Cantilever Retaining Wall Design

Stability & reinforcement — TSC 2019 · Rankine earth pressure · Seismic thrust · SI units (kN, m, MPa)

📚 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 = 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. Serviceability stability checks: overturning (FS ≥ 1.5), sliding (FS ≥ 1.5), bearing (q_max ≤ q_a)
3. Apply factored loads for strength design per TSC 2019 (stem and footing)
4. Select reinforcement (d, A_s,req) and verify shear (φV_c ≥ V_u)

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

TBDY 2018 §16.12.2.4 Earth Pressure Coefficient (Eq. 16.24)

The calculator uses the generalized formula from TBDY 2018 §16.12.2.4 for both static (θ = 0) and seismic cases. With δ = 0 it reduces to Rankine; with δ > 0 it yields the Coulomb coefficient.

TBDY Eq.(16.24) — K_a (vertical back face, α = 90°)

K_a = cos²(φ−θ) / [cos(θ+δ) · (1 + √(sin(φ+δ)·sin(φ−θ−β) / (cos(θ+δ)·cosβ)))²]
θ = 0 (static); δ = wall friction angle (0 for Rankine smooth-wall); β = backfill slope

For static Rankine (δ = 0, θ = 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.

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.

Factor of Safety against Overturning

FS_OT = M_r / M_o ≥ 1.5 (static) ≥ 1.3 (seismic, §16.12.1.1)

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

Sliding Stability

Factor of Safety against Sliding

Sliding resistance: R = μ · V_eff + F_p + F_p,key
μ = tan(φ_base) · V_eff = V - U (vertical resultant net of uplift)
FS_SL = R / H_total ≥ 1.5 (static) ≥ 1.1 (seismic)

Base friction angle φ_base = 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 = (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

TS 498 / TSC 2019 Load Combinations

Factored loads for strength design

Stem: U = 1.6H (lateral earth pressure governs)
Heel/Toe: U = 1.2G (vertical loads — soil reaction and self-weight)
Shear key: U = 1.6H (passive horizontal pressure, cantilever bending)
Water: U = 1.4F (hydrostatic, treated as fluid pressure)

§7.4–7.5 Flexural Design — One-Way Sections

All wall sections are designed as one-way elements (per unit length, b = 1 m). Rectangular stress block model is applied.

Required steel area (φ = 0.90)

R_u = M_u / (φ · b · d²)
ρ = (0.85f'c / f_y) · [1 - √(1 - 2R_u / 0.85f'c)]
A_s,req = max(ρ · b · d, A_s,min)

Effective depth d = h - cover - ½d_bar (8 mm assumed). b = 1000 mm.

Minimum steel — §7.4 / §7.5

A_s,min = max(0.25√f'c / f_y, 1.4/f_y) · b · d [MPa units]
Shrinkage & temperature: A_st = 0.0018 · b · h (§7.7)

§8.1 One-Way Shear (No Stirrups)

Concrete shear capacity — §8.1

V_cr = 0.65 · f_ctd · b · d [N, MPa, mm]
f_ctd = 0.35 · √f'c / 1.5 (design tensile strength)
Requirement: V_cr ≥ V_d (no stirrups in wall/footing sections)

Size effect factor λ_s is applied as reference (ACI 318-25 §22.5.5.1).

§7.4 Distribution Steel — Walls

Horizontal (transverse) distribution steel

ρ_t ≥ 0.0020 (d_b ≤ 16 mm or f_y ≥ 420 MPa) — placed at each face

Vertical front-face steel

ρ_l ≥ 0.0012 (d_b ≤ 16 mm or f_y ≥ 420 MPa)

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. Tension face toward retained soil.

Bending moment at fixed base

Toe key: M_d = 1.6 · K_p γ_s (h_toe d_k²/2 + d_k³/6)
Heel key: M_d = 1.6 · K_p γ_s (H_e d_k²/2 + d_k³/6)

TSC 2019 §16.12.2.1 Seismic Coefficients (Eq. 16.22)

TSC 2019 §16.12 governs the design of earth-retaining structures under earthquake action. The horizontal and vertical static-equivalent seismic coefficients are derived directly from the short-period design spectral acceleration S_DS and a displacement-dependent reduction factor r.

