Beginner — Rock Properties and Classification
Start here if you are new to rock mechanics.
Rock Properties and Classification
Intact rock properties are determined from laboratory tests on core samples. The uniaxial compressive strength (UCS) is the most fundamental parameter, measured by loading cylindrical specimens (height/diameter 2-3) to failure. Typical UCS ranges: granite 150-250 MPa, limestone 50-100 MPa, sandstone 30-80 MPa, shale 10-30 MPa. The Brazilian (indirect) tensile test measures tensile strength, typically 5-15% of UCS. The modulus ratio E/UCS varies from 200-500 for typical rocks. Poisson's ratio ranges from 0.15 (granite) to 0.35 (shale). The Schmidt hammer provides a field estimate of UCS.
Rocks are classified by origin: igneous (granite, basalt — high strength, low porosity), sedimentary (sandstone, limestone, shale — strength varies with cementation and porosity), and metamorphic (gneiss, marble, slate — foliated or non-foliated). Porosity affects strength and deformability: porosity of 1-5% for crystalline rocks, 10-30% for sandstones. Permeability of intact rock is low (10^-9 to 10^-12 m/s), but jointed rock mass permeability can be orders of magnitude higher. The Hoek-Brown failure criterion is widely used for rock: sigma1 = sigma3 + sigma_ci * (mb * sigma3 / sigma_ci + s)^a, where mb, s, and a are material constants reflecting rock mass quality.
Rock Mass Classification Systems
Rock mass classification integrates rock properties and discontinuity characteristics into a single index for engineering design. RQD (Rock Quality Designation) measures the percentage of intact core pieces longer than 10cm in a core run: RQD>90% excellent, 75-90% good, 50-75% fair, 25-50% poor, <25% very poor. The RMR (Rock Mass Rating) system by Bieniawski combines UCS, RQD, joint spacing, joint condition, and groundwater conditions into a rating from 0-100, classifying rock into 5 classes from very good to very poor. RMR is used for tunnel support estimation, slope design, and foundation bearing capacity.
The Q-system by Barton uses six parameters: RQD, joint set number Jn, joint roughness Jr, joint alteration Ja, joint water reduction Jw, and stress reduction factor SRF. Q = (RQD/Jn)(Jr/Ja)(Jw/SRF), with values ranging from 0.001 (exceptionally poor) to 1000 (exceptionally good). The Q-system provides specific recommendations for rock bolt spacing, shotcrete thickness, and steel set requirements. GSI (Geological Strength Index) by Hoek estimates rock mass strength by visually assessing blockiness and joint surface condition, bypassing the need for detailed discontinuity data. GSI is directly used in the Hoek-Brown failure criterion.
Rock Discontinuities
Discontinuities (joints, bedding planes, faults, shear zones, foliation) control the mechanical behavior of rock masses. Joint orientation is described by dip direction (compass direction of steepest descent) and dip angle (angle from horizontal). Stereographic projection plots discontinuity orientations as great circles or poles on a hemispherical projection, enabling identification of joint sets, kinematic analysis of slope failures, and visualization of orientation distributions. Contoured pole plots show concentration zones representing dominant joint sets.
Joint roughness affects shear strength: rough joints have higher peak friction angles than smooth or slickensided joints. Barton's JRC (Joint Roughness Coefficient) ranges from 1 (smooth planar) to 20 (rough undulating). Joint compressive strength JCS and basic friction angle phi_b are combined in Barton's shear strength model: tau = sigma_n * tan(JRC * log10(JCS/sigma_n) + phi_b). Joint persistence (the proportion of jointed area along a potential failure plane) critically affects rock mass strength. Water flow through joints is governed by the cubic law: Q proportional to aperture^3, making flow highly sensitive to stress-induced aperture changes.
Intermediate — Underground Excavations and Slopes
Build on fundamentals with excavation design and slope stability.
Tunnel Stability and Support Design
Tunnel excavation in rock redistributes in-situ stresses, creating a plastic (yielded) zone around the opening. The ground reaction curve (GRC) relates tunnel wall displacement to internal support pressure, derived from the convergence-confinement method. Support systems include: rock bolts (mechanical expansion shell, resin-grouted rebar, friction bolts like Swellex and Split Set), shotcrete (plain or steel fiber reinforced, 50-150mm thick, applied in multiple passes), steel sets (lattice girders, rigid I-beam ribs, TH-section yielding arches), and cast-in-place concrete linings (300-600mm thick for permanent support).
The New Austrian Tunneling Method (NATM) is a philosophy based on mobilizing the inherent strength of the surrounding rock mass through controlled deformation. The shotcrete lining is applied as a thin, flexible layer that deforms with the rock, monitoring convergence to verify design assumptions. Sequential excavation (heading and bench) in large tunnels, with temporary invert arches to close the ring. The observational method uses monitoring data (convergence, extensometer, load cells) to adjust support as construction proceeds. Empirical support design uses rock mass classification (RMR, Q-system) with standard support charts. Numerical modeling (FEM, DEM) provides optimized support for complex conditions.
