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Reliability-Based Robust Design of
Rock Slopes – A New Perspective
on Design Robustness
Hsein Juang
Glenn Professor
Glenn Department of Civil Engineering
Clemson University
2
ACKNOWLEDGMENTS
This study was supported by National Science Foundation through Grant CMMI-1200117 (“Transforming Robust Design Concept into a Novel Geotechnical Design Tool”). The results and opinions expressed in this paper do not necessarily reflect the views and policies of the National Science Foundation.
Outline of Presentation
• Introduction
• Framework for Reliability-based Robust
Geotechnical Design (RGD)
• Case History Overview: Sau Mau Ping rock slope
• Reliability-based RGD of Sau Mau Ping slope
• Reliability-based RGD of Rock Slope System
• Concluding Remarks
Introduction
The shear strength of discontinuities is often
difficult to measure. And budgetary constraints
for site investigation and field and laboratory
tests also limit the amount of data available to
an engineer.
The statistics of rock properties is difficult to
determine due to measurement error, small
sample size, transformation error, and spatial
variability.
Introduction
Traditional reliability-based rock slope designs
are often sensitive to variations in noise factors
such as rock shear properties.
Under- or over-estimation of the variation of
rock properties can lead to under- or over-
design with respect to target reliability.
Introduction
Address this dilemma by making design
insensitive to, or robust against, variation in
noise factors (such as rock shear properties)
Incorporate robustness explicitly in design
process
Introduction
Robust design focuses on achieving an optimal
design that is insensitive to the variation of
noise factors by carefully adjusting easy to
control design parameters of rock slope
Multi-objective optimization considering safety,
robustness, and cost is performed to obtain the
most optimal design or a Pareto Front (set of
optimal designs)
Reliability-based RGD Methodology
(1) Establish the deterministic computational model for
stability analysis of rock slope
• Identifying number of removable blocks based on a
proper characterization of rock mass structure
(2) Characterize uncertainty in variation of noise factors
and specify the design domain for rock slope
• “Easy to control” design parameters – slope height
and slope face angle; treated as discrete variables
in design space
• “Hard to control” noise factors – uncertain rock
properties
• Uncertainty of statistics of noise factors estimated
from limited data or published literatures
Reliability-based RGD Methodology
(3) Evaluate the variation of the failure probability for
robustness consideration
• Variation in the failure probability is evaluated using
Point estimate method (PEM) integrated with first-
order reliability method (FORM)
(4) Perform multi-objective optimization to establish a
Pareto front optimal for both robustness and cost
• Three criteria: safety, cost and robustness
• Cost approximated as volume of rock mass that
must be excavated
• Optimization performed using a fast elitist Non-
dominated sorting genetic algorithm (NSGA-II)
Multi-objective optimization
When conflicting objectives are enforced, it is likely that
no single best design exists.
However, a set of designs may exist that are superior to
all other designs in all objectives; but within the set, none
of them is superior or inferior to others in all objectives.
This set of optimal designs constitutes a Pareto Front.
Any solution design on the Pareto Front cannot be
improved in any one objective without worsening at least
one other objective.
The Pareto Front is established using NSGA-II through
Non-dominated sorting and crowding distance sorting.
Reliability-based RGD Methodology
(5) Determine feasibility robustness for each design on the
Pareto Front
• The feasibility robustness is the probability that a
design can remain “feasible” (acceptable in terms of
satisfying the safety requirements) even when the
system undergoes variations.
• Symbolically, this probability (and thus the feasibility
robustness index) is computed as:
Pr[( ) 0] ( )f Tp p
Case History: Sau Mau Ping slope
http://www.rocscience.com/hoek/
• Rock mass composed of
unweathered granite with
sheet joints
• Sheet joints formed by
exfoliation process during
cooling of granite
• Hoek (2006) simplified the
slope as a single unstable
block with a water-filled
tension crack with a single
plane failure mode.
