32-The Use of Elastic Theory for Settlement Prediction Under Three-Dimensional Conditions, Davis and...

DAVIS, E. H. and H. G. POULOS, 1968. Gioolechnique, 18 : 67-91

THE USE OF ELASTIC THEORY FOR SETTLEMENT PREDICTION UNDER THREE-DIMENSIONAL CONDITIONS

E. H. DAVIS, B.Sc.(Eng.), A.M.I.C.E.* and H. G. POULOS, B.E., Ph.D.*

SYNOPSIS

The results of recent research into the immediate and final settlement and rate of settlement of pad foundations on clay are given. The necessity for a three-dimensional approach is discussed and a method of predicting settlements is proposed, based on the use of elastic displacement theory in combina- tion with experimental soil parameters determined over a representative range of stress. The results of computations of elastic theory suitable for this method are outlined. The limitations of an elastic approach to the calculation of immediate settlements are discussed in relation to local yield within the soil. The results of laboratory tests on model footings and predictions by the proposed method are compared. The agreement is good with respect to the total final settlement, fair with respect to the immediate settlement, and good with respect to the rate of settle-

Les resultats des r&cents travaux de recherche portant sur le tassement immediat et final et de la vitesse de tassement des plaques de fondation sur I’argile sont donnes. La necessite d’une methode d’etude 8 trois dimensions est discutee et une mkthode de predire les tassements est proposee, basee sur l’utilisation de la theorie du deplacement Blastique combinee a des parametres relatifs au sol experimentaux Btablis pour une gamme representative des contraintes. Les resultats des calculs de theorie de l’elasticite convenant a cette methode sont esquissds. Les restrictions d’une methode d’etude de l’elasticite pour le calcul des tassements immedi- ats sont discutees par rapport L la limite Clastique locale a l’interieur du sol. Les resultats des essais de laboratoire portant sur des maquettes de semelles et les previsions par la methode proposee sont com- pares. La concordance est bonne en ce qui concerne le tassement final total, moyenne en ce qui concerne le tassement immediat, et excellente en ce qui

ment. concerne la vitesse de tassement.

INTRODUCTION

In a previous paper (Davis and Poulos, 1963) the Authors outlined their view that a general unified three-dimensional treatment is desirable in the analysis of the immediate, the final, and the rate of settlement of foundations on deep beds of compressible soil. Details were given of a form of triaxial compression apparatus and test procedure suitable for the determination of all the necessary soil parameters for such a settlement analysis. The present Paper summarizes the Authors’ research work to date on settlement of pad foundations on clay. The current research programme has recently been extended to include pile and raft foundations.

Under three-dimensional conditions in which a soil element is subject to strains in the three co-ordinate directions, the total settlement S,, of a foundation on a saturated clay soil is given

by s,, = S,+Scr . . . . . . . . (1)

where S, is the immediate or undrained settlement and So, is the final consolidation settlement.

At any time t after the application of the foundation load, the settlement S,, is

s,t = s,+u,.s,, . . . . . . . . (2)

where U, is the degree of consolidation settlement appropriate to three-dimensional conditions. Under these conditions it is not, in general, synonymous with the average degree of pore pressure dissipation np.

The conventional method of settlement calculation assumes one-dimensional conditions, i.e. that only vertical strains occur. The settlement can be calculated on this assumption from the results of oedometer tests and is therefore termed S,,,.

* University of Sydney. 67

68 E. H. DAVIS AND H. G. POULOS

Under true one-dimensional conditions S, = 0 and therefore

S,* = Soed and

. . (3)

STt = US& 1

U now being the degree of consolidation given by the ordinary Terzaghi one-dimensional theory. This, of course, is identical with the average degree of pore pressure dissipation.

When conditions are not even approximately one-dimensional it is still often assumed that either

or

S S,‘,”

Z s,+ US& = Sl+S,,, >

. . . . . . . (4)

S Tt ,” &+ ub%ed -‘%) S

. . . . . (5) TF z soed >

Equations (5) are perhaps used more frequently than equations (4) and will be the ones implied by the term ‘conventional approach’ in this Paper.

Skempton and Bjerrum’s (1957) method is a refinement of equation (4)

.s,t = s,+ UyS&

S,, = SU + Goed > @I

where p is a semi-empirical factor depending on the geometry of the problem and pore pressure coefficient A.

NEED FOR A THREE-DIMENSIONAL APPROACH TO SETTLEMENT ANALYSIS

Total @al and immediate settlements

It is now generally recognized that, unless the compressible soil is only present as a relatively thin layer at some depth below foundation level, the immediate or undrained settlement should be predicted as well as the long term settlement. Clearly, this undrained settlement can only be calculated by a three-dimensional strain approach because there can be no settlement of a saturated soil under one-dimensional strain conditions unless some water is expelled from the voids. In these circumstances, it is only logical to predict also the long term settlement by a three-dimensional approach since, theoretically, this should give more accurate results. The conventional settlement analysis is only one-dimensional with respect to the numerical integration of the strains vertically below the point for which the settlement calculation is being made ; the vertical stresses used in this calculation are nevertheless usually obtained from three- dimensional elastic stress distribution theory.

It is not only on the grounds of logical consistency that a three-dimensional approach is advocated. The Authors have considered all the available reliable published comparisons between observed and predicted foundation behaviour (see e.g., Skempton, Peck and McDonald, 1955 ; Helenelund, 1953 ; Jeppensen, 1948 ; Cooling, 1948(a and b), 1962 ; Simons, 1957). There are many reasons why, in individual cases, the predicted settlement differs appreciably from the observed settlement but, from a total of 16 cases which allowed a comparison between the conventional one-dimensional prediction and the observed total final settlement, there were nine cases in which the observed was 7-27% greater than the predicted. It is significant that this is the direction of the error that would be expected from three-dimensional theory. However, it must be admitted that the magnitude of the error is not very serious for many practical engineering purposes. On the other hand, gross disagreement between predicted and observed settlement is less likely to be reported than good agreement.

