Rock Mechanics and Rock Engineering · PDF fileRock Mechanics and Rock Engineering Rock mechanics involves characterizing the intact strength and the geometry and mechanical properties

Rock Mechanics and Rock Engineering

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OverviewOverview

Rock mechanics is the theoretical and applied science of the mechanical behaviour of rock and rock masses. Rock mechanics deals with the mechanical properties of rock and the related methodologies required for engineering design.

The subject of rock mechanics has evolved from different disciplines of applied mechanics. It is a truly interdisciplinary subject, with applications in geology and geophysics, mining, petroleum and geotechnical engineering.

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Rock Mechanics and Rock Mechanics and Rock EngineeringRock Engineering

Rock mechanics involves characterizing the intact strength and the geometry and mechanical properties of the natural fractures of the rock mass.

Rock engineering is concerned with specific engineering circumstances, for example, how much load will the rock support and whether reinforcement is necessary.

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Nature of RockNature of Rock

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A common assumption when dealing with the mechanical behaviour of solids is that they are:

· homogeneous · continuous · isotropic

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

· heterogeneous · discontinuous · anisotropic

Nature of RockNature of Rock

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sandstone

sandstone

shale fault

joints

strengthequal in

all directions

highstrengthvaries withdirection low

Homogeneous Continuous Isotropic

Heterogeneous Discontinuous Anisotropic

Rock as an Engineering MaterialRock as an Engineering Material

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One of the most important, and frequently neglected, aspects of rock mechanics and rock engineering is that we are utilizing an existing material which is usually highly variable.

intact ‘layered’ intact highly fractured

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 structure will be madeof rock

… or a structure will be built on the rock … or a structure will be built in the rock

In the context of the mechanics, we must establish:

… the properties of the material … the pre-existing stress state in the ground (which will be

disturbed by the structure) … and how these factors relate to the engineering objective

Influence of Geological FactorsInfluence of Geological Factors

We have the intact rock which is itself divided by discontinuitiesto form the rock structure.

We find then the rock is already subjected to an in situ stress.

Superimposed on this fundamental mechanics circumstance are the influence of pore fluid/water flow and time.

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Five primary geological factors can be viewed as influencing a rock mass. In the context of the mechanics problem, we should consider the material and the forces applied to it.

In all of these subjects, the geological history has played its part, altering the rock and the applied forces.

Influence of Geological Factors Influence of Geological Factors –– Intact Intact RockRock

The most useful description of the mechanical behaviour of intact rock is the complete stress-strain curve in uniaxial compression.

From this curve, several features of interest are derived:

· the deformation modulus · the peak compressive strength· the post-peak behaviour

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Influence of Geological Factors Influence of Geological Factors –– Intact Intact RockRock

high stiffness

high strength

very brittle

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medium stiffness

medium strength

medium brittleness

low stiffness

low strength

brittle

low stiffness

low strength

ductile

Influence of Geological Factors Influence of Geological Factors ––Discontinuities and Rock StructureDiscontinuities and Rock Structure

The result in terms of rock fracturing is to produce a geometrical structure (often very complex) of fractures forming rock blocks. The overall geometrical configuration of the discontinuities in the rock mass is termed rock structure. It is often helpful to understand the way in which discontinuities form. There are three ways in which a fracture can be formed:

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Mode 1(tensile)

Mode 2(in-plane shear)

Mode 3(out-of-plane shear)

Influence of Geological Factors Influence of Geological Factors ––Discontinuities and Rock StructureDiscontinuities and Rock Structure

In practice, failure is most often associated with discontinuities which act as pre-existing planes of weakness. Some examples of the way in which the discontinuity genesis leads to differing mechanical properties are:

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… open joint which will allow free flow of

water.

… stylolitic discontinuity with high shear

resistance.

… slickensided fault surface with low shear

resistance.

Influence of Geological Factors Influence of Geological Factors ––PrePre--Existing In Existing In Situ Situ Rock StressRock Stress

In some cases, such as a dam or nuclear power station foundation, the load is applied to this.

In other cases, such as the excavation of a mine or tunnel, no new loads are applied but the pre-existing stresses are redistributed.

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When considering the loading conditions imposed on the rock structure, it must be recognized that an in situ pre-existing state of stress already exists in the rock.

