failure citerion.pdf

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9/15/2015Rock Mechanics Course

By Hasabelrsool E.A.Elsadig, MSc ‐Mining @ UofK

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1. Mohr - Coulomb failure criterion2. Hoek - Brown failure criterion

Rock Mechanics Course

Hasabelrsool E.A.Elsadig, MSc (Geomechanics ‐ ITB)

Sudan University of Science and Technology

Faculty of Petroleum Engineering



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Over the years, comprehensive laboratory studies have yielded a

variety of failure criterion to describe rock strength in compression.

Ultimate expression of failure criteria is to predict rock strength.

Rock failure criteria is determined based on experimentation data.

Rock failure criteria is determined based on empirical and

theoretical approaches.

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Assumption for Failure Criterion of Rock

The expression of this criteria consists of one or more parameters ofmechanical properties of rock and becomes simple if it is calculatedin 2D with assumption of plane strain or plane stress.

On plane strain, if the following condition prevails: σ1 > σ2 > σ3, theintermediate principal stress σ2 is a function of other two principalstresses or the failure criteria only function on the two principalstresses (σ1 & σ3).

On plane stress, only the two principal stresses influence to thefailure criteria because another principal stress is zero.



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OC = Uniaxial Compressive Strength

OT = Uniaxial Tensile Strength

CM = Triaxial Test

Stress Space

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1. Coulomb Failure Criterion

• The simplest, and still most widely used, failure criterion is that ofCoulomb (1773).

• Based on his extensive experimental investigations into friction,Coulomb assumed that failure in a rock or soil takes place along aplane due to the shear stress τ acting along that plane.

Motion is assumed to be resisted by:

1. A frictional-type force whose magnitude equals the normalstress σn acting along this plane, multiplied by some constantfactor μ. { ∅ .

2. An internal cohesive force of the material.{C}



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These considerations lead to the mathematical criterion thatfailure will occur along a plane if the following condition issatisfied:

∅

Conversely, failure will not occur on any plane for which:

∅



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The parameter μ is known as the coefficient of internal friction, as

it applies along an imaginary surface that is internal to the rock

before failure occurs.

• The form of criterion suggests that the Mohr’s circle construction

will be useful in its analysis.

∅



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A stress state whose Mohr’s circle lies below the line AL will not

give rise to failure on any plane.

If the principal stresses are such that the circle touches the failure line,

the rock will fail in shear.



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It would be convenient to express the Coulomb failure criterion

directly in terms of the principal stresses {σ1, σ3}.

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φsin 1

φsin 1σ

φsin 1

φ cos2Cσ

φsin σσ2

1φ cos Cσσ

2

1

φsin σσ2

1φcot Cσσ

2

1

φsin OCAOCP

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3131

3131

The failure criterion can also be written in many seemingly

different but equivalent forms, each of which is convenient in

certain circumstances.9/15/2015

Rock Mechanics Course By Hasabelrsool E.A.Elsadig, MSc ‐Mining

@ UofK

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MOHR’S Hypothesis

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• The Coulomb theory predicts a relatively modest ratio of

compressive to tensile strength (5.83). Experimental values

of this ratio, however, tend to be on the order of 10 or so.

• Roughly, this deficiency can be expressed by saying that the

Coulomb failure line extends too far into the tensile region of

the {σ , τ} plane.

• According to Coulomb’s theory, failure will occur on a plane when

the normal and shear stresses acting on that plane satisfy the

failure criterion.


@ UofK

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• In the {σ, τ} plane, this condition appears as a straight line with slope

μ = tan φ.

• The Mohr’s circle corresponding to any state of stress that leads to

failure will be tangent to this line.

• This theory ignores the effect of the intermediate principal stress.

However, in principle, Coulomb’s theory could be expected to apply

to stress states in which σ2 = σ3.

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• Coulomb’s theory also predicts that the compressive stress

required to cause failure, σ1, will increase linearly with the

confining stress, σ3. Experiments typically show that σ1 at

failure increases at a less-than-linear rate with σ3.

In order to correct these deficiencies, Mohr (1900) suggested

that Coulomb’s equation be replaced by a more general,

possibly nonlinear, relation of the form:

|τ|= f(σ)


@ UofK

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In principle, the curve can be determined experimentally as the

envelope of all of the Mohr’s circles that correspond to states of stress

that cause failure.

Aside from the fact that may now be a nonlinear function, the

basic ideas of Coulomb’s model are retained. Specifically, failure is

supposed to occur if one of the Mohr’s circles touches the curve.9/15/2015


@ UofK15



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An Application of Mohr‐ Coulomb Criterion(S.F = Safety Factor):

Safety factor = S.F = A/Bcos ∅ 2 sin∅

2

………(1)

A

B



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Hoek-brown

failure criterion

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Generalised Hoek‐Brown criterion

• ’ and ’ : Max. and min. effective stresses at failure

• mb : Hoek-Brown constant m for the rock mass

• s and a : Constants

• : UCS of the intact rock pieces.

a

ci

3bci31 sσ

'σm σ 'σ 'σ



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In order to use the Hoek-Brown criterion for estimating the

strength and deformability of jointed rock masses, three ‘properties’

of the rock mass have to be estimated :

of the intact rock pieces

Hoek-Brown constant mi for these intact rock pieces

Geological Strength Index GSI for the rock mass

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Intact rock properties

• The relationship between the principal stresses at failure for a givenrock is defined by two constants, sci and mi.

