Technical Paper

Ahsana Parammel Vatteri DOI: 10.1002/suco.201500081

K. Balaji Rao*
Anoop M. Bharathan

Time-variant reliability analysis of RC bridge girders

subjected to corrosion – shear limit state
Chloride-induced corrosion of reinforced concrete (RC) bridge ies during the past few decades, the nature of the problem
girders has led to a huge loss of national resources. One of the demands further research and development of strategies
important concerns affecting RC bridge girders is corrosion of the to address the problem. The process of corrosion of rein-
stirrups, which can even cause the failure mechanism to change forcement in RC members is complex and beset with un-
from a ductile flexural mode to a brittle shear mode. Hence, anal- certainties. Sources of uncertainties are numerous, such as
ysis of the reduction in shear capacity overtime is essential in the those associated with the phenomenon of corrosion itself,
reliability assessment of bridge girders, which is the topic of the measurement and quantification of environmental factors
paper. This paper proposes a stochastic modelling approach for affecting corrosion, time-dependent variations, physical
estimating the time-variant shear capacity and reliability within and microstructure properties of the concrete members,
the framework of a Monte Carlo simulation, which assists in the effect of corrosion on the structural behaviour, etc. to
sustainability-based service life design of bridge girders. Such
name just a few. A number of investigations have been
modern design concepts require methodologies for estimating
reported in the literature and deal with the mechanism of
whole life cost at the design stage itself. The development of
corrosion initiation and propagation [1–5], how corrosion
such methodologies would provide the designer with various op-
influences the structural behaviour of RC elements [6–9],
tions for arriving at an optimum design having the desired perfor-
the influence of micro-environments [10, 11] and preven-
mance level during the service life. The proposed approach takes
into account: 1) the randomness in basic variables, 2) the effect of tive techniques and repair and rehabilitation strategies for
micro-environments and the spatial variation of corrosion, 3) the corroded elements [12–14]. Chloride-induced corrosion of
number of stirrups resisting web shear failure, and 4) the ductile reinforcement reduces the diameter of the sound steel bar
to brittle transition of stirrup steel as corrosion propagates. The over time. In natural chloride-laden environments, con-
incorporation of this transition is found to have a significant influ- crete with zero initial chloride concentration (i.e. concrete
ence on the time-variant reliability of the girder. Although PFA not contaminated by chlorides during mixing) undergoes
concrete is known to have better durability characteristics than pitting-type corrosion, where corrosion of the steel bar is
OPC concrete, this paper gives a framework for its quantification initiated from the face closest to the chloride-laden en-
in terms of time-variant reliability. vironment. This leads to stress concentrations at the pit
locations [15], calling for additional consideration of unfa-
Keywords:  time-variant reliability, chloride-induced corrosion, shear capacity
vourable effects due to stress concentration. The corrosion
products occupy a much greater volume than the original
1 Introduction steel. In general, the volume of corrosion products can be
approximately 6–8 times the volume of the original rein-
Corrosion of the reinforcing steel has been identified as forcing bar [16].These products exert hoop pressure on the
one of the major causes of the premature rehabilitation concrete surrounding the reinforcement, which can result
of reinforced concrete structures. For (RC) bridge girders, in cracking/spalling of the concrete cover in localized ar-
especially in chloride-rich environments such as coastal eas. Apart from this, a reduction in the steel area reduces
zones, chloride-induced corrosion has led to a huge loss the structural capacity of the concrete member subjected
of national resources, thus hampering national econom- to corrosion. In general, a reduction in the longitudinal
ic growth. Even though the chloride-induced corrosion steel area causes a reduction in the flexural capacity and a
phenomenon and effects on RC structures have been re- reduction in the shear reinforcement area causes a reduc-
searched by conducting experimental and analytical stud- tion in the shear capacity of the member.
One of the important corrosion-governed failure
modes of RC bridge girders is that due to corrosion of
shear reinforcement/stirrups [6, 17]. Investigations have
* Corresponding author: balaji@serc.res.in
shown that corrosion of stirrups can even change the
Submitted for review: 11 June 2015; revision: 07 September 2015; accepted failure mechanism from a ductile flexural mode to a brittle
for publication: 12 October 2015. Discussion on this paper must be submitted
within two months of the print publication. The discussion will then be
shear mode. Hence, an analysis of the reduction in shear
published in print, along with the author’s closure, if any, approximately nine capacity over time is an important factor in the reliability
months after the print publication. assessment of RC bridge girders, which is the topic of this

