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8/8/2019 Limit State Handout[1]
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1
RELIABILITY-BASED DESIGN OF PIPELINES
LIMIT STATE DESIGN
When a structure is under certain loadings, it will experience stress/strain and deformin some way. To ensure structural integrity, designers have to assess all the possible
failure modes of the structure. A 'limit state' is a condition beyond which the structure
is viewed as failure. In another word, the 'violation' of a limit state can be defined as
an undesirable condition for the structure. Limit states for structures can be divided
into three categories: (1) ultimate limit state, which defines the collapse of structures.
(2) damage limit state, which defines the damage of structures. (3) serviceability limit
state, which defines disruption of normal use of structures.
Deterministic Design - safety factor
This is the conventional design methodology. It is also called 'working stress' design.In this design, a safety factor is introduced to ensure certain safety margin in the
design. The format of strength assessment is
[ ]n
y = (1)
where is stress in structure, [ ] is allowable stress, y is yield stress, n is safetyfactor. So this safety factor includes all the uncertainties in the design.
Deterministic limit state design - partial factor
The uncertainties in different design variables are obviously different. So a single
safety factor is not a good way to cover these different uncertainties. So partial factors
are introduced in deterministic limit state design. This means that different design
variables could have different partial factors. A general expression for strength
assessment is:
RLL ++ ...2211 (2)
where 1 , 2 and are partial factors, 1L , 2L are load effect, R is resistance.
Reliability-based limit state design
The same safety format as in deterministic limit state design (Eq. 2) is used in
reliability-based limit state design. The partial factors (also called partial safety
factors) are calculated by reliability analysis. In this way, target reliability can be
achieved by using this method.
This is why some people thought limit state design was reliability-based limit state
design. This method will be explained in the next section in details.
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RELIABILITY THEORY AND ITS APPILICATION
Structural reliability is concerned with lifetime structural safety, how much is required
and how this can be achieved in design. Structural reliability analysis will take
uncertainties, which are related to the process of structural design, into consideration.
Structural reliability is different from 'risk analysis' or 'reliability engineering' in the
following ways:
* structural engineering only has to do with failures of the structure resulting from
excessive service loads and or too low structural strength.
* risk analysis aims to account for all possible failure scenarios in the operation of
the structure. Hence structural reliability is only one aspect of risk analysis.
Probabilistic method is a more rational approach than deterministic method because:
* it enables uncertainties to be handled in a rational way in design and
assessment; in particular it enables the sensitivity of the reliability to variousdesign variables and decision to be determined.
* it follows that it also provides a more rational basis for decision making than is
possible with purely deterministic analysis.
* Reliability index is invariant for a limit state, but safety factor is not.
* less load combinations need to be considered.
Probabilistic approach is not a replacement but a complement of deterministic
approach.
Uncertainties in Structural Engineering
Structural reliability is concerned with uncertainties, so we will classify the
uncertainties in the structural design process. The uncertainties we need to consider
are those associated with the predicted stillwater and environmentally-induced load
effects on structural elements and with the predicted resistance of these elements to
the various limit states. Broadly, formal uncertainties may be classified into two
groups: physical uncertainties and knowledge-based uncertainties. The later may be
reduced at a cost by collecting more information more carefully, or by adopting more
realistic and/or sophisticated models. A third less formal group is human
uncertainties- particularly important, as we shall see.
Physical uncertainties
This is also referred to as 'inherent, intrinsic or fundamental uncertainty'. It is related
to the natural randomness of a basic variable and can be reduced but not be
eliminated. Examples in this category are:
* the variation in yield strength
* the variation of dimensions of a structure
* the variation of loads including wind load, wave-induced load, etc.
Knowledge Based uncertainties
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Uncertainties of this type can be reduced by some techniques and judgement. They
are sometimes referred to as subjective uncertainties.
