The shear strength of a soil

Here's the core idea:

* Shear strength is the soil's ability to resist sliding or shearing.
* Normal stress is the force pressing down on the soil (perpendicular to the shearing plane). Think of it as the weight pushing the soil particles together.
* Angle of internal friction (φ) represents how much the soil particles resist sliding against each other due to friction. A higher angle means more friction.

The relationship:

The shear strength (τ) of a soil is described by the Mohr-Coulomb equation:
τ = c + σ * tan(φ)
where:
c = cohesion of the soil
σ = normal stress
φ = angle of internal friction

Explanation of each option:
A: Is directly proportional to the angle of internal friction of the soil
* This is correct. As the angle of internal friction (φ) increases, the shear strength (τ) also increases because tan(φ) increases.
B: Is inversely proportional to the angle of internal friction of the soil
* This is incorrect. Inversely proportional means one increases as the other decreases, which is the opposite of what happens.
C: Decreases with increase in normal stress
* This is incorrect. The formula (τ = c + σ * tan(φ)) shows that as normal stress (σ) increases, the shear strength (τ) also increases.
D: Decreases with decrease in normal stress
* This is incorrect. If the normal stress (σ) decreases, the shear strength (τ) decreases as well.

Therefore, the correct answer is A.