A Simpliﬁed Method for Reliability and Integrity-based Design of Engineering Systems and Its...

8/11/2019 A Simplied Method for Reliability and Integrity-based Design of Engineering Systems and Its Application to Offshore Mooring Systems

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A simplied method for reliability- and

integrity-based design of engineering systems

and its application to offshore mooring systems

Mir Emad Mousavi a,*, Paolo Gardoni b

a Texas A&M University, TAMU-3136 College Station, TX 77843, USAb University of Illinois at Urbana-Champaign, Urbana, IL, USA

a r t i c l e i n f o

Article history:

Received 15 February 2013

Received in revised form 26 December 2013

Accepted 4 February 2014

Keywords:

Reliability

Integrity

Design

System

Structure

Offshore

Mooring

Optimization

Probability

a b s t r a c t

This paper presents a simplied method for the reliability- and the

integrity-based optimal design of engineering systems and its

application to offshore mooring systems. The design of structural

systems is transitioning from the conventional methods, which arebased on factors of safety, to more advanced methods, which

require calculation of the failure probability of the designed system

for each project. Using factors of safety to account for the un-

certainties in the capacity (strength) or demands can lead to sys-

tems with different reliabilities. This is because the number and

arrangement of components in each system and the correlation of

their responses could be different, which could affect the system

reliability. The generic factors of safety that are specied at the

component level do not account for such differences. Still, using

factors of safety, as a measure of system safety, is preferred by

many engineers because of the simplicity in their application. The

aim of this paper is to provide a simplied method for design of

engineering systems that directly involves the system annual

failure probability as a measure of system safety, concerning sys-

tem strength limit state. In this method, using results of conven-

tional deterministic analysis, the optimality factors for an

integrity-based optimal design are used instead of generic safety

factors to assure the system safety. The optimality factors, which

estimate the necessary change in average component capacities,

are computed especially for each component and a target system

annual probability of system failure using regression models that

* Corresponding author. Present address: Aker Solutions Inc., 3010 Briarpark Dr., Suite 500, Houston, TX 77042, USA.

Tel.:1 713 981 2047.

E-mail address: [email protected](M.E. Mousavi).

Contents lists available atScienceDirect

Marine Structures

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l oc a t e /

m a r s t r u c

0951-8339/$ see front matter 2014 Elsevier Ltd. All rights reserved.

http://dx.doi.org/10.1016/j.marstruc.2014.02.001

Marine Structures 36 (2014) 88104
mailto:[email protected]://www.sciencedirect.com/science/journal/09518339http://www.elsevier.com/locate/marstruchttp://www.elsevier.com/locate/marstruchttp://dx.doi.org/10.1016/j.marstruc.2014.02.001http://dx.doi.org/10.1016/j.marstruc.2014.02.001http://dx.doi.org/10.1016/j.marstruc.2014.02.001http://dx.doi.org/10.1016/j.marstruc.2014.02.001http://dx.doi.org/10.1016/j.marstruc.2014.02.001http://dx.doi.org/10.1016/j.marstruc.2014.02.001http://www.elsevier.com/locate/marstruchttp://www.elsevier.com/locate/marstruchttp://www.sciencedirect.com/science/journal/09518339http://crossmark.crossref.org/dialog/?doi=10.1016/j.marstruc.2014.02.001&domain=pdfmailto:[email protected]


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estimate the effect of short and long term extreme events on

structural response. Because in practice, it is convenient to use the

return period as a measure to quantify the likelihood of extreme

events, the regression model in this paper is a relationship be-

tween the component demands and the annual probability density

function corresponding to every return period. This method ac-

counts for the uncertainties in the environmental loads and

structural capacities, and identies the target mean capacity of

each component for maximizing its integrity and meeting the

reliability requirement. In addition, because various failure modes

in a structural system can lead to different consequences

(including damage costs), a method is introduced to compute

optimality factors for designated failure modes. By calculating the

probability of system failure, this method can be used for risk-

based decision-making that considers the failure costs and con-

sequences. The proposed method can also be used on existing

structures to identify the riskiest components as part of inspection

and improvement planning. The proposed method is discussed

and illustrated considering offshore mooring systems. However,

the method is general and applicable also to other engineering

systems. In the case study of this paper, the method is rst used to

quantify the reliability of a mooring system, then this design is

revised to meet the DNV recommended annual probability of

failure and for maximizing system integrity as well as for a

designated failure mode in which the anchor chains are the rst

components to fail in the system.

