Aalborg Universitet

Probabilistic investigations into the value of information

A comparison of condition-based and time-based maintenance strategies
Zou, Guang; Banisoleiman, Kian; Arturo, González; Faber, Michael Havbro

Published in:
Ocean Engineering

DOI (link to publication from Publisher):

10.1016/j.oceaneng.2019.106181

Creative Commons License

CC BY-NC-ND 4.0

Publication date:
2019

Document Version
Accepted author manuscript, peer reviewed version

Link to publication from Aalborg University

Citation for published version (APA):

Zou, G., Banisoleiman, K., Arturo, G., & Faber, M. H. (2019). Probabilistic investigations into the value of
information: A comparison of condition-based and time-based maintenance strategies. Ocean Engineering, 188,
Article 106181. https://doi.org/10.1016/j.oceaneng.2019.106181

General rights
Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners
and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.

- Users may download and print one copy of any publication from the public portal for the purpose of private study or research.
- You may not further distribute the material or use it for any profit-making activity or commercial gain
- You may freely distribute the URL identifying the publication in the public portal -
Take down policy
If you believe that this document breaches copyright please contact us at vbn@aub.aau.dk providing details, and we will remove access to
the work immediately and investigate your claim.

Downloaded from vbn.aau.dk on: November 13, 2024

 Probabilistic investigations into the value of information: a

comparison of condition-based and time-based maintenance

strategies

1,2 1 2 3
Guang Zou ; Kian Banisoleiman ; Arturo González ; Michael H. Faber
1
Lloyd’s Register Global Technology Center, Lloyd’s Register Group Limited, Southampton, UK
2
School of Civil Engineering, University College Dublin, Dublin, Ireland
3
Departement of Civil Engineering, Aalborg University, Aalborg, Denmark

Keywords: Integrity management; maintenance planning; probabilistic approach; reliability-based

optimization; value of information; life cycle approach

Abstract

Optimal maintenance of marine structures is challenging due to numerous fatigue-prone components,

serious failure consequences, high maintenance costs, harsh sea environments, difficult access, and
uncertainties in fatigue loading, resistance, and inspection and maintenance activities. Time-based
maintenance (TBM) is convenient to implement. However, condition-based maintenance (CBM) is
proved to be more cost-effective. This paper assesses the value of information (VoI) by inspections
in CBM, compared to TBM, and investigates the conditions for CBM to outperform TBM in terms of
fatigue reliability. A probabilistic maintenance optimization method and a life cycle cost analysis
framework are established to derive optimal CBM and TBM strategies with the objective of maximizing
lifetime reliability and evaluating their life cycle costs. The advantages of CBM, in comparison to TBM,
and the VoI depends strongly on the inspection time. The CBM can achieve higher reliability with
fewer maintenance costs than the TBM. An illustrative example is provided using the established
probabilistic method and framework to support optimal maintenance planning. This example serves
as a basis to explain the benefits of CBM to lifetime fatigue reliability and cost reduction, the conditions
when CBM is more beneficial than TBM, the conditions for beneficial, ineffective and unbeneficial
repair, and the VoI by inspections.

1 Introduction

Fatigue and fracture are common deterioration phenomena across many industrial fields, e.g. marine
structures, airplanes and nuclear plants, etc., and it is of paramount importance that strategies for
design, inspection and maintenance are targeted and optimized to ensure that the associated failure
risks are managed efficiently and kept within acceptable limits. It is well known that even under a
stress level much lower than material tensile strength, structural details can still fail if exposed to cyclic
loading for a sufficient time, which defines fatigue life. Fatigue failure may initially be local, e.g., occurs
in some stress concentration areas, especially in the vicinity of welds, however, failures of some
critical details can lead to sudden rupture of the whole structural system and thus present huge risks
to assets, human lives and the environment (Moan 2011). Welded details are especially prone to
fatigue cracks due to welding flaws, material heterogeneity, and complicated local geometries, etc.,
which typically shorten fatigue life greatly (Fricke 2003). To mitigate the failure risks, fatigue life is
assessed in the structural design stage to ensure that the designed fatigue life is longer than the
required service life with a high confidence level. Depending on the application areas, several fatigue
design approaches such as the well-known safe-life approach and damage-tolerate approach are
available (Zerbst et al. 2014). Apart from the design stage, fatigue is a lifetime matter for structural
management that needs continuous attention and measures, given that structural performance is in
essence time-variant and subject to uncertainties associated with loading and material characteristics,
geometries and modelling methods (Biondini and Frangopol 2016). Fatigue failure probability and risk
assessed at the design stage need to be re-assessed and validated during operation due to several
reasons, e.g. human errors in design and fabrication, discrepancies between design and as-built
condition, changes of operational modes and loading conditions, and other hazards that were not
foreseen or had not been taken into account at the design stage. Following re-assessment,
maintenance actions may be needed to recover structural integrity and to improve reliability. The costs
of maintenance are often justified by the huge loss associated with failure, not only financially, but
also environmentally and socially (Moan 2011).

The benefits and costs of maintenance are dependent on maintenance strategies, such as inspection
times and methods, repair criteria and repair methods, and these need to be optimized and planned
well in advance to improve maintenance effectiveness, which is of great significance for structural
systems with a substantial number of welded details, e.g. marine structures (Moan 2011, Soliman,
Frangopol, and Mondoro 2016, Ventikos, Sotiralis, and Drakakis 2018). Traditionally maintenance
activities have been reactive and corrective, in which a maintenance action is taken after a failure is
observed. This can be rather risky depending on the failure consequences of a structural detail. To
avoid significant failure consequences, preventive maintenance planning approaches have been
developed, based on metrics such as service age (time), reliability, risk, damage condition, etc. Time-
based maintenance (TBM) is a classical preventive maintenance approach and widely applied in
engineering practices due to its simplicity in decision-making and implementation (Cullum et al. 2018).
Maintenance actions are scheduled at specific points in time to prevent significant failures during
lifetime, and maintenance times and methods are optimized to reduce lifetime maintenance costs by
time-variant reliability/risk analysis utilizing as-built and operational structural information (Temple and
Collette 2015, Rinaldi et al. 2017, Wang et al. 2018)

Nowadays, condition-based maintenance (CBM) is gaining a wide range of attention as a result of

developments in non-destructive (NDT) testing techniques, sensing and monitoring technology,
system identification algorithms, and data science. Information on structural responses and damage
conditions is collected during operations to assist rational maintenance decision-making. This
information can be gathered via periodic inspections, which have been investigated for maintenance
planning of general deteriorating systems (Yang, Zhao, and Ma 2018), ships (Ventikos, Sotiralis, and
Drakakis 2018), marine machineries (Emovon, Norman, and Murphy 2016), process industry (Kallen
and van Noortwijk 2005), etc. Although periodic inspections are simple to implement, they may not be
as economically-efficient as non-periodic inspections. Similar to periodic inspections, non-periodic
inspections have been applied to maintenance of general deteriorating structural systems (Zhao et al.
2010, Jiang 2010), ships (Guo et al. 2012, Dong and Frangopol 2016, Yang and Frangopol 2018),
offshore assets and installations (Faber, Straub, and Goyet 2003, Faber et al. 2005, Straub and Faber
2006), etc. A more complete, yet costly, information collection approach, may be continuous health
monitoring. Optimization frameworks for health monitoring strategies based on probabilistic analysis
have been developed with applications in general structural systems (Zhou, Xi, and Lee 2007,
Memarzadeh, Pozzi, and Kolter 2016), marine and offshore structures (Lu et al. 2018, Wang et al.
2018, Sabatino and Frangopol 2017), nuclear plants (Baraldi et al. 2011), railway networks (Verbert,
De Schutter, and Babuška 2017), etc. In this paper, the information collection process by surveys,
NDT and monitoring are referred to as inspection.

