RELIABILITY ANALYSIS OF GEOTECHNICAL SYSTEMS

Dr. G L Sivakumar BabuDepartment of Civil EngineeringIndian Institute of ScienceBangalore, India

Contents

MotivationShallow foundationsPile foundations Unsaturated soil slopesRetaining systemsBuried pipesConclusions

Acknowledgments

• Seshagiri Rao • D S N Murthy • Sumanta Haldar • Munwar Basha • Amit Srivatsava

Motivation

• Steel and concreteManufacturedControlled conditionsMaterial behaviour

• SoilNatural materialFormed through complex processesComplicated material behaviour (non-

linear and stress dependent, numerous widely accepted transformation models)

Contd..

Contd..

Is it appropriate to neglect such high Is it appropriate to neglect such high degree of soil property variations degree of soil property variations associated with mean design associated with mean design parameter???parameter???

Motivation

MotivationContd..

Resistance factors are functions of variability in LRFD design

Reliability analysis

)()0)(( β−Φ=<= XgPp f

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

+

−Φ−=

22

)(1SR

SRfp

σσμμ

)()0)(( β−Φ=<= XgPp f

Performance function is Z=R-S

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

+

−=

22

)(

SR

SR

σσμμβ

( )⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

++

++=−Φ= −

)1)(1ln(

})1/()1()ln{(1

22

22

1

SR

SRS

R

fpδδ

δδμμ

β

⎥⎥⎦

⎤

⎢⎢⎣

⎡

++

++Φ−=

)1)(1ln(

)1)(1()/ln(1

22

22

SR

SRSRfp

δδ

δδμμ

(USACE 1999)

Probability density function of safety margin (R-S)

f–N diagram adopted by Hong Kong Planning Department for planning purposes

Whitman (1985)

Practical implications of effect of variability on design of shallow foundations (Lacasse 2001)

Probability density function of FS=R/S

Seismic stability of slopes

CoVc,φ=10%

Contd..

Inherent variability Inhomogeneous Anisotropic

Measurement uncertaintyTransformation uncertainty due to use simplified

mathematical correlations

Variability

Under this uncertain environment, how far solutions based on deterministic approaches produce realistic estimates of safety in designs??

ReliabilityThe probability that a system performs satisfactorily the intended function under specified operating conditions, during its design period

Input parametersMoments (mean, variance, etc.)Distributions (Normal and Lognormal etc.)Auto-correlation

Probabilistic Analysis

2

1])([

11 ∑

=ℜ ℜ

−=

n

iiz

nσ

t

vocμσ ℜ

ℜ =...

)(11

i

n

it zt

n ∑=

=μ

The mean of the soil property is defined as mean of the trend function fitted to the experimental data

The standard deviation of variability is

The coefficient of variation of variability is:

Evaluation of spatial variability - Autocorrelation

Indication of distance within which the property values show relatively strong correlation

The sample autocorrelation function is

∑

∑

=

−

=+

−−

−−−−= n

iYi

jn

iYjiYi

j

Yn

YYjn

1

2

1

)(1

1

))((1

1

)(μ

μμτρ

Evaluation of spatial variability - Variance reduction function

For theoretical triangular fit to sample autocorrelation function

For theoretical exponential fit to sample autocorrelation function

For theoretical double exponential fit to sample autocorrelation function

a, b, d are the autocorrelation distances, and T is the averaging distance, the domain over which the soil properties are averaged

Commonly used theoretical fits to sample autocorrelation functions (vanmarcke, 1983)

Inherent Soil Variability

USEFULNESS OF CPT BASED PROBABILISTIC USEFULNESS OF CPT BASED PROBABILISTIC ANALYSIS OF SOIL PROFILESANALYSIS OF SOIL PROFILES

•• WHY CPT BASED METHOD?WHY CPT BASED METHOD?– Simple, fast and continuous.– Analysis is based on well established concepts– Less average cost compared to soil boring (LTRC, 1999).– Laboratory tests can be avoided.– Provides a format for quantifying information regarding subsurface

condition of a particular site.

•• ADVANTAGES in Reliability Based DesignADVANTAGES in Reliability Based Design– To quantify variability from CPT data that map into load-settlement behaviour

and integrate with the design of shallow/pile foundations.– The pile-soil interface parameters can be calculated from undrained shear

strength values obtained from CPT data.– Propose reliability based design methodologies for foundations considering

Ultimate Limit State (ULS) and Serviceability Limit State (SLS).

• The soil properties are measured by physical means. This measurement process introduces variability.

• Measured soil property (Ym(z)) can be described as :

• Where Y(z) is the in-situ soil property, e(z) is the measurement uncertainty.

• The expanded form of above equation as :

• In the published literature the range of measurement error i.e. for ECPT is generally 5%-15%.

)()()( ZezYzYm +=

)()()()( zezztzYm +ℜ+=

eCoV

Characterization of uncertainty-

Measurement uncertainty

• A transformation model is required to relate the test measurement to an appropriate design property.

