CELE Geotechnical Engineering — Soil Properties and Phase RelationshipsStudy Notes

Soil consists of three distinct phases arranged in a vertical phase diagram: **Phase Composition:** • **Solids (Ws, Vs):** Mineral particles forming the soil skeleton • **Water (Ww, Vw):** Pore water occupying void spaces • **Air (Va):** Pore air in voids not occupied by water Total volume V = Vs + Vv, where void volume Vv = Vw + Va. Total weight W = Ws + Ww (air weight is negligible). The phase diagram is a visual tool that organizes these quantities vertically, with solids at the base, water above it, and air at the top. This schematic representation helps engineers visualize and calculate the relationships between phases. **Why the Phase Diagram Matters:** In Filipino practice, NSCP 2015 Section 2.3 requires geotechnical investigations to characterize soil phases before foundation design. The phase diagram is the standardized method for organizing this information. When solving board exam problems, always sketch the phase diagram first — it prevents calculation errors and demonstrates systematic thinking.

**Void Ratio (e):** Defined as the ratio of void volume to solid volume: e = Vv / Vs = (Vw + Va) / Vs Physical meaning: For every unit volume of solids, there are 'e' units of void space. Void ratio ranges from ~0.4 (dense sand) to >1.0 (soft clay). Unlike porosity, void ratio can exceed 1.0. **Porosity (n):** Defined as the ratio of void volume to total volume: n = Vv / V = Vv / (Vs + Vv) Physical meaning: The fraction of total soil volume occupied by voids. Porosity ranges from 0 to 1 (or 0% to 100%). Always ≤ 1.0. **Relationship Between e and n:** These two ratios describe the same physical property (void fraction) but reference different denominators. They are interconvertible: e = n / (1 - n) n = e / (1 + e) Derivatation: From definitions, e/n = (Vv/Vs) ÷ (Vv/V) = V/Vs = (Vs + Vv)/Vs = 1 + e/n, which rearranges to the formulas above. **Board Exam Tip:** Many candidates confuse e and n or forget the conversion. Always verify: if e = 0.6, then n = 0.6/1.6 = 0.375 (37.5%). Dense sands have e ≈ 0.5–0.7 (n ≈ 33–41%); soft clays have e ≈ 0.8–1.2 (n ≈ 44–55%).

**Water Content (w):** Water content is the ratio of water weight to solid weight: w = Ww / Ws This is a weight-based ratio, not volume-based. Values are commonly expressed as percentages (e.g., w = 18% means 18 kg of water per 100 kg of solids). Unlike void ratio, water content can theoretically exceed 1.0 for very wet soils (e.g., marine clays may have w = 1.5 or 150%). **Physical Interpretation:** High water content indicates moisture-rich soil (clays, silts). Low water content indicates relatively dry conditions (sands, gravels). Water content is crucial for assessing construction workability and consolidation settlement. **Degree of Saturation (S):** Degree of saturation is the ratio of water volume to void volume: S = Vw / Vv = Vw / (Vw + Va) Ranges from S = 0 (completely dry, no pore water) to S = 1.0 (completely saturated, all voids filled with water, no air). Expressed as a decimal or percentage. **Three Saturation States:** • S = 0: Dry soil (no pore water, all air) • 0 < S < 1: Partially saturated (both water and air in voids) • S = 1: Saturated (all voids filled with water) **Master Relationship — The Critical Identity:** Se = wGs where Gs is the specific gravity of solids (typically 2.65–2.70). This equation is the single most important formula in soil phase relationships. It connects saturation, void ratio, water content, and specific gravity. Every board exam includes problems requiring this formula. Derivation: From phase diagram, Vw = S·Vv and Ww = w·Ws. Also, Vs = Ws/ρs·g and Vv = e·Vs. Substituting Vw = S·e·Vs and Ww = w·Ws into Vw·ρw = Ww/g yields S·e·(Ws/Gs·ρw) = w·Ws, simplifying to Se = wGs. **Critical Board Exam Insight:** If you know any three of {S, e, w, Gs}, you can always solve for the fourth using Se = wGs. This makes it a powerful constraint for completing partial information.

