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Soil Mechanics – Complete Study Notes

GATE ESE / IES SSC JE State PSC RRB JE

Comprehensive chapter-wise notes covering Soil Exploration, Index Properties & Classification, Permeability & Seepage, Consolidation, Shear Strength, Earth Pressure Theories, Stress Distribution, and Geosynthetics. All IS codes, key formulae, Mohr circles, pressure diagrams, and exam-focused tables for GATE, ESE & SSC JE.

Ch 1 · Soil Exploration Ch 2 · Properties of Soil Ch 3 · Soil Classification Ch 4 · Tests & Interrelationships Ch 5 · Permeability & Seepage Ch 6 · Compressibility & Consolidation Ch 7 · Shearing Resistance Ch 8 · Earth Pressure & Stress Distribution Ch 9 · Geosynthetics ★ Quick Revision & Mnemonics

1Soil Exploration – Planning & Methods►

1.1 Purpose & Planning

Soil exploration (also called site investigation) is carried out to determine the nature, extent, and engineering properties of subsurface soils and to establish groundwater conditions before designing foundations, embankments, or retaining structures.

Planning Steps

Collect existing data: geological maps, previous borehole records, topographic surveys
Reconnaissance survey: visual inspection of terrain, rock outcrops, nearby excavations
Decide spacing & depth of borings based on project type
Select appropriate investigation method
Conduct laboratory testing on recovered samples
Prepare subsurface profile and geotechnical report

Depth & Spacing Guidelines

Structure Type	Typical Boring Depth	Typical Spacing
Light buildings	3–6 m or 1.5× least foundation width	15–30 m
Heavy buildings / multi-storey	10–30 m	15–30 m
Highways & embankments	3–6 m	30–150 m
Dams & large structures	To hard stratum or 1.5× base width	15–30 m (varies)
Deep foundations (piles)	3–5 m below pile tip	At each pile location or 10–15 m grid

IS 1892:1979 governs subsurface investigation for foundations in India. Minimum one boring per 230 m² of building area for initial investigation.

1.2 Methods of Soil Exploration

Method	Principle / Tool	Suitable For	Depth
Trial Pits / Test Pits	Open excavation; direct visual inspection	Shallow soils; identification of strata; undisturbed sampling	Up to 3 m (safe without shoring)
Auger Boring	Hand/mechanical auger rotated into ground	Soft to stiff clays, silts above water table	Up to 6 m (hand); 30 m (power auger)
Wash Boring	Water jet + chopping bit; cuttings returned in wash water	Sandy soils; locating strata changes	30–60 m
Rotary Drilling	Rotating bit with drilling fluid; core recovery	Hard soils, rocks; good core samples	Any depth; most versatile
Percussion Drilling	Heavy chisel/bit dropped repeatedly	Boulders, hard rock	Any depth
Shell & Auger (Cable Tool)	Shell (in soft) or auger (in stiff soil) lowered on cable	Soft to medium soils	60 m

1.3 Sampling Methods

Sample Type	Description	Use
Disturbed sample	Structure destroyed during collection (wash boring, auger)	Classification tests (grain size, Atterberg limits, specific gravity)
Undisturbed sample	Natural structure preserved (thin-walled tube sampler)	Strength tests, consolidation tests, permeability
Representative sample	Mix representative of stratum; moisture content preserved	Compaction, classification

Thin-Walled (Shelby) Tube Sampler

Area Ratio (AR) = (D_e² − D_i²) / D_i² × 100 (%)
AR < 10% → undisturbed; AR < 15% for soft sensitive clays

Inside Clearance Ratio (ICR) = (D_i − D_s) / D_s × 100 (%)
ICR = 0.5–1.0% recommended (reduces friction on sample)

Outside Clearance Ratio (OCR) = (D_e − D_w) / D_w × 100 (%)
OCR = 0–2% (prevents soil heave into sampler)

1.4 In-Situ Tests

Standard Penetration Test (SPT)

Split-spoon sampler (OD = 51 mm, ID = 35 mm)
Hammer: 63.5 kg dropped from 760 mm
N = blows for last 300 mm penetration (first 150 mm = seating drive, discarded)

Corrections to N:
N₆₀ = N × (E_m / 60%) [energy correction]
Overburden correction: N₁ = N × √(100/σ'_v) [Liao & Whitman]
Dilatancy correction (Terzaghi): if N > 15 in fine sands below water table:
N_corrected = 15 + 0.5(N − 15)

N value	Sand Density	Clay Consistency	qu (kPa)
0–4	Very loose	Very soft	<25
4–10	Loose	Soft	25–50
10–30	Medium dense	Medium stiff	50–200
30–50	Dense	Stiff	200–400
>50	Very dense	Very stiff / Hard	>400

Static Cone Penetration Test (SCPT / CPT)

Cone: 60° apex, base area = 10 cm²
Cone resistance: q_c = Q_c / A_cone
Friction ratio: R_f = f_s / q_c × 100 (%)
Sand: R_f < 1%; Clay: R_f > 4%
φ (sand): q_c ≈ σ'_v × N_q → φ estimated from charts
c_u (clay): c_u = (q_c − σ_v) / N_k; N_k = 15–20 (typical)

Vane Shear Test (VST)

Used for soft to medium clays in-situ
c_u = T / [π × D² × (H/2 + D/6)] [for H/D = 2]
where T = torque at failure, D = vane diameter, H = vane height
Correction: c_u(corrected) = μ × c_u(measured); μ = Bjerrum's correction factor

Plate Load Test (PLT)

Rigid circular plate (dia = 300–750 mm) loaded incrementally
Settlement at every load increment; load vs settlement curve plotted
Modulus of subgrade reaction: k_s = q / δ (kN/m³)
For clay: q_u(foundation) ≈ q_u(plate) [size independent]
For sand: q_u(foundation) = q_u(plate) × (B_f / B_p)
Settlement: S_f = S_p × [B_f(B_p + 0.3) / B_p(B_f + 0.3)]² [IS 8009]

Pressure-meter Test & Dilatometer

The pressure-meter (Ménard) expands a cylindrical probe in a borehole to measure soil stiffness and lateral stress. The flat dilatometer (DMT) is pushed into soil and the membrane expanded to determine horizontal stress index, material index, and soil type.

