Rowe and Booker (1981) CGJ 18(2) 250-264

The behaviour of footings resting on a non-homogeneous soil mass with a crust. Part I. Strip footings

R. K . ROWE Faculty of Etlgineerit~g Scierrce. University of Wester11 Otlturio, Lot~ljotl, Otlt., Catlada N6A 5B9

AND

J . R. BOOKER Scllool of Civil Etrgineeritlg, Utziversity of Sydr~ey, Sydtley, A~rstralia

Received June 30, 1980 Accepted December 30,1980

A convenient and economical finite layer method of analysing the behaviour of footings on a non-homogeneous soil is outlined. This method of analysis is used to investigate the elastic response of strip footings for a wide range of soil profiles involving a weathered crust overlying a main deposit. The stiffness of the crust is consiclered to be either constant or to decrease with depth while the stiffness of the underlying deposit increases with depth.

The effect of layer depth, crust depth, and the rate of variation of stiffness within the crust and the underlying soil are examined. The influence of this non-homogeneity upon settlement profile, differential settlement, and consolidation settlement are disc~~ssed and the results obtained from this detailed analysis are compared with results obtained from the approximate use of existing homogeneous elastic solutions.

The results of this sttidy are presented in the form of influence charts which may be used in hand calculations to estimate the settlement of footings for a wide range of practical cases. The applicability of the solutions and the determination of soil parameters is briefly discussed.

L'article prCsente une mCthode h couches finies pratique et Cconomique pour analyser le comportement de semelles sur ttn so1 non homogkne. Cette mCthode d'analyse est utilisie pour Ctudier la rCponse Plastique de semelles filantes sur un large Cventail de profils geotechniques comportant une croDte altCrCe sur un dCpBt principal. La rigidite de la croDte est supposie constante ou dkcroissante avec la profondeur, alors que la rigidite du dCpBt sous jacent aug- mente avec la profondeur.

Les effets des Cpaisseurs de co~~ches et de croi3te et du taux de variation de la rigidit6 dans la croDte et le sol sous jacent sont CtudiCs. L'influence de cette non homog6nCitC sur le profil de tassement, le tassement dityirentiel et le tassement de consolidation est discutCe et les resultats obtenus de cette analyse ciCtaillCe sont compares avec les rCsultats dicoulant de l'usage de solutions Clastiques homogenes approchkes.

Les rCsultats de cette Ctude sont prCsentds sous la forme d'abaques d'influence qui peuvent Stre utilises dans des calculs manuels pour Cvaluer le tasse~nent de semelles dans un grand nonlbre de cas pratiques. L'applicabilitC des,solutions et la dktermination des paramktres du sol sont brikvement discuties.

[Traduit par la revue] Can. Geotech. J., 18,250-264 (1981)

1. Introduction The behaviour of surface footings resting upon a

homogeneous soil mass has been extensively studied and published solutions (e.g., Poulos and Davis 1974) have found wide application for estimating the displacement of shallow foundations. It has been recognised that the assumption of homogeneity does not truly represent many soil profiles and some attempt has been made to analyse the behaviour of footings resting upon a soil whose modulus increases linearly with depth. This problem has been studied analytically by Gibson (1967) and Brown and Gibson (1972) who have considered a loaded half-space, while Gibson et al. (1971) and Brown and Gibson

(1979) have considered the behaviour of a finite layer whose modulus increases linearly with depth. Others (e.g., Carrier and Christian 1973) have performed parametric studies for circular footings on a soil with a linearly varying modulus using the finite element technique.

A problem of some practical importance to geotechnical engineers is that of predicting the settlement of foundations resting on a weathered crust which overlies the main soil deposit. Typically, the soil stiffness will decrease with depth in the crust and then increase with depth below the crust (e.g., Lo and Becker 1980). The simplest idealisation of this soil profile is that of an elastic soil whose modulus varies

0008-3674/81/020250-15$01.00/0 @ 1981 National Research Council of Canada/Conseil national de recherches du Canada

ROWE AND BOOKER: I

Frc. 1 . Soil profiles for parametric study.

linearly with depth within both the crust and the main deposit as shown in Fig. 1. This non-homogeneous elastic profile could be analysed using finite element techniques; however, an accurate study would require very careful attention to the details of the finite element mesh and would be expensive. No comprehensive study appears to have ever been attempted.

In this paper an alternative approach to the analysis of non-homogeneous soils is outlined. This approach, which has been coded in the program FLANS (finite layer analysis of non-homogeneous soils), provides a convenient means of obtaining accurate solutions using relatively little input data and is computationally more economical than the finite element method. This technique has been used to perform a parametric study of the effect of non- homogeneity upon the settlement of surface footings. Because of the number of variables involved, this study will be presented in two parts. In this paper, attention will be restricted to strip footings resting on a non-homogeneous isotropic soil mass. A subsequent paper (Rowe and Booker 1981) will consider the behaviour of circular footings on both isotropic and anisotropic soil deposits.

