Rock Mechanics and Rock Engineering

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OverviewOverview

Rock mechanics is t he theoret ical and applied science of t he mechanical behaviour of rock and rock masses. Rock mechanics deals with the mechanical propert ies of rock and the related methodologies required f or engineering design.

The subject of rock mechanics has evolved f rom dif f erent disciplines of applied mechanics. I t is a t ruly interdisciplinary subject , with applicat ions in geology and geophysics, mining, pet roleum and geotechnical engineering.

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Rock Mechanics and Rock Mechanics and Rock EngineeringRock Engineering

Rock mechanics involves characterizing the intact st rength and the geomet ry and mechanical propert ies of t he natural f ractures of the rock mass.

Rock engineering is concerned with specif ic engineering circumstances, f or example, how much load will t he rock support and whether reinf orcement is necessary.

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Nature of RockNature of Rock

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A common assumpt ion when dealing with the mechanical behaviour of solids is t hat t hey are:

· homogeneous · cont inuous · isot ropic

However, rocks are much more complex than this and their physical and mechanical propert ies vary according to scale. As a solid material, rock is of ten:

· heterogeneous · discont inuous · anisot ropic

Nature of RockNature of Rock

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sandstone

sandstone

shale fault

joints

strength

equal in

all directions

highstrength

varies with

directionlow

Homogeneous Cont inuous I sot ropic

Heterogeneous Discont inuous Anisot ropic

Rock as an Engineering MaterialRock as an Engineering Material

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One of t he most important , and f requent ly neglected, aspects of rock mechanics and rock engineering is t hat we are ut ilizing an exist ing material which is usually highly variable.

int act ‘layered’ intact highly f ractured

Rock as an Engineering MaterialRock as an Engineering Material

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Rock as an engineering material will be used either:

… as a building material so the st ructure will be madeof rock

… or a st ructure will be built on t he rock … or a st ructure will be built in t he rock

I n the context of t he mechanics, we must establish:

… the propert ies of t he material … the pre - exist ing st ress state in the ground (which will be

disturbed by the st ructure) … and how these f actors relate to the engineering obje ct ive

I nf luence of Geological FactorsI nf luence of Geological Factors

We have the intact rock which is it self divided by discont inuit iesto f orm the rock st ructure.

We f ind then the rock is already subjected to an in sit u st ress .

Superimposed on this f undamental mechanics circumstance are the inf luence of pore f luid/ water f low and t ime .

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Five primary geological f actors can be viewed as inf luen cing a r ock mass. I n the context of t he mechanics problem, we sho uld conside r the material and the f orces applied t o it .

I n all of t hese subjects, t he geological history has p layed it s part , alt ering the rock and the applied f orces.

I nf luence of Geological Factors I nf luence of Geological Factors –– I ntact I ntact RockRock

The most usef ul descript ion of t he mechanical behaviour of intact rock is the complete st ress - st rain curve in uniaxial compression.

From this curve, several f eatures of interest are derived:

· the def ormat ion modulus · t he peak compressive st rength· the post - peak behaviour

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I nf luence of Geological Factors I nf luence of Geological Factors –– I ntact I ntact RockRock

high st if f ness

high st rength

very brit t le

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medium st if f ness

medium st rength

medium brit t leness

low st if f ness

low st rength

brit t le

low st if f ness

low st rength

duct ile

I nf luence of Geological Factors I nf luence of Geological Factors ––Discont inuit ies and Rock St ructureDiscont inuit ies and Rock St ructure

The result in t erms of rock f racturing is t o produce a geomet ric al st ructure (of ten very complex) of f ractures f orming rock blocks. The overall geomet rical conf igurat ion of t he discont inuit i es in t he rock mass is t ermed rock st ructure . I t is of ten helpf ul t o understand the way in which discont inuit ies f orm. There are three ways in w hich a f rac ture can be f ormed:

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Mode 1(t ensile)

Mode 2(in- plane shear)

Mode 3(out - of - plane shear)

I nf luence of Geological Factors I nf luence of Geological Factors ––Discont inuit ies and Rock St ructureDiscont inuit ies and Rock St ructure

I n pract ice, f ailure is most of ten associated with disc ont inuit i es which act as pre - exist ing planes of weakness. Some examples of the way i n which the discont inuit y genesis leads to dif f ering m echanical pr opert ies are:

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… open joint which will allow f ree f low of

wat er.

