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Lenhardt Rock Mechanics/ Basics 1
Rock MechanicsRock Mechanics--
BasicsBasicsby by
Wolfgang A. Wolfgang A. LenhardtLenhardt
[email protected]@zamg.ac.at
Department of GeophysicsDepartment of GeophysicsCentral Institute for Meteorology and GeodynamicsCentral Institute for Meteorology and Geodynamics
HoheHohe WarteWarte 3838AA--1190 Vienna1190 Vienna
AustriaAustria
Phone: +43 1 36 026 ext. 2501Phone: +43 1 36 026 ext. 2501TelefaxTelefax: +43 1 368 6621: +43 1 368 6621
www.zamg.ac.atwww.zamg.ac.at
Lenhardt Rock Mechanics/ Basics 2
Content
TerminologyStress & Strain
Mohr CircleState of Stress
Stress ConcentrationsStrain Energy Density
ClosureDiscontinuities
Excess Shear Stress
Lenhardt Rock Mechanics/ Basics 3
Youngs ModulusE - modulus of elasticity[GPa]
normalstress
normal strain
Terminology
Uniaxial Compressive StrengthUCS stress at failure[MPa]
Shear ModulusG - modulus of rigidity[GPa]
shearstress
shear strain
slope = G
slope = E
Lenhardt Rock Mechanics/ Basics 4
Terminology - contd
Poissons Ratio - Ratio of radial/axial deformation[1]
CohesionSo - Intrinsic shear strength[MPa]
=-x/z
AdhesionTo Tensile strength[MPa]
Tonormal stress
shearstress
SoCoefficient of Friction Tangent of angle between shear and normal stress[1]
slope =
Lenhardt Rock Mechanics/ Basics 5
Stress and Strain
d1
l1
lo
do
Area
Force
Stress = Force / Area
Radial strain = r = (do- d1)/doAxial strain = a = (lo- l1)/lo
< 0in compression !
> 0
Poissons ratio = - r / aYoungs modulus E = a / a
72 GPa0.22
Typical values of Witwatersrand quartzite
E
Note: Poissons ratio and the Youngs modulus are material-dependent and may vary in different directions.Special cases of : 0.25 ... purely elastic, 0.5 ... liquid (incompressible)
Lenhardt Rock Mechanics/ Basics 6
Mohrs Circle
The Mohr-Circle diagram connects the state of deformation and the currently applied stress regime in a graphical way.
Let us consider a cube, which is exposed to external forces per area or stresses. Within this cube we would like to determine these stresses. For reasons of simplicity, lets assume, that the largest stress is acting vertical, and the smallest stress is acting horizontal:
Definition:These major two stresses are called the
1. Major principal stress, and2. Minor principal stress
The stress, which acts orthogonal to these stresses, is named intermediate stress.
In these three distinct directions, only normal stresses are acting. No shear stresses are present. Why?
Each stress-system can be represented by three principal (normal) stresses and their directions only! We will see now, how this works.
Lenhardt Rock Mechanics/ Basics 7
Mohrs Circle Derivation contdLets consider a pressure p acting at an arbitrary angle on an inclined plane in respect to a Cartesian co-ordinate system.
The force-equilibrium
requires, the following:
Consider a stress px in x-direction due to a stress acting at an angle to the inclined surface A:
px = x . . . . cos + yx . sin
Lenhardt Rock Mechanics/ Basics 8
Mohrs Circle Derivation contd
Now, these components can be separated into a normal and a shear stress acting on the plane:
is the angle between the normal stress on a plane and the major principal stress. Rotation of the arbitrary chosen coordinate system by would eliminate the shear stresses.The angle , at which shear stresses vanish, is given by:
xy= yx
Since the forces exerted on a square must be in equilibrium, we set:
Lenhardt Rock Mechanics/ Basics 9
Mohrs Circle StressesThe derived relation between shear- and normal stresses can be expressed graphically by a circle the Mohr Circle, which can be completely described by the principal stresses.
Lenhardt Rock Mechanics/ Basics 10
Mohrs Circle Strains
Note: Only half of the shear strains are plotted in the diagram, for the other half is consumed by rotation.