Horizontal & vertical seismic coefficients

S_DS = F_S · S_S    (short-period design spectral acceleration)
k_h = 0.4 · S_DS / r    (Eq. 16.22)
k_v = 0.5 · k_h    (Eq. 16.22)

F_S: site-class-dependent short-period site coefficient — TSC 2019 Table 2.3. r: see Table 16.7 below.

TSC 2019 Table 16.7 Reduction Factor r — Allowable Displacement

The factor r reduces k_h when the wall is permitted to displace during the earthquake. Larger allowable displacement → larger r → smaller seismic demand; serviceability and residual stability must still be verified.

r factor selection (TSC 2019 §16.12.2.1 Table 16.7)

Retaining Structure Type	r
Gravity wall, displacement ≤ 120·S_DS (mm)	2.0
Gravity wall, displacement ≤ 80·S_DS (mm)	1.5
Anchored walls / non-yielding gravity walls	1.0

For saturated soils where significant excess pore-water pressure may develop, r shall not be taken greater than 1.0 (§16.12.2.2).

TSC 2019 §16.12.2 Total Earth Pressure & Dynamic Increment

The total (static + dynamic) active thrust is obtained from a Mononobe–Okabe-type coefficient. The dynamic increment is the difference between the total and the static (ψ = 0) coefficients.

Seismic inertia angle ψ — §16.12.3.2 Eq. (16.26)

Water below base: ψ = arctan( k_h / (1 - k_v) ), γ* = γ
Water above base (impermeable): ψ = arctan( γ_d/(γ_d - γ_su) · k_h/(1 - k_v) ), γ* = γ_d - γ_su

Total active coefficient K (vertical wall, α = 90°) — Eq. (16.24)

K = sin²(α + φ - ψ) / [cos ψ · sin²α · sin(α - ψ - δ) · (1 + √( sin(φ+δ)·sin(φ-ψ-β) / (sin(α-ψ-δ)·sin(α+β)) ))²]
Static coefficient: same expression with ψ = 0 (§16.12.2.7)

α = wall back-face inclination (90° for vertical cantilever stem); β = backfill slope; δ = wall friction (active: δ ≤ ⅔φ per §16.12.2.9).

Total thrust & dynamic increment — Eq. (16.23) / §16.12.2.7

P_t = K (1 - k_v) (½ γ* H² + q H) (total static + dynamic)
ΔP_ae = [ K_total(1 - k_v) - K_static ] (½ γ* H² + q H)
Point of application: mid-height H/2 from base (§16.12.2.8)
Overturning: M_o,seis = M_o,static + ΔP_ae · (H/2)

Per §16.12.2.8 the dynamic-increment resultant acts at the mid-height of the wall (not the 0.6H Seed–Whitman point).

TSC 2019 Table 2.3 Short-Period Site Coefficient F_S

Site	S_S≤0.25	S_S=0.50	S_S=0.75	S_S=1.00	S_S=1.25	S_S≥1.50
ZA	0.8	0.8	0.8	0.8	0.8	0.8
ZB	0.9	0.9	0.9	0.9	0.9	0.9
ZC	1.3	1.3	1.2	1.2	1.2	1.2
ZD	1.6	1.4	1.2	1.1	1.0	1.0
ZE	2.4	1.7	1.3	1.1	0.9	0.8

Safety Factor Limits under Seismic Loading

Check	Static FS	Seismic FS	Reference
Overturning	≥ 1.5	≥ 1.3	TSC 2019 §16.12.1.1 (R_dev ≥ 1.3)
Sliding	≥ 1.5	≥ 1.1	TSC 2019 §16.8.4
Bearing capacity	q ≤ q_a	q ≤ 1.25 q_a	TSC 2019 §16.8.3 / geotech. report

Design Codes & Standards

Standard	Topic
TSC 2019 §16.12	Earth-retaining structures — seismic design rules
TSC 2019 Tables 2.3–2.4	Local site coefficients F_S and F₁
TS EN 1997-1	Geotechnical design — overturning, sliding, bearing capacity checks

Key References

Source	Reference
AFAD (2018)	Turkish Seismic Code — TSC 2019 (§16.12 Earth-retaining structures). Disaster and Emergency Management Authority, Ankara.
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.

