Shear Strength of DiscontinuitiesSlope Stability Analysis

Lecture 5

Earth 691B: Rock Engineering

Outline

• Shear Strength of Discontinuities (Chapter 4, Hoek, 2000)

– Testing– Field Estimates

• Sau Mau Ping Slope Example(Chapters 7 and 8, Hoek, 2000)

– Assumed shear strength values– Test results

– Accounting for data variability

Shear Strength of Joints

Hoek, 2000

Shear Strength of Joints

Hoek, 2000

Shear Testing Machine

Hoek, 2000

Shear Testing Machine

Hoek, 2000

Shear Strength of Saw Tooth SpecimenPatton (1966)

Hoek, 2000

Shear Strength of Saw Tooth SpecimenPatton (1966)

Hoek, 2000

Shear Strength of Discontinuities

JRC = Joint Roughness CoefficientJCS = Joint Wall Compressive Strength

Hoek, 2000: after Barton and Choubey, 1977)

Hoek, 2000: Barton, 1982

Hoek, 2000: Deere and Miller, 1966

Influence of Scale on JRC and JCS002.0

00

JRC

nn L

LJRCJRC

003.0

00

JRC

nn L

LJCSJCS

Where:JRC0, JCS0 and L0 refer to 100 mm lab scale specimens

JRCn, JCSn and Ln refer to insitu block sizes

Instantaneous Cohesion and Friction

Hoek, 2000

Instantaneous Cohesion and Friction

0

5

10

15

20

25

30

35

0 5 10 15 20 25 30 35 40 45 50

Normal Stress, Sigman

Shea

r Str

engt

h, T

au

29 degrees16.996

0.36 MPa

Normal Stress,

Sigma n

Shear Strength,

Tau

Friction Angle,

phi'

Cohesive Strength,

c'MPa MPa Degrees MPa0.36 0.99 1.65 58.82 0.390.72 1.54 1.42 54.91 0.511.44 2.48 1.21 50.49 0.732.88 4.07 1.03 45.85 1.115.76 6.78 0.87 41.07 1.76

11.52 11.34 0.73 36.22 2.9123.04 18.97 0.61 31.33 4.9546.07 31.53 0.50 26.40 8.67

INPUT Parameters:

dTau / dSigma n

Barton Shear Strength Criterion

Basic friction angle, phib

Joint roughness coefficient, JRCJoint compressive strength, JCSMinimum normal stress, Sigman min

c' (MPa) (degrees) c' (MPa) (degrees)Basalt Clayey basaltic breccia, wide variation from clay to

basalt content0.24 42

Bentonite seam in chalk 0.015 7.5Thin layers 0.09 to 0.12 12 to 17Triaxial tests 0.06 to 0.1 9 to 13Triaxial tests 0 to 0.27 8.5 to 29Direct shear tests 0.03 8.5

Clays Over-consolidated, slips, joints and minor shears 0 to 0.18 12 to 18.5 0 to 0.003 10,5 to 16Triaxial tests 0.06 32Stratification surfaces 0 19 to 25

Coal measure rocks Clay mylonite seams, 10 to 25 mm 0.012 16 0 11 to 11.5Dolomite Altered shale bed, +/- 150 mm thick 0.04 14.5 0.02 17Diorite, granodiorite, porphyry Clay gouge (2% clay, PI = 17%) 0 26.5

Clay filled faults 0 to 0.18 24 to 45Sandy loam fault filling 0.05 40Tectonic shear zone, schistose and broken granites, disintegrated rock and gouge

0.24 42

Greywacke 1 - 2 mm clay in bedding planes 0 216 mm clay layer 0 1310 - 20 mm clay fillings 0.1 13 to 14< 1 mm clay filling 0.05 to 0.2 17 to 21Interbedded lignite layers 0.08 38Lignite / marl contact 0.1 10

Limestone Marlaceous joints, 20 mm thick 0 25 0 15 to 24Lignite Layer between lignite and clay 0.014 to 0.03 15 to 17.5Montmorillonite 0.36 14 0.08 11Bentonite Clay 0.016 to 0.02 7.5 to 11.5

100-150 mm thick clay filling 0.03 to 0.08 32Stratification with thin clay 0.61 to 0.74 41Stratification with thick clay 0.38 31

Slates Finely laminated and altered 0.05 33Quartz / kaolin / pyrolusite Remoulded triaxial tests 0.042 to 0.09 36 to 38

Shear Strength of Filled Discontinuities (Barton, 1974)

80 mm seams of bentonite clay in chalk

Peak Residual

Bentonite

Bentonitic Shale

Schists, quartzites and siliceous schists

Rock Description

Clay Shale

Granite

Limestone

Limestone, marl and lignites

Slope Stability Calculations

• Preliminary estimate of joint strength with sensitivity analysis of Factor of Safety (Recall from Lecture 2)

• Joint strength assessment from laboratory testing with Factor of Safety Calculation

