Mohrcoulomb TheoryEdit
Mohr-Coulomb theory, commonly known as the Mohr-Coulomb criterion, is one of the most enduring and widely used shear-strength models in geotechnical engineering. It provides a simple, linear relationship between shear strength on a potential failure plane and the normal stress acting on that plane. The core idea is that soil and rock fail when the shear stress reaches a threshold that increases with confining pressure, captured by a small set of material parameters. In its standard form, the shear strength along a plane is given by the equation τ = c' + σ'n tan φ', where τ is the shear stress, σ'n is the normal effective stress on the plane, c' is the effective cohesion, and φ' is the effective angle of internal friction. This compact expression has proven extremely practical for design and analysis across earthworks, foundations, slopes, and many underground excavations, and it remains a baseline model in codes and practice worldwide. It rests on combining Mohr's circle concepts of stress with Coulomb's friction principle, and it works best for drained, quasi-static conditions in relatively homogeneous materials.
Foundations and formulation
Mohr-Coulomb theory sits at the intersection of three ideas: the representation of stress on a plane by Mohr's circle, the observation that friction along rough surfaces governs how materials fail, and the translation of that friction into a probabilistic, engineering-friendly envelope. The theory implies that when a material is subjected to increased normal stress, it can carry more shear before failure, but that relationship is linear in the standard formulation. The effective-stress version uses σ'n, the normal stress corrected for pore water pressure, to reflect the actual stresses acting on the solid skeleton within saturated soils. For convenience, many engineers interpret σ'n as the stress that remains supporting the soil skeleton once pore pressures are accounted for. See Mohr's circle for the geometric basis of this idea and Coulomb criterion for the frictional physics that inform the linear envelope.
The two key parameters, c' and φ', summarize a material's stability characteristics under the Mohr-Coulomb framework. Cohesion c' represents the inherent strength of the material when there is no normal stress, while φ' (the effective angle of internal friction) measures how rapidly strength increases with additional normal stress. In practice, these parameters are not universal constants; they depend on soil type, density, moisture content, stress history, and loading conditions. Their typical interpretation is straightforward: c' is most relevant to clays and cemented soils, whereas φ' tends to be a dominant factor for granular materials such as sands and gravels. See cohesion and internal friction angle for related concepts.
Material parameters and testing
Determining c' and φ' involves lab testing under controlled conditions, most commonly via direct shear tests and triaxial tests. Direct shear tests provide a direct measurement of shear strength at various normal stresses, from which c' and φ' can be back-calculated. Triaxial compression tests, performed with drained or undrained conditions, likewise yield strength data from which the same parameters can be inferred, often with attention to pore pressures and consolidation. The effective-stress framework emphasizes that the interpretation of test results must consider pore-water effects, drainage state, and specimen preparation. See direct shear test and triaxial compression test for more detail. For the broader concept of how stresses are balanced in soils, see effective stress.
In practice, φ' is typically higher for well-graded, dense granular materials and lower for loose or highly structured soils, while c' tends to be more important for cohesive soils with cementation or clay minerals. The material’s density, relative compaction, and current moisture regime can shift these parameters enough to influence stability assessments significantly. See also soil mechanics for the broader field that underpins these measurements.
Applications
The Mohr-Coulomb criterion underpins a wide range of design and evaluation tasks in geotechnical engineering. In slope stability analyses, it is used to estimate the shear strength along potential slip surfaces and to determine factor of safety against failure under given loading and water conditions. In foundation engineering, the bearing capacity of shallow and deep foundations is frequently predicted by comparing the applied stresses to the Mohr-Coulomb strength envelope. Retaining walls and earth-retaining structures rely on similar assessments of soil strength to resist sliding, overturning, and bearing failure. For related topics, see slope stability, bearing capacity, and retaining wall.
In rock mechanics, the Mohr-Coulomb criterion is often applied in a modified form to estimate shear strength along rock joints or intact rock under certain confining conditions. However, for highly anisotropic, fragmented, or highly stressed rock masses, engineers commonly turn to alternative or augmented models such as the Hoek-Brown criterion to capture nonlinear, scale-dependent behavior. See also rock mechanics for the broader field.
Assumptions, limitations, and debates
Like any simplified model, Mohr-Coulomb theory rests on a set of assumptions that constrain its applicability. It presumes:
- Linear failure envelope in the τ–σ'n space, which implies a constant φ' and c' independent of stress level, strain history, and loading rate.
- Homogeneous, isotropic material behavior with a single set of parameters representative of the whole mass, or at least for the portion of interest.
- Drained, quasi-static loading conditions where pore pressure remains in a predictable relationship with effective stresses.
- A focus on shear failure along planar or approximately planar surfaces, which may not capture complex three-dimensional failure modes or dilatant/contractive behavior.
In soils that exhibit significant dilatancy, strain-hardening/softening, or rate effects, the Mohr-Coulomb envelope can misrepresent when and how failure occurs. Similarly, in sandy soils at high confining pressures or in very stiff rock masses with pronounced jointing and block behavior, a straight-line envelope may be too simple, leading to conservative or non-conservative predictions depending on the context. For rock masses, practitioners often use adjustments or alternative criteria, such as the Hoek-Brown criterion, to better capture the nonlinearity and scale effects inherent in fractured geology.
Controversies in the field typically center on whether a simple, linear, constant-parameter model remains appropriate for modern design, especially as projects push into more demanding conditions (deep foundations, long-span slopes, seismic loading). Proponents of Mohr-Coulomb emphasize its elegance, transparency, and regulatory acceptance: it provides conservative, checkable, and widely understood results that align with historical data and many practical experiences. Critics argue that, in some cases, the model is overly simplistic and can obscure complex mechanical behavior, potentially leading to misjudged safety margins if used without caution or without supplemental analyses. The contemporary engineering approach often blends Mohr-Coulomb with more advanced models or uses it as a baseline, applying safety factors and site-specific calibrations to reflect uncertainties. Some critics, particularly from more progressive or risk-averse perspectives, contend that review and update of design codes are overdue to incorporate modern constitutive models; defenders counter that the added complexity and data requirements must be justified by demonstrable improvements in reliability and cost-effectiveness. In any case, effective practice depends on understanding both the strengths and the limits of the criterion, and on pairing the model with appropriate testing, site characterization, and engineering judgment.
From a practical standpoint, Mohr-Coulomb remains the workhorse for many projects because it yields transparent, auditable, and defensible designs that engineers and regulators can reasonably check. Where more complex behavior is anticipated—such as in highly anisotropic rock masses, aggressive environmental loading, or situations with significant rate dependence—engineers may supplement with additional criteria or use alternative models to guide decisions without abandoning the foundational intuition that Mohr-Coulomb provides. See limit equilibrium analysis for a methodological context and critical state soil mechanics for a broader view of how researchers model soil behavior beyond the linear Mohr-C coulomb envelope.