Agricultural Engineering · Engineering Sciences · also Geotechnical Engineering
Soil as an engineering material: its phases, flow of water through it, compaction, consolidation, shear strength and stability.
In the exams: GATE AG Section 4 (soil mechanics asked in 6 of 8 papers 2019–26); ICAR AIEEA-PG Unit IV; NCEES PE soils and foundations
Soil mechanics is the branch of engineering that treats soil as a material: a loose assembly of mineral grains whose pores hold water and air. It asks how much a soil weighs and how much of it is void, how easily water moves through it, how much it squeezes under a load and how long that takes, and how much shear it can carry before it slides. Every answer comes back to one idea, Terzaghi's principle of effective stress: the grains carry only the part of the total stress that the pore water does not.
For an agricultural engineer the subject is not abstract. Farm ponds, earthen check dams, bunds, canal embankments and tank bunds are built of compacted soil and fail by seepage, piping or slope sliding. Silos, godowns and farm buildings need foundations checked for bearing capacity and settlement, retaining walls hold back ramps and drop structures, and drains depend on permeability.
Examinations test the subject heavily through calculation. GATE Agricultural Engineering places soil mechanics under Soil and Water Conservation Engineering and has asked it in most recent papers, especially phase relations, permeability, lateral earth pressure and the factor of safety of an infinite slope. The ICAR AIEEA-PG paper lists soil mechanics in Unit IV. In the United States the NCEES PE Agricultural and Biological Engineering exam draws on soils and foundations for structures and embankments, and licensing examinations in Nepal, the Philippines and Canada include soil mechanics in their syllabi. Standards differ between countries (Indian Standard IS 1498 and IS 2720, ASTM in the United States, British Standard BS 1377 and Eurocode 7 in Europe), so this manual states the physics once and notes where test details differ. The chapters follow the usual textbook order: phases and index properties, water in soil, deformation, then strength and its uses.
Soil is formed by the weathering of rock. Physical weathering breaks rock into gravel, sand and silt that keep the parent minerals; chemical weathering alters feldspars and micas into clay minerals such as kaolinite, illite and montmorillonite. A soil that stays where it formed is residual; one carried away is transported and named by its agent: alluvial, lacustrine, marine, aeolian (such as loess), glacial or colluvial.
A soil sample is idealised as three phases: solids, water and air. The phase diagram separates them into blocks. Volumes are total volume V, volume of solids Vs, volume of water Vw, volume of air Va and volume of voids Vv = Vw + Va. Masses (or weights) are those of solids, water and the total; the mass of air is taken as zero. A soil with Va = 0 is saturated (two phases, solids and water); a soil with Vw = 0 is dry. Most engineering soils above the water table are partly saturated, and all three phases must be tracked.
The whole of phase relations is bookkeeping on this diagram. The quickest method in an examination is to assume Vs = 1 (or total volume = 1), fill in every block from the given ratios, and then read off whatever is asked. Memorised formulas are shortcuts for the same bookkeeping and it is safer to be able to rebuild them.
Void ratio e is the volume of voids divided by the volume of solids, e = Vv/Vs. It can exceed 1; soft clays often have e between 1 and 3. Porosity n is the volume of voids divided by total volume, n = Vv/V, always less than 1 and often written as a percentage. The two are linked: n = e/(1 + e) and e = n/(1 − n). Degree of saturation S is Vw/Vv, from 0 for dry soil to 1 (100%) for saturated soil. Air content ac = Va/Vv, and percentage air voids na = Va/V = n·ac.
Water content (moisture content) w is the mass of water divided by the mass of dry solids, w = Mw/Ms, not by the total mass. It is found by oven-drying a sample at 105 to 110 °C to constant mass; lower temperatures (about 60 °C) are used for soils with gypsum or organic matter, because higher temperatures drive off water of crystallisation or burn organic matter. Water content can exceed 100% in peats and very soft clays.
Specific gravity of solids G (also written Gs) is the density of the solid grains divided by the density of water. Most inorganic soils lie between about 2.6 and 2.8; quartz sand is close to 2.65 and clays are commonly 2.7 to 2.8, while organic soils are lower. It is measured with a density bottle (for fine soils) or a pycnometer, with air removed by vacuum or boiling.
Unit weight γ is weight per unit volume. The bulk unit weight is γ = W/V. The dry unit weight is γd = Ws/V. The saturated unit weight γsat applies when S = 1. The submerged or buoyant unit weight is γ′ = γsat − γw, where γw is the unit weight of water, 9.81 kN/m³ (often rounded to 10 kN/m³ in examinations; always check which the question uses). Density ρ is the mass counterpart, with ρw = 1000 kg/m³ = 1 g/cm³.
With Vs = 1 the phase diagram gives Vv = e, V = 1 + e, Vw = Se, mass of solids = Gρw and mass of water = wGρw. Equating the two expressions for water volume gives the most used identity in the subject: Se = wG. For a saturated soil e = wG.
From the same diagram: bulk unit weight γ = (G + Se)γw/(1 + e); dry unit weight γd = Gγw/(1 + e); saturated unit weight γsat = (G + e)γw/(1 + e); submerged unit weight γ′ = (G − 1)γw/(1 + e). Also γd = γ/(1 + w), which converts a field bulk density to dry density once the water content is known, and γd = (1 − na)Gγw/(1 + wG), which gives the dry unit weight at a stated air-voids content and is the basis of the air-voids lines on a compaction curve.
Density index or relative density Dr describes how dense a coarse soil is relative to its own loosest and densest states: Dr = (emax − e)/(emax − emin), or in terms of dry unit weight Dr = [(γd − γd,min)/(γd,max − γd,min)]·(γd,max/γd). Common descriptive bands are very loose below about 15%, loose 15 to 35%, medium 35 to 65%, dense 65 to 85% and very dense above 85%; different textbooks draw the bands slightly differently. Relative density is meaningful only for free-draining sands and gravels; it is not used for clays.