Unit weight is the weight per unit volume of soil. Four types are essential for board exams: **Dry Unit Weight (γd):** The weight of solids per unit total volume (air-filled voids are included in the volume): γd = Ws / V = Gs·γw / (1 + e) Alternative form (direct from phase): γd = (Ws/Vs) / (1 + e) = Gs·γw / (1 + e) Physical meaning: Maximum unit weight achievable at a given void ratio (all pore space is air, no water). Typical range: 14–18 kN/m³ for most soils. Dry unit weight increases with compaction (void ratio decreases). **Moist (Bulk) Unit Weight (γ):** The weight of solids plus pore water per unit total volume: γ = (Ws + Ww) / V = Gs·γw + S·e·γw / (1 + e) Simplified form: γ = γd·(1 + w) This simplified form is very useful on exams. Derivation: γ = (Ws + Ww)/V = Ws/V + Ww/V = γd + (w·Ws)/V = γd + w·(Ws/V) = γd + w·γd = γd(1+w). Physical meaning: Moist unit weight of a field soil sample. Typical range: 16–22 kN/m³. Moist unit weight increases with water content (more pore water adds weight) and with compaction (smaller void ratio). **Saturated Unit Weight (γsat):** The weight of solids plus pore water (all voids filled with water, S = 1.0) per unit total volume: γsat = (Gs + e)·γw / (1 + e) Physical meaning: Unit weight of soil below water table (all voids saturated). Typical range: 18–23 kN/m³. Saturated unit weight is independent of water content (all voids are filled with water regardless of how much we pour in). **Submerged (Buoyant/Effective) Unit Weight (γ'):** The effective weight under water, accounting for buoyancy: γ' = γsat − γw = (Gs + e)·γw / (1 + e) − γw = (Gs − 1)·γw / (1 + e) Alternative form (cleaner): γ' = (Gs − 1)·γw / (1 + e) Physical meaning: Apparent weight of soil below groundwater table. This is the unit weight used in effective stress calculations (see Chapter 5). Typical range: 8–13 kN/m³. Submerged unit weight is always positive because Gs > 1 for all soils. **Quick Reference — All Four Unit Weights:** | Type | Formula | Typical Range | Condition | |------|---------|---------------|----------| | Dry | γd = Gs·γw / (1+e) | 14–18 kN/m³ | All voids air-filled | | Moist | γ = γd(1+w) | 16–22 kN/m³ | Partial saturation | | Saturated | γsat = (Gs+e)·γw / (1+e) | 18–23 kN/m³ | All voids water-filled | | Submerged | γ' = (Gs−1)·γw / (1+e) | 8–13 kN/m³ | Below water table | **Important Notes:** 1. γw = 9.81 kN/m³ is the unit weight of water (use 10 kN/m³ for quick estimates). 2. Moist unit weight γ always falls between γd and γsat (it's the intermediate state). 3. Submerged unit weight γ' is used for effective stress; never use moist or saturated unit weight for earth pressure below water table. 4. If water content w and Gs are known, always calculate γd first, then γ = γd(1+w).

Solving phase relationship problems requires a systematic approach. This section integrates all formulas and presents the standard calculation sequence used by professional engineers and board examiners. **The Complete Formula Set:** Void and Porosity Relations: • e = n / (1−n) • n = e / (1+e) Saturation and Water Content: • S = Vw / Vv (saturation ranges 0 to 1) • w = Ww / Ws (water content, weight ratio) • Se = w·Gs (master identity) Unit Weights (with γw = 9.81 kN/m³): • γd = Gs·γw / (1+e) • γ = γd(1+w) or γ = [(Gs + S·e)·γw] / (1+e) • γsat = [(Gs + e)·γw] / (1+e) • γ' = γsat − γw = [(Gs − 1)·γw] / (1+e) **Standard Problem-Solving Sequence:** 1. **Sketch the phase diagram** — label known and unknown quantities (V, Vs, Vv, Vw, Va, W, Ws, Ww). 2. **List given data** — Typically 3–4 values from {Gs, e, n, S, w, γd, γ, γsat, γ'}. Note: 0 < S ≤ 1, 0 < e, 0 ≤ w, 2.6 < Gs < 2.8 for most soils. 3. **Apply constraint equations** in order: - If e is unknown and n is given: e = n/(1−n). - If Se = w·Gs can be applied with three known values, solve for the fourth. - If γd is unknown, use γd = Gs·γw/(1+e) with known e and Gs. - If γ is needed from w and γd: γ = γd(1+w). - If γsat is needed: γsat = (Gs+e)·γw/(1+e). - If γ' is needed: γ' = γsat − γw or γ' = (Gs−1)·γw/(1+e). 4. **Check against physical limits:** - 0 < S ≤ 1.0 (saturation must be realistic) - γd < γ < γsat (moist unit weight must be intermediate) - γ' > 0 and γ' < γsat (buoyancy effect reduces weight) - e > 0 and n > 0 (void ratio and porosity must be positive) 5. **Verify units** — Ensure all weights are in kN/m³ (or compatible), volumes in m³, and unit weight of water γw = 9.81 kN/m³. **Decision Tree for Phase Problems:** Start with data given → Which three parameters do we have? → Apply appropriate master equations → Solve unknowns → Check limits → Report answer in requested units. **Common Problem Types on Board Exams:** Type A: Given e, w, Gs, S → find γd, γ, γsat, γ', n Type B: Given γd, w, Gs → find γ, e (and sometimes S if more info available) Type C: Given γsat, Gs, S=1.0 → find γd, e, γ' (saturated soil problem) Type D: Reverse problem — given γ, w, Gs → find γd, e, and other unknowns **Sample calculation strategy for Type B:** Input: γd = 16.0 kN/m³, w = 15%, Gs = 2.68 Step 1: Calculate γ = γd(1+w) = 16.0 × 1.15 = 18.4 kN/m³ Step 2: From γd = Gs·γw/(1+e), solve e: 1+e = Gs·γw/γd = 2.68 × 9.81 / 16.0 = 26.29 / 16.0 = 1.643 e = 0.643 Step 3: Find S from Se = w·Gs: S = w·Gs/e = 0.15 × 2.68 / 0.643 = 0.625 = 62.5% Step 4: Calculate γsat = (Gs+e)·γw/(1+e) = (2.68+0.643) × 9.81 / 1.643 = 3.323 × 9.81 / 1.643 = 19.87 kN/m³ Step 5: Find γ' = γsat − γw = 19.87 − 9.81 = 10.06 kN/m³ Verification: 16.0 < 18.4 < 19.87 ✓ and 0.625 = 62.5% is reasonable for partially saturated soil ✓

