2.5 Consolidation & Compressibility Analysis

2.5 Consolidation & Compressibility Analysis

1. Fundamentals of Soil Consolidation

When a saturated cohesive soil layer is subjected to a load increment ($\Delta\sigma$), the incompressible pore water initially bears the entire stress increase as excess pore water pressure ($u_e = \Delta\sigma$) due to low permeability. Over time, as pore water drains out, $u_e$ dissipates and stress transfers to the mineral soil matrix, increasing effective stress ($\Delta\sigma'$) and producing volumetric compression.

Total Settlement Components:

$S_{total} = S_i + S_c + S_s$ Where:

• $S_i$: Immediate (Elastic) Settlement: Occurs rapidly without volume change (constant $w$).
• $S_c$: Primary Consolidation Settlement: Time-dependent volume change due to drainage of pore water under excess pore pressure gradients.
• $S_s$: Secondary Compression (Creep) Settlement: Time-dependent volume change occurring at constant effective stress after complete $u_e$ dissipation.

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2. One-Dimensional Oedometer Test & Stress History

The compressibility parameters of fine-grained soils are determined using a 1D Oedometer (Consolidation) Test. Incremental vertical loads are applied to a laterally constrained cylindrical soil specimen, and equilibrium deformation is recorded at each step.

Plotting the $e - \log\sigma'$ Compression Curve

Plotting void ratio $e$ against the logarithm of vertical effective stress $\log_{10}\sigma'_v$ reveals the soil's stress history:

1. Preconsolidation Stress ($\sigma'_p$): The maximum historical effective vertical stress the soil has ever experienced. Determined graphically using Casagrande’s Method (finding point of maximum curvature, drawing horizontal line, tangent line, bisector, and intersecting with virgin line projection).
2. **Overconsolidation Ratio ($OCR$): $OCR = \frac{\sigma'_p}{\sigma'_{v0}}$
   • Normally Consolidated (NC) Soil ($OCR = 1.0$): In-situ effective stress $\sigma'{v0}$ is the maximum historical stress ($\sigma'{v0} = \sigma'_p$).
   • Overconsolidated (OC) Soil ($OCR > 1.0$): Past stress exceeds current effective stress ($\sigma'p > \sigma'{v0}$). Caused by desiccation, glacial loading, eroded overburden, or groundwater table drops.
   • Underconsolidated Soil ($OCR < 1.0$): Soil is currently consolidating under existing overburden (e.g., recent hydraulic fills).

Key Compressibility Indices:

• Compression Index ($C_c$): Slope of virgin consolidation curve: $C_c = \frac{e_1 - e_2}{\log_{10}(\sigma'_2 / \sigma'_1)}$ Empirical estimation for remolded/undisturbed clays (Terzaghi & Peck): $C_c \approx 0.009 (LL - 10)$
• Recompression (Swell) Index ($C_r$ or $C_s$): Slope of recompression/rebound curve: $C_r \approx \frac{1}{5} \text{ to } \frac{1}{10} C_c$

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3. Primary Consolidation Settlement Formulations

Case 1: Normally Consolidated Clays ($\sigma'_{v0} = \sigma'_p$)

The entire stress increment ($\Delta\sigma'$) acts along the steep virgin compression curve ($C_c$):

$S_c = \frac{C_c \cdot H_0}{1 + e_0} \log_{10}\left( \frac{\sigma'_{v0} + \Delta\sigma'}{\sigma'_{v0}} \right)$

Case 2: Overconsolidated Clays ($\sigma'_{v0} < \sigma'_p$)

Subcase A: Final stress remains below preconsolidation stress ($\sigma'_{v0} + \Delta\sigma' \le \sigma'_p$)

Deformation stays entirely on the flatter recompression curve ($C_r$):

$S_c = \frac{C_r \cdot H_0}{1 + e_0} \log_{10}\left( \frac{\sigma'_{v0} + \Delta\sigma'}{\sigma'_{v0}} \right)$

Subcase B: Final stress exceeds preconsolidation stress ($\sigma'_{v0} + \Delta\sigma' > \sigma'_p$)

Deformation re-compresses along $C_r$ up to $\sigma'_p$, then transitions to virgin compression along $C_c$:

$S_c = \frac{C_r \cdot H_0}{1 + e_0} \log_{10}\left( \frac{\sigma'_p}{\sigma'_{v0}} \right) + \frac{C_c \cdot H_0}{1 + e_0} \log_{10}\left( \frac{\sigma'_{v0} + \Delta\sigma'}{\sigma'_p} \right)$

Coefficient of Volume Compressibility Method ($m_v$)

Alternatively, using $m_v = \frac{a_v}{1 + e_0} = \frac{\Delta e}{\Delta\sigma' (1 + e_0)}$: $S_c = m_v \cdot H_0 \cdot \Delta\sigma'$

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4. Time Rate of 1D Consolidation (Terzaghi Differential Equation)

Terzaghi derived the partial differential equation governing 1D excess pore pressure dissipation:

$C_v \frac{\partial^2 u_e}{\partial z^2} = \frac{\partial u_e}{\partial t}$

Where the Coefficient of Consolidation ($C_v$) is: $C_v = \frac{k}{m_v \cdot \gamma_w} = \frac{k (1 + e_0)}{a_v \cdot \gamma_w}$

Dimensionless Time Factor ($T_v$)

The relationship between time $t$, drainage distance $H_{dr}$, and consolidation progress is given by:

$T_v = \frac{C_v \cdot t}{H_{dr}^2} \implies t = \frac{T_v \cdot H_{dr}^2}{C_v}$

Drainage Path Length ($H_{dr}$):

• Double Drainage: Permeable strata (sand/gravel) above AND below the clay layer $\implies H_{dr} = \frac{H_{clay}}{2}$.
• Single Drainage: Permeable stratum on ONE side, impermeable boundary (bedrock/geomembrane) on the other $\implies H_{dr} = H_{clay}$.