E X P O N E N T I A L APPROXIMATION rc FINITE L A Y E R S W H I C H N E E D NOT BE S M A L L

FIG. 2. Typical non-homogeneous soil profile with exponential approximation.

The results from this study are presented in the form of influence charts which may readily be used in hand calculations for estimating the settlement of footings resting on a non-homogeneous soil. The use of these influence charts is illustrated by an example in the second paper.

2. Method of Analysis The behaviour of elastic horizontally layered soil

deposits can be analysed efficiently using finite layer techniques (Peutz et al. 1968; Wardle and Fraser 1975; Cheung 1976). These methods are most suited for application to deposits which consist of several distinct layers, and deposits which vary continuously must be approximated by a "staircase" profile. Re- cently, the authors (Rowe and Booker 1980) have developed a method which is better suited to the treatment of a continuously varying deposit. In this method it is assumed that the deposit may be divided into a number of distinct layers z,; 5 z 5 zk+~ in which Poisson's ratio v/; is assumed to be constant and the elastic modulus is assumed to vary exponen- tially with depth

[ I ] E = Qk exp [2+g(z - zg)]

where Qk is the modulus at z = zk and L J . ~ describes the variation of modulus with depth.

The adoption of this exponential variation in modulus within a layer significantly simplifies the form of the governing equations and permits the use of a finite layer technique. The exponential form given by [l] may be used to approximate any variation in modulus with depth by appropriate selection of layer depths and the parameters Qs and pk. An example of this approximation for a material whose modulus varies linearly with depth is shown in Fig. 2.

This method, like the conventional finite layer

252 CAN. GEOTECH. J. VOL. 18, 1981

approach, makes use of a Fourier transform to tions it is found that the displacements may be represent the displacement and stresses as expressed in the form

Hookes law may be used to express the transformed stress components S,,, S,,, S,, in terms of the transformed displacements U,, U,, U,

where pj and V j are known functions of the soil parameters (details of their calculation are given by Rowe and Booker (1980)) and E j are 6 constants t o be determined.

The constants E j may be calculated in terms of the 6 node plane displacements Uk, Uk+l. Using these expressions [4] may be substituted into [3] to obtain the stiffness relation

ox: Szz = - (1 f vk)

ox: where Th = +(S,,, S,,, S,,),T is the traction on the s,, = - (1 + vx:) exp L2p*(~ - node plane z = Z* ; T*+l = - (S,,, S,,, is the

DI traction on the node plane z = ~ l ; + ~ ; Uk,k+~ =

Slk exp [2pk(z - 2k)I (U,, U,, U,)k,k+lT are the displacements on the node Szz = planes z = z*,~+I; and the matrices Ah, Bk, Cx: are

(1 + vlc)(l - 2vk) all known. The conditions of displacement compatibility and

uv*u3 + PvkU, + (I - traction equilibrium at the node planes may now be used to obtain the behaviour of an n layer system and

If eqs. [2] are substituted into the equilibrium equa- it is found that

where for simplicity it has been assumed that the layered system rests on a rough rigid base and is subject to surface loading.

Equation [6] may now be used to calculate the node plane displacements Ul, . . . Un for any value of the integration parameters a, (3 and so the actual displacements can be obtained by evaluating [2a] numerically. The stress components may be calculated similarly. (For plane strain or axi-symmetric conditions, the double integration indicated in [2] may be replaced by a single integration in terms of a.)

3. Numerical Details and Validation of the Analysis he method of analysis is essentially analytic in

nature; however, some numerical approximations are involved. The first approximation arises from the use of an exponential function to represent the linear increase in modulus with depth. The accuracy of this approximation depends upon the variation in modulus between the top and bottom of each layer. If the ratio is considered to be too large for a particular layer then the layer may be split into sublayers. The results presented in this and the subsequent paper

ROWE AND BOOKER: I 253

were obtained for a maximum ratio of a modulus between the top and bottom of each layer of 1.25. This ensures that the error in the representation of the soil modulus was always less than 0.4%. This is a very conservative numerical restriction but was adopted to ensure consistently accurate results. In practice one may perform analyses for a modulus ratio of 1.5 without appreciably altering the results and a much higher modulus ratio may be adopted without significant error for layers more than 2 footing widths (diameters) below the footing.

