The shear strength of cohesion-less soil is

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Option C: Proportional to the tangent of the angle of shearing resistance

Explanation & Concepts:
The shear strength of cohesion-less soil (like sand) is governed by the Mohr-Coulomb failure criterion, which states that shear strength ($	au$) is directly proportional to the normal stress ($\sigma$) and the tangent of the angle of internal friction ($\phi$). Since cohesion ($c$) is zero for cohesion-less soil, the strength is defined solely by the frictional resistance between particles.

Working Principle: In cohesion-less soils, resistance to shear is purely frictional. When a normal force is applied, the particles are pressed together, and their ability to slide over one another is limited by inter-particle friction. The shear strength developed is proportional to the normal stress acting on the failure plane multiplied by $	an(\phi)$.

Key Formulas: $	au = \sigma 	an(\phi)$ — Mohr-Coulomb equation for cohesion-less soil; $	au = c + \sigma 	an(\phi)$ — General Mohr-Coulomb equation where $c=0$

Key Points: Cohesion-less soils (c=0) derive their strength entirely from inter-particle friction.; The angle of shearing resistance ($\phi$) is a measure of the frictional properties of the soil grains.; Shear strength increases linearly with effective normal stress and the tangent of the friction angle.; As $\phi$ increases, the shear resistance of the granular skeleton increases.

Advantages: Predictable behavior under drained conditions; High shear strength at higher confining pressures

Disadvantages: No intrinsic strength without confining pressure; Highly dependent on void ratio and relative density

Applications: Foundation design on sandy soils; Slope stability analysis for granular embankments

Additional Info: Typical values of $\phi$ for loose sand range from 28° to 30°, while for dense sand, they can exceed 40°.. Option B is incorrect because $	an(\phi)$ is an increasing function for $0 \le \phi

Answer: C is correct — The shear strength of cohesion-less soil is expressed as $	au = \sigma 	an(\phi)$, making it directly proportional to the tangent of the angle of shearing resistance.

Exam Tip: Always remember that for cohesion-less soils, $c=0$; never confuse the proportional relationship of $	an(\phi)$ with the angle $\phi$ itself, as the function is non-linear.

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Option A: Proportional to the angle of shearing resistance

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Option B: Inversely proportional to the angle of shearing resistance

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Option D: None of above

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