Transcript Acceptability Criteria for Engineering Design

Shear Strength of Discontinuities
Slope 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
 p  c   n tan
Hoek, 2000
Shear Strength of Joints
 r   n tan b
Hoek, 2000
Shear Testing Machine
Hoek, 2000
Shear Testing Machine
Hoek, 2000
Shear Strength of Saw Tooth Specimen
Patton (1966)
Hoek, 2000
Shear Strength of Saw Tooth Specimen
Patton (1966)
   n tanb  i 
Hoek, 2000
Shear Strength of Discontinuities

 JCS  
 
   n tanb  JRC log10 


 n 

JRC = Joint Roughness Coefficient
JCS = 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 JCS
 Ln 
JRC n  JRC0  
 L0 
 Ln 
JCSn  JCS0  
 L0 
0.02 JRC0
0.03 JRC0
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
  

i  arctan 
  n 

 JRC  2 
 

JCS
JCS
 tan JRC log10
 b  
 b   1
tan  JRC log10
 n
n
n

 180ln10 

 
ci     n tan i
Barton Shear Strength Criterion
INPUT Parameters:
Basic friction angle, phib
Joint roughness coefficient, JRC
Joint compressive strength, JCS
Minimum normal stress, Sigman min
Shear
Strength,
Tau
MPa
0.99
1.54
2.48
4.07
6.78
11.34
18.97
31.53
dTau /
dSigma n
Friction
Angle,
phi'
Degrees
58.82
54.91
50.49
45.85
41.07
36.22
31.33
26.40
35
1.65
30
1.42
Shear Strength, Tau
Normal
Stress,
Sigma n
MPa
0.36
0.72
1.44
2.88
5.76
11.52
23.04
46.07
29
16.9
96
0.36
1.21
1.03
25
0.87
0.73
0.61
20
0.50
degrees
MPa
Barton Shear Strength Criterion
Cohesive
Strength,
c'
MPa
0.39
0.51
0.73
1.11
1.76
2.91
4.95
8.67
15
10
5
0
0
5
10
15
20
25
30
Normal Stress, Sigman
35
40
45
50
Shear Strength of Filled Discontinuities (Barton, 1974)
Rock
Description
Basalt
Bentonite
Bentonitic Shale
Clays
Clay Shale
Coal measure rocks
Dolomite
Diorite, granodiorite, porphyry
Granite
Greywacke
Limestone
Limestone, marl and lignites
Limestone
Lignite
Montmorillonite
Bentonite Clay
Schists, quartzites and
siliceous schists
Slates
Quartz / kaolin / pyrolusite
Peak
c' (MPa)
 (degrees)
0.24
42
Clayey basaltic breccia, wide variation from clay to
basalt content
Bentonite seam in chalk
0.015
Thin layers
0.09 to 0.12
Triaxial tests
0.06 to 0.1
Triaxial tests
0 to 0.27
Direct shear tests
Over-consolidated, slips, joints and minor shears
0 to 0.18
Triaxial tests
0.06
Stratification surfaces
Clay mylonite seams, 10 to 25 mm
0.012
Altered shale bed, +/- 150 mm thick
0.04
Clay gouge (2% clay, PI = 17%)
0
Clay filled faults
0 to 0.18
Sandy loam fault filling
0.05
Tectonic shear zone, schistose and broken
0.24
granites, disintegrated rock and gouge
1 - 2 mm clay in bedding planes
6 mm clay layer
10 - 20 mm clay fillings
0.1
< 1 mm clay filling
0.05 to 0.2
Interbedded lignite layers
0.08
Lignite / marl contact
0.1
Marlaceous joints, 20 mm thick
0
Layer between lignite and clay
0.014 to 0.03
80 mm seams of bentonite clay in chalk
0.36
0.016 to 0.02
100-150 mm thick clay filling
0.03 to 0.08
Stratification with thin clay
0.61 to 0.74
Stratification with thick clay
0.38
Finely laminated and altered
0.05
Remoulded triaxial tests
0.042 to 0.09
Residual
c' (MPa)
 (degrees)
7.5
12 to 17
9 to 13
8.5 to 29
12 to 18.5
32
16
14.5
26.5
24 to 45
40
42
13 to 14
17 to 21
38
10
25
15 to 17.5
14
7.5 to 11.5
32
41
31
33
36 to 38
0.03
0 to 0.003
8.5
10,5 to 16
0
0
0.02
19 to 25
11 to 11.5
17
0
0
21
13
0
15 to 24
0.08
11
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 Kong
Hoek, 2000
Sau Mau Ping Road, Hong Kong
Hoek, 2000
Hoek, 2000
Sau Mau Ping Road, Hong Kong
Hoek, 2000
Planar Failure
Hoek and Bray, 1981
Release surfaces
Conditions?
Failure Plane
Hoek and
Bray, 1981
For sliding: f > p > 
T otal force resisting sliding
Factor of Safety 
T otal force tending to induce sliding
cA  (W  cos p  U  V  sin p ) tan 
F
W  sin p  V  cos p
W  weight of sliding block (see next two pages)
c,   Mohr Coulombstrength factors for slidingsurface
 f  dip angle of slope face
 p  dip angle of slidingsurface
A  area  ( H  z )  cosec p
U  water pressure uplift force
 1  w  z w  ( H  z )  cosec p
2
V  water pressure in tension crack  1  w  z 2w
2
Tension Crack in Upper Surface
Hoek and
Bray, 1981
2
 z 
W  1 2  w H 2 [(1    ) cot p  cot f ]
H 
Tension Crack in Slope Face
Hoek and
Bray, 1981
W
1
2
[(1 

