In this, the second part of their paper, O'Brien and Sharp discuss the proposed method and in an appendix give a worked example of calculation of total settlement. A notation table was published with the first part of this paper in the October issue of GE.
Discussion It is recognised that the stress strain behaviour of soils is highly complex 20 .The proposed method does not attempt to be comprehensive in the way it models soil behaviour, or mathematically rigorous, since this would preclude a simple approach appropriate for routine applications.
Nevertheless, it is believed that the proposed method allows a rational framework to be developed to take account of non-linear stress strain behaviour.The method is based on conventional theories and terminology which should be familiar to experienced geotechnical engineers.The approach can be readily utilised within a computer program; the iterative calculations are generally performed within a few seconds.
With experience in using the method, further refinements can be readily taken into account; for example, the magnitude of drained Young's modulus could be related to mean effective stress raised to a power of between 0.5 and 1.0 dependent on strain magnitude (to account for the observation that at small strains Young's modulus is typically a function of the square root of mean effective stress, whereas at large strains Young's modulus is proportional to mean effective stress 21,22 ), the influence of anisotropy could also be studied in more detail 15 .An example of the calculation procedure is presented in Appendix A (see page 53).
This comprises the calculation of total settlement for a circular footing. As discussed below, the key assumptions are believed to be valid for most sub-structures which have an adequate factor of safety (in excess of two) against bearing capacity failure.
The key assumptions may be summarised as follows:
l isotropic linear elastic theory provides reasonable predictions of stress changes within the soil mass;
l for undrained loading horizontal and shear strains are negligible compared with vertical compressive strains.
The influence of anisotropy is relatively small and ignoring it provides predictions which are conservative, but not excessively so;
l for drained loading, the one-dimensional method provides realistic predictions of total settlement for most practical situations which usually involve increasing stiffness with depth and stiffness anisotropy.
Stress changes A wide range of solutions have been published for calculating stress changes within elastic media. These solutions normally assume the soil is:
a semi-infinite compressible layer.
Several researchers have assessed the errors associated with the above assumptions.These include:
onset of yielding, ie plasticity, Morgernstern and Phukan 1968 23 ; l non-linear elasticity, Jardine et al 1986 13 ; soil layering, eg stiffer layer overlying softer layer, Poulos and Davis 1974 15 ; l stiffness increasing with depth, Gibson 1971 24 ; anisotropy, Gerrard and Harrison 1970 25 .The main conclusions from these studies may be summarised as follows:
Where the Poisson's ratio is equal to 0.5 (ie undrained loading), then the vertical and horizontal stress changes calculated by conventional linear elastic theory are reasonably accurate.The exceptions are if a stiff layer overlies a less stiff layer and if the stiffness ratio G9 vh /E9 48 vdeparts significantly from the isotropic value of 0.4. For the former case published solutions are available which can be readily utilised, Poulos and Davis 1974.For the latter case, as discussed later, compensating errors mean that any inaccuracies associated with isotropic linear elastic theory will lead to an insignificant overall error in calculating settlement (or heave).Typically such errors are less than about 10%.
For drained loading, when the Poisson's ratio is less than 0.5 (generally between 0.1 and 0.2 for overconsolidated clays), the horizontal stress changes given by conventional theories are significantly in error.
In particular, for the common situation of soil stiffness increasing with depth, the actual horizontal stress changes are much smaller than those given by conventional linear elastic theory.Horizontal stress changes during drained loading are not used by the proposed method.However, the vertical stress changes (which are used by the proposed method for calculating total deformation) are reasonably accurate for most real situations.
Undrained loading Lee and Rowe 1989 26 have reviewed the role of anisotropy in undrained settlement predictions. For foundations with an adequate factor of safety against bearing capacity failure, they demonstrated that the ground deformation mechanism is related primarily to vertical compressive strain. Horizontal and shear strains are generally quite small, with a localised area of moderately high shear strains adjacent to the edge of the loaded area.For overconsolidated clays, the error involved in the assumption of isotropy would lead to an overprediction of settlement of less than 20%, and typically less than 10%.Hence the assumptions inherent in the use of equation 9 that vertical strain is the dominant strain component (and hence the effect of other strain components can be ignored for the purposes of iteration) and that the soil is isotropic should lead to insignificant error.
Burland et al 1977 27 demonstrated that the one-dimensional method gives predictions of total settlement (for vertically loaded foundations on overconsolidated clays) which are as good as and usually better than more sophisticated methods, for example Figure 7 after Nyaoro 1983 28 .For elastic soils, the one-dimensional method gives total (ie undrained plus time dependent) settlement if the drained elastic modulus is used.
