CELE Geotechnical Engineering — Soil Properties and Phase RelationshipsRevision Notes

Formulas

Exam Tips

• Draw the phase diagram for every problem — label Vs, Vw, Va, Ws, Ww. This prevents errors in identifying which volumes and weights to use.
• The master identity Se = wGs is the single most-tested equation — memorise it forward and backward.
• When a soil is described as 'saturated,' immediately set S = 1.0, simplifying Se = wGs to e = wGs.
• When a soil is described as 'dry,' set S = 0 and w = 0, meaning Vw = 0 and γ = γdry.
• Always check: S ≤ 1.0 after computing S = wGs/e. If S > 1.0, recheck your inputs — a physical impossibility signals an arithmetic error.

Key Points

• Soil is composed of three phases: solid particles (S), water (W), and air (A). The phase diagram separates these three components by volume and weight.
• Total volume: V = Vs + Vv, where Vv = Vw + Va (void volume = water volume + air volume).
• Total weight: W = Ws + Ww (air has negligible weight in engineering practice).
• Void ratio e = Vv / Vs: ratio of void volume to solid volume. Range: loose sands ~0.8; dense sands ~0.4; soft clays can exceed 1.5.
• Porosity n = Vv / V: ratio of void volume to total volume. Always between 0 and 1 (or 0% to 100%).
• Water content w = Ww / Ws: weight of water divided by weight of solids — NOT volumes. Expressed as a decimal or percentage. Can exceed 100% in very soft clays.
• Degree of saturation S = Vw / Vv: fraction of voids filled with water. S = 0 for completely dry soil; S = 1.0 (100%) for fully saturated soil.
• Specific gravity of solids Gs = γs / γw = ρs / ρw. Typical range: 2.60–2.80; most mineral soils use 2.65–2.70.
• The MASTER IDENTITY: S·e = w·Gs. This single equation links the four most-tested parameters.

Definitions

Section Title

The Phase Diagram and Basic Volume–Weight Ratios

Common Mistakes

• Confusing void ratio e (can exceed 1.0) with porosity n (always less than 1.0) — never substitute one for the other directly.
• Expressing water content as Vw/Vs or Vw/Vv — water content is ALWAYS weight-based: w = Ww/Ws.
• Forgetting to convert w and S to decimals before applying Se = wGs — using percentages gives wrong answers.
• Assuming Gs = 1.0 (the value for water) instead of 2.65–2.70 for solid mineral grains.
• Using Gs = 2.65 when the problem explicitly states a different value — always use the given Gs.

Formulas

Exam Tips

• Memorise the three principal formulas: γdry = Gs·γw/(1+e), γsat = (Gs+e)·γw/(1+e), γ' = γsat − γw. These cover 90% of board questions on unit weight.
• Notice that γsat − γdry = e·γw/(1+e) = n·γw — this is a useful cross-check in multi-part problems.
• If a problem gives γ and w but not e, find γdry = γ/(1+w) first, then back-calculate e = Gs·γw/γdry − 1.
• For a quick sanity check: γ' ≈ γsat − 9.81 should give a value between 8 and 13 kN/m³ for most natural soils.
• Board problems often give three of the four quantities {S, e, w, Gs} and ask for the fourth — apply Se = wGs directly.

Key Points

• Four unit weights are used in geotechnical engineering: moist (bulk), dry, saturated, and submerged (buoyant/effective).
• Dry unit weight γdry = Gs·γw / (1+e): derived by setting Ww = 0 in the phase diagram. Indicator of compaction state.
• Moist unit weight γ = (Gs + Se)·γw / (1+e), equivalently γ = γdry·(1+w). Use when partial saturation exists.
• Saturated unit weight γsat = (Gs + e)·γw / (1+e): obtained by setting S = 1.0 in the moist formula.
• Submerged (buoyant) unit weight γ' = γsat − γw: represents the effective weight of soil submerged under water. Used in effective stress calculations below the water table.
• Typical values (kN/m³): γdry = 14–18; γmoist = 17–21; γsat = 18–23; γ' = 8–13.
• Unit weight of water γw = 9.81 kN/m³ (use this exact value unless the problem specifies 10 kN/m³).
• The relationship between dry and moist unit weight: γdry = γ / (1+w). This is a board-exam shortcut for back-calculating void ratio from field measurements.

Definitions

Section Title

Unit Weights of Soil

Common Mistakes

• Subtracting γw from γmoist instead of γsat when computing buoyant unit weight — γ' is ALWAYS γsat − γw.
• Forgetting to add 1 to e in the denominator of the unit weight formulas — (1+e) is in the denominator, not e alone.
• Using γw = 10 kN/m³ when the problem does not specify this simplification — default is 9.81 kN/m³.
• Mixing up γdry and γmoist in effective stress calculations — use γmoist above the water table and γsat below.
• Computing γ' by subtracting γw from γmoist — this is physically incorrect; buoyancy applies only to saturated submerged soil.