… st ylolit ic discont inuit y wit h high shear

resist ance.

… slickensided f ault surf ace wit h low shear

resist ance.

I nf luence of Geological Factors I nf luence of Geological Factors ––PrePre-- Exist ing I n Exist ing I n Sit u Sit u Rock St ressRock St ress

I n some cases, such as a dam or nuclear power stat ion f oundat ion, t he load is applied to t his.

I n other cases, such as the excavat ion of a mine or tunnel, no new loads are applied but t he pre -exist ing st resses are redist ributed.

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When considering the loading condit ions imposed on t he rock st ru cture, it must be recognized that an in sit u pre- exist ing state of st ress already exist s in t he rock.

I nf luence of I nf luence of St ructure & St ructure & I n I n Sit u Sit u Rock Rock

St ress TogetherSt ress Together

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… t ypes of f ailure which occur in dif f erent rock masses under low and

high in sit u st ress levels.

I nf luence of Geological Factors I nf luence of Geological Factors ––Pore Fluids and Water FlowPore Fluids and Water Flow

Many rocks in t heir int act st at e have a very low permeabilit y compared t o t he durat ion of t he engineeri ng const ruct ion, but t he main wat er f low is usually via secondary permeabilit y, (i. e. pre - exist ing f ractures). Thus t he st udy of f low in rock masses will generally be a f unct ion of t he discont inuit ies, t heir connect ivit y and t he hydrogeological environment .

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A primary concern is when t he wat er is under pressure, which in t urn act s t o reduce t he ef f ect ive st ress and/ or induce inst abilit ies. Ot her aspect s, such as groundwat er chemist ry and t he alt erat ion of rock and f racture surf aces by f luid movement may also be of concern.

I nf luence of Geological Factors I nf luence of Geological Factors –– TimeTime

Rock as an engineering material may be millions of years old, however our engineering const ruct ion and subsequent act ivit ies are generally only designed f or a century or less.

Thus we have two t ypes of behaviour: the geological processes in which equilibrium will have been established, with current geological act ivit y superimposed; and the rapid engineering process.

The inf luence of t ime is also important given such f actors as the decrease in rock st rength through t ime, and the ef f ects of creep and relaxat ion

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Scalars, Vectors and TensorsScalars, Vectors and Tensors

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There is a f undamental dif f erence, both conceptually and mathema t ically, between a tensor and the more f amiliar quant it ies of scal ars andvectors:

Scalar: a quant it y with magnitude only (e. g. t emperature, t ime, mass).

Vector: a quant it y with magnitude and direct ion (e. g. f orce, velocit y, accelerat ion).

Tensor: a quant it y with magnitude and direct ion, and with ref ere nce to a plane it is act ing across (e. g. st ress, st rain, per meabilit y).

Both mathemat ical and engineering mistakes are easily made if t h is crucial dif f erence is not recognized and understood.

Normal and Shear St ress Components Normal and Shear St ress Components

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On a real or imaginary plane through a material, t here can b e normal f orces and shear f orces . These f orces create the st ress tensor. The normal and shear st ress components are the normal and she ar f orc es per unit area.

I t should be remembered that a solid can sustain a shear f orce, whereas a liquid or gas cannot . A liquid or gas contains a pressu re, whi ch acts equally in all direct ions and hence is a scalar quant it y.

Force and St ress Force and St ress

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We are now in a posit ion to obtain an init ial idea of t he crucia l dif f erence between f orces and st resses.

When the normal f orce component , Fn, is f ound in a direct ion θ f rom F, t he value is F cos θ (i. e. Fn = F cos θ ).

