calcsheetConcreteCantilever retaining wall
Sign in
CSA A23.3

Cantilever retaining wall

Verified against CFEM 5th ed., NBC 2020 Div B Part 4 and CSA A23.3-24, 2026-10-09

Stability to CFEM and the stem, toe and heel to CSA A23.3, with surcharge, backslope, passive toe, key and a Mononobe-Okabe case.

Given

Jump to results
Loads
\(q\) \(\mathrm{kPa}\) 0-100
\(\mathrm{PGA}\) 0-1
Stem
\(h_{s}\) \(\mathrm{m}\) 1-12
\(t_{\mathrm{top}}\) \(\mathrm{mm}\) 200-1,500
\(t_{\mathrm{bot}}\) \(\mathrm{mm}\) 200-2,000
Base
\(L_{\mathrm{toe}}\) \(\mathrm{m}\) 0.1-6
\(L_{\mathrm{heel}}\) \(\mathrm{m}\) 0.1-10
\(t_{b}\) \(\mathrm{mm}\) 200-2,000
Key and embedment
\(D_{f}\) \(\mathrm{m}\) 0-5
\(d_{\mathrm{frost}}\) \(\mathrm{m}\) 0-5
\(d_{\mathrm{key}}\) \(\mathrm{mm}\) 0-1,500
\(t_{\mathrm{key}}\) \(\mathrm{mm}\) 200-1,500
Backfill
\(\gamma\) \(\mathrm{kN/m³}\) 10-25
\(\phi\) 20-45
\(\beta\) 0-40
\(\delta\) 0-30
Foundation soil
\(\delta_{b}\) 5-40
\(q_{a}\) \(\mathrm{kPa}\) 25-2,000
\(q_{\mathrm{ult}}\) \(\mathrm{kPa}\) 25-3,000
Materials
\(f_{c}\) \(\mathrm{MPa}\) 20-60
\(f_{y}\) \(\mathrm{MPa}\) 300-500
Reinforcement
\(c_{\mathrm{formed}}\) \(\mathrm{mm}\) 25-100
\(c_{\mathrm{cast}}\) \(\mathrm{mm}\) 50-150
\(\mathrm{bar}_{\mathrm{stem}}\)
\(s_{\mathrm{stem}}\) \(\mathrm{mm}\) 75-600
\(\mathrm{bar}_{\mathrm{toe}}\)
\(s_{\mathrm{toe}}\) \(\mathrm{mm}\) 75-600
\(\mathrm{bar}_{\mathrm{heel}}\)
\(s_{\mathrm{heel}}\) \(\mathrm{mm}\) 75-600
\(\mathrm{bar}_{\mathrm{key}}\)
\(s_{\mathrm{key}}\) \(\mathrm{mm}\) 75-600
Df = 1 m ttop = 300 mm hs = 3.5 m tb = 500 mm δb = 23° B = 3.4 m Ltoe = 0.6 m tbot = 400 mm Lheel = 2.4 m
Section through the wall, to scale, with each input drawn where it acts: the lengths, the grade line, the backfill slope and surcharge, and the friction angles on the virtual back and the base.
Wbase = 40.0 kN/m Wsoil = 151.2 kN/m Wrect = 24.7 kN/m Wwedge = 4.1 kN/m Ph = 40.7 kN/m ΔPh = 7.0 kN/m PI = 26.4 kN/m Vs = 219.9 kN/m Fsl = 93.4 kN/m Fs = 2.29 qs = 65.4 kPa xs = 1.682 m es = 17.8 mm
Service loads on the wall after CFEM Fig. 20.10: the weights at their centroids, the thrusts on the virtual back, passive resistance, base friction, the resultant and the pressure under the base, each coloured by the check it governs. Dashed arrows are the seismic increment and the wall inertia.

Title block

Optional. Printed under the title of the PDF.

This printed copy omits the derivation. The full report, with every step, is the PDF at https://calc.struct.work/calc/retaining-wall.pdf.

