Compressibility & Consolidation - Civil Engineering SSC JE (Technical) - Civil Engineering

Coefficient Of Compressibility (A_v)

The coefficient of compressibility describes the change in void ratio or volume of a soil mass per unit change in effective stress. It is used to estimate settlement due to consolidation.

• Void ratio notation: e₁ = void ratio at one effective stress, e₂ = void ratio at another effective stress.

Common symbols and meanings:

• 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).

Important reminder: Consolidation settlement is governed by changes in effective stress, not by changes in total stress.

Coefficient Of Compression (C_c)

The coefficient of compression C_c is the slope of the virgin compression (primary consolidation) portion of the e - log σ' curve and is used to calculate primary consolidation settlement.

Empirical relations and commonly used approximations for C_c:

• (b) C_c = 0.009 (W_L - 10) for undisturbed clays of low to medium sensitivity, where W_L is the liquid limit (%)
• (c) C_c = 0.007 (W_L - 7) for remoulded soils of low sensitivity
• (d) C_c = 0.40 (e₀ - 0.25) for undisturbed soils of medium sensitivity, where e₀ is initial void ratio
• (e) For remoulded soil of low sensitivity: C_c = 1.15 (e₀ - 0.35)
• (f) C = 0.115 w, where w = water content (%)

Use direct measurement from consolidation test (e-log σ′ curve) where possible; use empirical relations only when test data are not available.

Over-consolidation Ratio (O.c.r.)

The over-consolidation ratio (O.C.R.) indicates whether a soil has been subjected to higher effective stresses in the past.

• O.C.R. = σ'_p / σ'_v0, where σ'_p = preconsolidation effective stress and σ'_v0 = current vertical effective stress.
• O.C.R. > 1 → overconsolidated soil.
• O.C.R. = 1 → normally consolidated soil.
• O.C.R. < 1 → under-consolidated condition (rare; usually indicates transient excess pore pressures).

Differential Equation Of 1-d Consolidation

The one-dimensional consolidation of a saturated soil column is governed by a diffusion-type equation:

• ∂u / ∂t = C_v ∂²u / ∂z²

where:

• u = excess pore water pressure (function of z and t)
• ∂u / ∂t = rate of change of pore pressure with time
• C_v = coefficient of consolidation
• ∂²u / ∂z² = rate of change of pore pressure with depth (second derivative)

Coefficient Of Volume Compressibility (M_v)

The coefficient of volume compressibility relates volumetric change to change of effective stress and is defined as:

• m_v = ΔV / (V₀ Δσ')
• In terms of void ratio: m_v = Δe / ((1 + e₀) Δσ')

where e₀ is the initial void ratio and Δσ′ is the change in effective stress.

Compression Modulus (E_c)

The compression modulus is the reciprocal of the coefficient of volume compressibility for small strains under one-dimensional conditions:

• E_c = 1 / m_v

This modulus is used in elastic estimates of settlement and to relate stress to strain in compressible soils.

Degree Of Consolidation (U)

The degree of consolidation describes the fraction (or percentage) of primary consolidation settlement completed at time t.

• U (%) = (settlement at time t / final primary settlement) × 100
• Using pore pressures: U = (U_i - U) / U_i × 100, where U is current excess pore pressure and U_i is initial excess pore pressure.

Boundary conditions give:

• At t = 0, U = 0% (no consolidation completed).
• At t = ∞, U = 100% (primary consolidation complete).

Void-ratio form:

• U = (e₀ - e) / (e₀ - e_f)

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

Relation in terms of settlements:

• U = Δh(t) / ΔH

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

Time Factor (T_v)

The time factor relates elapsed time to the consolidation rate and is defined as:

• T_v = C_v t / d²

where:

• C_v = coefficient of consolidation (cm²/s or m²/s)
• t = time (s)
• d = length of drainage path
• d = H₀/2 for two-way drainage
• d = H₀ for one-way drainage
• H₀ = thickness (depth) of the consolidating layer

Standard approximate values used for solutions of the consolidation equation:

• 
  If U ≤ 60%: T₅₀ = 0.196
• 
  If U > 60%: other values from tables or numerical solutions (e.g., T₉₀ ≈ 0.848 for 90% consolidation)

Method To Find C_v

Coefficient of consolidation can be evaluated from laboratory consolidation test results by curve-fitting methods.

