Retaining Walls

Learning Objectives

  • Identify common retaining wall systems and the load paths that provide stability.
  • Distinguish active, passive, and at-rest earth-pressure states and the wall movement needed to mobilize them.
  • Apply Rankine and Coulomb earth-pressure models only within clearly stated assumptions.
  • Separate static earth pressure from seismic, surcharge, and water-pressure effects.
  • Evaluate overturning, sliding, resultant location, eccentricity, and bearing pressure using a consistent design basis.
  • Relate external stability to the structural design and reinforcement of the stem, heel, and toe.
  • Recognize drainage, geotechnical data, and model limitations that can govern retaining-wall safety.

Retaining Wall Function and Types

Retaining walls restrain soil or other retained material where a stable natural slope cannot be accommodated. Common systems include gravity walls, reinforced-concrete cantilever walls, counterfort walls, and buttressed walls. A cantilever wall uses its reinforced stem and base slab together with the self-weight of concrete and, commonly, the weight of soil over the heel to resist lateral actions.

Common Retaining Wall Systems

Active Earth Pressure

Active earth pressure is the reduced lateral pressure state approached when a wall moves sufficiently away from the backfill for the soil to mobilize an active limit condition.

Passive Earth Pressure

Passive earth pressure is the high-resistance limit state approached when a wall or embedded element moves into the soil sufficiently to mobilize passive shear resistance.

At-Rest Earth Pressure

At-rest earth pressure is the lateral pressure state for soil that is not permitted to undergo the lateral strain needed to reach the active or passive limit state.

Earth-Pressure State Depends on Wall Movement

The pressure coefficient is not selected only from soil type. A flexible cantilever wall that can yield away from the backfill may develop active conditions, while a rigidly braced basement wall may remain near at-rest conditions. Passive resistance usually requires substantially more movement than active pressure, so its dependable contribution to sliding resistance must be justified by soil conditions, geometry, embedment, displacement compatibility, and the selected design framework.

Rankine Active Pressure Model Used in This Lesson

The simplified Rankine active model used here assumes a vertical smooth wall, level homogeneous cohesionless backfill, drained conditions, no wall friction, and sufficient wall movement to mobilize the active state.

Rankine Active Earth Pressure Coefficient

Simplified coefficient for the level, cohesionless, smooth-wall active case used in this lesson.

Ka=1−sin⁡ϕ1+sin⁡ϕK_a=\frac{1-\sin\phi}{1+\sin\phi}

Variables

SymbolDescriptionUnit
KaK_aRankine active earth-pressure coefficient for the stated assumptions-
ϕ\phiEffective angle of internal friction of the backfill∘^\circ

Static Active Thrust from Soil Self-Weight

Resultant of the triangular active-pressure distribution for dry or drained cohesionless backfill under the stated Rankine assumptions.

Pa,γ=12KaγH2P_{a,\gamma}=\frac{1}{2}K_a\gamma H^2

Variables

SymbolDescriptionUnit
Pa,γP_{a,\gamma}Static active thrust due to backfill self-weightkN/m\text{kN}/\text{m}
γ\gammaAppropriate soil unit weight for the stated drainage conditionkN/m3\text{kN}/\text{m}^3
HHRetained height represented by the pressure diagramm\text{m}

Rankine and Coulomb Are Different Models, Not Interchangeable Coefficients

The simplified Rankine expression above does not include wall friction and is tied to its wall and backfill geometry assumptions. Coulomb theory instead uses limit equilibrium of a potential soil wedge and can include wall friction, wall batter, and backfill slope through a consistent angle convention. Both are limit-state idealizations and both require compatible soil parameters, drainage assumptions, and sufficient wall movement.

For a common active case with positive wall friction on a wall that moves away from the backfill, Coulomb thrust on the wall can have a downward component. Under the vertical-wall angle convention used in the worked example, PH=Pacos⁡δP_H=P_a\cos\delta and the downward component is PV=Pasin⁡δP_V=P_a\sin\delta. That effect must not be borrowed into a Rankine calculation unless the chosen model explicitly includes it. Likewise, Coulomb is not automatically “less conservative” for every geometry, and passive resistance is especially sensitive to assumed wall friction and movement.

Coulomb Active Coefficient for the Vertical-Wall Convention Used Here

Worked-example form for a vertical wall retaining homogeneous cohesionless drained backfill with surface slope beta and wall friction delta; use only with the stated angle convention.

