Deterioration of Strength of RC Beams due to Corrosion and

Its Influence on Beam Reliability

Dimitri V. Val1

Abstract: This paper examines the effect of corrosion of reinforcing steel on flexural and shear strength, and subsequently on reliability,
of reinforced concrete beams. Two types of corrosion—general and pitting—are considered, with particular emphasis on the influence of
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pitting corrosion of stirrups on the performance of beams in shear. Variability of pitting corrosion along a beam is considered and the
possibility of failure at a number of the beam cross sections is taken into account. Probabilities of failure are evaluated using Monte Carlo
simulation. Uncertainties in material properties, geometry, loads, and corrosion modeling are taken into account. Results show that
corrosion of stirrups, especially pitting corrosion, has a significant influence on the reliability of reinforced concrete beams.
DOI: 10.1061/共ASCE兲0733-9445共2007兲133:9共1297兲
CE Database subject headings: Concrete beams; Concrete, reinforced; Corrosion; Flexural strength; Reliability; Shear strength;
Deterioration.

Introduction cording to tests of RC beams subjected to corrosion, most of the

beams, especially those with high reinforcement ratio, failed in
Corrosion of embedded reinforcing steel is one of the main causes shear 共Rodriguez et al. 1997兲. This can be explained by smaller
of deterioration of reinforced concrete 共RC兲 structures. Reinforc- diameters of stirrups compared to those of longitudinal reinforc-
ing steel is normally passive in concrete due to high alkalinity of ing bars so that relative loss of the cross-sectional area due to
the concrete pore solution, however, penetration into the concrete corrosion in the stirrups is much more significant than that in the
of chlorides or carbonation destroys this inhibitive property of the longitudinal bars.
concrete and leads to corrosion. Two types of corrosion—general The present paper examines the influence of corrosion of re-
and pitting—are possible. General corrosion affects a substantial inforcing steel 共both general and pitting兲 on flexural and shear
area of reinforcement with more or less uniform metal loss over strength and corresponding reliability of RC beams, with particu-
the perimeter of reinforcing bars. It also causes cracking and
lar emphasis on the effect of pitting corrosion on the performance
eventually spalling/delamination of the concrete cover and pro-
of the beams in shear. Since the mechanics of pitting corrosion is
duces rust staining on the concrete surface so it can be detected
still not completely understood it is modeled by a Gumbel distri-
quite easily during inspection of a structure. Pitting 共or localized兲
corrosion, in contrast to general corrosion, concentrates over bution using an empirical approach based on statistical character-
small areas of reinforcement. Pitting corrosion often does not ization of maximum pit depths. It is assumed that maximum pit
cause disruption of the concrete cover and produces little 共or depths in different reinforcing bars are statistically independent.
none兲 rust staining on the concrete surface and so it may be dif- Variability of pitting corrosion along a beam is considered and
ficult to be discovered during inspection. possibility of failure 共flexural and shear兲 at a number of the beam
Most studies to date have considered the effect of corrosion cross sections 共and not only at the most loaded one兲 is taken into
共both general and pitting兲 on the flexural strength 共and corre- account. Other parameters affecting load bearing capacity of RC
sponding reliability兲 of RC members 共e.g., Val and Melchers beams such as concrete strength, effective depth of cross section,
1997; Enright and Frangopol 1998; Vu and Stewart 2000; Li and steel strength may also vary along the beam length 共or, as
2004; Stewart 2004兲. A few studies, which have also considered steel strength, between different reinforcing bars兲. Currently, there
the influence of corrosion on shear strength, have modeled only are not sufficient data to quantify their variability. In order to
general corrosion 共e.g., Frangopol et al. 1997兲. Meanwhile, ac- estimate a possible effect of this variability on reliability of RC
beams, two limit cases are considered. First, concrete strength
1
Senior Lecturer, Dept. of Structural Engineering and Construction along the beam length as well as steel strength of different
Management, Faculty of Civil and Environmental Engineering,
reinforcing bars are treated as fully correlated within a beam.
Technion–Israel Institute of Technology, Haifa 32000, Israel. E-mail:
val@tx.technion.ac.il Second, these parameters are treated as completely statistically
Note. Associate Editor: Shahram Sarkani. Discussion open until independent.
February 1, 2008. Separate discussions must be submitted for individual Probabilities of failure are evaluated using Monte Carlo simu-
papers. To extend the closing date by one month, a written request must lation. Uncertainties in material properties, geometry, loads, and
be filed with the ASCE Managing Editor. The manuscript for this paper corrosion modeling are taken into account. Results show that cor-
was submitted for review and possible publication on April 5, 2006;
approved on January 9, 2007. This paper is part of the Journal of Struc- rosion of stirrups 共especially, pitting corrosion兲 and its effect on
tural Engineering, Vol. 133, No. 9, September 1, 2007. ©ASCE, ISSN shear strength may have significant influence on the reliability of
0733-9445/2007/9-1297–1306/$25.00. RC beams.