1.5 Geophysical Methods

Method	Principle	Output	Application
Seismic Refraction	P-wave travel time across layers	Layer velocities and depths	Depth to rock; rippability
Electrical Resistivity	Wenner/Schlumberger array; current through ground	Resistivity profiles	Depth to water table, clay layers, contamination
Ground Penetrating Radar (GPR)	Electromagnetic pulses reflected from interfaces	Subsurface profiles	Utility detection, shallow stratigraphy
Cross-hole / Down-hole seismic	S-wave and P-wave velocities between boreholes	G_max, dynamic modulus	Liquefaction assessment, earthquake engineering

2Properties of Soil►

2.1 Three-Phase System

Soil is a three-phase material: solid grains + water + air. All volumetric and gravimetric relationships stem from this model.

2.2 Volumetric Relationships

Void ratio: e = V_v / V_s (can be >1 for soft clays)
Porosity: n = V_v / V × 100 (%) → n = e/(1+e)
Degree of saturation: S = V_w / V_v × 100 (%) [0–100%]
Air content (a_c): a_c = V_a / V = n(1−S)

Relationship: e·S = w·G_s ← very important for numericals

2.3 Gravimetric (Mass) Relationships

Water content: w = M_w / M_s × 100 (%) [oven-dry at 105–110°C, 24 hr]
Bulk unit weight: γ = W/V = G_s·γ_w(1+w) / (1+e)
Dry unit weight: γ_d = γ / (1+w) = G_s·γ_w / (1+e)
Saturated unit weight: γ_sat = (G_s + e)·γ_w / (1+e) [S=100%]
Submerged / buoyant unit weight: γ' = γ_sub = γ_sat − γ_w
Specific gravity: G_s = M_s / (ρ_w · V_s) [IS: 2.60–2.80 for most soils]

Typical γ_w = 9.81 kN/m³ ≈ 10 kN/m³ (for quick calculations)

2.4 Typical Values

Soil Type	G_s	e (range)	γ_d (kN/m³)	n (%)
Gravel	2.65–2.68	0.3–0.6	16–20	25–40
Sand (loose)	2.65	0.6–0.9	14–16	37–47
Sand (dense)	2.65	0.4–0.6	16–19	29–37
Silt	2.68–2.72	0.6–1.1	12–16	38–52
Soft clay	2.70–2.80	0.9–2.0	9–13	47–67
Peat / Organic	1.40–1.70	2.0–10	4–8	70–90

2.5 Atterberg Limits (Consistency Limits)

Applicable to fine-grained soils (silts and clays). Defined on soil–water mixtures.

Liquid Limit (LL / w_L): min. water content at which soil flows under its own weight (Casagrande cup; 25 blows)
Plastic Limit (PL / w_P): min. water content at which soil can be rolled into 3 mm thread without crumbling
Shrinkage Limit (SL / w_S): water content below which further drying causes no volume change

Plasticity Index: PI = LL − PL
Liquidity Index: LI = (w − PL) / PI [LI > 1 → liquid; LI < 0 → semisolid]
Consistency Index: CI = (LL − w) / PI = 1 − LI
Activity: A = PI / (% clay fraction <2μm) [Skempton]
A < 0.75 = inactive clay; 0.75–1.25 = normal; >1.25 = active (e.g. montmorillonite)

LI	Consistency State	Field Description
<0	Semi-solid / Solid	Hard, brittle
0–0.25	Very stiff	Can be indented by thumbnail
0.25–0.50	Stiff	Moulded under strong pressure
0.50–0.75	Medium	Moulded under moderate pressure
0.75–1.00	Soft	Easily moulded
>1.00	Very soft / Liquid	Flows under its own weight

2.6 Particle Size Distribution

Sieve analysis: for coarse-grained soils (gravel, sand) — IS 460 sieves
Hydrometer analysis: for fine-grained soils (silt, clay) — Stokes' law
Stokes' law: v = (G_s − 1)·γ_w·D² / (18η) where η = dynamic viscosity of water

D₁₀: effective size (10% finer); D₃₀, D₆₀: 30%, 60% finer
Coefficient of Uniformity: C_u = D₆₀ / D₁₀
C_u < 2 = uniform; C_u > 6 for well-graded sand; C_u > 4 for well-graded gravel
Coefficient of Curvature: C_c = D₃₀² / (D₁₀ × D₆₀)
Well-graded (GW/SW): C_u > 6 (sand) or 4 (gravel) AND 1 ≤ C_c ≤ 3

2.7 Relative Density

D_r = (e_max − e) / (e_max − e_min) × 100 (%)
Also: D_r = (γ_d − γ_d,min) × γ_d,max / [(γ_d,max − γ_d,min) × γ_d] × 100 (%)

D_r < 15% = very loose; 15–35% = loose; 35–65% = medium; 65–85% = dense; >85% = very dense
Used for granular soils only (not applicable to cohesive soils)

3Classification of Soil►

3.1 IS Soil Classification System (IS 1498:1970 – BIS)

India follows the IS Classification System, which is essentially the Unified Soil Classification System (USCS) with minor modifications. Two-letter symbols used.