• The correlation between the undrained shear strength and tip resistance is:

where su is the undrained shear strength; NK is the empirical constant, qc is the cone tip resistance , total overburden stress

)( vocKk

vocu qD

Nqs σσ

−=−

=

KK N

D 1=

voσ

Characterization of uncertainty - Transformation Uncertainty

Total variability

( )εξξ ,md T=

22

22

22

2εξ ε

SDTSDeTSD

wTSD ewd ⎟

⎠⎞

⎜⎝⎛

∂∂

+⎟⎠⎞

⎜⎝⎛

∂∂

+⎟⎠⎞

⎜⎝⎛

∂∂

=

( )εξ ,ewtTd ++=

The uncertainty associated with design soil property such as cone tip resistance is a function of inherent soil variability (w), measurement error (e) and transformation uncertainty ε.

The design soil property is predicted from test measurement using the following transformation model using second-moment statistics

Design property and measurement are related byDesign property, related to inherent variability, measurement

and transformation is given by

( ) 22

22

222

2εξ ε

SDTSDeTSDL

wTSD ewa ⎟

⎠⎞

⎜⎝⎛

∂∂

+⎟⎠⎞

⎜⎝⎛

∂∂

+Γ⎟⎠⎞

⎜⎝⎛

∂∂

=

Studies on shallow foundations

Moments of cone tip resistance-Shear failure criterion-NGES data

Moments of design parameters-Shear failure criterion-NGES data

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+=

avo

acTC

p

pq

/

/log0.116.17 10σ

φ

Analysis of allowable pressureDeterministic approach

Probabilistic approach for system reliability index of three

Allowable bearing pressure-Keswick clay

(Skempton 1951)

For undrained conditions

For footing with Df/B=1.1

Spatial averaging CoV of Su or cu

Total CoV of Suor cu

All three components of uncertainty

Bearing capacity – Keswick clay

Factor of safety Vs. Reliability index shows that lower FS can be allowed.

Bearing capacity of clays– Power plant site -India

A proposed 445 MW Konaseema EPS Oakwell gas-fired combined cycle power plant on the East coast in Indian state of Andhra Pradesh

All three components of uncertainty

Bearing capacity-Power plant clay site

Effect of anisotropic spatial correlation

Variance reduction factors for 2-D space, Lv=2 m and Lh=7 m

Effect of anisotropic spatial correlation

Coefficient of variation of bearing capacity (autocorrelation distance in the vertical direction=0.19 m)

Effect of anisotropic spatial correlation

Assumption of isotropic correlation structure influences reliability

RELIABILITY BASED DESIGN OF PILE RELIABILITY BASED DESIGN OF PILE FOUNDATIONSFOUNDATIONS

CONVENTIONAL DESIGN METHODOLOGYCONVENTIONAL DESIGN METHODOLOGY

•• VERTICALLY LOADED PILEVERTICALLY LOADED PILE– Ultimate axial load carrying capacity:– Qu = p x L x qus + Ab x qub - W– D: Pile diameter– L: Length of pile – p: Pile perimeter = pi x D– Ab: Area of pile base– qus: Ultimate unit skin frictional resistance– qub: Ultimate unit end bearing resistance– W: Weight of pile– Design load capacity: Qu/FOS; FOS varies from 2-3– qus and qub are the functions of shear strength of soil

qsu

qsb

qu

D

LW qsu

qsb

qu

D

LW

qus

qub

qusqsu

qsb

qu

D

LW qsu

qsb

qu

D

LW

qus

qub

qus

•• INADEQUECIES IN THE PRESENT APPROACHINADEQUECIES IN THE PRESENT APPROACH– It is not unique and varies significantly over a wide range– In-situ behaviour of the pile foundation is considerably influenced by

variability in soil properties. – Handles ultimate state and serviceability states separately

EVIDENCE OF VARIABILITYEVIDENCE OF VARIABILITY•• 4 PILE LOAD TEST RESULTS 4 PILE LOAD TEST RESULTS

(TEJCHMAN & GWIZDALA, 1977)(TEJCHMAN & GWIZDALA, 1977)– For 4 pile load tests:– Pile Diameter: 1.5 m– Pile length: 12 m– Load tests are from the same site– If the allowable settlement is 0.02 m,

the allowable load varies from 3200 kN3200 kN--5020 kN5020 kN

– This variation indicates the randomness of pile-soil interface properties

0

1000

2000

3000

4000

5000

6000

7000

0 0.01 0.02 0.03 0.04 0.05Settlement (m)

Load

(kN

)

Case S/7

Case S/1

Case S/3

Case S/6

0

1000

2000

3000

4000

5000

6000

7000

0 0.01 0.02 0.03 0.04 0.05Settlement (m)

Load

(kN

)

Case S/7

Case S/1

Case S/3

Case S/6

•• SOURCES OF VARIABILITYSOURCES OF VARIABILITY– Inherent soil variability: in-situ variation in soil strength parameters depth wise– Measurement error: due to the process of measurement of field data– Transformation uncertainty: use of various transformation model to estimate

soil parameters (say e.g. undrained shear strength from CPT data)

EFFECT OF SPATIAL VARIABILITY ON PILEEFFECT OF SPATIAL VARIABILITY ON PILE•• SPATIAL AVERAGINGSPATIAL AVERAGING

– The fluctuation in the soil property tends to cancel in the process of spatial averaging.

– Spatial averaging length, which is equal to the failure zone, needs to be considered in the reliability analysis of foundations.

– The larger the length over which the property is averaged, higher is the fluctuation that tends to cancel in the process of spatial averaging. This causes reduction in standard deviation.

