Eurocodesand design of structures Philosophy of partial ...

Eurocodes and design of structures

Philosophy of partial factors calibrationPhilosophy of partial factors calibration

Jean-Armand Calgaro

Ingénieur général des ponts et chaussées e.r.

Chairman of CEN/TC250 (2007-2013)

Marcus Vitruvius Pollio (born c. 80–70 BC, died after c. 15 BC), Roman

author, architect, civil engineer and military engineer during the 1st century

BC, known for his multi-volume work entitled De Architectura.

Vol. III – Section 1:

“The design of Temples depends on symmetry, the rules of which Architects

should be most careful to observe. Symmetry arises from proportion (…). should be most careful to observe. Symmetry arises from proportion (…).

Proportion is a due adjustment of the size of the different parts to each other

and to the whole; on this proper adjustment symmetry depends. Hence no

building can be said to be well designed which wants symmetry and

proportion. In truth they are as necessary to the beauty of a building as to that

of a well formed human figure”.

Fra Luca Bartolomeo de Pacioli (c. 1447–1517) was an Italian

mathematician, Franciscan friar, collaborator with Leonardo da Vinci, and

a seminal contributor to the field now known as accounting

Pacioli published several works on mathematics, and more specifically De

divina proportione (written in Milan in 1496–98, published in Venice in

1509). The subject was mathematical and artistic proportion, especially

the mathematics of the golden ratio and its application in architecture. the mathematics of the golden ratio and its application in architecture.

Leonardo da Vinci, (15 April 1452 – 2 May 1519) was an Italian

polymath. His areas of strength included painting, sculpting, architecture,

science, music, mathematics, engineering, invention, anatomy, geology,

astronomy, botany, writing, history, and cartography.

Leonardo da Vinci

The Vitruvian Man

(c. 1485) Accademia,

Venice

The Golden Ratio

61803,12

51 ≅+=x

6 November 1953

CEB was founded

1970

International

Recommendations

CEB/FIP

A brief history of the reliability approach

For construction works

1906

First French

Technical « regulation »

Concrete constructions

1964

Practical

Recommendations

For constructors

1978

Model-Code 78

CEB/FIP

No specialist of geotechnical design participated to the discussions concerning

the problems of structural safety and there was no chapter in Part I of the

Model-Code 1978 devoted to concrete structures dealing with foundations. Model-Code 1978 devoted to concrete structures dealing with foundations.

This explains that the “semi-probabilistic format” has been, in reality,

“imposed” to geotechnical experts (and other experts …).

Eurocodius : Ratio firmitatisEurocodius : Ratio firmitatisEurocodius : Ratio FirmitasEurocodius : Ratio Firmitas

Latin versionLatin version

EN 1990 – EUROCODE

BASIS OF BASIS OF STRUCTURAL DESIGN

EN 1990 – 1.5 Terms and definitions (1)

1.5.2.17

reliability

ability of a structure or a structural member to fulfil the specified requirements, including

the design working life, for which it has been designed. Reliability is usually expressed in

probabilistic terms

NOTE Reliability covers safety, serviceability and durability of a structure.NOTE Reliability covers safety, serviceability and durability of a structure.

1.5.2.18

reliability differentiation

measures intended for the socio-economic optimisation of the resources to be used to

build construction works, taking into account all the expected consequences of failures

and the cost of the construction works

EN 1990 – 1.5 Terms and definitions (2)1.5.2.8

design working life

assumed period for which a structure or part of it is to be used for its intended purpose with anticipated

maintenance but without major repair being necessary

1.5.2.12

limit states

states beyond which the structure no longer fulfils the relevant design criteria

1.5.2.131.5.2.13

ultimate limit states

states associated with collapse or with other similar forms of structural failure

NOTE They generally correspond to the maximum load-carrying resistance of a structure or structural

member.

1.5.2.14

serviceability limit states

states that correspond to conditions beyond which specified service requirements for a structure or

structural member are no longer met

� Identification of undesired phenomena and situations (limit-states).

� Evaluation of the risks attached to these phenomena and situations.

The probabilistic approach of structural safety:

The heritage

� Adoption, for a construction work, of a design such that the probability of the

mentioned risks is limited to a low value in order to remain acceptable.

Two questions :

- What is a low value ?

- By whom this « low » value would be accepted ?

