Settlement theory — the ultimate (primary consolidation) settlement of a normally consolidated clay layer is directly proportional to which quantity or quantities?

Introduction / Context:
Estimating consolidation settlement is essential for foundation design on clay deposits. For normally consolidated clays, the one-dimensional consolidation equation gives a simple relationship among settlement, layer thickness, compressibility, and stress change. Recognizing proportionalities helps with preliminary screening and back-of-the-envelope checks before detailed calculations.

Given Data / Assumptions:

• Soil is normally consolidated (no preconsolidation rebound).
• One-dimensional drainage and loading conditions apply.
• Stress increases from sigma1' to sigma2' are known from foundation loads.

Concept / Approach:

Primary consolidation settlement S for a layer of thickness H is often computed as S = (Cc / (1 + e0)) * H * log10(sigma2' / sigma1'). Thus, for fixed initial void ratio and stress ratio, S is directly proportional to the compression index Cc and to the thickness H of the compressible stratum. While e0 influences magnitude, settlement is not linearly proportional to e0 alone; it enters as a divisor (1 + e0) and through stress history effects.

Step-by-Step Solution:

Verification / Alternative check:

Field performance and oedometer back-analyses often show linear sensitivity of S to Cc and H, validating the proportional relationship.

Why Other Options Are Wrong:

(c) Initial void ratio affects S but not as a direct proportional factor alone. (e) contradicts the consolidation equation.

Common Pitfalls:

Ignoring stress range dependence; applying normally consolidated equations to overconsolidated soils without modification; mixing natural log and log10 without conversion.

Final Answer:

both (a) and (b)