Soil Mechanics Toolkit

Laboratory test analysis and engineering calculations for soil mechanics courses. Each module shows the full calculation path.

Developed by Dr Ivan Gratchev, Associate Professor of Geotechnical Engineering. Worked examples and explanations: Soil Mechanics and Engineering Geology on YouTube.

e = wG_s/SPhase relationships

Water content, void ratio, porosity, saturation, unit weights and densities from three known parameters.

D₆₀ / D₁₀Grain size distribution

Sieve analysis: percent passing, gradation curve, D₁₀, D₃₀, D₆₀, C_u and C_c, sent straight to the classifier.

LL · PL · PIAtterberg limits

Liquid limit from Casagrande cup or fall cone readings, plastic limit trials, plasticity index and flow curve.

GP-GMUSCS classification

USCS group symbol and group name (ASTM D2487) from gradation and Atterberg limits.

A · LI · S_t · D_rSoil indices

Activity, liquidity and consistency index, sensitivity and relative density, each placed on its classification scale.

σ′ = σ − uIn-situ stresses

Total and effective vertical stress, pore water pressure and horizontal stresses in a layered profile.

Δσ_z = qIStress under loads

Vertical stress increase below point, strip, circular and rectangular loads (Boussinesq), compared with the 2:1 method.

σ₁, σ₃, τ_maxMohr circle

Stresses on any inclined plane of a soil element, principal stresses and the pole method.

w_opt · γ_d,maxCompaction

Proctor test analysis from laboratory readings, with zero-air-voids line and field compaction control.

k = QL / AhtPermeability

Constant-head and falling-head tests, and equivalent permeability of layered soils.

σ′ = γ′z ∓ izγ_wSeepage and effective stress

Upward and downward flow through a soil layer below a lake or water table, or a laboratory column: critical gradient, heave and boiling, flow rate.

S = C_cH/(1+e₀)·logConsolidation settlement

Primary settlement of normally and overconsolidated clay, and time–settlement from T_v–U.

τ_f = c′ + σ′ tan φ′Shear strength

Direct shear and triaxial (CD, CU, UU) results: failure envelope, c′ and φ′, with Mohr circles.

How to use this toolkit

Formulas in number boxes. Start with = to calculate, as in a spreadsheet: =7-1 gives 6, =pi*10^2/4 gives 78.54. Press Enter or move to the next box. Formulas can use + − × ÷ ^, brackets, pi, sqrt(), log() and ln().
One unit switch. Choose SI or US customary units at the top of the page; all modules follow, entered values are converted, and the choice is remembered on this device.
Copy your work. “Copy results and calculation path” in each module copies the results and every step as text for reports and assignments.
Linked modules. Send results onwards with one click: grain size and Atterberg limits to the classifier, and stress under loads to consolidation.
Report an error. If a result looks wrong, use “Report it” at the bottom of the module. Paste the copied calculation path so the problem can be checked.
Theory and video. Open “Theory and video” in any module for the key equations and a link to video explanations.

Grain Size Distribution

Sieve analysis from the mass retained on each sieve: percent passing, gradation curve, characteristic sizes and coefficients of uniformity and curvature.

Theory and video

Sieve analysis separates the coarse particles by size. The percent passing each sieve is 100 minus the cumulative percent retained.
The gradation curve is plotted against the logarithm of particle size. D₁₀, D₃₀ and D₆₀ are the sizes at which 10, 30 and 60% of the soil (by mass) is finer.
Coefficient of uniformity C_u = D₆₀/D₁₀ measures the range of sizes; coefficient of curvature C_c = D₃₀²/(D₁₀D₆₀) measures the shape of the curve.
Well-graded soils (C_u ≥ 4 for gravel or ≥ 6 for sand, and 1 ≤ C_c ≤ 3) contain a good spread of sizes and compact to a dense state.
When more than about 10% passes the 75 µm sieve, the fine fraction is analysed with a hydrometer test.

Videos

Sieve analysis: testing procedure and data analysis

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Test setup

Sieve set Total dry mass of sample (g)

Leave the total mass blank to use the sum of the masses retained (including the pan). If entered, the difference is reported as mass loss. Sieve sizes can be edited; list them from largest to smallest.

