Inspection and maintenance planning for offshore wind structural components: Integrating fatigue failure criteria with Bayesian networks and Markov decision processes

Abstract

Exposed to the cyclic action of wind and waves, offshore wind structures are subject to fatigue deterioration processes throughout their operational life, therefore constituting a structural failure risk. In order to control the risk of adverse events, physics-based deterioration models, which often contain significant uncertainties, can be updated with information collected from inspections, thus enabling decision-makers to dictate more optimal and informed maintenance interventions. The identified decision rules are, however, influenced by the deterioration model and failure criterion specified in the formulation of the pre-posterior decision-making problem. In this paper, fatigue failure criteria are integrated with Bayesian networks and Markov decision processes. The proposed methodology is implemented in the numerical experiments, specified with various crack growth models and failure criteria, for the optimal management of an offshore wind structural detail under fatigue deterioration. Within the experiments, the crack propagation, structural reliability estimates, and the optimal policies derived through heuristics and partially observable Markov decision processes (POMDPs) are thoroughly analysed, demonstrating the capability of failure assessment diagram to model the structural redundancy in offshore wind substructures, as well as the adaptability of POMDP policies.

Full text

Inspection and maintenance planning for offshore wind structural
components: Integrating fatigue failure criteria with Bayesian networks and
Markov decision processes
Nandar Hlainga*, Pablo G. Moratoa, Jannie S. Nielsenb, Peyman Amirafsharic, Athanasios
Kolioscand Philippe Rigoa
aNaval & Offshore Engineering, ArGEnCo, University of Liege, Liege, Belgium; bDepartment of the
Built Environment, Aalborg University, Aalborg, Denmark; cDepartment of Naval Architecture,
University of Strathclyde, Glasgow, United Kingdom
ARTICLE HISTORY
Compiled April 23, 2022
ABSTRACT
Exposed to the cyclic action of wind and waves, offshore wind structures are subject to
fatigue deterioration processes throughout their operational life, therefore constituting a struc-
tural failure risk. In order to control the risk of adverse events, physics-based deterioration
models, which often contain significant uncertainties, can be updated with information col-
lected from inspections, thus enabling decision-makers to dictate more optimal and informed
maintenance interventions. The identified decision rules are, however, influenced by the dete-
rioration model and failure criterion specified in the formulation of the pre-posterior decision-
making problem. In this paper, fatigue failure criteria are integrated with Bayesian networks
and Markov decision processes. The proposed methodology is implemented in the numerical
experiments, specified with various crack growth models and failure criteria, for the optimal
management of an offshore wind structural detail under fatigue deterioration. Within the ex-
periments, the crack propagation, structural reliability estimates, and the optimal policies de-
rived through heuristics and partially observable Markov decision processes (POMDPs) are
thoroughly analysed, demonstrating the capability of failure assessment diagram to model the
structural redundancy in offshore wind substructures, as well as the adaptability of POMDP
policies.
KEYWORDS
Offshore wind turbines; Inspection and maintenance planning; Fracture mechanics; Failure
criteria; Failure assessment diagram; Partially observable Markov decision processes
1. Introduction
An optimal and rational management of offshore wind substructures is becoming increasingly
important due to the growth of offshore wind installations, with a trend towards larger wind
turbines, often located far offshore. Exposed to harsh marine environmental conditions, the
degradation of offshore wind substructures is accentuated, thus inducing a risk of structural
failure. Additionally, inspection and maintenance interventions may become more complex and
expensive far offshore. In this context, inspection and maintenance (I&M) planning methods
offer a framework for minimising life-cycle costs, controlling structural failure risks by optimally
allocating inspections and maintenance actions.
Already in the 1990s, early I&M planning methods address the decision-making problem by
exploiting Bayesian decision analysis with the objective to identify optimal strategies for struc-
tures subjected to fatigue deterioration (Fujita, Schall, & Rackwitz, 1989; Madsen, Sørensen, &
*CONTACT Nandar Hlaing Email: nandar.hlaing@uliege.be

Olesen, 1990), with many applications focused on I&M planning for offshore structures (Faber,
Sørensen, & Kroon, 1992; Goyet, Maroini, Faber, & Paygnard, 1994). By defining the I&M
policies based on a set of predefined decision rules, the computational complications associated
with solving an extensive pre-posterior analysis were alleviated, enabling the identification of ra-
tional strategies within a reasonable computational time (Straub, 2004; Straub & Faber, 2006).
Heuristic-based I&M planning methods have been widely applied to the management of fatigue-
sensitive structures, planning inspections at periodic intervals or immediately after a specified
failure probability threshold is exceeded (Moan, 2005; Straub & Faber, 2005). More recently,
the integration of discrete dynamic Bayesian networks (DBNs) into I&M methodologies has en-
abled the efficient evaluation of more sophisticated heuristic decision rules (Nielsen & Sørensen,
2018). For instance, Luque and Straub (2019) has proposed an I&M planning approach for
structural systems, evaluating system-level heuristic decision rules in a DBN simulation envi-
ronment. Other existing I&M research works consider multiple conflicting objectives within the
policy optimisation (Frangopol & Kim, 2019; Soliman, Frangopol, & Mondoro, 2016), planning
maintenance actions, in some cases, based on specified thresholds (Kim, Ge, & Frangopol, 2019).
Relying also on dynamic Bayesian networks, single- and multi-objective optimisation methods
provide robust Bayesian inference and enable the evaluation of advanced decision rules, e.g.
adaptive repair thresholds (Yang & Frangopol, 2018). Through multi-objective policy optimisa-
tion methods, decision-makers can operate under budget constraints and/or control maintenance
delays.
Even if heuristic decision rules alleviate the computational complexity of the I&M decision-
making problem, as mentioned before, the obtained I&M strategies are constrained by the
number of evaluated pre-defined rules out of the vast available policy space. Instead, I&M
planning methods that rely on Markov decision processes determine adaptive policies, providing
a mapping from the dynamically updated deterioration state to the optimal actions. Various
I&M planning applications for deteriorating structures modelled the decision-making problem
via Markov decision processes, e.g. (Corotis, Ellis, & Jiang, 2005; Memarzadeh & Pozzi, 2016;
Robelin & Madanat, 2007). The benefits offered by adaptive I&M policies are substantiated
by Yang and Frangopol (2022), and Morato, Papakonstantinou, Andriotis, Nielsen, and Rigo
(2022) have demonstrated that partially observable Markov decision process (POMDP) solved
via point-based algorithms can efficiently determine optimal I&M policies. An overview of state-
of-the-art point-based solvers and their applicability to infrastructure management can be found
in Papakonstantinou, Andriotis, and Shinozuka (2017).
As explained before, I&M planning methods aim at controlling arising structural failure
risks by timely allocating inspection and maintenance actions. Essentially, the failure risk of
a structural component represents both the probability of a failure event and its associated
economic, societal, and environmental consequences. The estimation of the failure probability
is governed by a failure criterion, which is specified by the decision-maker. Within the context
of fatigue deteriorating structures, a through-thickness failure criterion is normally prescribed,
e.g. asset management of offshore pipelines and containers. This conventional criterion might
be over-conservative for redundant structures such as the tubular joints of jacket-type offshore
wind turbines (OWTs), as tubular connections have the capacity to sustain through-thickness
cracks until the loading exceeds the resistance of the cracked structure. In such applications, the
fatigue failure limit state can be, instead, formulated via a failure assessment diagram (FAD).
The failure assessment diagram, originally proposed by Dowling and Townley (1975) and
Harrison, Milne, and Gray (1981), describes the interaction between brittle fracture and plastic
failure through a two-parameter failure criterion. The specification of FAD as the governing
failure criterion has recently gained attention in offshore wind applications. Among them, Fa-
juyigbe and Brennan (2021) and Mai, Sørensen, and Rigo (2016) showcased the evaluation of
flaw acceptability in offshore wind support structures using a failure assessment diagram of
BS7910 (British Standards, 2015). In parallel with the reported FAD research work, several
probabilistic fatigue studies and I&M planning methods for offshore wind substructures have
still specified fatigue limit states based on the conventional through-thickness failure criterion,
potentially drawing over-conservative conclusions.
2