Influence of Influence of Structure & Structure & In In Situ Situ Rock Rock

Stress TogetherStress Together

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… types of failure which occur in different rock masses under low and

high in situ stress levels.

Influence of Geological Factors Influence of Geological Factors ––Pore Fluids and Water FlowPore Fluids and Water Flow

Many rocks in their intact state have a very low permeability compared to the duration of the engineering construction, but the main water flow is usually via secondary permeability, (i.e. pre-existing fractures). Thus the study of flow in rock masses will generally be a function of the discontinuities, their connectivity and the hydrogeological environment.

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A primary concern is when the water is under pressure, which in turn acts to reduce the effective stress and/or induce instabilities. Other aspects, such as groundwater chemistry and the alteration of rock and fracture surfaces by fluid movement may also be of concern.

Influence of Geological Factors Influence of Geological Factors –– TimeTime

Rock as an engineering material may be millions of years old, however our engineering construction and subsequent activities are generally only designed for a century or less.

Thus we have two types of behaviour: the geological processes in which equilibrium will have been established, with current geological activity superimposed; and the rapid engineering process.

The influence of time is also important given such factors as the decrease in rock strength through time, and the effects of creep and relaxation

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Scalars, Vectors and TensorsScalars, Vectors and Tensors

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There is a fundamental difference, both conceptually and mathematically, between a tensor and the more familiar quantities of scalars andvectors:

Scalar: a quantity with magnitude only (e.g. temperature, time, mass).

Vector: a quantity with magnitude and direction (e.g. force, velocity, acceleration).

Tensor: a quantity with magnitude and direction, and with reference to a plane it is acting across (e.g. stress, strain, permeability).

Both mathematical and engineering mistakes are easily made if this crucial difference is not recognized and understood.

Normal and Shear Stress Components Normal and Shear Stress Components

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On a real or imaginary plane through a material, there can be normal forces and shear forces. These forces create the stress tensor. The normal and shear stress components are the normal and shear forces per unit area.

It should be remembered that a solid can sustain a shear force, whereas a liquid or gas cannot. A liquid or gas contains a pressure, which acts equally in all directions and hence is a scalar quantity.

Force and Stress Force and Stress

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We are now in a position to obtain an initial idea of the crucial difference between forces and stresses.

When the normal force component, Fn, is found in a direction θ from F, the value is F cos θ (i.e. Fn = F cos θ ).

However, when the normal stress component, σn, is found in the same direction, the value is σ cos2 θ (i.e. σn = σ cos2 θ ).

Force and Stress Force and Stress

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The reason for this is that it is only the force that is resolved in the first case (i.e. vector), whereas, it is both the force and the area that are resolved in the case of stress (i.e. tensor).

In fact, the strict definition of a second-order tensor is a quantity that obeys certain transformation laws as the planes in question are rotated. This is why the conceptualization of the stress tensor utilizes the idea of magnitude, direction and “the plane in question”.

Stress as a Point Property Stress as a Point Property

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We can now consider the stress components on a surface at an arbitrary orientation through a body loaded by external forces (e.g. F1, F2, …, Fn).

Consider now the forces that are required to act in order to maintain equilibrium on a small area of a surface created by cutting through the rock. On any small area ∆A, equilibrium can be maintained by the normal force ∆N and the shear force ∆S.

Stress as a Point Property Stress as a Point Property

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Because these forces will vary according to the orientation of ∆A within the slice, it is most useful to consider the normal stress (∆N/∆A)and the shear stress (∆S/∆A) as the area ∆Abecomes very small, eventually approaching zero.

Although there are practical limitations in reducing the size of the area to zero, it is important to realize that the stress components are defined in this way as mathematical quantities, with the result that stress is a point property.

Intact RockIntact Rock

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Uniaxial Uniaxial Compression TestCompression Test

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… typical record from a uniaxial compression test. Note that the force and displacement have been scaled respectively to stress (by dividing by the original cross-sectional area of the specimen) and to strain (by dividing by the original length).

Stages of StressStages of Stress--Strain BehaviourStrain Behaviour

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As the rock is gradually loaded, it passes through several stages:

Stage I - Existing cracks preferentially aligned to the applied stress will close (σcc).

Stage II - Near linear elastic stress-strain behaviour occurs.

