Relationship Between the SPT values with Angle of shearing resistance, Compactness, and Relative Density

The relationship between the standard penetration test(SPT or N) values with angle of shearing resistance(ɸ), compactness, and relative density are as follows:

Swedish circle method for analyzing the slope stability:

This method was first introduced by Fellenius (1926) for the analysis of slope stability of C-φ soil. In this method, the soil mass above the assumed slip circle is divided into a number of vertical slices of equal width. The forces between the slices are neglected and each slice is considered to be an independent column of soil of unit thickness. The location of the center of the failure arc (o) is assumed.

The weight of slice [latex] (w) = \gamma bh [/latex] Where,

[latex] \gamma [/latex] = Bulk unit weight of soil;

h = Average height of slice &

b = Width of slice.

The weight of each slice (w) is plotted as a vector and it passes through the midpoint of each slice and is divided into normal & tangential components N & T respectively.

Then,

N=W cosα

T=W sinα

Considering the whole slip surface AB of length ‘L’. If ‘c’ is the unit cohesion, then the resisting force, from Coulomb’s equation is equal to (c L + N tanφ’):

Now,

Driving force = ΣT Resisting force =c L + tanφ’ ΣN

Driving moment =(ΣT) × R Resisting moment = c L R+ (tanφ’ ΣN × R).

ΣT = Sum of all tangential components.

ΣN = Sum of all normal components.

So, the Factor of safety (FOS) against sliding is

[latex] F =\frac{Resisting Moment}{Driving Moment}= \frac{c L + \tan \phi \sum N}{\sum T} [/latex]

Putting, N=W cosα, and T=W sinα

[latex] F = = \frac{c L + \tan \phi \sum W \cos \alpha }{\sum W \sin \alpha } [/latex]

Following are the 9 assumptions of Coulomb’s wedge theory or Coulomb’s wedge failure theory

1. The backfill is dry and cohesionless.

2. The backfill is also homogenous and isotropic.

3. The backfill surface is a plane surface and can be inclined.

4 The back of the wall is vertical and can be inclined.

5. The failure surface is a planner surface that passes through the heel of the wall.

6. The backfill is elastically non-deformable but may be breakable.

7. The position and the line of action of the earth’s pressure are known.

8. Friction is considered between the wall and backfill soil, Therefore, the contact surface is considered rough.

9. The sliding wedge is considered to be a rigid body and the earth pressure is obtained by considering the limiting equilibrium of the sliding wedge as a whole.

Relation Between Submerged Unit Weight, Specific Gravity, Void Ratio and Unit Weight of Water

In this article, we shall make a formula or relation between the submerged unit weight([latex]{\gamma }'[/latex]), specific gravity(G), void ratio(e) and unit weight of water/ density of water([latex]\gamma_{w} [/latex]).

Typical Values of Permeability for Different Soils

The permeability or coefficient of permeability can be defined as the rate of flow of water through a unit cross-sectional area of the soil under a unit hydraulic gradient at a temperature of [latex]10^{\circ}[/latex]C. Typical values of permeability or coefficient of permeability for different soils are as follows:

Relation Between Dry Unit Weight, Specific Gravity, Percentage of Air Voids, and Water Content

In this article, we shall make the formula or relation between thedry unit weight([latex]\gamma _{d}[/latex]), specific gravity(G), percentage of air voids, and water content(w).

[ We know, [latex] \gamma_{s} = \frac{W_{s}}{V_{s}} [/latex], and Specific Gravity(G) [latex] = \frac{\gamma_{s}}{\gamma_{w} } = \frac{W_{s}}{V_{s}\gamma_{w} }[/latex] ]

[From this equation, [latex]G = \frac{W_{s}}{V_{s}\gamma_{w} }[/latex], we can write [latex] V_{s} = \frac{W_{s}}{G\gamma _{w}} [/latex]. Now, place the value of [latex] V_{s} [/latex] in the above equation.]

5 Empirical Formula for Determination of the Coefficient of Permeability

The coefficient of permeability of soil(K) can also be computed from several empirical formulae, which are given below:

1. Jaky’s Formula

Jakey in 1944 found an empirical formula of ‘K’ for all soils; [latex] K = 100 D_{m}^{2} [/latex] , Where [latex] D_{m} [/latex] is grain size (in cm) that occurs with the greatest frequency.

