Study Notes/Civil Engineering/Geotechnical Engineering

Geotechnical Engineering

Soil mechanics, classification, and foundation design

1. Soil Composition and Properties

Three-Phase Soil System

Soil consists of solid particles, water, and air. Understanding phase relationships is fundamental to geotechnical engineering.

Phase Diagram Components

Phase	Volume	Weight
Solids	V_s	W_s
Water	V_w	W_w
Air	V_a	0 (negligible)
Total	V = V_s + V_v	W = W_s + W_w

Volume-Weight Relationships

Void Ratio (e)

e = V_v / V_s

Volume of voids to volume of solids

Porosity (n)

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

Volume of voids to total volume

Degree of Saturation (S)

S = V_w / V_v × 100%

S = 0% (dry), S = 100% (saturated)

Water Content (w)

w = W_w / W_s × 100%

Weight of water to weight of solids

Unit Weights

Bulk Unit Weight: γ = W / V
Dry Unit Weight: γ_d = W_s / V = γ / (1 + w)
Saturated Unit Weight: γ_sat = (G_s + e)γ_w / (1 + e)
Submerged Unit Weight: γ' = γ_sat - γ_w
Specific Gravity: G_s = γ_s / γ_w (typically 2.65-2.75)

Important Relationships

Se = wG_s

Fundamental relationship between saturation, void ratio, water content, and specific gravity

2. Atterberg Limits

Consistency Limits

Atterberg limits define the water content boundaries at which fine-grained soils change consistency states.

Soil States and Limits

SOLID

Brittle

← SL →

SEMI-SOLID

Crumbly

← PL →

PLASTIC

Moldable

← LL →

LIQUID

Flows

Increasing Water Content →

Liquid Limit (LL)

Water content at which soil flows under its own weight

Test: Casagrande cup - 25 blows to close groove

Plastic Limit (PL)

Minimum water content at which soil can be rolled into 3mm thread without crumbling

Test: Rolling thread method

Shrinkage Limit (SL)

Water content below which volume remains constant upon drying

Test: Mercury displacement

Index Properties

Index	Formula	Purpose
Plasticity Index (PI)	PI = LL - PL	Range of plastic behavior
Liquidity Index (LI)	LI = (w - PL) / PI	Current consistency state
Consistency Index (CI)	CI = (LL - w) / PI	CI = 1 - LI
Activity (A)	A = PI / (% clay)	Clay mineral activity

Activity Classification

Inactive: A < 0.75 (Kaolinite)
Normal: 0.75 < A < 1.25 (Illite)
Active: A > 1.25 (Montmorillonite)

3. Soil Classification Systems

Particle Size Classification

Soil Type	USCS Size	AASHTO Size
Gravel	75 mm - 4.75 mm (#4)	75 mm - 2 mm
Sand	4.75 mm - 0.075 mm (#200)	2 mm - 0.075 mm
Silt	< 0.075 mm (passes #200)
Clay	< 0.002 mm

USCS (Unified Soil Classification System)

First Letter: Predominant Grain Size

G = Gravel, S = Sand, M = Silt, C = Clay, O = Organic, Pt = Peat

Coarse-Grained Soils

>50% retained on #200 sieve

GW, SW: Well-graded (good gradation)
GP, SP: Poorly-graded (uniform or gap)
GM, SM: Silty (fines are silt)
GC, SC: Clayey (fines are clay)

Fine-Grained Soils

>50% passing #200 sieve

ML: Low plasticity silt (LL < 50)
MH: High plasticity silt (LL ≥ 50)
CL: Low plasticity clay (LL < 50)
CH: High plasticity clay (LL ≥ 50)

Gradation Coefficients

Coefficient of Uniformity (C_u)

C_u = D₆₀ / D₁₀

Well-graded: C_u > 4 (gravel), C_u > 6 (sand)

Coefficient of Curvature (C_c)

C_c = (D₃₀)² / (D₁₀ × D₆₀)

