Hoek-Brown Calculator — Rock Mass Strength Envelopes, Free

GSI, honestly

The Geological Strength Index is a number between 0 and 100 that you read off a chart by looking at a rock face. One axis is blockiness — intact, blocky, very blocky, disintegrated, laminated — and the other is the condition of the discontinuity surfaces, from very good to very poor. You find the box your rock mass sits in and read the diagonal contour through it.

It is a judgement, not a measurement, and the people who invented it say so. It is not derived from RQD, it is not an RMR score with the water and orientation terms removed, and it is not something a borehole log can hand you on its own. Two competent engineers looking at the same face routinely differ by five to ten points, and that is regarded as normal. Which means the right way to use this page is to run your envelope at GSI 40 and again at GSI 50 and see whether the answer you are about to sign changes — not to type one number and print it.

Where GSI drops below about 25 the criterion is at the edge of its calibration. The exponent a climbs past 0.53, s becomes vanishingly small, and the rock mass has almost no unconfined or tensile strength left to speak of. The tool flags those rows rather than refusing them, because very weak and sheared rock masses are real and the criterion is still the best available answer — it just stops being a precise one.

mi, and where to get it

mi is a property of the intact rock and describes how much its strength rises with confinement — the curvature of the envelope, in effect. It runs from about 4 for some claystones to about 33 for granite and diorite, with the frictional, well-cemented rocks at the top and the weak, fine-grained ones at the bottom. Carbonates sit in the middle; a limestone is around 8 to 12, a sandstone around 17, a quartzite around 20.

The right way to get it is to fit it to your own triaxial data. In the absence of that, the published tables of typical values by rock type are the standard fallback and are what most projects actually use. Either way it is an intact-rock property: it does not change with GSI or D, which only touch mb.

D — the parameter everyone gets wrong

The disturbance factor runs from 0 for an undisturbed in-situ rock mass to 1 for one badly damaged by blasting and stress relief. It was added in 2002 because slope designs in large open pits were coming out consistently optimistic, and it is by far the most influential number on this page relative to how casually it is usually chosen. Take the shipped example: the same granodiorite at σci 50 MPa, GSI 45, mi 10 goes from an equivalent friction angle of about 47° at D = 0 to about 32° at D = 0.7. Nothing about the rock changed. Only how it was excavated.

The failure mode is not usually picking the wrong value — it is applying the right value to the wrong volume. Hoek and Brown's 2018 edition names this outright: assuming D applies to the entire rock mass in which the excavation sits is "a common error" that produces "an extremely conservative and inappropriate design". D belongs to a damaged zone of a stated thickness. Their own worked cases assign D = 1 with a linear decay to zero over the first 3 m behind a badly blasted 8 m tunnel, and D = 0.5 over 1 to 2 m behind a pre-split slope face. Beyond that zone, the rock mass is undisturbed.

A practical consequence: if you are drawing one envelope for a whole tunnel, D = 0 with careful blasting or a TBM is defensible, and a blanket D = 0.8 usually is not. If you need the damaged zone represented, model it as its own material — which on this page means a second row.

Why σ′3max decides your c′ and φ′

The Hoek-Brown envelope is a curve; Mohr-Coulomb is a straight line. Fitting one to the other means choosing a confining-stress range to fit over, and the answer you get depends entirely on that choice. Fit over a narrow range near the origin and you get a steep line with little cohesion; fit out to high confinement and the line flattens and lifts. Both are "correct". Neither is transferable to a different problem.

That is why this page asks what you are building. For a tunnel, σ′3max comes from the relationship fitted so that the Hoek-Brown and Mohr-Coulomb solutions give equivalent characteristic curves — it needs the depth below surface and the rock unit weight. For a slope, it comes from the parallel study using Bishop circular analyses matched for equivalent factor of safety and failure-surface geometry, and it needs the slope height. Choose General and you get a plain fraction of σci, 0.25 by default, which is the range the global rock-mass strength is defined over.

One more caution, and it is the 2018 edition's: most modern stress and slope-stability software takes the Hoek-Brown criterion directly, so the equivalent parameters are now a compatibility device rather than the main event. Where you do use them, they must not be used without a tension cutoff — which is why σt is drawn on every envelope here and printed in every results row.

What the two panels show

The left panel is the criterion as it is defined: major principal stress against minor. Each material gets its own line style, the open circle at the bottom-left is the tensile cutoff σt = −s·σci/mb, and the filled marker at the top-right is σ′3max — the far end of the range the Mohr-Coulomb fit was made over. Past that marker the dashed straight line is an extrapolation and should be treated as one.

The right panel is the same curve in shear and normal stress, which is the form a limit-equilibrium program wants. It is not a separate calculation and it is not a re-fit: it is the exact parametric transformation published by Balmer in 1952, sweeping σ′3 and computing σ′n and τ from the envelope and its slope at each point. The two panels always agree because they are the same numbers.