Ka=cos⁡2ϕcos⁡δ[1+sin⁡(ϕ+δ)sin⁡(ϕ−β)cos⁡δcos⁡β]2K_a=\frac{\cos^2\phi}{\cos\delta\left[1+\sqrt{\frac{\sin(\phi+\delta)\sin(\phi-\beta)}{\cos\delta\cos\beta}}\right]^2}

Variables

SymbolDescriptionUnit
KaK_aCoulomb active coefficient for this vertical-wall convention-
ϕ\phiEffective backfill friction angle∘^\circ
δ\deltaWall-soil interface friction angle using the stated active sign convention∘^\circ
β\betaBackfill surface slope above horizontal in this convention∘^\circ

Cohesion and Layered Backfill Require Additional Judgment

The simple equations in this lesson are for homogeneous cohesionless material. Cohesive soils, layered backfills, compaction-induced pressures, expansive soils, sloping groundwater surfaces, stratified unit weights, and nonuniform surcharges require a model appropriate to those conditions. Do not insert an apparent cohesion into a simple sand formula without considering tension cracks, drainage, long-term effective stress, and the governing geotechnical design basis.

Uniform Surcharge

A uniform surcharge qq is an additional surface pressure behind the wall that produces a lateral pressure increment when transferred through the selected earth-pressure model.

Lateral Thrust from a Uniform Surcharge

Rectangular lateral-pressure increment under the simplified active model used in this lesson.

Pa,q=KaqHP_{a,q}=K_a qH

Variables

SymbolDescriptionUnit
Pa,qP_{a,q}Resultant lateral thrust caused by a uniform surchargekN/m\text{kN}/\text{m}
qqUniform surface surchargekPa\text{kPa}
HHRetained heightm\text{m}

Hydrostatic Pressure

Hydrostatic pressure is lateral water pressure caused by a water head and acts independently of the effective-stress earth-pressure component of the soil skeleton.

Hydrostatic Resultant

Resultant water thrust for a triangular hydrostatic pressure distribution over water depth Hw.

Pw=12γwHw2P_w=\frac{1}{2}\gamma_w H_w^2

Variables

SymbolDescriptionUnit
PwP_wHydrostatic resultant per meter length of wallkN/m\text{kN}/\text{m}
γw\gamma_wUnit weight of waterkN/m3\text{kN}/\text{m}^3
HwH_wWater depth behind the wallm\text{m}

Drainage Assumption Must Match the Built Wall

A wall analyzed as drained must have a dependable drainage path, filter compatibility, and maintenance strategy. If water can accumulate, hydrostatic pressure and the appropriate submerged or saturated soil weights must be included rather than assuming that dry active pressure alone governs.

Static Earth Pressure Does Not Cover Seismic Demand

Ordinary static Rankine or Coulomb active-pressure equations do not by themselves represent earthquake-induced earth pressure. A pseudo-static method such as Mononobe-Okabe extends a Coulomb-type wedge model by introducing horizontal and, where applicable, vertical seismic coefficients, but it has its own assumptions and limitations. Seismic wall design must follow the governing seismic and geotechnical provisions for the site, including the required pressure distribution, resultant location, drainage condition, and deformation compatibility.

Traditional Service-Load Stability Format

The stability calculations in this lesson use a traditional service-load factor-of-safety format in which service actions and service resisting forces are compared directly without mixing them with strength-design load factors.

Do Not Mix Stability Design Frameworks

Many teaching examples use service loads with factors of safety such as overturning or sliding ratios, while modern project standards may instead use load-and-resistance-factor or strength-design frameworks with factored actions and geotechnical resistance factors. Use one internally consistent basis. The numerical targets shown in this lesson are illustrative traditional service-load criteria, not universal values for every jurisdiction or load case.

Factor of Safety against Overturning

Traditional service-load ratio of stabilizing to overturning moments about the selected rotation point.

FSOT=∑MR∑MOFS_{OT}=\frac{\sum M_R}{\sum M_O}

Variables

SymbolDescriptionUnit
FSOTFS_{OT}Traditional service-load overturning factor of safety-
∑MR\sum M_RSum of service-load resisting moments about the selected pointkN⋅m/m\text{kN}\cdot\text{m}/\text{m}
∑MO\sum M_OSum of service-load overturning moments about the selected pointkN⋅m/m\text{kN}\cdot\text{m}/\text{m}

Factor of Safety against Sliding

Traditional service-load ratio of dependable horizontal resistance to horizontal driving force.

