Lateral earth pressure:

• The pressure exerted by soil in horizontal direction.

Lateral earth pressure = K * over burden stress

or, σ_h = K* σ_v

or, σ_h = K* γZ

where,

K = Coefficient of lateral pressure

Types of lateral earth pressure:

1. Earth pressure at rest

• When soil mass is not subjected any lateral yielding or movement the pressure in this condition is known as earth pressure at rest.

2. Active earth pressure

• Occurs when soil mass yields in such a way that it tends to stretch horizontally.
• It is the state of plastic equilibrium as the entire soil mass is on verge of failure.

3. Passive earth pressure

• When retaining wall moves inward to the backfill the soil gets compressed and failure due to upward movement of wedge occurs.

1.2 Rankine’s earth pressure theory for active and passive state

Rankine’s earth pressure theory:

Assumption

• Soil is homogenous and semi-infinite.
• Back of retaining wall is vertical and smooth.
• Ground surface is plane which may be horizontal or inclined.
• Soil is dry and cohesion less.
• Wall movement is sufficient so that plastic equilibrium is fulfilled.

Rakine’s various backfill condition are:

A. Rankine theory for cohesionless soil (C=0)

1. Rankine theory for horizontal backfill

a. Active earth pressure

Consider,

• σ_v = γZ
• Intially there is no lateral movement.

i.e σ_h = K_o* σ_v

• As the wall moves away from the soil σ_v remains same but σ_h decreases till failure occurs i.e σ_h  → σ_a.

As wall moves away

σ_a = K_a σ_v

Now, for expression of K_a

References:

• Terzaghi, Karl, Peck, R.B & John, Wiley (1969) Soil mechanics in engineering practice, New York.
• Arora , K.R (2008), Soil mechanics and foundation engineering, Delhi: Standard Publisher Distribution.

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Gravitational water:

• Percolates through the soil under gravity
• Pore water pressure at GWT = 0
• Soil above GWT is saturated by capillary action (pore water pressure is tensile and negative = -ϒwh)
• Above GWT – impervious formation above- local saturation
• Water occurring at local saturation – perched water

Held water:

• Water held in soil pores other than gravity.

Types of Held Water:

Structural Water

• Remains in the crystal structure of soil minerals chemically, cannot removed by normal drying (105-110⁰C)

Hygroscopic Water

• Removed by normal drying, replaced back

• Capillary Water
• Due to surface tension

Height of Capillarity = 4T/ϒwd

Note:

Soil Suction: in case of capillary tube, water above WT has a ‘-‘ pressure, soil remains in a state of reduced pressure, known as soil suction (P_F = log₁₀(h_c))

Permeability:

• Flow of water through interconnected voids.

Darcy’s Law :

Laminar flow:

• Flow follows a well-defined path and doesn’t cross the path of other particles (Re ≤ 2000)

Turbulent flow:

• Fluctuates with the time both in magnitude and direction (Re ≥ 2000)

v ∝ i

v = ki

q = vA = Aki

therefore, q= Aki

Seepage Velocity:

Seepage Velocity > Avg. Velocity (v = q/A; v_s = q/A_s)

q = v*A = v_s*A_s

v_s =v* A/A_s

     = v V/v_s

     = v/n

     = ki/n

K = cd² * (e³/(1+e)) * (ϒ_w/μ)

K ∝ c

K ∝ (1/μ) 

{(K_T/K₂₀) = (μ₂₀/μ_T)}

K ∝ e

K ∝ T (temp^r)

Note:

• Impurities (salts, alkalis) increases, K decreases
• Entrapped air, k decreases

Determination of coefficient of Permeability:

(A) Laboratory Method:

Constant Head Method (k > 10^-4)

• Coarse grained soil
• Constant head of water at supply tank
• Vol. of water flowing out of permeameter per unit time = q

H= head water level – tail water level

k_stone > k_soil

From Darcy’s law

q = Aki

   = Ak h/L  

 Where L = soil specimen ht.

Therefore,

k = qL/Ah

= qL/(A*h*t)   

Where,

k = (k1+K2+ ………………Kn)/n

Falling head method (10^-7 <K< 10^-4)

• Undisturbed sample
• Fine-grained soils
• A standpipe of area ‘a’ above the cylinder
• Variable head (h1 to h2) at time t

 rate of change of head = -dh/dt

Acc. Darcy’s Law

q = A*k*i

a*v = A*k*i

-a* (dh/dt) = A*k* (h/L)

After Integrating from 0 to t and from h1 to h2;

K = 2.303*(q/A) *(L/t) *log₁₀(h1/h2)

Note:

• soil in the permeameter should be fully saturated.

Determination of Coeff. Of Permeability (Field):

(B) In-situ method:

• More reliable than the lab method

(a) Pumping test (pumping out)

• Continuous pumping from test well and observation in bore wells (observation well) till steady state.
• Min. two observation wells
• Well should penetrate full depth of water-bearing strata

Confined Aquifer – upper and lower surface impervious

Unconfined Aquifer- no overburden lying over them, top most water-bearing strata.

Unconfined Aquifer:

Observation wells – (r1 and r2 apart test well)

Ht. of observations wells – (z1 and z2)

i = dh/dr

z =ht. of water level at a distance, r

A= 2π*r*z

Darcy’s law;

q = kAi

   = 2π*r*z* dh/dr

On integrating;

K = {2.303*q*log10(r2/r1)}/ {π*(z_z² – z₁²)}

If only one observation well;

K= {2.303*q*log10(R/rw)}/ {π*(H² – hw²)}

R = Radius of influence

H = depth to the bottom of the aquifer from WT

Rw = radius of the test well

Hw = ht. of water in the test well

 Confined Aquifer:

r, r1, r2 – same as above

z.z1.z2 – ht. from the bottom impervious layer (same as above)

b= spacing between two layers

i= dz/dr

A = 2π * r* b

q= kAi

= k * dz/dr * 2π*r*b

On integrating,

K = {q*log10(R/rw)}/ {2.727*b*(h – hw)}

R = 3000*d*√k, R in m

d = draw down, m

k= Coeff. of permeability, m/sec

(b) Bore Hole Test

Constant head method:

• Water    is allowed to flow through the bottom of the bore
• Low end of the casing should not be less than 5d from the top and bottom of the stratum.
• Water level in borehole = constant

Therefore, k = q/(2.7*d*h)

Where d = dia. of well

H = head above GWT

Variable head method:

• Drop from h1 to h2
• For D ≤ 1.5 m

k = (πd/11t) * log(h1/h2)

• For D > 1.5 m

K= (πd²/81t) * log (2L/d) * log (h1/h2)

Insitu measurement of seepage velocity:

• Two trial pits A & B
• Dye inserted in A
• Observe in t time in B

h= level diff. between A and B

i = h/AB

v_s (seepage velocity) =AB/t

We know;

v_s =ki/n

Therefore,

AB/t =(k*h)/(n*AB)

k = (AB²*n)/(t*h)

Permeability on layered deposit:

h= h1+h2

q= q1+q2

q = kz * A * h/(h1+h2)

= k1*A*h1/h1

=k2*A*h2/h2

Therefore,

k =(h1+h2)/{(h1/k1)+(h2/k2)}

In direction of bedding;

i = i1 =i2

q = q1+q2

A*k_x*i = A*k1*i + A*k2*i

k = (k1*h1+k2*h2)/(h1+h2)

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