Mohr-Coulomb equation, abbreviated as \( \tau \) (Greek Symbol tau), also called Mohr-Coulomb failure criterion, is a foundational mathematical model in civil and geotechnical engineering that describes the shear strength of soils, rocks, and other brittle materials under combined normal and shear stresses. The criterion establishes a linear relationship between the shear stress at failure and the applied normal stress on a potential failure plane. This relationship forms the failure envelope on a plot of shear stress versus normal stress, allowing engineers to predict when a material will reach its limiting state and fail in shear, which is essential for analyzing the stability of earth structures such as slopes, foundations, retaining walls, and embankments.
Mohr-Coulomb Equation formula |
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\( \tau \;=\; c + \sigma \cdot tan( \varphi ) \) (Mohr-Coulomb Equation) \( c \;=\; \tau - \sigma \cdot tan( \varphi ) \) \( \sigma \;=\; \dfrac{ \tau - c }{ tan( \varphi ) } \) |
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| Symbol | English | Metric |
| \( \tau \) (Greek Symbol tau) = Shear Stress of the Soil | \(lbf \;/\; ft^2\) | \(Pa\) |
| \( c \) = Cohesion of the Soil | \(lbf-sec\;/\;ft^2\) | \(Pa-s \) |
| \( \sigma \) = Normal Stress on the Plane | \(ft\;/\;sec\) | \(m\;/\;s\) |
| \( \varphi \) = Angle of Internal Friction | \(deg\) | \(rad\) |
The Mohr-Coulomb equation serves as the basis for limit equilibrium methods in slope stability analysis and bearing capacity calculations, where engineers compare the mobilized shear stress along a potential failure surface against the available strength defined by the criterion. Cohesionless granular materials, such as clean sands, exhibit and rely solely on the frictional term for strength, while cohesive soils like clays derive additional resistance from the intercept. The criterion assumes a linear envelope and isotropic behavior, which holds well for many practical applications but may be extended or modified for nonlinear or anisotropic conditions in advanced analyses. Its widespread adoption stems from its simplicity, verifiability through standard soil testing protocols, and alignment with observed failure modes in field and laboratory data across countless engineering projects worldwide.
The Mohr–Coulomb criterion is used extensively in the analysis and design of geotechnical systems such as foundations, retaining walls, slopes, and embankments. It provides a means to estimate shear strength parameters from laboratory tests such as direct shear tests and triaxial compression tests. Although it is an idealization and assumes linear behavior and constant strength parameters, it remains one of the most established and widely accepted models in civil engineering for describing shear failure in soils and rocks.
