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Geotechnical Engineering Simulator

Rankine Earth Pressure Simulator — Active and Passive Earth Pressure on Retaining Walls

Compute active Pa, passive Pp and at-rest Ko earth pressures in real time using Rankine theory for vertical walls and horizontal backfill. Vertical pressure distribution and Ka/Kp/Ko vs phi curves update live.

Parameters
Friction angle phi
°
Unit weight gamma
kN/m³
Wall height H
m
Surcharge q
kPa
Results
—
Active coefficient Ka
—
Passive coefficient Kp
—
Active force Pa
—
Passive force Pp
Wall and pressure distribution
Earth pressure coefficient K vs friction angle phi
Theory & Key Formulas

$$K_a = \tan^2\!\left(45^\circ - \tfrac{\phi}{2}\right) = \frac{1-\sin\phi}{1+\sin\phi}$$

Active earth pressure coefficient. The minimum lateral pressure ratio when the wall moves away from the backfill.

$$K_p = \tan^2\!\left(45^\circ + \tfrac{\phi}{2}\right) = \frac{1+\sin\phi}{1-\sin\phi}$$

Passive earth pressure coefficient. The maximum resistance when the wall pushes into the soil; $K_p = 1/K_a$.

$$K_0 = 1 - \sin\phi$$

At-rest earth pressure coefficient (Jaky). Used for rigid walls and normally consolidated soils.

$$P_a = \tfrac{1}{2}K_a\,\gamma\,H^2 + K_a\,q\,H,\quad P_p = \tfrac{1}{2}K_p\,\gamma\,H^2 + K_p\,q\,H$$

Total active and passive force per unit wall length. The first term is from self-weight, the second from surcharge.

What is the Rankine Earth Pressure Simulator

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What kind of force is actually pushing on a retaining wall? It is just the soil leaning against it, right?
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Good question. Soil behind a wall is always being pulled down by gravity, and part of that load gets pushed sideways onto the wall as earth pressure. This tool uses Rankine theory, which assumes a vertical wall and a flat ground surface, and gives you active Pa, passive Pp, and at-rest Ko. Try sliding the friction angle phi from 30 to 40 degrees and watch Ka drop sharply.
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Wait, increasing phi reduces the pressure? I thought it would be the other way around.
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A common mix-up. Phi measures how well the soil holds itself up. Higher phi means the soil grains interlock and the soil leans on the wall less. Imagine wet sand: it can stand at a steeper slope on its own. Conversely, passive Kp grows quickly with phi because Kp = 1/Ka. At phi=30 the ratio Kp/Ka is about 9, and at phi=40 it is around 20.
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Then what is the surface load q for? Like a warehouse or a truck on top of the backfill?
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Exactly. A uniform load q on top of the backfill adds a constant Ka*q to the pressure at every depth, so the diagram becomes a triangle plus a rectangle. Slide q from 0 to 50 kPa and you will see the rectangular block grow from the top of the wall. Traffic loads are usually idealised as 10 to 20 kPa surcharges in practice.
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There is also K0 in the chart. Is it different from Ka and Kp?
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Sharp eye. Ka and Kp assume the wall actually moves, but for rigid walls, like a basement or a stiff abutment, the wall does not displace and the soil never reaches the active state. There you use the at-rest pressure K0 = 1 - sin(phi). At phi=30 that gives 0.5, larger than Ka = 0.333. Ignoring this and using Ka would underestimate the design load.

Physical model and key formulas

Rankine theory considers a semi-infinite soil mass at limit equilibrium. Once the so-called Rankine state is reached, the ratio between horizontal effective stress $\sigma_h$ and vertical effective stress $\sigma_v$ becomes a constant coefficient.

In the active state (wall moves away from the backfill) $\sigma_h = K_a \sigma_v$, and in the passive state (wall pushes into the soil) $\sigma_h = K_p \sigma_v$. For a vertical wall, horizontal backfill, and zero wall friction, the coefficients are:

$$K_a = \tan^2(45^\circ - \phi/2),\quad K_p = \tan^2(45^\circ + \phi/2),\quad K_0 = 1 - \sin\phi.$$

The active horizontal pressure at depth $z$ is $\sigma_a(z) = K_a (\gamma z + q)$. Integrating along the wall height H gives the total active force per unit wall length, $P_a = \tfrac{1}{2}K_a \gamma H^2 + K_a q H$. The passive side follows by replacing $K_a$ with $K_p$.

Real-world applications

Retaining wall stability: Sliding, overturning, and bearing checks for highway and residential walls use the active Rankine pressure as the lateral load on the back face. Underestimating phi leads to overdesign; overestimating it puts the wall on the unsafe side.

Sheet piles and soldier piles: The required embedment depth comes from balancing active pressure on one side against passive resistance on the other. The reliability of Kp in particular drives the safety margin.

Basement walls and bridge abutments: Walls that cannot displace are designed using K0 instead of Ka. At phi=30 degrees that means about 1.5 times more load than the active case, which is a meaningful difference.

Sanity check for FEM models: Before running a soil-structure interaction analysis in PLAXIS or Midas GTS, this tool gives a quick hand-calc estimate to verify that the FEM results have the right order of magnitude.

Common misconceptions and pitfalls

Mixing Rankine and Coulomb: Rankine assumes zero wall friction (delta = 0); Coulomb can include delta > 0. Real walls usually have delta ≈ phi/2 to 2phi/3. Using Rankine instead of Coulomb is conservative on the active side, but for inclined walls or sloped backfill Rankine is not applicable and you must switch theories.

Over-trusting Kp: Computed Kp values can be many times Ka, but the full passive resistance only develops at large wall displacements. Practice typically applies a safety factor of 2 to 3 to Kp before using it in design.

Ignoring groundwater: Below the water table you must use the buoyant unit weight gamma' = gamma - gamma_w and add hydrostatic pressure separately. This tool models a dry-soil idealisation, so projects with high groundwater require additional terms.

Frequently asked questions

Rankine assumes a vertical wall, horizontal backfill, and zero wall friction; coefficients come from the limit stress state of the soil. Coulomb uses a limit-equilibrium wedge analysis that allows wall friction and inclined geometry, so it is more general but more involved.
Active pressure Pa is the minimum pressure when the wall moves away from the backfill, used as a load on retaining wall stability checks. Passive pressure Pp is the maximum resistance when the wall pushes into the soil, used as side resistance on the toe or for embedded sheet piles.
Use Ko for rigid walls that do not displace, such as basement walls, box culverts, and stiff bridge abutments. Jaky's formula Ko = 1 - sin(phi) is the standard for normally consolidated soils; for overconsolidated soils multiply by OCR raised to sin(phi).
A uniform surcharge q adds a constant overburden Ka*q at every depth, producing a rectangular block stacked on the triangular self-weight diagram. Total active force gains an extra Ka*q*H term. Warehouse and traffic loads are usually idealised this way.

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