SETTLEMENT PREDICTION UNDER THREE-DIMENSIONAL CONDITIONS 69

For an ideal two-phase soil consisting of a linearly elastic soil skeleton and an incompressible fluid filling the voids, the error in the one-dimensional approach can be analysed. The settlements are given by elastic displacement theory and require values of Young’s modulus, E, and Poisson’s ratio, V. The correct three-dimensional value of the total final settlement, S,,, is given by the displacement theory by putting E = E’ and v = v’, where E’ and v’ are values for the skeleton. On the other hand, the conventional one-dimensional settlement, Soed, is given by the displacement theory by putting v = 0 and

The ratio S,,,/S,, is plotted for all depths of an elastic layer subject to uniform loading on the surface over a circular area in Fig. 1. It can be seen that this ratio is close to unity unless v’ is greater than about 0.25. This probably accounts for the fact that the one-dimensional approach has frequently been more successful than might have been expected. However, values of v’ greater than 0.25 may well be encountered. The Authors (Davis and Poulos, 1963) obtained 0.43 for kaolin and about 0.30 for a sandy clay. Fig. 1 shows that the errors in the one-dimensional approach increase rapidly as V' increases from 0.25. For high values of V' even quite shallow layers deserve three-dimensional treatment.

Analysis of the behaviour of the ideal two-phase soil may also be used to indicate the likely importance of immediate settlement. The immediate or undrained settlement, S,, will be given by elastic displacement theory by putting E = E, = 3E’/2(1 +v') and v = vu = 0.5. The ratio .5+,/S,, is plotted in Fig. 2 for the same surface loading and full range of layer depths as in Fig. 1. Again, the importance of Poisson’s ratio of the soil skeleton is emphasized. Two expected trends are apparent: the immediate settlement contributes a much higher proportion to the total final settlement when V' is high, and for shallow layers, the immediate settlement is relatively small.

Figures 1 and 2 could be used for practical cases to predict S, and S,, from Soed using an estimated value of v'. However, this procedure is likely to be inaccurate. Although the

Fig. 1 (left). Errors in the conventional one-dimensional approach for total final settlement

Fig. 2 (right). Relative importance of immediate settlement


theory for an ideal two phase elastic soil will give the order of three-dimensional effects on settlements, the non-elastic behaviour of real soil produces considerable departures from the theoretical relationships and hence the actual magnitudes of the settlements should be more reliably predicted by the methods advocated in the section on the use of elastic displacement theory.

Rate of settlement

The review of published information mentioned also included comparisons between observed and predicted rates of consolidation settlement. Out of 14 cases in which a reasonably reliable comparison could be made between the rate predicted from Terzaghi’s one-dimensional consolidation theory and the observed rate, eight showed that the observed rate was faster, four being very much faster. In the six remaining cases the observed rate was either very close to or only slightly slower than the predicted, and, further, the thickness of the clay and its depth below the foundation were such that the conditions of the problem were close to being one-dimensional.

In a further four cases in this review, the rate was predicted by three-dimensional theory and in three of these cases the predicted rate was reasonably close to the observed rate.

There is thus good practical evidence that the conventional one-dimensional theory will underestimate the rate of settlement under all conditions which are not close to being one- dimensional. This trend is also to be expected theoretically. Under three-dimensional conditions, excess pore pressures are able to dissipate laterally as well as vertically. The theoretical importance of this effect is shown in Fig. 10.

USE OF ELASTIC DISPLACEMENT THEORY

Total jinal settlement

In three-dimensional problems involving several strata of different types, the total final settlement may be obtained by summation of the strains in the individual strata, in a fashion similar to the conventional method of calculating Soed. Thus

STF = & (a,-v’u,--v’u,)Sh . . . .

where E’ and v’ are the parameters for the soil skeleton appropriate to the changes of stress in each layer ; a,, ay and CT~ are the stress increases due to the foundation, estimated from elastic stress distribution theory and Sh is the thickness of each stratum or layer.

If, however, the clay stratum is reasonably homogeneous, and appropriate average values of E’ and V' can be assigned to the clay for the whole depth of the stratum, S,, may be calculated from elastic displacement theory as

S @I TF = E’ . . . . . . . (9)

where q is the foundation pressure, B is some convenient dimension of the foundation, and I is the influence factor given by elastic displacement theory for v = v'.

In a one-dimensional situation, the analogous equation to equation (9) is

S OBd = m,qBI . . _ . . . . . (10)

where m, is a value of coefficient of volume decrease appropriate to the whole depth of the stratum and I is the displacement influence factor for v = 0.

In principle, the use of equation (8) for S,, and the similar equation (12) for S, corresponds to the effective stress path method proposed by Lambe (1964).


Immediate OY undrained settlement

The appropriate version of equation (9) for the immediate settlement, S,, is

su=q;l . . . . . . . . (11) ll

where I is the influence factor for Y = vu = 0.5 and E, is the modulus determined from undrained triaxial or unconfmed compression tests.

It is common practice to determine E, over a range of deviator stress equal to half the ultimate (E, = E,,), but it is probably a worthwhile refinement to determine E, over the range of stress relevant to the particular problem, since the experimental strain-strain relationship may not be linear even for stresses less than 5Oo/o of the ultimate. The value of E, used in equation (12) must also be a suitably chosen average value with respect to variations with depth. In a deep stratum of normally consolidated clay E, will increase with depth as markedly as the undrained shear strength.

A numerical integration approach analogous to equation (8) can be made and may be worthwhile in dealing with several strata of different types :

-0~5(0,+461z . . . . . . (12)

Rate of consolidation settlement

As shown in the section on rate of settlement, both the coefficient of consolidation, cV, and the relationship between degree of consolidation settlement, U, and the time factor, TV, are greatly affected by a change from one to two or three-dimensional strain conditions. The proposed unified three-dimensional method of settlement prediction therefore includes the use of appropriate rate theory and the determination of c, from the same triaxial tests as those used to determine E’ and V’ and E,, these tests being made over a range of stress appropriate to the problem.

APPlicability of elastic theory to soil

It will be seen that the proposed method of settlement prediction relies heavily on the displacement theory for a homogeneous isotropic elastic body and the use of constant soil parameters E’, V’ and E,. This does not imply any belief that real soil behaves as an ideal elastic solid. The stiffness of real soil increases as the average effective compressive stress increases,l but may decrease as the shear stresses increase, i.e. as failure is approached. However, the stress distribution produced by local surface loading and, consequently, the surface displacements due to that loading, are virtually impossible to analyse theoretically for such a non- linear material. Nevertheless, in a reasonably homogeneous mass of such material, the stresses will generally decrease away from the loading in the same broad manner as in the ideal elastic mass. The elastic stress distribution is at least statically permissible even if it does not pre- cisely satisfy the strain compatibility requirements of the non-linear material. Furthermore, the vertical surface displacement in the elastic theory is obtained by integration of strain down a vertical line and, hence, can equally be considered as an integration of the stresses. Minor departures from the stress distribution of the elastic theory will therefore tend to be smoothed out in this integration process. Thus it can be seen that elastic displacement theory should frequently be sufficiently accurate for practical settlement prediction. The overriding re- quirement is that the elastic ‘constants’ used in the theory should be experimental values

1 Although E’ increases with increase in 0’, there is indication (Davis and Poulos. 1963) that v’ is much less stress dependent.


determined for a range of stress appropriate to the problem. The initial and final states of stress in this range should both be fair average states representative of the whole soil mass and the stresses produced by the particular foundation.

