Module 12 · Lesson 12.1

Earth pressure

Why the load on a retaining wall depends on the wall, and why water is the thing that actually breaks them.

Why this matters

Every other structure in this course has a load that exists independently of the structure. A retaining wall does not: the pressure it attracts depends on how far it moves, and a wall that cannot move attracts half as much again. Get that wrong and the wall is under-designed by 50% before any calculation begins — and no check anywhere will reveal it, because the calculation will be perfectly consistent with the wrong assumption.

By the end of this lesson you should be able to

State the three pressure states and the movement each requires
Compute the active, at-rest and passive coefficients
Explain why water is more severe than soil
Identify which quantities come from the ground investigation

What you should already know

Mohr's circle and principal stresses (Structural Analysis Fundamentals, Module 13)
Bearing pressure and the middle third (Module 11)

Three states, decided by movement

Soil behind a wall is not in one fixed state. What it does depends on what the wall does.

At rest. The wall has not moved. The soil is in the state it was left in by deposition and compaction, with horizontal stress a fixed fraction of vertical stress. K₀ ≈ 1 − sin φ.

Active. The wall yields away from the soil. The soil expands laterally, mobilises its shear strength, and the horizontal stress falls to a minimum. K_a = (1 − sin φ)/(1 + sin φ).

Passive. The wall pushes into the soil. The soil is compressed laterally, its shear strength again resists, and the horizontal stress rises to a maximum. K_p = 1/K_a.

For a 30° soil these are 0.50, 0.33 and 3.00. The spread is enormous — a factor of nine between active and passive — and which one applies is decided entirely by movement.

Rankine active coefficient

K_{a} = \frac{1 - sin φ}{1 + sin φ} = tan^{2} (4 5^{\circ} - \frac{φ}{2})

What it calculates: The ratio of horizontal to vertical effective stress in the active state.

$φ$: Angle of shearing resistance of the soil (degrees)
K_a: Active earth pressure coefficient (—)

This assumes

Cohesionless soil, or cohesion conservatively ignored
Smooth vertical wall back and horizontal ground surface
The wall has moved enough to mobilise the active state
DERIVED from the Mohr circle at failure — not empirical

In plain terms: The two forms are identical, and the second is worth knowing because the 45° − φ/2 is the angle of the failure plane in the soil. The coefficient is not a fudge factor: it is the ratio of the two principal stresses when a soil element is on the point of failing.

Try it

Check a retaining wall

Three stability checks at once. Move one slider and watch them trade against each other — then raise the water table, which is what most retaining walls actually fail on.

Retained height H5.00 m

Base width B3.2 m

Toe length0.9 m

Soil friction angle φ30 degrees

Surcharge q10 kPa

Water table above base0.00 m

Wall movement

Coefficient (active): 0.333
Horizontal force: 91.7 kN/m
Vertical load: 253.2 kN/m
Overturning moment: 167 kNm/m
Restoring moment: 497 kNm/m
Overturning factor, need 2.0: 2.98
Sliding factor, need 1.5: 1.38
Resultant in middle third?: yes
Peak bearing pressure: 123 kPa

Inadequate, governed by sliding. Overturning 2.98 against 2.0; sliding 1.38 against 1.5.

Things worth trying

Raise the water table to the top. The overturning moment nearly doubles and the wall fails outright — no amount of reinforcement would save it. Most retaining walls that fail, fail this way.
Widen the base. Sliding and overturning both improve, because more soil sits on the heel to hold the wall down. Base width is the cheapest variable a retaining wall has.
Switch to a propped wall. The at-rest coefficient is half as much again as the active one, so a basement wall designed for active pressure is under-designed by that margin.
Raise H by one metre. The overturning moment goes with the CUBE of the height, so it is a far bigger change than it looks.
Move the toe forward and back at fixed B. Sliding barely changes, but the resultant shifts and the bearing pressure redistributes — that is what the toe length is for.

Why water is the thing that breaks retaining walls

Soil pushes with K γ z. Water pushes with γ_w z — no coefficient at all.

That is not an oversight. A coefficient less than one exists because soil has shear strength: it can carry some of its own weight sideways through friction between grains. Water has no shear strength whatsoever, so it pushes with its full weight in every direction. K = 1, always.

Compare the two for a typical soil:

	Effective horizontal push per metre depth
Drained soil, K_a γ	0.33 × 18 = 6.0 kN/m³
Water, γ_w	9.81 kN/m³

Water pushes more than half again as hard as the soil it is sitting in. And when the water table rises, both act: the soil below it becomes buoyant, which relieves some of the earth pressure, but nowhere near enough to compensate.

A wall that is comfortable when drained can fail outright when the drainage blocks. Most retaining walls that fail, fail this way — not because the reinforcement was wrong, but because a filter drain silted up.

This is why the drainage behind a retaining wall is a structural detail, not a landscaping one.

Assumption

These are geotechnical inputs, not structural ones

The friction angle φ, the soil unit weight γ, the base friction coefficient and the allowable bearing pressure all come from a ground investigation. They are not values a structural engineer assumes.

This course uses φ = 30°, γ = 18 kN/m³ and a base friction of 0.5 throughout because they are representative of a medium-dense granular fill and make the worked examples consistent. They are not defaults to be reached for.

The same applies to the factors of safety used here — 2.0 on overturning and 1.5 on sliding. These are the traditional values and they make the physics visible. A Eurocode 7 design applies partial factors to actions and to the soil's material properties instead. The mechanics is identical; the bookkeeping differs, and the course does not pretend the traditional presentation is the only one.

Predict first

A drained 5 m wall has an overturning moment of 167 kNm/m. The drainage blocks and the water table rises to the top. Roughly what does the overturning moment become?

Practice

A soil has a friction angle of 35°. What is the active earth pressure coefficient K_a?

Practice

For that same 35° soil, what is the at-rest coefficient K₀?

Practice

A 5 m high wall retains soil with K_a = 1/3 and γ = 18 kN/m³. What is the horizontal force per metre run, in kN/m?

Check yourself

Why does water pressure use no earth-pressure coefficient?

Summary

Three states — at rest, active, passive — decided by how far the wall moves
For a 30° soil: K₀ = 0.50, K_a = 0.33, K_p = 3.00
Active needs about 0.1% of the height in rotation; a propped wall cannot deliver it
K_a is DERIVED from the Mohr circle, not fitted
Water uses K = 1 because it has no shear strength
Flooding roughly doubles the overturning moment, buoyancy notwithstanding
φ, γ, base friction and bearing resistance are geotechnical inputs, not assumptions

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This is educational material. It uses simplified examples to teach principles, and must not be relied on for real design or safety-critical decisions. Module overview and checkpoint