In the reported I&M planning methods that formulate the fatigue failure limit state via
a failure assessment diagram (Amirafshari, Brennan, & Kolios, 2021; Hlaing et al., 2020), the
identified I&M policies are, however, based on heuristic decision rules. In this paper, we origi-
nally integrate the modelling of stochastic fatigue deterioration processes and the specification
of FAD-based limit states via dynamic Bayesian networks, and we introduce the necessary for-
mulation for modelling the overarching I&M decision-making problem as a partially observable
Markov decision process (POMDP). The proposed method is flexible and can be easily adopted
by other applications whose limiting failure criterion is also defined as a function of multiple
failure parameters. The applicability and efficacy of the proposed approach is verified through
numerical experiments, in which optimal I&M strategies are determined for the specific case
of an offshore wind tubular joint. Within the numerical experiments, the fatigue deterioration
of offshore wind structural details is modelled by one-dimensional and two-dimensional frac-
ture mechanics methods as well as various failure criteria, thoroughly investigating the effect
of model selection on the identified I&M strategies. The results reveal that the choice of failure
criteria and the optimality of the implemented I&M planning methods significantly affect the
resulting I&M policies. In particular, the benefits of adopting FAD criteria for offshore wind
substructures and the cost savings provided by POMDP-based policies are both meticulously
discussed.
2. Background: Risk-based inspection and maintenance planning
Risk-based inspection and maintenance planning is based on pre-posterior decision analysis inte-
grated with deterioration modelling, inspection and repair modelling, and cost modelling. This
section presents deterioration modelling including failure criteria and inspection modelling. Cost
modelling along with policy optimisation methods is discussed in Section 4. The probabilistic
fatigue deterioration model is used as reference and fracture mechanics models are calibrated
to the fatigue model. The through-thickness failure criterion has been commonly used in the
I&M planning of offshore wind structures (Luque & Straub, 2016; Morato, 2021). In this paper,
the failure assessment diagram criterion is also addressed and integrated into I&M planning.
Procedures to obtain a FAD and failure assessment points are also presented.
2.1. Deterioration modelling
2.1.1. Probabilistic SN model
Offshore wind turbine support structures are subjected to a large number of environmental
load cycles (e.g. waves) and other operational loading in their lifetime. For such structures, the
long-term stress range distribution can be represented by a Weibull distribution, described by
a scale parameter qand a shape parameter h. The shape parameter hfor offshore wind (OW)
substructures as recommended by DNV standards (DNV, 2019) is 0.8 and the scale parameter
qis computed such that the cumulative lifetime damage of the structural component designed
for Tdyears is equivalent to the damage limit corrected by the fatigue design factor (F DF ), i.e.
DSN (t=Td) = 1/F D F where the temporal fatigue evolution DSN is defined as:
DSN (t) = nt "qm1
k1
γ1 1 + m1
h;S1
qh!+qm2
k2
γ2 1 + m2
h;S1
qh!#, t = 0,1, ...Td(1)
m1, m2, k1, k2, S1are parameters of the bi-linear SN curve. nis the number of stress cycles per
year. γ1and γ2are the upper and lower incomplete gamma functions.
Probabilistic fatigue analysis can then be based on Palmgren-Miner’s rule with the long-term
3

stress range distribution. The limit state function applied in the fatigue deterioration model is:
gSN (t)=∆−DSN (t) (2)
where ∆ is the fatigue limit beyond which failure happens and is considered as a random variable
owing to the uncertainty of Palmgren–Miner’s rule. DSN (t) is the accumulated fatigue damage
at year tcalculated as in Equation (1).
The design of offshore structures is based on SN curves but the inspection information, i.e.
the presence of crack or the crack size measurement, cannot be directly used to update the
SN-based reliability. Inspection and maintenance planning therefore demands the use of frac-
ture mechanics (FM) models. Different FM models are described in Section 2.1.2. It becomes
necessary to relate the two deterioration models and such a relation can be attained by cal-
ibrating the FM models to the SN model which includes all the information from the design
stage. The calibration is performed such that a similar fatigue life is calculated by fracture
mechanics models as that of S-N test data. Whereas the FM models compute the crack growth,
the only information that contains in the SN model is the failure or survival of the hotspot
through fatigue damage. Therefore, the calibration between SN and FM models has been based
on the probability of failure along the lifetime (DNV, 2019). Typically, FM parameters with the
largest influence on the crack growth and/or with the least available information are calibrated
(Straub, 2004).
2.1.2. Fracture mechanics models
Two-dimensional crack growth model
Possible sites of fatigue crack initiations
(undercuts)
Growing fatigue crack
Figure 1.: Illustration of fatigue crack initiation.
In offshore wind support structures, fatigue cracks initiate from manufacturing imperfections
and welding defects as illustrated in Figure 1. The severity increases over the time as the cracks
grow under the cyclic loading of wind and waves. Paris-Erdogan’s law has been widely used in
linear elastic fracture mechanics (LEFM) to model the crack growth (Paris & Erdogan, 1963):
d
dnd=Cd(∆Kd)m(3)
d
dnc=Cc(∆Kc)m(4)
where dis the crack depth growing through the thickness of the structural member and cis the
crack length growing along the surface. ncorresponds to the number of stress cycles. Cd, Ccand
mare Paris law parameters, also called crack growth parameters and ∆Kis the stress intensity
4

factor range at the crack tip calculated as:
∆Kd= ∆σYd(d, c)√πd (5)
∆Kc= ∆σYc(d, c)√πd (6)
Ydand Ycare stress intensity correction factors and theoretically, are dependent on the geometry
of the component, welded joint detail and time-varying two-dimensional crack size. The applied
stress range ∆σis assumed to be composed of membrane and bending stress components. The
two components are quantified by the ratio of bending stress to total stress, denoted as the
degree of bending Db. Stress concentration due to weld geometry is also incorporated as the
stress magnification factor Mk. The stress intensity factor ranges can then be described as:
∆Kd= ∆σ[YmdMkmd (1 −Db) + YbdMkbd Db]√πd (7)
∆Kc= ∆σ[YmcMkmc (1 −Db) + YbcMkbc Db]√πd (8)
The subscripts d, c refer to crack depth and crack length and m, b refer to membrane and
bending stress components respectively. Geometry functions Ymd, Ybd, Ymc, Ybc and stress mag-
nification factors Mkmd, Mkbd, Mkmc , Mkbc can be solved by using parametric equations, for
instance, as in BS7910 (British Standards, 2019). Alternatively, one can perform finite element
analysis of the cracked structure and directly compute the stress intensity factors Kdor Kcat
each stress cycle (Fajuyigbe & Brennan, 2021).
One-dimensional crack growth model
In the one-dimensional fracture mechanics model, the crack propagation is considered only in
the direction of the member’s thickness. Therefore, the stress intensity correction factor Yd
simply becomes a function of crack depth only. Additionally, if it is further simplified such that
the stress intensity correction factor Yddoes not depend on the time-varying crack depth and
is approximated as a constant value over the lifetime, an explicit solution of the crack growth
can then be obtained as follows:
d(t) = h1−m
2CdYm
dπm/2(∆σ)mn+d1−m/2
t−1i(1−m/2)−1
(9)
Through this simplification, the crack propagation can be analytically computed attenuating
the computation associated to solving the coupled equations. For small cracks situated far away
from the boundaries of the structural member, Ydcan be taken as 1 with sufficient accuracy
(Ditlevsen & Madsen, 2007). In probabilistic deterioration modelling, Ydis assigned as a time-
invariant random variable to introduce the model uncertainties associated to the simplifications
of the stress intensity correction factor (JCSS, 2011).
2.1.3. Failure criteria
Through-thickness failure criterion
As illustrated in Figure 1, a crack grows both through the thickness and along the surface of the
structural component taking a semi-elliptical shape. Assuming that the thickness is smaller than
the length and the width of the member, the crack is likely to penetrate the whole thickness
first. The failure criterion can be formulated depending on the capacity of the structure to
further resist the applied load after through-thickness penetration.
5

