Lythos Rock — examples
Every output is from a real run. The starter project: a sandstone rock mass.
| Input | Value |
|---|---|
| Intact rock | sandstone, σci = 60 MPa, mi = 17, MR = 275 (Ei = 16 500 MPa) |
| Rock mass | GSI = 55, D = 0 |
| Application | general (σ3max = σci/4); for a tunnel or slope γ = 26 kN/m³, H = 100 m |
| Instantaneous strength | at σ3 = 1 MPa |
| Triaxial tests | 2 uniaxial, 4 triaxial at σ3 = 5 … 20 MPa |
1. A sandstone rock mass
lythos-rock example -o project.rock
lythos-rock run project.rockROCK MASS STRENGTH — GENERALISED HOEK–BROWN
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Intact rock
rock type Sandstone
σci and mi from values entered
σci 60.0 MPa
mi 17.00
Ei 16,500 MPa (MR = 275)
Rock mass
GSI 55.0 (entered)
D 0.00
Hoek–Brown constants
mb 3.4078
s 0.0067379
a 0.50405
Rock mass strength
tensile strength σt -0.1186 MPa
uniaxial compressive strength σc 4.826 MPa
global strength σcm 14.98 MPa
Equivalent Mohr–Coulomb strength
application General (σ3max = σci/4)
σ3max 15.00 MPa
cohesion c' 3.769 MPa
friction angle φ' 36.59°
uniaxial strength of the line 14.98 MPa
tensile strength of the line -3.792 MPa
Deformation modulus Erm
Hoek & Diederichs 2006, generalised * 6,737 MPa
Hoek & Diederichs 2006, simplified 13,965 MPa
Hoek, Carranza-Torres & Corkum 2002 10,329 MPa
Instantaneous strength at σ3 = 1.00 MPa
σ1 15.96 MPa
σn 2.711 MPa
τ 4.761 MPa
instantaneous friction angle φi 50.46°
instantaneous cohesion ci 1.477 MPa
Residual strength (Cai et al. 2007)
GSIr 26.3
mr, sr, ar 1.224, 0.0002783, 0.5286
cohesion c' 2.366 MPa
friction angle φ' 27.86°
Equivalent Mohr–Coulomb strength by application
General (σ3max = σci/4) * σ3max = 15.0 MPa c' = 3.77 MPa φ' = 36.59°
Tunnel σ3max = 1.36 MPa c' = 0.876 MPa φ' = 55.48°
Slope σ3max = 2.19 MPa c' = 1.13 MPa φ' = 52.12°Reading the output. The same rock mass gives c′ = 3.77 MPa, φ′ = 36.6° over the general range but c′ = 0.88 MPa, φ′ = 55.5° around a tunnel at 100 m: the Hoek–Brown envelope is curved, and a straight line through it depends on the range of σ3 it is fitted over. That is why every application is listed. The three modulus estimates differ by a factor of two; the generalised one, which uses Ei, is the one to prefer.
2. A blasted tunnel
The same rock mass around a tunnel at 100 m depth, with the disturbance of blasting rising from none to heavy:
from lythosrock import forms
from lythosrock.web.session import Session
session = Session(lang="en")
values = forms.defaults() # sandstone, σci = 60 MPa, mi = 17, GSI = 55
values.update(application="tunnel", unit_weight=26.0, depth=100.0)
for D in (0.0, 0.3, 0.5, 0.8):
values["D"] = D
r = session.analyse(values)
print(f"D = {D:.1f} c' = {r['c']:5.2f} MPa φ' = {r['phi']:5.2f}° Erm = {r['Erm']:6,.0f} MPa")D = 0.0 c' = 0.88 MPa φ' = 55.48° Erm = 6,737 MPa
D = 0.3 c' = 0.76 MPa φ' = 53.55° Erm = 4,490 MPa
D = 0.5 c' = 0.68 MPa φ' = 51.73° Erm = 3,337 MPa
D = 0.8 c' = 0.55 MPa φ' = 47.64° Erm = 2,070 MPaD reduces the modulus much faster than the strength: at D = 0.8 the rock mass is about a third as stiff, while φ′ has dropped by eight degrees. Hoek's guidance for D — good controlled blasting, poor blasting, mechanical excavation — is offered as a picker; the number stays editable.