The second approximation arises from the necessity to perform the integrations of [2] numerically. Gauss quadrature was used to perform these integrations. For the plane strain (strip footing) and axi-symmetric (circular footing) analyses reported in this and the subsequent paper, the integrations were performed between the limits of -60B < a < 60B. Further increasing the range of integration had no appreciable effect upon the results and a much smaller range would have been adequate in many cases. In general, Gauss quadrature involving the use of a polynomial of order 19 was adequate although in some cases it was necessary to use a polynomial of order 39 or, alternatively, reduce the integration subinterval.

The analysis was validated by comparison with published solutions and the results of the authors' finite element analyses. Good agreement was obtained with the published results for footings on a homogeneous soil (e.g., solutions published in Poulos and Davis (1974)) and for footings upon a soil with a linearly increasing modulus with depth (e.g., Brown and Gibson 1979). Agree~llent to better than lY0 was also obtained between this method and finite element calculation for a number of non-homogeneous soil profiles. It should be noted that the finite element results required considerably greater data prepara- tion time and were comparatively more expensive. The advantage of the authors' approach over finite elements increased rapidly with increasing layer depth.

4. Soil Profile Corlsidered Many soil deposits consist of a weathered, possibly

overconsolidated crust overlying the main deposit (e.g., Soderman et al. 1961; Lo and Becker 1980). Typically, the crust stiffness may be constant or may decrease with depth while the stiffness of the soil below the crust will increase with depth. It is convenient to assume that the Young's modulus of the soil varies linearly with depth in each of these layers. Thus for an isotropic soil, Young's modulus E(z) at a depth z in the crust is given by the expression

E(z) = Eo - pcz for z < z,

where EO is Young's modulus at the soil surface; p, is the rate of decrease of modulus with depth (within the crust); and z, is the depth of the crust.

Below the crust, Young's n~odulus is given by

E(z) = Eo - pczc + p(z - z,) for z > z, where p is the rate of increase in modulus with depth (in the main deposit) and the other terms are as defined above.

It is assumed that Poisson's ratio Y is constant in both layers. The general profile described above is illustrated for a deposit of depth D in Fig. Ic. For the special case where there is no crust (2, = 0) this profile reduces to a "Gibson soil" as shown in Fig. I(/. Other limiting cases of some interest involve, firstly, a weathered crust underlain by a relatively homogeneous soil (p = 0) as shown in Fig. le and, secondly, a homogeneous crust (p, = 0) underlain by a soil whose modulus increases with depth as shown in Fig. If.

For a given footing of width B, the crusted soil profile described above may be characterized by: (1) the dimensionless crust depth zc/B; (2) the dimensionless decrease in stiffness with depth in the crust, p,B/Eo; and (3) the dimensionless increase in stiffness with depth below the crust, pB/Eo.

The variation in stiffness with depth may be non- dimensionalised with respect to the modulus at any selected depth below the footing provided that this reference depth is used consistently. However, it is most convenient to take the reference depth to be just below the footing and to non-dimensionalise the stiffness variations with respect to the corresponding "surface" modulus Eo. It is recognised that in practice it may be difficult to determine this modulus directly; however, in these cases the value of EO may be esti- mated from the measured modulus EL at some depth zk and the measured changes in stiffness with depth (p and p,). Thus Eo = EL + pzh provided that zl, < z,.

This "surface" modulus is a direct consequence of the assumed linear variation in modulus with depth and its calculation in this manner does not introduce any additional error; the errors involved in the soil idealisation are independent of the choice of reference modulus. Clearly, as in any geotechnical analysis, judgement must be exercised in the selection of appropriate soil parameters.

The values of the soil parameters p, p,, and Eo will depend on the type of soil, its stress history, and the rate of loading. For footings on clay, one would normally expect both drained and undrained values of Eo in the range from 1 to 40 MPa, p, in the range from 0 to 5 MPa/m, and p in the range from 0 to 2 MPa/m (for drained analyses, p will generally lie


between 0.2 and 0.8 MPa/m). The geometry of the footing is highly variable with values of B typically ranging from 0.5 to 20 m. From physical arguments, it follows that pcB/Eo must be less than BIZ,. Con- sidering typical values of p, B, and Eo, it can be seen that the range of pB/Eo from 0 to 5 corresponds to many practical situations. (Cases where pB/Eo is greater than 5 may be envisaged but will be less fre- quently encountered for a crusted clay profile.) It is noted that the parameter p is usually not subject to wide variations and that the range of values for pB/Eo largely reflects variation in geometry and crust stiffness.

An example of a crusted soil profile reported by Lo and Becker (1980) for a site at Wallaceburg, Ontario, has typical parameters Eo = 25 MPa, p, = 2.8 MPa/m, z, = 6.75 m, and p = 1.4 MPa/m.

