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Compressibility & Consolidation of Soils - Notes | Study Soil Mechanics - Civil Engineering (CE)

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 Compressibility and Consolidation 5

  1. Coefficient of Compressibility (av)
    av = e- e2- σ1
    e1 = Void ratio at effective stress Compressibility & Consolidation of Soils - Notes | Study Soil Mechanics - Civil Engineering (CE)
    e2 =Void ratio at effective stress Compressibility & Consolidation of Soils - Notes | Study Soil Mechanics - Civil Engineering (CE)Compressibility & Consolidation of Soils - Notes | Study Soil Mechanics - Civil Engineering (CE)ΔV/V0 = ΔH/H0
    ΔV = Change in volume in m3, or cm3
    V0 = Initial volume in m3 or cm3.
    ΔH = Change in depth in 'm' or 'cm'.
    H0 = original depth in 'm' or 'cm'.
  2. Coefficient of Compression (Cc)
    (i) Compressibility & Consolidation of Soils - Notes | Study Soil Mechanics - Civil Engineering (CE) 
     
    Compressibility & Consolidation of Soils - Notes | Study Soil Mechanics - Civil Engineering (CE)
    (ii) Cc = 0.009(WL-10)
    For undisturbed soil of medium sensitivity.
    WL = % liquid limit.
    (iii) C= 0.009(WL-7)
    For remolded soil of low sensitivity
    (iv) Cc = 0.40(e0-0.25)
    For undisturbed soil of medium sensitivity e0 = Initial void ratioCompressibility & Consolidation of Soils - Notes | Study Soil Mechanics - Civil Engineering (CE)(v) For remoulded soil of low sensitivity.
    Cc = 1.15(e- 0.35)
    (vi) Cc = 0.115w where, w = Water content
  3. Over consolidation ratio
    O.C.R = Maximum effective stress applied in the past/Existing effective stress
    O.C.R > 1 For over consolidated soil.
    O.C.R = 1 For normally consolidated soil.
    O.C.R < 1 For under consolidated soil.

Compressibility & Consolidation of Soils - Notes | Study Soil Mechanics - Civil Engineering (CE) 

Differential Equation of 1-D Consolidation

Compressibility & Consolidation of Soils - Notes | Study Soil Mechanics - Civil Engineering (CE) where, u = Excess pore pressure.

∂u / ∂t  = Rate of change of pore pressure
Cv = Coefficient of consolidation
∂u / ∂z = Rate of change of pore pressure with depth.

  1. Coefficient of volume compressibility mv = av/1+e0 where, e0 = Initial void ratio
    mv = Coefficient of volume compressibility
    Compression modulus
    Ec = 1/mwhere, Ec =Compression modulus.
  2. Degree of consolidation
    (i) %U = (1-(U/U1)x100) where,
    %U = % degree of consolidation.
    U = Excess pore pressure at any stage.
    U1 = Compressibility & Consolidation of Soils - Notes | Study Soil Mechanics - Civil Engineering (CE) = Initial excess pore pressure
    at t = 0, u = u1 ⇒ %u = 0%
    at t = ∞, u = 0 ⇒ %u = 100%
    (ii) %u = (e0-e/e0-ef)x100 where,
    ef = Void ratio at 100% consolidation.
    i.e. of t = ∞
    e = Void ratio at time 't'
    e0 = Initial void ratio i.e., at t = 0
    (iii) %u = (Δh / ΔH) x 100 where,
    ΔH = Final total settlement at the end of completion of primary consolidation i.e.,
    at t = ∞
    Δh = Settlement occurred at any time 't'.
  3. Time factor
    Tv = Cv.(t/d2) where, TV = Time factor
    CV = Coeff. of consolidation in cm2/sec.
    d = Length of drainage path
    t = Time in 'sec'
    d = H0/2 For 2-way drainage
    d = H0 For one-way drainage.
    where, H0 = Depth of soil sample.
    (i) Tv = (π/4)(u)2 ... if u ≤ 60% T50 = 0.196
    (ii) Tv = -0.9332log10(1-u)-0.0851... 
     if u > 60%

