Stewart Paper

8/10/2019 Stewart Paper

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International Forum on Engineering Decision MakingSecond IFED Forum, April 26-29, 2006, Lake Louise, Canada

Utility of Spatially Variable Damage Performance Indicators forImproved Safety and Maintenance Decisions of Deteriorating

Infrastructure

Mark G. StewartProfessor, Centre for Infrastructure Performance and Reliability, School of Engineering,University of Newcastle, Australia.

and

John A. Mullard and Brendan J. Drake

Graduate Research Students, University of Newcastle, NSW, Australia

Abstract

Corrosion of concrete and structural steel is a primary cause of deterioration of builtinfrastructure. This deterioration can cause reduced load capacity and on-going and costlymaintenance and repair. Typically this corrosion can be seen on structures to be spatiallyvariable. Few studies have considered the effect of spatial variability of corrosion onstructural performance and its effect on structural reliability. Random fields may be used toconsider the temporal and spatial deterioration effects on structural performance and

performance indicators may include probability of extent of damage or structural reliability.In the present paper, three case studies showing the benefits of using spatially variable

damage performance indicators are discussed. Limitations of random field modelling will bepresented also. It will be shown how spatially variable damage performance indicators willallow for more informed decision-making about the level of safety and the selection ofoptimal maintenance and repair strategies.

1. Introduction

The deterioration of materials and structures is not homogenous, both at the componentand structure levels. The variability of deterioration depends on the spatial and temporalvariability of material properties, structure dimensions and orientation, exposure toaggressive agents, loading and maintenance. Although there is little question of the spatialand temporal variability of deterioration processes, such considerations have often not been

translated to deterioration models and its effect on structural performance and structuralreliability. This is not surprising as1.

spatial data from field and experimental deterioration studies are often not reported,which makes quantifying spatial variability difficult.

2.

closer inspection of more field and experimental data tends to reveal evidence ofadditional spatial variability (i.e., more parameters are revealed to be spatiallyvariable the closer one looks at the data). What is an appropriate scale to modelspatial variability?

3.

time-dependent reliability analyses for deteriorating structures are already complexand CPU intensive without the added complexity of spatial modelling.

The present paper will describe how random fields may be used to consider the temporal andspatial deterioration effects on structural performance. A probabilistic analysis that

incorporates spatial variability can generate new performance indicators, such as probabilityof extent of damage, as well as the more conventional structural reliability. In the present


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paper, the benefits of using spatially variable damage performance indicators are discussedand the limitations of random field modelling will be presented also. Spatial time-dependentreliability analyses and results are reviewed for the following example applications:

1.

Strength and safety predictions of RC beams subject to pitting corrosion.2. Corrosion damage and time to first maintenance for RC bridge decks (Stewart and

Mullard, 2006)

3.

Life-cycle performance and cost-effectiveness of maintenance strategies (Stewart,2006)

It will be shown how spatially variable damage performance indicators will allow for moreinformed decision-making about the level of safety and the selection of optimalmaintenance and repair strategies. To be sure, other work has recently focused on spatialvariability and its effect on structural performance (Sterritt, et al., 2001;Chryssanthopoulos and Sterritt, 2002; Li, et al., 2004; Malioka and Faber, 2004; Vu andStewart, 2005) which provides other examples of the utility of spatially variable damageindicators.

2. Random Field Modelling

Spatial variability of continuous media can be represented by the use of random fields, in thecase of a large surface a 2D random field would be used, or in the case of beam elements a1D random field (also known as a random process) may be more appropriate. Variousmethods of discretisation of random fields have been proposed (Sudret and Der Kiureghian,2000; Matthies et al., 1997) and can be classified into three main categories: pointdiscretisation, average discretisation and series expansion methods.

The midpoint method, a point discretisation method, is often preferred due to its ease ofcomputation, numerically stable results and is applicable to Gaussian and non-Gaussianrandom fields. In this method the random field needs to be discretised into k elements ofidentical size and shape. The random field within each element is represented by a single

random variable defined as the value of the random field at the centroid of the element. Forthe present analysis the random field is discretised into equal square elements of size .

