Which of the following is coulomb’s strength equation?

Question

Examoogle · Accepted Answer

The correct answer is Option C: $S = C + \sigma 	an \phi$. Coulomb's Law (or the Mohr-Coulomb failure criterion) states that the shear strength of a soil at failure is a linear function of the normal stress acting on the failure plane. It is expressed as the sum of cohesion ($C$) and the product of normal stress ($\sigma$) and the tangent of the internal angle of friction ($\phi$).

Step-by-Step Solution & Concepts:
Coulomb's Law (or the Mohr-Coulomb failure criterion) states that the shear strength of a soil at failure is a linear function of the normal stress acting on the failure plane. It is expressed as the sum of cohesion ($C$) and the product of normal stress ($\sigma$) and the tangent of the internal angle of friction ($\phi$).

Working Principle: The shear strength is composed of two components: the intrinsic cohesion ($C$) which represents the bonding between soil particles, and the frictional component ($sigma 	an \phi$) which arises from the intergranular contact and mechanical interlocking of soil particles under normal stress. As normal stress increases, the frictional resistance increases proportionally until failure occurs.

Key Formulas: $S = C + \sigma 	an \phi$ — The Mohr-Coulomb failure criterion equation

Key Points: Cohesion ($C$) is the shear strength at zero normal stress.; The internal angle of friction ($\phi$) represents the shear strength contributed by inter-particle friction.; The equation assumes a linear relationship between shear strength and normal stress.; This criterion is foundational for stability analysis of slopes and earth retaining structures.

Advantages: Simple mathematical model for complex soil behavior; Easily derived from standard laboratory tests like Direct Shear or Triaxial tests

Disadvantages: Assumes linear behavior which may not hold for all soil types at high pressures; Does not account for pore water pressure unless effective stress is explicitly considered

Applications: Bearing capacity analysis of foundations; Slope stability assessment; Lateral earth pressure calculations in retaining walls

Additional Info: In terms of effective stress, the equation is expressed as $s = c' + \sigma' 	an \phi'$, where $c'$ and $\phi'$ are effective stress parameters.. Option A is incorrect due to missing the normal stress variable.. Option B is incorrect as it transposes the shear strength ($S$) and cohesion ($C$) variables.

Answer: C is correct — The Coulomb strength equation describes the shear strength $S$ as $S = C + \sigma 	an \phi$.

Exam Tip: Always remember the difference between total stress parameters and effective stress parameters; in geotechnical problems, failure is usually governed by effective stress, $\sigma'$.

Examoogle · Answer

Option A: $S = C + 	an \phi$

Examoogle · Answer

Option B: $C = S + \sigma 	an \phi$

Examoogle · Answer

Option D: $S = 	an \phi$

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