Lenhardt Rock Mechanics/ Basics 11
Mohrs Circle Summary
Stress acting normalto the plane
Shear stress acting indirection of the plane
Lenhardt Rock Mechanics/ Basics 12
Mohrs Circle Summary contd
2
2
Stress
Strain
General case:3 normal stresses (x ,,,,y ,,,,z) or (xx ,,,,yy ,,,,zz) 3 shear stresses (xy , , , ,yz ,,,,zx) or (xy , , , ,yz ,,,,zx)
Transformed:3 principal stresses (1... ... ... ... 3)3 principal directions (1... ... ... ... 3)
or strains ...
Lenhardt Rock Mechanics/ Basics 13
Mohrs Circle Examples
1 = 200 MPa2 = 0 MPa3 = 0 MPa
1 = 400 MPa2 = 50 MPa3 = 0 MPa
1 = 350 MPa2 = 10 MPa3 = 10 MPa
Lenhardt Rock Mechanics/ Basics 14
Mohrs Circle Examples
1 = 50 MPa2 = 5 MPa3 = -10 MPa
1 = 0 MPa2 = 0 MPa3 = 0 MPa
1 = 50 MPa2 = 50 MPa3 = 50 MPa
Lenhardt Rock Mechanics/ Basics 15
Mohrs Circle Exercise 1
1 = 210 MPa
2 =10 MPa = 60
Task:
Determine the stresses normal and shear stresses acting on the inclined plane!
Hint:1. Determine 2. Plot principal stresses on x-axis3. Draw Mohrs Circle4. Introduce in Mohrs Circle5. Calculate the average principal stress6. Calculate the maximum shear stress7. Determine normal and shear stress
Solution:Normal stress =average stress +
cos(2)*maximum shear stressShear stress = sin(2)*maximum shear stress
Lenhardt Rock Mechanics/ Basics 16
Mohrs Circle Exercise 1 Solution
Result:
= average + cos(2) * max = 60 MPa
= sin(2) * max = 86.6 MPa
Solution:
= = 60 (just in this case, for the angle was given from 2, which is the same as the angle between the plane normal and the maximum principal stress!)
average = (1 + 2) / 2 = 110 MPa
max = (1 - 2) / 2 = 100 MPa
normal stress1 = 210 MPa
2 = 10 MPa
= ? = ? = ? = ?
=? =? =? =?
2 = 120
shear stress
max
shear stress
Lenhardt Rock Mechanics/ Basics 17
Mohrs Circle Exercise 2
y = 100 MPa
x = 50 MPa
Task:
Determine the magnitude and orientation of the principal stresses!
Hint:1. Plot normal stresses on x-axis2. Determine average normal stress3. Plot shear stress on y-axis4. Draw Mohrs Circle5. Determine 6. Calculate principal stresses
Solution:Orientation = = arctan (yx/(y- average))/2Determine average normal stressDetermine maximum shear stress1 = average stress + max2 = average stress - max
yx = 25 MPa
Note sign convention!
Lenhardt Rock Mechanics/ Basics 18
Mohrs Circle Exercise 2 Solution
normal stress
1 ????2 ???? 2
x
y
xy
yx
shear stress
shear stress
Result:
= (arctan(yx/(y- average)))/2 = 22.5
1111 = average + max = 110.4 MPa
2222 = average - max = 39.6 MPa
2 =39.6 MPa
= 22.51 =110.4 MPa
Solution:xy = - yx = - 25 MPa
average = (x + y) / 2 = 75 MPa
max = = 35.35 MPa
Lenhardt Rock Mechanics/ Basics 19
3-D Mohr Space
From: Jaeger & Cook (1978). In Pollard, D.D. & Fletcher, R.C. 2005. Fundamentals of Structural Geology. Cambridge.
