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Home β€Ί Formulas β€Ί Geotechnical

Geotechnical Engineering
Formula Reference

Essential equations for soil mechanics, seepage, consolidation, shear strength, earth pressure, bearing capacity, and pile design. Aligned with IS codes for Indian practice.

🧫 Soil Classification & Index Properties
βˆ’
Void Ratio
e = Vv / Vs
Ratio of volume of voids to volume of solids. Key indicator of soil density.

Vv = Volume of voids

Vs = Volume of solids

Sand: e = 0.4–0.8 | Clay: e = 0.6–1.5
Porosity
n = Vv / V = e / (1 + e)
Ratio of void volume to total volume. Always expressed as decimal or percentage.

V = Total volume

Degree of Saturation
S = Vw / Vv
Percentage of voids filled with water. S = 0 (dry) to S = 1 (fully saturated).

Vw = Volume of water

Water Content
w = (Ww / Ws) Γ— 100%
Weight of water to weight of solids. Determined by oven-drying at 105–110Β°C.

Ww = Weight of water

Ws = Weight of solids

Relative Density (Sand)
Dr = (emax βˆ’ e) / (emax βˆ’ emin)
Measure of compactness of granular soil. 0% = loosest, 100% = densest state.

emax = Void ratio at loosest state

emin = Void ratio at densest state

Dr < 35% Loose | 35–65% Medium | > 65% Dense
Consistency Index (Clay)
Ic = (wL βˆ’ w) / (wL βˆ’ wP)
Indicates whether clay is in liquid, plastic, or solid state.

wL = Liquid limit

wP = Plastic limit

w = Natural water content

πŸ’§ Permeability & Seepage
+
Darcy's Law Fundamental
v = k Γ— i
Flow velocity through soil is proportional to hydraulic gradient. Valid for laminar flow.

v = Discharge velocity

k = Coefficient of permeability (cm/s)

i = Hydraulic gradient = Ξ”h / L

Clay: k β‰ˆ 10⁻⁷ cm/s | Sand: k β‰ˆ 10⁻² cm/s | Gravel: k β‰ˆ 1 cm/s
Seepage Force
j = i Γ— Ξ³w
Force per unit volume exerted by flowing water. Causes piping in foundations.

i = Hydraulic gradient

Ξ³w = Unit weight of water (9.81 kN/mΒ³)

Critical condition: ic = (Gβˆ’1)/(1+e) for upward flow (quick sand)
Coefficient of Permeability (Kozeny-Carman)
k = (Ξ³w / ΞΌ) Γ— (Cs Γ— eΒ³) / (1 + e)
Theoretical equation linking permeability to void ratio and grain properties.

ΞΌ = Dynamic viscosity of water

Cs = Shape/size constant

⏳ Consolidation & Settlement
+
Compression Index (Cc) Skempton
Cc = 0.009 (wL βˆ’ 10)
Empirical relation for normally consolidated clays. From liquid limit.

wL = Liquid limit (%)

For remoulded: Cc = 0.007 (wL βˆ’ 10)
Total Consolidation Settlement
Sc = (Cc Γ— H) / (1 + eβ‚€) Γ— log₁₀(Οƒ'β‚€ + Δσ') / Οƒ'β‚€
Settlement of a normally consolidated clay layer due to applied load.

H = Thickness of clay layer

eβ‚€ = Initial void ratio

Οƒ'β‚€ = Initial effective stress

Δσ' = Increase in effective stress

Degree of Consolidation (Time)
U = f(Tv)
Time-dependent consolidation progress. Tv = Time factor.

Tv = (cv Γ— t) / dΒ²

cv = Coefficient of consolidation

d = Drainage path length

U β‰ˆ 60% when Tv = 0.848 (for U > 60%, Tv β‰ˆ 1.781 βˆ’ 0.933 log(100βˆ’U%))
Immediate (Elastic) Settlement
Si = q Γ— B Γ— (1 βˆ’ Ξ½Β²) / Es Γ— If
Immediate undrained settlement for clay under foundation load.

q = Net applied pressure

B = Width of footing

Es = Modulus of elasticity

If = Influence factor

Over-Consolidation Ratio (OCR)
OCR = Οƒ'c / Οƒ'β‚€
Ratio of pre-consolidation pressure to current effective stress. OCR > 1 = over-consolidated.

Οƒ'c = Pre-consolidation pressure

Οƒ'β‚€ = Present effective overburden

βœ‚οΈ Shear Strength of Soils
+
Mohr-Coulomb Criterion Fundamental
Ο„f = c + Οƒ' tan Ο†
Shear strength is the sum of cohesion and friction components on the failure plane.

Ο„f = Shear strength

c = Cohesion intercept

Οƒ' = Normal effective stress

Ο† = Angle of internal friction

Undrained Shear Strength (Cu)
Ο„f = cu (Ο†u = 0)
In undrained conditions (short term, clay), only cohesion contributes. Friction angle = 0.

cu = Undrained cohesion (from UU test or vane shear)

cu β‰ˆ 0.22 Γ— Οƒ'β‚€ for normally consolidated clay (Skempton)
Effective Stress (Terzaghi)
Οƒ' = Οƒ βˆ’ u
Total stress minus pore water pressure. Governs drained (long-term) strength.

Οƒ = Total stress

u = Pore water pressure

SPT N-Value Correction
Ncor = N Γ— CN
Correct observed SPT blow count for overburden pressure.

