Loksewa 7th Level -- Civil Engineering -- Geotechnical Engineering
Geotechnical Engineering -- Complete Formula Chart & Quick Revision
Every formula from Soil Mechanics and Foundation Engineering in one place -- phase relations, permeability, effective stress, compaction, consolidation, shear strength, bearing capacity, earth pressure, piles, slope stability, and ground improvement. Built to scan fast the night before the exam.
How to use this chart
Every formula is in a table: what it is -> the formula -> what the symbols mean / when to use it. Use the jump chips below or Ctrl+F to find any quantity in seconds. Full explanations of foundations, earth pressure and ground improvement are in the linked chapter posts -- this page is the pure formula reference.
Quick Formula Locator
Void ratio
e = Vv / Vs
Darcy's law
v = k . i
Effective stress
sigma' = sigma - u
Shear strength
tau = c' + sigma'.tan(phi')
Consolidation settlement
Sc = Cc.H/(1+e0) . log(sf'/s0')
Bearing capacity
qu = c.Nc + q.Nq + 0.5.gamma.B.Ngamma
Active earth pressure
Ka = (1-sin phi)/(1+sin phi)
Pile capacity
Qu = Qp + Qs
1. Phase Relations
2. Atterberg Limits
3. Classification
4. Permeability
5. Effective Stress
6. Compaction
7. Consolidation
8. Shear Strength
9. Bearing Capacity
10. Earth Pressure
11. Pile Foundations
12. Slope Stability
13. Ground Improvement
14. Constants to Memorize
15. Units Cheat Sheet
Formula Recall Quiz
01Phase Relations (Index Properties)
Foundation of every numerical question
Soil has three phases: solids, water, air. Weight-volume relationships link them together.
| Quantity | Formula | Notes |
|---|---|---|
| Water content | w = Ww / Ws | Weight of water / weight of solids, expressed as % |
| Void ratio | e = Vv / Vs | Volume of voids / volume of solids |
| Porosity | n = Vv / V | Volume of voids / total volume; n = e/(1+e) |
| Degree of saturation | S = Vw / Vv | 0 for dry soil, 1 (100%) for saturated soil |
| Specific gravity of solids | G = Ws / (Vs.gamma_w) | Typically 2.65-2.75 for most soils |
| Air content | ac = Va / Vv | ac + S = 1 |
| Key relation | S.e = w.G | Links water content, voids and saturation -- memorize this one |
| Bulk (moist) unit weight | gamma = W / V | Total weight / total volume |
| Dry unit weight | gamma_d = Ws / V = gamma / (1+w) | Used in compaction control |
| Saturated unit weight | gamma_sat = (G + e).gamma_w / (1+e) | When S = 1 (all voids filled with water) |
| Submerged (buoyant) unit weight | gamma' = gamma_sat - gamma_w | Used below the water table for effective stress |
| Dry unit weight (in terms of G, e) | gamma_d = G.gamma_w / (1+e) | Alternative form, no water content needed |
| Relative density | Dr = (emax - e) / (emax - emin) | Describes compactness of cohesionless (sandy) soil, 0 to 1 |
Memorize
gamma_w (unit weight of water) = 9.81 kN/m3 (or 1000 kgf/m3 = 1 g/cc). Most formulas above reduce to simple algebra once gamma_w and G are known.