• Using @Risk to include consideration of material variability

Sau Mau Ping Road, Hong KongHoek, 2000

Sau Mau Ping Road, Hong Kong

Hoek, 2000

Hoek, 2000

Hoek, 2000

Sau Mau Ping Road, Hong Kong

Planar Failure

Hoek and Bray, 1981

Conditions?Release surfaces

Failure Plane

For sliding: f > p >

Hoek and Bray, 1981

Tension Crack in Upper Surface

]cotcot)1[( 2

22

1 fpw H

zHW

Hoek and Bray, 1981

Tension Crack in Slope Face

)]1tan(cotcot)1[( 222

1 fppw H

zHW

Hoek and Bray, 1981

Sensitivity AnalysisHoek, 2000

Slope Stability Calculations

• Preliminary estimate of joint strength with sensitivity analysis of Factor of Safety (Recall from Lecture 2)

• Joint strength assessment from laboratory testing with Factor of Safety Calculation

• Using @Risk to include consideration of material variability

Testing Results

Test data from Hencher and Richards, 1982

Hoek, 2000

Factor of Safety Against Sliding• 14 foot tension crack half filled with water

• Earthquake acceleration = 0.08g

• Preliminary Field Estimate based on general published rock shear strength data: = 35, c = 10 FofS = 1.04 (used in Chapter 8) = 38, c = 12.5 FofS = 1.2 (interpreted from

graph on previous page)

• Based on re-analysis with shear strength data from Hencher and Richards (1982): = 48, c = 0 FofS = 1.22 (as noted by Hoek, 2000) = 56, c = 0 FofS = 1.63

Slope Stability Calculations

• Preliminary estimate of joint strength with sensitivity analysis of Factor of Safety (Recall from Lecture 2)

• Joint strength assessment from laboratory testing with Factor of Safety Calculation

• Using @Risk to include consideration of material variability

Probabilistic Analysis• Mathematical method for inclusion

of uncertainty and variability in deterministic slope stability analysis

Hoek, 2000

Definitions

• Probability Density Function

• Cummulative Distribution Function

• Sample Mean

• Probability Distributions

• Sampling Techniques

Probability Density Function

Hoek, 2000

Cumulative Distribution Function

Hoek, 2000

Sample Mean

• Assuming that there are n individual test values xi , the mean x is given by:

• Example application is to analyze results from laboratory uniaxial compression test data.

Measures of Data Distribution

Normal Distribution

• Most common type of distribution.

• Generally used for probabilistic studies in geotechnical engineering, unless there are good reasons for selecting a different distribution.

• Generally the best estimates for the true mean, , and the true standard deviation, , are given by the sample mean and standard deviation: = x and = s.

x -for 2

21

exp

f :ion) Distribut(Normal PDF

2

x

x

x

Other Distributions

Occurrence of earthquakes or rockburstsLength of joints in a rockmass

Lifetime of devices in reliability testsPoint load tests on rock core (in which high values rarely occur)

Weibul

Other Statistical Distribution Functions

Very versatile

Crushing of aggregates (multiplicative mechanism)

Beta

Exponential

Lognormal

Sampling Techniques

Analysis involves sampling a distribution function.

• Monte Carlo: uses random numbers to sample distributions. If sufficient numbers used, generate a distribution of values for the end product (i.e. factor of safety calculation).

• Latin Hypercube: stratified sampling with random selection within each stratum. Comparable answers to Monte Carlo with fewer samples.

Sampling Techniques

• Generalized Point Estimate method: – Two point estimates are made at one standard deviation

on either side of the mean ( ) from each distribution representing a random variable.

– The factor of safety is calculated for every possible combination of point estimates, producing 2 n solutions, where n is the number of random variables involved.

– The mean and the standard deviation of the factor of safety are then calculated from these 2 n solutions.

Probabilistic Analysis of the Sau Mau Ping Slope

1. Fixed dimensions:Overall slope height, H = 60 mOverall slope angle, f = 5Failure plane angle, p = 3Unit weight of rock, r = 2.6 tonnes/m3

Unit weight of water, w = 1.0 tonnes/m3

2. Mean values of Random variables:Friction angle on joint surface, = 3Cohesive strength of joint surface, c = 10 tonnes/m2

Depth of tension crack, z = 14 mDepth of water in tension crack, zw = z/2Horizontal earthquake:gravitational acceleration, = 0.08

Factor of Safety Calculation

Hoek, 2000

Overall slope height H = 60 metres zcalc = 14.0 metres

Overall slope angle psif = 50 degrees A = 80.2 m2

Failure plane angle psip = 35 degrees W = 2392.4 tonnes

Unit weight of rock gammar = 2.6 t/m3 U = 561.6 tonnes

Unit weight of water gammaw = 1 t/m3 V = 98.1 tonnesReinforcing force T = 0 tonnes Capacity = 1664.5 tonnesReinforcing angle theta = 0 degrees Demand = 1609.4 tonnes

Factor of Safety = 1.03

Slope Stability AnalysisSau Mau Ping Road: with a water filled tension crack above the slope crest

Fixed quantities: Calculated quantities:

Probability of Failure CalculationHoek, 2000

Probability of Failure Calculation

(7.6) tancot1 pfHz

)tan/tan1(max fpHz

Hoek, 2000

Probability of Failure CalculationHoek, 2000

Probabilistic Data Analysis

“For many applications it is not necessary to use all of the information contained in a distribution function and quantities summarised only by the dominant features of the distribution may be adequate.”