H
2 w
z 2
) cot p (cot p  tan f  1)]
H
Sensitivity Analysis
Hoek, 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
Hoek, 2000
Test data from Hencher and Richards, 1982
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
–  = 38, c = 12.5
FofS = 1.04 (used in Chapter 8)
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
–  = 56, c = 0
FofS = 1.22 (as noted by Hoek, 2000)
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:
1
x
n
n
x
i
i 1
• Example application is to analyze results
from laboratory uniaxial compression test
data.
Measures of Data Distribution
n

1
2
xi  x 
Sample Variance, s 
n  1 i 1
2
Standard Deviation, s  s2
s
Coefficien t of Variation, COV 
x
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.
2


1 x 
exp  
 
 2    
PDF (Normal Distribution) : f x x  
for -   x  
 2
Other Distributions
Other Statistical Distribution Functions
Beta
Very versatile
Exponential
Occurrence of earthquakes or rockbursts
Length of joints in a rockmass
Lognormal
Crushing of aggregates (multiplicative mechanism)
Weibul
Lifetime of devices in reliability tests
Point load tests on rock core (in which high values rarely
occur)
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 m
Overall slope angle, f = 50
Failure plane angle, p = 35
Unit 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, = 35
Cohesive strength of joint surface, c = 10 tonnes/m2
Depth of tension crack, z = 14 m
Depth of water in tension crack, zw = z/2
Horizontal earthquake:gravitational acceleration, a= 0.08
Factor of Safety Calculation
Hoek, 2000
Slope Stability Analysis
Sau Mau Ping Road: with a water filled tension crack above the slope crest
Fixed quantities:
Calculated quantities:
zcalc =
Overall slope height
H=
60
metres
14.0
metres
2
psif =
Overall slope angle
50
degrees
A=
80.2
m
psip =
Failure plane angle
35
degrees
W=
2392.4
tonnes
3
gammar =
Unit weight of rock
2.6
U=
561.6
tonnes
t/m
gammaw =
Unit weight of water
1
V=
98.1
tonnes
t/m3
Reinforcing force
T=
0
tonnes
Capacity =
1664.5
tonnes
Reinforcing angle
theta =
0
degrees
Demand =
1609.4
tonnes
Factor of Safety =
1.03
Probability of Failure Calculation
Hoek, 2000
Probability of Failure Calculation
Hoek, 2000

z  H 1  cot f tan p

(7.6)
zmax  H (1  tan p / tan f )
Probability of Failure Calculation
Hoek, 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.”