Structural Safety 29 (2007) 16–31

www.elsevier.com/locate/strusafe

STRUCTURAL

SAFETY

Spatial time-dependent reliability analysis of corrodingpretensioned prestressed concrete bridge girders

M. Sigit Darmawan a,b, Mark G. Stewart a,*

a Centre for Infrastructure Performance and Reliability, School of Engineering, The University of Newcastle,

Newcastle, NSW 2308, Australiab Department of Civil Engineering, Surabaya Institute Technology, Indonesia

Received 18 April 2005; received in revised form 31 October 2005; accepted 18 November 2005Available online 6 March 2006

Abstract

Accelerated pitting corrosion tests have been performed to obtain spatial and temporal maximum pit-depth data forprestressing wires. This data is then used to develop probabilistic models of pitting corrosion and strength capacity of7-wire strands. The probabilistic model of pitting corrosion for strands is then combined with a non-linear Finite ElementAnalysis and probabilistic models of corrosion initiation and propagation to study the spatial and temporal effects of pit-ting corrosion on a typical pretensioned prestressed concrete bridge girder. The limit states considered are flexural strengthand serviceability. The spatial time-dependent reliability analysis takes into account the uncertainties and variabilitiesrelated to material properties, dimensions, loads and corrosion parameters as well as the spatial variability of pitting cor-rosion of prestressing strands. Including the spatial variability of pitting corrosion in the reliability analysis increased boththe probability of strength and serviceability failure when compared with a mid-span sectional analysis.� 2006 Elsevier Ltd. All rights reserved.

Keywords: Pitting corrosion; Prestressed concrete; Structural reliability; Deterioration; Stochastic finite element analysis

1. Introduction

Corrosion of reinforcing and prestressing steel due to chloride contamination is one of the primarycauses of deterioration of concrete structures. In general, corrosion is of most concern because of theassociated reduction in steel cross-sectional area, spalling and loss of bond, which over time will leadto reductions of strength and serviceability. In the case of pretensioned prestressed concrete (PSC) struc-tures, the corrosion of prestressing strands may trigger structural collapse due to higher stress levels in thesteel.

0167-4730/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.strusafe.2005.11.002

* Corresponding author. Tel.: +61 2 4921 6027; fax: +61 2 4921 6991.E-mail address: mark.stewart@newcastle.edu.au (M.G. Stewart).

M.S. Darmawan, M.G. Stewart / Structural Safety 29 (2007) 16–31 17

Clearly, chloride-induced pitting corrosion is a spatial and temporal variable. Therefore, ignoring spatialvariability of pitting corrosion in structural prediction models is not very realistic. However, most studiesto date have modelled the strength and reliability of reinforced concrete (RC) and PSC flexural memberssubject to deterioration by focusing on homogeneous material and deterioration properties and sectionalcapacities at regions of peak or critical sections (e.g., [22,3,27,35]). However, Stewart [30] has undertaken astructural reliability analysis of RC structures considering the spatial and temporal effect of pitting corrosion.This study found that including the spatial variability of pitting corrosion can lead to significant decreases instructural reliability for flexural RC members.

The present paper considers the effect of temporal and spatial variability of pitting corrosion on the time-dependent performance and reliability of prestressed concrete beams in flexure with bonded tendons. Thepaper describes briefly the development of probabilistic models of pitting corrosion for prestressing wires sub-ject to pitting corrosion. The probabilistic model of pitting corrosion is developed from maximum pit-depthdata obtained from accelerated pitting corrosion tests in a chloride-contaminated concrete environment. Theprobabilistic model of pitting corrosion of strands is then combined with non-linear finite element analysis(FEA) as well as models of corrosion initiation and propagation to study the effect of pitting corrosion onthe strength and serviceability of prestressed concrete beams. This will allow time to failure and time-depen-dent structural reliabilities to be calculated. Including the effect of spatial variability allows the analysis to con-sider failure of strands at any location along the beam, as well as progressive failure of multiple strands asfailure of one strand will increase the stress in remaining strands thus leading to possible rupture of otherhighly stressed or corroded strands. Three alternate prestressing strand failure modes are considered: stresscorrosion cracking, brittle fracture and yielding. The proposed method is independent of the pretensionedPSC structure being considered, but for illustrative purposes a typical PSC bridge girder located at a coastalenvironment is selected as the application.

2. Probabilistic model of pitting corrosion

The actual mechanism of pitting corrosion is not yet fully understood. However, the use of extremevalue theory gives promising results when modelling pitting corrosion in aluminium and steel (e.g.,[5,12,28]).

