Use of monitoring extreme data for the performance prediction of structures: General approach

Engineering Structures 30 (2008) 3644–3653

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Engineering Structures

journal homepage: www.elsevier.com/locate/engstruct

Use of monitoring extreme data for the performance prediction of structures:General approachDan M. Frangopol, Alfred Strauss ∗,1, Sunyong KimDepartment of Civil and Environmental Engineering, ATLSS Center, Lehigh University, 117 ATLSS Dr., Bethlehem, PA 18015-4729, USA

a r t i c l e i n f o

Article history:Received 17 August 2007Received in revised form5 June 2008Accepted 11 June 2008Available online 26 July 2008

Keywords:Monitoring programsStructural degradationPrediction functionsMonitored extreme dataPerformance assessmentAcceptance sampling considerationsReliability profiles

a b s t r a c t

Engineering structures are subjected to time-dependent loading and strength degradation processes. Themain purpose of both designer and owner is to keep these processes under control. Several numericalapproaches based on mechanical, physical, chemical or combined models have been recently proposedto describe time-dependent processes of engineering structures. Most of them require considerations ofboth aleatory and epistemic uncertainties. The inclusion of such uncertainties demands intensive studiesin space and time of engineering structures under environmental and mechanical stressors. Existingmechanical models for structural performance assessment can be validated by using structural healthmonitoring. The use of monitored extreme data allows (a) the reduction of uncertainties associated withnumerical models, and (b) the validation and updating of existing predictionmodels and, sometimes, thecreation of novel models. This paper presents a general approach for the development of performancefunctions based on monitored extreme data and the estimation of possible monitoring interruptionperiods. An existing bridge in Wisconsin is used as an example for the application of the proposedapproach.

© 2008 Elsevier Ltd. All rights reserved.

1. Introduction

Physical quantities of engineering structures are subjected tochanges in both time and space. Some of these changes do not af-fect the serviceability and the ultimate capacity as defined in struc-tural specification [1,2], but others can have serious impacts onthe remaining life of an existing structure [5,22,24]. The uncertain-ties associated with structural degradation processes, their asso-ciated descriptive parameters, and the limitation of financial re-sources for maintenance, repair, and replacement require (a) thedevelopment of sophisticatedmodels for the reliability assessmentof structures [8,9,21], (b) the probabilistic description of degrada-tion processes [6,7], and (c) the development of lifetime predic-tion functions and cost optimized intervention planning strategies[11,15,20]. However, these models require the current structuralconditions obtained by inspection methods. Inspections are usu-ally visual and are discrete in space and time. The outcomes ofvisual inspections only allow a subjective assessment of the per-formance of a structure [13]. Novel monitoring techniques, suchas instrumented-based non-destructive inspection methods [18],

∗ Corresponding author. Tel.: +1 431476545254; fax: +1 431476545299.E-mail addresses: [email protected] (D.M. Frangopol),

[email protected] (A. Strauss), [email protected] (S. Kim).1 On leave fromDepartment of Civil Engineering and Natural Hazards, Universityof Natural Resources and Applied Life Sciences, Vienna, A-1190, Austria.

0141-0296/$ – see front matter© 2008 Elsevier Ltd. All rights reserved.doi:10.1016/j.engstruct.2008.06.010

have become more attractive during the past decade. These tech-niques are based on the measurement of physical quantities inspace and time and they provide, consequently, an objective andmore realistic assessment of structural performance.Sensors of monitoring systems only provide physical quanti-

ties at specified locations, but their continuous combination inspace and time allows the assessment of the space- and time-dependent system performance. Therefore, data obtained bymon-itoring systems have to serve as reliable information for the eval-uation of structures. These data have also to serve in reliability as-sessment and in optimized intervention planning of maintenanceactions. There are apparently two purposes to obtain monitoredextreme data: (a) for the assessment of structures and the devel-opment of prediction functions from data sets collected at differ-ent times, and (b) for the identification of parameters characteriz-ing random variables and processes of the structural models. Theapplication of monitoring systems on structures is associated withvarious cost types such as costs for the first installation, costs forthe maintenance and renewal of the systems, and costs for thestaff operating the systems. These costs have a significant effecton cost optimization for intervention planning [13,20]. Monitoringsystems are cost effective if they provide a reduction in the totallife-cycle cost of a structural system (i.e., the expected life cyclecost of the structure including the monitored system has to be lessthan the expected life-cycle cost of the structure without monitor-ing). This paper focuses on a general approach for the performance

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D.M. Frangopol et al. / Engineering Structures 30 (2008) 3644–3653 3645

Fig. 1. Monitored physical quantity during three time periods.

prediction of structures. The purpose of this paper is the presenta-tion of (a) a procedure for the establishment of prediction functionsusing monitored extreme data, and (b) a concept for the definitionof necessarymonitoring periods and possible interruption periods.The proposed approach is applied to an existing I-39 NorthboundBridge over the Wisconsin River, Wisconsin, USA. The companionpaper [25], using the same bridge as an example, is concernedwithincorporation of Bayesian updating in the performance predictionof structures.

