Estimation of the Uncertainty of the Robertson and Wride Model for

Reliability Analysis of Soil Liquefaction

C. Hsein Juang and Susan H. YangClemson University

This study was sponsored by the National

Science Foundation through Grants No.

CMS-0218365.

ACKNOWLEDGMENT

Objectives of the Research

Develop a procedure for estimating model uncertainty of limit state modelsExplore the use of FORM for calculating reliability index and probability of liquefactionExamine the robustness of Bayesian mapping approach

Outline of the Presentation

Robertson and Wride Model

First Order Reliability Method

Parameter and Model Uncertainties

Bayesian Mapping Approach

Estimation of the Model Uncertainty

Conclusions

Review of the Robertson and Wride Model

Liquefaction loading: cyclic stress ratio (CSR)

v max7.5 d

v

aCSR 0.65 (r )/MSF

g

Liquefaction resistance: cyclic resistance ratio (CRR)CRR7.5 = 93(qc1N,cs/1000)3+0.08, if 50qc1N,cs<160

CRR7.5 = 0.833(qc1N,cs/1000)+0.05, if qc1N,cs< 50

CRR = f(qc, fs, v, v)

See Robertson and Wride (1998) for detail

Overview of the First Order Reliability Method

First order second moment (FOSM)

Advanced first order second moment (AFOSM)

First order reliability method (FORM)

Reliability Index in the Reduced Variable Space

X1

X2

1

2

g1(x*) = 0 (limit state fcn. in reduced variable space)

g2(x*) = 0 (linearized limit state fcn.)

is defined as the shortest distance between the limit state surface and the origin in the reduced variable space

Limit State Function for Reliability Analysis of Soil Liquefaction

When model uncertainty is not considered:

g( ) = CRR CSR = g(qc, fs, v, v, amax, Mw)

When model uncertainty is considered:

g( ) = c1CRR CSR = g(c1, qc, fs, v, v, amax, Mw)

Model Uncertainty: the uncertainty in the limit state function

CSR is used as a reference in the development of the CRR model through calibration with field cases;Whatever uncertainty there is in the CSR model is eventually passed along to the uncertainty in the CRR model. The effect of the uncertainty associated with the CSR model is realized in the CRR model.

Parameter uncertainties in the first baseline reliability analysis

Mean to nominal = 1.0

Uncertainty of a parameter is characterized with a coefficient of variation (COV)

COV_qc = 0.08 COV_fs = 0.12

COV_v = 0.10 COV_v = 0.10

COV_amax = 0.10 COV_Mw = 0.05

Coefficient of correlation of the input parameters

qc fs

v

v amax Mw

qc 1.00 0.37 0.25 0.25 0.00 0.00

fs 0.37 1.00 0.54 0.52 0.00 0.00

v 0.25 0.54 1.00 0.93 0.00 0.00

v 0.25 0.52 0.93 1.00 0.00 0.00

amax 0.00 0.00 0.00 0.00 1.00 0.90

Mw 0.00 0.00 0.00 0.00 0.90 1.00

Probabilities of Liquefaction

By means of notional probability concept

1 (β)fp

By means of a Bayesian mapping function obtained through calibration of the calculated using a database of field observations

L

(β)P (Liq | β)

(β) (β)L

rL NL

fP

f f

Results of the reliability analysis of the CPT-based case histories

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

-5 -4 -3 -2 -1 0 1 2 3 4 5Reliability index, 1

Pro

bab

ility

of

liqu

efac

tio

n P

L

BayesianNotional

Estimation of Model Uncertainty: Premise

The premise:

A fully calibrated Bayesian mapping function is a true probability or at least a close approximation of the true probability.

This premise stands as the calibration of the calculated reliability index is carried out with a sufficiently large database of case histories.

Estimation of Model Uncertainty: Methodology

Model uncertainty is obtained through a trial and error process.The “correct” model uncertainty:The one that produces the probabilities matching best with those produced by the Bayesian mapping function that has been calibrated with a database of field observations.