Rock Slope Stability
Rock slope failures are controlled by the geometry and strength of discontinuities. Failure modes: planar sliding (sliding along a single discontinuity plane day-lighting on the slope face), wedge sliding (sliding along the intersection line of two discontinuities), toppling (overturning of slabby rock columns), and circular failure (through weak rock masses or heavily jointed rock). Kinematic analysis using stereonets identifies which failure modes are possible based on the orientation of discontinuities relative to the slope face and slope angle.
Limit equilibrium analysis for planar sliding: FoS = (c*A + W*cos(beta)*tan(phi)) / (W*sin(beta)), where c is joint cohesion, phi is joint friction angle, W is block weight, beta is failure plane dip, and A is failure plane area. Water pressure in tension cracks and on the failure plane reduces stability significantly. Wedge stability analysis considers the intersection line orientation and friction on both planes. Stabilization methods: rock bolts/dowels (increase normal force on sliding surface), cable anchors (post-tensioned to provide direct resistance), shotcrete (surface protection against ravelling), drainage (weep holes, horizontal drains to reduce water pressure), mesh and rock traps for surface protection of roads and infrastructure below slopes.
Rock Foundations
Foundations on rock masses transfer structural loads to the rock. Bearing capacity of jointed rock is estimated using the Hoek-Brown failure criterion with appropriate GSI values. For massive rock with widely spaced joints, allowable bearing pressures of 5-15 MPa are typical. For jointed rock, the bearing capacity is controlled by joint orientation, spacing, and strength. Settlement of rock foundations is typically small (5-15mm for most rock types) but differential settlement due to variably weathered zones or filled joints must be considered. The elastic modulus of rock mass Em is estimated from intact modulus Ei using reduction factors based on RMR or GSI.
Rock socketed piles (drilled shafts into rock) derive capacity from end bearing on rock and side friction along the rock socket. End bearing capacity: qp = N_phi * sigma_ci (for intact rock) or based on Hoek-Brown for jointed rock. Socket friction (side resistance) ranges from 0.5-2.0 MPa depending on rock strength and socket roughness. Required socket length is determined from load test data or empirical correlations (for UCS > 5 MPa, socket length L/D = 2-6 typically). Foundation preparation includes: cleaning loose material, dental concrete for over-excavated zones, foundation grouting (consolidation grouting to improve rock mass properties, curtain grouting to reduce permeability), and proof rolling of exposed rock surfaces.
Advanced — Numerical Modeling and Design
For senior students and practicing engineers.
Numerical Modeling in Rock Mechanics
Numerical modeling is essential for complex rock engineering problems. Continuum methods (FEM, FDM) treat the rock mass as an equivalent continuum with equivalent properties: Phase2 (2D FEM for underground excavations and slopes), FLAC (2D/3D finite difference with explicit time-stepping for large deformations and progressive failure). Discontinuum methods (DEM) model rock as an assembly of blocks or particles interacting through contact laws: UDEC (Universal Distinct Element Code for 2D blocky rock), 3DEC (3D block modeling), and PFC (Particle Flow Code for granular rock and fragmentation). Hybrid methods combine continuum and discontinuum elements for efficient modeling.
Input parameters for numerical models include: rock mass modulus (Em from GSI reduction of Ei), strength parameters (Hoek-Brown mb, s, a from GSI and mi), joint stiffness (kn, ks from lab testing or empirical correlations), and in-situ stress (magnitude and orientation from regional stress data or measurement). Modeling excavation sequences simulates the construction process: stress redistribution, support installation timing, and ground response monitoring. Verification against analytical solutions (Kirsch solution for circular openings, closed-form tunnel convergence) and validation against field monitoring data (extensometer, convergence, load cell readings) are critical for reliable design.
In-Situ Stress Measurement
In-situ stress is the most important and most uncertain parameter in deep rock engineering. The vertical stress sigma_v is typically assumed equal to gamma * depth (overburden). Horizontal stresses sigma_h are more variable, with stress ratios k = sigma_h / sigma_v ranging from 0.5 (normal faulting) to 3.0+ (thrust faulting) depending on tectonic regime. The World Stress Map database compiles stress measurements globally. Measurement methods include: overcoring (CSIRO HI cell, USBM borehole deformation gauge — measures complete stress tensor in 3D at depths up to 500m), hydraulic fracturing (deepest method, up to 3km+, measures minimum principal stress from fracture reopening pressure, maximum principal stress from breakdown pressure), and flat jack testing (in tunnels, measures stress normal to slot).