02Atterberg Limits & Consistency
| Quantity | Formula | Notes |
|---|---|---|
| Plasticity index | PI = LL - PL | LL = liquid limit, PL = plastic limit |
| Liquidity index | LI = (w - PL) / PI | LI > 1 = soil behaves like a liquid; LI < 0 = very stiff/brittle |
| Consistency index | CI = (LL - w) / PI | CI = 1 - LI |
| Shrinkage index | SI = PL - SL | SL = shrinkage limit |
| Activity of clay | A = PI / (% clay size fraction, <2 micron) | A < 0.75 inactive, 0.75-1.25 normal, > 1.25 active clay (e.g. montmorillonite) |
Quick states of consistency (increasing water content)
Solid-> SL ->Semi-solid-> PL ->Plastic-> LL ->Liquid
03Soil Classification (USCS / Grain Size)
| Quantity | Formula | Notes |
|---|---|---|
| Uniformity coefficient | Cu = D60 / D10 | Cu > 4 (gravel) or > 6 (sand) indicates well-graded soil |
| Coefficient of curvature | Cc = (D30)^2 / (D10 . D60) | Well graded if Cc is between 1 and 3 |
| A-line (plasticity chart) | PI = 0.73 (LL - 20) | Separates clay (above line) from silt (below line) in USCS |
| U-line (upper limit) | PI = 0.9 (LL - 8) | Practical upper bound of PI-LL data on the plasticity chart |
| USCS symbol | Meaning |
|---|---|
| G / S | Gravel / Sand |
| M / C / O | Silt / Clay / Organic |
| W / P | Well graded / Poorly graded |
| H / L | High plasticity / Low plasticity |
| Example: CH | Clay of high plasticity |
| Example: SW | Well-graded sand |
04Permeability & Seepage
| Quantity | Formula | Notes |
|---|---|---|
| Darcy's law | v = k . i | v = discharge velocity, k = coefficient of permeability, i = hydraulic gradient |
| Flow rate | q = k . i . A | A = cross-sectional area of flow |
| Hydraulic gradient | i = dh / dL | Head loss per unit length of flow path |
| Seepage velocity | vs = v / n | Actual velocity through voids; always greater than discharge velocity v |
| Constant head test | k = (Q.L) / (A.h.t) | Q = total flow in time t, h = constant head, L = sample length -- used for sandy (high-k) soils |
| Falling head test | k = (a.L)/(A.t) . ln(h1/h2) | a = standpipe area, A = sample area -- used for clayey (low-k) soils |
| Equivalent k, flow parallel to layers | kx = (k1.H1 + k2.H2 + ...) / H | Horizontal flow through stratified soil |
| Equivalent k, flow perpendicular to layers | kz = H / (H1/k1 + H2/k2 + ...) | Vertical flow through stratified soil -- always kz < kx for the same layers |
| Flow net discharge | q = k . H . (Nf / Nd) | H = head loss, Nf = number of flow channels, Nd = number of equipotential drops |
05Effective Stress
| Quantity | Formula | Notes |
|---|---|---|
| Principle of effective stress | sigma = sigma' + u | Total stress = effective stress + pore water pressure |
| Effective stress | sigma' = sigma - u | Effective stress controls soil strength and compressibility, NOT total stress |
| Pore water pressure (hydrostatic) | u = gamma_w . hw | hw = depth below water table |
| Total vertical stress | sigma = gamma.z (above WT) + gamma_sat.z (below WT) | Add layer by layer for multiple strata |
| Critical hydraulic gradient | ic = (G - 1) / (1 + e) | When i = ic, effective stress becomes zero -- quicksand condition |
| Factor of safety against boiling / quicksand | FS = ic / i | FS < 1 means quicksand / piping failure risk |
Quicksand -- 5 mark favourite
Quicksand is NOT a soil type, it is a CONDITION: when upward seepage pressure equals the submerged weight of soil, effective stress becomes zero, and the soil loses all shear strength. Occurs typically in fine sands with upward flow (e.g. around sheet piles, near excavations).
06Compaction
| Quantity | Formula | Notes |
|---|---|---|
| Dry unit weight from field data | gamma_d = gamma / (1 + w) | Used to plot the compaction curve (gamma_d vs w) |
| Zero air voids line | gamma_d(zav) = G.gamma_w / (1 + w.G) | Theoretical upper bound curve for S = 100%; the compaction curve always stays below it |
| Relative compaction | RC = (gamma_d field / gamma_d max) x 100% | Field quality control check against Proctor MDD |
OMC and MDD
OMC (Optimum Moisture Content) is the water content at which MDD (Maximum Dry Density) is achieved for a given compactive effort. Standard Proctor uses lower energy than Modified Proctor -- Modified Proctor gives higher MDD and lower OMC.