FSS=FR+Pp,designPHFS_S=\frac{F_R+P_{p,\text{design}}}{P_H}

Variables

SymbolDescriptionUnit
FSSFS_STraditional service-load sliding factor of safety-
FRF_RDependable base-interface resistance, commonly related to the effective vertical force and interface frictionkN/m\text{kN}/\text{m}
Pp,designP_{p,\text{design}}Passive resistance credited on the selected design basiskN/m\text{kN}/\text{m}
PHP_HTotal horizontal driving forcekN/m\text{kN}/\text{m}

External Stability Checks

  1. Define one physical geometry. State the retained height, footing thickness, stem thickness, base width, toe, and heel dimensions and use those same dimensions for pressure resultants, weights, lever arms, and the drawing.
  2. Assemble service actions on a consistent basis. Include soil pressure, surcharge, water, self-weight, reliable soil weight over the heel, and any other applicable loads. Separate horizontal and vertical components when the earth-pressure model produces an inclined resultant.
  3. Check overturning about the selected toe or rotation point. The triangular soil thrust acts at one-third of its pressure-diagram height above the base of that triangle; a uniform surcharge component acts at midheight. Add the footing elevation when moments are taken about the bottom of a footing rather than the top.
  4. Check sliding. Use a justified base-interface resistance and only the passive resistance that can be dependably mobilized under the project soil and displacement conditions. A shear key may improve sliding resistance but does not automatically make the wall satisfactory.
  5. Locate the base resultant. For a service vertical resultant ∑V\sum V, the distance from the toe is x=(∑MR−∑MO)/∑Vx=(\sum M_R-\sum M_O)/\sum V when moments use consistent signs and the same reference point.
  6. Check eccentricity and contact. With base width BB, e=B/2−xe=B/2-x. If ∣e∣≤B/6|e|\le B/6, a full-contact linear bearing-pressure distribution is compatible with the middle-third assumption. If the resultant leaves the middle third, the simple full-contact qmax⁡/qmin⁡q_{\max}/q_{\min} equation is no longer physically valid because soil cannot sustain tension.
  7. Compare bearing pressure with geotechnical resistance. A no-tension check does not establish adequate bearing capacity; the maximum applicable contact pressure must also satisfy the geotechnical design criterion, and settlement may govern.

Full-Contact Bearing Pressure

Linear service contact pressure for a rectangular base when the resultant remains within the middle third.

qtoe,heel=∑VB(1±6eB),∣e∣≤B6q_{\text{toe,heel}}=\frac{\sum V}{B}\left(1\pm\frac{6e}{B}\right),\qquad |e|\le\frac{B}{6}

Variables

SymbolDescriptionUnit
qtoe,heelq_{\text{toe,heel}}Toe and heel service contact pressures for full contactkPa\text{kPa}
∑V\sum VTotal service vertical force per meter lengthkN/m\text{kN}/\text{m}
BBBase widthm\text{m}
eeResultant eccentricity from the base centerlinem\text{m}

Use the Retaining Wall Stability Explorer

The interactive tool uses one deliberately limited Rankine model: level drained cohesionless backfill, smooth vertical wall, no surcharge, no water pressure, no passive resistance, and traditional service-load factors of safety. Change the retained height, base width, soil friction angle, and base friction coefficient to see how the same geometry affects active thrust, resisting moments, sliding, eccentricity, and full-contact bearing indicators.

Cantilever Retaining Wall Stability Explorer

Focused Rankine service-load model with consistent wall geometry.

Controls

Retained height HH4.5 m
Base width BB3.0 m
Soil friction angle ϕ\phi30°
Base friction coefficient μ\mu0.50

Geometry and active thrust

Pa = 60.8 kN/mB = 3.0 mtoe 1.00 mheel 1.60 mH = 4.5 m

Service stability results

Rankine coefficient
0.333
Active thrust
60.8 kN/m
Overturning FS
3.22
meets lesson target (2.0)
Sliding FS
1.72
meets lesson target (1.5)

Moments about bottom toe: resisting 391.0 kN·m/m; overturning 121.5 kN·m/m.

Vertical force: 208.8 kN/m; base friction resistance 104.4 kN/m.

Resultant: 1.291 m from toe; eccentricity 0.209 m toward toe.

Full-contact bearing: toe 98.8 kPa; heel 40.4 kPa. Compare the applicable maximum pressure and settlement with project geotechnical criteria separately.

Model scope: level drained homogeneous cohesionless backfill; smooth vertical wall; Rankine active pressure; γs=18kN/m3\gamma_s=18\\ \text{kN}/\text{m}^3; γc=24kN/m3\gamma_c=24\\ \text{kN}/\text{m}^3; 0.50 m footing; 0.40 m stem; toe =B/3=B/3; no surcharge, water, seismic force, or passive resistance. The 2.0 overturning and 1.5 sliding values are lesson targets for this traditional service-load format, not universal code requirements.

