Choose a solve mode to calculate Rankine passive pressure coefficient Kp from friction angle φ, or find φ from Kp. Use 0 ≤ φ < 90° or Kp ≥ 1; Kp = 1 gives φ = 0°.

Rankine model: smooth vertical wall, level homogeneous cohesionless backfill and fully mobilized passive resistance. A coefficient alone does not establish safe retaining-wall design.

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.

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).