The random field is characterised by its mean , standard deviation and correlationfunction (). The mean and standard deviation are inferred from samples taken acrossall elements within the random field. These statistics are readily available from existingreliability studies. The correlation function () defines the correlation coefficient between

two elements separated by distance and is representative of the spatial correlation between

elements. As the distance between correlated elements becomes smaller the correlationcoefficient approaches unity, and likewise as the distance increases the correlationcoefficient reduces. The triangular and Gaussian (or squared exponential) correlationfunctions are expressed as

Triangular : () =

1 x

2

ax2+

y2

ay2+

z2

az2

forx

2

ax2+

y2

ay2+

z2

az2 1

0 forx

2

ax2+

y2

ay2+

z2

az2 1

(1)

Gaussian : () =exp xdx

2

y

dy

2

zdz

2

(2)

where x=xi-xj, y=yi-yjand z=zi-zjare the distances between the centroid of element i and j

in the x, y and z directions respectively; ax=x, ay=y, az=z; dx=x/, dy=y/, dz=z/


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where is the scale of fluctuation (or correlation length). The scale of fluctuation for a 1D

random field is defined by Vanmarcke (1983) as

= ()d

+

(3)

Therefore, for a triangular correlation function the scale of fluctuation is the distance at

which correlation equals zero. By using this definition for the scale of fluctuation differentcorrelation functions can be compared. Vanmarcke (1983) describes different practicalmethods to evaluate the correlation function and scale of fluctuation. Correlation functionsfor anisotropic 2D random fields are shown in Figures 1 and 2 where x=0.5 m and y=0.75

m. The covariance (matrix) between points can then be calculated by

Ci,j =i, j2 (4)

The midpoint method has the advantage that the covariance matrix Ci,j is convenient tocalculate and can be used for non-Gaussian distributions. However, a disadvantage is that thediscretisation size needs to be relatively small in relation to the scale of fluctuation toconsider the random field within an element to be constant. For example, Sudret and Der

Kiureghian (2000) show that the error estimator for a given element of a random field isless then 5% if the ratio /d


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The cumulative probability of failure anytime during this time interval is

pf(0,tL) =1Pr G t1 (X) > 0G t 2 (X) > 0 .....G t k (X) > 0[ ] t1 < t 2 < ....< t k tL (6)

For deteriorating structures the deterioration process will reduce structural resistance and sostructural resistance is time-dependent. This represents a first passage probability. Theupdated probability that a structure will fail in t subsequent years given that it has survived Tyears of loads is referred to herein as pf(t|T) and is expressed as

pf(t | T) =pf(T+ t) pf(T)

1pf(T)(7)

where pf(T+t) and pf(T) are defined by Eqn. (6). This conditional probability may also bereferred to as a hazard function or hazard rate. For assessment purposes it is often moreconvenient to compare probabilities of failure for a fixed reference period. If a reference

period of one year is selected then this is referred to as an updated annual probability offailure. See Stewart (2001) for further details.

A spatial time-dependent reliability analysis is conducted for RC office floor beams.Member span is 6 m, Y16 reinforcement, where the beam is subject to four point loadingwith the central loads spaced at 1.6 m, time to corrosion initiation is 15 years and thecorrosion rate is 2 A/cm2. Considering only one limit state instead of limit states for eachelement along a member (spatial variability) is clearly going to underestimate structuralvulnerability. For example, Figure 4 shows that the governing (critical) limit state is notalways at midspan or region of peak actions, for 2Y16 bars and L=8 m. Since the middle20% of the bending moment diagram is uniform and =100 mm then eight elements eitherside of mid-span (elements 32-47) experience peak moments. However, elements adjacentto these also have a likelihood, albeit reduced, of containing a governing limit state. Figure5 shows that the resulting time-dependent probabilities of failure p f(0,t) for times sincecorrosion initiation considering spatial variability of pitting corrosion are up to 250%

higher then probabilities of failure obtained from a non-spatial analysis.