In the through-thickness criterion, the failure happens when the crack depth reaches the
thickness of the structural member which is also denoted as the critical crack size dcrit. This
common criterion is particularly adopted for structures containing pressurised containment, e.g.
pipelines, pressure vessels, etc. and is conservative for redundant structures such as OW jacket
foundations. The following limit state function is employed for the through-thickness failure:
gF M (t) = dcrit −d(t) (10)
in which d(t) is the crack propagation over time. The probability of failure PF(t) is then the
probability of the limit state function being negative such that PF(t) = P(gFM (t)≤0).
Failure assessment diagram
When a crack propagates through a structural member, ultimately the crack size may reach a
critical size which corresponds to a critical stress intensity factor, usually taken as the charac-
teristic value of the fracture toughness Kmat at which brittle fracture happens. Alternatively,
if the applied load is substantially high compared to the material tensile strength, the mem-
ber may reach its tensile capacity and fail by plastic collapse. In between brittle fracture and
plastic collapse is an elastoplastic failure mode, where the failure occurs before reaching the
plastic capacity or fracture toughness. The failure assessment diagram was therefore introduced
to combine the two failure modes (Dowling & Townley, 1975).
The most rigorous method to obtain a FAD for a particular application is to perform an
elastic-plastic J-integral analysis (Anderson, 2005). Since it can be cumbersome, simplified ap-
proximations are available. For instance, BS7910 (British Standards, 2019) provide three alter-
native FAD options which have been frequently used in offshore wind applications (Fajuyigbe
& Brennan, 2021; Mai et al., 2016). They are of increasing complexity in terms of the required
material properties and stress analysis data but also provide results of increasing accuracy with
less conservatism. An example of the FAD is plotted in Figure 2.
In a FAD, the ordinate plots the fracture ratio Kr, also called the crack-driving parameter
which represents the structure’s susceptibility to brittle fracture. The abscissa plots the load
ratio Lrwhich measures how close the structure containing the crack is to plastic collapse.
The load ratio Lris equal to 1 at the yield limit, however plastic collapse happens at a higher
value which is equal to Lr,max. The failure of a structural component is then defined by means
of a failure assessment line (FAL). If an assessment point lies inside the envelope below the
assessment line, the component is assumed to be safe. If it falls on or outside the FAL, it is
assumed to be failed. The failure assessment line (FAL) is in fact a plot of the critical values of
fracture ratio Kr,crit for a range of load ratio, i.e. 0 ≤Lr≤Lr,max. The cut-off value for plastic
collapse Lr,max according to BS7910 (British Standards, 2019) is:
Lr,max =σY+σU
2σY
(11)
where σYand σUare the design yield strength and ultimate strength of the material used.
Kr,crit is equal to 1 for fully brittle fracture and declines as the load ratio increases towards the
collapse load as in Figure 2. In addition, as it is illustrated, the FAD can be partitioned into
three different zones: Zone I is the fracture dominant zone, Zone II is the elastoplastic zone and
Zone III is the plastic collapse dominant zone (Hlaing et al., 2020).
When the FAD is used as a limit state function, the failure occurs when the applied load
exceeds the reduced capacity of the cracked structure. It becomes necessary to consider the
combined influence of applied loads and non-monotonic strength deterioration of the cracked
structure. Therefore, evaluation of the failure probability with a FAD requires to apply time-
variant reliability methods which are extremely time-consuming. Instead, a simplified criterion
proposed by JCSS (2011) has been used in this work as an alternative to FAD. In this case,
the failure is expected if the interaction of the crack-driving parameter Krand the load ratio
6

0 0.5 1 1.5 2 2.5
Lr
0
0.5
1
1.5
2
2.5
Kr
I
II
III
BS7910 (Option 1)
JCSS 2011
Rf
Figure 2.: Failure assessment diagram (British Standards, 2019) and the simplified criterion
(JCSS, 2011).
Lrexceeds a normalised resistance parameter Rf, see Figure 2. The concept of the normalised
resistance parameter and the recommended values are described in Dijkstra (1991) and JCSS
(2011). Then, the limit state equation and assessment points Krand Lrare reformulated as:
gF M (t) = Rf−pK2
r(t) + L2
r(t) (12)
Kr=KI
Kmat
+ρ, Lr=σref
σY
(13)
where Rfis the normalised resistance parameter. Kmat is the fracture toughness of the material.
KIis the stress intensity factor at the crack tip and can be computed for a particular crack size
as follows:
KI=σYd(d, c)√πd (14)
where σis the maximum applied stress. The plasticity correction factor ρ≥0 reflects the
interaction between the applied primary loads and the secondary loads, e.g. residual stress Rs.
Plasticity correction is important when the secondary loads are high which, for example, is the
case of welded joints. In such case, ρincreases as the crack size becomes larger, representing
the reduced load carrying capacity of the deteriorated structure driven by plasticity interaction
effects. The plasticity correction can be evaluated according to the procedures in JCSS (2011)
or British Standards (2019). σYis the material’s yield stress and σref is the net section stress
or reference stress of the cracked structure. For a surface crack at the weld toe, σref can be
evaluated as in the following equation (British Standards, 2019).
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σref =
(Db·σ) + (Db·σ)2+ 9 ((1 −Db)·σ)2(1 −µ00)20.5
3(1 −µ00)2
where µ00 =((d/t)
1+(t/c),if W≥2(c+t)
2d
t
c
W,if W < 2(c+t)(15)
where W and t are the width and the thickness of the structural member.
2.2. Inspection modelling
d1d2d3
Detection threshold ( መ
𝑑s,th)
f( መ
𝑑s|d1)
f( መ
𝑑s|d2)
f( መ
𝑑s|d3)
Crack size (log scale)
Signal response (log scale)
POD(d1)
POD(d2)
POD(d3)
Figure 3.: Probability of detection, adapted from signal response method (Chung et al., 2006).
The mean values of the signal response fall on a regression line with the regression parameters
β0and β1.εis associated with variability of imperfect inspection and is assumed normally
distributed with a zero mean and a standard deviation σε.
The information gathered during the operational lifetime can be used to update the uncer-
tainties in the deterioration model. For instance, the probability distribution of the crack size
can be updated after an inspection is performed. However, it is necessary to take into account
the measurement quality of the observation model. The probability of detection (P OD ) curves
have been adopted to characterise the quality of several non-destructive inspection techniques
such as ultrasonic testing (UT), magnetic particle inspection (MPI) and Eddy current (EC)
inspection (Berens & Hovey, 1981). The P OD depends on the crack size and the detection
threshold, as illustrated in Figure 3. A signal response above the detection threshold will give
an inspection outcome of crack detection and below the threshold results in a no-detection out-
come. Accordingly, the P O D is defined in detection theory (Macmillan & Creelman, 2004) as
8

follows:
P OD(d) = Z+∞
ˆ
ds,th
fsignal(ˆ
ds|d)dˆ
ds(16)
where dis the true crack size. ˆ
ds,th is the detection threshold and fsignal is the probability density
function of the signal response ˆ
ds. Given the regression parameters β0,β1and the variability
σεof an inspection technique, Equation (16) can be derived as:
P OD(d)=1−Φ ˆ
ds,th −(β0+β1·ln (d))
σε!(17)
where Φ is the cumulative distribution function of the standard normal distribution. For a
particular detection threshold ˆ
ds,th, the theoretical probability of detection curve according to
Equation (17) is a monotonically increasing function of the crack size. Adjusting the detection
threshold ˆ
ds,th, the shape of a P OD curve can be changed so that P OD = 1 when ˆ
ds,th → −∞
and P OD = 0 when ˆ
ds,th → ∞ for any crack size. Based on Hong (1997) and Straub (2004),
the following limit state function is used for the event of crack detection at time t:
gD(t) = u−P OD(d(t)) (18)
where uis a uniformly distributed random variable in the interval [0,1]. The P OD of the crack
depth at time tis computed according to Equation (17).
The additional information obtained from inspections can be used to update the reliability
through conditional failure probability (Lotsberg, Sigurdsson, Fjeldstad, & Moan, 2016). For
instance, given no-detection outcome of inspection at year tins, the updated failure probability
for t≥tins is:
PF(t) = P(gF M (t)≤0|gD(tins)>0) = P(gF M (t)≤0∩gD(tins )>0)
P(gD(tins)>0) (19)
3. Stochastic deterioration modelling through dynamic Bayesian networks
Bayesian networks (BN), introduced by (Pearl, 1988), is a graphical formalism to represent joint
probability distributions of a set of random variables. Dynamic Bayesian networks (DBNs) are
temporal repetitions of BNs which have been increasingly used in engineering structural reli-
ability and risk analysis (Nielsen & Sørensen, 2018; Straub, 2009; Zhu & Collette, 2015). To
implement DBNs in I&M planning, the continuous random variables involved in the deteriora-
tion model need to be discretised for prediction and exact inference tasks. This step is crucial
since the accuracy of the results and the computational efficiency are influenced by the number
of intervals and the discretised boundaries. Theoretically, the discretisation error tends to 0
as the size of the intervals approaches 0. In practical applications, the discretisation scheme is
preferred to provide sufficient accuracy with maximum computational efficiency.
The state space Sin DBNs is the domain of the discretised variables. In a stochastic de-
terioration process, the belief which is the probability distribution over the state space P(st)
transitions from one time step to the next one according the conditional probability P(st+1 |st),
also denoted as the transition matrix. Markovian property is assumed here, i.e. the state at time
t+ 1 depends only on the state at tand not on the past ones. Additionally, the transition ma-
trix is time-invariant meaning that P(st+1 |st) is the same for any two consecutive time steps.
Evidence from observations (e.g. inspections) can also be incorporated through Bayes’ rule such
9

