HGE Formulas 02

Geotechnical Engineering

Moisture Content

w = w_w / w_s

w_w = weight of water

w_s = weight of solids (dry soil)

Void Ratio

e = v_v / v_s

v_v = volume of voids

v_s = volume of soil solids

Degree of Saturation

S = v_w / v_v

v_w = volume of water

v_v = volume of voids

Porosity

n = v_v / v_T

v_v = volume of voids

v_T = total volume

Porosity, given void ratio

n = e / (1 + e)

Void Ratio, given porosity

e = n / (1 - n)

Relationship between Specific Gravity, Moisture Content, Degree of Saturation, and Void Ratio

Se = wG_s

Air Void Ratio

a_c = 1 - S = v_a / v_v

Dry Unit Weight

γ_d = γ_w G_s / (1 + e)

γ_d = γ_m / (1 + w)

γ_d = w_d / v_T

Moist Unit Weight

γ_m = γ_w G_s (1 + w) / (1 + e)

γ_m = γ_w (G_s + Se) / (1 + e)

γ_m = γ_d (1 + w)

γ_m = w_soil / v_T

Saturated Unit Weight

γ_sat = γ_w (Gs + e) / (1 + e)

Submerged Unit Weight

γ’ = γ_w (Gs - 1) / (1 + e)

γ’ = γ_sat - γ_w

Unit Weight of Water

γ_w = 62.4 lb/ft³ = 9810 N/m³

Specific Gravity from Laboratory Testing

G_s = (w_f+s - w_s) / ((w_f+w - w_f) - (w_f+s+w - w_f+s))

Relative Density

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

D_r = (1 / γ_d,min - 1 / D_d) / (1 / γ_{d, min} - 1 / γ_{d, max}) × 100%

Relative Compaction

R_c = γ_d / γ_{d, max} × 100%

Coefficient of Uniformity

C_u = D₆₀ / D₁₀

Coefficient of Curvature

D₃₀² / (D₁₀ × D₆₀)

Sorting Coefficient

S₀ = √(D₇₅ / D₂₅)

Suitability Number

S_n = 1.7√ (3 / D₅₀² + 1 / D₂₀² + 1 / D₁₀²)

Shrinkage Limit

SL = (m₁ - m₂)/m₂ - ρ_w (v₁ - v₂) / m₂

SL = e / G_s

m = mass

v = volume

ρ_w = density of water = 1 g/cm³

Shrinkage Ratio

SR = m₂ / ρ_wv₂

Relationship between Shrinkage Ratio and Shrinkage Limit

1 / G_s = 1 / SR - SLL

Liquidity Index

LI = (w - PL) / (LL - PL)

w = moisture content

PL = plastic limit

LL = liquid limit

Consistency Index

CI = (LL - w) / (LL - PL)

Plasticity Index

PI = LL - PL

Shrinkage Index

SI = PL - SL

Flow Index

FI = (w₁ - w₂) / log (N₂ / N₁)

w = moisture content

N = number of blows (Liquid Limit Test)

Toughness Index

TI = PI / FI

Activity of Clay

A = PI / %Clay

%Clay = F₂₀₀ or Fraction of Soil passing Sieve No. 200

USDA Classification

Gravel: > 2 mm

Sand: 0.05 mm - 2.00 mm

Silt: 0.002 mm - 0.005 mm

Clay: < 0.002 mm

AASHTO Classification

Gravel: 2 mm - 76.2 mm

Sand: 0.075 mm - 2 mm

Silt and Clay: Classified as Fines → < 0.075 mm

Group Index (GI) = (F - 35) [0.2 + 0.005(LL - 40)] + 0.01(F - 15)(PI - 10)

F = % Fines

LL = Liquid Limit

PI = Plasticity Index

Zero Air Voids Unit Weight

γ_zav = γ_wG_s / (1 + wG_s)

Discharge Velocity

v = ki

k = coefficient of permeability

i = hydraulic gradient

Hydraulic Gradient

i = h / L

h = head of sample

L = length of sample

Seepage Velocity

v_s = ki / n

Constant Head Permeability Test

k = VL / Aht

V = volume of water

L = length of sample

A = cross-sectional area of sample

h = head of sample

t = time of collection of water

Falling Head Permeability Test

k = aL / At × ln (h₁ / h₂)

a = cross-sectional area of standpipe

L = length of sample

A = cross-sectional area of sample

t = time of collection of water

h = head of sample before and after time was recorded

Unconfined Aquifer

k = Q ln(R₁ / R₂) / (π(h₁² - h₂²))

Q = Discharge

R₁ = Radial distance from the center of the well to observation point 1
R₂ = Radial distance from the center of the well to observation point 2

H₁ = Hydraulic Head of Groundwater at R₁

H₂ = Hydraulic Head of Groundwater at R₂

Discharge

Q = kiA = Av

Confined Aquifer

k = Q ln(R₁ / R₂) / (2πt (h₁ - h₂))

Q = Discharge

R₁ = Radial distance from the center of the well to observation point 1
R₂ = Radial distance from the center of the well to observation point 2

H₁ = Hydraulic Head of Groundwater at R₁

H₂ = Hydraulic Head of Groundwater at R₂

t = saturated thickness of the confined aquifer

Equivalent Coefficient of Permeability

k_|| = Σ(kh) / Σh

k_⟂ = Σh / (Σ (h/k))

Transmissivity

T = kt

Seepage Force

j = i γ_w

Seepage Flow Rate (Flow Nets)