Anisotropic stress-strain properties of real soil strata may also need to be taken into account if the maximum precision in prediction of settlement is to be achieved. However, the use of elastic ‘constants’ determined from tests, in which the major principal stress direction in the soil specimen coincides with that in the ground beneath the centre of the foundation, should at least partly counteract the errors of using isotropic theory for an anisotropic situation. In any case, the degree of anisotropy of stress-strain properties common in natural soil deposits is at present unknown.

ELASTIC THEORY USEFUL IN SETTLEMENT CALCULATIONS

In order to apply elastic theory to the calculation of settlements as outlined in the previous section, a detailed knowledge of the surface displacements of an elastic mass for all values of V' is required. In cases where it is more convenient to calculate S, and S,, by numerical integration of the strains, the three normal stresses uX, cry and u, are required. Furthermore, some knowledge (although not necessarily detailed) of the distributions of vertical and horizontal stress is needed in order to choose the representative range of stress over which the elastic parameters of the soil are to be determined by triaxial tests (see the section on determination of soil parameters). Finally, the distribution of bulk stress 0 throughout the soil mass may be needed to give the initial pore pressures in calculations of pore pressure dissipation during the consolidation process.

Solutions for stresses and displacements in a semi-infinite mass are well known and are frequently referred to collectively as the Boussinesq solutions since all depend on the relatively simple integration of the Boussinesq equations. However, the approximation of a real soil profile by a homogeneous, isotropic semi-infinite elastic mass is often obviously unsatisfactory. A full coverage of numerical solutions for a multi-layered non-linear system is impractical because of the large number of parameters involved, but many real situations will be sufficiently well approximated by either a layer resting on a rough rigid base or a semi-infinite mass the compressibility of which decreases with depth.

No general solutions for the latter case appear to have been published but mathematical solutions for the former case have been obtained, chiefly by Burmister (1943, 1956), and also by Marguerre (1931), Biot (1935), Pickett (1938) and Lemcoe (1960). From the Burmister solutions numerical answers have been obtained (Poulos, 1967a) which enable any stresses anywhere within the elastic layer and any displacement of any point of the surface due to vertical loading over any shaped area at the surface to be calculated with sufficient accuracy. These results have been given as curves, called ‘sector curves’, in which the influence factors for stress and displacement beneath the apex of a uniformly loaded sector are plotted against sector radius. The stress or displacement due to any shape of loaded area may then be calculated by dividing the area into a number of sectors, all emanating from the point at which the displacement is to be calculated, and adding the influence of each sector to obtain the displacement of the entire loaded area. This method of superposition of sectors may be used to calculate all stresses and displacements in an elastic layer and has been fully described by Poulos

(1967b). It is also possible to calculate the stresses and displacements due to a loaded area by the

method of superposition of rectangles. This has obvious advantages when the area itself is rectangular or is composed of rectangles. However, it has the drawback in relation to the sector method that an extra geometric parameter (the proportions of the rectangle) is involved and this makes the presentation of the values of stress at every point in the layer very volu- minous.


Davis and Taylor (1962) have published influence values for the vertical and horizontal surface displacements of the corner of a uniformly loaded rectangle on a finite layer for both vertical and horizontal loading.

Rigidity of footing

The elastic theory dealt with in the preceding paragraphs is concerned with stresses and displacements due to uniform loading of an area of the surface of the elastic layer or medium. However, pad foundations impose conditions closer to uniform displacement than to uniform loading. For accurate settlement predictions it is desirable to allow for this rigidity of actual foundations at least in the use of elastic displacement theory.a

Analytical solutions are available for the uniform displacement of a rigid circle and a rigid infinitely long strip on a semi-infinite medium (Boussinesq, 1335 ; Borowicka, 1936, 1938; Timoshenko, 1934). Numerical solutions have been obtained by Davis and Taylor (1961) for a rigid rectangle on a finite layer and by Poulos (1967c) for a rigid circle on a finite layer. However, it may often be necessary to obtain an approximation to the uniform displacement of a rigid footing from the maximum and minimum displacements of a uniformly loaded area of the same shape as the footing. This approximation can be obtained in two stages. First, the rigid footing displacement is known to be close to the mean displacement of the uniformly loaded area (Fox, 1948). Second, the approximate mean displacement can be obtained by assuming the displaced surface under the loaded area to be parabolic, in which case the centre and edge or corner displacements define the mean displacement completely (Davis and Taylor, 1961). These approximations give :

For a circle, Mean settlement cz $(Scentra+Sedge) . . (13)

For a rectangle, Mean settlement z @S,entre+Scorner) . . (14)

For a strip, Mean settlement z &(Scentre +Sedge) . . (15)

For rectangles, Davis and Taylor (1961) found this approximate approach gave answers within k 59/, of the correct answer for v = 0 and within +5% to - 10% of the correct answer for Y = 0.5. A similar order of accuracy was found by Poulos (1967c) for the circle except for the case of very shallow layers, where the approximate approach underestimated the answers by about 400/,.

Buried footings

All the foregoing elastic theory is for the case of loading on the surface of a mass. How- ever, in practice, footings are generally founded at some depth below the surface, and due to the resistance to deformation provided by the material above foundation level, a buried footing will not settle as much as the same footing at the surface.

The reduction in settlement is likely to be slight in case (a) in Fig. 3 and this is supported by the photoelastic determinations of stress distribution made by Skopek (1961). The solution of case (b) would be given to a close approximation by an analysis of displacement due to loading within an elastic layer but this would involve considerable computational difficulties. For relatively shallow footings, it is simpler and sufficiently accurate to use a displacement reduction factor derived in a manner similar to that used by Fox (1948). The ratio of the displacement of a point of a loaded area at some depth below the surface of a semi-infinite medium to the displacement of the same point of the loaded area when it is at the surface of the medium

a For estimating the distribution of bulk stress and of initial pore pressure within the medium in order to analyse rate of consolidation, it is generally accurate enough to assume uniform foundation loading. This assumption is also adequate for the estimation of a suitable range of stresses to use in triaxial testing.

