ENGINEER - Vol. XXXX, No. 03, pp. 16-24,2007© The Institution of Engineers, Sri Lanka

Cost and Reliability Interaction in Bridge MaintenanceP.B.R. Dissanayake and P.A.K. Karunananda

Abstract: This paper presents the relation between the cost of maintenance activities of a bridge andits reliability. Initially, based on fundamentals of reliability theory, a general methodology on bridgemaintenance is introduced. Based on this maintenance strategy, the cost associated with differentmaintenance activities and its reliability increment is described. Having this kind of maintenancestrategy is useful in bridge maintenance for a better understanding of the associated cost expenditures.

Keywords: Bridge maintenance, Reliability index, Failure probability, Cost

1. Introduction

In today's world, the quality of human life andeconomic progress of a particular countrydepend on the quantity, quality and efficiency ofits infrastructures system [1]. However, many ofthe engineering structures in today's world aregetting old and a large number of these is inneed of maintenance, rehabilitation orreplacement [2]. Among these, Highways andassociated structures play an important role inthe ground transportation systems and itconstitutes a considerable investment ininfrastructure and directly affects theproductivity of the road transport industry of acountry. Hence, condition estimation and propermaintenance of bridges have a widespreadsignificance for continuous research.

The history of developing a maintenancestrategy for bridges is probably a few decadesold. However, in the process of bridgemaintenance, a lack of a sufficient theoreticalbackground has led to the non-availability of awidespread of accepted methods. As most of thecurrent accepted strategies are based on /isualinspections with varying time intervals [3], theuncertainty of the human factor is adisadvantage. In fact, human experience shouldalways be backed with sufficient scientificknowledge. It is only then that any strategygives an acceptable performance. Howeverprevailing uncertainties in current proceduresmake a whole set of problems. The results of thisinadequate knowledge in bridge maintenanceare greater financial expenditure due to frequentmaintenance activities, unforeseen bridgedamages and accidental bridge failures. Theselosses in terms of financial, physical, reliabilityand other resources are not acceptableirrespective of the wealth of the country.Furthermore, when there is maintenance to be

carried out with different options, there is nodefined method of selecting an optionconsidered in relation to its cost and theincreased service life of the bridge. But the costconsideration is very essential in bridgemaintenance. The complexity of life predictionand bridge maintenance has hindered a lot ofresearchers in formulating a defined method oflife prediction and a related cost optimizedbridge maintenance strategy for bridges.

Thus, any future methodologies on conditionestimation and maintenance should be based onresults of field studies as well as scientificinferences to gain widespread acceptanceamong practicing engineers. Furthermore, sincethere are many uncertainties associated withcurrent procedures, it is essential that anymethodology proposed for the future be basedon a probabilistic approach which caters fordifferent uncertainties that have not beenaddressed by current ones. In this situation,reliability based methods can provide therational approach to use scarce resourcesefficiently while maintaining a prescribed levelof reliability of a structure throughout itsdesignated service life.

While deterministic structural analyses provideoutput that is precise and exact, it ignores theuncertainties of the output. These uncertaintiesmust be taken into account to assess the safetyand the performance of the structure.

A reliability analysis can be used to explore thefailure process and show which failure path ismost likely to occur. Uncertainty modeling in

Eng. (Dr.) P.B.R. Dissanayake, C. Eng., MIE(SL), B.Sc. Eng. (Hons)Peradeniya), M.Sc.. (Ehime), PhD. (Ehime), Senior Lecturer, Departmentof Civil Engineering, University of Peradeniya, Peradeniya.

Eng. P.A.K. Karunananda, AMIE(SL), B.Sc. Eng. (Hons) Peradeniya),M. Phil (Peradeniya), Research Student, Department of CivilEngineering, University of Peradeniya, Peradeniya.

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structural evaluation can incorporate both theon-site inspection results and nondestructiveevaluation data to provide a better estimation ofthe possibility of failure.

The reliability approach quantifies that risk inprobabilistic terms by accounting for therandomness and correlation of all relevantvariables in the analysis [4]. This probabilisticconsideration of variables makes structuralreliability more accurate over otherconventional deterministic approaches. Theconcept of structural reliability has been usedfor over four decades in condition estimation ofstructures including bridges [5]. However, thedata obtained from these research studies havenot been incorporated into the bridgemanagement databases in an effective way.Hence, there is an urgent need to combinereliability based research studies with conditionestimation and maintenance of bridges tooptimize resources expenditures in bridgemanagement [6].

