Geotechnical Engineering Formula Sheet

Void ratio: e = V_v / V_s

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

e = n / (1−n)

Degree of saturation: S = V_w / V_v (as decimal; S = 1 for saturated)

Air voids ratio: a_v = V_a / V = n(1−S)

Water content: w = M_w / M_s (decimal; multiply by 100 for %)

Specific gravity: G_s = ρ_s / ρ_w

Fundamental relation: S e = w G_s

For saturated soil (S = 1): e = w G_s

Bulk: γ = (G_s + Se) γ_w / (1+e)

Dry: γ_d = G_s γ_w / (1+e) = γ / (1+w)

Saturated: γ_sat = (G_s+e) γ_w / (1+e)

Submerged: γ′ = γ_sat − γ_w = (G_s−1) γ_w / (1+e)

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

States: Very loose < 15 % < Loose < 35 % < Medium < 65 % < Dense < 85 % < Very dense

Plasticity index: PI = LL − PL

Liquidity index: LI = (w − PL) / PI

Consistency index: CI = (LL − w) / PI = 1 − LI

Toughness index: I_T = PI / I_F (I_F = flow index from Casagrande test)

Activity: A = PI / (% clay fraction, particles < 2 μm)

Tension crack depth: z_c = 2c_u / γ (for φ_u = 0)

Critical height of vertical cut: H_c = 4c_u / γ = 2 z_c

Skempton’s empirical: C_c = 0.009 (LL − 10) for undisturbed NC clay

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 %

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: C_u ≥ 4 AND 1 ≤ C_c ≤ 3

  SW: C_u ≥ 6 AND 1 ≤ C_c ≤ 3

C_u = D₆₀/D₁₀; C_c = D₃₀²/(D₁₀ × D₆₀)

Hazen’s formula: k = C × D₁₀² (k in cm/s; D₁₀ in mm; C = 100 for uniform sand)

Organic criterion: LL_oven-dried/LL < 0.75 → OL or OH

γ_d = γ / (1+w)

ZAV line (S=100%): γ_d,ZAV = G_s γ_w / (1 + wG_s)

Degree of compaction: DC = (γ_d,field / γ_{d,max,Proctor}) × 100 %

Compaction energy: E = (W × H × N_b × N_L) / 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)

Darcy’s Law: v = k i; Q = k i A; v_s = 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: k_H = (k₁H₁ + k₂H₂ + …) / H

Stratified — perpendicular flow: k_V = H / (H₁/k₁ + H₂/k₂ + …)

Always: k_H ≥ k_V

Critical gradient (quicksand): i_cr = (G_s−1)/(1+e) = γ′/γ_w

Seepage flow net: q = kH(N_f/N_d) per unit length

Exit gradient: i_e = Δh/l (last square); FOS = i_cr/i_e ≥ 3–5

Anisotropic: transform x′ = x√(k_z/k_x); k_eff = √(k_xk_z)

Seepage force per unit volume: j = i γ_w

Effective stress principle: σ′ = σ − u

Terzaghi’s equation: ∂u_e/∂t = c_v ∂²u_e/∂z²

c_v = k / (γ_w m_v); m_v = a_v/(1+e_o)

Time factor: T_v = c_v t / H_dr²

H_dr = H/2 (double drainage); H_dr = H (single drainage)

Degree of consolidation U:

  U ≤ 60 %: T_v = (π/4)(U/100)²

  U > 60 %: T_v = 1.781 − 0.933 log(100−U%)

Settlement — NC clay: S_c = [C_cH/(1+e_o)] log[(σ_o′+Δσ)/σ_o′]

Settlement — OC clay (if final stress ≤ σ_c′): use C_s instead of C_c

Settlement — OC clay (if final stress > σ_c′): two-stage (C_s then C_c)

Alternative: S_c = m_v Δσ H

OCR = σ_c′ / σ_o′

Secondary: S_s = C_α H log(t₂/t₁)

c_v from Taylor’s method: c_v = 0.848 H_dr² / t₉₀

c_v from Casagrande’s method: c_v = 0.197 H_dr² / t₅₀

Mohr-Coulomb: τ_f = c′ + σ′ tanφ′ (effective); τ_f = c_u for φ_u=0 (undrained saturated)

