The coefficient of passive earth pressure

Explanation:

To understand why the coefficient of earth pressure of passive is given by σv/σh, we need to first understand the concept of effective stress and the principles of soil mechanics.

Effective Stress:
Effective stress is the stress that is transmitted between soil particles and is responsible for shear strength and deformation characteristics of the soil. It is defined as the difference between total stress and pore water pressure.

Principles of Soil Mechanics:
In soil mechanics, the concept of effective stress principle is used to determine the behavior of soil under different loading conditions. It states that the shear strength and deformation characteristics of soil depend on the effective stress acting on the soil particles.

Earth Pressure of Passive:
The earth pressure of passive refers to the lateral pressure exerted by soil on a retaining wall or structure when the soil is in a state of rest or equilibrium. It is influenced by the vertical and horizontal stresses acting on the soil element.

Coefficient of Earth Pressure of Passive:
The coefficient of earth pressure of passive is a dimensionless quantity that relates the lateral earth pressure to the vertical stress acting on the soil element. It is commonly denoted as Kp.

Derivation:
The coefficient of earth pressure of passive can be derived using the concept of effective stress.

- The total horizontal stress (σh) acting on the soil element is composed of the effective horizontal stress (σ'h) and the pore water pressure (u). Therefore, σh = σ'h + u.
- The total vertical stress (σv) acting on the soil element is composed of the effective vertical stress (σ'v) and the pore water pressure (u). Therefore, σv = σ'v + u.

From the above equations, we can determine the effective horizontal stress (σ'h) and the effective vertical stress (σ'v) as follows:

- σ'h = σh - u
- σ'v = σv - u

The coefficient of earth pressure of passive (Kp) is defined as the ratio of the effective horizontal stress (σ'h) to the effective vertical stress (σ'v). Therefore, Kp = (σh - u) / (σv - u).

Since the pore water pressure (u) is usually small compared to the total stresses, it can be neglected. Therefore, the coefficient of earth pressure of passive can be approximated as Kp ≈ σh / σv.

Hence, the correct answer is option C: σv / σh.