Retaining wall stability calculator

What it computes

Inputs are the stem height, base width, base thickness, stem thickness, toe length, the backfill (loose, medium or dense sand, given as a friction angle) and an optional surcharge. The sheet works out the weights of the stem, base and fill on the heel, the thrust of the fill and surcharge, the overturning and resisting moments about the toe, and the pressure under the base. Two sentences under the results say whether overturning and sliding meet the required factor of safety of 1.5, which is a constant in the sheet that you can change.

The formulas

• Active pressure coefficient K_a = tan^2(45 degrees - phi / 2).
• Thrust of the fill P_s = 1/2 K_a gamma H^2, acting H/3 above the underside of the base. H is the stem height plus the base thickness, because the thrust acts on the vertical plane through the heel.
• Surcharge q adds a thrust P_q = K_a q H at H/2, and its weight on the heel.
• Overturning: M_o = P_s H/3 + P_q H/2 about the toe, resisted by the weights times their lever arms. FS_o = M_r / M_o.
• Sliding: friction coefficient mu = tan(2 phi / 3), resistance mu W, FS_s = mu W / P_h.
• Base pressure: the resultant crosses the base at x_r = (M_r - M_o) / W from the toe, eccentricity e = B/2 - x_r, and q = W/B (1 plus or minus 6e/B).

Worked example

Defaults: a 3 m stem on a 2.4 m by 0.4 m base, a 0.4 m stem, a 0.6 m toe, medium sand (32 degrees), no surcharge, concrete at 24 kN/m^3 and fill at 18 kN/m^3. Engine results:

Overturning is comfortable at 4.95. Sliding is closer, at 1.56 against 1.5 required. The resultant is within the middle third of the base, so the whole base is in compression and the toe and heel pressures above apply.

Turn on the surcharge

Ticking the surcharge box (10 kPa on the fill) raises the horizontal thrust to 42.41 kN. Overturning drops to 3.76, and sliding to 1.3, which is below 1.5, so the sheet now prints that the factor of safety is too low and suggests widening or thickening the base. The toe pressure rises to 80.51 kPa. Sliding governs, and the sheet shows it immediately.

What this is, and is not

The sheet source

# Retaining wall: sliding and overturning

A template with inputs. Choose **Input** in the results bar to fill in the form. A cantilever
retaining wall of concrete (a stem on a base slab) holds back a level fill; the sheet takes a strip of wall one metre long.
The active earth pressure is $K_a = \tan^2(45^\circ - \phi/2)$, acting on the vertical plane through
the heel, so the thrust is $P_a = \tfrac12 K_a \gamma H^2$ at $H/3$ above the underside of the base. A
surcharge $q$ on the fill adds $K_a q H$ at $H/2$ and its weight on the heel. The wall slides on a
base friction of $\tan(2\phi/3)$, and overturns about the toe. The soil in front of the toe is ignored.

## Inputs
H_s:   ? 3 m | height of the stem above the base slab
B:     ? 2.4 m | width of the base slab
t_b:   ? 0.4 m | thickness of the base slab
t_s:   ? 0.4 m | thickness of the stem
L_t:   ? 0.6 m | length of the toe, in front of the stem
soil:  ? (Loose sand = 28 °, *Medium sand = 32 °, Dense sand = 36 °) | backfill, friction angle
q_s:   ? 10 kPa | surcharge on the fill, when it is on
sur:   ? off | surcharge on the fill

## Constants
gamma_c: 24 kN/m^3 | unit weight of concrete
gamma_s: 18 kN/m^3 | unit weight of the fill
b_w:     1 m | length of wall in the strip
FS_min:  1.5 | required factor of safety, for sliding and for overturning

## Geometry
phi: soil | friction angle of the fill
H:   H_s + t_b -> m | height of the thrust plane, above the underside of the base
L_h: B - L_t - t_s -> m | length of the heel, behind the stem
q:   sur * q_s -> kPa | surcharge in use

## Earth pressure
K_a: tan(45 ° - phi / 2)^2 | active pressure coefficient
P_s: K_a * gamma_s * H^2 / 2 * b_w -> kN | thrust of the fill
P_q: K_a * q * H * b_w -> kN | thrust of the surcharge
P_h: P_s + P_q -> kN | total horizontal thrust

## Weights and their lever arms about the toe
W_stem: gamma_c * t_s * H_s * b_w -> kN | stem
x_stem: L_t + t_s / 2 -> m | its lever arm
W_base: gamma_c * B * t_b * b_w -> kN | base slab
x_base: B / 2 -> m | its lever arm
W_fill: gamma_s * L_h * H_s * b_w -> kN | fill on the heel
W_sur:  q * L_h * b_w -> kN | surcharge on the heel
x_heel: B - L_h / 2 -> m | lever arm of the fill and the surcharge
W: W_stem + W_base + W_fill + W_sur -> kN | total vertical load

## Overturning
M_o: P_s * H / 3 + P_q * H / 2 -> kN*m | overturning moment about the toe
M_r: W_stem * x_stem + W_base * x_base + (W_fill + W_sur) * x_heel -> kN*m | resisting moment about the toe
FS_o: M_r / M_o | factor of safety against overturning
if FS_o >= FS_min: "Overturning: OK." else: "Overturning: the factor of safety is too low, widen the base."

## Sliding
mu:   tan(2 * phi / 3) | friction coefficient of the base on the ground
R_h:  mu * W -> kN | sliding resistance
FS_s: R_h / P_h | factor of safety against sliding
if FS_s >= FS_min: "Sliding: OK." else: "Sliding: the factor of safety is too low, widen the base or make it thicker."

## Bearing pressure under the base
x_r: (M_r - M_o) / W -> m | where the resultant crosses the base, from the toe
e:   B / 2 - x_r -> m | eccentricity from the middle of the base
q_max: W / (B * b_w) * (1 + 6 * e / B) -> kPa | pressure at the toe
q_min: W / (B * b_w) * (1 - 6 * e / B) -> kPa | pressure at the heel
if abs(e) <= B / 6: "The resultant is inside the middle third: the whole base is in compression." else: "The resultant is outside the middle third: part of the base lifts off, and the pressures above no longer apply."

How to use it

1. Open the template. Choose Open this template on this page. The sheet opens in a new tab of CivilKit Calcs, with the inputs set to the values used on this page.
2. Switch to Input mode. In the results bar choose Input. The input lines become boxes, lists, buttons and ticks.
3. Enter your values. Change the inputs to your own. The wall dimensions, the backfill and the surcharge are the inputs. Every change recalculates the whole page.
4. Read and export the result. Read the results and the OK or not OK lines under them. Use Report, then Print PDF, Word or Excel, to keep the calculation with its workings.