limit state handout[1]

8/8/2019 Limit State Handout[1]

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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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( )( ) 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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10

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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11

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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14

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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15

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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18

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

1. DNV, (1996): Rules for Submarine Pipeline Systems, Det Norske Veritas,DNV'1996, Norway (Reported in Jiao, G., et al, The Superb Project: Wall

Thickness Design Guideline for Pressure Containment of Offshore Pipelines,

Offshore Mechanics and Arctic Engineering Conference, OMAE 1996, Florence,

Italy.

2. Faulkner, D. (1984): On Selecting a Target Reliability For Deep Water Tension

Leg Platforms, Proceeding of 11th IFIP Conference on System Modelling and

Optimisation, Copenhagen, Denmark, July 25-29, pp. 490-513.

3. Hopkins, P. and Haswel, J. (1997): ' THE PRACTICAL APPLICATION OFSTRUCTURAL RELIABILITY THEORY AND LIMIT STATE DESIGN

CONCEPTS TO NEW AND IN-SERVICE TRANSMISSION PIPELINES',

International Seminar on Industrial Application of Structural Reliability Theory,

ESReDA, Paris, France, October 2-3 1997.

4. Mansour, A.E., Wirsching, P.H. (1996): Safety Assessment and TargetReliabilities for Floating Structures, Proceeding of International Workshop on

Very Large Floating Structures, Hayama, Japan, Nov. 25-28, pp.283-291.

5. Sotberg, T., Moan, T., Bruschi, R., Jiao, G. and Mork, K.J., (1997) : 'TheSUPERB Project: Recommended Target Safety Levels for Limit State Based

Design of Offshore Pipelines', Proc. of OMAE;97.

limit state handout[1]

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