Earth pressure and slope stability

Civil Engineering (B.E./B.Tech) · Geotechnical engineering · free study notes for exams, placements and interviews

In short

Earth pressure is the lateral stress that a soil mass exerts on any structure retaining it, and slope stability is the study of whether an unsupported soil mass will slide along a failure surface. The two are one subject: both ask how far a soil mass is from mobilising its full shear strength, and both are solved by limit-equilibrium analysis. Together they govern retaining walls, basement and excavation supports, bridge abutments, embankments, cuttings, earth dams, hill roads and mine dumps. In Indian practice the analyses follow IS 2720 shear parameters with IS 14458, IRC:6, IRC:75 and IS 7894; they appear every year in GATE Civil, in state PSC/PSU papers (NHAI, RITES, L&T, AFCONS, Tata Projects, Gammon, consultancies such as AECOM and WSP) and in interviews for geotechnical, highway and dam design roles.

Explained

States of stress and wall movement

At rest (K₀): no lateral strain; the soil has never been allowed to expand or compress sideways. Applies to braced basements, unyielding bridge abutments, culvert walls and thick rigid walls on rock.
Active (Kₐ): the wall moves *away* from the soil; the soil expands laterally, mobilises shear strength in its own favour and the thrust falls to a minimum. The soil is the pushing (actuating) agent.
Passive (K_p): the wall is pushed *into* the soil; the soil is compressed laterally and resists, giving the maximum thrust. The soil is the resisting agent.
Order: Kₐ < K₀ < K_p always. For φ′ = 30°, Kₐ = 0.33, K₀ ≈ 0.50, K_p = 3.0.
Movement needed: active requires only about 0.001H (dense sand) to 0.004H (loose sand), and more in clay; full passive needs roughly 0.01H–0.05H. Hence passive resistance is usually taken with a large factor of safety or reduced, because the displacement to develop it is unacceptable for the structure.

Rankine's theory

Assumptions: semi-infinite, homogeneous, isotropic, dry/drained soil; vertical smooth wall (no wall friction, δ = 0); plane failure surfaces; the soil is in a state of plastic equilibrium throughout.
Derivation outline: at failure the Mohr circle touches the strength envelope; with σ′₁ and σ′₃ as the vertical and lateral stresses, σ′₁ = σ′₃ tan²(45 + φ/2) + 2c tan(45 + φ/2). Inverting gives the active case; reversing principal stresses gives the passive case.
Failure planes: active wedge at (45 + φ/2)° to the horizontal; passive wedge at (45 − φ/2)°.
Cohesive soil: the active pressure is negative (tension) down to depth z₀ = 2c/(γ√Kₐ). Soil cannot take tension, so a tension crack forms; the accepted practice is to ignore the negative part and, for a crack that can fill with rainwater, add full hydrostatic pressure over that depth.
Unsupported (critical) height of a vertical cut in clay: H_c = 4c/(γ√Kₐ) = 2z₀; for φ = 0, H_c = 4c_u/γ.
Sloping backfill, surcharge, layers, water: Rankine is extended by (a) a modified Kₐ for backfill slope β, (b) adding Kₐq for a uniform surcharge q, (c) computing the pressure layer by layer using the K of the layer but the σ′_v carried down from above, and (d) using effective stress with γ′ below the water table plus separate full hydrostatic water pressure. Line and point loads are handled by Boussinesq-based charts.

Coulomb's wedge theory and graphical methods

Assumptions: rigid failure wedge sliding on a plane; wall friction δ included; wall face may be inclined; backfill may be sloping; failure surface found by trial to give the maximum active (minimum passive) thrust.
Pₐ acts at δ to the normal to the wall face, i.e. inclined to the horizontal — this is Coulomb's main advantage over Rankine; δ is usually taken as (1/2 to 2/3)φ′, and is zero for a smooth or lubricated face.
Rankine is a special case of Coulomb with δ = 0, vertical wall, plane ground.
Culmann's graphical method: plots the wedge weights on a φ-line and reads the thrust from the maximum ordinate to the "Culmann line"; it handles irregular backfill, surcharges, line loads and layered fill — commonly asked as a procedure question.
Rebhann's construction gives the same result graphically for plane backfill.
Limitation of Coulomb's passive case: the plane-surface assumption badly over-estimates K_p for large δ; curved-surface solutions (Caquot–Kerisel, Sokolovski, log-spiral) are used instead.