Sieve data

Leave a mass blank (or 0) if nothing was retained.

Gradation curve

Percent passingD₁₀, D₃₀, D₆₀

Calculation path

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Atterberg Limits

Liquid limit from a multipoint Casagrande cup or fall cone test, plastic limit from thread-rolling trials, and the plasticity and liquidity indices.

Theory and video

As water content increases, a fine-grained soil passes from solid to semi-solid, plastic and liquid states. The boundaries are the shrinkage, plastic and liquid limits.
Liquid limit (LL): the water content at which a groove closes after 25 blows of the Casagrande cup, or at which the 80 g cone penetrates 20 mm.
Plastic limit (PL): the water content at which a 3 mm thread of soil begins to crumble when rolled.
Plasticity index PI = LL − PL is the range of water content over which the soil behaves plastically. LL and PI locate the soil on the plasticity chart.

Videos

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Test setup

Liquid limit method

Casagrande cup Fall cone

Water content entered as

Water content (%) Masses of containers (g)

Liquid limit test

Plastic limit trials Non-plastic (a thread cannot be rolled, or PL ≥ LL)

Natural water content, w (%), optional

Flow curve

Plasticity chart

Calculation path

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USCS Soil Classifier

Group symbol and group name per ASTM D2487 from sieve analysis and Atterberg limits. The classification path is shown step by step.

Theory and video

More than 50% retained on the 75 µm (No. 200) sieve: coarse-grained soil (gravel G or sand S, depending on which fraction is larger). Otherwise: fine-grained soil.
Coarse soils with less than 5% fines are classified on grading (W or P); with more than 12% fines, on the plasticity of the fines (M or C); with 5–12% fines, a dual symbol is used.
Fine-grained soils are classified on the plasticity chart: above the A-line clay (C), below it silt (M); LL below 50 low plasticity (L), 50 or more high plasticity (H).
The group name adds the secondary components: “with sand/gravel” for 15–29% and “sandy/gravelly” for 30% or more (fine soils), or “with sand/gravel” at 15% or more (coarse soils).

Videos

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Classification path

Plasticity chart

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Phase Relationships

Enter any three independent parameters. The remaining volume–weight relationships are calculated and each equation is shown.

Theory and video

Soil is a three-phase material: solids, water and air. Mass and volume ratios between the phases describe its state.
Water content w = M_w/M_s; void ratio e = V_v/V_s; porosity n = V_v/V = e/(1 + e); degree of saturation S = V_w/V_v.
Key relationships: Se = wG_s; γ = (G_s + Se)γ_w/(1 + e); γ_d = G_sγ_w/(1 + e) = γ/(1 + w).
Any three independent quantities (or two, if G_s is assumed) define the state completely. A phase diagram drawn for V_s = 1 or V = 1 makes the relationships easy to derive.

Videos

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Mode

Known parameters Textbook problems

Enter three known parameters to obtain the remaining soil properties.

Units

SI (kN/m³) US customary (pcf)

Known parameters (choose three)

Parameter 1 Value

Parameter 2 Value

Parameter 3 Value

Some combinations are not independent, e.g. e with n, or γ, γ_d and w together. The page says when to choose another parameter.

Quick estimate Assume G_s (no G_s data available)

Assumed G_s

Typical values: 2.65 for quartz sands, 2.70–2.80 for clays. Only two parameters are then needed.

Calculation path

Phase diagram

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In-situ Stresses

Total and effective vertical stress, pore water pressure and horizontal stresses in a layered soil profile under geostatic, hydrostatic conditions.

Theory and video

Total vertical stress is the weight of everything above a point: σ_v = Σγh, using γ above the water table and γ_sat below it.
Pore water pressure under hydrostatic conditions: u = γ_w(z − z_w) below the water table.
Terzaghi's principle of effective stress: σ′ = σ − u. Effective stress controls the strength and compressibility of soil.
Horizontal effective stress at rest: σ′_h = K₀σ′_v, with K₀ = 1 − sin φ′ (Jaky) for normally consolidated soil, increasing with OCR.