IS Particle Size Ranges

Fraction	Size Range	Sub-division
Gravel (G)	4.75 mm – 80 mm	Coarse: 20–80 mm; Fine: 4.75–20 mm
Sand (S)	0.075 mm – 4.75 mm	Coarse: 2–4.75 mm; Medium: 0.425–2 mm; Fine: 0.075–0.425 mm
Silt (M)	0.002 mm – 0.075 mm	—
Clay (C)	< 0.002 mm	—
Peat (Pt)	Organic; fibrous texture	—

Classification Flowchart Logic

Group Symbol	Description	Key Criteria
GW	Well-graded gravel	>50% retained #4.75mm; C_u ≥ 4; 1 ≤ C_c ≤ 3
GP	Poorly-graded gravel	>50% retained #4.75mm; fails GW criteria
GM	Silty gravel	>12% fines; fines plot below A-line or PI < 4
GC	Clayey gravel	>12% fines; fines plot above A-line and PI > 7
SW	Well-graded sand	>50% pass #4.75mm; C_u ≥ 6; 1 ≤ C_c ≤ 3
SP	Poorly-graded sand	>50% pass #4.75mm; fails SW criteria
SM	Silty sand	>12% fines; fines below A-line or PI < 4
SC	Clayey sand	>12% fines; fines above A-line and PI > 7
ML	Silt of low plasticity	>50% pass #0.075mm; LL < 50; below A-line
CL	Clay of low plasticity	>50% pass #0.075mm; LL < 50; above A-line; PI > 7
OL	Organic silt/clay, low plasticity	LL < 50; organic; LL_oven/LL_wet < 0.75
MH	Silt of high plasticity	LL > 50; below A-line
CH	Clay of high plasticity	LL > 50; above A-line
OH	Organic clay/silt, high plasticity	LL > 50; organic
Pt	Peat	Fibrous; highly organic; dark; odour

Casagrande's Plasticity Chart (A-Line)

A-line equation: PI = 0.73 × (LL − 20)
U-line (upper boundary): PI = 0.9 × (LL − 8)

Points above A-line → Clay (C prefix)
Points below A-line → Silt or Organic (M or O prefix)
Hatched zone: 4 < PI < 7 and LL 15–25 → borderline CL-ML

3.2 AASHTO Classification

Used for highway subgrade classification in India (IRC guidelines). Soils classified as A-1 through A-7; each group has a Group Index (GI).

GI = (F − 35)[0.2 + 0.005(LL − 40)] + 0.01(F − 15)(PI − 10)
where F = % passing 0.075 mm sieve; GI = 0 → good subgrade; GI > 20 → poor

A-1-a: Gravel, stone fragments (best)
A-1-b to A-3: Coarse sand
A-4 to A-5: Silty soils
A-6 to A-7: Clayey soils (worst; A-7-6 most plastic)

3.3 Field Identification Tests

Test	Procedure	Result Interpretation
Dilatancy test	Pat soil; squeeze; observe water at surface	Quick reaction → Silt/fine sand; No reaction → Clay
Dry strength test	Dry pat and try to break with fingers	High strength → Fat clay (CH); Low → ML or MH
Toughness test	Roll near PL; check toughness of thread	Tough thread at PL → CH; Weak → CL or ML
Shine test	Cut dry pat with knife	Shiny surface → Clay; Dull → Silt

4Various Tests & Interrelationships►

4.1 Laboratory Index Tests

Test	IS Code	Method/Equipment	Output
Water content	IS 2720-2	Oven drying (105–110°C), Calcium carbide method, Pycnometer	w (%)
Specific gravity	IS 2720-3	Pycnometer (fine soil) or gas jar (coarse)	G_s
Grain size analysis	IS 2720-4	Sieve analysis + Hydrometer	PSD curve, D₁₀, C_u, C_c
Atterberg limits	IS 2720-5	Casagrande cup (LL), Rolling thread (PL), Mercury displacement (SL)	LL, PL, PI, SL
Field density	IS 2720-28/29	Core cutter, Sand replacement, Rubber balloon	γ, γ_d in field
Compaction (Proctor)	IS 2720-7/8	Standard (2.5 kg, 300 mm), Modified (4.9 kg, 450 mm)	OMC, MDD
Permeability	IS 2720-17	Constant head (coarse), Falling head (fine)	k (m/s)
Consolidation (Oedometer)	IS 2720-15	Oedometer/consolidometer; incremental loading	C_c, C_v, m_v, e-log p curve
Direct Shear	IS 2720-13	60×60 mm box; strain-controlled	c', φ'
Triaxial Shear	IS 2720-11/12	UU, CU, CD tests; cylindrical sample	c, φ, pore pressure u
Unconfined Compression	IS 2720-10	No cell pressure; σ₃ = 0	q_u; c_u = q_u/2
CBR	IS 2720-16	Penetration load at 2.5 mm and 5 mm vs standard	CBR (%); pavement design

4.2 Compaction Test (Proctor)

Standard Proctor: 2.5 kg hammer, 300 mm drop, 3 layers, 25 blows/layer → E = 605 kJ/m³
Modified Proctor: 4.9 kg hammer, 450 mm drop, 5 layers, 25 blows/layer → E = 2700 kJ/m³

Zero Air Voids (ZAV) line: γ_d = G_s·γ_w / (1 + w·G_s) [S = 100%]
ZAV always plots to the right of dry density curve (curve never crosses ZAV line)

Effect of compaction energy: Higher energy → higher MDD, lower OMC
Effect of soil type: Coarser soils → higher MDD, lower OMC

Degree of compaction: = (γ_d,field / γ_d,max) × 100 (%) [≥95–98% typical specification]

4.3 Critical Interrelationships (Master Equations)

e · S = w · G_s ← Universal; derived from 3-phase diagram

γ_d = γ / (1+w) and γ = G_s · γ_w · (1+w) / (1+e)

n = e / (1+e) and e = n / (1−n)

At full saturation (S = 1): w = e/G_s
Dry state (S = 0, w = 0): γ_d = G_s·γ_w / (1+e)