162 © 2016 Ernst & Sohn Verlag für Architektur und technische Wissenschaften GmbH & Co. KG, Berlin · Structural Concrete 17 (2016), No. 2
 A. P. Vatteri/K. B.Rao/A. M. Bharathan · Time-variant reliability analysis of RC bridge girders subjected to corrosion – shear limit state

paper. This paper proposes a stochastic modelling ap- empirical coefficient that relates the reduction in yield
proach for estimating the time-variant shear capacity and strength to corrosion area loss. It was found that the prob-
reliability within the framework of a Monte Carlo simula- ability of failure assuming brittle reinforcement behaviour
tion (MCS). is up to 450 % higher than that obtained assuming ductile
The paper is organized as follows: Section 2 gives a behaviour. As pointed out by Bertolini [32] and others,
brief review of the literature related to the shear capac- micro-environmental effects on the process of corrosion
ity reduction of beams due to corrosion of rebars and at different locations on the same girder could be signifi-
a summary of the corrosion initiation and propagation cant considering the large sizes of girders. Hence, there is
models adopted. Stochastic analysis of reduction of shear a need to consider these effects when estimating the time-
capacity, computation of the capacity of ductile and brittle variant reliabilities of RC beams.
elements and time-variant reliability analysis are elabo- Most of the investigations presented in the above dis-
rated in section 3. An example problem is defined and cussion focused on the structural behaviour of the beams/
the proposed methodology is applied in section 4. Section girders with corroding reinforcement, assuming that cor-
5 discusses the results and conclusions are presented in rosion of reinforcement had already been initiated. Two
section 6. possible practical cases addressed by this assumption are:
a) existing structures that have already started corroding,
2 Literature review and b) structures in which chlorides are introduced dur-
ing mixing itself (due to use of beach sand or seawater for
Many researchers have carried out experimental and ana- mixing). However, for the design of important structures
lytical studies on the shear capacity reduction and flexural such as RC bridges and tall RC buildings, the codes rec-
capacity reduction of RC beams subjected to chloride-in- ommend that care should be taken to ensure an adequate
duced corrosion [6, 7, 9, 17, 18]. Chlorides are introduced service life by avoiding beach sand and using potable
into concrete either by mixing chlorides into the concrete water. For example, bridge codes such as IRS code 1997
mix or by diffusion from the external environment. The [22] restrict the amount of chlorides in the concrete mix to
majority of the experimental work concentrated on the avoid excess chloride concentrations. Thus, in the service
structural behaviour of RC members after the initiation life design of such structures, it is more appropriate to as-
of corrosion, and thus involved the first method, where sume that no added chlorides are present in the concrete
chlorides are already present in the concrete mix. In the mix at the time of concreting. When there are no added
case of environment-assisted corrosion, the diffusion of chlorides in the concrete, the presence of chlorides in the
chlorides into the concrete cover leads to corrosion initia- concrete is attributed to the transportation of chlorides
tion. This is a finite time and in most cases when designing from the surface of a concrete member to the inside by dif-
new structures, the initiation time is considered as the life fusion. It is known that consideration of the corrosion ini-
of the structure. Some of the investigations available in tiation time is important in the service life design of a RC
the literature with respect to shear capacity reduction due structure. In a probabilistic/stochastic service life design
to corrosion of stirrups are presented below. paradigm, the uncertainty associated with the initiation
Many experimental investigations reported in the time has to be estimated and needs to be propagated into
literature (viz. [17]) have reported that shear failure mode a propagation period to estimate realistically the uncer-
dominates in the case of RC beams where stirrups are sub- tainty associated with the service life of an RC member.
jected to corrosion. Corrosion was induced in RC beams Based on the review of the literature, the following
by mixing calcium chloride into the mixing water and by aspects are investigated in this paper:
applying an accelerated corrosion procedure. Coronelli 1) Developing a framework for reliability-based sustain-
and Gambarova [19] and Bhargava et al. [20] attempted to able design against shear failure for RC bridge girders
model the corrosion phenomenon taking into considera- subjected to corrosion of reinforcement in line with
tion the experimental findings of Rodriguez et al. [17]. In Stewart’s [6] formulation, additionally incorporating
developing a time-dependent reliability model, Bhargava randomness in corrosion initiation time and the micro-
et al. [20] considered generalized corrosion, with a reduc- environment effect.
tion in steel area as well as concrete area due to spalling 2) Illustrating the need for considering the transition from
as corrosion progressed. In order to develop modelling ductile to brittle behaviour of stirrups with the progress
guidelines for the structural assessment of corroded RC of corrosion when estimating the residual shear capaci-
beams, Coronelli and Gambarova [19] considered pitting- ties of a girder at different times.
type corrosion by considering a factor defined by the ratio 3) Quantifying the improvement in performance of RC
of area of pit to nominal bar area in the model. These girders when using PFA concrete instead of OPC con-
studies focus on the structural behaviour of the beams/ crete, with respect to reliability against shear failure.
girders with corroding reinforcement without giving con-
sideration to corrosion initiation time. The importance of One of the aspects of sustainable design is the choice of
modelling the spatial variation in degradation phenomena more durable materials for construction. The framework
in the service life assessment of RC structures was studied developed in this paper works well in this aspect, as it
by Ying and Vrouwenvelder [21]. Stewart [6] developed a gives the designer an option to compare the durability
spatial time-dependent reliability model for pitting-type performance of the RC girder, by using different material
corrosion which takes into account the transition of the compositions, at the design stage itself, thus enabling a
mechanical behaviour of reinforcement in RC beams. The sustainable design.
effects of pitting are considered by a pitting factor and an