2.1. Prestressing wires

To obtain data on pitting phenomena, an accelerated corrosion testing regime was carried out at TheUniversity of Newcastle. Corrosion rates of 150–420 lA/cm2 were introduced in steel wires and strandsembedded in chloride-contaminated concrete slab specimens in order to measure pit depths along 1.5 mlengths of cold-drawn prestressing 7-wire strands (each outer wire of 4.3 mm diameter). The tests consistedof six concrete slabs of dimensions 1.5 m · 1.0 m · 0.25 m, each with a different corrosion rate and period ofexperiment. Nine strands of 100 mm equal spacing were placed in each slab, each with a 70 mm concretecover. The experimental set-up is shown in Fig. 1. Load tests on the corroded wires and examination ofthe fracture surface reveal that the mode of failure is yielding with reduced ultimate failure strains, andnot ultimate tensile failure, stress corrosion cracking (SCC) or brittle fracture [9]. For the 7-wire strandsthe maximum pit-depth was measured using a micrometer gage for each unwound wire was measuredfor each 650 mm length (excluding the inner-wire of the strand). To avoid edge effects the end 100 mmof each wire/strand was excluded from measurements. From visual observation of the pitting corrosionand the measurement of pit-depths, it was apparent that for the 7-wire strands pitting only formed onthe exposed surface of the wire (approximately half of the total surface of the six outer wires). The Gumbel(EV-Type I) distribution provided the best fit to maximum pit depth data for prestressing wires. Table 1shows the statistical parameters obtained from one slab specimen, for an accelerated corrosion rateof icorr-exp = 186 lA/cm2, period of experiment of T0-exp = 0.038 years (14 days) and wire lengthL0-exp = 650 mm. In this case, one strand was removed for further observation and statistical data obtainedfrom the remaining eight strands.

Power Supply &Current regulator

_ ++

Wire/Strand 5% Na Cl Solution

Stainless steel Plate

Fig. 1. Schematic and photo of experimental set-up.

Table 1Gumbel parameters for maximum pit-depths for a single wire in a strand

T0-exp (years) icorr-exp (lA/cm2) L0-exp (mm) a (mean) (mm) a (COV) l0-exp ao-exp No. of samples

0.03836 186 650 0.91 0.17 0.84 8.10 96

18 M.S. Darmawan, M.G. Stewart / Structural Safety 29 (2007) 16–31

The following assumptions are made in developing a more general probabilistic model for pitting corrosion:

(i) homogeneous environment along the wire under consideration (corrosion rate assumed constant alongwire);

(ii) after an initial period of corrosion, the number of pits formed is assumed constant, length of pit is heldconstant and pit depth continues to increase; and

(iii) at any cross-section of the wire only one pit can form.

The predicted Gumbel distribution of maximum pit depth (a in mm) at any time T (years), corrosion rateicorr(1) in lA/cm2 at start of corrosion propagation and wire length L (mm) is thus [8]:

faðT ; icorr; LÞ ¼a

k0:54e�a a

k0:54�l

� �e�e

�a ak0:54

�l

� �T > T i ð1Þ

where

M.S. Darmawan, M.G. Stewart / Structural Safety 29 (2007) 16–31 19

k ¼D2

0 � D0 � 0:0232icorrð1Þ 1þ jhþ1½ðT � T iÞhþ1 � 1�

n o� �2� �

D20 � D0 � 0:0232icorrð1Þ 1þ j

hþ1½T hþ1

0 � 1�n o� �2

� � ð2Þ

T 0 ¼ exp1

hþ 1ð Þ lnhþ 1ð Þ icorr- expT 0- exp

� �þ j� h� 1ð Þ icorrð1Þð Þ

jicorrð1Þ

� � �ð3Þ

l ¼ l0- exp þ1

a0- exp

lnL

L0- exp

� a ¼ a0- exp ð4Þ

icorrðT � T iÞ ¼ icorrð1Þ � jðT � T iÞh T � T i P 1 year ð5Þ

and Ti is time to corrosion initiation (years), l0-exp and a0-exp are the parameters of the Gumbel distribution asobtained from statistical analysis of maximum pit depths recorded from the accelerated corrosion tests (seeTable 1), D0 is the initial diameter of the wire (mm), and j and h are corrosion rate empirical factors. If cor-rosion rate reduces with time then j = 0.85 and h = � 0.29 [35]. Otherwise, if corrosion rate is constant withtime (time-invariant) then j = 1 and h = 0. The geometric model proposed by Val and Melchers [33] is thenused to predict the loss of cross-sectional area for a pit size of depth a, see Fig. 2. Failure of a wire occurs whenthe tensile load at the cross-section of maximum pitting exceeds the yield capacity of the cross-section(although at significantly reduced ductility), which was shown by Darmawan and Stewart [9] to most closelymatch tensile test results of corroded wires. Since the distribution of maximum pitting is a random variablegiven by Eq. (1) then time to failure of a single wire is a dependent variable as shown, for example, in Fig. 3.

Predictive results (using statistical parameters in Table 1) for other corrosion rates and period of experi-ments compared favourably with other slab specimen data [8], and so provides some verification of the modelsdeveloped herein.

a

b

Do

Fig. 2. Pit configuration.