Well-graded: 1 < C_c < 3

Plasticity Chart (A-Line)

A-Line: PI = 0.73(LL - 20)

• Above A-line = Clay (C)
• Below A-line = Silt (M) or Organic (O)
• LL < 50 = Low plasticity (L)
• LL ≥ 50 = High plasticity (H)

AASHTO Classification

A-1 to A-3: Granular materials (excellent to good subgrade)
A-4 to A-7: Silt-clay materials (fair to poor subgrade)
Group Index (GI): GI = (F-35)[0.2+0.005(LL-40)] + 0.01(F-15)(PI-10)
Higher GI = Poorer subgrade quality

4. Soil Compaction

Compaction

Mechanical process of increasing soil density by reducing air voids. Not to be confused with consolidation (time-dependent compression).

Laboratory Compaction Tests

Parameter	Standard Proctor	Modified Proctor
Hammer Weight	5.5 lb (2.5 kg)	10 lb (4.5 kg)
Drop Height	12 in (305 mm)	18 in (457 mm)
Number of Layers	3	5
Blows per Layer	25	25
Energy	600 kN-m/m³	2700 kN-m/m³

Compaction Curve

Optimum Moisture Content (OMC)

Water content at which maximum dry density is achieved

• Dry side: Flocculated structure, higher strength, more permeable
• Wet side: Dispersed structure, lower strength, less permeable

Maximum Dry Density (γ_d,max)

Highest dry unit weight achievable at given compactive effort

Modified Proctor produces higher γ_d,max at lower OMC

Zero Air Voids (ZAV) Line

γ_zav = G_sγ_w / (1 + wG_s)

Theoretical maximum density (S = 100%)

Compaction curve always lies below ZAV line

Field Compaction Control

Relative Compaction (RC)

RC = (γ_d,field / γ_d,max) × 100%

Typical specifications: RC ≥ 95% for structural fills

Field Density Tests

• Sand cone method (ASTM D1556)
• Nuclear density gauge
• Rubber balloon method

Relative Density (D_r)

D_r = (e_max - e) / (e_max - e_min) × 100%

Used for granular soils (sands and gravels)

Very Loose: 0-15%

Loose: 15-35%

Medium: 35-65%

Dense: 65-85%

Very Dense: 85-100%

5. Permeability and Seepage

Darcy's Law

Describes laminar flow of water through saturated soil. Valid for most soils except very coarse gravel.

q = kiA

v = ki

q = flow rate, k = coefficient of permeability

i = hydraulic gradient (Δh/L), A = cross-sectional area

v = discharge velocity (not seepage velocity)

Seepage Velocity

v_s = v/n = q/(nA)

Actual velocity through soil pores

Typical Permeability Values

Soil Type	k (cm/s)	Drainage
Clean Gravel	10⁰ - 10²	Good
Clean Sand	10⁻³ - 10⁰	Good
Silty Sand	10⁻⁵ - 10⁻³	Poor
Silt	10⁻⁷ - 10⁻⁵	Poor
Clay	10⁻¹⁰ - 10⁻⁷	Practically impervious

Laboratory Tests

Constant Head Test

k = QL / (Aht)

For coarse-grained soils (sand, gravel)

Falling Head Test

k = (aL/At) ln(h₁/h₂)

For fine-grained soils (silt, clay)

Equivalent Permeability

Horizontal Flow (Parallel)

k_h,eq = Σ(k_iH_i) / ΣH_i

Vertical Flow (Series)

k_v,eq = ΣH_i / Σ(H_i/k_i)

k_h > k_v for layered soils

6. Shear Strength of Soils

Mohr-Coulomb Failure Criterion

The fundamental equation defining soil shear strength based on cohesion and internal friction.