Figure 5 also shows updated annual probabilities of failure obtained from Eqn. (7). If there isno deterioration then as the structure ages (and survives) it becomes more service proven,resulting in lower updated annual probabilities of failure later in its service life. On the otherhand, for deteriorating structures the effect of corrosion quickly negates the benefits ofservice proven performance, resulting in increased updated annual probabilities of failure.The updated annual probabilities of failure are up to 270% higher for spatial analyses whencompared to non-spatial analyses. The effect of including spatial variability in thereliability analysis is significant.

If a lifetime target reliability index T=3.8 (probability of failure of 7.210-5) is selected

from ISO 2394 (1998), then Figure 5 shows that if there is no deterioration this probabilityis not exceeded over the lifetime of the structure. However, Figure 5 shows that this targetreliability is reached within 31 and 34 years for spatial and non-spatial analyses,respectively. Evidently, a non-spatial analysis will predict safe behaviour when in reality itwas unsafe for a period of three years. This leads to non-conservative safety assessmentsof remaining service lives which is a distinct disadvantage of non-spatial analyses.

3.2 Corrosion Damage and Time to First Maintenance for RC Bridge Decks (Stewart andMullard, 2006)

The expansive nature of rust and other corrosion products of steel reinforcement inconcrete can cause tensile stresses in the concrete and hence cover cracking, delaminationand spalling. This type of corrosion damage is costly to repair and disruptive to asset

owners and users. When the extent of this corrosion damage exceeds a threshold value givenas a percentage of total surface area then this is often the criteria for the time to first


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maintenance. In the present case, a typical repair strategy is patch repair and the criterionfor the onset of a patch repair is based on 1% of a concrete surface exhibiting severecracking, i.e., Xrepair=1% (Directoraat-Gerneraal Rijkswaterstaat, 2000).

A spatial time-dependent reliability analysis is developed for a RC bridge deck exposed to anaggressive marine environment. The analysis considers corrosion initiation and

propagation, and then the initiation and propagation of corrosion-induced cover cracking.A 2D random field is applied to the RC bridge deck considering the spatial variability o fconcrete cover, concrete compressive strength and surface chloride concentration. Thechloride diffusion coefficient, w/c ratio and corrosion rate are dependant variables on theconcrete compressive strength and/or cover and are thus also spatially variable. Theoutcomes are presented in terms of the proportion of surface subject to severe cracking (d)and the probability that at least x% of a concrete surface has severely cracked (Pr(d(t)x%). The likelihood of time to first maintenance is thus Pr(d(t)Xrepair%).

The example considered herein is a RC bridge deck located adjacent to the coast and so issubject to atmospheric sea-spray. The RC bridge deck has a total exposed area (A) of 900m2. Reinforcing bars of 16 mm diameter are spaced at 250 mm centres both longitudinally

and transversely. Cover is 50 mm and concrete water-cement ratio is 0.5. The 2D randomfield is discretised into square elements of size 0.5 m, resulting in 3,600 elements. Thecorrelation lengths in x and y directions for concrete compressive strength, cover and thechloride surface concentration are x=y=2.0 m.

Results from the spatial time-dependent reliability analysis are shown as probabilitycontours in Figure 6. The probability contours represent Pr(d(t)Xrepair%) and can be used to

predict the probability of cracking damage for any repair threshold. Figure 6 shows, forexample, that for Xrepair=1% there is a 90% probability that damage will occur between 21years and 47 years (probability of occurrence between 0.05 and 0.95). Hence, the time tofirst repair could be as little as 21 years. Selecting a less stringent repair threshold(Xrepair=2.5%) will defer repair actions by at least 9 years. A reliability analysis that cannot

predict the extent of damage has little utility for the purposes of predicting time to firstrepair for many types of corrosion damage. Other repair or maintenance thresholds can beselected based on how repair or maintenance strategies are influenced by the extent of thedamage. For example, widespread corrosion damage may result in rehabilitation of theentire structural component rather then a patch repair.