Statistical uncertainties
This is caused by a limited number of observations. Statistical estimators, such assample mean and higher moments, are derived from available data using standard
procedures. These are then used to suggest an appropriate probability density function
and associated parameters. But, generally, the observations of the variable do not
perfectly represent it. In addition, different sample data sets will usually produce
different statistical estimators and this causes statistical uncertainty.
Modelling uncertainties
Structural design and analysis make use of simplified mathematical models to
represent the real phenomenon or behaviour of the structure. The uncertainty relatedto this model is defined as model uncertainty.
response(modelled)predicted
responseactualXm =
The so-called actual response is not known, but in practice the experimental
(measured) data is used as actual response. But actually experiments have also
uncertainties, which are not counted in the practice.
At this stage three things are worth noting:* modelling uncertainty is usually by far the largest uncertainty in both loading
(actions) and strength (resistance) with a cov (mX
V ) typically in the range 15%
to 30% or more.
*mX
V alone is often loosely referred to as the 'modelling uncertainty', but the
mean biasmX
can be equally important to assess - especially when using lower
bound strength equations.
* the predicted model may be analytical, numerical, or based on physical tests, all
aimed at representing real actions and resistance.
Phenomenological uncertainty
It arises because an apparently "unimaginable" phenomenon occurs to cause structural
failure.
Human errors
Most of recorded structural collapse or losses are attributed to human errors rather
than insufficient prescribed safety in design. About 50% to 90% (from differentstatistical resources) of accidents were caused by human errors. 85% has been quoted
as a typical value for marine structures.
Notional reliability
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Because the failure probability caused by human errors is not normally considered in
structural reliability analysis, there is large difference between the actual (real) riskand the predicted failure probability. At present this gap is typically 1 to 3 orders of
magnitude. So the predicted failure probability is referred to as being notional or
nominal.
Reliability Analysis Levels
Reliability methods are normally classified in three levels:
Level 1: In this method reliability based partial safety factor (PSF) are applied to
characteristic value of load components and resistance factors in the safety checkequations used in design. This is a deterministic format most commonly advocated for
limit states design codes at present.
Level 2: In this method the information of mean and variance of random variables areused in the analysis. It is called first order and second moment (FOSM) method, [oradvanced first order and second moment (AFOSM) method].
Level 3: In this level the integration of the multi-dimensional joint probabilitydistributions of the design variables is used to calculate the 'exact' failure probability.
This method is sometimes called 'exact method'.
A fourth (level 4) is sometimes referred to (Melchers, 1987) as incorporating
engineering economic analysis to give design for minimum total cost or maximumutility; but this is really a decision method not related to the complexity of thereliability method used.
According to how the randomness of the structures is considered, the methods can bedivided into:
(1). random field method;(2). random process method
(3). random variable method
(a) time-independent variable(b) time-dependent variable
Random Variable Reliability Methods
In this course we will concentrate on time-independent variable methods, but these
methods are generally not restricted to this case. They can be equally applied to (or atleast can be extended to) time-dependent variable.
Following methods can be applied for structural reliability analysis:
1). FOSM (First Order Second Moment) method, (or called Advanced First OrderSecond Moment Method). In the development of reliability method, a method is
called Mean Value First Order Second Moment method (MVFOSM), which is not
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correct. Actually it is only suitable for linear safety margin equation. When limit stateequation is not linear, this method will give wrong results (not accurate).
2). SORM (Second Moment Method)
3). Simulation-based methods(a). Monte-carlo simulation method
(b). importance sampling method(c). directional simulation
4). Response surface methods
5). Integration method (level 3)
Special Case
A special case in reliability analysis is when the limit state equation can be expressedas:
LR = (3)
where M safety margin, R is resistance of structure, L is load effect of structure. If both R and L obeys normally distribution and are independent, M should obeys
normal distribution because it is a linear combination of R and L. So the failure
probability of this limit state is:
[ ] ( )=
=
==
M
M
M
Mf
00MPP (4)
where Pfis failure probability, P(.) is probability of an event. ( ) is standard normaldistribution function. MM , are mean and standard deviation of M. is equal to
M
M
and is called reliability index, or safety index.
