Reliability assessment of deteriorating structures:Challenges and (some) solutions

Daniel StraubEngineering Risk Analysis GroupTechnische Universitat Munchen, Germany

ABSTRACT: The assessment of deteriorating structural systems has been one of the major applications ofprobabilistic analysis and structural reliability theory. Nevertheless, looking back on over 40 years of researchand development in this area, the outcome has been mixed. While a lot of progress has been made, importantquestions and challenges still have not been answered. Structural reliability methods still play a minor role in themanagement of existing deteriorating structures in practice. There are encouraging signs, e.g. the developmentof standards and guidelines for existing structures that rely at least partly on structural reliability methods. Butthere remain significant open challenges to the structural reliability community, some of which I discuss in thiscontribution.

1 INTRODUCTION

The assessment of deteriorating structural systemshas been one of the major applications of probabilis-tic analysis and structural reliability theory, for multi-ple reasons. Firstly, with the exception of fatigue, thequantitative assessment of deterioration is not consid-ered in current codes based on the partial safety factorformat. Hence, it is challenging to demonstrate the re-liability of deteriorating structural systems with stan-dard (semi-probabilistic) safety concepts alone (Faber2000, JCSS 2001, Melchers 2001, Ellingwood 2005).Secondly, deterioration is often assessed for exist-ing structures in which damages have been observed.For such structures, data is typically available frompast inspections, monitoring or tests performed duringthe construction and operation of the structure. Thesedata can be used to update the parameters of struc-tural models, deterioration models or the reliability it-self. Such an integration is best performed through aprobabilistic analysis. Thirdly, in addition to deterio-ration, existing structures are often subject to changesin demand or loss of capacity, which may cause non-compliance of the structure with current code require-ments. In some cases, a probabilistic assessment candemonstrate that a structure is nevertheless safe forfuture usage.

All these applications of structural reliability re-quire a proper probabilistic description of the struc-tural model and its parameters. A major challengethereby is that the data available from existing struc-tures cannot be explained with the simple models as-

sumed in classical structural design and simple relia-bility analysis. Instead, the data reflect the real behav-ior of the structure and the spatial variability of theparameters. For this reason, the assessment of existingstructures often necessitates more sophisticated struc-tural and probabilistic models than the design of newstructures. Thereby, the proper modeling choices arecrucial, as overly simple models can lead to entirelywrong results, whereas overly complicated modelslead to an unnecessary effort for analysis and – be-cause of the difficulties in reliability analysis – arealso more error-prone.

The challenges associated with the modeling are re-lated to:

• Deterioration modeling

• Structural system modeling

• Probabilistic modeling

• Dependence modeling

• Modeling of inspection and monitoring

In this contribution, I review these challenges and out-line strategies for handling the modeling in Sections3 to 6. The aim is to discuss when simple modelsare sufficient, and when more advanced models ofthe structural system and spatially variable parame-ters are required. The selected modeling approach hasdirect implications on the computational aspect of the

reliability analysis. I will highlight recent computa-tional developments that facilitate the application ofadvanced models in engineering practice.

Finally, main challenges in the assessment of de-teriorating structures are associated with the manage-ment and the organization of owners and operators.These include data and model availability, develop-ment of codes and standards as well as general orga-nizational aspects. I will address these throughout themanuscript.

The paper starts out with the presentation of an ide-alized example and a review of the basic quantities inthe reliability assessment of structures over their ser-vice life (Section 2). This example should allow thereader to follow and – if desired – to replicate someof the analyses presented in this paper.

2 RELIABILITY ANALYSIS OFDETERIORATING STRUCTURES

2.1 An idealized structure

To illustrate the concepts, methods and challenges as-sociated with the assessment of deteriorating struc-tures, I introduce a highly idealized example, whichnevertheless features most of the relevant aspects ofa real-life structural system. The structural system,shown in Figure 1, is a frame structure with idealplastic material behavior. It has been studied in mul-tiple text books and papers previously, without con-sidering deterioration (Madsen et al. 1986, Der Ki-ureghian 2005). The frame is characterized by theplastic moment capacities at locations 1−5. There aretwo loads: the permanent load V and the time-variantload H . The latter is described by the distribution ofits annual maxima Hj , with the index j denoting theyear.

The structure is subject to deterioration, which ismodeled at the element level i = 1, . . . ,5 as:

Di(t) = 1−Φ(t−Bi

0.2Bi

)(1)

Φ is the cumulative distribution function (CDF) of thestandard normal distribution, Bi is the deteriorationmodel parameter.

The resulting moment capacity Ri(t) of element iat time t is

Ri(t) = Mi ·Di(t) (2)

with Mi being the initial moment capacity.The occurrence of the three failure mechanisms de-

picted in Figure 1 in year j is described by the follow-ing limit state functions (Madsen et al. 1986):

ga,j(X) = R1(tj) +R2(tj) +R4(tj) +R5(tj)

−5Hj (3)

gb,j(X) = R2(tj) + 2R3(tj) +R4(tj)− 5V (4)

(3)

H

V

5m 5m

5m

(4)

(5)

(2)

(1)

Failure mechanisms:(a)

(b) (c)

Figure 1: Frame structure and its three main failure mechanisms.

gc,j(X) = R1(tj) + 2R3(tj) + 2R4(tj) +R5(tj)

−5Hj − 5V (5)

wherein tj is the time associated with the jth year. Ifthe structure is installed at time t= 0, then it is simplytj = j.

The event of failure of the structural system dur-ing year j is then described by the system limit statefunction:

gj(X) = min[ga,j(X), gb,j(X), gc,j(X)] (6)

with failure in year j defined as:

Fj = {gj(X) ≤ 0} (7)

The probability of failure of the undamaged struc-ture in the first year of service (with Ri ' Mi) isPr(F0) = 9.95× 10−4.

