CEE 340 · Advanced Foundation Engineering · Ch. 8

Earth Retaining Structures

Last chapter told you how hard the soil pushes. This one is about surviving that push. A retaining wall has exactly three ways to lose — tip over, slide out, or sink — and design is nothing more than proving, with numbers, that none of them happen.

4 Wall Types Overturning Sliding Bearing & Eccentricity
01

Three Ways a Wall Loses

A retaining wall is a structure that holds back soil (the backfill) and resists the lateral earth pressure it generates. It has no bracing, no floor slab, nothing tying it to another structure. It stands up purely by being heavy enough and wide enough. So the whole design reduces to one question asked three different ways: is it heavy and wide enough?

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Picture yourself bracing a heavy door that's swinging toward you. There are exactly three ways you lose. You get tipped over — your feet stay planted but your body rotates backwards. Your feet slide out from under you across the floor. Or the floor itself crumbles beneath your shoes. Nothing else. A retaining wall faces the identical three, and design is just proving each one won't happen by a comfortable margin.

The Three Checks, Visually

1. Overturning 2. Sliding 3. Bearing failure

Rotation about the toe, translation along the base, and crushing of the foundation soil. Every number you compute in this chapter serves one of these three.

Why it matters: These three checks are the entire midterm-exam workflow for retaining wall design, and they're the same three an engineer runs on a real wall before it gets built. Miss one and the wall may stand for years and then fail the first time the backfill saturates.
02

The Four Wall Types

All four do the same job. They differ in where the resisting weight comes from — and that single difference drives cost, buildability, and the height each type can economically reach.

Type 1 — Gravity

Wins by sheer mass

Plain concrete or stone masonry, thick and heavy. No reinforcement — the section is made big enough that it never goes into tension. Simple to build, but concrete volume grows fast with height, so it stops being economical past roughly 3 m.

Type 2 — Semi-Gravity

Mass, trimmed with a little steel

A gravity wall with light reinforcement added so the section can be slimmed down. A middle ground: less concrete than pure gravity, less steel detailing than a cantilever.

Type 3 — Cantilever

Uses the backfill's own weight

A reinforced-concrete T or L: a thin vertical stem on a wide base slab. The clever part is the heel — the slab extends back under the backfill, so the soil sitting on it becomes stabilizing weight. The most common type, economical to about 6–8 m.

Type 4 — Counterfort

Cantilever plus internal ribs

For tall walls, the stem's bending moment becomes brutal. Counterforts — thin concrete ribs tying the stem back to the heel at intervals — turn the stem into a slab spanning horizontally between supports. Used above roughly 7–8 m.

Core Idea

A gravity wall is a sumo wrestler — it wins on mass alone. A cantilever wall is a judo throw: it makes the soil's own weight do the work of holding the soil back. That's why it needs so much less concrete for the same height.

03

Preliminary Dimensions

Stability checks tell you whether a wall works — they can't tell you what to try first. So design starts with proportioning rules of thumb, then iterates: size it, check it, adjust, re-check. These proportions get most walls close on the first pass.

Stem top thickness≥ 0.3 m (12 in) — for concrete placement
Base slab thickness DH/12 to H/10
Base width B0.5H to 0.7H
Toe projection≈ B/3
Minimum embedment Df≥ 0.6 m, below frost/scour depth
Stem bottom thicknesstapered up from base for moment
Why the base must be buried: passive resistance at the toe only exists if there's soil in front of the wall. Leave the base exposed — or let someone excavate in front of it later — and the sliding check you passed on paper quietly becomes false in the field.
04

The Five Failure Modes

Three of these belong to the wall. Two belong to the ground the wall happens to be standing on — and those two are the ones that surprise people, because a perfectly designed wall can still ride a failing hillside down.

Mode 1

Overturning about the toe

The wall rotates forward about its front bottom edge. Resisted by the weight of the wall and the backfill on the heel.

Mode 2

Sliding along the base

The whole wall translates outward. Resisted by friction and adhesion under the base slab, plus passive pressure at the toe.