Method to find 'Cv'

  1. Square Root of Time Fitting Method
    Cv = (T90.d2)/t90 where,
    T90 = Time factor at 90% consolidation
    t90 = Time at 90% consolidation
    d = Length of drainage path.
  2. Logarithm of Time Fitting Method
    Cv = T50.d2/t50
    where, T50 = Time factor at 50% consolidation
    t50 = Time of 50% consolidation.

Compression Ratio

  1. Initial Compression Ratio
    Compressibility & Consolidation of Soils - Notes | Study Soil Mechanics - Civil Engineering (CE) 
    where, Ri = Initial reading of dial gauge.
    R= Reading of dial gauge at 0% consolidation.
    Rf = Final reading of dial gauge after secondary consolidation.
  2. Primary Consolidation Ratio
    Compressibility & Consolidation of Soils - Notes | Study Soil Mechanics - Civil Engineering (CE) 
    where, R100 = Reading of dial gauge at 100% primary consolidation.
  3. Secondary Consolidation Ratio
    Compressibility & Consolidation of Soils - Notes | Study Soil Mechanics - Civil Engineering (CE) ri+rp+rs = 1

Total Settlement

S = S+ Sp + Ss where, Si = Initial settlement
Sp = Primary settlement
S= Secondary settlement

  1. Initial Settlement
    Compressibility & Consolidation of Soils - Notes | Study Soil Mechanics - Civil Engineering (CE)
    For cohesionless soil.
    where, Cs = 1.5(Cr0)
    where, Cr = Static one resistance in kN/m2
    H0 = Depth of soil sample Compressibility & Consolidation of Soils - Notes | Study Soil Mechanics - Civil Engineering (CE) For cohesive soil.
    where, It = Shape factor or influence factor
    A = Area.
  2. Primary Settlement
    (i) Compressibility & Consolidation of Soils - Notes | Study Soil Mechanics - Civil Engineering (CE)
    (ii)Compressibility & Consolidation of Soils - Notes | Study Soil Mechanics - Civil Engineering (CE)
    (iii)Compressibility & Consolidation of Soils - Notes | Study Soil Mechanics - Civil Engineering (CE)
    (iv)Compressibility & Consolidation of Soils - Notes | Study Soil Mechanics - Civil Engineering (CE)Compressibility & Consolidation of Soils - Notes | Study Soil Mechanics - Civil Engineering (CE) 
    Compressibility & Consolidation of Soils - Notes | Study Soil Mechanics - Civil Engineering (CE) = Settlement for over consolidated stage
    Compressibility & Consolidation of Soils - Notes | Study Soil Mechanics - Civil Engineering (CE)= Settlement for normally consolidation stage 
    Compressibility & Consolidation of Soils - Notes | Study Soil Mechanics - Civil Engineering (CE)
  3. Secondary Settlement
    Compressibility & Consolidation of Soils - Notes | Study Soil Mechanics - Civil Engineering (CE) 
    where, H0∼H100
    100 = Thickness of soil after 100% primary consolidation.
    e100 = Void ratio after 100% primary consolidation.
    t2 = Average time after t1 in which secondary consolidation is calculated

