Robust Geotechnical Design

- Methodology and Applications

Outline of Presentation

1. Introduction

2. Robust Geotechnical Design of

Drilled Shafts in Sand

3. Robust Geotechnical Design of

Shallow Foundations in Sand

4. Robust Geotechnical Design of

Braced Excavations in Clay

5. Concluding Remarks

2

1. Introduction

3

Introduction

Uncertainty in geotechnical engineering is

unavoidable and quite significant due to the

nature of geological material, measurement

error, empirical model uncertainties and so on.

Reliability-based design is a method to explicitly

consider these uncertainty, however,

quantification of uncertainties in soil parameters

and geotechnical models is a prerequisite for a

reliability-based design.

4

Introduction

Due to the budget constraint, only limited site

investigation can be conducted at site of

concern.

Difficulty in estimating statistics of soil

properties with limited data in practice.

Under or over estimation of variation of soil

properties leads to under or over design.

5

What is Robust Design?

Robust Design, originated in the field of Industrial

Engineering by Taguchi (1986), aims to make the

product of a design insensitive to (or robust

against) “hard-to-control” input parameters (called

“noise factors”) by adjusting “easy-to-control”

input parameters (called “design parameters”).

The essence of this design approach is to

consider robustness explicitly in the design

process along with safety and economic

requirements. 6

Robust Design Concept

7

Robust Geotechnical Design

(RGD)

Focuses on achieving an optimal design that is insensitive to, or robust against, variation in uncertain soil parameters (noise factors) by carefully adjusting design parameters.

Seeks the most preferred design by considering safety, robustness, and cost in a multi-objective optimization.

8

Key concepts in RGD

A) Design parameters & Noise factors

B) Measures of design robustness

C) Optimization and Pareto front

9

A) Design parameters & Noise factors (Using excavation in clay as an example)

Design parameters:

Wall length (L), Wall

thickness (t), Vertical

spacing of the struts (S),

Strut stiffness (EA)

10

GL -2 m-1 m

-7 mGL -8 m

GL -4 m-3 m

GL -6 m-5 m

GL -10 m

Clay

Clay

Noise factors: undrained

shear strength ( ),

horizontal subgrade reaction

( ), and surcharge

behind the wall (qs) h vk

B) Measures of design robustness

Variation of system response (in terms of

factor of safety, failure probability,

deformation)

Signal to Noise Ratio (SNR)

Feasibility Robustness

11

Find d to optimize: [C(d), R(d,z)]

Subject to: gi(d,z) ≤ 0, i = 1,..,n

C-cost;

R-robustness measure;

d-design parameters;

z-noise factors;

g-constraint functions (safety).

12 Multi-objective optimization

Cost estimation

Reliability analysis

Robust design

C) Robust design optimization

Illustration of Pareto front in 2-D

13

Optimization may not yield a single best design with respect to all objectives.

Rather, a set of optimal designs may be obtained that are “non-dominated” by any other designs. This set of optimal designs collectively forms a Pareto front.

Pt

Qt

Pt+1

F2

F1

F3

Rt

Rejected

Non-dominated

sorting Crowding

distance

sorting

Non-dominated Sorting Genetic Algorithm

14

2. Robust Geotechnical Design

of Drilled Shafts in Sand

15

ULS: Kulhawy 1991

Noise Factors:

COV of ’ = 7%

COV of K0 = 50%

Correlation = 0.75

B {0.9m, 1.2m, 1.5m}

D {2.0m, 2.2m, … , 8.0m}

SLS: Normalized load-

settlement curve

ULS side tipQ Q Q W

0.625

b

aSLS ULS

yQ a Q

B

0.0047SLS

Tp (Critical)

Design Example

D

B

0

3

o

0

50

γ 20kN/m

μ 32

μ 1.0

(K/K ) 1.0

F 800kN

y 25mm

K

n

a

RGD of Drilled Shafts in Sand

1.E-10

1.E-08

1.E-06

1.E-04

1.E-02

1.E+00

2 3 4 5 6 7 8

Depth of Drilled Shaft, D (m)

Pro

bab

ilit

y o

f S

LS

Fai

lure

B=0.9m

B=1.2m

B=1.5m

(a)

0.0047SLS

Tp

1.E-10

1.E-08

1.E-06

1.E-04

1.E-02

1.E+00

2 3 4 5 6 7 8

Depth of Drilled Shaft, D (m)

Pro

bab

ilit

y o

f U

LS

Fai

lure

B=0.9m

B=1.2m

B=1.5m

(b)

