PE Geotechnical Lateral Earth Pressure: Coulomb vs Rankine vs At-Rest

Three theories, three different results. Coulomb, Rankine, and at-rest each give you a different lateral earth pressure for the same wall and the same backfill — and the PE Civil Geotechnical exam tests whether you can pick the right one for the wall geometry, the backfill slope, the wall friction, and the strain state. Use Rankine on a sloping wall with significant wall friction and you'll be off by 10–15% on the active force. Use Coulomb passive on a wall with high wall friction and you'll overestimate passive resistance by a factor of 2 or more, because Coulomb's planar-failure assumption breaks down at large δ. The math itself is short. The pain is method selection.

Lateral earth pressure also splits cleanly by strain state. Active pressure develops when the wall yields away from the soil (the soil expands laterally and shears down). Passive pressure develops when the wall pushes into the soil (the soil compresses laterally and shears up). At-rest pressure exists when the wall doesn't move at all (basement walls, in-situ stress before excavation). Three different physics, three different formulas. The exam expects you to recognize which strain state the problem describes from the wall description alone.

This post walks through the four lateral-pressure problem types NCEES tests on the PE Geotechnical exam, with three fully solved worked examples (Rankine vs. Coulomb active for a sloping-backfill wall, at-rest pressure for an OC clay below the water table, and Coulomb passive with wall friction). Every formula traces to its section in the NCEES PE Civil Reference Handbook (§3.12 Earth Retention) and to NAVFAC DM-7.02 / FHWA NHI-06-088 where applicable.

Why lateral earth pressure dominates retaining-wall problems

Per the April 2024 NCEES PE Civil Geotechnical specification, Topic 8 (Retaining Structures, ASD or LRFD) carries 10–15 questions out of 80 — tied with Deep Foundations as the most heavily weighted topic on the exam. Within Topic 8, lateral earth pressure is the upstream input to every wall-design calculation: stability checks (sliding, overturning, bearing), structural design (moment and shear in stem and footing), and serviceability (deflection, stress on tieback anchors). A confident command of Coulomb / Rankine / at-rest puts a substantial fraction of a passing score on the table before you even get to the wall-design questions.

Beyond Topic 8, lateral earth pressure shows up in adjacent blocks: deep foundations (lateral pile capacity uses passive pressure on the leading face), excavation support (sheet piles, soldier-pile-and-lagging), and earth structures (slope stability often wraps a retaining-wall problem). The skill transfers everywhere on the geotechnical side of the exam.

Core concepts you must master

Active, passive, and at-rest pressure regimes

Three distinct physics depending on how the wall is moving relative to the soil:

• Active (K_a): Wall yields away from the backfill. Soil expands laterally and approaches its shear strength on a downward-sloping failure plane. K_a < 1; lower than at-rest.
• Passive (K_p): Wall pushes into the backfill. Soil compresses laterally and approaches its shear strength on an upward-sloping failure plane. K_p > 1; higher than at-rest.
• At-rest (K₀): Wall doesn't move; soil stays in its in-situ stress state. K₀ sits between K_a and K_p, typically 0.4–0.6 for normally consolidated soils, higher for OC soils.

Required wall movement to mobilize each regime: ~0.001H for active, ~0.05H for passive (for sands; even more for clays). This is why "at-rest" applies to non-yielding walls like basement walls and braced excavations.

Rankine theory (vertical wall, no wall friction)

The simplest and most commonly cited form. For a vertical wall, level backfill, no wall friction, cohesionless soil:

For a sloping backfill at angle β from horizontal (still with vertical wall, no wall friction):

The Rankine resultant for sloping backfill acts parallel to the backfill slope, not horizontally — you have to project to get the horizontal component for sliding/overturning checks.

Coulomb theory (allows wall friction and wall batter)

More general than Rankine. Coulomb assumes a planar failure surface and allows wall friction δ, wall batter α (angle of wall back from vertical), and backfill slope β. For a vertical wall (α = 0), level or sloping backfill, wall friction δ:

The Coulomb resultant acts inclined at angle δ from the wall normal — it has both horizontal and vertical components. Typical wall friction values: δ = (1/2 to 2/3)·φ for sand-on-concrete; δ = 0 if the wall has a smooth membrane or compressible joint.