The event of failure up to time tj is defined as

F (tj) = F1 ∪ F2 ∪ . . .∪ Fj (8)

The corresponding probability of failure of thestructure up to time tj is

Pr[F (tj)] = Pr(F1 ∪ F2 ∪ . . .∪ Fj) (9)

Pr[F (tj)] is the quantity that should be utilized forthe assessment of the reliability. Unfortunately, thiscorresponds to a system reliability problem amongthe failure events in different years (in addition to thesystem reliability problem describing failure in each

Table 1: Stochastic model of the example structure.Parameter Unit Distribution Mean St. dev.Hj (annual max.) kN Gumbel 50 20V kN gamma 60 12M1 – M5 kNm lognormal 200 30B1 – B5 yr lognormal 100 50ρM – det. 0.3ρA – det. 0.6

year, Eq. 6). In many instances, therefore, the proba-bility of Eq. 9 is approximated by one of the followingbounds:

Pr(Fj) ≤ Pr[F (tj)] ≤ 1−j∏

i=1

[1−Pr(Fi)] (10)

Efficient solutions to computing the exactPr[F (tj)] are presented in (Straub et al. 2019).

2.2 Lifetime reliability

To assess the reliability of structures over their life-time, the reliability is best expressed by the probabil-ity of failure up to time t, which is equal to the CDFof the time to failure TF (Barlow and Proschan 1996,Rausand and Høyland 2004):

Pr[F (t)] = FTF(t) = Pr(TF ≤ t) (11)

The reliability at time t is

Rel(t) = 1−Pr[F (t)] (12)

The probability density function (PDF) of the life-time TF is

fTF(t) =

dFTF(t)

dt= −dRel(t)

dt(13)

The hazard function (failure rate) is defined as

hF (t) =fTF

(t)

Rel(t)(14)

The probability of failure of the example structureis shown in Figure 2; the corresponding hazard func-tion is shown in Figure 3. The effect of the deteri-oration is evident. Without deterioration, the hazardfunction of the structure is slightly decreasing overtime, from 10−3 at time t = 0 to 0.89 · 10−3 at timet = 50yr. This is caused by a proof load effect: Sur-vival at earlier years is an indication that the structurehas a certain minimum capacity and that the perma-nent load V is not excessively large.

Note that the probability of failure of this structureis artificially inflated; some effects are more clearlyvisible at large probability values. Most real structureshave much smaller failure probabilities.

The risk associated with structural failure over thelifetime of the structure is a function of the lifetimePDF fTF

:

Risk =∫ ∞0CF exp[−γt]fTF

(t)dt (15)

10 20 30 40 500

0.05

0.1

0.15

0.2

0time [yr]

Pr[F

(t)] with deterioration

without

Figure 2: Probability of failure of the example structure.

10 20 30 40 50010-4

10-3

10-2

10-1

time [yr]

failu

re ra

te [y

r-1]

with deterioration

without

Figure 3: Hazard function (failure rate) of the example structure.

wherein CF is the cost associated with a failure and γis the discount rate. Risk is the net present value ofthe risk.

Eq. 15 assumes that the structure is utilized untilfailure. If only a finite service lifetime TSL is consid-ered, the upper limit of the integral is replaced by TSL.Figure 4 shows the net present risk of the examplestructure in function of the considered service life pe-riod TSL, with γ = 2% andCF = 106. It is evident thatthe possibility of failures beyond 100yr in the futuredoes not significantly affect the net present value ofthe risk; in the case of the non-deteriorating structureit is Risk(100yr) = 38 · 103 vs. Risk(∞) = 42 · 103.The reason lies in the discounting of future failurecosts to t = 0.

100 150 2000

0.05

0.1

0.15

0.2

0.25

0 50

with deterioration

withoutlifet

ime

risk

[106 €

]

lifetime [yr]

Figure 4: Lifetime risk (net present value) of the example struc-ture in function of the intended service life.

A proper life-cycle costing should also consider ob-solescence and replacement of the structure (Rack-witz 2000). However, almost all studies on optimalasset integrity management ignore the possibility ofobsolescence. Also, estimating the probability of ob-solescence appears challenging, since it requires pre-dictions on future technological progress and socioe-conomic developments.

2.3 The example structure vs. real life structures

In the following sections, the various aspects of as-sessing the system reliability of deteriorating struc-tures will be discussed. The example structure willbe utilized to illustrate points made in the discussion.The idealized example structure has multiple conve-nient features not encountered in real-life applica-tions, as every engineer who has worked on the as-sessment of deteriorating structures can testify. I willaddress these features in the subsequent sections. Myaim is to demonstrate that the reduction to an ide-alized model can often provide the solution path formore complex problems. But I will also show that tosome challenges no easy solutions are available.

3 DETERIORATION MODELING

3.1 Model availability and accuracy

There is a substantial amount of literature on stochas-tic deterioration models for structures (Lin and Yang1985, DuraCrete 1998, Stewart and Rosowsky 1998,Melchers 1999a, Frangopol et al. 2004). Nevertheless,good deterioration models exist only for few commonphenomena, in particular for fatigue in metallic struc-tures and for reinforcement corrosion in RC struc-tures. And even in these areas, the prediction abil-ity of the models can be quite poor outside of labo-ratory conditions. In one of the few examples of in-service validation of deterioration models, Aker Off-shore Partner (1999) compare predictions of fatiguelife with observed fatigue crack rates in offshore steelstructures. Their conclusions are rather discouraging,indicating a limited predictive power of fatigue mod-els. Figure 5 shows results from this study, in whichthe observed frequencies of fatigue cracks are sum-marized for groups of fatigue details with the samepredicted fatigue reliability at the time of inspection.The data shown in Figure 5 is from older structuresbuilt prior to 1975. Results from the same study indi-cate that the observed frequency of fatigue cracks innewer structures is significantly lower, yet the predic-tive power of the fatigue model is not better.