0.00

0.02

0.04

0.06

0.08

0.10

0 20 40 60 80 100

Prob

abili

ty D

ensi

ty

Time to Fail since Corrosion Initiation ( years )

7-wire Strand

Wire

icorr

(1) = 1 μA/cm2

L = 1 m σ = 1000 MPa

Fig. 3. Distributions of time to wire and strand failure.

20 M.S. Darmawan, M.G. Stewart / Structural Safety 29 (2007) 16–31

The pitting corrosion statistical parameters l0-exp and a0-exp are indicative only and increased confidence inpredictions will be obtained if these parameters are based on tests which more closely represent field condi-tions, i.e., longer T0-exp and lower icorr-exp. Further, the use of extreme value statistics to extrapolate distribu-tions of maximum pit depths for data significantly different from that used in the tests may not be totallyappropriate, e.g., if wire lengths (L) are extrapolated to very large values.

2.2. Prestressing strands

It is assumed that the outer six wires of a 7-wire strand corrode independently of each other. There is alsoredistribution (increase) in wire stress after failure of each successive wire. Failure of a single wire will not nor-mally result in strand failure since remaining wires will be able to sustain the increased load after redistributionof stresses. Strand failure will only occur when all wires in the strand fail. This means that strand performancecan be modelled as a ‘‘perfectly brittle’’ parallel system. In general, the strength of such a parallel system com-prising n elements (wires) is (e.g., [31])

RstrandðT Þ ¼ R7 þmax nR1ðT Þ; ðn� 1ÞR2ðT Þ; . . . ; 2 Rn�1ðT Þ;RnðT Þf g R1 < R2 < � � � < Rn ð6Þ

where R1(T), R2(T), . . . ,R6(T) are the strengths of the corroding outer wires at time T, n = 6 and R7 is the strengthof the non-corroding inner wire. The strength of a wire (assuming mode of failure is non-ductile yielding) at time T is

RnðT Þ ¼ fpy½A0 � ApitðT Þ� ð7Þ

where fpy is the yield stress, A0 is the initial cross-sectional area of the wire and Apit(T) is the cross-sectionalarea of the wire due to pitting.

Fig. 3 shows the distributions of time to failure for a corroding wire and strand, for icorr(1) = 1 lA/cm2,L = 1 m, r = 1000 MPa in all wires, and a yielding failure criterion of fpy = 1565 MPa. The time to failureof a single wire is higher than the minimum time to failure of six wires since in six wires there is higher like-lihood of deeper pitting and so reduced time to first wire failure. Progressive failure of other wires in the strandis then likely to rapidly occur as a result of stress redistribution (i.e., stress in remaining 5 wires increases to1200 MPa). Thus, the time to failure for a strand is typically lower than that of a single wire. For further detailof the experimental results and probabilistic model development see Refs. [8,9].

3. Stochastic finite element analysis

A three dimensional finite element analysis (FEA) of a typical simple span PSC bridge girder is employed inthis study using commercially available software ABAQUS [2]. The model includes eight-node solid elementsfor concrete and truss elements for the prestressing steel. The truss elements are embedded in the concrete ele-ment, which means bond-slip effects are not considered in the analysis. This assumption is reasonable sincepitting corrosion is localised and is less likely to cause the disruption of concrete cover and hence no reductionof bond strength around the pits [33].

The length of steel elements used to represent prestressing strands is taken as twice the development lengthof the strand embedded in concrete (i.e. 2Lp). This approach is based on the assumption that when failure ofprestressing steel occurs, there will be loss of capacity a distance Lp at either side of the rupture (assuming therupture occurs in the middle of the element), e.g., [30]. This is an approximate failure criterion since in realitystrand rupture could occur near to an element with higher actions or near to an element with lower actions,resulting in reduced and increased structural reliabilities, respectively. Nonetheless, it is anticipated that reli-abilities for non-central strand ruptures will average out producing reliabilities similar to that obtained fromthe proposed failure criterion. For prestressing strands with nominal diameter of 12.7 mm and cover of at least40 mm Lp is approximately 500 mm [10] and so the length of steel elements is L = 1.0 m.

Uniaxial elastic–plastic material models are used for both steel and concrete stress–strain relationships[7,16]. However, to model more realistically the effect of corrosion the steel stress–strain relationship is mod-ified which leads to a reduced ultimate strain at failure [8,4], but this has a negligible effect on structural per-formance for the PSC bridge girder considered herein. For concrete, the linear tension stiffening effect aftercracking is included.

M.S. Darmawan, M.G. Stewart / Structural Safety 29 (2007) 16–31 21

3.1. Structural reliability

The limit states considered are flexural strength and serviceability. In principle, other limit states (such asshear strength) could also be included if they are a governing design limit state or if deterioration is consideredto produce a governing limit state over the service life of the structure.

