Rock Mechanics formula solvers

Categories — 73 of them, 1787 solvers in all

Acoustics & Noise Additive Manufacturing Aerospace & Flight Air Quality & Dispersion Algebra Astronomy & Gravitation Axle Loads & Pavement Billiards & Cue Sports Biomedical & Clinical Brewing & Fermentation Calculus Ceramics & Glass Chemical Engineering Chemistry Civil & Surveying Coastal & Ocean Waves Computer Science Concrete Corrosion & Protection Crop Production Cryptography & Key Sizing Ecology & Biodiversity Electrical Trade Electricity & Magnetism Epidemiology & Diagnostic Accuracy Erosion & Sediment Everyday & Health Farm Operations Fire Protection & Fire Dynamics Fluid Mechanics Forestry & Forest Ecology Freight & Shipping Games & Ratings Geometry Heat Transfer HVAC & Hydronics Indoor & Greenhouse Industrial Engineering Livestock & Feed Machine Design Machining & Turning Mechanics Metallurgy & Heat Treatment Modern Physics Money & Business Navigation & Position Optics Packaging & Box Strength Photogrammetry & Remote Sensing Physics Plant Breeding & Genetics Polymers & Plastics Processing Prices & Markets Probability Rock Mechanics Seismology Semiconductors & Chips Snow & Ice Soil Mechanics Solar & Wind Power Statistics Storm & Sky Strength of Materials Textiles & Fabric Thermodynamics Trades & Construction Traffic Flow & Signals Trigonometry Vectors & Matrices Water & Wastewater Water Treatment Waves & Oscillations Welding & Joining

All 1787 solvers →

A B E G H K M O P R S T

Airflow to Dilute a Gas

Q = \frac{q_{gas}}{C_{limit} - C_{intake}}

Rock MechanicsHow much air a working place needs to keep a contaminant below its limit: the volume of gas being made, divided by the concentration headroom between the limit and whatever the intake air already carries. It is a mass balance and nothing more, and its one assumption — perfect mixing — is the one a real heading breaks.

Airway Resistance from Geometry

R = \frac{k \, O \, L}{A^{3}}

Rock MechanicsEverything an airway does to the air, collapsed into one number. Atkinson's equation splits neatly into a part that depends only on the opening — its lining, its rubbing surface, its length and its area — and a part that depends only on the airflow. This is the first part, and it is what a ventilation network model stores for every branch in the mine.

Atkinson's Equation for Airway Pressure Drop

p = \frac{k \, O \, L \, Q^{2}}{A^{3}}

Rock MechanicsJohn Atkinson's 1854 equation for the pressure a mine airway takes out of the ventilating current: the friction factor of the lining, times the rubbing surface, times the square of the airflow, divided by the cube of the cross-section. It is Darcy-Weisbach rearranged for a passage described by its perimeter and area rather than by a diameter, and after 170 years it is still how mine ventilation is done.

Barton Q-System Rock Mass Quality

Q = \frac{RQD}{J_n} \cdot \frac{J_r}{J_a} \cdot \frac{J_w}{SRF}

Rock MechanicsBarton, Lien and Lunde's 1974 tunnelling quality index, fitted to some 200 case records at the Norwegian Geotechnical Institute. Six ratings multiplied in three pairs: how blocky the rock is, how well the blocks interlock, and what the water and the stress field are doing to them. It spans six orders of magnitude, from 0.001 for squeezing ground to 1000 for massive unjointed rock.

Barton Roof Support Pressure (simplified)

P_{roof} = \frac{2}{J_r}\,Q^{-1/3}

Rock MechanicsThe form of Barton's roof support pressure used when the rock mass has three or more joint sets, where the block size term has saturated and drops out. Two ratings carry the whole estimate: the tunnelling quality index Q and the joint roughness number Jr. Everything true of the fuller form is true of this one — it is a fit to case records, and the leading 2 is 2 kg/cm² rather than a pure number.

Barton Roof Support Pressure (with Jn)

P_{roof} = \frac{2\sqrt{J_n}}{3\,J_r}\,Q^{-1/3}

Rock MechanicsBarton, Lien and Lunde's 1974 estimate of the permanent support pressure a tunnel roof will need, from the rock mass quality Q, the joint set number and the joint roughness. This is the form for masses with FEWER than three joint sets, where the block size term still matters; at Jn = 9 it becomes the simplified form exactly. It is a curve fitted through some 200 case records, not a mechanics result.