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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

TBDY 2018 §16.12.2.4 Earth Pressure Coefficient (Eq. 16.24)

Submerged / Hydrostatic Water Table

Passive Pressure Shear Key

Overturning Stability

Sliding Stability

Bearing Pressure

Base Width Proportioning Rules of Thumb

TS 498 / TSC 2019 Load Combinations

§7.4–7.5 Flexural Design — One-Way Sections

§8.1 One-Way Shear (No Stirrups)

§7.4 Distribution Steel — Walls

Shear Key Design

TSC 2019 §16.12.2.1 Seismic Coefficients (Eq. 16.22)

TSC 2019 Table 16.7 Reduction Factor r — Allowable Displacement

TSC 2019 §16.12.2 Total Earth Pressure & Dynamic Increment

TSC 2019 Table 2.3 Short-Period Site Coefficient F_S

Safety Factor Limits under Seismic Loading

Design Codes & Standards

Key References

Site	S_S≤0.25	S_S=0.50	S_S=0.75	S_S=1.00	S_S=1.25	S_S≥1.50
ZA	0.8	0.8	0.8	0.8	0.8	0.8
ZB	0.9	0.9	0.9	0.9	0.9	0.9
ZC	1.3	1.3	1.2	1.2	1.2	1.2
ZD	1.6	1.4	1.2	1.1	1.0	1.0
ZE	2.4	1.7	1.3	1.1	0.9	0.8

Site	S_S≤0.25	S_S=0.50	S_S=0.75	S_S=1.00	S_S=1.25	S_S≥1.50
ZA	0.8	0.8	0.8	0.8	0.8	0.8
ZB	0.9	0.9	0.9	0.9	0.9	0.9
ZC	1.3	1.3	1.2	1.2	1.2	1.2
ZD	1.6	1.4	1.2	1.1	1.0	1.0
ZE	2.4	1.7	1.3	1.1	0.9	0.8

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

TBDY 2018 §16.12.2.4 Earth Pressure Coefficient (Eq. 16.24)

Submerged / Hydrostatic Water Table

Passive Pressure Shear Key

Overturning Stability

Sliding Stability

Bearing Pressure

Base Width Proportioning Rules of Thumb

TS 498 / TSC 2019 Load Combinations

§7.4–7.5 Flexural Design — One-Way Sections

§8.1 One-Way Shear (No Stirrups)

§7.4 Distribution Steel — Walls

Shear Key Design

TSC 2019 §16.12.2.1 Seismic Coefficients (Eq. 16.22)

TSC 2019 Table 16.7 Reduction Factor r — Allowable Displacement

TSC 2019 §16.12.2 Total Earth Pressure & Dynamic Increment

TSC 2019 Table 2.3 Short-Period Site Coefficient FS

Safety Factor Limits under Seismic Loading

Design Codes & Standards

Key References

Related Calculators & Guides

TSC 2019 Table 2.3 Short-Period Site Coefficient F_S

Site	S_S≤0.25	S_S=0.50	S_S=0.75	S_S=1.00	S_S=1.25	S_S≥1.50
ZA	0.8	0.8	0.8	0.8	0.8	0.8
ZB	0.9	0.9	0.9	0.9	0.9	0.9
ZC	1.3	1.3	1.2	1.2	1.2	1.2
ZD	1.6	1.4	1.2	1.1	1.0	1.0
ZE	2.4	1.7	1.3	1.1	0.9	0.8