However, when the normal st ress component , σn, is f ound in the same direct ion, t he value is σ cos2 θ (i. e. σn = σ cos2 θ ).

Force and St ress Force and St ress

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The reason f or t his is t hat it is only t he f orce that is resolve d in t he f irst case (i. e. vector), whereas, it is both the f orce and the area that are resolved in t he case of st ress (i. e. t ensor).

I n f act , t he st rict def init ion of a second - order tensor is a quant it y that obeys certain t ransf ormat ion laws as the planes in quest i on are rotated. This is why the conceptualizat ion of t he st ress tenso r ut ilizes t he idea of magnitude, direct ion and “ the plane in quest ion ”.

St ress as a Point Propert y St ress as a Point Propert y

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We can now consider the st ress components on a surf ace at an arbit rary orientat ion through a body loaded by external f orces (e. g. F1, F2, …, Fn).

Consider now the f orces that are required to act in order to maintain equilibrium on a small area of a surf ace created by cut t ing through the rock. On any small area ∆A, equilibrium can be maintained by the normal f orce ∆N and the shear f orce ∆S.

St ress as a Point Propert y St ress as a Point Propert y

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Because these f orces will vary according to the orientat ion of ∆A within the slice, it is most usef ul t o consider t he normal st ress (∆N/ ∆A)and t he shear st ress (∆S/ ∆A) as the area ∆Abecomes very small, eventually approaching zero.

Although there are pract ical limitat ions in reducing th e size of t he area to zero, it is important t o realize that t he st ress compone nts are def ined in this way as mathemat ical quant it ies, with t he result t hat st ress is a point propert y .

I nt act RockI ntact Rock

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Uniaxial Uniaxial Compression TestCompression Test

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… typical record f rom a uniaxial compression test . Note that t he f orce and displacement have been scaled respect ively t o st res s (by div iding by t he original cross - sect ional area of t he specimen) and to st rain (by dividing by the original length).

Stages of St ressStages of St ress -- St rain BehaviourSt rain Behaviour

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As the rock is gradually loaded, it passes through seve ral stage s:

Stage I - Exist ing cracks pref erent ially aligned t o t he applied st ress will close ( σcc).

Stage I I - Near linear elast ic st ress - st rain behaviour occurs.

Stage I I I - I nit iat ing cracks propagat e in a st able f ashion ( σci).

Stage I V - Cracks begin t o coalesce and propagat e in an unstable f ashion ( σcd)

σaxial

T o t al

Measured

∆ V/V

Calculated

Crack Volumetric

StrainCrack

Closure

Crack

Growth

εax ial ε later a l

Dil

ati

on

Co

ntr

acti

on

∆V

/Vσ cd

σ ucs

σ cc

ax ial

σ ci

crack closure t hreshold

crack initiation threshold

crack damage t hreshold

peak

strength

Elast ic Constant sElast ic Constant s

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Focussing on the interval of near linear behaviour , we can draw analogies to the ideal elast ic rock represented by our elast ic c ompliance mat rix. Remembering that t he Young’s modulus, E, is def ined as the rat io of st ress to st rain (i. e. 1/ S 11), it can be determined in two ways:

Tangent Young’s modulus, E T – t aken as t he slope of t he axial σ- ε curve at some f ixed percent age, generally 50%, of t he peak st rengt h.

Secant Young’s modulus, E S – t aken as t he slope of t he line joining t he or igin of t he axial σ- ε curve t o a point on t he curve at some f ixed percent age of t he peak st rengt h.

Elast ic Constant sElast ic Constant s

… dif f erent iat ion between elast ic and plast ic st rains, w ith def init ion of t he Young’s modulus, E, and Poisson’s rat io, ν.

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Elast ic Constant sElast ic Constant s

… typical values of Young’s modulus and Poisson’s rat io f or various rock t ypes

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Compressive St rengthCompressive St rength

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Another important parameter in the uniaxial compression test is t he maximum st ress that t he test sample can sustain. Under u niaxial loading condit ions, t he peak st ress is ref erred to as the uniaxial compressive st rength, σc.