5. The Settlement of a Smooth, Uniformly Loaded Footing on a Non-homogeneous Soil

In this section consideration will be given to the settlement of a uniformly loaded, smooth strip footing. The settlement of rigid footings will be discussed in a subsequent section.

The entire surface displacement profile (including the central displacement) caused by a uniformly loaded strip may be deduced by superposition of elastic solutions for the settlement at the edge of the footing. Conversely, the edge settlement of a footing of width B may be shown to be one half the central settlement calculated for a footing with a (ficticious) width 2B. Thus solutions need only be given for the central settlement of the strip footing. Edge settlements may be deduced from the central settlement solutions as described above.

The settlement of a footing on a non-homogeneous soil can be conveniently expressed in terms of the settlement of an equivalent footing on a homogeneous soil mass, multiplied by correction factors for the effect of non-homogeneity. Thus the settlement 6, at the centre of a uniformly loaded strip resting on a non-homogeneous soil of depth D and with Poisson's ratio v would be given by

[71 6c = (qB/Eo)IcRc

where q is the pressure acting over a width B; Eo is Young's modulus at the soil surface; I, is the influence factor for the central settlement of a footing on a homogeneous soil of depth D and Poisson's ratio v; R, is the correction factor for the effect of non-homogeneity and is defined by the ratio: R, = (centre displacement for a non-homogeneous soil, v = 0.3)/(centre displacement for a homogeneous soil, E = Eo, v = 0.3).

CENTRE

FIG. 3. Centre settlement factor for a uniformly loaded strip on a homogeneous soil.

Thus the correction factor R, reflects the increase or decrease in settlement for a non-homogeneous soil relative to the settlement of a homogeneous soil with a reference modulus equal to Eo.

The solutions for the settlement of a strip on a finite homogeneous layer are well known (e.g., Poulos 1967); however, for conlpleteness, the influence factor I, for the centre displacement is shown in Fig. 3.

5.1. Soil Crust with Decreasing Stzffness with Depth Unclerlain by Homogeneous Material

The behaviour of a footing on a non-homogeneous soil will depend upon the depth of the crust and its rate of decrease in stiffness with depth, as well as the variation in stiffness of the soil beneath the crust. In this section the influence of the soil crust will be isolated by considering the material below the crust to be homogeneous, as indicated in Fig. le.

The correction factor R, given in Figs. 4 and 5 shows the effect of a soil crust upon footing displacement for a range of layer depths. The results given in Fig. 4 are relevant to problems involving wide footings or shallow crusts (i.e., z,/B ( 1). The results in Fig. 5 are for relatively narrow footings on crusts which typically extend to a depth of more than one footing width. The correction factor is greater than

ROWE AND BOOKER: I 255

FIG. 4. Non-homogeneity correction factor R, for centre displacement. Shallow non-homogeneous crust on an otherwise homogeneous deposit.

unity in all cases because the solutions have been normalised with respect to the maximum modulus of the soil.

The influence of the crust upon the footing response is relatively insensitive to the layer depth for crusts which extend to a depth less than half a footing width (z,/B < 0.5) and for crusts which show only slight reduction in stiffness with depth (i.e., p,B/Eo < 0.25). As might be expected, the response of footings on these soils is governed by the modulus of the soil below the crust and a typical engineering approximation, ignoring the crust and assuming the soil to be completely homogeneous with a modulus E = Eo - p,z,, would give results to an accuracy of better than 10% in these cases.

Footings on soil with a crust extending to a depth of more than half a footing width are noticeably affected by the nature of the crust. The influence of the crust is greatest for footings on shallow layers and decreases appreciably with increasing layer depth.

In situations where there is a weathered crust extending a significant distance below the surface, a

FIG. 5. Non-homogeneity correction factor for a soil with a deep non-homogeneous crust on an otherwise homogeneous deposit.

common approximate approach using existing homogeneous solutions would be to determine the soil modulus at a representative depth z, which lies in the range 0.25-0.330 up to a maximum depth of 0.9B (Davis and Poulos 1968). This representative modulus would then be used in conjunction with the solutions for a homogeneous soil. Under many cir- cumstances this approximate approach would yield reasonable settlement predictions; however, success- ful application of this approach requires considerable judgement in selecting a suitable "representative" modulus. A more satisfactory approach would be to predict the settlement using [7] in conjunction with Figs. 3-5. The correction factor R, is given for a number of combinations of crust parameters. To sufficient accuracy, the values of R, for intermediate combinations of crust parameters may be obtained by linear interpolation.

