2.3 Shear Strength of Soil and Stability of Slopes

2.3 Shear Strength of Soil and Stability of Slopes

Introduction to Shear Strength and Slope Stability

• The shear strength of soil is its most critical engineering property, defining its ability to resist sliding along internal surfaces.

• This resistance governs the stability of all geotechnical structures, from natural slopes and earth dams to retaining walls and foundations.

• Understanding the factors that contribute to shear strength—cohesion and internal friction—and how they are quantified through laboratory tests is essential.

• This knowledge is directly applied to analyze the stability of slopes, a common and often hazardous geotechnical problem.

• This unit connects the fundamental theory of shear strength with the practical methods used to assess the factor of safety against slope failure.

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1. Concept of Shear Strength

• Definition: The maximum shear stress a soil mass can withstand along an internal plane before failure (sliding) occurs.

• Components: Soil shear strength ($\boldsymbol{\tau_f}$) is derived from two primary components:

  • Cohesion ($\boldsymbol{c}$): The inherent "stickiness" or bonding between soil particles, primarily in fine-grained soils (clays). It is independent of the normal stress.

  • Internal Friction ($\boldsymbol{\phi}$): The resistance due to interlocking of soil particles and the frictional forces at their contacts. It is proportional to the normal stress ($\boldsymbol{\sigma}$) acting on the failure plane.

• Mathematical Representation (Mohr-Coulomb Failure Criterion):

  $\boldsymbol{\tau_f = c + \sigma \tan \phi}$

  where:

  • $\boldsymbol{\tau_f}$ = Shear strength at failure.

  • $\boldsymbol{\sigma}$ = Total normal stress on the failure plane (for total stress analysis).

  • $\boldsymbol{c}$ = Cohesion intercept.

  • $\boldsymbol{\phi}$ = Angle of internal friction.

• Effective Stress Principle: In terms of effective stress (for long-term, drained analysis), the equation becomes:

  $\boldsymbol{\tau_f = c' + \sigma' \tan \phi'}$

  where $\boldsymbol{c'}$ and $\boldsymbol{\phi'}$ are the effective cohesion and friction angle, and $\boldsymbol{\sigma'}$ is the effective normal stress.

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2. Principal Planes and Principal Stresses

• Principal Planes: These are the planes on which the shear stress is zero. At any point in a stressed body, there are three mutually perpendicular principal planes.

• Principal Stresses: The normal stresses acting on the principal planes. They are denoted as:

  • Major Principal Stress ($\boldsymbol{\sigma_1}$): The largest normal stress.

  • Intermediate Principal Stress ($\boldsymbol{\sigma_2}$): In most geotechnical 2D problems, this is ignored or assumed equal to $\boldsymbol{\sigma_3}$.

  • Minor Principal Stress ($\boldsymbol{\sigma_3}$): The smallest normal stress.

• Significance: In soil mechanics, failure is often analyzed in terms of principal stresses. The Mohr's circle construction graphically relates normal and shear stresses on any plane to the principal stresses.

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3. Mohr-Coulomb Theory of Shear Strength

• Postulate: The theory states that failure occurs when the shear stress on a plane reaches a value that depends linearly on the normal stress on that same plane.

• Graphical Representation (Mohr's Circle):

  • Stress states at a point can be represented by a circle on a $\boldsymbol{\tau}$ vs. $\boldsymbol{\sigma}$ plot.

  • The Mohr-Coulomb failure criterion is drawn as a straight line with intercept $\boldsymbol{c}$ and slope $\boldsymbol{\tan \phi}$. This is the failure envelope.

• Failure Condition: Failure occurs when the Mohr's circle for a given stress state just touches (becomes tangent to) the failure envelope.

• Angle of the Failure Plane:

  • The failure plane is inclined at an angle $\boldsymbol{\theta_f}$ to the plane of the major principal stress ($\boldsymbol{\sigma_1}$).

  • From geometry of the Mohr circle at failure:

    $\boldsymbol{\theta_f = 45^\circ + \frac{\phi}{2}}$

  • This theoretical failure plane is observed in direct shear tests on sands.

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4. Calculation of Normal and Shear Stresses at Different Planes

• Using Mohr's Circle:

  • Given principal stresses $\boldsymbol{\sigma_1}$ and $\boldsymbol{\sigma_3}$, the center of the circle is at $\boldsymbol{\frac{\sigma_1 + \sigma_3}{2}}$ and the radius is $\boldsymbol{\frac{\sigma_1 - \sigma_3}{2}}$.

