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2. RELIABILITY OF STEEL STRUCTURES

35

Chapter 2

RELIABILITY OF STEEL STRUCTURES

2.1. GENERAL ASPECTS

In order to check the safety of a structure it is necessary to assess whether a

dangerous situation, able to make the structure unusable, might be reached due to

some extreme events. There are three types of methods to make the analysis of

steel structure reliability:

deterministic methods, which consider all parameters with their deterministic

values;

probabilistic methods, which consider all parameters and the relations among

them as random variables; they are difficult to carry on and they need a very

sophisticated mathematical procedure; they also need a great amount of data

about loads, material properties etc.;

semi-probabilistic methods, which use probabilistic models to establish the

values for actions and capacities but they compare them using deterministic

models; most of present day design codes for steel structures use such methods.

Generally, when checking the safety of a structural element or of a whole

structure, the following requirements are to be satisfied:

strength requirement;

stiffness requirement.

In some cases, like seismic design, ductility requirements need also to be fulfilled.

2.2. ALLOWABLE STRESS METHOD (DETERMINISTIC METHOD)

In this method the strength requirementis expressed by the following relation:

all ( 2.1 )

In this equation (2.1) the allowable stress allis given by:

c

fyall= ( 2.2 )

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36

where c is a global safety coefficient taking into account the following possibilities:

actual nominal loads considered in calculating the effective stress in equation

(2.1) could be greater than assumed;

actual nominal yielding stress fyin equation (2.2) could be lower than presumed; fabrication and/or erection may produce unfavourable effects.

The stiffness requirementis expressed by the following equation (same as

(1.2)):

a ( 2.3 )

where and aare the calculated and the allowable deformation respectively.

Critical remarkThe method considers only a simultaneous increase of the loads that can

unfavourably affect a correct analysis of the reliability, especially when permanent

loads (dead loads) are significantly smaller than the imposed ones (live loads).

2.2. PROBABILISTIC ANALYSIS OF RELIABILITY

2.2.1. Probabilistic bases

A more rational approach to analyse the problem of structural reliability safety

is a probabilistic one. In such a model of analysis, all the parameters whose

uncertainty can influence the reliability of structures, especially those ones

concerning resistance and loads, are considered as random variables. Generally, the

management of the reliability of a construction is governed by codes like EN 1990

[10]or ISO 2394:2015 [36].

2.2.2. Resistance randomness

The resistance R(s) of a structural member with respect to a certain internal

force S (N, M, V, T) may be expressed in a general form by:

( ) ( )dR,fsR = ( 2.1 )

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where is the cross-sectional characteristic corresponding to the internal force S,

i.e.:

= A for members in tension;

= W for members in bending.For industrially fabricated steel structural members, the cross sectional

characteristic may be considered as a deterministic value. The yield stress fymust

be considered as a random variable.

The following steps are to be followed to define the random variable x = fy:

consider the results on a sample of n = nitensile specimen tests (i.e. n values of

yield stress fy);

according to the values given in table 2.1, draw the histogram in figure 2.1,noticing that the normalized area of any rectangle on the histogram represents

the ratio:

==

i

iii

n

n

n

nf ( 2.2 )

where niis the number of samples satisfying the condition:

fy,i< x fy,i+ fy ( 2.3 )

where fy= 20 N/mm2

as shown in figure 2.1.

Table 2.1.Example of values of the yielding limit fy

Results association Frequency ofresults

Calculation

mean value xm(N/mm2)

dispersion D(N2/mm4)

Interval ofassociation

Intervalcentral

values xi

Absoluteni

Relativefi fixi (xi xm)2 fi(xi xm)2

220 240 230 20 0.0500 11.500 4140.923 207.0461

240 260 250 19 0.0475 11.875 1966.923 93.4288

260 280 270 59 0.1475 39.825 592.923 87.4560

280 300 290 140 0.3500 101.500 18.923 6.6228

300 320 310 101 0.2525 78.275 244.923 61.8429

320 340 330 40 0.1000 33.000 1270.923 127.0923

340 360 350 21 0.0525 18.375 3096.923 162.5884

n= 400 fi= 1,0 xm= 294.35 D=746.0775s= (D)0,5= 27.3144

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38

05% 5%

15%

35%25%

10%5%

00.10.2

0.30.4

220 240 260 280 300 320 340 360

Fig. 2.1.Histograms corresponding to the values in table 2.1

It is to observe that any rectangle f i represents the relative frequency of the

results (simple probability) and in this case the normalized area of the whole

histogram is:

1fi = ( 2.4 )

calculate the mean value:

=

=n

1i

iim xfx ( 2.5 )

(for the case in table 2.1, xm= 294N/mm2)

calculate the dispersion:

( )=

==n

1i

2

mii2 xxfsD ( 2.6 )

(for the case in table 2.1, D = 746N2/mm4)

calculate the standard deviation:

( )=

=n

1i

2

mii xxfs ( 2.7 )

(for the case in table 2.1, s = 27,3N/mm2

)The values xmand s define the random variable.

The histogram in figure 2.1 may be represented by the normal (Gaussian)

function of probability density described by (Fig. 2.2):

( )

2m

s

xx

2

1

e2s

1xf

= ( 2.8 )

The characteristic value of the yield stress fymay be defined in a probabilistic

manner by the following relation:

fy

fi

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skff m,yk,y = ( 2.9 )

Codes usually accept k = 2, which represents a probability of 2,28% (inferior

fractil p) that the yield stress will not be inferior to fy,k. It means:

s2ff m,yk,y = ( 2.10 )

The fractil p is defined as that value of the yield stress for which there is a probability

p for the yield stress to be inferior to that value.

By noting:

mx

sv = ( 2.11 )

where vis the coefficient of variation, equation (2.10) becomes:

( )v21ff m,yk,y = ( 2.12 )

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

Fig. 2.2.Gaussian function of probability density for the yielding limit randomness

The European code EN 1990 [10]distinguishes between resistanceand strength.

The resistance is defined in EN 1990 [10] (def. 1.5.2.15) as the capacity of a

member or component, or a cross-section of a member or component of a structure,

to withstand actions without mechanical failure e.g. bending resistance, buckling

resistance, tension resistance. Strength is used in EN 1990 [10] (def. 1.5.2.16) to

express the mechanical property of a material indicating its ability to resist actions,

usually given in units of stress.

f(x)

inferior fractil

( p = 2.28% )

x = fy

fy,k ks

fy,m

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2.2.3. Force randomness

The internal force S(Fk) in a certain cross-section of a structural member, with

regard to the type of load and the structural model of calculation, may be written as:

S(Fk) = (L) ( 2.13 )

where:

L represents the acting loads;

are formulas derived from accepted principles of structural model of calculation.

Example:

For a simply supported beam, the maximum bending moment is:

8DqMS

2

max,Ed ==

In this case, the load L = q is considered to be the random variable:

x = L = q

( ) x8

Dx

2

=

A histogram may be drawn in the same way as described for steel randomness,

determining the mean value Fmand the standard deviation sfor loads (Fig. 2.3).

0

0.005

0.01

0.015

0.02

Fig. 2.3.Gaussian function of probability density for force randomness

Accepting the formula as deterministic, equation (2.13) becomes:

S(Fk) = S(L) ( 2.14 )

The characteristic value Fk, depending on the loads, may be written as:

skFF mk += ( 2.15 )

f(F)

superior fractil

F

Fm Fkks

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Codes usually accept k = 1,645, corresponding to a 5% probability for the value Fkto

be exceeded (superior fractil p).

2.2.4. Reliability Safety analysis

Basically, to assess the reliability safety of a structure in the probabilistic

concept means to check that the probability pof exceeding a given limit state is not

greater than an a priori chosen probability pu, depending on the consequences of

reaching that limit state (Fig. 2.4).

p pu ( 2.16 )

0

0.005

0.01

0.015

0.02

Fig. 2.4.Example of reliability safety analysis

2.2.5. Probabilistic methods

Basically, three methods are to be considered:

the semi-probabilistic method (level 1);

the reliability index method (level 2);

the exact probabilistic method (level 3).