For many deterioration phenomena, quantitativepredictive models are lacking entirely. For example,in Germany significant resources are spent on ensur-ing the reliability of prestressed concrete bridges builtmainly in the 1960s and 70s that are subject to stress

-3.5 -2.5 -1.5 -0.5 0.5 1.5 2.5 3.50

0.2

0.4

0.6

0.8

1

Predicted reliability index β

Freq

uenc

y of

fatig

ue c

rack

s

predicted

observed

Figure 5: Observed frequency of fatigue cracks in structural de-tails in function of the calculated fatigue reliability of these de-tails. The predicted frequency is the one according to theory, i.e.Φ(−β). Figure adapted from (Aker Offshore Partner 1999).

corrosion cracking (SCC). Because no suitable pre-dictive model for SCC exists, authorities are requiredto take a highly conservative approach based on sub-stantial inspection efforts (Lingemann et al. 2010).

Overall, stochastic modeling of deterioration forthe purpose of assessing the reliability of a structure ischallenging. The associated uncertainty signifies thatin most applications, models can give useful predic-tions only if accompanied by data collection throughtests, inspections and monitoring. However, modelsare necessary for making any kind of predictions, anddespite trials to do otherwise, it is doubtful a quantita-tive demonstration of the reliability of a deterioratingstructure is possible without an underlying deteriora-tion model. The increased availability and quality ofmonitoring and other type of data provides an oppor-tunity to develop more realistic deterioration modelsin the future. For this to happen, it is however im-portant that infrastructure owners and operators de-fine and impose standards for the collection of suchdata.

3.2 Deterioration modeling at the system level

With few exceptions, deterioration in structural sys-tems is modeled at the element level, in analogy tothe example structure of Section 2.1. This is due tothe uniqueness of structures, which does not allow formodels to be easily transferred from one structure toanother. Exceptions to the rule are some empirically-based models that are based on past inspection data,e.g. models utilized in bridge management systems(BMS) (Scherer and Glagola 1994, Thompson et al.1998). A main limitation to developing such empir-ical models has been the lack of high-quality data.Existing data comes from traditional bridge inspec-tions, which suffer from reporting standards that arenot aimed at developing deterioration models. For ex-ample, in most cases, only the degree of damages ina part of the structure is reported, often without sys-tematically noting the type of damage nor the mech-anism causing it. Furthermore, the assessment by in-

spectors has a substantial subjective component andhas also shown to be subject to significant uncertaintyand variability (Phares et al. 2001).

Nevertheless, empirical models both at the subsys-tem and system level are likely to improve in the fu-ture. In particular efforts of infrastructure owners andoperators to implement BIM (Building InformationManagement) standards should substantially improvethe data collection, as is the progress in sensor, moni-toring and communication technology.

4 STRUCTURAL SYSTEM MODELING

Since deterioration is typically modeled at the ele-ment level, a model of the structural system is nec-essary for understanding the effect of deterioration onthe structural reliability. In the following, different ap-proaches to handle this challenge are presented, in theorder of increasing accuracy and complexity.

4.1 Limiting the analysis to element failure events

In Eurocode, and most other structural codes, the sys-tem reliability is not modeled explicitly. Instead, it isrequired that all structural elements comply with thereliability requirement, by demonstrating compliancewith the design limit state for each cross section. Inmany cases, this is a conservative approach, but fornew-built structures the approximation is often justi-fied by the difficulty and cost of a more realistic sys-tem model.

The example structure of Section 2.1 is ductile andhence has significant reserves beyond yielding of thefirst member, as demonstrated in the following. If fail-ure at the element level is defined as reaching the plas-tic limit (in accordance with a static equilibrium ap-proach as used in Eurocode), the corresponding limitstate functions are:

ge,i,j(X) = Ri(tj)− |MS,i(X, j)| (16)

wherein MS,i,j is the maximum resulting moment atlocation i in year j. The stiffness of the structureis constant throughout the structure and is describedby the parameters E = 200GPa, A = 0.02m2 andI = 10−4m4. With these parameters, a linear-elasticstructural analysis results in the moments as follows:

MS,1,j = 1.56m ·Hj − 0.5m · Vj (17)

MS,2,j = −0.94m ·Hj + 1m · Vj (18)

MS,3,j = 1.5m · Vj (19)

MS,4,j = −0.94m ·Hj − 1m · Vj (20)

MS,5,j = 1.56m ·Hj + 0.5m · Vj (21)

10 20 30 40 500

0.2

0.4

0.6

0.8

0

time [yr]

Pr[F

(t)]

Cross-section based analysis(equilibrium approach)

System-based analysis(kinetic approach)

Figure 6: Probability of system failure computed with anelement-based model in accordance with the classical designmethodology in Eurocode, compared to the probability of fail-ure evaluated with a system-based model (the same as in Figure2).

If failure of the structural system is defined as theviolation of any of the resulting 5 limit states, a cor-responding system reliability estimate can be com-puted. The limit state functions ge,i,j(X) give signif-icantly lower values than the ones describing the fail-ure mechanisms (Eqs. 3-5). Therefore, the static equi-librium approach results in much higher probabilityof failure estimates than the system analysis approachof Section 2.1, which corresponds to a kinematic limitanalysis. Figure 6 compares the probabilities of fail-ure as computed with these two approaches. Assess-ing system performance through Eq. 16 can thereforebe highly conservative. Such a model is neverthelessutilized in the assessment of existing and deteriorat-ing structures because it complies with the safety for-mat of common standards and is readily implementedwith standard software tools, e.g. (Mix 2016).

4.2 Full system reliability analysis

A system reliability analysis, as it is performed on theexample structure in Section 2.1, is not commonlyapplied for the assessment of structures in practice.In research, such analyses are performed and thereseems to be no fundamental difficulty in performingthem, even if some challenges remain. In practice,these analyses have been hindered by the significantcomputational demands and the practical difficultiesassociated with the need of coupling a structural anal-ysis model with a structural reliability code. Today’scomputer performance and the large number of avail-able structural reliability software solutions alleviatethese problems.

