Learn: Rankine passive earth pressure coefficient
Rankine’s passive coefficient relates limiting horizontal effective stress to vertical effective stress in a simplified soil state. Passive resistance develops when a wall moves into the soil.
What the formula is saying
First calculate sin φ using the effective friction angle. Combine 1 + sin φ and 1 − sin φ in the order shown. The active and passive formulas are reciprocals for this same idealized Rankine case.
Read the symbols in plain language
- φ′
- Effective friction angle
Angle defining the frictional part of drained effective-stress shear strength; enter degrees here, not its tangent.
degAngles are entered in degrees; multiply by π/180 for trigonometric calculations in radians.
- Kp
- Result to find
Rankine passive earth pressure coefficient. Apply the exact expression for this stress state; the other earth-pressure states use different relationships.
ratio / no unit
Sort out the units first
φ is in degrees and must be converted to radians before the sine function. The coefficient has no unit and multiplies a compatible effective stress; water pressure is added separately where appropriate.
Assumptions before calculating
Assume cohesionless homogeneous soil, level backfill, a vertical smooth wall and sufficient movement to reach the specified Rankine limit.
Let’s solve one together
Read the given values, follow each operation, then check what the result means.
Read the supplied values as one complete study case. Find Kp and explain the result in the stated output unit.
- φ′ · Effective friction angle
- 30 deg
Find the sine of the friction angle
Convert the degree input to radians before evaluating the sine.
sin((30) × π ÷ 180) = 0.5Calculate the passive coefficient
Apply the exact expression for this stress state; the other earth-pressure states use different relationships.
(1 + (0.5)) ÷ (1-(0.5)) = 3
Does this worked answer make sense?
At φ = 0 the coefficient is 1. For φ > 0, Ka is below 1 and Kp is above 1; their product is 1 in the stated model.
A second worked example — different values
A second case uses different data. Predict which way the answer will change, then calculate it without reusing the first answer.
- φ′ · Effective friction angle
- 35 deg
Find the sine of the friction angle
Convert the degree input to radians before evaluating the sine.
sin((35) × π ÷ 180) ≈ 0.5735764364Calculate the passive coefficient
Apply the exact expression for this stress state; the other earth-pressure states use different relationships.
(1 + (0.5735764364)) ÷ (1-(0.5735764364)) ≈ 3.690172332
Now try your own values
Change a value or its unit. The same method will show your calculation, step by step.
Results update only when you calculate. The lesson example above stays unchanged.
Your turn — check your understanding
Solve this separate case yourself. Use only the values below; the two worked examples use different data. Give the requested result in the selected unit.
- φ′ · Effective friction angle
- 25 deg
Find: Learn: Rankine passive earth pressure coefficient
A hint, not the answer
First calculate sin φ using the effective friction angle. Combine 1 + sin φ and 1 − sin φ in the order shown. The active and passive formulas are reciprocals for this same idealized Rankine case.
φ is in degrees and must be converted to radians before the sine function. The coefficient has no unit and multiplies a compatible effective stress; water pressure is added separately where appropriate.
Show the full practice solution
Compare the steps with your work; revealing a solution does not mark the lesson complete.
Find the sine of the friction angle
Convert the degree input to radians before evaluating the sine.
sin((25) × π ÷ 180) ≈ 0.4226182617Calculate the passive coefficient
Apply the exact expression for this stress state; the other earth-pressure states use different relationships.
(1 + (0.4226182617)) ÷ (1-(0.4226182617)) ≈ 2.463912811
Avoid the common trap
Do not mix active, passive and at-rest states merely because the same φ is used. Do not put φ itself in place of sin φ or add pore pressure into the effective-stress coefficient.
When this method applies — and when it does not
Require 0 ≤ φ < 90° mathematically; realistic soil parameters must still be justified. Wall friction, sloping backfill, cohesion, layered soils, seismic loading and limited displacement require other models. Passive resistance in particular must be mobilizable before it is relied on.
For study and understanding, not approval of a real structure, site operation or design. Apply the correct standard, National Annex and professional review to actual engineering work.
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Sources & further reading
References open in a new tab and explain the underlying principles. The teaching text and examples here are SimpleFlick’s own; the source organisations have not endorsed this calculator.
Reading focus: Rankine passive earth pressure coefficient. Read the relevant soil phase, seepage, earth-pressure, settlement or foundation topic. Effective stress, drainage and idealized geometry determine whether the relationship applies.
Lesson updated: · Both examples and the separate practice case are checked against an independent high-precision numerical implementation. This verifies arithmetic for the stated model, not engineering certification.