[ ] [ ] [ ] [ ] LRMLERELREME ==== (5)
( ) [ ] 2L2
R
2
M
2
M MVARME +=== (6)
( ) 2/12L2RLR
M
M
+
=
= (7)
A geometric explanation of the special case is shown in Figs. 1 and 2. A fewimportant points are worth noting:
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From Eq. 7 the reliability index can be calculated by the mean and standarddeviation of R and L. We don't need to know the probability distribution functions
of them.
The failure probability can be easily calculated once reliability index is know byEq.(4). This is only true when R and L are normally distributed.
Fig.2 shows that the reliability index is equal to the number of standard deviationby which M exceeds zero.
Fig.1
Fig. 2
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Based on Fig.1, another expression of failure probability can be derived as:
( ) ( )[ ] ( ) ( )
=== dxxfxF0LRP0MPP LRf (8)
It should be pointed out that this expression is valid when R and L have other kinds of
distribution functions.
Reliability S is defined as:
fP1S = (9)
General Cases
If there are n random variables { }n21 x,,x,x =X , and the limit state equation is:
{ } 0x,,x,xfM n21 == (10)
M0 means safe. In this general case the failure probability
is:
( )( )
n21
0f
n21x,...,x,xf dx...dxdxx,...,x,xfP n21
=X
(11)
where ( )n21x,...,x,x x,...,x,xf n21 is the joint probability density function of n variables.In practice, it is impossible to calculate the above integration except for some special
cases. So approximate methods have to be used in engineering calculations.
FIRST ORDER SECOND MOMENT METHOD
This method is widely used in engineering calculation. Even in this category there are
various algorithms. Among them, Fiessler and Rackwitz's algorithm, which show
good accuracy, efficiency and robustness, is the best. Therefore the procedure is
briefly described below.
If { }n21 x,...,x,x=X are the n independent variables involved in a structural designproblem, a general expression for any limit state function of a structure is:
M = ( )n21 x,...,x,xg (12)
When M < 0, the structure fails, and M > 0 the structure is safe. The failure surface is
given by M = 0.
A linear approximation of M can be found by using Taylor series expansion.
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( ) ( ) ( )=
+=n
1i
*'
i
*
ii
*
n
*
2
*
1 gxxx,...,x,xgM X (13)
where
( ) ( )i
**'
ix
gg
=X
X
{ }*n*2*1* x,...,x,x=X is the unknown design point.
If i and i represent the mean and standard deviation of the variable ix , the meanvalue of M is:
( ) ( )==
n
1i
*'
i
*
iiM
gx X (14)
and the standard deviation is:
( ){ }2/1
n
1i
2
i
*'
iM g
=
=
X (15)
M can be expressed as a linear combination of i as follows:
( )= =n
1i
i*'
iiM g X (16)
where
( )
( ){ }2/1
n
1j
2
j
*'
j
i
*'
ii
g
g
=
=
X
X(17)
are referred as sensitivity factors since they reflect the relative influence of each
design variable on the reliability index. The larger the sensitivity factor, the more
influential the variable is.
Hence the reliability index is:
( ) ( )
( )
=
=
=
=
n
1i
i
*'
ii
n
1i
*'
i
*
ii
M
M
g
gx
X
X
(18)
From Eq. (18), one gets
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9
( )( ) 0xgn
1i
ii
*
ii
*'
i ==
X (19)
The solution of this equation is:
iii
*
ix = for all i (20)
Because the design point is not known, iteration is needed to get the solution.
Finally, the probability of failure of the structure is:
( )=fP (21)
where is the standard normal distribution function.
If any of the design variables has non-normal distribution, a transformation is
necessary.