2. Processing monitored data for prediction functions

Monitoring systems applied to engineering structures provide,in general, an immense amount of recorded data. The data amountdepends mainly on the monitoring frequency, data acquisitionsystems, and the number of sensors [14]. The factors affectingthe monitoring frequency and the number of sensors are (a)the physical quantities of the structure to be investigated andrecorded, (b) the expected variation of the physical quantities intime and space, (c) the planned monitoring period, and (d) thebudgetary constraints. Monitored extreme data contain differenttypes of information, and, therefore, allow the evaluation of variousstructural characteristics (e.g., yield strength, fatigue strength,deformation). Since monitored extreme data are variable in timeand space, the interest in the development of prediction functionsfp for structural characteristics is on extreme values. The proposedprediction functions focus on the development of structural strainproperties at a particular location in the structure, such as thestrain in a mechanical high loaded (e.g., by dead weight, liveload) steel detail. The proposed prediction function fp relates pastand present extreme strain or stress quantities Ep, derived fromsensor measured voltages, to future strain or stress behavior at aparticular considered location. There is no strict requirement touse all monitored extreme data. Only extreme values Ep,i collectedover defined monitoring time periods Pi, as shown in Fig. 1, areof interest. The grouping of extreme values in sample sets withinPi (e.g., days) allows an approximated location of fp with respectto a threshold fT of the considered physical quantity (e.g., yieldstrength), see Fig. 1.The central and variability tendencies of those sample sets

can be taken into account by chart methods as proposed byLevine et al. [16]. These methods indicate if the dispersion ofthe consecutive sample sets is caused by chance or by otherprocesses, or in other terms, if the monitored processes are stableor unstable, respectively. A stable process is a process that onlycontains common cause variation. Common cause variation is

Fig. 2. Updating of a prediction function fp for a period P (ti) based on monitoredextreme data.

that which is normal to the process and doesn’t change overtime. Since in engineering there are many environmental factorsinfluencing the monitored extreme data (e.g., temperature, solarradiation, and traffic), this check will yield in most cases unstableprocesses. Therefore, the single application of chart methods, ingeneral used for production control, for the derivation of predictionfunctions, is not efficient. Extreme values Ep,i per monitoring timeperiod Pi of the sample sets, as shown in Fig. 1, provide morepowerful information for the computation of prediction functions.There are numerous prediction functions for the description ofstructural degradation processes that do not take into account theinformation available frommonitored extreme data. Most of thesefunctions are based on advanced analytical formulations [23,27].However, the use of monitored extreme data is necessary for amore accurate estimation of prediction functions fp. Polynomialapproaches of first, second, or higher order can be generally usedfor defining prediction functions, such as

fP =w∑k=0

ak · tk w = 1, 2, 3 (1)

where ak = coefficients, w = order of the polynomial function,and t = time.The coefficients ak can be obtained by using three successive

steps as follows:First step: Finding the necessary overall monitoring period P (ti)

associated with the prediction function fp. The duration of thenecessary overall monitoring period P (ti) (see Figs. 1 and 2) canbe computed based on an accepted probability p of monitoredextreme data to overcross the prediction function fp (i.e., f > fp)per time interval Pi and the confidence level C associated with thisprobability.Therefore, it is necessary to define in advance the probability p

and the confidence level C = 1−λ, where λ is the probability thatthe monitored extreme data indicating f > fp is actually not largerthan fp (i.e., f ≤ fp) (see Fig. 3). The overall period P (ti) is divided inequal time intervals, e.g., monitoring time periods Pi−1 = Pi = Pi+1(see Figs. 1 and 2). The definition of the duration of these intervalsdepends on the monitoring frequency, the characteristics of therecorded data, the mathematical formulation of the predictionfunction, and the duration of the overall monitoring period.The required magnitude of P (ti) can be computed by an

acceptance sampling approach as proposed in [3]:√

P (ti) ·[Φ−1 (p)− Φ−1 (m)