  • To find stresses on a plane inclined at angle $\boldsymbol{\theta}$ (measured counterclockwise from the plane of $\boldsymbol{\sigma_1}$):

    $\boldsymbol{\sigma_\theta = \frac{\sigma_1 + \sigma_3}{2} + \frac{\sigma_1 - \sigma_3}{2} \cos 2\theta}$

    $\boldsymbol{\tau_\theta = \frac{\sigma_1 - \sigma_3}{2} \sin 2\theta}$

• Using Pole (Origin of Planes) Method:

  • A powerful graphical technique where a unique point (the Pole) on the Mohr circle has the property that a line drawn through the Pole parallel to any physical plane will intersect the circle at the point representing the stresses on that plane.

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5. Principle Stress at Failure Condition

• At failure, the Mohr's circle is tangent to the failure envelope.

• Relationship between $\boldsymbol{\sigma_1}$ and $\boldsymbol{\sigma_3}$ at failure can be derived from the geometry of the tangent circle:

  $\boldsymbol{\sigma_1 = \sigma_3 N_\phi + 2c \sqrt{N_\phi}}$

  or conversely,

  $\boldsymbol{\sigma_3 = \sigma_1 / N_\phi - 2c / \sqrt{N_\phi}}$

  where,

  $\boldsymbol{N_\phi = \frac{1 + \sin \phi}{1 - \sin \phi} = \tan^2(45^\circ + \phi/2)}$

• Terms:

  • $\boldsymbol{N_\phi}$ is called the flow value.

  • $\boldsymbol{45^\circ + \phi/2}$ is the theoretical angle of the failure plane.

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6. Types of Shear Tests

• Laboratory tests determine the shear strength parameters ($\boldsymbol{c}$ and $\boldsymbol{\phi}$) for design.

6.1 Direct Shear Test (IS 2720 Part XIII)

• Procedure: A soil specimen is placed in a split box. A normal load ($\boldsymbol{N}$) is applied. The lower half is moved relative to the upper half, inducing shear on a predetermined horizontal plane.

• Output: A plot of shear stress ($\boldsymbol{\tau}$) vs. horizontal displacement. Peak stress is taken as $\boldsymbol{\tau_f}$.

• Analysis: Multiple tests at different normal stresses are conducted. A plot of $\boldsymbol{\tau_f}$ vs. $\boldsymbol{\sigma}$ gives $\boldsymbol{c}$ (intercept) and $\boldsymbol{\phi}$ (slope).

• Advantages: Simple, fast, good for granular soils and discontinuities.

• Disadvantages: Failure plane forced, non-uniform stress/strain, drainage control is difficult.

6.2 Triaxial Compression Test (IS 2720 Part XI)

• Procedure:

  1. A cylindrical specimen is enclosed in a rubber membrane inside a pressure chamber.

  2. It is subjected to an all-around cell pressure ($\boldsymbol{\sigma_3}$).

  3. Additional deviator stress ($\boldsymbol{\sigma_1 - \sigma_3}$) is applied axially until failure.

• Types (Based on Drainage Conditions):

  • UU (Unconsolidated-Undrained): Quick test, no drainage allowed. Gives undrained strength ($\boldsymbol{c_u}$, $\boldsymbol{\phi_u = 0}$). Used for short-term stability of saturated clays.

  • CU (Consolidated-Undrained): Drainage allowed during cell pressure application (consolidation), then sheared undrained. Often measures pore pressure. Gives $\boldsymbol{c'}$ and $\boldsymbol{\phi'}$ (effective) or total stress parameters.

  • CD (Consolidated-Drained): Slow test with drainage allowed throughout. Gives drained or effective strength parameters ($\boldsymbol{c'}$ and $\boldsymbol{\phi'}$). Used for long-term stability.

• Analysis: Mohr circles are drawn for each test at failure. A common tangent to these circles gives $\boldsymbol{c}$ and $\boldsymbol{\phi}$.

• Advantages: Uniform stress/strain, controlled drainage, can measure pore pressure, failure occurs on weakest plane.

• Disadvantages: Complex, time-consuming (especially CD).

6.3 Unconfined Compression Test (IS 2720 Part X)

• Procedure: A special case of triaxial test with $\boldsymbol{\sigma_3 = 0}$. Axial load is applied to an unsaturated or saturated clay specimen without lateral support.

• Output: Undrained shear strength, $\boldsymbol{s_u}$ ($\boldsymbol{q_u / 2}$ at failure). For saturated clays ($\boldsymbol{\phi_u = 0}$), $\boldsymbol{q_u = 2 c_u}$.

• Use: Quick estimate of undrained strength of cohesive soils.