The limit-state concept

The semi-probabilistic verification format of construction works is based on

rules, partly empirical or historical, which introduce safety at three levels :

� Adoption of representative values of random variables (actions and

resistances) defined on clear bases;

The semi-probabilistic verification format of

construction works

resistances) defined on clear bases;

� Application of partial factors to actions and resistances;

� Introduction of safety margins more or less apparent in the various

models : models of actions, models of action effects and models of resistances.

6.4 Ultimate limit states

6.4.1 General

(1)P The following ultimate limit states shall be verified as relevant :

EQU : Loss of static equilibrium of the structure or any part of it considered as a rigid body, where :

minor variations in the value or the spatial distribution of actions from a single source are significant,

and the strengths of construction materials or ground are generally not governing ;

EN 1990 – Eurocode : Basis of Structural Design

and the strengths of construction materials or ground are generally not governing ;

STR : Internal failure or excessive deformation of the structure or structural members, including

footings, piles, basement walls, etc., where the strength of construction materials of the structure

governs ;

GEO : Failure or excessive deformation of the ground where the strengths of soil or rock are significant

in providing resistance ;

FAT : Fatigue failure of the structure or structural members.

EN 1990 – Basis of Structural Design - 6.3 Design valu es

dd RE ≤Ed is the design value of the effect of actions such a s internal force, moment or a vector representing several internal fo rces or moments ;

The semi-probabilistic format for the verification of construction works (ultimate limit states)

{ } 1≥= iaFEE direpiFd ;,,γ

Rd is the design value of the corresponding resistance.

{ }direpiFd ,,

= diM

ikid a

XRR ;

,

,

γη

6.5.1 Verifications

(1)P It shall be verified that :

The semi-probabilistic format for the verification of construction works (serviceability limit states)

dd CE ≤where :

Cd is the limiting design value of the relevant serviceability criterion.

Ed is the design value of the effects of actions specified in the serviceability

criterion, determined on the basis of the relevant combination.

∑ ∑≥ >

+++1 1

,,0,1,1,,, """"""j i

ikiiQkQPjkjG QQPG ψγγγγ

Expression (6.10)

6.4.3 - Ultimate limit-states STR/GEO - Fundamental combination for

persistent and transient design situations

EN1990 - Eurocode : Basis of structural design

≥ >1 1j i

Expressions (6.10a) and (6.10b)

+++

++

∑∑

∑∑

>≥

≥≥

1

,,0,1,1,

1

,,

1

,,0,

1

,,

""""""

""""

i

ikiiQkQP

j

jkjGj

i

ikiiQP

j

jkjG

QQPG

QPG

ψγγγγξ

ψγγγ

0,85 ≤≤≤≤ ξξξξ ≤≤≤≤ 1,00

∑>

++++1

,,01,inf,sup, 5,1""5,1""""""35,1i

ikikmkjkj QQPGG ψ

Expression (6.10)

Expressions for the fundamental combination of actions based

on recommended values of partial factors

for buildings


++++

+++

∑

∑

>

≥

1

,,01,inf,sup,

1

,,0inf,sup,

5,1""5,1""""""15,1

5,1""""""35,1

i

ikikmkjkj

i

ikimkjkj

QQPGG

QPGG

ψ

ψ


Accidental design situations : expression 6.11b

∑ ∑≥ >

++++1 1

,,21,1,21,1, "")(""""""j i

ikikdjk QQouAPG ψψψ


∑ ∑≥ ≥

+++1 1

,,2, """"""j i

ikiEdjk QAPG ψ

Seismic design situations : expression 6.12b

6.5.3 Serviceability limit states : combinations of actions

� Characteristic Combination (irreversible SLS)

∑ ∑≥ >

+++1 1

,,01,, """"""j i

ikikjk QQPG ψ


� Frequent Combination (reversible SLS)

� Quasi-permanent Combination (reversible SLS)

≥ >1 1j i

∑ ∑≥ >

+++1 1

,,21,1,1, """"""j i

ikikjk QQPG ψψ

∑ ∑≥ ≥

++1 1

,,2, """"j i

ikijk QPG ψ

Overview of reliability methods (EN 1990 – Annex C)

Deterministic

methods

Probabilistic methods

Historical methods Empirical methods

FORM (Level II)

Full probabilistic (Level III)

Calibration Calibration Calibration Calibration Calibration Calibration

Semi-probabilistic

methods (Level I)

Method c

Method a Partial factor design

Method b

The Eurocodes have been primarily based on method a. Method c or equivalent

methods have been used for further development of the Eurocodes.

NOTE An example of an equivalent method is design assisted by testing (see annex D).

