Application · Rock and tunnels

Rock entered as a geologist writes it down: σci, mi, GSI and D.

The strength of a rock mass is a curve. Fitting a Mohr–Coulomb line to it is correct over a narrow band of confining stress and wrong outside it in both directions. KATAI 2D implements the generalised Hoek–Brown criterion directly, and verifies it where it matters — around an opening. A circular tunnel under 28 MPa of in-situ stress lands on the closed-form elasto-plastic solution to 0.59% inside the plastic annulus and 0.72% outside it.

Install KATAI 2D See the verified numbers

The engineering problem

Near an excavation boundary the confining stress falls to zero; a few radii away it is the full in-situ stress. A linear strength envelope cannot be right at both, and the part of the rock that decides whether the tunnel needs support is the part where it is most wrong. A tangent fit at low confinement overstates the strength at depth; a secant fit over the whole range overstates it at the wall.

The Hoek–Brown criterion is also the form in which the data arrive. A site investigation yields the uniaxial strength of intact core, a petrographic constant, and a Geological Strength Index from the exposure. Converting those to an equivalent c′ and φ′ is an extra step, with a stress range that must be chosen, done by hand, outside the model — and not recorded in it.

The formulation

σ1′=σ3′+σci(mb σ3′σci+s)a\sigma'_1 = \sigma'_3 + \sigma_{ci}\left(m_b\,\frac{\sigma'_3}{\sigma_{ci}} + s\right)^{a}The generalised Hoek–Brown criterion (Hoek, Carranza-Torres & Corkum 2002), compression positive.
mb=mi exp⁡ ⁣(GSI−10028−14D),s=exp⁡ ⁣(GSI−1009−3D),a=12+16(e−GSI/15−e−20/3)m_b = m_i\,\exp\!\left(\frac{\mathrm{GSI} - 100}{28 - 14D}\right), \quad s = \exp\!\left(\frac{\mathrm{GSI} - 100}{9 - 3D}\right), \quad a = \frac{1}{2} + \frac{1}{6}\left(e^{-\mathrm{GSI}/15} - e^{-20/3}\right)Rock-mass constants from the intact-rock parameter, the Geological Strength Index and the disturbance factor.
ln⁡rR=1mb (1−a)[(mb σrσci+s)1−a−(mb piσci+s)1−a]\ln\frac{r}{R} = \frac{1}{m_b\,(1-a)}\left[\left(m_b\,\frac{\sigma_r}{\sigma_{ci}} + s\right)^{1-a} - \left(m_b\,\frac{p_i}{\sigma_{ci}} + s\right)^{1-a}\right]Radial stress in the plastic annulus around a circular tunnel of radius R under support pressure pi — the closed form case KV-CST-016 is checked against.

The full statement of what is solved, element by element and model by model, is on the theory and constitutive models pages.

What KATAI 2D does

  • Direct input. hoek_brown(E, nu, sigma_ci, mi, gsi, D) — the rock-mass modulus and the four numbers of the classification. No c′/φ′ fit.
  • Stress-dependent dilatancy that decays from ψ at zero confinement to zero at a stated confining stress.
  • Optional tension cut-off below the criterion’s own tensile limit.
  • A consistent tangent, so an ill-conditioned boundary value problem iterates to equilibrium rather than only integrating correctly at a point.
  • Plane strain and axisymmetry. A deep circular tunnel reduces exactly to an axisymmetric radial strip; the shipped example uses that.
  • Staged unloading of the support pressure traces the ground reaction curve of the convergence–confinement method.

Verified

The numbers, with their sources.

Each row is a case in the generated verification matrix, asserted by an automated test on every build of release 0.9.0.

