52

Chapter 3 Structural probability modelling and sensitivity analysis methods

3.1 General The following is the general procedure for modelling of a physical problem in a probability manner:

a) Identify physical modes to describe all relevant or significant failure modes (see Section 3.2).

b) Formulate the limit state function for each of the chosen failure modes (see Section 3.3 and Chapter 4).

c) Select the basic variables, including sensitivity analysis etc (see Section 3.4). d) Model uncertainty in a consistent way by probability distributions (see Sections

3.5, 1.3 and 4.6). e) Choose statistical distribution types and distribution parameters for all uncertain

variables (see Sections 3.5, 1.3 and 4.6). f) Integrate probability distributions with the physical mode in a limit state

function formulation to form the basis for a reliability analysis (see Chapter 2). g) Identify target reliability indices (see Section 3.6). h) Analyse and assess structural reliability (see Chapter 4, 5). i) Check and update.

3.2 Identification of problem Structural systems are composed of individual structural components. Well-designed structures are often redundant, so failure of an individual component does not usually constitute collapse. The overall goal for a structural design is to achieve some target reliability for the total structure, and the goal for the structural reliability analysis is then to document that this target reliability is achieved. All significant modes of failure for the structure (see section 1.4) should be identified. This includes failures of the individual structural components where each component may often fail in more than one ways. Component-based design should have safety considerations with respect to consequences for the system. Target reliabilities depend on the type and consequence of failure and are differentiated accordingly. Reliability methods are based on structural analysis models, such that target reliabilities depend on the applied analysis model and the distributions assumed. 3.3 Establishment of limit state function To ensure the safety of the designed structure, all significant modes of failure for the structure should be identified. Then the models for physical representation of the failure characteristics must be chosen. For each considered failure mode, a failure criterion must be formulated. The failure criterion may, for example, be expressed in terms of a deformation, which exceeds a critical threshold, or a load which exceed a capacity.

53

When a failure criterion has been formulated, analytical models for physical representation of the failure characteristics must be chosen. The models should be taken as same state-of-the-art deterministic models used in a corresponding deterministic analysis of the structural component. The chosen models give the characteristics used in the failure criterion as mathematical functions of governing basic variables. Each relevant limit state function should be established. Limit state function is a comprehensive function, by which all kinds of uncertainties, such as structure strength, loads acting on the question, etc, are reflected in failure probability of structure in question. The following formula is a basic form of limit state function:

)()()( XSXRXG −= where )(XG is limit state function, see section 2.2. X is stochastic variables.

)(XR is the resistance of structure in question. )(XS is the loads or stresses etc. acting on the structure.

In general, )(XR and )(XS should be consistent with the corresponding state-of-the-art methods of deterministic modes as mentioned above. 3.4 Sensitivity analysis and selections of random variables 3.4.1 introduction Selection and determination of basic random variables are important part of structural reliability assessments. One benefit of sensitivity studies is the identification of the model parameters that have the most effect on the estimated reliability or safety index. The other benefit is being able to identify those parameters that can be taken as fixed values and need not be considered as random variables in reliability models. In this section, the basic concepts and methods of sensitivity analysis are described. According to the sensitivity analysis, the basic random variables should be selected for the structure in question. 3.4.2 Deterministic sensitivity factor Before giving the concept of probabilistic sensitivity factor, this section first presents deterministic sensitivity factor. Define a response variable, R, as a function of a vector, Y, of design factors,

niYi ,,2,1, L= : )(YRR = (3.4.1)

Define the deterministic sensitivity, iζ , of R with respect to iY as the change in R with respect to a change in iY :

ii Y

R∂∂

=ζ at 0YY = (3.4.2)

where 0Y is a reference value of Y. Alternatively, define a relative sensitivity coefficient

as the fractional sensitivity of R with respect to a fractional change in Y:

54

∂∂

=RY

YR i

iRi

ζ at 0YY = (3.4.3)

where the subscript ‘R’ indicates relative sensitivity. This is the percentage change in the response relative to the percentage change in the input variable. 3.4.3 Probabilistic sensitivity factor The following are the few traditional methods (definitions) of calculating probabilistic factor (Zhang, 1998): 1) Probabilistic sensitivity factor as the gradient of limit state function at mean values:

λ

∂∂

σ

∂∂

σ

ii P

x

k Px

k

n

Gx

Gx

i

k

=

−

=∑

*

*

2

1

(3.4.4)

where P* is the mean value point. 2) Probabilistic sensitivity factor as the gradient of limit state function at design point:

λ

∂∂

σ

∂∂

σ

ii P

x

k Px

k

n

Gx

Gx

i

k

=

−

=∑

*

*

2

1

(3.4.5)

where P* is the design point. 3) Consider the response variable )(YRR = . Let Y be a random vector. The CDF of each iY is assumed to be known and is denoted as )(yFi . As a function of Y, R will be a random variable; let )(rFR denote the cumulative distribut ion function (CDF) of R. A commonly used sensitivity measure is

iY

Ri σ

σζ

∂∂

= (3.4.6)

where Rσ is the standard deviation of R and iYσ is the standard deviation of iY . This is

a measure of the degree of response uncertainty as a function of input uncertainty. 4) Partial sensitivity factors (Mansour, 1995):

δ∂β∂µ

σii

i= , δ∂β∂µ

µii

i' = (3.4.7)

η∂β∂σ

σii

i= , η∂β∂νi

i

' = , νσµi

i

i= (3.4.8)

The following presents another method to calculate sensitivity factor (Zhang, 1998).

55

3.4.4 Adaptive probabilistic sensitivity factor In this chapter, the following notations are used and the others are defined when used: x x xn1 2, , ,L are random variables and denoted by a vector ),,,( 21 nxxxX L=

sometimes. G x x xn( , , , )1 2 L is limit state function, 0)( >XG for X in safe set of structure. 0)( =XG for X on limit state surface. 0)( <XG for X in failure set.

)(Xf = f x x xX n( , , , )1 2 L is the probability function of X. Pf is failure probability of structure in question,

P f X dXf XG X

=≤

∫ ( )( ) 0

Exi , Sxi are mean values of ix and Exi* ? Sxi

* are reference values of Exi , Sxi .

iSx , Sxi are standard deviation of ix and *iSx ? Sxi

* are reference values of iSx , Sxi .

In order to quantify the effect of each random variable on the estimated reliability or safety index, the following definition is introduced.

Definition 1:

a) Sensitivity gradient: a gradient vector of Pf to ExSx

i

i*

? SxSx

i

i* :

( iα ,iη ) where α

∂∂i

xx

PE

Si

i= * , η

∂∂i

xx

PS

Si

i= *

b) Sensitivity module: sP

ExSx

P

SxSxi

f

ii

f

ii= +( ) ( )* *

∂

∂

∂

∂2 2

c) Sensitivity factor: λ ii

kk

n

s

s=

=∑

1

About the notes of these definitions, refer to Zhang, 1998.