07Consolidation & Settlement
| Quantity | Formula | Notes |
|---|---|---|
| Compression index (empirical, undisturbed clay) | Cc = 0.009 (LL - 10) | Skempton's correlation, LL in % |
| Settlement, normally consolidated clay | Sc = [Cc.H / (1+e0)] . log10(sigma0' + d.sigma / sigma0') | H = layer thickness, e0 = initial void ratio, d.sigma = stress increase |
| Settlement, overconsolidated clay (recompression) | Sc = [Cr.H / (1+e0)] . log10(sigmaf' / sigma0') | Cr = recompression index, much smaller than Cc |
| Coefficient of volume compressibility | mv = av / (1 + e0) | av = coefficient of compressibility (slope of e vs sigma' curve) |
| Settlement using mv | Sc = mv . d.sigma . H | Quick alternative form for settlement |
| Coefficient of consolidation | Cv = (Tv . H_dr^2) / t | H_dr = length of drainage path (half thickness if double drainage) |
| Time factor (avg. degree of consolidation U) | Tv = (pi/4).(U/100)^2 for U < 60% | Different approximation used for U > 60% |
| Degree of consolidation | U = (Sc at time t) / (Sc final) x 100% | 0% at start, 100% at end of primary consolidation |
Normally vs Over-consolidated
OCR (Over-Consolidation Ratio) = sigma_c' (pre-consolidation pressure) / sigma_0' (present effective stress). OCR = 1: normally consolidated. OCR > 1: over-consolidated (has carried a higher load before, e.g. removed glacier/ overburden).
08Shear Strength of Soil
| Quantity | Formula | Notes |
|---|---|---|
| Mohr-Coulomb failure criterion (total stress) | tau = c + sigma.tan(phi) | c = cohesion, phi = angle of internal friction |
| Mohr-Coulomb (effective stress) | tau = c' + sigma'.tan(phi') | Correct form for long-term / drained strength analysis |
| Relation between principal stresses at failure | sigma1' = sigma3'.tan^2(45 + phi/2) + 2c.tan(45 + phi/2) | Used in triaxial test interpretation |
| Unconfined compressive strength | qu = 2 . c (for phi = 0, i.e. undrained clay) | So undrained shear strength Cu = qu / 2 |
| Sensitivity of clay | St = qu(undisturbed) / qu(remoulded) | High St = soil loses a lot of strength when disturbed (sensitive/quick clay) |
| Angle of failure plane | theta = 45 + phi/2 | Measured from the plane of major principal stress |
| Test | Drainage condition | Gives |
|---|---|---|
| Unconsolidated Undrained (UU) | No drainage allowed at any stage | Undrained strength Cu (phi = 0 in terms of total stress) |
| Consolidated Undrained (CU) | Consolidated first, then sheared undrained | Total and effective stress parameters (with pore pressure measurement) |
| Consolidated Drained (CD) | Fully drained throughout | Effective stress parameters c', phi' -- slowest test |
09Bearing Capacity of Soil
| Quantity | Formula | Notes |
|---|---|---|
| Terzaghi's ultimate bearing capacity (strip footing) | qu = c.Nc + q.Nq + 0.5.gamma.B.Ngamma | Nc, Nq, Ngamma depend only on phi (Terzaghi's bearing capacity factors) |
| Terzaghi's for square footing | qu = 1.3.c.Nc + q.Nq + 0.4.gamma.B.Ngamma | Shape factor applied to cohesion and last terms |
| Terzaghi's for circular footing | qu = 1.3.c.Nc + q.Nq + 0.3.gamma.B.Ngamma | B = diameter for circular footing |
| Net ultimate bearing capacity | qnu = qu - gamma.Df | Subtracts the original overburden pressure |
| Safe bearing capacity | qsafe = qu / FS | FS typically 2.5 to 3 |
| Net safe bearing capacity | qns = qnu / FS | Use for settlement-sensitive design |