The results presented in Figs. 4 and 5 are for Poisson's ratio v = 0.3. The effect of Poisson's ratio upon the value of R, was investigated and was found to generally be less than 7% relative to the value for v = 0.3. Thus to sufficient accuracy, the secondary


effect of Poisson's ratio upon the correction factor R, may be ignored.

5.2. Soil Crust Underlain by n Soil Whose StifSness Increased with Depth

The results of the previous section indicate that a soil crust may have an appreciable effect upon the elastic response of an otherwise homogeneous soil. These results are applicable to a number of practical situations; however, it is well known that the stiffness of many soils increases with the depth below the soil crust. Accordingly, in this section consideration will be given to the general case of a soil with a profile as shown in Fig. lc.

5.2.1. Effect of Layer Depth (D/B) Figure 6 shows the variation in the correction

factor R, with increasing layer depth for a number of crust parameters (z,/B; pcB/Eo) and underlying soil parameters (pB/Eo). Three special cases may be identified. The results shown for pcB/Eo = 0 are for a homogeneous crust (see Fig. If); the results for zc/B = 0 are for crustless soil (see Fig. Id); and the results for pB/Eo = 0 are for a homogeneous soil beneath the crust (Fig. l e ) as discussed in the previous section.

For a given crust, the importance of the rate of increase in stiffness with depth (pB/Eo) below the crust is a function of the layer depth. For example, consider the curves shown for p,B/Eo = 0.75, zc/B =

0.75. At a depth D = B the stiffness parameter pB/Eo alters the central footing displacement by less than 27% for pB/Eo < 20. However, as the layer depth increases the soil below the crust dominates the response of the footing. Thus for D = 10B there is more than a fourfold difference in settlement for stiffness parameters in the range 0 < pB/Eo < 20. It is noted that the edge settlement is even more sensi- tive to the value of pB/E~ and D/B than is the centre settlement just discussed.

In practice, it is sometimes difficult to define the "layer depth" for soil profiles which change gradually from a soft or firm soil to a very stiff soil or weak rock and it is of interest to see how important the definition of layer depth is for these cases. To study this aspect of behaviour, the ratio of settlement of a strip footing on a layer of depth D = 1OB to that footing on a layer of depth D = 2B was calculated for a number of soil profiles as indicated in Table 1.

For a homogeneous soil, it is well known that settlement increases significantly with the depth of the layer. However, an increase in stiffness with depth can be seen to greatly reduce the importance of layer depth. This is particularly so for soils with an appreciable crust. For instance, given a soil with a

TABLE 1. Variation in settlement with layer depth for a uniformly loaded strip ( y = 0.3)

Ratio of settlement for D / B = 10 to that for D / B = 2

PCB/EO zc/B PBIEO Centre Edge

crust (z,/B = 0.75, pcB/Eo = 0.75) a fivefold increase in layer depth from D = 2B to D = 10B increases the footing settlement by less than 30% for moderate increases in stiffness with depth (pB/Eo 2 1) and by less than 3y0 for a highly non-homogeneous soil (@B/Eo = 20). Clearly, for practical problems where there is a significant increase in soil stiffness with depth (pB/Eo > 1) the selection of a "layer depth" for use in settlement predictions is not particularly critical. However, layer depth is very important for soil profiles consisting of a crust underlain by soil with only a slight increase in stiffness with depth (i.e., pB/Eo < 1).

In the previous section it was found that reasonabte estimates of settlement could be obtained using elastic solutions for a homogeneous soil in conjunction with an appropriate "representative" modulus. However, this is not true for the more general soil profile considered here. The "representative" modulus may be defined as the modulus of the soil which would give the same central displacement of a footing on a homogeneous soil as that observed for the true non-homogeneous profile. The location of the representative depth at which this modulus should be determined varies appreciably with the choice of soil parameters and a poor estimate of the representative depth may lead to significant errors in predicted settlement. This approximate approach will give a good estimate of settlement for certain combinations of parameters, but this good fortune cannot generally be relied upon. It is perhaps worth noting that non- homogeneity not only alters the magnitude of settlement, but also changes the shape of the displacement profile. Thus the differential displacement predicted using homogeneous results will usually not be appropriate for a non-homogeneous soil even if the central displacements are identical. This will be discussed in more detail in a later section.

ROWE AND BOOKER: I

FIG. 6. Variation in R, with layer depth.

