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

 Compressibility and Consolidation 5

1. Coefficient of Compressibility (a_v)
   a_v = e₁ - e₂/σ₂ - σ₁
   e₁ = Void ratio at effective stress
   e₂ =Void ratio at effective stress 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 (C_c)
   (i)  
   ↓ 
   
   (ii) C_c = 0.009(W_L-10)
   For undisturbed soil of medium sensitivity.
   W_L = % liquid limit.
   (iii) C_c = 0.009(W_L-7)
   For remolded soil of low sensitivity
   (iv) C_c = 0.40(e₀-0.25)
   For undisturbed soil of medium sensitivity e_{0 = Initial void} ratio(v) For remoulded soil of low sensitivity.
   C_c = 1.15(e₀ - 0.35)
   (vi) C_c = 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.

Differential Equation of 1-D Consolidation

 where, u = Excess pore pressure.

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

1. Coefficient of volume compressibility m_v = a_v/1+e₀ where, e₀ = Initial void ratio
   m_v = Coefficient of volume compressibility
   Compression modulus
   E_c = 1/m_v where, E_c =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 =  = Initial excess pore pressure
   at t = 0, u = u₁ ⇒ %u = 0%
   at t = ∞, u = 0 ⇒ %u = 100%
   (ii) %u = (e₀-e/e₀-e_f)x100 where,
   e_f = 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
   T_v = C_v.(t/d₂) where, T_V = Time factor
   C_V = 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) T_v = (π/4)(u)² ... if u ≤ 60% T₅₀ = 0.196
   (ii) T_v = -0.9332log₁₀(1-u)-0.0851... 
    if u > 60%

Method to find 'Cv'

1. Square Root of Time Fitting Method
   C_v = (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
   C_v = T₅₀.d²/t₅₀
   where, T₅₀ = Time factor at 50% consolidation
   t₅₀ = Time of 50% consolidation.

Compression Ratio

1. Initial Compression Ratio
    
   where, R_i = Initial reading of dial gauge.
   R₀ = Reading of dial gauge at 0% consolidation.
   R_f = Final reading of dial gauge after secondary consolidation.
2. Primary Consolidation Ratio
    
   where, R₁₀₀ = Reading of dial gauge at 100% primary consolidation.
3. Secondary Consolidation Ratio
    r_i+r_p+r_s = 1

Total Settlement

S = S_i + S_p + S_s where, S_i = Initial settlement
S_p = Primary settlement
S_s = Secondary settlement

1. Initial Settlement
   
   For cohesionless soil.
   where, C_s = 1.5(C_r/σ₀)
   where, C_r = Static one resistance in kN/m²
   H₀ = Depth of soil sample  For cohesive soil.
   where, I_t = Shape factor or influence factor
   A = Area.
2. Primary Settlement
   (i) 
   (ii)
   (iii)
   (iv) 
    = Settlement for over consolidated stage
   = Settlement for normally consolidation stage 
   
3. Secondary Settlement
    
   where, H₀∼H₁₀₀
   H­₁₀₀ = 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 = KiAWhere, 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 = H_L/L where, H_L = Head loss = (H₁ – H₂)
   i = tanθ(dy/dx)
   (ii) Seepage velocity
   Vs = V/n where, V_s = Seepage velocity (m/sec)
   n = Porosity & V = discharge velocity (m/s)
   (iii) Coefficiency of percolation
   K_P = K/n where, K_P = coefficiency of percolation and n = Porosity.
3. Constant Head Permeability Test
   K = QL/tH_LA 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.
4. Falling Head Permeability Test or Variable Head Permeability Test
   K = 2.303aL/At(log₁₀)(h₁ / 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
   h₂ = level of upstream edge after 't'.
5. Konzey-Karman Equation
    
   Where, C = Shape coefficient, ∼5mm for spherical particle
   S = Specific surface area = Area/Volume
6. For spherical particle.
    
   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)
   γ_w = unit weight of water in kN/m³
   
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 = C_vm_vγ_w
   Where, C_v = Coefficient of consolidation in cm²/sec
   m_v = Coefficient of volume Compressibility in cm²/N
9. Capillary Permeability Test i = h₀ + h_c/x where, S = Degree of saturation
   K = Coefficient of permeability of partially saturated soil.
   
   where h_c = remains constant (but not known as depends upon soil)
   = head under first set of observation,
   n = porosity, h_c = capillary height
   Another set of data gives,
     = head under second set of observation
   For S = 100%, K = maximum. Also, k_u ∝ S.
10. Permeability of a stratified soil
    (i) Average permeability of the soil in which flow is parallel to bedding plane,
    k_eq∼k_x(ii) Average permeability of soil in which flow is perpendicular to bedding plane.
    k_eq∼k_z(iii) For 2-D flow in x and z direction
    
    (iv) For 3-D flow in x, y and z direction k_eq = (k_x.k_y.k_z)^1/3
    Coefficient of absolute permeability (k₀)
     

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
   P_S = hγ_ω where, P_s = Seepage pressure
   γ_ω = 9.81 kN/m³
   Here, h = head loss and z = length
   (i) F_S = hAγ_ω where, F_s = Seepage force
   (ii) f_s = iγ_ω where, f_s = 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
   i_c = (G - 1)/(1 + e) where, i_c = Critical hydraulic gradient.
   2.65 ≤ G ≤ 2.70 0.65 ≤ e ≤ 0.70
   To Avoid Floating Condition
   i < i and F.O.S = i_c/i > 1

Laplace Equation of Two Dimensional Flow and Flow Net: Graphical Solution of Laplace Equation
(i) 
where, ∅ = Potential function = kH
H = Total head and k = Coefficient of permeability
(ii)  … 2D Laplace equation for Homogeneous soil.
where, ∅ = k_X H and ∅ = k_y H for Isotropic soil, k_x= k_y

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

• Shape factor = N_f/N_d
• N_f = N_ψ - 1
  where, N_f = Total number of flow channels
  N_ψ = Total number of flow lines.
• N_d = N_∅ - 1
  where, N_d = Total number equipotential drops.
  N_∅ = Total number equipotential lines.
• Hydrostatic pressure = U = h_wγ_w
  where, U = Pore pressure h_w = Pressure head
  h_w = Hydrostatic head – Potential head
• Seepage Pressure
  P_s = h'γ_w where, h' = h - (2h/N_d)

• Exit gradient,
   where, size of exit flow field is b x b.
  and ΔH = h/N_d 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 Filter 
   Phreatic line (Top flow line).
   ↓
   Follows the path of base parabola
   CF = Radius of circular arc =
   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) 
   (iii) 
2. Phreatic Line without Filter (i) For ∝ < 30°
   q = k a sin² ∝
   
   (ii) For ∝ > 30°
   q = k a sin ∝ tan ∝ and