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J. Struct. Eng. 2007.133:1297-1306.

 Corrosion Modeling

General Corrosion
A quantitative description of corrosion propagation is usually
given in terms of the corrosion rate, which is defined as the loss
of metal per unit of surface area per unit of time. Most nonde-
structive techniques currently used for monitoring corrosion are
based on electrochemical measurements, with the corrosion rate
estimated in terms of a corrosion current density, icorr. In the case
of general corrosion, the latter can be transformed directly into
the loss of metal by the use of Faraday’s law of electrochemical
equivalence, which indicates that a constant corrosion current
density of 1 ␮A / cm2 corresponds to a uniform corrosion penetra-
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tion of 11.6 ␮m per year. Thus, assuming a constant corrosion

rate, the reduction in the diameter of a corroding reinforcing bar
⌬D after t years since corrosion initiation can be estimated 共in
millimeters兲 as
⌬D共t兲 = 0.0232icorrt 共1兲
If to assume that the corrosion current density is the same for a
group of n reinforcing bars of the same diameter D0, their cross-
sectional area after t years of general corrosion will be
␲关D0 − ⌬D共t兲兴2
As共t兲 = n ⱖ0 共2兲
4

Pitting Corrosion Fig. 1. Pit configuration

Pitting corrosion, in contrast to general corrosion, concentrates
over small areas of reinforcement. As a result, a corroding area of

冋冉 冊 冏 冏册
a reinforcing bar may be much smaller than the area associated 2
with the measurement of icorr. Thus, icorr cannot be converted di- 1 D0 D0 p共t兲2
A1 = ␪1 −a − 共5兲
rectly into loss of cross-sectional area of a corroding bar. Accord- 2 2 2 D0
ing to results of Gonzales et al. 共1995兲, the maximum penetration

冋 册
of pitting, Pmax, on the surface of a rebar is about 4–8 times the
average penetration, Pav, of general corrosion. The results were 1 p共t兲2
A2 = ␪2 p共t兲2 − a 共6兲
obtained for 125-mm-long specimens of 8 mm diameter reinforc- 2 D0
ing bars. These results are in broad agreement with those reported

冑 冋 册
by Tuutti 共1982兲, who received the ratio R = Pmax / Pav between 4 2
and 10 for 5 and 10 mm reinforcing bars of 150– 300 mm length. p共t兲
a = 2p共t兲 1− 共7兲
Thus, the depth of a pit, p 共which is equivalent to the maximum D0
penetration of pitting兲 after t years since corrosion initiation can

冉 冊 冉 冊
be evaluated as
a a
p共t兲 = 0.0116icorrtR 共3兲 ␪1 = 2 arcsin , ␪2 = 2 arcsin 共8兲
D0 2p共t兲
In order to estimate the loss of a cross-sectional area of a rein-
forcing bar due to pitting, additional assumptions about the form For a group of n reinforcing bars of the same diameter D0 the
of a pit are needed. In this study a hemispherical model of a pit cross-sectional area after t years of pitting corrosion can be esti-
suggested by Val and Melchers 共1997兲 is employed 共see Fig. 1兲. mated as
Based on this model, the cross-sectional area of a pit, A p, in a
n
reinforcing bar with a diameter D0 after t years of corrosion can ␲D20
be calculated as As共t兲 = n
4
− 兺 A p,i共t兲 ⱖ 0 共9兲

冦
i=1
D0
A1 + A2 , p共t兲 ⱕ
冑2 As indicated by the results of Gonzales et al. 共1995兲 and Tuutti
共1982兲, there is significant uncertainty associated with R, i.e., the
␲D20 D0
冑2 ⬍ p共t兲 ⱕ D0
A p共t兲 = − A1 + A2 , 共4兲 ratio between the maximum pit depth and the average corrosion
4 penetration. At the same time, a popular approach to modeling pit
␲D20 depth is based on statistical characterization of maximum pit
, p共t兲 ⬎ D0 depth using extreme value theory, in particular, the Gumbel dis-
4
tribution 共e.g., Turnbull 1993兲. Thus, in this study R is treated as
where a random variable modeled by the Gumbel distribution

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 The flexural and shear strengths of a RC beam are evaluated in
accordance with ACI 共2005兲. The flexural strength, M R, of a rect-
angular cross section is calculated as

冉
M R = As f y d −
As f y
1.7f cb
冊 共16兲

where As⫽cross-sectional area of reinforcing steel;

