2.2 Rock Mechanics, Stress Analysis & Strata Control

2.2 Rock Mechanics, Stress Analysis & Strata Control

Quick Answer: Rock mechanics provides the quantitative foundation for underground excavation design and strata control. Rock mass classification systems — including Bieniawski's Rock Mass Rating (RMR), Barton's Q-System, and Hoek's Geological Strength Index (GSI) — combine intact rock properties with discontinuity parameters to evaluate overall rock mass competence. In-situ stresses ($ \sigma_v = \gamma z $, $ K = \sigma_h / \sigma_v $) concentrate around underground openings according to elastic stress fields (e.g., Kirsch equations), creating tangential boundary stresses ($ \sigma_\theta $) that govern roof stability, sidewall spalling, and potential rockburst hazards.

Empirical Rock Mass Classification Systems

Rock masses differ fundamentally from intact laboratory rock specimens because discontinuities (faults, joints, bedding planes, shears) control overall strength and deformation behavior. Geotechnical engineers rely on standardized classification systems to characterize ground conditions for stope design and support selection.

1. Bieniawski's Rock Mass Rating (RMR89)

Bieniawski's RMR system rates a rock mass on a scale from 0 to 100 by summing five basic parameters:

$\text{RMR}_{\text{basic}} = P_1 + P_2 + P_3 + P_4 + P_5$

1. Uniaxial Compressive Strength (UCS) of Intact Rock ($P_1$): Rated 0 to 15 (from <1 MPa up to >250 MPa).
2. Rock Quality Designation (RQD) ($P_2$): Rated 3 to 20 (percentage of sound core pieces >10 cm in total core run length).
3. Spacing of Discontinuities ($P_3$): Rated 5 to 20 (from <60 mm up to >2 m).
4. Condition of Discontinuities ($P_4$): Rated 0 to 30 (evaluating roughness, separation, weathering, infilling, and persistence).
5. Groundwater Conditions ($P_5$): Rated 0 to 15 (completely dry to damp, wet, dripping, or flowing).

An orientation adjustment factor ($P_6$, ranging from 0 to -12 for drives, -24 for foundations) is subtracted based on whether joint strikes and dips are favorable or unfavorable relative to excavation alignment.

2. Barton's Q-System (NGI)

Developed by Barton, Lien, and Lunde (1974), the Q-System ranks rock mass quality on a logarithmic scale from 0.001 (Exceptionally Poor) to 1,000 (Exceptionally Good). The value of Q is computed from three numerical quotients:

$Q = \left( \frac{\text{RQD}}{J_n} \right) \times \left( \frac{J_r}{J_a} \right) \times \left( \frac{J_w}{\text{SRF}} \right)$

• Block Size Quotient ($ \text{RQD} / J_n $): RQD divided by Joint Set Number ($J_n$), representing spatial block size.
• Inter-Block Shear Strength ($ J_r / J_a $): Joint Roughness Number ($J_r$) divided by Joint Alteration Number ($J_a$), reflecting friction along joint surfaces.
• Active Stress State ($ J_w / \text{SRF} $): Joint Water Reduction Factor ($J_w$) divided by Stress Reduction Factor (SRF), representing water pressure and active stress conditions.

3. Hoek-Brown Geological Strength Index (GSI)

Hoek and Brown introduced GSI (ranging from 10 to 100) to account for heavily jointed rock masses where individual discontinuity mapping is impractical. GSI maps rock mass structure (blocky, disintegrated, foliated) against joint surface conditions (smooth, slickensided, weathered) to modify the non-linear Hoek-Brown failure criterion:

$\sigma_1' = \sigma_3' + \sigma_{ci} \left( m_b \frac{\sigma_3'}{\sigma_{ci}} + s \right)^a$

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Intact Rock Mechanics & Laboratory Testing

Determining intact rock material strength requires standard ISRM/ASTM laboratory tests:

1. Uniaxial Compressive Strength (UCS, $\sigma_c$): Cylindrical core specimen ($L/D = 2.0 - 2.5$) loaded axially until brittle failure. Measures peak compressive strength $\sigma_c$ and Young's Modulus ($E$).
2. Triaxial Compression Test: Confining pressure $\sigma_3$ is applied via hydraulic cell while axial load $\sigma_1$ increases to failure. Plotting Mohr circles at varying confining pressures establishes the Mohr-Coulomb shear strength envelope: $\tau = c + \sigma_n \tan \phi$ where $c$ is cohesion and $\phi$ is the internal friction angle.
3. Brazilian Tensile Test: Indirect tensile strength measurement. A disk-shaped rock core ($t/D \approx 0.5$) is loaded diametrically in compression. Failure occurs in tension along the vertical diameter. Tensile strength $\sigma_t$ is calculated as: $\sigma_t = \frac{2P}{\pi D t}$ where $P$ is peak failure load (N), $D$ is specimen diameter (m), and $t$ is specimen thickness (m).

