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Qwestrum Engineering360 · Mining & Metallurgy · Rock Mechanics

Rock Strength Criteria

Mohr-Coulomb τ = c + σ tanφ governs shear failure on planes; Hoek-Brown captures rock mass non-linearity via GSI and σ_ci. Triaxial tests yield c and φ or Hoek-Brown parameters for slope and tunnel design.
Exam tip: keep SI units consistent end-to-end, write the governing relation symbolically before substituting, and sanity-check magnitude and sign.

Key formulas & points

Skim these first — then read the full notes below.

  • Cohesion c and friction angle φ
  • GSI geological strength index for HB
  • Brittle vs ductile failure modes

Topic details

Introduction

Rock mass failure rarely follows intact lab specimen behaviour — Hoek-Brown with GSI reduction is standard in Indian slope stability software (Slide, Phase2). Mohr-Coulomb still appears in undergraduate exams for triaxial data interpretation.

Scope in B.Tech and GATE syllabus

Brittle failure (sudden) vs ductile (yield) affects support choice — coal pillars show post-peak softening; hard rock tunnels may spall without warning.

Why this topic matters in practice

Hartman & Mutmansky provides Hoek-Brown parameter tables by rock type — match GSI from field mapping.

Key relations & formulas

Formulas (Indian textbook notation)

  • Mohr−Coulomb:τ=c+σtanϕMohr-Coulomb: \tau = c + \sigma tan\phi

Formulas (Indian textbook notation)

  • Hoek−Brown:σ1=σ3+σci(mbσ3σci+s)aHoek-Brown: \sigma_{1} = \sigma_{3} + \sigma_{ci}(m_{b} \frac{\sigma_{3}}{\sigma_{ci}} + s)^a

Formulas (Indian textbook notation)

  • uniaxialσcfromtriaxialtestuniaxial \sigma_{c} from triaxial test

Notation and sign conventions

Relation 1 —
Mohr−Coulomb:τ=c+σtanϕMohr-Coulomb: \tau = c + \sigma tan\phi

Formulas (Indian textbook notation)

  • Mohr−Coulomb:τ=c+σtanϕMohr-Coulomb: \tau = c + \sigma tan\phi
Write this relation with symbols exactly as in Jaeger Cook Rock Mechanics — Standard reference before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.
Relation 2 —
Hoek−Brown:σ1=σ3+σciHoek-Brown: \sigma_{1} = \sigma_{3} + \sigma_{ci}

Formulas (Indian textbook notation)

  • Hoek−Brown:σ1=σ3+σci(mbσ3σci+s)aHoek-Brown: \sigma_{1} = \sigma_{3} + \sigma_{ci}(m_{b} \frac{\sigma_{3}}{\sigma_{ci}} + s)^a
Write this relation with symbols exactly as in Jaeger Cook Rock Mechanics — Standard reference before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.
Relation 3 —
uniaxialσcfromtriaxialtestuniaxial \sigma_{c} from triaxial test

Formulas (Indian textbook notation)

  • uniaxialσcfromtriaxialtestuniaxial \sigma_{c} from triaxial test
Write this relation with symbols exactly as in Jaeger Cook Rock Mechanics — Standard reference before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.

Fundamentals and definitions

Mohr-Coulomb: shear strength τ = c + σ_n tanφ on failure plane. From triaxial: plot σ₁ vs σ₃ circles, draw envelope — c and φ from intercept and slope. Effective stress analysis: use σ' not total σ when pore water present.

Governing relations in practice

Hoek-Brown: σ₁ = σ₃ + σ_ci (m_b σ₃/σ_ci + s)^a — m_b, s, a from GSI and disturbance factor D. σ_ci intact uniaxial compressive strength from lab. Rock mass σ_ci reduced dramatically at low GSI.

Design and analysis considerations

Uniaxial σ_c special case σ₃ = 0: σ_c = σ_ci s^a approx for HB. Compare to applied stress for FS.

Advanced theory and extensions

Brittle: peak strength followed by drop — strain localisation. Ductile: gradual yield — plastic zone around tunnel. Support must accommodate convergence in ductile squeezing ground (Himalayan tunnels).

Assumptions and validity limits

State assumptions explicitly before using any relation for rock strength criteria — steady state, uniform properties, linear elastic material, ideal gas, incompressible flow, etc., as applicable.
Wrong assumptions invalidate the entire solution even when the formula is correct. In Rock Mechanics viva and GATE descriptive questions, listing valid assumptions often earns separate marks.

Step-by-step problem approach

1. Read the question and list given data with SI units (common in Rock Mechanics papers).
2. Draw a neat labelled diagram where applicable — examiners in Indian universities award diagram marks even when arithmetic slips.
3. Identify which relation from this topic applies to rock strength criteria.
4. Use equation 1:
Mohr−Coulomb:τ=c+σtanϕMohr-Coulomb: \tau = c + \sigma tan\phi
.
5. Use equation 2:
Hoek−Brown:σ1=σ3+σciHoek-Brown: \sigma_{1} = \sigma_{3} + \sigma_{ci}
.
6. Substitute values, compute, and verify units and sign (direction).
7. State conclusion in one line — e.g. safe/unsafe, stable/unstable, feasible/infeasible.

Applications & exam relevance

Rock Strength Criteria appears in tunnels and slopes in mines. In Indian mining curricula this topic is tested because it connects theory to strength and support of rock mass.
GATE and semester exams often combine rock strength criteria with earlier units — revise prerequisites before attempting mixed problems.
Industry interview panels sometimes ask: "Where did you use rock strength criteria?" — answer with a lab, mini-project, or plant visit example if possible.

Common mistakes in exams

• Using intact σ_ci for heavily jointed rock mass without GSI reduction
• Total stress Mohr-Coulomb in saturated slope without pore pressure
• φ > peak friction from lab polish — use peak vs residual correctly
• Hoek-Brown parameters from wrong disturbance factor D

Quick revision checklist

Before attempting rock strength criteria problems, confirm you can:
1. Cohesion c and friction angle φ
2. GSI geological strength index for HB
3. Brittle vs ductile failure modes
Revise the solved examples in Jaeger Cook Rock Mechanics — Standard reference and one previous-year GATE or university paper for this unit.

Worked examples

Try the problem first — open the solution when you are ready to check.

Guided practice — Rock Strength Criteria

Problem

A standard Rock Mechanics numerical on rock strength criteria supplies given data in SI units. Using Mohr-Coulomb: τ = c + σ tanφ and Hoek-Brown: σ₁ = σ₃ + σ_ci, find the unknown quantity and state whether the result is physically reasonable.

Solution

1. List all given quantities with units (convert to SI if needed).
2. Draw a neat labelled diagram — diagram marks are common in Indian B.Tech papers.
3. Select
Mohr−Coulomb:τ=c+σtanϕMohr-Coulomb: \tau = c + \sigma tan\phi
and write it symbolically before substitution.
4. Substitute values, compute, and attach correct units.
5. Sanity-check: magnitude, sign, and direction must match strength and support of rock mass.
Cross-check with solved examples in your Rock Mechanics textbook.

Conceptual check — Rock Strength Criteria

Problem

In a Rock Mechanics semester or GATE paper you are asked: "State the main assumption, the governing relation, and one practical consequence of rock strength criteria." What should a complete answer include?

📖 Standard books (India)

  • Jaeger Cook Rock Mechanics — Standard reference

    Read: Syllabus unit

    Referenced in Indian B.Tech syllabus

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