Fig. 2. Beam layout and division into elements b⫽cross-sectional width; d⫽effective depth of cross section;
f c⫽compressive strength of concrete; and f y⫽yield strength of

再 冋 册冎
reinforcing steel. The flexural strength should also not exceed the
共R − ␮兲 moment capacity for compression failure 共e.g., MacGregor 1997兲
F共R兲 = exp − exp − 共10兲
␣ 1
M R ⱕ f cbd2 共17兲
where ␣ and ␮⫽parameters of the distribution. According to
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3
Stewart 共2004兲, these parameters can be determined by assuming,
In probabilistic analysis a normal random variable with a mean of
based on the results of Gonzales et al. 共1995兲, that for an 8 mm
1.10 and a coefficient of variation of 0.12 is introduced to take
diameter bar of 125 mm length R = 4 and R = 8 represent the fifth
into account model uncertainty associated with Eqs. 共16兲 and 共17兲
and 95th percentiles of the distribution, respectively. The mean
共MacGregor et al. 1983; Lu et al. 1994兲. The shear strength, VR,
and coefficient of variation are then 5.65 and 0.22, respectively,
which includes the strength provided by concrete, Vc, and the
which corresponds to the parameters of the Gumbel distribution
strength provided by stirrups, Vs, is calculated as
␮0 = 5.08 and ␣0 = 1.02. For a reinforcing bar with different di-
mensions the parameters of the Gumbel distribution are deter- VR = Vc + Vs 共18兲
mined as 共Turnbull 1993兲
冑f c
␮ = ␮0 +
1
␣0
ln
A
A0
, 冉 冊 ␣ = ␣0 共11兲
Vc =
6
bd, Vs =
Av f yd
s
共19兲

where Av⫽cross-sectional area of the stirrups and s⫽stirrup spac-

where A⫽surface area of the given bar; and A0⫽surface area of
ing. Model uncertainty associated with Eq. 共18兲 is represented by
an 8 mm diameter bar of 125 mm length.
a normal random variable with a mean of 1.20 and a coefficient of
variation of 0.112 共Mirza and MacGregor 1979b; Lu et al. 1994兲.
Probabilistic analysis is carried out using Monte Carlo simu-
Probabilistic Modeling of RC Beam Failure lation. This allows us to take into account quite easily the possi-
bility of different failure modes, variability of pitting corrosion
In order to account for spatial variability of pitting corrosion and,
along the length of a beam, and the influence of time, in particu-
as a result, possibility of failure at different cross sections along
lar, the effect of the previous performance of a beam on its reli-
the length of a beam 共i.e., not necessarily the most loaded one兲 the
ability. Load acting on a beam is comprised of dead and live
beam is divided into a number of elements along its length 共see
loads. Dead load is treated as a normal random variable with a
Fig. 2兲. For each element two possible modes of failure—in flex-
mean of 1.05 times its nominal value, Gn, and coefficient of varia-
ure and in shear—are considered. The corresponding limit state
tion of 0.10 共Ellingwood et al. 1980兲. Live load is comprised of
functions for the ith element can be written as
sustained and extraordinary 共transient兲 components. Sustained
G M,i共t兲 = M R,i共t兲 − M S,i共t兲 共12兲 live load is modeled by a gamma distribution with mean of
0.30Qn 共Qn denotes a nominal value of live load兲 and coefficient
of variation 0.60 共Chalk and Corotis 1980兲. It is assumed that its
GV,i共t兲 = VR,i共t兲 − VS,i共t兲 共13兲 mean duration is eight years. Extraordinary live load is also mod-
where M R,i共t兲⫽flexural strength of the ith element at time t after eled by a gamma distribution with annual mean of 0.19Qn and
corrosion initiation; M S,i共t兲⫽bending moment in the middle of coefficient of variation of 0.66 共Philpot et al. 1993兲.
this element at the same time; VR,i共t兲⫽shear strength of the ith In the analysis for each simulation run values of the random
element; and VS,i共t兲⫽shear force in the ith element at time t. variables representing structural properties, dead load, and the
Thus, the probability of failure of a beam between t j and t j+1 year parameter of pitting corrosion R are generated at time t = 0. It is
after corrosion initiation can be presented as assumed that the maximum pitting depths in different reinforcing
bars, in the same reinforcing bar within different elements along