Average Degree of Consolidation ($U%$) and $T_v$ Relations:

• For $U < 60%$: $T_v = \frac{\pi}{4} \left( \frac{U\%}{100} \right)^2 \approx 0.785 \left( \frac{U\%}{100} \right)^2$
• For $U > 60%$: $T_v = 1.781 - 0.933 \log_{10}(100 - U\%)$

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5. Secondary Compression Settlement (Creep)

Secondary compression occurs after primary excess pore water pressure has fully dissipated ($U = 100%$), driven by plastic adjustment of clay micro-structure under constant effective stress.

$S_s = \frac{C_\alpha}{1 + e_p} H_0 \log_{10}\left( \frac{t_2}{t_1} \right) = C_{\alpha\epsilon} \cdot H_0 \log_{10}\left( \frac{t_2}{t_1} \right)$

Where:

• $C_\alpha = \frac{\Delta e}{\log_{10}(t_2 / t_1)}$: Secondary compression index.
• $e_p$: Void ratio at the end of primary consolidation.
• $t_1$: Time to end of primary consolidation ($U = 95 - 100%$).
• $t_2$: Target design lifetime (e.g., 30 or 50 years).

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6. Comprehensive Worked Example

Problem Description: A 4.0 m thick saturated clay layer is buried between two permeable sand layers (Double drainage condition). The properties of the clay are:

• Initial void ratio: $e_0 = 0.90$
• Compression Index: $C_c = 0.36$
• Recompression Index: $C_r = 0.06$
• Coefficient of consolidation: $C_v = 3.2 \text{ m}^2/\text{year}$
• Initial effective vertical overburden stress at layer mid-height: $\sigma'_{v0} = 80 \text{ kPa}$
• Preconsolidation stress: $\sigma'_p = 120 \text{ kPa}$ ($OCR = 1.50$)

A wide embankment surcharge increases vertical stress at mid-height by $\Delta\sigma' = 100 \text{ kPa}$.

Calculate:

1. Preconsolidation status and total primary consolidation settlement ($S_c$).
2. Time required in days to achieve 50% primary consolidation ($U = 50%, T_v = 0.197$).
3. Time required in years to achieve 90% primary consolidation ($U = 90%, T_v = 0.848$).

Detailed Solution:

1. Calculate Primary Consolidation Settlement ($S_c$):

   • Check stress range: $\sigma'_{v0} = 80 \text{ kPa}$ $\sigma'_p = 120 \text{ kPa}$ $\sigma'_f = \sigma'_{v0} + \Delta\sigma' = 80 + 100 = 180 \text{ kPa}$

   • Since $\sigma'_f (180 \text{ kPa}) > \sigma'_p (120 \text{ kPa})$, this is an Overconsolidated Case 2 (Subcase B) calculation.

   • Apply the two-part formula: $S_c = \frac{C_r \cdot H_0}{1 + e_0} \log_{10}\left(\frac{\sigma'_p}{\sigma'_{v0}}\right) + \frac{C_c \cdot H_0}{1 + e_0} \log_{10}\left(\frac{\sigma'_f}{\sigma'_p}\right)$ $S_{c1} = \frac{0.06 \times 4.0 \text{ m}}{1 + 0.90} \log_{10}\left(\frac{120}{80}\right) = \frac{0.24}{1.90} \log_{10}(1.50) = 0.1263 \times 0.1761 = 0.0222 \text{ m} = 2.22 \text{ cm}$ $S_{c2} = \frac{0.36 \times 4.0 \text{ m}}{1 + 0.90} \log_{10}\left(\frac{180}{120}\right) = \frac{1.44}{1.90} \log_{10}(1.50) = 0.7579 \times 0.1761 = 0.1335 \text{ m} = 13.35 \text{ cm}$ $S_c = S_{c1} + S_{c2} = 2.22 \text{ cm} + 13.35 \text{ cm} = 15.57 \text{ cm} \quad (0.1557 \text{ m})$

2. Calculate Time for 50% Consolidation ($t_{50}$):

   • Double drainage $\implies H_{dr} = \frac{H_{clay}}{2} = \frac{4.0 \text{ m}}{2} = 2.0 \text{ m}$.
   • Using $t = \frac{T_v \cdot H_{dr}^2}{C_v}$: $t_{50} = \frac{0.197 \times (2.0 \text{ m})^2}{3.2 \text{ m}^2/\text{year}} = \frac{0.197 \times 4.0}{3.2} = \frac{0.788}{3.2} = 0.24625 \text{ years}$
   • Convert to days: $t_{50} = 0.24625 \text{ years} \times 365.25 \text{ days/year} = 89.9 \text{ days} \approx 90 \text{ days}$

3. Calculate Time for 90% Consolidation ($t_{90}$): $t_{90} = \frac{0.848 \times (2.0 \text{ m})^2}{3.2 \text{ m}^2/\text{year}} = \frac{0.848 \times 4.0}{3.2} = \frac{3.392}{3.2} = 1.06 \text{ years}$