Seismic and special loading

Mononobe–Okabe extends Coulomb by adding horizontal (k_h) and vertical (k_v) inertia on the wedge through the seismic inertia angle ψ = tan⁻¹[k_h/(1 − k_v)]; the active thrust rises and the passive resistance falls.
The dynamic *increment* is conventionally applied higher up the wall (around 0.5H–0.67H), so overturning moments increase more than the thrust itself.
Indian codes: IRC:6 gives seismic earth pressure for bridge abutments; the detailed retaining-wall provisions of the older IS 1893:1984 are still widely quoted in textbooks, but IS 1893 (Part 1):2016 restricts itself to buildings — the applicable part/edition should be checked for any live design.
Compaction-induced pressure in the upper fill can exceed K₀ values; swelling/frost pressure from expansive clay backfill is avoided by specifying free-draining granular fill with weep holes and a drainage blanket (IS 14458 Part 2 for hill-road walls).

Types of slopes and modes of failure

Infinite slope: length ≫ depth of slip, failure is a plane parallel to the surface (translational slide, hill soil mantle over rock).
Finite slope: embankments, cuttings, dam faces; failure is usually along a curved (approximately circular) surface.
- Face/slope failure: surface emerges above the toe — steep slopes, strong soil at depth.
- Toe failure: passes through the toe — the common case.
- Base failure: passes below the toe through a weak stratum — flat slopes, soft foundation soil; governed by the depth factor D = depth to hard stratum / slope height.
Other mechanisms: translational slide on a weak seam, wedge failure, flow slide/liquefaction, creep, rotational failure in soft clay beneath an embankment.
Causes of failure: increase in shear stress (steepening, surcharge at crest, toe erosion/excavation, earthquake, construction loads) or decrease in shear strength (rise in pore pressure during monsoon, loss of apparent cohesion on saturation, weathering, progressive failure, strain-softening to residual strength).

Stability analysis methods

Factor of safety definitions: F_s with respect to shear strength (= available strength / mobilised strength, the standard definition); F_c with respect to cohesion; F_φ with respect to friction; F_H with respect to height. Only F_s = F_c = F_φ at the true critical condition.
Total vs effective stress: use total stress (φ_u = 0, c_u) for short-term/end-of-construction stability of saturated clay (undrained), and effective stress (c′, φ′, u) for long-term stability, sand and any case where pore pressures can be estimated. Rapid drawdown and sudden drawdown in dams are undrained-critical cases.
φ = 0 analysis (Swedish circle, total stress): moment of the disturbing weight about the circle centre balanced against c_u × arc × radius; the simplest hand method, exact for undrained clay.
Friction circle method: for c–φ soil with a circular arc; the resultant of the frictional component is assumed tangent to a circle of radius R·sinφ; solved by force triangle.
Method of slices: the wedge is divided into vertical slices; equilibrium is written slice by slice so that varying strength, pore pressure and geometry can be handled.
- Fellenius/Swedish (ordinary) method: inter-slice forces neglected; simple, always conservative (F typically 5–20% low, worse at high pore pressure).
- Bishop's simplified method: inter-slice shear neglected but normal forces retained; satisfies moment equilibrium; needs iteration on F; the industry workhorse, accurate to ~2%.
- Rigorous methods: Spencer, Morgenstern–Price, Janbu generalised and Sarma satisfy both force and moment equilibrium; used in software (GeoStudio SLOPE/W, PLAXIS, GEO5) with non-circular surfaces located by automatic search.
Charts: Taylor's stability number charts (homogeneous slope, no seepage), Bishop–Morgenstern charts (with pore pressure ratio r_u), and Culmann's plane-wedge method for steep slopes and vertical cuts.
Typical required factors of safety (values vary with code, consequence and data quality — verify against the governing code): about 1.3–1.4 for end-of-construction and temporary works, 1.4–1.5 for long-term permanent slopes and embankments, and about 1.1–1.25 for seismic/pseudo-static and rapid drawdown; IS 7894 for earth dams and IRC:75 for high road embankments give the specific values.
Stabilisation measures: flatten the slope or add berms, toe weighting/counterfort berm, surface and sub-surface drainage (the cheapest and most effective single measure), soil nails, anchors, retaining/breast walls, gabions, reinforced soil, lime/cement treatment, vegetation/bio-engineering and instrumented monitoring (inclinometers, piezometers) — guidance in IS 14680 and in MoRTH/IRC hill-road practice.