Blast Burden (Konya's Formula)

B = 3.15 \, D_e \left( \frac{SG_e}{SG_r} \right)^{1/3}

Rock MechanicsHow far back from the free face a blast hole should be drilled, from Konya's formula as published in FHWA-HI-92-001. The burden is the most important single dimension in a bench blast: get it right and the rock moves, get it too large and the shot is confined, too small and the energy goes into the air as noise, flyrock and airblast.

Blast Vibration: Scaled Distance and Peak Particle Velocity

PPV = K \left( \frac{D}{\sqrt{W}} \right)^{-\beta}

Rock MechanicsThe standard blast vibration prediction: peak particle velocity falls off as a power of the square-root scaled distance, which is the distance to the point of interest divided by the square root of the charge fired on one delay. The FORM comes from Siskind and colleagues at the US Bureau of Mines; the two CONSTANTS come from your own site and from nowhere else.

Bond's Law and the Work Index

E = 10 \, W_i \left( \frac{1}{\sqrt{P_{80}}} - \frac{1}{\sqrt{F_{80}}} \right)

Rock MechanicsFred Bond's 1952 relation for the energy it takes to grind ore: the specific energy depends on the difference in the reciprocal square roots of the product and feed sizes, scaled by a work index measured for that ore. It is the backbone of every grinding circuit sized in the last seventy years, and it is a correlation rather than a mechanism.

Elastic Tunnel Convergence

u = \frac{(p_0 - p_i)\,R}{2G}

Rock MechanicsHow far the wall of a circular tunnel moves inwards while the ground around it is still elastic: the difference between the in-situ stress and whatever the support is pushing back with, times the radius, over twice the shear modulus. It is the straight elastic portion of the ground reaction curve, it assumes a hydrostatic stress field, and it is the one part of convergence-confinement that closes in a single line.

Equivalent Dimension (De = span / ESR)

D_e = \frac{B}{ESR}

Rock MechanicsThe horizontal axis of Barton's support chart: the span, diameter or wall height of the opening divided by an excavation support ratio that says how much risk the opening's PURPOSE allows. Two tunnels of identical size in identical rock get different equivalent dimensions, and therefore different support, because one is a temporary mine drift and the other carries passengers.

Generalized Hoek–Brown Failure Criterion (2002)

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

Rock MechanicsThe strength of a jointed rock mass at a given confinement, in the 2002 edition of Hoek, Carranza-Torres and Corkum. Unlike Mohr–Coulomb, the envelope is CURVED: rock gains strength with confinement quickly at first and then far more slowly, which is why a straight line fitted at one stress level misleads badly at another.

Heim Ratio and Runout Distance (Fahrböschung)

L = \frac{H}{\left(H/L\right)}

Soil MechanicsRock MechanicsHorizontal runout of a rock avalanche or debris slide from its vertical drop and the Heim ratio H/L — the fall height divided by the travel distance, also read as the tangent of the travel angle a straight line from crown to toe makes with the horizontal.

Hoek–Brown Constants s and a (2002)

s = \exp\!\left( \frac{GSI - 100}{9 - 3D} \right), \qquad a = \frac{1}{2} + \frac{1}{6}\left( e^{-GSI/15} - e^{-20/3} \right)

Rock MechanicsThe other two constants of the 2002 generalized criterion. s measures how much of the intact rock's cohesion the mass has kept — 1 for genuinely intact rock, falling to a ten-thousandth for a poor mass — and a is the curvature of the envelope, fixed by GSI alone and never by anything else.

Hoek–Brown Rock Mass Constant m_b (2002)

m_b = m_i \, \exp\!\left( \frac{GSI - 100}{28 - 14D} \right)

Rock MechanicsHow the intact rock's frictional constant m_i is knocked down to the rock mass value m_b, in the 2002 edition. Two judgement calls go in — the Geological Strength Index and the disturbance factor — and both of them make the rock look better if you are generous.

Kuz–Ram Mean Fragment Size

X_m = A \left( \frac{V_0}{Q_e} \right)^{0.8} Q_e^{1/6} \left( \frac{115}{E} \right)^{19/20}

Rock MechanicsCunningham's 1983 combination of Kuznetsov's fragmentation equation with the Rosin–Rammler distribution: it predicts the MEAN fragment size a blast will produce from the rock, the geometry and the explosive. It is the standard first estimate in the industry, and its accuracy is entirely hostage to one input that nobody can measure.

