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Qwestrum Engineering360 · Mining & Metallurgy · Rock Mechanics

Stress in Rock Mass

Vertical stress σ_v = γH increases with depth; horizontal stress K_0σ_v often exceeds unity in Indian shield regions. Excavation redistributes stress — concentration at tunnel crown and pillar corners drives support 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.

  • In-situ stress measurement overcoring
  • Stress concentration around excavations
  • Redistribution arching in tunnels

Topic details

Introduction

Rock mechanics underpins slope, pillar, and tunnel design. Indian peninsular gneiss often has K_0 > 1 — horizontal stress causes buckling in thin tunnels if ignored. Overcoring and hydrofracture measure in-situ stress in research and major projects.

Scope in B.Tech and GATE syllabus

Stress concentration factor (e.g. 3× at circular opening in elastic rock) guides initial support pressure estimate before numerical modelling.

Why this topic matters in practice

Singh & Singh and Hartman geotechnical chapters link σ_v to pillar and opening stability.

Key relations & formulas

Formulas (Indian textbook notation)

  • σv=γ×HverticalstressatdepthH\sigma_{v} = \gamma\times H vertical stress at depth H

Formulas (Indian textbook notation)

  • K0=σhσvearthpressurecoefficientatrestK_{0} = \frac{\sigma_{h}}{\sigma_{v}} earth pressure coefficient at rest

Formulas (Indian textbook notation)

  • principalstressesσ1≥σ2≥σ3principal stresses \sigma_{1} \ge \sigma_{2} \ge \sigma_{3}

Notation and sign conventions

Relation 1 —
σv=γ×HverticalstressatdepthH\sigma_{v} = \gamma\times H vertical stress at depth H

Formulas (Indian textbook notation)

  • σv=γ×HverticalstressatdepthH\sigma_{v} = \gamma\times H vertical stress at depth H
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 —
K0=σhσvearthpressurecoefficientatrestK_{0} = \frac{\sigma_{h}}{\sigma_{v}} earth pressure coefficient at rest

Formulas (Indian textbook notation)

  • K0=σhσvearthpressurecoefficientatrestK_{0} = \frac{\sigma_{h}}{\sigma_{v}} earth pressure coefficient at rest
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 —
principalstressesσ1≥σ2≥σ3principal stresses \sigma_{1} \ge \sigma_{2} \ge \sigma_{3}

Formulas (Indian textbook notation)

  • principalstressesσ1≥σ2≥σ3principal stresses \sigma_{1} \ge \sigma_{2} \ge \sigma_{3}
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

σ_v = γ H: overburden unit weight γ ≈ 25–27 kN/m³ for rock, 18–20 for soil overburden. At H = 500 m, σ_v ≈ 12.5–13.5 MPa before considering tectonic component.

Governing relations in practice

K_0 = σ_h/σ_v at rest (no lateral strain). Normally consolidated sediments K_0 ≈ 1−sinφ; tectonic history raises K_0 in many Indian mining districts. High K_0 squeezes openings — circular shape favourable.

Design and analysis considerations

Principal stresses σ₁ ≥ σ₂ ≥ σ₃ — orientation relative to excavation determines failure mode. Spalling when σ₁ parallel to free face exceeds tensile or shear strength.

Advanced theory and extensions

Redistribution: excavated zone stress drops to zero at boundary; load transfers to surrounding rock (arching) or pillars. Elastic Kirsch solution for circular tunnel gives tangential stress 3σ_h − σ_v at crown for hydrostatic far-field — check tension if negative.

Assumptions and validity limits

State assumptions explicitly before using any relation for stress in rock mass — 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 stress in rock mass.
4. Use equation 1:
σv=γ×HverticalstressatdepthH\sigma_{v} = \gamma\times H vertical stress at depth H
.
5. Use equation 2:
K0=σhσvearthpressurecoefficientatrestK_{0} = \frac{\sigma_{h}}{\sigma_{v}} earth pressure coefficient at rest
.
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

Stress in Rock Mass 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 stress in rock mass with earlier units — revise prerequisites before attempting mixed problems.
Industry interview panels sometimes ask: "Where did you use stress in rock mass?" — answer with a lab, mini-project, or plant visit example if possible.

Common mistakes in exams

• Using soil γ for deep hard rock stress calculation without justification
• Assuming K_0 = 0.5 everywhere in India
• Confusing far-field stress with boundary stress on tunnel wall
• Ignoring pore pressure reducing effective stress in saturated slopes

Quick revision checklist

Before attempting stress in rock mass problems, confirm you can:
1. In-situ stress measurement overcoring
2. Stress concentration around excavations
3. Redistribution arching in tunnels
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.

Vertical stress at depth

Problem

Find σ_v at H = 400 m with γ = 26 kN/m³.

Solution

σ_v = γH = 26 × 400 = 10400 kPa = 10.4 MPa

Conceptual check — Stress in Rock Mass

Problem

In a Rock Mechanics semester or GATE paper you are asked: "State the main assumption, the governing relation, and one practical consequence of stress in rock mass." 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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