K₀ (at rest pressure coefficient) = 1 − sin φ' [Jaky's formula, NC clays]
K₀(OC) = K₀(NC) × OCR^(sin φ') [Mayne & Kulhawy]

4.4 Sensitivity and Thixotropy

Sensitivity: S_t = q_u(undisturbed) / q_u(remoulded)
S_t < 2 = insensitive; 2–4 = slightly sensitive; 4–8 = medium; 8–16 = very sensitive; >16 = quick clay

Thixotropy: Regain of strength after remoulding (if left undisturbed at same water content)
Thixotropic hardening important in pile installation and deep mixing

5Permeability & Seepage►

5.1 Darcy's Law

v = k · i [Darcy velocity / discharge velocity]
Q = k · i · A
where k = coefficient of permeability (m/s); i = hydraulic gradient = Δh / L

Seepage velocity: v_s = v / n [n = porosity]
Valid only for laminar flow: Reynolds number Re = v·D₁₀/ν < 1–10

k depends on: grain size, void ratio, degree of saturation, fluid viscosity, soil structure

5.2 Laboratory Permeability Tests

Constant Head Test (coarse soils)

k = Q · L / (A · h · t)
Q = volume collected; L = sample length; A = cross-section area;
h = constant head difference; t = time

Falling Head Test (fine soils)

k = (a · L) / (A · t) × ln(h₁/h₂) = 2.303 × (a·L)/(A·t) × log(h₁/h₂)
a = area of standpipe; h₁, h₂ = initial and final head in standpipe

5.3 Empirical Correlations

Allen Hazen (clean sands): k = C · D₁₀² [k in cm/s; D₁₀ in cm; C = 100–150]
Kozeny-Carman: k ∝ e³/(1+e) [fundamental; valid for all soils]
Temperature correction: k_T = k₂₇ · (η₂₇/η_T) [standard at 27°C in India]

5.4 Equivalent Permeability in Stratified Soils

Horizontal (flow parallel to layers):
k_H = (k₁H₁ + k₂H₂ + … + k_nH_n) / (H₁ + H₂ + … + H_n) [weighted average]

Vertical (flow perpendicular to layers):
k_V = (H₁ + H₂ + … + H_n) / (H₁/k₁ + H₂/k₂ + … + H_n/k_n) [harmonic mean]

Always: k_H > k_V (anisotropy in natural deposits)

5.5 Typical k Values

Soil Type	k (m/s)	Drainage
Clean gravel	10⁻¹ – 10⁻²	Good
Coarse sand	10⁻² – 10⁻³	Good
Fine sand, loose silt	10⁻³ – 10⁻⁵	Fair
Silty clay	10⁻⁵ – 10⁻⁸	Poor
Clay	< 10⁻⁸	Very poor / impervious

5.6 Seepage Analysis – Flow Nets

A flow net is a graphical solution of the Laplace equation for 2D seepage flow consisting of flow lines and equipotential lines forming curvilinear squares.

Laplace equation: ∂²h/∂x² + ∂²h/∂z² = 0 (for isotropic, homogeneous soil)

Seepage flow: Q = k · H · (N_f / N_d)
N_f = number of flow channels; N_d = number of equipotential drops; H = total head

Seepage pressure: p_s = i · γ_w · V (per unit volume: i · γ_w)
Exit gradient: i_e = Δh / a (a = length of last equipotential square at exit)

Critical hydraulic gradient: i_cr = (G_s − 1) / (1 + e) = γ' / γ_w
Factor of Safety against piping: F.S. = i_cr / i_e (must be > 3–4)

5.7 Quick Sand Condition & Piping

Quicksand occurs when upward seepage force = submerged weight of soil
i_e ≥ i_cr → piping / heave → effective stress = 0

Critical gradient: i_cr = (G_s − 1)/(1+e) ≈ 1 for most soils (G_s = 2.65, e = 0.65 → i_cr ≈ 1)

Creep ratio (Lane's weighted creep):
C_L = (1/3 × L_H + L_V) / H [C_L ≥ 3 for fine sand up to 18 for soft clay]

6Compressibility & Consolidation►

6.1 Settlement Components

Total Settlement: S_total = S_i + S_c + S_s
S_i = immediate (elastic) settlement [occurs during loading; sand dominant]
S_c = consolidation (primary) settlement [slow; clay dominant; pore pressure dissipation]
S_s = secondary compression (creep) [after consolidation; organic soils]

6.2 Consolidation Theory (Terzaghi, 1925)

Terzaghi's 1D consolidation assumes: saturated soil, incompressible solid grains and water, Darcy's law valid, small strain, constant k and m_v.

Governing equation: ∂u/∂t = c_v · ∂²u/∂z²

Coefficient of consolidation: c_v = k / (m_v · γ_w) [m²/year]
Coefficient of compressibility: a_v = −Δe / Δσ' [m²/kN]
Coefficient of volume change: m_v = a_v / (1 + e₀) [m²/kN]

Compression index: C_c = Δe / Δlog(σ') [slope of e-log p curve, virgin compression]
Skempton: C_c = 0.009(LL − 10) [for normally consolidated clays]
Recompression index: C_s ≈ C_c / 5 to C_c / 10 [swelling/rebound line]

Consolidation settlement:
For NC clay: S_c = C_c/(1+e₀) × H × log(σ'_f / σ'_0)
For OC clay (σ'_f ≤ σ'_p): S_c = C_s/(1+e₀) × H × log(σ'_f / σ'_0)
For OC clay (σ'_f > σ'_p): S_c = C_s·H/(1+e₀)·log(σ'_p/σ'_0) + C_c·H/(1+e₀)·log(σ'_f/σ'_p)

6.3 Time Rate of Consolidation

Time factor: T_v = c_v · t / H_dr²
H_dr = drainage path [= H/2 for double drainage; = H for single drainage]

Degree of consolidation (U) vs T_v:
U < 60%: T_v = π/4 × U² (parabolic approximation)
U > 60%: T_v = 1.781 − 0.933 × log(100 − U%)

