2.2 Effective Stress & Pore Water Pressure Dynamics

2.2 Effective Stress & Pore Water Pressure Dynamics

1. Terzaghi’s Effective Stress Principle

In 1923, Karl Terzaghi established the foundational framework of modern soil mechanics by identifying that soil behavior is governed not by total overburden pressure, but by the stress transferred through the solid mineral matrix.

The Effective Stress Equation:

$$\sigma' = \sigma - u$$

Where:

• $\sigma$: Total normal stress acting on a plane within the soil mass.
• $\sigma'$: Effective normal stress (intergranular stress carried by solid skeleton contacts).
• $u$: Pore water pressure acting in all directions within the fluid-filled void space.

Key Physical Consequences:

1. Shear Strength Control: Water has zero shear resistance. Therefore, the shear strength of soil depends strictly on effective normal stress: $$\tau_f = c' + \sigma' \tan\phi' = c' + (\sigma - u) \tan\phi'$$
2. Volume Change and Settlement: Soil compression occurs due to the rearrangement of mineral grains, which is driven entirely by changes in effective stress ($\Delta\sigma'$). An increase in total stress $\sigma$ accompanied by an equal increase in pore pressure $u$ results in zero effective stress change ($\Delta\sigma' = 0$) and zero consolidation settlement.

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2. Total Stress and Pore Pressure in Layered Soil Stratigraphy

To evaluate effective stress at a depth $z$ below the ground surface, the total vertical stress and hydrostatic pore pressure are summed across soil layers:

Total Vertical Stress ($\sigma_v$):

$$\sigma_v = \sum_{i=1}^{n} \gamma_i \cdot h_i + q$$

Where $\gamma_i$ and $h_i$ are the total unit weight and thickness of layer $i$, and $q$ is an applied surface surcharge load.

Hydrostatic Pore Water Pressure ($u$):

For static ground water conditions (no vertical seepage):

$$\nu = \begin{cases} 0 \& \text{above the water table (ignoring capillarity)} \ \gamma_w \cdot z_w \& \text{at depth } z_w \text{ below the phreatic surface} \end{cases}$$

Effective Vertical Stress ($\sigma'_v$):

$$\sigma'_v = \sigma_v - u$$

In fully submerged soil layers under static conditions, effective stress can be directly computed using the submerged unit weight ($\gamma' = \gamma_{sat} - \gamma_w$):

$$\sigma'_v = \sum \gamma_{dry/moist} \cdot h_{above\,WT} + \sum \gamma' \cdot h_{below\,WT} + q$$

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3. Seepage Forces and Dynamic Pore Pressure Dynamics

When water flows through soil, viscous drag forces are transferred from the fluid to the soil skeleton. This fluid-solid interaction alters the pore pressure distribution and effective stress state.

Consider one-dimensional vertical flow through a soil column of length $L$ under a total head loss $\Delta h$: The hydraulic gradient is $i = \frac{\Delta h}{L}$.

Case A: Downward Vertical Seepage

Downward flow exerts a drag force in the direction of gravity, increasing effective stress:

• Seepage force per unit volume: $j = i \cdot \gamma_w$ (acting downward)
• Pore pressure at depth $z$: $u = \gamma_w z - i z \gamma_w$
• Effective stress at depth $z$: $$\sigma'_v = \gamma' z + i z \gamma_w = (\gamma' + i \gamma_w) z$$

Case B: Upward Vertical Seepage

Upward flow exerts a drag force opposing gravity, reducing effective stress:

• Pore pressure at depth $z$: $u = \gamma_w z + i z \gamma_w$
• Effective stress at depth $z$: $$\sigma'_v = \gamma' z - i z \gamma_w = (\gamma' - i \gamma_w) z$$

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4. Critical Hydraulic Gradient ($i_{cr}$) & Quicksand / Boiling

In upward seepage, if the upward hydraulic gradient $i$ increases to the point where effective stress becomes zero ($\sigma'_v = 0$):

$$\sigma'_v = (\gamma' - i_{cr} \gamma_w) z = 0 \implies i_{cr} = \frac{\gamma'}{\gamma_w}$$

Substituting $\gamma' = \frac{G_s - 1}{1 + e} \gamma_w$:

$$\ni_{cr} = \frac{G_s - 1}{1 + e}$$

Engineering Implications:

• For typical soils ($G_s \approx 2.65, e \approx 0.65$), $i_{cr} \approx \frac{1.65}{1.65} = 1.0$.
• When $i \ge i_{cr}$, coarse cohesionless soils completely lose shear strength ($\tau_f = 0 \cdot \tan\phi' = 0$), behaving as a heavy liquid. This phenomenon is termed quicksand, boiling, or heave.
• The factor of safety against quicksand / piping instability is defined as: $$FS_{piping} = \frac{i_{cr}}{i_{exit}}$$ A minimum factor of safety of $FS \ge 3.0 \text{ to } 4.0$ is mandatory in geotechnical design for excavation support and dam foundations.