The strength limit state is exceeded when actual load effects exceed flexural resistance of the member deter-mined by the singularity of the global stiffness matrix in which the FEA analysis does not converge. If it isassumed that k load events takes place at time intervals DT within the time period (0,T) at times ti (i = 1,2, . . . ,k and ti = iDT), the cumulative probability of structural failure of service proven structures anytime dur-ing the time interval (0,T) is

pf ð0; T Þ ¼ 1� Pr½GU1> 0 \ GU2

\ � � � \ GUk > 0� t1 < t2 < � � � < tk 6 T ð8Þ

where GUi is the strength limit state function at time ti and GUi > 0 represents convergence of the FEA. Thisrepresents a first passage probability.

For the serviceability limit state, the peak deflection is of interest (i.e. mid-span deflection). The cumulativeprobability of serviceability failure during the time interval (0, T) is

psð0; T Þ ¼ 1� Pr½GS1> 0 \ GS2

\ � � � \ GSk > 0� t1 < t2 < � � � < tk 6 T ð9Þ

where GSiðX Þ ¼ ðDallow � DmidðtiÞÞ, Dallow is the allowable deflection and Dmid(ti) is the peak deflection due tothe ith load effect at time ti. It should be noted that deflection predicted by the FEA is monitored up to timeTf � DT, where Tf is the time to collapse. Hence, in this study ps is defined as the cumulative probability ofserviceability failure prior to collapse.

3.2. Computational procedure

The spatial time-dependent reliability analysis is complicated due to non-linear and non-explicit limit states,time-dependent random variables, dependent and correlated variables, and more importantly, the inclusion oftemporal and spatial variability of pitting corrosion. Hence, closed form solutions are intractable and soMonte Carlo event-based simulation is employed as the computational tool.

The Monte–Carlo event-based simulation analysis considers the variability and uncertainty of loads, mate-rial properties, dimensions and deterioration processes. For each simulation run the time to corrosion initia-tion and corrosion rate for each layer of strands is calculated. At each time increment (DT), the pit-depth foreach wire in each strand and the peak live load is generated. The cross-sectional area and strength of the wiresare then inferred, and strength of the strands are calculated from Eq. (6). The strand forces are then calculatedfrom the FEA. If the actual strand force is higher than the strength of strand for any element (which will varyspatially along the length of the strand and decrease with time due to strand corrosion) then the strand is

Time

Corrosion Initiation

Str

uct

uar

l Cap

acit

y

1st strand failure

2nd strand failure

3rd strand failure

Collapse

Fig. 4. Schematic of deterioration process of PSC structures.

22 M.S. Darmawan, M.G. Stewart / Structural Safety 29 (2007) 16–31

removed from that element and actions in the remaining strands are recalculated (i.e. stress redistribution).The event-based process continues for successive time increments, leading to more strand failures until eitherthe flexural capacity of the bridge girder is exceeded or until its service life is reached, see Fig. 4. If the deflec-tion exceeds the allowable deflection during simulation, this will be recorded as serviceability failure and thesimulation continues.

4. Illustrative example

4.1. Prestressed concrete (PSC) bridge girder

The bridge considered in this study is a typical simple span PSC bridge, which has a span of 21 m and aclear roadway width of 8.4 m. The bridge consists of four precast prestressed AASHTO Type IV girders(see Fig. 5) with equal spacing of 2.3 m and a 200 mm thick cast-in-place concrete deck. The girder wasdesigned according to the AASHTO LRFD [1] Bridge Design Specifications assuming bonded tendons, unsh-ored construction and no composite action between the girder and the cast-in-place slab. The specified con-crete strength F 0c of the girder is 35 MPa and the nominal ultimate tensile strength of the prestressing steel(fpk) is 1750 MPa. A stress of 65% of fpk is applied to the girder as an initial pre-stress. A total of 26 7-wirestrands (12.7 mm diameter) were required to carry the total design loads. Six centrally located strands (posi-tioned in three rows) are harped at the third span of the girder and only twenty strands remain horizontal intwo levels of ten strands each. Concrete covers for prestressing strands are 50 mm (level 1) and 100 mm (level2).

For convenience, the present analysis ignores other sources of spatial variability even though it is recogni-sed that concrete quality, concrete cover, chloride exposure, etc. do vary spatially (e.g., [11,36]). These spatialvariables, however, will not have as significant an influence on structural reliability of flexural members as pit-ting corrosion.

Three different components of dead load are considered: precast concrete, cast-in-place deck and 80 mmasphalt overlay. Axle spacings and distribution of axle loads are calculated based on a US HS-20 truckand the truck is located on the bridge to cause peak flexural actions. The service life of the structure consideredin this study is taken as 100 years.

The PSC bridge girder is exposed to an atmospheric marine environment on the coastline. The predictivemodel for corrosion rate for a relative humidity of 80% is

203

152

584

229

203

660

3 x 50

508

1371

harped strands

level 2 prestressing strands

level 1 prestressing strands

level 3 prestressing strands

Fig. 5. AASHTO type IV bridge girder.