Typical errors are dividing water mass by total mass instead of dry mass, using n for e, and mixing kN/m³ with g/cm³. A useful check is that γd < γ < γsat for a partly saturated soil, and that S must come out between 0 and 1.
Coarse soils (gravel and sand) are graded by sieving; fine soils (silt and clay) by sedimentation, using a hydrometer or pipette. Sedimentation rests on Stokes' law: a sphere of diameter D settling in a fluid of viscosity μ reaches a terminal velocity v = (γs − γw)D²/(18μ). The law assumes spheres, laminar settling and no interference between particles, and is used for sizes roughly between 0.2 mm and 0.2 µm. The result is an equivalent diameter, not a true size, because clay particles are plates.
Size limits differ between systems. In the Indian Standard (IS 1498) gravel is 80 to 4.75 mm, sand 4.75 to 0.075 mm, silt 0.075 to 0.002 mm and clay finer than 0.002 mm. The Unified Soil Classification System (USCS, ASTM D2487) uses 75 mm, 4.75 mm (No. 4 sieve) and 0.075 mm (No. 200 sieve) and treats all material finer than 0.075 mm as fines, classified by plasticity rather than size.
A grain-size distribution curve plots percentage finer against particle size on a log scale. D10, D30 and D60 are the sizes at which 10, 30 and 60% of the soil by mass is finer. D10 is called the effective size. The coefficient of uniformity is Cu = D60/D10 and the coefficient of curvature (gradation) is Cc = D30²/(D10·D60). In the USCS a gravel is well graded when Cu ≥ 4 and 1 ≤ Cc ≤ 3, and a sand when Cu ≥ 6 and 1 ≤ Cc ≤ 3; the IS system uses the same limits. A soil failing either test is poorly graded; uniformly graded (Cu close to 1) and gap-graded (missing sizes, Cc outside 1 to 3) are both forms of poor grading.
A fine-grained soil changes from solid to semi-solid, plastic and liquid states as its water content rises. The boundaries were defined by Atterberg and standardised by Casagrande. The liquid limit wL is the water content at which the soil just begins to flow; in the Casagrande cup it is the water content at which a standard groove closes over 13 mm (half an inch) after 25 blows, read from a flow curve of water content against log number of blows. The cone penetrometer method, preferred in British and Eurocode practice and also standardised in India, takes the water content at which a standard cone penetrates 20 mm. The plastic limit wP is the water content at which a thread of soil begins to crumble when rolled to 3 mm diameter. The shrinkage limit ws is the water content below which further drying causes no further reduction in volume.
Derived indices follow. The plasticity index Ip = wL − wP is the range of water content over which the soil is plastic. The liquidity index IL = (w − wP)/Ip places the natural water content w on that range: IL near 0 means the soil is at its plastic limit and stiff; IL near or above 1 means a soft, sensitive soil close to flowing. The consistency index Ic = (wL − w)/Ip = 1 − IL. The flow index If is the slope of the flow curve and the toughness index It = Ip/If. Activity A = Ip/(percentage by mass finer than 2 µm) measures how plastic the clay fraction is: kaolinite is inactive (below about 0.75), illite normal (about 0.75 to 1.25) and montmorillonite highly active (above about 1.25, often several). Active clays swell and shrink most.
The shrinkage limit can be computed from the shrinkage ratio R, the ratio of the dry density of the dried pat to the density of water: ws = (1/R − 1/G).
Classification sorts soils into groups with similar engineering behaviour so that experience can be transferred. The USCS and the Indian Standard system (IS 1498, which follows the same logic) first split by the fraction passing 0.075 mm. If more than 50% is retained on the 0.075 mm sieve the soil is coarse-grained: G (gravel) if more of the coarse fraction is gravel, S (sand) otherwise. With less than 5% fines it is W (well graded) or P (poorly graded) on the Cu and Cc tests; with more than 12% fines it is M (silty) or C (clayey) according to where the fines plot on the plasticity chart; between 5 and 12% it takes a dual symbol such as SW-SM.
If 50% or more passes 0.075 mm the soil is fine-grained and is placed on Casagrande's plasticity chart of Ip against wL. The A-line, Ip = 0.73(wL − 20), separates clays (C, above) from silts and organic soils (M and O, below). In the USCS the second letter is L (low plasticity) for wL below 50 and H (high plasticity) for wL of 50 or more. The IS system divides into three bands: L below 35, I (intermediate) from 35 to 50, and H above 50. Organic soils are identified by odour, colour and a large drop in liquid limit after oven drying; highly organic soils and peat are Pt. The U-line, Ip = 0.9(wL − 8), is the practical upper bound of natural soils; points above it should be rechecked. The AASHTO system, used for farm and rural roads, groups subgrade soils from A-1 (best) to A-7 with a group index. Classification tells a designer that a CH clay will be compressible and nearly impermeable, and an SW sand free-draining, but it does not replace tests for a real design.
Water flows through soil from a point of higher total head to a point of lower total head. Total head is the sum of elevation head, pressure head and velocity head; in soil the velocity head is negligible, so total head is elevation plus pressure head. The hydraulic gradient i is the loss of total head per unit length of flow path.
Darcy's law, from Darcy's 1856 experiments on sand filters, states that for laminar flow the discharge velocity is proportional to the gradient: v = ki, and the discharge is q = kiA, where A is the gross cross-sectional area of soil (solids plus voids). The constant k is the coefficient of permeability or hydraulic conductivity, with units of velocity (m/s or cm/s). Because water actually moves only through the voids, the average seepage velocity is vs = v/n, larger than v. It can break down in coarse gravels where flow becomes turbulent.