Maximum Unsupported Span (Barton)

B_{max} = 2\,ESR\,Q^{0.4}

Rock MechanicsBarton's estimate of the largest span that will stand with no support at all, from the rock mass quality Q and the excavation support ratio the opening's purpose earns. The leading 2 is 2 METRES, not a pure number: this is a metric equation, and the answer an imperial reader sees has been converted from one.

Mine Air Power

P_{air} = p \, Q

Rock MechanicsThe power actually delivered to the air: pressure times volume flow. It is the useful output of a mine fan and the floor under its electricity bill — the shaft power is this divided by the fan efficiency, and the motor draw is that divided by the drive and motor efficiencies again.

Obert–Duvall Pillar Strength

S_p = S_1 \left( 0.778 + 0.222 \, \frac{W_p}{h} \right)

Rock MechanicsObert and Duvall's 1967 linear fit for the strength of a hard-rock pillar, scaled from the strength of a cubical specimen by the pillar's width-to-height ratio. Squat pillars are stronger than slender ones because the platens confine the middle of them, and this is the simplest honest expression of that fact.

Pillar Factor of Safety

FS = \frac{S_p}{\sigma_p}

Rock MechanicsPillar strength divided by pillar stress: the number the whole room-and-pillar exercise exists to produce. Both terms come from correlations with real scatter in them, so the factor of safety is a way of carrying that uncertainty explicitly rather than a promise about any one pillar.

Plastic Zone Radius (Mohr–Coulomb)

R_p = R \left[ \frac{2\left(p_0(k-1) + \sigma_{cm}\right)}{(1+k)\left((k-1)p_i + \sigma_{cm}\right)} \right]^{\frac{1}{k-1}}

Rock MechanicsHow far out from a circular tunnel the ground has failed, for a Mohr–Coulomb rock mass in a hydrostatic stress field. This is the classical closed-form convergence–confinement result, and it is the page that shows what support actually buys: in the plastic range, a modest confinement shrinks the failed ring sharply, which is the opposite of what happens in the elastic range.

Point Load Strength Index and the UCS it Implies

\sigma_c \approx k \, I_{s(50)}, \qquad I_{s(50)} = \left( \frac{D_e}{50\,\mathrm{mm}} \right)^{0.45} \frac{P}{D_e^{2}}

Rock MechanicsThe ISRM Suggested Method's field strength test: break a lump of rock between two conical platens, divide the breaking load by the square of the equivalent core diameter, correct it to a 50 mm standard size, and multiply by a conversion factor to estimate the uniaxial compressive strength. The test is cheap and portable. The conversion factor is where all the uncertainty lives, so it is an input here and not a constant.

Powder Factor

PF = \frac{m_e}{V_r}

Rock MechanicsThe mass of explosive spent per unit volume of rock broken — the single number a blast is judged and costed by. Everywhere except North America it is kilograms per cubic metre; in North America it is pounds per cubic yard, or per short ton, and the three are quoted interchangeably by people who then wonder why the figures do not agree.

Rock Avalanche Volume-Mobility Relation

\frac{H}{L} = k\,V^{-n}

Soil MechanicsRock MechanicsThe empirical power law behind large-landslide mobility: the Heim ratio H/L falls as the event volume rises, so bigger rock avalanches travel disproportionately further than friction allows. Both coefficients are regional fits and are entered by the reader.

Rock Mass Deformation Modulus (Hoek–Diederichs)

E_{rm} = 100000 \left[ \frac{1 - D/2}{1 + e^{(75 + 25D - GSI)/11}} \right] \ \mathrm{MPa}

Rock MechanicsThe stiffness of a jointed rock mass, from Hoek and Diederichs' 2006 simplified expression: a sigmoid in GSI, knocked down by the disturbance factor. This is the number that governs how much a tunnel converges or a foundation settles — strength decides whether the ground fails, stiffness decides how much it moves before it does.

Rock Quality Designation (RQD)

RQD = \frac{\sum L_{i \ge 100\,\mathrm{mm}}}{L_t} \times 100\%

Rock MechanicsDeere's 1964 index: of the core you pulled out of a run, what fraction came up as sound pieces at least 100 mm long. It is the oldest and cheapest measure of rock quality still in daily use, it feeds both the Q-system and RMR, and it is far cruder than the confidence usually placed in it.

