Module 15 · Lesson 15.2
The two limits
Where the calculation and the measurements part company.
Why 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.
The limiting shaft stress. Measured unit shaft friction in sand is observed to stop increasing beyond roughly 15 to 20 diameters of embedment, whatever the calculation says. The conventional response is to cap the effective stress used in the shaft term at a limiting value.
The limiting base pressure. Likewise, measured base resistance in sand does not keep climbing with depth. Limiting values of the order of 10 to 15 MPa are commonly applied.
Both caps make a large difference. For a 25 m driven pile in sand with γ′ = 10 kN/m³ and β = 0.577, the uncapped shaft resistance is 2834 kN; capping the effective stress at 100 kPa brings it to 1814 kN — a 36% reduction, on the term that was going to carry half the pile.
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
This is educational material. It uses simplified examples to teach principles, and must not be relied on for real design or safety-critical decisions. Module overview and checkpoint