Question

Let \( \sigma_{\nu}' \) and \( \sigma_{h}' \) denote the effective vertical stress and effective horizontal stress, respectively. Which one of the following conditions must be satisfied for a soil element to reach the failure state under Rankine's passive earth pressure condition?

• A. \( \sigma_{\nu}' < \sigma_{h}' \)
• B. \( \sigma_{\nu}' > \sigma_{h}' \)
• C. \( \sigma_{\nu}' = \sigma_{h}' \)
• D. \( \sigma_{\nu}' + \sigma_{h}' = 0

Hint:

In Rankine's passive earth pressure condition, the effective horizontal stress must be greater than the effective vertical stress for failure to occur.

Answer 1

The Correct Option is A

Under Rankine's passive earth pressure condition, the failure occurs when the effective vertical stress \( \sigma_{\nu}' \) is less than the effective horizontal stress \( \sigma_{h}' \). This is because in the passive state, the soil is resisting the lateral movement, and the horizontal stress exceeds the vertical stress at the point of failure.

Step 1: Understand Rankine's Earth Pressure Theory.
According to Rankine's theory, the soil element reaches failure under passive conditions when: \[ \sigma_{\nu}' < \sigma_{h}'. \]

Step 2: Conclusion.
Thus, the condition for failure under Rankine's passive earth pressure condition is \( \sigma_{\nu}' < \sigma_{h}' \).

Final Answer: \[ \boxed{\sigma_{\nu}' < \sigma_{h}'} \]