In Sri Lanka, an expansion and development ofhighways happened in the 1960's and 1970's.Now, most of these structures, at least 30 yearsold, need more attention than they needed in theoriginal state to obtain service for more years tocome. The condition of a bridge deteriorates dueto two major effects; load increment over timeand the aging effects. In a Sri Lankan context,bridge infrastructure is not subjected to heavyloads as in most of the developed countries.However, the aging effects resulting fromdegradation processes such as corrosion shouldnot be unaccounted. Continuous actions of suchdegradation processes have created some of theworst problems at some bridge sites in thenational bridge network of Sri Lanka. Theultimate impact of these processes is thecomplete collapse of bridges. A well knowncatastrophe is the collapse of Paragastota bridgein Bandaragama in the year 2000. Had properinspection and maintenance been carried out onthat bridge, the collapse could have beenavoided. There are maintenance activitiesvarying in different degrees to reduce the effectsof these degradation processes on bridgeinfrastructures. However, the constraint forthese activities is their related high costs. Hence,there is a need to study the relation between costand reliability of bridges in terms of bridgemaintenance.

Sri Lanka being a developing country and thegovernment controlled Road DevelopmentAuthority (RDA) being the organizationresponsible for national bridge network, a largeamount of public money has to be spent on theroad sector every year. From this amount, asignificant amount of money goes on bridgemaintenance. In this context, a carefulunderstanding of cost and reliability interactionwill drastically reduce resource expenditure onbridge maintenance.

The central issue in bridge maintenance is, howeffective a particular maintenance activity interms of the money spent is. Such aconsideration is of extreme importance whenthere are a number of options of maintenanceactivities. It is normal that maintenanceactivities with a higher reliability incrementhave to be done at higher costs. An increase inreliability should be looked at with the intendedincrease in service life of the bridge in order tounderstand the effectiveness achieved at thehigher cost [7,8].

2. Methodology

2.1 Development of Reliability Expressions

For a simple structural member selected atrandom from a population with a knowndistribution function FR of ultimate strength R insome specified mode of failure, the probabilityof failure, P. under the action of a single knownload effect S is as follows,

(1)

where R and 5 represent resistance and loadvariables and r and 5 denote values that thosevariables can have. In general, if the load effectS is a random variable, with distributionfunction , Fs Equation 1 can be replaced by

Pf=P(R-S<0)=lFR(X)fs(X)dx (2)

where FR (x) is the distribution function forresistance variable and fs(x) is the densityfunction of the load variable. Under thecondition that R and 5 are statisticallyindependent, Equation 2 can be best understoodby plotting the density functions of R and 5denoted byy^ (r) andy^, (5) respectively. It shouldbe noted that and must necessarily have the

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same dimensions e.g. loads and load-carryingcapacities (e.g. failure strength), or bendingmoments and flexural strength etc.

Equation 2 gives the total probability of failureP as the product of the probabilities of twoindependent events, summed over all possibleoccurrences; namely the probability Pt where 5lies in the range from x to x+dx and theprobability P2 where R is less than or equal to x.It is clear that

(3)

(4)and P2=FR(x)

Under these conditions the reliability, ̂ , that isthe probability that the structure will survivewhen the load is applied, is given by

(5)

From the consideration of symmetry, it can beseen that the reliability may also be expressed as

(6)

where Fs (x) represent the distribution functionof load variable andy^, (x) represents the densityfunction of the resistance variable.

For the general case, close-form solutions do notexist for the integrals in Equations 5 and 6, butthese two equations are very important becausethey represent the mathematical background tothe reliability analysis. There are, however, anumber of special simplified cases, and with theuse of these, failure probabilities of structurescan also be calculated.

2.2 Simplification to Fundamental Case

(R and 5 are independent normally distributedvariables)

The condition that R and 5 are independentnormally distributed variables is a simplifiedapproximation to the reliability analysis. Failureprobability P can be found only if variables arenormally distributed. In case of non normal

•variables, these have to be converted toequivalent normal distributed variables andhave to follow the procedure as mentionedbelow.

pf = P(M < o)

where M = R — S

Thus U.,, = UK -

(8)

=Var[M]=ff2R+(72

s (10)and

giving

where fj,^ HR and ns are the probabilistic meanvalues of safety margin, resistance, and loadrespectively. Similarly aM, aR and as are thestandard deviations of safety margin, resistanceand load respectively. Var [M] is the variance ofthe safety margin. Since and are normallydistributed and M is a linear function of andsafety margin M is also normally distributedand the quantity (M - p.M) / aM is the unitstandard normal variable of M. According toEquation 3.7, failure probability can be foundwhen M is less than or equal to zero, thus, itwould give

(12)'M

where 0 is the standard normal distributionfunction and ,uy jj.^ as and OR are the probabilisticmean values and standard deviations of randomvariables R and 5.

