Probabilistic Estimates of Hoek Brown Intact Parameters...

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PROBABILISTIC ESTIMATES OF INTACT HOEK BROWN PARAMETERS

Transcript of Probabilistic Estimates of Hoek Brown Intact Parameters...

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 (Hoek’s Rock Engineering Notes)

But for intact rock s = 1 and a = 0.5

After multiplying and rearranging the equation can be rewritten as:

Note the last term is squared, there is a typo in Hoek’s NotesSubstituting  and X = gives:

Which is the familiar equation for a straight line

Mathematical Fun with Hoek-BrownTwo Conclusions:Linear Regression in Excel can be used if  is plotted against  (or Y against X)

And the slope of the fitted line is:

And 

GROUND RULES – What is Probability

• Epistemic – Uncertainty• Aleatory – Randomness• Degrees of Belief • Bayesian

GROUND RULES – Useful Definition (Grant et al)

• ‘Assume that if a large number of trials are made under the same essential conditions, the ratio of trials in which a certain event happens to the total number of trials will approach a limit as the total number of trials are indefinitely increased. This limit is called the probability that the event will happen under these conditions’

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 can’t 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

Major Prin

cipa

l Stress (MPa)

Minor Principal Stress (MPa)

Probabilistic HB Example – Best Fit Curve

0

20

40

60

80

100

120

0 2 4 6 8 10 12

Major Prin

cipa

l Stress (M

Pa)

Minor Principal Stress (MPa)

σci=36 MPami = 8

Probabilistic HB Example – Eyeball Percentiles

0

20

40

60

80

100

120

140

0 2 4 6 8 10 12

Major Prin

cipa

l Stress (MPa)

Minor Principal Stress (MPa)

σci= 65 MPa, mi = 10

σci= 50 MPa, mi = 9

σci= 36 MPa, mi = 8

σci= 20 MPa, mi = 7

σci= 7 MPa, 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

Major Prin

cipa

l Stress (M

Pa)

Minor Principal Stress (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

Major Prin

cipa

l Stress (M

Pa)

Minor Principal Stress (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

Major Prin

cipa

l Stress (M

Pa)

Minor Principal Stress (MPa)

95%: σci= 65 MPa, mi = 25

75%: σci= 45 MPa, mi = 16

50%: σci= 32 MPa, mi = 10

25%: σci= 19 MPa, mi = 3

5%: σci= 0.1 MPa, mi = ‐5

Probabilistic HB Example – Lognormal Distribution Fit

0

20

40

60

80

100

120

140

0 2 4 6 8 10 12

Major Prin

cipa

l Stress (M

Pa)

Minor Principal Stress (MPa)

95%: σci= 65 MPa, mi = 240

75%: σci= 50 MPa, mi = 20

50%: σci= 22 MPa, mi = 4

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

0

20

40

60

80

100

120

140

160

180

200

0 2 4 6 8 10 12

Major Prin

cipa

l Stress (M

Pa)

Minor Principal Stress (MPa)

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

95%: σci= 70 MPa, mi = 30

75%: σci= 44 MPa, mi = 13

50%: σci= 29 MPa, mi = 6

25%: σci= 17 MPa, mi = 35%: σci= 7 MPa, 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

ability Den

sity Fun

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

ability Den

sity Fun

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

Correlation Coefficient = ‐0.2

Conventional Multi-Stage Testing Pros and Cons

For• Same sample used at different confinements

• More efficient use of samples• HB curve from single sample• Probabilistic Advantages as presented earlier

Against• Too much damage between stages

• Manual picking of stages imprecise

Servo Controlled Automated Triaxial Machine

• Completely automated testing• Stage end picked based on Poisson’s 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 - µe

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• Same sample used at different confinements

• More efficient use of samples• HB curve from single sample• Probabilistic Advantages as presented earlier

• Limited damage between stages• Automated picking of stages

Against• Too much damage between stages

• Manual picking of stages imprecise

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

ability Den

sity Fun

ction

GSI

Norm Beta

Average = 41St Dev = 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

ability Den

sity Fun

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

ability Den

sity Fun

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

ability Den

sity Fun

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

ability Den

sity Fun

ction

c (kPa)

Norm Beta

Average = 164St Dev = 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

ability Den

sity Fun

ction

phi (deg)

Norm Beta

Average = 22St Dev = 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

SV Slope – 3D

SV Slope – 3D

SV Slope – 3D

References

• Grant, E. L., Ireson, W. G., & Leavenworth, R. S. (1990). Principles of Engineering Economy (éd. 8th). Toronto: JohnWiley and Sons.

• Hoek E., 2007. Rock Engineering Course Notes. www.Rocscience.com• Hoek E., Carranza‐Torres C., Corkum B., 2002. Hoek‐Brown Failure Criterion 2002 Edition. www.Rocscience.com

• Harr M.E., 1987. Reliability‐Based Design in Civil Engineering