Permeability

  1. Permeability of Soil
    The permeability of a soil is a property which describes quantitatively, the ease with which water flows through that soil.
  2. 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)
    qα(Δh/L)A → q = KiACompressibility & Consolidation of Soils - Notes | Study Soil Mechanics - Civil Engineering (CE)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
    i = HL/L where, HL = Head loss = (H1 – H2)
    i = tanθ(dy/dx)
    (ii) Seepage velocity
    Vs = V/n where, Vs = Seepage velocity (m/sec)
    n = Porosity & V = discharge velocity (m/s)
    (iii) Coefficiency of percolation
    KP = K/n where, KP = coefficiency of percolation and n = Porosity.
  3. Constant Head Permeability Test
    K = QL/tHLA 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.Compressibility & Consolidation of Soils - Notes | Study Soil Mechanics - Civil Engineering (CE)
  4. Falling Head Permeability Test or Variable Head Permeability Test
    K = 2.303aL/At(log10)(h/ h2)
    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
    h= level of upstream edge after 't'.Compressibility & Consolidation of Soils - Notes | Study Soil Mechanics - Civil Engineering (CE)
  5. Konzey-Karman Equation
    Compressibility & Consolidation of Soils - Notes | Study Soil Mechanics - Civil Engineering (CE) 
    Where, C = Shape coefficient, ∼5mm for spherical particle
    S = Specific surface area = Area/Volume
  6. For spherical particle.
    Compressibility & Consolidation of Soils - Notes | Study Soil Mechanics - Civil Engineering (CE) 
    R = Radius of spherical particle.
    S = 6/√ab
    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)
    γw = unit weight of water in kN/m3
    Compressibility & Consolidation of Soils - Notes | Study Soil Mechanics - Civil Engineering (CE)
  7. Allen Hazen Equation
    K=C.D210 Where, D10 = Effective size in cm. k is in cm/s C = 100 to 150
  8. Lioudens Equation
    log10KS2= a + b.n
    Where, S = Specific surface area
    n = Porosity.
    a and b are constant.
    Consolidation equation K = Cvmvγw
    Where, C= Coefficient of consolidation in cm2/sec
    mv = Coefficient of volume Compressibility in cm2/N
  9. Capillary Permeability TestCompressibility & Consolidation of Soils - Notes | Study Soil Mechanics - Civil Engineering (CE) i = h+ hc/x where, S = Degree of saturation
    K = Coefficient of permeability of partially saturated soil.
    Compressibility & Consolidation of Soils - Notes | Study Soil Mechanics - Civil Engineering (CE)
    where hc = remains constant (but not known as depends upon soil)
    = head under first set of observation,
    n = porosity, hc = capillary height
    Another set of data gives,
    Compressibility & Consolidation of Soils - Notes | Study Soil Mechanics - Civil Engineering (CE)  = head under second set of observation
    For S = 100%, K = maximum. Also, ku ∝ S.
  10. Permeability of a stratified soil
    (i) Average permeability of the soil in which flow is parallel to bedding plane,
    Compressibility & Consolidation of Soils - Notes | Study Soil Mechanics - Civil Engineering (CE)keq∼kxCompressibility & Consolidation of Soils - Notes | Study Soil Mechanics - Civil Engineering (CE)(ii) Average permeability of soil in which flow is perpendicular to bedding plane.
    Compressibility & Consolidation of Soils - Notes | Study Soil Mechanics - Civil Engineering (CE) keq∼kzCompressibility & Consolidation of Soils - Notes | Study Soil Mechanics - Civil Engineering (CE)(iii) For 2-D flow in x and z direction
    Compressibility & Consolidation of Soils - Notes | Study Soil Mechanics - Civil Engineering (CE)
    (iv) For 3-D flow in x, y and z direction keq = (kx.ky.kz)1/3
    Coefficient of absolute permeability (k0)
    Compressibility & Consolidation of Soils - Notes | Study Soil Mechanics - Civil Engineering (CE) 