2014 Elsevier Ltd. All rights reserved.

1. Introduction

The extensive costs and damages induced by the 2010 oil spill in the Gulf of Mexico as well as other

offshore failures during recent years have intensied the efforts to improve the reliability and integrity

of offshore systems. The relatively high rates of offshore failure are more than what is considered

acceptable, which implies a need for improved methods for better safety of current and future offshore

structures. For example, more than twenty-three permanent mooring systems have failed since 2000;

1500 mooring lines were either repaired or replaced. The damage cost of a single mooring failure event

was approximately $1.8 billion[1]. The costs of loss of lives or damages to the environment cannot be

quantied. As offshore drilling and production sites move to deeper and more challenging environ-

ments, the safety of offshore systems becomes even more important, demanding technology devel-opment that accounts for their inherent uncertainties. In response, the design methods seem to be in a

transition from the conventional methods, which use factors of safety (FoS), to more advanced

methods, which include the system reliability and risk assessments.

The simplicity of using FoS to account for the uncertainties has made their application favorable

to many engineers; however, as highlighted by the API RP 2SK[2], several studies have shown that

the current design practice results in systems with inconsistent failure probabilities, and thus,

reliabilities. It is because the safety of a system does not solely depend on the FoS of its compo-

nents. In turn, structural reliability methods can be used to design an offshore system for a target

probability of failure. The DNV OS E301[3] already facilitate the application of structural reliability

methods for design of mooring systems by providing a target annual probability of failure, as an

alternative to its FoS-based method; however, such application requires practical methods that are

feasible using the available data and decent amount of engineering and computing resources.

Developing simplied methods that can quantify and target the offshore system reliability is an

important step toward the successful advancements in offshore engineering, including the design

of offshore mooring systems.

M.E. Mousavi, P. Gardoni / Marine Structures 36 (2014) 88104 89


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The aim of this paper is to propose a simplied method that can be used for the design of reliable

mooring systems, which is also applicable to other engineering systems. The proposed method is based

on the denition of integrity-based optimal design[4]. In assessing the system reliability in this paper,

any failure in the intact system is assumed undesirable, so this method targets the optimal design for

maximum system integrity given a target annual probability of any component failure in the intact

system. This denition of system failure complies with the ultimate limit-state (ULS) criterion providedfor the target annual probability of failure by DNV OS E301. The computation of the annual failure

probability of such a system and its components and determining the optimal average strength of its

components under extreme environmental conditions to meet the target reliability and achieve a

maximum integrity or designated failure mode are discussed in the paper. The scope of this paper is

limited to the considerations related to an ultimate-strength failure of the structural components.

Therefore, other design considerations (e.g. the fatigue failure, geometrical limitations) are beyond the

scope of this work and should be checked separately.

After this introduction, the paper is organized into two main sections. The next section describes the

proposed method. Next, the proposed method is illustrated in a case study about mooring systems.

2. Methodology

This section rst discusses the calculation of the probability of failure of a series system. Then, a

simplied method is introduced to calculate the annual probability of failure for structural systems.

Next, the proposed method is used to derive simplied equations for an integrity-based optimal design.

Finally, the adjustments in the method for favoring a designated failure mode are presented.

A closed-form solution for the probability of failure of a mooring system requires a relationship

between the probability of failure of the system and its components. In this study, by assuming the

independency of the component failures under conditioned demand, a more realistic approach is used.

