Lythos Pile — reference

Shaft friction

Q_s = Σ f_s·p·Δz; the shaft is cut into slices no thicker than 0.25 m that never straddle a layer boundary.

Granular layers: f_s = K·σ′_v·tan δ, K = (K/K₀)·K₀, K₀ = 1 − sin φ′, δ = (δ/φ′)·φ′. K/K₀ is 1.0 for a bored pile, 1.2 for a small-displacement and 1.4 for a large-displacement driven pile, unless entered. With the critical depth on, σ′_v in sand is held at its value at z_c = 15·D.

Cohesive layers:

Method	fs
API RP 2A (1987)	α·cu, α = 0.5·ψ−0.5 (ψ ≤ 1), 0.5·ψ−0.25 (ψ > 1), ψ = cu/σ′v, α ≤ 1
Kulhawy & Phoon (1993)	α·cu, α = 0.21 + 0.26·pa/cu ≤ 1
Sladen (1992)	α·cu, α = C·(σ′v/cu)0.45 ≤ 1; C = 0.4 bored, 0.5 driven
β — Burland (1973)	(1 − sin φ′)·tan φ′·√OCR·σ′v
λ — Vijayvergiya & Focht (1972)	λ·(σ′v + 2cu), λ from the pile's penetration

SPT (Meyerhof 1976): f_s = 0.02·p_a·N₆₀ (large displacement), 0.01·p_a·N₆₀ otherwise.

Base resistance

Soil	Method	qb
Sand	Meyerhof (1976)	σ′v·Nq* ≤ 0.5·pa·Nq*·tan φ′
Sand	Vesić (1977)	σ′v·Nq*(Irr)
Sand	Janbu (1976)	σ′v·Nq, Nq = (tan φ′ + √(1 + tan²φ′))²·e2η′·tan φ′
Clay	Skempton / Meyerhof	9·cu
Clay	Vesić	Nc·cu, Nc = 4/3·(ln Ir + 1) + π/2 + 1
Clay	Janbu, φ = 0	Nc* = 2 + 2η′ (5.14 at η′ = 90°)
Either	SPT (Meyerhof 1976)	0.4·pa·N60·Lb/D ≤ 4·pa·N60

A weaker layer within 3·D below the tip is reported.

Weight, capacity and the check

W = Ab·[γp·(length above the water table) + (γp − γw)·(length below it)]
Qult = Qs + Qb,   Qult,net = Qult − W,   Qall = Qult,net / FS

Every combination of a shaft method with a base method is reported; the chosen pair makes the checks.

Groups

B_g × L_g = [(n₁ − 1)s_x + D] × [(n₂ − 1)s_y + D].

Method	η
Converse–Labarre	1 − θ·[(n1 − 1)n2 + (n2 − 1)n1]/(90·n1·n2), θ = arctan(D/s) [°]
Los Angeles Group	1 − D/(π·s·n1·n2)·[n1(n2 − 1) + n2(n1 − 1) + √2(n1 − 1)(n2 − 1)]
Seiler–Keeney	1 − [36s/(75s² − 7)]·(n1 + n2 − 2)/(n1 + n2 − 1) + 0.3/(n1 + n2)
Feld	1 − (number of neighbours, straight and diagonal)/16, averaged over the group

Block failure: the group as one block; f_s = c_u in clay and K₀·σ′_v·tan φ′ in sand; the base with Skempton's N_c in clay.

Qg,ult = min(η·n·Qult, Qblock),   Qg,all = (Qg,ult − n·W) / FS,   Q ≤ Qg,all

Settlement

Case	Method
Single pile	Vesić (1977): s1 (shaft shortening) + s2 (tip) + s3 (along the shaft)
Group	equivalent raft at 2/3·L, 2:1 spread; clays consolidate with Cc, Cr, e0, OCR, the rest compress elastically
Group	Vesić s·√(Bg/D)
Group	Meyerhof's SPT rule

Rock socket

Quantity	Method
Side shear	12 correlations: Rosenberg & Journeaux, Horvath & Kenney, Meigh & Wolski, Williams et al., Reynolds & Kaderabek, Gupton & Logan, Rowe & Armitage, Carter & Kulhawy, Toh et al., Zhang & Einstein, O'Neill & Reese / AASHTO, Kulhawy et al.; qu ≤ f′c; the weak-rock rules left out above a limit
Base	Coates, Rowe & Armitage, Carter & Kulhawy (Hoek–Brown), Zhang & Einstein, AASHTO, CFEM
Socket length	Qs/FSside + Qb/FSbase − W = Q, by bisection, per correlation and for the design
Rock mass modulus	from RQD (Gardner), from GSI (Hoek & Diederichs), or entered
Settlement	Randolph & Wroth with and without the base; Vesić; plus the shortening through the overburden

The design statistic design ∈ {mean, median, lower, upper} or a single correlation; the base base_design ∈ {none, min, mean} or a single method.

Inputs

Group	Fields
Pile	circular / square, D, L, depth of the pile head, installation (bored, driven_low, driven_high), γp, Ep
Load	the vertical load on the group at the underside of the cap
Group	piles along B and L, spacings, the efficiency method, block failure on / off (1 × 1 is a single pile)
Soil profile	thickness, granular / cohesive, γ, γsat, φ′, cu, OCR, N60, E, ν, Cc, Cr, e0
Methods	clay method, base method, K/K0 and δ/φ′ in sand, critical depth, Janbu's η′, Sladen's C, the SPT rule; weight subtracted / buoyant
Settlement	the method for the check, the raft depth, the load spread, the distribution of the shaft friction
Criteria	FS, allowable settlement, the length search
Rock socket	diameter and length, head and rock surface depths, load; qu, modulus (RQD / GSI / direct), GSI, mi, D, ν, joint spacing and aperture; f′c, Ec; design statistic and factors of safety

Modules

File	Content
profile.py	The layered column and its stresses, slices and averages
axial.py	Unit shaft friction and base resistance, per method
group.py	Group layout, efficiencies, Skempton's Nc of the block
settlement.py	Vesić's single-pile settlement, the equivalent raft, the group rules
socket.py	Rock sockets: correlations, base, length, Randolph & Wroth
engine.py	The pile analysis: shaft, base, weight, group, settlement, length
study.py, report.py, web/	Studies, report, interface

Validation

Every formula is checked against a hand calculation: the α, β and λ methods, Meyerhof's table and limit, Vesić's and Janbu's factors, the four efficiencies, Feld's count, Vesić's settlement term by term, the consolidation of a clay, the equivalent raft, each of the twelve socket correlations and the six base methods, Randolph & Wroth's rigid limits. The engine is tested on simple cases worked by hand (a clay pile, a sand pile, the weight, the water table, a block failure, the required length).

Details: docs/theory.md.