CN = 2 / (1 + Οƒ'β‚€/100) [Peck & Bazaraa]

Also apply corrections for dilatancy, borehole diameter, rod length, and sampler type
🧱 Earth Pressure (Retaining Walls)
+
Active Earth Pressure Coefficient (Rankine)
Ka = (1 βˆ’ sin Ο†) / (1 + sin Ο†) = tanΒ²(45Β° βˆ’ Ο†/2)
Minimum lateral pressure when wall moves away from soil.

Ο† = Angle of internal friction

For Ο† = 30Β°: Ka = 1/3 | For Ο† = 0Β°: Ka = 1.0
Passive Earth Pressure Coefficient (Rankine)
Kp = (1 + sin Ο†) / (1 βˆ’ sin Ο†) = tanΒ²(45Β° + Ο†/2)
Maximum lateral pressure when wall pushes into soil.

Ο† = Angle of internal friction

Kp = 1/Ka. For Ο† = 30Β°: Kp = 3.0
Active Pressure with Surcharge
pa = Ka Γ— (Ξ³z + q)
Active pressure at depth z accounting for uniform surcharge load q on the surface.

Ξ³ = Unit weight of soil

z = Depth below surface

q = Surcharge pressure

Coulomb's Active Pressure
Ka = sinΒ²(Ξ±+Ο†) / [sinΒ²Ξ± Γ— sin(Ξ±βˆ’Ξ΄) Γ— {1+√...}Β²]
Accounts for wall friction (Ξ΄), wall inclination (Ξ±), and backfill slope. More realistic for design.

Ξ΄ = Wall friction angle

Ξ± = Angle of wall back face with horizontal

πŸ—οΈ Bearing Capacity (Shallow Foundations)
+
Terzaghi's Ultimate Bearing Capacity IS 6403
qu = cNc + Ξ³DfNq + 0.5Ξ³BNΞ³
Ultimate bearing capacity for strip footing. Three terms: cohesion, surcharge, self-weight.

Nc, q, Ξ³ = Bearing capacity factors (depend on Ο†)

Df = Depth of foundation

B = Width of footing

Safe Bearing Capacity
qs = qu / FOS + Ξ³Df
Ultimate capacity divided by factor of safety (typically 2.5–3.0), plus overburden relief.

FOS = Factor of Safety (3 for static, 2 for seismic)

Meyerhof's Bearing Capacity (General)
qu = cNcscdcic + Ξ³DfNqsqdqiq + 0.5Ξ³BNΞ³sΞ³dΞ³iΞ³
Extended Terzaghi formula with shape (s), depth (d), and inclination (i) factors. For square, rectangular, and circular footings.

s, d, i = Shape, depth, and load inclination factors

Net Safe Bearing Capacity
qns = (qu βˆ’ Ξ³Df) / FOS
The net increase in pressure that the soil can safely carry above the existing overburden.

qu = Ultimate bearing capacity

Used when settlement controls design rather than shear failure.
πŸ”© Pile Capacity IS 2911
+
Ultimate Pile Capacity (Static)
Qu = qb Γ— Ab + fs Γ— As
Sum of end-bearing and skin friction components.

qb = End-bearing pressure

Ab = Cross-sectional area at toe

fs = Unit skin friction

As = Shaft surface area

End Bearing in Clay
qb = 9 Γ— cu
End-bearing capacity in cohesive soil. cu at pile tip level.

cu = Undrained cohesion at tip

Skin Friction in Clay (Ξ±-Method)
fs = Ξ± Γ— cu
Adhesion factor Ξ± ranges from 0.3 (stiff) to 1.0 (very soft clay).

Ξ± = Adhesion factor (see IS 2911 Fig. 1)

cu = Average undrained cohesion along shaft

Skin Friction in Sand
fs = K Γ— ΟƒΜ„'v Γ— tan Ξ΄
Friction along pile shaft in granular soil.

K = Earth pressure coefficient (0.5–1.5)

ΟƒΜ„'v = Average effective vertical stress

Ξ΄ = Angle of wall friction β‰ˆ 0.75Ο†

ENR Driving Formula
Qu = (W Γ— H) / (S + C)
Estimate pile capacity from driving records. Approximate only.

W = Hammer weight

H = Height of fall

S = Set (penetration per blow)

C = 25 mm (drop), 2.5 mm (single-acting)

Group Efficiency
Ξ·g = Qgroup / (n Γ— Qsingle)
Ratio of group capacity to sum of individual capacities.

n = Number of piles

Clay: Ξ·g < 1 (block failure) | Sand: Ξ·g β‰₯ 1 (densification)
⛰️ Slope Stability
+
Factor of Safety (Slope)
FOS = Ο„f / Ο„mobilized
Ratio of available shear strength to mobilized shear along the potential slip surface.

Ο„f = Available shear strength

Ο„mobilized = Required shear for equilibrium

Minimum FOS: 1.3 (temporary), 1.5 (permanent), 1.1 (earthquake)
Infinite Slope (C-Ο† soil)
FOS = (c + γ'z cos²β tanφ) / (γsatz cosβ sinβ)
Stability of long slopes in homogeneous soil with parallel failure surface.

Ξ² = Slope angle

z = Depth of failure plane

Ξ³' = Submerged unit weight

Taylor's Stability Number
Sn = c / (Ξ³ Γ— H Γ— FOS)
Stability number from Taylor's charts. Used for preliminary assessment of slope stability.

H = Height of slope

⚠️
Disclaimer These formulas are provided for quick reference and educational purposes. Always verify with the relevant IS code edition and consult a qualified geotechnical engineer for actual design. Site-specific conditions may require modifications to standard equations.