Despite the popularity of CBM, there are additional (direct and indirect) costs and efforts in relation to
TBM, involving the collection of information, which can be substantial, especially for marine structures
with a larger number of fatigue-prone details, difficult access and high loss as a result of interventions
to normal operation. Questions remaining to be addressed are related to whether the costs associated
with information collection in CBM are paid off by its benefits to risk reduction, i.e., “is the CBM a more
beneficial maintenance strategy than the TBM for marine structures and under what conditions?”.
While some theoretical algorithms for Value of Information (VoI) analysis have been proposed in civil
and structural engineering (Sebastian 2018, Malings and Pozzi 2018, Konakli, Sudret, and Faber
2015), the concept has rarely been applied in the context of marine engineering. There is a lack of
research on a direct comparison of CBM and TBM maintenance strategies, and on where the VoI
comes from, both of which are worthy topics for a marine engineering field currently dominated by
TBM (Cullum et al. 2018). This paper aims to contribute to developing maintenance planning and VoI
assessment for marine structures by comparative studies of TBM and CBM strategies, both of which
are optimized for maximizing lifetime fatigue reliability taking probabilistic modelling of fatigue
deterioration and expected maintenance costs into account. The remainder of this paper is structured
as follows: Section 2 defines the fracture mechanics model employed for fatigue deterioration in terms
of crack propagation, where the associated uncertainties are characterized probabilistically; Section
3 describes the maintenance strategies under investigation, a probabilistic maintenance optimization
method for the maintenance strategies based on a reliability metric, and a lifetime cost analysis
framework; Section 4 applies the maintenance optimization method and lifetime cost analysis
framework to a typical fatigue-prone structural detail, that serves as example to illustrate and discuss
the benefits of CBM and the VoI; and finally, Section 5 draws conclusions for maintenance planning
of marine structures.

2 Probabilistic fatigue modelling

Fatigue analysis for marine structures is typically based on either the S-N approach or fracture
mechanics (FM) approach. The S-N approach has been widely used in fatigue design, codes and
regulations by virtue of its simplicity and solid experimental basis. The objective of a fatigue analysis
based on the S-N approach is normally to ensure that the designed fatigue life is longer than a
required service life with a relatively high confidence level. However, the S-N approach may not be
suitable for providing a theoretical basis for maintenance planning, which requires taking details on
crack dimensions into account. On the other hand, the FM approach addresses crack propagation
explicitly. The fatigue process is understood as crack evolution and can be divided into three stages
as shown by Figure 1: crack initiation, crack propagation and final fracture, where the vertical axis
labelled a denotes crack size and the horizontal axis N represents the number of cycles. NI is the
number of cycles required for the crack propagation stage to start, and NF is the number of cycles
until the final fracture or fatigue life. Although the exact mechanism for crack initiation is still
controversial, it is widely acknowledged that it relates largely to the mechanical behaviour of the
material in the scale of grain size and to surface treatment techniques (Zerbst et al. 2014). The crack
size in the crack initiation stage may not be critical for structural safety, as it is typically rather small
and is hardly detectable by common NDT methods. In practice, the time spent by the crack in the
crack initiation stage is often negligible compared with the crack propagation stage due to the
presence of initial flaws/cracks introduced by the welding process. Also, the final fracture usually
occurs very quickly, and the crack propagation stage thus is the focus of structural integrity
management in terms of maintenance intervention.

Figure 1. Three stages of crack development

2.1 FM approach

FM approach is employed herein since it provides a means of modelling the crack propagation
explicitly and thus allows for reliability updating based on observations of crack growth. Equation (1)
is Paris’ law (Paris and Erdogan 1963), which correlates the crack propagation rate with the range of
stress intensity factor for one-dimensional crack propagation.

𝑑𝑎
𝑑𝑁
= 𝐶∆𝐾 𝑚 , ∆𝐾𝑡ℎ ≤ ∆𝐾 ≤ 𝐾𝑚𝑎𝑡 (1)

where 𝑑𝑎⁄𝑑𝑁 is crack propagation rate; 𝐶 and 𝑚 are material parameters; 𝐾𝑚𝑎𝑡 is material fracture
toughness; ∆𝐾 is stress intensity factor range, and; ∆𝐾𝑡ℎ is the threshold value for the stress
intensity factor range. Figure 2 illustrates Equation (1) using a logarithmic scale.

Figure 2. Stress intensity factor range

A typical stress intensity factor range ∆𝐾 in the crack propagation period can be derived using
Equation (2) for one-dimensional crack propagation.

∆𝐾 = ∆𝜎𝑌(𝑎)√𝜋𝑎 (2)

where 𝑌(𝑎) is a geometry function and ∆𝜎 is stress range.

The allowed stress range ∆𝜎 can be established by an S-N design curve such as Equation (3).

𝑁 ∆𝜎 𝑚1 = ̅̅̅
𝑎1 𝑁𝐹 ≤ 107
{ 𝐹 𝑚2 (3)
𝑁𝐹 ∆𝜎 = ̅̅̅
𝑎2 𝑁𝐹 ≥ 107

where 𝑁𝐹 is the fatigue life, 𝑚1 and 𝑚2 are fatigue strength exponents, and ̅̅̅
𝑎1 and ̅̅̅
𝑎2 are fatigue
strength coefficients. The fatigue strength exponents and coefficients are obtained from a statistical
analysis of specimen fatigue strength test data.

By integration of Equation (1), the number of cycles for the crack to develop from an initial crack size
𝑎0 to the critical size 𝑎𝑐 , i.e., the crack propagation life 𝑁𝑃 , can be obtained vis Equation (4).
Depending on the initial crack size, the crack propagation life 𝑁𝑃 may be shorter than the fatigue life
𝑁𝐹 .

1 𝑎 𝑑𝑎
𝑁𝑃 = 𝜋𝑚⁄2 𝐶∆𝜎𝑚 ∫𝑎 𝑐 𝑎𝑚⁄2 𝑌(𝑎)𝑚 (4)
0

If the geometry function 𝑌(𝑎) was known, it is also possible to obtain the crack size 𝑎(𝑡) at time 𝑡
when the structural detail has been exposed to 𝑁(𝑡) cycles of fatigue loading.