Videos

Stresses in soil: total, vertical and horizontal stresses and pore water pressure

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Units

SI (m, kN/m³, kPa) US customary (ft, pcf, psf)

Layers, top to bottom

γ is used above the water table and γ_sat below it. K₀ = 1 − sin φ′ (Jaky); for OCR > 1, K₀ = (1 − sin φ′)·OCR^{sin φ′}. An entered K₀ overrides both.

Water table and surcharge

Depth to water table below ground surface, z_w (m) Surface surcharge, q (kPa)

Leave z_w blank if the water table is below the profile. Surcharge is optional (uniform, infinite extent).

Point of interest Depth, z (m)

The calculation path is shown for this depth. If blank, the base of the profile is used.

Soil profile

Calculation path

Stress table

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Compaction

Proctor compaction test: compaction curve, optimum water content and maximum dry density or unit weight from laboratory readings or from water content and dry density/unit weight, followed by field compaction control.

Theory and video

Compaction increases the dry density of soil by expelling air, improving strength and reducing compressibility and permeability.
The Proctor test compacts soil at several water contents. Dry density rises to a maximum at the optimum water content (OMC) and then falls as water fills the voids.
The zero-air-voids line (S = 100%) is the theoretical upper limit of dry density at each water content; no test point can lie above it.
Modified Proctor uses about 4.5 times the energy of Standard Proctor, giving a higher maximum dry density at a lower OMC.
In the field, relative compaction RC = ρ_d,field/ρ_d,max is compared with the specification, together with the water content range.

Videos

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Test setup

Test

Units

SI US customary

Report as

Dry density, ρ_d Dry unit weight, γ_d

G_s (for zero-air-voids line)

Dry density (Mg/m³ = t/m³) follows AS 1289, BS 1377 and EN 13286-2. Choose dry unit weight for US practice (ASTM, kN/m³ or pcf); entered values convert automatically.

Test points

Laboratory readings Water content and bulk density/unit weight

Mould volume, V (cm³) Mass of mould, M_m (g)

Common moulds: 944 cm³ (ASTM, 4-in), 1000 cm³ (AS 1289; BS 1377 one-litre), 2124 cm³ (ASTM, 6-in). Container masses for the water content may be in any consistent unit; grams are shown.

Select a point number to show its calculation path. Empty rows are ignored.

Compaction curve

Optimum from

Show lines of constant saturation (S = 80%, 90%)

Selected point (calculation path shown)Other test pointsFitted curveZero air voids (S = 100%)

Calculation path

Field compaction control

Field dry unit weight Field water content, w (%) Required relative compaction (%) Allowed w below OMC (%) Allowed w above OMC (%)

Field values from a sand cone, nuclear gauge or similar test. The required relative compaction and water content limits come from the project specification and differ between projects; the defaults (95%, OMC ± 1.5%) can be changed. The acceptance zone is shaded on the chart.

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Permeability

Coefficient of permeability (hydraulic conductivity) from constant-head and falling-head tests, and the equivalent permeability of layered soils.

Theory and video

Darcy's law: v = ki, where i is the hydraulic gradient (head loss per unit length) and k the coefficient of permeability (hydraulic conductivity).
The constant-head test suits coarse soils with high k; the falling-head test suits fine soils with low k.
k ranges over many orders of magnitude: about 1 to 10⁻³ cm/s for clean sands and gravels, and below 10⁻⁷ cm/s for clays.
In layered soils, horizontal flow is controlled by the most permeable layer and vertical flow by the least permeable one, so k_h > k_v.

Videos

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Units for the laboratory tests

Cross-sections given as Time unit

Lengths and heads in mm, volumes in mL (cm³).

Constant-head test

Coarse-grained soils (sands and gravels)

Volume of water collected, Q (mL) Time of collection, t (s)

Specimen length, L (mm) Head difference, h (mm)

Specimen diameter, D (mm)

Falling-head test

Fine-grained soils (silts and clays)

Standpipe diameter, d (mm) Specimen diameter, D (mm)

Specimen length, L (mm) Elapsed time, t (s)

Initial head, h₁ (mm) Final head, h₂ (mm)

Layered soils

Equivalent permeability for flow parallel (horizontal) and perpendicular (vertical) to the layers

Permeability unit

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Seepage and Effective Stress

Effective stress in a soil layer under a lake or water table with upward (artesian) or downward flow, or in a laboratory soil column; hydraulic gradient, seepage force, critical gradient and the factor of safety against heave or boiling.