The reliability index P may now be defined asthe ratio nM/aMor the number of standarddeviations by which ̂ exceeds zero as shown inFigure 1. Here m represents a general value thatM can take.

Figure 1: Illustration of the reliability index (P)

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Hence -&L =</>(- J3) ....................... (13)M

(14)

2.3 Graphical Representation of ReliabilityIndex

The reliability index /? is defined by Equation 14for the fundamental case in Equation 8. For thegeneral linear safety margin case it can beobtained by a simple geometrical interpretation.Hasofer and Lind proposed this graphicalrepresentation method of reliability index in1974 and it is sometimes referred to as Hasoferand Lind reliability index.

Consider the fundamental case with independentbasic variables R and S and the safety margin M= R- S. Let the mean values be HR and fj,s and thestandard deviations be a,, and a. Introduce

A 5

normalized random variables such that,

p.-K= o.-o= (15)

Then the failure surface f(r, s) = r - s = 0 will betransformed into a straight line in thenormalized (r', s") coordinate system as shownin Figure 3. Lower case letters (r', s} representthe values that variables R' and S' can take. Bysubstituting Equation 14 to Equation 8, thefailure surface in the (r', 5") coordinate systemcan be obtained, and it is given by,

(16)

The shortest distance from the origin to thislinear failure surface is equal to

(17)

Therefore, an alternative geometrical definition ofthe reliability index )3is the shortest distance fromthe origin to the linear failure surface as shown inFigure 2. This geometrical interpretation is shownhere for a linear safety margin with only two ba-sic variables, but it can be extended easily to a lin-ear safety margin with w basic variables.

Failure region

Figure 2: Graphical representation of reliability index

Proposed Reliability Maintenance Strategy

Studies have been done on critical failure criteriafor different types of bridges. The inspection andmaintenance strategy for different bridges can bederived from the reliability relationships intro-duced in the referenced literature [9]. In fact, theinitial reliability index of a bridge is on the de-crease as the bridge is exposed to traffic and tonature.

Reliability index

\r^ current )With maintenance

Time/(years)

Figure 3: Graphical presentation of the maintenance requirement of a bridge

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As illustrated in Fig. 3, the reliability index goesdown with time. Initially, the value of thereliability index (/3g) after construction is a highvalue ( > 6.0) and for a time period Tl it remainsthe same. This is mainly because there is sometime needed to initiate the degradationprocesses. After time Tl, the reliability indexreduces with the increase of time. During thisperiod, the important aspect of study is the ratereduction of the reliability index (a). The valueof a is dependent on the how the degradationprocess has affected the particular bridge. Forexample, a steel truss bridge located on a coastalenvironment has a higher a than a steel bridgelocated on land under the condition that, otherfactors such as load effects being the same thesalt environment triggers the corrosion on steel.When the reliability index reaches target value,"Target Reliability Index" (Ptarge), themaintenance activity should be done.Maintenance activity such as replacing a heavilycorroded girder in a steel girder concrete girderbridge. Such Essential Maintenance (EM) willincrease the reliability index by an amount 4/3.The value of increased Aft should alwaysaccompany cost. Hence, it is generally essentialto optimize increase of reliability index with cost(C). This increase of reliability index keepsconstant a for period of time T3. After that,reliability index goes down with a reduced rateof degradation of a - 8. 8 is called the reductionof deterioration rate due to maintenance activity.Its effect continues for a time period T4. Then,degradation rate becomes a again. When itreaches to fltar et, maintenance should be doneagain.

2.4 Target Reliability Index

Many studies have been carried out fordevelopment of existing reliability design codes.The American Association of State and HighwayTransportation Officials guide specification forsteel bridges (AASHTO 1990) bases theremaining life of steel bridges on the use of areliability oriented approach. In this guide, twolevels of risk are considered. For a redundantbridge, a reliability index of 2.0 is used, and fora nonredundant bridge, a reliability index of 3.0is used. However, more research should becarried out to find out target reliability index tosuit a Sri Lankan context.

3. Case Study on a masonry arch bridge

3.1 Development of a Reliability Model forMasonry Arch Bridges

During the time most of the masonry archbridges were constructed, the original designershad no idea about what kind of loading thesearch bridges would have to sustain in future. Inparticular, they had no reason to predict presentloading as they were not exposed to high trafficdensity that is prevailing at present.

With the heavy traffic both in magnitude andthe number at present, it is very much essentialto have some kind of evaluation of thesemasonry arch bridges on the basis of loadcarrying capacity that these could bear.Therefore, in the reliability analysis of masonryarch bridges, failure criterion is derived usingthe loads exerted on the bridge.

