Compressibility & Consolidation of Soils | Soil Mechanics - Civil Engineering (CE) PDF Download

Compressibility and Consolidation - notes and formulae

Coefficient of Compressibility (a_v)

a_v = (e₁ - e₂)/(σ₂ - σ₁)

Here, e₁ is the void ratio corresponding to effective stress σ₁ and e₂ is the void ratio corresponding to effective stress σ₂. The coefficient of compressibility expresses change of volume (or void ratio) per unit change in effective stress.

Relationship between volume change and thickness change:

ΔV/V₀ = ΔH/H₀

• ΔV = change in volume (m³ or cm³).
• V₀ = initial volume (m³ or cm³).
• ΔH = change in depth (m or cm).
• H₀ = original depth (m or cm).

Coefficient of Compression (C_c)

The coefficient of compression (C_c) is used with void ratio-log stress plots (e-log σ′) to estimate primary consolidation settlement.

• Empirical and approximate relations (useful for quick estimates):
  • C_c = 0.009(W_L - 10) for undisturbed soil of medium sensitivity, where W_L is liquid limit (%) .
  • C_c = 0.009(W_L - 7) for remoulded soil of low sensitivity.
  • C_c = 0.40(e₀ - 0.25) for undisturbed soil of medium sensitivity, where e₀ is initial void ratio.
  • C_c = 1.15(e₀ - 0.35) for remoulded soil of low sensitivity.
  • C_c = 0.115 w where w is water content (expressed in the same units as required by the empirical relation).

Overconsolidation Ratio (O.C.R.)

O.C.R. = Maximum past effective stress (preconsolidation stress) / Current effective stress

• O.C.R. > 1 → overconsolidated soil.
• O.C.R. = 1 → normally consolidated soil.
• O.C.R. < 1 → underconsolidated (usually not encountered in static natural soils).

Differential Equation of One-Dimensional Consolidation

The governing differential equation for one-dimensional consolidation (Terzaghi) is written in terms of excess pore pressure u(z,t):

∂u/∂t = C_v ∂²u/∂z²

where:

• u = excess pore pressure (function of depth z and time t).
• ∂u/∂t = rate of change of pore pressure with time.
• ∂²u/∂z² = curvature of pore pressure distribution with depth.
• C_v = coefficient of consolidation (m²/s or cm²/s).

Coefficient of Volume Compressibility and Compression Modulus

Coefficient of volume compressibility m_v is related to a_v by

m_v = a_v / (1 + e₀)

where e₀ is the initial void ratio.

Compression modulus E_c is the reciprocal of m_v:

E_c = 1 / m_v

Degree of Consolidation (U)

• Formulation from excess pore pressure:

  %U = (1 - u/u₁) × 100%

  Here u is the excess pore pressure at time t and u₁ is the initial excess pore pressure (at t = 0). At t = 0, %U = 0%; as t → ∞, u → 0 and %U → 100%.

  
• Formulation using void ratio:

  %U = [(e₀ - e)/(e₀ - e_f)] × 100%

  where e is void ratio at time t, e₀ is initial void ratio (t = 0) and e_f is void ratio at 100% primary consolidation (t = ∞).

• Formulation using settlement:

  %U = (Δh / ΔH) × 100%

  where Δh is settlement at time t and ΔH is final total settlement due to primary consolidation (settlement at t = ∞).

Time Factor, T_v, and Relations with Degree of Consolidation

The time factor T_v is defined as:

T_v = C_v t / d²

where:

• C_v = coefficient of consolidation (m²/s or cm²/s).
• t = time (s).
• d = length of drainage path (m or cm). For two-way drainage d = H₀/2; for one-way drainage d = H₀.
• H₀ = thickness of consolidating layer or sample.

Approximate relations between T_v and degree of consolidation U (useful for hand calculations and laboratory curve fitting):

• For U ≤ 60%: T_v ≈ (π/4) U². A commonly used value is T₅₀ ≈ 0.196 for U = 50%.
• For U > 60%: T_v ≈ -0.9332 log₁₀(1 - U) - 0.0851.

Methods to Determine C_v from Consolidation Tests

• Square-root of time fitting method:

  C_v = (T₉₀ × d²) / t₉₀

  where T₉₀ is the time factor corresponding to 90% consolidation and t₉₀ is the observed time for 90% consolidation. d is the drainage path.

