Soil Mechanics formula solvers

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A B C D E F G H I L M N P R S T V W

Active Thrust on a Retaining Wall

P_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)

K_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

N_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

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.

Compactive Viscosity of Snow (Kojima)

\eta = \eta_0 \, e^{f \rho_s}

Snow & IceSoil MechanicsStrength of MaterialsThe compactive viscosity of snow, rising exponentially with its own density — Kojima's 1967 relation, in the form Anderson's SNTHERM and most land-surface schemes have used since. It is why snow settles fast on the first night and then almost stops: the act of compacting is what makes it resist compacting.

Compressive Strength of Sintered Snow

\sigma_c = \sigma_i \left( \frac{\rho_s}{\rho_i} \right)^{\! n}

Snow & IceStrength of MaterialsSoil MechanicsUnconfined compressive strength of sintered snow as a power law in density, normalised to solid ice. It is an empirical fit with very wide scatter, not a law, and it leaves out the variable that matters as much as density does: how long the snow has been sitting undisturbed.

Consolidation Settlement of Normally Consolidated Clay

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.

Culmann Planar Wedge Factor of Safety

FS = \frac{2c'\sin\beta}{\gamma H\sin(\beta-\theta)\sin\theta} + \frac{\tan\phi'}{\tan\theta}

Soil MechanicsMechanicsFactor of safety of a rigid triangular wedge sliding on a single plane beneath a steep cut of height H and face angle β — Culmann's 1866 analysis, the oldest slope calculation still in use and the right one for a steep face where the infinite-slope assumption fails.

Darcy's Law for Groundwater Flow

Q = 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.

Debris Flow Impact Pressure

p = a\,\rho\,v^{2}

Soil MechanicsFluid MechanicsHydrodynamic impact pressure of a debris flow on an obstruction, from the bulk density of the mixture and the flow velocity, with an empirical coefficient that carries the whole disagreement in the literature.

Degree of Saturation (Se = wGs)

S = \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

\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

\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)

\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

k_{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 = \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.

Green-Ampt Infiltration Rate

f = K\left(1 + \frac{\psi \, \Delta\theta}{F}\right)

Water & WastewaterCivil & SurveyingSoil MechanicsThe Green-Ampt model: infiltration rate under ponded conditions, derived by applying Darcy's law across a sharp wetting front that advances into the soil like a piston. Unlike Horton's curve it is not a fit — every parameter is a measurable soil property.

Heim Ratio and Runout Distance (Fahrböschung)

L = \frac{H}{\left(H/L\right)}

Soil MechanicsRock MechanicsHorizontal runout of a rock avalanche or debris slide from its vertical drop and the Heim ratio H/L — the fall height divided by the travel distance, also read as the tangent of the travel angle a straight line from crown to toe makes with the horizontal.

Horton Infiltration Rate

f = f_c + (f_0 - f_c) \, e^{-kt}

Water & WastewaterCivil & SurveyingSoil MechanicsHorton's empirical infiltration curve: the rate starts at a dry-soil capacity, decays exponentially as the surface layer saturates, and settles to a steady final rate. It is a fitted description of what soils are observed to do, not a derivation from soil physics.

Hydraulic Gradient

i = \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.

Infinite Slope Factor of Safety with Cohesion and Pore Pressure

FS = \frac{c' + \left(\gamma z\cos^{2}\beta - u\right)\tan\phi'}{\gamma z\sin\beta\cos\beta}

Soil MechanicsMechanicsThe general infinite-slope factor of safety: Mohr-Coulomb strength on a slope-parallel plane at depth z, with cohesion in the numerator and the pore pressure eating the effective normal stress that friction acts on.

Infinite Slope Factor of Safety with Slope-Parallel Seepage

FS = \frac{\gamma'}{\gamma_{sat}}\cdot\frac{\tan\phi'}{\tan\beta}

Soil MechanicsFluid MechanicsFactor of safety of a long shallow slide in cohesionless soil with the water table at the ground surface and seepage running parallel to the slope — the wet-season case, worse than the dry one by the ratio of buoyant to saturated unit weight.

Infinite Slope Factor of Safety, Dry Cohesionless Soil

FS = \frac{\tan\phi'}{\tan\beta}

Soil MechanicsMechanicsFactor of safety of a long shallow slide in dry cohesionless soil, where the whole answer is the ratio of two tangents: the friction angle over the slope angle. Depth and unit weight cancel out completely.

Liquidity Index

LI = \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

\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

q_{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 - 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 = \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

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

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 = \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

D_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.

Rock Avalanche Volume-Mobility Relation

\frac{H}{L} = k\,V^{-n}

Soil MechanicsRock MechanicsThe empirical power law behind large-landslide mobility: the Heim ratio H/L falls as the event volume rises, so bigger rock avalanches travel disproportionately further than friction allows. Both coefficients are regional fits and are entered by the reader.

Rockfall Kinetic Energy at a Barrier

E = \tfrac{1}{2} m v^{2}

Soil MechanicsMechanicsTranslational kinetic energy of a falling or bouncing block at the point it reaches a barrier — the number a rockfall net, fence or embankment is selected against, and the number that barrier ratings are published in.

Saturated Unit Weight

\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

v_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

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 = \frac{100}{n}

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

Snow Avalanche Alpha-Beta Runout (Lied-Bakkehøi)

\alpha = k\,\beta + c

Soil MechanicsRock MechanicsThe alpha-beta statistical model for extreme snow avalanche runout: the runout angle α from the release area to the distal limit is a regression on the beta angle, the angle to the point where the path first flattens to 10°. The coefficients are regional and must be entered.

Snowpack Settlement (Viscous Compaction)

\rho_s = \rho_0 \, \exp\!\left( \frac{\sigma \, t}{\eta} \right)

Snow & IceSoil MechanicsWater & WastewaterHow much a layer of snow densifies under a constant overburden, treating snow as a Newtonian fluid in compaction — Kojima's model, in the form land-surface schemes have used since Anderson wrote SNTHERM. It is the reason a snowpack is shallower in the morning than the depth board said last night, with the same water still in it.

SPT Overburden Correction (Liao–Whitman)

(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

\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

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)

q_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

T_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%)

T_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)

\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 = \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 = \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.