Checks

Stability

Surcharge within limit\(\htmlClass{sym-P_q}{P_{q}} \leq 0.3 \cdot \htmlClass{sym-P_h}{P_{h}} \quad \Rightarrow \quad \htmlClass{sym-P_q}{0\ \mathrm{N/m}} \leq 0.3 \cdot \htmlClass{sym-P_h}{40.71\ \mathrm{kN/m}}\) 0.00 PASS
Middle third\(\htmlClass{sym-e_abs_s}{e_{\mathrm{abs},s}} \leq \frac{\htmlClass{sym-B}{B}}{6} \quad \Rightarrow \quad \htmlClass{sym-e_abs_s}{17.75\ \mathrm{mm}} \leq \frac{\htmlClass{sym-B}{3.4\ \mathrm{m}}}{6}\) 0.03 PASS
Sliding fs\(\htmlClass{sym-F_s}{F_{s}} \geq \htmlClass{sym-FS_lim}{\mathrm{FS}_{\mathrm{lim}}} \quad \Rightarrow \quad \htmlClass{sym-F_s}{2.293} \geq \htmlClass{sym-FS_lim}{1.5}\) 0.65 PASS
Bearing service\(\htmlClass{sym-q_s}{q_{s}} \leq \htmlClass{sym-q_a}{q_{a}} \quad \Rightarrow \quad \htmlClass{sym-q_s}{65.37\ \mathrm{kPa}} \leq \htmlClass{sym-q_a}{150\ \mathrm{kPa}}\) 0.44 PASS
Sliding factored\(\htmlClass{sym-P_hu}{P_{\mathrm{hu}}} \leq \htmlClass{sym-R_u}{R_{u}} \quad \Rightarrow \quad \htmlClass{sym-P_hu}{61.07\ \mathrm{kN/m}} \leq \htmlClass{sym-R_u}{67.22\ \mathrm{kN/m}}\) 0.91 PASS
Bearing factored\(\htmlClass{sym-q_u}{q_{u}} \leq \htmlClass{sym-q_ult}{q_{\mathrm{ult}}} \quad \Rightarrow \quad \htmlClass{sym-q_u}{84.18\ \mathrm{kPa}} \leq \htmlClass{sym-q_ult}{225\ \mathrm{kPa}}\) 0.37 PASS
Middle third factored\(\max\left(\htmlClass{sym-e_abs_uo}{e_{\mathrm{abs},\mathrm{uo}}}, \htmlClass{sym-e_abs_u}{e_{\mathrm{abs},u}}\right) \leq \frac{\htmlClass{sym-B}{B}}{6} \quad \Rightarrow \quad \max\left(\htmlClass{sym-e_abs_uo}{182.3\ \mathrm{mm}}, \htmlClass{sym-e_abs_u}{67.11\ \mathrm{mm}}\right) \leq \frac{\htmlClass{sym-B}{3.4\ \mathrm{m}}}{6}\) 0.32 PASS
Bearing overturning\(\htmlClass{sym-q_uo}{q_{\mathrm{uo}}} \leq \htmlClass{sym-q_ult}{q_{\mathrm{ult}}} \quad \Rightarrow \quad \htmlClass{sym-q_uo}{65.21\ \mathrm{kPa}} \leq \htmlClass{sym-q_ult}{225\ \mathrm{kPa}}\) 0.29 PASS