Another hindrance in performing an integratedsystem-wide analysis lies in the analysis workflow,which is still mostly element-based. The deteriorationis evaluated at the element level (e.g. fatigue lives arecalculated for structural details), and it is often prefer-able to assess the reliability element by element. Inprinciple, such an approach can also ensure the re-liability of the system, but it necessitates that relia-bility acceptance criteria at the element level are de-

Figure 7: System model of (Straub and Der Kiureghian 2011) for determining reliability acceptance criteria at the structural elementlevel based on the n− 1 redundancy.

rived, which are consistent with the requirements tothe structural system. This is outlined in the next sec-tion.

4.3 Deriving reliability requirements at the elementlevel based on a simplified representation ofredundancy by an n− 1 analysis

For some types of deterioration in structures, e.g. fa-tigue, it can be sufficiently accurate to consider an ele-ment as either fully intact or completely failed. In thiscase, the element performance can be modeled at thesystem level by a Bernoulli random variable with bi-nary outcome states. On this basis, it is possible to de-rive reliability acceptance criteria at the element levelbased on overall reliability levels at the system level(Straub and Der Kiureghian 2011).

To this end, simplified system models were pro-posed in the past for the analysis of offshore steelstructures (Moan 1999, Faber et al. 2000, Moan 2005,Straub and Faber 2005a). They are based on evalu-ating the reduction in overall system capacity uponfailure of individual elements and then estimating theprobability of system failure conditional on these fail-ures. These analyses are in analogy to n− 1 contin-gency analyses, which are commonly applied to testsystem redundancy, in structures (e.g., Frangopol andCurley 1987) as well as other engineering systems,e.g. power grids (Stott et al. 1987).

A measure to assess the element importance in thisway is the probability of failure of the structure withelement i removed1: Pr(F0,−i). This quantity is com-puted for the non-deteriorated structure, i.e. all mem-bers other than i are intact.

To understand the redundancy, Pr(F0,−i) shouldbe compared to the intact reliability. This can beachieved with the Single Element Importance (SEI)measure, with Pr(F0) being the probability of fail-ure of the intact structure (Straub and Der Kiureghian2011):

SEIi = Pr(F0,−i)−Pr(F0) (22)

For the example system, the resulting SEI are givenin Table 2. Element 3 is the one with the highest SEI,

1Note that this is not identical to the conditional probabilityof structural failure given failure of element i. In computing thisconditional failure probability, one has to consider the stochasticdependence among the elements, which is not included in thedefinition used here.

Table 2: Element importance in the example structure.

Element i 1 2 3 4 5Pr(F0,−i) 0.010 0.009 0.128 0.051 0.010Pr(F0,−i)/Pr(F0) 10 9.5 129 51 10SEIi 0.009 0.008 0.127 0.050 0.009

hence it contributes most to the overall failure proba-bility.

To address the effect of element failure on the sys-tem reliability, the following model has been appliedfrequently:

Pr[F (t)] ≈ Pr(F0) +n∑

i=1

Pr(F0,−i) ·Pr[Fe,i(t)] (23)

Fe,i(t) is the event of fatigue failure of element i up totime t. The advantage of this system model is its sim-plicity, as it does not require to consider element in-teractions. Hence the model allows a treatment of de-terioration reliability at the element level. The modelhas e.g. been used in the reassessment of offshoresteel platforms (Moan 1999, Faber et al. 2000, Moan2005).

In (Straub and Der Kiureghian 2011) we showedthat the model of Eq. 23 can be oversimplifying instructures with significant redundancy, because it ne-glects the interactions and dependence among ele-ments. Critically, it can lead to strong overestimationof the reliability of a damaged structure.

To mitigate the problem, without losing modelingand computational advantages, we proposed to extendthe model by construction of a conceptual structure,in which the structural element is part of one or mul-tiple Daniels systems that are connected in series, asshown in Figure 7. The Daniels system is the sim-plest instance of load sharing structures (Gollwitzerand Rackwitz 1990). For each element, we estimatethe number of elements in the associated Daniels sys-tem and the total number of Daniels systems in series,based on the Pr(F0,−i) values and the reliability of theundamaged structure. The model of Figure 7 can beevaluated cheaply, hence it is suitable for determiningthe reliability requirements for each element, whichensure that the overall system has sufficient reliabil-ity.

The model was examined in (Straub and Der Ki-ureghian 2011) by considering example structureswith a target reliability of β = 4.2. When determining

the necessary element reliabilities based on the sim-ple model of Eq. 23, the resulting structural systemreliability achieved is in the range of β = 2.5− 3.8,well below the target of 4.2. With the model of Figure7, the system reliability is β = 4.0− 4.1.

4.4 Research needs

In the 1980-90s, substantial research efforts weremade towards improving structural system reliabil-ity models (an overview is provided in Melchers1999b). Since the 2000s, this has pretty much seizedto be an active field of research, with a few ex-ceptions (e.g., Lee and Song 2011). Solutions havebeen found for specific applications, e.g. as discussedabove. But generally satisfactory solutions, in partic-ular approaches that can be included in standard andcodes, are still lacking. Current design codes do notrigorously address the topic. For example, Eurocode0 is mostly based on demonstrating the reliability atthe cross-section level as in Section 4.1 above. Struc-tural redundancy is addressed only qualitatively andreliability targets are not consistent at the structuralsystem level. These considerations were explicitly ex-cluded at the time of writing the code (DIN 1981).However, the currently ongoing revision of the Eu-rocode does not improve upon this situation, due tothe lack of suitable approaches that lend themselvesto codification. For deteriorating structures this is es-pecially critical.

5 DEPENDENCE AND CORRELATION INSTRUCTURAL CONDITION ANDPERFORMANCE

Deterioration processes at different locations in astructure are mutually dependent. In particular, vari-ably and uncertain common influencing factors causesuch dependence. For example, the same material andexecution quality is encountered throughout a struc-ture; the system as a whole is subject to the sameload; and the same maintenance regime is applied forthe entire structure. Only a limited number of studieshave investigated these effects, but they show clearevidence of such dependences (e.g., Vrouwenvelder2004, Li et al. 2004, Malioka et al. 2006, Luque et al.2017).