2.2.6. The semi-probabilistic limit states method (level 1)

2.2.6.1. The Eurocodes

f(S)f(R)

S, R

f(S) f(R)

P

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Structural EUROCODESis a set of harmonised technical rules for the design

of construction works in Europe. In a first stage, they were intended to be an

alternative to the national design codes and, at present, they are replacing the

former national rules. There are ten families of standards, each one consisting of

several parts:

EN 1990 Eurocode 0: Basis of structural design

EN 1991 Eurocode 1: Actions on structures

EN 1992 Eurocode 2: Design of concrete structures

EN 1993 Eurocode 3: Design of steel structures

EN 1994 Eurocode 4: Design of composite steel and concrete structures

EN 1995 Eurocode 5: Design of timber structures

EN 1996 Eurocode 6: Design of masonry structures

EN 1997 Eurocode 7: Geotechnical design

EN 1998 Eurocode 8: Design of structures for earthquake resistance

EN 1999 Eurocode 9: Design of aluminium structures

2.2.6.2. Limit states

A limit state can be defined as the state beyond which the structure no

longer fulfils the relevant design criteria (EN 1990 [10] (def. 1.5.2.12)).

There are two categories of limit states:

1. ultimate limit states, which are states associated with collapse or with other

similar forms of structural failure and they generally correspond to the maximum

load-carrying resistance of a structure or structural member (EN 1990 [10] (def.

1.5.2.13)). Ultimate limit states are related to the safety of people and/or the

safety of the structure (EN 1990 [10]). It is to consider here:

loss of equilibrium of the structure or any part of it, considered as a rigid

body;

failure by excessive deformation, transformation of the structure or any

part of it into a mechanism, rupture, loss of stability of the structure or any

part of it, including supports and foundations;

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43

failure caused by fatigue or other time-dependent effects (EN 1990 [10]).

The following ultimate limit statesshall be verified as relevant:

a) 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;

b) 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;

c) GEO: Failure or excessive deformation of the ground where the strengths

of soil or rock are significant in providing resistance;

d) FAT: Fatigue failure of the structure or structural members (EN 1990 [10])

2. serviceability limit states, which refer to the normal use of the structure and

correspond to conditions beyond which specified service requirements for a

structure or structural member are no longer met (EN 1990 [10] (def. 1.5.2.14)).

Serviceability limit statesare related to:

the functioning of the structure or structural members under normal use;

the comfort of people;

the appearance of the construction works.

NOTE 1 In the context of serviceability, the term "appearance" is concerned with

such criteria as high deflection and extensive cracking, rather than aesthetics.

NOTE 2 Usually the serviceability requirements are agreed for each individual

project (EN 1990 [10]).

Two types of serviceability limit states can be mentioned:

irreversible serviceability limit states(EN 1990 [10] (def. 1.5.2.14.1))

serviceability limit states where some consequences of actions exceeding

the specified service requirements will remain when the actions are

removed;

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44

reversible serviceability limit states (EN 1990 [10] (def. 1.5.2.14.2))

serviceability limit states where no consequences of actions exceeding the

specified service requirements will remain when the actions are removed.

The verification of serviceability limit states should be based on criteria concerning the

following aspects:

a) deformations that affect the comfort of people;

the appearance,

the comfort of users, or the functioning of the structure (including the functioning of machines or services),

or that cause damage to finishes or non-structural members;b) vibrations;

that cause discomfort to people, or

that limit the functional effectiveness of the structure;

c) damage that is likely to adversely affect

the appearance, the durability, or

the functioning of the structure(EN 1990 [10]).

2.2.6.3. Actions

An action(F) (EN 1990 [10] (def. 1.5.3.1)) can be:

a) Set of forces (loads) applied to the structure (direct action);

b) Set of imposed deformations or accelerations caused for example, by

temperature changes, moisture variation, uneven settlement or earthquakes

(indirect action).

The effect of an action (E) (EN 1990 [10] (def. 1.5.3.2)) designates effect of

actions (or action effect) on structural members, (e.g. internal force, moment, stress,

strain) or on the whole structure (e.g. deflection, rotation).

Actions shall be classified, according to EN 1990[10], by their variation in time as follows: permanent actions (G), e.g. self-weight of structures, fixed equipment and road surfacing, and

indirect actions caused by shrinkage and uneven settlements;

variable actions (Q), e.g. imposed loads on building floors, beams and roofs, wind actions orsnow loads;

accidental actions (A), e.g. explosions, or impact from vehicles(EN 1990 [10]).