that:
P(st+1 |ot+1)∝P(ot+1 |st+1)P(st+1) (20)
whereas the likelihood P(ot+1 |st+1) quantifies the quality of the observation.
3.1. Deterioration rate DBNs adopting a through-thickness criterion
Dynamic Bayesian networks (DBNs) have been frequently used to model engineering deterio-
ration processes in risk analysis, often through combination of random variables to reduce the
dimension of the state space and the computation time (Straub, 2009). When more complex
deterioration model and failure criterion are used, the state space becomes high-dimensional and
non-combinative, e.g. the Weibull scale parameter qcannot be combined with other variables
since the crack size is conditional on it (through ∆σ) and so are the failure assessment points
Krand Lr(through σ). In addition, the necessity of high-dimensional conditional probabilities
for propagating the belief and computing the failure probability, such as P(dt+1 |dt, ct, q, Ca),
P(ct+1 |dt, ct, q, Cc), P(Kr,t |dt, ct, q, Kmat , Rs), P(Lr,t |dt, ct, q, σY), P(gF M |Lr, Kr, Rf)
increases computational complexity.
Another DBN representation, denoted here as “deterioration rate” DBN, represents a
stochastic deterioration process as a function of the deterioration rate. The graphical repre-
sentation of such DBNs adopting a through-thickness criterion is shown in Figure 4. The crack
evolution is traced by the nodes dtand is dependent on the deterioration rate τt. The node τt
is a one-hot (one-zero) vector indicating the current deterioration rate. Unless any maintenance
action is taken, it transitions one deterioration rate, i.e. τito τi+1, at every time step. Note that
the component may have the same deterioration rate for different time steps in the lifetime.
For example, the crack may return to its initial deterioration rate τ0after a perfect repair or
jump a number of deterioration rates back after an imperfect repair. The inspection model is
considered within the observation nodes ot. The nodes Ftindicate the probability of failure. In
the through-thickness failure criterion, Ftis dependent only on the crack size dt. The failure
subspace SF⊆ S is therefore defined based on the discretisation scheme of dtto compute the
failure probability.
𝜏𝑡
𝑑𝑡
𝒐𝒕
𝜏𝑡+1
𝑑𝑡+1
𝒐𝒕+𝟏
𝜏𝑇𝑁
𝑑𝑇𝑁
𝒐𝑻𝑵
𝐹𝑡𝐹𝑡+1 𝐹𝑇𝑁
Figure 4.: Deterioration rate DBNs adopting a through-thickness criterion. The nodes dtdescribe
the crack evolution dependent on the deterioration rate τt. The nodes otrepresent the imperfect
observations (inspections) conditional on dt, and Ftindicates the probability of a failure event.
The initial belief b0(s) corresponds to a joint probability distribution of the initial crack
size and deterioration rate P(d0, τ0). The belief transitions from each time step tto the next
10

t+ 1 according to the predefined conditional matrix as follows:
P(dt+1, τt+1 |o0, ..., ot) = X
dtX
τt
P(dt+1, τt+1 |dt, τt)P(dt, τt|o0, ..., ot) (21)
When the evidence is available, the estimation of the updated belief can be done through
the normalisation of:
P(dt+1, τt+1 |o0, ..., ot+1 )∝P(ot+1 |dt+1)P(dt+1 , τt+1 |o0, ..., ot) (22)
3.2. Deterioration rate DBNs adopting a FAD criterion









 
 
Figure 5.: Deterioration rate DBNs adopting a FAD criterion. The nodes dtdescribe the crack
evolution dependent on the deterioration rate τt. The nodes otrepresent the imperfect observa-
tions (inspections) conditional on dt, and Ftindicates the probability of a failure event through
the nodes gF M t.
A method to implement a FAD criterion in deterioration rate DBNs is presented here. The
DBN model with the FAD criterion is illustrated in Figure 5. In this case, the probability of
a failure event cannot be obtained only from the nodes dtsince it requires the crack length
as well as other time-invariant variables σY, Kmat, Rs, Rfto evaluate the failure. Alternatively,
one can include additional nodes gF M t, denoted here as the limit state variable and computed
from Equations (12-15) in the DBNs to allow the direct estimation of the failure probability
Ft. In this I&M planning problem, the observation nodes otare conditional on the crack depth
and the nodes dtstill need to be tracked. Therefore, the belief space b0(s) becomes a joint
distribution of the deterioration rate, the crack size and the limit state variable P(dt, τt, gF M t).
The deterioration evolution over the subsequent time steps can be computed through tran-
sition and estimation steps. However, the computational complexity significantly increases due
to the larger state space size with this failure criterion. The transition of the crack length has
been implicitly considered in the DBNs through the nodes gF M t. If the component returns to its
initial belief P(τ0, d0, gF M 0) after the perfect repair, both the crack depth and the crack length
consistently return to the initial condition.
11

4. Policy optimisation methods
4.1. I&M planning through heuristics
The objective of I&M planning is to identify the optimal inspection strategy which provides the
minimum total expected cost E[CT]. It is theoretically feasible to obtain optimal inspection and
maintenance plans by means of the pre-posterior decision theory, however it becomes computa-
tionally intractable as the branches of the decision tree exponentially increase with time. One
approach to circumvent this problem is to impose predefined decision rules in order to reduce
the policies which have to be evaluated. Some of the decision rules that have been frequently
applied in risk-based inspection planning of offshore structures are:
(i) Inspections are planned either periodically or before an annual failure probability threshold
∆PFis reached. The optimal interval and optimal annual failure probability threshold are
then identified.
(ii) A perfect repair action is immediately performed if the inspection gives an outcome of
crack detection (gD(tins)≤0).
(iii) After the perfect repair, it is assumed that the component goes back to its initial state
thus forming a new decision tree with a lifetime equal to TN−tins.
I&M strategies are evaluated through Monte Carlo simulations to compute the total expected
cost E[CT]. The expected failure cost E[CF] is the sum of annual failure probabilities multiplied
by the failure cost CF. The expected cost of inspection E[CI] is the product of inspection cost
CIand the number of inspections. The expected cost of repair E[CR] is the product of the repair
cost CRand the number of repairs performed. All the costs are discounted by a factor γ∈[0,1]
to take into account the time value of money. The total expected cost E[CT] is the averaged
sum of the failure, inspection and repair costs over Nsim simulations.
E[CT] = 1
Nsim
Nsim
X
n=1 

TN
X
t=1
CF∆PF(t)γt+
tIn
X
t=tI1
CIγt+
tRn
X
t=tR1
CRγt
(23)
where TNis the planned lifetime of the structure. ∆PF(t) is the annual failure probability for
year t.Inand Rnrepresent the number of inspections and repairs performed in each simulation.
4.2. I&M planning through POMDPs
In the following section, a brief description of partially observable Markov decision processes
(POMDPs) is presented with its particular implementation in offshore wind I&M planning
problem. A POMDP is a generalisation of a Markov decision process (MDP) in which the agent
takes probabilistic actions in a stochastic environment and imperfect observations. In the 7-
tuple hS,A,O, T, Z, R, γiprocess, the agent takes an action a∈ A thereby transitioning the
belief state b(s) according to the transition model T(s0, a, s) = P(s0|s, a). The agent then
receives an imperfect observation o∈ O with the probability Z(o, s0, a) = P(o|s0, a) and also
collects the reward R(b, a) for taking the action a.
An inspection and maintenance planning problem can be formulated as a POMDP through
proper definition of its elements hS,A,O, T, Z, R, γi. A concise explanation is provided below,
and more details can be found in Morato et al. (2022), Papakonstantinou and Shinozuka (2014a,
2014b).
•States: As already described before, the implementation of DBNs/POMDPs requires
efficient and effective discretisation of continuous random variables. The first element of
POMDP tuple Scan be directly defined from the domain of the discretised intervals. For
example, the through-thickness criterion POMDP consists of |S|=|Sd|·|Sτ|states and
that of FAD criterion POMDP is |S|=|Sd| · |Sτ| · |SgF M |. The initial belief b0(s) is the
12