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A. P. Vatteri/K. B.Rao/A. M. Bharathan · Time-variant reliability analysis of RC bridge girders subjected to corrosion – shear limit state

A brief overview of the corrosion initiation and Vu(t) = Vc(t) + Vs(t) (4)
propagation processes is carried out in the next section.
That is followed by the estimation of the shear capacity of The shear capacity of the concrete (without shear stirrups)
girders according to code provisions. depends on the grade of concrete and longitudinal steel
area, and is given by
2.1 Corrosion initiation and propagation
Vc(t) = τ c(t) ⋅ bw ⋅ d (5)
The ingress of chlorides into concrete is assumed to be
due to the diffusion of chlorides from the environment where:
through the concrete cover. Most of the models for initia- τc(t) permissible shear stress in concrete in girder with-
tion of chloride-induced corrosion are based on Fick’s sec- out shear reinforcement
ond law of diffusion (e.g. [9], [23]). Corrosion in stirrups bw width of web
is assumed to initiate when chloride concentration at the d effective depth of girder
location of stirrups reaches a critical value ccr. The time of
initiation is evaluated as follows: The value of τc(t) is obtained from the Eurocode 2 [26]
formula. The Eurocode is used instead of the IS code
−2 because the latter does not include any explicit formula
c 2  −1  c0 − ccr  
Ti = erf   (1) showing the relation between concrete grade and longitu-
4D   c0   dinal steel area. As the area of longitudinal steel reduces
with time due to corrosion, τc(t) is estimated with the
where: available area of longitudinal steel at any time t by using
Ti corrosion initiation time in years the Eurocode2 formula for τc [25, 26]:
c̃ concrete cover to stirrups in cm
D chloride diffusion coefficient of concrete in cm2/year  0.18 1 
τ c(t) =  ⋅ K ⋅ (100 pt(t)fcl ) 3 + 0.15 ⋅ σ cp
c0 chloride concentration at concrete surface as percent-  γc  (6)
age weight of concrete 3 1
ccr critical chloride concentration to initiate corrosion ≥ (0.035 ⋅ K2 ⋅ fcl 2
as percentage weight of concrete
where:
The first phase of corrosion, i.e. the corrosion initiation, is
 200 
mainly affected by the aforementioned factors, which are K = 1 +  ≤2
random variables. Hence, the time to initiation of corro-  d 
sion is also a random variable.
After initiation, corrosion propagates in the stirrups,
Ast(t)
causing a reduction in the cross-sectional area of the stir- pt(t) longitudinal steel ratio =
bw d
rups. This corresponds to the second phase. The reduced
diameter after corrosion is given by σcp axial stress in the case of prestressed girders
fcl characteristic cylinder compressive strength of con-
∅(t) = ∅(0) − rcorr(t − Ti ) (2) crete

where: The girder is analysed for its ultimate capacity and so the
∅(t) remaining diameter at time t partial safety factor for concrete γc is set to 1.
∅(0) initial diameter of stirrup The resistance offered by shear reinforcement Vs at
Ti corrosion initiation time any time t should be calculated as follows [24]:
rcorr rate of corrosion penetration, given by
σ sv(t)Asv(t)d (7)
Vs(t) =
rcorr = 0.0115 Icorrα (3) s