Definition 2: Sensitivity vector: a vector is composed of sensitivity gradient and sensitivity factors:

( ∂∂

PE

Sx

xi

i

* , ∂∂

PS

Sx

xi

i

* , λ i ),i.e.,

(α i , η i ,λ i ).

Definition 3: Sensitivity matrix: a n × 3 matrix made of sensitivity vectors of all random variables, i.e.,

56

nnn ληα

ληα

ληα

222

111

LLL

3.4.5 Approximate calculation method In Section 3.4.4, a sensitivity matrix is suggested to estimate (analyse) the sensitivity of

each variable to structural failure. From the definition above, λ ii

kk

ns

s=

=∑

1

can be easy to

get if (α i ,η i ) is known. The following give an approximate method to calculate

(α i ,η i ). According to the differential definition, α i and η i can be calculate as follows:

α∂

∂ii

x Ex

Exi

ix Ex

Exi

ix

PEx

SPEx

SP P

ExS

ii

ii

i= = =

−→ →

* * *lim lim∆ ∆

∆∆∆ ∆0 0

(3.4.9)

η∂

∂ii

x Sx

Sxi

ix Sx

Sxi

ix

PSx

SPSx

SP P

SxS

ii

ii

i= = =

−→ →

* * *lim lim∆ ∆

∆∆∆ ∆0 0

(3.4.10)

where P is the structural failure probability; P Exi∆ is the failure probability of structure when Exi

is replaced by E Ex xi i+ ∆ .

P Sxi∆ is the failure probability of structure when Sxi is replaced by S Sx xi i

+ ∆ .

So, α i , η i can be obtained by:

α iExi

ix

P PEx

Si

≈−∆

∆* (3.4.11)

η iSxi

ix

P PSx

Si

≈−∆

∆* (3.4.12)

3.5 Uncertainties and probability distributions In Section 1.3, an overview of uncertainties and probabilistic distributions has been given. This section will give a general guidance further. Structural uncertainties in reliability analysis are represented by random variables modelling the governing variables. The degree of exactness of distribution of random variables should affect directly the exactness of calculation results of reliability analysis of structures in question. Uncertainties associated with an engineering problem and its physical representation in an analysis have various sources which may be grouped as follows:

57

a) Physical uncertainty, also known as intrinsic or inherent uncertainty, is a natural randomness of a quantity, such as the uncertainty in the yield stress of steel as caused by a production variability, or the variability in wave and wind loading.

b) Measurement uncertainty is uncertainty caused by imperfect instructions and sample disturbance when observing a quantity by some equipment.

c) Statistical uncertainty is uncertainty due to limited information such as a limited number of observations of a quantity.

d) Model uncertainty is uncertainty due to imperfections and idealisations made in physical model formations for load and resis tance as well as in choices of probability distribution types for representation of uncertainties.

However, one should be aware that other types of uncertainties may be present, such as uncertainties related to human errors which are not covered here. Transactions between the quoted different uncertainty types may exist. Uncertainties are represented in reliability analysis by modelling the governing variables as random variables. The corresponding probability distributions can be defined based on statistical analyses of variable observations of the individual variables, providing information on their mean values, standard deviations, correlation with other variables, and in some cases also their distribution types. Variables whose uncertainties are judged to be important, e.g. by experience or by a sensitivity study, should be represented as random variables in a reliability analysis. Their respective probability distributions should be documented as far as possible, based on a statistical analysis of available background data. Correlation between variables may appear and should be accounted for. Correlation coefficients can be estimated by statistical analysis. Model uncertainties in a physical model for representation of load and/ or resistance quantities can be described by random factors, each defined as the ration between the true quantity and the quantity as predicted by the model (see equation 1.3.1). A mean value not equal to 1.0 expresses a bias in the model. The standard deviation expresses the variability of the predictions by the model. An adequate assessment of a model uncertainty factor may be available from sets of field measurements and predictions. Subjective choices of the distribution of a model uncertainty factor will, however, often be necessary. The importance of a model uncertainty may vary from case to case and should be studied by interpretation of parametric sensitivities. The probability distribution for a random variable represents the uncertainty in that variable. The probability dis tribution is most conveniently given in terms of a standard distribution type with some distribution parameters. Distribution types usually used in engineering structures, in general, can be obtained by mathematical statistics methods. Table 3.5.1 gives some distributions usually used. Regressions of available observations of a quantity will not always given enough information to allow for interpretation of the distribution type for the uncertainty quantity, and a choice of the distribution type has to

58

be made. The results of a reliability analysis may be very sensitive to the tail of the probability distribution, so a proper choice of the distribution type will often be crucial.

Table 3.5.1 Commonly used distributions No. Types No. Types 1 Normal 7 Shifted Rayleigh 2 Uniform 8 Type I largest value 3 Lognormal 9 Type I smallest value 4 Gamma 10 Type II largest value 5 Beta 11 Type II smallest value 6 Shifted exponential

Normal or log-normal distributions should normally be used when no detailed information is available. The log-normal distribution is required for load variables, whereas the normal distribution is required for resistance variables. However, a variable which is known to never take on negative values should usually be assigned a long-normal distribution rather than a normal distribution. The following definitions apply in this context:

a) A variable is considered a strength variable if it is unfavourable with respect to failure when its value is less than the mean.

b) A variable is considered a load variable if it is unfavourable with respect to failure when its value is greater than the mean.

Mean values and standard deviations should normally be obtained from recognised data sources. In general, the following procedure is required for determination of the distribution type and estimation of the associated distribution parameters (DNV, Classification Notes, No.30.6, 1992):

a) Choose a set of possible distributions Based on experience from similar types of problems, physical knowledge or analytical results, choose a set of possible distributions. In general, the traditional distributions (see table 3.5.1) are considered first. Especially when there are not sufficient documents or data associated with the variable in question, normal and lognormal distributions should be used. The theoretical basis of choosing normal consideration is the central- limit statistical theorem. If there are several possible distribution choices, the following techniques can be used for acceptance or rejection of distribution choices:

1) Visual identification by plot of data on probability paper 2) By comparison of moments (a Hermite distribution covers a wide range

of combinations) 3) Statistical tests such as Kolmogorov and Chi-square

59

4) Asymptotic behaviour for extreme value distributions. If two types of distributions give equally good fits, it is recommended, particularly for load variables, to choose the distribution with most probability content in the tail, unless one of the distributions fits possible data observations in the tail better than other.

b) Estimate the relevant distribution parameters The relevant distribution parameters are estimated in these distributions by statistical analysis of available observations of the uncertain quantities. Here statistical analysis methods are used to deal with available observations of the uncertainty quantities. Regressions may be based on:

1) Moment estimators 2) Least-square fit methods 3) Maximum likelihood methods 4) Visual inspections of data plotted on probability paper

It is important that, when distributions are chosen by the above methods, such choices, including the steps leading to the choices, are satisfactorily documented. The distribution parameter estimates can be uncertain themselves, especially if they are based on regressions of very few data. This uncertainly is called statistical uncertainty, and it should be assessed as to whether it can be neglected or not. 3.6 System reliability 3.6.1 Introduction System reliability can be defined simply as the reliability that contains more than one limit state function. System reliability is a complex and extensive problem, and includes the following aspects to be evaluated:

a) Load re-distribution, i.e. redundancy; b) Multiple failure modes, i.e. complexity; c) Correlation between safety margins for different failure events or types of

failure.

