Explaining the Mohr-Coulomb Failure Criterion

Understanding why soils and rocks fail is at the heart of geotechnical engineering — and few tools are as fundamental to that understanding as the Mohr-Coulomb Failure Criterion. This widely adopted model offers a simple yet powerful way to predict when geomaterials will fail under shear stress, making it indispensable for the safe design of slopes, foundations, tunnels, and retaining structures. By combining the effects of cohesion and friction, the Mohr-Coulomb approach bridges theoretical soil mechanics with practical field engineering. Whether assessing the risk of a landslide or designing the foundations of a skyscraper, this criterion remains a cornerstone of geotechnical decision-making.

Before delving into the Mohr-Coulomb Failure Criterion, it is important to outline the concept of shear strength. As opposed to tensile failure (by stretching) and compressive failure (by squashing), shear failure involves sliding or tearing along a plane due to opposing but offset forces. Shear strength refers to the maximum shear stress that a material can withstand before failing in shear. Soils are a granular medium with virtually zero tensile strength and very high compressive strength in most applications. Shear strength is somewhere in between and is usually the most critical.

Failure occurs by sliding and rotation of soil particles against each other, resisted by interlock and friction between them. Rocks are composed of larger blocks with planar joints between them. The joints also have virtually zero tensile strength, very high compressive strength and a shear strength somewhere in between. In geotechnical terms, shear strength is crucial for determining the stability of both soil and rock masses under various loading conditions.

Shear strength can be divided into two primary components:

Cohesion (c): Cohesion binds soil and rock grains together. Intact blocks of rock have high cohesion due to natural cementation between the grains. Rock joints and soil particles may have slight cohesion due to some weak natural cementation but in general they have zero cohesion. Clay soils have an apparent cohesion due to electromechanical forces between particles and water between them.

Friction Angle (φ): The friction angle represents the resistance to sliding along a plane within the material. It comes from interlock and friction between soil particles and at rock joints. In general, it increases with particle size. So, clays, silts, sand and gravels have gradually increasing values, but it depends on other factors too.

These components form the basis for the Mohr-Coulomb Failure Criterion, providing a framework to evaluate geomaterial stability under shear stress.

The Mohr-Coulomb Failure Criterion: Mathematical Formulation

The Mohr-Coulomb Failure Criterion is mathematically represented as a linear equation that relates shear stress (τ) and normal stress (σ) along a potential failure plane. The equation is given byThe Mohr-Coulomb Failure Criterion is mathematically represented as a linear equation that relates shear stress (τ) and normal stress (σ) along a potential failure plane. The equation is given by:

τ = c + σ tan φ

Where:

• τ is the shear stress at failure

• c is the cohesion of the material

• σ is the normal stress acting perpendicular to the shear plane

• φ is the internal friction angle of the material

The Mohr’s Circle Representation

To visualise the stress state in a material, geotechnical engineers often use Mohr’s circles, a graphical representation of stress in all directions at a point. Mohr’s Circle aids in understanding the relationship between normal and shear stresses, and it is instrumental in applying the Mohr-Coulomb Failure Criterion.

Practical Application of the Mohr-Coulomb Criterion

The Mohr-Coulomb Failure Criterion is extensively used in geotechnical engineering to analyse and design various structures:

Slope Stability Analysis

In slope stability analysis, the Mohr-Coulomb Criterion is employed to assess the potential failure of slopes under different loading conditions. By evaluating the shear strength parameters, engineers can predict the likelihood of landslides and design appropriate mitigation measures.

Foundation Design

When designing foundations, understanding the shear strength of the soil is crucial for ensuring the stability of structures. The Mohr-Coulomb parameters are used in the calculation of the load-bearing capacity of the soil, facilitating the design of safe and effective foundations systems.

Retaining Wall Design

Retaining walls are structures designed to hold back soil and prevent erosion. The Mohr-Coulomb Criterion is used to evaluate the forces acting on retaining walls, enabling engineers to design walls that can withstand the stresses imposed by the retained soil.

Limitations of the Mohr-Coulomb Failure Criterion

Despite its widespread use, the Mohr-Coulomb Failure Criterion has certain limitations. It assumes a linear relationship between shear stress and normal stress, which may not always hold true for all geomaterials, particularly rock masses. Additionally, the criterion does not account directly for time-dependent and strain-dependent changes in strength.

Advancements and Alternatives

In response to these limitations, geotechnical engineers have developed alternative models and methods to better predict material behaviour under complex conditions. These include the Hoek-Brown Criterion, which is a non-linear failure criterion more suited to rock masses.

Conclusion: The Significance of the Mohr-Coulomb Criterion in Geotechnical Engineering

The Mohr-Coulomb Failure Criterion remains a cornerstone in geotechnical engineering, offering a robust framework for assessing the shear strength of geomaterials. While it has its limitations, its simplicity and effectiveness make it suitable for a wide range of applications, from slope stability analysis to foundation and retaining wall design.