Actions shall also be classified:

by their origin, as direct or indirect,

by their spatial variation, as fixed or free, or

by their nature and/or the structural response, as static or dynamic.

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45

1.5.3.3permanent action(G)

action that is likely to act throughout a given reference period and for which the variation in

magnitude with time is negligible, or for which the variation is always in the same direction

(monotonic) until the action attains a certain limit value

1.5.3.4variable action (Q)

action for which the variation in magnitude with time is neither negligible nor monotonic

1.5.3.5accidental action (A)

action, usually of short duration but of significant magnitude, that is unlikely to occur on a given

structure during the design working life

NOTE 1 An accidental action can be expected in many cases to cause severe consequences unless appropriate measures are

taken.

NOTE 2 Impact, snow, wind and seismic actions may be variable or accidental actions, depending on the available

information on statistical distributions.

1.5.3.6seismic action (AE)action that arises due to earthquake ground motions

1.5.3.8fixed actionaction that has a fixed distribution and position over the structure or structural member such that the

magnitude and direction of the action are determined unambiguously for the whole structure or

structural member if this magnitude and direction are determined at one point on the structure or

structural member

1.5.3.9free actionaction that may have various spatial distributions over the structure

1.5.3.11static actionaction that does not cause significant acceleration of the structure or structural members

1.5.3.12dynamic actionaction that causes significant acceleration of the structure or structural members

1.5.3.13quasi-static action

dynamic action represented by an equivalent static action in a static model.(EN 1990 [10])

2.2.6.4. Values of actions

P The characteristic value (Fk)

of an action is its main representative value and shall be specified: as a mean value, an upper or lower value, or a nominal value (which does not refer to a

known statistical distribution) (see EN 1991);

in the project documentation, provided that consistency is achieved with methods given in EN1991.

1.5.3.16combination value of a variable action (0Qk)

value chosen - in so far as it can be fixed on statistical bases - so that the probability that the effects

caused by the combination will be exceeded is approximately the same as by the characteristic value

of an individual action. It may be expressed as a determined part of the characteristic value by using a

factor 01

1.5.3.17frequent value of a variable action (1Qk)

value determined - in so far as it can be fixed on statistical bases - so that either the total time, withinthe reference period, during which it is exceeded is only a small given part of the reference period, or

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46

the frequency of it being exceeded is limited to a given value. It may be expressed as a determined

part of the characteristic value by using a factor 11

1.5.3.18quasi-permanent value of a variable action (2Qk)

value determined so that the total period of time for which it will be exceeded is a large fraction of the

reference period. It may be expressed as a determined part of the characteristic value by using a factor

21

1.5.3.19accompanying value of a variable action (Qk)value of a variable action that accompanies the leading action in a combination

NOTE The accompanying value of a variable action may be the combination value, the frequent value or the quasi-

permanent value.

1.5.3.20representative value of an action (Frep)value used for the verification of a limit state. A representative value may be the characteristic value

(Fk) or an accompanying value (Fk).(EN 1990 [10])

The design value Fdof an action is expressed by:

repfd FF = (EN 1990 [10], rel. (6.1a)) ( 2.17 )

where:

krep FF = (EN 1990 [10], rel. (6.1b)) ( 2.18 )

Fk the characteristic value of that action (2.15);

Frep the relevant representative value of that action;

f a partial factor for the action which takes account of the possibility of

unfavourable deviations of the action values from the representative values.

is either 1,00 or 0, 1or 2 (EN 1990 [10]).

2.2.6.5. Load combinations (combinations of actions)

1. According to the Romanian code STAS 10101/0A-77, two design situations areconsidered:

Fundamental combination

++ iigiiii VnnCnPn ( 2.21 )

Specialcombination

1idiii EVnCP +++ ( 2.22 )

In equations (2.21) and (2.22):

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ng is a factor taking into account the probability of simultaneous action of a

number of variable actions (Vi) at their highest intensity:

ng= 1 for one Vi;

ng= 0,9 for two or three Vi; ng= 0,8 for four or more Vi.

nid is a factor representing the long lasting part of a variable action; nid< 1.

