Learn · worked example

Cantilever retaining wall to SANS 10100

A 3 m cantilever retaining wall with a 10 kPa surcharge: earth pressure, overturning, sliding, bearing, and the steel and shear in the stem, toe and heel. Every number below comes from the free STATIX retaining wall calculator and was checked by hand.

1. The problem

Input	Value
Retained height (top of backfill to underside of base)	H = 3.0 m
Stem	350 mm thick, vertical, 2.6 m high above the base
Base	400 mm thick: toe 0.7 m + stem 0.35 m + heel 2.0 m = B = 3.05 m
Backfill	level, φ = 30°, γ = 18 kN/m³, no cohesion, no wall friction, no water
Surcharge on the backfill	q = 10 kPa
Soil in front of the toe	0.6 m deep, φ = 30°, γ = 18 kN/m³
Base friction and bearing	μ = 0.5, q_all = 150 kPa
Materials	f_cu = 30 MPa, f_y = 450 MPa, concrete 25 kN/m³, cover 50 mm (stem) and 60 mm (base)
Targets	overturning 2.0, sliding 1.5

Everything is per metre run of wall.

Wall cross-section with the stem, toe and heel bars, the bearing pressure under the base, the earth pressure on the plane through the heel and the critical slip circle (F = 1.54), from the STATIX retaining wall calculator — The STATIX retaining wall calculator with these inputs: the cross-section with its bars, the bearing pressure, the earth pressure on the vertical plane through the heel (21.3 kPa at the base) and the critical slip circle.

2. Earth pressure (Rankine)

With a level backfill and no wall friction, Rankine gives the active pressure. The thrust acts on a vertical plane through the back of the heel, over the full height H.

K_a = (1 − sinφ)/(1 + sinφ) = (1 − 0.5)/(1 + 0.5) = 0.333 Soil: P_s = ½ K_a γ H² = 0.5 × 0.333 × 18 × 3.0² = 27.0 kN/m at H/3 = 1.00 m Surcharge: P_q = K_a q H = 0.333 × 10 × 3.0 = 10.0 kN/m at H/2 = 1.50 m P_ah = 37.0 kN/m M_o = 27.0 × 1.00 + 10.0 × 1.50 = 42.0 kNm/m (about the toe)

The pressure at the base is K_aγH + K_aq = 18.0 + 3.3 = 21.3 kPa.

3. Weights and restoring moment

Lever arms are measured from the front edge of the toe.

Item	Weight (kN/m)	Arm (m)	Moment (kNm/m)
Stem 0.35 × 2.6 × 25	22.75	0.875	19.9
Base 3.05 × 0.4 × 25	30.50	1.525	46.5
Soil on heel 2.0 × 2.6 × 18	93.60	2.050	191.9
Surcharge on heel 2.0 × 10	20.00	2.050	41.0
Total with surcharge	166.85		299.3
Total without surcharge	146.85		258.3

The surcharge pushes the wall over, but the part of it standing on the heel also holds the wall down. You cannot count on it for both. SANS 10160-1 puts a factor of 0 on a variable action where it helps, so the stability checks leave it off the heel and keep it on the driving side.

4. Overturning

FoS = M_r / M_o = 258.3 / 42.0 = 6.15 ≥ 2.0 OK (with the surcharge wrongly counted on the heel it would be 299.3 / 42.0 = 7.13)

5. Sliding

Friction under the base and passive resistance of the soil in front of the toe hold the wall back.

Friction: μ N = 0.5 × 146.85 = 73.4 kN/m Passive: K_p = 1/K_a = 3.0 P_p = ½ K_p γ D² = 0.5 × 3.0 × 18 × 0.6² = 9.7 kN/m R = 73.4 + 9.7 = 83.1 kN/m FoS = R / P_ah = 83.1 / 37.0 = 2.25 ≥ 1.5 OK

This example counts full passive pressure over the 0.6 m in front of the toe (the calculator lets you switch it off). That soil can be dug out for services or washed away, and passive pressure needs movement to develop. Many engineers ignore it, or the top 0.5 m of it. Without it here: 73.4 / 37.0 = 1.98, still above 1.5. The friction factor μ = 0.5 is an input; tan(⅔φ) = 0.36 is a common cautious value for concrete cast on this soil.

6. Bearing pressure

Bearing is the one check where the surcharge on the heel is unfavourable: it adds load. So check it with the surcharge on as well as off, and take the worse.

With the surcharge on the heel: x̄ = (M_r − M_o) / N = (299.3 − 42.0) / 166.85 = 1.542 m e = B/2 − x̄ = 1.525 − 1.542 = −0.017 m (towards the heel) q = N/B (1 ± 6e/B) = 54.7 × (1 ± 0.034) q_heel = 56.5 kPa, q_toe = 52.9 kPa ≤ 150 kPa OK Without it: e = 1.525 − (258.3 − 42.0)/146.85 = +0.052 m (towards the toe) |e| ≤ B/6 = 0.508 m in both cases: the whole base is in contact OK

Without the surcharge on the heel the pressure runs from 53.1 kPa at the toe to 43.2 kPa at the heel. That case is used for the heel design below, because it gives the least upward help to the heel.

7. Stem (SANS 10100 cl. 4.3.3 and 4.3.4)

The stem is a vertical cantilever loaded over its own height of 2.6 m. The calculator uses the SANS 10160-1 STR factors: 1.2 on the soil pressure (permanent) and 1.6 on the surcharge pressure (variable).