M.S. Darmawan, M.G. Stewart / Structural Safety 29 (2007) 16–31 23

icorrð1Þ ¼ 27ð1� wcÞ�1:64=C ð10Þ

where C is concrete cover in mm and wc is the water–cement ratio. Fick’s second law of diffusion is used topredict corrosion initiation and the corrosion rate is assumed constant with time (j = 1.0 and h = 0). Thewater–cement ratio, chloride diffusion coefficient and corrosion rate are dependent variables on the concretecompressive strength. See Ref. [35] for further details of the stochastic deterioration models. Clearly, concretedurability design specifications such as cover and w/c ratio will influence time to corrosion initiation and cor-rosion rate. There is a wide range of stochastic deterioration models proposed in the literature, the presentdeterioration models are simply used to illustrate the applicability of the novel approach developed hereinto representing the spatial and temporal corrosion damage to PSC structures.

Each longitudinal strand in the 21 m span girder is divided into 21 steel elements since the length of the steelelement is 1 m (see Section 3). The concrete element length is also 1 m. Fig. 6 shows the FEA mesh comprising294 concrete elements and 546 steel elements. It is assumed that wires in a strand will corrode independently ofeach other and pitting between adjacent elements is also statistically independent. It is assumed that eachstrand in the same layer (i.e. same concrete cover) has the same time to corrosion initiation and corrosion rate.The total loss of prestress is estimated based on AASHTO specifications as only long-term behaviour is con-sidered for this study. The mode of strand failure is yielding (fpy) of the wires. For two lane bridges, the criticalload effect usually occurs when two heavily loaded trucks are side by side and have fully correlated weights[23]. It is assumed that the number of fully correlated trucks is 600 trucks/year [23]. The extreme value distri-bution of the weight of the heaviest truck for any time interval is then readily inferred (e.g., [35]). A summaryof statistical parameters representative of PSC bridge girders in the US is given in Table 2. In the results tofollow the time interval (DT) is taken as 2 years.

4.2. Results

4.2.1. Corrosion parameters

The statistical parameters for the time to corrosion initiation Ti and corrosion rate icorr for level 1 prestress-ing strands obtained from Monte Carlo simulation are given in Table 3. Although the mean time to corrosioninitiation is high, its high variability means that there is a reasonable likelihood of time to corrosion initiation

Fig. 6. FEA mesh and elements.

Table 2Statistical parameters for PSC bridge girder [8]

Parameters Mean COV Distribution Reference

f 0cyl, concrete cylinder strength F 0ca + 7.5 MPa r = 6 MPa Lognormal Stewart [29]

ki, in situ concrete strength factor 1.2-0.0082 · meanðf 0cylÞ 0.1 Normal Stewart [29]MEf – concrete tensile strength (ft) 1.0 0.12 Normal Mirza et al. [19]ME – concrete elastic modulus (Ec) 1.0 0.12 Normal Mirza et al. [19]fpy, yield strength 0.88 fpk

b 0.025 Normal Mirza et al. [21]Ep, steel elastic modulus 195000 MPa 0.02 Normal Mirza et al. [21]Total prestress loss kc = 1.0 0.30 Normal JCSS [15]Cb

d, bottom cover Cbnom r = 7.9 mm Normal Mirza and MacGregor [20]H, beam depth (mm) Hnom + 0.8 r = 3.6 mm Normal Mirza and MacGregor [20]C0, surface chloride concentration 3.05 kg/m3 0.79 Normal Vu and Stewart [36]ME – diffusion coeff. (D) 1.0 0.20 Normal Vu and Stewart [35]Cr

e, threshold chl. concentration 3.35 kg/m3 0.375 Normal Val and Stewart [34]ME – corrosion rate (icorr) 1.0 0.20 Normal Vu and Stewart [35]D1, dead load precast 1.03 Dn 0.08 Normal Nowak et al. [26]D2, dead load cast-in-place 1.05 Dn 0.10 Normal Nowak et al. [26]D3, dead load asphalt 80 mm 0.30 Normal Nowak et al. [26]Single truck load 240 kN 0.40 Normal Nowak and Hong [24]Impact factor 1.15 0.10 Normal Hwang and Nowak [13]Girder distribution factor k = 0.93 0.12 Normal Nowak and Grouni [25]Pitting model l0-exp = 0.84 a0-exp = 8.1 Gumbel Darmawan [8]

a F 0c ¼ specified ðcharacteristicÞ concrete compressive strength.b fpk = characteristic tensile strength of prestressing steel.c k = bias factor.d Truncated at 10 mm.e Truncated at 0.35 kg/m3.f Model Error.

Table 3Statistical parameters for time to corrosion initiation Ti and corrosion rate icorr for level 1 prestressing strands

Parameters Mean COV

Ti time to corrosion initiation (years) 300 1.17icorr corrosion rate (lA/cm2) 1.76 0.39

24 M.S. Darmawan, M.G. Stewart / Structural Safety 29 (2007) 16–31

being only 20–30 years. It is this high variability of predictions of deterioration processes that can significantlyeffect the structural reliability of a corroding structure.

