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2. RELIABILITY OF STEEL STRUCTURES

33

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 requirement is expressed by the following relation:

all ( 2.1 )

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

c

fyall = ( 2.2 )

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


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actual nominal loads considered in calculating the effective stress in equation

(2.1) could be greater than assumed;

actual nominal yielding stress fy in equation (2.2) could be lower than presumed;

fabrication and/or erection may produce unfavourable effects.The stiffness requirement is expressed by the following equation (same as

(1.2)):

a ( 2.3 )

where and a are the calculated and the allowable deformation respectively.

Critical remark

The method considers only a simultaneous increase of the loads that canunfavourably affect a correct analysis of the reliability, especially when permanent

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

2.3. PROBABILISTIC ANALYSIS OF RELIABILITY

2.3.1. Probabilistic bases

A more rational approach to analyse the problem of structural safety is a

probabilistic one. In such a model of analysis, all the parameters whose uncertaintycan influence the reliability of structures, especially those ones concerning

resistance and loads, are considered as random variables.

2.3.2. Resistance randomness

The resistance is defined in EN 1990 [10] (1.5.2.15) as the capacity of a member or component, or across-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] (1.5.2.16) to express the mechanical property of a material indicating its ability to resist actions,

usually given in units of stress.The resistance R(s) of a structural member with respect to a certain internal

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

( ) ( )dR,fsR = ( 2.4 )

where is the cross-sectional characteristic corresponding to the internal force S,i.e.:

= A for members in tension;

= W for members in bending.


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For industrially fabricated steel structural members, the cross sectional

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

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 = ni tensile specimen tests (i.e. n values of

yield stress fy);

according to the values given in table 2.1, draw the histogram in figure 2.3,

noticing that the normalized area of any rectangle on the histogram represents

the ratio:

==i

iii

n

n

n

nf ( 2.5 )

where ni is the number of samples satisfying the condition:

fy,i < x fy,i + fy ( 2.6 )

where fy = 20 N/mm2

as shown in figure 2.3.

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 fi xi (xi xm)

2fi (xi xm)

2

220 240 230 20 0.05 11.5 4140.923 207.0461

240 260 250 19 0.0475 11.875 1966.923 93.42882

260 280 270 59 0.1475 39.825 592.9225 87.45607

280 300 290 140 0.35 101.5 18.9225 6.622875

300 320 310 101 0.2525 78.275 244.9225 61.84293

320 340 330 40 0.1 33 1270.923 127.0923340 360 350 21 0.0525 18.375 3096.923 162.5884

n = 400 fi = 1,0 xm= 294.35 D=746.0775

s = (D)0,5

= 27.31442


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05%

5%

15%

35%25%

10%5%

00.1

0.20.30.4

220 240 260 280 300 320 340 360

Fig. 2.3. Histograms corresponding to the values in table 2.1

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

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

histogram is:

1fi = ( 2.7 )

calculate the mean value:

=

=n

1i

iim xfx ( 2.8 )

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

calculate the dispersion:

( )= ==n

1i

2mii

2 xxfsD ( 2.9 )

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

calculate the standard deviation:

( )=

=n

1i

2

mii xxfs ( 2.10 )

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

The values xm and s define the random variable.

The histogram in figure 2.3 may be represented by the normal (Gaussian)

function of probability density described by (Fig. 2.4):

( )

2m

s

xx

2

1

e2s

1xf

= ( 2.11 )

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

manner by the following relation:

skff m,yk,y = ( 2.12 )


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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 = ( 2.13 )

The fractil p is defined as that value of the yield stress for which there is a probabilityp for the yield stress to be inferior to that value.

By noting:

mx

sv = ( 2.14 )

where v is the coefficient of variation, equation (2.13) becomes:

( )v21ff m,yk,y = ( 2.15 )

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

Fig. 2.4. Gaussian function of probability density for the yielding limit randomness

2.3.3. 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) ( 2.16 )

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:

f(x)

inferior fractil

( p = 2.28% )

x = fy

fy,k ks

fy,m


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8

DqMF

2

max

==

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 Fm and the standard deviation s for loads (Fig. 2.5).