•• ESSENTIAL PAREMETERSESSENTIAL PAREMETERS– Vertical scale of fluctuation: indicates the distance, within which soil property

shows strong correlation.– Averaging length: for pile shaft it is length of the pile (L)(L) and and for pile base it is

the failure zone at the pile toe i.e. (r(rtt + r+ rbb)).

rrt t = D e= D etan(tan(πφπφ))

rrbb= D e= D eφφtan(tan(φφ) ) cos(cos(φφ))

For clay, rt and rb = D

0

2

4

6

8

10

12

14

16

18

20

0 5 10qc (MPa)

Dep

th (m

)

0

2

4

6

8

10

12

14

16

18

20

0 5 10qc (MPa)

Dep

th (m

)

DLL

Typical CPT profile

Failure zone

RELIABILITY BASED DESIGN APPROACHRELIABILITY BASED DESIGN APPROACH

•• STEPSTEP--1: CONE TIP RESISTANCE PROFILE AND 1: CONE TIP RESISTANCE PROFILE AND DETERMINATION OF SOIL PARAMETERSDETERMINATION OF SOIL PARAMETERS– CPT profile from Konaseema site (India)– Soil parameters:– where DK=1/NK is the empirical constant, is the total overburden stress,

su is the undrained shear strength of soil, qc is the cone tip resistance.– Averaged su over a length of pile considered for skin friction.– Averaged su over failure zone near pile tip considered for end bearing.

)( vocKk

vocu qD

Nq

s σσ

−=−

=

voσ

Field load-settlement data

Fitted by t-z model

0

500

1000

1500

2000

2500

3000

0 0.01 0.02 0.03 0.04Settlement (m)

Axi

al lo

ad (k

N)

0

500

1000

1500

2000

2500

3000

0 0.01 0.02 0.03 0.04Settlement (m)

Axi

al lo

ad (k

N)

Field load-settlement data

Fitted by t-z model

0

500

1000

1500

2000

2500

3000

0 0.01 0.02 0.03 0.04Settlement (m)

Axi

al lo

ad (k

N)

0

500

1000

1500

2000

2500

3000

0 0.01 0.02 0.03 0.04Settlement (m)

Axi

al lo

ad (k

N)

•• STEPSTEP--2: INTERFACE PARAMETERS2: INTERFACE PARAMETERS– Interface parameters, (i) average shear modulus

of pile-soil interface (ii) ultimate soil-pile interface shear strength and (iii) end bearing soil elastic modulus: Obtained by fitted with load test data.

– Interface parameters = constant x undrained shear strength

– Undrained shear strength: from CPT data

– Mean and standard deviation of constants are obtained by fitting several numbers of field pile load-settlement test data.

– Statistical estimates of soil shear strength are obtained from CPT data.

•• STEPSTEP--3: EVALUATION OF VARIABILITY3: EVALUATION OF VARIABILITY– The spatially averaged combined COV is described as (Phoon & Kulhawy,

1999):( ) 2

2

2222

1

)(tr

t

vo

mis COV

COVCOVLCOV

au+

⎟⎟⎠

⎞⎜⎜⎝

⎛−

+Γ≈

μσ

– where is the average total overburden pressure over the averaging length L, is the mean value of over a depth L, COVi is the COV of inherent variability, COVtr is the COV of transformation uncertainty and COVm is the COV of measurement error.

– is the variance reduction function given by Vanmarcke (1983):

voσtμ cq

( )•Γ 2

⎟⎟⎠

⎞⎜⎜⎝

⎛+−⎟⎟

⎠

⎞⎜⎜⎝

⎛= − usz

u

u L

sz

sz eLL

L δ

δδ

Γ /22

2 122

2)(

uszδ– is the vertical scale of fluctuation

– Determination of vertical scale of fluctuation and variance reduction

-5000 0

5000

10000

15000

05

1015

20D

epth (m)

qc (kPa)

t= 478.25zt=trend function; z=deptht= 478.25zt=trend function; z=depth

Linear trendLinear trend

Residual

Residual

Cone tip resistance

Cone tip resistance

COVi

0.0

0.2

0.4

0.6

0.8

1.0

0 1 2 3 4 5Lag distance, τ (m)

Auto

corr

elat

ion,

ρsu

Autocorrelation

Fitted autocorrelation functionρsu = exp(-2τ/0.85) ; R2 = 0.9

hence, δz = 0.85 m

Exponential fit

( )•Γ 2

•• STEPSTEP--4: RELIABILITY ANALYSIS4: RELIABILITY ANALYSIS– Basic random variables: undrained shear strength near pile shaft, near pile tip,

constants and allowable settlement / serviceable settlement.– Standard deviation, scale of fluctuation and variance reduction of undrained

shear strength is obtained from CPT data.– COV of measurement error : COVm = 15 % (Phoon & Kulhawy, 1999)– COV of transformation variability : COVtr = 29 % (Phoon & Kulhawy, 1999)– COV of serviceable settlement : 58.3 % (Zhang et al., 2005)– Random variables follow log-normal distribution.