Stage III - Initiating cracks propagate in a stable fashion (σci).

Stage IV - Cracks begin to coalesce and propagate in an unstable fashion (σcd)

σaxial

T o t alMeasured

∆ V/V

CalculatedCrack Volumetric

StrainCrack

ClosureCrack

Growth

εax ial ε later a l

Dil

atio

nC

ontr

acti

on

∆V

/Vσ cd

σ ucs

σ cc

ax ial

σ ci

crack closure t hreshold

crack initiation threshold

crack damage t hreshold

peakstrength

Elastic ConstantsElastic Constants

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Focussing on the interval of near linear behaviour, we can draw analogies to the ideal elastic rock represented by our elastic compliance matrix. Remembering that the Young’s modulus, E, is defined as the ratio of stress to strain (i.e. 1/S11), it can be determined in two ways:

Tangent Young’s modulus, ET – taken as the slope of the axial σ-ε curve at some fixed percentage, generally 50%, of the peak strength.

Secant Young’s modulus, ES – taken as the slope of the line joining the origin of the axial σ-ε curve to a point on the curve at some fixed percentage of the peak strength.


… differentiation between elastic and plastic strains, with definition of the Young’s modulus, E, and Poisson’s ratio, ν.

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… typical values of Young’s modulus and Poisson’s ratio for various rock types

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Compressive StrengthCompressive Strength

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Another important parameter in the uniaxial compression test is the maximum stress that the test sample can sustain. Under uniaxial loading conditions, the peak stress is referred to as the uniaxial compressive strength, σc.

It is important to realize that the compressive strength is not an intrinsic property. Intrinsic material properties do not depend on the specimen geometry or the loading conditions used in the test: the uniaxial compressive strength does.

Compressive StrengthCompressive Strength

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The compressive strength is probably the most widely used and quoted rock engineering parameter and therefore it is crucial tounderstand its nature. In other forms of engineering, if the applied stress reaches σc, there can be catastrophic consequences. This is not always the case in rock engineering as rock often retains some load bearing capacity in the post peak region of the σ-ε curve.

Whether failure beyond σc is to be avoided at all costs, or to be encouraged, is a function of the engineering objective, the form of the complete stress-strain curve for the rock (or rock mass), and the characteristics of the loading conditions. These features are crucial in the design and analysis of underground excavations.

Effects of Specimen SizeEffects of Specimen Size

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Having described how the complete σ-ε curve can be obtained experimentally, we can now consider other factors that affect the complete σ-ε curves of laboratory tested rock.

If the ratio of sample length to diameter is kept constant, both compressive strength and brittleness are reduced for larger samples. Rock specimens contain microcracks: the larger the specimen, the greater the number of microcracks and hence the greater the likelihood of a critical flaw and effects associated with crack initiation and propagation.

Effects of Loading ConditionsEffects of Loading Conditions

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Intact rock strength is dependent on the types of stresses applied to it. In other words, rock has strength in tension, compression and shear.

… these different strengths may be tested either directly (e.g. uniaxial tension test, direct shear test, etc.) or indirectly (e.g. Brazilian tensile test, triaxial compression test, etc.).

Effects of Loading ConditionsEffects of Loading Conditions

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With the application of a confining load an additional energy input is needed to overcome frictional resistance to sliding over a jagged rupture path. Most rocks are therefore strengthened by the addition of aconfining stress.

As the confining pressure is increased, the rapid decline in load carrying capacity after the peak load is reached becomes less striking until, after a mean pressure known as the brittle-to-ductile transition pressure, the rock behaves in a near plastic manner.

Pore Pressure EffectsPore Pressure Effects

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Some rocks are weakened by the addition of water, the effect being a chemical deterioration of the cement or binding material. In most cases, however, it is the effect of pore water pressure that exerts the greatest influence on rock strength. If drainage is impeded during loading, the pores or fissures will compress the contained water, raising its pressure. The resulting effect is described by Terzaghi’s effective stress law:

… as pore pressure “P” increases the effective normal stresses are reduced and the Mohr circles are displaced towards failure.

TimeTime--Dependent EffectsDependent Effects

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We have indicated that during the complete σ-ε curve, microcracking occurs from the very early stages of loading. Through these processes, four primary time-dependent effects can be resolved:

Relaxation – a decrease in strain occurs when the applied stress is held constant.