Lenhardt Rock Mechanics/ Basics 20
Failure Criteria
normal stress
max= So
shear stress
No failure
Failure
MAXIMUM SHEAR STRESS
normal stress
shear stress
No failure
Failure
COULOMB-NAVIER
normal stress
shear stress
No failure
Failure
HOEK-BROWN
= So + n . tan
1111 = = = = 3((((m. UCS. 3 + UCS s)m, s... describe the integrity of the rock mass (Hoek & Brown parameters), s = integrity (0-1 = degree of fracturing), m = type of rock
Lenhardt Rock Mechanics/ Basics 21
Failure Criteria contd
normal stress
shear stress
No failure
Failure
COULOMB-NAVIER
minor stress
major stress
No failure
Failure
Compressive StrengthUCS
Tensile StrengthTo
Intrinsic StrengthSo
Lenhardt Rock Mechanics/ Basics 22
Failure Criteria contd
The following relations between the different parameters are useful:
Lenhardt Rock Mechanics/ Basics 23
Pre- and Post Failure Behaviour
Lenhardt Rock Mechanics/ Basics 24
Pre- and Post Failure Behaviour contdWhether the rock behaves brittle or ductile depends mainly on the confining stress.
After: Wawersik, W.R. & Fairhurst C.A, 1970: A study of brittle rock fracture in laboratory compression experiments. Int. J. Rock Mech. Min. Sci. 7, 561 675.
Confin
ing
Stres
s
Yielding(resisting constant stress)
Not yielding(loosing load
carrying capacity)
Lenhardt Rock Mechanics/ Basics 25
Pre- and Post Failure Behaviour contdIf the confining stress is low such as in slim pillars shear failure is likely to take place, especially when transgressive fractures (due to blasting or geology) are present.
Lenhardt Rock Mechanics/ Basics 26
Strain Rate
Materials do respond different to high and low strain rates:
UCS... Uniaxial compressive strengthE... Youngs modulusTd... Duration of test
From: Chong & Boresi, 1990: Strain rate properties of New Albany Reference shale. Int. J. Rock Mech. Min. Sci. & Geomech. Abstr., Vol. 27, No.3, 199 - 205.
Lenhardt Rock Mechanics/ Basics 27
State of Stress
k= horizontal st ressvert ical st ressdepth
surface
k < 1 (deep)
k > 1 (shallow)
Lenhardt Rock Mechanics/ Basics 28
State of Stress contd
From: Brady, B.H.G. & Brown, E.T., 1985. Rock Mechanics for Underground Mining, George Allen & Unwin Publishers.
Terminology:
VIRGIN STRESS Original stress prior to miningINDUCED STRESS Additional change in stress due to miningFIELD STRESS VIRGIN + INDUCED STRESS = actual stress acting around the excavation
k = 0.5 (pxx + pyy) / pzz
Lenhardt Rock Mechanics/ Basics 29
State of Stress 2D Plane Stress
An example of a plane stress regime is the surface of an underground excavation. Perpendicular to the free surface at the stope face, no stress is acting only strain.
The basic equations for calculating the principal strains and stresses during the linear-elastic state of deformation are given here:
An example of a plane strain condition is a section through a tunnel. In direction of the axis of a tunnel only the stress remains, - the strain equals zero. The equivalent formulas can be derived from the 3D state of stress.
Lenhardt Rock Mechanics/ Basics 30
State of Stress 3D
The general case in three dimensions for a linear-elastic state of deformation is given by:
Lenhardt Rock Mechanics/ Basics 31
Stress ConcentrationsStress concentrations do occur around all excavations. The actual field stress depends on:
1. Depth2. Shape of excavation3. Mining layout4. Support5. Rock characteristics
The rock mass has a very low tensile strength. Whether an excavation is subjected to tensile stresses depends on its shape (e.g. height to width ratio) and the ratio of the horizontal to the vertical stress (k - ratio).
Lenhardt Rock Mechanics/ Basics 32
Stress Concentrations in a Tunnel Exercise
Lenhardt Rock Mechanics/ Basics 33
Stress Concentrations in a Tunnel contd
Tensile stresses may easily develop in the crown of a haulage (rockfall !) if the
k-ratio is low and/or theheight/width ratio is small
After: Brady, B.H.G. & Brown, E.T., 1985. Rock Mechanics for Underground Mining, George Allen & Unwin Publishers.
Lenhardt Rock Mechanics/ Basics 34
Stress Concentrations contd
High stresses such as inside or below a pillar may lead to disking of borehole-samples. The example shown, was taken from a pillar 3000 m belowsurface, intersecting the reef (pebbles). The disk at the right end originated from the edge of the pillar.