τ_f = c + σ tan φ

τ_f = c' + σ' tan φ' (effective stress)

τ_f = shear strength, c = cohesion

σ = normal stress, φ = angle of internal friction

Primed values (') = effective stress parameters

Effective Stress Principle

σ' = σ - u

Effective stress = Total stress - Pore water pressure

Soil strength is governed by effective stress, not total stress

Laboratory Shear Tests

Direct Shear Test

• Simple and quick
• Predetermined failure plane
• Cannot control drainage
• Results: c and φ directly

Triaxial Test Types

Test Type	Consolidation	Shear	Use
UU	Undrained	Undrained	Rapid loading (short-term)
CU	Drained	Undrained	Staged construction
CD	Drained	Drained	Long-term stability

Undrained Shear Strength

s_u = c_u = q_u/2

For saturated clay under undrained loading (φ_u = 0)

q_u = unconfined compressive strength

Typical Soil Parameters

Soil Type	φ' (degrees)	c' (kPa)
Loose Sand	28-32°	0
Dense Sand	35-45°	0
Soft Clay	20-25°	10-25
Stiff Clay	25-30°	50-100

7. Bearing Capacity

Foundation Bearing Capacity

The maximum load per unit area that soil can support without shear failure.

Terzaghi's Bearing Capacity Equation

Strip Footing (Continuous)

q_u = cN_c + qN_q + 0.5γBN_γ

Square Footing

q_u = 1.3cN_c + qN_q + 0.4γBN_γ

Circular Footing

q_u = 1.3cN_c + qN_q + 0.3γBN_γ

Where:

• q_u = ultimate bearing capacity
• c = soil cohesion
• q = γD_f = overburden pressure at foundation level
• γ = unit weight of soil below foundation
• B = width of foundation
• N_c, N_q, N_γ = bearing capacity factors (function of φ)

Bearing Capacity Factors

φ (°)	N_c	N_q	N_γ
0	5.14	1.0	0
20	14.8	6.4	5.0
25	20.7	10.7	10.9
30	30.1	18.4	22.4
35	46.1	33.3	48.0
40	75.3	64.2	109.4

Allowable Bearing Capacity

q_a = q_u / FS

Typical Factor of Safety: FS = 2.5 to 3.0

Also check settlement limitations!

Net Bearing Capacity

q_net = q_u - q = q_u - γD_f

Accounts for overburden removed during excavation

Groundwater Effects

Modify unit weights when water table is present:

Case 1: Water table at foundation level (D_w = D_f)
Use γ' = γ_sat - γ_w for N_γ term
Case 2: Water table above foundation (D_w < D_f)
Modify both q and N_γ terms
Case 3: Water table below foundation within B
Interpolate for N_γ term

8. Settlement Analysis

Types of Settlement

Total settlement = Immediate + Primary Consolidation + Secondary Compression

Immediate Settlement (Elastic)

S_i = qB(1-μ²)I_f/E_s

• Occurs immediately upon loading
• Significant in sands and gravels
• I_f = influence factor (depends on shape, rigidity)

Consolidation Settlement

Normally Consolidated Clay (σ'₀ = σ'_p)

S_c = (C_cH)/(1+e₀) × log(σ'_f/σ'₀)

Overconsolidated Clay (σ'₀ < σ'_p)

Case 1: σ'_f ≤ σ'_p

S_c = (C_rH)/(1+e₀) × log(σ'_f/σ'₀)

Case 2: σ'_f > σ'_p

S_c = (C_rH)/(1+e₀) × log(σ'_p/σ'₀) + (C_cH)/(1+e₀) × log(σ'_f/σ'_p)

Where:

• C_c = compression index (virgin compression)
• C_r = recompression index (≈ C_c/5 to C_c/10)
• σ'₀ = initial effective stress at center of layer
• σ'_f = final effective stress = σ'₀ + Δσ
• σ'_p = preconsolidation pressure
• OCR = σ'_p/σ'₀ (Overconsolidation Ratio)

Time Rate of Consolidation

T_v = c_vt / H²_dr

T_v = time factor, c_v = coefficient of consolidation

H_dr = drainage path (H for single, H/2 for double drainage)

