Geotechnical Engineering Formula Sheet
All key formulas for GATE CE and university exams — index properties, compaction, permeability, consolidation, shear strength, bearing capacity, earth pressure, slope stability, and pile foundations
Last Updated: March 2026
How to Use This Sheet
- All formulas follow IS codes and standard geotechnical practice unless noted otherwise.
- Symbol conventions: γ = unit weight (kN/m³); σ = total stress; σ′ = effective stress; u = pore water pressure; all in kPa unless stated.
- Water unit weight γw = 9.81 kN/m³ (use 10 kN/m³ only if problem states so).
- GATE-critical values are highlighted in the Quick Reference Table at the end.
- Use this sheet alongside individual topic pages for derivations and worked examples.
1. Index Properties & Phase Relationships
1.1 Volume Ratios
Void ratio: e = Vv / Vs
Porosity: n = Vv / V = e / (1+e)
e = n / (1−n)
Degree of saturation: S = Vw / Vv (as decimal; S = 1 for saturated)
Air voids ratio: av = Va / V = n(1−S)
1.2 Water Content & Specific Gravity
Water content: w = Mw / Ms (decimal; multiply by 100 for %)
Specific gravity: Gs = ρs / ρw
Fundamental relation: S e = w Gs
For saturated soil (S = 1): e = w Gs
1.3 Unit Weights
Bulk: γ = (Gs + Se) γw / (1+e)
Dry: γd = Gs γw / (1+e) = γ / (1+w)
Saturated: γsat = (Gs+e) γw / (1+e)
Submerged: γ′ = γsat − γw = (Gs−1) γw / (1+e)
1.4 Relative Density
Dr = (emax − e) / (emax − emin) × 100 %
States: Very loose < 15 % < Loose < 35 % < Medium < 65 % < Dense < 85 % < Very dense
2. Atterberg Limits & Derived Indices
Plasticity index: PI = LL − PL
Liquidity index: LI = (w − PL) / PI
Consistency index: CI = (LL − w) / PI = 1 − LI
Toughness index: IT = PI / IF (IF = flow index from Casagrande test)
Activity: A = PI / (% clay fraction, particles < 2 μm)
Tension crack depth: zc = 2cu / γ (for φu = 0)
Critical height of vertical cut: Hc = 4cu / γ = 2 zc
Skempton’s empirical: Cc = 0.009 (LL − 10) for undisturbed NC clay
2.1 Plasticity Chart — A-line and Classification
A-line: PI = 0.73 (LL − 20)
U-line (upper boundary): PI = 0.9 (LL − 8)
LL < 50 % → Low plasticity (L); LL ≥ 50 % → High plasticity (H)
Above A-line → Clay (C); Below A-line → Silt/Organic (M or O)
CL: above A-line, LL < 50 %, PI ≥ 7 | CH: above A-line, LL ≥ 50 %
ML: below A-line, LL < 50 % | MH: below A-line, LL ≥ 50 %
3. Soil Classification — IS 1498 (USCS)
Primary split: > 50 % passing 75 μm → Fine-grained; ≥ 50 % retained → Coarse-grained
G vs S: ≥ 50 % of coarse fraction retained on 4.75 mm → Gravel (G); else Sand (S)
Well-graded criteria:
GW: Cu ≥ 4 AND 1 ≤ Cc ≤ 3
SW: Cu ≥ 6 AND 1 ≤ Cc ≤ 3
Cu = D60/D10; Cc = D30²/(D10 × D60)
Hazen’s formula: k = C × D10² (k in cm/s; D10 in mm; C = 100 for uniform sand)
Organic criterion: LLoven-dried/LL < 0.75 → OL or OH
4. Soil Compaction
γd = γ / (1+w)
ZAV line (S=100%): γd,ZAV = Gs γw / (1 + wGs)
Degree of compaction: DC = (γd,field / γd,max,Proctor) × 100 %
Compaction energy: E = (W × H × Nb × NL) / V
Standard Proctor: 2.6 kg rammer, 310 mm drop, 3 layers, 25 blows = 605 kJ/m³
Modified Proctor: 4.9 kg rammer, 450 mm drop, 5 layers, 25 blows = 2726 kJ/m³
Higher energy → higher MDD, lower OMC (curve shifts up and left)
