Stand a vertical wall against a bank of soil and the soil does not just sit there. It leans on the wall, and it leans hard. That sideways push is what topples basement walls during construction, bows out retaining structures over the years, and blows out unbraced excavations without warning. Soil behaves partly like a fluid — the deeper you go, the harder it pushes — but unlike water, the pressure also depends on the soil's own internal friction and on which way the wall is moving.
This article explains lateral earth pressure: where it comes from, how Rankine's theory turns soil properties into a pressure you can design for, a full worked example on a granular backfill, and the mistakes that most often lead to an underdesigned wall.
Why this calculation matters
Lateral earth pressure is the load case behind a long list of structures: retaining walls, basement and foundation walls, bridge abutments, sheet-pile cofferdams, and braced excavations. Every one of them holds back a body of soil, and every one of them must be sized for the horizontal thrust that soil delivers. Underestimate the pressure and the wall slides, overturns, or cracks. Overestimate it and the wall becomes needlessly massive and expensive.
What makes earth pressure more subtle than a simple fluid load is that its magnitude depends on movement. A wall that yields slightly away from the soil lets the backfill relax into its active state, the lowest pressure it can exert. A wall pushed into the soil mobilises the passive state, the highest resistance the soil can offer. A wall held perfectly rigid sees the at-rest pressure, somewhere in between. Knowing which state applies — and designing for it — is central to getting a retaining structure right.
The core formula
Rankine's theory, published in 1857, gives a clean way to estimate earth pressure for the idealised case of a smooth vertical wall retaining horizontal backfill. The pressure at any depth is the vertical stress multiplied by an earth pressure coefficient that depends on the soil friction angle phi.
For the active case — the wall yielding away from the soil — the coefficient is:
Ka = (1 - sin(phi)) / (1 + sin(phi))
The passive coefficient, for a wall pushed into the soil, is the reciprocal:
Kp = (1 + sin(phi)) / (1 - sin(phi))
The contrast is large. A soil with a friction angle of 30 degrees gives Ka = 0.333 and Kp = 3.0 — a ninefold difference. That is why passive resistance in front of a wall toe can be a major stabilising force, and why losing it, through erosion or excavation, is so dangerous.
For a dry granular backfill with no surcharge, the active pressure increases linearly with depth z:
p = Ka*gamma*z
where gamma is the soil unit weight. The pressure forms a triangular distribution, zero at the top and largest at the base, p = Ka*gamma*H. Because the distribution is triangular, the total active thrust per unit length of wall is the area of that triangle:
Pa = 0.5*Ka*gamma*H^2
and it acts at the centroid of the triangle, one third of the wall height above the base.
A worked example
Take a smooth vertical wall retaining a dry granular backfill. The soil has a friction angle phi = 30 degrees and a unit weight gamma = 18 kN/m^3, and the wall retains the soil to a height H = 4 m.
Step 1 — compute the active earth pressure coefficient.
Ka = (1 - sin(30 degrees)) / (1 + sin(30 degrees))
Ka = (1 - 0.5) / (1 + 0.5)
Ka = 0.5 / 1.5 = 0.333
Step 2 — find the active pressure at the base. The pressure grows linearly with depth and reaches its maximum at z = H:
p = Ka*gamma*H = 0.333 * 18 * 4 = 24 kPa
Step 3 — compute the total active thrust per metre of wall. This is the area of the triangular pressure diagram:
Pa = 0.5*Ka*gamma*H^2
Pa = 0.5 * 0.333 * 18 * 16
Pa = 48 kN/m
Step 4 — locate the thrust. The resultant acts at the centroid of the triangle, one third of the height above the base:
H/3 = 4 / 3 = 1.33 m above the base
So this wall must resist a horizontal thrust of 48 kN per metre of length, applied 1.33 m above its base. That lever arm matters as much as the force itself: the overturning moment about the wall toe is the product of the two, and it is the moment, not the force alone, that decides whether the wall tips.
Common mistakes
Designing for the wrong pressure state. Using the at-rest coefficient where the wall can yield wastes material; using the active coefficient on a rigid, unyielding wall underestimates the load. The wall's restraint and expected movement determine which coefficient applies.
Placing the thrust at mid-height. Because the active pressure distribution is triangular, not rectangular, its resultant acts at one third of the height from the base — not at the middle. Putting it at mid-height understates the overturning moment.
Ignoring water behind the wall. Rankine's dry-backfill formula leaves out pore water entirely. If drainage fails and the backfill saturates, full hydrostatic water pressure adds to the soil pressure, and water pushes with a coefficient of 1.0, far higher than Ka. Poor drainage is one of the most common causes of retaining wall failure.
Forgetting surcharge loads. Traffic, stockpiled material, or a building near the top of the wall adds a vertical surcharge that raises the lateral pressure at every depth. A surcharge q adds a roughly uniform pressure of Ka*q across the full wall height.
Counting on passive resistance that may not be there. Passive pressure in front of the wall toe is a genuine stabilising force, but only while the soil that provides it stays in place. Future excavation, scour, or erosion can remove it, so it is often discounted or used cautiously.
Try the interactive NovaSolver calculator
Hand calculation is the right way to learn the method, but comparing cases — different friction angles, wall heights, and surcharges — is far faster interactively. The Rankine Earth Pressure Simulator on NovaSolver computes lateral earth pressure for a vertical wall with horizontal backfill — set the soil friction angle, unit weight, wall height, and surcharge, and it returns the active and passive coefficients Ka and Kp, the active and passive forces Pa and Pp, and a live diagram of the vertical pressure distribution along with the coefficient curves against friction angle.
Related calculators
- Retaining wall calculator — turns the earth thrust into checks on sliding, overturning, and bearing for a complete wall design.
- Slope stability analysis — for deciding whether soil can stand at its natural angle before a wall is even needed.
- Bearing capacity calculator — to confirm the soil beneath the wall base can carry the vertical load the wall delivers.
You can browse the full set in the geotechnical tools hub.
Closing note
Lateral earth pressure is the load that defines retaining structures, and Rankine's theory gives an honest first estimate of it from just two soil properties — friction angle and unit weight — plus the wall height. The key ideas are worth carrying with you: pressure grows linearly with depth, the active state is far gentler than the passive state, the resultant thrust sits one third of the way up the wall, and water behind a wall changes everything. Compute the thrust, find its line of action, account for drainage and surcharge, and the retaining wall design that follows rests on solid ground.
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