Failure plane angle: α_f = 45° + φ′/2 (to major principal plane)

Principal stress at failure: σ₁′ = σ₃′ N_φ + 2c′√N_φ

N_φ = tan²(45° + φ′/2)

Mohr circle: centre = (σ₁+σ₃)/2; radius = (σ₁−σ₃)/2

UCT: c_u = q_u/2 (unconfined compressive strength divided by 2)

Pore pressure: Δu = B[Δσ₃ + A(Δσ₁−Δσ₃)]; B=1 for saturated soil

Sensitivity: S_t = c_{u,undisturbed} / c_u,remoulded

Vane shear: c_u = T / [π D²H/2 (1 + D/3H)] ≈ T/(3.66D³) for H=2D

CD test → c′, φ′ (drained); CU test → c′, φ′ and c_cu, φ_cu; UU test → c_u, φ_u=0

Terzaghi strip: q_u = c N_c + q N_q + 0.5 γ B N_γ

Square: q_u = 1.3c N_c + q N_q + 0.4γB N_γ

Circular: q_u = 1.3c N_c + q N_q + 0.3γB N_γ

q = γ D_f (overburden at foundation level)

Net ultimate: q_net,u = q_u − q = cN_c + q(N_q−1) + 0.5γBN_γ

Net safe: q_ns = q_net,u / FOS (FOS = 3 general; 2.5 local shear)

For φ = 0 (undrained): q_net,u = 5.14 c_u (strip); 6.68 c_u (square)

IS 6403: q_u = cN_cs_cd_ci_c + qN_qs_qd_qi_q + 0.5γBN_γs_γd_γi_γ

Water table at foundation level: replace γ in N_γ-term with γ′

Water table at surface: replace γ in both q-term and N_γ-term with γ′

At-rest: K₀ = 1 − sinφ′ (Jaky, NC soil); K_0,OC = K_0,NC × OCR^0.5

Rankine active: K_a = (1−sinφ)/(1+sinφ) = tan²(45°−φ/2)

Rankine passive: K_p = (1+sinφ)/(1−sinφ) = tan²(45°+φ/2) = 1/K_a

Active pressure at depth z (c-φ soil): σ_a = K_aγz − 2c√K_a

Passive pressure: σ_p = K_pγz + 2c√K_p

Total active force (cohesionless, level backfill): P_a = ½K_aγH² at H/3 from base

With surcharge q: P_surcharge = K_aqH at H/2 from base

Tension crack depth: z_c = 2c/(γ√K_a)

Critical height: H_c = 4c/(γ√K_a) = 2z_c

Below WT: add hydrostatic pressure u = γ_w(z−z_w) separately to effective earth pressure

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′)/m_α] / ΣWsinα (iterative)

m_α = cosα + (tanφ′sinα)/FOS

Infinite slope (dry cohesionless): FOS = tanφ′ / tanβ

Infinite slope (saturated, seepage parallel to slope): FOS = (γ′/γ_sat)(tanφ′/tanβ)

Taylor stability: FOS = c / (γH S_n); S_n from chart

Critical height (φ=0, vertical cut): H_c = 4c_u/γ

Pore pressure ratio: r_u = u/(γz)

Ultimate capacity: Q_u = Q_s + Q_b

Skin friction — clay (α method): f_s = α c_u; Q_s = α c_u π D L

Skin friction — sand (β method): f_s = K σ_v′ tanδ

End bearing in clay: Q_b = 9 c_u A_b (deep pile, L/D ≥ 4)

End bearing in sand: Q_b = σ_v,crit′ N_q A_b (capped at 5 MPa)

Safe capacity: Q_safe = Q_u/FOS (FOS = 2.5 with load test; 3.0 from formula)

ENR formula: Q_u = W_hh/(s+C)

Group efficiency: η = Q_group/(n Q_individual)

Block failure: Q_block = c_u × perimeter × L + 9c_u,base × A_block

Group capacity = min(n Q_individual, Q_block)

NSF: Q_n = α c_u π D H_c; total load = structural + NSF ≤ Q_safe

Minimum pile spacing (driven): 3D or 1.0 m (IS 2911)