• Logarithm-of-time fitting method:

  C_v = (T₅₀ × d²) / t₅₀

  where T₅₀ is the time factor for 50% consolidation and t₅₀ is the observed time for 50% consolidation.

Compression Ratios

• Initial Compression Ratio

  Defined from dial gauge readings; typical notation uses initial reading R_i, zero consolidation reading R₀ and final reading R_f after secondary compression.

  
• Primary Consolidation Ratio

  Measured as the change in dial gauge reading corresponding to 100% primary consolidation (R₁₀₀).

  
• Secondary Consolidation Ratio

  Ratios for initial, primary and secondary components satisfy r_i + r_p + r_s = 1 for partitioning of total compression.

  

Total Settlement

Total settlement S is the sum of initial, primary and secondary settlements:

S = S_i + S_p + S_s

• Initial settlement (S_i)

  For cohesionless soils, initial settlement may be estimated using empirical bearing capacity or elastic approaches. An approximate factor given is C_s = 1.5 (C_r/σ₀) where C_r is a static resistance and σ₀ reference stress. H₀ denotes the depth of influence.

  

  For cohesive soils, initial elastic settlement may be estimated using a shape (influence) factor I_t and loaded area A.

  
• Primary settlement (S_p)

  Primary consolidation settlement is obtained from e-logσ′ relations or from m_v:

  S = ΔH = m_v H₀ Δσ′ for normally consolidated ranges, or by integrating changes in void ratio from initial to final effective stress.

  
  
  
  
  
  
  = Settlement for normally consolidation stage
  

  Note: separate plots and formulae are used for normally consolidated and overconsolidated stages; the images above correspond to common consolidation and settlement diagrams.

• Secondary settlement (S_s)

  Secondary compression (creep) occurs after primary consolidation is essentially complete. It is often estimated from a secondary compression index and logarithmic time relations. Typical symbols include H₀ ~ H₁₀₀ (thickness after 100% primary consolidation), e₁₀₀ (void ratio after 100% consolidation) and t₂ the period over which secondary compression is calculated.

  

Permeability of Soils

Definition

Permeability is a measure of the ease with which water flows through a soil. The coefficient of permeability is usually denoted by K (or k).

Darcy's Law

For laminar flow in a porous medium (Darcy conditions): the discharge per unit time is proportional to the hydraulic gradient and cross-sectional area and inversely proportional to the length.

Rate of flow: q = K i A

• q = rate of flow (m³/s).
• K = coefficient of permeability (m/s).
• i = hydraulic gradient = H_L/L = (H₁ - H₂)/L.
• A = cross-sectional area of flow.

Seepage velocity and percolation coefficient

• Seepage velocity V_s = V / n where V is the average pore water velocity and n is porosity.
• Coefficient of percolation K_P = K / n.

Laboratory permeability tests

• Constant head test (suitable for coarse-grained soils):

  K = Q L / (t H_L A)

  Q = volume collected in time t; L = sample length; H_L = head loss; A = cross sectional area.

  
• Falling head (variable head) test (suitable for finer soils):

  K = (2.303 a L) / (A t) log₁₀(h₁/h₂)

  a = cross-sectional area of standpipe; A = area of specimen; t = time interval; L = length of specimen; h₁, h₂ = head readings at start and end of interval.

  

Kozeny-Carman equation and particle geometry

The Kozeny-Carman type relations connect permeability with particle shape, specific surface and porosity. The general form uses a shape coefficient C and specific surface S:

• C is a shape coefficient (order of magnitude depends on particle shape; a value around a few mm is often used in empirical forms).
• S = specific surface (surface area per unit volume or mass).

For spherical particles, relations for specific surface and characteristic dimensions are often simplified; R denotes particle radius. The input images show geometric relations and typical substitutions for S and particle sizes.

Empirical relations

• Hazen (Allen Hazen) equation:

  K = C × D₁₀²

  D₁₀ = effective grain size (cm). C ≈ 100 to 150 for clean sands (K in cm/s).

• Lioudens equation:

  log₁₀(K S²) = a + b n

  S = specific surface; n = porosity; a and b are empirical constants.

• Relation with consolidation parameters:

  K = C_v m_v γ_w

  This relates permeability with coefficient of consolidation C_v, coefficient of volume compressibility m_v and unit weight of water γ_w.