]= Φ−1 (1− λ) (2)

or, alternatively√

P (ti) ·[Φ−1 (q)− Φ−1 (m)

]= Φ−1 (κ) (3)

3646 D.M. Frangopol et al. / Engineering Structures 30 (2008) 3644–3653

Fig. 3. Probability λ that a monitored sample indicating violating a predictionfunction does not violate it, and probability κ that a monitored sample indicatingnon-violating of a prediction function does violate it.

where Φ−1(.) = the inverse of the standard normal cumulativedistribution function, κ (see Fig. 3) is the probability that themonitored extreme data indicating f < fp is actually not less thanfp (i.e., f ≥ fp),

q = 1− p, (4)

m = SP/P (ti) = acceptable fraction of violations during P (ti),and SP represents the number of allowable violating samples SP(i.e., f > fp) within the overall monitoring period. Eq. (4) can berearranged as follows:

√

P (ti) =Φ−1 (1− λ)

Φ−1 (p)− Φ−1(Sp/P (ti)

) . (5a)

For example, an accepted daily probability p = 0.10, λ = 0.05(i.e., C = 0.95), and Sp = 1 results in√P (ti)o =

Φ−1 (0.95)

Φ−1 (0.10)− Φ−1(1/P (ti)o

) . (5b)

Therefore, if the accepted probability p = 0.10 is during 5 daysinstead of one, the required monitoring period is P (ti) = 20.135×5 = 100.675 days instead of 20.135 days. Finally, if during 5 days,p = 0.05 instead of 0.10, the required monitoring period will be36.284×5 = 181.42 days.Second step: Finding the prediction function fp. Once the overall

monitoring period P (ti) is computed, the coefficients a′k of Eq. (1)can be obtained. First a mean square fitting of Eq. (1) [19] to themonitored extreme data of the monitoring periods Pi providesthe coefficients ak (see Fig. 2.). These coefficients applied to Eq.(1) represent the tendency of the monitored extreme data Ep,i, asfollows:

fp = f ′p =w∑k=0

a′k · tk, w = 1, 2, 3 (6a)

To match the previously defined criterion (i.e., Sp = 1) for thecomputation of the overall monitoring period P (ti) the predictionfunction fp must be moved (see Fig. 2). This updating is carried outvia a new set of coefficients as follows:

fP = f ′′P =w∑k=0

a′′k · tk, w = 1, 2, 3. (6b)

Eq. (6b) represents a translation of the prediction function Eq. (6a)towards the threshold fT of the investigated physical quantity (seeFigs. 1 and 2).Third step: Restriction of themagnitude of the violating extreme

data. The above defined procedure for the location of fp does notrestrict the magnitude ζ of the violating extreme values Ep,i(seeFig. 2). The constraint on ζ within each Pi can be performed byusing chart methods [16].

Fig. 4. Assessment of ζ based on the process1f (ti)p of the monitoring period P (ti) .

These methods provide the possibility to adjust the predictionfunction again (if necessary) with respect to the magnitudes ζ ,so that the differences between Ep,i and fp can be considered ascaused by chance (i.e., stable process). The approach requires thecomputation of the differences between the prediction functionsfp (see Eq. (6b)) and the monitored extreme data Ep indicated by fm(see Figs. 2 and 4) as follows:

1fp = fp − fm for fp ≥ 0 (7a)

or

1fp = fm − fp for fp < 0. (7b)

The 1fp process within the periods Pi serves for the assessmentof the magnitudes ζ . Two kinds of charts are essential for thisassessment: (a) the range chart R, which captures the variability ofa process, and (b) the mean chart X , which captures the tendencyof a process. The center line of the range chart can be located as

CLRange = R =

(k∑i=1

Ri

)/k (8)

where Ri = individual ranges of the computed 1fp,i and k =number of Pi within P (ti). The range is defined as the differencebetween the largest and smallest extreme values in the set Pi. Theassociated upper and lower bounds are given as:

UCLRange = R+ 3sR (9)

LCLRange = R− 3sR (10)

where sR = the standard deviation of R, defined as

sR = (d3/R)/d2 (11)

d3 and d2 are control chart factors documented in [16]. A stableprocess of the variability of the monitored extreme values Ep isgiven if the ranges of the samples 1fp,i are within the boundsUCLRange and LCLRange. Fig. 4 shows the center line, CLMean, of themean chart resulting from the monitored extreme data of theperiod P (ti). The center line indicates the tendency of a process andcan be generated according to the mean chart X method as follows