2. Allen Hazen’s Formula

[latex] K = C D_{10}^{2} [/latex] , where C is a constant. The value of C is taken approximately equal to 100, when [latex] D_{10} [/latex] is expressed in cm.

3. Terzaghi’s Formula

Terzaghi (1955) developed the following formula for fairly uniform sands, which reflects the effects of grain size and void ratio.

[latex] K = 200 D_{e}^{2} . e^2 [/latex] , Where [latex] D_{e} [/latex] = Effective grain size.

4. Kozney’s Formula

[latex] K = \frac{1}{K_{k} \eta S_{s}^{2}} \times \frac{n^3}{1-n^3} [/latex] , where [latex] S_{s} [/latex] = Specific surface of particles, [latex] \eta [/latex] = Viscosity, and [latex] K_{k} [/latex] = Constant( that is equal to 5 for spherical particles).

5. Louden’s Formula

[latex] \Rightarrow log_{10} (KS_{s}^{2}) = a + bn [/latex]

Where a and b are constants, the value of which are 1.365 and 5.15 respectively for permeability at 10°C.

In the previous articles, We have discussedFalling Head Permeameter and Constant Head Permeameterfor the determination of the coefficient of permeability of the soil. In this article, we are going to discuss the capillarity permeability test or the horizontal capillarity test, which is another method used for determining the coefficient of permeability as well as the value of the capillarity height of the soil sample.

In this test, a sample of dry soil, at the desired density is placed in a transparent glass or lucite tube, about 4cm in diameter and 35 cm long. Water is allowed to flow through the soil sample from one end, under a constant head [latex] h_0 [/latex] and the other end is kept open to the atmosphere through an air vent tube.

At any time interval ‘t’ after the beginning of the test, let the capillary water travels through a distance x from point A to B. At the point A, there is a pressure head [latex] h_0 [/latex], while at the point B, there is a pressure deficiency [latex] h_c [/latex] ( i.e., a negative pressure head ).

Head lost in causing flow from A to B (x distance apart)

[latex] = h_{0} - (-h_{c}) = h_{0} + h_{c} [/latex]
Hydraulic gradient
[latex] i = \frac{h_{0} + h_{c}}{x} [/latex]
From Darcy's law
[latex] v = K.i [/latex] [Where [latex] v_{s} = \frac{v}{n} [/latex] ]
Or, [latex] n.v _{s} = K.i [/latex]
Where [latex] v [/latex] = Discharge velocity,
[latex] v_{s} [/latex] = Seepage velocity
and [latex] n [/latex] = Porosity
But seepage velocity = [latex] \frac{dx}{dt} [/latex]

If the coefficient of permeability is designated as [latex] K_u [/latex] at a partial saturation [latex] S [/latex], then the above expression may be rewritten as:

Integrating between the limit [latex] x_1 [/latex] and [latex] x_2 [/latex] for [latex] x [/latex], and corresponding values of time [latex] t_1 [/latex] and [latex] t_2 [/latex], we get

In the above equation, there are two unknowns: [latex] k [/latex] and [latex] h_c [/latex]. In order to overcome this difficulty, the test is performed in two stages, with two different heads of water [latex] h_0 [/latex]. The first set of observations is taken under a head [latex] h_{(0)1} [/latex] up to capillary saturation penetrating about half the length of the test tube.

As the capillarity saturation proceeds, the value of [latex] x [/latex] is recorded at various time intervals. A graph between [latex] x^2 [/latex] and [latex] t [/latex] will give a straight line, and its slope is given as:

In the second stage, for the next half of the tube, is performed with head [latex] h_{(0)2} [/latex], and the recorded values of x and t are again used to plot a graph between [latex] x^2 [/latex] and [latex] t [/latex]. The slope of this line, say [latex] m_2 [/latex], is now determined.

The degree of saturation S can also be found by measuring the wet mass of the soil sample at the end of the experiment. The porosity is computed from the known dry mass, volume and specific gravity.