In the Level II procedures, an alternative measure of reliability is conventionally defined

by the reliability index β which is related to Pf by :

β−Φ=

where Φ is the cumulative distribution function of the standardised Normal distribution.

The relationship between the target reliability index for one year, , and , the value for

the design working life equal to n years, is:

)( β−Φ=fP

1β nβ

[ ]n

n )( 1

1 ββ ΦΦ= −

Consequences

ClassDescription

Examples of buildings and

civil engineering works

Reliability

class

Recommended minimum

value for the reliability index

1 year

reference

period

50 years

reference

period

CC3

High consequence for loss of

human life, or economic, social

or environmental consequences

very great

Grandstands, public

buildings where

consequences of failure are

high (e.g. a concert hall)RC3 5,2

4,3

(= 3,8 + 0,5)

Definition of consequences classes and associated reliability classes

CC2

Medium consequence for loss

of human life, economic, social

or environmental

consequences considerable

Residential and office

buildings, public buildings

where consequences of

failure are medium (e.g. an

office building)

RC2 4,7 3,8

CC1

Low consequence for loss of

human life, and economic,

social or environmental

consequences small or

negligible

Agricultural buildings where

people do not normally enter

(e.g. storage buildings),

greenhousesRC1 4,2

3,3

(= 3,8 – 0,5)

In the « Common unified rules for different types of construction and material »

elaborated by JCSS in 1976, a target value for the probability of failure in one year,

Pf,1 = 1.10-5 was proposed for the reference case with serious economic consequences

and a medium number of persons endangered (the « acceptable » probability of

failure).

This corresponds to Pf,50 = 5.10-4 and the equivalent β value is 3.29.

Finally, the target value for β50, adopted for the reference case and the reference period R

= 50 years, is 3.8, which corresponds to Pf,50 = 7,2.10-5

For a reference period of 1 year, the corresponding β1 value is 4.7.

Model Code CEB 78 : Failure probability in 50 years (ULS)

PROBABILITY ASSOCIATED TO LIMIT STATES

Average num-ber of endan-gered people

Economic consequences

Low Medium High Low (< 0,1) 10-3 10-4 10-5

Medium 10-4 10-5 10-6 High (> 10) 10-5 10-6 10-7

Model Code CEB 78 : Failure probability in 50 years (ULS)

It should be stressed that the beta-values and the corresponding

failure probabilites are formal or notional numbers, intended

primarily as a tool for developing consistent design rules, rather thanprimarily as a tool for developing consistent design rules, rather than

giving a measure of the structural failure frequency. The CEB MC 78

should not be misinterpreted.

EN 1990 - Annex C

Basis for Partial Factor Design and Reliability Analysis

Target reliability index ββββ for Class RC2 structural members 1)

Limit state Target reliability index

1 year 50 years 1 year 50 years

Ultimate 4,7 3,8 Fatigue 1,5 to 3,8 2) Serviceability (irreversible) 2,9 1,5

1) See Annex B 2) Depends on degree of inspectability, reparability and damage tolerance.

Hypotheses :

R and E are independently Normally distributed with the usualparameters (µ R, σσσσR) and (µE, σσσσE)

⇒⇒⇒⇒ Z = R – E is also Normally distributed with the followingparameters :

Reliability - Basic approach

22)()( ERZERZ

Z

Z µµµµz

zF σσσσ

+=−=−Φ=

)(2

1

2

1)0(

_

2

0

_

2

)( 2

2

2

Z

Z

tx

Z

f

µdtedxeZPP

Z

Z

Z

Z

σππσ

σµ

σµ

−Φ===<= ∫∫

−

∞

−

∞

−−

∫∞

−=Φ

x t

dtex_

2

2

2

1)(

π

Zσ

(cumulative standardized Normal

distribution function)

Reliability index (Rjanitzyne-Cornell)

Pf = ΦΦΦΦ(- ββββ)

22

ER

ER

z

z

σσσµβ

+µ−µ==

ER

ey

rx

σσ==

E

EC

R

RC yx

σσµ=µ=

Geometrical interpretation (1)

Standardized coordinates

Design point

βαβα

ECP

RCP

yy

xx

−=−=

Rσα =

Likely state

C22

22

ER

EE

ER

RR

σσσα

σσσα

+−=

+=

E

E

R

R ey

rx

σσµ−=µ−=

Geometrical interpretation (2)