5.2.2. Eflect of Crust Depth (z,/B) values of p,B/Eo greater than 0.5 the increase in The effect of crust depth upon settlement factor R, settlement with increasing crust depth depends largely

is illustrated in Figs. 7 and 8 for layer depth of 2B and upon the nature of the soil below the crust (i.e., 10B respectively. The settlement of a footing in- pB/Eo). The variation in the actual magnitude of the creases with increasing crust depth in an almost correction factor R, with the depth of the crust is linear fashion for values of p,B/Eo of 0.5 or less. For significant for these soils and it would be unwise to


FIG. 7. Variation in R, with crust depth ( D / B = 2).

use solutions for a crustless non-homogeneous soil (e.g., Brown and Gibson 1979) if the footing does not rest directly upon such a soil; the influence of the crust should not be neglected.

5.2.3. Effect of Crust Stzffness (p,B/Eo) The influence of the crust stiffness upon settlement

is implicit in Figs. 7 and 8. The effect is greatest for deep crusts on a soil whose modulus does not increase appreciably with depth (i.e., low pB/Eo). In these cases the variation of R, with crust stiffness (p,B/Eo) is highly nonlinear; however, a reasonable estimate of R, can be obtained for intermediate values of pcB/Eo by linear interpolation between the nearest values given in these figures. Alternatively, more accurate results could be obtained by plotting the values of R, given in Figs. 7 and 8 against the parameter pcB/Eo.

5.2.4. Eflect of Underlying Soil Stzffness (pB/Eo) The effect of underlying soil stiffness (pB/Eo) is

partially apparent from Figs. 7 and 8; however, this parameter is quite important and consequently the variation in R, with pB/Eo is shown in Figs. 9 and 10

zc - B

FIG. 8. Variation in R, with crust depth ( D / B = 10).

for a number of cases. For a given soil crust, the settlement of a footing will decrease with increasing stiffness of the underlying soil. As previously indicated, the effect of pB/Eo depends upon the values of the crust parameters, but in general the major reduction in settlement occurs for values of pB/Eo in the range from 0 to 5.

The effect of increasing stiffness beneath the soil crust could be neglected; however, these results indicate that this would be extremely conservative for many cases. For instance, consider a 9 m wide footing on a soil with dimensionless parameters pcB/Eo = 1, z,/B = 0.75, pB/Eo = 0.5, D/B = 2. (This is an idealisation of a soil profile near Wallaceburg in Ontario (Lo and Becker 1980).)The crust is the major factor influencing the settlement of this footing; however, the neglect of even this small increase in stiffness with depth below the crust would lead to an overestimate of the central and edge settlements by 36 and 56% respectively. In a deeper layer, this small increase in stiffness of the soil beneath the crust would be even more significant.

5.2.5. Influence of Poisson's Ratio The nature of the response of homogeneous and

ROWE AND BOOKER: I

FIG. 9. Variation in R, with non-homogeneity for D / B = 2.

non-homogeneous soils is different, and this difference is reflected in the variation of the settlement correction factor R, with Poisson's ratio. This variation increases with the non-homogeneity of the soil beneath the crust (pB/Eo). For most practical cases, where pB/Eo is less than 10, the influence of Poisson's ratio upon the correction factor is relatively modest (generally less than 10%) and may be neglected. However, it should be noted that for values of pB/Eo greater than 10, the correction factor does change appreciably with varying Poisson's ratio.

5.2.6. Displacenzent Projle The results of the previous subsections suggest that

there may be a change in the nature of the settlement profile for different levels of soil non-homogeneity. This suggestion is further supported by direct consideration of the displacement profile for a number of cases. For example, Fig. 11 shows the surface dis-

placement caused by a uniformly loaded strip on a layer of depth 2B and 1OB for several soil profiles. This figure has been plotted for the case of the same surface nlodulus Eo but different crust depths z,/B and soil parameters p,B/Eo and pB/Eo. As might be expected, soils whose stiffness increases with depth exhibit lower levels of settlement than soils whose stiffness decreases with depth. Of greater interest is the shape of the different settlement profiles. To facilitate a direct comparison of the different deformation profiles, the results given in Fig. 11 have been normalised with respect to the central settlement and are shown in Fig. 12. (This is equivalent to assuming that Eo is selected for each soil such that the central settlement is equal to unity in each case.)

Considering firstly, a crustless soil with an increase in stiffness with depth, it will be noted that increasing the non-homogeneity (i.e., pB/Eo) tends to give a

CAN. GEOTECH. J. VOL. 18, 1981

FIG. 10. Variation in R, with non-homogeneity for D / B = 10.

more uniform settlement profile beneath most of the footing, but results in smaller edge settlements and much smaller settlements outside the footing. For this case, the differential settlement for DIB = 2 increases by up to 30% for stiffness parameter pB/Eo in the range from 0 to 20. The effect becomes more pronounced for deeper layers with an increase in differential settlements by 110% for D/B = 10 over the same range in stiffness. (Clearly, finite rigidity of the footing would tend to reduce these variations.)