| Rankine's minimum depth of foundation | Df = (q/gamma) . [(1-sin phi)/(1+sin phi)]^2 | q = safe bearing capacity of soil |
10Earth Pressure
| Quantity | Formula | Notes |
|---|---|---|
| At-rest coefficient (Jaky's equation) | K0 = 1 - sin(phi) | Normally consolidated soils; no wall movement |
| Rankine's active earth pressure coefficient | Ka = (1 - sin phi) / (1 + sin phi) | Minimum lateral pressure; wall moves away from soil |
| Rankine's passive earth pressure coefficient | Kp = (1 + sin phi) / (1 - sin phi) | Maximum lateral pressure; wall moves toward soil. Kp = 1/Ka |
| Active pressure at depth h (cohesionless) | sigma_a = Ka.gamma.h | Triangular distribution, resultant at H/3 above base |
| Passive pressure at depth h (cohesionless) | sigma_p = Kp.gamma.h | Triangular distribution, resultant at H/3 above base |
| Active pressure, c-phi soil | sigma_a = Ka.gamma.h - 2c.sqrt(Ka) | Tension crack depth zc = 2c/(gamma.sqrt(Ka)) |
| Total active thrust (no surcharge) | Pa = 0.5.Ka.gamma.H^2 | Area of the pressure triangle, per metre length of wall |
| Order to remember | Kp > K0 > Ka | Always true for the same soil |
Rankine vs Coulomb, one line each
Rankine: vertical wall, horizontal backfill, no wall friction -- simple, analytical.
Coulomb: inclined wall/backfill allowed, wall friction (delta) included -- more general, matches field conditions better.
11Pile Foundations
| Quantity | Formula | Notes |
|---|---|---|
| Ultimate pile capacity | Qu = Qp + Qs | Qp = end/point bearing, Qs = skin friction resistance |
| Point bearing resistance | Qp = Ap . qp | Ap = area of pile tip, qp = unit end bearing resistance |
| Skin friction resistance | Qs = As . fs | As = surface area of pile shaft, fs = unit skin friction |
| Allowable pile load | Qa = Qu / FS | FS typically 2.5 |
| Group efficiency (Converse-Labarre, concept) | Eg = Qg(actual) / (n . Qu single) | Eg usually < 1 for closely spaced piles |
12Slope Stability
| Quantity | Formula | Notes |
|---|---|---|
| Factor of safety (general) | FS = resisting force (or moment) / driving force (or moment) | FS > 1 = stable; FS = 1 = at the verge of failure |
| Infinite slope, cohesionless soil, dry | FS = tan(phi) / tan(beta) | beta = slope angle; independent of depth for dry cohesionless soil |
| Infinite slope, c-phi soil | FS = [c + (gamma.z.cos^2(beta)).tan(phi)] / [gamma.z.sin(beta).cos(beta)] | z = depth of failure plane below surface |
| Infinite slope with seepage parallel to slope | FS = (gamma'/gamma_sat) . (tan phi / tan beta) | Seepage roughly halves the factor of safety versus dry conditions |
| Taylor's stability number | Sn = c / (gamma.H.FS) | H = critical height of slope; read from Taylor's charts using phi and slope angle |
13Ground Improvement -- Key Values
| Quantity | Formula | Notes |
|---|---|---|
| Area replacement ratio (stone columns) | as = Ac / A | Ac = stone column area, A = tributary soil area per column |
| Radial consolidation (Barron's theory) | Uh = 1 - exp(-8.Th / mu) | Governs the rate of consolidation with vertical (sand/PVD) drains |
| Combined degree of consolidation | (1 - U) = (1 - Uv)(1 - Uh) | Uv = vertical drainage component, Uh = radial drainage component |
Full explanation of stone columns, preloading, grouting and other techniques is in the dedicated Ground Improvement chapter post.