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In-Situ Stress Fields & Boundary Stress Concentration

Prior to excavation, rock mass experiences virgin in-situ principal stresses: vertical stress $\sigma_v$ and horizontal principal stresses $\sigma_H$ and $\sigma_h$.

Vertical Stress ($\sigma_v$)

Vertical stress increases linearly with depth $z$ under gravity: $\sigma_v = \gamma z$ For average rock unit weight $\gamma = 0.027 \text{ MPa/m}$ ($27 \text{ kN/m}^3$), vertical stress at depth $z = 500 \text{ m}$ is $\sigma_v = 13.5 \text{ MPa}$.

Stress Ratio ($K$)

The ratio of average horizontal stress to vertical stress is defined as: $K = \frac{\sigma_h}{\sigma_v}$ Under purely elastic gravitational loading, $ K_0 = \frac{\nu}{1-\nu} \approx 0.25 - 0.33 $. However, tectonic compression in active mountain belts like the Philippine archipelago frequently creates high horizontal stresses where $ K = 1.0 - 2.5 $ at shallow to moderate depths.

Kirsch Equations for Circular Openings

When a circular tunnel of radius $a$ is driven through an elastic rock mass, surrounding stresses are redistributed. At the boundary ($r = a$), radial stress $\sigma_r = 0$, and tangential stress $\sigma_\theta$ varies with angle $\theta$ relative to horizontal:

$\sigma_\theta = \sigma_v \left[ (1+K) + 2(1-K)\cos(2\theta) \right]$

• Tangential Stress at the Roof/Crown ($\theta = 90^\circ$): $\sigma_{\theta,\text{roof}} = (3K - 1)\sigma_v$
• Tangential Stress at the Sidewall ($\theta = 0^\circ$): $\sigma_{\theta,\text{sidewall}} = (3 - K)\sigma_v$

For hydrostatic stress ($K = 1$), boundary tangential stress is uniform around the circular opening at $\sigma_\theta = 2\sigma_v$.

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Rockburst Mechanisms & Strata Control

In deep underground mines (>500 m) with hard, highly brittle rock masses (e.g., quartzites, dense diorites), high differential stress concentration can trigger rockbursts — sudden, violent expulsions of rock driven by rapid release of strain energy.

• Mechanisms: Occurs when boundary stress exceeds rock mass compressive strength ($\sigma_1 / \sigma_{ci} > 0.4$), or when sudden slip occurs along fault planes near excavations.
• Strata Control Mitigation: Destress blasting to soften rigid pillars, microseismic monitoring networks for continuous seismic hazard tracking, and purpose-designed dynamic support systems, such as yielding energy-absorbing bolts or cables combined with heavy mesh.

From Classification to a Defensible Ground-Control Design

RMR and Q summarize observations; they do not directly prove that an excavation is stable. Preserve the underlying mapping, core photographs, laboratory results, discontinuity orientations, groundwater evidence, stress data, scale, and uncertainty. Two headings with the same classification score can have different failure modes because one is structurally controlled, another stress-damaged, and another weakens when wet. Classification correlations should be applied only within their empirical scope and checked against local performance.

A ground-control workflow defines excavation purpose and life, develops geotechnical domains, identifies credible gravity-, structure-, stress-, and time-dependent mechanisms, and selects analysis proportionate to consequence. Kinematic screening tests whether planes or intersections can daylight; limit-equilibrium or numerical models test forces and deformation under stated assumptions; empirical support charts provide comparison with case histories. None substitutes for construction mapping and observational verification.

Support design must specify demand and capacity over the intended life. Record bolt type, length, spacing, bond, plate and mesh compatibility, shotcrete thickness and toughness, corrosion and dynamic requirements, installation timing, quality tests, rehabilitation rules, and exclusion during curing or hazardous conditions. Inspect actual geology after every advance and compare deformation, damage, water and support loads with trigger levels. If observations fall outside the design basis, restrict exposure, investigate, revise the model and support, and document the change. The exam-safe principle is to connect measured ground behavior to a failure mechanism and a verified control rather than select support from one rock-mass number alone.