Soil mechanics – general relationship for bulk unit weight: Given specific gravity of solids (G), unit weight of water (gamma_w), degree of saturation (Sr), and void ratio (e), what is the standard formula for the bulk unit weight gamma of a partially saturated soil mass?

Introduction / Context:
Bulk unit weight connects soil phase relationships (solids, water, air) with measurable engineering properties. For a partially saturated soil, gamma depends on the solids present and the portion of voids filled with water, quantified by degree of saturation Sr, plus the geometric void ratio e.

Given Data / Assumptions:

• Specific gravity of soil solids G (dimensionless).
• Unit weight of water gamma_w.
• Void ratio e = Vv / Vs.
• Degree of saturation Sr = Vw / Vv, 0 ≤ Sr ≤ 1.
• Standard phase-relationship assumptions (no dissolved gases effect on volume; constant G, e, Sr over the specimen).

Concept / Approach:
For a representative volume where Vs = 1 (reference), Vv = e, total volume V = 1 + e. Weight of solids Ws = G * gamma_w * Vs. Weight of water Ww = Sr * Vv * gamma_w = Sr * e * gamma_w. Bulk unit weight is total weight divided by total volume.

Step-by-Step Solution:

Verification / Alternative check:
Special cases: Sr = 0 (dry) gives gamma_d = [G / (1 + e)] * gamma_w. Sr = 1 (saturated) gives gamma_sat = [(G + e) / (1 + e)] * gamma_w, matching standard dry/saturated formulas.

Why Other Options Are Wrong:

• Option B omits water contribution; only valid for dry soil.
• Option C misplaces Sr and e.
• Option D double-counts Sr with G but omits water in voids correctly.
• Option E corresponds to saturated condition (Sr = 1), not general Sr.

Common Pitfalls:
Using Sr in percent instead of decimal; confusing gamma with unit weight of solids; forgetting to divide by total volume (1 + e).

Final Answer:
gamma = [(G + Sr * e) / (1 + e)] * gamma_w