Cantilever Retaining Wall Design
Stability & reinforcement — EC7 (EN 1997-1) · EC2 (EN 1992-1-1) · EC8-5 seismic · Rankine earth pressure · SI units (kN, m, MPa)
Input Parameters
kN · mGeometry
m
m
Stem profile
m
m
m
m
m
Soil — Backfill
kN/m³
°
kPa
°
Soil — Foundation
kPa
kN/m³
Sliding resistance — EC7 §6.5.3
°
kPa
Kp = (1 + sin φ) / (1 − sin φ) =
Loads
Materials
Material Properties
MPa
MPa
mm
Design Criteria — EC7 Partial Factors
EC7 Design Approach 1 (DA1): EQU limit state for overturning, GEO (DA1-C2) for sliding.
EQU — Overturning (EN 1997-1 Annex A)
GEO DA1-C2 — Sliding (EN 1997-1 Annex A)
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. Stability limit-state checks: EQU overturning (EC7), GEO sliding (EC7 DA1-C2), bearing (qmax ≤ qa)
- 3. Apply factored loads for strength design per EN 1990 / EC2 (stem and footing)
- 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
EC8-5 Annex E Mononobe–Okabe Seismic Coefficient
Under seismic loading, the Mononobe–Okabe method (EC8-5 Annex E) extends Coulomb's formula by rotating the gravity vector by the seismic inertia angle ψ. With ψ = 0 it reduces to the static Rankine coefficient.
M-O coefficient Kae (vertical back face, α = 90°)
Kae = cos²(φ−ψ) / [cos(ψ) · cos(ψ+δ) · (1 + √(sin(φ+δ)·sin(φ−ψ−β) / (cos(ψ+δ)·cosβ)))²]ψ = seismic inertia angle = atan(kh/(1−kv)); δ = 0 (Rankine, smooth-wall); β = backfill slope
For static Rankine (δ = 0, ψ = 0): Ka = cos β · (cos β − √(cos²β − cos²φ)) / (cos β + √(cos²β − cos²φ)).
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.
EC7 EQU limit state — Overturning (EN 1997-1 §2.4.7.2)
γG,dst · Mo,k ≤ γG,stb · Mr,kEC7 DA1: γG,dst = 1.10 (Annex A Table A.1) · γG,stb = 0.90
Ratio = (γG,stb · Mr) / (γG,dst · Mo) ≥ 1.00 (verified)
Uplift pressure (hydrostatic beneath footing) reduces Mr,eff = Mr - Muplift when a water table is present.
Sliding — EC7 §6.5.3 (GEO DA1-C2)
GEO DA1-C2 sliding resistance
Rd = (V'k · tan δd + α · c'd · Ak) / γRhδd = φ'cv,d = atan(tan φ'k / γφ') · c'd = c'k / γc'
γφ' = 1.25 · γc' = 1.25 · γRh = 1.0 (EC7 Annex A, DA1)
Ratio = Rd / Ed,h ≥ 1.00 (verified)
EC7 §6.5.3(6): for cast-in-place concrete directly on soil, δk = φ'cv,k (full interface friction). 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
| Parameter | Typical Range | Note |
|---|---|---|
| Base width B | 0.45 – 0.70 × H | Start with 0.5H; adjust for stability |
| Toe length Lt | 0.15 – 0.20 × B | Increases bearing eccentricity control |
| Footing thickness tf | 0.08 – 0.10 × H (min 300 mm) | Controls shear without stirrups |
| Stem thickness ts | 0.06 – 0.10 × H (min 200 mm) | Tapered walls save concrete |
EN 1990 (STR) Load Combinations for Strength Design
Factored design loads
Permanent actions (earth, self-weight): γG = 1.35Variable actions (surcharge): γQ = 1.50
Water pressure (permanent): γG = 1.35
Seismic action (accidental, EC8): γAEd = 1.00
EN 1990 Expression (6.10): Ed = Σ γG,j·Gk,j + γQ,1·Qk,1. Seismic uses accidental combination per EN 1990 §6.4.3.4.
EC2 §6.1 Flexural Design — One-Way Sections
All wall sections are designed as one-way elements (per unit length, b = 1 m). EC2 rectangular stress block model is applied with αcc = 1.0.
Required steel area — EC2 §6.1
μ = MEd / (b · d² · fcd) (normalized moment)ξ = 1 − √(1 − 2μ) (neutral axis depth factor, ξ ≤ ξlim)
z = d · (1 − 0.4ξ) (lever arm)
As,req = MEd / (fyd · z) (≥ As,min)
Design strengths: fcd = fck/γc (γc=1.5) · fyd = fyk/γs (γs=1.15). Effective depth d = h − cover − 8 mm.
Minimum steel — EC2 §9.2.1.1
As,min = max(0.26 · fctm/fyk · b·d, 0.0013·b·d)fctm = 0.30 · fck2/3 (MPa, EC2 Table 3.1)
Retaining wall faces: As,min = 0.0015 · b · h (EC2 §9.8.5)
EC2 §6.2.2 One-Way Shear (No Stirrups)
VRd,c — concrete shear resistance
VRd,c = max( CRd,c·k·(100·ρl·fck)1/3·bw·d, vmin·bw·d )k = 1 + √(200/d) ≤ 2.0 · CRd,c = 0.18/γc = 0.12
vmin = 0.035·k3/2·√fck · Requirement: VRd,c ≥ VEd
No stirrups are required when VEd ≤ VRd,c. EC2 §6.2.1(4) requires minimum shear reinforcement in beams; wall elements may be exempt.
