Back to Soil Mechanics	Based on part of the GeotechniCAL reference package by Prof. John Atkinson, City University, London

Basic mechanics of soils

• Analysis of stress and strain
• Strength
• Stiffness
• Material behaviour

Loads from foundations and walls apply stresses in the ground. Settlements are caused by strains in the ground. To analyse the conditions within a material under loading, we must consider the stress-strain behaviour. The relationship between a strain and stress is termed stiffness. The maximum value of stress that may be sustained is termed strength.

• Special stress and strain states
• Mohr circle construction
• Parameters for stress and strain

Stresses and strains occur in all directions and to do settlement and stability analyses it is often necessary to relate the stresses in a particular direction to those in other directions.

normal stress s = Fn / A shear stress t = Fs / A

normal strain e = dz / zo shear strain g = dh / zo

Note that compressive stresses and strains are positive, counter-clockwise shear stress and strain are positive, and that these are total stresses (see effective stress).

Special stress and strain states	Analysis of stress and strain
In general, the stresses and strains in the three dimensions will all be different. There are three special cases which are important in ground engineering:	General case	princpal stresses
Axially symmetric or triaxial states Stresses and strains in two dorections are equal. s'x = s'y and ex = ey Relevant to conditions near relatively small foundations, piles, anchors and other concentrated loads.
Plane strain: Strain in one direction = 0 ey = 0 Relevant to conditions near long foundations, embankments, retaining walls and other long structures.
One-dimensional compression: Strain in two directions = 0 ex = ey = 0 Relevant to conditions below wide foundations or relatively thin compressible soil layers.
Uniaxial compression s'x = s'y = 0 This is an artifical case which is only possible for soil is there are negative pore water pressures.

Values of normal stress and shear stress must relate to a particular plane within an element of soil. In general, the stresses on another plane will be different. To visualise the stresses on all the possible planes, a graph called the Mohr circle is drawn by plotting a (normal stress, shear stress) point for a plane at every possible angle.
	There are special planes on which the shear stress is zero (i.e. the circle crosses the normal stress axis), and the state of stress (i.e. the circle) can be described by the normal stresses acting on these planes; these are called the principal stresses s'1 and s'3 .

	stress	strain
mean	p' = (s'a + 2s'r) / 3 s' =  s'a + s'r) / 2	ev = DV/V = (ea + 2er) en = (ea + er)
deviator	q' = (s'a - s'r) t' =  (s'a - s'r) / 2	es = 2 (ea - er) / 3 eg = (ea - er)

In the Mohr circle construction t' is the radius of the circle and s' defines its centre.

Note: Total and effective stresses are related to pore pressure u:

p' = p - u

s' = s - u

q' = q

t' = t

• Types of failure
• Strength criteria
• Typical values of shear strength

The shear strength of a material is most simply described as the maximum shear stress it can sustain: When the shear stress t is increased, the shear strain g increases; there will be a limiting condition at which the shear strain becomes very large and the material fails; the shear stress t_f is then the shear strength of the material. The simple type of failure shown here is associated with ductile or plastic materials. If the material is brittle (like a piece of chalk), the failure may be sudden and catastrophic with loss of strength after failure.

Materials can ‘fail’ under different loading conditions. In each case, however, failure is associated with the limiting radius of the Mohr circle, i.e. the maximum shear stress. The following common examples are shown in terms of total stresses:

Shearing
Shear strength = t_f
s_nf = normal stress at failure

Uniaxial extension
Tensile strength s_tf = 2t_f

Uniaxial compression
Compressive strength s_cf = 2t_f

Note:
Water has no strength t_f = 0.
Hence vertical and horizontal stresses are equal and the Mohr circle becomes a point.

A strength criterion is a formula which relates the strength of a material to some other parameters: these are material parameters and may include other stresses.

For soils there are three important strength criteria: the correct criterion depends on the nature of the soil and on whether the loading is drained or undrained.

In General, course grained soils will "drain" very quickly (in engineering terms) following loading. Thefore development of excess pore pressure will not occur; volume change associated with increments of effective stress will control the behaviour and the Mohr-Coulomb criteria will be valid.

Fine grained saturated soils will respond to loading initially by generating excess pore water pressures and remaining at constant volume. At this stage the Tresca criteria, which uses total stress to represent undrained behaviour, should be used. This is the short term or immediate loading response. Once the pore pressure has dissapated, after a certain time, the effective stresses have incresed and the Mohr-Coulomb criterion will describe the strength mobilised. This is the long term loading response.

• Tresca criterion
• Mohr-Coulomb (c’=0) criterion
• Mohr-Coulomb (c’>0) criterion

The strength is independent of the normal stress  since the response to loading simple increases the pore water pressure and not the effective stress.

The shear strength t_f is a material parameter which is known as the undrained shear strength s_u.

t_f = (s_a - s_r) = constant

The strength increases linearly with increasing normal stress and is zero when the normal stress is zero.
t'_f = s'_n tanf'
f' is the angle of friction

In the Mohr-Coulomb criterion the material parameter is the angle of friction f and materials which meet this criterion are known as frictional. In soils, the Mohr-Coulomb criterion applies when the normal stress is an effective normal stress.

The strength increases linearly with increasing normal stress and is positive when the normal stress is zero.
t'_f = c' + s'_n tanf'
f' is the angle of friction
c' is the 'cohesion' intercept

In soils, the Mohr-Coulomb criterion applies when the normal stress is an effective normal stress. In soils, the cohesion in the effective stress Mohr-Coulomb criterion is not the same as the cohesion (or undrained strength s_u) in the Tresca criterion.

Often the value of c' deduced from laboratory test results (in the shear testing apperatus) may appear to indicate some shar strength at s' = 0. i.e. the particles 'cohereing' together or are 'cemented' in some way. Often this is due to fitting a c', f' line to the experimental data and an 'apparent' cohesion may be deduced due to suction or dilatancy.