2.2 Uncertainty modelling

Given that the FM approach builds upon the physics of crack propagation for fatigue deterioration
modelling, being somewhat more sophisticated than the S-N approach, it is important to consider
various sources of uncertainties surrounding FM. Main sources of uncertainties associated with the
FM approach include modelling uncertainty when using Paris’ law for crack growth prediction,
measurement and statistical uncertainty associated with material properties, 𝐶 and 𝑚 , and initial
crack size, 𝑎0 , and modelling uncertainty associated with the calculation of stress range ∆𝜎 (Souza
and Ayyub 2000). Probabilistic modelling allows to explicitly address the uncertainties associated with
these parameters, to derive distributions of crack growth predictions, and to calculate failure
probability caused by fatigue and fatigue reliability.

The material properties, 𝐶 and 𝑚, are typically obtained by statistical analysis of results from
specimen tests. The uncertainties associated with 𝐶 and 𝑚 are believed to be originated from the
inhomogeneities in material, measurement method, procedure and statistical method for parameter
estimation (Lassen and Recho 2013). Although typically understood as material properties, 𝐶 and 𝑚
are also influenced by environmental and loading conditions and they are correlated. Common
practice is to assume that they are mutually independent, e.g., 𝑚 is fixed and 𝐶 is a variable. In a
probabilistic analysis, 𝐶 is typically assumed to be lognormally distributed (Guedes Soares and
Garbatov 1998, Dong and Frangopol 2016, Faber et al. 2005, Lotsberg et al. 2016).

Representative statistical data on the initial crack size 𝑎0 for specific applications is often hard to
obtain due to challenges in sampling and measuring. A comprehensive review of the literature about
the initial crack size can be found in (Zou, Banisoleiman, and González 2016). The initial crack size
depends on many factors in design and manufacture that may not be easy to fully control, e.g.
materials, welding techniques, NDT methods, quality control procedure and human factors, etc. The
parameter 𝑎0 is often modelled as a variable with a lognormal distribution (Kim and Frangopol 2011)
or exponential distribution (Dong and Frangopol 2016, Lotsberg et al. 2016).

The uncertainty associated with the stress range ∆𝜎 should be assessed on the basis of the
uncertainty in the applied stress level, which includes the uncertainty in fatigue loading and in stress
analysis method. These uncertainties can be quantified by systematic analysis of structural response
data collected by measurement and calculation. In probabilistic fracture mechanics analysis, an
additional variable can normally be introduced as a multiplication factor for the applied stress level
(Lassen and Recho 2015).

3 Probabilistic frameworks for lifetime reliability and cost analysis

Table 1 summarizes the three maintenance strategies for structural integrity management under
investigation. The first strategy labelled ‘Case 1’, is no action. The probability of fatigue failure is thus
determined solely by design plan and manufacture quality control. This is the basic case. The second
strategy, denoted by ‘Case 2’, consists of time-based maintenance without any inspection, e.g. time-
based repair or replacement. The time for repair, 𝑡𝑟 , is optimized for maximizing lifetime fatigue
reliability. If the structure has survived at the planned repair time, a repair will be implemented. The
third strategy, i.e., ‘Case 3’, is condition-based maintenance, where damage condition is examined
by inspection before a repair decision is made. The time for the inspection, 𝑡𝑖 , is optimized with the
same objective as ‘Case 2’. If the structure has survived at the planned inspection time, an inspection
will be carried out first. If a crack is detected, it will then be repaired. It is assumed that the repair is
implemented shortly after detection, i.e., without delay, and that after repair, the structure returns to
its initial state.

Table 1. Maintenance strategies under investigation.

Case Maintenance strategy
Case 1 No action
Case 2 TBM (no inspection)
Case 3 CBM (inspection before repair)

3.1 Probabilistic maintenance optimization method

The maintenance strategies in Table 1 are optimized with the objective of maximizing lifetime fatigue
reliability index. The fatigue reliability calculations without maintenance (Case 1) are performed via
Monte Carlo simulation. The TBM and CBM strategies (Cases 2 and 3) leading to maximum reliability
indexes are regarded as optimum ones. As both the structural damage state and inspection result are
probabilistic at the decision analysis point in time, a decision tree analysis is implemented for Cases
2 and 3 before the fatigue reliability index with maintenance can be calculated. An exhaustive search
algorithm can be employed for this optimization problem. Alternatively, if there are many optimization
parameters, some optimization techniques can be adopted to reduce the time for deriving optimum
solutions (Kim, Soliman, and Frangopol 2013).

3.1.1 Initial design fatigue reliability

The initial fatigue reliability is defined relative to the failure probability caused by fatigue without any
maintenance (Case 1), which can be calculated by the probability of exceedance of a limit state
signifying fatigue failure. The limit-state function is formulated based on the crack size in the thickness
direction, as Equation (5).

𝑀(𝑡) = 𝑎𝑐 − 𝑎(𝑡) (5)

where 𝑎𝑐 is the critical crack size and 𝑎(𝑡) is the crack size at time 𝑡. Fatigue failure is defined as
the occurrence of through-thickness crack, i.e., the critical crack size, 𝑎𝑐 , is set to be equal to the
plate thickness 𝑇.

As both 𝑎𝑐 and 𝑎(𝑡) can be expressed in number of cycles, the above limit state function can be re-
written as Equation (6).

𝑀(𝑡) = 𝑁𝐹 − 𝑁(𝑡) (6)

where 𝑁𝐹 is the fatigue capacity and 𝑁(𝑡) is the fatigue loading by time 𝑡.

The probability of fatigue failure 𝑃𝑓 (𝑡) and the fatigue reliability index 𝛽 are given by Equations (7)
and (8) respectively.

𝑃𝑓 (𝑡) = 𝑃(𝑀(𝑡) ≤ 0) (7)

𝛽(𝑡) = −Φ−1 [𝑃𝑓 (𝑡)] (8)

where Φ−1 [∙] is the inverse function of the standard normal cumulative density function.

3.1.2 Probability of repair

Decision tree analysis is implemented for the maintenance strategies involving maintenance actions
(Cases 2 and 3), illustrated by Figures 3 and 4 respectively. In the figures, 𝐹, 𝐷, 𝑅 represent failure,
detection and repair respectively; 𝐹̅ , 𝐷
̅ , 𝑅̅ represent survival, no detection and no repair respectively;
𝑡𝑖 is the time for inspection and 𝑡𝑟 the time for repair. In the two cases, one maintenance intervention
is planned to clearly present the differences between CBM and TBM in terms of their benefits and
costs, and the VoI in CBM.

Figure 3. Illustration of a decision tree analysis for Case 2

 Figure 4. Illustration of a decision tree analysis for Case 3

In Case 2, no inspection is involved. Equation (9) gives the probability, 𝑃𝑟𝑒𝑝 (𝑡) , of repair at time
𝑡, which is equal to the probability that the structure has survived at the time.

𝑃𝑟𝑒𝑝 (𝑡) = 1 − 𝑃𝑓 (𝑡) (9)

In Case 3, the probability, 𝑃𝑖𝑛𝑠𝑝 (𝑡), of inspection at time 𝑡, is equal to the probability that the
structure has survived at that time, given by Equation (10).