Key T_v values:
U = 50% → T_v = 0.197
U = 90% → T_v = 0.848
U = 100% → T_v = ∞

OCR (Overconsolidation Ratio): OCR = σ'_p / σ'_0
OCR = 1 → Normally Consolidated (NC); OCR > 1 → Overconsolidated (OC)

6.4 Secondary Compression

S_s = C_α · H · log(t₂ / t₁)
C_α = secondary compression index = Δe / Δlog(t)
C_α / C_c ≈ 0.02–0.10 (higher for organic soils and peats)

6.5 Preloading & Vertical Drains

Preloading: Apply surcharge equal to or greater than design load before construction; allows consolidation to complete
Vertical (sand/wick) drains: Reduce drainage path from H to half the drain spacing → dramatically reduce consolidation time
Equivalent diameter of sand drain: D_e = 1.05·s (triangular grid) or 1.13·s (square grid)

7Shearing Resistance►

7.1 Mohr-Coulomb Failure Criterion

Shear strength: τ_f = c' + σ' tan φ' [in terms of effective stress]
τ_f = c + σ tan φ [in terms of total stress]
where c = cohesion intercept; φ = angle of internal friction

Effective stress: σ' = σ − u [u = pore water pressure]

At failure on Mohr circle:
σ'₁ = σ'₃ tan²(45 + φ'/2) + 2c' tan(45 + φ'/2)
Major principal stress: σ'₁; Minor: σ'₃

Failure plane angle: θ = 45 + φ/2 (to major principal plane)

7.2 Triaxial Test Types

Test Type	Abbreviation	Drainage Condition	Applicable To	Parameters
Unconsolidated Undrained	UU / Quick	No drainage (consolidation or shear)	Short-term stability of clay; immediate bearing capacity	φ_u = 0; c_u = q_u/2
Consolidated Undrained	CU (with pore pressure)	Consolidated; undrained shear	Embankment end-of-construction; slope analysis	c', φ' and c_u, φ_u
Consolidated Drained	CD / Slow	Full drainage; slow loading	Long-term stability; effective parameters	c' (≈0 for NC sand), φ'

7.3 Shear Strength Parameters

Soil	φ' (°)	c' (kPa)	Remarks
Loose sand	28–34	0	c' = 0 for all sands (no true cohesion)
Dense sand	35–45	0	Dilation effect increases φ
Gravel	35–45	0	—
Soft NC clay	20–30	0	Effective; c' ≈ 0 for NC
Stiff OC clay	25–30	10–50	Peak; c' due to cementation
Residual (clay)	10–20	≈0	Used for long-term slope analysis

7.4 Pore Pressure Parameters (Skempton)

Δu = B [Δσ₃ + A (Δσ₁ − Δσ₃)]
B = pore pressure parameter = 1.0 for fully saturated soil
A = Skempton's A parameter at failure:
  Heavily OC clay: A = −0.5 to 0
  Normally consolidated clay: A = 0.5–1.0
  Very sensitive/loose sand: A > 1.0

7.5 Unconfined Compression Test

σ₃ = 0 (no cell pressure); equivalent to UU with σ₃ = 0
q_u = unconfined compressive strength
c_u = q_u / 2 (undrained cohesion; φ_u = 0 assumption)

Stress at failure: σ_f = P_f / A_c; A_c = A₀ / (1 − ε) [corrected area]

7.6 Critical State Soil Mechanics

At critical state: soil shears at constant volume (e = e_cs = constant)
Critical State Line (CSL) in e-log p' space: e = e_Γ − λ log p'
In q-p' space: q = M·p' where M = 6sinφ'/(3−sinφ') [triaxial compression]

Loose of CSL → contractive; Dense of CSL → dilative

8Earth Pressure Theories & Stress Distribution in Soil►

8.1 Types of Lateral Earth Pressure

State	Condition	Pressure Coeff.	Relative Magnitude
At-rest (K₀)	No wall movement; lateral strain = 0	K₀ = 1 − sinφ' (Jaky)	Intermediate
Active (Ka)	Wall moves away from soil; soil expands	K_a = tan²(45 − φ/2)	Minimum; mobilises full shear
Passive (Kp)	Wall pushed into soil; soil compressed	K_p = tan²(45 + φ/2) = 1/K_a	Maximum; large displacement needed

Wall movement required: Active ≈ 0.001–0.005 × H (dense sand); Passive ≈ 0.02–0.1 × H — much larger movement for passive pressure to develop.

8.2 Rankine's Earth Pressure Theory

Assumptions: smooth, vertical wall; no wall friction; horizontal backfill; isotropic, homogeneous soil.

Active pressure at depth z (cohesionless):
σ_a = K_a · γ · z; K_a = (1−sinφ)/(1+sinφ) = tan²(45−φ/2)

For c-φ soil:
σ_a = K_a · γ · z − 2c√K_a
Tension crack depth: z_c = 2c / (γ√K_a)

Passive pressure (c-φ soil):
σ_p = K_p · γ · z + 2c√K_p

Total active force (cohesionless): P_a = ½ K_a γ H² [acts at H/3 from base]
With surcharge q: P_a = ½ K_a γ H² + K_a q H [surcharge adds rectangle]

8.3 Coulomb's Earth Pressure Theory

Considers wall friction (δ), inclined backfill (β), and inclined wall (α). More general than Rankine.

