Lythos Rock — reference

Stresses are in MPa and compression is positive; the tensile strength therefore comes out negative, as Hoek writes it. Unit weights are in kN/m³ and depths in m.

The generalised Hoek–Brown criterion

Hoek, Carranza-Torres & Corkum (2002):

σ1 = σ3 + σci·(mb·σ3/σci + s)^a

mb = mi·exp((GSI − 100)/(28 − 14·D))
s  = exp((GSI − 100)/(9 − 3·D))
a  = 1/2 + (exp(−GSI/15) − exp(−20/3))/6

GSI = 100 and D = 0 give m_b = m_i, s = 1, a = 1/2 — the original criterion for intact rock. The criterion describes a rock mass that behaves isotropically; where a single discontinuity governs, analyse it with Lythos Kinematic. The program warns above GSI = 85.

Rock mass strengths

tensile    σt  = −s·σci/mb
uniaxial   σc  = σci·s^a
global     σcm = σci·(mb + 4s − a(mb − 8s))·(mb/4 + s)^(a−1) / (2(1 + a)(2 + a))

σ_cm is the strength of the rock mass as a whole — the uniaxial strength of the Mohr–Coulomb line fitted over σ_t < σ₃ < σ_ci/4 — rather than the stress at which failure starts at a boundary (σ_c).

Equivalent Mohr–Coulomb strength

The least-squares line through the Hoek–Brown curve over σ_t < σ₃ < σ_3max, in closed form (Hoek et al. 2002), with σ_3n = σ_3max/σ_ci:

φ′ = asin[ 6a·mb·(s + mb·σ3n)^(a−1) / (2(1 + a)(2 + a) + 6a·mb·(s + mb·σ3n)^(a−1)) ]

c′ = σci·[(1 + 2a)s + (1 − a)mb·σ3n]·(s + mb·σ3n)^(a−1)
     / { (1 + a)(2 + a)·√[1 + 6a·mb·(s + mb·σ3n)^(a−1) / ((1 + a)(2 + a))] }

Application	σ3max
General	σci/4
Tunnel	σcm·0.47·(σcm/γH)−0.94, H the depth of the tunnel
Slope	σcm·0.72·(σcm/γH)−0.91, H the height of the slope
Custom	the value given

Every application is reported side by side.

Instantaneous strength

At any σ₃, with k = ∂σ₁/∂σ₃ = 1 + a·m_b·(m_b·σ₃/σ_ci + s)^a−1 (Balmer 1952):

σn = (σ1 + σ3)/2 − (σ1 − σ3)/2 · (k − 1)/(k + 1)
τ  = (σ1 − σ3)·√k/(k + 1)
φi = asin((k − 1)/(k + 1)),   ci = τ − σn·tan φi

Deformation modulus

Estimate	Erm
Hoek & Diederichs (2006), generalised	Ei·(0.02 + (1 − D/2)/(1 + e(60 + 15D − GSI)/11))
Hoek & Diederichs (2006), simplified	100 000·(1 − D/2)/(1 + e(75 + 25D − GSI)/11) MPa
Hoek, Carranza-Torres & Corkum (2002)	(1 − D/2)·√(σci/100)·10(GSI − 10)/40 GPa, the root 1 for σci > 100 MPa

E_i is entered or taken as MR·σ_ci. The generalised estimate cannot exceed E_i; the program warns when the one chosen does.

GSI from other classifications

Route	GSI	Source
Chart	the structure and surface classes, placed on the quantified chart	Hoek, Carter & Diederichs (2013)
JCond89, RQD	1.5·JCond89 + RQD/2	Hoek, Carter & Diederichs (2013)
RMR89	RMR89 − 5 (dry, no orientation adjustment; not below RMR89 = 23)	Hoek, Kaiser & Bawden (1995)
Q′	9·ln Q′ + 44	Hoek et al. (1995)
Vb, Jc	(26.5 + 8.79·ln Jc + 0.9·ln Vb)/(1 + 0.0151·ln Jc − 0.0253·ln Vb), Vb in cm³	Cai et al. (2004)

Triaxial fit and residual strength

With s = 1 and a = 1/2 the criterion becomes a straight line, (σ₁ − σ₃)² = m_i·σ_ci·σ₃ + σ_ci² (Hoek & Brown 1980); least squares gives σ_ci = √b, m_i = m/σ_ci and r². Uniaxial tests enter at σ₃ = 0, tensile tests at σ₃ = −σ_t with σ₁ = 0.

Residual strength (Cai et al. 2007): GSI_r = GSI·e^{−0.0134·GSI}, and the residual m_b, s, a, c′ and φ′ from GSI_r over the same σ_3max.

Inputs

Group	Fields
Intact rock	source (values / triaxial tests), rock type (fills typical mi and MR), ISRM strength grade (fills a typical σci), σci, mi
Triaxial tests	one row per test: name, σ3, σ1
Rock mass	GSI source (direct, chart, jcond, rmr, q, cai) and its inputs; D, with Hoek's excavation cases as a picker
Application	general, tunnel, slope (γ, H) or custom (σ3max)
Deformability	the estimate; Ei entered or MR·σci
Instantaneous strength	the σ3 at which to read it
Study	method (one at a time, Latin hypercube, Monte Carlo), samples, seed, variables as ranges or normal / lognormal / uniform distributions

Project files (.rock) are JSON; missing entries keep their defaults.

Modules

File	Content
hoekbrown.py	The criterion: constants, strengths, the Mohr–Coulomb fit, σ3max, Balmer's envelope, the moduli
gsi.py	GSI from the chart, JCond89/RQD, RMR89, Q′, Vb/Jc
labfit.py	σci and mi fitted to triaxial tests
tables.py	Typical values: mi and MR by rock type, σci grades, D cases, the chart's classes
engine.py	The analysis: intact rock, GSI, constants, strengths, every application, moduli, residual, warnings
study.py, study_plots.py	Parametric and probabilistic studies, 5 % fractiles, Spearman sensitivities, CSV / XLSX
report.py, pdf.py	Calculation report: one HTML assembly, exported as PDF / HTML / DOCX
web/	The local HTTP server, the session and the browser interface

Validation

The tests check the constants and strengths against hand calculations, the closed-form Mohr–Coulomb fit against a numerical least-squares fit of the Hoek–Brown curve, the global strength against the uniaxial strength of the general fit, Balmer's tangent against the geometry of the Mohr circle, each modulus and GSI route against its expression, the laboratory fit against exact data, the report in all three formats and the interface (session and HTTP layer).

pip install -e ".[dev]"
pytest -q

Limits

• The criterion is for rock masses that behave isotropically; blocky rock with a few persistent sets needs a discontinuity analysis.
• Reading GSI from a chart is good to about ±5; run a study rather than trust one value.
• The tunnel and slope σ_3max relations were fitted to numerical analyses; they are estimates of the confinement, not measurements.
• Typical m_i, MR and σ_ci values are where an estimate starts when no test says otherwise, not design values.

Full derivations and sources: docs/theory.md.