Rosin–Rammler Passing Fraction

P(x) = 1 - \exp\!\left[ -\left( \frac{x}{x_c} \right)^{n} \right]

Rock MechanicsRosin and Rammler's 1933 distribution for broken and ground material: the fraction of a sample that passes a screen of size x, from just two parameters — a characteristic size and a uniformity index. It is the second half of Kuz-Ram, it is how a P80 is defined, and it is the curve behind every screen analysis in mineral processing.

RQD from Volumetric Joint Count

RQD = 115 - 3.3\,J_v

Rock MechanicsPalmstrom's 1982 estimate of RQD for rock you can see but cannot drill: count the joints crossing a cubic metre of the mass, and the correlation returns the RQD a core run would probably have shown. Useful in a face, a trench or an outcrop, and never a substitute for core where core exists.

Snow Avalanche Alpha-Beta Runout (Lied-Bakkehøi)

\alpha = k\,\beta + c

Soil MechanicsRock MechanicsThe alpha-beta statistical model for extreme snow avalanche runout: the runout angle α from the release area to the distal limit is a regression on the beta angle, the angle to the point where the path first flattens to 10°. The coefficients are regional and must be entered.

Squeezing Competence Factor

N_c = \frac{\sigma_{cm}}{\sigma_v}

Rock MechanicsRock mass strength divided by in-situ stress: the first screening question in any deep tunnel. Below about one the ground cannot carry its own overburden elastically and will squeeze into the opening; well above one it behaves, and the problem becomes structurally controlled blocks instead. It is a ratio, not a prediction, and the boundary is a smear rather than a line.

Stemming, Subdrilling and Spacing from the Burden

X = f_B \cdot B

Rock MechanicsOnce the burden is fixed, the rest of a bench pattern follows from it as simple multiples: stemming about 0.7 B, subdrilling about 0.3 B, spacing between about 1.15 and 1.4 B for delayed holes. Enter the burden and the ratio you are working to and this returns the dimension. The ratios themselves come from FHWA-HI-92-001 and are starting points, not laws.

Terzaghi Rock Load Height

H_p = k\,(B + H_t)

Rock MechanicsTerzaghi's 1946 estimate of how deep a block of loosened rock hangs above a tunnel: a load factor read from his rock condition table, multiplied by the sum of the tunnel's width and height. It was calibrated on drill-and-blast tunnels held up by steel sets on timber blocking, and it is deliberately conservative.

Terzaghi Support Pressure

p_v = \gamma\,H_p

Rock MechanicsThe weight of Terzaghi's loosened rock block, expressed as a pressure on the support: the rock load height multiplied by the unit weight of the rock. This is where a rock load height becomes a number a steel set or a lining can be designed against, and it is the second half of the 1946 method.

The Airway Square Law

p = R \, Q^{2}

Rock MechanicsThe single most consequential fact in mine ventilation: pressure drop rises with the SQUARE of the airflow. Doubling the air through an opening quadruples the pressure the fan must raise, and because power is pressure times flow, it octuples the power bill. Plot it and you have the mine characteristic curve that a fan curve is read against.

Tributary Area Pillar Stress

\sigma_p = \sigma_v \frac{(W_p + B)^{2}}{W_p^{2}}

Rock MechanicsThe load a room-and-pillar layout puts on each pillar: every pillar carries the full weight of the rock over its own area plus the rock over the openings around it. It is a statics argument with no rock mechanics in it at all, which is both why it is trustworthy and why it is crude.

Two Airways in Parallel

\frac{1}{\sqrt{R_{eq}}} = \frac{1}{\sqrt{R_1}} + \frac{1}{\sqrt{R_2}}

Rock MechanicsWhen the air can choose between two openings it takes both, dividing itself so that the pressure drop across each is the same. The equivalent resistance that results is far below either branch — and unlike electrical resistors, the combination goes as the reciprocal SQUARE ROOT, because the airway law is quadratic rather than linear.

Two Airways in Series

R_{eq} = R_1 + R_2

Rock MechanicsAir that must pass through one opening and then the next carries the same quantity through both, so the pressure drops add and the resistances add with them. It is the simplest relation in ventilation network analysis and the one that makes the pointlessness of enlarging the wrong airway obvious.