Permeability ranges over many orders of magnitude. Clean gravel is typically more than about 10⁻² m/s, clean sands about 10⁻³ to 10⁻⁵ m/s, silts about 10⁻⁶ to 10⁻⁸ m/s and intact clays below about 10⁻⁸ or 10⁻⁹ m/s; published bands differ. Permeability rises with grain size (Hazen's empirical rule for clean uniform sands is k in cm/s ≈ C·D10² with D10 in cm and C commonly taken near 100, though reported values of C range widely), with void ratio, and with the unit weight and falling viscosity of the permeant: k = K·γw/μ, where K is the intrinsic permeability of the soil (units m²). Warm water therefore flows faster; lab results are usually corrected to 20 °C.
The constant-head permeameter suits coarse soils. A steady head difference h is kept across a sample of length L and area A, and the volume Q collected in time t is measured: k = QL/(Aht).
The falling-head (variable-head) permeameter suits fine sands, silts and clays, where too little water passes to measure under constant head. Water drops in a standpipe of area a from head h1 to h2 in time t: k = (aL/(At))·ln(h1/h2) = 2.303(aL/(At))·log10(h1/h2). A common slip is using the sample area where the standpipe area belongs. For clays, k is also back-calculated from consolidation tests through k = cv·mv·γw.
Field pumping tests give the average permeability of a large volume of ground. For an unconfined aquifer pumped at a steady rate q, with water-table heights h1 and h2 above the impermeable base in observation wells at radii r1 and r2, k = q·ln(r2/r1)/[π(h2² − h1²)]. For a confined aquifer of thickness b, k = q·ln(r2/r1)/[2πb(h2 − h1)]. Both assume steady radial flow and fully penetrating wells (Dupuit–Thiem). Auger-hole tests measure k at a point below the water table for drainage design.
Natural and compacted deposits are layered, and the effective permeability depends on the flow direction. For flow parallel to the layers the gradient is the same in every layer and the flows add: kh = Σ(ki·Hi)/ΣHi, a thickness-weighted average dominated by the most permeable layer. For flow perpendicular to the layers the flow is the same through each layer and the head losses add: kv = ΣHi/Σ(Hi/ki), a harmonic mean dominated by the least permeable layer. Always kh ≥ kv, and in layered alluvium or compacted fills kh is often several times kv.
This is why a thin clay seam controls vertical drainage through a profile while a thin sand seam controls horizontal seepage under a dam. In drainage design and flow-net construction for anisotropic ground the geometry is transformed by scaling horizontal distances by √(kv/kh), and the net is drawn with the equivalent permeability k′ = √(kh·kv).
For steady two-dimensional flow through a homogeneous, isotropic, saturated soil in which Darcy's law applies and neither soil nor water is compressed, continuity leads to Laplace's equation for the total head h: ∂²h/∂x² + ∂²h/∂z² = 0. Its solution can be drawn as a flow net: two families of curves that cross at right angles. Flow lines are paths of water particles; equipotential lines join points of equal total head. The space between adjacent flow lines is a flow channel carrying equal discharge; the drop in head between adjacent equipotentials is equal.
A good flow net is drawn with curvilinear squares, so that each field has equal mean width and length. Then the seepage per unit width is q = k·H·(Nf/Nd), where H is the total head loss across the structure, Nf the number of flow channels and Nd the number of equipotential drops. The boundaries fix the net: the upstream and downstream bed surfaces of a weir or dam are equipotentials, and an impervious base or the underside of a sheet pile or concrete floor is a flow line.
The flow net also gives pore pressures. At any point the total head is the upstream head minus the number of drops passed times the head per drop (H/Nd); pressure head is total head minus elevation head. Integrating pore pressure along the base of a concrete weir gives the uplift force, which the weight of the floor must resist.
When the soil is anisotropic, horizontal distances are scaled by √(kz/kx) to form a transformed section, the net is drawn on it as for an isotropic soil, and discharge is computed with k′ = √(kx·kz). In an earth dam the top flow line (phreatic line) is found by Casagrande's parabola.
Seepage exerts a drag force on the soil skeleton equal to i·γw per unit volume, in the direction of flow. When flow is upward the seepage force opposes gravity. If the upward gradient reaches the critical hydraulic gradient ic = γ′/γw = (G − 1)/(1 + e), the effective stress becomes zero, a cohesionless soil loses all strength and it boils. This is the quick (quicksand) condition; it is a hydraulic state, not a soil type, and for typical sands ic is close to 1. The factor of safety against boiling at the exit of a flow net is ic divided by the exit gradient, and values of 3 to 4 or more are commonly required because the exit gradient is sensitive to how the net is drawn.
Piping is the progressive backward erosion of soil from the exit point, forming a pipe that grows upstream until the structure fails. It is a leading cause of failure of small earth dams and weirs on permeable foundations. It is resisted by lengthening the seepage path (upstream impervious blankets, cutoffs and sheet piles), by weighting the exit with filters, and by controlling the exit with toe drains and relief wells. Bligh's and Lane's creep methods and Khosla's method are used for weir floors.
A protective filter must be fine enough to stop the base soil washing through and coarse enough to drain freely. Terzaghi's classic criteria are D15 of the filter no more than about 4 to 5 times D85 of the base soil (retention) and D15 of the filter at least about 4 to 5 times D15 of the base soil (permeability). Agencies have refined these rules for different base soils, so the governing standard should be checked in design.
Terzaghi's principle of effective stress states that σ = σ′ + u, where u is the pore water pressure and σ′ is the effective stress. It is the effective stress, not the total stress, that controls the volume change and shear strength of soil. A change in total stress that is carried entirely by the pore water produces no immediate change in strength or volume; a change in pore pressure at constant total stress changes both.
