Soil Mechanics & Foundation : Principle of Effective Stress & Darcy's Law and 1D-Permeability

By Sajal Gupta|Updated : October 8th, 2021

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Effective Stress Principle, Capillarity & Seepage

Stresses in the Ground

Total Stress
When a load is applied to soil, it is carried by the solid grains and the water in the pores. The total vertical stress acting at a point below the ground surface is due to the weight of everything that lies above, including soil, water, and surface loading. Total stress thus increases with depth and with unit weight.

Vertical total stress at depth z, sv = g.Z

Below a water body, the total stress is the sum of the weight of the soil up to the surface and the weight of water above this. sv = g.Z + gw.Zw

 
The total stress may also be denoted by sz or just sIt varies with changes in water level and with excavation.

Pore Water Pressure
The pressure of water in the pores of the soil is called pore water pressure (u). The magnitude of pore water pressure depends on:

  • the depth below the water table.
  • the conditions of seepage flow.

Under hydrostatic conditions, no water flow takes place, and the pore pressure at a given point is given by 
u = gw.h

where h = depth below water table or overlying water surface

It is convenient to think of pore water pressure as the pressure exerted by a column of water in an imaginary standpipe inserted at the given point.

The natural level of ground water is called the water table or the phreatic surface. Under conditions of no seepage flow, the water table is horizontal. The magnitude of the pore water pressure at the water table is zero. Below the water table, pore water pressures are positive. 

Principle of Effective Stress

The principle of effective stress was enunciated by Karl Terzaghi in the year 1936. This principle is valid only for saturated soils, and consists of two parts:

1. At any point in a soil mass, the effective stress (represented by σ' or s) is related to total stress (s) and pore water pressure (u) as

σ's - u 

Both the total stress and pore water pressure can be measured at any point.

2. All measurable effects of a change of stress, such as compression and a change of shearing resistance, are exclusively due to changes in effective stress.

Compression = f1')
Shear Strength = f2')

In a saturated soil system, as the voids are completely filled with water, the pore water pressure acts equally in all directions.

The effective stress is not the exact contact stress between particles but the distribution of load carried by the soil particles over the area considered. It cannot be measured and can only be computed.

If the total stress is increased due to additional load applied to the soil, the pore water pressure initially increases to counteract the additional stress. This increase in pressure within the pores might cause water to drain out of the soil mass, and the load is transferred to the solid grains. This will lead to the increase in effective stress. 

Effective Stress in Unsaturated Zone

Above the water table, when the soil is saturated, pore pressure will be negative (less than atmospheric). The height above the water table to which the soil is saturated is called the capillary rise, and this depends on the grain size and the size of pores. In coarse soils, the capillary rise is very small.

Between the top of the saturated zone and the ground surface, the soil is partially saturated, with a consequent reduction in unit weight . The pore pressure in a partially saturated soil consists of two components: 
Pore water pressure = uw
Pore air pressure = ua

Water is incompressible, whereas air is compressible. The combined effect is a complex relationship involving partial pressures and the degree of saturation of the soil. 

Effective Stress Under Hydrodynamic Conditions

There is a change in pore water pressure in conditions of seepage flow within the ground. Consider seepage occurring between two points P and Q. The potential driving the water flow is the hydraulic gradient between the two points, which is equal to the head drop per unit length. In steady state seepage, the gradient remains constant.

Hydraulic gradient from P to Q, i = dh/d


As water percolates through soil, it exerts a drag on soil particles it comes in contact with. Depending on the flow direction, either downward of upward, the drag either increases or decreases inter-particle contact forces.

A downward flow increases effective stress.

In contrast, an upward flow opposes the force of gravity and can even cause to counteract completely the contact forces. In such a situation, effective stress is reduced to zero and the soil behaves like a very viscous liquid. Such a state is known as quick sand condition. In nature, this condition is usually observed in coarse silt or fine sand subject to artesian conditions.

At the bottom of the soil column, 

During quick sand condition, the effective stress is reduced to zero.

where icr = critical hydraulic gradient

This shows that when water flows upward under a hydraulic gradient of about 1, it completely neutralizes the force on account of the weight of particles, and thus leaves the particles suspended in water. 

