Effective stress in unsaturated soils
As defined previously, the effective stress is the stress that controls changes in both the volume and the strength of a soil.
The effective equation for saturated soils was evolved by Terzaghi in 1923 and is described in Chapter 3
a' = a — u where a' — the effective stress a = the total applied stress u = pore water pressure.
As there are two void fluids, liquid and gas, in an unsaturated soil the direct use of Terzaghi's effective stress equation is obviously not possible.
It should also be noted that an unsaturated soil, or a saturated soil that becomes unsaturated, can experience the problems of shrinkage, swelling and structural collapse, described in the previous section, as well as the problems of variation in strength and consolidation characteristics associated with saturated soils.
12.6.1 The x parameter
It was felt by most soil workers in the 1950s and early 1960s that it must be possible to apply the principle of effective stress to partially saturated soils. In 1955, Bishop suggested an effective stress formula for unsaturated soils which combined the total stress, a, the pore air pressure, ua, and the pore water pressure, uw, into a single effective stress tensor, a'\
where x is a parameter related to the degree of saturation and, to some extent, the soil structure having a value of 0.0 for a dry soil and 1.0 for a fully saturated one.
Only two years later, Jennings and Burland (1962) reported on the results they had obtained from consolidation tests carried out on samples of artificially prepared silty sand, silt and silty clay and questioned the validity of Bishop's equation.
Virgin consolidation curves were prepared for each soil by (i) consolidating from a slurry, and (ii) soaking a set of the prepared oedometer samples under various normal pressures. These virgin consolidation curves were compared with consolidation curves obtained from tests on unsoaked samples of each soil.
It was found that, for each soil type, the effective stress equation only applied over a range of Sr values. For the silty clay the range was as small as 15 per cent.
The oedometer tests also dramatically illustrated how the effective stress principle, in its proposed form, could not be used to predict the possibility of an unsaturated soil collapse. At the end of the dry tests on the unsaturated samples the oedometers were filled with water and the samples allowed to soak.
It is well known that soaking a soil reduces the suction forces within it to zero. This decrease in suction must mean that the effective stress also decreases. A decrease in effective stress should cause expansion yet, in every case, the samples, when soaked under constant loading, suffered further volume reduction, i.e. collapse. Collapse phenomenon is exactly the reverse of what should happen according to the effective stress principle which clearly did not apply.
Since the early 1960s, research into the determination of satisfactory x values were carried out with invariably disappointing results. In some tests X values greater than 1.0 and, in others, x values less than 0.0 were obtained.
After some time the reason for the erratic values obtained for x began to be understood. The problem is caused by the presence of 'menicus water' and a good description of its effects has been given by Wheeler and Karube in the introduction to their 1995 paper. The following is a summary.
In a saturated soil, no free air exists, every void within the soil being full of water. Pore water, in this state, is often referred to as 'bulk water' although it is possible that it contains a small amount of dissolved air.
If a saturated soil is slowly dried out the outer limits of the bulk water tend to evaporate and the outer soil voids begin to empty of water and take in air. Voids of an unsaturated soil are therefore filled either with water, a water and air mixture or simply air.
With air and water filled voids small lenses of water form menisci around the particle contacts (see Fig. 12.7d). With clays there will also be adsorbed water, so strongly attached to the soil particles that it can be regarded as being part of the soil skeleton.
Although the volume of meniscus water in an unsaturated soil may be very small it can have a dramatic effect on the mechanical behaviour of the soil, an effect that cannot be estimated.
A further problem is the inability of a single effective stress tensor, a', to replicate both swelling and collapse effects. If a dry soil is inundated with water its suction decreases and the soil will either collapse or swell, depending upon the relative magnitude of the applied loading.
The inability to determine whether a decrease in suction acts like an increase or a decrease in the effective stress of a saturated soil means that, for an unsaturated soil, it is not possible to combine a — ua and ua — uw into a practical single value for a'. This fact was generally accepted by the end of the 1980s (Alonso et al., 1990).
12.6.2 The effect of degree of saturation on unsaturated soils
Barden, in his paper of 1965, was concerned with the consolidation of unsaturated clays and, in the absence of an effective stress equation, his work can help to give some guidance to the soils engineer. Barden suggested that the effects of a varying degree of saturation in clay could best be studied by dividing the range of Sr into a set of five increments. His classification provides an understandable description of how the behaviour of an unsaturated soil changes as it progresses from Sr < 5 per cent to Sr > 95 per cent.
