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Active Thrust on a Retaining Wall

Pa=12 Ka γ H2P_a = \tfrac{1}{2}\,K_a\,\gamma\,H^{2}

Soil MechanicsMechanicsTotal Rankine active thrust per unit length of wall from a dry cohesionless backfill, acting at one third of the wall height above the base.

At-Rest Earth Pressure Coefficient (Jaky)

K0=1−sin⁡ϕK_0 = 1 - \sin\phi

Soil MechanicsMechanicsJaky's 1944 empirical coefficient of earth pressure at rest for a normally consolidated soil that is not permitted to strain laterally.

Bearing Capacity Factor Nc

Nc=(Nq−1)cot⁡ϕN_c = (N_q - 1)\cot\phi

Soil MechanicsMechanicsPrandtl's cohesion bearing capacity factor Nc derived from Nq and the friction angle, tending to 5.14 as the friction angle goes to zero.

Bearing Capacity Factor Nq

Nq=eπtan⁡ϕ tan⁡2 ⁣(45∘+ϕ2)N_q = e^{\pi\tan\phi}\,\tan^{2}\!\left(45^\circ + \frac{\phi}{2}\right)

Soil MechanicsMechanicsPrandtl–Reissner surcharge bearing capacity factor Nq from the friction angle, the value tabulated by Meyerhof, Hansen and Vesic.

Consolidation Settlement of Normally Consolidated Clay

Sc=Cc H1+e0log⁡10 ⁣σf′σ0′S_c = \frac{C_c\,H}{1 + e_0}\log_{10}\!\frac{\sigma'_f}{\sigma'_0}

Soil MechanicsStrength of MaterialsPrimary consolidation settlement of a normally consolidated clay layer from its compression index, thickness and the stress increase applied.

Darcy's Law for Groundwater Flow

Q=k i AQ = k\,i\,A

Soil MechanicsFluid MechanicsDarcy's law for laminar flow through soil: discharge equals hydraulic conductivity times hydraulic gradient times gross cross-sectional area.

Degree of Saturation (Se = wGs)

S=w GseS = \frac{w\,G_s}{e}

Soil MechanicsStrength of MaterialsDegree of saturation from water content, specific gravity of solids and void ratio, using the phase identity Se = wGs.

Dry Unit Weight from Gs and Void Ratio

γd=Gs γw1+e\gamma_d = \frac{G_s\,\gamma_w}{1 + e}

Soil MechanicsStrength of MaterialsDry unit weight of a soil from the specific gravity of its solids and its void ratio, the phase-diagram route used to back out e in the lab.

Dry Unit Weight from Moist Unit Weight

γd=γ1+w100\gamma_d = \frac{\gamma}{1 + \dfrac{w}{100}}

Soil MechanicsStrength of MaterialsStrips the pore water out of a measured bulk unit weight to give the dry unit weight used for compaction control and phase work.

Effective Stress (Terzaghi, σ′ = σ − u)

σ′=σ−u\sigma' = \sigma - u

Soil MechanicsMechanicsTerzaghi's effective stress principle: the grain-to-grain stress that controls soil strength equals total stress minus pore water pressure.

Equivalent Horizontal Permeability of Layered Soil

keq=k1H1+k2H2H1+H2k_{eq} = \frac{k_1 H_1 + k_2 H_2}{H_1 + H_2}

Soil MechanicsFluid MechanicsThickness-weighted equivalent permeability for flow parallel to the bedding of two soil layers, the parallel-resistance case of stratified seepage.

Factor of Safety Against Sliding

FS=Wtan⁡δPhFS = \frac{W\tan\delta}{P_h}

Soil MechanicsMechanicsFactor of safety of a retaining structure against base sliding, comparing frictional resistance under its weight with the driving horizontal thrust.

Hydraulic Gradient

i=ΔhLi = \frac{\Delta h}{L}

Soil MechanicsFluid MechanicsHydraulic gradient as the loss of total head divided by the length of the flow path, the dimensionless driving force behind all seepage.

Liquidity Index

LI=w−PLPILI = \frac{w - PL}{PI}

Soil MechanicsStrength of MaterialsLiquidity index locating the natural water content of a clay between its plastic limit and liquid limit, a direct index of consistency.

Mohr–Coulomb Shear Strength

τf=c′+σ′tan⁡ϕ′\tau_f = c' + \sigma'\tan\phi'

Soil MechanicsMechanicsMohr–Coulomb failure criterion giving the shear strength of soil from effective cohesion and the friction mobilised by effective normal stress.

Net Allowable Bearing Pressure

qall=qu−qFSq_{all} = \frac{q_u - q}{FS}

Soil MechanicsMechanicsNet allowable bearing pressure for a footing, the ultimate capacity less the existing overburden, divided by the chosen factor of safety.

Plasticity Index (PI = LL − PL)

PI=LL−PLPI = LL - PL

Soil MechanicsStrength of MaterialsPlasticity index of a fine-grained soil as the liquid limit minus the plastic limit, the width of the moisture range where clay behaves plastically.

Pore Water Pressure (u = γw zw)

u=γwzwu = \gamma_w z_w

Soil MechanicsFluid MechanicsHydrostatic pore water pressure at a point below a static water table, from the depth of water standing above it.

