Probabilistic Estimates of Hoek Brown Intact Parameters Fun with Hoek-Brown...

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Transcript of Probabilistic Estimates of Hoek Brown Intact Parameters Fun with Hoek-Brown...

  • PROBABILISTIC ESTIMATES OF INTACT HOEK BROWN

    PARAMETERS

  • CONTENTS Mathematical Fun with Hoek-Brown Ground Rules Deterministic Hoek-Brown Estimation (Single Stage) Probabilistic Hoek-Brown Estimation (Single Stage) Probabilistic Hoek-Brown Estimation (Multi Stage) Normal Distribution Probabilistic Hoek-Brown Estimation (Multi Stage) Log Normal Distribution Probabilistic Hoek-Brown Estimation (Multi Stage) Beta Distribution Automated Servo Controlled Triaxial Testing Conclusion References

  • Mathematical Fun with Hoek-Brown (Hoeks Rock Engineering Notes)

    Butforintactrocks=1anda=0.5

    Aftermultiplyingandrearrangingtheequationcanberewrittenas:

    Notethelasttermissquared,thereisatypoinHoeksNotesSubstituting andX= gives:

    Whichisthefamiliarequationforastraightline

  • Mathematical Fun with Hoek-BrownTwoConclusions:LinearRegressioninExcelcanbeusedif isplottedagainst (orYagainstX)

    Andtheslopeofthefittedlineis:

    And

  • GROUND RULES What is Probability

    Epistemic Uncertainty Aleatory Randomness DegreesofBelief Bayesian

  • GROUND RULES Useful Definition (Grant et al)

    Assumethatifalargenumberoftrialsaremadeunderthesameessentialconditions,theratiooftrialsinwhichacertaineventhappenstothetotalnumberoftrialswillapproachalimitasthetotalnumberoftrialsareindefinitelyincreased.Thislimitiscalledtheprobabilitythattheeventwillhappenundertheseconditions

  • GROUND RULES What is Probability Really?

  • GROUND RULES The distribution of inputs does not always equal the

    distribution of outputs Based on the Principle of Maximum Entropy, input

    distributions to be selected to maximize Entropy (Harr, 1987)

    Correlations cannot be ignored Beware of Procrustean Errors (rocks cant do math) Truncated Normal Distributions Normally Distributed

  • GROUND RULES PoF is not just varying strength parameters Mechanisms understood Important driving factors identified Important factors included in PoF

  • Probabilistic HB Example - Data

    0

    20

    40

    60

    80

    100

    120

    0.00 2.00 4.00 6.00 8.00 10.00 12.00

    MajorPrin

    cipa

    lStress(MPa)

    MinorPrincipalStress(MPa)

  • Probabilistic HB Example Best Fit Curve

    0

    20

    40

    60

    80

    100

    120

    0 2 4 6 8 10 12

    MajorPrin

    cipa

    lStress(M

    Pa)

    MinorPrincipalStress(MPa)

    ci=36MPami=8

  • Probabilistic HB Example Eyeball Percentiles

    0

    20

    40

    60

    80

    100

    120

    140

    0 2 4 6 8 10 12

    MajorPrin

    cipa

    lStress(MPa)

    MinorPrincipalStress(MPa)

    ci=65MPa,mi=10

    ci=50MPa,mi=9

    ci=36MPa,mi=8

    ci=20MPa,mi=7

    ci=7MPa,mi=6

  • Probabilistic HB Example What if Multi Stage Tests could Work?

    0

    20

    40

    60

    80

    100

    120

    0 2 4 6 8 10 12

    MajorPrin

    cipa

    lStress(M

    Pa)

    MinorPrincipalStress(MPa)

  • Probabilistic HB Example What if Multi Stage Tests could Work?

    0

    20

    40

    60

    80

    100

    120

    0 2 4 6 8 10 12

    MajorPrin

    cipa

    lStress(M

    Pa)

    MinorPrincipalStress(MPa)

  • Probabilistic HB Example Normal Distribution Fit

    20

    0

    20

    40

    60

    80

    100

    120

    140

    160

    180

    0 2 4 6 8 10 12

    MajorPrin

    cipa

    lStress(M

    Pa)

    MinorPrincipalStress(MPa)

    95%:ci=65MPa,mi=25

    75%:ci=45MPa,mi=16

    50%:ci=32MPa,mi=10

    25%:ci=19MPa,mi=3

    5%:ci=0.1MPa,mi=5

  • Probabilistic HB Example Lognormal Distribution Fit

    0

    20

    40

    60

    80

    100

    120

    140

    0 2 4 6 8 10 12

    MajorPrin

    cipa

    lStress(M

    Pa)

    MinorPrincipalStress(MPa)

    95%:ci=65MPa,mi=240

    75%:ci=50MPa,mi=20

    50%:ci=22MPa,mi=4

    25%:ci=10MPa,mi=0.85%:ci=3MPa,mi=0.1

  • 0

    20

    40

    60

    80

    100

    120

    140

    160

    180

    200

    0 2 4 6 8 10 12

    MajorPrin

    cipa

    lStress(M

    Pa)

    MinorPrincipalStress(MPa)

    Probabilistic HB Example Beta Distribution Fit (Satisfies Principle of Maximum Entropy)

