Retaining Walls: Active, Passive and At-Rest Earth Pressure

NCEES FE Civil spec: Geotechnical Engineering, E. Stability of retaining structures (e.g., active/passive/at-rest pressure). FE Reference Handbook: Civil Engineering chapter, Geotechnical section (Horizontal Stress Profiles and Forces, Retaining Walls).

What the FE Civil Spec Covers

Section E covers lateral earth pressure on retaining structures and the basic stability checks. Expect to compute an earth pressure coefficient, the resultant thrust per unit length of wall and its location, and sometimes a factor of safety against sliding or overturning. Surcharges and water tables add a layer of bookkeeping but the method stays the same.

Key Concepts

At-rest (K0): no wall movement. Pressure is between active and passive.
Active (Ka): the wall moves slightly away from the backfill; pressure drops to its minimum.
Passive (Kp): the wall is pushed into the soil; pressure rises to its maximum. Much more movement is needed to mobilize full passive resistance.
Rankine theory assumes a smooth vertical wall and level backfill. Coulomb theory adds wall friction and sloping surfaces.
Below the water table, use effective stress for the soil pressure and add the full hydrostatic water pressure separately, because K applies only to the soil skeleton.

Core Formulas

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

K_{p} = tan^{2} (4 5^{\circ} + \frac{ϕ}{2}) = \frac{1}{K _{a}} K_{0} \approx 1 - sin ϕ

σ_{h}^{'} = K σ_{v}^{'} P = \frac{1}{2} K γ H^{2} (acts at H /3)

P_{q} = K q H (acts at H /2)

F S_{s l i d in g} = \frac{\sum V tan δ + other resistance}{\sum H} F S_{o v er t u r nin g} = \frac{\sum M _{R}}{\sum M _{O}}

For overconsolidated soil the at-rest coefficient increases with the overconsolidation ratio. For cohesive backfill, Rankine active pressure is reduced by a cohesion term, which creates a tension zone near the top; the FE usually keeps backfill cohesionless.

Stability checks for a gravity or cantilever wall

Overturning: take moments about the toe. Resisting moments come from the wall weight and soil on the heel; the overturning moment comes from the lateral thrust.
Sliding: compare base friction (vertical load times tan of the base friction angle, plus any adhesion and passive resistance at the toe) with the total horizontal thrust.
Bearing: find where the resultant crosses the base. Its eccentricity from the base center is e = B/2 minus the distance from the toe to the resultant.

q_{ma x, min} = \frac{\sum V}{B} (1 \pm \frac{6 e}{B})

Keeping $e \leq B /6$ (the resultant in the middle third of the base) keeps the whole base in compression. The maximum toe pressure is then compared with the allowable bearing pressure.

Where It Is in the FE Reference Handbook

Search the Civil Engineering chapter, Geotechnical section, for Horizontal Stress Profiles and Forces. It shows the layered active force diagram with water, the Rankine Ka and Kp expressions and the at-rest coefficient. The Retaining Walls entry right after Ultimate Bearing Capacity gives the overturning, sliding and bearing checks.

Worked Example

A smooth vertical wall 6 m high retains level, drained sand with $γ = 18$ kN/m³ and $ϕ = 3 2^{\circ}$ . A uniform surcharge of 10 kPa acts on the backfill. Find the total active thrust per meter of wall and its height above the base.

Step 1: active coefficient.

K_{a} = tan^{2} (4 5^{\circ} - 1 6^{\circ}) = tan^{2} (2 9^{\circ}) = (0.5543)^{2} = 0.307

Step 2: thrust from soil weight (triangle).

P_{a} = \frac{1}{2} (0.307) (18) (6)^{2} = 99.6 kN/m at 2.0 m

Step 3: thrust from surcharge (rectangle).

P_{q} = (0.307) (10) (6) = 18.4 kN/m at 3.0 m

Step 4: total thrust and location.

P_{t o t a l} = 99.6 + 18.4 = 118.0 kN/m

\overset{y}{ˉ} = \frac{( 99.6 ) ( 2.0 ) + ( 18.4 ) ( 3.0 )}{118.0} = \frac{254.4}{118.0} = 2.16 m

For comparison, if the wall were braced so it could not move, the at-rest coefficient would be 1 minus sin 32 degrees = 0.470, and the soil thrust alone would rise to 0.5 x 0.470 x 18 x 36 = 152 kN/m, about 53 percent more than the active value. The passive coefficient for the same sand is 3.25.

Common Mistakes

Using 45 plus phi/2 for the active case. Active uses minus, passive uses plus.
Applying K to the water pressure. Water pushes equally in all directions, so add it with a coefficient of 1.
Using total instead of effective vertical stress below the water table.
Placing the surcharge resultant at H/3 instead of H/2.
Forgetting that results are per unit length of wall, in kN/m or lb/ft.

Practice Problems

Try these from the free PolyCalc FE Civil problem bank. Each one has a full worked solution.

For cohesionless soil with friction angle φ = 30°, what is the approximate Rankine active earth-pressure coefficient K_a for level backfill and a smooth vertical wall? See the worked solution or practice the Soils & Foundations deck
A rigid retaining wall 5 m high retains cohesionless backfill with unit weight 18 kN/m³ and friction angle 30°. The backfill is level and drained. The total active thrust per metre of wall is most nearly: See the worked solution or practice the Soil Mechanics deck
A 5-m retaining wall holds drained soil with γ = 18 kN/m³ and Ka = 1/3. The active force on the wall is: See the worked solution or practice the Soil Mechanics deck
A 5-m retaining wall holds drained, level granular backfill with no surcharge. The resultant of the triangular active earth pressure acts: See the worked solution or practice the Soil Mechanics deck
A 5-m retaining wall has drained backfill with Ka = 1/3. A uniform surcharge of 20 kPa acts on the backfill. The additional horizontal force on the wall due to the surcharge is most nearly: See the worked solution or practice the Soil Mechanics deck
A wall has a vertical base force of 400 kN/m, a base friction angle of 25°, and a horizontal driving force of 120 kN/m. Ignoring passive resistance and adhesion, the factor of safety against sliding is most nearly: See the worked solution or practice the Soil Mechanics deck

Related Calculators and Examples

Frequently Asked Questions

When should I use active, passive or at-rest pressure?

Use active pressure when the wall can move away from the soil, as with a cantilever wall. Use at-rest pressure when the wall is restrained and cannot move, such as a basement wall braced by floors. Use passive pressure for soil being pushed by the wall, such as soil in front of the toe.

Where does the active thrust act on the wall?

For a triangular pressure distribution from the soil weight, the resultant acts at one third of the wall height above the base. A uniform surcharge produces a rectangular distribution whose resultant acts at mid-height.

What factors of safety are typical for retaining walls?

Common targets are about 1.5 against sliding and 2.0 against overturning, but always use the values given in the problem.

Why is drainage behind a retaining wall so important?

If water builds up behind the wall, the wall must resist full hydrostatic pressure in addition to the soil pressure, and the soil pressure itself only drops by about a third to a half. The total load can roughly double. Weep holes, drains and free-draining backfill prevent this.

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