Stem

Stem flexure\(\htmlClass{sym-M_f_stem}{M_{f,\mathrm{stem}}} \leq \htmlClass{sym-M_r_stem}{M_{r,\mathrm{stem}}} \quad \Rightarrow \quad \htmlClass{sym-M_f_stem}{54.55\ \mathrm{kN} \cdot \mathrm{m}} \leq \htmlClass{sym-M_r_stem}{148.5\ \mathrm{kN} \cdot \mathrm{m}}\) 0.37 PASS
Stem shear\(\htmlClass{sym-V_f_stem}{V_{f,\mathrm{stem}}} \leq \htmlClass{sym-V_c_stem}{V_{c,\mathrm{stem}}} \quad \Rightarrow \quad \htmlClass{sym-V_f_stem}{46.75\ \mathrm{kN}} \leq \htmlClass{sym-V_c_stem}{192.7\ \mathrm{kN}}\) 0.24 PASS
Stem min steel\(\htmlClass{sym-A_s_stem}{A_{s,\mathrm{stem}}} \geq \htmlClass{sym-A_s_min_stem}{A_{s,\mathrm{min},\mathrm{stem}}} \quad \Rightarrow \quad \htmlClass{sym-A_s_stem}{1333\ \mathrm{mm}^{2}} \geq \htmlClass{sym-A_s_min_stem}{0\ \mathrm{m}^{2}}\) 0.00 PASS
Stem spacing\(\htmlClass{sym-s_stem}{s_{\mathrm{stem}}} \leq \htmlClass{sym-s_max_stem}{s_{\mathrm{max},\mathrm{stem}}} \quad \Rightarrow \quad \htmlClass{sym-s_stem}{150\ \mathrm{mm}} \leq \htmlClass{sym-s_max_stem}{500\ \mathrm{mm}}\) 0.30 PASS
Stem crack control\(\htmlClass{sym-z_stem}{z_{\mathrm{stem}}} \leq \htmlClass{sym-z_lim}{z_{\mathrm{lim}}} \quad \Rightarrow \quad \htmlClass{sym-z_stem}{24.07\ \mathrm{MN/m}} \leq \htmlClass{sym-z_lim}{25\ \mathrm{MN/m}}\) 0.96 PASS
Stem ductility\(\frac{\htmlClass{sym-c_stem}{c_{\mathrm{stem}}}}{\htmlClass{sym-d_stem}{d_{\mathrm{stem}}}} \leq \htmlClass{sym-cd_lim}{\mathrm{cd}_{\mathrm{lim}}} \quad \Rightarrow \quad \frac{\htmlClass{sym-c_stem}{32.27\ \mathrm{mm}}}{\htmlClass{sym-d_stem}{342\ \mathrm{mm}}} \leq \htmlClass{sym-cd_lim}{0.5091}\) 0.19 PASS

Toe

Toe flexure\(\htmlClass{sym-M_f_toe}{M_{f,\mathrm{toe}}} \leq \htmlClass{sym-M_r_toe}{M_{r,\mathrm{toe}}} \quad \Rightarrow \quad \htmlClass{sym-M_f_toe}{14.17\ \mathrm{kN} \cdot \mathrm{m}} \leq \htmlClass{sym-M_r_toe}{182.5\ \mathrm{kN} \cdot \mathrm{m}}\) 0.08 PASS
Toe shear\(\htmlClass{sym-V_f_toe}{V_{f,\mathrm{toe}}} \leq \htmlClass{sym-V_c_toe}{V_{c,\mathrm{toe}}} \quad \Rightarrow \quad \htmlClass{sym-V_f_toe}{17.8\ \mathrm{kN}} \leq \htmlClass{sym-V_c_toe}{223.5\ \mathrm{kN}}\) 0.08 PASS
Toe min steel\(\htmlClass{sym-A_s_toe}{A_{s,\mathrm{toe}}} \geq \htmlClass{sym-A_s_min_toe}{A_{s,\mathrm{min},\mathrm{toe}}} \quad \Rightarrow \quad \htmlClass{sym-A_s_toe}{1333\ \mathrm{mm}^{2}} \geq \htmlClass{sym-A_s_min_toe}{1000\ \mathrm{mm}^{2}}\) 0.75 PASS
Toe spacing\(\htmlClass{sym-s_toe}{s_{\mathrm{toe}}} \leq \htmlClass{sym-s_max_toe}{s_{\mathrm{max},\mathrm{toe}}} \quad \Rightarrow \quad \htmlClass{sym-s_toe}{150\ \mathrm{mm}} \leq \htmlClass{sym-s_max_toe}{500\ \mathrm{mm}}\) 0.30 PASS
Toe crack control\(\htmlClass{sym-z_toe}{z_{\mathrm{toe}}} \leq \htmlClass{sym-z_lim}{z_{\mathrm{lim}}} \quad \Rightarrow \quad \htmlClass{sym-z_toe}{24.07\ \mathrm{MN/m}} \leq \htmlClass{sym-z_lim}{25\ \mathrm{MN/m}}\) 0.96 PASS
Toe ductility\(\frac{\htmlClass{sym-c_toe}{c_{\mathrm{toe}}}}{\htmlClass{sym-d_toe}{d_{\mathrm{toe}}}} \leq \htmlClass{sym-cd_lim}{\mathrm{cd}_{\mathrm{lim}}} \quad \Rightarrow \quad \frac{\htmlClass{sym-c_toe}{32.27\ \mathrm{mm}}}{\htmlClass{sym-d_toe}{417\ \mathrm{mm}}} \leq \htmlClass{sym-cd_lim}{0.5091}\) 0.15 PASS