冦 冧
m
the beam length, and in legs of a stirrup are mutually independent.
艛 关共G M,i共␶兲 ⬍ 0兲 艛 共GV,i共␶兲 ⬍ 0兲兴
p f 共t j,t j+1兲 = Pr i=1 共14兲 Thus, for all these segments of reinforcing bars values of R are
generated independently from the Gumbel distribution with the
␶ 傺 共t j,t j+1兲
parameters determined using Eq. 共11兲.
where m⫽number of elements along the beam length. The cumu- For each of the segments the maximum pit depth is then cal-
lative probability of failure up to the tLth year after corrosion culated by Eq. 共3兲, and the remaining cross-sectional area of re-
initiation, P f 共tL兲 is then equal to inforcing steel by Eqs. 共4兲–共9兲. For simplicity, it is assumed that
corrosion rates expressed in terms of the corrosion current den-
L
sity, icorr, are deterministic and do not change with time. In the
P f 共tL兲 = 兺
j=1
p f 共t j−1,t j兲 共15兲 case of general corrosion, uniform reduction of the cross-sectional
area of reinforcing bars is estimated by Eqs. 共1兲 and 共2兲. Using the
where t0 = 0. calculated cross-sectional areas of longitudinal reinforcement and

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J. Struct. Eng. 2007.133:1297-1306.

 Table 1. Statistical Properties of Random Variables
Parameter Mean COY Distribution Reference
Concrete compressive strength 共MPa兲 26.2 0.18 Log-normal Mirza et al. 共1979兲
Model uncertainty 共concrete shear strength兲 1 0.156 Normal Mirza et al. 共1979兲
Steel strength 共MPa兲 490 0.10 Log-normal Mirza and MacGregor 共1979a兲
Flexural model uncertainty 关Eqs. 共16兲 and 共17兲兴 1.10 0.120 Normal MacGregor et al. 共1983兲, Lu et al. 共1994兲
Shear model uncertainty 关Eq. 共18兲兴 1.20 0.112 Normal Mirza and MacGregor 共1979b兲, Lu et al. 共1994兲
Effective depth of cross section Nominal 0.02 Normal Östlund 共1991兲, Lu et al. 共1994兲
Parameter of pitting corrosion 共R兲 5.65a 0.22a Gumbel Stewart 共2004兲
Dead load 1.05Gn 0.10 Normal Ellingwood et al. 共1980兲
Sustained live load 0.3Qn 0.60 Gamma Chalk and Corotis 共1980兲
Extraordinary live load 0.19Qn 0.66 Gamma Philpot et al. 共1993兲
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a
For a 8-mm diameter 125-mm long bar.

stirrups and generated values of the other structural parameters, As stated previously, the corrosion rates are treated as deter-
the flexural and shear strengths of each element are evaluated and ministic and time independent. According to the classification
compared with the corresponding load effects 共shear force or suggested by BRITE/EURAM 共1995兲, corrosion rates between
bending moment兲, which are estimated in accordance to generated 0.1 and 0.5 ␮A / cm2 correspond to low corrosion intensity, be-
values of the loads. The procedure is carried out for successive tween 0.5 and 1.0 ␮A / cm2 correspond to moderate, and above
annual time increments until one of the beam elements fails or in 1.0 ␮A / cm2 correspond to high. Four different corrosion rates:
shear, or in flexure. A “failed” structure is not considered in sub- 0.5, 1.0, 2, and 3 ␮A / cm2, which represent a variety of corrosion
sequent time increments. After the completion of all simulation intensities, are considered in this study. For comparison, results
runs, the annual probabilities of failure defined by Eq. 共14兲 are for noncorroded beams will also be presented.
evaluated by dividing the number of beams failed in a particular Initially, it was assumed that material properties such as the
year by the total number of simulation runs. The cumulative prob- strength of concrete and of reinforcing steel are fully correlated
abilities of failure are then calculated using Eq. 共15兲. Note that the within a beam. Analysis is carried out for the following uniform
use of Monte Carlo simulation allows to estimate separately prob- loads: dead load Gn = 20 kN/ m and live load Qn = 35 kN/ m
abilities of failure in shear and in flexure. 共Qn / Gn = 1.75兲. Beams designed for this load combination have
the following reinforcement: longitudinal, nine No. 8 bars 共the
effective depth is 710 mm兲 or three No. 14 bars 共the effective
Results of Reliability Analysis of RC Beams depth is 740 mm兲; and shear, No. 3 stirrups placed every
200– 2,100 mm from the supports and every 350 mm for the rest
Analyses are carried out for simply supported RC beams. The of the beam 共in this case near the supports the ratio of the shear
beams’ dimensions are: length, 10 m; depth, 0.8 m; and width, strength provided by stirrups to that provided by concrete Vs / Vc
0.35 m. The specified compressive strength of concrete used for ⬇ 1兲.
the beams was 27.6 MPa 共4,000 psi兲. The specified yield strength Results for the beams with noncorroded reinforcement are pre-
of reinforcing steel was 414 MPa 共60,000 psi兲. Reinforcing bars sented in Fig. 3. As can be seen, for the noncorroded beams the
of two sizes, No. 8 共25.4-mm diameter兲 and No. 14 共43-mm di- probability of failure is mainly dependent on flexural failures,
ameter兲, were used for longitudinal reinforcement; stirrups were while the contribution of shear failures to the total failure prob-
made of No. 3 共9.5-mm diameter兲 rebars. The beams were uni- ability is insignificant. The results also show that the beams with
formly loaded by dead and live loads. For the purpose of analysis longitudinal reinforcement of Nos. 8 and of 14 bars have similar
the beams were divided into 13 elements along their length 共i.e., probabilities of failure. Results for the beams with corroding re-
m = 13兲; the length of each element was 0.77 m, which is approxi- inforcement are presented in Fig. 4. According to the results, there
mately equal to the effective depth of the beam cross section. is no significant difference between the total probabilities of fail-
Statistical properties of the random variables used in the analysis ure for the noncorroded beams and those for the beams with the
are summarized in Table 1. corrosion current density icorr = 0.5 ␮A / cm2 共for this reason cor-