Thus, for a given series system that is subject to a system demand (environmental load),dE, assuming

that the component failures are conditionally independent, the conditional probability of system

failure can be written as[7]

Pgs 0jdE 1 YN

j 1

n1 P

gj 0

dEo 1 YNj 1

1 P

cj dj

dE (1)wheregsis the system limit-state function, Nis the total number of components, gs 0 indicates the

system failure[8], P($) denotes the probability,gjis the component limit-state function,

gj cjdj (2)

where cj is the component capacity (strength) and a random variable, dj is the component demand

(maximum internal force) that is determined through a deterministic analysis (e.g. Finite Element)

under eachdE, and thereforegj 0 indicates the component failure. Then, using the total probabilityrule[9], the probability of the system failure can be calculated as

Pgs 0

ZB

Pgs0jdEfdEddE

ZB

241 YNj 1

1 P

cj dj

dE35 fdEddE (3)

where f(dE) is the probability density function (PDF) and B is the domain ofdE. Similarly, the probability

of failure of each component can be computed as:

Pgj 0 ZB Pcj djdEfdEddE (4)Eqs.(3) and (4) suggest that calculating the probability of failure of a component or the system

requires computing Pcj djdE, which depends on the cumulative distribution function ofcj, CDFcj,

and on f(dE). Please note that depending on the availability of statistical data, the length of each

M.E. Mousavi, P. Gardoni / Marine Structures 36 (2014) 8810490


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component in this analysis could equal the full length of the actual mooring components or the lengthy

components could be divided into several shorter components for this reliability analysis. A ner

component size would improve the accuracy of this reliability assessment provided that the statistical

data are also based on similar size test samples. A change in the length of structural components would

not alter the equations in this paper because the system is treated as a series system and thus using the

same formulation with relevant number of components and capacity statistical data and componentdemand information are applicable.

The determination of CDFcj and f(dE) are discussed in the following sections. Please note that the

above formulations are general; however, in the following sections, f(dE) will be determined and

discussed for the annual failure probabilities and therefore will be the annual probability density

function of the environmental demands.

2.1. Estimating CDFcj

The capacities of the mooring system components approximately follow a lognormal distribution

[5,10]. Therefore, we can write

CDFcj

dj

F

"ln

dj

mjsj

# F

26664ln

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1COV2

j

pEcj=dj

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ln

1COV2j

r37775 F

2664ln ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1COV2j

q =lj

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ln

1COV2j

r3775 (5)

where mjis the mean of the natural logarithm ofcj

mj ln

E

cj

1

2ln

1

Var

cj

E

cj2!

(6)

andsjis the standard deviation of the natural logarithm ofcj

sj

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiln

(1

Var

cj

E

cj2)vuut (7)

and E[cj] and Var(cj) are the mean and the variance ofcj respectively. E[cj] can be estimated by the

sample mean of the component capacities,bcj, and Var(cj) can be estimated by the sample variance ofthe component capacities, Varbcj. In Eq.(5),COVjis the coefcient of variation ofcjdened as

COVj

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiVar

cj

q

Ecj (8)andljis the mean safety factor of component j, which we dene as

lj E

cj

dj(9)

Eq. (5)calculates the conditional probability of failure of component j given dj. This equation is

visualized in Fig.1. Because the coefcient of variation for a class of components is usually similar and if

the component capacity follows a lognormal probability distribution, given such information, thisgure

can be used to estimate this conditional probability of failure for a target mean safety factor. Note that, as

expected, asljapproaches 1 the probability of failure approaches 0.5 irrespective of the value ofCOVj.

2.2. Estimating f(dE) and calculating the component and system annual failure probabilities

It is common to use the mean recurrence time, usually known as the (mean) return period of

an extreme event, T, as a measure to quantify its likelihood. T is the average time interval



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between two consecutive occurrences of such an event and 1=T is the average rate of its occur-

rence in one year [9]. It is common to compute the likelihood that an event with a return

period ofTwill not be exceeded within t 1 year by assuming it as a Poisson Process and using

an exponential function as et=T, in which tis the reference duration (i.e. 1 year)[9]. Thus, ifdj isthe internal force in component j in response to the environmental condition dE that

corresponds to TdE, then f(dE) is the derivative of the cumulative distribution function ofTdE with

respect to dE

fdE d

dTdE

e1=TdE

dTdEddE

1

TdE2

e1=TdEdTdEddE

(10)

Note that TdE>0 (years). Eq. (10)is the rst step for a change of variable from dE to TdE in the

integral in Eqs. (3) and (4). An advanced reliability analysis of an offshore structural system involves the

development of the combined probability distribution of environmental events and uses them to

dene the probability corresponding to any structural response. However, in this paper, by decouplingthe analysis of environmental conditions with various return periods from the structural response

analysis, which are usually performed by different specialists in the industry, f(dE) is expressed in terms

ofTdEas a simplied practical approach that allows engineers who use the metocean data to analyze or

design the structure to complete a reliability-based design using the same analysis results that they

typically produce.

