Rankine Active Earth Pressure Coefficient

William Rankine's 1857 analysis asks what horizontal stress a soil mass needs in order to be on the verge of failing by spreading outwards. The answer depends only on the friction angle: at φ = 30°, $K_a$ = tan²(30°) = 1/3, so a wall that yields slightly away from its backfill feels only a third of the vertical stress pushed sideways at it. At φ = 35° it drops to 0.271, at 40° to 0.217: compacted granular backfill is cheap insurance.

The trap is the word "active". These pressures only develop if the wall actually moves, and the movement required is small but real: roughly 0.1% of the wall height for dense sand, 0.4% for loose. A basement wall braced by the floor slabs cannot move, so it sees at-rest pressure K₀ ≈ 0.5, half again as much, and designing it for $K_a$ is a classic and expensive error. The second trap is the assumptions behind the tidy formula: vertical smooth wall, level backfill, no cohesion, no water. Any wall friction, backslope or surcharge, and you need Coulomb's theory or a wedge analysis; any water behind the wall and the hydrostatic pressure will dwarf the earth pressure entirely, which is why drainage, not thickness, is what keeps retaining walls standing.