Calculate the Rankine passive earth pressure coefficient Kp from the soil friction angle φ, or find φ from Kp, with active Ka and at-rest K₀ shown.

Typical φ: loose sand about 28–30°, medium sand 30–35°, dense sand 35–40°. Use test data for design.

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Passive Pressure Coefficient Formula

This Rankine formula applies to a smooth vertical wall and level, homogeneous, cohesionless backfill at the fully mobilized passive limiting state. Passive resistance requires wall movement; a coefficient alone does not determine safe retaining-wall design.

Kp = (1 + sin(φ)) / (1 - sin(φ))

Variables:

  • Kp is the Passive Pressure Coefficient
  • φ is the angle of internal friction in degrees

To calculate the Passive Pressure Coefficient, convert the angle of internal friction from degrees to radians. Then, apply the formula to determine Kp.

How do you calculate passive pressure and wall force?

Kp is a dimensionless coefficient, not a pressure or a force. For dry, homogeneous, cohesionless soil against a smooth vertical wall with level ground, the fully mobilized passive pressure at depth z is:

pp = Kp × γ × z

Here γ is soil unit weight (weight force per volume). Pressure increases linearly with depth. Over a soil height H, the triangular pressure distribution gives a resultant force per unit wall length:

Pp = ½ × Kp × γ × H²

This resultant acts at H/3 above the base. Using γ in kN/m³ and z or H in meters gives pressure in kPa and force per wall length in kN/m. These equations exclude groundwater, surcharge, cohesion, wall friction, sloping ground and seismic effects. Full passive resistance also requires enough wall movement into the soil; it is not automatically available for a stationary wall.

What is Passive Pressure Coefficient?

The Passive Pressure Coefficient (Kp) is a dimensionless factor used in geotechnical engineering to describe the ratio of lateral earth pressure to vertical earth pressure in a soil mass. It is particularly relevant in the context of retaining walls and other structures that interact with soil. The value of Kp depends on the angle of internal friction of the soil, which is a measure of the soil's shear strength. For the same vertical effective stress under this model, higher Kp implies greater limiting passive lateral resistance. Water pressure, cohesion, wall friction and sloping backfill require separate treatment.

How to Calculate Passive Pressure Coefficient?

The following steps outline how to calculate the Passive Pressure Coefficient (Kp).

  1. First, determine the angle of internal friction (φ) of the soil.
  2. Next, convert the angle from degrees to radians by multiplying by π/180.
  3. Finally, use the formula Kp = (1 + sin(φ)) / (1 - sin(φ)) to calculate the Passive Pressure Coefficient.
  4. After inserting the values and calculating the result, check your answer with the calculator above.

Example Problem : 

Use the following variables as an example problem to test your knowledge.

Angle of internal friction (φ) = 30 degrees

Kp = (1 + sin 30°) / (1 − sin 30°) = 1.5 / 0.5 = 3 (dimensionless).