4

Fig. 3

rack

(a) (b) (c)

Buried Footings Equivalent Surface

Footing

Known from

- PLASTIC THEORY

Strip Footing.

4

Fig. 4. Unknown elasto-plastic

A\ I load-settlement curve

\ /A ldaal

p, -6 elasto -plastic

I&--

purely cohesive

Ph material.

initial stata of stress p”-p,,=zfc

-1dfGl

Settlamant.

Fig. 5. Occurrence of local yield under undrained conditions-strip footing


can be derived analytically from Mindlin’s equation for displacement due to a point load applied below the surface. This ratio can then be used as a reduction or correction factor to apply to the displacement calculated for the equivalent surface footing, this equivalent surface footing being taken as resting on a layer of depth equal to the depth of material below the foundation level of the actual buried footing (Fig. 3). Reduction factors for a full range of Y have been given by Poulos (196713) in the form of sector curves.

Calculations in connexion with pile and pier foundations show that the above approximation is not satisfactory for rough footings which are buried to a depth greater than the breadth of the footing. For such footings, the major part of the applied load is, at least theoretically, taken by side adhesion and the theoretical settlement of such footings is considerably less than would be calculated from this approximation.

LOCAL YIELD

The use of elastic theory for the prediction of settlements becomes inappropriate once the load to cause local yield within the soil mass has been passed. Further increase in load produces growth of plastic regions until they are sufficiently extensive to permit continuous settlement, at which point the ultimate bearing capacity is reached. During the elasto-plastic phase between wholly elastic conditions and full plastic failure, the stress distribution departs in- creasingly from the elastic distribution. In the plastic regions, the strains at any point are no longer determined solely by the stresses at the point, so that even if the stress distribution can be approximately estimated in an elasto-plastic situation, the settlement can no longer be predicted by analytical or numerical integration of the stresses in the manner described in the section on the use of elastic displacement theory. The dilemma is illustrated in Fig. 4.

The development of a sound method of predicting settlements under loads significantly greater than that for local yield must await the outcome of theoretical elasto-plastic analyses of the type being undertaken by M.I.T. (Whitman and Hoeg, 1965). In the meantime it is important to examine the magnitude of the footing pressure qy at which local yield first occurs. This depends on the initial stress state. Understandably, qy decreases if the initial stress state moves towards either active or passive failure.3

Unless the foundation load is applied very slowly, local yield under undrained conditions is more likely than under partly or fully drained conditions. Attention is therefore confined to the former. It is then convenient to define the initial vertical stress minus the initial horizontal stress as 2fc where c is the undrained cohesion (cJ, assumed isotropic, and f is a factor lying between - 1 and + 1.

Considering the case of a strip footing with a uniform footing pressure of q on a semi-infinite purely cohesive isotropic homogeneous elasto-plastic mass, the result that when f = 0 the maximum shear stress occurs on the semi-circle with the footing as diameter and has a magnitude of q/n is well known. Further, it can be shown from elastic theory that when f > 0, the maximum shear stress is q/r(l -f) an d occurs on the centre line (point B in Fig. 4), whereas when f < 0, it is q/n( 1 -f2) an d occurs at the edges of the footing (points A in Fig. 4). Thus the foundation pressure for first yield, qy, is given by

qy = ~(1 -f)c for O<f<l and

’ ’ ’ * . . (16)

qy = 7r(l-f2)cfor -l<f<O 1

Equations (16) are plotted in Fig. 5(a) to give the theoretical minimum factor of safety against ultimate bearing capacity failure required to ensure that local yield does not occur.

3 The ultimate bearing capacity for a given shape and depth of footing is dependent only on the strength properties of the soil and is independent of the initial state of stress except in so far as this affects the strength properties.


From this figure it can be seen that with the commonly used minimum factor of safety of 3, it is necessary to have -0.33 <f < + 0.45 if local yield is not to play a possibly significant part in determining the magnitude of the immediate settlement. Although the curve in Fig. 5(a) is for a strip footing, it is likely to be a sufficiently accurate approximation for other footing shapes of common interest. For example, the maximum shear stress beneath a uniformly loaded circle is also 411~ when f = 0.

It is simple to link the factor f in the analysis to two quantities commonly quoted for natural clay deposits: the coefficient of earth pressure at rest with respect to effective stress, KIO and the ratio c/p or c,/p,_.

1 -K’, 2f = -

CIP . . . . . . . . (17)

Equation (17) is plotted in Fig. 5(b). F rom Fig. 5 it can be seen that the majority of normally consolidated clays will have values off greater than about +0.3. Also, since decrease in plasticity index involves, with the most normally consolidated clay soils, a decrease in both K’, and c/p, there will be a tendency for low plasticity indices to correlate with high positive values off, and hence increased likelihood of early local yield. On the other hand, increasing overconsolidation reduces f, possibly into the negative range, and unless the clay is very heavily overconsolidated, the likelihood of local yield under the design footing load is reduced.

Figure 5 and this discussion assume that the strength properties are isotropic. For greater accuracy it may be necessary to take into account anisotropy, especially as this will vary with the degree of overconsolidation. A detailed discussion of anisotropic strength properties of clays is given by Duncan and Seed (1966).

The foregoing analysis is for a footing with a uniform foundation pressure. Most actual footings approach the condition of full rigidity so that the theoretical elastic distribution of foundation pressure involves infinite values at the edges of the footing. Some local yield is therefore bound to occur even with the smallest footing load. However, this type of local yield is likely to be of minor importance in modifying the elastic load-settlement relationship, compared with the effect of the type of local yield which is considered in the foregoing analysis and which occurs after the foundation pressures under a rigid footing approach a uniform distribution. Some justification for this contention is provided by the fact that, as mentioned in the paragraphs on rigidity of footing, the theoretical elastic settlement of a rigid footing of a certain shape is close to the mean settlement of a uniformly loaded area of the same shape and with the same total load. For example, in the case of a circle on a semi-infinite mass, the former settlement is only 8% less than the latter.

DETERMINATION OF SOIL PARAMETERS

Selection of an appropriate stress range

The stresses of major importance are clearly those vertically beneath the centre of the foundation. It might at first be thought that the change from the initial to final states at the mid- depth of the clay stratum would be a suitable stress range for determination of fair average soil properties. However, this neglects the fact that, due to overburden effects, the soil is more compressible in the upper levels, and that also the increase in stress due to the foundation is greater in these upper levels. Thus the top half of the soil stratum contributes a greater proportion of the settlement than the bottom half, and therefore the stress changes occurring at some level above mid-depth are more representative of the average changes than those at mid-depth.

