ” Druckverteilung im Baugrunde” (” Pressure distribution in foundations “).

Springer, Vienna. HETENYI, M., 1946. “Beams on an elastic foundation.” Univ. Michigan Press, Ann Abor. MICHELL, J. H., 1900. “The stresses in an Aelotropic solid with an infinite plane boundary.” Proc.

London Math. Sot., 32 : 247. VESIC. A. B., 1961. “Beams on elastic subgrade and the Winkler hypothesis.” Proc. 5th I+zt. Conf. Soil

Mech., Paris. VOGT, F., 1925. “Uber die Berechnung der Fundamentdeformation” (“On the determination of deforma-

tion of foundations “) . Avhandlinger det kgl. Norske Videnskapsakademi, Oslo. WESTERGAARD, H. M., 1938. “A problem of elasticity suggested by a problem in soil mechanics.” Timo-

shenko 60th Anniversary Volume 268-277. Macmillan, New York.

barden - 1962 - distribution of contact pressure under foundations

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