Research ArticlePredicting Future Deterioration of HydraulicSteel Structures with Markov Chain andMultivariate Samples of Statistical Distributions

Guillermo A Riveros and Elias Arredondo

US Army Engineer Research and Development Center Vicksburg MS 39180-6199 USA

Correspondence should be addressed to Guillermo A Riveros guillermoariverosusacearmymil

Received 30 January 2014 Revised 7 April 2014 Accepted 13 April 2014 Published 6 May 2014

Academic Editor Mohamad Alwash

Copyright copy 2014 G A Riveros and E Arredondo This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

Combined effects of several complex phenomena cause the deterioration of elements of steel hydraulic structures on the nationrsquoslock systems loss of protective systems corrosion cracking and fatigue impacts and overloads This paper presents examplesof deterioration of steel hydraulic structures A method for predicting future deterioration based on current conditions is alsopresented This paper also includes a procedure for developing deterioration curves when condition state data is available

1 Introduction

In the absence of a mechanistic-based deterioration modelthat requires quantitative contribution of these complexphenomena based on environmental effects andmaintenanceconstraints steel hydraulic structurersquos (SHS) inspection datacan be used to determine the need for rehabilitation orreplacement and prioritize the order of work and fundingThis can be accomplished by the use of deterioration models[1ndash4]

Information on current and future conditions of naviga-tion or flood-control SHS is essential for maintenance andrehabilitation of navigation infrastructure Current condi-tions of navigation infrastructure are measured by periodicand detailed inspections following recommendations from[5ndash7]

The accuracy of these conditions depends on the type ofinspection performed On occasions detailed inspections areconducted when the operators perceive a problem In somecases the deterioration of the SHS has been found to becritical and emergency repairs and contingencies have beenconducted This reactive approach will usually incur morecostsThese emergency repairs are avoidable with a proactiveapproach (eg a deteriorationmodel) and used to predict thefuture condition of the structureThe prediction will indicate

when the structure will fall below a satisfactory performancelevel and when its condition may become severe if thestructure is not maintained properly Accurate predictionsof the condition of the structure in the future are essentialto maintain the inventory on a safe and reliable level ofperformance

Methods for predicting infrastructure deterioration canbe categorized into deterministic- and probabilistic-basedmodels Deterministic-based models are those in whichno randomness is involved in the development of futuredeterioration states of the system These models calculatethe condition of the system as a precise value based onmathematical formulations of the actual deterioration [8]Probabilistic-based models judge the deterioration states ofthe system as random variables and they are modeled byunderlying probability distributions [9]

2 Deterioration Examples of Steel HydraulicStructures

The following examples illustrate the potential results ofcasual inspections combined with inattention to deteriora-tion of different components of SHS

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2014 Article ID 360532 8 pageshttpdxdoiorg1011552014360532

2 Journal of Applied Mathematics

Figure 1 Corrosion on a miter gate

Figure 2 Corrosion inside a lock miter gate compartment

Figures 1 and 2 show some particularly bad corrosionoccurring in miter gate compartments that are normallyabove the water line In Figure 1 the coating has not been keptin good condition thus allowing general corrosion to occur

Figure 2 illustrates the adverse effects of corrosion insidea lock miter gate compartment This figure shows that thisparticular miter gate has not had an impressed currentcathodic protection system for many years If the protectivesystem is not preserved and repairs are not performedperiodically it will lead to a significant amount of section lossdue to corrosion that may require an emergency closure forrepairs and maintenance

Quoin block deterioration analysis conducted by [10]demonstrated that deterioration in the quoin block (Figure 3)could drastically affect the state of stresses on the elementtransferring loads to the pintle and the pintle connection

If the deterioration is severe the stresses can reachundesirable levels The location of the stress concentrationsdepends on the quoin deterioration area if the deteriorationoccurs in the pintle area (bottom section of quoin block)the maximum stresses will be generated in the pintle zoneIf the deterioration is in the upper region of the quoinblock the maximum stress will be generated in the elementsnear the quoin block effective area end This deteriorationwill cause some elements such as the thrust diaphragmthrust diaphragm stiffeners end diaphragms and the pintleconnection to be overloaded from the redistribution of the

Figure 3 Quoin block failure

Figure 4 Barge impact at Belleville Locks and Dam

forces not being transmitted to the wall when the gate is inthe miter position In some cases some of these elementshave shown buckling failureswhen severe deterioration of thequoin block is present

Barge impact is one of the main concerns regardingnavigation infrastructures (Figure 4) since they occurwithoutprior warning

Figure 5 shows a Tainter gate with strut arm damage frombarge impact before and after repairs

Failure of the project operating systems can render lockand flow control gates inoperable causing delays to rivertraffic or possible overtopping of the project Structuralfailure of a lock gate could severely impede or stop rivertraffic Catastrophic failure of a spillway gate dewateringbulkhead or a lock gate could cause uncontrolled releaseandor loss of pool resulting in loss of life [6]

Additionally it would be necessary to close that sectionof the river to navigation traffic disrupting the movement ofcoal shipments to power companies and affecting the towingindustry If the impact generates a long closure of the lockthe industry may have to find alternative routes or sources of

Journal of Applied Mathematics 3

(a) (b)

Figure 5 Tainter gate with strut arm damage from barge impact before and after repairs

Figure 6 Fatigue crack in diaphragm flanges of miter gate

transportation decreasing production and causing lost salesand loss in revenue in addition to the extra cost for extralabor hours for the repairs

In many cases the primary forms of distress have beenfatigue damage and fracture The most common causesof fatigue cracking have been a lack of proper detailingduring design poor weld quality during fabrication andpoor detailing and execution of repairs Recent inspectionsindicated that a significant number of stop logs and bulkheadshad deficient welds that required repairs to bring them up tostandards and proper operating conditions

Many of these deficiencies were the result of ineffectivequality control during the original fabrication welding of thestructures (Figures 6 and 7)

3 Condition States for SteelHydraulic Structures

Infrastructure conditions represent discrete condition states[2] Condition states expressly define the condition of indi-vidual components of bridges and sewer pipes [9 11ndash13] NewYork State Department of Transportation uses a rating system(condition states) from 1 to 7 where 7 represents near-perfect

Figure 7 Fatigue crack in girder flanges of miter gate

conditions and 1 represents a state of failure [2] Reference[12] recommends a rating system from 1 to 5 where 1 is near-perfect condition and 5 represents a state of failure

Reference [14] developed a condition rating system sim-ilar to that in [12] for SHS that uses an ordinal integer-value scale from 1 to 5 This system indicates relative healthof the infrastructure elements for the four most commondeteriorations encountered in SHS protective systems cor-rosion fatigue and fracture and impacts or overloads Theoverall condition rating of the entire structure is computedby a weighted average of the individual element conditionratings and is a function of selected weights The selection ofappropriate weights is driven by sound engineering reasonssuch as the importance of fracture-critical members primarymembers and pintle

The following stages describe corrosion and section loss

(1) A protective coating protects the member or othermeans or it has not been subjected to corrosive actionThe member is in like-new or as-built condition andhas no deterioration

(2) The member has lost some of its protection or hasbeen subjected to corrosive action and is beginning todeteriorate (corrode) but has no measurable section

4 Journal of Applied Mathematics

loss Deterioration does not affect functionThis stateis bounded minimally by the onset of corrosion andmaximally by section loss that is not measurable forexample pitting notmeasurable by simple hand tools

(3) Themember continues to deteriorate andmeasurablesection loss is present but not to the extent that itaffects its function The upper bound of this state isfor example pitting to a depth less than 15875mm(00625 in) or total loss of section thickness less than3175mm (0125 in)

(4) Themember continues to deteriorate and section lossincreases to the point where functionmay be affectedAn evaluation may be necessary to determine if thestructure can continue to function as intended ifrepairs are necessary or if its use should be restrictedThe upper bound is a function of member strengthmember load andmember use but it could be cappedat 10 percent of the total section loss for ease of andconsistency in reporting

(5) Themember continues to deteriorate and section lossincreases to the point where the member no longerserves its intended function and it affects the safetyAn evaluation may be necessary to determine if thestructure can continue to function safely

The five general condition states are listed in Table 1 [14]

4 Markov Chain Prediction Model Applied toSteel Hydraulic Structures

The literature reveals thatMarkovmodels are extensively usedto predict infrastructure deterioration [2 15 16] with bridgesbeing a frequent candidate [9] followed by pavements [8] andsewer pipes [15 17] The Markov chain prediction model isa stochastic process that is discrete in time has a finite statespace and establishes that the future state of the deteriorationprocess depends only on its present state

Applying the Markov process to predict the deteriorationof navigation structures involves the following observationsand assumptions First the deterioration process of a struc-ture is continuous in time However to render it discrete intime the condition is usually analyzed at specific periodsFor SHS these periods correspond to periodic and detailedinspections Second the condition of a structure can have aninfinite number of states but in reality the condition of a SHSis defined by a finite set of numbers [14] such as 1 2 3 4 and 5where 1 represents the structure in its best possible conditionand 5 represents imminent failure of the structure Finallythe future condition of a SHS depends only on its presentcondition and not on its past conditions

Markov chain is defined as follows

119875 (119883119905+1= 119894119905+1| 119883119905= 119894119905 119883119905minus1= 119894119905minus11198831= 1198941 1198830= 1198940)

= 119875 (119883119905+1= 119894119905+1| 119883119905= 119894119905)

(1)

where 119875 is a function of 119883 representing the probability tochange from state 119894 to state 119895 at time 119905+1 for all deteriorationstates 119894

0 1198941 119894

119905minus1 119894119905 119894119905+1

and all 119905 ge 0

Markov chain is considered to be homogeneous if theprobability 119901

119894119895of going from state 119894 at time 119905 to state 119895 at

time (119905 + 1) is independent of 119905For all states 119894 and 119895 and all 119905

119875 (119883119905+1= 119895 | 119883

119905= 119894) = 119901

119894119895 (2)

The expressed transition probabilities are an 119898 times 119898 matrixcalled the transition probability matrix The transition prob-ability matrix 119875 is defined as

119875 =

[[[[

[

1199011111990112sdot sdot sdot 119901

1119898

1199012111990122sdot sdot sdot 119901

2119898

11990111989811199011198982sdot sdot sdot 119901119898119898

]]]]

]

(3)

The probability that the system goes from state 119894 to state 119895after 119905 periods can be obtained by multiplying the probabilitymatrix 119875 by itself 119905 times Thus

119875119905= 119875119905 (4)

If 1198760is the initial state vector

1198760= [1199021 1199022 119902

119898] (5)

And 119902119894represents the probability of being in state 119894 at time 0

then the state vector 119876119905 representing the state at time 119905 can

be expressed as

119876119905= 1198760sdot 119875119905 (6)

If the system is in the first state at time zero 1198760can be

expressed as

1198760= [1 0 0 0 0] (7)

indicating that the probability of the system being in the firststate is equal to 1 (or 100) and the probability of any otherstate is 0