Beneath them, one row per material: mb, s, a, the tensile strength, the rock-mass unconfined strength σc = σci·s^a, the global rock-mass strength σ′cm, the σ′3max actually used, the equivalent c′ and φ′, and the rock-mass modulus Erm — with a footnote marking which of the three modulus routes each row took.

Rock-mass modulus, three ways

Deformation modulus is the other thing a design needs and the other thing you cannot measure at rock-mass scale without a very expensive in-situ test. The tool uses Hoek and Diederichs (2006), fitted to a large set of Chinese and Taiwanese plate and jacking tests. If you give it a measured intact modulus Ei, it applies the Ei-based sigmoid. If you give it a modulus ratio MR instead, it estimates Ei = MR × σci first — MR is tabulated by rock type, around 400 for granodiorite and 350 for siltstone. If you give it neither, it falls back to the simplified GSI-only equation, which needs no intact property at all.

The three routes do not give identical answers and the table says which one each row used, because a modulus derived from a ratio and a modulus derived from a laboratory test deserve different amounts of trust.

Who uses this

Tunnel engineers converting a GSI estimate into support design inputs. Mining and civil geotechnical engineers running slope stability where the failure surface passes through rock rather than along a single joint. Engineering geologists writing the ground model that everything downstream is argued from. Students and lecturers, who need the equations visible and cited rather than buried in a licensed binary. And anybody who has been sent a spreadsheet that produces a c′ and a φ′ with no statement of the stress range it was fitted over.

It is a strength-parameter tool, not a design suite. There is no factor of safety here, no support calculation, no numerical model. What it gives you is the correctly computed, correctly cited set of parameters those calculations start from — and everything runs in this browser tab, so nothing about your project goes anywhere.

Common questions

Is this free? What is the catch?

It is free, with no account, no trial timer and no gated options. Enter your own materials, change every setting, export the figure as SVG or PNG. Free exports carry a small watermark, as they do everywhere on this site. There is no catch — the criterion is four published equations and this page exists partly because it is the kind of thing people link to.

Which edition of the criterion is this?

The 2002 edition (Hoek, Carranza-Torres & Corkum, NARMS-TAC 2002, 267–273) for mb, s, a, the σ′3max relationships and the closed-form c′ and φ′, together with the 2018 update (Hoek & Brown, Journal of Rock Mechanics and Geotechnical Engineering 11(3), 445–463), which leaves those equations unchanged and sharpens the guidance on D and on when the Mohr-Coulomb fit is appropriate. The shear–normal transformation is Balmer (1952) and the modulus is Hoek & Diederichs (2006). All four are cited on the figure itself.

How do I know the arithmetic is right?

The 2002 paper states a worked example outright: σci 50 MPa, mi 10, GSI 45 gives φ′ = 47.16° and c′ = 0.58 MPa for an undisturbed rock mass around a tunnel 100 m down, and φ′ = 27.61° with c′ = 0.35 MPa for the same rock at D = 1 in a 100 m slope. Both are reproduced by the shipped sample data and both are pinned in the test suite, along with the limit case GSI = 100, D = 0 giving mb = mi, s = 1 and a = ½ exactly.

Can I put my triaxial test results in and get σci and mi out?

No — σci and mi are inputs here, not outputs. Fitting them to test data is a regression with its own choices about weighting and about which specimens to include, and it belongs in a step before this one. If you have a fit, type the result in; if you do not, use a published mi for your rock type and a UCS from your laboratory report.

What unit weight should I use for a tunnel?

0.027 MN/m³ — about 2700 kg/m³ — is the usual first estimate for hard rock and is the default here. It matters because the tunnel relationship for σ′3max works in terms of σ′cm/γH, so the depth and the unit weight together set the confinement the Mohr-Coulomb fit is made at. Where the horizontal stress exceeds the vertical, the 2002 paper says to use the horizontal stress in place of γH.

Why is my tensile strength so small?

Because rock masses have almost none. σt = −s·σci/mb, and s falls off exponentially with GSI: at GSI 45 in a 50 MPa granodiorite it is about −0.08 MPa, which is a hundredth of a per cent of the intact strength. That is not a bug in the arithmetic, it is the physical claim the criterion makes — a jointed rock mass pulls apart at the joints. It is also exactly why the 2018 edition insists the Mohr-Coulomb fit is never used without the cutoff drawn.

Can I plot more than three materials?

Yes. Add rows and each gets its own line style — dash pattern first and colour second, so the figure survives a greyscale printer. The results table prints up to twelve materials in full and summarises beyond that. The most useful figure is usually a small number: the same rock at two disturbance factors, or your design unit bracketed by a GSI five points either side.

Does anything leave my browser?

No. The whole calculation runs in this tab, there is no account and there is no upload. For pre-tender ground models on somebody else's licence area, that is usually the entire question.