Learn: At-rest earth pressure — Jaky approximation
Soil restrained from lateral movement can retain an at-rest horizontal stress different from the active and passive limits. Jaky’s empirical expression estimates the effective at-rest coefficient for normally consolidated soil.
What the formula is saying
First calculate sin φ using the effective friction angle. Subtract it from 1; this is an empirical normally consolidated at-rest relationship rather than a limiting failure solution.
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.
- K0
- Result to find
At-rest earth pressure — Jaky approximation. 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
Use a normally consolidated soil under one-dimensional loading with the applicable empirical at-rest approximation.
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 K0 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 at-rest coefficient
Apply the exact expression for this stress state; the other earth-pressure states use different relationships.
1-(0.5) = 0.5
Does this worked answer make sense?
At φ = 0 the coefficient is 1. Increasing φ reduces this normally consolidated K0 estimate.
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 at-rest coefficient
Apply the exact expression for this stress state; the other earth-pressure states use different relationships.
1-(0.5735764364) ≈ 0.4264235636
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: At-rest earth pressure — Jaky approximation
A hint, not the answer
First calculate sin φ using the effective friction angle. Subtract it from 1; this is an empirical normally consolidated at-rest relationship rather than a limiting failure solution.
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 at-rest coefficient
Apply the exact expression for this stress state; the other earth-pressure states use different relationships.
1-(0.4226182617) ≈ 0.5773817383
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. Overconsolidation, stress history, compaction and restraint can change K0; the simple expression does not model those effects.
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: At-rest earth pressure — Jaky approximation. 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.