Similarly if the system is in the second state 1198760can be

expressed as

1198760= [0 1 0 0 0] (8)

indicating that the probability of the system being in thesecond state is equal to 1 (or 100) and the probability of anyother state is 0

The time when the available data changes from state 1 tostate 2 state 2 to state 3 state 3 to state 4 and so forth isobtained from the linear regression equation of the conditionstate data shown in Figure 8 and Table 2

Defining a normalized vector of time when the measuredcondition states change as follows

119877 = [0 025 050 075 100] (9)

the time when the condition rating changes state after 119905periods is calculated as

119877119875119905= 119876119905sdot 1198771015840 (10)

Journal of Applied Mathematics 5

Table 1 Five condition states

Number Condition Description1 Protected Member is sound functioning properly and lacking in deficiency

2 Exposed Members show beginning signs of deficiency but are still sound and functioning as intended There is noimpact on performance or reliability

3 Attacked Deficiency has advanced and the member still functions as intended but if continued unabateddeterioration will lead to the next condition state

4 Damaged Deficiency has advanced to the point that function may be impaired

5 Failed Deficiency has advanced to the point that the member no longer serves its intended function and safety isimpacted

Table 2 Time at which condition states change (from Figure 8)

State Time of change (years)1 02 63113 72614 109115 14561

where

1198771015840= 119877119879 (11)

When using the process to simulate deterioration the follow-ing condition applies

119901119894119895= 0 for 119894 gt 119895 (12)

This is because the condition of a deteriorating elementcannot return to a previous state (a better condition) withoutexternal intervention That is the probability of an elementreturning to a previous condition is always zero

When an element reaches its worst state (failure state) thefollowing condition applies

119901119898119898= 1 (13)

This indicates the element has deteriorated to the pointof failure and will remain in that state Consequently thegeneral form of the transition probability matrix defined fora deteriorating element is

119875 =

[[[[[[

[

119901111199011211990113sdot sdot sdot 1199011119898

0 1199012211990123sdot sdot sdot 1199012119898

0 0 11990133sdot sdot sdot 1199013119898

0 0 0 sdot sdot sdot 1

]]]]]]

]

(14)

A further restriction allowing the condition to deteriorate byno more than one state in one rating cycle is commonly usedin deterioration modeling The transition probability matrixis indicated as

119875 =

[[[[[[

[

11990111119901120 sdot sdot sdot 0

0 1199012211990123sdot sdot sdot 0

0 0 11990133sdot sdot sdot 0

0 0 0 sdot sdot sdot 1

]]]]]]

]

(15)

However some SHS inspection reports have shown that thestructure has changed by more than one state during theinspection period therefore the transition probabilitymatrixdefined in (11) may better fit actual inspection data

5 Derivation of Transition Probabilities

There are several methods for deriving a transition prob-ability matrix The methods include expert opinion linearregression and Poisson regression [2] Since the availabledata containing condition states are limited for navigationstructures the development of a probabilistic method is pro-posed that can be updated as data will become available Themain goal is to develop a method and verifying this methodas actual data becomes available This allows confidence inthe use of the method to predict future deterioration ofhydraulic steel structures The New York State Departmentof Transportation provided the data used in the developmentof this method This data was applicable not only because itwas accessible but also because it represents condition statedata for thousands of steel bridge elements over a periodof eighty years of inspections Additionally the same effectsthat cause the deterioration of navigation structures (loss ofa protective system corrosion cracking and fatigue impactand overloads) cause steel bridge element deteriorationFigure 8 shows the data used

Fluctuations in the data as can be seen between 30 and 40years occur because the data represent the average conditionstate of many elements To eliminate the fluctuations andmake the data more manageable a linear regression equationwas calculated as follows

119910 = 00274119909 + 10104 (16)

where 119909 is the age in years and 119910 is the condition stateThe authors calculated condition state values at ten-year

intervals by using (13)The calculated values were used as theaverage condition state at each interval Using Weibull dis-tribution and a Latin hypercube simulation (LHS) syntheticrandom condition state values were generated to represent arange of condition states at each ten-year intervalThe authorsused Weibull distribution parameters for each interval toyield approximately the same average values represented inFigure 8

Figure 9 shows the synthetic values (vertical points)generated to simulate a range of condition states at each ten-year interval The authors generated one-thousand random

6 Journal of Applied Mathematics

NY Department of Transportation data

y = 00274x + 10104

Average condition ratingLinear regression

0 101

2

3

4

5

20 30 40 50 60 70 80

Age (years)

Con

ditio

n st

ate

Figure 8 New York Department of Transportation condition statedata

values at each interval The diagonal line crosses through theaverage value of each ten-year interval

Figure 10 shows the distribution of values generatedfor the 20-year interval These superimposed values are inFigure 9 In addition similar distributions of values weregenerated for each of the other intervals

Using the generated condition state values for eachinterval the transition probabilities were calculated as

119901119894119895=

119873119894119895

119873119894

(17)

where 119873119894119895

is the number of elements that change fromcondition 119894 to condition 119895 after one interval and 119873

119894is the

number of elements that were in condition 119894 in the previousinterval

Table 3 shows the transition probability valuesApplying Markov chain

119875 =

[[[[[

[

0973 0027 0 0 0

0 0972 0028 0 0

0 0 0972 0028 0

0 0 0 0973 0027

0 0 0 0 1000

]]]]]

]

(18)

Express the initial state of a new element as

1198760= [1 0 0 0 0] (19)

The normalized vector of time when the measured conditionstates change is as follows

119877 = [0 025 050 075 100] (20)

010

20

30

40

50

10 20 30 40 50 60 70 80 90 100

Age (years)Condition states (Weibull distribution)

Con

ditio

n st

ate

Figure 9 Synthetic condition state values

Now to calculate the time for the next rating cycle (119905 = 2)applying Markov chain we obtain

1198752= 1198752

1198752=

[[[[[

[

09469 00524 00007 0 0

0 09456 00536 00008 0

0 0 09447 00545 00008

0 0 0 09460 00540

0 0 0 0 10000

]]]]]

]

(21)

Applying (6) vector 119876119905representing the condition state after

two periods is calculated as

1198762= 1198760sdot 1198752= [09469 00524 00007 0 0] (22)

And the normalized time using (10) is calculated as

1198771198752= 1198762sdot 1198771015840= 00135 (23)

Continuing in a similar fashion the time at which the stateschange (Table 4) is set In addition plotting this informationwe obtain the stepwise graph shown in Figure 11 Figure 11also shows the upper and lower bounds for the deteriorationof navigation steel structures

6 Conclusions

The aim of this study was to present deterioration examplesof navigation steel structures and to develop a results-based model of current inspections Because of this studya deterioration model was developed with the assistance ofreal inspection data provided by the New York Departmentof Transportation The results presented in Figure 11 indicatethat there is a marked correlation between the proposeddeteriorationmodel and the actual data presented in Figure 8For example the model indicates that the condition statereaches level 2 at 3571 years and state 3 at 7479 yearsCompared with Figure 8 it indicates that the condition statereaches state 2 at 3612 years and state 3 at 7261 years Thedifference between the model and the actual data was about1

These results suggest that using the method presentedin this paper you may develop a deterioration model that

Journal of Applied Mathematics 7

0

10

20

30

40

50

60

70

1 2 3 4

Condition state

Freq

uenc

y

Condition state distribution

Figure 10 Distribution of condition state values generated for 20-year interval

Table 3 Transition probabilities

Condition state 1 2 3 4 51 0973 00272 0972 00283 0972 00284 0973 00275 1000

Table 4 Time at which the states change

State Time (years)1 02 36693 74694 127045 30000

reflects actual results Therefore predicting future deterio-ration of these structures is possible by using this modelThe model represents a vital starting point in predictingdeterioration which can continue updating and recalibratingas more data becomes available

For this study the authors used the Weibull distributionto generate synthetic data for condition states However if thecurrent data curve has different morphology they could useother (normal lognormal) distributions

In addition to predicting the deterioration themodel canbe useful in scheduling inspections Currently inspectionsare programmed without any consideration of anticipateddeterioration If a deterioration model is used as a refer-ence these inspections can be programmed by the modelsuggesting better anticipation of when the structure must bemaintained to avoid emergency repairs

The deterioration model is applicable in combinationwith a life cycle analysis on the prediction and optimal repaircosts Additionally you can determine the optimum pointwhen inspections are necessary to maintain the structuralsystem in a reliable condition

1

2

3

4

5

0 25 50 75 100 125 150 175 200 225 250 275 300

Con

ditio

n st

ate

Time (years)Deterioration model

Upper bound

Lower bound

Figure 11 Markov chain deterioration model

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The Navigation Systems Research Program of the US ArmyCorps of Engineers and the Information Technology Labo-ratory of the US Army Engineer Research and DevelopmentCenter funded the development of this research The authorsoffer their sincere thanks to Courtney Tuminello for theeditorial reviews during the preparation of the paper whichwere instrumental in its publication

References

[1] S Bulusu and K C Sinha ldquoComparison of methodologies topredict bridge deteriorationrdquo Transportation Research Recordno 1597 pp 34ndash42 1997

[2] S Madanat R Mishalani and W H W Ibrahim ldquoEstimationof infrastructure transition probabilities from condition ratingdatardquo Journal of Infrastructure Systems vol 1 no 2 pp 120ndash1251995

[3] S MMadanat M G Karlaftis and P S McCarthy ldquoProbabilis-tic infrastructure deteriorationmodels with panel datardquo Journalof Infrastructure Systems vol 3 no 1 pp 4ndash9 1997

[4] G Morcous H Rivard and A M Hanna ldquoModeling bridgedeterioration using case-based reasoningrdquo Journal of Infrastruc-ture Systems vol 8 no 3 pp 86ndash95 2002

[5] Headquarters US Army Corps of Engineers ldquoPeriodic inspec-tion and continuing evaluation of completed civil works struc-turesrdquo Engineer Regulation 1110-2-100 Headquarters US ArmyCorps of Engineers Washington DC USA 1995

[6] Headquarters US Army Corps of Engineers ldquoResponsibilityfor hydraulic steel structuresrdquo Engineer Regulation 1110-2-8157Headquarters US Army Corps of Engineers Washington DCUSA 2009

[7] Headquarters US Army Corps of Engineers ldquoInspection eval-uation and repair of hydraulic steel structuresrdquo Engineer Man-ual 1110-2-6054 Headquarters US Army Corps of EngineersWashington DC USA 2001

8 Journal of Applied Mathematics

[8] J J Ortiz-Garcıa S B Costello and M S Snaith ldquoDerivationof transition probability matrices for pavement deteriorationmodelingrdquo Journal of Transportation Engineering vol 132 no2 pp 141ndash161 2006

[9] A K Agrawal A Kawaguchi and G Qian Bridge Deterio-ration Rates New York State Department of TransportationTransportation Infrastructure Research Consortium AlbanyNY USA 2008

[10] G A Riveros J L Ayala-Burgos and J Perez NumericalInvestigation of Miter Gates ERDCITL-TR-09-1 US ArmyEngineer Research and Development Center Vicksburg MissUSA 2009