The ultimate limit states are usually examined considering the effects of the

design values of actions, while for serviceability limit states the characteristic

values of actions are generally used.

2. EN 1990[10]uses design situationsto express the requirements to be fulfilled

for each limit state. Design situations(EN 1990 [10] (def. 1.5.2.2)) are sets of

physical conditions representing the real conditions occurring during a certain

time interval for which the design will demonstrate that relevant limit states are

not exceeded.

Thedesign working life(EN 1990 [10] (def. 1.5.2.8)) is the 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. Values of the

design working life are given in table 2.2. Design situations are defined as

follows:

a persistent design situation (EN 1990 [10] (def. 1.5.2.4)) is a design

situation that is relevant during a period of the same order as the design

working life of the structure; it refers to the conditions of normal use of the

structure;

a transient design situation (EN 1990 [10] (def. 1.5.2.3)) is a design

situation that is relevant during a period much shorter than the design working

life of the structure and which has a high probability of occurrence; it refers to

temporary conditions applicable to the structure, e.g. during execution or

repair;

an accidental design situation (EN 1990 [10] (def. 1.5.2.5)) is a design

situation involving exceptional conditions of the structure or its exposure, in-

cluding fire, explosion, impact or local failure; it refers to exceptional

conditions applicable to the structure or to its exposure, e.g. to fire, explosion,

impact or the consequences of localised failure;

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a seismic design situations (EN 1990 [10] (def. 1.5.2.7)) is a design

situation involving exceptional conditions of the structure when subjected to a

seismic event.

Table 2.2.Indicative design working life(EN 1990 [10] Tab. 2.1)

Designworking

lifecategory

Indicativedesign

working life(years)

Examples

1 10 Temporary structures (1)

2 10 to 25 Replaceable structural parts, e.g. gantry girders,bearings

3 15 to 30 Agricultural and similar structures

4 50 Regular buildings and other regular structures

5 100 Monumental building structures, bridges, and othercivil engineering structures

(1) Structures or parts of structures that can be dismantled with a view to being re-used should not be considered as temporary.

According to EN1990 [10], three types of combinations of actions are to be

considered when designing steel members:

for persistent and transient design situations (fundamental

combinations), the most unfavourable of:

>

1i

i,ki,0i,Q1,k1,01,QP

1j

j,kj,G QQPG

(EN 1990 [10], rel. (6.10a)) ( 2.19a )

>

1i

i,ki,0i,Q1,k1,QP

1j

j,kj,Gj QQPG

(EN 1990 [10], rel. (6.10b)) ( 2.19b )

for accidentaldesign situations

( ) >

1i

i,ki,21,k1,21,1d

1j

j,k QQorAPG

(EN 1990 [10], rel. (6.11b)) ( 2.20 )

for seismicdesign situations

>

1i

i,ki,2Ed

1j

j,k QAPG

(EN 1990 [10], rel. (6.12b)) ( 2.21 )

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In relations (2.19), (2.20), (2.21) the meanings are as follows:

= the combined effect of;

= combined with;

Gk,j= characteristic value of permanent actionj;

P = relevant representative value of a prestressing action;

Qk,1= characteristic value of the leading variable action 1;

Qk,i= characteristic value of the accompanying variable action i;

Ad = design value of an accidental action;

AEd= design value of seismic action EkIEd AA = ;

AEk= characteristic value of seismic action;

I = importance factor, given in EUROCODE 8 (EN 1998-1)[11];

G,j = partial factor for permanent actionj;

P = partial factor for prestressing actions;

Q,i = partial factor for the variable action i;

0 = factor for combination value of a variable action;

1 = factor for frequent value of a variable action;

2 = factor for quasi-permanent value of a variable action;

= a reduction factor for unfavourable permanent actions G.The value for and factors may be set by the National annex. Some examples

of recommended values of factors for buildings are given in table 2.3. The

values adopted in the Romanian National Annex are given in table 2.4.