The analysis of a realistic structural system, even within a deterministic framework, can be a considerable task. Usually it is facilitated by simplifications and idealizations in each of (i) applied loads and load sequencing (load modelling), (ii) structural system and its components and connections between components (system modelling), and (iii) material response and strength characteristics (material modelling). Criteria for limit state violation also need to be specified – in conventional design usually a permissible stress criterion is adopted but other criteria may have greater validity. In theory, the main difference between simple reliability and system reliability lies in the complexity of limit state functions. Marine structures, however, involve several modes of failure, i.e., there is a possibility that a structure may fail in one or more of several possible failure scenarios. Two main

60

sources of ‘system effects’ are identified (SSC-351, 1990). The first is due to possible multiplicity of failure modes of a component or a structural member. For example, a beam under bending and axial loads may fail in buckling, flexure or shear. Each one of these modes can be defined by one limit state equation. Even though in this case, system reliability methods must be used in order to combine the possible failure modes and to obtain an assessment of the total risk of failure of the beam. Another example of multiplicity of failure modes is the primary behaviour of a ship hull. In the primary behaviour, one treats the ship as a single beam subjected to weight, buoyancy and wave loads which induce sagging and hogging bending moments. The hull may fail (or reach a limit state) in one of several possible modes, e.g., buckling of deck or bottom panels or grillages, yielding of deck or bottom plating, etc. Here again, system reliability methods must be used to combine these different modes of failure and to obtain a total probability of failure. Multiple modes of failure of a member are usually modelled in system reliability analysis as a series system. A series system is one that is composed of links connected in series such that the failure of any one or more of these links constitute a failure of the system, i.e., ‘weakest link’ system. In the case of the primary behaviour of a ship hull, for example, any one of the failure modes discussed earlier will constitute failure of the hull (or a limit state to be prevented) and therefore can be considered as a series system. The second source of ‘system effects’ is due to redundancy in multi-component engineering structures. In such structures, the failure of one member or component dose not constitute failure of the entire system. Usually members must fail to form a ‘failure path’ before the entire structure fails. The failure of each member is defined by at least one limit state equation and a corresponding probability of failure. These individual member probabilities of failure must be combined to get the probability of failure of the system for a particular ‘failure path’. Thus, system reliability methods must be used to determine the reliability of a redundant structure. An example of a multi-component redundant structure in which system effects are important is a fixed offshore platform. For such a platform to fail, several members must fail to form a failure path. The probability of failure of the system in this case is usually modelled as a parallel system in which all links along the failure path of the system must fail for the entire structure to fail. Moreover, there will be several possible paths of failure, any of which will constitute failure of the entire platform. Therefore each failure path and the associated probability of failure can be considered as a link in a series system since failure of any link constitutes a failure of the system in the series model. The total offshore platform can be thus modelled as several parallel subsystems each of which represents a failure path connected together in series, since any of them will constitute a failure of the platform. Parallel systems and general systems, considering series and parallel subsystems will be discussed in later sections of this chapter. 3.6.2 General calculation methods

61

The exact system reliability problem, taking into consideration possible time-dependent random variables, is an “outcrossing” problem. If the time-dependant loads or response of the structure exceeds (outcrosses) one or more of several possible failure modes (surfaces), failure of the structure occurs. The problem formulated in terms of stochastic processes however is difficult to solve. Only a few cases of very simple structures, with certain load history models, can be evaluated in this manner and the reliability of the structure at any time during its life be calculated. For a single time-varying load it is possible to treat the peaks as a random variables and its extreme-value distribution may be formulated to perform the reliability calculation. At present, the general problem is formulated as a time- independent problem, which is sufficient only for the evaluation of an instantaneous reliability. As such, the form of the equation to evaluate the system reliability is the same as that of component reliability (equation 2.2.1) except that, now, the multiple integration is carried out over all possible limit state functions corresponding to the potential modes of failure. For k modes of failure, and n random variables, the system probability of failure can be written as:

22121

,,2,10)(

),,,( dxdxdxxxxfP nX

kiXG

f

i

LLL

L=≤

∫ ∫= (3.6.1)

where ),,,( 21 nX xxxf L is the joint probability density function of the n random variables and ),,,()( 21 nii xxxGXG L= is the k limit state function. The domain of

integration in equation (3.6.1) is over the entire space where each of the ‘k’ limit state functions are negative or zero. In addition to the difficulties encountered in the computation of component reliability, the domain of integration over all possible modes of failure in equation (3.6.1) will present additional numerical difficulties. For these reasons, this general exact formation is not used, and instead of determining the combined total probability of failure of the system as given by (3.6.1) only an upper and lower bounds on that system probability are determined. These upper and lower bounds are usually determined by considering the structure to be a series system or a parallel system or a combination of both (general system). It should be noted that, in principle, simulation methods and the Monte Carlo technique can be used to solve equation (3.6.1) in the similar manner presented in chapter 2. Reduced variate techniques and other methods for improving convergence may be used also. 3.6.3 Bounds on the probability of failure of a series system A series system is one which fails if any one or more of its components fails. Such a system has no redundancy and is also known as ‘weakest link’ system. Schematically a series system is represented as in figure 3.6.1.