Key points

K₀ - at-rest coefficient, no lateral strain; Jaky: K₀ ≈ 1 − sin φ′, higher for over-consolidated soil.
Kₐ, K_p - minimum and maximum lateral coefficients; K_p = 1/Kₐ for Rankine with δ = 0, β = 0.
Wall movement - active needs ~0.001H–0.004H, passive 0.01H–0.05H, so passive is rarely fully mobilised.
Rankine - smooth vertical wall, plane failure, thrust parallel to the ground surface; simple and slightly conservative for active.
Coulomb - rigid wedge with wall friction δ, thrust inclined at δ; over-estimates passive pressure badly.
Failure plane - (45 + φ/2)° for active, (45 − φ/2)° for passive, measured from the horizontal.
Tension crack - depth z₀ = 2c/(γ√Kₐ) in cohesive backfill; ignore the negative pressure and allow for water filling the crack.
Critical cut height - H_c = 4c/(γ√Kₐ); for φ = 0, H_c = 4c_u/γ.
Water table - always use γ′ for effective overburden and add the full hydrostatic pressure separately; a submerged backfill more than doubles the thrust, which is why drainage is mandatory.
Surcharge - a uniform q adds a rectangular pressure Kₐq over the full height.
Infinite slope in sand - F = tan φ′/tan β, independent of depth; with seepage parallel to the surface it drops to (γ′/γ_sat)(tan φ′/tan β), roughly halving F.
Angle of repose - the limiting stable angle of a dry cohesionless slope, numerically ≈ φ.
Taylor's stability number - S_n = c/(Fγ H), dimensionless; for φ_u = 0 it ranges from about 0.18 (flat slopes, base failure) to 0.261 (vertical cut).
Depth factor D - decides whether failure is a toe circle or a base (mid-point) circle.
φ = 0 analysis - F = c_u·L_a·R/(W·x); applies to short-term undrained clay stability.
Bishop's simplified method - iterative, satisfies moment equilibrium, 2% accurate; Fellenius is non-iterative but conservative.
Pore pressure ratio - r_u = u/(γh); the single parameter that carries seepage into chart solutions.
Critical condition - clay embankments are usually critical at end of construction (undrained) and cuttings in clay in the long term, as negative pore pressures dissipate.

Formulas and examples

K₀ = 1 − sin φ′ (Jaky, normally consolidated); K₀(OC) ≈ K₀(NC)·√OCR; elastic form K₀ = ν/(1 − ν). ν = Poisson's ratio, OCR = over-consolidation ratio.

Kₐ = (1 − sin φ)/(1 + sin φ) = tan²(45° − φ/2); K_p = (1 + sin φ)/(1 − sin φ) = tan²(45° + φ/2) — dimensionless; φ in degrees.

Cohesionless, level fill: pₐ = Kₐ γ z (kN/m²); Pₐ = ½ Kₐ γ H² (kN per metre run) acting at H/3 above the base. γ = unit weight (kN/m³), H = wall height (m).

Cohesive soil: pₐ = Kₐ γ z − 2c√Kₐ; p_p = K_p γ z + 2c√K_p; z₀ = 2c/(γ√Kₐ); H_c = 4c/(γ√Kₐ). c = cohesion (kN/m²).

Surcharge q (kN/m²): Δpₐ = Kₐ q over the whole height, resultant Kₐ q H at H/2.

Rankine, sloping backfill at β (c = 0): Kₐ = cos β · [cos β − √(cos²β − cos²φ)] / [cos β + √(cos²β − cos²φ)], thrust parallel to the slope.

Coulomb active: Kₐ = sin²(α + φ) / { sin²α · sin(α − δ) · [1 + √( sin(φ + δ)·sin(φ − β) / ( sin(α − δ)·sin(α + β) ) )]² }. α = angle of wall face to the horizontal, δ = wall friction angle, β = backfill slope.