𝑃𝑖𝑛𝑠𝑝 (𝑡) = 1 − 𝑃𝑓 (𝑡) (10)

In Case 3, the repair decision is dependent on the inspection result. The probability of repair at time
𝑡 is equal to the probability that an inspection is implemented at that time with the inspection result
being detection. If the detectable crack size of an NDT method (e.g. magnetic particle inspection) is
𝑎𝑑 , then the limit state function for an inspection event can be formulated as:

𝐷(𝑡) = 𝑎𝑑 − 𝑎(𝑡) (11)

The function is negative when a crack is detected and is otherwise positive. The probability of repair
is given by Equation (12).

𝑃𝑟𝑒𝑝 (𝑡) = 𝑃(𝑀(𝑡) > 0⋂𝐷(𝑡) < 0) (12)

3.1.3 Fatigue reliability with maintenance

Let 𝑃𝑓1 (𝑡) designate the failure probability with maintenance and 𝛽1 (𝑡) the fatigue reliability index
with maintenance. When 𝑡 ≤ 𝑡𝑟 , the planned maintenance action has not been implemented yet, and
hence, the failure probability 𝑃𝑓1 (𝑡) will be equal to the initial failure probability without maintenance
(Equation (13)).

𝑃𝑓1 (𝑡) = 𝑃𝑓 (𝑡) (13)

When 𝑡 > 𝑡𝑟 , the influence of planned maintenance on the failure probability should be taken into
account based on the decision tree analysis shown by Figures 3 and 4. The failure probability with
maintenance for Case 2 and Case 3 are given by Equations (14) and (15) respectively, while the
fatigue reliability index with maintenance, 𝛽1 (𝑡), is given by Equation (16).

𝑃𝑓1 (𝑡) = 𝑃𝑓 (𝑡𝑟 ) + 𝑃(𝑀(𝑡𝑟 ) > 0⋂𝑀(𝑡 − 𝑡𝑟 ) < 0) (14)

𝑃𝑓1 (𝑡) = 𝑃𝑓 (𝑡𝑟 ) + 𝑃(𝑀(𝑡𝑟 ) > 0⋂𝐷(𝑡𝑟 ) > 0⋂𝑀(𝑡) < 0) + 𝑃(𝑀(𝑡𝑟 ) > 0⋂𝐷(𝑡𝑟 ) < 0⋂𝑀(𝑡 − 𝑡𝑟 ) < 0) (15)
𝛽1 (𝑡) = −Φ−1 [𝑃𝑓1 (𝑡)] (16)

The method can be applied to multiple inspections and repairs by doing decision tree analysis for a
sequence of times at which inspections or repairs are scheduled. The number of branches of the
decision trees in Figures 3 and 4 would increase exponentially with the number of inspections and
repairs. The probability and reliability calculation would be more complex but can be done by common
structural reliability calculation software or by programming.

3.2 Lifetime cost analysis framework

This paper focuses on structural integrity management at the operation stage. The time point of
decision analysis is the beginning of the service life. This means that the structural integrity baseline
has been established and the main tasks for integrity management are to develop a maintenance
programme to maintain structural integrity. Equation (17) divides the life cycle costs (C) into inspection
costs (𝐶𝐼 ), repair costs (𝐶𝑅 ) and failure cost (𝐶𝐹 ).

𝐶 = 𝐶𝐼 + 𝐶𝑅 + 𝐶𝐹 (17)

Inspection, repair and failure costs are variables subjected to uncertainties associated with material
and loading characteristics, inspection times and qualities, repair criteria and repair qualities. Lifetime
cost analysis is based on the expected values of the inspection, repair and failure costs, and these
costs are adjusted at the time of the cost analysis by an annual discounting rate of interest. Therefore,
inspection and repair costs can be defined by Equations (18) and (19) respectively.

𝑛 𝑘 𝑘 1
𝑖
𝐶𝐼 = ∑𝑘=1 𝑃𝑖𝑛𝑠𝑝 ∙ 𝐶𝑖𝑛𝑠𝑝 ∙ 𝑘 (18)
(1+𝑟)𝑡𝑖

𝑛 𝑘 𝑘 1
𝑟
𝐶𝑅 = ∑𝑘=1 𝑃𝑟𝑒𝑝 ∙ 𝐶𝑟𝑒𝑝 ∙ 𝑘 (19)
(1+𝑟)𝑡𝑟

𝑘 𝑘
where 𝑛𝑖 and 𝑛𝑟 are numbers of inspections and repairs in the life cycle; 𝐶𝑖𝑛𝑠𝑝 and 𝐶𝑟𝑒𝑝 are costs

𝑘 𝑘
for the kth inspection and repair activity respectively; 𝑃𝑖𝑛𝑠𝑝 and 𝑃𝑟𝑒𝑝 are the probabilities of the kth

inspection and repair actually being performed; 𝑡𝑖𝑘 and 𝑡𝑟𝑘 are the timing of the kth inspection and
repair; and 𝑟 is the annual discounting rate of interest.

The failure cost, CF, is given by Equation (20).

1
𝐶𝐹 = 𝑃𝑓𝑛 ∙ 𝐶𝑓𝑎𝑖𝑙 ∙ (1+𝑟)𝑇𝑆𝐿 (20)

where 𝐶𝑓𝑎𝑖𝑙 is the consequence of structural failure in terms of monetary loss; 𝑇𝑆𝐿 is the required
service life, and; 𝑃𝑓𝑁 is the probability of structural failure considering the planned inspections and
repairs.

4 An illustrative example

The structural detail subjected to cyclic fatigue loading used as an example is a stiffened plate
comprising of typical T joints. Figure 5 shows the geometry and critical location that were chosen for
this joint. There are a large number of such joints in marine and offshore structures. Those areas
where stiffeners are welded to the plate are critical as they are prone to crack initiation and
propagation. The stability of the plate may be improved with stiffeners, but cracks are likely to initiate
and propagate along the weld toes of joints due to welding notch, residual stresses, material
inhomogeneity, etc. Fatigue reliability of such joints is thus an outstanding problem that needs to be
addressed during the life cycle of the detail.
 Figure 5. A typical stiffened plate with welded T joints.

First, the values of the parameters of the model defined in Section 2 are established for the structural
detail based on existing literature. Then, the three maintenance strategies of Section 3 are tested. As
mentioned in Section 3.2.1, Case 1 represents the initial fatigue reliability, which is determined by the
structural plan and execution of manufacture quality control, without any operational maintenance. At
the beginning of service life, the probabilistic maintenance optimization method and lifetime cost
analysis framework are adopted to support the development of a maintenance program. Two different
maintenance strategies are tested comparatively: Case 2 reflects the influence of time-based
maintenance on lifetime fatigue reliability, while Case 3 reflects the influence of both inspection and
repair. The time for the inspection, 𝑡𝑖 , and for the repair, 𝑡𝑟 , in Cases 2 and 3 are optimized for
maximizing the lifetime fatigue reliability. The optimum maintenance strategies for Cases 2 and 3 are
evaluated using the metric of life cycle costs, which is the sum of the financial costs associated with
failure and the costs associated with the maintenance intervention as per Section 3.2.

