1

Calibration of Resistance Factor for

Design of Pile Foundations

Considering Feasibility Robustness

Hsein Juang

Glenn Professor of Civil Engineering

Clemson University

2

3

Outline of Presentation

• Background

• Traditional Resistance Factor Calibration

• Calibration Considering Robustness

• Design Example and Further Discussion

• Summary

4

Foundations Design Methodologies

19th – Early 20th century

Empirical Design

Early 20th century - now

Allowable Stress Design

Late 20th century - now

Reliability-based Design

development of soil mechanics

and analysis methods

following the lead of

structural design practice

5

Allowable Stress Design (ASD)

The factor of safety (FS) is introduced and applied to the

geotechnical capacity as:

ni nQ R FS

The FS is used to account for all uncertainties in

• Load and material properties

• Design models

• Construction effects etc.

FS = 2 – 3 is adequate for foundations

6

FS = “True” Safety Level ?

0 10 20 30 40 50 600

0.05

0.10

0.15

0.20

0.25

μQ

μR

R

f R(R

) o

r f Q

(Q)

Resistance or Load (R, Q)

Q

mean FS = 2.5

P[R < Q] = 0.0002 (β = 3.6)

0 10 20 30 40 50 600

0.05

0.10

0.15

0.20

0.25

μQ μ

R

R

f R(R

) o

r f

Q(Q

)Resistance or Load (R, Q)

Q

mean FS = 2.5

P[R < Q] = 0.073 (β = 1.5)

400 times less safe

The same FS may imply very different safety margins

7

FS = “True” Safety Level ?

A larger FS does not necessarily mean a smaller level of risk

0 10 20 30 40 50 600

0.05

0.10

0.15

0.20

0.25

μQ

μR

R

f R(R

) or f

Q(Q

)Resistance or Load (R, Q)

Q

mean FS = 3.0

P[R < Q] = 0.0036 (β = 2.7)

20 times less safe

0 10 20 30 40 50 600

0.05

0.10

0.15

0.20

0.25

μQ

μR

R

f R(R

) or

f Q(Q

)

Resistance or Load (R, Q)

Q

mean FS = 2.5

P[R < Q] = 0.0002 (β = 3.6)

8

Reliability-based Design (RBD)

• Full-probabilistic approach

e.g., Expanded RBD (Wang et al. 2011)

• Semi-probabilistic approach

e.g., Partial Factor Approach

Load and Resistance Factor Design (LRFD)

Multiple Resistance and Load Factor Design (Phoon et al. 2003)

Quantile Value Method (Ching and Phoon 2011)

Robust-LRFD (Gong et al. 2016)

Reliability analysis is used and probability of failure (P(R<Q))

is introduced to measure the design risk.

9

Load and Resistance Factor Design (LRFD)

Under the LRFD approach, design must satisfy the equation:

γQQ

n=R

n/γ

R

Qn R

n

R

f R(R

) o

r f Q

(Q)

Resistance or Load (R, Q)

Q

Qi ni nQ R Qi ni n RQ R or (AASHTO) (Eurode 7)

Resistance factor, γR=(1/φ) ≥1,

accounts for variabilities in soil

properties, design models and

construction

Load factors, γQi ≥1, accounts

for variability in loads

10

Reliability Concept in LRFD

( ) ( 0) ( ) ( )g

f

g

p p R Q p g

Assuming R and Q are lognormally distributed, performance function

can be described as g = ln(R) – ln(Q), which follows normal distribution:

2 2

2 2

ln 1 1

ln 1 1

R Q Q R

Q R

COV COV

COV COV

2 2

2 2

ln 1 1

ln 1 1

R Q Q RR Q

Q R

COV COV

COV COV

2 2ln 1 1g R Q Q RCOV COV

, ,Q n n R R R n Q Q nQ R R Q

2 2ln 1 1g Q RCOV COV

≥ βT

11

Selection of βT

(U.S. Army Corps

of Engineers 1997)

References βT

Meyerhof 1970 3.1-3.7

Phoon et al. 1995 3.2

Canadian Building Code 1995 3.5

AASHTO 1997 2.0-3.5

Paikowsky et al. 2004 2.33 for redundant piles

3.0 for non-redundant piles

12

Calibration of Resistance Factor

Load factors (γQ) and load statistics (λQ and COVQ) developed

in the structural design codes are adopted.