0 0.01 0.02 0.03 0.04

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

Settlement (m)

Axi

al lo

ad (k

N)

Mean load-settlement curve

MCS generated curves

0 0.01 0.02 0.03 0.04

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

Settlement (m)

Axi

al lo

ad (k

N)

0 0.01 0.02 0.03 0.04

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

Settlement (m)

Axi

al lo

ad (k

N)

0 0.01 0.02 0.03 0.04

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

Settlement (m)

Axi

al lo

ad (k

N)

Mean load-settlement curve

MCS generated curves

– Load-settlement curves are generated by t-z method from mean and standard deviations of interface parameters using Monte Carlo simulations.

– For an applied load Q, number of sample realizations that exceed the ultimate load as well as allowable settlement are computed and expressed in terms of probability of failure.

– 5000 Monte Carlo samples are used.

– Ultimate Limit State (ULS): When the applied load is greater or equal to pile ultimate load carrying capacity, the probability of failure due to applied load is estimated by Monte Carlo Simulation (MCS):

samplesofnumberTotalQloadunderdsettlementtoscorrespondloadultiamtetheexceedingsamplesofNumber

p f)05.0(

1 =

– Reliability index corresponding to ultimate limit state criteria: – Serviceability Limit State (SLS): When the settlement is greater or equal to

serviceable limit (SSER), the probability of failure due to serviceable criteria is estimated at any axial load by MCS:

( )1

11fULS p−Φ= −β

samplesofnumberTotalQloadunderSsettlementallowableeserviceabltheexceedingsamplesofNumber

p SERf =2

– Reliability index due to serviceable limit state criteria:– System reliability:

( )2

11fSLS p−Φ= −β

)()()()( SERufSERfufSERuf SSQQpSSpQQpSSQQp ≥∩≥−≥+≥=≥∪≥

( )SERfSERufSERfuf SSpSSQQpSSpQQp ≥≥≥−≥+≥= )|()()(

System reliability index: ( )( )SERufSYS SSQQp ≥∪≥−Φ= − 11β

•• DESIGN LOADDESIGN LOAD– Conventional FOS design gives

design load, Q = 413 Q = 413 kNkN–– For For ββSYSSYS = 2,= 2,–– Q = 350 Q = 350 kNkN ((SSSERSER = 0.015 m= 0.015 m))–– Q = 410 Q = 410 kNkN ((SSSERSER = 0.025 m= 0.025 m))–– Q = 452 Q = 452 kNkN ((SSSERSER = 0.030 m= 0.030 m))–– For For ββSYSSYS = 2.5,= 2.5,–– Q = 320 Q = 320 kNkN ((SSSERSER = 0.015 m= 0.015 m))–– Q = 375 Q = 375 kNkN ((SSSERSER = 0.025 m= 0.025 m))–– Q = 422 Q = 422 kNkN ((SSSERSER = 0.030 m= 0.030 m))

00.20.40.60.8

11.21.41.61.8

2

1.5 2 2.5 3

Target reliability index, β

Req

uire

d pi

le d

iam

eter

(m)

Serviceable settlement = 0.015mServiceable settlement = 0.025mServiceable settlement = 0.030m

•• CHOICE OF PILE DIAMETERCHOICE OF PILE DIAMETER– If the pile is designed for the target

reliability indices of 2.0, 2.5, and 3.0, required diameters are 0.8 m0.8 m, 1.2m1.2m, 1.7m1.7m for SSER = 0.015m.

– For the same reliability, the required pile diameters are 0.7 m0.7 m, 1.0 m1.0 m, 1.4 m1.4 m for SSSERSER = 0.025= 0.025 mm and 0.5 m0.5 m, 0.7 m0.7 m and 1.0 m1.0 m respectively if SSSERSER = 0.030 m= 0.030 m.

1

1.5

2

2.5

3

3.5

4

200 250 300 350 400 450 500

Design load (kN)

Syst

em re

liabi

lity,

βS

YS

Conventional FACTOR OF SAFETY Approach

DP = 0.8 mLP = 15 mCOV = 38 %

SSER =0.015 m

SSER =0.025 m

SSER =0.03 m

LATERALLY LOADED PILESLATERALLY LOADED PILES•• DESIGN OF LATERALLY LOADED PILEDESIGN OF LATERALLY LOADED PILE

– Maximum lateral displacement at pile head.– Maximum bending moment

Mmax

ohpp

o

PkIEP

32.0)(707.066.0

4/3*25.0

*

−=

δδ

δ

94.0

92.0

92.0*

02.0

max )()(

)()(

52.0h

opp

kPIE

M ⋅=δ

δP0 – Load Maximum lateral displacement relation:

– Load Maximum bending moment relation:

– kh is the coefficient of lateral subgrade reaction– d is the pile diameter– EpIp is the uniform flexural rigidity of the pile

δ* is the yield displacement of soilRef: Hsuing and Chen (1997)

– The coefficient of lateral subgrade reaction: kkhh = = κ κ ssuu/d/dκκ is the correlation parameteris the correlation parameter

EVALUATION OF VARIABILITYEVALUATION OF VARIABILITY•• RANDOM VARIABLESRANDOM VARIABLES

– Undrained shear strength of soil, ssuu (mean value: average over the pile length)

– Correlation constant, κκ– Coefficient of lateral subgrade reaction, kkhh

– COV of kkh h : : 22

κCOVCOVCOVuh sk += 10 %

From CPT data•• MEAN AND VARIANCE OF MEAN AND VARIANCE OF

RESPONSERESPONSE– Mean maximum lateral displacement : ( ) ohpp

o

PkIEP

32.0)(707.066.0

4/3*25.0

*

−=

δδδ

– Variance of maximum lateral displacement : 22

2h

hh

k

kkathk

σδ

σδ ⋅⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂=

=

– Mean maximum bending moment :

– Variance of maximum bending moment :

( )( ) ( )( ) ( ) ( ) ( )2

2

4/325.02

04/325.0

202 .