I t is important to realize t hat t he compressive st rength is not an int r insic propert y . I nt rinsic material propert ies do not depend on the specimen geomet ry or the loading condit ions used in the test : t he uniaxial compressive st rength does.

Compressive St rengthCompressive St rength

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The compressive st rength is probably the most widely used and quoted rock engineering parameter and theref ore it is crucial t ounderstand it s nature. I n other f orms of engineering, if t he applied st ress reaches σc, t here can be catast rophic consequences. This is not always the case in rock engineering as rock of ten retains some load bearing capacit y in the post peak region of the σ- ε curve.

Whether f ailure beyond σc is to be avoided at all costs, or to be encouraged, is a f unct ion of t he engineering object ive , t he f orm of t he complete st ress - st rain curve f or the rock (or rock mass), and the characterist ics of the loading condit ions. These f eatures ar e crucial in t he design and analysis of underground excavat ions.

Ef f ect s of Specimen SizeEf f ect s of Specimen Size

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Having described how the complete σ- ε curve can be obtained experimentally, we can now consider other f actors that af f ect th e complete σ- ε curves of laboratory tested rock.

I f t he rat io of sample length to diameter is kept constant , both compressive st rength and brit t leness are reduced f or larger samples . Rock specimens contain microcracks : the larger the specimen, the greater the number of microcracks and hence the greater the likelihood of a crit ical f law and ef f ects associated with crack init iat ion and propagat ion.

Ef f ect s of Loading Condit ionsEf f ect s of Loading Condit ions

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I ntact rock st rength is dependent on the t ypes of st resses appli ed t o it . I n other words, rock has st rength in t ension , compression and shear .

… these dif f erent st rengths may be tested either direct ly (e. g. uniaxial t ension test , direct shear t est , et c. ) or indirect ly (e. g. Brazilian t ensile t est , t r iaxial compression test , et c. ).

Ef f ect s of Loading Condit ionsEf f ect s of Loading Condit ions

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With the applicat ion of a conf ining load an addit ional ene rgy in put is needed to overcome f r ict ional resistance to sliding over a jagged rupture path. Most rocks are theref ore st rengthened by the add it ion of aconf ining st ress.

As the conf ining pressure is increased, the rapid decline in load carrying capacit y af ter the peak load is reached becomes less st riking unt il, af ter a mean pressure known as the brit t le - t o- duct ile t ransit ion pressure, t he rock behaves in a near plast ic manner.

Pore Pressure Ef f ect sPore Pressure Ef f ect s

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Some rocks are weakened by t he addit ion of wat er, t he e f f ect bei ng a chemical det eriorat ion of t he cement or binding mat erial. I n mo st cases, however, it is t he ef f ect of pore wat er pressure t hat exert s t he gre at est inf lu ence on rock st rengt h. I f drainage is impeded during loading, t he p ores or f i ssures will compress t he cont ained wat er, raising it s pressure. Th e result in g ef f ect is described by Terzaghi’s ef f ect ive st ress law :

… as pore pressure “P” increases t he ef f ect ive normal st resses a re reduced and t he Mohr circles are displaced t owards f ailure.

TimeTime -- Dependent Ef f ect sDependent Ef f ect s

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We have indicated that during the complete σ- ε curve, microcracking occurs f rom the very early stages of loading. Through these proc esses, f our primary t ime - dependent ef f ects can be resolved:

Relaxat ion – a decrease in st rain occurs when t he applied st ress is held const ant .

Fat igue – an increase in st rain occurs due t o cyclic changes in st ress.

St rain - rate - t he σ- ε curve is a f unct ion of t he applied st rain rat e.

Creep – st rain cont inues when t he applied st ress is held const ant .

