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.
52
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:
G ( X ) = R( X ) − S ( X )
where G ( X ) is limit state function, see section 2.2.
X is stochastic variables.
R( X ) is the resistance of structure in question.
S ( X ) is the loads or stresses etc. acting on the structure.
In general, R( X ) and S ( X ) 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,
Yi , i = 1,2,L , n :
R = R(Y )
(3.4.1)
Define the deterministic sensitivity, ζ i , of R with respect to Yi as the change in R with
respect to a change in Yi :
∂R
ζi =
at Y = Y0
(3.4.2)
∂Yi
where Y0 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:
53
∂R  Yi 
(3.4.3)
  at Y = Y0
∂Yi  R 
where the subscript ‘R’ indicates relative sensitivity. This is the percentage change in
the response relative to the percentage change in the input variable.
ζ Ri =
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:
∂G
−
σx
∂xi P * i
(3.4.4)
λi =
2
n
 ∂G



σ
x
 ∂x * k 


k
k =1
P
∑
where P * is the mean value point.
2) Probabilistic sensitivity factor as the gradient of limit state function at design
point:
∂G
−
σx
∂xi P * i
(3.4.5)
λi =
2
n
 ∂G



σ
x
 ∂x * k 


k P
k =1
∑
where P * is the design point.
3) Consider the response variable R = R(Y ) . Let Y be a random vector. The CDF of
each Yi is assumed to be known and is denoted as Fi ( y ) . As a function of Y, R will be
a random variable; let FR (r ) denote the cumulative distribut ion function (CDF) of R. A
commonly used sensitivity measure is
∂σ R
(3.4.6)
ζi =
∂σ Yi
where σ R is the standard deviation of R and σ Yi is the standard deviation of Yi . This is
a measure of the degree of response uncertainty as a function of input uncertainty.
4) Partial sensitivity factors (Mansour, 1995):
∂β
∂β
σ i , δ 'i =
µi
∂µ i
∂µ i
∂β
∂β
σ
ηi =
σ i , η 'i =
, νi = i
∂σ i
∂ν i
µi
δi =
(3.4.7)
(3.4.8)
The following presents another method to calculate sensitivity factor (Zhang,
1998).
54
3.4.4 Adaptive probabilistic sensitivity factor
In this chapter, the following notations are used and the others are defined when used:
x1 , x 2 ,L , x n are random variables and denoted by a vector X = ( x1 , x 2 , L, x n )
sometimes.
G ( x1 , x 2 ,L , xn ) is limit state function,
G ( X ) > 0 for X in safe set of structure.
G ( X ) = 0 for X on limit state surface.
G ( X ) < 0 for X in failure set.
f ( X ) = f X ( x 1 , x 2 , L, x n ) is the probability function of X.
Pf is failure probability of structure in question,
Pf =
∫f
X
G ( X ) ≤0
( X ) dX
Exi , Sxi are mean values of x i and Exi* ? Sxi* are reference values of Exi , Sxi .
Sxi , Sxi are standard deviation of x i and Sx*i ? Sxi* are reference values of Sxi , 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
Ex i
Sxi
?
:
*
Sxi*
Sx i
( α i , η i ) where α i = ∂P S x* , η i = ∂P S x*
∂Ex
∂Sx
i
i
i
i
b) Sensitivity module: si = (
c) Sensitivity factor: λ i =
∂Pf
∂Ex i
Sxi* ) 2 + (
∂Pf
∂Sx i
Sxi* ) 2
si
n
∑s
k =1
k
About the notes of these definitions, refer to Zhang, 1998.
Definition 2:
Sensitivity vector: a vector is composed of sensitivity gradient and sensitivity factors:
( ∂ P S x* , ∂ P S x* , λ i ),i.e.,
∂E x
∂S x
i
i
i
i
( α i , η i , λ i ).
Definition 3:
Sensitivity matrix: a n × 3 matrix made of sensitivity vectors of all random variables,
i.e.,
55
α 1 η 1