EC2 §9.3.1 Distribution Steel — Walls & Slabs
Secondary (transverse) reinforcement — EC2 §9.3.1.1
As,secondary ≥ 0.20 · As,mainMax spacing: min(3h, 400 mm) (EC2 §9.3.1.1)
Wall distribution steel — EC2 §9.6
ρh ≥ 0.001 (horizontal) · ρv ≥ 0.002 (vertical, per face)
Shear Key Design
The shear key is modelled as a cantilever fixed at the footing base. Load is the triangular/trapezoidal passive earth pressure on the key face. Section thickness = wk; b = 1000 mm.
Bending moment at fixed base (γG = 1.35)
Toe key: MEd = 1.35 · Kp γs (htoe dk²/2 + dk³/6)Heel key: MEd = 1.35 · Kp γs (He dk²/2 + dk³/6)
EC8-5 §7.3.2.2 Seismic Coefficients
EC8-5 §7.3.2.2 defines the horizontal seismic coefficient kh for the pseudo-static design of earth-retaining structures. A displacement reduction factor r allows kh to be reduced when the wall is permitted to deflect during the earthquake.
Horizontal seismic coefficient — EC8-5 Eq.(7.1)
kh = α · S / r where α = ag/g (PGA on type A ground)kv = ±0.5 · kh (EC8-5 §7.3.2.2; vertical component)
S: soil amplification factor from EC8-1 Table 3.1 (see table below)
Both signs of kv should be examined. The critical direction for overturning is kv upward (reduces resisting weight); for earth pressure kv downward may be critical.
EC8-5 Table 7.1 Reduction Factor r
r factor — allowable wall displacement dr
| Retaining Structure | dr limit | r |
|---|---|---|
| Free-to-rotate gravity wall | ≤ 300·ag/g mm | 2.0 |
| Free-to-rotate gravity wall | ≤ 200·ag/g mm | 1.5 |
| Non-yielding / rigid wall | — | 1.0 |
r = 1.0 must be used for saturated soils that may develop excess pore pressure, anchored walls, or where wall movement must be minimized (EC8-5 §7.3.2.2).
EC8-1 Table 3.1 Soil Amplification Factor S
| Ground type | S (Type 1) | S (Type 2) |
|---|---|---|
| A — Rock (Vs,30 > 800 m/s) | 1.00 | 1.00 |
| B — Dense sand/gravel (360–800 m/s) | 1.20 | 1.35 |
| C — Medium-dense sand (180–360 m/s) | 1.15 | 1.50 |
| D — Loose soil / soft clay (< 180 m/s) | 1.35 | 1.80 |
| E — Alluvium over rock (5–20 m) | 1.40 | 1.60 |
EC8-5 Annex E Mononobe–Okabe Dynamic Increment
Seismic inertia angle ψ
Drained: ψ = arctan(kh / (1 − kv))Submerged: ψ = arctan( (γsat/(γsat−γw)) · kh/(1−kv) )
Dynamic earth pressure increment ΔPae
ΔPae = [Kae(1−kv) − Ka,static] · (½γH² + qH)Point of application: mid-height H/2 from base (EC8-5 §7.3.2.2)
Overturning: Mo,seis = Mo,static + ΔPae · (H/2)
EC7/EC8-5 limit state checks use the same characteristic partial factors (γG,dst=1.10, γG,stb=0.90) under both static and seismic combinations.
Design Codes & Standards
| Standard | Topic |
|---|---|
| EN 1990:2002+A1 (EC0) | Basis of structural design — load combinations, partial factors |
| EN 1992-1-1:2004 (EC2) | Design of concrete structures — flexure, shear, minimum steel |
| EN 1997-1:2004 (EC7) | Geotechnical design — EQU overturning, GEO sliding, bearing capacity |
| EN 1998-5:2004 (EC8-5) | Seismic design — geotechnical aspects, retaining walls §7.3.2 |
| EN 1998-1:2004 (EC8-1) | Seismic ground types A–E, soil amplification factor S (Table 3.1) |
Key References
| Source | Reference |
|---|---|
| CEN (2004) | EN 1997-1: Eurocode 7 — Geotechnical design, Part 1: General rules. European Committee for Standardisation, Brussels. |
| CEN (2004) | EN 1998-5: Eurocode 8 — Design of structures for earthquake resistance, Part 5: Foundations, retaining structures. CEN, Brussels. |
| CEN (2004) | EN 1992-1-1: Eurocode 2 — Design of concrete structures, Part 1-1: General rules. CEN, Brussels. |
| Rankine, W.J.M. (1857) | On the stability of loose earth. Phil. Trans. Royal Society, 147, 9–27. |
| 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). |
| Das, B.M. (2019) | Principles of Foundation Engineering, 9th ed. Cengage. |