Temperat ure Ef f ect sTemperat ure Ef f ect s

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Only a limited amount inf ormat ion is available indicat ing the ef f ect of t emperature on the complete σ- ε curve and other mechanical propert ies of intact rock.

The limited test data does show though, t hat increasing temperatures reduces the elast ic modulus and compressive st rength, whilst increasing the duct ilit y in t he post - peak region.

Failure Crit er ionFailure Crit er ion

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Rock f ails t hrough an ext remely complex process of microcrack init iat ion and propagat ion that is not subject t o convenient characterizat ion through simplif ied models. Building on the history of material t est ing, it was natural to express the st rength of a material in term s of th e st ress present in t he test specimen at f ailure (i. e. phenomenological approach ).

Since uniaxial and t r iaxial test ing of rock are by f ar the most common laboratory procedures, t he most obvious means of expre ssing a f a ilure crit erion is:

St rength = ƒ (σ1, σ2, σ3)

Or with the advent of st if f and servo - cont rolled test ing machines:

St rength = ƒ (ε1, ε2, ε3)

MohrMohr -- Coulomb Crit er ionCoulomb Crit er ion

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The Mohr - Coulomb f ailure crit erion expresses the relat ionship between the shear st ress and the normal st ress at f ailur e along a hypothet ical f ailure plane. I n two - dimensions, t his is expressed as:

τpeak = c + σn t an φ

Where:φ is called the angle of internal f r ict ion (equivalent t o

the angle of inclinat ion of a surf ace suf f icient t ocause sliding of a block of similar material);

c is t he cohesion (and represents the shear st rength of t he rock when no normal st ress is applied); and

τpeak is the peak shear st rength .

MohrMohr -- Coulomb Crit er ionCoulomb Crit er ion

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This can be presented graphically using a Mohr circle diagram :

MohrMohr -- Coulomb Crit er ionCoulomb Crit er ion

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The Mohr - Coulomb crit er ion is most suit able at high conf ining pressures w hen rock generally f ails t hrough t he development of shear planes. Ho wever, some limit at ions are :

- it implies t hat a major shear f ract ure exist s at peak st re ngt h, at a specif ic angle, which does not always agree wit h experi mental ob servat ions;

- it predict s a shear f ailure in uniaxial t ension (at 45 - φ/ 2 wit h σ3) whereas f or rock t his f ailure plane is perpendicular t o σ3. A t ension cutof f has been int roduced t o t he Mohr - Coulomb cr it er ion t o predict t he proper or ient at ion of t he f ailure plane in t ension.

- experiment al peak st rengt h envelopes are generally non - linear. They can be considered linear only over limit ed ranges of conf ini ng pressure s.

Despit e t hese dif f icult ies, t he Mohr - Coulomb f ailure cr it er ion remains one of t he most commonly applied f ailure cr it er ion, and is es pecially s ignif icant and valid f or discont inuit ies and discont inuous rock masses.

TheThe HoekHoek-- Brown Empir ical Failure Crit er ionBrown Empir ical Failure Crit er ion

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The Hoek- Brown empirical crit erion was developed f rom a best - f it curve t o experimental f ailure data plot t ed in σ1- σ3 space. Since this is one of t he f ew techniques available f or est imat ing in situ rock mass st rength f rom geological data, t he crit erion has become widely u sed in ro ck mechanics analysis.

σ1 = σ3 + (m σc σ3+ sσc2)0. 5

where σc is t he int act compressive st rengt h, sis a rock mass const ant (s=1 f or int act rock, s<1 f or broken rock), and m is a const ant (charact erist ic of t he rock t ype where values range f rom 25, f or coarse grained igneous and met amorphic rocks t o 7 f or carbonat e rocks).

Discont inuit iesDiscont inuit ies

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I t is t he exist ence of discont inuit ies in a rock mass t hat makes rock mechanics a unique subject . The word ‘ discont inuit y ’ denot es any separat ion in t he rock cont inuum having ef f ect ively zero t ensile st rengt h and is used wit hout any generic connot at ion (e. g. j oint s and f ault s are t ypes of discont inuit ies f ormed in dif f erent ways).