The corrosion rate rcorr is a quantity derived from ex- where:

perimental studies on corrosion current density Icorr σsc(t) permissible tensile stress in shear reinforcement
and pitting factor a. The pitting factor incorporates the Asv(t) total cross-sectional area of stirrup legs
effects of pitting-type corrosion. Owing to randomness s spacing of stirrups
in Icorr and a, the rate of corrosion rcorr, and hence ∅(t),
the reduced diameter of the reinforcing bars, are random In this paper the shear resistance of the cross-section is
variables. computed using the EC2 formula (without material safety
factor) for estimating the contribution of concrete to the
2.2 Computation of girder shear capacity shear resistance. The shear resistance offered by the steel
is calculated using IS 456. The contribution of steel to the
The ultimate shear capacity of a girder with shear rein- shear resistance of the cross-section, estimated using the
forcement (Vu(t)) comprises the shear capacities of the Eurocode formula (calculations not presented here), is
concrete (Vc(t)) and the shear reinforcement (Vs(t)) at any found to be almost the same as the IS code result. In this
time t (IS 456 [24]): investigation the formula given in IS456 is used for esti-

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 A. P. Vatteri/K. B.Rao/A. M. Bharathan · Time-variant reliability analysis of RC bridge girders subjected to corrosion – shear limit state

mating the shear capacity of the cross-section. It should

also be noted that the overall shear resistance can be cal-
culated using the fib Model Code for Concrete Structures
2010 [27] level III approximation. From calculations (not
presented here) it is found that the detailed equations of
fib Model Code 2010 and the equations used to compute
the shear resistance give similar results. However, a study Fig. 1.  Discretization of the girder for the right shear span from the continuous
similar to the one presented in this paper can be carried support
out using the shear formula of fib Model Code 2010. The
estimation of the time-dependent shear capacity of a gird-
er subjected to chloride-induced corrosion is discussed in ometrical influences of corrosion pits on the rebar surface
the next section. and the corresponding strain concentrations at such pits
[15]. It is also reported in the literature that the ductility of
3 Stochastic analysis of reduction in shear capacity corroded rebars is a function of the non-uniform distribu-
tion of corrosion pits along the bar length and the local
Stirrups are the outermost reinforcing bars and so cor- penetration depths [28]. In the present study, a value of
rosion is supposed to initiate in them first, followed by 0.005 is considered for b, following the suggestions by Du
longitudinal bars later. It is assumed in this study that et al [28] and Stewart [6]. Thus, this paper considers the
the reduction in shear capacity is due to the reduction effect of pitting and the stress concentration associated
in the area of stirrups and the transition of mechanical with pitting through this variable b and a pitting factor a
behaviour of stirrup steel from ductile to brittle as cor- (as introduced in section 2.1 and also in [6]).
rosion propagates. The total shear capacity of a girder is As pointed out by Stewart [6], a bar can be considered
the summation of the shear capacities of the reinforced brittle when the reduction in area exceeds 20 %. In addition,
concrete section and the steel stirrups. The contribution when > 20 % of the stirrups in an element has become brit-
of concrete to shear capacity depends on the concrete tle, the element is conservatively assumed to be brittle in the
cross-sectional area and the area of longitudinal steel. In present study. The shear capacity of an element is evaluated
this study, the reduction in capacity due to the reduction considering it as either ductile or brittle, as explained later.
in concrete cross-section is not considered; however, When one or more corroded stirrups fail under load, load
the reduction due to loss of area of longitudinal steel is redistribution is assumed to take place among the remaining
considered. Hence, corrosion of longitudinal steel should stirrups within the element until the maximum capacity of
also be taken into account, along with corrosion of stir- the element is reached.
rups. As mentioned before, corrosion initiation time is With bridge girders, being large in size, the size of
considered as a random variable for both the stirrups and single element will be considerable. Hence, variations in
the longitudinal bars to account for the randomness in properties associated with the elements (corrosion initiation
the cover, diffusion coefficient of concrete, surface and time, properties of concrete and steel, etc.) can be expected
critical chloride concentrations. After corrosion is initi- from element to element. The variables are thus treated as
ated, propagation is governed by the rate of corrosion as independent and identically distributed (i.i.d.) random vari-
given by Eq. (2), in which Icorr and a are considered to be ables from element to element. The properties are assumed
random variables. to remain the same within an element. In general, each ele-
Stirrup corrosion can be happening simultaneously ment can be considered as an identically distributed random
(but independently of each other) in many stirrups along variable with respect to the properties mentioned above.
the span of the girder and shear failure can occur any- In this investigation, the effect of a micro-environment
where along the shear-critical span. Hence, it is necessary is considered at the corrosion propagation stage by consid-
to discretize the shear-critical span of the girder and ering two different ranges of Icorr for elements. Since each
analyse the capacity of discretized elements separately. In element length is significant, i.e. the level of RH, availability
the present study, the shear span is considered as a series of oxygen and magnitude of temperature may vary within
system of Ne elements, each of length equal to the effective an element, and also that the stirrups are discrete within
depth d of the girder, such that failure of any one element an element, the Icorr and a of each stirrup in an element are
would lead to the total failure of the girder. considered to be i.i.d. random variables. The general assump-
Each element consists of a number of stirrups ns tions made in the present reliability analysis are summarized
that form a parallel system (Fig. 1). The shear capacity of below:
each element depends on the mechanical behaviour of 1. The analysis-design framework is developed for new
the stirrups in the element, which change from ductile to structures. The framework is intended to assist in the
brittle as corrosion propagates. Pitting is found to reduce sustainable design of structures with respect to shear
the ultimate strain at failure and thus the ductility of the capacity, enabling the designer to choose an appropri-
steel bars. Severe stress concentrations can be expected at ate material composition at the design stage itself.
the pits [15]. A regression factor b [6, 28, 29] is used in this 2. Deterministic dead load, live load and impact loads are
study which takes care of the effect of pitting in the me- considered for designing a bridge girder for shear.
chanical behaviour of rebars by linking the reduction in 3. Stress concentrations due to pitting-type corrosion are
yield strength to the corrosion (area) loss of rebars. There taken in to account by two factors: a, the pitting factor,
is general agreement in the literature that brittle behav- and b, an empirical regression factor that defines the
iour of corroded rebars may be mainly attributed to the ge- loss of yield strength with corrosion area loss.