CLMean = X =

(k∑i=1

Xi

)/k (12)

where Xi = the value of1fp,i during Pi according to Eqs. (7a) and(7b). The associated upper and lower bounds of the center line aregiven by

UCLMean = X + 3sX (13)


Fig. 5. Updating of prediction functions f (ti−1, ti)p using polynomial functions of firstorder.

and

LCLMean = X − 3sX (14)

with

sX = X/(d2 · n1/2) (15)

where d2 and n are control chart factors documented in Levine et al.[16]. A stable process in the tendency of the monitored extremedata Ep is obtained, if the samples1fp,i are within the UCLMean andLCLMean bounds. The ζ value violating the prediction function fp,according to Eq. (7), indicates negative values in Fig. 4, since theadjusted prediction function represents the zero-line. Therefore,the stability of a process in the tendency of the ζ values associatedwith themonitoring periods P (ti) can be checked bymoving CL andLCL lines to the new positions CL′ and LCL′, respectively, (see Fig. 4).An alternativemethod for the validation of a stable tendency of theζ values is the check of their location in the 3σ distance from theCL line, where σ = standard deviation of the monitored extremedata of the considered monitoring period, as proposed by Levineet al. [16] (see Fig. 4). In case of an unstable process, indicatedby the magnitude ζ violating the limit defined by the LCL′ line,or violating the 3σ distance, the coefficients a′′k of the predictionfunction f (ti)p , have to be updated again. This updating finally yieldsthe prediction function for the period P (ti) as follows

fp = f (ti)p =w∑k=0

a(ti)k · tk, w = 1, 2, 3. (16)

In the case of a stable process, the magnitude ζ is within thedefined limits, and no further updating is necessary.The definition and updating of prediction functions f (ti)p as

described previously are based on single monitored periods P (ti).Monitoring is a continuous process allowing access to the pastand the current structural performance assessment. Therefore, toaccount for these aspects, the updating of the prediction functionsmust be extended. For instance, a polynomial function of firstorder will require at least two monitoring periods to determine aprediction function f (ti−1,ti)p using the past, ti−1, and the current, ti,monitoring information. Therefore, the previous described processfor the computation of ak should be based on a first orderpolynomial f (ti−1,ti)p spanning at least two periods P (ti−1) and P (ti),see Fig. 5.Important questions associated with monitoring programs are

about the necessary frequency and duration of data acquisition.Optimizing the frequency and duration of monitoring is a crucialstep in reducing the overall cost of structural monitoring. Theduration and the interruption of monitoring can be related to the

reliability index profile β(t). This reliability index is associatedwith the difference fT−f

(ti)p , where fT = time-dependent threshold

of the investigated physical quantity, and f (ti)p = predictionfunction, as indicated in Fig. 2. The reliability index profile β(t)and the target reliability index β0 can be used for computingthe necessary duration of monitoring periods. This approach isbased on the number of violations, β(t) < β0, per monitoringunit Pi which are defined by the ratio α/n, where α =

number of violations during the monitoring period and n =number of unit time intervals included in the monitoring period.Assuming that the number of violations α over the monitoringperiods P (ti) (i = 1, 2, . . . , n) is a Poisson process, the time T1until the occurrence of a new violation can be expressed by anexponential distribution. Therefore, the probability of no violationuntil t (e.g., associated with P (ti+1)) is given by

P(T1 > t) = P(X = 0) = e−νt (17)

where ν = α/n, T1 = recurrence time (i.e., time betweentwo consecutive violations). Therefore, the probability distributionfunction of T1 is given as

FT1(t) = P(T1 ≤ t) = 1− e−ν·t . (18)

Assuming that during the periods P (1) to P (ti), there have been αviolations (i.e., β < β0), the probability that violations will occurin the next time period P (ti+1) can be easily computed. Hence, thepossible monitoring interruption period PI can be based on theprobability of a violation of a target reliability indexwithin the nexttime period P (ti+1).

3. Application to the I-39 Northbound Bridge

The I-39 Northbound Bridge over the Wisconsin River wasbuilt in 1961 in Wausau, Wisconsin, USA. The bridge carriesthe northbound traffic of the interstate I-39, see Fig. 6. It is afive span continuous steel plate girder bridge. The alignments ofthe horizontal curved girders are symmetric with respect to thecenter-line of the third span. The spans of the bridge are indicatedin Fig. 7. The total length of the bridge is 188.81 m (619.45 ft).Details of this bridge are provided in [17].