[11] Federal Highway Administration ldquoRecording and coding guidefor the structure inventory and appraisal of the nationrsquos bridgesrdquoReport No FHWA-PD-96-001 Office of Engineering BridgeDivision Bridge Management Branch Washington DC USA1995

[12] American Association of State Highway and TransportationOfficials Guide for Commonly Recognized Structural Elementsand Its 2002 Interim Revisions American Association of StateHighway and Transportation Officials Washington DC USA2002

[13] P D Thomson and R W Shepard ldquoAASHTO commonly-recognized bridge elements successful applications and lessonslearnedrdquo in Proceedings of the National Workshop on CommonlyRecognizedMeasures for Maintenance American Association ofState Highway and Transportation Officials Washington DCUSA June 2000

[14] P Sauser and G Riveros A System for Collecting and Com-piling Condition Data for Hydraulic Steel Structures for Usein the Assessment of Risk and Reliability and Prioritization ofMaintenance and Repairs Miter gates Report 1 ERDCITL TR-09-04 US Army Engineer Research and Development CenterVicksburg Miss USA 2009

[15] T Micevski G Kuczera and P Coombes ldquoMarkov modelfor storm water pipe deteriorationrdquo Journal of InfrastructureSystems vol 8 no 2 pp 49ndash56 2002

[16] P D Destefano and D A Grivas ldquoMethod for estimatingtransition probability in bridge deterioration modelsrdquo Journalof Infrastructure Systems vol 4 no 2 pp 56ndash62 1998

[17] H-S Baik H S Jeong and D M Abraham ldquoEstimatingtransition probabilities in Markov chain-based deteriorationmodels for management of wastewater systemsrdquo Journal ofWater Resources Planning and Management vol 132 no 1 pp15ndash24 2006

Journal of Applied Mathematics 3

(a) (b)

Figure 5 Tainter gate with strut arm damage from barge impact before and after repairs

Figure 6 Fatigue crack in diaphragm flanges of miter gate

transportation decreasing production and causing lost salesand loss in revenue in addition to the extra cost for extralabor hours for the repairs

In many cases the primary forms of distress have beenfatigue damage and fracture The most common causesof fatigue cracking have been a lack of proper detailingduring design poor weld quality during fabrication andpoor detailing and execution of repairs Recent inspectionsindicated that a significant number of stop logs and bulkheadshad deficient welds that required repairs to bring them up tostandards and proper operating conditions

Many of these deficiencies were the result of ineffectivequality control during the original fabrication welding of thestructures (Figures 6 and 7)

3 Condition States for SteelHydraulic Structures

Infrastructure conditions represent discrete condition states[2] Condition states expressly define the condition of indi-vidual components of bridges and sewer pipes [9 11ndash13] NewYork State Department of Transportation uses a rating system(condition states) from 1 to 7 where 7 represents near-perfect

Figure 7 Fatigue crack in girder flanges of miter gate

conditions and 1 represents a state of failure [2] Reference[12] recommends a rating system from 1 to 5 where 1 is near-perfect condition and 5 represents a state of failure

Reference [14] developed a condition rating system sim-ilar to that in [12] for SHS that uses an ordinal integer-value scale from 1 to 5 This system indicates relative healthof the infrastructure elements for the four most commondeteriorations encountered in SHS protective systems cor-rosion fatigue and fracture and impacts or overloads Theoverall condition rating of the entire structure is computedby a weighted average of the individual element conditionratings and is a function of selected weights The selection ofappropriate weights is driven by sound engineering reasonssuch as the importance of fracture-critical members primarymembers and pintle

The following stages describe corrosion and section loss

(1) A protective coating protects the member or othermeans or it has not been subjected to corrosive actionThe member is in like-new or as-built condition andhas no deterioration

(2) The member has lost some of its protection or hasbeen subjected to corrosive action and is beginning todeteriorate (corrode) but has no measurable section

4 Journal of Applied Mathematics

loss Deterioration does not affect functionThis stateis bounded minimally by the onset of corrosion andmaximally by section loss that is not measurable forexample pitting notmeasurable by simple hand tools

(3) Themember continues to deteriorate andmeasurablesection loss is present but not to the extent that itaffects its function The upper bound of this state isfor example pitting to a depth less than 15875mm(00625 in) or total loss of section thickness less than3175mm (0125 in)

(4) Themember continues to deteriorate and section lossincreases to the point where functionmay be affectedAn evaluation may be necessary to determine if thestructure can continue to function as intended ifrepairs are necessary or if its use should be restrictedThe upper bound is a function of member strengthmember load andmember use but it could be cappedat 10 percent of the total section loss for ease of andconsistency in reporting

(5) Themember continues to deteriorate and section lossincreases to the point where the member no longerserves its intended function and it affects the safetyAn evaluation may be necessary to determine if thestructure can continue to function safely

The five general condition states are listed in Table 1 [14]

4 Markov Chain Prediction Model Applied toSteel Hydraulic Structures

The literature reveals thatMarkovmodels are extensively usedto predict infrastructure deterioration [2 15 16] with bridgesbeing a frequent candidate [9] followed by pavements [8] andsewer pipes [15 17] The Markov chain prediction model isa stochastic process that is discrete in time has a finite statespace and establishes that the future state of the deteriorationprocess depends only on its present state

Applying the Markov process to predict the deteriorationof navigation structures involves the following observationsand assumptions First the deterioration process of a struc-ture is continuous in time However to render it discrete intime the condition is usually analyzed at specific periodsFor SHS these periods correspond to periodic and detailedinspections Second the condition of a structure can have aninfinite number of states but in reality the condition of a SHSis defined by a finite set of numbers [14] such as 1 2 3 4 and 5where 1 represents the structure in its best possible conditionand 5 represents imminent failure of the structure Finallythe future condition of a SHS depends only on its presentcondition and not on its past conditions

Markov chain is defined as follows

119875 (119883119905+1= 119894119905+1| 119883119905= 119894119905 119883119905minus1= 119894119905minus11198831= 1198941 1198830= 1198940)

= 119875 (119883119905+1= 119894119905+1| 119883119905= 119894119905)

(1)

where 119875 is a function of 119883 representing the probability tochange from state 119894 to state 119895 at time 119905+1 for all deteriorationstates 119894

0 1198941 119894

119905minus1 119894119905 119894119905+1

and all 119905 ge 0

Markov chain is considered to be homogeneous if theprobability 119901

119894119895of going from state 119894 at time 119905 to state 119895 at

time (119905 + 1) is independent of 119905For all states 119894 and 119895 and all 119905

119875 (119883119905+1= 119895 | 119883

119905= 119894) = 119901

119894119895 (2)

The expressed transition probabilities are an 119898 times 119898 matrixcalled the transition probability matrix The transition prob-ability matrix 119875 is defined as

119875 =

[[[[

[

1199011111990112sdot sdot sdot 119901

1119898

1199012111990122sdot sdot sdot 119901

2119898

11990111989811199011198982sdot sdot sdot 119901119898119898

]]]]

]

(3)

The probability that the system goes from state 119894 to state 119895after 119905 periods can be obtained by multiplying the probabilitymatrix 119875 by itself 119905 times Thus

119875119905= 119875119905 (4)

If 1198760is the initial state vector

1198760= [1199021 1199022 119902

119898] (5)

And 119902119894represents the probability of being in state 119894 at time 0

then the state vector 119876119905 representing the state at time 119905 can

be expressed as

119876119905= 1198760sdot 119875119905 (6)

If the system is in the first state at time zero 1198760can be

expressed as

1198760= [1 0 0 0 0] (7)

indicating that the probability of the system being in the firststate is equal to 1 (or 100) and the probability of any otherstate is 0

Similarly if the system is in the second state 1198760can be

expressed as

1198760= [0 1 0 0 0] (8)

indicating that the probability of the system being in thesecond state is equal to 1 (or 100) and the probability of anyother state is 0

The time when the available data changes from state 1 tostate 2 state 2 to state 3 state 3 to state 4 and so forth isobtained from the linear regression equation of the conditionstate data shown in Figure 8 and Table 2

Defining a normalized vector of time when the measuredcondition states change as follows

119877 = [0 025 050 075 100] (9)

the time when the condition rating changes state after 119905periods is calculated as

119877119875119905= 119876119905sdot 1198771015840 (10)

Journal of Applied Mathematics 5

Table 1 Five condition states

Number Condition Description1 Protected Member is sound functioning properly and lacking in deficiency

2 Exposed Members show beginning signs of deficiency but are still sound and functioning as intended There is noimpact on performance or reliability

3 Attacked Deficiency has advanced and the member still functions as intended but if continued unabateddeterioration will lead to the next condition state

4 Damaged Deficiency has advanced to the point that function may be impaired

5 Failed Deficiency has advanced to the point that the member no longer serves its intended function and safety isimpacted

Table 2 Time at which condition states change (from Figure 8)

State Time of change (years)1 02 63113 72614 109115 14561

where

1198771015840= 119877119879 (11)

When using the process to simulate deterioration the follow-ing condition applies

119901119894119895= 0 for 119894 gt 119895 (12)

This is because the condition of a deteriorating elementcannot return to a previous state (a better condition) withoutexternal intervention That is the probability of an elementreturning to a previous condition is always zero

When an element reaches its worst state (failure state) thefollowing condition applies

119901119898119898= 1 (13)

This indicates the element has deteriorated to the pointof failure and will remain in that state Consequently thegeneral form of the transition probability matrix defined fora deteriorating element is

119875 =

[[[[[[

[

119901111199011211990113sdot sdot sdot 1199011119898

0 1199012211990123sdot sdot sdot 1199012119898

0 0 11990133sdot sdot sdot 1199013119898

0 0 0 sdot sdot sdot 1

]]]]]]

]

(14)

A further restriction allowing the condition to deteriorate byno more than one state in one rating cycle is commonly usedin deterioration modeling The transition probability matrixis indicated as

119875 =

[[[[[[

[

11990111119901120 sdot sdot sdot 0

0 1199012211990123sdot sdot sdot 0

0 0 11990133sdot sdot sdot 0

0 0 0 sdot sdot sdot 1

]]]]]]

]

(15)

However some SHS inspection reports have shown that thestructure has changed by more than one state during theinspection period therefore the transition probabilitymatrixdefined in (11) may better fit actual inspection data

5 Derivation of Transition Probabilities

There are several methods for deriving a transition prob-ability matrix The methods include expert opinion linearregression and Poisson regression [2] Since the availabledata containing condition states are limited for navigationstructures the development of a probabilistic method is pro-posed that can be updated as data will become available Themain goal is to develop a method and verifying this methodas actual data becomes available This allows confidence inthe use of the method to predict future deterioration ofhydraulic steel structures The New York State Departmentof Transportation provided the data used in the developmentof this method This data was applicable not only because itwas accessible but also because it represents condition statedata for thousands of steel bridge elements over a periodof eighty years of inspections Additionally the same effectsthat cause the deterioration of navigation structures (loss ofa protective system corrosion cracking and fatigue impactand overloads) cause steel bridge element deteriorationFigure 8 shows the data used