(i) Square-Root-of-Time Fitting Method

Procedure outline:

• Plot settlement (or degree of consolidation) against √t.
• Find the time corresponding to U = 90% (or other specified U) on the √t plot.
• Use C_v = T₉₀ d² / t₉₀, where T₉₀ is the time factor for 90% consolidation and d is drainage path.

Here, T₉₀ is the time factor at 90% consolidation and t₉₀ is the time at 90% consolidation.

(ii) Logarithm-of-Time Fitting Method

Procedure outline:

• Plot settlement versus log(t).
• Determine t₅₀, the time corresponding to 50% consolidation, using the straight-line portion of the secondary curve intersection method.
• Use C_v = T₅₀ d² / t₅₀, where T₅₀ is the time factor at 50% consolidation.

Remember: Square-root-of-time fitting is generally preferred when secondary compression is significant; logarithm-of-time fitting is widely used because T₅₀ values are well established.

Compression Ratio

(i) Initial Compression Ratio

In consolidation tests the compression ratios are measured from dial gauge readings or settlement readings.

• R_i = initial reading of dial gauge
• R₀ = reading at 0% consolidation
• R_f = final reading after secondary consolidation

(ii) Primary Consolidation Ratio

R₁₀₀ = dial gauge reading at 100% primary consolidation.

(iii) Secondary Consolidation Ratio

Total Settlement

S = S_i + S_p + S_s

• S_i = initial (or immediate) settlement
• S_p = primary consolidation settlement
• S_s = secondary (creep) settlement

(i) Initial Settlement

Initial settlement (also called immediate settlement) occurs rapidly on application of load and is associated with elastic compression of the soil skeleton and rapid adjustment of pore pressures in non-fully saturated or granular soils.

Estimation approaches depend on soil type and available parameters:

• For cohesionless soils, elastic theory and bearing capacity or in situ tests (e.g., cone resistance, static penetrometer) are used to estimate immediate settlement.
• For cohesive soils, elastic compression using modulus and influence factors is used.

Symbols appearing in laboratory/empirical formulae:

• C_r = static cone resistance (kN/m²)
• H₀ = thickness of compressible layer

For cohesive soil methods:

• I_t = shape factor or influence factor
• A = footing area
• Remember: For a square footing of width B, A = B² and I_t = 1

For strip footings, use the corresponding influence factor for a strip geometry as given in standard references.

(ii) Primary Consolidation Settlement

Primary consolidation settlement is due to expulsion of pore water and reduction in void ratio under increased effective stress. It is calculated from changes in void ratio or directly from measured compression curves.

• s_c1 = settlement for overconsolidated stage (when increase of stress does not enter virgin compression)
• s_c2 = settlement for normally consolidated stage (when stress increase follows the virgin compression line)

Useful relation (void-ratio form):

• ΔH = H₀ (e₀ - e_f) / (1 + e₀)

where H₀ is initial thickness, e₀ is initial void ratio and e_f is void ratio after primary consolidation.

(iii) Secondary Settlement

Secondary settlement (also called creep) occurs after primary consolidation is complete and is usually a function of time at nearly constant effective stress.

• Secondary settlement may be estimated from laboratory oedometer tests by the slope of the void-ratio versus log(time) curve after primary consolidation.
• Typical expression used for secondary settlement over a time interval t₁ to t₂ is based on the secondary compression index and layer thickness after primary consolidation.

Notation used in secondary settlement expressions:

• H₁₀₀ ≈ thickness of soil after 100% primary consolidation
• e₁₀₀ = void ratio after 100% primary consolidation
• t₂ = average time after t₁ for which secondary consolidation is calculated

Final remarks: For practical design and exam-level problems, always state the assumptions (drainage conditions, layer thickness, boundary conditions), use the correct drainage path (d = H/2 for two-way drainage; d = H for one-way), and prefer laboratory consolidation data (e-log σ′ curves and oedometer test results) to empirical estimates. Retain unit consistency throughout calculations.