6.4 Vane Shear Test

• Procedure: A four-bladed vane is inserted into soft clay and rotated. The torque required to cause a cylindrical shear failure is measured.

• Use: In-situ test for very soft to medium clays to determine undrained shear strength ($\boldsymbol{s_u}$).

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7. Stability of Slopes

7.1 Types of Slope Failures

• Rotational (Slip Circle) Failure:

  • Common in homogeneous clays.

  • Failure surface is circular or near-circular in cross-section.

  • Types: Base failure, toe failure, slope failure.

• Translational (Planar) Failure:

  • Occurs along a weak plane (e.g., layer of soft clay, bedding plane).

  • Common in layered soils or rock slopes.

• Compound Failure: Combination of circular and planar surfaces.

• Wedge Failure: In rock slopes with intersecting joints.

7.2 Factor of Safety (FOS)

• Definition: The ratio of the available shear strength of the soil to the mobilized shear stress required for equilibrium along a potential failure surface.

  $\boldsymbol{ \mathrm{FOS} = \frac{\mathrm{Shear\ Strength\ of\ Soil}} {\mathrm{Mobilized\ Shear\ Stress}} }$

• Interpretation:

  • FOS > 1: Stable.

  • FOS = 1: At limiting equilibrium (failure is imminent).

  • FOS < 1: Unstable (failure occurs).

7.3 Methods of Analysis for Rotational Failures

• Swedish Circle Method / Method of Slices:

  • The most common method for analyzing circular slip surfaces, especially in non-homogeneous soils.

  • The sliding mass above the trial failure arc is divided into vertical slices.

  • The forces on each slice are analyzed (weight, normal and shear forces on the base, side forces).

• Ordinary Method of Slices (Fellenius Method):

  • Simplest method. Assumes the side forces between slices are zero.

  • Factor of Safety:

    $\boldsymbol{FOS = \frac{\sum [c'l + (W \cos \alpha - u l) \tan \phi']}{\sum W \sin \alpha}}$

    where:

    • $\boldsymbol{W}$ = weight of slice.

    • $\boldsymbol{\alpha}$ = inclination of slice base to horizontal.

    • $\boldsymbol{l}$ = length of slice base.

    • $\boldsymbol{u}$ = pore water pressure at slice base.

• Bishop's Simplified Method:

  • More accurate than Ordinary Method.

  • Assumes the vertical side forces are equal and opposite (cancel), but ignores the shear forces between slices.

    $\boldsymbol{FOS = \frac{\sum \frac{1}{m_\alpha} [c' b + (W - u b) \tan \phi']}{\sum W \sin \alpha}}$

    where:

    $\boldsymbol{m_\alpha = \cos \alpha + \frac{\tan \phi' \sin \alpha}{FOS}}$

  • Note: FOS appears on both sides, requiring an iterative solution.

7.4 Analysis for Planar Failures (Infinite Slopes)

• Assumption: Slope is long and uniform, failure plane parallel to slope surface.

• For a cohesionless soil ($\boldsymbol{c' = 0}$):

  $\boldsymbol{FOS = \frac{\tan \phi'}{\tan \beta}}$

  where $\boldsymbol{\beta}$ is the slope angle.

  • Stability is independent of slope height.

• For a cohesive soil with seepage parallel to the slope:

  • Critical condition often involves seepage forces.

    $\boldsymbol{FOS = \frac{c'}{\gamma_{sat} z \sin \beta \cos \beta} + \frac{\gamma' \tan \phi'}{\gamma_{sat} \tan \beta}}$

    where $\boldsymbol{z}$ is depth to failure plane, $\boldsymbol{\gamma_{sat}}$ is saturated unit weight, $\boldsymbol{\gamma'}$ is submerged unit weight.

7.5 Stabilization Measures

• Geometry Modification: Flattening the slope, providing benches (berms).

• Drainage: The most effective and economical measure. Includes surface drains, horizontal drains, and drainage blankets to reduce pore water pressure.

• Retaining Structures: Gravity walls, cantilever walls, gabion walls at the toe.

• Reinforcement: Soil nailing, geosynthetic reinforced soil (GRS) slopes.

• Internal Strengthening: Grouting, stone columns, soil mixing.

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Conclusion

The journey from the fundamental Mohr-Coulomb failure criterion to the analysis of a complex slope embodies the core of geotechnical engineering. Determining accurate shear strength parameters through appropriate laboratory testing is the first critical step. Applying these parameters within stability analysis frameworks, while accounting for the ever-present influence of water, allows engineers to design slopes, embankments, and excavations with a quantifiable margin of safety, protecting both infrastructure and life.

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