Figure 3.6.1 Schematic representation of a series system

Fn-1 Fn F1 F2

62

An example of a series system is a beam or an element which may fail in any of several possible modes of failure each of which may depend on the loading condition of the beam. A ship hull girder in its ‘primary behaviour’ is such a system with the additional combination that failure may occur in hogging or sagging condition. Each condition includes several modes of failure. If iF denotes the ith event of failure, i.e., the event that 0)( ≤Xg i , and iS represents the corresponding safe event, i.e., the event that 0)( >Xg i , then the combined system failure event sF is denoted as the union ‘U’ of all individual failure events iF as

iis FF ∪= (3.6.2)

The corresponding probability of system failure is ( ) ( )iiiis SPFPFP ∩−=∪= 1)( (3.6.3)

where ∩ represents the intersection or mutual occurrence of events. The calculation of the probability of system failure for a series system using equation (3.6.3) is generally difficult and requires information on correlation of all failure events. Approximations are therefore necessary and upper and lower bounds on the system probability of failure are constructed instead of evaluating the exact value. Two types of bounds can be constructed: first and second order bounds. 3.6.3.1 First order bounds These are bounds on the reliability of system failure which require no information on the correlation between the events of failure. In other words, the user of such bounds does not need any information on the correlation between the events of failure which, in many cases, are not available. They are constructed as follows. If the events of failure of a series system are assumed to be perfectly correlated, the probability of system failure is simply the maximum of the individual probabilities of failure. For positively correlated failure events, this assumption leads to the lower non-conservative bound on the actual system probability, i.e.,

( ) )(max siiFPFP ≤ (3.6.4)

On the other hand, if the events of failure are assumed to be statistically independent, an upper bound (conservative) can be determined. In this case, for independent failure events of a series system, the right hand side of equation (3.6.3) reduces to

( ) ( ) ( )[ ]∏∏==

−−=−=∩−k

ii

k

iii

iFPSPSP

11

1111 (3.6.5)

where ( )∏=

k

iiSP

1

represents the product of the probabilities of survival. The result given

by equation (3.6.5) represents an upper bound on the true probability of system failure, i.e.,

( ) ( )[ ]∏=

−−≤k

iis FPFP

1

11 (3.6.6)

63

Combining equations (3.6.4) and (3.6.6), an upper and lower bounds are obtained as follows

( ) ( ) ( )[ ]∏=

−−≤≤k

iisi

iFPFPFP

1

11max (3.6.7)

Although the upper bound in equation (3.6.7) is not difficult to evaluate, it can be further simplified by noticing that

( )[ ] ( )∑∏==

≤−−k

ii

k

ii FPFP

11

11 (3.6.8)

therefore, equation (3.6.7) can be written as

( ) ( ) ( )∑=

≤≤k

iisi

iFPFPFP

1

max (3.6.9)

Equation (3.6.9) gives the final result for the bounds of a series system and states the obvious conclusion that the actual probability of system failure lies between the maximum of the individual probabilities and the sum of all individual probabilities. These bounds are narrow if one mode of failure is dominant, i.e., if one of the individual probabilities of failure is much larger than the others. If not, these bounds may be too wide to be useful. In such cases a more narrow set of bounds, second order bounds, should be considered. 3.6.3.2 Second order bounds These bounds were developed in references (Cornell, 1967; Kounias, 1968; Hunter, 1976; Ditlevsen, 1979; Ang, 1984) and are given in terms of pair-wise dependence between failure events, therefore, are called second order bounds. The original bounds for k potential modes of failure are given as (Cornell, 1967; Kounias, 1968):

( ) ( ) ( ) ( )

( ) ( )ji

k

iij

k

ii

s

k

i

i

jjii

FFPFP

FPFFPFPFP

∑∑

∑ ∑

=<

=

=

−

=

−≤

≤

−+

21

2

1

11

max

0;max (3.6.10)

where ( )1FP is the maximum of the individual probabilities of failure and ( )21FFP is the probability of intersection (mutual occurrence) of two events of failure, 1F and 2F . The bounds given by equation (3.6.10) depend on the ordering of the failure modes and different ordering may correspond to wider or narrower bounds. Therefore, bounds corresponding to different ordering may have to be evaluated to determine the narrowest bounds. The evaluation of the joint probability ( )21FFP required in equation (3.6.10) remains difficult. A weakened version of these bounds (more relaxed bounds) was proposed by Ditlevsen in 1979 as follows. In the lower bound of equation (3.6.10), ( )21FFP is replaced by (Ang, 1984):

( ) )()(21 BPAPFFP += (3.6.11) whereas, in the upper bound, the same term is replaced by

64

( ) )](),(max[21 BPAPFFP = (3.6.12)

where ( )

−

−−Φ−Φ=

21)(

ρ

ρβββ ij

iAP (3.6.13)

( )

−

−−Φ−Φ=

21)(

ρ

ρβββ ji

jBP (3.6.14)

and ( ).Φ is the standard normal cumulative distribution function and iβ are the individual safety indices (Hasofar-Lind). ρ is the correlation coefficient between two

failure events (or modes). Such a correlation coefficient between the failure events { }0)(: ≤= XgXF ii and { }0)(: ≤= XgXF jj can be evaluated (Ditlevsen in 1979):

ji

ji

gg

jigg

gg

σσρ

),cov(, = (3.6.15)

where

∑=

∂

∂

∂∂

=n

m m

j

m

iji x

g

xg

gg1 *

'*

'),cov( (3.6.16)

2/1

1

2

*'

∂∂

= ∑=

n

m m

ig x

gi

σ (3.6.17)

2/1

1

2

*'

∂∂

= ∑=

n

m m

ig x

gi

σ (3.6.18)

In equations (3.6.16) to (3.6.18), ''2

'1 ,,, nxxx L are the reduce random variables and the

derivations are evaluated at the most likely failure points as discussed in chapter 2. The proposed bounds by Ditlevsen (1979) apply only for normally distributed random variables. Narrower bounds than the second order bounds can be constructed, but they involve intersection of more than two failure events and much more complicated. 3.6.4 Bounds on the probability of failure of a parallel system A parallel system is one which fails only if all its components fail, i.e., failure of one component only will not necessarily constitute failure of the system. Schematically, such a system can be represented as shown in figure 3.6.2.

Fn

F2

Fn-1

Figure 3.6.2 Schematic representation of a parallel system

F1

65

If iF denotes again the ith event of failure, i.e., the event that 0)( ≤Xg i , and iS represents the corresponding safe event, i.e. the event that 0)( >Xg i , then the combined system failure event of a parallel system pF of k components (i.e., failure

events) is the intersection or mutual occurrence of all failure events iF , i.e.,

iip FF ∩= (3.6.14)

The corresponding probability of system failure is ( ) ( )iiiip SPFPFP ∪−=∩= 1)( (3.6.15)

Equation (3.6.15) for failure of a parallel system should be compared with equation (3.6.3) for failure of a series system. It is clear that the failure of a series system is the union (any) of the component failure, whereas, the failure of a parallel system is the intersection (all) of the component failures. As in a series system, the evaluation of equation (3.6.15) for determining the exact system failure of a parallel system is generally difficult, and approximation by constructing bounds is usually necessary. Simple first order lower and upper bounds can be constructed using similar arguments as for the series system. Now however, perfect correlation between all failure events ( 1=ρ ) corresponds to the upper bound and no correlation between any pair corresponds to lower bound. Thus, for positively correlated failure events, these bounds are:

( ) ( ) )(min1

ii

p

k

ii FPFPFP ≤≤∏

=

(3.6.16)

Unfortunately, the bounds given by equation (3.6.16) on the probability of failure of a parallel system are wide and no second order bounds are available. In some special cases, however, the exact system failure can be evaluated. For example, Thoft-Christensen and Baker (1982) evaluated the probability of parallel system failure under deterministic loading and other restrictive conditions. 3.6.5 General System A general system is one that consists of a combination of series and parallel subsystems. A useful general system from an application point of view, is one that consists of parallel subsystems connected together in a series. An example application for such a general system would be an offshore platform (or, in general a statically indeterminate structure) where each failure path can be modelled as a parallel subsystem and all possible failure paths (parallel subsystems) are connected together in a series since any of them constitute failure of the platform. This representation is called ‘minimal cut set’ since no component failure event in the parallel subsystem (a failure path) can be excluded without changing the state of the structure from failure to safe. A schematic representation of parallel subsystems connected together in a series is shown in figure 3.6.3.