A sufficiently accurate guide to this representative depth can be obtained from a one- dimensional analysis. The variation of m, with effective stress is derived from the usual


straight line relationship between void ratio and the logarithm of the effective stress. It is then possible to calculate values of Soed from both equation (10) and the conventional method of summation of layers, and to find at what depth the stress range gives a value of m, for use as the single value in equation (10) that yields agreement in the two values of Soed.

The results of this analysis can be summarized by an approximate rule. The representative depth is between 2 and 3 of the full stratum depth according to whether the distribution with depth of vertical stress due to the foundation is more nearly triangular or uniform. The maximum error in settlement arising from the approximate nature of this rule is about + 10%. The rule breaks down for very deep strata when the distribution with depth of the vertical stress due to the foundation is far from linear. Taking such cases into account, it appears that the representative depth should never be taken as greater than 0.9 times the foundation breadth.

During consolidation, Poisson’s ratio, V, of a saturated soil (with respect to total stress) will change from the undrained value of 0.5 to the final value of v’, resulting in a change in the stress distribution within the soil and an increase in deviator stress beneath the centre of the foundation, Therefore, in a three-dimensional problem, there are strictly two representative ranges of stress increase. The first, appropriate to the undrained condition, is calculated for v = 0.5, and the second, appropriate to the end of the drained phase of the test, is calculated for v = v’. Ideally, the stresses should be gradually altered from the first to the second range as consolidation during the drained test proceeds, but in view of the approximate nature of the approach, this additional refinement appears unwarranted, and the use of the stress increases relevant to the undrained phase of the test throughout the whole test should generally be satisfactory.

Triaxial test procedure

The details of a form of triaxial apparatus and test procedure suitable for the determination of the soil parameters necessary for all aspects of the proposed method of settlement analysis have already been published by Davis and Poulos (1963).

In outline, the main features of the apparatus are:

(i) A special triaxial cell which can either be used under K, conditions, i.e. no lateral strain, or under the usual conditions of controlled lateral pressure.

(ii) Gauges to measure both the lateral pressure developed during K, tests and pore water pressures.

(iii) An accurate method of measuring the volume of pore water expelled during consolidation, the pore water system being kept under a constant back pressure to prevent air coming out of solution.

(iv) The application of constant axial stress to the specimen by hanger loading.

The test procedure involves three stages as indicated in Fig. 6.

Stage I. In this stage the ‘undisturbed’ specimen is given a preliminary consolidation under K, conditions and with an axial stress equal to the effective overburden pressure at the sampling depth.4 This procedure aims at putting the specimen back as closely as possible to its original state of effective stress in the ground, assuming that the horizontal effective stress in the ground was the same as that produced by the laboratory K, condition. If reliable independent evidence of the actual value of the horizontal effective stress in the ground is available, it is obviously better to apply this as the lateral pressure during Stage I.

* If the axial stress is applied in increments, the ordinary one-dimensional parameters m, and c,~ can be determined during Stage I. Values have been obtained in this way which agree with those from the ordinary oedometer apparatus.


, -‘/a . . -/ii,, -a I

. ..rrrt.t .r” Ml..,.* mr.,r.l.“ld.l *trw”

(a) Stress Distributions

Fig. 6 (above). Triaxial test procedure

Fig. 7 (top opposite). Typical result of K. test. Normally consolidated Hurstville Clay. Same specimen as Figs 8 and 9

Fig. 8 (middle opposite). Typical result of undrained triaxial test. Normally consolidated Hurstville Clay. Same specimen as Figs 7 and 9

Fig. 9 (bottom opposite). Typical results of triaxial consolidation test. NormaIly consolidated Hurstville Clay. Same specimen as Figs 7 and 8

Stage II. During this stage, expulsion of pore water from the specimen is prevented. The lateral pressure and applied axial stress are both controlled to constant values correspond- ing to the representative stress changes given by the method outlined in the section on the selection of an appropriate stress range, and the resulting axial strain enables E, to be calculated.

Stage III. With the lateral and applied axial stresses unchanged, the drainage cocks are opened so that the specimen consolidates. The consequent final volume strain and the axial strain combined with the previous axial strain of Stage II, enable E’ and V' to be calculated. Measurements of the rate at which these consolidation strains occur enable cVg to be determined.

A typical set of results is given in Figs 7,8 and 9 which also show the method of interpreting these results to obtain the required soil parameters.

RATE OF SETTLEMENT

The rate of settlement of any foundation on an ideal elastic two-phase soil can, in principle, be predicted by solving the basic equations of three-dimensional consolidation derived by Biot (1941). Because the consolidation process under three-dimensional conditions generally involves stress redistribution as well as pore pressure dissipation, the use of Biot’s theory for particular problems, such as local surface loading of a layer of finite depth, presents considerable mathematical difficulty. It is intended to present an approximate method of obtaining such solutions, together with experimental evidence from tests on model footings, in a further paper. However, to complete the general treatment of the theoretical effects of three- dimensional conditions on settlement behaviour with which this Paper is concerned, it is appropriate to indicate here the order of these effects on rates of consolidation settlement.

TIME (min.) 100 1000 10000

,. TIME (mid 100 1000 loocx3 I I

Initial efkctive stressconditions q’=3&5 Ib/sq.in. (before undrslned test) &;.,7.g ‘a ‘. ‘*

‘. Tot& strain : strain during undrained test Fmal effective stress conditions 4’1 39.5 *. ,. I.

x +straln during drained test

&, = .lm98+0062 r.0150

ti- &, = 0 +‘0079 =.0079

_~~ __-

z6 E3E,= ;tz- E,)=-.0036

E

2 y’; qAo; - <A$ , .0416 E,CAS+AU;, -?&Au,

Z-way end drainage

So E. H. DAVIS AND H. G. POULOS

It can be shown (Gibson and Lumb, 1953) that the equation governing the rate of dissipation of pore pressure obtained from Biot’s theory is

where zl = @=

C v3=

v2z

au aa at = C&4+g~ . . . . . * *

excess pore pressure

bulk total stress due to the foundation, = a,+a,+a,

three-dimensional coefficient of consolidation, = kE’/3y,( 1 - 2~‘)

(P/W) + p/ayy + (a2p.9)

(18)

For two and one dimensional strain conditions, equations of the same form as equation (18) can be obtained, but the coefficients cV2 and cV1 are not the same as c,~. They are related by the equation

C”1 = 2(1-v’)c,, = 3(1-v’)c,,/(l+v’) . . . . . (19)

Thus only as V' approaches 0.5 do they become equal and, for the other extreme of V’ = 0, C “1 = 2CV2 = 3CV3. This difference can be important. For example, the theoretical curve relating degree of consolidation to time for a hydrostatic triaxial consolidation test, in which drainage is permitted from the ends only, has the same shape as the ordinary curve for the one- dimensional oeclometer test, but the coefficient of consolidation, obtained by fitting the experimental to the theoretical curve, is cVg in the former test and cV1 in the latter test.