4.1 Probabilistic models

The fatigue resistance of the structural detail is categorised as ‘F’ class by (DNV 2014), in which a bi-
linear S-N model as Equation (3) with the parameters given in Table 2 is proposed to give its fatigue
resistance. The required service life, 𝑇𝑆𝐿 , is assumed to be 20 years. The plate thickness, 𝑇 , is
adopted to be 25 mm. The critical crack size, 𝑎𝑐 , is set to be equal to the plate thickness, i.e., 𝑎𝑐 =
25 mm. The frequency of typical wave loading is approximately 0.16 Hz, which corresponds to 𝑁0 =
5 × 106 cycles per year (Lotsberg et al. 2016).
Table 2. Parameters used in reliability analysis.
Parameter Unit Value Parameter Unit Value
𝑇𝑆𝐿 year 20 𝑚1 - 3
𝑁0 cycle 5 × 106 𝑚2 - 5
log10 ̅̅̅
𝑎1 N 4 ∙ mm−6 11.855 𝑎𝑐 mm 25
log10 ̅̅̅
𝑎2 N 4 ∙ mm−6 15.091 𝑎𝑑 mm 0.89

Table 3 provides the distributions and statistical characteristics for 𝐶 and 𝑎0 following (Lotsberg et
al. 2016). The uncertainties associated with loads and stress calculations are modelled with a
normally distributed variable 𝐵 (Lassen and Recho 2015). Magnetic particle inspection is adopted for
Case 3.

Sensitivity analysis of life cycle costs to the monetary cost of failure, 𝐶𝑓𝑎𝑖𝑙 , the cost of one repair,
𝐶𝑟𝑒𝑝 , and the cost of one inspection, 𝐶𝑖𝑛𝑠𝑝 , is carried out based on 9 sets of cost ratios (CR), which are
referred to (Straub and Faber 2006, Kulkarni and Achenbach 2007, Breysse et al. 2009) and listed in
Table 4. Herein CR1 is considered as baseline of cost values. The annual discounting rate of interest
is taken as 𝑟 = 0 so that the life cycle costs are determined only by the structural plan, fatigue loading,
maintenance activities and associated uncertainties, i.e., ignoring social-economic factors.

Table 3. Uncertainty modelling used in reliability analysis.

Parameter Distribution Unit Mean Standard Deviation
𝑎0 Exponential mm 0.043 0.043
log10 𝐶 Normal N −4 ∙ mm5.5 -12.74 0.11
𝐵 Normal - 1 0.15

Table 4. Life cycle costs for Case 1, 2, 3 under different cost ratios.
CR1 CR2 CR3 CR4 CR5 CR6 CR7 CR8 CR9
𝐶𝑖𝑛𝑠𝑝 1 1 1 2 0.5 0.1 2 1 1
𝐶𝑟𝑒𝑝 10 20 5 10 10 20 4 10 10
𝐶𝑓𝑎𝑖𝑙 100 100 100 100 100 100 100 1000 50
𝐶𝑖𝑛𝑠𝑝 ⁄𝐶𝑟𝑒𝑝 0.1 0.05 0.2 0.2 0.05 0.005 0.5 0.1 0.1
𝐶𝑟𝑒𝑝 ⁄𝐶𝑓𝑎𝑖𝑙 0.1 0.2 0.05 0.1 0.1 0.2 0.04 0.01 0.2
𝐶𝑐𝑎𝑠𝑒 1 13.23 13.23 13.23 13.23 13.23 13.23 13.23 132.7 6.64
𝐶𝑐𝑎𝑠𝑒 2 10.78 20.74 5.81 10.78 10.78 20.74 4.81 18.39 10.37
𝐶𝑐𝑎𝑠𝑒 3 4.54 7.66 2.99 5.55 4.05 6.76 3.68 8.52 4.33
 𝐶𝑐𝑎𝑠𝑒 3 ⁄𝐶𝑐𝑎𝑠𝑒 2 0.42 0.37 0.51 0.51 0.38 0.33 0.77 0.46 0.42
𝐶𝑐𝑎𝑠𝑒 3 ⁄𝐶𝑐𝑎𝑠𝑒 1 0.34 0.58 0.23 0.42 0.31 0.51 0.28 0.06 0.65

4.2 Results and discussion

The probabilities, reliability indexes, and expected costs below are calculated with Monte Carlo
simulations, with 5 × 106 samples for each variable. It is checked that more samples do not lead to
much changes in the results. Figure 6 shows the decrease of fatigue reliability β with service year for
Cases 1 and 2. It can be seen that with the adoption of TBM (e.g. a repair or replacement is planned
at 𝑡𝑟 = 10 years), the lifetime fatigue reliability index increases from 1.12 to 2.40.

Figure 6. Fatigue reliability index β against service year

Figure 7 presents the influence of maintenance intervention time on lifetime fatigue reliability. Figures
8 – 10 show the influence of maintenance intervention time on life cycle costs.
Figure 7. Fatigue reliability index against maintenance intervention time

Figure 8. Life cycle costs against maintenance intervention time (CR1)

 Figure 9. Life cycle costs against maintenance intervention time (CR6)

Figure 10. Life cycle costs against maintenance intervention time (CR8)

Table 5 summarizes the optimum maintenance strategies derived for Cases 2 and 3, the probability
of inspection and, the probability of repair associated with the optimum strategies. The lifetime fatigue
reliability index and life cycle costs (CR1) for Cases 1, 2 and 3 are also listed in Table 4, and they are
evaluation metrics for the three maintenance strategies. The life cycle costs for Cases 1, 2 and 3
under all cost ratios are listed in Table 4. In Table 4, the efficiency of CBM is signified by 𝐶𝑐𝑎𝑠𝑒 3 ⁄𝐶𝑐𝑎𝑠𝑒 1 ,
as in Case 1, no maintenance intervention involves; while the advantage of CBM in terms of cost
reduction in comparison to TBM is signified by 𝐶𝑐𝑎𝑠𝑒 3 ⁄𝐶𝑐𝑎𝑠𝑒 2 .

It should be noted that the results of lifetime fatigue reliability, probability of inspection and probability
of repair are independent on cost ratios (as can be seen from the formulations in Section 3), and life
cycle costs are dependent on cost ratios. Sensitivity of maintenance efficiency and the advantage of
CBM to cost ratios are analysed below:

• The efficiencies of both TBM and CBM increase with decrease of 𝐶𝑟𝑒𝑝 ⁄𝐶𝑓𝑎𝑖𝑙 . This conclusion
is clearly shown by Table 4 and by comparison of Figure 8 to Figure 10. The conclusion
indicates that it is more important and efficient to implement maintenance when the costs of
repair is low compared with the costs of failure.
• The advantage of CBM in terms of cost reduction (in comparison to TBM) is more pronounced
with decrease of 𝐶𝑖𝑛𝑠𝑝 ⁄𝐶𝑟𝑒𝑝 . This conclusion can be seen from Table 4 and from Figure 9, in
comparison to Figure 8. In reality, the value of 𝐶𝑖𝑛𝑠𝑝 ⁄𝐶𝑟𝑒𝑝 is typically very small, as it is more
convenient to do an inspection than to carry out a repair, which requires much more resources,
e.g. money, materials, manhours, instrumentation, etc. In this regard, the advantage of CBM
would be widely acknowledged with the development and popularity of inspection and
monitoring techniques.