32.0707.0

35.0hkhPP

hPP

kCOVkIEPkIE

Ph⎟

⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛⋅

−

−=

∗

∗

δ

δσδ

94.0

92.0

92.0*

02.0

max )()(

)()(

52.0h

opp

kPIE

M ⋅=δ

( ) ( )( ) ( )

( )22

94.192.0

92.00

02.02 .5.0

max hk

k

PPM kCOV

kPIE

h⋅⎟

⎟⎠

⎞⎜⎜⎝

⎛ −=

∗δσ

Mean of kh

–– For an applied lateral load pile foundation is considered to be For an applied lateral load pile foundation is considered to be satisfactory:satisfactory:

–– Lateral displacement at pile head does not exceed allowable Lateral displacement at pile head does not exceed allowable displacement (1% diameter of pile) displacement (1% diameter of pile)

–– Maximum bending moment does not exceed moment capacity of pile Maximum bending moment does not exceed moment capacity of pile section.section.

–– Performance functions:Performance functions:( ) δδδ −= aG1

δαδ δ )1( COVa ⋅+=

( ) maxmax2 MMMG R −=

PYR ZF ⋅=Μ

RELIABILITY ANALYSIS OF LATERALLY LOADED RELIABILITY ANALYSIS OF LATERALLY LOADED PILEPILE

Allowable lateral displacement

Mean of δ

δa

δ

α σδ

Resisting moment of pile section

MRMmax

Pro

babi

lity

dens

ity fu

nctio

n

–– CPT data: Konaseema area CPT data: Konaseema area (SCPT(SCPT--9) [Clay site]9) [Clay site]

DESIGN APPROACHDESIGN APPROACH

t= 478.25zt=trend function; z=depth

-5000 0

5000

10000

15000

05

1015

20D

epth (m)

qc (kPa)

Cone tip resistance

Residual

Linear trend

t = 23.01zt=trend function; z=depth

-200 0

200

400

600

800

05

1015

20

Undrianed shear strength(kPa)

Undrained shear strength

Residual

Linear trend

D =

0.8

mD

= 0

.8m

L=10

m

•• PARAMETERS OBTAINED FROM PARAMETERS OBTAINED FROM CPT PROFILECPT PROFILE– Mean value of subgrade reaction kh

= 36750 kN/m3

– Inherent variability, COVi = 37%– Scale of fluctuation : 0.85 m– Spatial COV of undrained shear

strength, COVsu = 38 %

•• ASSUMED PARAMETERSASSUMED PARAMETERS– Yield displacement of soil, δ* = 0.015

m– Allowable lateral displacement, δa =

0.008 m– Pile resisting moment, MR = 208

kNm– COVm = 15 %– COVtr = 29 %

•• CONVENTIONAL DESIGNCONVENTIONAL DESIGNδa = 0.008 m Lateral load = 424 kN

MR = 208 kNm Lateral load = 224 kNMin Lateral load = 224 kN FOS

= 2.75Design Lateral load = 82 kN

•• RELIABILITY BASED DESIGNRELIABILITY BASED DESIGNCOVkh= 39%

0

100

200

300

400

500

600

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014

Lateral displacement (m)

Late

ral l

oad,

P0 (

kN)

α = 4

α = 8

Deterministic load-displacement curve

(1+α.COVδa)δ

lines

a = 0

.008

mPdet = 424 kN

Pall = 248 kN

Pall = 150 kN

βδ = 2.8

βδ = 4.3

(a)

βδ = 2.8

0

50

100

150

200

250

300

350

0 100 200 300 400

Maximum bending moment (kNm)

Late

ral l

oad,

P0 (

kN)

Deterministic load-maximum moment curve

MR

= 20

8 kN

m

Pdet = 224 kN

Pall = 150 kN βmom = 2.5

System reliability index β = 2.5 for α = 8

(b)

P0 = 248 kN βmom = 0.1

System reliability index β = 0.1 for α = 4

Mm

ax =

248

kN

m

Mm

ax =

97

kNm

Assume αObtain mean (δ) &

variance (σδ) of displacement

Obtain mean (Mmax) & Variance of

maximum moment for the design load

Plot (δ + α.σδ) δa = 0.008

βmom

Design load& βδ

MR = 208 kNmβsystem=βtarget

βsystemYES

Fina

l Des

ign

load

NO

Reliability based design lateral load obtained

150 kN > 82 kN (from conventional FOS design)

CONCLUDING REMARKSThe study shows that the probabilistic analysis of soil profile provides a format for quantifying the information about the subsurface condition of the site and it also provides the basis for predicting the reliability of the pile foundations.

Depending upon the uncertainty level and spatial variability of soil, allowable load can be suggested.

The study shows that, it is useful to choose a suitable value for serviceability limit, so that the combined reliability index is ensured from both the considerations of ULS and SLS.

