2 determining the value of G0 using seismic wave techniques (either in a triaxial cell using specialist bender elements, or in-situ as described in Chapters 6 and 7) and combining this with a normalised G/G0 versus γ relationship (e.g. Equation 5.11; see Atkinson, 2000 for further details).
Of these methods, the second is usually the cheapest and quickest to implement in practice. In principle, the value of G can also be estimated from the curve relating principal stress difference and axial strain in an undrained triaxial test (this will be described in Section 5.4). Without small-strain sample measurements, however, the data are only likely to be available for γ> 0.1%
(see Figure 5.4(d)). Because of the effects of sampling disturbance (see Chapter 6), it can be prefer- able to determine G (or E) from the results of in-situ rather than laboratory tests.
Soil behaviour in shear
limiting strength of soils was frictional, imagining that if slip (plastic failure) occurred along any plane within an element of closely packed particles (soil), then the slip plane would be rough due to all of the individual particle-to-particle contacts. Friction is commonly described by:
T = mN
where T is the limiting frictional force, N is the normal force acting perpendicular to the slip plane and μ is the coefficient of friction. This is shown in Figure 5.5(a). In an element of soil, it is more useful to use shear stress and normal stress instead of T and N, so that
tf =
(
tanf)
swhere tanϕ is equivalent to the coefficient of friction, which is an intrinsic material property related to the roughness of the shear plane (i.e. the shape, size and angularity of the soil particles).
The frictional relationship in terms of stresses is shown in Figure 5.5(b).
While Coulomb’s frictional model represented loosely packed particle arrangements well, if the particles are arranged in a dense packing then additional initial interlocking between the particles can cause the frictional resistance τf to be higher than that predicted considering friction alone.
If the normal stress is increased, it can become high enough that the contact forces between the individual particles cause particle breakage, which reduces the degree of interlocking and makes slip easier. At high normal stresses, therefore, the interlocking effect disappears and the material behaviour becomes purely frictional again. This is also shown in Figure 5.5(b).
In accordance with the principle that shear stress in a soil can be resisted only by the skeleton of solid particles and not the pore water, the shear strength (τf) of a soil at a point on a particular plane is normally expressed as a function of effective normal stress (σ′) rather than total stress, as it was for stiffness in Section 5.2.
The Mohr–Coulomb model
As described in Section 5.1, the state of stress in an element of soil is defined in terms of the normal and shear stresses applied to the boundaries of the soil element. States of stress in two dimensions Figure 5.5 (a) Frictional strength along a plane of slip, (b) strength of an assembly of
particles along a plane of slip.
can be represented on a plot of shear stress (τ) against effective normal stress (σ′). The stress state for a 2-D element of soil can be represented either by a pair of points with coordinates (sz¢, τzx) and (s¢x, τxz), or by a Mohr circle defined by the effective principal stresses s1¢ and s¢3, as shown in Figure 5.6. The stress points at either end of a diameter through a Mohr circle at an angle of 2θ to the horizontal represent the stress conditions on a plane at an angle of θ to the minor principal stress (s3¢). The circle therefore represents the stress states on all possible planes within the soil element. The principal stress components alone are enough to fully describe the position and size of the Mohr circle and so are often used to describe the stress state, as it reduces the number of stress variables from three (s¢x, sz¢, τzx) to two (s1¢, s3¢). When the element of soil reaches failure, the circle will just touch the failure envelope, at a single point. The failure envelope is defined by the frictional model described above; however, it can be difficult to deal with the non-linear part of
Figure 5.6 Mohr–Coulomb failure criterion.
Soil behaviour in shear the envelope associated with interlocking, so that it is common practice to approximate the failure envelope by a straight line described by:
tf = ¢ + ¢c s tanf¢ (5.12)
where c′ and ϕ′ are shear strength parameters referred to as the cohesion intercept and the angle of shearing resistance, respectively. Failure will thus occur at any point in the soil where a critical combination of shear stress and effective normal stress develops. It should be appreciated that c′ and ϕ′ are simply mathematical constants defining a linear relationship between shear strength and effective normal stress. Shearing resistance is developed mechanically due to inter-particle contact forces and friction, as described above; therefore, if effective normal stress is zero then shearing resistance must also be zero (unless there is cementation or some other bonding between the particles) and the value of c′ would be zero. This point is crucial to the interpretation of shear strength parameters (described in Section 5.4).