K_a(Coulomb) = sin²(α+φ) / {sin²α · sin(α−δ) · [1 + √(sin(φ+δ)sin(φ−β)/sin(α−δ)sin(α+β))]²}

For vertical wall (α=90°), horizontal fill (β=0°), no friction (δ=0°):
→ Coulomb = Rankine

Wall friction angle δ: δ = 0 for smooth; δ = 2φ/3 for rough walls (typical design)
Coulomb gives resultant inclined at δ to wall normal; Rankine gives horizontal resultant

8.4 Special Cases of Lateral Pressure

Submerged Backfill

σ_a = K_a · γ' · z + γ_w · z [effective earth pressure + water pressure]
Water pressure acts on both sides; total lateral force often governed by water
Drain backfill → use K_a · γ · z (no water pressure if drainage ensured)

Layered Soils

Calculate pressure at bottom of each layer using appropriate K_a and γ for that layer. At layer interfaces, two pressure values exist (using respective K_a values).

Backfill with Uniform Surcharge (q)

Additional lateral pressure: Δσ_a = K_a · q [uniform rectangle over full height]

8.5 Retaining Wall Stability Checks

Overturning: F.S. = ΣM_stabilising / ΣM_overturning ≥ 1.5–2.0
Sliding: F.S. = (W·tan δ_b + c_b·B) / P_a ≥ 1.5
(δ_b = base friction angle ≈ 2φ/3; c_b = base adhesion; B = base width)
Bearing capacity: q_max = (ΣV/B)(1 + 6e/B) ≤ q_allowable
(e = eccentricity of resultant; for no tension: e ≤ B/6 = middle third rule)

8.6 Stress Distribution in Soil

Geostatic (In-Situ) Vertical Stress

σ_v = γ₁H₁ + γ₂H₂ + … (sum of overburden layers)
Effective vertical stress: σ'_v = σ_v − u
Above water table: u = 0 → σ'_v = σ_v
Below water table: u = γ_w · h_w → σ'_v = σ_v − γ_w · h_w

Stress Due to Surface Loads – Boussinesq's Theory

Assumes: elastic, isotropic, homogeneous, semi-infinite mass. Most commonly used in practice.

Point load (Q):
σ_z = (3Q / 2πz²) · [1 / (1 + (r/z)²)]^(5/2)
= Q/z² · I_B [I_B = Boussinesq influence factor]

Uniformly loaded circular area (radius R, intensity q):
σ_z = q[1 − (1/(1+(R/z)²))^(3/2)] [on axis below centre]

Rectangular load (B × L, intensity q) — at corner:
σ_z = q · I_r [I_r from Fadum's chart; function of m = B/z, n = L/z]
For point not at corner → superposition (add/subtract rectangles)

Line load (q per unit length):
σ_z = (2q/π) · z³ / (x² + z²)²

Westergaard's Theory

For layered / anisotropic soils (thin horizontal layers); gives lower σ_z than Boussinesq
σ_z = (Q / z²) · I_W [I_W = Westergaard factor]
Assuming ν = 0: I_W = (1/π) / [2(r/z)² + 1]^(3/2)

2:1 Load Distribution Method (Approximate)

σ_z = q · B · L / [(B + z)(L + z)] [stress spreads at 2V:1H from foundation edges]
Quick estimate for settlement calculations; conservative for shallow depths

8.7 Newmark's Influence Chart

Graphical method for irregular-shaped loaded areas. Plot area to scale, count influence units enclosed, multiply by 0.005 × q (for 200-division chart) to get σ_z at any point at depth z.

9Properties & Uses of Geosynthetics►

9.1 Definition & Classification

Geosynthetics are planar, polymeric (synthetic or natural) products used with soil, rock, or other geotechnical-related material as an integral part of a civil engineering structure. Manufactured from polyethylene (PE), polypropylene (PP), polyester (PET), polyamide (nylon), or PVC.

Type	Structure / Manufacture	Primary Function
Geotextile (GTX)	Woven, Nonwoven, Knitted fabric; permeable	Filtration, separation, drainage, reinforcement
Geogrid (GGR)	Apertures (openings); Uniaxial or Biaxial; stiff ribs	Reinforcement of soil (tensile)
Geomembrane (GMB)	Impermeable sheet; HDPE, PVC, EPDM	Barrier / containment (liquid, gas)
Geocomposite (GC)	Combination of two or more geosynthetics	Multi-function (e.g., drain + filter + separator)
Geonet (GN)	Open netting; ribbed structure	In-plane drainage (high transmissivity)
Geosynthetic Clay Liner (GCL)	Bentonite sandwiched between geotextiles	Hydraulic barrier (liner for landfills)
Geofoam	EPS foam blocks	Lightweight fill, thermal insulation, vibration reduction
Geocell	3D honeycomb cellular structure	Confinement of fill; slope protection; base reinforcement

9.2 Primary Functions

Function	Description	Typical Application	Key Product
Separation	Prevents intermixing of dissimilar soils	Subgrade/subbase interface in roads	Nonwoven geotextile
Filtration	Allows fluid flow; retains soil particles	Drainage blankets, retaining wall drains, erosion control	Nonwoven/woven geotextile
Drainage	Transmits fluids within plane of geosynthetic	Pavement edge drains, chimney drains in embankments	Geonet, geocomposite drain
Reinforcement	Provides tensile strength to soil mass	MSE walls, reinforced slopes, unpaved road subgrade	Geogrid, woven geotextile
Containment / Barrier	Impedes fluid or gas movement	Landfill liner, pond liner, canal lining	Geomembrane, GCL
Erosion control	Protects surface from water/wind	Slopes, riverbanks, coastal protection	Erosion control mat, geocell
Protection	Cushion against mechanical damage	Geomembrane protection layers	Thick nonwoven geotextile

9.3 Geotextile Properties & Design

Key Engineering Properties

Apparent Opening Size (AOS) / O₉₅: size at which 95% of openings are smaller
Filtration criterion: O₉₅ ≤ D₈₅ of soil [to retain soil particles]
Permeability criterion: k_GTX ≥ k_soil × 10 [to allow free drainage]

Permittivity (Ψ): Ψ = k_n / t [flow perpendicular to plane; s⁻¹]
Transmissivity (θ): θ = k_p × t [flow in plane of fabric; m²/s]