Mononobe–Okabe: ψ = tan⁻¹[k_h/(1 − k_v)]; Pₐₑ = ½ γ H² (1 − k_v) Kₐₑ, with Kₐₑ the Coulomb coefficient computed with φ and the geometry adjusted by ψ. k_h, k_v = horizontal and vertical seismic coefficients.

Infinite slope: cohesionless, dry → F = tan φ′/tan β; with seepage parallel to the surface → F = (γ′/γ_sat)(tan φ′/tan β); c′–φ′ soil, slip at vertical depth H → F = c′/(γ H cos²β tan β) + tan φ′/tan β (replace γ by γ_sat and tan φ′ by (γ′/γ_sat)tan φ′ under steady seepage).

Taylor: S_n = c/(F γ H) → H_c = c/(S_n γ) for F = 1. Culmann (plane wedge): H_c = (4c/γ)·[sin β cos φ/(1 − cos(β − φ))].

φ = 0 circular analysis: F = c_u L_a R/(W x). L_a = arc length = Rθ (θ in radians), x = horizontal lever arm of W from the centre.

Fellenius: F = [c′ L_a + tan φ′ Σ(W cos α − u l)] / Σ W sin α. Bishop simplified: F = Σ{[c′b + (W − u b) tan φ′]/m_α} / Σ W sin α, with m_α = cos α (1 + tan α tan φ′/F) — iterate on F. α = base inclination of a slice, b = slice width, l = base length, u = pore pressure. r_u = u/(γ h).

Example 1 — Rankine active and passive with surcharge. Wall 5 m high, level cohesionless backfill, γ = 18 kN/m³, φ′ = 30°, surcharge q = 20 kN/m², no water table.

Kₐ = tan²30° = 1/3; K_p = 3.
Soil thrust = ½ × (1/3) × 18 × 5² = 75.0 kN/m at 5/3 = 1.667 m above the base.
Surcharge thrust = (1/3) × 20 × 5 = 33.33 kN/m at 2.5 m.
Total Pₐ = 108.33 kN/m; moment about base = 75 × 1.667 + 33.33 × 2.5 = 125.0 + 83.3 = 208.3 kN·m/m → resultant acts at 208.3/108.33 = 1.92 m above the base.
Passive resistance of the same soil = ½ × 3 × 18 × 25 = 675 kN/m, i.e. K_p/Kₐ = 9 times the active value.

Example 2 — cohesive backfill and the tension crack. H = 6 m, γ = 18 kN/m³, c = 15 kN/m², φ = 20°.

Kₐ = tan²35° = 0.490; √Kₐ = 0.700.
z₀ = 2 × 15/(18 × 0.700) = 30/12.60 = 2.38 m; H_c = 2z₀ = 4.76 m (so a 6 m vertical cut in this soil is unsafe unsupported).
pₐ at base = 0.490 × 18 × 6 − 2 × 15 × 0.700 = 52.95 − 21.01 = 31.9 kN/m².
Ignoring the tension zone: Pₐ = ½ × (6 − 2.38) × 31.9 = 57.8 kN/m at (6 − 2.38)/3 = 1.21 m above the base.
If the full triangle including the negative part were taken: ½Kₐγ H² − 2cH√Kₐ = 158.9 − 126.0 = 32.8 kN/m — only 57% of the correct design value, a classic exam trap.
If the crack fills with water: add ½ × 9.81 × 2.38² = 27.8 kN/m → design thrust ≈ 85.6 kN/m.

Example 3 — water table in the backfill. H = 6 m, water table 3 m below the top; γ = 18 kN/m³ above and γ_sat = 20 kN/m³ below; φ′ = 32°, c′ = 0.

Kₐ = (1 − sin 32°)/(1 + sin 32°) = 0.4701/1.5299 = 0.307.
At 3 m: σ′_v = 54 → pₐ = 16.6 kN/m². At 6 m: σ′_v = 54 + (20 − 9.81)×3 = 84.6 → pₐ = 26.0 kN/m². Water pressure at 6 m = 29.4 kN/m².