Dependence among deterioration effects at differ-ent locations of the structure is relevant for two rea-sons: (a) In redundant systems, stochastic dependenceamong element capacities reduces the overall reliabil-ity of the structural system. This is further discussedin Section 5.2 below. (b) When managing the reliabil-ity by means of inspections, dependence signifies thatan inspection of one element provides information onthe deterioration at other elements as well (Straub andFaber 2005b). This is illustrated in Figure 8 and dis-cussed in Section 6.

10 20 30 40 500

0.005

0.01

0.015

0.02

0time [yr]

Pr[M

5(t)<1

] no inspectionsinspection of element 1inspection of elements 1- 4

Probability of deterioration failure of element 5 with:

Figure 8: Probability of deterioration failure of element 5 in theexample structure. The a-priori probability of failure is com-pared to (a) the case where inspection results from element 1 areavailable and (b) the case where inspection results are availablefrom elements 1–4. In both cases, inspections are performed in 5year intervals and it is assumed that all inspections reveal intactelements. The inspection is modeled following Section 6.1. Thisillustrates the effect of information obtained from correlated el-ements on the reliability of an un-inspected element.

5.1 Modeling dependence

Probabilistic deterioration models are developedmainly at the structural element level, resulting ina prediction of deterioration Di(t) at element i.Stochastic dependence can then be modeled by intro-ducing a correlation among the Di(t)s, or among theparameters of the models describingDi(t), such asBi

in the example of Section 2.1. The two most commonclasses of models to describe these correlations arehierarchical models and random field models.

Because a (if not the) main source for dependenceare common influencing factors, a hierarchical modelis a natural model for representing stochastic depen-dence. It is based on defining the deterioration Di(t)or its parameters by means of conditional probabil-ity distributions, which are conditioned on hyperpa-rameters α that are common to a group of elements(Maes et al. 2008). These hyperparameters can bephysical parameters (e.g. common environmental fac-tors), joint model uncertainties or simply empiricallydetermined parameters. It is also possible to introducemultiple hierarchies, as in the model of Figure 9 fromLuque et al. (2017). There are a large number of ap-plications of the hierarchical model described in theliterature (e.g. Maes 2002, Straub et al. 2009, Qin andFaber 2012).

The hierarchical model has computational advan-tages. In particular, for fixed values of the hyperpa-rameters, the Di(t)s are conditionally independent,which can facilitate computation. For this reason,the model can also be employed purely for com-putational reasons. In particular, the case of equi-correlation among the Di(t)s can be readily repre-sented by a hierarchical model with a single hyper-parameter α (Song and Kang 2009, Straub 2018).This has been exploited when modeling deterioratingstructures through Bayesian networks (e.g., Schnei-

Fleet

Vessel

Compartment

Frame

Structural element

Plates

Figure 9: Hierarchical model for corrosion in ship structures.At each level of the hierarchy, common factors are introducedto represent the dependence among elements belonging to thesame instance. At the lowest level (e.g. plates in a compartment),the model allows for a random field. The model is learned fromthickness measurements in (Luque et al. 2017).

der et al. 2015, Luque and Straub 2016). For exam-ple, the equi-correlation among the deterioration pa-rameters Bi of the example structure can be modeledby defining the Bis through conditional distributions(see Chapter 9 of Straub 2018):

FBi|Uc(b|uc) = Φ

[F−1Bi

(b)− uc√ρ

√1− ρUc

](24)

wherein Uc is the hyperparameter with standard nor-mal distribution, F−1Bi

is the inverse CDF of Bi andρUc is the equivalent correlation coefficient among thenormal-transformed Bi as used in the Nataf transfor-mation (Der Kiureghian and Liu 1986). (In the exam-ple of Section 2.1, ρB = 0.6 translates to ρUc = 0.626.)

For continuously spatially distributed deterioration,e.g. corrosion on a surface, random field models areutilized (Hergenroder and Rackwitz 1992, Ying andVrouwenvelder 2007, Stewart and Mullard 2007).The main parameter of these models, besides themarginal distributions, is the correlation length. Un-fortunately, only few studies exist that measure cor-relation lengths for relevant parameters in real struc-tures (e.g., Malioka et al. 2006).

For computational purposes, random fields must bediscretized (e.g. Li and Der Kiureghian 1993, Betzet al. 2014). The resulting models have a significantnumber of input random variables, but this can behandled by state of the art reliability methods (e.g.,Papaioannou et al. 2015, Allaix and Carbone 2016).A bigger challenge is the difficulty of lay engineersto understand the concept of random fields and theirimplications. One reason for this is the notorious dif-ficulty in graphically representing 2D or 3D randomfields, because a single plot can only show marginalinformation (e.g. a map of the mean or standard devi-ation) or one realization of the random field. While itis possible to show multiple realizations jointly, thisis often confusing to non-experts. Good solutions for

0.2 0.4 0.6 0.8 1.03

3.5

4

4.5

0

Correlation coe�cient among fatigue failure margins

Syst

em re

liabi

lity

inde

x β S

Figure 10: Reliability of a frame structure whose elements aresubject to fatigue deterioration. The system reliability dependsstrongly on the correlation among fatigue failures in the elements(from Straub and Der Kiureghian 2011).

representing spatially dependent random variables arestill sought.

While dependence caused by common factors (rep-resented by hierarchical models) should generally beincluded in the assessment, spatial variability as mod-eled by random fields can in some cases be repre-sented by an equivalent random variable. However,explicit consideration of random fields is necessaryif measurements are taken, in which case the corre-lation between the measured locations cannot be ig-nored (see Section 6). The random field must further-more be included in the analysis when the dependencehas a pronounced effect on the reliability, as discussedin the next section.