Analysis of unsaturated slopes• Shear strength of unsaturated soils (Fredlund and Rahardjo 1993)• τ= c' + (ua-uw) tanφb+(σn-ua) tanφ'• Where

c' is effective cohesion (ua-uw) is matric suction, ua is pore-air pressure, uw is pore water pressure, φb is the angle indicating the rate of increase in shear strength relative to the increase in matric suction, σn is the total stress normal to the sloping surface, and φ' is effective friction angle

β=slope anglez=depth of failure plane

Surficial stability of unsaturated infinite slope model (Cho and Lee 2002)

Stability of unsaturated slopes Contd..

Suction variation with depth

Stability of unsaturated slopesContd..

Variation of FS with depth of failure plane for different elapsed periods

Stability of unsaturated slopesContd..

Variation of reliability index with depth of failure plane for different elapsed periods

Stability of unsaturated slopes

Influence of saturated hydraulic conductivityVariation of reliability index with depth of failure plane for elapsed time = 5 days

Variation of reliability index with depth of failure plane for elapsed time = 10 days

Variation of reliability index with depth of failure plane for elapsed time = 15 days

Influence of saturated hydraulic conductivity

Variation of reliability index with depth of failure plane for elapsed time = 20 days

ANALYSIS OF GRAVITY RETAINING ANALYSIS OF GRAVITY RETAINING WALLS BY RELIABILITY BASED WALLS BY RELIABILITY BASED

DESIGN OPTIMIZATIONDESIGN OPTIMIZATION

STABILITY ASSESSMENT OF GRAVITY WALLSSTABILITY ASSESSMENT OF GRAVITY WALLS

– The stability assessment of gravity retaining walls is characterized by many sources of uncertainty and variability

– The retaining wall system is modeled as a series-parallel combination of failure modes.

– The first order reliability method (FORM) is applied to estimate the component reliability indices of each failure mode and to assess the effect of uncertainties in design parameters.

– The analysis is performed by treating back fill and foundation soil properties, geometric properties of wall, reinforcement properties and concrete properties as random variables.

Optimum wall proportions for gravity retaining structures by targeting various system reliability indices needs to be computed

OBJECTIVEOBJECTIVE

STABILITY ASSESSMENTSTABILITY ASSESSMENT

aP

)90cos( δη +−aP

)90sin( δη +−aP

H

hL

( )hft LbSbLB ++++=

S

t

δ

1w

2w

3w

4w

F G H J

M KNOP

V

b

L

3H

2γ1φ ch

5w

η

( )η−90

)90(

δη +−

I

6w

Q R U

1γ φ

ηtL fb

FAILURE MODES CONSIDERED FAILURE MODES CONSIDERED

1. Overturning 1. Overturning failure failure

2. Sliding failure2. Sliding failure

3. Eccentricity 3. Eccentricity failurefailure

4. Bearing failure4. Bearing failure

FAILURE MODES CONSIDEREDFAILURE MODES CONSIDERED contdcontd……

5. Toe Shear failure5. Toe Shear failure

6. Toe moment failure6. Toe moment failure

7. Heel shear failure7. Heel shear failure

8. Heel moment failure8. Heel moment failure

Performance functionsPerformance functions1. Overturning Failure mode1. Overturning Failure mode

( )1 1R

O

Mg x

M= −∑

∑2. Sliding Failure mode2. Sliding Failure mode

( )2 1R

D

Fg x

F= −∑

∑3. Eccentricity Failure mode3. Eccentricity Failure mode

4. Bearing Failure mode4. Bearing Failure mode

( ) ( )3

/ 61

Bg x

e= −

( )4max

1uqg xq

= −

5. Toe shear Failure mode5. Toe shear Failure mode

6. Toe moment Failure mode6. Toe moment Failure mode

7. Heel shear Failure mode7. Heel shear Failure mode

8. Heel moment Failure mode8. Heel moment Failure mode

( )5 1c

vtoe

g x ττ

= −

( )6 1toe

utoe

MRg xM

= −

( )7 1c

vheel

g x ττ

= −

( )8 1heel

uheel

MRg xM

= −

( ) ( )( ) ( )

1,2,3,4 1 2 1

5,6,7,8 1 2 1

, , , , , , , , , ,

, , , , , , , , , , , , , , /( ), /( )

t h f

t h f c ck y stoe sheel

g x f c L L S b b t

g x f c L L S b b t f f A pt A pt

γ φ γ φ

γ φ γ φ γ

⎧ =⎪⎨

=⎪⎩

1γ = unit weight of backfill soil

φ

2γ

1φ

c tL

hL

S

fb

b

tcγ

ckf

yf /( )stoeA pt

/( )sheelA pt

= friction angle of backfill soil

= unit weight of foundation soil

= friction angle of foundation soil

= cohesion of foundation soil

= unit weight of concrete

= compressive strength of concrete

= yield strength of HYSD bars = steel reinforcement ratio in the toe slab= steel reinforcement ratio in the heel slab

= width of stem at top of wall

= batter width of front face of wall

= batter width of back face of wall

= length of toe slab

= length of heel slab

= Width of stem at top of wall

Parameters to optimizeParameters to optimize

Reliability indices satisfying all the constraints in the form oReliability indices satisfying all the constraints in the form of f performance functions as given belowperformance functions as given below

( ) ( ) ( ) ( )( ) ( ) ( ) ( )