Suppose that the variable ix has density function ( )ix xf i and distribution function
( )ix xF i . The basic idea of the transformation is to let the original density function and
distribution function of the variable ix be equal to that of a normal variable at the
design point. That is:
( ) = Ni
N
i
*
i*ix xxF i (22)
( )
=
N
i
N
i
*
i
N
i
*
ix
x1xf
i(23)
From Eqs.(22) and (23), the equivalent mean and standard deviation are expressed as:
( )( ) Ni*ix1*iNi xFx i = (24)
( ){ }( )( )*ix
*
ix
1
N
ixf
xF
i
i
= (25)
where is the standard normal probability density function.
The flowchart for the above algorithm is shown as follows:
Step 1
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give an initial approximation of design point. Mean values are normally used as the
initial design point, eg. { }n,2,1)0*( ...,=X
Step 2
Calculate the following:
*
n
*
2
*
10 x,...,x,xgg =
( ) ( )i
**'
i
'
ix
ggg
==X
X
Step 3
Transform the non-normal variables to equivalent normal variables
N
i
*
ix
1*
i
N
i xFx i =
( ){ }( )( )*ix
*
ix
1
N
ixf
xF
i
i
=
Step 4
Calculate the following:
( ) *in
1i
*'
i xgx =
= X
( ) in
1i
*'
ix g = =
X
( ){ }2/1
n
1i
2
i
*'
ix g
=
=
X
x
i
*'
ii
g
=X
x
x0gx
=
Step 5
Calculate the design point for the next iteration
iii
)1m(*
ix =+
Step 6
Check if the iteration converges. The criterion for this is
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27.088.19267324500088.192673
xxx)2*(
2
)1*(
2
)2*(
2 == >
So results don't converge, continue iteration.
For iteration 2:
( ) 019.86148.53857988.19267384.2x,x,xgg *3*2*10 ===
88.192673xx
g *2
1
==
84.2xx
g *1
2
==
1x
g
3
=
x = 838.5558078.53857988.19267384.284.288.192673xg3
1i
*
i
'
i =+==
( ) ( ) ( ) 64.898821375000124500084.2388.192673g
3
1i
i
'
ix =++== =
( ) ( ) ( ) ( ) 102223
1i
2
i
'
i
2
x 101301648.17500012450084.215.088.192673g =++== =
214.106309x =
272.0214.106309
15.088.192673g
x
1
'
11 =
=
=
655.0
214.106309
2450084.2g
x
2
'
22 =
=
=
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705.0214.106309
750001g
x
3
'
33 =
=
=
308.3214.106309
64.898821019.8614838.555807gx
x
x0 =
=
=
865.215.0308.3272.03x 111)3(*
1 ===
87.19191424500308.3655.0245000x 222)3(*
2 ===
5.54991075000308.3705.0375000x 333)3(*
3 =+==
check stop criteria
009.0865.2
84.2865.2
x
xx)3*(
1
)2*(
1
)3*(
1 =
=
<
004.087.191914
88.19267387.191914
x
xx)3*(
2
)2*(
2
)3*(
2 =
=
<
021.05.549910
8.5385795.549910
x
xx)3*(
3
)2*(
3
)3*(
3 =
=
>
The results don't converge, so continue
Iteration 3
( ) 397.745.54991087.191914865.2x,x,xgg *3*2*10 ===
87.191914xx
g *2
1
==
865.2xx
g *1
2
==
1x
g
3
=
x = 705.5497615.54991087.191914865.2865.287.191914xg3
1i
*
i
'
i =+==
( ) ( ) ( ) 61.902669750001245000865.2387.191914g3
1i
i
'
ix=++==
=
( ) ( ) ( ) ( ) 102223
1i
2
i
'
i
2
x 10138.175000124500865.215.087.191914g =++== =
325.106680x =
270.0325.106680
15.087.191914g
x
1
'
11 =
=
=
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658.0325.106680
24500865.2g
x
2
'
22 =
=
=
703.0
325.106680
750001g
x
3
'
33 =
=
=
( )307.3
325.106680
61.902669397.74705.549761gx
x
x0 =
=
=
866.215.0307.327.03x 111)4(*
1 ===
853.19168724500307.3658.0245000x 222)4(*
2 ===
575.54936175000307.3703.0375000x 333)4(*
3 =+==
check stop criteria
0003.0866.2
865.2866.2
x
xx)4*(
1
)3*(
1
)4*(
1 =
=
<
001.0853.191687
87.191914853.191687
x
xx)4*(
2
)3*(
2
)4*(
2 =
=
<
001.0575.549361
5.549910575.549361
x
xx)3*(
3
)3*(
3
)4*(
3 =
=
<
001.0307.3
308.3307.3)4*(
)3*()4*(
=
=
<
So the iteration converges. The reliability index is 3.307. The design point is*
1x =2.866,*
2x =191687.853,*
3x =549361.575.