3.1. Monitoring program

The monitoring program for this bridge included the assess-ment of the strain of specified structural components and, for theentire structure, a controlled testing and long-term assessment.Strain gages as well as linear variable differential magnetic basedtransformers (LVDTs) were used for the monitoring program [17].The data acquisition throughout the controlled testing and long-term monitoring had been performed by a Campbell ScientificCR9000 Data Logger. More details about the aim and results of themonitoring program are given in [17].In the following, the proposed approach is applied to the

monitored data of the sensor CH15. This sensor was mounted onthe bottom flange of the Northbound Bridge girder (see Fig. 7). Thestrain gage was installed in span two (first lateral field) at P.P.9 onthe girder G4, as shown in Fig. 7. The sensor was located at thisposition, since the stress concentrations and crack initiations dueto the welded flange cover plates associated with field splices aresignificant at that detail [17].


Fig. 6. Map view of the I-39 North Bound Bridge (adapted from Google Map 2007,March 28. 2007).

3.2. Prediction function for the monitoring period P (1)

The duration of the long term monitoring for the I-39Northbound Bridge was 97 days [17]. The chart method [16] wasapplied to the entire monitored data of the sensor CH15. Themethod indicated an unstable process of the monitored data, sincemonitored data violate the upper mean center line (UCL) as well asthe lower mean center line (LCL), see Fig. 8, associated with thecomputed mean center line (CL). Prediction functions fp providea more robust way to describe the variability and tendency ofmonitored data.

Fig. 8. Monitored extreme values of the sensor CH15 and the associated centerlines obtained by a chart method.

In order to obtain prediction functions fp it is first necessary tocompute the duration of monitoring P . An accepted probability ofviolation p = 0.10 per day, with a confidence level C = 0.975,based on a single violating sample Sp = 1 yields according toEq. (5a) to a monitoring period P (1) = 22.3 days, see Fig. 9. Theupdating of the first-order prediction function f (1)p to the extremevalues of the monitoring period P (1) results, according to Eq. (6a),in the coefficients a′0 = 24.347 and a′1 = 0.1042. The linearfunction a′0 + a

′

1 · t divides the monitored extreme data in twogroups: violating values Sp and non-violating values Ep,i. Finally,using a′′0 = 32.5 satisfies the constraint of only one violatingsample Sp = 1 within the monitoring period P (1). The adjustedprediction function f (1)p is located by this procedure between theviolating values Sp and the others, see Fig. 9. Nevertheless, thisprocedure allows numerous solutions for the location of f (1)p . The

Fig. 7. I-39 North Bound Bridge, Instrumentation plan of the strain gage, CH15 (adapted from [17]).


Fig. 9. Prediction function f (1)p of the monitoring period P (1) extracted from themonitored extreme values.

Fig. 10. Process1f (1)p for the assessment of the extreme value E(1)p according to thechart method of the monitoring period P (1) .

number of solutions can be reduced by the previously discussedchart method applied to the computed differences 1f (1)p (see Eqs.(7a) and (7b)) as shown in Fig. 10. The magnitude of the stressζ associated with the first monitoring period P (1) violates boththe LCL(1) and LCL′(1) lines. Since the first monitoring period isin general influenced by uncertainties in the treatment of data,the limitation for the first period has been extended to the 3σrestriction, as shown in Fig. 10, as the line CL(1) − 3σ . Since thisconstraint is not violated the process is stable. The predictionfunction fp and the threshold of the investigated physical quantityfT are required for the computation of the reliability index βp ateach point in time. The extreme samples Sp are located in theregion between fp and fT . It is appropriate to divide the domainof the monitored physical quantity with respect to the predictionfunction fp in two parts: (a) the region where the extreme valuesare not violating the prediction function, fm ≤ fp, and (b) the regionwhere the extreme values are violating the prediction function,fm > fp (see Figs. 9–11).