0

0.005

0.01

0.015

0.02

Fig. 2.5. Gaussian function of probability density for force randomness

Accepting the formula as deterministic, equation (2.16) becomes:

S(Fk) = S(L) ( 2.17 )

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

skFF mk += ( 2.18 )

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

be exceeded (superior fractil p).

2.3.4. Safety analysis

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

means to check that the probability p of 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. 2.6).

p pu ( 2.19 )

f(F)

superior fractil

F

Fm

Fk

ks


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0

0.005

0.01

0.015

0.02

Fig. 2.6. Example of safety analysis

2.3.5. 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).

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

2.3.6.1. The Eurocodes

Structural EUROCODES is a programme for establishing 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 in

the end, they will replace the national rules. The Structural Eurocode programme

comprises the following 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 resistanceEN 1999 Eurocode 9: Design of aluminium structures

f(S)f(R)

S, R

f(S) f(R)

P


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2.3.6.1. Limit states

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

longer fulfils the relevant design criteria (1.5.2.12 in EN1990 [10]).

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 (1.5.2.13 in EN1990 [10]). Ultimate limit states are related to the safety

of people and/or the safety of the structure.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;

failure caused by fatigue or other time-dependent effects.

The following ultimate limit states shall be verified as relevant:

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

minor variations in the value or the spatial distribution of actions from a single sourceare 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 constructionmaterials of the structure governs;

c) GEO: Failure or excessive deformation of the ground where the strengths of soil or rockare significant in providing resistance;

d) FAT: Fatigue failure of the structure or structural members.

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 (1.5.2.14 in EN1990 [10]). Serviceability limit states

are 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.

Two types of serviceability limit states can be mentioned:


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irreversible serviceability limit states (1.5.2.14.1 in EN1990 [10]) serviceability

limit states where some consequences of actions exceeding the specified service

requirements will remain when the actions are removed;

reversible serviceability limit states (1.5.2.14.2 in EN1990 [10]) 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.

2.3.6.2. Actions

An action (F) (1.5.3.1 in EN1990 [10]) 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) (1.5.3.2 in EN1990 [10]) 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, andindirect 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.

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.

1.5.3.3 permanent action(G)


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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

1.5.3.4 variable action (Q)

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

1.5.3.5 accidental 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 aretaken.

NOTE 2 Impact, snow, wind and seismic actions may be variable or accidental actions, depending on the availableinformation on statistical distributions.

1.5.3.6seismic action (AE)

action that arises due to earthquake ground motions

1.5.3.8 fixed action

action 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

1.5.3.9 free action

action that may have various spatial distributions over the structure

1.5.3.11 static action

action that does not cause significant acceleration of the structure or structural members

1.5.3.12 dynamic action

action that causes significant acceleration of the structure or structural members

1.5.3.13 quasi-static action

dynamic action represented by an equivalent static action in a static model

2.3.6.3. 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 aknown statistical distribution) (see EN 1991); in the project documentation, provided that consistency is achieved with methods given in EN

1991.

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

factor0 1

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, within

the reference period, during which it is exceeded is only a small given part of the reference period, or


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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 factor1 1

1.5.3.18 quasi-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

2 1

1.5.3.19 accompanying 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.

1.5.3.20 representative 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)

The design value Fd of an action is expressed by:

k,ii

i

d FF = ( 2.20 )

where:

Fi,k is the characteristic value of that action (2.18);

i is the partial safety factor for the action F i, being Fi = (Pi, Ci, Vi, Ei).

2.3.6.4. Load combinations (combinations of actions)1. According to the Romanian code STAS 10101/0A-77, two design situations are considered:

Fundamental combination

++ iigiiii VnnCnPn ( 2.21 )

Specialcombination

1i

d

iii EVnCP +++ ( 2.22 )In equations (2.21) and (2.22):

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; n i

d < 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.


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

for each limit state. Design situations (1.5.2.2 in EN1990 [10]) 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 Design situations shall be

classified as follows:

persistent design situations, which refer to the conditions of normal use; transient design situations, which refer to temporary conditions applicable to the structure,

e.g. during execution or repair;

accidental design situations, which refer to exceptional conditions applicable to the structureor to its exposure, e.g. to fire, explosion, impact or the consequences of localised failure;

seismic design situations, which refer to conditions applicable to the structure when subjectedto seismic events.