The above conclusions are the same as expected. It is more interesting to look at the results of lifetime
fatigue reliability and probability of repair in Cases 2 and 3. The below discussions and conclusions
are made mainly based on the results of lifetime fatigue reliability and probability of repair. When life
cycle costs are mentioned, they are referred to CR1. Based on engineering experience and the
references (Straub and Faber 2006, Kulkarni and Achenbach 2007, Breysse et al. 2009), it is believed
that for most structural components, 𝐶𝑖𝑛𝑠𝑝 ⁄𝐶𝑟𝑒𝑝 is smaller than 0.1 and thus the advantage of CBM
in cost reduction is more pronounced than shown by Figure 8.

Table 5. Optimum maintenance strategies

Notation meaning Case 1 Case 2 Case 3
𝛽 Reliability index 1.12 2.40 2.62
𝑡𝑖 (year) Inspection time n/a n/a 9
𝑡𝑟 (year) Repair time n/a 10 9
𝑃𝑖 Inspection probability n/a n/a 0.998
𝑃𝑟 Repair probability n/a 0.996 0.311
𝐶(CR1) Lifecycle costs 13.23 10.78 4.54

Based on Figures 6, 7, 8 and Table 5, the following points can be made about the structure detail
under investigation:
 • The optimum time for repair in Case 2 is approximately the middle of its service life (Figure 7).
The lifetime fatigue reliability index increases to 2.40 from 1.12 in Case 1 (Figure 6), due to
repair, by which the structure is physically changed. The life cycle costs drop slightly from 13.23
in Case 1 to 10.78 in Case 2 (Figure 8).
• With the adoption of inspection and possible repair (if detected), the lifetime fatigue reliability
index increases significantly from 1.12 in Case 1 to 2.62 in Case 3 (Figure 7), and the life cycle
costs drop significantly from 13.23 in Case 1 to 4.54 in Case 3 (Figure 8). The saving in life
cycle costs benefits from both repair and inspection.
• By comparing Case 3 with Case 2, it is worth to highlight that more repairs do not necessarily
lead to higher lifetime fatigue reliability. The probability of repair in Case 2 is much higher than
in Case 3 (0.996 versus 0.311) as well as the lifetime total costs (10.78 versus 4.54). However,
the lifetime fatigue reliability index in Case 2 is lower than that in Case 3 (2.40 versus 2.62)
(Table 4). Hence, in certain conditions repair can be less beneficial to lifetime fatigue reliability
compared with ‘do nothing’ and thus a waste of money. This is explained by the fact that
damage extent can be mitigated by repair, but the uncertainties in material property and in
stress range cannot be decreased. On the other hand, the information of no detection collected
by an inspection implies slow deterioration rate and favourable material property and stress
range. The failure probability may be decreased more significantly by the utilization of the
information than by repair.
• The VoI provided by the inspection in Case 3 comes from two aspects. On the one hand, if the
fatigue deterioration rate is fast, cracks would be detected and then repaired, by which the
structure detail would be physically changed, and thus the failure probability is decreased, and
failure risk is mitigated. In this circumstance, repair is beneficial to fatigue reliability. On the
other hand, if the fatigue deterioration rate is slow, the most probable inspection result would
be no detection. By utilization of the information, the failure probability is also decreased. In
this circumstance, repair is ineffective or even unbeneficial to lifetime fatigue reliability.
Therefore, an inspection can help to identify beneficial repair, unbeneficial repair and ineffective
repair.

Even further, Figures 7 and 8 highlight the importance of optimizing inspection time in CBM and show
when CBM strategy can be more beneficial than TBM strategy. Both the lifetime fatigue reliability and
life cycle costs are strongly dependant on the inspection time. Based on the differences in Case 2
and Case 3 in terms of lifetime fatigue reliability (Figure 7) and life cycle costs (Figure 8), it is possible
to distinguish three periods for inspection scheduling:
 • An inspection scheduled at the late stage of service life, e.g. 𝑡𝑖 > 13 years in this example,
can identify and eliminate ineffective repair. The repair in Case 2 is regarded as ineffective, as
it results in the same lifetime fatigue reliability as ‘do nothing’ in Case 3, in case of no detection
at the late stage (Figure 7). The reason is that in case of no detection at the late stage, failure
probability caused by fatigue is approximately zero, whether repaired or not. Thus, the life cycle
costs in Case 3 is less than Case 2 (Figure 8), due to the elimination of ineffective repair by
virtue of an inspection scheduled at the late stage.
• An inspection scheduled near the interim of service life, e.g. 7 years < 𝑡𝑖 < 13 years in this
example, can identify and eliminate unbeneficial repair. The repair in Case 2 is regarded as
unbeneficial, as it leads to lower lifetime fatigue reliability than ‘do nothing’ in Case 3, in case
of no detection in the interim (Figure 7). The reason is that no detection in the interim implies
slow deterioration rate, and thus favourable material property and stress range. In such
circumstances, the failure probability after repair can be higher than that of the original structure.
Thus, the life cycle costs in Case 3 are much less than Case 2 (Figure 8), due to the elimination
of unbeneficial repair and the lowest failure costs (the highest fatigue reliability), by virtue of an
inspection scheduled near the interim of service life.
• An inspection scheduled at the early stage of service life, e.g. 𝑡𝑖 < 7 years in this example,
is likely to eliminate beneficial repair, although decreases life-cycle costs. In case of no
detection at the early stage (Figure 7), The repair in Case 2 is regarded as beneficial, as it
results in higher lifetime fatigue reliability than ‘do nothing’ in Case 3. The reason is that the
implications of no detection at the early stage on lifetime failure probability are probably very
weak compared with a repair. Thus, although the costs in Case 3 are less than Case 2 (Figure
8), due to less repair, the lifetime fatigue reliability in Case 3 is lower than Case 2 (Figure 7).