4.2.2. Failed strands

Fig. 7 shows the number and timing of surviving (non-failed) strands in the mid-section of the girder fortypical realisations from three Monte Carlo simulation runs. As seen in this figure, the progressive loss ofall level 1 strands due to pitting corrosion occurs within a short time period of first strand failure (often withinseveral years due to stress redistribution), and generally leads to instantaneous failure (i.e., in the same timeinterval) of the remaining (Level 2) strands causing the sudden collapse of the girder. This observation is sup-ported by reference to Fig. 8 which shows the distribution of the average number of failed strands that initiatesthe collapse of the girder. Fig. 8 also shows that the average number of failed strands decreases toward thegirder supports. This trend is expected since the stress in strands located in the mid-section is higher than thatlocated close to the supports.

4.2.3. Time to collapse (Tf)Fig. 9 shows the simulation histogram of time to collapse (Tf) of the girder. Collapse can occur within as

little time as 12 years. From intermediate results [8] it is observed that collapse of the girder generally occursdue to the combined action of very early time to corrosion initiation (less than 40 years), and slightly aboveaverage corrosion rate and live load. Fig. 10 shows the cumulative probabilities of strength and serviceabilityfailure.

level 3 strands0

2

4

6

8

10

12

14

16

18

20

22

24

26

0 20 40 60 80 100

Num

ber

of S

urvi

ving

Str

ands

Time T (years)

prog

ress

ive

failu

rein

stan

tane

ous

failu

re

level 1 strands

level 2 strands

Fig. 7. Timing of individual strand failure for PSC bridge girder in the middle of the span.

0

2

4

6

8

10

12

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

level 1level 2

Ave

rage

Num

ber

of F

aile

d St

rand

s

Position Along the Girder (m)

Fig. 8. Distribution of average number of failed strands along the girder that initiates the collapse of the girder.

Fig. 9. Time to collapse of PSC bridge girder.

M.S. Darmawan, M.G. Stewart / Structural Safety 29 (2007) 16–31 25

Fig. 10 shows that at 100 years the probability of strength failure is 0.096, which is equal to a reliabilityindex (b) of 1.30. Akgul and Frangopol [3] performed a time-variant reliability analysis of corroding PSCbridge girders assuming general corrosion and considered only critical (mid-span) section analysis. They foundthat for mean icorr = 4.9 lA/cm2 the reliability index of the girders decreased to a value of b = 1.5–2.0 after 75years. This clearly shows that the structural reliability results obtained herein are, albeit low, comparable to

0.00

0.02

0.04

0.06

0.08

0.10

0 20 40 60 80 100

Prob

abili

ty o

f Fa

ilure

Time T (years)

serviceability

strength

Fig. 10. Time-dependent probability of failure of PSC bridge girder.

26 M.S. Darmawan, M.G. Stewart / Structural Safety 29 (2007) 16–31

available existing reliability studies. Note that Akgul and Frangopol [3] ignored the spatial aspect of pittingcorrosion in their analysis, which is expected to lead to a lower probability of failure than a spatial analysis,such as considered herein. The purpose of the reliability analyses is comparative only. The calculated reliabil-ities should thus be considered as nominal or indicative values rather than being considered as absolute values.

4.2.4. Live load deflectionAASHTO [1] specifies that the maximum deflection of a bridge (Dallow) under live load should not exceed 1/

800 of its span. Fig. 10 shows that the probability of serviceability failure prior to collapse is less than theprobability of strength failure. This trend is caused by the following factors: (i) in the FEA the deflectioncan only be measured up to time Tf � DT and (ii) it is assumed that when the girder collapses at time Tf itis recorded only as a strength failure and not as a serviceability failure. In reality, a strength failure might alsobe considered as the serviceability failure (i.e. collapse means a large deflection).

Fig. 11 shows the simulation histogram of deflection due to live load in the middle of the span in the timeincrement preceding collapse (Tf � DT) conditional on girder collapse. It shows that approximately 45% ofcollapses are not preceded by excessive deflection. This is a likely consequence of the observation that progres-sive strand failure generally occurs within a short time interval (see Fig. 7) so there is little warning of collapse.This suggests that deflection monitoring during service life might not always be useful for health monitoring ofPSC pretensioned bridges.

4.2.5. Comparison with mid-section analysis

Most studies model the strength and reliability of structural members subject to deterioration byfocusing on mid-section capacities, regions of peak actions or other critical sections. This conventional, but

0.00

0.02

0.04

0.06

0.08

0 5 10 15 20 25 30 25 40 45 50 55

Prob

abili

ty D

ensi

ty

Live Load Deflection (mm)

Δallow

=Ls/800

Fig. 11. Simulation histogram of live load deflection at time Tf � DT conditional on girder collapse.

0.00

0.02

0.04

0.06

0.08

0.10

0 20 40 60 80 100

Prob

abili

ty o

f St

reng

th F

ailu

re p

f(0,T

)

Time T (years)

spatial

mid-section (general corrosion)

mid-section (pitting corrosion)

Fig. 12. Probability of strength failure pf for a spatial and a mid-section analysis.