Resistance factor (γR) is calibrated using:

2 2

2 2

exp ln 1 1

1 1

Q T Q R

R

R Q Q R

COV COV

COV COV

λR and COVR = the mean and the COV of the resistance bias

factor, which are estimated from a load test database

13

Challenges

• The resistance bias factor statistics are hard to ascertain,

uncertainty is inherent in the derived statistical parameters

of the resistance bias factor

• The resistance factor calibrated for LRFD is very sensitive

to the uncertainty in the resistance bias factor

• Consequently, a design obtained using the calibrated

resistance factor may not achieve βT (i.e., the design is not

feasible) if the variation in the resistance bias factor is

underestimated.

14

Goal of This Study

• To propose a new approach for resistance factor

calibration that considers explicitly the feasibility

robustness of design

Feasibility robustness is a measure of robustness, indicating the extent

that a system remains feasible even when it undergoes variation.

• Resistance factor is re-calibrated considering variation

in the resistance bias factor.

• Design using the re-calibrated resistance factor will

always satisfy the βT requirement to the extent defined

by the designer in the face of uncertainty in the

computed capacity

15

Outline of Presentation

• Background

• Traditional Resistance Factor Calibration

• Calibration Considering Robustness

• Design Example and Further Discussion

• Summary

16

Traditional Resistance Factor Calibration Li, J.P., Zhang, J., Liu, S.N., & Juang, C.H. (2015). Reliability-based code revision for design

of pile foundations: Practice in Shanghai, China. Soils and Foundations, 55(3), 637-649.

Pile Types

driven piles

bored piles

Design Methods

load test-based method

design table method

CPT-based method

Uncertainty in Capacity

within-site variability

cross-site variability

Design Equation in Shanghai is written as:

nD Dn L Ln

R

RQ Q

17

Uncertainty Analysis of Design Methods

Computed

capacity (Rn)

1 2n nR NR N N R

where N1 and N2 are bias factors accounting for within-site variability and

cross-site variability, respectively; and N is lumped bias factor.

The actual capacity (R) can be expressed as:

subjected to

within-site

variability

cross-site

variability

the variation of the soil properties within a site

the regional variation of the soil properties

the construction error associated with the site-specific workmanship

the construction error associated with workmanship in a region

18

Statistics of Resistance Bias Factors

As the uncertainties associated with N1 and N2 are from different sources, it

might be reasonable to assume that N1 and N2 are statistically independent.

It can be show that:

1 2R R R

2 2

1 2R R RCOV COV COV

where λR, λR1, and λR2 are the means of N, N1, and N2, respectively;

COVR, COVR1, COVR2 are the COVs of N, N1, and N2, respectively.

The within-site variability can be characterized by comparing the measured

and the predicted bearing capacities of piles within a site, while the cross-

variability can be characterized by comparing the measured and the

predicted bearing capacities of piles from different sites.

19

Calibration Database

A database consisting of 146 piles from 32 sites and another

database comprising 37 piles from 10 sites were used to

characterize the within-site variability for driven piles and

bored piles, respectively.

DATABASE driven piles

bored piles

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Characterization of Within-site and

Cross-site Variabilities

• Characterization of within-site variability

λR1=1 since within-site is unbiased

The COVR1 values vary from site to site, and the computed

mean of the COVR1 is used, i.e., COVR1 = 0.087 and COVR1 =

0.093 are adopted for the analysis of driven and bored piles,

respectively.

• Characterization of cross-site variability

The values of λR2 and COVR2 are taken based on the previous

design code SUCCC (2000)

SUCCC. (2000). Foundation Design Code, Shanghai Urban Construction and

Communications Commission (SUCCC), Shanghai (in Chinese).

