Cantilever Retaining Wall Design

Stability & reinforcement — ACI 318-25 · Rankine / Coulomb · Mononobe–Okabe seismic · SI units (kN, m, MPa)

Input Parameters

kN · m
Geometry
m
m
Stem profile
m
m
m
m
Soil — Backfill
kN/m³
°
kPa
°
Soil — Foundation
kPa
Passive pressure coefficient Kp
Loads
Materials
Material Properties
MPa
MPa
mm
Design Criteria
Static
Earth pressure method
Wall Cross-Section
Schematic cross-section — updates with inputs
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Enter inputs and press Calculate to see stability checks and reinforcement design.

📚 Design Background & Code References

Cantilever Retaining Wall — Structural System

A cantilever retaining wall consists of a vertical stem, a base footing (with toe and heel), and optionally a shear key beneath the footing. The stem acts as a vertical cantilever fixed at the top of the footing. The footing is a T-shaped horizontal cantilever: the heel projects toward the retained soil and carries the weight of the backfill above it; the toe projects toward the front and is loaded by bearing pressure from below.

The retained soil mass above the heel moves with the wall as a rigid body — this is the key insight of cantilever wall design. The active pressure acts on a virtual back plane at the rear of the heel rather than on the stem face.

Sign Convention & Geometry

  • B = total base width = Lt + ts + Lh
  • He = effective height of retained soil = H + tf (for Rankine virtual back plane)
  • e = eccentricity of resultant from base centroid; kern limit = B/6
  • Positive x measured from toe edge; overturning moment taken about toe

Design Workflow

  • 1. Compute earth pressures (Ka, Kp) from chosen method
  • 2. Serviceability stability checks: overturning (FS ≥ 1.5), sliding (FS ≥ 1.5), bearing (qmax ≤ qa)
  • 3. Factored loads for ACI 318-25 strength design of stem and footing sections
  • 4. Select reinforcement (d, As,req) and verify shear (φVc ≥ Vu)

Rankine (1857) Active & Passive Coefficients

Rankine's theory assumes a smooth (frictionless) wall-soil interface. The active coefficient Ka gives the ratio of horizontal-to-vertical effective stress in a soil mass on the verge of active failure.

Horizontal backfill (β = 0)
Ka = tan²(45° − φ/2)    Kp = tan²(45° + φ/2)
Pa = ½ Ka γs He²  (+ Ka q He for surcharge)
Acts at He/3 from base for triangular pressure; He/2 for uniform surcharge component.
Sloped backfill (β > 0) — Rankine oblique resultant
Ka = cos β · [cos β − √(cos²β − cos²φ)] / [cos β + √(cos²β − cos²φ)]
Resultant Pa acts parallel to backfill slope at He/3 from base
Horizontal component: Pa,h = Pa · cos β  ·  Vertical: Pa,v = Pa · sin β

Coulomb (1776) Wall Friction

Coulomb's method accounts for wall-soil friction angle δ (typically δ = 0.5φ to 0.67φ). The active thrust acts at angle δ from the wall normal.

Coulomb Ka (vertical back face, sloped fill)
Ka = sin²(α+φ) / [sin²α · sin(α−δ) · (1 + √((sin(φ+δ)·sin(φ−β))/(sin(α−δ)·sin(α+β))))²]
α = wall-back inclination (90° for vertical), β = backfill slope, δ = wall friction

Submerged / Hydrostatic Water Table

Effective stress with water table at top of retained height
Pa uses γs,eff = γsat − γw    (buoyant unit weight)
Hydrostatic: Pw = ½ γw He²    (acts separately, load factor 1.4)

Passive Pressure Shear Key

Passive resistance of shear key (Rankine, smooth face)
Kp = tan²(45° + φ/2)
Toe key:   Fp,key = Kp γs dk (htoe + dk/2)  ·  Mkey = Kp γs (htoe dk²/2 + dk³/6)
Heel key:   Fp,key = Kp γs (He dk + dk²/2)  ·  arm from key centroid
Shear key passive resistance adds to the total sliding resistance. Key depth dk measured from footing base.