The next step before the integrals in Eqs. (3) and (4)can be used is developing a method for esti-

matingdjgivenTdE. For any structural componentj,djis usually measured or computed using physical

or numerical models under a limited number of environmental conditions corresponding to a few

return periods. However, in reality, dj is a continuous random variable. A regression model can estimate

dj as a function ofTdEbased on the available data ofdj versus TdE. Review of available data from dynamic

(frequency domain) analysis of mooring systems under environmental conditions with return periodsof 1, 10, 20, 50 and 100 years suggests that for all the mooring components, dj has a highly linear

relationship with the logarithm (in base 10) ofTdE. Based on this observation, following the general

formulation for probability demand models [1113], we suggest the following regression model to

estimatedjas a function ofTdE:

Fig. 1. A visualization of Eq.(5).



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dj qj1 qj2log

TdE

sdj 3dj (11)

where Qj qj;s2dj

indicates the unknown model parameters in which qj qj1; qj2, 3dj is a

random variable with zero mean and unit variance, and sdj is the standard deviation of the model error

sdj 3dj. Please note that under very large TdE, the proposed linear regression model in Eq.(11)may nolonger be accurate; however, as implied by Eq. (10), the contribution of such large TdEis not signicant.

Still, estimating the model parameters using results of deterministic analysis with larger return periods

or possible use of nonlinear regression models could improve the accuracy of the model. Eq. (11) can be

written in a matrix form as

dj Hqj sdj 3 (12)

where dj is the vector of component demands, H is the matrix of regressors (1 and the logarithm ofTdE),

and 3is a vector standard normal random variable. For example, ifk data points are available for djversus TdE, we can write

2664 dj1

dj2

djk

3775 26641 logTdE1

1 log

TdE2

1 log

TdEk3775 qj1qj2

sdj

26643dj13dj2

3djk

3775 (13)The posterior distribution ofQj as suggested by Box and Tiao[14]is

pQj

djfps2j s2djpbqjqj; s2djpQj (14)where p:is the probability of the event inside the parenthesis and

bqj H0H1H0djs2j

1y

dj

bdj0djbdjy k 2bdj Hbqj

(15)

Assuming independency ofqjand s2dj

we can write[14]

pQj

p

qj

ps2dj

fs2dj (16)

and thus rewrite Eq.(14)as

pqj; s

2dj

djfps2djs2j pqjbqj;s2dj (17)where the marginal posterior distribution ofs2

djis the inverse chi-square distribution ys2jc

2y with mean

equal to ys2j=y 2 and the variance equal to 2y2s4j=y 2

2y 4. The marginal posterior distri-

bution ofqjis a multivariate tdistribution,tkbqj; s2jH

0H1; y, which is written as

pqj

dj Gy22jH0Hj1=2 s2G122Gy2y

1

qbqjH0Hqbqj

ys2

y2=2N


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P

gj 0z

Z Tk0

F

8>>>:

ln ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1COV2j

q =bljffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ln

1 COV2j

rdE

9>>=

>>;

e1=TdE

TdE2

dTdE

Z N

Tk

F

8>>>:ln ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1COV2

jq =bljffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiln

1 COV2j

r dE9>>=>>;

e1=TdE

TdE2

dTdE (19)

The two integrals in Eq. (19) correspond to the interpolation and extrapolation regions of the

domain of environmental loads in this analysis model becausebljin both the regions is estimated basedon the regression model based on response data from 1 to Tk years return periods:

blj Ecj

bdjz

bcjbq1j

bq2jlog

TdE

(20)

The two integrals in Eq. (19) can be used for assessing the contribution of each region in thecomputed annual probability of system failure. For such calculation, larger Tk, which is the largest

return period that is used as input in Eq.(13), will improve the accuracy of the analysis. Therefore, we

summarize Eq.(19)as:

