Advanced Civil Engineering Handbook › Chapter 6

Chapter 6 of 14 · 42 formulas · 80 one-liners

Soil Mechanics & Foundation Engineering — formulas & one-liners

Phase relations, classification, permeability, compaction, consolidation, shear strength, earth pressure, bearing capacity, piles, slopes and soil exploration, with references to IS 1498, IS 2720, IS 6403, IS 1888, IS 1904, IS 2131 and IS 2911.

6.1 Phase Relations, Index Properties & Classification

Void ratio and porosity

e = \frac{n}{1 - n}, n = \frac{e}{1 + e}

e = V_v/V_s; n = V_v/V

n is always < 1; e can exceed 1 (soft clays)

Basic phase relation

S e = w G

S = degree of saturation; e = void ratio; w = water content; G = specific gravity of solids

For a saturated soil S = 1, so e = wG

Bulk unit weight

γ = \frac{( G + S e ) γ _{w}}{1 + e}

γ_w = 9.81 kN/m³

S = 1 gives γ_sat; S = 0 gives γ_d

Submerged (buoyant) unit weight

γ^{'} = γ_{s a t} - γ_{w} = \frac{( G - 1 ) γ _{w}}{1 + e}

γ_sat = saturated unit weight

Relative density (density index)

D_{r} = \frac{e _{ma x} - e}{e _{ma x} - e _{min}} \times 100

e_max, e_min = loosest and densest void ratios; e = natural void ratio

Used for cohesionless soils only

Liquidity and consistency index

I_{L} = \frac{w - w _{P}}{I _{p}}, I_{C} = \frac{w _{L} - w}{I _{p}}

w = natural water content; I_p = w_L − w_P (plasticity index)

I_L + I_C = 1; I_L = 0 at plastic limit and 1 at liquid limit

Uniformity and curvature coefficients

C_{u} = \frac{D _{60}}{D _{10}}, C_{c} = \frac{D _{30}^{2}}{D _{10} D _{60}}

D_x = particle size than which x % of soil is finer

Well graded: gravel C_u > 4, sand C_u > 6, and C_c between 1 and 3 (IS 1498)

IS 1498 particle size ranges: boulder > 300 mm, cobble 80–300 mm, gravel 80–4.75 mm, sand 4.75–0.075 mm, silt 0.075–0.002 mm, clay < 0.002 mm.
A soil is coarse-grained under IS 1498 when more than half of it by weight is retained on the 75 µm sieve.
Fine-grained soils are of low (L), intermediate (I) and high (H) compressibility for liquid limits below 35 %, 35–50 % and above 50 % respectively (IS 1498).
Coarse-grained soils with 5–12 % fines are given dual symbols such as SW-SM or GP-GC.
The A-line of the plasticity chart is I_p = 0.73(w_L − 20); clays plot above it, silts and organic soils below it.
In the Casagrande apparatus the cup falls 10 mm at 2 drops per second, and the liquid limit is the water content at which the groove closes at 25 blows.
Plastic limit is the water content at which a soil thread just begins to crumble when rolled to 3 mm diameter.
Shrinkage limit is the lowest water content at which the soil is still fully saturated; below it there is no further volume change on drying.
Activity (I_p ÷ % clay finer than 2 µm) is below 0.75 for inactive, 0.75–1.25 for normal and above 1.25 for active clays (Skempton).
Kaolinite is a 1:1 clay mineral with low swelling, while montmorillonite is a 2:1 mineral with very high swelling; black cotton soils owe their expansiveness to montmorillonite.

6.2 Permeability, Seepage & Effective Stress

Darcy's law

q = k i A, v = k i

k = coefficient of permeability; i = h/L hydraulic gradient; A = gross area

Valid for laminar flow

Falling head permeameter

k = 2.303 \frac{a L}{A t} lo g_{10} \frac{h _{1}}{h _{2}}

a = standpipe area; A = sample area; L = sample length; h₁, h₂ = heads at start and end of time t

Equivalent permeability of layered soil

k_{h} = \frac{\sum k _{i} H _{i}}{\sum H _{i}}, k_{v} = \frac{\sum H _{i}}{\sum ( H _{i} / k _{i} )}

k_i, H_i = permeability and thickness of layer i

k_h is always greater than k_v

Seepage through a flow net

q = k H \frac{N _{f}}{N _{d}}

H = total head loss; N_f = number of flow channels; N_d = number of equipotential drops