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A. P. Vatteri/K. B.Rao/A. M. Bharathan · Time-variant reliability analysis of RC bridge girders subjected to corrosion – shear limit state

4. The mechanical behaviour of stirrups is assumed to where Vs(i)(t) = Rns(i)(t), as obtained by either Eq. (10) or
change from ductile to brittle when the loss of stirrup (11) and Vc(i)(t) is the shear capacity of concrete without
area exceeds 20 % of the initial nominal area [6]. Ad- stirrups at time t, as obtained with Eqs. (5) and (6), which
ditionally, an element is considered to be brittle when reduces with the reduction in longitudinal steel area with
> 20 % of its stirrups has become brittle. time; V is the time-invariant shear demand. The uncertain-
ties associated with the resistance model are taken care of
Details of the computation of shear capacity reduction, by modelling errors associated with the computation of
considering the elements as either ductile or brittle and reduced diameter f(t), reduced yield strength fy(i)j(t) and
the time-variant reliability estimation are discussed in the permissible shear stress in concrete without shear rein-
following sections. forcement τc(t) (Table 3).

3.1 Capacities of perfectly ductile and perfectly brittle 3.2 Estimation of probability of failure
elements
The probability of failure is the probability that Vus(t) ≤ 0 at
The shear capacity of one element when it is ductile (ac- time t. The probability is evaluated by the relative frequency
cording to the criteria mentioned before) is obtained by approach, i.e. probability of failure at time t is
adding the individual shear capacities of each stirrup at
every instant of time. Let i denote the element number pf(t) = (number of realizations in which Vus(t)
and j the stirrup number. The individual shear capacity ≤ 0)/1000) (14)
r(i)j(t) of the jth stirrup in the ith element at time t is given
by The methodology for estimating the time-variant probabil-
ity of failure against shear capacity of a girder subjected
r(i) j(t) = Asv
(i) (t) ⋅ f (i) (t)
j y j (8) to reinforcement corrosion is summarized in a flowchart
(Fig.2). Some of the codes of practice (viz. IS 456) specify
(i)
where A sv j(t) is the available reduced area of the same deemed-to-satisfy clauses for the durability-based service
stirrup, given by life design of RC structures. However, fib Model Code
2010 gives an explicit procedure for such design. This
(t − T )
2
(i) (t) = 2 ⋅ π  (i) (i)
Asv ∅ (0) − rcorr(i) (9) methodology, incorporating aspects related to sustaina-
j
4 j i
bility-based design, is presented in Fig. 3. Accordingly, the
structure should be redesigned if the failure probability
(i)
and f y j(t) is the yield strength of the same stirrup at time obtained is greater than an allowable failure probability at
t, given by the given time (say, expected service life).