Lenhardt Rock Mechanics/ Basics 35
Stress Shadow
The presence of nearby mining openings affect the field stresses and thus the intensity and orientation of induced fractures in e.g. tunnels.
stope
From: An Industry Guide to Methods of Ameliorating the Hazards of ROCKFALL and ROCKBURSTS, Chamber of Mines of South Africa, 1988.
Lenhardt Rock Mechanics/ Basics 36
Mining Conditions
Conditions in deep mines differ from shallow mines in many ways.
From: An Industry Guide to Methods of Ameliorating the Hazards of ROCKFALL and ROCKBURSTS, Chamber of Mines of South Africa, 1988.
Lenhardt Rock Mechanics/ Basics 37
Mining Conditions contd
From: An Industry Guide to Methods of Ameliorating the Hazards of ROCKFALL and ROCKBURSTS, Chamber of Mines of South Africa, 1988.
Lenhardt Rock Mechanics/ Basics 38
Strain Energy DensityStrain energy density represents the energy stored in the rock per volume:
W=i= 1
3 i i
2
i... principal stressi... principal strain
The total stored strain energy U in the rock mass with volume V is therefore
U = W * V
The major principal stress dominates the stored strain energy density once the k-ratio is less than 0.5 (error < 10 %, exact if k = 0):
Example:Uniaxial Compressive Test (k-ratio = 0)
220 MPa at failure
E = 72 GPa(Youngs modulus)
/2 = W
Strain Energy Density:
Lenhardt Rock Mechanics/ Basics 39
Strain Energy - ExerciseCalculate the strain energy density as function of depth, while considering Poissons ratio, the k-ratio and the material properties.
Lenhardt Rock Mechanics/ Basics 40
Strain Energy Exercise contd
Lenhardt Rock Mechanics/ Basics 41
Closure
From: An Industry Guide to Methods of Ameliorating the Hazards of ROCKFALL and ROCKBURSTS, Chamber of Mines of South Africa, 1988.
Closure is the amount by which the original stoping width is reduced.
Closure = Elastic deformation + Inelastic deformation
convergence
Principle of beamBed separation
Opening of jointsSwelling
Slip along plane
Inelastic deformation exceeds by far elastic deformation!
Examples:
Lenhardt Rock Mechanics/ Basics 42
Closure contd
Deformation of the rock mass around deep stopes.
From: Jager, A.J. & Ryder, J.A. 1999. A Handbook for Rock Engineering Practice for tabular hard rock mines. The Safety in Mines Research Advisory Committee (SIMRAC), Johannesburg, South Africa.
Lenhardt Rock Mechanics/ Basics 43
Closure contd
Inelastic deformation manifests itself in face-parallel fracturing of the footwall.
Face advance
Stope face
Direction of advance
Old mine workings
Lenhardt Rock Mechanics/ Basics 44
Closure contd
Timber support in an old stope, at which total closure took place.
Lenhardt Rock Mechanics/ Basics 45
Rock MassThe description of discontinuities is a matter of scale!
Lenhardt Rock Mechanics/ Basics 46
Types of Fractures
Triaxial compression Shear(offset between surfaces)
Special caseCross Shear
Uniaxialcompression
Uniaxial tension
Special casesIntrusion, Uplift
Longitudinal splitting during perfect compression (no end-effects)(irregular surface)
Extension(no shear-offset between
surfaces)
Lenhardt Rock Mechanics/ Basics 47
Rock Quality
Barton, N. 2007. Rock quality, seismic velocity, attenuation and anisotropy. Taylor & Francis/Balkema, The Netherlands, 729 pp.
Lenhardt Rock Mechanics/ Basics 48
DiscontinuitiesRock characteristics depend on the number of discontinuities existing inside of the rock mass. A very small sample (laboratory test) usually contains no discontinuities. A largerock sample may contain numerous discontinuities, however.