joint probability distribution of those random variables at t= 0.
•Action-Observation: Several maintenance actions a∈ A can be defined herein such
as “perfect-repair”, “imperfect-repair” or “do-nothing”. Observations o∈ O refer to
different types of inspection techniques described in Section 2.2. Note that monitoring
can also be modelled as an observation through systematic post-processing of continuous
data stream into discrete observations.
•Transition probabilities: A transition matrix T(s0, a, s) for each maintenance action a∈ A
is defined as the probability of the component changing from the state s∈ S to the state
s0∈ S.
For the action “do-nothing”, the transition matrix T(s0, aD N , s) follows the stochastic
deterioration process since no maintenance action is performed. Therefore, P(dt+1, τt+1 |
dt, τt) and P(dt+1, τt+1, gF M t+1 |dt, τt, gF Mt ) become the transition models for POMDPs
with different failure criteria.
The transition model for a “perfect-repair” action T(s0, aP R , s) is constructed such
that the component holding any belief b(s) returns to its initial condition b0(s) (Morato
et al., 2022). Despite being briefed to only two actions in this paper, other repair transition
matrices can also be defined for different types of maintenance actions (Papakonstantinou
& Shinozuka, 2014a).
•Observation probabilities: An observation matrix Z(o, s0, a) defines the probability of
collecting an observation o∈ O for the component being in state s0∈ S after taking
the action a. Frequently used ones in offshore I&M planning are “no-inspection”,
“binary-indication” and “continuous-indication”, etc.
•Rewards: After taking an action a∈ A every time step, the agent collects the reward
R(b, a) which is a weighted sum of the belief b(s) and the state reward R(s, a). One
needs to define the state reward R(s, a) for each action-observation combination of RBI
planning.
The reward of “do-nothing/no-inspection” is the failure risk computed from the failure
cost −CFassigned to the failure states within ¯
R(s, aDN −N I ) and the transition probability
as follows:
R(s, aDN −N I ) = P(s0|s, a)¯
R(s0, aDN −N I )−¯
R(s, aDN −N I ) (24)
The reward of “do-nothing/inspection” is one inspection cost −CIadditional to the
reward of “Do-nothing/no-inspection” such that:
R(s, aDN −I) = R(s, aDN −N I )−CI(25)
The reward of “perfect-repair/no-inspection” is simply equal to the repair cost −CR
for any state:
R(s, aP R−N I ) = −CR(26)
The objective of I&M planning being to identify the optimal policy which minimises the
total expected cost can be rephrased, within the POMDP framework, as to obtain a sequence
of actions that maximises the total expected reward. In an MDP policy (π:S → A), the
current state can prescribe which action to be taken. Since the agent cannot fully observe the
current state in POMDPs, action decisions are planned based on the belief. A POMDP policy
(π:B→ A) therefore maps a belief bto the prescribed action and the objective is to identify
the optimal policy π∗(b) which maximises the expected sum of the rewards. The value of the
13

optimal policy π∗is described by the value function:
V∗(b) = max
a∈A "X
s∈S
b(s)R(s, a) + γX
o∈O
P(o|b, a)V∗(b0)#(27)
Recently, efficient point-based solvers have been developed which solve high-dimensional state
space POMDPs based on a representative set of belief points (Kurniawati, Hsu, & Lee, 2008;
Spaan & Vlassis, 2004). In the point-based solvers, the value function in Equation (27) is
parametrised by a set of α-vectors each of which is associated to an action. For a certain belief
b(s), the optimal action is the one corresponding to the α-vector which maximises the value
function:
V∗(b) = max
α∈ΓX
s∈S
α(s)b(s) (28)
Since the belief is updated after every action and observation, as in Equations (21) and (22),
the value function is therefore recomputed to choose sequential optimal actions over time.
5. Numerical experiments: Application to a tubular joint
With the objectives of implementing the presented I&M planning methods integrated with
various failure criteria, as well as exploring the effects of failure criteria, deterioration and
inspection models on the identified I&M strategies, a set of numerical experiments are conducted
here for the particular case of an offshore tubular joint subjected to fatigue deterioration. Table
1 lists all the conducted experiments, classified by the implemented failure criterion and fracture
mechanics model.
Table 1.: List of analysed cases for RBI planning.
Experiment 1 - Fixed detection threshold
Option Case name Deterioration model Failure criterion
1 1D-Thick-Fixed 1-D FM Through-thickness
2 2D-Thick-Fixed 2-D FM Through-thickness
3 2D-FAD-Fixed 2-D FM FAD
Experiment 2 - Varied detection threshold
Option Case name Deterioration model Failure criterion
3 2D-FAD-Varied 2-D FM FAD
First, the I&M planning is performed with an inspection model in which the detection
threshold ˆ
ds,th (Section 2.2) is fixed. The effects of fracture mechanics models and failure criteria
on the crack propagation, reliability updating and optimal I&M plan are thoroughly analysed.
The optimal I&M strategies identified by different optimisation methods for each case are also
compared. Afterwards in Experiment 2, only one combination of 2-D FM model and FAD
criterion is considered while the detection threshold ˆ
ds,th of the inspection technique is varied
within a range to demonstrate how the I&M policies adapt with different inspection models.
Detailed explanation of the deterioration models, inspection and cost models is provided in the
following sections.
14

5.1. Deterioration models
5.1.1. SN Model
The fatigue deterioration is first estimated by computing the cumulative fatigue damage fol-
lowing the design recommendations provided by DNV standards (DNV, 2016, 2018, 2019).
Considering that the tubular joint is located just above the mean waterline, which is an accessi-
ble area for inspections, a fatigue design factor F DF of 2 is assigned in this case. Assuming the
structural component is designed to the limit for a lifetime of 20 years, the scale parameter of
the Weibull stress range distribution is found to be q= 6.4839 from Equation (1). The variables
used in the SN approach are listed in Table 2. The reliability over the lifetime according to
SN-Miner’s rule is computed by crude Monte Carlo simulations with one million samples.
Table 2.: Variables used in SN model.
Variable Distribution Mean Std References for Std (CoV)
(Median) (CoV)
m1Determ. 3
m2Determ. 5
∗log10(k1) Normal 12.48 0.2 DNV (2016, 2019)
∗log10(k2) Normal 16.13 0.2 DNV (2016, 2019)
S1Determ. 67.09
nDeterm. 3.5·107
qNormal 6.4839 (0.2) DNV (2016)
hDeterm. 0.8
∆ Lognormal (1) (0.3) DNV (2016, 2019); JCSS (2011)
Determ. = Deterministic
*log10(k1) and log10(k2) are fully correlated.
5.1.2. FM models
For each considered setting, the initial crack size d0and crack growth parameter Cdare cali-
brated to render a similar reliability in both SN and FM models. Calibration is performed by
the least-square fitting of the normalised failure probability. In Option 3 cases, the through-
thickness failure criterion is still used for the calibration since it is assumed that the cracks fail
when they penetrate the thickness during SN tests. The FAD criterion is only used for relia-
bility analysis and I&M planning. Figure 6 shows the goodness-of-fit of the calibrations. The
calibration for the 2-D FM model shows some discrepancies in the high reliability region. Yet
the probabilities of failure in this region are very small so that they are assumed not to affect
the optimal decision. The calibrated parameters together with all other parameters used in FM
models are listed in Table 3. The normalised resistance parameter Rffor the FAD criterion is
taken as recommended by JCSS (2011).
Incorporation of residual stress
When the failure assessment diagram criterion is used for the case of welded joints, it is neces-
sary to take into account the residual stress as a consequence of weld metal contraction being
restrained by the base material (Anderson, 2005). The presence of residual stress in welded joints
contributes as secondary stress component in the stress intensity factor such that KI=KP
I+KS
I.
However, secondary stress does not contribute in the plastic collapse since it has no significant
effect on the tensile strength (British Standards, 2019).
Realistic estimates of the residual stress are possible by finite element simulations of the
considered welded detail. Alternatively, the residual stress can be conservatively assumed to be
uniform. In the experiments, the values recommended in JCSS (2011) are used for lognormal
15

0 5 10 15 20
Time (years)
2
2.5
3
3.5
4
4.5
5
Reliability index
SN model
1-D FM model
2-D FM model
Figure 6.: Calibration between SN and FM approaches. The reliability index is the normal
inverse cumulative distribution function of the failure probability β(t) = −Φ−1(PF(t)).
distribution of uniform residual stress Rs, see Table 3. The applied primary stress is considered to
be fully reversed, i.e. the primary mean stress is zero and therefore, the value of stress amplitude
is used when the primary stress intensity factor KP
Iand the load ratio Lrare computed.
Tensile strength and fracture toughness
Material properties are usually considered as uncertain variables due to production variability.
The tensile strength of a structural material is often described by a lognormal distribution.
Fracture toughness is a quantitative description of material’s resistance to fracture failure be-
yond which the crack propagation becomes unstable. A three-parameter Weibull distribution is
proposed to describe the fracture toughness Kmat as in the following equation:
FKmat (k)=1−exp−k−K0
AkBk(29)
The shape parameter Bkis 4 and the recommended value of the threshold parameter K0is 20
MPa√m(JCSS, 2011). The scale parameter Akis computed according to the following equation
(British Standards, 2019). The resulting fracture toughness is in MPa√m.
Ak=11 + 77 exp T−T0−TK
52 25
t0.25 ln 1
1−p0.25
(30)
where Tis the temperature at which Kmat is to be determined (in ◦C). T0is the temperature
for a median toughness of 100 MPa√min 25 mm thick specimens and calculated as T0=
T27J−18◦C. T27Jis the temperature for 27J measured in a standard Charpy V specimen. TK
is the temperature term that describes the scatter in the Charpy versus fracture toughness
correlation. For Std = 15◦C and 90% confidence, TKis +25◦C. tis the thickness of the material
for which an estimate of Kmat is required (in mm), and pis the probability of Kmat being less
than estimated and 5% is recommended without experimental evidence (Wallin, 2011). In this
paper, the tubular joint is considered to be made of EN10025 −S355 −J R structural steel and
the required values for material properties are obtained as follows: T= 10◦C, T27J= 20◦C and
σY= 355 MPa (Igwemezie, Mehmanparast, & Kolios, 2018).
16

