Coefficient of compressibility (a_c): Given initial void ratio e_0, final void ratio e, initial effective stress σ′_0, and final effective stress σ′, choose the correct expression for a_c over the stress increment.

Introduction / Context:
The coefficient of compressibility a_c (often denoted a_v) quantifies the change in void ratio per unit increase in effective vertical stress in one-dimensional consolidation. It is a fundamental parameter for settlement predictions and for interpreting oedometer test results.

Given Data / Assumptions:

• One-dimensional consolidation framework.
• Effective stresses σ′_0 → σ′, with corresponding void ratios e_0 → e.
• Small to moderate stress increment over which a_c is evaluated.

Concept / Approach:

By definition, a_c = Δe / Δσ′ with the understanding that e decreases as σ′ increases. Using e_0 at σ′_0 and e at σ′: Δe = e_0 − e and Δσ′ = σ′ − σ′_0. Thus a_c = (e_0 − e) / (σ′ − σ′_0). This form is consistent with positive a_c for compressive loading that reduces void ratio.

Step-by-Step Solution:

Verification / Alternative check:

Units: a_c is dimensionless per unit stress (e.g., per kPa). The sign convention yields positive values for consolidation (e decreases as σ′ increases).

Why Other Options Are Wrong:

(b) reverses the numerator sign; (c) inverts the ratio; (d) and (e) are not the standard definition for a_c across a stress increment.

Common Pitfalls:

Mixing e vs. void ratio change with volumetric strain; forgetting to use effective stress rather than total stress in consolidation.

Final Answer:

a_c = (e_0 − e) / (σ′ − σ′_0)