Shear Strength - Theory & Concepts

Shear Strength of Soil

Learning Objectives

Understand the Mohr-Coulomb failure criterion.
Differentiate between effective stress and total stress analysis for shear strength.
Calculate change in pore water pressure using Skempton's parameters.
Compare and interpret laboratory shear strength tests (Direct Shear, Triaxial, Unconfined Compression).
Determine undrained shear strength from the in-situ Vane Shear Test.

The shear strength of a soil is its resistance to shearing stresses. It is a fundamental property required to analyze the stability of slopes, the bearing capacity of foundations, and the lateral earth pressure on retaining walls.

Shear Strength ( $\tau_f$ )

The maximum internal resistance per unit area that a soil mass can offer to resist failure and sliding along any plane inside it.

Mohr-Coulomb Failure Criterion

The shear strength ( $\tau_f$ ) is typically described by the Mohr-Coulomb Failure Criterion, which states that failure occurs when the shear stress on any plane reaches a critical value dependent on the normal stress on that plane.

Shear Strength Equation

Shear strength equations differ depending on whether the loading is drained (effective stress) or undrained (total stress).

Effective Stress Analysis (Drained)

Mohr-Coulomb shear strength for drained (slow) loading conditions, where pore pressures have fully dissipated; gives the long-term strength.

\tau_f = c' + \sigma' \tan \phi'

Variables

Symbol	Description	Unit
$\tau_f$	Shear strength at failure	-
$c'$	Effective cohesion intercept	kPa
$\sigma'$	Effective normal stress on the failure plane	kPa
$\phi'$	Effective angle of internal friction	degrees

Effective Cohesion Intercept

$c'$ is ideally zero for sands and normally consolidated clays.

Total Stress Analysis (Undrained)

For saturated clays loaded rapidly, pore pressures do not dissipate.

Total Stress Analysis (Undrained)

Mohr-Coulomb shear strength for rapid (undrained) loading in saturated clays; excess pore pressures are not measured and are combined into the total stress parameters.

\tau_f = c_u + \sigma \tan \phi_u

Variables

Symbol	Description	Unit
$\tau_f$	Undrained shear strength	-
$c_u$	Undrained cohesion intercept (S_u)	-
$\sigma$	Total normal stress	-
$\phi_u$	Undrained angle of internal friction	-

Undrained Parameters

$\phi_u \approx 0$ for saturated clays.
$c_u = S_u$ (Undrained shear strength).

Interactive Mohr's Circle

Interactive Simulation

Visualize the state of stress and the failure envelope. Adjust the principal stresses ( $\sigma_1, \sigma_3$ ) and soil properties ( $c, \phi$ ) to see when failure occurs.

Mohr-Coulomb Failure Criterion

Stable

Major Stress (σ₁)100

Minor Stress (σ₃)40

Cohesion (c)10

Friction Angle (φ)30 °

If the circle touches or crosses the red failure envelope, the soil fails in shear. The radius of the circle represents the maximum shear stress ( $\\tau_{max} = (\\sigma_1 - \\sigma_3)/2$ ).

Pore Pressure Parameters (Skempton)

To predict how pore water pressure (

u

) will change in a saturated clay when subjected to changes in total principal stresses (

\Delta \sigma_1, \Delta \sigma_3

), engineers use Skempton's Pore Pressure Parameters (

A

and

B

Skempton's Equation

The change in pore water pressure ( $\Delta u$ ) during undrained loading is governed by Skempton's equation.

Skempton's Pore Pressure Equation

Predicts the change in pore water pressure induced by changes in total principal stresses during undrained loading, using empirical parameters A and B.

\Delta u = B [ \Delta \sigma_3 + A (\Delta \sigma_1 - \Delta \sigma_3) ]

Variables

Symbol	Description	Unit
$\Delta u$	Change in pore water pressure	-
$A$	Skempton's pore pressure parameter A	-
$B$	Skempton's pore pressure parameter B	-
$\Delta \sigma_1$	Change in major principal stress	-
$\Delta \sigma_3$	Change in minor principal stress (confining pressure)	-

Skempton's Parameter Ranges

Parameter B: For fully saturated soils, $B \approx 1.0$ . For dry soils, $B = 0$ .
Parameter A: For NC clays, $A$ is typically positive (0.5 to 1.0). For heavily OC clays, $A$ can be negative (-0.5 to 0).

Laboratory Tests

To determine the shear strength parameters ( $c, \phi$ ), several laboratory tests are used.

Direct Shear Test

A sample is placed in a split box and sheared along a predetermined horizontal plane.

Procedure: Apply normal load ( $N$ ), then apply shear force ( $T$ ) until failure. Repeat for different normal loads.
Advantages: Simple, inexpensive, good for sands.
Disadvantages: Failure plane is forced, stress distribution is non-uniform, drainage is hard to control.

Triaxial Test

A cylindrical sample is encased in a rubber membrane and subjected to confining pressure ( $\sigma_3$ ). An axial load ( $\Delta \sigma_d$ ) is increased until failure.

UU (Unconsolidated-Undrained): Quick test. Simulates end-of-construction stability for saturated clays. ( $\phi_u = 0, c_u$ ).
CU (Consolidated-Undrained): Sample consolidated under $\sigma_3$ , then sheared undrained. Pore pressure ( $u$ ) is measured to get effective strength parameters ( $c', \phi'$ ).
CD (Consolidated-Drained): Slow test. Excess pore pressure dissipates completely. Simulates long-term stability. ( $c', \phi'$ ).

Unconfined Compression Test (UCT)

A special case of the triaxial test where confining pressure $\sigma_3 = 0$ .

$q_u$ : Unconfined compressive strength.
$c_u = q_u / 2$ : Undrained shear strength.
Suitable only for cohesive soils (clays).

Field Tests

In addition to laboratory testing, shear strength can be estimated directly in the field.

Vane Shear Test (VST)

The Vane Shear Test is used for soft to stiff clays. A four-bladed vane is pushed into the soil and rotated to determine the undrained shear strength in-situ.

Vane Shear Strength

Calculates undrained shear strength from the torque required to rotate a four-bladed vane; used for soft to stiff clays in-situ.

c_u = \frac{T}{\pi \left( \frac{d^2 h}{2} + \frac{d^3}{6} \right)}

Variables

Symbol	Description	Unit
$c_u$	Undrained shear strength	-
$T$	Maximum torque applied	-
$d$	Diameter of vane	-
$h$	Height of vane	-

Key Takeaways

Shear Strength depends on cohesion ( $c$ ) and friction angle ( $\phi$ ).
Mohr's Circle is used to represent the stress state at a point. Failure occurs when the circle touches the Failure Envelope.
Effective Stress Parameters ( $c', \phi'$ ) govern long-term stability and drained conditions.
Total Stress Parameters ( $c_u, \phi_u$ ) govern short-term stability in saturated clays (End-of-Construction).
Skempton's Parameters ( $A$ and $B$ ) are essential for predicting undrained pore pressure responses to loading.
Triaxial Tests (UU, CU, CD) provide the most comprehensive data on soil strength and pore pressure behavior.