Particle-size distribution indices — the coefficient of curvature (Cc, also called coefficient of gradation) for a soil gradation curve is defined as:

Introduction / Context:
Gradation characteristics govern engineering behavior of sands and gravels, affecting permeability, shear strength, and compaction. Two widely used indices are the coefficient of uniformity (Cu) and the coefficient of curvature (Cc). Correctly defining Cc ensures proper soil classification (e.g., well-graded vs. poorly graded) under systems like IS, ASTM, or USCS.

Given Data / Assumptions:

• D10, D30, and D60 are particle sizes corresponding to 10%, 30%, and 60% finer by weight from the semi-logarithmic gradation curve.
• Standard sieve analysis has been performed with consistent units.
• Soil is primarily coarse-grained where gradation indices are most informative.

Concept / Approach:

The formal definition is Cc = (D30)^2 / (D10 * D60). This index reflects the curvature of the gradation curve around the 30% passing ordinate. For well-graded sands, typical acceptance criteria include Cu >= 6 and 1 <= Cc <= 3; for gravels, Cu >= 4 and 1 <= Cc <= 3. Meeting both Cu and Cc thresholds helps avoid pronounced gaps in particle sizes, promoting denser packing and better drainage control.

Step-by-Step Solution:

Verification / Alternative check:

Cross-check Cu = D60 / D10 simultaneously. If both Cu and Cc meet specified ranges, the soil is typically well-graded; otherwise it is poorly graded (uniform) or gap-graded.

Why Other Options Are Wrong:

Options expressing D10/D60 or D60/D10 represent Cu or its inverse, not Cc. Other algebraic forms lack the squared D30 term central to the curvature definition.

Common Pitfalls:

Reading D-values off a linear x-axis; mixing units (e.g., mm and microns) between D-values; rounding too aggressively leading to borderline misclassification.

Final Answer:

Cc = (D30)^2 / (D10 * D60)