Coulomb passive caveat: the planar-failure assumption overestimates passive resistance when δ > φ/3. Real failure surfaces curve into log-spirals at high wall friction, giving smaller K_p than Coulomb predicts. NAVFAC DM-7.02 and FHWA NHI-06-088 publish log-spiral / Caquot–Kerisel charts for design; use those when δ is large or when the question explicitly says to.

At-rest pressure (K₀)

For non-yielding walls (basement walls, braced excavations, the moment a wall is first cast against in-situ soil) the strain state is at-rest. The Jaky empirical formula for normally consolidated soils:

For overconsolidated soils, the Mayne–Kulhawy (1982) refinement raises K₀ as OCR increases:

For heavily OC clays (OCR > 4), K₀ can exceed 1.0 — meaning the horizontal effective stress is higher than the vertical. That's a counterintuitive but well-documented result worth recognizing on sight in basement-wall and braced-excavation problems.

Pressure distribution and resultant force

For a homogeneous soil with no surcharge or water table, lateral pressure varies linearly with depth: σ_h(z) = K · γ · z. The resultant force per unit length of wall is:

acting at H/3 from the base (centroid of the triangular distribution). When the soil is below the water table, switch to effective unit weight γ′ = γ_sat − γ_w in the soil-pressure term, and add the hydrostatic water force P_w = ½ · γ_w · H² separately. The water force never sees K — water pushes equally in all directions.

The 4 types of lateral-pressure problems on the PE exam

Type 1: Active pressure on a cohesionless backfill

Given wall height, backfill γ and φ, and either wall friction δ or none. If δ = 0 and the wall is vertical with level backfill, Rankine and Coulomb give the same answer — use Rankine for speed. Otherwise compute both and report whichever the question asks for. Worked below.

Type 2: Passive pressure on a cohesionless soil

Given embedded wall depth and soil parameters. For passive, wall friction matters more (and the Coulomb overestimation caveat applies). If the question explicitly says Coulomb, compute it; otherwise default to Rankine for safety, or to a log-spiral chart from NAVFAC DM-7.02 if available. Worked below.

Type 3: At-rest pressure on a non-yielding wall

Common scenarios: basement wall before backfill is placed, braced excavation between bracing levels, or below-grade structural wall with rigid floor diaphragms preventing wall yielding. Compute K₀ from Jaky (NC) or Mayne–Kulhawy (OC). For OC clays below the water table, you also need to add hydrostatic pressure separately. Worked below.

Type 4: Sloping backfill (Coulomb required)

If the backfill slopes upward at angle β > 0 and the wall has wall friction δ > 0, Coulomb is the only theory that handles both. Rankine-with-sloping-backfill gives an answer but ignores wall friction. Coulomb is more general and almost always closer to physical reality.

Worked example: Rankine vs. Coulomb active on a sloping-backfill wall

Worked example: At-rest pressure on an OC clay below the water table

Worked example: Coulomb passive with wall friction

Common errors that cost points

Using Rankine when wall friction matters

Rankine assumes δ = 0. If the question gives a wall friction angle (or describes a sand-on-concrete interface where δ ≈ 2/3·φ is implicit), Coulomb is the right theory. Rankine overestimates the active force by 5–15% in that case. For passive, Coulomb's overestimation goes the other way (passive force is too large in Coulomb but too small in Rankine without friction).

Forgetting backfill slope in K_a

The simple Rankine K_a = (1−sin φ)/(1+sin φ) is only valid for level backfill. When the backfill slopes up at angle β, you need the modified Rankine form (with the cos²β − cos²φ radical) or Coulomb. For backfill slopes greater than ~5°, the difference is meaningful (5–20% higher K_a for upward-sloping backfill).

