Settlement at working load is usually the critical design consideration for vertically loaded foundations on stiff overconsolidated clays. Foundation settlement is usually calculated by assuming that the ground behaves as a linear elastic material. The reliability of this calculation is mainly dependent on an appropriate choice of elastic modulus.
However, modern laboratory testing techniques have identified the highly non-linear stress strain characteristics of overconsolidated clays, for example, refer to Jardine et al 1984 1and Figure 1 2. This explains the wide range of values for 'linear elastic' moduli, which have been reported in the technical literature; for example E u/C uratios of between 150 and 1500 have been quoted in CIRIA SP 27 3. As a result the selection of an appropriate value for a linear elastic modulus becomes a matter of considerable engineering judgement. The recent Ground Engineering article 4exemplifies some of the difficulties that engineers face in selecting appropriate deformation moduli.
An additional problem is that linear elasticity incorrectly predicts the pattern of settlement adjacent to and beneath a loaded area. For example, if total settlement of a structure is correctly predicted, settlement at depth or remote from the structure will be incorrectly predicted. In view of these problems, the applicability of linear elastic calculations for overconsolidated clays is limited. However, the alternatives to conventional linear elastic calculations, such as non-linear finite element techniques, can be complex, expensive and time consuming, requiring high level expertise and considerable engineering interpretation.
Hence, there is a need for a simplified method which enables the engineer to gain an understanding of foundation deformation behaviour under loads of varying intensity. This paper describes a method which enables the non-linear stress strain behaviour of overconsolidated clays to be modelled in a manner which is relatively simple and is appropriate for routine design calculations. The calculations are most conveniently undertaken by computer, however, the method is readily amenable to hand calculation. Undrained and total settlement (or heave) can be calculated under foundation loading of any shape and of varying intensity using a varying ground stiffness - depth profile.
Background to proposed method
Field observations of ground movement During the 1970s, the BRE performed several large scale plate loading tests. Figure 2 shows settlement recorded beneath a 0. 9 m diameter plate test on London Clay. The data is shown in dimensionless form (subsurface settlement divided by plate settlement at founding level) and can be compared with the prediction of settlement within a homogeneous linear elastic medium. The observed subsurface settlements are quite localised, most settlement occurs within a depth of about 0. 6 times the plate diameter.
In contrast the linear elastic calculation predicts a significantly greater depth of influence. Figure 2 also shows normalised subsurface settlement recorded beneath a large building founded on glacial till of low plasticity 6. The pattern of settlement with depth is similar to that for the plate test on London Clay, with most settlement occurring within a depth of about 0. 6 times the building width. An interesting feature of this field data is that the building settlement increases by more than a factor of 2. 5, when bearing pressure increases by a factor of only about 1. 8. Also as bearing pressure is increased, normalised sub surface settlement becomes concentrated closer to the building foundation. Linear elastic calculations would not predict this pattern of behaviour.
Figure 3 summarises some settlement and heave observations for relatively large rigid structures founded on overconsolidated clays. Reviews of published case histories of settlement of buildings constructed on overconsolidated clays by, for example, Simons and Som 1970 7, and Morton and Au 1975 8indicate that settlement at the end of construction is about 60% of the total settlement. Good case histories of time dependent heave are, by comparison, relatively rare.
However, comparing the settlement case histories with those for heave, it is apparent that for comparable foundation type, ground conditions and net change in foundation pressure, time dependent heave due to unloading is greater than time dependent settlement due to loading. The fundamental difference in behaviour between the development of settlement and heave is highlighted by comparing the ratio of the end of construction movement to the time dependent movement, R = d u/d td . The settlement records typically indicate R to vary between 1. 4 and 2. 0. However, the records of heave indicate that R varies between 0. 4 and 0. 8. Although incomplete records of time dependent heave, the rates of heave described by Mettyear 9and Pierpoint 10 (for 24m and 10m deep excavations in London Clay and Oxford Clay respectively) provide additional evidence of the lower deformation moduli mobilised beneath excavations.
It is observed that compared to settlement, time dependent heave appears to develop over significantly longer periods of time. At the Shell Building, in central London, the rate of time dependent heave shows little sign of decreasing even though the excavation took place over 30 years ago, Burford 1992 11,12 . In contrast, time dependent settlement is generally found to be complete within about five to 10 years.
Previous analytical studies
Figure 4 summarises the results of non-linear finite element studies for a rigid footing, carried out by Jardine et al 1986 13 . Compared to linear elastic theory, the non-linear analysis predicts that settlement will reduce more rapidly with depth, Figure 4(a). Also the non-linear model predicts that as the factor of safety against bearing capacity failure reduces, the normalised settlement beneath the rigid footing becomes concentrated closer to the loaded boundary. This pattern of behaviour is similar to that identified from field observations, Figure 2. Figure 4(b) shows normalised surface settlement adjacent to the rigid footing. Normalised surface settlements predicted by non-linear elasticity are concentrated much closer to the rigid footing than those predicted by linear elasticity.