Table 2.3.Values of factors for buildings (EN 1990 [10] Tab. A.1.1)

Action 0 1 2

Imposed loads in buildings, category (see EN 1991-1-1)Category A: domestic, residential areasCategory B: office areasCategory C: congregation areasCategory D: shopping areasCategory E: storage areasCategory F: traffic area

vehicle weight 30kNCategory G: traffic area

30kN < vehicle weight 160kN

Category H: roofs

0,70,70,70,71,00,7

0,7

0,71)

0,50,50,70,70,90,7

0,5

0

0,30,30,60,60,80,6

0,3

0

Snow loads on buildings (see EN 1991-1-3)* Finland, Iceland, 0,7 0,5 0,2

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Norway, SwedenRemainder of CEN Member States, for sites located at altitude H> 1000 m a.s.l.Remainder of CEN Member States, for sites located at altitude H_ 1000 m a.s.l.

0,7

0,5

0,5

0,2

0,4

0

Wind loads on buildings (see EN 1991-1-4) 0,6 0,2 0Temperature (non-fire) in buildings (see EN 1991-1-5:2005) 0,6 0,5 0

NOTE The values may be set by the National annex.

* For countries not mentioned below, see relevant local conditions.

Table 2.4.Values of factors for buildings (EN 1990 [10] Tab. NA A.1.1)

Action 0 1 2

Imposed loads in buildings, category (see SR EN 1991-1-1:2004

and its National Annex)Category A: domestic, residential areasCategory B: office areasCategory C: congregation areas

C1: Areas with tablesC1.1 areas in schools, reading roomsC1.2 medical laboratories and offices, computer rooms etc.C1.3 cafs, restaurants, dining halls, receptions

C2 Areas with fixed seatsC3 Areas without obstacles for moving peopleC4 Areas with possible physical activitiesC5 Areas susceptible to large crowds

Category D: shopping areasCategory E: storage areasCategory F: traffic area

vehicle weight 30kNCategory G: traffic area

30kN < vehicle weight 160kN

Category H: roofs

0,70,70,7

0,71,00,7

0,7

0,71)

0,50,50,7

0,70,90,7

0,5

0

0,30,30,6

0,60,80,6

0,3

0

Snow loads on buildings (see SR EN 1991-1-3:2005 and itsNational Annex)

All sites

0,7 0,5 0,4

Wind loads on buildings (see SR EN 1991-1-4:2006 and itsNational Annex)

0,7 0,2 0

Temperature (non-fire) in buildings (see SR EN 1991-1-5:2005)* * *

1)See SR EN 1991-1-1:2004, 3.3.2(1).

* Values of factors will be available after the completion of SR EN 1991-1-5:2005National Annex.

According to the American codes ASCE 798 [3] (the latest version is from 2010)

and LRFD[4], the following combinations shall be investigated:

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( )

( ) ( ) ( )

( ) ( )

( )

H6,1E0,1D9,0

H6,1W6,1D9,0

S2,0L5,0E0,1D2,1

RorSorL5,0L5,0W6,1D2,1

W8,0orL5,0RorSorL6,1D2,1

RorSorL5,0HL6,1TFD2,1

FD4,1

r

r

r

++

++

+++

+++

++

+++++

+

( 2.26 )

being:

D = dead load (Pi+ Ci)

F = load due to fluids with well-defined pressures and maximum heights

Fa = flood load

H = load due to lateral earth pressure, ground water pressure or pressureof bulk materials

L = live load (Viimposed loads)

Lr = roof live load

W = wind load

S = snow load

T = self-straining force

E = earthquake loadR = rain water or ice

3. According to the American code ASCE/SEI 710 [37]:

( )

( ) ( )

( )

E0,1D9,0W0,1D9,0

S2,0LE0,1D2,1

RorSorL5,0LW0,1D2,1

W5,0orLRorSorL6,1D2,1

RorSorL5,0L6,1D2,1

D4,1

r

r

r

++

+++

+++

++

++

( 2.22 )

being:

D = dead load (Pi+ Ci)

F = load due to fluids with well-defined pressures and maximum heights

Fa = flood load

L = live load (Viimposed loads)

Lr

= roof live loadW = wind load

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S = snow load

E = earthquake load

R = rain water or ice

2.3.6.5. Material design properties

The design value Rdof a material property is generally defined as:

M

kd

fR

= ( 2.27 )

where:

fk = characteristic value of the considered material property;

M = partial safety factor for the considered material property.For the design strength Rof a structural steel, equation (2.27) becomes:

M

kfR

= ( 2.28 )

being ( )v21ff mk = (see equation (2.15)).