Phase relationships are not abstract calculations — they directly influence foundation behavior and design decisions. Understanding their physical meaning is critical for engineering judgment. **Dry Unit Weight (γd) and Compaction:** Dry unit weight is the primary control variable in soil compaction (required by NSCP 2015 Section 2.3 for earth embankments). Higher γd means denser soil: • Reduced compressibility → smaller settlement • Increased shear strength → better slope stability • Reduced permeability → better seepage control Compaction specifications typically require γd ≥ 95% of maximum γd (from Standard Proctor or Modified Proctor test). For example, if maximum γd = 17.5 kN/m³, the contractor must achieve γd ≥ 16.6 kN/m³. **Water Content (w) and Workability:** Water content affects both construction ease and final soil properties: • Too dry (low w): Difficult to compact, clumps form, poor contact • Optimal w: Soil flows during compaction, particles rearrange easily, maximum γd achieved • Too wet (high w): Excess water weakens inter-particle bonds, γd decreases, soft/muddy consistency Field water content is measured by oven-drying: w = (Wmoist − Wdry) / Wdry × 100%. **Void Ratio (e) and Compressibility:** Higher void ratio means more compressible soil. Settlement of a clay layer is proportional to change in void ratio (Δe) and layer thickness. For foundations on soft clays, reducing e (by preloading or surcharge) is a key mitigation strategy. **Degree of Saturation (S) and Stability:** Partially saturated soils (0 < S < 1) are more stable than saturated soils because: • Air-water menisci in pores create negative pore pressure (matric suction) • Matric suction provides apparent cohesion, increasing shear strength • When S → 1 (wet season, rainfall), suction vanishes and strength drops This is why embankments often fail during rainy seasons (S increases from 0.70 to 1.0). **Submerged Unit Weight (γ') and Effective Stress:** In foundations below the water table, the effective weight is γ', not γsat. The difference γsat − γ' = γw is the buoyancy force. Effective stress controls soil strength: • σ' = σ − u (effective stress = total stress − pore pressure) • Larger effective stress → larger shear strength • Below water table, effective stress is smaller, so shear strength is smaller This is why deep underwater foundations are inherently weaker than equivalent shallow foundations. **Saturated Unit Weight (γsat) and Stability Below Water Table:** For earth dams, embankments, and underground excavations below water table, γsat is used for weight calculations in slope stability analyses (not γd or γ). Using the wrong unit weight is a common board exam trap. **Philippine Context — NSCP 2015 Requirements:** NSCP 2015 Section 2.3.2 (Soil Investigation) requires determination of: • Water table location and seasonal variation (affects γ vs. γsat choice) • In-situ void ratio and water content (measured or estimated from SPT/CPT correlations) • Compaction specifications for embankments (minimum γd and w) • Consolidation properties (related to void ratio change) Phase relationships are the quantitative foundation for all these investigations.