x3 is the most influential variable because the corresponding sensitivity factor has the
largest absolute value.
RELIABILITY-BASED LIMIT STATE DESIGN OF PIPELINES
The basic idea and procedure of reliability-based limit state design of pipelines will be
described.
Reliability-based limit state design has been used by some design codes of pipelines,
such as the recent DNV code (1996) for pipelines.
Procedure of Reliability-Based Limit State Design
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Reliability-based limit state design involves the following tasks:
identifying failure modes of the structure defining design format and limit state functions determining the uncertainties of all the design variables calculating the failure probability setting up target reliability levels for each failure modes calibrating partial safety factors for all limit states evaluating results of the designIdentifying failure modes of the structure
To ensure the integrity of pipelines following limit states must be checked in design:
out of roundness for serviceability bursting due to internal pressure, longitudinal force and bending buckling/collapse due to pressure, longitudinal force and bending fracture of welds due to bending / tension low-cycle fatigue due to shutdowns ratcheting due to reeling and shutdowns accumulated plastic strain
Defining design format and limit state functions
This is also called level 1 method. The combination of characteristic values and
partial safety factors is used to ensure certain level of safety of the structures.
A typical code format for safety checking might be
kmkfc RL (26)
where c and f are load effect partial safety factors > 1.0, and m is a resistancePSF < 1.0. They are illustrated in Fig. 4. Fig. 5 shows the design point in design
variable space.
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Fig.4
Fig.5
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The number of PSFs is decided by code writers. In principle, every design variable
can have a PSF, but this is not necessary because some design variables only have
marginal effect on reliability of the structure (shown by sensitivity factors). Hence
PSFs are only introduced to important design variables. In practice, if too few PSFs
(say 2 or 3), are used in a code, the resulting spread in reliability for a variety of
structural components will be unnecessarily large and wasteful. Five PSFs are thecommon number nowadays.
Lkand Rkare characteristic values of load effect and resistance respectively.
Characteristic Values
Characteristic values are defined as a fractile of the probability distribution of the
variable. For resistive variables the characteristic values are defined on the low side
of the mean resistance.
( ) RRRRRRk kVk1R == (27)
where R , R and RV are mean, standard deviation and C.O.V. of resistancerespectively. kR is characteristic resistance. Rk is a constant appropriate for the
fractile chosen and is determined from the standard normal distribution function. For
example if kR is to be the value of resistance below which 5% of samples will fail
Rk is evaluated from:
0.05 = ( )Rk (28)
[Q 0.05= ( ) ( )( ) ( )RR
RRRRRRRk k
kkRPRRP =
=
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Lower bound 5% values are often assumed for yield stress, and (incorrectly) for
strength curves used in offshore design. 2 standard deviation values (2.32%
probability) are generally used for fatigue strength design.
Determining the uncertainties of all the design variables
Uncertainties needs to be determined by statistical methods.
Calculating the failure probability
Many methods for reliability analysis can be used for this purpose. The First Order
and Second Moment (FORM ) method is normally used because of its simplicity and
reasonable accuracy.
Setting up target reliability levels for each failure modes
An important task in probability-based design is to determine the target reliability.