V = ½ × 0.333 × 18 × 2.6² + 0.333 × 10 × 2.6 = 20.3 + 8.7 = 28.9 kN/m M = 20.3 × 2.6/3 + 8.7 × 2.6/2 = 17.6 + 11.3 = 28.8 kNm/m V_u = 1.2 × 20.3 + 1.6 × 8.7 = 38.3 kN/m M_u = 1.2 × 17.6 + 1.6 × 11.3 = 39.2 kNm/m (calculator: 38.2 and 39.1) d = 350 − 50 − 12/2 = 294 mm K = M/(b d² f_cu) = 39.1×10⁶ / (1000 × 294² × 30) = 0.015 ≤ K′ = 0.156 z = d [0.5 + √(0.25 − K/0.9)] = 0.98d, limited to 0.95d = 279 mm A_s,req = 39.1×10⁶ / (0.87 × 450 × 279) = 358 mm²/m A_s,min = 0.13% × 1000 × 350 = 455 mm²/m ← governs Provide Y12 at 225 mm on the soil face = 503 mm²/m OK v = V_u/(b d) = 38.2×10³ / (1000 × 294) = 0.130 MPa v_c = (0.75/1.4)(100A_s/bd)^⅓(400/d)^¼(f_cu/25)^⅓ = 0.536 × 0.537 × 1.080 × 1.063 = 0.330 MPa OK

For v_c the calculator takes A_s as the steel required (455 mm²/m), not the 503 mm²/m provided. That gives a slightly lower v_c, on the safe side. It does the same for the toe and the heel below.

Under SANS 10160-1 earth pressure is a permanent action (factor 1.2) and the surcharge a variable one (factor 1.6). The single 1.5 is on the safe side unless the surcharge is larger than about γH, which only happens on short walls under heavy traffic.

8. Toe (bottom steel)

The toe is a cantilever pushed up by the ground, so its tension face is the bottom. For bending the calculator takes the highest pressure at the toe edge, 53.1 kPa, from the surcharge-free case. For shear it takes the case that puts the most upward load on the toe, which here is the one with the surcharge on: 52.9 kPa at the edge and 53.7 kPa at the stem face.

M_u = 16.8 kNm/m (the calculator, ultimate bearing pressure on the toe less the slab weight) d = 400 − 60 − 12/2 = 334 mm, K = 0.005 A_s,req = 135 mm²/m; A_s,min = 0.13% × 1000 × 400 = 520 mm²/m ← governs Provide Y12 at 200 mm on the bottom face = 565 mm²/m Shear at stem face: V_u = 38.8 kN/m v = 38.8×10³ / (1000 × 334) = 0.116 MPa ≤ v_c = 0.320 MPa OK

Minimum steel governs the toe by a wide margin.

9. Heel (top steel)

The heel carries the soil, surcharge and its own weight down, and the ground pushes it up. The load down wins, so the heel bends with tension on top. The calculator factors the load down (1.2 on the soil and slab, 1.6 on the surcharge) and takes off the bearing pressure up under the same load case.

w_down = 1.2 × (18 × 2.6 + 25 × 0.4) + 1.6 × 10 = 1.2 × 56.8 + 16 = 84.2 kPa M_u at the stem face (load down less pressure up) = 82.6 kNm/m d = 400 − 60 − 16/2 = 332 mm K = 82.6×10⁶ / (1000 × 332² × 30) = 0.025 ≤ 0.156 z = 0.95d = 315 mm A_s,req = 82.6×10⁶ / (0.87 × 450 × 315) = 669 mm²/m Provide Y16 at 300 mm on the top face = 670 mm²/m OK V_u at the stem face = 80.5 kN/m v = 80.5×10³ / (1000 × 332) = 0.242 MPa ≤ v_c = 0.350 MPa OK

9b. Stability to SANS 10160 (design values)

Besides the global factors of safety above, the calculator checks the wall with the SANS 10160 partial factors on the soil and the actions (GEO and EQU), the bearing resistance of the ground to EN 1997-1 Annex D, and a slip circle through the soil behind and under the wall:

Sliding, GEO: E_d = 49.1 ≤ R_d = 0.40 × 146.8 + 7.9 = 66.7 kN/m (74 %) ← governs Overturning, EQU: M_dst = 62.6 ≤ M_stb = 251.7 kNm/m Bearing (EN 1997-1 D.4): 142.5 ≤ 419.6 kPa Slip circle (Bishop): 113.9 ≤ 175.2 kNm/m (factor 1.54)

10. Result

Check	Result
Overturning	FoS 6.15 ≥ 2.0
Sliding	FoS 2.25 ≥ 1.5 (1.98 without passive)
Bearing	56.5 kPa ≤ 150 kPa, whole base in contact
Stem	Y12 at 225 (soil face), minimum steel governs; shear 0.130 ≤ 0.330 MPa
Toe	Y12 at 200 (bottom face), minimum steel governs; shear 0.116 ≤ 0.320 MPa
Heel	Y16 at 300 (top face); shear 0.242 ≤ 0.350 MPa
Sliding, GEO (SANS 10160-5)	49.1 ≤ 66.7 kN/m, the most highly used check at 74%

Sliding is the check to watch. A shear key under the base, or a longer heel, are the usual fixes when sliding fails.

Run this example yourselfThe free STATIX retaining wall calculator opens with exactly these inputs. Change the height, the soil, the surcharge or the water table and every check, bar and schedule updates, with a printable calc sheet.
Open the free retaining wall calculator

A worked example for learning, not a design. Check every value against the current code and your own project, and have the design checked and signed by a registered professional engineer.

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