What is the Rankine passive earth pressure coefficient?

Kp = tan²(45° + φ/2) is a dimensionless ratio from Rankine's theory for cohesionless soils. It relates vertical effective stress to the maximum horizontal passive pressure. A higher friction angle gives a higher Kp, meaning greater passive resistance.

What does "passive" earth pressure mean?

Passive pressure develops when the wall moves toward the soil, compressing the backfill laterally. It represents the maximum possible resistance the soil can provide and is much larger than active pressure for the same soil.

How does Kp relate to Ka?

For the same friction angle φ, Kp = 1/Ka. For example, if Ka = 0.333 then Kp = 3.0. The passive coefficient is always greater than 1 for any φ > 0°, while Ka is always less than 1.

Does this calculator account for cohesion (c)?

No. The Rankine formula here assumes c = 0 (cohesionless soil: sand, gravel). For cohesive soils, a modified Rankine or Coulomb method including the 2c√Kp term is required.

How is the resultant passive force calculated?

The resultant horizontal passive force per unit width is: P_p = ½·Kp·γ·H² (triangular soil component) + Kp·q·H (rectangular surcharge component). The triangular component acts at H/3 from the base; the rectangular at H/2.

Can I use imperial units (pcf, feet)?

Yes. Select "Imperial" in the unit toggle. Enter unit weight in pcf (lb/ft³) and height in feet; results will be in psf and lb/ft.

Passive Earth Pressure Calculator

Compute the Rankine passive earth pressure coefficient (Kp), total lateral passive pressure, and resultant force on a retaining wall from soil friction angle, unit weight, wall height, and surcharge.

Soil & Wall Parameters

Units:

Internal Friction Angle φ (degrees)

Typical range: 28° – 42° for granular soils.

Soil Unit Weight γ (kN/m³)

Typical: 17–21 kN/m³ (108–133 pcf).

Embedded Wall Height H (m)

Surcharge Load q (kN/m²) (optional)

Uniform load on the passive soil surface (e.g. fill, slab).

Kp (Rankine)

—

dimensionless

Resultant Force / width

—

kN/m

Passive Pressure Diagram

Lateral pressure (kPa)

Depth	Soil Pressure	Surcharge Press.	Total Pressure

Force Breakdown (per unit width)

Triangular soil force (P_soil = ½·Kp·γ·H²) —

Rectangular surcharge force (P_q = Kp·q·H) —

Total passive resultant P_p —

Soil component acts at — from base (H/3).

Surcharge component acts at — from base (H/2).

Combined resultant acts at — from base.

Kp vs Ka Comparison

Ka (Active)

—

Kp (Passive)

—

Kp / Ka

—

Passive pressure is always larger than active pressure for the same soil. The ratio Kp/Ka illustrates the difference in resistance vs. driving force.

Enter soil parameters and click Calculate.

Kp and the passive pressure diagram will appear here.

Summary

This tool applies Rankine's theory to compute the passive earth pressure coefficient Kp = tan²(45° + φ/2), where φ is the effective internal friction angle of cohesionless backfill. It calculates the passive lateral pressure at any depth, the contribution of a uniform surcharge, and the total resultant horizontal passive force per unit length of wall. Passive pressure resists wall movement into the soil and is the maximum lateral earth pressure state.

How it works

Enter the soil internal friction angle (φ) in degrees.
Enter the soil unit weight (γ) in kN/m³ or pcf.
Enter the embedded wall height (H) in meters or feet.
Optionally enter a uniform surcharge load (q) acting on the soil surface.
The calculator applies Kp = tan²(45° + φ/2) (Rankine passive theory).
Results show Kp, the pressure diagram, and total horizontal passive resultant force per unit width.

Use cases

Checking passive resistance for retaining wall sliding stability.
Designing sheet pile walls and calculating toe resistance.
Geotechnical coursework and exam problems.
Comparing Rankine Kp with active Ka to assess overall wall equilibrium.
Estimating passive resistance behind deadman anchors.
Quick sanity checks before running full numerical models.

Frequently Asked Questions

Last updated: 2026-05-23 · Reviewed by Nham Vu