Rankine Active Earth Pressure Coefficient

Ka=tan⁡2 ⁣(45∘−ϕ2)K_a = \tan^{2}\!\left(45^\circ - \frac{\phi}{2}\right)

Soil MechanicsMechanicsRankine coefficient of active earth pressure for a smooth vertical wall retaining level cohesionless backfill that has yielded away from the soil.

Rankine Passive Earth Pressure Coefficient

Kp=tan⁡2 ⁣(45∘+ϕ2)K_p = \tan^{2}\!\left(45^\circ + \frac{\phi}{2}\right)

Soil MechanicsMechanicsRankine coefficient of passive earth pressure, the resistance mobilised when a wall or footing is pushed into level cohesionless soil.

Relative Compaction (Percent Proctor)

R=γd,fieldγd,max×100R = \frac{\gamma_{d,field}}{\gamma_{d,max}}\times 100

Soil MechanicsStrength of MaterialsRelative compaction of placed fill as the field dry unit weight divided by the Proctor maximum dry unit weight, in percent.

Relative Density of a Granular Soil

Dr=emax−eemax−emin×100D_r = \frac{e_{max} - e}{e_{max} - e_{min}}\times 100

Soil MechanicsStrength of MaterialsRelative density of a sand or gravel, placing its in-situ void ratio on the scale between its loosest and densest laboratory states.

Saturated Unit Weight

γsat=(Gs+e) γw1+e\gamma_{sat} = \frac{(G_s + e)\,\gamma_w}{1 + e}

Soil MechanicsStrength of MaterialsUnit weight of a soil whose voids are completely full of water, from the specific gravity of the solids and the void ratio.

Seepage Velocity from Discharge Velocity

vs=vnv_s = \frac{v}{n}

Soil MechanicsFluid MechanicsActual seepage velocity through the pores, obtained by dividing Darcy's fictitious discharge velocity by the porosity of the soil.

Shrinkage: Compacted Volume from Bank Volume

VC=VB(1−Sh100)V_C = V_B\left(1 - \frac{S_h}{100}\right)

Civil & SurveyingSoil MechanicsConverts bank volume into the compacted volume it fills in an engineered embankment, using the soil's percent shrinkage.

Slope Ratio (H:V) to Percent Grade

G=100nG = \frac{100}{n}

Civil & SurveyingSoil MechanicsConverts an embankment slope quoted as n horizontal to one vertical into the equivalent percent grade, and back.

SPT Overburden Correction (Liao–Whitman)

(N1)60=N60paσv′(N_1)_{60} = N_{60}\sqrt{\frac{p_a}{\sigma'_v}}

Soil MechanicsMechanicsCorrects a field SPT blow count to a reference overburden of one atmosphere using the Liao and Whitman square-root factor CN.

Submerged (Buoyant) Unit Weight

γ′=γsat−γw\gamma' = \gamma_{sat} - \gamma_w

Soil MechanicsFluid MechanicsEffective or buoyant unit weight of soil below the water table, the saturated unit weight less the uplift of the water it displaces.

Swell: Loose Volume from Bank Volume

VL=VB(1+S100)V_L = V_B\left(1 + \frac{S}{100}\right)

Civil & SurveyingSoil MechanicsConverts undisturbed bank volume into the loose volume the same soil occupies once excavated, using its percent swell.

Terzaghi Ultimate Bearing Capacity (Strip Footing)

qu=c Nc+q Nq+12 γ B Nγq_u = c\,N_c + q\,N_q + \tfrac{1}{2}\,\gamma\,B\,N_\gamma

Soil MechanicsMechanicsTerzaghi's three-term ultimate bearing capacity of a shallow strip footing, summing the cohesion, surcharge and footing-width contributions.

Time Factor for Consolidation

Tv=cv tHdr2T_v = \frac{c_v\,t}{H_{dr}^{2}}

Soil MechanicsFluid MechanicsDimensionless time factor of Terzaghi consolidation theory, with the coefficient of consolidation entered in m²/s and the longest drainage path.

Time Factor from Degree of Consolidation (U ≤ 60%)

Tv=π4(U100)2T_v = \frac{\pi}{4}\left(\frac{U}{100}\right)^{2}

Soil MechanicsFluid MechanicsTerzaghi's parabolic approximation relating the time factor to the average degree of consolidation, valid for U of 60 percent or less.

Total Vertical Stress (σ = γz)

σv=γz\sigma_v = \gamma z

Soil MechanicsMechanicsTotal vertical stress at depth in a uniform soil layer, the weight of the overburden column standing on one unit of area.

Void Ratio and Porosity (e = n/(1 − n))

e=n1−ne = \frac{n}{1 - n}

Soil MechanicsStrength of MaterialsConverts between void ratio, the void volume per unit of solid, and porosity, the void volume per unit of total soil volume.

Water (Moisture) Content

w=MwMs×100w = \frac{M_w}{M_s}\times 100

Soil MechanicsStrength of MaterialsGravimetric water content of a soil as the mass of pore water divided by the mass of oven-dry solids, expressed as a percentage.