3.3. Prediction functions for successive monitoring periods

The prediction function f (1)p was adjusted to the monitoredextreme data of the first monitoring period P (1). In order totake into account the past monitored information, the predictionfunction f (1,2)p has to be based at least on the monitoring periodsP (1) and P (2), and the next prediction function f (2,3)p on the periodsP (2) and P (3), see Figs. 11(a)–(e). Therefore, the updating of thecoefficients a0 and a1 has been performed by considering themonitored extreme data of the two associated monitored periods.Figs. 11(c) and (d) show the previously defined steps for f (2,3)p asfollows: (a) fitting the polynomial first order to the monitoredextreme data of P (2) and P (3), (b) shifting the polynomial towardsthe monitored extreme values Sp of P (2) and P (3) by updating a′0,and (c) verifying the magnitude ζ associated with1f (2,3) by usingthe chart method, (see Fig. 11(d)). It has to be noted that due to thesmall differences between f (1,2)p and f (2,3)p the prediction functionf (2,3)p is replaced by f (1,2)p . Fig. 11(e) shows the prediction functionsof the monitored periods P (ti) associated to the sensor CH15 forthe whole monitoring program of the I-39 Northbound Bridge. Thefunctions represent the development of the observed stress field.

4. Reliability profile associated with yield strength

The monitored sensor data and the design data of the I-39 Northbound Bridge provide the basis for the probabilisticassessment with respect to steel yielding. The assessment isstrongly influenced by the steel grade. The steel used in the girdersof the I-39 Northbound Bridge is M270 Grade 50 W. The nominalyield strength of this steel is 345 MPa (50 ksi). The probabilisticanalysis requires the mean value and standard deviation of allrandom variables. For the above mentioned steel, there havealready been performed extensive examinations of probabilisticmodels for the yield strength, the tensile strength, and theircorrelation [26]. The probabilistic descriptors of the yield strengthfor the steel girder fy of the I-39 Northbound Bridge are derivedfrom these investigations which yield to a mean value of 380 MPa(55.11 ksi) and a standard deviation of 26.6 MPa (3.86 ksi). Theseprobabilistic descriptors serve in the following as mean µR andstandard deviation σR of the steel yield strength.The long term monitored data displays the variability of the

stresses caused by traffic, temperature, shrinkage, creep andstructural changes. The stresses from the dead weight of the steelstructure and the concrete deck are not included in the measureddata. Therefore, the computation of the reliability index profileβp associated with the prediction function fp has to be based onadditional information. The reliability index βp associatedwith themonitored data of the sensor CH15, can be computed as follows

βp =µR − µS − µC − γp × µp√σ 2R + σ

2S + σ

2C +

(γp × σp

)2 (19)

where µp, σp = mean and standard deviation of the stressassociated with the prediction function fp, respectively; µS , σS =mean and standard deviation of the stress caused by the deadweight of steel, respectively; µC , σC = mean and standarddeviation of the stress caused by the dead weight of concrete,respectively; and γp is a factor assigned to the data provided bysensors.The stresses associatedwith sensor CH15 are not themaximum

stresses representative for the yield strength assessment. Fig. 7shows that the sensor CH15 is located out of themiddle (maximumstress domain) of the girder in the second lateral span. Severalsimulations according to different load combinations showed that


Fig. 11. (a) Prediction functions f (1)p and f (1,2)p of the monitoring periods P (1) and P (2) extracted from monitored extreme values, (b) Processes 1f (1)p to 1f (1,2)p for theassessment of the extreme values E(1)p and E

(2)p according the chartmethod of themonitoring periods P (1) to P (2) , (c) Prediction functions f

(1)p , f

(1,2)p and f (2,3)p of themonitoring

periods P (1) to P (3) extracted frommonitored extreme values, (d) Processes1f (1)p to1f (2,3)p for the assessment of the extreme values E(1)p to E(3)p according the chart method

of the monitoring periods P (1) to P (3) , and (e) Prediction functions f (1)p , f(1,2)p to f (3,4)p of the monitoring periods P (1) to P (4) extracted from monitored extreme values.

the traffic load located in the second and fourth span producesthe maximum stress in the bottom of the steel girder. Thefactor γp assigned to the measured sensor data and the stressesto be expected in the middle of the second lateral field, as

derived from the numerical simulations, is 1.15. This factor allowsthe computation of the minimum reliability index. The designspecifications µR = 380 MPa and σR = 380 × 0.07 = 26.6 MPa,µS = 116.3 MPa and σS = 116.3 × 0.04 = 4.65 MPa, and


µC = 108.8 MPa and σC = 108.8 × 0.04 = 4.35 MPa of thestresses of the structural dead weight yield to