Benchmark Reference KATAI 2D Case
Circular tunnel in Hoek–Brown rock, σ0 = 28 MPa — radial stress profile closed-form elasto-plastic solution (Carranza-Torres & Fairhurst 1999; Carranza-Torres 2004) 0.59% in the plastic annulus  ·  0.72% outside KV-CST-016
Triaxial specimen from a project file — plateau stress at two cell pressures the Hoek–Brown envelope −0.0000% at both KV-CST-015
Material point — constants, uniaxial and tensile strengths, envelope the 2002 criterion, evaluated independently 0.000 (identical arithmetic) KV-CST-014
Degeneration to Tresca as mb → 0 this engine’s verified Mohr–Coulomb return difference ∝ mb: 1.8×10−4 → 1.8×10−8 kPa KV-CST-014

The rock mass and stress state of the tunnel case are case D1 of Hoek, Carranza-Torres, Diederichs & Corkum (2008): a graphitic phyllite at 1100 m depth from the Yacambú–Quibor tunnel. The closed form is derived from radial equilibrium in the test itself and checked by reduction to the published scaled forms at a = ½, so it does not rest on a remembered formula.

The complete record, with every tolerance band, is on the verification page.

In Python

The analysis as a script.

rock = prj.materials.hoek_brown(
    "Graphitic phyllite",
    E=7.5e6,           # kN/m2 -- the ROCK MASS modulus, not the intact core's
    nu=0.25,
    sigma_ci=50.0e3,   # kN/m2, intact uniaxial compressive strength
    mi=7.0,            # intact rock parameter
    gsi=48.0,          # Geological Strength Index, 0..100
    D=0.0)             # disturbance factor, 0 undisturbed .. 1 heavily blasted

These are the parameters of verification case KV-CST-016, which the suite asserts in test_hoek_brown_tunnel. For this rock mass the closed form gives a critical support pressure of 14.14 MPa — above it the rock stays elastic — and a plastic radius of 2.55 R when the tunnel is left unsupported.

The whole scripting surface is documented on the Python API page.

Stated limits

What this analysis does not do, in the same place as what it does. A limit you know about is an engineering judgement; one you find out about later is an error.

  • Four analyses are refused for rock rather than approximated, each with a message that names the way around it: Undrained (B) and (C), a Safety phase, EC7 design approaches that reduce a characteristic strength, and interface strength left at the schema default.
  • The rock mass is an equivalent continuum. Discrete joints, blocks and wedges are a different class of model.
  • The wall node of a tunnel carries the weakest stress a displacement method produces — a recovered value at a free boundary. It is reported and deliberately not asserted (measured −0.10%).

Questions

Why not just convert Hoek–Brown parameters to Mohr–Coulomb?

Because the conversion is a straight line through a curve. It matches over a narrow band of confining stress and is wrong on both sides of it, and around a tunnel the confining stress runs from zero at the wall to the full in-situ stress a few radii away. KATAI 2D integrates the Hoek–Brown criterion itself.

Which version of the Hoek–Brown criterion is implemented?

The generalised criterion of Hoek, Carranza-Torres and Corkum (2002), with m_b, s and a computed from m_i, GSI and the disturbance factor D, including the GSI-dependent exponent a.

Can I compute a factor of safety for a rock slope with it?

Not with a Safety phase in this release: strength reduction of a curved envelope by one factor is not equivalent to reducing c and tan φ, and the engine refuses rather than approximates. Deformation and collapse analyses with the criterion are supported.

Related

Other analyses on the same engine.

Application

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Prandtl N_c: +0.6% Read more → Application

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Walls, prestressed anchors, struts and interfaces through a real construction sequence.

beam moments +0.00% vs theory Read more → Application

Slope stability

Factor of safety by φ–c reduction; the failure mechanism is found, not assumed.

FoS 1.384 vs 1.380 charted Read more →

Install it, run a published benchmark, check the number.

KATAI 2D is free under Apache‑2.0, installs in one line and ships the input files of the benchmarks it is verified against. The fastest way to judge it is to rerun one.

Install KATAI 2D See the verification record Source on GitHub