The target reliability is determined by social and economic considerations. The social
considerations are dominant for assessing the acceptable risks of collapse of primary
structural components, which could have serious consequences on lives or the
environment (i.e. with reference to the ultimate limit states). The economic
considerations are dominant for assessing the acceptable risks of loss of quality of the
structure, increased maintenance and repair costs, permanent or temporary
interruption of normal service operations (i.e. with reference to the serviceability limit
states).
Faulkner [1984] has studied the target reliability for various steel structures, which are
shown in Fig. 5. It is observed that the reliability for merchant ships and British
frigates vary in a very large range. A value of = 3.0 for frigates and 3.0 to 4.0 formerchant ships was recommended.
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Fig. 6 Target reliability
Based on a recent study [Mansour et al, 1996], the suggested target reliability for
ships is shown in Table 2. It is seen that the target reliability indices for collapse ofthe entire structure (primary failure mode) is greater than that of a non-critical welded
detail relative to fatigue.
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Table 2 Recommended target safety indices
[Mansour and Wirsching, 1996]
Failure mode Target reliability index
Primary (initial yield) 5.0
Primary (ultimate) 4.0Secondary 3.0
Tertiary 2.5
Category 1 (not serious) 2.5
Category 2 (serious) 3.0
Category 3 (very serious) 3.5
In the context of pipelines, target reliability level needs to be evaluated considering
the implied safety level in the existing codes and rules. Sotberg et al (1997) proposed
target reliability level as follows:
Limit state Safety class
Low Normal High
SLS 10-1 - 10-2 10-2 -10-3 10-2 -10-3
ULS 10-2 -10-3 10-3 -10-4 10-4 -10-5
FLS 10-3 10-4 10-5
ALS 10-4 10-5 10-6
where SLS is Serviceability Limit States, ULS is Ultimate Limit States, FLS is
Fatigue Limit State, ALS is accidental Limit State.
Calibrating partial safety factors for all limit states
The PSFs are evaluated using level 2 reliability methods:
i ,ik
*
ii
x
x= (30)
substitute Eqs. (20) and (27 or 29) into Eq. (30)
i ,ii
iic
ik
*i
iVk1
V1
x
x
== (31)
where i are loading PSFs usually 1.0 and the second denominator term is + ve,
i are resistance PSFs usually 1.0 and the second term in denominator is ve. For
each variable ix ,*
ix is the design point, ikx is characteristic value, c is the target
reliability index for the code. i is sensitivity factor.
Example 2
Follow example 1, calculate the partial safety factors of the design variables.
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Recall the results in example 1,
The reliability index is 3.307. The design point and sensitivity factors are:*
1x =2.866,*
2x =191687.853,*
3x =549361.575.
270.01
= 658.02
= 703.03
=
For 1x :
5% is used in characteristic values. Because this is a resistive variable,
111k1 kx = =3 - 1.6450.15 = 2.735
041.1753.2
866.2
x
xPSF
k1
*
11 ===
For 2x :
This is a resistive variable too, so
222k2 kx = = 245000 1.64524500 = 204679.5
936.05.204697
853.191687
x
xPSF
k2
*
22 ===
For 3x :
This is a loading variable, so
333k3 kx += = 375000 + 1.64575000 = 498375.0
102.1498375
575.549361
x
xPSF
k3
*
33 ===
The above is the procedure to calculate the PSFs. The example means that if a safety
check format
( )( ) ( )k33k22k11 xPSFxPSFxPSF
is used, the designed structure will have a safety index =3.307. This is the idea of aprobability-based design code. In the design, as long as the characteristic values and
PSFs are used, target reliability is ensured implicitly (e.g. reliability analysis is not
carried out.).
Evaluating results of the design
The obtained design needs to be evaluated by various methods.
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The above mentioned reliability-based limit state design methodology can be applied
to new design and inspection and maintenance of pipelines. Hopkins and Jaswel
(1997) applied this method to uprating pipelines (i.e. their pressure increased beyond
their original design pressure).
REFERENCES
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