CELE Geotechnical Engineering — Soil Properties and Phase RelationshipsCheat Sheet

Formula

e = V_v / V_s

Meaning

e = void ratio; V_v = volume of voids (air + water); V_s = volume of solids

Watch Out

V_v ≠ V_w alone — it includes both air and water. Also e can exceed 1 for loose soils.

When To Use

Any problem needing void geometry or as bridge between porosity and saturation

Formula

n = V_v / V = e / (1 + e)

Meaning

n = porosity (decimal); V = total volume; conversion formula shows e ↔ n

Watch Out

n is always < 1; common error to use e and n interchangeably without conversion

When To Use

When problem gives or asks for porosity; easier for percent expression

Formula

e = n / (1 - n)

Meaning

Inverse conversion: given n, find e instantly

Watch Out

Denominator (1 - n) is small for high porosity (e.g., n = 0.9 → e = 9). Errors compound here.

When To Use

Reverse direction from n to e

Formula

w = W_w / W_s

Meaning

w = water content (mass ratio, unitless); W_w = mass of water; W_s = mass of solids. CAN EXCEED 1.

Watch Out

w is weight-based NOT volume-based. For clay/silt w often 0.20–0.60 (20–60%). Not a percentage automatically.

When To Use

Any saturation, permeability, or consolidation problem

Formula

S = V_w / V_v

Meaning

S = degree of saturation (0 = dry, 1 = fully saturated). Decimal or percent.

Watch Out

Must be ≤ 1. If Se/w·G_s > 1, soil is overspecified (inconsistent input data).

When To Use

Identifies if soil is dry (S ≈ 0), partially saturated (0 < S < 1), or saturated (S = 1)

Formula

G_s = ρ_s / ρ_w = γ_s / γ_w

Meaning

G_s = specific gravity of solids; ρ = density; γ = unit weight; w subscript = water

Watch Out

G_s ≈ 2.7 for most soils; clay may be 2.70–2.80; organic/peat soils lower (~2.0–2.4)

When To Use

Standardized property, usually given. Typical values 2.65–2.75 (silica-rich soils)

Value

2.65–2.67

Symbol

G_s

Quantity

Specific gravity of quartz/silica soils

Value

2.70–2.80

Symbol

G_s

Quantity

Specific gravity of clay minerals

Value

1.50–2.40

Symbol

G_s

Quantity

Specific gravity of organic soils / peat

Value

9.81 kN/m³ (or 1000 kg/m³, 62.4 lb/ft³)

Symbol

γ_w

Quantity

Unit weight of water (standard)

Value

15–50% (0.15–0.50)

Symbol

w

Quantity

Typical water content (clays)

Value

0.4–0.8

Symbol

e

Quantity

Typical void ratio (medium-dense soil)

Term

Phase

Example

A saturated clay has only solids and water; a dry sand has only solids and air.

Definition

One of three components: solids (mineral grains), water (pore fluid), air (pore gas).

Term

Void Ratio (e)

Example

e = 0.6 means 0.6 volumes of void per 1 volume of solid (60% voidage relative to solids).

Definition

Ratio of volume of voids to volume of solids; uniquely defines soil compaction state.

Term

Porosity (n)

Example

n = 0.4 means 40% of total soil volume is void space.

Definition

Ratio of volume of voids to total volume; always between 0 and 1.

Term

Water Content (w)

Example

w = 0.25 means 250 g of water per 1000 g of dry soil (not 250 mL).

Definition

Ratio of mass of water to mass of dry solids; often 20–50% for natural clays.

Term

Degree of Saturation (S)

Example

S = 0.7 in a partially saturated sand means 70% of pores hold water, 30% hold air.

Definition

Fraction of voids filled with water; 0 (dry) to 1 (saturated) or 0–100%.

Term

Specific Gravity of Solids (G_s)

Example

Most silica-rich soils G_s ≈ 2.65; determined experimentally, not calculated.

Definition

Density of soil solids relative to water; standard property measured by pycnometer.

