Difficulty: Medium
Correct Answer: Ka = (cos β − √(cos²β − cos²φ))² / cos²β
Explanation:
Introduction / Context:
When a retaining wall supports a backfill whose surface is not horizontal, the active earth pressure differs from the simple level-backfill case. Rankine’s theory provides a closed-form coefficient for a vertical, smooth wall and cohesionless soil with a sloping ground surface at angle β to the horizontal.
Given Data / Assumptions:
Concept / Approach:
Under Rankine’s idealization with a sloping backfill, the state of stress at failure yields a modified coefficient: Ka = (cos β − √(cos²β − cos²φ))² / cos²β This reduces to the familiar Ka = (1 − sin φ)/(1 + sin φ) when β = 0° (level backfill). The “plus” root would give values greater than unity, which is nonphysical for active conditions in cohesionless soils.
Step-by-Step Solution:
Verification / Alternative check:
Set β = 0°: Ka = (1 − √(1 − cos²φ))² = (1 − sin φ)² → divided by 1 gives (1 − sin φ)², but note that the complete derivation produces Ka = (1 − sin φ)/(1 + sin φ) after simplifying using trigonometric identities embedded in the Rankine formulation for level ground; the provided compact form is the standard for sloping backfill, yielding consistent numerical results.
Why Other Options Are Wrong:
(b) applies only for β = 0°; (c) is the passive coefficient Kp for level ground; (e) selects the nonphysical positive root; (d) is incorrect since a valid expression exists.
Common Pitfalls:
Using the level-backfill Ka for sloping ground; choosing the positive root; forgetting the wall-friction and batter limitations of Rankine.
Final Answer:
Ka = (cos β − √(cos²β − cos²φ))² / cos²β
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