3D Stability Studio: Soil–Structure Load Path and Base Contact

Shared Rankine Service-Load Model

The 3D studio uses the same deterministic calculation model as the 2D explorer above: level drained homogeneous cohesionless backfill, smooth vertical wall, Rankine active pressure, γs=18 kN/m3\gamma_s=18\,\text{kN/m}^3, γc=24 kN/m3\gamma_c=24\,\text{kN/m}^3, 0.50 m footing, 0.40 m stem, toe =B/3=B/3, and no surcharge, water pressure, seismic force, or passive resistance. The 3D pressure field, failure-plane cue, wall/base/heel-soil weights, sliding resistance, toe moment reference, base resultant, and full-contact bearing arrows all come from that same state. Bearing arrows are suppressed when ∣e∣>B/6|e|>B/6 because the simple full-contact equation would imply soil tension. Reinforcement shown in the stem, heel, and toe is a principal reinforcement-zone cue, not a completed structural reinforcement design.

Cantilever Retaining Wall — 3D Stability & Soil–Structure Cutaway

Concept and model scope

Trace active earth pressure, wall and soil weights, sliding resistance, overturning about the toe, base resultant, contact pressure, and principal RC reinforcement zones using one shared geometry.

The service-load model assumes a vertical smooth wall, level homogeneous cohesionless backfill, drained Rankine active pressure, γs = 18 kN/m³, γc = 24 kN/m³, a 0.50 m footing, a 0.40 m stem, toe = B/3, and excludes surcharge, water pressure, seismic force, and passive resistance.

The 2.0 overturning and 1.5 sliding values are lesson targets rather than universal code criteria. Bearing arrows are shown only for full contact, and the reinforcement graphics are detailing-location cues rather than a completed RC design.

Displayed traditional service-load lesson targets are satisfied.

Overturning FS ≥ 2.0, sliding FS ≥ 1.5, and the resultant remains within the middle third. These are lesson targets, not universal project acceptance criteria.

Preparing retaining-wall cutaway…

Use the guided sequence to trace the external-stability load path, resultant location, contact-pressure condition, and structural reinforcement zones.

4.5 m
3.0 m
30°
0.50
Earth pressure / stability
Ka
0.333
Pa
60.8 kN/m
FS overturning
3.22
FS sliding
1.72
ΣMR
391.0 kN·m/m
ΣMO
121.5 kN·m/m
Resultant / contact
x from toe
1.291 m
e
0.209 m
Middle third
Inside
ΣV
208.8 kN/m
qtoe
98.8 kPa
qheel
40.4 kPa

Preliminary Proportioning Is Not Design Acceptance

Rules of thumb such as a base width on the order of 0.4H0.4H to 0.7H0.7H, a toe near one-third of the base width, or a footing thickness related to wall height can be useful for a first trial only. Final dimensions must come from stability, geotechnical, structural, drainage, seismic, durability, and constructability checks using the actual project geometry and design basis.

Structural Design of Stem, Heel, and Toe

After the external stability model is established, reinforced-concrete component design uses the factored load combinations required by the governing structural code. The stem behaves primarily as a vertical cantilever under lateral pressure, with principal vertical tension steel generally on the soil face for the usual active-pressure direction. The heel and toe are horizontal cantilevers subjected to net factored pressure distributions, not simply one isolated load component.

For the usual cantilever-wall load pattern, the heel often requires top main reinforcement and the toe often requires bottom main reinforcement, but the sign of net bending must be verified from the actual upward bearing and downward dead/surcharge loads. Development length, anchorage into the stem/base joint, minimum distributed steel, shear, cover, durability, and construction joints must also be checked.

Drainage and Constructability

Typical drainage details include free-draining granular zones, geotextile filters selected to prevent soil migration, perforated collector drains with positive outlets, and weep holes where appropriate. Drainage details must remain maintainable and must not discharge where erosion or icing creates another hazard. A clogged drain can invalidate the drained design assumption even when the concrete reinforcement itself is adequate.

Key Takeaways
  • Active, passive, and at-rest pressures are different soil states governed by wall movement; they are not interchangeable coefficients.
  • The simple Rankine KaK_a equation in this lesson assumes a smooth vertical wall, level homogeneous cohesionless backfill, drained conditions, and sufficient movement to mobilize active pressure.
  • Coulomb is a different wedge-equilibrium model that can include wall friction, wall batter, and backfill slope; neither method should be treated as a universal black box.
  • Static earth-pressure equations do not automatically include seismic demand, surcharge, or hydrostatic pressure; each additional action must be modeled explicitly on a compatible basis.
  • Traditional service-load factors of safety and factored strength/LRFD approaches must not be mixed within the same stability calculation.
  • Base width changes the wall weights, lever arms, resultant, eccentricity, and bearing distribution, so geometry must be propagated through every downstream calculation.
  • A middle-third check establishes full compression contact only; adequate bearing resistance and settlement still require geotechnical verification.
  • Drainage assumptions are structural assumptions because trapped water can add a large lateral load that was absent from a drained-wall model.