Case History: Sau Mau Ping slope
• Deterministic limit equilibrium model for plane
failure developed by Hoek and Bray (1981)
• This is a 2-D analysis and dimensions refer to
a 1 metre thick slice through the slope
• Factor of safety (FS) is defined as the ratio of
the forces resisting sliding to the forces tending
to induce sliding along the slip surface:
[ (cos sin ) sin ]tan
(sin cos ) cos
cA W U VFS
W V
Case History: Sau Mau Ping slope
• Initial condition before remediation:
• H = 60 m , θ= 50º, ψ= 35º , γ = 2.6 ton/m3
αW
W
U
V
H
θ
ψ
z
wz
Water pressure
distribution
Tension crack
Failure surface
Reliability-based Design
• Hoek (2006): Five random variables considered
• Cohesion of rock discontinuities, c
• Friction angle of rock discontinuities, φ
• Tension crack depth, z
• Percentage of tension crack filled with water, iw
• Gravitational acceleration coefficient, α
Random variables
Probability distribution
Mean Std. dev.
c Normal 10 ton/m2 2 ton/m2
Normal o35 o5
z Normal 14 m 3 m
iw Exponential with mean 0.5, truncated to [0, 1] Exponential with mean 0.08, truncated to [0, 0.16]
Reliability-based Design
• The design space should be specified before
performing the reliability-based design
• For construction practicality, these parameters
are modeled as discrete variables in the design
space.
H {50m, 50.2m, 50.4m,…, 60m }
θ {44º, 44.2º, 44.6º,…, 50º} Totally 1581 pairs of H and θ
Reliability-based Design
• FORM procedure to calculate pf
1.E-06
1.E-05
1.E-04
1.E-03
1.E-02
1.E-01
44 45 46 47 48 49 50
Slope Angle, θ (°)
Pro
bab
ilit
y o
f F
ailu
re
H = 50 m H = 52 m
H = 54 m H = 56 m
H = 58 m H = 60 m
0.00621Tp
Reliability-based Design
• The reliability-based design is performed by
minimizing the cost while satisfying the reliability
constraint (pf<pT= 0.0062).
• Target reliability index 2.5 (Low 2008)
• The least cost design yields H = 54.8 m and
θ = 50º with a design cost of 91.1 units
Reliability-based Design
• For Sau Mau Ping slope, shear properties were
estimated based on published information for
similar rocks (Hoek 2006).
• The COV and ρ may vary, reported ranges from
literatures (Lee et al. 2012, Low 2008),
10%<COV[c]<30%
10%<COV[φ]<20%
-0.2< ρc,φ <-0.8
-0.2< ρz,iw <-0.8
What will happen with uncertain COV and ρ?
Reliability-based Design with different assumed COVs
[ ]COV c [ ]COV H (m) (°) Cost (units)
0.10 0.10 57.6 50.0 39.3
0.10 0.14 56.0 50.0 68.1
0.10 0.20 52.8 50.0 132.0
0.20 0.10 55.8 50.0 71.8
0.20 0.14 54.8 50.0 91.1
0.20 0.20 52.2 50.0 145.0
0.30 0.10 51.6 50.0 158.3
0.30 0.14 51.0 50.0 171.8
0.30 0.20 50.0 49.2 225.2
Reliability-based Design
• The optimal design obtained from traditional
reliability-based design method is sensitive to the
assumed COVs of rock properties
• Under the lowest uncertainty level of rock
properties, the least-cost design costs 39.3 units
• Under the highest uncertainty level, the least-cost
design costs 225.2 units
• Cost becomes much higher when uncertainty
in rock properties increases
Reliability-based Design
[ ]COV c [ ]COV H (m) (°) fp
0.10 0.10 54.8 50.0 8.68E-04
0.10 0.14 54.8 50.0 3.23E-03
0.10 0.20 54.8 50.0 1.23E-02
0.20 0.10 54.8 50.0 3.70E-03
0.20 0.14 54.8 50.0 6.04E-03
0.20 0.20 54.8 50.0 1.36E-02
0.30 0.10 54.8 50.0 1.33E-02
0.30 0.14 54.8 50.0 1.52E-02
0.30 0.20 54.8 50.0 2.22E-02
The optimal design (H = 54.8 m and θ = 50º ) may no
longer be satisfactory if COVs are underestimated
Reliability-based Design
• Statistical characterization carries its own
uncertainty:
• Even at highest level of uncertainty, an
acceptable design (for example, H = 50 m and
θ = 50º can be selected that meet pT
• Thus some designs, although at a higher cost,
can be chosen to ensure robustness against
variation in rock slopes.