Heel

Heel flexure\(\htmlClass{sym-M_f_heel}{M_{f,\mathrm{heel}}} \leq \htmlClass{sym-M_r_heel}{M_{r,\mathrm{heel}}} \quad \Rightarrow \quad \htmlClass{sym-M_f_heel}{50.83\ \mathrm{kN} \cdot \mathrm{m}} \leq \htmlClass{sym-M_r_heel}{193.8\ \mathrm{kN} \cdot \mathrm{m}}\) 0.26 PASS
Heel shear\(\htmlClass{sym-V_f_heel}{V_{f,\mathrm{heel}}} \leq \htmlClass{sym-V_c_heel}{V_{c,\mathrm{heel}}} \quad \Rightarrow \quad \htmlClass{sym-V_f_heel}{36.95\ \mathrm{kN}} \leq \htmlClass{sym-V_c_heel}{233\ \mathrm{kN}}\) 0.16 PASS
Heel min steel\(\htmlClass{sym-A_s_heel}{A_{s,\mathrm{heel}}} \geq \htmlClass{sym-A_s_min_heel}{A_{s,\mathrm{min},\mathrm{heel}}} \quad \Rightarrow \quad \htmlClass{sym-A_s_heel}{1333\ \mathrm{mm}^{2}} \geq \htmlClass{sym-A_s_min_heel}{1000\ \mathrm{mm}^{2}}\) 0.75 PASS
Heel spacing\(\htmlClass{sym-s_heel}{s_{\mathrm{heel}}} \leq \htmlClass{sym-s_max_heel}{s_{\mathrm{max},\mathrm{heel}}} \quad \Rightarrow \quad \htmlClass{sym-s_heel}{150\ \mathrm{mm}} \leq \htmlClass{sym-s_max_heel}{500\ \mathrm{mm}}\) 0.30 PASS
Heel crack control\(\htmlClass{sym-z_heel}{z_{\mathrm{heel}}} \leq \htmlClass{sym-z_lim}{z_{\mathrm{lim}}} \quad \Rightarrow \quad \htmlClass{sym-z_heel}{24.07\ \mathrm{MN/m}} \leq \htmlClass{sym-z_lim}{25\ \mathrm{MN/m}}\) 0.96 PASS
Heel ductility\(\frac{\htmlClass{sym-c_heel}{c_{\mathrm{heel}}}}{\htmlClass{sym-d_heel}{d_{\mathrm{heel}}}} \leq \htmlClass{sym-cd_lim}{\mathrm{cd}_{\mathrm{lim}}} \quad \Rightarrow \quad \frac{\htmlClass{sym-c_heel}{32.27\ \mathrm{mm}}}{\htmlClass{sym-d_heel}{442\ \mathrm{mm}}} \leq \htmlClass{sym-cd_lim}{0.5091}\) 0.14 PASS

Key

Key flexure\(\htmlClass{sym-M_f_key}{M_{f,\mathrm{key}}} \leq \htmlClass{sym-M_r_key}{M_{r,\mathrm{key}}} \quad \Rightarrow \quad \htmlClass{sym-M_f_key}{0\ \mathrm{N} \cdot \mathrm{m}} \leq \htmlClass{sym-M_r_key}{137.2\ \mathrm{kN} \cdot \mathrm{m}}\) 0.00 PASS
Key shear\(\htmlClass{sym-V_f_key}{V_{f,\mathrm{key}}} \leq \htmlClass{sym-V_c_key}{V_{c,\mathrm{key}}} \quad \Rightarrow \quad \htmlClass{sym-V_f_key}{0\ \mathrm{N}} \leq \htmlClass{sym-V_c_key}{183.1\ \mathrm{kN}}\) 0.00 PASS