 (i)(0) − A(i) (t)

Asv 4 Example
sv j
fy(i) j(t) = fy(i)(0) ⋅ 1 − β  (10)
 (i)(0)
Asv 
A two-span continuous T-beam bridge girder is considered
[30]. The length of each span is 21 m. The girder cross-sec-
(i) (i)
where ∅(i)(0), Ti(i), f y (0) and Asv (0) are the initial dia­ tion is shown in Fig.4. Maximum shear force is experienced
meter, corrosion initiation time, initial yield strength and in the intermediate (continuous) support region, where the
initial area of all stirrups in ith element respectively, rcorr(i)j flanges of the T-beam are in tension. The maximum shear
is the rate of corrosion penetration of the jth stirrup in the force (due to dead load, live load and impact load) is con-
same element and β is an empirical coefficient, taken as sidered to be acting uniformly over a span L/3 = 7  m on
0.005 [6], details of which are explained in section 3. both sides of the continuous support, as shown in Fig. 5.
When an element is ductile, element capacity due to
stirrups alone is the summation of the individual stirrup 4.1  Deterministic and random variables:
capacity of ns stirrups, given by
The nominal dimension of the girder with an assumed
service life tL = 50 years [22] is given in Fig. 4. Variables con-
∑ j=1 Asv(i) j(t) ⋅ fy(i) j(t)
ns (11)
Rns (i)(t) =
sidered to be deterministic are: overall depth of girder D1 =
1447.8 mm, width of web bw = 457.2 mm, width of flange bf
When the element is brittle, the capacity of the element = 2134 mm, depth of flange Df = 190.5 mm and spacing of
due to stirrups alone will be as follows: stirrups = 155 mm c/c. Variables considered to be random
are given in Table1 (properties of girder), Table 2(variables
Rns (i)(t) = max ns ⋅ r(i) 1(t),(ns − 1) ⋅ r(i) 2(t), ..., r(i) n (t) (12) related to corrosion initiation time) and Table 3 (variables
 s 
related to corrosion propagation). Modelling errors consid-
ered in this paper are also presented in Table 3.
where r(i) 1(t) < r(i) 2(t) <... r(i) ns(t) are the individual stirrup
capacities. 4.2 Initiation of corrosion in stirrups and longitudinal bars
Since the elements are in series, the limit state func-
tion for shear of the girder is defined by The girder is assumed to be exposed to a marine environ-
ment. The shear-critical spans (spans continuous over sup-
e
(
Vus(t) = min i =1:N Vs(i)(t) + Vc(i)(t) − V ) (13) port) are the focus of this investigation. It is known that

166 Structural Concrete 17 (2016), No. 2

 A. P. Vatteri/K. B.Rao/A. M. Bharathan · Time-variant reliability analysis of RC bridge girders subjected to corrosion – shear limit state

Fig. 2.  Flowchart for estimating the time-variant probability of failure against shear capacity due to corrosion

Fig. 4.  Nominal cross-section of bridge girder at continuous support [30]

Once a critical chloride concentration is attained at the

level of the tension bars, corrosion initiates. Propagation
Fig. 3.  Flowchart for sustainability-based service life design with respect to of corrosion in tension bars and stirrups is assumed to
shear capacity – continuation of Fig. 2 occur independently of each other once corrosion has
started in both.
the flanges of a T-beam at the support will be in tension.
It is assumed that the longitudinal tension reinforce- 4.3 Propagation of corrosion
ment in the flange corrodes due to ingress of chlorides
from the underside, and the stirrup reinforcement in the It is known that the local environment can vary not only
web (near the support) also undergoes corrosion due to along the span of a bridge girder, but also from girder to
ingress of chlorides from the side surfaces of the web. girder over the width of the bridge span. The variations in

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A. P. Vatteri/K. B.Rao/A. M. Bharathan · Time-variant reliability analysis of RC bridge girders subjected to corrosion – shear limit state

tinuous support. Hence, the range of Icorr is taken to be

between 1 and 5 mA/cm2 for the first two elements adja-
cent to the support, and i.i.d. variables within one element
[4]. For the elements further away from the support, the
range of Icorr is taken to be between 0.2 and1mA/cm2, and
is not considered to vary for stirrups within an element,
as variability in Icorr within an element is assumed to be
insignificant in more remote elements.
The Monte Carlo simulation technique with 1000 sim-
ulations is employed for the analysis of shear capacity and
reliability evaluation with time. The step-by-step procedure
explained in the flowchart in Fig.2 is followed to obtain
the results, which are discussed in the next section. For the
Fig. 5.  Approximate shear force diagram of loaded girder sustainability-based service life design for the shear limit
state, the procedure has to be extended to the flowchart
given in Fig.3 (which is self-explanatory).