Per unit length of structure

Critical hydraulic gradient (quick condition)

i_{c} = \frac{G - 1}{1 + e} = \frac{γ ^{'}}{γ _{w}}

G = specific gravity; e = void ratio

About 1.0 for most sands; FOS against piping = i_c ÷ exit gradient

Effective stress (Terzaghi)

σ^{'} = σ - u

σ = total stress; u = pore water pressure

The constant head test suits coarse-grained soils and the falling head test suits fine-grained soils of low permeability.
Hazen's formula k = C·D₁₀², with k in cm/s, D₁₀ in cm and C ≈ 100, applies to clean uniform sands.
Typical permeability: clean gravel above 1 cm/s, clean sands about 1 to 10⁻³ cm/s, clays below 10⁻⁷ cm/s.
Seepage velocity equals discharge velocity divided by porosity, so it is always greater than the Darcy velocity.
Permeability varies directly as the unit weight of water and inversely as its viscosity, so k increases with temperature.
Flow lines and equipotential lines cross at right angles, and each field of a flow net is approximately square.
Quick sand is a hydraulic condition, not a soil type: it occurs in cohesionless soil when upward seepage reduces effective stress to zero.
Shear strength and compressibility of soil are governed by effective stress, not total stress.
Raising the water table reduces effective stress, while lowering it increases effective stress and can cause settlement.
Pore water pressure in the capillary zone above the water table is negative, which increases effective stress.

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6.3 Compaction

Dry unit weight from bulk unit weight

γ_{d} = \frac{γ}{1 + w}

γ = bulk unit weight; w = water content (decimal)

Dry unit weight for given air voids

γ_{d} = \frac{( 1 - n _{a} ) G γ _{w}}{1 + w G}

n_a = air voids (decimal); G = specific gravity

n_a = 0 gives the zero-air-voids (100 % saturation) line

Compactive energy per unit volume

E = \frac{W h N _{b} N _{l}}{V}

W = rammer weight; h = drop; N_b = blows per layer; N_l = number of layers; V = mould volume

Standard Proctor ≈ 593 kJ/m³; modified ≈ 2700 kJ/m³ (about 4.5 times)

Standard (light) Proctor test, IS 2720 Part 7: 2.6 kg rammer, 310 mm drop, 3 layers, 25 blows per layer in a 1000 cm³ mould.
Modified (heavy) Proctor test, IS 2720 Part 8: 4.9 kg rammer, 450 mm drop, 5 layers, 25 blows per layer in a 1000 cm³ mould.
Increasing the compactive effort raises the maximum dry density and lowers the optimum moisture content.
Clay compacted dry of optimum has a flocculated structure with higher strength, higher permeability and more swelling.
Clay compacted wet of optimum has a dispersed structure with lower permeability, more shrinkage and better flexibility.
Sheepsfoot rollers suit cohesive soils, vibratory rollers suit cohesionless soils, and smooth-wheel rollers are used for finishing and granular bases.
Cohesionless soils are best compacted by vibration and are controlled by relative density rather than a Proctor curve.

6.4 Consolidation, Stress Distribution & Settlement

Consolidation settlement of NC clay

S_{c} = \frac{C _{c} H}{1 + e _{0}} lo g_{10} \frac{σ _{0}^{'} + Δ σ}{σ _{0}^{'}}

C_c = compression index; H = clay thickness; e₀ = initial void ratio; σ₀' = initial effective stress at mid-depth; Δσ = stress increase

Empirical compression index (Terzaghi–Peck)

C_{c} = 0.009 (w_{L} - 10)

w_L = liquid limit in %

For undisturbed normally consolidated clays of low to medium sensitivity; Skempton's 0.007(w_L − 10) applies to remoulded clays