P

gj 0z

Z N

0F

8>>>:ln ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 COV2j

q =bljffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ln

1COV2j

rdE9>>=>>;

e1=TdE

TdE2

dTdE (21)

Similarly, Pgs 0can be computed as

Pgs 0z

Z N

0

26641 YNj 1

26641 F8>>>:ln ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 COV2

jq =bljffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiln

1COV2j

r dE9>>=>>;37753775 e1=TdE

TdE2

dTdE (22)

In the derivation of Eqs.(21) and (22), it is assumed thatbq1j andbq2jand thusbljare estimated usingthe results of a model in which the environmental conditions corresponding to eachTdEare simulated

in all directions, but some analysis models only simulate unidirectional environmental loads. In turn,

hurricane conditions are usually associated with various possible loading directions on a oating

structure because of their cyclic nature and because the offshore platforms are usually located far from

any major barriers. If the hurricane model already applies the realistic hurricane loading corresponding

to each return period, then Eqs. (21) and (22)can be used directly for computing the probability of

failure of components or the system. However, if a simplied modeling in which all the possible loading

directions are applied in unidirectional loading models are used, the conditional annual probability of

failure can be computed for each direction,a. For the special case where under each return period, the

occurrence of the dominant environmental load is equally possible in all the directions, we can write

the joint PDF ofdEand a as

fdE;a 1

TdE2

e1=TdEdTdEddE

da

2p(23)

Thus, we can modify Eqs.(21) and (22)as

P

gj 0z

1

2p

Z 2p0

Z N

0F

8>>>:ln ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1COV2jq =bldirj ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ln

1COV2j

r dE9>>=>>;

e1=TdE

TdE2

dTdEda (24)



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hurricane lines) and anchor foundations. Basic properties of the mooring components are given inTable 1. The lines are arranged as four groups of three lines with about 15 angle between the lines. The

payout (horizontal distance between the two ends) of each lines in about 8500 ft (2591 m) and the

pretensions of the conventional and hurricane lines are about 200 kips (890 kN) and 250 kips

(1112 kN), respectively. Because this system has a total of 40 components, for Strategy B,12 components

will be considered as fuse and the remaining components as non-fuse.

3.2. Environmental loads

Table 2 shows the wave, wind, and current data that were used for environmental conditions in this

study. 72 load directions (every 5) were considered for each return period (therefore, a total of 288

load cases were used for ve return periods of 1-, 10-, 20-, 50-, and 100-year hurricane.) The winddirection was taken to be 15 different from the wave and current directions in all the load cases. The

NPD (Norwegian Petroleum Directorate) and the JONSWAP (Joint North Sea Wave Analysis Program)

energy spectrums were used for winds and waves respectively. The peakedness parameter of 2 and

spreading parameters of 2.5 were assumed for the waves. No kinematic reduction factor was applied.

3.3. Random variables

The uncertainty in the component loads (demand) and ultimate-strengths (capacity) are consid-

ered. The statistical properties of component capacities are given in Table 3, which are based on

suggestions from Refs.[7,10,16]. In this case study we descritized the segments so that every two chain

components (links) have a length of six times its diameter and each cable component has a length of32.5 times its diameter[17].

The regression parameters for the component demands are determined using dynamic analysis

(accounting for mean, wave frequency, and low frequency motions using a decoupled frequency-

domain modeling) under environmental conditions corresponding to 10-, 20-, 50-, and 100-year

Table 1

Basic properties of the mooring system components.

Mooring lines Line components Anchor

capacity (kipa)Type Stiffness

(103 kipa)

Wet weight

(lb/fta)

Friction

factor

Min break

strength (kipa)

Length (fta)

1, 5, & 8 Anchor chain 135 87.3 1.0 1450 41384145 733

Rig wire 137 18.8 0.6 1452 61076144

2, 4, & 6 Anchor chain 135 87.3 1.0 1450 40724293 1729

Rig wire 137 18.8 0.6 1452 59806208

3 & 7 Anchor chain 135 87.3 1.0 1450 39624253 1064

Rig wire 137 18.8 0.6 1452 60376248

9, 10, 11, & 12 Anchor chain 135 87.3 1.0 1570 3200 1385

Insert wire 168 23.1 0.6 1825 6500

Rig chain 135 87.3 1.0 1850 457600

a 1 kip is about 4.45 kN, 1 ft is about 0.30 m and 1 lb/ft is about 1.49 kg/m.