Fig. 3. Cumulative probabilities of failure for noncorroded beams 共Vs / Vc ⬇ 1兲

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Fig. 4. Cumulative probabilities of failure for corroding beams 共Vs / Vc ⬇ 1兲

rosion rates lower than 0.5 ␮A / cm2 were not considered in this bars 共see Fig. 3, icorr = 2 and 3 ␮A / cm2兲, however, at that time the
study兲. Corrosion at this rate results mainly in an increase in the probabilities of failure are so high that this result is hardly of any
probability of shear failure, which is initially much lower than practical interest. In the case of general corrosion the relative
that of flexural failure and its increase during the considered time influence of shear and flexural types of failure on the beam reli-
period is not sufficiently large to make a noticeable effect on the ability depends also on the diameter of longitudinal reinforce-
total failure probability. As can also be seen, the influence of ment. For example, for icorr = 1 ␮A / cm2 the probability associated
general corrosion in this case is slightly higher than that of pitting
with shear failure becomes higher than that due to flexural failure
corrosion.
after about 80 years when the longitudinal reinforcement is made
For higher corrosion rates the influence of shear failure on the
beam reliability increases significantly, especially when pitting of No. 14 and after 90 years when it is made of No. 8 bars 共in the
corrosion takes place. In the case of pitting corrosion shear failure latter case there is actually no significant difference between the
becomes the dominant type of failure after approximately 50 two probabilities兲; for icorr = 2 ␮A / cm2 this happens after about 40
years when icorr = 1 ␮A / cm2, after 25 years when icorr years with No. 14 bars and after 50 years with No. 8 bars. It can
= 2 ␮A / cm2, and after 17 years when icorr = 3 ␮A / cm2. As corro- also be seen that for the corrosion rates of 1 ␮A / cm2 and higher
sion propagates flexural failure may become dominant again, es- pitting corrosion has a stronger effect on the beam reliability than
pecially when the longitudinal reinforcement consists of No. 8 general corrosion.

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Fig. 5. Cumulative probabilities of failure in noncorroded beams: 共a兲 Vs / Vc ⬇ 0.5; 共b兲 Vs / Vc ⬇ 1.5