Fluctuations in the data as can be seen between 30 and 40years occur because the data represent the average conditionstate of many elements To eliminate the fluctuations andmake the data more manageable a linear regression equationwas calculated as follows

119910 = 00274119909 + 10104 (16)

where 119909 is the age in years and 119910 is the condition stateThe authors calculated condition state values at ten-year

intervals by using (13)The calculated values were used as theaverage condition state at each interval Using Weibull dis-tribution and a Latin hypercube simulation (LHS) syntheticrandom condition state values were generated to represent arange of condition states at each ten-year intervalThe authorsused Weibull distribution parameters for each interval toyield approximately the same average values represented inFigure 8

Figure 9 shows the synthetic values (vertical points)generated to simulate a range of condition states at each ten-year interval The authors generated one-thousand random

6 Journal of Applied Mathematics

NY Department of Transportation data

y = 00274x + 10104

Average condition ratingLinear regression

0 101

2

3

4

5

20 30 40 50 60 70 80

Age (years)

Con

ditio

n st

ate

Figure 8 New York Department of Transportation condition statedata

values at each interval The diagonal line crosses through theaverage value of each ten-year interval

Figure 10 shows the distribution of values generatedfor the 20-year interval These superimposed values are inFigure 9 In addition similar distributions of values weregenerated for each of the other intervals

Using the generated condition state values for eachinterval the transition probabilities were calculated as

119901119894119895=

119873119894119895

119873119894

(17)

where 119873119894119895

is the number of elements that change fromcondition 119894 to condition 119895 after one interval and 119873

119894is the

number of elements that were in condition 119894 in the previousinterval

Table 3 shows the transition probability valuesApplying Markov chain

119875 =

[[[[[

[

0973 0027 0 0 0

0 0972 0028 0 0

0 0 0972 0028 0

0 0 0 0973 0027

0 0 0 0 1000

]]]]]

]

(18)

Express the initial state of a new element as

1198760= [1 0 0 0 0] (19)

The normalized vector of time when the measured conditionstates change is as follows

119877 = [0 025 050 075 100] (20)

010

20

30

40

50

10 20 30 40 50 60 70 80 90 100

Age (years)Condition states (Weibull distribution)

Con

ditio

n st

ate

Figure 9 Synthetic condition state values

Now to calculate the time for the next rating cycle (119905 = 2)applying Markov chain we obtain

1198752= 1198752

1198752=

[[[[[

[

09469 00524 00007 0 0

0 09456 00536 00008 0

0 0 09447 00545 00008

0 0 0 09460 00540

0 0 0 0 10000

]]]]]

]

(21)

Applying (6) vector 119876119905representing the condition state after

two periods is calculated as

1198762= 1198760sdot 1198752= [09469 00524 00007 0 0] (22)

And the normalized time using (10) is calculated as

1198771198752= 1198762sdot 1198771015840= 00135 (23)

Continuing in a similar fashion the time at which the stateschange (Table 4) is set In addition plotting this informationwe obtain the stepwise graph shown in Figure 11 Figure 11also shows the upper and lower bounds for the deteriorationof navigation steel structures

6 Conclusions

The aim of this study was to present deterioration examplesof navigation steel structures and to develop a results-based model of current inspections Because of this studya deterioration model was developed with the assistance ofreal inspection data provided by the New York Departmentof Transportation The results presented in Figure 11 indicatethat there is a marked correlation between the proposeddeteriorationmodel and the actual data presented in Figure 8For example the model indicates that the condition statereaches level 2 at 3571 years and state 3 at 7479 yearsCompared with Figure 8 it indicates that the condition statereaches state 2 at 3612 years and state 3 at 7261 years Thedifference between the model and the actual data was about1

These results suggest that using the method presentedin this paper you may develop a deterioration model that

Journal of Applied Mathematics 7

0

10

20

30

40

50

60

70

1 2 3 4

Condition state

Freq

uenc

y

Condition state distribution

Figure 10 Distribution of condition state values generated for 20-year interval

Table 3 Transition probabilities

Condition state 1 2 3 4 51 0973 00272 0972 00283 0972 00284 0973 00275 1000

Table 4 Time at which the states change

State Time (years)1 02 36693 74694 127045 30000

reflects actual results Therefore predicting future deterio-ration of these structures is possible by using this modelThe model represents a vital starting point in predictingdeterioration which can continue updating and recalibratingas more data becomes available

For this study the authors used the Weibull distributionto generate synthetic data for condition states However if thecurrent data curve has different morphology they could useother (normal lognormal) distributions

In addition to predicting the deterioration themodel canbe useful in scheduling inspections Currently inspectionsare programmed without any consideration of anticipateddeterioration If a deterioration model is used as a refer-ence these inspections can be programmed by the modelsuggesting better anticipation of when the structure must bemaintained to avoid emergency repairs

The deterioration model is applicable in combinationwith a life cycle analysis on the prediction and optimal repaircosts Additionally you can determine the optimum pointwhen inspections are necessary to maintain the structuralsystem in a reliable condition

1

2

3

4

5

0 25 50 75 100 125 150 175 200 225 250 275 300

Con

ditio

n st

ate

Time (years)Deterioration model

Upper bound

Lower bound

Figure 11 Markov chain deterioration model

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The Navigation Systems Research Program of the US ArmyCorps of Engineers and the Information Technology Labo-ratory of the US Army Engineer Research and DevelopmentCenter funded the development of this research The authorsoffer their sincere thanks to Courtney Tuminello for theeditorial reviews during the preparation of the paper whichwere instrumental in its publication

References

[1] S Bulusu and K C Sinha ldquoComparison of methodologies topredict bridge deteriorationrdquo Transportation Research Recordno 1597 pp 34ndash42 1997

[2] S Madanat R Mishalani and W H W Ibrahim ldquoEstimationof infrastructure transition probabilities from condition ratingdatardquo Journal of Infrastructure Systems vol 1 no 2 pp 120ndash1251995

[3] S MMadanat M G Karlaftis and P S McCarthy ldquoProbabilis-tic infrastructure deteriorationmodels with panel datardquo Journalof Infrastructure Systems vol 3 no 1 pp 4ndash9 1997

[4] G Morcous H Rivard and A M Hanna ldquoModeling bridgedeterioration using case-based reasoningrdquo Journal of Infrastruc-ture Systems vol 8 no 3 pp 86ndash95 2002

[5] Headquarters US Army Corps of Engineers ldquoPeriodic inspec-tion and continuing evaluation of completed civil works struc-turesrdquo Engineer Regulation 1110-2-100 Headquarters US ArmyCorps of Engineers Washington DC USA 1995

[6] Headquarters US Army Corps of Engineers ldquoResponsibilityfor hydraulic steel structuresrdquo Engineer Regulation 1110-2-8157Headquarters US Army Corps of Engineers Washington DCUSA 2009

[7] Headquarters US Army Corps of Engineers ldquoInspection eval-uation and repair of hydraulic steel structuresrdquo Engineer Man-ual 1110-2-6054 Headquarters US Army Corps of EngineersWashington DC USA 2001

8 Journal of Applied Mathematics

[8] J J Ortiz-Garcıa S B Costello and M S Snaith ldquoDerivationof transition probability matrices for pavement deteriorationmodelingrdquo Journal of Transportation Engineering vol 132 no2 pp 141ndash161 2006

[9] A K Agrawal A Kawaguchi and G Qian Bridge Deterio-ration Rates New York State Department of TransportationTransportation Infrastructure Research Consortium AlbanyNY USA 2008

[10] G A Riveros J L Ayala-Burgos and J Perez NumericalInvestigation of Miter Gates ERDCITL-TR-09-1 US ArmyEngineer Research and Development Center Vicksburg MissUSA 2009

[11] Federal Highway Administration ldquoRecording and coding guidefor the structure inventory and appraisal of the nationrsquos bridgesrdquoReport No FHWA-PD-96-001 Office of Engineering BridgeDivision Bridge Management Branch Washington DC USA1995

[12] American Association of State Highway and TransportationOfficials Guide for Commonly Recognized Structural Elementsand Its 2002 Interim Revisions American Association of StateHighway and Transportation Officials Washington DC USA2002

[13] P D Thomson and R W Shepard ldquoAASHTO commonly-recognized bridge elements successful applications and lessonslearnedrdquo in Proceedings of the National Workshop on CommonlyRecognizedMeasures for Maintenance American Association ofState Highway and Transportation Officials Washington DCUSA June 2000

[14] P Sauser and G Riveros A System for Collecting and Com-piling Condition Data for Hydraulic Steel Structures for Usein the Assessment of Risk and Reliability and Prioritization ofMaintenance and Repairs Miter gates Report 1 ERDCITL TR-09-04 US Army Engineer Research and Development CenterVicksburg Miss USA 2009

[15] T Micevski G Kuczera and P Coombes ldquoMarkov modelfor storm water pipe deteriorationrdquo Journal of InfrastructureSystems vol 8 no 2 pp 49ndash56 2002

[16] P D Destefano and D A Grivas ldquoMethod for estimatingtransition probability in bridge deterioration modelsrdquo Journalof Infrastructure Systems vol 4 no 2 pp 56ndash62 1998

[17] H-S Baik H S Jeong and D M Abraham ldquoEstimatingtransition probabilities in Markov chain-based deteriorationmodels for management of wastewater systemsrdquo Journal ofWater Resources Planning and Management vol 132 no 1 pp15ndash24 2006

4 Journal of Applied Mathematics

loss Deterioration does not affect functionThis stateis bounded minimally by the onset of corrosion andmaximally by section loss that is not measurable forexample pitting notmeasurable by simple hand tools

(3) Themember continues to deteriorate andmeasurablesection loss is present but not to the extent that itaffects its function The upper bound of this state isfor example pitting to a depth less than 15875mm(00625 in) or total loss of section thickness less than3175mm (0125 in)

(4) Themember continues to deteriorate and section lossincreases to the point where functionmay be affectedAn evaluation may be necessary to determine if thestructure can continue to function as intended ifrepairs are necessary or if its use should be restrictedThe upper bound is a function of member strengthmember load andmember use but it could be cappedat 10 percent of the total section loss for ease of andconsistency in reporting

(5) Themember continues to deteriorate and section lossincreases to the point where the member no longerserves its intended function and it affects the safetyAn evaluation may be necessary to determine if thestructure can continue to function safely

The five general condition states are listed in Table 1 [14]

4 Markov Chain Prediction Model Applied toSteel Hydraulic Structures

The literature reveals thatMarkovmodels are extensively usedto predict infrastructure deterioration [2 15 16] with bridgesbeing a frequent candidate [9] followed by pavements [8] andsewer pipes [15 17] The Markov chain prediction model isa stochastic process that is discrete in time has a finite statespace and establishes that the future state of the deteriorationprocess depends only on its present state