5.2 Effect of dependence on the reliability of thestructure

In redundant systems, dependence among compo-nents can (severely) reduce the system reliability(Grigoriu and Turkstra 1979, Gollwitzer and Rack-witz 1990). Since most structural systems exhibit atleast some redundancy, neglecting dependency canlead to a strong underestimation of the risk, as illus-trated in Figure 10. This has been recognized, but in-cluding the dependence in the assessment has beenhindered by the limited availability of models anddata on dependence of deterioration in the structure.

Without inspection results, spatially varying deteri-oration is often modeled by means of a homogeneousrandom field (i.e. the marginal statistics are the samethroughout the domain). In this case, a conservativeapproximation for reliability analysis is to considerthe deterioration as identical throughout the domain(i.e. model it by means of a single random variable)with the marginal distribution of the random field.In many instances, this gives sufficiently accurate re-sults. If a more accurate description is necessary, itmight be possible to find an equivalent distribution ofthe representative random variable that leads to thesame probability of failure estimate as a complete re-liability analysis with the random field. This approach

is applied in geotechnics (e.g., Griffiths et al. 2009).Finding similar approximations for non-homogenousrandom fields is challenging (Papaioannou and Straub2017). It therefore remains to be investigated how tohandle spatial variability in simplified reliability as-sessments when inspection and measurement data areavailable.

6 MAINTENANCE, INSPECTION ANDMONITORING

Maintenance, inspection and monitoring are an es-sential part of the management of deteriorating struc-tures. Maintenance regimes for technical systems canbe classified as in Figure 11. For structural systems,which are almost always safety critical, correctivemaintenance is not typically an alternative. However,for many structures, in particular smaller structures,it is common to adopt a do-nothing policy and actonly upon the indication of a damage (which predom-inantly is not a failure but an indication of a dam-age). Such an approach can be interpreted as an ad-hoc condition-based maintenance strategy.

In contrast, professional owners and operators ofinfrastructure mostly have a systematic or condition-based maintenance policy, whereby maintenance andrepairs are performed at regular intervals, combinedwith inspections. Upon indication of a potentially crit-ical damage or deterioration process, a predictive ap-proach is typically implemented, in which structuraland deterioration models are utilized to predict the de-velopment of the damage. These predictions are ide-ally made with probabilistic models. In most cases,inputs to these models are provided based on tests andinspections on the structure.

As discussed earlier, deterioration models are of-ten subject to large uncertainty. Hence a commonlyadopted strategy is to combine predictive models withan inspection and monitoring plan. In a probabilisticsetting, the data from inspection and monitoring canbe directly included in the prediction using a Bayesiananalysis, as outlined in the following subsection.

6.1 The effect of inspection and monitoring on thereliability

Arguably the most common approach to dealing withstructures for which a potential deterioration problemhas been identified is to perform inspections, possiblycombined with monitoring. The effect of inspectionon the reliability can be quantified consistently withBayesian analysis (Tang 1973, Madsen 1987, Sindeland Rackwitz 1998, Faber 2000). Recent develop-ments in Bayesian computation make the applicationto structural reliability problems rather straightfor-ward from a computational point of view (e.g., Jensenet al. 2013, Straub and Papaioannou 2015). However,the application of the method in practice is still lim-ited. Besides the general aversion of many engineers

to probabilistic methods, there are a number of addi-tional challenges. One is modeling the quality of theinspection data, i.e. the likelihood function describingthe measurement data. Difficulties here involve theunderstanding of dependence among measurement re-sults (Simoen et al. 2013, Goulet and Smith 2013), aswell as the need for a model that connects the mea-surements with the quantity of interest. For example,a crack in a concrete structure can be an indication ofa damage, but a model connecting this observation tothe structural parameters is not necessarily available.

Another challenge in Bayesian updating is the needfor a prior model. Because often no or only crude de-terioration models are available, engineers are reluc-tant to employ a prior model. However, if the inspec-tion is at all informative, then the inspection data willeventually dominate the reliability if weakly informa-tive priors are used. This is illustrated in the followingnumerical example.

Consider the example structure. To reduce theprobability of failure, it is decided to perform regu-lar inspections at a 5 year interval. If a critical degreeof damage is identified at the inspection, repair ac-tion would be initiated. The identification of a criticaldamage can be mathematically described by a prob-ability of detection (PoD) function. For this applica-tion, the PoD is

PoD(Di) = Φ(

0.5−Di

0.2

)(25)

At an inspection, all elements are checked. It is hereassumed that all inspections (i.e. at all elements at alltimes) result in no-detection of a critical damage.

The conditional probability of failure given the in-spection results can be computed with Bayesian up-dating. Here, the BUS approach is utilized to performthis updating (Straub and Papaioannou 2015, Straubet al. 2016). Figure 12 shows the resulting conditionalprobabilities of failure. Results are computed for theoriginal model of Section 2.1 and the same structurewith a modified deterioration model, mimicking a sit-uation in which there is large uncertainty on the dete-rioration model.

The results of Figure 12 nicely show that the ef-fect of the prior model is only limited and the updatedreliability is ultimately dominated by the inspectionresults. The reliability of the (pessimistic) alternativemodel is initially low because of the large uncertaintyassociated with the deterioration process. However,the inspections, which reduce the uncertainty, ensurea minimum reliability that is quite constant through-out the service life. With increasing number of inspec-tions the predicted reliability approaches that of theoriginal, more informative model. For that model, theinitial inspections have no effect because deteriora-tion is not expected to have an effect before year 20anyway.

Maintenance

corrective(wait until failure is observed)

preventive(maintain before failure occurs)

systematic(at regular time intervals)

condition-based(based on current condition)

predictive(based on predicted condition)

. upon failure replace or switch to stand-by unit

. perform maintenance at �xed times

. using inspection and/or monitoring data

. maintenance decided based on thresholds or classi�cation (diagnostics)

. using monitoring and/or inspection data

. maintenance decided based on prediction of future condition

. requires prognostic model

Figure 11: Summary of maintenance regimes (from Straub 2018).