Water carries no shear, so the shear stress on a plane is carried wholly by the skeleton, and shear strength depends on σ′. In partly saturated soils the pore air pressure also matters and Bishop's extended expression, with a parameter χ that depends on the degree of saturation, is used; this manual deals with saturated soils unless stated otherwise. Every later calculation of consolidation, shear strength, earth pressure and slope stability rests on this principle.
In level ground the vertical total stress at depth z is the weight of everything above a unit area: σv = Σγi·zi. Below a static water table the pore pressure is hydrostatic, u = γw·zw, where zw is the depth below the water table. The effective stress is the difference. For a uniform saturated layer with the water table at the surface, σv′ = γsat·z − γw·z = γ′·z, so the submerged unit weight gives the effective stress directly.
Lowering the water table (by pumping, drainage or drought) raises effective stress in the soil that was below it, because pore pressure falls while total stress changes little. That increase in effective stress is why ground pumping and subsurface drainage of soft clays cause settlement. Raising the water table, for example by flooding a field or impounding a reservoir, lowers effective stress and can reduce the strength of slopes and foundations.
Above the water table, water held by capillarity is under negative pressure (suction). In the capillary fringe, if the soil is saturated, u = −γw·hc at the top of the rise, and effective stress is larger than total stress. The height of capillary rise in a tube of diameter d is hc = 4T·cosα/(γw·d); with T about 0.073 N/m for water at 20 °C and α ≈ 0 this gives hc in metres of about 0.03/d, with d in millimetres. In soils Hazen's empirical form hc = C/(e·D10) is used, with C an empirical constant. Fine soils have high capillary rise but slow rates of rise; coarse soils the opposite. Capillarity gives damp sand its apparent cohesion.
When water flows vertically through soil, the pore pressure is no longer hydrostatic. For upward flow with gradient i through a layer, the effective stress at depth z below the top of the layer is σ′ = γ′z − i·γw·z; downward flow gives σ′ = γ′z + i·γw·z. Upward seepage therefore reduces effective stress and, when i reaches γ′/γw, drives it to zero, which is the quick condition of the previous chapter. Downward seepage increases effective stress and is sometimes used deliberately, for instance by drainage pumping, to consolidate soft ground. In examinations draw a sketch with the total stress, pore pressure and effective stress at each boundary, compute each separately and subtract. The commonest errors are using γsat above the water table, forgetting the head in a piezometer when artesian pressure is present, and applying γ′ where the pore pressure is not hydrostatic.
Compaction is the densification of soil by mechanical energy that expels air, not water. It is different from consolidation, which is the slow expulsion of water from a saturated soil under sustained load. Compaction raises shear strength, lowers compressibility and lowers permeability, which is why embankments, pond linings, farm roads and building platforms are compacted in layers.
Proctor showed in 1933 that for a given compactive effort the dry density achieved depends on the water content. At low water content the grains resist rearrangement; adding water lubricates them and dry density rises. Beyond a certain water content the added water takes up space that grains could otherwise occupy and dry density falls. The peak of this curve gives the maximum dry density (γd,max) and the optimum moisture content (OMC). The curve can never cross the zero-air-voids line, γd = Gγw/(1 + wG), which is the dry density at full saturation for each water content; lines of constant air voids, γd = (1 − na)Gγw/(1 + wG), run parallel below it. At optimum most soils still have a few per cent air voids, commonly quoted as about 3 to 8% although this varies.
Greater compactive effort shifts the curve upward and to the left: higher maximum dry density at lower optimum water content. The line joining the peaks for different efforts roughly follows a line of constant air voids. Coarse, well-graded soils reach high dry densities with flat, well-defined peaks; heavy clays give low maximum densities at high optimum water contents; clean uniform sands often give no clear peak and are specified by relative density instead.
The standard Proctor test (ASTM D698; AASHTO T 99) compacts soil in three layers with 25 blows per layer of a 2.5 kg (5.5 lb) rammer falling 305 mm (12 in) into a mould of about 944 cm³, delivering about 600 kN·m/m³ (12,400 ft·lbf/ft³). The modified Proctor test (ASTM D1557; AASHTO T 180), introduced for heavier airfield and highway plant, uses five layers with 25 blows of a 4.5 kg (10 lb) rammer falling 457 mm (18 in), delivering about 2,700 kN·m/m³ (56,000 ft·lbf/ft³), roughly 4.5 times the standard energy. The Indian Standard light compaction test (IS 2720 Part 7) uses a 2.6 kg rammer, 310 mm drop, three layers of 25 blows in a 1000 cm³ mould, and the heavy compaction test (IS 2720 Part 8) a 4.9 kg rammer, 450 mm drop and five layers. Compactive energy per unit volume is computed as (number of blows per layer × number of layers × rammer weight × drop height)/volume of mould.
In the test, each compacted specimen is weighed to give bulk density, a sample is oven-dried for water content, and dry density is computed from γd = γ/(1 + w).
Compaction water content changes clay structure. Dry of optimum a clay has a flocculated structure: higher strength and stiffness at the as-compacted water content, higher permeability, more swelling on wetting and a more brittle response. Wet of optimum it has a more dispersed structure: lower permeability, lower as-compacted strength, more flexibility and less swelling but more shrinkage. For the impervious core of a small dam or a pond liner, clay is usually placed slightly wet of optimum to reduce permeability and the risk of cracking; for a strong subgrade a water content near or slightly dry of optimum is common. Local specifications govern.
Field plant is matched to soil: smooth-wheeled rollers for gravels and for finishing surfaces; pneumatic-tyred rollers for a wide range of soils; sheepsfoot and padfoot rollers for cohesive soils, where they knead the soil and compact from the bottom of the lift upward; vibratory rollers and plate compactors for sands and gravels, where vibration rearranges grains.