FRACTURE plane which separates the rock materialJOINT break of geological origin, no shear displacement visibleFAULT fracture with identifiable shear displacementDYKE long narrow vertical intrusionSILL near horizontal intrusionBEDDING PLANE separates sedimentary rocks into beds or strataSHEAR ZONE bonds of material in the order of metres in which local shear
displacement took place
Lenhardt Rock Mechanics/ Basics 49
Discontinuities contd
Shear fracture Joints & plumoses
Lenhardt Rock Mechanics/ Basics 50
Discontinuities contd
Dyke-contact
Dyke
Lenhardt Rock Mechanics/ Basics 51
Discontinuities contd
Fault Movement along a dyke-contact, as seen through the
shot (gunned) concrete
Dyke Contact contd
Dyke Quartzite
After: Cook, N.G.W. et al. 1966.52
Lenhardt Rock Mechanics/ Basics 53
Types of Faults
Reverse or thrust faultNormal fault
Strike-slip fault
There are
Normal,Reverse/thrust andStrike-slip faults
Sometimes, their effects daylight on surface.
Lenhardt Rock Mechanics/ Basics 54
Types of FaultsThree types of faults can be distinguished:
Reverse or thrust faultNormal fault Strike-slip fault
Lenhardt Rock Mechanics/ Basics 55
Characteristics of FaultsCoefficients of sliding-friction differ significantly (based on faults at shallow depth, < 1000 m below surface):
From: Jaeger, J.,C. & Cook, N.G.W. 1979. Fundamentals of Rock Mechanics, 3rd edition, Chapman and Hall, London, pages 377-379 and 425 427.
Under mining conditions faults may slip reverse to their geological sense of displacement!
> 10Extremely highAAA1 10Very highAA0.1 1HighA
0.01 0.1ModerateB
0.001 0.01LowC< 0.001Extremely lowD
Slip rate (cm/year)Fault activityclass
From: Bonilla M. G. 1982. Evaluation of potential surface faulting and other tectonic deformation. 8Th World Conference on Earthquake Engineering, Vol.1, 65. Note: No distinction is made here for aseismic slip (creep).
0.202 (-8.0)0.235 (+6.9)0.3830.220Strike-slip faulting0.381 (-10.9)0.476 (+11.4)0.7900.427Thrust faulting0.521 (-16.6)0.746 (+19.6)0.8480.625Normal faulting0.471 (-9.6)0.577 (+10.5)0.7860.522Combined data
Lower(per cent)
Upper(per cent)
Correlation coefficient
Coefficient of friction from
regression line
98 per cent confidence limits
Lenhardt Rock Mechanics/ Basics 56
Tectonic UpliftPost-glacial rebound leads to a dome-shaped uplift of the crust, along with an alteration of the prevailing stress-regime in the dome and the surrounding forebuldge.
From: Muir Wood, R. 1995. Deglaciation seismotectonics: A principal influence on intraplate seismogenesis at high latitudes? In Proc. of 2nd France-United States Workshop Earthquake Hazard Assessment in Intraplate Regions (G. Mohammadioun, ed.), Quest Editions, Press
Acamedique, 85 103.
Lenhardt Rock Mechanics/ Basics 57
Excess Shear Stress - ESS
For details see: Ryder, J.A. 1988. Excess shear stress in the assessment of geologically hazardous situations. J.S.Afr.Inst.Min.Metall., Vol.88, pages 27-39.
All parameters in this Mohr-diagram refer to conditions along a geological feature and not to solid (undisturbed) rock.
ESS =
normal stress
shearstress
So slope =
ESS
Stress prior to slip
Stress after slip
Lenhardt Rock Mechanics/ Basics 58
Excess Shear Stress contd
ESS-lobes above and below a stope
From: An Industry Guide to Methods of Ameliorating the Hazards of ROCKFALL and ROCKBURSTS, Chamber of Mines of South Africa, 1988.
Lenhardt Rock Mechanics/ Basics 59
Blasting contd
Blasting induced fractures are orientated perpendicular
to the minor principal stress and the borehole-axis, or follow the stratification of the rock
mass.
drill hole
1 - major principal stress
3 -minor principal stress
Lenhardt Rock Mechanics/ Basics 60
Tectonic versus Induced/Trigggered Events
Tectonic Induced /TriggeredVirgin stresses High Low
Short term stress changes Low HighDepth < 3 km > 3 kmMechanism Shear Slip Fracture and Shear SlipMax. magnitude 9.4 6.5 (?)Rupture length 1.400 km Few kms
Lenhardt Rock Mechanics/ Basics 61
SummaryTerminology
Stress & StrainMohr Circle
State of StressStress ConcentrationsStrain Energy Density
ClosureDiscontinuities
Excess Shear Stress