Influence of Load Level Vs / Vc ⬇ 0.5 are presented in Fig. 6. When icorr = 0.5 ␮A / cm2 the
results are very similar to the results without corrosion—the total
Obviously, the effect of corrosion of stirrups on the beam reliabil-
failure probabilities depend almost completely on flexural failure,
ity should depend on the load level. At lower load levels the main
contribution to the shear strength of a RC beam is provided by and their increase compared to those for the noncorroded beams
concrete, therefore, reduction of the cross-sectional area of stir- is insignificant. For higher corrosion rates the influence of corro-
rups due to corrosion has a relatively small effect on the shear sion on the beam reliability becomes significant; however, the
strength of the beam and subsequently on the beam reliability. relative influence of shear and flexural failures on the total failure
The contribution of stirrups to the shear strength increases along probabilities differs from that for the beams with Vs / Vc ⬇ 1. In the
with an increase in the load level 共at least, under the condition case of general corrosion, shear failure does not represent any real
that the beam cross section remains the same兲 and as a result the danger for the beam reliability except for the corrosion rate of
beam reliability becomes more sensitive to loss of the cross- 3 ␮A / cm2 and the longitudinal reinforcement consisting of No.
sectional area of stirrups caused by corrosion. 14 bars 共in this case shear failure starts to make a noticeable effect
In order to investigate the influence of the load level on reli- on the total failure probability after about 60 years兲. In the case of
ability of RC beams subjected to corrosion two additional load pitting corrosion, the influence of shear failure on the beam reli-
combinations are considered: 共1兲 Gn = 14 kN/ m, Qn = 24.5 kN/ m; ability depends on the size of longitudinal reinforcing bars. When
and 共2兲 Gn = 25 kN/ m, Qn = 43.75 kN/ m 共for both the combina- No. 8 bars are used, the probability of shear failure exceeds that
tions Qn / Gn = 1.75兲. Beams designed for the first load combina- of flexural failure just for relatively short periods of time 共be-
tion have the following reinforcement: longitudinal, six No. 8 tween 80 and 100 years for icorr = 1 ␮A / cm2, between 40 and 60
bars or 2 No. 14 bars; shear, No. 3 stirrups placed every 350 mm years for icorr = 2 ␮A / cm2, and between 25 and 40 years for icorr
共in this case Vs / Vc ⬇ 0.5兲; for the second: longitudinal, 12 No. 8 = 3 ␮A / cm2兲, and even within these periods it does not become
bars or 4 No. 14 bars; shear, No. 3 stirrups placed every 140 mm completely dominant, i.e., the contribution of flexural failure to
to 2,350 mm from the supports and every 350 mm for the rest of the total failure probability is still noticeable. When the longitu-
the beam 共in this case Vs / Vc ⬇ 1.5兲. dinal reinforcement consists of No. 14 bars, the results are quite
Results for the beams with noncorroded reinforcement are similar to those for Vs / Vc ⬇ 1, although shear failure becomes the
shown in Fig. 5. The results are similar to those for the beams dominant type of failure at later times 共after 75 years for icorr
with Vs / Vc ⬇ 1 共see Fig. 3兲—flexural failure is the dominant type = 1 ␮A / cm2, after 40 years for icorr = 2 ␮A / cm2, and after 25
of failure, while the influence of shear failure is even less signifi- years for icorr = 3 ␮A / cm2; in the last case flexural failure be-
cant than for Vs / Vc ⬇ 1. Relatively high probabilities of failure for comes dominant again after about 70 years兲.
the beams with Vs / Vc ⬇ 1.5 关Fig. 5共b兲兴 can be explained by two Figure 7 shows results for the beams with corroding reinforce-
factors: first, a higher level of live load increases the total load ment and Vs / Vc ⬇ 1.5. The results are quite similar to those for
variability and leads to an increase in the failure probability 共as Vs / Vc ⬇ 1. For icorr = 0.5 ␮A / cm2 the total failure probabilities are
can be seen, the probabilities of failure for the beams with very similar to those for the noncorroded beams and clearly de-
Vs / Vc ⬇ 0.5 are also lower than those for Vs / Vc ⬇ 1兲; second, the pend mostly on flexural failure with the dependence even stronger
cross-sectional area of the longitudinal reinforcement is slightly than for Vs / Vc ⬇ 1 共this is explained by the high probabilities of
less than that required by design, which is done in order to avoid flexural failure for the noncorroded beams兲. For the corrosion
overreinforcement 共i.e., compression failure in flexure兲. rates of 1 ␮A / cm2 and higher in the case of pitting corrosion 共and
Results for the beams with corroding reinforcement and 2 ␮A / cm2 and higher in the case of general corrosion兲, shear

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Fig. 6. Cumulative probabilities of failure for corroding beams 共Vs / Vc ⬇ 0.5兲

failure becomes the dominant type of failure after a number of strength of reinforcing steel are fully correlated within a RC
years of corrosion 共the time when it happens depends, of course, beam. However, the analyzed beams were still divided into m
on the corrosion rates兲. However, after that, in contrast to the 共=13兲 elements along their length in order to account for spatial
results for Vs / Vc ⬇ 1, the probabilities of flexural failure continue variability of pitting corrosion. This allowed accounting for fail-
to remain lower than those of shear failure for the rest of the time ures at different cross sections along the length of a beam and not
considered. just at the most loaded one. Fig. 8 shows the relative failure
The presented results clearly demonstrate that corrosion 共espe- probabilities at the different elements along the beam length for
cially, pitting corrosion兲 of reinforcing steel can change com- flexural and shear types of failure 共i.e., the probabilities of flex-
pletely the relative influence of shear and flexural types of failure ural or shear failure at the elements divided by the total failure
on the beam reliability—for nondeteriorated RC beams the flex- probabilities of the corresponding failure modes兲 for different cor-
ural type of failure is usually dominant but as corrosion propa- rosion rates. As can be seen, due to the spatial variability of
gates the shear type of failure becomes the one that controls the pitting corrosion failure in flexure at the corrosion rates of
beam reliability. 1 ␮A / cm2 and higher may occur not only at the midspan of the
beam 共Element 7兲, but also at the adjacent elements as well. This
Influence of Spatial Variability result has been also demonstrated by Stewart 共2004兲. Shear fail-
The results previously presented were obtained based on the as- ure mainly occurs in the most loaded elements near the beam
sumption that material properties such as strength of concrete and supports 共Elements 1 and 13兲, and the influence of spatial vari-