Table 3.: Variables used in FM models.
Variable Option Distribution Mean Std References for Std (CoV)
(Median) (CoV )
d01 Exponential 0.1235 *calibrated
2,3 Exponential 0.1603 *calibrated
log(Cd) 1 Normal -27.7903 0.3473 *calibrated
2,3 Normal -27.6302 0.4599 *calibrated
n1,2,3 Determ. 3.5.107
h1,2,3 Determ. 0.8
m1,2,3 Determ. 3
q1,2,3 Normal 6.4839 (0.2) DNV (2016)
dcrit 1,2 Determ. 16
Yd1 Lognormal (1) (0.1) JCSS (2011); Morato (2021)
d0/c02,3 Determ. 0.2
Db2,3 Determ. 0.81
Cd/Cc2,3 Determ. 1
Rs3 Lognormal (300) (0.2) JCSS (2011)
σY3 Lognormal (355) (0.07) Igwemezie et al. (2018); JCSS (2011)
Kmat 3 3P-W JCSS (2011)
Rf3 Lognormal (1.7) (0.18) JCSS (2011)
Determ. = Deterministic; 3P-W = Three-parameter Weibull distribution
5.2. Inspection models
P OD curves of different inspection methods frequently used for OWTs are provided in DNV
(2019). Eddy current (EC) inspection has become a common inspection method for offshore wind
structures as it can be used to detect fatigue cracks without removing coating. The EC inspection
in the normal working conditions is used as a reference inspection model in Experiment 1 -
Fixed detection threshold. The parameters of signal response method β0,β1,σεand ˆ
ds,th as
in Equation (17), are therefore calibrated to provide an equivalent P OD curve as the chosen
inspection technique, see Figure 7a.
In Experiment 2, the risk-based I&M planning is conducted for a range of detection thresh-
olds. The parameters of the inspection models used in the RBI experiments are shown in Table
4. The shape of the P OD curve changing with the detection threshold is illustrated in Figure
7b.
Table 4.: Parameters of inspection models.
β0β1σεˆ
ds,th
Experiment 1 - Fixed detection threshold 7.3074 2.092 4.189 5.4898
Experiment 2 - Varied detection threshold 7.3074 2.092 4.189 0 ≤ˆ
ds,th ≤10
5.3. Modelling I&M planning in POMDPs
The I&M planning experiments are formulated as POMDPs through the deterioration rate
DBNs. As discussed in Section 3, the continuous random variables need to be discretised to
implement the DBNs and the discretisation scheme should be an optimal compromise between
the accuracy of the results and the computational performance. Since the discretisation is ar-
bitrary and case-specific, several attempts are made on the selection of the number of intervals
and boundary values, and the discretisation schemes shown in Table 5 with the number of
17

0 5 10 15 20
Crack depth (mm)
0
0.2
0.4
0.6
0.8
1
Probability of detection
EC - Normal (DNV)
Experiment 1
(a) Experiment 1 inspection model, ˆ
ds,th =
5.4898.
0 5 10 15 20
Crack depth (mm)
0
0.2
0.4
0.6
0.8
1
Probability of detection
(b) Experiment 2 inspection model, ˆ
ds,th ∈
[0,10].
Figure 7.: Illustration of inspection models.
states |Sd|= 40,|Sτ|= 21,|SgF M |= 30, are selected as the relevant ones. Accordingly, the
deterioration rate DBNs with the through-thickness criterion has overall 840 states and that of
the FAD criterion has 25,200 states with the additional limit state variable gF M . Following the
discretisation schemes, the initial belief b0and transition matrices T(s0, a, s) for each case are
defined from one million simulations of crack size and FAD assessment points.
Table 5.: Discretisation schemes utilised in the numerical experiments.
Option Case name Variable Interval boundaries
1 1D-Thick-Fixed a 0, d0,mean :dcrit −d0,mean
|Sd| − 2:dcrit,∞
τ0 : 1 : 20
2 2D-Thick-Fixed a 0, d0,mean :dcrit −d0,mean
|Sd| − 2:dcrit,∞
τ0 : 1 : 20
3 2D-FAD-Fixed/2D-FAD-Varied a 0, d0,mean :dcrit −d0,mean
|Sd| − 2:dcrit,∞
τ0 : 1 : 20
gF M −∞,0 : 2
|SgF M | − 2: 2,∞
Since the existing point-based solvers are set up for the solution of infinite horizon POMDPs
and the I&M planning is desired for 20 years (finite horizon), the state space of deterioration rate
DBNs is augmented by encoding the time in the state space and adding a terminal state. For the
detailed explanation about state augmentation, the reader is referred to Papakonstantinou and
Shinozuka (2014a, 2014b). Finally, the state space of finite horizon POMDP with the through-
thickness criterion has 9240 states and that of the FAD criterion has 277,200 states.
For all case studies, three action-observation pairs are considered: (1) do-nothing/no-
inspection (DN-NI) (2) do-nothing/inspection (DN-I) and (3) perfect-repair/no-inspection (PR-
NI). The consequence of a failure event is associated with a cost CFof 106monetary units. The
cost of corrective repair and the risk of system failure conditional on component failure are
taken into account in the failure cost CFof the joint. The cost of inspection CIindependent of
18

the detection threshold is 103monetary units and the repair cost CRis 1.2·104monetary units.
The discount factor γ= 0.94 is considered. SARSOP point-based solver is used for solving the
POMDPs to obtain optimal I&M policies (Kurniawati et al., 2008).
Rf
(a) Mean crack depth and crack length.
(b) Mean Krand Lr(2-D FM).
Figure 8.: Illustration of crack deterioration.
5.4. Results and discussion: Experiment 1 - Fixed detection threshold
Crack growth
As mentioned in the previous sections, both 1-D and 2-D fracture mechanics models are applied
to estimate the crack deterioration. In both models, the crack propagation rate is influenced by
the Paris law parameters and the stress intensity correction factor. To represent one-dimensional
crack growth, Equation (9) is used where the stress intensity correction factor Ydis assigned
as a time-invariant random variable, see Table 3. In two-dimensional crack growth, the stress
intensity correction factors Ydand Ycbecome functions of time-varying crack size and are re-
computed at every time step. The geometry functions Ymd, Ybd, Ymc , Ybc and stress magnification
factors Mkmd, Mkbd, Mkmc , Mkbc are evaluated by parametric equations following the procedures
of Newman and Raju (1981) and DNV (2019).
A crude Monte Carlo Simulation (MCS) containing 1 million samples was run to estimate
the stochastic crack evolution. The comparison of mean crack propagation between 1-D and
2-D FM models can be seen in Figure 8a. The crack length is not comparable between the two
models as the 1-D model only measures the crack depth. The crack depth grows faster in the
2-D model than in the 1-D model. In fact, this is due to different initial crack depths d0and
crack growth parameters Cdof the two models since they are calibrated to the same target
reliability, see Table 3. However, it successively implies different crack propagation by the two
19