1 2 3 4

5 6 7 8

0; 0; 0; 0

0; 0; 0; 0

g x g x g x g x

g x g x g x g x

⎫≤ ≤ ≤ ≤ ⎪⎬

≤ ≤ ≤ ≤ ⎪⎭in the standard normal space U as in the standard normal space U as

2

1; 1 8 1

n

k ii

Minimize u k to and i to nβ=

= = =∑Reliability index corresponding to each limit state equation can be obtained using non-linear constrained optimization technique such as the method of Lagrange multipliers and is given by

Lagrange function 2

1( ) ; 1 8 1

n

k i k ki

L u g u k to and i to nλ=

⎛ ⎞= + = =⎜ ⎟

⎝ ⎠∑

The stationary points can be found by solving the following equations

( ) 0k iL u∂ ∂ = ( ) 0k iL λ∂ ∂ = where k = 1 to 8 and i = 1 to n

Statistics Random variable

Mean )( iμ

Coefficient of variation

)( iCOV Distribution

1γ 18 kN/m3 7% Normal φ 30o 5% and10% Log-Normal

2γ 19 kN/m3 7% Normal 1φ 20o 5% Log-Normal

c 30 kN/m2 5% to 20% Log-Normalcγ 24 kN/m3 5% Normal

ckf 20 kN/m2

(M20 concrete is assumed for the present study) 10% Normal

yf 415 kN/m2 (Fe 415 steel HYSD bars) (Fe 415 steel is assumed for the present study) 5% Normal

⎟⎟⎠

⎞⎜⎜⎝

⎛pt

Astoe 0.5% Normal

⎟⎟⎠

⎞⎜⎜⎝

⎛pt

Asheel 0.5% Normal

tL 0.5% Normal

hL 0.5% Normal S 0.5% Normal

fb 0.5% Normal b 0.5% Normal t

Mean values of wall proportions and area of reinforcement in toe and heel slab should be

obtained from the optimizion for target system reliability indices

0.5% Normal

Statistics of input parametersStatistics of input parameters

Identification of MPP in FORMIdentification of MPP in FORM

In the standard normal space, the point on the first order limit state function at which the distance from the origin is minimum is the Most Probable Point of failure (MPP) and the shortest distance corresponding to MPP is called as reliability index ( )

β

β

SeriesSeries--Parallel Combination Model ConsideredParallel Combination Model Considered

System reliability based optimizationSystem reliability based optimizationOverall stability of gravity retaining wall system is influencedby overturning, sliding, eccentricity, bearing, toe shear, toe moment, heel shear and heel moment failure modes.

Toe slab failure sequence is a parallel system of its toe shear and moment failure events as shown in above Figure. Probability of failure of toe slab is given by

( )( ) ( )( ) ( )( ){ } ( )( ){ } ( ){ } ( ){ }_ 5 6 5 60 0 0 0f toe tsh tmP P g u g u P g u P g u β β⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤= < ∩ < = < < = Φ − Φ −⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎣ ⎦

probability of failure of heel slab is given by

( )( ) ( )( ) ( )( ){ } ( )( ){ } ( ){ } ( ){ }_ 7 8 7 80 0 0 0f heel hsh hmP P g u g u P g u P g u β β⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤= < ∩ < = < < = Φ − Φ −⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎣ ⎦

Assuming that the overturning, sliding, eccentricity, bearing, toe slab and heel slab failure modes are statistically independent, Probability of failure of the wall system having series-parallel combination model can be computed as follows

( )( ) ( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )( ) ( )( )

1 2 3 4

_5 6 7 8

0 0 0 0

0 0 0 0f system

g u g u g u g uP P

g u g u g u g u

⎡ ⎤< ∪ < ∪ < ∪ <⎢ ⎥=⎢ ⎥⎡ ⎤ ⎡ ⎤∪ < ∩ < ∪ < ∩ <⎣ ⎦ ⎣ ⎦⎣ ⎦

( ){ } ( ){ } ( ){ } ( ){ }( ){ } ( ){ }_

1 1 1 11

1 1ot sli e b

f systemtoe heel

Pβ β β β

β β

⎡ ⎤− Φ − − Φ − − Φ − − Φ −⎢ ⎥= −⎢ ⎥− Φ − − Φ −⎣ ⎦

System reliability index of gravity retaining wall is

( )1_1sys f systemPβ −= Φ −

COMPONENT RELIABILITY VS SYSTEM RELIABILITY

Fig. 5. Variation of component reliability indices ( iβ ) and system reliability index( sysβ ) with batter width of back face ( /b H ) of gravity retaining wall for COV of φ ,c and 1φ = 5%, COV of 1γ and 2γ = 7 % and /stoeA pt = 0.10% and /sheelA pt =0.26%

Variation of batter width of back face ( /b H ) and front face( /fb H ) of gravity retaining wall with target system reliabilityindex ( sysβ ) for COV of φ = 5% & 10% and COV of c = 5%

OPTIMUM WALL PROPORTIONSOPTIMUM WALL PROPORTIONS/S H = 0.05, /tL H = 0.07, /hL H = 0.07, /t H = 0.07

Area of HYSD steel bars in the toe slab ( /( )stoeA pt ) = 0.10 % Area of HYSD steel bars in the heel slab ( /( )sheelA pt ) = 0.26 %

_ argsys t etβ

2.5 3.0 3.5 4.0 4.5

/fb H /b H /fb H /b H /fb H /b H /fb H /b H /fb H /b H

0.28 0.02 0.28 0.048 0.28 0.08 0.28 0.12 0.28 0.17

- - 0.30 0.02 0.30 0.052 0.30 0.09 0.30 0.15

- - - - - - 0.32 0.06 0.32 0.12

- - - - - - 0.34 0.02 0.34 0.10

- - - - - - - - 0.36 0.06

- - - - - - - - 0.38 0.02

The areas of cross section from optimized sections are lesser than those obtained from the specifications.