Note

This is THE KEY EQUATION. Ties saturation, void ratio, water content, and specific gravity. If three are known, solve for the fourth.

Equation

S × e = w × G_s

Conditions

Master identity; always valid for any soil state (dry, moist, saturated)

Note

Use to convert between e and n without getting lost in algebra.

Equation

e + 1 = (1 + n) / (1 - n) equivalent to n / (1 + e) × (1 + e) = n

Conditions

Purely algebraic; derived from definitions of e and n

Formula

γ_dry = (G_s × γ_w) / (1 + e)

Meaning

Dry unit weight (kN/m³); γ_w = 9.81 kN/m³ standard

Watch Out

This is the MOST important formula in geotechnical engineering. Denominator (1+e) varies with compaction; looser soil → larger e → lower γ_dry.

When To Use

Any problem involving dry or compacted soil. Foundation design, embankments.

Formula

γ_sat = ((G_s + e) × γ_w) / (1 + e)

Meaning

Saturated unit weight (kN/m³); applies when S = 1 (all voids filled with water)

Watch Out

Numerator is (G_s + e), not G_s. Common mistake: forget to add e. Also γ_sat is ALWAYS ≥ γ_dry.

When To Use

Submerged foundations, underground excavations, below water table

Formula

γ = (G_s + S×e) × γ_w / (1 + e)

Meaning

Moist (bulk) unit weight for any saturation S; general form

Watch Out

S is a decimal (0 to 1), not percent. At S = 0.5, result is average-ish between γ_dry and γ_sat.

When To Use

Partially saturated soil above water table. Set S = 1 for γ_sat, S = 0 for γ_dry.

Formula

γ = γ_dry × (1 + w)

Meaning

Moist unit weight from water content and dry unit weight; compact form

Watch Out

This assumes the soil is partially saturated (S < 1). Do NOT use this for saturated soils without checking S first.

When To Use

Field measurements: if γ and w are measured, find γ_dry instantly

Formula

γ' = γ_sat - γ_w

Meaning

Buoyant (effective/submerged) unit weight (kN/m³); effective stress concept

Watch Out

SUBTRACT exactly one γ_w (not two, not zero). γ' = (G_s - 1)γ_w / (1+e) is equivalent; useful check.

When To Use

Effective stress, stability of submerged slopes, footing beneath water table

Formula

γ' = (G_s - 1) × γ_w / (1 + e)

Meaning

Alternative form for buoyant unit weight; derived directly

Watch Out

Numerator (G_s - 1), not G_s. Since G_s ≈ 2.65, numerator ≈ 1.65; result typically 8–12 kN/m³.

When To Use

Quick check or when γ_sat not yet computed

Value

15–17 kN/m³

Symbol

γ_dry

Quantity

Dry unit weight (typical medium sand)

Value

14–16 kN/m³

Symbol

γ_dry

Quantity

Dry unit weight (typical clay)

Value

19–21 kN/m³

Symbol

γ_sat

Quantity

Saturated unit weight (sand)

Value

17–20 kN/m³

Symbol

γ_sat

Quantity

Saturated unit weight (clay)

Value

9–12 kN/m³

Symbol

γ'

Quantity

Buoyant unit weight (most soils)

Term

Unit Weight (γ)

Example

γ = 18 kN/m³ for moist sand means 18 kilonewtons of mass per cubic meter.

Definition

Weight per unit volume (kN/m³ or lb/ft³); includes all three phases.

Term

Dry Unit Weight (γ_dry)

Example

γ_dry = 16 kN/m³ for a compacted clay is typical; lower value = looser soil.

Definition

Weight of solids per unit volume (S = 0); directly reflects compaction state.

Term

Saturated Unit Weight (γ_sat)

Example

γ_sat = 21 kN/m³ for a clay; always > γ_dry by amount ρ_w × (e/(1+e)).

Definition

Weight per unit volume when all voids are water-filled (S = 1); below water table.