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Fig. 7. Cumulative probabilities of failure for corroding beams 共Vs / Vc ⬇ 1.5兲

ability of pitting corrosion on this type of failure is insignificant. ment within a RC beam was assumed. Results of the analysis
It is worth noting that according to the results the probability of 共including comparison with the previously obtained results based
shear failure in the adjacent Elements 2 and 12 is not zero but on the assumption of full correlation兲 are shown in Fig. 9. As can
very low 共less than 10−4兲. be seen, there is no significant difference between the new results
Finally, in order to examine the influence of spatial variability and those obtained using the assumption about full correlation. It
of material properties within a RC beam 共i.e., compressive is worth noting that similar results 共not presented in the paper兲
strength of concrete along the beam length and yield strength of were observed for the beams with longitudinal reinforcement
steel between different reinforcing bars兲, analysis of the previ- made of No. 14 bars and for corrosion rates of 0.5 and 2 ␮A / cm2.
ously considered beams 共with loads Gn = 20 kN/ m and Qn This indicates that the use of the assumption about full correlation
= 35 kN/ m and No. 8 longitudinal bars兲 has been carried out as- is justified.
suming that the compressive strength of concrete in different el-
ements along the beam length is statistically independent as well Conclusions
as the yield strength of different reinforcing bars. This is opposite
to the previous analyses, in which full correlation of the compres- The paper considered the effect of general and pitting corrosion
sive strength of concrete as well as the yield strength of reinforce- on the reliability of RC beams. Two possible modes of failure—in

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Fig. 8. Relative probabilities of failure in shear and flexure in elements along the beam length 共Vs / Vc ⬇ 1, longitudinal reinforcement: No. 8 bars兲

flexure and in shear—were taken into account. To model variabil- liability of the considered beams. For higher rates 共1 ␮A / cm2 and
ity of pitting corrosion along the length of a beam, the beam was higher兲 the effect of corrosion becomes significant. At these rates
divided into elements and flexural and shear resistances were pitting corrosion is more dangerous than general corrosion. More-
checked for each of the elements. Failure of any element in over, at these rates the reduction of shear resistance due to corro-
flexure or in shear was considered as failure of the beam. The sion of stirrups, especially pitting corrosion, has a major effect on
analysis was carried out for RC beams subjected to corrosion at
the beam reliability. The results demonstrate that assessment of
different levels of intensity. The results showed that corrosion
the performance of RC beams in corrosive environments should
共both general and pitting兲 with low and moderate to low rates
共0.5 ␮A / cm2 and lower兲 had a insignificant influence on the re- include consideration of the effect of corrosion of stirrups on the

Fig. 9. Cumulative probabilities of failure in corroding beams with and without taking into account spatial variability of material properties
within the beams

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 shear resistance of a beam, otherwise reliability of the beam may BRITE/EURAM. 共1995兲. “The residual service life of reinforced concrete
be significantly overestimated. structures.” Final Technical Rep. BRUE-CT92-0591, Brite/EURAM,
European Union.
Chalk, P. L., and Corotis, R. B. 共1980兲. “Probability model for design live
Acknowledgments loads.” J. Struct. Div., 106共10兲, 2017–2033.
Ellingwood, B. R., Galambos, T. V., MacGregor, J. G., and Cornell, C. A.
共1980兲. “Development of a probability based load criterion for Ameri-
This research was supported by the Fund for the Promotion of
can National Standard A58.” National Bureau of Standards special
Research at Technion. publication 577, U.S. Government Printing Office, Washington D.C.
Enright, M. E., and Frangopol, D. M. 共1998兲. “Probabilistic analysis of
resistance degradation of reinforced concrete bridge beams under cor-
Notation rosion.” Eng. Struct., 20共11兲, 960–971.
Frangopol, D. M., Lin, K.-Y., and Estes, A. C. 共1997兲. “Reliability of
The following symbols are used in the paper: reinforced concrete girders under corrosion attack.” J. Struct. Eng.,
A p ⫽ cross-sectional area of a pit; 123共3兲, 286–297.
Downloaded from ascelibrary.org by New York University on 05/18/15. Copyright ASCE. For personal use only; all rights reserved.