Mode 3

Bearing capacity failure

The foundation soil is crushed by the pressure under the base — worst at the toe, because the load is eccentric.

Mode 4

Deep-seated shear failure

A slip surface passes beneath the entire wall. This is a slope-stability problem, not a wall problem — covered in the slope stability chapter.

Mode 5

Excessive settlement

The wall doesn't collapse, it just sinks or tilts enough to crack, misalign, or lose serviceability. A settlement problem, checked separately.

The pattern

Strength vs. serviceability

Modes 1–3 are strength checks with factors of safety. Modes 4–5 ask a different question: is the site stable, and can it tolerate the movement?

05

Check 1 — Overturning

Intuition

A seesaw pivoted at the toe

The earth pressure pushes the wall's upper body outward, trying to spin it about the front bottom edge. Everything heavy sitting on the base — concrete and backfill alike — pushes down at some distance behind that pivot and spins it the other way.

Mechanics

Moments about the toe

Take moments about the toe. The driving moment is the horizontal active force times its lever arm (H'/3 above the base). The resisting moment is each weight times its horizontal distance from the toe. The heel soil dominates — it's usually the single biggest term.

The Math

Factor of safety

\[ FS_{OT} = \frac{\sum M_R}{\sum M_O} \ge 2 \] \[ \sum M_O = P_a\cos\alpha \cdot \frac{H'}{3} \] \[ \sum M_R = \sum W_i x_i \]

FS ≥ 2 for granular backfill, ≥ 3 for cohesive.

Core Idea

Widening the heel is almost always the cheapest fix for a failing overturning check — it adds a large soil weight at a long lever arm, and costs only a bit more slab.

06

Check 2 — Sliding

Intuition

Friction under your shoes

Nothing is bolted down. The only thing stopping the wall from being shoved outward is friction between the underside of the base slab and the soil — and friction is proportional to how hard the wall presses down. Heavier wall, more grip.

Mechanics

Why we discount the strength

The interface is concrete cast against soil, not soil against itself, so it can't mobilize the soil's full φ' and c'. Practice uses reduction factors k₁, k₂ ≈ ½ to ⅔. Passive resistance at the toe may be added, but conservatively it's often ignored.

The Math

Factor of safety

\[ FS_{SL} = \frac{\sum F_R'}{P_a\cos\alpha} \ge 1.5 \] \[ \sum F_R' = (\sum V)\tan(k_1\phi'_2) + k_2 c'_2 B + P_p \]
When sliding won't pass: the standard fix is a shear key — a downward projection under the base that forces the potential slip surface to pass through soil-on-soil (full φ') rather than concrete-on-soil, and mobilizes extra passive resistance in front of the key.
07

Check 3 — Bearing & the Middle Third

Here's the subtlety that makes this check different from an ordinary footing: the resultant force under a retaining wall is not centered. The earth pressure is constantly trying to rotate the wall forward, which shifts the load toward the toe. So the pressure under the base is a trapezoid, not a uniform block — and the toe carries the peak.

🪑
Stand on a bathroom scale and lean forward. Your weight hasn't changed, but the pressure under your toes shoots up while the pressure under your heels drops. Lean far enough and your heels lift off entirely — no contact, no pressure. A wall base does exactly this, and the point where the heel "lifts off" is the middle-third rule.
Eccentricity & pressure
\[ e = \frac{B}{2} - \frac{\sum M_R - \sum M_O}{\sum V} \] \[ q_{max,min} = \frac{\sum V}{B}\left(1 \pm \frac{6e}{B}\right) \]
The two acceptance rules
\[ e \le \frac{B}{6} \quad\text{(no uplift at heel)} \] \[ FS_{BC} = \frac{q_{ult}}{q_{max}} \ge 3 \]

qult from Meyerhof, using the effective width B' = B − 2e.

Core Idea

If e > B/6, the formula returns a negative qmin — soil "pulling down" on the heel, which is impossible. The heel has physically lifted off. Keep the resultant inside the middle third of the base and the whole base stays in compression.

08

Live Stability Calculator

A complete cantilever wall check. Change any dimension and watch which check fails first — try shrinking the base width B, or widening the heel, and see how differently each factor of safety responds.