0.00069ULS

Tp

Least Cost Design

B = 0.9 m and D = 5.6 m

This design may no longer

meet reliability requirement

if COVs are underestimated

[ ]COV 0[ ]COV K SLS

fp

0.05 0.2 2.55E-04

0.05 0.5 3.23E-05

0.05 0.9 5.98E-05

0.07 0.2 1.06E-02

0.07 0.5 4.02E-03

0.07 0.9 4.16E-03

0.1 0.2 6.51E-02

0.1 0.5 4.79E-02

0.1 0.9 4.60E-02

17

Traditional Reliability-Based Design

[ ]COV 0[ ]COV K 0,K B (m) D (m) Cost (USD)

SLS failure

probability SLS

fp

0.05 0.2 -0.6 0.9 5.4 1395 0.00188

0.05 0.5 -0.6 0.9 5.2 1343 0.00356

0.05 0.9 -0.6 1.2 3.6 1392 0.00395

0.07 0.2 -0.75 0.9 6.0 1550 0.00232

0.07 0.5 -0.75 0.9 5.6 1447 0.00402

0.07 0.9 -0.75 0.9 5.6 1447 0.00416

0.1 0.2 -0.9 0.9 6.8 1757 0.00328

0.1 0.5 -0.9 0.9 6.2 1602 0.00207

0.1 0.9 -0.9 0.9 6.0 1550 0.00207

The optimal design obtained from the traditional reliability-based

design is sensitive to estimated COV of noise factors.

18

19

A major challenge in the traditional reliability-based

design (RBD) is to estimate COVs of soil parameters

accurately.

How to conduct reliability-based design in the face of

uncertainty in the estimated COVs?

Reliability-based RGD, where the effect of

uncertainty in the estimated COVs on the system

response (pf) is eliminated or reduced by enforcing

robustness against the variation of the system

response.

Reliability-based RGD Framework

Inner loop:

Yes

Yes

Outer loop: Repeat M times

Complete

repetitions for each

of M designs?

Identify all possible designs

in the design domain

Assign mean and std. dev.

of each noise factor for

each design based on PEM

sampling requirement

Compute the failure

probability for each design

using FORM

Determine mean and std.

dev. of failure probability for

each design using PEM

No Repeat N times

START

Multi-objective optimization using NSGA to establish Pareto Front considering

safety, robustness, and cost

DESIGN DECISION

Characterize the

uncertainties in the

sample statistics of the

noise factors

Complete N

repetitions as per

PEM?

Determine feasibility robustness for each design on Pareto Front

No

Estimation of variation in statistics of noise factors

based on its typical range using three-sigma rule

Mean=(HCV+LCV)/2 Std. dev. = (HCV-LCV)/4

Uncertainty in statistics of noise factors

Lower

Bound

Upper

Bound Mean Std. dev.

[ ]COV 0.05 0.10 0.07 0.0125

0[ ]COV K 0.2 0.9 0.5 0.175

0,K -0.9 -0.6 -0.75 0.075

RGD: Multi-objective Optimization

Find d = [B, D]

Subject to: B {0.9m, 1.2m, 1.5m} and D {2m, 2.2m, 2.4m, … , 8m}

0.00069ULS ULS

p Tp

0.0047SLS SLS

p Tp

Objectives: Minimizing the standard deviation SLS failure probability ( p )

Minimizing the cost for drilled shaft.

22

(design parameters)

(safety – SLS requirement)

(maximize robustness)

(minimize cost)

(safety – ULS requirement)

23

Pareto Front for Drilled Shaft

All designs satisfy

safety requirement

Robustness

Cost

The design with feasibility robustness is the

design that can remain “feasible” in a

predefined constraint for certain probability even

when it undergoes variation.

Feasibility is measured in terms of probability,

and the robustness index is defined as:

0Pr[( ) 0] ( )SLS SLS

f Tp p P

24

Feasibility Robustness Concept

25

Cost versus Feasibility Robustness

Cost

Robustness

All designs satisfy

safety requirement

0P B (m) D (m) Cost (USD)

1 84.13% 0.9 6.2 1602

2 97.72% 0.9 6.8 1757

3 99.87% 0.9 7.6 1963

4 99.997% 1.2 6.6 2552

26

Selected final designs at various

feasibility robustness levels

27

Summary of RGD of Drilled Shaft

The final design obtained from the traditional

reliability-based design is sensitive to

estimated COV of noise factors.

The reliability-based design is incorporated

into RGD framework to deal with uncertainty

in the estimated parameter COVs.

The Pareto Front obtained through

optimization depicts a trade-off relationship

between cost and robustness. The feasibility

robustness concept is further introduced as

the measure of design robustness.

3. Robust Geotechnical Design of

Shallow Foundations in Sand

28

RGD of Shallow Foundations

B = L= ?