local environment (viz. local temperature, relative humid- 5 Results and discussion
ity) will change the vulnerability of the beam with regard 5.1 Corrosion initiation time for stirrups
to corrosion propagation. This micro-environment effect
can be significant for tropical countries and for bridges It is assumed that the quality of workmanship would be
located in coastal areas. For example, the portion of the nominally the same and there should not be any compel-
intermediate girder located on an abutment is more prone ling need to consider it as a sequence of random variables
to corrosion than the portion remote from it. Thus, the within an element. The consequence of this assumption
corrosion propagation life changes. For this reason, rcorr is is – for a given element – that Ti is only one random vari-
expected to be higher for the elements closer to the con- able. However, from element to element, Ti variables are

Table 1.  Random variables associated with girder properties [9], [25]

Variable Unit Mean COV Type of distribution

Yield strength of steel (Fe415) fy MPa 415 0.12 lognormal

Compressive strength of concrete (M30) fck MPa 30 0.18 lognormal
Diameter of stirrup f mm 12 0.02 lognormal
Diameter of longitudinal bar f mm 30 0.02 lognormal
Clear cover c̃1, side of girder cm 5.0 0.1 lognormal
Clear cover c̃2,slab cm 8.75 0.1 lognormal

Table 2.  Variables associated with corrosion initiation time [18]

Variable Unit Mean COV Type of distribution

Chloride concentration on surface c0,OPC % by wt. of conc. 0.211–0.401 0.18 uniform

Chloride concentration on surface c0, PFA % by wt. of conc. 0.405–0.647 0.13 uniform
Critical chloride concentration ccr, OPC % by wt. of conc. 0.073–0.182 0.25 uniform
Critical chloride concentration ccr, PFA % by. Wt. of conc. 0.05–0.15 0.29 uniform
Diffusion coefficient D, OPC cm2/year 4.888 0.257 lognormal
Diffusion coefficient D, PFA cm2/year 0.1989 0.057 lognormal

Table 3.  Random variables considered corresponding to corrosion propagation phase

Variable Unit Mean COV Type of distribution

Icorr(1) near support mA/cm2 1–5 0.38 uniform

Icorr (2) remote from support mA/cm2 0.2–1 0.38 uniform
Alpha, a [31] – 4–6 0.12 uniform
Modelling error (me1)-associated with prediction – 1 0.1 lognormal
of f(t) and fy(t)
Modelling error (me2) associated with prediction – 1 0.2 lognormal
of τc(t)

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 A. P. Vatteri/K. B.Rao/A. M. Bharathan · Time-variant reliability analysis of RC bridge girders subjected to corrosion – shear limit state

considered to be i.i.d. variables. Thus, Ti is the same for tribution is found to be the best fit with the distribution
all stirrups in one element. A comparative study is car- above as verified by a K-S test.
ried out with a 30 % PFA concrete girder having the same
cross-sectional and strength properties, using the random 5.1.2 30 % PFA concrete
variables listed in Table 2.
The mean corrosion initiation time for stirrups is about 38
5.1.1 OPC concrete years for a T-girder made from30 % PFA concrete, which
is more than six times that of the OPC concrete girder;
The mean value of Ti for stirrups in the OPC concrete standard deviation varies from 13.8 to 14.7 years from
girder is about 6 years; standard deviation varies from element to element for the five elements in the shear span
6.99 to 7.36 years for the five elements of the shear span. considered, as shown in Fig. 7. The average value of COV
The average value of COV in the five elements is 1.156. for Ti in the five elements is 0.3639. The distribution is
The range space of Ti is from zero to 80 years consider- skewed to the right. The range space of the distribution is
ing all the elements. From the histograms of Ti of five from 10 to 120 years in most of the elements and can be
elements in Fig. 6, the distribution is found to be heavily considered general. In the case of 30 % PFA concrete, Ti is
skewed towards the right. A two-parameter Weibull dis- found to follow a lognormal distribution.

Fig. 6.  Corrosion initiation time of stirrups in different elements for OPC concrete girder

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A. P. Vatteri/K. B.Rao/A. M. Bharathan · Time-variant reliability analysis of RC bridge girders subjected to corrosion – shear limit state

Fig. 7.  Corrosion initiation time of stirrups in different elements for 30 % PFA concrete girder