Theory and video

Water flowing through soil exerts a seepage force j = iγ_w per unit volume in the direction of flow.
Downward flow increases effective stress (σ′ = γ′z + iγ_wz); upward flow reduces it (σ′ = γ′z − iγ_wz).
When σ′ falls to zero under upward flow, the soil loses its strength: a quick condition or boiling. This occurs at the critical hydraulic gradient i_cr = γ′/γ_w, about 1 for most soils.
Excavations and cofferdams in sand are designed with a factor of safety against boiling, FS = i_cr/i.
In the field, upward flow occurs when a permeable layer below the soil is under artesian pressure: water in a piezometer installed in that layer rises above the lake level or water table. A piezometric level below it causes downward flow.

Videos

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Units

SI (m, kN/m³, kPa) US customary (ft, pcf, psf)

Problem type

Field profile (lake or water table) Laboratory tank

Water above the soil layer

Lake No lake: water table in the soil

Depth of lake water, H_w (m)

Soil layer

Thickness of the soil layer, H (m) γ_sat of soil (kN/m³)

Piezometric level in the permeable layer below, h_p (m above the lake bed) Depth below ground surface, z (m), optional

If z is blank, the base of the soil layer (z = H) is used.

Flow rate (optional)

Permeability, k (m/s) Cross-sectional area, A (m²)

Calculation path

Soil profile

Stress table

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Stress Under Loads

Vertical stress increase in soil below surface loads from elastic theory (Boussinesq), with the 2:1 approximation for comparison.

Theory and video

Boussinesq's solution gives the stresses in a homogeneous, elastic half-space below a point load. Solutions for strip, circular and rectangular loads are obtained by integration.
The stress increase decreases with depth and with distance from the load. Below a square footing it falls to about 10% of q at a depth of about 2B; below a strip, at about 6B.
Stresses below any point can be found by superposition: below a rectangle's centre, the value is four times the corner value of a quarter rectangle.
The 2:1 method spreads the load at 2 vertical to 1 horizontal. It is a quick estimate of the average stress increase, not the value below the centre.

Videos

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Stress increase with depth

Boussinesq (elastic)2:1 method

Calculation path

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Consolidation Settlement

Primary consolidation settlement of a clay layer (normally or overconsolidated) and the time for consolidation from Terzaghi's one-dimensional theory.

Theory and video

When a saturated clay is loaded, excess pore pressure dissipates slowly as water drains out, and the clay settles over time: primary consolidation.
The e–log σ′ curve has a flat recompression part (slope C_r) up to the preconsolidation pressure σ′_p and a steep virgin compression part (slope C_c) beyond it.
OCR = σ′_p/σ′₀. Overconsolidated clay settles much less while the stress stays below σ′_p.
Terzaghi's theory gives the rate: t = T_vH_dr²/c_v, where H_dr is the longest drainage path (H/2 for double drainage).
Secondary compression (creep) after primary consolidation is not included here.

Videos

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Units

Length Stress

Clay layer

Thickness, H (m) Initial void ratio, e₀

Compression index, C_c Recompression index, C_r

Normally consolidated σ′_p given OCR given

Preconsolidation pressure, σ′_p (kPa)

C_r is needed only for overconsolidated clay.

Stresses

σ′₀ at mid-layer given From the soil profile

Initial effective stress at mid-layer, σ′₀ (kPa) Stress increment, Δσ (kPa)

Void ratio – effective stress (log scale)

Calculation path

Time rate of consolidation

Coefficient of consolidation, c_v c_v unit

Double drainage (top and bottom) Single drainage

Time for a given U U after a given time

Average degree of consolidation, U (%) Time unit

Uses the clay thickness H above. Settlement at time t = U × final settlement, when the settlement has been calculated.