Some illustrative calculations of the proposed method compared with other techniques To illustrate the application of the proposed method, some illustrative calculations have been performed and are compared with other calculation methods.The parameters which have been utilised are summarised on Figure 8 and Table 1. In particular, comparisons are made with the geostructural mechanism approach for the prediction of undrained movement of clays which has been developed at Cambridge University, Bolton and Sun 1991 29 .This uses the concept of mobilised strength and a power law curve is used to represent the nonlinear stress strain behaviour of the soil. The power law constants used for the geostructural calculations provide a similar stress strain curve to NL1 (Sun 1991 30 ).The curve NL1 may be regarded as 'typical'whereas curve NL2 approaches the upper bound of the data reported by O'Brien et al, 1992, Figure 1. Predictions of total settlement by the proposed method are compared with a non-linear method proposed by Stroud, 1988 31 .The geostructural mechanism approach was developed from a programme of research which included the back analysis of undrained movements observed during model foundation loading experiments performed within a centrifuge.The non-linear method proposed by Stroud was based on the back analysis of case histories of total settlement.
Undrained settlement of rigid raft The relationship between undrained settlement and bearing pressure predicted by various methods is given on Figure 9 for a relatively large (20 m x 20 m) rigid raft founded at the surface of an overconsolidated clay layer. There are two methods which attempt to model non-linear stress strain behaviour, the proposed method and the geostructural mechanism approach; the others assume linear elastic behaviour. The proposed method (using curve NL1) gives similar predictions to the geostructural mechanism approach for bearing pressures less than 150kN/m 2.For pressures less than about 125kN/m 2the linear elastic methods overpredict settlement compared with the non-linear methods, and for pressures less than about 50kN/m 2the linear elastic methods overpredict settlement by more than a factor of two.The predictions for the stress strain curve NL2 emphasises the fact that current empirically derived linear elastic methods and their associated parameters can be grossly conservative for some overconsolidated clays.
Total settlement of rigid raft Figure 10 summarises the predicted relationship between bearing pressure and total settlement. The two non-linear methods (Stroud, 1988 and the proposed method) give similar predictions of total settlement, which are generally less than those given by linear elastic methods, particularly for pressures less than 75kN/m 2.For relatively stiff clays (eg NLD2), the total settlements predicted by the non-linear methods are about 25% to 50% of those given by the linear elastic methods, for pressures less than 100kN/m 2.In routine practice, the 'allowable' bearing pressure (q a) is often defined as the bearing pressure at which total settlement equals 25mm. Using this criterion, the linear elastic and Skempton and Bjerrum methods would suggest a q avalue of about 30kN/m 2, whereas the proposed method would indicate a q avalue of about 60kN/m 2(using curve NLD1) and about 85kN/m 2(using curve NLD2).
Settlement beneath and adjacent to a rigid raft Figures 11 and 12 summarise normalised undrained subsurface settlement beneath a rigid raft and normalised total surface settlement adjacent to a rigid raft. The proposed method correctly predicts the features of behaviour identified from field observations and from non-linear finite element studies; refer to Figures 2 and 3.These are:
settlement is concentrated closer to the loaded boundary than linear elastic methods predict;
as bearing pressure is increased, settlement increases by a proportionately greater amount;
as bearing pressure is increased, normalised settlements become concentrated closer to the loaded boundary.
Heave compared with settlement Table 2 compares the heave calculated for a net reduction in stress of 160kN/m 2with the settlement calculated for a net increase of stress of 160kN/m 2(at the centre of a 20m wide, infinitely long strip). The calculated total heave is about 150mm compared with 100mm total settlement, and the time dependent heave is 110mm compared with 60mm time dependent settlement. The ratio R = d u/d td is significantly greater for settlement than for heave, similar to that observed in practice. This effect is purely due to the differences in average mean effective stress which are prevalent during drained unloading compared with drained loading, and which, in turn, influence the mobilised drained Young's modulus.Hence, application of the proposed method assists in explaining much of the differences which have been observed in the way heave and settlement develop.Stress path and recent stress history effects have not been taken into account, although in practice these may also lead to differences in the magnitude of heave and settlement for a given change in net bearing pressure. In this comparison, it has been assumed that there has been no change in equilibrium pore water pressure. It should be noted that the magnitude of settlement or heave is sensitive to this assumption. If it is believed that the drainage boundary conditions may change significantly as a result of foundation construction, then it may be prudent to carry out seepage analyses, so that the change in equilibrium pore water pressure can be quantified and incorporated within the analysis.
Negative skin friction on piles The application of the proposed method to a soil-structure interaction problem, that of negative skin friction on 'floating'piles, is illustrated on Figure 13.This is a common situation when bridge abutments are founded within overconsolidated clays.Due to near surface weathering, skin friction piles are often required in order to found at depth in unweathered material.Adjacent embankment loading causes some negative skin friction to develop. Intuitively, experienced engineers would anticipate that the magnitude of negative skin friction due to settlement of overconsolidated clays would be negligible. However, conventional linear elastic methods predict that significant negative skin friction forces will develop. For the example shown on Figure 13, the negative skin friction forces predicted by the proposed method are about 7.5 times smaller than those forces predicted by linear elastic methods. The difference is mainly due to the fact that the proposed method calculates subsurface settlement decreasing more rapidly with depth than the linear elastic method.Hence, the 'neutral point' (ie the depth at which pile settlement equals ground settlement) is predicted by the proposed method to be at a shallower depth than that predicted by linear elastic methods.