5 Conclusions

Current maintenance methods in the marine industry are still mainly corrective maintenance and time-
based preventive maintenance (TBM). However, condition-based maintenance (CBM) has
increasingly been a hot research topic, especially in industries such as wind power plants, nuclear
plants, bridge engineering, etc. One factor, among many, impeding adoption of the new CBM strategy
in marine engineering is probably lack of explicit and conclusive evidence of the benefits of CBM. This
paper has carried out an investigation into the implications of a rational maintenance planning for a
marine structure detail. A probabilistic maintenance optimization method and a lifetime cost analysis
framework has been built upon life cycle analysis, probabilistic modelling and decision tree analysis.
The method and the framework have enabled direct modelling and integrated management of the
uncertainties affecting fatigue deterioration and maintenance activities and can be used to support
rational and optimal maintenance planning under uncertainty. Employing the method and the
framework, two maintenance strategies (TBM and CBM) have been optimized and evaluated based
on the metrics of lifetime fatigue reliability and total costs. By comparison to TBM, the benefits of the
CBM strategy to lifetime fatigue reliability and cost reduction, the conditions when the TBM strategy
can be more beneficial than the CBM, and when a repair can be beneficial, unbeneficial or ineffective
to lifetime fatigue reliability have been discussed. Based on the classification of repair, the value of
information (VoI) provided by inspection in the CBM strategy and, the conditions when the VoI can be
realized and maximized have been discussed. In summary:

1) Compared with ‘do nothing’, repair can be less beneficial to lifetime fatigue reliability when
there is a high degree of uncertainties in material property and in stress range. In such
conditions, repairing relatively small cracks, which would be implemented under the TBM
strategy but can be avoided under the CBM strategy, would be unbeneficial to lifetime fatigue
reliability. If a CBM strategy was to repair detected cracks, it is not recommended to use a very
accurate inspection method (with very small detectable crack size) for inspections scheduled
at the late stage, to avoid unbeneficial repair.
2) Repair is classified into beneficial, ineffective and unbeneficial repair, according to their
benefits to lifetime fatigue reliability. Classification of repair is important for making clear the
VoI provided by inspection in the CBM strategy, maximizing the VoI and thus improving the
efficiency of a maintenance strategy. Inspection can help to identify beneficial, unbeneficial
and ineffective repair, in addition, to identifying cracks.
3) The VoI provided by inspection in the CBM strategy comes from two aspects. On the one hand,
if the fatigue deterioration rate is fast, cracks would be identified by the inspection, and
subsequently repaired. After repair, the lifetime fatigue reliability of the structure detail would
be higher than prior to repair due to the elimination of serious damages. On the other hand, if
the fatigue deterioration rate is slow, the most likely inspection result would be no detection.
Utilizing this additional information, the lifetime fatigue reliability would be higher than before
inspection.
4) The VoI is maximized when the unbeneficial repair can be identified and eliminated. Based on
an illustrative example, to reap the maximum VoI, it is required that an inspection method with
appropriate quality is adopted and that the inspection is scheduled near the interim of service
life. The maximum VoI is achieved by virtue of elimination of unbeneficial repair and the lowest
failure costs. Ineffective repair can be avoided with an inspection scheduled at the late stage
of service life. The reason is that lifetime fatigue reliability would be the same in case of no
 detection at the late stage of service life, whether repaired or not.
5) Based on an illustrative example, (a) if inspection was scheduled near the interim of service
life, the CBM strategy is more beneficial than the TBM strategy in terms of both lifetime fatigue
reliability and costs; (b) if inspection was scheduled at the late stage of service life, the CBM
strategy is more beneficial than the TBM strategy in terms of life cycle costs and is the same
as the TBM strategy in terms of lifetime fatigue reliability; and (c) if inspection was scheduled
at the early stage of service life, the CBM strategy can be superior or inferior (when 𝐶𝑟𝑒𝑝 ⁄𝐶𝑓𝑎𝑖𝑙
is very small, as shown by Figure 10) to the TBM strategy in terms of life cycle costs, but is
less beneficial than the TBM strategy in terms of lifetime fatigue reliability.

In future work, the methodology can be extended to maintenance strategy of multiple inspections and
repairs by doing decision tree analysis for a sequence of times at which inspections or repairs are
scheduled. The branches in the decision tree would increases exponentially with the number of
inspections or repairs in lifetime. The life cycle costs can still be calculated by Equation (17) – (20). It
is expected that unbeneficial and ineffective maintenance are more likely be identified by CBM with
multiple inspections.

ACKNOWLEDGEMENTS
The authors would like to express their gratitude to the European Union’s Horizon 2020 research
and innovation programme for their funding toward this project under the Marie Sklodowska-Curie
grant agreement No. 642453 (http://trussitn.eu).

