Common Soil Mechanics Equations

Equations Need A Soil Model

Soil-mechanics equations are compact descriptions of a physical model. The equation is only defensible when its drainage condition, stress basis, geometry, parameter source, and unit system match the problem. Before substituting a number, state whether the analysis is total stress or effective stress, short term or long term, and whether the soil is uniform enough for the simplified relationship.

This reference groups common equations by mechanism and adds the checks that prevent the most frequent setup errors. It is useful for FE and PE review, calculation checking, and preliminary educational work. It is not a substitute for project-specific subsurface data or an applicable design standard.

Weight-Volume Relationships

Use a phase diagram with weights on one side and volumes on the other. Air and water occupy the void volume, while solids occupy the solids volume.

Dimensionless quantities must be treated consistently. Water content of 20 percent is 0.20 in an equation. Porosity is always less than 1, while void ratio can exceed 1 in very loose or highly organic material.

Weight-Volume Check

If $G_s=2.70$ and $e=0.65$, the three unit weights are:

The sequence is physically consistent: $\gamma_d<\gamma_{sat}$ and $\gamma^{\prime}<\gamma_{sat}$.

Grain Size And Plasticity

Gradation coefficients are:

$D_{10}$ is the particle size at which 10 percent is finer by mass on the gradation curve. These coefficients support USCS classification of clean sands and gravels but do not replace the full classification sequence.

Plasticity index and the USCS A-line are written as:

Use the fines content, liquid limit, plasticity index, position relative to the A-line, and organic observations together.

Activity is sometimes expressed as $A=PI/(\text{percent clay-size fraction})$. Treat activity as an indicator, not a project design parameter. Mineralogy, stress history, fabric, and pore-fluid chemistry also affect expansive behavior.

Vertical Total And Effective Stress

At depth $z$ in a layered profile:

The effective-stress principle connects pore pressure to soil skeleton stress. An increase in pore pressure under unchanged total stress reduces effective stress. That reduction affects drained shear resistance and can influence settlement, uplift, and stability.

Below the water table, an effective-stress increment can be computed using gamma_prime. Use either total stress minus pore pressure or the submerged-unit-weight path. Using both adjustments subtracts water twice.

For a surface surcharge extending over a very large area, Delta sigma_v is approximately q with depth. Local strip, rectangular, circular, or point loads require an appropriate stress-distribution method and geometry.

Seepage And Hydraulic Head

Total head and hydraulic gradient are:

Velocity head is commonly negligible in soils, leaving elevation head plus pressure head.

Darcy flow, discharge velocity, and seepage velocity are:

Hydraulic conductivity $k$ depends on soil fabric, void ratio, saturation, fluid properties, and test conditions.

For a flow net in isotropic soil:

$N_f$ is the number of flow channels and $N_d$ is the number of equipotential drops. For transformed anisotropic sections, follow the governing method rather than inserting directional conductivities casually.

Critical hydraulic gradient is:

Exit gradients near an excavation or hydraulic structure require geometry-specific evaluation because localized piping can occur before a simple average-gradient check appears critical.

Compaction Relationships

Relative compaction and field dry unit weight are:

Use $w$ as a decimal in the dry-unit-weight equation.

The line of zero air voids can be derived from the saturated unit-weight relationship. It represents full saturation for the selected Gs and should lie above realistic compaction-test points. A field density result must be interpreted with the specified laboratory procedure, oversize correction, moisture criterion, lift thickness, and test-location representativeness.

Relative density for granular soil is:

It is sometimes expressed using dry unit weights in an equivalent form. Relative density and relative compaction describe different reference states.

One-Dimensional Consolidation

For a normally consolidated layer under one-dimensional loading, primary consolidation settlement is often written:

For an overconsolidated soil, use recompression behavior up to preconsolidation stress and virgin compression beyond it.

The time factor relationship is:

The maximum drainage path $H_{dr}$ equals the full layer thickness for single drainage and half the thickness for double drainage. Required degree of consolidation determines $T_v$ through the applicable theoretical relationship.

Stress increase is rarely perfectly uniform through a thick compressible layer. Divide the layer into sublayers where needed and calculate representative initial stress, stress increase, and compressibility for each. Settlement estimates also need immediate and secondary components when those mechanisms are material.

Shear Strength

The drained Mohr-Coulomb relationship is:

Effective normal stress acts on the prospective shear plane. The parameters $c^{\prime}$ and $\phi^{\prime}$ must come from a test or interpretation representative of stress range, drainage, material, and strain level.

An undrained total-stress idealization for saturated clay often uses $\tau_f=s_u$ with $\phi_u=0$. Undrained strength is not a universal soil constant; it depends on stress history, anisotropy, strain rate, sample disturbance, and loading path.

Peak strength may be appropriate for first-time failure in intact material, while residual or softened strength may govern reactivated surfaces. Selecting a strength model is a larger engineering decision than inserting values into the equation.

Lateral Earth Pressure

For Rankine conditions with level cohesionless backfill:

The basic resultants and their locations are:

The model assumes the wall can mobilize the selected movement condition. Sloping backfill, wall friction, cohesion, seismic loading, compaction, line loads, and staged bracing require a method that includes those effects.

Bearing Capacity

A general shallow-foundation form before method-specific correction factors is:

The factors depend on $\phi$ and on the selected bearing-capacity theory. The surcharge $q$ is the overburden pressure at footing level in the basic equation.

Shape, depth, load inclination, base inclination, ground slope, eccentricity, local shear, layering, and groundwater can change the result. Define gross versus net capacity consistently, then distinguish ultimate resistance from allowable pressure.

Settlement and bearing capacity are separate checks. A foundation may have ample shear resistance and still experience excessive total or differential settlement.

Infinite Slope Screening

For a soil mantle of vertical thickness $z$ on a long uniform slope angle $\beta$, one common effective-stress screening form is:

The pore pressure $u$ must correspond to the assumed failure plane.

For dry cohesionless soil, the expression reduces to $FS=\tan\phi^{\prime}/\tan\beta$. The apparent simplicity can be misleading. Infinite-slope analysis is intended for shallow translational movement approximately parallel to the ground. It does not represent deep rotational failure, toe instability, complex stratigraphy, or localized seepage without additional modeling.

Dimensional Checks That Catch Errors

Dimensional consistency is one of the strongest calculation checks.

• Unit weight times height gives stress or pressure.
• Pressure times wall height gives force per unit wall length.
• Force times lever arm gives moment per unit wall length.
• Hydraulic conductivity times gradient times area gives volume per time.
• Settlement has length units.
• Factor of safety, void ratio, porosity, degree of saturation, and earth-pressure coefficients are dimensionless.

Use calculators and spreadsheets to preserve a calculation trail, but keep a sketch and assumptions beside the numbers. The value of a soil-mechanics calculation lies in its model, not its decimal places.