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)

📚 Design Background & Code References

Theory↑ Top

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 = L_t + t_s + L_h
H_e = effective height of retained soil = H + t_f (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 (q_max ≤ q_a)
3. Apply factored loads for strength design per EN 1990 / EC2 (stem and footing)
4. Select reinforcement (d, A_s,req) and verify shear (φV_c ≥ V_u)

Earth Pressure↑ Top

Rankine (1857) Active & Passive Coefficients

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

Horizontal backfill (β = 0)

K_a = tan²(45° - φ/2) K_p = tan²(45° + φ/2)
P_a = ½ K_a γ_s H_e² (+ K_a q H_e for surcharge)

Acts at H_e/3 from base for triangular pressure; H_e/2 for uniform surcharge component.

Sloped backfill (β > 0) — Rankine oblique resultant

K_a = cos β · [cos β - √(cos²β - cos²φ)] / [cos β + √(cos²β - cos²φ)]
Resultant P_a acts parallel to backfill slope at H_e/3 from base
Horizontal component: P_a,h = P_a · cos β · Vertical: P_a,v = P_a · 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 K_ae (vertical back face, α = 90°)

K_ae = 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(k_h/(1−k_v)); δ = 0 (Rankine, smooth-wall); β = backfill slope

For static Rankine (δ = 0, ψ = 0): K_a = cos β · (cos β − √(cos²β − cos²φ)) / (cos β + √(cos²β − cos²φ)).

Submerged / Hydrostatic Water Table

Effective stress with water table at top of retained height

P_a uses γ_s,eff = γ_sat - γ_w (buoyant unit weight)
Hydrostatic: P_w = ½ γ_w H_e² (acts separately, load factor 1.4)

Passive Pressure Shear Key

Passive resistance of shear key (Rankine, smooth face)

K_p = tan²(45° + φ/2)
Toe key: F_p,key = K_p γ_s d_k (h_toe + d_k/2) · M_key = K_p γ_s (h_toe d_k²/2 + d_k³/6)
Heel key: F_p,key = K_p γ_s (H_e d_k + d_k²/2) · arm from key centroid

Shear key passive resistance adds to the total sliding resistance. Key depth d_k measured from footing base.

Stability↑ Top

Overturning Stability

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

EC7 EQU limit state — Overturning (EN 1997-1 §2.4.7.2)

γ_G,dst · M_o,k ≤ γ_G,stb · M_r,k
EC7 DA1: γ_G,dst = 1.10 (Annex A Table A.1) · γ_G,stb = 0.90
Ratio = (γ_G,stb · M_r) / (γ_G,dst · M_o) ≥ 1.00 (verified)

Uplift pressure (hydrostatic beneath footing) reduces M_r,eff = M_r - M_uplift when a water table is present.

Sliding — EC7 §6.5.3 (GEO DA1-C2)

GEO DA1-C2 sliding resistance

R_d = (V'_k · tan δ_d + α · c'_d · A_k) / γ_Rh
δ_d = φ'_cv,d = atan(tan φ'_k / γ_φ') · c'_d = c'_k / γ_c'
γ_φ' = 1.25 · γ_c' = 1.25 · γ_Rh = 1.0 (EC7 Annex A, DA1)
Ratio = R_d / E_d,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 = (M_r - M_o) / V_eff)
If e ≤ B/6 (kern): q_max,min = V_eff/B · (1 ± 6e/B) [trapezoidal]
If e > B/6: q_max = 2V_eff / (3·x̄) [triangular, heel lifts off]
Requirement: q_max ≤ q_a (allowable bearing capacity)

Bearing check is a serviceability (unfactored) check. q_a 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 L_t	0.15 – 0.20 × B	Increases bearing eccentricity control
Footing thickness t_f	0.08 – 0.10 × H (min 300 mm)	Controls shear without stirrups
Stem thickness t_s	0.06 – 0.10 × H (min 200 mm)	Tapered walls save concrete

Reinforcement (EC2)↑ Top

EN 1990 (STR) Load Combinations for Strength Design

Factored design loads

Permanent actions (earth, self-weight): γ_G = 1.35
Variable actions (surcharge):            γ_Q = 1.50
Water pressure (permanent):            γ_G = 1.35
Seismic action (accidental, EC8):        γ_AEd = 1.00

EN 1990 Expression (6.10): E_d = Σ γ_G,j·G_k,j + γ_Q,1·Q_k,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 α_cc = 1.0.

Required steel area — EC2 §6.1

μ = M_Ed / (b · d² · f_cd)    (normalized moment)
ξ = 1 − √(1 − 2μ)    (neutral axis depth factor, ξ ≤ ξ_lim)
z = d · (1 − 0.4ξ)    (lever arm)
A_s,req = M_Ed / (f_yd · z) (≥ A_s,min)

Design strengths: f_cd = f_ck/γ_c (γ_c=1.5) · f_yd = f_yk/γ_s (γ_s=1.15). Effective depth d = h − cover − 8 mm.