βp =380− 116.3− 108.8− 1.15× µp√26.62 + 4.652 + 4.352 +

(1.15× σp

)2=

155− 1.15× µp√27.3512 +

(1.15× σp

)2 (20)

assigned to bottom of the girder in themiddle of the second lateralspan. The previously adjusted prediction functions f (1)p to f (3,4)p(see Figs. 11(a)–(e)) of the monitoring periods 1 to 4, respectively,are necessary for the determination of µp and σp. Therefore, thereliability indices relate to the sensor measurements. They enablethe computation of the βp profile as shown in Fig. 12(a) accordingto Eq. (20). Theβp profile serves for the assessment of themeasuredphysical quantity in time and can also be used as reliabilityprediction function for a defined time horizon. Fig. 12(a) forinstance, shows the β(1)p profile, based on the monitored extremedata obtained from the first 22.3 days. This profile indicates aviolation after 57 days of βp = 4.0 (see Fig. 12(b)). This means,that a continuously operation of the structure with a constraint ofβp > 4.0 requires at least one maintenance action within the first57 days.The β(0)p profile, shown in Fig. 12, is generated by using only

the maximum extreme values for the determination of µp andσp in the monitoring period P (1). Since no other monitoring dataare considered to update the function, the slope of this profileis constant, The β(0)p = 3.95 profile can be treated as firstapproximation of a prediction without monitoring information.Figs. 12(b)–(e) show the βp profiles, according to the monitoredextreme values of the four periods P (1) to P (4). It is obvious thatthese βp profiles are associated with a higher safety margin thanthe β(0)p profile. This insight allows the definition of miscellaneousupdating processes according to the prediction functions suchas: (a) updating the βp profile in each monitoring period (seeFig. 12(a), continuously updated profile), or (b) updating only insome monitoring periods (see Fig. 12(a), e.g., updating in P (1) andP (4), discrete updated profile).

5. Monitoring interruption

The βp profiles, shown in Fig. 12 provide the possibility tocomputemonitoring interruption periods. It is assumed (a) that thecomputed βp profile violates the level β = 4.2 twice during 22.3days (i.e., monitoring period P (1)), and (b) that the occurrence ofthese events follows a Poisson process. Therefore, the question ofinterest is: what is the probability that such a violationwill reoccurwithin a given time period of 22.3 days (i.e., P (2))? This questioncan be answered by using Eq. (17). The number of violations α = 2(e.g., 2 days violating the level β = 4.2) within the monitoringperiod P (1) = 22.3 days yield to ν = 2/22.3 = 0.0897. Theprobability that no violations occur in the next 22.3 days (i.e., from22.3 to 44.6 days) is given by P(T1 > t) = e−(νt) = 0.1353.Fig. 13(a) shows the probability of recurrence of a violation forthree different cases: (A) considering two violations in 22.3 days,(B) considering two violations in 44.6 days, and (C) considering twoviolations in 66.9 days. The probability of recurrence is increasingwith time t . The decision to interrupt or reduce the durationof a monitoring program can be based on the probability ofrecurrence of a violation with respect to a defined reliability level.In addition, case (B) of Fig. 13(a) indicates at the end of period P (3)a probability of recurrence of 0.64 for two times (α = 2) violatingthe threshold β = 4.2. This information serves to decide on a

Table 1Probability of recurrence of violating the reliability index threshold β = 4.2 after22.3 days, for α = 1, α = 2 and α = 4 during 22.3 days

Days Number of violations during 22.3 days1 2 4Probability of recurrence

22.3 0.000 0.000 0.00027.3 0.201 0.361 0.59232.3 0.361 0.592 0.83437.3 0.490 0.740 0.93242.3 0.592 0.834 0.97247.3 0.674 0.894 0.98952.3 0.740 0.932 0.99557.3 0.792 0.957 0.99862.3 0.834 0.972 0.99967.3 0.867 0.982 1.00072.3 0.894 0.989 1.00077.3 0.915 0.993 1.00082.3 0.932 0.995 1.00087.3 0.946 0.997 1.00092.3 0.957 0.998 1.00097.0 0.965 0.999 1.000


Days Number of violations during 44.6 days1 2 4Probability of recurrence

44.6 0.000 0.000 0.00049.6 0.106 0.201 0.36154.6 0.201 0.361 0.59259.6 0.286 0.490 0.74064.6 0.361 0.592 0.83469.6 0.429 0.674 0.89474.6 0.490 0.740 0.93279.6 0.544 0.792 0.95784.6 0.592 0.834 0.97289.6 0.635 0.867 0.98294.6 0.674 0.894 0.98997.0 0.695 0.915 0.993

possible interruption or a reducedmonitoring period. For instance,by accepting a recurrence probability = 0.64 monitoring can beinterrupted or reduced in the period P (3), which starts at 44.6 daysand ends at 66.9 days.Fig. 13(a) also shows for two violations during 22.3 days (i.e.,