5. Permeability & Seepage
Darcy’s Law: v = k i; Q = k i A; vs = v/n (seepage velocity)
Hydraulic gradient: i = Δh / L
Constant head: k = QL / (Aht)
Falling head: k = (aL/At) ln(h1/h2) = 2.303(aL/At) log(h1/h2)
Stratified — parallel flow: kH = (k1H1 + k2H2 + …) / H
Stratified — perpendicular flow: kV = H / (H1/k1 + H2/k2 + …)
Always: kH ≥ kV
Critical gradient (quicksand): icr = (Gs−1)/(1+e) = γ′/γw
Seepage flow net: q = kH(Nf/Nd) per unit length
Exit gradient: ie = Δh/l (last square); FOS = icr/ie ≥ 3–5
Anisotropic: transform x′ = x√(kz/kx); keff = √(kxkz)
Seepage force per unit volume: j = i γw
6. Consolidation & Settlement
Effective stress principle: σ′ = σ − u
Terzaghi’s equation: ∂ue/∂t = cv ∂²ue/∂z²
cv = k / (γw mv); mv = av/(1+eo)
Time factor: Tv = cv t / Hdr²
Hdr = H/2 (double drainage); Hdr = H (single drainage)
Degree of consolidation U:
U ≤ 60 %: Tv = (π/4)(U/100)²
U > 60 %: Tv = 1.781 − 0.933 log(100−U%)
| U (%) | Tv | U (%) | Tv |
|---|---|---|---|
| 10 | 0.008 | 60 | 0.287 |
| 20 | 0.031 | 70 | 0.403 |
| 30 | 0.071 | 80 | 0.567 |
| 40 | 0.126 | 90 | 0.848 |
| 50 | 0.197 | 95 | 1.129 |
Settlement — NC clay: Sc = [CcH/(1+eo)] log[(σo′+Δσ)/σo′]
Settlement — OC clay (if final stress ≤ σc′): use Cs instead of Cc
Settlement — OC clay (if final stress > σc′): two-stage (Cs then Cc)
Alternative: Sc = mv Δσ H
OCR = σc′ / σo′
Secondary: Ss = Cα H log(t2/t1)
cv from Taylor’s method: cv = 0.848 Hdr² / t90
cv from Casagrande’s method: cv = 0.197 Hdr² / t50
7. Shear Strength
Mohr-Coulomb: τf = c′ + σ′ tanφ′ (effective); τf = cu for φu=0 (undrained saturated)
Failure plane angle: αf = 45° + φ′/2 (to major principal plane)
Principal stress at failure: σ1′ = σ3′ Nφ + 2c′√Nφ
Nφ = tan²(45° + φ′/2)
Mohr circle: centre = (σ1+σ3)/2; radius = (σ1−σ3)/2
UCT: cu = qu/2 (unconfined compressive strength divided by 2)
Pore pressure: Δu = B[Δσ3 + A(Δσ1−Δσ3)]; B=1 for saturated soil
Sensitivity: St = cu,undisturbed / cu,remoulded
Vane shear: cu = T / [π D²H/2 (1 + D/3H)] ≈ T/(3.66D³) for H=2D
CD test → c′, φ′ (drained); CU test → c′, φ′ and ccu, φcu; UU test → cu, φu=0
8. Bearing Capacity
Terzaghi strip: qu = c Nc + q Nq + 0.5 γ B Nγ
Square: qu = 1.3c Nc + q Nq + 0.4γB Nγ
Circular: qu = 1.3c Nc + q Nq + 0.3γB Nγ
q = γ Df (overburden at foundation level)
Net ultimate: qnet,u = qu − q = cNc + q(Nq−1) + 0.5γBNγ
Net safe: qns = qnet,u / FOS (FOS = 3 general; 2.5 local shear)
For φ = 0 (undrained): qnet,u = 5.14 cu (strip); 6.68 cu (square)
| φ° | Nc | Nq | Nγ |
|---|---|---|---|
| 0 | 5.14 | 1.00 | 0.00 |
| 10 | 8.35 | 2.47 | 1.22 |
| 20 | 14.83 | 6.40 | 5.39 |
| 25 | 20.72 | 10.66 | 10.88 |
| 30 | 30.14 | 18.40 | 22.40 |
| 35 | 46.12 | 33.30 | 48.03 |
| 40 | 75.31 | 64.20 | 109.41 |
IS 6403: qu = cNcscdcic + qNqsqdqiq + 0.5γBNγsγdγiγ
Water table at foundation level: replace γ in Nγ-term with γ′
Water table at surface: replace γ in both q-term and Nγ-term with γ′
9. Earth Pressure
At-rest: K0 = 1 − sinφ′ (Jaky, NC soil); K0,OC = K0,NC × OCR0.5
Rankine active: Ka = (1−sinφ)/(1+sinφ) = tan²(45°−φ/2)
Rankine passive: Kp = (1+sinφ)/(1−sinφ) = tan²(45°+φ/2) = 1/Ka
Active pressure at depth z (c-φ soil): σa = Kaγz − 2c√Ka