The air phase is continuous throughout the soil mass and any water is in the form of highly viscous adsorbed water firmly attached to the skeleton. As the air voids are interconnected only air will be expelled as the soil consolidates and, as Sr is very small, the value of the effective stress, a', can be taken as equal to the applied stress less the air pressure i.e. cr — ua. However ua is generally the atmospheric pressure and, in that case, can be assumed to have a value of 0.0 making a' = a.
Barden only included this extremely low range of Sr for completeness acknowledging that compacted clay fill would never be placed at such low Sr values.
As more water is added to a soil there is a gradual transition and the behaviour of the soil becomes more affected by the free water than the adsorbed. Towards the middle and higher reaches of the range it becomes possible to obtain measured values for ua and uw.
With consolidation, although air will be expelled, the value of uw will rarely become positive and the suction term (us — uw) is still large enough to ensure that very little water flows from the soil.
For his discussion Barden assumed that an Sr value of 90 per cent corresponded to the optimum water content although he was careful to point out that this value was not to be considered as applicable to all clays.
This range is a transition stage in which the value of (ua - uw) may drop low enough for uw to be greater than zero. If this happens then only water can drain from the soil.
(4) Wet of optimum (90 per cent < Sr < 95 per cent)
Air can no longer exist in a free state and is said to be occluded. There is no way of measuring ua and any air remaining is mainly static, trapped in the skeleton and unable to flow as a free fluid. However some air may remain in the soil in the form of water bubbles which, although having little effect on the value of uw, can make the pore fluid highly compressible.
Occlusion can occur very quickly. A good example is Vicksburg silty clay where the air can be continuous at a water content 4 per cent below optimum and occluded at 3 per cent below optimum.
For very wet clays it can be concluded that only trapped air is present and that, due to the lack of air bubbles, any further water flowing from the soil will be fairly incompressible.
Because of the high value of Sr the effective stress can usually be taken as a' — a — uw.
Conclusions
For clayey soils, with a relatively high degree of saturation, from about 90 per cent, the air in the soil is occluded and can often be assumed to have little effect on the pore water pressure. In such a case the unsaturated soil will tend to behave as if it were saturated and the effective stress can be assumed to be equal to o - uw. The exception is a fine grained soil near to but on the dry side of optimum where the air may not be occluded. In this case the effective stress will not even be approximately equal to a — uw.
For most fine grained soils, when sr is equal to or less than 5 per cent, the soil can be assumed to be dry and the effective stress taken as equal to the applied stress, a.
When dealing with sands and gravels above ground water level it should be remembered that suction effects are fairly negligible and the effective stress can often be taken as simply equal to the overburden pressure, any possible increase due to negative pore water pressure being ignored.
It is hoped that these conclusions will be of some assistance to the civil engineer confronted with a design problem involving an unsaturated soil. Some problems require considerable thought. For example, when considering the collapse mechanism of clayey soils, the general practice of compacting such soils to the dry side of optimum, rather than the wet side, may not necessarily be the best procedure and the inclusion of inundation test results in the site investigation report would be an advantage.
12.7 More recent research work
12.7.1 Two stress state variables
In an undrained soil there are three stress parameters, a — ua, a — uw and ua - uw. Fredlund and Morgenstern, in their paper of 1977, agreed with the approach adopted by Coleman (1962) and Bishop and Blight (1963) in that only two of the three stress variables are necessary to define the stress state of an unsaturated soil. The common choice is to use the net stress, a — ua and the matric soil suction, ua — uw as the two independent stress state variables. This is mainly because ua is the atmospheric pressure and can usually be assumed to have a value of zero. With this approach constitutive models of unsaturated soil behaviour, both for volumetric change and for shear strength, have been proposed and examined since the 1970s.
A possibly useful formula for the shear strength of an unsaturated soil was presented by Fredlund et al. in 1978:
r — c' + (ct - ua) tan <j>' + (ua - uw) tan 4>h where c' and are the cohesive and angle of friction for an equivalent saturated soil and 0b the angle of friction with respect to changes in suction.
However many authors subsequently showed that, although <p' is more or less constant for most soils, <f>h is not. Escario and Juca (1989) suggested that the equation should be rewritten as:
r = c' + (a' - ua) tan 4> + f(ua - uw) where f(ua - uw) is a nonlinear function of suction.