Effect of the COV component of model uncertainty onβ

3 = 1

R2 = 1

-5-4-3-2-1012345

-5 -4 -3 -2 -1 0 1 2 3 4 5

1

3

3 = 0.92

1 - 0.01

R2 = 1.00

-5-4-3-2-1012345

-5 -4 -3 -2 -1 0 1 2 3 4 5

1

3

3 = 0.85

1- 0.02

R2 = 1.00

-5-4-3-2-1012345

-5 -4 -3 -2 -1 0 1 2 3 4 5

1

3

3 = 0.77

1 - 0.03

R2 = 1.00

-5-4-3-2-1012345

-5 -4 -3 -2 -1 0 1 2 3 4 5

1

3

(a) c1 = 1.0, COV = 0% (b) c1 = 1.0, COV = 10%

(c) c1 = 1.0, COV = 15% (d) c1 = 1.0, COV = 20%

Effect of the c1 component of model uncertainty onβ

3 = 0.92

1 - 0.83

R2 = 1.00

-5-4-3-2-1012345

-5 -4 -3 -2 -1 0 1 2 3 4 5

1

3

3 = 0.92

1 - 0.40

R2 = 1.00

-5-4-3-2-1012345

-5 -4 -3 -2 -1 0 1 2 3 4 5

1

3

3 = 0.92

1 - 0.01

R2 = 1.00

-5-4-3-2-1012345

-5 -4 -3 -2 -1 0 1 2 3 4 5

1

3

3 = 0.92

1 + 0.67

R2 = 1.00

-5-4-3-2-1012345

-5 -4 -3 -2 -1 0 1 2 3 4 5

1

3

(c) c1 = 1.0, COV = 10% (d) c1 = 1.2, COV = 10%

(a) c1 = 0.8, COV = 10% (b) c1 = 0.9, COV = 10%

Effect of the COV component of model uncertainty on PL

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

Bayesian probability, PL1

No

tio

nal

pro

bab

ilit

y, P

L3

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

Bayesian probability, PL1

No

tio

nal

pro

bab

ilit

y, P

L3

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

Bayesian probability, PL1

No

tio

nal

pro

bab

ilit

y, P

L3

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

Bayesian probability, PL1

No

tio

nal

pro

bab

ilit

y, P

L3

(a) c1 = 1.0, COV = 0% (b) c1 = 1.0, COV = 10%

(c) c1 = 1.0, COV = 15% (d) c1 = 1.0, COV = 20%

Effect of the c1 component of model uncertainty on PL

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

Bayesian probability, PL1

No

tio

nal

pro

bab

ility

, PL

3

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

Bayesian probability, PL1

No

tio

nal

pro

bab

ility

, PL

3

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

Bayesian probability, PL1

No

tio

nal

pro

bab

ility

, PL

3

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

Bayesian probability, PL1

No

tio

nal

pro

bab

ility

, PL

3

(a) c1 = 0.8, COV = 10% (b) c1 = 0.9, COV = 10%

(c) c1 = 1.0, COV = 10% (d) c1 = 1.2, COV = 15%

Notional probability versusBayesian probability

The model uncertainty is

characterized by c1 = 0.94

and COV = 15%

The parameter uncertainty

is at the same level as in

the first baseline analysis

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

Bayesian probability, PL1

No

tio

nal

pro

bab

ility

, PL

3

Notional probability versusBayesian probability (con’d)

The model uncertainty is characterized by c1 = 0.94

and COV = 15%,

The parameter uncertainty is at the same level as in the second baseline analysis.0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

Bayesian probability, PL1

No

tio

nal

pro

bab

ility

, PL

3

Conclusions

The model uncertainty c1 can be characterized

with two statistical parameters, the mean (c1),

and the coefficient of variation (COV).

The results show that the c1 component has a

more profound impact than does the COV component.

Conclusions (con’d)

The first order reliability method (FORM) is shown to be able to estimate accurately and PL, provided that the correct parameter and

model uncertainties are incorporated in the analysis.

Conclusions (con’d)

Robustness of the Bayesian mapping approach is demonstrated in this study.

In the situation where the fully calibrated Bayesian mapping function is available, the model uncertainty may be estimated using the Bayesian mapping function.

Conclusions (con’d)

The Robertson and Wride model is estimated to have a model uncertainty that is characterized with a c1 of 0.94 and a COV of 15%.