Chapter 5 — In-Situ & Induced Stresses Around Underground Openings¶

5.1 The Virgin In-Situ Stress Field¶

Before an excavation is created, the subterranean rock mass is subjected to a state of natural equilibrium termed the virgin (or in-situ) stress field.

Vertical Stress (\(\sigma_v\))¶

Vertical stress is generated by the gravity weight of the overlying rock column (overburden depth \(z\), unit weight \(\gamma \approx 0.027\text{ MN/m}^3\)):

\[\sigma_v = \int_0^z \gamma(z)\, dz \approx \gamma z \approx 0.027 z \quad (\text{MPa for } z \text{ in meters})\]

Horizontal Stress Ratio (\(k_0\))¶

Tectonic plate motions, thermal contraction, and erosion history cause horizontal stresses (\(\sigma_h, \sigma_H\)) to deviate sharply from simple hydrostatic assumptions:

\[k_0 = \frac{\sigma_h}{\sigma_v}\]

Based on worldwide measurements compiled by Hoek and Brown, at shallow depths (\(< 500\text{ m}\)), \(k_0\) frequently exceeds \(1.5\) to \(3.5\) due to locked-in tectonic remnant stresses:

\[\frac{100}{z} + 0.3 \le k_0 \le \frac{1500}{z} + 0.5\]

5.2 The Kirsch Analytical Solution for Circular Tunnels¶

Excavating an underground cavity eliminates internal tractions, forcing stresses to flow around the void. For a circular tunnel of radius \(a\) in an isotropic elastic medium subjected to vertical stress \(p\) and horizontal stress \(k p\):

Tangential (Hoop) Stress (\(\sigma_\theta\))¶

\[\sigma_\theta = \frac{p}{2} \left[ (1 + k)\left(1 + \frac{a^2}{r^2}\right) - (1 - k)\left(1 + 3\frac{a^4}{r^4}\right)\cos 2\theta \right]\]

Radial Stress (\(\sigma_r\))¶

\[\sigma_r = \frac{p}{2} \left[ (1 + k)\left(1 - \frac{a^2}{r^2}\right) + (1 - k)\left(1 - 4\frac{a^2}{r^2} + 3\frac{a^4}{r^4}\right)\cos 2\theta \right]\]

At the Tunnel Boundary (\(r = a\)):¶

Radial stress is zero: \(\sigma_r = 0\).
Crown / Invert (\(\theta = \pi/2\)): \(\sigma_{\theta,\text{crown}} = p (3k - 1)\).
Sidewalls (\(\theta = 0\)): \(\sigma_{\theta,\text{sidewall}} = p (3 - k)\).

Case A: Hydrostatic Stress (k = 1.0)
Boundary Tangential Stress: sigma_theta = 2p everywhere (2x concentration).

Case B: High Horizontal Tectonic Stress (k = 2.0)
Crown Stress: sigma_theta = p(3(2) - 1) = 5p  (Severe compressive crushing / spalling).
Sidewall Stress: sigma_theta = p(3 - 2) = 1p.

Case C: Low Horizontal Stress (k = 0.25)
Crown Stress: sigma_theta = p(3(0.25) - 1) = -0.25p (Tensile opening / block dropouts).
Sidewall Stress: sigma_theta = p(3 - 0.25) = 2.75p.

5.3 Stress Concentration & Failure Modes Around Caverns¶

When induced tangential stress \(\sigma_\theta\) exceeds the rock mass compressive strength (\(\sigma_{cm}\)): 1. Brittle Spalling / Slabbing: Stable thin flakes peel off parallel to the excavation boundary in massive hard rock (\(UCS > 100\text{ MPa}\)). 2. Plastic Yielding Zone: A circular plastic annulus of radius \(r_p > a\) develops around the opening, redistributing peak stresses away from the perimeter into the solid interior. 3. Tensile De-Stressing: In tensile roof zones, joint friction drops to zero (\(\sigma_n' \le 0\)), causing unbonded rock blocks to drop under gravity unless secured with pattern rockbolts.

5.4 In-Situ Stress Measurement Methods¶

Technique	Operating Principle	Depth Range
Overcoring (CSIRO HI Cell)	Measures stress relief strains when a pilot hole is overcored by a larger diamond barrel	Up to \(30 - 50\text{ m}\) from underground faces
Hydraulic Fracturing (Hydrofrac)	Isolates a borehole interval with straddle packers, pressurizes with water until rock fractures perpendicular to \(\sigma_3\)	Surface boreholes \(> 1,000\text{ m}\)
Borehole Breakout Analysis	Caliper and acoustic televiewer logs detect compressive spalling diametrally aligned with minimum horizontal stress direction	Deep exploration drillholes

5.5 Canonical Terminology¶

English Term	Canonical Vietnamese Translation	Definition
In-situ stress / Virgin stress	Ứng suất nguyên sinh	Natural stress field existing prior to excavation
Induced stress	Ứng suất kích ứng	Re-distributed stress field surrounding an excavated cavity
Kirsch solution	Lời giải Kirsch	Closed-form elastic stress equations around a circular borehole
Tangential stress / Hoop stress	Ứng suất tiếp vòng biên	Stress acting tangent to the excavation perimeter
Overcoring	Phương pháp khoan chụp đo biến dạng giải phóng	Stress measurement technique drilling over an instrumented pilot core