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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)

Framework for

Reliability-based

Robust

Geotechnical

Design

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


(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)

Illustration of Pareto Front (with conflicting

objectives)

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.


(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 (秀茂坪)

rock slope design

http://www.rocscience.com/hoek/

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.


• 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


• 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]


• 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


• 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


• 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


• 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


• 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


[ ]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


• 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


• 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


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




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


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


• 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).

Cost versus feasibility robustness

• 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


• 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]


• 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



0.000

0.005

0.010

0.015

0.020

40 42 44 46 48 50


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


• Mean and std. dev. of failure probability can be


0.000

0.001

0.002

0.003

0.004

0.005

40 42 44 46 48 50


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


• 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.


60 designs are selected onto Pareto Front

Pareto Front in a bi-objective space

Cost versus feasibility robustness

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.


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].


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.

Thank You !

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