Coefficient of volume compressibility

m_{v} = \frac{a _{v}}{1 + e _{0}}, a_{v} = \frac{Δ e}{Δ σ ^{'}}

a_v = coefficient of compressibility

Settlement S_c = m_v Δσ' H

Coefficient of consolidation

c_{v} = \frac{k}{m _{v} γ _{w}}

k = permeability

Time factor

T_{v} = \frac{c _{v} t}{d ^{2}}

d = length of drainage path (H/2 for double, H for single drainage); t = time

Time factor for U up to 60 %

T_{v} = \frac{π}{4} U^{2}

U = average degree of consolidation (decimal)

For U > 60 %: T_v = 1.781 − 0.933 log₁₀(100 − U%)

Boussinesq vertical stress under a point load

σ_{z} = \frac{3 Q}{2 π z ^{2}} [\frac{1}{1 + ( r / z ) ^{2}}]^{5/2}

Q = point load; z = depth; r = radial distance

Directly below the load σ_z = 0.4775 Q/z²

Time to reach a given degree of consolidation varies as the square of the drainage path, so double drainage takes one quarter of the time of single drainage.
Time factor is 0.197 at 50 % and 0.848 at 90 % average consolidation, the basis of the log-time and root-time fitting methods for c_v.
At the instant of loading a saturated clay, the whole applied stress is carried as excess pore water pressure.
The oedometer test (IS 2720 Part 15) uses a load increment ratio of 1, each load being double the previous one.
Preconsolidation pressure is found from the e–log σ' curve by Casagrande's graphical construction.
Over-consolidation ratio is preconsolidation pressure ÷ present effective overburden; it equals 1 for a normally consolidated clay.
Primary consolidation is due to expulsion of pore water; secondary compression is creep of the soil skeleton at constant effective stress.
Immediate settlement governs in sands, while consolidation settlement governs in saturated clays.
Boussinesq's theory assumes a semi-infinite, homogeneous, isotropic, linearly elastic medium; Westergaard's theory suits clays interlaid with thin unyielding layers.
By the 2:1 dispersion method, the stress at depth z under a B × L footing is qBL ÷ [(B + z)(L + z)].
Newmark's influence chart has 200 units, each with an influence value of 0.005.
Under a rigid footing contact pressure is maximum at the edges on clay and maximum at the centre on sand.

6.5 Shear Strength

Mohr–Coulomb failure criterion

τ_{f} = c^{'} + σ^{'} tan ϕ^{'}

c' = effective cohesion; φ' = effective angle of shearing resistance; σ' = effective normal stress

Inclination of failure plane

θ_{f} = 4 5^{\circ} + \frac{ϕ}{2}

Measured from the major principal plane

Principal stress relation at failure

σ_{1} = σ_{3} tan^{2} (4 5^{\circ} + \frac{ϕ}{2}) + 2 c tan (4 5^{\circ} + \frac{ϕ}{2})

σ₁, σ₃ = major and minor principal stresses at failure

Skempton's pore pressure equation

Δ u = B [Δ σ_{3} + A (Δ σ_{1} - Δ σ_{3})]

A, B = pore pressure parameters

B = 1 for saturated soil, 0 for dry soil

Vane shear strength (both ends shearing)

τ = \frac{T}{π D ^{2} ( \frac{H}{2} + \frac{D}{6} )}

T = torque at failure; D = vane diameter; H = vane height

In unconsolidated undrained (UU) tests on saturated clay all Mohr circles have the same diameter, giving φ_u = 0.
UU strength is used for end-of-construction stability of saturated clays; CD strength for long-term stability.
Pore pressure parameter A at failure is about 0.5 to 1 for normally consolidated clay and negative for heavily over-consolidated clay.
Dense sand dilates during shear and shows a peak strength; loose sand contracts and shows no marked peak.
In the direct shear test the failure plane is forced to be horizontal and drainage cannot be controlled.
The unconfined compression test is a triaxial test with zero cell pressure; for saturated clay c_u = q_u ÷ 2.
Standard triaxial specimens have a length-to-diameter ratio of 2, for example 38 mm × 76 mm.
Sensitivity is the ratio of undisturbed to remoulded unconfined strength at the same water content; thixotropy is the regain of strength of a remoulded clay with time.
The vane shear test is used in the field for soft and sensitive clays.
Liquefaction is the loss of strength of loose saturated sand when cyclic loading builds up pore pressure until effective stress becomes zero.