Passive pressure: σp = Kpγz + 2c√Kp
Total active force (cohesionless, level backfill): Pa = ½KaγH² at H/3 from base
With surcharge q: Psurcharge = KaqH at H/2 from base
Tension crack depth: zc = 2c/(γ√Ka)
Critical height: Hc = 4c/(γ√Ka) = 2zc
Below WT: add hydrostatic pressure u = γw(z−zw) separately to effective earth pressure
10. Slope Stability
FOS = resisting moment / driving moment = Σ(c′l + N′tanφ′) / ΣW sinα (Fellenius)
N′ = W cosα − ul (pore pressure correction)
Bishop simplified: FOS = Σ[(c′b+(W−ub)tanφ′)/mα] / ΣWsinα (iterative)
mα = cosα + (tanφ′sinα)/FOS
Infinite slope (dry cohesionless): FOS = tanφ′ / tanβ
Infinite slope (saturated, seepage parallel to slope): FOS = (γ′/γsat)(tanφ′/tanβ)
Taylor stability: FOS = c / (γH Sn); Sn from chart
Critical height (φ=0, vertical cut): Hc = 4cu/γ
Pore pressure ratio: ru = u/(γz)
11. Pile Foundations
Ultimate capacity: Qu = Qs + Qb
Skin friction — clay (α method): fs = α cu; Qs = α cu π D L
Skin friction — sand (β method): fs = K σv′ tanδ
End bearing in clay: Qb = 9 cu Ab (deep pile, L/D ≥ 4)
End bearing in sand: Qb = σv,crit′ Nq Ab (capped at 5 MPa)
Safe capacity: Qsafe = Qu/FOS (FOS = 2.5 with load test; 3.0 from formula)
ENR formula: Qu = Whh/(s+C)
Group efficiency: η = Qgroup/(n Qindividual)
Block failure: Qblock = cu × perimeter × L + 9cu,base × Ablock
Group capacity = min(n Qindividual, Qblock)
NSF: Qn = α cu π D Hc; total load = structural + NSF ≤ Qsafe
Minimum pile spacing (driven): 3D or 1.0 m (IS 2911)
12. GATE Quick-Reference Table
The most-tested values and relationships in GATE CE Geotechnical Engineering — memorise these.
| Item | Value / Formula |
|---|---|
| Fundamental relation | Se = wGs |
| Dry unit weight from bulk | γd = γ/(1+w) |
| e–n relationship | n = e/(1+e); e = n/(1–n) |
| ZAV line | γd = Gsγw/(1+wGs) |
| Standard Proctor energy | 605 kJ/m³; Modified: 2726 kJ/m³ |
| Critical hydraulic gradient | icr = (Gs–1)/(1+e) ≈ 1.0 for most sands |
| Flow net seepage | q = kH(Nf/Nd) |
| Tv for U = 50 % | 0.197 |
| Tv for U = 90 % | 0.848 |
| cv from oedometer (Taylor) | cv = 0.848 Hdr²/t90 |
| Time ratio (single vs double drainage) | tsingle = 4 × tdouble (for same U) |
| Failure plane angle | αf = 45° + φ/2 |
| φu for saturated clay (UU) | φu = 0; τf = cu |
| Ka for φ = 30° | 0.333 |
| Kp for φ = 30° | 3.00 |
| Ka × Kp | = 1 always (Rankine) |
| Jaky’s K0 | K0 = 1 – sinφ′ |
| Active force (cohesionless) | Pa = ½KaγH² at H/3 |
| Tension crack depth | zc = 2cu/γ |
| qnet,u for φ=0 strip | 5.14 cu |
| qnet,u for φ=0 square | 1.3×5.14×cu = 6.68 cu |
| FOS for bearing capacity | 3.0 (general shear); 2.5 (local shear) |
| End bearing in deep clay pile | Qb = 9 cu Ab |
| FOS for pile (with load test) | 2.5; without test: 3.0 |
| Infinite slope (dry) FOS | tanφ/tanβ |
| Infinite slope (saturated) FOS | (γ′/γsat) tanφ/tanβ |
| Critical height vertical cut (φ=0) | Hc = 4cu/γ |
13. Common Mistakes
Mistake 1: Using w as a Percentage Instead of a Decimal in Phase Relationship Formulas
What happens: w = 20 % is substituted as 20 in Se = wGs, giving Se = 20Gs instead of the correct 0.20Gs. This produces void ratios that are unrealistically large.