The Importance of Effective Stress

  • At any point within the soil mass, the magitudes of both total stress and pore water pressure are dependent on the ground water position. With a shift in the water table due to seasonal fluctuations, there is a resulting change in the distribution in pore water pressure with depth.

  • Changes in water level below ground result in changes in effective stresses below the water table. A rise increases the pore water pressure at all elevations thus causing a decrease in effective stress. In contrast, a fall in the water table produces an increase in the effective stress.

  • Changes in water level above ground do not cause changes in effective stresses in the ground below. A rise above ground surface increases both the total stress and the pore water pressure by the same amount, and consequently effective stress is not altered.

  • If both total stress and pore water pressure change by the same amount, the effective stress remains constant.
  • Total and effective stresses must be distinguishable in all calculations.Ground movements and instabilities can be caused by changes in total stress, such as caused by loading by foundations and unloading due to excavations. They can also be caused by changes in pore water pressures, such as failure of slopes after rainfall. 

Permeability

Permeability of Soil

The permeability of a soil is a property which describes quantitatively, the ease with which water flows through that soil.

Darcy's Law : Darcy established that the flow occurring per unit time is directly proportional to the head causing flow and the area of cross-section of the soil sample but is inversely proportional to the length of the sample.

(i) Rate of flow (q)

image074

image075

Where, q = rate of flow in m3/sec.

 K = Coefficient of permeability in m/s

 I = Hydraulic gradient

 A = Area of cross-section of sample

image076 

where, HL = Head loss = (H1 – H2)

image077

(ii) Seepage velocity

image078 

where, Vs = Seepage velocity (m/sec)

n = Porosity & V = discharge velocity (m/s)

(iii) Coefficiency of percolation

image079 

where, KP = coefficiency of percolation and n = Porosity.

Constant Head Permeability Test

image080

 where, Q = Volume of water collected in time t in m3.

Constant Head Permeability test is useful for coarse grain soil and it is a laboratory method.

image081

Falling Head Permeability Test or Variable Head Permeability Test

image082

a = Area of tube in m2

A = Area of sample in m2

t = time in 'sec'

L = length in 'm'

h1 = level of upstream edge at t = 0

h2 = level of upstream edge after 't'.

image083

Konzey-Karman Equation

image084

Where, C = Shape coefficient, ∼5mm for spherical particle

S = Specific surface area = image086

For spherical particle.

image087

R = Radius of spherical particle.

image088

When particles are not spherical and of variable size. If these particles passes through sieve of size 'a' and retain on sieve of size 'n'.

e = void ratio

μ = dynamic viscosity, in (N-s/m2)

image089 = unit weight of water in kN/m3

image090

Allen Hazen Equation

image091 

Where, D10 = Effective size in cm. k is in cm/s C = 100 to 150

Lioudens Equation

image092

Where, S = Specific surface area

 n = Porosity.

a and b are constant.

Consolidation equation 

image093

Where, Cv = Coefficient of consolidation in cm2/sec

mv = Coefficient of volume Compressibility in cm2/N

Capillary Permeability Test

image094

image095 

where, S = Degree of saturation

K = Coefficient of permeability of partially saturated soil.

image098

where hc = remains constant (but not known as depends upon soil)

image099 = head under first set of observation,

n = porosity, hc = capillary height

Another set of data gives,

image100

image099 = head under second set of observation

  • For S = 100%, K = maximum. Also, ku ∝ S.

Permeability of a stratified soil

(i) Average permeability of the soil in which flow is parallel to bedding plane,

image100image101

image102

(ii) Average permeability of soil in which flow is perpendicular to bedding plane.

image103image104

image105

(iii) For 2-D flow in x and z direction

image106

(iv) For 3-D flow in x, y and z direction image107

Coefficient of absolute permeability (k0)

image108

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Sajal GuptaSajal GuptaMember since Mar 2020
B.E- Civil, Thapar University, Patiala Cracked Gate-2017, 2018, 2020 1 yr work exp at Wapcos, MoWR
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