Reliability-based RGD Design
• Statistics of rock properties are difficult to
ascertain. In robust design of rock slope, the
effect of uncertain statistics are explicitly
considered.
• Therefore COV[c], COV[φ], ρc,φ , treated as RVs
• In addition ρz,iw treated as RV.
• Based on their typical ranges,
μCOV[c]= 0.20; δCOV[c]= 0.17
μCOV[φ]= 0.14; δCOV[φ]= 0.12
μρc,φ= -0.50; δρc,φ = 0.25
μρz,iw= -0.5; δρz,iw= 0.25
Reliability-based RGD Design
• Mean and std. dev. of the failure probability can be
obtained with PEM integrated with FORM procedure
1.E-06
1.E-05
1.E-04
1.E-03
1.E-02
1.E-01
44 45 46 47 48 49 50
Slope Angle, θ (°)
Mea
n P
rob
abil
ity
of
Fai
lure
H = 50 m H = 52 m
H = 54 m H = 56 m
H = 58 m H = 60 m
0.00621Tp
Reliability-based RGD Design
• Mean and std. dev. of the failure probability can be
obtained with PEM integrated with FORM procedure
1.E-06
1.E-05
1.E-04
1.E-03
1.E-02
1.E-01
44 45 46 47 48 49 50
Slope Angle, θ (°)
Std
. D
ev.
of
Pro
bab
ilit
y o
f F
ailu
re
i
H = 50 m H = 52 m
H = 54 m H = 56 m
H = 58 m H = 60 m
Reliability-based RGD Design
• A multi-objective optimization is set up as follows:
Find d = [H, ]
Subjected to: H {50m, 50.2m, 50.4m,…, 60m }
{44°, 44.2°, 44.6°,…, 50°}
0.0062p Tp
Objectives: Minimizing the standard deviation failure probability ( p )
Minimizing the cost for rock slope design.
0
100
200
300
400
500
1.E-04 1.E-03 1.E-02
Std. Dev. of Probability of Failure
Cost
(unit
s)
Acceptable designswith cost < 200 units
o
Optimal Design H=50 m, θ = 50
Pareto Front in a bi-objective space
Reliability-based Robust Geotech Design
• Designs are optimized to both cost and robustness
• Safety is guaranteed by a reliability constraint
• 89 designs are selected into Pareto Front
• An apparent trade-off relationship between cost and
robustness exists for designs on Pareto Front
• If the maximum acceptable cost for the designer, for
example, is 200 units, then the design with the
smallest σp while in the acceptable cost range will
be the best design (H = 50 m and θ = 50º).
• Feasibility robustness can aid in making decision
• When a target feasibility robustness level is
selected, the least cost design is readily identified
Selected final designs at various
feasibility robustness levels
T
0P H (m) (°) Cost (units)
0.5 69.15% 54.0 50.0 107.0
1.0 84.13% 52.8 50.0 132.0
1.5 93.32% 51.4 49.8 170.6
2.0 97.72% 50.4 48.4 247.6
2.5 99.38% 50.0 45.4 378.9
Rock slope with multiple failure modes
RGD of Rock Slope System
Rock slope is composed of two blocks separated by a
vertical tension crack; location of tension crack is random,
either at the slope top or slope face
H
fψ pψ
zz H
hh H
FI AB
A
B
ww zz h
AA XX XBB XX X
cot pX H
(a)
H
fψ pψ
zz H
hh HFI AB
A
B
AA XX XBB XX X
cot pX H
ww zz h
(b)
H
fψ pψ
zz H
hh H
FI AB
A
B
ww zz h
AA XX XBB XX X
cot pX H
(a)
H
fψ pψ
zz H
hh HFI AB
A
B
AA XX XBB XX X
cot pX H
ww zz h
(b)
System reliability approach (Jimenez-Rodriguez et al. 2006)
Disjoint cut-set formulation for system reliability evaluation
1 0g 1 0g 1 0g
1 0g
3 0g 3 0g 2 0g
2 0g
7 0g 6 0g 5 0g 4 0g
Failure Mode 1 Failure Mode 2 Failure Mode 3 Failure Mode 4
1
( )C
k
N
ii C
k
P E P E
For a given cut set, ( , ) β R
kk
i Ci C
P E
Four failure modes depending on interactions between
two blocks and location of tensile crack
RGD of Rock Slope System
• Seven random variables considered
• Cohesion and friction angle of rock discontinuities
• Location of tension crack
• Percentage of tension crack filled with water
Random
variables
Probability
distribution Mean Std. dev.
