Effective Stress

Effective Stress

Learning Objectives

Understand the Principle of Effective Stress and its implications for soil strength.
Differentiate between total stress, pore water pressure, and effective stress.
Calculate effective stress under static and seepage conditions.
Identify the conditions that lead to quick conditions (boiling) and frost heave.

The Principle of Effective Stress, first proposed by Karl Terzaghi in 1923, is the most fundamental concept in soil mechanics. It states that the strength and deformation characteristics of a soil are governed by the effective stress (

\sigma'

), not the total stress (

\sigma

The intergranular stress carried by the soil skeleton (particle-to-particle contact points), which governs the strength and compressibility of the soil.

Stress Components

In a saturated soil mass, the total vertical stress at any point is distributed between the soil skeleton and the fluid in the pore spaces.

Total Stress

The total vertical stress at a given depth due to the total weight of all overlying soil layers, water, and any surface surcharge.

Pore Water Pressure

The neutral hydrostatic or hydrodynamic pressure exerted by the water in the soil voids.

Total Stress ( $\sigma$ )

The total vertical stress at a depth $z$ due to the weight of everything above it (soil + water + surcharge).

Total Stress

Vertical stress at a given depth due to the total weight of all overlying soil layers and any surface surcharge.

\sigma = \sum \gamma H + q

Variables

Symbol	Description	Unit
$\sigma$	Total vertical stress	-
$\gamma$	Unit weight of soil layer	-
$H$	Thickness of soil layer	-
$q$	Uniform surcharge at the surface	-

Pore Water Pressure ( $u$ )

The neutral stress carried by the water in the voids. It acts equally in all directions (hydrostatic).

Pore Water Pressure

Hydrostatic water pressure at a given depth below the groundwater table under static (no-flow) conditions.

u = \gamma_w h_w

Variables

Symbol	Description	Unit
$u$	Pore water pressure	-
$\gamma_w$	Unit weight of water	9.81 kN/m³
$h_w$	Depth below the groundwater table (piezometric head)	-

Hydrostatic Assumption

This formula assumes hydrostatic conditions (no seepage or flow).

Effective Stress ( $\sigma'$ )

The stress transmitted through the soil skeleton (particle-to-particle contact points).

Effective Stress

The intergranular stress carried by the soil skeleton; the fundamental driver of strength, compressibility, and volume change in soils.

\sigma' = \sigma - u

Variables

Symbol	Description	Unit
$\sigma'$	Effective stress	-
$\sigma$	Total stress	-
$u$	Pore water pressure	-

Important Principles of Effective Stress

Effective stress cannot be measured directly; it is always calculated.
An increase in effective stress leads to compression (settlement) and increased shear strength.

Interactive Stress Profile

Interactive Simulation

Visualize how the total stress, pore water pressure, and effective stress vary with depth and water table position.

Effective Stress Profile

Parameters

Water Table Depth (m)5.0

Layer 1 Unit Weight (kN/m³)16.0

Layer 2 Unit Weight (kN/m³)19.0

Layer 1: 0-4m (Sand)
Layer 2: 4-10m (Clay)
Observe how raising the water table increases pore pressure ( $u$ ) and decreases effective stress ( $\\sigma'$ ).

Seepage Effects

When water flows through soil, the viscous drag (seepage force) alters the effective stress by either increasing or decreasing it, depending on flow direction.

Quick Condition (Boiling)

A state where upward seepage forces equal the effective weight of the soil, reducing effective stress to zero and causing the soil to lose all shear strength and behave like a fluid.

Upward Seepage

Water flowing upward exerts a drag force on soil particles, opposing gravity. This reduces the effective stress.

Effective Stress with Upward Seepage

Effective stress in a soil layer experiencing upward seepage; the drag force of water reduces the effective stress, potentially causing a quick condition.

\sigma' = z\gamma' - i z \gamma_w

Variables

Symbol	Description	Unit
$\sigma'$	Effective stress	-
$z$	Depth	-
$\gamma'$	Effective (submerged) unit weight of soil	-
$i$	Hydraulic gradient (h/L)	-
$\gamma_w$	Unit weight of water	-

Quick Condition (Boiling): Occurs when the upward seepage force equals the effective weight of the soil, reducing effective stress to zero ( $\sigma' = 0$ ). The soil loses all strength and behaves like a fluid.

Critical Hydraulic Gradient ( $i_{cr}$ ):

Critical Hydraulic Gradient

The upward hydraulic gradient at which upward seepage forces exactly balance the submerged weight of the soil, causing a quick (boiling) condition.

i_{cr} = \frac{\gamma'}{\gamma_w} = \frac{G_s - 1}{1+e}

Variables

Symbol	Description	Unit
$i_{cr}$	Critical hydraulic gradient	-
$\gamma'$	Effective unit weight	-
$\gamma_w$	Unit weight of water	-
$G_s$	Specific gravity of soil solids	-
$e$	Void ratio	-

Typical Value

Typically the critical hydraulic gradient is approximately $i_{cr} \approx 1.0$ .