66

A general system may also consist of a series of subsystems connected together in parallel (minimal link set). Such systems, however, have less potential for application to structural reliability and therefore will not be discussed further.

The failure event ‘ gF ’ of a general system consisting of parallel subsystems connected

together in a series (minimal cut set) is given by the union (series) of intersection (parallel) of individual failure events, i.e.,

)( ijij

g FF ∩∪= (3.6.17)

where )( ijF is the ith component failure in the jth failure path. The probability of failure

of such a system is thus determined from

∩∪= )()( ijijg FPFP (3.6.18)

Exact evaluation of (3.6.18) is difficult and requires information of the joint dependence of failure events. Similarly, bounds on the probability of failure given by (3.6.18) are not available in general. If however, one is able to determine the probability of failure of each parallel subsystem (for example, under restrictive conditions), then first or second order bounds can be determined using equations (3.6.9) or 3.6.10) for the remaining series system. 3.6.6 Summaries of the other calculation methods The analysis of realistic structural systems even within a deterministic framework can be a considerable task. Usually it is facilitated by simplifications and idealizations in each of (Robert E. Melchers, 2001)

(i) applied loads and load sequencing (load modelling) (ii) structural system and its components and connections between components

(system modelling) (iii) material response and strength characteristics (material modelling).

For system reliability analysis, there are, in principal at least, two complementary approaches which can be adopted (Bennett and Ang, 1983). These are the ‘failure modes’ approach and the ‘survival modes’ approach.

Figure 3.6.3 Schematic representation of parallel subsystems connected in a series ( minimal cut set)

F1 F2

F3

F4 F5

67

The failure mode approach is based on the identification of all possible failure modes for the structure. The basic calculation formula is similar to equation (3.6.3). Since failure through any one failure path implies failure of the structure, the event ‘structural failure’ sF is the union of all m possible failure modes:

∪==

=)()(

1i

m

isf FPFPP (3.6.19)

where iF is the event ‘failure in the ith mode’. For each such mode, a sufficient number

of members (or structural ‘nodes’) must fail; thus

∩=

=)()(

1ji

j

n

i FPFP (3.6.20)

where jiF is the event ‘failure of the jth member in the ith failure mode’ and in represent

the number of members required to form the ith failure mode. The survival mode approach is based on identifying various states (or modes) under which the structure survives. Survival of the structure requires survival in at least one survival mode, or

∪==

=)()(

1i

i

k

Ss SPSPP (3.6.21)

where SS is the event ‘structural survival’ and iS the event ‘structural survival in mode i’, ki ,,2,1 L= , with k not equal to the final node index. From (3.6.21) it follows that

∩=

=)(

1i

i

k

f SPP (3.6.22)

where iS is the event ‘structure does not survive in survival mode i’. Clearly, to attain

survival in any particular survival mode all the members contributing to that survival mode must survive. It follows that failure to survive in given survival mode is equivalent to failure of a sufficient number of the contributing members, or

∪=

=)()(

1 jij

l

i SPSPi

(3.6.23)

where jiF is the event ‘failure of the jth member in the ith survival mode’ and where il

represent the number of members required to ensure survival of the ith survival mode. Some results have been given for structural systems composed of ideal rigid-plastic members (Bennet and Ang, 1983). The survival mode approach has received much less attention in the literature than the failure modes approach, perhaps in part owing to difficulties in conceptualisation of survival modes and in formulating the limit state equations and in part owing to the difficulty of generating a truly lower bounds stress field to satisfy the requirements for the survival mode. In addition to the above methods, the integral method and the Monte Carlo method are still important methods of estimating system reliability. The integral method is the same

68

as the reliability analysis of component with one failure mode (see chapter 2). The basic Monte Carlo method is also in Chapter 2. The following is a simple introduction of importance sampling (Monte Carlo) method of series system reliability analysis. The probability of failure represented by (3.6.1) can be written as:

∫ ∫ ∪∫ ∫

==

==

≤

dXXfXGIdxdxdxxxxfP Xi

k

inX

kiXG

f

i

)()(),,,(1

22121

,,2,10)(

LLLL

L

(3.6.24)

where I[ ] is the indicator function for a series system

≤=

∪∪ =

= others the 0

0)( 1)( 1

1

XGXGI i

k

ii

k

i

(3.6.25)

where )(XGi represents the ith limit state function, ki ,,2,1 L= .

The integration of (3.6.24) using importance sampling was described in Chapter 2 for one limit state function. Where there are k different limit state functions as in a structural system, it is not sufficient to use a uni-model sampling density function. Very large errors can be introduced this way (Melchers, 1991). Instead, a useful approach is to use the multi-model sampling function (Melchers, 1984, 1990):

( ) ( ) ( ) ( )kVkVVV lalalal +++= L

21 21 (3.6.26)

with

11

=∑=

k

iia (3.6.27)

where the ia are weighing coefficients. Each component ( )iVl is selected for the ith

limit state function in the same way as for an individual limit state, with most interest being the regions contribut ing the greatest probability density for the limit state. Normally, not all limit states will be of equal importance for a reliability analysis. This can be taken into account by appropriate selection of the weighing coefficients ia . In

particular the calculations will be simplified if those limit states which contribute in only a minor way to fP can be identified. One way in which this can be done is with

reference to FOSM concepts. The suggested algorithm runs as follows (Melchers, 1984, 1990):

a) For each limit state i determine *ix , the point in n-dimensional X space having

the highest probability density ( )Xf consistent with ( ) 0≤iG .

b) For each *ix , calculate ( )

5.0

1

2*

= ∑

=

n

jiji yδ with iy )( * given by

( ) XjXjjj xy σµ /** −= (i.e., a ‘standardized’ space might be visualized in which

the relative importance of each limit state function is considered). c) Ignore all limit state functions for which Li δδ > where Lδ is some arbitrarily

chosen limit. As a first-order approximation, the error in fP associated with any

limit state which is ignored in this way is given by ( )LerrorP δ−Φ≈ .

69

d) For the remaining limit states, use (3.6.26) as the sampling function in (3.6.24) with ia chosen on the basis of the iδ values.