Tensile Strength (wide-width strip test, ASTM D4595):
T_ult = ultimate tensile strength [kN/m]
T_allowable = T_ult / (RF_ID × RF_CR × RF_CBD × RF_W)
RF_ID = installation damage; RF_CR = creep reduction; RF_CBD = chemical/biological degradation; RF_W = weathering

9.4 Geogrid – Reinforced Soil

Mechanism: interlocking of soil into apertures → confinement + tensile resistance

Mechanically Stabilised Earth (MSE) Wall design:
Horizontal spacing: S_v from stability analysis
T_max = K_a × γ × z × S_v [tension in reinforcement at depth z]
Embedment length: L_e = T_max / (2 × σ'_v × f* × R_c)
f* = pullout interaction coefficient; R_c = coverage ratio

Total length: L = L_a + L_e [L_a = length in active zone; L_e in resistant zone]

9.5 Geomembrane – Landfill Liner

HDPE (High-Density Polyethylene): Most common; excellent chemical resistance; k ≈ 10⁻¹³ m/s; minimum 1.5 mm thick (IS/CPCB)
LLDPE: More flexible; used on steeper slopes; better UV resistance
PVC: Flexible; susceptible to plasticiser migration; older landfills
Seaming: extrusion welding or fusion welding; seam tested by air pressure, spark, or vacuum box
Double-liner system (CPCB for hazardous waste): primary HDPE + leak detection layer + secondary HDPE

9.6 Geosynthetic Clay Liner (GCL)

A factory-manufactured hydraulic barrier containing sodium bentonite. When hydrated, bentonite swells and k reduces to ≈ 5×10⁻¹¹ m/s. Advantages: consistent quality, thin (≈ 5–10 mm), self-healing, lower cost than compacted clay liner (CCL). Used as alternative to 600 mm thick CCL in composite liner systems.

9.7 IS Codes for Geosynthetics

IS Code	Topic
IS 15462:2004	Classification and identification of geosynthetics
IS 13162 (Parts)	Geotextiles — methods of test
IS 14322:1995	Geotextiles — determination of mass per unit area
BIS SP 36	Handbook on construction of embankments using geosynthetics
IRC: SP-59 (2019)	Guidelines for use of geosynthetics in road pavement
CPCB MSW Rules 2016	Liner specifications for MSW landfills

★Quick Revision, Formulae & Mnemonics►

A. Essential Formulae at a Glance

Three-Phase Relationships

Atterberg & Consistency

PI = LL − PL | LI = (w−PL)/PI | CI = 1−LI | Activity A = PI/(% <2μm)
C_c = 0.009(LL−10) [Skempton, NC clay]

Permeability

Consolidation

S_c = C_c·H/(1+e₀)·log(σ'_f/σ'₀) [NC] | T_v = c_v·t/H_dr²
U<60%: T_v = π/4·U² | U=50%: T_v=0.197 | U=90%: T_v=0.848
c_v = k/(m_v·γ_w) | m_v = a_v/(1+e₀)

Shear Strength

τ_f = c' + σ'tanφ' | σ₁ = σ₃·tan²(45+φ/2) + 2c·tan(45+φ/2)
c_u = q_u/2 | Δu = B[Δσ₃ + A(Δσ₁−Δσ₃)] | K₀ = 1−sinφ'

Earth Pressure

K_a = tan²(45−φ/2) = (1−sinφ)/(1+sinφ) | K_p = 1/K_a = tan²(45+φ/2)
P_a = ½K_aγH² [cohesionless; at H/3] | Tension crack: z_c = 2c/(γ√K_a)
σ_a = K_aγz − 2c√K_a | σ_p = K_pγz + 2c√K_p

Stress Distribution

Boussinesq point load: σ_z = (3Q/2πz²)·[1/(1+(r/z)²)]^(5/2)
Circular load (on axis): σ_z = q[1−(1/(1+(R/z)²))^(3/2)]
2:1 method: σ_z = qBL/[(B+z)(L+z)]

B. Exam-Angle Summary

Topic	GATE Focus	ESE Focus	SSC JE Focus
Soil Properties	Numerical on 3-phase; e·S=w·G_s; relative density	All index properties; compaction curves; ZAV construction	Definitions; typical values; Atterberg limits
Classification	USCS/IS group symbols; A-line equation; C_u & C_c criteria	Full classification with borderline cases; AASHTO GI computation	IS particle size ranges; GW/GP/ML/CH identification
Permeability	Constant/falling head formulae; equivalent k; i_cr; flow net Q	Kozeny-Carman; seepage force; piping FS; Lane's creep	Darcy's law; typical k table; Hazen formula
Consolidation	S_c formula; T_v-U relationship; c_v calculation; OCR	Full settlement analysis (immediate + consolidation + secondary); vertical drains	Primary vs secondary; T_v values; NC vs OC distinction
Shear Strength	Mohr-Coulomb; triaxial test types; c_u from UCS; Skempton A, B	Mohr circle construction; CU test pore pressures; residual strength	Definitions; test types; typical φ' for sand/clay
Earth Pressure	K_a, K_p; P_a numericals; tension crack depth; surcharge effect	Coulomb's K_a; retaining wall FS (overturning, sliding, bearing); MSE walls	Rankine's formula; pressure diagram shapes; K_a for clean sand
Stress Distribution	Boussinesq point load; circular & rectangular loads; 2:1 method	Newmark chart; Westergaard; stress isobars; settlement from σ_z	2:1 method; concept of stress distribution; bulb of pressure
Geosynthetics	Functions (SFDREB); geogrid aperture interlocking; AOS criterion	MSE wall design; GCL vs CCL; geomembrane seaming; transmissivity	Types of geosynthetics; function and application; geomembrane uses
Soil Exploration	SPT N corrections; area ratio; CPT friction ratio; vane shear c_u	Full site investigation program; all in-situ tests; geophysics	SPT N interpretation; types of borings; disturbed vs undisturbed

