PE Exam (Civil) Lateral Earth Pressure: Rankine, Coulomb, At-rest
Last updated: May 2, 2026
Lateral Earth Pressure: Rankine, Coulomb, At-rest questions are one of the highest-leverage areas to study for the PE Exam (Civil). This guide breaks down the rule, the elements you need to recognize, the named traps that catch most students, and a memory aid that scales to test day. Read it once, then practice the same sub-topic adaptively in the app.
The rule
Lateral earth pressure on a wall depends on wall movement, backfill geometry, and shear strength. For a wall that yields away from the backfill, use active pressure with $K_a$; for a wall pushed into the backfill, use passive pressure with $K_p$; for a non-yielding wall (basement, braced cut, integral abutment), use at-rest pressure with $K_0$. Rankine theory (NCEES Reference Handbook, Geotechnical §Lateral Earth Pressure) assumes a smooth vertical wall and a planar failure surface, while Coulomb theory accounts for wall friction $\delta$, wall batter $\theta$, and a sloped backfill. At-rest pressure for normally consolidated soil is estimated by Jaky's equation $K_0 = 1 - \sin\phi'$.
Elements breakdown
At-Rest Coefficient $K_0$
Coefficient when no lateral wall movement is permitted, applicable to non-yielding structures.
- Use Jaky for normally consolidated cohesionless
- $K_0 = 1 - \sin\phi'$ for NC soils
- $K_0 = (1 - \sin\phi') \cdot OCR^{\sin\phi'}$ for OC soils
- Apply to basements, braced cuts, abutments
- Larger than $K_a$, smaller than $K_p$
- Resultant acts at $H/3$ above base
Rankine Active Coefficient $K_a$
Minimum lateral pressure when a wall translates or rotates outward enough to mobilize backfill shear strength.
- Smooth vertical wall, planar slip surface assumed
- Horizontal backfill: $K_a = \tan^2(45^{\circ} - \phi'/2)$
- Sloped backfill (angle $\beta$): use Rankine sloped form
- No wall friction included ($\delta = 0$)
- Cohesion reduces pressure: subtract $2c'\sqrt{K_a}$
- Tension crack depth $z_c = 2c'/(\gamma\sqrt{K_a})$
Rankine Passive Coefficient $K_p$
Maximum lateral pressure when wall is pushed into backfill, mobilizing shear resistance fully.
- $K_p = \tan^2(45^{\circ} + \phi'/2)$
- Reciprocal of $K_a$ for cohesionless: $K_p = 1/K_a$
- Requires much larger displacement than active
- Passive resistance contributes to sliding stability
- Apply factor of safety $\ge 2$ on $K_p$
Coulomb Theory
Wedge equilibrium method that accommodates wall friction, wall batter, and sloped backfill.
- Inputs: $\phi'$, $\delta$ (wall friction), $\theta$ (wall batter), $\beta$ (backfill slope)
- Active: failure wedge slides down and outward
- Passive: $K_p$ overestimates due to planar surface assumption
- Use $\delta \approx \frac{2}{3}\phi'$ for concrete-soil interface
- Resultant inclined at angle $\delta$ from wall normal
Pressure Distribution and Resultant
Procedure to convert coefficients into design force.
- Effective stress: $\sigma'_h = K \cdot \sigma'_v$
- Triangular distribution for homogeneous soil
- Resultant: $P = \frac{1}{2}K\gamma H^2$
- Resultant location: $H/3$ above base
- Surcharge $q$: add $K \cdot q$ uniform pressure (resultant at $H/2$)
- Submerged: use $\gamma'$ and add hydrostatic separately
Common patterns and traps
The Non-Yielding Wall Trap
The problem describes a basement wall, integral abutment, or braced cut, but the candidate defaults to $K_a$ because the geometry looks like a retaining wall. Non-yielding walls cannot mobilize active conditions because the wall does not translate enough. The correct coefficient is Jaky's $K_0 = 1 - \sin\phi'$, which produces a thrust roughly 1.4 to 1.6 times the active value.
A distractor computed using $K_a = \tan^2(45^{\circ} - \phi'/2)$ when the stem clearly states 'rigidly braced' or 'basement floor slab cast against wall'.