Table 2

Metocean data for the project site.

Environment Hurricane return period (years)

10 20 50 100

Wind 1-min Speed, knots (m/s) 73.8 (38.0) 89.5 (46.1) 101.0 (52.0) 108.6 (55.9)1-h Speed, knots (m/s) 59.8 (30.8) 71.1 (36.6) 79.1 (40.7) 84.3 (43.4)

Wave Signicant height, ft (m) 36.1 (11.0) 42.7 (13.0) 47.2 (14.4) 50.2 (15.3)

Peak period, sec 12.06 12.78 13.32 13.59

Surface current speed, knots (m/s) 1.9 (1.0) 2.1 (1.1) 2.9 (1.5) 3.3 (1.7)

http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-


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return periods. The review of the results conrm that the mean of 3djis about zero and its coefcient of

variation (C.O.V.) is about 0.02, which indicates that the regression model error is small.

3.4. Deterministic results

Deterministic results indicated that both the intact and damaged (1 broken line) conditions of the

mooring system of this study meet the API RP 2SK requirements. The minimum safety factor of the

mooring lines under the 10-year peak wave (hurricane) environment ranged from 2.3 to 3.3 in differentdirection in the intact system and from 1.8 to 2.3 in the damaged system. The anchors also met the

requirement with safety factors ranging from 1.81 to 4.86. Under this 10-year extreme environment,

the vessel offset remained close to 10% of water depth in all the directions at the intact system and

between 13% and 14% of the depth in the damaged condition.

3.5. Probabilistic results and discussion

Table 4presents the annual probability of failure and the Integrity Index of the mooring system

using the method proposed in this paper as well as the result by Mousavi and Gardoni [4]. The results

indicate that the mooring system Integrity Index of 0.78 is smaller than that of Mousavi and Gardoni. By

further evaluating the effect of each of the differences between the two studies, we observed that by

maintaining the size of the reliability analysis components same as that of Mousavi and Gardoni, the

Integrity Index was increased to 0.89, which shows that by reducing the size of the elements in reli-

ability analysis, we are capturing more detailed information about the mooring system that cannot be

captured using larger size elements. In turn, the annual failure probability in this study is computed to

be 1.71 103, which is slightly smaller but in a same order of magnitude of the estimation of Mousavi

and Gardoni, which was 5.05 103. Mousavi and Gardoni estimated the PDF of the demands using a

four-point PMF but in this study, a more advance model is used to estimate the PDF of the environ-

mental loads. For example, they assumed an annual probability of 0.9 for the maximum demands with

the return periods corresponding to 10-year extreme event response. Such method is likely to over-

estimate the annual probability of failure of the system for that return period. However, their method is

likely to underestimate the contribution of the environmental loads with greater than 100-year return

periods as their analysis was limited to the 100-year demands. In turn, the discretization of the

mooring segments to smaller components in this study has reduced the probability of failure because

under each environmental condition, the maximum demand is smaller in a major part of each long

segment, which is more realistic and unbiased compared to when lengthy components are considered

as individual elements. Computation also shows that if complete independency was assumed for the

failure of the mooring components, the annual probability of system failure in this study would in-

crease to 5.72 103, which is almost four times the computed annual probability of failure using Eq.

(3). This result highlights the signicance of the conditional independency assumption compared to

Table 3

Statistical properties of the capacity random variables.

Random variable PDF Mean/nominal C.O.V.

Anchor capacity Lognormal 1.1 0.15

Chain strength Lognormal 1.1 0.05

Wire strength Lognormal 1.1 0.05

Table 4

Summary of the integrity and reliability analysis.

Method Probability of failure Integrity Index

Mousavi and Gardoni[4] 5.05 103 0.89

This study 1.71 103 0.78



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A Simpliﬁed Method for Reliability and Integrity-based Design of Engineering Systems and Its...

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