Foundation Engineering / Pile capacity in sand — the β-method
The two limits
Where the calculation and the measurements part company.
Lesson review: expert verification remains open
This lesson has received a full-context editorial review and a check of its technical explanations. Selected arithmetic has been checked separately. This is not independent engineering sign-off.
Corrections checked in this pass
- Honoured an explicit zero resistance cap and rejected negative or non-finite caps.
Checks needed for project use
- A geotechnical reviewer must confirm drainage and stress basis, bearing-factor formulation, correlations and settlement influence factors. Partial factors and project-specific ground conditions are outside these teaching checks.
- Site- and installation-specific beta, Nq and limiting shaft/base resistance require justification. Additional arbitrary invalid geometry/stress input validation is not fully implemented.
These are educational models and exercises. Check their assumptions before using the results. The full technical content and every diagram have not yet been independently verified.
Concrete and steel calculations retain their stated first-generation Eurocode teaching basis. Use the standards and amendments specified for your project; do not mix generations without an appropriate design basis.
Read the lesson register and outstanding review notesWhy this matters
The β-method says unit shaft friction grows without limit as σ′v grows with depth, and that base resistance grows in proportion too. Measurements on long piles say otherwise. What to do about that disagreement is one of the genuinely unsettled questions in foundation engineering.
Limiting effective vertical stress in this example. Some empirical pile methods cap the effective stress used in the shaft term; others model depth and installation differently. A fixed critical-depth rule is not universal.
Limiting base pressure. A selected method may also impose a base-resistance limit. Use limits supported by that method and ground conditions.
For the stated 25 m pile, integrating the uncapped stress profile gives about 2834 kN of shaft resistance. Capping σ′v at 100 kPa gives about 1814 kN, a 36% reduction. The 100 kPa cap is on vertical effective stress, not directly on unit shaft friction.
Worked example
A long driven pile in sand, with both limits applied
Given
- Driven pile, D = 0.5 m, embedded 25 m
- Sand throughout, γ′ = 10 kN/m³ — water table at the surface
- K = 1.0, δ = 30°, so β = 0.577
- Limiting shaft stress 100 kPa; deep Nq = 100; limiting base pressure 10 MPa
Find
The capacity, and how much the two limits removed
Check yourself
Why is a long pile in sand far less shaft-dominated than a comparable pile in clay?
Try it
Shaft friction with depth
The β-method says friction grows without limit. Measurements say otherwise. Set the cap and see the cost.
Still disputed
- Whether the 'critical depth' is a soil effect or an artefact of how the tests were done is not resolved.
- The practical response is the same either way: cap the calculation and say what you capped it at.
- Base resistance is subject to the same observation and is capped here at 10 MPa.
- β = K·tan δ
- 0.577
- Shaft, uncapped
- 2834 kN
- Shaft, capped
- 1814 kN
- Removed by the shaft cap
- 36 %
- Base, uncapped
- 4909 kN
- Base, capped at 10 MPa
- 1963 kN
- Ultimate, both caps
- 3777 kN
- Shaft share
- 48 %
- Shaft
- Base
- Removed by the limits
The two limits between them removed 51% of the calculated capacity. In sand the limits are not a refinement — they are most of the answer.
Summary
- β = K·tan δ: K is set by installation, δ by the interface, neither by a soil test
- A bored pile's β was 40% of a driven pile's in the same sand
- The deep Nq for a pile base is several times the shallow-foundation value — never use the shallow one
- Capping shaft stress cut 36% and capping base pressure cut 60% from the example
- The pile finished 48% shaft, against 85% for the clay pile in Module 14