Component	Force (kN/m)	Lever arm above base (m)	Moment (kN·m/m)
Upper triangle ½×16.6×3	24.9	4.00	99.6
Rectangle 16.6×3	49.8	1.50	74.7
Lower triangle ½×9.4×3	14.1	1.00	14.1
Water ½×29.4×3	44.1	1.00	44.1
Total	132.9	1.75	232.5

The same wall with a fully drained backfill (γ = 18 throughout) would carry ½ × 0.307 × 18 × 36 = 99.6 kN/m — the water adds 33%, and a backfill saturated to the top would add far more. This is the numerical argument for weep holes and a granular drainage layer.

Example 4 — infinite slope. β = 20°, c′ = 10 kN/m², φ′ = 28°, γ = 19 kN/m³, potential slip plane at a vertical depth of 3 m.

cos²20° = 0.883; tan 20° = 0.364; tan 28° = 0.5317.
Cohesion term = 10/(19 × 3 × 0.883 × 0.364) = 10/18.32 = 0.546; friction term = 0.5317/0.364 = 1.461 → F = 2.01.
If cohesion is lost on weathering (c′ = 0, dry): F = 1.46.
Under monsoon steady seepage parallel to the surface with γ_sat = 20 kN/m³ (γ′ = 10.19): friction term = (10.19/20)(1.461) = 0.744; cohesion term = 10/(20 × 3 × 0.883 × 0.364) = 0.519 → F = 1.26; with c′ = 0 it falls to 0.74, i.e. failure. This is why most hill-road slides occur in the rains.

Example 5 — undrained circular failure and Taylor check. Soft clay embankment, c_u = 40 kN/m², φ_u = 0, γ = 18 kN/m³. Trial circle: R = 12 m, arc subtending 80°, sliding mass W = 1,400 kN/m with its centroid 4.2 m horizontally from the centre.

L_a = 12 × (80 × π/180) = 12 × 1.3963 = 16.76 m.
Resisting moment = c_u L_a R = 40 × 16.76 × 12 = 8,042 kN·m/m; disturbing moment = 1,400 × 4.2 = 5,880 kN·m/m.
F = 8,042/5,880 = 1.37 — acceptable for end-of-construction, marginal for a permanent slope; other circles must be tried to find the minimum.
Chart cross-check for a 60° slope in the same clay (S_n = 0.191 for φ_u = 0, toe circle): H_c = 40/(0.191 × 18) = 11.6 m; for F = 1.5, safe height = (40/1.5)/(0.191 × 18) = 7.8 m.

Aspect	Rankine	Coulomb
Wall friction δ	Neglected (δ = 0)	Included
Wall face	Vertical, smooth	Inclined, rough
Failure surface	Plane, fixed at 45 ± φ/2	Plane, found by trial
Thrust direction	Parallel to backfill surface	At δ to the wall normal
Active result	Slightly conservative (higher Pₐ)	More realistic, lower Pₐ
Passive result	Conservative	Unsafe — over-estimates K_p

Method	Equilibrium satisfied	Surface	Accuracy / use
φ = 0 (Swedish circle)	Moment	Circular	Exact for undrained clay; hand calculation
Friction circle	Force + moment (graphical)	Circular	c–φ soils, classical exam method
Fellenius (ordinary slices)	Moment only, inter-slice forces ignored	Circular	Simple, conservative by 5–20%
Bishop simplified	Moment, normal inter-slice forces	Circular	±2%; standard design method
Spencer / Morgenstern–Price	Force + moment	Any shape	Rigorous; software, critical projects
Taylor / Bishop–Morgenstern charts	—	Circular	Rapid preliminary checks

Where it is used

Highways and railways: cut-slope and high-embankment design on NHAI/NHIDCL and Indian Railways projects, including the Konkan Railway and Himalayan corridors; IRC:75 for embankments above 6 m and IS 14458 for hill-road breast and retaining walls.
Urban basements and metro works: earth pressure on diaphragm walls, secant piles and sheet piles with strut/anchor design for Delhi, Mumbai, Chennai and Bengaluru metro station boxes; at-rest pressure governs braced walls.
Dams and water resources: upstream/downstream slope stability of earth and rockfill dams for steady seepage, rapid drawdown and seismic cases (IS 7894), tailings dams and canal banks — CWC and state irrigation departments.
Landslide management: Himalayan and Western Ghats slides, Char Dham road slopes, NDMA and GSI landslide-hazard work, soil nailing and anchored shotcrete remediation.
Mining, ports and industry: overburden dump and ultimate pit-slope stability (Coal India, NMDC), quay-wall and reclamation bunds, ash-pond dykes at NTPC stations.
Job roles: geotechnical design engineer, site/QA engineer on earthworks, instrumentation and monitoring engineer, slope-stability analyst using GeoStudio/PLAXIS, and forensic investigation of failures.