Fatigue – an increase in strain occurs due to cyclic changes in stress.

Strain-rate - the σ-ε curve is a function of the applied strain rate.

Creep – strain continues when the applied stress is held constant.

Temperature EffectsTemperature Effects

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Only a limited amount information is available indicating the effect of temperature on the complete σ-ε curve and other mechanical properties of intact rock.

The limited test data does show though, that increasing temperatures reduces the elastic modulus and compressive strength, whilst increasing the ductility in the post-peak region.

Failure CriterionFailure Criterion

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Rock fails through an extremely complex process of microcrack initiation and propagation that is not subject to convenient characterization through simplified models. Building on the history of material testing, it was natural to express the strength of a material in terms of the stress present in the test specimen at failure (i.e. phenomenological approach).

Since uniaxial and triaxial testing of rock are by far the most common laboratory procedures, the most obvious means of expressing a failure criterion is:

Strength = ƒ (σ1, σ2, σ3)

Or with the advent of stiff and servo-controlled testing machines:

Strength = ƒ (ε1, ε2, ε3)

MohrMohr--Coulomb CriterionCoulomb Criterion

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The Mohr-Coulomb failure criterion expresses the relationship between the shear stress and the normal stress at failure along a hypothetical failure plane. In two-dimensions, this is expressed as:

τpeak = c + σn tan φ

Where:φ is called the angle of internal friction (equivalent to

the angle of inclination of a surface sufficient tocause sliding of a block of similar material);

c is the cohesion (and represents the shear strength of the rock when no normal stress is applied); and

τpeak is the peak shear strength.


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This can be presented graphically using a Mohr circle diagram:


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The Mohr-Coulomb criterion is most suitable at high confining pressures when rock generally fails through the development of shear planes. However, some limitations are :

- it implies that a major shear fracture exists at peak strength, at a specific angle, which does not always agree with experimental observations;

- it predicts a shear failure in uniaxial tension (at 45-φ/2 with σ3) whereas for rock this failure plane is perpendicular to σ3. A tension cutoff has been introduced to the Mohr-Coulomb criterion to predict the proper orientation of the failure plane in tension.

- experimental peak strength envelopes are generally non-linear. They can be considered linear only over limited ranges of confining pressures.

Despite these difficulties, the Mohr-Coulomb failure criterion remains one of the most commonly applied failure criterion, and is especially significant and valid for discontinuities and discontinuous rock masses.

TheThe HoekHoek--Brown Empirical Failure CriterionBrown Empirical Failure Criterion

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The Hoek-Brown empirical criterion was developed from a best-fit curve to experimental failure data plotted in σ1- σ3 space. Since this is one of the few techniques available for estimating in situ rock mass strength from geological data, the criterion has become widely used in rock mechanics analysis.

σ1 = σ3 + (m σc σ3+ sσc2)0.5

where σc is the intact compressive strength, sis a rock mass constant (s=1 for intact rock, s<1 for broken rock), and m is a constant (characteristic of the rock type where values range from 25, for coarse grained igneous and metamorphic rocks to 7 for carbonate rocks).

DiscontinuitiesDiscontinuities

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It is the existence of discontinuities in a rock mass that makes rock mechanics a unique subject. The word ‘discontinuity’ denotes any separation in the rock continuum having effectively zero tensile strength and is used without any generic connotation (e.g. joints and faults are types of discontinuities formed in different ways).

Discontinuities have been introduced into the rock by geological events, at different times and as a result of different stress states. Very often, the process by which a discontinuity has been formed may have implications for its geometrical and mechanical properties.

DiscontinuitiesDiscontinuities

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In fact, all rock masses are fractured, and it is a very rare case where the spacings between discontinuities are appreciably greater than the dimensions of the rock engineering project. Very often major discontinuities delineate blocks within the rock mass, and within these blocks there is a further suite of discontinuities.

Thus, we might expect that a relation of the form:

should exist.

Geometrical Properties of DiscontinuityGeometrical Properties of Discontinuity

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The main features of rock mass geometry include spacing and frequency, orientation (dip direction/dip angle), persistence (size and shape), roughness, aperture, clustering and block size.

