Rock Mechanics - Theory & Concepts

Rock Mechanics

State of Stress in Rock Mass

In-situ stresses govern rock mass behavior during excavation.

Before excavation, rock is subjected to natural in-situ stresses:

Checklist

Vertical Stress ( $\sigma_v$ ): Primarily due to the weight of the overlying rock (overburden). $\sigma_v = \gamma z$ , where $\gamma$ is the unit weight and $z$ is depth.
Horizontal Stress ( $\sigma_h$ ): More complex, influenced by Poisson's ratio effect, tectonic history, and topography. Often quantified by the lateral earth pressure coefficient $k = \sigma_h / \sigma_v$ . In rock, $k$ can frequently exceed 1.0 (horizontal stress greater than vertical) due to locked-in tectonic stresses.

Rock Strength

The behavior of rock materials under engineering stress.

Rock Mechanics is the theoretical and applied science of the mechanical behavior of rock and rock masses. It is the branch of mechanics concerned with the response of rock to the force fields of its physical environment. A critical distinction in this field is between Intact Rock and the Rock Mass.

Intact Rock

The unfractured, continuous, and uniform blocks of rock material that exist between structural discontinuities. This is what is typically tested in the laboratory (e.g., Uniaxial Compressive Strength or UCS testing on core samples).

Rock Mass

The entire in-situ volume of rock, which includes both the intact rock blocks and all the geological discontinuities (fractures, joints, bedding planes, faults). These discontinuities significantly reduce the overall strength, stiffness, and stability of the mass compared to the intact rock.

Failure Criteria

Mohr's Circle for Plane Stress

Input Stresses (MPa)

Normal Stress X (

\sigma_x

): +80

Normal Stress Y (

\sigma_y

): +20

Shear Stress (

\tau_{xy}

): +40

Principal Results

Center (C)

50.0

Radius (R)

50.0

Max Principal (

\sigma_1

)

100.0

Min Principal (

\sigma_2

)

0.0

Max In-Plane Shear (

\tau_{max}

)

50.0

Loading chart...

Red dots indicate principal stresses, max shear stresses, the center, and the original state of stress faces (X and Y).

To predict when a rock mass will fail under complex stress states, engineers rely on mathematical failure criteria.

Mohr-Coulomb Criterion

The Mohr-Coulomb criterion is the most widely used linear empirical model for rock failure under shear stress. It is mathematically simple and widely implemented in basic geotechnical software. However, it often overestimates rock strength at high confining pressures because real rock failure envelopes are typically curved, not straight.

\tau = c + \sigma_n \tan \phi

Where:

Checklist

$\tau$ = Shear Strength (MPa)
$c$ = Cohesion (MPa) - The inherent shear strength of the rock when there is zero normal stress clamping it together.
$\sigma_n$ = Normal Stress (MPa) - The stress acting perpendicularly across the potential failure plane, pushing the surfaces together.
$\phi$ = Angle of Internal Friction ( $^\circ$ ) - A measure of the frictional resistance between the rock surfaces sliding past one another.

Hoek-Brown Criterion

The industry standard for modeling the non-linear failure of highly fractured rock masses.

Unlike Mohr-Coulomb, the Hoek-Brown failure criterion is an empirical, non-linear relationship developed specifically for rock masses. It explicitly accounts for the degree of fracturing (via the Geological Strength Index, GSI) and the quality of the intact rock pieces.

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

Where:

Checklist

$\sigma_1'$ and $\sigma_3'$ are the major and minor effective principal stresses at failure.
$\sigma_{ci}$ is the uniaxial compressive strength (UCS) of the intact rock material.
$m_b$ , $s$ , and $a$ are empirical constants derived from the rock type and the Geological Strength Index (GSI) of the rock mass. Intact, massive rock has $s = 1$ and $a = 0.5$ . Highly crushed, poor-quality rock has $s \approx 0$ .

Rock Mass Classification

Empirical systems for quantifying the quality of complex geology.

Because it is nearly impossible to mathematically model every single joint in a large rock mass, engineers have developed several empirical classification systems. These systems convert qualitative geological observations into quantitative ratings used for design, particularly for estimating tunnel support requirements.

1. Rock Mass Rating (RMR)

Developed by Z.T. Bieniawski (1973), the Geomechanics Classification or RMR system assigns point ratings (summing from 0 to 100) based on six fundamental parameters:

Checklist

1. Intact Rock Strength: Evaluated via UCS or point-load testing. (Max 15 points)
2. RQD (Rock Quality Designation): The percentage of intact core pieces longer than 10 cm recovered from a drill run. (Max 20 points)
3. Spacing of Discontinuities: How far apart the joints are. (Max 20 points)
4. Condition of Discontinuities: Accounts for joint roughness, separation, continuity, weathering, and infilling materials (like weak clay gouge). (Max 30 points)
5. Groundwater Conditions: Ranging from completely dry to flowing under high pressure. (Max 15 points)
6. Orientation of Discontinuities: A negative adjustment factor applied based on how favorable or unfavorable the strike and dip are relative to the tunnel axis or slope face.

2. Q-System (Barton)

Developed by N. Barton, R. Lien, and J. Lunde (1974) at the Norwegian Geotechnical Institute (NGI), the Q-system is specifically tailored for designing rock support (rockbolts and shotcrete) in tunnels and underground caverns. The values range logarithmically from 0.001 (exceptionally poor) to 1000 (exceptionally good).

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

This equation represents three distinct quotients representing different physical parameters of the rock mass:

Checklist

Block Size ( $\text{RQD}/J_n$ ): Represents the relative block size. $J_n$ is the Joint Set Number (e.g., massive rock = 0.5, two joint sets = 4, crushed rock = 20).
Inter-Block Shear Strength ( $J_r/J_a$ ): Represents friction. $J_r$ is joint roughness (rough = 3, slickensided = 0.5). $J_a$ is joint alteration or clay filling (tight and clean = 0.75, thick swelling clay = 15).
Active Stress Environment ( $J_w/\text{SRF}$ ): $J_w$ is the joint water reduction factor (dry = 1.0, high pressure = 0.1). $\text{SRF}$ is the Stress Reduction Factor, accounting for loosening loads in shallow tunnels or rock burst potential in deep, highly stressed tunnels.

Tunneling in Rock

Excavation methods and stability in underground construction.

Stand-Up Time

Common excavation methods include:

Checklist

Drill and Blast (D&B): The traditional, cyclical method used in hard rock. It is highly flexible for changing tunnel shapes and variable ground conditions but involves discontinuous progress (drill, load, blast, ventilate, muck, support).
Tunnel Boring Machine (TBM): A massive machine providing continuous excavation, often installing pre-cast concrete support segments simultaneously. TBMs are exceptionally fast for long tunnels in relatively uniform, predictable ground but lack flexibility if unexpected fault zones are encountered.

Key Takeaways

While Intact Rock is tested in the laboratory, the large-scale behavior of the Rock Mass in the field is governed almost entirely by geological discontinuities (joints, faults, bedding).
The Mohr-Coulomb criterion is the fundamental linear model describing rock shear strength as a function of inherent cohesion, internal friction, and applied normal stress.
The non-linear Hoek-Brown criterion is widely used to estimate the rock mass strength based on the intact rock strength and the Geological Strength Index (GSI).
Empirical classification systems like RMR and the Q-System are essential engineering tools for quantifying complex geological conditions.
The choice of tunneling method (e.g., Drill & Blast vs. TBM) heavily depends on the expected ground conditions, the required flexibility, and the tunnel's total length.

PreviousStructural Geology - Examples & Applications

Quiz Me

NextRock Mechanics - Examples & Applications