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Load Statistics and Load Factors

Typical load statistics used in different studies (after Li et al. 2015)

References λD λL COVD COVL

Ellingwod et al. (1980) 1.00 1.05 0.10 0.18

Ellingwood and Tekie (1999) 1.05 1.0 0.1 0.25

Nowak (1999) and ASSHTO (2007) 1.08 1.13 1.15 0.18

Nowak (1994) and FHWA (2001) 1.03-1.05 0.08-0.10 1.1-1.2 0.18

Li et al. (2015) based on MOC (2002) 1.00 1.00 0.07 0.29

• Load Statistics

• Load Factors

γD =1.0 and γL = 1.0 used in MOC (2002) are adopted

MOC, (2002). Code for Design of Foundations (GB 50007-2002). Ministry of

Construction (MOC) of China, Beijing. (In Chinese).

22

2 2

2 2

exp ln 1 1

1 1

Q T Q R

R

R Q Q R

COV COV

COV COV

Calibration of Resistance Factor

= =Q D Dn L Ln D L

Q D Dn L Ln D L

Q Q

Q Q

= =0.2Ln DnQ Q

2 2 21=

1+Q D LCOV COV COV

2 2

2 2

exp ln 1 COV 1 COV

1 1

T R QD L

R

R D LQ RCOV COV

Load factors Driven piles Bored piles

LT method DT method CPT method LT method DT method

γL=1.0

γD=1.0 1.53 1.93 1.72 1.56 2.26

• Calibration Equation

• Calibration Results

Calibrated resistance factors (γR) for load-carrying capacity (βT = 3.7)

23

Variation in COVR1 (Background study)

0 0.05 0.10 0.15 0.20 0.25 0.300

0.2

0.4

0.6

0.8

1.0

Driven Piles

Lognormal Distribution

Cum

ula

tive

Fre

quen

cy

COVR1

0 0.05 0.10 0.15 0.20 0.25 0.300

0.2

0.4

0.6

0.8

1.0

Bored Piles

Lognormal Distribution

Cu

mu

lati

ve

Fre

qu

ency

COVR1

• Sort the COVR1 values in ascending order;

• Rank the values from i = 1 to n;

• Compute the cumulative probability pi = i/(n+1);

• Establish cumulative distribution function.

Cumulative frequency of the observed COVR1 with fitted lognormal CDF

24

Effect of the Variation in COVR1

1 2 3 4 5 60

0.02

0.04

0.06

0.08

0.466

βT

Driven Piles LT Method

Histogram

Rel

ativ

e F

requ

ency

Reliability index, β

0

0.25

0.50

0.75

1.00

Cumulative Frequency

Cum

ula

tiv

e F

requ

ency

5000 random samples of

COVR1 are generated and

the corresponding β values

are computed with calibrated

γR using:

2

2

2 2

1 COVln

1 COV=

ln 1 COV 1 COV

QR R D L

D L R

R Q

Relative and cumulative frequency of β

associated with calibrated γR

The β values distribute in wide ranges and many of the designs cannot

achieve βT = 3.7 (i.e., the designs are not feasible); the probability of

(β < βT) can be obtained from the cumulative frequency curve of β.

25

How to deal with the effect of

variation in COVR1?

26

Outline of Presentation

• Background

• Traditional Resistance Factor Calibration

• Calibration Considering Robustness

• Design Example and Further Discussion

• Summary

27

Robust Design

Robust design, originated from the field of Quality Engineering

(Taguchi 1986), seeks an optimal design by selecting controllable

design parameters so that the system response of the design is

insensitive to, or robust against, the variation of noise factors.

Robust design has recently been applied to geotechnical

problems (Juang et al. 2013), and examples of geotechnical

design with LRFD approach considering robustness have been

reported (Gong et al. 2016).

This study is aimed at introducing the robustness concept into the

LRFD calibration.

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Robustness Measures

(from Khoshnevisan et al. 2014)

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Feasibility Robustness

The feasibility robustness (Parkinson et al. 1993) is adopted

herein to measure the robustness of partial-factor design with

respect to uncertain parameters (i.e., COVR1), and is defined

as the probability that βT can still be satisfied even with the

variation in COVR1.