Field control compares the dry density achieved with the laboratory maximum. Relative compaction RC = γd,field/γd,max × 100%. Specifications commonly require values in the range of about 90 to 100% of a stated Proctor test, depending on the use and the agency. Field density is measured by the sand-replacement (sand cone) method, the core cutter method in soft cohesive soils, the rubber balloon method, or a nuclear density gauge, with water content by oven drying or a rapid method checked against oven drying.
When a saturated clay is loaded, it can compress only by expelling pore water. Because clay permeability is low, this takes time: months to decades in the field. The process is consolidation. Settlement of a foundation on clay has three parts: immediate (elastic, undrained distortion), primary consolidation (volume change as excess pore pressure dissipates) and secondary compression (creep of the skeleton after excess pore pressure is essentially gone).
The oedometer (consolidometer) test loads a laterally confined specimen, usually doubling the load each day, and records void ratio against effective stress. On a plot of e against log σ′ the curve has a flat recompression part and a steeper virgin compression part. The break marks the preconsolidation pressure σ′p, the largest effective stress the soil has carried; Casagrande's graphical construction estimates it. The slope of the virgin line is the compression index Cc = Δe/Δlog σ′; the unloading–reloading slope is the swelling or recompression index Cs or Cr, typically a small fraction of Cc. For normally consolidated clays of moderate sensitivity Skempton's correlation Cc ≈ 0.009(wL − 10) is widely quoted (with wL in per cent), and Cc ≈ 0.007(wL − 10) for remoulded clay; correlations are estimates, not substitutes for tests.
The overconsolidation ratio OCR = σ′p/σ′0, where σ′0 is the present effective overburden. A normally consolidated clay has OCR = 1; an overconsolidated clay (unloaded by erosion, desiccation or a falling water table) has OCR above 1 and is much stiffer on reloading up to σ′p. Other compressibility measures are the coefficient of compressibility av = −Δe/Δσ′ and the coefficient of volume compressibility mv = av/(1 + e0), the volumetric strain per unit stress increase.
Primary consolidation settlement of a layer of thickness H is computed from the strain: S = H·Δe/(1 + e0). For a normally consolidated clay loaded from σ′0 to σ′0 + Δσ′: S = [Cc·H/(1 + e0)]·log10[(σ′0 + Δσ′)/σ′0]. For an overconsolidated clay whose final stress stays below σ′p: S = [Cr·H/(1 + e0)]·log10[(σ′0 + Δσ′)/σ′0]. If the final stress passes σ′p, the settlement is the recompression part to σ′p plus the virgin part beyond it. With mv, S = mv·Δσ′·H, valid over the stress range for which mv was measured.
Stresses are taken at the middle of the layer, and thick layers are split into sublayers. The stress increase under a finite foundation falls with depth; it is computed from Boussinesq's elastic solution or approximated by the 2:1 method, in which the load spreads at one horizontal to two vertical, so that under a B × L footing at depth z the increase is Δσ = Q/[(B + z)(L + z)].
Secondary compression, after primary consolidation, is estimated with the coefficient Cα = Δe/Δlog t; it is significant in organic clays and peats, which can settle considerably for years after primary consolidation is complete.
Terzaghi's theory (1925) assumes a homogeneous, saturated clay; incompressible water and grains; one-dimensional flow and compression; Darcy's law; k and mv constant over the load increment; and small strains. It gives the governing equation ∂u/∂t = cv·∂²u/∂z², where u is excess pore pressure and cv = k/(mv·γw) is the coefficient of consolidation (units m²/year or mm²/min).
The solution is written in terms of the dimensionless time factor Tv = cv·t/Hdr², where Hdr is the longest drainage path: half the layer thickness if drained at top and bottom (double drainage), and the full thickness if drained on one face only (single drainage). The average degree of consolidation U depends only on Tv for a given initial pore-pressure distribution. For uniform initial excess pore pressure, Tv = (π/4)U² for U up to about 60%, and Tv = 1.781 − 0.933·log10(100 − U%) above that. Two values to remember: Tv = 0.197 for U = 50% and Tv = 0.848 for U = 90%.
Because t = Tv·Hdr²/cv, time for a given degree of consolidation is proportional to the square of the drainage path. Halving the drainage path (double drainage instead of single) cuts the time to a quarter; this is the principle of vertical sand drains and prefabricated wick drains. It also lets a laboratory result be scaled to the field: t_field/t_lab = (Hdr,field/Hdr,lab)², if cv is the same.
cv is found from the oedometer by Casagrande's log-time method (fitting t50) or Taylor's root-time method (t90 found where a line with abscissae 1.15 times those of the initial straight portion cuts the curve).
Soils fail in shear. At a point, the stresses on planes of every orientation are represented by Mohr's circle, drawn from the major and minor principal stresses σ1 and σ3. The centre is at (σ1 + σ3)/2 and the radius (σ1 − σ3)/2. On a plane inclined at θ to the major principal plane, σn = (σ1 + σ3)/2 + [(σ1 − σ3)/2]·cos2θ and τ = [(σ1 − σ3)/2]·sin2θ. The maximum shear stress equals the radius and acts at 45° to the principal planes.
Combining Coulomb's linear strength law with Mohr's failure envelope, and writing strength in terms of effective stress as Terzaghi's principle requires, gives the Mohr-Coulomb criterion τf = c′ + σ′·tanφ′, where c′ is the effective cohesion intercept and φ′ the effective angle of shearing resistance. At failure the circle touches the envelope, and the failure plane is inclined at 45° + φ′/2 to the major principal plane (not at 45°, where shear stress is highest but so is the normal stress). The principal stresses at failure are related by σ′1 = σ′3·tan²(45° + φ′/2) + 2c′·tan(45° + φ′/2); for a cohesionless soil σ′1/σ′3 = tan²(45° + φ′/2) = (1 + sinφ′)/(1 − sinφ′).
c′ and φ′ are fitting parameters over a stress range, not fixed material constants. For normally consolidated clays and clean sands c′ is close to zero. For sands φ′ typically lies between about 28° and 45°, rising with density, angularity and grading.