Term

Effective (Submerged) Unit Weight (γ')

Example

γ' = 11 kN/m³ for submerged soil means 11 kN/m³ additional weight above water effect.

Definition

Unit weight relative to water; accounts for buoyancy; used in effective stress calculations.

Term

Moist Unit Weight (γ)

Example

γ = 18 kN/m³ at w = 12% and γ_dry = 16 kN/m³ checks: 16(1+0.12) = 17.92 ≈ 18 ✓

Definition

Unit weight of partially saturated soil above water table; interpolates between dry and saturated.

Note

Master formula. All specific unit weight formulas derive from this. Understand the structure, not just the values.

Equation

γ = (G_s + S×e)×γ_w / (1+e)

Conditions

General formula for any saturation state S. Set S=0 → γ_dry; S=1 → γ_sat

Note

Recognize all three in exam. The middle form shows why γ' is always positive (since G_s > 1).

Equation

γ' = γ_sat − γ_w = [(G_s + e)/(1+e) − 1]×γ_w = (G_s−1)/(1+e)×γ_w

Conditions

Three equivalent forms; all correct

Formula

S×e = w×G_s ⟹ solve for unknown

Meaning

Master identity. Given any three, find the fourth.

Watch Out

Check that S ≤ 1 and w ≥ 0. If result violates, input data is inconsistent (overspecified).

When To Use

FIRST STEP in almost every phase-relationship problem

Formula

e ↔ n: e = n/(1−n); n = e/(1+e)

Meaning

Convert between void ratio and porosity instantly

Watch Out

ALWAYS convert to same metric before comparing. Do not mix e from one formula with n from another.

When To Use

Problem mixes e and n; use appropriate conversion

Formula

γ_dry = (G_s×γ_w)/(1+e)

Meaning

Solve for e if γ_dry, G_s, γ_w given; rearrange: e = (G_s×γ_w/γ_dry) − 1

Watch Out

Rearrangement is straightforward algebra. Verify e is positive and realistic (e > 0, usually e < 2).

When To Use

Lab test gives γ_dry; back-calculate e for quality control

Formula

γ = γ_dry(1+w)

Meaning

If γ and w measured, find γ_dry: γ_dry = γ/(1+w)

Watch Out

This formula assumes S < 1. For saturated soil, use γ_sat = (G_s+e)/(1+e)×γ_w instead.

When To Use

Field unit weight γ known with water content w; separate dry weight

Term

Given / Find / Assume Strategy

Example

Given: w=20%, S=0.8, G_s=2.70. Find: e, γ. Assume: γ_w=9.81 kN/m³.

Definition

List all known values (Given), identify unknowns (Find), and note standard assumptions (G_s, γ_w, unit system).

Term

Degree of Freedom

Example

If e, w, G_s given, soil state is fully defined (use Se=wG_s to find S). If only e and w given, need G_s to find S.

Definition

Number of independent variables needed to fully specify soil state. Typically 4 (e, w, S, G_s) with 1 master equation → 3 degrees of freedom.

Note

Follow this sequence on every exam problem. Do not skip checking the result.

Equation

Step 1: List Given (e.g., w, S, G_s). Step 2: Use S×e=w×G_s to find missing ratio. Step 3: Use unit-weight formula for γ. Step 4: Check (S≤1, e>0).

Conditions

Systematic approach; prevents errors

Formula

ΔH = H_i × Δe / (1 + e_i)

Meaning

Settlement ΔH (m) from void-ratio change Δe; H_i = initial height, e_i = initial void ratio

Watch Out

Δe is negative (compression). Common error: forget the negative sign or confuse Δe with e ratio.

When To Use

Primary consolidation; relates compressibility to void ratio

Formula

C_c = (e_i − e_f) / log₁₀(σ'_f / σ'_i)

Meaning

Compression index; slope of virgin consolidation curve on e-log(σ') plot

Watch Out

σ' is effective stress (not total). Log is base-10, not natural log. Different clays: 0.2–1.0.