Discont inuit ies have been int roduced int o t he rock by geological event s, at dif f erent t imes and as a result of dif f erent st ress st at es. Very of t en, t he process by which a discont inuit y has been f ormed may have implicat ions f or it s geomet rical and mechanical propert ies.

Discont inuit iesDiscont inuit ies

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I n f act , all rock masses are f ractured, and it is a very rare case where the spacings between discont inuit ies are appreciably greater t han the dimensions of t he rock engineering project . Very of ten major discont inuit ies delineate blocks within the rock mass , and withi n t hese blocks there is a f urther suite of discont inuit ies.

Thus, we might expect that a relat ion of t he f orm:

should exist .

Geomet rical Propert ies of Discont inuit yGeomet rical Propert ies of Discont inuit y

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The main f eatures of rock mass geomet ry include spacing and f requency, orientat ion (dip direct ion/ dip angle), persistence (size and shape), roughness, aperture, clustering and block size.

There is, however, no standardized method of measuring and characterizing rock st ructure geomet ry, because the em phasis andaccuracy with which the separate parameters are specif ied will d epend on the engineering object ives.

Discont inuit y Spacing and FrequencyDiscont inuit y Spacing and Frequency

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Spacing is t he distance between adjacent discont inuit y interse ct ions with the measuring scanline . Frequency (i. e. t he number per unit distance) is the reciprocal of spacing (i. e. t he mean o f these intersect ion distances).

… quant if ying discont inuit y occurrence along a sampling line, where f requency λ=N/ L m- 1 and mean spacing x=L/ N m.

Rock Qualit y Designat ionRock Qualit y Designat ion

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A natural clustering of discont inuit ies occurs through the genet ic process of superimposed f racture phases, each of which could have a dif f erent spacing dist r ibut ion. An important f eature f or engineering is t he overall qualit y of t he rock mass cut by these superimposed f racture systems. For this reason, the concept of t he RQD was developed.

Discont inuit y Orientat ionDiscont inuit y Orientat ion

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I f we assume t hat a discont inuit y is a planar f eat ure, t hen it s or ient at ion can be uniquely def ined by t wo paramet ers: dip direct ion and dip angle . I t is of t en usef ul t o present t his dat a in a graphical f orm t o aid visualizat ion and engineering analysis.

I t must be remembered t hough, t hat it may be dif f icult t o dist inguish which set a part icular discont inuit y belongs t o or t hat in some cases a single discont inuit y may be t he cont rolling f act or as opposed t o a set of discont inuit ies.

Discont inuit y PersistenceDiscont inuit y Persistence

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Persistence ref ers t o t he lateral extent of a discont inuit y plane, either the overall dimensions of t he plane, or in terms of w hether it c ontains ‘rock bridges’. I n pract ice, t he persistence is almos t always me asured by the one dimensional extent of the t race lengths as exposed on rock f aces. This obviously int roduces a degree of sampling bias that must be accounted f or in the interpretat ion of result s.

Discont inuit y RoughnessDiscont inuit y Roughness

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The word ‘roughness’ is used t o denote deviat ion of a discont inuit y surf ace f rom perf ect planarit y , which can rapidly become a complex mat hemat ical procedure ut ilizing 3 - D surf ace characterizat ion t echniques (e. g. polynomials, Fourier series, f ract als).

From t he pract ical point of view, only one t echnique has received some degree of universalit y – the Joint Roughness Coef f icient (JRC). This met hod involves comparing discont inuit y surf ace prof iles t o st andard roughness curves assigned numerical values.

The geomet rical roughness is naturally relat ed t o various mechanical and hydraulic propert ies of discont inuit ies.

Discont inuit y ApertureDiscont inuit y Aperture

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The aperture is the distance between adjacent walls of a discont inuit y. This parameter has mechanical and hydraulic importance, and a dist ribut ion of apertures f or any give n discont inuit y and f or dif f erent discont inuit ies with in the same rock mass is expected.