The direct shear (shear box) test forces failure along a horizontal plane in a split box under a known normal load; plotting peak shear stress against normal stress for several specimens gives c and φ. It is simple, but drainage cannot be controlled or pore pressure measured.
The triaxial test encloses a cylindrical specimen in a membrane in a cell pressure σ3 and loads it axially to failure with a deviator stress σ1 − σ3. Drainage can be allowed or prevented and pore pressure measured. The unconsolidated undrained (UU) test gives the undrained strength cu of saturated clay, with φu = 0: all circles have the same diameter. The consolidated undrained (CU) test with pore-pressure measurement gives c′ and φ′ quickly. The consolidated drained (CD) test, sheared slowly enough for no excess pore pressure, gives c′ and φ′ directly but takes long in clays.
The unconfined compression test is a UU test with σ3 = 0; for saturated clay cu = qu/2, where qu is the unconfined compressive strength. The laboratory or field vane shear test measures cu in soft clays from the torque needed to rotate a four-bladed vane: for a vane of height H and diameter D with both ends shearing, T = π·D²·cu·(H/2 + D/6). Sensitivity St is the ratio of undisturbed to remoulded unconfined strength; clays range from insensitive (St close to 1) through sensitive to quick clays with St above 16, and the band limits differ slightly between classifications.
Whether a soil behaves drained or undrained depends on how fast it is loaded relative to how quickly it can drain. Sands usually drain as fast as they are loaded, so drained (effective-stress) parameters apply. Saturated clays loaded quickly, as by a new embankment or footing, behave undrained, and short-term stability is checked with cu and φ = 0 (total-stress analysis). As excess pore pressures dissipate, the long-term case is checked with c′ and φ′. For a soft clay under a new embankment the end of construction is usually critical; for a cutting in clay the long term is usually critical, because unloading creates negative excess pore pressures that slowly dissipate and strength falls with time.
Skempton's pore-pressure parameters express the pore-pressure change under undrained loading: Δu = B[Δσ3 + A(Δσ1 − Δσ3)]. B is close to 1 for saturated soil and falls sharply with small amounts of air. A depends on the soil and its stress history: roughly 0.5 to 1 at failure for normally consolidated clays, and small or negative for heavily overconsolidated clays, which tend to dilate.
Loose sands and normally consolidated clays contract when sheared; dense sands and heavily overconsolidated clays dilate after an initial contraction. Dilation gives dense sand a peak strength above its constant-volume (critical-state) strength.
Soil behind a wall pushes sideways with a pressure that depends on whether and how the wall moves. If the wall does not move, the soil is in the at-rest state and the lateral effective stress is σ′h = K0·σ′v. For normally consolidated soils Jaky's expression K0 ≈ 1 − sinφ′ is widely used, giving about 0.4 to 0.5 for sands; overconsolidated soils have higher K0.
K0 is the right coefficient for rigid walls propped against movement, such as culvert and bunker walls. If the wall moves away from the soil, the lateral stress falls until the soil behind reaches failure: the active state, with the lowest lateral pressure. If the wall is pushed into the soil, the lateral stress rises until failure in the passive state, with the highest lateral pressure. Very little wall movement is needed to reach the active state; far more is needed to mobilise full passive resistance, so passive resistance is usually reduced in design.
Rankine considered a semi-infinite mass of cohesionless soil with a level surface, a smooth vertical wall and a state of plastic equilibrium throughout. In the active state the vertical stress is the major principal stress and σ′h = Ka·σ′v, where Ka = (1 − sinφ′)/(1 + sinφ′) = tan²(45° − φ′/2). In the passive state the horizontal stress is the major principal stress and Kp = (1 + sinφ′)/(1 − sinφ′) = tan²(45° + φ′/2) = 1/Ka. For φ′ = 30°, Ka = 1/3 and Kp = 3.
For dry cohesionless backfill of unit weight γ and wall height H, the active pressure rises linearly from zero at the top to Ka·γ·H at the base, and the resultant thrust is Pa = ½Ka·γ·H² per metre run, acting at H/3 above the base. The active failure planes are inclined at 45° + φ′/2 to the horizontal; the passive failure planes at 45° − φ′/2.
A uniform surcharge q on the surface adds a constant lateral pressure Ka·q over the full height, giving an extra thrust Ka·q·H acting at H/2. When the backfill contains a water table, the lateral pressure below it is Ka·γ′·z′ (effective) plus the full hydrostatic water pressure γw·z′, because water acts equally in all directions (its K is 1). A wall with a saturated backfill and no drainage can therefore carry roughly twice the thrust of the same wall with a drained backfill, and weep holes and drains behind walls exist for this reason. For layered backfills, Ka changes at each interface, so the pressure diagram has a step there; compute the vertical effective stress continuously and multiply by the Ka of the layer in which the point lies.
For a backfill with a sloping surface at angle β, Rankine's theory gives a thrust parallel to the ground surface with a modified coefficient; Coulomb's wedge theory (1776) further allows for wall friction δ and a battered back. With large wall friction Coulomb's passive values are unsafe, so log-spiral methods are used for passive resistance.
For a soil with cohesion c (use c′ and φ′ for effective-stress analysis, or cu with φu = 0 for undrained clay), Rankine's active pressure becomes σa = Ka·γ·z − 2c·√Ka and the passive pressure σp = Kp·γ·z + 2c·√Kp. Near the top the active expression is negative, implying tension that soil cannot sustain, so cracks open to a depth zc = 2c/(γ·√Ka). For undrained clay (Ka = 1) zc = 2cu/γ.