Degree of consolidation with time

Calculation path

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Mohr Circle

Normal and shear stresses on any plane through a soil element, principal stresses and planes, and the pole (origin of planes) construction.

Theory and video

The stresses on all planes through a point plot on a circle in σ–τ space, with centre (σ₁ + σ₃)/2 and radius (σ₁ − σ₃)/2.
Principal planes carry no shear stress; the maximum shear stress τ_max = R acts on planes at 45° to them.
A rotation θ of the plane in the soil element corresponds to a rotation 2θ around the circle.
Pole method: a line through a known stress point, parallel to the plane it acts on, meets the circle at the pole. A line from the pole parallel to any plane gives the stresses on that plane.

Videos

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Known stresses

σ_y, σ_x, τ_xy σ₁, σ₃, orientation

σ_y on horizontal plane σ_x on vertical plane

τ_xy on vertical plane (+ counterclockwise)

The horizontal plane carries the same shear magnitude with the opposite sense. Check the arrows on the element diagram.

Stress unit

Inclined plane

θ is measured counterclockwise from the horizontal plane (0–180°).

Sign convention: compression positive; shear positive when it produces counterclockwise rotation of the element, plotted upward on the Mohr circle.

Calculation path

Soil element

Mohr circle and pole construction

Plane through H (horizontal) and V (vertical)Chosen plane, P → QPrincipal planes

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Shear Strength

Mohr–Coulomb strength parameters from direct shear or triaxial test results, with the failure envelope fitted by least squares.

Theory and video

Mohr–Coulomb failure criterion: τ_f = c′ + σ′ tan φ′, where c′ is the effective cohesion and φ′ the effective friction angle.
Direct shear tests force failure on a horizontal plane; triaxial tests apply a cell pressure σ₃ and increase the axial stress until failure, giving a Mohr circle per test.
Drainage condition: CD tests give effective parameters directly; CU tests with pore pressure measurement give both total and effective parameters; UU tests on saturated clay give the undrained strength s_u with φ_u = 0.
The failure plane is inclined at 45° + φ′/2 to the major principal plane.

Videos

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Soil Indices

Activity, liquidity index, sensitivity and relative density. Each card is independent; the Atterberg limits below are shared by activity and liquidity index.

Theory and video

Activity A = PI/(% clay fraction finer than 2 µm) reflects the clay mineral: low for kaolinite, high for montmorillonite (swelling clays).
Liquidity index LI = (w − PL)/PI shows the state at the natural water content; consistency index I_c = (LL − w)/PI = 1 − LI.
Sensitivity S_t is the ratio of undisturbed to remoulded strength; highly sensitive (quick) clays lose most of their strength when disturbed.
Relative density D_r describes how dense a coarse soil is between its loosest (e_max) and densest (e_min) states.

Videos

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Atterberg limits (used for activity and liquidity index)

Liquid limit, LL (%) Plastic limit, PL (%)

Activity

Skempton (1953)

Clay fraction, % finer than 2 µm

Typical activities (Skempton, 1953): kaolinite 0.38, illite 0.9, Ca-montmorillonite 1.5, Na-montmorillonite 7.2.

Liquidity and consistency index

State of a fine-grained soil at its natural water content

Natural water content, w (%)

Sensitivity

Ratio of undisturbed to remoulded undrained strength

Undisturbed strength Remoulded strength

Use the same test and units for both (q_u or s_u, e.g. kPa).

Classification scale

Relative density

Coarse-grained soils

Void ratios Dry unit weights

e e_max e_min

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How to use this toolkit

Gradation curve

Calculation path

Flow curve

Plasticity chart

Calculation path

Classification path

Plasticity chart

Calculation path

Phase diagram

Soil profile

Calculation path

Stress table

Compaction curve

Any point on the curve

Method comparison

Calculation path

Constant-head test

Falling-head test

Layered soils

Calculation path

Soil profile

Stress table

Stress increase with depth

Calculation path

Soil profile

Void ratio – effective stress (log scale)

Calculation path

Degree of consolidation with time

Calculation path

Calculation path

Soil element

Mohr circle and pole construction

Failure envelope

Calculation path

Activity

Liquidity and consistency index

Sensitivity

Relative density