The proposed reliability expression must have aterm for load capacity that simulates the strengthvariable and it must also have a term thatexpresses the current loading. With the selectionof Provisional Axle Load (PAL) as the strengthvariable and Actual Axle Load (AAL) as the loadvariable, the reliability expression or the safetymargin concerning the masonry arch bridge canbe built. The PAL is the maximum permissibleload on the selected masonry arch bridge and theAAL is the load that is applied on the bridge at aselected time. Hence, the reliability model of themasonry arch bridges can be proposed as,

M = PAL -AAL

where M is the safety margin, PAL is theProvisional Axle Load in kN and AAL is theActual Axle Load in kN.

Both axle loads are not deterministic quantitiesin the true sense. In this regard, those areassumed to behave as random variables withsome probabilistic parameters. Therefore, thoseare modeled as variables having someprobabilistic distributions.

3.2 Selection of a Masonry Arch Bridge

Taking into consideration all the above facts, asingle spanned stone masonry arch bridge (71 /1) situated close to Hatton town in A7 road wasselected for the case study with the help of theRoad Development Authority. This bridge was

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constructed in 1918 and it has been in operation

ever since. It has the following geometric details,

Bridge length = 14 OT/

Clear span (L) = 8.8 m,

Thickness of the barrel (d) = 0.55 m,

Height of the compacted fill from the crest of thebarrel (A) = 1.50 m.

Those geometric details are as shown in Figure 4

and a photograph of the selected bridge isshown in Figure 5.

Road Surface

1.50m

Figure 4: Geometric details of the selected arch bridge

Figure 5: A view of the selected arch bridge at 71/1along A7road

The current state of the bridge is visually

satisfactory except for a few signs of water

seepage through masonry blocks in the inner

barrel of the arch bridge as shown in Figure 6.

If this is continued, blocks may become loose

and would tend to behave as individual blocks.

In fact, this was one of the main reasons for the

selection of this masonry arch bridge as a case

study. More importantly, there is a significant

increment of the traffic flow over this particular

bridge since the upgrading of road A7 and this

could significantly reduce the life of the bridge.

Hence, proper attention should be paid to the

current condition of the bridge.

3.3 Probabilistic Parameters of Actual AxleLoad (AAL) and Provisional Axle Load(PAL)

Axle load measurements over the bridge were

carried out and it was found that nearly 2570

vehicles had passed over this bridge under

consideration. From this population, a random

sample of 293 vehicles was selected as a sample

population. Axle loads were measured for thissample population.

From Axle load measurements, it is found that

AAL has a mean of 50.7 kN and standard

deviation of 29.03 kN. From the central limit

theorem, it can be concluded that AAL

measurements obey normal distribution even if

the parent distribution is not normal.

In the determination of PAL, the modified

MEXE method was used. Initial value of PAL

was measured as 184.03 kN. With the geometric

details five adjustment factors in the modified

MEXE method were calculated. Using those, the

modified PAL was calculated as 233.91 kN. In

fact, this value itself has an uncertainty. Hence,

to counter this uncertainty, three values of

Coefficient of Variation (COV) were used.

3.4 Calculation of Reliability Index and FailureProbability for the Selected Bridge

Having found the probabilistic parameters of

the PAL and AAL, it is possible to calculate the

reliability index and the failure probability.

Tabulated values of probabilistic parameters of

PAL and AAL are given in Table 1.

Table 1: Probabilistic parameters of PAL and AAL

Figure 6: Seepage through blocks

Variable

PALAAL

Mean/(kN)

233.9150.7

Standard Deviation/(kN)

COV= 0

0

COV= 0.1

23.39

COV= 0.246.78

29.03

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3.5 Results

Having known the probabilistic parameters ofthe PAL and AAL, the reliability index and thefailure probability can be obtained. These valuesare given in Table 2.

Table 2: Reliability index and failure probabilitiesfor the selected bridge.

Case

Case I: whenCOVofPAL = 0

Case II: whenCOVofPAL = 0.1

Case III: whenCOVofPAL = 0.2

ReliabilityIndex /($

6.31

4.91

3.32

FailureProbability/^)

1.5 x lO'10

4.8xlO-7

4.5x10^

From the results shown in Table 2, for first andsecond cases, the failure probabilities are lessthan 10"5. The third case represents the extremepossibility and it shows a failure probability ofmore than 10'5. Since the target failureprobability is 10"5, as a conclusion, it can beexpressed that the selected bridge is currently ina safe condition but fair attention is needed infuture regarding the axle loads on the bridge.