Even when the foundation load remains constant, the term a@/% in equation (18) is not in general zero because of stress redistribution which takes place within a three-dimensional mass during consolidation. Equation (18) cannot therefore be solved exactly without recourse to the full theory of Biot. However, it can be proved that as V' tends to 0.5, M/at tends to zero, and, furthermore, even when v’ -C 0.5, the overall change in the distribution of 0, from start to finish of consolidation, is not very great in many cases, so that as a first crude approximation the term ES/at may be ignored. Equation (18) then becomes

at4 at= C,~V% . . , . . . . . (20)

Equation (20) can be solved as an ordinary diffusion equation using numerical finite difference methods (Gibson and Lumb, 1953). From such solutions the average degree of pore pressure dissipation 0, on the centre line beneath the foundation can be calculated for any time t.

.q= l_JSZ . . . . . . . . (21) 0

where ut = excess pore pressure at time t

zto = initial excess pore pressure

A series of curves for UP for different ratios of the depth of layer, h, to radius, a, of a circular footing obtained in the above manner are shown in Fig. 10 for the hydraulic boundary conditions of a permeable base to the layer and a permeable footing and upper surface. Within the limits of accuracy of the numerical calculations, the curves in Fig. 10 should be correct for V' = 0.5 and approximately correct for V' -c 0.5, the possible error increasing as the ratio h/a decreases. For small values of lz/a the error can be largely removed by substituting cV1 for c,, in the definition of the time factor. Since the time factor TV is defined in terms of the depth of layer, Fig. 10 can be regarded as giving the effect of changing the layer depth on the rate of consolidation beneath a footing of constant radius. It can be seen that the three- dimensional effects are very strong, the ability of the pore pressures to dissipate laterally as well as vertically producing a very considerable increase in the rate of consolidation even for


quite shallow layers. Only when the layer is less than about one footing radius deep, does the one-dimensional consolidation curve give a reasonable estimate.

According to the Biot theory, 0, is not in general equal to the degree of settlement US. However, comparison with analytical solutions for a semi-infinite mass (de Jong, 1957 ; McNamee and Gibson, 1960) reveals that, with the time factor defined in terms of cvar UP for V' = 0.5 is always likely to be less than U, for any value of v', the difference being at most O-16 and generally considerably smaller. For shallow layers 0, must be almost identical with U, since for one-dimensional conditions the two degrees of consolidation are exactly equal for all values of v'. For such layers, cVg in the definition of time factor in Fig. 10 should be re- placed by c,r. Thus the curves for oP in Fig. 10 can be regarded as lower bounds to the

TIME FACTOR T,

0.0001 0001 DOI @I

Fig. 10. Three-dimensional effects on rate of consolidation

degree of settlement U, for all v', and they should at least indicate the general effects of three- dimensional conditions on rate of settlement with reliability.

Discussion on the improvement in the above approximate approach that can be obtained by making a simple arbitrary allowance for the change in the bulk total stress @ during consolidation, will be included in a further paper.

MODEL FOOTING TESTS

To determine the order of accuracy with which settlements and rates of settlement can be predicted under conditions which are as close as possible to ideal, a series of model footing tests was carried out. In these tests, the geometry, hydraulic boundary conditions and the initial state of stress in the soil could be controlled with much greater precision than they could be assessed for any actual foundation in the field.

a2 E. H. DAVIS AND H. G. POULOS

PORE WATER SYSTEM I CELL WATER SYSTEM

C@xrc WlUrn.? c@“9C

Apparatus

Fig. 11. Layout of apparatus for model footing tests

The layout of the apparatus used in these tests is shown in Fig. 11 and the details of the pressure vessel in which the tests were actually carried out are shown in Fig. 12. The vessel consisted of three sections, constructed of brass; a centre section D of 12 in. internal diameter in which the soil specimen A was placed (see Fig. 11), a base section F, and an upper section E in which water under pressure was contained. The upper and centre sections were separated by three rubber membranes each with silicone grease between, and the footing B was placed in position on the soil beneath the membrane. To allow drainage from the upper surface, a sheet of 100 mesh gauze and a sheet of filter paper were placed on top of the soil and directly above the footing. The gauze and filter paper were in contact with a pocket of sand, C, from which connexions led to the mercury control, M, and volume gauges N and 0. Drainage from the base, when required, was obtained by means of gauze and filter paper, the expelled water passing through cocks g, e and f to the volume gauges N and 0. Leakage between each section

Fig. 12. Details of pressure vessel


was prevented by 13 in. dia. O-rings, the upper of which also served to hold the rubber membranes in place. The footing B was loaded through plunger G by means of dead loading on hanger I and the consequent settlements were recorded on dial gauge H.

The ancillary equipment was divided into a pore pressure system and a cell water system. Details of these systems are similar to those described in the triaxial tests by Davis and Poulos (1963).

Test procedure

The main body of the pressure vessel was filled with well slurried clay at a moisture content a few per cent below the liquid limit. After installation of the footing and assembly of the vessel, a pressure was applied to the upper section and the soil bed allowed to consolidate. The vessel thus acted as a large oedometer and a homogeneous bed of normally consolidated clay was produced. After completion of consolidation, and with the consolidation pressure remaining unaltered, the footing was loaded and its settlement behaviour observed.

In interpreting the test results, calibration tests were made to determine what part of the directly measured settlement was due to compression of the rubber membrane, the gauze sheets, the plunger and the footing itself. This apparatus correction was found to be between 5 and ISo/, of the measured settlements.

Summary of test results

Model footing tests were made on two types of remoulded normally consolidated clay: kaolin (LL = 55, PL = 33, K, = O-51, c/p = O-44), a clay of relatively high permeability, and Hurstville Clay (LL = 43, PL = 27, K, = 0.51, c/p = 0*46), a clay of much lower permeability than kaolin. In each test, the soil was initially consolidated under K, conditions to an effective pressure of 30 lb/sq. in., with a back pressure of 20 lb/sq. in. on the pore water system.