Mechanical Propert ies of Discont inuit iesMechanical Propert ies of Discont inuit ies

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The mechanical behaviour of discont inuit ies is generally plot t ed in t he f orm of st ress -displacement curves, wit h t he result t hat we can measure discont inuit y st if f ness(t ypically expressed in unit s of MPa/ m) and st rengt h .

I n compression , t he rock surf aces are gradually pushed t oget her, wit h an obvious limit when t he t wo surf aces are closed. I n t ension , by def init ion, discont inuit ies can sust ain no load. I n shear , t he st ress -displacement curve looks like t hat f or compression of int act rock, except of course f ailure is localized along t he discont inuit y.

Mechanical Propert ies Mechanical Propert ies -- St rengthSt rength

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I t is normally assumed that t he shear st rength of disco nt inuit ie s is a f unct ion of t he f r ict ion angle rather than the cohesion. This is done by using t he Mohr - Coulomb f ailure crit erion, τ = c + σt anφ, and set t ing the cohesion to zero.

Mechanical Propert ies Mechanical Propert ies -- St rengthSt rength

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The bi - linear f ailure cr it er ion int roduces t he idea t hat t he irregularit y of discont inuit y surf aces could be approximated by an asperit y angle i ont o which t he basic f r ict ion angle is superimposed .

Thus, at low normal st resses, shear loading causes t he discont inuit y surf aces t o dilate giving an ef f ect ive f r ict ion of (φ+i). As t he shear loading cont inues, t he shear surf aces become damaged as asperit ies are sheared and t he t wo surf aces r ide on t op of one anot her, giving a t ransit ion zone bef ore t he f ailure locus st abilizes at an angle of φ.

Rock MassesRock Masses

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Building on our examinat ion of f irst intact rock behavi our and t hen discont inuit y behaviour, we can now concent rate on ext ending the se ideas to provide a predict ive model f or t he def ormabilit y and st rength of rock masses.

Rock Mass Def ormabilit yRock Mass Def ormabilit y

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As an init ial step in determining the overall def ormabilit y of a rock mass, we can f irst consider the def ormat ion of a set of par allel discont inuit ies under the act ion of a normal st ress, as suming li near elast ic discont inuit y st if f nesses .

To calculate the overallmodulus of def ormat ion , the applied st ress is divided by the total def ormat ion. We will assume that def ormat ion is made up of two components: one related to t he intact rock; the other to the discont inuit ies.

Rock Mass Def ormabilit yRock Mass Def ormabilit y

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The cont r ibut ion made by t he int act rock t o t he def or mat ion, δI , is σL/ E (i. e. st rain mult iplied by lengt h). The cont r ibut ion made by a s ingle discont inuit y t o t he def ormat ion, δD, is σ/ ED (remembering t hat ED relates t o displacement direct ly). Assuming a discont inuit y f req uency of λ, t here will be λL discont inuit ies in t he rock mass and t he t ot al cont r ib ut ion mad e by t hese t o t he def ormat ion will be δD

t , which is equal t o σ λL / ED. Hence, t he t ot al displacement , δT, is:

Hence, t he t ot al displacement , δT, is:

Wit h t he overall st rain being given by:

Finally, t he overall modulus, EMASS, is given by:

Rock Mass Def ormabilit yRock Mass Def ormabilit y

x

… variat ion of in sit u rock def ormabilit y as a f unct ion of t he discont inuit ies (f or t he idealized case of a single set of discont inuit ies).

Rock Mass St rengthRock Mass St rength

x

I n the same way as we considered the def ormabilit y of a rock mas s, expressions can be developed indicat ing how st rength is af f ected by the presence of discont inuit ies, start ing with a single d iscont inuit y and then extending to any number of discont inuit ies.

… scale dependent st rength of a single discont inuit y.

