Cristina de H.C. Tsuha & Nelson AokiUniversity of São Paulo / Brazil

RISK EVALUATION ACCORDING TO STANDARDS

Risk“The term risk implies a combination (the product) of theprobability of an event occurring and the consequences

of the event should it occur”“Probability of failure is a measure of risk only if all failure modes

result in the same consequences”Lacasse & Nadim (1998)

Probability and costs of foundation problems

Their impactsLabor, Materials , Equipment,

Business Costs, EnvironmentalCosts , Social Costs , Deaths, etc.

Risk cost = x

• Structural collapse• Excessive settlements

probability of failure cost of failure

1

Risks associated with pile foundation

Design of pile foundations involves manylimitations and uncertanties

DESIGN PROBLEM

GOAL : minimize the risks(acceptable level/ economical)

Limited calculation models

Limited ground investigation

Uncertanties in ground parameters

Spatial variability

Bauduin (2003)

2

R1 RnRiR4

R3

R2

Variability of pile resistance in a construction project

Resistance (kN)

Example of aconstruction project

R mean = 3295 kN

Standard deviation = 483 kNCoefficient of variation = 14,7 %

Dynamic measurements(CAPWAP) on 74 piles

Freq

uenc

y(%

)

3

R1 RnRiR4

R3

R2

E1Ei EnE4E3

E2

Action effect : E

E = action effectsE , E

vE = E / E

Coef. de variation of action effects4

R-E should be > 0Mathematically:pile does not fail

Margin of safety (M = R-E)

M

σM

mM

pf

y

0

Probability of failure and Reliability index β

β = µZ / σZ

22ER

ER

Normally distributed random variable

5

β

Lognormal distribution

)1(1ln

1

1ln

22

2

2

RE

R

E

E

R

vv

v

v

6

Probability of failure and Reliability index β

Formulations (Freudenthal) involved convolution functions(R and E distribution) to obtain pf

Probability of failure and Reliability index β

7

Probabilistic Deterministic

x

y

µR

vR2

µE

vE

µR

‘

vR3vR1

µRµR

Allowable Stress Design

E

RF calc

S

constant Fs

same Fs ≠ pf

Allowable Stress Design

8

Safety factor and probability of failure

The factor of safety is therefore not a sufficient indicator of safety margin because theuncertainties in the analysis parameters affect probahility of failure

uncertainties do not interven in the deterministic calculation of safety factor.

9

Lacasse & Nadim (1998)

Reliability levels of a construction project

Level zero: deterministic methodsrandom variables are taken as deterministic and uncertainties are taken intoaccount by a global safety factor (based on past experience)

Level I: semi-probabilistic methodsdeterministic formulas are applied to representative values of RVs multiplied bypartial SFs. The characteristic values are calculated based on statisticalinformation/ the partial SFs are based on level II or level III reliability methods

Level II: approximate probabilistic methodsRVs are characterised by their distribution and statisticalparameters probabilisticevaluation of safety achieved using approximate numerical techniques

Level III: full probabilistic methodsTechniques that take into account all of the probabilistic characteristics of the RVs

Level IV: risk analysisprobabilistic characteristics & consequences of failure are taken into account

The level of accuracy depends on the way that uncertainties are considered in the design(Teixeira et al. 2012)

10

Load and Resistance Factor Design (LRFD)

LRFD is appropriate for geotechnical designs because:the variabilities and uncertainties associated with natural systems (the ground in this

case) are much greater than those associated with well-controlled engineeredsystems

The specifications were calibrated based on a combination of simplistic reliabilityanalysis, fitting to WSD and engineering judgment.

11

Lacasse & Nadim (1998)

(Paikowsky 2004)

Load and Resistance factor design

Separate uncertanties in loading from uncertanties in resistance

Use procedures from probability theories

LRFD requires a selection of a set of target reliability levels (β)

LRFD formulation – Pile foundations

12

Traditional design

E

RF calc

S

Single (Global) Safety Factor(margin for error and uncertainty in actions and resistances)

Design value ofaction effect

LRFD, Partial factor method (Eurocode 7)Limit state design concept with partial factors andcharacteristics values

Ed Rd

Design value ofresistance

to obtain appropriate levelsof reliability (RBD methods)

related to a specificcalculation model

LRFD equations – Pile Foundations

13

Ed Rd

Compressive resistance

. " " . partial factoron pileresistance(European)

characteristic pileresistance

partial factors of permanent and variableaction effect ; + ;

base shaftCharacteristic pile resistance Rk:• Uncertanties related to calculation method

• Variability over the construction site

∅. reductionfactor(other codes)

Calibration of partial factors

14

Eurocode (EN 1990)

Partial factors linked to reliability index β

15

Low probabilityof failure

Ex: β = 3.8, Pf = 7.2 x 10-5

Partial factors (gvalues)

Reliability levels for representative structures asclose as possible to the target reliability indexbT

reliability index βRelated to a probability of failure

Quantity to evaluate “safety”

Density functionsof R and EE, E , vE

R, R , vR

Partial factors linked to reliability index β

16

FOSM reliability formulas

Lognormal distribution is often used:Sensitivity factors

E and vE ?R and vR ?