Additional to the above methods, theresponse surface method is also a calculation method of system reliability (see Chapter 2). 3.7 Target reliability 3.7.1 general In order to assess structural reliability, target reliability is given the same importance as the reliability analysis. Target reliability is a standard that has to be met in design or in service in order to ensure that certain safety levels are achieved. The overall safety goal of a structure design is to achieve some target reliability for the total structure and one of the goals of structural reliability ana lysis is then to document that this target reliability is achieved. A reliability analysis can be used to verify whether such a target reliability is achieved for a structure or structural element. One of the difficulties is that the uncertainties included in a structural reliability analysis will deviate from those encountered in real life. This is because (DNV, 1992):

a) The reliability analysis does not include gross errors which may occur in real life.

b) The reliability analysis, due to the lack of knowledge, includes statistical uncertainty and model uncertainty in addition to the physical uncertainty (often referred to as epistemic) which is present in real life.

c) The reliability analysis may include uncertainty in the probabilistic model due to distribution tail assumptions.

Target reliability depends on the type and consequence of failure and the applied analysis model and the distribution assumed. Target reliability of components based design should also have safety considerations with respect to consequence for the system. So a reliability index calculated by a reliability analysis is only a nominal value or operational value, dependent on the analysis model and the distribution assumptions, rather than a true reliability value which may be given a frequency interpretation. Calculated reliabilities can therefore usually not be compared with required target reliability values, unless the latter are based on similar assumptions with respect to analysis models and probability distributions. This is a limitation which implies that target reliability indices cannot, normally, be specified on a general basis, but only case-by-case for individual examples. All of the above ingredients should have to be considered in determining the target reliability of structure. 3.7.2 Definitions of target values To establish probability-based design criteria, it is necessary to define a maximum allowable risk (or probability of failure), 0P . Define

0P = target risk, or probability of failure

fP = the probability of failure (as estimated from analysis)

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Then, for a safe design,

0PPf ≤ (3.7.1)

Alternatively, the safety index can be used. In fact, its use is more common for design criteria development. Define

0β = target safety index β = safety index (as estimated from analysis)

( )01

0 P−Φ−=β ( )fP1−Φ−=β (3.7.2)

Φ is the standard normal cumulative distribution function. Then, for a safe design, 0ββ ≤ (3.7.3)

As described in Section 3.7.1, the selection of target reliabilities is difficult task (Payer et al., 1994). These values are not readily available and need to be generated or selected. Also, these levels might vary from one industry to another, due to factors such as the implied reliability levels in currently used design practices by industries, failure consequences, public and media sensitivity, or response to failures that can depend on the industry type, types of users or owners, design life of a structure, and other political, economic, and social factors. 3.7.3 Methods of selecting target values The following is the general procedure to determine the target reliabilities:

a) Analysis of sequence and nature of structure failure in question. b) Reliability calculation and analysis of established relevant structure.

Minimum values of target reliabilities depend on the consequence and nature of failure, and to the extent possible, should be calibrated against well-established cases that are known to have adequate safety. In cases where well established structures are not available for the calibration of target reliabilities, such target reliabilities may be derived by comparison of safety levels established for similar existing structural design solutions that may be satisfactorily considered as being transferable.

c) Experienced and acceptable decision techniques. If there is no possibility of establishing target reliabilities by calibration against existing, well-established structures, or using similar design-transferable target reliabilities, then the minimum target reliability values may be based upon accepted decision analysis techniques. Table 3.7.1 gives general acceptable failure probabilities and reliability indices. In this stage, the determination of structural target reliability index should be discussed with experts of relevant academic and industrial bodies. It is also worth determining the current technique/expertise level of manufacture and production of whole structure and related components. When target reliabilities are taken as those values stated in Table 3.7.1, the following listed considerations should be assessed (DNV, Classification Notes, N0.30.6, 1992):

71

[1] The evaluation of the consequence of failure considers the use of the structure and the nature of the relevant surroundings to such structure, i.e., the relative extent of the possibilities for: personnel injuries, physical damage and / or pollution. (for a consequence of failure to be described as being less serious the risk to life upon failure is normally to be considered as being relatively negligible).

[2] The evaluation of the less serious considers the type of structural failure (i.e., the possibility for timely warning of failure and the possible development of such failure).

[3] The stated values for acceptable target reliabilities (β -values) are to be

further increased if a failure situation may result in catastrophic consequences).

d) Updating. According to the above procedure, there are three methods which have been employed: (1) The code writers and / or the profession agrees upon a reasonable or

acceptable value. This method is used for novel structures where there is no prior history.

(2) Code calibration (calibrated reliability levels that are implied in currently used codes). The level of risk is estimated for each provision of a successful code. Safety margins are adjusted to eliminate inconsistencies in the requirements. This method has been commonly used for code revisions.

(3) Economic value analysis (cost benefit analysis). Target reliabilities are chosen to minimize total expected costs over the service life of the structure. In theory, this would be the preferred method, but it is impractical because of the data requirements for the model.

Table 3.7.1 General acceptable failure probabilities and reliability indices

Consequence of failure Class of failure

Less serious Serious

I-Redundant structure 310−=FP

( 09.3=tβ )

410−=FP ( 71.3=tβ )

II-Significant warning before the occurrence of failure in a non-redundant structure

410−=FP ( 71.3=tβ )

510−=FP ( 26.4=tβ )

III-No warning before the occurrence of failure in a non-redundant structure

510−=FP ( 26.4=tβ )

610−=FP ( 75.4=tβ )

The second approach was commonly used to develop reliability-based codified design. The target reliability levels, according to this approach, are based on calibrated values

72

of implied levels in a currently used design practice. The argument behind this approach is that a code represents a documentation of an accepted practice. Therefore, since it is accepted, it can be used as a launching point for code revision and calibration. Any adjustments in the implied levels should be for the purpose of creating consistency in reliability among the resulting designs according to the reliability-based code. Using the same argument, it can be concluded that target reliability levels used in one industry might not be usefully applicable to another industry. The third approach is based on cost-benefit analysis. This approach was used effectively in dealing with designs for which failures result in only economic losses and consequences. Because structural failures might result in human injury or loss, this method might be very difficult to be used because of its need for assigning a monetary value to human life. Although this method is logical on an economic basis, a major shortcoming is its need to measure the value of human life. Consequently, the second approach is favoured for this study. An important consideration in the choice of design criteria is the consequences of failure. Clearly the target reliability relative to collapse of the hull girder should be larger than that of a non-critical welded detail relative to fatigue. 3.7.4 Recommended target reliabilities Recommended target safety indices for hull girder (primary), stiffened panel (secondary) and unstiffened plate (tertiary) modes of failure and the corresponding notional probabilities of failure are summarized in table 3.7.2 (SSC-398). These lifetime values are based on professional judgement in view of the extensive reliability analysis performed in that project (SSC-398) together with the values reviewed in the literature.