10 20 30 40 500

0.05

0.1

0.15

0.2

0time [yr]

Pr[F

(t)]

no inspection(alt model)

with inspection

no inspection

with inspection(alt model)

Figure 12: Probability of failure of the example structure, withand without inspections. Results are computed for the originalmodel and an alternative model with increased uncertainty in thedeterioration model. This alternative model corresponds to theoriginal model with a modified standard deviation of Bi of 200instead of 50.

Figure 13: Locations (black dots) of thickness measurements inan inspection campaign on a ship structure (from Luque et al2017).

6.2 Spatially distributed inspection data

Inspection and monitoring data is typically availableat varying spatial locations. Exemplary, Figure 13shows measurement locations in a ship structure. Insome structures, data is available in a spatially dis-tributed form, e.g. for reinforcement corrosion in con-crete structures where continuous measurements ofcover depth or half-cell potentials are made (Gehlenand von Greve-Dierfeld 2010).

Bayesian updating of structural reliability with spa-tially distributed inspection data has been consideredfor some time (Hergenroder and Rackwitz 1992). By

modeling the correlation among deterioration at dif-ferent locations, the effect of inspecting one compo-nent on the condition estimate at other locations canbe computed, as illustrated in Figure 8. This effect iswell acknowledged by engineers, who make an as-sessment of the entire structure by samples taken atselected locations only, thus implying such a corre-lation. However, it is still rarely evaluated quantita-tively.

The large number of parameters arising from therandom field discretization combined with the largeamount of data arising from spatially distributed mea-surement leads to computational challenges that stillhave not been solved satisfactorily at the fundamen-tal level. However, for most practical purposes, so-lutions can be found. In some cases, analytical so-lutions are available, notably for lognormal randomfields (Straub 2011). In other cases, partial solutionscan be found by using some of the data only to updatethe marginal distributions, but not the joint distribu-tion of the random field. This is e.g. employed for ob-taining the spatially distributed estimate of corrosionshown in Figure 14 (Straub et al. 2018).

As discussed in Section 5.2 above, an exact repre-sentation of the spatial random field may not be neces-sary for the reliability analysis, and the posterior ran-dom field may be reduced to a single random variable.However, for the purpose of assessing the durabilityand planning maintenance actions, a spatially explicitrepresentation, such as shown in Figure 14, can be ofgreat value.

6.3 Planning and optimization of inspection andmonitoring actions

For many structures, inspections contribute signifi-cantly to the total life-cycle cost. Hence there is aninterest in optimizing inspection efforts, by findingthe optimal trade-off between the cost of inspectionsand the risk of failure (Figure 16). The effect of theinspection lies in the reduction of uncertainty in thestructural condition. This in turn enables an improved(condition-based or predictive) planning of repair and

14m

40m

1

0.8

0.6

0.4

0.2

0

Pr(c

orro

sion

initi

atio

n)

Figure 14: Probability of corrosion initiation (depassivation) in the reinforcement of a RC parking deck. Probabilities are evaluatedby Bayesian updating of the corrosion model with spatially distributed inspection data from half-cell potential measurements, coverdepth measurements and chloride profiles. From (Straub et al. 2018), based on data from (Gehlen and von Greve-Dierfeld 2010).

maintenance actions.In fact, reliability- or risk-based inspection (RBI)

planning has been one of the successful applica-tions of structural reliability assessment in practice.Following the Alexander J Kieland disaster in 1980(Almar-Naess et al. 1984), RBI planning has been de-veloped and implemented for offshore structures sub-ject to fatigue (Skjong 1985, Thoft-Christensen andSorensen 1987, Madsen et al. 1990, Goyet et al. 1994,Faber et al. 2005). In RBI planning, Bayesian updat-ing is utilized to quantify the effect of inspections onthe uncertainty in the structural condition and on thereliability estimate (see Figures 8 and 12).

RBI planning is a special case of a sequential deci-sion problem under uncertainty (Raıffa and Schlaifer1961, Kochenderfer 2015). When planning an inspec-tion for the next time step, the entire past history aswell as all potential future decisions and outcomesmust be considered. The corresponding decision treeis as shown in Figure 15. To facilitate practical solu-tions, RBI planning was mostly based on a heuristicsolution to the sequential decision planning, wherebyinspections are planned following simple criteria. Forexample, an element is inspected whenever its prob-ability of failure exceeds a threshold. This thresholdis then optimized to find the balance between risk andcost, as in Figure 16.

A challenge remains in the optimization of inspec-tion efforts at the system level, which has been ad-dressed explicitly only by a few publications (Strauband Faber 2005b, Papakonstantinou and Shinozuka2014, Memarzadeh and Pozzi 2016). In (Luque andStraub 2018, Bismut and Straub 2018, Schneider et al.2018) we propose an efficient framework based on adirect policy search, in analogy to the heuristic ap-proach applied in practice for optimizing inspectionsat the structural element level.

Optimization of monitoring systems is conceptu-ally simpler, as it is only necessary to compare theexpected lifetime cost with or without the monitor-ing system. This difference corresponds to the valueof information (VOI) of the monitoring system (Pozziand Der Kiureghian 2011, Straub 2014, Thons et al.

2015). However, the task also requires to quanti-tatively (probabilistically) predict future monitoringdata, as well as the diagnostics based on these data.For most monitoring systems, which provide data onmultiple parameters with high frequency, such mod-els are not (yet) available. In the meantime, pragmaticapproaches to appraise the VOI based on a mixtureof expert judgment and simple models can be utilized(e.g. Zonta et al. 2014).

7 DISCUSSION

Assessment of deteriorating and existing structures iscommonly seen as a major application area for struc-tural reliability methods (Faber 2000, Ellingwood2005). However, around 40 years after modern struc-tural reliability analysis was invented, the number ofreported practical applications of the theory to the as-sessment of deteriorating structures is still limited. Inmany countries, the assessment of deteriorating struc-tures is mostly based on simplified and conservativeengineering considerations. Among the applicationsthat have been reported, most deal with the reliabil-ity of bridge structures (e.g. Faber et al. 2003, Strausset al. 2009, Maljaars and Vrouwenvelder 2014). In ad-dition, as reported in Section 6, reliability methodshave to some degree found their place in the planningand optimization of inspections, where deterministicmethods have difficulty in making any kind of quan-titative statements.