Applying the Markov process to predict the deteriorationof navigation structures involves the following observationsand assumptions First the deterioration process of a struc-ture is continuous in time However to render it discrete intime the condition is usually analyzed at specific periodsFor SHS these periods correspond to periodic and detailedinspections Second the condition of a structure can have aninfinite number of states but in reality the condition of a SHSis defined by a finite set of numbers [14] such as 1 2 3 4 and 5where 1 represents the structure in its best possible conditionand 5 represents imminent failure of the structure Finallythe future condition of a SHS depends only on its presentcondition and not on its past conditions

Markov chain is defined as follows

119875 (119883119905+1= 119894119905+1| 119883119905= 119894119905 119883119905minus1= 119894119905minus11198831= 1198941 1198830= 1198940)

= 119875 (119883119905+1= 119894119905+1| 119883119905= 119894119905)

(1)

where 119875 is a function of 119883 representing the probability tochange from state 119894 to state 119895 at time 119905+1 for all deteriorationstates 119894

0 1198941 119894

119905minus1 119894119905 119894119905+1

and all 119905 ge 0

Markov chain is considered to be homogeneous if theprobability 119901

119894119895of going from state 119894 at time 119905 to state 119895 at

time (119905 + 1) is independent of 119905For all states 119894 and 119895 and all 119905

119875 (119883119905+1= 119895 | 119883

119905= 119894) = 119901

119894119895 (2)

The expressed transition probabilities are an 119898 times 119898 matrixcalled the transition probability matrix The transition prob-ability matrix 119875 is defined as

119875 =

[[[[

[

1199011111990112sdot sdot sdot 119901

1119898

1199012111990122sdot sdot sdot 119901

2119898

11990111989811199011198982sdot sdot sdot 119901119898119898

]]]]

]

(3)

The probability that the system goes from state 119894 to state 119895after 119905 periods can be obtained by multiplying the probabilitymatrix 119875 by itself 119905 times Thus

119875119905= 119875119905 (4)

If 1198760is the initial state vector

1198760= [1199021 1199022 119902

119898] (5)

And 119902119894represents the probability of being in state 119894 at time 0

then the state vector 119876119905 representing the state at time 119905 can

be expressed as

119876119905= 1198760sdot 119875119905 (6)

If the system is in the first state at time zero 1198760can be

expressed as

1198760= [1 0 0 0 0] (7)

indicating that the probability of the system being in the firststate is equal to 1 (or 100) and the probability of any otherstate is 0

Similarly if the system is in the second state 1198760can be

expressed as

1198760= [0 1 0 0 0] (8)

indicating that the probability of the system being in thesecond state is equal to 1 (or 100) and the probability of anyother state is 0

The time when the available data changes from state 1 tostate 2 state 2 to state 3 state 3 to state 4 and so forth isobtained from the linear regression equation of the conditionstate data shown in Figure 8 and Table 2

Defining a normalized vector of time when the measuredcondition states change as follows

119877 = [0 025 050 075 100] (9)

the time when the condition rating changes state after 119905periods is calculated as

119877119875119905= 119876119905sdot 1198771015840 (10)

Journal of Applied Mathematics 5

Table 1 Five condition states

Number Condition Description1 Protected Member is sound functioning properly and lacking in deficiency

2 Exposed Members show beginning signs of deficiency but are still sound and functioning as intended There is noimpact on performance or reliability

3 Attacked Deficiency has advanced and the member still functions as intended but if continued unabateddeterioration will lead to the next condition state

4 Damaged Deficiency has advanced to the point that function may be impaired

5 Failed Deficiency has advanced to the point that the member no longer serves its intended function and safety isimpacted

Table 2 Time at which condition states change (from Figure 8)

State Time of change (years)1 02 63113 72614 109115 14561

where

1198771015840= 119877119879 (11)

When using the process to simulate deterioration the follow-ing condition applies

119901119894119895= 0 for 119894 gt 119895 (12)

This is because the condition of a deteriorating elementcannot return to a previous state (a better condition) withoutexternal intervention That is the probability of an elementreturning to a previous condition is always zero

When an element reaches its worst state (failure state) thefollowing condition applies

119901119898119898= 1 (13)

This indicates the element has deteriorated to the pointof failure and will remain in that state Consequently thegeneral form of the transition probability matrix defined fora deteriorating element is

119875 =

[[[[[[

[

119901111199011211990113sdot sdot sdot 1199011119898

0 1199012211990123sdot sdot sdot 1199012119898

0 0 11990133sdot sdot sdot 1199013119898

0 0 0 sdot sdot sdot 1

]]]]]]

]

(14)

A further restriction allowing the condition to deteriorate byno more than one state in one rating cycle is commonly usedin deterioration modeling The transition probability matrixis indicated as

119875 =

[[[[[[

[

11990111119901120 sdot sdot sdot 0

0 1199012211990123sdot sdot sdot 0

0 0 11990133sdot sdot sdot 0

0 0 0 sdot sdot sdot 1

]]]]]]

]

(15)

However some SHS inspection reports have shown that thestructure has changed by more than one state during theinspection period therefore the transition probabilitymatrixdefined in (11) may better fit actual inspection data

5 Derivation of Transition Probabilities

There are several methods for deriving a transition prob-ability matrix The methods include expert opinion linearregression and Poisson regression [2] Since the availabledata containing condition states are limited for navigationstructures the development of a probabilistic method is pro-posed that can be updated as data will become available Themain goal is to develop a method and verifying this methodas actual data becomes available This allows confidence inthe use of the method to predict future deterioration ofhydraulic steel structures The New York State Departmentof Transportation provided the data used in the developmentof this method This data was applicable not only because itwas accessible but also because it represents condition statedata for thousands of steel bridge elements over a periodof eighty years of inspections Additionally the same effectsthat cause the deterioration of navigation structures (loss ofa protective system corrosion cracking and fatigue impactand overloads) cause steel bridge element deteriorationFigure 8 shows the data used

Fluctuations in the data as can be seen between 30 and 40years occur because the data represent the average conditionstate of many elements To eliminate the fluctuations andmake the data more manageable a linear regression equationwas calculated as follows

119910 = 00274119909 + 10104 (16)

where 119909 is the age in years and 119910 is the condition stateThe authors calculated condition state values at ten-year

intervals by using (13)The calculated values were used as theaverage condition state at each interval Using Weibull dis-tribution and a Latin hypercube simulation (LHS) syntheticrandom condition state values were generated to represent arange of condition states at each ten-year intervalThe authorsused Weibull distribution parameters for each interval toyield approximately the same average values represented inFigure 8

Figure 9 shows the synthetic values (vertical points)generated to simulate a range of condition states at each ten-year interval The authors generated one-thousand random

6 Journal of Applied Mathematics

NY Department of Transportation data

y = 00274x + 10104

Average condition ratingLinear regression

0 101

2

3

4

5

20 30 40 50 60 70 80

Age (years)

Con

ditio

n st

ate

Figure 8 New York Department of Transportation condition statedata

values at each interval The diagonal line crosses through theaverage value of each ten-year interval

Figure 10 shows the distribution of values generatedfor the 20-year interval These superimposed values are inFigure 9 In addition similar distributions of values weregenerated for each of the other intervals

Using the generated condition state values for eachinterval the transition probabilities were calculated as

119901119894119895=

119873119894119895

119873119894

(17)

where 119873119894119895

is the number of elements that change fromcondition 119894 to condition 119895 after one interval and 119873

119894is the

number of elements that were in condition 119894 in the previousinterval

Table 3 shows the transition probability valuesApplying Markov chain

119875 =

[[[[[

[

0973 0027 0 0 0

0 0972 0028 0 0

0 0 0972 0028 0

0 0 0 0973 0027

0 0 0 0 1000

]]]]]

]

(18)

Express the initial state of a new element as

1198760= [1 0 0 0 0] (19)

The normalized vector of time when the measured conditionstates change is as follows

119877 = [0 025 050 075 100] (20)

010

20

30

40

50

10 20 30 40 50 60 70 80 90 100

Age (years)Condition states (Weibull distribution)

Con

ditio

n st

ate

Figure 9 Synthetic condition state values

Now to calculate the time for the next rating cycle (119905 = 2)applying Markov chain we obtain

1198752= 1198752

1198752=

[[[[[

[

09469 00524 00007 0 0

0 09456 00536 00008 0

0 0 09447 00545 00008

0 0 0 09460 00540

0 0 0 0 10000

]]]]]

]

(21)

Applying (6) vector 119876119905representing the condition state after

two periods is calculated as

1198762= 1198760sdot 1198752= [09469 00524 00007 0 0] (22)

And the normalized time using (10) is calculated as

1198771198752= 1198762sdot 1198771015840= 00135 (23)

Continuing in a similar fashion the time at which the stateschange (Table 4) is set In addition plotting this informationwe obtain the stepwise graph shown in Figure 11 Figure 11also shows the upper and lower bounds for the deteriorationof navigation steel structures

6 Conclusions

The aim of this study was to present deterioration examplesof navigation steel structures and to develop a results-based model of current inspections Because of this studya deterioration model was developed with the assistance ofreal inspection data provided by the New York Departmentof Transportation The results presented in Figure 11 indicatethat there is a marked correlation between the proposeddeteriorationmodel and the actual data presented in Figure 8For example the model indicates that the condition statereaches level 2 at 3571 years and state 3 at 7479 yearsCompared with Figure 8 it indicates that the condition statereaches state 2 at 3612 years and state 3 at 7261 years Thedifference between the model and the actual data was about1

These results suggest that using the method presentedin this paper you may develop a deterioration model that

Journal of Applied Mathematics 7

0

10

20

30

40

50

60

70

1 2 3 4

Condition state

Freq

uenc

y

Condition state distribution

Figure 10 Distribution of condition state values generated for 20-year interval

Table 3 Transition probabilities

Condition state 1 2 3 4 51 0973 00272 0972 00283 0972 00284 0973 00275 1000

Table 4 Time at which the states change

State Time (years)1 02 36693 74694 127045 30000

reflects actual results Therefore predicting future deterio-ration of these structures is possible by using this modelThe model represents a vital starting point in predictingdeterioration which can continue updating and recalibratingas more data becomes available

For this study the authors used the Weibull distributionto generate synthetic data for condition states However if thecurrent data curve has different morphology they could useother (normal lognormal) distributions

In addition to predicting the deterioration themodel canbe useful in scheduling inspections Currently inspectionsare programmed without any consideration of anticipateddeterioration If a deterioration model is used as a refer-ence these inspections can be programmed by the modelsuggesting better anticipation of when the structure must bemaintained to avoid emergency repairs

The deterioration model is applicable in combinationwith a life cycle analysis on the prediction and optimal repaircosts Additionally you can determine the optimum pointwhen inspections are necessary to maintain the structuralsystem in a reliable condition

1

2

3

4

5

0 25 50 75 100 125 150 175 200 225 250 275 300

Con

ditio

n st

ate

Time (years)Deterioration model

Upper bound

Lower bound

Figure 11 Markov chain deterioration model

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The Navigation Systems Research Program of the US ArmyCorps of Engineers and the Information Technology Labo-ratory of the US Army Engineer Research and DevelopmentCenter funded the development of this research The authorsoffer their sincere thanks to Courtney Tuminello for theeditorial reviews during the preparation of the paper whichwere instrumental in its publication