The Surcharge-At-H/3 Mistake
A uniform surcharge $q$ produces a uniform horizontal pressure $K \cdot q$, whose resultant acts at the mid-height $H/2$, not at $H/3$. Candidates frequently lump the surcharge resultant with the soil resultant and place both at $H/3$, which under-predicts overturning moment.
A wrong overturning moment that uses $(P_a + P_q)(H/3)$ instead of $P_a(H/3) + P_q(H/2)$.
The Submerged Backfill Confusion
When the water table is within the backfill, the soil contributes effective-stress lateral pressure using $\gamma'$ (buoyant unit weight), and water contributes a separate hydrostatic pressure using $\gamma_w$. A common error is to apply $K_a$ to the total unit weight, double-counting the water or under-counting it.
A choice computed as $\frac{1}{2}K_a\gamma_{sat}H^2$ instead of $\frac{1}{2}K_a\gamma'H^2 + \frac{1}{2}\gamma_w H_w^2$.
The Cohesion Tension-Crack Oversight
Cohesive backfills produce a theoretical negative pressure to depth $z_c = 2c'/(\gamma\sqrt{K_a})$. Because soil cannot sustain tension against a wall, the active force is computed only over the depth below the crack, but candidates include the negative zone and underestimate the resultant.
A choice that uses the full triangular minus-cohesion distribution from $z = 0$ rather than starting integration at $z = z_c$.
The $K_p$ Rankine-vs-Coulomb Overshoot
Coulomb passive coefficients including wall friction can be 2-3 times larger than Rankine values due to the assumption of a planar slip surface. Real passive failure surfaces are curved (log-spiral), so using Coulomb $K_p$ uncritically overestimates passive resistance.
A passive resistance value that exceeds the Rankine result by an unrealistic factor because Coulomb $K_p$ was applied with $\delta = \frac{2}{3}\phi'$ and not capped.
How it works
Start by asking how the wall will move — that decides the coefficient. A 12 ft cantilever retaining wall with $\phi' = 32^{\circ}$ and $\gamma = 120 \text{ pcf}$ on a horizontal granular backfill will translate outward, so use Rankine active. Compute $K_a = \tan^2(45^{\circ} - 16^{\circ}) = \tan^2(29^{\circ}) \approx 0.307$. The active resultant per foot of wall is $P_a = \frac{1}{2}(0.307)(120)(12)^2 = 2{,}652 \text{ lb/ft}$, acting at $H/3 = 4 \text{ ft}$ above the heel. Always verify units cancel: $\text{(dimensionless)}(\text{lb/ft}^3)(\text{ft}^2) = \text{lb/ft}$. If the same wall were a basement wall braced by a slab, switch to $K_0 = 1 - \sin 32^{\circ} = 0.470$, giving $P_0 = 4{,}061 \text{ lb/ft}$ — a 53% increase that often controls basement wall design.
Worked examples
The Reyes Bridge Replacement Project includes a 14 ft tall reinforced concrete cantilever retaining wall supporting a level granular backfill. The backfill is a clean medium sand with effective friction angle $\phi' = 34^{\circ}$ and moist unit weight $\gamma = 122 \text{ pcf}$. The water table is well below the base of the wall, and the wall is free to translate at the top. No surcharge is applied. The geotechnical engineer requests the lateral active thrust per foot of wall length, computed using Rankine theory with no wall friction.
Most nearly, what is the Rankine active thrust per foot of wall length?
- A $1{,}680 \text{ lb/ft}$
- B $3{,}380 \text{ lb/ft}$ ✓ Correct
- C $5{,}480 \text{ lb/ft}$
- D $11{,}960 \text{ lb/ft}$
Why B is correct: Compute $K_a = \tan^2(45^{\circ} - 34^{\circ}/2) = \tan^2(28^{\circ}) = 0.283$. The Rankine active resultant per foot is $P_a = \frac{1}{2}K_a\gamma H^2 = \frac{1}{2}(0.283)(122 \text{ pcf})(14 \text{ ft})^2 = 3{,}383 \text{ lb/ft}$. Unit check: $(\text{lb/ft}^3)(\text{ft}^2) = \text{lb/ft}$, which matches a force per unit wall length.