It can be argued that the main role of reliabilityanalysis is to serve as the basis for modern codesand standards. However, crucial aspects for the as-sessments of deteriorating and existing structures arenot well developed in structural codes. As Elling-wood (2005) puts it, ”current codes of practice pro-vide little guidance for the proper evaluation of ex-isting facilities for continued service, since their fo-cus is on new construction.” One major issue is thatthe simplified treatise of structural systems in semi-probabilistic code formats is not suitable for assess-ing existing structures (Ghosn et al. 2016). It does not

Deterioration process

Inspection decision

Inspection outcomes

I 1,1

I 1,nI

Repair decision

z1,1

z1,nZ

d0,1

d0,nd

System condition

d1,1

d1,nd

R 1,1

R 1,nR

Year 1

Deterioration process

System condition

Year 2

Figure 15: Schematic decision tree for risk-based inspection planning (from Bismut and Straub 2018).

10-4 10-50

0.1

0.2

0.3

0.4

0.5

10-3

total

inspection

expe

cted

life

time

cost

[% fa

ilure

cos

t]

failure rate threshold [yr-1]

risk

repair

Figure 16: Expected lifetime cost in function of the thresholdon the failure rate. Inspections are planned just before the fail-ure rate estimate exceeds the threshold (after Straub and Faber2005).

enable a proper understanding of the effect of dete-rioration on the structural reliability, as discussed inSection 4. Furthermore, with the exception of fatigue,in many design codes deterioration is not consideredby means of explicit limit state functions. Rather, Eu-rocode requires structures to be in an as-new condi-tion over its entire lifetime, which is reasonable formost new designs, but is of little help when deteriora-tion is already present.

Current efforts to develop specific codes and stan-dards for existing structures are a step in the right di-rection (Luchinger et al. 2015). These standards willbe more closely linked to structural reliability meth-ods than the design standards, they can include the ef-fect of deterioration on the reliability and they shouldaim at better addressing the need for more realisticsystem modeling. However, as discussed in Section4, it is not actually straightforward to do so in a cod-ified format, and future research should address thisquestion. Standards for existing structures will also becloser aligned with a risk-based philosophy, thus en-abling a more optimal management of deterioratingstructures.

A main reason for the small role of reliability-basedmethods in the assessment of deteriorating structureslies in the fact that serviceability criteria often deter-mine the need for intervention. This commonly in-volves bringing the structure back to a state withoutdeterioration, and there is no need for a detailed as-sessment of the effect of deterioration on the systemreliability. Another reason is that many structures be-come obsolete before deterioration plays a major role.

Many engineers argue that current practice is doingjust fine without an increased utilization of structuralreliability analysis. In recent times, there have beenonly a limited number of failures of structures causedby deterioration. Current practice has evolved over along time period, and any aggregation of structuralfailures has triggered investigations and eventually anadjustment of the design and assessment rules. Thatevolution has ensured that structures today are reli-able.

While one can state with some degree of cer-tainty that current practice for designing and manag-ing structures for durability does indeed lead to ac-ceptable reliability, it is unclear to what degree it isoptimal. Investigations into the optimality of possiblyconservative models and assessment procedures arelacking. It is challenging to quantitatively investigatethe degree of conservatism contained in current stan-dards (Teichgraber et al. 2018). But there is a needfor research in this direction, since regulating bodiesand standardization bodies will not support changestowards more efficient assessment procedures withoutthoroughly understanding their effect on the safety ofstructures.

The good news is that the principles of structuralsystem reliability and risk-based decision making areindeed applied in engineering design and manage-ment on a daily basis. For example, designers will au-tomatically direct more attention to critical elementsof a structure or rules for inspection planing distin-guish between primary, secondary and tertiary mem-

bers. Yet all this is limited to what is intuitively un-derstood by the engineers. In more complex situa-tions, the intuition of the engineer can fail and as aresult possibly over-conservative solutions are imple-mented. I have witnessed multiple instances of struc-tural rehabilitation projects where a proper stochas-tic modeling and reliability analysis might have savedsubstantial financial resources, but where these werenot applied due to the lack of understanding or accep-tance of these methods. In most, if not all, of these ex-amples, a relatively simple model would have sufficedto assess whether or not a structure is sufficiently safe.However, to find the right simple model is challeng-ing and requires insights into the topics discussed inthis article. Up to now, the structural reliability com-munity has not been very successful in bringing thesemessages across and increased efforts are needed toachieve this.

8 CONCLUSIONS

Structural reliability assessment provides a solidfoundation for optimal management of deterioratingstructures. However, there is still a long way aheaduntil reliability-informed planning and assessment be-comes the norm rather than the exception. Researchefforts are needed in particular on an improved mod-eling of system reliability that is compatible withstandard structural assessment approaches, and on theunderstanding of the real reliability associated withcurrent conservative modeling assumptions. Opportu-nities arise from improved IT, sensor and communi-cation technology, which should be embraced to en-hance our models. Most of all, we need to work onchanging the current prescriptive approach to man-agement and assessment of structures, in order to pro-vide incentives to the structural engineering commu-nity for more realistic and optimal predictions in lieuof conservative assumptions.

9 ACKNOWLEDGMENTS

Discussions with and input from Elizabeth Bismut,Ronald Schneider, Max Teichgraber, Hyun-JoongKim, Sebastian Geyer and Iason Papaioannou duringthe preparation of this manuscript are gratefully ac-knowledged. Financial support was provided by theGerman Science Foundation (DFG) through grantsSTR 1140/3-2 and STR 1140/5-1, the Bundesanstaltfur Wasserbau (BAW) and the Deutsches Institut furBautechnik (DIBt).

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