Mis-applying the water-table effect (effective vs. total stress)

When soil sits below the water table, the lateral effective pressure uses effective unit weight γ′ = γ_sat − γ_w. The hydrostatic water pressure must be added separately — water doesn't get multiplied by K (it pushes equally in all directions). Using total unit weight times K and forgetting hydrostatic gives too small a force; using saturated unit weight without separating water is also wrong. The right answer is K · γ′ for soil + γ_w for water, summed.

Using Coulomb passive at high wall friction without the caveat

Coulomb passive K_p can exceed 2–3× the Rankine value when δ is large. The Coulomb planar-failure assumption breaks down because real failure surfaces in the passive zone curve into log-spirals at high friction. NAVFAC DM-7.02 publishes correction charts (Caquot–Kerisel). For preliminary calculations or when the question is silent, default to Rankine passive for safety; use Coulomb passive only when the question explicitly directs.

Assuming the at-rest formula is K₀ = 1 − sin φ for OC soils

Jaky's K₀ = 1 − sin φ′ applies only to normally consolidated soils. For OC soils, use Mayne–Kulhawy: K₀ = (1 − sin φ′) · OCR^{sin φ′}. For high OCRs (> 4), K₀ can exceed 1.0, and missing this is a categorical error on basement-wall and braced-excavation problems where the soil has been unloaded by erosion or excavation.

How to study lateral earth pressure for the PE Geotechnical exam

Phase 1 — Theory-selection fluency (Week 1)

Read handbook §3.12 (Earth Retention) end-to-end. Practice writing Rankine K_a/K_p, Coulomb K_a/K_p, Jaky K₀, and Mayne–Kulhawy K_0,OC from a blank page. Drill until you can pick the right theory in under 30 seconds based on wall geometry, backfill slope, wall friction, and strain state.

Phase 2 — Worked-problem drills (Weeks 2–3)

Work fifteen problems across the four problem types: four active (Rankine and Coulomb compared), four passive (with Rankine baseline + Coulomb caveat), four at-rest (NC and OC), and three sloping-backfill cases. Time yourself: four to six minutes per problem on the exam. PEwise's Module 11 (Lateral Earth Pressure Fundamentals) covers Rankine, Coulomb, and at-rest theories with worked examples for active, passive, and water-table-affected cases.

Phase 3 — Integration with retaining-wall stability (Week 4)

Solve five integration problems where you start from the wall geometry and backfill conditions, compute the lateral pressure, then check the wall for sliding, overturning, and bearing capacity. That chain (geometry → pressure theory → resultant force → stability check) is the realistic Topic-8 pattern on the exam, and it brings together lateral earth pressure with bearing capacity (Topic 9) into a single multi-step question.

Quick reference: K_a, K_p, K₀ formulas and typical values

Rankine (vertical wall, level backfill, no wall friction)

Computed from Rankine and Jaky closed-form expressions. The NCEES PE Civil Reference Handbook §3.12 reproduces equivalent tables for exam-day reference.

Equation summary

Connecting this to your overall PE Geotechnical exam strategy

Lateral earth pressure feeds directly into retaining-wall stability and structural design (the rest of Topic 8), and it shares effective-stress and shear-strength fundamentals with slope stability (Topic 5) and bearing capacity (Topic 9). Get the K_a/K_p/K₀ selection automatic, and the wall-design problems collapse to a sequence of clean steps. For the broader Topic 1 / Topic 2 fundamentals plus the full 24-module curriculum, our geotechnical PE exam study guide walks the syllabus end-to-end. For the bearing-capacity checks that close out a wall footing once the lateral pressure is fixed, see our foundations and bearing-capacity post. For slope-stability problems that draw on the same effective-stress and shear-strength fundamentals, see the slope-stability problem-types post.

Final thoughts

Lateral earth pressure problems reward engineers who treat theory selection as the first 30 seconds of every wall question: yielding wall (active), pushing wall (passive), or non-yielding wall (at-rest)? Wall friction zero or non-zero? Backfill level or sloping? Once those three answers are fixed, the formula is fixed, and the calculation is mechanical. The candidates who pass make those three calls without thinking. The candidates who don't second-guess between Rankine and Coulomb at every problem and burn time on the wrong setup. Drill the theory-selection check until it's automatic.