5.2 Shear capacity reduction 5.2.1 OPC concrete girder

Initially, all the stirrups in the shear span are uncorroded, As discussed earlier, out of the series system of elements,
and the mechanical behaviour of the stirrup steel is duc- the shear capacity of the weakest element governs the
tile. As the stirrups start to corrode, the shear capacity shear capacity of the girder (Eq.(12)). Shear capacity
reduces from the original capacity, but the stirrups could distribution at different ages is studied to establish the
still exhibit ductile behaviour if the area loss is less than decrease in capacity over time as shown in Fig. 8. Mean
the critical limit of 20 % at which the mechanical behav- values of capacity at ages of 5, 10, 20, 30, 40 and 50 years
iour of the stirrup is assumed to become brittle. Only are 1130, 1020, 780, 619, 523 and 459 kN respectively.
when >  20 % of the stirrups within an element become The drastic fall in shear capacity of the girder beyond 20
brittle is the element itself considered to be brittle, as years (1020 to 780 kN) can be attributed to the transition
explained in section 3. After initiation of corrosion, the in the mechanical behaviour, as it can be shown from the
time-variant shear capacities of the girder are estimated deterministic calculation of the time required for a 20%
using the variables listed in Tables 1 and 3. The resulting reduction in stirrup area at which this transition occurs
distributions at various ages are presented in Figs. 8 and 9 is 21 years. If the transition in the mechanical behaviour
for OPC and 30 % PFA concrete respectively. from ductile to brittle is not taken into account, and the

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 A. P. Vatteri/K. B.Rao/A. M. Bharathan · Time-variant reliability analysis of RC bridge girders subjected to corrosion – shear limit state

Fig. 8.  Distribution of shear capacity at different ages for OPC concrete girder

stirrups are assumed to be ductile irrespective of the ex- The influence could be more predominant in structures
tent of corrosion, the mean values at the aforementioned with a longer service life than the 50-year life considered.
ages are approx. 1130, 1030, 823, 658, 544 and 467  kN Mean values of capacity at ages of 5, 10, 20, 30, 40 and
respectively. It can be seen that avoiding the brittle behav- 50 years are 1130, 1130, 1130, 1080, 944 and 789 kN
iour with the propagation of corrosion in stirrups can lead respectively (Fig. 9). The elements start becoming brittle
to overestimating the shear capacity, which is not desir- at an average age of 50 years, hence the sudden drop in
able due to the nature of the failure of the system. shear capacity between 40 and 50 years. If the transition
in the mechanical behaviour is not considered, mean
5.2.2 30 % PFA concrete values of capacities at the given ages are approximately
1130, 1130, 1130, 1085, 962 and 817kN respectively,
Similar observations to those for the OPC concrete which are overestimates, as mentioned in the case of the
girder are made in the case of 30 % PFA concrete. Since OPC concrete girder.
the corrosion initiation time is about six times higher for
the 30 % PFA concrete girder compared with the OPC 5.3 Time-variant probabilities of failure
concrete girder, the shear capacity reduction is com-
paratively lower for the 30 %PFA girder. Hence, the time The observations made above regarding reduction in
taken for the stirrups to become brittle will also be large. shear capacities over time are reflected in the correspond-

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A. P. Vatteri/K. B.Rao/A. M. Bharathan · Time-variant reliability analysis of RC bridge girders subjected to corrosion – shear limit state

Fig. 9.  Distribution of shear capacity at different ages for 30 % PFA concrete girder

ing probabilities of failure for OPC and 30 % PFA con- 30 % PFA concrete was carried out within the scope of
crete girders. The latter gives a more durable performance a Monte Carlo simulation (Fig. 2). The 30 % PFA con-
compared with the former, as is to be expected from its crete was found to give a more durable performance
low chloride diffusion coefficient and high corrosion over OPC concrete, and can be used as a sustainable
initiation time. Fig. 10 compares the time-variant prob- alternative to OPC. The reliability analysis took into
abilities of failure of the girder when the transition in the account the spatial variability of the corrosion process
mechanical behaviour is avoided and when it is taken into along the length of the girder, the micro-environment
account. Clearly, the first case, which is often followed, near the supports and the transition in the mechanical
gives lower probabilities of failure, which can lead to non- behaviour of stirrups from ductile to brittle, as explained
conservative design decisions. The difference in probabili- in section 3.
ties of failure of both approaches is more pronounced in More detailed studies have to be carried out to iden-
the case of 30 % PFA concrete. tify perfectly the transition in the mechanical behaviour
from ductile to brittle, as it is conservatively assumed to
6 Conclusions occur at an area loss of 20 % in the present study. Further,
the spatial variation in corrosion along the length of a
A stochastic analysis of the reduction in shear capac- stirrup and randomness in the loads, not addressed in this
ity of RC bridge girders made from OPC concrete and study, are identified as scope for future work.

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 A. P. Vatteri/K. B.Rao/A. M. Bharathan · Time-variant reliability analysis of RC bridge girders subjected to corrosion – shear limit state

Fig. 10.  Time-variant probabilities of failure

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