Seismic

Seismic middle third\(\htmlClass{sym-e_abs_E}{e_{\mathrm{abs},E}} \leq \frac{\htmlClass{sym-B}{B}}{6} \quad \Rightarrow \quad \htmlClass{sym-e_abs_E}{319.8\ \mathrm{mm}} \leq \frac{\htmlClass{sym-B}{3.4\ \mathrm{m}}}{6}\) 0.56 PASS
Seismic bearing\(\htmlClass{sym-q_E}{q_{E}} \leq \htmlClass{sym-q_ult}{q_{\mathrm{ult}}} \quad \Rightarrow \quad \htmlClass{sym-q_E}{79.67\ \mathrm{kPa}} \leq \htmlClass{sym-q_ult}{225\ \mathrm{kPa}}\) 0.35 PASS
Seismic sliding\(\htmlClass{sym-P_hE}{P_{\mathrm{hE}}} \leq \htmlClass{sym-R_E}{R_{E}} \quad \Rightarrow \quad \htmlClass{sym-P_hE}{74.15\ \mathrm{kN/m}} \leq \htmlClass{sym-R_E}{74.69\ \mathrm{kN/m}}\) 0.99 PASS
Seismic stem flexure\(\htmlClass{sym-M_E_stem}{M_{E,\mathrm{stem}}} \leq \htmlClass{sym-M_r_stem}{M_{r,\mathrm{stem}}} \quad \Rightarrow \quad \htmlClass{sym-M_E_stem}{53.46\ \mathrm{kN} \cdot \mathrm{m}} \leq \htmlClass{sym-M_r_stem}{148.5\ \mathrm{kN} \cdot \mathrm{m}}\) 0.36 PASS
Seismic stem shear\(\htmlClass{sym-V_E_stem}{V_{E,\mathrm{stem}}} \leq \htmlClass{sym-V_c_stem}{V_{c,\mathrm{stem}}} \quad \Rightarrow \quad \htmlClass{sym-V_E_stem}{40.02\ \mathrm{kN}} \leq \htmlClass{sym-V_c_stem}{192.7\ \mathrm{kN}}\) 0.21 PASS

Results

Earth pressure

\(\htmlClass{sym-K_a}{K_{a}}\) Active earth pressure coefficient 0.2827
\(\htmlClass{sym-P_h}{P_{h}}\) Horizontal active thrust 40.71 \(\mathrm{kN/m}\)
\(\htmlClass{sym-K_AE}{K_{\mathrm{AE}}}\) Seismic active earth pressure coefficient 0.3605
\(\htmlClass{sym-P_AE}{P_{\mathrm{AE}}}\) Seismic active thrust 47.76 \(\mathrm{kN/m}\)

Stability

\(\htmlClass{sym-F_s}{F_{s}}\) Factor of safety against sliding 2.293
\(\htmlClass{sym-e_s}{e_{s}}\) Eccentricity of the resultant, service 17.75 \(\mathrm{mm}\)
\(\htmlClass{sym-q_s}{q_{s}}\) Service bearing pressure 65.37 \(\mathrm{kPa}\)
\(\htmlClass{sym-q_u}{q_{u}}\) Factored bearing pressure 84.18 \(\mathrm{kPa}\)

Stem

\(\htmlClass{sym-M_r_stem}{M_{r,\mathrm{stem}}}\) Factored flexural resistance of the stem 148.5 \(\mathrm{kN} \cdot \mathrm{m}\)
\(\htmlClass{sym-V_c_stem}{V_{c,\mathrm{stem}}}\) Factored shear resistance of the stem 192.7 \(\mathrm{kN}\)

Derivation

This is a Pro calculation. Its step-by-step derivation, and its .tex and .pdf exports, come with a Pro subscription. The checks, results and figures above are free. Pro is CA$9.99 a month. Subscribe to Pro.

Questions

How does the calc get K_a, the active earth pressure coefficient?

Coulomb's K_a with a vertical virtual back through the heel, which reduces to Rankine's tan^2(45 - phi/2) for level backfill and no wall friction (CFEM 20.2.2). The sliding check, sliding_fs, uses the thrust it gives.

How is the seismic thrust P_AE computed?

By Mononobe-Okabe: K_AE from CFEM 18.7.1.1 with k_h equal to the peak ground acceleration and k_v two thirds of it, the increment over the static thrust acting at 0.6 H, plus the horizontal inertia of the wall and the soil over the heel.

Revision history

fix(examples): revalidate the A23.3 calcs against CSA A23.3-24
ValueWasNowUnit
k_beta230\(\mathrm{mm}\)
V_f_heel32.902636.6528\(\mathrm{kN}\)
x_v_heel397.8\(\mathrm{mm}\)
q_vh83.1613\(\mathrm{kPa}\)

Related calculations

Request a calculation · Report an error in this one