References

[1] S Bulusu and K C Sinha ldquoComparison of methodologies topredict bridge deteriorationrdquo Transportation Research Recordno 1597 pp 34ndash42 1997

[2] S Madanat R Mishalani and W H W Ibrahim ldquoEstimationof infrastructure transition probabilities from condition ratingdatardquo Journal of Infrastructure Systems vol 1 no 2 pp 120ndash1251995

[3] S MMadanat M G Karlaftis and P S McCarthy ldquoProbabilis-tic infrastructure deteriorationmodels with panel datardquo Journalof Infrastructure Systems vol 3 no 1 pp 4ndash9 1997

[4] G Morcous H Rivard and A M Hanna ldquoModeling bridgedeterioration using case-based reasoningrdquo Journal of Infrastruc-ture Systems vol 8 no 3 pp 86ndash95 2002

[5] Headquarters US Army Corps of Engineers ldquoPeriodic inspec-tion and continuing evaluation of completed civil works struc-turesrdquo Engineer Regulation 1110-2-100 Headquarters US ArmyCorps of Engineers Washington DC USA 1995

[6] Headquarters US Army Corps of Engineers ldquoResponsibilityfor hydraulic steel structuresrdquo Engineer Regulation 1110-2-8157Headquarters US Army Corps of Engineers Washington DCUSA 2009

[7] Headquarters US Army Corps of Engineers ldquoInspection eval-uation and repair of hydraulic steel structuresrdquo Engineer Man-ual 1110-2-6054 Headquarters US Army Corps of EngineersWashington DC USA 2001

8 Journal of Applied Mathematics

[8] J J Ortiz-Garcıa S B Costello and M S Snaith ldquoDerivationof transition probability matrices for pavement deteriorationmodelingrdquo Journal of Transportation Engineering vol 132 no2 pp 141ndash161 2006

[9] A K Agrawal A Kawaguchi and G Qian Bridge Deterio-ration Rates New York State Department of TransportationTransportation Infrastructure Research Consortium AlbanyNY USA 2008

[10] G A Riveros J L Ayala-Burgos and J Perez NumericalInvestigation of Miter Gates ERDCITL-TR-09-1 US ArmyEngineer Research and Development Center Vicksburg MissUSA 2009

[11] Federal Highway Administration ldquoRecording and coding guidefor the structure inventory and appraisal of the nationrsquos bridgesrdquoReport No FHWA-PD-96-001 Office of Engineering BridgeDivision Bridge Management Branch Washington DC USA1995

[12] American Association of State Highway and TransportationOfficials Guide for Commonly Recognized Structural Elementsand Its 2002 Interim Revisions American Association of StateHighway and Transportation Officials Washington DC USA2002

[13] P D Thomson and R W Shepard ldquoAASHTO commonly-recognized bridge elements successful applications and lessonslearnedrdquo in Proceedings of the National Workshop on CommonlyRecognizedMeasures for Maintenance American Association ofState Highway and Transportation Officials Washington DCUSA June 2000

[14] P Sauser and G Riveros A System for Collecting and Com-piling Condition Data for Hydraulic Steel Structures for Usein the Assessment of Risk and Reliability and Prioritization ofMaintenance and Repairs Miter gates Report 1 ERDCITL TR-09-04 US Army Engineer Research and Development CenterVicksburg Miss USA 2009

[15] T Micevski G Kuczera and P Coombes ldquoMarkov modelfor storm water pipe deteriorationrdquo Journal of InfrastructureSystems vol 8 no 2 pp 49ndash56 2002

[16] P D Destefano and D A Grivas ldquoMethod for estimatingtransition probability in bridge deterioration modelsrdquo Journalof Infrastructure Systems vol 4 no 2 pp 56ndash62 1998

[17] H-S Baik H S Jeong and D M Abraham ldquoEstimatingtransition probabilities in Markov chain-based deteriorationmodels for management of wastewater systemsrdquo Journal ofWater Resources Planning and Management vol 132 no 1 pp15ndash24 2006

Journal of Applied Mathematics 5

Table 1 Five condition states

Number Condition Description1 Protected Member is sound functioning properly and lacking in deficiency

2 Exposed Members show beginning signs of deficiency but are still sound and functioning as intended There is noimpact on performance or reliability

3 Attacked Deficiency has advanced and the member still functions as intended but if continued unabateddeterioration will lead to the next condition state

4 Damaged Deficiency has advanced to the point that function may be impaired

5 Failed Deficiency has advanced to the point that the member no longer serves its intended function and safety isimpacted

Table 2 Time at which condition states change (from Figure 8)

State Time of change (years)1 02 63113 72614 109115 14561

where

1198771015840= 119877119879 (11)

When using the process to simulate deterioration the follow-ing condition applies

119901119894119895= 0 for 119894 gt 119895 (12)

This is because the condition of a deteriorating elementcannot return to a previous state (a better condition) withoutexternal intervention That is the probability of an elementreturning to a previous condition is always zero

When an element reaches its worst state (failure state) thefollowing condition applies

119901119898119898= 1 (13)

This indicates the element has deteriorated to the pointof failure and will remain in that state Consequently thegeneral form of the transition probability matrix defined fora deteriorating element is

119875 =

[[[[[[

[

119901111199011211990113sdot sdot sdot 1199011119898

0 1199012211990123sdot sdot sdot 1199012119898

0 0 11990133sdot sdot sdot 1199013119898

0 0 0 sdot sdot sdot 1

]]]]]]

]

(14)

A further restriction allowing the condition to deteriorate byno more than one state in one rating cycle is commonly usedin deterioration modeling The transition probability matrixis indicated as

119875 =

[[[[[[

[

11990111119901120 sdot sdot sdot 0

0 1199012211990123sdot sdot sdot 0

0 0 11990133sdot sdot sdot 0

0 0 0 sdot sdot sdot 1

]]]]]]

]

(15)

However some SHS inspection reports have shown that thestructure has changed by more than one state during theinspection period therefore the transition probabilitymatrixdefined in (11) may better fit actual inspection data

5 Derivation of Transition Probabilities

There are several methods for deriving a transition prob-ability matrix The methods include expert opinion linearregression and Poisson regression [2] Since the availabledata containing condition states are limited for navigationstructures the development of a probabilistic method is pro-posed that can be updated as data will become available Themain goal is to develop a method and verifying this methodas actual data becomes available This allows confidence inthe use of the method to predict future deterioration ofhydraulic steel structures The New York State Departmentof Transportation provided the data used in the developmentof this method This data was applicable not only because itwas accessible but also because it represents condition statedata for thousands of steel bridge elements over a periodof eighty years of inspections Additionally the same effectsthat cause the deterioration of navigation structures (loss ofa protective system corrosion cracking and fatigue impactand overloads) cause steel bridge element deteriorationFigure 8 shows the data used

Fluctuations in the data as can be seen between 30 and 40years occur because the data represent the average conditionstate of many elements To eliminate the fluctuations andmake the data more manageable a linear regression equationwas calculated as follows

119910 = 00274119909 + 10104 (16)

where 119909 is the age in years and 119910 is the condition stateThe authors calculated condition state values at ten-year

intervals by using (13)The calculated values were used as theaverage condition state at each interval Using Weibull dis-tribution and a Latin hypercube simulation (LHS) syntheticrandom condition state values were generated to represent arange of condition states at each ten-year intervalThe authorsused Weibull distribution parameters for each interval toyield approximately the same average values represented inFigure 8

Figure 9 shows the synthetic values (vertical points)generated to simulate a range of condition states at each ten-year interval The authors generated one-thousand random

6 Journal of Applied Mathematics

NY Department of Transportation data

y = 00274x + 10104

Average condition ratingLinear regression

0 101

2

3

4

5

20 30 40 50 60 70 80

Age (years)

Con

ditio

n st

ate

Figure 8 New York Department of Transportation condition statedata

values at each interval The diagonal line crosses through theaverage value of each ten-year interval

Figure 10 shows the distribution of values generatedfor the 20-year interval These superimposed values are inFigure 9 In addition similar distributions of values weregenerated for each of the other intervals

Using the generated condition state values for eachinterval the transition probabilities were calculated as

119901119894119895=

119873119894119895

119873119894

(17)

where 119873119894119895

is the number of elements that change fromcondition 119894 to condition 119895 after one interval and 119873

119894is the

number of elements that were in condition 119894 in the previousinterval

Table 3 shows the transition probability valuesApplying Markov chain

119875 =

[[[[[

[

0973 0027 0 0 0

0 0972 0028 0 0

0 0 0972 0028 0

0 0 0 0973 0027

0 0 0 0 1000

]]]]]

]

(18)

Express the initial state of a new element as

1198760= [1 0 0 0 0] (19)

The normalized vector of time when the measured conditionstates change is as follows

119877 = [0 025 050 075 100] (20)

010

20

30

40

50

10 20 30 40 50 60 70 80 90 100

Age (years)Condition states (Weibull distribution)

Con

ditio

n st

ate

Figure 9 Synthetic condition state values

Now to calculate the time for the next rating cycle (119905 = 2)applying Markov chain we obtain

1198752= 1198752

1198752=

[[[[[

[

09469 00524 00007 0 0

0 09456 00536 00008 0

0 0 09447 00545 00008

0 0 0 09460 00540

0 0 0 0 10000

]]]]]

]

(21)

Applying (6) vector 119876119905representing the condition state after

two periods is calculated as

1198762= 1198760sdot 1198752= [09469 00524 00007 0 0] (22)

And the normalized time using (10) is calculated as

1198771198752= 1198762sdot 1198771015840= 00135 (23)

Continuing in a similar fashion the time at which the stateschange (Table 4) is set In addition plotting this informationwe obtain the stepwise graph shown in Figure 11 Figure 11also shows the upper and lower bounds for the deteriorationof navigation steel structures

6 Conclusions

The aim of this study was to present deterioration examplesof navigation steel structures and to develop a results-based model of current inspections Because of this studya deterioration model was developed with the assistance ofreal inspection data provided by the New York Departmentof Transportation The results presented in Figure 11 indicatethat there is a marked correlation between the proposeddeteriorationmodel and the actual data presented in Figure 8For example the model indicates that the condition statereaches level 2 at 3571 years and state 3 at 7479 yearsCompared with Figure 8 it indicates that the condition statereaches state 2 at 3612 years and state 3 at 7261 years Thedifference between the model and the actual data was about1

These results suggest that using the method presentedin this paper you may develop a deterioration model that

Journal of Applied Mathematics 7

0

10

20

30

40

50

60

70

1 2 3 4

Condition state

Freq

uenc

y

Condition state distribution

Figure 10 Distribution of condition state values generated for 20-year interval

Table 3 Transition probabilities

Condition state 1 2 3 4 51 0973 00272 0972 00283 0972 00284 0973 00275 1000

Table 4 Time at which the states change

State Time (years)1 02 36693 74694 127045 30000

reflects actual results Therefore predicting future deterio-ration of these structures is possible by using this modelThe model represents a vital starting point in predictingdeterioration which can continue updating and recalibratingas more data becomes available