D = ?

G = 2000 kN

Q = 1000 kN

ULS: Vesic model

SLS: Normalized load-

settlement curve

( / )

( / )

ULS tSLS

t

R s BR

a s B b

COV of G 10%

COV of Q 18%

Homogeneous dry sand

with ten effective friction

angles from triaxial tests

Design Example

B {1.0m, 1.1m, … , 5.0m}

D {1.0m, 1.1m, … , 2.0m}

29

Test

No. o( )

1 33.0

2 35.0

3 33.5

4 32.5

5 37.5

6 34.5

7 36.0

8 31.5

9 37.0

10 33.5

Complete required

N times repetitions?

1 2

Original Sample,

, ,..., k

A

A a a a

* *

1 2

Compute Statistics of the Bootstrap Sample

= ( ), = ( )j jX A X A

Yes

No

*

* * * *

,1 ,2 ,

Bootstrap Sample,

= , ,...,

j

j j j j k

A

A a a a

Random Sampling with Replacement

Determine Mean, Standard Deviation

of Each Statistics iX

Bootstrapping for soil

data processing

Both soil and model parameters are considered as noise factors :

Soil property: effective friction angle

ULS model parameter: bias factor (calibrated from loads test)

SLS curve fitting parameters: a and b (calibrated from loads test)

Bootstrapping estimation of statistics of noise factors

RGD of Shallow Foundations

oMean value, ( )S

30 32 34 36 38 400.0

0.2

0.4

0.6

0.8

1.0

Pro

bab

ilit

y d

ensi

ty

Histogram

Normal pdf

(a)

oStandard deviation, ( )S

0 1 2 3 40.0

0.5

1.0

1.5

Pro

bab

ilit

y d

ensi

ty

Histogram

Normal pdf

(b)

oMean value, ( )S

30 32 34 36 38 400.0

0.2

0.4

0.6

0.8

1.0

Pro

bab

ilit

y d

ensi

ty

Histogram

Normal pdf

(a)

oStandard deviation, ( )S

0 1 2 3 40.0

0.5

1.0

1.5

Pro

bab

ilit

y d

ensi

ty

Histogram

Normal pdf

(b)

31

Find d = [B, D]

Subject to: B {1.0m, 1.1m, 1.2m, … , 5.0m }

D {1.0m, 1.1m, 1.2m, … , 2.0m}

57.2 10ULS ULS

p Tp

Objectives: Minimizing the std dev of ULS failure probability ( p )

Minimizing the cost for shallow foundation.

(safety – ULS requirement)

(maximize robustness)

(minimize cost)

(design parameters)

32

RGD: Multi-objective Optimization

33

Pareto Front for Shallow Foundation

Robustness

Cost

All designs satisfy

safety requirement

34

Cost versus Feasibility Robustness

Cost

Robustness

All designs satisfy

safety requirement

Table 8. Selected final designs at various feasibility robustness levels

0P B (m) D (m) Cost (USD)

1 84.13% 2.1 1.9 1200.1

2 97.72% 2.3 2.0 1423.7

3 99.87% 2.6 2.0 1763.7

4 99.997% 3.1 2.0 2409.8

Selected final designs at various

feasibility robustness levels

35

36

Effect of Spatial Variability More conservative without considering spatial variability

Cost

Robustness

37

Summary of RGD of Shallow Foundation

The RGD framework is refined to achieve

design robustness against uncertainty in

statistics of both soil parameters and model

uncertainty parameters.

Bootstrapping technique is used to characterize

the uncertainty in the sample statistics derived

from a small sample.

Multi-objective optimization is performed

considering safety, cost and robustness.

The effect of spatial variability on the robust

design is explored.

4. Robust Geotechnical Design of

Braced Excavations in Clay

38

Design Example

Noise Factors:

su/’v Mean 0.32 & COV 0.2

kh/’v Mean 48 & COV 0.5

Correlation 0.7

qs Mean 1 ton/m & COV 0.2

Design Parameters: Wall length (L), Wall thickness (t),

Vertical spacing of the struts (S),

Strut stiffness (EA)

Deterministic Model: Winkler model, finite element

code based on beam-on-elastic

foundation theory (TORSA) GL -2 m-1 m

-7 mGL -8 m

GL -4 m-3 m

GL -6 m-5 m

GL -10 m

Clay

Clay

39

RGD of Braced Excavations

40

A simple flowchart

for RGD

of braced

excavations

Outer loop:

Inner loop:

Complete the repetitions for each of

M possible designs?

Complete N times repetitions as required

by PEM?

