SlipSurface IQ Rock — theory

Stresses are in MPa and compression is positive; the tensile strength therefore comes out negative, as Hoek and RocLab write it. Unit weights are in kN/m³ and depths in m; γH is turned into MPa where it is used.

1. 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

σci is the uniaxial compressive strength of the intact rock, mi its Hoek–Brown constant, GSI the Geological Strength Index and D the disturbance factor (0 undisturbed, 1 heavily disturbed). For intact rock, GSI = 100 and D = 0 give mb = mi, s = 1 and a = 1/2 — the original criterion of Hoek & Brown (1980).

The criterion describes a rock mass that behaves isotropically: jointed closely enough, compared with the size of the structure, that no single discontinuity governs. Where one does — a blocky rock mass with a few persistent sets, a slope that slides on one plane — analyse the discontinuity instead (SlipSurface IQ Kinematic does). The program warns above GSI = 85.

2. 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))

σt is the biaxial tensile strength, which Hoek shows to equal the uniaxial one for brittle rock. σcm is the uniaxial strength of the Mohr–Coulomb line fitted over σt < σ3 < σci/4 — the strength of the rock mass as a whole rather than the stress at which failure starts at a boundary (σc). The test suite checks that the two definitions agree.

3. Equivalent Mohr–Coulomb strength

The line σ1 = σcm′ + k·σ3 fitted to the Hoek–Brown curve over σt < σ3 < σ3max, with σ3n = σ3max/σci (Hoek et al. 2002):

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))] }

The closed form is the least-squares line through the curve over that range; the test suite fits the line numerically and compares.

The upper limit σ3max

The fit depends on the range, and the range on what loads the rock mass (Hoek et al. 2002):

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

The tunnel and slope relations were fitted by Hoek et al. to the results of numerical analyses; they give the confinement the rock around a deep tunnel, or under a slope, actually mobilises. The general range suits a first estimate and bearing capacity. Every application is reported side by side, so the effect of the choice is in plain view.

4. Instantaneous strength

At any σ3 the tangent to the envelope in the Mohr plane has, with k = ∂σ1/∂σ3 = 1 + a·mb·(mb·σ3/σci + 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

These are the parameters to use where the stress in the rock mass is known, and the envelope drawn in the τ–σn figure is the same construction over a range of σ3.

5. 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

Ei is the intact modulus, entered or taken as MR·σci with the modulus ratio MR of the rock type (Deere 1968; Palmström & Singh 2001). The generalised estimate is the one to prefer when Ei is known; it cannot exceed Ei, whereas the others can — the program warns when the estimate chosen does.

6. GSI from other classifications

route	GSI	source
chart	the chart's 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, with a dry groundwater rating and no orientation adjustment; not for RMR89 < 23	Hoek, Kaiser & Bawden (1995)
Q′	9·ln Q′ + 44, Q′ = RQD/Jn · Jr/Ja	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, Jc = JW·JS/JA	Cai et al. (2004)

The chart's classes stand for these points of the quantified chart:

structure	RQD	surface	JCond89
intact or massive	90	very good	27
blocky	70	good	21
very blocky	50	fair	15
blocky / disturbed / seamy	30	poor	9
disintegrated	10	very poor	3

Reading GSI from a chart is an estimate, and a range of ±5 is the precision the chart supports; a study with GSI as a variable shows what that range does to the result.

7. σci and mi from triaxial tests

With s = 1 and a = 1/2 the criterion squares into a straight line (Hoek & Brown 1980):

(σ1 − σ3)² = mi·σci·σ3 + σci²

A least-squares line y = m·x + b through the tests gives σci = √b and mi = m/σci; r² says how well the criterion describes them. Uniaxial tests enter at σ3 = 0, tensile tests at σ3 = −σt with σ1 = 0. Hoek recommends at least five tests over 0 ≤ σ3 ≤ σci/2: a narrower range leaves mi uncertain, a wider one may reach the brittle–ductile transition where the criterion no longer applies. The program warns about both, and about r² < 0.9.

8. Residual strength

Cai et al. (2007) reduce GSI to its residual value

GSIr = GSI·e^(−0.0134·GSI)

and take the residual mb, s, a from GSIr with the intact rock's σci and mi; the residual c′ and φ′ are fitted over the same σ3max as the peak ones.

9. Typical values

mi by rock type after Marinos & Hoek (2000) and Hoek (2007); MR after Hoek & Diederichs (2006); the ISRM field strength grades as given by Hoek (2007); the disturbance factor after Hoek et al. (2002), Table 1. They are where an estimate starts when no test says otherwise, not design values.

10. Studies

A study varies any input in use — σci, mi, GSI (or the inputs it is converted from), D, MR or Ei, γ, H, σ3max, the σ3 of the instantaneous strength — over a range or by a normal, lognormal or uniform distribution given by its mean and coefficient of variation, sampled one at a time, by Latin hypercube or by Monte Carlo. Every sample is a complete analysis. Reported for each output: mean, standard deviation, CoV, the 5, 50 and 95 % fractiles — the 5 % fractile of a strength being its characteristic value in the sense of EN 1997-1 — and the Spearman rank correlation with every input, tied values sharing their average rank.

References

• Balmer, G. (1952). A general analytical solution for Mohr's envelope. Proc. ASTM 52, 1260–1271.
• Cai, M., Kaiser, P.K., Uno, H., Tasaka, Y. & Minami, M. (2004). Estimation of rock mass deformation modulus and strength of jointed hard rock masses using the GSI system. Int. J. Rock Mech. Min. Sci. 41(1), 3–19.
• Cai, M., Kaiser, P.K., Tasaka, Y. & Minami, M. (2007). Determination of residual strength parameters of jointed rock masses using the GSI system. Int. J. Rock Mech. Min. Sci. 44(2), 247–265.
• Deere, D.U. (1968). Geological considerations. In Rock Mechanics in Engineering Practice, Wiley.
• Hoek, E. (2007). Practical Rock Engineering. RocScience.
• Hoek, E. & Brown, E.T. (1980). Empirical strength criterion for rock masses. J. Geotech. Eng. Div. ASCE 106(GT9), 1013–1035.
• Hoek, E., Carranza-Torres, C. & Corkum, B. (2002). Hoek–Brown failure criterion — 2002 edition. Proc. NARMS-TAC, Toronto, 267–273.
• Hoek, E., Carter, T.G. & Diederichs, M.S. (2013). Quantification of the Geological Strength Index chart. 47th US Rock Mech. / Geomech. Symp., ARMA 13-672.
• Hoek, E. & Diederichs, M.S. (2006). Empirical estimation of rock mass modulus. Int. J. Rock Mech. Min. Sci. 43(2), 203–215.
• Hoek, E., Kaiser, P.K. & Bawden, W.F. (1995). Support of Underground Excavations in Hard Rock. Balkema.
• Marinos, P. & Hoek, E. (2000). GSI: a geologically friendly tool for rock mass strength estimation. Proc. GeoEng2000, Melbourne, 1422–1440.
• Palmström, A. & Singh, R. (2001). The deformation modulus of rock masses — comparisons between in situ tests and indirect estimates. Tunnelling and Underground Space Technology 16(2), 115–131.