Partial factors linked to reliability index β

17

R = P . Rcal

Bias of the resistance funcion

E = E . Ek

Bias for the actionVE and VR ?

Model uncertainty (P and Vp )

Variability of R over the site ( )

Variability of effects of execution (monitoring)

Model uncertanty

18

Bauduin (2003)

Model uncertanty

Random variable B

VB

mB

R = P . Rcal

coefficient of variationcalculated resistance

If load tests were performedp% of measured would be lower

than prediction

Model factor m

Reliability of the calculation model

Model uncertanty

19

Bauduin (2003)

Model factor mod

For normal distribution

For lognormal distribution

Partial factors (calculation model uncertanty)

20

Design value of resistance Rd

.; + ;; + ; . 1

mod

P and Vp (different types of pile in different types of ground)

Correlation factor : spatial variability

21

Characteristic pile resistance

Ground tests resultsStatic load testsSpatial variability :

; + ; . 1.

Stiffness of the structure and monitoring

22

stiffness

Bauduin (2003)

transfer loads from weaker to stronger pilesFavorable effect

monitoring

Reduce uncertanty related to installation effects

Actions (permanent and variable loads)

23

Load factor: F Bias factor

ACTION EFFECT

Partial factors linked to reliability index β

24

FOSM

Targetreliability

ULS ocurrence: Ed = Rd VE VR

reductionfactor

(AASHTO)∅.

Brazilian code: NBR 6122 (2010) Recognize the risks involved in Foundations

Introduces the concept of correlation factor (static load tests &ground tests) to deal with the spatial variability of pile resistance

25

Static load tests (number and type of piles)

5 dynamic(CAPWAP)for 1 static

Estimation of vR using the Brazilian code

Correlation factors (static tests)

4

min,

3

,,

)(,

)(

mcmeanmc

kc

RRMinR

Variability of pile resistance (vR or R)

645.1

)( ,, kcmeanmcR

RR Reliability index

pf = 1-()Probability of failure

22ER

ER

simple closed formnormal reliabilitycalculation formula

E and R assumed to be normally distributed

uncertanty of calculations models (based on SPT test) ? ? ?

“ No information about bias of calculation models”

Estimation of β and Pf

26

P ???

R density distribution

27

RObtain R density distribution from:static tests, dynamic tests, dynamic formula, etc.VR

Example

Resistance (kN)

R measured Numberof piles

mean(kN) COV (%)

Static tests 4 3756 12,3Capwap 74 3295 14,7Dynamic formula 2506 3231 16,0

Best fit distribution for R

Formulations (Freudenthal) involvedconvolution functions (R and E distribution)to obtain and Pf

Update during and after construction

(normal, lognormal, beta, etc.)

Christian 2004, Baecher & Christian 2005, Phoon et al 1995, and Phoon et al. 2003):

• The uncertainties in geotechnical engineering are largely inductive: starting fromlimited observations, judgment, knowledge of geology, and statistical reasoning areemployed to infer the behavior of a poorly-defined universe.

• The probabilistic methods help to relieve the foundation engineer from the ill-suitedtask of assessing the complex relationship between uncertainties and risks intuitively,while at the same time emphasizing the importance of engineering judgment andexperience on the other design aspects that are currently beyond the scope ofmathematical analysis.

• The geotechnical engineer’s role is not solely to provide judgment on selection ofparameters, methods of calculations and resulting safety, but also to take an activepart in the evaluation of hazard, vulnerability and risk.

• Communication of risk within a transparent and rational framework is necessary inview of increasing interest in code harmonization public involvement in definingacceptable risk levels, and risk-sharing among client, consultant, insurer, and financier.

Advantages of employing RBD

• The probability theory can provide a formal framework for developing design criteriathat would ensure that the probability of "failure" (to refer to exceeding of anyprescribed limit state) is acceptably small.

28

Formalizationof uncertantyand risk

Partialfactors

BrazilianStandard

29

Conclusions

UncertantiesDesign of pile

foundations