M.S. Darmawan, M.G. Stewart / Structural Safety 29 (2007) 16–31 27

non-conservative approach, is referred to herein as a ‘‘mid-section’’ analysis. In the present paper a mid-sec-tion analysis will consider only the pitting corrosion of the central 1 m element and the rest of the beam isassumed not affected by corrosion.

Fig. 12 shows a comparison between a spatial and a mid-section analysis for the strength limit state. Itshows that including the spatial variability of pitting corrosion in the reliability analysis increases the proba-bility of failure by only 10%. When compared with a mid-section analysis assuming general corrosion, includ-ing the spatial variability of pitting corrosion leads to a 20% increase in the probability of failure. These smalldifferences can be explained from Fig. 8, which shows that the number of failed strands is higher in the regionclose to the mid-section than in the region close to the supports. This means that even for a spatial analysis thecollapse of the girder is still governed mainly by failure of strands located in the mid-section region as assumedin a mid-section analysis. Further, the observation made by Stewart [30], for RC beams in flexure, that as thenumber of reinforcing bars increases the variability of cross-sectional area reduces leading to increased struc-tural reliabilities is also pertinent for PSC beams. In this case, the effect of spatial variability of pitting corro-sion reduces as the number of prestressing strands increase.

Given the extensive computational effort associated with a spatial time-dependent reliability analysis, theresults presented herein suggest that it may be acceptable to ignore spatial variability of non-central sections,at least for the structural configuration considered herein.

Fig. 13 shows the comparison between a spatial and a mid-section analysis for the serviceability limit state.It shows that the effect of the spatial variability of pitting corrosion is more significant on the serviceabilitylimit state. This is expected since the loss of strands due to pitting corrosion will lead to higher tensile stress

0.00

0.02

0.04

0.06

0.08

0 20 40 60 80 100Prob

abili

ty o

f Se

rvic

eabi

lity

Failu

re p

s(0,T

)

Time T (years)

spatial

mid-section (pitting corrosion)

mid-section (general corrosion)

no deterioration

Fig. 13. Probability of serviceability failure ps for a spatial and a mid-section analysis.

28 M.S. Darmawan, M.G. Stewart / Structural Safety 29 (2007) 16–31

in the concrete, which lead to concrete cracking. The formation of cracking will then reduce significantly thestiffness of the girder. Loss of capacity (and bond) along non-central strands will also affect serviceability per-formance to a much greater extent than strength performance.

4.2.6. Concrete durability design specifications

To examine the effect of different concrete durability design specifications on the performance of a PSCbridge girder, three durability design specifications are examined:

(i) Cover = 30 mm; F 0c ¼ 30 MPa(ii) Cover = 50 mm; F 0c ¼ 35 MPa

(iii) Cover = 70 mm; F 0c ¼ 50 MPa

Fig. 14 shows that concrete durability design specifications influence significantly the probability ofstrength failure. For example, an improved durability design specification will increase time to corrosion ini-tiation and reduce the corrosion rate leading to significantly reduced probabilities of failure.

4.2.7. Modes of prestressing failure

The result obtained from the accelerated pitting corrosion test of prestressing wires in a concrete-chlorideenvironment is that the mode of failure of corroded cold-drawn prestressing wires under static tensile load is‘‘non-ductile’’ yielding [9]. However, the possibility of other possible modes of failure (e.g. brittle fracture orSCC) cannot be ruled out, especially with the old type quenched and tempered wires [18].

Brittle fracture and stress corrosion cracking (SCC) modes of failure can be obtained from a Linear-ElasticFracture Mechanics approach where the stress field ahead of a sharp crack can be characterised by a stressintensity factor KI. When the value of KI reaches a certain threshold level, failure occurs. The stress intensityfactor for the pit configuration shown in Fig. 2 is determined using the theoretical formula [6]:

KI ¼X4

i¼0;i 6¼1

X3

j¼0

Cija

D0

� i ab

� �jrffiffiffiffiffiffipap

ð11Þ

where r is the applied stress and the values of the coefficient Cij are obtained from Aztiz [6] and a and b aredefined from Fig. 2. The statistical parameters for threshold values for SCC and brittle fracture are given inTable 4. These values correspond well with values obtained earlier by Toribio and Lancha [32]. Note that cor-rosion pits may not be as sharp as cracks, which means that Eq. (11) may over-estimate stress intensity factors.

Hence, Fig. 15 shows the effect of different modes of failure on the reliability of a corroding PSC girder. Itshows that compared with yielding, both brittle fracture and SCC lead to significantly higher probabilities ofstrength failure.

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0 20 40 60 80 100

Prob

abili

ty o

f S

tren

gth

Failu

re p

f(0,T

)

Time T (years)

F'

c = 30 MPa; cover = 30 mm

F'

c = 50 MPa; cover = 70 mm

F'

c = 35 MPa; cover = 50 mm

Fig. 14. Effect of different concrete durability specifications on probability of strength failure pf.