09

Worked Example

Example — Full Stability Check

Cantilever Wall: Overturning, Sliding & Bearing

Problem: A cantilever retaining wall has stem height Hs = 6 m, base thickness D = 0.7 m, base width B = 4.0 m, toe projection 1.0 m, stem tapering from 0.7 m at the bottom to 0.3 m at the top. Backfill: γ = 18 kN/m³, φ' = 32°, horizontal surface. Foundation soil: φ'₂ = 28°, c'₂ = 30 kPa, qult = 450 kPa. Concrete γc = 23.6 kN/m³. Compute all three factors of safety.

1
Driving force. Total height on the vertical plane through the heel: \( H' = 6 + 0.7 = 6.7\text{m} \). \( K_a = \tan^2(45-16) = 0.307 \), so \( P_a = \tfrac12(0.307)(18)(6.7)^2 = 124.1\text{kN/m} \) acting at \( H'/3 = 2.23\text{m} \) above the base.
2
Overturning moment. \( \sum M_O = 124.1 \times 2.23 = 277.2\text{kN·m/m} \)
3
Weights and resisting moments (moment arms measured from the toe). Heel width = 4.0 − 1.0 − 0.7 = 2.3 m.
ComponentW (kN/m)x (m)M (kN·m/m)
Stem (rectangle)42.51.5565.8
Stem (taper)28.31.2735.9
Base slab66.12.00132.2
Soil on heel248.42.85707.9
Σ385.3941.8
Note the soil on the heel is 64% of the total resisting weight — that's the cantilever wall doing its judo.
4
Check 1 — Overturning. \( FS_{OT} = 941.8/277.2 = 3.40 \ge 2 \) ✓
5
Check 2 — Sliding (using k₁ = k₂ = ⅔, passive resistance neglected). \( \sum F_R' = 385.3\tan(\tfrac23 \cdot 28^\circ) + \tfrac23(30)(4.0) = 130.2 + 80.0 = 210.2\text{kN/m} \). \( FS_{SL} = 210.2/124.1 = 1.69 \ge 1.5 \) ✓
6
Check 3 — Eccentricity & bearing. \( \bar{x} = (941.8-277.2)/385.3 = 1.725\text{m} \), so \( e = 2.0 - 1.725 = 0.275\text{m} \). Middle third: \( B/6 = 0.667\text{m} > 0.275 \) ✓ (whole base in compression). \( q_{max} = \tfrac{385.3}{4.0}(1 + \tfrac{6(0.275)}{4.0}) = 136.1\text{kPa} \), \( q_{min} = 56.6\text{kPa} \). \( FS_{BC} = 450/136.1 = 3.31 \ge 3 \) ✓
Answer: FSOT = 3.40, FSSL = 1.69, FSBC = 3.31, e = 0.275 m < B/6. All three checks pass — the wall is adequate. Sliding is the governing check, with the least margin.
Read the result, don't just report it. Sliding governs here at FS = 1.69 against a 1.5 requirement. That's the check to watch if anything changes — a wetter backfill, a shallower embedment, or a contractor trimming the base width will push it under 1.5 long before overturning or bearing become a concern.
10

Quick Reference & Quick Check

FS overturningΣMR / ΣMO ≥ 2
FS slidingΣFR' / Pacosα ≥ 1.5
FS bearingqult / qmax ≥ 3
Eccentricitye = B/2 − (ΣMR−ΣMO)/ΣV
Middle-third rulee ≤ B/6 (no heel uplift)
Base pressuresq = (ΣV/B)(1 ± 6e/B)
Base friction reductionk₁, k₂ ≈ ½ to ⅔
Effective width (bearing)B' = B − 2e

1. In a cantilever wall, what usually provides the largest share of the resisting moment?

2. Your calculation returns e = 0.9 m for a base width B = 4.0 m. What does this tell you?

3. Why are the base friction parameters reduced by k₁, k₂ ≈ ⅔?

4. A wall fails its sliding check. Which fix targets that specific failure mode most directly?