α 2 η 2
L L

α n η n
λ1 

λ2 
L

λn 
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 i =
si
n
can be easy to
∑ sk
k =1
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:
∆PExi *
P −P *
∂P *
αi =
S xi = lim
S xi = lim ∆Exi
S xi
(3.4.9)
∆
Ex
→
0
∆
Ex
→
0
∂Exi
i
∆Ex i
i
∆Exi
∆PSxi *
P −P *
∂P *
ηi =
S xi = lim
S xi = lim ∆Sxi
S xi
(3.4.10)
∆Sxi → 0 ∆Sx
∆Sxi →0
∂Sx i
∆Sxi
i
where
P is the structural failure probability;
P∆Exi is the failure probability of structure when E x
P∆Sxi is the failure probability of structure when S x
i
i
is replaced by E xi + ∆E xi .
is replaced by Sx i + ∆S x i .
So, α i , η i can be obtained by:
P∆ Exi − P *
Sx
∆Exi
P −P *
η i ≈ ∆Sxi
Sx
∆Sxi
αi ≈
i
i
(3.4.11)
(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:
56
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
57
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.
No.
1
2
3
4
5
6
Table 3.5.1 Commonly used distributions
Types
No.
Types
Normal
7
Shifted Rayleigh
Uniform
8
Type I largest value
Lognormal
9
Type I smallest value
Gamma
10
Type II largest value
Beta
11
Type II smallest value
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 longnormal 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
58
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
59
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
60
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:
Pf = ∫ L ∫ f X ( x1 , x 2 ,L , xn ) dx1dx 2 L dx 2
(3.6.1)
Gi ( X )≤0
i =1, 2 ,L, k
where f X ( x1 , x2 , L, xn ) is the joint probability density function of the n random
variables and Gi ( X ) = Gi ( x1 , x 2 ,L , x n ) 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.
F1
F2
Fn-1
Fn
Figure 3.6.1 Schematic representation of a series system
61
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 Fi denotes the ith event of failure, i.e., the event that g i ( X ) ≤ 0 , and Si represents the
corresponding safe event, i.e., the event that g i ( X ) > 0 , then the combined system
failure event Fs is denoted as the union ‘U’ of all individual failure events Fi as
Fs = ∪ Fi
(3.6.2)
i
( )
( )
The corresponding probability of system failure is
P( Fs ) = P ∪ Fi = 1 − P ∩ S i
i
i
(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 nonconservative bound on the actual system probability, i.e.,
max P(Fi ) ≤ P ( Fs )
(3.6.4)
i
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
k
i =1
i =1
1 − P ∩ S i = 1 − ∏ P(S i ) = 1 − ∏ [1 − P (Fi )]
i
(3.6.5)
k
where
∏ P( S ) represents the product of the probabilities of survival. The result given
i
i =1
by equation (3.6.5) represents an upper bound on the true probability of system failure,
i.e.,
k
P( Fs ) ≤ 1 − ∏ [1 − P(Fi )]
(3.6.6)
i =1
62
Combining equations (3.6.4) and (3.6.6), an upper and lower bounds are obtained as
follows
k
max P( Fi ) ≤ P(Fs ) ≤ 1 − ∏ [1 − P( Fi )]
i
(3.6.7)
i =1
Although the upper bound in equation (3.6.7) is not difficult to evaluate, it can be
further simplified by noticing that
k
k
i =1
i =1
1 − ∏ [1 − P( Fi )] ≤ ∑ P( Fi )
(3.6.8)
therefore, equation (3.6.7) can be written as
k
max P( Fi ) ≤ P( Fs ) ≤ ∑ P (Fi )
i
(3.6.9)
i =1
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):
k
i −1
 
