2010-02-19 sm2 stresses

COURSE TITLE: Geotechnics 2

COURSE CODE: GEOT221B

ELEMENT NAME: Soil Mechanics 2

Recap Permeability is a measure of the ease of which water flows

through rocks and soil. Darcy’s law: Q = k A H

t l The ratio H/l is known as the hydraulic gradient and is

denoted as i. The constant head permeameter for determining coefficient

of permeability of coarse grained soils.

Permeability

It should be noted that water flows through fine grained soils for example clay, at a much slower rate than through coarse material, such as sand.

The constant head permeameter is not practical to use for fine grained soils as it is not possible to obtain a measurable amount of water within a reasonable time.

In this case a variable head permeameter is used.

Permeability – Fine-grained soils

Variable head permeameter


Variable head permeameter When the tap is opened, water will pass through the sample

and the level in the standpipe will fall. Once steady conditions have been obtained, two readings of

H are taken H1 and H2 at a time interval t.

During any increment of time dt the variation in head is –dH and hence the quantity of water flowing through the sample in time dt

Q = -a dH


Variable head permeameter From Darcy’s Law

-a dH = k A H

dt l

-a dH = k A H dt

l

or dt = - al dH

Ak H



Integrating between limits 0 to t and H1 and H2

-t = -al loge H1

Ak H2

or

k = a x l x 2.3 log10 H1

A t H2

In this equation all the terms may readily be found and hence k can be calculated.


Variable head permeameter Three readings of H should be taken H1 , H2 and H3 such that

the time for the head to drop from H1 to H2 is the same as the time for the head to drop from H2 to H3 .

Since in the permeability equation derived k, a, A and l are constant and t has also been made the same, then:

log10 H1 = H2

H2 H3



Or H1 = H2

H2 H3

Therefore H2 = √ H1 H3


Example

In a falling head permeameter test on a silty clay sample, the following results were obtained: sample length 120mm, sample diameter 80mm, initial head 1100mm, final head 420mm, time for fall in head 6 minutes; standpipe diameter 4mm.

Determine from first principles the coefficient of permeability of the soil.


Example

On close investigation of the sample it was found to be in 3 layers 20mm,60mm and 40mm thick, each of permeabilities 3 x 10-3 mm/s, 5 x 10-4mm/s and 17 x 10-4 mm/s respectively.

Check the average permeability through the sample in the laboratory test and estimate the permeability of this sample in a direction at right-angles to sampling.

Find the ratio kH /kV and comment on the result.


Determination of permeability on site

1. Borehole techniques

An estimate of the permeability of a soil may be made using the boreholes driven during the site investigation. There are many empirical techniques for determining permeability in this way and, given the general inaccuracy of determining permeability, these are reasonably satisfactory methods.

On site permeability


Generally, if the stratum being tested is above the water table, water is pumped into the bore-hole and the rate of flow to maintain a constant head is measured. If the stratum is below the water table, either pumping in or pumping out tests may be used, in conjunction with a casing to the bore-hole extending into the permeable stratum.



2. Well –point techniques:

If a well point method of ground-water lowering is used it is possible to determine the co-efficient of permeability in the field. When water is pumped from a well point the water is lowered adjacent to the point, giving a cone of depression. This cone of depression will form even in relatively impervious soils after sufficient time has elapsed.


At any horizontal section and depth z in a soil profile, the total downward pressure is due to the weight of soil above the section.

Consider two layers, one above the water table with bulk density γ and depth z1 and another below the water table with saturated density γsat and depth z2.

Resistance to this pressure is provided, partly by the soil grains, and if the section is below the water table, partly by the upward pressure of the water.

Neutral & Effective Stress

Total load at depth z per unit area

σ = z1 γ + z2 γsat

This is resisted by the intergranular σ’ which is referred to as the effective stress, and by the upward water pressure u, which is referred to as the neutral stress and equals z2 γw

Total downward load per unit area = intergranular pressure + upward water pressure.

Total downward load per unit area = effective stress + neutral stress

σ = σ’ + u

Neutral & Effective Stress

A borehole on a building site has the soil profile shown:

Find the effective stress at the bottom of the clay:

a. Under normal conditions,

b. If the ground water level is lowered 2.4m by pumping (assume the sand remains saturated with capillary water up to the original level).

Example

sand density = 1930kg/cu.m

4.8m

W.T.

3.6m saturated

sand

saturated density = 2010 kg/cu.m

2.4m clay

When water is seeping through the pores of a soil, total head is dissipated as viscous friction producing a frictional drag, acting in the direction of flow, on the solid particles.

A transfer of energy thus takes place from the water to the solid particles and the force corresponding to this energy transfer is called seepage force.

Influence of Seepage

Seepage forces act on the particles of a soil in addition to gravitational forces and the combination of the forces on a soil mass due to gravity and seeping water is called the resultant body force.

It is the resultant body force that governs the effective normal stress on a plane within a soil mass through which seepage is taking place.

Influence of Seepage

• Consider the special case of seepage vertically upwards. When the critical hydraulic gradient (ic), the effective normal stress on any plane will be zero, gravitational forces having been cancelled out by upward seepage forces. In the case of sands the contact forces between particles will be zero and the soil will have no strength.

The quick condition

• The soil is then said to be in a quick condition (quick meaning ‘alive’) and if the critical gradient is exceeded the surface will appear to be ‘boiling’ as the particles are moved around in the upward flow of water.

• It should be realized that ‘quicksand’ is not a special type of soil but simply sand through which there is an upward flow of water under a hydraulic gradient equal to or exceeding ic. It is possible for clays to have strength at zero effective normal stress.

The quick condition

• It has been shown that active pressure is associated with lateral expansion of the soil and is a minimum value; passive pressure is associated with lateral compression of the soil and is a maximum value. If the lateral strain in the soil is zero the corresponding lateral pressure is called the earth pressure at-rest and is usually expressed in terms of effective stress by the equation: PO=KOy’zwhere KO is defined as the coefficient of earth pressure at-rest, in terms of effective stress.

Earth pressure at rest

Since the at-rest condition does not involve failure of the soil (it represents a state of ‘elastic’ equilibrium) the Mohr circle representing the vertical and horizontal stresses does not touch the failure envelope and the horizontal stress cannot be evaluated. The value of KO, however, can be determined experimentally by means of a triaxial test in which the axial stress and the all-round pressure are increased simultaneously such that the lateral strain in the specimen is maintained at zero.

Earth pressure at rest

2. EFFECTIVE STRESS Revision of effective stress, total stress.

Effective stress profiles when there is seepage. Coefficient of earth pressure at rest K0. Calculation of horizontal effective stress profiles.

3. STRESS and STRAIN Definitions of stresses and strains. Sign

conventions. Principal stresses in axi-symmetric and plane

strain problems.