In keeping with common foundation practice, the load on each footing was fixed at not more than one third of its ultimate bearing capacity. This value of undrained bearing capacity was calculated from values of c,(+, = 0) determined from undrained triaxial tests. The footing sizes, applied footing pressures and factors of safety against general undrained failure and local yield in these footing tests are summarized in Table 1.

A typical experimental time-settlement curve is given in Figs 13 and 14, which show also the method of interpreting the results. The observed settlements from the footing tests are summarized in Tables 2 and 3. For the tests on Hurstville Clay, and the 5 in. footing tests on kaolin, the values given are the mean of two tests which were nominally identical except that different hydraulic boundary conditions were imposed in each test. For the one and two inch footing tests on kaolin, repeat tests were made for the boundary conditions of permeable upper

Table 1. Summary of footing sizes, applied footing pressures and factors of safety in model footing tests

Footing Applied footing Factor of safety Factor of safety Soil diameter pressure against general against local

in. lb/sq. in. undrained failure yield

Kaolin 1

20 4.1 0,97 1.94

5 3.79

Hurstville Clay 1 1 I 20 I 4.3 I 1.11 2 15 5.8 1.47

E. H. DAVIS AND H. G. POULOS

TIME : MIN.

@I I IO 100

Fig. 13. Results of model footing test on Hurstville Clay

vGG ( Si) o”

., 2 / 4 6 8 10 72 .i

I

Fig. 14. Results of model footing test. Root time plot

Table 2. Comparison between observed and predicted settlements for model footing tests on normally consolidated kaolin. Overburden pressure = 30 lb/sq. in.

Footing Applied diameter footing

in. pi-l%XXIY Ib/sq. in.

1

& s CF ~~

0~0190 0.0143 0.0177 0~0092 0.0106 0.0076

_____-


Table 3. Comparison between observed and predicted settlements for model footing tests OP normally consolidated Hurstville Clay. Overburden pressure = 30 lb/sq. in.

Footing Applied diameter footing

in. pressure

Observed values Three-dimensional (in.) displacement method

_~

-

Predicted values (in.)

Skempton and Bjerrum’s

method

S” s CF S TF S CP j S*, _______ 0.0152 0.0181 0.0333 0.0150 0.0302 0.0170 0.0240 0.0410 0.0181 0.0351

L

Conven- tional

method

S osll

0.0267 0.0312

surface and impermeable layer base only, as the rate of settlement would have been too rapid for accurate time-settlement measurements if a permeable layer base had been used. The tests were quite consistent within themselves, the greatest variation between S,, from two similar tests being about 3% of the mean value.

Comparisons between observed and firedieted settlements

The settlements were predicted by the proposed three-dimensional displacement method and these predictions, together with those obtained from Skempton and Bjerrum’s method and the conventional method of analysis, are compared with the observed values in Tables 2 and 3. The soil parameters used in the settlement predictions were determined from triaxial tests of the type described by Davis and Poulos (1963). The representative stress ranges for the three footings did not differ greatly and no very significant difference could be detected in the results of the triaxial tests. The mean values from all tests (8 for kaolin, 5 for Hurstville Clay) were used for settlement predictions. These values are given in Table 4.

In calculating the theoretical settlements, the footings were taken to be rigid, and the approximation to the settlement of a rigid footing was employed, using the sector curves obtained by Poulos (1967a) to calculate the centre and edge settlements of a flexible footing on the surface of a finite layer. For settlement predictions by the conventional method and Skempton and Bjerrum’s method, it was thought desirable to use the parameters and influence coefficients which would normally be used in such calculations, in order to make a fairer comparison with the settlements given by the three-dimensional method. Therefore, displacement influence factors were calculated from the Steinbrenner approximation since it (or its equivalent, the integrated Boussinesq stresses), would normally be used for both methods of calculation.

Table 4. Soil parameters used in settlement predictions for model footing tests

Kaolin Hurstville

Clay

Initial stress Stress state increment

lb/sq. in. lb/sq. in.

0; 1 oj do; / 00;

30.5 4.5 30.5 4.0

El E' lb/sq. in. lb/sq. in

315 272 640 370


Fig. 15. Summary of comparisons between observed 8 TF in model footing tests and values predicted by three-dimensional displacement method

a Tests on nurrtvlllr Clay Case PTPB ‘Q-“” II ” PTIB

0 Tests .n Kaolin csso PTPB 1, I * u * PTIB

Fig. 16. Surnmary of comparisons between observed S,, in model footing tests and values predicted by Skempton and Bjerruxn’s method

l-01 0 I 0 1 2

h/a 3 4 5

Al Tast on Hurstvilla Clay Casa PTPB E) ,I II PTIB 0 a 11 Kao;;n Cm% PT;B A ,I I, II II PTIB

Fig. 17. Summary of comparisons between observed S, in model footing tests and values predicted by three-dimensional displacement method


From Tables 2 and 3, the following comments may be made :

(i) For the tests on kaolin, the discrepancy between the experimental values of S,, and those predicted by the three-dimensional displacement method ranges between 13 and 5%, while for the tests on Hurstville Clay, the range of discrepancy is + 12 to -3%. The comparisons between the observed and calculated values of S,, are summarized in Fig. 15 in which all tests are plotted. The limits of agreement of the predicted values are clearly shown as 115 to 95% of the observed values.

(G) For the tests on kaolin, Skempton and Bjerrum’s method overestimates S,, considerably, while for the Hurstville Clay, it slightly overestimates S,, for the one inch diameter footing, but underestimates S,, for the 2 in. diameter footing by about 17%. The comparisons between the observed S,, and values calculated by this method are shown in Fig. 16.

(iii) The ratio of predicted to observed immediate settlement S, for all tests is plotted in Fig. 17. In every case, the predicted value is greater than the observed, the ratio of the two values being between 1.03 and 1.73. For the one inch footing on kaolin, this result is somewhat surprising since the factor of safety against local yield in this test was slightly less than one and it might have been expected that an analysis based on elastic theory would therefore underestimate the settlement. However, the effects of local yield were probably quite minor compared with discrepancies between prediction and observation due to different amounts of undrained creep in the triaxial and footing tests. In the undrained phase of the triaxial test, the strain for the calculation of E, was taken when the pore pressure had become stationary. This time was of the order of 80 min for the kaolin and 1800 min for the Hurstville Clay. In this time, a considerable amount of undrained creep would have occurred since the applied deviator stress was 70% of the ultimate for the kaolin and 63% for the Hurstville Clay. In the footing tests, the same amount of creep could not have occurred under undrained conditions since primary consolidation was virtually complete within the above times. In fact, the method used to determine E, was more appropriate to full-scale foundations than to small- scale laboratory footings.