Description of proposed non-linear method The estimation of settlement or heave involves three basic considerations:
(a) the magnitude and distribution of stresses set up in the soil mass by the foundation loading;
(b) the immediate and long term stress-strain properties of the soil mass in both depth and lateral extent;
(c) the linking of (a) and (b) above, in order to calculate strains, and hence displacements, throughout the soil mass affected by foundation loading.
An outline of the proposed non-linear method is presented below.
Calculation of total settlement, or heave Total settlement, or heave, d T, is defined as follows:
dT = d u +d td (1) Total settlement, or heave, is calculated from a modified version of the one-dimensional method: dT =m vDs9 vH(2) If the compressible stratum is divided into n layers, then for layer i: dTi =m viDs viHi (3) From Henkel, 1971: 14 mvi= k i (4)E iRearranging (3) and utilising (4): evi = dTi = k iDs9 vi= k i(Ds vi- Du i) (5) HiE9 iE9 iThe change in net vertical total stress, Ds vi, is calculated from conventional isotropic linear elastic theory (for example Poulos and Davis 1974 15 ), Du i is the change in equilibrium pore water pressure within layer i (due to, for example, the installation of drainage measures within the proposed foundation).
The drained secant Young's modulus, E9 i, is assumed to be dependent on the average mean effective stress during the load increment (or decrement) and the magnitude of vertical strain which the layer experiences. For the latter, it is necessary to iterate equation 5 until the vertical strain calculated for the layer is compatible with the strain assumed for estimating E9 i. Following a successful iteration, the final value of E9 iis the mobilised drained secant Young's Modulus for layer i, E9 i(mob) . From equation 5, e9 viis calculated following a successful iteration, and then the cumulative total vertical displacement is calculated by summing for all layers:
i=n dT= S (e9 viHi) (6) i=1 Calculation of undrained settlement, or heave The undrained settlement, or heave, d uis calculated from a modified version of the classical elasticity equation: duv = 1 [Ds v - v u(Ds h1+ Ds h2)]H (7) E uDividing the compressible strata into n layers, then for layer i: duvi = 1 [Ds vi- v u(Ds h1i+ Ds h2i)]H i(8) E uFrom Equation 8: Euvi = d ui= 1 [Ds vi- v u(Ds h1i+ Ds h2i)] (9) HiEui The undrained secant Young's modulus, E ui, is assumed to be a function, only, of the vertical strain which layer i experiences. Mean effective stress during undrained loading (or unloading) is assumed to be constant. Equation 9 has to be iterated until the vertical strain calculated for the layer is compatible with the strain assumed for estimating E ui. Following a successful iteration, the final calculated value of E ui is then the mobilised undrained secant Young's modulus for layer i, E ui(mob) . Following a successful iteration e uvi is determined and the cumulative undrained vertical displacement is calculated for the compressible strata: i=n du= S (e uviHi) (10) i=1 Mobilised ground stiffness The key element of the proposed method is the facility to allow for the dependence of mobilised secant Young's modulus on the magnitude of strain (and additionally, in the case of drained modulus, on the magnitude of mean effective stress).
For the proposed method, the characterisation of ground stiffness comprises two main elements:
(a) definition of the variation of secant Young's modulus (at a particular strain magnitude) with depth;
(b) definition of the change in secant Young's modulus with changes in the magnitude of vertical strain.