Wheeler and Karube (1995) have prepared a review of several recent models some of which use more complex parameters.
12.7.2 State surfaces
The volume change characteristics of an unsaturated soil can be expressed as the variation of the void ratio, e, and the variation of the degree of saturation, Sr. The relationships between these variations and the stress parameters (a - ua) and (ua - uw) can be shown by 'state surfaces' obtained from three-dimensional plots of e and the stress parameters and of Sr and the stress parameters.
Several authors have adopted this approach. Matyas and Radhakrishna (1968) presented experimental data to prepare the state surface plots shown in Fig. 12.8.
It is seen that the state surface for e is warped and is therefore able to account for the swelling (volume increase) that the soil will experience if it is flooded at a low value of net stress and its collapse (its volume decrease) when it is flooded at a high value of net stress.
- Fig. 12.8 State surface plots (based on Matyas and Radhakrishna, 1968).
12.7.3 Elasto-plastic critical state models for unsaturated soil
Critical state modelling is described in Chapter 13 and this approach is the most recent line of research in unsaturated soils. It is an attempt to allow for the possible occurrence of irreversible plastic strains in an unsaturated soil and to link them to the volumetric and/or shear behaviour of the soil.
Such an elasto-plastic critical state model for unsaturated soil was presented in a qualitative form by Alonso, Gens and Hight in 1987 and the model was developed into a full mathematical form by Alonso et al. (1990). Wheeler and Sivakumar (1995), using experimental data they obtained from controlled suction triaxial tests on compacted kaolin, proposed further modifications to the model.
There is little doubt that the strength and volume change behaviour of unsaturated soils together with the associated problems of swelling and collapse will be important areas of research for some years to come.
12.8 Testing techniques for unsaturated soils
The techniques for testing partially saturated soils are (or should be) considerably different from those for saturated soils. The difference is because of the need to measure (and differentiate between) the pore air and the pore water pressures. There are various papers that describe these methods and a good summary, which has stood the test of time, is that given in Bishop and Henkel's textbook (1962).
Briefly, pore water pressures can be measured through saturated, fine pored ceramic discs which, due to their high air entry values, act as filters and remain saturated at all times, so that the water in the pores of the disc is in equilibrium with the pore water in the soil.
Pore air pressure values can be measured through coarse pored ceramic discs, or glass-fibre filters, which have a moisture retention capacity so low that they are unable to draw water from the soil sample, i.e. its suction value is less than the ua — uw value operating within the soil. With this situation the disc remains practically dry. The air contained in the pores of this coarse filter remains in equilibrium with the pore air in the soil sample throughout the test.
It is well known that the equalisation of the pore fluid pressure throughout a triaxial sample is important if pore pressure measurements are to be taken and it is this condition that governs the rate at which a sample of saturated soil can be sheared. Unsaturated soils are generally less permeable than when saturated so that very low rates of shear are often necessary when conducting shear tests on such soils.
In this connection it must be appreciated that a transference of air from the soil sample to the water within the triaxial cell can take place by diffusion through the rubber membrane. Hence, if an 'undrained' test is to last for more than a few hours, a special type of triaxial cell, in which an inner perspex ring allows the membrane to be surrounded by mercury, should be used.
12.8.1 Common errors in testing unsaturated soils
In many soils commercial laboratories, consolidated undrained tests are still carried out on compacted partially saturated soil samples in order to determine c' and <p'. The technique is to use one measurement of pore pressure, u, in the hope that this will somehow be an equivalent pore pressure in the soil so that Terzaghi's expression of a' — a - u may be used. This procedure can produce two extreme results:
(1) Test using a coarse pored filter at base of sample
Although the operator may think that he is measuring pore water pressure, because of the grade of filter used in the test, he is actually measuring ua and his effective stress equation is actually cr' — a — ua. The result is that, when plotted, the strength will have a large 'cohesive' intercept (Fig. 12.9).
(2) Test using a fine pored filter at base of sample
Pore pressure measurements with this method will represent the pore water pressure, uw, within the soil sample and an apparent effective stress value will be obtained from the equation a' — a — uw and the strength envelope can have a large cohesive intercept (Fig. 12.10).
The correct result lies somewhere between these two results but it should be noted that <j>' is more or less correctly measured when either fine or coarse discs are used at the base of the sample. It is c' that is largely indeterminate unless a more exact procedure is used.
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