The computational times associated with a Monte-Carlo simulation analysis for a randomfield comprising large areas are not excessively high, i.e., approximately 90 minutes CPUfor 10,000 runs on an AMD Athlon64 PC for the example presented herein.

3.3 Life-Cycle Performance and Cost-Effectiveness of Maintenance Strategies (Stewart,2006)

The spatial time-dependent reliability analysis described in the previous section may beextended to times beyond time to first repair. This means, for example, that even thoughsome regions of a concrete surface may have been repaired other regions will continue t odeteriorate and are likely to require repair at other times. Further, repaired sections will alsodeteriorate and so for sufficiently long service lives some repaired sections may requiremultiple repairs. This allows, for the first time, a more accurate time-dependentrepresentation of the repair, renewal and deterioration of concrete structures. The

probability that multiple repair actions will be needed during the life of a structure can becalculated. Thus, the expected timing and extent of repairs can be predicted, for variousinspection intervals, repair thresholds. Damage costs will be influenced by the extent ofdamage; for example, Figure 7 shows how such a damage function may be idealised for patchrepair. When combined with a life-cycle cost analysis, this predictive capability enables the

extent of future repair and rehabilitation costs to be more realistically estimated and theoptimal maintenance strategies determined.


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Figure 8 shows a schematic of a typical patch repair strategy. The first repair will occurwhen the extent of damage is observed to exceed Xrepair=1% of total surface area. Thisrepaired area may be returned to its initial as new condition. Hence the remainingconcrete surface will continue to deteriorate. Note that the threshold X repairwill be reachedsometime between inspections so that by the time of inspection the extent of damage may

be more than the threshold value. Thus, the repair costs will be higher. On the other hand, a

greater time between inspections results in reduced monitoring costs and delays in repairsmean reduced present value costs.

The expected repair or rehabilitation cost during service life T can be estimated as

ESF (T) = Pf,m (it) Pf,m(it t)[ ] CSF

1+ r( )it

i=1

T / t

m=1

T / t

(8)

where t is the time between inspections, m is the number of repairs/rehabilitations, i the

inspection number, Pf,m(t) the probability of the mth

repair/rehabilitation before time t, CSFis the failure cost and r the discount rate.

It is assumed that damage is always detected when the structure is visually inspected. repair carried out immediately after extent of damage has been observed to exceed

Xrepair. repair is conducted for damaged area only (surrounding areas not repaired). repair returns damaged area to as new condition and will then commence to

deteriorate at same rate as original concrete when t=0. damage may re-occur during the remaining service life of the structure, i.e., multiple

repairs may be needed. total repaired areas during service life may exceed original surface area (A) since

some surfaces with high rates of deterioration may be subject to multiple repairs.

The probability of damage between inspections is cumbersome to express in closed formsolution since probabilities and extent of damage are recursive and dependent on prior repairhistory for each element on the concrete surface. For the timing of the first repair, P f,m() is

expressed as Pf,1(t) = Pr d(t) X repair( ) and the failure cost is

CSF = Pr d(t)=xmd(t t) < X repair( )x m=X cr

100%

C F (x) (9)

where xmis the extent of damage that is repaired (xmXrepair) and CF(x) is the cost of a singlerepair (see Figure 7). The conditional probability distribution given in Eqn. (9) can becalculated from Bayes Theorem.

Results are obtained for a RC bridge deck similar to that described in Section 3.2 for aservice life of T=100 years. The time for first repair is also derived from techniquesreviewed in Section 3.2. The subsequent performance is derived from binomial theory where

performance for a small area (say 9 m2) is assumed statistically independent and is

extrapolated to larger areas, however, this will not fully capture the spatial variability ofdamage to large areas and so will tend to underestimate the actual extent of damage. Event-

based Monte-Carlo simulation will be more accurate, although at the expense of increasedcomputational times.