    95%:ci=70MPa,mi=30

    75%:ci=44MPa,mi=13

    50%:ci=29MPa,mi=6

    25%:ci=17MPa,mi=35%:ci=7MPa,mi=2

  • Probabilistic HB Example mi Distributions

    0

    0.02

    0.04

    0.06

    0.08

    0.1

    0.12

    0.14

    0.16

    0 10 20 30 40 50

    Prob

    abilityDen

    sityFun

    ction

    mi

    Norm LogNorm Beta

  • Probabilistic HB Example Sigmaci Distributions

    0

    0.005

    0.01

    0.015

    0.02

    0.025

    0.03

    0.035

    0 20 40 60 80 100 120

    Prob

    abilityDen

    sityFun

    ction

    Sigmaci

    Norm LogNorm Beta

  • A Note on Correlation

    For single stage testing it is assumed that mi and Sigmaci is correlated with a coefficient of 1 (i.e. they are really the same variable)

    Using multistage testing the true correlation can be estimated For this example it is -0.2

  • A Note on Correlation

    0

    10

    20

    30

    40

    50

    60

    70

    0 5 10 15 20 25 30 35 40

    Sigm

    aci(MPa)

    mi

    CorrelationCoefficient=0.2

  • Conventional Multi-Stage Testing Pros and Cons

    For Samesampleusedatdifferentconfinements

    Moreefficientuseofsamples HBcurvefromsinglesample ProbabilisticAdvantagesaspresentedearlier

    Against Toomuchdamagebetweenstages

    Manualpickingofstagesimprecise

  • Servo Controlled Automated Triaxial Machine

    Completely automated testing Stage end picked based on Poissons ratio (0.4 suggested) More consistent stage transitions More precise stage yield points Results much better at simulating single stage testing

  • Servo Controlled Automated Triaxial Machine

  • Servo Controlled Automated Triaxial Machine

  • Servo Controlled Automated Triaxial Machine

  • Servo Controlled Automated Triaxial Machine

    0

    50

    100

    150

    200

    250

    300

    350

    400

    -2000 -1000 0 1000 2000 3000 4000 5000

    Axial

    Stre

    ss (M

    Pa)

    Strain - e

  • Servo Controlled Automated Triaxial Machine

  • Servo Controlled Automated Triaxial Machine

    0

    34

    68

    102

    136

    170

    204

    238

    272

    0 34 68 102 136 170 204 238 272 306

    Shea

    r Stre

    ss M

    Pa

    Normal Stress MPa

    0.4 Poisson's Ratio - Mohr Circle Plot12.01 MPa 24.00 MPa 35.99 MPa 48.00 MPa 59.99 MPa Envelope

  • Servo Controlled Automated Triaxial Machine

    0

    53

    106

    159

    212

    265

    318

    371

    424

    0 53 106 159 212 265 318 371 424 477

    Shea

    r Stre

    ss M

    Pa

    Normal Stress MPa

    Calculated Peak Stress Mohr Circle Plot12.01 MPa 24.00 MPa 35.99 MPa 48.00 MPa 59.99 MPa Envelope

  • Automated Servo Controlled Multi-Stage Testing Pros and Cons

    For Samesampleusedatdifferentconfinements

    Moreefficientuseofsamples HBcurvefromsinglesample ProbabilisticAdvantagesaspresentedearlier

    Limiteddamagebetweenstages Automatedpickingofstages

    Against Toomuchdamagebetweenstages

    Manualpickingofstagesimprecise

  • Core Log

  • Slope Design Parameters

    0

    0.005

    0.01

    0.015

    0.02

    0.025

    0.03

    0.035

    0.04

    0 20 40 60 80 100 120

    Prob

    abilityDen

    sityFun

    ction

    GSI

    Norm Beta

    Average=41StDev=12

  • Slope Design Parameters

    0

    0.5

    1

    1.5

    2

    2.5

    3

    3.5

    0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

    Prob

    abilityDen

    sityFun

    ction

    mb

    Norm Beta

  • Slope Design Parameters

    0

    500

    1000

    1500

    2000

    2500

    3000

    3500

    0 0.0005 0.001 0.0015 0.002 0.0025 0.003

    Prob

    abilityDen

    sityFun

    ction

    s

    Norm Beta

  • Slope Design Parameters

    0

    10

    20

    30

    40

    50

    60

    0.5 0.55 0.6 0.65 0.7

    Prob

    abilityDen

    sityFun

    ction

    a

    Norm Beta

  • Slope Design Parameters

    0

    0.0005

    0.001

    0.0015

    0.002

    0.0025

    0.003

    0.0035

    0.004

    0 100 200 300 400 500 600

    Prob

    abilityDen

    sityFun

    ction

    c(kPa)

    Norm Beta

    Average=164StDev=108

  • Slope Design Parameters

    0

    0.005

    0.01

    0.015

    0.02

    0.025

    0.03

    0.035

    0.04

    0 20 40 60 80 100

    Prob

    abilityDen

    sityFun

    ction

    phi(deg)

    Norm Beta

    Average=22StDev=12

  • Slope Design Parameters Correlation Coefficients

    mi Sci GSI mb s a c phimi 1 -0.20 -0.04 0.79 -0.01 0.12 0.26 0.51Sci 1 0.09 -0.09 0.01 -0.12 0.49 0.48GSI 1 0.42 0.75 -0.96 0.71 0.51mb 1 0.32 -0.36 0.53 0.67s 1 -0.57 0.68 0.32a 1 -0.61 -0.47c 1 0.88phi 1

  • SV Slope Results - Deterministic

    FoS =1.225

  • SV Slope Normal Distribution Inputs

    POF=36%

  • SV Slope Log Normal Distribution Inputs

    POF=45%

  • SV Slope Beta Distribution Inputs

    POF=33%

  • SV Slope Spatial Variability

  • SV Slope Spatial Variability

  • SV Slope 3D

  • SV Slope 3D

  • SV Slope 3D