Integrating the active pressure from the surface to depth H, the net thrust is zero at a depth of 2zc; a vertical cut in cohesive soil should theoretically stand unsupported to a critical height Hc = 4c/(γ·√Ka), or 4cu/γ for undrained clay. In practice cracks fill with water after rain, adding hydrostatic pressure, and the unsupported height is reduced by a factor of safety; many deaths in trench collapses happen in cuts that stood for a time and then failed. Design normally ignores the tension zone and may add water pressure in the cracks.
Retaining walls are checked for sliding (horizontal resisting force against thrust), overturning (resisting moment about the toe against overturning moment), bearing pressure at the base and overall stability. Minimum factors of safety are set by codes and vary by country; values of about 1.5 for sliding and 2 for overturning are common in working-stress practice, while Eurocode 7 uses partial factors instead.
A slope is called infinite when the potential slip surface is parallel to the ground and shallow compared with the slope's length, as on long hillsides, the faces of canal banks and bund surfaces, or where a thin weathered layer rests on firm rock. The factor of safety F is the ratio of available shear strength to the shear stress needed for equilibrium.
For a dry cohesionless soil the result is independent of depth: F = tanφ′/tanβ, where β is the slope angle. A dry sand slope can therefore stand at any angle up to φ′, the angle of repose for loose sand. When the soil is fully submerged with no seepage the same result holds, because γ′ appears both in the driving and the resisting forces.
When water seeps parallel to the slope with the water table at the ground surface, the pore pressure reduces the effective normal stress, and F = (γ′/γsat)·(tanφ′/tanβ). Since γ′/γsat is roughly one half, seepage parallel to the slope roughly halves the factor of safety, which is why saturated slopes slide after prolonged rain and why the downstream faces of earth dams need drains.
For a soil with both c′ and φ′, and a slip plane at depth z (dry, unit weight γ): F = [c′ + γ·z·cos²β·tanφ′]/[γ·z·sinβ·cosβ]. With parallel seepage and the water table at the surface, replace γ by γ′ in the numerator and γsat in the denominator. Here the factor of safety falls with depth, and the depth at which F = 1 is the critical depth; for c′ > 0 a limited depth of soil can stand steeper than φ′.
Slopes of limited height, such as embankments, dams and cuttings, usually fail on a curved surface. For a saturated clay under undrained conditions (φu = 0) the Swedish circle or φ = 0 method takes moments about the centre of a trial circle of radius R: F = cu·La·R/(W·x), where La is the length of the slip arc, W the weight of the sliding mass and x the horizontal distance from the centre to the line of action of W. Many trial circles are analysed to find the minimum F. Taylor's stability charts give the stability number Sn = cu/(F·γ·H) (some texts use its reciprocal) for simple slopes; for a vertical cut in undrained clay Sn is about 0.26, giving a critical height near 3.85cu/γ, lower than the 4cu/γ from Rankine's plane-failure argument.
Culmann's method assumes a plane failure surface through the toe; the critical height of a slope of angle β in a c–φ soil is Hc = 4c·sinβ·cosφ/[γ(1 − cos(β − φ))]. It gives reasonable answers for near-vertical slopes only.
For soils with friction the normal stress on the slip surface must be known, so the sliding mass is divided into vertical slices: the method of slices. The ordinary (Fellenius) method ignores interslice forces: F = Σ[c′·l + (W·cosα − u·l)·tanφ′]/Σ(W·sinα), where α is the base inclination of each slice and l its base length. Bishop's simplified method includes horizontal interslice forces and is more accurate; F appears on both sides and is found by iteration.
For small earth dams and farm ponds the stability cases to check are the end of construction (upstream and downstream faces, total-stress analysis for clay fill), steady seepage with a full reservoir (downstream face, effective stress with pore pressures from the flow net), and rapid drawdown (upstream face). Rapid drawdown is often critical: the water that supported the upstream face is removed quickly while the fill stays saturated, so the driving weight rises without a matching drop in pore pressure.
Stability is improved by flattening slopes, adding a berm or toe weight, lowering the phreatic line with toe drains, chimney drains and filters, compacting fill to specification, and protecting faces from erosion with grass or stone pitching. Minimum factors of safety depend on the loading case and agency; values of about 1.3 to 1.5 for long-term steady seepage are typical, and codes such as IS 7894 in India and agency manuals elsewhere give the required values. Many small-dam failures come from overtopping or piping along outlet pipes.
A shallow foundation (footing or raft) is one whose depth Df is not large compared with its width B; a common working definition is Df ≤ B. Its design must satisfy two conditions: the soil must not fail in shear, and the settlement must be tolerable. The allowable bearing pressure is the lower of the pressure that gives an adequate factor of safety against shear failure and the pressure that keeps settlement within limits.
Terms: the gross ultimate bearing capacity qu is the gross pressure at the base at which the soil fails in shear. The net ultimate bearing capacity qnu = qu − γ·Df is the extra pressure over the existing overburden that causes failure. The net safe bearing capacity is qnu/F, where F is a factor of safety (commonly 2.5 to 3 in working-stress practice), and the gross safe bearing capacity is qnu/F + γ·Df.
Vesic described three failure modes. General shear failure occurs in dense sand and stiff clay: a well-defined slip surface reaches the ground surface, the soil heaves beside the footing and the load–settlement curve shows a clear peak. Local shear failure occurs in medium-dense sand and medium clay: slip surfaces do not fully develop and settlement is large. Punching shear failure occurs in loose sand and soft clay, and where a footing is deep: the footing punches down with little heave.