α = 2) that the recurrence time associated with an acceptedprobability of recurrence of 0.50 is for monitored case (A) 7.7 days,(B) 15.4 days, and (C) 23 days. If the number of violations remainsthe same during an increased time interval, the increase in theprobability of recurrence in a successive time interval is less thanin the previous one. Fig. 13(b) presents the dependency of theprobability of recurrence on the number of violations α and thetime of their occurrence. The cases A, B and C (see Tables 1–3) display the increase of the probability of recurrence withincreasing number of violations α for the threshold β = 4.2. Itcan be concluded, that the probability of recurrence, as presentedin Fig. 13, is a basic decision tool for the interruption of monitoringprocesses.

6. Conclusions

The objectives of monitoring programs for structures are (a)detecting possible structural degradation, (b) detecting potentialfailure occurrence, and (c) providing a decision support tool foroptimum maintenance planning. Monitoring programs have thecapability of providing informed decisions on life extension ofstructures or structural components and the capability to detect


Fig. 12. Reliability profile (a) final profile, (b) based on the prediction function f (1)p , (c) based on the prediction functions f(1)p and f (1,2)p , (d) based on the prediction functions

f (1)p to f (2,3)p , and (e) based on the prediction functions f (1)p to f (3,4)p .


Days Number of violations during 66.9 days

1 2 4

Probability of recurrence

66.9 0.000 0.000 0.00071.9 0.072 0.139 0.25876.9 0.139 0.258 0.45081.9 0.201 0.361 0.59286.9 0.258 0.450 0.69891.9 0.312 0.526 0.77697.0 0.361 0.592 0.834

unexpected damages and, therefore, to reduce the probability ofstructural failures. The use of monitored extreme data also allows(a) the reduction of uncertainties associated with numerical mod-els, (b) the validation and updating of existing prediction mod-els, and sometimes, the creation of novel prediction models. Thispaper presented a general approach for the development of predic-tion functions and a procedure for the performance assessment ofstructures based on monitored extreme data. The updating of pre-diction functions is based on mean square fitting to extreme dataassigned to monitoring periods, while the necessary monitoringperiods are computed from acceptance sampling considerations.The proposed performance prediction functions based onmonitor-ing extreme data provide the following benefits:


Fig. 13. Probability of recurrence of violating the reliability index threshold β = 4.2 for (a) α = 2, and for (b) α = 1, α = 2 and α = 4.

• instantaneous inclusion of environmental and degradationprocesses in the structural reliability assessment;• reduction of time in the processing of monitored data of theobserved physical quantity;• flexible updating of performance functions associated withthe reliability index or to any performance indicator by usingacceptance criteria applied to monitored extreme data; and• optimum intervention planning based on updated performanceprofiles.

In addition to these benefits, it must be noted that monitoringsystems involve costs similar to costs for intervention andmaintenance of structures. A cost optimum lifetime planningof structures requires the efficient application of monitoring inspace and in time [12]. Therefore, there is a huge interest inthe definition of possible monitoring interruption periods. Theproposed procedure for finding prediction functions provides thebasic tool for the definition ofmonitoring interruption periods. Theapproach discussed in this paper provides an efficient support toolto both designer and owner for the optimum lifetime planningof deteriorating structural systems. This approach can also usesystem reliability [10] and Bayesian updating techniques [4,25].

Acknowledgments

The support by grants from the Commonwealth of Penn-sylvania, Department of Community and Economic Develop-ment, through the Pennsylvania Infrastructure Technology Al-liance (PITA) is gratefully acknowledged. Also, the support of theNational Science Foundation through grants CMS-0638728 andCMS-0639428 to Lehigh University is gratefully acknowledged. Fi-nally, the writers gratefully acknowledge the support from theFederal Highway Administration under cooperative agreementDTFH61-07-H-0040. The writers want to express their profoundthanks to Mr. Ian Hodgson, Lehigh University, and thank him forhis constructive comments and suggestions related to monitoringof bridges. The opinions and conclusions presented in this paperare those of the authors and do not necessarily reflect the views ofthe sponsoring organizations.

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Use of monitoring extreme data for the performance prediction of structures: General approach