Q = kH N_f / N_d

N_f = number of flow channels

N_d = number of drops

Mnemonic: Kathryn, Hindi Na Forever Ni Daniel

Total Stress

σ = Σγh

Porewater Pressure

µ = γ_wh_w

Effective Stress

σ’ = σ - µ

Capillary Height

h_c = c / eD₁₀

c = Capillary constant

e = void ratio

D₁₀ = effective grain size of soil

Porewater pressure at capillary zone

µ = Sγ_wh_z

S = degree of saturation

h_z = height from GWT

Seepage: Head Loss per Drop

∆h = H / N_d

Seepage: Pressure Head

h_p = H ± h_z

Seepage: Total Head

H_j = H - ∆h(N_d)_j

Seepage: Porewater Pressure

u_j = (h_p)_j γ_w

Seepage: Uplift Pressure

p_w = Σu_j ∆x_j

Primary Settlemnt: Normally Consolidated Soil

S_c = C_cH / (1 + e) × log ((P_o + ∆P) / P_o)

C_c = Compression index

H = height to middle of clay layer

e = void ratio of clay

P_o = Effective stress at middle of clay layer

∆P = Load applied above ground surface

Primary Settlement: P_o + ∆P < P_c

S_c = C_sH / (1 + e) × log ((P_o + ∆P) / P_o)

C_s = Swell index

H = height to middle of clay layer

e = void ratio of clay

P_o = Effective stress at middle of clay layer

∆P = Load applied above ground surface

Primary Settlement: When P_o + ∆P > P_c

S_c = C_sH / (1 + e) × log (P_c / P_o) + C_cH / (1 + e) × log ((P_o + ∆P) / P_c)

C_c = Compression index

C_s = Swell index

H = height to middle of clay layer

e = void ratio of clay

P_o = Effective stress at middle of clay layer

∆P = Load applied above ground surface

P_c = Preconsolidation pressure

Swell Index (if not given)

C_s = C_c / 5

Compression Index

C_c = 0.009 (LL - 10)

C_c = (e₁ - e₂) / log (P₂ / P₁)

Overconsolidation Ratio

OCR = P_c / P_o

Coefficient of Compressibility

a_v = (e₁ - e₂) / (P₂ - P₁)

Coefficient of Volume Compressibility

m_v = a_v / (1 + e_ave)

Coefficient of Consolidation

C_v = H_dr² T_v / t

H_dr = drainage height*

T_v = time factor for consolidation

t = time of consolidation

* Single drainage: H_dr = H; Double drainage: H_dr = H / 2

Hydraulic Conductivity

k = C_v m_v γ_w

Mohr’s Circle for Stresses in Soil

C = ½ (σ_x + σ_y)

R = √ (½ (σ_x - σ_y)² + τ_xy²)

σ₁ = C + R; σ₂ = C - R; τ_max = R

C = center of circle

R = radius

σ_x and σ_y = stress along x-face and y-face

σ₁ and σ₂ = principal normal stresses

Sign Convention: (+) = Compression, CCW shear; (-) = Tension, CW Shear

Triaxial Test

σ₃ + ∆σ = σ₁

θ = 45° + Φ/2

σ₃ = minimum principal stress (confining pressure)

∆σ = additional /deviator stress

σ₁ = maximum principal stress

θ = angle of failure in shear

Φ = angle of internal friction

Lateral Earth Pressure: Compoments

P_s = kγ’h

P_c = 2c√k

P_q = kq

P_w = γ_wh_w

Lateral Earth Pressure: Active Soil Pressure

Active → away from soil

Active: (-) P_c

Lateral Earth Pressure: Passive Soil Pressure

Passive → push towards soil

Passive: (+) P_c

Lateral Earth Pressure: At-Rest Earth Pressure

k_o = 1 - sinΦ

Terzaghi’s Bearing Capacity: General Shear Failure

Circular Footing: q_ult = 1.3cN_c + qN_q + 0.3γBN_γ

Square Footing: q_ult = 1.3cN_c + qN_q + 0.4γBN_γ

Strip Footing: q_ult = cN_c + qN_q + 0.5γBN_γ

Terzaghi’s Bearing Capacity: Local Shear Failure

Circular Footing: q_ult = 1.3c’N_c + qN_q’ + 0.3γBN_γ’

Square Footing: q_ult = 1.3c’N_c + qN_q’ + 0.4γBN_γ’

Strip Footing: q_ult = c’N_c + qN_q’ + 0.5γBN_γ’

c’ = 2/3 × c

Piles in Sand

Q_friction = (A_{pressure diagram}) k tanα P

Q_tip = q N_q A_tip

Q_total = Q_friction + Q_tip

D_c = Critical Depth = 10 x (size of pile) for loose sand; 20 x (size of pile) for dense sand

P = Perimeter of pile

k = coefficient of lateral pressure

N_q = soil bearing factor

Piles in Clay

Q_friction = α C_u P L

Q_tip = c_tip N_c A_tip

Q_total = Q_friction + Q_tip

α = friction factor = f / c

f = adhesion between pile and soil

c = cohesion

P = perimeter of pile

L = embedded length of pile

N_c = soil bearing factor

Braced Cuts

Soft to Medium Clay: γH / c > 4

Stiff Clay: γH / c < 4

S = M_max / σ_all

M_max = wL² / 8 from shear/moment diagram

Infinite Slope: Factor of Safety

c = cohesion

ß = angle of backfill from horizontal

Φ = angle of internal friction

H = thickness of soil layer

Fixed Slope

F_f = Friction Force

F_c = Cohesion Force

W = weight of soil above trial plane

c = cohesion

Φ = angle of internal friction