Detailed monitoring of the places where seepageis observed in the inner surface of the arch barrelshould also be carried out. The number of suchseepage paths should be counted periodicallyand if possible their average sizes should bemeasured. Any increase of such seepage pathsand path sizes should be thoroughlyinvestigated.

4. Cost Consideration in bridgemaintenance

4.1 Cost and reliability index interaction

In the maintenance management of bridges,there are two improvements happening in termsof the reliability of the bridge after maintenanceis carried out; increase of reliability index andreduction of the rate of deterioration ofreliability index. Hence, simplified cost functioncan be written as,

C(f) is the cost function and f3(f) is the reliabilityindex at time / and A/3(f) is the increase ofreliability index and is the reduction of thedeterioration rate of reliability index. In thiscontext, cost related to inspection stage has notbeen taken into consideration. The value of /3(f)is dependent on cost function in such a way thatlower values of fi(f) require higher values for C.

As the first case, interaction between the costand reliability index is studied. Thisconsideration of reliability index with the cost isof importance in setting a target reliabilityindex. The corresponding relationship betweencost and reliability index can be set as follows,

(19)

Where C-(l) is the cost associated with currentreliability index; Cg is the fixed cost irrespectiveof current reliability index value; /?(/) is thecurrent reliability index, p and q are the modelparameters.

Figure 7: variation of cost and reliability of a bridge.

By substituting the values at common point asthe calibration point, the parameter p can befound.

P = (Cc-C0)//3e (20)

C(t} = (18)

4.2 Cost and increase of reliability indexinteraction

The interaction between the cost and thereliability index is important to study effectivebridge maintenance. From all three parametersthat affect the bridge reliability aftermaintenance, this is the most importantinteraction. In general terms, increase ofreliability Afl(f) index is higher for Essential

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Maintenance (EM) than in the case of PreventiveMaintenance (PM). Hence the cost associatedwith Essential maintenance is higher thanpreventive maintenance. Cost function for thisinteraction can be represented as follows,

For Essential maintenance,

(21)

For Preventive maintenance,

(22)

Where CA^ft} is the cost associated with increase ofthe reliability index at the time /. C is the costcomponent independent of increase of thereliability index. Afi(t) is the increase of thereliability index at time t. P and y are the costparameters. In the case of preventivemaintenance, the fixed cost is small compared tothe latter component and hence cost function inthe case of preventive maintenance is representedby variable cost associated with reliabilityincrease. The parameter value / for Essentialmaintenance is higher than that of preventivemaintenance resulting in higher value for the costfunction in the case of EM than in PM.

4.3 Cost and reduction of deterioration rate ofreliability index

Due to the maintenance activity carried out, itinserts a reduction of deterioration of reliabilityindex 6 ft) on the deterioration rate of reliabilityindex aft) at the time t. Normally, for effect ofd(t) on the bridge maintenance, EM makes asignificant contribution whereas contributionfrom PM is not significant. The relationshipbetween 8ft) and the cost C is such that forhigher values, 8ft) the associated cost is higher.Hence cost function can be illustrated as follows,

(23)

Where CS(I) is the cost associated with thereduction of the deterioration rate of thereliability index. X and y are the cost parametersassociated with this case.

4.4 Expression for the Total cost of maintenance

Total cost for maintenance activity either PM orEM can be developed. In this study, it hasassumed that component costs are independent

with no correlation between them. Hence, totalcost of maintenance for maintenance activity ofEM can be expressed as follows,

+C0ther (24)

Where C is the total cost and C^^ is the fixedadditional cost such as inspection cost etc. Thusthe expression can be expanded fromrelationships obtained as above.

c= C0

+ Cother ................................................................................................... (25)

In a similar way, the cost associated with PM canbe expressed as follows,

C=C0+/>(/?(0)? (26)

Equations 25 and 26 can be read as anoptimization problem and thereby minimizationof the cost can be obtained while maximizingAfi(t) and 8(t). That will give the optimummaintenance to the bridge.

5. Conclusion

This study was focused on how the reliabilityindex and cost are dependent on each other fordifferent maintenance activities such as PM andEM. It also illustrated when maintenance isrequired and how a maintenance activity canaffect the bridge reliability. With this theoreticalunderstanding as illustrated better bridgemaintenance can be carried out in the nationalroad network of Sri Lanka.

Acknowledgement

Authors of this paper like to express theirsincere acknowledgement to the NationalScience Foundation of Sri Lanka for the financialsupport given through the research grant no:RG/ 2002/E/01 & NSF/SCH/ 2005/02.

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