Comparison of the displacements for a crustless soil with those for a soil with a homogeneous crust indicates that settlements are generally less uniform and the differential settlements are a little greater for the soil with a crust than they were for a soil with a similar value of pB/Eo but no crust. Further comparison of the results for a soil with a homogeneous and non-homogeneous crust shows that the nature

of the crust does not greatly affect the displacement profile (i.e., to less than 5%) for soils with an appreciable increase in stiffness below the crust. However, for soil with a crust of decreasing stiffness underlain by a relatively homogeneous deposit, increasing non- homogeneity of the crust (i.e., p,B/Eo) tends to give a more uniform displacement profile (i.e., lowers differential displacements) but this also corresponds to an increase in settlements which is particularly apparent at points outside the footing.

It is of some interest to look at the implications of the above findings with regard to the current, approximate, engineering approach of using elastic results for a homogeneous soil in conjunction with a representative modulus. Assume that for a non- homogeneous soil an appropriate representative modulus can be selected such that the central settlement is correctly predicted using homogeneous elas-

ROWE AND BOOKER: I 26 1

FIG. 11. Displacement profile due to a uniformly loaded strip.

TABLE 2. Ratio of rigid footing displacement to central displacement of a uniformly loaded strip ( v = 0.3)

Rigid displacement D/B pcB/Eo zC/Eo pB/Eo Central displacement

allows convenient determination of the average settlement of a uniformly loaded strip and also the settlement of a rigid strip. It is found that the average settlement of a uniformly loaded footing (i.e., half the centre and edge settlements) gives a reasonable prediction of the settlement of a rigid footing. The average settlement generally exceeds the rigid settlement by less than 8%. The discrepancy between the average flexible and rigid settlement is greatest for homogeneous soils or soils with a deep crust. The average settlement is very close to the rigid settlement for soils with a small crust and a high increase in stiffness with depth.

A comparison of the settlement of a perfectly rigid footing with the central settlement of a flexible footing for a range of soil profiles, indicates that a reasonable prediction of the rigid settlement would be obtained by taking it to be 84% of the central settlement of a flexible footing as given in the previous sections. The value of 0.84 represents the average factor for a number of cases. In fact, the ratio of rigid settlement to central flexible settlement depends upon the layer depth, the degree of non-homogeneity, and Poisson's ratio. Increasing the layer depth tends to increase this ratio, whereas increasing non- homogeneity tends to decrease the ratio as indicated in Table 2. The rigid settlement obtained by multi- plying the central settlement by 0.84 is considered to be accurate to within 7% for typical values of Poisson's ratio, zc/B, pcB/Eo, and pB/Eo.

tic results. Then the settlement predicted at other points using the same elastic modulus and homogeneous elastic solutions would differ from the true values in two respects. Firstly, homogeneous results would tend to underestimate the differential settlement of a perfectly flexible footing by up to 170% for the range of cases considered here. Sec- ondly, homogeneous results would significantly overestimate the settlement of points outside the footing for a highly non-homogeneous soil.

6. Settlement of a Rigid Footing on a Non-homogeneous Soil

To this point, consideration has been restricted to a uniformly loaded footing. The method of analysis

CAN. GEOTCH. J. VOL. 18. 1981

HOMOGENEOUS

HOMOGENEOUS

0

0 .8 /' HOMOGENEOUS

/

o l i ' r i h + i r c ' / - # -

-10 - PB - 0 .2 - z- - 20,.*- Eo ,, - - - Zc 6 0.4 - - = 0 . 7 5 .- - B / / I ,----

HOMOGENEOUS Eo 0.8 -

FIG. 12. (a) Displacement profile due to a uniformly loaded strip ( D / B = 2). (b) Displacement profile due to a uniformly loaded strip ( D / B = 10).

7. Applicability The solutions presented in the previous sections

were derived on the basis of an assumed linear-elastic material whose modulus varies linearly with depth. The applicability of these solutions to a particular geotechnical problem will therefore depend upon how accurately these assumed conditions represent the behaviour of the soil in question for the proposed loading. Potentially, the solutions could be used for estimating the immediate, undrained settlement and the final, drained settlement (neglecting secondary consolidation which, as is current practice, must be considered separately).

The use of the undrained modulus profile in conjunction with a value of Poisson's ratio Y, = 0.5 would give an estimate of immediate settlement. The relevance of this calculation to a particular situation

will depend upon the nature of the soil and the rate of loading. Clearly, if the rate of loading is slow with respect to the consolidation characteristics of the soil (e.g., see Leroueil et al. 1978) then the relevance of the immediate settlement calculations must be ques- tioned. For footings on soft soil, the occurrence of local yield could also result in settlements greater than predicted by elastic theory and in these cases it may be necessary to apply a local yield correction such as proposed by D'Appolonia et al. (1971).