14Constants & Typical Values to Memorize
| Quantity | Typical value |
|---|---|
| Unit weight of water, gamma_w | 9.81 kN/m3 (1000 kgf/m3, 1 g/cc) |
| Specific gravity of soil solids, G | 2.65 - 2.75 (most inorganic soils) |
| Bulk unit weight of soil | 16 - 20 kN/m3 (typical range) |
| FS for shallow foundation bearing capacity | 2.5 - 3.0 |
| FS for pile capacity | 2.5 |
| FS against sliding (retaining wall) | 1.5 |
| FS against overturning (retaining wall) | 1.5 - 2.0 |
| FS for slope stability (permanent slope) | 1.5 |
| Resultant active earth pressure location | H/3 above the base |
| A-line equation | PI = 0.73 (LL - 20) |
15Units Cheat Sheet (SI)
| Quantity | SI Unit |
|---|---|
| Unit weight (gamma) | kN/m3 |
| Stress / pressure (sigma, u, q) | kN/m2 (kPa) |
| Cohesion (c) | kN/m2 (kPa) |
| Coefficient of permeability (k) | m/s or cm/s |
| Coefficient of consolidation (Cv) | m2/year or cm2/s |
| Angle of friction (phi), wall friction (delta) | degrees |
| Void ratio, porosity, degree of saturation | dimensionless (or %) |
| Bearing capacity factors (Nc, Nq, Ngamma) | dimensionless |
QFormula Recall Quiz -- 20 Questions
Cover the answer, write the formula from memory, then expand to check.
1. Write the formula for void ratio.
e = Vv / Vs
2. Write the key relation linking S, e, w and G.
S.e = w.G
3. Write the formula for submerged unit weight.
gamma' = gamma_sat - gamma_w
4. Write the formula for plasticity index.
PI = LL - PL
5. Write the formula for liquidity index.
LI = (w - PL) / PI
6. Write the uniformity coefficient formula.
Cu = D60 / D10
7. Write Darcy's law.
v = k.i
8. Write the equivalent permeability formula for flow parallel to soil layers.
kx = (k1.H1 + k2.H2 + ...) / H
9. Write the effective stress equation.
sigma' = sigma - u
10. Write the critical hydraulic gradient formula.
ic = (G - 1) / (1 + e)
11. Write Skempton's empirical formula for compression index.
Cc = 0.009 (LL - 10)
12. Write the settlement formula for normally consolidated clay.
Sc = [Cc.H / (1+e0)] . log10(sigma0' + d.sigma / sigma0')
13. Write the Mohr-Coulomb failure criterion in terms of effective stress.
tau = c' + sigma'.tan(phi')
14. Write the relation between unconfined compressive strength and cohesion.
qu = 2c, so Cu = qu/2
15. Write Terzaghi's bearing capacity equation for a strip footing.
qu = c.Nc + q.Nq + 0.5.gamma.B.Ngamma
16. Write Rankine's active and passive earth pressure coefficients.
Ka = (1-sin phi)/(1+sin phi); Kp = (1+sin phi)/(1-sin phi)
17. Write Jaky's formula for the at-rest coefficient.
K0 = 1 - sin(phi)
18. Write the ultimate pile capacity formula.
Qu = Qp + Qs
19. Write the factor of safety formula for an infinite dry cohesionless slope.
FS = tan(phi) / tan(beta)
20. Write the area replacement ratio formula for stone columns.
as = Ac / A
Related chapter posts
This page is the compressed formula reference. For full explanations, diagrams, model answers and mock tests on these topics, see the Foundation Engineering & Earth Pressure post and the Ground Improvement Techniques post on this blog.
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