Failure mode Commercial ships Naval ships Primary (initial yield) 5.0 ( )7109.2 −× 6.0 ( )9100.1 −×

Primary (ultimate) 3.5 ( )4103.2 −× 4.0 ( )5102.3 −×

Secondary 2.5 ( )3102.6 −× 3.0 ( )3104.1 −×

Tertiary 2.0 ( )2103.2 −× 2.5 ( )3102.6 −×

Table 3.7.2 Recommended target safety indices (Failure probabilities) for ultimate strength

The consequences of the ultimate strength failure are considered as follows: primary ultimate, very serious; secondary, serious and tertiary, not serious. The primary initial yield failure mode is listed here only because it represented state-of-the-art design practice. The probabilities of failure associated with the β values given in Table 3.7.2 were

determined using the standard Gaussian cumulative distribution function (see equation (3.7.2)).

73

Recommended target safety indices for fatigue are summarized in Table 3.7.3 (SSC-398). These are considered to be lifetime values, i.e., related to be the probability of failure during the intended service life, as predicted prior to service. These values are based on professional judgement supported by the analysis reported in SSC-398, as well as a comprehensive review of the literature.

Description Commercial ships

Naval ships

Category 1 A significant fatigue crack is not considered to be dangerous to the crew, will not compromise the integrity of the ship structure, will not result in pollution; repairs should be relatively inexpensive.

1.0 ( )1106.1 −×

1.5 ( )2107.6 −×

Category 2 A significant fatigue crack is not considered to be immediately dangerous to the crew, will not immediately compromise the integrity of the ship structure, will not result in pollution; repairs should be relatively expensive.

2.5 ( )3106.2 −×

3.0 ( )3104.1 −×

Category 3 A significant fatigue crack is considered to compromise the integrity of the ship and put the screw at risk and / or will result in pollution. Service economic and political consequences will result in from significant growth of the crack.

3.0 ( )3104.1 −×

3.5 ( )4103.2 −×

Table 3.7.3 Recommended target safety indices (probabilities of failure) for fatigue design

The target reliabilities defined in Table 3.7.2 and 3.7.3 can be used as a design goal. A designer performing a comprehensive reliability assessment, relative to the failure modes addressed, can compare these results with the suggested targets. These values can be also be used to derive safety check expressions for use in a structural design code. These values may be used directly in design rather than assessment of an existing design. The following is some recommended target reliabilities of general structure or component (SSC-398). While the specific reliabilities will be a function of the strength criteria needed for specified materials and load combinations which are experienced by designated structures, it is useful to have an indication of the range of possible target reliability levels. Ellingwood et al. (1980) presented ranges for reliability levels for

74

metal structures, reinforced and prestressed concrete structures, heavy timber structures, and masonry structures, as well as discussions of issues that should be considered when making the calibration. Table 3.7.4 provided typical values for target reliability levels. This table was developed based on values provided by Ellingwood et al. (1980). The target reliability levels shown in Table 3.7.5 were also used by Ellingwood and Galambos (1982) to demonstrate the development of partial safety factors. The 0β

values in Table 3.7.4 and 3.7.5 are for structural members designed for 50 years of service.

Structural type Target reliability level ( 0β )

Metal structures for buildings (dead, live, and snow loads)

3

Metal structures for buildings (dead, live, and wind loads)

2.5

Metal structures for buildings (dead, live, and snow, and earthquake loads)

1.75

Metal connections for buildings (dead, live, and snow loads)

4 to 4.5

Reinforced concrete for buildings (dead, live, and snow loads) Ductile failure Brittle failure

3 3.5

Table 3.7.4 Target reliability levels

Member, limit state Target reliability level ( 0β )

Structural steel Tension member, yield Beams in flexure Column, intermediate slenderness

3.0 3.0 3.5

Reinforced concrete Beam in flexure Beam in shear Tied column, compressive failure

3.0 3.0 3.5

Masonry, unreinforced Wall in compression, inspected Wall in compression, uninspected

5.0 7.5

Table 3.7.5 Target reliability levels There are still the other recommended target reliabilities, such as Canadian Standard Association (CSA) Deliberations, National Building Code of Canada, A.S. Veritas Research, and so on.

75

As described in Section 3.7.3, recommendations on target reliabilities are a comprehensive problem. They are established on the basis of information from the following three aspects: a) A synthesis and interpretation of the results of the reliability analysis of past

successful design practice. b) Experiences from other systems. The results of other exercises in which the level of

risk has been estimated for large structures will be helpful in calibrating the figure that are presented

c) Professional judgement. 3.8 Time dependent models 3.8.1 General In general the basic variables X will be functions of time. This comes about, for example, because loading changes with time (even if it is quasi-static, such as is the case for most floor loading) and because material strength properties change with time, either as a direct result of previously applied loading or because of some deterioration mechanism. Fatigue and corrosion are typical examples of strength deterioration. The elementary reliability problem in time-variant terms with a resistance R(t) and a load effect S(t), at time t becomes

( ) ( ) ( )[ ]tStRPtP f ≤= (3.8.1) If the instantaneous probability density functions ( )tf R and ( )tf S of R(t) and S(t) respectively are known, the instantaneous failure probability ( )tP f can be obtained

from the convolution integral. Strictly, (3.8.1) only has meaning if the load effect S(t) increases in value at time t (otherwise failure would have occurred earlier) or if the random load (effect) is re-applied precisely at this time. Failure could not occur precisely at any exact instant of time t (assuming, of course, that at time less than t the member was safe). Thus, in general, a change in load or load effect is required. This is assured if:

a) There are discrete load changes; b) For continuous time-varying- loads, an arbitrary small increment tδ , in time, is

considered instead of instantaneous time t. With this interpretation, it follows that

( ) ( ) ( )[ ] [ ][ ]∫ ≤

=≤=0)( )( )()(

tXG tXf tdXtXftStRPtP (3.8.2)

As before, X(t) is a vector of basic variables. In principal, the instantaneous failure probability given by (3.8.1) or (3.8.2) can be integrated over an interval of time 0 – t to obtain the failure probability over that period. In practice, however, the instantaneous value of ( )tP f usually is correlated to the value

( ) 0, →+ tttP f δδ , since typically the processes X(t) themselves are correlated in time.