3. GSI five ways
With the starter project's inputs to every route — very blocky / fair on the chart, JCond89 = 15 and RQD = 60, RMR89 = 60, Q′ = 5, Vb = 1 000 cm³ and Jc = 1:
from lythosrock import forms
from lythosrock.web.session import Session
session = Session(lang="en")
values = forms.defaults()
for source in ("direct", "chart", "jcond", "rmr", "q", "cai"):
values["gsi_source"] = source
r = session.analyse(values)
print(f"{source:7s} GSI = {r['GSI']:5.1f} c' = {r['c']:5.2f} MPa φ' = {r['phi']:5.2f}°")direct GSI = 55.0 c' = 3.77 MPa φ' = 36.59°
chart GSI = 47.5 c' = 3.38 MPa φ' = 34.33°
jcond GSI = 52.5 c' = 3.64 MPa φ' = 35.84°
rmr GSI = 55.0 c' = 3.77 MPa φ' = 36.59°
q GSI = 58.5 c' = 3.96 MPa φ' = 37.63°
cai GSI = 39.6 c' = 3.01 MPa φ' = 31.95°The routes spread from 39.6 to 58.5, and c′ with them from 3.0 to 4.0 MPa. They were fitted to different data sets; knowing which one the site investigation supports is what decides. A reading of GSI is good to about ±5, which a study turns into a range of c′ and φ′ (example 5).
4. σci and mi from triaxial tests
Switching the intact rock source to the laboratory ("source": "lab" in the project file, or the picker in the interface) fits the six tests of the starter project:
lythos-rock run lab.rockIntact rock
rock type Sandstone
σci and mi from fit to triaxial tests
σci 59.8 MPa
mi 17.19
Ei 16,435 MPa (MR = 275)
laboratory fit 6 tests, r² = 0.9985, σ3 = 0 … 20 MPaThe fit is a straight line through (σ1 − σ3)² against σ3. Hoek recommends at least five tests over 0 ≤ σ3 ≤ σci/2; the program warns when the tests cover a narrower or a wider range, and when r² < 0.9.
5. A probabilistic study
The starter project defines a Latin hypercube study of 500 samples: σci lognormal with CoV 0.25, GSI normal with CoV 0.10, mi normal with CoV 0.15.
lythos-rock study project.rock -o samples.csvPARAMETRIC / PROBABILISTIC STUDY
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Sampling: Latin hypercube · samples: 500 · analysed: 500
Statistics of the outputs
n mean std CoV P5 P50 P95
c' [MPa] 500 3.779 1.033 0.273 2.348 3.644 5.665
φ' [°] 500 36.48 2.258 0.062 32.57 36.58 40.02
Erm [MPa] 500 6859 2687 0.392 3221 6491 1.192e+04
σcm [MPa] 500 15.09 4.482 0.297 8.8 14.56 23.53
σc [MPa] 500 5.084 2.223 0.437 2.321 4.656 9.338
σt [MPa] 500 -0.1321 0.07236 0.548 -0.2605 -0.1188 -0.05348
σ3max [MPa] 500 15 3.749 0.250 9.73 14.55 21.8
mb 500 3.487 0.9206 0.264 2.124 3.4 5.141
s 500 0.008115 0.005423 0.668 0.002468 0.00674 0.01826
a 500 0.5043 0.001725 0.003 0.5021 0.504 0.5076
ci [MPa] 500 1.501 0.2922 0.195 1.121 1.462 2.04
φi [°] 500 49.99 2.632 0.053 45.39 50.18 54.18
Characteristic values (lower 5 % fractile)
c' [MPa] P5 = 2.348 (mean 3.779)
φ' [°] P5 = 32.57 (mean 36.48)
σcm [MPa] P5 = 8.8 (mean 15.09)
Erm [MPa] P5 = 3221 (mean 6859)
Sensitivities — Spearman ρ
c' φ' Erm
Rock mass · GSI +0.344 +0.792 +0.752
Intact rock · mi +0.181 +0.659 +0.083
Intact rock · σci +0.925 +0.013 +0.649Reading the output. c′ is governed by σci (ρ = 0.93), φ′ by GSI and mi; the modulus by GSI and σci. So a better estimate of σci — more uniaxial tests — narrows c′, while narrowing φ′ needs a better reading of GSI. The 5 % fractiles are characteristic values in the sense of EN 1997-1.