1.3 Effective and Total Stresses

The Principle of Effective Stress

Karl Terzaghi's principle of effective stress is arguably the most important concept in geotechnical engineering. Total stress ($\sigma$) at a point within a soil mass is the total pressure resulting from the weight of all overlying materials (soil, water, and surface loads). However, soil behavior—such as its shear strength and compressibility—is not governed by total stress, but by effective stress ($\sigma'$).

Effective stress is the portion of the total stress that is carried by the physical contact points between the solid soil particles (the soil skeleton). The pore water, which occupies the voids between the particles, carries the rest of the total stress in the form of pore water pressure ($u$).

The fundamental equation for effective stress is:

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

where:

• $\sigma'$ = Effective stress
• $\sigma$ = Total stress
• $u$ = Pore water pressure

For PE Construction candidates, recognizing that water cannot carry shear forces is critical. As pore water pressure increases, effective stress decreases, causing the soil particles to lose contact force, which directly reduces the soil's frictional shear strength.

Vertical Geostatic Stress

To analyze effective stress at a depth $z$ below the ground surface, you must first calculate the total vertical geostatic stress ($\sigma_v$). This is computed by summing the weights of the distinct soil layers above the point of interest:

$$\sigma_v = \sum (\gamma_i \times H_i)$$

where:

• $\gamma_i$ = Total unit weight of layer $i$
• $H_i$ = Thickness of layer $i$

The pore water pressure ($u$) under static (hydrostatic) groundwater conditions is simply the depth below the water table ($z_w$) multiplied by the unit weight of water ($\gamma_w = 62.4$ pcf or $9.81$ kN/m$^3$):

$$\nu = \gamma_w z_w$$

Therefore, the vertical effective stress under hydrostatic conditions can be written as:

$$\sigma_v' = \sigma_v - u$$

Alternatively, for a fully submerged soil layer, you can use the buoyant (or effective) unit weight ($\gamma'$), where $\gamma' = \gamma_{sat} - \gamma_w$. The effective stress is then calculated directly by multiplying the buoyant unit weight by the submerged thickness.

Seepage Conditions and Hydraulic Gradient

The effective stress equation $\sigma' = \sigma - u$ applies directly when groundwater is static. However, during construction, activities like dewatering, pumping, or cofferdam excavation often cause water to flow through the soil. When water flows through soil, it exerts a frictional drag on the soil particles known as seepage force.

The direction of seepage profoundly affects the effective stress:

• Downward Seepage: Water flowing downward pushes in the same direction as gravity, increasing the contact forces between soil particles. This increases the effective stress and the soil's shear strength.
• Upward Seepage: Water flowing upward exerts an upward drag force that counteracts gravity, reducing the contact forces between particles. This decreases the effective stress.

The change in effective stress due to seepage is governed by the hydraulic gradient ($i$), which is the head loss ($h$) divided by the length of the flow path ($L$):

$$\ni = \frac{\Delta h}{L}$$

The magnitude of the change in effective stress due to seepage is given by $\pm i \gamma_w z$, where $z$ is the depth of the soil layer experiencing flow. Downward seepage adds to the effective stress, while upward seepage subtracts from it.

The Quicksand Condition (Boiling)

If upward seepage is strong enough, the upward drag force of the water can perfectly balance the downward buoyant weight of the soil particles. At this precise point, the effective stress drops to zero ($\sigma' = 0$).

When $\sigma' = 0$ in a cohesionless granular soil, the soil loses all its shear strength and behaves like a viscous liquid. This phenomenon is known as the quicksand condition or boiling. It is a major hazard in excavations below the water table, particularly at the bottom of cofferdams or braced trenches.

The critical hydraulic gradient ($i_c$) at which boiling occurs is:

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

where:

• $G_s$ = Specific gravity of soil solids
• $e$ = Void ratio

For typical sands, the critical hydraulic gradient is approximately 1.0. If the actual upward hydraulic gradient approaches this value, boiling is imminent, and mitigating measures (like deeper sheet piling or pressure relief wells) are mandatory.

Worked Example: Effective Stress Calculation

Problem: A soil profile consists of 8 ft of dry sand ($\gamma_{dry} = 105$ pcf) overlying 12 ft of saturated clay ($\gamma_{sat} = 122$ pcf). The water table is located exactly at the interface between the sand and clay. Assuming hydrostatic conditions, calculate the total stress, pore water pressure, and effective vertical stress at the bottom of the clay layer.

Solution:

1. Calculate Total Stress ($\sigma_v$) at the bottom (depth = 20 ft):

   • Contribution from dry sand: $105 \text{ pcf} \times 8 \text{ ft} = 840 \text{ psf}$
   • Contribution from saturated clay: $122 \text{ pcf} \times 12 \text{ ft} = 1464 \text{ psf}$
   • $\sigma_v = 840 + 1464 = 2304 \text{ psf}$

2. Calculate Pore Water Pressure ($u$) at the bottom:

   • The water table is at a depth of 8 ft, so the depth of water $z_w = 12$ ft.
   • $u = \gamma_w \times z_w = 62.4 \text{ pcf} \times 12 \text{ ft} = 748.8 \text{ psf}$

3. Calculate Effective Stress ($\sigma_v'$):

   • $\sigma_v' = \sigma_v - u = 2304 - 748.8 = 1555.2 \text{ psf}$

(Alternative Check using buoyant unit weight for the clay):

• $\gamma' = 122 - 62.4 = 59.6 \text{ pcf}$
• $\sigma_v' = (105 \times 8) + (59.6 \times 12) = 840 + 715.2 = 1555.2 \text{ psf}$. (Matches exactly)

Answer: At the bottom of the clay layer, the total stress is 2,304 psf, the pore water pressure is 748.8 psf, and the effective stress is 1,555.2 psf.

Exam Tips

• Capillary rise (negative pore water pressure) can sometimes exist above the water table in fine-grained soils. Negative pore pressure increases the effective stress because $\sigma' = \sigma - (-u) = \sigma + u$. This creates apparent cohesion in damp sand, allowing sandcastles to stand.
• Always accurately locate the groundwater table in a soil profile diagram. A mistake in the water table depth will invalidate the entire effective stress calculation.
• Remember that structural loads on the surface (like a footing) increase total stress, which immediately increases pore pressure in saturated clays (excess pore pressure) until consolidation occurs.