(ia) Since the calculated values of S, are greater than the observed, the values of SCF calculated by the three-dimensional displacement method are correspondingly somewhat lower than the observed values in all cases. In contrast, Skempton and Bjerrum’s method overestimates So, for the tests on kaolin, but underestimates Se, for the tests on Hurstville Clay.

(n) In all cases, the settlement predicted by the conventional method of analysis lies between the observed S,, and STF values.

Com$arisons between predicted and observed rates of settlement

Detailed comparison between theoretical and experimental rates of settlement is outside the scope of this Paper and it is intended to deal with them in a further paper specifically concerned with rates under three-dimensional conditions. It is sufficient to report here that the agreement between theoretical and experimental rates was quite good in all cases. A typical comparison is shown in Fig. 18. In general, at the theoretical time for 50% consolidation settlement, the experimental value lay between 40 and 60%. In obtaining theoretical rates of settlement, a rigorous solution of the Biot equations was not attempted but the approximate approach discussed in the section on rate of settlement, with the added improvement of an approximate allowance for stress redistribution during consolidation, was used.

Tests using a constant rate of settlement

In addition to the constant load tests described, some tests were made in which the footing was subjected to a very slow constant rate of settlement, i.e. deformation of the soil occurred

88 E. H. DAV

TIME : MIN.

0.1 I 10

,H

Impcrmcablc base Permeabl; top

-Ewerimmral - - - Predicted

Fig. 18

Fig. 18. Comparison between predicted and observed rate of settlement in model footing test

Fig. 19. Test at constant rate of settlement



S AND H. G. POULOS

2’ dlu. fmtlng on Kaolin

Fig. 20

Fig. 19

2 ;

8 t

2”dia.footing co Hurstv~lle Clay


entirely under drained conditions. The aim of these tests was to confirm that the magnitude of S,, is the same whether the load is applied instantaneously or so slowly that no appreciable excess pore pressures are developed. Under the latter conditions, the observed movement of the footing at any time is the total final settlement. The load-settlement curves obtained from these tests are shown in Figs 19 and 20, together with the values of S,, from the constant load tests. The value of S,, from the constant load test is in both cases slightly greater than the value for the same footing load in the test made at a constant rate of strain. Nevertheless, the agreement between the two values of S,, in each test is quite satisfactory, the discrepancy being about 10% of the mean value for the test on kaolin and 2% of the mean value for the test on Hurstville Clay. This agreement suggests that the effect of any local yield within the soil mass in the constant load tests was not significant.

A remarkable feature of Figs 19 and 20 is the linearity of the relationship between load and settlement over the entire range of load. A further test was made on a 4& in. diameter footing (Fig. 21) in which the applied pressure had risen to 576 lb/sq. in. before the limit of travel of the apparatus was reached. Even at this very high applied pressure, the load-settlement behaviour of the footing was still substantially linear. However, the settlement of the footing at an applied stress of 576 lb/sq. in. was about O-3 in., i.e. the footing had penetrated into the soil to a depth of one eighth of the original thickness of the layer, or O-6 times the diameter of the footing. Although & and cd for each soil were not measured, taking possible values of I$& = 30”, cd = 0 for both soils, the ultimate bearing capacity under drained conditions was found from Terzaghi’s theory to be 625 lb/sq. in. for a 4 in. diameter footing and 700 lb/sq. in. for a 2 in. diameter footing.

Although this aspect of the experimental work was not pursued, the tests described did indicate that, under conditions of very slow loading, soil can behave overall as a linear elastic solid and the results go some way in refuting those who are sceptical of any use of elastic theory for soil.

CONCLUSIONS

(i) The proposed method of predicting settlements makes extensive use of elastic displacement and stress distribution theory. The use of this theory for such a non-elastic material as soil is justified, to the accuracy required in practical problems, provided the ‘elastic’ soil parameters are determined over a range of stress representative of the changes in stress state occurring in the actual problem, and provided the foundation load at which local yield first occurs is not seriously exceeded.

(ii) In three-dimensional problems Poisson’s ratio, v’, of the soil skeleton has a major influence in determining the error in the conventional one-dimensional estimate of total final settlement. When V’ is small the error is also small. As V’ approaches O-5 the errors increase rapidly.

(ii;) The relative importance of the immediate settlement as part of the total final settlement depends markedly on the value of V’ as well as on the geometry of the problem.

(iv) The effect of three-dimensional conditions is most important in relation to rate of consolidation settlement. For deep beds of clay the time for a given degree of settlement may be only a small fraction of the time given by one-dimensional theory.

(v) For a given factor of safety against ultimate bearing capacity failure, the possibility of increased immediate or undrained settlement due to local yield depends primarily on the initial state of stress. With overconsolidated soils, this possibility appears to be remote unless the soil is so heavily overconsolidated that the initial stress state is close to that of passive failure. With some normally-consolidated clays, local yield may occur at footing pressures as low as 10% of the ultimate bearing capacity. Thus research into the extra immediate settlement due to plastic redistribution of stress assumes considerable practical importance for such soils.


(vi) Laboratory tests on model footings showed satisfactory agreement between the observed total final settlement and that predicted by the proposed method.

(vii) The agreement between the observed and predicted immediate settlements of the model footings was only fair. This was largely due to difficulty in obtaining reliable values of the undrained modulus E,.

(viii) Both the soils used in the model footing experiments showed negligible secondary consolidation under the one-dimensional conditions of the oedometer. However, in the triaxial consolidation tests they both showed marked secondary consolidation or creep in the axial strain after excess pore pressures had dissipated. This long term creep was reproduced in the settlement behaviour of the footings and indicates that, under three-dimensional conditions generally, creep or secondary consolidation may be much more significant than is sug- gested by oedometer test results.

ACKNOWLEDGEMENTS

The major part of the work described in this Paper was made possible by the award of an Australian Commonwealth Research Scholarship to Dr Poulos. The Authors did the computational work on SILLIAC within the Basser Computing Department of the School of Physics, Sydney University. The help and advice of Mr Harry Taylor, Executive Officer of the Austra- lian Road Research Board, with several aspects of the elastic analysis is gratefully acknow- ledged. The Authors also thank Mr J. L. Clift for help in constructing the apparatus used in the model footing tests.

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