REFERENCES
Baraldi, Piero, Roberto Canesi, Enrico Zio, Redouane Seraoui, and Roger Chevalier. 2011. "Genetic algorithm-
based wrapper approach for grouping condition monitoring signals of nuclear power plant components."
Integrated computer-Aided engineering 18 (3):221-234.
Biondini, Fabio, and Dan M Frangopol. 2016. "Life-cycle performance of deteriorating structural systems under
uncertainty." Journal of Structural Engineering 142 (9):F4016001.
Breysse, Denys, Sidi Mohammed Elachachi, Emma Sheils, Franck Schoefs, and Alan O’connor. 2009. "Life cycle
cost analysis of ageing structural components based on non-destructive condition assessment."
Australian Journal of Structural Engineering 9 (1):55-66.
Cullum, Jane, Jonathan Binns, Michael Lonsdale, Rouzbeh Abbassi, and Vikram Garaniya. 2018. "Risk-Based
Maintenance Scheduling with application to naval vessels and ships." Ocean Engineering 148:476-485.
DNV, GL. 2014. DNVGL-RP-0005: Fatigue design of offshore steel structures. Det Norske Veritas AS, Oslo,
Norway.
Dong, Y., and D. M. Frangopol. 2016. "Incorporation of risk and updating in inspection of fatigue-sensitive details
of ship structures." International Journal of Fatigue 82:676-688. doi: 10.1016/j.ijfatigue.2015.09.026.
Emovon, Ikuobase, Rosemary A. Norman, and Alan J. Murphy. 2016. "An integration of multi-criteria decision
making techniques with a delay time model for determination of inspection intervals for marine
machinery systems." Applied Ocean Research 59:65-82. doi: https://doi.org/10.1016/j.apor.2016.05.008.
Faber, Michael Havbro, Daniel Straub, and Jean Goyet. 2003. "Unified approach to risk-based inspection planning
for offshore production facilities." Journal of Offshore Mechanics and Arctic Engineering 125 (2):126-
131. doi: 10.1115/1.1555116.
Faber, Michael Havbro, John D. Sørensen, Jesper Tychsen, and Daniel Straub. 2005. "Field implementation of RBI
for jacket structures." Journal of Offshore Mechanics and Arctic Engineering 127 (3):220-226. doi:
10.1115/1.1951777.
Fricke, Wolfgang. 2003. "Fatigue analysis of welded joints: state of development." Marine Structures 16 (3):185-
200. doi: http://dx.doi.org/10.1016/S0951-8339(02)00075-8.
Guedes Soares, C., and Y. Garbatov. 1998. "Reliability of maintained ship hull girders subjected to corrosion and
fatigue." Structural Safety 20 (3):201-219. doi: 10.1016/S0167-4730(98)00005-8.
Guo, Jinting Jeffrey, Ge George Wang, Anastassios N Perakis, and Lyuben Ivanov. 2012. "A study on reliability-
based inspection planning–Application to deck plate thickness measurement of aging tankers." Marine
Structures 25 (1):85-106.
Jiang, Ren-yan. 2010. "Optimization of alarm threshold and sequential inspection scheme." Reliability
Engineering & System Safety 95 (3):208-215.
Kallen, Maarten-Jan, and Jan M van Noortwijk. 2005. "Optimal maintenance decisions under imperfect
inspection." Reliability Engineering & System Safety 90 (2-3):177-185.
Kim, Sunyong, and Dan M Frangopol. 2011. "Optimum inspection planning for minimizing fatigue damage
detection delay of ship hull structures." International Journal of Fatigue 33 (3):448-459.
Kim, Sunyong, Mohamed Soliman, and Dan M. Frangopol. 2013. "Generalized Probabilistic Framework for
Optimum Inspection and Maintenance Planning." Journal of Structural Engineering 139 (3):435-447.
doi: 10.1061/(ASCE)ST.1943-541X.0000676.
Konakli, Katerina, Bruno Sudret, and Michael H Faber. 2015. "Numerical investigations into the value of
information in lifecycle analysis of structural systems." ASCE-ASME Journal of Risk and Uncertainty in
Engineering Systems, Part A: Civil Engineering 2 (3):B4015007.
Kulkarni, Salil S, and Jan D Achenbach. 2007. "Optimization of inspection schedule for a surface-breaking crack
Probabilistic Engineering Mechanics 22 (4):301-312.
subject to fatigue loading."
Lassen, Tom, and Naman Recho. 2013. Fatigue Life Analyses of Welded Structures: Flaws : John Wiley & Sons.
Lassen, Tom, and Naman Recho. 2015. "Risk based inspection planning for fatigue damage in offshore steel
structures." ASME 2015 34th International Conference on Ocean, Offshore and Arctic Engineering.
Lotsberg, Inge, Gudfinnur Sigurdsson, Arne Fjeldstad, and Torgeir Moan. 2016. "Probabilistic methods for
planning of inspection for fatigue cracks in offshore structures." Marine Structures 46:167-192. doi:
http://dx.doi.org/10.1016/j.marstruc.2016.02.002.
Lu, Yang, Liping Sun, Xinyue Zhang, Feng Feng, Jichuan Kang, and Guoqiang Fu. 2018. "Condition based
maintenance optimization for offshore wind turbine considering opportunities based on neural network
approach." Applied Ocean Research 74:69-79. doi: https://doi.org/10.1016/j.apor.2018.02.016.
Malings, C, and M Pozzi. 2018. "Value-of-information in spatio-temporal systems: Sensor placement and
scheduling." Reliability Engineering & System Safety 172:45-57.
Memarzadeh, Milad, Matteo Pozzi, and J Zico Kolter. 2016. "Hierarchical modeling of systems with similar
components: A framework for adaptive monitoring and control." Reliability Engineering & System
Safety 153:159-169.
Moan, T. 2011. "Life-cycle assessment of marine civil engineering structures." Structure and Infrastructure
Engineering 7 (1-2):11-32.
Paris, PC, and Fazil Erdogan. 1963. "A critical analysis of crack propagation laws." Journal of Basic Engineering
85 (4):528-533.
Rinaldi, G, PR Thies, R Walker, and L Johanning. 2017. "A decision support model to optimise the operation and
maintenance strategies of an offshore renewable energy farm." Ocean Engineering 145:250-262.
Sabatino, Samantha, and Dan M Frangopol. 2017. "Decision making framework for optimal SHM planning of ship
structures considering availability and utility." Ocean Engineering 135:194-206.
Sebastian, Thöns. 2018. "On the Value of Monitoring Information for the Structural Integrity and Risk
Management." Computer-Aided Civil and Infrastructure Engineering 33 (1):79-94. doi:
doi:10.1111/mice.12332.
Soliman, Mohamed, Dan M. Frangopol, and Alysson Mondoro. 2016. "A probabilistic approach for optimizing
inspection, monitoring, and maintenance actions against fatigue of critical ship details." Structural
Safety 60:91-101. doi: 10.1016/j.strusafe.2015.12.004.
Souza, GFM, and BM Ayyub. 2000. "Probabilistic fatigue life prediction for ship structures using fracture
mechanics." Naval engineers journal 112 (4):375-397.
Straub, Daniel, and Michael H Faber. 2006. "Computational aspects of risk‐based inspection planning."
Computer‐Aided Civil and Infrastructure Engineering 21 (3):179-192.
Temple, DW, and MD Collette. 2015. "Minimizing lifetime structural costs: Optimizing for production and
maintenance under service life uncertainty." Marine Structures 40:60-72.
Ventikos, NP, P Sotiralis, and M Drakakis. 2018. "A dynamic model for the hull inspection of ships: The analysis
and results." Ocean Engineering 151:355-365.
Verbert, Kim, Bart De Schutter, and R Babuška. 2017. "Timely condition-based maintenance planning for multi-
component systems." Reliability Engineering & System Safety 159:310-321.
Wang, Haibin, Elif Oguz, Byongug Jeong, and Peilin Zhou. 2018. "Life cycle cost and environmental impact
analysis of ship hull maintenance strategies for a short route hybrid ferry." Ocean Engineering 161:20-
28.
Wang, Peng, Xinliang Tian, Tao Peng, and Yong Luo. 2018. "A review of the state-of-the-art developments in the
field monitoring of offshore structures." Ocean Engineering 147:148-164.
Yang, David Y, and Dan M Frangopol. 2018. "Probabilistic optimization framework for inspection/repair planning
of fatigue-critical details using dynamic Bayesian networks." Computers & Structures 198:40-50.
Yang, Li, Yu Zhao, and Xiaobing Ma. 2018. "Multi-level maintenance strategy of deteriorating systems subject to
two-stage inspection." Computers & Industrial Engineering 118:160-169.
Zerbst, U., R. A. Ainsworth, H. T. Beier, H. Pisarski, Z. L. Zhang, K. Nikbin, T. Nitschke-Pagel, S. Munstermann, P.
Kucharczyk, and D. Klingbeil. 2014. "Review on fracture and crack propagation in weldments - A fracture
mechanics perspective." EngineeringFracture Mechanics 132:200-276. doi:
10.1016/j.engfracmech.2014.05.012.
Zhao, Xuejing, Mitra Fouladirad, Christophe Bérenguer, and Laurent Bordes. 2010. "Condition-based
inspection/replacement policies for non-monotone deteriorating systems with environmental covariates."
Reliability Engineering & System Safety 95 (8):921-934.
Zhou, Xiaojun, Lifeng Xi, and Jay Lee. 2007. "Reliability-centered predictive maintenance scheduling for a
continuously monitored system subject to degradation." Reliability Engineering & System Safety 92
(4):530-534.
Zou, Guang, Kian Banisoleiman, and Arturo González. 2016. "Methodologies for crack initiation in welded joints
applied to inspection planning." World Academy of Science, Engineering and Technology 10 (11):1406-
1413.