When To Use

Predict settlement; C_c ≈ 0.009(w_L − 10%) for clay (Terzaghi approximation)

Value

0.5–1.0

Symbol

C_c

Quantity

Compression index (soft clay)

Value

0.2–0.5

Symbol

C_c

Quantity

Compression index (medium clay)

Term

Consolidation (Preview)

Example

A clay layer under a building foundation slowly compresses as pore water drains; settlement occurs.

Definition

Process of expulsion of pore water and reduction of void ratio under load application over time.

Values

• V_voids / V_solids
• V_voids / V_total

Property

Definition

Values

• > 0 (no upper limit, often e < 2)
• 0 < n < 1 (always < 1)

Property

Range

Values

• Yes (loose sand e ≈ 1.5)
• No (max n ≈ 0.5–0.6)

Property

Can exceed 1?

Values

• e = n / (1 − n)
• n = e / (1 + e)

Property

Conversion

Values

• Compact formulas, γ_dry = G_s γ_w / (1+e)
• Intuitive; percentage voids easier to visualize

Property

Exam use

Values

• (G_s γ_w) / (1+e)
• G_s, e only (NOT w or S)
• 15–17 (sand); 14–16 (clay)

Property

Dry γ_dry

Values

• γ_dry(1+w) or (G_s+S·e)γ_w/(1+e)
• G_s, e, w or S
• 16–19 (typical field)

Property

Moist γ

Values

• ((G_s+e)γ_w) / (1+e)
• G_s, e only (S=1 always)
• 19–21 (sand); 17–20 (clay)

Property

Saturated γ_sat

Values

• γ_sat − γ_w = (G_s−1)γ_w/(1+e)
• G_s, e only
• 9–12 (almost constant ~10)

Property

Buoyant γ'

Values

• 0
• 0 (S = 0)
• 0.6
• Desert sand

Property

Oven-dry

Values

• 2–5%
• ≈ 0.01–0.02
• 0.6
• Compacted fill, low rainfall

Property

Dry field

Values

• 8–15%
• 0.3–0.7
• 0.6
• Natural soil above water table

Property

Moist (partial sat.)

Values

• w = e/G_s = e/2.70
• 1.0
• 0.6
• Soil below water table or flooded

Property

Saturated (S=1)

Values

• 0.30–0.60
• 1.0
• 0.8–1.5
• Soft marine clay

Property

Saturated clay (high e)

Values

• e and n are NOT the same (e can > 1, n always < 1)
• Always convert: e = n/(1−n) or n = e/(1+e) before mixing in formulas

Property

Using e and n interchangeably

Values

• If S_calc > 1, soil state is impossible (overspecified data)
• Check S = w·G_s / e ≤ 1. If not, re-read problem or flag inconsistency

Property

Forgetting S must be ≤ 1

Values

• w is mass ratio (can be > 1); S is volume ratio (0 to 1). Completely different.
• Use S·e = w·G_s to relate them. If S=1, then w = e/G_s only.

Property

Confusing w (water content) with S (saturation)

Values

• Formula assumes S < 1 (partial saturation). For S=1, use γ_sat = (G_s+e)γ_w/(1+e).
• Check S first. If S = 1, use saturated formula; if S < 1, γ = γ_dry(1+w) works.

Property

Using γ = γ_dry(1+w) for saturated soil

Values

• Only γ_sat − γ_w gives correct γ'. Subtracting from dry or moist unit weights is wrong.
• γ' = γ_sat − γ_w only. Equivalent: γ' = (G_s−1)γ_w/(1+e). Typical result ≈ 10 kN/m³.

Property

Subtracting γ_w from γ_dry or γ_moist for buoyancy

Values

• Common algebra slip: γ_dry = G_s γ_w NOT G_s γ_w (without denominator)
• γ_dry = (G_s γ_w) / (1 + e). Denominator MUST be there. Loose soil (e=1) gives half the density.

Property

Forgetting the (1+e) denominator in γ_dry formula