U = 50%: T_v ≈ 0.197

U = 60%: T_v ≈ 0.287

U = 70%: T_v ≈ 0.403

U = 80%: T_v ≈ 0.567

U = 90%: T_v ≈ 0.848

U = 95%: T_v ≈ 1.129

Secondary Compression

S_s = C_αH × log(t₂/t₁)

Creep after primary consolidation complete

C_α = secondary compression index (typically 0.01-0.05)

Stress Distribution (2:1 Method)

Δσ = qBL / (B+z)(L+z)

For rectangular foundation at depth z

Also use Boussinesq or Westergaard solutions for more accuracy

Key Takeaways for CE Board Exam

Must-Know Formulas

✓ Se = wG_s (fundamental relationship)
✓ PI = LL - PL (Plasticity Index)
✓ A-Line: PI = 0.73(LL - 20)
✓ Darcy's Law: q = kiA
✓ Mohr-Coulomb: τ = c + σ tan φ
✓ Terzaghi's equation for bearing capacity
✓ Consolidation settlement formulas

Critical Concepts

✓ Effective stress principle: σ' = σ - u
✓ USCS classification (above/below A-line)
✓ Standard vs Modified Proctor differences
✓ Triaxial test types (UU, CU, CD)
✓ NC vs OC clay settlement
✓ Factor of Safety = 2.5 to 3.0
✓ Time rate of consolidation (T_v)

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1. Soil Composition and Properties

Three-Phase Soil System

Phase Diagram Components

Volume-Weight Relationships

Void Ratio (e)

Porosity (n)

Degree of Saturation (S)

Water Content (w)

Unit Weights

Important Relationships

2. Atterberg Limits

Consistency Limits

Soil States and Limits

Liquid Limit (LL)

Plastic Limit (PL)

Shrinkage Limit (SL)

Index Properties

Activity Classification

3. Soil Classification Systems

Particle Size Classification

USCS (Unified Soil Classification System)

Coarse-Grained Soils

Fine-Grained Soils

Gradation Coefficients

Coefficient of Uniformity (Cu)

Coefficient of Curvature (Cc)

Plasticity Chart (A-Line)

AASHTO Classification

4. Soil Compaction

Compaction

Laboratory Compaction Tests

Compaction Curve

Optimum Moisture Content (OMC)

Maximum Dry Density (γd,max)

Zero Air Voids (ZAV) Line

Field Compaction Control

Relative Compaction (RC)

Field Density Tests

Relative Density (Dr)

5. Permeability and Seepage

Darcy's Law

Seepage Velocity

Typical Permeability Values

Laboratory Tests

Constant Head Test

Falling Head Test

Equivalent Permeability

Horizontal Flow (Parallel)

Vertical Flow (Series)

6. Shear Strength of Soils

Mohr-Coulomb Failure Criterion

Effective Stress Principle

Laboratory Shear Tests

Direct Shear Test

Triaxial Test Types

Undrained Shear Strength

Typical Soil Parameters

7. Bearing Capacity

Foundation Bearing Capacity

Terzaghi's Bearing Capacity Equation

Where:

Bearing Capacity Factors

Allowable Bearing Capacity

Net Bearing Capacity

Groundwater Effects

Modify unit weights when water table is present:

8. Settlement Analysis

Types of Settlement

Immediate Settlement (Elastic)

Consolidation Settlement

Normally Consolidated Clay (σ'0 = σ'p)

Overconsolidated Clay (σ'0 < σ'p)

Where:

Time Rate of Consolidation

Secondary Compression

Stress Distribution (2:1 Method)

Key Takeaways for CE Board Exam

Must-Know Formulas

Critical Concepts

Table of Contents

Coefficient of Uniformity (C_u)

Coefficient of Curvature (C_c)

Maximum Dry Density (γ_d,max)

Relative Density (D_r)

Normally Consolidated Clay (σ'₀ = σ'_p)

Overconsolidated Clay (σ'₀ < σ'_p)