0 5 10 15 20
Time (years)
0
5
10
15
Crack depth (mm)
Mean
95% reference interval
(a) Crack distribution by 1-D FM model.
0 5 10 15 20
Time (years)
0
5
10
15
Crack depth (mm)
Mean
95% reference interval
(b) Crack distribution by 2-D FM model.
Figure 9.: Crack propagation by 1-D and 2-D FM models. To illustrate the difference between
the two models, the same initial crack size d0∼Exp(0.1603) and crack growth parameter
log(Cd)∼ N(−27.6302,0.4599) are used.
models. To examine this, crack propagation is computed by the two models using the same initial
crack size d0∼Exp(0.1603) and the crack growth parameter log(Cd)∼ N(−27.6302,0.4599).
The mean crack depth over the lifetime and 95% reference interval (between 2.5% and 97.5%
quantiles) are plotted in Figure 9. Note that a cut-off point is considered at dcrit = 16 mm
and all bigger cracks remain at 16 mm in Monte Carlo simulations. It is observed that the two
models give different variability in the crack distribution. The variability of the 1-D FM model
rapidly increases compared to 2-D model. This effect is important in the structural reliability
aspect such that the 1-D FM model with higher model uncertainty gives higher probability of
failure than the 2-D FM model. And therefore, when calibrated to have the same reliability, the
2-D FM model results in a higher mean and standard deviation of Cd.
Using the 2-D FM model, the deterioration of the tubular joint can also be described by the
fracture ratio Krand the load ratio Lrcomputed from Equations (13-15). The mean values of
Krand Lrare plotted in Figure 8b. Before year 10, the FAD assessment point mainly increases
in the Kraxis and the load ratio Lris initially less sensitive to the crack size. With small values
of crack size, the tubular joint has a sufficient intact area and therefore it is not subjected to
high net section stress σref . However, the load ratio starts increasing as the crack depth and
crack length rapidly grow after year 10.
Updating reliability
The effect of failure criteria and FM models on the updated failure probability after an inspection
is examined here. Assuming an inspection is performed at year t= 11 and no crack is detected
during the inspection, the reliability is updated for different cases using Equation (19) through
Monte Carlo simulations. In the deterioration rate DBNs, the updated failure probability is the
probability of being in the failure states after performing the transition and estimation steps.
As shown in Figure 10, consistent PFvalues are obtained by the MCS and the DBNs, which
verifies the proper discretisation of the state space variables. On the other hand, the updated
failure probabilities after the inspection are different among the analyzed cases. The 2D-FAD-
Fixed case gives the smallest failure probabilities in all years. This is expected simply due to
the assumption of the capacity to hold the through-thickness cracks when the FAD criterion is
used.
The effect of fracture mechanics models on reliability updating can be analysed through
comparison of 1D-Thick-Fixed and 2D-Thick-Fixed cases. As discussed before, the two FM
20

2 4 6 8 10 12 14 16 18 20
Time (years)
10-4
10-3
10-2
10-1
Probability of failure
1D-Thick-Fixed (MCS)
2D-Thick-Fixed (MCS)
2D-FAD-Fixed (MCS)
1D-Thick-Fixed (DBN)
2D-Thick-Fixed (DBN)
2D-FAD-Fixed (DBN)
Figure 10.: The updated cumulative failure probability given no-detection in the inspection at
year 11.
models provide different initial crack sizes d0and crack growth parameters Cdwhen calibrated
to the SN failure probabilities. Consequently, the crack depth belongs to different probability
distributions with respect to each FM model, as illustrated with the crack size histograms in
Figure 11a. Firstly, the variation in prior distributions influences the updated failure probability
given no crack detection. However, this is not obvious just after the no-detection event, in Figure
11b. Since the probability of no-detection and probability of crack distribution are very low in
the failure bin, i.e. d≥dcrit, the difference in the updated PFis extremely small when normalised
by the overall no-detection probability according to Equation (19). As the posterior distribution
further propagates accumulating and increasing the probability in the failure bin, the difference
in the updated PFcan then be clearly observed as in Figure 11c and 11d. The more-detailed
2-D FM finally achieves higher reliability at the end of the lifetime. And vice versa, the decision-
maker may use simple models with higher uncertainty, e.g. 1-D FM but may take a higher risk
than using more precise ones or perform more inspections to remain at the same reliability and
risk.
Secondly, the overall probability of detection with the same inspection model is different
between the two cases. The 2-D FM model with its higher crack growth parameters results in
a higher probability of detection. In heuristic-based inspection planning, maintenance decision
rules are often prescribed based on inspection outcomes, e.g. repair is performed if a crack is
detected. In this case, using the 2-D FM will result in higher maintenance costs than the 1-D
FM for the same inspection strategy.
Optimal I&M strategies
I&M planning is performed through traditional heuristic-based methods as well as through the
formulation as POMDPs. The SARSOP point-based POMDP solver is used for the computation
of the optimal I&M policies (Kurniawati et al., 2008). Two sets of heuristic decision rules -
equidistant inspections (EQ-INS) and inspections planned before an annual failure probability
threshold is exceeded (THR-INS) - have been evaluated in the simulation environment through
DBNs. If the inspection indicates the presence of crack, a perfect repair is immediately performed
and after, the component goes back to its initial condition. The identified optimal I&M strategies
21

5 10 15
Crack depth (mm)
10-6
10-4
10-2
100
Probabililty
1-D FM
2-D FM
(a) Prior distribution of crack depth (t=11).
5 10 15
Crack depth (mm)
10-6
10-4
10-2
100
Probabililty
1-D FM (POD t=11 = 0.2796)
2-D FM (POD t=11 = 0.4147)
(b) Posterior distribution of crack depth
(t=11).
5 10 15
Crack depth (mm)
10-6
10-4
10-2
100
Probabililty
1-D FM (POD t=11 = 0.2796)
2-D FM (POD t=11 = 0.4147)
(c) Posterior distribution of crack depth
(t=15).
5 10 15
Crack depth (mm)
10-6
10-4
10-2
100
Probabililty
1-D FM (POD t=11 = 0.2796)
2-D FM (POD t=11 = 0.4147)
(d) Posterior distribution of crack depth
(t=19).
Figure 11.: Propagation of crack distribution. (a) The prior crack distribution at year t= 11 is
estimated by the two FM models. (b) The posterior PFat the year of inspection is almost equal
for 1-D and 2-D models whereas the overall detection probability is different. (c, d) The updated
PFw.r.t the two models becomes different as the posterior crack distribution propagates.
from heuristics and POMDPs are evaluated in a simulation environment. The resulting total
expected costs E[CT] as well as the numerical confidence intervals over 105simulations are
reported in Table 6.
In all combinations of deterioration models and failure criteria, POMDP policies outperform
traditional heuristics with significant differences in the total expected cost. Comparing among
different options, 2D-FAD-Fixed case results in less expected cost than the other two options
which rely on the through-thickness criterion. When the FAD criterion is used, it is assumed
that the through-thickness cracks can grow further in the length until the critical value of the
stress intensity factor is reached. It is also worth mentioning that the fracture toughness of
the material considered for the tubular joint is high enough so that the component does not
fail before the crack reaches the thickness and can hold the through-thickness crack. Therefore,
the failure probabilities over the lifetime with the FAD criterion are smaller than the other
cases, as already discussed before. It results in a significant reduction of failure risk as well
as less observations and maintenance actions can be generally expected. Random realisations
are presented to visualise the policies prescribed by different approaches. Figure 12a, 12c and
22

Table 6.: Experiment 1 - Fixed detection threshold: Comparison of the total expected cost.
E[CT] 95% C.I
Option 1: 1D-Thick-Fixed
Finite horizon POMDP - SARSOP 6267.20 ±20.49
Heuristic EQ-INS (∆ins = 7) 6892.72 ±22.462
Heuristic THR-INS (∆PF th = 9 ·10−4) 6775.92 ±20.045
Option 2: 2D-Thick-Fixed
Finite horizon POMDP - SARSOP 5387.88 ±22.43
Heuristic EQ-INS (∆ins = 11) 6066.32 ±15.681
Heuristic THR-INS (∆PF th = 1.2·10−3) 6066.32 ±15.681
Option 3: 2D-FAD-Fixed
Finite horizon POMDP - SARSOP 3599.45 ±16.39
Heuristic EQ-INS (∆ins = 11) 4118.25 ±17.462
Heuristic THR-INS (∆PF th = 5 ·10−4) 4118.25 ±17.462
12e represent the finite horizon POMDP policy realisations and Figure 12b, 12d and 12f show
the realisations of the equidistant heuristic. As manifested in the POMDP policies, only one
inspection is required in the 2D-FAD-Fixed case if the first inspection outcome is no-detection
thanks to the low failure risk. Contrarily, more inspections are conducted in a number of years
in the cases in which the through-thickness criterion is used.
In the heuristic-based policies, an immediate repair action is prescribed after detection, but
the POMDP policies may opt to perform subsequent inspections in case of detection. A repair
action is planned only if the subsequent inspections also give detection outcomes. In Figure 12a,
POMDP policies plan one more inspection after a detection event and a repair is performed
as the second inspection also gives detection. However, POMDP policies also consider more
subsequent inspections when the deterioration model has lower uncertainty. For the 2D-Thick-
Fixed case, POMDP plans up to three consecutive inspections as in Figure 12c. Since the last
inspection declares no-detection, no maintenance action is taken. Therefore, POMDP solutions
can provide adaptive policies for different scenarios such that, as in this example, it is plausible
to take advantage of the more precise 2-D model to avoid an expensive repair.
In contrast, traditional heuristic approaches which only follow the predefined decision rules
are not able to capture such aspects. Utilising the 2-D FM instead of the 1-D FM affects
the optimal heuristic policies such that the number of inspections is reduced, which opposes
the pattern in the POMDP policies. In the equidistant inspection approach, two inspections
(∆ins = 7) are performed in the 1D-Thick-Fixed whereas only one inspection (∆ins = 11) is
conducted in the 2D-Thick-Fixed. As discussed before, using the 2-D FM model is a matter of
reducing the risk but on the other hand increasing the maintenance cost. And in this case, the
increased maintenance cost is more than the reduced risk, therefore one inspection has been
reduced to adjust the optimal policy.
5.5. Results and discussion: Experiment 2 - Varied detection threshold
Optimal I&M strategies
Hereafter, only one combination with 2-D FM model and FAD criterion is considered. The
optimisation is performed only through POMDPs as it has been demonstrated in the previous
experiment that POMDP outperforms traditional heuristic approaches for any combination of
deterioration model and failure criteria. In this experiment, the RBI planning is repeatedly
performed for several values of detection thresholds 0 ≤ˆ
ds,th ≤10. The transition models
and the rewards models of the POMDP remain the same as the 2D-FAD-Fixed case from the
previous experiment. Only the observation model Z(o, s0, s) is modified for each threshold value.
The breakdown of the total expected cost evaluated over 105simulations are reported in
23