There is, however, no standardized method of measuring and characterizing rock structure geometry, because the emphasis andaccuracy with which the separate parameters are specified will depend on the engineering objectives.

Discontinuity Spacing and FrequencyDiscontinuity Spacing and Frequency

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Spacing is the distance between adjacent discontinuity intersections with the measuring scanline. Frequency (i.e. the number per unit distance) is the reciprocal of spacing (i.e. the mean of these intersection distances).

… quantifying discontinuity occurrence along a sampling line, where frequency λ=N/L m-1 and mean spacing x=L/N m.

Rock Quality DesignationRock Quality Designation

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A natural clustering of discontinuities occurs through the genetic process of superimposed fracture phases, each of which could have a different spacing distribution. An important feature for engineering is the overall quality of the rock mass cut by these superimposed fracture systems. For this reason, the concept of the RQD was developed.

Discontinuity OrientationDiscontinuity Orientation

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If we assume that a discontinuity is a planar feature, then its orientation can be uniquely defined by two parameters: dip direction and dip angle. It is often useful to present this data in a graphical form to aid visualization and engineering analysis.

It must be remembered though, that it may be difficult to distinguish which set a particular discontinuity belongs to or that in some cases a single discontinuity may be the controlling factor as opposed to a set of discontinuities.

Discontinuity PersistenceDiscontinuity Persistence

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Persistence refers to the lateral extent of a discontinuity plane, either the overall dimensions of the plane, or in terms of whether it contains ‘rock bridges’. In practice, the persistence is almost always measured by the one dimensional extent of the trace lengths as exposed on rock faces. This obviously introduces a degree of sampling bias that must be accounted for in the interpretation of results.

Discontinuity RoughnessDiscontinuity Roughness

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The word ‘roughness’ is used to denote deviation of a discontinuity surface from perfect planarity, which can rapidly become a complex mathematical procedure utilizing 3-D surface characterization techniques (e.g. polynomials, Fourier series, fractals).

From the practical point of view, only one technique has received some degree of universality – the Joint Roughness Coefficient (JRC). This method involves comparing discontinuity surface profiles to standard roughness curves assigned numerical values.

The geometrical roughness is naturally related to various mechanical and hydraulic properties of discontinuities.

Discontinuity ApertureDiscontinuity Aperture

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The aperture is the distance between adjacent walls of a discontinuity. This parameter has mechanical and hydraulic importance, and a distribution of apertures for any given discontinuity and for different discontinuities within the same rock mass is expected.

Mechanical Properties of DiscontinuitiesMechanical Properties of Discontinuities

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The mechanical behaviour of discontinuities is generally plotted in the form of stress-displacement curves, with the result that we can measure discontinuity stiffness(typically expressed in units of MPa/m) and strength.

In compression, the rock surfaces are gradually pushed together, with an obvious limit when the two surfaces are closed. In tension, by definition, discontinuities can sustain no load. In shear, the stress-displacement curve looks like that for compression of intact rock, except of course failure is localized along the discontinuity.

Mechanical Properties Mechanical Properties -- StrengthStrength

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It is normally assumed that the shear strength of discontinuities is a function of the friction angle rather than the cohesion. This is done by using the Mohr-Coulomb failure criterion, τ = c + σtanφ, and setting the cohesion to zero.

Mechanical Properties Mechanical Properties -- StrengthStrength

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The bi-linear failure criterion introduces the idea that the irregularity of discontinuity surfaces could be approximated by an asperity angle i onto which the basic friction angle is superimposed.

Thus, at low normal stresses, shear loading causes the discontinuity surfaces to dilate giving an effective friction of (φ+i). As the shear loading continues, the shear surfaces become damaged as asperities are sheared and the two surfaces ride on top of one another, giving a transition zone before the failure locus stabilizes at an angle of φ.

Rock MassesRock Masses

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Building on our examination of first intact rock behaviour and then discontinuity behaviour, we can now concentrate on extending these ideas to provide a predictive model for the deformability and strength of rock masses.

Rock Mass DeformabilityRock Mass Deformability

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As an initial step in determining the overall deformability of a rock mass, we can first consider the deformation of a set of parallel discontinuities under the action of a normal stress, assuming linear elastic discontinuity stiffnesses.