 
P( F1 ) + ∑ max   P(Fi ) − ∑ P(Fi F j ) ;0 ≤ P( Fs )
 
i= 2
j =1
 
(3.6.10)
≤ ∑ P( Fi ) − ∑ max P(Fi F j )
k
k
i =1
i= 2
j <i
where P( F1 ) is the maximum of the individual probabilities of failure and P( F1 F2 ) is
the probability of intersection (mutual occurrence) of two events of failure, F1 and F2 .
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 P( F1 F2 ) 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), P( F1 F2 ) is replaced by (Ang, 1984):
P( F1 F2 ) = P( A) + P( B)
(3.6.11)
whereas, in the upper bound, the same term is replaced by
63
P( F1 F2 ) = max[ P( A), P( B)]
(3.6.12)
 β j − ρβ i 

P( A) = Φ (− β i )Φ  −
(3.6.13)
2 

1
−
ρ


 β i − ρβ j 

P( B ) = Φ (− β j )Φ  −
(3.6.14)
2 

1
−
ρ


and Φ ()
. is the standard normal cumulative distribution function and β i are the
individual safety indices (Hasofar- Lind). ρ is the correlation coefficient between two
where
failure events (or modes). Such a correlation coefficient between the failure events
Fi = {X : g i ( X ) ≤ 0} and F j = {X : g j ( X ) ≤ 0} can be evaluated (Ditlevsen in 1979):
ρ gi , g j =
cov( g i , g j )
σ gi σ g j
(3.6.15)
where
n
 ∂g   ∂g 
cov( g i , g j ) = ∑  'i   ' j 
m=1  ∂x m  *  ∂x m  *
σ gi
 n  ∂g  2 
= ∑  'i  
 m=1  ∂x m  * 
(3.6.16)
1/2
(3.6.17)
1/2
 n  ∂g  2 
σ gi = ∑  'i  
(3.6.18)
 m=1  ∂x m  * 
In equations (3.6.16) to (3.6.18), x1' , x2' , L, xn' 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.
F1
F2
Fn-1
Fn
Figure 3.6.2 Schematic representation of a parallel system
64
If Fi denotes again the ith event of failure, i.e., the event that g i ( X ) ≤ 0 , and Si
represents the corresponding safe event, i.e. the event that g i ( X ) > 0 , then the
combined system failure event of a parallel system Fp of k components (i.e., failure
events) is the intersection or mutual occurrence of all failure events Fi , i.e.,
Fp = ∩ Fi
(3.6.14)
( )
(3.6.15)
i
( )
The corresponding probability of system failure is
P( F p ) = P ∩ Fi = 1 − P ∪ S i
i
i
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
( ρ = 1 ) corresponds to the upper bound and no correlation between any pair
corresponds to lower bound. Thus, for positively correlated failure events, these bounds
are:
∏ P (F ) ≤ P(F ) ≤ min P(F )
k
i
i =1
p
i
i
(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, ThoftChristensen 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.
65
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.
F2
F1
F3
F4
F5
Figure 3.6.3 Schematic representation of parallel subsystems
connected in a series ( minimal cut set)
The failure event ‘ Fg ’ 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.,
Fg = ∪ ∩ ( Fij )
(3.6.17)
j
th
where ( Fij ) is the i
i
component failure in the j th failure path. The probability of failure
of such a system is thus determined from
P( Fg ) = P ∪ ∩( Fij ) 
(3.6.18)
 j i

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.
66
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’ Fs is the union of all m possible failure modes:
m

(3.6.19)
Pf = P( Fs ) = P ∪( Fi ) 
i
=
1


where Fi is the event ‘failure in the ith mode’. For each such mode, a sufficient number
of members (or structural ‘nodes’) must fail; thus
n

(3.6.20)
P( Fi ) = P ∩ ( F ji ) 
j
=
1


where F ji is the event ‘failure of the j th member in the ith failure mode’ and n i 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
k

(3.6.21)
Ps = P( S S ) = P ∪ ( S i )
i
=
1


where S S is the event ‘structural survival’ and Si the event ‘structural survival in mode
i’, i = 1,2,L , k , with k not equal to the final node index.
From (3.6.21) it follows that
k

(3.6.22)
Pf = P ∩( S i ) 
i
=
1


where Si 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
 li

P( S i ) = P  ∪ ( S ji )
(3.6.23)
 j =1

where F ji is the event ‘failure of the j th member in the ith survival mode’ and where l i
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
67
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:
 k