6.6 Lateral Earth Pressure

Earth pressure at rest (Jaky)

K_{0} = 1 - sin ϕ^{'}

φ' = effective angle of shearing resistance

For normally consolidated soil

Rankine active and passive coefficients

K_{a} = \frac{1 - sin ϕ}{1 + sin ϕ} = tan^{2} (4 5^{\circ} - \frac{ϕ}{2}), K_{p} = \frac{1}{K _{a}}

φ = angle of shearing resistance of backfill

Smooth vertical wall, horizontal backfill

Active pressure in c–φ soil

p_{a} = K_{a} γ z - 2 c K_{a}

γ = unit weight; z = depth; c = cohesion

Passive: p_p = K_p γ z + 2c√K_p

Depth of tension crack

z_{c} = \frac{2 c}{γ K _{a}}

Depth where active pressure becomes zero

For φ = 0, z_c = 2c/γ

Total active thrust from a dry cohesionless backfill is ½K_aγH², acting at H/3 above the base of the wall.
A uniform surcharge q adds a constant lateral pressure K_a·q over the full height, with its resultant at H/2.
For submerged backfill the lateral pressure is K_aγ'z plus the full water pressure γ_w z.
For φ = 30°, K_a = 1/3 and K_p = 3; in general K_p/K_a = tan⁴(45° + φ/2).
K_a < K₀ < K_p, and the passive state needs a much larger wall movement than the active state.
Critical height of an unsupported vertical cut in clay is twice the tension crack depth, i.e. 4c/(γ√K_a), or 4c/γ when φ = 0.
Rankine's theory assumes a smooth vertical wall and ignores wall friction; Coulomb's wedge theory allows for wall friction δ and an inclined wall face.

6.7 Bearing Capacity of Shallow Foundations

Terzaghi ultimate bearing capacity (strip footing)

q_{u} = c N_{c} + γ D_{f} N_{q} + 0.5 γ B N_{γ}

c = cohesion; D_f = depth of foundation; B = width; N_c, N_q, N_γ = bearing capacity factors

Square: 1.3cN_c + γD_fN_q + 0.4γBN_γ; circular: 1.3cN_c + γD_fN_q + 0.3γBN_γ (B = diameter)

IS 6403 net ultimate bearing capacity

q_{n u} = c N_{c} s_{c} d_{c} i_{c} + q (N_{q} - 1) s_{q} d_{q} i_{q} + 0.5 γ B N_{γ} s_{γ} d_{γ} i_{γ} W^{'}

q = effective overburden at base; s, d, i = shape, depth, inclination factors; W' = water table factor

Net safe and safe bearing capacity

q_{n s} = \frac{q _{n u}}{F}, q_{s} = \frac{q _{n u}}{F} + γ D_{f}

q_nu = q_u − γD_f; F = factor of safety

F is commonly 2.5–3

Plate load test: settlement of footing on sand

S_{f} = S_{p} [\frac{B ( B _{p} + 0.3 )}{B _{p} ( B + 0.3 )}]^{2}

S_p = plate settlement; B, B_p = footing and plate widths in m

On clay S_f = S_p B/B_p

For φ = 0, Terzaghi's factors are N_c = 5.7, N_q = 1, N_γ = 0, while IS 6403 uses N_c = 5.14.
IS 6403 shape factors: rectangle s_c = s_q = 1 + 0.2B/L, s_γ = 1 − 0.4B/L; square 1.3, 1.2, 0.8; circle 1.3, 1.2, 0.6.
In IS 6403, W' = 1 when the water table stays at or below D_f + B, W' = 0.5 when it is at or above the footing base, and is interpolated between.
General shear failure occurs in dense sand and stiff clay, local shear in loose sand and soft clay, and punching shear in very loose or very compressible soil.
For local shear failure, reduced parameters c_m = (2/3)c and tan φ_m = (2/3)tan φ are used.
In the plate load test (IS 1888) the pit is at least five times the plate width and a seating pressure of 7 kN/m² is applied and released before loading.
Ultimate bearing capacity from a plate load test is taken as equal for the footing on clay but proportional to width on sand.
Base pressure under eccentric load is (P/A)(1 ± 6e/B), so no tension develops when the eccentricity does not exceed B/6.

