Active earth pressure in cohesionless backfill (horizontal ground): In the active state of plastic equilibrium for a cohesionless soil with a horizontal ground surface, which principal stress is vertical?

Introduction / Context:
Active earth pressure corresponds to the state where lateral confinement is reduced until the soil mass reaches limiting equilibrium and lateral stress becomes the minor principal stress. Understanding the orientation of principal stresses clarifies wall design and interpretation of K-active relationships.

Given Data / Assumptions:

• Backfill is cohesionless with a level surface.
• Retaining wall yields sufficiently to mobilize active conditions.
• Gravity loading yields vertical overburden stress sigma_v = gamma * z.

Concept / Approach:
Under active conditions, the horizontal earth pressure equals sigma_h = Ka * sigma_v, where Ka < 1. Thus sigma_h is the minor principal stress, and sigma_v is the major principal stress. Conversely, in passive conditions, sigma_h becomes the major principal stress. For horizontal ground and no surcharge, principal stresses align with vertical and horizontal directions.

Step-by-Step Solution:

Verification / Alternative check:
From Mohr–Coulomb, active condition occurs when the Mohr circle is tangent to the failure envelope on the “active” side, giving sigma_1 = sigma_v and sigma_3 = sigma_h.

Why Other Options Are Wrong:

• Option A reverses roles of sigma_v and sigma_h.
• Option B mislabels the minor principal stress as vertical.
• Option D is incorrect for horizontal backfill; principal directions align with vertical and horizontal axes.

Common Pitfalls:
Confusing active and passive states; forgetting that sufficient wall movement is required to mobilize active pressure; mixing sign conventions.

Final Answer:
Major principal stress is vertical