(1) The stress-strain behaviour of overconsolidated clays is highly non-linear, such that the selection of appropriate 'linear elastic'moduli for calculations of settlement or heave is very difficult.
(2) A method is proposed in this paper which allows the non-linear stress strain characteristics of overconsolidated clays to be modelled in a rational manner, and enables undrained and total settlement (or heave) to be calculated. Uniform and non-uniform (soil-layering) increases of stiffness with depth, foundations of varying shape, rigidity and loading intensity can be analysed. Surface and subsurface settlement (or heave) adjacent to and beneath the foundations can be assessed. For predictions of total settlement or heave, the effect of changes in equilibrium pore water pressure (due to, for example, construction of drains) can be readily assessed.
(3) Field observations and non-linear finite element studies indicate that:
surface and subsurface settlements adjacent to and beneath foundations are concentrated closer to the loaded boundary than linear elastic methods predict;
as bearing pressure is increased, foundation settlement increases by a proportionately greater amount;
as bearing pressure increases normalised settlement is concentrated closer to the loaded boundary;
for a given net change in effective pressure, time dependent heave due to unloading is significantly greater than time dependent settlement due to loading.
(4) The features of behaviour described in (3) above are correctly predicted by the proposed method. In addition, the proposed method is quite flexible in its range of application. These include soil-structure interaction problems, such as negative skin friction on piles. For this type of problem the proposed method predicts significantly smaller (and it is believed more realistic) negative skin friction forces than linear elastic methods.Other problems, such as heave induced pile tension, can be readily assessed.Compared with other 'simplified' non-linear methods, the proposed method can be applied to a significantly wider range of problems.
(5) The proposed method allows modern ground investigation test data to be utilised for routine design purposes.
The proposed method is considered to be a practical means of overcoming some of the difficulties which are currently encountered by foundation engineers. These difficulties include conventional methods which assume unrealistic ground behaviour (ie linear elastic) and complex non-linear finite element techniques which can be difficult to understand and validate, and time consuming to use.In overcoming these difficulties, it is believed that the proposed method is particularly suited to the fields of design and build, and value engineering where there is a need for cost-effective, rapid (ie simple) and realistic evaluations of ground movements.
The application of the intended method, as with any calculation technique, needs to be carefully considered. However, it is believed that it will be a useful additional tool for foundation engineers. As with more sophisticated 'mathematically rigorous'numerical techniques, the reliability of the calculation is mainly dependent upon an appropriate choice of input parameters. The inaccuracies associated with some of the simplifying assumptions are unlikely to be significant for the majority of substructures founded on overconsolidated clays, which usually have a factor of safety against bearing capacity failure in excess of two.
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23 NR Morgernstern and ALT Phukan (1968).Stresses and displacements in a homogeneous non-linear foundation.Proc Int Symp Rock Mech, Madrid, pp313 to 320.
24 RE Gibson and GC Sills (1975).Some results concerning the plane deformation of a non-homogeneous elastic half-space.Proc Roscoe Memorial Symp on Stress-Strain Behaviour of Soils, Cambridge, Foulis, pp564 to 572.
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27 JB Burland, BB Broms and VFB DeMello (1977).Behaviour of foundations and structures.Proc 9th ICSMFE, Tokyo, Vol 2, pp495 to 546.
28 FK Nyaoro (1983).Msc Dissertation, Imperial College.
29 MD Bolton and HW Sun (1991).The behaviour of bridge abutments on spread foundations.Proc 10th ICSMFE, Florence.
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31 MA Stroud (1988). The Standard Penetration Test - its application and interpretation. Penetration Testing in the UK, Proc ICE Conf, Birmingham, pp29 to 49.
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Acknowledgements The authors wish to thank their colleagues at Mott MacDonald, particularly Bill Rankin, Ron Williams and Alan Powderham for their comments on an early draft of the paper.Special thanks to John McCallum for developing modern software to replace the antique created by the authors, and to Rob Jessep and Jeanette Chang for reviewing and testing the software.Finally, thanks to Debbie Tassell for drafting the figures.
Errata Figure 4, amended.
In the first part of this paper (GE October 2001) elements of the key for Figure 4 were transposed.The correct version appears above.
In the same issue, Figure 6, Box 8 on the right-hand flow chart should read: Calculate profile of correct E' cwith depth, at normalising strain magnitude and average mean effective stress. Box 9 should read: Calculate strain with each of 'n' layers.