The failure condition for a soil can be expressed in terms of limiting shear stress, called __________

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Option A: Principal stresses and Shear strength

Explanation & Concepts:
The failure of soil is determined by the Mohr-Coulomb failure criterion, which states that shear failure occurs when the shear stress on a plane reaches a limiting value that is a function of the normal stress acting on that plane. This limiting shear stress is defined by the shear strength parameters (cohesion $c$ and angle of internal friction $\phi$) relative to the acting principal stresses.

Working Principle: In a soil mass, the state of stress is defined by the major principal stress ($\sigma_1$) and the minor principal stress ($\sigma_3$). Failure occurs when the Mohr circle representing these stresses becomes tangent to the failure envelope defined by $	au_f = c + \sigma_n 	an(\phi)$. Thus, the failure condition is fundamentally tied to the principal stress state and the inherent shear strength parameters of the soil.

Key Formulas: $	au_f = c + \sigma_n 	an(\phi)$ — Mohr-Coulomb failure criterion; $\sigma_1 = \sigma_3 	an^2(45^\circ + \frac{\phi}{2}) + 2c 	an(45^\circ + \frac{\phi}{2})$ — Relationship between principal stresses at failure

Key Points: Shear strength is the internal resistance per unit area that the soil mass can offer to resist failure and sliding.; The failure envelope is generally represented as a straight line in the $	au$ vs $\sigma_n$ plot.; For cohesionless soils ($c=0$), the failure envelope passes through the origin.

Advantages: Predicts shear failure in slopes and foundations.; Provides a clear mathematical basis for stability analysis.

Disadvantages: Assumes a linear failure envelope which may not hold for high confining pressures.; Does not account for time-dependent deformation (creep).

Applications: Foundation design (bearing capacity calculation).; Slope stability analysis of embankments and dams.

Additional Info: The term 'Shearing resistances' is a physical property, but the 'failure condition' itself is mathematically expressed via the relationship between principal stresses and strength parameters.. Option B is incomplete because it defines the resistance but not the condition of limit equilibrium.

Answer: A is correct — The failure condition of soil is mathematically expressed by relating the acting principal stresses to the soil's shear strength parameters, $c$ and $\phi$.

Exam Tip: Always remember that at failure, the Mohr circle is tangent to the failure envelope; therefore, the failure condition is implicitly a relationship between $\sigma_1$, $\sigma_3$, and the shear parameters.

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Option B: Shearing resistances

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Option C: None of the mentioned

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Option D: All of the mentioned

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