Pf = ∫ L ∫ f X ( x1 , x 2 ,L , x n ) dx1dx 2 L dx 2 = ∫ L ∫ I ∪ Gi ( X )  f X ( X )dX (3.6.24)
 i =1

Gi ( X )≤0
i =1, 2 ,L, k
where I[ ] is the indicator function for a series system
k

Gi ( X ) ≤ 0
k
 1
∪
I ∪ G i ( X ) = 
i =1
 i=1
 
the others
0
th
where Gi ( X ) represents the i limit state function, i = 1,2,L , k .
(3.6.25)
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):
lV ( ) = a1lV1 ( ) + a 2 lV2 ( ) + L + a k l Vk ( )
(3.6.26)
with
k
∑a
i
=1
(3.6.27)
i =1
where the a i are weighing coefficients. Each component lVi (
)
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 a i . In
particular the calculations will be simplified if those limit states which contribute in
only a minor way to Pf 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 x *i , the point in n-dimensional X space having
the highest probability density f X (
b) For
each
(
*
i
x ,
)
y *j = x *j − µ Xj / σ Xj
) consistent with
Gi ( ) ≤ 0 .
0 .5
n
2
calculate δ i = ∑ y *j i 
with ( y * ) i given by
 j=1

(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 δ i > δ L where δ L is some arbitrarily
chosen limit. As a first-order approximation, the error in Pf associated with any
limit state which is ignored in this way is given by Perror ≈ Φ (− δ L ) .
68
d) For the remaining limit states, use (3.6.26) as the sampling function in (3.6.24)
with a i chosen on the basis of the δ i 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 limitatio n which implies that
target reliability indices cannot, normally, be specified on a general basis, but only caseby-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), P0 . Define
P0 = target risk, or probability of failure
Pf = the probability of failure (as estimated from analysis)
69
Then, for a safe design,
Pf ≤ P0
(3.7.1)
Alternatively, the safety index can be used. In fact, its use is more common for design
criteria development. Define
β 0 = target safety index
β = safety index (as estimated from analysis)
β 0 = −Φ −1 (P0 )
β = −Φ −1 (Pf )
(3.7.2)
Φ is the standard normal cumulative distribution function. Then, for a safe design,
β ≤ β0
(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 wellestablished 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 designtransferable 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):
70
[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
−3
PF = 10
PF = 10 −4
I-Redundant structure
( β t = 3.09 )
( β t = 3.71 )
II-Significant
warning
before
the P = 10 −4
F
occurrence of failure in a non-redundant
( β t = 3.71 )
structure
−5
III-No warning before the occurrence of PF = 10
failure in a non-redundant structure
( β t = 4.26 )
PF = 10 −5
( β t = 4.26 )
PF = 10 −6
( β t = 4.75 )
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
71
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
−7
5.0 (2 .9 × 10 )
6.0 (1 .0 × 10 −9 )
Primary (initial yield)
3.5 (2 .3 × 10 − 4 )
4.0 (3 .2 × 10 − 5 )
Primary (ultimate)
2.5 (6 .2 × 10 −3 )
3.0 (1 .4 × 10 −3 )
Secondary
2.0 (2 .3 × 10 − 2 )
2.5 (6 .2 × 10 −3 )
Tertiary
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)).
72
Recommended target safety indices for fatigue are summarized in Table 3.7.3 (SSC398). 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 Naval ships
ships
Category 1
A significant fatigue crack is not 1.0
1.5
−1
(6.7 × 10 −2 )
considered to be dangerous to the (1 .6 × 10 )
crew, will not compromise the
integrity of the ship structure, will not
result in pollution; repairs should be
relatively inexpensive.
Category 2
A significant fatigue crack is not 2.5
3.0
−3
considered to be immediately (2 .6 × 10 ) (1 .4 × 10 −3 )
dangerous to the crew, will not
immediately
compromise
the
integrity of the ship structure, will not
result in pollution; repairs should be
relatively expensive.
Category 3
A significant fatigue crack is 3.0
3.5
−3
considered to compromise the (1 .4 × 10 ) (2 .3 × 10 − 4 )
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.
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
73
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 β 0
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 ( β 0 )
Metal structures for buildings (dead, live, and 3
snow loads)
Metal structures for buildings (dead, live, and 2.5
wind loads)
Metal structures for buildings (dead, live, and 1.75
snow, and earthquake loads)
Metal connections for buildings (dead, live, and 4 to 4.5
snow loads)
Reinforced concrete for buildings (dead, live, and
snow loads)
Ductile failure
3
Brittle failure
3.5
Table 3.7.4 Target reliability levels
Member, limit state
Target reliability level ( β 0 )
Structural steel
Tension member, yield
3.0
Beams in flexure
3.0
Column, intermediate slenderness
3.5
Reinforced concrete
Beam in flexure
3.0
Beam in shear
3.0
Tied column, compressive failure
3.5
Masonry, unreinforced
Wall in compression, inspected
5.0
Wall in compression, uninspected
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.
74
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
P f (t ) = P [R (t ) ≤ S (t )]
(3.8.1)
If the instantaneous probability density functions f R (t ) and f S (t ) of R(t) and S(t)
respectively are known, the instantaneous failure probability P f (t ) 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 reapplied 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 δ t , in time, is
considered instead of instantaneous time t.
With this interpretation, it follows that
P f (t ) = P [R (t ) ≤ S (t )] =
∫
G [X ( t )]≤0
f X (t ) [ X (t ) ]dX (t )
(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 P f (t ) usually is correlated to the value
P f (t + δ t ), δ t → 0 , since typ ically 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
75
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):
(3.8.3)
Z (t ) = R (t ) − S (t )
and to establish the probability that Z(t) becomes zero or less in the lifetime t L 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 Z (t ) ≤ 0 occurs during t L 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 firstpassage 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 Fmax Y ,T ( y ) 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 Y (t ) = Y ( X (t ), t ) 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
 1