Why each wrong choice fails:
- A: This value comes from using $H = 14 \text{ ft}$ but only the linear pressure $K_a\gamma H = 488 \text{ psf}$ multiplied by $\frac{1}{2} \cdot 7$, effectively halving the height twice. The candidate forgot that $P = \frac{1}{2}K\gamma H^2$ already contains the integration over height. (The Unit-Cancellation Check)
- C: This value uses $K_0 = 1 - \sin 34^{\circ} = 0.441$ instead of $K_a$. The wall is a free-translating cantilever, which mobilizes active conditions, so $K_a$ is correct. (The Non-Yielding Wall Trap)
- D: This value uses the Rankine passive coefficient $K_p = \tan^2(45^{\circ} + 17^{\circ}) = 3.54$. Passive applies only when the wall is pushed into the backfill, which is the opposite of the loading described.
The Liu Civic Center underground parking structure has a 16 ft tall reinforced concrete basement wall cast monolithically against the ground floor slab and the mat foundation. The wall cannot translate or rotate. The retained soil is a normally consolidated silty sand with effective friction angle $\phi' = 30^{\circ}$ and moist unit weight $\gamma = 118 \text{ pcf}$. The water table is below the mat. A uniform surface surcharge of $q = 250 \text{ psf}$ from a future plaza is applied at the ground surface. Use Jaky's equation for the soil thrust.
Most nearly, what is the total lateral thrust per foot of wall length, including surcharge?
- A $5{,}040 \text{ lb/ft}$
- B $7{,}550 \text{ lb/ft}$
- C $9{,}550 \text{ lb/ft}$ ✓ Correct
- D $11{,}580 \text{ lb/ft}$
Why C is correct: Because the wall is non-yielding, use $K_0 = 1 - \sin 30^{\circ} = 0.500$. Soil thrust: $P_{soil} = \frac{1}{2}K_0\gamma H^2 = \frac{1}{2}(0.500)(118)(16)^2 = 7{,}552 \text{ lb/ft}$. Surcharge thrust: $P_q = K_0 \cdot q \cdot H = (0.500)(250)(16) = 2{,}000 \text{ lb/ft}$. Total: $7{,}552 + 2{,}000 = 9{,}552 \text{ lb/ft} \approx 9{,}550 \text{ lb/ft}$. Units: $\text{(dimensionless)}(\text{psf})(\text{ft}) = \text{lb/ft}$, consistent.
Why each wrong choice fails:
- A: This value uses $K_a = \tan^2(45^{\circ} - 15^{\circ}) = 0.333$ instead of $K_0$, ignoring that the basement wall is fully restrained against translation. Active conditions are not mobilized. (The Non-Yielding Wall Trap)
- B: This value computes only the soil thrust ($7{,}552 \text{ lb/ft}$) and forgets to add the surcharge contribution $K_0 \cdot q \cdot H$. (The Surcharge-At-H/3 Mistake)
- D: This value applies $K_0$ to the surcharge as $\frac{1}{2}K_0 q H$ doubled or treats the surcharge with the wrong distribution geometry, inflating the surcharge component to roughly $4{,}000 \text{ lb/ft}$. (The Surcharge-At-H/3 Mistake)
A 10 ft tall gravity retaining wall on the Okafor Industrial Park site has a vertical back face and supports a horizontal cohesionless backfill. The backfill has $\phi' = 36^{\circ}$, moist unit weight $\gamma = 125 \text{ pcf}$, and the wall-soil interface friction is $\delta = 24^{\circ}$ ($\delta \approx \frac{2}{3}\phi'$). The wall has translated enough to mobilize active conditions. Using Coulomb theory with vertical wall ($\theta = 90^{\circ}$ measured from horizontal) and horizontal backfill ($\beta = 0$), the Coulomb active coefficient evaluates to $K_{aC} = 0.235$. Compare this to the Rankine value to determine the horizontal component of the active thrust.
Most nearly, what is the horizontal component of the Coulomb active thrust per foot of wall length?