Overturning Stability

All moments are taken about the toe of the footing. The resisting moment Mr includes the weight of the wall, footing, backfill above heel, and the vertical component of active thrust (if any). The overturning moment Mo is due to horizontal earth pressure and surcharge.

Factor of Safety against Overturning
FSOT = Mr / Mo    ≥ 1.5 (static)   ≥ 1.1 (seismic)
Uplift pressure (hydrostatic beneath footing) reduces Mr,eff = Mr − Muplift when a water table is present.

Sliding Stability

Factor of Safety against Sliding
Sliding resistance: R = μ · Veff + Fp + Fp,key
μ = tan(φbase)  ·  Veff = V − U (vertical resultant net of uplift)
FSSL = R / Htotal    ≥ 1.5 (static)   ≥ 1.1 (seismic)
Base friction angle φbase = ⅔φ for concrete on soil (default). Passive pressure on key counted only when key is enabled.

Bearing Pressure

Trapezoidal / Triangular Bearing Distribution
e = B/2 − x̄    (eccentricity; x̄ = (Mr − Mo) / Veff)
If e ≤ B/6 (kern):   qmax,min = Veff/B · (1 ± 6e/B)   [trapezoidal]
If e > B/6:   qmax = 2Veff / (3·x̄)   [triangular, heel lifts off]
Requirement: qmax ≤ qa (allowable bearing capacity)
Bearing check is a serviceability (unfactored) check. qa is the allowable bearing capacity input by the user.

Base Width Proportioning Rules of Thumb

ParameterTypical RangeNote
Base width B0.45 – 0.70 × HStart with 0.5H; adjust for stability
Toe length Lt0.15 – 0.20 × BIncreases bearing eccentricity control
Footing thickness tf0.08 – 0.10 × H (min 300 mm)Controls shear without stirrups
Stem thickness ts0.06 – 0.10 × H (min 200 mm)Tapered walls save concrete

ACI 318-25 §5.3.1 Load Combinations

Factored loads for strength design
Stem:     U = 1.6H   (lateral earth pressure governs)
Heel/Toe: U = 1.2D   (gravity loads — bearing pressure and soil weight)
Shear key: U = 1.6H   (horizontal passive pressure, cantilever bending)
Water:    U = 1.4F   (hydrostatic, treated as fluid pressure F)

ACI 318-25 §7.5 / §9.5 Flexural Design — One-Way Sections

All wall sections are designed as one-way (per-unit-length, b = 1 m) members. The Whitney rectangular stress block model is used.

Required steel area (tension-controlled, φ = 0.90)
Ru = Mu / (φ · b · d²)
ρ = (0.85f'c / fy) · [1 − √(1 − 2Ru / 0.85f'c)]
As,req = max(ρ · b · d,   As,min)
Effective depth d = h − ccover − ½dbar (8 mm assumed for estimate). b = 1000 mm.
Minimum reinforcement — §7.6.1 / §9.6.1.2
As,min = max(0.25√f'c / fy,   1.4/fy) · b · d   [MPa units]
Shrinkage & temperature: Ast = 0.0018 · b · h   (§24.4.3.2, fy = 420 MPa)

ACI 318-25 §22.5 One-Way Shear (No Stirrups)

Shear capacity of concrete — §22.5.5.1 (members without minimum shear reinforcement)
φVc = φ · 0.17 · λs · √f'c · b · d    [N, MPa, mm]    φ = 0.75
λs = √(2 / (1 + d/250)) ≤ 1.0   (size effect factor; d in mm)
Requirement: φVc ≥ Vu   (no stirrups in wall/footing sections)
λs < 1.0 for d > 250 mm — penalises deeper sections without transverse reinforcement. New in ACI 318-25.