(3.8.4)
Fmax Y ,T ( y ) ≈ P0 exp  −
ν (t )dt 
∫
0
 P0

in which P0 = P (Y ( 0 ) < y ) and ν (t ) is the mean up-crossing rate of the process Y (t )
through the level y at time t, or equivalently the mean zero down-crossing rate of the
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process G ( t ) = G ( X ( t ), t ) = y − Y (t ) . The rate ν (t ) can be calculated by Rice’s formula
(1994) or as the parallel system parametric sensitivity factor (Hagen et al, 1991):
ν (t ) =
∂
P (G (t ) < 0 ∩ G ( t ) + G (t )θ < 0)θ = 0
∂θ
(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 τ i . The reference period T (e.g. one year) is divided into
n i intervals of length τ i = T / n i and n i 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 F X ( X i ) , the extreme value distribution in the reference period T
i
is determined as
Fmax X i ( X i ) = F X i ( X i )
ni
(3.8.6)
T
When combinations of load processes X 1 , X 2 , L , X k are considered it is assumed that
the loads are stochastically independent with positive integer repetition numbers n i ,
where
n1 ≤ n2 ≤ L ≤ n k
(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:
ni
=N+
ni − 1
(3.8.8)
The extreme value distribution of the combination X 1 , X 2 is now given by:


Fmax X1 , X 2 ( X 1 , X 2 ) = ∫ Fmax X1 ( X 1 X 2 ) f X 2 ( X 2 ) dX 2 
T

T

(3.8.9)
This equation is then applied recursively fo r each additional load process (e.g. X 3 ).
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:
Y1 = max {X 1 (t )} + X 2 (t * ) + L + X k (t * )
( )
( )
Y 2 = X 1 t * + max {X 2 (t )}+ L + X k t *
LL L
( )
(3.8.10)
( )
Y k = X 1 t * + X 2 t * + L + max {X k (t )}
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 Y k ,
where Y k is to be applied in the computation of reliability. If Y k represents different
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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:
78
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.
79