A state of stress represented by a stress point that plots above the failure envelope, or by a Mohr circle, part of which lies above the envelope, is impossible.
With reference to the general case with c′> 0 shown in Figure 5.6, the relationship between the shear strength parameters and the effective principal stresses at failure at a particular point can be deduced, compressive stress being taken as positive. The coordinates of the tangent point are τf and sf¢ where
tf = 12
(
s1¢ - ¢s3)
sin2q (5.13)¢ =
(
¢ + ¢)
+(
¢ - ¢)
sf 1 s s s s q
2
1
2 2
1 3 1 3 cos (5.14)
and θ is the theoretical angle between the minor principal plane and the plane of failure. It is apparent that 2θ= 90° + ϕ′, such that
q= ° +f¢
45 2 (5.15)
Now
sin
cot
¢ =
(
¢ - ¢)
¢ ¢ +
(
¢ + ¢)
f s s
f s s
1
2 1
2
1 3
1 3
c Therefore
¢ - ¢
(
s1 s3)
=(
s1¢ + ¢s3)
sinf¢ +2c¢cosf¢ (5.16a) or′ = ′ ° + ′
+ ′ ° + ′
s s f f
1 3 2 45
2 2 45
tan c tan 2 (5.16b)
Equation 5.16 is referred to as the Mohr–Coulomb failure criterion, defining the relationship between principal stresses at failure for given material properties c′ and ϕ′.
For a given state of stress it is apparent that, because s1¢ =s1-u and s3¢ =s3-u (Equation 3.1), the Mohr circles for total and effective stresses have the same diameter but their centres are separated by the corresponding pore water pressure u, as shown in Figure 5.7. Similarly, total and effective stress points are separated by the value of u.
Effect of drainage conditions on shear strength
The shear strength of a soil under undrained conditions is different from that under drained conditions. The failure envelope is defined in terms of effective stresses by ϕ′ and c′, and so is the same irrespective of whether the soil is under drained or undrained conditions; the difference is that under a given set of applied total stresses, in undrained loading excess pore pressures are generated which change the effective stresses within the soil (under drained conditions excess pore pressures are zero as consolidation is complete). Therefore, two identical samples of soil which are subjected to the same changes in total stress but under different drainage conditions (i.e. at different velocities) will have different internal effective stresses and therefore different strengths according to the Mohr–Coulomb criterion. Rather than have to determine the pore pressures and effective stresses under undrained conditions, the undrained strength can be expressed in terms of total stress, as an alternative description of the strength of the soil in these conditions.
The failure envelope will still be linear, but will have a different gradient and intercept; a Mohr–
Coulomb model can therefore still be used, but the shear strength parameters are different and denoted by cu and ϕu (= 0, see Section 5.6), with the subscripts denoting undrained behaviour.
The drained strength is expressed directly in terms of the effective stress parameters c′ and ϕ′
described previously.
In deciding whether to use drained or undrained strength parameters to subsequently analyse geotechnical constructions in practice, the principal consideration is the rate at which the changes Figure 5.7 Mohr circles for total and effective stresses.
Soil behaviour in shear in total stress (due to construction operations) are applied in relation to the rate of dissipation of excess pore water pressure (consolidation), which in turn is related to the permeability of the soil as described in Chapter 4. In fine-grained soils of low permeability (e.g. clay, silt), loading in the short term (e.g. of the order of weeks or less) will likely be undrained, while in the long-term, conditions will ultimately be drained. In coarse-grained soils (e.g. sand, gravel) both short- and long-term loading will result in drained conditions due to the higher permeability, which allows consolidation to take place rapidly. Under dynamic loading (e.g. earthquakes), loading may be fast enough to generate an undrained response in coarse-grained material. ‘Short-term’ is often taken to be synonymous with ‘during construction’, while ‘long-term’ usually relates to the design life of the construction (usually many tens of years). Within this chapter, simple models for estimat- ing loading rates in standard laboratory tests will be presented for achieving a response with a given amount of consolidation, to ensure that the desired strength (and stiffness) parameters are being measured. In Part 2 (Chapter 8 onwards), similar models will be presented to link dura- tion of loading in geotechnical constructions with an appropriate definition of soil strength (and stiffness).