ACI 318-25 §11.6 Distribution Reinforcement — Walls

Horizontal (transverse) distribution bars — §11.6.1
ρt ≥ 0.0020   (db ≤ 16 mm or fy ≥ 420 MPa)    — placed on both faces
Vertical (longitudinal) front-face bars — §11.6.2
ρl ≥ 0.0012   (db ≤ 16 mm or fy ≥ 420 MPa)

Shear Key Design

The shear key is modelled as a vertical cantilever fixed at the footing base. The loading is the triangular/trapezoidal passive earth pressure on the key face. Section thickness = wk (key width); b = 1000 mm. Tension face is toward the retained soil.

Bending moment at fixed base (cantilever)
Toe key:   Mu = 1.6 · Kp γs (htoe dk²/2 + dk³/6)
Heel key:   Mu = 1.6 · Kp γs (He dk²/2 + dk³/6)

Mononobe–Okabe (1926/1929) Seismic Earth Pressure

The Mononobe–Okabe (MO) method extends Coulomb's theory to pseudo-static seismic conditions by introducing a seismic coefficient kh. The total seismic active thrust PAE includes both the static and dynamic incremental components.

MO Active Coefficient KAE
ψ = arctan(kh)    (seismic inertia angle)
KAE = cos²(φ − ψ − α) / [cos ψ · cos²α · cos(δ + α + ψ) · (1 + √(sin(φ+δ)·sin(φ−β−ψ)/cos(δ+α+ψ)·cos(α−β)))²]
PAE = ½ γs He² (1 − kv) KAE
α = back-face inclination (90° for vertical); β = backfill slope; δ = wall friction; kv = 0 (not implemented).
Dynamic Increment ΔPAE
ΔPAE = PAE − PA    (seismic increment above static)
Point of application: 0.6He from base (ASCE 7 / Seed & Whitman recommendation)
Used in overturning: Mo,seis = Mo,static + ΔPAE · (0.6He − tf)

ASCE 7-16 Site Coefficients

Design spectral acceleration → kh
SDS = Fa · Ss    (short-period)    Sa(T=0) = SDS
kh = SDS / (2 · Rd)    Rd = 1.5 (rigid wall, conservative)
Fa amplification from ASCE 7-16 Tables 11.4-1/11.4-2 based on Site Class and Ss.

Seismic Factor of Safety Limits

CheckStatic FSSeismic FSReference
Overturning≥ 1.5≥ 1.1AASHTO / common practice
Sliding≥ 1.5≥ 1.1AASHTO / common practice
Bearingq ≤ qaq ≤ 1.33 qaIBC / ASCE 7

Design Codes & Standards

StandardTopic
ACI 318-25 §5.3Load combinations — U = 1.2D + 1.6H
ACI 318-25 §7.5–7.7Wall flexural design, minimum reinforcement, temperature steel
ACI 318-25 §9.5–9.6Slab/footing flexural design and minimum reinforcement
ACI 318-25 §11.6Distribution reinforcement in walls (S2, S3 bars)
ACI 318-25 §22.5.5.1Concrete shear strength φVc = 0.75 × 0.17 × λs × √f'c × b × d   (λs = size effect factor)
ACI 318-25 §25.3.2Footing transverse distribution bar spacing ≤ min(3h, 450 mm)
ASCE 7-16 §11.4Seismic ground motion parameters, site coefficients Fa, Fv

Key References

Author / SourceReference
Rankine, W.J.M. (1857)On the stability of loose earth. Phil. Trans. Royal Society, 147, 9–27.
Coulomb, C.A. (1776)Essai sur une application des règles des maximis et minimis. Mém. Acad. Roy. Div. Sav., 7.
Mononobe, N. & Matsuo, H. (1929)On the determination of earth pressures during earthquakes. Proc. World Eng. Conf., 9.
Okabe, S. (1926)General theory of earth pressure. J. Japan Soc. Civil Eng., 12(1).
Seed, H.B. & Whitman, R.V. (1970)Design of earth retaining structures for dynamic loads. ASCE Lateral Stresses Conf.
Das, B.M. (2019)Principles of Foundation Engineering, 9th ed. Cengage.
Bowles, J.E. (1996)Foundation Analysis and Design, 5th ed. McGraw-Hill.