0.00

0.05

0.10

0.15

0 20 40 60 80 100

yielding

Prob

abili

ty o

f St

reng

th F

ailu

re p

f(0,T

)

Time T (years)

SCC

brittle fracture

Fig. 15. Effect of different modes of failure on probability of strength failure pf.

Table 4Statistical parameters for stress corrosion cracking and brittle fracture failure modes [14]

Parameters Mean COV Distribution

KSCC (stress corrosion cracking) 43 MPa m0.5 0.10 LognormalKc (brittle fracture) 86 MPa m0.5 0.05 Lognormal

M.S. Darmawan, M.G. Stewart / Structural Safety 29 (2007) 16–31 29

4.2.8. Correlation of pitting corrosion

In practice, it is expected that there may be some degree of correlation between wires and strands subject tosimilar corrosive environments. However, structural reliabilities were not significantly affected by assuming acorrelation coefficient of up to 0.5 for correlations of: (i) pitting of wires within a strand or (ii) pitting betweenstrands in adjacent elements. The only exception being serviceability reliabilities for correlation of pitting cor-rosion between adjacent strands. This is not surprising, since this correlation is likely to lead to loss of bond

Table 5Influence of various parameters on the variability of time to strength and serviceability failures

Parameter Xi Influence (%)

Strength Serviceability

Truck live load 17.93 9.70Pitting model parameter l0 17.28 15.79Concrete cover level 1 strands 15.62 32.91In-situ concrete strength factor (ki) 13.81 1.71Girder distribution factor 12.05 4.79Impact factor 4.67 3.94Concrete cover level 2 strands 4.12 11.61Prestressing steel elastic modulus (Ep) 2.48 3.82ME for diffusion coefficient (D) 2.29 2.65Dead load due to asphalt (D3) 1.64 4.96Beam depth 1.45 0.00Prestress loss 1.44 1.53Yield strength of prestressing steel (fpy) 1.25 0.03Dead load due to cast-in-place element (D2) 1.06 1.38Dead load due to precast element (D1) 0.97 0.25Concrete cylinder strength ðf 0cylÞ 0.92 4.39Chloride threshold level (Cr) 0.43 0.27Surface chloride concentration (C0) 0.30 0.02ME for concrete elastic modulus (Ec) 0.26 0.02ME for concrete tensile strength (ft) 0.01 0.00ME for corrosion rate (icorr) 0.01 0.23

ME = model error.

30 M.S. Darmawan, M.G. Stewart / Structural Safety 29 (2007) 16–31

for longer lengths of strand which will most adversely affect serviceability performance. Although not explic-itly studied herein, it would be expected that correlation of pitting between the 26 strands would significantlyreduce structural reliabilities. This is an area for further study.

4.2.9. Sensitivity analysis

A sensitivity analysis is performed to study the relative importance of the variability of each random var-iable on the calculated response (e.g. time to strength and serviceability failure). In this study, the approachproposed by Melchers and Ahammed [17] is used.

The sensitivity analysis identified truck live load, pitting model parameters, concrete cover, concretestrength and girder distribution factor as the most important parameters influencing the time to failure, seeTable 5. These results correspond well to intermediate results which show that the time to corrosion initiationis the variable that most significantly influences the probability of failure. The time to corrosion initiation isclearly influenced by both concrete cover and concrete compressive strength. Vu and Stewart [35] have alsoidentified concrete cover and truck live load as among the most important parameters influencing the reliabil-ity of corroding RC slab bridges.

Following the results obtained herein, it is expected that structural reliabilities are also most influenced bythese important parameters. Hence, in order to obtain more accurate predictions of structural reliability, moreeffort should be directed at improving the accuracy of these parameters, including the pitting model statisticalparameters.

Further details of model development and results are described elsewhere [8].

5. Conclusions

This paper described the development of probabilistic models to predict the spatial distribution of maxi-mum depths of pitting for prestressing strands subjected to pitting corrosion. The models were then appliedto a stochastic FEA of a typical prestressed concrete bridge girder located at a coastal environment. The influ-ence of pitting corrosion on structural strength, time to failure and structural reliability were estimated. Ingeneral, including the spatial variability of pitting corrosion in the reliability analysis of a pretensionedPSC AASHTO bridge girder has increased the probability of strength failure by only 10% when comparedwith a mid-section analysis. However, the effect of the spatial variability of pitting corrosion is more significanton the serviceability limit state. It was also shown that approximately 45% of collapses have not been precededby excessive deflection. It was noted that the high variability of predictions of deterioration processes also sig-nificantly effects the structural reliability of a corroding structure. The probabilistic approach developed in thisstudy also allows for a more realistic representation of service life prediction.

Acknowledgement

The support of the Australian Research Council under grant DP0451871 is gratefully acknowledged.

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