Minimum steel — EC2 §9.2.1.1

A_s,min = max(0.26 · f_ctm/f_yk · b·d, 0.0013·b·d)
f_ctm = 0.30 · f_ck^2/3 (MPa, EC2 Table 3.1)
Retaining wall faces: A_s,min = 0.0015 · b · h (EC2 §9.8.5)

EC2 §6.2.2 One-Way Shear (No Stirrups)

V_Rd,c — concrete shear resistance

V_Rd,c = max( C_Rd,c·k·(100·ρ_l·f_ck)^1/3·b_w·d, v_min·b_w·d )
k = 1 + √(200/d) ≤ 2.0 · C_Rd,c = 0.18/γ_c = 0.12
v_min = 0.035·k^3/2·√f_ck · Requirement: V_Rd,c ≥ V_Ed

No stirrups are required when V_Ed ≤ V_Rd,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

A_s,secondary ≥ 0.20 · A_s,main
Max spacing: min(3h, 400 mm) (EC2 §9.3.1.1)

Wall distribution steel — EC2 §9.6

ρ_h ≥ 0.001 (horizontal) · ρ_v ≥ 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 = w_k; b = 1000 mm.

Bending moment at fixed base (γ_G = 1.35)

Toe key: M_Ed = 1.35 · K_p γ_s (h_toe d_k²/2 + d_k³/6)
Heel key: M_Ed = 1.35 · K_p γ_s (H_e d_k²/2 + d_k³/6)

Seismic (EC8-5)↑ Top

EC8-5 §7.3.2.2 Seismic Coefficients

EC8-5 §7.3.2.2 defines the horizontal seismic coefficient k_h for the pseudo-static design of earth-retaining structures. A displacement reduction factor r allows k_h to be reduced when the wall is permitted to deflect during the earthquake.

Horizontal seismic coefficient — EC8-5 Eq.(7.1)

k_h = α · S / r where α = a_g/g (PGA on type A ground)
k_v = ±0.5 · k_h (EC8-5 §7.3.2.2; vertical component)
S: soil amplification factor from EC8-1 Table 3.1 (see table below)

Both signs of k_v should be examined. The critical direction for overturning is k_v upward (reduces resisting weight); for earth pressure k_v downward may be critical.

EC8-5 Table 7.1 Reduction Factor r

r factor — allowable wall displacement d_r

Retaining Structure	d_r limit	r
Free-to-rotate gravity wall	≤ 300·a_g/g mm	2.0
Free-to-rotate gravity wall	≤ 200·a_g/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 (V_s,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(k_h / (1 − k_v))
Submerged: ψ = arctan( (γ_sat/(γ_sat−γ_w)) · k_h/(1−k_v) )

Dynamic earth pressure increment ΔP_ae

ΔP_ae = [K_ae(1−k_v) − K_a,static] · (½γH² + qH)
Point of application: mid-height H/2 from base (EC8-5 §7.3.2.2)
Overturning: M_o,seis = M_o,static + ΔP_ae · (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.

References↑ Top

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.

Cantilever Retaining Wall Design

Input Parameters

Calculation Report

📚 Design Background & Code References

Cantilever Retaining Wall — Structural System

Sign Convention & Geometry

Design Workflow

Rankine (1857) Active & Passive Coefficients

EC8-5 Annex E Mononobe–Okabe Seismic Coefficient

Submerged / Hydrostatic Water Table

Passive Pressure Shear Key

Overturning Stability

Sliding — EC7 §6.5.3 (GEO DA1-C2)

Bearing Pressure

Base Width Proportioning Rules of Thumb

EN 1990 (STR) Load Combinations for Strength Design

EC2 §6.1 Flexural Design — One-Way Sections

EC2 §6.2.2 One-Way Shear (No Stirrups)

EC2 §9.3.1 Distribution Steel — Walls & Slabs

Shear Key Design

EC8-5 §7.3.2.2 Seismic Coefficients

EC8-5 Table 7.1 Reduction Factor r

EC8-1 Table 3.1 Soil Amplification Factor S

EC8-5 Annex E Mononobe–Okabe Dynamic Increment

Design Codes & Standards

Key References

Cantilever Retaining Wall Design

Input Parameters

Calculation Report

Project Information

📚 Design Background & Code References

Cantilever Retaining Wall — Structural System

Sign Convention & Geometry

Design Workflow

Rankine (1857) Active & Passive Coefficients

EC8-5 Annex E Mononobe–Okabe Seismic Coefficient

Submerged / Hydrostatic Water Table

Passive Pressure Shear Key

Overturning Stability

Sliding — EC7 §6.5.3 (GEO DA1-C2)

Bearing Pressure

Base Width Proportioning Rules of Thumb

EN 1990 (STR) Load Combinations for Strength Design

EC2 §6.1 Flexural Design — One-Way Sections

EC2 §6.2.2 One-Way Shear (No Stirrups)

EC2 §9.3.1 Distribution Steel — Walls & Slabs

Shear Key Design

EC8-5 §7.3.2.2 Seismic Coefficients

EC8-5 Table 7.1 Reduction Factor r

EC8-1 Table 3.1 Soil Amplification Factor S

EC8-5 Annex E Mononobe–Okabe Dynamic Increment

Design Codes & Standards

Key References

Related Calculators & Guides