2.2.6.6. Design resistance

The ability of the cross-section, member or structure to withstand the effects

of loads is expressed by means of the resistance that includes the strength of the

material (fk). The design resistance Rdis generally defined as:

M

kd

RR

= (EN 1990 [10], rel. (6.6c)) ( 2.23 )

where:

Rk characteristic value of the resistance;

M partial factor for a material property.

2.2.6.7. Ultimate limit state

In the limit state method (also called the method of extreme values), the

probabilistic condition in equation (2.16) p < puis replaced by:

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53

SdRd ( 2.29 )

EdRd (EN 1990 [10], rel. (6.8)) ( 2.24 )

which means that the maximum probable internal design effort Eddoes not exceed

the minimum probable design resistance capacity Rd. In equation (2.24):Sd= S(niFi) is the internal design effort, calculated using design values of actions

and taking into account respectively the load combinations in eqs.

(2.21) and (2.22) or (2.23), (2.24) and (2.25) or (2.26), depending on

the code;

Rd = R(Rk/M) is the corresponding design resistance, calculated using the design

strength of steel.

where:Ed the design value of the effect of actions such as internal force or moment,

resulted from load combinations like (2.19), (2.20), (2.21) or (2.22), depending

on the design code that is used;

Rd the design value of the corresponding resistance.

2.2.6.8. Serviceability limit state

Depending on the serviceability criterion that is checked, the following

checking relation is used:

EdCd (EN 1990 [10], rel. (6.13)) ( 2.25 )

where:

Ed the design value of the effect of actions specified in the serviceability criterion,

resulted from appropriate load combinations;

Cd the limiting design value of the relevant serviceability criterion.

The most common serviceability limit state to be checked is the deformation

check. It will be verified that:

da ( 2.26 )

where:

d= (Fi) is the design deformation, calculated using the characteristic(nominal)

appropriate values of actions;

ais an allowable deformation given in codes or requested by the owner.

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54

2.3.6.8. Conclusive remarks

1. At present, the limit state method is the design method provided in most of the

important codes.

2. It represents a more accurate model compared to the allowable stress method

because it separates the material randomness from the load randomness and it

accepts different approaches for different types of loads.

2.3.7. The reliability index method (level 2)

In a general form, equation (2.25) becomes at limit:

Sd= Rd ( 2.27 )

Equation (2.31) may be written:

in the subtract model Rjanitin Cornell as:

E = Sd Rd= 0 ( 2.28 )

in the logarithmic model Freudenthal Rosenblueth as:

0R

S

lnE d

d

== ( 2.29 )

In equations (2.28) and (2.29) E = 0 is the reliability function, expressing (Fig. 2.5):

E < 0 : safety range;

E > 0 : unsafe range;

E = 0 : the border between safety and unsafe range.

Fig. 2.5.The reliability index method (level 2)

Xi

Xj

Xn

Unsafe rangeE > 0

SafetyrangeE < 0

space E

limit hypersurface E = 0

mE

fE

EsE

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55

In the case of a simple internal effort S(= N, M or Q), the reliability index Eis

defined as the reverse of the coefficient of variation vEof the function E:

E

E

E

Es

m

v

1== ( 2.30 )

Equation (2.30) may be written as:

0sm EEE =+ ( 2.31 )

In equations (2.30) and (2.31) mEand sEare the mean value and, respectively, the

standard deviation of the function E.

Figure 2.5 shows the physical significance of the reliability index E which

represents in hyper-space E the distance calculated in standard deviations sE

between the point with the abscissa mE and the point with the abscissa E = 0,located on the random hyper-surface which defines the border between safe and

unsafe behaviour, corresponding to a certain probability pu= p(E).

The properties of the main statistic characteristics for two variables, X1 and

X2, are given in table 2.5.