Identify all possible designs in the design space

and quantify the uncertainty in noise factors

Assign a sampled value of noise factors

based on PEM

Generate new FEM*.i input files for each set of

sampled noise factors for TORSA analysis

Use PEM to determine the mean and standard

deviation of system response for each design

Yes

No

Repeat

N times

Repeat

M times

Yes

No

START

DESIGN DECISION

Multiple-objective optimization considering safety, robustness and cost to obtain a Pareto

Front, and identify the knee point on Pareto Front

Construct an initial

FEM model and generate FEM*.i

input file

Extract the system response from the FEM*.o

output file corresponding to each input file

Defined the braced excavation problem and classify design parameters and noise factors

41

Multi-objective Optimization Formulation

Find Values of Design Parameters:

t (wall thickness), L (wall length), S (strut spacing), EA (strut stiffness)

Subject to Constraints:

t {0.5 m, 0.6 m, 0.7 m, 0.8 m, …, 1.3 m} S {1.5 m, 2 m, 3 m, 6 m}

L {20 m, 20.5 m, 21 m, 21.5 m, …, 30 m} EA {H300, H350, H400, 2@H350, 2@H400}

Mean factor of safety for the push- in and basal heave 1.5

Mean maximum wall deflection 7 cm (0.7%Hf)

Objective:

Minimizing the standard deviation of the maximum wall deflection (cm)

Minimizing the cost for the supporting system (USD)

(design parameters)

(safety - stability)

(robustness)

(cost)

(safety - serviceability)

Pareto Front for Excavation Design

0 1 2 3 4Standard deviation of maximum wall deflection (cm) as a measure of robustness

0.0

0.5

1.0

1.5

2.0C

ost

of

support

ing s

yst

em (

10

6U

SD

)

Pareto Front

42

zP

Objective 1

Objective 1

1.E-07

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

0 0.

5

1 1.

5

2 2.

5

3

Depth of drilled shaft, D(m)

Mea

sure

in

Ob

ject

ive

2

ULS

SLS

Overall

Solution Space

Pareto Front

Reflex Angle

Pareto Front

(a)

Objective 1

Objective 1

1.E-07

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

0 0.

5

1 1.

5

2 2.

5

3

Depth of drilled shaft, D(m)

Mea

sure

in

Obje

ctiv

e 2

ULS

SLS

Overall

Solution Space

Pareto Front

z * P

z*

z

n̂

(Knee Point)

Boundary Line

A

B

(b)

zP

Objective 1

Objective 1

1.E-07

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

0 0.

5

1 1.

5

2 2.

5

3

Depth of drilled shaft, D(m)

Mea

sure

in O

bje

ctiv

e 2

ULS

SLS

Overall

Solution Space

Pareto Front

Reflex Angle

Pareto Front

(a)

Objective 1

Objective 1

1.E-07

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

0 0.

5

1 1.

5

2 2.

5

3

Depth of drilled shaft, D(m)

Mea

sure

in O

bje

ctiv

e 2

ULS

SLS

Overall

Solution Space

Pareto Front

z * P

z*

z

n̂

(Knee Point)

Boundary Line

A

B

(b)

Search for Knee Point on Pareto Front

43

Search for Knee Point on Pareto Front

0 1 2 3 4Standard deviation of maximum wall deflection (cm) as a measure of robustness

0.0

0.5

1.0

1.5

2.0C

ost

of

support

ing s

yst

em (

10

6U

SD

)

Pareto Front

Knee Point

Boundary Line

maximum distance

44 Knee point : t = 0.6 m, L = 20 m, S = 1.5 m, EA = H400

45

Summary of RGD of Braced Excavation

RGD is further refined by treating the

variation of maximum wall deflection caused

by uncertainties in soil parameters and

surcharges as a robustness measure.

Multi-objective optimization is used to derive

Pareto Front, which describes a trade-off

relationship between cost and robustness at

a given safety level.

The knee point concept is used to select the

single most preferred design based on the

“sacrifice-gain” relationship on Pareto Front.

5. Concluding Remarks

46

Concluding Remarks

Robust Geotechnical Design, a new design

paradigm, has been demonstrated as an effective

tool to obtain optimal designs that are robust

against variation in noise factors (e.g., uncertain

geotechnical parameters).

RGD with multi-objective optimization can consider

safety, cost, and robustness simultaneously and

effectively.

RGD has been shown as an effective design tool

for many geotechnical problems.

47

Recommendations

Robust design of geothermal piles and off-shore

structure foundations.

Robust design and real-time updating of

underground constructions and ground

improvement operations.

Robust maintenance framework for geotechnical

systems. Possible integration of life-cycle

performance optimization within the robust

maintenance framework may be explored.

48

Thank You !

49