Stability Assessment of Buried pipesStability Assessment of Buried pipes

Optimum diameter to thickness ratio and thickness of steel pipe for buried flexible pipes by targeting various reliability indices considering four failure criteria buckling, crushing, deflection and handling flexibility (FWHA 2001).

OBJECTIVEOBJECTIVE

– Owing to the uncertainties in soil friction angle and unit weight of the backfill, modulus elasticity of soil, modulus of elasticity and yield strength of steel pipe, the assessment of stability of buried flexible pipes needs to be on rational basis considering variability in design parameters.

STABILITY ASSESSMENTSTABILITY ASSESSMENT

Limit states Considered Limit states Considered 1. Limit state for Buckling failure 1. Limit state for Buckling failure

2. Limit state for Crushing failure2. Limit state for Crushing failure

Limit States Considered Limit States Considered

3. Limit state for Deflection failure 3. Limit state for Deflection failure

4. Limit state for Handling flexibility failure4. Limit state for Handling flexibility failure

Performance functionsPerformance functions1. Buckling Failure mode1. Buckling Failure mode

( )1Allowable buckling pressure ( ) 1

External pressure due to Marston's load ( )a

b

Pg xP

= −

2. Crushing Failure mode2. Crushing Failure mode

( ) ( )( )2

Yield stress of the pipe material 1

Ring compressive strength or Bending stress y

A e

fg x

f or f= −

3. Deflection Failure mode3. Deflection Failure mode

4. Handling flexibility Failure mode4. Handling flexibility Failure mode

( ) ( )( )3

Allowable deflection (5% of diameter of pipe)1

Horizontal deflection of pipe a

g xx

Δ= −

Δ

( ) ( )( )

max4

Maximum permissible flexibility factor FF1

flexibility factor FFg x = −

Methodology for optimization Methodology for optimization 1. Assume a trial thickness of the steel pipe and find the diameter to thickness ratio of steel pipe for desired target reliability index against buckling failure using the formulationgiven below

( )T

minimize

subjected to

buck

buck

g u

u u β

⎧⎪⎨

=⎪⎩

2. Thickness of the steel pipe is needed to evaluate for the computed value of diameter to thickness ratio in the step 1 for the desired target reliability index against crushing failure using the formulation given below

( )T

minimize

subjected to

crush

crush

g u

u u β

⎧⎪⎨

=⎪⎩

Methodology for optimization Methodology for optimization Verify whether the thickness computed in step 2 is equal to the assumed trial thickness value, if not then again modify the thickness of the steel pipe and then go back to step 1 to evaluate the diameter to thickness ratio and iterate the process.

3. Reliability indices against deflection failure and handling flexibility failure are needed for the established diameter to thickness ratio and thickness of the steel pipe in steps 1 and 2 for the desired target reliability indices using the formulations given below

( )

Tminimize

subjected to def

def

u u

g u

β⎧ =⎪⎨⎪⎩ ( )

Tminimize

subjected to flex

flex

u u

g u

β⎧ =⎪⎨⎪⎩

Verify whether the reliability indices and computed in step 3 are equal to the desired target reliability indices, if not then iterate the entire process (starting from the computation of diameter to thickness ratio in step (1) until the criterion is met.

Statistics Random variable

Mean )( iμ

Coefficient of variation

)( iCOV Distribution

γ 18 kN/m3 7% Gaussian φ 30o 10% Log-Normal

soilE 1537.8 kN/m2 5%, 10%, 15% and 20% Gaussian

E 82.1374 10× kN/m2 5% Gaussian yf 228000 kN/m2 5% Gaussian

ν 0.3 0 - H 5.0 m 0.5% Gaussian

dB 2.0 m 0.5% Gaussian q 7000 kN/m2 30% Gaussian /D t

ratio

0.5% Gaussian

t

Mean value of pipe diameter to thickness ratio ( /D t ) and thickness of steel pipe should be obtained from the Target reliability based design optimization (TRBDO) for the target component reliability indices 0.5% Gaussian

Statistics of input parametersStatistics of input parameters

ConclusionsThe probabilistic analysis of the soil data and soil profiles provides a format for quantifying the information about the subsurface condition of the site.it also provides the basis for obtaining the response statistics which are useful in the the reliability analysis of geotechnical structures.Reliability based optimization is useful in the design of geotechnical structures

G. L. G. L. SivakumarSivakumar BabuBabuDepartment of Civil EngineeringDepartment of Civil EngineeringIndian Institute of Science Indian Institute of Science Bangalore Bangalore

Email:gls@civil.iisc.ernet.inEmail:gls@civil.iisc.ernet.in

Thank you for your Thank you for your attentionattention