Downward Seepage

Water flowing downward exerts a drag force in the direction of gravity. This increases the effective stress.

Effective Stress with Downward Seepage

Effective stress in a soil layer experiencing downward seepage; the drag force of water increases the effective stress, enhancing stability.

\sigma' = z\gamma' + i z \gamma_w

Variables

Symbol	Description	Unit
$\sigma'$	Effective stress	-
$z$	Depth	-
$\gamma'$	Effective (submerged) unit weight of soil	-
$i$	Hydraulic gradient (h/L)	-
$\gamma_w$	Unit weight of water	-

Capillary Rise and Frost Heave

Frost Heave

The destructive upward expansion of the soil surface caused by the formation of continuous ice lenses in cold climates due to capillary action.

Above the water table, pore water interactions can cause complex, detrimental phenomena such as apparent cohesion from capillary suction and devastating frost heave in cold regions.

Capillary Zone

In fine-grained soils (silts and clays) above the water table, surface tension pulls water upward into the voids, creating a zone of capillary rise. Within this zone, the pore water pressure is negative (suction). This suction provides "apparent cohesion" to moist sands and silts (the reason you can build a sandcastle with damp sand, but not dry sand).

Negative Pore Pressure (Capillary Suction)

Calculates the negative pore water pressure (suction) at a height above the water table due to capillary rise.

u = -\gamma_w h_c

Variables

Symbol	Description	Unit
$u$	Pore water pressure (suction)	-
$\gamma_w$	Unit weight of water	-
$h_c$	Height of capillary rise above the water table	-

Because the pore pressure is negative, the effective stress principle still applies, but results in a net increase in effective stress:

Effective Stress in Capillary Zone

Effective stress calculation in the capillary zone, demonstrating how negative pore pressure increases the effective stress.

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

Variables

Symbol	Description	Unit
$\sigma'$	Effective stress	-
$\sigma$	Total stress	-
$u$	Magnitude of the negative pore pressure (suction)	-

Frost Heave

In cold climates, the freezing of pore water can cause devastating upward expansion of the soil surface.

Mechanism: As freezing temperatures penetrate the ground, capillary water is drawn upward from the unfrozen soil below to form continuous ice lenses.
Because water expands 9% by volume when it freezes, and continuous ice lenses draw massive amounts of water, the soil physically heaves upward, destroying pavements, slabs, and light foundations.

Conditions Required for Frost Heave

Freezing Temperatures: Must penetrate the soil over an extended period.
A Source of Groundwater: Typically a high water table providing water for the ice lenses.
A Frost-Susceptible Soil: Silts are the most dangerous. Clean sands are not susceptible (pores are too large to support capillary rise). Dense clays are generally not susceptible (permeability is too low to feed water fast enough to form ice lenses).

Key Takeaways

Effective Stress ( $\sigma'$ ) controls the mechanical behavior of soil (strength and compression).
Total Stress ( $\sigma$ ) is the weight of everything above a point; Pore Pressure ( $u$ ) is the hydrostatic pressure.
$\sigma' = \sigma - u$ is the defining equation.
Upward seepage reduces effective stress and can lead to a Quick Condition (boiling) if $i \ge i_{cr}$ .
Capillarity causes negative pore pressure (suction) above the water table, increasing effective stress.
Frost Heave requires freezing temperatures, groundwater, and a frost-susceptible soil (primarily silts), resulting in the formation of destructive ice lenses.

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Effective Stress

Learning Objectives

Effective Stress

Stress Components

Total Stress

Pore Water Pressure

Total Stress (σ\sigmaσ)

Total Stress

Pore Water Pressure (uuu)

Pore Water Pressure

Hydrostatic Assumption

Effective Stress (σ′\sigma'σ′)

Effective Stress

Important Principles of Effective Stress

Interactive Stress Profile

Interactive Simulation

Effective Stress Profile

Parameters

Seepage Effects

Quick Condition (Boiling)

Upward Seepage

Effective Stress with Upward Seepage

Critical Hydraulic Gradient

Typical Value

Downward Seepage

Effective Stress with Downward Seepage

Capillary Rise and Frost Heave

Frost Heave

Capillary Zone

Negative Pore Pressure (Capillary Suction)

Effective Stress in Capillary Zone

Frost Heave

Conditions Required for Frost Heave

Total Stress ( $\sigma$ )

Pore Water Pressure ( $u$ )

Effective Stress ( $\sigma'$ )