Ac (kPa) Normal 20 4
Bc (kPa) Normal 18 4
A (°) Normal 36 3.015
B (°) Normal 32 3.015
AB (°) Normal 30 3.015
BX Beta distribution, q = 3, r = 4, a = 0, b = 1
wi Exponential with mean 0.25,
truncated to [0, 0.5]
RGD of Rock Slope System
• In robust design of rock slope system, seven
statistics of rock properties are considered
random variables
• COV[cA], COV[cB], COV[φA], COV[φB], COV[φAB],
ρcA,φB , ρcA,φB,
• Based on their typical ranges,
δCOV[cA]= δCOV[cB] = 0.17
δCOV[φA]=δCOV[φB] = δCOV[φAB]= 0.12
δρcA,φA= δρcB,φB= 0.25
• Mean and std. dev. of the failure probability can be
obtained with PEM integrated with FORM procedure
0.000
0.005
0.010
0.015
0.020
40 42 44 46 48 50
Slope Angle, θ (°)
Mea
n P
rob
abil
ity
of
Fai
lure H = 20 m H = 21 m
H = 22 m H = 23 m
H = 24 m H = 25 m
0.00621Tp
RGD of Rock Slope System
• Mean and std. dev. of failure probability can be
obtained with PEM integrated with FORM procedure
0.000
0.001
0.002
0.003
0.004
0.005
40 42 44 46 48 50
Slope Angle, θ (°)
Std
. D
ev.
of
Pro
bab
ilit
y o
f F
ailu
re
i
H = 20 m H = 21 m
H = 22 m H = 23 m
H = 24 m H = 25 m
RGD of Rock Slope System
• A multi-objective optimization is set up as follows:
Find d = [H, ]
Subjected to: H {20m, 20.2m, 20.4m,…, 25m }
{40°, 40.2°, 40.6°,…, 50° }
0.0062p Tp
Objectives: Minimizing the standard deviation failure probability ( p )
Minimizing the cost for rock slope design.
RGD of Rock Slope System
60 designs are selected onto Pareto Front
Summary and Concluding Remarks
A Robust Geotechnical Design methodology is
presented to achieve optimal designs that are
robust against the variation in noise factors (rock
properties).
Without considering design robustness, the
traditional reliability-based design method may
produce designs that are unsatisfactory due to
underestimation of variation in noise factors.
Summary and Concluding Remarks
Multi-objective optimization is used to identify
designs optimal to cost and robustness while
satisfying safety constraint.
Results from multi-objective optimization are
usually presented in a Pareto Front.
A trade-off relationship between cost and
robustness exists for designs on Pareto Front
[greater design robustness can only be attained
at the expense of a higher cost].
Summary and Concluding Remarks
The feasibility robustness provides an easy-to-
use quantitative measure for selecting the best
design from the Pareto Front.
The significance of the proposed RGD is
demonstrated with two design example of rock
slopes.
Related Papers
Wang, L., Hwang, J.H., and Juang, C.H., and Sez Atamturktur,
“Reliability-based design of rock slopes – A new perspective on
design robustness,” Engineering Geology, Vol. 154, 2013, pp. 56-63.
Xu, C., Wang, L. Tien, Y.M., Chen, J.M., Juang, C.H., “Robust design
of rock slopes with multiple failure modes - Modeling uncertainty of
estimated parameter statistics with fuzzy number,” Environmental
Earth Sciences, Springer, October 2014, Volume 72, Issue 8, pp
2957-2969. DOI 10.1007/s12665-014-3201-1.
Wang, L., Gong, W., Luo, Z., Khoshnevisan, S., and Juang, C.H.,
“Reliability-based robust geotechnical design of rock bolts for slope
stabilization,” Geotechnical Special Publication No. 256, ASCE,
Proceedings of IFCEE 2015, pp. 1926-1935.