C. Mnemonics & Memory Aids

Three-Phase Master Formula: "eSe = wG"
e × S = w × G_s → the single most-tested soil relationship

Atterberg Limits order: "Soil Lives Permanently in a Liquid"
SL < PL < LL → increasing water content; soil goes from solid → plastic → liquid

K_a and K_p: "Active is Away, Passive is Pushed"
Active → wall moves away from soil (minimum pressure); Passive → wall pushed into soil (maximum)
K_a = tan²(45−φ/2); K_p = tan²(45+φ/2); K_p = 1/K_a

Triaxial test types: "UU CU CD → Quick → Consolidated-Quick → Slow"
UU = no drainage at all; CU = consolidate then shear undrained; CD = fully drained throughout

Consolidation: "T_v at 50 and 90"
U=50% → T_v = 0.197 ≈ 0.2; U=90% → T_v = 0.848 ≈ 0.85 → "Point two, point eight-five"

Critical gradient: "G minus one over one plus e"
i_cr = (G_s−1)/(1+e) ≈ 1.0 → soil floats at i=1 (quicksand!)

IS classification prefixes: "Gravel & Sand are Coarse; Silt & Clay are Fine"
G/S → sieve analysis (coarse); M/C → hydrometer (fine); L = low plasticity (<50); H = high (>50)

Geosynthetics functions: "SFDREB-C"
Separation – Filtration – Drainage – Reinforcement – Erosion control – Barrier – Containment

SPT N value rules: "D4, D3, M2, VD5"
Very loose <4; Loose 4–10; Medium 10–30; Dense 30–50; Very dense >50

Boussinesq vs Westergaard: "Boussinesq is Bigger"
Boussinesq (isotropic elastic) gives higher σ_z than Westergaard (layered); conservative for design

Preconsolidation pressure σ'_p: "OC is Over-stressed in the Past"
OCR = σ'_p/σ'₀ > 1 → soil was once under higher stress (glaciation, erosion, desiccation)

D. Important IS Codes – Soil Mechanics

IS Code	Subject
IS 1498:1970	Classification and identification of soils (IS/USCS)
IS 1892:1979	Code of practice for subsurface investigation for foundations
IS 2131:1981	Method for SPT
IS 4434:1978	In-situ vane shear test
IS 4968:1976	DCPT and SCPT
IS 2720 (Parts)	Methods of test for soils (water content, Atterberg, compaction, permeability, consolidation, shear)
IS 6403:1981	Determination of bearing capacity of shallow foundations
IS 8009:1976	Settlement of shallow foundations
IS 15462:2004	Classification of geosynthetics
IRC SP-59:2019	Geosynthetics in road construction

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Soil Mechanics – Complete Study Notes

1.1 Purpose & Planning

Planning Steps

Depth & Spacing Guidelines

1.2 Methods of Soil Exploration

1.3 Sampling Methods

Thin-Walled (Shelby) Tube Sampler

1.4 In-Situ Tests

Standard Penetration Test (SPT)

Static Cone Penetration Test (SCPT / CPT)

Vane Shear Test (VST)

Plate Load Test (PLT)

Pressure-meter Test & Dilatometer

1.5 Geophysical Methods

2.1 Three-Phase System

2.2 Volumetric Relationships

2.3 Gravimetric (Mass) Relationships

2.4 Typical Values

2.5 Atterberg Limits (Consistency Limits)

2.6 Particle Size Distribution

2.7 Relative Density

3.1 IS Soil Classification System (IS 1498:1970 – BIS)

IS Particle Size Ranges

Classification Flowchart Logic

Casagrande's Plasticity Chart (A-Line)

3.2 AASHTO Classification

3.3 Field Identification Tests

4.1 Laboratory Index Tests

4.2 Compaction Test (Proctor)

4.3 Critical Interrelationships (Master Equations)

4.4 Sensitivity and Thixotropy

5.1 Darcy's Law

5.2 Laboratory Permeability Tests

Constant Head Test (coarse soils)

Falling Head Test (fine soils)

5.3 Empirical Correlations

5.4 Equivalent Permeability in Stratified Soils

5.5 Typical k Values

5.6 Seepage Analysis – Flow Nets

5.7 Quick Sand Condition & Piping

6.1 Settlement Components

6.2 Consolidation Theory (Terzaghi, 1925)

6.3 Time Rate of Consolidation

6.4 Secondary Compression

6.5 Preloading & Vertical Drains

7.1 Mohr-Coulomb Failure Criterion

7.2 Triaxial Test Types

7.3 Shear Strength Parameters

7.4 Pore Pressure Parameters (Skempton)

7.5 Unconfined Compression Test

7.6 Critical State Soil Mechanics

8.1 Types of Lateral Earth Pressure

8.2 Rankine's Earth Pressure Theory

8.3 Coulomb's Earth Pressure Theory

8.4 Special Cases of Lateral Pressure

Submerged Backfill

Layered Soils

Backfill with Uniform Surcharge (q)

8.5 Retaining Wall Stability Checks

8.6 Stress Distribution in Soil

Geostatic (In-Situ) Vertical Stress

Stress Due to Surface Loads – Boussinesq's Theory

Westergaard's Theory

2:1 Load Distribution Method (Approximate)

8.7 Newmark's Influence Chart

9.1 Definition & Classification

9.2 Primary Functions

9.3 Geotextile Properties & Design

Key Engineering Properties

9.4 Geogrid – Reinforced Soil

9.5 Geomembrane – Landfill Liner

9.6 Geosynthetic Clay Liner (GCL)

9.7 IS Codes for Geosynthetics

A. Essential Formulae at a Glance

Three-Phase Relationships

Atterberg & Consistency

Permeability

Consolidation

Shear Strength