Consistency index I_c (Atterberg limits): I_c is defined as which of the following ratios for a cohesive soil at natural water content w_n?

Introduction / Context:
The consistency index I_c indicates the relative consistency of a cohesive soil in the plastic range. It complements the liquidity index I_L and helps engineers infer the likely undrained behavior (stiff, soft) at the in-situ water content.

Given Data / Assumptions:

• Atterberg limits: LL (liquid limit), PL (plastic limit), PI = LL − PL.
• Natural water content w_n is known.
• Index applies to fine-grained soils.

Concept / Approach:

Definitions: liquidity index I_L = (w_n − PL)/PI and consistency index I_c = (LL − w_n)/PI. I_c close to 1 indicates a soil near the plastic limit (stiff), while I_c near 0 indicates a soil near the liquid limit (soft). Negative values imply w_n > LL (flowing).

Step-by-Step Solution:

Verification / Alternative check:

Check that I_c + I_L = 1, which is a useful consistency relation.

Why Other Options Are Wrong:

(a), (d) use addition incorrectly; (c) is the liquidity index; (e) is not a standard index definition.

Common Pitfalls:

Interchanging I_L and I_c; forgetting that PI must be positive and defined only if LL > PL.

Final Answer:

I_c = (LL − w_n) / PI