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5. Capillary Rise & Unsaturated Negative Pore Pressure

Above the phreatic surface (water table), capillary tension draws water upward into soil pores.

Capillary Height ($h_c$):

$$h_c = \frac{C}{e \cdot D_{10}}$$

Where $C$ is an empirical constant ($10 - 50 \text{ mm}^2$).

Pore Pressure in Capillary Zone:

Water in the capillary zone is under tension, resulting in negative pore water pressure (matric suction):

$$\nu = - S \cdot \gamma_w \cdot h_c$$

Where $S$ is the degree of saturation in the capillary fringe ($S \approx 1.0$ in fully saturated capillary zone, tapering off above).

Effective Stress Increase:

Because $u$ is negative ($u < 0$):

$$\sigma' = \sigma - (-u) = \sigma + |u|$$

Capillary suction increases effective stress and imparts apparent cohesion ($c_{app} = |u| \tan\phi'$). This allows vertical cuts in moist sand to stand temporarily. However, inundation saturates the soil ($u \to 0$), destroying apparent cohesion and causing collapse.

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6. Comprehensive Worked Example

Problem Profile: Consider a soil profile with three layers:

• Layer 1 (0 to 3.0 m): Moist sand above water table, $\gamma = 18.0 \text{ kN/m}^3$.
• Capillary Fringe (2.0 m to 3.0 m): Saturated capillary zone, $\gamma = 19.0 \text{ kN/m}^3$, full capillary rise $h_c = 1.0 \text{ m}$ above WT. Water table located at depth $z = 3.0 \text{ m}$.
• Layer 2 (3.0 to 8.0 m): Saturated clay below water table, $\gamma_{sat} = 20.0 \text{ kN/m}^3$.
• A uniform surface surcharge $q = 25 \text{ kPa}$ is applied at ground surface.

Calculate total stress $\sigma$, pore pressure $u$, and effective stress $\sigma'$ at:

1. Ground surface ($z = 0$)
2. Top of capillary zone ($z = 2.0 \text{ m}$)
3. Water table level ($z = 3.0 \text{ m}$)
4. Bottom of clay layer ($z = 8.0 \text{ m}$)

Detailed Solution:

1. At $z = 0 \text{ m}$ (Ground Surface):

   • Total stress: $\sigma = q = 25.0 \text{ kPa}$
   • Pore pressure: $u = 0 \text{ kPa}$
   • Effective stress: $\sigma' = 25.0 - 0 = 25.0 \text{ kPa}$

2. At $z = 2.0 \text{ m}$ (Top of Capillary Fringe, $1.0\text{ m}$ above WT):

   • Total stress: $\sigma = q + \gamma_{layer1} \cdot 2.0 = 25.0 + (18.0 \times 2.0) = 61.0 \text{ kPa}$
   • Pore pressure (capillary suction, $h_c = 1.0 \text{ m}$): $$u = -\gamma_w \cdot h_c = -9.81 \times 1.0 = -9.81 \text{ kPa}$$
   • Effective stress: $$\sigma' = \sigma - u = 61.0 - (-9.81) = 70.81 \text{ kPa}$$

3. At $z = 3.0 \text{ m}$ (Water Table Level):

   • Total stress: $\sigma = 61.0 + (\gamma_{cap} \cdot 1.0) = 61.0 + (19.0 \times 1.0) = 80.0 \text{ kPa}$
   • Pore pressure: $u = 0 \text{ kPa}$ (at phreatic surface)
   • Effective stress: $\sigma' = 80.0 - 0 = 80.0 \text{ kPa}$

4. At $z = 8.0 \text{ m}$ (Bottom of Layer 2, $5.0\text{ m}$ below WT):

   • Total stress: $\sigma = 80.0 + (\gamma_{sat,clay} \cdot 5.0) = 80.0 + (20.0 \times 5.0) = 180.0 \text{ kPa}$
   • Pore pressure ($z_w = 5.0 \text{ m}$): $u = \gamma_w \cdot z_w = 9.81 \times 5.0 = 49.05 \text{ kPa}$
   • Effective stress: $$\sigma' = \sigma - u = 180.0 - 49.05 = 130.95 \text{ kPa}$$

Check using submerged unit weight ($\gamma' = 20.0 - 9.81 = 10.19 \text{ kN/m}^3$):

$$\sigma' = 25.0 + (18.0 \times 2.0) + (19.0 \times 1.0) + (10.19 \times 5.0) = 25.0 + 36.0 + 19.0 + 50.95 = 130.95 \text{ kPa} \quad \checkmark$$