Table 1 shows some intermediate statistics obtained from the analysis, for time betweeninspections of 1, 2, 5 and 10 years. The mean time to first repair is approximately 55years. However, the mean time to subsequent repairs reduces significantly after the firstrepair with mean time between repairs of 2 to 5 years. Hence, the total number of repairsover the remaining 45 years service life is relatively high (see Table 1). There is high


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variability of times to repair due mainly to the high temporal and spatial variability of thedeterioration process.

As the time between inspections (t) increase the probability that the extent of repaired

damage is greater than Xrepair% increases. In other words, delaying an inspection means thatif an inspection does observe damage the extent of damage is likely to be higher than if

more frequent inspections were conducted. For example, conditional probabilities of extentof repaired damage obtained from Eqn. (9) are shown in Figure 9. This figure shows thatincreasing the time between inspections increases the likelihood of increased observeddamage. Further, intermediate results presented in Table 1 show that as t increases from 1

year to 10 years the average extent of repaired damage exceeds the repair threshold by asmuch as 270% and 40% for Xrepair=1% and Xrepair=5%, respectively.

Expected repair costs (ESF) evaluated from Eqns. (8) to (9) are also shown in Table 1, for adiscount rate (r) of 2%. Expected repair costs are not insignificant, totalling up to100,000 over 100 years. Table 1 shows that as inspection interval increases the expectedrepair costs reduce for all repair threshold levels. Thus it appears that the benefits ofdelaying the timing of repairs (reduced present value costs) outweighs the likelihood of

increased extent of damage and so increased damage costs CF(x). There is much scope forfurther work, such as accounting for the efficiency and effectiveness of repairs and assessinghow delays in repairs, selection of area to be repaired, etc, may effect life-cycle

performance.

4. Conclusions

Examples have been given to illustrate how spatial variability of deterioration can beincorporated into a structural reliability analysis, for pitting corrosion of reinforcement andcorrosion damage to concrete cover. The spatial time-dependent reliability analyses

produced significantly higher probabilities of failure and developed new spatially variabledamage performance indicators. This will allow for more informed decision-making about

the level of safety and improved modelling of the timing, extent and cost of maintenance,which can be optimised in a life-cycle cost analysis.

5. Acknowledgements

The support of the Australian Research Council (DP0451871) is gratefully acknowledged.

6. References

Chryssanthopoulos, M.K and Sterritt, G. (2002), Integration of Deterioration Modellingand Reliability Assessment for Reinforced Concrete Bridge Structures, First ASRANet

International Colloquium, Glasgow (CD-ROM).

Directoraat-Gerneraal Rijkswaterstaat (2000), Management and Maintenance System (inDutch), The Netherlands.

ISO 2394 (1998), General Principles on Reliability for Structures.Li, Y., Vrouwenvelder, T. , Wijnants, G.H. and Walraven, J. (2004), Spatial Variability of

Concrete Deterioration and Repair Strategies, Structural Concrete, 5(3): pp. 121-130.Malioka, V. and Faber, M.H. (2004), Modeling of the Spatial Variability for Concrete

Structures, Bridge Maintenance, Safety, Management and Cost, IABMAS04, E.Watanable, D.M. Frangopol, T. Utsunomiya (eds.), A.A. Balkema, Rotterdam (CD-ROM).

Matthies, G., Brenner, C., Bucher, C. and Guedes Soares, C. (1997), Uncertainties inProbabilistic Numerical Analysis of Structures and Solids Stochastic Finite Elements,Structural Safety, 19(3), pp. 283-336.

Sterritt, G., Chryssanthopoulos, M.K and Shetty, N.K. (2001), Reliability-based InspectionPlanning for RC Highway Bridges, Proc Int Conf on Safety, Risk and Reliability, IABSE,

pp. 1001-1007.


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Stewart, M.G. (2001), Risk-Based Approaches to the Assessment of Ageing Bridges,Reliability Engineering and System Safety, 74(3), pp. 263-273.

Stewart, M.G. (2004), Spatial Variability of Pitting Corrosion and its Influence on StructuralFragility and Reliability of RC Beams In Flexure, StructuralSafety, 26(4), pp. 453-470.