- A $1{,}340 \text{ lb/ft}$ ✓ Correct
- B $1{,}470 \text{ lb/ft}$
- C $1{,}620 \text{ lb/ft}$
- D $1{,}780 \text{ lb/ft}$
Why A is correct: The Coulomb resultant is $P_{aC} = \frac{1}{2}K_{aC}\gamma H^2 = \frac{1}{2}(0.235)(125)(10)^2 = 1{,}469 \text{ lb/ft}$, inclined at $\delta = 24^{\circ}$ from the wall normal (i.e., from horizontal for a vertical wall). The horizontal component is $P_{aC,h} = P_{aC}\cos\delta = 1{,}469 \cdot \cos 24^{\circ} = 1{,}469 \cdot 0.9135 = 1{,}342 \text{ lb/ft} \approx 1{,}340 \text{ lb/ft}$. Unit check: $(\text{lb/ft})(\text{dimensionless}) = \text{lb/ft}$.
Why each wrong choice fails:
- B: This value is the total Coulomb resultant $P_{aC} = 1{,}469 \text{ lb/ft}$ without resolving into horizontal and vertical components. The stem asks specifically for the horizontal component. (The Unit-Cancellation Check)
- C: This value uses Rankine $K_a = \tan^2(45^{\circ} - 18^{\circ}) = 0.260$ instead of the given Coulomb $K_{aC} = 0.235$, producing $P_a = 1{,}625 \text{ lb/ft}$. The problem explicitly directs the candidate to use Coulomb theory, which yields a smaller active coefficient when wall friction is included.
- D: This value comes from adding rather than projecting the wall friction component, computing $P_{aC}/\cos\delta = 1{,}469/0.9135 \approx 1{,}608 \text{ lb/ft}$ and then mistakenly inflating with the vertical component, ending near $1{,}780 \text{ lb/ft}$. (The $K_p$ Rankine-vs-Coulomb Overshoot)
Memory aid
WALK: Wall movement decides K. Away → Active. Locked → at-rest ($K_0$). Knocked-into → Passive. If you can't picture the wall moving, you can't pick the coefficient.
Key distinction
$K_0$ vs. $K_a$ hinges on whether the wall is allowed to translate roughly $0.001H$ to $0.004H$. A flexible cantilever wall mobilizes active; a rigid basement wall, integral bridge abutment, or braced excavation does not, and design with $K_a$ will under-predict the load by 30-60%.
Summary
Pick the coefficient ($K_0$, $K_a$, or $K_p$) from wall movement first, then compute $P = \frac{1}{2}K\gamma H^2$ with effective stresses, surcharges, and water added separately.
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Start your free 7-day trialFrequently asked questions
What is lateral earth pressure: rankine, coulomb, at-rest on the PE Exam (Civil)?
Lateral earth pressure on a wall depends on wall movement, backfill geometry, and shear strength. For a wall that yields away from the backfill, use active pressure with $K_a$; for a wall pushed into the backfill, use passive pressure with $K_p$; for a non-yielding wall (basement, braced cut, integral abutment), use at-rest pressure with $K_0$. Rankine theory (NCEES Reference Handbook, Geotechnical §Lateral Earth Pressure) assumes a smooth vertical wall and a planar failure surface, while Coulomb theory accounts for wall friction $\delta$, wall batter $\theta$, and a sloped backfill. At-rest pressure for normally consolidated soil is estimated by Jaky's equation $K_0 = 1 - \sin\phi'$.
How do I practice lateral earth pressure: rankine, coulomb, at-rest questions?
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What's the most important distinction to remember for lateral earth pressure: rankine, coulomb, at-rest?
$K_0$ vs. $K_a$ hinges on whether the wall is allowed to translate roughly $0.001H$ to $0.004H$. A flexible cantilever wall mobilizes active; a rigid basement wall, integral bridge abutment, or braced excavation does not, and design with $K_a$ will under-predict the load by 30-60%.
Is there a memory aid for lateral earth pressure: rankine, coulomb, at-rest questions?
WALK: Wall movement decides K. Away → Active. Locked → at-rest ($K_0$). Knocked-into → Passive. If you can't picture the wall moving, you can't pick the coefficient.
What's a common trap on lateral earth pressure: rankine, coulomb, at-rest questions?
Using $K_a$ on a non-yielding wall when $K_0$ is required
What's a common trap on lateral earth pressure: rankine, coulomb, at-rest questions?
Forgetting to subtract tension crack zone in cohesive backfill
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