Typically in the absence of advanced insitu or laboratory test data, the engineer can define the change in secant Young's modulus with changes in strain magnitude by utilising published literature. Such data is frequently normalised to allow data obtained for different sites, depths and types of test to be compared in a rational way. This also allows data for different soil types to be compared. The most common normalising parameter is undrained shear strength, that is, it is often assumed that undrained, or drained, secant Young's modulus (at a particular strain magnitude) is proportional to C u. Therefore, the engineer may utilise a profile of C uwith depth in order to derive a corresponding profile of secant Young's modulus (at a particular strain level). However, it should be noted that stiffness profiles obtained by using C uas a normalising parameter may be less reliable than those which use mean effective stress or the product of specific volume and mean effective stress as the normalising parameter. These normalising parameters have been used by Jardine 1and O'Brien et al, 1992 Modification of drained secant Young's modulus for average mean effective stress during loading The drained secant Young's modulus is dependent upon the average mean effective stress during the load increment (or decrement). Therefore, the variation of the initial drained secant Young's modulus, E9 owith depth (for a particular strain magnitude) needs to be modified to take account of the increase (or decrease) in mean effective stress (with depth) which occurs during drained loading (or unloading). The average mean effective stress during the load increment (or decrement) is defined as follows:
p' a= p' o+ p' f(11) 2For isotropic elastic materials under one-dimensional loading, the change in horizontal effective stress is related to the change in vertical effective stress by the Poisson's ratio: Ds' h1= Ds' h2 = v' Ds' v(12) (1-v') The increment or decrement of mean effective stress, Dp', can be related to the change in vertical and horizontal effective stress as follows: Dp' = 1 (Ds' v+ Ds' h1+ Ds' h2) (13) 3 Then from Equation 12, substituting for Ds' h1and Ds' h2in Equation 13: Dp' = (1 + v') Ds' v(14) 3(1-v') Hence the mean effective stress during a load increment (or decrement) for layer i may be expressed in terms of the change in vertical effective stress for layer i as follows:
Rearranging Equation 11: p' ai= p' oi+ Dp' i(15) 2From equation 14, substituting for Dp' in Equation 15: p' ai= p' oi+ (1 + v') Ds' vi(16) 6(1 - v') Hence, p' aican be calculated from a knowledge of only p' oiand Ds' vi. In order to modify the initial drained secant Young's modulus to allow for the increase (or decrease) in mean effective stress, the initial drained secant Young's modulus can be pro-rated as follows:
E' ci= E' oi~p' ai! (17) oiFrom Equation 17, the variation E' ci of with depth under the specified loading can be defined. It should be noted that the magnitude of E' ciis not usually unduly sensitive to the magnitude of p' oassumed.
Variation of secant Young's modulus with vertical strain
The variation of undrained, or drained secant Young's modulus with vertical strain is given by a user defined curve plotted as relative stiffness versus log vertical strain, Figure 5. This curve may be derived from modern laboratory or insitu testing or from data published in the technical literature. The magnitude of normalised secant Young's modulus at a particular strain magnitude is used as a basis for identifying the relative change in secant Young's modulus with changes in strain.
For example, in Figure 1, the value of E u/C ufor curve NL1 at a strain of 0. 1% is 450. Similarly, at a strain level of 0. 005%, E u/C uequals 1542 and at 0. 5% strain, E u/C uequals 167. Then, dividing these values with respect to the value for the particular strain level of 0. 1%, a relative stiffness versus log strain curve is derived for the complete range of strain magnitudes, Figure 5 (curve NL1). For example, the relative stiffnesses are 3. 43 (1542/450) and 0. 37 (167/450) for strain levels of 0. 005% and 0. 5% respectively.
The absolute value of the normalising strain magnitude (0. 1% in this case) is not critical, it merely allows the iterative calculation to be commenced. However, by choosing a normalising strain value within the range of expected strains for the problem, the number of iterations can be minimised. The iterative solution is commenced by using the value of E'ci (or Euoi ) (at the normalising strain magnitude) in equation 5 (or equation 9). The strain in each layer is calculated and compared with the assumed strain magnitude (used to derive E'ci or Euoi ). Using the user-defined relative stiffness - log strain curve the value for E'ci (orEuoi ) is adjusted to be compatible with the average of the assumed and calculated strain magnitudes from the first iteration. The adjusted value is defined as E' i(cal) (or Eui(cal) ). The calculation is repeated in subsequent iterations until the layer strain e'vi (or e'uvi ) is compatible with the average strain value used to derive E' i(cal) (or Eui(cal) ). To avoid numerical instability, maximum and minimum threshold relative stiffness values have to be defined, Figure 5.
Corrections to calculated settlement/heave The corrections which are necessary for refinement of the calculated settlement or heave depend on the level of sophistication which is applied in using the methods described above. At the most rudimentary level, stress changes within the soil mass may be calculated from simple Boussinesq theory for a uniformly distributed load on a perfectly flexible foundation resting on the surface of a semi-infinite medium. If this approach is adopted then for most real foundations corrections are required to take account of:
l depth of foundation beneath surface; l rigidity of foundation It is generally recognised that Fox's 16 depth correction factors will lead to an under estimate of the actual settlement or heave for most foundation substructures (except for piles), hence the depth factors recommended by Burland 17 should be utilised. For perfectly rigid foundations or those of intermediate stiffness appropriate correction factors are available, for example Poulos and Davis 1974, Fraser and Wardle 1976, Hooper 1975 15,18,19 . If vertical deformations adjacent to or beneath structures of finite stiffness are required then a more sophisticated approach is required, either: l predict stress changes within soil mass from appropriate elastic solutions for structures of finite stiffness, Poulos and Davis 1974; l use the concept of displacement compatibility along the structure to predict the variation of contact stress at the sub structure/soil interface. This contact stress distribution (simplified as a series of uniformly loaded areas of varying intensity) can then be used to calculate displacements within the soil mass, by using the principles of superposition to calculate stress changes at a particular point.
A simplified flow chart which summarises the key steps in calculating both undrained and total vertical deformation by the proposed method is presented in Figure 6.
Part two of this paper will be published in the November issue.
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