## C

Fig. 3.9 A cohesive coil, subjected to undrained conditions and zero total normal stress will still exhibit a shear stress, cu.

where

Tf = shear stress at failure, i.e. the shear strength c = unit cohesion a = total normal stress on failure plane (ยก> = angle of shearing resistance.

The equation gave satisfactory predictions for sands and gravels, for which it was originally intended, but it was not so successful when applied to silts and clays. The reasons for this are now well known and are that the drainage conditions under which the soil is operating together with the rate of the applied loading have a considerable effect on the amount of shearing resistance the soil will exhibit. None of this was appreciated in the 18th century and this lack of understanding continued more or less until 1925 when Terzaghi published his theory of effective stress.

Note It should be noted that there are other factors that affect the value of the angle of shearing resistance of a particular soil. They include the effects of such items as the amount of friction between the soil particles, the shape of the particles and the degree of interlock between them, the density of the soil, its previous stress history, etc.

Effective stress, a'

The principle of effective stress was introduced in Chapter 2. Terzaghi first presented the concept of effective stress in 1925 and, again in 1936, at the First International Conference of Soil Mechanics and Foundation Engineering, at Harvard University. He showed, from the results of many soil tests, that when an undrained saturated soil is subjected to an increase in applied normal stress, Atx, the pore water pressure within the soil increases by Au, and the value of Au is equal to the value of Act. This increase in u caused no measurable changes in either the volumes or the strengths of the soils tested and Terzaghi therefore used the term neutral stress to describe u, instead of the now more popular term pore water pressure.

Terzaghi concluded that only part of an applied stress system controls measurable changes in soil behaviour and this is the balance between the applied stresses and the neutral stress. He called these balancing stresses the effective stresses.

If a soil mass is subjected to the action of compressive forces applied at its boundaries then the stresses induced within the soil at any point can be estimated by the theory of elasticity, described in Chapter 4. For most soil problems, estimations of the values of the principal stresses, ai, 02 and <73 acting at a particular point are required. Once these values have been obtained, the values of the normal and shear stresses acting on any plane through the point can be computed.

At any point in a saturated soil each of the three principal stresses consists of two parts:

(1) u, the neutral pressure acting in both the water and in the solid skeleton in every direction with equal intensity;

(2) the balancing pressures (ctj - u), (<r2 - u) and (03 - u).

As explained above, Terzaghi's theory is that only the balancing pressures, i.e. the effective principal stresses, influence volume and strength changes in saturated soils:

Principal effective stress = Principal normal stress โ Pore water pressure i.e.

where the prime represents 'effective stress'.

Terzaghi explained that if a saturated soil fails by shear, the normal stress on the plane of failure, a, also consists of the neutral stress, u, and an effective stress which led to the equation known to all soils engineers:

This equation has stood the test of time and is accepted as applicable to all saturated soils. The problem of an effective stress equation for unsaturated soils is discussed in Chapter 12.

3.6 Modified Coulomb's law

Shear strength depends upon effective stress and not total stress. Coulomb's equation must therefore be modified in terms of effective stress and becomes:

Tf = c' + a tan where c' = unit cohesion, with respect to effective stresses a' = effective normal stress acting on failure plane

4> = angle of shearing resistance, with respect to effective stresses.

It is seen that, dependent upon the loading and drainage conditions, it is possible for a clay soil to exhibit purely frictional shear strength (i.e. to act as a 'c' = 0' or '</>" soil), when it is loaded under drained conditions or to exhibit only cohesive strength (i.e. to act as a = 0' or 'cu' soil) when it is loaded under undrained conditions. (See Example 3.8, Fig. 3.26.) Obviously, at an interim stage the clay can exhibit both cohesion and frictional resistance (i.e. to act as a V โ 0" soil). The same situation also applies to granular soils.

### 3.7 The Mohr-Coulomb yield theory

Over the years various yield theories have been proposed for soils. The best known ones are: the Tresca theory, the von Mises theory, the Mohr-Coulomb theory and the critical state theory. The first three theories have been described by Bishop (1966) and the critical state theory by Schofield and Wroth (1968).

Only the Mohr-Coulomb theory is discussed in this chapter. The theory does not consider the effect of strains or volume changes that a soil experiences on its way to failure nor does it consider the effect of the intermediate principal stress, (72- Nevertheless satisfactory predictions of soil strength are obtained and, as it is simple to apply, the Mohr-Coulomb theory is widely used in the analysis of most practical problems which involve soil strength.

The Mohr strength theory is really an extension of the Tresca theory which, in turn, was probably based on Coulomb's work, hence the title. The theory assumes that the difference between the major and minor principal stresses is a function of their sum, i.e. (ctj โ 0-3) = ffcri + a3). Any effect due to (72 is ignored.

The Mohr circle has been discussed earlier in this chapter and a typical example of a Mohr circle diagram is shown in Fig. 3.10. The intercept on the shear stress axis of the strength envelope is the intrinsic pressure, i.e. the strength of the material when under zero normal stress. As we know, this intercept is called cohesion in soil mechanics and given the symbol c.

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