Rock Mass St rengthRock Mass St rength

x

The init ial approach is via the ‘single plane of weakness’ theory, whereby the st rength of a sample of intact rock contain ing a sin gle discont inuit y can be established. Basically, t he st ress applied t o t he sample is resolved into the normal and shear st resses on the plane of weakness and the Mohr - Coulomb f ailure crit erion applied to consider t he possibilit y of slip.

Given the geomet ry of t he applied loading condit ion:

Rock Mass St rengthRock Mass St rength

x

The st rength of t he sample thus depends on the orientat ion of the discont inuit y. I f t he discont inuit y is, f or example, parallel orperpendicular to the applied loading, it will have no e f f ect on the sample st rength. At some angles, however, t he discont inuit y willsignif icant ly reduce the st rength of t he sample.

The lowest st rengthoccurs when the discont inuit y normal is inclined at an angle of 45° + ( φ°/ 2) t o t he major applied principal st ress.

Rock Mass St rengthRock Mass St rength

x

The plot of rock st rength and the discont inuit y angle s at which the sample st rength becomes less than that f or intact rock can be de rived by subst it ut ing the ‘single discont inuit y’ normal and shear st ress relat ionships into t he Mohr - Coulomb crit erion :

Subst it uted into

| τ| = cw + σnt anφw gives:

Where cw and φw are t he cohesion and f r ict ion f or t he discont inuit y.

Rock Mass St rengthRock Mass St rength

x

An alternat ive presentat ion of t he ‘single plane of weakness’ rock st rength theory is via t he Mohr’s circle representat ion. The Mohr - Coulomb f ailure loci f or both intact rock and the discont inuit y are given.

Circle A – case where t he f ailure locus f or t he discont inuit y is j ust reach ed, i. e. f or a discont inuit y at t he angle 2βw=90°+ φw.

Circle B – case when f ailure can occur along t he discont inuit y f or a range of angles.

Circle C – case where t he Mohr circle t ouches t he int act rock f ai lure locus , i. e. where f ailure occurs in t he int act rock.

Rock Mass St rengthRock Mass St rength

x

We can consider, on the basis of t his single plane of weakness t heory, what would happen if t here were two or more discont inuit ies at dif f erent orientat ions present in the rock sample. Eac h discont i nuit y would weaken the sample as shown below, but t he angular posit ion of t he st rength minima would not coincide.

As a result t he rock is weakened in several dif f erent direct ions simultaneously . With increasing f ractures, t he material t ends to become isot ropic in st rength, like a granular soil.

Rock Mass St rength Rock Mass St rength –– HoekHoek-- BrownBrown

x

A methodology of assessing rock mass st rength that does not depend on the ‘single plane of weakness’ t heory is t he Hoek- Brown f ailure crit er ion . The crit erion is especially powerf ul in it s applicat ion to rock masses due to t he constants m and sbeing able to take on values which permit predict ion of the st rengths of a wide range of rock masses.

Rock Mass Rock Mass St rengthSt rength

x

… Hoek - Brown representat ion and summary of rock mass condit ions, test ing methods and theoret ical considerat ions.

Rock Mass St rength Rock Mass St rength –– HoekHoek-- BrownBrown

x

For intact rock, t he Hoek- Brown crit erion may be expressed as:

The more general f orm, however, was derived to take into account t he f ractured nature of the rock mass through the parameters s and a.

Rock Mass Rock Mass St rength St rength ––

HoekHoek-- BrownBrown

x

… Hoek - Brown ‘m’ values f or dif f erent rock t ypes.

Rock Mass Rock Mass St rength St rength –– HoekHoek--

BrownBrown

x

… est imat ion of Hoek -Brown constants and rock mass def ormat ion constants based on rock mass st ructure and discont inuit y surf ace condit ions.

Rock Mass St rength Rock Mass St rength –– HoekHoek-- BrownBrown

x

… the Hoek - Brown empirical crit erion applied to a sandstone rock mass. The crit erion represents best - f it curves t o experimental f ailure data plot t ed in σ1- σ3 space.