Interview questions

Q: Why is the active earth pressure coefficient lower than the at-rest value? A: In the active state the wall yields, the soil strains laterally and mobilises its shear strength to help support itself, so the pressure needed from the wall drops to a minimum; at rest no strain occurs and no strength is mobilised.
Q: Which earth pressure would you use for a braced basement wall? A: At-rest pressure (K₀ ≈ 1 − sin φ′), because the bracing prevents the movement needed to develop the active state; apparent-pressure envelopes are used for strut loads.
Q: Why do codes not allow the full Coulomb passive resistance? A: The plane-surface assumption over-estimates K_p when wall friction is high, and the 1–5% of H movement required to mobilise passive resistance is usually unacceptable; curved-surface (Caquot–Kerisel) values with a large factor of safety are used.
Q: What is the depth of the tension crack and why does it matter? A: z₀ = 2c/(γ√Kₐ); the soil cannot take tension there, so the negative pressure is ignored — and if the crack fills with rainwater, full hydrostatic pressure must be added, which can govern the design.
Q: Why is granular backfill specified behind retaining walls? A: It is free-draining, so pore pressures do not build up, it has a high φ′ giving a low Kₐ, and it does not swell or shrink like clay.
Q: For a long cohesionless slope, does the factor of safety depend on the depth of the slip surface? A: No — F = tan φ′/tan β is independent of depth, because both the shear stress and the normal stress increase in proportion to depth.
Q: When is a clay embankment most critical, and when is a clay cutting? A: The embankment at end of construction, when pore pressures are highest and undrained strength governs; the cutting in the long term, as the negative pore pressures set up by excavation dissipate and strength falls.
Q: Difference between Fellenius and Bishop's simplified method? A: Fellenius ignores all inter-slice forces and gives a non-iterative but conservative F; Bishop retains the inter-slice normal forces, satisfies moment equilibrium and needs iteration, giving results within about 2% of rigorous methods.
Q: What does the pore pressure ratio r_u represent? A: r_u = u/(γh), the ratio of pore pressure to total vertical overburden at a point; it is the single parameter used in stability charts to represent seepage, typically 0 to 0.5.
Q: How does an earthquake change the earth pressure on a wall? A: Horizontal inertia on the failure wedge increases the active thrust and reduces passive resistance (Mononobe–Okabe), and the dynamic increment acts higher up the wall, so moments grow faster than forces.
Q: A slope analysis gives F = 1.05. What would you do first? A: Check the input strength and pore pressure assumptions and the critical surface search, then look at drainage — lowering the water table is normally the cheapest effective measure — before considering berms, nails or walls.
Q: Why is the angle of repose roughly equal to φ for dry sand but not for clay? A: A dry cohesionless slope stands only by friction, so its limiting angle equals φ; clay has cohesion, which lets it stand vertically up to H_c, so its stable angle depends on height.

Common mistakes

Using total unit weight below the water table instead of γ′ plus separate hydrostatic pressure — or double-counting the water by using γ_sat *and* adding u.
Computing the cohesive active thrust as the full triangle including the negative (tension) zone, and so under-estimating the design thrust.
Mixing up the failure-plane angles: (45 + φ/2) for active and (45 − φ/2) for passive, not the reverse.
Applying the surcharge as a triangle, or forgetting that its resultant acts at mid-height while the soil thrust acts at H/3.
Taking a single trial circle as the answer in a slope problem instead of searching for the minimum factor of safety.
Using undrained c_u with effective-stress pore pressures, or using c′, φ′ for a short-term loading case — the commonest conceptual error in both exams and practice.
Forgetting the drainage provisions (weep holes, filter, chimney drain) on an otherwise correct design, and ignoring that soil parameters from IS 2720 tests must match the drainage condition analysed.