Formulas

Exam Tips

• In a 5-minute board problem, spend 30 seconds drawing the phase diagram — it saves time in the long run by preventing sign and assignment errors.
• The sequence for most problems: Step 1 — Se = wGs (find missing parameter); Step 2 — γdry = Gs·γw/(1+e); Step 3 — apply γ = γdry(1+w) or γsat = (Gs+e)·γw/(1+e).
• Unit weight problems often have three parts in sequence: the answer from Part 1 feeds into Part 2. Carry full decimal precision until the final answer.
• Typical answer ranges: e = 0.4–1.5; n = 0.28–0.60; w = 5–80%; S = 0–100%; γsat = 17–23 kN/m³. Results outside these ranges usually indicate an error.
• When the problem says 'determine the degree of saturation,' it is almost always asking you to use S = wGs/e after finding e from another given.

Key Points

• Problem 1 — Saturated soil: Given w = 18%, Gs = 2.70, S = 100% (saturated). Find e, n, γsat, γ'.
• Problem 2 — Partially saturated soil: Given Gs = 2.65, e = 0.60, w = 12%. Find γdry, γmoist, S.
• Problem 3 — Back-calculation from field data: Given γ = 19 kN/m³, w = 15%, Gs = 2.70. Find γdry, e, S.
• Problem 4 — From porosity: Given n = 0.40, Gs = 2.68, S = 0.80. Find e, w, γmoist.
• Systematic approach: (1) Draw phase diagram; (2) Apply Se = wGs; (3) Compute unit weight formula; (4) Check S ≤ 1.0.
• Always write out given data, identify the unknown, select the governing equation, substitute, and box the answer with units.

Definitions

Section Title

Worked Board-Style Problems

Common Mistakes

• Skipping the phase diagram and making volume/weight assignment errors that propagate through the entire solution.
• Not converting percentage values of w and S to decimal form before substituting into formulas.
• In Problem 3 type questions, computing e using γmoist instead of γdry in the formula e = Gs·γw/γdry − 1.
• Mixing up n and e when given porosity — always convert n to e first using e = n/(1−n).
• Reporting degrees of saturation greater than 100% without recognising the arithmetic error.

Formulas

Exam Tips

• Create a personal formula sheet with the five core equations: Se = wGs; e = n/(1−n); γdry = Gs·γw/(1+e); γsat = (Gs+e)·γw/(1+e); γ' = γsat − γw.
• Practice converting between all four variables in Se = wGs in all four directions until it becomes automatic.
• For multi-step problems, label every computed quantity and carry 4 significant figures throughout; round only at the final answer.
• When given γsat and asked for γ', the answer is always γsat − 9.81 — this is a one-second computation, do not overthink it.
• Review the solved examples in NSCP 2015 Volume I Appendix discussions and Punzalan/Gillesania review books for additional Filipino board-style practice.

Key Points

• The four most-tested quantities are e, w, S, and Gs — tied by Se = wGs. Mastering this one equation handles half the phase-relationship problems.
• Porosity n and void ratio e are different — they are linked by e = n/(1−n) and n = e/(1+e). Never write e when you mean n.
• Water content is WEIGHT-based (Ww/Ws), not volume-based. This is a deliberate distractor in many board problems.
• Submerged unit weight ALWAYS uses γsat, not γmoist: γ' = γsat − γw.
• Gs for most mineral soils is 2.65–2.70; if not given, assume Gs = 2.70 for conservative calculations in saturated clay problems.
• γw = 9.81 kN/m³ is the standard value. Some problems use γw = 10 kN/m³ for simplicity — use whichever is given.
• A higher void ratio means a looser, more compressible, more permeable soil — qualitative reasoning questions also appear on the board.
• When a clay shrinks (desiccation or consolidation), e decreases, n decreases, and γdry increases — Gs remains constant.

Definitions

Section Title

Common Board-Exam Pitfalls and Key Takeaways

Common Mistakes

• Applying γ' = γsat − γw to soil above the water table — buoyancy only acts on fully submerged soil.
• Using the saturated unit weight formula when the soil is only partially saturated — use the moist formula with the given S value.
• Forgetting that e can exceed 1.0 for soft clays — this is physically valid and not an error.
• Reporting final answers without units — board examiners expect kN/m³ for unit weight, % or decimal for w, S, and n.
• Rounding intermediate results to 2 decimal places too early, causing propagated errors in multi-step problems.