Permeability of partially saturated soil (capillary permeability)

In partially saturated soils the hydraulic behaviour depends on degree of saturation S; capillary heights and retained suction influence flow. Typical relations include a head distribution i = (h₀ + h_c)/x for certain simplified geometries where h_c is capillary height and x is distance. Permeability K generally increases with saturation and reaches maximum at S = 100%.

Note: K_u (permeability for unsaturated soil) ∝ S; for S = 100% K = maximum.

Permeability of stratified soils

• When flow is parallel to layers (bedding), the equivalent permeability k_eq can be approximated by a weighted arithmetic mean; the expressions in the images give the appropriate averaging.
• When flow is perpendicular to the layers, k_eq is influenced by harmonic averaging of layer permeabilities; the images illustrate the formula forms.
• For 2-D and 3-D flows similar averaging rules apply; for 3-D isotropic combination k_eq = (k_x·k_y·k_z)^1/3 is commonly used.

Effective Stress, Capillarity and Seepage

Seepage pressure and seepage force

Seepage pressure is the pressure exerted by moving water on soil particles as a result of flow. Seepage force acts in the direction of flow and is due to frictional drag of water on the soil skeleton.

Seepage pressure: P_s = h γ_w, where h is head loss and γ_w is unit weight of water (≈ 9.81 kN/m³).

Seepage force over a volume: F_s = h A γ_w.

Seepage force per unit volume: f_s = i γ_w, where i is hydraulic gradient (i = h/z).

Quicksand condition

When upward seepage through a cohesionless soil produces a hydraulic gradient large enough to counteract the effective weight of soil, the net effective vertical stress becomes zero and the soil may behave as quick sand.

Critical hydraulic gradient is given by:

i_c = (G - 1)/(1 + e)

• G is specific gravity of solids (typical range 2.65-2.70).
• e is void ratio (typical 0.65-0.70 in the example values given).

To avoid a floating (piping/quick) condition the actual gradient i must satisfy i < i_c. A factor of safety against quick condition can be expressed as F.O.S. = i_c/i > 1.

Two-Dimensional Flow: Laplace Equation and Flow Nets

The potential function ∅ and total head H are related for seepage in homogeneous isotropic soil: ∅ = k H.

For isotropic homogeneous soil the 2-D Laplace equation applies to H or ∅ and flow nets (orthogonal families of streamlines and equipotential lines) provide graphical solutions.

Seepage discharge from flow net

For unit length in the out-of-plane direction, seepage discharge q is given by:

q = k h (N_f / N_d)

• h = total head difference between upstream and downstream.
• N_f = number of flow channels (≈ number of flow nets between upstream and downstream boundaries).
• N_d = number of equipotential drops.
• Shape factor = N_f/N_d.
• N_f = N_ψ - 1, where N_ψ is total number of flow lines.
• N_d = N_∅ - 1, where N_∅ is total number of equipotential lines.

Hydrostatic (pore) pressure at a point: U = h_w γ_w, where h_w is pressure head.

An expression for seepage pressure using equipotential drops: P_s = h′ γ_w, where h′ = h - (2h/N_d) in the specific diagrammatic formulation shown in the images.

Exit gradient

The exit gradient at the downstream face of a porous medium (e.g., in an exit flow field of dimensions b × b shown in the figure) is approximated from flow net geometry. The equipotential drop ΔH = h / N_d is used in that evaluation.

Phreatic Line

The phreatic line is the topmost streamline in an earth dam seepage profile; it represents the free surface of flow and follows a parabolic path in many simple geometries. Pressure on the phreatic line is atmospheric, while pressure below it is hydrostatic.

Phreatic line with filter (permeable blanket)

The phreatic line follows the base parabola; positions and radii of characteristic arcs (e.g., CF as radius of a circular arc) are shown in the figure. Typical notations: C = entry point of base parabola, F = junction of permeable and impermeable surfaces, S = focal length (distance between focus and directrix), FH = S, etc.

Relevant relations from the diagrams and analysis include discharge per unit length q = k s (where s is the seepage path length component shown), and other geometric relations shown in the images.

Phreatic line without filter

For a simple dam geometry with downstream slope angle α the approximate discharge expressions shown in the diagrams are:

• For α < 30°: q = k a sin²α (diagram and derivation shown).
• For α > 30°: q = k a sin α tan α (diagram and derivation shown).