Stress orientation affects tunnel stability: tunnels oriented parallel to the major principal stress experience the least roof instability. Stress-induced damage in deep tunnels (>500m depth) includes spalling (tensile splitting parallel to excavation boundary), rockburst (violent strain energy release in massive brittle rock), and slabbing. The ratio of maximum tangential stress to intact rock UCS (stress/strength ratio) predicts spalling potential: values >0.5 indicate minor spalling, >0.8 indicate severe spalling, and >1.0 indicate extreme conditions requiring special support. Destressing techniques include stress relief by pilot tunnels and predetermined break lines in the excavation profile.
Rock Reinforcement Design
Rock reinforcement design ensures that support elements work together with the rock mass to create a stable arch. Rock bolt types: fully grouted (deformed rebar with cement or resin grout, develops bond strength along entire length), mechanical expansion shell (quick installation, good for temporary support, works in competent rock), friction bolts (Split Set — slotted tube compressed during installation, Swellex — folded tube expanded by water pressure, both provide immediate support). Cable bolts (multi-strand steel cables grouted into boreholes, 3-20m length) provide deep reinforcement for large excavations. Bolt pattern design: spacing typically 1-2m, length determined by suspension effect (L > bolt spacing) or beam building theory (L = 1.2-1.6 * bolt spacing * rock quality factor).
Shotcrete design: plain shotcrete (20-40 MPa compressive strength, 0.25-0.50 MPa bond strength), steel fiber reinforced shotcrete (SFRS — 30-60 kg/m3 steel fibers, improved toughness and ductility, reduced rebound). Design thickness from ground reaction curve: 50-100mm for good rock, 100-200mm for fair rock, 200-300mm for poor rock. Steel sets: lattice girders (light, easy to install, good for combining with shotcrete), TH-section yielding arches (for squeezing ground, allow controlled deformation while maintaining support resistance). The ground-support interaction diagram plots the ground reaction curve against the support reaction curve — the intersection defines the equilibrium displacement and support pressure, which must be within the support system capacity with adequate factor of safety.
Practice Exercises
Exercise 1: Rock Mass Classification
A tunnel is being driven through granite: UCS=120MPa, RQD=75%, joint spacing=0.3m, joint condition=slightly rough with <1mm aperture, slightly weathered, groundwater condition=dry. Calculate the RMR and Q-system ratings. Determine the rock mass class, stand-up time, and preliminary support recommendations per the Q-system support chart.
Exercise 2: Planar Wedge Stability Analysis
A rock slope has a bedding plane dipping 45 degrees toward the slope face (slope face dip 70 degrees). Joint cohesion c=50kPa, friction angle phi=30 degrees, rock unit weight 26kN/m3, slope height H=20m. Calculate the factor of safety for planar sliding. Determine if rock bolts inclined at 15 degrees above horizontal with 300kN capacity per bolt are sufficient to achieve FoS=1.5.
Exercise 3: Tunnel Convergence Analysis
A circular tunnel radius 3m is excavated at 200m depth in rock with sigma_ci=30MPa, GSI=40, mi=10, unit weight 27kN/m3. Using the convergence-confinement method, estimate the plastic zone radius, tunnel wall displacement, and required support pressure. Assume hydrostatic in-situ stress conditions and Hoek-Brown failure criterion.
Exercise 4: Rock Foundation Bearing Capacity
A 3m x 3m foundation rests on limestone rock mass: UCS=50MPa, GSI=50, mi=12, unit weight 26kN/m3. Using the Hoek-Brown failure criterion, calculate the allowable bearing pressure for a factor of safety of 3. Foundation depth Df=1.5m. Compare with the empirical bearing capacity values from RMR classification.
Related Calculators
Slope Stability Calculator
Calculate factor of safety for slope stability using various methods.
Soil Bearing Capacity Calculator
Calculate bearing capacity of soil using Terzaghi, Meyerhof, and Hansen methods.
Soil Permeability Calculator
Calculate flow rate and hydraulic conductivity using Darcy's law.
Pile Foundation Calculator
Calculate pile load capacity using static formulas for driven and bored piles.
SPT Analysis Calculator
Analyze Standard Penetration Test data and correlate soil properties.
Sheet Pile Calculator
Design cantilever and anchored sheet pile walls for excavation support.
References
- Hoek, E. and Brown, E.T. Underground Excavations in Rock. Institution of Mining and Metallurgy, 1980.
- Goodman, R.E. Introduction to Rock Mechanics. 2nd ed., Wiley, 1989.
- Hudson, J.A. and Harrison, J.P. Engineering Rock Mechanics. Pergamon, 1997.
- Brady, B.H.G. and Brown, E.T. Rock Mechanics for Underground Mining. 3rd ed., Kluwer, 2004.
- Bieniawski, Z.T. Engineering Rock Mass Classifications. Wiley, 1989.
- Civil Engineering Handbook — Rock mechanics and tunnel design chapter.
- Engineering Formula Library — Hoek-Brown, bearing capacity formulas.
- Engineering Standards Reference — ASTM rock testing standards.
- Engineering Glossary — Definitions of rock mechanics terms.