Foundation Engineering / Pile capacity in sand — the β-method
The β-method
Shaft friction that grows with depth — in the calculation, at least.
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 below assumes drained loading in sand. Confirm that drainage is appropriate for the permeability, drainage length and loading rate; rapid or cyclic loading can require a different assessment. The method relates shaft resistance to effective vertical stress through an empirical coefficient.
By the end of this lesson you should be able to
- Compute shaft resistance from β and the effective stress profile
- Separate K from δ and say which installation changes
- Compute base resistance with a deep bearing factor
What it calculates: Drained shaft resistance from the vertical effective stress profile
- K
- Ratio of horizontal to vertical effective stress against the shaft (—)
- Interface friction angle between pile and soil (°)
- Vertical effective stress at the middle of the layer (kPa)
This assumes
- Fully drained behaviour, which sand supplies
- K and δ from installation and interface judgement, not from a soil test
In plain terms: Friction on the shaft is proportional to the normal stress pressing the sand against it, and that stress is a fraction K of the vertical effective stress. Because σ′v rises with depth, the calculated unit friction rises with depth too — a claim the second lesson revisits.
K is an installation effect. In undisturbed ground the horizontal stress is roughly K₀·σ′v. Driving a pile pushes sand outwards and locks in a much higher horizontal stress, so K can approach or exceed 1. Boring removes support and lets the sand relax inwards, so K falls below K₀.
δ is an interface property. It is the friction angle between the pile surface and the sand, typically 0.7 to 1.0 of φ′ for concrete or steel, and higher for a rough cast-in-place shaft than for smooth steel.
Put together: a driven pile with K = 1.0 and δ = 30° has β = 0.577. A bored pile in the same sand with K = 0.5 and δ = 25° has β = 0.233 — about 40% of the driven value, which is the number Module 12 quoted for why installation method is a design decision.
In the method illustrated here, base resistance is qb=Nqσ′v at the toe. The chosen pile-resistance method must define Nq, the stress basis and any limiting pressure.
Do not transfer a shallow-footing factor into a pile calculation without checking that method. Different pile methods treat installation, confinement and limiting resistance differently; a universally applicable deep-pile multiplier is not supplied here.
Check yourself
Two identical piles are installed in the same sand, one driven and one bored. Which parameter differs most, and why?
Practice
A driven pile has K = 1.0 against sand with an interface friction angle δ = 30°. What is β?