As ⫽ cross-sectional area of reinforcement; Gonzales, J. A., Andrade, C., Alonso, C., and Feliu, S. 共1995兲. “Compari-
Av ⫽ cross-sectional area of stirrups; son of rates of general corrosion and maximum pitting penetration on
b ⫽ width of cross section; concrete embedded steel reinforcement.” Cem. Concr. Res., 25共2兲,
257–264.
D0 ⫽ initial diameter of a reinforcing bar;
Li, C. Q. 共2004兲. “Reliability based service life prediction of corrosion
⌬D ⫽ reduction in diameter of a reinforcing bar due
affected concrete structures.” J. Struct. Eng., 130共10兲, 1570–1577.
to corrosion; Lu, R. H., Luo, Y. H., and Conte, J. P. 共1994兲. “Reliability evaluation of
d ⫽ effective depth of cross section; reinforced concrete beams.” Struct. Safety, 14共4兲, 277–298.
f c ⫽ compressive strength of concrete; MacGregor, J. G. 共1997兲. Reinforced concrete: Mechanics and design,
f y ⫽ yield strength of reinforcing steel; 3rd Ed., Prentice-Hall, Upper Saddle River, N.J.
G M ⫽ limit state function for failure in flexure; MacGregor, J. G., Mirza, S. A., and Ellingwood, B. 共1983兲. “Statistical
GV ⫽ limit state function for failure in shear; analysis of resistance of reinforced concrete and prestressed concrete
Gn ⫽ nominal value of dead load; members.” J. Am. Concr. Inst., 80共3兲, 167–176.
icorr ⫽ corrosion current density; Mirza, S. A., and MacGregor, J. G. 共1979a兲. “Variability of mechanical
M R ⫽ flexural strength; properties of reinforcing bars.” J. Struct. Div., 105共5兲, 921–937.
M S ⫽ bending moment at beam cross section; Mirza, S. A., and MacGregor, J. G. 共1979b兲. “Statistical study of shear
m ⫽ number of elements along beam length; strength of reinforced concrete slender beams.” J. Am. Concr. Inst.,
n ⫽ number of reinforcing bars; 76共11兲, 1159–1177.
Mirza, S. A., MacGregor, J. G., and Hatzinikolas, M. 共1979兲. “Statistical
Pav ⫽ average corrosion penetration;
descriptions of strength of concrete.” J. Struct. Div., 105共6兲, 1021–
P f ⫽ cumulative probability of failure; 1037.
Pmax ⫽ maximum penetration of pitting corrosion; Östlund, L. 共1991兲. “An estimation of ␥ values.” Bulletin d’Information
p ⫽ pit depth; No. 202, Comite Euro-International du Beton, Lausanne, Switzerland,
p f ⫽ annual probability of failure; 37–97.
Qn ⫽ nominal value of live load; Philpot, T. A., Rosowsky, D. V., and Fridley, K. J. 共1993兲. “Serviceability
R ⫽ ratio of maximum penetration of pitting design in LRFD for wood.” J. Struct. Eng., 119共12兲, 3649–3667.
corrosion to average corrosion penetration; Rodriguez, J., Ortega, L. M., and Casal, J. 共1997兲. “Load carrying capac-
t ⫽ time; ity of concrete structures with corroded reinforcement.” Constr. Build.
Vc ⫽ shear strength provided by concrete; Mater., 11共4兲, 239–248.
VR ⫽ shear strength; Stewart, M. G. 共2004兲. “Spatial variability of pitting corrosion and its
influence on structural fragility and reliability of RC beams in flex-
VS ⫽ shear force at beam cross section;
ure.” Struct. Safety, 26共4兲, 453–470.
Vs ⫽ shear strength provided by stirrups; Turnbull, A. 共1993兲. “Review of modeling of pit propagation kinetics.”
␣ ⫽ parameter of Gumbel distribution; and Br. Corros. J., London, 28共4兲, 297–308.
␮ ⫽ parameter of Gumbel distribution. Tuutti, K. 共1982兲. “Corrosion of steel in concrete.” Fo 4.82, Swedish
Cement and Concrete Research Institute, Stockholm, Sweden.
Val, D. V., and Melchers, R. E. 共1997兲. “Reliability of deteriorating RC
References slab bridges.” J. Struct. Eng., 123共12兲, 1638–1644.
Vu, K. A. T., and Stewart, M. G. 共2000兲. “Structural reliability of concrete
American Concrete Institute 共ACI兲. 共2005兲. “Building code requirements bridges including improved chloride-induced corrosion models.”
for structural concrete.” ACI 318-05, ACI, Detroit. Struct. Safety, 22共4兲, 313–333.

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