Terzaghi (1943) extended Prandtl's plasticity solution to a strip footing on a c–φ soil, treating the soil above the base as a surcharge q = γDf. For general shear failure of a strip footing, qu = c·Nc + q·Nq + 0.5·γ·B·Nγ. The three terms are the contributions of cohesion, surcharge and the soil's own weight below the base; Nc, Nq and Nγ are bearing capacity factors that depend only on φ. For square footings Terzaghi used qu = 1.3c·Nc + q·Nq + 0.4γ·B·Nγ, and for circular footings of diameter B, 1.3c·Nc + q·Nq + 0.3γ·B·Nγ.
For a saturated clay under undrained loading (φu = 0), Terzaghi's factors are Nc = 5.7, Nq = 1 and Nγ = 0, so a strip footing has qu = 5.7cu + γDf. Prandtl's exact plasticity value used by Skempton and later writers is Nc = π + 2 = 5.14. Skempton's charts give Nc rising with depth and with footing shape (about 6.2 for a square footing at the surface).
Later general equations (Meyerhof, Hansen, Vesic, and the form in IS 6403 and Eurocode 7) use Nq = e^(π tanφ)·tan²(45° + φ/2) and Nc = (Nq − 1)·cotφ, and multiply each term by shape, depth and inclination factors. Nγ differs more between authors; Vesic's Nγ = 2(Nq + 1)tanφ is widely used. The factors rise steeply with φ, so a few degrees of error in φ change the capacity greatly.
For local shear failure Terzaghi reduced the strength parameters: c_m = (2/3)c and tanφ_m = (2/3)tanφ, then used the same equations with factors for φ_m.
A water table near the foundation reduces bearing capacity because effective unit weight falls from γ to γ′. If the water table is at or above the base, use γ′ in the 0.5γBNγ term, and in the surcharge term for the soil below the water table. If it is at a depth of about B or more below the base, it has little effect. Between these, interpolate; Indian and other codes give reduction factors. For a sand with the water table rising to the ground surface, capacity can fall to roughly half.
Field tests supplement laboratory strength. The plate load test loads a steel plate (commonly 0.3 to 0.75 m square) in a pit at foundation level and records load against settlement. For clay the ultimate bearing capacity is taken as independent of size, qu,footing ≈ qu,plate. For sand it is taken as proportional to width, qu,footing ≈ qu,plate·(Bf/Bp). Settlement on sand is scaled by Terzaghi and Peck's relation Sf = Sp·[Bf(Bp + 0.3)/(Bp(Bf + 0.3))]² with widths in metres. It tests only soil within about twice the plate width.
The standard penetration test (SPT) counts blows N of a 63.5 kg hammer falling 760 mm to drive a split-spoon sampler 300 mm, after a 150 mm seating drive. N values are corrected for overburden and hammer energy and correlated with relative density, φ′ and allowable pressure on sand. Light farm buildings on expansive black cotton soils fail more often by seasonal swelling and shrinkage than by bearing failure, and are founded below the zone of seasonal moisture change or on under-reamed piles.
Treat phase relations as bookkeeping and practise them until they are automatic, because almost every other numerical problem starts with a unit weight or void ratio. Draw the phase diagram, set Vs = 1, and check that S lies between 0 and 1. For effective stress and earth pressure, always sketch the soil profile with total stress, pore pressure and effective stress at each boundary before multiplying by any coefficient. Learn the handful of numbers that recur: Tv = 0.197 and 0.848, Ka = 1/3 at φ = 30°, Nc = 5.7 in Terzaghi's φ = 0 case, and the Proctor test energies. In the laboratory course, do the Atterberg limits, Proctor and permeability tests yourself; the feel of a soil at its plastic limit teaches more than the definition.
A young engineer on a watershed or minor irrigation programme uses this subject on site: choosing borrow soil for a farm pond or check dam from classification and Proctor results, setting lift thickness and moisture for compaction, checking relative compaction with a sand-replacement or core-cutter test, sizing a toe filter so the downstream face does not pipe, and flattening an upstream slope that must survive rapid drawdown. For godowns and small silos the task is to estimate safe bearing pressure from a plate load or SPT result, check the water table in the monsoon, and avoid founding on the seasonal-movement zone of expansive clay. Knowing when to call a geotechnical specialist is part of the job.
With long experience the engineer judges risk rather than solving textbook problems. The decisions are which of hundreds of small dams in a district need spillway enlargement or seepage control after an extreme monsoon, what compaction specification contractors can actually meet with local plant and still give an impervious core, whether a cheaper homogeneous section with a toe drain is safe on a given foundation, and how to write site-investigation scopes that are proportionate to the value and consequence of failure. The senior engineer builds lessons from failure investigations into standard designs and inspection checklists.
Research at this level addresses unsaturated soil mechanics (suction and its effect on strength and volume change, central to agricultural soils above the water table), the mechanics of tillage and traction, compaction of subsoils by heavy machinery and its effect on root growth and drainage, seepage and internal erosion in embankment dams, and the behaviour of expansive and dispersive soils. Methods include constitutive modelling (critical-state models and their extensions), finite-element seepage–deformation coupling, centrifuge modelling and field instrumentation with piezometers and inclinometers. A senior researcher also helps national standards bodies revise test methods and design codes, and connects agricultural soil physics with geotechnical engineering, which often use different vocabulary for the same phenomena.
This manual covers the soil mechanics in an agricultural engineering degree and the exams named. It does not cover deep foundations (piles, under-reamed piles beyond a mention, wells), stress distribution beyond Boussinesq and the 2:1 method in outline, unsaturated soil mechanics beyond a mention, soil dynamics and liquefaction analysis, ground improvement, geosynthetics, or rock mechanics. Bearing capacity factor tables are not reproduced because published Nγ values differ between authors; questions supply the factors they need. Code-specific factors of safety and test details vary by country and edition and should be checked against the current national standard before design.
Original practice questions written to the syllabus and pattern of the exams named in each question's examTags. They are not past papers and do not reproduce any board's questions. Every numerical answer was computed by script; the working is in each explanation.
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