For this study the authors used the Weibull distributionto generate synthetic data for condition states However if thecurrent data curve has different morphology they could useother (normal lognormal) distributions

In addition to predicting the deterioration themodel canbe useful in scheduling inspections Currently inspectionsare programmed without any consideration of anticipateddeterioration If a deterioration model is used as a refer-ence these inspections can be programmed by the modelsuggesting better anticipation of when the structure must bemaintained to avoid emergency repairs

The deterioration model is applicable in combinationwith a life cycle analysis on the prediction and optimal repaircosts Additionally you can determine the optimum pointwhen inspections are necessary to maintain the structuralsystem in a reliable condition

1

2

3

4

5

0 25 50 75 100 125 150 175 200 225 250 275 300

Con

ditio

n st

ate

Time (years)Deterioration model

Upper bound

Lower bound

Figure 11 Markov chain deterioration model

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The Navigation Systems Research Program of the US ArmyCorps of Engineers and the Information Technology Labo-ratory of the US Army Engineer Research and DevelopmentCenter funded the development of this research The authorsoffer their sincere thanks to Courtney Tuminello for theeditorial reviews during the preparation of the paper whichwere instrumental in its publication

References

[1] S Bulusu and K C Sinha ldquoComparison of methodologies topredict bridge deteriorationrdquo Transportation Research Recordno 1597 pp 34ndash42 1997

[2] S Madanat R Mishalani and W H W Ibrahim ldquoEstimationof infrastructure transition probabilities from condition ratingdatardquo Journal of Infrastructure Systems vol 1 no 2 pp 120ndash1251995

[3] S MMadanat M G Karlaftis and P S McCarthy ldquoProbabilis-tic infrastructure deteriorationmodels with panel datardquo Journalof Infrastructure Systems vol 3 no 1 pp 4ndash9 1997

[4] G Morcous H Rivard and A M Hanna ldquoModeling bridgedeterioration using case-based reasoningrdquo Journal of Infrastruc-ture Systems vol 8 no 3 pp 86ndash95 2002

[5] Headquarters US Army Corps of Engineers ldquoPeriodic inspec-tion and continuing evaluation of completed civil works struc-turesrdquo Engineer Regulation 1110-2-100 Headquarters US ArmyCorps of Engineers Washington DC USA 1995

[6] Headquarters US Army Corps of Engineers ldquoResponsibilityfor hydraulic steel structuresrdquo Engineer Regulation 1110-2-8157Headquarters US Army Corps of Engineers Washington DCUSA 2009

[7] Headquarters US Army Corps of Engineers ldquoInspection eval-uation and repair of hydraulic steel structuresrdquo Engineer Man-ual 1110-2-6054 Headquarters US Army Corps of EngineersWashington DC USA 2001

8 Journal of Applied Mathematics

[8] J J Ortiz-Garcıa S B Costello and M S Snaith ldquoDerivationof transition probability matrices for pavement deteriorationmodelingrdquo Journal of Transportation Engineering vol 132 no2 pp 141ndash161 2006

[9] A K Agrawal A Kawaguchi and G Qian Bridge Deterio-ration Rates New York State Department of TransportationTransportation Infrastructure Research Consortium AlbanyNY USA 2008

[10] G A Riveros J L Ayala-Burgos and J Perez NumericalInvestigation of Miter Gates ERDCITL-TR-09-1 US ArmyEngineer Research and Development Center Vicksburg MissUSA 2009

[11] Federal Highway Administration ldquoRecording and coding guidefor the structure inventory and appraisal of the nationrsquos bridgesrdquoReport No FHWA-PD-96-001 Office of Engineering BridgeDivision Bridge Management Branch Washington DC USA1995

[12] American Association of State Highway and TransportationOfficials Guide for Commonly Recognized Structural Elementsand Its 2002 Interim Revisions American Association of StateHighway and Transportation Officials Washington DC USA2002

[13] P D Thomson and R W Shepard ldquoAASHTO commonly-recognized bridge elements successful applications and lessonslearnedrdquo in Proceedings of the National Workshop on CommonlyRecognizedMeasures for Maintenance American Association ofState Highway and Transportation Officials Washington DCUSA June 2000

[14] P Sauser and G Riveros A System for Collecting and Com-piling Condition Data for Hydraulic Steel Structures for Usein the Assessment of Risk and Reliability and Prioritization ofMaintenance and Repairs Miter gates Report 1 ERDCITL TR-09-04 US Army Engineer Research and Development CenterVicksburg Miss USA 2009

[15] T Micevski G Kuczera and P Coombes ldquoMarkov modelfor storm water pipe deteriorationrdquo Journal of InfrastructureSystems vol 8 no 2 pp 49ndash56 2002

[16] P D Destefano and D A Grivas ldquoMethod for estimatingtransition probability in bridge deterioration modelsrdquo Journalof Infrastructure Systems vol 4 no 2 pp 56ndash62 1998

[17] H-S Baik H S Jeong and D M Abraham ldquoEstimatingtransition probabilities in Markov chain-based deteriorationmodels for management of wastewater systemsrdquo Journal ofWater Resources Planning and Management vol 132 no 1 pp15ndash24 2006

6 Journal of Applied Mathematics

NY Department of Transportation data

y = 00274x + 10104

Average condition ratingLinear regression

0 101

2

3

4

5

20 30 40 50 60 70 80

Age (years)

Con

ditio

n st

ate

Figure 8 New York Department of Transportation condition statedata

values at each interval The diagonal line crosses through theaverage value of each ten-year interval

Figure 10 shows the distribution of values generatedfor the 20-year interval These superimposed values are inFigure 9 In addition similar distributions of values weregenerated for each of the other intervals

Using the generated condition state values for eachinterval the transition probabilities were calculated as

119901119894119895=

119873119894119895

119873119894

(17)

where 119873119894119895

is the number of elements that change fromcondition 119894 to condition 119895 after one interval and 119873

119894is the

number of elements that were in condition 119894 in the previousinterval

Table 3 shows the transition probability valuesApplying Markov chain

119875 =

[[[[[

[

0973 0027 0 0 0

0 0972 0028 0 0

0 0 0972 0028 0

0 0 0 0973 0027

0 0 0 0 1000

]]]]]

]

(18)

Express the initial state of a new element as

1198760= [1 0 0 0 0] (19)

The normalized vector of time when the measured conditionstates change is as follows

119877 = [0 025 050 075 100] (20)

010

20

30

40

50

10 20 30 40 50 60 70 80 90 100

Age (years)Condition states (Weibull distribution)

Con

ditio

n st

ate

Figure 9 Synthetic condition state values

Now to calculate the time for the next rating cycle (119905 = 2)applying Markov chain we obtain

1198752= 1198752

1198752=

[[[[[

[

09469 00524 00007 0 0

0 09456 00536 00008 0

0 0 09447 00545 00008

0 0 0 09460 00540

0 0 0 0 10000

]]]]]

]

(21)

Applying (6) vector 119876119905representing the condition state after

two periods is calculated as

1198762= 1198760sdot 1198752= [09469 00524 00007 0 0] (22)

And the normalized time using (10) is calculated as

1198771198752= 1198762sdot 1198771015840= 00135 (23)

Continuing in a similar fashion the time at which the stateschange (Table 4) is set In addition plotting this informationwe obtain the stepwise graph shown in Figure 11 Figure 11also shows the upper and lower bounds for the deteriorationof navigation steel structures

6 Conclusions

The aim of this study was to present deterioration examplesof navigation steel structures and to develop a results-based model of current inspections Because of this studya deterioration model was developed with the assistance ofreal inspection data provided by the New York Departmentof Transportation The results presented in Figure 11 indicatethat there is a marked correlation between the proposeddeteriorationmodel and the actual data presented in Figure 8For example the model indicates that the condition statereaches level 2 at 3571 years and state 3 at 7479 yearsCompared with Figure 8 it indicates that the condition statereaches state 2 at 3612 years and state 3 at 7261 years Thedifference between the model and the actual data was about1

These results suggest that using the method presentedin this paper you may develop a deterioration model that

Journal of Applied Mathematics 7

0

10

20

30

40

50

60

70

1 2 3 4

Condition state

Freq

uenc

y

Condition state distribution

Figure 10 Distribution of condition state values generated for 20-year interval

Table 3 Transition probabilities

Condition state 1 2 3 4 51 0973 00272 0972 00283 0972 00284 0973 00275 1000

Table 4 Time at which the states change

State Time (years)1 02 36693 74694 127045 30000

reflects actual results Therefore predicting future deterio-ration of these structures is possible by using this modelThe model represents a vital starting point in predictingdeterioration which can continue updating and recalibratingas more data becomes available

For this study the authors used the Weibull distributionto generate synthetic data for condition states However if thecurrent data curve has different morphology they could useother (normal lognormal) distributions

In addition to predicting the deterioration themodel canbe useful in scheduling inspections Currently inspectionsare programmed without any consideration of anticipateddeterioration If a deterioration model is used as a refer-ence these inspections can be programmed by the modelsuggesting better anticipation of when the structure must bemaintained to avoid emergency repairs

The deterioration model is applicable in combinationwith a life cycle analysis on the prediction and optimal repaircosts Additionally you can determine the optimum pointwhen inspections are necessary to maintain the structuralsystem in a reliable condition

1

2

3

4

5

0 25 50 75 100 125 150 175 200 225 250 275 300

Con

ditio

n st

ate

Time (years)Deterioration model

Upper bound

Lower bound

Figure 11 Markov chain deterioration model

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The Navigation Systems Research Program of the US ArmyCorps of Engineers and the Information Technology Labo-ratory of the US Army Engineer Research and DevelopmentCenter funded the development of this research The authorsoffer their sincere thanks to Courtney Tuminello for theeditorial reviews during the preparation of the paper whichwere instrumental in its publication

References

[1] S Bulusu and K C Sinha ldquoComparison of methodologies topredict bridge deteriorationrdquo Transportation Research Recordno 1597 pp 34ndash42 1997

[2] S Madanat R Mishalani and W H W Ibrahim ldquoEstimationof infrastructure transition probabilities from condition ratingdatardquo Journal of Infrastructure Systems vol 1 no 2 pp 120ndash1251995

[3] S MMadanat M G Karlaftis and P S McCarthy ldquoProbabilis-tic infrastructure deteriorationmodels with panel datardquo Journalof Infrastructure Systems vol 3 no 1 pp 4ndash9 1997

[4] G Morcous H Rivard and A M Hanna ldquoModeling bridgedeterioration using case-based reasoningrdquo Journal of Infrastruc-ture Systems vol 8 no 3 pp 86ndash95 2002

[5] Headquarters US Army Corps of Engineers ldquoPeriodic inspec-tion and continuing evaluation of completed civil works struc-turesrdquo Engineer Regulation 1110-2-100 Headquarters US ArmyCorps of Engineers Washington DC USA 1995

