Shear Strength - Theory & Concepts

Shear Strength of Soil

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.

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

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

$c'$ 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	-

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

Interactive Mohr's Circle

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 given by:

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)	-

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

Vane Shear Test (VST)

Used for soft to stiff clays. A four-bladed vane is pushed into the soil and rotated.

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.

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