Fix: In all phase relationship and unit weight formulas, w must be a decimal fraction (20 % → 0.20). Only report the final answer in percentage form.
Mistake 2: Applying Double Drainage Hdr = H/2 to a Singly Drained Layer
What happens: Hdr = H/2 is used for a clay layer that drains only at the top (e.g., overlying rock or very stiff impermeable clay below). This underestimates consolidation time by a factor of 4.
Fix: Carefully identify drainage boundaries: permeable layer above AND below → double drainage (Hdr = H/2). Permeable on one side only → single drainage (Hdr = H).
Mistake 3: Using Nc = 5.14 for Pile Tip in Clay (Should be 9)
Fix: Shallow foundation bearing capacity (Terzaghi, φ=0): Nc = 5.14. Deep pile tip in clay (IS 2911): Nc = 9. These are different conditions — confined failure at depth gives higher Nc.
Mistake 4: Forgetting to Subtract Overburden When Computing Net Bearing Capacity
Fix: qnet,u = qu − q (where q = γDf). The factor of safety is applied to qnet,u, not qu. Using qs = qu/FOS underestimates the safe bearing capacity.
Mistake 5: Using Total Stress Active Pressure When Water Table is Present
Fix: Below the water table: horizontal pressure = effective earth pressure (Kaγ′z contribution) + hydrostatic pressure (γw(z−zw)). Calculate separately and add — never use Kaγsatz as the total horizontal pressure.
14. Frequently Asked Questions
Q1. Which geotechnical formulas are most commonly tested in GATE CE?
Based on analysis of GATE CE papers from 2018 to 2025, the most frequently tested formulas are: (1) consolidation settlement (Sc = CcH/(1+eo) log(…) and Tv–U relationships); (2) shear strength from Mohr’s circle and triaxial test data; (3) bearing capacity of shallow foundations (Terzaghi + water table corrections); (4) effective stress and pore pressure (Se = wGs; σ′ = σ−u); (5) earth pressure (Ka, Pa, tension crack). Each year at least one numerical question comes from consolidation and one from bearing capacity. Index properties and permeability appear as conceptual MCQs.
Q2. How is the effective stress principle applied in every geotechnical calculation?
Terzaghi’s effective stress principle (σ′ = σ − u) is the thread running through all geotechnical analysis. In consolidation: settlement occurs because σ′ increases as excess pore pressure u dissipates. In shear strength: the Mohr-Coulomb criterion uses σ′ (effective normal stress on the failure plane). In bearing capacity: ultimate bearing capacity depends on c′, φ′ (effective stress strength parameters) in drained conditions. In earth pressure: the effective horizontal pressure Kaγ′z acts on the wall, while hydrostatic pressure u = γwz acts separately. In slope stability: Bishop’s method computes N′ = Wcosα − ul to get the effective normal force. Understanding where and why effective stress is used is more important than memorising individual formulas.
Q3. What is the single most important concept to master for maximum GATE marks in geotechnical?
Consolidation theory — specifically the time rate of consolidation — consistently produces the highest-mark numerical questions in GATE CE geotechnical. A thorough understanding of: (a) the Tv–U relationship and which formula to use at what U value; (b) the critical role of Hdr (drainage path) and the Hdr² dependence of consolidation time; (c) computing cv from given test data (t50 or t90); and (d) the three-case settlement calculation for NC and OC clays — will alone account for 3–5 marks in most GATE CE papers. Master consolidation first, then bearing capacity, then shear strength.
Q4. How do the geotechnical topics interconnect — what is the logical learning sequence?
The optimal sequence, reflecting the logical dependency chain, is: Index Properties (the vocabulary) → Atterberg Limits (consistency of clays) → Soil Classification (predicting behaviour from index tests) → Compaction (improving soil) → Permeability (flow through soil) → Seepage (2D flow, piping, uplift) → Consolidation (time-dependent settlement driven by pore pressure dissipation, governed by permeability) → Shear Strength (the Mohr-Coulomb criterion, tested in CD/CU/UU conditions) → Bearing Capacity (shallow foundations: applies shear strength to failure under footings) → Earth Pressure (lateral pressure on walls: applies Mohr-Coulomb to incipient failure of soil mass) → Slope Stability (applies shear strength to slope failure surfaces) → Pile Foundations (deep foundations: integrates shear strength along the pile shaft and at the tip). Each topic builds on the previous, so gaps in earlier concepts create compounding confusion in later ones.