Module 15 · Lesson 15.1

The β-method

Shaft friction that grows with depth — in the calculation, at least.

Why this matters

Sand drains as fast as it is loaded, so there is no undrained case to fall back on and no s_u to multiply. Everything must be done in effective stress, which makes the method look more fundamental than the α-method. In one sense it is. In another it simply moves the judgement into different letters.

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

Shaft resistance in sand — the β-method

Q_{s} = \sum β σ_{v}^{'} π D Δ L, β = K tan δ

What it calculates: Drained shaft resistance from the vertical effective stress profile

K: Ratio of horizontal to vertical effective stress against the shaft (—)
$δ$: Interface friction angle between pile and soil (°)
$σ_{v}^{'}$: 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.

Base resistance uses q_b = N_q·σ′v at the base, and the N_q here is not the one from Module 4.

A shallow footing's failure surface can break out to the ground surface. A pile toe many diameters down is completely confined, and the resulting mechanism gives bearing factors several times larger at the same φ′. Where Module 4's N_q at φ′ = 35° is 33.3, a deep-failure value for a driven pile is of the order of 100.

The course does not supply a chart of deep N_q, because the appropriate value depends on installation method and on which of several competing solutions is being used, and a number lifted without that context is worth little. What it does insist on is that the shallow value must never be used here — that error understates nothing and overstates nothing in a comforting direction; it simply produces a base resistance that bears no relation to the mechanism.

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?

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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