[6] Headquarters US Army Corps of Engineers ldquoResponsibilityfor hydraulic steel structuresrdquo Engineer Regulation 1110-2-8157Headquarters US Army Corps of Engineers Washington DCUSA 2009

[7] Headquarters US Army Corps of Engineers ldquoInspection eval-uation and repair of hydraulic steel structuresrdquo Engineer Man-ual 1110-2-6054 Headquarters US Army Corps of EngineersWashington DC USA 2001

8 Journal of Applied Mathematics

[8] J J Ortiz-Garcıa S B Costello and M S Snaith ldquoDerivationof transition probability matrices for pavement deteriorationmodelingrdquo Journal of Transportation Engineering vol 132 no2 pp 141ndash161 2006

[9] A K Agrawal A Kawaguchi and G Qian Bridge Deterio-ration Rates New York State Department of TransportationTransportation Infrastructure Research Consortium AlbanyNY USA 2008

[10] G A Riveros J L Ayala-Burgos and J Perez NumericalInvestigation of Miter Gates ERDCITL-TR-09-1 US ArmyEngineer Research and Development Center Vicksburg MissUSA 2009

[11] Federal Highway Administration ldquoRecording and coding guidefor the structure inventory and appraisal of the nationrsquos bridgesrdquoReport No FHWA-PD-96-001 Office of Engineering BridgeDivision Bridge Management Branch Washington DC USA1995

[12] American Association of State Highway and TransportationOfficials Guide for Commonly Recognized Structural Elementsand Its 2002 Interim Revisions American Association of StateHighway and Transportation Officials Washington DC USA2002

[13] P D Thomson and R W Shepard ldquoAASHTO commonly-recognized bridge elements successful applications and lessonslearnedrdquo in Proceedings of the National Workshop on CommonlyRecognizedMeasures for Maintenance American Association ofState Highway and Transportation Officials Washington DCUSA June 2000

[14] P Sauser and G Riveros A System for Collecting and Com-piling Condition Data for Hydraulic Steel Structures for Usein the Assessment of Risk and Reliability and Prioritization ofMaintenance and Repairs Miter gates Report 1 ERDCITL TR-09-04 US Army Engineer Research and Development CenterVicksburg Miss USA 2009

[15] T Micevski G Kuczera and P Coombes ldquoMarkov modelfor storm water pipe deteriorationrdquo Journal of InfrastructureSystems vol 8 no 2 pp 49ndash56 2002

[16] P D Destefano and D A Grivas ldquoMethod for estimatingtransition probability in bridge deterioration modelsrdquo Journalof Infrastructure Systems vol 4 no 2 pp 56ndash62 1998

[17] H-S Baik H S Jeong and D M Abraham ldquoEstimatingtransition probabilities in Markov chain-based deteriorationmodels for management of wastewater systemsrdquo Journal ofWater Resources Planning and Management vol 132 no 1 pp15ndash24 2006

Journal of Applied Mathematics 7

0

10

20

30

40

50

60

70

1 2 3 4

Condition state

Freq

uenc

y

Condition state distribution

Figure 10 Distribution of condition state values generated for 20-year interval

Table 3 Transition probabilities

Condition state 1 2 3 4 51 0973 00272 0972 00283 0972 00284 0973 00275 1000

Table 4 Time at which the states change

State Time (years)1 02 36693 74694 127045 30000

reflects actual results Therefore predicting future deterio-ration of these structures is possible by using this modelThe model represents a vital starting point in predictingdeterioration which can continue updating and recalibratingas more data becomes available

For this study the authors used the Weibull distributionto generate synthetic data for condition states However if thecurrent data curve has different morphology they could useother (normal lognormal) distributions

In addition to predicting the deterioration themodel canbe useful in scheduling inspections Currently inspectionsare programmed without any consideration of anticipateddeterioration If a deterioration model is used as a refer-ence these inspections can be programmed by the modelsuggesting better anticipation of when the structure must bemaintained to avoid emergency repairs

The deterioration model is applicable in combinationwith a life cycle analysis on the prediction and optimal repaircosts Additionally you can determine the optimum pointwhen inspections are necessary to maintain the structuralsystem in a reliable condition

1

2

3

4

5

0 25 50 75 100 125 150 175 200 225 250 275 300

Con

ditio

n st

ate

Time (years)Deterioration model

Upper bound

Lower bound

Figure 11 Markov chain deterioration model

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The Navigation Systems Research Program of the US ArmyCorps of Engineers and the Information Technology Labo-ratory of the US Army Engineer Research and DevelopmentCenter funded the development of this research The authorsoffer their sincere thanks to Courtney Tuminello for theeditorial reviews during the preparation of the paper whichwere instrumental in its publication

References

[1] S Bulusu and K C Sinha ldquoComparison of methodologies topredict bridge deteriorationrdquo Transportation Research Recordno 1597 pp 34ndash42 1997

[2] S Madanat R Mishalani and W H W Ibrahim ldquoEstimationof infrastructure transition probabilities from condition ratingdatardquo Journal of Infrastructure Systems vol 1 no 2 pp 120ndash1251995

[3] S MMadanat M G Karlaftis and P S McCarthy ldquoProbabilis-tic infrastructure deteriorationmodels with panel datardquo Journalof Infrastructure Systems vol 3 no 1 pp 4ndash9 1997

[4] G Morcous H Rivard and A M Hanna ldquoModeling bridgedeterioration using case-based reasoningrdquo Journal of Infrastruc-ture Systems vol 8 no 3 pp 86ndash95 2002

[5] Headquarters US Army Corps of Engineers ldquoPeriodic inspec-tion and continuing evaluation of completed civil works struc-turesrdquo Engineer Regulation 1110-2-100 Headquarters US ArmyCorps of Engineers Washington DC USA 1995

[6] Headquarters US Army Corps of Engineers ldquoResponsibilityfor hydraulic steel structuresrdquo Engineer Regulation 1110-2-8157Headquarters US Army Corps of Engineers Washington DCUSA 2009

[7] Headquarters US Army Corps of Engineers ldquoInspection eval-uation and repair of hydraulic steel structuresrdquo Engineer Man-ual 1110-2-6054 Headquarters US Army Corps of EngineersWashington DC USA 2001

8 Journal of Applied Mathematics

[8] J J Ortiz-Garcıa S B Costello and M S Snaith ldquoDerivationof transition probability matrices for pavement deteriorationmodelingrdquo Journal of Transportation Engineering vol 132 no2 pp 141ndash161 2006

[9] A K Agrawal A Kawaguchi and G Qian Bridge Deterio-ration Rates New York State Department of TransportationTransportation Infrastructure Research Consortium AlbanyNY USA 2008

[10] G A Riveros J L Ayala-Burgos and J Perez NumericalInvestigation of Miter Gates ERDCITL-TR-09-1 US ArmyEngineer Research and Development Center Vicksburg MissUSA 2009

[11] Federal Highway Administration ldquoRecording and coding guidefor the structure inventory and appraisal of the nationrsquos bridgesrdquoReport No FHWA-PD-96-001 Office of Engineering BridgeDivision Bridge Management Branch Washington DC USA1995

[12] American Association of State Highway and TransportationOfficials Guide for Commonly Recognized Structural Elementsand Its 2002 Interim Revisions American Association of StateHighway and Transportation Officials Washington DC USA2002

[13] P D Thomson and R W Shepard ldquoAASHTO commonly-recognized bridge elements successful applications and lessonslearnedrdquo in Proceedings of the National Workshop on CommonlyRecognizedMeasures for Maintenance American Association ofState Highway and Transportation Officials Washington DCUSA June 2000

[14] P Sauser and G Riveros A System for Collecting and Com-piling Condition Data for Hydraulic Steel Structures for Usein the Assessment of Risk and Reliability and Prioritization ofMaintenance and Repairs Miter gates Report 1 ERDCITL TR-09-04 US Army Engineer Research and Development CenterVicksburg Miss USA 2009

[15] T Micevski G Kuczera and P Coombes ldquoMarkov modelfor storm water pipe deteriorationrdquo Journal of InfrastructureSystems vol 8 no 2 pp 49ndash56 2002

[16] P D Destefano and D A Grivas ldquoMethod for estimatingtransition probability in bridge deterioration modelsrdquo Journalof Infrastructure Systems vol 4 no 2 pp 56ndash62 1998

[17] H-S Baik H S Jeong and D M Abraham ldquoEstimatingtransition probabilities in Markov chain-based deteriorationmodels for management of wastewater systemsrdquo Journal ofWater Resources Planning and Management vol 132 no 1 pp15ndash24 2006

8 Journal of Applied Mathematics

[8] J J Ortiz-Garcıa S B Costello and M S Snaith ldquoDerivationof transition probability matrices for pavement deteriorationmodelingrdquo Journal of Transportation Engineering vol 132 no2 pp 141ndash161 2006

[9] A K Agrawal A Kawaguchi and G Qian Bridge Deterio-ration Rates New York State Department of TransportationTransportation Infrastructure Research Consortium AlbanyNY USA 2008

[10] G A Riveros J L Ayala-Burgos and J Perez NumericalInvestigation of Miter Gates ERDCITL-TR-09-1 US ArmyEngineer Research and Development Center Vicksburg MissUSA 2009

[11] Federal Highway Administration ldquoRecording and coding guidefor the structure inventory and appraisal of the nationrsquos bridgesrdquoReport No FHWA-PD-96-001 Office of Engineering BridgeDivision Bridge Management Branch Washington DC USA1995

[12] American Association of State Highway and TransportationOfficials Guide for Commonly Recognized Structural Elementsand Its 2002 Interim Revisions American Association of StateHighway and Transportation Officials Washington DC USA2002

[13] P D Thomson and R W Shepard ldquoAASHTO commonly-recognized bridge elements successful applications and lessonslearnedrdquo in Proceedings of the National Workshop on CommonlyRecognizedMeasures for Maintenance American Association ofState Highway and Transportation Officials Washington DCUSA June 2000

[14] P Sauser and G Riveros A System for Collecting and Com-piling Condition Data for Hydraulic Steel Structures for Usein the Assessment of Risk and Reliability and Prioritization ofMaintenance and Repairs Miter gates Report 1 ERDCITL TR-09-04 US Army Engineer Research and Development CenterVicksburg Miss USA 2009

[15] T Micevski G Kuczera and P Coombes ldquoMarkov modelfor storm water pipe deteriorationrdquo Journal of InfrastructureSystems vol 8 no 2 pp 49ndash56 2002

[16] P D Destefano and D A Grivas ldquoMethod for estimatingtransition probability in bridge deterioration modelsrdquo Journalof Infrastructure Systems vol 4 no 2 pp 56ndash62 1998

[17] H-S Baik H S Jeong and D M Abraham ldquoEstimatingtransition probabilities in Markov chain-based deteriorationmodels for management of wastewater systemsrdquo Journal ofWater Resources Planning and Management vol 132 no 1 pp15ndash24 2006