A retaining wall holds back soil that would otherwise slide or settle, and the load it must resist depends on how the wall moves. When a wall tilts or translates away from the backfill, the soil mobilizes its minimum resistance and the pressure drops to the active state. That state, called active earth pressure, is the lower bound of lateral load used in most cantilever and gravity wall designs. The full analysis sequence, from wall types to drainage details, is covered in this review of retaining wall engineering, which pairs the theory here with sheet pile walls and drainage systems for earth retention.
Rankine and Coulomb Theories of Active Pressure
Two classical theories dominate active pressure analysis. Rankine’s theory assumes a smooth, vertical wall with horizontal backfill, so the lateral stress on a soil element is the minimum principal stress while the vertical stress is the maximum principal stress. Coulomb’s theory adds wall friction and allows sloping backfill and inclined walls, which fits real geometries more closely. Designers compare the two when the wall face is rough, the backfill slopes, or the wall leans, and the differences between Rankine and Coulomb theories show up most clearly in those situations.
Assumptions Behind Each Theory
- Rankine: smooth wall face, no wall-soil friction, vertical wall, horizontal or uniformly sloping backfill
- Coulomb: wall friction included, any wall inclination, sloping backfill, wedge-based force equilibrium
- Both: the soil reaches the Mohr-Coulomb failure criterion and the wall moves enough to mobilize full shear strength
Neither theory captures creep, partial drainage, or compaction-induced locked-in stresses, so measured field pressures usually sit between the at-rest and active values.
When to Apply Rankine versus Coulomb
Rankine suits preliminary sizing when the wall is vertical, the backfill is flat, and wall friction can be ignored conservatively. Coulomb suits final checks when the backfill slopes away from the wall or the wall is battered. For typical cantilever walls with drained granular backfill, the two methods agree within about 10 percent, which is close enough for design.
Design Practice Guidance
Keep one method for the whole design. Mixing Rankine for the stem and Coulomb for the base produces inconsistent thrust locations and can undersize the footing. Record the chosen coefficient, the assumed wall roughness, and the backfill slope in the calculation set.
Cases of Active Earth Pressure on Cohesionless Backfill
Active pressure problems are classified by the condition of the backfill, and each case changes how the lateral stress is built up. The standard cases are dry or moist backfill, submerged backfill, partly submerged backfill, backfill with a uniform surcharge, and backfill with a sloping surcharge. A worked treatment of active earth pressure walks through the same cases with diagrams and numeric examples.
Dry or Moist Backfill
Consider an element at depth z below the ground surface. The vertical pressure comes from the weight of the soil above it, and when the wall moves away from the backfill, the lateral pressure falls to the minimum principal stress. The active lateral stress equals Ka times the vertical stress, so the pressure diagram is triangular, growing from zero at the surface to Ka times gamma times H at the base of a wall of height H.
Submerged Backfill
When the sand behind the wall is saturated, the lateral pressure has two components. The first comes from the submerged unit weight of the soil, and the second comes from the hydrostatic water pressure acting over the full wall height. Water adds load through both components, so a drained wall with weep holes or a gravel blanket usually carries far less thrust than the undrained calculation suggests.
Partly Submerged Backfill
A partly submerged condition means the backfill is moist down to depth H1 and saturated below that level. The pressure diagram is a triangle through the moist zone and a steeper composite line below, where the soil component switches to the submerged unit weight and the water component begins. The kink in the diagram sits at the water table, and the total thrust is the sum of the two areas.
Uniform Surcharge and Sloping Backfill
If the backfill is horizontal and carries a uniform surcharge of intensity q, the vertical pressure at every depth increases by q and the lateral pressure increases by Ka times q. The surcharge contribution is constant with depth, so it adds a rectangle to the pressure diagram. A sloping surcharge, such as a ramp or stockpile above the wall, is handled with the Coulomb wedge method, which resolves the slope weight into the sliding wedge.
Earth Pressure Coefficients: At Rest, Active, and Passive
On a small element at depth z in the backfill, two pressures act: the vertical earth pressure from the soil above and the horizontal earth pressure from the wall’s restraint. The ratio between the two defines the coefficient, and the value depends entirely on how the wall moves.
Coefficient of Earth Pressure at Rest
When the wall does not move at all, the ratio of lateral pressure to vertical pressure is the coefficient of earth pressure at rest, K0. For normally consolidated sands, K0 is often estimated as 1 minus sin phi, which gives values around 0.4 to 0.5 for typical sands. Rigid basements, braced excavations, and walls cast against rock operate near the at-rest state because movement is too small to mobilize the active condition.