0[( ) 0]TP P

where P[(β−βT) ≥ 0] is the probability that βT is satisfied; and P0 is a pre-

defined acceptable level of this probability (i.e., feasibility robustness).

Feasibility robustness is formulated as (Juang et al. 2013):

30

Calculation of Feasibility Robustness

• Monte Carlo Simulation (MCS)

1 2 3 4 5 60

0.02

0.04

0.06

0.08

0.466

βT

Driven Piles LT Method

Histogram

Normal Distribution

Rel

ativ

e F

requen

cy

Reliability index, β

0

0.25

0.50

0.75

1.00

Cumulative Frequency

Cum

ula

tive

Fre

quen

cy

For driven piles with LT method, when using calibrated γR = 1.53

P[(β− βT) ≥ 0] = 1-0.466 = 0.534

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Calculation of Feasibility Robustness

0[( ) 0]=T

TP P

• Point Estimation Method (PEM)

7 72 2

1 1

= , = ( )i i i i

i i

P P

Assuming β follows normal distribution:

By using PEM (Zhao and Ono 2000):

Approach Driven piles Bored piles

LT method DT method CPT method LT method DT method MCS 0.534 0.489 0.515 0.558 0.500

PEM 0.541 0.479 0.506 0.578 0.464

Feasibility robustness of calibrated partial factors in Li et al. (2015)

obtained from MCS and PEM

[( ) 0]= ( )T RP G

32

Resistance Factor Calibration Considering

Feasibility Robustness

0[( ) 0]=TP P

The procedure to evaluate feasibility robustness of a design using the existing

γR actually is the inverse of the task of resistance factor calibration considering

robustness, which is a process of determining value of γR such that the

resulting design can achieve the pre-defined feasibility robustness level.

• Trail-and-error approach

A trail γR MCS P[(β−βT) 0] = P0 ?

• Solving equation P[(β−βT) 0] = G(γR) = P0 based on PEM

33

Resistance Factor Calibration Results

P0 Driven piles Bored piles

LT method DT method CPT method LT method DT method 0.5 1.52 1.95 1.72 1.53 2.30 0.6 1.55 1.98 1.75 1.57 2.34 0.7 1.59 2.01 1.79 1.62 2.38 0.8 1.64 2.05 1.83 1.69 2.44 0.9 1.72 2.11 1.90 1.81 2.52 0.99 2.04 2.27 2.10 2.39 2.76

Calibrated resistance factors (γR) for load-carrying capacity at different

feasibility robustness levels (γL = 1.0 and γD = 1.0; βT = 3.7)

To achieve the same P0, the DT method requires larger γR, as it is associated

with greater uncertainties. On the other hand, the required γR for the LT

method is smaller due to the lower uncertainties involved with the LT method.

34

Values of μβ and σβ at various feasibility

robustness levels

P0

Driven piles Bored piles

LT method DT method CPT method LT method DT method

μβ σβ μβ σβ μβ σβ μβ σβ μβ σβ

0.5 3.70 0.67 3.70 0.30 3.70 0.43 3.70 0.83 3.70 0.30

0.6 3.88 0.70 3.79 0.30 3.80 0.44 3.90 0.89 3.79 0.31

0.7 4.11 0.74 3.87 0.31 3.95 0.46 4.18 0.93 3.86 0.31

0.8 4.38 0.79 3.98 0.32 4.10 0.47 4.54 1.01 3.98 0.32

0.9 4.81 0.87 4.13 0.33 4.36 0.50 5.14 1.13 4.12 0.33

0.99 6.32 1.13 4.53 0.36 5.04 0.56 7.56 1.65 4.54 0.36

For the same P0, piles designed with the LT method have the largest μβ;

while piles designed with the DT method have the smallest μβ.

Both the values of μβ and σβ increase with increasing P0. Thus feasibility

robustness is primarily affected by μβ.