Table 2.5.Main statistic characteristics

Y mY DY vY

X1 mX1 DX1 vX1

C C 0 0

CX1 CmX1 C2DX1 vX1

X1 C mX1C DX1Cm

vm

1X

!X1X

X1+ X2 mX1+ mX2 DX1+ DX22X1X

2

2X

2

2X

2

1X

2

1X

mm

vmvm

+

+

X1 X2 mX1 mX2 DX1+ DX2

2X1X

2

2X

2

2X

2

1X

2

1X

mm

vmvm

+

X1X2 mX1mX22X

22X1X

21X DmDm +

2

2X

2

1X vv +

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56

X1/ X2 mX1/ mX2 2X2

1X1X

2

2X2

2X

DmDmm

1+ 2

2X

2

1X vv +

For the two models presented above, the reliability index , taking intoaccount the relations in table 2.5, becomes:

SR

SRRS

DD

mm

+

= ( 2.32 )

2S

2R

S

R

R

Sln vv

m

mln

+

= ( 2.33 )

Table 2.6 shows a correspondence between the index and the probability pu of

losing the safety for SR(S and R normal distributions) and lnS/Rrespectively (S

and R lognormal distributions).

The American code ASCE/SEI 710 [37] provides the reliability lnS/R index

(2.33) and the following targets were selected:

loadingearthquake+live+deadunder75,1

loadingwind+live+deadunder5,2

loadingsnowand/orlive+deadundersconnectionfor5,4

loadingsnowand/orlive+deadundermembersfor3

=

=

=

=

( 2.34 )

Table 2.6.Correspondence between the index and the probability pu

pu SR; lnS/R SR; lnS/R pu

10-1 1,29 1,0 1,59 10-1

10-2 2,33 1,5 6,68 10-2

10-3 3,09 2,0 2,27 10-2

10-4 3,72 2,5 6,21 10-3

10-5 4,27 3,0 1,35 10-3

10-6 4,75 3,5 2,33 10-4

10-7 5,20 4,0 3,17 10-5

10-8 5,61 4,5 3,40 10-6

10-9 6,00 5,0 2,90 10-7

10-10 6,35 5,5 1,90 10-8

Example 2.1.

Calculate the index SRand lnS/Rfor the beam in figure 2.6:

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57

Fig. 2.6.Example 2.1

Given:

for the loading:

mean value: mq= qm= 20kN/m

variation factor: vq= 0,1

for the steel in use:

mean value: mRc= Rm= 294N/mm2

dispersion: DRc= 744N2/mm4

Calculate for the loading q (S):

2

3

22

qM mmN5,1691035412

600020

W12

Lm

W

mm =

=

==

42222q

2qq mmN41,020vmD ===

42

622

4

q

222

mmN3,28741035412

6000D

W12

Lq

W12

LDD =

=

=

=

1,05,169

3,287

m

Dv ===

Calculations for the material (R):

mRc= 294N/mm2

DRc= 744N2/mm4

093,0294

744

m

Dv

Rc

Rc ===

Calculate the index SR(2.32):

0,3877,33,287744

5,169294

DD

mm

DD

mm

Rc

Rc

SR

SRRS >=

+

=

+

=

+

=

Calculate the index lnS/R(2.33):

L = 6m

12

LqM

2=

I24; Wy= 354cm3

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0,3033,41,0093,0

5,169

294ln

vv

m

mln

vv

m

mln

2222Rc

Rc

2S

2R

S

R

R

Sln

>=+

=+

=+

=

Remarks1. In this method, the general condition p pu(2.16) is replaced by:

u ( 2.35 )

which expresses the condition E > 0 (S > R); uis a risk a priori accepted.

2. At present, this method is used especially to calibrate the partial safety factors in

the limit state method and the coefficients ni in the load combinations; in the

future it is to be expected that the index method will replace the limit state

method.3. In order to improve the index method two tendencies are to be observed in

scientific works:

a more adequate location of points on the hyper-surface E = 0;

an extension of the method to various non-normal distributions.

2.3.8. The probabilistic method (level 3)

In this method the reliability analysis is based on the general condition p pu

(2.16), where p is the probability of E > 0, being:

( ) 0R,,R,R;S,,S,SE n21n21 =KK ( 2.36 )

a function of random variables Siand Riand puan accepted risk, depending on the

consequences.

At present this method is used only in scientific works.

Documents

Cap 2 2015