Effective Stress, Capilarity, Seepage

  1. Seepage Pressure and Seepage Force
    Seepage pressure is exerted by the water on the soil due to friction drag. This drag force/seepage force always acts in the direction of flow.
    The seepage pressure is given by
    PS = hγω where, Ps = Seepage pressure
    γω = 9.81 kN/m3
    Here, h = head loss and z = length
    (i) FS = hAγω where, Fs = Seepage force
    (ii) fs = iγω where, fs = Seepage force per unit volume.
    i = h/z where, I = Hydraulic gradient.
  2. Quick Sand Condition
    It is condition but not the type of sand in which the net effective vertical stress becomes zero, when seepage occurs vertically up through the sands/cohesionless soils.
    Net effective vertical stress = 0
    ic = (G - 1)/(1 + e) where, ic = Critical hydraulic gradient.
    2.65 ≤ G ≤ 2.70 0.65 ≤ e ≤ 0.70
    To Avoid Floating Condition
    i < i and F.O.S = ic/i > 1

Laplace Equation of Two Dimensional Flow and Flow Net: Graphical Solution of Laplace Equation
(i) Compressibility & Consolidation of Soils - Notes | Study Soil Mechanics - Civil Engineering (CE)
where, ∅ = Potential function = kH
H = Total head and k = Coefficient of permeability
(ii) Compressibility & Consolidation of Soils - Notes | Study Soil Mechanics - Civil Engineering (CE) … 2D Laplace equation for Homogeneous soil.
where, ∅ = kX H and ∅ = ky H for Isotropic soil, kx= ky

Seepage discharge (q)
q = kh.(Nf/Nd) where, h = hydraulic head or head difference between upstream and downstream level or head loss through the soil.

  • Shape factor = Nf/Nd
  • Nf = Nψ - 1
    where, N= Total number of flow channels
    Nψ = Total number of flow lines.
  • Nd = N∅ - 1
    where, Nd = Total number equipotential drops.
    N∅ = Total number equipotential lines.
  • Hydrostatic pressure = U = hwγw
    where, U = Pore pressure hw = Pressure head
    hw = Hydrostatic head – Potential head
  • Seepage Pressure
    Ps = h'γw where, h' = h - (2h/Nd)

Compressibility & Consolidation of Soils - Notes | Study Soil Mechanics - Civil Engineering (CE)

  • Exit gradient,
    Compressibility & Consolidation of Soils - Notes | Study Soil Mechanics - Civil Engineering (CE) where, size of exit flow field is b x b.
    and ΔH = h/Nd is equipotential drop.

Phreatic Line

It is top flow line which follows the path of base parabola. It is a stream line. The pressure on this line is atmospheric (zero) and below this line pressure is hydrostatic.

  1. Phreatic Line with FilterCompressibility & Consolidation of Soils - Notes | Study Soil Mechanics - Civil Engineering (CE) 
    Compressibility & Consolidation of Soils - Notes | Study Soil Mechanics - Civil Engineering (CE)Phreatic line (Top flow line).

    Follows the path of base parabola
    CF = Radius of circular arc = Compressibility & Consolidation of Soils - Notes | Study Soil Mechanics - Civil Engineering (CE)
    C = Entry point of base parabola
    F = Junction of permeable and impermeable surface
    S = Distance between focus and directrix
    = Focal length.
    FH = S
    (i) q = ks where, q = Discharge through unit length of dam.
    (ii) Compressibility & Consolidation of Soils - Notes | Study Soil Mechanics - Civil Engineering (CE)
    (iii) Compressibility & Consolidation of Soils - Notes | Study Soil Mechanics - Civil Engineering (CE)
  2. Phreatic Line without FilterCompressibility & Consolidation of Soils - Notes | Study Soil Mechanics - Civil Engineering (CE) (i) For ∝ < 30°
    q = k a sin2
    Compressibility & Consolidation of Soils - Notes | Study Soil Mechanics - Civil Engineering (CE)
    (ii) For ∝ > 30°
    q = k a sin ∝ tan ∝ andCompressibility & Consolidation of Soils - Notes | Study Soil Mechanics - Civil Engineering (CE)  
The document Compressibility & Consolidation of Soils - Notes | Study Soil Mechanics - Civil Engineering (CE) is a part of the Civil Engineering (CE) Course Soil Mechanics.
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