• Wherever possible the values of these constants should be determinedby statistical analysis of the results of a set of triaxial tests on carefullyprepared core samples.

0.5

ci

3ici31 1σ

'σm σ 'σ 'σ

• The range of minor principal stress (s3’) values over which thesetests are carried out is critical in determining reliable values for thetwo constants.



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• In deriving the original values of and mi, Hoek and Brown(1980) used a range of { 0 < < . .}

• In order to be consistent, it is essential that the same range beused in any laboratory triaxial tests on intact rock specimens.

• At least five data points should be included in the analysis.

• Once the five or more triaxial test results have been obtained,they can be analysed to determine the and the mi as describedby Hoek and Brown (1980).

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231

3

cici

0.5

ci

3ici31

'σ 'σ y

'σx

sσ xmσ y

1σ

'σm σ 'σ 'σ

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Short-term laboratory tests on very hard brittle rocks tend tooverestimate the in situ rock mass strength.

Laboratory tests and field studies on excellent quality Lac du Bonnetgranite, reported by Martin and Chandler (1994), show that the insitu strength of this rock is only about 70% of that measured inthe laboratory.

This appears to be due to damage resulting from micro-cracking of the

rock which initiates and develops critical intensities at lower stress

levels in the field than in laboratory tests carried out at higher loading

rates on smaller specimens.



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• Hence, when analysing the results of laboratory tests on these types

of rocks to estimate the values of sci and mi , it is prudent to reduce

the values of the major effective principal stress at failure to 70% of

the measured values.

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Field estimates of mi

Rock type Class GroupTexture

Coarse Medium Fine Very fine

Sedimentary

Clastic

Conglomerate(22)

Sandstone(19)

Siltstone(9)

Claystone(4)

Greywacke(18)

Non‐Clastic

Organic

Chalk(7)

Coal(8‐21)

CarbonateBreccia(20)

SpariticLimestone

(10)

MicriticLimestone

(8)

ChemicalGypstone

(16)Anhydrite

(13)

Rock Mechanics Course By Hasabelrsool E.A.Elsadig, MSc ‐Mining @ UofK




Metamorphic

Non foliatedMarble(9)

Hornfels(19)

Quartzite(24)

Slightly foliatedMigmatite

(30)Amphibolite

(25‐31)Mylonites

(6)

FoliatedGneiss(33)

Schists(4‐8)

Phyllites(10)

Slate(9)


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Igneous

Light

Granite(33)

Rhyolite(16)

Obsidian(19)

Granodiorite(30)

Dacite(17)

Diorite(28)

Andesite(19)

Dark

Gabbro(27)

Dolerite(19)

Basalt(17)

Norite(22)

ExtrusivePyroclactic type

Agglomerate(2)

Breccia(18)

Tuff(15)


Geological Strength Index GSI

The strength of a jointed rock mass depends on the properties of the intact rock pieces and also upon the freedom of these pieces to slide and rotate under different stress conditions.

This freedom is controlled by the geometrical shape of the intact rock pieces as well as the condition of the surfaces separating the pieces.

The Geological Strength Index (GSI), introduced by Hoek (1995) and Hoek,Kaiser and Bawden (1995) provides a system for estimating the reduction in rock mass strength for different geological conditions.


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′ ′′

s, mb, and a are constants dependent on the rock mass properties, computed from (GSI), Hoek et al. 1995, using the following relations for undisturbed rock masses:

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For GSI > 25

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100GSI exp m m ib

200

GSI0.65a and 0 s

For GSI < 25

0.5a and 9

100GSI exp s

Rock Mass properties

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Geological Strength Index GSI

Angular rock pieces with clean, rough discontinuity surfaces will result in a much stronger rock mass than one which contains rounded particles surrounded by weathered and altered material.

Once the Geological Strength Index has been estimated, the parameters that describe the rock mass strength characteristics, can be calculated.


GSI


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GSI


GSI – Mohr‐Coulomb parameters


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GSI – Mohr‐Coulomb parameters


Rock Mechanics Course By Hasabelrsool E.A.Elsadig, MSc ‐Mining @ UofK38

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Rock Mass Long Term Strength


Illinois, USA, 50 years


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Pennsylvania, USA, 50 years


Witbank, South Africa, 20‐100 years


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Koomfontein, South Africa, 80 years


Sasolburg, South Africa, 5 years


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failure citerion.pdf

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