To calculate the overallmodulus of deformation, the applied stress is divided by the total deformation. We will assume that deformation is made up of two components: one related to the intact rock; the other to the discontinuities.


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The contribution made by the intact rock to the deformation, δI, is σL/E (i.e. strain multiplied by length). The contribution made by a single discontinuity to the deformation, δD, is σ/ED (remembering that ED relates to displacement directly). Assuming a discontinuity frequency of λ, there will be λL discontinuities in the rock mass and the total contribution made by these to the deformation will be δD

t, which is equal to σ λL /ED. Hence, the total displacement, δT, is:

Hence, the total displacement, δT, is:

With the overall strain being given by:

Finally, the overall modulus, EMASS, is given by:


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… variation of in situ rock deformability as a function of the discontinuities (for the idealized case of a single set of discontinuities).

Rock Mass StrengthRock Mass Strength

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In the same way as we considered the deformability of a rock mass, expressions can be developed indicating how strength is affected by the presence of discontinuities, starting with a single discontinuity and then extending to any number of discontinuities.

… scale dependent strength of a single discontinuity.


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The initial approach is via the ‘single plane of weakness’ theory, whereby the strength of a sample of intact rock containing a single discontinuity can be established. Basically, the stress applied to the sample is resolved into the normal and shear stresses on the plane of weakness and the Mohr-Coulomb failure criterion applied to consider the possibility of slip.

Given the geometry of the applied loading condition:


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The strength of the sample thus depends on the orientation of the discontinuity. If the discontinuity is, for example, parallel orperpendicular to the applied loading, it will have no effect on the sample strength. At some angles, however, the discontinuity willsignificantly reduce the strength of the sample.

The lowest strengthoccurs when the discontinuity normal is inclined at an angle of 45° + (φ°/2) to the major applied principal stress.


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The plot of rock strength and the discontinuity angles at which the sample strength becomes less than that for intact rock can be derived by substituting the ‘single discontinuity’ normal and shear stress relationships into the Mohr-Coulomb criterion:

Substituted into|τ| = cw + σntanφw gives:

Where cw and φw are the cohesion and friction for the discontinuity.


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An alternative presentation of the ‘single plane of weakness’ rock strength theory is via the Mohr’s circle representation. The Mohr-Coulomb failure loci for both intact rock and the discontinuity are given.

Circle A – case where the failure locus for the discontinuity is just reached, i.e. for a discontinuity at the angle 2βw=90°+φw.

Circle B – case when failure can occur along the discontinuity for a range of angles.

Circle C – case where the Mohr circle touches the intact rock failure locus, i.e. where failure occurs in the intact rock.


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We can consider, on the basis of this single plane of weakness theory, what would happen if there were two or more discontinuities at different orientations present in the rock sample. Each discontinuity would weaken the sample as shown below, but the angular position of the strength minima would not coincide.

As a result the rock is weakened in several different directions simultaneously.With increasing fractures, the material tends to become isotropic in strength, like a granular soil.

Rock Mass Strength Rock Mass Strength –– HoekHoek--BrownBrown

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A methodology of assessing rock mass strength that does not depend on the ‘single plane of weakness’ theory is the Hoek-Brown failure criterion. The criterion is especially powerful in its application to rock masses due to the constants m and sbeing able to take on values which permit prediction of the strengths of a wide range of rock masses.

Rock Mass Rock Mass StrengthStrength

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… Hoek-Brown representation and summary of rock mass conditions, testing methods and theoretical considerations.


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For intact rock, the Hoek-Brown criterion may be expressed as:

The more general form, however, was derived to take into account the fractured nature of the rock mass through the parameters s and a.

Rock Mass Rock Mass Strength Strength ––HoekHoek--BrownBrown

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… Hoek-Brown ‘m’ values for different rock types.

Rock Mass Rock Mass Strength Strength –– HoekHoek--

BrownBrown

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… estimation of Hoek-Brown constants and rock mass deformation constants based on rock mass structure and discontinuity surface conditions.


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… the Hoek-Brown empirical criterion applied to a sandstone rock mass. The criterion represents best-fit curves to experimental failure data plotted in σ1- σ3 space.

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Rock Mechanics and Rock Engineering · PDF fileRock Mechanics and Rock Engineering Rock mechanics involves characterizing the intact strength and the geometry and mechanical properties