Coefficient of Active Earth Pressure
When the wall moves away from the backfill, the lateral pressure falls until the soil reaches failure, and the ratio at that point is the active coefficient Ka. For cohesionless soil, Ka equals (1 minus sin phi) divided by (1 plus sin phi), which is the same as tan squared of (45 degrees minus phi over 2). A sand with a 30-degree friction angle gives Ka of 0.333, so the wall sees only one third of the vertical pressure as lateral load.
Coefficient of Passive Earth Pressure
Pushing the wall toward the backfill compresses the soil and raises the lateral pressure to the passive state, with Kp equal to (1 plus sin phi) divided by (1 minus sin phi). Passive resistance anchors the toe of a cantilever wall, and the passive earth pressure theory, calculation methods, and practical applications explain how much of that resistance can be counted on safely.
Typical Coefficient Values for Cohesionless Soils
| Friction angle (deg) | K0 at rest | Ka active | Kp passive |
|---|---|---|---|
| 25 | 0.58 | 0.41 | 2.46 |
| 30 | 0.50 | 0.33 | 3.00 |
| 35 | 0.43 | 0.27 | 3.69 |
| 40 | 0.36 | 0.22 | 4.60 |
The passive values look large, but they require the wall to move toward the soil by roughly 1 to 5 percent of the wall height, and partial mobilization under service loads is much smaller.
Step-by-Step Active Pressure Calculation
A hand calculation for a simple gravity or cantilever wall takes about six steps and produces the thrust used for sliding and overturning checks. The same sequence is used in design offices before any finite element model is opened, because the closed-form result gives a fast sanity check for the computer output.
Step-by-Step Procedure
- Collect soil properties: unit weight gamma, friction angle phi, and cohesion c, which is zero for clean sand.
- Compute the active coefficient Ka from the Rankine or Coulomb formula.
- Find the vertical stress at depth z as gamma times z, plus any surcharge q.
- Multiply by Ka to get the active lateral stress at that depth.
- Draw the pressure diagram and integrate over the wall height to get the total thrust.
- Add water pressure and surcharge rectangles where those cases apply.
Worked Example: Four-Meter Wall With Dry Backfill
Take a 4 m wall with dry sand backfill at 18 kN per cubic meter and a friction angle of 30 degrees. Ka is 0.333, so the lateral stress at the base is 0.333 times 18 times 4, which equals 24 kPa. The pressure diagram is a triangle, so the total thrust is half of 24 kPa times 4 m, or 48 kN per meter of wall, acting at one third of the height above the base.
Checking the Result
Compare the computed thrust with the at-rest value, which for the same wall is K0 times gamma times H squared over 2, or about 72 kN per meter. The at-rest earth pressure theory and calculation methods used in this comparison give the upper bound, and if the wall cannot tolerate the movement needed to reach the active state, the design should use a value between the two.
Design Considerations: Drainage, Surcharge, and Coefficient Selection
The biggest error in retaining wall design is usually water, not the coefficient. A wall designed for drained backfill can fail when the drainage clogs, because the hydrostatic component can exceed the soil component. Understanding the lateral earth pressure coefficient in retaining structure design is the starting point, but the drainage detail decides whether the wall ever sees that number.
Drainage and Water Pressure
Weep holes at 1 to 2 m spacing, a gravel drainage blanket behind the wall, and a perforated collector pipe at the base keep the water table below the wall. When the water table rises to the surface, the total thrust on a 4 m wall roughly doubles, which is why submerged backfill calculations are always paired with permanent drainage.
Surcharge From Adjacent Traffic and Stockpiles
Vehicles, stockpiles, and future fills act as uniform surcharges on the backfill surface. A 20 kPa surcharge, about the load of a loaded truck, adds Ka times 20 kPa of constant lateral stress over the full wall height. Include it in the diagram before integrating, because it changes both the total thrust and the location of the resultant.
Verifying Assumptions on Site
Field conditions rarely match the textbook. The backfill is often clayey, the drainage blanket is missing, or compaction equipment pushes the wall before the concrete cures. For rigid walls and basement conditions where movement is restricted, the at-rest earth pressure branch of the analysis provides the upper-bound check that active theory cannot.