Back to Compression and shear	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.

• Change of size: bulk modulus
• Change of shape: shear modulus
• Uniaxial loading: Young's modulus and Poisson's ratio
• Relationships between stiffness moduli

As stresses are increased or decreased a material body will tend to change size and shape as strains occur: stiffness is the relationship between changes of stress and changes of strain.

The stiffness E' is the gradient of the stress-strain curve. Stiffnesses may be described by

a tangent modulus E'_tan = ds' / de

or by a secant modulus E'_sec = Ds' / De

Note: If the material is linearly elastic the stress-strain curve is a straight line and E'_tan = E'_sec.

Change of size: bulk modulus

Back to Stiffness

As the mean stress increases materials compress (reduce in volume). The bulk modulus K' relates the change in stress to the volumetric strain.

Note:
In soils volumetric strains are due to changes of effective stress.
In triaxial tests K' = dp' / de_v

As the shear stress increases materials distort (change shape). The shear modulus G' relates the change in shear stress to the shear strain.

Note:
In triaxial tests G' = ¹/₃ (dq' / de_s)

• Typical values of E

Young's modulus and Poisson's ratio are measured directly in uniaxial compression or extension tests, i.e. tests with constant (or zero) residual stress.

Young's modulus


Poisson's ratio
n' = - de_r / de_a

Note:
If the material is incompressible so e_v = 0 Poisson’s ratio is n = 0.5.

In bodies of elastic material the three stiffness moduli (E', K' and G') are related to each other and to Poisson’s ratio (n'). It assumed that the material is elastic and isotropic (i.e. linear stiffness is equal in all directions). The following relationships can be demonstrated (for proofs refer to a text on the strength of materials).

G' = E' / 2(1 + n')

K' = E' / 3(1 - 2n')

• Elasticity
• Perfect plasticity
• Elasto-plasticity

The stress-strain curve has features which are characteristic of different aspects of material behaviour and which are represented by different theories.

Shear modulus G' = dt' / dg Bulk modulus K' = dp’ / dev or Young’s modulus E' = ds'a / dea (where ds'r = 0) Poisson’s ratio n' = - der / dea (where ds'r = 0)

Yield Normality OA - rigid AB - plastic

Yield	Back to Perfect plasticity
Yield stress is the stress at the end of elastic behaviour (in a perfectly plastic material this is the same as the failure stress). In a 2-dimensional stress system the combination of yield stresses forms a yield curve.

The relationship between the ratio of plastic strains is such that the plastic strain vector is normal to the yield curve.

Normality is also known as associated flow.

There are simultaneous elastic and plastic strains and the plastic strains cause the yield stress to change. There are two cases:

Strain hardening The plastic strain dep causes an increase in the yield stress. The hardening law is ds'y / dep Soils with loosely packed grains are strain hardening because the disturbance during sharing causes the grains to move closer together.
Strain softening The plastic strain dep causes a decrease in the yield stress. Soils with densly packed grains are strain softening because disturbance during sharing causes the grains to move apart.