0 5 10 15 20
Time (years)
10-4
10-3
10-2
10-1
PF
ND
D
Repair
(a) 1D-Thick-Fixed (FH-POMDP).
0 5 10 15 20
Time (years)
10-4
10-3
10-2
10-1
PF
ND
D
Repair
(b) 1D-Thick-Fixed (Heuristic EQ-INS).
0 5 10 15 20
Time (years)
10-4
10-3
10-2
10-1
PF
ND
D
Repair
(c) 2D-Thick-Fixed (FH-POMDP).
0 5 10 15 20
Time (years)
10-4
10-3
10-2
10-1
PF
ND
D
Repair
(d) 2D-Thick-Fixed (Heuristic EQ-INS).
0 5 10 15 20
Time (years)
10-4
10-3
10-2
10-1
PF
ND
D
Repair
(e) 2D-FAD-Fixed (FH-POMDP).
0 5 10 15 20
Time (years)
10-4
10-3
10-2
10-1
PF
ND
D
Repair
(f) 2D-FAD-Fixed (Heuristic EQ-INS).
Figure 12.: Policy realisations of Experiment 1 - Fixed detection threshold. Inspection outcomes
are represented by a circle (for no-detection) or a five-pointed star (for detection). A red bar
denotes that a perfect repair is performed.
24

Figure 13. In general, moderate to high detection thresholds 5 ≤ˆ
ds,th ≤10 provide the lowest
expected costs. The policy realisations of some representative cases are also presented. Figure 14a
0 1 2 3 4 5 6 7 8 9 10
0
1000
2000
3000
4000
5000
6000
Expected cost
Figure 13.: Experiment 2 - Varied detection threshold: Comparison and breakdown of the total
expected cost.
and 14b show random realisations of inspection planning with low detection threshold ˆ
ds,th = 0
and Figure 14c and 14d show the realisations with high threshold ˆ
ds,th = 10. An interesting
pattern of inspection planning can be discovered in the POMDP prescribed policies. When the
inspection model has a lower threshold, POMDP tends to plan several subsequent inspections in
case of detection. The shape of the P OD curve plateaus earlier with lower detection threshold,
e.g. P OD curve of ˆ
ds,th = 0 becomes flat at around d= 5 mm in Figure 7b. If a detection
outcome is obtained, one may not simply presume bigger crack size and/or higher probability
failure, noting that even the small cracks which do not cause failure may also be detected.
Therefore, the probability of failure is only slightly increased and successive inspections are
planned instead of taking an expensive repair action, see Figure 14a. In the case of no-detection
from the low threshold inspection, a small crack can be assured since bigger cracks d > 5
mm have almost zero probability of no-detection in the P O D curve. The probability of failure
therefore drops drastically, see Figure 14b.
On the other hand, the P OD curve with a high detection threshold slants up with the crack
size, see ˆ
ds,th = 10 in Figure 7b. Bigger cracks are more likely to be detected than smaller
ones. If this inspection model gives a detection outcome, a severe crack can be expected with
a higher confidence, thus resulting in a high jump of failure probability as in Figure 14c. The
repair action is also performed immediately after one detection. Nevertheless, detection events
are less frequent in this case and the expected repair cost is still remarkably lower compared to
the low threshold inspection.
6. Conclusions
In this paper, the effects of failure criteria, deterioration and inspection models on the optimal
inspection and maintenance (I&M) strategies are examined. Two-dimensional fracture mechan-
ics model and failure assessment diagram (FAD) criterion have been successfully integrated
with dynamic Bayesian networks (DBNs), therefore allowing the formulation and optimisation
of I&M planning via partially observable Markov decision processes (POMDPs). Various dete-
rioration, inspection and failure criteria settings were tested for the optimal management of a
structural detail subjected to fatigue deterioration, revealing the following findings:
25

0 5 10 15 20
Time (years)
10-4
10-3
10-2
10-1
PF
ND
D
Repair
(a) 2D-FAD- ˆ
ds,th = 0 (FH-POMDP).
0 5 10 15 20
Time (years)
10-4
10-3
10-2
10-1
PF
ND
D
Repair
(b) 2D-FAD- ˆ
ds,th = 0 (FH-POMDP).
0 5 10 15 20
Time (years)
10-4
10-3
10-2
10-1
PF
ND
D
Repair
(c) 2D-FAD- ˆ
ds,th = 10 (FH-POMDP).
0 5 10 15 20
Time (years)
10-4
10-3
10-2
10-1
PF
ND
D
Repair
(d) 2D-FAD- ˆ
ds,th = 10 (FH-POMDP).
Figure 14.: Policy realisations of Experiment2 - Varied detection threshold. Inspection outcomes
are represented by a circle (for no-detection) or a five-pointed star (for detection). A red bar
denotes that a perfect repair is taken.
•The failure assessment diagram might be preferred as the failure criterion to model redun-
dant structures with capacity to sustain through-thickness cracks since it offers significant
savings in the total expected cost, especially through advanced optimisation methods such
as POMDPs. However, one needs the knowledge of material properties, compulsorily yield
strength and fracture toughness to implement the FAD criterion.
•2-D fracture mechanics models are more robust than 1-D models since the effects of time-
dependent crack size, geometry of the structure and welded detail are inherently considered
in the stress intensity correction factors Ydand Yc. The shortcoming is that it requires
FEM analysis or the use of parametric equations, both of which demand computational
resources.
•The observation model, specified often through probability of detection (P OD) curves,
can be adjusted by varying the detection threshold of the inspection technique. Generally,
very low detection thresholds are not recommended due to its flat and high P OD curve,
causing frequent detections, inspections and/or repairs.
Throughout the experiments, it is demonstrated that I&M polices provided by finite horizon
POMDPs outperform heuristic-based polices for any combination of deterioration models and
failure criteria. POMDPs also reveal adaptability in the policy patterns depending on the models
26

specified in the I&M planning. The main limitation associated with implementing the 2-D
fracture mechanics model and FAD criterion in discrete DBNs/POMDPs is the high-dimensional
state space. Computational intractability becomes a constraint to apply such complex models
and failure criterion for the case of multiple components or longer horizon lengths. To overcome
this curse of dimensionality, further research efforts are suggested towards the integration of
the FAD criterion with POMDP-based deep reinforcement learning (DRL) approaches. The
capability of POMDP-based DRL approaches to efficiently provide optimal I&M strategies for
large state space problems has been demonstrated in Andriotis and Papakonstantinou (2021).
Acknowledgments
This research has been developed at the University of Li`ege, Belgium, in collaboration with the
University of Strathclyde, UK, and Aalborg University, Denmark. The financial support pro-
vided by the Belgian Energy Transition Fund (FPS Economy) through the projects PhairywinD
and MaxWind is greatly appreciated. Dr. Morato would further like to acknowledge the support
granted by the National Fund for Scientific Research in Belgium F.R.I.A. - F.N.R.S.
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