35

Outline of Presentation

• Background

• Traditional Resistance Factor Calibration

• Calibration Considering Robustness

• Design Example and Further Discussion

• Summary

36

A Bored Pile Design Example

Suggested side and toe resistances of bored

piles in different soil layers (after SUCCC 2010)

Soil description Depth (m) ƒs (kPa) qt

(kPa)

Grayish yellow clay 0-4 15 -

Very soft gray clay 4-20 15~30 150~250 Gray silty sand 20-35 55~75 1250~1700

Gray fine sand with silt 35-60 55~80 1700~2550 Gray fine, medium or coarse sand 60-100 70~90 2100~3000

QDn= 2500 kN, QLn= 500 kN, pile diameter D=0.85m, pile length L needs to be determined by using DT method

n s t p si i t tR R R U f l q A

n R D Dn L LnR Q Q

37

Bored Pile Design Results

0.5 0.6 0.7 0.8 0.9 1.044

46

48

50

52

54

Pil

e l

en

gth

, L

(m

)

Feasibility robustness level , P0

Design results at various feasibility robustness levels

Design with high robustness against variation in the computed capacity can

always be achieved at the expense of cost efficiency.

38

Optimization between Robustness and Cost

A design with higher feasibility robustness and relatively lower

cost is desired. A tradeoff decision is thus needed based on an

optimization of γR performed with respect to two objectives,

design robustness and cost efficiency.

Find: An optimal γR compatible with γL = 1.0 and γD = 1.0

Subject to: G(γR) = P0 {0.5, 0.51, 0.52, …, 0.99}

Objectives: Maximizing the feasibility robustness (in terms of P0)

Minimizing the construction cost (in terms of γR)

39

Optimization between Robustness and Cost

0.5 0.6 0.7 0.8 0.9 1.01.4

1.6

1.8

2.0

2.2

2.4Driven Piles LT Method

Resi

stan

ce f

acto

r, γ

R

Feasibility robustness, P0

A tradeoff exists between design robustness and cost efficiency; the tradeoff

relationship is presented here as a Pareto front. The knee point on the Pareto

front conceptually yields the best compromise among conflicting objectives.

40

Determination of Knee Point

Reflex Angle approach, Normal Boundary Intersection approach,

Marginal Utility Function approach and Minimum Distance approach

Approaches:

Minimum Distance approach

Gong et al. (2016)

• Perform a single-objective optimization with

respect to each objective function of concern

• Determine the corresponding maximum value

of each objective function

• Normalize the objective functions into values

ranging from 0.0 to 1.0

• Compute the distance from the normalized

utopia point to the normalized objective

functions

• The point that has the minimum distance from

the “utopia point” is taken as the knee point.

Objective 1

Ob

ject

ive

2

Feasible designs

Knee point

Utopia design

Most optimal

with respect to Objective 2

Most optimal

with respect to Objective 1

Minimum distance

41

On the left side of knee point, a

slight reduction in γR (i.e., cost)

would lead to a large decrease in

design robustness P0.

0.5 0.6 0.7 0.8 0.9 1.01.4

1.6

1.8

2.0

2.2

2.4Driven Piles LT Method

Knee Point

Resi

stan

ce f

acto

r, γ

R

Feasibility robustness, P0

On the other side of the knee point, a

slight gain in robustness P0 requires

a large increase in γR, rendering it

cost inefficient.

The knee point represents the best compromise between design

robustness and cost efficiency.

Determination of Knee Point

42

Calibrated partial factors obtained from

knee point of Pareto front

Piles Design

method

Knee point Suggested

P0 γR P0 γR

Driven

piles

LT method 0.86 1.68 0.85 1.67

DT method 0.84 2.07 0.85 2.08

CPT method 0.85 1.86 0.85 1.86

Bored

piles

LT method 0.88 1.78 0.85 1.74

CPT method 0.84 2.46 0.85 2.47

(γL = 1.0 and γD = 1.0 ; βT = 3.7)

43

Outline of Presentation

• Background

• Traditional Resistance Factor Calibration

• Calibration Considering Robustness

• Design Example and Further Discussion

• Summary

44

Summary

• A new approach for calibration of resistance factors

for design of pile foundations considering feasibility

robustness is proposed.

• Design using re-calibrated resistance factors is

robust against the uncertainty in computed

capacity.

• The methodology is demonstrated effective for

design of pile foundations in Shanghai, China.

45

Thank You