The use of a drained modulus profile in conjunction with the drained Poisson's ratio would give an estimate of the final (excluding creep) settlement of a footing. The result obtained from this analysis will be most relevant for soils with an overconsolidated weathered crust whose stiffness decreases with depth, underlain by an overconsolidated main deposit

ROWE AND

whose modulus increases with depth. Consideration might also be given to the use of the solutions to obtain an approximate estimate of settlement when the main deposit is normally consolidated, provided that the stress changes in this deposit are not high and provided that the modulus values adopted are secant values determined over an appropriate stress range. Clearly, considerable engineering judgement must be exercised in such a situation.

In some cases. the soil modulus could be determined from field tests. More commonly, the parameters would be obtained from laboratory tests on undisturbed samples. The relevant parameters may be determined from KO consolidated triaxial tests performed over appropriate stress ranges (e.g., see Davis and Poulos 1963) on samples taken at various depths within the deposit. In general, such a detailed investigation would not be practical and in these cases an estimate of the modulus variation with depth may be obtained from conventional triaxial tests performed at a limited number of depths by corre- lating this modulus with the variation in shear strength with depth (e.g., from a field vane test) or with the variation in coefficient of volume decrease (from an oedometer test) with depth.

The form of the solutions presented in these papers is particularly suited for use in conjunction with normalised soil parameters (e.g., Ladd and Foote 1974; Ladd et al. 1977), provided that the nature of the soil is such as will allow the determination of normalised parameters.

In conclusion, it should be noted that any limita- tions regarding the use of non-homogeneous elastic solutions presented herein apply equally to the use of homogeneous elastic solutions and that the solutions presented in this paper do allow convenient consideration of a non-homogeneity encountered in many soil deposits which cannot be directly considered using existing elastic solutions.

8. Conclusions A convenient and economical method of analysing

the behaviour of footings on a non-homogeneous soil has been outlined. This analysis was used to investigate the elastic response of strip footings for a wide range of soil profiles involving a weathered crust overlying a deposit whose stiffness increases with depth. The results of this study are presented in the form of influence charts which may be used in hand calculations to estimate the settlement of footings for a wide range of practical cases. It is considered that these solutions would provide a convenient means of checking more elaborate analyses

and would provide the field engineer with a means of quickly assessing the importance of changes in soil profile. The solutions may also be used to indicate the sensitivity of the foundation behaviour to the accuracy with which parameters are determined.

A number of general conclusions regarding the effect of non-homogeneity upon footing response may be drawn from this study.

(1) The settlement of footings resting on a weathered crust underlain by a relatively homogeneous deposit is noticeably affected by the nature of the crust. It is found that approximate engineering ap- proaches using homogeneous elastic solutions in conjunction with a "representative" modulus may give errors in central displacement of f 30% for typical weathered crust profiles. Although these errors are not particularly large, a better estimate of the settlement may be obtained by use of the influence charts given in this paper.

(2) The settlement of footings on a weathered crust underlain by a deposit whose stiffness increases with depth is significantly affected by the nature of the crust and the rate of increase in stiffness below the crust. Even a relatively moderate increase in stiffness may have an appreciable influence upon the settlement profile. For these soils, approximate ap- proaches using elastic solutions for a homogeneous soil mass lead to large errors in the predicted settlement. In particular, the settlement profile and thus the differential settlement is highly dependent upon the changes in soil stiffness with depth.

(3) Definition of the actual depth of the soil deposit is important in predicting the settlement of footings on soils involving a weathered crust underlain by a homogeneous deposit. However, for soil deposits with a moderate increase in stiffness with depth, such as where there is a gradual change from soft-firm to very stiff soil or weak rock, the selection of "layer depth" for use in settlement prediction is not particularly critical.

(4) Published solutions for a crustless non-homogeneous soil should not be used if the footing does not rest directly upon the non-homogeneous material whose strength increases with depth. The influence of the crust between the footing and the main deposit should not be neglected.

( 5 ) A comparison of the results obtained for a uniformly loaded and rigid strip footing indicates that the rigid displacement may be taken to be 0.84 times the central displacement of a uniformly loaded strip. This ratio is an average value for a range of cases and is considered to be accurate to within 7% for typical values of crust and main deposit parameters.

264 CAN. GEOTECH.

Acknowledgements The work described in this paper was supported by

grant No. A1007 from the Natural Sciences and Engineering Research Council of Canada. The authors also wish to acknowledge the value of dis- cussions with Dr. K. Y. Lo and Dr. R. M. Quigley.

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