6.8 Pile Foundations

Ultimate capacity of a pile in clay (IS 2911)

Q_{u} = A_{p} N_{c} c_{p} + \sum α \overset{c}{ˉ}_{i} A_{s i}

A_p = base area; N_c = 9; c_p = cohesion at tip; α = adhesion factor; c̄_i = average cohesion of layer i; A_si = shaft surface area

Ultimate capacity of a pile in sand (IS 2911)

Q_{u} = A_{p} (\frac{1}{2} D γ N_{γ} + P_{D} N_{q}) + \sum K P_{D i} tan δ A_{s i}

D = pile diameter; P_D = effective overburden at tip; K = earth pressure coefficient; δ = pile–soil friction angle

P_D is limited to the value at the critical depth: 15D for φ ≤ 30°, rising to 20D for φ ≥ 40°

Engineering News formula (safe load)

Q_{a} = \frac{W H}{6 ( S + C )}

W = hammer weight (kg); H = drop (cm); S = set per blow (cm); C = 2.5 cm for drop hammer, 0.25 cm for steam hammer

Factor of safety 6 is built in

Group efficiency (Converse–Labarre)

η_{g} = 1 - \frac{θ}{90} \cdot \frac{n ( m - 1 ) + m ( n - 1 )}{m n}

m = rows; n = piles per row; θ = tan⁻¹(d/s) in degrees; d = pile diameter; s = spacing

IS 2911 Part 1 covers concrete piles: Section 1 driven cast in-situ, Section 2 bored cast in-situ, Section 3 driven precast and Section 4 precast piles in prebored holes.
Minimum pile spacing is 2.5 times the shaft diameter for end-bearing piles on a hard stratum and 3 times the diameter for friction piles (IS 2911).
Negative skin friction develops when the soil around a pile settles more than the pile, for example in fresh fill over soft clay or after lowering the water table; it adds load to the pile.
Group efficiency of friction piles in clay is less than 1, but driven piles in loose sand can show efficiency above 1 because driving densifies the sand.
From an initial vertical load test, safe load is the lesser of 2/3 of the load at 12 mm settlement and 1/2 of the load at settlement equal to 10 % of pile diameter (IS 2911 Part 4).
Routine load tests on working piles are carried to 1.5 times the working load.
Under-reamed piles (IS 2911 Part 3) for expansive soils have bulbs of 2.5 times the shaft diameter spaced 1.25–1.5 times the bulb diameter apart.

6.9 Slope Stability & Soil Exploration

Infinite slope in dry cohesionless soil

F = \frac{tan ϕ}{tan i}

φ = angle of shearing resistance; i = slope angle

With seepage parallel to slope: F = (γ'/γ_sat)(tan φ/tan i), about half the dry value

Area ratio of a sampler

A_{r} = \frac{D _{o}^{2} - D _{i}^{2}}{D _{i}^{2}} \times 100

D_o = outer diameter; D_i = inner diameter of cutting edge

Generally 10 % or less for undisturbed samples in soft clays; up to about 20 % is accepted in stiff soils

Taylor's stability number is S_n = c/(FγH), from which the safe height of a slope in cohesive soil is found.
The Swedish slip circle method with φ = 0 gives F = c_u L R ÷ (W x̄); the method of slices extends it to c–φ soils, and Bishop's method gives a higher F than Fellenius' method.
The upstream slope of an earth dam is critical under sudden drawdown, and the downstream slope under steady seepage.
Standard penetration test (IS 2131): 63.5 kg hammer, 750 mm free fall, split-spoon driven 450 mm, and N is the blow count for the last 300 mm.
The SPT N value is corrected first for overburden pressure and then for dilatancy.
Dilatancy correction N' = 15 + ½(N − 15) applies when N exceeds 15 in saturated fine or silty sand.
A good sampler has an inside clearance of 1–3 % and an outside clearance of 0–2 %.
Depth of exploration below a footing should generally be at least 1.5 times its width (IS 1892).
Rock quality designation (RQD) is the total length of core pieces 100 mm or longer as a percentage of the run length; above 90 % is excellent and below 25 % very poor.

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Values follow the current Indian Standards and IRC / Indian Railways documents named in each item. Always check the latest amendment of a code before using a value in design.