The classical approach is to consider the integration transferred to the load or load effect process, which is then assumed to be representable, over the total time period, by an

76

extreme value distribution. The resistance is assumed essentially time invariant. This approach (also called ‘classical’ reliability) formed the basis of discussion in Chapter 2. A refinement is to consider shorter periods of time, such as the duration of a storm, or a year, and to apply extreme value theory within that period. Simple ideas akin to the concept of the return period then can be used to determine the failure probability over the lifetime of the structure. This approach is quite popular for practical reliability analysis of major structures such as offshore platforms, towers, etc., which are subject to definable and discrete loading events. A somewhat different way of looking at the problem is to consider the safety margin associated with (3.8.1):

( ) ( ) ( )tStRtZ −= (3.8.3) and to establish the probability that Z(t) becomes zero or less in the lifetime Lt of the

structure. This constitutes a so-called ‘crossing’ problem. The time at which Z(t) becomes less than zero for the first time is called the ‘time to failure’ and is a random variable. The probability that ( ) 0≤tZ occurs during Lt is called the ‘first-passage’ probability. The first-passage concept is more general than the classical approaches. In particular, there is no restriction on the form of G(X). However, the determination of the first-passage probability and a proper understanding of the concept require some knowledge of stochastic processes. If the elementary reliability problem (3.8.1) is to be made to cope with more than one load or load effect, as is required, for example, in design code applications, it is necessary to combine two or more loads effects into one equivalent load effect. 3.8.2 Time dependent models in marine structures Recommended models for the maximum value of combined time dependent loads are (DNV, Classification Notes No. 30.6, 1992):

a) Out-crossing models. b) The Ferry Borges-Castanheta load model. c) Turkstra’s rule.

The distribution ( )yF TY ,max of the maximum value in a given reference period [0, T] of

time dependent loads, can be expressed in terms of the mean out-crossing rate from a safe region of a stochastic load vector process. Let the individual loads be given as the components of the load vector process X(t), and let ( ) )),(( ttXYtY = be an in general

non- linear, explicit time-dependent combination of the load processes. The distribution of the maximum of Y is then approximately given as

( )

−≈ ∫00

0,max )(1exp dttP

PyF TY ν (3.8.4)

in which ( )yYPP <= )0(0 and )(tν is the mean up-crossing rate of the process )(tY

through the level y at time t, or equivalently the mean zero down-crossing rate of the

77

process ( ) )(),()( tYyttXGtG −== . The rate )(tν can be calculated by Rice’s formula

(1994) or as the parallel system parametric sensitivity factor (Hagen et al, 1991):

( ) ( )( ) 00)()(0 =<+∩<∂∂

= θθθ

ν tGtGtGPt (3.8.5)

Equation (3.8.5) can be determined using standard methods from time independent reliability theory (such as Robert E. Melchers, 2001, etc.). The Ferry Borges-Castanheta load model is a simplified load model where the mathematical problems associated with the estimation of the extreme of a sum of loading processes is facilitated. For each load process it is assumed that the load changes after equal, so-called, elementary intervals of time iτ . The reference period T (e.g. one year) is divided into

in intervals of length ii nT /=τ and in is called the ‘repletion’ number. The loads in

the elementary intervals are further assumed to be constant in each interval and statistically independent from interval to interval. For a load process having the marginal distribution ( )iX XF

i, the extreme value distribution in the reference period T

is determined as ( ) ( ) i

iT

iX

niXi XFXF =max (3.8.6)

When combinations of load processes kXXX ,,, 21 L are considered it is assumed that the loads are stochastically independent with positive integer repetition numbers in ,

where knnn ≤≤≤ L21 (3.8.7)

and where the number of repetitions to be applied on each of the load processes is +N , where +N is the set of positive natural numbers such that:

+=−

Nn

n

i

i

1 (3.8.8)

The extreme value distribution of the combination 21 , XX is now given by:

( )

= ∫ 2221max21max )()(,

212,1dXXfXXFXXF X

TX

TXX

(3.8.9)

This equation is then applied recursively fo r each additional load process (e.g. 3X ).

Turkstra’s rule is an approximation to determine the largest maximum for the sum of loads or load effects. Using Turkstra’s rule, the maximum value, X, over time is replaced by k stochastic variables, namely:

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

( ) ( ) ( ){ }tXtXtXY

tXtXtXY

tXtXtXY

kk

k

k

max

max

)()(max

*2

*1

*2

*12

**211

+++=

+++=

+++=

L

LLLL

L

(3.8.10)

where *t is an arbitrary point in time. If the load processes are statistically dependent, the conditional distribution for the ‘non- leading’ loads should be used, conditioning on the leading load at its maximum. The largest load is determined as the largest of kY , where kY is to be applied in the computation of reliability. If kY represents different

78

load effects, all combinations should be checked according to the corresponding failure criteria. By this rule the reliability of a structure is only checked at those points in time where one individual load processes reaches its maximum value. Therefore, the reliability will be overestimated. However, it has been shown that this overestimation is usually very small (DNV, Classification Notes, No.30.6). Alternatively to the above independence-based formulations, a series system may be utilized to account for correlation between the event safety margins as assumed in the two combination rules above. Adding the absolute values of extremes of several load variables, assuming that these occur simultaneously, is conservative. 3.9 Conclusions In this chapter, the general procedure for modelling of a physical problem in a probability manner is represented. First, identification of problem is introduced. Then, to ensure the safety of the designed structure, all significant modes of failure for the structure are identified. For each considered failure mode, a failure criterion must be formulated. The failure criterion may, for example, be expressed in terms of a deformation which exceeds a critical threshold, or a load which exceed a capacity. When a failure criterion has been formulated, analytical models for physical representation of the failure characteristics must be chosen. The models should be taken as same state-of-the-art deterministic models used in a corresponding deterministic analysis of the structural component. The chosen models give the characteristics used in the failure criterion as mathematical functions of governing basic variables. Each relevant limit state function should be established. Third, the sensitivity analysis method is represented. One benefit of sensitivity studies is the identification of the model parameters that have the most effect on the estimated reliability or safety index. The other benefit is being able to identify those parameters that can be taken as fixed va lues and need not be considered as random variables in reliability models. In this section, the basic concepts and methods of sensitivity analysis are described. According to the sensitivity analysis of structural reliability, the basic random variables should be selected for a structural reliability analysis. Fourth, an overview of uncertainties and probabilistic distributions is given. In general, the following procedure is required for determination of the distribution type and estimation of the associated distribution parameters: a) Choose a set of possible distributions; b) Estimate the relevant distribution parameters. The relevant distribution parameters are estimated in these distributions by statistical analysis of available observations of the uncertain quantities. Here statistical analysis methods are used to deal with available observations of the uncertainty quantities. Regressions may be based on:

79

Moment estimators; Least-square fit methods; Maximum likelihood methods; Visual inspections of data plotted on probability paper. Fifth, system reliability is introduced. System reliability can be defined simply as the reliability that contains more than one limit state function. In this section, the calculation methods of system reliability are represented. Sixth, in order to assess structural reliability, target reliability determination is the same importance as the reliability analysis. They are established on the basis of information from the following three aspects: a) A synthesis and interpretation of the results of the reliability analysis of past successful design practice; b) Experiences on other systems. The results of other exercises in which the level of risk has been estimated for large structures will be helpful in calibrating the figure that are presented; c) Professional judgement. Finally, time dependent models are introduced simply.