22

22

ER

EE

ER

RR

σσσα

σσσα

+−=

+=

P : Design point

αβ−=OP

)1(

)1(

RRRRRRP

EEEEEEP

VR

VE

βαβσαβαβσα

−µ=−µ=−µ=−µ=

Coordinates of the Design Point

RRRRRRP

Design values of actions and resistances

EEEdRRRd ER σαβµσαβµ 00 −=−=

0βTarget value of the reliability index :

0ββ ≥⇔≥ dd ER

)()(1

1)(Pr1)(Pr

00 βαβασ

EE

E

Eddd

µEEEobEEob

Φ=−Φ−=

−Φ−=≤−=>

)()(Pr 0βασ R

R

Rdd

µRRRob −Φ=

−Φ=≤

Design values and associated probabilities

Numerical approach

R

E

RERREER

EEERRRdd

with

t

ER

σσ

σβσσβ

σαβσαβ

= t

1 2

0

22

0

00

+−µ−µ=+−µ−µ=

+µ−−µ=−

Comparison between and T = 0,8 + 0,7t21 tT +=

0

1

2

3

4

5

6

7

T=0,8+0,7t

00 1 2 3 4 5 6 7

)7,0()8,0(

)7,08,0(

00

0

EERR

R

ERERdd ER

σβµσβµσσσβµµ

+−−=

+−−≅−

The choice of αE = -0,70 and αR = +0,80 leads to a safe-sided design point P with:

that, together with a lower limit

gives a tolerance field for practical applications, that is defined by the limits:

05,48,380,070,0' 22 =×+=β

55,380,305,4

80,3" =×=β

63,716,0 ==R

E

R

E MaxMinσσ

σσ

RR σσ

Conclusions

Dominating actions and resistances

Non-dominating actions

EEdRRd ER σβµσβµ 00 7,08,0 +=−=

)8,0()(Pr

)7,0()(Pr

0

0

ββ

−Φ=≤−Φ=>

d

d

RRob

EEob

)28,0()7,04,0()(Pr 00 ββ −Φ=−Φ=> xEEob d

Partial factors

00d

k

EEE

k

dF

EE

E σβαµγ 0−==k

FFFSd

k

dSdF

FF

F σβαµγγγ 0−==

RRR

k

d

kM

R

R

R

σβαµγ

0−==

fRf

kRd

d

kRdM

f

f

f

σβαµηγηγγ

0−==

Effects of actions and

Resistances

Individual actions and

resistances

Central safety factorE

R

µµµµµµµµγγγγ ====

Coefficients of variation

E

EE

R

RR VV

µµµµσσσσ

µµµµσσσσ ========

RRRk

EEEk

kµR

kµE

σσσσσσσσ

−−−−====++++====

Characteristicsafety factor

EE

RR

EEE

RRR

k

k

kVk

Vk

k

k

E

R

++++−−−−××××====

++++−−−−========

1

1γγγγσσσσµµµµσσσσµµµµγγγγ

)( ββββ−−−−ΦΦΦΦ====fp

22

ER

ER

Z

Zµ

σσσσσσσσµµµµµµµµ

σσσσββββ

++++−−−−========

)(1

22222ββββγγγγ

γγγγ

γγγγσσσσσσσσµµµµµµµµββββ f

VV REER

ER ====⇒⇒⇒⇒++++

−−−−====++++

−−−−====

Verification with partial factors

kMF

M

kkF

RE γγγγγγγγγγγγ

γγγγγγγγ ≤≤≤≤××××⇒⇒⇒⇒≤≤≤≤

Reliability index

Mγγγγ

kMFMFk γγγγγγγγγγγγγγγγγγγγγγγγγγγγββββ ≤≤≤≤××××→→→→→→→→→→→→ /),(

Global approach

E

E

σσσσ

CCmm

E

mE

σσσσ

P Design

point

Boundary between safety

and failure

ββββββββE

dE

σσσσ γγγγγγγγFF

γγγγγγγγ

CCkk

CCdd

kd

RR

γγγγ====

kFd EE γγγγ=====β

E

EEm

E

k kEE

σσσσσσσσ

σσσσ++++====

DDff

DDss

Reliability index

R

R

σσσσ

Eσσσσ

R

dR

σσσσ

γγγγγγγγMM

R

mR

σσσσM

dRγγγγ

====

R

RRm

R

k kRR

σσσσσσσσ

σσσσ−−−−====

M

kkF

RE

γγγγγγγγ ≤≤≤≤

Interpretation of partial factors for permanent actionsγγγγG,sup = 1,35 γγγγG,inf = 1,00

In general (simplification): Gk = Gm

With:

)7,01(sup, VGG md β+=

)7,01(sup,

sup, VG

GSd

m

d

SdG βγγγ +==

05.020.18.3 === VγβWith:

Where favourable, a permanente action may be considered as a resistance. Thus:

05.020.18.3 === VSdγβ

mmd GGG 36.1)05.08.37.01(20.1sup, =××+=

mmmd GGGG ≅=××−= 02.1)05.08.38.01(20.1inf,

Interpretation of partial factors for variable actions

(buildings)

γγγγQ = 0 or 1,50

Qk is assessed on the basis of a return period equal to 50 years (climatic actions and imposed

loads on buildings). In fact, the real return period is more 100 to 200 years if we take into account the

safety margins included in the models.

In any case, it was generally admitted that the design values of the leading actions, γSd

excluded, did correspond to a return period of about 4000 years (in fact, a return period of 1.4xk

between 1000 and 10 000 years).between 1000 and 10 000 years).

Example: X follows a Gümbel law (annual maxima). Coefficient of variation: V

Characteristic value: xk with F(xk) = 0,98 (return period T(xk) = 50 years)

Return period T(1.4xk) = 5720 years for V = 0,15

2433 years for V = 0,20

1457 years for V = 0,25

Finally:

γQ = 1,50 = γq × γSd =1,40 x1,10 = 1,54 # 1,50

Interpretation of partial factor for concrete

Traditional interpretation (CEB)

γγγγC = 1,5 = 1,10 x 1,10 x 1,24

• 1,24 : transition from the characteristic fractile 0,05 (5%) to the 0,005 (0,5%) fractile for a

coefficient of variation circa 0,15 with a Gaussian scatter and circa 0,23 with a log-normal

scatter ;

• 1,10 : average value of the conversion factor (in general between 1,0 and 1,20) from strength

of control samples to the effective strength in the structure (η factor) ;

• 1,10 : covers those uncertainties which cannot be accounted for in a purely probabilistic way

(model uncertainties in the design modelfor the resistance, inaccuracies in the execution of the

structure, more particularly deviations from the theoretical location of the reinforcement,

etc.).

Interpretation of partial factors for steel

Traditional interpretation (CEB)

Concrete reinforcement and prestressing tendonsγS = 1,15 = 1,05 x 1,05 x 1,05

· 1,05 : transition from the characteristic fractile 0,05 (5%) to the 0,005 (0,5%) fractile, for a coefficient

of variation circa 0,05 ;

· 1,05 : little loss of cross-sectional area due to long-term corrosion ;

· 1,05 : covers uncertainties on the flexural resistance of a member due to the real location of the · 1,05 : covers uncertainties on the flexural resistance of a member due to the real location of the

reinforcement.

Structural steel

γM0 = 1,0 : Cross-sections susceptible to reach plasticizing

γM1 = 1,1 : Cross-sections the resistance of which is limited by local instability.

The models of resistance of steel structures, in particular for the design of joints, are more precise than

the models of resistance of the concrete structures.

Interpretation of partial factors for material strength:

interpretation based on reliability methods

)exp(

)exp(

RR

f

d

kM

V

kV

R

R

βαµµ

γ−

−==

8,08,3645,1 === Rk αβ

222

fGmR VVVV ++=

)645,104,3exp( fR

d

kM VV

R

R −×== ηγ

Vm model uncertainty

VG geometrical uncertainty

Vf uncertainty on the property (strength)

Material Vm VG Vf VR γγγγconv γγγγM

Concrete 0,05 0,05 0,15 0,166 1,15 1,49

Partial factors for resistances

Reliability-based interpretation

Concrete 0,05 0,05 0,15 0,166 1,15 1,49

Reinforcement 0,05 0,05 0,05 0,087 1 1,20

Structural steel 0,03 0,03 0,03 0,052 1 1,115

• Partial factors are intended to ensure an appropriate reliability level to the majority of

construction works and parts of construction works, in order to limit the frequency of

accidents of structural origin.

•Partial factors (and reliability index) have not been defined by Heaven !

• Partial factors are not calibrated, and are not intended, to cover gross human errors.

• But, in fact, they cover minor human errors. Where is the limit between minor and gross

human errors (good question) ?

Conclusion: a few basic ideas

human errors (good question) ?

• And how a distinction between errors due to design, calculation (models, use of

sophisticated software) and execution may be distinguished ?

Thank you for your attention, andThank you to my cat which slept

during the preparation of my speech

Eurocodesand design of structures Philosophy of partial ...

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