The design working life(1.5.2.8 in EN1990 [10]) 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 majorrepair being necessary. Based on this, the design situations are defined as follows:

apersistent design situation(1.5.2.4 in EN1990 [10]) is a design situation that is relevantduring a period of the same order as the design working life of the structure;

atransient design situation(1.5.2.3 in EN1990 [10]) is a design situation that is relevantduring a period much shorter than the design working life of the structure and which has a

high probability of occurrence;

anaccidental design situation(1.5.2.5 in EN1990 [10]) is a design situation involvingexceptional conditions of the structure or its exposure, including fire, explosion, impact or

local failure;

aseismic design situation(1.5.2.7 in EN1990 [10]) is a design situation involvingexceptional conditions of the structure when subjected to a seismic event;Table 2.1 - Indicative design working life

Design working

life category

Indicative design

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 other civil engineeringstructures

(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:

Forpersistent and transient design situations, the most unfavourable of:

>

1i

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

1j

j,kj,G QQPG ( 2.23a )


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>

1i

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

1j

j,kj,Gj QQPG ( 2.23b )

foraccidental design situations

( ) >

1i

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

1j

j,k QQorAPG ( 2.24 )

forseismic design situations

>

1i

i,ki,2Ed

1j

j,k QAPG ( 2.25 )

In relations (2.23), (2.24), (2.25) 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) [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 2.2.


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Table 2.2. Examples of recommended values of factors for buildings [10]

Table A1.1 - Values offactors for buildings

Action 0 1 2Imposed loads in buildings, category (see EN 1991-1-1)Category A: domestic, residential areas

Category B: office areasCategory C: congregation areas

Category D: shopping areasCategory E: storage areas

Category F: traffic area

vehicle weight 30kNCategory G: traffic area

30kN < vehicle weight 160kNCategory H: roofs

0,7

0,70,7

0,71,0

0,7

0,7

0,71)

0,5

0,50,7

0,70,9

0,7

0,5

0

0,3

0,30,6

0,60,8

0,6

0,3

0

Snow loads on buildings (see EN 1991-1-3)* Finland, Iceland, Norway, Sweden

Remainder 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

0,7

0,5

0,5

0,5

0,2

0,2

0,4

0Wind loads on buildings(see EN 1991-1-4) 0,6 0,2 0

Temperature (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 NA A1.1 - Values offactors for buildings

Action 0 1 2Imposed loads in buildings, category (see SR EN 1991-1-1:2004 and its National

Annex)Category A: domestic, residential areas

Category B: office areasCategory C: congregation areas

C1: Areas with tablesC1.1 areas in schools, reading rooms

C1.2 medical laboratories and offices, computer rooms etc.C1.3 cafs, restaurants, dining halls, receptions

C2 Areas with fixed seats

C3 Areas without obstacles for moving peopleC4 Areas with possible physical activitiesC5 Areas susceptible to large crowds

Category D: shopping areasCategory E: storage areas

Category F: traffic areavehicle weight 30kNCategory G: traffic area

30kN < vehicle weight 160kNCategory H: roofs

0,70,7

0,7

0,7

1,00,7

0,7

0,71)

0,50,5

0,7

0,7

0,90,7

0,5

0

0,30,3

0,6

0,6

0,80,6

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 its National 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 1991-1-5:2005 National Annex.


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

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

( )( ) ( ) ( )

( ) ( )

( )

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 heightsFa = flood load

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

of bulk materials

L = live load (Vi imposed loads)

Lr = roof live load

W = wind load

S = snow loadT = self-straining force

E = earthquake load

R = rain water or ice

4. According to the American codesASCE 710:

( )

( ) ( )

( )

E0,1D9,0

W0,1D9,0

S2,0LE0,1D2,1

RorSorL5,0LW0,1D2,1

W5,0orLRorSorL6,1D2,1

RorSorL5,0L6,1D2,1

D4,1

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

L = live load (Vi imposed loads)


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Lr = roof live load

W = wind load

S = snow load

E = earthquake load

R = rain water or ice

2.3.6.5. Material design properties

The design value Rd of 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 R of a structural steel, equation (2.27) becomes:

M

kfR

= ( 2.28 )

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

2.3.6.6. Ultimate limit state

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

probabilistic condition in equation (2.19) p < pu is replaced by:

Sd Rd ( 2.29 )

which means that the maximum probable internal design effort Sd does not exceed

the minimum probable design resistance capacity Rd. In equation (2.29):

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.