Stewart, M.G. and Mullard, J.A. (2006), Reliability-Based Assessment of the Influence o fConcrete Durability on the Timing of Repair of RC Bridges, IABMAS'06 - Third

International Conference on Bridge Maintenance, Safety and Management, Porto.Stewart, M.G. (2006), Spatial Variability of Damage and Expected Maintenance Costs for

Deteriorating RC Structures, Structure and Infrastructure Engineering(in press).Sudret, B. and Der Kiureghian, A. (2000), Stochastic Finite Element Methods and

Reliability: A State of the Art Report, Report No. UCB/SEMM-2000/08, Department ofCivil & Environmental Engineering, University of California, Berkeley.

Vanmarcke, E. (1983), Random Fields: Analysis and Synthesis, The MIT Press, Cambridge,Massachusetts.

Vu, K.A.T. and Stewart, M.G. (2005), Predicting the Likelihood and Extent of RCCorrosion-Induced Cracking, Journal of Structural Engineering,ASCE, 131(11), pp.1681-1689.

Xrepair t=1 year t=2 years t=5 years t=10 years

1% 16.3 13.7 8.7 5.4Average number of repairs(nm) 2% 8.8 8.0 6.4 4.6

5% 3.4 3.3 3.0 2.6

1% 1.2% 1.4% 2.2% 3.7%Average repaired area perrepair (xm) 2% 2.2% 2.4% 3.0% 4.3%

5% 5.2% 5.4% 6.0% 7.0%

1% 19.7% 19.7% 19.7% 19.7%Average total repaired area(xm) 2% 19.3% 19.3% 19.5% 19.6%

5% 17.8% 17.9% 18.1% 18.5%ESF(T) r=2% 1% 99.8E3 96.0E3 87.7E3 80.1E3

2% 87.7E3 86.3E3 83.1E3 80.0E3

5% 72.0E3 71.5E3 70.4E3 68.6E3

Table 1: Intermediate Results and Expected Repair Costs for Repair Strategies.

Figure 1: Triangular Correlation Function, for x=0.5 m and y=0.75 m.


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Figure 2: Gaussian Correlation Function, for x=0.5 m and y=0.75 m.

j=1

j=2

element j

j=m

n reinforcing barsL

Figure 3: Discretisation of RC Beam.

2 6 10 14 18 22 26 30 34 38 42 46 50 54 58 62 66 70 74 78

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Proportion

ofmin[G

t(X)]

Element j

peak actionsj=32-47

80

Bending Moment Diagram(=20%)

Figure 4: Distribution of Spatial Position of Governing Limit State over 50 years of PittingCorrosion, for L=8 m.


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10-8

10-7

10-6

10-5

0.0001

0.001

0.01

0 10 20 30 40 50

No Deterioration

Spatial Analysis

Non-Spatial Analysis

Probability

ofFailure

Time t (years)

pf(1|t)

pf

(0,t)

T=3.8

Figure 5: Cumulative and Updated Annual Probabilities of Failure for a RC Beam Subject to

Pitting Corrosion.

0 10 20 30 40 50 60 70 80 90 100 110 120

0

2.5

5

7.5

10

12.5

Time t (years)

Repai

rThreshold

X

repair

% 0.95

0.05

0.5A=900 m

2

Xrepair

=1%

Xrepair

=2.5%

Figure 6: Probability Contours for Pr(d(t)Xrepair%).

5,000

Extent of Damage (x%)

Repair Cost CF(x)

100%

2,000/m2

0%

Figure 7: Idealised Damage Function For a Single Patch Repair.


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first repair

(m=1) at time t1t

inspectioninterval

Xrepair%

deterioration of

repaired concrete

deterioration of original

+ repaired concrete

deterioration of

original concrete

continued deterioration of

original (unrepaired) concrete

second repair

at time t2

Extent of damaged(t)

Time (t)

first repair

d(t1) Xrepair

d(t1-t)

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