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2.3.6.7. Serviceability limit state

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

check. It will be verified that:

d

a( 2.30 )

where:

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

values of actions;

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

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 (2.29) becomes at limit:

Sd = Rd ( 2.31 )

Equation (2.31) may be written:

in the subtract model Rjanitin Cornell as:

E = Sd Rd = 0 ( 2.32 )

in the logarithmic model Freudenthal Rosenblueth as:

0R

SlnE

d

d == ( 2.33 )

In equations (2.32) and (2.32) E = 0 is the reliability function, expressing (Fig. 2.8):

E < 0 : safety range;

E > 0 : unsafe range;

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


18/21


50

Fig. 2.8. The reliability index method (level 2)

In the case of a simple internal effort S (= N, M or Q), the reliability index E isdefined as the reverse of the coefficient of variation vE of the function E:

E

E

E

Es

m

v

1== ( 2.34 )

Equation (2.34) may be written as:

0sm EEE =+ ( 2.35 )

In equations (2.34) and (2.35) mE and sE are the mean value and, respectively, the

standard deviation of the function E.

Figure 2.8 shows the physical significance of the reliability index E whichrepresents 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 2.3.

Table 2.3. Main statistic characteristics

Y mY DY vY

X1 mX1 DX1 vX1

C C 0 0

CX1 CmX1 C2 DX1 vX1

Xi

Xj

Xn

Unsafe range

E > 0

SafetyrangeE < 0

space E

limit hypersurface E = 0

mE

fE

EsE


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51

X1 C mX1 C 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

+

X1 X2 mX1 mX22X

2

2X1X

2

1X DmDm + 2 2X2

1X vv +

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 2.3, becomes:

SR

SRRS

DD

mm

+

= ( 2.36 )

2

S

2

R

S

R

R

Sln vv

m

mln

+

= ( 2.37 )

Table 2.4 shows a correspondence between the index and the probability pu oflosing the safety forSR (S and R normal distributions) and lnS/R respectively (Sand R lognormal distributions).

The American code provides the lnS/R index (2.37) and the following targetswere selected:

loadingearthquake+live+deadunder75,1

loadingwind+live+deadunder5,2

loadingsnowand/orlive+deadundersconnectionfor5,4

loadingsnowand/orlive+deadundermembersfor3

=

=

=

=

( 2.38 )

Table 2.4. Correspondence between the index and the probability pupu 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


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52

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 2.4.

Calculate the index SR and lnS/R for the beam in figure 2.9:

Fig. 2.9. Example 2.4

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/mm

4

Calculate for the loading q (S):

2

3

22

qM

mmN5,1691035412

600020

W12

Lm

W

mm =

=

==

42222

q

2

qq 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/mm

2

L = 6m

12

LqM

2=

I24; Wy = 354cm


21/21


DRc = 744N2/mm

4

093,0294

744

m

Dv

Rc

Rc ===

Calculate the index SR (2.36):

0,3877,33,287744

5,169294

DD

mm

DD

mm

Rc

Rc

SR

SRRS >=

+

=

+

=

+

=

Calculate the index lnS/R (2.37):

0,3033,41,0093,0

5,169

294ln

vv

m

mln

vv

m

mln

2222

Rc

Rc

2

S

2

R

S

R

R

Sln

>=+

=+

=+

=

Remarks

1. In this method, the general condition p pu (2.19) is replaced by:

u ( 2.39 )

which expresses the condition E > 0 (S > R); u is 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 statemethod.

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

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

( ) 0R,,R,R;S,,S,SE n21n21 =KK ( 2.40 )

a function of random variables Si and Ri and pu an accepted risk, depending on the

consequences.

At present this method is used only in scientific works.

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